Anal. Chem. 198% 61, 471-482
supporting electrolyte) without affecting the binding of heavy metals. Although these advantages are presented within the framework of continuous-flow ASV, silica-immobilized algae columns could greatly benefit other on-line analytical schemes, e.g., flow-injection atomic absorption spectroscopy. While on-line separation is the focus of the present study, the utility of algae columns as preconcentration tools can be easily envisioned. Because of their foundation on bioaccumulation processes, algae columns should be extremely useful for on-line metal speciation measurements. LITERATURE CITED Ruzlcka. J.: Hansen. E. A. Flow Inlectbn Analvsis: Wilev: New York. 1988. Olsen, S.; Pessenda, L. C. R.; Ruzlcka, J. Hansen, E. A. Analyst (London) 1983, 108, 905. Marshal, M. A.; Mottola, H. A. Anal. Chem. 1985, 57, 729. Maiamas, F.; Bengtsson, M.; Johansson, G. And. Ch/fn. Acta 1984, 160, 1.
47 1
(5) Wang, J.; Dewald. H. D. Anal. Chem. 1983, 55, 933. (8) Yang, X.; Rislnger, L.; Johansson, G. Anal. Chim. Acta 1987, 792, 1. (7) Darnall. D. W.; Qreene. B.; Hosea, M.; McPherson, R. A.; Henzi, M.; Alexander, D. In Trace Metal Removal Irom Aqueous Solutions; Thompson, R., Ed.; Royal Society of Chemistry: London, 1986; Special Publication No. 81, p 1. (8) Watklns. J. W., Elder, R. C.; Oreene, B.: Darnall. D. W. Inorg. Chem. 1987, 26. 1147. (9) Greene, B.; Henzl, M.; Hosea, M.; Darnall, D. W. Bbtechnol. Bioeng. 1988, 26, 764. (10) Darnall, D. W.; Greene, B.; Henzl, M.; Hosea, M.; McPherson, R. A,; Sneddon, J.; Alexander, D. Envlron. Scl. Techno/. 1988, 20, 206. (11) Gardea-Torresdey, J.; Darnall, D.; Wang, J. Anal. Chem. 1988. 60, 72.
(12) Wang, J. Stripping Analysis : Principles , InstrumntaMOnand Applica fbns ; VCH Publishers: Deerfield BeechlWeinheim, 1985.
-
RECEIVED for review September 12,1988. Accepted December 1,1988. This work was supported by the National Institutes of Health (Grant No. GM 30913-05) and the National Science Foundation (Grant No. CBT-8610461).
Theory of Homopolymer and Oligomer Separation and Its Application to Gradient Elution Liquid Chromatography Richard E. Boehm* and Daniel E. Martire
Department of Chemistry, Georgetown University, Washington, D.C. 20057
A slmpler, more general derlvatlon is developed for a prevlously Introduced theory of homopolymer fractionation by gradient elutlon high-performance liquld chromatography (HPLC). Improved treatments are given for the determlnatlon of the solvent-stationary phase adsorption Isotherm and the soivent-entralnedsolute expansion factors. Retentlon tlmes are evaluated for (hypothetical) homologous serles of flexible, chalntlke, chemlcalty homogeneous solutes In gradlent elutlon HPLC. The feasiblllty of HPLC fractionation of flexible oilgomers and homopolymers is confirmed. The theory also Investigates the signHicant ellects that sample concentratlon has on the retentlon behavior of hlgh molecular welght homopolymers. For example, the molecular weight dependence of the variation of the capacity factor with mobile phase composition Is predicted to depend strongly on the sample concentration in the elution band. The theory Is compared wlth the empirical linear solublllty strength theory proposed by Snyder and co-workers.
INTRODUCTION Successful fractionation of high molecular weight homopolymers using gradient elution high-performance liquid chromatography (HPLC) has bees reported (1-4,5-11). For example, polystyrene homopolymers and/or oligomers in a molecular weight range 105 I molecular weight I lo7 have been separated efficiently and rapidly by means of gradient elution high-performance liquid chromatography (HPLC) utilizing either a C-18 or C-8 chemically bonded stationary phase (CBSP) and a methylene chloride-methanol mixed mobile phase (1-4). Other mixed mobile phases such as H,O-THF have also been employed (5,11). Separation of styrene oligomers also has been achieved by supercritical fluid chromatography with a density-programmed supercritical n-pentane mobile phase (12,13). A considerable variety of other homopolymers (2-4) and biopolymers (6-10,14-16)also 0003-2700/89/0361-047 1$0 1.50/0
have been fractionated effectively through gradient elution HPLC in various mixed mobile-stationary phase combinations. A statistical thermodynamic theory of homopolymer fractionation and its application to gradient elution HPLC has been developed by Boehm, Martire, Armstrong, and Bui (hereafter designated BMAB-1) to describe the retention behavior of an isolated flexible homopolymer molecule distributed between a binary mixed mobile phase and an idealized stationary phase consisting of sorbed solvent (s) on a homogeneous planar surface (17). In this analysis the Flory-Huggins lattice model approach is applied to an isolated polymer molecule and entrained solvent molecules in each chromatographic phase and nearest neighbor interactions are included in the Bragg-Williams random-mixing approximation (18,19). The BMAB-1 theory successfully accounts for the observed experimental trends of an increasingly abrupt transition from very high to very low retention as the degree of polymerization, M , of the homopolymer increases and as the mobile phase becomes sufficiently enriched to a critical composition of the better solvent for the polymer. The critical composition represents that mobile phase composition where the capacity factor becomes unity. The critical composition is predicted to be a monotonically increasing function of M for flexible homopolymers of sufficiently large M (M L 15) and this dependence emanates primarily from the molecular flexibility of the polymeric solute which allows size and shape alterations by solvent uptake or expulsion in response to its solvent and/or surface environment (18,19). The dependence of the critical composition on M also generates the opportunity for polymer fractionation by gradient HPLC. The BMAB-1 theory was subsequently extended to include intermolecular polymer segment-polymer segment interactions and thus be applicable to semidilute polymer solutions. This extension is hereafter referred to as the BMAB-2 theory (20). The onset of the semidilute regime where different flexible polymer molecules begin to interpenetrate appreciably occurs in good solvents for a volume fraction of chain monomers given 0 1989 American Chemical Society
472
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
by E M-4/5(21). Hence departures from infinite dilution can be anticipated for small values of a, when M is large (e.g. when lo3 5 M 5 lo5, 4 X 2 0 I1 X lo-*). When solute sample concentrations during elution fall in the semidilute solution range, concentration effects on chromatographic retention behavior must be included for meaningful comparisons between theory and experiment. The BMAB-2 theory predicts that the transition region between high and low polymeric solute retention becomes more gradual as the concentration increases. Glockner and co-workers have formulated a precipitation model to describe the chromatographic retention behavior of polymeric solutes (22-26). This analysis asserts that a transition from essentially complete to negligible retention occurs abruptly whenever the polymeric solute becomes soluble in the mobile phase whose composition is progressively enriched with the better polymer solvent through gradient elution. The BMAB-1 and BMAB-2 analyses respectively treat isolated polymer molecules and semidilute polymer solutions and are entirely incapable of describing bulk precipitation phenomena interpreted as a collective aggregation of polymer solute onto the stationary phase. Thus the Glockner and BMAB analyses apply to distinctly different situations, and there is no connection between them. Snyder and co-workers have interpreted the chromatographic retention of polymeric solutes in terms of an empirical linear solvent strength theory (LSS) (6, 11, 27, 28). The advocates of LSS theory suggest that “conventional” chromatographic concepts which apply to small-molecule separations are completely adequate for the comprehension of the chromatography of polymeric solutes for sufficiently small sample concentrations although precipitation is observed to occur at higher sample concentrations (27). One purpose of the present paper is to develop a more general and transparent derivation of the BMAB-1 theory. This is achieved by application of the McMillan-Mayer theory of dilute polymer solutions toward the derivation of a more general expression for the capacity factor 12 for flexible chainlike polymer molecules distributed between mobile and stationary chromatographic phases. In particular, the present generalization of the original BMAB theory given in the next section is anticipated to be more reliable for lower molecular weight oligomen since more general expressions for the solvent entrained polymer coil expansion factors are introduced which apply for smaller values of M than the asymptotic large M results used in the original BMAB analysis (29). Additionally a more detailed treatment of the solvent-stationary phase adsorption isotherm is provided. The analysis is initially applied to the theoretical prediction of partition coefficients of oligomers and polymers between immiscible solvent phases. Then, explicit expressions are obtained for the chromatographic capacity factors and corresponding retention times in gradient elution for flexible, chainlike oligomers and polymers. The effects of solvent and/or surface environment on the configurational behavior of the chainlike solute molecules are included in the analysis. For comparison we also determine the corresponding chromatographic properties for perhaps hypothetical chainlike solutes which assume average configurations that are independent of their solvent environment. We refer to these solutes as structurally rigid to indicate that solvent swelling effects play no role in determining their average configurational behavior. Furthermore we also investigate the effects of including nonlinear (i.e. quadratic) contributions to the mobile phase composition dependence of In k on chromatographic retention times in a linear gradient elution. The nonlinear contributions are found to be more important for the lower M solutes.
In the results section we present examples of the critical compositions, retention times, capacity factors, and solute swelling ratios as a function of M for hypothetical homologous series of both flexible and structurally rigid, chainlike oligomers and homopolymers. One typical set of values of the relevant energy parameters required to implement the theoretical calculations is employed throughout. For purposes of comparison, calculations for the flexible solutes are performed by using both the asymptotic and the more general version of the solute expansion factors given in ref 29. Retention times calculated for the flexible oligomers and homopolymers are found to increase monotonically with M for all M I3 in a linear solvent gradient, which suggests the feasibility of fractionation of homologues by molecular weight. The retention times calculated for the structurally rigid solutes, however, possess a maximum when considered as a function of M for M 100 and this suggests severe difficulties for the fractionation of the large M homologues. The section dealing with the effects of sample concentration upon retention combines the current analysis with the BMAB-2 theory in order to determine the quantity S 5 -[d In k / d ~ ] k which = ~ represents the rate of change of In k with the mobile phase composition, x , when k = 1. In general S is a function of M and C which represents the average concentration of polymer chain monomers in the solute elution band. When C 0 and M is large, the limiting slope S is predicted to be proportional to M . However, as C increases, S changes more gradually and becomes a somewhat weaker, although more complicated, function of M . Finally for sufficiently large C where semidilute solution behavior is approached, S becomes independent of M. These results emphasize the importance of consideration of sample concentration, even when it is low enough t o guarantee solubility, whenever comparison between theory and experiment is contemplated for polymeric solute chromatography. Finally, we briefly compare the present theory with the LSS theory of homopolymer retention. We conclude that the present statistical thermodynamic treatment provides a molecular description of the chromatographic retention behavior of homopolymers and/or oligomers which accounts for the observed experimental trends. The LSS approach fits empirical relationships to experimental observation.
-
-
THEORY In this section we derive expressions for the partition coefficient and capacity factor of a flexible, chainlike solute consisting of M chemically homogeneous monomers distributed respectively between two immiscible solvents or mobile and stationary chromatographic phases. The immiscible solvent or chromatographic mobile phases are generally binary solvent mixtures and the stationary phase consists of a homogeneous planar adsorption surface and sorbed solvents. Size exclusion effects on solute elution are not considered here. The solute concentration in each phase is assumed to be infinitely dilute so that intermolecular solute-solute interactions can be neglected. This assumption permits advantageous application of the McMillan-Mayer theory of dilute solutions in the subsequent analysis (19). Let N,, (i = 1,2) and N3a,respectively, denote the number of solvent molecules of species 1 and 2 and solute molecules (3) in phase a and let pra(i = 1,2,3) represent the chemical potentials of each component in phase a. The grand canonical partition function for phase a is
Z(btaI, Va, P) = CCCQa(Nla,Nza,
N3a,
Va,
PI
X
N13dh 2
n ~ i a ~ b a ~ = 3 aC+Nh(hla, ~ h X2a,
r=l
Va,
(1)
Nh
where V, is the volume (or area) of phase a, 6 = l/kBT, where
T i s the absolute temperature and k~ is Boltzmann's constant, Qa(Nla, N2,,Na,Va) is the corresponding canonical partition function with the p dependence suppressed, and Xi, exp(&J (i = 1, 2, 3) denote the corresponding activities and where Pa(Va*) represents the probability that the solute-entrained solvent system occupies a volume Va*. The separation of the canonical partition functions in eq 8 into a product of partition functions which apply to regions of volume Va - Va* and Va* corresponds to decoupling the interactions between them and regarding them as independent systems. This implies neglect of surface tension contributions between the solute-entrained solvent system and the external solvent environment. This factorization is expected to be reasonable provided the intermolecular interactions are short-ranged and
The average number of solute molecules in phase a is
va>> va*.
It is again convenient to invoke the maximum term approximation in eq 8 by finding the value of Va* Nauwhich maximizes the generic term of the summation over Va*
where
' ~ ~ a ( I l ~ i a - ' ~ ) ~ a (xN a ) 1=1 QaWiat 1,Na)Qa(Nia" - li - Nia, 0, Va - N a v ) Qa(Nia + XiaMt O,Na)Qa(NiaO - N la . - XiaM, 0, Va - N a u )
(N3a)i1i2 =
and Qa(Nla, N2a9 N 3 a = n, Val E Q a W i a , n, Val (n = 0, 1; i = 1,2) are the canonical partition functions for a binary solvent mixture and a single solute molecule-binary solvent system, respectively. We employ the maximum term approximation for the by setting a t constant Va and p
and 2 +la
Qa(NiL,1, Va)IIXiaN"' i=l
(6)
Furthermore, if NiaO - NIL = 1, (i = 1, 2 ) where the li are nonnegative integers with l1 + l2 = M for incompressible fluid phases, then from eq 4 and 6 2 (N3a)l1lZ =
X3a(~Xia"i)Qa(Niao - li, 1, Va)/Qa(Niao, 0, Va) (7)
where the 1112 subscript indicates that addition of a solute molecule of M repeating units in phase a at fixed Va displaces li type i monomeric solvent molecules. Since the Helmholtz free energy Fa is related to the canonical partition function by @Fa = -In Qa, eq 7 can also be expressed in terms of the free energy difference for the process of introducing a single solute molecule in phase a with the concomitant removal of li type i solvent molecules with I, + l2 = M
In
(N3a)111z 2
=
- iClipia) - P(Fa(NiaO - l i t 1, Va) - Fa(Niao, 0, Va)) =l Whenever M is small, irrespective of the solvent environment, and for solutes in a poor solvent environment, irrespective of the value of M, a solute molecule behaves as a compact particle and occupies a volume, V* = M u , where u is the volume per monomer. Under these circumstances eq 7 can be employed directly in the subsequent analysis. However, for chainlike oligomer and polymer solutes with intermediate and large values of M, respectively, in good solvent environments, the effective dimensions of the solute expand by solvent uptake. Hence the volume, Va*, occupied by the oligomeric or polymer solute and entrained solvent molecules exceeds Mu and is subject to fluctuations. For this situation it is expedient to rewrite eq 7 as
(9) Here Na = N,, + N,, + M represents the number of sites occupied by the solute and entrained solvent molecules in an incompressible fluid and zia = Nia/(Na- M)(i = 1, 2) represents the solvent volume fractions. We further assume that the composition of the solvent entrained within the solute is identical with that in the bulk solution. For polymeric solutes in good solvents the volume fraction of monomers varies as M-4/6in spherical coils (M-'l2 in disks) and is very low and induces only small composition variations in the entrained solvent from the bulk value. Utilization of eq 9 requires explicit expressions for the canonical partition functions and Pa(Na). We apply the familiar Flory-Huggins lattice model theory of polymer solutions to a single isolated polymer molecule and a binary solvent mixture to express (18, 19) Q a W i a , 1, N a ) = Naca(ca 1)M-2N,-M(Na!/N~a!N2a!)qlaN1aq2aNhq3, exp(-PEa) (10) where ca is the coordination number of the lattice in phase a, qia (i = 1,2,3) are the molecular partition functions for each respective species, and BEais the reduced interaction energy of the solute-entrained solvent system. We further express qa = qa'MVa where qa' is the internal, temperature-dependent segmental partition function for the solute, and the factor Va appears from the translational partition function of the center of mass of the solute. The solvent molecules and polymer segments are assumed to interact with themselves and each other through nearest-neighbor pair interactions, and within the framework of the Bragg-Williams random-mixing approximation, we set
@Ea= (Pca/2Na)(N1,2w11 + N 2 a 2 ~ 2 2+ 2N1aN2aw12) + ( P ~ a M / N a ) ( N l a ~ 1 3+ N2aw23 + M w 3 3 / 2 ) + P(Niaeia* + Nzaeza* + Me3a*)aas (11) Here the wi, (i, j = 1-3) represent the nearest neighbor pair interaction energies between solvent molecules and/or polymer segments and the era*are surface-solvent molecule and surface-solute segment adsorption energies which may appear in a stationary chromatographic phase (a = s) and 6,, is the Kronecker delta. The remaining canonical partition functions in eq 9 which do not involve the solute are assumed to be given generically by
474
ANALYTICAL CHEMISTRY, VOL. 61, NO. 5, MARCH 1, 1989
where 2
with @E,' = (PC./~N,')(N~,'~W~~ + Nb"4022 + 2NlalN2a'w12) /3(Nllela* + N2;eza*)Gaa. The Helmholtz free energy is
+
2
PF,/N,'
Cxi, In xia + XlaXZaCa(w12 - ( ~ 1 + 1 ~22)/2)2 Cxi, In (qi, exp(-P,wi;/2) exp(-Pe;,*&J)
= -In Q,/N,' =
i=l
i=l
with xia = Nia'/N,' (i = 1,2). This is the Bragg-Williams free energy of a two-component regular solution mixture (19). For application in eq 9 the NiL and N,' in eq 12 assume the following sets of values (Ni,', N,'): (Ni, + xi&, A',), (NiaO Ni, - li, V,/u - N,) and (Niao- Ni, - xiaM, V,/u - N,). Substitution of eq 10,11, and 12 into eq 9 and taking the thermodynamic limit V, m and N,/Va 0 gives after considerable simplification
-
-
(1- M/Na)-(Na-M exp(P( C lieia*- Me3a*)) i=l 2
exd-pcaCli(wi3 - wii)) exp(llx2a(X13a+ x12' - x23') + i=l
bla(~2+ 3 ~~ 1 2 '- ~13')) exp((W/N,) x (XlaX13, + X2aX23a - XlaX2aX12a)) (l3) where xija= @c,(wij - (wii wjj)/2) (i, j = 1-3) represents the reduced interchange energy required to form an i-j nearest neighbor pair from an i-i and a j-j nearest neighbor pair. The volume occupied by the polymeric (or oligomeric) solute and entrained solvent molecules, N,u, is determined from the maximum term condition a In (N3,)lllZ/aNa= 0
+
+
d In P,(N,)/dN, N;l - In (1- M/N,) - M/N, (M/Na)2(xla~13a+ x2ax2sa - ~ l a ~ ~ a ~ 1=2 0' ) (14) Solution of eq 14 requires specification of P,(N,). If M is sufficiently large and the solute manifests the characteristics of a solvent-swollen spherical coil in phase a, then the normalized Gaussian approximation
P,(N,) = ( 3 / 2 ~ M b ~exp(-3R,2/2Mb2) )~/~ = (3/ 2 ~ M b ~ e)~~p (/- ~3 N , 2 / ~2M) / (15) where N, (R,/b)3 is appropriate. In eq 15, R, represents the diameter of an equivalent spherical coil of uniform segment density and u = rb3/6 is selected to represent the volume of a monomer unit or single lattice site so that N, = rR:/6u = (Ra/b)3. The mean square end-to-end distance of an unperturbed (i.e. nonswollen) polymer coil is (R2)o= Mb2 and the average number of lattice sites occupied by the unperturbed coil is N," = (R,2)03/2/b3 = M I 2 . Hence the ratio N,/N," = CY: or N , = a,3MI2will provide a measure of the expansion ( a , > 1) or contraction ( a , < 1) of an isolated polymer molecule relative to its unperturbed dimensions in response to its solvent environment. Insertion of eq 15 into eq 14 and substitution of a, = (yaM1/2)-1/3 gives y,1/3M-4/3+ y,( 1 - 1/M) + In (1 - y,) 2
Y:(xla(l
- XlJX12'
- X ~ i a x i 3 ' ) = 0 (16) i=l
When ya 2 a nontrivial r ) # 0 solution with 0 < r) I1 exists and corresponds to phase separation. In terms of r ) eq 23 becomes
M-l In
k3&
+
= r)(X23 - X13) - (5/2)M-4/5(g,2/5 - gb2/5) M-' In 4 (3/5)M-l In (g,/gb) (23')
where ga = g'- ~ ~ x 1 2 + / 4 dxz3 - x13)/2; gb = g'- v2x12/4 d X 2 3 - x13)/2 with g' = (1 - Xl3 - x23 + x12/2). Phase preference in phase a or b for polymer molecules of large M essentially depends on the sign of the right-hand side of eq 23' and the phase richer in the better solvent (Le. the one with the smaller xi3) is strongly favored. For the following comments we consider 0 as constant and set 4 = 1. If 1 is the better solvent, then ~ 2 -3 x i 3 > 0 and k&,b > 1indicating solute preference for phase a which is richer in solvent 1. For large M solutes, k3,b = exp(Mr)(xZ3- ~ 1 3 ) ) which can be either very large or very small depending on whether ~ 2 -3 xi3 > 0 or ~ 2 -3 xi3 < 0 and to a lesser extent on the magnitude of r). For temperatures slightly below the upper critical solution temperature, T,, r ) = (3AT/T,)1/2where AT T, - T 1 0 in the mean-field approximation. Hence for homopolymers of relatively large M and for small AT/ T,, In k3ab = M(3AT/T,)1/2(~23 - ~ 1 3 where ) ~ 2 and 3 xi3 are determined at T = T,. More generally, we anticipate that r ) = (AT/TJ@ and In It&,), = M(AT/T,J@(x, - ~ 1 3 for ) large M where the critical exponent p = 1/3 differs from the mean-field value of 1 / 2 (33). The preceding comments presuppose that the presence of solute does not perturb the phase separation behavior near T,. When a and b, respectively, represent stationary (a s) and mobile (b m) chromatographic phases, and the stationary phase is assumed to consist of a planar adsorption surface with adsorbed solvents, then the fi in eq 22 become, with c, - c, = 1
-
-
+ ((1 - xiJ(Ca/Cm)-
(1 - Xi,)) (~i3, + ~ 1 - ~j3,)) 2 ~ (i = 1,j = 2; i = 2, j = l ) , where pAe(i, 3; s) = @(ei,- e3, - wii + wi3) (i = 1 , 2 ) with ei, 1 eb* - kBT In (qk/qim). Furthermore if eq 15 and 18 are employed for P,(N,) and P,(N,) and we utilize the asymptotic approximations N , = g,3/5M.s and N , = g,'/2M.5 which apply for large M polymers in good solvent
fi
= Xis exp(pAe(i, 3; S)
(Exi,exp(-@(Ae(i, 3; s) + i=l
k3ab = (vaca(c, - 1)M-2Napa(l- hf/Na)M-Na/ vbc,(cb 1)M-2N&,(l - M/Nb)M-Nb)eXp((l/2 i=l
475
where J = dc,(c, - 1)2(2n/3)1/2/c,(c, - 112. The solvent compositions xlm and xla are related by the equilibrium requirement that plm - pZm= plS- pzs where @pia = 4In Qa/dNk(a = m, s). Utilization of eq 12 for Q, generates the isotherm (x1,/(1 - d )exp(xlzs(1- 2x1,) + @(el,-e2,)) = (xlm/(1 - XI,)) exp(x12m(l - 2x1,) + P(wii - w22)/2) (25) In many applications of interest @(el,- e2,) > 1 and x2, T, (low retention) where T, kBX1k/(Cm(W13 - (wll + satisfies Fo = F'(x = 1, ~ 1 3 , ) = M-' In ko = 0. In the limit M-- m, T," = -Ae(l, 3; S)/(kg ln ((c, - l)/(cs - 1))since the last three terms on the right hand side of eq 36 vanish and k" = expM(1n ((c, - l)/(c, - l))((T,"/T) - 1)). Clearly high (low) retention occurs with T < T,"( T > T,")and the transition becomes increasingly abrupt as M increases but becomes discontinuous only in the limit M m a t T = T,", where the solute molecule itself possesses an infinite number of degrees of freedom. For large but finite M in the vicinity
-
Of X13c
(37) k" = ~ X P ( M A I O ( X-Ix13c)) ~ where AI0 = dF0/dxl3, = Ae(1, 3; s)/kBTx13, + (cS/cm). ('g,*M)-l/Z - gm*-3/5M-4/5- cs/2cds*M + (3/5)/gm*M* An explicit expression for the retention time, tR, as a function of M for linear gradient elution has been previously derived for the case where In k = JAIJM(x,- x ) (17)
tR =
t,
477
+
(IAlIMB)-' In (1 + IAlIMBtm exp(JAIIM(x, - ~ ( 0 ) ) ) )
(38) where t, is the time required for unretained solute to pass through the column, and the mobile phase composition at the column inlet is assumed to vary linearly in time according to the relation x ( t ) = x(0) + Bt, where x ( t ) , x ( O ) , and B are the mobile phase compositions a t the inlet at time t ( t 2 O), a t time t = 0, and the time rate of change of the composition gradient respectively. Whenever JAIJMBt, exp(JAIJM(x,~ ( 0 ) )>>) 1, eq 38 reduces to
t~ = t ,
+ B-'(x, - ~ ( 0 )+) (IA,IMB)-'
In (IAIIMBt,)
(38') = t, + ( x , - x(O))/B=
Clearly for large M and x ( 0 ) < x,, t R t , + t, where t, is the time required for the mobile phase composition at the inlet to develop to xc when a linear gradient is sustained. Also utilization of eq 32 in eq 38' allows one to express asymptotically for large M that In tR = In tRm- const M-'/2 where tRm= t , + B-'(x," - x ( 0 ) ) . Equations 38 or 38' provide a relationship between t R and M and predict that a very dilute homopolymer mixture can be separated into molecular weight fractions by linear gradient elution HPLC when x ( 0 ) is less than the critical composition of the lowest M homopolymer present. The order of elution then proceeds successively from the lowest to the highest molecular weight constituent since x,(M) - x ( 0 ) monotonically increases with M. If, however, x ( 0 ) > x,(M*) where x,(MI) is the critical composition for some intermediate M = MI, then eq 38 reduces to t~ = t,(l + exp(-JA,IM(x(O)- x,))) = t , for all polymer components with M < MI. Thus separation for all components with M < MI should be difficult, if not impossible, since they proceed essentially unretained through the column provided their M values are sufficiently large to render exp(-(Al(M(x(0) - x , ) ) 0 and no transition from high to low retention is apparent for values of x in the range x , 5 x I1 in disagreement with experiment. The LSS theory is also incapable of predicting that the mobile phase composition, x,, at the transition, where e.g. k(x,) = 1,depends on and increases monotonically with M. Hence the LSS approach cannot theoretically forecast (although it might experimentally anticipate) homopolymer fractionation by molecular weight through gradient elution HPLC. If the substitution k(x,) = 1is employed to eliminate k, in the LSS formulation, then k = e ~ p ( 0 . 2 2 ~ / ~-( xx),) which, at least, renders k 2 1for 0 5 x 5 x,and k < 1for x,< x = 1. However, the LSS approach fails to provide a theoretical prescription for the determination of x , and its depencence on M , the chromatographic environment and the structural and chemical nature of the solute homologues. The polymeric attribute of flexibility plays an important role in determining the dependence of x , and the retention time on M . The absence of such flexibility in small and/or structurally rigid solute molecules leads to a decidedly different dependence of tR on M from the flexible polymer behavior (see Figure 2). This comparison challenges the LSS contention that retention of high polymers can be understood in terms of small molecule retention. The present analysis assumes regular solution behavior and utilizes a rather idealized planar surface model for the stationary phase. More sophisticated models can be adopted to describe the statistical thermodynamics of the solutions (34) and the stationary phase structure (35-38). As M increases, however, the effects of local irregularities of surface structure (e.g. pores and slits) should become less significant in determining retention behavior and the planar surface approximation should become more viable. The proposed improvements can be incorporated readily into the general theoretical formulation expressed in eq 1-9. Despite its limitations the present analysis provides a molecular description of the chromatographic retention behavior of flexible homopolymers in reversed-phase HPLC which is consistent with the observed experimental behavior and trends, particularly when the sorbed solvent and polymeric solute possess comparable affinities for the surface. Much of the present analysis can be transcribed directly to the fractionation of oligomers and homopolymers using supercritical fluids and density programming to consistently increase the density and hence upgrade the quality of the supercritical fluid mobile phase. Here the supercritical fluid and the “holes” respectively play the roles of good and poor solvent for the solute homologues. Separation of styrene oligomers by supercritical n-pentane using capillary or C-18 packed columns serve as a prototype system (12). Finally extension of the present analysis to investigate the chromatographic retention behavior of chemically heterogeneous chainlike and rodlike polymers and oligomers seems worthwhile. -+
B2m)
Bzs
b C
P Ea eis*
eia
Ae(i, 3; S)
Ae(i) Fa F
fi
f(u,M) ga ga* g’
H(X) J k , k’ k” kE*
ko k3ab
kB
M Na Nao Nia (N3a) P a (Na) Qa
4is 43:
Rm, Rs
s, so
GLOSSARY A,
a
B
= ( j ! ) - l a ~ F ( ~ , ) /0’a = x ~1, 2, the j t h partial derivative of the free energy of transfer of solute with respect to the critical composition divided by j!. A:, A / , and A:* are similarly defined phase label (e.g. a = m and a = s correspond respectively to the mobile and stationary phase) linear time rate of change of the mobile phase composition e-)
T T,, T,”
AT
481
second osmotic virial coefficient of polymeric solute in the mobile and stationary phases, respectively length of a solute monomer unit concentration of polymeric solute (mg/cm3) in the elution band lattice coordination number in phase a maximum thickness of a polymer disk adsorbed on a planar surface internal energy in phase a surface-solvent (i = 1, 2) and surface-polymer segment (i = 3) adsorption energies =eis* - kBT In (qis/qim) (i = 1-3) surface-solvent molecule and surface-polymer segment adsorption free energies =cis - e3a- wii + wi3; i = 1, 2 solute-solvent displacement energy at the stationary phase =Ae(i, 3; S) kBT In ( ( C , - l ) / ( C a - 1)) Helmholtz free energy in phase a =M-’ In k free energy of transfer of a solute molecule between the mobile and stationary phases per monomer of solute. Similar definitions apply to F’, F“, and F W function defined after eq 21 of text function defined by eq 28 =1/2 - C i = 1 2 ~ i a ~ i + 3 a xlaxzax12a measure of the quality of a binary solvent for a polymeric solute =1/2 - ~ 1 3 ~ =1 - x13 - x 2 3 + X 1 d 2 Heaviside unit steD function: H = 1. X 1 0: H =
+
o,x
