SOME E M P I R I C A L R E L A T I O N S H I P S FOR T R A N S P O R T PROCESSES IN L I Q U I D S AND S O L I D S J.
H. M A G I L L ’ A N D R . J . G R E E T 2
Carnegie-Mellon University, Pittsburgh, Pa. 15213
Some phenomenological theories of liquid viscosity and diffusion are briefly reviewed. The broad underlying principles embodied in these theories are discussed. The validity of correlative relations for transport, particularly between log 7 or log D and T,/T, is tested for organic compounds, small molecules and molten salts. High pressure-viscosity data on organic molecules are used to test the efficacy of T,/T as a reduced parameter, but this is only partially successful. Correlations are found between T, and the apparent activation energy for viscous flow and diffusion for metals (polycrystalline and liquid) and ionic melts. The results are discussed in relation to some prevalent molecular models of transport for solids and liquids.
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VARIETY of phenomenological theories of viscosity exists in the literature. Some have been discussed in a paper on corresponding states relationships for liquid viscosity (Greet and Magill, 1967), where a relationship between log 7 and Tm/T (7 is the shear viscosity and T , the melting temperature) was forwarded. The relation enabled characterization of a wide variety of materials according to their chemistry. Furthermore, this formulation could be used in testing the consistency of new viscosity data. It seemed useful for estimating liquid viscosit,y outside the existing experimental range of available data. In these respects, the general patterns of characterization were of practical importance. More extensive tests of the T,/T relationship have been made for an even wider range of materials and the relationship is extended to examine viscous systems measured under constant volume, a t a variety of applied pressures. An extensive survey of the literature, made to rationalize the corresponding states of relationships forwarded in the earlier paper, has revealed that linear correlations exist between the Arrhenius activation energy for transport and the melting points of chemically related materials. Diffusion and viscosity data on metals (liquid and crystalline), ionic melts, and ceramics have been examined in some detail. This investigation suggests that several serious discrepancies may exist in the reported values of some of these properties for common materials. Phenomenological Theories of Viscosity and Diffusion
We attempt an outline of viscosity models which are pertinent to the interdependent relationships used here and in the earlier paper (Greet and Magill, 1967). I n viscous flow, transfer of momentum between “layers” of molecules may occur through collision processes, akin to vibrations limited by the repulsion barriers due to the proximity of nearest neighbors. Since the distance of separation between molecules in the liquid state is of the same order as in the solid, it seems meaningful to relate transport processes in liquids with melting parameters, particularly for the close-packed metallic materials which exhibit lower volume changes on fusion (-2%) than the covalent crystals (-10%). Among the molecular theories which have some relevance here are Present address, Department of Metallurgical and Materials Engineering, University of Pittsburgh, Pittsburgh, Pa. 15213. Mechanical Engineering Department, New Haven College, P. 0. Box 1306, New Haven, Conn. 06505.
those due to Frenkel (1946), Andrade (1934a, b ) , Batschinski (1913), and Eyring (Glasstone et al., 1941). Frenkel (1946) considered that the constituent particles of the liquid acquire their translational energy (for jumping) after several vibrations for a time r around arbitrary equilibrium positions. This theory gives the correct form for the temperature dependence of the transport coefficients, though the magnitude is not correct. Andrade’s equation for metals (1934a, b ) also considers viscosity from the viewpoint of molecular vibrations, the difference between the solid and liquid being one of amplitude of the vibrations. This model is useful for calculating the viscosity for close-packed liquids a t the melting point and has the form: 7 = A (MT,)1’2V-2’3 (1) where 7 is the viscosity, A is a constant, T , is the melting temperature, V is the atomic volume a t T,, and M is the molecular weight. Grosse (1961, 1963) has further examined the Andrade relationship for metals and finds that there is a fairly good correlation between log Allq,the activation energy associated with viscous flow, and log T,. Recent and independent work by Nachtrieb (1965) has evolved a direct relationship between the activation energy for self-diffusion, AED,and T , for liquid metals. This seems to contrast with the work of Grosse, who found a logarithmic relation. Bockris and Richards (1965), Bockris et al. (1965), and Nanis and Bockris (1963) have also shown that AE, and AED may be correlated with melting temperature. They examined data for a variety of liquids and deduced that AH 3.74RTm. The enthalpy, A H ) is considered to be comprised of two terms, the heat content change in the formation of a mole of holes and the heat of activation associated with the movement of a particle from one site to another. The latter term is usually small except for bonded or structured liquids. For solid-state diffusion, h’achtrieb et al. (1959) rationalized the temperature dependence of self-diffusion in polycrystalline lead from 1- to 8000-atm. pressure on the basis of a dynamical theory based on the normal vibration modes of the crystal lattice. They showed that an almost linear relationship exists between log D and Tm/T for polycrystalline lead. Rice and Nachtrieb (1959) indicated that this model can be associated with thermodynamic functions and showed that a correlation exists between the volume of activation and enthalpy of activation for transport. Earlier investigations by
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Lawson (1957), Keyes (1958), and Lawson et al. (1959) had presented related formulations which embodied the expansion coefficient and compressibility coefficient. With the equation r] = A exp ( A E , , / R T )for the temperature dependence of the viscosity (empirically suggested by GuzmBn in 1913), Barrer (1943) noted a correlation between log A and A E , / T . Dienes (1950) has shown that, for selfand intermetallic diffusion in the polycrystalline solid state, log DO from the relationship D = Doexp ( - A E D / R T ) plotted against A E D I T , is almost linear. Following Mott (1948), Dienes proposed that the activation step in the process be associated with local melting. This concept has physical plausibility below, but not above, T,. A partially successful theory establishes Dienes’ relationship. Ward (1937) found that the ratio of AE,,/AH, (AH,,, the heat of fusion) is roughly constant for materials of the same structural type, though Barrer commented that some of the experimental data used by Ward required revision. In any case, the idea seems reasonable and there has been a substantial effort to find usable viscosity and/or diffusion relations of practical as well as theoretical value. Implied in Eyring’s rate theory, with (Ewe11 and Eyring, 1937) and without (Eyring and Rhee, 1961) considerations for the presence of holes in the liquid, is the assumption that AE,, is proportional to AH,,,, the energy of vaporization. A correction factor (
