FLUCTUATIONS AND TRANSPORT IN BINARY MIXTURES

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FLUCTUATIONS AND TRANSPORT IN BINARY MIXTURES

J. Thompson

To cite this version:

J. Thompson. FLUCTUATIONS AND TRANSPORT IN BINARY MIXTURES. Journal de Physique

Colloques, 1974, 35 (C4), pp.C4-367-C4-369. �10.1051/jphyscol:1974468�. �jpa-00215659�

JOURNAL DE PHYSIQUE

Colloque C4, suppliment au no 5, Tome 35, Mai 1974, page C4-367

FLUCTUATIONS AND TRANSPORT IN BINARY MIXTURES

J. C. THOMPSON

Physics Department, University of Texas at Austin Austin, Texas 78712 U. S. A.

R6sum6. - La formation de composbs dans les melanges binaires s'accompagne d'une rkduction des fluctuations de composition. Cette reduction fournit une indication pour la comprkhension des anomalies souvent observees des coefficients de transport de ces matkriaux.

Abstract. - The formation of compounds in binary mixtures is accompanied by a reduction in composition fluctuations. This reduction provides a clue for the understanding of the anomalies often observed in transport coefficients in such materials.

This paper is concerned with the relationship of concentration fluctuations and transport properties in binary metallic mixtures. The concentration- concentration correlation function Scc(k) and its long-wavelength limit are of particular interest.

Scc, most recently discussed by Bhatia and Thorn- ton [I], carries quite a bit of information on the ten- dencies toward compound formation (or chemical reaction) in binary mixtures and permits speculation on the origin of the negative temperature coefficient of resistivity often observed in binary alloys. At zero wavevector Scc is proportional to the mean square composition fluctuation :

where Ax is the fluctuation in concentration x. The growth of all fluctuations near a critical point is well known and large values of Scc(0) are observed there.

The present case concerns a composition range wherein a chemical reaction takes place so that a compound is formed or wherein there is a tendency toward the formation of a compound which is nevertheless unstable. In such a range, fluctuations away from the stoichiometry of the compound are inhibited and

< (Ax)' > must be minimal as Bhatia, et al. [2]

have recently established for a very simple model.

Darken [3] has reached the same conclusions from quite a different route. The depth of the minimum in Scc(0) is related to the equilibrium constant of the compound. There is a sum rule [I]

00

[Scc(k) - x(1 - x)] k2 dk = 0

0

(2)

where ,u is the chemical potential of the species of concentration x. Chemical potentials can, of course, be obtained from vapor pressures or cell emf's.

Figure 1 shows values of S,-,(O) taken from the lite- rature [4] which demonstrate the expected behavior.

The dip at Cd,As2 is deeper than that at the less stable compound CdAs2. When there are several possible compounds as in Fe-Si alloys the minima are joined.

which states that about x(' - x)7 the _{Fm. 1 . - Scc(0)} for several alloys. The dashed curves denote in an mixture. determination the ideal value for ^Scc(0). See ref. ^[4] for original citations. The of Scc(0) is based on the relation [I] large negative deviations from ideality are associated with compound formation ; the steep rise in the TI-rich alloys is due Scc(0) = N ~ B T [ ( a ~ / a x ) ~ , ~ / ( l - x)]-l

(3) to phase separation.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974468

C4-368 J. C. THOMPSON Clearly Cu,Sn is less stable than TeT1,. Figure 2

shows some data on metal-ammonia solutions taken in part in my laboratory [4]. Dips may be seen at the concentrations expected from solvation numbers [5].

CONCENTRATION (MPM)

FIG. 2. -

for several M-NH3 solutions : -. ^- Li- NH3 ; - Na-NH3 ; and -. .- for Cs-NH3. The dashed

curve denotes the ideal value.

The implications for transport coefficients may be seen if the Faber-Ziman [6] theory of alloy resistivity is rewritten in terms of Scc and its companion corre- lation functions [I]. In the resistivity integral, the density correlation function SNN(k) is multiplied by the weighted sum of the component pseudopotentials (form factors) Scc(k) by the dzfference in form factors, and the density-composition correlation SNc(k) by the product of sum and difference of pseudopotential form factors :

The effect of Scc may be ascertained as follows.

The sum rule (eq. (2)) requires that the trends of SCc(k1) for some k' well away from zero be opposite to those of S,,(O). That is, if Scc(0) is approaching x(l - x) from below, as in figure 3, then Scc(kt) must be approaching x(1 - x) from above. If kt is at or near 2 k, then trends in the resistivity (which is very sen- sitive to the integrand at 2 k,) must also be opposite to those of Scc(0). For example, the dip in SC,(O) found near compounds would result in a peak in the resistivity at the compound composition. This is just the obser- vation [7] in alloys such as Mg-Bi or Te-T1.

The temperature dependence of p may be treated in the same way. The equilibrium will always shift towards dissociation as the temperature increases.

This will cause < ^{AX)^} > to rise toward the ideal mixture value. As before, the behavior at or near 2 k, will be opposite : Scc(kt) will decrease. Therefore,

FIG. 3. - Mean square fluctuations as a function of wave number. The shift from solid to dashed curve schematically illustrates the consequences of the sum rule. If the solid curve denotes the fluctuations at T I , then the dashed curve might represent a higher temperature Tz. The arguments of the text

require 2 k~ to be near k'.

the resistivity is expected to also decrease with increas- ing temperature, as is often observed. Indeed, since the equilibrium is exponentially dependent on the temperature one may even find

where A is the enthalpy of reaction. In this way the alloy mimics a semiconductor [7].

These speculations concerning the origin of

ano- malies

in alloy transport coefficients require the pseudopotential form factors to be significantly different near 2 k,. This condition must apply in the alloy, at the composition of interest, e. g., Mg'Bi,.

Contributions to the resistivity from SNN and SNC must also be considered. The trend of SNN is much like that of the structure factor in single component liquids and must be expected to contribute a term in the resistivity which, e. g., increases with temperature in most cases. The kinds of form factors which differ at 2 k, so as to enhance the effect of Scc must at the same time tend to weaken the effect of S,,. If the compound is but loosely bound then the effect of SNN will be to shift dpldT toward positive values- perhaps cause a sign change - as in Li-NH, solu- tions [a]. The influence of SNc is expected to be weak in any case [I].

The generality of the arguments presented here is a part of their strength. The observation of a negative dp/dT is sufficiently widespread as to require an explanation which is nonspecific. One need only look to the form factor difference and to < AX)^ >.

The apparent anomalies in the resistivity of binary mixtures thus arise from pecularities of structure imposed by compound formation. As the mixtures.

seek the local order required by the stoichiometry a

peak in Scc at the equivalent wave vector is produced

FLUCTUATIONS AND TRANSPORT IN BINARY MIXTURES C4-369 Should that peak become strong enough even a band

gap is possible. No comments are made concerning ionicity, non-additive forces, etc. rather the structural manifestations of compound formation are empha- sized. One may infer nevertheless that the Faber- Ziman formalism is capable of encompassing the transport processes which accompany approaching compound formation, and must be abandoned only when strong scattering occurs.

Acknowledgments. - Conversations with J. B.

Swift, K. Ichikawa, and P. R. Antoniewicz have been exceedingly useful. This work has received direct support from the R. A. Welch Foundation (Texas) and the U. S. NSF. In addition, it has benefited from support of research on amorphous semiconductors at U. T.-Austin by the Advanced Research Project Agency of the U. S. Do D., monitored by the U. S.

ARO under grant No. DA-ARO-D-31-124-73-G81.

References

[I] BHATIA, A. B. and THORNTON, D. E., Phys. Rev. B 2 (1970) [5] GARRAWAY, A. N. and CQTTS, R. M., Phys. Rev. A 7 (1973)

3004; 4 (1971) 2325. 635.

[2] BHATIA, A. B., HARGRO~E, W. H. and THORNT~N, D. E. [61 FABER, T. E.,

Intro. Theory of Liquid Metals

(cambridge

Phys. Rev. B 9 (1974) 435. Univ. Press, Cambridge) 1972.

[7] ENDERBY, J. E. and COLLINGS, E. W., J. Non-Cryst. Sol. 4 [3] DARKEN, L. S., Trans. Met. Soc. AIME 239 (1967) 80. (1970) 161.

[4] ICHIKAWA, K. and THOMPSON, J. C., J. Chem. Phys. 59 [8] SCHROEDE~, R. L., THOMPSON, J. C. and OERTEL, P. L.,

(1973) 1680. Plzys. Rev. 178 (1969) 298.