diff --git a/benchmarks/benchmark2.ipynb b/benchmarks/benchmark2.ipynb index e6f346ea0..b23ee9501 100644 --- a/benchmarks/benchmark2.ipynb +++ b/benchmarks/benchmark2.ipynb @@ -173,13 +173,14 @@ "stress the numerical solver while not making the problem intractable.\n", "\n", "In this benchmark problem, the free energy of the system is based on\n", - "the formulation presented in Ref. [ZHU][] and is expressed as\n", + "a more general formulation of that presented in Ref. [ZHU][]. Specifically, the free energy here includes the concentration gradient energy as outlined in [WBM1][] and is expressed as\n", "\n", "$$\n", "F=\\int_{V}\\left(f_{chem}\\left(c,\\eta_{1},...\\eta_{p}\\right)+\\frac{\\kappa_{c}}{2}|\\nabla\n", "c|^{2}+\\sum_{i=1}^{p}\\frac{\\kappa_{\\eta}}{2}|\\nabla\\eta_{i}|^{2}\\right)dV\n", "$$\n", "\n", + "where the concentration gradient energy is \n", "where $\\kappa_{c}$ and $\\kappa_{\\eta}$ are the gradient energy\n", "coefficients for $c$ and $\\eta_{i}$, respectively. While the model in\n", "Ref. [ZHU][] follows the Kim-Kim-Suzuki ([KKS][]) formulation for\n", @@ -268,6 +269,7 @@ "[ZHU]: http://dx.doi.org/10.1016/j.actamat.2004.02.032\n", "[KKS]: https://doi.org/10.1103/PhysRevE.60.7186\n", "[WBM]: https://doi.org/10.1103/PhysRevA.45.7424\n", + "[WBM1]: https://doi.org/10.1103/PhysRevE.47.1893\n", "[CAHN]: http://dx.doi.org/10.1016/0001-6160(61)90182-1\n", "[ELLIOT]: http://dx.doi.org/10.1007/BF01396363\n", "[ALLEN]: http://dx.doi.org/10.1016/0001-6160(79)90196-2\n" @@ -1437,7 +1439,7 @@ "metadata": { "anaconda-cloud": {}, "kernelspec": { - "display_name": "Python [default]", + "display_name": "Python 3 (ipykernel)", "language": "python", "name": "python3" }, @@ -1451,7 +1453,7 @@ "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", - "version": "3.6.1" + "version": "3.6.4" } }, "nbformat": 4, diff --git a/benchmarks/benchmark2.ipynb.raw.html b/benchmarks/benchmark2.ipynb.raw.html index 860ca4263..0c492bc76 100644 --- a/benchmarks/benchmark2.ipynb.raw.html +++ b/benchmarks/benchmark2.ipynb.raw.html @@ -209,12 +209,13 @@
In this benchmark problem, the free energy of the system is based on -the formulation presented in Ref. ZHU and is expressed as
+a more general formulation of that presented in Ref. ZHU. Specifically, the free energy here includes the concentration gradient energy as outlined in WBM1 and is expressed as$$ F=\int_{V}\left(f_{chem}\left(c,\eta_{1},...\eta_{p}\right)+\frac{\kappa_{c}}{2}|\nabla c|^{2}+\sum_{i=1}^{p}\frac{\kappa_{\eta}}{2}|\nabla\eta_{i}|^{2}\right)dV $$
-where $\kappa_{c}$ and $\kappa_{\eta}$ are the gradient energy +
where the concentration gradient energy is +where $\kappa_{c}$ and $\kappa_{\eta}$ are the gradient energy coefficients for $c$ and $\eta_{i}$, respectively. While the model in Ref. ZHU follows the Kim-Kim-Suzuki (KKS) formulation for interfacial energy, we use the @@ -3185,10 +3186,10 @@