Interactive Gel Networks I. Treatment of Simple Complexation and Masking Phenomena David H. Freeman Znstitute f o r Materials Research, National Bureau of Standards, Washington, D.C. 20234 Chemical interactions provide an unlimited basis for extending the separation capabilities provided by gel chromatography. This paper concerns the theory of simple acceptor : donor complexation equilibria, which are treated as idealized associative processes. Equations are developed to predict and interpret chromatographic measurements involving a diffusible solute and solvent, and the nondiffusible gel functional group. Systems treated include: I. Inert species, II. Simple solute : gel complexation, 111. Solute : gel complexation with solvent : gel masking, IV. Solute :gel complexation with solvent: solute masking and, finally, consideration of the effects of solute dimerization and solute difunctionality. The results are directly applicable to the experimentally measured solute distri bution coefficient.
AN INTERATIVE GEL NETWORK,as recently described ( I ) , exhibits selectivity of a specific kind in a fluid containing dissolved solutes. The application requirements of such gels are analogous to ion exchange resins where solvent swelling t o form the gel state and solute permeability are needed. Gel selectivity derives from a network structure that offers a n interactive functional group that is able to undergo weak and reversible association with a certain type of solute. The complexation processes are especially clear cut and amenable t o chromatographic treatment, as will be developed here. The complexation prucess is a n acid-base interaction whose origin is traced to the availability of, or affinity for, electron density of n or T type, as in the Lewis acid-base definition, or similar proton donor, acceptor, or amphoteric properties within the BrOnsted definition. These concepts have been widely studied in organic solvents as summarized by Davis ( 2 ) , and by Mulliken and Person (3). The details of acid-base theory need not be explored in depth in order to obtain a helpful relationship between a given complexation process and the corresponding influence upon the chromatographic retention behavior. The origin of chromatographic selectivity requires a permissive factor, solution permeation of the gel phase, followed by the causative chemical interactions among the gel, solute, and solvent functionalities. The present work seeks t o explore the stoichiometry of the elementary processes. Inert Solvent, Inert Gel. When no interaction occurs among solute, solvent, and gel species, partitioning depends only upon the effective molecular size of the solute relative to the steric occultative properties of the swollen gel network. Gel permeability varies oppositely with the molecular volume of the solute ( 4 ) . A partial collapse of the network structure (1) D. H. Freeman and D. P. Enagonio, Nature, 230, 135-36 (1971). (2) M. M. Davis, “Acid-Base Behavior in Aprotic Organic Solvents,” Nut. Bur. Stand. (US.),Moiiogr. 105. (3) R. S. Mulliken and W. B. Person, A m . Rec. Phys. Cliem., 13, 107 (1962). (4) W. B. Smith and A. Kollmansberger, J. Pliys. Chem., 69, 4157 (1965).
by deswelling (5) or the increase of solute size by chemical association (6) promote solute exclusion. Guidelines are needed for predicting solute-solvent-gel interactions. It is useful to review Brgnsted’s discussion (7) of specific ion interactions including the “avidity” of interactions and the uniform or - - interactions which are of a weaker nature. It is assumed now that acceptor (A) and donor (D) combinations of the type A D are interactive while A A or D D combinations are not interactive. Amphoteric species are both A and D so these are assumed t o interact with both A and D. The foregoing implies the further assumption that the A property is non-D and conversely, that D is non-A. Finally, the inert (I) species is inert in any case. The schematic for non-interaction is shown in Table I. Now, solute partitioning between the gel phase and the external liquid phase is subject to the thermodynamic requirement that diffusible species tend to reach constant chemical potential at chemical equilibrium with a multiphase system and, further, that the species thermodynamic activities are well defined. The present treatment will ignore all activity coefficient variations and the molar concentrations of the solute, the solvent, and the gel functional group will represent the respective activities. This parallels the treatment of interactions in liquid solution; a discussion by R. Foster (8) of charge transfer complexation processes is of interest in this regard. Also, it is possible to correct for deviations from ideality (9). The two-phase partitioning equilibrium of solute species L is given by:
++
L*L
+
(1)
where (-) refers to the gel phase or, without it, to the external liquid phase. Then, the equilibrium distribution coefficient for a non-interactive species is defined Do
ME ML
= -
Equation 2 is the model for any non-interactive solute and gel system. The value of Do should have the range 0 < Do < 4 where 4 is the volume fraction of solvent in the gel (IO). To a first approximation, this applies to all gels, including crosslinked polystyrene, the dextrans, ion exchange resins, etc. The inclusion of activity coefficients in Equation 2 leads to the thermodynamic constant, Kth = D o y ~ / y ~AS . stated ( 5 ) J. T. Ayres and C. K. Mann, ANAL.CHEM., 38, 861 (1966). (6) J. G. Hendrickson and J. C. Moore, J . Polym. Sci., 4, 2898 (1963). (7) J. N. Bronsted, J . Amer. Cliem. Soc., 44, 877 (1922). (8) R. Foster, “Organic Charge Transfer Complexes,” Academic Press, New York, N.Y., 1969, Chap. 6. (9) H. A. Benesi and J. H. Hildebrand, J . Amer. Cliem. Soc., 70, 3978 (1948). (10) J. Porath, Pure Appl. Cliem., 6,233 (1963).
ANALYTICAL CHEMISTRY, VOL. 44, NO. 1, JANUARY 1972
117
earlier, in the expression, as in the text to follow, we assume that the activity coefficient ratios are constant. Solute: Gel Interaction. The preceding discussion of gel permeation describes the ability of a gel to absorb certain diffusible solute species. Next is considered the interaction between absorbed solute and the functional group structure in the gel network. The occurrence of so1ute:gel complexation is the specific phenomenon that causes chromatographic affinity. Any difference in the strength of the interaction is tantamount to the origin of interactive selectivity among the solutes. Assume the formation of a 1 :1 so1ute:gel complex. Two symmetrical cases arise as shown in Table 11. The observed 12:poly(styrene/divinylbenzene) interaction in chloroform solvent is a n example ( I ) of simple so1ute:gel interaction. A similar interaction is superimposed upon the steric exclusion mechanism in the results reported by Ayres and Mann (5). Assume that the solute-gel complexation occurs in accordance with the equilibrium
Table I. Conditions for Solute : Gel Non-Interaction Case Solute Solvent Gel 1(0) I I, A or D I, A or D la A I, A or D A l b D I, A o r D D Table 11. Conditions for Solute: Gel Interaction (Without Masking) Case Solute Solvent Gel 2a A I D 2b D I A Table 111. Conditions for Solvent :Gel Interaction (Masking) Solute Solvent Gel Case A D 3a A D A 3b D
(3)
Table IV. Conditions for So1vent:Solute Interaction (Masking) Case Solute Solvent Gel 4a A D D 4b D A A
The molar equilibrium expression is obtained : (4) Equations 3 and 4 d o not depend upon the polarity of the assignment to L or G as acceptor or donor. It is required that L and G be conjugate, meaning that both acceptor and donor properties occur so that the L:G complex is a donor: acceptor or acceptor :donor combination. Assume that no further association occurs. The distribution coefficient for L is then given by
D =
ML
+ MLG
Since SG formation implies depletion of the available interactive network sites, two mass balance expressions are obtained: M O g = Mg M ~ and s M"G = MG M m . The equalities are treated here with the assumed absence of solute, or by extrapolation to that as a limiting condition. Substitution in Equation 8 gives
+
+
ME
(9)
Equations 2, 4, and 5 are combined to give D
=
Do(l
+~MG)
Equation 9 is quadratic in Mm ; the solution is given by (6)
Equation 6 represents the combination of non-interactive (Do)and interactive (kMc) contributions. A similar expression holds for gas chromatography (ZZ). As will be considered later, the expression for D is ready for substitution into the elution volume expression given in Equation 26. The evaluation of Equation 6 can be approached by referring to a n inert ( k = 0) compound with the same molecular volume and, therefore, the same Do as the interactive solute. Solvent: Gel Interaction (Masking). It is given now that solute :gel complexation may occur in the presence of a n interactive solvent. Two major types of behavior may arise. For the first of these, if the solvent and gel are conjugate, solvent :gel masking is expected according to Table 111. Since the net effect is t o reduce the effective concentration of the interactive sites in the gel, the masking process is directly influential toward decreased solute :gel affinity with reference to a non-masking solvent. The 1 :1 association of solvent (S) and gel functionality (G) is described by -
S+G+SG
(7)
The molar equilibrium coefficient is described by
ME
l/*(T - (T2 - ~ M " s M " G ) " ~ )
(10)
whereT = M " g + M o ~ l+/ k g SinceM=
