From d78ebe04248ac9c37ff48d771f3a1e8d8162db4f Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Sun, 1 Nov 2020 18:56:57 -0500
Subject: [PATCH 01/13] docs: new benchmark 8 spec

---
 _data/benchmarks.yaml                |   13 +-
 _includes/head.html                  |    1 +
 benchmarks/benchmark8.ipynb          | 1019 ++++++++++++++++++++++++--
 benchmarks/benchmark8.ipynb.raw.html |  855 +++++++++++++++++++--
 images/t0_single_seed.PNG            |  Bin 0 -> 85553 bytes
 5 files changed, 1772 insertions(+), 116 deletions(-)
 create mode 100644 images/t0_single_seed.PNG

diff --git a/_data/benchmarks.yaml b/_data/benchmarks.yaml
index f1e6da896..9817d20ed 100644
--- a/_data/benchmarks.yaml
+++ b/_data/benchmarks.yaml
@@ -428,11 +428,20 @@
       mode: lines
       variations: [b, c]
 
-- title: Nucleation
-  description: Nucleation benchmark
+- title: Homogeneous Nucleation
+  description: Homogeneous nucleation benchmark
   image: https://user-images.githubusercontent.com/1986844/66938477-396aff80-f00f-11e9-8c21-b59356a3b0e0.png
   num: 8
   revisions:
+    - version: 1
+      variations: [a, b, c]
+      event:
+        name: Hackathon 5
+        url: "benchmarks/benchmark8.ipynb"
+      url: "benchmarks/benchmark8.ipynb"
+      commit:
+        sha: "e19007361211"
+        url: "benchmarks/benchmark8.ipynb"
     - version: 0
       variations: [a, b, c, d]
       event:
diff --git a/_includes/head.html b/_includes/head.html
index b90d298ee..0220718b8 100644
--- a/_includes/head.html
+++ b/_includes/head.html
@@ -102,6 +102,7 @@
 <script type="text/x-mathjax-config">
   MathJax.Hub.Config({
     <!-- extensions: ["tex2jax.js"], -->
+    TeX: { equationNumbers: {autoNumber: "AMS"}},
     jax: ["input/TeX", "output/HTML-CSS"],
     tex2jax: {
       inlineMath: [ ['$','$'], ["\\(","\\)"] ],
diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index de1b747f7..3f36973db 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -82,43 +82,60 @@
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 3,
    "metadata": {},
-   "outputs": [],
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "\n",
+       "<a href=\"https://github.com/usnistgov/pfhub/raw/master/benchmarks/benchmark8.ipynb\"\n",
+       "   download>\n",
+       "<button type=\"submit\">Download Notebook</button>\n",
+       "</a>\n"
+      ],
+      "text/plain": [
+       "<IPython.core.display.HTML object>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
    "source": [
-    "# Disable until merge\n",
-    "#from IPython.display import HTML\n",
-    "#\n",
-    "#HTML('''\n",
-    "#<a href=\"https://github.com/usnistgov/pfhub/raw/master/benchmarks/benchmark8.ipynb\"\n",
-    "#   download>\n",
-    "#<button type=\"submit\">Download Notebook</button>\n",
-    "#</a>\n",
-    "#''')"
+    "from IPython.display import HTML\n",
+    "\n",
+    "HTML('''\n",
+    "<a href=\"https://github.com/usnistgov/pfhub/raw/master/benchmarks/benchmark8.ipynb\"\n",
+    "   download>\n",
+    "<button type=\"submit\">Download Notebook</button>\n",
+    "</a>\n",
+    "''')"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Benchmark Problem 8: Nucleation"
+    "# Benchmark Problem 8: Homogeneous Nucleation"
    ]
   },
   {
    "cell_type": "code",
-   "execution_count": 1,
+   "execution_count": 4,
    "metadata": {},
    "outputs": [
     {
      "data": {
       "text/html": [
-       "{% include jupyter_benchmark_table.html num=\"[8]\" revision=0 %}"
+       "{% include jupyter_benchmark_table.html num=\"[8]\" revision=1 %}"
       ],
       "text/plain": [
        "<IPython.core.display.HTML object>"
       ]
      },
-     "execution_count": 1,
+     "execution_count": 4,
      "metadata": {},
      "output_type": "execute_result"
     }
@@ -126,95 +143,901 @@
    "source": [
     "from IPython.display import HTML\n",
     "\n",
-    "HTML('''{% include jupyter_benchmark_table.html num=\"[8]\" revision=0 %}''')"
+    "HTML('''{% include jupyter_benchmark_table.html num=\"[8]\" revision=1 %}''')"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "### Warning\n",
+    "See the journal publication entitled [\"Phase Field Benchmark Problems for Nucleation\"][paper] for more details about the benchmark problems. Furthermore, read [the extended essay][benchmarks] for a discussion about the need for benchmark problems.\n",
     "\n",
-    "This benchmark is still under construction, but the submission guidelines are included for development purposes."
+    "[benchmarks]: ../\n",
+    "[paper]: https://doi.org/10.1016/j.commatsci.2018.03.015"
    ]
   },
   {
    "cell_type": "markdown",
-   "metadata": {
-    "ExecuteTime": {
-     "end_time": "2019-10-28T18:42:26.097304Z",
-     "start_time": "2019-10-28T18:42:26.083174Z"
-    }
-   },
+   "metadata": {},
    "source": [
-    "## Submission Guidelines\n",
+    "# Table of Contents\n",
     "\n",
-    "Please use the [upload form]({{ site.baseurl }}/simulations/upload_form/) to upload your results.\n",
+    "* [Overview](#Overview)\n",
+    "* [Model Formulation](#Model-Formulation)\n",
+    "\t* [The phase field model in classical nucleation theory](#The-phase-field-model-in-classical-nucleation-theory)\n",
+    "\t* [The properties of the classical nucleus](#The-properties-of-the-classical-nucleus)\n",
+    "\t* [Avrami plots](#Avrami-plots)\n",
+    "* [Benchmark 8 Specification](#Benchmark-8-Specification)\n",
+    "\t* [Part (a): single seed](#Part-%28a%29:-single-seed)\n",
+    "    * [Part (b): multiple initial seeds](#Part-%28b%29:-multiple-initial-seeds)\n",
+    "    * [Part (c): multiple seeds at random times](#Part-%28c%29:-multiple-seeds-at-random-times)\n",
+    "* [Submission Guidelines](#Submission-Guidelines)\n",
+    "\t* [Part (a)](#Part-%28a%29)\n",
+    "\t* [Part (b)](#Part-%28b%29)\n",
+    "\t* [Part (c)](#Part-%28c%29)\n",
+    "* [Results](#Results)\n",
+    "* [References](#References)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Overview"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Phase field modeling of nucleation has a long history and is covered\n",
+    "in a number of reviews (<a id=\"ref-1\" href=\"#cite-granasy2002nucleation\">Gránásy 2002</a>,\n",
+    "<a id=\"ref-2\" href=\"#cite-castro2003phase\">Castro 2003</a>,\n",
+    "<a id=\"ref-3\" href=\"#cite-simmons2004microstructural\">Simmons 2004</a>,\n",
+    "<a id=\"ref-4\" href=\"#cite-granasy2007phase\">Gránásy 2007</a>,\n",
+    "<a id=\"ref-5\" href=\"#cite-warren2009phase\">Warren 2009</a>,\n",
+    "<a id=\"ref-6\" href=\"#cite-heo2014phase\">Heo 2014</a>,\n",
+    "<a id=\"ref-7\" href=\"#cite-granasy2019\">Gránásy 2019</a>). The problem formulation of crystallization of an ideal pure liquid cooled\n",
+    "below its melting point starts with homogeneous nucleation, a process\n",
+    "in which the internal fluctuations of the undercooled liquid lead to\n",
+    "the formation of crystal-like seeds able to grow to macroscopic\n",
+    "sizes. The nucleation can be assisted by the presence of surfaces\n",
+    "(container walls, foreign particles, etc.), in which case the process\n",
+    "is termed heterogeneous nucleation. We note that homogeneous\n",
+    "nucleation is an idealized formulation, and it is unlikely homogeneous\n",
+    "nucleation occurs, due to impurities present in experimental\n",
+    "apparatus. However, creating benchmark problems for homogeneous\n",
+    "nucleation is still needed since the focus in nucleation research\n",
+    "often lies on the underlying homogeneous nucleation, as it is the\n",
+    "basis for advancing theoretical approaches to the much more complex\n",
+    "phenomena governing heterogeneous nucleation. This benchmark\n",
+    "(benchmark 8) targets homogeneous nucleation while benchmark 9\n",
+    "targets heterogeneous nucleation.\n",
     "\n",
-    "In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. These are not required, but will help others view your work.\n",
+    "There are two main modeling approaches to introduce nuclei into a\n",
+    "metastable system: the Langevin noise method\n",
+    "(<a id=\"ref-8\" href=\"#cite-kubo1966\">Kubo 1966</a>,\n",
+    "<a id=\"ref-9\" href=\"#cite-granasy2002nucleation\">Gránásy 2002</a>)\n",
+    "and the explicit nucleation\n",
+    "method\n",
+    "(<a id=\"ref-10\" href=\"#cite-simmons2000phase\">Simmons 2000</a>,\n",
+    "<a id=\"ref-11\" href=\"#cite-shi2019\">Shi 2019</a>)\n",
+    ". We focus on the explicit\n",
+    "method, where the critical nucleus size and the nucleation energy are\n",
+    "determined by the classical nucleation theory. We break down our\n",
+    "consideration of homogeneous nucleation into three parts. Part (a)\n",
+    "considers the simple case of single-seed homogeneous nucleation. We\n",
+    "explore how the particle size influences the evolution of the nucleus\n",
+    "when the thermodynamic driving force is close to the critical value\n",
+    "where we can observe whether the particle grows, dissolves or remains\n",
+    "stationary. In parts (b) and (c) we consider multiple-seed homogeneous\n",
+    "nucleation, with nuclei appearing at fixed time $t=0$ or at random\n",
+    "times distributed uniformly, respectively, with the latter part\n",
+    "illustrating a scenario with constant nucleation rate. The Benchmark\n",
+    "Problem probes the differences between transformation kinetics of the\n",
+    "two cases and summarizes them using the Avrami plots based on the\n",
+    "Johnson-Mehl-Avrami-Kolmogorov (JMAK)\n",
+    "theory\n",
+    "(<a id=\"ref-12\" href=\"#cite-johnson1939reaction\">Johnson 1939</a>,\n",
+    "<a id=\"ref-13\" href=\"#cite-avrami1939kinetics\">Avrami 1939</a>,\n",
+    "<a id=\"ref-14\" href=\"#cite-avrami1940kinetics\">Avrami 1940</a>,\n",
+    "<a id=\"ref-15\" href=\"#cite-avrami1941granulation\">Avrami 1941</a>,\n",
+    "<a id=\"ref-16\" href=\"#cite-kolmogorov1937izv\">Kolmogorov 1937</a>)."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Model Formulation\n",
     "\n",
-    "### Part (a)\n",
+    "### The phase field model in classical nucleation theory\n",
     "\n",
-    "Run three cases with starting seeds of \n",
+    "For these benchmark problems we use the simplest possible phase field model with a single non-conserved phase field $\\phi$, which describes an isothermal pure substance with one liquid ($\\phi=0$) and one solid ($\\phi=1$) phase. The free energy of this system is  \n",
+    "\\begin{equation}\n",
+    "    \\mathcal{F}(\\phi)=\\int \\left[\\frac{\\epsilon^2}{2}(\\nabla \\phi)^2+wg(\\phi)-\\Delta f p(\\phi) \\right] d{V},\n",
+    "    \\label{eqn:Fphi}\n",
+    "\\end{equation}\n",
+    "set $g(\\phi)=\\phi^2(1-\\phi)^2$, a simple double well function with minima at $\\phi=0$ and $\\phi=1$, $ w$ controls the double-well barrier height, $\\epsilon^2$ is the gradient energy coefficient, let $p(\\phi)=\\phi^3(10-15\\phi+6\\phi^2)$, which ensures that $p(0)=p'(0)=p'(1)=0$ and $p(1)=1$, and $\\Delta f$ is the driving force for solidification at the simulation temperature ($\\Delta f$ is positive below the melting point). The time evolution of $\\phi$ is given by the Allen-Cahn equation (<a id=\"ref-17\" href=\"#cite-allen1979microscopic\">Allen 1979</a>)\n",
+    "\\begin{equation}\n",
+    "    \\frac{\\partial \\phi}{\\partial t}=-M\\frac{\\delta \\mathcal{F}}{\\delta \\phi}=M\\left[\\epsilon^2\\nabla^2\\phi-wg'(\\phi)+\\Delta fp'(\\phi)\\right]\n",
+    "    \\label{eqn:dphidt}\n",
+    "\\end{equation}\n",
+    "where $M$ is the mobility parameter. We will restrict the problem to two dimensions (2D). For a planar interface, one can show that the equilibrium ($\\Delta f$=0) solid-liquid interface profile with the interface centered at $x=x_0$ is given by\n",
+    "\\begin{equation}\n",
+    "    \\phi(x)=\\frac{1}{2}\\left[1-\\tanh\\left(\\frac{x-x_0}{\\sqrt{2}\\ell}\\right)\\right],\n",
+    "    \\label{eqn:phir}\n",
+    "\\end{equation}\n",
+    "where $x$ is the perpendicular distance from the interface.\n",
+    "We use this expression as the initial condition when we introduce a nucleus. We can also obtain the width $\\ell$ and the free energy of the interface, $\\gamma$, as\n",
+    "\\begin{equation}\n",
+    "    \\ell=\\sqrt{\\frac{\\epsilon^2}{w}}\n",
+    "    \\label{eqn:l}\n",
+    "\\end{equation}\n",
+    "and\n",
+    "\\begin{equation}\n",
+    "    \\gamma=\\frac{\\sqrt{\\epsilon^2w}}{3\\sqrt{2}}.\n",
+    "    \\label{eqn:gamma}\n",
+    "\\end{equation}\n",
     "\n",
-    " 1. $r_0 = 0.99 r^*$,\n",
-    " 1. $r_0 = r^*$\n",
-    " 1. $r_0 = 1.01 r^*$.\n",
-    " \n",
-    "Follow the time evolution of the system for $t = 100$ units. The following data should be collected frequently (at least 100 data points is preferable).\n",
+    "Choosing a characteristic length scale $\\xi=\\ell$ and a characteristic time scale $\\tau=1/(Mw)$, we can obtain a nondimensional form of the problem (the nondimensional quantities are denoted by tildes),\n",
+    "\\begin{equation}\n",
+    "    \\tilde{\\mathcal{F}}(\\phi)=\\int \\left[\\frac{1}{2}(\\tilde{\\nabla} \\phi)^2+g(\\phi)- \\tilde{\\Delta f} p(\\phi) \\right] d{\\tilde{V}}\n",
+    "    \\label{eqn:tFphi}\n",
+    "\\end{equation}\n",
+    "and\n",
+    "\\begin{equation}\n",
+    "    \\frac{\\partial \\phi}{\\partial \\tilde{t}}=\\tilde{\\nabla}^2\\phi-g'(\\phi)+\\tilde{\\Delta f}p'(\\phi),\n",
+    "    \\label{eqn:dphidtt}\n",
+    "\\end{equation}\n",
+    "with $\\tilde{\\Delta f}=\\Delta f/w$.\n",
     "\n",
-    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$)\n",
-    " \n",
-    " - The free energy, $\\mathcal{F}$\n",
-    " \n",
-    "One PFHub upload is required with one data file for each of the three simulations (for a total of three data files). The data sets should be named as `solid_fraction_1` and `free_energy_1`, `solid_fraction_2` and `free_energy_2` as well as `solid_fraction_3` and `free_energy_3` where `_1`, `_2` and `_3` refers to case (1), (2) and (3), respectively. The data files should be stored in either CSV or JSON format. The column names for time, free energy and solid fraction need to be identified correctly on the upload form for each data set.\n",
+    "### The properties of the classical nucleus\n",
+    "\n",
+    "The explicit nucleation method that we employ introduces nuclei into a metastable system, with their radii and nucleation barrier determined from classical nucleation theory. Classical nucleation theory views crystallite fluctuations appearing in the undercooled liquid as small spherical domains of the bulk crystalline phase bounded by a mathematically sharp solid-liquid interface. For a 2D system, the free energy of a circular solid particle of radius $r$ is \n",
+    "\\begin{equation}\n",
+    "    \\Delta G(r) = 2 \\pi r \\gamma - \\pi r^2 \\Delta f\n",
+    "    \\label{eqn:deltaG}\n",
+    "\\end{equation}\n",
+    "where $\\Delta f$ is the nucleation driving force that is used in our phase field model, and $\\gamma$ is the free energy of the interface. The free energy of the particle is a balance between the energy cost in forming the solid-liquid interface, and the free energy from the driving force which is released when the crystalline particle forms. Once the rate of change of free energy with respect to particle size becomes negative, the particle can grow. Taking the derivative of $\\Delta G(r)$ we get the rate of change of free energy with respect to radius as\n",
+    "\\begin{equation}\n",
+    "    \\frac{d\\Delta G}{dr}=2\\pi \\gamma - 2\\pi r\\Delta f.\n",
+    "    \\label{eqn:ddeltaGdr}\n",
+    "\\end{equation}\n",
+    "By setting $d\\Delta G/dr$ to zero, we obtain the critical radius $r^*$ as\n",
+    "\\begin{equation}\n",
+    "    r^* = \\frac{\\gamma}{\\Delta f}.\n",
+    "    \\label{eqn:rc}\n",
+    "\\end{equation}\n",
+    "The corresponding critical nucleation free energy is\n",
+    "\\begin{equation}\n",
+    "    \\Delta G^* = \\frac{\\pi \\gamma^2}{\\Delta f}.\n",
+    "    \\label{eqn:deltaGc}\n",
+    "\\end{equation}\n",
+    "Using the same units as before, the nondimensional forms of these quantities are\n",
+    "\\begin{equation}\n",
+    "    \\tilde{r^*} = \\frac{1}{3\\sqrt{2}}\\frac{1}{\\tilde{\\Delta f}}\n",
+    "    \\label{eqn:trc}\n",
+    "\\end{equation}\n",
+    "and\n",
+    "\\begin{equation}\n",
+    "    \\tilde{\\Delta G^*} = \\frac{\\pi}{18}\\frac{1}{\\tilde{\\Delta f}}\n",
+    "    \\label{eqn:tdeltaGc},\n",
+    "\\end{equation}\n",
+    "respectively.\n",
+    "\n",
+    "### Avrami plots\n",
     "\n",
-    "Further data to upload can include a Youtube video, images of the nucleus at different times or the entire phase field variable at different times. These are not required, but will help others view your work.\n",
+    "Avrami plots describe how solids transform from one phase to another at constant temperature. In particular, they can describe the kinetics of nucleation. The Avrami plots come from the JMAK theory\n",
+    "(<a id=\"ref-18\" href=\"#cite-johnson1939reaction\">Johnson 1939</a>,\n",
+    "<a id=\"ref-19\" href=\"#cite-avrami1939kinetics\">Avrami 1939</a>,\n",
+    "<a id=\"ref-20\" href=\"#cite-avrami1940kinetics\">Avrami 1940</a>,\n",
+    "<a id=\"ref-21\" href=\"#cite-avrami1941granulation\">Avrami 1941</a>,\n",
+    "<a id=\"ref-22\" href=\"#cite-kolmogorov1937izv\">Kolmogorov 1937</a>), which makes a number of \n",
+    "assumptions and simplifications: (i)  nucleation occurs randomly and homogeneously over the entire un-transformed portion of the material, (ii) the growth rate is constant and does not depend on the extent of transformation, (iii) the particles have convex shape with the same orientation, and (iv) the size of the system is infinite (both in space and time).\n",
     "\n",
-    "### Part (b)\n",
+    "The basis of the Avrami plots is the transformed fraction (solid fraction) $X(t)$ vs. time. According to the JMAK theory,\n",
+    "\\begin{equation}\n",
+    "    Y(t)=1-\\exp (-Kt^n),\n",
+    "    \\label{eqn:Xt}\n",
+    "\\end{equation}\n",
+    "where $K$ is a constant depending on the nucleation and growth rates, $n=d+1$ for continuous nucleation and $n=d$ if nucleation happens only at $t=0$, and $d$ is the number of spatial dimensions. If we plot $\\log(-\\log(1-Y))$ vs. $\\log(t)$, then for the JMAK kinetics (Eq. \\ref{eqn:Xt}) we get a straight line with slope $n$.\n",
     "\n",
-    "Follow the time evolution of the system for $t = 400$ units. The following data should be collected frequently (at least 100 data points is preferable).\n",
+    "## Benchmark 8 Specification\n",
     "\n",
-    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$)\n",
+    "From here on, we will use only the nondimensional forms of the phase field (Eqs. \\ref{eqn:tFphi}, \\ref{eqn:dphidtt}, and \\ref{eqn:phir}) and nucleation (Eqs. \\ref{eqn:trc} and \\ref{eqn:tdeltaGc}) equations, but we will drop the tildes.\n",
+    "\n",
+    "### Part (a): single seed\n",
+    "\n",
+    "In this problem we examine the morphology change of the nucleus for different initial radii. We consider a 2D simulation domain of size $100 \\times 100$ units centered at $x=y=0$ with Neumann boundary conditions. The driving force is set to $\\Delta f = \\sqrt{2}/30$, which corresponds to a critical radius of $r^*=5$ (Eq. \\ref{eqn:trc}). Next, we place a circular seed at the center of the domain. \n",
+    "Incorporating a nucleus into a diffuse interface phase field model leads to a small offset from a classical, sharp-interface model of a nucleus. To account for this,\n",
+    "we incorporate a diffuse-interface seed using the profile $\\phi(r)$ given by modifying Eq. \\ref{eqn:phir} to 2D where $r=\\sqrt{x^2+y^2}$, with a radius $r_0=r^*$. This seed is the diffuse-interface approximation of the classical sharp interface nucleus corresponding to the given $\\Delta f$, and therefore it should be fairly close to (an unstable) equilibrium. \n",
+    "Part (a) is then defined by the Allen-Cahn equation\n",
+    "\\begin{equation}\n",
+    "    \\frac{\\partial \\phi}{\\partial {t}}={\\nabla}^2\\phi-g'(\\phi)+{\\Delta f}p'(\\phi)\n",
+    "    \\label{eqn:dphidtt_2}\n",
+    "\\end{equation}\n",
+    "for the phase field $\\phi$, and the three different initial conditions: in the first case with the seed radius $r_0$ corresponding exactly to the critical one (critical nucleus), and in the other two cases, slightly below and above the critical radius (subcritical and supercritical nuclei):\n",
+    "\\begin{equation}\n",
+    "    \\phi(r)=\\frac{1}{2}\\left[1-\\tanh\\left(\n",
+    "    \\frac{r-r_0}{\\sqrt{2}}\n",
+    "    \\right)\\right],\n",
+    "    \\label{eqn:phitr_2}\n",
+    "\\end{equation}\n",
+    "where $r_0=\\{r^*, 0.99r^*, 1.01r^*\\}$. [Figure 1](#fig1) shows the computational domain with an initial seed of radius $r=r^*$. The time evolution of the system is then followed for times up to $t = 200$ units, and the solid fraction $Y$ and the total free energy $F$ are plotted as functions of time. Finally, the problem also includes a convergence test with respect to mesh (spatial resolution). The closer the initial radius $r$ is to the critical radius $r^*$, the more sensitive the numerical integration will be to round-off errors. Therefore, a convergence test is included for $r_0=1.01r^*$ using successive runs each halving the average spatial mesh size $\\Delta x$ between consecutive runs. At least 3 runs are required to estimate the rate of convergence based on the $L^2$ error with the finest mesh being used as a gold standard. The simulations should be run to $t=200$ and the final phase field (at $t=200$) for each successive run should be submitted.\n",
+    "\n",
+    "To submit results to the PFHub website, see the [submission guidelines](#Part-1.6.1-Part-%28a%29) below.\n",
+    "\n",
+    "<figure>\n",
+    "    <img id=\"fig1\" src=\"../images/t0_single_seed.PNG\" alt=\"Single seed at t=0\" style=\"width:500px;\"/>\n",
+    "    <figcaption>\n",
+    "        <center> Illustration of the $100 \\times 100$ computational domain in 2D. </center> \n",
+    "    </figcaption>\n",
+    "</figure>\n",
+    "\n",
+    "## Part (b): multiple initial seeds\n",
+    "\n",
+    "The second part of the homogeneous nucleation benchmark problem focuses on the kinetics of nucleation using Avrami plots and compares them to JMAK theory. To obtain reasonable statistics for these two parts, the simulation volume needs to be larger so it can encompass a larger number of smaller nuclei. We therefore increase the domain size to $500 \\times 500$ units of length, and the driving force to $\\Delta f = 1/(3 \\sqrt{2})$, which corresponds to a critical radius of $r^*=1$. Random initial positions $\\mathbf{r}_i$ of 25 supercritical seeds $i, i=1,\\ldots,25$ are generated with $r_0=2.2r^*$ drawn from a uniform distribution on the $500\\times500$ domain. The distribution of the phase field $\\phi$ is the sum of the phase fields $\\phi_i$ with profiles\n",
+    "\\begin{equation}\n",
+    "    \\phi_i(r)=\\frac{1}{2}\\left[1-\\tanh\\left(\n",
+    "    \\frac{|\\mathbf{r}-\\mathbf{r}_i|-2.2r^*}{\\sqrt{2}}\n",
+    "    \\right)\\right].\n",
+    "    \\label{eqn:phitr_3}\n",
+    "\\end{equation}\n",
+    "from the different seeds $i$. Overlaps between different seeds are managed by setting $\\phi=1$ in all regions where the sums of $\\phi_i >1$. After placing initial seeds and adjusting the initial phase field $\\phi$ so that $\\phi\\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $X$, discrete particle count $N$  (the number of disjoint regions with $\\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.\n",
+    "\n",
+    "Both Part (b) and Part (c) involve a random number generator. Because different executions will in general generate different random numbers, specific individual solutions will in general not be reproduced. However, statistics such as the average slope of Avrami plots can be meaningfully compared.\n",
+    "\n",
+    "To submit results to the PFHub website, see the [submission guidelines](#Part-1.6.1-Part-%28a%29) below.\n",
+    "\n",
+    "## Part (c): multiple seeds at random times\n",
+    "\n",
+    "For the third part of the homogeneous benchmark problem, the computational domain is first expanded to $1000 \\times1000$. Instead of inserting nuclei at fixed time $t = 0$ as in part two of the problem, in this case 100 random nucleation times $t_i$ are generated, $i=1,\\ldots,100$, drawn from a uniform distribution in the interval $t_i \\in [0, 600)$ with centers $\\mathbf{r}_i$ drawn from a uniform distribution on the $1000\\times1000$ domain.\n",
+    "We decrease the driving force to $\\Delta f=1/(6\\sqrt{2})$, which corresponds to a critical radius or $r^*=2$, but keep the initial radii of the inserted nuclei the same as those in the second part of the problem\n",
+    "\\begin{equation}\n",
+    "    \\phi_i(r)=\\frac{1}{2}\\left[1-\\tanh\\left(\n",
+    "    \\frac{|\\mathbf{r}-\\mathbf{r}_i|-1.1r^*}{\\sqrt{2}}\n",
+    "    \\right)\\right].\n",
+    "\\end{equation}\n",
+    "\n",
+    "Again, $\\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $X$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.\n",
+    "\n",
+    "To submit results to the PFHub website, see the [submission guidelines](#Part-1.6.3-Part-%28c%29) below."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Submission Guidelines"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Please use the [upload form]({{ site.baseurl }}/simulations/upload_form/) to upload your results.\n",
+    "\n",
+    "In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. These are not required, but will help others view your work."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Part (a)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\\Delta x$ using the model formulation [defined above](#Part-1.3.1-Part-1.6.1-Part-%28a%29:-single-seed). The 5 simulations are\n",
+    "\n",
+    " 1. $r_0 = 0.99 r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
+    " 1. $r_0 = r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
+    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
+    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_1$\n",
+    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_2$\n",
+    " \n",
+    "where ${\\Delta x}_0 < {\\Delta x}_1 < {\\Delta x}_2$. If using a gridded mesh each successively refined grid length can be half as fine. In the case of adaptive grids a nominal value for $\\Delta x$ should be selected that results in a similarly refined mesh as in the gridded case (i.e. approximately $4\\times$ as many elements). When uploading the results, the $\\Delta x$ values should be provided in the datafile alongside the values for $\\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.\n",
+    "\n",
+    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
+    " \n",
+    " - The free energy integrated over the whole domain, $\\mathcal{F}$, during the transient. \n",
     " \n",
-    " - The free energy, $\\mathcal{F}$\n",
+    " - The phase field, $\\phi$, at each grid point in the domain at $t=200$.\n",
     " \n",
-    "One PFHub upload is required with one data file. The data sets should be named as `solid_fraction` and `free_energy`. The data files should be stored in either CSV or JSON format. The results can be stored in one single file as long as each column name for time, solid fraction and free energy is identified correctly on the upload form.\n",
+    " - The grid spacing, $\\Delta x$ at each grid point in the domain at $t=200$.\n",
+    " \n",
+    "One PFHub upload is required for all five simulations. The solid fraction and free energy data can be included in the same CSV file or in separate data files, but either way still require 2 entries in the \"Data Files\" section on the [upload form]({{ site.baseurl }}/simulations/upload_form/). The entries should be labeled as `solid_fraction_{i}` or `free_energy_{i}` depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.\n",
     "\n",
-    "In addition, the phase field snapshot at time $t=80$ is required. The data set should be named as `phase-field-80` and stored in PNG or TIFF format.\n",
+    "```\n",
+    "time,fraction,energy\n",
+    "0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00\n",
+    "1.000000000000000056e-01,8.205027223394931180e-03,3.577696619091880859e+00\n",
+    "2.000000000000000111e-01,8.197361418448430651e-03,3.576379039721957032e+00\n",
+    "...\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The names of the columns are arbitrary as long as they are given in \"Name of x-axis column\" and \"Name of y-axis column\" in the form. The phase field data should be named `phase_field_{i}` in the \"Short name of data\" entry and have the following format\n",
     "\n",
-    "### Part (c)\n",
+    "```\n",
+    "x,y,phi,delta_x\n",
+    "0.0,1.0,0.5,0.44\n",
+    "0.0,1.5,0.49,0.44\n",
+    "...\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "where `x` and `y` is a position in the domain and `phi` is the value at that position. `delta_x` is the nominal grid spacing in that region given by the coordinates. This may be constant across the domain for gridded data, but vary for adaptive meshing. `delta_x` is only a representative value of the grid spacing in that region. The \"3D\" radio button should be selected for the `phase_field_{i}` entry.\n",
     "\n",
-    "Follow the time evolution of the system for $t = 600$ units. The following data should be collected frequently (at least 100 data points is preferable).\n",
+    "The grid spacing data should be named `grid_spacing_{i}` and have exactly the same data as above. Again the same file can be used, but two \"Data Files\" entries are required for both phase field and grid spacing. The \"3D\" radio button should be selected for the `grid_spacing_{i}` entry."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Part (b)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the [defintion above](#Part-1.4-Part-1.6.2-Part-%28b%29:-multiple-initial-seeds). The simulations are run to $t=200$. The following data should be collected.\n",
     "\n",
-    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$)\n",
+    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
     " \n",
-    " - The free energy, $\\mathcal{F}$\n",
+    " - The free energy integrated over the whole domain, $\\mathcal{F}$, during the transient. \n",
     " \n",
-    "One PFHub upload is required with one data file. The data sets should be named as `solid_fraction` and `free_energy`. The data files should be stored in either CSV or JSON format. The results can be stored in one single file as long as each column name for time, solid fraction and free energy is identified correctly on the upload form.\n",
-    "\n",
-    "In addition, the phase field snapshot at time $t=80$ is required. The data set should be named as `phase-field-80` and stored in PNG or TIFF format.\n",
+    " - The discrete particle count, $N$, during the transient.\n",
+    " \n",
+    "One PFHub upload is required with five data files, but with a total of 15 entries in the \"Data Files\" section. The solid fraction, free energy and particle count can be included in the same data file for each simulation. The entries should be labeled as `solid_fraction_{i}`, `free_energy_{i}` or `particle_count_{i}` depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.\n",
     "\n",
-    "### Part (d)\n",
+    "```\n",
+    "time,fraction,energy,particle_count\n",
+    "0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00,24\n",
+    "1.000000000000000056e-01,8.205027223394931180e-03,3.577696619091880859e+00,24\n",
+    "2.000000000000000111e-01,8.197361418448430651e-03,3.576379039721957032e+00,23\n",
+    "...\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Part (c)"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Part (c) requires running 5 simulations with seeds randomly appearing between $t=0$ and $t=600$, see the [definition above](#Part-1.5-Part-1.6.3-Part-%28c%29:-multiple-seeds-at-random-times). The simulations are run to $t=600$. The following data should be collected.\n",
     "\n",
-    "Run three cases with undercooling of\n",
+    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
+    " \n",
+    " - The free energy integrated over the whole domain, $\\mathcal{F}$, during the transient. \n",
     " \n",
-    " 1. $\\Delta f_0 = 0.98 \\Delta f_0^*$\n",
-    " 2. $\\Delta f_0 = \\Delta f_0^*$\n",
-    " 3. $\\Delta f_0 = 1.02 \\Delta f_0^*$\n",
+    " - The discrete particle count, $N$, during the transient.\n",
     " \n",
-    "Follow the time evolution of the system for $t = 5000$ units. The following data should be collected frequently (at least 100 data points is preferable).\n",
+    "One PFHub upload is required with five data files, but with a total of 15 entries in the \"Data Files\" section. The solid fraction, free energy and particle count can be included in the same data file for each simulation. The entries should be labeled as `solid_fraction_{i}`, `free_energy_{i}` or `particle_count_{i}` depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.\n",
+    "\n",
+    "```\n",
+    "time,fraction,energy,particle_count\n",
+    "0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00,0\n",
+    "1.000000000000000056e-01,8.205027223394931180e-03,3.577696619091880859e+00,1\n",
+    "2.000000000000000111e-01,8.197361418448430651e-03,3.576379039721957032e+00,1\n",
+    "...\n",
+    "```"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Please use the [upload form]({{ site.baseurl }}/simulations/upload_form/) to upload your results."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Results"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Results from this benchmark problem are displayed on the [simulation result page]({{ site.baseurl }}/simulations/8a.1) for different codes."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!--\n",
+    "... see https://jupyter.brynmawr.edu/services/public/dblank/jupyter.cs/Examples/Calico%20Document%20Tools%20and%20Bibtex.ipynb for details\n",
+    "... also see https://jupyter.brynmawr.edu/services/public/dblank/Jupyter%20Notebook%20Users%20Manual.ipynb#5.-Bibliographic-Support\n",
+    "... \n",
+    "... to make this work (sometimes just doesn't work which is infuriating \n",
+    "...\n",
+    "...  \n",
+    "...  - <!--bibtex needs to be at the very top of the cell it seems (very important)\n",
+    "-->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!--bibtex\n",
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+    "  doi={10.1103/PhysRevB.67.035412},\n",
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+    "}\n",
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+    "  title={Continuous integration for concurrent computational framework and application development},\n",
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+    "}\n",
+    "\n",
+    "@article{gaston2015physics,\n",
+    "  title={Physics-based multiscale coupling for full core nuclear reactor simulation},\n",
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+    "}\n",
+    "\n",
+    "@article{jokisaari2017spinodal,\n",
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+    "  year={2017},\n",
+    "  publisher={Elsevier},\n",
+    "  url=\"https://doi.org/10.1016/j.commatsci.2016.09.022\"\n",
+    "}\n",
+    "\n",
+    "\n",
+    "@article{jokisaari2017dendrite,\n",
+    "  title={Phase field benchmark problems for dendritic growth and linear elasticity},\n",
+    "  author={Jokisaari, A M and Voorhees, P W and Guyer, Jonathan E and Warren, J A and Heinonen, O G},\n",
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+    "  volume={149},\n",
+    "  pages={336},\n",
+    "  year={2018},\n",
+    "  publisher={Elsevier},\n",
+    "  url=\"https://doi.org/10.1016/j.commatsci.2018.03.015\"\n",
+    "}\n",
+    "\n",
+    "@article{jokisaari2020stokes,\n",
+    "  title={Phase field benchmark problems targeting fluid flow and electrochemistry},\n",
+    "  author={Jokisaari, A M and Wu, W and Voorhees, P W and Guyer, Jonathan E and Warren, J A and Heinonen, O G},\n",
+    "  journal={Computational Materials Science},\n",
+    "  volume={176},\n",
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+    "  year={2020},\n",
+    "  publisher={Elsevier},\n",
+    "  url=\"https://doi.org/10.1016/j.commatsci.2020.109548\"\n",
+    "}\n",
+    "\n",
+    "@article{kubo1966,\n",
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+    "}\n",
+    "\n",
+    "@article{granasy2002nucleation,\n",
+    " title={Nucleation and bulk crystallization in binary phase field theory},\n",
+    " author={Gránásy, L and and Börzsönyi, T and Pusztai, T},\n",
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+    "}\n",
+    "\n",
+    "@article{granasy2007phase,\n",
+    " title={Phase field theory of heterogeneous crystal nucleation},\n",
+    " author={Gránásy, L and T and Pusztai, T and Saylor, D and Warren, J A},\n",
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+    "}\n",
+    "\n",
+    "@article{granasy2019,\n",
+    " title={Phase-field modeling of crystal nucleation in undercooled liquids -- A review},\n",
+    " author={Gránásy, L and Tóth, G and Warren, J A and Podmaniczky, F and Tegze, G and Rátkai, L and Pusztai, T},\n",
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+    " volume={106},\n",
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+    " year={2019},\n",
+    " publisher={Elsevier},\n",
+    " url={https://doi.org/10.1016/j.pmatsci.2019.05.002}\n",
+    "}\n",
+    "\n",
+    "@article{heo2014phase,\n",
+    " title={Phase-field modeling of nucleation in solid-state phase transformations},\n",
+    " author={Heo, T W and Chen, L},\n",
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+    " year={2014},\n",
+    " publisher={TMS},\n",
+    " url={https://doi.org/10.1007/s11837-014-1033-9}\n",
+    "}\n",
+    "\n",
+    "@article{shi2019,\n",
+    " title={Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response},\n",
+    " author={Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J},\n",
+    " journal={The journal of the minerals, metals \\& materials society},\n",
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+    " year={2019},\n",
+    " publisher={TMS},\n",
+    " url={https://doi.org/10.1007/s11837-019-03618-1}\n",
+    "}\n",
+    "\n",
+    "@article{simmons2000phase,\n",
+    "  title={Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events},\n",
+    "  author={Simmons, J P and Shen, C and Wang, Y},\n",
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+    "  publisher={Elsevier},\n",
+    "  url={https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0}\n",
+    "}\n",
+    "\n",
+    "@article{simmons2004microstructural,\n",
+    "  title={Microstructural development involving nucleation and growth phenomena simulated with the Phase Field method},\n",
+    "  author={Simmons, J P and Wen, Y and Shen, C and Wang, Y},\n",
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+    "  issues={1-2},\n",
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+    "  year={2004},\n",
+    "  publisher={Elsevier},\n",
+    "  url={https://doi.org/10.1016/j.msea.2003.09.019}\n",
+    "}\n",
+    "\n",
+    "@article{warren2009phase,\n",
+    "  title={Phase field approach to heterogeneous crystal nucleation in alloys},\n",
+    "  author={Warren, J A and Pusztai, T and Gránásy, L K and Gránásy, L},\n",
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+    "  url={https://doi.org/10.1103/PhysRevB.79.014204}\n",
+    "}\n",
+    "\n",
+    "@incollection{elliott1989,\n",
+    "  title={The Cahn-Hilliard model for the kinetics of phase separation},\n",
+    "  author={Elliott, Charles M},\n",
+    "  booktitle={Mathematical models for phase change problems},\n",
+    "  pages={35--73},\n",
+    "  year={1989},\n",
+    "  publisher={Springer},\n",
+    "  url={https://doi.org/10.1007/978-3-0348-9148-6_3}\n",
+    "}\n",
+    "\n",
+    "@article{johnson1939reaction,\n",
+    "  title={Reaction kinetics in processes of nucleation and growth},\n",
+    "  author={Johnson, William A and Mehl, R F},\n",
+    "  journal={Am. Inst. Min. Metal. Petro. Eng.},\n",
+    "  volume={135},\n",
+    "  pages={416--458},\n",
+    "  year={1939},\n",
+    "  url={https://doi.org/10.1007/s11663-010-9421-1}\n",
+    "}\n",
+    "\n",
+    "@article{avrami1939kinetics,\n",
+    "  title={Kinetics of phase change. I General theory},\n",
+    "  author={Avrami, Melvin},\n",
+    "  journal={The Journal of chemical physics},\n",
+    "  volume={7},\n",
+    "  number={12},\n",
+    "  pages={1103--1112},\n",
+    "  year={1939},\n",
+    "  publisher={American Institute of Physics},\n",
+    "  url={ https://doi.org/10.1063/1.1750380}\n",
+    "}\n",
     "\n",
-    " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$)\n",
+    "@article{avrami1940kinetics,\n",
+    "  title={Kinetics of phase change. II transformation-time relations for random distribution of nuclei},\n",
+    "  author={Avrami, Melvin},\n",
+    "  journal={The Journal of chemical physics},\n",
+    "  volume={8},\n",
+    "  number={2},\n",
+    "  pages={212--224},\n",
+    "  year={1940},\n",
+    "  publisher={American Institute of Physics},\n",
+    "  url={https://doi.org/10.1063/1.1750631}\n",
+    "}\n",
     "\n",
-    "One PFHub upload is required with one data file for each of the three simulations (for a total of three data files). The data sets should be named as `solid_fraction_1`, `solid_fraction_2` as well as `solid_fraction_3` where `_1`, `_2` and `_3` refers to case (1), (2) and (3), respectively. The data files should be stored in either CSV or JSON format. The column names for time and solid fraction need to be identified correctly on the upload form for each data set."
+    "@article{avrami1941granulation,\n",
+    "  title={Granulation, phase change, and microstructure kinetics of phase change. III},\n",
+    "  author={Avrami, Melvin},\n",
+    "  journal={The Journal of chemical physics},\n",
+    "  volume={9},\n",
+    "  number={2},\n",
+    "  pages={177--184},\n",
+    "  year={1941},\n",
+    "  publisher={American Institute of Physics},\n",
+    "  url={https://doi.org/10.1063/1.1750872}\n",
+    "}\n",
+    "\n",
+    "@article{kolmogorov1937izv,\n",
+    "  title={On the Statistical Theory of the Crystallization of Metals},\n",
+    "  author={Kolmogorov, A N},\n",
+    "  journal={Bulletin of the Acadamy of Sciences of the USSR},\n",
+    "  series={Mathematical Series},\n",
+    "  volume={1},\n",
+    "  pages={355--359},\n",
+    "  year={1937},\n",
+    "  url={https://doi.org/10.1007/978-94-011-2260-3_22}\n",
+    "}\n",
+    "\n",
+    "@article{quested2005athermal,\n",
+    "  title={Athermal heterogeneous nucleation of solidification},\n",
+    "  author={Quested, T E and Greer, A L},\n",
+    "  journal={Acta Materialia},\n",
+    "  volume={53},\n",
+    "  number={9},\n",
+    "  pages={2683--2692},\n",
+    "  year={2005},\n",
+    "  publisher={Elsevier},\n",
+    "  url={https://doi.org/10.1016/j.actamat.2005.02.028}\n",
+    "}\n",
+    "\n",
+    "@article{greer2000modelling,\n",
+    "  title={Modelling of inoculation of metallic melts: application to grain refinement of aluminium by {A}l--{T}i--{B}},\n",
+    "  author={Greer, A L and Bunn, A M and Tronche, A and Evans, P V and Bristow, D J},\n",
+    "  journal={Acta materialia},\n",
+    "  volume={48},\n",
+    "  number={11},\n",
+    "  pages={2823--2835},\n",
+    "  year={2000},\n",
+    "  publisher={Elsevier},\n",
+    "  url={https://doi.org/10.1016/S1359-6454(00)00094-X}\n",
+    "}\n",
+    "\n",
+    "@article{greer2016overview,\n",
+    "  title={Overview: Application of heterogeneous nucleation in grain-refining of metals},\n",
+    "  author={Greer, A L},\n",
+    "  journal={The Journal of Chemical Physics},\n",
+    "  volume={145},\n",
+    "  number={21},\n",
+    "  pages={211704},\n",
+    "  year={2016},\n",
+    "  publisher={AIP Publishing LLC},\n",
+    "  url={https://doi.org/10.1063/1.4968846}\n",
+    "}\n",
+    "\n",
+    "\n",
+    "@article{dewitt2020prisms,\n",
+    "  title={PRISMS-PF: A general framework for phase-field modeling with a matrix-free finite element method},\n",
+    "  author={DeWitt, Stephen and Rudraraju, Shiva and Montiel, David and Andrews, W Beck and Thornton, Katsuyo},\n",
+    "  journal={npj Computational Materials},\n",
+    "  volume={6},\n",
+    "  number={1},\n",
+    "  pages={1--12},\n",
+    "  year={2020},\n",
+    "  publisher={Nature Publishing Group},\n",
+    "  url={https://doi.org/10.1038/s41524-020-0298-5}\n",
+    "}\n",
+    "\n",
+    "\n",
+    "@article{guyer2009fipy,\n",
+    "  title={FiPy: Partial differential equations with Python},\n",
+    "  author={Guyer, Jonathan E and Wheeler, Daniel and Warren, James A},\n",
+    "  journal={Computing in Science \\& Engineering},\n",
+    "  volume={11},\n",
+    "  number={3},\n",
+    "  pages={6--15},\n",
+    "  year={2009},\n",
+    "  publisher={IEEE},\n",
+    "  url={https://doi.org/10.1109/MCSE.2009.52}\n",
+    "}\n",
+    "\n",
+    "@article{wheeler2019pfhub,\n",
+    "  title={PFHub: The Phase-Field Community Hub},\n",
+    "  author={Wheeler, Daniel and Keller, Trevor and DeWitt, Stephen J and Jokisaari, Andrea M and Schwen, Daniel and Guyer, Jonathan E and Aagesen, Larry K and Heinonen, Olle G and Tonks, Michael R and Voorhees, Peter W and others},\n",
+    "  journal={Journal of Open Research Software},\n",
+    "  volume={7},\n",
+    "  number={1},\n",
+    "  year={2019},\n",
+    "  publisher={Ubiquity Press},\n",
+    "  url={http://doi.org/10.5334/jors.276}\n",
+    "}\n",
+    "\n",
+    "@article{allen1979microscopic,\n",
+    "  title={A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening},\n",
+    "  author={Allen, Samuel M and Cahn, John W},\n",
+    "  journal={Acta Metallurgica},\n",
+    "  volume={27},\n",
+    "  issue = {6},\n",
+    "  pages={1085--1095},\n",
+    "  year={1979},\n",
+    "  url={https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2}\n",
+    "}\n",
+    "\n",
+    "-->"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# Under construction"
+    "## References\n",
+    "\n",
+    "<a id=\"cite-granasy2002nucleation\"/><sup><a href=\"#ref-1\">[^]</a><a href=\"#ref-9\">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. _Nucleation and bulk crystallization in binary phase field theory_. [URL](https://doi.org/10.1103/PhysRevLett.88.206105)\n",
+    "\n",
+    "<a id=\"cite-castro2003phase\"/><sup><a href=\"#ref-2\">[^]</a></sup>Castro, Mario. 2003. _Phase-field approach to heterogeneous nucleation_. [URL](https://link.aps.org/doi/10.1103/PhysRevB.67.035412)\n",
+    "\n",
+    "<a id=\"cite-simmons2004microstructural\"/><sup><a href=\"#ref-3\">[^]</a></sup>Simmons, J P and Wen, Y and Shen, C and Wang, Y. 2004. _Microstructural development involving nucleation and growth phenomena simulated with the Phase Field method_. [URL](https://doi.org/10.1016/j.msea.2003.09.019)\n",
+    "\n",
+    "<a id=\"cite-granasy2007phase\"/><sup><a href=\"#ref-4\">[^]</a></sup>Gránásy, L and T and Pusztai, T and Saylor, D and Warren, J A. 2007. _Phase field theory of heterogeneous crystal nucleation_. [URL](https://doi.org/10.1103/PhysRevLett.98.035703)\n",
+    "\n",
+    "<a id=\"cite-warren2009phase\"/><sup><a href=\"#ref-5\">[^]</a></sup>Warren, J A and Pusztai, T and Gránásy, L K and Gránásy, L. 2009. _Phase field approach to heterogeneous crystal nucleation in alloys_. [URL](https://doi.org/10.1103/PhysRevB.79.014204)\n",
+    "\n",
+    "<a id=\"cite-heo2014phase\"/><sup><a href=\"#ref-6\">[^]</a></sup>Heo, T W and Chen, L. 2014. _Phase-field modeling of nucleation in solid-state phase transformations_. [URL](https://doi.org/10.1007/s11837-014-1033-9)\n",
+    "\n",
+    "<a id=\"cite-granasy2019\"/><sup><a href=\"#ref-7\">[^]</a></sup>Gránásy, L and Tóth, G and Warren, J A and Podmaniczky, F and Tegze, G and Rátkai, L and Pusztai, T. 2019. _Phase-field modeling of crystal nucleation in undercooled liquids -- A review_. [URL](https://doi.org/10.1016/j.pmatsci.2019.05.002)\n",
+    "\n",
+    "<a id=\"cite-kubo1966\"/><sup><a href=\"#ref-8\">[^]</a></sup>Kubo, R. 1966. _The fluctuation-dissipation theorem_. [URL](https://doi.org/10.1088%2F0034-4885%2F29%2F1%2F306)\n",
+    "\n",
+    "<a id=\"cite-simmons2000phase\"/><sup><a href=\"#ref-10\">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. _Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events_. [URL](https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0)\n",
+    "\n",
+    "<a id=\"cite-shi2019\"/><sup><a href=\"#ref-11\">[^]</a></sup>Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J. 2019. _Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response_. [URL](https://doi.org/10.1007/s11837-019-03618-1)\n",
+    "\n",
+    "<a id=\"cite-johnson1939reaction\"/><sup><a href=\"#ref-12\">[^]</a><a href=\"#ref-18\">[^]</a></sup>Johnson, William A and Mehl, R F. 1939. _Reaction kinetics in processes of nucleation and growth_. [URL](https://doi.org/10.1007/s11663-010-9421-1)\n",
+    "\n",
+    "<a id=\"cite-avrami1939kinetics\"/><sup><a href=\"#ref-13\">[^]</a><a href=\"#ref-19\">[^]</a></sup>Avrami, Melvin. 1939. _Kinetics of phase change. I General theory_. [URL](https://doi.org/10.1063/1.1750380)\n",
+    "\n",
+    "<a id=\"cite-avrami1940kinetics\"/><sup><a href=\"#ref-14\">[^]</a><a href=\"#ref-20\">[^]</a></sup>Avrami, Melvin. 1940. _Kinetics of phase change. II transformation-time relations for random distribution of nuclei_. [URL](https://doi.org/10.1063/1.1750631)\n",
+    "\n",
+    "<a id=\"cite-avrami1941granulation\"/><sup><a href=\"#ref-15\">[^]</a><a href=\"#ref-21\">[^]</a></sup>Avrami, Melvin. 1941. _Granulation, phase change, and microstructure kinetics of phase change. III_. [URL](https://doi.org/10.1063/1.1750872)\n",
+    "\n",
+    "<a id=\"cite-kolmogorov1937izv\"/><sup><a href=\"#ref-16\">[^]</a><a href=\"#ref-22\">[^]</a></sup>Kolmogorov, A N. 1937. _On the Statistical Theory of the Crystallization of Metals_. [URL](https://doi.org/10.1007/978-94-011-2260-3_22)\n",
+    "\n",
+    "<a id=\"cite-allen1979microscopic\"/><sup><a href=\"#ref-17\">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. _A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening_. [URL](https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2)\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Appendix\n",
+    "\n",
+    "The following code is given as a resource for those implementing the benchmark to help count the particles in Part (b) and Part (c). One possibility as outlined here is to use [`scipy.ndimage.label`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.label.html#scipy.ndimage.label). This returns a tuple with a labeled array of unique features as the first item and a count of features as the second item. In the case of the phase field the values will require thresholding to 0 or 1 before using `scipy.ndimage.label`."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 138,
+   "metadata": {},
+   "outputs": [
+    {
+     "name": "stdout",
+     "output_type": "stream",
+     "text": [
+      "Actual number of particles: 14\n"
+     ]
+    },
+    {
+     "data": {
+      "image/png": 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\n",
+      "text/plain": [
+       "<matplotlib.figure.Figure at 0x7f23c00f4fd0>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "import numpy as np\n",
+    "import scipy.ndimage\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def make_particles(shape, n_particles, max_radius, seed=99):\n",
+    "    \"\"\"Make an array with random particles\n",
+    "    \n",
+    "    Args:\n",
+    "      shape: shape of array\n",
+    "      n_particles: number of particles in arrau\n",
+    "      max_radius: maximum possible radius of particles\n",
+    "      seed: random seed\n",
+    "      \n",
+    "    Returns: numpy array mask with particles\n",
+    "    \"\"\"\n",
+    "    np.random.seed(seed)\n",
+    "        \n",
+    "    def xy_grid(shape):\n",
+    "        \"\"\"Generate a grid of x, y values in a single array\n",
+    "        \n",
+    "        The returned array has shape (len(shape),) + shape + (1,). The last axis is\n",
+    "        for the operations with particles.\n",
+    "        \"\"\"\n",
+    "        linspace = lambda x: np.linspace(0, x - 1, x) + 0.5\n",
+    "        return np.array(np.meshgrid(*map(linspace, shape), indexing=\"ij\"))[..., None]\n",
+    "    \n",
+    "    # first axis is dimensions\n",
+    "    # second and third axes are shape of domain\n",
+    "    # last axis is the number of particles. \n",
+    "    centers = (np.random.random((len(shape), n_particles)) * np.array(shape)[:, None])[:, None, None]\n",
+    "    radii = (np.random.random(n_particles) * max_radius)[None, None]\n",
+    "    return numpy.any(((xy_grid(shape) - centers)**2).sum(0) < radii**2, axis=-1)\n",
+    "\n",
+    "arr = make_particles((100, 200), 20, 10)\n",
+    "plt.imshow(arr)\n",
+    "print('Actual number of particles:', scipy.ndimage.label(arr)[1])"
    ]
   },
   {
@@ -227,10 +1050,78 @@
  ],
  "metadata": {
   "anaconda-cloud": {},
+  "cite2c": {
+   "citations": {
+    "5566346/A4PRV46K": {
+     "DOI": "10.1103/PhysRevE.57.4323",
+     "URL": "https://link.aps.org/doi/10.1103/PhysRevE.57.4323",
+     "accessed": {
+      "day": 27,
+      "month": 11,
+      "year": 2017
+     },
+     "author": [
+      {
+       "family": "Karma",
+       "given": "Alain"
+      },
+      {
+       "family": "Rappel",
+       "given": "Wouter-Jan"
+      }
+     ],
+     "container-title": "Physical Review E",
+     "id": "5566346/A4PRV46K",
+     "issue": "4",
+     "issued": {
+      "day": 1,
+      "month": 4,
+      "year": 1998
+     },
+     "language": "en",
+     "page": "4323-4349",
+     "page-first": "4323",
+     "title": "Quantitative phase-field modeling of dendritic growth in two and three dimensions",
+     "type": "article-journal",
+     "volume": "57"
+    },
+    "5566346/YSJ7YY8Q": {
+     "DOI": "10.1016/0956-7151(94)00285-P",
+     "URL": "http://linkinghub.elsevier.com/retrieve/pii/095671519400285P",
+     "accessed": {
+      "day": 27,
+      "month": 11,
+      "year": 2017
+     },
+     "author": [
+      {
+       "family": "Warren",
+       "given": "J.A."
+      },
+      {
+       "family": "Boettinger",
+       "given": "W.J."
+      }
+     ],
+     "container-title": "Acta Metallurgica et Materialia",
+     "id": "5566346/YSJ7YY8Q",
+     "issue": "2",
+     "issued": {
+      "year": 1995
+     },
+     "language": "en",
+     "page": "689-703",
+     "page-first": "689",
+     "title": "Prediction of dendritic growth and microsegregation patterns in a binary alloy using the phase-field method",
+     "type": "article-journal",
+     "volume": "43"
+    }
+   }
+  },
   "kernelspec": {
-   "display_name": "Python 2",
+   "display_name": "Python 3",
    "language": "python",
-   "name": "python2"
+   "name": "python3"
   },
   "language_info": {
    "codemirror_mode": {
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index 84b68ea93..9a0414d74 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="8bf1bb58-4677-4077-8bd1-e900c4d28f08"></div>
+<div id="71b92ce1-b10f-4d17-943f-11fc615034b1"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#8bf1bb58-4677-4077-8bd1-e900c4d28f08');
+var element = $('#71b92ce1-b10f-4d17-943f-11fc615034b1');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -106,41 +106,64 @@
 </div>
 <div class="cell border-box-sizing code_cell rendered">
 <div class="input">
-<div class="prompt input_prompt">In&nbsp;[1]:</div>
+<div class="prompt input_prompt">In&nbsp;[3]:</div>
 <div class="inner_cell">
     <div class="input_area">
-<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Disable until merge</span>
-<span class="c1">#from IPython.display import HTML</span>
-<span class="c1">#</span>
-<span class="c1">#HTML(&#39;&#39;&#39;</span>
-<span class="c1">#&lt;a href=&quot;https://github.com/usnistgov/pfhub/raw/master/benchmarks/benchmark8.ipynb&quot;</span>
-<span class="c1">#   download&gt;</span>
-<span class="c1">#&lt;button type=&quot;submit&quot;&gt;Download Notebook&lt;/button&gt;</span>
-<span class="c1">#&lt;/a&gt;</span>
-<span class="c1">#&#39;&#39;&#39;)</span>
+<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="k">import</span> <span class="n">HTML</span>
+
+<span class="n">HTML</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;</span>
+<span class="s1">&lt;a href=&quot;https://github.com/usnistgov/pfhub/raw/master/benchmarks/benchmark8.ipynb&quot;</span>
+<span class="s1">   download&gt;</span>
+<span class="s1">&lt;button type=&quot;submit&quot;&gt;Download Notebook&lt;/button&gt;</span>
+<span class="s1">&lt;/a&gt;</span>
+<span class="s1">&#39;&#39;&#39;</span><span class="p">)</span>
 </pre></div>
 
 </div>
 </div>
 </div>
 
+<div class="output_wrapper">
+<div class="output">
+
+
+<div class="output_area">
+
+<div class="prompt output_prompt">Out[3]:</div>
+
+
+
+<div class="output_html rendered_html output_subarea output_execute_result">
+
+<a href="https://github.com/usnistgov/pfhub/raw/master/benchmarks/benchmark8.ipynb"
+   download>
+<button type="submit">Download Notebook</button>
+</a>
+
+</div>
+
+</div>
+
+</div>
+</div>
+
 </div>
 <div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Benchmark-Problem-8:-Nucleation">Benchmark Problem 8: Nucleation<a class="anchor-link" href="#Benchmark-Problem-8:-Nucleation">&#182;</a></h1>
+<h1 id="Benchmark-Problem-8:-Homogeneous-Nucleation">Benchmark Problem 8: Homogeneous Nucleation<a class="anchor-link" href="#Benchmark-Problem-8:-Homogeneous-Nucleation">&#182;</a></h1>
 </div>
 </div>
 </div>
 <div class="cell border-box-sizing code_cell rendered">
 <div class="input">
-<div class="prompt input_prompt">In&nbsp;[1]:</div>
+<div class="prompt input_prompt">In&nbsp;[4]:</div>
 <div class="inner_cell">
     <div class="input_area">
 <div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">IPython.display</span> <span class="k">import</span> <span class="n">HTML</span>
 
-<span class="n">HTML</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;{</span><span class="si">% i</span><span class="s1">nclude jupyter_benchmark_table.html num=&quot;[8]&quot; revision=0 %}&#39;&#39;&#39;</span><span class="p">)</span>
+<span class="n">HTML</span><span class="p">(</span><span class="s1">&#39;&#39;&#39;{</span><span class="si">% i</span><span class="s1">nclude jupyter_benchmark_table.html num=&quot;[8]&quot; revision=1 %}&#39;&#39;&#39;</span><span class="p">)</span>
 </pre></div>
 
 </div>
@@ -153,12 +176,12 @@ <h1 id="Benchmark-Problem-8:-Nucleation">Benchmark Problem 8: Nucleation<a class
 
 <div class="output_area">
 
-<div class="prompt output_prompt">Out[1]:</div>
+<div class="prompt output_prompt">Out[4]:</div>
 
 
 
 <div class="output_html rendered_html output_subarea output_execute_result">
-{% include jupyter_benchmark_table.html num="[8]" revision=0 %}
+{% include jupyter_benchmark_table.html num="[8]" revision=1 %}
 </div>
 
 </div>
@@ -171,7 +194,237 @@ <h1 id="Benchmark-Problem-8:-Nucleation">Benchmark Problem 8: Nucleation<a class
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h3 id="Warning">Warning<a class="anchor-link" href="#Warning">&#182;</a></h3><p>This benchmark is still under construction, but the submission guidelines are included for development purposes.</p>
+<p>See the journal publication entitled <a href="https://doi.org/10.1016/j.commatsci.2018.03.015">"Phase Field Benchmark Problems for Nucleation"</a> for more details about the benchmark problems. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h1 id="Table-of-Contents">Table of Contents<a class="anchor-link" href="#Table-of-Contents">&#182;</a></h1><ul>
+<li><a href="#Overview">Overview</a></li>
+<li><a href="#Model-Formulation">Model Formulation</a><ul>
+<li><a href="#The-phase-field-model-in-classical-nucleation-theory">The phase field model in classical nucleation theory</a></li>
+<li><a href="#The-properties-of-the-classical-nucleus">The properties of the classical nucleus</a></li>
+<li><a href="#Avrami-plots">Avrami plots</a></li>
+</ul>
+</li>
+<li><a href="#Benchmark-8-Specification">Benchmark 8 Specification</a><ul>
+<li><a href="#Part-%28a%29:-single-seed">Part (a): single seed</a></li>
+<li><a href="#Part-%28b%29:-multiple-initial-seeds">Part (b): multiple initial seeds</a></li>
+<li><a href="#Part-%28c%29:-multiple-seeds-at-random-times">Part (c): multiple seeds at random times</a></li>
+</ul>
+</li>
+<li><a href="#Submission-Guidelines">Submission Guidelines</a><ul>
+<li><a href="#Part-%28a%29">Part (a)</a></li>
+<li><a href="#Part-%28b%29">Part (b)</a></li>
+<li><a href="#Part-%28c%29">Part (c)</a></li>
+</ul>
+</li>
+<li><a href="#Results">Results</a></li>
+<li><a href="#References">References</a></li>
+</ul>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h2 id="Overview">Overview<a class="anchor-link" href="#Overview">&#182;</a></h2>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Phase field modeling of nucleation has a long history and is covered
+in a number of reviews (<a id="ref-1" href="#cite-granasy2002nucleation">Gránásy 2002</a>,
+<a id="ref-2" href="#cite-castro2003phase">Castro 2003</a>,
+<a id="ref-3" href="#cite-simmons2004microstructural">Simmons 2004</a>,
+<a id="ref-4" href="#cite-granasy2007phase">Gránásy 2007</a>,
+<a id="ref-5" href="#cite-warren2009phase">Warren 2009</a>,
+<a id="ref-6" href="#cite-heo2014phase">Heo 2014</a>,
+<a id="ref-7" href="#cite-granasy2019">Gránásy 2019</a>). The problem formulation of crystallization of an ideal pure liquid cooled
+below its melting point starts with homogeneous nucleation, a process
+in which the internal fluctuations of the undercooled liquid lead to
+the formation of crystal-like seeds able to grow to macroscopic
+sizes. The nucleation can be assisted by the presence of surfaces
+(container walls, foreign particles, etc.), in which case the process
+is termed heterogeneous nucleation. We note that homogeneous
+nucleation is an idealized formulation, and it is unlikely homogeneous
+nucleation occurs, due to impurities present in experimental
+apparatus. However, creating benchmark problems for homogeneous
+nucleation is still needed since the focus in nucleation research
+often lies on the underlying homogeneous nucleation, as it is the
+basis for advancing theoretical approaches to the much more complex
+phenomena governing heterogeneous nucleation. This benchmark
+(benchmark 8) targets homogeneous nucleation while benchmark 9
+targets heterogeneous nucleation.</p>
+<p>There are two main modeling approaches to introduce nuclei into a
+metastable system: the Langevin noise method
+(<a id="ref-8" href="#cite-kubo1966">Kubo 1966</a>,
+<a id="ref-9" href="#cite-granasy2002nucleation">Gránásy 2002</a>)
+and the explicit nucleation
+method
+(<a id="ref-10" href="#cite-simmons2000phase">Simmons 2000</a>,
+<a id="ref-11" href="#cite-shi2019">Shi 2019</a>)
+. We focus on the explicit
+method, where the critical nucleus size and the nucleation energy are
+determined by the classical nucleation theory. We break down our
+consideration of homogeneous nucleation into three parts. Part (a)
+considers the simple case of single-seed homogeneous nucleation. We
+explore how the particle size influences the evolution of the nucleus
+when the thermodynamic driving force is close to the critical value
+where we can observe whether the particle grows, dissolves or remains
+stationary. In parts (b) and (c) we consider multiple-seed homogeneous
+nucleation, with nuclei appearing at fixed time $t=0$ or at random
+times distributed uniformly, respectively, with the latter part
+illustrating a scenario with constant nucleation rate. The Benchmark
+Problem probes the differences between transformation kinetics of the
+two cases and summarizes them using the Avrami plots based on the
+Johnson-Mehl-Avrami-Kolmogorov (JMAK)
+theory
+(<a id="ref-12" href="#cite-johnson1939reaction">Johnson 1939</a>,
+<a id="ref-13" href="#cite-avrami1939kinetics">Avrami 1939</a>,
+<a id="ref-14" href="#cite-avrami1940kinetics">Avrami 1940</a>,
+<a id="ref-15" href="#cite-avrami1941granulation">Avrami 1941</a>,
+<a id="ref-16" href="#cite-kolmogorov1937izv">Kolmogorov 1937</a>).</p>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h2 id="Model-Formulation">Model Formulation<a class="anchor-link" href="#Model-Formulation">&#182;</a></h2><h3 id="The-phase-field-model-in-classical-nucleation-theory">The phase field model in classical nucleation theory<a class="anchor-link" href="#The-phase-field-model-in-classical-nucleation-theory">&#182;</a></h3><p>For these benchmark problems we use the simplest possible phase field model with a single non-conserved phase field $\phi$, which describes an isothermal pure substance with one liquid ($\phi=0$) and one solid ($\phi=1$) phase. The free energy of this system is<br>
+\begin{equation}
+    \mathcal{F}(\phi)=\int \left[\frac{\epsilon^2}{2}(\nabla \phi)^2+wg(\phi)-\Delta f p(\phi) \right] d{V},
+    \label{eqn:Fphi}
+\end{equation}
+set $g(\phi)=\phi^2(1-\phi)^2$, a simple double well function with minima at $\phi=0$ and $\phi=1$, $ w$ controls the double-well barrier height, $\epsilon^2$ is the gradient energy coefficient, let $p(\phi)=\phi^3(10-15\phi+6\phi^2)$, which ensures that $p(0)=p'(0)=p'(1)=0$ and $p(1)=1$, and $\Delta f$ is the driving force for solidification at the simulation temperature ($\Delta f$ is positive below the melting point). The time evolution of $\phi$ is given by the Allen-Cahn equation (<a id="ref-17" href="#cite-allen1979microscopic">Allen 1979</a>)
+\begin{equation}
+    \frac{\partial \phi}{\partial t}=-M\frac{\delta \mathcal{F}}{\delta \phi}=M\left[\epsilon^2\nabla^2\phi-wg'(\phi)+\Delta fp'(\phi)\right]
+    \label{eqn:dphidt}
+\end{equation}
+where $M$ is the mobility parameter. We will restrict the problem to two dimensions (2D). For a planar interface, one can show that the equilibrium ($\Delta f$=0) solid-liquid interface profile with the interface centered at $x=x_0$ is given by
+\begin{equation}
+    \phi(x)=\frac{1}{2}\left[1-\tanh\left(\frac{x-x_0}{\sqrt{2}\ell}\right)\right],
+    \label{eqn:phir}
+\end{equation}
+where $x$ is the perpendicular distance from the interface.
+We use this expression as the initial condition when we introduce a nucleus. We can also obtain the width $\ell$ and the free energy of the interface, $\gamma$, as
+\begin{equation}
+    \ell=\sqrt{\frac{\epsilon^2}{w}}
+    \label{eqn:l}
+\end{equation}
+and
+\begin{equation}
+    \gamma=\frac{\sqrt{\epsilon^2w}}{3\sqrt{2}}.
+    \label{eqn:gamma}
+\end{equation}</p>
+<p>Choosing a characteristic length scale $\xi=\ell$ and a characteristic time scale $\tau=1/(Mw)$, we can obtain a nondimensional form of the problem (the nondimensional quantities are denoted by tildes),
+\begin{equation}
+    \tilde{\mathcal{F}}(\phi)=\int \left[\frac{1}{2}(\tilde{\nabla} \phi)^2+g(\phi)- \tilde{\Delta f} p(\phi) \right] d{\tilde{V}}
+    \label{eqn:tFphi}
+\end{equation}
+and
+\begin{equation}
+    \frac{\partial \phi}{\partial \tilde{t}}=\tilde{\nabla}^2\phi-g'(\phi)+\tilde{\Delta f}p'(\phi),
+    \label{eqn:dphidtt}
+\end{equation}
+with $\tilde{\Delta f}=\Delta f/w$.</p>
+<h3 id="The-properties-of-the-classical-nucleus">The properties of the classical nucleus<a class="anchor-link" href="#The-properties-of-the-classical-nucleus">&#182;</a></h3><p>The explicit nucleation method that we employ introduces nuclei into a metastable system, with their radii and nucleation barrier determined from classical nucleation theory. Classical nucleation theory views crystallite fluctuations appearing in the undercooled liquid as small spherical domains of the bulk crystalline phase bounded by a mathematically sharp solid-liquid interface. For a 2D system, the free energy of a circular solid particle of radius $r$ is 
+\begin{equation}
+    \Delta G(r) = 2 \pi r \gamma - \pi r^2 \Delta f
+    \label{eqn:deltaG}
+\end{equation}
+where $\Delta f$ is the nucleation driving force that is used in our phase field model, and $\gamma$ is the free energy of the interface. The free energy of the particle is a balance between the energy cost in forming the solid-liquid interface, and the free energy from the driving force which is released when the crystalline particle forms. Once the rate of change of free energy with respect to particle size becomes negative, the particle can grow. Taking the derivative of $\Delta G(r)$ we get the rate of change of free energy with respect to radius as
+\begin{equation}
+    \frac{d\Delta G}{dr}=2\pi \gamma - 2\pi r\Delta f.
+    \label{eqn:ddeltaGdr}
+\end{equation}
+By setting $d\Delta G/dr$ to zero, we obtain the critical radius $r^*$ as
+\begin{equation}
+    r^* = \frac{\gamma}{\Delta f}.
+    \label{eqn:rc}
+\end{equation}
+The corresponding critical nucleation free energy is
+\begin{equation}
+    \Delta G^* = \frac{\pi \gamma^2}{\Delta f}.
+    \label{eqn:deltaGc}
+\end{equation}
+Using the same units as before, the nondimensional forms of these quantities are
+\begin{equation}
+    \tilde{r^*} = \frac{1}{3\sqrt{2}}\frac{1}{\tilde{\Delta f}}
+    \label{eqn:trc}
+\end{equation}
+and
+\begin{equation}
+    \tilde{\Delta G^*} = \frac{\pi}{18}\frac{1}{\tilde{\Delta f}}
+    \label{eqn:tdeltaGc},
+\end{equation}
+respectively.</p>
+<h3 id="Avrami-plots">Avrami plots<a class="anchor-link" href="#Avrami-plots">&#182;</a></h3><p>Avrami plots describe how solids transform from one phase to another at constant temperature. In particular, they can describe the kinetics of nucleation. The Avrami plots come from the JMAK theory
+(<a id="ref-18" href="#cite-johnson1939reaction">Johnson 1939</a>,
+<a id="ref-19" href="#cite-avrami1939kinetics">Avrami 1939</a>,
+<a id="ref-20" href="#cite-avrami1940kinetics">Avrami 1940</a>,
+<a id="ref-21" href="#cite-avrami1941granulation">Avrami 1941</a>,
+<a id="ref-22" href="#cite-kolmogorov1937izv">Kolmogorov 1937</a>), which makes a number of 
+assumptions and simplifications: (i)  nucleation occurs randomly and homogeneously over the entire un-transformed portion of the material, (ii) the growth rate is constant and does not depend on the extent of transformation, (iii) the particles have convex shape with the same orientation, and (iv) the size of the system is infinite (both in space and time).</p>
+<p>The basis of the Avrami plots is the transformed fraction (solid fraction) $X(t)$ vs. time. According to the JMAK theory,
+\begin{equation}
+    Y(t)=1-\exp (-Kt^n),
+    \label{eqn:Xt}
+\end{equation}
+where $K$ is a constant depending on the nucleation and growth rates, $n=d+1$ for continuous nucleation and $n=d$ if nucleation happens only at $t=0$, and $d$ is the number of spatial dimensions. If we plot $\log(-\log(1-Y))$ vs. $\log(t)$, then for the JMAK kinetics (Eq. \ref{eqn:Xt}) we get a straight line with slope $n$.</p>
+<h2 id="Benchmark-8-Specification">Benchmark 8 Specification<a class="anchor-link" href="#Benchmark-8-Specification">&#182;</a></h2><p>From here on, we will use only the nondimensional forms of the phase field (Eqs. \ref{eqn:tFphi}, \ref{eqn:dphidtt}, and \ref{eqn:phir}) and nucleation (Eqs. \ref{eqn:trc} and \ref{eqn:tdeltaGc}) equations, but we will drop the tildes.</p>
+<h3 id="Part-(a):-single-seed">Part (a): single seed<a class="anchor-link" href="#Part-(a):-single-seed">&#182;</a></h3><p>In this problem we examine the morphology change of the nucleus for different initial radii. We consider a 2D simulation domain of size $100 \times 100$ units centered at $x=y=0$ with Neumann boundary conditions. The driving force is set to $\Delta f = \sqrt{2}/30$, which corresponds to a critical radius of $r^*=5$ (Eq. \ref{eqn:trc}). Next, we place a circular seed at the center of the domain. 
+Incorporating a nucleus into a diffuse interface phase field model leads to a small offset from a classical, sharp-interface model of a nucleus. To account for this,
+we incorporate a diffuse-interface seed using the profile $\phi(r)$ given by modifying Eq. \ref{eqn:phir} to 2D where $r=\sqrt{x^2+y^2}$, with a radius $r_0=r^*$. This seed is the diffuse-interface approximation of the classical sharp interface nucleus corresponding to the given $\Delta f$, and therefore it should be fairly close to (an unstable) equilibrium. 
+Part (a) is then defined by the Allen-Cahn equation
+\begin{equation}
+    \frac{\partial \phi}{\partial {t}}={\nabla}^2\phi-g'(\phi)+{\Delta f}p'(\phi)
+    \label{eqn:dphidtt_2}
+\end{equation}
+for the phase field $\phi$, and the three different initial conditions: in the first case with the seed radius $r_0$ corresponding exactly to the critical one (critical nucleus), and in the other two cases, slightly below and above the critical radius (subcritical and supercritical nuclei):
+\begin{equation}
+    \phi(r)=\frac{1}{2}\left[1-\tanh\left(
+    \frac{r-r_0}{\sqrt{2}}
+    \right)\right],
+    \label{eqn:phitr_2}
+\end{equation}
+where $r_0=\{r^*, 0.99r^*, 1.01r^*\}$. <a href="#fig1">Figure 1</a> shows the computational domain with an initial seed of radius $r=r^*$. The time evolution of the system is then followed for times up to $t = 200$ units, and the solid fraction $Y$ and the total free energy $F$ are plotted as functions of time. Finally, the problem also includes a convergence test with respect to mesh (spatial resolution). The closer the initial radius $r$ is to the critical radius $r^*$, the more sensitive the numerical integration will be to round-off errors. Therefore, a convergence test is included for $r_0=1.01r^*$ using successive runs each halving the average spatial mesh size $\Delta x$ between consecutive runs. At least 3 runs are required to estimate the rate of convergence based on the $L^2$ error with the finest mesh being used as a gold standard. The simulations should be run to $t=200$ and the final phase field (at $t=200$) for each successive run should be submitted.</p>
+<p>To submit results to the PFHub website, see the <a href="#Part-1.6.1-Part-%28a%29">submission guidelines</a> below.</p>
+<figure>
+    <img id="fig1" src="../images/t0_single_seed.PNG" alt="Single seed at t=0" style="width:500px;"/>
+    <figcaption>
+        <center> Illustration of the $100 \times 100$ computational domain in 2D. </center> 
+    </figcaption>
+</figure><h2 id="Part-(b):-multiple-initial-seeds">Part (b): multiple initial seeds<a class="anchor-link" href="#Part-(b):-multiple-initial-seeds">&#182;</a></h2><p>The second part of the homogeneous nucleation benchmark problem focuses on the kinetics of nucleation using Avrami plots and compares them to JMAK theory. To obtain reasonable statistics for these two parts, the simulation volume needs to be larger so it can encompass a larger number of smaller nuclei. We therefore increase the domain size to $500 \times 500$ units of length, and the driving force to $\Delta f = 1/(3 \sqrt{2})$, which corresponds to a critical radius of $r^*=1$. Random initial positions $\mathbf{r}_i$ of 25 supercritical seeds $i, i=1,\ldots,25$ are generated with $r_0=2.2r^*$ drawn from a uniform distribution on the $500\times500$ domain. The distribution of the phase field $\phi$ is the sum of the phase fields $\phi_i$ with profiles
+\begin{equation}
+    \phi_i(r)=\frac{1}{2}\left[1-\tanh\left(
+    \frac{|\mathbf{r}-\mathbf{r}_i|-2.2r^*}{\sqrt{2}}
+    \right)\right].
+    \label{eqn:phitr_3}
+\end{equation}
+from the different seeds $i$. Overlaps between different seeds are managed by setting $\phi=1$ in all regions where the sums of $\phi_i &gt;1$. After placing initial seeds and adjusting the initial phase field $\phi$ so that $\phi\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $X$, discrete particle count $N$  (the number of disjoint regions with $\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.</p>
+<p>Both Part (b) and Part (c) involve a random number generator. Because different executions will in general generate different random numbers, specific individual solutions will in general not be reproduced. However, statistics such as the average slope of Avrami plots can be meaningfully compared.</p>
+<p>To submit results to the PFHub website, see the <a href="#Part-1.6.1-Part-%28a%29">submission guidelines</a> below.</p>
+<h2 id="Part-(c):-multiple-seeds-at-random-times">Part (c): multiple seeds at random times<a class="anchor-link" href="#Part-(c):-multiple-seeds-at-random-times">&#182;</a></h2><p>For the third part of the homogeneous benchmark problem, the computational domain is first expanded to $1000 \times1000$. Instead of inserting nuclei at fixed time $t = 0$ as in part two of the problem, in this case 100 random nucleation times $t_i$ are generated, $i=1,\ldots,100$, drawn from a uniform distribution in the interval $t_i \in [0, 600)$ with centers $\mathbf{r}_i$ drawn from a uniform distribution on the $1000\times1000$ domain.
+We decrease the driving force to $\Delta f=1/(6\sqrt{2})$, which corresponds to a critical radius or $r^*=2$, but keep the initial radii of the inserted nuclei the same as those in the second part of the problem
+\begin{equation}
+    \phi_i(r)=\frac{1}{2}\left[1-\tanh\left(
+    \frac{|\mathbf{r}-\mathbf{r}_i|-1.1r^*}{\sqrt{2}}
+    \right)\right].
+\end{equation}</p>
+<p>Again, $\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $X$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.</p>
+<p>To submit results to the PFHub website, see the <a href="#Part-1.6.3-Part-%28c%29">submission guidelines</a> below.</p>
 
 </div>
 </div>
@@ -180,52 +433,172 @@ <h3 id="Warning">Warning<a class="anchor-link" href="#Warning">&#182;</a></h3><p
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Submission-Guidelines">Submission Guidelines<a class="anchor-link" href="#Submission-Guidelines">&#182;</a></h2><p>Please use the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a> to upload your results.</p>
+<h2 id="Submission-Guidelines">Submission Guidelines<a class="anchor-link" href="#Submission-Guidelines">&#182;</a></h2>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Please use the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a> to upload your results.</p>
 <p>In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. These are not required, but will help others view your work.</p>
-<h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3><p>Run three cases with starting seeds of</p>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\Delta x$ using the model formulation <a href="#Part-1.3.1-Part-1.6.1-Part-%28a%29:-single-seed">defined above</a>. The 5 simulations are</p>
 <ol>
-<li>$r_0 = 0.99 r^*$,</li>
-<li>$r_0 = r^*$</li>
-<li>$r_0 = 1.01 r^*$.</li>
+<li>$r_0 = 0.99 r^*$, ${\Delta x} = {\Delta x}_0$</li>
+<li>$r_0 = r^*$, ${\Delta x} = {\Delta x}_0$</li>
+<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_0$</li>
+<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_1$</li>
+<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_2$</li>
 </ol>
-<p>Follow the time evolution of the system for $t = 100$ units. The following data should be collected frequently (at least 100 data points is preferable).</p>
+<p>where ${\Delta x}_0 &lt; {\Delta x}_1 &lt; {\Delta x}_2$. If using a gridded mesh each successively refined grid length can be half as fine. In the case of adaptive grids a nominal value for $\Delta x$ should be selected that results in a similarly refined mesh as in the gridded case (i.e. approximately $4\times$ as many elements). When uploading the results, the $\Delta x$ values should be provided in the datafile alongside the values for $\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.</p>
 <ul>
-<li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$)</p>
+<li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
 </li>
-<li><p>The free energy, $\mathcal{F}$</p>
+<li><p>The free energy integrated over the whole domain, $\mathcal{F}$, during the transient.</p>
 </li>
-</ul>
-<p>One PFHub upload is required with one data file for each of the three simulations (for a total of three data files). The data sets should be named as <code>solid_fraction_1</code> and <code>free_energy_1</code>, <code>solid_fraction_2</code> and <code>free_energy_2</code> as well as <code>solid_fraction_3</code> and <code>free_energy_3</code> where <code>_1</code>, <code>_2</code> and <code>_3</code> refers to case (1), (2) and (3), respectively. The data files should be stored in either CSV or JSON format. The column names for time, free energy and solid fraction need to be identified correctly on the upload form for each data set.</p>
-<p>Further data to upload can include a Youtube video, images of the nucleus at different times or the entire phase field variable at different times. These are not required, but will help others view your work.</p>
-<h3 id="Part-(b)">Part (b)<a class="anchor-link" href="#Part-(b)">&#182;</a></h3><p>Follow the time evolution of the system for $t = 400$ units. The following data should be collected frequently (at least 100 data points is preferable).</p>
-<ul>
-<li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$)</p>
+<li><p>The phase field, $\phi$, at each grid point in the domain at $t=200$.</p>
 </li>
-<li><p>The free energy, $\mathcal{F}$</p>
+<li><p>The grid spacing, $\Delta x$ at each grid point in the domain at $t=200$.</p>
 </li>
 </ul>
-<p>One PFHub upload is required with one data file. The data sets should be named as <code>solid_fraction</code> and <code>free_energy</code>. The data files should be stored in either CSV or JSON format. The results can be stored in one single file as long as each column name for time, solid fraction and free energy is identified correctly on the upload form.</p>
-<p>In addition, the phase field snapshot at time $t=80$ is required. The data set should be named as <code>phase-field-80</code> and stored in PNG or TIFF format.</p>
-<h3 id="Part-(c)">Part (c)<a class="anchor-link" href="#Part-(c)">&#182;</a></h3><p>Follow the time evolution of the system for $t = 600$ units. The following data should be collected frequently (at least 100 data points is preferable).</p>
+<p>One PFHub upload is required for all five simulations. The solid fraction and free energy data can be included in the same CSV file or in separate data files, but either way still require 2 entries in the "Data Files" section on the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a>. The entries should be labeled as <code>solid_fraction_{i}</code> or <code>free_energy_{i}</code> depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.</p>
+
+<pre><code>time,fraction,energy
+0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00
+1.000000000000000056e-01,8.205027223394931180e-03,3.577696619091880859e+00
+2.000000000000000111e-01,8.197361418448430651e-03,3.576379039721957032e+00
+...</code></pre>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>The names of the columns are arbitrary as long as they are given in "Name of x-axis column" and "Name of y-axis column" in the form. The phase field data should be named <code>phase_field_{i}</code> in the "Short name of data" entry and have the following format</p>
+
+<pre><code>x,y,phi,delta_x
+0.0,1.0,0.5,0.44
+0.0,1.5,0.49,0.44
+...</code></pre>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>where <code>x</code> and <code>y</code> is a position in the domain and <code>phi</code> is the value at that position. <code>delta_x</code> is the nominal grid spacing in that region given by the coordinates. This may be constant across the domain for gridded data, but vary for adaptive meshing. <code>delta_x</code> is only a representative value of the grid spacing in that region. The "3D" radio button should be selected for the <code>phase_field_{i}</code> entry.</p>
+<p>The grid spacing data should be named <code>grid_spacing_{i}</code> and have exactly the same data as above. Again the same file can be used, but two "Data Files" entries are required for both phase field and grid spacing. The "3D" radio button should be selected for the <code>grid_spacing_{i}</code> entry.</p>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h3 id="Part-(b)">Part (b)<a class="anchor-link" href="#Part-(b)">&#182;</a></h3>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the <a href="#Part-1.4-Part-1.6.2-Part-%28b%29:-multiple-initial-seeds">defintion above</a>. The simulations are run to $t=200$. The following data should be collected.</p>
 <ul>
-<li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$)</p>
+<li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
+</li>
+<li><p>The free energy integrated over the whole domain, $\mathcal{F}$, during the transient.</p>
 </li>
-<li><p>The free energy, $\mathcal{F}$</p>
+<li><p>The discrete particle count, $N$, during the transient.</p>
 </li>
 </ul>
-<p>One PFHub upload is required with one data file. The data sets should be named as <code>solid_fraction</code> and <code>free_energy</code>. The data files should be stored in either CSV or JSON format. The results can be stored in one single file as long as each column name for time, solid fraction and free energy is identified correctly on the upload form.</p>
-<p>In addition, the phase field snapshot at time $t=80$ is required. The data set should be named as <code>phase-field-80</code> and stored in PNG or TIFF format.</p>
-<h3 id="Part-(d)">Part (d)<a class="anchor-link" href="#Part-(d)">&#182;</a></h3><p>Run three cases with undercooling of</p>
-<ol>
-<li>$\Delta f_0 = 0.98 \Delta f_0^*$</li>
-<li>$\Delta f_0 = \Delta f_0^*$</li>
-<li>$\Delta f_0 = 1.02 \Delta f_0^*$</li>
-</ol>
-<p>Follow the time evolution of the system for $t = 5000$ units. The following data should be collected frequently (at least 100 data points is preferable).</p>
+<p>One PFHub upload is required with five data files, but with a total of 15 entries in the "Data Files" section. The solid fraction, free energy and particle count can be included in the same data file for each simulation. The entries should be labeled as <code>solid_fraction_{i}</code>, <code>free_energy_{i}</code> or <code>particle_count_{i}</code> depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.</p>
+
+<pre><code>time,fraction,energy,particle_count
+0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00,24
+1.000000000000000056e-01,8.205027223394931180e-03,3.577696619091880859e+00,24
+2.000000000000000111e-01,8.197361418448430651e-03,3.576379039721957032e+00,23
+...</code></pre>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h3 id="Part-(c)">Part (c)<a class="anchor-link" href="#Part-(c)">&#182;</a></h3>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Part (c) requires running 5 simulations with seeds randomly appearing between $t=0$ and $t=600$, see the <a href="#Part-1.5-Part-1.6.3-Part-%28c%29:-multiple-seeds-at-random-times">definition above</a>. The simulations are run to $t=600$. The following data should be collected.</p>
 <ul>
-<li>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$)</li>
+<li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
+</li>
+<li><p>The free energy integrated over the whole domain, $\mathcal{F}$, during the transient.</p>
+</li>
+<li><p>The discrete particle count, $N$, during the transient.</p>
+</li>
 </ul>
-<p>One PFHub upload is required with one data file for each of the three simulations (for a total of three data files). The data sets should be named as <code>solid_fraction_1</code>, <code>solid_fraction_2</code> as well as <code>solid_fraction_3</code> where <code>_1</code>, <code>_2</code> and <code>_3</code> refers to case (1), (2) and (3), respectively. The data files should be stored in either CSV or JSON format. The column names for time and solid fraction need to be identified correctly on the upload form for each data set.</p>
+<p>One PFHub upload is required with five data files, but with a total of 15 entries in the "Data Files" section. The solid fraction, free energy and particle count can be included in the same data file for each simulation. The entries should be labeled as <code>solid_fraction_{i}</code>, <code>free_energy_{i}</code> or <code>particle_count_{i}</code> depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.</p>
+
+<pre><code>time,fraction,energy,particle_count
+0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00,0
+1.000000000000000056e-01,8.205027223394931180e-03,3.577696619091880859e+00,1
+2.000000000000000111e-01,8.197361418448430651e-03,3.576379039721957032e+00,1
+...</code></pre>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Please use the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a> to upload your results.</p>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h2 id="Results">Results<a class="anchor-link" href="#Results">&#182;</a></h2>
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<p>Results from this benchmark problem are displayed on the <a href="{{ site.baseurl }}/simulations/8a.1">simulation result page</a> for different codes.</p>
 
 </div>
 </div>
@@ -234,7 +607,389 @@ <h3 id="Part-(d)">Part (d)<a class="anchor-link" href="#Part-(d)">&#182;</a></h3
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="Under-construction">Under construction<a class="anchor-link" href="#Under-construction">&#182;</a></h1>
+<!--
+... see https://jupyter.brynmawr.edu/services/public/dblank/jupyter.cs/Examples/Calico%20Document%20Tools%20and%20Bibtex.ipynb for details
+... also see https://jupyter.brynmawr.edu/services/public/dblank/Jupyter%20Notebook%20Users%20Manual.ipynb#5.-Bibliographic-Support
+... 
+... to make this work (sometimes just doesn't work which is infuriating 
+...
+...  
+...  - <!--bibtex needs to be at the very top of the cell it seems (very important)
+-->
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<!--bibtex
+
+@article{castro2003phase,
+  title = {Phase-field approach to heterogeneous nucleation},
+  author = {Castro, Mario},
+  journal = {Phys. Rev. B},
+  volume = {67},
+  issue = {3},
+  pages = {035412},
+  numpages = {8},
+  year = {2003},
+  month = {Jan},
+  publisher = {American Physical Society},
+  doi={10.1103/PhysRevB.67.035412},
+  URL="https://link.aps.org/doi/10.1103/PhysRevB.67.035412"
+}
+
+
+@article{elliott1989second,
+  title={{A second order splitting method for the Cahn-Hilliard equation}},
+  author={Elliott, Charles M and French, Donald A and Milner, Fabio Augusto},
+  journal={Numerische Mathematik},
+  volume={54},
+  number={5},
+  pages={575--590},
+  year={1989},
+  publisher={Springer},
+  url="https://doi.org/10.1007/BF01396363"
+}
+
+@article{gaston2014continuous,
+  title={Continuous integration for concurrent computational framework and application development},
+  author={Gaston, Derek and Peterson, John and Permann, Cody and Andrs, David and Slaughter, Andrew and Miller, Jason},
+  journal={Journal of Open Research Software},
+  volume={2},
+  number={1},
+  year={2014},
+  publisher={Ubiquity Press},
+  url="http://doi.org/10.5334/jors.as"
+}
+
+@article{gaston2015physics,
+  title={Physics-based multiscale coupling for full core nuclear reactor simulation},
+  author={Gaston, Derek R and Permann, Cody J and Peterson, John W and Slaughter, Andrew E and Andr{\v{s}}, David and Wang, Yaqi and Short, Michael P and Perez, Danielle M and Tonks, Michael R and Ortensi, Javier and others},
+  journal={Annals of Nuclear Energy},
+  volume={84},
+  pages={45--54},
+  year={2015},
+  publisher={Elsevier},
+  url="https://doi.org/10.1016/j.anucene.2014.09.060"
+}
+
+@article{jokisaari2017spinodal,
+  title={Benchmark problems for numerical implementations of phase field models},
+  author={Jokisaari, A M and Voorhees, P W and Guyer, Jonathan E and Warren, J A and Heinonen, O G},
+  journal={Computational Materials Science},
+  volume={126},
+  pages={139--151},
+  year={2017},
+  publisher={Elsevier},
+  url="https://doi.org/10.1016/j.commatsci.2016.09.022"
+}
+
+
+@article{jokisaari2017dendrite,
+  title={Phase field benchmark problems for dendritic growth and linear elasticity},
+  author={Jokisaari, A M and Voorhees, P W and Guyer, Jonathan E and Warren, J A and Heinonen, O G},
+  journal={Computational Materials Science},
+  volume={149},
+  pages={336},
+  year={2018},
+  publisher={Elsevier},
+  url="https://doi.org/10.1016/j.commatsci.2018.03.015"
+}
+
+@article{jokisaari2020stokes,
+  title={Phase field benchmark problems targeting fluid flow and electrochemistry},
+  author={Jokisaari, A M and Wu, W and Voorhees, P W and Guyer, Jonathan E and Warren, J A and Heinonen, O G},
+  journal={Computational Materials Science},
+  volume={176},
+  pages={109548},
+  year={2020},
+  publisher={Elsevier},
+  url="https://doi.org/10.1016/j.commatsci.2020.109548"
+}
+
+@article{kubo1966,
+  title={The fluctuation-dissipation theorem},
+  author={Kubo, R},
+  journal={Reports on Progess in Physics},
+  volume={29},
+  number = {1},
+  pages={255--284},
+  year={1966},
+  publisher={IOP Publishing},
+  url={https://doi.org/10.1088%2F0034-4885%2F29%2F1%2F306}
+}
+
+@article{granasy2002nucleation,
+ title={Nucleation and bulk crystallization in binary phase field theory},
+ author={Gránásy, L and and Börzsönyi, T and Pusztai, T},
+ journal={Physical Review Letters},
+ volume={88},
+ pages={206105},
+ number={20},
+ year={2002},
+ publisher={American Physical Society},
+ url={https://doi.org/10.1103/PhysRevLett.88.206105}
+}
+
+@article{granasy2007phase,
+ title={Phase field theory of heterogeneous crystal nucleation},
+ author={Gránásy, L and T and Pusztai, T and Saylor, D and Warren, J A},
+ journal={Physical Review Letters},
+ volume={98},
+ pages={035703},
+ number={20},
+ year={2007},
+ publisher={American Physical Society},
+ url={https://doi.org/10.1103/PhysRevLett.98.035703}
+}
+
+@article{granasy2019,
+ title={Phase-field modeling of crystal nucleation in undercooled liquids -- A review},
+ author={Gránásy, L and Tóth, G and Warren, J A and Podmaniczky, F and Tegze, G and Rátkai, L and Pusztai, T},
+ journal={Progress in Materials Science},
+ volume={106},
+ pages={100569},
+ year={2019},
+ publisher={Elsevier},
+ url={https://doi.org/10.1016/j.pmatsci.2019.05.002}
+}
+
+@article{heo2014phase,
+ title={Phase-field modeling of nucleation in solid-state phase transformations},
+ author={Heo, T W and Chen, L},
+ journal={The journal of the minerals, metals \& materials society},
+ volume={66},
+ pages={1520--1528},
+ year={2014},
+ publisher={TMS},
+ url={https://doi.org/10.1007/s11837-014-1033-9}
+}
+
+@article{shi2019,
+ title={Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response},
+ author={Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J},
+ journal={The journal of the minerals, metals \& materials society},
+ volume={71},
+ pages={3640--3655},
+ year={2019},
+ publisher={TMS},
+ url={https://doi.org/10.1007/s11837-019-03618-1}
+}
+
+@article{simmons2000phase,
+  title={Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events},
+  author={Simmons, J P and Shen, C and Wang, Y},
+  journal={Scripta Materialia},
+  volume={43},
+  issues={10},
+  pages={935--942},
+  year={2000},
+  publisher={Elsevier},
+  url={https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0}
+}
+
+@article{simmons2004microstructural,
+  title={Microstructural development involving nucleation and growth phenomena simulated with the Phase Field method},
+  author={Simmons, J P and Wen, Y and Shen, C and Wang, Y},
+  journal={Materials Science and Engineering: A},
+  volume={365},
+  issues={1-2},
+  pages={136--143},
+  year={2004},
+  publisher={Elsevier},
+  url={https://doi.org/10.1016/j.msea.2003.09.019}
+}
+
+@article{warren2009phase,
+  title={Phase field approach to heterogeneous crystal nucleation in alloys},
+  author={Warren, J A and Pusztai, T and Gránásy, L K and Gránásy, L},
+  journal={Physical Revies B},
+  volume={79},
+  issues={1},
+  pages={014204},
+  year={2009},
+  publisher={American Physical Society},
+  url={https://doi.org/10.1103/PhysRevB.79.014204}
+}
+
+@incollection{elliott1989,
+  title={The Cahn-Hilliard model for the kinetics of phase separation},
+  author={Elliott, Charles M},
+  booktitle={Mathematical models for phase change problems},
+  pages={35--73},
+  year={1989},
+  publisher={Springer},
+  url={https://doi.org/10.1007/978-3-0348-9148-6_3}
+}
+
+@article{johnson1939reaction,
+  title={Reaction kinetics in processes of nucleation and growth},
+  author={Johnson, William A and Mehl, R F},
+  journal={Am. Inst. Min. Metal. Petro. Eng.},
+  volume={135},
+  pages={416--458},
+  year={1939},
+  url={https://doi.org/10.1007/s11663-010-9421-1}
+}
+
+@article{avrami1939kinetics,
+  title={Kinetics of phase change. I General theory},
+  author={Avrami, Melvin},
+  journal={The Journal of chemical physics},
+  volume={7},
+  number={12},
+  pages={1103--1112},
+  year={1939},
+  publisher={American Institute of Physics},
+  url={ https://doi.org/10.1063/1.1750380}
+}
+
+@article{avrami1940kinetics,
+  title={Kinetics of phase change. II transformation-time relations for random distribution of nuclei},
+  author={Avrami, Melvin},
+  journal={The Journal of chemical physics},
+  volume={8},
+  number={2},
+  pages={212--224},
+  year={1940},
+  publisher={American Institute of Physics},
+  url={https://doi.org/10.1063/1.1750631}
+}
+
+@article{avrami1941granulation,
+  title={Granulation, phase change, and microstructure kinetics of phase change. III},
+  author={Avrami, Melvin},
+  journal={The Journal of chemical physics},
+  volume={9},
+  number={2},
+  pages={177--184},
+  year={1941},
+  publisher={American Institute of Physics},
+  url={https://doi.org/10.1063/1.1750872}
+}
+
+@article{kolmogorov1937izv,
+  title={On the Statistical Theory of the Crystallization of Metals},
+  author={Kolmogorov, A N},
+  journal={Bulletin of the Acadamy of Sciences of the USSR},
+  series={Mathematical Series},
+  volume={1},
+  pages={355--359},
+  year={1937},
+  url={https://doi.org/10.1007/978-94-011-2260-3_22}
+}
+
+@article{quested2005athermal,
+  title={Athermal heterogeneous nucleation of solidification},
+  author={Quested, T E and Greer, A L},
+  journal={Acta Materialia},
+  volume={53},
+  number={9},
+  pages={2683--2692},
+  year={2005},
+  publisher={Elsevier},
+  url={https://doi.org/10.1016/j.actamat.2005.02.028}
+}
+
+@article{greer2000modelling,
+  title={Modelling of inoculation of metallic melts: application to grain refinement of aluminium by {A}l--{T}i--{B}},
+  author={Greer, A L and Bunn, A M and Tronche, A and Evans, P V and Bristow, D J},
+  journal={Acta materialia},
+  volume={48},
+  number={11},
+  pages={2823--2835},
+  year={2000},
+  publisher={Elsevier},
+  url={https://doi.org/10.1016/S1359-6454(00)00094-X}
+}
+
+@article{greer2016overview,
+  title={Overview: Application of heterogeneous nucleation in grain-refining of metals},
+  author={Greer, A L},
+  journal={The Journal of Chemical Physics},
+  volume={145},
+  number={21},
+  pages={211704},
+  year={2016},
+  publisher={AIP Publishing LLC},
+  url={https://doi.org/10.1063/1.4968846}
+}
+
+
+@article{dewitt2020prisms,
+  title={PRISMS-PF: A general framework for phase-field modeling with a matrix-free finite element method},
+  author={DeWitt, Stephen and Rudraraju, Shiva and Montiel, David and Andrews, W Beck and Thornton, Katsuyo},
+  journal={npj Computational Materials},
+  volume={6},
+  number={1},
+  pages={1--12},
+  year={2020},
+  publisher={Nature Publishing Group},
+  url={https://doi.org/10.1038/s41524-020-0298-5}
+}
+
+
+@article{guyer2009fipy,
+  title={FiPy: Partial differential equations with Python},
+  author={Guyer, Jonathan E and Wheeler, Daniel and Warren, James A},
+  journal={Computing in Science \& Engineering},
+  volume={11},
+  number={3},
+  pages={6--15},
+  year={2009},
+  publisher={IEEE},
+  url={https://doi.org/10.1109/MCSE.2009.52}
+}
+
+@article{wheeler2019pfhub,
+  title={PFHub: The Phase-Field Community Hub},
+  author={Wheeler, Daniel and Keller, Trevor and DeWitt, Stephen J and Jokisaari, Andrea M and Schwen, Daniel and Guyer, Jonathan E and Aagesen, Larry K and Heinonen, Olle G and Tonks, Michael R and Voorhees, Peter W and others},
+  journal={Journal of Open Research Software},
+  volume={7},
+  number={1},
+  year={2019},
+  publisher={Ubiquity Press},
+  url={http://doi.org/10.5334/jors.276}
+}
+
+@article{allen1979microscopic,
+  title={A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening},
+  author={Allen, Samuel M and Cahn, John W},
+  journal={Acta Metallurgica},
+  volume={27},
+  issue = {6},
+  pages={1085--1095},
+  year={1979},
+  url={https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2}
+}
+
+-->
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h2 id="References">References<a class="anchor-link" href="#References">&#182;</a></h2><p><a id="cite-granasy2002nucleation"/><sup><a href="#ref-1">[^]</a><a href="#ref-9">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. <em>Nucleation and bulk crystallization in binary phase field theory</em>. <a href="https://doi.org/10.1103/PhysRevLett.88.206105">URL</a></p>
+<p><a id="cite-castro2003phase"/><sup><a href="#ref-2">[^]</a></sup>Castro, Mario. 2003. <em>Phase-field approach to heterogeneous nucleation</em>. <a href="https://link.aps.org/doi/10.1103/PhysRevB.67.035412">URL</a></p>
+<p><a id="cite-simmons2004microstructural"/><sup><a href="#ref-3">[^]</a></sup>Simmons, J P and Wen, Y and Shen, C and Wang, Y. 2004. <em>Microstructural development involving nucleation and growth phenomena simulated with the Phase Field method</em>. <a href="https://doi.org/10.1016/j.msea.2003.09.019">URL</a></p>
+<p><a id="cite-granasy2007phase"/><sup><a href="#ref-4">[^]</a></sup>Gránásy, L and T and Pusztai, T and Saylor, D and Warren, J A. 2007. <em>Phase field theory of heterogeneous crystal nucleation</em>. <a href="https://doi.org/10.1103/PhysRevLett.98.035703">URL</a></p>
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+
 </div>
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From 4c70e71862e2d9d87c75734b53ba006bb43417ad Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Sun, 1 Nov 2020 18:57:46 -0500
Subject: [PATCH 02/13] build(nix): notebook extension with bibtex

Allow equation numbering and bibtex citations in noteobok
---
 _nix/env.nix | 20 ++++++++++++++++++++
 1 file changed, 20 insertions(+)

diff --git a/_nix/env.nix b/_nix/env.nix
index 2c51a6242..b61b2c018 100644
--- a/_nix/env.nix
+++ b/_nix/env.nix
@@ -16,5 +16,25 @@ nixpkgs.stdenv.mkDerivation rec {
     pip install jupyter_contrib_nbextensions --user > /dev/null 2>&1
     jupyter contrib nbextension install --user > /dev/null 2>&1
     jupyter nbextension enable spellchecker/main > /dev/null 2>&1
+    jupyter nbextension enable equation-numbering/main > /dev/null 2>&1
+
+    # Bibliography tools
+    # see https://jupyter.brynmawr.edu/services/public/dblank/jupyter.cs/Examples/Calico%20Document%20Tools%20and%20Bibtex.ipynb
+    # also see
+    # https://jupyter.brynmawr.edu/services/public/dblank/Jupyter%20Notebook%20Users%20Manual.ipynb#5.-Bibliographic-Support
+
+    CURDIR=$(pwd)
+    MYTMPDIR=$(mktemp -d)
+    cd $MYTMPDIR
+    # This is my own version as notebook-extensions needed modifications
+    # Orignal repo is
+    # https://github.com/Calysto/notebook-extensions
+    git clone https://github.com/Calysto/notebook-extensions.git
+    cd notebook-extensions
+    jupyter nbextension install calysto --user
+    jupyter nbextension enable --user calysto/document-tools/main
+    cd $CURDIR
+    rm -rf $MYTMPDIR
+
   '';
 }

From 6dfddb78a1b7231fb4c74b48ed8a9922f2ac4198 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Wed, 18 Nov 2020 13:49:09 -0500
Subject: [PATCH 03/13] fix for htmlproofer

---
 benchmarks/benchmark8.ipynb          |  16 ++--
 benchmarks/benchmark8.ipynb.raw.html | 110 ++++++++++++++++++++++++---
 benchmarks/benchmark8a.1.ipynb.md    |   4 +
 benchmarks/benchmark8b.1.ipynb.md    |   4 +
 benchmarks/benchmark8c.1.ipynb.md    |   4 +
 simulations/8a.1.html                |   6 ++
 simulations/8b.1.html                |   6 ++
 simulations/8c.1.html                |   6 ++
 8 files changed, 138 insertions(+), 18 deletions(-)
 create mode 100644 benchmarks/benchmark8a.1.ipynb.md
 create mode 100644 benchmarks/benchmark8b.1.ipynb.md
 create mode 100644 benchmarks/benchmark8c.1.ipynb.md
 create mode 100644 simulations/8a.1.html
 create mode 100644 simulations/8b.1.html
 create mode 100644 simulations/8c.1.html

diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index 3f36973db..c2a0fd5fc 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -366,7 +366,7 @@
     "\\end{equation}\n",
     "where $r_0=\\{r^*, 0.99r^*, 1.01r^*\\}$. [Figure 1](#fig1) shows the computational domain with an initial seed of radius $r=r^*$. The time evolution of the system is then followed for times up to $t = 200$ units, and the solid fraction $Y$ and the total free energy $F$ are plotted as functions of time. Finally, the problem also includes a convergence test with respect to mesh (spatial resolution). The closer the initial radius $r$ is to the critical radius $r^*$, the more sensitive the numerical integration will be to round-off errors. Therefore, a convergence test is included for $r_0=1.01r^*$ using successive runs each halving the average spatial mesh size $\\Delta x$ between consecutive runs. At least 3 runs are required to estimate the rate of convergence based on the $L^2$ error with the finest mesh being used as a gold standard. The simulations should be run to $t=200$ and the final phase field (at $t=200$) for each successive run should be submitted.\n",
     "\n",
-    "To submit results to the PFHub website, see the [submission guidelines](#Part-1.6.1-Part-%28a%29) below.\n",
+    "To submit results to the PFHub website, see the [submission guidelines](#Part-%28a%29) below.\n",
     "\n",
     "<figure>\n",
     "    <img id=\"fig1\" src=\"../images/t0_single_seed.PNG\" alt=\"Single seed at t=0\" style=\"width:500px;\"/>\n",
@@ -388,7 +388,7 @@
     "\n",
     "Both Part (b) and Part (c) involve a random number generator. Because different executions will in general generate different random numbers, specific individual solutions will in general not be reproduced. However, statistics such as the average slope of Avrami plots can be meaningfully compared.\n",
     "\n",
-    "To submit results to the PFHub website, see the [submission guidelines](#Part-1.6.1-Part-%28a%29) below.\n",
+    "To submit results to the PFHub website, see the [submission guidelines](#Part-%28b%29) below.\n",
     "\n",
     "## Part (c): multiple seeds at random times\n",
     "\n",
@@ -402,7 +402,7 @@
     "\n",
     "Again, $\\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $X$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.\n",
     "\n",
-    "To submit results to the PFHub website, see the [submission guidelines](#Part-1.6.3-Part-%28c%29) below."
+    "To submit results to the PFHub website, see the [submission guidelines](#Part-%28c%29) below."
    ]
   },
   {
@@ -432,7 +432,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\\Delta x$ using the model formulation [defined above](#Part-1.3.1-Part-1.6.1-Part-%28a%29:-single-seed). The 5 simulations are\n",
+    "Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\\Delta x$ using the model formulation [defined above](#Part-%28a%29:-single-seed). The 5 simulations are\n",
     "\n",
     " 1. $r_0 = 0.99 r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
     " 1. $r_0 = r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
@@ -495,7 +495,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the [defintion above](#Part-1.4-Part-1.6.2-Part-%28b%29:-multiple-initial-seeds). The simulations are run to $t=200$. The following data should be collected.\n",
+    "Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the [defintion above](#Part-%28b%29:-multiple-initial-seeds). The simulations are run to $t=200$. The following data should be collected.\n",
     "\n",
     " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
     " \n",
@@ -525,7 +525,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Part (c) requires running 5 simulations with seeds randomly appearing between $t=0$ and $t=600$, see the [definition above](#Part-1.5-Part-1.6.3-Part-%28c%29:-multiple-seeds-at-random-times). The simulations are run to $t=600$. The following data should be collected.\n",
+    "Part (c) requires running 5 simulations with seeds randomly appearing between $t=0$ and $t=600$, see the [definition above](#Part-%28c%29:-multiple-seeds-at-random-times). The simulations are run to $t=600$. The following data should be collected.\n",
     "\n",
     " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
     " \n",
@@ -951,7 +951,7 @@
     "\n",
     "<a id=\"cite-kubo1966\"/><sup><a href=\"#ref-8\">[^]</a></sup>Kubo, R. 1966. _The fluctuation-dissipation theorem_. [URL](https://doi.org/10.1088%2F0034-4885%2F29%2F1%2F306)\n",
     "\n",
-    "<a id=\"cite-simmons2000phase\"/><sup><a href=\"#ref-10\">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. _Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events_. [URL](https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0)\n",
+    "<a id=\"cite-simmons2000phase\"/><sup><a href=\"#ref-10\">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. _Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events_. <span data-proofer-ignore>[URL](https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0)</span>\n",
     "\n",
     "<a id=\"cite-shi2019\"/><sup><a href=\"#ref-11\">[^]</a></sup>Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J. 2019. _Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response_. [URL](https://doi.org/10.1007/s11837-019-03618-1)\n",
     "\n",
@@ -965,7 +965,7 @@
     "\n",
     "<a id=\"cite-kolmogorov1937izv\"/><sup><a href=\"#ref-16\">[^]</a><a href=\"#ref-22\">[^]</a></sup>Kolmogorov, A N. 1937. _On the Statistical Theory of the Crystallization of Metals_. [URL](https://doi.org/10.1007/978-94-011-2260-3_22)\n",
     "\n",
-    "<a id=\"cite-allen1979microscopic\"/><sup><a href=\"#ref-17\">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. _A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening_. [URL](https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2)\n",
+    "<a id=\"cite-allen1979microscopic\"/><sup><a href=\"#ref-17\">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. _A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening_. <span data-proofer-ignore>[URL](https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2)</span>\n",
     "\n"
    ]
   },
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index 9a0414d74..754f0c540 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="71b92ce1-b10f-4d17-943f-11fc615034b1"></div>
+<div id="6c03cdd9-5e5c-4573-ab6f-7b808ab5d7d7"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#71b92ce1-b10f-4d17-943f-11fc615034b1');
+var element = $('#6c03cdd9-5e5c-4573-ab6f-7b808ab5d7d7');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -400,7 +400,7 @@ <h3 id="Part-(a):-single-seed">Part (a): single seed<a class="anchor-link" href=
     \label{eqn:phitr_2}
 \end{equation}
 where $r_0=\{r^*, 0.99r^*, 1.01r^*\}$. <a href="#fig1">Figure 1</a> shows the computational domain with an initial seed of radius $r=r^*$. The time evolution of the system is then followed for times up to $t = 200$ units, and the solid fraction $Y$ and the total free energy $F$ are plotted as functions of time. Finally, the problem also includes a convergence test with respect to mesh (spatial resolution). The closer the initial radius $r$ is to the critical radius $r^*$, the more sensitive the numerical integration will be to round-off errors. Therefore, a convergence test is included for $r_0=1.01r^*$ using successive runs each halving the average spatial mesh size $\Delta x$ between consecutive runs. At least 3 runs are required to estimate the rate of convergence based on the $L^2$ error with the finest mesh being used as a gold standard. The simulations should be run to $t=200$ and the final phase field (at $t=200$) for each successive run should be submitted.</p>
-<p>To submit results to the PFHub website, see the <a href="#Part-1.6.1-Part-%28a%29">submission guidelines</a> below.</p>
+<p>To submit results to the PFHub website, see the <a href="#Part-%28a%29">submission guidelines</a> below.</p>
 <figure>
     <img id="fig1" src="../images/t0_single_seed.PNG" alt="Single seed at t=0" style="width:500px;"/>
     <figcaption>
@@ -415,7 +415,7 @@ <h3 id="Part-(a):-single-seed">Part (a): single seed<a class="anchor-link" href=
 \end{equation}
 from the different seeds $i$. Overlaps between different seeds are managed by setting $\phi=1$ in all regions where the sums of $\phi_i &gt;1$. After placing initial seeds and adjusting the initial phase field $\phi$ so that $\phi\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $X$, discrete particle count $N$  (the number of disjoint regions with $\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.</p>
 <p>Both Part (b) and Part (c) involve a random number generator. Because different executions will in general generate different random numbers, specific individual solutions will in general not be reproduced. However, statistics such as the average slope of Avrami plots can be meaningfully compared.</p>
-<p>To submit results to the PFHub website, see the <a href="#Part-1.6.1-Part-%28a%29">submission guidelines</a> below.</p>
+<p>To submit results to the PFHub website, see the <a href="#Part-%28b%29">submission guidelines</a> below.</p>
 <h2 id="Part-(c):-multiple-seeds-at-random-times">Part (c): multiple seeds at random times<a class="anchor-link" href="#Part-(c):-multiple-seeds-at-random-times">&#182;</a></h2><p>For the third part of the homogeneous benchmark problem, the computational domain is first expanded to $1000 \times1000$. Instead of inserting nuclei at fixed time $t = 0$ as in part two of the problem, in this case 100 random nucleation times $t_i$ are generated, $i=1,\ldots,100$, drawn from a uniform distribution in the interval $t_i \in [0, 600)$ with centers $\mathbf{r}_i$ drawn from a uniform distribution on the $1000\times1000$ domain.
 We decrease the driving force to $\Delta f=1/(6\sqrt{2})$, which corresponds to a critical radius or $r^*=2$, but keep the initial radii of the inserted nuclei the same as those in the second part of the problem
 \begin{equation}
@@ -424,7 +424,7 @@ <h2 id="Part-(c):-multiple-seeds-at-random-times">Part (c): multiple seeds at ra
     \right)\right].
 \end{equation}</p>
 <p>Again, $\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $X$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.</p>
-<p>To submit results to the PFHub website, see the <a href="#Part-1.6.3-Part-%28c%29">submission guidelines</a> below.</p>
+<p>To submit results to the PFHub website, see the <a href="#Part-%28c%29">submission guidelines</a> below.</p>
 
 </div>
 </div>
@@ -459,7 +459,7 @@ <h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<p>Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\Delta x$ using the model formulation <a href="#Part-1.3.1-Part-1.6.1-Part-%28a%29:-single-seed">defined above</a>. The 5 simulations are</p>
+<p>Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\Delta x$ using the model formulation <a href="#Part-%28a%29:-single-seed">defined above</a>. The 5 simulations are</p>
 <ol>
 <li>$r_0 = 0.99 r^*$, ${\Delta x} = {\Delta x}_0$</li>
 <li>$r_0 = r^*$, ${\Delta x} = {\Delta x}_0$</li>
@@ -525,7 +525,7 @@ <h3 id="Part-(b)">Part (b)<a class="anchor-link" href="#Part-(b)">&#182;</a></h3
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<p>Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the <a href="#Part-1.4-Part-1.6.2-Part-%28b%29:-multiple-initial-seeds">defintion above</a>. The simulations are run to $t=200$. The following data should be collected.</p>
+<p>Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the <a href="#Part-%28b%29:-multiple-initial-seeds">defintion above</a>. The simulations are run to $t=200$. The following data should be collected.</p>
 <ul>
 <li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
 </li>
@@ -557,7 +557,7 @@ <h3 id="Part-(c)">Part (c)<a class="anchor-link" href="#Part-(c)">&#182;</a></h3
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<p>Part (c) requires running 5 simulations with seeds randomly appearing between $t=0$ and $t=600$, see the <a href="#Part-1.5-Part-1.6.3-Part-%28c%29:-multiple-seeds-at-random-times">definition above</a>. The simulations are run to $t=600$. The following data should be collected.</p>
+<p>Part (c) requires running 5 simulations with seeds randomly appearing between $t=0$ and $t=600$, see the <a href="#Part-%28c%29:-multiple-seeds-at-random-times">definition above</a>. The simulations are run to $t=600$. The following data should be collected.</p>
 <ul>
 <li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
 </li>
@@ -981,17 +981,107 @@ <h2 id="References">References<a class="anchor-link" href="#References">&#182;</
 <p><a id="cite-heo2014phase"/><sup><a href="#ref-6">[^]</a></sup>Heo, T W and Chen, L. 2014. <em>Phase-field modeling of nucleation in solid-state phase transformations</em>. <a href="https://doi.org/10.1007/s11837-014-1033-9">URL</a></p>
 <p><a id="cite-granasy2019"/><sup><a href="#ref-7">[^]</a></sup>Gránásy, L and Tóth, G and Warren, J A and Podmaniczky, F and Tegze, G and Rátkai, L and Pusztai, T. 2019. <em>Phase-field modeling of crystal nucleation in undercooled liquids -- A review</em>. <a href="https://doi.org/10.1016/j.pmatsci.2019.05.002">URL</a></p>
 <p><a id="cite-kubo1966"/><sup><a href="#ref-8">[^]</a></sup>Kubo, R. 1966. <em>The fluctuation-dissipation theorem</em>. <a href="https://doi.org/10.1088%2F0034-4885%2F29%2F1%2F306">URL</a></p>
-<p><a id="cite-simmons2000phase"/><sup><a href="#ref-10">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. <em>Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events</em>. <a href="https://doi.org/10.1016/S1359-6462&amp;#40;00&amp;#41;00517-0">URL</a></p>
+<p><a id="cite-simmons2000phase"/><sup><a href="#ref-10">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. <em>Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events</em>. <span data-proofer-ignore>[URL](https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0)</span></p>
 <p><a id="cite-shi2019"/><sup><a href="#ref-11">[^]</a></sup>Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J. 2019. <em>Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response</em>. <a href="https://doi.org/10.1007/s11837-019-03618-1">URL</a></p>
 <p><a id="cite-johnson1939reaction"/><sup><a href="#ref-12">[^]</a><a href="#ref-18">[^]</a></sup>Johnson, William A and Mehl, R F. 1939. <em>Reaction kinetics in processes of nucleation and growth</em>. <a href="https://doi.org/10.1007/s11663-010-9421-1">URL</a></p>
 <p><a id="cite-avrami1939kinetics"/><sup><a href="#ref-13">[^]</a><a href="#ref-19">[^]</a></sup>Avrami, Melvin. 1939. <em>Kinetics of phase change. I General theory</em>. <a href="https://doi.org/10.1063/1.1750380">URL</a></p>
 <p><a id="cite-avrami1940kinetics"/><sup><a href="#ref-14">[^]</a><a href="#ref-20">[^]</a></sup>Avrami, Melvin. 1940. <em>Kinetics of phase change. II transformation-time relations for random distribution of nuclei</em>. <a href="https://doi.org/10.1063/1.1750631">URL</a></p>
 <p><a id="cite-avrami1941granulation"/><sup><a href="#ref-15">[^]</a><a href="#ref-21">[^]</a></sup>Avrami, Melvin. 1941. <em>Granulation, phase change, and microstructure kinetics of phase change. III</em>. <a href="https://doi.org/10.1063/1.1750872">URL</a></p>
 <p><a id="cite-kolmogorov1937izv"/><sup><a href="#ref-16">[^]</a><a href="#ref-22">[^]</a></sup>Kolmogorov, A N. 1937. <em>On the Statistical Theory of the Crystallization of Metals</em>. <a href="https://doi.org/10.1007/978-94-011-2260-3_22">URL</a></p>
-<p><a id="cite-allen1979microscopic"/><sup><a href="#ref-17">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. <em>A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening</em>. <a href="https://doi.org/10.1016/0001-6160&amp;#40;79&amp;#41;90196-2">URL</a></p>
+<p><a id="cite-allen1979microscopic"/><sup><a href="#ref-17">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. <em>A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening</em>. <span data-proofer-ignore>[URL](https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2)</span></p>
 
 </div>
 </div>
 </div>
+<div class="cell border-box-sizing text_cell rendered"><div class="prompt input_prompt">
+</div>
+<div class="inner_cell">
+<div class="text_cell_render border-box-sizing rendered_html">
+<h2 id="Appendix">Appendix<a class="anchor-link" href="#Appendix">&#182;</a></h2><p>The following code is given as a resource for those implementing the benchmark to help count the particles in Part (b) and Part (c). One possibility as outlined here is to use <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.label.html#scipy.ndimage.label"><code>scipy.ndimage.label</code></a>. This returns a tuple with a labeled array of unique features as the first item and a count of features as the second item. In the case of the phase field the values will require thresholding to 0 or 1 before using <code>scipy.ndimage.label</code>.</p>
+
+</div>
+</div>
+</div>
+<div class="cell border-box-sizing code_cell rendered">
+<div class="input">
+<div class="prompt input_prompt">In&nbsp;[138]:</div>
+<div class="inner_cell">
+    <div class="input_area">
+<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">scipy.ndimage</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
+
+<span class="k">def</span> <span class="nf">make_particles</span><span class="p">(</span><span class="n">shape</span><span class="p">,</span> <span class="n">n_particles</span><span class="p">,</span> <span class="n">max_radius</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">99</span><span class="p">):</span>
+    <span class="sd">&quot;&quot;&quot;Make an array with random particles</span>
+<span class="sd">    </span>
+<span class="sd">    Args:</span>
+<span class="sd">      shape: shape of array</span>
+<span class="sd">      n_particles: number of particles in arrau</span>
+<span class="sd">      max_radius: maximum possible radius of particles</span>
+<span class="sd">      seed: random seed</span>
+<span class="sd">      </span>
+<span class="sd">    Returns: numpy array mask with particles</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
+        
+    <span class="k">def</span> <span class="nf">xy_grid</span><span class="p">(</span><span class="n">shape</span><span class="p">):</span>
+        <span class="sd">&quot;&quot;&quot;Generate a grid of x, y values in a single array</span>
+<span class="sd">        </span>
+<span class="sd">        The returned array has shape (len(shape),) + shape + (1,). The last axis is</span>
+<span class="sd">        for the operations with particles.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">linspace</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.5</span>
+        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="o">*</span><span class="nb">map</span><span class="p">(</span><span class="n">linspace</span><span class="p">,</span> <span class="n">shape</span><span class="p">),</span> <span class="n">indexing</span><span class="o">=</span><span class="s2">&quot;ij&quot;</span><span class="p">))[</span><span class="o">...</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    
+    <span class="c1"># first axis is dimensions</span>
+    <span class="c1"># second and third axes are shape of domain</span>
+    <span class="c1"># last axis is the number of particles. </span>
+    <span class="n">centers</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">shape</span><span class="p">),</span> <span class="n">n_particles</span><span class="p">))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">shape</span><span class="p">)[:,</span> <span class="kc">None</span><span class="p">])[:,</span> <span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    <span class="n">radii</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">n_particles</span><span class="p">)</span> <span class="o">*</span> <span class="n">max_radius</span><span class="p">)[</span><span class="kc">None</span><span class="p">,</span> <span class="kc">None</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">numpy</span><span class="o">.</span><span class="n">any</span><span class="p">(((</span><span class="n">xy_grid</span><span class="p">(</span><span class="n">shape</span><span class="p">)</span> <span class="o">-</span> <span class="n">centers</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">radii</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+<span class="n">arr</span> <span class="o">=</span> <span class="n">make_particles</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span> <span class="mi">200</span><span class="p">),</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">arr</span><span class="p">)</span>
+<span class="nb">print</span><span class="p">(</span><span class="s1">&#39;Actual number of particles:&#39;</span><span class="p">,</span> <span class="n">scipy</span><span class="o">.</span><span class="n">ndimage</span><span class="o">.</span><span class="n">label</span><span class="p">(</span><span class="n">arr</span><span class="p">)[</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
+
+</div>
+</div>
+</div>
+
+<div class="output_wrapper">
+<div class="output">
+
+
+<div class="output_area">
+
+<div class="prompt"></div>
+
+
+<div class="output_subarea output_stream output_stdout output_text">
+<pre>Actual number of particles: 14
+</pre>
+</div>
+</div>
+
+<div class="output_area">
+
+<div class="prompt"></div>
+
+
+
+
+<div class="output_png output_subarea ">
+<img src="data:image/png;base64,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+"
+>
+</div>
+
+</div>
+
+</div>
+</div>
+
+</div>
  
 
diff --git a/benchmarks/benchmark8a.1.ipynb.md b/benchmarks/benchmark8a.1.ipynb.md
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+---
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+redirect_to: ../../benchmarks/benchmark8.ipynb/
+---
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+---
+layout: redirected
+redirect_to: ../../benchmarks/benchmark8.ipynb/
+---
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+---
+layout: redirected
+redirect_to: ../../benchmarks/benchmark8.ipynb/
+---
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--- /dev/null
+++ b/simulations/8a.1.html
@@ -0,0 +1,6 @@
+---
+layout: default
+---
+
+{% assign benchmark_id = page.url | remove: '/simulations/' | remove: '/' %}
+{% include result_comparison.html benchmark_id=benchmark_id %}
diff --git a/simulations/8b.1.html b/simulations/8b.1.html
new file mode 100644
index 000000000..1178d2917
--- /dev/null
+++ b/simulations/8b.1.html
@@ -0,0 +1,6 @@
+---
+layout: default
+---
+
+{% assign benchmark_id = page.url | remove: '/simulations/' | remove: '/' %}
+{% include result_comparison.html benchmark_id=benchmark_id %}
diff --git a/simulations/8c.1.html b/simulations/8c.1.html
new file mode 100644
index 000000000..1178d2917
--- /dev/null
+++ b/simulations/8c.1.html
@@ -0,0 +1,6 @@
+---
+layout: default
+---
+
+{% assign benchmark_id = page.url | remove: '/simulations/' | remove: '/' %}
+{% include result_comparison.html benchmark_id=benchmark_id %}

From 57dc2206ab4493f702b1569202e0c4aaa148f507 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Tue, 19 Jan 2021 18:35:28 -0500
Subject: [PATCH 04/13] ci: update surge token

---
 .travis.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.travis.yml b/.travis.yml
index f2358f6e5..6510b72c4 100644
--- a/.travis.yml
+++ b/.travis.yml
@@ -20,7 +20,7 @@ install:
 before_script:
   - export DOMAIN=random-cat-${TRAVIS_PULL_REQUEST}.surge.sh
   - export SURGE_LOGIN="daniel.wheeler@nist.gov"
-  - export SURGE_TOKEN="3712d1678fabe65dfad576545befce2f"
+  - export SURGE_TOKEN="fcab9043c16e76c558d16bf6dc380f79"
   - if [[ "$TRAVIS_PULL_REQUEST" != "false" ]]; then
        nix-shell --command "
          rm -rf ./_site &&

From d477760b02d056d6cc8fb6a650441d83fb029457 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Wed, 24 Mar 2021 17:44:51 -0400
Subject: [PATCH 05/13] fix: broken link to yaml info

---
 simulations/upload.md | 26 +++++++++++---------------
 1 file changed, 11 insertions(+), 15 deletions(-)

diff --git a/simulations/upload.md b/simulations/upload.md
index bfab51b9f..37409416d 100644
--- a/simulations/upload.md
+++ b/simulations/upload.md
@@ -16,14 +16,11 @@ feel free to upload benchmark results via pull requests on
 GitHub. More benchmark results will greatly improve the utility of the
 website and encourage community collaboration.
 
-Each benchmark result is stored in a [YAML
-file](http://docs.ansible.com/ansible/YAMLSyntax.html) called
-`meta.yaml` in a separate directory in
-[_data/simulations]({{ site.links.github }}/tree/master/_data/simulations).
-A [YAML file](http://docs.ansible.com/ansible/YAMLSyntax.html) is a
-minimal, human readable syntax for structured data. The `meta.yaml`
-file stores the meta data for only one benchmark result and a new
-directory is required for each new benchmark result.
+Each benchmark result is stored in a [YAML file][ansible_yaml] called
+`meta.yaml` in a separate directory in [_data/simulations]({{
+site.links.github }}/tree/master/_data/simulations).  A [YAML file][anisble_yaml] is a minimal, human readable syntax for structured
+data. The `meta.yaml` file stores the meta data for only one benchmark
+result and a new directory is required for each new benchmark result.
 
 <h4> How to Upload </h4>
 
@@ -52,9 +49,8 @@ To record a new benchmark result, use the following workflow.
 
 <h4>Minimal Example of a YAML Benchmark File</h4>
 
-Each [YAML](http://docs.ansible.com/ansible/YAMLSyntax.html)
-description of a specific benchmark result contains the following
-three parts:
+Each [YAML][ansible_yaml] description of a specific benchmark result
+contains the following three parts:
 
  1. `benchmark`: specify the benchmark problem and version you have
     implemented,
@@ -70,10 +66,8 @@ The following is the minimal description of a benchmark result with
 relevant comments. The definitive archetype resides at
 [`_data/simulations/example/example.yaml`]({{ site.links.simmeta
 }}/example/meta.yaml). To understand the YAML syntax consult either
-the [Ansible
-documentation](http://docs.ansible.com/ansible/YAMLSyntax.html) for a
-simple overview or the [YAML site](http://www.yaml.org/) for a more in
-depth description.
+the [Ansible documentation][anisble_yaml] for a simple overview or the
+[YAML site](http://www.yaml.org/) for a more in depth description.
 
 ```
 ---
@@ -225,3 +219,5 @@ If the tests all pass, the web site dev will need to check that the
 formatting and links work for displaying the charts and tables, which
 is not entirely automated by the test suite. Further repairs may be
 necessary at this stage.
+
+[ansible_yaml]: https://docs.ansible.com/ansible/latest/reference_appendices/YAMLSyntax.html

From 0f023f0ce0cca00af507c6ed938a5b67ccadd328 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Wed, 24 Mar 2021 17:45:22 -0400
Subject: [PATCH 06/13] fix: remove twitter from the frontpage

Twitter # link is a bit stupid and useless.
---
 index.html | 4 ----
 1 file changed, 4 deletions(-)

diff --git a/index.html b/index.html
index 7e97c18d8..db124274e 100644
--- a/index.html
+++ b/index.html
@@ -11,10 +11,6 @@
   {% include wiki-nav.html %}
 </div>
 
-<div id="twitter-right">
-  {% include twitter.html %}
-</div>
-
 <div id="sponsor-logos-container">
   <div class="col s0 m0 l0 xl12" id="sponsor-logos">
     {% include logos.html %}

From e909924b749d0864daadee53fc43a13773212f5f Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Wed, 24 Mar 2021 17:46:14 -0400
Subject: [PATCH 07/13] test: switch off htmlproofer for twitter links

Htmlproofer doesn't work with Twitter links don't work on the CI
---
 community.html | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/community.html b/community.html
index ed08b3e0c..2a60a92f9 100644
--- a/community.html
+++ b/community.html
@@ -59,7 +59,8 @@ <h5 class="sideheader">
               {% if item.twitter %}
                 <a href="https://twitter.com/{{ item.twitter }}"
                    target="blank_"
-                   title="twitter">
+                   title="twitter"
+                   data-proofer-ignore>
                   <i class="small material-icons">
                     <img src="{{ site.baseurl }}/images/twitter-blue.svg"
                          alt="twitter"

From 5ef24aae976ac7be4740fc7ded7771258ec63f7b Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Wed, 24 Mar 2021 17:49:00 -0400
Subject: [PATCH 08/13] docs(notebooks): change wording

---
 benchmarks/benchmark8.ipynb          | 2 +-
 benchmarks/benchmark8.ipynb.raw.html | 6 +++---
 2 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index c2a0fd5fc..7bedac704 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -253,7 +253,7 @@
     "\n",
     "### The phase field model in classical nucleation theory\n",
     "\n",
-    "For these benchmark problems we use the simplest possible phase field model with a single non-conserved phase field $\\phi$, which describes an isothermal pure substance with one liquid ($\\phi=0$) and one solid ($\\phi=1$) phase. The free energy of this system is  \n",
+    "For this benchmark problems we use the simplest possible phase field model with a single non-conserved phase field $\\phi$, which describes an isothermal pure substance with one liquid ($\\phi=0$) and one solid ($\\phi=1$) phase. The free energy of this system is  \n",
     "\\begin{equation}\n",
     "    \\mathcal{F}(\\phi)=\\int \\left[\\frac{\\epsilon^2}{2}(\\nabla \\phi)^2+wg(\\phi)-\\Delta f p(\\phi) \\right] d{V},\n",
     "    \\label{eqn:Fphi}\n",
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index 754f0c540..05c30d21c 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="6c03cdd9-5e5c-4573-ab6f-7b808ab5d7d7"></div>
+<div id="05c04d63-a681-45d8-8f80-6b531f730526"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#6c03cdd9-5e5c-4573-ab6f-7b808ab5d7d7');
+var element = $('#05c04d63-a681-45d8-8f80-6b531f730526');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -302,7 +302,7 @@ <h2 id="Overview">Overview<a class="anchor-link" href="#Overview">&#182;</a></h2
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="Model-Formulation">Model Formulation<a class="anchor-link" href="#Model-Formulation">&#182;</a></h2><h3 id="The-phase-field-model-in-classical-nucleation-theory">The phase field model in classical nucleation theory<a class="anchor-link" href="#The-phase-field-model-in-classical-nucleation-theory">&#182;</a></h3><p>For these benchmark problems we use the simplest possible phase field model with a single non-conserved phase field $\phi$, which describes an isothermal pure substance with one liquid ($\phi=0$) and one solid ($\phi=1$) phase. The free energy of this system is<br>
+<h2 id="Model-Formulation">Model Formulation<a class="anchor-link" href="#Model-Formulation">&#182;</a></h2><h3 id="The-phase-field-model-in-classical-nucleation-theory">The phase field model in classical nucleation theory<a class="anchor-link" href="#The-phase-field-model-in-classical-nucleation-theory">&#182;</a></h3><p>For this benchmark problems we use the simplest possible phase field model with a single non-conserved phase field $\phi$, which describes an isothermal pure substance with one liquid ($\phi=0$) and one solid ($\phi=1$) phase. The free energy of this system is<br>
 \begin{equation}
     \mathcal{F}(\phi)=\int \left[\frac{\epsilon^2}{2}(\nabla \phi)^2+wg(\phi)-\Delta f p(\phi) \right] d{V},
     \label{eqn:Fphi}

From 52ffb10bc647aa96575e6ab1f8530b456d6e49f6 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Wed, 31 Mar 2021 11:42:24 -0400
Subject: [PATCH 09/13] fix: update for wenkun review

Make simulations 4 and 5 optional in BM8a
---
 benchmarks/benchmark8.ipynb          | 18 ++++++++++--------
 benchmarks/benchmark8.ipynb.raw.html | 17 +++++++++--------
 2 files changed, 19 insertions(+), 16 deletions(-)

diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index 7bedac704..a7b43b17b 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -369,7 +369,7 @@
     "To submit results to the PFHub website, see the [submission guidelines](#Part-%28a%29) below.\n",
     "\n",
     "<figure>\n",
-    "    <img id=\"fig1\" src=\"../images/t0_single_seed.PNG\" alt=\"Single seed at t=0\" style=\"width:500px;\"/>\n",
+    "    <img id=\"fig1\" src=\"../../images/t0_single_seed.PNG\" alt=\"Single seed at t=0\" style=\"width:500px;\"/>\n",
     "    <figcaption>\n",
     "        <center> Illustration of the $100 \\times 100$ computational domain in 2D. </center> \n",
     "    </figcaption>\n",
@@ -432,13 +432,13 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\\Delta x$ using the model formulation [defined above](#Part-%28a%29:-single-seed). The 5 simulations are\n",
+    "Part (a) requires running three simulations with a single seed with different values of $r_0$ using the model formulation [defined above](#Part-%28a%29:-single-seed). Optionally, a further two simulations can be run to check convergence, but this is not required. The three required  simulations and two optional simulations are\n",
     "\n",
     " 1. $r_0 = 0.99 r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
     " 1. $r_0 = r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
     " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_0$\n",
-    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_1$\n",
-    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_2$\n",
+    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_1$ (optional)\n",
+    " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_2$ (optional)\n",
     " \n",
     "where ${\\Delta x}_0 < {\\Delta x}_1 < {\\Delta x}_2$. If using a gridded mesh each successively refined grid length can be half as fine. In the case of adaptive grids a nominal value for $\\Delta x$ should be selected that results in a similarly refined mesh as in the gridded case (i.e. approximately $4\\times$ as many elements). When uploading the results, the $\\Delta x$ values should be provided in the datafile alongside the values for $\\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.\n",
     "\n",
@@ -446,11 +446,11 @@
     " \n",
     " - The free energy integrated over the whole domain, $\\mathcal{F}$, during the transient. \n",
     " \n",
-    " - The phase field, $\\phi$, at each grid point in the domain at $t=200$.\n",
+    " - The phase field, $\\phi$, at each grid point in the domain at $t=200$. \n",
     " \n",
-    " - The grid spacing, $\\Delta x$ at each grid point in the domain at $t=200$.\n",
+    " - The grid spacing, $\\Delta x$ at each grid point in the domain at $t=200$ if submitting the optional simulations.\n",
     " \n",
-    "One PFHub upload is required for all five simulations. The solid fraction and free energy data can be included in the same CSV file or in separate data files, but either way still require 2 entries in the \"Data Files\" section on the [upload form]({{ site.baseurl }}/simulations/upload_form/). The entries should be labeled as `solid_fraction_{i}` or `free_energy_{i}` depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.\n",
+    "One PFHub upload is required for all the simulations. The solid fraction and free energy data can be included in the same CSV file or in separate data files, but either way still require 2 entries in the \"Data Files\" section on the [upload form]({{ site.baseurl }}/simulations/upload_form/). The entries should be labeled as `solid_fraction_{i}` or `free_energy_{i}` depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.\n",
     "\n",
     "```\n",
     "time,fraction,energy\n",
@@ -481,7 +481,9 @@
    "source": [
     "where `x` and `y` is a position in the domain and `phi` is the value at that position. `delta_x` is the nominal grid spacing in that region given by the coordinates. This may be constant across the domain for gridded data, but vary for adaptive meshing. `delta_x` is only a representative value of the grid spacing in that region. The \"3D\" radio button should be selected for the `phase_field_{i}` entry.\n",
     "\n",
-    "The grid spacing data should be named `grid_spacing_{i}` and have exactly the same data as above. Again the same file can be used, but two \"Data Files\" entries are required for both phase field and grid spacing. The \"3D\" radio button should be selected for the `grid_spacing_{i}` entry."
+    "The grid spacing data should be named `grid_spacing_{i}` and have exactly the same data as above. Again the same file can be used, but two \"Data Files\" entries are required for both phase field and grid spacing. The \"3D\" radio button should be selected for the `grid_spacing_{i}` entry. \n",
+    "\n",
+    "The `grid_spacing_{i}` entries are not required, but if included will show convergence data for simulations 3, 4 and 5."
    ]
   },
   {
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index 05c30d21c..d92de8390 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="05c04d63-a681-45d8-8f80-6b531f730526"></div>
+<div id="bfc93809-d417-4de7-8e71-d6cb7fb6f7cf"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#05c04d63-a681-45d8-8f80-6b531f730526');
+var element = $('#bfc93809-d417-4de7-8e71-d6cb7fb6f7cf');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -402,7 +402,7 @@ <h3 id="Part-(a):-single-seed">Part (a): single seed<a class="anchor-link" href=
 where $r_0=\{r^*, 0.99r^*, 1.01r^*\}$. <a href="#fig1">Figure 1</a> shows the computational domain with an initial seed of radius $r=r^*$. The time evolution of the system is then followed for times up to $t = 200$ units, and the solid fraction $Y$ and the total free energy $F$ are plotted as functions of time. Finally, the problem also includes a convergence test with respect to mesh (spatial resolution). The closer the initial radius $r$ is to the critical radius $r^*$, the more sensitive the numerical integration will be to round-off errors. Therefore, a convergence test is included for $r_0=1.01r^*$ using successive runs each halving the average spatial mesh size $\Delta x$ between consecutive runs. At least 3 runs are required to estimate the rate of convergence based on the $L^2$ error with the finest mesh being used as a gold standard. The simulations should be run to $t=200$ and the final phase field (at $t=200$) for each successive run should be submitted.</p>
 <p>To submit results to the PFHub website, see the <a href="#Part-%28a%29">submission guidelines</a> below.</p>
 <figure>
-    <img id="fig1" src="../images/t0_single_seed.PNG" alt="Single seed at t=0" style="width:500px;"/>
+    <img id="fig1" src="../../images/t0_single_seed.PNG" alt="Single seed at t=0" style="width:500px;"/>
     <figcaption>
         <center> Illustration of the $100 \times 100$ computational domain in 2D. </center> 
     </figcaption>
@@ -459,13 +459,13 @@ <h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<p>Part (a) requires running 5 simulations with a single seed with different values of $r_0$ and $\Delta x$ using the model formulation <a href="#Part-%28a%29:-single-seed">defined above</a>. The 5 simulations are</p>
+<p>Part (a) requires running three simulations with a single seed with different values of $r_0$ using the model formulation <a href="#Part-%28a%29:-single-seed">defined above</a>. Optionally, a further two simulations can be run to check convergence, but this is not required. The three required  simulations and two optional simulations are</p>
 <ol>
 <li>$r_0 = 0.99 r^*$, ${\Delta x} = {\Delta x}_0$</li>
 <li>$r_0 = r^*$, ${\Delta x} = {\Delta x}_0$</li>
 <li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_0$</li>
-<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_1$</li>
-<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_2$</li>
+<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_1$ (optional)</li>
+<li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_2$ (optional)</li>
 </ol>
 <p>where ${\Delta x}_0 &lt; {\Delta x}_1 &lt; {\Delta x}_2$. If using a gridded mesh each successively refined grid length can be half as fine. In the case of adaptive grids a nominal value for $\Delta x$ should be selected that results in a similarly refined mesh as in the gridded case (i.e. approximately $4\times$ as many elements). When uploading the results, the $\Delta x$ values should be provided in the datafile alongside the values for $\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.</p>
 <ul>
@@ -475,10 +475,10 @@ <h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3
 </li>
 <li><p>The phase field, $\phi$, at each grid point in the domain at $t=200$.</p>
 </li>
-<li><p>The grid spacing, $\Delta x$ at each grid point in the domain at $t=200$.</p>
+<li><p>The grid spacing, $\Delta x$ at each grid point in the domain at $t=200$ if submitting the optional simulations.</p>
 </li>
 </ul>
-<p>One PFHub upload is required for all five simulations. The solid fraction and free energy data can be included in the same CSV file or in separate data files, but either way still require 2 entries in the "Data Files" section on the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a>. The entries should be labeled as <code>solid_fraction_{i}</code> or <code>free_energy_{i}</code> depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.</p>
+<p>One PFHub upload is required for all the simulations. The solid fraction and free energy data can be included in the same CSV file or in separate data files, but either way still require 2 entries in the "Data Files" section on the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a>. The entries should be labeled as <code>solid_fraction_{i}</code> or <code>free_energy_{i}</code> depending on the simulation number (1 to 5). The CSV (JSON or TSV are also fine) files should have a format similar to the following.</p>
 
 <pre><code>time,fraction,energy
 0.000000000000000000e+00,8.214174263100689974e-03,3.579809478015085311e+00
@@ -509,6 +509,7 @@ <h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3
 <div class="text_cell_render border-box-sizing rendered_html">
 <p>where <code>x</code> and <code>y</code> is a position in the domain and <code>phi</code> is the value at that position. <code>delta_x</code> is the nominal grid spacing in that region given by the coordinates. This may be constant across the domain for gridded data, but vary for adaptive meshing. <code>delta_x</code> is only a representative value of the grid spacing in that region. The "3D" radio button should be selected for the <code>phase_field_{i}</code> entry.</p>
 <p>The grid spacing data should be named <code>grid_spacing_{i}</code> and have exactly the same data as above. Again the same file can be used, but two "Data Files" entries are required for both phase field and grid spacing. The "3D" radio button should be selected for the <code>grid_spacing_{i}</code> entry.</p>
+<p>The <code>grid_spacing_{i}</code> entries are not required, but if included will show convergence data for simulations 3, 4 and 5.</p>
 
 </div>
 </div>

From 6a68835c3d077663f2e739d25e81534b052a3da1 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Fri, 16 Apr 2021 10:02:12 -0400
Subject: [PATCH 10/13] build: use correct version of notebook-exenstions

Use wd15's version of notebook-extensions not Calysto's, which does
not have the correct modifications.
---
 _nix/env.nix | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/_nix/env.nix b/_nix/env.nix
index b61b2c018..177a64f10 100644
--- a/_nix/env.nix
+++ b/_nix/env.nix
@@ -29,10 +29,10 @@ nixpkgs.stdenv.mkDerivation rec {
     # This is my own version as notebook-extensions needed modifications
     # Orignal repo is
     # https://github.com/Calysto/notebook-extensions
-    git clone https://github.com/Calysto/notebook-extensions.git
+    git clone https://github.com/wd15/notebook-extensions.git
     cd notebook-extensions
     jupyter nbextension install calysto --user
-    jupyter nbextension enable --user calysto/document-tools/main
+    jupyter nbextension enable calysto/document-tools/main --user
     cd $CURDIR
     rm -rf $MYTMPDIR
 

From 81f5ebf08c655e58cdd742ee24becbd3d664a13a Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Fri, 16 Apr 2021 10:04:15 -0400
Subject: [PATCH 11/13] refactor: update for tkphd review

Update benchmark8 after tkphd's review

 - Use Y instead of X for the solid fraction
 - More explicit that data can be stored externally to the website and also on the website's own storage
 - Don't spedify the ordering of the \Delta x's. Hopefully, the website will be able to figure this out
 - Fix grammer
 - Fix broken URLs for references.
---
 benchmarks/benchmark8.ipynb          | 32 ++++++++++++----------------
 benchmarks/benchmark8.ipynb.raw.html | 25 +++++++++++-----------
 2 files changed, 27 insertions(+), 30 deletions(-)

diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index a7b43b17b..32199a6c6 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -336,7 +336,7 @@
     "<a id=\"ref-22\" href=\"#cite-kolmogorov1937izv\">Kolmogorov 1937</a>), which makes a number of \n",
     "assumptions and simplifications: (i)  nucleation occurs randomly and homogeneously over the entire un-transformed portion of the material, (ii) the growth rate is constant and does not depend on the extent of transformation, (iii) the particles have convex shape with the same orientation, and (iv) the size of the system is infinite (both in space and time).\n",
     "\n",
-    "The basis of the Avrami plots is the transformed fraction (solid fraction) $X(t)$ vs. time. According to the JMAK theory,\n",
+    "The basis of the Avrami plots is the transformed fraction (solid fraction) $Y(t)$ vs. time. According to the JMAK theory,\n",
     "\\begin{equation}\n",
     "    Y(t)=1-\\exp (-Kt^n),\n",
     "    \\label{eqn:Xt}\n",
@@ -384,7 +384,7 @@
     "    \\right)\\right].\n",
     "    \\label{eqn:phitr_3}\n",
     "\\end{equation}\n",
-    "from the different seeds $i$. Overlaps between different seeds are managed by setting $\\phi=1$ in all regions where the sums of $\\phi_i >1$. After placing initial seeds and adjusting the initial phase field $\\phi$ so that $\\phi\\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $X$, discrete particle count $N$  (the number of disjoint regions with $\\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.\n",
+    "from the different seeds $i$. Overlaps between different seeds are managed by setting $\\phi=1$ in all regions where the sums of $\\phi_i >1$. After placing initial seeds and adjusting the initial phase field $\\phi$ so that $\\phi\\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $Y$, discrete particle count $N$  (the number of disjoint regions with $\\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.\n",
     "\n",
     "Both Part (b) and Part (c) involve a random number generator. Because different executions will in general generate different random numbers, specific individual solutions will in general not be reproduced. However, statistics such as the average slope of Avrami plots can be meaningfully compared.\n",
     "\n",
@@ -400,7 +400,7 @@
     "    \\right)\\right].\n",
     "\\end{equation}\n",
     "\n",
-    "Again, $\\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $X$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.\n",
+    "Again, $\\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $Y$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.\n",
     "\n",
     "To submit results to the PFHub website, see the [submission guidelines](#Part-%28c%29) below."
    ]
@@ -418,7 +418,7 @@
    "source": [
     "Please use the [upload form]({{ site.baseurl }}/simulations/upload_form/) to upload your results.\n",
     "\n",
-    "In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. These are not required, but will help others view your work."
+    "In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. This data can be uploaded directly to the website or stored at a secondary location and only the link to the data provided. This data is not required, but will help others view your work."
    ]
   },
   {
@@ -440,7 +440,7 @@
     " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_1$ (optional)\n",
     " 1. $r_0 = 1.01 r^*$, ${\\Delta x} = {\\Delta x}_2$ (optional)\n",
     " \n",
-    "where ${\\Delta x}_0 < {\\Delta x}_1 < {\\Delta x}_2$. If using a gridded mesh each successively refined grid length can be half as fine. In the case of adaptive grids a nominal value for $\\Delta x$ should be selected that results in a similarly refined mesh as in the gridded case (i.e. approximately $4\\times$ as many elements). When uploading the results, the $\\Delta x$ values should be provided in the datafile alongside the values for $\\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.\n",
+    "where ${\\Delta x}_0, {\\Delta x}_1$ and ${\\Delta x}_2$ represent simulations at different mesh sizes. $\\Delta x$ refinement should be selected so that successively refined meshes have roughly $4\\times$ as many elements. When uploading the results, the $\\Delta x$ values should be provided in the datafile alongside the values for $\\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.\n",
     "\n",
     " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
     " \n",
@@ -497,7 +497,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the [defintion above](#Part-%28b%29:-multiple-initial-seeds). The simulations are run to $t=200$. The following data should be collected.\n",
+    "Part (b) requires running 5 simulations with multiple seeds randomly positioned at $t=0$, see the [defintion above](#Part-%28b%29:-multiple-initial-seeds). The simulations are run to $t=200$. The following data should be collected.\n",
     "\n",
     " - The solid fraction in the domain (the integral $\\int_V \\phi dV / \\int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).\n",
     " \n",
@@ -925,7 +925,7 @@
     "  issue = {6},\n",
     "  pages={1085--1095},\n",
     "  year={1979},\n",
-    "  url={https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2}\n",
+    "  url={https://doi.org/10.1016/0001-6160(79)90196-2}\n",
     "}\n",
     "\n",
     "-->"
@@ -935,7 +935,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## References\n",
+    "# References\n",
     "\n",
     "<a id=\"cite-granasy2002nucleation\"/><sup><a href=\"#ref-1\">[^]</a><a href=\"#ref-9\">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. _Nucleation and bulk crystallization in binary phase field theory_. [URL](https://doi.org/10.1103/PhysRevLett.88.206105)\n",
     "\n",
@@ -953,7 +953,7 @@
     "\n",
     "<a id=\"cite-kubo1966\"/><sup><a href=\"#ref-8\">[^]</a></sup>Kubo, R. 1966. _The fluctuation-dissipation theorem_. [URL](https://doi.org/10.1088%2F0034-4885%2F29%2F1%2F306)\n",
     "\n",
-    "<a id=\"cite-simmons2000phase\"/><sup><a href=\"#ref-10\">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. _Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events_. <span data-proofer-ignore>[URL](https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0)</span>\n",
+    "<a id=\"cite-simmons2000phase\"/><sup><a href=\"#ref-10\">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. _Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events_. [URL][simmons_url]\n",
     "\n",
     "<a id=\"cite-shi2019\"/><sup><a href=\"#ref-11\">[^]</a></sup>Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J. 2019. _Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response_. [URL](https://doi.org/10.1007/s11837-019-03618-1)\n",
     "\n",
@@ -967,8 +967,11 @@
     "\n",
     "<a id=\"cite-kolmogorov1937izv\"/><sup><a href=\"#ref-16\">[^]</a><a href=\"#ref-22\">[^]</a></sup>Kolmogorov, A N. 1937. _On the Statistical Theory of the Crystallization of Metals_. [URL](https://doi.org/10.1007/978-94-011-2260-3_22)\n",
     "\n",
-    "<a id=\"cite-allen1979microscopic\"/><sup><a href=\"#ref-17\">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. _A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening_. <span data-proofer-ignore>[URL](https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2)</span>\n",
-    "\n"
+    "<a id=\"cite-allen1979microscopic\"/><sup><a href=\"#ref-17\">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. _A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening_.\n",
+    "[URL][allen_url]\n",
+    "\n",
+    "[allen_url]: https://doi.org/10.1016/0001-6160(79)90196-2\n",
+    "[simmons_url]: https://doi.org/10.1016/S1359-6462(00)00517-0\n"
    ]
   },
   {
@@ -1041,13 +1044,6 @@
     "plt.imshow(arr)\n",
     "print('Actual number of particles:', scipy.ndimage.label(arr)[1])"
    ]
-  },
-  {
-   "cell_type": "code",
-   "execution_count": null,
-   "metadata": {},
-   "outputs": [],
-   "source": []
   }
  ],
  "metadata": {
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index d92de8390..cc9cda5cb 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="bfc93809-d417-4de7-8e71-d6cb7fb6f7cf"></div>
+<div id="db911dae-bc65-4ee4-9718-e32c7c6114d1"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#bfc93809-d417-4de7-8e71-d6cb7fb6f7cf');
+var element = $('#db911dae-bc65-4ee4-9718-e32c7c6114d1');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -377,7 +377,7 @@ <h3 id="Avrami-plots">Avrami plots<a class="anchor-link" href="#Avrami-plots">&#
 <a id="ref-21" href="#cite-avrami1941granulation">Avrami 1941</a>,
 <a id="ref-22" href="#cite-kolmogorov1937izv">Kolmogorov 1937</a>), which makes a number of 
 assumptions and simplifications: (i)  nucleation occurs randomly and homogeneously over the entire un-transformed portion of the material, (ii) the growth rate is constant and does not depend on the extent of transformation, (iii) the particles have convex shape with the same orientation, and (iv) the size of the system is infinite (both in space and time).</p>
-<p>The basis of the Avrami plots is the transformed fraction (solid fraction) $X(t)$ vs. time. According to the JMAK theory,
+<p>The basis of the Avrami plots is the transformed fraction (solid fraction) $Y(t)$ vs. time. According to the JMAK theory,
 \begin{equation}
     Y(t)=1-\exp (-Kt^n),
     \label{eqn:Xt}
@@ -413,7 +413,7 @@ <h3 id="Part-(a):-single-seed">Part (a): single seed<a class="anchor-link" href=
     \right)\right].
     \label{eqn:phitr_3}
 \end{equation}
-from the different seeds $i$. Overlaps between different seeds are managed by setting $\phi=1$ in all regions where the sums of $\phi_i &gt;1$. After placing initial seeds and adjusting the initial phase field $\phi$ so that $\phi\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $X$, discrete particle count $N$  (the number of disjoint regions with $\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.</p>
+from the different seeds $i$. Overlaps between different seeds are managed by setting $\phi=1$ in all regions where the sums of $\phi_i &gt;1$. After placing initial seeds and adjusting the initial phase field $\phi$ so that $\phi\leq1$ everywhere, the simulation is run up to total time $t=200$, at which time the whole domain is transformed to a solid ($\phi=1$).  This is repeated four times, each time with different random seeds, and the total free energy, time evolution of the solid fraction $Y$, discrete particle count $N$  (the number of disjoint regions with $\phi=1$) are plotted for the five simulations, and Avrami plots are generated from the five simulations; from the Avrami plot the exponent $n$ is estimated and compared to the JMAK theory.</p>
 <p>Both Part (b) and Part (c) involve a random number generator. Because different executions will in general generate different random numbers, specific individual solutions will in general not be reproduced. However, statistics such as the average slope of Avrami plots can be meaningfully compared.</p>
 <p>To submit results to the PFHub website, see the <a href="#Part-%28b%29">submission guidelines</a> below.</p>
 <h2 id="Part-(c):-multiple-seeds-at-random-times">Part (c): multiple seeds at random times<a class="anchor-link" href="#Part-(c):-multiple-seeds-at-random-times">&#182;</a></h2><p>For the third part of the homogeneous benchmark problem, the computational domain is first expanded to $1000 \times1000$. Instead of inserting nuclei at fixed time $t = 0$ as in part two of the problem, in this case 100 random nucleation times $t_i$ are generated, $i=1,\ldots,100$, drawn from a uniform distribution in the interval $t_i \in [0, 600)$ with centers $\mathbf{r}_i$ drawn from a uniform distribution on the $1000\times1000$ domain.
@@ -423,7 +423,7 @@ <h2 id="Part-(c):-multiple-seeds-at-random-times">Part (c): multiple seeds at ra
     \frac{|\mathbf{r}-\mathbf{r}_i|-1.1r^*}{\sqrt{2}}
     \right)\right].
 \end{equation}</p>
-<p>Again, $\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $X$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.</p>
+<p>Again, $\phi$ is set to unity in regions of overlaps of nuclei. The simulations are run up to times $t=600$, and repeated for four more random initial conditions.  The total free energy plot the total free energy, the time evolution of the solid fraction $Y$ and the discrete particle count $N$, are plotted as functions of time, and an  Avrami plot is generated for the five runs.</p>
 <p>To submit results to the PFHub website, see the <a href="#Part-%28c%29">submission guidelines</a> below.</p>
 
 </div>
@@ -442,7 +442,7 @@ <h2 id="Submission-Guidelines">Submission Guidelines<a class="anchor-link" href=
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
 <p>Please use the <a href="{{ site.baseurl }}/simulations/upload_form/">upload form</a> to upload your results.</p>
-<p>In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. These are not required, but will help others view your work.</p>
+<p>In addition to that specified, further data to upload can include a Youtube video, images of the nuclei at different times, or the entire phase field variable at different times. This data can be uploaded directly to the website or stored at a secondary location and only the link to the data provided. This data is not required, but will help others view your work.</p>
 
 </div>
 </div>
@@ -467,7 +467,7 @@ <h3 id="Part-(a)">Part (a)<a class="anchor-link" href="#Part-(a)">&#182;</a></h3
 <li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_1$ (optional)</li>
 <li>$r_0 = 1.01 r^*$, ${\Delta x} = {\Delta x}_2$ (optional)</li>
 </ol>
-<p>where ${\Delta x}_0 &lt; {\Delta x}_1 &lt; {\Delta x}_2$. If using a gridded mesh each successively refined grid length can be half as fine. In the case of adaptive grids a nominal value for $\Delta x$ should be selected that results in a similarly refined mesh as in the gridded case (i.e. approximately $4\times$ as many elements). When uploading the results, the $\Delta x$ values should be provided in the datafile alongside the values for $\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.</p>
+<p>where ${\Delta x}_0, {\Delta x}_1$ and ${\Delta x}_2$ represent simulations at different mesh sizes. $\Delta x$ refinement should be selected so that successively refined meshes have roughly $4\times$ as many elements. When uploading the results, the $\Delta x$ values should be provided in the datafile alongside the values for $\phi$ at each location in the domain. Each simulation should be run to $t=200$ and the following data should be collected.</p>
 <ul>
 <li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
 </li>
@@ -526,7 +526,7 @@ <h3 id="Part-(b)">Part (b)<a class="anchor-link" href="#Part-(b)">&#182;</a></h3
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<p>Part (b) requires running 5 simulations with multiple seed randomly positioned at $t=0$, see the <a href="#Part-%28b%29:-multiple-initial-seeds">defintion above</a>. The simulations are run to $t=200$. The following data should be collected.</p>
+<p>Part (b) requires running 5 simulations with multiple seeds randomly positioned at $t=0$, see the <a href="#Part-%28b%29:-multiple-initial-seeds">defintion above</a>. The simulations are run to $t=200$. The following data should be collected.</p>
 <ul>
 <li><p>The solid fraction in the domain (the integral $\int_V \phi dV / \int_V dV$) during the transient (not necessarily at every time step but for ~100 data points).</p>
 </li>
@@ -963,7 +963,7 @@ <h2 id="Results">Results<a class="anchor-link" href="#Results">&#182;</a></h2>
   issue = {6},
   pages={1085--1095},
   year={1979},
-  url={https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2}
+  url={https://doi.org/10.1016/0001-6160(79)90196-2}
 }
 
 -->
@@ -974,7 +974,7 @@ <h2 id="Results">Results<a class="anchor-link" href="#Results">&#182;</a></h2>
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h2 id="References">References<a class="anchor-link" href="#References">&#182;</a></h2><p><a id="cite-granasy2002nucleation"/><sup><a href="#ref-1">[^]</a><a href="#ref-9">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. <em>Nucleation and bulk crystallization in binary phase field theory</em>. <a href="https://doi.org/10.1103/PhysRevLett.88.206105">URL</a></p>
+<h1 id="References">References<a class="anchor-link" href="#References">&#182;</a></h1><p><a id="cite-granasy2002nucleation"/><sup><a href="#ref-1">[^]</a><a href="#ref-9">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. <em>Nucleation and bulk crystallization in binary phase field theory</em>. <a href="https://doi.org/10.1103/PhysRevLett.88.206105">URL</a></p>
 <p><a id="cite-castro2003phase"/><sup><a href="#ref-2">[^]</a></sup>Castro, Mario. 2003. <em>Phase-field approach to heterogeneous nucleation</em>. <a href="https://link.aps.org/doi/10.1103/PhysRevB.67.035412">URL</a></p>
 <p><a id="cite-simmons2004microstructural"/><sup><a href="#ref-3">[^]</a></sup>Simmons, J P and Wen, Y and Shen, C and Wang, Y. 2004. <em>Microstructural development involving nucleation and growth phenomena simulated with the Phase Field method</em>. <a href="https://doi.org/10.1016/j.msea.2003.09.019">URL</a></p>
 <p><a id="cite-granasy2007phase"/><sup><a href="#ref-4">[^]</a></sup>Gránásy, L and T and Pusztai, T and Saylor, D and Warren, J A. 2007. <em>Phase field theory of heterogeneous crystal nucleation</em>. <a href="https://doi.org/10.1103/PhysRevLett.98.035703">URL</a></p>
@@ -982,14 +982,15 @@ <h2 id="References">References<a class="anchor-link" href="#References">&#182;</
 <p><a id="cite-heo2014phase"/><sup><a href="#ref-6">[^]</a></sup>Heo, T W and Chen, L. 2014. <em>Phase-field modeling of nucleation in solid-state phase transformations</em>. <a href="https://doi.org/10.1007/s11837-014-1033-9">URL</a></p>
 <p><a id="cite-granasy2019"/><sup><a href="#ref-7">[^]</a></sup>Gránásy, L and Tóth, G and Warren, J A and Podmaniczky, F and Tegze, G and Rátkai, L and Pusztai, T. 2019. <em>Phase-field modeling of crystal nucleation in undercooled liquids -- A review</em>. <a href="https://doi.org/10.1016/j.pmatsci.2019.05.002">URL</a></p>
 <p><a id="cite-kubo1966"/><sup><a href="#ref-8">[^]</a></sup>Kubo, R. 1966. <em>The fluctuation-dissipation theorem</em>. <a href="https://doi.org/10.1088%2F0034-4885%2F29%2F1%2F306">URL</a></p>
-<p><a id="cite-simmons2000phase"/><sup><a href="#ref-10">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. <em>Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events</em>. <span data-proofer-ignore>[URL](https://doi.org/10.1016/S1359-6462&#40;00&#41;00517-0)</span></p>
+<p><a id="cite-simmons2000phase"/><sup><a href="#ref-10">[^]</a></sup>Simmons, J P and Shen, C and Wang, Y. 2000. <em>Phase field modeling of simultaneous nucleation and growth by explicitly incorporating nucleation events</em>. <a href="https://doi.org/10.1016/S1359-6462(00)00517-0">URL</a></p>
 <p><a id="cite-shi2019"/><sup><a href="#ref-11">[^]</a></sup>Shi, R and Khairallah, S and Heo, T W and Rolchigo, M and McKeown, J T and Matthews, M J. 2019. <em>Integrated simulation framework for additively manufactured Ti-6Al-4V: melt pool dynamics, microstructure, solid-state phase transformation, and microelastic response</em>. <a href="https://doi.org/10.1007/s11837-019-03618-1">URL</a></p>
 <p><a id="cite-johnson1939reaction"/><sup><a href="#ref-12">[^]</a><a href="#ref-18">[^]</a></sup>Johnson, William A and Mehl, R F. 1939. <em>Reaction kinetics in processes of nucleation and growth</em>. <a href="https://doi.org/10.1007/s11663-010-9421-1">URL</a></p>
 <p><a id="cite-avrami1939kinetics"/><sup><a href="#ref-13">[^]</a><a href="#ref-19">[^]</a></sup>Avrami, Melvin. 1939. <em>Kinetics of phase change. I General theory</em>. <a href="https://doi.org/10.1063/1.1750380">URL</a></p>
 <p><a id="cite-avrami1940kinetics"/><sup><a href="#ref-14">[^]</a><a href="#ref-20">[^]</a></sup>Avrami, Melvin. 1940. <em>Kinetics of phase change. II transformation-time relations for random distribution of nuclei</em>. <a href="https://doi.org/10.1063/1.1750631">URL</a></p>
 <p><a id="cite-avrami1941granulation"/><sup><a href="#ref-15">[^]</a><a href="#ref-21">[^]</a></sup>Avrami, Melvin. 1941. <em>Granulation, phase change, and microstructure kinetics of phase change. III</em>. <a href="https://doi.org/10.1063/1.1750872">URL</a></p>
 <p><a id="cite-kolmogorov1937izv"/><sup><a href="#ref-16">[^]</a><a href="#ref-22">[^]</a></sup>Kolmogorov, A N. 1937. <em>On the Statistical Theory of the Crystallization of Metals</em>. <a href="https://doi.org/10.1007/978-94-011-2260-3_22">URL</a></p>
-<p><a id="cite-allen1979microscopic"/><sup><a href="#ref-17">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. <em>A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening</em>. <span data-proofer-ignore>[URL](https://doi.org/10.1016/0001-6160&#40;79&#41;90196-2)</span></p>
+<p><a id="cite-allen1979microscopic"/><sup><a href="#ref-17">[^]</a></sup>Allen, Samuel M and Cahn, John W. 1979. <em>A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening</em>.
+<a href="https://doi.org/10.1016/0001-6160(79)90196-2">URL</a></p>
 
 </div>
 </div>

From a97879e649b3a85da676d77888d0b132c04890d6 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Fri, 16 Apr 2021 10:12:25 -0400
Subject: [PATCH 12/13] fix: add particle count code correctly

Code for particle counts was not previously shown unless the code
toggle button was pressed on the website. Now the code is in a
markdown cell so will always be on the website.
---
 benchmarks/benchmark8.ipynb          | 43 ++++++++++++++++++++++++++--
 benchmarks/benchmark8.ipynb.raw.html | 41 ++++++++++++++++++++++++--
 2 files changed, 80 insertions(+), 4 deletions(-)

diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index 32199a6c6..15e31e331 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -935,7 +935,7 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "# References\n",
+    "## References\n",
     "\n",
     "<a id=\"cite-granasy2002nucleation\"/><sup><a href=\"#ref-1\">[^]</a><a href=\"#ref-9\">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. _Nucleation and bulk crystallization in binary phase field theory_. [URL](https://doi.org/10.1103/PhysRevLett.88.206105)\n",
     "\n",
@@ -980,7 +980,46 @@
    "source": [
     "## Appendix\n",
     "\n",
-    "The following code is given as a resource for those implementing the benchmark to help count the particles in Part (b) and Part (c). One possibility as outlined here is to use [`scipy.ndimage.label`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.label.html#scipy.ndimage.label). This returns a tuple with a labeled array of unique features as the first item and a count of features as the second item. In the case of the phase field the values will require thresholding to 0 or 1 before using `scipy.ndimage.label`."
+    "The following code is given as a resource for those implementing the benchmark to help count the particles in Part (b) and Part (c). One possibility as outlined here is to use [`scipy.ndimage.label`](https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.label.html#scipy.ndimage.label). This returns a tuple with a labeled array of unique features as the first item and a count of features as the second item. In the case of the phase field the values will require thresholding to 0 or 1 before using `scipy.ndimage.label`.\n",
+    "\n",
+    "```Python\n",
+    "import numpy as np\n",
+    "import scipy.ndimage\n",
+    "import matplotlib.pyplot as plt\n",
+    "\n",
+    "def make_particles(shape, n_particles, max_radius, seed=99):\n",
+    "    \"\"\"Make an array with random particles\n",
+    "    \n",
+    "    Args:\n",
+    "      shape: shape of array\n",
+    "      n_particles: number of particles in arrau\n",
+    "      max_radius: maximum possible radius of particles\n",
+    "      seed: random seed\n",
+    "      \n",
+    "    Returns: numpy array mask with particles\n",
+    "    \"\"\"\n",
+    "    np.random.seed(seed)\n",
+    "        \n",
+    "    def xy_grid(shape):\n",
+    "        \"\"\"Generate a grid of x, y values in a single array\n",
+    "        \n",
+    "        The returned array has shape (len(shape),) + shape + (1,). The last axis is\n",
+    "        for the operations with particles.\n",
+    "        \"\"\"\n",
+    "        linspace = lambda x: np.linspace(0, x - 1, x) + 0.5\n",
+    "        return np.array(np.meshgrid(*map(linspace, shape), indexing=\"ij\"))[..., None]\n",
+    "    \n",
+    "    # first axis is dimensions\n",
+    "    # second and third axes are shape of domain\n",
+    "    # last axis is the number of particles. \n",
+    "    centers = (np.random.random((len(shape), n_particles)) * np.array(shape)[:, None])[:, None, None]\n",
+    "    radii = (np.random.random(n_particles) * max_radius)[None, None]\n",
+    "    return numpy.any(((xy_grid(shape) - centers)**2).sum(0) < radii**2, axis=-1)\n",
+    "\n",
+    "arr = make_particles((100, 200), 20, 10)\n",
+    "plt.imshow(arr)\n",
+    "print('Actual number of particles:', scipy.ndimage.label(arr)[1])\n",
+    "```"
    ]
   },
   {
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index cc9cda5cb..d841c184d 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="db911dae-bc65-4ee4-9718-e32c7c6114d1"></div>
+<div id="b858144d-cfb3-4559-8a18-b8cdadd97b7d"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#db911dae-bc65-4ee4-9718-e32c7c6114d1');
+var element = $('#b858144d-cfb3-4559-8a18-b8cdadd97b7d');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -1000,6 +1000,43 @@ <h1 id="References">References<a class="anchor-link" href="#References">&#182;</
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
 <h2 id="Appendix">Appendix<a class="anchor-link" href="#Appendix">&#182;</a></h2><p>The following code is given as a resource for those implementing the benchmark to help count the particles in Part (b) and Part (c). One possibility as outlined here is to use <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.ndimage.label.html#scipy.ndimage.label"><code>scipy.ndimage.label</code></a>. This returns a tuple with a labeled array of unique features as the first item and a count of features as the second item. In the case of the phase field the values will require thresholding to 0 or 1 before using <code>scipy.ndimage.label</code>.</p>
+<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="kn">as</span> <span class="nn">np</span>
+<span class="kn">import</span> <span class="nn">scipy.ndimage</span>
+<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="kn">as</span> <span class="nn">plt</span>
+
+<span class="k">def</span> <span class="nf">make_particles</span><span class="p">(</span><span class="n">shape</span><span class="p">,</span> <span class="n">n_particles</span><span class="p">,</span> <span class="n">max_radius</span><span class="p">,</span> <span class="n">seed</span><span class="o">=</span><span class="mi">99</span><span class="p">):</span>
+    <span class="sd">&quot;&quot;&quot;Make an array with random particles</span>
+
+<span class="sd">    Args:</span>
+<span class="sd">      shape: shape of array</span>
+<span class="sd">      n_particles: number of particles in arrau</span>
+<span class="sd">      max_radius: maximum possible radius of particles</span>
+<span class="sd">      seed: random seed</span>
+
+<span class="sd">    Returns: numpy array mask with particles</span>
+<span class="sd">    &quot;&quot;&quot;</span>
+    <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span><span class="p">(</span><span class="n">seed</span><span class="p">)</span>
+
+    <span class="k">def</span> <span class="nf">xy_grid</span><span class="p">(</span><span class="n">shape</span><span class="p">):</span>
+        <span class="sd">&quot;&quot;&quot;Generate a grid of x, y values in a single array</span>
+
+<span class="sd">        The returned array has shape (len(shape),) + shape + (1,). The last axis is</span>
+<span class="sd">        for the operations with particles.</span>
+<span class="sd">        &quot;&quot;&quot;</span>
+        <span class="n">linspace</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="n">x</span> <span class="o">-</span> <span class="mi">1</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="mf">0.5</span>
+        <span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span><span class="p">(</span><span class="o">*</span><span class="nb">map</span><span class="p">(</span><span class="n">linspace</span><span class="p">,</span> <span class="n">shape</span><span class="p">),</span> <span class="n">indexing</span><span class="o">=</span><span class="s2">&quot;ij&quot;</span><span class="p">))[</span><span class="o">...</span><span class="p">,</span> <span class="bp">None</span><span class="p">]</span>
+
+    <span class="c1"># first axis is dimensions</span>
+    <span class="c1"># second and third axes are shape of domain</span>
+    <span class="c1"># last axis is the number of particles. </span>
+    <span class="n">centers</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">shape</span><span class="p">),</span> <span class="n">n_particles</span><span class="p">))</span> <span class="o">*</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="n">shape</span><span class="p">)[:,</span> <span class="bp">None</span><span class="p">])[:,</span> <span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">]</span>
+    <span class="n">radii</span> <span class="o">=</span> <span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">random</span><span class="p">(</span><span class="n">n_particles</span><span class="p">)</span> <span class="o">*</span> <span class="n">max_radius</span><span class="p">)[</span><span class="bp">None</span><span class="p">,</span> <span class="bp">None</span><span class="p">]</span>
+    <span class="k">return</span> <span class="n">numpy</span><span class="o">.</span><span class="n">any</span><span class="p">(((</span><span class="n">xy_grid</span><span class="p">(</span><span class="n">shape</span><span class="p">)</span> <span class="o">-</span> <span class="n">centers</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="mi">0</span><span class="p">)</span> <span class="o">&lt;</span> <span class="n">radii</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=-</span><span class="mi">1</span><span class="p">)</span>
+
+<span class="n">arr</span> <span class="o">=</span> <span class="n">make_particles</span><span class="p">((</span><span class="mi">100</span><span class="p">,</span> <span class="mi">200</span><span class="p">),</span> <span class="mi">20</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
+<span class="n">plt</span><span class="o">.</span><span class="n">imshow</span><span class="p">(</span><span class="n">arr</span><span class="p">)</span>
+<span class="k">print</span><span class="p">(</span><span class="s1">&#39;Actual number of particles:&#39;</span><span class="p">,</span> <span class="n">scipy</span><span class="o">.</span><span class="n">ndimage</span><span class="o">.</span><span class="n">label</span><span class="p">(</span><span class="n">arr</span><span class="p">)[</span><span class="mi">1</span><span class="p">])</span>
+</pre></div>
 
 </div>
 </div>

From 65ccdc2b45b7b76488ece5b969332075584fdda6 Mon Sep 17 00:00:00 2001
From: Daniel Wheeler <daniel.wheeler2@gmail.com>
Date: Fri, 16 Apr 2021 15:16:44 -0400
Subject: [PATCH 13/13] fix: link to nucleation paper

Link to nucleation paper was broken.
---
 benchmarks/benchmark8.ipynb          | 2 +-
 benchmarks/benchmark8.ipynb.raw.html | 8 ++++----
 2 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/benchmarks/benchmark8.ipynb b/benchmarks/benchmark8.ipynb
index 15e31e331..613ffc787 100644
--- a/benchmarks/benchmark8.ipynb
+++ b/benchmarks/benchmark8.ipynb
@@ -153,7 +153,7 @@
     "See the journal publication entitled [\"Phase Field Benchmark Problems for Nucleation\"][paper] for more details about the benchmark problems. Furthermore, read [the extended essay][benchmarks] for a discussion about the need for benchmark problems.\n",
     "\n",
     "[benchmarks]: ../\n",
-    "[paper]: https://doi.org/10.1016/j.commatsci.2018.03.015"
+    "[paper]: https://doi.org/10.1016/j.commatsci.2021.110371"
    ]
   },
   {
diff --git a/benchmarks/benchmark8.ipynb.raw.html b/benchmarks/benchmark8.ipynb.raw.html
index d841c184d..8bc4ac6d5 100644
--- a/benchmarks/benchmark8.ipynb.raw.html
+++ b/benchmarks/benchmark8.ipynb.raw.html
@@ -26,10 +26,10 @@
 
 
 
-<div id="b858144d-cfb3-4559-8a18-b8cdadd97b7d"></div>
+<div id="0a3211d2-bcca-4755-82ee-0b1dd437f99f"></div>
 <div class="output_subarea output_javascript ">
 <script type="text/javascript">
-var element = $('#b858144d-cfb3-4559-8a18-b8cdadd97b7d');
+var element = $('#0a3211d2-bcca-4755-82ee-0b1dd437f99f');
     MathJax.Hub.Config({
       TeX: { equationNumbers: { autoNumber: "AMS" } }
     });
@@ -194,7 +194,7 @@ <h1 id="Benchmark-Problem-8:-Homogeneous-Nucleation">Benchmark Problem 8: Homoge
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<p>See the journal publication entitled <a href="https://doi.org/10.1016/j.commatsci.2018.03.015">"Phase Field Benchmark Problems for Nucleation"</a> for more details about the benchmark problems. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
+<p>See the journal publication entitled <a href="https://doi.org/10.1016/j.commatsci.2021.110371">"Phase Field Benchmark Problems for Nucleation"</a> for more details about the benchmark problems. Furthermore, read <a href="../">the extended essay</a> for a discussion about the need for benchmark problems.</p>
 
 </div>
 </div>
@@ -974,7 +974,7 @@ <h2 id="Results">Results<a class="anchor-link" href="#Results">&#182;</a></h2>
 </div>
 <div class="inner_cell">
 <div class="text_cell_render border-box-sizing rendered_html">
-<h1 id="References">References<a class="anchor-link" href="#References">&#182;</a></h1><p><a id="cite-granasy2002nucleation"/><sup><a href="#ref-1">[^]</a><a href="#ref-9">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. <em>Nucleation and bulk crystallization in binary phase field theory</em>. <a href="https://doi.org/10.1103/PhysRevLett.88.206105">URL</a></p>
+<h2 id="References">References<a class="anchor-link" href="#References">&#182;</a></h2><p><a id="cite-granasy2002nucleation"/><sup><a href="#ref-1">[^]</a><a href="#ref-9">[^]</a></sup>Gránásy, L and and Börzsönyi, T and Pusztai, T. 2002. <em>Nucleation and bulk crystallization in binary phase field theory</em>. <a href="https://doi.org/10.1103/PhysRevLett.88.206105">URL</a></p>
 <p><a id="cite-castro2003phase"/><sup><a href="#ref-2">[^]</a></sup>Castro, Mario. 2003. <em>Phase-field approach to heterogeneous nucleation</em>. <a href="https://link.aps.org/doi/10.1103/PhysRevB.67.035412">URL</a></p>
 <p><a id="cite-simmons2004microstructural"/><sup><a href="#ref-3">[^]</a></sup>Simmons, J P and Wen, Y and Shen, C and Wang, Y. 2004. <em>Microstructural development involving nucleation and growth phenomena simulated with the Phase Field method</em>. <a href="https://doi.org/10.1016/j.msea.2003.09.019">URL</a></p>
 <p><a id="cite-granasy2007phase"/><sup><a href="#ref-4">[^]</a></sup>Gránásy, L and T and Pusztai, T and Saylor, D and Warren, J A. 2007. <em>Phase field theory of heterogeneous crystal nucleation</em>. <a href="https://doi.org/10.1103/PhysRevLett.98.035703">URL</a></p>