7584 Langmuir 00, 18, 7584-7593 The Coupling between the Hyration an ouble Layer Interactions Eli Ruckenstein* an Marian Manciu epartment of Chemical Engineering, State University of New York at Buffalo, Buffalo, New York 1460 Receive May 8, 00. In Final Form: July 31, 00 The electric potential an the polarization between two charge, flat surfaces immerse in water are calculate without the usual assumption that the polarization is proportional to the electric fiel. The new constitutive equation accounts for an aitional interaction, ue to the orientational correlation of the water ipoles, which is a result of the mutual interaction between neighboring ipoles. The surfaces shoul be characterize not only by their charges or potentials but also by their ipole ensities. Because both the ouble layer an the hyration forces are epenent in the present moel on the polarization, the repulsion cannot be separate into two aitive terms, one being the traitional ouble layer repulsion (LVO theory an the other the structural repulsion (hyration. In the absence of surface ipoles, the repulsion between charge surfaces becomes stronger than that preicte by the LVO theory, particularly at high ionic strengths. The total repulsion can be increase or even ecrease by the presence of ipoles on the surfaces, which contraicts the aitivity of the repulsions. The repulsion between uncharge surfaces that possess ipoles was foun to epen on the electrolyte concentration, an to be extene over a much longer istance than the conventional exponential ecay, particularly at high ionic strengths. As a consequence of the coupling between the ouble layer an hyration, the ecay length of the repulsive force becomes larger than those of the two conventional repulsions an at high ionic strength the ifference becomes increasingly larger. I. Introuction Gouy 1 an Chapman, who were the first to preict the istribution of electrolyte ions in water aroun a charge flat surface, emonstrate that the ions form a iffuse layer (the electric ouble layer in the liqui near the interface. The interaction between two charge surfaces, ue to the overlapping of the ouble layers, was calculate much later by eryaguin an Lanau 3 an Verwey an Overbeek. 4 The stability of the collois was successfully explaine by them in terms of a balance between the ouble layer an van er Waals interactions (the LVO theory. 3,4 However, it is well-known that the LVO theory is reliable only for a limite range of electrolyte concentrations. eryaguin himself has note that this theory is vali between approximately 10-3 an 5.0 10 - M. There have been numerous attempts to improve the LVO theory, by accounting for the saturation of the molecular polarizability at high fiels, 5 the image forces, 6 finite ion sizes, 7 or the correlation between ions, 8 to cite only a few. Experiments with uncharge lipi bilayers in water 9 an the restabilization of some collois at high ionic strengths 10 inicate that another, non-lvo repulsion is also * Corresponing author: e-mail aress, feaeliru@ acsu.buffalo.eu; phone, (716 645-911 (ext 14; fax, (716 645-38. (1 Gouy, G. J. Phys. Raium 1910, 9, 457. ( Chapman,. L. Philos. Mag. 1913, 5, 475. (3 eryaguin, B. V.; Lanau, L. Acta Physicochim. URSS 1941, 14, 633. (4 Verwey, E. J.; Overbeek, J. Th. G. Theory of stability of lyophobic collois; Elsevier: Amsteram, 1948. (5 Henerson,.; Lozaa-Cassou, M. J. Colloi Interface Sci. 1986, 114, 180. (6 Jönnson, B.; Wennerström, H. J. Chem. Soc., Faraay Trans. 1983, 79, 19. (7 Levine, S.; Bell, G. M. iscuss. Faraay Soc. 1966, 4, 69. Ruckenstein, E.; Schiby,. Langmuir 1985, 1, 61. (8 Wennerström, H.; Jönnson, B.; Linse, P. J. Chem. Phys. 198, 76, 4665. (9 Ran, R. P.; Parsegian, V. A. Biochim. Biophys. Acta 1989, 988, 351. present. This repulsion was relate to the structuring of the solvent aroun the interface an is known (when water is the solvent as the hyration force. While the existence of the hyration force is unispute, its origin is still a matter of ebate. Marcelja an Raic showe that the exponential repulsion observe experimentally can be obtaine if a suitable Lanau free energy ensity, epenent on an unknown orer parameter, is associate with the correlation of the water molecules in the vicinity of the surface. 11 Later, Schiby an Ruckenstein 1 an Gruen an Marcelja 13 presente two ifferent moels, both involving the polarization of the water molecules. Gruen an Marcelja consiere that the electric an polarization fiels are not proportional in the vicinity of a surface an that while the electric fiel has the ion concentrations as its source, the source of the polarization fiel is provie by the Bjerrum efects. The couple equations for the electric an polarization fiels were erive through a variational metho. Attar et al. 14 conteste the Gruen-Marcelja moel because, to obtain an exponential ecay of the repulsion, the nonlocal ielectric function was assume to have a simple monotonic epenence upon the wavelength (eq 33 in ref 13. This was foun to be inconsistent with the exact expression for multipolar moels. 14 In aition, the characteristic ecay length for polarization (enote ξ in eq 18, ref 13 is inversely proportional to the square of the (unknown concentration of Bjerrum efects in ice. While at large concentrations of Bjerrum efects the isorere ice becomes similar to water an the traitional Poisson- (10 Healy, T. W.; Homola, A.; James, R. O.; Hunter, R. J. Faraay iscuss. Chem. Soc. 1978, 65, 156. (11 Marcelja, S.; Raic, N. Chem. Phys. Lett. 1976, 4, 19. (1 (a Schiby,.; Ruckenstein, E. Chem. Phys. Lett. 1983, 95, 435. (b Schiby,.; Ruckenstein, E. Chem. Phys. Lett. 1983, 100, 77. (13 Gruen,. W. R.; Marcelja, S. J. Chem. Soc., Faraay Trans. 1983, 79, 11, an 5. (14 Attar, P.; Wei,. Q.; Patey, G. N. Chem. Phys. Lett. 1990, 17, 69. 10.101/la00435v CCC: $.00 00 American Chemical Society Publishe on Web 09/06/00
Coupling between Hyration an ouble Layer Interactions Langmuir, Vol. 18, No. 0, 00 7585 Boltzmann equation is recovere, a low concentration of Bjerrum efects provie a much too large polarization ecay length (ref 13 inicate as an example the value ξ 33 Å, which is by more than orers of magnitue larger than the ecay length of the hyration force. The moel of Schiby an Ruckenstein 1a preicte that the surface ipoles inuce a polarization, even in the absence of a ouble layer, which ecaye exponentially from the surface. The electroynamics of continuous meia preicts that the fiel generate by a planar surface with an uniform ipole ensity immerse in a flui of uniform ielectric constant vanishes outsie the surface. However, at a molecular scale, the flui is not uniform. The interactions between remote charges or ipoles can be consiere screene by the intervening meium which has a large ielectric constant (ɛ 80 for water; however, the interaction between ajacent ipoles is much less screene (ɛ 1 represents no screening. 16 Consequently, a net electric fiel is generate by the surface ipoles which polarizes the nearby water molecules. These ipoles generate in turn electric fiels in the neighboring molecules an so on. The ecay length for polarization, calculate from this moel, was foun in goo agreement with the values etermine from experiment on neutral lipi bilayers. 9 Schiby an Ruckenstein also suggeste a new constitutive equation that relate the polarization to a local electric fiel, which inclue the interaction between neighboring ipoles. 1b Both moels preicte a monotonic ecay of the polarization from the surface. Because the two moels coul not explain the oscillatory profile of the average polarization obtaine by Monte Carlo simulations, it was suggeste that the ensity of hyrogen boning is a more suitable orer parameter. 15 Recently, it was, however, shown 17 that the Schiby-Ruckenstein moel can lea to an oscillatory polarization, if the water in the vicinity of a flat surface is assume organize in icelike layers. The oscillations are smoothe out when the surface is rough (or fluctuating or if some isorer is assume to exist in the icelike structure. It was also shown that the continuum approximation, base on a secon orer ifferential equation, can escribe well the average interaction. 17 One cannot yet rule out that other interactions contribute to the hyration, such as the isruption of the hyrogen bon networks when two surfaces approach each other. However, at least a part of this isruption is alreay containe in the ipole-ipole interactions inclue in the polarization moel. In aition, the polarization moel of hyration can relate the magnitue of the hyration force to the ensity of ipoles on the surface. This can explain the epenence of the hyration repulsion on the surface ipolar potential 18 or the restabilization of some collois at high ionic strength 16 observe experimentally. 10 It is usually assume that the total repulsion is the sum between a ouble layer repulsion, ue to the charges on the interface, an a hyration repulsion, ue to the structuring of water in the vicinity of the interface, an that the two effects are inepenent of each other. This is, however, not accurate when the hyration is inuce by the orientational correlation of neighboring ipoles, because both forces epen on polarization. The presence of ipoles on a surface free of charge generates an electric fiel an a polarization, epenent (15 Kjellaner, R.; Marcelja, S. Chem. Scr. 1985, 5, 73. Attar, P.; Batchelor, M. T. Chem. Phys. Lett. 1988, 149, 06. (16 Manciu, M.; Ruckenstein, E. Langmuir 001, 17, 7061. (17 Manciu, M.; Ruckenstein, E. Langmuir 001, 17, 758. (18 Simon, S. A.; McIntosh, T. J. Proc. Natl. Aca. Sci. U.S.A. 1989, 86, 963. on the istance from the surface. The graient of polarization affects, via the Poisson equation, the macroscopic electric fiel an, hence, the ouble layer, even in the absence of a surface charge. If ions are present in the system (because of the aition of an electrolyte, they interact with the electric fiel an hence affect both the electric fiel an the polarization. Therefore, the total repulsion shoul epen on the electrolyte concentration, even for uncharge surfaces. On the other han, in the absence of surface ipoles, the correlation between the water ipoles inuce by a surface charge is expecte to increase the repulsion as compare to the LVO theory. This is ue to the tenency of the water ipoles to orient in the same irection the ajacent water ipoles, thus increasing the ecay length of the polarization. Consequently, even when only a surface charge is present (an the surface ipole ensity is zero, the repulsion is not accurately escribe by the LVO theory (particularly at high ionic strength, as will be shown later because it isregars the orientational correlation of neighboring ipoles. In aition, even for uncharge surfaces, the repulsion inuce by the surface ipoles is not accurately escribe by the traitional hyration force, inepenent of the electrolyte concentration, because the graient of polarization generates an electric fiel which is affecte by the ionic strength. The problem is further complicate when surface charges as well as surface ipoles are present. Can the total repulsive interaction still be escribe by a superposition between two inepenent interactions, the ouble layer an the hyration repulsions? The purpose of this paper is to present a moel that accounts in an unitary manner for both the ouble layer an hyration repulsions. The moel is an extension of an earlier treatment of Schiby an Ruckenstein. 1 The equation coupling the polarization an the electric potential are erive here using an analysis base on the Lorentz moel for the polarization an also a variational treatment (see Appenix. It will be shown that if the mutual interaction between neighboring water ipoles is taken into account, the ouble layer repulsion increases in the absence of surface ipoles, when compare to the LVO theory, particularly at high ionic strengths. Similarly, in the absence of a surface charge, but the presence of a surface ipole ensity, the repulsion is increase by the aition of an electrolyte, an effect that is important particularly at high ionic strengths. For a surface whose charge is generate by the issociation of a surfactant, an the surface ipoles are provie by the nonissociate surfactant molecules, it will be shown that the hyration an the ouble layer repulsions are not only nonaitive but that the presence of surface ipoles can ecrease the repulsion. II.1. The Basic Equations We will employ here the polarization moel evelope in refs 1 an 17. The water molecules are consiere ipoles localize at the site of a lattice; however, the fiels generate by the ajacent ipoles are treate ifferently from those of the remote ipoles, because the latter are screene by the intervening water molecules. Starting from the ipoles of the surface, the polarization propagates through water because of the interactions between the neighboring ipoles. When two external surfaces approach each other, the overlap of the polarization layers ecreases the ipole moments an hence increases the free energy of the system, thus generating a repulsion.
7586 Langmuir, Vol. 18, No. 0, 00 Ruckenstein an Manciu The electric fiel E P at a site of the layer i of a water lattice, situate at z i, generate by all the other ipoles of magnitue m i of the same layer an those of magnitue m i(1 of the ajacent layers (the contribution of the other layers is neglecte because of the screening can be expresse as 17 E P (z i C 1 m i-1 + C 0 m i + C 1 m i+1 C 0 m(z i + C 1 (m(z i-1 + m(z i+1 (C 0 + C 1 m(z i + C 1 m(z C 1 m(z zzi (1 zzi where is the istance between the water layers, m(zis a continuous function which is equal in every layer (locate at z i i to the average polarization of the water molecules in that layer (m(z zzi m i, an C j (j 0, 1 are interaction coefficients (see ref 17 C 0-3.766 4πɛ 0 ɛ l 3 C 1 1.87 4πɛ 0 ɛ l 3 ( with l the istance between the centers of two ajacent water molecules, ɛ 0 the vacuum permittivitty, an ɛ the ielectric constant for the interaction between neighboring molecules, which is expecte to be nearer to unity than to the ielectric constant of water, ɛ 80. In aition to the fiel generate by the ajacent ipoles, there is a macroscopic fiel E ue to the presence of charges an of the average polarization P of the meium. In the Lorentz treatment of polarization, for a constant macroscopic fiel in a linear an homogeneous meium of ielectric constant ɛ (hence satisfying P ɛ 0 (ɛ - 1E, the local fiel E local at a site of a selecte ipole is relate to the macroscopic fiel E via 19 the macroscopic fiel being generate by all the charges an ipoles present, while the local fiel is generate by all the charges an ipoles, except the selecte ipole. Extening the Lorentz relation when the aitional fiel E P is present, one obtains The average ipole moment m(z P(zv 0, where v 0 is the volume of a water molecule, is relate to E local via m(z γ( E + E P + m(z 3ɛ 0 v 0 w E local E + P 3ɛ 0 (3 E local E + P 3ɛ 0 + E P (4 m(z where γ is the molecular polarizability. At constant polarization, E P (C 0 + C 1 m = 0. Hence eq 5 becomes m γ( E + γ 1 - γ 3ɛ 0 v 0 (E + E P (5 m 3ɛ 0 v 0 (6 The macroscopic relation between the polarization an (19 Frankl,. R. Electromagnetic Theory; Prentice Hall: Englewoo Cliffs, NJ, 1986. the electric fiel, m ɛ 0 v 0 (ɛ - 1E can be employe in eq 6 to etermine the value of γ. Assuming that the molecular polarizability remains the same in a nonconstant fiel, eqs 1 an 5 lea to m(z ɛ 0 v 0 (ɛ - 1E(z + ɛ 0 v 0 (ɛ - 1C 1 m(z (7 The Poisson equation - E(z ψ(z - F + 1 m(z ɛ 0 ɛ 0 v 0 where F is the local charge ensity, given, as in the traitional theory, by F(z -ec E(exp( eψ(z - exp(- eψ(z where c E is the bulk electrolyte concentration, e is the protonic charge, k is the Boltzmann constant, an T is the absolute temperature, can be rewritten as ψ(z ec E ɛ 0 Equations 7 an 10 constitute a complete system of equations for ψ(z an m(z, which replaces the traitional equations of the ouble layer. II.. The Linear Approximation For the sake of simplicity, in what follows it will be consiere that the ouble layer potential is sufficiently small to allow the linearization of the Poisson-Boltzmann equation (the ebye-hückel approximation. The extension to the nonlinear cases is (relatively straightforwar; however, it will turn out that the ifferences from the LVO theory are particularly important at high electrolyte concentrations, when the potentials are small. In this approximation, the istribution of charge insie the ouble layer is given by where λ (ɛɛ 0 /e c E 1/ is the ebye-hückel length. In the linear approximation, the system of eqs 7 an 10 becomes where λ m ɛ 0 v 0 (ɛ - 1C 1. In the absence of an electrolyte (c E 0, λ, eq 1a leas to (8 -ec E sinh( eψ(z (9 sinh ( eψ + 1 m(z ɛ 0 v 0 (10 F-ec E sinh ( eψ c E =-e ψ - ɛɛ 0 λ ψ (11 ψ(z ɛ λ ψ + 1 m(z ɛ 0 v 0 λ m m(z m(z + ɛ 0 v 0 (ɛ - 1 ψ(z m(z ɛ 0 v 0 ψ(z (1a (1b + constant (13 The constant is zero since the average polarization shoul
Coupling between Hyration an ouble Layer Interactions Langmuir, Vol. 18, No. 0, 00 7587 vanish when the electric potential is constant (this can be seen better from eq 1b, since a constant polarization an constant electric potential imply a vanishing polarization. Introucing the result (with constant 0 in eq 1b yiels m(z λ m m ɛ λ m(z H (14 which preicts a monotonic ecay of the polarization from the surfaces, with a ecay length λ H (ɛ 0 v 0 C 1 (ɛ - 1/ ɛ 1/. The solution of eq 14, which accounts for the antisymmetry of the polarization (m(- m 0, m( -m 0, is m(z -m 0 sinh(z/λ H sinh(/λ H (15 where z is measure from the mile istance an is the istance between the plates. Employing the values l.76 Å for the istance between the centers of two ajacent water molecules in an icelike structure, ɛ 1 for the ielectric constant for the interaction between neighboring molecules (selecte as in refs 16 an 17, 4 / 3 l 3.68 Å for the istance between two ajacent water layers in an icelike structure, v 0 30 Å 3 for the volume occupie by one water molecule, an ɛ 80, one obtains λ m 14.9 Å an λ H λ m /ɛ 1/ 1.67 Å, which is in goo agreement with the hyration length experimentally etermine for neutral lipi bilayers. 9 This exponential epenence of the polarization was foun also (in the continuum approximation by earlier analysis. 1,16,17 It shoul also be note that using eq 14 (which is vali only in the absence of electrolyte, the free energy of the system (iscusse in etail in section II.5 reuces to the form employe in previous polarization-base treatments of hyration forces. 16,17 However, as will be shown below, the aition of electrolyte affects the hyration even in the absence of surface charges. II.3. The Characteristic ecay Lengths In the presence of an electrolyte (λ *, we will seek solutions of the type ψ ψ 0 exp(z/λ an m m 0 exp(z/λ for the homogeneous system of the two linear equations (1a,b. The conition for existence of nontrivial solutions leas to the characteristic equation ( 1 - ɛ λ λ ( λ m λ - 1 - ɛ - 1 0 S λ 4 - λ (λ + λ m λ + λ λ m 0 (16 ɛ The solutions of eq 16, (λ 1 an (λ are always real, since the ielectric constant of the meium is higher than the vacuum ielectric constant (the iscriminant λ 4 + λ m4 + λ λ m (1 - /ɛ > 0 for ɛ > 1. The epenence of the ecay lengths λ 1 an λ on the electrolyte concentration (λ for ɛ 80 an λ m 14.9 Å are presente in Figure 1. At low electrolyte concentrations (λ. λ m, the ecay lengths λ 1 an λ are well approximate by λ an λ H λ m /ɛ 1/ ; however, when λ becomes comparable to λ m, λ 1, an λ iffer markely from λ an λ H. Figure 1. The epenence of the characteristic ecay lengths λ 1, λ of the system on the ebye-hückel length λ (λ m 14.9 Å, λ H λ m/ɛ 1/ 1.67 Å. II.4. Bounary Conitions for Two Ientical Surfaces Immerse in Water The symmetry of the system implies that the potential is symmetric an the average polarization is antisymmetric with respect to the mile istance; hence, the general solutions of the system (1a,b are ψ(z a 1 cosh( λ z + a ( cosh z λ m(z aj 1 sinh( z λ 1 + aj ( sinh z λ where the constants aj 1 an aj are relate to the constants a 1 an a via eq 1a aj 1 a 1 ɛ 0 v 0 λ 1( 1 λ 1 - aj a ɛ 0 v 0 λ ( 1 λ - (17a (17b The remaining two inepenent constants, a 1 an a can be etermine using the bounary conitions for the electrical potential an polarization at the surfaces. For constant surface potential ψ 0, the bounary conition is For constant surface charge ensity σ, the conition of overall electroneutrality σ - - F(z z leas to The ouble layer charge istribution (eq 11 was employe in the erivation of the last equation. When the value of the polarization at the bounary (m 0 m(- is known, the corresponing bounary conition is A more realistic approach 17 is to assume that the average polarization of the water molecules of the first water layer near the surface is proportional to the local fiel, generate by the surface charges, surface ipoles, an the water ipoles of the first two water layers. ɛ λ ɛ λ a 1 cosh( λ 1 + a cosh ( λ ψ 0 a 1 λ 1 sinh( λ 1 + a λ ( sinh λ λ σ ɛɛ 0 aj 1 sinh( - λ 1 + aj sinh ( - λ m 0 (18a (18b (19a (19b (0a
7588 Langmuir, Vol. 18, No. 0, 00 Ruckenstein an Manciu The average electric fiel generate by surface ipoles with a surface ensity of 1/A an with a ipole moment p normal to the surface, whose centers are locate at a istance from the center of the first water layer, is given by 17 E S ( p ɛ 1 1 πɛ 0 ( A (1 π + 3/ where ɛ is the local ielectric constant for the fiel generate by the surface ipoles in the neighboring water molecules. The macroscopic fiel at the interface is relate to the electric potential, via the expression E - ψ(z/, while the fiel generate by the water ipoles is 17 E SP C 0 m 1 + C 1 m ( where m 1 an m are the average polarization of the water molecules of the first an secon water layer near the surface, respectively, an C 0 an C 1 are given by eq. Hence, the bounary conition for polarization has the form m 1 γ( E + m 1 3ɛ 0 v 0 + E S + E SP ɛ 0 v 0 (ɛ - 1(E + E S + E SP (3 where eq 6 was employe to erive the last equality. Assuming that the positions of the centers of the first water layers are at the istances (, an using eqs 1a an 1b, eq 3 becomes a 1[ λ 1 ɛ 0 v 0( 1 λ 1 - ɛ λ ( (1 - ɛ 0 v 0 (ɛ - 1C 0 sinh ( - λ 1 - ɛ 0 v 0 (ɛ - 1C 1 sinh( - - λ 1 + ɛ 0 v 0 (ɛ - 1 λ 1 sinh( - λ 1 ] + a [ λ ɛ 0 v 0( 1 λ - ɛ λ ( (1 - ɛ 0 v 0 (ɛ - 1C 0 sinh ( - λ - ɛ 0 v 0 (ɛ - 1 ɛ 0 v 0 (ɛ - 1C 1 sinh( - - λ + λ sinh( - λ ] ( ɛ p v 0 (ɛ - 1 π 1 ( A π + 3/ (0b II.5. The Free Energy of the System In the LVO framework, the free energy of a system of two overlappe ouble layers is compose of an electrostatic energy, an entropic contribution ue to the ions in the ouble layer, an a chemical term, where applicable. 4 The electrostatic energy per unit area of the ouble layer is provie by the familiar expression 19,0 F el 1 1 - F (zψ(z z - (zψ(z z 1 (zψ(z z z- - 1 - (z ψ(z z 1 1 - (ze(z z - (ze(z z (4 where F is the total charge ensity (which inclues the charge istribute between surfaces, F, an the surface charge ensity, σ, ψ is the electric potential, E is the macroscopic electric fiel, z is the istance measure from the mile between the surfaces an ɛ 0 E + P is the isplacement fiel. The above expression accounts for the fact that the fiel is nonvanishing only between the surfaces, locate at (; the Poisson equation F an the relation E - ψ were also employe. The excess entropy contribution (with respect to the bulk per unit area ue to the ions of an electrolyte, calculate as for an ieal solution, is given by 1 -T S i where c i is the actual (ouble layer concentration of ions of species i, c i0 is the concentration at large istances, an the subscript i runs over all ion species. For an 1-1 electrolyte of concentration c E, which obeys the Boltzmann istribution, the above expression becomes The entropic contribution to the free energy (per unit area becomes in the ebye-hückel approximation The chemical contribution to the free energy, per unit area, ue to the asorption of n molecules (per unit area of charge q on each surface of potential ψ S is 4 at constant surface potential, µ being the change in the chemical part of the electrochemical potential of a molecule at its asorption from the bulk on the surface, an σ the surface charge ensity. At equilibrium, the electrochemical potential is constant through the system ( µ -qψ S. At constant surface charge the chemical contribution to the free energy is zero. 4 In aition to these well-known free energy contributions, one has to consier another one, which accounts for the mutual interactions between neighboring ipoles where eq 1 was employe. It shoul be note that the above expression contains both the energy an the entropy of the ipole, in the weak fiel (linear approximation. (0 Schwinger, J.; eraa, L. L., Jr.; Milton, K. A.; Tsai, W.-Y. Classical Electroynamics; Perseus Books: Reaing, MA, 1998. (1 Overbeek, J. T. G. Collois Surf. 1990, 51, 61. - -T S -c E - i1, ( c i ( c ln i c i0 - c i i0 + c z (5 ( (-1i eψ exp( (-1i+1 eψ exp( (-1i+1 eψ + - 1 z (6 -T S c E e - (ψ(z z ɛɛ 0 λ - (ψ(z z (7 F ch n µ -nqψ S -σψ S (8 F m - 1 me P -C 1 m m (9
Coupling between Hyration an ouble Layer Interactions Langmuir, Vol. 18, No. 0, 00 7589 Consequently, the total electrostatic free energy per unit area (accounting for all the interactions between charges an ipoles is provie by the expression F el - 1 - [ 1 E(ɛ 0E + P - 1 P] PE z [( ɛ m(z m(z 0E(z + v 0 E(z - E v P 0 (z] z (30 The free energy of the surface layer forme by surface ipoles an the water molecules between them is assume to be inepenent of the istance. At constant surface potential, the free energy per unit area of a system of two ientical charge surfaces at istances ( from the mile istance can be written as the sum between a surface term (the chemical energy an an integral over the ensity of the entropy of the mobile ions an all the electrostatic interactions between charges an ipoles F ψ -σψ S + 1 - [( ɛ ψ 0 - m v 0 ψ + ɛɛ 0 λ (ψ - C 1 The same results (eqs 1a,b an 31 were obtaine using a variational metho. The etails are given in Appenix. At constant surface charge, the free energy is given by F σ 1 - [( ɛ ψ 0 - m v 0 ψ + ɛɛ 0 λ (ψ - III.1. The Influence of the ipole-ipole Interactions on the ouble Layer Force At low electrolyte concentrations, because the ebye- Hückel length is large, the polarization preicte by the LVO theory is slowly varying in space. Therefore, when the ipole ensity on the surface is negligible, one expects the aitional interaction, ue to the mutual interaction between neighboring ipoles, to be also small, its ensity being proportional to m( m/ (m /λ. Consequently, the contribution of the interaction between neighboring ipoles to the total free energy becomes negligible, an the LVO theory is recovere. However, when λ is sufficiently small an becomes comparable to λ m, a coupling between the two effects is expecte to occur. As a consequence, a larger ecay length (λ 1 > λ appears in the system (see Figure 1. There are two main reasons for the eparture of the present moel from the LVO theory. First, the constitutive equations, which relate the polarization to the electric potential, are ifferent. Secon, the bounary conitions are ifferent, since the average polarization in the LVO theory is irectly relate to the surface charge, while in the present treatment it epens also on the surface ipole ensity. Let us first investigate the effect of the new equations alone, by using for both the LVO theory an the present equations the same bounary conitions. For the surface charge ensity the constant value σ 5 10-4 C/m was employe (the value selecte is low enough for the linear approximation to be accurate for all the electrolyte concentrations investigate here, while the polarization v 0 C 1 v 0 ] m m z (31a ] m m z (31b at the surface was consiere inuce by the surface charge only, as in the LVO theory m 0 -ɛ 0 v 0 (ɛ - 1 ψ(z z- (3a In parts a an b of Figure, the electric potential an the polarization, respectively, calculate using eqs 1a,b are compare to those preicte by LVO, with the bounary conitions (19b an (3a, using for the parameters the values v 0 30 Å 3, ɛ 80, T 300 K, λ m (ɛ 0 v 0 (ɛ - 1C 1 1/ 14.9 Å, an the values 0 Å an λ 10 Å. The force per unit area between surfaces, Π -[ F(]/ [ (], with F( given by eq 31, at constant surface charge σ 5 10-4 C/m an for a polarization m 0 given by eq 3a, is compare to that preicte by the LVO theory in parts c an of Figure, for various electrolyte concentrations (λ 100, 30, 10, an 3 Å. As expecte, at low electrolyte concentrations, the interaction is well escribe by the LVO theory. At large electrolyte concentrations, the repulsion is, however, markely larger than that provie by the LVO theory, because the interactions between neighboring ipoles attenuate the ecay of the polarization. To examine the effect of the surface ipole ensity, we consier that the polarization on the surface acquires the value m 0 -ɛ 0 v 0 (ɛ - 1 ψ(z z- + δm (3b where the change δm of the average ipole moment of the water molecules at the interface is generate by the surface ipoles. It shoul be note that the bounary conition (3b affects the value of both ψ an m on the surface an hence m 0 * m 0 + δm. Let us consier that the surface charge arises via the issociation of surfactant molecules asorbe on the interfaces an that the surface ipoles are ue to the unissociate surfactant molecules asorbe. In this case, the electric fiel inuce by the surface ipoles is opposite to that generate by the surface charge (δm < 0. Hence, the presence of surface ipoles actually ecreases the repulsion. The effect is illustrate in parts c an of Figure, for δm -0.. Therefore, in this case the total repulsion cannot be obtaine (as usually assume by aing two inepenent terms, a ouble layer force ue to the surface charges an a hyration force ue to the surface ipoles. The secon important ifference with the LVO theory arises from the bounary conition for the polarization; while the classical theory ignores the interactions between neighboring ipoles, eq 0b takes them into account. The electrical potential an the polarization calculate with eqs 1a,b an the bounary conitions eqs 19b an 0b with σ 5 10-4 C/m an p/ɛ 0orp/ɛ -0.1 are compare in parts a an b of Figure 3 with the LVO preictions (the value A 100 Å was selecte for the area occupie by a surface ipole. The repulsion forces, per unit area, at constant surface charge ensity are presente in Figure 3c an Figure 3. The ifferences from the LVO theory become again important at large electrolyte concentrations, an the presence of surface ipoles ecreases the repulsion. At high ionic strengths, the range of the interaction is much longer than that preicte by the LVO theory. At large separation istances, the first term of eqs 17a an 17b, with the ecay length λ 1 (λ 1. λ, ominates both the electrical potential
7590 Langmuir, Vol. 18, No. 0, 00 Ruckenstein an Manciu Figure. (a The electric potential an (b the average polarization of a water molecule between two surfaces with σ 5 10-4 C/m, separate by a istance 40 Å, as a function of the position from the mile istance. The average polarization of the water molecules from interface, m 0, is calculate from eq 3a. A perturbation δm -0. illustrates the effect of surface ipoles. (c, The interaction force calculate, at various electrolyte concentrations, from eqs 1a,b with the bounary conitions (19b an (3a, versus the separation istance, compare to the LVO preictions. A perturbation δm -0. illustrates the effect of surface ipoles. an the polarization, respectively. Consequently, at large istances, the force epens mainly on the term containing the ecay length λ 1. For λ 0, one obtains from eq 16 the minimum of λ 1 λ 1 (λ 0 lim λ f0( ( 1 λ + λ m + ( λ 4 + λ 4 m + λ λ m ( 1 - ɛ 1/ 1/ λ m (33 with λ m 14.9 Å for the values of the parameters employe here. Therefore, a long-range interaction, with a ecay length larger than λ m, is always present in the system, at any electrolyte concentration. The magnitue of this longrange interaction epens markely on the ionic strength (see Figure 3 an is much larger than the ebye an hyration ecay lengths. When the interaction ue to the surface charge is large, the presence of a low ipole ensity on the surface (generate by surface charge association with counterions, as iscusse above ecreases the repulsion, since the electric fiel generate by the surface ipoles is opposite to the electric fiel generate by the charges. However, a sufficiently high surface ipole ensity, which still inuces an electric fiel opposite to that generate by the charge, can eventually lea to an increase in the repulsion (the regime when the hyration ominates. This effect is illustrate in Figure 3e, for weakly charge surfaces (σ -0.00 C/m an λ 3Å. In this case, the increase of the ipole strength ecreases initially the repulsion (when 0 < p/ɛ < 0.7 ; the repulsion is, however, increase for p/ɛ > 0.7. When p/ɛ > 1.7, the total repulsion becomes larger than that in the absence of surface ipoles. III.3. The Repulsion between Uncharge Surfaces The potential ψ(z an the ipole moment m(z provie by the system of eqs 1a,b are presente in parts a an b of Figure 4, for σ 0, p/ɛ 1 an λ 3, 10, 30, an 100 Å. The forces between surfaces at various electrolyte concentrations (with the other parameters unchange are plotte in Figure 4c. At large separation istances, the first term in eqs 17a an 17b, which ecays much slower, becomes ominant, with λ 1 λ at low ionic strengths an λ 1 λ m at high ionic strengths. At short istances an low ionic strengths, the first term is, however, small, an the force is well escribe by an exponential hyration force, with constant preexponential factor an ecay length (λ = λ H λ m /ɛ 1/. In the limiting cases σ 0 an λ f, the free energy expression eq 31 leas (because of eq 13 to F H - 1 - [ C 1 v 0 m m ] z (34
Coupling between Hyration an ouble Layer Interactions Langmuir, Vol. 18, No. 0, 00 7591 Figure 3. (a The electric potential an (b the average polarization of a water molecule between two surfaces with σ 5 10-4 C/m, separate by a istance 40 Å, as a function of the position from the mile istance. The average polarization of the water molecules from interface, m 1, was calculate using eq 0b, for p/ɛ 0 an p/ɛ -0.1, an A 100 Å. (c, The interaction force calculate, at various electrolyte concentrations, from eqs 1a,b with the bounary conitions (19b an (0b, versus separation istance, compare to the LVO preictions. (e The interaction force for λ 3 Å an σ -0.00 C/m for various ipole strengths (0 < p/ɛ < 3,A 100 Å. The repulsion initially ecreases an then increases with the increasing strength of the surface ipoles. where m is given by eq 15 an the hyration free energy from the previous work 16,17 is recovere. At high ionic strength, both terms, with ecay length λ 1 an λ, are important an the force is smaller at short separation istances an larger at long separation istances when compare to the force for λ f (see Figure 4c. The long-range repulsion inuce by the electrolyte concentration (see Figure 4c can be explaine as follows. At zero surface charge, the total charge of the electrolyte ions between the surfaces (therefore the integral over potential (eq 11 between - an vanishes. This can occur only if the potential changes sign between the surfaces, hence if the coefficients a 1 an a in eq 17a have opposite signs, with the ratio of their magnitue etermine by electroneutrality. The rapily varying, positive potential near the surfaces (we assume a positive surface charge is compensate by a slowly varying, negative potential in the mile range (see Figure 4a. This generates a polarization, which ecays with a ecay length
759 Langmuir, Vol. 18, No. 0, 00 Ruckenstein an Manciu Figure 4. (a The electric potential an (b the average polarization of a water molecule between two neutral surfaces (σ 0, separate by a istance 40 Å an λ 3, 10, 30, an 100 Å, as a function of the position from the mile istance, calculate using eqs 1a,b. The average polarization of the water molecules at the interface, m 1, was calculate from eq 0b, for p/ɛ 1 an A 100 Å. (c The interaction force between two neutral surfaces calculate, at various electrolyte concentrations, using eqs 1a,b with the bounary conitions (19b an (0b for p/ɛ 1 an A 100 Å versus the separation istance. >λ m, which is much larger than λ H (see Figure 4b. It shoul be, however, note that the magnitue of the long-range repulsive force is small at low ionic strengths (see Figure 4c. IV. Conclusions The interaction between two charge surfaces immerse in a liqui was traitionally escribe by assuming that the polarization of the liqui is proportional to the electric fiel. However, an aitional interaction, cause by the structuring of the liqui in the vicinity of a surface, shoul be also taken into account. This can be inclue, at least partially, via the mutual interactions between the water ipoles. This inuces a polarization that propagates, from the surfaces, through water, being generate by both the surface charge an surface ipoles. The polarization graient prouces an electric fiel, which interacts both with the ipole moments of the water molecules an with the charges of the electrolyte. In this paper a moel was presente, which allowe one to calculate both the electric potential an the polarization between two surfaces, without assuming, as in the traitional theory, that the polarization an the macroscopic electric fiel are proportional. An aitional local fiel, ue to the interaction between neighboring ipoles, was introuce in the constitutive equation which relates the polarization to the local fiel. The basic equations were also erive using a variational approach. It was shown that the interaction between ipoles increases markely the repulsion at high ionic strength an large separation istances, when compare to the LVO theory. When both charges an ipoles are present on the surface, the repulsion is not provie by the sum of two inepenent repulsions, a ouble layer an a hyration repulsion. The presence of ipoles on the surface can even ecrease the repulsion. It was also shown that the presence of an electrolyte generates a long-range repulsion, even at zero surface charge, if the interfaces carry a surface ipole ensity. At low ionic strength, this repulsion can be escribe, in the vicinity of the surface, by an exponential with both the ecay length an the preexponential factor almost inepenent of electrolyte concentration, as usually consiere for the hyration forces. The long-range interactions between the neutral surfaces is in this case small. However, at high ionic strengths, the repulsion between neutral surfaces iffers markely from this escription, the force being smaller at short istances an larger at large istances than that at zero electrolyte concentration. Appenix. erivation of the Equations for m an ψ from a Variational Principle The Maxwell equation of electrostatics in a vacuum for a system with planar xy symmetry ( ψ/ -E, E/ F /ɛ 0, F being the total charge ensity present in the system, can be erive as extremals of the electrostatic
Coupling between Hyration an ouble Layer Interactions Langmuir, Vol. 18, No. 0, 00 7593 free energy functional 0 F el V ( F ψ + ɛ 0 E ψ z + ɛ 0 E V V J V (A.1 using the Euler-Lagrange equations J J z( ( ψ ψ J J z( ( E E (A. Let us try to fin a free energy functional of a polarizable meium, which can be extene to any constitutive relation between E an P. Since the Poisson equation in a meium is (ɛ 0 E F - P, a natural choice for this functional is P F el V (( F - ψ + ɛ 0 E ψ + ɛ 0 E + f(p V (A.3 The Euler-Lagrange equations of this functional, with respect to ψ an E, are ( /(ɛ 0 E + P F an E - ψ/, respectively, provie that F an the arbitrary function f(p o not epen on either E or ψ. Let us now obtain a functional, which represents the free energy ensity of a linear, homogeneous, an isotropic meium, that satisfies the constitutive equation P ɛ 0 (ɛ - 1E. To obtain the classical result for the free energy ensity, (1/E(ɛ 0 E + P, the function f(p must acquire the form In this case, the conitions of extremum of the functional given by eq A.3 with respect to ψ, E, an P, consiere as inepenent functions (the Euler-Lagrange equations, lea to the Maxwell equations an the equation that relates the polarization to the fiel. It shoul be note that the above equations imply that F is inepenent of E an P. Of course, this assumption is not vali in the presence of an electrolyte. For two overlapping ouble layers, the free energy per unit area F LVO can be written as the sum between the electrostatic energy, the chemical energy, an the entropic term of the electrolyte ions. Since the total charge ensity F is compose of the surface charges, σ, an the charge ensity F istribute between the surfaces, the later obeying (in the linear approximation eq 11, the electrostatic energy per unit area becomes F el σψ S + - f(p ( ɛɛ 0 - P ɛ 0 (ɛ - 1 (A.4 λ ψ - P ψ + ɛ 0 E ψ + ɛ 0 P E + z (A.5 ɛ 0 (ɛ - 1 where ψ S is the surface potential. The chemical energy per unit area (eq 8 is given by F ch -σψ S (A.6 at constant surface potential an the entropy contribution (per unit area, eq 7 is Aing eqs A.5, A.6, an A.7 an writing E - ψ/, one obtains F LVO,ψ - The extremals of ψ(z an P(z, obtaine through the Euler-Lagrange equations are given by which represent the linear Poisson-Boltzmann equation an the proportionality relation between polarization an electric fiel, respectively. Until now, the classical LVO results have been recovere. Let us suppose that another interaction, whose free energy ensity is -RP[ P(z/ ] is also present. An integration by parts leas to -R - Hence, the free energy at constant potential F ψ per unit area is given by an the extremals function ψ(z, P(z satisfy the Euler equations Since P(z m(z/v 0 an RC 1 /, the system of eqs A.13 an A.14 becomes ientical to the system of eqs 1a,b, while the free energy eq A.1 coincies with eq 31a. Equation 31b, for constant surface charge, can be erive in a similar manner. LA00435V ( P(z P(z F ψ -RP P z + z- - -T S ɛɛ 0 λ - (ψ(z z (A.7 - ɛ 0 ( ψ(z - ɛɛ 0 λ (ψ(z - ψ P(z + ψ(z ɛ λ ψ + 1 P(z ɛ 0 P(z -ɛ 0 (ɛ - 1 ψ(z z -RP P z + z- - P (z z (A.8 ɛ 0 (ɛ - 1 (A.9 (A.10 R ( P(z z (A.11 ( ɛ 0 - ( ψ - ɛɛ 0 λ (ψ - ψ P + P ɛ 0 (ɛ - 1 +R ( P ψ(z ɛ λ ψ + 1 P(z ɛ 0 R P(z P(z ɛ 0 (ɛ - 1 + ψ(z (z z (A.1 (A.13 (A.14