The Development,analysis And Algorithm For The Thermodynamically Consistent Phase Field Model

Non-equilibrium phenomena are widely found in science and engineer,the development of models that satisfy the laws of thermodynamics are what we desired.In addition to this,the models also need to meet other physical laws such as volume conservation,mass conservation,atomic conservation and so on.In recent years,phase field methods are more and more popular to describe the non-equilibrium phenomena.We call such model as the thermodynamically consistent phase field model.Allen-Cahn and Cahn-Hillard models are two commonly used models,Allen-Cahn model does not guarantee the conservation of physical quantity as the Cahn-Hillard does.However,the order of the space operator in the Cahn-Hillard equation is higher than that in the Allen-Cahn equation,which may lead to more time cost during simulation.In this paper,the modified Allen-Cahn model can meet the conservation properties by adding non-local constraints to the Allen-Cahn equation.We call it as the Allen-Cahn model with non-local constraints or nonlocal Allen-Cahn model.In this paper,we developed two Allen-Cahn models with non-local constraints,which can be proved that satisfy the energy dissipation and the conservation properties.We also performed linear stability analysis on the Allen-Cahn model with nonlocal constraints at near equilibrium state and compared it with Allen-Cahn model and Cahn-Hillard model.We used the strategy of the energy quadratic form in time to derive energy stable semi-discrete numerical algorithms.By applying second order finite difference methods on cell-centered grids in space the fully discrete schemes can be obtained afterwards.The unconditional energy dissipation and uniqueness of the solution for all seme-discrete and full-discrete schemes were proved in this paper.We gave several benchmark numerical examples to assess the performance of the schemes designed by energy quadratization(EQ)and scalar auxiliary variable(SAV)methods.Meanwhile,we compared the dynamics obtained using Allen-Cahn models with nonlocal constraints with those using the classical Cahn-Hilliard as well as the Allen-Cahn model,respectively.Some tricks to enhance the performance of practical implementation for the linear energy stable schemes designed by EQ are discussed in the paper.Besides these,we also developed a model to describe the transport process and chemical reaction process in the chemical system by using generalized Onsager principle.For the homogeneous chemical system,we can prove the Helmholtz free energy is always dissipative.For the inhomogeneous chemical system,we can prove the Helmholtz free energy can also be dissipative with suitable boundary conditions and assumptions.From the view of generalized Onsager principle,the reaction-transport processes can be described by an Allen-Cahn equation plus a Cahn-Hillard equation.The analysis shows the phase separation in an inhomogenous chemical system depends on both the chemical reaction process and the diffusion process.At last,we simulated the reaction-transport processes in a three-component polymer solution to investigate the effect of reaction on the transport process.