Langmuir 1988,4, 559-564
559
Association of Counterions with Adsorbed Potential-Determining Ions at a Solid/Solution Interface. 1. Theoretical Analysis7 Nikola Kallay* and Melanija TomiC* Laboratory of Physical Chemistry, Faculty of Science, University of Zagreb, 41001 Zagreb, P.O. Box 163, Yugoslavia Received June 24, 1987. I n Final Form: November 18, 1987 The association of counterions with surface charged groups at the solid/solution interface was analyzed by using the Boltzmann distribution function. The effect of a local electrostatic field is introduced by superposing Coulomb’s potential, caused by the presence of a fixed central ion at the surface, on the electrical double-layer potential as obtained from the Gouy-Chapman theory. The electrostatic potential and, consequently, the probability of ion pairing were found to be higher within the electrical double layer than in the bulk of solution. This finding explains why ions of strong electrolytes associate in the double layer despite their almost complete dissociation in the bulk of solution. The analysis also introduces the ionic specificitythrough ionic size parameter, i.e., the distance of closest approach of two ions. Thus,the lyotropic series effect can also be explained. The surface association equilibrium constants depend on the distance of closest approach, surface potential, ionic strength, temperature, and permittivity of the medium. Other common approaches, such as the triple-layer model, are also discussed.
Introduction The equilibrium a t the metal solid/solution interface is usually described in terms of surface reactions. In the case of metal (hydrous) oxides in contact with aqueous media, the hydrated surface sites behave as amphotheric and the charge is created through the following reactions: MOH + H+ + MOHZ’ (1) MOH * MO- + H+ (2) where M denotes a surface metal atom and MOH is the amphotheric surface site. In this study the following simplified description will be used:
where S is a surface active site, K is the apparent equilibrium constant, I’ is the surface concentration, and a is the activity in the bulk of the solution. The charged groups may associate with counterions according to
SH’ + A- + SH’A;
~SH.A
KSH.A
=-
(5)
~sH+~A-
where M+and A- represent any counterion, cation, and anion, respectively. The above equilibrium constants are designated as “apparent” since their values depend on the composition of the surface. They CM be related to the thermodynamic equilibrium constants by introducing activity ~oefficients.’~ Alternately, the deviation from ideality may be taken into account by considering the electrostatic contribution to
* Author to whom correspondence should be addressed. +Supported by U.S.-YU Joint Fund, DOE, PN 741. Water Chemistry Program, University of f Present address: Wisconsin, Madison, WI 53706. 0743-7463/88/2404-0559$01,50/0
the Gibbs energy of the surface reactions. The approaches based on the surface complexation or site-binding model introduced so-called “intrinsic” equilibrium constants (I(int):&7
KSH+ =
@A+
exp(-&e/kT)
(7)
where e is the elementary charge, k the Boltzmann constant, T the thermodynamic temperature, and +o the electrostatic potential affecting the SOH- and SH+ charged sites, which are assumed to be located in the same zero plane. Associated counterions are considered to be located in the p plane, with $@ being the potential influencing their energy state. The composition of the surface may be ex(1)Parks,G.A.; de Bruyn, P. L. J. Phys. Chem. 1962,66,967. (2)Yaks, D. E.; Levine, S.; Healy, T. W. J. Chem. SOC.,Faraday Trans. 1 1974,70,1807. (3)Peaam, J. W.;Hunter, R. J.; Wright, H. J. L. A u t . J. Chem. 1974, 27,461. (4)Bowden, J. W.; Poaner, A. M.; Quirk, J. P.A w t . J.Soil Res. 1977, 15, 121. (5) Davis, J. A,; James, R. 0.;Leckie, J. 0. J. Colloid Interface Sci. 1978,63,480. (6) Davis, J. A.; Leckie, J. 0. J. Colloid Interface Sci. 1978,67,90. (7)Davis, J. A.; Leckie, J. 0. J. Colloid Interface Sci. 1980,74,32. (8)Westall, J.; Hohl, H. Adu. Colloid Interface Sci. 1980,12,265. (9)Hohl, H.; Sigg, L.; Stumm, W. in Particulates in Water; Kavanaugh, M. C., Leckie, J. O., Edgs.; Advances in Chemistry 189;American Chemical Society: Washington, DC, 1980. (10) Schindler, P. W. in Adsorption of Inorganics at SolidlLiquid Interfaces; Anderson, M. A., Rubin, J. A., Eds.;Ann Arbor Science: Ann Arbor, MI, 1981;p 1. (11)James, R. 0.;Parks,G. A. In Surface and Colloid Chemistry; MatijeviE, E., Ed.; Plenum: New York, 1982;Vol. 12, p 119. (12)Hingston, F.J. In Adsorption of Inorganics at SolidlLiquid Interfaces; Anderson, M. A., Rubin, J. A., Eds.; Ann Arbor Science: Ann Arbor, MI, 1981;p 51. (13)Morel, F. M. M. In Adsorption of Inorganics at SolidlLiquid Interfaces; Anderson, M. A., Rubin, J. A., Eds.; Ann Arbor Science: Ann Arbor, MI,1981;p 263. (14)Sposito, G . J. Colloid Interface Sci. 1983,91, 329.
0 1988 American Chemical Society
Kallay and TomiE
560 Langmuir, Vol. 4, No. 3, 1988 pressed in terms of mole fractions (e.g. XSH+ = FsH+/rtot); if the standard state is defined by x o = 1 and = 0, the intrinsic values correspond to the standard (thermodynamic) equilibrium constants. Yates, Levine, and Healy2 analyzed the equilibria of ion-pairing reactions by considering a surface ion pair (e.g., SH’A) as an oriented dipole in the electric field. I t was that their approach led to a result consistent with eq 5, 6, 9, and 10. The next step in modeling the interfacial equilibria is to define certain regions (planes) in the double layer and to introduce proper charge-potential relationships. In the most developed triple-layer model5-’ the following expressions are used: (11) 00 = Le(rSH+ + I1SH.A - rS0H.M - FSOH-)
+
a# = Le(rS0H.M - rSH.A)
(12)
where L is the Avogadro constant. Surface charge density of the diffuse layer is (13) a d = -(go + a#) and the corresponding capacitances are “0 c1 = -
*o
- *&4
For a planar diffuse layer and 1:l electrolyte system, the Gouy-Chapman theory yields16J7
“neutral” counterions such as NO3-. The reason for the discrepancy may be due to the assumption according to which associated counterions are located in the /3 plane of the capacitor C1 rather than being distributed statistically around fixed charged surface sites. The aim of this study is to discuss the association of counterions a t the interface as affected by their distribution around the surface charged groups. Their distribution depends on the local electric field of the central ion and on the double-layer field caused by the presence of all ions in the interfacial layer.
Theoretical Association of Ions in the Electrolyte Solutions. The association of ions with charged surface groups will be considered in a manner similar to Bjerrum’s treatment of ion pairing in electrolyte solution.20 Bjerrum’s model assumed the distribution of ions of type i around a central ion of type j according to the Boltzmann function: ni(r)= nioexp(-zie+(r)/k7‘) (18) where ni(r) and nioare densities (number concentrations) of ions of type i (of charge number zi) a t the center to center distance a t r and infinity from the central ion j, respectively. The electrostatic potential a t distance r, +(r),with respect to the potential at infinite distance, can be calculated in the dilute systems as J / ( r )= zje/4?rtr (19)
The application of eq 3-17 for fitting the potentiometric titration Gats involves seven adjustable parameters, Kih+, p t SOH-, P&A, P{&.M, rtot,C1,and Cz, which obviously enables a successful regression analysis regardless of the physical reality of the model applied.14 The procedure may not distinguish between different variations of the double-layer Therefore, efforts should be directed toward developing new experimental techniques and the theoretical considerations to reduce the number of adjustable parameters or toward improving the physical basis of the present models. With the present models one faces two problems. The intrinsic constants for ion association reactions are often found to depend on the ionic ~trength,’~J* which is inconsistent with their definition. Furthermore, according eq 5,6,9, and 10 the association may take place only if Pg > 0. Under the same conditions the oppositely charged ions should also form pairs in solution, which in reality need not take place. For example, pairs like H+-N03- and OH--Na+ do not associate in the solution bulk but will be paired in the interfacial layer. However, spectroscopic resultslg do not provide any evidence for specific chemical bonding a t the surface for
where E is permittivity of the medium. According to eq 18 the probabiltiy of finding an ion of type i in a spherical shell between radii r and r + dr is2I G ( r ) dr = 4?rnior2exp(-zie+(r)/kT) dr (20) By combining eq 19 and 20 and setting the derivative of G(r)with respect to r equal to zero, one obtains the minimum value of the probability function G(r)at the Bjerrum critical distance zizje2 dB = -(21) 8mkT For 1:l electrolytes in water a t 25 O C this distance is 3.5 A. Bjerrum postulated that ions closer than dBare associated. By introducing the distance of closest (center to center) separation b, and by integrating eq 20 from r = b to r = dg, the “probability” (average number of ions in the shell of radii b and dB)can be calculated. The Bjerrum theory explains why some electrolytes do not form ion pairs; the hydrated ions are too large and cannot approach close enough to be within critical distance dg. The latter (dg = 3.5 A) is shorter than the closest separatioll for ions of strong electrolytes, which is estimated to be22in the range 4-5 A. The applicability of Bjerrum’s theory is demonstrated by measurements of ionic association equilibrium constants in water in a mixture with different organic solvents as glycine, ethanol, dioxane, acetone, etc.22 Association of Ions at Solid/Electrolyte Solution Interface. The application of the Boltzmann distribution law to ion association in the electrical double layer requires some modifications as compared to applications concerning
(15) Blesa, M. A.; Kallay, N. Adu. Colloid Interface Sci., in press. (16) Gouy, G. J. Phys. 1910, 9, 457. (17) Chapman, D.L.Philos. Mag. 1913,25,475. (18) Regazzoni, A. E.; Blesa, M. A.; Maroto, A. J. J. Colloid Interface Sci. 1983, 91, 560. (19) Tejedor-Tejedor, M. I.; Anderson, M. A. Langmuir, 1986,2,203.
(20) Bjerrum, N. Ergeb. Exakten Naturwiss. 1926, 6, 125. (21) Fuoss,R. M. Trans. Faraday SOC.1934,30,967. (22) Robinson, R. A.; Stokes, R. H. Electrolyte Solutions; Butterworths: London, 1955. (23) Hesleitner, P.; BabiE, D.; K d a y , N.; MatijeviC, E. Langmuir 1987, 3, 815.
where E is the permittivity and I the ionic strength. In addition, the following mass balance is used for the surface sites: (17) rtot = rS + r S H + + rSH.A + rSOH- + rS0H.M
Langmuir, Vol. 4, No. 3,1988 561
Counterion Association at SolidISolution Interface
.L
Y
J
Figure 1. Choice of coordinate. Central ion is in the origin; the surface is defined by the X-Y plane. Positive side of Z-axis is in the liquid phase. The coordinates of the spherical systems are r and 8.
the bulk of solution. At the solid/solution interface, only a part of the space is available for counterion distribution. Furthermore, in addition to the electrostatic potential caused by the central surface charged site ($I&). one should also take into account the double layer potential ($d, caused by the distribution of all ions in the electrical double layer. Consequently, the total potential t o be considered in eq 18 is
Figure 2. Cmas section of the assoeiation space in the X-Z plane. Top: the probability function (lower part) for the (E) electrical double layer and the ( 8 )absence of the electrical double layer: b = 3 A; b < dB < d(B=O);Bb = u/2. is defined as d(BJ = b. Association in the double layer is stronger than in the bulk of solution. Bottom: association probability (defined as in eq 20) 88 a function of the center to center distance between ions: in the presence (upper curve) and absence (lower curve) of the double layer.
+ $dl (22) Figure 1describes the situation at the surface (X-Yplane). The local potential is defined by eq 19, and it depends on the radial coordinate r. The double-layer potential $d may be calculated by using the Gouy-Chapman t h e ~ r y ~ ~ ~ ’ ’ $=h
a l
which for symetrical z:z electrolytes reads
where exp(ze$o/2kr) go = exp(ze$o/2kr)
-1
+1
(24)
and the reciprocal Debye-HUckel distance is K
= (2e2LI/rkT)1/2
(25)
where L is Avogadro’s constant, I the ionic strength, and $o the electrostatic potential at the surface. Since $a depends on the distance from the surface 2 (= r cos 8; see Figure 1)and Ad on the distance from the center of the “central” ion r, the probability of finding an ion of tvoe i in a volume element depends on both r and 0 (see eq 19,22, and 23). The critical distance d(0) for ionic association in the double layer is defmed in &e same manner as for the bulk of an electrolytic solution, i.e., by the position of the minimum of the probability function G(r), which for an angle @ is given by
By evaluating the derivative of the expression in the brackets a t r = d(0) and substituting eq 21 (with zi = -q), one obtains the relationship from which d(@)can be calCulated
Same 88 Figure 2, but for the association in the double layer only: b = 6 A; dB < b < d(B=O): 0 < OB < u/2.
Figure 3.
critical distance dm At 0 < n (locations in the liquid phase), the critical distance is always larger than dB, because the second term in eq 27 is always positive. Accordingly, ion association may indeed take place at the interface as established experimentally. The value of d(@) may be obtained by applying an iterative procedure to eq 27. The space in which association occurs,V,, is bounded by a sphere of radius b (the distance of closest approach) and by a surface described by d(0). Figures 2 and 3 show the cross sections of this space in the X - 2 plane and the asaociation probabilities. Figure 2 applies when ion pairing can occur in the bulk (b < dB),while Figure 3 demonstrates the caae where ions are completely dissociated in the bulk of solution and yet are significantly associated a t the interface. The average number of ions of type i (their centers) in volume V, is = n,
5”-e-&W
dV
(28)
For the angle 0 = n/2, corresponding to location on the surface plane (see Figure 1).the critical distance for the association of counterions with adsorbed potential-determining ions (charged surface groups) is equal to Bjerrum’s
Introducing eq 19,21,22, and 23 in eq 28 and taking into account the definition of V, and d V = 2m3 sin 0 d(@)d r yield
Kallay and TomiE
562 Langmuir, Vol. 4, No. 3, 1988 -3
The procedure for partial solution of the integral in eq 29 is given in the Appendix. The association equilibrium constant (in terms of concentration) as defined by eq 5 and 6 and may be calculated by K,,, = LN/nio (30) It should be noted that the equilibrium constants in eqs 5 and 6 include activities of counterions in bulk. The model is based on the assumption of high dilution, which implies that the activity coefficients of bulk ions (y) approach unity and a = y c / c o i= c/co (31) where co = 1 mol dm-3. Consequently, the equilibrium constant, as defined by eq 5 and 6, is expressed as
K,,, = lOOON ( L / m ~ l - ' ) / ( n ~ ~ / m - ~ ) (32) Approximate Relationship. The solution of eq 29 (see also Appendix) is not a simple task. To make the use of this theory easier, we introduce a simple approximate expression, obtained on the basis of numerical solutions of eq A18. The value of the surface association equilibrium constant for a z:z ionic pair system a t a given temperature T, for solvent of permittivity e, depends on the surface potential (or charge), ionic strength, and characteristic ionic size parameter b. For 1:l systems in water a t 25 O C (t = 6.954 X 10-lo F m-l; e, = 78.54) In (K,,,) = A l ( l - e-A2ua/C-m-2)-
where A1 = 11.772 - 7.106/(b/A)
(34)
+ 35.466/(b/A)
(35)
A2 = 4.367
Surface charge density a,/C-m-2 is expressed in coulombs per square meter and is related to surface potential through the Gouy-Chapman16~"equation:
= 2 ( 2 k T ~ L I ) ~ / ~ s h ( e l C . ~ / 2 k T ) (36) The quantity u, is the net surface charge density due to the adsorbed potential-determining ions which are not associated. It is equal in magnitude to the charge density of the diffuse layer ad: ( r e = -ad (37) is taken here as the potential on the surface that also determines the equilibrium in the diffuse part of the double layer; i.e., it is equal to $d in eq 16. The procedure for the calculation of the surface ion pair association constant by means of eq 33-35 is simple enough to be used in the interpretation of adsorption data by regression analysis. Figures 4 and 5 display the comparison of exact (numerical) solution of eq A18 and 32 and approximate solutions of K,, (eq 33-35) for two different distances of closest approach, 3.6 and 6 A, as a function of surface potential t,bo for different ionic strengths. It is obvious that the value of K,, becomes higher as the surface potential increases. The effect of ionic strength is also significant; the addition of electrolytes reduces the K,, value. As a*
_ I
3.6
-1
3F-
I
/ I
I
100
0
200
0
100
200
Yo/mV YoImV Figure 4. Logarithm of the surface association equilibrium as a function of the surfacepotential ($o) for ionic constant and mol dm-3. The calculations were strength (I)of performed for a 1:l aqueous electrolyte system (c, = 78.54) at 25 "C,and the distances of closest approach were b = 3.6 and 6 A by exact solution (numerical integration of A18) (-) and approximation of eq 33 (-). The exact solution leads to K , = 0
(Ka
at surface potentials lower then the critical potential, which is denoted by an arrow. The numbers associated with arrows designate the corresponding b values (A). 2
3.6
a
Y
cn
2
-1 -2 0
100
%ImV Vr',/mV Figure 5. Same as Figure 4 for ionic strength lo-' and 1 mol dn-~-~.
expected, the association equilibrium constant is higher for "smaller" ions, characterized by shorter distance of closest approach b and, consequently, larger association space (V,). The approximate expressions 33-35 produced K , values reasonably close to the exact values in the region of relatively high surface potentials. In the region of low surface potential, only a small portion of surface charged groups is associated, so that error in K, does not significantly influence results of the calculations of the double-layer equilibrium. The discrepancy is more pronounced for higher b values and becomes significant a t high ionic strength (1mol dm-3). Note that in this region the model is also far from initial assumptions; expressions 24-32 were derived for diluted systems and for low surface density of charged groups.
Discussion The analysis applied in this study took the discrete charge effect into account in solving the equilibrium a t solid/liquid interfa- by distinguishing between associated and freely distributed ions in the diffuse layer. Such a treatment was also applied by Yates et a1.2 and Davis et al."' A somewhat different approach was introduced by M i r r ~ i kand ~ ~Manning,% who assumed a Debye-Huckeltype distribution of ions around the fixed central ions. Manning also considered the mutual influence of ionic clouds surrounding central ions. (24) Mirnik, M. Croat. Chem. Acta 1970, 42, 161. (25) Manning, G. S. Q.Reu. Biophys. 11 1978, 2, 179.
Langmuir, Vol. 4, No. 3, 1988 563
Counterion Association at SolidlSolution Interface Kjellander and MarEelja26*27 introduced the effect of finite ionic size on double-layer equilibrium through ionion correlations in hypernetted chain approximation. The difference between the approach developed in this work and the previously described models is in the assumption made about the location of the associated counterions. In the triple-layer model”’ the apparent association equilibrium constant is a product of the intrinsic constant, K!!:, and an exponential factor which includes the potential of the plane locating associated counterions (eq 9 and This study assumes that associated counterions are not located a t fixed distance from the surface; instead they are distributed in the “association space”, V,, surrounding the central adsorbed ion. The probability of finding a counterion in that volume depends on the surface potential, ionic strength, and counterion charge and size. The triple-layer modelk7 assumes a parallel plate capacitor, with charged planes 0 and p, of constant capacitance C1.This assumption requires planes 0 and ,t?to be separated by a constant distance determined by the size of the ions forming the pair. However, it is shown here that associated counterions are distributed around the fixed potential-determining ions and that the “average” distance depends also on the surface potential and ionic strength. Consequently, no constant values of the capacitance, C1,and of K !:: may be expected. According to this study, the values of the surface ionpairing constants depend on an ionic size parameter, i.e., on the distance of closest approach b. Since this parameter was found to be independent of permittivity,22it may be obtained by means of the Bjerrum theory for ionic association in an electrolyte solution. Ions of strong electrolytes do not associated in pure water, but in mixtures with less polar solvents (e.g., ethanol, dioxan, etc.) the association becomes significant. Consequently, the necessary value of b can be determined from studies in mixed solvents. The distance of closest approach is also a parameter in the Debye-Huckel theory; thus for strong electrolytes it may be obtained by fitting the experimental activity coefficients. The parameter b, evaluated from the behavior of electrolyte solutions, may be used in the interpretation of the double-layer equilibria using the present model. However, in doing so, one assumes that the hydration of ions a t the interface is the same as in the bulk of solution. Such an assumption is supported by IR spectr~scopy.’~ Finally, one may simply treat b as an adjustable parameter in the interpretation of adsorption data. By applying the proposed model the number of adjustable parameters is reduced to four FgAH-, rbt, b) compared to seven in the triple-layer model. When the point of zero charge (pzc) is known, one less parameter is needed because the following relationship applies: PH(PZC)= -Yz log (&&-i-Kw/W%+) (38) where K, is the equilibrium constant of water dissociation. Further reduction of adjustable parameters can be done by using the results of tritium-exchange measurements to determine rtot. In the final analysis only g&+remains as an adjustable parameter (assuming that b is evaluated independently), while in the triple-layer model one still must fit E&+, &%.A, &%H.M, C1, and C2. The values of counterion/charged surface group association constants, as obtained by the method presented
in this study, could be used for the interpretation of data on the adsorption of the potential-determining ions. This method could be especially useful for the examination of the effect of permittivity (mixed solvents), because the invariable distance of the closest approach is assumed. This study explains why two ions can associate in the interfacial layer, despite their complete separation in the bulk of solution. The effects of lyotropic series in adsorption and coagulation phenomena could be quantitatively treated by considering the size of (solvated) ions, i.e., by means of distance of closest approach. The simulation of adsorption data for a model system is presented in the second part of this work.28
Acknowledgment. We are deeply indepted to Professor Eytan Barouch for his assistance in solving some of the mathematical problems. Appendix Equation 29 was solved by introducing a new system of variables Z = r cos 8 (AI) t = cos 8 (A21 with Jacobian l / t and by defining the function (A3) g(t) = d W According to eq 27, d(t) is
Equation 29 can be rewritten as
where
z2e2 - 2 d ~ 47rrkT Note that the symbol a in this Appendix does not mean the activity. Expression A5 corresponds to a=--
(@A+,
(26) Kjellander, R.; MarEelja, 5. J. Phys. Chem. 1986, 90,1230. (27) Kjellander, R.; MarEelja, S. Chem. Phys. Lett. 1986, 127, 402.
Integration by parts gives
(AlO) The second term in brackets is equal to Na/6rnio. Since eq A l l is valid (28) TomiE, M.; Kallay, N. Langmuir,following paper in this issue.
564 Langmuir, Vol. 4, No. 3, 1988
Kallay and Tomit
or integration of eq A10 results in
where t(g) is expressed by use of eq A3 and A4 as
All three terms on the right-hand side of eq A12 are integrals of the same type:
where hl = 1 / 2
h2
= t(g)/g;
h3
= l/b
(A15)
Ei(ahj) is the exponential integral defined as Ei(x) =
X'
dx'= C
+ In x + k = l
Xk
- (A16) k*k!
eatbi( a 1
+
91
dq) (AM)
The exponential integral Ei(x) is tabulated in mathematical handbooks, so that eq A17 can be used to avoid the summations in eq A18. A further step requires numerical integration of either eq A17 or eq A18. The integration boundaries must first be found. They are defined as follows: (1)the critical distance in the direction perpendicular to the X-Y plane, given by the iterative solution of
The use of eq A14-Al6 enables eq A12 to be expressed 85 (2) the distance of closest approach, b; and (3) the product of the distance of closest approach b and the cosine of the angle Ob: btb = b
COS Ob
(-420)
The value of tb can be obtained (for a given value of b) by iterative solution of (2b - u)(erbtb- goPe-xbtb) tb =
4go~b2
(A21)
