J . Pkys. Ckem. 1992, 96, 25-33
25
FEATURE ARTICLE Contact Potentials of Solution Interfaces: Phase Equilibrium and Interfacial Electric Fields Lawrence R. Pratt Chemical and Laser Sciences Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Received: November 20, 1990)
Calculations of the electric fields at the water liquid-vapor interface are used to motivate a general discussion of the electrostatic potential differences between conducting phases in equilibrium with respect to independent transport of charged species. A formula which requires only structural information on the interfacial charge density profile is presented for this contact (or surface) potential. In contrast to the purely thermodynamic basis for introduction of surface potentials, this surface structural perspective suggests that surface potentials are generally measurable quantities. The consistency between thermodynamic and structural definitions of the contact potential is discussed. Although a solvent contact potential can be defined and, in principle, measured, this solvent contact potential generally differs from the infinite dilution electrostatic potential difference involved in the thermodynamics of electrolyte solutions: ionic contributions to thermodynamic contact potentials do not vanish in the infinite dilution limit. In fact, if the solvent conductivity is negligible, then the limiting infinite dilution value of the contact potential depends on the compositional path chosen for the approach to infinite dilution. Further, although the contact potential can be identified with the total dipole moment of the charges in the interfacial regions, the defined solvent contribution to the contact potential is generally not determined only by the interfacial density profile of molecular dipole moments. Finally, it is suggested that electron (and positron) reflectivity experiments would provide the most direct attempt to measure contact potentials of solution interfaces, and some primitive aspects of such experiments are considered.
Introduction A contact potential (or interface or surface potential) is an electrostatic potential difference across an interface separating chemically dissimilar conducting phases at equilibrium. It is associated with charge separation and nonzero average electric fields in the interfacial region. Those are statements about interfacial structure. However, surface potentials are also involved in discussions of thermodynamic phase equilibrium from an electrochemical point of view. From the initial introductions of a surface potential in this context, its legitimacy as a characteristic of bulk phase equilibrium was doubted because the measurability of surface potentials by bulk thermodynamic techniques was doubted.’** This article uses recent theoretical results on fluid interfaces involving liquid water as a motivation to renew the general discussion of surface potential^.^-^ The basic point of view adopted here is that surface potentials might be introduced on the basis of two different types of information: (i) high-mlution structural information on interfacial charge densities or (ii) knowledge of (1) The issues of thermodynamic determination of electrical potential differences are discussed in: Collected Works of J . Willard Gibbs; Yale University Press: New Haven, 1948; pp 332-349 and especially 425-434. In view of the fact that the latter segment is a fragment not published by Gibbs himself, contentions about Gibbs’s own conclusions regarding the measurability of junction potentials are speculative. However, this fragment does clearly support the view that Gibbs seriously concerned himself with this issue but had not arrived at a conclusion at the time of the writing of that fragment. (2) Guggenheim, E. A. J. Phys. Chem. 1929.33, 842. Guggenheim, E. A. In A Commentary of the Scientific Writings of J. Willard Gibbs; Yale University Press: New Haven, 1936; Vol. 1, Chapter E ‘Osmotic and Membrane Equilibria, Including Electrochemical Systems”, Sections 14 and 15. Guggenheim, E. A. Thermodynamics, An Advanced Treatmentfor Chemists and Physicists; John Wiley & Sons: New York, 1967; Sections 8.02 and 8.03. (3) References 4 and 5 provide accessible general discussions of surface potentials in solution chemistry. (4) See: Gaines, G. L. Insoluble Monolayers at Liquid-Gas Interfaces; Interscience: New York, 1965; Sections 2.111 and 3.1V.C. (5) Llopis, J. In Modern Aspects of Electrochemistry; Bockris, J. O M . , Conway, B. E., Eds.; Plenum: New York, 1971; Vol. 6, pp 91-154.
single-ion activities. It is the difficulty in devising measurements of singleion activities which leads to a view that surface potentials are unmeasurable. The general discussion below analyzes the first alternative for definition of surface potentials. A formula is set forth for the surface potential of a planar interface. Although this formula appears to involve only interfacial structural information, it also transparently offers a “mathematical mechanism” for the surface potential as determined by bulk singleion activities to be consistent with the surface structural result. Together with the thermodynamic view, this surface structural formula also makes it clear that the definition of a surface potential for fluid dielectrics, Le., nonconductors, requires a more restrictive physical argument. We anticipate the development below by noting one particularly striking example of the difficulties which can arise: For an electrolyte solution at low ionic concentration, the limiting value of the surface potential will depend on the compositional path chosen in the approach to infinite dilution (or zero conductivity) unless alternative physical arguments, which do alter the problem, are interjected. The general discussion here will consider some of those restrictions which complicate more sweeping views of surface potentials. However, precisely because of those complications we plan first to give simple examples of interfacial structural information which lead to the question of surface potentials. These examples will serve to motivate the development and, it is hoped, help to avoid the mysteriousness which theoretical discussions can sometimes attain. Those examples include an explicit calculation of the electrostatic fields present at the water liquid-vapor interface. After those examples, we take up the theoretical discussion by giving a brief exposition of the thermodynamic introduction, via single-ion Gibbsian electrochemical potentials, of the electrostatic potential of a phase. We next show that the algorithm used to obtain the surface potential within the molecular dynamics calculations yields a compact alternative theoretical definition of the surface potential which involves only measurable interfacial charge
0022-365419212096-25%03.00/0 0 1992 American Chemical Society
26 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
.
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Figure 1. Surface tension of the n-heptane-water interface at 320 K as a function of concentration of sodium alkyl sulfate solute. The data are taken from ref 7. The molar surfactant concentration is denoted by p .
densities. As a matter of principle, that density is measurable. On the basis of that formula, it is argued that the surface potential is a measurable quantity, in principle.6 The surface structural formula possesses some counterintuitive or “paradoxical” properties, particularly in limiting c a w where the conductivity of the system vanishes. These paradoxical properties are directly relevant to the objective of obtaining a surface potential for a pure solvent to the extent that the solvent in question is a nonconductor. These paradoxical properties are discussed. Examination of the ionic contributions to contact potentials permits a contrast of the thermodynamic with the surface structural identification of the contact potential. The consistency between the two alternative definitions of the junction potential is discussed next, particularly emphasizing the nontrivial condition that the mean electric fields in the interior of the coexisting phases be zero. This condition can be regarded as an identification of conductors, but it will not hold for nonconductors, generally. Physical ideas for extraction of the solvent surface potential between isotropic nonconducting phases are then discussed, emphasizing that the surface potential obtained on the basis of such arguments is generally different from the electrostatic potential difference between conducting phases even in the limit of infinite dilution of mobile charge carriers. Finally, we consider types of measurements which should provide direct determinations of surface potentials: electron (and positron) reflectivity from solution interfaces. These are “in principle” experiments which will typically be exceedingly difficult. But they are expected to be practical in favorable cases, and they serve to make definite the “inprinciple” argument that surface potentials are measurable quantities. The concluding section summarizes the major points of the paper.
Motivation Figure 1 gives an example of the type of phenomena which motivated this work. Shown there is a plot of the dependence of the surface tension of a n-heptane-water interface as a function of the surfactant concentration for a simple homologous family of ionic surfactant^.^ These curves show behaviors which have simple qualitative interpretations. That the surface tension decreases with increasing concentration through a low concentration region gives a demonstration that the surfactant is positively absorbed a t the interface. The near leveling off at higher concentrations is a signature of micelle formation. In spite of the naturalness of these interpretations, a quantitative description of these curves in terms of molecular forces and structures is not currently available. These behaviors are of interest for their own sake. But they are also of interest because of the expectation that data of this sort should teach us about the molecular interactions which govern (6) Oppenheim, I. J . Phys. Chem. 1964,68, 2959. (7) Kling, W.; Lang, H. In Proceedings of the 2nd International Congress of Surface Actiuity; Butterworths: London, 1957; pp 295-305.
0 -6
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-3
-4
-2
In [ p (moldliter) 1 Figure 2. Model calculation of the dependence on sodium alkyl sulfate concentration of the surface tension of n-heptane-water interface at 320 K. The curves are the theoretical calculations of ref 8. The circles are the data of ref 7 also shown in Figure 1.
I I
4
6
8
10
12
(A, Figure 3. Molecular dynamics results for the variation of the concen-
tration profile of an Na+ near the water vapor-liquid interface located at z = 0 with the vapor on the left. The dot-dash curve is a continuum model for the same quantity. The vertical dashed line indicates the position of the dielectricjump in the model, approximately two molecular layers inside the liquid phase. See ref 11. more complex molecular organizations such as micellization and the folding of globular proteins. Often theoretical studies of data such as that shown in Figure 1 adopt the standard interpretation of one feature of such a curve and then attempt to formulate a model which permits a better quantification of that feature. An example of such an approach is the treatment of the limiting, low-concentration behavior of the surface tensions of Figure 1 given some time ago by Nichols and Pratt.s Their results in this case are reproduced in Figure 2. These “topdown” modelistic approaches are helpful, but they often beg the question of the fidelity of the model to the molecular reality under study. For the case shown in Figure 2, the theoretical calculation adopted a drastically simplified model of the hydrophobic interactions between the tail of the amphiphile and the hydrocarbon phase and a drastically simplified model of the forces on ionic species in the interfacial region. The correspondence of the calculated results with experiment teaches something about the correctness of the model. On the basis of calculations like those depicted in Figure 2, it was suggested that the tail unit of an amphiphilic ion would have to be at least as hydrophobic as the n-butyl group for such an ion to be judged surface active? Independently, this point was elegantly demonstrated by studies of second harmonic generation at aqueous solution surfaces.1° Augmenting the traditional comparison of theory and experiment, however, computer simulations provide (8) Nichols, A. L., 111; Pratt, L. R. Faraday Symp. Chem. SOC.1982.17, 1291. (9) Wilson, M. A.; Nichols, A. L., 111; Pratt, L. R.J . Chem. Phys. 1984, 81, 579. (10) Bhattcharyya, K.; Castro, A,; Sitzmann, E. V.; Eisenthal, K. B. J . Chem. Phys. 1988,89, 3376.
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 21
Feature Article
3
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,,.,,
,,."'
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-5
6, Figure 4. Variation of electrostatic potential with progress through the
water vapor-liquid interface. The vapor phase is to the left and the mid-interfacial region, as judged by the oxygen density profile is near z = -10 A. The lower (solid) curve corresponds to the direct calculation with the TIP4P model using those partial charges to determine the electric field at each configuration. The other curves correspond to adjustment of that charge distribution as discussed in the text and in ref 15.
an alternative source of information on the molecular-level correctness of theoretical models. The work described here grew out of molecular dynamics calculations of model aqueous solutions which aimed a t just such a testing of models. Recent molecular dynamics studies of simple ions interacting with the water liquid-vapor interface provide straightforward examples of this checking into molecular aspects of conventional Some of those results are reproduced here as Figure 3." That work11*12 showed that a common continuum model for the interaction of simple ions with a solution interface begins to break down significantly, in the case of Na+, when the ion reaches two solvent diameters from the nominal water liquid-vapor surface as located by the oxygen equimolar surface. That conclusion is consistent with experimental informationI4 and with the model calculations leading to Figure 2, which used just such a continuum model to treat the necessary ionic interactions. The continuum model tested in that comparison ascribes the work of positioning an ion a particular distance from the interface to work against the polarization induced by the ion in the dielectric medium. The numerical simulation results show the significance of a molecular treatment of ion solvation. However, the solvent molecules in the interfacial region will be preferentially oriented in the absence of the ion, and this polarization is of interest, too. The variation of the electrostatic potential associated with that solvent interfacial polarization, in the absence of ionic species, is shown in Figure 4.15 Those calculations, however, led to renewed questionsI6 regarding the objective existence, Le., the measurability, of the electrostatic potential difference (or Galuani potential difference3-5,14,17) between chemically dissimilar materials. This has been an issue of interest and contention for ~ o m e t i m e . ' + ~ For J~,'~ example, Guggenheim2 states definitely that surface (or contact) potentials "can never be measured". In spite of such statements, efforts to measure surface potentials have not ceased. Furthermore, very reasonable discussions seem to conclude that surface potentials can be m e a ~ u r e d ~ but * ~ ~that . ~ 'a high confidence cannot be established for the interpretation of extant measurements.14 A most important aspect of molecular-level simulation of complex materials is a requirement of concrete molecular-level (1 1) Wilson, M. A.; Pohorille, A.; Pratt, L. R. Chem. Phys. 1989,129,209. (12) Wilson, M. A,; Pohorille, A. Interaction of Monovalent Ions with the Water Liquid-Vapor: A Molecular Dynamics Study. J. Chem. Phys., in
press. (13) Benjamin, I. Theoretical Study of Ion Solvation at the Water Liquid-Vapor Interface. J . Chem. Phys., in press. (14) Randles, J. E. B. Phys. Chem. Liq. 1977, 7, 107. (15) (a) Wilson, M. A.; Pohorille, A.; Pratt, L. R. J . Phys. Chem. 1987, 91,4873. (b) Wilson, M. A.; Pohorille, A,; Pratt, L. R. J . Chem. Phys. 1988, 88, 3281. (16) Zhou, Y.; Friedman, H. L.; Stell, G. J . Chem. Phys. 1988,89,3836. (1 7 ) Parsons, R . In Standard Potential in Aqueous Solution; IUPAC: New York, 1985; Chapter 2.
definitions of quantities to be computed. It is not emphasized enough that the activity of carrying out computer experiments helps to redefine, or even to eliminate, ill-defined concepts which sometimes arise in heuristic modeling of materials. This virtue of molecular-level calculations can be independent of the actual numerical values obtained in a particular case. This is exemplified by the surface potential in the present discussion. The molecular models which are used in the available calculations are sufficiently crude that they are unlikely to give accurate values for the surface potential as judged by experimental reality. But computing the surface potential for a particular model requires a definition first. Then one can hope to learn, even from crude models, which aspects of the model influence the features of the computed results. For the surface potentials, in particular, it is important that the issue of definition be clearly resolved at a basic level because surface potentials have some counterintuitive properties. For example, although the surface potential can be identified with the total dipole moment of the charge in the interfacial region, that part of the electrostatic potential change which is due to neutral solvent molecules generally would be nonzero even under the very special conditions that the molecular orientations were isotropic and the solvent molecules displayed no permanent or induced dipole moments. The presence of ionic components in the solution raises a second counterintuitive feature which is associated with the long-ranged nature of ionic interactions. Ionic contributions to the surface potential do not vanish, generally, as the ionic concentrations tend to zero. This point has been clearly articulated recently by Zhou, Friedman, and Stell.I6 The ionic contribution is there even though the ions are not. These counterintuitive properties should seem less paradoxical after the theoretical discussion which is initiated in the next section. However, we conclude this motivation section by returning to the interfacial phenomena typified by the results shown in Figure 1. Precision in our understanding of the molecular basis of interfacial electric fields may be of practical importance in studies of insoluble films spread on solution interfaces: measurements of certain potential differences4' are a common method for characterization of those molecular structure^.'^^'^
Theory Consideration of electrostatic potential differences between coexisting phases makes sense if those phases are conductors. If this is not true, nonzero macroscopic electric fields may be present in the bulk phases and the electrostatic potential may not be constant over the interior of the phases. Pyroelectric and ferroelectric materials, e.g., barium titanate, provide straightforward examples of this possibility.20 Insistence upon this restriction makes the conceptual basis of the present subject simpler. Average macroscopic electric fields must be zero, and the average electrostatic potential must be spatially constant in the bulk phases. In that case, for example, the conventional, but always somewhat arbitrary, distinction between interfacial electrostatic fields and average fields far from an interface is not necessary. In particular, it is then unnecessary separately to introduce inner (or Galuani) and outer (or Volta) potentials for a phase. This is true because the outer potential is traditionally associated with the electrostatic fields in nonconducting regions outside the material system of interest. (It should be emphasized, however, that manipulation of inner potentials may utilize detection of external electric fields.) Many materials can be considered conductors over sufficiently long time scales.20 This restriction, therefore, begs the question of the time scale probed by any particular experiment. This issue is important in analyzing experiments, but we will set it aside in the discussion which follows. Thermodynamics. The electrostatic potential of a equilibrium phase also arises in studies of electrochemical reaction equilibria. (18) Miller, A.; Helm, C. A,; Mohwald, H. J . Phys. (Paris) 1987, 48, 693. (19) Vogel, V.; Mobius, D. J. Colloid Interface Sci. 1988,126,408. Vogel, V.; Mobius, D. Thin Solid Films 1988, 159, 73. (20) See: Landau, L. D.; Lifshitz, E. M. Electrodynamics ojContinuous Media; Pergamon: New York, 1960; pp 60-61.
28 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992
Pratt
Indeed, the measurement of electric potential differences provides an important method of determining free energies of redox reactions.2’ The electric potential of the a t h phase, @(a), is incorporated into the chemical potential by
electric fields in the interior of these phases are expected to be zero. The observed electric fields were consistent with this expectation. Therefore, although this simulated system would be considered a nonconductor, this is a system for which the question of the electrostatic potential difference between the two phases is reasonable. This point will be discussed further below. An operational way to think about the interfacial electric fields can be based upon the idea that the electric field due to a planar sheet of charge is independent of the distance from the sheet and proportional to the surface density of charge on the sheet; in particular
= fi?)
+ qj@(d
(1)
Here qj is the charge of thejth component of the material under study. If electrochemical phase equilibrium is excluded from need not be considered either and consideration, then piu) =
fi)a)(p,T,~I(“) ,...,x,‘“))
(2)
is, to within a constant, the familiar chemical potential for that case, a function of the pressure, temperature, and the bulk composition of phase a. That composition is here described by the mole fractions ( x , ( ~ )..., , x,‘“)). A common terminology is that the first term on the right side of eq 1 is the “chemical” part of the electrochemical potential, p?); the second term is the “electrical” part. This definition already raises the basic question in a primitive form. Most of chemistry and materials science can be interpreted in terms of simple electrical and magnetic models analyzed via quantum mechanics. How then is the “chemical” part of the electrochemical potential to be separated from the “electrical” part? This ambiguity is reflected in the language whereby the decomposition equation (1) is referred to as a ”nonoperational” separation.22 The thermodynamic development gives two partial responses to this question. The first response is that separation of @(a) from the conventional bulk thermodynamic variables (p, T , xlca),..., x,‘*)) makes good physical sense: can be manipulated with (p,T) fixed while making vanishingly small changes in the bulk thermodynamic composition variables. This is done, in principle, by positioning charges in regions exterior to and on the surface of the material under study, i.e., by manipulation of external electric fields. The second thermodynamic response to the issue of the ambiguity of is that the difference in the electric potential, A@ = @(a) - @(r),between two phases which are chemically identical, Le., are at the same bulk thermodynamic state (p, T, x , ( ~ )..., , %(*I), is thermodynamically measurable. In this case, the “chemical” part of the electrochemical potential does not contribute to the difference in the electrochemical potential between the two phases:
E, = 27q sgn (z - zo)
where q is the surface density of charge and it is assumed that the sheet is perpendicular to the z axis with altitude zo. This relation permits us to calculate the surface potential because the net charge profile in the interfacial region can be viewed as a stack of planar sheets of charge. Using the symbol @(z) for the mean electrostatic potential at elevation z, from eq 4 we obtain d@(z)/az = 27(Q+(z) - Q-(z))/area where the quantity Q+(z) (Q-(z)) is the average net charge above (below) the plane parallel to the interface a t altitude z. The cross-sectional area of the system is “area”, i.e., the surface area. Equation 5 was used to obtain the surface potential associated with the water liquid-vapor interface at T = 320 K as described by the TIP4P model. That is the result shown as the lowest curve of Figure 4. The charges involved in eq 5 can be expressed in terms of the charge density profile p,(z) as Q+(z) = area ILt2dz’pq(z) This assumes, as is the case for the computer-simulated system with periodic boundary conditions, that the upper boundary of the system along the z axis is at L/2; the lower boundary is located at -L/2. For a system which has no net charge Q+(z) = -Q-(z). Therefore, eqs 5 and 6 become L/2
@(z) - @(-L/2) = 4 7 1 ’ d z ’ 1 , dz”pq(z”) -L/2
(21) This is discussed in many good thermodynamics textbooks. See, for
example: Lewis, G. N.; Randall, M. Thermodynamics;McGraw-Hill: New York, 1961; Chapter 24; revised by K. S.Pitzer and L. Brewer. (22) Kirkwood, J. G.; Oppenheim, I. Chemical Thermodynamics; McGraw-Hill: New York, 1961; Chapter 13.
z
or
(3)
This quantity is a thermodynamic driving force for material transport and can be measured by thermodynamic techniques. Since the chemical part iip)does not vanish when equilibrium of chemically dissimilar phases is considered, this discussion leads us to the suggestion that the potential drop between chemically dissimilar phases cannot be measured by thermodynamic techniques based upon the separation of eq 1. However, if we insist on the identification of as an electrostatic potential, then we can characterize with the help of Poisson’s equation and a determination of the average density of electrical charge.6 This charge density is expected to be nontrivial, Le., nonzero, in the interfacial region. We will refer to this procedure as the mechanical procedure to distinguish it from the alternative which relies on thermodynamic analysis of the phase equilibrium. We proceed next to codify that mechanical alternative more thoroughly. Mechanical Formula for A@. The results of Figure 4 were obtained from an observation of the mean electric fields throughout a simulated two-phase (water liquid-vapor) system. Because of the isotropic symmetry of the phases involved and because of the symmetry of the computer simulation geometry, the average
(4)
@(z) - @(-L/2) = 4 7 1-L/2 ‘ dz”(z”- z)p,(z’)
(7)
Further assuming the limit A@ 1imLdm [@(L/2) - @(-L/2)], we have
A@ = 4 7 1-m- d ~zp,(z)
(8)
This can be viewed as a “gravitational formula” for A@ which is thereby identified with the dipole moment of the charges in the interfacial region.23 From the point of view offered by this result, the experimental determination of A@ can be reduced to the measurement of a mechanical density, p,(z). This formula does have some unexpected properties, however. Two of those surprising properties are discussed next. Some Properties of the Mechanical Definition of A@. The mechanical definition (8) relates the surface potential directly to the dipole moment of the charges in the interfacial region. In considering this formula, we first defer consideration of free ionic species and inquire how the solvent contribution to the surface potential is related to molecular dipole moments. An analysis of this question was given in ref 24 with the result that A@
@(a)
- @(7) = 4 7 l’P,(z) dz - (4*/3)A(Tr Q) (9)
(23) See: Landau, L. D.; Lifshitz, E. M. Electrodynamics of Continuous Media; Pergamon: New York, 1960; Section 22.
Feature Article where P,(z) is the z component of the density of molecular dipole moments at the elevation z, with the z axis pointing from phase y to phase a,and A(Tr Q) is the difference between the traces of the molecular quadrupole moment densities in the two phases, CY and y.24 For example, if the molecular constituents were rigid so that any molecular quadrupole moment, Qmol,would be the same in each phase, then we would have A(Tr Q) = (Tr Qmo,)Ap with Ap = p(") - p ( Y ) the density difference between the two phases. It was initially a surprise to find out that the density of molecular dipole moments does not fully determine the solvent part of A@. This formula notes that an interface separating fluids composed of nonpolar molecules can support a sizable electrostatic potential difference. The three upper curves of Figure 4 show how the computed surface potential for the TIP4P model water at 320 K would change with changes in the molecular quadrupole moment if the simplifying assumption were made that alteration of the surface structure with changes in the quadrupole moment can be neglected.Isb This assumption will not be correct in detail. However, the reasonable conclusion which may be drawn from the additional results of Figure 4 is that modest alterations of the molecular charge distribution of the TIP4P model of water could bring the computed surface potential into agreement with estimates of that quantity based on available experiment^.^^^^^^**^ (Recent work26 has presented a comparison of the computer experimental results with more traditional calculations.) The quadrupolar contribution of eq 9 appears not to contain any specific information about the interfacial region but instead reflects the overall density change. If the molecular polarization vanishes, for example if strictly nonpolar molecules are considered, then this is an accurate view. It should be particularly noted, however, that each of the terms on the right side of eq 9 generally depends on the origin of the moleculefvred axis system from which the molecular multipole moments are reckoned. Equation 8 makes it plain that the sum of these terms is independent of that molecular origin. For the water liquid-vapor interface, a natural choice of that molecular origin is the position of the oxygen atom of the H 2 0 molecule. When this choice is adopted, the two terms on the right side of eq 9 are comparable in magnitude, but they contribute oppositely. This provides a natural interpretation of the small values for the solvent part of A@ which has been repeatedly suggested from experimental work on the free surface of liquid water.14J5b~2s Furthermore, together with the small values estimated for the solvent contribution to A 9 , the establishment of this quadrupolar contribution greatly bolsters our confidence in the conclusion that the sense of the molecular polarization (utilizing the oxygen atom as the molecular origin) of this surface orients the molecular dipoles away from the vapor phase and toward the liquid phase.2630 The significance of the quadrupolar contribution is consistent with the results of second harmonic generation studies of the orientation of water molecules at this surface.30 However, the influence of ionic species on A@ is important. In fact, recently it has been clearly stated that ionic contributions to A@ do not vanish, in general, as the ionic concentration tends to zero.I6 This is the second unexpected property of the basic formula (8) which we consider. Just as with the contribution of molecular quadrupole moments which was discussed above, this behavior can be established on general grounds.31 However, in
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 29 12
+ 10
-
_*___._..._..-.
a6-
...' .-...........----.--
41 -3
-2
-1
0
1
2
3
U
Figure 5. Model calculation of ionic concentration profiles over a length scale K-I, the screening length for the ionic correlations, which is much larger than molecular sizes. A simple 1-1 salt is considered here. See ref 32.
the interest of simplicity we discuss here the particular case of a generic 1-1 electrolyte MX in solution at a thermodynamic state of coexistence of two fluid phases, CY and y. We anticipate that any ionic contribution to the surface potential will be of special interest, and we note that the conditions of equilibrium as they relate to the solute concentrations are At sufficiently low concentrations, these chemical potentials can be expressed in the form pp)=
kBT In
x?) + Apia)(p,T ) + qj@(a)
(1 1)
In solving the conditions of equilibrium, eqs 10, with this form we must remember that the compositions will render the bulk phases electrically neutral so that xM+(") = xx-("),in the present case, and similarly for the phase y. Therefore, if one of eqs 10 is solved for the equilibrium compositions, say for the ratio
and then this information is used in the second member of eqs 10, that second member of eqs 10 will not be satisfied unless A@ has a special value. That special value can be obtained in this case simply by subtracting the second of eqs 10 from the first. We then obtain
or (3 1) A more general view of the process which leads to eq 12 for the special case is obtained by supping that the system contains c constituents,generally charged species, and Y phases in coexistence. The set of equations expressing the conditions of equilibrium can be augmented by the conditions that the bulk composition of each coexisting phase conform to electrical neutrality, that is E
zq,xp = 0
j- 1
(24) Wilson, M. A.; Pohorille, A,; Pratt, L. R. J. Chem. Phys. 1989, 90, 521 1. This paper provides references to a host of related molecular calcula-
tions on aqueous solution interfaces. (25) Farrel, J. R.; McTigue, P. J . Electroanal. Chem. 1982, 139, 37; 1984, 163, 129. Borazio, A.; Farrel, J. R.; McTigue, P. Ibid. 1985, 192, 103. (26) Yang, B.; Sullivan, D. E.; Tjipto-Margo, B.; Gray, C. G. Molecular Orientational Structure of the Water Liquid/Vapor Interface. Preprint, 1991. (27) Fletcher, N. H. Philos. Mag. 1962, 7, 255; 1963,8, 1425; Sci. Prog. (Oxford) 1966, 54, 227; Philos. Mag. 1968, 18, 1287. (28) Stillinner, F. H.; Ben-Naim. A. J. Chem. Phvs. 1967. 47. 4431. (29) Croxton, C. A. Phys. Lett. 1979, 74A, 325; Physica A 1981; 106A. 239. See also: Croxton, C. A. Statistical Mechanics of the Liquid Surface; Wiley: New York, 1980 Chapter 7. (30) Goh, M. C.; Hicks, J. M.; Kemnitz, K.; Pinto, G. R.; Bhattacharyya, K.; Eisenthal, K. B.; Heinz, T. F. J. Phys. Chem. 1988, 92, 5075.
for each phase a. In counting the variables to be determined, the potential in each phase should then be considered, too. Because of the linearity of eq 1 with respect to the solutions of these equations depend only on the differences AcP between phases. With this accounting, the phase rule appears as f = c - Y + 1. This can be understood either as inclusion of (Y 1) additional variables together with Y additional constraints, f = c - Y + 2 ( y - 1) - Y , or as a reduction by one of the number of independent composition variables in each phase, f = (c - 1) - Y 2 . For example, a mixture of H+ and 02- ions, Le. HzO,can support no more than three coexisting phases at once: c = 2,f = 3 - Y. If none of the species is charged, then the condition of electrical neutrality provides no constraint. But these arguments do lead us to the conclusion that the difference in the electrostatic potentials of conducting phases in equilibrium with respect to exchange of charged species depends only on the thermodynamic state.
+
+
30 The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 A@ 5 1 = -(Aji,+(”(P,T) 2e
@(a) - @(Y)
- AjiM+(U)(p,T)- Aji,-(Y)(P,T)
+
APx-(a)(P,T)) (12 )
A common, idealized point of view is that the solvent molecules are permanent neutral units and ionic species are more dilute solutes. From that point of view, it can be observed that eq 12 is nontrivial even in the limit of infinite dilution: This ionic effect is there “even though the ions are not”. Notice also that the size of this ionic contribution depends essentially on the chemical forces on the ionic species in both phases. Therefore, there is little of a general nature that can further be concluded about this influence of ionic species on the contact potentiaL3I The result for the surface potential as extracted from the conditions of equilibrium and the neutrality of bulk compositions3’ does not depend directly on the structure of the solution interface.I6 If the composition of a molecular liquid is described with the concentrations of ionic species which can form neutral solvent molecules by combination, then the approach exemplified by eq 12 provides a separate way to calculate the surface potential. Such a description can be validly constructed whatever the positive extent of dissociation of solvent molecules. A more physical statement of the argument above is the following: Although the chemical forces on an ion and its counterions bear no simple general relation to each other, the relative solubilities of ions and counterions between two phases must precisely track each other according to the simple general prescription of electrical neutrality. Nature achieves this remarkable trick by sorting ions in the interfacial region in order to set up just the A@ which will permit the bulk compositions of both phases to be electrically neutral. Figure 5 gives an example of a model calculation of this behavior, a calculation drawn from a study of the concentration dependence of the surface tension of an electrolyte solution.32 In that previous work, this phenomenon of a nonzero ionic contribution to A 0 in the infinite dilution limit was used to establish that the dependence of the surface tension, u, on the salt concentration, x, was generally as u - no a x ’ / ~not , as u uo a x In x which is an often considered a l t e r n a t i ~ e . ~Here ~ uo is the surface tension of the neat solvent. The results of Figure 5 also exemplify the expected consistency between the two methods of determining the surface potential, namely, either obtaining high-resolution structural information on the solution interface and using eq 8 or determining single-ion activities and relying on the derivation which leads to eq 12. The paradoxical statement that the ionic contribution is there “even though the ions are not” is clearly slightly misleading. This outcome is achieved by the ionic interfacial concentration profiles extending to a range comparable to the screening length for ionic correlations, K - ~ , as the concentration is reduced. At the same time, the magnitude of the bulk concentrations diminish with x. Because of the two factors of z“ in the integrand of eq 8, when the concentration is low enough the overall concentration dependence of that integral comes down to x / K ~ ,which does not vanish at infinite dilution. Thus, physically infinitesimal levels of the ionic impurities can have finite effects on a contact potential. A similar comment applies to the solvent composition profiles: We expect those solvent interfacial profiles to be altered over ranges proportional to K - I by amounts proportional to the ionic concentration, x . The question “which ions are not there?” arises immediately after we note that the ionic contribution to the contact potential is there “even though the ions are not”. Consideration of this question leads to the conclusion that the limiting value of AO will generally depend on the compositional path chosen in the approach (32) Nichols, A. L., 111; Pratt, L. R. J . Chem. Phys. 1984, 80, 6225. (33) It is worth additional emphasis that the distinction between these two alternatives for the limiting law was tied directly to the equilibrium of ionic material between coexisting fluid phases, a situation that can be of practical importance. For examples, see: The Inregace Structure and Electrochemical Processes at the Boundary between Two Immiscible Liquids; Kazarinov, V. E., Ed.; Springer-Verlag: New York, 1987.
Pratt to infinite dilution. This conclusion is clearly consistent with the physical picture presented above. The formal character of these conclusions regarding ionic influences on contact potentials should be underscored. A finite effect arises from an infinitesimal cause, in this case, because ranges of integration which appear, for example, in eq 8 are formally extended to infinity. This does not conform literally to the physical reality of actual experiments. This will be important when K-I becoma appreciable to a linear dimension of the system in which case the size and shape of the system must be considered. This more global but explicit view then reinforces the physical point that electrical conduction provides the general mechanism for equalization of the electrostatic potential throughout the interior regions of the system. CooPisteocy of the Meiclnojcpl pnd Thermodynarmcal Definitiaop of the Surface Potential. In this section we comment more specifically upon the consistency between the two approaches for introduction of surface potentials. The objective is to identify more clearly the fact that the two approaches develop from a single physical assumption: that the macroscopic electric field vanishes in the interior of the bulk phases. We will see that the two different approaches to the surface potential develop this assumption is different ways, however. For conducting phases, the assumption that the electric field vanishes throughout the interior of the material can be supported by thermodynamic considerations. For the present discussion, however, we will regard absence of a macroscopic electric field in the bulk merely as an assumption. This assumption, together with Poisson’s equation, leads to the structural result eq 8 when a planar geometry is specified for the interface.23 Additionally, when we use the assumption that the macroscopic electric field vanishes throughout the interior, Poisson’s equation 47rp,(r) = V.E(r) requires that the bulk composition of the coexisting phases be neutral. The latter observation gives a justification of the electroneutrality condition which was used in the thermodynamic derivation of the contact ~ t e n t i a l . ~If’ the assumed form of the chemical potential, eq 1, is shown to be correct, and if the phases in question are in equilibrium with respect to charged particle transport, then the thermodynamic development of the contact potential will have been derived from the condition that the mean electric fields are zero in the bulk phases. But the correctness of the form eq 1 is rather transparent. This is easiest to discuss explicitly on the basis of the potential distribution theorem of classical statistical mec h a n i c ~ , ~although ~ , ~ ~ more general treatments are possible. According to this basic principle, chemical potentials can be determined from the formula Aj is the thermal de Broglie wavelength of a particle of type j , and pj(O) is the number density of that species in phase a. The right side of this equation is to be understood in the following way: AU is the change in the potential energy of the system due to the insertion of a test particle of type j at an arbitrary point on the homogeneous bulk phase a;the brackets indicate the average over the thermal motion of the system unaffected by the test particle. That this formula takes the form of eq 1 can be noted explicitly by considering the formal evaluation of the average via a cumulant e ~ p a n s i o n . ~The ~ first quantity to be considered in that calculation, ( AU), will include q,O@),Le., the electrostatic potential energy of a particle of charge qj when that particle is treated as a test particle. In a detailed calculation, the potential for each of the configurations visited by that thermal motion would be determined from the positions of real charged species together with boundary conditions via Poisson’s equatione6 (34) Widom, B. J . Chem. Phys. 1%3,39,2808. Widom, B. J . Stat. Phys. 1978, 19, 563. Widom, B. J . Phys. Chem. 1982,86, 869. (35) Jackson, J. L.; Klein, L. S. Phys. Fluids 1963, 7, 279. (36) Kubo, R. J . Phys. SOC.Jpn. 1962, 17, 1100.
Feature Article In bringing this discussion of consistency to a close, we note that there are physical situations where there is important and well-founded interest in a contact potential but where the thermodynamic development above cannot be expected to be consistent with the mechanical formula. One case is Lippman's electrometer where, although both phases are conductors, equilibrium with respect to independent transport of charged species is not produced-the impressed potential difference is "too weak to produce a lasting current".37 In that case the electrostatic potential difference between the phases can be varied, and the appropriate thermodynamic treatment leads to the Lippman-Gibbs equation of ele~trocapillarity.~~-~~ But the mechanical formula still is expected to be appropriate. A second case occurs when the phases in question are not conductors, but it is established by some extraneous process that the macroscopic fields are zero in the interior of the phases. Fluid dielectric phases often fall into this category: just as in the first case, equilibrium with respect to independent transport of charged species between the two phases is not achieved. The thermodynamic development above does not apply directly, but the mechanical formula is still expected to be appropriate. Physical Identification of the sdvent Contributionto tbe Surface Potential The discussion above separately considered solvent and ionic contributions to the contact potential. From the point of view of the mechanical formula (8), such a decomposition is quite natural. The charge density can be separated into several contributions, and provided that the integral of eq 8 still makes sense for each of those contributions, the desired contact potential is the sum of the potentials due to each contribution. In contrast, a separation into solvent and ionic contributions seems much less natural when the thermodynamic point of view is adopted. In the thermodynamic view, the contact potential is established by solvation of the ions in the bulk phases. Peculiarities of the interfacial structure of the solvent in the absence of conducting solute materials does not seem directly relevant. Here we argue that the solvent contribution to the surface potential could be isolated by physical measurements under the circumstances that the ionic solutes are extremely dilute. However, a t the same time it must be emphasized that this solvent contribution is not the A@ which appears in thermodynamic formulas such as eq 12 even in the limit of infinite dilution. The physical idea for separating distinct contributions to the contact potential is that, when the electrolyte concentration is low, ionic contributions can be distinguished by the spatial ranges involved. Nonthermodynamic measurements of the potential drop across a solution interface will typically determine the potential drop across a layer of finite thickness, call it 1. For such a measurement to make sense, 1 must be considerably larger that the interfacial width, w,identified with ionic contributions excluded. For sufficiently low concentrations, this identification of w is possible because the natural length scale for ionic correlations is much larger, K - I >> w. The electrostatic potential difference acras a layer of thickness 1 should be insensitive to that thickness when I is substantially larger than molecular sizes. In the limit of infinite dilution this insensitivity should also obtain when K-I >> 1 >> w. The measured potential difference can, in that case of infinite dilution, be identified with the surface potential of the solvent. The delicacies associated with ionic contributions to surface potentials lead to the pragmatic view that ionic species present an extreme instance of a commonplace aspect of the study of surfaces: levels of impurities which have negligible influence on bulk behavior can have a nonnegligible effect on surface properties. There are at least two practical steps which can be taken to mitigate this difficulty for electrolyte solutions. The first step is to add background salt. This ensures that the system will have (37) Collected Works ofJ. Willard Gibbs; Yale University Press: New Haven, 1948; pp 335-337. (38) See: Landau, L. D.; Lifshitz, E. M.Electrodynamics of Continuous Media; Pergamon: New York, 1960; Section 24. (39) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, Fundamentals and Applications; Wiley: New York, 1980; Chapter 12.
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 31
En=-K2k2
Figure 6. A generic electron reflectivity experiment. In the interpretation of these experiments by analogy with optical reflectivity:' n(Eo)is the index of refraction of electrons of incident energy Eo.
an appreciable conductivity and that the electrostatic potentials will be relatively insensitive to the presence of uncontrolled impurities provided those impurities are not in some other way surface active. Attention to levels of added background salt has the second but related effect of controlling the spatial range over which ionic contributions come into play. The second step which can be taken to deal with ionic contributions to surface potentials is to devise experiments which probe electrostatic potential changes within a layer of finite but variable spatial thicknesses surrounding the interface. In the next section we discuss nonthermodynamic methods for measurement of surface potentials.
Measurement of A@ To make the theoretical arguments above more definite, we now consider how measurements of A@ might be designed. We note that Oppenheim proposed one such experiment some time ago.6 Although we focus on the mechanical formula and leave aside the thermodynamical defmition of contact potentials, an important question for such experiments is whether the potential differences obtained are only a property of the bulk thermodynamic conditions or whether they depend on factors specific to the interfacial regions. If the surface structures can be changed without altering the bulk thermodynamic state, then the measured potential differences should be unchanged if the two theoretical formulations are to be consistent. The developments above lead to the conclusion that a measurement of the interfacial charge density profile can provide the surface potentials which have been discussed. The measurement of the intensity of particles reflected from an interface provides an important general approach to the determination of interfacial density profiles. and X-ray4W8 reflectivity are well-recognized methods for probing interfacial structure. The scattering of neutrons from interfaces is also a well-appreciated technique for the structural studies of surfaces.4+54 The discussion (40) Huang, J. S.; Webb, W. W. J. Chem. Phys. 1969,50,3677. Wu, E. S.; Webb, W. W. Phys. Rev. A 1973,8, 2065,2077. Wu, E. S.; Webb, W. W. J. Phys. C 1972,-33,149. (41) Katyl, R. H.; Ingard, U. In In Honor of Philip M . Morse; Fwhback, H., Ingard, K. U., Eds.; MIT Press: Cambridge, MA, 1969; pp 70-87. (42) Zollweg, J.; Hawkins, G.; Smith, I. W.; Giglio, M.; Benedek, G. B. J . Phys. C 1972, 33, 135. (43) Bouchiat, M. A.; Meunier, J. J . Phys. C 1972, 33, 141. (44) Langevin, D. J. Chem. Soc., Faraday Trans. 1 1974, 70, 95. (45) Byme, D.; Earnshaw, J. C. J . Phys. D Appl. Phys. 1979,12,1133. (46) Braslau, A,; Deutsch, M.;Pershan, P. R.; Weiss, A. H.; Als-Neilsen, J.; Bohr, J. Phys. Rev. Lett. 1985, 54, 114. Als-Neilsen, J. Physica 1986, 140A, 376. Als-Neilsen, J. In Structure and Dynamics of Surfaces II; Schommers, W., von Blanckenhagen, P., Eds.; Springer-Verlag: New York, 1987; pp 181-221. Kjaer, K.; Als-Nielsen, J.; Helm, C. A.; Tippman-Krayer, P.; Mohwald, H. J . Phys. Chem. 1989, 93, 3200. (47) Rice, S. A. In Springer Series in Surface Sciences; de Wette, F. W., Ed.; Springer-Verlag: Berlin, 1988; pp 129-136. Rice, S. A. Proc. Natl. Acad. Sci. U.S.A. 1987, 84, 4709. (48) Liang, K. S.; Eisenberger, P.Mater. Rec. SOC.Symp. 1987,82,493. (49) Koester, L. In Neutron Physics; Springer-Verlag: New York, 1977; Section 3.6. (50) Werner, S. A.; Klein, A. G. In Methods of Experimental Physics; Skold, K., Price, D. L., Eds.; Academic Press: New York, 1986; Vol. 23, Part A, Chapter 4. ( 5 1) Farnoux, B. In Neutron Scattering in the Nineties; IAEA: Vienna, 1985; pp 205-209. ( 5 2 ) Sinha. S. K.: Sirota, E. B.; Garoff, S.; Stanley, H. B. Phys. Rev. B 1988, 38, 2297.
32 The Journal of Physical Chemistry, Vol. 96, No. 1 , I992 here focuses on the use of electrons (or positrons) for reflection measurements because these particles directly scatter from the electrical potentials which are to be studied. When the surface reflectivities are studied with charged particles, those reflectivity experiments might be considered an adaptation of ionizing electrode methods for investigation of surface potential^.^^^^'^ Of course, electron diffraction is a well-known experimental method in the crystallography of solid surfaces in good vacuum^.^^-^^ The primitive idea for electron reflectivity experiments is depicted in Figure 6. In an ideal case, we might expect total external reflection of elastically scattered electrons when the scattering angle 8 is less than the critical angle BC = (-eA@/Eo)1'2
Pratt
1
1
1
(13)
provided A@ < 0. The magnitude of the electronic charge is e. If A@ > 0, then the experiment should be done with positrons; this changes the spin appearing on the right in eq 13. Positron sources are well-known, and the necessity of considering reflection of positrons also is, therefore, not an insurmountable obstacle.58 However, for simplicity of discussion we will continue to refer to the scattered particles as electrons. Equation 13 is valid when the scattering angles are small. If the electron energies are in the range 1-10 keV, then critical angles will be in the range 0.1-0.001 '. Since electron penetration depths vary with kinetic energy Eo,it is important that experiments such as these investigate the dependence of the inferred critical angle on Eo,i.e., verify that the inferred surface potential is independent of the energies of the incident electrons.s9 It would be important to vary the incident energy and the scattering angle in the low-angle regime for the additional reason that the ideal interpretation is unliiely to be generally satisfactory. Generally, it will be essential to account for multiple scattering, inelasticities, and losses of several kinds, including those associated with ionization. Corrections for these effects will depend on the incident energy and scattering angle. Therefore, data over a range of incident energies and scattering angles should be helpful in making the appropriate corrections. Furthermore, an important task in implementing these experiments is to identify an energy range in which the necessary corrections are the least troublesome. For example, if the energies are too low, inelastic events are likely to dominate the scattering. Relatively high electron energies are likely to be preferred for this reason as well as for the longer penetration depth probed; considerations like these led to the suggestion above that the 1-10-keV energy range should be
2.
-10
-5
0
5
10
z (au)
Figure 7. Contact potential for a jellium model of a simple metal bilayer.64 The continuous curve of the upper panel is the electron density for the model; the step is the idealized positive charge density of the model. Parameters are chosen for an AI-Cu junction.
(53) Felcher, G. P. Proc. SPIE-Int. Soc. Opt. Eng. 1989, 983, 2. (54) Anastasiadis, S. H.; Russell, T. P.; Saioja, S. K.; Majkrzak, C. F. Phys. Reu. Lett. 1989, 62, 1852. Composto, R. J.; Stein, R. S.; Kramer, E. J.; Jones, R. A. L.; Mansour, A.; Karim, A.; Felcher, G.P. Physica B 1989, 156, 157,434. Karim, A,; Mansour, A,; Felcher, G. P.; Russell, T. P. Physica B 1989, 156, 157, 430. Russell, T. P.; Karim, A.; Mansour, A,; Felcher, G. P. Macromolecules 1988, 21, 1890.
( 5 5 ) Pendry, J. B. Low-EnerRy -. Electron Diffraction; Academic: New York, 1974. 156) Demuth. J. E. J. Colloid Interface Sci. 1977. 58. 184. i 5 7 j Laeallv.'M. G. A d . Surf. Sii. 1982. 13. 260. Leeallv. M. G. In Sohd StarpPh&s: Surfkes; Pa;k, R. L., Lagally, M. G., ids.: Academic: New York, 1985; Vol. 22, p 237. (58) Positron annihilation is a well-developed method for the study of materials. It is interesting to note a previous consideration of the use of positrons, positron annihilation in particular, to the study of liquid interfaces: Percus, J. K. In Positron Annihilation Studies of Fluids; World Scientific: Teaneck, NJ, 1988; pp 96-1 15. (59) Characterization of an "inner potential" can be an integral part of quantitative LEED experiments on crystals. For example: Lander, J. J. In Progress in Solid Stare Chemistry; Reiss, H., Ed.; Pergamon: New York, 1965; Vol. 2, pp 26-1 16. See particularly Section 4.9. It should be emphasized that the electric potential sought is not generally the binding energy of an electron divided by the electronic charge, neither is it the chemical potential of the electrons, the work function, or the Fermi energy. The foundations of these other energy parameters have been discussed recently by: Reiss, H. J. Phys. Chem. 1985.89.3783. Reiss, H. J . Electrochem.Soc. 1988,135,247C. Both "chemical" and "electrostatic" effects contribute to work function values, for example. The objective of the present discussion is to separate these contributions. To this end, one point of using high-energy electrons at low angles of incidence is to minimize chemical or relaxation effects which are likely to be more severe for the scattering of low-energy electrons near normal incidence.
Figure 8. Some experimental geometries for electron reflectivity studies of liquid interfaces.
considered initially.60 However, ionization of the material will become significant when the electron energies are too high. The necessary compromise, therefore, should be sought in an intermediate energy range. As a probe of condensed matter structure, electron scattering is a more difficult experiment than the scattering of X-rays or neutrons. The general reason for this is that electrons are scattered (60) Seah,
M.P.; Dench, W. A. Surf. Interface Anal. 1979, I ,
2.
Feature Article
The Journal of Physical Chemistry, Vol. 96, No. 1, 1992 33
very much more strongly than X-rays or neutrons. Aside from the corrections indicated above which must be considered for electron scattering data, it is essential that the path lengths of electron beams through condensed materials be extremely short.60 In the general case, the difficulties of these experiments are no greater than the (considerable) difficulties of studies of condensed matter structure with electron scattering.6143 These would typically be very difficult experiments. For this reason, we will not here further discuss more technical aspects of these experiments, including issues of data correction. It should be emphasized, however, that low vapor pressure materials with good conductivities can be expected to be optimum candidates for such experiments. Liquid metals might be suitable materials as well as solid metals prepared in layered structures. Firm estimates are available for the contact potentials of simple metal junctions. Figure 7 gives an example of a straightforward calculation for a jellium model of a AI-Cu bilayer structure.64 These calculations again exemplify the consistency which can be expected between the mechanical and thermodynamical definitions of the contact potential. Calculations such as these suggest that positrons should be used in order to see total external reflection from a liquid metal surface. The limitation to short path lengths probably eliminates the possibility of studying interfaces of near-critical fluids. For other cases, circumvention of this general limitation will typically require ingenuity in experimental design. Figure 8 gives an indication of some possible approaches for getting electrons to a liquid surface and then away again. The upper design of that figure takes advantage of the fact that electrons beams can be focused on small spots.65 The liquid can then be confined to a narrow pore. This will help to maintain the vacuums necessary to achieve acceptable beam current through the target region. The middle drawing of Figure 8 suggests how it might be possible to study liquid-liquid interfaces with this arrangement. The upper layer of liquid would be progressively pumped away during the course of an experiment. The analysis of this data would require the consideration of reflections from both interfaces as a function of that layer thickness. Finally, the lower drawing of Figure 8 shows a possibility for the study of solid-liquid interfaces for cases like SiOz in which the solid can be etched away to a thin film. The liquid phase would be located on the backside of the film. The vapor pressure of the liquid is then not a problem, but the mechanical stability of the solid layer would be the first concern. Again, it is important to note that the cross-sectional area of the interface could be rather Such a design would provide a very natural way to study, at an atomic level, the structural chemistry of contacts between glass electrodes and water.
Conclusions Consideration of differences between electrostatic potentials of coexisting phases on the basis of eq 1, or the general development3I which leads to eq 12 in that specific case,has sometimes led to the doubt that surface potentials are measurable quantiIf we were to agree that electrostatic potentials between chemically dissimilar phases were not measurable by thermodynamic manipulations, then we could not justify an interest in surface potentials on the basis of interpretation of thermodynamic measurements. However, manipulation of charges species at interfaces is an intrinsic feature of many chemical processes, and this fact provides a straightforward justification for the study of interfacial electric fields. This perspective leads to eq 8, which suggests that electrostatic potential differences between coexisting phases can be determined by interfacial structural information which is accessible in principle. Measurements of these surface potentials can be formulated in terms of the determination of a mechanical density, the interfacial charge density profile. Interfacial density profiles are measured by X-ray and neutron reflectivity experiments. In particular, X-ray reflection experiments can produce the electron interfacial density profiles and neutron reflection data can yield the interfacial density profiles for nuclear positions. Those alternative methods are presently better developed as general probes of condensed matter structure than is electron scattering. Therefore, those alternatives are probably presently better choices for general studies of the molecular-level structure of solution interfaces. However, elastic reflection of electrons directly interrogates electric potentials and is, therefore, of special interest if information on interfacial electric fields is sought. Because electrons are so strongly scattered by condensed matter, the use of electron reflectivity for the study of liquid surfaces has not been pursued previously. The electron reflectivity experiments discussed above would be very difficult typically. But it is also clear that they are feasible experiments in the favorable cases, particularly where the interface under study separates a liquid from a low-pressure vapor. With respect to the particular motivations for the general work discussed above, it is worth emphasizing that electrostatic interactions are a common feature of surfactant coatings between water and other immiscible liquid phases such as hydrocarbon liquids. However, the localization of the head groups of ionic surfactants in an interfacial region is a result of a compromise between hydrophobic forces which would tend to draw the tail group, considered in separation, into the bulk hydrocarbon phase and the opposing hydrophilic forces which would tend to draw the head group into the bulk aqueous phase. It is clear that better molecular-resolution characterizations of hydrophilic and hydrophobic forces, both separately and together, would significantly boost our ability helpfully to m o d e P 7 j scientifically and technically important chemical processes at solution interfaces. Acknowledgment. The calculations discussed above were carried out during recent years with several skillful collaborators: A. Pohorille, A. L. Nichols 111, and M. A. Wilson. I thank these collaborators for their contributions. Ideas for experimental determinations of interfacial electric fields have been tried out on several experimentalist colleagues, particularly W. P. Ellis and J. A. Martin. Their good suggestions have been incorporated into several aspects of the experimental ideas discussed. I thank them for their time and thought. I also thank H. C. Andersen and J. C. Wheeler for helpful discussions of these issues.
(61) A short review of the history of electron diffraction studies of liquid structure is given in: Kalman, E. Inst. Phys. Con$ 1978, 41, 423. (62) Kalman, E.; Palinkas, G.; Kovacs, P. Mol. Phys. 1977, 34, 505. Palinkas, G.;Kalman, E.; Kovacs, P. Mol. Phys. 1977, 34, 525. Kabisch, G.; Kalman, E.; Palinkas, G.; Radnai, T. Chem. Phys. Lett. 1984, 107, 463. Kalman, E.; Serke, I.; Palinkas, G.; Zeidler, M. D.; Wiesmann, F. J.; Bertagnolli, H.; Chieux, P. 2.Naturforsch. 1983, 38.4, 231. (63) Brah, A. S.; Dobson, P. J.; March, N. H.; Unvala, B. A.; Virdhee, L. Inst. Phys. ConJ 1978, 41,430. (64) Pratt, L. R. Unpublished work, 1990. These results were produced with the optimized Thomas-Fermi model of Pratt, Hoffman, and Hams. See: Pratt, L. R.; Hoffman, G. G.; Harris, R. A. J . Chem. Phys. 1988,88, 1818. Hoffman, G. G.; Pratt, L. R. In Proceedings of the International Workshop on Quantum Simulation of Condensed Matter Phenomena; Doll, J. D., Gubernatis. J. E., Eds.;World Scientific Publishing: Teaneck, NJ, 1990; pp 105-11554. Pratt, L. R.; Hoffman, G. G.; Harris, R. A. J. Chem. Phys. 1990, 92, 6687. Hoffman, G . G.; Pratt, L. R. Statistical Theories of Electron Densities: Multiple Scattering Perturbation Theory. Proc. Roy. Soc. A, in press. The results of Figure 7 were obtained from the study of a standard jellium model utilizing an Slater X a electron density functional Hamiltonian. Calculations such as these are commonplace. For another example see: Lang, N . D.; Kohn, W. Phys. B 1971, 3, 1215. (65) It is also worth noting a practical advantage that these electron scattering experiments have over X-ray or neutron scattering: The electron scattering experiments can be carried out with long available electron microscopy equipment. Large centralized facilities are not required.
ties.192314*16
(66) Dill, K. A.; Flory, P. L. Proc. Natl. Acad. Sci. U.S.A. 1980, 7,3115. (67) van der Ploeg, P.; Berendsen, H. J. C. J. Chem. Phys. 1982,76,3271. (68) Haile, J. M.; OConnell, J. P. J . Phys. Chem. 1984,88, 6363. Haile, J. M.; OConnell, J. P. J . Phys. Chem. 1986, 90, 1875. (69) Vacatello, M.; Yoon, D. Y. J . Chem. Phys. 1990, 92, 757. (70) Owenson, B.; Pratt, L. R. J. Phys. Chem. 1984.88, 6048. Teleman, 0.J . Chem. Phys. 198685,2259. (71) Jonsson, B.; Edholm, 0.; (72) Watanabe, K.; Ferrario, M.; Klein, M. L. J . Phys. Chem. 1988, 92, 891. Watanabe, K.; Klein, M. L. J . Phys. Chem. 1989, 93, 6897. (73) Egbert, E.; Berendsen, H. J. C. J . Chem. Phys. 1988, 89, 3718.
