In this chapter the behavior of the phase field model in its asymptotic limit is discussed, as described by Bhate et al. . The analysis shows that in the limit of an infinitely thin interface the phase field model equations converge to the equations of the sharp interface model. The interface thickness, described in Section 3.6, is controlled by the parameter and, therefore, the asymptotic limit to an infinitely thin interface is accomplished by driving this parameter against zero .
For an easier handling the governing equations are rewritten in a dimensionless form :
where in both the gradient of the chemical potential and the therm due to the electric field are added and
where the connection between the dimensionless quantities and the quantities with units
are given by
There is a characteristic length, is a characteristic electric field strength, and is a characteristic strain energy. Furthermore, is the volume of an atom in the lattice, is the surface energy of the metal/void interface, and is the diffusion coefficient at the metal surface.
In the further discussion the tilde sign above the dimensionless values will be omitted for easier reading. If a distinction is needed, it will be explicitly pointed out. The parameters in (A.1) and (A.3) are given by
where the quantities and are dimensionless numbers characteristic for the formulated problem.
For the derivation a local coordinate system is chosen, as already used for the sharp interface model. This coordinate system is extended to the region of the diffuse interface and splits up the simulation domain into three regions. The regions are the metal region, the void region, and the interface region separating the two former (cf. Figure A.1).
The derivations for the asymptotic limit is carried out in two steps. First the formulas are transformed into the local coordinate system. This step is followed by the introduction of the Taylor expansion in of the functions and a splitting of the equations in terms of -orders.
For the local coordinate system, some definitions are required regarding the calculation of the normal vector
and the curvature for the contour.
The subscript is used for the common differential operators. Furthermore, the normal velocity of the interface is expressed by the time derivative of the contour (interface)
and the -coordinate of the local coordinate system is normalized as
As the asymptotic expansion is carried out in the local coordinate system all functions have to be expressed in this coordinate system as well:
For the time derivative the chain rule gives
where the indices after the comma stand for the derivative
By again employing the chain rule the operator can be expressed in the new coordinate system by
The last needed differential operator is the Laplace operator given by
in the new coordinate system, where the first term, containing only functions differentiated with respect to , is defined by
The boundary conditions for the and contour for the phase field function (cf. Figure A.1) are given by
as there the pure metal or the pure void starts and
as the phase field function has to be a smooth function everywhere and, therefore, also at the boundary contours between the metal and the interface and between the void and the interface. The flux at the interface has to be limited to the interface leading to a third boundary condition of zero flux from the interface into the metal or the void given by
The inner expansion of the order parameter and the chemical potential, where the EM therm is included, is the Taylor expansion with respect to the interface thickness controlling parameter
The constant multipliers from the Taylor expansion are absorbed into the functions and . First the differential operators (A.18)-(A.21) for functions in the local coordinate system in (A.1) are employed, resulting in
where the derivatives of the phase field function with respect to the tangential direction are zero as in this direction the phase field function is constant (zero). Furthermore, and are orthogonal to each other and due to the normalization of , the inner product of with itself is equal to one. Using (e.g. ), derived from (A.16), results in
Inserting the inner expansions (A.26) and (A.28) defined above and taking only terms to the first order in into account leads to the equation
Reordering the equation by collecting the terms with the same order of and leaving away the terms of the second order leads to the equation
and the different orders of can be handled separately giving the following set of equations:
The first over-brace in (A.33) shows that the term is constant in . With the definition of the diffusion coefficient
it can be concluded that
and the first over-brace in (A.34) is zero as is independent of and the second over-brace in (A.34) can be handled like the first over-brace in (A.33).
Applying the same procedure of the transformation into the local coordinate system and applying the Taylor expansion to the chemical potential results in the equation
and a separation of the equation into a set of equations ordered by the order in gives
where the double obstacle function defined in Section 3.6 was used for the bulk free energy defined by
Setting the term of the order equal zero leads to the differential equation
with the solutions
where, due to the boundary conditions (A.23) and (A.24), equals one and equals zero and the thickness of the interface in the coordinate is . Taking the terms of the zeroth order of of (A.39) and rearranging them leads to
where is not a function of . This equation has the same structure as the derivative of (A.43) with respect to . Therefore only the trivial solution can meet the boundary conditions given by (A.24) and
Integrating (A.45) in direction over the whole interface region results in
where the under-braces give the results of the integrals. With the assumption of zero elastic strain energy in the void this leads to
This is the same equation as (3.47) and shows that the chemical potential in the asymptotic limit converges to the sharp interface model. Coming back to (A.32), taking the terms of first order in , and again integrating over the interface in gives
where for the third term on the right hand side the zero flux condition was employed, resulting in the equation for the normal velocity of the sharp interface model
These evaluations show that the phase field model for going to zero converges to the sharp interface model and can therefore be used for the simulations of voids as long as is chosen carefully. The upper limit is in the order of the smallest curvature occurring at the surface of the voids. The lower limit is given by the meshing resolution. From one side of the boundary region to the other a minimum of five meshing elements is needed to guarantee the stability of the FEM simulation, as was found by empirical studies.« PreviousUpNext »Contents