J. PhyS. Chem. 1980, 84, 1203-1205
(3) N. F. Mott, “Metal-Insulator Transitions”, Taylor & Francis, London, 1974. (4)N. F. Mott iii ”The Metal-corMehl TransiticKls in Disordered Systems”,
Proceedings of the 19th Scottish Universities Summer School in Physics, SUSSP publication, University of Edinburgh, 1978,p 149. (5) N. F. Mott, M. Pepper, S. Polli, R. H. Wallis, and C. W. Adkins, Proc. R . SOC. London, Ser A , 345, 196 (1975). (6) N. Koboyatshi, S.Ikehata, S. Koboyashl, and W. Sasaki, Solid State Commun., 24, 67 (1977). (7) J. R. Marko, J. P. Harrison, and J. D. Quirt, Phys. Rev. B , 10,2448
(1974). (8) M. Pepper, Commun. Phys., 1, 147 (1976). (9) M. Pepper, J . Noncryst. Solids, 32, 181 (1979). (10) G. Bikubski, Phil. Iwsg., in press. (11) D. C. Licciardello and D. J. Thouless, J Phys. C,8,4157 1975); Phys. Rev. Lett., 35, 1475 (1975). (12) S. F. Edwards, J . hloncryst. Solids, 32, 113 (1979). (13) F. J. Wegner, Z . Phys. B., 23, 327 (1976). (14) W. Gotre, J . Phys. C , 12, 1279 (1979). (151 D. Weaire and B. Kramer. J. Noncwst. Solids, 32. 131 1979). (l6j N. F. Mott, Commun. Phys., 1, 203 (1976). (17) M. H. Cohen and J. Jortner, J. Phys. Chem., 79, 2900 (1Y75). (18) Y. Nakamura, U. Horie, and M. Shimoji, J . Chem. SOC.,Faraday Trans. I, 70, 1376 (1974). (19) P. Damay and P. Chieux, paper in this issue. (20) W. Sasakl, S.Ikehata, N. Kobayashi, and S. Kobayashi, “Physics of Semiconductors”, Conference Series No. 43,Institute of Physics, London, 1!348,p 923. (21) P. P. Edwards and M. J. Sienko, Phys. Rev. B, 17, 2575 (1978).
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(22) W. F. Brinkman and T. M. Rice, Phys. Rev. 8 , 7, 1508 (1973). (23)N. F. Mott, Phil. Mag., 37, 377 (1978). (24)B. T. Debney, J. Phys. C , 10, 4719 (1977). (25)T. H. Geballe and G. W. Hail, Phys. Rev., 98, 940 (1953). (26) P. J. Mole, Ph.D. Thesis, Cambridge, 1978. (27)F. Cyrot-Lackmann and J. P. Gaspard, J . Phys. C,7, 1829 (‘1974). (28) W. Freylend, J. Noncryst. Solids, 35, 36, 1313 (1980). (29) W. F. Brinkman and T. M. Rice, Phys. Rev. D , 2, 1324 (1970). (30) J. C.Thompson, “Electrons in Liquid Ammonia”, Oxford University Press, Oxford, 1976. (31) R. Catterall and M. R. C. Symons, J. Chem. SOC.A , 13, part XXIV (1966). (32) L. Onsager, Rev. Mod. Phys., 40,709 (1968). (33) M. H. Cohen and J. C. Thompson, Adv. Phys., 17, 857 (1968). (34)J. P. Lelieur, P Damay, and G. Lepoutre in “Electrons in Fluids”, Colloque Weyl 111, J. Jortner and N. R. Kestner, Ed., Springer Verlag, West Berlin, 1973,p 203. (35) S. Lamis, C. Schleirker, 8. K. Chakraverty, R. Buder, and M. Marezio, Phys. Rev. 8 , 14, 1429 (1976). (36) M. Gourmala, C. Schlenker, and M. Mercier, “Physics of Semiconductors”, B. L. H. Wilson, Ed., Institute of Physics Iierles No. 43, 1978,p 469. (37) E. Schindewolf, paper in this issue. (38) A. Robertson, J. V. Acrivos, A. Thompson, M. P. Klein, and K. Hathaway, paper in this Issue. (39) J. Acrivos and N. F. Mott, Phil. Mag., 24 (1971). (40) L. Friedman, J . Noncryst. Solids, 6, 329 (1971). (41)Papers by various authors in Phil. Mag., 38, 463-549 (197’8). (42) C. J. Adkins, J . Phys. C , 11, 857 (1978).
Influence of Critical Fluctuations of Concentration on the Transport Properties of Metal-Ammonia Solutions Pierre Damay’ Laboratoire des Surfaces nt Interfaces LA 253, Ecole des Mutes Etudes Industrielles, 59046 Lille, France
and Pierre Chieux Instnut Laue-Langevin, 156X, 38042 Grenoble-C%ex, France (Received July 17, 1979)
The metal-to-nonmetaltransition in metal-ammonia solutions is too difficult to be completely and quantitatively understood from a microscopic point of view (Mott). Jortner and Cohen supposed that the solution was not homogeneous in order to solve the problem from a classical standpoint by use of macroscopic methods. We first analyze the nonhomogeneous model. The structure of the solution must be proposed from specific experiments able to determine a structure and not to interpret transport properties. From structural and thermodynamic considerationswe show that the Jortner-Cohen model is unlikely. From recent data obtained by neutron scattering we consider the influence of critical concentration fluctuations on the transport properties of these solutions.
The metal-to-nonmetal transition in metal-ammonia solutions (MAS) is a very difficult topic for which no quantitative description has been given yet by a microscopic the0ry.l A large amount of work has already been devoted to the problem (see Thompson2 for an exhaustive review), the most elaborated approaches being those of Mott1p3 and J~rtner--@ohen.~ We make here some comments based on recent experimental observations about this last model. Jortner and Cohen made the hypothesis that the solution was not homogeneous or bimodal for concentration between 2.3 and 9% of metal. In that concentration range the system was considered as a mixture of microscopic regions of either 9 or 2.3% concentration. A percolation problem was then posed. The size of the microticopic regions was quite large (about 30 A across); the percolation could be treated classically by a refinement of the effective medium theory (EMT).4 Numerous transport properties were accounted for by this 0022-3654/80/2084-1203$0 1.OO/O
model in the above concentration range. The question raised then was to find experimental support for such a structure where the local concentration fluctuates around two poles and the microscopic regions have rather well-defined boundaries, Le., a micelle type of model. Such information may easily be obtained by light, X-ray, or neutron scattering, the last technique being more adapted for experimental reasons. We shall now review the most recent results obtained by small angle neutron scattering (SANS) and see how they could support the Jortner-Cohen model. In the second part we will show how the structural information presently available may influence the electronic transport properties. Microstructure of MAS as Studied by Neutron Scattering SANS experiments were performed on Na and Li ijolutions in liquid ND3 at concentrations near 4% in imole 0 1980 American Chemical Society
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The Journal of Physical Chemistry, Vol. 84, No. 10, 1980
Darnay and Chieux
T "K
-I0
II
- 40
-30
T("C)
d
-20
Figure 2. Thermoelectric power (TEP) as a function of temperature along the critical isochore for the sodium-arnmonia system. TEP departs from linearity for T - T, N 4.5 OC.
I 3
I
I
4
5
I 6
I
7
mole % metal Flgure 1. Liquid-liquid phase separation and is0 [ line for the Na-ND, system. Is0 t lines are given on a reduced scale, [/to,wah Eo = 5.07
A.
fraction from the liquid-liquid critical temperature up to room temperature. The whole set of informations collected up to now could be entirely attributed to concentration fluctuations as predicted on thermodynamical grounds near a critical point. The correlation lengths [ for fluctuations which are large near T, decrease as a function of temperature following a power law5 of the type
where Y = 0.5 for T > T, + 4 "C. We have shown6 that at concentrations between 2 and 10% the SANS results were in good quantitative agreement with the thermodynamic description of the MAS. On that basis a full description of the concentration and temperature dependence of [ could be obtained from a combination of detailed SANS results obtained on the critical isochore and along the coexistence line, and the available thermodynamic data. Figure 1summarizes the findings. The plot is in the reduced scale, [/to.Lines of constant [ value have been drawn. The tovalue is 5.07 A for Na-ND3. We should emphasize that the experimental setup was sensitive to diffusing species of any size from about 7 to 1.000 A; we mean any large molecule, aggregate, or clusters within these sizes would have been detected providing of course it was of sufficient concentration and contrast. We would, however, have had different temperature laws as it is observed for clustering effects in some binary metal alloys. Again nothing of that type was detected in MAS. Of course at high concentration, say above 10%)a peak at low momentum transfer is observed in the structure factor but this is attributed to the solvated ion. Therefore all the inhomogeneities detectable in the intermediate concentration range are typically of unimodal type with a distribution of the local concentration around its mean value. We have already7 given a detailed description of the distribution of the concentration around their mean value. For more precision one should add to that description what is known today about the concentration dependence of the coorelation length,8 or consider that the previous results are correct along is0 [ lines.
Effect of Concentration Fluctuations on the Electrical Conductivity Paraconductiuity. In bimetallic liquid mixtures, the electrical resistivity along the critical isochore decreases linearly with decreasing temperature far above T, but it decreases faster in the vicinity of the critical pointagThere is a strong peak in the temperature coefficient of the re,. sistivity at the critical point. Typically, the temperature coefficient can be ten times larger at T, than in the normal mixture far from T,. This result is at first sight quite surprising. Indeed one expects more scattering and an increase of the electrical resistivity at the approach of the critical point due to the contribution of the large concentration fluctuations. The electrical resistivity is related to the structure via the Faber-Ziman transport equation
( A U )Scc(i)] ~ d(
$)
where SNN, SNc, and SCCrefer to the partial structure factors as introduced by Bathia and Thorntonlo and ii is the average pseudopotentiel. The temperature coefficient of the resistivity is related to the variation of the partial structure factors with temperature. The linear part of this coefficient can be accounted for by the density-density partial structure factor SNN(k) which is not much affected by the vicinity of the critical point. The nonlinear part must be accounted for by Scc(k) which depends on the concentration fluctuations. Ruppersbergll calculated the contribution of Scc(k) to the temperature coefficient of the electrical conductivity when the Orstein-Zernike theory applies. A very interesting result is that there is no contribution to the temperature coefficient in the classical or mean-field case (critical indice y = 1)but that there is a strong peak at T,in the nonclassical or tridimensional Ising case (y = 1.23).
It must be pointed out that in the case of bimetallic liquid mixtures this nonlinear effect on the electrical conductivity is small. At the critical temperature the departure of the resistivity from linearity is less than 1% . No such an effect has been reported for MAS in the literature. I t is not surprising since it is not sure that anybody has ever carefully looked for it. Furthermore, because of the vicinity of the metal-to-nonmetal transition (the conductivity varying so fast with concentration) it is not sure that such a small effect can even be detected. Moreover, the MAS deviate from mean-field theory only quite near TC,l2say from T , to T , + 4 OC,and the mea-
Transport Properties of Metal-Ammonia Solutions
surements should be made under very severe conditions of temperature control and stabilized as closed as possible from T,.. A nonlinear effect has been detected by one of us on the thermoelectric power (TEP).13 It is well known that this quantity is more sensitive to small details of the structure factor than the electrical conductivity. For the sodiumammonia system on the critical isochore, the TEP decreases linearly from -20 to -37.5 "(;: it decreases faster between -37.6 "C and the critical point at -41.4 "C. The nonlinear part is to be accounted for by the variation of Scc(k) with temperature. It is important to note that the variation of T'EP with temperature becomes nonlinear at the same temiperature apart from T, where the systems goes from a classic critical behavior to a nonclassical one. I t is what is expected from the Ruppersberg calculation. It can be argued that the Faber-Ziman formula should not be used for systems with such a small conductivity. We can alternatively take the interpretation of Friedman14 who suggests that the energy barrier to conduction at the Fermi level diminishes as the critical point is approached. The Friedman interpretation was applied to the case of semiconducting super-critical mercury where a similar drops in the thermoelectric power is observed at the approach of the critical point. Nevertheless one notice that in MAS at 4% the thermoelectric power has already a quite metallilc value (less than 10 pV/deg). In any case, independently of the exact formula used to describe the transport properties, it is important to remember from Ruppersberg's result that no anomaly will be found in the temperature derivative of those properties as long as the system is of mean-field type. This result is illustrative of a more general phenomenon relative to the difference existing between the mean-field and the critical regime. The mean-field theory is derived from a first-order expansion of the local free energy and this theory applies when the local fluctuations of the order parameter (the concentration for binary mixtures) are negligible as compared to the order parameter itself. Near a critical point fluctuations become more important and the mean-fielid theory breaks down5 As we have shown from the SANS results, the MAS deviate from mean field
The Journal of Physical Chemisfry, Vol. 84, No. 10, 1980 1205
for [/to> 7,, It is only in this restricted region in the temperature--concentration diagram that the effect of' the concentration fluctuations on transport properties should be observed. Conc 1usi on All the thermodynamic and structural informations on MAS in the intermediate concentration range are contradictory with the existence of a micellar or bimodal type of structure. It seems that a very static view has been adopted by Jortner and Cohen requiring a well-defiined subdivision of the volume into regions of different structure or concentration. In that sense their model corresponds more to what one would expect from a segregating glass than from a liquid. The effect of the critical fluctuations themselves on the transport properties is confined to a small region of the concentration temperature diagram bounded by the is0 [ line [/to= 7 , and thus we confirm the earlier hypothesis of Mott. This idea is supported by the results of Schindewolf15who found the same shape for the conductivity as a function of concentration at temperatureia far removed from the critical point.
References and Notes N. F. Mott, J. Phys. Chem., 79, 2915 (1975). J. C. Thompson, "Electrons in Liquld Ammonia", Ciarendon Press, Oxford, 1976. N. F. Mott, "Metal Insulator Transitions", Taylor and Francis, London 1974. M. H. Cohen and J. Jortner, Phys. Rev. Lett., 30, 699 (1973); J . Chem. phys., 58,5170 (1973); J. phys., 35, C4-345 (1974); J. Phys. Chem., 79, 2900 (1975); phys. Rev. B, 13, 1548 (1976); J. Chem. Phys., 84, 2013 (1976). L. P. Kadanoff et ai., Rev. Mod. Phys., 39, 395 (1967). P. Chieux and P. Damay, Chem. Phys. Left., 58, 619 (1978). P. Damay and P. Schettler, J . Phys. Chem., 79, 2930 (1975). P. Damay and P. Chiew, submitted for publication to J. Chim. Lphvs. H. K. Schurmann and R. D. Parks, Phys. Rev. Len., 27, 1790 (1971). A. B. Bhatia and D. E. Thornton, Phys. Rev., B2, 3004 (1970). H. Ruppersberg and W. Knoll, Z. Nafurforsch. A, 32, 1374 (1977). P. Damay and P. Chleux, to be submitted for publicatlon. P. Damay, Thesis, Lille, 1972. L. Friedman, Phil. Mag., 145 (1973). S. Hahne, P. Krebs, and U. Schlndewolf, Ber. Bunsenges. Phys. Chem., 80, 804 (1976).
