Stress development and relaxation during crystal growth in glassforming liquids
ABSTRACT We analyze the effect of elastic stresses on the thermodynamic driving force and the rate of crystal growth in glassforming 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 glassforming 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 glasstransition 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 glassforming melt.
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Article: Nucleation Theory and Applications
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ABSTRACT: Abstract The peculiarity of an ensemble of repelling particles at low temperatures, where interac tion between nearest neighbors dominates its properties, is its polycrystalline structure with a density of particles lower than that of the crystal lattice. This result follows from computer simulations, model experiments with filling a container with hard balls, and the virial theorem. Moreover a polycrystalline structure is observed directly in a colloid solution. The question arises of whether such a polycrystalline state of an ensemble of repulsed atoms at low temperatures is thermodynamically stable, or is a metastable, glassy state that results from the kinetics of relaxation of this ensemble to low tempera tures and high densities. The disk model for repelling particles and the cell model for the configurations of these particles show that the glassy state of this system under indicated conditions is thermodynamically favorable. 6.1 Introduction We consider ensembles of particles with a sharply varying repulsive potential be  SourceAvailable from: sciencedirect.com01/1965; Pergamon Press.
 SourceAvailable from: Edgar Zanotto[show abstract] [hide abstract]
ABSTRACT: This article reviews recent findings on internal and surface nucleation in silicate glasses. The internal homogeneous nucleation rates sharply decrease and the induction times increase with the Tg/TL ratio (Tg is the glasstransition temperature and TL is the liquidus temperature). Only systems that have Tg/TL < 0.58 display measurable internal nucleation rates on a laboratory timescale. Numerous tests of the classical nucleation theory have demonstrated that the theory fails to describe nucleation rates in glasses quantitatively. Possible explanations for this failure are tested and discussed. Surface nucleation depends strongly on the surface quality, e.g. tips, cracks and scratches, elastic stresses, foreign particles and surrounding atmosphere. The mechanisms of surface nucleation are still not fully understood, but some of the key factors are now known and the surfacenucleation density can thus be controlled for the development of sintered glasses or glass ceramics.Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences 01/2003; 361:591612. · 2.38 Impact Factor
Page 1
Stress development and relaxation during crystal growth in
glassforming 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,
13565905 Sa ˜o Carlos, SP, Brazil
cInstitute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
dS.I. Vavilov State Optical Institute, ul. Babushkina 361, 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 glassforming 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 glassforming 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 glasstransition 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 glassforming 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 glasstransi
tion temperature, Tg, glassforming 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
glassforming liquids, in particular. Indeed, a variety of
experimental results, summarized in Refs. [5,6], demon
00223093/$  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.
Email address: dedz@power.ufscar.br (E.D. Zanotto).
www.elsevier.com/locate/jnoncrysol
Journal of NonCrystalline Solids 352 (2006) 434–443
Page 2
strates that elastic stresses may have a significant influence
on the course of phase transformations in glassforming
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 glassforming melts, when the segre
gating component has a diffusivity much higher than the
basic building units of the glassforming 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 glassforming
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 onecomponent 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 selfdiffusion 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 glassforming 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 crystalgrowth 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 timescales, sG/sR, where
sGis the time required to form one monolayer of the newly
evolving crystalline phase in steadystate growth, and sRis
a characteristic (Maxwellian) relaxation time of the matrix.
If the (shortrange) 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
glassforming liquids – decoupling of shortrange 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 glassforming 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 xaxis 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 NonCrystalline 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 selfdiffusion
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
steadystate conditions (defined by dx/dt = constant) the
effective stress parameter, e, is also a constant. So, the
conditions for steadystate growth are
e0¼
d ¼vm? vc
vc
;
ð9Þ
dx
dt¼ const;
e ¼ const.
ð10Þ
Here we restrict the analysis to such steadystate 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 timescale 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 steadystate 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 NonCrystalline 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 glassforming 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 glassforming 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 liquidlike to solidlike. At Td, viscous relax
ation and molecular motion decouple, the ratio of the char
acteristic timescales 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 NonCrystalline 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 glasstransition temperature equals Tg=
728 K. For the Poisson ratios of glassforming 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 glassforming 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 timelag. 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 liquidlike 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 timescales (note that the
glasstransition 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 NonCrystalline 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 timelag 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 wellknown 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 viscositydata
(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 timelag 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 0.80.91.0
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.
0.380.400.420.44 0.46 0.480.50 0.520.540.56
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.
0.380.400.42 0.440.460.48 0.500.52 0.540.56
0.00E+000
5.00E022
1.00E021
1.50E021
2.00E021
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).
0.50.6 0.70.80.9 1.0
1E19
1E17
1E15
1E13
1E11
1E9
1E7
1E5
1E3
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 timelag
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 NonCrystalline 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 timelag 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 timelag 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 timelag 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 timelag 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
0.52 0.530.54 0.55 0.560.570.580.59
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 timelag 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 NonCrystalline 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 glassforming 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 glassforming liquids, one can expect that the
effect of stresses on nucleation will be, in most cases, much
more significant than on growth. Indeed, the steadystate
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 glasstransition 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,
08 1012
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
b
Td
Tg
1exp(ΔμΔμ / kBT)
Δ ΔHm/kBTm
05101520 2530
0.0
0.2
0.4
0.6
0.8
1.0
a
Tm
T/Tm
1exp(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 NonCrystalline 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 2Dsurface
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 nearcritical 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 glassforming 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 socalled
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 sizedepen
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 Nabarrotype 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 Nabarrotype 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 glassform
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 03/126170;
99/008712; 2003/035752), CNPq and Pronex for financial
support. The critical comments of M. L. F. Nascimento of
LaMaVUFSCar are fully appreciated.
442
J.W.P. Schmelzer et al. / Journal of NonCrystalline Solids 352 (2006) 434–443
Page 10
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