Self-energy in Adsorption JAGADISHWAR MAHANTY Department of Theoretical Physics BARRY W. NINHAM* Department of Applied Mathematics, Research School of Physical Sciences, The Australian National University, Canberra, A.C.T. 2600, Australia AND Receiued 10th December, 1974 The concept of self-energy of a molecule, the dispersion analogue of the Born self-energy of an ion, is developed. The use of this concept in theories of adsorption and interfacial energies is discussed. When molecular size is taken into account, the theories of Lifshitz, Brunauer, Emmett and Teller, restricted adsorption and the Hill theory all emerge as special cases. In general a simple power law of the isotherm is not always appropriate for multilayer adsorption. The concept of self-energy takes on significance whenever an object has a finite extent or is delocalised.For then the abstraction that the object can be considered separately from its surroundings becomes philosophically tenuous, as one part of the object can consider its other parts to belong to the rest of the world: hence, perhaps, the uncertainty principle. No difficulty occurs if the environment or object are immutable. If the opposite holds, the reaction of the (changed) environment to the object will be different and the sev-energy due to this reaction field will be different. The shift in self-energy due to radiative corrections to energy levels is a central problem of quantum electrodynamics. The Born 9 (electrostatic) self- energy of an ion is important in electrolyte theory, physical adsorption and in the migration of ions through membranes.Debye-Huckel theory results from the change of Born energy due to other ions. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager’s theory of the dielectric constant for dipolar fluids. Equally, in any theory for the surface energy of water, or adsorption of dipolar molecules, the self-energy of a dipole as a function of its distance from the interface must be invoked. In adsorption proper the same self-energy for an ion appears in the partition function. Once this is determined, so is the adsorption isotherm, and the change in interfacial tension due to dissolved ions.5 The point of this preamble is to stress that new developments reviewed here, which may appear mathematically abstruse or obtuse, stem from one fundamental physical concept.We shall focus attention on consequences for dispersion forces and adsorption which come about provided we admit that molecules are not points. Semi-classical aspects of such radiative effects have received a fair amount of attention lately: and the desirability of further work is clear. To fix ideas, consider physisorption from a gas. The self-energy problem arises as follows. Recall that for many layer adsorption continuum theory gives an isotherm ’-lo P l (noti-retarded); In - cc - (retarded), P , l4 1314 SELF-E NER G Y IN ADS 0 R P TI ON where AH is interpreted as in Lifshitz t h e ~ r y . ~ This form had been in dispute, but recent comparison of experiment l1 and theory l2, l3 for liquid helium leave little doubt that Lifshitz theory is quantitatively correct here, in spite of continuing com- putational difficulties due to incomplete spectral data.Additional support comes from a priori calculations on spreading of hydrocarbons on water 14* l5 and on sapphire.16 However, for very thin layers continuum theory breaks down, and a microscopic theory is necessary. 7-1 We require the molecular partition function exp( - E,/kT) where EI is the energy required to bring an adsorbate molecule from the jth absorbed layer to the gas phase. Because of the divergence in dispersion energy ( c ~ l Z / ~ ) near an interface, the molecular partition function had to be treated as a phenomenological term, severely limiting predictive capabilities. B.E.T. theory gives ln(P/P,) GC 1 /Z, and although it has seen good service, appears impossible to reconcile with eqn (1). The divergence difficulty is not peculiar to adsorption.Theories of interfacial tension, interaction energies (via either macroscopic or micro- scopic approaches), transport in inhomogeneous media, polymer adsorption and configurational problems, all require resolution of this problem. The introduction of finite size into the description of a molecule leads to a finite value for dispersion energy and the difficulty disappears. A parameter free theory of adsorption which reconciles existing theories can then be constructed. MOLECULAR SIZE A N D SELF-ENERGY The dispersion self-energy of a molecule of finite size can be defined 20-22 as the change in its energy due to its coupling with the electromagnetic field, or equivalently, as the change in zero point energy of the field due to its coupling with the oscillating dipole moment it induces on the molecule.We need to characterise molecular size, a notion which is hardly new. London 2 3 extended his theory of dispersion forces between point molecules to include extended electronic oscillators. More recently dispersion forces between large molecules has received much attention.24 We consider a single atom centred at R interacting with the radiation field, and follow an approach developed in several earlier papers.2o* 21 In Lorentz gauge, after Fourier transformation with respect to time, Maxwell’s equations become 1 (3) To close the equations we need a relation between the dipole moment density p(r, w) and electric field E(r, co), which for point molecules within the framework of linear response theory is E(P, co) = a(o)E(R, o)6(r-R), (4) where a(o) is the polarisability tensor of the molecule.This relation cannot be strictly correct, because the dipole is spread out over a region of the order of the volume of the molecule and in a semi-classical formulation it can be shown 20* 2 1 that the actual relation is ( 5 ) where ar(r-R, w ) is peaked around R with a range of the order of the size of the molecules, and can be given an explicit expression in terms of a sum over matrix p(r, a) = a(r-R, w)E(R, a),J . MAHANTY A N D B . W. NINHAM 15 elements of atomic wavefunctions. That is the end of the matter, for the solution of (2) is now G(r-r’, w)a(r’-R; co)E(R, co) d3r‘ s 4nio A(r,co) = -- C with G(r-r’ ; co) the diadic Green function solution of i.e. Putting (6) into (3), and taking r = R, we have a secular equation for the perturbed frequencies of the electromagnetic field : II+4n6(R9R; o)l = 0; 6 ( R , R ; o) = S d 3 r ” ~ , + V r V r ) G ( r r ” ; co)a(r”-r’; co).(9) Immediately then, the dispersion self-energy of the atom is d odw - lnlI+4nQ(R, R; o)l N 2fi d( trace 6 ( R , R; it). (10) fi ”=Gf du;, The effect of the spread in polarisation density is to make B(R, R ; is) convergent, and it is this spread, analogous to the current distribution in a two-level atom 2 5 which gives convergent results for radiative corrections to atomic energy levels. Now assume, for computational convenience only, that the atom is isotropic and that a(r, co) = la(co)f(r) where the form factorf(r) is a peaked function and take (a is the size of the atomic system) f ( r ) = In the non-retarded limit, we have r m - 1 exp( - 5) +a a2 * L I C 4 d< a(i5) N - rydberg (H atom) J n E, E! ~ n3a3 J o so that the dispersion self-energy is of the same order of magnitude as the binding energy, but the opposite sign.The same formalism permits extension to two or more atoms. The interaction energy, the difference between the complete energy of the coupled system and sum of dispersion self-energies of two isolated atoms reduces to London-Casimir results for IR1-Rzl % a, but remains finite at zero separation. For like atoms this energy is of the order of the binding energy of the molecule formed by them.The same concepts can be used to develop a simple semi-classical estimate of the Lamb shift in hydrogen,26 and to explain the differences in binding energy (face centred cubic versus hexagonal close packed) of rate gas crystals. 9 * A slightly different approach can be based on charge rather than polarisation distribution, and the formalism16 SELF-ENERGY IN ADSORPTION extended to include quadrupole and octopole effects.” Interestingly, for a three- dimensional oscillator form factor, the net potential curve appears to show a deepening of the “ bowl ” part of the potential which is necessary to fit the thermodynamic pro- perties of argon calculated by Monte Carlo For a many body system the same formalism holds. The sole difference is that the complete expression for dispersion energy now becomes 21 a 3Nx 3N matrix, the 0th submatrix of which Gi(Ri, Rj; co).If the molecules are considered to constitute a material medium, the same kind of analysis can be carried through to work out the dispersion energy, and interaction energy between different parts of the system in terms of the macro- scopic fields in a dielectric medium. The entire N-particle dispersion energy can be shown to be 21 E,(N) = - 47ci d o In1 I - 4n6‘M)I, where WM) is a 3N x 3N matrix with 3 x 3 elements 6iM)(R,, Rj ; a) constructed from Maxwell’s equations. We can now construct a theory of surface 2o and interaction 21 energies which goes beyond and substantiates those of other authors. 30-33 Liftshitz theory emerges in the limit that separation of bodies is greater than the size of any molecules, and the rigorous result that the Lifshitz energy between two bodies is the difference between the surface energy of the whole system and the sum of surface energies of the bodies taken in isolation follows.SELF-ENERGY OF A MOLECULE NEAR AN INTERFACE In an inhomogeneous medium, self-energy depends on position, and this change with position provides a force field acting on a molecule. To lowest order in WM> the expansion of eqn (13) gives E,(N as a sum of self-energies of individual molecules, that of the Zth being (14 E,(E) 21 - - 4dco trace 6(M)(Rl, R,; 0). i J The general expression is much more complicated, and this potential is analogous to that used in the theory of the inhomogeneous electron 35 For illustration, we exhibit this potential for a molecule near an interface of two dielectrics (1,2), in the non-retarded limit.Take the z axis perpendicular to the interface and medium 1 to the left. Then if r’ denotes source and r field point, we have ( 1 9 WM)(r, r‘ ; w) = -V,VptJ G(M)(~, r”)cI(r”-r’ ; co) d3r” with V2GCM)(r, Y‘) = Id(r-r’). Since we are now concerned with macroscopic fields, the boundary conditions are continuity of G(M) and ~ 3 G ( ~ ) / d z , and explicitly with 6(z) and sgn(z) step and signum functions and AI2 = ( E ~ - E ~ ) / ( E ~ + E ~ ) . This result is familiar. Recall that - 4 ~ G ( ~ ) ( r , r‘) is the electrostatic potential at r due to a unit charge at r‘, which includes direct and image potential, so that -44nV,V,GM)(r, r’) * This extension has been carried out by D.D. Richardson, J. Phys. A, 1975, 8, 1828.J . MAHANTY AND B. W. NINHAM 17 is just the field at Y due to a dipole at P'. the interface we find For a gaussian molecule at distance z froin E,(z) = k3 P a so dr . ( i e ) ~ ( ~ + 6 ~ ) - A ~ 2 ( ~ - 6 ~ ) x { exp( - $) - '+ erfc (t) + G3 ~ ' " " (exp - ( t 2 ) - exp - la)^) dt}] (1 8) n3a3 .s2 0 2A * .(it). dt-, ( z + +a) 1 1 1 --s -+ n+a EJr d t u(i<){-(-+-)-G(L-')}; 2 E l E2 3 E2 E l (z = 0). (19) Further, for large z, izl & a, the force on the molecule is the well-known result, e.g. ref. (36) It turns out that this is a good asymptotic representation when z 2 2a, but for z -+ 0 force and energy tend to a finite value. Such expressions provide very easily the change in interfacial tension at a liquid interface due to dissolved (non-ionic) molecules.PHYSICAL ADSORPTION We return to physical adsorption and the reconciliation of Langmuir, B.E.T. l7 and Lifshitz isotherms, and follow the formulation of de Boer 38 illustrated in fig. 1-3. Following an earlier paper 39 for a molecule MI in the first layer, with the zero of energy the self-energy in the gas phase at z = co, we have from eqn (18) A1 3 Ei(a) = -- tic' Jm d< a3(i&--, 7ra3 E3 where c1 z 0.2 is a constant whose value depends weakly on the form factor. the alternative model of fig. 3 eqn (1 9) gives the corresponding result. For The difference FIG. 1.-Schematic representation of adsorbed layers. Molecules MI, M2, M3 . . . sit on filled layers with dielectric properties of the bulk adsorbate18 SELF-ENERGY I N ADSORPTION Qint = -(Ead-Egas) = E:l)(O)-E,'l)(a) gives the activation energy which must be taken up by an adsorbed molecule to allow it to move over the surface, and can be estimated using typical U.V.data. For molecule Mj in the jth layer, we assign the Mi E3 E3 U € 2 E l zi = z j = ( 2 j - l ) a v 1 f , = O M I € 1 FIG. 2.-Model for calculation of self-energies. FIG. 3.-An alternative model. Here the adsorbed molecule nestles closely into the bulk adsorbate or liquid layer. underlying adsorbate dielectric properties of the bulk liquid. The corresponding self-energy which includes multiple imaging is much more complicated, but to lowest order in cx consists of (1) the energy of adsorption from the gasfphase onto an infinite medium 2, and (2) the interaction energy of a molecule 3 with medium 1 across a medium 2 of thickness (2j- 1)a.The last term gives the l/Z3 isotherm for thick films and is absent from B.E.T. theory. The first term does not appear in the theories of Halsey and Tgnoring entropic effects, the isotherm can be written in de Boer's notation 38 as j The symbols have their usual meaning and Q j = - Eik) is the heat of adsorption in thejth layer. With as above, it can be shown 39 that for largej, Q j comprises 4 terms: (a) the contribution to the surface energy/molecule of medium 2 at the interface [12]; (b) at the interface [23]; (c) the energy of interaction/molecule of a molecule at the surface of medium 3 with 1 across 2 ; (d) the energy of condensation required to take thejth molecule from gas to bulk liquid phase.We now do the sums and find (I) the Lifshitz isotherm emerges automatically under the condition I % ln(p,/p) & 1 / &, where k = 119 depends on molecular size. The second inequality is automatically satisfied by a continuum theory, but if it is reversed, and the conditions under which this can happen are clear, then (11) the isotherm becomes identical with B.E.T. theory : J . MAHANTY AND B . W. NINHAM 0- K The difference is that present theory contains only one parameter, go, the number of adsorption sites, rather than three. Evidently, depending on molecular size, pressure and dielectric properties, a wide range of behaviour can be expected in principle even for the special case of dispersion forces, and attempts to fit isotherms to a power law may be inappropriate.In general the more complicated theoretical expressions should be used. (111) At lower pressures restricted adsorption can, but need not necessarily emerge automatically. A step-wise isotherm need not indicate composite surfaces and depends on polarisability, size and dielectric properties in a predictable manner. (IV) We have ignored lateral interactions in the first layer. If the entropy of the first layer is included, and we make two extreme assumptions: (a) neglect terms in Eil) which involve the influence of the substrate, and (b) assume that at coverage 8 = o/ao the adsorbed layer has dielectric properties 6; = (1-8)+& where g2 is the bulk adsorbate dielectric constant, then the Hill isotherm also emerges.In general the Hill isotherm 37 is too drastic an approximation, and the formalism above permits the inclusion of the nature of the substrate which strongly affects the strength of the lateral interaction and condensation. We remark further on this point in the concluding paragraph. ELECTROLYTES AND CONDUCTION PROCESSES For completeness we remark briefly on effects of mobile ions or charge carriers (metals) in adsorption on electrically neutral surfaces. (For charged surfaces, while some answers have been obtained,40 much work remains to be done.) At finite temperature eqn (1 3) becomes the free energy arising out of dispersion interaction : 2nnkT C,, = T , 00 F = kT c' In D(i5,); n = O where D(iC,) is the secular determinant.With free charges present, the Green function which occurs in D(i5,) must be modified; e.g. in electrolytes if the source oscillates with high frequency, the ions are too massive to follow the field and con- tribute to the dielectric constant, but at low frequencies the ions respond to the field and modify dispersion interactions. For electrolytes, only the n = 0 term is modified. Many such problems can be handled through the linearised Poisson-Boltzmann For the term in zero frequency rather than (16) we have to satisfy Poisson's equation where the sum is over all ionic species of charge e, and density n, = n,(O) exp( - e+(r)/kT.20 SELF-ENERGY I N ADSORPTION Linearisation yields (27) Repetition of the same analysis then gives the change in interaction free energy between two molecules due to electrolyte as 22 (V" - rc;)+(v, v') = - 4n6(r- r'), l/rcD = Debye length.{6[exp( - ~ D R ) - 11 f AF(R) = -- kTa2(0) 2&2(0)R6 and a temperature independent change in the self-free energy of a neutral molecule which is 22 Electrostatic contributions to the free energy of an ion can be discussed within the same framework. This self-energy is not due to dispersion interactions, but due to interactions of the ionic charge cloud with itself in the presence of other ions, and is where p v is a form factor for the ion. Together with (27) this expression gives immediately the Born energy and Debye-Hiickel theory. With the interface problem, the self-energy of an ion at distance z can easily be calculated, and leads to an extension of Onsager-Samaras theory for the change in surface tension of water due to dissolved electrolyte.CONCLUSION In general in adsorption problems, both electrostatic (in double layers) and dispersion self-energies occur in the molecular partition function and are often equally important. It is usual to ignore one, or the other. The dispersion self- energy concept provides a unification of existing theories of physical adsorption. It goes somewhat further, for the distinction between physisorption and chemisorption now becomes much less clear, and there is some possibility that semi-classical ap- proaches of the sort outlined, with decoration, may provide a useful quantitative picture complementary to present theories of chemical binding. Dispersion self- energy has been used with some considerable success to predict theta points in polymer solutions in terms of dielectric proper tie^,^^ and phase changes in polymer adsorp- t i ~ n .~ ~ Finally we remark that few layer, lateral interaction, 2-dimensional con- densation problems in adsorption are accessible by these techniques. It will be normally sufficient to recognise that the first few layers have anisotropic dielectric properties. Extension of the formalism using formulae which give the anisotropy as a function of coverage, derived el~ewhere,~~ will provide the appropriate isotherms- * M. Born, Z. Phys., 1920,1,45. R. A. Robinson and R. H. Stokes, Efectrolvte Solutions (Butterworth, London, 2nd edn., 1959). P. Debye and E. Huckel, Phys. Z., 1923, 24, 185.L. Onsager, J. Amer. Chern. SOC., 1936,58, 1486. L. Onsager and N. T. Samaras, J. Chem. Phys., 1934,3, 528. M. 0. Scully and M. Sargent. Phys. Today, 1972, 25, 38. G. Halsey, J. Chem. Phys.. 1948, 16, 25. M. 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