Langmuir 2007, 23, 1507-1517
1507
Structure of Electrolyte Solutions Sorbed in Carbon Nanospaces, Studied by the Replica RISM Theory A. Tanimura,† A. Kovalenko,‡ and F. Hirata*,† Institute for Molecular Science, Okazaki, 444-8585, Japan, National Institute for Nanotechnology, National Research Council of Canada, and Department of Mechanical Engineering, UniVersity of Alberta, 11421 Saskatchewan DriVe, Edmonton, AB, T6G 2M9, Canada ReceiVed June 6, 2006. In Final Form: October 16, 2006 The replica RISM theory is used to investigate the structure of electrolyte solutions confined in carbonized polyvinylidene chloride (PVDC) nanoporous material, compared to bulk electrolyte solution. Comparisons are made between the models of electrolyte solution sorbed in the carbonized PVDC material and a single carbon nanosphere in bulk electrolyte solution. Particular attention is paid to the chemical potential balance between the species of the sorbed electrolyte solution and the bulk solution in contact with the nanoporous material. As a result of the strong hydrophobicity of the carbonized PVDC material in the absence of activating chemical groups, the densities of water and ions sorbed in the material are remarkably low compared to those in the ambient bulk solution. The interaction between water molecules and cations becomes strong in nanospaces. It turns out that, in carbon nanopores, a cation adsorbed at the carbon surface is fully surrounded by the hydration shell of water molecules which separates the cation and the surface. Distinctively, an anion is adsorbed in direct contact with the carbon surface, which squeezes a part of its hydration shell out. The tendency increases toward smaller cations, which are characterized as “positive hydration” ions. In the bulk, cations are not hydrated so strongly and behave similarly to anions. The results suggest that the specific capacitance of an electric double-layer supercapacitor with nanoporous electrodes is intimately related to the solvation structure of electrolyte solution sorbed in nanopores, which is affected by the microscopic structure of the nanoporous electrode.
1. Introduction Porous materials play an important role in technological advances related to gas storage, separation, purification, and electric devices. In particular, the electric energy storage device called the electric double layer capacitor (EDLC) is in the focus of attention due to its importance in industrial applications. The device utilizes the charge-discharge process in the electric double layer formed on the huge specific internal surface of nanoporous electrodes. Huge values of the specific capacitance of nanoporous EDLC have been achieved, millions of times higher than the capacity of a parallel plate condenser (PPC). However, the study of nanoporous EDLC just started, and the origin of its high capacitance is not sufficiently understood. In the previous paper,1 we proposed realistic models for the carbon nanoporous electrode using the carbonized polyvinylidene chloride (PVDC) and activated carbon (AC) porous materials. On the basis of the replica reference interaction site model (replica RISM) theory,2-5 we have reproduced the capacitance comparable to that of the EDLC devices. The study revealed that the mechanism of nanoporous EDLC is different from that of PPC. The capacitance of nanoporous EDLC is governed essentially by the solvation chemical potentials of the ions in the porous electrode, whereas that of ordinary PPC relates simply to the area of the Helmholtz layer. These chemical potentials result from the interplay of the microscopic electric field, the porous electrode structure, and the distribution of electrolyte solution on the carbon surface. * To whom correspondence should be sent. E-mail:
[email protected]. † Institute for Molecular Science. ‡ National Research Council of Canada and University of Alberta. (1) Tanimura, A.; Kovalenko, A.; Hirata, F. Chem. Phys. Lett. 2003, 378, 638. (2) Kovalenko, A.; Hirata, F. J. Chem. Phys. 2001, 115, 8620. (3) Kovalenko, A.; Hirata, F. Condens. Matter Phys. 2001, 4, 643. (4) Kovalenko, A.; Hirata, F. J. Theor. Comput. Chem. 2002, 2, 381. (5) Kovalenko, A. J. Comput. Theor. Nanosci. 2005, 1, 398.
The Stern-Gouy-Chapman model has been known as a model of the electrolyte solution structure formed on a planar electrode surface, but this model disregards the microscopic structure of solvent. Considerable effort has been made to take the solvent structure into account, either by using the integral equation method6,7 or by extending the Poisson-Boltzmann approach.8-11 However, those approaches do not account for the microscopic structure of an electrode. As distinct from a double layer on a planar electrode facing bulk solution, the microscopic structure of a nanoporous electrode couples with the microscopic solvation structure of ions, and they both play a crucial role in the formation of a double layer inside narrow inner spaces of the electrode. Therefore, it is important to clarify the solvation structure of ions affected by the microscopic structure of confined spaces inside a nanoporous electrode. In 1957, Samoilov12 and Frank and Wen13 proposed two models of the hydration of ions in bulk solution, which give essentially the same picture of the ion hydration. According to these models, the mobility of water molecules in the vicinity of an ion is determined by the interplay of two competing effects. One is the electrostatic field of the central ion which forces water dipoles to orient toward the field direction. The other is the orientation inherent in the hydrogen-bonding network of water molecules. When the size of the ion is small, water molecules are bound firmly to the ion. Samoilov referred to the state of hydration as “positive hydration”. When the size of an ion is so large that the two competing effects become comparable, water molecules in (6) Carnie, S. L.; Chan, D. Y. C. J. Chem. Phys. 1980, 73, 2949. (7) Torrie, G. M.; Kusalik, P. G.; Patey, G. N. J. Chem. Phys. 1989, 90, 4513. (8) Schiby, D.; Ruckenstein, E. Chem. Phys. Lett. 1984, 95, 435. (9) Lyklema, J. Solid-Liquid Interfaces; Fundamentals of Interface and Colloid Science Series, Vol. II; Academic Press: London, 1995; p 768. (10) Ruckenstein, E.; Manciu, M. Langmuir 2002, 18, 7584. (11) Manciu, M.; Ruckenstein, E. AdV. Colloid Interface Sci. 2004, 112, 109. (12) Samoilov, O. Ya. Discuss. Faraday Soc. 1957, 24, 141. (13) Frank, H. S.; Wen, W. Y. Discuss. Faraday Soc. 1957, 24, 133.
10.1021/la061617i CCC: $37.00 © 2007 American Chemical Society Published on Web 12/20/2006
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the vicinity of the ion become more mobile than those in bulk water. Samoilov called such situation a “negative hydration”. The robustness of the model has been verified by a number of experimental as well as theoretical studies and is on duty in the interpretation of the hydration of biomolecules.14-16 One of our focuses in the present work is to clarify how the standard picture of the ion hydration in the bulk changes when the electrolyte solution is confined in nanopores. In this paper, we employ the replica RISM theory2-5 to investigate the structure of electrolyte solution confined in the carbonized PVDC nanoporous material, compared to that of the bulk electrolyte solution. This theory is capable of predicting the molecular solvation structure of sorbed electrolyte solution affected by the microscopic structure of the host material. We examine the effect of confinement on the solvation structure, thermodynamics, and density of the electrolyte solution. The structure of various aqueous electrolyte solutions of simple ions is studied in order to elucidate the correlation between the ion size and the hydration structure. The main purpose of the paper is to clarify the mechanism of formation of the high specific capacitance of nanoporous EDLC compared to PPC. In order to make a direct comparison, we model PPC by a single carbon sphere (SCS) of a large, nanosize radius. In this study, we ensure the chemical balance between the sorbed electrolyte solution and the bulk solution in contact with the nanoporous material. We achieve the balance by adjusting the densities of the sorbed species to reach equality between the chemical potentials of the solution species sorbed in nanopores and those in the bulk solution. The procedure brings the chemical potentials of solution species (particularly those of ions) sorbed in the carbonized PVDC material down to the values in the bulk solution.
2.1. Replica RISM Theory. The replica RISM integral equation method2-5 for the structure and thermodynamics of quenched-annealed molecular systems treats the molecular fluid of species 1 with temperature T1 sorbed in a porous matrix of species 0 with temperature T0. The average free energy of the sorbed fluid can be calculated as a statistical average of the fluid free energy with canonical partition function Z1(q0) over the ensemble of all matrix spatial configurations q0
(1)
where Aid 1 is the ideal gas term and kB is the Boltzmann constant. Calculation of the statistical average of a logarithm is difficult and is obviated by employing the so-called replica identity relating the logarithm to the analytic continuation of moments Zs
(2)
For integer s, the moment average takes the form of the equilibrium canonical partition function
〈Zs1(q0)〉q0 ) Zrep(N0, N1 ) . . . ) Ns)
(4)
Provided the free energy Arep(s) is available in a close analytical form, the derivative in eq 4 can be taken analytically, which yields the thermodynamics of sorbed fluid. In the case of rigid intermolecular bonds, we write Chandler and Andersen’s site-site Ornstein-Zernike equations or RISM integral equations for the site-site correlation functions of the annealed (s + 1)-component molecular mixture. With allowance for the symmetry of the s liquid replicas, the RISM equations take the form 00 00 00 00 00 01 00 00 10 ) ωRµ *c00 hRγ µν * ωνγ + ωRµ * cµν * F0hνγ + sωRµ * cµν * F1hνγ
(5) 10 11 00 11 10 00 hRγ ) ωRµ * c10 µν * ωνγ + ωRµ * cµν * F0hνγ + 11 10 11 12 10 * c11 ωRµ µγ * F1hνγ + (s - 1)ωRµ * cµν * F1hνγ (6) 01 00 11 00 00 01 hRγ ) ωRµ * c01 µν * ωνγ + ωRµ * cµν * F0hνγ + 00 11 00 01 12 * c01 ωRµ µγ * F1hνγ + (s - 1)ωRµ * cµν * F1hνγ (7) 11 11 11 11 10 01 hRγ ) ωRµ * c11 µν * ωνγ + ωRµ * cµν * F0hνγ + 11 11 11 12 * 12 * c11 ωRµ µγ * F1hνγ + (s - 1)ωRµ * cµν F1hγν (8)
11 12 11 12 11 * c11 ωRµ µγ * F1hνγ + ωRµ * cµγ F1hνγ + 11 * 12 * c12 (s - 2)ωRµ µν F1hγν (9) ij where hRγ is the intermolecular part of the total correlation ij function between interaction sites R and γ of species i and j, cRγ 12 is the site-site direct correlation function, the terms hRγ and 12 cRγ mean the site-site correlations between molecules from ij is the equivalent but different replicas of the liquid, ωRγ intermolecular matrix, Fi is the number density of molecular species i ) 0, 1 (bearing in mind that F1 ) F2), and an asterisk (*) means convolution in the direct space and summation over repeating site indices µ, ν. Assuming that there is no replica symmetry breaking in the analytic continuation of these equations to s ) 0, we obtain the set of the replica RISM integral equations
00 00 00 00 00 00 ) ωRµ * c00 hRγ µν * ωνγ + ωRµ * cµν * F0hνγ
s
dZ ln Z1 ) lim s f 0 ds
dArep(s) sf0 ds
A1 ) lim
12 11 11 11 10 01 ) ωRµ * c12 hRγ µν * ωνγ + ωRµ * cµν * F0hνγ +
2. Theory and Models
A1 ) Aid 1 - kBT1〈ln Z1(q0)〉q0
of the molecular fluid is obtained as the analytic continuation of the free energy of the annealed replicated system Arep(s)
(3)
of a fully annealed (s + 1)-component liquid mixture comprising the matrix species and s equivalent replicas of the liquid with no interaction between replicas. Then, the average free energy (14) Chong, S. H.; Hirata, F. J. Phys. Chem. B 1997, 101, 3209. (15) Collins, K. D. Biophys. J. 1997, 72, 65. (16) Hribar, B.; Southall, N. T.; Vlachy, V.; Dill, K. A. J. Am. Chem. Soc. 2003, 124, 12302.
(10)
10 11 00 11 10 00 ) ωRµ * c10 hRγ µν * ωνγ + ωRµ * cµν * F0hνγ + 11 10 * c(c) ωRµ µγ * F1hνγ (11) 00 01 11 00 00 01 hγ01 R ) ωRµ * cµν * ωνγ + ωRµ * cµν * F0hνγ + 00 (c) ωRµ * c01 µγ * F1hνγ (12) 11 11 11 11 10 * 01 ) ωRµ * c11 hRγ µν * ωνγ + ωRµ * cµν F0hνγ + 11 11 11 (b) (c) ωRµ * c(c) µγ * F1hνγ + ωRµ * cµν * F1hνγ (13) (c) 11 11 11 (c) (c) ) ωRµ * c(c) hRγ µν * ωνγ + ωRµ * cµν * F1hνγ
(14)
where the site-site total and direct correlation functions between fluid molecules are subdivided into the connected and discon-
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Langmuir, Vol. 23, No. 3, 2007 1509
nected (or blocking) parts denoted respectively by superscripts (c) and (b) 11 hRγ
)
(c) hRγ
+
(b) hRγ
(15)
11 (c) (b) cRγ ) cRγ + cRγ
(16)
(b) (b) and cRγ obtained in the limit and the blocking correlations hRγ sf0 from the correlations between different replicas of liquid, (12) (12) hRγ and cRγ . To appropriately complement the replica RISM equations with closure approximations, we employed the approximation proposed by Kovalenko and Hirata17,18 (KH closure) for the matrixmatrix, fluid-matrix, and fluid-fluid correlations
ij ij ij ) exp(dRγ (r)) for dRγ (r) e 0 gRγ ij ij (r) for dRγ (r) > 0 ) 1 + dRγ
ij dRγ (r) ) -
ij uRγ (r) ij ij + hRγ (r) - cRγ (r) kBT
(17a)
ij hRγ
(17) Kovalenko, A.; Hirata, F. J. Chem. Phys. 1999, 110, 10095. (18) Hirata, F., Ed. Molecular Theory of SolVation; Understanding Chemical Reactivity Series, Vol. 24; Mezey, P. G., Ed.; Kluwer Academic Publishers: Dordrecht, 2003; pp 1-60, 178-184. (19) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic: London, 1986. (20) Given, J.; Stell, G. Physica A 1994, 209, 495. (21) Given, J.; Stell, G. In Condensed Matter Theories; Blum, L., Malik, F. B., Eds.; Plenum: New York, 1993; Vol. 8, pp 395-410.
(18)
with the bridge correction based on the Verlet functional form,22 which is modified as23 (b) (r) ) bRγ
(b) (b) 0.5[hRγ (r) - cRγ (r)]2
(19)
(b) (b) 1 + 0.8‚max [hRγ (r) - cRγ (r), 0]
The modified Verlet correction (eq 19) provides a reasonable first approximation to the nonlinearity of the solutions due to the blocking correlations in a quenched-annealed system with charged species. Notice that the closure (eq 18) does not contain an interaction potential, because the blocking correlations are obtained in the limit sf0 from the correlations between different replicas of liquid which do not interact with each other. 2.2. Chemical Potential of Sorbed Species. The chemical potential µs of sorbed solution species s consists of the ideal gas contribution µid s ) kBT1 ln(FsΛs) and the excess term ∆µs due to the intermolecular interactions inside the porous material
µs ) µid s + ∆µs
(17b)
where (r) ) (r) + 1 is the site-site radial distribution ij (r) is the site-site interaction potential. The function, and uRγ KH closure (eq 17) combines the hypernetted chain (HNC) ij approximation19 for density depletion regions of hRγ (r) < 0 and the mean spherical approximation (MSA)19 for density enrichment ij (r) > 0. The distribution function and its first regions of hRγ ij (r) ) 0 by derivative are continuous at the joint point dRγ construction. For bulk water, simple ions in aqueous solution, and molecular ions in a polar molecular solvent, the KH closure slightly reduces and widens high peaks of the distribution functions as compared to the HNC closure, but it still has less influence on the coordination number of solvation shell. The linearization in the rich density region prevents exponential rise of the distribution function that can bring about the divergence of the RISM equations in the case of a strong attractive potential. The KH closure efficiently combines the advantages of the HNC and MSA closures and enables one to describe associating molecular fluids and solutions in the whole density range from gas to liquid. It predicts fluid phase diagrams of complex molecular fluids,4,18 including those of complex fluids sorbed in nanoporous media.4 We applied the KH closure to the matrixmatrix, matrix-fluid, and fluid-fluid correlations. In this context, the KH closure is suitable to reproduce a given structure of pores and to describe the correlation of electrolyte solution in charged electrodes, which is of coupled long-range electrostatic and shortrange associating character. Proper account of the blocking correlations requires a closure to eq 13 that goes beyond such linearized approximations as the MSA.20,21 We found that the HNC approximation strongly overestimates the blocking correlations in the presence of charged species and eventually leads to divergence of the replica RISM equations for electrolyte solution sorbed in nanoporous material. Therefore, we used the closure ij gRγ
(b) (b) (b) (b) hRγ (r) + 1 ) exp[hRγ (r) - cRγ (r) + bRγ (r)]
(20)
where Fs is the number density of species s, and Λs )
x2πp2/(mskBT1) is the de Broglie thermal wavelength of ideal
monatomic particles with molecular weight ms. The excess chemical potential ∆µs of sorbed fluid can be decomposed into the contributions of the fluid-matrix, fluid-fluid, and blocking correlations2-5
∆µs ) ∆µs0 + ∆µs1 - ∆µ(b)
(21)
In the HNC approximation to the RISM equations, the solvation chemical potential can be calculated from the site-site correlation functions by using the Singer-Chandler analytical expression.24 The KH closure approximation (eq 17) leads to a similar analytical form of the excess chemical potential. The fluid-matrix and fluid-fluid terms of the excess chemical potential of sorbed liquid in the KH approximation read2-5
∆µsj ) 4πFskBT1
[
1
∫r2 dr 2 θ(-hRγsj (r))(hRγsj (r))2 ∑ R,γ
]
1 sj sj sj (r) - hRγ (r)cRγ (r) (22) cRγ 2 where Fs is the number density of solvent species s, and the index j ) 0, 1 assumes the values meaning matrix and solvent, respectively. The blocking term of the excess chemical potential, ∆µ(b), is obtained in the approximate analytical form4,5 which is available for the modified Verlet closure (eqs 18-19) in the assumption of unique functionality of the correlations when “switching on” the interactions in the system.25 2.3. Model of Nanoporous Carbonized PVDC. A TEM image of PVDC reveals that it has a uniform nanoporous texture. Spherical cavities close in size are connected into wormlike pores and separated by walls formed of connected carbon particles shaped as overlapping spheres similar in size to cavities. This resembles a close-packing structure of a binary mixture, which is disordered by displacement of spheres from the periodic lattice positions and by their random choice as either a carbon (22) Verlet, L. Mol. Phys. 1980, 41, 183. (23) Kinoshita, M.; Imai, T.; Kovalenko, A.; Hirata, F. Chem. Phys. Lett. 2001, 348, 337. (24) Singer, S. J.; Chandler, D. Mol. Phys. 1985, 55, 621. (25) Lee, L. L. J. Chem. Phys. 1992, 97, 8606.
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Figure 1. Model structure of nanoporous carbonized material (PVDC) and the single carbon sphere (SCS). Black and white spheres represent carbon nanoparticles and portions of pore voids, respectively.
nanoparticle or a cavity (nanopore). In order to model PVDC, two kinds of spheres, “black” and “white” ones, were introduced to represent carbon nanoparticles and cavities of nanoporous material. Its structure was modeled by a fluid of these repulsive spheres at a very high packing fraction close to the hexagonal close packing (HCP) (Figure 1). The interaction potential is the same for “black” and “white” particles and is chosen in the form of the Weeks-Chandler-Andersen (WCA) repulsive part19 of the 2-Yukawa potential
[(
u00 (r) ) 0 4 exp
) ]
(σ0 - r) 2(σ0 - r) - exp +1 w w
) 0 for r g σ0
Figure 2. Radial distribution function between carbon spheres, and pore size distribution of PVDC model. Theory (solid line) and experimental data26 (dashed line). Table 1. Properties of the Carbonized PVDC Nanoporous Material expt26 (g/cm3)
for r < σ0 (23)
For the size, energy, and core edge width parameters, we chose the values σ0 ) 17 Å, 0 ) 10 kcal/mol, and w ) 1 Å, respectively. The diameter σ0 determines the mean distance between nanospheres in contact, whereas the energy and width parameters of the initial 2-Yukawa form, 0 and w, determine the steepness of the repulsive potential (eq 23) and thus the spread of the distances around σ0. Carbon nanopores are modeled by setting the interaction potential between “white” spheres and solution species to be zero and thus making these cavity spheres “invisible” for the solution. The number densities of “carbon” and “cavity” spheres are set as F0crb ) 1.27 × 10-4 Å-3 and F0cav ) 1.43 × 10-4 Å-3. To make up for overlap and connectedness of carbon nanoparticles in the PVDC material, the diameter σ0 is chosen to be somewhat smaller than the carbon nanosphere diameter D0 ) 19 Å “seen” by solution which enters the interaction potential (eq 27) between carbon nanospheres and solution species. The correlation functions of spheres in the model porous material was obtained by solving the Ornstein-Zernike integral equation19 complemented with the KH closure for the binary mixture of “black” and “white” nanoparticles at the temperature chosen for simplicity to be the same as that of the sorbed solution, T0 ) T1 ) 300 K. The radial distribution functions of carbon g00(r) shown in Figure 2 have a very sharp first peak and slowly diminishing long-range oscillations lasting up to 200 Å. The positions of the first and subsequent peaks correspond to the size of spheres in the PVDC model. We adjusted the above parameters so as to fit the pore size distribution, the density, porosity, and surface area of the PVDC
density porosity (cc/cm3) surface area (m2/cm3) average pore size (Å)
PDVC model
0.9 630 11
0.9 0.60 630 10
model as probed by a test particle of size 3.5 Å to the experimental data for the PVDC material26 (Table 1). The porosity of a model material p(σhs) is defined as a fraction of its volume available for insertion of a hard sphere probe with diameter σhs, which “sees” the “black” nanoparticles as hard bodies of size D0 and does not “see” the “white” ones. Using Widom’s test particle theory19 that relates the void volume in an ensemble of hard body particles to the excess chemical potential of a test particle ∆µ(σhs), we calculate the porosity as1,4,5
p(σhs) ) exp[-∆µ(σhs)/(kBT0)]
(24)
where ∆µ(σhs) is obtained by solving the solute-solvent Ornstein-Zernike integral equation complemented with the Verlet closure22 for the test particle at infinite dilution in this binary mixture of “black” and “white” nanoparticles. The Verlet closure approximation is adequate in this case, as it was elaborated for hard-sphere fluid.22 The distribution of pores over their size is then obtained by differentiation of the porosity with respect to the probe size
V(σhs) )
dp dσhs
(25)
The specific surface area of pores is determined as a change of the void volume fraction with covering quenched particles by a thin hard layer, which is given by the derivative (26) Takeda, T.; Endo, M. Tanso 1999, 198, 179.
Electrolytes Sorbed on C Nanospaces
s(σhs) )
∂p ∂R0
Langmuir, Vol. 23, No. 3, 2007 1511
|
(26)
g00(r),σhs
The pore size distribution of the PVDC model is shown in Figure 2. The position of this pore peak is fitted to the experimental curve taken from Figure 10 in ref 26. A carbon composite nanosphere of diameter D0 ) 19Å is treated as an aggregate of uniformly distributed carbon atoms interacting with solution species by the 12-6 Lennard-Jones interaction potential. This leads to the Kaminsky-Monson interaction potential27 between the carbon sphere and interaction sites of the solution species
(r) u(KM) R )
[
12 (r6 + 21r4R20/5 + 3r2R40 + R60/3)σRC 16 3 πR0FCRC 3 (r2 - R2)0
]
0
6 σRC
(r2 - R20)3
for R0 < r < ∞
) ∞ for r e R0
(27)
where FC is the number density of carbon atoms constituting a composite sphere, and R0 ) D0/2 is the radius of the carbon composite sphere. The potential goes to infinity as the interaction site of the fluid approaches the surface of the composite sphere, rfR0. The composite sphere density FC is set equal to that of graphite with physical density 2.27 g/cm3. The Lennard-Jones size and energy parameters of the carbon atom, σC and C, and those of fluid interaction sites, σR and R, are given in Table 2. The parameters of the Lennard-Jones potential between unlike species are obtained from the standard mixing rules: σRC ) (σR + σC)/2 and RC ) xRC. For a nanoporous carbon electrode with the specific electric charge qext and the number density of carbon nanospheres F0crb, the electric charge per carbon nanosphere is Q0 ) qextF0crb, and we include the electrostatic potential of a conducting sphere with charge Q0 in the interaction between the nanosphere and solution species site R with partial charge qR (KM) (r) + u10 R (r) ) uR
(r) + ) u(KM) R
Q0qR for R0 < r < ∞ r
Q0qR for r e R0 R0
(28)
In order to mimic nanopores, we set u10 R (r) ) 0 for the interaction between “white” nanospheres and electrolyte solution species. The site-site interaction potential between solution species comprises the 12-6 Lennard-Jones potential and the Coulomb interaction between their site charges 11 (r) ) 4Rγ uRγ
[( ) ( ) ] σRγ r
12
-
σRγ r
6
+
Q0qR r
(29)
The parameters for unlike site pairs are obtained from the standard mixing rules. We used the SPC water model28 and the LennardJones parameters of the simple ions taken from ref 29, which are summarized in Table 2. (27) Kaminsky, R. D.; Monson, P. A. Langmuir 1994, 10, 530. (28) Berendsen, H. J. C.; Postma, J. P. M.; van Gunsteren, W. F.; Hermans, J. In Intermolecular Forces; Pullman, B., Ed.; Reidel: Dordrecht, 1981; p 331. (29) Koneshan, S.; Rasaiah, J. C.; Lynden-Bell, R. M.; Lee, S. H. J. Phys. Chem. B 1998, 102, 4193.
Table 2. Lennard-Jones Potential Parameters of the Water Oxygen and Hydrogen Interaction Sites, Simple Ions, and Carbon Atoms Constituting the Nanoporous Material atom
σ [Å]
[kcal/mol]
O H Na+ K+ Cs+ ClBrIC
3.166 1.0 2.583 3.331 3.883 4.401 4.539 5.167 3.400
0.1554 0.046 0.100 0.100 0.100 0.100 0.100 0.100 0.0556
2.4. Model of a Parallel Condenser. The main purpose of this paper is to clarify the microscopic origin of the large difference in the capacitance exhibited by two types of EDLC: with electrodes made of nanoporous carbon and with parallel plate electrodes. To facilitate a direct comparison, the model with parallel plate electrodes should be as close as possible to the model of nanoporous carbon, still retaining the essential feature of a planar electrode which faces bulk solution or a macroscopically thick solution layer. From this consideration, we modeled PPC by a single carbon sphere (SCS) of diameter 19 Å. In the actual implementation for the binary mixture described in the previous subsection, the SCS is realized simply by making one of the spheres “black” and the rest “white”, as illustrated in Figure 1. The interaction between SCS and solution species is described by the potential (eq 28) with the same parameters as those of the carbonized PVDC model.
3. Results and Discussion 3.1. Equilibrium Conditions for Sorbed Electrolyte Solution. When the electrolyte solution is confined in nanospace, it should be equilibrated with the bulk solution with which the nanoporous carbon is in contact. In other words, the chemical potential of each solution species in nanopores should be equal to that in the bulk solution. This thermodynamic requirement imposes a condition on the density (and concentration) of each species of the solution in nanospace, which is entirely different from the bulk. In the present study, we adjusted the densities of sorbed solution so that the chemical potentials of the species in nanopores and in ambient bulk solution are equal. This gives density of the electrolyte solution sorbed in nanopores in agreement with the experimental findings30 and molecular simulations.31,32 Aqueous solutions sorbed in hydrophobic nanopores resemble a gaseous phase of comparably high density. Alcaniz-Monge et al.30 investigated the density of water sorbed in activated carbon fiber by measuring the adsorption isotherms. They revealed that the density of sorbed water is less than that of liquid-state water. Segarra and Glandt31 found by molecular simulations that the amount of water vapor at p/p0 ) 0.8 adsorbed in a hydrophobic nanoporous material is less than 10-3 of water adsorbed by commercial carbon. McGrother and Gubbins32 used molecular simulations to describe the water adsorption isotherm in a carbon nanopore and showed that the density of sorbed water is small. Our findings are consistent with these experiments and simulations. Table 3 shows the density of water and ions of the electrolyte solutions sorbed in the carbonized PVDC nanoporous electrode at zero charge in chemical equilibrium with bulk solution at (30) Alcaniz-Monge, J.; Linares-Solano, A.; Rand, B. J. Phys. Chem. B 2002, 106, 3209. (31) Segarra, E. I.; Glandt, E. D. Chem. Eng. Sci. 1994, 49, 2953. (32) McGrother, S. C.; Gubbins, K. E. Mol. Phys. 1999, 97, 955.
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Table 3. Density of Species of Bulk Electrolyte Solutions at Ambient Conditions, with a Single Carbon Sphere (SCS)a
Table 4. Coordination Numbers Nrγ of the First Solvation Shell of Species γ around Species r in NaCl Electrolyte Solutiona SCS
Bulk Solution with SCS electrolyte NaCl KCl CsCl KBr KI
water
[g/cm3
0.9794 0.9704 0.9570 0.9360 0.9549
]
ions [g/cm ]
concentration [M]
R-γ pair
0.0603 0.0730 0.1689 0.1190 0.1555
1.032 0.980 1.003 1.000 0.937
O-O O-H Na+-O Cl--O C-O C-H C-Na+ C-Cl-
3
Solution Sorbed in PVDC electrolyte
water [g/cm3]
ions [g/cm3]
concentration [M]
NaCl KCl CsCl KBr KI
0.2839 (0.4732) 0.2836 (0.4727) 0.2806 (0.4677) 0.2797 (0.4662) 0.1026 (0.1710)
2.911 × 10-8 (4.852 × 10-8) 1.882 × 10-6 (3.137 × 10-6) 0.9086 × 10-4 (1.514 × 10-4) 0.6916 × 10-5 (1.153 × 10-5) 0.6478 (1.0797)
4.87 × 10-7 2.34 × 10-5 4.81 × 10-4 5.28 × 10-5 3.90
a Average density of species of sorbed electrolyte solutions per macroscopic volume of the carbonized PVDC nanoporous material. Density of sorbed electrolyte solutions normalized per accessible volume of nanopores (values in parentheses). Concentration of the bulk and sorbed solutions.
ambient conditions, as well as the densities of the corresponding species in the bulk solution with a single carbon sphere (SCS). It turns out that the densities of water in PVDC are remarkably low, less than 1/3 compared to those in bulk. Taking into account that the porosity of the model PVDC material is about 60% (Table 1), the normalized density of the sorbed solution per volume of pores is about half of that in bulk solution. Hence, water and ions in nanopores have a gaslike rather than liquid structure. The hydrophobic matrix of PVDC material suppresses the concentration of ions in the sorbed solution by several orders of magnitude, resulting in the salting-out effect. An exception is the aqueous solution of KI, for which the opposite effect of salting-in is observed. Because the large I- anions are attracted to carbon nanospheres stronger than the smaller Cl- or Br- (see Figure 9 below), the ionic density considerably increases and depletes the water density. The pores get filled essentially with the KI electrolyte which reaches a concentration close to that in the saturated bulk solution. Notice that the salting-out effect is not observed for the CsCl aqueous solution. Although the attraction of the large cation Cs+ to a carbon nanosphere in bulk solution is much stronger compared to that of smaller Na+ and K+, this enhancement for Cs+ disappears and even changes to the opposite trend for solution sorbed in PVDC material (see Figure 8 below). The latter can be explained by the stronger hydration of Cs+ in nanospace, and so the enhanced hydration shell holds Cs+ apart from carbon nanospheres in PVDC. The above results lead to the following two considerations. First, the hydration numbers of ions become small, because the number of water molecules available to hydrate the ions is reduced. Second, the interaction between an ion and water becomes strong, because the competition between water molecules weakens. Ohkubo et al.33-35 reported that the hydration number of Rb+ is smaller and the distance between Rb+ and water molecules is shorter in nanospace. They concluded that the interaction between Rb+ and water molecules is stronger in nanospace, compared to those in bulk. Moreover, such a state of low density is apparently close to the supercritical condition. The structure (33) Ohkubo, T.; Konishi, T.; Hattori, Y.; Kanoh, H.; Fujiwara, T.; Kaneko, K. J. Am. Chem. Soc. 2002, 124, 11860. (34) Ohkubo, T.; Kanoh, H.; Hattori, Y.; Konishi, T.; Kaneko, K. Stud. Surf. Sci. Catal. 2003, 146, 61. (35) Ohkubo, T.; Hattori, Y.; Kanoh, H.; Konishi, T.; Fujiwara, T.; Kaneko, K. J. Phys. Chem. B 2003, 107, 13616.
PVDC
rmin [Å]
NRγ
rmin [Å]
NRγ
4.20 2.30 3.25 4.65 13.40 13.65 15.15 13.65
9.90 1.40 5.03 13.04 159.1 352.6 4.75 3.13
5.05 2.30 3.35 5.15 13.85 14.25 15.45 14.15
8.87 0.74 3.67 9.38 67.7 154.3 2.8 × 10-6 1.9 × 10-6
a The size of the solvation shell is defined by the first minimum of the corresponding radial distribution functions gRγ(r) at separation rmin. Bulk solution at concentration 1 M around a single carbon sphere (SCS), and solution sorbed in the carbonized PVDC nanoporous material in chemical equilibrium with the bulk solution.
of aqueous electrolyte solutions in dense gas at supercritical conditions has been studied by Kubo et al.36 They calculated the hydration number of an ion in supercritical solution by means of integration of the local density distribution and reported that the hydration shell around the ion persists even at a gaslike density due to the very strong ion-water interaction. It is expected that the hydration shell remains present in nanospace as well, despite such a low density of sorbed solution. ij The hydration number NRγ of site γ of species j with number density Fj around site R of a labeled molecule of species i is calculated by integrating the site-site radial distribution function gRgγij (r) up to its first minimum at separation rmin ij NRγ (rmin) ) 4πFj
∫0r
min
ij r2 dr gRγ (r)
(30)
(Note that the hydration number is a somewhat ambiguous indicator which depends on a definition of the hydration shell; we use the common one based of the first minimum of the radial distribution function.) The results are summarized in Table 4. In carbonized PVDC nanopores, almost all the first minima considerably shift to a larger distance, compared to bulk solution. Although the average density of water sorbed in nanopores amounts to only 1/3 of that in bulk solution, the water oxygenoxygen coordination numbers remain almost the same as in the bulk due to the stronger O-O correlations (see the next subsection) and the expansion of the first hydration shell in size (see the values of rmin in Table 4). However, the first minimum rmin of the O-H correlations remains at precisely the same distance; hydrogen bonds do not form between water molecules at larger O-O separations occurring in the expanded first hydration shell. The reduced water density thus decreases the O-H coordination number in PVDC, compared to bulk solution. At the same time, this change is in part cancelled by the considerably increased strength of hydrogen bonds (see the next subsection). As a result, the water oxygen-hydrogen coordination number in PVDC amounts to 1/2 of the value in bulk solution, reflecting the depletion of the water hydrogen-bonding network confined in nanospace. The ion-water coordination numbers decrease in PVDC by just 28%. This finding confirms the observation that nanospace confinement strengthens the hydration of ions and keeps the local density of their hydration shells close to the water density in bulk solution. The carbon sphere-water coordination numbers decrease to 0.43 of those in the bulk solution around SCS, which still corresponds to water density enhancement around PVDC (36) Kubo, M.; Levy, R. M.; Rossky, P. J.; Matubayasi, N.; Nakahara, M. J. Phys. Chem. B 2002, 106, 3979.
Electrolytes Sorbed on C Nanospaces
Figure 3. Radial distribution functions between water O, water H, and Na+ and Cl- ions in aqueous NaCl electrolyte solution sorbed in nanoporous PVDC material, versus those of the bulk solution with a single carbon sphere (SCS).
nanospheres. The carbon sphere-ion coordination numbers strongly diminish in the PVDC material with the drastic drop of the density of sorbed ions. Thus, water molecules and ions sorbed in carbonized PVDC nanopores show a tendency to concentrate in small clusters or nanodroplets with the local density strongly enhanced in comparison to the gaseous mean density of sorbed solution. 3.2. Structure of Electrolyte Solution Confined in Carbon Nanopores. In this section, we discuss the structure of electrolyte solutions confined in the nanoporous PVDC material in terms of the site-site radial distribution functions (RDFs). Our main focus here is the effect of nanoporous confinement on the RDFs. In order to clarify the effect, we compare the results for PVDC with those for the SCS. 3.2.1. Water-Water and Ion-Water Distributions. In Figure 3are shown the site-site RDFs between solution species: water oxygen and hydrogen sites, and sodium and chloride ions. (Notice that the distance axis is the logarithmic scale). Figure 4 zooms in the region of small oscillations of the same RDFs at a distance. We make a comparison of the RDFs in the sorbed solution (solid lines) with those in the bulk solution (dashed lines). Averaged over the whole volume, the RDFs in the bulk solution are not affected by the presence of the SCS which is at infinite dilution. A deviation from these bulk RDFs indicates the effect of nanopores on the structure of the aqueous solution.
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Figure 4. Long-range oscillations of the radial distribution functions given in Figure 3 for the sorbed and bulk solutions (the same line nomenclature as in Figure 3). Shown also is the connected part of the correlations between solution species in PVDC (thin solid lines).
As can be seen in Figure 3a, the water oxygen-oxygen RDF in PVDC has two salient features. First, the value of the O-O RDF at the first minimum runs to the region of local density enrichment, gO-O(rmin) > 1, whereas in bulk solution, it corresponds to local density depletion, gO-O(rmin) < 1. In conventional liquid, the first minimum of g(rmin) < 1 is a manifestation of the packing effect for liquidlike densities and shifts up into the region of strong local density enhancement in a supercritical or nearly critical regime. This strongly suggests that the water sorbed in nanopores behaves more like supercritical or critical fluid and exists as an equilibrium state between a range of phases, from liquid nanodroplets to critical and gaseous state. Our findings for the coordination numbers in the previous subsection are in agreement with this conclusion. The second characteristic of the O-O RDF of water sorbed in the carbonized PVDC material is the long-range tail, which extends well over 100 Å, as seen from Figure 4a. It is caused by the characteristic long-range correlations in the nanostructure of the PVDC host material (see Figure 2). Since the average radius of the pore is about 5.5 Å, this tail of the O-O RDF of sorbed water reflects the correlation between water molecules in different pores, or the interpore correlation. The latter is allowed for in the replica RISM equations by the blocking part of the correlation functions, introduced in the subdivisions (eqs 15 and 16). For comparison, the connected parts of the RDFs which correspond to the correlations within a pore are also shown in
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Figure 4 by thin solid lines. As is clearly seen, the oscillations of the connected part of the RDFs decay even faster than the corresponding RDFs in the bulk solution. The peak of the water oxygen-hydrogen RDF in Figure 3b corresponds to the typical configuration of the hydrogen-bond structure. As discussed above, the hydrogen-bonding peak in the sorbed water is considerably higher than in the bulk solution, which diminishes the effect of the reduced average density of sorbed water on the water oxygen-hydrogen coordination number (see Table 4). The long-range oscillations of the O-H RDF in the sorbed solution (Figure 4b) are similar in phase and amplitude to those of the O-O RDF (Figure 4a). The changes in the ion-water RDFs in PVDC compared to bulk solution exhibit basically the same trend as the water O-O RDF. Shown in Figure 3c,d are the RDFs for the Na+-O and Cl--H pairs, respectively. In each case, the first minima of the RDFs are not as deep as those in the bulk solution, although they are somewhat more pronounced than that in the O-O RDF. The results are indicative of the existence of a more stable hydration shell around the ions. Much as the in water-water correlations, the ion-water RDFs acquire the long-range tails characteristic of the interpore correlations caused by the long-range correlations of the PVDC nanostructure (Figure 4c,d). The amplitudes of the oscillations are larger than those of the water-water interpore correlations, as expected for the ion-water interaction, which is stronger than that between water molecules. The connected part of the ion-water RDFs again decays even faster than the RDFs in the bulk solution. Being quite small in amplitude, the long-range oscillations of the RDFs between the sorbed solvent species induced by the PVDC correlations noticeably affect the hydration numbers of the subsequent solvation shells. The running excess hydration ij number ∆NRγ (r) of site γ of species j within the distance r around site R of a labeled molecule of species i is defined as an ij excess of the hydration number NRγ (r) given by eq 30 over the uniform distribution of species j with number density Fj
4 ij ij ∆NRγ (r) ) NRγ (r) - πr3Fj ) 4πFj 3
∫0r r′2 dr′ hRγij (r′)
(31)
ij In the limit ∆NRγ (r f ∞), it obviously gives the Kirkwood-Buff ij integral GRγ. Figure 5 makes a comparison of the running excess hydration numbers in the sorbed solution with those in the bulk solution. Both the water-water and ion-water excess hydration numbers (Figure 5a,b) significantly increase at a distance, as a result of the strong enhancement of the corresponding RDFs at the distance from 5 to 20 Å (see Figure 3). Next, follow the long-range oscillations of considerable amplitude which last up to 500 Å. The connected part of the excess hydration numbers (eq 31) which is defined according to the subdivision (eq 15) is shown in Figure 5 with dotted lines. For both the water-water and ion-water excess hydration numbers, their connected parts have neither long-range enhancement nor long-range oscillations. The connected part of the Kirkwood-Buff values, G(c) ) ∆N(c)(r f ∞), for water around ions in the sorbed solution (in particular, for Na+-O shown by the dotted line in Figure 5b) are different from those in the bulk solution, as a result of the different water density. It is the values G(c) that enter the compressibility of solution sorbed in nanoporous material,2,3 whereas the full RDFs and correspondingly the excess hydration numbers ∆N(r) show up in experiment as the structure factors of sorbed solution. 3.2.2. Ion-Ion Distributions. In Figure 6, plotted are the ionion RDFs in the solution sorbed in PVDC, compared to the bulk solution with SCS at infinite dilution. In the latter case, these are typical results of the RISM theory for bulk solutions, and we
Figure 5. Running excess hydration numbers in the sorbed and bulk solutions of NaCl (solid and dashed lines, respectively). Connected part of the correlations (dotted lines). Running excess hydration numbers of water oxygens around a water oxygen (part a) and around a Na+ ion (part b).
provide only a brief summary of these results. The RDF for the unlike ion pair (+ -) exhibits a large first peak due to the electrostatic attraction, which is identified as the so-called ionic atmosphere. On the other hand, the interactions between the like pairs (+ +) or (- -) are essentially repulsive, and their RDFs do not have such a large first peak as the RDF for the unlike pair. However, the RISM theory as well as molecular simulation for anions in aqueous solution produces a well-defined first peak in the anion-anion RDF, just like that of the Cl--Cl- RDF in Figure 6c. The peak is attributed to two anions bridged by a water molecule via hydrogen bonds. It appears that the height of the first peak between two anions in aqueous solution strongly depends on the potential parameters used in the calculation. It is still a controversial issue whether such a stable pair of waterbridged anions in aqueous solution is real or just an artifact of the force field employed. The ion-ion RDFs in the solution sorbed in PVDC show a behavior entirely different from that in bulk solution. The RDFs for like pairs Na+-Na+ and Cl--Cl- are zero at distances r up to about 10 Å (Figure 6a,c). It means there are essentially no co-ions within a pore, because the average radius of the pore is about 10 Å. On the other hand, the RDF for the Na+-Cl- pair shows a very large and slowly decaying curve, the tail of which extends farther than 100 Å (Figure 6b). The behavior of the RDFs for both the like and unlike pairs looks quite similar to that following from the Poisson-Boltzmann equation for the primitive electrolyte model but with the Debye length κ-1 much larger than in the bulk solution, where κ ) x8πI/(kBT1) and I
Electrolytes Sorbed on C Nanospaces
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Figure 6. Radial distribution functions between ions in NaCl aqueous electrolyte solution sorbed in nanoporous PVDC material, versus those of the bulk solution with a single carbon sphere (SCS).
) 1/2∑iFiz2i is the ionic strength, is the dielectric constant of the solvent, and zi is the charge of ion i. This behavior can be understood in terms of the reduced density and concentration of the electrolyte solution sorbed in carbon nanopores, the conclusion drawn in the previous subsection. It is a novel finding of this study that the distribution of ions in electrolyte solution sorbed in nanoporous material has a screening length which extends over the region including many nanopores. 3.2.3. Carbon Solution Distributions. The RDFs between carbon nanospheres and the species of NaCl aqueous solution are shown in Figure 7. The carbon-water oxygen and hydrogen RDFs are plotted in Figure 7a,b, respectively. In comparison to the SCS case, the plot for solution sorbed in PVDC exhibits a higher and broader first peak for both the C-O and C-H RDFs. The broadening of the peak has apparently two origins. The first one is the reduced solution density which makes the distribution more supercritical or vaporlike, much as for the water O-O RDF shown in Figure 3a. The other reason for the broadening is the overlap of the hydration shell around the labeled carbon nanospheres with those around other carbon nanospheres of the nanoporous material. The distance between carbon nanospheres and water molecules is thus likely to have dispersion. Unlike the case of the water O-O RDF, the curves have more pronounced first minima, which drop below 1, gCO(rmin) < 1, and also have a small second peak. This suggests that a liquidlike phase of water exists right at the carbon surface, but its density is not as high as in bulk solution. The C-Na+ RDF of the solution sorbed in PVDC exhibits a distinctive feature (Figure 7c). Na+ has a little affinity to a carbon wall even for SCS, as is indicated by the small first peak of the C-Na+ RDF. This is due to the existence of the stable hydration shell, or the “positive hydration”, which is observed in Figure 3c. It leaves only a small chance for the ion to access the carbon wall, manifesting in a low prepeak/hump on the C-Na+ RDF
Figure 7. Radial distribution functions between PVDC carbon nanospheres and sorbed aqueous NaCl solution (water O and H sites, and Na+ and Cl- ions), versus those between a single carbon sphere (SCS) and the bulk solution.
in the bulk solution. However, even this small chance diminishes in PVDC, as the first peak strengthens at the position corresponding to a hydration shell of water molecule separating the ion from the carbon nanosphere. The broadening of the first peak of the C-Na+ RDF in PVDC compared to SCS can be explained by the same reason as for the carbon-water RDFs in Figure 6a,b, namely, in terms of dispersion of the distance between the ion and carbon nanospheres making nanopores. In contrast, the C-Cl- RDF has a large first peak at the distance of contact with the carbon surface in PVDC as well as with SCS. It indicates that the Cl- ion can directly access the carbon wall, stripping off a part of the ion hydration shell. This behavior can be understood in terms of Cl- being an ion with “negative hydration”. That is, water molecules in the vicinity of the Clion have greater mobility than those in the water bulk. Therefore, the Cl- ion is more easily stripped of water molecules to bring it into contact with the carbon surface. The slight shift of the first peak of the C-Cl- RDF in PVDC toward a larger distance compared to that for SCS can be attributed to the increased ion-water interaction due to the reduced solution density in PVDC, which has already been pointed out in the discussion of Figure 3d. The first peak of the C-Cl- RDF in PVDC is broadened, compared to SCS in the bulk solution, much as for the C-Na+ RDF.
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Figure 8. Radial distribution functions between PVDC carbon nanospheres and cations of sorbed electrolyte solutions (part a), versus those between a single carbon sphere (SCS) and cations in the bulk solutions (part b).
Figure 9. Radial distribution functions between PVDC carbon nanospheres and anions of sorbed electrolyte solutions (part a), versus those between a single carbon sphere (SCS) and anions in the bulk solutions (part b).
To summarize the discussion for the results concerning the site-site RDF, we can draw the following picture for the structure of sorbed electrolyte solution. In carbon nanopores, the density of solution species becomes much smaller compared to that of the bulk as a result of chemical equilibrium between the solution sorbed inside the nanopores and the bulk solution outside the nanoporous material. Due to the reduced density, water molecules cannot make the hydrogen-bond network typical of bulk water, but instead make small clusters or nanodroplets coexisting with regions of water vapor, the whole nanoheterogeneous phase of sorbed solution behaving similarly to supercritical fluid. Closer to the surface of carbon nanopores, the local density of water approaches liquidlike values and apparently forms nanodroplets. For both cations and anions, the ion-water interaction in PVDC is enhanced due to the reduced density of the sorbed water. The ion-ion distribution in nanopores becomes more like a PoissonBoltzmann distribution, but with an amplified scale of the Debye screening length which extends over a large number of nanopores. A pore with radius of 5.5 Å can include only a pair of ions on average, a cation and an anion, which may or may not make a contact ionic pair. A small cation which is a “positively hydrated” ion is mainly prevented by its hydration shell from contacting the carbon surface, and it is distributed more or less evenly over the nanopore space. On the other hand, an anion which is a “negatively hydrated” ion can directly access the carbon wall and stays in contact with the nanopore surface. 3.2.4. Effect of Ionic Type on the SolVation Structure in PVDC. It is of great interest to examine how the structure of solution sorbed in nanopores depends on the type of ionic species. In Figure 8, the carbon-ion RDFs in PVDC and in the bulk solution (parts 6a and 6b, respectively) are plotted for several simple ions from the Hoffmeister series: the Na+, K+, and Cs+ cations and the anion fixed to Cl-. The concentration of the bulk solution is chosen to be about 1 M for each system, and the corresponding number densities of the solution species at ambient conditions are taken from the literature.37 The number densities of the solution species sorbed in PVDC are determined from the conditions of chemical equilibrium with the bulk solution outside the nanoporous material, satisfied for each solution species. This gives the concentrations of the electrolyte solutions sorbed in PVDC.
Table 3 lists the number densities of water and ions and the corresponding concentrations for all the systems. A striking dependence of the carbon-ion RDF on the cation type is observed for SCS in the bulk electrolyte solution: the first peak attributed to the ion in contact with the carbon surface grows with increasing ion size (Figure 8b). This is due to the well-known transition of the hydration scheme of ions from “positive” to “negative”, somewhere in between Na+ and K+. In contrast, the RDFs of ions sorbed in PVDC do not show such diversity (Figure 8a). The distributions keep essentially the same shape with some shift toward a larger distance due to the increased ion size and with a small rise at the ion-carbon contact separation. This behavior can be explained in terms of the increased tendency of the “positive hydration” in a nanopore, which in turn is caused by the reduced solvent density. The carbon-ion RDFs of Cl-, Br-, and I- anions are shown in Figure 9, with the cation fixed at K+. As distinct from the cations, the RDF shapes and the position of the first maximum for different anions change much less, both in PVDC and for SCS. This can be understood, since all the anions examined are “negatively hydrated” ions. For such ions, the carbon-ion distribution is dominated by the first peak at the ion-carbon contact separation, and there occurs only a variation due to the difference in the ion size, with a considerable increase of the first hydration shell peak for the largest I- anion.
(37) Handbook of Chemistry and Physics, 68th ed.; CRC Press: Boca Raton, 1987.
4. Conclusion We have investigated the structure of aqueous electrolyte solutions confined in a carbon nanoporous electrode by means of the replica RISM theory. The randomly distributed carbon nanopores were modeled by a binary mixture of nanospheres, in which one sort of nanospheres represents carbon nanoparticles and the other makes up for nanopores of the carbon material. The latter sort does not interact with solution species (water molecules and ions) and thus renders the corresponding space as pores accessible for the solution. The densities of solution species inside nanopores are determined from the chemical equilibrium condition that the chemical potentials of solution species sorbed in the nanoporous material are equal to those in the bulk solution in contact with the nanoporous material. The density of sorbed water so determined constitutes about one-third of bulk ambient water,
Electrolytes Sorbed on C Nanospaces
Figure 10. Illustration of the arrangement of NaCl aqueous solution species sorbed in a nanopore of carbonized PVDC material.
while the electrolyte concentration in the sorbed aqueous solution is much lower than that in the bulk electrolyte solution. These findings are in agreement with the results of experiment and molecular simulations. The structure of the electrolyte solution is deduced from the site-site radial distribution functions obtained. In comparison to bulk solution, the structure of the electrolyte solution sorbed in carbon nanopores exhibits substantial changes, illustrated in Figure 10. Due to the reduced density of sorbed solution, water molecules cannot form the hydrogen-bond network typical of its bulk, but make small clusters coexisting with regions of water vapor. The whole nanoheterogeneous phase of sorbed aqueous
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solution behaves similarly to supercritical fluid. At the carbon surface, water becomes somewhat denser, making a condensed phase similar to liquid. For both cations and anions, the ionwater interaction is enhanced due to the reduced density of solution. The ion-ion distributions in nanopores become more like a Poisson-Boltzmann distribution but with an amplified scale of the Debye screening length which extends over a distance including a large number of pores. A pore with a radius of 10 Å can include only a pair of ions on average, a cation and an anion, which may or may not make a contact ionic pair. A small cation which is “positively hydrated” is expelled by its hydration shell from the carbon surface and is adsorbed at the surface mainly in the solvent-separated arrangement. Cations are distributed more or less evenly over the pore space. On the other hand, an anion which is “negatively hydrated” can access the carbon surface and stays in direct contact with the surface. The above-described peculiarities of the structure of electrolyte solution sorbed in carbonized PVDC material strongly affect the thermodynamics and electric capacitance of a nanoporous carbon electrode. Our subsequent study for electrolyte solution sorbed in a charged nanoporous carbon electrode subject to the chemical equilibrium condition with the bulk solution is in progress and will be presented in the next paper. Acknowledgment. This work was supported by the grant from the NAREGI Nanoscience Project of the Japanese Ministry of Education, Culture, Sports, Science and Technology (Monbukagakusho). A.K. acknowledges the support of the National Research Council (NRC) of Canada. LA061617I
