Polar Phase Field model

With the help of the PolarPhaseFieldAction users can set up the three phase polar phase field model from Momeni and Levitas (2014).

Ginzburg-Landau equations

The model consists of the following main equations (eqs. (14) and (15) in the 2014 paper)

$var element = document.getElementById("moose-equation-005634a5-e03e-4c1c-bb37-129250ee3f99");katex.render("\\underbrace{\\frac1{L_\\Upsilon}\\frac{\\partial\\Upsilon}{\\partial t} }_{\\text{SusceptibilityTimeDerivative}} = \\underbrace{-\\frac{\\partial\\psi^l}{\\partial\\Upsilon}}_{\\text{PolarPFMDerivative}} \\underbrace{-\\frac{\\beta^{21}}2\\frac{\\partial\\phi(\\Upsilon,a_\\phi,a_0)}{\\partial\\Upsilon}|\\nabla\\vartheta|^2}_{\\text{PolarPFMGradient}} + \\underbrace{\\nabla\\cdot\\left[\\beta^{S0}(\\vartheta)\\nabla\\Upsilon\\right]}_{\\text{MatDiffusion}}", element, {displayMode:true,throwOnError:false});$

$var element = document.getElementById("moose-equation-bb162fe8-04c0-4755-843d-b6dc81e7c7ac");katex.render("\\underbrace{\\frac1{L_\\vartheta}\\frac{\\partial\\vartheta}{\\partial t} }_{\\text{SusceptibilityTimeDerivative}} = \\underbrace{-\\frac{\\partial\\psi^l}{\\partial\\vartheta}}_{\\text{PolarPFMDerivative}} \\underbrace{-\\frac12\\frac{\\partial\\beta^{S0}(\\vartheta)}{\\partial\\vartheta}|\\nabla\\Upsilon|^2}_{\\text{PolarPFMGradient}} + \\underbrace{\\nabla\\cdot\\left[\\beta^{21}\\phi(\\Upsilon,a_\\phi,a_0)\\nabla\\vartheta\\right]}_{\\text{MatDiffusion}}", element, {displayMode:true,throwOnError:false});$

This model requires the implementation of two new kernels PolarPFMDerivative and PolarPFMGradient for the terms indicated. The remaining terms can be modeled using the existing SusceptibilityTimeDerivative and MatDiffusion kernels.

Sub-terms appearing the equations above are implemented as materials

Material	Term
PolarPFMBetaS0	$var element = document.getElementById("moose-equation-4e53d813-6f1a-4312-a727-778f8021f7dd");katex.render("\\beta^{S0}(\\vartheta)", element, {displayMode:false,throwOnError:false});$
PolarPFMPhi	$var element = document.getElementById("moose-equation-2a4203f2-69bb-407a-94bb-12398fdbba2b");katex.render("\\beta^{21}\\phi(\\Upsilon,a_\\phi,a_0)", element, {displayMode:false,throwOnError:false});$
PolarPFMPsiL	$var element = document.getElementById("moose-equation-d3e95f36-7412-4675-a403-75031164e4af");katex.render("\\psi^l", element, {displayMode:false,throwOnError:false});$

Physical meaning

The model implements a two sold phase system with a third interfacial melt phase. The $var element = document.getElementById("moose-equation-3612b4d1-0f2e-4511-84c6-b51d96f124ea");katex.render("\\vartheta", element, {displayMode:false,throwOnError:false});$ order parameter switched between solid1 ( $var element = document.getElementById("moose-equation-f9fbb176-4bf4-4468-b5b5-cdca17bbfd41");katex.render("\\vartheta=0", element, {displayMode:false,throwOnError:false});$ ) and solid2 ( $var element = document.getElementById("moose-equation-b28a6fa9-3098-496a-98d8-b96334ccac10");katex.render("\\vartheta=0", element, {displayMode:false,throwOnError:false});$ ). The $var element = document.getElementById("moose-equation-7bf8719d-a746-47c9-8caf-32299ea231d3");katex.render("\\Upsilon", element, {displayMode:false,throwOnError:false});$ order parameter switches between solid ( $var element = document.getElementById("moose-equation-48ec5f05-9c94-4462-acfe-9e15b714156d");katex.render("\\vartheta=0", element, {displayMode:false,throwOnError:false});$ - governed by $var element = document.getElementById("moose-equation-1f7d0714-b49b-44a5-a3ca-04f287bb49dc");katex.render("\\vartheta", element, {displayMode:false,throwOnError:false});$ ) and interfacial melt ( $var element = document.getElementById("moose-equation-21a0971a-5ee0-4750-9741-5c1c63cdb2eb");katex.render("\\vartheta=1", element, {displayMode:false,throwOnError:false});$ ). The indices $var element = document.getElementById("moose-equation-abf53e75-52e3-45a0-bbbf-659f12ba072b");katex.render("s", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-84759553-2454-454f-9ac7-05a5b5157d6a");katex.render("1,2", element, {displayMode:false,throwOnError:false});$ refer to an unspecified and specified solid phase respectively, while the index $var element = document.getElementById("moose-equation-a1af62d2-40f9-4413-9833-4b41e89365f5");katex.render("0", element, {displayMode:false,throwOnError:false});$ refers to the interfacial melt phase.

Both order parameters $var element = document.getElementById("moose-equation-547d780c-da89-46e7-8463-4ec61932d1cd");katex.render("\\vartheta", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-85a4aca7-4272-4853-9b77-e56e244b5a69");katex.render("\\Upsilon", element, {displayMode:false,throwOnError:false});$ are non-conserved \emph{Allen-Cahn like} phase order parameters.

The parameters in the test and example files are using the parameterization for HMX from the original publication.

References

Kasra Momeni and Valery I. Levitas. Propagating phase interface with intermediate interfacial phase: Phase field approach. Physical Review B, May 2014. URL: https://link.aps.org/doi/10.1103/PhysRevB.89.184102 (visited on 2020-01-24), doi:10.1103/PhysRevB.89.184102.

@article{Momeni2014,
    author = "Momeni, Kasra and Levitas, Valery I.",
    title = "Propagating phase interface with intermediate interfacial phase: {Phase} field approach",
    volume = "89",
    issn = "1098-0121, 1550-235X",
    shorttitle = "Propagating phase interface with intermediate interfacial phase",
    url = "https://link.aps.org/doi/10.1103/PhysRevB.89.184102",
    doi = "10.1103/PhysRevB.89.184102",
    language = "en",
    number = "18",
    urldate = "2020-01-24",
    journal = "Physical Review B",
    month = "May",
    year = "2014"
}

Ginzburg - Landau equations
Physical meaning
References