J. MOLLER et al.: Kinetics of Segregation and Crystallization with Stress 49

phys. stat. sol. (b) 196, 49 (1996)

Subject classification: 64.75

Fachbereich Physik der Universitiit Rostock') (a) and Znstitute of Physical Chemistry, Bulgarian Academy of Sciences, Sofia2) (b )

Kinetics of Segregation and Crystallization with Stress Development and Stress Relaxation

BY J. MOLLER (a), J. SCHMELZER (a), and I. AVRAMOV (b)

(Received March 15, 1996)

The kinetics of cluster growth in segregation and crystallization processes in highly viscous materials taking into account stress development and stress relaxation is investigated theoretically. Differen t models of stress development are considered for the case that stresses arise due to the formation and growth of the newly evolving phase. The theory is formulated in terms of differential equations de- scribing the evolution of the cluster size, the change in time of the thermodynamic driving force of cluster growth and other related quantities. Based on the numerical solution of these equations several effects are analyzed which may occur as the result of stress development and stress relaxation.

1. Introduction Processes of phase separation in solid solutions are accompanied, in general, by the de- velopment of elastic strains. The strains can be of different type and magnitude in de- pendence on the growth rate and may be described by different model approaches.

If the ambient and the newly formed phases have the same composition, the only reason for the evolution of the elastic strains is the difference of the specific volumes or densities in both considered solids (coherency strains). In this case, the total energy of elastic deformation connected with the formation of a new phase increases linearly with the volume of the new phase (see Nabarro [l], Christian [2], Schmelzer [3], and Moller et al. [4, 51).

Another type of strains occurs if in a multi-component system one or several of the components segregate to form clusters of a new phase. Here, starting with some initial value of the volume, a quadratic increase of the elastic strains with a further increase of the volume is to be expected ([6, 71). While in the first considered case the strains mod- ify the crystallization kinetics only quantitatively, in the second one strains always re- sult in a total inhibition of crystallization and also in a qualitative modification of the kinetics of Ostwald ripening as shown in detail for the first time in [3] (see also [7, 81).

However, in viscoelastic materials, like polymeric systems or glass-forming melts in the vicinity of the temperature of vitrification, in addition to the elastic response, some re- laxation of the matrix may take place. This relaxation results in a decrease of the inhib- iting effect of elastic strains on the crystallization kinetics and, consequently, in a further growth, at a rate which is determined by the velocity of relaxation.

') August-Bebel-Str. 55, D-18051 Rostock, Federal Republic of Germany. ') BG-1113 Sofia, Bulgaria.

4 physica (b) 196/1

50 J. MOLLER, J. SCHMELZER, and I. AVRAMOV

In the framework of a particular model, such relaxation effects and their influence on cluster growth and Ostwald ripening have been already discussed in [3] and [6]. In the present paper, these results are extended in order to develop a general formalism allow- ing to describe processes of formation and growth of a new phase in viscoelastic media for any type of rheological behavior arid mechanisms of evolution of elastic strains. The general method is illustrated for special cases, and possible further developments are sketched.

2. Basic Equations

In general, the rate of growth of an aggregate of a newly evolving phase having macro- scopic dimensions is a function of the thermodynamic driving force of the phase transfor- mation. Ap (see, e.g., Gitzow and Schmelzer [9]), i.e.,

R is a characteristic length parameter used to describe the size of the newly evolving aggregate. When not specified otherwise we identify R with the radius of a spherical aggregate of the newly evolving phase. Sometimes the term “cluster” will be used synon- ymously.

Taking into account elastic strains the thermodynamic driving force of cluster gr0wt.h can be expressed as

AP = APCh - Aps t r . (2) Here ApcLh denotes the difference between the chemical potentials of the particles in both considered phases in the absence of strains, Ap,,, is the term describing the influence of el&ic st,rains on cluster growth; for the cases considered here elastic strains inhibit, in general, crystallization and Ap,,, is a positive qi1antit.y. It can be determined in the following way.

Neglecting variations of the state of the ambient phase in the course of the transfor- mation, the change of the Gibbs free energy AG due to the formation of a new phase consisting of j particles may be written as

AG = - j Apcl, + OA + dE) . (3)

Here u is t,he specific interfacial energy, A the area of the interface between both phases, and dE) the total energy of elastic deformations evolving as the result of the formation of the new phase.

The critical size of an aggregate of the new phase can be determined by deriving AG with respect to j or R, giving, for spherical aggregates, the result

ca is the density of particles in the evolving phase. Two conclusions can be drawn from (4). Firstly, they give an additional verification of

(2 ) , second, it becomes evident that, once dp) is known, Apgtr may be determined by partially deriving &) with respect to j .

All three quantities entering (2) are, in general, functions of time. As will be shown below, the equations describing the rate of change of these quantities can be expressed

Kinetics of Segregation and Crystallization with Stress Development 51

primarily in terms of first-order differential equations. Therefore, as the next step, we have to go over from (2) to a relation connecting the derivatives of these quantities with respect to time.

We get for the time derivative of Ap according to (2)

Provided that the analytical expressions for the terms on the right-hand side of (5) are known, the coupled set of equations (1) and (5) may be solved simultaneously to deter- mine the time dependence of the cluster size R.

The difference of the contribution Ap,, to the chemical potential difference Ap, which is determined by the chemical composition of the system and the thermodynamic bound- ary conditions, depends on the actual thermodynamic state of the system only, in parti- cular, on the size and spatial distribution of the new phase. It may be calculated by thermodynamic methods (see e.g. [3, 61).

The situation is different with respect to the contribution Apstr resulting from the evolu- tion of elastic strains in the course of cluster growth. The time dependence of Apstr is determined by two different processes. One is connected with the relaxation of elastic: strains for a constant size of the aggregate of the new phase, while the other describes changes of Apstr connected with the further growth or shrinkage of the cluster. In this case, we have to take into account an explicit time dependence due to relaxation processes.

In a mathematical form we may describe these two mechanisms of change by two different terms as

The first term on the right-hand side of (6) accounts for relaxation processes due to viscous flow. This meaning is specified by the term “relaxation”. For simplicity of the notations, we omit this subscript, in general, in the further equations.

The subscript “g/d” in the second term stands for growth-dissolution. This second term on the right-hand side of (6) is not a genuine partial time derivative but has to be understood as

In such a form we will iise it in the subsequent analysis. In order to apply the general equations, in addition to the dependence growth rate 2’

versus chemical potential difference Ap (expressed by the function fi(Ap), cf. (1)) a description of the rheological properties of the viscous melt has to be given. In a good approximation, one may assume that the rate of relaxation is a well-defined function of ApStr, i.e., that

holds. The function f2 (Apstr) describes specific rheological properties of the substance where the transformation takes place. For different possibilities concerning the analytic form of this function see, e.g., [16, 171.

I *

52 J. MOILER, J. SCHMELZER, and I. Avrwhrov

An integration of (6), taking into consideration (7) and ( B ) , yields

t o is some (in general, arbitrarily chosen) initial moment of time.

(9)) are known, the growth rate at time t may be determined by (cf. (1)) Once Apcl, (calculated by thermodynamic methods) and bstr (calculated according to

In this way, the set of equations for the determination of Apch, AvytS, and (10) have to be solved simultaneously to determine the time dependence of the cluster size R(t), as well as V, and Apstr.

On the other hand, we can apply (1) to express Ap in terms of the growth rate 'u. A derivation of (1) with respect to time yields

01'

After performing the differentiation of the function fi with respect to Ap the latter quantity can be replaced by the rate of growth w applying (1) or similar relations.

This second-order ordinary differential equation is equivalent to the set of two first- order differential equations (1) and (5) (or (9) and (10)) and can be used alternatively in order to determine the time evolution of the size of the aggregates for different mod- els of growth.

3. A Special Case: Normal Growth and Maxwellian Relaxation

Assuming normal growth, the linear rate w E dR/dt of crystal growth depends on Ap (see, e.g. [9 to 141) in the form

where 6 is the mean intermolecular distance, Icn the Boltzmann constant, t the average time interval required for a jump of a building unit of the crystal across the interface, and w accounts for the relative concentration of the active places on the interface. w is almost a constant having values of the order 112.

Evidently, the upper limit of the growth rate vo, attained for large values of Ap, is given by

wdo 710 = -

z (14)

The influence of elastic strains on growth may be accounted for in the above equa- tions by the appropriate determination of the two parameters Ap and t (or 710). t is a

Kinetics of Segregation and Crystallization with Stress Development 53

measure of the mobility of the segregating particles, i.e., of the viscosity of the melt. It is affected by the evolving elastic fields through its pressure dependence.

The pressure dependence of the shear viscosity of undercooled melts can be deter- mined by methods outlined in [14]. Similarly it is possible to specify the influence of elastic fields on the diffusion coefficient of segregating particles (see, e.g. [15]).

Although the theoretical approaches developed in [14, 151 give a sound basis for i i

calculation of the dependence of t on the strength of elastic strains, for the sake of simplicity, at the present stage of investigation we assume that t is a constant. In this way, the problem is reduced to the determination of the influence of the elastic fields on the thermodynamic driving force of the transformation, 4, and its dependence on time t.

We now substitute Ap in the form as given by (2) into (13) giving the growth rate for normal growth. As the result we obtain

In a number of applications it can be assumed that Apt,, does not change considerably with time in the course of the transformation. Such a simplification will be used in the following applications for an illustration of the method.

Assuming, moreover, a Maxwell-type relaxation mechanism (see also Moller [IS]) we have as a special case fi(Apstr) = -(Ap,,,/t~) (see (8)). For this special case, the total change of Apstr with time is given by

Here ZR is a characteristic relaxation time of the ambient phase. The expressions for (a Ap,,/aR) remain to be determined, they depend on the partic-

ular model used for the description of elastic strains. Save for this circumstance, (13) and (16), along with the assumption that Ap,, is constant, represent a complete set of differential equations for the description of the time evolution of the newly evolving phase.

The kinetics of growth depends significantly on the value of the parameter ' t ~ entering the expression for Maxwellian relaxation. Its connection with viscosity q and elastic moduli of the material under consideration will be discussed in the following section.

Finally, we have to mention that also the alternative approach based on (12) can be used equivalently for a description of cluster growth. For normal growth and Maxwel- lian relaxation we have, in general,

Generalizations for other modes of growth and other types of rheological behavior can be carried out straightforwardly.

4. Estimates for the Maxwellian Relaxation Time

As a spherical cluster is compressed hydrostatically, we can assume that only the ambi- ent phase is subject to viscous flow. In order to give estimates for the magnitude of ZR,

54 J. MOLLEH, J. SCH.lltl,XI.:R, a d I. AVRAMOV

we start with the general expression for the stress tensor at:) resulting from elastic deformations (see, e.g. [19]),

and the one (o!?') that is due to viscous flow,

The parameters E (Young's modulus) and y (Poisson ratio) describe the elastic proper- ties of the considered substance, while v and 5 are the shear and bulk viscosities, respec- tively. d a k is the Kronecker symbol. In the latter and subsequent equations it is supposed that the sum has to be taken over the same indices.

Since all accelerations of the different masses involved are small, we do not have to take them into account in formulating the equilibrium conditions. In this way, we get

For clusters in an elastic matrix, having the same elastic constants as the cluster phase, the relation

l L / / = 0 (21)

holds (see [ 5 ] ) . Consequently, there is no compressional flow in the system and the reln- tion

U,l = 0 (22)

is also fulfilled for all times. The latter equation is equivalent to the continuity equation in the Navier-Stokes relations.

From the equilibrium conditions we find for the considered situation

~ ?L,k - 2v'& = 0 . E

l+Y By solving this relation as an ordinary differential equation we obtain

Equations (23) and (24) yield

A substitution of (24) into (18) gives

For the process of relaxation of the density of elastic energy q(') we find

Kinetics of Segregation and Crystallization with Stress Development 55

or

with

In this way, the Maxwellian relaxation time q t is determined mainly by the ratio of shear viscosity 11 and Young's modulus E.

5. A First Application: Segregation in a Multi-Component Viscoelastic Solid

In earlier papers, Schmelzer et al., see [3, 61, have analyzed the strain that develops during segregation processes in multi-component systems if one (or some) of the segre- gating components have considerably higher mobilities compared with the ambient phase particles. It follows from their model consideration that in case relaxation pro- cesses do not occur the total energy of elastic deformations grows with increasing vol- ume of the aggregate as

V" is the volume of a hole which would remain in the matrix if the cluster were removed from it. It is determined both by the actual cluster size and by the relaxation processes in the remaining matrix taking place in the course of growth and shrinkage of the clus- ter and viscous flow. V is the actual volume of the aggregate ([3, 61).

According to (4) we have for this model in the general case, when viscous relaxation occurs,

In the considered model Vu is a parameter which may change by relaxation processes only. Consequently, the relation

holds and (31) yields

In (33), V cannot exceed & considerably, otherwise the cluster growth is stopped immediately. In this way, save for differences (V - &) we may replace V by fi and vice versa as in the case of a purely elastic Hookean solid.

Hence, (33) simplifies to

56 J. MOLLER, J. SCHMELZER, and I. AVIMAMOV

Provided that the size of the cluster is fixed and Apstr behaves according to (16) (hlaxwellian relaxation) then V, tends to V with time. The parameter V, describing the size of the spherical hole, where the cluster grows, consequently, changes in time also when additional particles are added to it. The partial derivative of Apstr with respect to time is, due to the relaxation model, assumed to be proportional to Apstr, hence

When switching from cluster volumes to cluster radii R and the radius of the hole & ils variables for the description of the growth process we get

instead of (34), resulting in

In this way, the derivation of all necessary relations for the description of the growth of the aggregates of the newly evolving phase is completed. Equation (17), for example, takes the form

dt - kBT

R.esults of the numerical solution of this or equivalent sets of equations will be given in Section 7.

6. A Second Application: Elastic Strains and Relaxation in Crystallization Processes

In crystallization processes, when the matrix and the newly evolving phase have the same composition, elastic strains are generated by differences in the molar or specific volumes of both phases, wmatrk (ambient phase) and wcluster (newly evolving phase). For simplicity, we neglect here possible anisotropies of the solids considered.

For isotropic Hookean bodies, when cluster and matrix are characterized by the same values of elastic constants, the elastic strains resulting from the formation of a cluster of volume V are known to be of the form (see, e.g. [l, 3, 5, 181)

To generalize the above equations to the case that viscous properties of the matrix have to be taken into account we realize that (39) may be rewritten in an equivalent form as

Kinetics of Segregation and Crystallization with Stress Development 57

In this model, Vo denotes the volume of the hole in the ambient phase, which would remain in the matrix when the cluster is removed from it. It is determined both by the actual size (or volume V ) of the cluster and relaxation processes in the matrix taking place in the course of growth and shrinkage of the clusters.

In the considered alternative model, V” changes with the size of the cluster according to the relation

From (4) and (40) we then obtain

The partial derivative (86/OR) in (42) may be expressed as

[v (Z) - .] (Z) dd aR - V2 _ -

(see [18]) and (43) takes the form

3 (&) = (do - 6 ) .

A substitution into the second of equations (42) yields

If there are no relaxation processes, i.e., the matrix is a Hookean elastic solid, then 6 and 6 0 coincide. In this case, Apst,tr is a constant and independent of the cluster size.

7. Method of Solution of the Basic Equations For the numerical evaluation of the growth equations, we consider here the general case that there is an initial “pore” size, RP’, in which a cluster of initial size zero starts to grow. It is assumed that only after the cluster has reached this size we have to take into account the viscoelastic response of the matrix. The notation “pore” in above-mentioned sense means either a real pore in a solid viscoelastic matrix or the lowest size of the cluster at which elastic effects become of importance.

We analyze the effects of elastic strains on growth in a viscoelastic material by com- paring the kinetics with the cases when either strains do not develop at all or the matrix behaves as a purely elastic Hookean solid with no relaxation processes to occur.

For the numerical calculations the values for ZR, R y ) , 60, the elastic properties, etc. were chosen such that the plots reveal the characteristic behavior of the plotted quanti- ties as clearly as possible. The values of the parameters are: T = 1000 K, do = m, eit) = m, E = 0.7 x 10” Nm-’, y = 0.3, ca = ( 3 0 0 ~ * / ( k ~ T ) ) , for both sets of fig- ilres in common, ZR = 4 x 10-12s, ApChcm = lo-’’ J for the first model (Fig. 1 to 3), as well as ZR = 1.6 x lO-’s, Apchcm = 5 x J, 60 = - ( l / l O ) for the second one (Fig. 4 to 6).

58 J. M O i . i , ~ i t , J. SCHMELZER, and I. AVRAMOV

The integration of the set of differential equations was carried out by simply adding

- The equation of growth yielded the increase of dR = R dt. - Then R was incrernented R + dR + R. - Now V = (4n/3) @ was computed. - After that, Vo was determined according to (34) when being solved for &I, and,

- The determination of 6 according to (40) followed. - With that we were able to obtain (a ApStr/aR). - The partial time derivative of Apstr was obtained by means of (35). - Apstr was integrated by means of (16). - After that, time was incremented again, and so forth.

small differences. In a loop, time was incrernented by a sniall difference as follows:

accordingly, by the first of equations (42) in conjunction with (41).

All steps beginning from the computation of Vo on were carried out only after R had exceeded the initial value of &.

8. Results

In Fig. 1, the chemical potential in the first model (elastic strains and relaxation in seg- regation in multi-component solutions) is represented. In the purely elastic case (lower curve), Ap is a monotonically decreasing function of time until a constant value is reached. The decrease of the chemical potential is due to the buildup of elastic strains beginning from the moment of time in which the cluster encounters the size of the pore. In the viscoelastic case (upper curve), 4 decreases first, too, but then goes through a minimum. After that. the initial value of the chemical potential is reached, again. This type of behavior is due to an interplay between amplification of the elastic strains and processes of matrix relaxation.

Fig. 2 shows the influence of the elastic strains on the change in time of the radius of the cluster for different models of rheological behaviour of the matrix. With curve 4 the initial pore size is marked, at which an elastic response of the matrix may start, to take place. Curve 1 represents the growth without any elastic strains. In the purely elastic case, where the elastic energy grows quadratically with the relative misfit 6, the chemi-

10

8

=: s 6 E: v

$ 4

2

0

\

Fig. 1. Results for the first model: Cluster growth in segregation in a multi-component solution (cf. Section 5 ) . Time dependence of the chemical p u tential difference A,u(t) is shown both for the elw- tic (lower curve) and the viscoelastic cases (upper curve). The values of the parameters, used for the numerical evaluation, are specified in Section 7

0 0.7 1.4 t s)

Kinetics of Scgrcgation and Crystallization with Stress Development, 50

6

5 -

I Fig. 2. Time dependence of the radius of the clus- ter for the first model (cf. Fig. 1). The diffcrcnt curves refer to: (1) growth without elastic influ- ence, (2) growth in a viscoelastic matrix with Maxwellian relaxation bchavior, (3) growth in ii

purely elastic solid, (4) initial sizc of the clnster, at which elastic strains become significant

4 - h

E

0 0.7 I .4 t (lo-'" s)

cal part in the chemical potential is soon cornpensated by strains leading to a terniina- tion of the growth. Such behavior is represented by curve 3.

In the viscoelastic case, Ap approaches its initial value over time (cf. Fig. l), because the slopes of the R = R(t) dependences for curve 1 and the viscoelastic case, curve 2, tend to the same value. In this way, the elastic response of the matrix results in this case exclusively in some time lag in growth.

The change of the time lag with time is shown in Fig. 3. For the time interval consid- ered, the time lag behaves as Tgrowtl, 0: In t.

For our second model (elastic strains and relaxation in crystallization), the behavior of the chemical potential, as shown in Fig. 4, is essentially the same as that in the first; case (cf. Fig. 1). As can be learned from the equations, this case contains, in some way, the first case. However, Apstr was chosen such that in the purely elastic case, the growth does not stop, but the cluster continues to grow at a smaller rate (see also Fig. 5, curve 3).

In Fig. 4, there remains a value of Ap greater than zero, a situation which is not possible in the first case. Hence, the general behavior of the upper curve in Fig. 4 can be

h Ln - - I

- Fig. 3. Time dependence of the time lag as ob- tained for the first considered modcl. The defi- nition of the timc lag tgroulll in cluster growth

I I 1 is specified in Fig. 2 1.4 2.8

t s)

60

5

4

h

- 3

0 3 . 2

x v

Q

1

n

1

0 4 8 t ( s)

Fig. 4

J. MOLLER, J . SCHMELZER, and I. AvRAhiov

t ( I O P s)

Fig. 5

Fig. 4. Results for the second rnodcl: Elastic strains and relaxation in crystallization processes (cf. Section 6). Time dependence of the chemical potential difference Ap in the elastic (lower curve) and the viscoelastic cases (uppcr curve). In contrast to the first model, Ap approaches a finite nonzero value. Note, however, that also for this rnodcl the elastic term may inhibit the growth cornpletely. 111 such n situation, the chemical potential diffcrcnce drops to zero, again

Fig. 5 . Time dependence of the radius of the cluster for the second model (cf. Fig. 4). The different curves refcr to: (1) growth without an elastic influence, (2) growth in a viscoelastic matrix with Maxwellian relaxation behavior, (3) growth in a purely clastic solid, (4) initial size of the cluster at which clastic strains become significant

obtained by a simple reasoning. The purely elastic case is marked by a constant value of Apstr (cf. the lower curve in Fig. 4). Hence, ApBtr in the viscoelastic case had to be mono- tonically decreasing, resulting in a monotonic increase of Ap (cf. Fig. 4). Not just so obvious by reasoning, but evident after the simulation, is that A,uStr tends to zero in both considered viscoelastic cases.

I I I I

Kinetics of Segregation and Crystallization with Stress Dcveloprncnt 6 1

Fig. 6 shows again the dependence of the time lag on time for the second model con-

Finally, we would like to mention that the purely elastic cases were partly discussed sidered.

in ([8, 201) already.

9. Discussion

A first conclusion, which can be drawn from the results outlined above is the following: Provided the viscous relaxation of the matrix is described by Maxwell’s equation, the growth of the clusters is inhibited temporarily but not terminated totally. For large cluster radii, the growth proceeds as if elastic effects do not play a role at all, the only effect being some retardation of the growth kinetics (time lag in growth).

On the other hand, experimental evidence shows that elastic strains, indeed, may stop the growth of single clusters and ensembles of clusters (cf. [6, 71) in highly viscous glass- forming melts. Since such a behavior may be excluded, according to the theoretical in- vestigations, for Maxwell-type viscous relaxation, the mentioned experimental results give a strong indication of the importance of non-Maxwellian relaxation in glassforming melts (cf. [9, 171).

More generally, the theoretical method outlined in the present paper allows a descrip tion of the growth of clusters of a newly evolving phase for any mechanism of growth and any type of rheological behavior of the matrix. It should allow, in particular, the specification of the type of rheological behavior by a comparison of experimental and theoretical results concerning the rates of growth or dissolution of aggregates of a newly evolving phase.

Moreover, the knowledge of the growth rates permits also the determination of the expressions for the nucleation rates, the determination of the kinetic coefficients for the set of kinetic equations describing nucleation and growth in the framework of classical nucleation theory [9].

An interesting extension of the outlined results consists in the analysis of the kinetics both of nucleation and of Ostwald ripening in viscoelastic materials with arbitrary types of rheological behavior. The results of such an analysis will be reported in a forthcoming contribution.

Acknowledgements One of the authors gratefully acknowledges the financial s u p port of the Bulgarian Ministry of Sciences (I. A.), the two others (J. S., J. M.) the long- standing support from the Deutsche Forschungsgemeinschaft (DFG; projects Kr 1732/2- 1; 436 BUL-113/83/2; 436 BUL-111/5/95) which allowed to carry out a number of coni- mon research projects including the present one.

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