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J . Phys. Chem. 1986, 90, 3892-3894
Systematic Variations in Volume and Conflguratlonal Energy of Pure Liquid Metals C. T. Lira and C. A. Eckert* Department of Chemical Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 (Received: May 2, 1986) Although widely applied to characterize the thermodynamic properties of liquid metals, the nearly free electron method is limited in its extension to the excess properties of liquid alloys because of the loss of accuracy inherent in differences in large, somewhat uncertain numbers. This work demonstrates interrelationshipsbetween the molar volumes, valence, and configurational energies of metals and suggests an alternate reference state. Such results may prove useful in the development and application of improved thermodynamic models of liquid metal mixtures.
In recent years, several novel metallurgical processes have been for which process development requires an ability to model the thermodynamic behavior of liquid metal alloys. Although there have been many empirical models developed, the most desirable modeling techniques are based upon first principles, since these methods incorporate the most realistic assumptions into model development and may be extended to multicomponent modeling with the greatest confidence. Most first principle modeling approaches are based upon the nearly free electron (NFE) concept. The N F E approach has represented successfully many properties of pure metals; however, researchers attempting to extend modeling capabilities to alloy behavior have reported mixed s u c c e ~ s . ~ The ~ ' N F E model represents the internal energy referenced to an ideal ionized gas. The magnitude of these energies range from 100 kcal/mol for liquid alkali metals to 3000 kcal/mol for liquid bismuth and antimony. The excess energies of mixing for alloys with nonspecific interactions are typically less than 1 kcal/mol. This comparison of magnitudes means that the N F E energy calculations must be accurate to better than 1% if the e x u s energies are to be modeled without a fortuitous cancellation of errors. This Letter introduces evidence of a relationship between volume, valence, and configurational energy. The configurational energy is referenced to a nonionized gas state rather than an ionized gas state. The configurational energies of liquid metals range from 40 to 80 kcal/mol, and therefore they may provide a more reasonable basis for modeling excess properties. The Nearly Free Electron Model The valence electrons in liquid metals are delocalized and cannot be associated with any particular atomic center. This delocalization of electrons gives rise to cohesive forces which are not pairwise additive. Historically, the mathematical description of the cohesive forces has led to a separation of the energy into structure-independent and structure-dependent parts. The electron-ion interactions in the metal are included in the structureindependent portion and are expressed in terms of a net effective potential (pseudopotential), which is so weak that perturbation theory may be used for mathematical modeling. A basic understanding of the pseudopotential led to the development of model potential in the 1960s. Introductory reviews of the historical development are provided by Hehe.* Because the net effect of these forces is small, the interactions may be treated as a perturbation to free electron behavior. The internal energy of a liquid metal may be represented by (1) Anderson, R. N.; Parlee, N. A. D. US.Patent 3794482, 1974. (2) Anderson, R. N.; Parlee, N. A. D. J. Vac. Sci.-Technol. 1976,13, 526. ( 3 ) Bakshani, N.; Parlee, N. A. D.; Anderson, R. N. I d . Res. Deu. 1979, 21, 122. (4) Eckert, C. A.; Irwin, R.; Graves, C. Ind. Eng. Chem. Process Des. Deu. 1984, 23, 210. (5) Lira, C. T. M.S. Thesis, University of Illinois at Urbana-Champaign, 1984.
(6)Yokoyama, I.; Stott, M. J.; Watabe, M.; Young, W. H.; Hasegawa, M. J . Phys. F 1979, 9, 207. (7) Ball, M. A.; Islam, Md., M. J. Phys. F 1980, 10, 1943. (8) Heine, V. In Solid State Physics; Ehrenreich, H., Seitz, F., Turnbull, D., Eds.; Academic: New York, 1970; Vol. 24.
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where Ef, is the energy of unperturbed free electrons, E,, and E, are the energies introduced by the exchange and correlation effects due to the quantum nature of electrons, E, models the energies due to electron-electron interactions, Eei models the electron-ion interactions, and E,, is the structurally dependent energy which represents the ion-ion interactions and the effects of electron screening upon the ion-ion interactions. The first five terms of eq 1 are structurally independent. A simplified N F E electron model will be introduced to help in discussing the limitations of the N F E approach. The energy Ere is represented by considering the electrons to be a highly degenerate electron gas. This calculation is usually illustrated in any introductory solid-state physics text or quantum mechanics text, and it is not repeated here. The exchange and correlation energies are usually calculated from an interpolation formula; the work of Nozieres and Pinesg has been selected for this discussion. The electron-electron interactions may be modeled by making a continuum assumption regarding electron distribution and modeling the interactions due to electrons within each atomic sphere by solving Poisson's equation within each atomic sphere. The resulting potential is shown as the upper line in Figure 1. The electron-ion energy may be modeled with a model pseudopotential such as the empty-core pseudopotential proposed by Ashcroft and Langreth.lo Ashcroft's model pseudopotential has been shown to permit reasonably accurate modeling behavior of many metallic properties even though it is somewhat limited by the one adjustable parameter." Ashcroft's empty-core pseudopotential is illustrated by the lower curve in Figure 1. The cancellation of the true potential is represented by a complete cancellation of the potential within a core of radius R,, where R, is the adjustable parameter. The structure-dependent energy is the next higher order perturbation. The resulting mathematical form of the N F E energy equation is
E N F E= 2513
359.3-
v1~3
2413
- 206.7-
351.2(ZR,)2/v
u1~3
- 36.352 - 3.232 In
z
z
V
u1~3
- - 406.3-
+
+ i J m g ( r , u ( r , Z ) )u ( r , Z ) 4 d dr kcal/mol (2)
where E,, is represented by the last term in the equation. The volume, v, is in cm3/mol. The screened ion-ion interactions are represented by an effective pair potential u(r,Z),where the valence is important for both the Coulombic interaction and the screening. The radial distribution function is given by the term g(r,u(z,Z)). Heine and Weairel* and ShimojiI3 have shown that neglecting this last term does not affect significantly the calculation of energy. (9) Nozieres, P.; Pines, D. Phys. Rev. 1958, 111, 442. (10) Ashcroft, N.; Langreth, D. Phys. Rev. 1967, 155, 682. (1 1) Kumaravadivel, R. J. Phys. F. 1983, 13, 1607. (12) Heine, V.; Weaire, D. In Solid State Physics; Ehrenreich, H., Seitz, F., Turnbull, D., Eds.; Academic: New York, 1970; Vol. 24. (13) Shimoji, M. Liquid Metals; Academic: New York, 1978.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 17, 1986 3893
Letters
Figure 1. Electron-electron and electron-ion potentials for proposed model. R, is the radius of an atomic sphere. TABLE I: Core Parameters and NFE Energies of Metals at Their Melting Points -ENFE(calcd), -ENFE(exptl), metal R,, bohrs kcal kcal Ag AI
Bi Cd
cs cu
Ga Hg In
K Na Pb Rb Sb Sn
TI Zn
0.9486 1.1567 1.5329 1.2689 2.7255 0.7721 1.1812 1.3225 1.3382 2.2694 1.7330 1.4841 2.4384 1.4841 1.4038 1.4404 1.0754
202 1305 2809 592 112 220 1289 578 1168 127 151 1894 121 2897 1967 1138 650
236 1300 3521 620 107 25 1 1382 686 1270 120 143 2033 114 3525 2217 1339 658
-Econf (kcal/gmole)
Figure 3. Correlation of interatomic distance, valence, and configurational energy.
The N F E energy calculated without the structural term will be called the structureless energy. Table I compares the representation of the structureless N F E energy with experimental measurements when the core radius is fit to represent accurately the volume derivative, (aE/aV)p Literature data for this derivative were compiled based upon velocity of sound data compiled by Webber and StephensI4 and heat capacity data compiled by Hultgren and c o - w o r k e r ~ ;the ~ ~ calculations are summarized elsewhere.I6 The structureless energies are approximately correct, but it is necessary to introduce the structure to improve the agreement with experimental measurements. The experimental measurement data for the N F E energy are calculated from the formula ENFE
= - c a p - cz
(3)
where F a P is the energy of vaporization and the sum is over the ionization potentials Z of all the valence electrons. Introduction of the structure into the N F E may be achieved either by using experimental measurements of the radial distribution function or by using the Perm-Yevick equation. The Percus-Yevick equation requires at least two more parameters-a hard-sphere radius and the temperature dependence of the hard-sphere radius. A realistic temperature dependence of the energy is desirable for modeling, because metallurgical process may occur over significant temperature ranges. Further, the structural term must be included to represent accurately the heat capacity of liquid metals, because models based upon the structureless NFE energy contain only the ideal gas heat capacity of the reference state. If volumetric (14) Webber, G.; Stephens, P. In Physical Acoustics; Mason, W. P., Ed.; Academic: New York, 1968; Vol. 4, Part B. (15) Hultgren, R.;Desai, P.; Howkin, D.; Gleisen, M.; Kelley, K.; Wagman, D.Selecrrd Values of rhe Thermodynamic Properties of the Elements; American Soaiety of Metals: Metals Park, OH, 1973. (16) Lira, C. T. Ph.D. Thesis, University of Illinois at UrbanaChampign, 1986.
Atomic Number
Figure 2. Systematic behavior of valence electron density of liquid metals at 1000 K.
behavior is to be modeled over temperature ranges, the representation of the heat capacity is imperative
( d / d T ( d E / W T ) V = (acV/au),
(4)
Refinement of the N F E energy approach may be capable of representing the energy accurately, and certainly more rigorous models have been proposed. The approach seems to be somewhat limited by the magnitude of the terms which make up the N F E equation. The reader may understand this point by calculating the magnitude of the terms of eq 1. The alkali metals have volumes of roughly 15-80 cm3/mol, the alkali earth metals have volumes of roughly 15-40 cm3/mol, and most other nontransition elements have molar volumes between 10 and 20 cm3/mol. New Relationships In the search for alternate fundamental relationships, an understanding must be developed for the systematic variations which naturally occur in the liquid metals. Figure 2 shows that a systematic variation occurs in the valence electron density of the nontransition elements. The molar volumes used for calculating the valence electron densities are all taken at 1000 K so data for some metals are extrapolated to a hypothetical liquid state. Another relationship exists between the valence, volume, and configurational energy. Figure 3 shows that a relationship exists that is dependent upon the group in the periodic table. The quantity plotted on the abscissa, Z / d , is the valence divided by the interatomic distance. The interatomic distance is calculated from the formula
d = 2(3~/4?rN,)'/~
(5)
where differences in structure and packing efficiency are neglected. The configurational energy for each metal is calculated from the relationship -Em"' = EVaP = HVaP - RT (6)
J. Phys. Chem. 1986, 90, 3894-3895
3894
All metals seem to fall onto curves which are specific to their groups in the periodic table with the exception of the alkali earth elements. Volumetric data for this figure were once again taken at 1000 K, and configurational energy data were calculated at the melting points. Although a different basis could be justified, the relationships would not be affected greatly, since both quantities are not strongly temperature dependent. The heat of vaporization is calculated from the Clausius-Clapeyron equation by differentiating vapor pressure data. Some of the scatter in Figure 3 may be due to the temperatures used in the calculations, but scatter may also be due to inaccurate heat of vaporization data used in eq 6. The magnitude of the electron density and configurational
energy of gold deviate from the trends exhibited by the other elements in both Figure 2 and Figure 3 due to interactions of the inner d orbitals. In these respects gold seems to behave more like a transition metal. These figures demonstrate that the configurational energies, molar volumes, and valence are interrelated. Studies and correlations which neglect these relationships should be regarded with some skepticism. Further studies to understand the configurational energy behavior on a fundamental level should permit mathematical modeling. These figures indicate that an ideal gas reference state might be considered for fundamental study rather than the ionized ideal gas reference state used in the NFE approach.
Damping of Capillary Waves by Polymeric Monolayers. Comparison with Hydrodynamic Theory K. Dysthe,? G. Rovner, Center for Studies of Nonlinear Dynamics,$La Jolla Institute, La Jolla, California 92037
and Y. Rabin* Chemical Physics Department, The Weizmann Institute of Science, Rehovot, Israel 76100 (Received: May 19, 1986)
A combination of the hydrodynamic theory of damping of capillary waves by compressiblemonolayers and of the experimental equilibrium pressure vs. area curves was used to calculate the wave damping coefficient vs. area curves for poly(dimethylsiloxane)s. The results are in qualitative agreement with the complex wave damping behavior observed by Garrett and Zisman.
The hydrodynamic theory of damping of short surface waves by insoluble monolayers is based on the notion that the main effect of the thin film is to modify the tangential stress boundary condition at the liquid-vapor interface.’ In the simplest version of the theory which will be referred to as the elastic hydrodynamic theory (EHT), one neglects the effects of film viscosity and assumes that the only film parameter entering the expressions for the damping rate is its equilibrium compressibility (or the inverse compressibility, e.g. the elastic modulus)
where A is the area per film (surfactant) molecule and Il is the film pressure given by the difference between the surface tensions in the absence and in the presence of the film, respectively. It can be shown2 that an excellent approximation to the (temporal) damping coefficient y is given by 1
y = -kSw
2
1
+ 2@k8(@- 1) ( @ -1 ) 2 + 1
(2)
+
where k is the wavenumber and w = [gk (a/p)k3]’/* is the wave frequency (g is the acceleration of gravity, a is the surface tension in the presence of the film, and p is the density of the bulk liquid). In deriving the above expression we have neglected terms of order (kQ2(but kept terms of order @(kQ2which give the correct pure liquid limit) where S is the width of the viscous shear layer3 which is much smaller than the wavelength in both the capillary and gravity wave regimes. For a given wavenumber, the dependence on film properties comes through the relation between w Permanent address: Institute of Mathematical and Physical Sciences, De artment of Physics, University of Tromso, Tromso, Norway. PAffiliated with the University of California, San Diego.
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and k (e.g., a) and, more importantly, through the dimensionless parameter @ @=(2~)’/~pC,u~/~k-~
(3) where v is the kinematic viscosity of the bulk liquid. Notice that /3 varies from @ m in the absence of the film to @ 0 for an incompressible monolayer. The striking feature of eq 2 is the existence of a maximum at 0 = 1 which goes against the intuitive expectation that the damping is largest for “solid”, incompressible films. For frequencies in the capillary range (lo2-lo3 SI), maximal damping occurs when the compressibility is approximately 0.15 cm/dyn which, in many cases, corresponds to the transition region between the “gasesous” and “liquid-expanded” states of the monolayer: This maximum has been observed in monolayers of simple surfactants.l In other cases involving polymeric monolayers, highly complex variation of the damping rate as a function of area per film molecule has been o b ~ e r v e d . ~This complex behavior was attributed to conformational transitions in the film and, although no explicit analysis has been made, the experimentalists felt that the hydrodynamic theory could not predict the observed phenomena. In this Letter we report the results of the EHT analysis of damping of capillary waves on water by monomolecular films of linear poly(organosi1oxane)s(PDMQ5 In order to determine the film parameter p (eq 3), we have computed the compressibility as a‘function of the area per film molecule ( A ) from the exper-
-
-
(1) Lucassen-Reynders, E. H.; Lucassen, J. Adu. Colloid Interface Sci. 1969, 2, 347, and references therein. (2) Dorrestein R. Proc. K.Ned. Akad. Wet., Ser. B: Palaeontol., Geol., Phys., Chem., Anthropol. 1951, B54, 260, 350. ( 3 ) Landau, L. M.; Liftshitz, E. M. Fluid Mechanics; Pergamon: London, 1959. (4) Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience: New York, 1966. (5) Garrett, W. D.; Zisman, W. A. J . Phys. Chem. 1970, 74, 1796.
0 1986 American Chemical Society
