Anal. Chem. 1994,66,440-449
Electrostatic Retention ,Model for I on-Exchange Chromatography Jan Stdhlberg' Quality Control, Astra Pharmaceutical Production AB, S- 15 1 85 SWedaJe, Sweden A theoretical framework for the effect of eluting salt concentrationon the capacity factor of small ions in ion-exchange and ion chromatography is described. The model is based on the Gouy-Chapman theory for the electrical double layer complemented with the possibility for specific adsorption of the counterions as well as the analyte ions to the chromatographic surface. Because of the complex dependence of the capacity factor on parameters such as net charge density of the stationaryphase, eluent salt concentration, and electrostatic potential of the surface, numerical evaluation of the model is needed in the general case. The calculationsshow, in agreement with the general experimental observation, that a log k'vs log (eluent concentration of 1:l salt) plot is linear with a slope value close to -1 and -2 for mono- and dicharged analytes, respectively. It is also shown that the linearity as well as the slope is insensitive to the type of analyte ion, eluent counterion, and type of stationary phase and its charge density. The theory therefore offers a physically consistent approach to the analysis of retention data without resorting to the unrealistic stoichiometric models which has mainly been used so far in the ionexchange chromatography of small ions. The first investigation of the ion-exchange phenomenon was made in 1850 by Thompson, who studied the adsorption behavior of ammonium ion to soils.1 The regular use of ion exchangers in analytical chemistry started in 1939, when Samuelson introduced organic ion-exchange resinsa2 Since then ion-exchange chromatography has been widely used for separation and quantification of ions in the analytical laboratory. Many papers have appeared wherein different theoretical approaches to the description of ion-exchange equilibria have been presented. During these years two mainstreams have developed: the Donnan potential concept introduced by Mattson in his investigation of ion-exchange equilibria in soils3and the stoichiometric theory based on the mass action law. Of these two, the latter seems to have gained higher popularity among analytical chemists, probably because of its simplicity and the fact that predictions made from it very often agree with experimental data. A review of the theories for ion-exchange equilibria developed before 1960 and a thorough discussion of the Donnan potential concept is found in the classical book by Helfferi~h.~ A presentation of the current view on retention models in ion chromatography, focusing on stoichiometric models, is found in the book by * Present address: Department of Analytical Chemistry, University of Stockholm, S-10691 Stockholm, Sweden. (1) Thompson, H. S. J. R. Agric. SOC.Engl. ISSO, 11, 68. (2) Samuelson, 0. 2.Anal. Chem. 1939, 116, 328. (3) Mattson, S. Soil Sci. 1929, 28, 179. (4) Helfferich, F. Ion Exchange; McGraw-Hill: New York, 1962. 440
Analytical Chemistry, Vol. 66, No. 4, February 15, 1994
Haddad and J a ~ k s o n .Recently, ~ de B o k and Boots developed a theory.for the selectivity in ion-exchange chromatography based on the experimental observation that enthalpy-entropy compensation occurs in the exchange process;6 they also proposed a general thermodynamic framework for enthalpyentropy compensation.' In the stoichiometric model for ion-exchange chromatography, retention of a solute ion i is described as an exchange process between the solute and electrolytecounterionsj, where the latter are assumed to be bound as a 1:l complex to the chemically bound charged groups at the surface. The exchange process is illustrated in eq 1 for the case of positively
monocharged ions. By assigning an equilbrium constant for this process, neglecting activity coefficients, a relation is obtained between the capacity factor for the solute and the counterion concentration in the eluent, cj. For the general case in which the charge of i and j is zi and zj, respectively, the relation between the capacity factor and counterion concentration, cj, is given by eq 2,5 'i
log k' = const - - log cj
(2)
'j
where the constant inter alia includes the equilibrium constant for the exchange process. Indeed, plots of logarithmic capacity factor versus the logarithmic salt concentration in the eluent generally yield straight lines with a slope value close to the quotient (-zi/zj) .5 However, the agreement between the experimental results and the stoichiometric theory does not imply conclusively that the description is valid from a physical point of view. The main problem associdted with this theory is that it neglects the activity coefficient of both the solute and the counterion in the resin phase. Furthermore the activity coefficient is determined by long-range electrostatical forces, implying that it is a multibody interaction and therefore cannot be described by stoichiometric relations. The theory developed in this paper estimates the activity coefficient of the solute ion in the resin phase as a function of, for example, the concentration of counterion in the eluent by using well-known relations for electrostatic interactions. The departure of electrolytes from ideal solution laws may to large extent by accounted for in terms of electrostatic ( 5 ) Haddad, P. R.; Jackson, P. E. Ion Chromatography; Elsevicr: Amsterdam, 1990. (6) de B o k , P. K.; Boots, H, M. J. J. Phys. Chem. 1989, 93, 8243. (7) Boots, H. M. J.; de B o k , P. K. J. Phys. Chem. 1989, 93, 8240.
0003-2700/94/0366-0440$04.50/0
0 1994 Amerlcan Chemical Society
interactions between the ions and in our case also between the ions and the charged surface. The strength of the electrostatic interaction is determined by the electrostatic potential, which therefore is the relevant parameter to find. In this paper we calculate the activity coefficient of a solute present in trace concentration in the resin from the Gouy-Chapman theory in which the electrostatic potential in the diffuse double layer is given as a function of parameters such as salt concentration, surface concentration of charges, etc. Because of the connection between the activity coefficient and the electrostatic potential the ensuing discussion will concentrate on electrostatic potential rather than on activity constants. Besides the effect of nonspecific electrostatic interactions as described by the Gouy-Chapman theory, the possibility of specific interactions between the charged surface and counterions as well as solute ions is also considered. Modelingof the electrostatic effects in ion-exchangeresins started with the use of the Donnan potential concept, which considers the equilibrium distribution of ions between two identical homogenous phases with different electrostatic potentials. The main problems with this approach when applied to ion-exchange chromatography is that it does not give any information about how different parameters influence the potential differenceand that the charge distribution within the two phases is assumed to be homogenous. The study of the interaction between a charged surface and the ions constituting the electrolyte is an old but still very active subject in physical and surface chemistry, and the level of sophistication of the descriptions used may vary somewhat depending on the problem considered. In the simpler case, the starting point is the Poisson-Boltzmann equation, which is a second-order differential equation derived from statistical mechanical and electrostatical principles by making the simplifying assumptions of mean field interactions; the ions are considered as point charges and the solvent is assumed to be a continuum with constant polarizability. Solutions of this equation have often been applied in colloid and surface chemistry for treating electrostatic interactions between colloid particles as well as the interaction between electrolyte ions and charged surfaces. The literature in this area is vast, and a comprehensive summary of the state of the art is found in ref 8. For a charged planar surface in contact with an electrolyte solution, the Poisson-Boltzmann equation gives the well-known Gouy-Chapman (G-C) theory for the diffuse double layer. The original G-C theory can be modified in several ways depending on the propertiesof the studied system; e.g., in the Stern theory (see Figure l), a correction is made for specific adsorption and the nonzero size of the counterions by dividing the charged space into two regions. In the Stern layer, near to the surface, the interaction is determined by short-range interactions between the ions, the wall, and the adjoining dipoles. The diffuse part of the double layer, where the Poisson-Boltzmann equation is expected to hold, starts at a plane outside the Stern layer. The dividing plane between the Stern layer and the diffuse double layer is often called the outer Helmholtz plane and is situated 2-3 A out from the surface. The introduction of such a plane may be physically meaningful for molecularly smooth surfaces but becomes (8) Hunter, R. J. Foundationsof Col1oidScience;Oxford UniversityPress: Oxford, UK, 1992; Vol I and 11.
d-
Helmholtz plane
0
0 0
0 0
0
0 Bulk solution
0
0 0
pk
0 Stern layer
Figure 1. Schematic illustration of the double-layer model including the Stern layer. In the theory proposed in this paper, the thickness of the Stern layer is zero so that the Gouy-Chapman theory describes the electrostatic potential and counterion concentration up to the surface, i.e., to x = 0.
ambigous when applied to the highly irregular surfaces used in chromatography. The Stern theory was used by Cantwell and co-workers in a series of papers discussing retention of ions in ion-exchange and ion pair chr~matography.~-*~ They considered the experimentally measured capacity factor as a sum of two separate contributions: one contribution is ion exchange in the diffuse part of the double layer and the other is adsorption on the chromatographic surface, i.e., in their case in Stern layer. The key point in their theory is the assumption of a difference in the standard chemical potential, po, for an ion between the diffuse part of the double layer and the bulk of the mobile phase. According to Cantwell and co-workers, the exchange of ions between the bulk and the diffuse part of the double layer is very significant. Based on their model, numerical values for the equilibrium constants as high as 900 were estimated for the exchange between p-nitrobenzenesulfonate and chloride ions.1° From their investigations they conclude that ion exchange between the bulk and the diffuse layer is the dominating contribution to retention at low ionic strengths or high surface p0tentia1s.l~Since podoes not include any electrostatic effects, the model proposed by Cantwell and co-workers assumes a significant difference in solvent-analyte interaction in the bulk and the diffuse layer, respectively. Thermodynamically they describe the transfer of an ion between the bulk and the diffuse part of the double layer as if it were a transfer between two different phases. Furthermore, they assume that the standard chemical potentials of (9) Cantwell, F. F.; Puon, S. Anal. Chem. 1979, 51, 623. (10) Afrashtehfar, S.; Cantwell, F. F. Anal. Chem. 1982, 54, 2422. (1 1) R. A. Hux; Cantwell, F. F. Anal. Chem. 1984, 56, 1258. (12) Cantwell, F. F. In Ion ExchangeandSolvent Extractions: ASeriesofAdvances; Marinsky, J. A., Marcus, Y., Eds.; Marcel Dekker: New York, 1985. (13) Liu, H.; Cantwell, F. F. Anal. Chem. 1991, 63, 993. (14) Liu, H.; Cantwell, F. F. Anal. Chem. 1991, 63, 2032.
AnaIyticalChemistry, Vol. 66, No. 4, February 15, 1994
441
the counterion and the solute ion are either independent of distance from the charged surface or vary proportionally to one another with distance from the charged surface within the diffuse double layer. A theory for ion pair ~hromatographyl~ based on the GouyChapman theory has been used to describe quantitatively changes in retention when different chromatographic parameters are varied.16-18 The role of the electrostatic surface potential in chromatography of ions has also been discussed by Weber.’9.20 Recently we proposed that the Coulombic interaction between a protein molecule and the charged stationary phase in ion-exchange chromatography can be described by using the solution of the linearized PoissonBoltzmann equation for two charged planar surfaces in contact with a buffered salt solution.21 At sufficiently high salt concentrations in the mobile phase, the van der Waals interaction becomes important and is combined with the Coulombic interactions to a generalized retention theory for proteins.22 A recent application of the Stern layer concept to describe the equilibrium between electrolyte ions and an ionexchange resin has been presented by Horst et As has been pointed out above, the stoichiometric theory predicts the correct slope value for a log k’vs log Cj plot despite the fact that such a theory is not physically consistent. The purpose of this paper is to investigate whether a theory based on a more solid physical view gives the same agreement with experiments as the stoichiometric theory does. Here a theory is developed for retention of small ions in ion-exchange chromatography based on the Gouy-Chapman theory and including the possibility of specific interactions between the solute ions and the salt counterions with the charged surface. It is found that the proposed theory predicts slope values that are almost independent of the magnitude of the physical parameters introduced and with numerical values that agree with the experimentally found values. Since the Gouy-Chapman theory is used to calculate the activity coefficients of the solute and electrolyte ion in the resin phase, the stationary-phase surface is regarded as a rigid wall with a fixed number of charged groups per unit surface area. This implies that the theory is best suited to describe the retention behavior for rigid column materials and shall therefore not be applied to soft gel ion exchangers which swell and shrink with ionic strength. Theories describing retention and selectivity in the latter type of exchangers are usually based on the Donnan concept complemented with osmotic pressure coefficients.4~2~ THEORY General Aspects. Consider a surface on which negatively charged groups are chemically bound and in contact with an electrolyte solution. The positively charged counterions are in general not bound as a stoichiometric 1: 1 complex with the (IS) Stihlberg, J. J. Chromofogr.1986, 356, 231 (16) Stihlberg, J.; Furingen, A. Chromarographta 1987, 24, 78. (17) Stihlberg, J.; Hagglund, I. Anal. Chem. 1988, 60, 1958. (18) Bartha, A.; Stihlberg, J.; Szokoli, F. J. Chromorogr. 1991, 552, 13. (19) Weber, S.G.; Orr, J. D.J . Chromafogr. 1991, 332, 433. (20) Weber, S.G. Talnnra 1989, 36, 99. (21) Stihlberg, J.; Jdnsson, B.; Horvith, C. Anal. Chem. 1991, 63, 1867. (22) Stihlberg, J.; J6nsson. B.; Horvith, C. Anal. Chem. 1992, 64, 31 18. (23) Horst, J.; Hdll, W. H.; Eberle, S.H. React. Polym. 1990, 13, 209. (24) Marinsky, J. A. J. Chromarogr. 1980, 201, 5 .
442
Anaiyticai Chemistry, Vol. 66,No. 4, February 15, 1994
fixed charges, but will be distributed in a layer close to the surface, the diffuse double layer. The distribution of electrolyte ions in the double layer can be estimated by using the G-C theory, where the charges on the stationary-phase surface are considered as evenly smeared out and the electrolyte ions are approximated as point charges. Compared to the Stern model we therefore assume that the Poisson-Boltzmann equation holds from the bulk solution and up to the surface; Le., no inner region close to the surface is assumed (see Figure 1 and the legend). In Figure 2 are shown calculations made from the G-C theory of the counterion concentration as a function of the distance from the planar surface for different concentrations of an 1: 1 electrolyte and for a constant surface charge density on the surface. The concentration of counterions decreases monotonically with the distance from the oppositely charged surface; the profile is a result of the electrostatic attraction of the counterions to the surface, the way the counterions ”shield” each other, and the “smearing out” effect of entropy. The electrolyteco-ions are expelled from the double layer (not shown in Figure 2), and it is furthermore assumed that they do not adsorb onto the chromatographic surface. In thermodynamic terms, the concentration profile is the result of the fact that the electrolyte ions are distributed so that their (electro) chemical potential is equal at every point in the system. The fixed charge groups on the surface create a difference in electrostatic potential between the surface and the ionic bulk solution which is included in the term for the (e1ectro)chemicalpotential. It is thechange in the electrostatic potential as a function of the distance from the surface that determines the concentration profile. In Figure 3 is shown this potential as a function of the distance from the surface calculated from the G-C theory for the salt concentrations and surface charge density used in Figure 2;negative numerical values for the electrostatic potential are obtained because the surface is assumed to be negatively charged. The G-C theory is a general theory for the behavior of point charges with no specific properties or binding to the surface. Different combinations of charged surface groups and counterions can give rise to different kinds of specific interactions; e.g., a proton will bind with different strengths to a surface of carboxylic groups than to a surface with sulfonate groups. It is evident that, for a given combination, it is the counterion concentration at the surfaceand the surface concentration of bound groups that determine the number of counterions specifically bound to the surface. Because of the local character of the specific interaction, a stoichiometric association constant can be assigned to describe the concentration dependence of the specific binding. In this paper is discussed the retention of trace amounts of solute ions in the presence of eluent ions. Electroneutrality requires that the addition of a solute ion to the double layersurface system expells a corresponding eluent ion from the system. The assumption of trace amounts of solute ions means, however, that their presence have a negligible effect on the concentration and potential profiles established by the eluent ions. The distribution of solute ions is therefore determined by the potential profile created by the eluent ions, and changes in retention are consequently due to changes in the electrostatic potential profile caused by changes in concentration and type of eluent ions.
So, the general approach of the theory presented in this paper is to calculate the electrostatic potential profile of the system consisting of the eluent and the charged stationary phase and then introduce trace amounts of solute ions into the system and calculate retention from the distribution of solute ions in the double layer and to the surface. The potential profile is dependent on the electrolyte concentration and on the net surface charge density, i.e., the number of fixed charges minus the number of adsorbed counterions per unit area. The concentration profile of the electrolyte is on the other hand determined by the potential profile, so that there is an interdependence between the concentration and potential profiles, respectively. When there is no specific adsorption of ions, this interdependence is solved in the Gouy-Chapman theory. However, when specific adsorption of counterions occurs, the surface charge density is not constant and the problem must be solved numerically. In this paper the theory is therefore developed at two different levels: first the general case obtained as numerical solutions is presented and then some approximations are introduced which make it possible to derive an analytical expression for the retention equation. Capacity Factor for DistanceDependentInteractions. The foregoing discussion shows that retention is due partly to the accumulation of solute ions in the diffuse layer close to the stationary-phase surface. Our treatment therefore departs from traditional chromatographic retention models which assume that retention is due to the distribution of the solute between two distinct layers. A detailed treatment of the thermodynamic interpretation of the capacity factor for distance-dependent interactions has been presented previouslyzz and only a summary is given here. It can be shown that the zone velocity, u, for an eluite can be written as the integral
Jvc(r)u(r) dr
(3)
U =
$#C).
dr
where c(r) and u(r) are the local concentration and velocity in the z-direction at a point r within the column with the volume V. Within the moving zone, there is an equilibrium distribution of solute as a result of the interaction with the chromatographic surface and the concentration at any given point c(r) is related to the concentration at a reference point, co, by c(r) = co exp(-AG(r)/RT) (4) where AG(r) is the change in free energy upon transferring 1 mol of solute from the reference point to point r. The retention time is the column length, L,divided by the velocity of the zone in the z-direction. Combining eqs 3 and 4 yields for the retention time the expression
40001 3000
I
\SO0
“0
mM
1
2
3 x nm
Flgurr 2. Counterionconcentration as a function of the distance from a surface with charge density -0.1 C/m2 for three different bulk concentrationsofa 1:l electrolyteat 500,100, and 1 mM, respectively, and E, set to 80.
column so that c(r) = co and the integral equation for to is
We now make the assumption that the volume available for the solute can be divided into a stagnant volume, V,, where u(r) = 0 and a mobile part, V, where u(r) # 0. It is furthermore assumed that AG(r) = 0 for the solute at all points within the volume Vm. Applying these assumptions to eq 5 and 6, the following expression for the capacity factor is obtained
For most practical applications the integral in the numerator can be transformed from volume dependence to distance dependence by the relation dV, = A, dx, where A, is the surface area of the stationary phase and x is the distance from the surface. The expression for the capacity factor for distancedependent interactions now becomes l(exp(-AG(x)/RT) - 1) dx
k‘= A,
(8) VO where VO (=V, + Vm) is the column dead volume and the value of x is formally chosen so that
A , t d V , = V, The retention time for a marker molecule, which does not interact with the stationary phase and has the same size as the solute ion, is the holdup time of an inert tracer in the column, to. For such a solute AG(r) = 0 at all points in the
(9)
The integral in the numerator in eq 8 represents a sum of surfaceexcesses of thesoluteover the distance from thesurface as is illustrated in Figure 4. For ion-exchange chromatography this means that we can relate the experimentally measured capacity factor to the distance-dependent distribution of solute. Ana&tlcalChemlshy, Vot. 66,No. 4, Februery 15, 1994
443
x nm 0
1
2
3
density (C/m2) and c, the bulk concentration of the z,-z, electrolyte (mol/m3). The differencein electrostatic potential, #O (V), between the surface and the bulk of the electrolyte as a function of surface charge density is obtained by inverting the sinh(x) function:
The potential in a point located a distance x from a surface is given by 1 + y exp(-Kx) 1 - y exp(-Kx) where J Figure 3. Electrostatic potentialas a function of the distance from a surface under the same conditions as in Figure 2.
y = tanh(F&/4RT)
(13)
and
1 / is~ usually called the Debye length and is an important parameter in describing the properties of the electrical double layer. Examples of the electrostatic potential as a function of the distance from the surface are shown in Figure 3 for three different values of cj. The (e1ectro)chemical potential of an ion j with charge 2, at a distance x from the surface is p , ( ~ )= p;
X
Figure 4. Concentration of a solute ion, CAN, relative to its bulk concentration, co,as a function of the distance from the surface. The shaded area corresponds to the integral in eq 8.
Electrical Double Layer. The fixed charged groups on the stationary-phase surface create an electrostatic potential difference between the surface and the bulk solution. By assuming that the electrolyte ions are point charges, the surface charges are smeared out, and neglecting ion correlation effects, the Gouy-Chapman theory solves the basic Poisson-Boltzmann equation for the interaction between a charged planar surface and a symmetrical z,-z, electrolyte. Equations that apply to asymmetrical electrolytes, e.g., 2: 1 electrolytes, have been derived by Grahame25but are more complicated to use, and since they do not change the theoretical principles presented here, the classical G-C theory will be used throughout this paper. A complete derivation of the G-C theory can be found e l ~ e w h e r e ,so ' ~ only ~ ~ ~a brief discussion of relevant equations is presented. The variation of the surface charge density with the surface potential is described by eq l0,where u is the surface charge (25) Grahame, D. C. J . Chem. Phys. 1953,21, 1054. (26) Hiemenz, P. C. Principlesof Colloid and Surface Chemistry; Marcel Dekker: New York, 1977.
444
Analytical Chemistry, Vol. 66, No. 4, February 15, 1994
+ RT In x,(x) + z ~ ( x )
(15)
where X,(x) is the mole fraction of the ion at point x. In dilute water solutions the mole fraction is c,/55500, where Cj is the ion concentration in moles per cubic meter. Thermodynamics gives that, at equilibrium, the (e1ectro)chemical potential of the ion is equal at all points so that the molar concentration of the ion as a function of the distance to the surface is described by the relation c,(x) = c, exp(-z,F$(x)/RT)
-
where Cj is the bulk concentration of the ion, Le., for x It is important to note that $ ( x ) is the electrostatic potential relative to the bulk solution; i.e., in the present theory there is no need to explicitly consider the activity coefficient for the ions in the bulk. In Figure 2 is shown c@) for a monocharged counterion calculated from the above set of equations for different values of cy Specific binding of the counterions to the surface may partially neutralize the surface charges so that neither u nor $0 is constant as the electrolyte concentration changes. The chromatographic selectivity between different ions of the same charge is due to different values of this specific binding constant in analogy of the description of specificity previously used in ion pair chromatography. The influence of specific binding effects on the electrostatic properties of the system is obtained through a thermodynamic analysis including the effect of the 1512'
~
~~~
~~
(27) Deelder, R. S.; van den Berg, J. H. M. J. Chromatogr. 1981, 218, 327
00.
electrostatic potential on the (e1ectro)chemical potential for the species involved. Consider the equilibrium
+
j(1) s j s (16) where j(1) and j s represent the counterion in the bulk of the mobile phase and specificallybound to the surface, respectively, and s is the part of the surface not occupied by counterions. The condition of equilibrium is Pj + P, = Pj,
-
-
P, = PO,
pjs = PO,,
+ R T In X,
(19)
+ R T In xis+ Z,F$~
(20) where X,and Xjsis the fraction of unoccupied and occupied surface, respectively, Le.:
x,+ xis= 1
(21) To be specific we assume that the maximum number of counterions that can adsorb to the surface equals the total number of fixed charges on the surface, a0 (C/m2), so for this case
X,= a/ao and Xis= uj/uo
(22)
Here ais the surface concentration of free fixed surface charges and aj the surface concentration of fixed charges to which a counterion is bound. By combining eqs 17-21 the association constant for the above equilibrium is obtained
K;, = exp((ccojS- PO, - P’,)/RT) =
Xjs/(X;U, ex~(-zf’$o/RT)) (23) where we note that X, exp(-zjF$o/RT) = Xj(O),i.e., the counterion concentration at the surface. Using eqs 22 and 23 and converting the mole fraction of counterions in the mobile phase to the corresponding volume concentration (m0l/m3), the surface concentration of free charges can be expressed as
= ~ 0 / ( 1+ Kafj ex~(-zF$o/RT)) (24) where K j = K$/55500 in water. From the two equations relating u to $0, eqs 10 and 24, we can determine these two parameters for a given set ofvalues for the association constant and counterion concentration in the electrolyte. Furthermore, from the obtainedvalue for $0, the function $(x) is determined from eqs 12-14. We can now calculate thecapacity factor for trace amounts of a solute ion, i, assuming that its concentration in the mobile phase is so low that the electrostatic surface potential and the surface charge density are determined by the ions in the electrolytic eluent. Under these conditions the arguments leading to eq 24 give for the solute ion that the specific a
(25)
where index i refer to the solute ion. The retention due to accumulation in the electrical double layer is obtained from its relation to the distance-dependent interaction, eqn 8, and recognizing that AG(x) = ziF$(x). The expression for the capacity factor for solute i is obtained by adding the contributions from specific adsorption and accumulation in the double layer:
k’= -
UKai
exP(-ziF$o/RT) -ziF
+
+ RT In xi
(18) where Xj is the mole fraction of counterions in the bulk of the mobile phase. The (e1ectro)chemical potential for the surface and the counterion in the adsorbed state is, respectively, p, = PO,
-niziF = ui = Kaiaciexp(-ziF$,/R7‘)
(17)
The (e1ectro)chemical potential for the counterion in the bulk is obtained from eq 15 by setting x so that $(x) 0 and we obtain Q)
adsorption is described by a linear relation
In conclusion, from eq 26 we see that the electrolyte concentration in the mobile phase changes the capacity factor of a solute through its influence on the double layer, the surface potential, and the surface concentration of free fixed surface charges, u. The equation can be used for any combination of solute and surface charge, i.e., it also describes expulsion of solute ions with the samesign of charge as thechromatographic surface. Because of the integral and the interdependence between $(x), u, and cj eq 26 in the general case can only be evaluated numerically. Analytical Solution for Low Surface Potentials. The final expression for the capacity factor, eq 26, is unwieldy because of the intergral and the complex interdependence between the parameters cj, $ ( x ) , and a. It is therefore desirable to find relations between these, so that eq 26 can be integrated and an analytical equation for the capacity factor is obtained. This is possible for the special case of low surface charge density on the stationary phase and high salt concentration in the eluent, by using the Debye-Huckel approximation. For low surface potentials it can be shown thatz6
To solve the integral in eq 26 we make a series expansion of the exponential so that
Physically this approximation means that the calculated drop in potential with the distance from the surface is more rapid and that the “tail” is cut off for distances larger than 1 / and ~ consequently the upper limit of integration is changed to 1 / ~ . Since retention mainly is due to accumulation in that part of the diffuse double layer which is closest to the surface and to specific binding to the surface, the approximation is valid for many cases. By introduction of these simplifications into eq 26, the integral can be solved and gives the following equation for the capacity factor:
aK,, exp( )),ZiF+O -ziF
(29)
The solution of the linearized Poisson-Boltzmann equation €or a planar charged surface gives a linear relation between Analytical Chemistry, Vol. 66,No. 4, February 15, 1994
445
surface potential and surface charge density $0
log C,(mol/m')
= Q/(KeOer)
(30)
Series expansion of the exponential term in the equation for the isotherm, eq 24, gives
Introducing eq 30 into eq 31 and solving the second-order equation for $0 gives $0
K,,
.-5.
5 103(m3/mol)
-0.054
=
RT(l
+ Ka,cj)7((1 + K a j ) 2 R 2 p- 4Ka,c~~RT(ao/~eoer))'/2 2Ka,cjz,F Kaj > 0 (32)
In this equation, the root with the minus sign is the physically correct one; this is seen from taking the limit Kaj 0 and series expanding the square root, which then gives the correct limit, Le., eq 30. It should also be noted that since zj and QO are of opposite sign, the root is always real. Equation 32 gives the desired relation between the electrostatic surface potential and the association constant of the eluent counterion, its concentration in the mobile phase, and the surface concentration of fixed charges on the stationary phase, ao. Inserting eqs 32 and 30 into eq 29 gives an analytical solution for the capacity factor as a function of electrolyte concentration in the eluent and the association constants for the analyte and the counterion in the eluent.
-0.10-(
i
5 . 1 0 4 1
-+
RESULTS AND DISCUSSION It is well-known from experiments that log k'vs log c, plots are linear with a numerical value of the slope close to the quotient (-z,/zj) for absolute values of zi and zj less than 3. It is also found experimentally that the value of the slope is rather insensitive to the nature of the ions i and j as well as to the stationary-phase properties. A theory for ion-exchange chromatography of small ions shall therefore give numerical results which are consistent with these general experimental findings. Recently it was pointed out by Lederer that the predictions made by the stoichiometric theory are not followed for ions with charge 3 or higher,28and we will therefore only discuss mono- and divalent analytes. According to the proposed-theory, the capacity factor of small ions in ion-exchange chromatography is calculated from eq 26. In this equation the fundamental parameters that may influence the slope of a log k'vs log c, plot are the surface concentration of fixed charges, the association constant, and the charge of both the analyte and the eluent counterion, respectively. The validity of the proposed model is tested by numerical evaluation of eq 26 for different values of these parameters and plotting the result in a log k'vs log cj plot. The numerical calculations have been performed on a PC using the Mathcad 3.0 software. Ion-exchange chromatography of small ions is performed with a great number of different stationary-phase materials, functional groups, and varying degree of exchange capacity. The parameters in eq 26 which are directly related to the (28) Lederer, M . J . Chromatogr. 1988, 452, 265.
446
AnaIjdcalChemistry, Vot. 66,No. 4, February 15, 1994
-0.13
Figure 5. Surface charge density as a function of the logarithm of the concentration of an 1:l electrolyte with the counterlon essoclation constant, K* as a parameter. The results were calculated with uoset to -0.15 C/m2.
properties of the column are the column phase ratio, A,/Vo, and the surface charge density, a, where only the latter can influence the slope of the log k'vs log cj plot. The value of adepends, however, on the concentration of eluent counterion, its association constant to the fixed charged groups on the surface, and the total charge density, ao. This is illustrated in Figure 5 , where a is plotted as a function of log cj in the range 1-1000 mM, for three different values of Kaj,keeping a0 constant to - 0 . 1 5 C/m2. The values for Ka, used in the figure are intentionally choosen to cover a wide range of u values and are therefore pertinent to use in order to illustrate the effect of the different parameters on the log k'vs log cJ plot. The following numerical values are used in all calculations in this paper: e, = 80 (water), A,/Vo = 1.5 X lo8 l / m , and T = 298 K. The dependence of the capacity factor for a monocharged solute ion on the eluent concentration of monocharged counterion, for different association constants of the counterion to the fixed surface charges, is illustrated in Figure 6 as log k'vs log c, plot. The curves are obtained through a numerical evaluation of eq 26 with the value for Kai set to 5 X 10-4 m3/mol for the monocharged solute ion. The values for Kaj and a0 used in these calculations are the same as in Figure 5 showing that the Kaj values represent a large variation in counterion-surface charge interactions. Despite the wide range of eluent counterion concentration used in the calculations (c, = 10-1000 mM), the plots are linear or very nearly so, giving a slope of -1 .Ofor Kaj values in the range 0-5 X 1V m3/mol and a slightly higher slope for higher Kaj values; e.g., -1.15 when Ka, = 5 X m3/mol. In the figure is also shown a reference line with slope -1. The physical interpretation of the result in Figure 6 is straightforward: increasing Kajvalues results in a decrease in the net surfacechargedensity (see Figure 5 ) and thereby also a decrease of the electrostatic surface potential. The effect is both a decrease in solute concentration in the double layer and a decrease in the number of unbound surface charges to which the solute can bind
log k'
..
2
4-
31-
21-
0-
K,l
,108
I
0 5,103
010'' -1
1
1
5.10" -1-
I
I
1.5
2
I 1 2.5 3 log C,(mol/m3)
Figure 6. Theoretlcally calculated log k' vs log c/ plots for a monocharged solute wlth K., = 5 X lo4 (m3/mol) wlth K,, as a parameter. The curves were calculated from eq 26 for a 1:l salt in the eluent and by setting the column parameter uoto -0.15 C/m2 and A,IVo to 1.5 X lo8 (llm). A reference line with slope -1 Is also shown.
specifically. The most important conclusion from Figure 6 is that the slope of the plots agrees with what is found experimentally, Le., around -1, and that it is almost independent of the Kaj value used. In the derivation of eq 26 we assumed that the maximum surface concentration of counterions is limited and equal to the surface concentration of fixed charges. This may not always be true when organic ions are used as an electrolyte in combination with a hydrophobic stationary phase; it is of course possible to introduce other isotherm equations instead of eq 24 in the derivation. There is no attempt in this paper to evaluate or relate the magnitude of Kar or Kaj to actual ionic properties. To illustrate the effect of the charge of the solute ion on a log k'vs log C j plot, eq 26 is evaluated for a solute ion charge of +2, keeping all the other parameters for the eluent ions as well as for the solute ions the same as in Figure 6 . The result is shown in Figure 7, and it is seen that linear relations are found with the slope varying between -1.9 for the lower Kaj value to -1.8 for the highest. In the figure is also drawn a reference line with slope -2. We can conclude that a doublecharged solute ion also give a slope which is nearly independent of the type of counterion used in the eluent. Some caution is however necessary in discussing multiply charged ions; it has been shown that the Poisson-Boltzmann approach may give erroneous results because of the neglect of ion correlation effects.* Separation between ions in ion-exchange chromatography iscaused by a differencein their respective associationconstant, Kai. In Figure 8 is shown log k'vs log cj plots for monocharged solutes with Kai as a parameter and for an eluent of monocharged ions with Kaj = 5 X 10-4 m3/mol. Also in this case, the theoretically calculated log k' vs log cj plots are linear for a wide concentration range of counterions in the eluent, giving slopes close to -1 .O for Kai values between 5 X and 5 X lo4 m3/mol and slightly higher dopes (=-1.15 when Kai = 0) and small nonlinearities for lower Kai values. That monocharged solutes with different specific interactions
'
-2
1
I
I
1.5
2
I
1
2.5 3 log C,(mol/rn3)
Figure 7. Theoretical log k'vs log q plots calculated from eq 26 for a double-charged solute with the remaining parameters as in Flgure 6. A reference line with slope -2 Is also shown. log k'
io3
10''
io5 I 1
I
I
I
I
1.5
2
2.5
3
log C,(mol/rn3)
Flgure 6. Theoretical log k'vs log c/ plots calculated from eq 26 for a monocharged solute with K., as a parameter. The value of K,, Is set to 5 X lo-' m3/moland the remalnlngparameters are as In Flgure 6.
with the charged surface give parallel log k'vs log cj plots is well-known and lends further support for the validity of the theory proposed here. From the presented theory it is clear that the surface charge densityof fixed charges on the stationary phase is a parameter which influences the retention of ions. In Figures 5-8, its value is fixed to -0.15 C/m2, corresponding to 107 A2 per charged surface group. The effect of changing this parameter to-0.05 C/m2 on the theoretical log k'vs log Cj plot is illustrated in Figure 9 where the values of Kat and Kgj are the same as in Figure 8 and, as expected, a decrease in surface charge density does lower the capacity factor. The calculated log k' vs log cj plot is linear in the counterion concentration interval 10-100 mM with a slope close to -1.0; slight nonlinearities appear at higher concentrations. The experimental fact that many different types of stationary phases give linear log k'vs log cj plots with a slope close to -1 for monocharged solutes, when a monocharged counterion is used, can therefore be explained from the suggested theory. Analytical ChemisW, Vol. 66, No. 4, February 15, 1994
447
loa k
log k ’
2-
1-
0104
10’5
-2
‘
1
I
I
I
1
1.5
2
2.5
3
log C,(mol/m’)
Flgure 9. Theoretical log k’vs log cl plots calculated from eq 26 for a monocharged solute and with uoset to -0.05 C/m2; the remaining parameters are as in Figure 8.
In Figure 10 is shown the result of calculations where the approximate analytical solution obtained by inserting eqs 32 and 30 into eq 29 is used, keeping all the pararneters the same as in Figure 9. Comparing these curves to those shown in Figure 9, we can see that there is a good agreement between these two sets of calculations when the counterion concentration in the eluent is in the range 100-1000 mM. When the counterion concentration decreases below 100 mM, the linear relation between surface potential and surface charge density, eq 30, results in too high surface potentials which causes an overestimation of the calculated capacity factor.
CONCLUSION The discussed theory for retention of small ions in ionexchange or ion chromatography is based on the GouyChapman theory for the diffuse double layer complemented with a mass action equation for specific adsorption of both the eluent and solute counterions to the charged surface. Because of the distance dependence in the interaction between the charged surface and the solute ion, the general definition of the capacity factor, eq 26, is used in the calculations. Although the Gouy-Chapman theory formally is restricted to an eluent containing a symmetrical z,-z, salt, it is used in this paper because of its convenience compared to the more general Grahame equations.23 Yet, the agreement with experimental data demonstrates that solutions of the PoissonBoltzmann equation correctly describe the salt concentration dependence of thecapacity factor. A limitation of the PoissonBoltzmann equation is the neglect of ion correlation effects so that the proposed theory may give erroneous results at high concentrations at the surface (Le., >5-10 M) of a double charged electrolyte counterion and for tricharged and higher charged solute ions. An advantage with the Gouy-Chapman theory is that theelectrostatic potentialat thesurfaceis relative to the electrostatic potential in the bulk of the electrolyte, which is set to zero, implying that the activity coefficient of neither the eluent electrolyte nor the solute ion in the bulk solution needs to be considered explicitly in the equations for the capacity factor. 448
Analytical Chemistry, Vol. 68, No. 4, February 15, 1994
-1
L
1
I
I
I
1.5
2
2.5
3
log C,(mol/m3)
Flyre 10. Theoreticallogk’vs log qplots calculatedfrom the analytical solution, eqs 29 and 32, with all parameters as in Flgure 9.
The theoretically calculated curves in Figures 6-9 show that there is a good agreement between the theory and the general experimental observation that a log k‘vs log cj plot is linear or nearly so with a slope value close to the quotient (-zt/zj). It is furthermore seen that the slope only slightly depends on the properties of the solute ion or the stationary phase, which is also in agreement with the experimental facts. The theory can therefore describe the dependence of the capacity factor on the concentration of eluting salt in ionexchange or ion chromatography of small solute ions under conditions of linear elution. Because of its firm physical basis, the proposed theory offers a consistent approach to the analysis of retention data without resorting to the physically invalid stoichiometric models that has mainly been used so far in describing retention in ion-exchange and ion chromatography of small ions.
ACKNOWLEDGMENT I thank Bengt Jonsson and A’kos Bartha for valuable comments during the preparation of the manuscript. GLOSSARY
surface area of the stationary phase, m2/g bulk concentration of solute ion i, mol/m3 bulk concentration of electrolyte ion j , mol/” concentration of k ( k = i or j ) a distance x from the surface, moI/m3 concentration of electrolyte ion j at the surface, Le., x-0
Faraday constant, C/mol Gibbs free energy, J/mol solute ion eluent or electrolyte ion capacity factor association constant for the association of solute ion i to the surface, m3/mol association constant for the association of electrolyte ion j to the surface, m3/mol association constant for the association of electrolyte ion j to thesurface, dimensionless, K,j = 55500Kajin dilute water solution
column length, m surface concentration of bound ions, k = i or j, mol/m2 gas constant, J/(mol K) a point in the column surface site temperature, K holdup time in the column for a nonretained species, s retention time, s mean velocity of a zone in the column, m/s velocity of the mobile phase in point r, m/s column dead volume (=V8+ J"), m3 volume of that part of the column with a moving mobile phase, m3 volumeof that part of the column where the mobilephase is stagnant, i.e., in pores, m3 mole fraction of ion j in the bulk of the mobile phase fraction of surface charges to which a counterion j is bound fraction of surface charges which are unoccupied by a counterion ion charge of solute ion i charge of electrolyte ion j
permittivity of vacuum, F/m dielectric constant of the mobile phase inverse Debye length, l/m standard chemical potential of ion k (k = i or j ) , J/mol electrochemicalpotential of ion k (k = i o r j ) when located a distance x from the surface surface density of free fixed charges on the stationary phase, C/mZ surface density of the total number of fixed charges on the stationary phase, C/mZ surfacedensity of charges on the stationary phaseto which a counterion is bound, Le., (a + aj = ao), C/m2 electrostatic potential of the surface relative to the bulk of the electrolyte, V electrostatic potential at a point a distance x from the surface, V
Received for review March 12, 1993. Accepted November 22, 1993.' Abstract published in Advance ACS Abstracts, January 1, 1994.
Analytical Chemism, Vol. 66,No. 4, February 15, 1994
449
