Analytics

Abstract

The chapter reviews the basic analytic solutions of phase-field models at the mesoscopic scale. First the phase-field equation is derived by the Clausius-Duhem relation applied to a simple phase-field functional. The concept of variational derivative, related to the gradient contribution in the phase-field functional, is introduced. The traveling wave solution, or solitonian wave solution, for the phase-field equation in 1-dimension is derived for two relevant potential functions. Finally, the relations between model parameters and physically defined material parameters are derived.

1 The Problem of Propagating a Wave Front on a Numerical Grid

In this chapter, we will recapture the analytical background of traveling-wave solutions in 1D with two of the most commonly used separating potentials: the so-called “double-well” and “double-obstacle” potentials. This solution, also known as the antisymmetric soliton solution (see previous Chap. 1), is the basic reason that phase fields work so well in a numerical solution: the solution is quite robust against perturbations, which are inevitable in numerical simulations. Of course this resembles the physical fact, that the wave front is robust against any perturbation—wind, a bird etc.—in reality. The front self-stabilizes while traveling over the numerical grid, just like a solitonian wave on the surface of water.

Let us start with an example. We define a wave front ϕ(x, t₀) at time t₀ in one dimension x:

$$\displaystyle \begin{aligned} {} \phi(x,t_0) = \frac 12 \left ( 1 - \tanh \left(\frac {3x}\eta \right)\right). \end{aligned} $$

(2.1)

We know the wave front from Fig. 1.4b: the solitonian wave, but now defined between 0 and 1, as we do nowadays in phase field. In the following we take this convention that the left-hand side of the wave is ϕ = 1 and its right-hand side ϕ = 0. The front is centered around x = 0, and the width η is a measure of the distance between ϕ = 0.05 and ϕ = 0.95. This “cutoff” is necessary since the hyperbolic tangent converges to 0 or 1 only at infinity, and we need a finite measure of the interface width.

We now want to propagate this wave with constant speed v₀. One writes:

$$\displaystyle \begin{aligned} {} \dot \phi(x,t) = \frac{\partial \phi}{\partial x} \frac{\partial x}{\partial t} = \frac{\partial \phi}{\partial x} v_0. \end{aligned} $$

(2.2)

You may try to solve this equation numerically in 1D (as we do in the class), applying the profile (2.1) as a starting condition. However, this will simply not work for propagating the wave for a reasonable distance while maintaining its profile! Equation (2.2) is called a hyperbolic transport equation, and it is prone to large numerical instability up to shock-waves, and we do not want this at all. The equation thus has to be regularized using an appropriate method. We will not go into the details of numerical regularization approaches in hydro-dynamical wave mechanics, but we will modify the equation for better solvability. We want to solve (2.2) exactly. The only things we can do to the equation are: (i) add 0, or (ii) multiply by 1. We will do the first. We add:

$$\displaystyle \begin{aligned} {} 0 = \frac{\partial^2 \phi}{\partial x^2} - \frac{72}{\eta^2}\phi^2(1-\phi)^2 \left (\frac 12-\phi \right ). \end{aligned} $$

(2.3)

Does this look strange? Let us do the derivation. We will prove that (2.3) has the solution (2.1) and that it is self-similar in time for a propagating planar wave, a wave in 1D. Therefore, it adds 0 to the original Eq. (2.2). We add it:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \dot \phi(x,t) & =&\displaystyle 0 + \frac{\partial \phi}{\partial x} v_0, \\ & =&\displaystyle \frac{\partial^2 \phi}{\partial x^2} - \frac{72}{\eta^2}\phi^2(1-\phi)^2 \left(\frac 12-\phi \right) + \frac{\partial \phi}{\partial x} v_0. \end{array} \end{aligned} $$

(2.4)

This is the famous “phase-field equation” we will be dealing with in the rest of the lectures.¹ Adding this strange 0 Eqs. (2.3)–(2.1) changes the type of the equation from hyperbolic to parabolic. It becomes a “diffusion equation,” which is easy to solve in a discretized manner on a numerical grid. Let us go through this step by step.

2 Equation of Motion for the Phase Field

We start with the phase-field functional density, Eq. (1.3). A functional, by its mathematical definition, is a mapping of an arbitrary set of functions (functional densities), within a given definition domain, onto the real numbers. These functional densities, functions in space and time, are very difficult to compare. If we map functions onto a scalar, the comparison becomes easy. The functions can then be ordered on a linear coordinate indicating whether they are “larger or smaller.” For a human, sadly speaking, “money” is an often-used measure; this measures the value of all goods that can be bought. In metallurgy, we like to use the free energy of the system because it is a thermodynamic functional of the state variables temperature and pressure, two quantities that are easily controllable in lab experiments. Other functionals are useful under different conditions. In thermodynamics, we use Legendre transformation to relate these functionals uniquely. Nevertheless, different functionals should be used for different applications.

To introduce “time,” we start from an entropy functional S. The second law of thermodynamics demands that entropy increases with time:

$$\displaystyle \begin{aligned} {} 0 < \frac{d S}{dt} = \frac{\partial \phi}{\partial t}\frac{\delta S}{\delta \phi}. \end{aligned} $$

(2.5)

Here, the symbol δ denotes the “variational derivative” of a functional, i.e., of a function of functions. This will be elaborated below in some detail. For a rigorous derivation, see Kiselev, Shnir, and Tregubovich’s book [2]. For now, let us simply take the message that Eq. (2.5) can easily be fulfilled if

$$\displaystyle \begin{aligned} {} \frac{\partial \phi}{\partial t}\propto\frac{\delta S}{\delta \phi} \quad \mathrm{or} \quad \frac{\partial \phi}{\partial t}\propto-\frac{\delta F}{\delta \phi}. \end{aligned} $$

(2.6)

This relation is termed the “Clausius–Duhem relation” in continuum mechanics. It is not a “physical law,” but it is the simplest possible Ansatz to ensure the positivity of entropy production; it works well and is generally accepted. It is termed the “relaxation Ansatz” because it has only a first derivative in time and no accelerations. Since the entropy S enters the Gibbs free energy F with a “−” sign, we accept that:

$$\displaystyle \begin{aligned} \frac{\delta S}{\delta \phi} = - \frac{\delta F}{\delta \phi}. \end{aligned}$$

F =∫_Ωd³xf is the total free energy within the 3D domain Ω with the free-energy density f. We have to work on the functional as an integral over the domain of interest since the free energy density f of a phase-field model is a function of gradients in ϕ, ∇ϕ, which are, by construction, not defined on a special point in space but need at least two different points. Let us treat the three parts of the free energy in (1.3) separately:

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_1 & {\,=\,}&\displaystyle \frac\epsilon2 (\nabla\phi)^2, {} \end{array} \end{aligned} $$

(2.7)

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_2 & {\,=\,}&\displaystyle \frac\gamma4\phi^2(1-\phi)^2, {} \end{array} \end{aligned} $$

(2.8)

$$\displaystyle \begin{aligned} \begin{array}{rcl} f_3 & {\,=\,}&\displaystyle m(\phi, T, c, \epsilon, \ldots) {\,=\,} h(\phi) \Delta g(T, c, \epsilon, \ldots){\,=\,}\frac {3\phi^2{-}2\phi^2}6\Delta g(T, c, \epsilon, \ldots). {} \quad \;\;\;\;\;\; \end{array} \end{aligned} $$

(2.9)

We start with the easiest one: f₂. We are not afraid of the phase-field function ϕ(x, t); we treat it as a normal variable at the point in space x and time t under consideration, and it should not depend on the values of the field at other locations regarding the function f₂. This will come later. Therefore, we can treat the variational derivative as a normal partial derivative. We calculate:²

$$\displaystyle \begin{aligned} {} \frac{\delta F_2}{\delta \phi} = \frac{\partial f_2}{\partial \phi}= \gamma\phi(1-\phi)\left (\frac 12-\phi \right ). \end{aligned} $$

(2.10)

As mentioned before, this part is only active in the interface. It changes its sign at $\phi =\frac 12$, i.e., in the center of the interface. It is the first derivative of the so-called double-well potential, also called the Landau potential, ϕ²(1 − ϕ)². Therefore, we call it the potential contribution.

In the third term, f₃ (2.9), we have already separated the phase-field-dependent part $h(\phi )=\frac {3\phi ^2-2\phi ^2}6$, the so-called coupling function, from the physical part, i.e., the Gibbs-energy-density difference between the phases as a function of temperature, composition, strain, and other parameters, Δg(T, c, 𝜖, …). Here, we will take this as a constant Δg =  Δg₀. Because this part also does not depend on non-local contributions, i.e., gradients of the phase field, the functional derivative can again be treated as a normal partial derivative:

$$\displaystyle \begin{aligned} {} \frac{\delta F_3}{\delta \phi} = \frac{\partial f_3}{\partial \phi}= \phi(1-\phi)\Delta g_0. \end{aligned} $$

(2.11)

As we can see, the special form of (2.9) has been chosen such that the free-energy difference acts only within the interface when 0 < ϕ < 1. Δg₀ is called the “driving force,” since it makes a positive or negative contribution to the evolution of the phase field $\dot \phi$, driving it to grow or to shrink.

The first part (2.7) is more involved because it contains a gradient in ϕ, (∇ϕ)², in 1D, $(\frac {\partial \phi }{\partial x})^2$. We will perform the variational differentiation in 3D so as not to increase the confusion of the problem with the dimensionality of the variational derivative. The 1D case works identically. Applying the chain rule, we find:

$$\displaystyle \begin{aligned} {} \frac{\delta F_1}{\delta \phi} = \frac{1}{\delta \phi} \delta \left( \int_\Omega d^3x \frac\epsilon2 (\nabla\phi)^2 \right)= \frac{1}{\delta \phi} \int_\Omega d^3x \left( \epsilon \nabla \phi \delta (\nabla\phi)\right). \end{aligned} $$

(2.12)

The difficulty lies in finding a way to treat the variation δ(∇ϕ) of the gradient operator. It helps to realize that all derivatives—regardless of whether they are functional derivatives δϕ, partial derivatives ∂ϕ, or partial derivatives in space ∇ϕ—are linear operators and are therefore commutable. What we will do is commute the operators δ and ∇ by a partial integration in space:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \int_\Omega d^3x \left( \epsilon \nabla \phi \delta (\nabla\phi)\right) & =&\displaystyle - \int_\Omega d^3x \left( \epsilon \nabla^2 \phi \right)\delta \phi + \mbox{boundary integral}, \end{array} \end{aligned} $$

(2.13)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\delta F_1}{\delta \phi} & =&\displaystyle - \epsilon \nabla^2\phi. {} \end{array} \end{aligned} $$

(2.14)

The boundary integral is omitted mostly with the reasoning that the interface should not touch the boundary of the domain. This will not be the case in most applications, so boundary conditions should always be treated with care. This topic, however, will only be touched on in some numerical exercises. The most important factor of the derivation is the change of sign in the gradient contribution (2.12) due to the partial integration. In the free-energy functional, both interface contributions (2.7) and (2.8) are positive penalties against forming interfaces; they will counteract with opposite signs in the equation of motion: The Laplace- or diffusion-operator will act as to smear out the profile, while the potential operator will act as to sharpen it, as we will see in detail. Collecting all terms defines the equation of motion of the phase field, the so-called phase-field Eq. (2.4), and it now has physical constants in Kobayashi’s notation. It is derived from the Clausius–Duhem relation (2.6) with the proportionality constant τ, which is called the relaxation constant.

$$\displaystyle \begin{aligned} {} \tau\frac{\partial \phi}{\partial t} = \epsilon \nabla^2\phi - \gamma\phi(1-\phi)\left (\frac 12-\phi \right) - \phi(1-\phi)\Delta g_0. \end{aligned} $$

(2.15)

For completeness, a common formal replacement of the functional derivative for theories with gradient contributions (quantum mechanics in general) has the following form, known as the “Euler–Lagrange equation,” in which ∇ϕ is treated as a symbolic entity in the denominator of the first differential operator:

$$\displaystyle \begin{aligned} \frac\delta{\delta\phi} F = \left \{-\nabla\frac\partial{\partial\nabla\phi}+\frac\partial{\partial\phi} \right \} f. \end{aligned}$$

3 Traveling-Wave Solution for the Double-Well Potential

A “traveling-wave solution,” as introduced in Chap. 1 (Fig. 1.4), has the special feature of self-similarity in time (for constant driving force Δg₀ and velocity v₀); i.e., we can define a coordinate ξ = x − v₀t to describe this solution in its own frame moving with constant velocity with regard to a resting frame. We derive this solution as the minimum solution of the functionals (1.3) or (1.4). Of course, the condition of self similarity $\frac {d\xi }{dt}=0$ and the minimum energy condition are related, because a minimum solution requires the vanishing of the first derivative of the phase-field $\frac {\delta \phi }{\delta t}\propto \frac {\delta F}{\delta \phi }=0$ in Eq. (2.15). Here, we will not try to derive the solution by applying some scheme to solve partial differential equations, but we will prove that the given solution from the literature does the job!³ The solution has already been introduced in (2.1) for the initial time step t = 0. We change x → ξ = x − v₀t to find the traveling solution:

$$\displaystyle \begin{aligned} {} \phi(x,t) = \frac 12 \left ( 1 - \tanh \left (\frac{3(x-vt)}\eta \right)\right). \end{aligned} $$

(2.16)

To prove that this is a solution of (2.15), we have to compute the first and second derivatives in the space coordinate x:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac {\partial \phi}{\partial x}& =&\displaystyle - \frac 6\eta \phi(1-\phi), \end{array} \end{aligned} $$

(2.17)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac {(\partial \phi)^2}{\partial x^2}& =&\displaystyle \frac{72}{\eta^2} \phi(1-\phi)(\frac 12-\phi). {} \end{array} \end{aligned} $$

(2.18)

We find:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \epsilon \nabla^2\phi - \gamma\phi(1-\phi)(\frac 12-\phi) & =&\displaystyle \left(\frac{72}{\eta^2}\epsilon-\gamma\right)\phi(1-\phi)\left (\frac 12-\phi \right) \\ {} & =&\displaystyle 0 \quad \mbox{for} \quad \frac{72}{\eta^2}\epsilon=\gamma. \end{array} \end{aligned} $$

(2.19)

This is the first important result, and all phase-field models agree: the minimum solution of ϕ for a phase-field functional (1.3) with Δg = 0, or with a self-similar solution and arbitrary but constant velocity v, demands an interface width

$$\displaystyle \begin{aligned} {} \eta = \sqrt{\frac{72\epsilon}\gamma}. \end{aligned} $$

(2.20)

Inserting this relation back into the phase-field Eq. (2.15), only the driving force part remains, and we have:

$$\displaystyle \begin{aligned} {} \tau\frac{\partial \phi}{\partial t} = - \phi(1-\phi) \Delta g_0=\frac\eta6\frac{\partial \phi}{\partial x}\Delta g_0. \end{aligned} $$

(2.21)

We compare this with the Gibbs–Thomson equation for a moving planar interface, i.e., where the capillarity term is 0, and the interface mobility is M^ϕ:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} v_0 & =&\displaystyle M^{\phi} \Delta g_0, \end{array} \end{aligned} $$

(2.22)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{\partial \phi}{\partial t} & =&\displaystyle M^{\phi} \frac{\partial \phi}{\partial x} \Delta g_0. \end{array} \end{aligned} $$

(2.23)

This relates the relaxation constant τ to the interface mobility M^ϕ:

$$\displaystyle \begin{aligned} {} \tau = \frac\eta{6 M^{\phi}}. \end{aligned} $$

(2.24)

Finally, we have to relate 𝜖 and γ to the interface energy σ. The interface energy (in units $\left [ \frac {J}{m^2}\right ]$) must come out if we integrate the free energy density (in units $\left [ \frac {J}{m^3}\right ]$) in the normal direction through the interface:

$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma & =&\displaystyle \int_{-\infty}^{+\infty} dx \left[ \frac\epsilon2 (\nabla\phi)^2 + \frac\gamma4\phi^2(1-\phi)^2 \right] \\ & =&\displaystyle \int_{-\infty}^{+\infty} dx \left(\frac{18}{\eta^2}\epsilon+\frac\gamma4\right)\phi^2(1-\phi)^2 \\ & =&\displaystyle \int_{-\infty}^{+\infty} dx \qquad \frac 12\gamma \qquad \phi^2(1-\phi)^2 \\ {} & =&\displaystyle - \int_{0}^{1} d\phi \frac{\partial x}{\partial \phi} \qquad \frac 12\gamma \quad \phi^2(1-\phi)^2 \end{array} \end{aligned} $$

(2.25)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & =&\displaystyle + \int_{0}^{1} d\phi \qquad \frac 1{12}\eta\gamma \quad \phi(1-\phi) \end{array} \end{aligned} $$

(2.26)

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} & =&\displaystyle \qquad \qquad \quad \frac 1{72}\eta\gamma \quad = \quad \frac\epsilon\eta. \end{array} \end{aligned} $$

(2.27)

We have used the first derivative of ϕ with respect to x, $\frac {\partial \phi }{\partial x}$, (2.17) twice, along with the relation between η, γ, and η (2.20), and we have substituted the integration over x by the integration over ϕ in (2.25). To summarize this derivation, we insert the relation between the model parameters in Kobayashi’s notation 𝜖, γ, and τ and the physical parameters M^ϕ, σ, and η into (2.15) to arrive at the final phase-field equation with a double-well potential:

$$\displaystyle \begin{aligned} {} \frac{\partial \phi}{\partial t} = M^{\phi} \left\{ \sigma \left[\nabla^2\phi - \frac{72}{\eta^2}\phi(1-\phi)\left(\frac 12-\phi \right )\right ]- \frac 6\eta\phi(1-\phi)\Delta g_0 \right \}. \end{aligned} $$

(2.28)

Going back to our original problem, to solve the propagation of a wave front with constant speed v₀ (2.2), we now can prove (2.3) just by inserting the second derivative (2.18). In fact, both terms cancel if the front is in the right contour. The first derivative (2.17) then motivates the ansatz for the driving force $m(\phi )\propto \frac \partial {\partial x}\phi \propto \phi (1-\phi )$ (2.9). Now we see that the hyperbolic transport Eq. (2.2) is turned into a parabolic equation with the second derivative in the space coordinate x. This type of equation—also termed a diffusion equation—is well behaved and easily solved on a computer. You can try this out using the program with which you tried to solve the hyperbolic Eq. (2.2).

4 Interpretation of the Phase-Field Equation

Let us now have a closer look at the phase-field Eq. (2.28) to determine the effect of each term on the phase-field profile. The first term is the Laplacian, or diffusion operator. This smoothens any profile, ensuring a smooth phase-field profile between 0 and 1. For the right-hand hyperbolic tangent profile, it contributes negative increments above $\phi > \frac {1}{2}$ and positive increments below $\phi < \frac {1}{2}$, as indicated in Fig. 2.1a. The second term stems from the double-well potential. Its first derivative changes its sign at $\phi = \frac {1}{2}$ because the potential function is symmetric around $\frac {1}{2}$. Now we see that the potential contribution has negative increments where the Laplacian has positive increments, and vice versa [Fig. 2.1b].

Fig. 2.1

The competition between spreading and contraction of the phase-field contour. Description see text

This competition between the two contributions guarantees the stable phase-field profile! If a numerical solution deviates from the correct profile, the contributions will not balance but push the contour back to the correct contour. Later, in Chap. 3, we will see that this balance is also violated for a curved interface in more than one dimension. This will be used to consider capillarity. Furthermore, Δg will not be a constant in real applications: it will be positive or negative, and it will vary from point to point in space and time. It will depend on transport of temperature, solute, and momentum, as will be explained in later chapters.

5 Phase-Field Equation and Traveling-Wave Solution for the Double-Obstacle Potential

As a general comment, we shall keep in mind that the potential function (2.8), which is called the double-well potential, is not a unique choice for any system. It was first introduced by Landau to explain the behavior of a ferromagnet at the critical point, which is called the Curie point in this system. Only at this critical point, which is a second-order phase transformation, is it justified to truncate the potential describing the interface energies—the so-called Landau potential—to this simple form (for details, see [6]). In phase field, however, we are dealing with a first-order phase transformation such as solidification or solid-state phase transformations, which are far away from a critical point. We also stick to the notion of a mesoscopic phase-field method (Sect. 1.4) whereby phenomena inside the interface cannot, and need not, be interpreted physically. Therefore, we have some freedom to choose a potential function in a different form than (2.8).

For reasons that will be explained in Chap. 6, the multi-phase-field method, as implemented in OpenPhase [3], applies the so-called double-obstacle potential:

$$\displaystyle \begin{aligned} {} f^{\mathrm{DO}} = \frac{\gamma^{\mathrm{DO}}}2\left|\phi(1-\phi)\right|. \end{aligned} $$

(2.29)

The “obstacle” is realized by the absolute signs, which flip the negative branches of the function ϕ(1 − ϕ) to positive values.⁴ Figure 2.2 shows a comparison between the double-well and double-obstacle potentials.

Fig. 2.2

The “double-well” and “double-obstacle” potentials

As we can see, the graphs of the two potentials are similar. They are determined such that: (i) the free energy goes to infinity f →±∞ for ϕ →±∞; (ii) they have two minima at f = 0 for ϕ = 0 and ϕ = 1; and (iii) the region of the so-called potential barrier between 0 and 1 has an area that is $\frac 12$ the interface energy σ. Other potentials can be used that fulfill above criteria, but they are little discussed in the phase-field literature. As long as we are speaking about a two-phase system, they should not show another minimum. More complex potentials, so-called “multi-well potentials” are used for multi-phase systems (e.g., [1, 4]).

The minimum solution of the free energy with the double-obstacle potential (2.29) is:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \phi^{\mathrm{DO}}(x,t) =\begin{cases} 1 & \mbox{for} \quad x-v_0t < -\frac{\eta}2, \\ \frac 12- \frac 12 \sin \left(\frac{\pi(x-v_0t)}{\eta}\right ) & \mbox{for} \quad -\frac{\eta}2\le x-v_0t < +\frac{\eta}2, \\ 0 & \mbox{for} \quad x-v_0t\ge +\frac{\eta}2 . \end{cases} \end{array} \end{aligned} $$

(2.30)

As before, we derive in the transition region 0 < ϕ < 1 (do this as an exercise!):

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac {\partial \phi}{\partial x}& =&\displaystyle - \frac \pi\eta \sqrt{\phi(1-\phi)}, \end{array} \end{aligned} $$

(2.31)

$$\displaystyle \begin{aligned} \begin{array}{rcl} \frac {(\partial \phi)^2}{\partial x^2}& =&\displaystyle \frac{\pi^2}{\eta^2} (\frac 12-\phi). {} \end{array} \end{aligned} $$

(2.32)

The final task is to find a proper coupling function m^DO corresponding to the first derivative in ϕ, (2.31), of the phase-field profile (2.30) for the double-obstacle potential. We only give here the function and leave the proof as an exercise:

$$\displaystyle \begin{aligned} {} m^{\mathrm{DO}} = \frac 14\left[(2\phi-1)\sqrt{\phi(1-\phi)}+\frac 12 \arcsin{2\phi-1)}\right] \Delta g_0. \end{aligned} $$

(2.33)

Repeating the analysis for the double-well potential, we find the relations between the model parameters 𝜖^DO, γ^DO, and τ^DO and the physical parameters η, σ, and M^ϕ:

$$\displaystyle \begin{aligned} {} \eta = \pi\sqrt{\frac{\epsilon^{\mathrm{DO}}}{\gamma^{\mathrm{DO}}}}; \qquad \sigma = \frac{\pi^2}8\frac{\epsilon^{\mathrm{DO}}}{\eta}=\gamma^{\mathrm{DO}}\frac\eta8; \qquad M^{\phi}=\frac{\pi^2}8\frac\eta{\tau^{\mathrm{DO}}}. \end{aligned} $$

(2.34)

Collecting all these pieces, we define the free-energy density in physical notation for the double-obstacle potential:

$$\displaystyle \begin{aligned} {} f = \frac 8{\pi^2} \frac\sigma\eta \left [ \eta^2(\nabla\phi)^2+\pi^2 \left |\phi(1-\phi)\right | \right] + \Delta g. \end{aligned} $$

(2.35)

The normalization of the interface contribution $\frac 8{\pi ^2}$ is necessary to fulfill the condition that the integral of this term in the normal direction through the interface becomes the interface energy σ [cf. (2.26)]. We also chose the notation where σ is divided by a length, the interface width η, to emphasize that this contribution is an energy density like Δg. The term in the brackets is now a dimensionless contour function indicating the interface position. In the physical parametrization, the phase-field equation with the double-obstacle potential reads:

$$\displaystyle \begin{aligned} {} \frac{\partial \phi}{\partial t} = M^{\phi} \left\{ \sigma \left[\nabla^2\phi - \frac{\pi^2}{\eta^2}\left (\frac 12-\phi\right )\right ] - \frac\pi\eta\sqrt{\phi(1-\phi)}\Delta g_0 \right \}. \end{aligned} $$

(2.36)

6 Gibbs–Thomson Limit of the Phase-Field Equation

Summarizing all contributions, we can return to the underlying physical equation of interest: the Gibbs–Thomson equation (1.5). We have already done this in 3D, relating the velocity of the interface $\vec v$ to the rate change of the phase field with the normal vector to the interface $\vec n=\frac {\vec \nabla \phi }{|\vec \nabla \phi |}$:

$$\displaystyle \begin{aligned} \begin{array}{rcl} {} \dot\phi & =&\displaystyle \sum_{i=1}^3\frac{\partial\phi}{\partial x_i}\frac{\partial x_i}{\partial t} = \vec \nabla \phi \vec v,\\ \vec v & =&\displaystyle \frac{\dot\phi}{|\vec\nabla \phi|}\vec n. \end{array} \end{aligned} $$

(2.37)

The curvature of the interface κ, which is given by the strange expressions in Fig. 2.3, will be derived in Chap. 3.

Fig. 2.3

The relations between the phase-field equations in Kobayashi’s notation and the Gibbs–Thomson equation in physical notation for the double-well and double-obstacle potentials

7 Exercises

Exercise

Prove the first and second derivatives of the traveling-wave solutions for the double-well and double-obstacle potentials (2.17), (2.18) and (2.31), (2.32).

Exercise

Prove that the coupling function in (2.33) does the job, i.e., that the derivative of this function with respect to ϕ is proportional to the first derivative of ϕ in the case where a double-obstacle potential is used.

Exercise

Prove the relation (2.34) between the model parameters and the physical parameters for the double-obstacle potential.

Exercise

Check the relations between the model parameters as indicated in Fig. 2.3.

Further Reading

• Appendix A in the review “Phase-field models in materials science” [5], where another potential function, the “top hat” function, is introduced.