## Abstract

The necessary heat treatment of single-crystal semi-insulating gallium arsenide (GaAs), which is deployed in micro- and optoelectronic devices, generates undesirable liquid precipitates in the solid phase. The appearance of precipitates is influenced by surface tension at the liquid–solid interface and deviatoric stresses in the solid.

The central quantity for the description of the various aspects of phase transitions is the chemical potential, which can be additively decomposed into a chemical and a mechanical part. In particular, the calculation of the mechanical part of the chemical potential is of crucial importance. We determine the chemical potential in the framework of the St. Venant–Kirchhoff law, which gives an appropriate stress–strain relation for many solids in the small-strain regime. We establish criteria that allow the correct replacement of the St. Venant–Kirchhoff law by the simpler Hooke law.

The main objectives of this study are the following: (i) we develop a thermo-mechanical model that describes diffusion and interface motions, which are both strongly influenced by surface tension effects and deviatoric stresses, (ii) we give an overview and outlook on problems that can be posed and solved within the framework of the model, and (iii) we calculate non-standard phase diagrams for GaAs above 1059 K, i.e. those that take into account surface tension and deviatoric stresses, and we compare the results with classical phase diagrams without these phenomena.

## 1. Introduction

Phase transitions in solids are usually strongly influenced by surface tension and non-isotropic stresses that give rise to non-zero stress deviators. An important example regards the nucleation and the growth of liquid droplets in semi-insulating gallium arsenide (GaAs; Mori *et al*. 1990; Oda *et al*. 1992, 1994; Steinegger 2001; Steinegger *et al*. 2001). These processes are accompanied by deviatoric stresses resulting from the liquid–solid misfit (Gokcen 1989; Brice 1990). In the classical treatment, the nucleation barrier is determined by the surface tension of the droplet (Thomson 1870; Gibbs 1878; Volmer 1939). However, due to deviatoric stresses in the neighbourhood of the droplet, and in particular at the liquid–solid interface, the nucleation barrier may be decreased.

A further mechanism that controls the evolution of liquid droplets in semi-insulating GaAs is diffusion in the vicinity of the droplet (Dreyer *et al*. 2004). The diffusion flux results from a competition of chemical and mechanical driving forces.

The quantity of central importance for the description of all these phenomena is the chemical potential. Its calculation in the presence of mechanical stresses is among the subjects of this study. We determine the chemical potential in the framework of the St. Venant–Kirchhoff law, which gives an appropriate stress–strain relation for many solids in the small-strain regime (Truesdell & Noll 1965). Subtle problems concerning the chemical potentials appear in the limiting case, where the St. Venant–Kirchhoff law is approximated by the classical Hooke law.

In a series of studies, we develop a thermodynamical model for the description of liquid–solid phase transitions that are accompanied by deviatoric stresses, diffusion and chemical reactions. The model is especially designed to describe phenomena that arise during heat treatments of GaAs wafers at elevated temperatures above the right eutectic line. However, a generalization of the model to other materials with dynamic precipitation phenomena may be undertaken. An example is the formation of cementite in steel.

This paper starts a series of studies with the formulation of the model and a description of various settings for the following tasks: (i) calculation of non-standard phase diagrams in the presence of surface tension and deviatoric stresses, (ii) diffusional processes in the presence of surface tension and deviatoric stresses (Dreyer *et al*. 2004), and (iii) determination of the size distributions of droplets by a generalized Becker/Döring model that takes mechanical stresses into account (Dreyer & Duderstadt 2006).

We have organized the paper as follows.

We describe in §2 the constitution of the phases of GaAs and, in particular, the semi-insulating solid phase. The central fact is the description of the sublattice structure with three sublattices. Experiments and preliminary theoretical considerations have motivated a special distribution of the constituents of GaAs on the sublattices that we will call the Freiberg model (M. Jurisch 2002, personal communication); see also Rudolph (2003).

Section 3 reminds the reader of some basic thermodynamics for fluids and solids with special emphasis on a correct description of strains and stresses within the nonlinear theory of elasticity. In this section, we define the chemical potentials for the constituents and we give rules for their calculation within a quite general framework.

In §4, we restrict ourselves to thermodynamic processes at constant and uniform temperature and at constant external pressure. For such processes, we derive and exploit the thermodynamic inequality that determines the dynamics of thermodynamic processes. The inequality serves here for various purposes. These consist of the following: (i) establishment of relations between driving forces and thermodynamic fluxes; among these there are driving forces that induce diffusion fluxes and interface motion and (ii) identification of mechanical, chemical, diffusional and phase equilibria and the determination of possible equilibrium states.

Section 5 addresses the special constitutive laws that we will use for the description of semi-insulating GaAs. We decompose the constitutive quantities into chemical and mechanical parts. The chemical parts rely on the well-established sublattice model that was formulated by Oates *et al.* (1995) and Wenzl *et al*. (1990, 1993). The mechanical parts rely on the St. Venant–Kirchhoff law that relates the Green strain tensor to the second Piola–Kirchhoff stress tensor. The mechanical parts of the chemical potentials are calculated here for the first time, and for this reason an extensive discussion and a comparison with those chemical potentials that appear in the literature is included in §7 in the electronic supplementary material.

In §8, in the electronic supplementary material, we solve the mechanical boundary-value problem for a misfitting liquid sphere in a solid surrounding.

Section 9 in the electronic supplementary material contains the first important application of the proposed model, *viz*. the calculation of phase diagrams in the presence of surface tension and stress deviators. A detailed comparison with standard phase diagrams is included.

In §6 and §10 in the electronic supplementary material we give a short summary and pose two further problems that can be treated by the model equations.

We conclude the electronic supplementary material with two appendices, which contain (i) technical proofs of some statements of the main text and (ii) certain material data regarding standard phase diagrams.

## 2. Constitution of the three phases of GaAs

### (a) Chemical constitution of semi-insulating solid GaAs

Semi-insulating GaAs is a single-crystal solid with the major substances gallium (Ga) and arsenic (As). The stoichiometric solid, i.e. equal amounts of Ga and As, has a zinc-blend structure with two face-centred cubic (FCC) sublattices, *α* and *β*, which are completely occupied by Ga atoms and As atoms, respectively. In order to fabricate semi-insulating GaAs, a small amount of excess As atoms and further trace elements are added (table 1). Among these may be oxygen (O), silicon (Si), boron (B) and carbon (C) in very small quantities. The constituents of semi-insulating GaAs are found either on *α* and *β* sublattice sites or on interstitial sites that form a third FCC sublattice, *γ*. The major substances of the sublattice *γ* are vacancies (Va). However, vacancies may also be found on the two other sublattices. A serious description of the physical and chemical properties of semi-insulating GaAs also needs to consider charged states of the introduced constituents and additionally free electrons and holes (Blakemore 1982; Wenzl *et al*. 1990, 1993; Hurle 1999, 2004; Gebauer 2000).

The so-called *Freiberg model* gives a complete list of the possible distribution of all constituents on the three sublattices for semi-insulating GaAs as it is fabricated at Freiberg Compound Materials (FCM, Flade *et al*. 1999). The notion Freiberg model refers, in particular, to the fact that Ga atoms exclusively live on the sublattice *α* (M. Jurisch 2002, personal communication); see also Rudolph (2003).

In fact, we have used this model to calibrate the various material parameters that appear in the model equations (Dreyer & Duderstadt in preparation).

However, for a simplified description of semi-insulating GaAs, a reduced chemical model is possible, which is described by table 2.

The reduced model embodies qualitatively the essential properties of the Freiberg model: (i) Ga atoms reside exclusively on sublattice *α*, (ii) the antisite As_{α} controls the semi-insulating behaviour of GaAs, (iii) vacancies are taken into account, and (iv) there is interstitial arsenic As_{γ} that drives the diffusion processes.

In summary, the solid GaAs of this study consists of seven chemically reacting substances. Their distribution on the sublattices is described by the mole densities , , , , , and .

### (b) Chemical constitution of the liquid and gaseous phases of GaAs

The solid phase of GaAs can coexist with a liquid phase and a gas phase. The liquid phase consists of Ga and As, with mole densities and in the liquid. The gaseous phase has four constituents, namely the molecules Ga, As, As_{2} and As_{4}. However, in the interesting range between 0.1 kPa and 2 MPa, the appearance of Ga and monatomic As can be ignored, so that we may deal with a gas phase that consists exclusively of As_{2} and As_{4} with mole densities and , see also Arthur (1967) and Stringfellow (2004).

## 3. Some thermodynamics of mixtures

This section reminds the reader of some basic facts of thermodynamics of mixtures, see Müller (1985, 2001) for an extensive treatment. Furthermore, we present some simple generalizations regarding solid mixtures. The general thermodynamic relations are formulated so that they can be applied to solids as well as to liquids and gases. However, the gaseous phase will not be considered here explicitly. It is introduced in Dreyer & Duderstadt (in preparation), where we rely on experiments, involving the gaseous phase, in order to determine the required material data.

### (a) The basic variables of the solid and liquid phases of GaAs

We consider a body *Ω*=*Ω*_{S}∪*Ω*_{L}, which may consist of solid and liquid phases, denoted by *Ω*_{S} and *Ω*_{L}, respectively. At any time *t*≥0, the thermodynamic state of the body *Ω* is described by a certain number of variables, which may be the functions of space *x*=(*x*^{1},*x*^{2},*x*^{3})∈*Ω*. In general, they are thus given by fields.

The variables of the solid phase are the seven mole densities *n*_{a}(*t*, *x*), *a*∈*a*_{S}={Ga_{α}, As_{α}, Va_{α}, As_{β}, Va_{β}, As_{γ}, Va_{γ}} and the mechanical displacement field *u*(*t*, *x*)=(*u*^{1}(*t*, *x*), *u*^{2}(*t*, *x*), *u*^{3}(*t*, *x*)). In the liquid phase, the variables are the two mole densities *n*_{a}(*t*, *x*), a∈a_{L}={Ga_{L}, As_{L}}, which determine the thermodynamic state of the liquid.

There are thus 7+3+2 unknowns, whose determination, for given temperature *T* and given outer pressure *p*_{0}, is the main objective of this study.

### (b) Detailed description of the constitution of the solid and liquid phases

In this section, we introduce further quantities that describe various aspects of the constitution of solid and liquid phases.

#### (i) Mass densities

The mole densities can be used to define the mass densities of the phases(3.1)The quantities *M*_{a} are the constant molecular weights of the constituents, *viz*. and .

#### (ii) Conservation law of mass

The mass density *ρ* satisfies the local conservation law of mass, which reads, in each of the phases,(3.2)The newly introduced quantity *υ*^{i} denotes the barycentric velocity of the mixture.

#### (iii) Vacancies

We assume that the vacancies are carriers of energy and entropy, but they have no mass, i.e. we set

#### (iv) Mole fractions and lattice occupancies in the solid phase

The solid phase is a crystal with three FCC sublattices *α*, *β* and *γ*, which have the same number of lattice sites. We now introduce their common mole density, *n*_{G}, and lattice occupancies according to(3.3)The original seven variables *n*_{a}, with *a*∈*a*_{S}, may be substituted by the seven variables *n*_{G} and *Y*_{b} with *b*∈*b*_{S}=*a*_{S}\{Ga_{α}}. There holds *n*_{b}=*Y*_{b}*n*_{G} for *b*∈*b*_{S} and . We indicate this transformation by with .

The distribution of the constituents on the lattice sites according to table 2 implies that the total amounts of lattice occupancies for the three groups of constituents are given by(3.4)with the side conditions(3.5)The density of the solid phase can now be written as(3.6)Note that the ratio of *ρ*_{S} and *n*_{G} does only depend on *Y*_{Ga} and *Y*_{As}. We express this fact by the definition of a mean molecular weight of the solid(3.7)It is often useful to represent the mole densities of the lattice sites and of the constituents, respectively, by the functions(3.8)Furthermore, we need mole fractions of the material constituents of the solid phase. These are defined by , with and .

#### (v) Mole fractions in the liquid phase

In the liquid phase, we change the variables from and to the total mole density of the liquid, *n*_{L}, and to the mole fraction *X*_{L} of the arsenic. We thus introduce(3.9)It follows that and , which can also be written as(3.10)The liquid mass density of (3.1) can now be written as(3.11)Note that the ratio *ρ*_{L}/*n*_{L} does only depend on *X*_{L} but not on *n*_{L}. We define the mean molecular weight of the liquid by(3.12)The following representations will become useful in the next subsections:(3.13)

### (c) Detailed description of motion and strain in the solid phase

In this section, we relate the displacement field to the motion and strain of a solid phase. We start with the introduction of a reference state in order to measure the motion of a material point of the solid phase.

Let *X*=(*X*^{1}, *X*^{2}, *X*^{3}) be the location of a material point in a reference state, whose location at time *t* is given by *x*=(*x*^{1}, *x*^{2}, *x*^{3}). The location *x* is determined by the function *x*=*Χ*(*t*, *X*)=(*Χ*^{1}(*t*, *X*), *Χ*^{2}(*t*, *X*), *Χ*^{3}(*t*, *X*)). We call *Χ*^{i}(*t*, *X*) the motion of the material points of the solid phase. The motion can be used to calculate the barycentric velocity of the mixture, . The displacement of a material point is defined by *U*^{i}(*t*, *X*)=*Χ*^{i}(*t*, *X*)−*X*^{i}. The gradients of *Χ*^{i}(*t*, *X*) and *U*^{i}(*t*, *X*) are called the deformation gradient and displacement gradient, respectively, and(3.14)The Jacobian of *F*^{ij} is denoted by *J*. If we assume that *J*>0, we may invert the motion *x*^{i}=*Χ*^{i}(*t*, *X*) at any time *t* with respect to the coordinates *X*^{i}. We write(3.15)We are now able to identify the displacement field *u*^{i}, which was introduced as a variable in the last section, by(3.16)This is a typical example for the representation of mechanical quantities with respect to actual coordinates. We call this representation the Euler or the spatial description, whereas the representation with respect to the reference coordinates is called the Lagrange or material description.

The barycentric velocity can likewise be given with respect to the coordinates *x*^{i}. We define , and this quantity has already appeared in the local conservation law for the mass density (3.2).

A similar definition for the mass density, *viz*. , is useful to integrate (3.2) to obtain(3.17)where *ρ*_{R} is the mass density for *F*^{ij}=*δ*^{ij}.

In this study, we prefer the Euler representation, and to this end we introduce the spatial displacement gradient and the inverse deformation gradient according to(3.18)We then have the relations(3.19)Further important objects for the description of the stretch are the right and the left Cauchy–Green tensor, *C*^{ij} and *B*^{ij}, and for the description of the strain we define the Green strain tensor, *G*^{ij},(3.20)These quantities may also easily be given with respect to the spatial representation.

Finally, we decompose the stretch of a body into a part that gives pure volume changes of the body and complementary part that describes pure changes of its shape. The pure changes of the volume are obviously given by the Jacobian *J*, whereas the unimodular tensor,(3.21)represents changes of the shape of a body. For details see Krawietz (1986).

### (d) Notations regarding Helmholtz energy, chemical potentials and stresses

In order to describe the various mentioned thermodynamic processes in GaAs, we need to introduce further quantities. These are the specific internal energy *u*, the specific entropy *s*, the specific Helmholtz energy *ψ*=*u*−*Ts* and the chemical potential of the constituents *μ*_{a} with *a*∈*a*_{L}∪*a*_{s}. Furthermore, we need to introduce two measures of stress, the Cauchy stress *σ*^{ij} and the second Piola–Kirchhoff stress *t*^{ik}. It holds that(3.22)The isotropic part of the Cauchy stress is related to the pressure *p*, which is defined by(3.23)Here, the angle brackets indicate the stress deviator, which represents the stress due to pure changes of the shape of a solid, whereas the pressure is related to pure changes of the volume of a body.

### (e) Constitutive model, part 1: general constitutive equations

The constitutive model establishes equations that allow us to calculate quantities that are not among the list of variables that were introduced in §3*a*.

The general constitutive model that we use for the description of the solid and the liquid phases of GaAs starts from a Helmholtz energy density *ρψ*, which we assume here to be given by the constitutive function(3.24)for *a*∈*a*_{S} and *a*∈*a*_{L} in the solid and liquid phase, respectively. In liquids, there is no dependence of the Helmholtz energy density on ** c** because there is no change of shape during deformations of liquids.

Entropy, stresses and chemical potentials are related to the Helmholtz energy via the Gibbs equation, which reads(3.25)Furthermore, we have the Gibbs–Duhem equation(3.26)

The proof of the Gibbs equation and of the Gibbs–Duhem equation for fluid mixtures and for solids consisting of a pure substance is given in Müller (1985). The extension to the corresponding equation (3.25) can be carried out along the same strategies that are explained in detail by Müller (1985).

The quantity *g*=*ψ*+*p*/*ρ* is called the specific Gibbs energy and *Mg* is identical to the chemical potential of a pure substance with the molecular weight *M*. The Gibbs equation implies(3.27)which can be read directly off from (3.25).

It is important to note that there is no unique definition of the chemical potentials in the literature. There are authors who define the chemical potentials at a given wall with a certain direction in the material. For example, Müller (1985) defines the chemical potential of constituent *a* as *the* quantity that continues at *a* semipermeable wall for constituent *a*. A further example is given by Rusanov (1996), who likewise introduces chemical potentials in the context of given walls. However, Müller exclusively considers fluids, so that he ends up with a scalar quantity, while Rusanov also takes solids into account and thus he has to define the chemical potentials as second-order tensors. We prefer to define the chemical potentials as scalars using the second equation of (3.27) in a local point of the bulk. Relying on that definition, we may then calculate the jump conditions at a given wall involving the chemical potentials. This discussion will be continued in §4*f*.

In order to relate the stress and the pressure to the Helmholtz energy density, we first change the variables.

In the solid, we substitute by . By means of the second equation of (3.8), we obtain(3.28)In the liquid, we change from to *T*, *X*_{L}, *ρ*_{L}. By means of the second equation of (3.13), we obtain(3.29)Inserting the second equation of (3.27) into (3.26) both for solids and liquids leads to(3.30)We prove this statement in two steps.

At first we insert (3.27)_{2} in (3.26) and carry out the indicated differentiations to obtain(3.31)According to (3.28) and (3.29), we calculate(3.32)and(3.33)By means of the identities(3.34)we finally obtain, from (3.31), the proposition (3.30).

In the liquid phase, the pressure is the only contribution to the stress, so that we may write here(3.35)

In the solid phase, there is an extra contribution to the stress that we calculate now. We read off from (3.28), (3.17), (3.20) and (3.21) the representation(3.36)and we rewrite the Gibbs equation (3.25) as a total differential for , which implies, by means of (3.26), the familiar result(3.37)Thus, the knowledge of the Helmholtz energy density *ρψ*(*T*, *n*_{a}, ** c**) is sufficient to calculate all quantities that were introduced in the last section.

In order to illustrate the similarity between solid and liquid, we introduce, in analogy to (3.36), for the liquid phase the representation(3.38)instead of (3.29). Herein *J*_{L} is defined by .

In §5, we will introduce an explicit constitutive model for GaAs. This model will turn out to appropriately describe the various phenomena in GaAs, which were listed in the introduction.

## 4. The approach of a thermodynamic system to equilibrium

### (a) The global laws of energy and entropy

Our objective is to study phase transitions for various systems with a common feature, so that their temperature is homogeneous and constant and that the total system is subjected to a constant outer pressure, for example, see figure 1. For the derivation of the relevant thermodynamic inequality, it is sufficient to consider the generic system, which is shown in figure 2. The adjustment of the results to the special systems is trivial and left to the reader.

The body *Ω* of the generic system consists of a solid phase *Ω*_{S} and a liquid phase *Ω*_{L}, which are separated by the interface I. The volume of *Ω* can be decomposed as *V*(*t*)=*V*_{S}(*t*)+*V*_{L}(*t*). The outer surfaces ∂*Ω*_{S}\I and ∂*Ω*_{L}\I are considered as material surfaces, i.e. they move with the barycentric velocity of the mixture, *υ*, so that there are no mass fluxes through ∂*Ω*_{S}\I and ∂*Ω*_{L}\I.

The interface I is parametrized by any two Gauss parameters (*ξ*^{1}, *ξ*^{2}), and a point *x*∈I is determined by the function , see for example Truesdell & Toupin (1960) or Aris (1962). We choose the parametrization so that its speed is given by *w*^{i}=*w*_{ν}*ν*^{i}. The unit normal of the interface, *ν*^{i}, points into the solid phase. The tangent vectors, i.e. the *ξ* derivatives of , are denoted by , *α*∈{1,2}. The mean curvature of the interface is denoted by *k*_{M}.

We study exclusively processes at constant outer pressure *p*_{0} and constant outer temperature *T*_{0}. Furthermore, we assume that the temperature *T* within *Ω* is constant with *T*=*T*_{0}.

We now apply the global balance laws of total energy *E* and entropy *S* to this system,(4.1)The quantity denotes the heat power, which may enter, , or leave, , the system, so that a constant temperature *T*_{0} is guaranteed. The equality sign of the second equation of (4.1) holds in equilibrium, while in non-equilibrium, the variation of the entropy is greater than the ratio of supplied heat and temperature. This statement expresses the Clausius version of the second law of thermodynamics (Clausius 1854).

The elimination of the heat power leads to the thermodynamic inequality(4.2)The newly defined quantity is called the available free energy or availability. We conclude that for arbitrary thermodynamic processes, which run at constant outer pressure, constant temperature and constant total mass, the availability must always decrease and assumes its minimum in thermodynamic equilibrium.

The availability contains the combination *Ψ*=*E*−*T*_{0}*S*. The total energy *E* is the sum of the internal energy and kinetic energy. In the following, we will neglect the kinetic energy, which implies that *Ψ* is identical to the Helmholtz energy. Note however, that =*Ψ*+*p*_{0}*V* is not the Gibbs energy *G* of the system. The available free energy coincides with the Gibbs energy only if there is an overall constant pressure *p*=*p*_{0} in the interior of the volume *V*. However, in general, we will not meet this case here.

The Helmholtz energy may be additively decomposed into three contributions that refer to the two phases and the interface. The volume is the sum of the solid and the liquid volume, *Ψ*=*Ψ*_{S}+*Ψ*_{L}+*Ψ*_{I} and *V*=*V*_{S}+*V*_{L}.

The three contributions to the Helmholtz energy can be represented by volume and surface integrals, *viz*.(4.3)The density of the interfacial free energy, which is also called surface tension, is denoted by *σ*. We assume that the surface tension depends only on temperature, and thus is a constant here.

The inequality (4.2) can now be written as(4.4)and is valid for arbitrary thermodynamic processes in *Ω*, which, however, are subjected to (i) constant temperature in *Ω*, (ii) constant outer pressure on ∂*Ω*, and (iii) constant mass in *Ω*, and several further side conditions, which will be introduced and discussed in §4*b*.

### (b) Side conditions

In this section, we consider various side conditions of different origin. They concern the sublattice structure of the solid phase, which restricts possible chemical reactions, and diffusion in the solid phase. Further side conditions result from the conservation of mass during the thermodynamic processes.

#### (i) Equal number of lattice sites of the three sublattices

The three FCC sublattices, *α*, *β*, *γ*, of solid GaAs have the the same number of lattice sites. This implies two restrictions on the seven mole densities in the solid, *viz*.(4.5)and there are thus only five independent mole densities in the solid phase.

#### (ii) Balance of particle numbers including chemical reactions

Let us start from the mole number balance for the constituent *a*,(4.6)The quantity is the velocity of constituent *a*, and *τ*_{a} denotes its chemical production rate.

An alternative form of (4.6), which is more suitable for our purposes, results by introducing the diffusion flux and the material time derivative (4.7)In §3*b*, we have already introduced the barycentric velocity *v* of the mixture. Here we note, that *v* can be calculated from the velocities of the constituents by(4.8)According to its definition, the weighted diffusion fluxes sum up to zero. Multiplication of the balance (4.7) by the molecular weights *M*_{a} yields that the sum of the weighted production rates *M*_{a}*τ*_{a} also gives zero,(4.9)Chemical reactions concern here the transfer of constituents between the three sublattices. According to the Freiberg model, the Ga atoms cannot leave the sublattice *α*. Consequently, there are only two independent chemical reactions *r*∈{1,2}, *viz*.(4.10)The stoichiometric coefficients, , can be read off from (4.10),(4.11)Thus, we may introduce production rates *Γ*^{r}, which measure, per unit volume, the number of reactions per second, and we write(4.12)According to (4.5), we require in *Ω*_{S}(4.13)which guarantees that there are only five independent mole balances in the solid phase for five independent mole densities as variables.

#### (iii) Conservation of mass flux through the interface

The one-sided mass fluxes through the interface I are defined by(4.14)where the ‘+’ and ‘−’ indicate the limiting values of any quantity approaching the interface from the solid, +, and the liquid phase, −, respectively.

The conservation of total mass implies(4.15)

#### (iv) Conservation of material mole flux across the interface

The one-sided mole fluxes across the interface are defined by(4.16)The conservation of the mole numbers of Ga and As when they cross the interface implies(4.17)The four classes of side conditions will now be used to exploit the inequality (4.4).

### (c) Intermediate representation of the thermodynamic inequality

In the electronic supplementary material, appendix A, we take care of all side conditions and evaluate the time derivative of available free energy, which appears in inequality (4.4). We obtain a quite explicit form, which is best suited for further evaluation.

In this section, we change from the general system shown in figure 2 to the single droplet system, which is shown on the r.h.s. of figure 1, so that there is no boundary ∂*Ω*_{L}\I. Furthermore, from now on we ignore diffusion in the liquid droplet.

Proposition:(4.18)This representation of the thermodynamic inequality will be derived in the electronic supplementary material, appendix A.

The inspection of the inequality (4.18) reveals four mechanisms that drive the system to thermodynamic equilibrium. The first two lines represent mechanical processes. The third line describes the two chemical reactions between the constituents of the three sublattices. Lines four to seven represent diffusion within the sublattices and the remaining lines eight to twelve represent mole fluxes across the interface, which drive the system to phase equilibrium.

There are thus four different contributions to d/d*t*. We write(4.19)in order to indicate the different processes. The identification of the newly introduced quantities may be easily read off from (4.18).

### (d) Necessary conditions for equilibrium, part 1: mechanical equilibrium

The integrands that contribute to are linear in the barycentric and the interface velocity. The Galilean invariance of the inequality requires that the corresponding coefficients of the velocities must vanish in order to avoid the possibility of violating the inequality. This leads to the conditions(4.20)We identify these conditions as the well-known necessary conditions for mechanical equilibrium. Insertion of the constitutive laws for and *p*_{L} will lead to an elliptic boundary-value problem, which will be solved explicitly in §8 in the electronic supplementary material, where we consider the problem of a spherical liquid droplet within a solid matrix.

Note that the reasoning that has lead to the conditions of (4.20) was also valid in non-equilibrium, so that the same conditions arise in non-equilibrium. At first glance, this is a surprising result. However, it is the fact that we have already ignored the kinetic energy of the mechanical motion that restricts us to the case of quasi-static mechanical equilibrium. In other words, the neglect of the kinetic energy is equivalent to the assumption that mechanical equilibrium is established with zero relaxation time.

### (e) General structure of the thermodynamic inequality and its exploitation

We proceed to discuss the remaining three contributions to the main inequality (4.18), *viz*.(4.21)At first, we restrict ourselves to boundary conditions at ∂*Ω*_{S}\I so that the surface integral vanishes, i.e.(4.22)In other words, we consider here the case that the solid is in contact with an inert gas so that Neumann conditions result. A different consideration is necessary if the solid is in contact with the gaseous phase of GaAs. We consider this case in Dreyer & Duderstadt (in preparation) and Dreyer *et al*. (2004).

An inspection of the explicit form of the r.h.s. of (4.21) reveals that it consists of volume and surface integrals with sums of binary products as integrands. The factors of these products are called fluxes and driving forces. The fluxes are , *α*∈{1,2}, , *α*∈{1, 2, …, 12}, and , *α*∈{1, 2, …, 5}, whereas the corresponding factors are the driving forces , and .

The generic form of the three contributions to (4.21) thus reads(4.23)The fluxes are the independent variations of an equilibrium state, where . We conclude that the necessary conditions for equilibrium are given by(4.24)In non-equilibrium is negative, so that its equilibrium value, which is zero, establishes a minimum of . Let us now assume that the fluxes are given as functions of the driving forces, i.e.(4.25)where *y* denotes quantities that are not among the driving forces. It follows that(4.26)See Gurtin & Vorhees (1996) for a similar consideration.

The simplest ansatz that satisfies (4.26) is the assumption that driving forces and fluxes are proportional to each other. The proportionality factors are called mobilities. In particular, if we ignore cross effects, the ansatz simply reads(4.27)with positive mobilities , , .

Next, we exploit the possible equilibria according to (4.26).

### (f) Necessary conditions for equilibrium, part 2: chemical equilibrium

The contribution of chemical reactions to the inequality (4.18) reads(4.28)In equilibrium, we have and vanishing chemical driving forces, i.e.(4.29)The conditions of (4.29) are necessary conditions for equilibrium and give two algebraic equations for the determination of the equilibrium mole densities of the constituents after the mechanical problem has been solved according to the mechanical boundary-value problem from §4*d*.

### (g) Necessary conditions for equilibrium, part 3: diffusional equilibrium

The contribution of diffusion processes to the inequality (4.18) reads(4.30)In equilibrium, we have and vanishing diffusional driving forces. Owing to (4.29), this does not give four but only two independent conditions, *viz*.(4.31)The quantities *c*_{As} and *c*_{Va} are constants that can be calculated from the boundary conditions.

### (h) Necessary conditions for equilibrium, part 4: interfacial equilibrium

The contribution of interface motion to the inequality (4.18) reads(4.32)In equilibrium, we have and vanishing interfacial driving forces. Owing to (4.29), this does not give five but only three independent conditions, *viz*.(4.33)The conditions of (4.33) are necessary ones for equilibrium and give three further algebraic equations for the determination of the equilibrium values of the mole densities after we have solved the mechanical boundary-value problem from §4*d*.

Jump conditions such as (4.33) can be written in an alternative manner. For example, we may write the first equation of (4.33) as with the definition of the second-order tensor as the chemical potential of Ga in the solid, which is thus continuous across the interface. We note that for a solid body, which is a pure substance, reduces in the Lagrangian coordinates to the Eshelby tensor, see Eshelby (1957). Owing to the fact that jump conditions cannot be written down independently of the material and of the regime of the process, we prefer to define chemical potentials as scalars in the bulk by using the second equation of (3.27) and hereafter to determine their properties at a given interface.

### (i) Assumption on chemical relaxation times

The rate at which a thermodynamic system approaches equilibrium is controlled by the mobilities via (4.27). These are inversely proportional to the relaxation times of the three different processes that drive the system to chemical, diffusional and interfacial equilibria. The corresponding relaxation times are denoted by *τ*^{C}, *τ*^{D} and *τ*^{I}. Mechanical processes do not appear here because they are by far the fastest processes, so that we always set the mechanical relaxation time equal to zero.

Chemical reactions concern here the transfer of atoms between the three sublattices. Owing to the high temperature range that we are considering and due to the locality of sublattice changes by particles, we assume that these processes run at the same time scale as mechanical processes, i.e. we set *τ*^{C}=0.

### (j) Diffusion laws and assumptions on diffusional and interfacial mobilities

We have to determine seven mole fractions in the solid and one mole fraction in the liquid. Between the solid mole fractions, there are three trivial restrictions, *viz*. (3.5), and the two algebraic equations in (4.29) describing local chemical equilibrium. Thus, there remain 2+1 mole fractions in the solid and the liquid, respectively. Their evolution is driven by the diffusional and interfacial driving forces. In this section, we establish the diffusion laws in the solid phase and the interface conditions for non-equilibrium processes.

Relying on (4.27) and with (4.29), we read off from (4.30) the two diffusion laws(4.34)In a similar manner we obtain the interface laws(4.35)The bulk mobilities *B*≥0 and *B*_{Va}≥0 can be related to the corresponding diffusion coefficients, which may be determined from measurements. In this manner, the bulk mobility *B* is calculated from diffusion data concerning the interstitial As_{γ}, see Dreyer *et al*. (2004) for details.

Experimentally, nothing is known about the second mobility *B*_{Va}. In order to proceed we assume *B*_{Va}≫*B*, so that the limiting case *B*_{Va}→∞ can be applied. In other words, it is assumed that vacancies approach local diffusional equilibrium in zero time and for this reason does not depend on space.

Thus, we end up with a single diffusion problem that relies on the diffusion flux in the first equation of (4.34).

The motion of the interface is controlled by the laws in (4.35), including the mobilities , and . The data basis for their determination is inadequate. Motivated by the assumption *B*_{Va}≫*B* in the bulk, we now assume and also consider here the limiting case , which implies at the interface and thus likewise in the bulk.

Concerning the two remaining mobilities and , there are two limiting cases that are called *diffusion controlled* and *interface controlled*, respectively, which will both be studied.

The interface motion is called diffusion controlled, if the diffusion flux in the first equation of (4.34) is used in combination with the condition that the interface is in local equilibrium. In other words, the three conditions of (4.33), which follow from the first two equations of (4.35) by setting and , control the interface. In this case, we may use the conservation law in the second equation of (4.17) for the determination of the normal speed *w*_{ν} of the interface. For high numerical accuracy, it is necessary to take into account the barycentric velocity, which appears in (4.17). Its calculation relies on the time-dependent density of total mass, which is due to the quasi-static evolution of the mechanical deformations according to (3.17). We refer to Dreyer *et al*. (2004) for the details of a subtle discussion of this point. However, for completeness we give here the result, which reads(4.36)Herein, *V*_{L} and *O*_{L} denote, respectively, the volume and the surface of a homogeneous liquid droplet in a solid matrix.

The interface motion is called interface controlled if its normal speed *w*_{ν} is determined by the laws of the first two equations of (4.35) with finite mobilities in combination with the limiting case *B*→∞ in the first equation of (4.34). By means of similar arguments that have led to (4.36), we conclude from (4.16) and . With (4.17) and (4.35) we thus obtain the equation(4.37)which is supplemented by the single growth law(4.38)We again refer to Dreyer *et al*. (2004) for the details.

There remains the determination of the interface mobilities and . In principle, this can be done by experiments. However, there exists a case where a kinetic reasoning leads to an explicit theoretical expression for the interface mobilities. If a liquid droplet is in contact with its vapour, the growth of the droplet due to the incoming gas particles with mole density *n* and molecular weight *M* is given by the number of hits of the droplet per time unit and surface unit. It follows in this case . This result can be transferred as a guess to the solid–liquid interface for the case at hand. We set(4.39)

## 5. Explicit constitutive model for GaAs

In this section, we will formulate explicit constitutive laws for solid and liquid GaAs. These laws regard the dependence of Helmholtz energy densities, chemical potentials and stresses to the variables and on temperature-dependent quantities that refer to special reference states, which will be introduced next.

### (a) Reference systems and reference configurations

We consider at first the situation that is depicted in figure 1*a*. A solid phase is in contact with a liquid phase and a gas. The order of the phases ought to indicate that the solid is under hydrostatic pressure. Moreover, there are no curved interfaces, so that no capillary forces appear and the three phases live under the same common pressure, which is equal to the outer pressure *p*_{0} established. In this case, we call the system in figure 1*a* the reference standard.

If the gas is an inert gas, which does not take part during phase changes between the liquid and solid phases, we call this system the *standard system*.

If the gas consists of Ga and As constituents, which may cross the interfaces, a triple phase equilibrium under the vapour pressure may be established. In this case, we call the system of figure 1 the *reference standard system*. A detailed introduction and an exploitation of the reference standard system are found in Dreyer & Duderstadt (in preparation).

In this study, the Helmholtz energy densities and the chemical potentials will be given with respect to the reference standard system. On the other hand, the stress–strain relations will be formulated with respect to a reference configuration, which is defined as follows. A liquid–solid body of GaAs is free of strain, i.e. , in the solid and in the liquid, if the body (i) is under uniform pressure and (ii) has the composition in the solid and in the liquid. The bar indicates that the corresponding quantity is measured in the reference standard system.

According to the representations (3.7) and (3.12) for the solid and liquid mass densities, we define(5.1)Note that gives the mass density of a solid whose lattice coincides with the lattice of the reference configuration, but the distribution of particles over the lattice sides is different from the distribution in the reference configuration. This state of the solid is needed for the description of elastic deformations, which are reversible deformations, and these are not accompanied by a redistribution of atoms over the lattice sites.

The changes of mass densities due to elastic deformations are thus given by(5.2)The mole densities in the reference configuration may be read off from the corresponding data tables that can be found in the literature, for details see Dreyer & Duderstadt (in preparation).

### (b) Decomposition of the Helmholtz energy and chemical potentials into chemical and mechanical parts

We start from the general constitutive model as it was described in §4*e* of this study. First we show that the introduction of a strain-free reference state implies a decomposition of the specific Helmholtz energy and the chemical potentials into chemical and mechanical parts,(5.3)The proof of validity of this decomposition starts from equations (3.29), (3.28), (3.38) and (3.36),(5.4)Correspondingly, we write the chemical potentials as(5.5)We now define the chemical parts of these quantities by setting , and *c*^{ij}=δ^{ij}, which characterizes a state of the body with and ,(5.6)(5.7)Consequently, the mechanical parts of the specific free energies result from(5.8)By analogy, we define(5.9)Recall that elastic deformations result from deviations from the state and , which thus may be called a reference state for elastic deformations. This treatment guarantees that the chemical composition is not related to elastic deformations.

### (c) Constitutive model, part 2: the chemical parts of the chemical potentials for the solid and liquid phases of GaAs

The modelling of the solid phase relies on the sublattice model, which was introduced and described by Wenzl *et al*. (1990, 1993), Oates *et al*. (1995), Hurle (1999, 2004). Furthermore, we assume that there are exclusively entropic contributions to the chemical potentials. Owing to the sublattice structure, these are not given in terms of the mole fractions *X*_{a}, see (3.9), but by lattice occupancies *Y*_{a}, which are defined by (3.3). Recall that the three sublattices have an equal number of lattice sites, so that the *Y*_{a} give the mole densities per sublattice site.

The reason to consider only entropic contributions is the fact that currently no data are available for the material constants of the energetic contributions. In this case, we write(5.10)The quantities refer to the equilibrium of the triple-phase system of figure 1, where surface tension is ignored and exclusively hydrostatic stresses may appear. We have called this system the reference standard system, and we have chosen the thermodynamic equilibrium states of this system as the reference states of the chemical parts. Accordingly, we denote the minimizers of the available free energy for the reference standard system by . The exploitation and determination of all quantities that refer to the reference standard system are found in Dreyer & Duderstadt (in preparation).

In the liquid phase, we consider entropic and energetic contributions to the chemical potentials of the two constituents As_{L} and Ga_{L}. We use the arsenic mole fraction *X*_{L}, see (3.9), to represent the chemical potentials of the two constituents of the liquid phase. These read, for Ga_{L},(5.11)and for As_{L},(5.12)As above, the functions and refer to the reference standard system, and represents the arsenic mole fraction, which minimizes the available free energy of the reference standard system. Data for these quantities can be read off from Dreyer & Duderstadt (in preparation). The constitutive laws of (5.11) and (5.12) are given by Oates *et al*. (1995).

### (d) Constitutive model, part 3: the St. Venant–Kirchhoff law for stresses and mechanical parts of the Helmholtz energy and the chemical potentials of the solid phase

While the chemical parts of the chemical potentials for the solid phase rely on the well-established sublattice model, their mechanical parts are introduced here for the first time in order to model and simulate the problem of the appearance and evolution of As-rich droplets in solid GaAs. The procedure is as follows. We first formulate the stress–strain relation according to the St. Venant–Kirchhoff law, see for example Truesdell & Noll (1965). Next, we calculate the Helmholtz energy density by an integration, which relies on the general law (3.37). Finally, we obtain the mechanical parts of the chemical potentials by differentiation of the Helmholtz energy density with respect to the mole densities according to the second equation of (3.27).

The St. Venant–Kirchhoff law assumes that the stress that results for small elastic strains, , is given by a linear representation for the second Piola–Kirchhoff stress. The special form of this representation, which will be given below, is suited to describe the response of GaAs to elastic deformations.(5.13)This version of the St. Venant–Kirchhoff law relies on the assumption that elastic deformations describe here exclusive deformations of the crystal lattice, so that the volumetric part, in particular, is given by the ratio . Furthermore, there is no misfit strain due to a rearrangement of the Ga and As atoms on the lattice sites. If we were to allow that Ga atoms may occupy the *β* and *γ* sublattices, such a misfit strain would appear because the lattice sites of both sublattices offer less space than the lattice sites of the *α* lattice. Finally, we mention that the absence of misfit strain due to thermal expansion results from the chosen reference state. In other words, thermal expansion is already included in and the temperature-dependent stiffness matrix *K*^{ijkl}.

By means of (5.2) we may rewrite (5.13) as(5.14)The St. Venant–Kirchhoff law is appropriate in the small-strain regime. The reference pressure is introduced here, so that the homogeneous deformation leads to . The complete linearization of the St. Venant–Kirchhoff law with respect to the spatial displacement gradient *h*^{ij} gives the spatial version of the classical Hooke law. The linearization of the Green strain (*C*^{ij}−*δ*^{ij})/2 gives the strain *e*^{ij}=(*h*^{ij}+*h*^{ji})/2.

We now insert (5.14) into the l.h.s. of (3.37) and calculate the Helmholtz energy density by integration. Next, we rewrite the function according to (3.36) and obtain the following representation of the mechanical part of the Helmholtz energy density,(5.15)We add this function to the chemical part of the Helmholtz energy density in order to calculate the chemical potentials by differentiating with respect to the mole densities according to the second equation of (3.27). Note that the dependence on the mole densities is contained in *J* and *J*^{*} via the functions from §5*a* and from (3.3).

We start the calculation of the chemical potentials with the Gibbs–Duhem equation (3.26), which reads, in the elastic strain free configuration (5.16)Multiplying this equation by *ρ* and with (5.15) and the first equation of (5.3), we obtain(5.17)Next we use the second equation of (3.27) to obtain the chemical potentials by differentiation. This results in(5.18)and(5.19)The calculation of relies on (5.2) where *n*_{G} is given by (3.3) and with . We finally obtain(5.20)In §7 in the electronic supplementary material of this paper, the reader will find a comparison with the literature. In particular, we shall discuss the treatment that is found in Landau & Lifschitz (1989) with a special focus on the role of the nonlinear terms.

### (e) Constitutive model, part 4: pressure and mechanical parts of the Helmholtz energy and the chemical potentials for the liquid phase

We describe the liquid phase as a compressible liquid that is linear in and whose strain-free state is realized under the pressure . We writeThe newly introduced function denotes the bulk modulus of the liquid.

The problem that we consider exclusively in this study regards the formation and evolution of liquid droplets in semi-insulating GaAs. Owing to this application, we may restrict ourselves to liquid GaAs mixtures with an arsenic mole fraction *X*_{L}>0.9. Thus, for simplification and due to lack of data for the function , we set *X*_{L}=1 within and write(5.21)The thermal expansion of the liquid is included in this law and represented by means of the dependence of and *k*_{L} on temperature, see Dreyer & Duderstadt (in preparation) for details.

The chemical part of the chemical potentials of the liquid phase is now calculated in a manner analogous to the solid phase. We obtain(5.22)

## 6. Summary and outlook

In this study, we have proposed a model that is designed to simulate the appearance of liquid droplets in semi-insulating GaAs. The main part of the study regards the incorporation of mechanical stresses in the thermodynamic equations that describe phase transitions without surface tension and deviatoric stresses.

Section 9 in the electronic supplementary material contains, as a first application, the calculation of non-standard phase diagrams. An extensive comparison with classical phase diagrams is included here.

Further applications of the model regard (i) the diffusion problem of an evolving liquid droplet within a solid matrix and (ii) the evolution of a many droplet system and, in particular, the determination of the size distributions of the droplets. These tasks have been described in Dreyer & Duderstadt (2006) and Dreyer *et al*. (2006).

## Acknowledgments

The presented model of semi-insulating GaAs at temperatures above the right eutectic line has been formulated in close collaboration with Stefan Eichler and Manfred Jurisch from FCM, and we are indebted for countless, most fruitful discussions of various aspects of the model. The classical basis of the model has been established by Alan Oates and Helmut Wenzl. Their comments and suggestions to our work are invaluable. This work was partly supported by BMBF grant under 03DRM3B5.

## Footnotes

Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2007.0205 or via http://journals.royalsociety.org.

- Received October 1, 2007.
- Accepted April 25, 2008.

- © 2008 The Royal Society