Weinberg 1988

Weinberg 1988

(Parte 1 de 4)

Journal of Non-Crystalline Solids 108 (1989) 9-108 9 North-Holland, Amsterdam

Michael C. WEINBERG and Edgar D. ZANOTTO 1 Department of Materials Science and Engineering, University of Arizona, Tucson, AZ 85721, USA

Received 14 March 1988 Revised manuscript received 29 September 1988

A comparison is made between the experimentally determined and predicted (Classical Nucleation Theory) temperature dependence of the homogeneous crystal nucleation rate in glass. Literature data for several glass-forming systems are analyzed. It is shown that the standard assumptions which are invoked to replace the free energy of activation for transport across the liquid-crystal interface, AGD, in favor of a more easily measured parameter (viscosity) are not required. It is demonstrated that AG D may be found experimentally by determining the temperature dependence of the nucleation transient time. Furthermore, it is shown that precise thermodynamic data are not necessarily required in order to make comparisons between the theoretically predicted and experimentally determined temperature variation of the nucleation rate. It is concluded that the temperature dependence of the classical homogeneous nucleation rate is still a matter of controversy.

1. Introduction

Crystal nucleation in glass is currently a topic of keen interest. This stems from the fact that understanding and control of the factors which affect crystallization are not only essential for many technological applications \[1-3\], but also are important for the synthesis of novel glasses \[4,5\].

During the past decade it has been recognized that the existing knowledge pertaining to crystal nucleation in glasses is quite incomplete. James \[6\] has recently presented an excellent review of the status of crystal nucleation in simple silicate glasses, in which the following pertinent points were made.

Nearly ten years ago Rowlands and James \[7\] and Neilson and Weinberg \[8\] compared experi- mental crystal nucleation data for Li20.2SiO 2 with that predicted by the Classical Nucleation Theory (CNT). Both groups concluded that there existed two discrepancies between the experimen- tal results and CNT. First, the magnitude of the

1 On sabbatical leave from Universidade Federal de Sao Carlos, Brazil.

02-3093/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) observed nucleation rates was many orders of magnitude larger than the predicted rates. Second, although the temperature dependence of the nucleation rate could be fitted to CNT at high temperatures, at the lowest temperatures where nucleation measurements were made (T_< T~) the experimental points fell below the theoretical curve.

Subsequently, crystal nucleation measurements were performed in other simple silicate glasses (BaO- 2SiO 2 and Na 20.2CaO • 3SiO 2) which ap- peared to nucleate homogeneously, and the experi- mental results were compared with CNT \[9-1\]. For the latter systems the predicted nucleation rates, once again, were found to be orders of magnitude too small, but the temperature depen- dences of the nucleation rates were described well by theory in all cases. These findings led Zanotto and James \[12\] to re-examine the temperature dependence of the nucleation rate in the LizO- 2SiO 2 system. They performed careful measure- ments of the temperature dependence of the viscosity, and found that use of their viscosity data led to substantial improvement between ex- perimental and CNT results for the temperature variation of the nucleation rate. Hence. it is the currently held belief that CNT provides a good

100 M.C. Weinbergo E.D. Zanotto / Temperature dependence of the classical nucleation rate description of the temperature dependence of the homogeneous crystal nucleation rate in glasses.

The discrepancy between the experimental and theoretical values for the magnitude of the rate persists. However, James \[6\] has indicated that a small variation with temperature of the liquid- crystal surface tension could produce agreement between CNT and experiment for the nucleation magnitude, without destroying the accord in tem- perature dependence. This suggestion appears quite plausible, but to data unverified.

In the present work attention is directed upon the temperature dependence of the nucleation rate. In order to appreciate the approach taken herein, it is necessary first to review a portion of the standard arguments used to obtain the form of the CNT expression applied to crystal nucleation in glasses. The expression for the steady-state, homo- geneous crystal nucleation rate, I °, in its unmod- ified form, is given by \[13\]

I ° =A(T) exp(- \[W* +AGD\]/RT), (1) where W*, the thermodynamic barrier to nuclea- tion, is equal to the free energy/mol of forming a

critical nucleus, A(T) is a weakly temperature-de- pendent prefactor, and AGD, a kinetic barrier to nucleation, is equal to the free energy of activa- tion/mol for atomic diffusion across the liquid- crystal interface. Unfortunately, theoretical ex- pressions for AG D are not available. Hence, it has been common practice to introduce two separate assumptions in order to relate AG D to an easily measured experimental parameter; i.e. the viscos- ity. First, it is presumed that AG D can be associ- ated with the free energy of activation for ordinary diffusion. Hence, one writes

D = D O exp(-AGD/RT), (2) where D o is constant and D is the ordinary diffu- sion coefficient. Next, one invokes the Stokes- Einstein equation \[14\], which predicts D - (kT/71) (where 7/ is the shear viscosity). Via these two assumptions one obtains that the nucleation rate is inversely proportional to the viscosity.

Although there is an "intuitive feeling" that one possesses that I ° should be inversely related to ,/, neither of the latter two assumptions has ever been well justified. The Stokes-Einstein rela- tionship is highly accurate for large particles in liquids and has been shown to have predictive capability even for atomic motions in simple fluids, but its applicability in complex network-forming liquids has not been established. Furthermore, the identification of AG D with ordinary liquid diffu- sion processes is questionable. In any event, the important point that should be noted here is that these assumptions are not necessary. AG D can be experimentally determined if in addition to nucleation data, one can measure the temperature dependence of the transient nucleation time. Hence, experimental nucleation data can be com- pared with CNT in its "pristine" form.

Here nucleation data are analyzed using this approach. First, a description is presented discuss- ing how AGD can be determined from the temper- ature dependence of the transient nucleation time. Next, the available nucleation data are analyzed. It is demonstrated that precise thermodynamic data may not be necessary in order to assess the temperature dependence of the nucleation rate. Finally, in the discussion, some of the unresolved problems are mentioned.

2. Theory

The classical nucleation theory for condensed systems was derived by Turnbull and Fisher in 1949 \[15\] and gives the steady state nucleation rate

I ° as a function of the equilibrium concentration of critical nuclei N e and of the rate of molecular rearrangement O *,

I ° = Z. re*. D*. (3) In eq. (3) Z is the Zeldovich factor, given by

W'Nail~2 1 Z= 3rrRT \] n* ' (4) where R is the gas constant, N a is Avogadro's number, and n* is the number of atoms in a nucleus of critical size. The number of critical nuclei is given by a Boltzmann expression

Are* = N o exp( - W*/RT), (5) where N o is the number of sites per unit volume available for homogeneous nucleation; i.e., the total number of unit cells per unit volume, W* is the thermodynamic barrier to form a tool of criti-

M.C Weinberg, E.D. Zanotto / Temperature dependence of the classical nucleation rate 101 cal nuclei, and T is the absolute temperature. D* is given by

D* kT0* \[ -AGD), (6) = -- exp\[ where O* is the number of atoms in the surface of a critical radius, k is Boltzmann's constant and h is Planck's constant. For isotropic, spherical nuclei

16rr 0 3 W * - -- (7)

3 (AGv) 2' where o is the interfacial free energy and AG v the thermodynamic driving force per unit volume.

The rate of molecular rearrangement is usually assumed to be given by eq. (2), but the identifica- tion of AG D with AG n (the activation energy for viscous flow) and the use of the Stokes-Einstein equation have no firm theoretical sanction•

A more rigorous way to determine the kinetic barrier is by using the transient times, r. Accord- ing to Kashchiev \[16\],

4 = (8)

,/r 3Z2D *

This equation has been recently tested and proved to be accurate by the independent com- puter simulations of Kelton et al. \[17\] and Volt- erra and Cooper \[18\].

Since the rates of molecular rearrangement in transient times and crystal nucleation are theoreti- cally identical \[13\], the combination of equations

(3,5 and 8) gives

-W* ,0 expi-- t,, (9) 7r3Zr

It should be reiterated that the form of the nuclea- tion expression given by eq. (9) does not rely upon the use of additional assumptions to eliminate AGD. The only approximation which was made was the use of eq. (8) for r, and as indicated above this expression for r appears to be nearly exact. One may take the logarithm of eq. (9) to obtain

N° 167r°3 (10) ln(/°r) = In ¢raZ \] 3RT(AGv) 2 .

Since the first term on the right-hand side of eq. (10) is nearly temperature-independent, a plot of ln(I°-r) versus 1/T.(AGv) 2 should give a straight line *. The intercept could then be com- pared to the theoretical value and the interfacial energy of the nucleus/parent phase, o, could be calculated from the slope. In any case, if the predicted temperature dependence of I °, as given by eq. (9), is correct, a straight line is obtained.

3. Analysis and results

Equation (10) may be rewritten in the simple form

C ln(I°r) = E- -- (1)

T(AG) 2' where E and C are positive constants. Hence, the Classical Theory predicts that ln(I°r) should be a linear function of T-1(AG) 2. In order to con- struct such plots the difference in free energy between the crystal and liquid must be known as a function of temperature. This is equivalent to re- quiring knowledge of ACp(T) (heating capacity per unit volume of crystal - that of liquid) since

ACp(T') (1 - T/T')\] (12) _ f/m dT' AS m

In eq. (12), T m is the melting temperature and ASm is the change in entropy per unit volume upon crystallization. If one introduces the reduced temperature, T~ = T/T m, and substitutes eq. (12) into eq. (1), then one obtains ln(I°r) = E - C'T(-t(1 - Tr) 2

×{1-(1- Tr) 1 fl ACp (Tr,) dTr, } 2 • (1 -- Tr/Tr' ) AS m

03) where C' = CTm3ASm 2.

* Rigorously ln(l°rAG2/¢T) should be plotted to take into account the temperature term in Z. However, calculations were performed including the temperature dependence of Z, and it was found that the inclusion of this temperature dependence did not change our results on a qualitative level and, also, only had a small quantitative effect.

102 M.C. Weinberg, E.D. Zanotto / Temperature dependence of the classical nucleation rate it is convenient to make the following definitions: x- T~-a(1 - T~)-2; (14a)

(Parte 1 de 4)