2854
Langmuir 1999, 15, 2854-2864
Adsorption Layer of Charged Surfaces: An Exact Nearest-Neighbor Model for the Linear Lattice Kenneth S. Schmitz Department of Chemistry, University of MissourisKansas City, Kansas City, Missouri 64110 Received May 13, 1998. In Final Form: January 14, 1999
The effects of finite ion size and polyion concentration on the adsorption isotherms are examined within the context of a one-dimensional lattice restricted to nearest-neighbor interactions. The nearest-neighbor interactions include that of the bound ion with adjacent lattice sites as well as with a neighbor bound ion. Both screened and unscreened Coulombic interactions are employed, as they are shown to exhibit different behaviors under high salt concentrations. For the screened interactions, previously adsorbed ions are released, whereas for the unscreened Coulombic interaction charge reversal is observed for sufficiently separated lattice sites. The effect of polyion concentration is to provide a “constant” degree of adsorption for added salt concentrations less than the concentration of counterions released by the polyion. The model is applied to experimental data on monovalent and divalent salts.
1. Introduction Adsorption of molecules onto surfaces is a topic that encompasses many fields, ranging from the adsorption of gases onto solid surfaces to the adsorption of small molecules onto the surfaces of larger molecules in solutions and colloidal suspensions. In regard to colloidal suspensions and polyelectrolyte solutions, the surface-bound ions can make a significant contribution to the electrokinetic properties of the system by their lateral movement.1,2 A nonuniform distribution of surface charge also affects the electrophoretic mobility of the macroions.3,4 Recent focus has been directed to the role of surface charge distribution effects on the interaction between charged particles.5-8 Rouzina and Bloomfield,5 for example, suggested that the mutual repulsion between the adsorbed ions results in a periodic arrangement of positive and negative “patches” on each surface, and the correlation of these patches leads to an electrostatic attraction between the two surfaces. Grønbech-Jensen et al. 6 and Ha and Liu7,8 address the interaction between charged rods. In these studies an attraction occurs when there is a correlation between the positions of the counterions adsorbed on each rod. Ha and Liu7,8 reported that the degree of counterion adsorption varied as the two rods approached each other, as would be expected for a chemical equilibrium in response to the change in the local concentration of ions. In regard to the dimensionality of the problem in solving the equations, the one-dimensional system can be solved exactly for a variety of situations, the exact solution for the two-dimensional Ising spin system has been achieved,9 and the more complicated multidimensional surfaces must (1) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 32. (2) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1986, 114, 45. (3) Anderson, J. L. J. Colloid Interface Sci. 1985, 105, 45. (4) Yoon, B. Y. J. Colloid Interface Sci. 1991, 142, 575. (5) Rouzina, I.; Bloomfield, V. A. J. Phys. Chem. 1996, 100, 9977. (6) Grønbech-Jensen, N.; Mashl, R. J.; Bruinsma, R. F.; Gelbert, W. M. Phys. Rev. Lett. 1997, 78, 2477. (7) Ha, B.-Y.; Liu, A. Phys. Rev. Lett. 1997, 79, 1289. (8) Ha, B.-Y.; Liu, A. Phys. Rev. Lett. 1998, 80, 1011. (9) Huang, K. Statistical Mechanics; (ohn Wiley & Sons: New York, 1963; p 349.
find consolation in approximate solutions to the equations. The Bragg-Williams approximation exemplifies the latter situation, in which an average pairwise interaction is assumed operable for all pair interactions and a binomial term is included to account for the degeneracy of the system.10 A popular model for the adsorption of ions onto a linear array of sites is that of Scatchard,11 which utilizes the Bragg-Williams model approximations in which a “random binding” assumption coupled with an “average nearest-neighbor” interaction between bound sites was employed. The reduced free energy change for the association of nb bound ligands to a naked lattice of np sites was given by
( ) [
]
mnb ∆F(nb,np) np! ) ln - ln n RT nb!(np - nb)! mpmf b nb ln(K) + wnb2 (1)
where RT is the molar thermal energy, mf is the molal concentration of the free ligand, mp is the molal concentration of polymer, mnb is the molal concentration of polymer with nb bound sites, K is the intrinsic binding constant at the site, and w is the reduced value of the average nearest-neighbor interaction. In the Scatchard approach the analytical expression for the adsorption isotherm was obtained using eq 1 by calculation of the free energy difference for the addition of one more ligand to the lattice, i.e., ∆F(∆nb)1) ) ∆F(nb+1,np) - ∆F(nb,np). Scatchard further assumed that there was no difference in the concentrations of polymers that differed by only one bound ligand, as manifested in the approximations ∆F(∆nb)1) ≈ 0 and mnb+1/mnb ≈ 1. The resulting expression for the isotherm thus obtained was
ln
[
θb
mf(1 - θb)
]
) ln(K′) - 2w′θb
(2)
where K′ is an apparent intrinsic equilibrium constant, (10) Hill, T. L. An Introduction to Statistical Thermodynamics; Dover Publications: New York, 1986; p 246. (11) Scatchard, G. Ann. N.Y. Acad. Sci. 1949, 51, 660.
10.1021/la9805746 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/12/1999
Adsorption Layer of Charged Surfaces
w′ is an effective average nearest-neighbor interaction, and θb ) 〈nb〉/np is the fraction of bound sites. It has been shown12 that eq 2 is strictly applicable to systems in which |w′| , 1 and that for |w′| > 1 a plot of ln[θb/mf(1 - θb)] vs θb contains no molecular information except at the midpoint of the curve.12,13 The error in the application of eq 2 in these strongly interacting systems lies in the calculation of the degeneracy of the states;12,13 i.e., the combinatorial factor in eq 1 does not distinguish between pair interactions on adjacent bound sites and the pair interactions or distal bound sites. It is clear that there is an increased emphasis on the molecular details of the physical and chemical processes that occur on charged surfaces. Thus it is imperative to evaluate in more detail the relative importance of the various terms that may contribute to the energy and entropy of the ion adsorption process for systems of realistic dimensions. These terms include, but are not limited to, the intrinsic distribution of binding sites on the surface, the finite size of the ion, multiple-neighbor interactions, bulk ion concentration dependence, and competition of different ion types for the surface positions. We have examined these effects using a matrix generation method. The present communication focuses on linear systems with nearest-neighbor interactions, where the matrix method is used to generate exactly the grand partition function for the system. 2. General Description of the System The system is composed of three solute particles: the linear polyion is component 1; the counterion is component 2; and the gegenion (added electrolyte) is component 3. The added electrolyte need not be a symmetric salt, but there is the constraint that one ion be the same as the counterion. The polyion is composed of ns sites, each with a charge of magnitude with sign Z1. Each of these sites can bind one counterion whose charge is of magnitude with sign Z2. The charge of the gegenion is Z3. The finite size of the ion is taken into consideration by limiting the value of the distance of closest approach of the ion to the surface.5,14 This distance is denoted by “a”. Ions that are found a distance greater than “a” from the surface are defined as the “free” ions, and those at a distance “a” are defined as the “bound” ions. The present model does not permit the gegenions to adsorb on the charged surface because of the large repulsion energy for a co-ion with the charged site. Nonetheless the gegenions do contribute to the ionic strength of the solution. Since the centers of the charge at site j and the bound counterion do not coincide for ions of finite size, the adsorption process results in a dipole rather than zero net charge at the bound site. The charged surface is represented as a linear array of charges separated by a distance “b”. Because this distance is assumed to represent actual distances in a real system rather than projected distances, there are two possible “bound states” for the adsorbed ions, located directly above the lattice site or between lattice sites. Therefore the two models presented herein are referred to as the “local site” model where the counterion coordinate along the polyion backbone is the same as the lattice site, and the “midway” model in which the counterion is offset a distance b/2 from the lattice site. (12) Schmitz, K. S. Biopolymers 1977, 16, 143. (13) Schmitz, K. S. Macroions in Solution and Colloidal Suspension; VCH Publishers: New York, 1993; p 22. (14) Gregor, H. P.; Gregor, J. M. J. Chem. Phys. 1977, 66, 1934.
Langmuir, Vol. 15, No. 8, 1999 2855
3. Interaction Energies The reference state for the system is the totally unbound lattice. For the nearest-neighbor case, there are three parameters that contain the interaction energies in the act of adsorbing the ions. The parameter K is a Boltzmann weighting factor that reflects the interaction of the adsorbed ion with the lattice site and is referred to as a “direct” interaction. The parameter S is the Boltzmann weighting factor for the interaction of the adsorbed ion with the neighboring lattice “site”. The parameter B is the interaction of the adsorbed ion with a neighboring “bound” ion. Although the interactions are limited to electrostatic interactions, it is not obvious that these interactions are screened or unscreened Coulombic interactions. Rouzina and Bloomfield5 employ unscreened Coulombic interactions in the calculation of their two-dimensional coupling constant, which we denote as ΓC, for the neighbor ion-ion interaction, which in the present notation is defined as
ΓC )
(Z2qe)2 ) Z22ξC kBTb
(3)
where qe is the magnitude of the electron charge, is the bulk dielectric constant, kB is the Boltzmann constant, and ξC is the ratio of the unscreened Coulombic energy for charges separated by the neighbor lattice distance b to the thermal energy
ξC )
WC (qe2/b) ) kBT kBT
(4)
Likewise, a screened Coulombic interaction has been employed in polyion adsorption theories.15,16 We therefore examine both mathematical forms. The mathematical development, however, is given in terms of the screened Coulombic interaction as this can easily be converted to the unscreened form by setting the screening variable parameter κ to zero, where
κ)
(
)
8πNAλBI 1000
1/2
(5)
where NA is Avogadro’s constant and the ionic strength I is
I)
1
Zi2Ci ∑ 2 i
(6)
where Ci is the molar concentration and the summation is over all of the small ions, viz., i * 1. “Local Site” Model. In the “local site” model the interaction of the ion with the jth site is manifested in the intrinsic association constant K. The interaction of the bound ion at site j with the adjacent lattice sites j - 1 and j + 1 is represented by the Boltzmann factor denoted by S, and the nearest-neighbor interaction between bound ions is represented by the Boltzmann factor B. Hence the weighting factor for an isolated ion bound at site j is the product (m2/55)KS2, where m2 is the molality of the “free” counterion in aqueous solution (see the Appendix). The (15) Satoh, M.; Komiyama, J.; Iijima, T. Macromolecules 1985, 18, 1195. (16) Reuben, J.; Shporer, M.; Gabbay, E. J. Proc. Natl Acad. Sci. U.S.A. 1975, 72, 245.
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explicit forms for the parameters K, S, and B in the local site model are
ln(K) ) -Z2Z1ξC
ln(S) ) -Z2Z1ξC
exp(-κa) rred
(7)
exp[-κb(1 + rred2)1/2] (1 + rred2)1/2
(8)
and
ln(B) ) -Z22ξC exp(-κb)
(9)
where rred ) a/b is a reduced separation distance. “Midway” Model. In this model the adsorbed counterions lie midway between the charged groups on the surface. Since the range of nearest-neighbor interaction is limited to the distance b, a counterion bound between the lattice sites j and j + 1 can interact only with the j and j + 1 lattice sites. The explicit form of the parameter S is therefore
ln(S) ) -Z2Z1ξC
exp[-κb(0.5 + rred2)1/2] (0.5 + rred2)1/2
(10)
The interaction between two neighboring bound counterions is given by eq 9. Since the counterion is not located “above” a lattice site, the association of a counterion ion on an isolated lattice is represented by the weighting factor (m2/55)S2 in aqueous solution. The charged groups along the lattice are finite in size as well as the counterions. It is assumed that the charges of both the groups at the lattice sites and the ions in solution are located at their centers. Since these calculations are aimed for real systems, the distance b cannot be smaller than the diameter of the counterions without imposing “excluded site” restrictions. Hence for the present set of calculations we employ values of b such that b g a, where a is the diameter of both the charged group and the counterion. These models and interaction parameters are schematically represented in Figure 1. 4. Generation of the States of the Surface
(
)
()
m2K 1 , 1 ‚Mns-1‚ 1 55
[ ]
m2KBS2 S m2K 55 ,1 ‚ ) m2KS 55 1 55
(
)
ns-1
‚
() 1 1
The grand partition function for the “midway” model is generated by
Ξmidway )
(
)
()
)
2
[ ]
ns-1
()
‚
1 1
(12)
5. Generation of the Adsorption Isotherms The adsorption isotherms were obtained by the standard finite difference expression for an infinitely long lattice, where the fraction of bound sites θb is calculated from the expression
log[λ+(m2 + δm2)] - log[λ+(m2 - δm2)] log[m2 + δm2] - log[m2 - δm2]
(13)
where λ+ is the largest eigenvalue of the generating matrix M (cf. eq A10) and δm2 ) 0.001m2 is the increment change in the molality of the counterion concentration used in the calculations presented herein. The computation variable m2 is the total counterion molality, which also contains the added symmetric salt contribution m3. For the general case of arbitrary valencies m2 is
(
Z1 1 +
(11)
where ns is the number of sites on the linear surface and m2 is the molality of the counterion in aqueous solution.
(
m2S2 1 , 1 ‚Mns-1‚ 1 55
m2BS2 1 m2S , 1 ‚ 552 ) 55 m2S 1 55
θb )
As shown in the Appendix, the grand partition function is generated by the matrix method with the reference state that of the naked lattice. The grand partition function for the “local site” model is generated by the expression
Ξlocal site )
Figure 1. Schematic diagram indicating the interaction energy parameters K, B, and S. The lattice sites (filled circles) are labeled as j - 3, j - 2, j - 1, j, and j + 1. The bound ions of radius a are indicated by the open circles. Local Site Model: K is the local site intrinsic binding parameter; B is the interaction between two nearest-neighbor bound ions; and S is the interaction of a bound ion with an adjacent lattice site. As indicated above, the addition of a bound site j - 1 to a preexisting lattice in which j - 2 is a bound site has a statistical weighting factor of (m2/55)KBS2 and the addition of a bound site j + 1 to a preexisting lattice in which site j is unbound has a statistical weighting factor of (m2/55)KS. Midway Model: Since there is no lattice site associated with the location of the counterion bound between surface sites, there is no local site intrinsic binding parameter K. In this case the “bound” unit added to the preexisting lattice is defined by the lattice site to the right of the counterion. In the above example the “bound units” are j - 2, j - 1, and j + 1. The addition of a bound unit j - 1 to a preexisting set of lattice sites with a bound unit at j - 2 has a weighting factor of (m2/55)S2B and the addition of a bound unit j + 1 to a preexisting lattice with an unbound unit at j is given by (m2/55)S2.
m2 ) -
)
Z2 θ m + Z3m3 Z1 b 1 ) Z2 Z1(1 - θC)m1 + Z3m3 (14) Z2
Adsorption Layer of Charged Surfaces
Figure 2. Local site model for monovalent ions: unscreened coulombic interactions. The intersite spacing is b ) 3 Å and the site concentration is m1 ) 0.001. The charges are Z1 ) -1, Z2 ) 1, and Z3 ) -1. The parameters of closest approach are a ) 3 Å (s), a ) 2 Å (- - -), a ) 0.5 Å (- - -), and a ) 0.2 Å (- - - -).
where m1 is the molality of the total number of polymer sites and θC is the fraction of neutralized charge on the polyion which is related to the fraction of bound sites by the equation θC ) -(Z2/Z1)θb. Notice that the counterion concentration, and therefore θb, enters both in the exponential prefactor and also in exponent via the calculation of the screening factor κ. An iterative procedure was used to calculate θb such that m2 simultaneously satisfies eqs 13 and 14 for fixed values of m1 and m3. Although the three charges may be varied independently in the program, the present calculations are restricted to Z1 ) -1 and Z2 ) -Z3 ) 1 or 2. The distance of closest approach of the ions was taken to be a ) 3 or 2 Å for “real” systems. However, values of a ) 0.5 and 0.2 Å were also used to indicate a trend to the “point ion” limit. We use the value m1 ) 0.001 to represent concentrations of monomer units actually used in experiments. In all calculations the temperature was taken to be T ) 298 K with ) 80. All programs used in this study were written with Mathematica, a system for doing mathematics on a computer.
Langmuir, Vol. 15, No. 8, 1999 2857
Figure 3. Local site model: screened and unscreened coulombic interactions. The intersite spacing is b ) 3 Å and the site concentration is m1 ) 0.001. The charges are Z1 ) -1, Z2 ) 1, and Z3 ) -1. The type of interaction and the parameter of closest approach are as follows: screened interaction, a ) 2 Å (s); screened interaction, a ) 0.5 Å (- - -); unscreened interaction, a ) 2 Å (- - -); unscreened interaction, a ) 0.5 Å (- - - -).
Figure 4. Local site model for screened coulombic interaction for monovalent salts with different site concentrations. The intersite spacing is b ) 3 Å and a ) 2 Å. The charges are Z1 ) -1, Z2 ) 1, and Z3 ) -1. The values of the site concentrations are m1 ) 0.001 (s), m1 ) 0.005 (- - -), m1 ) 0.01 (- - -), and m1 ) 0.1 (- - - -).
6. Results: Local Sites Model Shown in Figure 2 are the isotherms for Z2 ) 1, m1 ) 0.001, b ) 3 Å, and a ) 3 Å, 2, 0.5, and 0.2 Å for the case of screened Coulombic interactions. The interaction of the ion with a ) 3 Å shows very little adsorption onto the surface, and the trend is to saturate the lattice as the centers of the two charges merge. This is understandable from the unlimited increase in the interaction energy in the limit of a point charge. The decrease in the adsorbed fraction of sites is a direct result of electrostatic screening by other ions. As the added electrolyte concentration is increased, the effective electrostatic interaction is weakened and the ions are released from the surface. This effect is verified in Figure 3, where the isotherms for screened and unscreened electrostatic interactions are compared. In Figure 3 a comparison is made between the screened and unscreened Coulombic interactions for Z2 ) 1, m1 ) 0.001, b ) 3 Å, and a ) 2 and 0.5 Å. In the case of the unscreened interaction, there is a substantial increase in site binding for the “real” system. Saturation, however, is not attained even at 1 m concentration. The effect of polyion concentration on the adsorption isotherm is shown in Figure 4 for the screened interaction. The fixed parameters in these calculations are Z2 ) 1, b ) 3 Å, and a ) 2 Å. The concentrations of the monomer units in this figure are m1 ) 0.001, 0.005, 0.01, and 0.1. It is clear that the constant region of adsorption is due to the presence of the finite concentration of counterions that
Figure 5. Local site model for screened coulombic interaction for divalent salts with different site concentrations. The intersite spacing is b ) 3 Å and a ) 2 Å. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -2. The value of the site concentrations are m1 ) 0.001 (- - - - -), m1 ) 0.005 (- - -), m1 ) 0.01 (- - -), and m1 ) 0.05 (s).
originated from the charged surface. The degree of adsorption is strongly dependent upon the charge of the counterion, as shown in Figure 5. In Figure 5 are adsorption isotherms for Z2 ) 2, b ) 3 Å, and a ) 2 Å for m1 ) 0.001, 0.005, 0.01, and 0.05. Comparison of the low concentration region in Figure 5 with those in Figure 4 indicates that the effect of doubling the counterion charge more than doubles the extent of adsorption. An unusual feature introduced in these calculations is the negative adsorption exhibited by all curves for m3 > 10-3. This is an unphysical result and due to the characteristic of the dependence of the screening
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Figure 6. Local site model for unscreened coulombic interaction for divalent salts with different site concentrations. The intersite spacing is b ) 3 Å and a ) 2 Å. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -2. The values of the site concentrations are m1 ) 0.05 (- - - - -), m1 ) 0.01 (- - -), m1 ) 0.005 (- - -), m1 ) 0.001 (s). Figure 9. Local site model for screened coulombic interaction for divalent salts as a function of added salt and intersite distance. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -2, with a closest approach parameter a ) 2 Å. The value of the site concentration is m1 ) 0.01.
Figure 7. Local site model for unscreened coulombic interaction for divalent salts with variable intersite spacing. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -2, with a closest approach parameter a ) 2 Å. The value of the site concentration is m1 ) 0.001. The intersite spacings are b ) 2 Å (s), b ) 5 Å (- - -), b ) 7 Å (- - - -), and b ) 9 Å (- - - -).
Figure 10. Local site model for unscreened coulombic interaction for divalent salts as a function of added salt and intersite distance. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -2, with a closest approach parameter a ) 2 Å. The value of the site concentration is m1 ) 0.01.
Figure 8. Local site model for screened coulombic interaction for divalent salts with variable intersite spacing. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -2, with a closest approach parameter a ) 2 Å. The value of the site concentration is m1 ) 0.001. The intersite spacings are b ) 2 Å (s), b ) 5 Å (- - -), b ) 7 Å (- - - -), and b ) 9 Å (- - - -).
parameter on the concentration of the counterions. This is discussed in more detail in the appendix. A similar calculation for the unscreened interactions does not exhibit this behavior, as shown in Figure 6. The effect of the nearest-neighbor interaction on the adsorption isotherm is reflected in the dependence on the average spacing parameter b. Shown in Figure 7 are isotherms for the fixed monomer concentration m1 ) 0.001 for Z2 ) 2 and a ) 2 Å. The intersite spacing for these curves are b ) 3, 5, 7, and 9. In the low concentration regime, the nearest-neighbor interaction is primarily that of the ion with the adjacent surface sites. Hence the degree of adsorption is greater in this region for the more closely spaced sites (dominance by the parameter S). However,
as the adsorption process advances, the repulsion between adjacent bound sites plays an increasingly important role in the process. Now the more closely spaced sites are at the disadvantage, and further adsorption becomes more difficult (dominance by the parameter B). On the other hand, the larger values of b do not exhibit this phenomenon. As indicated in these calculations, at sufficiently high concentrations there may be a charge reversal on the surface. This charge reversal should obtain whenever |Z2| > |Z1| for sufficiently separated adsorption sites. Charge reversal does not occur for the screened Coulomb interaction, as shown by the isotherms given in Figure 8. Plots of the dependence of θb on m3 and b for the screened and unscreened Coulombic interactions, respectively, are shown in Figures 9 and 10. 7. Results: Midway Model According to eqs 11 and 12, the only difference between the local sites and midway model is the intrinsic binding constant K, which appears in the former model, and the
Adsorption Layer of Charged Surfaces
Langmuir, Vol. 15, No. 8, 1999 2859
Figure 11. Midway model: screened and unscreened coulombic interactions. The intersite spacing is b ) 3 Å and the site concentration is m1 ) 0.001. The charges are Z1 ) -1, Z2 ) 1, and Z3 ) -1. The type of interaction and parameter of closest approach are as follows: screened interaction, a ) 2 Å (s); screened interaction, a ) 0.5 Å (- - -); unscreened interaction, a ) 2 Å (- - -), unscreened interaction, a ) 0.5 Å (- - - -).
two different definitions of S. The absence of K in the midway model is partially compensated for by the larger values of S because of the closer proximity of the “nearestneighbor” lattice sites. Hence the calculations given in the previous section virtually apply also to the midway model. To illustrate, shown in Figure 11 are the calculations for the midway model using the same parameters as in Figure 3 for the local sites model. 8. Characteristics of the Finite Ion Size Nearest-Neighbor Model The local site and midway models employed in this study assume a rigid array of lattice sites and do not inflict the properties of point charges for either the ions or lattice sites. An important feature in the analysis for these systems is that the interaction between the lattice sites does not contribute to the energy change associated with the adsorption process. That is, these surface site-surface site interactions cancel when the thermodynamic functions are calculated using the bare lattice as the reference state. The only way that surface site-surface site interactions affect the adsorption isotherms under the present philosophy is if a conformational change occurs that alters the spacing between adjacent sites. In this regard the present model is also applicable to flexible polyelectrolytes with fixed distances between adjacent surface sites. A second consequence of using finite size charges is that the energy associated with the placement of two counterions directly onto adjacent sites on the surface may be either attractive or repulsive. Let us examine the total interaction of generating a lattice when the j - 1 site is bound and the newly formed j lattice site is also bound. The reduced energy for this addition is given as
(
)
Z1Z2 Eββ ) 2 ln(S) + ln(B) ) ξC 2 + Z22 2 1/2 kBT (1 + rred )
(15)
Since Z1 and Z2 are of opposite sign, the sum in the brackets in eq 15 may be either positive or negative, depending upon the values of the reduced separation distance rred and the values of the charges Z1 and Z2. The value of rred ) Rred for which this sum in the brackets is zero is
Rred )
[
]
4Z12 - Z22 Z22
1/2
(16)
Thus the net nearest-neighbor interaction for the addition of a new lattice location to an existing lattice is attractive
only if 2|Z1| > |Z2|. This does not mean, however, that this condition results in a clustering of bound sites. To determine whether clustering is a favorable condition, one must move together two separated bound states to nearest-neighbor status. The energy change associated with this process is ln(B), which is clearly a repulsive interaction in the present model. By limiting the range of the interaction to nearestneighbor only, there is the possibility that the adsorption of multivalent ions will lead to charge reversal for unscreened Coulombic interactions. Consider first an intersite spacing b such that the highly repulsive energies effectively render the neighboring sites as “excluded” sites. It is this “exclude site” effect that leads to the prediction that the polyion lattice is half occupied at saturation regardless of the valence of the multivalent ion. This is illustrated in the isotherms in Figure 6. Thus for counterions of valencies for which |Z2| > 2, half saturation always leads to charge reversal. In the case of divalent counterions, charge reversal may also result if more than half of the surface sites may be bound. As the ratio a/b is decreased, the magnitude of the repulsive nearestneighbor interaction decreases to the point that the lattice can be saturated, viz., θb f 1. This is illustrated in Figure 7 for the symmetric divalent salt. In the case of the screened Coulombic interaction, charge reversal does not occur due to the fact that the interactions become weaker as the salt concentration is increased. The release of bound counterions is also contained in the model of Satoh, Komiyama, and Iijima15 for the screened Coulomb interaction. These authors also find negative values of θb for b ) 5 Å with the salt concentration above 10-3 M. They interpret the decrease in θb as a loss in entropy due to condensation that predominates over the energy gain in the free solution. It is our conclusion, however, that the additional screening at the higher salt concentration weakens the interactions as the cause of the decrease in the fraction of bound sites. 9. Application of the Nearest-Neighbor Model to Experimental Data The present model is now employed in the analysis of experimental data for monovalent and divalent ion binding by linear surfaces. In both of these examples the polyion is the double-stranded DNA. With the interaction limited to nearest-neighbor only, the phosphate groups on opposite sides of the double strand do not interact with each other. Thus the one-dimensional system is represented by the array of phosphate groups along one side of the double strand. Instead of using the “projected” value of 1.7 Å for the average distance between phosphate groups as employed in condensation theories, we have used the more realistic value b ) 3.4 Å as the distance between base pairs. The concentration of accompanying counterions is calculated on the basis of the reported phosphate concentrations. Reuben-Shporer-Gabbay Data on Monovalent Cations. Reuben, Shporer, and Gabbay16 used NMR methods to study the association of the cations 23Na and 87Rb to DNA. It was concluded that these monovalent ions do not site bind but rather are “trapped” within an ion cloud. The basis of this conclusion was a “two-state” analysis of the longitudinal relaxation rates 1/T1 of 23Na and 87Rb in aqueous solution, viz.
(
)
mb 1 1 1 1 ) + T1 T1f mt T1b T1f
(17)
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Schmitz
Table 1. “Effective” Dissociation Constant for the Local Site Modela (m2(1 - θb)/θb) × 1000 m2 (molal)
m1 ) 0
m1 ) 0.0262
0.001 0.00316 0.01 0.0316
12.5 19.3 39.7 110.2
3.78 11.8 36.7 119.2
a Calculated from eq 32 where θ was calculated from eqs 3-10, b 12, and 13 with a ) 2 Å.
where the subscripts “t” and “b” refer to the total (concentration mt) and bound (concentration mb) states of the alkali metals, respectively. They assumed the association model was that of noninteracting bound particles with the dissociation constant KD given by
KD )
Pfmf 1 - θb ) mf mb θb
(18)
where Pf ) Po - Pb is the concentration of free lattice sites for a system of Po sites of which Pb are bound with the metal ion. The equation solved by these authors to obtain a value of mf was
2mb ) mt + Po + KD -[(mt + Po + KD)2 - 4Pomt]1/2 (19) where mt ) mb + mf is the total molality of the alkali metal ions. The concentration of free ions, Na+ or Rb+, was then calculated from eq 19 with a value of KD obtained with an iterative best fit of eq 17. Hence the values of mf and KD are subject to the validity of the adopted law-ofmass-action model of independent binding sites. Note that eq 18 obtains from eq 2 upon setting w′ ) 0. To compare the experimental results for KD with the present model, an “effective” dissociation constant, expressed as (1 - θb)m2/θb, is calculated from eqs 11 and 13. Numerical values of (1 - θb)m2/θb for selected concentrations that cover the reported experimental range of salt concentrations are given in Table 1 for the “local site” model with finite polyion concentration. Taking into consideration the differences in the two models used to obtain the KD values, the computed values of (1 - θb)m2/θb for the local site model are within the range of the “best fit” values to the experimental data of KD ) 10.9 mM for 23Na and KD ) 37.8 mM for 87Rb as obtained from the calculated estimates of the alkali concentrations and the random binding model assumption. This comparison between the effective and experimental values of KD clearly supports the notion that the interaction is largely, if not solely, electrostatic in nature. Granot-Kearns-Feigon Data on Divalent Cations. Kearns and co-workers17-19 have presented detailed studies on the interaction of divalent metal cations with DNA. These authors gave a well-defined definition of “bound” counterions, viz,19 “Henceforth, we shall use the term “bound” to refer to those metal ions that form either inner- or second-sphere complexes with the polyelectrolyte.” These “bound” ions are therefore “site localized” in the vicinity of the phosphate groups of the DNA and do not include ions that are “only held in an ion atmosphere around the polyion electrolyte by long-range, through space, electrostatic interactions.”19 Those ions that are (17) Granot, J.; Feigon, J.; Kearns, D. R. Biopolymers 1982, 21, 181. (18) Granot, J.; Kearns, D. R. Biopolymers 1982, 21, 203. (19) Granot, J.; Kearns, D. R. Biopolymers 1982, 21, 219.
Figure 12. Manganese binding by DNA. The values of the parameters used in this calculation were those reported by Kearns and co-workers on manganese binding to DNA. The intersite spacing is b ) 3.4 Å for the separation distance between base pairs along the DNA double helix. The reported site concentration is m1 ) 0.0045. The charges are Z1 ) -1, Z2 ) 2, and Z3 ) -1 for the DNA phosphate, Mn2+, and Cl-, respectively. The parameters of closest approach are based on the reported values of “inner” and “outer” bound cations, 3 and 6 Å, respectively: screened interaction, a ) 3 Å (s), screened interaction, a ) 6 Å (- - -); unscreened interaction, a ) 3 Å (- - -), unscreened interaction, a ) 6 Å (- - - -).
associated with the “inner shell” mode of binding were detected by 31P NMR methods,19 whereas the totals of “inner shell” and “outer shell” bound ions were detected by proton relaxation methods.18 The experiments by Kearns and co-workers that are of relevance to the present study are the “noncompetitive” experiments involving Mn2+ with chicken erythrocyte DNA.19 The DNA phosphate concentration used in these studies was 4.5 and 4.2 mM, and the range of Mn2+ concentration was 2.8 × 10-5 to 2.1 × 10-3 M. It was reported that θb ≈ 0.4 and was independent of the ionic strength. Since proton NMR was used, this fraction of bound sites represents both “inner” and “outer” complexed ions. Calculations were performed with the present theory for a ) 3 Å and a ) 6 Å, as given for the “inner” sphere and “outer” sphere complexes, respectively.19 The calculations were performed with the “mixed” electrolyte MnCl2 used in their studies. The concentration of the counterion was therefore calculated from eq 13 with Z1 ) -1, Z2 ) +2, and Z3 ) -1. The results of these calculations are given in Figure 12 for both the screened and unscreened Coulombic interaction models, where the phosphate concentrations (m1) were those of the Granot-Kearns study.19 Granot and Kearns19 reported that the total extent of Mn2+ binding was θb ) 0.43 ( 0.04 (corrected to the control results for acid precipitated DNA) and independent over the metal ion concentration range 2.8 × 10-5 M e CMn2+ e 2.1 × 10-3 M for DNA phosphate concentrations of 4.24.5 mM. The isotherm in Figure 12 for a ) 3 Å gives an ionic strength independent value of θb ) 0.26. Taking into consideration that the parameters used in these calculations were taken from the literature, the present model is in good agreement with the data of Granot and Kearns without having to invoke a reduction in the phosphate charges. As may be expected from the greater distance used in the “outer sphere” calculations, these “adsorbed” counterions to not contribute much to the overall adsorption isotherm when both states are combined. As in the monovalent case, the extent of adsorption in the infinite dilution calculations depends strongly on the ionic strength of the solvent. Again, the physical interpretation of these calculations is that the dominance of the counterions released from the polyion at finite concentrations is
Adsorption Layer of Charged Surfaces
responsible for the constant value of θb over the range of added electrolyte. 10. Discussion The present study parallels that of Zimm and Rice20 (ZR) on the helix-coil transition for one-dimensional polypeptides. They also use a matrix generation method for the grand partition function and employed “realistic” molecular parameters in their calculations. Their model parameters include (1) s, an equilibrium constant for hydrogen bonding that stabilizes the helix configuration, (2) σ, a “nucleation” parameter with a fixed value of 10-4 which is the statistical weighting factor for a helical region following a random coil segment, and (3) a “screened Coulomb” interaction between sites that are ionized. In their calculations the states of four contiguous sites is considered for the elements of the generation matrix, which is of dimensions 16 × 16. The only long-range interaction therefore occurs in the screened Coulomb term between the ionized hydrogen bonding sites within the subgroup. Hence the magnitude of the electrostatic interaction between two charged sites differs in accordance to their relative locations along the polypeptide and the configuration, helix or coil, of that subregion. The ZR calculations differ from those herein in several important aspects. First, the surface in the ZR calculation varies with the state of the polypeptide, where units in the helical state are closer than those in the coil state. These differences contribute to the free energy through the nucleation parameter σ and the differences in the Coulombic interaction through the distance differences for the two configurations. Because of the rigidity of the surface and the separation of charge centers, the surface characteristics do not contribute to the isotherms of the current study. Second, the ZR model gives complete neutralization of the charge upon the association of the hydrogen ions. This result obtains whether point charge approximations are used because the hydrogen association results in a covalent bond. Thus, at best, there is in the covalent bond formation a partial separation of charge that reflects the relative electron-drawing power of the constituent atoms in the bond. In contrast the association of counterions in the present study is via ionic bonds; hence there is a physical separation of positive and negative charge centers as dictated by the parameter representing the finite size of the ions. A third difference of the ZR paper is that their calculations are for the infinite dilution case of the polypeptide. Hence there is no conservation of mass equation as given by eq 14 in the ZR model. The shortcomings of the present model are similar to those of the ZR model. In particular the dielectric constant and ligand concentration in the vicinity of the adsorption surface may greatly differ from that of the bulk solution. As noted in the ZR paper, the accumulation of counterions in the vicinity of the polyion surface results in an effective screening length that is considerably smaller than that based on the bulk salt concentration. That is, there is a “local screening” effect. Likewise the dielectric constant for the polyion is less than that for the solvent for aqueousbased solutions. These effects are more important for the unscreened Coulomb interaction than for the screened form. This is because in the screened Coulomb form the dielectric constant and local concentration appear in both the preexponential factor and exponential argument of the matrix elements. These two terms therefore act as a mutually compensating feature for screened attractive interactions. (20) Zimm, B. H.; Rice, S. A. Mol. Phys. 1960, 3, 391.
Langmuir, Vol. 15, No. 8, 1999 2861
Stigter21 recently examined the effect of divalent ion association with the major and minor grooves of DNA on the bending of the DNA, where the nonlinear PoissonBoltzmann equation was employed. The DNA was modeled as a dielectric cylindrical core. Superimposed on this core was either a “discrete charge” model, in which a double helix configuration representing the phosphate groups of the DNA, or a “smeared charge” model, in which the DNA charge was uniformly distributed along the dielectric cylinder surface. The electrical interactions were taken to be a screened Coulomb interaction. The dielectric constant of the cylindrical core was taken as 4 while that of the bulk was 78.54, from which an “effective” dielectric constant was calculated for the determination of the potential about the DNA. Stigter found that for the cylindrical polyion with the smeared charge having radial symmetry the potential depended only on the bulk dielectric constant. In the case of discrete charge placement the surface potential was found to depend on the core dielectric constant. However, this dependence was relatively weak such that Stigter concluded that the “... simple dielectric model is quite satisfactory.”21 From the result of Stigter regarding the screened Coulomb interaction, the calculations using this form of the potential may be insensitive to the difference in the bulk and local dielectric constants. As previously mentioned, this is due to the fact that the dielectric constant appears in both the preexponential factor and exponential arguments, hence a “mutual cancellation” of the adjustments regarding the dielectric constant effects. The same cannot be said for the unscreened Coulomb interaction calculations. The effects of local concentration and local dielectric constant on the adsorption isotherms may be partially corrected by assuming a three-state model for the ion environment rather than the standard two-state model. To do this we assume an equilibrium between the three regions: the bulk solution, the local concentration, and the adsorption surface. The local concentration has no precise definition, but we take it to be the “excess” concentration associated with the region between the surface and the bulk solution. That is, it is the integrated concentration between the surface and the bulk solution. We assign this region as the “diffuse ion cloud” generally associated with charged surfaces. Hence the “modified” system is represented by the equilibrium between the three regions,
m2,b / m2,dc / m2,f
(20)
where m2,b is the local concentration of the bound ligand, m2,dc is the integrated local concentration of the free ligand in the surrounding ion cloud, and m2,f is the bulk concentration of the free ligand. Hence the concentration m2 is replaced by the concentration m2,dc ) m2,fKC, where KC is the Boltzmann weighting factor for the electrical work to transport the ligand from the bulk to the diffuse cloud. One may use the bulk dielectric constant in the calculation of KC while the parameters K, B, and S may employ the local value of the dielectric constant. Such a correction factor would require knowledge of the volume of the diffuse ion cloud, which is expected to vary with the bulk ionic strength of the medium. Since the relevant parameters are not available such corrections are not feasible at this time. However, one may anticipate that the net effect of using local rather than bulk values is to increase the fraction of bound sites for a given bulk concentration. (21) Stigter, D. Biopolymers 1998, 46, 503.
2862 Langmuir, Vol. 15, No. 8, 1999
One of the problems regarding ions adsorbed onto a surface is the mathematical form of the interactions, both with the surface and with other adsorbed ions. Rouzina and Bloomfield5 draw important conclusions based on an unscreened Coulombic interaction, with the apparent justification that the concentration of surface bound ions is considerably less than the bulk ionic strength and one can therefore use the infinite dilution limit, i.e., the unscreened Coulombic interaction. On the other hand the screened Coulombic interaction is more generally the choice of the pairwise interaction.15,20,21 In the present study both forms are examined, with differing results only under high salt conditions. According to the calculations represented in Figures 2-9, there are positive and negative features of both mathematical forms. In the case of unscreened interactions, an increase in the concentration of the counterion results in a continued increase in the degree of adsorption. For spacings b sufficiently far apart to reduce the dominance of nearest-neighbor interactions, the unscreened Coulombic interaction has the property of charge reversal. This is a physically realistic result as it represents the association of two simple ions of opposite charge and differences in magnitude. If the situation was reversed, i.e., the larger magnitude of charge was on the polyion, then there would be no charge reversal even though the same picture emerges for the case of large b. On the other hand the screened Coulombic interaction also exhibits characteristics that might be in line with one’s intuition. As the concentration of added electrolyte is increased, the law of mass action dictates that the amount of adsorbed ligands must also increase. However, it is generally accepted that the effect of added electrolyte is to reduce the electrostatic interaction between charged groups, i.e., attain the state of “theta solvent”. Hence one might expect that a continued increase in the ion concentration would weaken the hold that the surface had on the bound ions and, in the proverbial sense, set them free. However, a problem arises as the concentration of added electrolyte and/or counterion charge is increased. One obtains negative adsorption! It is our opinion that the negative adsorption is not a physically meaningful event and arises from the nearest-neighbor mathematical form of the eigenvalues. Hence the question of the true nature of electrostatic interactions, viz., screened or unscreened, is not resolved in the present study. Perhaps the most controversial conclusion among some practitioners pertains to the role of the counterions that come with the polyion. Under most experimental conditions the concentrations of the monomer units of the polyion are in the millimolar range. It is therefore a meaningless exercise to discuss adsorption isotherms for added electrolyte below the monomer unit concentration. This conclusion places more demands on the experimentalist to go to much lower polyion concentrations to match, or attempt to match, the theoretical constraints imposed on equations. As shown in Figures 2-7, the counterions released by the charged surface give isotherms that appear to be constant under low added salt conditions. Appendix General Considerations for the Grand Partition Function. The system is composed of a charged surface (component 1), counterions (component 2), gegenions (component 3), and solvent. The counterions and gegenions are collectively referred to as the solute particles. The
Schmitz
solution is partitioned into two regions, that which contains the “free” solute particles in the solvent (region I) and that which contains the charged surface and adsorbed counterions (region II). It is assumed that the solute particles in region I are sufficiently dilute that they obey Henry’s law, viz.
µ2 ) µ2o + kT ln(X2)
(A1)
where X2 is the mole fraction of component 2 in region I. The grand partition function for region II for a lattice of ns sites is given by the usual expression ns
Ξ)
λ2n Q(nb,ns) ∑ n )0
(A2)
b
b
where λ2 is the activity of the bound ligand that includes any accompanying change in bound solvent and Q(nb,ns) is the canonical partition function for a system of nb bound sites for a lattice of ns sites. The total energy for the jth configuration is symbolically given by the expression
Ej ) E(1) + E(1|2|) + E(1|3) + E(1|4) + E(1|5) + ... + E(2) + E(2|3) + E(2|4) + E(2|5) + E(2|6) + ... + ... + E(ns) (A3) where E(j) represents the intrinsic interaction energy of the ligand with the jth site and E(j|i) represents the pair interaction energy of the jth site with the ith site, where j < i. Restriction of the development to nearest-neighbor interactions in which each site can be found in two states (R or β), the four possible states for a neighboring pair of sites are
{
ER(j) + ER(j + 1) + ER,R(j|j + 1) ER(j) + Eβ(j + 1) + ER,β(j|j + 1) Epair ) Eβ(j) + ER(j + 1) + Eβ,R(j|j + 1) Eβ(j) + Eβ(j + 1) + Eβ,β(j|j + 1)
(A4)
where it is understood that the reference state energy has already been subtracted. It is clear that one need only to consider the energy for the addition of lattice site j + 1 to a preexisting lattice of j sites. Since the present development is limited to nearest-neighbor interactions, only the “new” energy of the lattice system can be represented by a 2 × 2 table in accordance with eq A4. The grand partition function is therefore generated by the matrix equation,
()
Ξ ) (XR,KR,XβKβ)‚Mns-1‚
1 1
(A5)
where the general form of the generating matrix is given by
[
X K S XKS M ) XRKRSR,R XβKβSR,β R R β,R β β β,β
]
(A6)
Eigenvalues of Generating Matrix. It is more convenient to generate the grand partition function using the largest eigenvalue of the generating matrix rather than eq A5. Let us assume the existence of the matrixes
Adsorption Layer of Charged Surfaces
Langmuir, Vol. 15, No. 8, 1999 2863
Figure 13. Ratio of eigenvalues for the local site model as a function of molality. The eigenvalues given by eqs A11 and A12 were evaluated using the values Z1 ) -1, Z2 ) 1, Z3 ) -1, intersite spacing b ) 3 Å, and the counterion radius is a ) 2 Å for the screened Coulombic interactions.
Figure 14. Ratio of eigenvalues for the local site model as a function of molality in region of the maximum. The eigenvalues given by eqs A11 and A12 were evaluated using the values Z1 ) -1, Z2 ) 1, Z3 ) -1, intersite spacing b ) 3 Å, and the counterion radius is a ) 2 Å for the screened Coulombic interactions. The above figure shows the region in which the absolute value of the ratio of eigenvalues is a minimum.
S and its inverse S-1 that diagonalizes M. We can now write eq A5 as
Ξ ) (XR,KR,XβKβ)‚[S‚S-1‚M‚S‚S-1]ns-1‚ ) λ+ns-1(XRKR,XβKβ)‚S‚
[
()
Figure 15. The eigenvalues for the local site model as a function of molality. The eigenvalues given by eqs A11 and A12 were evaluated using the values Z1 ) -1, Z2 ) 1, Z3 ) -1, intersite spacing b ) 3 Å, and the counterion radius is a ) 2 Å for the screened Coulombic interactions. The above figure shows the region in which the absolute value of the ratio of eigenvalues is a minimum.
Figure 16. The ratio of eigenvalues for the local site model as a function of the chain length for divalent salts. The eigenvalues given by eqs A11 and A12 were evaluated using the values Z1 ) -1, Z2 ) 2, Z3 ) -2, intersite spacing b ) 3 Å, and the distance of closest approach a ) 2 Å. The concentrations are m1 ) 0.001, and m2 ) m3 ) 0.001.
generating matrix for the local site model is therefore of the form,
[ ]
1 1
] ()
1 0 1 ‚S-1‚ 1 0 (λ-/λ+)ns-1
m2KBS2 S 55 M) m2KS 1 55
(A7)
where the eigenvalues are
(A10)
with eigenvalues
(
where
)
m2KBS2 + 1 (1 ( x1-4W) 55 λ( ) 2
(XRKRSR,R + XβKβSβ,β)(1 ( x1 - 4W) (A8) λ( ) 2
(A11)
where
W)
XRKRXβKβ(SR,RSβ,β - Sβ,RSR,β) (XRKRSR,R + XβKβSβ,β)
2
(A9)
In the models presented in the present communication the reference state is the fully charged naked lattice and identified with state β. Since the energies in eq A4 already have the reference energies subtracted out, and whose configuration is assumed not to change when the ions bind, one has for the parameters in aqueous solution for the local site model: Xβ ) 1; Kβ ) 1; Sβ,β ) 1; XR ) m2/55; KR ) K; Sβ,R ) SR,β ) S; and SR,R ) BS2, where K, S, and B are defined by eqs 7, 8, and 9, respectively. The
m2KS2 (B - 1) 55 W) 2 m2KBS2 +1 55
(
)
(A11)
Numerical Results for the Local Site Model with Screened Coulombic Interactions. Calculations were performed for the local site model with screened Coulombic interactions using the parameters employed in the main body of this communications, viz., a repeat unit spacing of b ) 3 Å and a counterion radius of a ) 2 Å. In contrast to the curves given in the main text, these calculations
2864 Langmuir, Vol. 15, No. 8, 1999
are plotted as a function of m2 rather than m3 since the latter was used to conform with experimental representations. Illustrated in Figure 13 is the ratio of eigenvalues λ+/λplotted as a function of log(m2) over the total range of solute concentrations examined in the main text. As shown in Figure 14 the ratio λ+/λ- does not approach zero but rather attains a maximum in the vicinity m2 ) 10-1 m. We now examine the concentration dependence of the individual eigenvalues λ+ and λ-. The values of λ+ and λare plotted as a function of log(m2) in Figure 15. It is clear that the difference in these eigenvalues, 2(1 - 4W)1/2, is at a maximum in the concentration range 10-4 < m2 < 10-3. The maximum in λ+ likewise explains the “downturn” in the isotherms shown in Figures 3-5.
Schmitz
The question of the applicability of the limiting expression for the grand partition function as defined by eq A8. As indicated in eq A7, the 2-2 matrix element contains the ratio of eigenvalues. Since this ratio is negative as shown above, a plot of this ratio as a function of ns will oscillate from positive to negative values. Since we are only interested in the relative values of these two quantities, we have plotted the negative of this ratio raised to the appropriate power, (-λ-/λ+)ns, as a function of the number of repeat units, ns, in Figure 16. Clearly the infinite chain length limiting expression for Ξ is valid for ns > 6. LA9805746
