J . Phys. Chem. 1993,97, 334-3349
3344
Dipole-Dipole Interactions near Interfaces M. Urbakh' and J. Klafter' School of Chemistry, Tel- Aviv University, Tel Aviv 69978, Israel Received: August 18, 1992
The nature of the pair interaction between dipoles embedded in a liquid in the vicinity of a nonmetallic interface is investigated in the continuum limit. It is shown that thedipole-dipole interaction can be significantly modified in the presence of a boundary in liquids characterized by a nonlocal dielectric function or in liquids whose dielectric properties change due to geometrical restrictions. Different limits are studied and relationships to experimental observables are discussed.
be written as
1. Introduction
Recent experimental and theoretical ~ t u d i e s l -demonstrated ~ that spatial restrictions can strongly affect the dynamic and thermodynamic properties of embedded liquids and molecules. The effects observed and predicted include the influence of interfaces on the pair-interaction energy between charged or uncharged particle^,^.^ translational and rotational diffusion of probe molecule^,^^^ and chemical reactions and energy transfer properties. The influence of boundaries on molecular properties in their vicinity is of course related to the nature of the intermolecular forces and to how they are modified near an interface. These modifications can directly influence processes such as adsorption and electron, or energy, transfer at interfaces. In this paper we investigate changes in the interaction between point dipoles embedded in a liquid near a nonmetallic interface when compared to the bulk liquid. The liquid is represented in the continuum approximation, in terms of a nonlocal dielectric function t(k,w), and we analyze the dependence of the dipoledipole interactionon the distance between thedipoles and between thedipoles and the boundary as well as on thedielectric parameters that characterize the interface region. Although the continuum framework does not explicitly include molecular level details, it enables the derivation of closed expressions that relate microscopic quantities to measurable observables such as dielectric functions.9-' * Continuum approximations and their modifications have been shown to be powerful in unraveling leading physical processes in complex systems. The electrostatic interaction energy between two dipoles is given by where T is the dipole-dipole interaction energy tensor and paand are the dipole moments of the participating molecules. When a uniform liquid is considered, this tensor is usually written in the following form
pd
(3) In our problem, therefore, the interaction tensor T in eq 1 is the sum (4) All the characteristic distances in the problem (the distance between the dipoles and the distances between the dipoles and the interface) are assumed to be smaller than the radiation wavelength in the liquid, so that retardation can be neglected and the Laplace equation applies for the calculation of electric field, E, and interaction tensor, T. We follow ref 13where we represented the liquid by a nonlocal dielectric function and calculated the change in the dielectric friction due to the presence of a boundary. In analogy to this previous work, the effect of the interface is introduced through the concept of additional boundary conditions.I4 The nonlocal nature of the liquid defines a length scale, A, which is a measure of spatial correlations in the liquid. This parameter can be estimated on the base of diffraction experimentsI5 and on molecular dynamic simulations of liquids.16J7 For instance, in aqueous solutions the correlation length, A, is of the order of the extension of local hydrogen-bonded clusters. The nonlocal description introduces short range order within the continuum model of the liquid, at least phenomenologically, and leads to dipole-dipole and dipole-boundary distance and pore size dependencies which do not appear in the case of a local dielectric function. In order to mimic the possibility that the liquid itself changes its dielectric behavior due to interaction with the b o ~ n d a r y , ~we % assume ~ ~ J ~ a region of modified liquid near the boundary. The influence of a modified surface layer of a liquid on ion-ion and dipole-dipole interactions at metal-electrolyte solution interface was previously considered and shown to provide interpretations to experimental observations of ionic adsorption in electrochemical systems which could not be understood in the framework of the traditional d e s c r i p t i ~ n . ~ . ~ ~ 2. The Model
Here tb(w) is the local dielectric constant of the liquid and r = (rr,r,, rz)is the vector connecting the two dipoles. The approach used here is similar to the one applied by us for studying dipole relaxation in liquids near boundaries,I3 an approach suitable for both staticanddynamic limits. Inorder toevaluate theinteraction energy near an interface, one has to find the electric field, E, produced by an oscillating dipole of moment p d at the point of location of other dipole of moment pa. The electric field, E,is the sum of a part arising from the dipole field directly, ECdiP),and a part, E(lnd),produced by the polarization due to the presence of an interface. The net electric field acting on the dipole pacan 0022-3654/93/2097-3344504.00/0
We now consider the model we use for the description of electromagnetic interaction of time-dependent point dipoles embedded in a liquid near a substrate. As in ref 13 we assume that the substrate is characterized by a local dielectric function, C,,b(W), and the liquid is described by the nonlocal dielectric function 1 -=--
1
e(k,w)
e.(~)
1
1
1
(=-=)=
(5)
Here t . ( ~ )is the short-wavelength dielectric constant of a bulk 0 1993 American Chemical Society
DipoleDipole Interactions near Interfaces
The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 3345
liquid, fb(W) = tb(k=O,w) and k is a wave vector. One usually assumes the Debye form of the local dielectric function, Cb(W) tb
- iws
fb(0) = 1 - ius where T is thecharacteristic relaxation time, although other forms may be chosen. A similar expression is used for c.(w). The quantity e. is close to unity and in the low frequency range which we consider in this paper the relationship It.(w)l > A. The situation is quite different at interfaces. The interaction tensor T(ind) which arises due to the polarization at the interfacial region is
I.
(1 1) where J0(klR) is the Bessel function of order zero. To follow the details of the dipoleaipole interaction near an interface, we will study the behavior of the interaction tensor, Tzz(ind), as a function of distances R, Zd and z,. In accordance with typical experimental conditions we consider the case where the distance between molecules in the direction along the interface, R, is larger than the other characteristic sizes, Zdr zarand A. In order to find the asymptotic behavior of Tzz(ind) for R >> Zd, za, A, it is convenient tocarry out in eq 1 1 a transformation from theoscillating function, Jo(klR), to the modified Bessel function of third kind Ko(rR) which decreases exponentially for large values of rR.22 Taking into account that Jo(k,R) = Re HO(l)(klR),where H o ( ~is) the Hankel function, and the fact that all functions in the integrand in eq 11 have no singularities in the region ( I m k l > 0, Re k l > 01 we can move the integration contour to the imaginary axis where Ho(l)(irR)= 2Ko(tR)/iu. After the transformation, eq 11 can be rewritten in the following form
2
In writing eq 12, it was taken into account that due to the presence of the exponentially decreasing function, Ko(rR), the integral in eq 12 converges for values r IR-1. Only the leading terms with respect to the parameter (rR) were conserved in the integrand in eq 12. Equations 8 and 12 have the following asymptotic behaviors of the interaction tensor TZ2:
Urbakh and Klafter
3346 The Journal of Physical Chemistry. Vol. 97, No. 13, I993 1. At Large Distances between Dipoles, R >> 4, z,, gA.
+
where g = Cb&b/{E.(&b Bb)]. As an example, a t the interface between silica and water €sub = 1 1.6, t b = 80 and t o = 1, and for low frequencies w < lo6 Hz the parameter g = 10. 2. In the Intermediate Region, a,z,, A z,, zd) if we add to expression 14 the term (8) which takes into account direct interaction between dipoles. The region of intermediate asymptotics, eq 14, with anomalously slow (Coulomb-like) decrease of dipole-dipole interaction with distance R exists only in systems with high values of the dielectric constants of the substrate, €sub, and of the liquid, €b, when t$ub/C. >> 1 and t b / t . >> 1, and therefore correspondingly the coefficient g >> 1. For instance at silica-water interface, as discussed in case 1, and for A 4 Asthe limit in eq 14 can apply for a wide range of distances 3 A < R < 20 A. Also at an interface between two immiscible liquids (for instance water-nitrobenzene system; for nitrobenzene 6, = 34.8) the slow R dependence of eq 14 can play an important role. 3. Dipoles Very Close to the Interface, z,, zd < A. When both dipoles are placed at small distances from the surface, Zd, z, < A, eqs 13 and 14 are simplified to
-
These equations are appropriate for the description of the interaction between dipoles located in the first few layers closest to the substrate, as well as the interaction between adsorbed molecules, z, = Zd LZ! rm,where rmis the molecular size. In the limit of c,,b 0 eqs 15 and 16 reduce to the results obtained for metal-electrolyte i n t e r f a ~ e .It~ should be noted that in this case the expression of the dipole-dipole interaction energy at large R is directly proportional to the bulk dielectric constant of a liquid, tb, in contrast to the traditional description obtained for a local representation of the liquid in contact with a substrate, in which the energy, HaT,20pd,is inversely proportional to c b
-
When the bulk dielectric constant of a liquid, tb, is larger than both tsub and e., eqs 15 and 16 can be written as
The different behavior in eqs 15-16 and 18-19 originates solely from our nonlocal description of the liquid and is essentially
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0 D 2 0 3 0 4 0 6 0 6 0 7 0 8 0 8 0 l o o
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Figure 2. Dependence of the energy of interaction between dipoles perpendicular to the interface, U = PdT::Pa, on the distance between them. Theenergy isnormalized by the function Lp = pdTz;Oh representing the traditional expression for the energy of dipole4ipole interaction (in pxpendicular orientation) inside a local medium near a substrate (T:;O is given by eq 17). Solid lines are the results of exact calculations on the base of eqs 8 and 12; dashed lines are long range asymptotes ( 1 5 ) and dotted lines are intermediate asymptotes (16) with account of direct interaction between dipoles (8). The calculations were carried out for the following values of parameters: Csub = 10, e. = 2, Cb = 80, Zd = z. = 1 A, (a, top) -A = 2 A, and (b, bottom) -A = 5 A.
insensitive to the details of the imposed boundary conditions.14q23 At large distances between dipoles (and Cb > &,b) the interaction tensor, eq 18, has the same form as the traditional expression, T,:, but with a reduced effective dielectric constant, e.. Our results demonstrate that nonlocal description of liquids leads to a nonuniform distribution of polarization fluctuations in the interfacial region reflected in the dependence of thedielectric response of the liquid on the distance from the substrate. A new characteristic length, the correlation length in a liquid A, appears in the problem. Comparison of eqs 15-1 8 with the resultsobtained in the model of the modified liquid layer at the substrate surface7 (see also the results of next section) shows that this effect corresponds to the formation of the interface layer with reduced dielectric constant, c.. The presence of such a layer reflects the structuring effect of a ~ubstrate.~J* The thickness of the layer is of the order of the characteristic liquid structure distance, A. Only when the dipoles are placed far beyond the interfacial layer, the traditional description of eq 17 applies. The structuring effect (interfacial hydration) gives rise also to hydration forces which are of crucial importance in the interaction and fusion of biological membranes and macromolecules. 1 2 * 1 8 We see that for all distances between dipoles our results differ from the corresponding local behavior, eq 17. The effect of nonlocality may lead toan enhancement in the interaction between dipoles. At large distances, R >> gA, the ratio of interaction tensors is T,,/T.,O = (€b/C.)2 which for water is of the order of
Dipole-Dipole Interactions near Interfaces 102-103. Similar nonlocal enhancement of dipole-dipole interaction a t large distances R was predicted in ref 24. The interaction a t the substrate-liquid interface can be larger than the interaction near a free substrate, as described by the following ratio, ( Tzz/ Tzzo(Cb = 1)) = Cb(Csub + I)/(e.2(caub + Cb))). We see the presence of a dielectric medium by no means weakens the dipole-dipole interaction. This is due to the pulling of electrostatic lines into the interfacial layer with the reduced dielectric constant, e.. It should also be mentioned that in a nonlocal medium instead of the general law (at R >> Zd,Za), eq 17, we have a more complicated behavior of the dipole-dipole interaction. This behavior shows a significant change of the form of T,,(R) a t a new characteristic length gA. The dependencies of the interaction on the distance between dipoles, R, calculated using eq 12 over the whole range of distances R and for different values of the system parameters are shown in Figure 2. Equations 13 and 14 describe also the interaction between dipoles placed beyond the microscopic surface layer with thickness A. For instance, when one of the dipoles c(d is inside this layer (Zd < A) and the second pa is outside (z, 1 A) we have
The Jourrd of Physical Chemistry, V O ~97, . NO. 13, 1993 3347
between dipoles with other directions of dipole moments can be considered in a similar way.
3. Substrate Modification There is some experimental and numerical evidence, that a liquid near a substrate may have modified p r ~ p e r t i e s . * . ~Close .~*.~~ to the interface the liquid may develop local structure due to interaction with the substrate, a structure which is characterized by different dielectric properties and can therefore manifest itself in different equilibrium and dynamic behaviors. Here we extend our results of the previous section and describe the liquid side by two dielectric functions which correspond to the modified liquid in the close vicinity of the boundary, cs(w), and the bulk solution, tb(W). Following the same type of calculations as in the previous section, we arrive at the following expression for the dielectric tensor, T,for dipoles in the surface layer
and
again a result which differs from eq 17. The interaction between dipoles with other orientations of the dipole moments p d and pacan be found similarly from eqs 8 and 9. In the case of dipoles with dipole moments parallel to the surface plane and to the vector R and placed at small distances from the substrate (zd = z, < A), the long range asymptotic behavior of the interaction has the form
where d is the layer thickness and
The asymptotic behavior of eq 23 can be studied along the same procedure as in the previous section. For instance, the interaction tensor for two dipoles perpendicular to the surface in the region of large distances R, between them, has the form
The first two terms in eq 2 1 constitute the classical result describing an attraction (for €sub > f b ) of two dipoles parallel to the interface between local media. The third term takes into account the influence of the liquid structure as mimicked by the length A. In contrast to the previous case, eq 15, here the interaction energy depends on the correlation length, A, even at large distances between dipoles. A comparison of eqs 15 and 21 shows that for large values of dielectric constants of the liquid, Cb, or of a substrate, €subr the interaction between dipoles with moments parallel to the surface is weaker than the interaction between dipoles perpendicular to the surface. In the limiting cases &b >> 1 or e b >> 1 eq 2 1 shows RS dependence of the interaction on distance between dipoles, different from the case of dipoles perpendicular to the surface where interaction energy decreases as R3.Again the inclusion of nonlocal dielectric properties leads to an enhancement of dipole-dipole interaction. But in this case the effect of enhancement is not so pronounced. In contrast to the case of perpendicular dipoles when for all values of parameters (cb, Csub, em, A, R, Zd, and z,) we had the usual repulsion between dipoles, now the type of interaction (repulsive or attractive) depends on these parameters. For given parameters characterizing the liquid and substrate (Cb, esub, e., A) the interaction energy may change sign as a function of the distance between dipoles, R. The typical dependences of the interaction between dipoles parallel to the surface on the distance R are presented in Figure 3. Interaction
which behaves as the corresponding eq 15 obtained in the framework of nonlocal description of a liquid. The comparison of these two equations shows again that the inclusion of a liquid structure even on a phenomenological level leads to the creation of the surface layer of modified liquid with dielectric constant C, and with the thickness d. This means that the nonlocal case discussed in the previous section leads essentially to an equivalent behavior but with a surface layer of a thickness of the order of the correlation length A with a reduced dielectric constant cS = e.. Other limits of eq 23 can be derived as well following the previous sections.
4. Conclusions We have investigated the influence of a nonmetallic interface on the interaction between point dipoles located near the interface in the liquid side. Both the liquid and the boundary are described in terms of the continuum approach by their dielectric properties. We have assumed that the substrate is given by a local dielectric function and the liquid by a nonlocal dielectric function which introduces a typical length A into the problem. The results show some new limits of the dipole-dipole interaction which originates from the nonlocal nature of the liquid. The results stronglydepend on the embedding and neighboring dielectric functions and display a rich range of behaviors which may be amenable to experimental
3348 The Journal of Physical Chemistry, Vol. 97, No. 13, 1993 I "
the F6rster typeZSand adsorption isotherms of molecules carrying dipole moments. The relationship between the energy transfer rate, wtr, from a donor molecule to an acceptor and the nature of the dipole-dipole interaction can be obtained on the basis of the golden rule expression (to be summed over all possible transitions)
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Figure 3. Dependence of the energy of interaction between dipoles parallel o n them. Theenergy tothe interface, U = p d T R ~ j ~ ~ ,thedistancebetween is normalized by the function Lp = jLdTRR'jLa representing the energy of interaction between dipoles in planar orientation inside a local medium near a substrate (TRR'is given by the first two terms in eq 21). Solid lines are the results of exact calculations on the base of eqs 8 and 9; dashed lines are long range asymptotes (21) and dotted lines are intermediate asymptotes. The calculations were carried out for the following values of parameters: €rub = 10, C* = 2, Cb = 80, Zd = za = 1 A, (a, top) -A = 2 A, and (b, bottom) -A = 5 A.
tests. In a more realistic case one has to account also for the finite size of molecules. This will introduce higher multipolar contributions. Estimates along this line have been done for interacting ionsZoshowing small corrections to the point ion approximation. Our studies demonstrate that in order to provide a correct description of the interaction between dipoles in a liquid near an interface, it is necessary to take into account the influence of the bulk liquid and not only the first few layers. There is a difference between dipoles interacting inside one monolayer of liquid molecules on a substrate-vacuum interface and in the first layer at a substrate-liquid interface. The polarization of the region in the liquid with thickness of the order of the distance between dipoles, R,m a y contribute significantly to the interaction between two dipoles at a surface. This fact should be taken into account in numerical simulations of the interfacial properties of liquid. The approach introduced in the paper can be used also in describing liquid-liquid interfaces where one takes into account the nonlocal properties of both liquids in terms of their structure parameter A. In such cases the functional form of the interaction tensor, eqs 13-16 is retained. On the basis of the calculations in section 3, we believe that the parameter A should, however, be replaced by an effective length characterizing the thickness of the surface layer of both liquids. Examples for cases where the modification of thedipole-dipole interaction due to the presence of a boundary directly related t o experimental observables are direct electronic energy transfer of
o
Here c(d is the dipole moment for the transition from the donor state ( q d i ) to the acceptor state I q d f ) , p a is the transition dipole moment of the acceptor; Er- Ei is the net change in total energy of the donor-acceptor pair and T is the dipole-dipole interaction energy tensor. All those limits discussed above which lead to an incoherent energy transfer process (which excludes the slowly decaying, Coulomb-like behavior) should apply to the energy transfer calculations through eq 26 and may therefore make it a method to relate the microscopic process of donor to acceptor energy transfer to the macroscopic dielectric behavior at the interface. However for most realistic cases the contribution of the dipoledipole interaction tensor to eq 26 will be in the highfrequency range where the effects of liquid structures and molecules are less pronounced. This may explain recent observation on energy transfer at silica interfaces which do not show marked differences when compared to energy transfer in bulk liquids.z6 The effect of the dipoledipole interaction on adsorption isothermscomes through the contribution to thechemical potential of the surface layer (A or d). For low concentrations one expects therefore that the slope of the isotherm (surface coverage vs concentration) will depend on the nature of the liquid through A and the dielectric functions in the interface region. For dipoles perpendicular to the surface the slope decreases as a result of the nonlocal nature of the liquid.
Appendix A The energy of pair interaction between dipoles is expressed with help of eq 7 through the potential @(e& induced by a unit point charge located at r = rd in a liquid. The potential @(rd,r) is a solution of Laplace equation which in a substrate side has a form
V2+(rd,r) = 0, and in solution can be written as
I
0
(A4)
In the substrate the potential has the form @(sub)(z,kl) = D exp(k,z), z
