# Stress development and relaxation during crystal growth in glass-forming liquids

**ABSTRACT** We analyze the effect of elastic stresses on the thermodynamic driving force and the rate of crystal growth in glass-forming liquids. In line with one of the basic assumptions of the classical theory of nucleation and growth processes it is assumed that the composition of the clusters does not depend significantly on their sizes. Moreover, stresses we assume to be caused by misfit effects due to differences in the specific volume of the liquid and crystalline phases, respectively. Both stress evolution (due to crystallization) and stress relaxation (due to the viscous properties of the glass-forming liquids) are incorporated into the theoretical description. The developed method is generally applicable independently of the particular expressions employed to describe the crystal growth rate and the rate of stress relaxation. We show that for temperatures lower than a certain decoupling temperature, Td, elastic stresses may considerably diminish the thermodynamic driving force and the rate of crystal growth. The decoupling temperature, Td, corresponds to the lower limit of temperatures above which diffusion and relaxation are governed by the same mechanisms and the Stokes–Einstein (or Eyring) equation is fulfilled. Below Td, the magnitude of the effect of elastic stresses on crystal growth increases with decreasing temperature and reaches values that are typical for Hookean elastic bodies (determined by the elastic constants and the density differences of both states of the system) at temperatures near or below the glass-transition temperature, Tg. By these reasons, the effect of elastic stress must be properly accounted for in a correct theoretical description of crystal nucleation (as some of us have shown in previous papers) and subsequent crystal growth in undercooled liquids. The respective general method is developed in the present paper and applied, as a first example, to crystal growth in lithium disilicate glass-forming melt.

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- Marcio Luis Ferreira Nascimento, Vladimir Mihailovich Fokin, Edgar Dutra Zanotto, Alexander S Abyzov[Show abstract] [Hide abstract]

**ABSTRACT:**We collect and critically analyze extensive literature data, including our own, on three important kinetic processes--viscous flow, crystal nucleation, and growth--in lithium disilicate (Li(2)O·2SiO(2)) over a wide temperature range, from above T(m) to 0.98T(g) where T(g) ≈ 727 K is the calorimetric glass transition temperature and T(m) = 1307 K, which is the melting point. We found that crystal growth mediated by screw dislocations is the most likely growth mechanism in this system. We then calculated the diffusion coefficients controlling crystal growth, D(eff)(U), and completed the analyses by looking at the ionic diffusion coefficients of Li(+1), O(2-), and Si(4+) estimated from experiments and molecular dynamic simulations. These values were then employed to estimate the effective volume diffusion coefficients, D(eff)(V), resulting from their combination within a hypothetical Li(2)Si(2)O(5) "molecule". The similarity of the temperature dependencies of 1/η, where η is shear viscosity, and D(eff)(V) corroborates the validity of the Stokes-Einstein/Eyring equation (SEE) at high temperatures around T(m). Using the equality of D(eff)(V) and D(eff)(η), we estimated the jump distance λ ~ 2.70 Å from the SEE equation and showed that the values of D(eff)(U) have the same temperature dependence but exceed D(eff)(η) by about eightfold. The difference between D(eff)(η) and D(eff)(U) indicates that the former determines the process of mass transport in the bulk whereas the latter relates to the mobility of the structural units on the crystal/liquid interface. We then employed the values of η(T) reduced by eightfold to calculate the growth rates U(T). The resultant U(T) curve is consistent with experimental data until the temperature decreases to a decoupling temperature T(d)(U) ≈ 1.1-1.2T(g), when D(eff)(η) begins decrease with decreasing temperature faster than D(eff)(U). A similar decoupling occurs between D(eff)(η) and D(eff)(τ) (estimated from nucleation time-lags) but at a lower temperatureT(d)(τ) ≈ T(g). For T > T(g) the values of D(eff)(τ) exceed D(eff)(η) only by twofold. The different behaviors of D(eff)(τ)(T) and D(eff)(U)(T) are likely caused by differences in the mechanisms of critical nuclei formation. Therefore, we have shown that at low undercoolings, viscosity data can be employed for quantitative analyses of crystal growth rates, but in the deeply supercooled liquid state, mass transport for crystal nucleation and growth are not controlled by viscosity. The origin of decoupling is assigned to spatially dynamic heterogeneity in glass-forming melts.The Journal of Chemical Physics 11/2011; 135(19):194703. · 3.12 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**During crystallization of glass-forming melts an important amount of stress energy may develop. The reason for this development is the difference in volume of the new system and that in the ambient phase. Therefore, the growth rate decreases with time. Here we develop an algorithm for the process of crystallization using Monte Carlo simulation techniques that takes into account the stress energy. We find that there is a short period of initial fast growth stage followed by a second period of much slower growth, controlled by the relaxation rate. This picture has also been recently observed experimentally. Additionally, we find that during the growth process the shape of the crystal is changing. Although we start from a highly symmetric crystal with flat lang10rang interfaces, a shape with a large number of facets is soon created.EPL (Europhysics Letters) 01/2010; · 2.26 Impact Factor -
##### Article: Diffusion-controlled crystal growth in deeply undercooled melt on approaching the glass transition

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**ABSTRACT:**Crystal-growth velocity in metallic melts has been reported by others to increase monotonically with undercooling. Nevertheless, such an observation is not predicted by conventional growth theory. In this work, the metallic melt of Zr50Cu50 is studied to address the problem by measuring the growth velocity over a wide range of undercooling up to 325 K. A maximum growth velocity is observed at an undercooling of 200 K instead of the monotonic increase reported in the literature. We find that the planar or dendrite growth theories can explain the value of the maximum growth velocity, but the predicted location of the maximum in undercooling is far less than that seen by experiment. With the assistance of current results, a general pattern of crystal growth is established for melts of a variety of substances, where all sluggish crystal-growth kinetics is explained by the diffusion-controlled mechanism at deep undercooling.Physical review. B, Condensed matter 01/2011; 83. · 3.77 Impact Factor

Page 1

Stress development and relaxation during crystal growth in

glass-forming liquids

Ju ¨rn W.P. Schmelzera,b, Edgar D. Zanottob,*, Isak Avramovc, Vladimir M. Fokinb,d

aInstitut fu ¨r Physik der Universita ¨t Rostock, Universita ¨tsplatz, 18051 Rostock, Germany

bVitreous Materials Laboratory, Department of Materials Engineering, Federal University of Sa ˜o Carlos, UFSCar,

13565-905 Sa ˜o Carlos, SP, Brazil

cInstitute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria

dS.I. Vavilov State Optical Institute, ul. Babushkina 36-1, 193171 St. Petersburg, Russia

Received 20 August 2004; received in revised form 11 May 2005

Available online 9 March 2006

Abstract

We analyze the effect of elastic stresses on the thermodynamic driving force and the rate of crystal growth in glass-forming liquids. In

line with one of the basic assumptions of the classical theory of nucleation and growth processes it is assumed that the composition of the

clusters does not depend significantly on their sizes. Moreover, stresses we assume to be caused by misfit effects due to differences in the

specific volume of the liquid and crystalline phases, respectively. Both stress evolution (due to crystallization) and stress relaxation (due

to the viscous properties of the glass-forming liquids) are incorporated into the theoretical description. The developed method is gener-

ally applicable independently of the particular expressions employed to describe the crystal growth rate and the rate of stress relaxation.

We show that for temperatures lower than a certain decoupling temperature, Td, elastic stresses may considerably diminish the thermo-

dynamic driving force and the rate of crystal growth. The decoupling temperature, Td, corresponds to the lower limit of temperatures

above which diffusion and relaxation are governed by the same mechanisms and the Stokes–Einstein (or Eyring) equation is fulfilled.

Below Td, the magnitude of the effect of elastic stresses on crystal growth increases with decreasing temperature and reaches values that

are typical for Hookean elastic bodies (determined by the elastic constants and the density differences of both states of the system) at

temperatures near or below the glass-transition temperature, Tg. By these reasons, the effect of elastic stress must be properly accounted

for in a correct theoretical description of crystal nucleation (as some of us have shown in previous papers) and subsequent crystal growth

in undercooled liquids. The respective general method is developed in the present paper and applied, as a first example, to crystal growth

in lithium disilicate glass-forming melt.

? 2006 Elsevier B.V. All rights reserved.

PACS: 64.43.Fs; 64.60.?i; 64.60.Qb; 64.70.Dv

Keywords: Crystallization; Viscosity and relaxation

1. Introduction

Elastic stresses are known to play an important role in

phase transformations in crystalline solids [1–4]. They

may change the course of the transformations both quanti-

tatively and qualitatively. In the vicinity of the glass-transi-

tion temperature, Tg, glass-forming liquids behave as

viscoelastic bodies. Hereby, the elastic properties become

increasingly dominant with a decrease of temperature.

Near, and especially below Tg, glasses display properties

that are typical for Hookean elastic solids. One thus

expects that, in the neighborhood of Tgand below, elastic

stresses may affect significantly the phase transformation

processes, in general, and crystallization processes of

glass-forming liquids, in particular. Indeed, a variety of

experimental results, summarized in Refs. [5,6], demon-

0022-3093/$ - see front matter ? 2006 Elsevier B.V. All rights reserved.

doi:10.1016/j.jnoncrysol.2006.01.016

*Corresponding author. Tel.: +55 16 3351 8527; fax: +55 16 3361 5404.

E-mail address: dedz@power.ufscar.br (E.D. Zanotto).

Journal of Non-Crystalline Solids 352 (2006) 434–443

Page 2

strates that elastic stresses may have a significant influence

on the course of phase transformations in glass-forming

liquids. In particular, it has been shown both experimen-

tally and theoretically [7–9] that elastic stresses may quali-

tatively change the kinetics of growth of single clusters and

of Ostwald ripening in glass-forming melts, when the segre-

gating component has a diffusivity much higher than the

basic building units of the glass-forming melt. This situa-

tion is similar to the results of Stephenson [10] in applica-

tion to spinodal decomposition.

However, up to now, the effects of stresses developing in

the course of crystallization are neglected in most analyses

dealing with crystal nucleation and growth in glass-forming

liquids. The common argument is that these stresses relax

too fast as to affect these phenomena. The above men-

tioned argumentation is based on several assumptions.

First, in line with classical nucleation theory, it is assumed

that the state of the crystallites does not depend on their

sizes and is widely identical to the state of the newly evolv-

ing crystal phase. In such cases, the kinetics of cluster

nucleation and growth can be treated similarly to nucle-

ation and growth in one-component systems with appro-

priately chosen values of the effective diffusion coefficient

and thermodynamic driving force [5,6,11,12]. Second, it is

assumed that elastic stresses are due to the difference

between the specific volumes of liquid and crystalline phase

first analyzed by Nabarro [1]. Both these assumptions we

will also employ in the present analysis. In addition it is

commonly assumed that the Stokes–Einstein/Eyring equa-

tion, which connects viscosity (governing relaxation pro-

cesses) and self-diffusion coefficients (determining the rate

of aggregation) holds and retains its validity for tempera-

tures near and below Tg. However, by different methods

of analysis – theoretical, computer simulation and experi-

mental techniques – it has been convincingly demonstrated

for a variety of liquids that at some temperature, Td,

decoupling of diffusion and viscous flow takes place (see

Refs. [5,6] for an overview). The value of the decoupling

temperature is frequently found at about Td= 1.2Tg, but

some authors report values of Tdffi Tg[13,14]. Above Td,

the Stokes–Einstein/Eyring equation is typically fulfilled,

but below Tdit is not.

By incorporating these ideas into the theoretical descrip-

tion of crystal nucleation in glass-forming liquids some of

us arrived at the conclusion that elastic stresses may have

a significant effect on critical nucleus formation in glass-

forming liquids [5,6]. These theoretical arguments were

then applied for the description of nucleation in lithium

disilicate melts and were able to explain a number of effects

which have not found a satisfactory explanation before

[15,16]. It is thus of significant interest to extend the anal-

ysis of the possible effects of stresses on crystal growth.

In the present paper we develop a general formalism

that allows us to describe growth processes of a new phase

in viscoelastic media taking into account both stress devel-

opment and relaxation. The stress energy, affecting the

growth rate of the crystalline phase in the liquid, results

from an interplay between the rate of stress development

(due to the propagation of the crystal-growth front

throughout the matrix) and stress dissipation (due to stress

relaxation in the viscous matrix). We show that the value of

the stress energy, which affects the growth kinetics, depends

on the ratio of two characteristic time-scales, sG/sR, where

sGis the time required to form one monolayer of the newly

evolving crystalline phase in steady-state growth, and sRis

a characteristic (Maxwellian) relaxation time of the matrix.

If the (short-range) interfacial rearrangements controlling

crystal growth are of the same nature as those involved

in viscous flow, then stresses relax comparatively fast and

have no effect on crystal growth. However, if – as demon-

strated to be a typical phenomenon for different classes of

glass-forming liquids – decoupling of short-range diffusion

and viscous flow occurs at some temperature, Td, then

stresses may have a significant effect on the crystal growth

kinetics for temperatures T 6 Td. As will be shown in the

present paper, the effect of stresses on the driving force of

crystallization increases with decreasing temperature reach-

ing values that are typical for Hookean (elastic) solids near

or below Tg.

The paper is organized as follows: in Section 2 a theoret-

ical approach is developed allowing one to determine the

effect of stresses on crystal growth. In Section 3 we show

that, in the range of temperatures where the Stokes–Ein-

stein (or Eyring) equation holds, elastic stresses do not play

any role in crystallization and glass-forming liquids behave,

with respect to crystal growth, as Newtonian liquids. How-

ever, below Td, elastic stresses may have a significant influ-

ence on crystal growth. The magnitude of this effect

depends on the values of the elastic constants of both

phases and on a misfit parameter characterizing the volume

changes in crystallization. The theory is applied in Section

4 for the description of crystal growth in lithium disilicate

glass. A discussion of the results (Section 5), possible mod-

ifications and extensions of the theory (Section 6) and a

summary of the conclusions completes the paper.

2. Theory: basic assumptions and results

We consider a planar crystallization front with an

interfacial area, A, moving into a direction specified by

the x-axis of an appropriately chosen system of coordinates

normal to the considered front. The number of particles in

the crystalline phase, n, can then be written as

n ¼ cAx;

c ffi2

d3

0

;

c ¼1

vc;

vc¼1

2d3

0.

ð1Þ

Here c is the volume concentration of the ambient phase

particles in the crystalline phase, vcthe volume per particle

in the crystalline phases, d0is a characteristic size parame-

ter (diameter) of the basic structural units of the system (cf.

[17]).

The growth rate, U = (dx/dt), can be connected with the

change of the characteristic thermodynamic potential, DU,

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

435

Page 3

or the difference of the chemical potential per particle in

both considered states of the system as [17,18]

?

Here kBis the Boltzmann constant and T the absolute tem-

perature, and f is a dimensionless parameter describing the

specific properties of the different growth modes. For sim-

plicity of the notation, we will assume here f = 1 corre-

sponding to the case of normal growth. But the main

results of this analysis are – as will be shown below – inde-

pendent of this assumption. Instead of the self-diffusion

coefficient of the ambient phase particles, D, we will use

the characteristic time of molecular motion, s, in the

description employing the relation d2

In the absence of elastic stresses, the change of the ther-

modynamic potential, connected with the transfer of n

ambient phase particles into the crystal phase, can be

expressed as

U ¼dx

dt¼ f

D

4d0

1 ? exp

?Dl

kBT

???

;

Dl ¼oDU

on

.

ð2Þ

0ffi Ds.

DUðnÞ ¼ ?nDl.

Elastic stresses can be incorporated into the above equa-

tion by adding a term, U(e), i.e. the total energy of elastic

deformations connected with the formation of a new phase

region with n particles. In such cases, instead of Eq. (3) we

get

ð3Þ

DUðnÞ ¼ ?nDl þ UðeÞðnÞ.

The driving force of crystallization in the absence of elastic

stresses, Dl, is a function of the temperature difference

(Tm? T) [17]. In the analysis we will employ either exper-

imentally determined data or, for the derivation of some

more general conclusions, the Volmer–Turnbull expression

?

where DHmis the enthalpy of melting per particle at the

melting temperature, Tm.

We consider stresses due to misfit effects between the

melt and the newly evolving crystalline phase. In this case,

we can write generally [1,3,5,6]

ð4Þ

DlðTÞ ¼ DHm 1 ?T

Tm

?

;

ð5Þ

UðeÞ¼ en.

A substitution of this expression into Eq. (4) leads to the

consequence that elastic stresses effectively result in a de-

crease of the thermodynamic driving force of crystalliza-

tion by the quantity e. Indeed, we can write Eq. (4) in the

form

ð6Þ

DUðnÞ ¼ ?nDlðeffÞ;

Consequently, for any value of e, from Eqs. (2), (4) and (6)

we get

?

For purely elastic solids, we have e = e0, where e0 is a

parameter that depends on the elastic constants and the

DlðeffÞ¼ Dl ? e.

ð7Þ

dx

dt¼

D

4d0

1 ? exp

?Dl ? e

kBT

???

.

ð8Þ

densities of both phases. In general, the stress parameter

e0can be written as [1,19]

EcEmvc

3½Ecðcmþ 1Þ ? 2Emð2cc? 1Þ?d2;

where E is Young’s modulus, c Poisson’s ratio, and v the

volume per particle. The subscripts (m) and (c) refer to

the parameters of melt and crystalline phase, respectively.

Generally, due to viscous relaxation, the inequality

e 6 e0holds and the growth rate and the values of the effec-

tive elastic stress parameter e must be determined in a self-

consistent way as functions of time. In order to do so, we

have to develop, in addition, an equation for the time-

dependence of e. In general, the solution of the resulting

set of equations for the time dependencies of the growth

rate and the rate of change of the effective stress parameter

e can be found only numerically (as it has been done for the

solution of related problems in Refs. [20,21]). However, for

steady-state conditions (defined by dx/dt = constant) the

effective stress parameter, e, is also a constant. So, the

conditions for steady-state growth are

e0¼

d ¼vm? vc

vc

;

ð9Þ

dx

dt¼ const;

e ¼ const.

ð10Þ

Here we restrict the analysis to such steady-state conditions

allowing one to analytically determine the basic factors

that affect the crystal growth rate.

In order to proceed with the derivation, let us denote by

sGthe time required to form one crystalline monolayer. We

assume further that the effective stress parameter, e, is given

by the solution of the stress relaxation equation via

?

for Maxwellian relaxation or

?

for the stretched exponential relaxation mechanism. In Eq.

(12), b is a parameter specifying the relaxation behavior of

the particular liquid analyzed. In other words, we assume

that the time of formation of one monolayer determines

the effective time-scale at which stress relaxation may

occur.

The time sGis determined by the growth rate Eq. (8).

Then, in order to allow for the formation of one mono-

layer, dx has to be set equal to dx ffi d0(d0is the diameter

of an ambient phase unit). Consequently, employing the

expression for the growth rate Eq. (8), sGcan be expressed

via

?

In this way, in the case of steady-state growth, Eqs. (11) or

(12) and Eq. (13) allow us to determine the two unknown

quantities e and sGprovided such solution does indeed exist.

e ¼ e0exp

?sG

sR

?

ð11Þ

e ¼ e0exp

?

sG

sR

?b

()

ð12Þ

d0

sG¼

D

4d0

1 ? exp

?Dl ? e

kBT

???

.

ð13Þ

436

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

Page 4

In order to prove the existence of such a solution, we

rewrite Eq. (13) in the form

?

The auxiliary function f(sG) is equal to zero for sG! 0, it

increases monotonicly with increasing sG and tends to

infinity for sG! 1. Consequently, for any set of parame-

ters (in particular, for any value of the ratio ð4d2

exists one and only one solution for sGand, according to

Eqs. (11) or (12), one and only one solution for the effective

stress parameter, e.

4d2

D

0

¼ fðsGÞ ¼ sG 1 ? exp ?Dl ? eðsGÞ

kBT

???

.

ð14Þ

0=DÞ), there

3. Decoupling, elastic stresses and crystal growth

In the analysis of the possible effect of elastic stresses on

crystal growth in glass-forming liquids, we consider first

the case that stress relaxation is governed by Maxwell’s

equation. Eqs. (11) and (13) yield then

?

Employing further the relation [22]

4d2

D

0

?

¼ ?sRln

e

e0

? ?

1 ? exp ?Dl ? e

kBT

????

.

ð15Þ

sR¼gð1 þ cmÞ

Em

ð16Þ

between Maxwell’s relaxation time, sR, and viscosity, g, we

arrive at

?

Eq. (17) allows one to determine the effective stress param-

eter e in dependence on temperature and, consequently, the

magnitude of the effect of elastic stresses on crystal growth

in glass-forming liquids.

As a first general result of the analysis of Eq. (17), we

conclude that, as far as the Stokes–Einstein (or Eyring)

equation [17],

D ¼kBT

4

Emd3

0

ð1þcmÞkBT

?

kBT

Dgd0

??

¼ ?ln

e

e0

? ?

1 ? exp ?Dl?e

kBT

??hi

.

ð17Þ

gd0;

ð18Þ

is fulfilled, elastic stresses cannot have any effect on crystal

growth. The ratio (e0/e) has to be sufficiently large in order

that above equation can be fulfilled, i.e. e must be small as

compared with e0.

Indeed, introducing the characteristic time of molecular

jumps, s, via the relation [17]

s ¼d2

we arrive with Eq. (16) at [6]

0

D;

ð19Þ

sR

s¼

kBTð1 þ cmÞ

Emd3

0

!

Dgd0

kBT

??

.

ð20Þ

For liquids of sufficiently low viscosity, the Stokes–Ein-

stein/Eyring equation is fulfilled and the characteristic

times of molecular motion, s, are of the same order of mag-

nitude as Maxwell’s relaxation time, sR. Hence, one can get

the following estimate for the Young modulus of the liquid,

Em(see also [23])

Emd3

ð1 þ cmÞffi kBT.

Substitution of Eqs. (18) and (21) into Eq. (17) yields

? ?

resulting in

? ?

In this way, the effect of elastic stresses can be ignored as

far as Eqs. (18) and (21) are fulfilled.

However, when the decoupling temperature, Td, is

approached in cooling the liquid, molecular motion in

the melt (which determines the rate of crystal growth)

changes from liquid-like to solid-like. At Td, viscous relax-

ation and molecular motion decouple, the ratio of the char-

acteristic time-scales sR/s increases exponentially with

decreasing temperature, the Stokes–Einstein/Eyring equa-

tion does not hold any more and, according to Eq. (20),

the ratio (kBT/Dgd0) tends to zero. In order to analyze

the dependence of the ratio e/e0in this alternative case,

we employ Eq. (20) and rewrite Eq. (17) as

?

Since, below Tdwith a further decrease of temperature, the

ratio (s/sR) decreases exponentially, the parameter e tends

to e0. This result is independent on any assumptions con-

cerning the temperature dependence of the stress parame-

ter, e0, itself, it is exclusively a consequence of the

breakdown of the Stokes–Einstein/Eyring equation below

Td.

A similar analysis, as outlined above in detail for normal

growth, can be made for any other modes of growth. The

only difference in the resulting equations is that the number

‘4’ in Eqs. (23) and (24) has to be replaced by (4/f). Since f

has generally finite positive values less than one, the con-

clusions remain the same. Note also that a similar analysis

with qualitatively equivalent results can be easily per-

formed for the case of stretched exponential relaxation,

as given by Eq. (12). Obviously, the only difference is that

a replacement lnðe0=eÞ ) ½lnðe0=eÞ?1=bhas to be made in Eq.

(17). We thus conclude that the results of the present anal-

ysis are independent on the specific mechanisms of crystal

growth and stress relaxation.

Analogousconclusionshavebeenderivedinourprevious

analyses of stress effects on nucleation. Similarly to crystal

growth, elastic stresses do not have any effect on nucleation

as far as the Stokes–Einstein/Eyring equation is fulfilled.

However, in the temperature range below the decoupling

0

ð21Þ

4 ffi ?ln

e

e0

1 ? exp ?Dl ? e

kBT

????

ð22Þ

?ln

e

e0

? 4i.e.

e ? e0e?4.

ð23Þ

4

s

sR

?

¼ ?ln

e

e0

? ?

1 ? exp

?Dl ? e

kBT

????

.

ð24Þ

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

437

Page 5

temperature, Td, elastic stress effects may be of considerable

importance. It follows that elastic stresses can be of signifi-

cance both for critical crystallite nucleation – as shown

earlier[5,6,15,16]–andforthedescriptionofcrystalgrowth,

in both cases, in the same range of temperatures, T 6 Td.

Similarly, as for nucleation, the approach developed here

allows one to account for a continuous transition from a

behavior of the ambient phase typical for a Newtonian

liquid (for temperatures at and above the temperature of

decoupling) to a behavior typical for a Hookean solid (for

temperatures near and below Tg). In order to demonstrate

these results, we will analyze in the next section, as an exam-

ple, stress effects on crystal growth in lithium disilicate glass.

4. Application: crystal growth rates in lithium

disilicate glass

According to Eq. (9), for lithium disilicate the effective

stress parameter e, determining the effect of elastic stresses

on crystal growth, does not exceed, in the vicinity of Tg, a

value of about 4% of the thermodynamic driving force, Dl.

By this reason, we are aware that the magnitude of the

effect of elastic stress on crystal growth will not be strong

for this particular system. Despite this disadvantage, the

general features of the theory developed above can be

clearly demonstrated. The choice of this particular system

has the advantages that crystal growth rate data are avail-

able over a wide range of temperature and, moreover, we

can compare stress effects on crystal growth with the results

of the analysis of stress effects on nucleation performed ear-

lier by us for this particular system [15,16]. The availability

of such additional information will allow us to derive some

additional general conclusions not only concerning the rel-

ative magnitudes of the effect of stress on nucleation, on

one side, and crystal growth, on the other, but also of more

general nature.

Assuming, again, that stress relaxation proceeds in

accordance with Maxwell’s law (cf. Eq. (11)), Eq. (17)

allows us to determine the ratio (e/e0) and, as a next step,

the value of e as a function of temperature. The parameters

of lithium disilicate glass are taken from the previous anal-

yses of the effect of elastic stresses on crystal nucleation

[15,16]. By this reason, we only present here the final results

without a detailed discussion on how the respective depen-

dencies were obtained.

The thermodynamic driving force of crystal growth in

the absence of stresses, Dl, was taken from the experimen-

tal investigations performed by Takahashi and Yoshi [29].

The molar mass equals M = 150 g mol?1, the density of the

glass and the crystalline phase are qglass= 2.35 g cm?3and

qcrystal= 2.45 g cm?3, respectively, resulting in a value of

the misfit parameter equal to d = 0.04255. The characteris-

tic size of the building units of the crystalline phase equals

d0= 5.88 · 10?10m. The melting temperature is Tm=

1307 K and the glass-transition temperature equals Tg=

728 K. For the Poisson ratios of glass-forming melt and

crystal, we take cm= cc= 0.23.

To the best of our knowledge, experimental data on the

dependence of Emon temperature are not available for

the system under consideration. By this reason, we have

set the Young’s modulus of the ambient phase equal to

the modulus of the crystal, Ec= 76 GPa. There exists a

variety of experimental data and general theoretical argu-

ments indicating the existence of a considerable increase

of Young’s modulus of glass-forming melts with decreasing

temperature near the respective temperature of vitrifica-

tion, Tg [6,17,24–28]. The incorporation of such effects

(as done, for example, in Ref. [6]) does not affect the results

of the present analysis and is, therefore, omitted here.

The temperature dependencies of the viscosity (in Pa s)

and the diffusion coefficient (in m2s?1) are interpolated

via the following equations

logg ¼ ?2:37 þ3248:6

and

T ? 500

ð25Þ

logD ffi ?7:57 ?

3941

T ? 452.

ð26Þ

In both equations, the temperature is given in Kelvin.

For the determination of the effective diffusion coeffi-

cient, which determines the rate of crystallization as a func-

tion of temperature, we employed measurements of the

nucleation time-lag. Such data are available for tempera-

tures in the range 693 K 6 T 6 763 K (Tg? 725 K). The

results were interpolated in such a way as to fulfill the

Stokes–Einstein/Eyring relation for temperatures above

Td= 1.2Tg. This approach is corroborated by the compar-

ison of diffusion coefficients calculated from viscosity via

the Stokes–Einstein/Eyring equation and from growth

rates of lithium disilicate crystals in the melt of the same

composition [30]. They indicate the possible existence of

decoupling of diffusion and relaxation near Td= 1.2Tg

(see also Fig. 4). The Eyring ratio (kBT/gDd0) versus tem-

perature, shown in Fig. 1, demonstrates significant devia-

tions from the Stokes–Einstein/Eyring behavior for low

temperatures.

The ratio (e/e0) and the effective stress parameter e ver-

sus temperature, obtained with Eqs. (25) and (26), are

shown in Figs. 2 and 3 by full curves. One first sees that,

for temperatures in the range 0.46 6 T/Tm6 0.49, the sys-

tem switches from a liquid-like behavior (where elastic

stresses are negligible) to a behavior that is typical for a

Hookean solid. However, this transition is found here in

a temperature range, where growth rates cannot be exper-

imentally measured in reasonable time-scales (note that the

glass-transition temperature for lithium disilicate is Tg/

Tmffi 0.557). Therefore, for the considered system, elastic

stresses seem to have no sizeable effect on crystal growth

rates, U, in the range of temperatures where U can be

measured.

However, in the present analysis we employed values of

the diffusion coefficient calculated from data on the time-

lag for nucleation. In this procedure, classical nucleation

438

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

Page 6

theory has been employed in order to determine the size of

the critical clusters as a function of undercooling and nor-

mal growth was assumed in describing aggregation pro-

cesses to clusters of critical sizes. Latter assumption has

no effect on the results of the computations of the temper-

ature dependence of effective stress parameter. Indeed,

employing time-lag data for the determination of the

kinetic coefficients and assuming other modes of growth,

actually the product Df in Eq. (2) is computed and in Eq.

(17), we have to replace D by Df as well leaving the results

of the computations of the stress parameters unchanged.

However, taking into account well-known limitations of

the classical nucleation theory in application to crystalliza-

tion [31–33], we have to check whether the estimates of

the effective diffusion coefficients, obtained in this way,

describe crystal growth rates correctly. For these purposes,

in Fig. 4 the linear growth rate U, determined by different

methods, is shown as a function of temperature. The

dashed curve (1) represents the results if viscosity-data

(according to Eq. (25)) are employed for the determination

of the diffusion coefficient and growth rates. The full curve

gives the respective data obtained with the diffusion coeffi-

cient – derived from time-lag measurements (cf. [15]) –

without (full curve (2)) and with (circles (3)) an account

of elastic stress effects. As we expected and in agreement

0.50.60.7 0.8 0.9 1.0

0

1

Tg

kBT/D do

T/Tm

Fig. 1. Eyring ratio, (kBT/gDd0), as a function of temperature. To a good

approximation, for T > Tdffi 1:2Tg, the Stokes–Einstein/Eyring equation

is fulfilled. For T 6 Td, significant deviations are found and the Eyring

ratio tends to zero with decreasing temperature.

0.38 0.40 0.420.440.460.480.500.520.540.56

0.0

0.2

0.4

0.6

0.8

1.0

ε ε / /ε εo o

T/Tm

Fig. 2. Ratio (e/e0) as a function of temperature. The full curve shows the

results of the computations if the diffusion coefficient is determined via Eq.

(26). The dashed curves correspond to possible corrections of the diffusion

coefficient by a factor 102(see text). In both cases, the approach of the

effective stress parameter to zero is found for temperatures, where Young’s

modulus retains values typical for the crystal.

0.380.400.420.440.460.480.500.520.540.56

0.00E+000

5.00E-022

1.00E-021

1.50E-021

2.00E-021

ε ε, J

T/Tm

Fig. 3. Effective elastic stress parameter e versus temperature. The full

curve shows the results of the computations if the diffusion coefficient is

determined via Eq. (26). The dashed curves correspond to possible

corrections of the diffusion coefficient by a factor 102(see text).

0.5 0.60.70.80.9 1.0

1E-19

1E-17

1E-15

1E-13

1E-11

1E-9

1E-7

1E-5

1E-3

Tg

U, m /s

T/Tm

1

2

3

4

Fig. 4. Crystal growth rate, U, versus temperature. Dashed curve: viscosity

(Eq. (25)) and the Stokes–Einstein/Eyring equation (Eq. (18)) are

employed for the determination of the diffusion coefficient; full curve:

data obtained with the diffusion coefficient – derived from time-lag

measurements (cf. [15]) – without (full curve) and with (circles) an account

of elastic stress effects. In addition, available experimental data are shown

[30]. In order to describe adequately the experimental data in the vicinity

of Tg, the diffusion coefficient should increase by a factor of the order 102.

Available experimental data denoted by stars.

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

439

Page 7

with the results of computations shown in Figs. 2 and 3,

elastic stresses do not have any sizeable effect on the growth

rate in the considered range of temperatures. In addition,

available experimental data on crystal growth rates are

shown [30].

Deviations of experimental crystal growth data and pre-

dictions in the range of temperatures 0.8 < T/Tm< 1,

employing viscosity measurements described by Eq. (25),

can be resolved easily: the authors, which performed the

analyses in this range, report lower (by a factor 1.5–2) val-

ues of viscosity for their systems. Consequently, employing

their viscosity data, their growth data are described appro-

priately by diffusion coefficients determined via the Stokes–

Einstein/Eyring equation. Fig. 4 shows also that, in order

to describe adequately the experimental growth rate data

in the vicinity of Tg, the diffusion coefficient – obtained

from time-lag data via classical theory – has to be increased

by a factor 102to reproduce the experimental results. The

origin of this deviation in the considered range of temper-

atures can be twofold. First, one can suppose that the clas-

sical nucleation theory does not give an appropriate

description of the properties and the parameters of the crit-

ical clusters. This point of view is supported by the dra-

matic deviations between the predictions of the classical

theory and experimental rates of critical crystallite forma-

tion (cf. Refs. [16,31–33]). Such point of view gets addi-

tional support also from a generalization of Gibbs’

approach as developed in recent years (cf. Refs. [34–36]

for an overview). On the other hand, the mentioned theo-

retical analyses and investigations by other authors,

discussed there, show that the kinetic parameters determin-

ing nucleation and growth processes may depend on cluster

size. In this way, the existing deviation in the estimates of

the effective diffusion coefficient can be considered a reflec-

tion of existing problems in the description of crystalliza-

tion kinetics which do not have found a satisfactory

general solution so far.

In analyzing the effect of elastic stresses on nucleation in

lithium disilicate, in Ref. [16] we determined the driving

force of crystallization and the specific interfacial energy

from experimental data on time-lag and nucleation rates.

In our analysis, both driving force and surface energy were

considered as unknown functions of temperature. These

results can be employed to determine the diffusion coeffi-

cient from time-lag data, again, but in an alternative way

as compared with classical nucleation theory. The resulting

dependence of diffusion coefficient on temperature is shown

in Fig. 5 by a dashed curve. It turns out that it is one order

of magnitude larger than the estimates obtained via the

classical nucleation theory.

Provided we assume that time-lag data and their inter-

pretation by the classical theory underestimate the diffu-

sion coefficient by the mentioned factor, then we have to

repeat the computations, but with higher values of the dif-

fusion coefficients. The results of such computations are

shown in Figs. 2 and 3 by dashed curves. Qualitatively,

the results of the previous computations, given in Figs. 2

and 3 by full curves, are not changed, however, the respec-

tive curves are moved into a range of considerably higher

temperatures (see the dashed curves in Figs. 2 and 3). Such

a modification of the values of the diffusion coefficients

leads, consequently, to a similar transition of the effect of

elastic stresses both on growth and also on nucleation (cf.

the analysis in Ref. [15]).

5. Discussion

In the present analysis, we developed a general method

to treat the effect of elastic stresses on crystal growth taking

into account both stress evolution and stress relaxation.

We assumed normal growth and Maxwellian relaxation.

However, the procedure developed can be similarly per-

formed for any other mechanism of crystal growth and/

or relaxation with qualitatively equivalent results.

We have shown that in the range of temperatures where

the Stokes–Einstein/Eyring equation is fulfilled, stresses

relax too fast to allow for a significant influence on the

growth rates. However, as soon as this condition is vio-

lated, elastic stresses may be important for the description

of crystal growth and have, in general, to be accounted for.

In the example analyzed here, elastic stress effects are of

importance only in temperature ranges where the growth

rate is too low to be experimentally detectable. In this

way, the question arises, for which classes of systems elastic

effects may be of particular importance?

As mentioned above, as soon as the Stokes–Einstein/

Eyring equation is fulfilled, elastic stresses cannot have a

significant effect on crystal growth. Consequently, the

higher the deviations from the Stokes–Einstein/Eyring

0.520.530.540.550.560.57 0.58 0.59

-24

-23

-22

-21

-20

-19

log(D, m

2/s)

T/Tm

Fig. 5. Effective diffusion coefficient, determining the rate of crystalliza-

tion, calculated from nucleation time-lag data employing the classical

nucleation theory (full curve) and data from a fit of both driving force of

crystallization and specific interfacial energy as performed in Ref. [16]

(dashed curve).

440

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

Page 8

equation immediately below the decoupling temperature,

the higher is the possible effect of elastic stresses. With

respect to the results shown in Fig. 1, we can reformulate

this statements as follows: the effect of elastic stresses

increases with increasing rate of approach to zero of the

Eyring ratio kBT/(Dgd0).

Second, the effect of elastic stresses depends significantly

on the ratio of the thermodynamic driving force Dl(Tg)

and the elastic stress parameter, e0. An overview of the

spectrum of possible values of this ratio for different classes

of glass-forming liquids is given in Ref. [37]. As it turns out,

the stress parameter e0can be comparable in magnitude

with Dl. This is a necessary condition for a significant

dependence of the growth rate on stress.

Third, despite the uncertainties connected with the

applicability of the classical nucleation theory to crystalli-

zation of glass-forming liquids, one can expect that the

effect of stresses on nucleation will be, in most cases, much

more significant than on growth. Indeed, the steady-state

nucleation rate depends exponentially on the driving force

squared via

Ist¼ I0exp

?

r3v2

c

kBTðDlÞ2

!

;

ð27Þ

where r is the nucleus/melt surface energy and I0a param-

eter determined mainly by kinetic factors. In contrast,

according to Eq. (2), the dependence of the growth rate

on the driving force and thus, on the effect of stresses, is

rather weak. Elastic stresses can have an effect on growth

only as soon as e0is comparable in magnitude with Dl.

For the case of lithium disilicate, for example, elastic stres-

ses can reduce the growth rate by less than 2% at 420 ?C

(this estimate results if one determines e0via Eq. (9) with

Em= Ecneglecting stress relaxation). However, as shown

in Refs. [15,16], the effect on the nucleation rate can reach

several orders of magnitude. Hence, if elastic stresses have

a significant effect on growth rates, homogeneous nucle-

ation in the bulk will be totally suppressed, in general. Vice

versa, in glasses which crystallize by homogeneous nucle-

ation (i.e. in glasses having a reduced glass-transition tem-

perature Tg/Tm6 0.6 (cf. [38])) elastic stresses will not

affect significantly the growth rates.

In more detail, the kind of dependence of the growth

rates on the thermodynamic driving force is illustrated in

Fig. 6. In this figure, the value of the thermodynamic

factor,

?

DHm

kBTm

Tr

Uth¼ 1 ? exp

?Dl

kBT

?

?

¼ 1 ? exp ?

?ð1 ? TrÞ

??

;

ð28Þ

in the expression for the growth rate, Eq. (2), is shown as a

function of (a) the reduced thermodynamic driving force,

Dl/(kBT), and (b) the reduced melting entropy, DSmr=

DHm/(kBTm). In Eq. (28), DHmis the melting enthalpy

per particle, Tr= T/Tmand the Volmer–Turnbull equation

(5) was employed. In Fig. 6(b), the full curve shows the

respective values of this quantity at Td, while the dashed

curve refers to the value of Uthat Tg. Here we assume

Tg= (2/3)Tm and Td= 1.2Tg. The graph presented in

Fig. 6(a) is valid for any value of the reduced melting entro-

py, DSmr= DHm/(kBTm). In order to find out the tempera-

ture, corresponding to a given point along the curve, one

has to know the value of DSmr(which is a characteristic

of the substance under consideration).

As we have shown above, elastic stresses can affect crys-

tal growth rates only at temperatures below the decoupling

temperature T 6 Tdffi 1.2Tg. It is evident from Fig. 6(a)

that, when Tdcorresponds to the right part of the plot

where the growth rate weakly depends on Dl, elastic stres-

ses can diminish the growth rate only if the values of the

stress parameter e0are close to or exceed the thermody-

namic driving force. However, if Tdcorresponds to the left

part of the plot, then the growth rate is highly sensitive to

slight changes of Dl and an effect of elastic stresses can be

expected to occur even then if the magnitude of e0is consid-

erably smaller as compared with Dl.

Since the temperature scale in Fig. 6(a) depends on DSmr

we plotted Uth at Td and Tg as a function of DSmr

(Fig. 6(b)). Hereby we estimated Tg/Tmas 2/3 taking into

account the above made conclusion. The interval of the

DSmrvariation, employed in Fig. 6(b), is typical for silicate

glasses.

According to the solid and dotted lines in Fig. 6(b), the

Uthvalues at Tdand Tgand, consequently, the position of

Tdand Tgon the plot of Fig. 6(a) strongly depend on DSmr.

Since the values of Uthin the range from 0.0 to 0.8 corre-

spond to a strong dependence of the growth rate on the

thermodynamic driving force (see Fig. 6(a)), the effect of

elastic stresses on growth rates can be expected to be signif-

icant in glasses with DSmrlower than about 5. Therefore,

081012

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

b

Td

Tg

1-exp(-Δμ Δμ / kBT)

Δ ΔHm/kBTm

051015202530

0.0

0.2

0.4

0.6

0.8

1.0

a

Tm

T/Tm

1-exp(-Δμ/kBT)

Δμ/kBT

246

Fig. 6. Value of the thermodynamic factor, 1 ? exp(?Dl/(kBT)), in Eq.

(2) versus (a) reduced thermodynamic driving force, Dl/(kBT), and (b)

reduced melting enthalpy, DHm/(kBT). In (b) the full curve shows the

respective values of this quantity at Td, while the dashed curve refers to its

value at Tg. Here Tgis taken to be equal to Tg= (2/3)Tmand Td= 1.2Tg.

J.W.P. Schmelzer et al. / Journal of Non-Crystalline Solids 352 (2006) 434–443

441

Page 9

the probability of having important elastic stress effects

increases with decreasing DSmr. Taking into account

Jackson’s [39] criterion for crystal growth mechanisms,

we can suppose that normal growth is more affected by

elastic stresses than screw dislocation and 2D-surface

nucleation mediated growth. By this reason, we performed

the respective analyses here explicitly for the case of normal

growth.

In the present analysis, we considered the effect of elastic

stresses on the growth of crystals of macroscopic sizes. The

results can also be employed for the analysis of growth of

crystallites of near-critical sizes taking into account, in

addition, interfacial contributions to the thermodynamic

potential. In this case, we have to add a term rA into

Eq. (4) where r is the specific interfacial energy and A

the surface area of the growing aggregate. As the result,

the effective driving force of crystal growth would contain

then an additional term due to interfacial effects propor-

tional to ?n1/3.

6. Possible modifications and generalizations

In the present and preceding papers [5,6,15,16], we ana-

lyzed the effect of elastic stress evolution and relaxation on

nucleation and crystal growth. This analysis was based on

two basic assumptions: (i) The state of the newly evolving

phase does not depend on the size of the aggregates consid-

ered and is widely equivalent to the state of the stable mac-

roscopic phase. (ii) Elastic stresses are due to misfit effects

and grow linearly with the size of the newly evolving phase

(Nabarro model). The basic result of the analysis is that –

as soon as the Stokes–Einstein/Eyring equation is not ful-

filled – elastic stresses may have a significant effect both

on nucleation and cluster growth. The magnitude of this

effect depends, of course, on a variety of additional factors,

however, the principal possibility of such effect has always

to be taken into account.

Posing the question about possible modifications and

generalization of the theory developed, one has to answer

the question whether the assumptions of the theory are

generally fulfilled or not. A detailed analysis of experimen-

tal data on nucleation of glass-forming silicate melts proves

that the classical nucleation theory leads to serious prob-

lems in treating nucleation data [31–33]. These problems

can be resolved in the framework of a newly developed

approach to the description of nucleation and growth pro-

cesses [34–36,40,41]. The basic advantages of this so-called

generalized Gibbs’ approach consists in its ability to deter-

mine theoretically the dependence of the state parameters

both of critical, sub- and supercritical clusters on supersat-

uration and cluster size. Hereby it turns out that the state

of the clusters is essentially cluster size and supersaturation

dependent. As a consequence, a variety of thermodynamic

and kinetic parameters (surface energy, diffusion coeffi-

cients, driving force for cluster formation, growth rates)

become cluster size dependent as well. Such a size-depen-

dence occurs (due to the variation of the bulk properties

with cluster size) even then if one neglects surface energy

terms in the thermodynamic description. In this way, the

problem arises to treat theoretically stress evolution and

stress relaxation going beyond the classical theory of nucle-

ation and growth and relying on mentioned generalized

Gibbs’ approach. A first attempt to proceed in this direc-

tion can be found in Ref. [16]. The analysis shows that

the effect of elastic stresses on nucleation is increased as

compared with the case when basic assumptions of classical

nucleation theory are employed in the analysis.

Another limitation of the present model is connected

with the assumption that stresses are due to misfit effects

and can be described by Nabarro-type dependencies. As

already mentioned, in the case of segregation in solutions,

when the segregating component has a much higher diffu-

sivity than the ambient phase particles, stresses may evolve

growing proportional to (V ? V0)2. Here V is the volume of

the cluster of the new phase and V0some initial volume,

where such kind of stresses become dominant. A detailed

analysis shows that such model of stress evolution may

be effective for sufficiently large clusters when the above

mentioned condition is fulfilled. In such situations, several

time or length scale parameters (connected with the large

differences in the partial diffusion coefficients of the differ-

ent components of the system) exist and decoupling of

relaxation and growth is not connected with the Stokes–

Einstein/Eyring equation, but with the differences in the

diffusivities. Consequently, in a further generalization of

the theory one has to specify more clearly the conditions

at which Nabarro-type stresses will be eventually replaced

by mechanisms of stress evolution resulting in a more rapid

increase of stress with the size of the clusters or regions of

the newly evolving phase.

7. Conclusions

As far as the Stokes–Einstein/Eyring equation is ful-

filled, elastic stresses do not have any effect on crystal

growth. However, below the temperature of decoupling

of diffusion and viscous flow, when the Stokes–Einstein/

Eyring equation breaks down, stresses may have a signifi-

cant influence on crystal growth. The present results there-

fore challenge the widespread argument that internal

stresses relax too fast to affect crystal growth in glass-form-

ing liquids and could help to explain the often observed

lack of agreement between model predictions (which do

not take stresses into account) and experimental crystal

growth data.

Acknowledgments

The authors would like to express their gratitude to the

Deutsche Forschungsgemeinschaft (DFG), the State of Sa ˜o

Paulo Research Foundation FAPESP (Grants 03/12617-0;

99/00871-2; 2003/03575-2), CNPq and Pronex for financial

support. The critical comments of M. L. F. Nascimento of

LaMaV-UFSCar are fully appreciated.

442

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Page 10

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