Structure and entropy change in amorphous materials

Journal of Non-Crystalline Solids 117/118 (1990) 183-186 North-Holland

183

Structure and Entropy Change in Amorphous Materials

Shalom Baer Department of Physical Chemistry, The Hebrew University 91904 Jerusalem, Israel

A variational relation between the equilibrium free energy and the pair potential is extended to changes of the system along paths of non-equilibrium states. Combined with a parametrized model for the radial distribution function, this relation leads to an expression for the configurational entropy change in an amorphous phase in terms of empirically determined radial distributions and a given pair potential. Applied in combination with the "structural diffusion" model to computer simulated amorphous Aluminium, it gives an estimate of the entropy difference between the liquid at melting and the amorphous solid at 273 K, which is of the order of the entropy of melting, implying a negligible residual entropy. The temperature dependence of the structural diffusion coefficient, as determined from structural data, exhibits a glass transition of the amorphous system. kBT, we suggest to present the configurational part of the energies of the system as a sum of pair contributions

1. Constraints and variational relations Formally, a system in a non-equilibrium state can always be viewed as a system in equilibrium under the constraint of certain fictitious external forces 1. This follows fi'om the principle of maximum entropy with constraints, which by the method of Lagrange multipliers produces the corresponding external potentials. Thus, let enumerate the sets of frozen-in states or configurations, given with probabilities p~, and let s enumerate the states of the system within each configuration, with E ~ the corresponding energies. The p~ being fixed but the states s being otherwise in thermal equilibrium, we have then P~ = TQ~ e-~¥~ '

~ = (ksr)-~ '

where the W~ are the external potentials of the fictitious constraining forces, Q~ = Y,exp(-l~E~,) $

~J(V~ + E~) --->~¢~(rij )

where the pair function t~(r) can be related to an efective pair potential u (r) by

~(r) = I~u (r).

•

(D

~.~ is the generalized partition function of the system, related to the free energy of the system, F, by I~F = -logQ. Without knowing ~ explicitly, we can make certain general assumptions about their nature. Viewing quenching from a liquid to a glass as a process producing constraints on molecular pair geometries, or more generally, bounds on pair potential energies relative to the thermal energy 0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)

(3)

Assuming (2), we have disregarded possible contributions of many body terms to the constraining field and have dumped all constraints into the pair function ~(r), or effective pair potential u (r). It follows from (2) that the configurational partition function, written in the classical form Q = (NI)-IS"'Sexp[- ~,¢~(rij)]drl""drN i
,

(4)

has the same functional dependence on the pair potential, whether the system is in free or in constrainted thermal equilibrium. Furthermore, this holds also for the variational relation 2

and Q = Zexp[-I~(v~+e~,)]

(2)

i
8(I~F)

=

~N[.pg(r,[ C~])SC~(r)d3r ,

(5)

where [~] denotes the functional dependence of the radial distribution g (r) on d~. Relation (5) holds necessarily for any variation 8(I) corresponding to a variation in the external field ~ or in T or in both. Thus, we can determine the variation of I~F along any path traversed by the system, whether in or out of equilibrium, provided we know the effective pair potential and can follow g (r) along this path. This becomes

184

S. Baer / Change in amorphous mater/Ms

feasable in particular if we can construct a model of g (r), paramtrized with a small set of parameters, and express the constrained functional variation ~b in terms of variations of these parameters.

2. Modelling the radial distribution The structural diffusion model3 provides a useful example of a concise parametrization of g (r). In this model the radial distribution is given explicitly in terms of a virtual lattice L in a 3 dimensional (or higher dimensional3c) "structure space" associated with the local atomic structure and a "width" function W(r) characterizing the loss of coherence between local structures at different points in space. In the simplest version of the model it is given by _ fD (r-to)

W(r) - [

0

, r > ro

, r
(6)

where r 0 is an exclusion diameter and D is a "structural diffusion" coefficient measuring the radial evolution of a diffusion-like process in structure space. The derived radial distribution function has two equivalent representations. One is a sum over the points of the reciprocal lattice to L: g (r) = ~ C v e -wb~ sinbvr v bvr

(7)

The other is a sum over the points of L: 1 pg(r) = (4~W) 2 is (4~ra is)-I { exp[-(r-ais)2/4W] + exp[-(r+a is)2/4W] } .

(7') From (7') it is clear that the short range peaks of g (r), positioned around a~t's, have each a dispersion of order 2W(ais). The transformation between (7) and (7') is given by the Poisson sum formula3a. The parameters implied in W and L can be varied so as to make a good fit of (7) with empirically obtained radial distribution functions of any condensed system. In principle (7) becomes asymptotically exact in the limit of L having a very large unit cell (encompassing the entire bulk system) and D becoming very small (reducing g (r) to a sum of 8-functions). In practice, a good fit is obtained with very simple lattices. An example of such a fit is given by Lopez and Silbert4 who used one and the same L (a distorted fee lattice) and a fixed r0 together with a variable D = D (T) to obtain a good fit of (7) to empirical g (r) data of Aluminium over a range of 1000K, from the normal liquid down through the supercooled liquid to a quenched amorphous phase5'6 .

3. Entropy change and variation of g (r) The variation of D over the given range of temperatures defines a path from the stable liquid to the metastable amorphous phase. Along this path we can put g ( r ) - g(r,D), ~ ( r ) - ~(r,D) and performing the variation (5) between two states (1 and 2) on the path, we obtain 2 :

1

ou

1

2

The second equality in (9) follows through integration by parts. The firstterm on the rhs of this equality is seen by (3) to be (~E)2-([3E) 1 (E - a configurational energy) and comparing with the lhs we obtain for the configurational entropy change 2 kB (r,D) d3r . (10) Note that with negligible error (10) holds for both a constant volume and a constant pressure change, because of the negligible difference between energy and enthalpy changes in a condensed system at low pressures. The path defined in (10) by a variation in D can be understood in a more general sense as including possible simultaneous variations of the lattice L or even of the effective pair potential7 . To obtain the value of the residual entropy of an amorphous material, we now substitute (7) (or (7'), depending on rates of convergence of the series) into (10) and obtain Og(r,D)/OD analytically. Then, given L, r0 and the parameters Do, tx determining D (T), the integrations in (10) can be performed numerically. Putting t~(r) = [3u (r) and performing first the the spatial integration, we can define a generalized heat capacity Cs(D) which is the differential increase in the energy (heat) content of the system per unit increase in D along the path connecting the amorphous and liquid state. Thus we have for the generalized specific heat per particle Cs(D) = 2 ~ S u ( r ) ~ p ~ , D ) 0

r2dr .

(11)

The values of this function, obtained for Aluminium by numerical integration of the rhs of (11), using u(r) obtained by Rautioahos from calculations based on pseudopotential theory, are plotted in fig. 1 for a range covering the entire set of simulation data from the amorphous solid to the liquid well above melting. We see here that Cs(D)

S. Baer / Change in amorphous materials changes smoothly throughout this range. By (11) and (10) we write now the entropy change per particle in the form

Dt

This is in full analogy to the expression for the entropy change in terms of the ordinary specific heat C (r) AS = rf -C -(T) dT . Tt" T

(12')

Moreover, as in (11), C(T) could be obtained from structural data by

C(T) = 2x'~u(r) Opg(r'T) r2dr . tIz o

(11')

However, our modelling of g (r) provides a convenient expression for g (r) = g (r,D) whereas it is very difficult to construct a comparable convenient expression for g (r ) = g (r,T), - the temperature dependence of the radial distribution. Obviously, we have dD c ( r ) = Cs(D) dT.

(13)

Examining the D (T) data one can recognize two distinct sets of points fitting a smooth curve obtained by combination of two different functions of T (see fig. 2) representing, respectively, a liquid and a solid-like branch. We have chosen as a fitting formula D = 1+ c Do

+ A exp o e ( r - r g l J

(14)

where Dola = 1.9.10 -3, c = 2.2, A = 5.4, b = 0.4 and Tg = 0.86 Tm, given the unit length of L, a = 2.8/~ and the melting temperature Tm =933 K. Oc(x) is an integrated "rounded" step function, obtained by integration of a Fermi-type distribution, and is defined by O~(x) = e log(1 + eX/e).

(15)

4. Results and discussion As seen from fig. 2, a certain structural change is recognizable in the temperature range around a T = Tg, and we can identify it with some kind of glass transition. This transition (corresponding also to a gradual change from a supercooled liquid to an amorphous solid) is characterized by a pronounced change in the slope dD/tiT. This change can serve as a criterion for a glass transition. Compared with the Wcndt-Abraham OVA) criterion9 of change of the

185

slope of gmin/gmax, the ratio of the minimnm to the maximum of g (r), as a function of T, we see that both criteria are based on characteristics of the temperature dependence of g (r). However whereas the WA criterion is based on a very restricted, though pronounced feature of g (r), the one offered here is based on the properties of the entire set of g (r) values as expressed by the function D (T). Using (14) we obtain riD/tiT analytically and hence from (13) the numerically evaluated C (T), which is plotted in fig. 3 in the temperature range of the available D (T) data. Having obtained C(T), we integrate (12) between T = 273 K to T = Tm so as to obtain the configurational entropy change of Aluminium from the amorphous solid to the liquid melt. We find AS = 2.9 Cal/K Mol, which is of the order of the entropy of melting ASm = 2.73 Cal/K Mol. This result is based on limited D (T) data and the interpolation formula (15). If we could ignore the errors involved in our estimate, as well as the differences between the contribution of vibrations in the amorphous and crystalline phase, we could obtain the absolute entropy of amorphous Aluminium at T = 273 K from ASm-AS. However, this gives a negative value which is an indication of the magnitude of the errors involved. One source of error may lie in the inaccuracy of our parametrization, which does not reproduce g (r) sufficiently faithfully over the wide range of temperatures covered by the given data. This can be seen also from the obtained C (T) curve (fig. 3). Although a rise in the heat capacity with decreasing T is a general feature of supercooled liquids, we have here a rise of C(T) to its maximum already around T = Tin. We believe that such errors could be rectified by a more refined treatment of the parametrization of g (r), which should take into account a possible change of the virtual lattice L and include also an r independent but T dependent Debye-Waller type term contribution to the width function W(r) (equ. (6)), due to the vibrational motion of the atoms. Another possible error could arise from the fact that all foregoing calculations are based on a single effective pair potential u (r) used for both the liquid and amorphous phase (see (11)). However by definition (3) the effective potential must include the effect of constraints imposed by the quenching process on the amorphous phase. It has been suggested by Jlickle1° that a "constrained" effective potential can be related to an unconstrained potential via a certain "fictive temperature" T*: For T > T* an unconstrained potential u (r) is effective while for T < T* a constrained potential (to be denoted by ~(r)) is effective and should correspond to a practically frozen configuration. By (4) and (5) this implies that ~b(r) should remain fixed in this temperature range. Thus, by (3),

186

S. Baer / Change in amorphous materials ~(r) = kBTCp(r) = T--~--u(r), .

"°,,,,

240 a~

Oleo o °et°eeeeoeo e

(~

°°°°e°eeooeoeo

X

(..) IZO

Ot -(3

l 50

1

I I00

Ola

150

1 200

x I0 -4

Fig. 1. Plot of Cs(D) (J/amolec) v. D/a (equ. (11)) for amorphous A1 (see text): L, D data from ref. 4, pair potential from ref. 8. a is the unit length of L. 8.5

n

t

T < T* .

(16)

T*

.

I

In terms of the parameter D we should have D = D * - D (T*) for T < T*. Although not strictly correct, fig. 2 confirms the trend towards a more slowly varying D in the low T range. Similarly, even if (16) is not strictly correct our findings conform with a conswained effective potential whose magnitude is smaller than that of the unconstrained u (r). Substitution of ~(r) from (16) for u (r) in (11) will produce a smaller value for Cs(D) in the low T range and correspondingly (12) will produce a smaller value for AS, or equivalently a larger value for the absolute entropy at T = 273K. Nevertheless, we conclude from our result for AS of amorphous Aluminium that its residual entropy (and possibly that of all close-packed one-component metal glasses) is negligibly small in comparison to the entropy of melting. Acknowledgement

'

The author is grateful to M.J. Lopez and M. Silbert for providing him with their results prior to publication and also to J. Jltckle for illuminating remarks concerning this work.

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References

1. T. Morita J. Math. Phys. 5 (1964) 1401.

3..5

I.O o . z.~

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i

055

0.85

2. See e.g.G. Stell in: C. Domb and M. S. Green, editors, Phase Transitions and Critical Phenomena, vol. 5b (Academic, New York, 1986) pp. 205-258.

I 1.15

J.45

3. S. Baer, (a) Physica 87A (1977) 569; (b) ibid. 91A (1978) 603; (c) ibid. 144B (1987) 145.

T/TIn

Fig. 2. Plot of DID o v. TITm: (a) points - data from ref. 4; (b) solid line - equ. (14).

4. L. M. Lopez and M. Silbert Solid State Comm. 69 (1989) 585. 5. K. J. Smolander Phys. Scripta 31 (1985) 427.

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/

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i

6. Y. Waseda, The Structure of Non-Crystalline Materials (McGraw-Hill, New York, 1980).

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•

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5.0 ° e°

C

~

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7. See S. Baer, J. Phys.: Cond. Matter, in print. 8. R. H. Rautioaho, Phys. Star. Sol. b 112 (1982) 83.

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eooe lee

9. H. R. Wendt and F. F. Abraham, Phys Rev. Lett. 41 (1978) 1244. 10. J. J~kle, private communication (1989).

0 025

I 0.55

I 0.85 T/Tm

I 1.15

I 1.45

Fig. 3. Plot of C(T) (cal/K Mol) v. T/Tm for amorphous A1 (see text). The vertical line is at T = Tm.