15700
J. Phys. Chem. C 2007, 111, 15700-15705
Simple Extension of a Field Theory Approach for the Description of the Double Layer Accounting for Excluded Volume Effects† Dung Di Caprio,*,‡ Mo´ nika Valisko´ ,§ Myroslav Holovko,| and Dezso _ Boda§,⊥ CNRS-ENSCP-PARIS6, 4, Pl. Jussieu, 75252 Paris Cedex 05, France, Department of Physical Chemistry, UniVersity of Pannonia, H-8201 Veszpre´ m, P.O. Box 158, Hungary, Institute for Condensed Matter Physics, National Academy of Ukraine, 1 SVientsitskii Str., 79011 LViV, Ukraine, and Department of Molecular Biophysics and Physiology, Rush UniVersity Medical Center, Chicago, Illinois 60612 ReceiVed: May 15, 2007; In Final Form: August 13, 2007
In this paper, we discuss an extension of a field theoretical approach for the description of the double layer containing ions of finite size. In particular, the density profile and the differential capacitance are presented in the low-temperature regime, and excluded volume effects are discussed. Simple approximations are presented which allow for analytic expressions. Our results are compared to Monte Carlo simulation data.
I. Introduction Recently, there has been a renewed interest in the study of the structure of the double layer,1-11 in particular, in the regime of low reduced temperatures. Experimental results for molten salts12-14 and frozen electrolytes15 as well as Monte Carlo (MC) simulation results for the primitive model of electrolytes1,2,10 show that the behavior of these systems differs from the predictions of the traditional approach of the Gouy-Chapman theory. Both MC simulations2,16 and theoretical considerations6,7,16 confirmed that electrolytes at an electrode show a negative adsorption at low temperatures (or, equivalently, at high ionic coupling). This phenomenon appears to be associated with another interesting feature: the anomalous behavior of the differential capacitance as a function of the temperature (see a review on simulation and theoretical results for the anomalous capacitance behavior in the paper of Valisko´ et al.10). A number of different theoretical approaches have been applied to account for this behavior: density functional theory,6 mean spherical approximation (MSA) with association,4 modified PoissonBoltzmann (MPB) theory,7 and the field theoretical approach.9 Among them, the field theoretical approach has been able to account for this phenomenon within a simple description of point ions. This shows that the anomalous behavior of the differential capacitance is a phenomenon essentially electrostatic in nature. In this paper, we study various extensions of the field theory for point ions to account for the effect of volume exclusion. The theoretical results will be compared with MC simulations of the restricted primitive model of electrolytes where the ions are modeled as charged hard spheres of equal diameter. In the following, we discuss various approximations of the field theory for point ions in order to account for these excluded volume effects. Our purpose is to obtain simple tractable expressions accounting for this effect instead of a detailed microscopic description of the structure of the double layer. The article is organized as follows. In the first part, we discuss the effect of the volume exclusion of the density profile and †
Part of the “Keith E. Gubbins Festschrift”. * Corresponding author. ‡ CNRS-ENSCP-PARIS6. § University of Pannonia. | Institute for Condensed Matter Physics. ⊥ Rush University Medical Center.
present different approximations. In the following part, we discuss a simple correction for the differential capacitance curve accounting for the volume exclusion. II. Field Theory for Point Ions A. Formalism. We consider a field theoretical description of a system of point ions.9,17,18 The grand potential in terms of the fields F+(r) and F-(r) representing the density for cations and anions, respectively, is written as a functional integral
Θ[F((r)] )
∫ DF((r) exp{-βH[F((r)]}
(1)
where β ) 1/(kBT) is the inverse temperature. We assume a symmetric electrolyte. The grand potential is then obtained as β(-pV + γA) ) -ln Θ[F(], where p is the pressure, V is the volume, γ is the surface tension, and A is the area of the electrodes. For the study of the Coulomb interactions, it is more convenient to use the charge density field q(r) ) F+(r) - F-(r) and the total density field s(r) ) F+(r) + F-(r). The Hamiltonian as a functional of these fields is written as
βH[q(r), s(r)] ) βHent[q(r), s(r)] + βHCoul[q(r)] -
∫βµss(r) dr - ∫βµqq(r) dr
(2)
where µs ) µ+ + µ-, µq ) µ+ - µ- with µ+ and µ- being the chemical potentials of the ions. The contributions from the ideal entropy and from the chemical potentials can be written as
βHent[q(r), s(r)] -
∫
∫βµss(r) dr - ∫βµqq(r) dr )
[(
) ]
s(r) + q(r) s(r) + q(r) ln - 1 dr + 2 2Fj+ s(r) - q(r) s(r) - q(r) ln - 1 dr (3) 2 2Fj-
∫
[(
) ]
where Fj( ) exp(-βµ()Λ3 with Λ being the de Broglie wavelength. The second term in the Hamiltonian is the Coulombic contribution
10.1021/jp0737395 CCC: $37.00 © 2007 American Chemical Society Published on Web 10/04/2007
Double Layer Accounting for Excluded Volume Effects
βHCoul[q(r)] )
J. Phys. Chem. C, Vol. 111, No. 43, 2007 15701
βσ2V + 2
∫
βe q(r) Vext(r) dr +
βe2 8π
∫
q(r) q(r′) dr dr′ (4) |r - r′|
where e is the elementary electric charge, and Vext(r) is the potential created by the charges σ and -σ placed on the walls as shown in Figure 1. This part of the Hamiltonian includes three terms. One is related to the external field, one is the interaction energy of the solution and the external field due to the charged surfaces, and finally there is the electrostatic energy of the solution. The expression for the external field is
Vext(r) ) -
σ (x - L1)
(5)
where the zero point for the electric potential is arbitrarily set at x ) L1. B. Results for the Point Ions. In ref 9, we have shown that the excess density profile at a neutral interface is known analytically
δs(1) ˜) ) elec(z
K3D I(z˜) ) -FηI(z˜) 8π
(6)
where η ) K3D/(8πF), F is the total bulk density of anions and cations, z˜ ) zKD is the reduced distance to the wall,
KD )
( ) e2F kBT
∫1∞ dt
1/2
(7)
exp(-2z˜t)
(t + xt2 - 1 )2
(8)
Realize that the total density profile in the limit of infinite dilution is a relatively simple function of the parameters F, η, and z˜, or, alternatively, F, KD, and z. The diameter of the ions does not appear in this formulation. Introducing a scaling factor d (which will be meaningful when we consider charged hard spheres; d will be the ionic diameter), we define the following reduced quantities: the reduced density F* ) Fd3, the reduced inverse Debye length K/D ) KDd, and the reduced temperature T* ) e/(4πkBTd)1/2. The parameter η can then be expressed in terms of these reduced quantities as η ) The contact value is
δs(1) elec(0)
xπF*/(T*)3.
KD3 η ) -F )24π 3
C/GC 1 + Rη
(11)
where R ) -ln 2 + 3/4 ≈ 0.0568 and where we use the capacitance in reduced units C* ) Cd/(4π). In the following, we intend to introduce volume exclusion effects into the theory. Here, we intend to discuss simple modifications of the results for the point ions which we will use as a starting point. This is suggested by the fact that this model is shown9 to grasp the phenomena in the low-temperature regime. Hereafter, we discuss some approximations for the density profile. III. Effect of the Volume Exclusion on the Density Profile A. Additive Contribution. The most basic way of accounting for the hard sphere (HS) effect is to consider that this effect is decoupled from the electrostatics. In this work, we shall consider low values of the density. In this regime, the excess profile due to the repulsive potential can be considered as monotonic and approximated by an exponential
δshs ) Fh0e-aHSz (9)
and satisfies the contact theorem for the pressure calculated at the level of the Debye approximation. The differential capacitance has been calculated for the slab from the following expression9
1 ) C e2 L - 2 Ak T B
approximation, the expression of the differential capacitance shows that there is a modification of the Gouy-Chapman differential capacitance C/GC ) K/D/(4π) (ref 9) given by
C* )
is the inverse Debye length, and
I(z˜) )
Figure 1. Electrolyte in a slab between two walls charged with vanishingly small charge density (σ; the dielectric constant of the pure solvent characterizes all of the space.
(12)
The coefficients can be obtained simply; h0 is derived from the contact theorem assuming the value of the bulk HS pressure. Using the Carnahan-Stirling expression for the pressure:19 2 3 βp 1 + ηp + ηp - ηp ) F (1 - η )3
(13)
p
we obtain
∫dr1 dr2(x1 - L1)(x2 - L1)〈q(r1) q(r2)〉
(10)
h0 )
2 - ηp βp - 1 ) 2ηp F (1 - η )3
(14)
p
Near the point of zero charge, we can consider the linear response regime, and the charge-charge correlations will be computed for the uncharged system σ ) 0. At the one loop
where ηp ) πF*/6 is the packing fraction. The parameter aHS is obtained from the condition that the adsorption for the
15702 J. Phys. Chem. C, Vol. 111, No. 43, 2007
Di Caprio et al. of the parameter η to ensure that the density value is positive. The agreement of eq 19 with the simulation results is quite good for η ) 0.05. The explanation is that, for this value of η and for the reduced densities considered, the reduced temperature is very large (greater than 1) and the hard sphere effects dominate over electrostatic effects. For larger η, the electrostatic interaction between ions is larger and the assumption of additivity employed in this approximation is not valid. The contact value for this approximation is
(K/D)3 s(1)(0) ) 1 + h0 F 24πF* Figure 2. Total density profiles (solid lines) compared with MC simulation results (symbols) as obtained from the various approximations for η ) 0.2 and F* ) 0.01. In this figure and all following ones, the surface charge is zero. The hard sphere (dotted lines) and electrostatic (dashed lines) parts of the total density profiles are also shown.
exponential profile corresponds to the adsorption of the HS fluid which is estimated by20,21
Γads 9ηp2 ) 2 F πd F(1 + 2ηp)
(15)
We then require that
Γads ) Fh0
∫0∞ e-a
HSz
dz ) F
h0 aHS
(16)
This gives
aHSd ) 4
(2 - ηp)(1 + 2ηp) 3(1 - ηp)3
(17)
in reduced diameter units. For comparison with electrostatics, we shall use reduced distances in relation with the Debye length / / K-1 D . We thus define aHS ) aHS/KD ) (aHSd)/KD. Finally,
δshs(z˜) ) Fh0 e-aHSz˜ *
(18)
The density profile for this approximation is
s (z˜) ) F + δshs(z˜) + (1)
δs(1) ˜) elec(z
For a given value of the reduced density (F* ) 0.03), we show the contact density values as a function of the reduced temperature in Figure 6 (dashed line). The agreement with MC data16 is poor, which also indicates the shortcomings of this approximation. Figures 3, 4, and 5 show that the field theoretical curve (thick dashed line) for the point ions appears to be the correct limiting curve for all values of η. Our simulations for even lower reduced densities (not shown) support this: with F* f 0 the simulation curves approach the point-ion curve. Indeed, for very low densities, we may reasonably expect that the excluded volume effects are negligible. The fact, that this limit is correct for all values of η indicates that the field theoretical treatment appropriately describes the electrostatics of the system even at high ionic couplings. B. Simple Modification of the Electrostatics by the Excluded Volume. An improvement of the previous approximation is to modify the electrostatic behavior for the point ions in order to account for the repulsive potential. A simple way of performing this is to replace the parameter KD with the MSA parameter 2Γ.23 This is a natural extension of the DebyeHu¨ckel theory for nonpoint ions. This correction couples the HS potential effect with electrostatics. The Γ parameter is given by
2Γ* ) x1 + 2K/D - 1
(21)
where Γ* ) Γd. In this case, the electrostatic profile is modified according to
˜) ) δs(2) elec(z (19)
Figure 2a shows the HS (1 + δshs(z˜)/F) and excess electrostatic δs(1) ˜)/F (we added 1 to it) terms of the density profile for an elec(z intermediate value of η ) 0.2 and a relatively high density F* ) 0.01. The electrode charge is zero in this and the next figures. The profiles are plotted starting from the contact position of the ions (d/2). It is seen that the HS contribution is positive, and it brings the point-ion electrostatic term closer to the MC values; however, the agreement is still not very satisfactory, and the nonmonotonic behavior of the MC profile near the wall is not reproduced (the description of our MC simulation method can be found in our previous papers1,22). The density profiles are plotted in Figures 3, 4, and 5 for η ) 0.05, 0.2, and 0.5, respectively. A certain panel shows the results as obtained from a given approximation (see later), while inside a panel results for reduced densities F* ) 0.001, 0.005, and 0.01 are plotted. To discuss the field theory at the one loop level approximation, we have considered small enough values
(20)
( )
(2Γ)3 2Γ* I z˜ / 8π KD
(22)
The density profile is
˜) s(2)(z˜) ) F + δshs(z˜) + δs(2) elec(z
(23)
Figure 2b shows that the electrostatic term is more positive than for point ions (Figure 2a) and the total density profile is much closer to the MC values than in the case of the additive approximation. In this approximation, the contact values are the following
(Γ*)3 s(2)(0) ) 1 + h0 F 3πF*
(24)
This approximation is an improvement for the contact density values with respect to the simulated profiles for all values of η (Figure 6, solid line). The agreement with simulation data is very good even for low values of T*. The F* dependence of
Double Layer Accounting for Excluded Volume Effects
J. Phys. Chem. C, Vol. 111, No. 43, 2007 15703
Figure 3. Total density profiles (lines) compared with MC simulation results (symbols) as obtained from the various approximations for η ) 0.05 and F* ) 0.001, 0.005, and 0.01. The limiting (F* f 0) electrostatic expression for point ions is also shown (thick dashed line).
Figure 4. Total density profiles (lines) compared with MC simulation results (symbols) as obtained from the various approximations for η ) 0.2 and F* ) 0.001, 0.005, and 0.01. The limiting (F* f 0) electrostatic expression for point ions is also shown (thick dashed line).
the contact value is also appropriately reproduced for T* ) 0.15 (Figure 7). The consequence is an improvement in the density profiles (see Figures 3b, 4b, and 5b). The behavior of the curves farther from contact, nevertheless, is still unsatisfactory for larger η values. C. Short-Range Fitting of the Profile for the Contact Value. In the previous approximation, the substitution of KD by 2Γ in the expression of the profile does not reproduce the short range behavior of the curves in the vicinity of the electrode. It does reproduce, however, the correct behavior at the electrode (see the contact values in Figures 6 and 7) and far from the electrode (see Figures 3b, 4b, and 5b), where, overall, the pointion behavior is reproduced. Our approximation should interpolate between these two limiting cases. Starting from the
expression in eq 22, we propose the following transformation of the electrostatic term:
δs(3) elec
( )
* K3D 2Γ* e-caHSz˜ I z˜ / - [(2Γ)3 - K3D] )8π 24π KD
(25)
The first term provides the long range behavior of the profile, while the second term provides a short range interpolation with the contact theorem. Because the scale length where there is a difference between the MC results and those obtained from the previous approximation seems to be of the order of the HS potential (Figures 3b, 4b, and 5b), we suggest a correction term related to the short range HS potential. It takes an exponential
15704 J. Phys. Chem. C, Vol. 111, No. 43, 2007
Di Caprio et al.
Figure 5. Total density profiles (lines) compared with MC simulation results (symbols) as obtained from the various approximations for η ) 0.5 and F* ) 0.001, 0.005, and 0.01. The limiting (F* f 0) electrostatic expression for point ions is also shown (thick dashed line).
Figure 6. Contact values of the total density profiles as a function of the reduced temperature. The dashed line represents approximation eq 19, while the full line represents approximation eq 23. The symbols denote MC data.16
form with a fitting parameter c which governs the relative scale of the correction term compared to the HS potential (c is basically of order 1). The value of the amplitude is constructed so that we restore the contact theorem with the MSA contact value (eq 24). The density profile is now given by
s(3)(z˜) ) F + δshs + δs(3) elec
(26)
Figure 2c shows that this correction modifies the behavior of the electrostatic term close to the electrode. The agreement of the total density profile with MC is excellent at this value of η and F*. A comparison with simulation results for a range of parameters considered previously (Figures 3c, 4c, and 5c) shows that the agreement is good. The parameter c ) 2 in all figures. The particular value of c does not notably influence our results as long as it is close to 1. With this last approximation, we obtain an analytic expression which can reasonably reproduce profiles for all values of the density and of η presented. It is satisfying to see that it can be obtained from simple modifications of the limit system of point
Figure 7. Contact values of the total density profiles as a function of the reduced density for T* ) 0.15. The symbols and dashed line are the MC and MPB data of Bhuiyan et al.,24 while the solid line represents approximation eq 23.
ions showing that an essential part of the phenomenon is captured by the correct treatment of the fluctuations due to the Coulomb potential within the field theoretical approach. The effect of the excluded volume is then essentially obtained by introducing the parameter 2Γ in an adequate way. IV. Differential Capacitance Curves Accounting for the Hard Sphere Effects In this section, we extend the expression for the capacitance curve for point ions eq 11 to include the effect of the excluded volume. The first straightforward correction is the one used in ref 25, where to compare simulation with ions with finite size, we have introduced the Stern layer, that is, the distance of closest approach d/2. The expression for the capacitance becomes
C* )
C/GC 2πC/GC + 1 + Rη
(27)
However, in this expression, the diffuse layer still describes point ions.
Double Layer Accounting for Excluded Volume Effects
J. Phys. Chem. C, Vol. 111, No. 43, 2007 15705 parameter in place of the Debye length. A number of simple approximations have been discussed which show that we must be cautious and cannot simply replace one length by the other. However, even for the most realistic approximations the expressions remain analytic and simple. Alternatively, even the least realistic approximations describe the low temperature behavior of the double layer qualitatively. This indicates that in this regime the volume exclusion effects can be considered as perturbations and the main physics is associated with Coulombic effects described by the point-ion model.
Figure 8. Differential capacitance as a function of the reduced temperature for F* ) (a) 0.04 and (b) 0.06. Dashed lines are computed using eq 27, and solid lines are computed using eq 28; the symbols correspond to the MC simulations.2,25
A further improvement of the theory could be on the same line as that previously mentioned, namely, to introduce the MSA parameter 2Γ to substitute KD. As for the Debye-Hu¨ckel theory in the bulk, this is a natural extension of the linear GouyChapman theory for the interface.26 This can be done in two terms: C/GC and η. We consider here the change in the first term, which corresponds to the lowest order approximation. We then simply replace the Gouy-Chapman capacitance K/D/(4π) by the MSA value Γ*/(2π). Note that, in using the MSA value of the capacitance, we do not need to introduce the Stern layer. The expression of the capacitance is then
C* )
C/MSA 1 + Rη
(28)
The Gouy-Chapman based theory, with Stern layer correction, and the correction introducing the MSA modified theory are shown in Figure 8. There is a very good agreement of the capacitance curves for F* ) 0.04. However, the result is underestimated for the higher F* ) 0.06. In both cases, accounting for the excluded volume by using Γ represents an improvement toward a quantitative theory. V. Conclusion In this paper, we have extended a field theoretical approach for a point-ion electrolyte to include excluded volume effects. We have focused particularly on the low-temperature regime, where Coulombic effects are increasingly important and where phenomena like the anomalous behavior of the capacitance appear. Starting from the point-ion model, we show that it is possible to introduce the excluded volume effects in very simple ways. This is essentially performed introducing the MSA range
Acknowledgment. M.V. and D.B. are grateful for the support of the Hungarian National Research Fund (OTKA K63322). M.V., D.B., and D.d.C. are grateful for the support of the Hungarian Academy of Science and of the Centre National de la Recherche Scientifique (CNRS), in the framework of Project No. 18983. M.H. and D.d.C. are grateful for the support of the National Academy of Science of Ukraine (NASU) and the CNRS, in the framework of Project No. 18964. References and Notes (1) Boda, D.; Henderson, D.; Chan, K. Y. J. Chem. Phys. 1999, 110, 5346. (2) Boda, D.; Henderson, D.; Chan, K. Y.; Wasan, D. T. Chem. Phys. Lett. 1999, 308, 473. (3) Mier-y-Teran, L.; Boda, D.; Henderson, D.; Quinones-Cisneros, S. E. Mol. Phys. 2001, 99, 1323. (4) Holovko, M.; Kapko, V.; Henderson, D.; Boda, D. Chem. Phys. Lett. 2001, 341, 363. (5) Pizio, O.; Patrykiejew, A.; Sokołowski, S. J. Chem. Phys. 2004, 121, 11957. (6) Reszko-Zygmunt, J.; Sokołowski, S.; Henderson, D.; Boda, D. J. Chem. Phys. 2005, 122, 084504. (7) Bhuiyan, L. B.; Outhwaite, C. W.; Henderson, D. J. Chem. Phys. 2005, 123, 034704. (8) Henderson, D.; Boda, D. J. Electroanal. Chem. 2005, 582, 16. (9) di Caprio, D.; Stafiej, J.; Borkowska, Z. J. Electroanal. Chem. 2005, 582, 41. (10) Valisko´, M.; Henderson, D.; Boda, D. J. Mol. Liq. 2007, 131, 179. (11) Bhuiyan, L. B.; Outhwaite, C. W.; Henderson, D. Langmuir 2006, 22, 10630. (12) Painter, K. R.; Ballone, P.; Tosi, M. P.; Grout, P. J.; March, N. H. Surf. Sci. 1983, 133, 89. (13) Ballone, P.; Pastore, M. P.; Tosi, M. P.; Painter, K. R.; Grout, P. J.; March, N. H. Phys. Chem. Liq. 1984, 13, 269. (14) March, N. H.; Tosi, M. P. Coulomb Liquids; Academic Press: London, 1984. (15) Hamelin, A.; Rottergamm, S.; Schmickler, W. J. Electroanal. Chem. 1987, 230, 281. (16) Henderson, D.; Boda, D.; Wasan, D. T. Chem. Phys. Lett. 2000, 325, 655. (17) di Caprio, D.; Stafiej, J.; Badiali, J. P. Mol. Phys. 2003, 101, 2545. (18) di Caprio, D.; Stafiej, J.; Badiali, J. P. Electrochim. Acta 2003, 48, 2967. (19) Carnahan, N. F.; Starling, K. E. J. Chem. Phys. 1969, 51, 635. (20) Henderson, D.; Abraham, F. F.; Barker, J. A. Mol. Phys. 1976, 31, 1291. (21) Holovko, M. F.; Vakarin, E. V. Mol. Phys. 1995, 84, 1057. (22) Boda, D.; Chan K. Y.; Henderson D. J. Chem. Phys. 1998, 109, 1362. (23) Blum, L. Mol. Phys. 1975, 81, 136. (24) Bhuiyan, L. B.; Outhwaite, C. W.; Henderson, D. J. Electroanal. Chem. 2007, 607, 54. (25) di Caprio, D.; Valisko´, M.; Holovko, M.; Boda, D. Mol. Phys. 2006, 104, 3777. (26) Blum, L. J. Phys. Chem. 1977, 30, 1529.
