3020
J. Phys. Chem. 1980, 84, 3620-3630
A Unified Theory of Retention and Selectivity in Liquid Chromatography. 1. Liquid-Solid (Adsorption) Chromatography Richard E. Boehm" and Daniel E. Martire Department of Chemistry, Georgetown University, Washington,D.C. 20057 (Received: April 18, 1980)
A statistical mechanical analysis of solute retention in liquid-solid chromatography is formulated. The model examines the competitive equilibrium at the molecular level between solvent and solute molecules distributed between generally nonideal mobile and stationary phases. The stationary phase is assumed to be a monolayer of solute and solvent molecules formed by adsorption on either a planar homogeneous or heterogeneous surface. The mobile phase is assumed to be a regular solution of solvent and solute molecules. We employ a cubic lattice model in conjunction with the Bragg-Williams random mixing approximation to enumerate the number of accessible configurations and the total nearest neighbor pair interaction energies in each phase. In the limit of infinite solute dilution, we obtain explicit expressions for the distribution coefficient for solutes having a wide variety of geometries and structures including monomers, homogeneous and heterogeneous rodlike and platelike molecules possessing many segments, and simple irregularly shaped molecules such as bent trimers. We examine the retention behavior of such solutes in both neat and binary solvent mobile and stationary phases. Whenever possible we compare our theoretical predictions with experiment or existing theories.
I. Introduction Theoretical determination of the optimum conditions for solute retention and selectivity by liquid-solid chromatography (LSC) ultimately requires comprehension at the molecular level of the thermodynamic properties of the solvent-solute mixture both in the presence (stationary phase) and in the absence (mobile phase) of a solid adsorption surface. In this paper we present a statistical mechanical analysis of solute retention in LSC which is based on a competitive equilibrium model between solvent and solute molecules distributed between generally nonideal mobile and stationary phases. The theory is applied to a wide variety of stationary phases and solute types. For example, in Section I1 we determine the distribution coefficients for solute retention between a nonideal binary solvent-solute mobile phase and stationary phases formed by monolayer adsorption of solvent and solute molecules on both energetically homogeneous and heterogeneous planar surfaces. The analysis in Section I1 is restricted to solute and solvent molecules having comparable molecular dimensions and is usually applied in the limit of infinite dilution of solute. In Section 111, we extend the analysis of Section I1 to determine the distribution coefficients for solutes possessing more complex molecular structure and geometry such as chemically homogeneous and heterogeneous rodlike or platelike molecular species. Whenever possible we compare the results derived in Sections I1 and I11 to results obtained in previous analyses. For example, Snyder1,2has developed a model of liquid adsorption chromatography which is limited to ideal mobile and adsorbed stationary phases. The Snyder approach generates a phenomenological, practical guide for determining the eluotropic strength of a series of mobile phase solvents and solvent mixtures in contact with specified surfaces. In mixed solvent systems, the Snyder model disregards the different environments and interactions which solvent and solute molecules experience in the stationary and mobile phases. Consequently the predicted solute distribution coefficients possess an oversimplified mobile and stationary phase composition dependence. The Snyder analysis has been recently extended to nonideal binary solvent mobile phases3 in the presence of either energetically homogeneous or heterogeneous4* ad0022-3654/80/2084-3620$0 1.OO/O
sorption surfaces. Differences in the molecular areas of solute and solvent molecules have also been included in some recent modifications of the m0de1.~*~ These extensions neglect nonideal behavior in the stationary adsorbed phase, and they generally utilize phenomenological thermodynamics, rather than a detailed microscopic approach, to obtain individual distribution coefficients. Hence a generally oversimplified composition dependence again persists.
Section I1 Here we develop a statistical mechanical treatment to analyze the equilibrium distribution of solute molecules between a mobile phase consisting of a binary solvent mixture and a stationary phase monolayer formed by adsorption of solvent and solute molecules at a planar homogeneous surface. We employ a cubic lattice model to enumerate the possible molecular arrangements in both the mobile and stationary phases, and we include both surface-molecule and nearest neighbor molecular interactions to determine the total interaction energy. The Bragg-Williams (BW) random mixing approximation9 is employed to estimate the number of molecular pairings of each type. Initially we investigate the case where the solvent and solute molecules are of comparable molecular dimensions. Later we extend the analysis to include solutes of more complex molecular structure (e.g., rigid rods) and also energetically heterogeneous surfaces. We assume that the planar adsorption surface consists of M, equivalent square lattice sites upon which N1, and N,, solvent molecules of types 1 and 2 and N3, solute molecules are singly adsorbed. The BW canonical partition function for the three-component adsorbed stationary phase is taken to be
where 3
M, = C Ni, i=l
(2)
The quantities qise-Bfh' (i = 1, 2, 3) represent the partition 1980 Arnerlcan Chemical Society
The Journal of Physical Chemistry, Vol. 84, No. 26, 7980 3821
Retention and Selectivity in Liquid Chromatography
functions for single surface adsorbed solvent or solute molecules with cis' denoting the surface-molecule adhesion energies. The wij (i, J = 1,2,3) values are nearest neighbor pair interaction energies between i- and j-type molecules. In calculating the totalpair interaction energy for adsorbed molecules, we have assumed for simplicity that the stationary phase composition extends into the layer directly above the adsorbed monolayer.'0 Similarly the BW canonical partition function for Nlm, Nzmsolvent, and N3msolute molecules arranged randomly on M , cubic lattice sites in the mobile phase is Qm({Nim),Mm,T)= M,!
r=l
3
(qimNim/Nim!) exp[-3PMm-'
from the mobile to the stationary phase with the concomitant displacement of a 1- or 2-type solvent molecule. Inverting eq 7 and its counterpart obtained by interchanging the roles of solvents 1and 2 and adding the two results gives eq 11.
In most applications of LSC, the solute is present in very low concentration in both the stationary and mobile phases so that the limit of infinite dilution of solute becomes an excellent approximation and eq 8 and 11 reduce to
3
2
C NimN;m~i,] is1 j=1 (3)
where 3
Mm = C N i m
(4)
i=l
and the qim(i = 1, 2, 3) denote the partition functions for isolated molecules in the mobile phase. After eq 2 and 4 are employed to eliminate respectively Nzaand Nh in eq 1and 3, the following chemical potential differences are readily derived = 1, 2) (see eq 9)
0' = 1, 2)
- Wj3)l
(i, j
(5)
(i = 1, 3) (6)
wj~))]]
where xis = Nis/Msand xim = Nim/Mm(i = 1,2,3) are the mole fractions of the solvents and the solute in the stationary and mobile phases. At equilibrium P(p3 - p2)s= P(p3 - p2)m,and from eq 5 and 6 with i = 3 we obtain X3s/X3rn
= 3
( x ~ s / ~ ~e m x ~) [ - P l e-~62s ~
+ rC (6xim - 5xis)(wiz- wi3))I =l
(7) for the equilibrium mole-fraction distribution coefficient for the solute. When the roles of solvents 1 and 2 are interchanged, an alternate expression for x3s/x3mis obtained which when added to eq 7 yields
(12)
= 1, 2, 3)
Equation 8a implies that the total solute retention in the stationary phase is a linear combination of two independent retention contributions involving either 3-1 or 3-2 exchanges between the mobile and stationary phases. Equation l l a indicates that solute retention is a sum weighted by mobile phase mole fractions of reciprocal distribution coefficients analogous to the total ohmic resistance encountered in a parallel circuit. At equilibrium P(pl - p2), = P(pl - p2)m,and employing eq 5 and 6 with i = 1provides a relationship between the stationary and the experimentally accessible mobile phase solvent compositions in the limit x% 0 and x3, 0. If the ternary system is ideal, all the 6 , (i, J = 1, 2, 3) vanish and eq 8a and l l a reduce formally to the results obtained by Snyder?
-
-
or 2
2 K3[1,2]
= X3s/x3m = (l - x 3 m ) - ? C
j=l
XjsK3(-f')1
(8)
where 3
K3(i) exp[-p(t3, - tjs
+ i = l (6xjm- 5xiJ (wij
tis
qS(T)=
cji -
- wi3))l
k T In (q,,/qjm)
~;~(n
X
0' = 1, 2)
0' = 1, 2, 3)
(9)
(10)
where the actually represent temperature-dependent surface free energies of adhesion. The quantities KSV)0' = 1, 2) represent Boltzmann factors which involve the energy change required for transfer of a solute molecule
The quantities K3+1 0' = 1 , 2 ) defined by eq 12 represent the mole fraction distribution coefficients at infinite solute dilution when only a single solvent is present and the adsorption surface is energetically homogeneous. The quantities AcCi,3;s) = t3,(T) - c j , ( T ) + w j j - wj3 0' = 1, 2) introduced in eq 12 represent the (free) energy changes for the processes 3(m) + j ( s ) + 3(s) j(m) 0' = 1, 2). Clearly when Ac < 0 (Ac > 0), solute (solvent) adsorption in the stationary phase is preferred. When any or all of the 6i. # 0, the ternary solvent-solute system is no longer idea{ and eq 8a and l l a no longer demonstrate a simple linear dependence on either the stationary or mobile phase composition. In Figure 1 we
+
3822
The Journal of Physical Chemistry, Vol. 84, No. 26, 7980
Boehm and Martire
a
1
"
t
b
I
0
"
"
"
"
"
0.2
"
'
"
'
l
-
1
0.6
0.4
0.8
1.0
'zm
xpmfor various choices of the set (PA$1,2;s),fi6127f613, Pi,?, K3m[2]): (1) (2, ' I 3 . 0,0,1); (2) (5, ' I 3 , 0,0,' 1 2 ) ; (3) (2, 14 0, 0, 1 3 ) ; (4) (2, ' 1 3 , 0, -1, 1); (5)(2, ' 1 4 , '13. 0, 1); (6) (2, '131 0, ' 1 3 , 1); (7)(0,'1 , 0, ' 1 3 , 1); (8)(0,' 1 3 , 0,0, 1); '/3, 0, 0.6065). (9) (0,' I 3 , ' I 3 , 0, 1); (10) (0.5, Flgure 1. Graphs of K,,
1 , 2 ~ - 1vs.
present graphs of K3m[1,21-1 vs. x2mfor various selections of /?At(l,B;s), the OSij, and K3,[j]-l using the relationship x2&1- xJ1 e x p [ - @ 1 0 ~ ~ ,=6 ~ ~ ] xzrn(1- X2m)F1 exp[P(Ac(l,2;s)- 12~2rn612)I obtained from eq 5 and 6 with i = 1 to eliminate x2s = 1
- Xis.
The value of PAt(l,2;s) selected measures the energy change for the process l(m) 2(s) l(s) 2(m). The K3.,(1,21-~ VS. xPrn curves depicted in Figure 1 were constructed with PAt(1,Z;s)I0, indicating that solvent 2 is either preferentially (@At > 0) or at least equally (@At = 0) adsorbed at the surface. One practical possibility for @At > 0 is to have a nonpolar (1) and polar (2) mixed
+
solvent in contact with a surface such as silica which possesses polar adsorption sites. A situation where @At 0 might occur is for solvents in contact with a rather inert carbon black surface which possesses limited selectivity for solvent adsorption. In the examples depicted in Figure 1 we have set Pal2 = 'I3 or 0.25. The choice /?al2 = 1/3 corresponds to the consolute temperature for phase ~eparation.~ We have included some xP8-xZrnadsorption isotherms for selected values of PAt(l,2;s) and Pd12 in Figure 2. Clearly when PAt(l,2;s) k 2 the surface becomes nearly saturated with solvent 2 even at very low xZm. The isotherms displayed for P612 = 1/3 are, of course, critical isotherms. Curves 1-5 represent examples where solvent 2 is preferentially adsorbed on the surface and the solute is preferentially retained at the surface with respect to solvent 1 (Le., K3,[11+> K3,[21) and 0613 k P823. These curves may exhibit a minimum at low x2rn (0 < x2rn < 0.1) and either a maximum (@Si3 = P623) or a monotonic increase (P613 > P623) with increasing xZm. The maxima in curves 1-3 (Le., retention minima) appear because solvent 2 is strongly adsorbed on the surface (see Figure 2), and the solute distribution in the two phases is not biased by differences in the average solvent-olute interaction since P613 = 0623. If P6,, = ll3, these maxima appear at x~~ = when P612 (curves 1 and 2), while it shifts to x2rn> < lI3(curve 3). Similar retention behavior is often observed for mixtures of a nonpolar and a polar solvent in the presence of a silica surface." In curves 4 and 5 the more favorable solvent 2-solute interaction determines the solute retention behavior. The minima at low x2,,, values (Le., retention maxima) results because xza>> xh and the more favorable 2-3 interactions enhance solute retention. The retention decreases as xZm increases because the stationary and mobile phases become more similar in composition. On the other hand curve 6 exemplifies the opposite situation where 86, > P613 @At(l,2;s)= 2). The maximum (i.e., retention minimum) results because the strongly adsorbed solvent 2 rejects solute relative to the mobile phase which has a higher concentration of the more favorable solvent 1. The retention increases as xZrn increases because the two phases then tend to provide similar solvent environments to the solute. When the surface is relatively inert to both solvents and solute ( t l s t28 t3s),solute retention in the stationary
+
- -
The Journal of Physical Chemlstv, Vol. 84, No. 26, 1980
Retention and Selectivity in Liquid Chromatography
phase is primarily determined by the relative values of &3 and PA,. Curves 7 -10 provide some illustrations and reveal that retention maxima and minima are both possiblq For any xZrnthe equilibrium distribution of solute can essentially be rationalized by comparing the averaged solventsolute interactions in the individual phases rather than the surfacesolute interaction. This feature has almost always been neglected in previous analyses.1-8 and For example, curve 7 corresponds to p823 > PAt(1,2;s) = 0 and the K3-[121-‘ - x2, curve passes through a minimum (Le., retention maximum) in contrast to the behavior depicted by 6. When @At(l,2;s)= 0, solvent 2 no longer is strongly adsorbed in the stationary phase (see Figure 2), and when xZm increases, the unfavorable 2-3 interactions in the mobile phase enhance solute retention. Recently Jaroniiec et al.4 have obtained an expression for the solute mole fraction distribution coefficient in a nonideal binary mobile phase and an ideal stationary phase. Their result (eq 25 of their paper) disagrees with eq 8a or l l a undoubtedly because of their conflicting assumptions of ideality and nonideality in the separate phases. When only a single solvent (e.g., 1)is present, eq 8 and 9 reduce to K3[1] = lim K3[1,2] = xa=xa=O ~XPI-PIA~(~ +,2(6 ~ ; ~-)5Ic3[11)~3m813)1 X (1 + x3,(ex~I-P{A41,3;~>+ 2(6 .- 5K3[1$x3mS13)1- 1)I-l (13) For an ideal solution 813 = 0 and eq 13 simplifies t o &[I]
= &[i]id. =
+ x3m(K3m[l]-
&-[l](l
(13a)
where (recall eq 9 and 10) x,-ox,-o
represents the distribution coefficient at a planar homogeneous surface with only one solvent present. Equations 13 and 13a reveal the dependence of K3[i1and K3[11id. on x3,, and we observe that K3fllid.= K3-[1l when x3, = 0 and K3[11id. = 1 when x3m= 1 and that K3[11id. either monotonically decreases (increases) with increasing x3, if K3m[11> 1 or At < 0 (K3m[11 < 1 or At > 0). The analysis leading to eq 8a or l l a can be easily generalized to include a heterogeneous planar surface consisting of a random mixture of single molecule adsorption sites of types A and B. Iff, = M J M , A , B with I d n = 1,represents the fraction of A- and B-type sites, then the distribution coefficient at infinite dilution of solute is 2
R
where xisn = Nisn/,Vn(i = 1,2 and n E A, B) are the representative surface fractions of solvent at A and B sites and
which is the generalization of eq 9 at infinite solute dilution for a heterogeneous surface. The quantities cis, tisn(T) (i = 1, 2, 3; n E A , B ) represent the generalization of eq 10 for the adhesion free energies of solvent and solute at an energetically heterogeneous surface. In terms of mobile phase compositions eq 15a may also be expressed as
K3m[l,Z;AB]-1=
2
E
1=1
n=A
ximfnK3m-l(;;n)
3623
(15b)
which generalizes eq l l a . Clearly eq 15a or 15b may be extended to a planar heterogeneous surface consisting of a random mixture of L different types of single molecule adsorption sites: 2
L
with
for each n. When only a single solvent (e.g., 1)is present, all xisn = 1 in eq 15a or its extension to L types of sites and L
K3m[ljri$,...L1
- n=A C f, exp[-PA4,3;n;s)l
(16)
where A&3;n;s) = t d T ) - h ( T ) + wll - ~ 1 3 n = A, B, ...,L with K3m[l;AB] = f A exp[-PAt(1,3;A;s)l + f E exp[-PAt(l,3;B;s)l (16’) and f A t fB = 1 for retention with a binary heterogeneous surface. The distribution coefficient given by eq 16’ varies linearly with f A (or f ~ ) .The intercepts at fA = 0 and fA = 1give the homogeneous type-B or type-A surface distribution coefficients and the slope d & m [ l ; ~ ~ ~ may / d f ~be either positive or negative depending upon whether exp[-P(At(1,3;A;s) - At(l,3;B;s))] exceeds unity or not.
Section 111 Here we investigate the retention of solutes of more complex structure using LSC by again applying the BW approximation and a cubic lattice model. In particular, we seek the distribution of homogeneous rigid rodlike molecules of length-to-breadth ratio p between a single monomeric (Le., p = 1)solvent mobile phase and a stationary phase imonolayer formed by adsorption of both solvent and solute molecules at a planar energetically heterogeneous solid surface. We ultimately avoid the rather formidable steric problems arising from solute-solute interactions by again assuming that solute is present in vanishingly small concentrations in both stationary and mobile phases. The present treatment can also be readily extended to considerably more complex solute-solventsurface systems provided infinite dilution of solute prevails. The molecular configurations of solvent and solute in each phase are again determined from random mixing on a cubic lattice. We require that adsorbed rodlike solute molecules have at least one terminal segment in physical contact with thre surface. A more sophisticated analysis might include multilayer adsorption of rodlike solute molecules hovering near to but not contacting the surface.1° We assume that the surface consists of a total of M, sites formed from a random mixture of M A type-A and MB type-B sites with MA+ MB = Ma. Each adsorption site can accommodate either a single solvent molecule or a solute segment. In general, an adsorbed rigid rodlike molecule may assume an infinitle number of configurations relative to the surface, and the rod-surface interaction is orientationally dependent. However the configurational problem is considerably simplified by restricting the rod orientations to
3624
The Journal of Physical Chemistty, Vol. 84, No. 26, f980
three mutually orthogonal directions: two parallel and one normal to the surface.12 We assume that solvent and normally adsorbed rodlike solute molecules respectively adsorb to an A or B site with surface adhesion free energies eld, E~ or el&, and given by an obvious generalization of eq 10 with q3sA, q3sB9 and q3rnrepresenting single solute segment partition functions in the stationary and mobile phases. Rods adsorbed parallel to the surface upon j type-A and p - j t y p e 4 sites adhere with a surface free energy (17) €11G,p-j) = ~ C B ~+A( p - j ) ~ 3 ~ ~0 5 j 5 P We let N&,p-j) denote the number of solute molecules adsorbed parallel to the surface on j A- and p - j B-type sites and let N3sA and N3,B respectively represent the number of normally adsorbed rods on A- and B-type sites. The total number of adsorbed rods is
Boehm and Martire
where N311= CfroN3,0’,p-j) and eq 18-20 have been employed to eliminate C k = A $ N 3 & , Nls(l) and Nls. The terms of O(N3,2/M9)involve nearest neighbor rod-rod segment interactions and may be neglected when N3,/Ms 0. In formulating eq 22 we have included solvent-solvent and solvent-solute segment pair interactions in the (p - 1) layers, directly above the adsorbed surface monolayer because normally adsorbed rods either interact with or displace solvent molecules in these layers. We employ the familiar maximum term method to estimate In Q, by seeking the N3,G,p-j) = 0, 1,...,p) and N3sA which maximize In Q, subject to the constraint imposed by eq 18:
-
P
In
Qsrnax
= (2Ms - (P - W
3 ,
C=o rO’) 4G,p-j)) In (Ma-
J
P
( P - UN3BC rO’) 4G,p-j)/2) - M8 In M , -
P
J=o
P
C rG) d4,p-j) In [ N ~ & J G , P - ~-) / ~ I
j=O
The number of adsorbed solvent molecules is
In
N3843A
D
N3s63A -
P
N3A1 - C rG) j=O
of which
P P
N18A(1)= MA- 2 C jN3,G,p-j)/2 - N3sA (19a)
J=o
J=o
( p - j)N3,G,p-j)/2 - N3sB
k=A,B
N3sk)
P D
(MA
P
Nlsk(l)€lskjl
exp[-PEINT1)/
fi=o [(N3so’,p-j)/2)!]2fi N3&!Nl&(l)!)](21) k=A
J
In (MA - N3,(? jrG)
J=o
~ G , P + )+ 4 3 A ) )
(M, - MA- N3&C ( P - j - 1)rG) d 4 , p - j ) j=O
where the lattice statistics developed by DiMarzio12for rigid rods has been utilized. The sum in eq 21 extends over all N3,G,p-j) and NBsk k A , B consistent with eq 18. We have also assumed that equal numbers of rods are adsorbed along the two mutually orthogonal directions parallel to the surface (isotropic parallel adsorption). In the limit N3,/M8 0 usually encountered in chromatographic applications, isotropic parallel adsorption prevail^.^^^^^ The quantity P E I N T represents the interaction energy between adsorbed solvent-solvent, solvent-solute, and solute-solute segment nearest neighbor pairs:
-
+ 1-
P 43A))
ln (Ma- MA- N d C ( p - j - UrG) $G,p-j) + j=O
(20)
Ct([(Ms- ( P - 1)j=O C N3,O’,p-j)/2)!I2x
k=A,B
J=o
j=O
P
The canonical partition function for N3, rigid rodlike molecules adsorbed on a planar, heterogeneous surface of M , sites may be expressed in the BW approximation as &8(N3s,M5,T)=
P
(19b)
are respectively adsorbed on A- and B-type sites. The number of solvent molecules located in the (p - 1)solvent layers directly above the surface monolayer is ( p - 1)Nl5= ( p - ~ ) ( M-s
In [N3,(1-
C rG) @G,P-~)- 43d1 - (MA- N3,(CjrG) 4G,p-j) + 43A))
P
NI,B(~) = ME - 2
PEINT
@G,P-J - 4 3 A )
P
1- 4 3 A ) ) - PN3s(C rO’) d’0’tp-j) J=o (€llG,p-j)- €388) + 4 3 A h 3 8 A - €398) +
t38B)
-
P
@(MA- N38(Cjrg’)$O‘,p-j) + 4 3 A ) ) € l s A J=o
-
P(Ms - MA -
P
N d jC (p-j-UrG)4G,p-j) + 1 - 43A))€lsB - PEINT(23) =O where rG) p!/j!(p - j)! is a degeneracy factor which enumerates the number of distinguishable ways a solute rod can occupy p contiguous adsorption sites of which j (p - j ) are type A (E)with A and B being randomly mixed. We have also introduced
N3,G,p-j) = r0’) 4G,p-j)N3, N3sA
0‘= 0, 1, ..., P )
= 43AN3s
(244 (24b)
-
which satisfy a In Q,,/aN,,G,p-j) = d In Q,,,,/dN3,~ = 0 0’ = 0, 1, p). When N3,/M9 0 the maximization equations reduce to
= P{(5(M5- N3s - ( p - 1)N31d2/2+ 3 ( p - 1)
r(k) +(k,p-k)/(2(1 - CrG) 4G,p-j) - 4 3 A ) ) = J=o r(k)(MA/Ms)k(MB/Ms)P-l-k eXP[-P(k(€3sA- ElsA) + 0, k - 1)($,~ - ti,^) (P - wii))] (25)
(Ms- (N3s
- N311))2)wll/Ms+
[(Ms - N3s - ( p - 1)N311) (2(P + 1)N311 + 4(N3s - N311)) + pN311 + ( 4 -~3W3, - N311)1w13/Ms + terms of O(N3,2/Ms)) (22)
e..,
-
P
P 4 3 ~ / ( 1-
TO’) 4G~p-j)- @3A) = J=o (MA/MB)exp[-@(e3sA- €388 +
ClsB
- ElsA)I (26)
The difference in chemical potentials of the solute and
The Journal of Physical Chemistty, Vol. 84, No. 26, 1980 3625
Retention and Selectivity in Liquid Chromatography
solvent in the limit x3# N3,/Ms P(p3 - pd8 =:
In xQa- In {
C
fk
k=A$
-+
0 is
exp[-@(At(l,3;k;s)-
6 ~ ~ +~ (4p 1 1+ 2 h d I
+
7.01
P
2
]=O
rO’)(fAexp[-PAt.(1,3;A,s)l)jX (fB
e~p[-PA€(l,3;B;s)])P-’ exp[-P(-6pwll + (4p + 2)w13)11 (27)
I i I
5.0
where f k = Mk/Ms (k = A, B ) and eq 25 and 26 have been utilized to eliminate the +(k,p-k) and 4 3 A . In order to obtain the equilibrium distribution coefficient for rigid rodlike solute molecules between the stationary and mobile phases, we must first calculate p(p3 pJm for the mobile phase and then equate the result to eq 27. For that purpose we assume the mobile phase is a binary regular solution with the rodlike solute molecules restricted to three mutually orthogonal directions and isotropically distributed among them. For N3mrods in volume M,, the canonical partition function in the BW approximation is Qm(N3m,Mm,T) = [((Mm - ( p - 1)N3m/3)!13/ ((Mm!)2((~3~/3)!)3(Mm - pN3m)!)] X exp[-P(3(Afm- P N ~ ~ ) ~ w +~ (I 4/ M +~ ~ 2)(N3m/Mm)(Jvm- PN3m)w13 .f Q(N3m2/Mm))1 (28) where the configurational contribution has again been estimated from the DiMarzio analysis.12 The solute segment and solvent molecule partition functions qsmand qlm have already been incorporated in the surface free energies of adhesion (see eq 10). We anticipate proceeding to the limit N3,/Mm 0 by not specifying explicitly the nearest neighbor solute-solute segment interactions in the Boltzmann factor. From eq 28 we obtain in the limit N3,/Mm 0
-
-
4.0
3.0
0.2
0
0.4
0.6
0.8
1.0
fA
Flgure 3. @)[,:Aslvs.
curves calculated from eq 30 for various values of pand /%(1,3;A,s) = 0.5 and oAt.(l,S;B,s) = -0.5. The value of Is independent of p for fA = (1 - e-BAf(1,3:E,*))/ (e-@At(l,3A!f!1C-$ Ad1,3:E,*))(fA= 0.3775 for @At(A) = -PA@) = 0.5.). fA
For all other values of f A separation of p-mers is theoretically possible.
phase with the dlisplacement of a solvent molecule. Clearly when -At. > 0 (-At. < 0), solute (solvent) adsorption is energetically preferred. If, however, - A 4 1,3;A,s) and -At.(1,3;B,s) are of opposite sign, the solute retention depends sensitively on the surface composition. Some @$;AB] vs. f A curves for various selections of p and AE(1,3;k,s) are displayed in Figure 3. When the su:rface is homogeneous, f A = 1 and eq 30 reduces to K@,I, =
where xQm N3,/Mm. Setting P(A - dS = P(PQ- p l ) m by using eq 24 and 29 leads to the equilibrium distribution coefficient for rigid rod solute molecules: K&?2rl;ABI=
ii% (~38/~3rn) = 2(
c
k=A,B
fk
ex~[-PAt.(l,3;k,s)l)~l (30)
where we have recalled the binomial expansion. The distribution coefficient KSDm)[l;AB1 may also be regarded as an equilibrium constant for the transfer process 3(m) .+-pl(s) + 3(s) pl(m)
+
Equation 30 reveals that K p $ ; A B ] consists of a sum of contributions each corresponding to a specific mode of solute retention by adsorption in the stationary phase with the concomitant transfer of p solvent molecules into the mobile phase. The first term in eq 30 corresponds to retention by normal adsorption of solute on either an Aor B-type site while the last term corresponds to retention by parallel adsorption on p contiguous A or B sites along the two allowed orthogonal directions. The quantities Ae(1,3;h,s) k 2 A, B represent the free energy differences for transfer of a solute segment between the mobile and either an A- or B-type site in the stationary
(%)[exp[-PA4,3;s)
1 + 2 exp [ - P ( q / a I)A 4 3 ; s ) I 1
(31)
where we have set p = all/a, and alland a, are the molecular cross sectional areas for parallel and normal adsorption. Clearly when -At. > 0 (-At. < 0) parallel (normal) adsorption is thle preferred mode of retention. When the solute is monomeric, eq 30 and.31 reduce to eq 14 as required. Also, eq 30 and 31 reveal that separation of a mixture of chemically homogeneous rigid rod homologues having different values of p (i.e., a l/a,) is predicted (except perhaps for a single value of f A (see Figure 3)) for a heterogeneous1 surface irrespective of the values of At.(1,3;k,s), k % A,& while At.(1,3;s)# 0 is required for separation on a homogeneous surface. It is possible to generalize eq 30 to render it applicable to homogeneous solute molecules possessing the geometry of a rectangular prism with molecular facial cross-sectional areas ai (i = 1,2,3). If the adsorption surface is composed of a random mixture of L different types of single molecular segment adsorption sites, then eq 32 represents the
@k$, AB...L]
3 (%I)
L
(
i=l k=A
fk
ex~[-pA€(l,3;k,s)l)~’ (32)
equilibrium dist.ribution coefficient. The solvent is again monomeric and the f k 3 Mk/M,, k E AB...L with c k = A f f k = 1, represent the surface site fractions of each distinctive type. In eq 32 we have again assumed that the molecular symmetry axes are restricted to three mutually orthogonal
3828
The Journal of Physical Chemisfry, Vol. 84, No. 26, 1980
directions, two parallel and one normal to the surface. I t is easy to demonstrate that K@/)1,AB,,,L1 as a function of the fk possesses no extrema. Assume @A~(1,3;A,s) > PAc(l,3;k,s) for h = B, C , ..., L (this is no specialization) and substitute fA = 1 - &=BLfk into eq 32 to obtain 3
Kk$,
AB...LI
=
(exp[-PA41,3;A,s)l
(73)
+
1=1
L k=B
fk(exp[-PA41,3;k,s)] - e~p[-PA41,3;A,s)]))"~ (32a)
Seeking the fk which render K&t)l,AB.,.L]stationary yields 3
~ K @ ' / { , A B . . .= L I0/ ~=~(73) ~
2=1
ui(exp[-PA€(l,3;A,s)l +
L k=B
fk(exp[-PAt(l,3;k,s)]
-
exp[-/3A~(1,3;A,s)I))~~-~ X
(exp[-PA41,3;k,s)l
-
exp[-PA4,3;A,s)l)
Since the first factor is positive, the derivative vanishes if and only if PAc(1,3;k,s) = PAc(l,3;A,s) which implies, contrary to the original hypothesis, that the Ath- and kth-type sites are indistinguishable with respect to solute segment and solvent molecule adsorption. Hence eq 32 possesses no stationary points with respect to changes in surface composition. Derivation of eq 32 by the same approach which leads to eq 30 is difficult because the configurational lattice statistics for regular prisms on a surface is required particularly when finite solute concentrations are important. Unfortunately, to our knowledge, this configurational problem remains to be solved. However, we present an alternate derivation of eq 30 using mathematical induction and a simple probabilistic argument which is readily extended to eq 32 and other interesting results. For a monomeric solute (infinite dilution)-single solvent solution retained on a heterogeneous planar surface of L distinctly different sites having coverage fractions f k = M k / M , , k = A, B, ..., L , eq 30 corresponds to eq 14 rewritten as K3m(1)= C k = A L d k where ek = fk exp[-PA41,3;k,s)l is termed a retention factor on site k. The distribution coefficient for a homogeneous dimer solute is
Boehm and Martire
by first summing over all possible single segment retention factors and then raising this sum to a power equal to the number of segments adhering to the surface in the particular adsorption mode considered. The total distribution coefficient is then a sum of individual contributions associated with each mode of retention with each term of the sum weighted by the configurational probability that the particular mode appears. This analysis should also apply to both chemically homogeneous and heterogeneous solutes of arbitrary molecular shape in the limit of infinite solute dilution. Clearly eq 32 is an obvious generalization of eq 30 when the inductive approach is applied. For instance, the inductive analysis can be applied to determine the distribution coefficient for a rigid rodlike solute of p segments which has a terminal segment chemically distinguishable from the remaining p - 1contiguous segments and is retained on a heterogeneous planar surface of L randomly mixed types of adsorption sites: L
K&$',%,..L]
(%)
k=A
fk(exP[-PAZ(1,3;rz,s)l(l-I-
L
4(
c fkt exp[-pA€(1,3;k',s)])P-l)
k'=A
+ exp[-PA41,3;k,s)]) (33)
where AZ(1,3;k,~)= z 3 k s - elks
+ W11 - IB13
(k = A, B,
represents the energy difference for stationary phase adsorption of the distinguishable segment and concomitant displacement of a solvent molecule. Equation 33 may provide a reasonable estimate for the LSC distribution coefficient for certain lyotropic molecules such as fatty acids and long-chain normal alcohols or esters mixed with simple low-molecular-weight solvents. The distribution coefficient for irregularly shaped multisegmented solute molecules may also be constructed by using induction. For example, the distribution coefficient K'$&]* for bent L-shaped, chemically homogeneous trimers retained at a homogeneous planar surface is 3
Kd?{,]= (y3) C exp[-j/3A41,3;s)l j=l
L
L
C 8k + 2(k = A 8,)'
k=A
where the first (second) term corresponds to normal (parallel) adsorption. The latter contribution to K3m(2) is proportional to a sum of all possible binary products of the individual solute segment retention factors. By mathematical induction one obtains the distribution coefficient for rigid rods of length-to-breadth ratio p - 1:
Finally for rods with p segments, the contribution from parallel adsorption on p contiguous lattice sites is
(34)
where we have set the molecular cross-sectional area for single segment adsorption equal to unity. By comparison for rigid rod trimers eq 31 gives
K&] = (l/g)[ex~[-PA~(1,3;s)l + 2 exp[-3PA4,3;s)ll (35) Provided A€ f 0 the separation factor a = K$?p]/KL?[1] f 1 and separation based on shape selectivity is predictable from the present model of LSC. Similar conclusions can also be obtained for chemically heterogeneous multisegmented solutes possessing different structural geometries. For example, the distribution coefficients for rigid rod and L-shaped trimers having one segment chemically different from the remaining two are (homogeneous surface) K53-f]]= (f/6)[4e-6(2Ae+Az) e-oAz e-BAf] (36a)
+
~ h 3m[- b1) =
+
(1/6)[4ed(2Ae+Az) + 2e-8AtI
K$?f]] = (f/6)[2e-8(2Ae+AZz) + e-B(Aa+Az)+ e-82Ae + e-flAz which generalizes eq 30 and completes an alternate proof. The preceding analysis suggests that the contribution to the distribution coefficient for solutes which are retained by multisegment modes of adsorption can be determined
L)
K&3-fJ1 = (f/6)[2e-8(2At+Az) + 2e-B(Ae+Az)+
(3W
+ e-mt] ( 3 6 ~ ) (3W
where structures 3-a and 3-b (3-c and 3-d) are rigid rod
Retention and Selectivity in Liquid Chromatography
The Journal of Physical Chemistry, Vol. 84, No. 26, 7980 3627
(L-shaped) trimers with a terminal and internal segment respectively being distinguishable. Clearly if AT # A6 # 0, separation by ILSC is again predicted. One can also generalize the inductive approach to construct equilibrium distribution coefficients for solutes of various geometries and chemical compositions which are present in binary monomeric solvent mobile and stationary phases. We illustrate initially by considering rigid rod solute molecules of p segments which possess a terminal segment chemically distinct from the remaining p - 1 segments and are retained at a homogeneous planar surface. The distribution coefficient is
K&$]
I=
l/s[4atat*@’intp-2 + at + at*]
(37)
where 2
2
j=l
i=l
- -
In the limit when xZm x~~ approximately as
1,eq 39’ can be expressed
= -In (Y2) - p In x~~ + In K&~$lK3~[21* In ~&+j!l 1)(813 823 812) + h* 823* P ( ( 6 ~ 2 m- 5% -
-
+ ( X Z s - X23[(2(P - 2) wZ3*]) In C - p In x~~
8121
-
+ 1)(w13 - 1023) + w13* ( x ~ 1,~zZs -t
1) (39”)
where C = (2/3)K3@m7i\K3m[21* is independent of composition. The latter form of eq 39” agrees with the result reported by Soczewinski7and Jaroniec et a1.8 for solute molecules having different (i.e., greater) molecular areas than the solvents in the limit when one solvent is in excess. If, on the other hand, normal adsorption with the distinct segment dominates, then
at E C xjS exp[-8(e3, - ejS + C (&xim- h i s ) x 2
(wji - wi3) -
T,A his 1:;1
- xim)wi3)]
(384
where C’I (‘/t6)K3,[21*. This form has been utilized frequently for monomeric solutes4 in binary solvents when one component predominates. However, when both parallel and normal adsorptions contribute corriparably to the retention of rodlike solutes, a simple composition dependence for In as xZm 1 and x~~ 1 is not obtainable. In Figure 4, a and b, we present some K3m[l,2]-’- X2m curves calculated from eq 39 and 40 for both homogeneous and heterogeneous rodlike solutes of length-to-breadth ratios p = 2,3, and 5 for PAl2 = P6,, = P613* = 1/3, 0613 = 0, and P623* = -1/3 and the two choices PAc(l,2;s) = 0 (Figure 4a) andl @A~(1,2;s) = 2 (Figure 4b). The quantities @Si; and @Gij* have been selected to physically simulate a system where solvent 1and a normal solute segment are nonpolar while solvent 2 and a distinct solute segment denoted by an asterisk are polar. The heterogeneous rodlike solutes might roughly simulate lyotropic molecules such as fatty acids, alcohols, or esters. The number and position (i.e., internal or terminal) of the chemically distinct segments have also been varied in order to simulate, albeit crudely, molecular species such as normal and is0 alcohols, glycols, diacids, diesters, ethers, etc. When PAe(l,2;s) = 0, the homogeneous adsorption surface is indifferent to both solvents and perhaps physically corresponds to an inert carbonaceous surface.15 Figure 4a depicts a homologous series of dimers, trimers, and pentamers which possess either identical normal segments (curves 1-3) or have either one (curves 4-6) or both (curves 7-9) terminal segments chemically distinct. Curves 10 and 11 correspond ito a trimer and a pentamer with a chemically distinct internal segment while curve 12 depicts a trimer with both an internal and terminal segment which is chemically distinct. The distribution coefficients for the homogeneous solutes increase and pass through maxima as xZmincreases - xZmcurves possess minima). The re(Le., the K3m[1,21-1 tention is enhanced because the unfavorable value of Pa23 (0823 = promotes parallel solute adsorption in the stationary phahie to diminish the number of solute segment-solvent 2 interactions. For small values of x~~there exists little or no difference in the retention of homologous series of similar solutes. This observation follows by comparing curves 1-3,4-6, and 7-9 in Figure 4a. These results are consistent with those obtained experimentally by Guoichon et al.15 for homologous series of solutes retained at carbon black surfaces. Solutes possehising one or more “polar” segments (curves 4-12 in Figure 4a) all display retention minima (i.e., maxima in the 1
