J. Phys. Chem. 1983, 87, 1045-1062
hence increases the rigidity of the polymer. (ii) In the second explanation, the assumption is made that the compressibility of PVA in the adsorbed state is the same as in solution; i.e., no structural changes occur on adsorption. However, in this model the adsorbed PVA hinders the surface chemical reaction between the surface sites (sulfate and sulfonate end groups) and counterions (H+) due to dipole-ion interactions between alcohol segments of the PVA and the end groups of the latex. Ottewill and Vincent26 made a study of adsorption and wetting behavior of 1-alkanols of polystyrene latex particles. From the adsorption isotherm of 1-butanol on the latex particles, they reached the conclusion that the interaction occurred between hydroxyl groups on the butanol molecule and the hydrophilic sites of the surface. They also concluded that adsorption of alcohol molecules on the charged hydrophilic sites leads to desorption of the counterions from the inner part of the double layer. In Figure 6, the solid line represents the compressibility of the latex (from Figure 3) while the points are the compressibility calculated from data with the PVA-coated latex assuming that the compressibility observed is of the right
1045
order of magnitude to be explained by the elimination of the relaxation compressibility arising from the surface chemical reaction. With the present experimental evidence, it is not possible to distinguish between the two possible explanations given above. Acknowledgment. We are indebted to Dr. C. A. Young, who carried out the adsorption isotherms of various PVA samples on polystyrene latex samples, to Mr. M. J. Castle and Dr. M. C. Wilkinson of the Chemical Defense Establishment, Porton Down, Salisbury, Wiltshire, England, for determination of the glass transition temperature of the latex A and sucrose density centrifuge density measurements on various latices. Latex A was prepared by one of the authors (M.E.G.) in Bristol University, U.K. A sample of this latex was used in the present work with the kind permission of Prof. D. H. Everett. This research has been supported by the Office of Naval Research. Registry No. Polystyrene, 9003-53-6; sodium styrenesulfonate-styrene copolymer, 39307-76-1; poly(viny1 acetate), 9003-20-7.
Unified Theory of Retention and Selectivity in Liquid Chromatography. 2. Reversed-Phase Liquid Chromatography with Chemically Bonded Phases Danlei E. Martire” and Richard E. Boehm oepertment of Chemlstty, Georgetown University, Washlngton, D.C. 20057 (Receivd: September 7, 1982; I n Final Form: October 27, 1982)
A lattice model is developed and used to study retention and selectivity in reversed-phase liquid chromatography (RPLC). The composition and the structure of the stationary phase are analyzed as a function of the chain length of the chemically bonded phase (CBP),the intrinsic chain stiffness, the surface coverage, and the nature of the mobile-phase solvent, for neat solvents and binary mixtures. The solute distribution process (retention mechanism) is investigated. Distribution constants are analyzed as a function of the above variables, the nature of the solute, and the temperature. The general behavior of the model system and the behavior of the special limiting cases of completely collapsed and fully extended CBP chains are considered, the former limiting case being particularly significant.
Introduction Reversed-phase liquid chromatography (RPLC), which may be broadly classified as involving the distribution of nonpolar or moderately polar solute between a polar mobile phase (eluent) and a relatively nonpolar stationary phase, has developed into a highly advantageous and successful separation technique. Indeed, it is estimated that 80-90’70 of modern liquid-chromatographic separations utilize the RPLC mode.’ I t is generally a~knowledged’-~ that, despite the enormous popularity of RPLC, there is a lack of understanding of the details of solute retention and selectivity. Such an understanding is crucial for informed control and manip(1) W. R. Melander and C. Horvith in “High Performance Liquid Chromatography, Advances and Perspectives”, Vol. 2, C. Horvath, Ed., Academic Press, New York, 1980, pp 113-319, and references therein. (2) H. Colin and G . Guiochon, J. Chromatogr., 141, 289 (1977), and references therein. (3) C. H. Lochmtiller and D. R. Wilder, J . Chromatogr. Sci., 17, 575 (1979). (4) R. M. McCormick and B. L. Karger, Anal. Chem., 52,2249 (1980). (5) G. E. Berendsen and L. de Galan, J.Chromatogr., 196,21 (1980).
ulation of separations, which ultimately require a sufficiently detailed description of how retention and selectivity depend on the various mobile- and stationary-phase variables. It is this problem that is addressed in the present paper through a molecular theory based on a lattice model. Before the theory and its application to RPLC systems are described, it is essential for proper perspective to review certain aspects of the current state of practical knowledge (below) and to outline the scope of the present study (next section). The most prevalent type of modern RPLC column packing or chemically bonded phase (CBP) is prepared by reacting uniformly small (5-10 hm in diameter) and porous (typical average pore diameter of -10 nm) silica gel particles with, e.g., dimethyl-n-octadecylchlorosilane or its n-octyl counterpart, generally followed by reaction with trimethylchlorosilane (TMCS) to eliminate accessible, residual silanols (“end capping”).1+2~e-s The bulk of this (6) R. E. Majors in “High Performance Liquid Chromatography, Advances and Perspectives”, Vol. 1, C. Horvath, Ed., Academic Press, New York, 1980, pp 75-111, and references therein.
O022-3654/83/2087-1045$0l.50/O0 1983 American Chemical Society
1046
The Journal of Physical Chemistty, Vol. 87,
No. 6, 1983
Martire and Boehm
surface modification, resulting in a monolayer of n-alkyl a “breathing” surface which adjusts itself to maintain a chains, takes place within the high-surface-area pores, relatively nonpolar character. The picture would become which, to first approximation, may be taken as cylindrically complete by considering the nature (and role) of the interface between the stationary and mobile phases (see ~ h a p e d . The ~ actual surface topology on the molecular level is not known, but the modified surface is believed to later). be rather uniform and nonpolar.116-8 There have been several studies of the sorption by nalkyl CBP’s of commonly used “organic modifiers” in The best available e v i d e n ~ e ’ ~ ~ , indicates ~ ~ ~ J O -that ~ ~ the RPLC e 1 ~ e n t s . ~ J ~ ,The ~ ~ tsalient ~ ~ - ~findings ~ may be minimum site requirement for the CBP n-alkyl chain is qualitatively summarized as follows: the more compatible approximately 0.4-0.5 nm2. This corresponds to 5 0 4 0 % the solvent is with n-alkanes, the more it is sorbed by the surface coverage compared to the area per head group in a compressed fatty acid m~nolayer.’~J~ On this scale, CBP; more compatible solvents displace previously sorbed, less compatible solvents from the CBP. surface coverages ranging from 20% to 60% have been reported and used,2,3,6,7J+13 with typical values being The nature of the solute distribution process in RPLC, i.e., the retention mechanism, has also been a topic of much 40-50% for current commercial CBP’s. study, discussion, and s p e ~ u l a t i o n . ’ - ~ ~ ~The J ~ Jmost ~~~~-~~ It is generally agreed1v9J2J5J6 that, if the surface coverage rigorous treatment to date is the solvophobic theory put is sufficiently high and TMCS end capping is carried out forth by Horvath and co-workers.1J8,28It focuses on the to the degree possible, then the amount of chromatoimportant role of the mobile phase, as was recognized at graphically accessible silanols should be minimal in most situations. Although there are some exceptions to t h i ~ , ’ * ~ , ~ J an ~ earlier stage by Lockeqn Solute distribution is modeled by invoking “solvophobic” interactions, i.e., exclusion of the accessible underlying surface (i.e., at the base of the the less polar solute molecule from the polar mobile phase CBP chains) will henceforth be regarded as relatively inert. with subsequent sorption by the nonpolar stationary phase. The structure and the composition of the stationary (The mobile phase “drives” the solute toward the staphase in the presence of different solvents have been tionary phase, rather than any inherently strong attraction subjects of much study and because between the solute and the stationary phase.) The basic of their direct bearing on the solute distribution process. premise of the theory is reasonable and agreement with What is clear at this point is that the extent to which these experiment is generally good, but, as has been pointed alkyl-modified surfaces can be swollen (Le., permit solvent the description is incomplete in that it does not provide penetration) depends on the n-alkyl chain length and the a sufficiently detailed explanation of the dependence of nature of the solvent. Solvents compatible with n-alkanes solute retention and selectivity on the stationary-phase tend to swell the surface and promote extension of the variable^.^' Moreover, it has been reported3 that, under bonded n-alkyl chains,21while those which are incomcertain chromatographic conditions (very polar mobilepatible tend to promote collapse of the chains upon each phase solvent and relatively long CBP n-alkyl chains), other and toward the underlying surface.22 At the exsolute distribution in RPLC appears to approach that of tremes, the former case (nonpolar solvents) presumably partitioning between two bulk liquid phases, suggesting leads to a stationary phase having a brushlike structure, quasi-liquidlikebehavior of unswollen CBP n-alkyl chains. where the extended chains are translationally restricted In summary, there is ample evidence that stationaryand oriented more or less normal to the surface, and where phase structure and composition depend upon the CBP there is full penetration by the solvent (and potentially full n-alkyl chain length and the nature of the mobile phase, penetration by the solute), and the latter (fairly polar and, perhaps, upon the CBP surface coverage and the solvents) to a quasi-liquidlike layer of recumbent alkyl intrinsic stiffness of the bonded chains.23Accordingly, any chains (still translationally restricted and possibly oriented attempt to model solute distribution in an RPLC system in some manner2,3),where there is negligible solvent penneeds first to address this question, so that a more cometration, but where the possibility of solute penetration plete description of the nature and role of the stationary is not denied.23 The picture that emerges then is that of phase may ultimately be obtained. Also, in addition to considering solute sorption at the interior of the stationary phase, the contribution to solute retention from adsorption (7)R. E.Majors, J . Chromatogr. Sci., 18,488 (1980). (8) We shall not discuss or attempt to model the less popular, so-called at the mobile phase-stationary phase interfa~e’,~J~ should ‘polymeric” CBP, which is formed by reaction with, e.g., dichloro- or be considered for completeness. trichlorosilanes. We shall focus on the monolayer-type CBP (as described in the text), which is better characterized, can be more uniformly swollen, and, in any event, yields more reproducible and efficient columns.1*2*6*7 (9)G. E. Berendsen, K. A. Pikaart, and L. de Galan, J . Liq. Chromaton.. 3. 1437 (1980). (16)F.Riedo, M. Czencz, 0. Liardon, and E. sz. Kovlts, Helu. Chim. Acta, 61,1912 (1978). (11)G. Korosi and E. sz. Kovlts, Colloids Surfaces. 2, 315 (1981). (12)P. Roumeliotis and K. K. Unger, J. Chromutogr., 149,211(1978). (13)H. Englehardt and G. Ahr, Chromatographia, 14, 227 (1981). (14)G. L. Gaines, Ed., “Insoluble Monolayers at the Liquid-Gas Interface”, Interscience, New York, 1965. (15)K. Karch, I. Sebestian, and I. Halasz, J. Chromatogr., 122, 3 (1976). (16)G.E. Berendsen and L. de Galan, J . Liq. Chromatogr., 1, 561 (1978). (17)R. P. W.Scott and P. Kucera, J. Chromatogr., 142,213 (1977). (18)C. Horvith and W. Melander, J. Chromatogr. Sci., 15,393(1977). (19)H. Hemetaberger, P. Behrensmeyer, J. Henning, and H. Ricken, Chromatographia, 12,71 (1979). (20)R. P. W.Scott and C. F. Simpson, J. Chromatogr., 197,ll (1980). (21)It is even possible to cocrystallize such surfaces with long-chain n-alkane solvents.’OJ1 (22)This may be explained by a ‘eolvophobic” effed;’J Le., the chains tend to aggregate in order to minimize the surface area in contact with a hostile solvent.
(23)We note that this picture, which is necessarily tentative in the absence of specific structural information, is consistent with, e.g., the following: (a) computer simulations of semiflexible chains attached to a planar surface, which indicate that the extended-chain conformation is favored by a good solvent, high surface coverage, and intrinsic chain Faraday Trans. 2,74, stiffness (A. T. Clark and M. Lal, J. Chem. SOC., 1857 (1978));(b) theoretical results on macromolecules grafted to an inert surface, which describe the dependence of conformation on surface coverage and solvent (P. G. de Gennes, Macromolecules, 13,1069 (1980),and references therein). (24)R. M. McCormick and B. L. Karaer, J . Chromatogr., 199,259 (1980). (25) R. P. W. Scott and C. F. Simpson, Symp. Faraday Soc., 15,69 (1980). (26)E.H. Slaats, W. Markovski, J. Fekete, and H. Poppe, J . Chromatogr., 207,299 (1981). (27)D.C. Locke, J. Chromatogr. Sci., 12,433 (1974). (28)C. Horvdth, W.Melander, and I. Molnar, J. Chromatogr., 125,129 (1976). (29)E. J. Kikta, Jr., and E. Grushka, Anal. Chem., 48,1098 (1976). (30)Currently held views on RPLC retention mechanisms are nicely summarized elsewhere.’-3 (31)It should be noted, however, that the theory is applicable to various modes of RPLC, such as ion pairing.’
Theory of Liquid Chromatography
Scope The present study is an advancement of previous ~ o r kwhich ~ ~ was * ~directed ~ toward liquid-solid (adsorption) chromatography, primarily with unmodified surfaces and in the normal-phase (NP) mode. As in our earlier treatment, the present theory will be presented in a generalized form, so that it may be applied to both NPLC and RPLC, although we concentrate here on ita application to the latter. Our aim in this series of studies is to develop a unified theory, based on a simple lattice model, which will provide a tractable and more complete basis for molecular-levelinterpretation and semiquantitative prediction of solute retention and selectivity in a variety of liquidchromatographic systems. A parallel aim is to obtain relations which reveal more clearly the functional dependence of retention and selectivity on the different mobileand stationary-phase variables. At the very least, this should provide guidance in determining optimum conditions for separations and in devising useful semiempirical procedure^.^^^^^ In common with our previous investigation, the present treatment examines the competitive equilibrium at the molecular level among solvent and solute molecules distributed between generally nonideal mobile and stationary phases, all components being nonelectrolyte^.^^ Both entropy and interaction energy effects are included. The modeling draws, in a major way, from related lattice statistics developed by us and others to treat liquid-crystalline sy~te1119,3’~~ fatty acid mono1ayers,43*u and the amorphous region in diblock copolymers.46 The analysis is simplified by assuming an inert underlying surface (see above) and neglecting the effects of surface curvature and external pressure.& The chemically homogeneous CBP chains are taken to be semiflexible and their sites of attachment randomly distributed on a planar surface. Chemically homogeneous and heterogeneous solvent molecules of different size are considered, as well as neat solvents and binary solvent mixtures?’ Chemically homogeneous and heterogeneous solutes of different size, shape, and flexibility are treated. Considering, in order, pure solvents and then solvent mixtures, we analyze the structure and composition of the stationary phase as a function of CBP chain length, chain “stiffness”, surface coverage, and the nature of the mobile-phase solvent. Subsequently, solute distribution (32)R. E.Boehm and D. E. Martire, J.Phys. Chem., 84,3620(1980). (33)D. E. Martire and R. E. Boehm, J. L ~ QChromatogr., . 3, 753 (1980). (34)H. Colin, G. Guiochon, and P. Jandera, Chromatographia, 15,133 (19821,and references therein. (35)P. Jandera, H.Colin, and G. Guiochon, Anal. Chem., 54, 435 (1982),and references therein. (36)Ion pairing, ligand exchange, and gel (exclusion) liquid chromatography are not within the scope of the present treatment. (37)E. A. DiMarzio, J. Chem. Phys., 35, 658 (1961). (38)M. A. Cotter and D. E. Martire, Mol. Cryst. L ~ QCryst., . 7,295 (1969). . 13,193 (1971). (39)R. Alben, Mol. Cryst. L ~ QCryst., (40)H. T.Peterson, D. E. Martire, and M. A. Cotter, J.Chem. Phys., 61, 3547 (1974). (41)G. I. Agren and D. E. Martire, J. Phys. (Orsay, FrJ, 36,CI-141 (1975). (42)F.Dowell and D. E. Martire, J. Chem. Phys., 69,2332 (1978);F. Dowell, ibid., 69,4012 (1978). (43)K. Motomura and R. Matuura, J. Colloid Interface Sci., 29,617 (1969). (44)R. E.Boehm and D. E. Martire, Mol. Phys., 43,351 (1981),and references therein. (45)E. A. DiMarzio, C. M. Guttman, and J. D. Hoffman, Macromolecules, 13, 1194 (1980). (46)An analysis in ref 1indicates that external pressure should have only a minor effect on equilibrium properties in typical RPLC systems. (47)Here we model only isochratic elution, Le., maintenance of constant mobile-phase solvent composition throughout the column.
The Journal of Physical Chemktry, Vol. 87, No. 6, 1983
1047
constants are determined, and their dependence on the above-mentioned variables, on the nature of the solute, and on temperature is studied. The general behavior of the model system and the behavior for the special limiting cases of completely collapsed and fully extended CBP chains are considered. Where possible, model results are compared with experimental results or trends.@ In particular, the dependence of retention and selectivity on on CBP surface solute structure and type,1-3,5113,15,16,29,4*~ coverage and chain length,1-3~5*’3J5J6Jg~4*51~55 and on mobile-phase c o m p ~ s i t i o n ’ ~are ~ ~examined. ~ + ~ ~ ~We ~ ~also ~ consider the related problems of determination of mobile-phase v o l ~ m and e ~ the~validity ~ ~ of~ the ~ RPLC ~ ~ method for estimating octanol-water partition coefficients. In the comparison of experiment and theory, it must be kept in mind that the measurable quantity is the capacity factor k’ (ratio of the number of moles of solute in the stationary phase to that in the mobile phase), which is related to the thermodynamically significant and theoretically derived distribution constant K (the corresponding ratio of solute concentrations in the two phases, on some scale) through the phase ratio a: k’ = Ck: = CKx(CSx/V,)= CKX@., (1) X
X
X
where x denotes the mode of solute retention or distribution (interior of the stationary phase, mobile-stationary phase interface, etc.), C , is the stationary-phase capacity for mode x (volume V,, interfacial area A,, etc.), and V , is the mobile-phase v0lume.3~There is eviden~e’,~J~” that, with increasing CBP chain length, up to -1/3 of the original pore space (i.e., with unmodified silica) becomes occupied or inaccessible, leading to increased V,, and reduced A, and v,. Also, the nature of the eluent and the surface coverage apparently affect ax(see later). Therefore, comparisons of It’ values with theoretically derived Rs strictly need to take into account possible changes in axwith the RPLC system variables.
Theory Here we develop a simple cubic lattice model to obtain the relevant equations for treating solute retention and selectivity in RPLC systems. First, we model the stationary phase, which is taken to consist of chemically homogeneous, semiflexible chains permanently bonded to (48)While there is not a dearth of RPLC data, more quantitative comparison between theory and experiment is often difficult, because (a) many of the earlier experiments were conducted on polymeric CBP’s,8 (b) experimental variables are sometimes not controlled systematically (e.g., varying CBP chain length at fiied surface coverage, and vice versa), and (c) the experimental results are usually not presented in sufficient detail. (49)H. Hemetsberger, M. Maasfeld, and H. Ricken, Chromatographia, 9,303 (1976). (50)H.Hemetaberger, M. Kellermann, and H. Ricken, Chromatographia, 10,726 (1977). (51)M.C. Hennion, C. Pickard, and M. Caude, J. Chromatogr., 166, 21 (1978). (52)H. Colin, N.Ward, and G. Guiochon, J. Chromatogr., 149,169 (1978). (53)S. A. Wise, W. J. Bonnett, F. R. Guenther, and W. E. May, J. Chromatogr. Sci., 19,457(1981). (54)W. R. Melander and C. Horvith, Chromatographia, 15,M (1982). (55)K. K.Unger, N. Becker, and P. Roumeliotis, J. Chromatogr., 125, 115 (1976). (56) P. J. Schoenmakers, H. A. H. Billiet, R. Tijssen, and L. de Galan, J. Chromatogr., 149,519 (1978). (57)P. J. Schoenmakers, H. A. H. Billiet, and L. de Galan, J . Chromatogr., 218,216 (1981). (58)P. J. Schoenmakers, H. A. H. Billiet, and L. de Galan, J. Chromatogr., 185,179 (1979). (59)J. H.Knox and E. sz.KovHts, S y m p . Faraday SOC.,15,177(1980). (60)F. Riedo and E. sz. Kovita, J. Chromatogr., 239, 1 (1982). (61)W. K.Al-Thamir, J. H. Purnell, C. A. Wellington, and R. J. Laub, J. Chromatogr., 173, 388 (1979). (62)G. E. Berendsen, P. J. Schoenmakers, L. de Galan, G. Vigh, Z. Varga-Puchony, and J. InczBdy, J . Liq. Chromatogr., 3, 1669 (1980).
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a solid surface plus those solvent and solute molecules residing in the same layers as the chain segments, all species being nonelectrolytic. The N , randomly placed chains each contain r, segments and each is attached at one end to a planar surface having a total of Ma potential attachment sites. The fractional surface coverage is given by t = N,/M,. For tractability we assume a uniform segment density, i.e., that each of the stationary-phase layers parallel to the surface contains the same number of chain segments.63 The solvent and solute molecules are taken to be either completely flexible or completely rigid r-mers. (Later, we shall also consider globular solute molecules, i.e., cubes in this lattice model.) In general, we allow for a binary solvent mixture containing N 1 molecules of component 1 and N 2 molecules of component 2, each containing rl and r2 segments, respectively, while the N3 solute molecules each contain r3 segments or structural units. (Eventually, we shall treat the chromatographically significant limit of solute infinite dilution, i.e., N 3 0.) Accordingly, the lattice of the stationary phase consists of M unit cells, where M = & N k r k k, = c, 1, 2, 3. If we designate the “film thickness” of, or number of layers in, the stationary phase by 7 (equilibrium value to be determined), then clearly M = 7M,, and the individual volume fractions are given by 4j = N j r j / M ,j = c, 1, 2, 3. The physical bounds on 7 are Er, I7 I rc,where the lower limit (7min)pertains to a completely “collapsed” bondedchain structure and the upper limit (7”) to a fully extended or brushlike one. The corresponding bounds on 4, are 1 2 4c 1 t, where, in general 4, = N,r,/M = tr,/7 = t u (2)
-
-
where u-l = r / r c = 717,- is the reduced film thickness (t I u-l 5 1). It follows that, as N3 0, then 0 I + 42 I 1 - E . As will be shown, the equilibrium bonded-chain structure (and segment density, 4J and, hence, the amount of admitted solvent (G1 + 42)depend primarily on E, the intrinsic stiffness of the bonded chains and the nature of the solvent. For the present, we shall disregard the interfacial layer between the stationary and mobile phases. We proceed by determining the total Helmholtz free energy, A , of the stationary phase in three steps: (a) the configurational free energy (Ao),(b) the packing entropy (SJ,. and (c) utilizing the Bragg-Williams approximati~n,~~ the interaction energy (Ein),where
Martire and Boehm
the fraction of bonds in each of the two mutually orthogonal directions parallel to the planar surface and t,, denotes the fraction of bonds perpendicular to the surface. As usual, fi = (kT)-l, where k is the Boltzmann constant. Since the chain in this model is restricted to starting at the surface and pressing forward until it reaches exactly the 7-th layer,43*45 eq 4 gives, using Stirling’s approximation In Wb,, = -rc[aacIn
+ a b c In (abceBEb/4)]
aac
(5)
where aac= abc
t,, = 7 / r , = u-l
= 1 - a,, = 1 - 7 / r c = 1 - u-l
(64 (6b)
and where aacis the fraction of bonds perpendicular to the surface and ab the fraction parallel to the surface. Clearly, the corresponding bounds on aacand ab, are E I a,, I 1 and 1 - t 2 a b , 2 0. The configurational free energy for N , bonded chains is then (7) @Ao,,= -In W0,,= -N, In Wb,, which, when combined with eq 5 and 6, yields
The total number of configurations Wo,ifor the solute and solvent components is approximated
Wo,i
(giNi)!/II(git,iNJ!
(9)
U
where u = x , y, z (the directions), i = 1 , 2 , 3 (the components), and Cut,. = 1. For rigid-rod solvents or solutesm1 gi = 1, while for (completely) flexible-chain solvents or solutes42gi = ri - 1. Letting aai = tZibe the fraction of molecules (rigid rods) or bonds (flexible chains) perpendicular to the surface, and ‘Ybi = 1 - aai be the fraction parallel to the surface, and using Stirling’s approximation, eq 9 gives for the configurational entropy
Combining eq 8 and 10, and recalling eq 2, we obtain the total configurational Helmholtz free energy: -PA0 =
The configurational Helmholtz free energy of the bonded chains Ao,,is determined from a modification of the Motomura-Matuura In this modification we introduce a Boltzmann factor involving a “bond-bending” energy Eb$2where Eb = 0 for a completely flexible chain and Eb= m for a completely rigid one. If the bond between the surface and the first segment of each chain is taken to be perpendicular to the surface, then the Boltzmann term reflects the intrinsic stiffness of the chain and the surface bias.42 Ebthen represents the energy required for a bond to bend in a direction parallel to the surface. Accordingly, the weighted configurations W,,, of a bonded semiflexible chain are W’o,,= ( r , ! / ~ [ t u c r , ] ! ) ( 2 e - ~ Et y ~~ ))~(c ~ ~ ~ ( 4 )
We note that eq 11 describes, in general, an ordered or anisotropic system, which would become isotropic only when 7 / r , = a,, = aai = ‘I3(see later). The packing entropy (S,) of a generally anisotropic system is familiar from previous studies on liquid-crystalline systems: 37-42
+
U
where u = x , y , 2, Cutuc= 1,and where t,, and t,, denote where (63) Determination of the segment density profile, as a function of
surface coverage, chain length, chain flexibility, and solvent, is a rather complex problem.23For tractability, we have little choice but to assume a uniform segment density.
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983 1049
Theory of Liquid Chromatography
Ppim
= In (Oi/ri) - (2ri + 1) In Qo - gi In 3 + l>m
The attractive interaction energy contribution (EJ,in the Bragg-Williams approximation, is also familiar: 32
PEi,/M = @2[(1/2)c4?wjj + I
4j4kwjkl
(14)
Ivk P k
where j , k = c, 1, 2, 3; z = 6 is the coordination number of a simple cubic lattice and w is a segmental pair interaction energy (negative for attractive interactions). Accordingly, substitution of eq 11,12, and 14 into eq 3 gives the total Helmholtz free energy in the form OA/M@ @ A / M = (rhs eq 11) + (rhs eq 12) + (rhs eq 14) (15) (rhs = right hand side). The equilibrium value of r (hence u ) is determined by first minimizing PA with respect to r37-42 [d(PA)/d71Tflj,aar = 0 (16) where j = c, 1,2,3; 1 = 1,2, 3; aal= 1- abl. Recalling eq 2, eq 15 and 16 give 1 (1- e) In Q, (2 e) In Qb e In ((u - 1)/4) + e&, + PZ[(l/2)c$?wjj + 4j4kwjkl = 0 (17)
+
+ +
+
j,k j>k
J
where j , k = c, 1, 2, 3. The chemical potential of component i (i = 1, 2, or 3) in the stationary phase p; is obtained in the usual manner:
[d(PA)/ d h ’ i l ~ , ~ ; , a j= PP?
(18)
where j = c, 1, 2, 3 and j ’ denotes the remaining j - 1 variables (i.e., excluding i). Accordingly @pis =
In (4i/ri) + ri + (ri - l)[aai In Qa + a b i In Qb] + gi[a& a a i + a b i In ( a b i / 2 ) ] + ri@zs(bjwij(19) J
A more convenient form of Pp; is obtained by multiplying eq 17 by ri and subtracting the result from the right-hand side of eq 19: Pp;
= In (4i/ri) + [(Ti - l)a&- ri(l - 41 In Q, + [(ri- 1 ) a b i - ri(2 e)] In Qb - rie In {(u - 1)/4} riePEb+ gi[a&In aai+ (Ybi In (“bi/2)] + $j$kXjk] + riz@wii/2 (20) ri[C 4jxij -
+
j#i
j,k j>k
where i = 1, 2, or 3; j , k = c, 1, 2, 3 and where the interaction parameters xij and Xjk are related to the respective segmental interchange energies (Aw); e.g. (21) xij = ~ P A w i = j ~ P [ w i-j (wii + wjj)/2] The chemical potential of component i (i = 1, 2, or 3) in the mobile phase pim is simply obtained by setting e = 0 in eq 20 and $c = 0 in eq 13. (Component c is not present in the mobile phase.) If we then designate the mobilephase volume fractions by 01, we have, for an isotropic mixture (a,l = a b l = 1/3) (64) Strictly, eq 8 (hence, eq 11) and 12 are valid only for sufficiently large re. In the derivation of the former equation, Stirling’s approximation is used for the individual bonded chain. In the derivation of the latter, it is implicitly assumed that molecules of type i are sufficiently small relative to r,, 80 that they are wholly contained, in any orientation, within the T layers. Also, eq 14 represents a mean-field approximation (volume fraction averaging) where the anisotropy of attractive interactions is n e g l e ~ t e d . ~ * ~ ‘ ~
where 1, m = 1, 2, 3, and where
Qo = 1 - ( 1 / 3 ) C W l - U/rl
(23)
1
Subtracting eq 22 from eq 20, we obtain the basic equations describing equilibrium between the two phases (i = 1, 2, or 3):
Phi” - pim) =
In ($i/6i) + [(ri - l)ad - ri(l - e)] In Q, + [(ri - 1 ) a b i ri(2 + e)] In Q b + [2ri + 11 In Qo + gi[a&In 3a,i + (Yhi In (3ahi/2)] - rie In ( ( u - 1)/41 - riePEb + l>m
(24)
We note that, although x has been formulated in terms of “energetics”, it is strictly a free energy quantity since it incorporate differences in the internal partition functions of the species.32 We now consider the limit of infinite dilution of component 3 (the solute), i.e., 43 0 + 41 + 4 2 l),6 3 0 (61 O2 l),and 43/63 constant. Recalling that 4c = eu (eq 2), eq 24 yields
-
P b l S-
+
--
-
-
=
- 41 In Qa + In (41/6J+ [(rl - 1 ) a a l [(rl - 1 ) a b l - r1(2 e)] In Qb + [2r1 + 11 In 8 0 + g1[aalIn 3aa1+ a b 1 In (3ab1/2)1 - rle In {(u - 1)/41 rlePEb + rl[42(1- 4Jx12 + (eu) X (1- b1)Xlc- (eU)42X2c - e?xl2i = 0 (25)
+
PG2s - PZm) =
In (42/62) + [(r2- 1)aaP - r2(1 - 41 In Qa + [(r2- 1)(Yb2 - r2(2 + e)] In Qb + [2r2+ 11 In QO + g2[ad In 3aa2 + a b 2 In (3ab2/2)1 r2e In ((u - 1)/4} - r2ePEb + r2[41(1 - 4Jx1-2 + (tu) X (1- 4 2 ) x z C - ( ~ u ) ~ i x-i c4 2 ~ i z 1= 0 (26)
In K3(1+2)= In (43/’93) = [r3(1 - 4 - (r3- 1)aaal In Qa + [r3(2+ e) - (r3- l ) a b 3 ] In Qb - [2r3 + 11 In Qo g3[aa3In 3aa3+ a b 3 In (3”b3/2)1 + r3e In Ku - 1)/4) + r3ePEb - r3[41X13 + 4 2 x 2 3 + (eu)xac - 4142x12 - (eu)dlXlc - (eu)42x2cl +
+ 82x23
(27) where K3(1+2)is the infinite-dilution solute (3) distribution constant with a binary mobile phase (1+2), and where, from eq 2, 13, and 23 Q, = 1 - (ac + 41aalR1 + 4 2 a a 2 R 2 ) (284 1 - ( 1 / 2 ) ( C ( U - l)Rc + 41abiRi + 42ab2R-2) (28b) Qb r3[e1x13
Qo = 1 - (1/3)(61R1 + 62Rz)
-ele2x121
(284
where R (r - l)/r. To apply eq 25-28, one needs to determine the equilibrium values of a& = 1- a b i (i = 1, 2, 3). This is done through a minimization condition of the same form as eq 16: [ W A )/e a a i l T J V , , ~ ~=’ 0 (29) where j = c, 1, 2, 3 and j ’ denotes the remaining j - 1 variables (i.e., excluding i). Recalling eq 2, eq 15 and 29 give42
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The Journal of Physical Chemistty, Vol. 87, No. 6, 1983
(ri
- 1) In (Qa/Qb)
g, ln (2"ai/"bi)
=0
(30)
where, again, gi = 1 for rigid-rod molecules and gi = ri 1 for flexible-chain ones. At this point, we let solvent component 2 be a flexible-chain rz-mer, so that g, = r2 - 1in eq 26 and 30. Also, we scale our system by taking solvent component 1 to be a monomer (rl = l), so that eq 25 and 28 simplify to (with "a1 = ab1/2 = '/3) = In (4JO1) - (1 - t) In Qa (2 + e) In Qb 3 In Qo - t In {(U - 1)/4) - t@& + [ 4 2 U - 41)XlZ + (cu)(l - 41)Xlc - ( ~ U h X 2 C- ~ Z 2 X 1 2 1= 0 (31)
@(PI* - plm)
+
+ 42aaZR2) = 1 - (1/2)(E(u 4z"bzRz) Qa
Qb
= 1- (cRc
80= 1 - (1/3)(@&)
(324 (32b) (32c)
Equations 26, 27, and 30-32, which constitute our set of fundamental equations, are not yet in convenient enough form for analysis. Let us first obtain the relevant equations for neat-solvent systems and then for binary mixed-solvent systems. With a pure monomer soluent (& = 0, r#q = 1- t u , O2 = 0), eq 31 and 32 give @(PIS- Plm) = In [ l - a]- (1 - E ) In [ l - tR,] - (2 1)RC/2]- t In [(U - 1)/4] - t @ E b
+
t)
In [l - t ( u = 0 (33)
(eU)2xlc
Therefore, selecting values for t, R,, @Eb,and xlc,one may determine the equilibrium value of u (hence, the "structure" and composition of the stationary phase). Also, eq 27 becomes In K3(1)= [r3(l - t) - (r3 - l)aa3] In [ l - tR,] + [r3(2 + t) - (r3 - l)ab3] 1n [1 - t ( U - 1)Rc/2] g3[",3 In 3 a a 3 + a b 3 In (3ab3/2)1 + rat In [ ( u - 1)/4] + r3t@Eb+ r3[(tu)(x13 - x3J + (1 - ~~)(~~)x1,1 (34) Therefore, with further selection of solute structure, r3and x13- xk, and with ad = 1- ab3from eq 30, the distribution constant may be evaluated. A second useful form for K3(1),which directly invokes the competitive equilibrium between components 1 and 332 (see later), is obtained by multiplying eq 33 by r3 and adding the result to eq 34: In K3(1)= r3 In (1 - tu) - (r3- l)aa3In [ l - ER,] (r3- l)ab3In [ l - 4 u - 1)Rc/21 - g3[aa3 In 3aa3 + a b 3 In (3ab3/2)1 + r3[(EU)(X13 - x 3 c + Xlc)] (35a) By induction,66we extend eq 35a to rigid cubic solutes:
In
K3(1) = r3 In (1 - eu) - [(r3- r32/3)/3] In [I - eR,] - [2(r3r32/3)/31In [1- 4 u - 1)Rc/21 + r3[(tU)(X13 - xSc + xdl
(35b)
where r3 = d33,d3 being the side dimension of the cubic solute molecule. With a pure r2-mer,flexible-chain solvent (g2 = r2 - 1, +1 = 0, 4, = 1 - tu, O1 = 0), eq 26 gives (65) The r3 segments of a rigid cubic solute must be isotropically Also, the correspondin distributed, so that aa3 = 4 2 = "interference"factor in the packing entropy for such solutes is r3 - r32fi instead of r3 - 141 (second and third terms on right-hand side of eq 35), where r2l3 is the area of a face of the cube.
In (1- t u ) + [(r2- l)aa2- r2(l- E)] X b 2 - r2(2 + 41 In Qb + [2r2 + 11 In Qo + (r2 - l)[aa2 In 3aa2 + a b 2 In (3abz/2)1 r2t In [ ( u - 11/41 - r&Eb + r2(tu)2~zc= 0 (36)
@(pz8- kZm)=
In
Qa
+ [ ( r 2- h
where Q,, Qb, and Qo are given by eq 32. Thus, selecting values for E, R,, @ E b , r2, and x2,, one may determine the equilibrium value of u from eq 30 and 36. If we here limit our treatment to flexible-chain solutes (g3 = r3 - l),eq 27 becomes In K3(2)= [r3(1- 4 - (r3 - 1)aa31 In Qa + [r3(2 + t) - (r3- l ) a b 3 ] In Qb - [2r3 + 11 In Qo - (r31 ) [ a a 3 In 3~ + a b 3 In (3ab3/2)1 + r3t In [ ( u - 1)/4] + r3t@Eb+ r3[(tu)(x23 - x3,) + (cu)(l - ~~)x~,l (37) where, from eq 30, ad = cya3. Also, with further selection of r3 and ~ 2 -3 x3,, the distribution constant may be evaluated. Another useful form for K3(2) may be obtained by multiplying eq 36 by r3/r2and adding the result to eq 37: In &(2) = (r3/r2) In (1 - EU) - [(r3/r2) - 1 l a a 3 1n Qa 80- [(r3/r2) Qb + [ ( r 3 / r 2 ) - 11 [ ( r 3 / r ~-) l I a b 3 1I[cya3 In 3aa3 + a b 3 In (3ab3/2)1 + r3[(tu)(x23 - x3, + XZ,)~ (38) where we have used a,, = aa3. Finally, we obtain the relevant equations for a binary mixed-solvent system consisting of, in general, a monomer (rl = 1) plus a flexible-chain r2-mer. Multiplying eq 31 by r2 and subtracting the result from eq 26 (with g, = r2- 1) yields In
+ ( ~ -2 l ) [ a a 2 In (3"azQa) + r2[(41 - 4z)Xiz+ (EU)(Xzc - Xi,)] = In (3"b&b/2)1 In (e2/slr2) + r2(B1- 02)x12+ (r2 - 1) In Qo (39)
(42/+ir2)
ab2
where, again, Q,, Qb,and Qoare given by eq 32, and where 41 dz = 1- t u and 8, O2 = 1. Accordingly, from eq 30 and 39, u and the stationary-phase composition may be determined. If solvent component 2 were also a monomer ( r , = I), eq 39 would simplify to 1n (42/41) + [h- &)x12 + (tv)(xzc - xdl = In (8,/e1) + (4 - 82)x12 (40)
+
+
Both eq 39 and 40 require assignment of XIZand xzC- xlc for evaluation. The general expression for the distribution constant with a binary solvent system is given by eq 27. We again restrict our analysis to flexible-chain solutes (g3 = r3 - 1). Two other useful equations may be derived. Multiplying eq 31 by r3 and adding the result to eq 27 yields In &(1+2) = r3 In (41/4) - (r3 - l ) [ a a 3 ln Qa + ab3
QbI
- (r3 - 1)["a3 In 3 a a 3 + a b 3 In (3ab3/2)1 + r3(4 - 42)(x23 - x13 - x12) + r3[(tu)(x13 - xsC + XI,)]
(r3 - 1) In
80
(41) which reduces to eq 35a when O2 = 0. Multiplying eq 26 by r3/r2and adding the result to eq 27, we derive ln K3(1+2)= ( ~ ln /(&/e,) 4 - [ ( r 3 / b ) - 11 x [ a a 3 ln Qa + a b 3 ln Qbl + [ ( r 3 / r z )- 11 In 80[(r3/r2)- 1 I [ a a 3 In 3 a a 3 + "b3 ln (3ab3/2)1 + - 4 1 k 1 3 - x23 - x12) + r3[(tU)(X23 - x3, + XZAI (42)
which reduces to eq 38 when 8, = 0.
The Journal of Physlcal Chemistty, Vol. 87, No. 6, 1983
Theory of LiquM Chromatography
Equations for Limiting Cases Here we derive the relevant equations for the limiting cases discussed earlier, viz., completely collapsed CBP 6 ) and fully extended CBP chains (case I v-l = a,, chains (case I1 v-l = a,, 1). It will be argued later that, with n-alkyl CBP's, the former case is particularly significant in that a fairly compressed structure is apparently induced by the most commonly used RPLC (polar) solvents and solvent mixtures. Also, increasing interest should develop in the latter case, as more rigid CBP's are synthesized and sed.^^^' We proceed as in the previous section: neat monomer solvents, neat r-mer solvents, and mixtures thereof. Examination of eq 33 reveals that r$l = 1- tu becomes small (case I, monomer small as xlc becomes large and @Eb solvent). A t 6 = 1/2 and R, = 1,for example, r$l = 0.01 for @Et,= 0, xic = 2.85; for @ E b = 2, xic = 3.85; and, in general, for xlc - (@Eb/2) = 2.85. Formally, as r$l 0 (4, = cv l ) , eq 34 gives for rigid-rod solutes (g3 = 1)and flexiblechain solutes (g3 = r3 - 1) In K3(1) = [r3(l- 4 - (r3- 1 ) ( ~In , ~Q,] + [r3(2 + e) (r3 - l ) a b 3 l In Qb - g3[",3 In 3% + a b 3 In (3"b3/2)1 + r36 In [(I - 4 / 4 4 + r3tPEb + r3[x13- x d (43) where Q, = 1 - cR, Qb = 1 - (1 - €)R,/2 (44) and where aa3= 1 - a b 3 may be determined from eq 30. For cubic solutes, we obtain from eq 33 and 35b In K3(1)= [r3(l- c) - (r3- r:I3)/3] In Q, + [r3(2 + e) - 2(r3- r32/3)/3]In Qb + r3t In [(I - 4 / 4 ~ 1+ r34Eb + r3[x13- XJ (45) As this limit is approached, there is virtual exclusion of solvent, and the solute distribution process approaches that of classical partitioning between two pure bulk phases, where one phase (stationary) is translationally restricted and anisotropic (see later). On the other hand, r$l approaches its maximum value (1- E ) when xlc is sufficiently small and @Eb sufficiently large (case 11, monomer solvent). At E = 1/2 and R, = 1, for example, it is determined from eq 33 that r$l comes to within 1% of its maximum value for xlc = 0, @Eb = 5.30; for xlc = 4, @Eb = 7.30; and, in general, for @Eb - (x1,/2) = 5.30. Formally, as $ J ~ 1- E (6, e), eq 35a gives for rigid-rod and flexible-chain solutes In K3(1) = r3 In (1- E ) - (r3- l)ag In (1 - tR,) g3["a3 In 3%3 + "b3 In (3"b3/2)1 + r3'[X13 - x 3 c + x l c l (46) where Q, = 1- &, and Qb = 1. For cubic solutes, eq 35b simplifies to In K3(1) = r3 In (1- E) - [(r3- r32/3)/3]In (1- ER,) +
--
-
-
-
-
r3t[X13
- x3c + x l c l (47)
A further simplification of eq 46 and 47 is possible for sufficiently large r,, Le., when R, = (r, - l)/r, = 1. Combining eq 30 and 46, we then obtain for rigid-rod and flexible-chain solutes, respectively In K3(1) = In [Y3(l - E ) + Y3(l - elr$] + r34x13 - x3, + XI,] (48) In K3(1) = r3 In [(3 - 2 ~ ) / 3 ]+ In [3(1 - e)/(3 -241 + r3c[x13 - x3c + xlcl (49) (66)N.Tanaka, Y.Tokuda, K. Iwaguchi, and M. Araki, J. Chromatogr., 239,761 (1982). (67)C.H.Lochmtiller, private communication.
1051
and for cubic solutes
In
= [@r3 + r32/3)/31In (1- 4
K3(1)
+ r 3 h 3 - x3, + x1,l
(50)
Considering case I for flexible-chain r2-mer solvents, it can be shown from eq 36 that the collapsed-chain limit is again approached for sufficiently high x2, and/or sufficiently low @Eb.At r2 = 2, = 1/2, and R, = 1,for example, r$2 = 0.01 for @Eb = 0, x2, = 1.22; for @EL = 2, xZc= 2.22; and, in general, for x2, - (@&,/2) = 1.22. As -* 0 (4, = cu l),eq 37 yields for flexible-chain solutes In &(2) = [r3(1 - 4 - (r3- 1)aa31 In Q, + [r3(2 + 4 - (r3 - 1)"b31 x In Qb - (r3- l)[aa3In 3aa3+ a b 3 In (3ab3/2)] - [2r3 + 11 80+ r3c In [(I- 4 / 4 4 + r 3 d E b + r3[x23 - X3cI (51) where Q, and Qb are given by eq 44, Qo = 1- (r2(from eq 32), and where CY& = ad may be determined from eq 30. (Note that, except for the In Qoterm, eq 43 and 51 have exactly the same form.) r$2 tends toward its maximum value under the same general conditions as in case I1 for monomer solvent. A t r2 = 2, E = 1/2, and R, 1, & is within 1% of 1- E for xZc = 0, @ E b = 5.20; for xzc = 4, @Eb = 7.20; and, in general, for pEb - (xZc/2)= 5.20. Also In K3(2)for flexible-chain solutes is given by eq 30, 32, and 38, with = 1- e, u = 1, and e2 = 1. For binary mixtures of the two solvents, we obtain from eq 27, 28 (both with v = & = 0, r$2 = 0), 43, 44, and 51, the following case I result:
-
In K3(1+2) = In K3(1) + 82 In K3(2)- 81e2r3X12 (52) where In Qo has been expanded through first order in R2 (error < 0.07). Although eq 52 has been derived for flexible-chain solutes, it can be shown to hold for any solute structure. Another useful case I form may be obtained from the same equations: In K3(1+2) = In K 3 ( 2 ) + 81r3[x13 - x23 - xl2l 81[(2r3 + 1)(r2- 1)/3r21 + 812r3~12 (53) Finally, we consider the special situation (see later) where the r2-mer solvent is strongly sorbed and the monomer solvent is virtually excluded (c#q = 0). From eq 42 we have
In
= In K3(2) + elr3[x13 - x23 - XlZl + 4[(r3 - r2)(r2- 1)/3r221 - (r3/r2)In (1- el) (54)
K3(1+2)
where In Qo has again been expanded.
Results and Discussion In this section we interpret, apply, and discuss the results of the model. Monomer solvent systems are considered first, including what is believed to be a significant limiting case in practice, viz., collapsed CBP chains. Since their behavior parallels that of monomer solvents, our treatment of pure r-mer solvents is less thorough. Binary monomer + r-mer solvent systems are considered in more detail, followed by other important applications of the theory to RPLC systems. Pure Monomer Solvent. General Behavior. Here we analyze the model RPLC system for small (monomeric) solvent molecules by applying eq 30-35, assuming that the CBP chain length is sufficiently high so that Rc = (r, - l)/r, = 1.@We defer discussion of chain length and temperature effects until later. From eq 33 it is clear that the equilibrium composition of the stationary phase depends on the CBP surface cov-
1052
The Journal of Physical Chemlsfty, Vol. 87, No. 6, 1983 I
I
_ _ _ _ _ _ Max.
I
0.5
I
Martire and Boehm
I
0.4 -
0.6
0.3 -
0.5 -
\a
41 0.4 -
0.2 -
41 0.1 -
0.3
-
0.0 -
0.2
-
0
1
2
3
4
0.1 -
x1c
Figure Variation of stationary-phase solvent volume frac _ ~ 1 4with xlcan- JEb (a: 0.0;b: 1.0; c: 2.0; d: 3.0) for t = ‘ / z and R , = 1.
0.0 0
1
2
3
4
x1c
t L--’---’
Figure 3. Variation of stationary-phase solvent volume fraction with xlcand t (a: 1/3; b: l/z; c: 2/3) for BEb = 2.0 and R , = 1 (-, maximum value).
Max.
’
V-1
0.7
0.9 -
0.6
0.8 -
0.5
u 0 1 2 3 4
0.7
-
V-1
0.6 -
x1c
Figure 2. Variation of stationary-phase reduced film thickness v-l with xlcand PEb (a: 0.0; b: 1.0; c: 2.0; d: 3.0) for e = ’ I z and R , = 1.
erage (e), the intrinsic stiffness of the CBP chains (BE!,), and the compatibility of the solvent with the CBP chains (xlc, where xlc= 0 represents ideal mixing). Shown in Figure 1is a plot of the volume fraction of sorbed solvent (&) as a function of xlcand PEb at a typical surface coverage (t = l / J . As the CBP chains become more rigid (larger P E b ) and as the compatibility between the solvent and a CBP chain segment increases (smaller xlc),the extent of solvent penetration or “swelling” increases. Also 41 rapidly approaches zero with increasing xlc (poorer solvents), even with relatively more rigid CBP chains. The variation of the reduced film thickneas, u-l= r/r-, with xlcand PEb is illustrated in Figure 2. With increasing DEb and decreasing xlc,the (more swollen) CBP chains becomes more fully extended and more ordered in the direction normal to the surface (a,,= I J - ~ ) ,while at higher xlcthe collapsed-chain limit is approached (u-l e). Shown in Figures 3 and 4 is the dependence of d1 and v-’, respectively, on t and xlc for @Eb = 2.0, which we take to be a reasonable model estimate for n-alkyl CBP chains.@ Note that, as xlcand e decrease, the volume fraction of sorbed solvent increasessg (maximum solvent capacity of
-
(68) In this simple cubic lattice model, the 90° bond angles and the 4:l statistical factor in favor of ‘bends” parallel to the surface (see eq 4) complicate assignment of p E b . The gauche-trans energy difference in n-alkyl chains is 500-800 cal mol-l, with a statistical ratio of 2:l. Accordingly, an effective @Ebof 1.5-2.0 in this model produces the actual gauche-trans population at room temperature. (69) There are exceptions to this at larger &?Cb values (>-3) and at y g h x1 In this regime, b1values are larger for larger c in the range -c5 Nevertheless, the b1’s are all relatively small and, as xlc decreases, the usual behavior (Figure 3) obtains.
0.5 -
0.31
I
I
,
,
I
0
1
2
3
4
XlC
Flgwe 4. Variation of stationary-phase reduced film thickness v-l with xlcand t (a: 1/3; b: l/z; c: 2/3) for PEb = 2.0 and R c = 1.
1 - E ) . At any surface coverage, c $ ~ approaches zero with increasing xlc.In Figure 4,more fully extended and, hence, more ordered CBP chains are evident at lower xlc and higher e. (Although C#J~ is larger at smaller t because of a greater inherent swelling capacity, the closer spacing of CBP chains at larger 6 leads to larger 19.) Again, the respective collapsed-chain limits are rapidly approached with increasing xlc. Therefore, the results are thus far consistent with experimental observations and with other model results on related s y s t e m ~ . ’ - ~ J ~ J ~ ~ ~ ~ Turning to the partition coefficients K3(1),we note from eq 35a and 35b that solute retention is governed by several factors. The last term in both relations, the interaction free energy contribution, which may be rewritten by using eq 21 r3[(eu)(x13- x3c + X l J l = r3zP[(tu)(w13+ wlc - ~3~ - wll)l (55) reflects the competitive equilibrium associated with the transfer of a solute segment from the mobile to the stationary phase and the concomitant reversed transfer of a (previously sorbed) solvent molecule. As discussed else(see also Appendix I), this term is expected to be positive (hence, retention enhancing) in RPLC systems
Theory of LiquM Chromatography
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983 I
t
la5
10.0
/.
Figure 5. Dependence of the natural logarithm of the distribution constant relative to that of a cubic solute, In CY,on solute molecular sire r 3 , with xk = 0 , x,, = x i 3 = 2, O E b = 2, e = ’I2,and R , = 1 (a: rigMrod solutes; b: flexlble-chain solutes).
and to be controlled primarily by the polarity of the mobile-phase so1vent,1-18~28~32~ with more polar solvents (more negative wI1) leading to longer retention (larger K3(1)).This retention contribution increases with increasing solute size (r3),which is proportional to its molecular volume, and increasing CBP chain-segment density (tu). The first term in eq 35a and 35b reflects the statistics of solvent displacement in this natural competitive-equilibrium model, i.e., the probability, (1 - tu)‘3, that there are r3 contiguous solvent molecules available in the stationary phase for displacement by a solute molecule. This term leads to reduced retention, more so with increasing r3 and increasing tu. (As will be seen, the last term outweighs the first term in RPLC systems.) The middle terms in these equations are quite often the less important ones on an absolute basis. They are related to the “packing“ or accommodation of the solute molecules in the stationary phase. They depend not only on the size of the solute molecule but also on its shape and flexibility, thus affording the possibility of RPLC selectivity based on solute structure (see below). Note that all of the terms in eq 35a and 35b also depend implicitly (through u in eq 33) On Xic, p&, and Regarding, at this stage, solute distribution between the mobile phase and the interior of the stationary phase as the only operative retention mode, the dependence of the distribution constants of rodlike and chainlike solutes relative to cubic solutes (a)on solute molecular size (r3) is illustrated in Figure 5 for representative values of xlc, &?&,and t. Here, the solute and the CBP chains are taken to be chemically homogeneous and similar in nature (e.g., aliphatic), so that xk = 0 and xlc = ~ 1 3 permitting , us to focus on the effects of solute shape and flexibility. Referring first to Figure 2, under the system conditions for Figure 5 approximately 55% of the CBP-chain bonds are oriented normal to the surface, thus presenting an anisotropic environment where, except for the unalignable cubes, normal orientation of the solute molecules is sterically favored. For r3 1 2, approximately 44% of the bonds of the flexible-chain solutes are aligned perpendicular to the surface (independent of r3), while, for the rigid-rod solutes, 44% of the molecules are so aligned at r3 = 2, increasing to 84% at r3 = 6. The packing terms follow the trend of solute orientational order in the stationary phase and, in turn, produce the general trend seen in Figure 5: rods (a) > chains (b) > cubes (c), at fixed r3. It is also apparent from Figure 5 that, here, solute shape is more important than solute flexibility. For flexible-chain solutes, the constancy of the order parameters ad leads to a linear increase in In K3(1) with increasing is.The rigid-rod solutes closely follow the flexible-chain ones,
8.0
,
,
,
,
,
1053
,
1
2
0
4
6
r3
Flgure 6. Dependence of In KX1)on @Eb, xlC, and c for flexiblachain Solutes, with X k = 0,X I , (a: XlC = 2, @ E b = 2, t = ’12; b: X i , = 2, @Eb = 0,t = /2, C: X I , = 1, @Eb= 2, t = d: XI, = 2, @Eb= 0, = ‘ / 3 ) .
7
diverging upward only at higher r3. However, the cubic solutes lie well below the chain-solute line. At q = 4,e.g., the relative K3(1)values are 2.26:1.931.00 (r0d:chain:cube). Such shape selectivity between more elongated and more globular molecules is well documented in the RPLC literature, e.g., separation of linear and branched alkane derivatives,2 and of isomers of polynuclear aromatic hydrocarbon~.~~ Demonstrated in Figure 6 is the dependence of In K311) on @&,xlc,and e. Without detracting from our analysis, we consider only flexible-chain solutes and, once more, the simplification x3c = 0, xi3 = xlc. The slopes are governed primarily by the first and last terms in eq 35a, and the intercepts entirely by the middle terms. The volume fraction of admitted solvent decreases with increasing xlo decreasing Eb, and increasing t (Figures 1 and 3). As discussed, each of these leads to reduced slopes (first term). The relative slopes of lines a m. b (varying ,dEb) and b vs. d (varying t) stem mainly from the relative availability of displaceable solvent. Examining the results for a vs. c (varying xlc),we find that the decrease in the first term with increasing xlcis accompanied by a larger increase in the last term, thus producing the greater slope for line a. Indeed, in RPLC practice it is well-known that for homologous series the slopes increase with increasing solvent polarity (see, e.g., ref 13). The intercepts are controlled by solute accommodation in the stationary phase, leading to the most negative intercept in example d.’O To simulate the behavior of n-alkyl homologous solute series in a given RPLC system (fixed solvent and CBP) containing n-alkyl bonded phases, we treat chemically heterogeneous, flexible-chain solutes consisting of r3 - 1 aliphatic segments (type s) and one polar segment (type 9). From Appendix I r3[eu(X13- x 3 c
+ Xlc)]
= (r3- 1)tu(2xos) +
4xOq + xoS- xqs) (56)
where 0 denotes the polar monomer solvent. Combining (70) We again bring to the reader’s attention that care must be exercised in comparing model results (K’s) with experiment (k’values). As mentioned earlier, changes in xlc (solvent), r,, BEb, and/or c may result in changes in the phase ratio, thus often precluding meaningful comparison of absolute values. However, normalization of results relative to some standard solute(s) would help to overcome this problem. We also note that, while increased chain rigidity per se should lead to increased retention, such stiffenin is achieved in practice by changing the chemical type of CBP “chain”.@SKiHence, the interaction parameters xlcand x~~ would also change.
1054
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
Martire and Boehm
TABLE I: Comparison of In K, ,) from Figure 6
l
3.0
(Complete) and from Eq 43 (Collapsed-Chain Limit) line a b C
d
Figure 6 1.678r3- 0.407 0.689r3 - 0.432 0.520r3 - 0.346 0.338r3 - 0.584
eq 43
2.0
l
,
i
eq 35a and 56, we obtain for homologous series an equation of the form
C'
1
b a'
0-
a
- 1.0 -
b
(71) D. E. Martire and M. M. Miller, to be submitted for publication. (Briefly, to each modifier was added n-hexadecane (HDA) and the equilibrium compositions of the modifier-rich and HDA-rich phases were determined a t 25 "C. For ME and ACN there was marked phase separation, while THF was found to be completely miscible with HDA. Using lattice-model equations for bulk solutions (see Appendix II), the xl;s were calculated for ME and ACN. From gas-liquid chromatographic determination of its activity coefficient in HDA, xlc for T H F was similarly calculated.) (72) For convenience, the solvent molecule was taken to be a monomer. Different xlc values would be obtained if the solvent were scaled as an r-mer, but the basic conclusion would remain the same.
-
-2.0 C
(57)
where A, B, and C are constants, where CAxo,,,, (second term on right-hand side of eq 56) represents the solute functional group (9) contribution, and where r3 - 1 is proportional to the number of carbon atoms n3. Equation 57 indicates that In vs. n3 plots for different homologous series in a given solvent should have the same If we exslope, but intercepts which differ by CAX~,,,~. amine the extremes, (a) when q = 0 (solute functional group as polar as the solvent), AX^,^,^ = 0; (b) when q = s (n-alkane solute series), Axo, = 2xos > 0. Therefore, more polar solutes are expectec! to have lower intercepts, and the intercepts are expected to be more sensitive to solute functionality than to stationary-phase structure and composition (see Figure 6). Also, changing the solvent would affect both the slope and the intercept, with more polar solvents (larger xOsand EU) giving longer retention but greater selectivity for adjacent homologues. These predictions are in full agreement with experimental findings.2J3J5,52,62 Moreover, eq 57 leads directly to the linear free energy r e l a t i ~ n s h i papplied ~~ with much success to analyzing retention in a variety of chromatographic systems. Pure Monomeric Solvents. Important Limit of Collapsed CBP Chains. In the previous section it was shown that, for ,BEb= 2,68 xlc values in excess of 3.85 produce minimal solvent sorption (41< 0.01) and promote collapse of the CBP chains. It is also of note that lines a, b, and d in Figure 6 (for which xlc = 2) lie very close to the lines obtained by assuming the collapsed-chain limit (eq 43). As seen in Table I, the slopes are within 0.01 and the intercepts within 0.03 of the respective limiting values. The compelling question at this point is where, in terms of xlc, do the common RPLC organic modifiers fall: methanol (ME), acetonitrile (ACN), and tetrahydrofuran (THF)? This question was recently addressed in our l a b ~ r a t o r y .From ~ ~ experiment and theory (lattice model) the following xlcvalues were determined: 4.2 (ME), 3.7 (ACN), and 0.7 (THF).72 From eq 33, with PEb = 2 and E = 'I2,the following c$~ values are calculated for ME, ACN, and THF, respectively: 0.007 (0.0064),0.012 (0.0115), and 0.33, where the values in parentheses are those found in the hydrocarbon-rich phase in the bulk e ~ p e r i m e n t .The ~ ~ implications are clear: except for THF (the least popular modifier), which lies midway between the two CBP structural limits (see Figures 1 and 21, the common organic solvents and, by extension,
,
d'
1.0 -
1.683rj - 0.442 0.683r3 - 0.442 0.683rj - 0.442 0.349r3 - 0.588
K3(1) = An3 + B + cAxO,q,s
,
- 4'0 5.0
L 0
1
2
3
4
5
r3
Figure 7. Difference between RPLC and classical in Kq1)(eq 43 minus eq 58), in the collapsed-chain limit, for flexible-chain solutes as a b: c = c: c = 1/2; d: function of r3, e, and @Eb(a: E = t = 2/3), with R , i= 1, and for of, = 0 (unprimed) and @Eb =2 (primed).
H 2 0 and their mixtures with H 2 0 appear to induce near-collapse of the CBP chains and to drive the distribution process to one akin to partitioning between two immiscible phases. If one accepts this evidence, then far more tractable equations may be applied to investigate the majority of nonelectrolytic RPLC systems. We start by comparing eq 43 for flexible-chain solutes and the corresponding expression for classical partitioning between the same solvent and a bulk-liquid n-alkane of the same chain length as the CBP (see Appendix 11): In K3(1) = r3(xI3- x3J - 0.811r3 - 0.405
(58)
As seen in Figure 7, the model RPLC values are lower than the classical partitioning values for = 0 and higher for PEb = 2, with the difference IA In K3(1)1increasing with increasing solute size and generally increasing with increasing CPB surface coverage. For = 1/3 (isotropic system), the intercept is exactly zero and the slope is c[ln ((1- 4 / 4 4 + The first term is due to the different configurational entropies of the respective hydrocarbon phases, the CBP chains being translationally restricted; the second, to the different configurational energies. At other surface coverages (anisotropic mixtures), differences in the packing entropy also play a role, affecting both the slope and the intercept. The crossover from negative to positive differences in Figure 7 occurs at PEb values in the general range found for real n-alkyl chains.6s This together with the uncertainty in the arrangement of chains in real CBP's makes comparison with experiment particularly difficult here. Also, adsorption of polar solutes on accessible silanols and at the mobile-stationary phase interface (see below) may cause anomalously high RPLC values. Colin et al.73and Lochmuller and Wilder3 have measured relative70 (to benzene) bulk-liquid partition coefficients and RPLC capacity factors for a variety of solutes using polar solvents (ME, ACN, and their binary mixtures with H20). Both positive and negative differences were observed, depending on the size and polarity of the solute. It merits mention that with fairly small and nonpolar solutes (methyl-substituted benzenes), longer CBP chains, and relatively low surface coverages (0.20-0.25), the classical-partitioning value is virtually attained3 (see Figure 7). (73) H. Colin, A. KrstuloviE, G. Guiochon, and Z. Yun, submitted t o J . Chromatogr.
The Journal of Physical Chemistry, Vol. 87, No. 6, 1983
Theory of Liquid Chromatography
4 0 ,
I
I
200.6*In
K3(V
k‘3(1)
-
a
0.2-
In k’3(1)
.R
0-
-
,
0.0-
3.8
10 -
b
o.4-
la’l
A’4
30-
0.8-
I
!
1055
1
2.0
I
I
I
0
5
10
I
I
I
15
20
25
‘C
03
252015
10
5 ‘C
Flgure 9. Dependence of (a) k’ql) and (b) in k,(’,on r c , for r 3 = 3 (flexible-chain solute), t = ’I2,Kqr)= 100 at r c 5 (arbitrary reference), 7 = 0.2 at r c = 25; prime designation: f ( 7 ) from eq 62b; no prime: f ( 7 ) = 1.
-
-
Flgure 8. Difference between In Kql) values at r c and as r c a, in the collapsed-CBP-chaln limit, for flexibie-chain solutes as a function of r c and r 3 (a: r 3 = 1; b: r 3 = 3; c: r 3 = 5), with t = ’I2.
We now address the effect of CBP chain length rc on to this point it has been assumed that r, is large enough to allow the approximation R, = (r, - l ) / r c = 1. Calculated by using eq 43 and 44 (for flexible-chainsolutes) and displayed in Figure 8 is the dependence of In K3(1)on r, at t = 1/2, represented as the difference, A In K3(1), relative to the value as rc 00 (R, 1). Note that, for r, > 10, In K3j1) is a comparatively weak function of rc, varying approximately as r;l. Continuing to let R, = 1, we consider the contribution to retention from solute adsorption at the mobile-stationary phase interface. Assuming that there is a monolayer of displaceable solvent at the interface, we have from eq 1
3.0 -
K3(1). Up
- -
-
k??? = k?(i)[1 + K$i)/K3(i)T]
(59)
where Kitl) is the interfacial distribution constant, T = V,/A,, and k\(l). and K3(1)are respectively the capacity factor and the distribution constant pertaining to the interior of the stationary phase. Clearly, if K$l.)/K3(1)T 5, as calculated) are apparently required for the extended-chain limit to be closely approached. However, since such CBP’s are now being ~ynthesized~fj’ and are potentially useful, a discussion of their expected behavior would be in order. Referring to eq 55 (with v = 1 here) and the discussion following it, and comparing eq 43 and 46, we note that, in contrast to the collapsed-chain limit, the competitive equilibrium process for solute distribution is retained in the present limit, due to the presence of (substantial) sorbed solvent in the stationary phase (as indicated by the xlc term). Note also that, in comparison to n-alkyl bonded phases, xlc should be less positive for wholly or partially aromatic in RPLC systems.70 The interfacial adsorption contribution in this limit is again estimated via eq 59. For flexible-chain solutes, eq 49 gives In K3(1)= (r3/2)Ax1,3,c= 0.406r3 - 0.288 (67)
+
and for parallel adsorption of such solutes on a now heterogeneous stationary-phase outer surface (i.e., at the top of the CBP chains), with a fraction t of the type-c sites and a fraction 1 - t of type-1 sites, we have32r33 In K;f1) = r3 In [l + exp(A~,,,,~/6)] - 0.693r3- 0.406 (68) where we have set t = ‘I2, and where A ~ 1 , 3 =, ~~ 1 -3x3c+ xlc. From the above, it is estimated that at T = 5 the interfacial contribution constitutes 20% or more of the total retention only when Ax1,3,cC -1. (In this limit, T - T, = rc for monolayer adsorption at a planar interface.) Therefore, having assumed the most favorable solute orientation for adsorption, having assigned a moderately low T value and even allowing xlc to be zero, we find, as before, that this secondary effect may be neglected except for highly polar solutes. The dependence of In K3(1)on r,, calculated from eq 46 and illustrated in Figure 11, is similar to, but less pronounced than, the collapsed-chain case (Figure 8). Again, for r, > -10, In K3(1)varies approximately as rC1 and is not highly dependent on rC.l5 (74) H.Abraham, J. Am. Chem. SOC.,104, 2085 (1982).
Theory of Liquld Chromatography
7 AInK3(,)"" 0.2 -
The Journal of Physical Chemistry, Vol. 87, No. 8, 1983 1057
1
A' a
-
0.0
-
1
00
252015
10
5 'C
Figure 11. I n the extendedCBP-chain limit; otherwise, see caption for Figure 8.
TABLE 11: In K 3 ( l )for Flexible-Chain Solutes in the Extended-CBP-Chain Limit at Different Surface Coveragesa
In K 3 1 , ) b , C
E
r3(l/6)(Ax- 0.708) - 0.065 r3(l/3)(Ax- 0.753) - 0.154 r3(1/2)(Ax- 0.812)- 0.288 r3(2/3)(Ax- 0.882)- 0.511
'I6 '13 '/2
z/3
Flgure 12. Dependence of In CY (relative to cubic solutes) on solute molecular size r 3 In the extended-CBP-limit, with 6 = ' I 2(a: rigid-rod solutes: b: flexlble-chain solutes).
"
0.4
Calculated from eq 49. A X = AX^,^,^ = x13- xJC + x l c . In k ' 3 ( l ) = In K 3 ( , )+ In Q , where the phase ratio cp is independent of
E.
Roughly modeling the variation in k j(l)with r, and E in = r c / r p At the present limit, q in eq 62 is given by r,-/ro fixed E , the dependence of k'3,1, (and In I Z $ ( ~ ) ) on r, is qualitatively similar to that shown in Figure 9, and the same conclusions apply as before. However, the dependence of In k3(1)on E at fixed r, is qualitatively and quantitatively different from that illustrated in Figure 10, for two reasons: (a) the interaction free energy term, reflecting the competitive equilibrium process, is now a function of E (compare eq 43 and 46), and (b) the phase ratio is independent of e (in eq 62, q is independent of E ) . Summarized in Table 11are the resulta for flexible-chain solutes (from eq 49) in the experimentally significant range 1/8 I c I2/3. Provided the flexible-chain solutes are fully contained within the stationary phase, and provided Ax1,3,, R 2 and r3 R 2, it is predicted that In kj,,) should increase with increasing E for completely extended CBP chains. In general, however, In K3(1)is a complicated function of E which depends on the size and structure of the solute and on the exchange free energy associated with solvent displacement (see eq 48-50 and below). The distribution constants of rodlike and chainlike solutes relative to cubic solutes at E = 1/2 are calculated from eq 48-50 and displayed in Figure 12. Comparison with Figure 5 (close to the collapsed-chain limit) indicates enhanced shape selectivity in the present limit and greater discrimination between rigid and flexible solute molecules. With a true brushlike CBP structure, the solute molecules must find a channel to enter the stationary phase and displace the solvent. Channels normal to the surface are less obstructed by the protruding CBP chains, thus favoring perpendicular penetration of rigid, rodlike solutes, more so for larger solutes and at higher surface coverages. (This is reflected in the solute orientational order parameters CY,^.) Thus, the purely structural component of se(75) Given that the two extremes of CBP structure have now been considered, it is tempting to generalize this statement. However, one would have to continue to &98umethat the solute is fully contained within the stationary phase and that surface curvature has but little effect on K3W.
0.0
I / /-
I/ /
0.5
I
I
I
1
I
1
2
3
4
5
'2
Figure 13. Variation of the statlonary-phase volume fraction 4 and reduced film tlckness Y-' with r for flexlble-chain solvents; @Eb= 2, 2, xps= 4. = ' I 2 ,R , = 1, x2, = xps/r2
lectivity increases with increasing r3 and E . Note, however, that, if the rodlike and chainlike molecules, e.g., are of different type, the exchange free energy term would play a role as well. Also, the steric effect leads to reduced retention with increasing r3 and E for all solutes (more so for larger cubic solutes) while the exchange free energy contribution acts in the opposite direction (see Table 11). The counterpart of eq 63 is In aij =
E~~@z[(wI ~ wli)
- (wj, - wit)]
(69)
Comparing the two, note that In a is reduced here by a factor of E , which may be traced to the presence of solvent (volume fraction 1 - E ) in the stationary phase. Moreover, if aromatic CBP's were to be utilized, the selectivity afforded by differential solute interactions with the solvent would be diminished to some extent by the difference in solute-CBP chain interactions. It would appear then that there is little to be gained by using such CBP's in the RPLC mode to effect separations based solely on interactional differences. The counterpart of eq 64 will not be reported. It is simply noted that the same general conclusions apply in the present limit, where eq 65 also obtains. For a homologous solute series eq 21 and 49 give [a In k'3(i)/a@l = ~ ~ G ( ~ E ) [ z+ A~ mO 3 , ,G]t [ ~ ( A h ,+ Aho, - Ah,,)] (70) The general discussion following eq 66 also applies to eq 70.
Martire and Boehm
The Journal of Physical Chemlstry, Vol. 87, No. 6, 1983
1058
TABLE 111: In K 3 ( 2 )for Flexible-Chain Solvent and Solute Moleculesu r2
x 2c b
1
4.000
2
1.000
3 4 5
0.444 0.250 0.160
O1az
0.333 (0.333) 0.478 (0.333) 0.531 0.554 0.569 (0.667)
In K 3 ( 2 ) 3.685r3- 0.438 (3.683r3- 0.442)c 0.925r3- 0.254 (0.520r3- 0.346)d 0.308r3- 0.245 0.128r3- 0.254 0.057r3- 0.263 (0.112r3- 0.452)e
xJC= 0 and x Z 3= xZc;see Figure 13 caption for remaining conditions. b xZc= xps/rz2= 4/rZZ (see text). Result for Collapsed-CBP-Chainresult from eq 43. monomer solvent with x l c = 1 and x l S = xlc, calculated from eq 3 5a, under the same conditions as in Figure 1, curve c. e Extended-CBP-chain result from eq 30, 32, and 38 (with Q~ = 1 - E = ' I 2 , u = 1, and e 2 = 1). (I
Pure r2-mer Solvents. Illustrated in Figure 13 is the effect of larger solvent molecules on the composition and structure of the stationary phase. The results are obtained from the general equations 30, 32, and 36. The solvent molecules are modeled as flexible chains consisting of one polar terminal segment (type p) and r2 - 1 aliphatic segments (type 8); thus, with an aliphatic CBP, x2c= xP/r? (see Appendix I). As the solvent chain becomes longer, solvent-CBP compatibility increases (lower xZc),leading to greater swelling (higher &) and more extended CBP chains (higher v-l). Concurrently, the orientational order parameter of the solvent (aa2)increases with increasing r2 (Table 111). With even longer solvent chains, the limit of fully extended CBP chains is approa~hed.'~ Shown in Table I11 are the corresponding distribution constants for chemically homogeneous, flexible-chain solutes, as calculated from eq 38. The solute segments are of type s (aliphatic), so that xk = 0 and xB = xzc. Observe that K,,,, increases with increasing r3 and increasing xzc (decreasing rz). For the monomer solvent (x2, = 4), the collapsed-CBP-chain result is approached; for the pentamer solvent (x2, = 0.16), the result falls closer to the extended-CBP-chain one. For comparison we have also included in Table I11 the result for a monomer solvent having xlc = 1. At the same xlcvalue, larger distribution constants are found with the dimer solvent, stemming primarily from more favorable statistics of solvent displacement (first term in eq 35a and 38); i.e., a chainlike solute molecule can more readily displace an equal number of solvent segments when the solvent segments are connected. In general and in the two limiting cases, the behavior with r-mer solvents closely parallels that found with monomer solvents. For example, at a given PEb and t, the difference between the classical-partitioning and collapsed-CBP-chain values are the same as those seen in Figure 7. Also, equations of the same form as eq 57,63-66, 69, and 70 apply, as well as the conclusions relating to the effects of rc, t, and interfacial adsorption. Accordingly, pure r2-mer solvents need not be considered in any more detail. Binary Solvent Mixtures. We begin by investigating the sorption isotherms for a binary solvent mixture in contact with a CBP. Analysis of eq 39 indicates that the isotherm does not exhibit a Langmuir form32,33 unless rl = r2 = 1, xlz = 0, and unless tu = 4cremains constant over the entire composition range of the mobile phase (volume fractions O1 and 02). The last condition is strictly satisfied only in (76) A liquid-to-solid phase transition, i.e., cocrystallization of the CBP,Z1is not obtained in this lattice model.
the two limiting cases. Therefore, we expect, in general, a variety of isotherm shapes, depending on the relative molecular volumes of the solvent molecules, deviations from ideal mixing (x12# 0), and the extent to which CBP structure varies with mobile-phase composition. The lattice model permits determination of the actual solvent composition in the stationary phase (volume fractions $ J ~and 42). However, only relative values or excess compositions (I', and r2)are experimentally accessible.26vmWe relate the two by adopting a symmetric convention,26,mrl = -r2,and defining r2as follows: rZ is the difference, per unit volume of CBP chains (VJ, between the volume of component 2 in the actual system (42V, + 02Vm)and in a reference system containing the same total volume of solvent as in the actual system (41V, b2V, Vm),but in which the volume fraction of component 2 is equal to that of the mobile phase in the actual system (e2). Accordingly rz = [(4zV, + 02Vm) - 02(41vB+ 42Vs+ Vm)l/Vc (714
+
+
rz = -rl = (6142 - 0241)/4c
U1b) where 41 42 = 1 # 1, and where V , and Vm are respectively the total volumes of the stationary and mobile phases. Equation 71 takes into account only the solvent sorbed at the interior of the stationary phase. Including the contribution from the (assumed) monolayer adsorbed on top of the CBP (i.e., at the interface), we have r z a d = - r l a d = (44zed- b # J l a d ) / 4 c T (72)
+
C#J~
rztotal = -rltotal
=
r2+ rZad
(73) where represents the volume fraction in the adsorbed monolayer 42ad= I), T is again the number of stationary-phase layers (excluding the adsorbed monolayer), and r2is a dimensionless quantity ("excess" volume of component 2/(volume of CBP chains))." To simulate a binary solvent mixture consisting of two highly polar substances (e.g., ME + H 2 0 or ACN + H20), which individually should promote collapse of n-alkyl CBP chains, values of xlC= 5 and xZc= 3 are assigned. On the basis of analysis of activity coefficients for the mixture ME H20,78we set x12= 1. For economy, we limit our treatment of the isotherms to the case rl = r2 = 1,noting that salient features of the isotherms will still be revealed and that less restrictive cases may be handled within the framework of the lattice model. The results for the contribution from the interior of the stationary phase, calculated by using eq 31 and 40, are shown in Figure 14. As expected, the less polar component (2) is more strongly sorbed and the total volume fraction (& + &) is small. As the mobile phase becomes richer in the more polar component (l),the less polar one is rapidly expelled from the stationary phase without a concomitant buildup of the more polar component. Thus, the virtually collapsed CBP chains become even more compacted at higher 41(here, a mole fraction). The excess composition of component 2 is everywhere positive and exhibits a skewed maximum (0, = 0.7). Comparing rZfrom Figure 14 with the calculated excess isotherms of ME HzO and ACN + H20,26we note the resemblance in shape and direction of skew of the maximum. However, the model results are approximately 1 order of magnitude smaller than those reported in ref 26.
+
+
+
(77) In ref 26 excess compositions are reported as excess moles of component 2/(g of CBP chains). The results from eq 71-73 may be converted to these units by assuming that the CBP chains have roughly the same intrinsic density as the corresponding bulk hydrocarbon. (78) M. Kato, H. Konishi, and M. Hirata, J.Chem. Eng. Data, 15,501 (1970).
The Journal of Physical Chemistty, Vol. 87, No. 6, 1983 lOSQ
Theory of Liquid Chromatography
“I
0.008
4
. QL
0.004
’/
r2
//
O.Oo0
I
I
0.2
0.0 I
I
I
0.4
0.6
1.0
4
-1
I
0.8
Figure 14. Isotherm (4)and the excess isotherm (r,)at the interior of the stationary phase for a binary mixture of hlghly polar solvents as a function of the mobilephase mole fraction of the more polar component (ei); conditions: xi, = 5, xZc= 3, x12= 1, j%b = 2, t = i/2, R , = 1 (a: 4 1; b: 42; c:
Flgve 16. Dependence of the solute distribution constant with a binary solvent mixture (In Kq1+,))on mobilephase volume fraction (e1), In the collapsed-CBP-chain llmk conditions: Kxl,/Kxz, = e’ and faxl2= 3. I
1
I
I
I
I
I
1
r,).
I
I
I
I
I
I
C’
0.06
1.0 @ad
\I
j0.00
0.80.6 -
0.4 -
0.0 0.2 0.4 0.6 0.8 1.0
6,
0.2 -
(r2)
Flgure 17. Isotherms (4) and the excess isotherm at the interior of the stationary phase, for a binary mixture of highly polar (1) and moderately polar (2) solvents, as a function of mobile-phase mole fraction (el):conditions: x,, = 5, xZc= 1, x12= 2, p b = 2, e = l/,, R , = 1 (a: 4 b: 42; c:
0.0 0.0 0.2 0.4 0.6 0.8 1.0
r,).
01
15. Interfaclal adsorption isotherms ( 4 9 and excess lsatherms for a binary mixture of highly polar solvents as a function of the conmobile-phase mole fraction of the more polar component ditions: xi, = 5, xZc= 3, x12= I, 7 = 5 (a: 4 ,@‘; b: 4 ,@‘; c: rZM; c’:
(r,)
(el);
r,”).
We interpret this apparent discrepancy as indicating that, with highly polar solvents, most of the sorption occurs at the stationary-mobile phase interface rather than at the interior of the stationary phase. Support for this interpretation may be found in ref 26, where it was noted in studying CBP’s of different n-alkyl chain lengths that r2 correlated well with the surface area of the CBP packing. Additional support is evident in Figure 15, where we show the composition of the adsorbed monolayer (curves a and b), ita contribution to the excess composition (curve c), and the total excess composition (curve c’). These results are obtained from eq 71-73 (with 7 = crc = 5) and from32*33 In (4zad/(1 - 42ad)l - 242adX12= In [62/(1 - 62)l + [(Xlc - x 2 c + Xl2)/61 - 202x12 (74) The curve for the total excess composition of component 2 remains skewed, now falls in the proper range of value^,'^ and is dominated by the interfacial-adsorption contribution. However, as noted earlier, this adsorbed monolayer (79) Recall that the possible effect of residual silanola has been neglected in this model but is probably relevant to the sorption isotherms of small, highly polar solvents. This could account for some of the differences in the details of the excess isotherms here and in ref 26.
appears to be incidental to the solute distribution process (i.e., the “retention mechanism”), except for rather polar solutes. With the indication that a binary mixture of highly polar solvents induces a near-collapse of n-alkyl CBP chains, we examine eq 52 and 53, which apply to that limit. Illustrated in Figure 16 is an example of the dependence of In K3(1+2)on volume fraction 61, calculated from eq 52, which predicts that such deviations from linearity should increase with increasing solute molecular volume and increasing nonideality of solvent mixing. The quadratic dependence on dl predicted by eq 52 and 53 is generally observed in RPLC systems employing ME H 2 0 and ACN H,O m i ~ t u r e s , ~provided ~ ~ . ~a~wide ~ ~ enough ,~ el range is investigated. Note that eq 52, which has the same form as the corresponding relation for classical partitioning (see Appendix 11),may be readily extended to an n-component solvent mixture:
+
In K3(1+2+ ...+n) =
1
+
In K,(,) - r3 C
o@mx1m
(75)
4m
m>i
where 1, m = 1, 2, ..., n. Moreover, since the phase ratio remains constant in this limit, Rs may be replaced by it’ values. To simulate a binary solvent mixture consisting of a highly polar solvent and a moderately polar one (e.g., THF + H20),values of xlc = 5 and xZc= 1 are assigned. Also, since THF + H 2 0 mixing should be more nonideal than
1080
No. 6, 1983
The Journal of Physical Chemistry, Vol. 87,
I 1
I
1.0 -
I
Martire and Boehm
TABLE IV: Difference between In K,(,+,) from Eq 54 (Approximate) and Eq 42 (Complete)a,b
]
0.3 0.6 a
0.4 -
0.0
,
0.2
I
1
I
I
0.4
0.6
0.8
1.0
4 Flgure 18. Interfacial adsorption isotherms (4&)as a function of ,, for the binary solvent mixture in Figure 17; additional condition: 7 = 5/4 (a: 4 lad; b: 4 2ad).
0.6 0.8
+ 0.011 + 0.015
O.llOr,
0.205r,
+ +
0.016 0.028
Equation 54 value minus eq 4 2 value, with r 1 = rz = 1
,-,
0.3
//
/
c'
\
/
\ \
A \
I
/
0.2
/
I
/
//
0.1
/
0.0
0.0
I
,
1
0.2
0.4
0.6
I
0.8
1.0
81
(r,)
Flgure 19. Excess isotherms as a function of 8,, for the binary solvent mixture in Figure 17; conditions: see Figures 17 and 18 (c: r2ad; c':
rial).
+
ME H 2 0 mixing, we set x12= 2. Again, our treatment of the isotherms is restricted to the case r1 = =. 1. Presented in Figure 17 are the results for the contribution from the interior of the stationary phase. The less polar component (2) is more strongly sorbed and, in contrast to Figure 14, its volume fraction in the stationary phase (&) remains fairly high (but not constant) up to O1 = 0.8, while never exceeds 0.007. Therefore, the structure of the CBP chains is between the two extremes, approaching the collapsed-chain limit only at much higher O1. The maximum in the excess composition of component 2 is much greater than in Figure 14 and occurs at a somewhat higher O1 value. The composition of the adsorbed interfacial monolayer, displayed in Figure 18, is calculated from eq 74. Since the outer surface of the stationary phase is now heterogeneous (approximate surface site fractions of & and &), the following estimatem is used in eq 74:
r2
X l C - xzc + XlZ = 4AXlC - X Z C + XlZ) + 242x12 (76) Note in Figure 18 that > 0.9 for O1 < 0.8. Only at very high O1 does the adsorbed monolayer become richer in component 1. In Figure 19 are shown rZad (curve c) and rzbtal (curve c'). In contrast to Figure 15, sorption at the interior contributes more to rzt0"than adsorption at the interface. Also higher maxima and greater asymmetry are evident. Here again, our earlier analysis indicates that the
(80) The 'effective"
adsorbed monolayer is incidental to the solute distribution process, except for small and fairly polar solutes. Equation 54 describes the solute distribution constant in a binary solvent mixture where one component is strongly sorbed and the other essentially excluded, as in Figure 17. In the derivation of eq 54 it is assumed that the uptake of component 2 by the CBP (&) maintains its value at O2 = 1over the entire mobile-phase composition range. This assumption will,of course, become invalid at sufficiently high O1 for any solvent mixture (In (1 - 0,) -m, as O1 1). However, considering the same conditions as in Figure 17, it would be of interest to examine the suitability of eq 54 in the experimentally significant range 0 5 8, I0.8. Shown in Table IV is the difference between the approximate relation (eq 54) and the complete expression (eq 42) in this composition region. Note that the approximation is indeed reasonable but becomes less so as r3 and 61 increase. Since the phase ratio should change but slightly in this region, these results suggest that RPLC systems employing solvent mixtures such as THF + H20 might be adequately represented by eq 54. Comparing eq 53 (collapsed-chain limit) and eq 54, we see that the linear terms in d1 differ only in the respective combinatorial contributions. Both equations predict the observed direction of curvature in In k'3(1+2)vs. O1 plots; and from both, it is apparent that In k$(1+2)should be a linear function of solute carbon number for homologous series at any mobile-phase composition. Determination of Mobile-Phase Volume. There is currently much interest in developing a rigorous method for determining the mobile-phase or "dead" volume in a liquid-chromatographics y ~ t e m . One * ~ method ~ ~ ~ ~which ~ appears to be gaining favor is the isotope method, which we now investigate in the light of the lattice-model results. The net retention volume of a solute in an LC system with a binary mobile phase (V3(1+2))is given b y '
-
-
0.4
and A X Z ~ .
0.078r, 0.107r,
other conditions see Figure 17.
0.0I 1
+2x12
0
0.2 0.4
x , =~ and x z 3= xzc (simulating n-alkane solutes); for
0.2 -
1;
0.0
xle and xzein eq 74 become respectively
+
+ K!fl+Z)(vs/7)
(77) + vm where, in the present situation (small polar solutes included), we allow for the possibility of interfacial solute adsorption. For convenience and without detracting from the ensuing analysis, we continue to assume that there is a monolayer of solvent, having a total volume of Vs/7, adsorbed at a planar stationary-mobile phase interface. If the solute is a deuterated isotope of solvent component 1 (designated 1*) at infinite dilution in the chromatographic system, then eq 41 gives In K1*(l+2)= In (41/4) + (02 - dd(x1.2 - x11. - XIZ) + v3(1+2)
= K3(1+2)vs
(tu)(Xii* - x i * C
+ xi,)
(78)
where r3 = rl* = p1 = 1. For a deuterated isotope of solvent component 2 (designated 2*), eq 42 gives In Kz*cl+z,= In (42/&) + 72*(4
- 41)(XlZ* - Xzz* - x12) + rz*(~u)(Xzz* - XZ'C
+ x2c) (79)
(81) L. R. Snyder and J. J. Kirkland, 'Introduction to Modern Liquid Chromatography",Wiley-Interscience, New York, 1979.
The Journal of Physical Chemlstiy, Vol. 87, No. 6, 1983 1081
Theory of Liquld Chromatography
where r3 = r2* = r2. If, compared to the first term on the right-hand side of eq 78 and 79, we reasonably neglect isotope effects on the interaction parameters, then eq 78 and 79 simplify to
elK1.(1+2)= 41 02K2*(1+2)= 42 Similarly, it can be shown that32i33 "2)
= 41ad
(80)
62K!(1+2)= 42ad
(81)
Substitution of eq 80 and 81 into eq 77 yields
count for the often large differences between K3(1+2)and KO,,, especially when very high B2 values are employed.@ These differences are clearly due to the effect of solvent composition (through the interaction-parameter and combinatorial terms) on K3(1+2).From the form of eq 85, it is suggested that the RPLC method can be improved by making measurements at several, well-spaced d2 values and extrapolating to O2 = 0. If need be, even a linear extrapolation (neglectingthe quadratic term) would be preferable to measurements a t a single, high &.
where component 2 is the organic modifier. The term f(r,,c,@Eb)presents no serious problems4 and cannot ac-
Summary and Concluding Remarks The prominent findings from this lattice-model treatment of RPLC are summarized below. (1)At fixed CBP surface coverage ( E ) and chain length (r,), the amount of solvent sorbed at the interior of the stationary phase (4) and the extension of the CBP chains ( u - l ) both increase with increasing CBP chain stiffness (larger @Eb)and increasing solvent-CBP chain compatibility (smaller xlc) (Figures 1, 2, and 13). (2) At fixed ,BEb and rc, 4 generally increases with decreasing xlc and E , and u-l generally increases with decreasing xlc and increasing e (Figures 3 and 4). (3) Our analysis indicates that relatively polar RPLC organic solvents, such as ME and ACN, induce virtual collapse of n-alkyl CBP chains; for slightly polar ones, such as THF, 4 is substantial and the structure of the CBP chains falls between the completely collapsed and fully extended limits. (4) The amount of solvent adsorbed a t the stationarymobile phase interface may be a substantial portion of the total amount of sorbed solvent (interface + interior; Figures 14, 15, and 17-19). However, it appears to be incidental to the solute distribution process, in that the contribution from interfacial solute adsorption is negligible except for small, polar solutes. (5) The solute distribution process in the collapsedCBP-chain limit resembles that of classical liquid-liquid partition chromatography.=* The difference between the distribution constants stems from the different configurational properties of, and the solute entropy in, the respective stationary phases (Figure 7). (6) Both 4 and u-l are relatively insensitive to r,. The logarithm of the solute distribution constant (In K ) increases linearly, but only slightly, with increasing r;l. If the phase ratio is proportional to r,, then the capacity factor (k') should be a linear function of r, (Figures 8, 9, and 11). (7) The dependence of In K and In k' on E is not straightforward (Figures 6 and 10; Table 11). (8) For homologous solute series, In K and In k'vary linearly with solute carbon number (n3). With a given solvent (or solvent mixture), the slopes are the same for all such series; they increase with increasing solvent polarity (Figure 6; Tables 1-111). (9) CBP's exhibit shape selectivity, which increases as the CBP chains become more fully extended. The predicted order of solute retention is as follows: rigid-rod solutes > flexible-chain solutes > globular solutes (Figures 5 and 12). (10) Both In K and In k' increase linearly with increasing reciprocal temperature. For a homologous solute series, the slopes should increase with increasing n3 and increasing solvent polarity.
(82)Y.B. Tewari, M. M. Miller, S. P. Wasik, and D. E. Martire, J . Chem. Eng. Data, 27,451 (1982),and references therein. (83)G. D.Veith, N. M. Austin, and R. T. Morris, Water Res., 13,43 (1979).
(84)Indeed, it can be argued that the anistropic CBP is a better model system for the lipophilic part of a membrane than l-octan~l.~~ (85)D. C. Locke and D. E. Martire, Anal. Chem., 39, 921 (1967). (86)D.E.Martire and D. C. Locke, Anal. Chem., 43,68 (1971).
( V * ) = 4V1.(1+2)+ 02V2.(1+2)= V l Vlad V 2 + VZad+ V , = Vmmm(82)
+
+
where V l + Vladand V 2+ V2adare respectively the total volume of components 1 and 2 sorbed by the stationary phase. Accordingly, eq 82 indicates that the isotope method, which is based on measurement of ( V * ) ,does not give the true mobile-phase volume (V,), but rather gives the total volume of solvent in the system (V,-, i.e., mobile plus sorbed solvent). Support for this analysis is found in ref 4, where it was observed that V21(2) = V,", independent of the nature of the organic modifier (component 2). Since the "weighing" method also gives Vmmm,an alternative route to the determination of V , is the procedure involving linearization of a homologous solute series.61-62As discussed earlier, there is theoretical justification for taking In K3(1+2)to be a linear function of solute carbon number, as is assumed in the linearization method. While the isotope method may not yield a true V,, it should, nevertheless, provide excess isotherms.26@' Manipulating eq 71-73, 80, and 81, we obtain77 r2total = -rltotal
vc
= m[V2*(1+2) - V1*(1+2)1/
(83)
Note that eq 82 and 83 are not restricted to solvent molecules of the same size and are in other respects general equations. RPLC Estimation of Octanol-Water Partition Coefficients. The tediousness of the traditional "shake-flask" method of determining octanol-water partition coefficients of solutes (KO/,)has prompted the development of alternative methods, direct and indirect.82 Among the latter is the use of RPLC with n-alkyl CBP's and, usually, ME + H20mixtures.= (Retention times with pure H20 as the mobile phase are often prohibitively long.) Given the environmental importance attached to the theory is applied here to examine the relationship between the RPLC distribution constant (K3(1+2))and KO,,. The collapsed-CBP-chain limit should apply to the RPLC measurement. Neglecting the effect of the OH group in l-octanol, the main difference between the RPLC distribution constant with pure water (K3(1))and KO,, is in the translational freedom and structure of the respective hydrocarbon phases (see Figure 7): In K3(1)=
Ko/w
+ f(r3,6,@Eb)
(84)
where f is a linear function of r3. From eq 52,53, and 84
K3(1+2)= In Kolw + f(r3,e,@Eb)+ 02r3[x23- x 1 3 &[(2r3 + l)(r2 - 1)/3r2I + &2r3Xi2 (85) Xi21
1062
Martire and Boehm
The Journal of Physlcal Chemistry, Vol. 87, No. 6, 1983
(11) With mixtures of two relatively polar solvents (e.g.,
ME + H 2 0 or ACN + H,O), the dependence of In K or In
k ’on mobile-phase composition follows a simple quadratic equation (eq 52 and Figure 16). (12) With mixtures of a highly polar solvent (1) and a slightly polar one (2) (e.g., THF + H,O), & does not diminish appreciably until the mobile phase becomes fairly rich in component 1 (high el). Equation 54 adequately describes the dependence of In K or In k’ on mobile phase composition for dl S 0.8 (Figure 17; Table IV). (13) The “isotope method” does not yield the true mobile-phase volume, but rather the maximum mobile-phase volume. It does, however, provide sufficient information to construct excess sorption isotherms. Further applications of the theory to such areas as a solvent-strength or eluotropic scale for RPLP4v81 and gradient-elution RPLCS8and more elaborate treatment of selected experimental data48 will be subjects of future studies. The present analysis should provide a reasonable basis for interpreting retention and selectivity in RPLC. I t should be kept in mind, however, that several simplifications were made: assuming a uniform CBP-chain segment density, neglecting any effects of residual silanols, and assuming complete containment of solute molecules within the stationary phase. Departure from the first two simplifications would present formidable problems in the lattice statistics, especially in the absence of any molecular-level guidance from experiment. By retention of these simplifications, it is unlikely that the model can be used to investigate, e.g., the possibility of phase transitions in the CBP.87,88 However, it appears feasible to refine the model to allow for possible effects from residual silanols and incomplete solute containment. Work in that direction is underway. Finally, the Bragg-Williams approximation has been exclusively employed here. For the species under consideration, the Bethe-Guggenheim or quasi-chemical app r o ~ i m a t i o nmay ~ ~ be more appropriate and accurate. Acknowledgment. This research was supported by a grant from the National Science Foundation. We also thank M. F. Burke, E. A. DiMarzio, G. Guiochon, B. L. Karger, E. sz. Kovlts, C. H.Lochmuller and D. C. Locke for many stimulating discussions, and G.G. and E.sz.K. for access to unpublished results.
Appendix I The monomer solvent molecule (component 1) is taken to have a type-0 segment. The n-alkyl CBP chain is taken to have r, segments of type s (aliphatic). Clearly X l c = xos (AI) We let the solute molecule (component 3) be chemically heterogeneous and have rG type-s segments and r type-q segments, where rSs rgq = r3: Using volume%action averaging for the segmental pair interactions (w)
+
w13
=
(r3a/r3)w0s w11
w33
=
(r3a/r3)2wss
+
x13
=
(r3a/r3)XOs
+ 2(r3ar3q/r32)wqs + h q / d 2 w q q
which, when substituted into eq 21, gives (87) D. Morel and J. Serpinet, J. Chromatogr., 200,95(1980);214,202 (1981). (88)R. K.Gilpin and J. A. Squires, J. Chromatogr. Sci., 19,195(1981). (89)T. L. Hill, “An Introduction to Statistical Thermodynamics”, Addison-Wesley, Reading, MA, 1980,Chapter 14.
(A2)
In the same manner we obtain =
(r3,/r3)2X,a
(-43)
+ x l c ) = 2r3sxOs + r3q(xOq + XOS - xqs)
(A4)
x3c
From eq Al-A3 r3(x13
- x3c
which ranges in value from 2r&x, (for q = 0) to 2r3xos(for q = s). If component 1 is a polar solvent, then xos is expected to be positive and to increase with increasing solvent polarity, with the sign and magnitude of xosbeing governed by woo. From eq A2 and A3
r3(x13- x
= rasxb + r3&xOq - xqs)
d
(AS)
as used in eq 64. If we let the chemically heterogeneous rz-mer solvent have rZa= r2 - 1 type-s segments and rPp= 1type-p segment, then by analogy to eq A3 xzc
=
(A6)
xps/r22
Appendix I1 Subtracting from eq 22 the standard chemical potential of component i (pi0 = &Ii = l)), we have for a bulk solution @(pi- p:) - In e, = In X i = (2rj + 1) In (QOO/Qoe) +
-
l>m
-
where X i is the activity coefficient of component i (Xi 1 as !Ii l),Q? = 1 - Ri/3, Q< = 1 - ClelR,/3, and R = ( r - l)/r. For solute (i = 3) at infinite dilution in a single solvent (j), equation B1 gives In = (2r3 + 1) In [ ( l - &/3)(1 - Rj/3)] + r3xj3 (B2) The solute distribution constant, for partitioning between two neat, immiscible solvents, 1 and c, is then
- -
- -
Letting rl 1 (R, 0) and rc m (R, l),we obtain eq 58. Expanding the logarithmic term in eq B2 through first order in R, we have In K3(1) = [(2r3 + 1 ) / 3 1 [ r ~-~q 1 1
+ r3(x13- x d
(B4)
- r2-’1 + r3(x23 - x d
035)
Similarly In &(2) = [(2r3 + 1)/31[r;’
From eq B1 In X;(1+2) = [(2r3+ 1)/3][e1(r3-1 - rl-l)
(r3q/r3)wqa
= woo
+ (r3q/r3)xOq - (r3ar3q/r32)xqs
r2-l)I
+ 02(r3-’
-
+ r3[&x13 + e2X23 - 4 ~ ~ x 1(B6) ~1
is the infinite-dilution solute activity coeffiwhere cient in a mixture of 1 2. With j = c in eq B2
+
In X& = [(2r3+ l)/3][r3-l - rcll + r3xgC (B7) Combining eq B4-B7 In K3(1+2)= ln
[X3m(1+2)/~3m(C)l O1
In
K3(1)
=
+ O2 In
which has the same form as eq 52.
- r3°1e2x12 (B8)
K3(2)
