**rodrigues 2007**

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(Parte **1** de 4)

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

Jurn W.P. Schmelzer a,b, Edgar D. Zanotto b,*, Isak Avramov c, Vladimir M. Fokin b,d a Institut fur Physik der Universitat Rostock, Universitatsplatz, 18051 Rostock, Germany b Vitreous Materials Laboratory, Department of Materials Engineering, Federal University of Sao Carlos, UFSCar, 13565-905 Sao Carlos, SP, Brazil c Institute of Physical Chemistry, Bulgarian Academy of Sciences, 1113 Soﬁa, Bulgaria d S.I. Vavilov State Optical Institute, ul. Babushkina 36-1, 193171 St. Petersburg, Russia

Received 20 August 2004; received in revised form 1 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 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 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 quantitatively 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 Tg and 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-

02-3093/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2006.01.016

* Corresponding author. Tel.: +5 16 3351 8527; fax: +5 16 3361 5404. E-mail address: dedz@power.ufscar.br (E.D. Zanotto).

w.elsevier.com/locate/jnoncrysol Journal of Non-Crystalline Solids 352 (2006) 434–443 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 experimentally and theoretically [7–9] that elastic stresses may qualitatively change the kinetics of growth of single clusters and of Ostwald ripening in glass-forming melts, when the segregating component has a diﬀusivity much higher than the basic building units of the glass-forming melt. This situation is similar to the results of Stephenson [10] in application 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 mentioned 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 evolving crystal phase. In such cases, the kinetics of cluster nucleation and growth can be treated similarly to nucleation and growth in one-component systems with appropriately chosen values of the eﬀective diﬀusion coeﬃcient and thermodynamic driving force [5,6,1,12]. 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 equation, which connects viscosity (governing relaxation processes) 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 Td ﬃ Tg [13,14]. Above Td, the Stokes–Einstein/Eyring equation is typically fulﬁlled, but below Td it 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 glassforming 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 analysis 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 development 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 sG is the time required to form one monolayer of the newly evolving crystalline phase in steady-state growth, and sR is 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 demonstrated 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 reaching 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–Einstein (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 inﬂuence 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 modiﬁcations 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

Here c is the volume concentration of the ambient phase particles in the crystalline phase, vc the volume per particle in the crystalline phases, d0 is a characteristic size parameter (diameter) of the basic structural units of the system (cf.

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 or the diﬀerence of the chemical potential per particle in both considered states of the system as [17,18]

U ¼ dx dt ¼ f D

Here kB is the Boltzmann constant and T the absolute temperature, and f is a dimensionless parameter describing the speciﬁc properties of the diﬀerent growth modes. For simplicity of the notation, we will assume here f = 1 corresponding to the case of normal growth. But the main results of this analysis are – as will be shown below – independent 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,i nt he description employing the relation d20 ﬃ Ds. 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

Elastic stresses can be incorporated into the above equation 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

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 experimentally determined data or, for the derivation of some more general conclusions, the Volmer–Turnbull expression

where DHm is 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]

UðeÞ ¼ en. ð6Þ

A substitution of this expression into Eq. (4) leads to the consequence that elastic stresses eﬀectively result in a decrease of the thermodynamic driving force of crystallization by the quantity e. Indeed, we can write Eq. (4) in the form

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

dt ¼ D

For purely elastic solids, we have e = e0, where e0 is a parameter that depends on the elastic constants and the densities of both phases. In general, the stress parameter e0 can be written as [1,19] e0 ¼ EcEmvc 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 e0 holds and the growth rate and the values of the eﬀective 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 timedependence 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

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 sG the 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 sG is 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 (d0 is the diameter of an ambient phase unit). Consequently, employing the expression for the growth rate Eq. (8), sG can be expressed via

In this way, in the case of steady-state growth, Eqs. (1) or (12) and Eq. (13) allow us to determine the two unknown quantities e and sG provided such solution does indeed exist.

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

In order to prove the existence of such a solution, we rewrite Eq. (13) in the form kBT

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 ð4d20=DÞ), there

exists one and only one solution for sG and, according to Eqs. (1) or (12), one and only one solution for the eﬀective stress parameter, e.

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. (1) and (13) yield then

Employing further the relation [2] sR ¼ gð1 þ cmÞ Em between Maxwell’s relaxation time, sR, and viscosity, g,w e arrive at

4 E d ð1þc Þk T k T

Eq. (17) allows one to determine the eﬀective stress parameter 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),w e conclude that, as far as the Stokes–Einstein (or Eyring) equation [17],

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]

we arrive with Eq. (16) at [6] sRs ¼ kBTð1 þ cmÞ

! Dgd0 kBT . ð20Þ

(Parte **1** de 4)