PHASE FIELD METHOD EQUATIONS DERIVED FROM THERMODYNAMIC EXTREMAL PRINCIPLE
The thermodynamic extremal principle (TEP) formulated in discrete parameters 1 represents a handy tool in modeling of thermodynamic processes in solids. As a further concept the phase field method (PFM) has gotten an established method for simulation of microstructure evolution and allows accounting parallel action of a number of thermodynamic processes in an extremely easy way. The PFM goes back to the seminal work by Ginzburg and Landau 2, dealing with the ordering of atoms within unit cells, see the discussions by Penrose and Fife 3 and Gurtin 4. In this contribution it is shown that the elementary equations of the PFM can be directly derived from the TEP. For doing this it is necessary to express the total Gibbs energy consisting of the chemical Gibbs energy of a mixture of individual phases (two different crystalline phases and one phase characterizing the amorphous structure in the interface) in the system and of a penalty term due to square of gradient of the order parameter. The order parameter also represents an indicator determining, how the thermodynamic quantities or kinetic parameters of individual components, e.g. the chemical potentials or diffusion coefficients, must be calculated (i.e. to which phase the thermodynamic quantities are addressed). If no diffusion is assumed in the system, the dissipation is expressed exclusively as quadratic form of the rate of the order parameter. As there exists a direct link between linear non-equilibrium thermodynamics and the TEP, then one can easily interpret the PFM parameters by means of standard thermodynamic parameters and express their values by standard thermodynamic quantities, if the PFM equations are derived from TEP. The TEP is used in the case at hand for the derivation of PFM equations for interface migration. A detailed PFM study is performed for the relaxation of the standing non-equilibrium interface to its equilibrium state and of the interface migration driven by a constant thermodynamic force. A PFM model of the thick interface allowing for the solute segregation and drag is also derived from TEP. The results of simulations based on the TEP model indicate two effects. The first one is the threshold value of the thermodynamic force for the interface migration, which depends in a numerical concept on the density of the fixed grid, relative to which the interface moves. Grids of low density provoke local minima occur in the dependence of the total Gibbs energy on the interface position. The coarser the grid, the higher is the threshold value of the thermodynamic force. The second effect stems from the non-linear relation between the driving force and the interface velocity yielding a qualitative change in the character of the order parameter profile and of the specific Gibbs energy profile, when the driving force exceeds a certain value. For sufficiently dense grids the proportionality relation between the interface velocity and the driving force does not hold for high values of the driving force, and the specific interface Gibbs energy drops to zero and even to negative values. For grids with low density the interface moves for high values of the driving force, but the interface properties are far from those for dense grids. One can conclude that the migration of the interface can be treated in agreement with laws of linear non-equilibrium thermodynamics only on a sufficiently dense grid and only for sufficiently low driving force. In that case the interface thickness as well as the specific interface Gibbs energy and interface mobility (thermodynamic parameters) can be considered as constants, and their required values can be obtained by proper values of the PFM parameters. Finally, the relations amongst the thermodynamic parameters and the PFM parameters can be provided e.g. in polynomial form. If simulations are performed by means of PFM, one should always check, whether the conditions consistent with assumptions of linear non-equilibrium thermodynamics are fulfilled for any driving force acting in the system.
thermodynamic extremal principle: phase field method modeling interface migration solute segregation and drag
J.Svoboda F.D.Fischer
Institute of Physics of Materials,Academy of Sciences of the Czech Republic,Zizkova 22,CZ-616 62,Brn Institute of Mechanics,Montanuniversitat Leoben,Franz-Josef-StraBe 18,A-6700 Leoben,Austria
国际会议
The Third International Conference on Heterogeneous Material Mechanics(第三届国际非均匀材料力学会议)
上海
英文
318-319
2011-05-22（万方平台首次上网日期，不代表论文的发表时间）