Molecular theory of chromatography for blocklike solutes in anisotropic

J. Phys. Chem. 1992,96, 3489-3504

1.3330)both eqs 17 and 4 give poor results (0.85and 0.94instead of 1.14)and this must be due to the particular structure of water which is sensitive to temperature effects. This qualitative analysis has been completed by the calculation of the relative deviations A = (Y(calcu1ated) - Y(experimental))/Y(experimental), where Y = (dn2/dd)p For the 75 compounds of Table I1 the mean values A of A are +4.1% (Eykman’s rule (4)) and -3.3% (our eq 17). The absolute mean relative deviations given in the same order are equal to 8.3% and 6.6%. After removal of the nine doubtful cases (seeabove) the A values for the 66 remaining solvents become equal to +6.1% and -1.6% and the values to 6.4% and 4.1%. These results show that the uncertainties are better distributed (A) in the case of the new equation, and the individual values are also in better agreement with the experimental results. A part of the residual discrepancies could be explained by taking into account polarizability anisotropy effects related to nonspherical molecular shapes. Unfortunately,

3489

such calculations need more accurate and reliable values on dn/dT and dd/dT values than those which are usually available in the literature. In conclusion, through a more general and consistent derivation than those which are usually performed, we have deduced a new expression of the DFRI term. First, the new equation gives results which are better distributed and with slightly lower deviations than those deduced from the empirical Eykman’s rule. But its main interest lies in its theoretical derivation which has been made by using the same assumptions as for the isothermal DFRI term in the Rayleigh light scattering t h e ~ r y . ~ ‘ JThese ~ good results give a new direct m n f i i t i o n of the validity of our theoretical method. Similar studies on dielectric constants and attempts to explain residual discrepancies are in progress. Acknowledgment. We express our gratitude to Prof. Martial Chabanel for his helpful comments.

Molecular Theory of Chromatography for Blocklike Solutes In Anisotropic Stationary Phases and Its Application Chao Yant and Daniel E.Martire* Department of Chemistry, Georgetown University, Washington, D.C. 20057-2222 (Received: September 26, 1991)

To develop a more complete and informative molecular theory of chromatography, DiMarzio’s lattice model, based on statistical thermodynamics, is extended and used to describe the equilibrium partitioning of blocklike molecules between an isotropic mobile phase and an anisotropic stationary phase. Both repulsive and attractive interactions are taken into account. The configurational partition functions for both the mobile and stationary phase systems are rigorously derived, and from them thermodynamic properties such as the Helmholtz free energy, the chemical potential, the entropy, and the enthalpy can be expressed in terms of state variables (temperature and density) and molecular parameters (interaction energies and molecular dimensions). A retention equation, applicable to gas, liquid, and supercritical fluid chromatography (GC, LC, and SFC), for blocklike solutes in an isotropic mobile phase and an anisotropic stationary phase is obtained. The solute distribution coefficient, K,is well represented by In K = QIAmin+ Q2Acf+ Q3Vw,where A ~ , ,Act, , and V, are, respectively, the minimum area, the effective contact area, and the van der Waals volume of the solute molecules, and Ql, Q2, and Q3 are related to the state variables and molecular parameters. A linear relationship between In K and for isomeric, polycyclic aromatic hydrocarbon solutes in anisotropic stationary phases is predicted. The theory is successfully applied to the interpretation and analysis of GC, LC, and SFC data.

1. Introduction

The separation and identification of polycyclic aromatic hydrocarbons (PAHs) are important and challenging tasks. The pioneering work of Kelker’ and Dewar and Schroeder* on the use of liquid crystals as stationary phases in gas chromatography (GC) opened a realm of new possibilities for the truly effective separation of structural isomers. It has been shown that liquid crystals are very effective stationary phases for the separation of isomers in complex PAH m i ~ t u r e s . ~ -In ~ liquid chromatography (LC), immobilized bonded p h a w with some anisotropic character, such as (4,4’-dipentylbiphenyl)dimethylsiloxane (55B),6 and octadecylsilane (C18)polymeric phase,’ also exhibit good shape selectivity for isomeric polycyclic aromatic hydrocarbons that is similar to the shape selectivityobserved with bulk liquid crystalline phases. The retention mechanism of PAHs, especially on anisotropic phases such as liquid crystalline and cl8 bonded phases, is still not fully understood. Sleight8 studied the structure-retention ‘Present address: Analytical Research and Development, Building 360/ 1034, Sandoz Pharma Ltd.. CH-4002 Basel, Switzerland. Corresponding author.

relationship of PAHs in LC on c18 columns and derived a simple linear relationship between the retention and the number of carbon atoms. Lockeg suggested that the basis of reversed-phase LC selectivity for PAHs is the relative solubility of the PAHs in the mobile phase. However, the solubilitydata alone do not adequately explain the unique chromatographic selectivity of many PAHs. In GC, Janini et al.lOJ’noted that the retention of PAHs on liquid-crystal stationary phases correlated with the shape of the (1) Kelker, H. 2.Anal. Chem. 1963, 198, 254. (2) Dewar, M. J . S.; Schroeder, J. P. J . Am. Chem. SOC.1964,86, 5235. (3) Janini, G. M.; Muschik, G.M.; Zielinski, W. L., Jr. Awl. Chem. 1976, 48, 1879. (4) Kelker, H. Advances in Liquid Crystals; Brown, G . H., Ed.; Academic Press: New York, 1978; Vol. 3, p 237. ( 5 ) Witkiewicz, Z.; Mazur, J. LC-GC 1990, 8, 224. (6) Lochmiiller, C. H.; Hunnicutt, M . L.; Mullaney, J. F. J . Phys. Chem. 1985.89, 5770. (7) Wise, S. A.; Sander, L. C.; Chang, H. C. K.; Markides, K. E.; Lee, M . L. Chromafographia 1988. 25, 473. (8) Sleight, R. B. J . Chromatogr. 1973, 83, 31. (9) Locke, D. C. J . Chromarogr. Sci. 1974, 12, 433. (10) Janini, G.M.; Johnston, K.; Zielinski, W. L., Jr. Anal. Chem. 1975, 47, 670. (11) Zielinski, W. L.;Janini, G.M. J . Chromatogr. 1979, 186, 237.

0022-365419212096-3489$03.00/00 1992 American Chemical Society

3490 The Journal of Physical Chemistry, Vol. 96,No. 8, 1992

molecules, specifically their length-to-breadth ratio (L/B). Radecki et discussed the relationship between G C retention indices on liquid-crystal phases and the shape of the PAHs. The connectivity index ( x ) and shape parameter (a) were used to develop this relationship. Wise et al.I3 described the relationship between the L/B of PAHs with their LC elution order on CI8bonded phases. The stdpe factor is the maximized ratio of the longer to the shorter des of a rectangle enclosing the planar representation of the molecular structures. Generally, retention increased as the L/B imeased. Exceptions included the isomers of nonplanar molecules such as benzo(c)phenanthrene. Wise and Sander proposed a qualitative slot modeli4 to explain the retention in terms of a nypothetical phase consisting of a number of narrow slots into which solute molecules can penetrate. Martire and Boehmis concluded that, in LC, chemically bonded phases exhibit shape selectivity which increases as the bonded chains become more fully extended and that rigid-rod solutes mould have greater retention than globular solutes. A model stationary phase was synthesized and used by Lochmiiller et aL6 10 test the theory. The results of the measurement of selectivity for flexible-chain, rigid-rod, and platelike solutes support the general trend predicted by the model with selectivity following the order: rods > plates > flexible chains. Later, Martire16 developed a molecular theory addressing the enhancement in selectivity experienced by structural isomers from the orientational order of nematic phases in GC. A variety of molecular structures of solutes were treated in the context of the model: thin and thick rods, plates, cubes, and semiflexible chains. This model considered oniy packing effects. A molecular structural descriptor, g = 3(vR - a R ) ,for the solute emerges from the theory, where uR is the reduced molecular volume and aRis the reduced molecular area. Whalen-Pedersen and Jursl’ described a quantitative structure-retention relationship (QSRR) for large data set of PAHs. Capillary GC retention indices of these compounds were regressed against numerical quantities (descriptors) derived from their molecular structures. In short, although there exist various empirical and theoretical treatments of PAH retention in GC, LC, and SFC, and extensive thermodynamic studies’* and attempts at defining a retention mechanism have been made,I9v2Oa satisfactory molecular theory of retention for PAH solutes in anisotropic phases is still lacking. An understanding of the retention mechanism is important for at least two reasons: (a) informed selection of the right combination of mobile and stationary phase variables for a particular separation problem, which was and still is accomplished by a t*ial-and-error approach; (b) developing the potential of chromatography for physicochemical measurements by exploiting its equilibrium distribution process to obtain various properties involving the solute, mobile and stationary phases. In the present paper, a lattice model based on DiMarzio’s statistics is used to describe the equilibrium partitioning of blocklike molecules in the model system. The model we develop is for a layered, completely aligned stationary phase where possible absorption of the mobile phase solvent is neglected, as a first approximation. Retention equations, applicable to gas, liquid, and supercritical fluid mobile phases, are derived. The theory is applied successfully to the interpretation and analysis of GC, LC and SFC data published by other Several puzzling chromatographic phenomena found in searching ( I 2) Radecki, A.; Lamparczyk, H.; Kaliszan, R. Chromatographia 1979.

12, 595. (13) Wise, S.A.; Bonnett, W. J.; Guenther, F. R.; May, W. E. J . Chromatogr. Sei. 1981, 19, 457. (14) Wise, S . A.; Sander, L. C. J . High Resolut. Chromatogr. Chromatogr. Commun. 1985, 8, 248. (15) Martire, D. E.; Boehm, R. E. J . Phys. Chem. 1983, 87, 1045. (16) Martire, D. E. J . Chromarogr. 1987. 406, 27. (17) Whalen-Pedersen, E. K.; Jurs, P. C. Anal. Chem. 1981, 53. 2184. (18) Oweimreen, G . A.; Martire, D. E. J . Chem. Phys. 1980, 72, 2500. (19) Martire, D. E.; Nikolif, A.; Vasanth, K. L. J . Chromatogr. 1979, 178,

401. (20) Luffer, D. R.; Ecknig. W.; Novotny. M. J . Chromatogr. 1990. 505, 79.

Yan and Martire the literature21.22can be resolved theoretically. A carefully designed SFC experiment mimicking the theoretical system has been carried out and a more quantitative test is reported in the following article.

2. Theory The canonical partition function, Q, serves as a bridge between the quantum mechanical energy states of a macroscopic system and the thermodynamic properties of that system. The Helmholtz free energy, A , can be derived in terms of Q: A(N,V,T) = -kT In Q(N,V,T) (1) where N is the number of molecules in the system, Vis the volume, T i s the absolute temperature, and k is the Boltzmann constant. In principle, all thermodynamic properties of the system can be determined from the canonical partition function for the system: Q(N,V,T) = E u j ( N , V exp[-Ej(N,V/kTl J

(2)

where EJ is the energy of the j t h energy level and uJ is the degeneracy of the jth energy level. If we consider only the external properties and assume that each molecule exists in the mean field produced by all the other molecules in the lattice system, the configurational partition function, Q,, in the Bragg-Williams approximation, is then given by Q,(N,M,T) = Q,(N,M) exp[-E,(N,M)/kTl (3) where M is the total number of cells or sites of the system, Q,(N,M) is the average distribution of configurations, and E,(N,M) is the average configurational energy. The most successful lattice model treatments of liquid crystals have been based on lattice statistics developed by D i M a r ~ i o . ~ ~ This type of model has been applied to describe nematic liquidcrystalline system^.^^-^' A related model has been employed by Her~feld’*-~~ to treat the configurational entropy of a population of rigid rectangular parallelepipeds. 2.1. Statistical Mechanics for Mean-Field Lattice Model of Isotropic Mixture of Blocklike Molecules (the Mobile Phase). The model is a three-dimensional mean-field lattice with M cells or sites of volume uo. The model system consists of No holes (each occupies one cell) and N , blocklike mobile phase molecules (each has qqr segments) and Nt solute molecules (each has abc segments). All of them are assumed to be isotropically distributed among the M sites ( M = No + qqrN, + abcN,). The total configurational partition function is divided into a steric term, Q,, and an attractive interaction-energy term, exp(-EcIkT). 2.1.1. Steric Term. We extent DiMarzio’s lattice statistics for rods23to a binary mixture of blocks. To determine Q,, the molecules are placed into the lattice, one by one, segment by segment. Let us consider the packing of the pure mobile phase molecules first. We will assume that only the three mutually perpendicular base-vector directions are orientations in which the mobile phase blocklike molecules lie (Figure 1). The number of mobile phase molecules that lie in orientation i will be denoted by N,,. We ask for the number of ways, g,(( ...N,, . .I, No), to pack the N, (21) Chang, H. K.; Markides, K . E.; Bradshaw, J. S.; Lee, M. L. J . Chromatogr. Sci. 1988, 26, 280. (22) Jinno, K.; Ibuki, T. J . Chromotogr. 1989, 461, 209. (23) DiMarzio, E. A. J . Chem. Phys. 1961, 35, 658. (24) Cotter, M. A.; Martire, D. E. Mol. Cryst. Lip. Crysr. 1969, 7, 295. (25) Cotter, M. A. Mol. Cryst. Liq. Cryst. 1976, 35, 33. (26) Alben, R. Mol. Cryst. 1971, 13, 193. (27) Peterson, H. T.; Martire, D. E.; Cotter, M. A. J . Chem. Phys. 1974, 61, 3547.

(28) Dowell, F.; Martire, D. E. J . Chem. Phys. 1978, 68, 1088. (29) Dowell, F.; Martire, D. E. J . Chem. Phys. 1978, 68, 1094. (30) Sokolova, E. P.; Vlasova, A. Y . 139.Cryst. 1990, 8, 47. (31) Dowell, F.; Martire, D. E. J . Chem. Phys. 1978, 69, 2332. (32) Herzfeld, J. J . Chem. Phys. 1982, 76, 4185. (33) Herzfeld, J. J . Chem. Phys. 1988, 88,2776.

The Journal of Physical Chemistry, Vol. 96, No. 8, 1992 3491

Molecular Theory of Chromatography

z

molecules such that Nmiof them lie in orientation i and there are No holes. We place the Nml,Nm2,and then Nm3molecules, one at a time, in orientation 1, 2, and 3, respectively (see Appendix for details). Orientation 1: Nml(rz,qx,qy)

4

Using DiMarzio's counting strategy, we obtain the number of ways, g,,, of placing N,, mobile phase molecules in orientation 1: gm, =

}r

lx

[M- N,,qr(q - I)]! l / q r M! N,,!(M - Nm,q2r)! [M- ",qr(q - l)]! I / r [M- Nm,q2(r- l)]! (4) M!

(

1..

1

orientation 3: N lil ( r

.

q q. )

Orientation 2: Nm2(rx,q,,,qz) Similarly, we obtain the total number of ways to pack Nm2 mobile phase molecules in orientation 2 given that N,, mobile phase molecules in orientation 1 have already been placed:

(M- Nm1q2r)! gmz =

2![M- q2r(N,I

Figure 1. Mobile phase molecules are allowed only three orientations, and the numbers lying in each orientation are N,,, Nm2,and N,,, respectively.

X

+ Nm2)]!

[M- N,,qr(q - 1) - N,,qr(q - l)]! [(M-Nmlqr(q - 1)1! [M- N,,qr(q - 1) - Nm2q2(r- l)]! [M- N m , q r ( q - 1)1!

where g,,, gt2,gt3,gt4,,gtS,and g16 are given in the Appendix. The total configurational partition function for the mobile phase system will be

Qc(m)(N,M,T) = Qc(m)(N,M) exP[-Ec(m)(NvM)/kTl

,

Orientation 3: Nm3(ry,qx,qz)

(9a)

Note that, with a = 6 = 1 (rod) and N , = 0, neglecting attractive interactions and keeping three mutually orthogonal orientations of the solute molecules, one can retrieve the configurational partition function derived by D i M a r ~ i ofor~ ~an isotropic distribution of rodlike molecules:

n, = [ M -N , ( c - l)]![M - N4(c - l)]![M-

N,(c ~)]!/N~!N,!N~!NS!(IW)~ (9h)

The total number of ways to pack Nm3mobile phase molecules in orientation 3 is

We now consider the attractive interaction-energy part of

Q

(N,M,T).

'fb.2. Attractive Energy Term. DiMarzio's lattice statistics can be extended25to determine the attractive energy term, exp[-E,(N,M)/kT], of the total configurational partition function. The average configurational energy E, is assumed to be a mean-field sum of pairwise segmental intermolecular attractive interactions. There is an attractive interaction energy, ti,, between two segments of different molecules if the two segments are directly adjacent. The average attractive energy E, can be expressed as k

E, = The product obtained from e q s 4-6 gives the steric configurational partition function, or the total number of ways, of packing the N , pure mobile phase molecules into the lattice system: Note that, with q = 1, we retrieve DiMarzio's expression for rigid rods (see eq 7 in ref 23). We will assume that the solute molecules can lie in six possible orientations (see Figure 2). The number of molecules that lie in orientation i will be denoted by Nti. We have 6

E N t i = N , (Nti = q N , ) i= 1 where ai is the average fraction of solute molecules in orientation i.

By packing the solute molecules into the lattice in which the mobile phase molecules have already been placed (see Appendix for details), the steric configurational partition function for the mobile phase system (including the solute) can be obtained: Rc(m)(N,M) = gmgtlgt~t3gl&iSgt6 (8)

I3

Ivijeij

ij=I

where RJ,is an average number of ij pairwise segmental interactions. The following pairwise segmental intermolecular attractive energies (t,(O) are defined: :,e, interaction energy between a segment of a solute molecule and a segment of a mobile phase molecule;: , ,e interaction energy between segments of mobile phase molecules; qt: interaction energy between segments of solute molecules. To calculate the average number of pairwise interactions 0.e tween a segment in a molecule of type i and one in a molecule of type j , it is necessary to know the number of segments i? molecules of type i available for contact and the conditional probability that any such segment will be directly contacted by the segments of type j . Let P,,be the conditional probability that the segments of type j are directly adjacent to any given segment of type i in orientation k. The number of segments of the so1l;t.e molecules in orientation 1 available for contact is N,,Aef= A , p , N , , where Aef is the effective contact area for a solute molecule. Hence, the conditional probability that the segments of mobile

3492 The Journal of Physical Chemistry, Vol. 96, No. 8, 1992

4

Yan and Martire

't

'f C

C

b

a

Y

a

i

Y

i

Z

f J

H

orlentation 6: N,, ( a b, c. ) (a. b. c. ) Figure 2. Blocklike solute molecules are allowed six orientations, and the numbers lying in each orientation are N,,,N,*, N,,, NL4. N,,, and N,6, respectively.

xwlentatlon 4: N,, (a, b. c. 1

orlentetlon 6: N.

phase molecules are, on average, directly adjacent to a solute segment in orientation 1 is

I )- NJabc - (aI+ a,)ac - (az+ a4)bc - (a3+ a5)ab] 3r (11)

The average number of solute-mobile phase., pairwise segmental interactions is then given by alNtAefPtm(1)

"(1)

(12)

2.1.3. Total Configurational Partition Function. From eqs 8, 9, and 17, the total configurational partition function for the system can be written Qc(m)(Ndf,n= Qc(m)(N,M)exP[Ec(m)(N,M)/kTl = g&tlgt&3gt4&5&6 eXP[-(Ntmttm + Rmm%nm + Nttk)/ k T ] (18)

We now can evaluate Qc(,,,)(N,M,gand from it obtain the configurational equilibrium thermodynamic properties for the system, such as AC

the Helmholtz free energy: - = In Qc kT

(19)

the chemical potential of component i:

We know that Rtm(2),Rtm(3), IVtm(4)$I V t m ( 5 ) v and IVtm(6) will have similar forms as Rtm( l), and Ptm(1 ) = P,,(2) = Ptm(3)= P,,(4) = P0(5) = Ptm(6)= PI,,,. Thus

R t m = NtAefPtm

(13)

Similarly, one obtains

the entropy:

S -k = In Q(N,M)

(21)

the dimensionless pressure-to-temperature ratio: where P,, = Pm. The factor 2 occurs in the denominator to avoid counting all pairs twice. Similarly, one obtains N,t = NtAefPt,/2 (15) where

P,,= { N l [ ( a l+ a&c

+ (a2+ a4)bc+ (a3+ a s ) o b ] } /

The average configurational energy E, can thus be obtained: &c(m) =

(Rtmctm

+ " n c m m + Nttttt)

(17)

Now let us consider the chemical potential for the solute component in the mobile phase. From eqs 18 and 19, we obtain the Helmholtz free energy for the system: --Ac(m) ln Qc(m) = In Qc(m)(N,M) - Ec(m)(N,M)/kT= kT In g m + In gtl + In gt2 + In 813 + In gl4 + In g15 + In g16 Ec(m,(N,M) / k T (23) According to eq 20, the partial derivative of the Helmholtz free energy with respect to the number of the solute molecules (NJ gives the chemical potential for the solute in the mobile phase system:

Molecular Theory of Chromatography Substituting the relevant terms of eq 23 into eq 24, using Stirling’s approximation, and then assuming an isotropic distribution of solute molecules (al= a2 = a3 = a4= a5= 0 6 = ‘ 1 6 ) and infinite dilution of the solute component in the mobile phase, we obtain the chemical potential for the solute in the mobile phase system:

The Journal of Physical Chemistry, Vol. 96,No. 8, 1992 3493 Orientation 1: NI~(axrbyrcz) The steric term for N l l molecules in orientation 1 is

--k ( m ) = kT

abc In (1 - e,)

+ (1 - abc) In

where abc is the volume of the blocklike solute molecule, Om is the segmental number density (or volume fraction) of the mobile phase molecules, and pI(,) is the molecular number density of the solute in the mobile phase. 2.2. Statistical Mechanics for Mean-Field Lattice Model for Anisotropic Mixture of Blocklike Molecules (Statio~ryPhase). The model system consists of N,’ blocklike stationary phase molecules, each having 021segments, and N,‘ blocklike solute molecules, each having abc segments, and N,,’ holes, each occupying a unit cell of volume uo. The stationary phase molecules are arranged in parallel layers with the center of every block in a particular layer lying in the same plane. Within each layer the long axes of the molecules are perfectly aligned in one direction (say, I). In principle, the empty cells can occur in two kinds of positions: in layers of “holes” between layers of molecules and in vacancies in the layer. In the thermodynamic limit, however, configurations containing layers of “holes” can be shown to make a vanishingly small contribution to In Qc.25 We shall assume, therefore, that all empty sites are located in the molecular layers. The total configurational partition function in eq 3 is divided into a steric term, a,, and an attractive interaction-energy term, exp(-E,l k 13. 2.21. Steric Term. To determine a,, the molecules are placed into the lattice, one by one, segment by segment. First, all the stationary phase molecules with only two equivalent orientations are placed, and then all the solute molecules with six orientations are placed. Let us pack the stationary phase molecules first. By a reasoning process analogous to that used in the last section, we obtain the following for g,, the steric configurational partition function of stationary phase molecules in one of the layers:

where T is the number of layers. To determine the steric configurational partition function, we have to pack the solute molecules into the lattice given that the stationary phase molecules have already been placed in the system. First, assume six orientations (see Figure 2). Later, we will prove that some particular orientations are preferred. The number of solute molecules that lie in orientation i will be denoted by Nw The packing approach in this section will be similar to that used in the previous section (see Appendix for details). However, the results will be different since we are dealing with an anisotropic phase (such as a smectic liquid crystal). Within a layer of the anisotropic phase, between two rodlike molecules aligned along the I axis, the probability of finding (c - 1) empty sites in the z direction will approach unity. In other words, if A is empty all sites above or below it in the layer will be empty. (This is a good assumption if the solute concentration is very low).

Orientation 2: N12(ayrbx,~z)

Orientation 3: N13(ax,bz,cy)

where

Orientation 4: (ay,bz,cx)


W ~ B=T M - N,d(W - 1) - N,Iac(b - 1) - N,$c(u - 1) Nt,ab(c - 1) - N , ~ ~ c-( u1)

Yan and Martire 2.2.2. Attractive Energy Term. The following pairwise segmental intermolecular attractive interaction energies are defined: eu is interaction energy between a segment of a solute molecule and a segment of a stationary phase molecule; and c is interaction energy between segments of stationary phase molecules. Neglecting end-end interaction, the segmental number of the solute molecules in orientation 1 available for contact by the stationary phase molecules can be obtained: N,IAefl= N,l(2ac + 2bc) = cylNt2(ac + bc) (36a)

where a1is the fraction of solute molecules in orientation 1 and Acll is the effective contact area of a solute molecule lying in orientation 1. Similarly, we have N,2Aen = N , ~ ( ~ + u c2 6 ~ ) C Z ~ N , ~+( Ubc) C (36b)

Orientation 5 : (a,,b,,c,)

Nt3Acf3= N,3(2ab + 2bc) = cy,N,Z(ab

+ bc) Nt4Aef4= N,4(2ab + 26c) = cy4Nt2(ab+ bc) Nt5Acf5 = Nt5(2ab+ 2ac) = cy5Nt2(ab+ uc) Nt6Acf6= N,6(2ab + 2aC) = a6Nt2(ab + UC)

(36c) (36d) (36e) (360

where a2,a,,a4,cy5, and asare the fractions of solute molecules in orientation 2, 3, 4, 5 , and 6, respectively. The probability that the segments of stationary phase molecules are, on average, directly adjacent to a segment of a solute molecule is

where Os is the segmental density or occupied volume fraction of the stationary phase molecules. The product of eq 37 with the sum of eqs 36a-f gives the average number of solute-stationary phase, pairwise segmental interactions: 6

R t

= P,t"iAefi

(38)

i= I

The segmental number of stationary phase molecules available for contact, neglecting the end-end effect, is 4N,wl. The probability that the segments of stationary phase molecules are, on average, directly adjacent to a segment of a stationary phase molecule is w6AE

=

M - N,021 - abc(N,I + Nt2 + N,,

(39)

+ N,4 + Nts + Nt6)

The average pairwise segmental number of stationary phasestationary phase interactions is 2NSw1~,( 1/ w ) Ivs = 1 - e,(i - i/w) Similarly, we can get the average number of solute-splute pairwise segmental interactions in the stationary phase, N,t. The total average configurational energy EC(,)is then given by

E,(,) = m s t c s t + &ess+

"et,)

(41)

2.2.3. Total Configurational Partition Function. Using eqs 35 with 34 and 41, one obtains the total configurational partition function for the stationary phase system (including the solutes): Q,(,)(NN,T)=

%(,)("I exp[-E,(,)(N,M)/kT]

Q,(,,(N,MT) = exp[-A,(,)/kTl = [g&tlgtflt3gtdt5gt61'exp([-T(Nstest + f l s % s + Nttett)l

(42)

/kg

(43) Substituting eqs 26, 28-33, 38, and 40 into 43, using Stirling's approximation, then taking the partial derivative of the Helmholtz free energy with respect to N,, and assuming infinite dilution of


Molecular Theory of Chromatography the solute component in the stationary phase, one obtains the chemical potential for the solute in a layer of the stationary phase system:

w n gt3)

g14)

aa

aa

+ a4)ac + (as+ a&] In (1 - e,) + - ab) + (a3+ a4)(l - ac) +

(as

[(a,+ a2)(1

+

1

(as a6)(1 - bc)] In 1

(

I):

- Os 1 - - -

where 0, is the segmental number density (volume fraction) of stationary phase molecules and pt is the molecular number density of solute. The chemical potential for the solute in the whole stationary phase system will be the same since pi is an intensive quantity. Let us assume that the equilibrium orientational distribution is symmetric about the axis of the preferred direction. (This assumption is certainly valid in our system). Then we have a1 = u2= a, u3 = a4 8, as= f f 6 p y. In this case, the chemical potential for the solute in the stationary phase can be written &s)

--

kT

1

- e,( 1 -

i)

1[

In (apt)

+ pt 2 In (Opt) - (ac + bc) In eS(l / w )

+ 4y(ac + ab)]$

1

-e,(,

(45)

Equation 45 is the general expression for the chemical potential of the solute in the anisotropic stationary phase. At this point it is convenient to single out direction z (the preferred direction) by further assuming that 8 y. (This assumption is not strictly true; however, we will see that it does not affect our final results.) Keep in mind that ffl ff2 = f f “3 = a4 = a5 = ff6 = @ (46) 2ff 4 8 = 1 @ = y4- y2a

2(bc

-i)

Pt

4@(bc+ ab)

(48e)

Summation of eqs 48a-e gives the left-hand side of eq 47:

+

= (2aab 2@ac+ Zybc) In (1 - 6,) + [2a(l - ab) + 28(1 - ac) + 2y(l - bc)] In [ l eS(l - l/w)l - Pa In (4 + 28 In (@d +

2pt(bc + ac - 2ab)It kT

[ l-:&,]l-

+ ac - 2 a b kT )k

(49)

-

+

where a and @ can be determined for given M , T , and N by minimizing the Helmholtz free energy with respect to ai,i.e., by solving the following equation a t equilibrium:

[:(2)] T.P=

O

Putting @ =

- a/2 into eq 50 and then rearranging it, we have 1

f f = -

2

+ 4R

where

(47)

Taking the logarithm, respectively, in eqs 26 and 28-33, using Stirling’s approximation, taking the partial derivative of those equation and eq 41 with respect to a, then assuming infinite dilution of the solute component, and for simplicity, further assuming bc L ac 2 ab > 1 , we obtain

[y].O

* -ac + - - abcb c > O (aIbIc) “ 2 2 1 - e,(i - i/w) 11 - e,

est

< 0 (attractive) ;.

0 IR I1, Le.,

!/6

Ia I!I2

If (ac/2) + (bc/2) - ab is large enough to make 4R > 1; then eq 45 simplifies to

(2) When the solute molecules are cubes, we have ( u c / ~ ) (bc/2) - ab = 0

+

Then 1 - e,

-1

[4a(ac

+ bc) + 4@(bc+ ab) + 4y(ac + ab)]$

where pt(s)is the molecular number density of solute in the stationary phase. From eq 25, we have the chemical potential for the solute component in the mobile phase:

and 1

r We also have LY = 8 = y is ki,(S)

--

kT

(59)

-.

1

In this case, the chemical potential

=c21n(1-8,)+(1 -c2)1n[1-8,(1-l/w)]r 1

In

2.3. Retention Equation (Distribution Coefficient). The chromatographic distribution coefficient, K,is defined as the ratio of the equilibrium concentration of solute in the stationary phase,

(34) Martire,

(y )

D.E.;Boehm, R. E. J . Phys. Chem. 1987, 91, 2433

(60)



TABLE I: Geometric Parameters of Solutes" dimensions, solute chrysene benz [a]anthracene triphenylene benzo [c]phenanthrene pyrene anthracene phenanthrene naphthalene p-terphenyl biphenyl

V,', A3 207.77 207.77 207.77 207.77 181.01 165.27 165.27 122.77 218.79 149.45

" VWis the van der Waals volume36 in A',

U

b

C

6.66 7.38 8.53 6.54 7.38 6.09 6.51 5.97 4.76 4.47

12.13 10.95 9.48 8.90 9.54 10.55 9.88 8.01 13.74 9.15

and Vw = Vfw/u0, Amin = ab/a,, and A,, = (ob + uc

ri(s)/kT = P i ( m ) / k T

(61)

where i denotes the ith component distributed between the stationary and the mobile phases. Applying eqs 61 with 59 and 60 to the solute component ( i = t ) , we obtain the retention equation for blocklike solutes in the anisotropic stationary phase:

I

+ ybc) In

r

abc ln

1-8,

1---39

B(bc + ab) + ~

[

re1 parameters

2.57 2.57 2.57 3.57 2.57 2.57 2.57 2.57 3.35 3.65

where pI(,) is the molecular density of solute in the mobile phase. The equilibrium condition at the operational temperature and pressure is given by

In (6K) = 2(aab + Bac

A Amin 5.16 5.71 6.60 7.03 5.71 4.72 5.04 4.62 4.80 4.92

Ad 38.90 38.54 38.29 37.65 34.3 1 32.26 32.06 25.21 38.33 27.31

+ bc)/a,, where uo = 6.06 A' and a. = 3.32 A2.

isotropically distributed in the anisotropic stationary phase: r 1 1 In K = -(ab ac bc) In 3

+ +

abc In

[

m] 1-8,

1----

39

1

1 - 8sl(l-a.l/w)] - 4[a(ac

VW 34.29 34.29 34.29 34.29 29.87 27.27 27.27 20.26 36.10 24.66

-

3r

-

+ bc) +

3r

+ ab)]-

( U C

For cubic solutes (a = b = c), eq 64 provides a useful reference point for the examination of the selectivity enhancement experienced by structural isomers in an anisotropic stationary phase (relative to an isotropic one),I6 as will be discussed in a subsequent paper. (2a In

-

CY

+2S In B

+ 27 in y)

(62)

-

2.3.2. Solutes Fully Aligned with the S t a t i o ~ t yPhase. With and B = y 0, eq 62 gives the distribution coefficient for blocklike solutes fully aligned with the anisotropic stationary phase: a

r

1

In ( 3 K ) = ab In

r

1

3. Application to CC, LC, and SFC Let us apply the theory to gas, liquid, and supercritical fluid chromatography and subject the final equation to preliminary experimental tests. 3.1. Calculation of Molecular Dimensions. To calculate the molecular dimensions, we need to scale the system. Martire and B~ehm treated ~ ~ the mobile phase molecules as flexible chains. The segmental number, rb, of COz was estimated from the ratio of a low-temperature solid density ( q b * , the close-packed density) to the critical density, pcr: pb*/pcr = 1

+ rb'/2

-

(65)

From the critical density of C02,pcr = 0.468 g/cm3, and the density of crystalline COz at -79 OC, pb* 1.56 g/cm3, one calculates rb = 5.40, which is close to the value assigned by V e i ~ e t t i . We ~ ~ will see later that the assignment of the segmental number of C 0 2does not affect the calculation of the stationary and mobile phase densities. From B ~ n d iC02 , ~ ~had van der Waals volume of 32.12 A3/ molecule. The unit segmental volume can be calculated: uo = Vw/rb = 6.06 A3 (66) Accordingly, the unit segmental area and length are lo = 1.82 A a. = uO2l3= 3.32 A2 Equation 63 is applied in section 3. 2.3.3. Solutes Isotropically Distributed. With CY = /3 = y = 'I6,eq 62 gives the distribution coefficient for blocklike solutes

(35) Vezzetti, D.J . Chem. Phys. 1982, 77, 1512; 1984,80, 866. (36) Bondi, A. J . Phys. Chem. 1964, 68,441.

(67)


We model a PAH molecule as a block with dimensions of ubc (a Ib Ic). To get molecular dimensions, a carbon-skeleton plane of each molecule has been constructed by fuing the bond distances D(C-C) = 1.40 A (in a phenyl ring) and D(C-C) = 1.50 A (for a single C-C bond). Then a rectangle with a minimum area which encloses the PAH molecule is constructed and expanded equally to mutually orthogonal directions on the plane in such a way that when the rectangular area is multiplied by the thickness, the van der Waals volume, V,, of the molecule is obtained. We will adopt Barman’s3’ approach by choosing the thickness of planar PAH as 2.57 A. The extra thickness, A, of nonplanar molecules can be estimated from X-ray crystallographic The rectangular block for each molecule can be represented by the computed dimensions a, 6, and c, such that V, = abc. To provide an example, a schematic for the evaluation of the molecular dimensions of chrysene is shown in Figure 3. The relative van der Waals volumes and areas of the molecules are obtained by scaling dimensions relative to the unit volume, uo, and unit area, up Table I provides the geometric parameters of some PAH and polyphenyl molecules. 3.2. Test of the Theory and Discllssioa Strictly, the retention equation is applicable to systems composed of blocklike solutes and a completely aligned and layered stationary phase. However, we anticipate that it may be qualitatively applicable to any anisotropic stationary phase (see later). We have already verified that, both entropically and energetically, orientations 1 and 2 for the solute molecules are the dominant orientations in a completely 1/2 and fl = y aligned stationary phase. In this ease, a 0, and eq 63 applies. 3.2.1. Effect of Solute Characteristics. Rearranging eq 63, we obtain

-

I

-

.

Yan and Martire

r

Then we have In (3K) = QIAmin + Q . 2 4 + Q~Jw ‘

(68)

(37) Barman, B. N . Ph.D. Dissertation, Georgetown University, Washington, DC, 1985. (38) Herbstein, F. H.; Schmidt, G . M . J. J . Chem. SOC.1954, 3302. (39) Charbonneau, G. P.; Delugeard, Y. Acfo Crysfollogr. 1977,33, 586. (40) Baudour, P. L.; Cailleau, Y. D. H. Acta Crystallogr. 1976, 832, 150.

(72)

For a specific chromatographic system at fixed operational conditions, Q,, Q2, and Q3 should all be constant if all solutes considered belong to the same “family” and have equivalent segmental interaction energies, est and emt. Note that QIis always negative since est < 0 and 8, I1; Q2 is usually positive because, in most chromatographic systems, est < emt < 0 and Os > 8,; Q3is positive since em I1. Accordingly, one can rewrite eq 72 as (73) In (3K) = -1QIIAmin + IQtlAer + IQ~VW For PAH solutes with the same number of rings (isomers) the

van der Waals volumes are the same and the effective contact areas are very close (see Table I). Therefore, we may assume that the sum of the second and third terms on the right-hand side of eq 73 is essentially a constant for the isomers. Then, we have In k’ = -IQIIAmin+ constant (74) Equation 74 predicts a linear relationship of In k’versus the minimum area (scaled), with a slope of -IQII and an intercept of Q2Acf Q3Vw- In 34, for isomeric PAH solutes. To confirm this prediction experimentally, we searched the literature for pertinent GC, LC, and SFC data. Although many studies on PAHs in anisotropic stationary phases have been undertaken, there are not many data available which provide the necessary experimental details. A few sets of data were identified, although some were in graphical form. Janini et aL3 synthesized a liquid-crystalline (BPhBT) phase and used it as a stationary phase in GC. They separated fiveand six-ring PAH isomers successfully. Retention data for five-ring isomers at 270 OC and six-ring isomers at 290 OC were detcnnined from the published graphs. Listed in Table I1 are the values of minimum area and the retention data for the five and six-ring PAHs on the liquid-crystalline phase. Linear least-squares analysis of the data in the Table 11, according to eq 74, yields In k’(5mcr)= -0.89Amin+ 8.09 (75)

+

In k’(6mcr)= -0.73Ami, + 8.51

Let us define

1

(76)

with correlation coefficients of 0.97 and 0.99, respectively. Illustrated in Figure 4 and 5 are the plots of In k‘vs Aminfor five- and sh-ring PAH isomers, respectively. The agreement between the theoretically predicted and experimental dependence of In k’on Aminis good. On the basis of observations of shape selectivity for liquidcrystalline phaseg in GC by Janini et a1.3q10and Radecki12et al., Wise and co-workers7J3 described a relationship between the length-to-breadth ratio (L/B) of PAHs and their reversed phase LC retention. Retention data on both a monomeric and a polymeric CI8phase for over 100 PAHs were compared, and a similarity in shape selectivity for cl8 bonded-phase LC and liquidcrystalline phase GC was observed. For the four-ring PAH isomers, the LC and GC elution orders are the same in the two chromatographic systems. The L/B ratio predicted the retention order with one exception, benzo[c]phenanthrene (with a larger L/B) eluting earlier than triphenylene. The anomalous behavior was attributed to the nonplanarity of benzo[c]phenanthrene, an interpretation with which we agree. For the fivering PAH isomers, the GC (liquid-crystalline phase) and LC (polymeric c18 phase) relative elution characteristics are nearly identical. The structures and minimum areas of the five-ring PAH isomers are illustrated in Figure 6. The retention, in general, increases with increasing L/B, and anomalies in elution order relative to L/B ratios are consistent in both chromatographic



TABLE II: Minimum Areas and GC Retention Data for Five- and Six-Ring PAH Isomers on BPhBT Liquid-CrystallineColumn

solute

structure

Amin

dibenz [o,c]anthracene

In k' 2.08

dibenz[a,h]anthracene

2.89

5.56

picene

3.47

5.25

pentacene

3.85

4.84

2.53

7.80

2.83

7.33

3.66

6.33

3.99

5.79

4,5,7,8-dibenzpyrene

6.78

k' is generated with ro = 0.50 min from ref 3 for five-ring isomers at 270 "C and six-ring isomers at 290 "C, respectively.

4.b

7 Tempratut = 270 g:

1.b! 4.5

5

e

5.5

0.5

I

AIM.

[Temperatlm = 290 4:

5.5

a

0.5

9m

7

7.5

Figure 4. Plot of In k'vs minimum area (Ah) for five-ring PAH isomers on BPhBT (liquid crystalline) phase at 270 "C in GC: M, experimental data generated from ref 3; linear fit.

Figure 5. Plot of In k'vs minimum area (Amin)for six-ring PAH isomers on BPhBT (liquid crystalline) phase at 290 "C in GC: m, experimental data generated from ref 3; -, linear fit.

systems. Two isomers, dibenz[a,c]anthracene (compound no. 2 in Figure 6 ) and benzo[a]naphthacene (compound no. 9 in Figure 6), have significantly longer retention in both LC and G C than predicted by their L/B values. Assuming that the isomers have the same van der Waals volume and roughly the same contact area, eq 74 predicts a linear relationship between In k'and Amin. Shown in Figure 7 is the plot of logarithm of the relative retention vs the minimum area for five-ring isomers on the liquid-crystalline phase using the data from ref 7. The linear correlation between the retention data and the minimum area is very good (coefficient 0.98).

It is very interesting, as Wise et al.I3pointed out, that the elution order of PAHs on alumina is identical to that on the G C liquid-crystalline or LC polymeric phase. Because of the crystalline structure of alumina, the adsorptive sites are arranged in a regular and linear fashion (similar to a liquid crystal), thus the PAHs tend to line up along the ~ u r f a c e . ~ ' 3.2.2. Effect of Mobile Phase Parameters. Once the stationary phase is fixed, the sum of the first and the third terms in eq 68

-.

(41) Snyder, L. R. Principles of Adsorption Chromatography; Marcel Dekker, Inc.: New York, 1968; pp 324-29.

& & &&

3500 The Journal of Physical Chemistry, Vol. 96, No. 8,1992

00

Yan and Martire

00

000

A m~n-S.73

A m k r 7.51

&y&&& 000

000

00

Amkr7.2S

0

Am(lr6.Ol

A

5.47

Figure 6. Structures and minimum areas (Amin)of five-ring PAH isomers.

O.b! 6

6.6

6

6.6

7

7.6

8

8.6

0

0.1

I

10

A*

Figure 7. Plot of natural logarithm of relative retention vs minimum area of solute for five-ring PAH isomers on SBsmectic (liquid crystalline) phase in SFC: 0,experimental data from ref 7; -, linear fit.

will be a constant for a particular solute at constant temperature. Then, eq 68 becomes

(77)

where In KO, a constant = the first term in eq 68 + the third term in eq 68 - In 3, is the limiting (zero density) value. The first term in eq 77, arising from the configurational entropy of the solute in the mobile phase, is independent of temperature. It leads to

an increase in In K (or In k') with increasing mobile phase density e(), for a particular solute. The second, usually dominant term reflects the mobile phase-solute attractive interactions and leads to a decrease in In K with increasing 6,. We can see, from the coefficient (slope) of the term, that the decrease in In K with increasing Om will be more rapid under the following conditions: (a) stronger mobile phase-solute interactions (more negative emJ; (b) larger contact area of the solute with the mobile phase; (c) lower temperature. If the second term in eq 77 is indeed the dominant term (usually so under the operational conditions), we should be able to confirm these predictions experimentally. In searching the literature to test the theory, we found several puzzling but interesting phenomena. Lee and co-workers2' studied the selectivity enhancement for petroleum hydrocarbons using a smectic liquid-crystalline phase and a methylpolysiloxane stationary phase in SFC. Two compounds, benzo[a]pyrene (BaP), which has a planar structure with MW 252, and tetrabenzo[o,cf,h]naphthalene (TBN), which has a nonplanar structure with MW 328, were used to examine the retention mechanism. TBN eluted later than BaP on the methylpolysiloxane phase at all temperatures studied. However, on the liquid-crystalline phase, BaP eluted later than TBN at C 0 2 mobile phase densities between 0.40 and 0.65 g/mL at 140 OC,while the elution order was reversed at 180 OC. At intermediate temperatures (150, 160, and 175 "C), the plots of capacity factor vs mobile phase density for BaP and TBN intersected. TBN eluted later than BaP at low densities and earlier than BaP at high densities (see Figure 8). They also found that the selectivity for BaP/TBN increased with increasing density at all operating temperatures and suggested that under isothermal conditions, the molecular shape selectivity of the liquid crystal is enhanced with increasing density as a result of pressure-induced ordering of the liquid-crystalline stationary phase. Now, this puzzle can be solved using our model: It is simply because TBN has a larger contact area and a larger van der Waals volume than does BaP. According to eq 77, the retention of TBN is longer in the lower mobile phase density region because of the larger value of the first term (the packing term). However, from the energetic (second term) point of view, the retention of TBN should decrease more rapidly because the six-ring compound has a larger contact area than does the five-ring BaP, therefore leading to a more negative slope of In K (or In k') versus mobile phase density (6,). On the other hand, BaP has a relatively shorter

Molecular Theory of Chromatography retention in the lower mobile phase density region but a less negative slope of In K vs 0,. Therefore, the crossover can occur. Of course, the increase of 8,’ from the entropic point of view, will make the retention increase through the van der Waals volume (as the slope), but this term is hardly dominant on a relative basis since the mobile phase density dependence is logarithmic in the first term. Since TBN has a larger slope than BaP does, the selectivity, aBap/TBN, would naturally increase with increasing density. Jinno and IbukiZ2used Fourier transform infrared spectroscopy (FT-IR), nuclear magnetic resonance spectroscopy (NMR), and differential scanning calorimetry (DSC) techniques in addition to chromatographic techniques to study the retention mechanism in reversed-phase LC for large PAH on a CI8polymeric bonded phase. They found a similar phenomenon (the crossover, see Figure 2 in ref 22). However, the elution order of different compounds changed with the change of mobile phase composition, instead of mobile phase density. Using our retention equation, this phenomenon can also be explained. Within the framework of the general model, a second mobile phase component (a modifier) can be readily included. Adding a second mobile phase component (dichloromethane, in this case), which can interact strongly with the solute molecules in the single-component mobile phase, is equivalent to strengthening the average mobile phase-solute interaction energy in a single mobile phase system and therefore to strengthening the solvating power of the mobile phase as a whole. According to eq 77, this effect is exerted on the solute retention through the solute contact area as the coefficient (the slope of the plot of In k’vs mobile phase composition). At low dichloromethane concentration, the retention of tetrabenzo[a,c,dj,m]perylene (with a relatively larger contact area) is longer (because of the stronger interaction with the stationary phase); however, the decrease of the retention for this solute with increasing dichloromethane concentration is much more rapid compared with coronene (with a relatively smaller contact area). Therefore, the crossover occurs. Note that Martire and B ~ e h m have ~ ~stated in their unified molecular theory that replacing the “poor” solvent by the “good” solvent in LC with a binary mobile phase through increasing the volume fraction of the latter is formally equivalent to replacing empty space by molecules in SFC with a single-component mobile phase through increasing the density (occupied volume fraction of the SF carrier). Therefore, the phenomenon observed in LC by Jinno and Ibuki is analogous to that observed in SFC by Lee and co-workers.*’ 3.2.3. Effect of Temperature. Equation 68 can be rearranged to give In ( 3 K ) = ab In

r

1 - 4 A

“s

1 1 - 0,(l - l / w )

1

1

-

The Journal of Physical Chemistry, Vol. 96, No. 8,1992 3501

1.4 1-

O b

0-

4.6,

Substituting eq 80 into eq 79, we obtain In K = - A H / R T AS/R

+

Comparing eq 81 with eq 78, we see clearly the connection of the macroscopic thermodynamic properties with the microscopic effects:

AH = 2(ac + bc)Lcsl

eS(1

/w)

1 - 0,(l - I/@)

-

1

where L is Avogadro’s number and where we have neglected (small) effects due to thermal expansion of the stationary phase. Now, with eqs 82 and 83 we can understand, at the molecular level, the significance of the enthalpy (AH) and the entropy (AS) transfer of the solute from the isotropic mobile phase to the anisotropic stationary phase. Furthermore, knowing the parameters in eq 82 and 83, we can (roughly) estimate the values of AH and AS. Equation 82 reveals that the so-called solute enthalpy of transfer is really the difference between the energetic term for a solute in the mobile phase (Hm)and the energetic term for the solute in the stationary phase (HE). Therefore, we can write AH=Hs-Hm

(84)

where H E = 2(ac + bc)r,,L

Recall that

AG= AH-ThS (79) The distribution coefficient, K, is related to the transfer free energy of the solute, AG, by AG = -RT In K (80)

(81)

1

- 0,(l

- l/w)


Substituting K = 4k'into (81), we obtain In k' = -AH/RT + M / R - In 4

(85)

For a specific chromatographic system at constant mobile-phase density, the van? Hoff plot (In k'vs 1 / T ) according to eq 86 should give a slope of -AH/R and intercept of ASIR - In 4. If AH is a constant throughout the temperature region, then the van't Hoff plot should be linear. If AH is not a constant, then a curve should be observed. Keep in mind that s l o p e = A H / R = (Hs - P ) / R

(86)

where i Y and Hm are always negative since both est and c, are negative. The following is a summary of the trends predicted by eq 78: (a) The slope of the van't Hoff plot is predicted to be constant (linear relationship) provided that AH is constant and there is no phase transition in the operational temperature range, and that there is no change of solute planarity. (b) The slope should be positive (AH I 0) in most chromatographic systems because the mobile phase density (e,,) is usually lower than the stationary phase density and crl I cmt in most chromatographic systems. (c) The slope in LC should be lower than the slope in GC and SFC since the mobile phase density in LC is higher. (d) Under the same conditions, an isomeric solute with a relatively larger contact area with the stationary phase should have a larger slope for the van't Hoff plot. (e) A mobile phase which can interact more strongly with the solute would make Hm more negative and, therefore, lead to a decrease in the slope. Note that, if a phase transition occurs in the stationary phase operational temperature range, we would expect a discontinuity (first-order phase transition) or a change in the slope (second-order phase transition) in such plots. Novotny and co-workers4*did an experimental study of PAH retention on both a SB-smectic phase and a SB-biphenyl phase in SFC. It was observed that anthracene exhibits a more negative AH than phenanthrene, which is in agreement with prediction d. Predictions a, b, c, and e are commonly followed in chromatographic experiments. In this and other lattice model systems34as applied to SFC, if effects due to absorption of the supercritical fluid by, and thermal expansion of, the stationary phase are neglected, then the enthalpy terms b m e internal energy (v) terms and AH = AU under conditions of constant mobile phase density. However, in experimental SFC systems, the slope of a constant-density van't Hoff plot requires correction to evaluate AH or AU.43

Yan and Martire plete penetration of the solute molecules into the stationary phase; (e) assuming infinite dilution of the solute component. The current theoretical approach can be extended to treat many kinds of chromatographic systems with different mobile phases, stationary phases, and solutes with different sizes, shapes, and degrees of flexibility. The sorption of mobile phase solvent by the stationary phase and partial alignment of the stationary phase can also be taken into account. Work in these directions is currently underway. Acknowledgment. This material is based upon work supported by the National Science Foundation under Grant CHE-8902735. We are also grateful to Richard E. Boehm for many helpful discussions. Appendix: Packing of the Solute Molecules

Orientation 1 : NII(uxrby,c,)

To estimate the number of ways to place the (rill + 1)th molecule into the lattice in orientation 1, given that N , mobile phase molecules and qlsolute molecules have already been placed, we must determine the probability that abc contiguous sites lying in this orientation are empty. Consider a contiguous pair of sites arbitrarily chosen except for the fact that the line determined by the centers of these sites lies along direction y. Label the sites A and B. Site A has a probability of being empty equal to the fraction of sites that are unoccupied by molecular segments since A can be thought of as chosen arbitrarily, i.e., the numbers of ways of A being empty is M - Nmq2r- ntlabc. The number of ways that site B is empty is 2(M - N,q2r - q,abc) since we can find B on either side of A in they direction. The number of ways of B being occupied, given A is empty, is 2[n,,ac (N,/3)(2qr q2)]. (Here we assume that the mobile phase molecules are isotropically distributed.) The total number of ways (the number of ways of being empty plus the number of ways of being occupied) of B is 2[M - N,q2r(l - 2 / 3 q - 1/3r) - n,,oc(b - l ) ] . The probability that site B is empty, given A is empty, is

+

+

N - Nmq2r- nIlabc (1-1) M - N,q2r(l - 2/3q - 1 / 3 r ) - n,,ac(b - 1) The probability of finding ( b - 1) contiguous empty sites in the y direction is

4. Conclusion

DiMarzio's lattice model was extended to treat blocklike molecules in both isotropic and anisotropic systems. Expressions for the thermodynamic properties (pi, A , G, S, H , etc.) were derived in terms of both the state variables ( p and )'7 and molecular parameters (V,, Amin,Aef, tij,etc.). The theory provides a bridge between the macroscopic properties and the microscopic parameters. A retention equation for blocklike solutes in anisotropic stationary phases, applicable to GC, LC, and SFC, was derived using the chemical potentials obtained. The preliminary tests using GC, LC, and SFC data published by other groups are very promising. Several puzzling phenomena found in the literature can be explained by the theory. However, most of the predictions remain to be further confirmed. A carefully designed SFC experiment mimicking the model system has been carried out and more quantitative tests are reported in the following article.44 It should be kept in mind that several simplifications were made: (a) assuming a clear boundary between the stationary phase and the mobile phase; (b) assuming a perfectly aligned anisotropic phase; (c) assuming a limited number of orientations for the blocklike molecules in the mean-field model; (d) assuming com~~~

~~

(42) Luffer, D.; Novotny, M. J . fhys. Chem. 1990, 94, 3161. (43) Roth, M.J . Ckromarogr. 1991, 543, 262. (44) Yan, C.; Martire, D. E. J . f h y s . Chem., following article in this issue

[

- N,q2r - nllabc - Nmq2r(l- 2 / 3 q - 1 / 3 r ) - n,,ac(b - 1 ) M

M

I

(1-2)

Similarly, the probability of finding ( a - 1 ) contiguous empty sites in the x direction is M

- N,q2r

- ntlabc

M - Nmq2r(1 - 2/3q - 1 / 3 r ) - n,,bc(o- 1 )

la-'

(1-3)

and the probability of finding (c - 1) contiguous empty sites in the z direction will be M - Nmq2r- n,,abc

M

- Nmq2r(1 - 2 / 3 q - 1 / 3 r ) - n,,ab(c - 1 )

I

(1-4)

Thus, the total number of ways to place the (rill f 1)th molecule into the lattice is ( M - Nmq2r- nllabc)obc Vn,,+l = X [ M - Nmq2r(l- 2 / 3 q - 1 / 3 r ) - n,,ac(b - l)]&I 1

[ M - Nmq2r(l- 2 / 3 q - 1 / 3 r ) - n&(a 1

[ M - Nmq2r(l- 2 / 3 q

X

- l)]b(a-l)

- 1 / 3 r ) - n,,ab(c - l)l4*('-I)

(1-5)


Molecular Theory of Chromatography In the mean-field approximation, the number of ways, g,, of placing N, indistinguishable molecules in orientation i into the lattice is

w2BT

M

= M / ( M - aN,)!

nro

( - -- ( - - )1

fq

(1-8)

=

w2DT

- N,,abc

(

VI== M - Nmq2r 1 -

(

W~CT M - Nmq2r 1 - fq -

WOIDT

M-N,q2r

- N,q2r

WiAE = M - N,q2r

-fq ir

)

$

N , , ~ c ( -o 1)

(

1 - -q:- -

ir)-

N,,ab(c - 1)

- Nt2ub(c- 1)

W3AE = M

- N,q2r - abc(N,, + N,, + Nt3)

W,BT = M - Nmq2r N , , ~ c ( u- 1) - N , ~ u ~-( 1) c

(

= M - N,q2r 1 - - - f q :r)

( 3', ir)-

ir

Orientation 3: Nt3(a,,bz,cy)

- - N,,bc(u- 1)

W~DT= M - N,q2r 1 - - - -

--

- Ntlbc(o- 1) - N,,ac(b - 1)

Here

=M

q:

- NI2bc(a- 1)

(1-7)

we have the expression for g,,, the total number of ways to pack N,, molecules into the lattice in orientation 1 given that N , mobile phase molecules have already been isotropically placed:

VIAE

; r ) - N,lac(b - 1)

W ~ C=TM - N,q2r 1

Substituting eq 1-5 into 1-6 and then using the approximation

E (M-

- N,q2r

=

W3BT

- N,,ac(b - 1) - N,,bc(a - 1)

N,,ab(c- 1)

where VIAE is the number of ways that A is empty before N,, solute molecules have been placed; WIAEis the number of ways that A is empty after N,, solute molecules have already been placed; W IBT is the total number of ways to place B before N,, solute molecules have been placed; and WIBTis the total number of ways to place B after N,, solute molecules have already been packed; and so on and so forth. By an exactly analogous reasoning process, we obtain g,,, gm3, g,,, gm5, and g,, for orientations 2-6, respectively. Orientation 2: Nt2(u,,rbxr~,)

W~CT =M

- N,q2r N , , u c ( ~- 1)

- N&(u

- 1)

- N,,bc(a - 1) - N,,ac(b - 1) W~DT =M

- N,q2r N&(c

- 1) - N,+(b - 1)

W3DT

M - Nmq2r( 1 -

52 - $) - N,,ab(c - 1) - N,,ab(c - 1 )

Orientation 4: (a,,,b,,c,)


Yan and Martire W5CT = M - Nmq2r

N , ~ u c -( ~1 ) - N , ~ u c -( ~l)N,,ab(c - 1)

(

i)

WSDT= M - Nmq2r 1 - - - - - N,,ab(c - 1 ) N&(c

q:

- 1 ) - N , ~ u c (-~1 ) - N14ac(b- 1 ) - N,&(u - 1)

W D ~ D T= M - Nmq2r N&(c

-

1)

- N , ~ u c -( ~1) - N,,UC(~- 1 )

Orientation 6: (a,,by,cx)

Orientation 5: (u2,bX,cY)

where WOSAE

= M - Nmq2r- abc(NtI+ Nt2 + N13 + N14)

Molecular theory of chromatography for blocklike solutes in anisotropic

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