Anal. Chem. 1994,66, 1718-1730
Retention Theory of Sedimentation Field-Flow Fractionation at Finite Concentrations Mauricfo Hoyos and Michel Martin' Ecole Supbrieure de Physique et Chimie Industrielles, Laboratoire de Physique et Mkanique des Milieux Hhtbroghes (URA CNRS 857), 10, rue Vauquelin, 75231 Paris Cedex 05, France
In field-flow fractionation (FFF) of concentrated suspensions, particle interactions result in deviations of the equilibrium concentration profile and velocity profile from their respective exponential and parabolic infinite dilution limits. Following the Einstein diffusion formalism, the sedimentationequilibrium concentration profile of a concentrated suspension is derived for hard spheres. The velocity profile is calculated by taking into account the concentration dependence of the viscosity. The retentionfactor is then found to depend on two parameters, the basic FFF parameter, as usual, and the average volume fraction of the suspension. While the effects of deviations of the concentration and velocity profiles on the retention factor act in opposite directions, it is found that the former is dominating and that the retention factor increases with increasing concentrations of the suspension. The predictions of this retention model are found in good agreement with available experimental data on concentrated particulate suspensions in sedimentation and other FFF techniques. Methods for taking van der Waals and electrostatic particleparticle interactions as well as particle-wall interactions into account in the model are given. Experimental FFF data on concentrated polymer solutions are discussed at the light of the model as well as its application to other modes of retention. Field-flow fractionation is a separation method applicable to macromolecules in solutions and particles in suspensions. The separation occurs during the transportation, by means of a carrier liquid, of the sample through a ribbonlike channel under the influence of an external field applied transversally.' Various FFF subtechniques can be distinguished depending on the nature of the external field. As a result of the force exerted by the field, a given sample species is displaced near one of the channel walls and travels along the direction of the channel main axis in the low velocity streamlines of the laminar flow. The average migration velocity of a given sample species and its average retention time in the channel depend on the axial velocity profile (which is assumed to be two-dimensional parabolic flow in isothermal channels with aspect ratios large enough for the infinite parallel plate assumption to be valid) as well as on the mean transverse distribution of the species in the channel. In properly operated systems, i.e. systems in which the transport occurs in quasi-equilibrium conditions, this mean transverse distribution is equal to the equilibrium concentration profile obtained in the absence of flow.2 When the equilibrium profile is controlled by the balance between the convective flux due to the field and the diffusive ( I ) Giddings, J. C. Sep. Sci. 1966, 1, 123-125. (2) Giddings, J. C. J . Chem. Phys. 1968, 49, 81-85.
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Analytical Chemistry, Vol. 66, No. 10, May 15, 1994
flux resulting from the accumulation of the species near one wall, one gets a concentration distribution profile exponentially decreasing from the accumulation wall, provided that the force exerted by the field on the molecules or particles of the species as well as its diffusion coefficient are constant through the channel thickness. In these conditions, one gets a relatively simple relationship between the retention factor, R, and the characteristic constant of the exponential concentration profile3 R = 6Ao[ coth(
&-) 2A0] -
R is the ratio of the mean axial migration velocity of the species to that of the carrier. ,&,the ratio of the characteristic constant of the exponential profile to the channel thickness, w, is related to the force, Fp,exerted by the field on individual particles or macromolecules through
kT
A, = lFJW
where k is the Boltzmann constant and T the absolute temperature. These important relationships allow optimization of the separation process as well as characterization of the analyzed species since & and Fp are related to a physicochemical property of the species (the nature of this property depends on the type of field used). This method has been largely used for characterization of a wide variety of macromolecular or particulate materials by thermal FFF, sedimentation FFF, or flow FFFaU The accuracy of the determination relies on the validity of therelationship on which it is based, and, hence, on thevalidity of the exponential distribution assumption which characterizes the Brownian (or normal) mode of ~ p e r a t i o n .Especially, ~ it assumes that the macromolecules or particles can be considered as pointlike and that there are no interactions between themselves or with the accumulation wall. Retention perturbations due to particle-wall8-I0 and particle-particle9Jo Coulombic and van der Waals interactions have been studied (3) Hovingh, M. E.; Thompson, G.H.; Giddings, J. C. Anal. Chem. 1970, 42, 195-203. (4) Giddings, J. C.; Caldwell, K. D.; Kesner, L. F. In Determination of Molecular Weight ; Cooper, A. R . , Ed.; Wiley-Interscience: New York, 1989; pp 337372. (5) Giddings, J. C.; Myers, M. N.; Moon, M. H.; Barman, B. N. In Particlesize Distribution II: Assessment and Characterization ; Provder, T.,Ed.; ACS Symp. Series No. 472, American Chemical Society: Washington, DC, 1991; pp 198-216. (6) Ratanathanawongs, S. K.;Giddings, J. C. In Particle Size Distribution 11: Assessment and Characterization ; Provder, T., Ed.; ACS Symp. Series No. 472, American Chemical Society: Washington, DC, 1991; pp 229-246. (7) Martin, M.; Williams, P. S . In Theoretical Aduancemenr in Chromatography and Related Separation Techniques ; Dondi, F., Guiochon, G., Eds.; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1992; pp 5 1 3-580. 0003-2700/94/0366-17 18$04.50/0
0 1994 American Chemical Society
in sedimentation FFF. However, even when such interactions are absent, the concentration profile can deviate from the exponential form because of steric (or excluded volume) interactions between particles, especially near the accumulation wall where the concentration can, depending on the field strength, be much larger than the average one in the channel. It is clear that, as the particles are driven toward this wall by the field, their volume fraction cannot exceed that corresponding to a packed bed, which is smaller than that of the most compact (hexagonal) crystal lattice, i.e. 0.74 (an exception is the case of highly deformable particles or coalescingdroplets for which the volume fraction can approach, but not exceed, unity). It is not always realized that, when applying the classical FFF retention equation, one implicitely assumes an exponential distribution in conditions where it may not be so. Indeed, it is easily shown that, in that case, the volume fraction at the accumulation wall, 40,at a given axial position is related to the average volume fraction in the channel cross section, ($), at the same axial position as 40
which, when
A0
=
(4)
(3)
AOI1 - exp(-1/A,)l
is low, Le. when lFplis large, becomes
(4) These equations may lead to unrealistic values for the volume fraction 40.For instance, when analyzing, by sedimentation FFF at 500 rpm (42g) in a 0.01 in. thick channel, a suspension of 0.4-pm silica particles with an average volume fraction of 0.556, one should get from eqs 3 and 4 an impossible 40value larger than 1 at the channel inlet (476% in this case). Deviations to the exponential distribution profile due to excludedvolume interactions will occur well before the volume fraction reaches its ceiling value. This is because the mean surface-to-surface distance, 6, between nearest neighboring particles in a region where the volume fraction is 4 decreases with increasing 4, as 4 0 zz ( $ ) / A 0
(5)
where dpis the diameter of the spherical particles. From eq 5, one sees that the interdistance is large for dilute suspensions, for instance 6/dp = 4.5 for 4 = 0.1%, decreases to 1.8 for a 1% suspension and becomes smaller than 1 when 4 is larger than 3.7%. In these latter situations, it is clear that the displacement of the particles toward or from the accumulation wall, induced by the field action and/or Brownian motion, is influenced by the presence of neighboring particles, which in turn influences the equilibrium concentration profile of the particle cloud. In these conditions, the hypothesis of an exponential distribution is no more correct. Clearly, the deviation to this ideal model increases with increasing concentrations of the sample species. It is the purpose of this paper to provide a retention model of FFF which takes into account finite concentration effects for suspensions of Brownian particles under a uniform field. ~
~
~
(8) Hansen, M. E.; Giddings, J. C. Anal. Chem. 1989, 61, 811-819. (9) Hansen, M. E.; Giddings, J. C.; Beckett, R. J . Colloid Interface Sci. 1989, 132, 3OC-312.
(10) Mori, Y.;Kimura, K.; Tanigaki, M.Anal. Chem. 1990, 62, 2668-2672.
Most often, in these conditions, for colloidal particles, electrostatic and van der Waals interactions influence the particle behavior as excluded volume interactions do. There are, however, some exceptions such as silica spheres stabilized with a thin organophilic layer and dispersed in cyclohexane11 which are considered as good models for hard spheres for which only the excluded volume interactions are present. In the following model, only these excluded volume interactions are taken into account. The reasons for restricting the study to these interactions are multiple: (1) the statistical physics of hard spheres is usually much simpler and much better understood than that of real particles; (2) it can lead to analytical expressions from which further computations relevant for describing the FFF behavior can be performed; (3) excluded volume interactions are always present in real suspensions and, therefore, the present model gives the specific contribution of these steric interactions to the retention behavior; (4) in some situations, the hard sphere interaction potential (infinite potential at center-to-center interparticle distances smaller than or equal to one particle diameter, zero potential at distances larger than dp) provides a satisfying approximation to the real potential; (5) rules exist to convert the real particle diameter into an effective particle diameter such that the behavior of a suspension of real particles is similar to that of a suspension of hard spheres with that effective diameter. Consequently, the resulting model can serve as a primary basis for understanding the behavior of real particles when adequately adapted.
THEORY General Expression of the Concentration Profile in FieldFlow Fractionation. One considers a suspension of identical Brownian particles submitted to the action of an external force field in a vessel. Let w be the distance between the two vessel walls perpendicular to the direction of the applied field (w is later to represent the thickness of the FFF channel). The vessel is assumed to have a large cross-sectional area in the direction perpendicular to the field so that side wall effects can be neglected. Let up be the volume of each individual particle of the suspension and Fp the force exerted by the external field on each particle. One assumes that this force Fpdoes not depend on the position of a given particle in the vessel and, hence, that the field is uniform. When no field is applied, the suspension is homogeneous. Due to the Brownian motion of the particles, a constant osmotic pressure is exerted on the walls of the vessel. When the field is applied, the particles move toward one of the walls perpendicular to the field direction. Let x be the distance from this wall. Their accumulation at this wall is counteracted by their Brownian motion. Accordingly a concentration gradient is established with the concentration decreasing with increasing x . When the concentration equilibrium is reached, the field force is exactly counterbalanced by the thermodynamic force responsible for the Brownian motion. The force exerted by the field on the particles per unit volume of suspension is Fp times the number of particles per unit volume, Le. Fptimes the number concentration, c, of the suspension. Following (11) Vrij, A.; Jansen, J. W.; Dhont, J. K. G.; Pathmamanoharan, C.; KopsWerkhoven, M. M.; Fijnaut, H. M. Faraday Discuss. 1983, 76, 19-36.
AnalyticalChemlstry, Vol. 66,No. 10,May 15, 1994
1719
the Einstein formalism for the derivation of the diffusion coefficient,12 also followed by Batchelor to derive the concentration dependence of the diffusion coefficient,13one notes that, if d n is the net osmotic pressure exerted across a layer of suspension of thickness dx, the thermodynamic force, acting in the direction of decreasing osmotic pressures and, hence, of decreasing concentrations, is, per unit volume, - d n / d x . Accordingly, at equilibrium, one has dn FPc-- = 0 dx Although a more correct analysis of the problem14shows that, in the case of a gravitational or centrifugal field, the two terms of the left-hand side of eq 6 have to be multiplied by (1 - 4), this equation is still correct since the multiplication factor is the same for the two terms. Accordingly, the theory developed here applies specifically to sedimentation FFF. One notes that, with the convention selected for the direction of the x axis, the field force is directed toward the accumulation wall and is therefore negative while the thermodynamic force is positive. Noting that dII _ - d-n-dc _
(7) dx dc d x one gets from eq 6 after rearrangement, with Fp = - lFpl
or (9)
with the dimensionless parameter X equal to
Equation 9 is the basic differential equation of the concentration distribution in the suspension. Its expression is very similar to that found in the FFF l i t e r a t ~ r e . ~Usually, ,~ X is assumed to be constant and the integration of this equation provides the classical exponential distribution. Here, X is rigorously given by eq 10 and is not necessarily constant, but, through a possible dependence on c, may vary with the distance x from the accumulation wall. Its specific dependence on c must be known in order to integrate eq 9. This dependence is discussed for suspensions of hard spheres at finite concentrations in the following. One can note that lFplwis the mechanical work done by the field to transport one particle along a distance w. Introducing the thermal energy, kT, one gets from eq 10
where XO is, since Fphas been assumed to be independent of the position of the particle, recognized as the basic FFF parameter given by eq 2. For a diluted suspension, van? Hoff has shown that the osmotic pressure increases linearly with the number concentration. Therefore, d(lI/kT)/dc is then equal to 1, X becomes equal to XO, and eq 9 is easily integrated (12) Einstein, A. Ann. Phys. 1906, 19, 371-381. (13) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1-29. (14) Martin, M.; Hoyos, M.; Lhuillier, D. Colloid Polym. Sci.. in press.
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Analyticai Chemistry, Voi. 66, No. 10, May 15, 1994
to give the classical exponential concentration distribution prevailing in the Brownian mode of FFF operation c = co exp(
-
e)
and AO is the basic FFF parameter. In the case of a concentrated suspension, the osmotic pressure is no longer proportional to the number concentration and the concentration profile is no longer exponential. Osmotic Pressure of a Concentrated Suspension. The infinite dilution limiting expression of the osmotic pressure (lI = ckT) is, for a suspension of particles in a liquid, equivalent to the equation of state of an ideal gas (PV = nRT or P = CMRT= c,kT, where n is the number of moles in the volume V, CM the molar concentration, and cm the molecular concentration). This is not surprising as particles in a diluted suspension behave independently of each other like individual molecules in an ideal gas and exert on the vessel walls a pressure (in that case, an excess pressure, the osmotic pressure). Deviations from this infinite dilution equation of state arise when intermolecular interactions in the gas phase cannot be neglected. Similarly, in a suspension, deviations from the osmotic pressure equation at infinite dilution are due to particle-particle interactions. Accordingly, theories of the dense gas state or of the liquid state can provide good models for studying the behavior of concentrated suspension^.^^ The development of an equation of state of a dense fluid requires the knowledge of the radial distribution function, g(r). Generally, it is not possible to get an analytical expression of this function. One exception concerns the case of hard sphere fluids. For these fluids, a semi-empirical equation of state was given by Carnahan and Starling16 in terms of the volume fraction, 4, of the hard spheres
Z i s thecompressibility factor. It was shown that this equation provides correctly the first seven virial coefficients and agrees with molecular dynamics r e ~ u l t s . ~ ~ItJ *is considered to satisfactorily describe the physical properties of homodisperse sols of spherical particles.19J0 The CarnahanStarling expression is used in the retention model developed below for concentrated suspensions. However, it is worth pointing out that one shortcoming of this expression is that it diverges for a volume fraction of 1 while it should diverge at some packing volume fraction around 0.6. Therefore, one must be careful to not extrapolate the results obtained by means of this expression tovolume fractions much larger than about 0.4. As previously discussed,14 other expressions of the compressibility factor are available near the packing volume fraction. ~
(1 5) Hunter, R. J. Foundations of ColloidScience; ClarendonPress: Oxford,1989; Vol. 11. (16) Carnahan, N. F.; Starling,K.E.J . Chem. Phys. 1969, 51, 635636. (17) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids, Academic Press: London, 1976, Chapter 4 (18) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions, Cambridge University Press: Cambridge, 1989. (19) Lyklema, J. Fundamentals of Interface and Colloid Science. Vol. I: Fundamentals, Academic Press: London, 1991, Chapter 6.
I I
0.1
0.3
0.2
0.4
@ Figure 1. Variations of X/X, versus the volume fraction,
4.
Expression of the FFF Parameter A. Noting that 4 = cup, one gets the dependence of X with 4 from eqs 11 and 13
have to be taken into account for describing the collective transport behavior, which is a very complex problem. The main interest of the treatment outlined above is that the dependence of the ratio D I U on the volume fraction can be derived, for hard spheres, with a fairly high accuracy in a large 4 range without knowing the individual 4 dependences of D and U. On a physical point of view, this arises from the fact that the osmotic pressure is an equilibrium property while diffusion and sedimentation are kinetic transport properties. Nevertheless, it is interesting to mention the present state of knowledge on hard spheres. The sedimentation velocity has been described to the first order in 4 by BatchelorZ2and, more recently, to the second order in 4 by taking into account threeparticle hydrodynamic interaction^,'^ which gives
U = Uo(l - 6.554 + 19.7r$2)
(17)
This agrees with simulation data up to 4 = 12-1 3%.24J5From eqs 14, 16, and 17, one gets the corresponding expression for the collective (or gradient) diffusion coefficient
(D
D = Do(1 + 1.454 - 2.742) (18) This expression, limited to the first order in 4, was first derived 240@ = X,
1
+ 378$1~+ 56047 + ...)
+ 44 + 442 + 443 + 44
(1 - 414 The variation of X/Xo with 4 is plotted on Figure 1. X is a rather strong function of the volume fraction of the suspension. For instance, when 4 = 10% X is 2.2 times larger than at infinite dilution. Therefore, only for highly diluted suspensions can the assumption of the constancy of A, which leads to an exponential concentration distribution, be correct. An alternative method, frequently used in FFF publications, for deriving h consists of writing the equilibrium condition expressed by eq 6, as resulting from the balance of the convective flux due to the field force, Uc, and of the diffusive flux expressed as -D dcldx, according to the Fick's law, where U and D are the field-induced velocity and the diffusion coefficient2' U c - D - dc =U$-D-=O d4 dx dx Then A, according to eq 9, becomes equal to
by Batchelor.26 Similar first-order results were obtained by means of different a p p r ~ a c h e s . ~ ~To - ~ lthe first-order approximations, eqs 17 and 18 were found to be in reasonable agreement with both s e d i m e n t a t i ~ n a l ~and ~ ~ diffusion~~-~~ a120336J7 experimental data on macromolecular and particulate species behaving as hard spheres. According to these equations, the sedimentation process is slowed down for concentrated suspensions, mostly because of the backflow of the suspending liquid while the diffusion process is speeded up. It thus appears that the concentration effects on both the sedimentation and diffusion processes converge to increase the h value. Accordingly, one expects that the relative concentration profile at finite concentrations will be less steep than it would be for diluted suspensions. It may look surprising that the diffusion coefficient is increasing with increasing concentration, as the opposite is sometimes believed. One should emphasize that the relevant coefficient involved in the derivation of the concentration profile is the collective (or gradient) diffusion coefficient, which, as discussed above, increases with 4 while the selfdiffusion coefficient is decreasing with $. Of course, in diluted suspensions, both coefficients converge to the same DOvalue.21
X/Xo represents the ratio (DIDO) 1 (U/Uo),where DOand UO are the diffusion coefficient and sedimentation velocity in the high dilution limit, such that Xo = Do 1 ((Uo(w ) . In principle, eq 14 could have been derived from eq 16 by taking into account the functional dependences of the diffusion coefficient and of the field-induced velocity with 4. However, these dependences are not known with accuracy in a large 4 domain, even for hard sphere suspensions. This is due to the fact that the velocity perturbation of a sphere moving under the influence of the field is slowly decreasing as the reciprocal of the distance from thesphere, so that many-body interactions
(22) Batchelor, G. K. J. Fluid Mech. 1972, 52, 245-268. (23) Clercx, H. J. H.; Schram, P. P. J. M. J . Chem. Phys. 1992,96,3137-3151. (24) Ladd, A. J. C. J . Chem. Phys. 1990, 93, 3484-3494. (25) Ladd, A. J. C. Phys. Fluids A 1993, 5, 299-310. (26) Batchelor, G. K. J . Fluid Mech. 1976, 74,1-29. (27) Felderhof, B. U. J. Phys. A: Marh. Gen. 1978, 11, 929-937. (28) Ohtsuki, T.; Okano, K. J . Chem. Phys. 1982, 77, 1443-1450. (29) Rallison, J. M.; Hinch, E. J. J. Fluid Mech. 1986, 167, 131-168. (30) Cichocki, B.; Felderhof, B. U.J. Chem. Phys. 1988,89, 1049-1054. (31) Jones, R. B.; Pusey, P. N . Annu. Rev. Phys. Chem. 1991, 42, 137-169. (32) Buscall, R.; G d w i n , 3. W.; Ottewill, R. H.; Tadros, T. F. J . Colloid Inrerface Sci. 1982, 85, 78-86. (33) Bacri, J.-C.; Frhois, C.; Hoyos, M.; Perzynski, R.; Rakotomalala, N.; Salin, D.Europhys. Leu. 1986, 2, 123-128. (34) Paulin, S.E.; Ackcrson, B. J. Phys. Rev. Leu. 1990, 64, 2663-2666. (35) Xuc, J.-Z.; Herbolzhcimer, E.; Rutgcn, M. A,; Russel, W. B.; Chaikin, P. M. Phys. Rev. Len. 1992, 69, 1715-1718. (36) Newman J.; Swinney, H. L.; Bcrkowitz, S. A.; Day, L. A. Biochemistry 1974,
(20) AI-Naafa, M. A.; Selim, M. S.Fluid Phase Equilib. 1993, 88, 227-238. (21) Russel, W. B. Annu. Rev. Fluid Mech. 1981, 13,425-455.
13,4832-4838. (37) Kops-Werkhoven, M. M.; Fijnaut, H. M. J . Chem. Phys. 1981, 74, 16181625.
AnalyticalChemistty, Vol. 66,No. 70, May 15, 1994
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Equilibrium Concentration Profile. The derivation of the equilibrium distribution profile for a hard sphere suspension was described in detail e1~ewhere.l~The main results are summarized here. Equation 9 is the basic differential equation of the concentration distribution in the channel. Noting that dc/c is equal to d$/$, this equation, in combination with eq 14 can be integrated to give the desired volume fraction distribution profile, 4(x/w), as the inverse function, x ( @ ) / w
-* XO
(19)
where $0 is the volume fraction at the accumulation wall. In the case of a diluted suspension for which the largest volume fraction, $0, is itself very small, the two last terms of the left-hand side of eq 19 vanish and this equation reduces to the classical exponential distribution given by eq 12. Generally, 40is not known but it depends on the two physical parameters of the system which are the average volume fraction, (4), in the channel cross section and the fieldinteraction parameter, XO. Noting that, according to eq 9 , 4 d(x/w) = -A d$ and that, according to eqs 1 1 and 13, X/Xo = a(4Z)/d$,one gets
where $1 is the volume fraction at the depletion wall, given by eq 19 for x/w = 1. 40 and can be determined in terms of (4) and XO by numerical resolution of the corresponding equation together with eq 20. For all cases of interest in FFF, when the field strength is large enough for retention to be significant, 40is much larger than $1,so that eq 20 can be simplified by neglecting all terms in (this amounts to say that n($~l),the osmotic pressure at the depletion wall, is zero). It thus appears that the maximum volume fraction in the channel, 90,depends only on the ratio ( ~ ) / X Obut not on the specific values of ( 4 ) and XO.
Flow Velocity Profile. In order to compute the retention factor, one needs to know the velocity profile in the channel cross section occupied by the sample zone. For very diluted suspensions of noninteracting spheres, the viscosity of the suspension can be considered as the same as that of the carrier liquid since the additional dissipation rate of energy produced by the particles is then very low. In that case, the viscosity is uniform through the channel thickness (in isothermal conditions) and the velocity profile is parabolic. For finite concentration suspensions, this is no more the case and the transversal nonuniformity of the concentration of the suspension leads a nonuniform viscosity profile which induces deviations of the velocity profile from its asymptotic parabolic shape. An explicit expression of the dependence of the viscosity, 7, of the suspension on the volume fraction is required to compute the velocity profile. In fact, there are theoretical arguments and experimental evidence that thevolume fraction is not the only parameter which affects the viscosity, but that 1722
Analytical Chemistry, Vol. 66, No. 10, May 15, 1994
other factors such as the shape and the size distribution of the particles, the electrostatic properties of the suspension, and the method used for measuring the viscosity play an important role. Even for the case of the hard sphere situation, considered here, of suspensions of identical, spherical particles without van der Waals or Coulombic interactions, there is not a single expression that gives the effective viscosity, 7/70,where 70 is the viscosity of the carrier liquid, as a function of the volume fraction. The Einstein r e l a t i ~ n s h i p 7/70 , ~ ~ = 1 + 2.54, can be applied satisfyingly only to diluted suspensions with 4 < 2%. The Batchelor extension39 to the second order in 4, obtained by taking into account pairwise interactions between Brownian particles, 7/70 = 1 2.54 + 6.242, is valid for volume fractions up to about 10%. Results from Stokesian dynamics simulation^^^ are consistent with e x p e r i m e n t ~ ~ l , ~ ~ until 4 = 40%. At higher volume fractions, lubrication interactions between particles dominate and cluster formation or agregation phenomena can induce non-Newtonian behaviors, i.e. the viscosity of the suspension becomes a function of the local shear velocity. One can find several semiempirical approaches with phenomenological coefficients which account for various experimental data, such as the expression used by Schure et al. for computing the velocity profile in focusing (or hyperlayer) FFF with a density gradient of the carrier mediume43 Some relationships are based on polynomial developments to higher orders in 4. Experimental results in a moderate 4 range can be fitted with such expressions. One must, however, be careful when extrapolating them to higher 4 values as they would give finite viscosityvalues at unrealistic volume fractions larger than 74%. This is why we selected the viscosity-volume fraction relationship derived by Graham et al. which provides an infinite viscosity for the ceiling 4 = 0.74 value44 and is given as
+
This relationship, derived on a semiempirical basis and reducing itself to the Einstein expression for vanishing volume fractions, does not contain any adjustable empirical parameter and appears to be in good agreement with viscosity data of Thomas41and of Gadala-Maria and A c r i v o ~on~hard ~ sphere suspensions in the entire 4 range up to 60%. Because volume fractions in our calculations do not exceed 40-45%, eq 21 is believed to correctly describe the viscous behavior of hard sphere suspensions. The basic steady-state equation of the unidirectional motion (in the z-direction) of an incompressible Newtonian fluid with a nonconstant viscosity is obtained from the general NavierStokes equation as46
where 7 is a function of x , a ( x / w ) . For the FFF channel (38) Einstein, A. Ann. Phys. 1906, 19, 289-306. (39) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97-117. (40) Brady, J. F.; Bossis, G. J. Fluid Mech. 1985, 155, 105-129. (41) Thomas, D. G. J . Colloid So'. 1965, 20, 267-277. (42) Pitzold, R. Rheol. Acfu 1980, 19. 322-344. (43) Schure, M. R.; Caldwell, K. D.; Giddings, J. C. AM^. Chem. 1986,58.1509-
1516.
geometry, the integration gives47,48
0.8T
with s = x/w and the constant C equal to
Here, dP/dz is the local axial pressure gradient. FFF experiments are generally performed with a constant carrier flow rate, Le. constant average flow velocity, ( u ) , rather than by imposing the pressure drop within the channel. Accordingly, the local pressure gradient in eq 23 can be eliminated by combining eq 26 with the expression of ( u ) obtained by integration of equation 23, as
It is clear that, since 7 depends, through eq 21, on 4 which itself depends, according to eq 19 and derived expressions for 40 and 41, on x/w, XO, and (@I),the relative velocity profile, u(x/w)/(u),dependsofcourseonx/w but alsoonXoand (4).
Retention Factor. The retention factor, R, is defined as the ratio of the average velocity of the species zone to the average velocity of the carrier liquid2
R appears to be a function not only of XO, as usual, but also of the average volume fraction in the zone. In addition, since that average volume fraction is changing with the distance traveled by the zone since the injection point, due to the axial dispersion, R is also changing with that distance. Accordingly, the relative retention time of the particulate sample in the channel must be computed from proper averaging of R along the channel length. Then, since, at a given time, the particle zone disturbs the velocity profile only in a relatively small fraction of the channel length, one may wonder whether the mean flow velocity, ( u ) , computed from eq 25 and that to be used in eq 26 are equal or not. In fact, most modern FFF instruments are operated in constant flow conditions rather than with a constant pressure drop along the channel. Therefore, these two mean flow velocities are identical and integration of eq 23 to obtain eq 25 allows elimination of the local pressure gradient, dP/dz, which of course is larger within the sample zone than within the regions of the channel length occupied by the pure carrier. COMPUTATIONAL PROCEDURE The first step in the computations is to calculate the volume fraction profile given by eq 19. To do this, for each couple (44) Graham, A. L.; Steele, R. D.; Bird, R. B. Ind. Eng. Chem. Fundam. 1984,23, 420-425. (45) Gadala-Maria, F.; Acrivos, A. J. Rheol. 1980, 24, 799-814. (46) Bird, R. B.; Stewart, W. E.; Lightfoot, E. N. Transport Phenomena; John Wilcy & Sons, Inc.: New York, 1960 Chapter 3. (47) Westermann-Clark, G. Sep. Sei. Technol. 1978, 13, 819-822. (48) Shah, A. B.; Reis, J. F. G.; Lightfoot, E. N.; Moore, R. E. Sep. Sci. Technol. 1919, 14,475-497.
@ Figure 2. Volume fraction profiles, plottedas xlwversus 6,for varlous average volume fractlons of the suspenslon: ( 4 ) = 1% , 2 % , 3%, 4 % , 5 % , 6%, 7 % , 8%, 9%, 10% from the lower curve to the upper curve. ho = 0.05.
of values of (4) and XO, 40and 41 are calculated numerically by solving the system of eq 20 with eq 19 derived in terms of 41 for x/w = 1. An alternative method is to calculate $0 directly from eq 20 by taking 41 = 0 in this equation. Indeed, even in the extreme case where XO = 0.1 and (4) = lo%, the error in $0 resulting from this approximation is lower than 0.04%. Then, from the resulting volume fraction profile, 4(x/w), obtained from eq 19, the viscosity profile, ?(x/w), is calculated by means of eq 21. The integrals entering eq 23 are numerically calculated using the composite Simpson rule for discrete values of x/w between 0 and 1 with a step of 0.01, allowing computation of the integrals in eq 25 and to obtain the relative velocity profile, u(x/w)/ ( u ) . Then this relative profile and the volume fraction profile are used to calculate the retention factor by integration of eq 26. The error arising from the discrete nature of the numerical procedure can be estimated by the relative difference in the retention factor obtained when performing the integrations with a step of 0.001 instead of 0.01 for x/w. In a typical case (XO = 0.01 and (4) = 3%), this relative difference is equal to 0.26%. It is much smaller than differences in retention factor obtained when changing the volume fraction from 2 to 3% or 3 to 4% (see below). The 0.01 step value for x/w was then chosen as a satisfying compromise between the accuracy of the calculations and the saving of computer time.
RESULTS AND DISCUSSION Volume Fraction Profile. The influence of the excluded volume interactidns between particles on the resulting sedimentation equilibrium concentration profiles has already been discussed.14 The main characteristics of these profiles can be summarized from Figure 2 in which the volume fraction profiles are plotted as x/w vs 4 for AO = 0.05 and various values of the average volume fraction. The deviation of these profiles from the exponential shape with increasing (6)is clearly apparent from this figure. In the low @ range, the profiles are nearly exponential. However, for a given Xovalue, the relative rate of variation of the volume fraction with the distance from the wall, Id In @/d(x/w)l, decreases as 4 increases. The volume fraction at the accumulation wall, 60, is consequently lower than the value, ( @ ) / [ A 0 (1 - e-l’h)], which would be obtained for an exponential profile. AnalyticalChemistry, Vol. 66, No. 10, May 15, 1994
1723
1.0-
I
I
I
I
I
I
I
I
x lw
XIW
0.8 -
C.6
0.6 -
0.4-
W$O
Figure 3. Relative volume fraction profiles, plotted as x / w versus #.I/&, for varlous average volume fractions of the suspension: (#.I) = 0%, 1%, 2%, 3 % ,4 % , 5 % , 6 % , ?%, 8 % , 9 % , 10% from the lower to the upper curve. ho = 0.05.
The distortion of the volume fraction profile from the ideal exponential profile is still more clearly apparent in the plots of the relative volume fraction profiles shown on Figure 3 for the same values of (4) and XO. For the sake of comparison, the exponential profile, corresponding to an infinitely diluted suspension ((4) = 0), is shown for the lower curve. If there was no effect of excluded volume interactions on the concentration profiles, this exponential curve would beobtained whatever the value of (4). The steadily increasing deviation of the various curves from this limiting law with increasing (4) reflects the influence of these interactions. As a consequence, the mean relative distance of the particles to the accumulation wall also increases significantly with increasing ( 4 ) . Accordingly, one expects that the mean axial velocity-and the retention factor-of the particles will increase with increasing concentrations of the suspension. Figure 3 shows that, when the suspension is very diluted, the relative volume fraction profile is nearly exponential and has a positive curvature (expressed as d24/d(x/w)2 or as d2(x/w)/d42). The profiles of slightly concentrated suspensions are also positively curved. For highly concentrated suspensions, it is clear that there is an upper limit for the volume fraction as one approaches the accumulation wall, since 4 cannot exceed the volume fraction of a dense packing (which varies from about 58% for the random packing found in sedimentation cake to 74% for the compact hexagonal lattice). If the average particle concentration in the vessel is quite large and Fpis also large (low XO), ope can even have the formation of a cake with a nearly constant volume fraction close to the packing value. In that case, the curvature of the relative volume fraction profile is negative near the accumulation wall. However, sufficiently far away from this wall, the suspension becomes diluted enough and the volume fraction decreases again nearly exponentially with increasing distances from the accumulation wall. Then the curvature of thevolume fraction profile is positive. Therefore, in that case, a position will be found in the intermediate region where the curvature is zero. It is easy to show from eq 9 that, in general, this inflection point corresponds to d In X/d In 4 = 1 and to dX/d(x/w) = -1. For the specific case of a hard sphere suspension, the volume fraction, @inn, corresponding to the 1724
Analytical Chemistry, Vol. 66,No. 10, May 15, 1994
v(X/W)/
Figure 4. Relative velocity profiles, plotted as x/ w versus H x / w ) / (v), for various average volume fractions of the suspension: (#.I) = 0% (solid Ilne, parabolic profile), 3% (dotted Ilne), 5 % (dashed line), 8% (dotdashed Ilne); ho = 0.03.
inflection point has been shown to be independent of A,-, and equal to 0.13044.14 Accordingly, an inflection in the volume fraction profile of the suspension will be found in the vessel if both $0 > $inn and 41 < $inn. In practice, since, as noted above, there is a monotonous relationship between $0 and ( ~ ) / X O ,the second of this condition will always be fulfilled while the first one amounts to say that an inflection in the concentration profile will be found if
(4) > 0.2272h0
(27)
Velocity Profile. As the viscosity of a suspension depends on the volume fraction, it is not constant throughout the thickness of the FFF channel but increases with decreasing distances from the accumulation wall. Consequently, the axial velocity profile in the zone of the flowing suspension is no longer parabolic, but becomes distorded. Compared to this isoviscous parabolic profile, the velocity is retarded near the accumulation walland, as the flow rateismaintainedconstant, is increased near the depletion wall. This can be seen on Figure 4 showing the relative velocity profile, u(x/w)/ (v), versusx/wfordifferent (4) at a given bvalue. Thedistortion is seen to increase with increasing (4), the position of the maximum velocity and the value of this maximum relative velocity both increase with (4). Similarly, as seen on Figure 5 , for a given average volume fraction of the suspension, the distortion of the velocity profile increases with decreasing A,-,, due to the concomitant increase of the volume fraction near the accumulation wall. Since, for a given flow rate, particles located at a given small distance from the accumulation wall move at a lower velocity if the suspension is concentrated than if it is diluted, this viscosity effect is expected to increase the particle retention, a trend opposite to that on the concentration profile. Retention of Concentrated Suspensions. The net effect of the excluded volume interactions between particles on the retention, or on the mean relative velocity, of the suspension is, according to eq 26, a composite result of the individual effects of these interactions on the concentration profile and on theviscosity profile. When the retention factor is computed, it appears that, for a given XO value, R increases steadily with
t 0.0
0.4
0.2
0.6
0.8
1.0
1.2
1.4
R
'
'
O
l
1.6
V(X/W)/
Flgure 5. Relative veloclty profiles, plottedas xl w versus qx/ w)/ ( v ) , for various hovalues: ho = 0.01 (dotdashed line), 0.03 (dashed Ilne), 0.06 (dotted line), (solid line, parabollc profile); (4) = 5 % .
0.00
0.04
0.08
0.12
A, Flgure 7. Variations of the retention factor, R,versus the basic FFF parameter Xo for various average volume fractlons of the suspension: (4) = 0 % , 0.5%, 1 % , 2%,3%, 4%, 5%, 6 % , 7 % , 8%, fromthe lower to the upper curve.
(R-R,)/R,
I."
J I
LO
o.o! 0
'
I
2
,
I
,
I
6
4
,
I
8
,
I
OS
10
%
Flgure 6. Varlatlons of the retention factor, R, versus the average volume fractlons of the suspenslon, (4), for varlous Xo values: ho = 0.01, 0.015, 0.02,0.03,0.04,0.05,0.06,0.07, 0.08,0.09,0.1,from the lower to the upper curve.
increasing average volume fractions of the suspension, as seen on Figure 6. Consequently, the effect of the excluded volume interactions on the concentration profile affects the retention factor more strongly than their effect on the viscosity profile does. This probably results from the fact that X increases faster with the volume fraction than the viscosity. Indeed, the first-order power series developments of eqs 14 and 21 give X = XO( 1 + 84) and 71 = qo( 1 2.54). In addition, one observes from Figure 6 that the rate of increase of R with (4) is larger at low (4) than at relatively large (4). Figure 7 shows the variations of the retention factor versus the basic FFF parameter X,for different values of the average volume fraction. For small ho values, these curves increase all the faster with XO, as the average volume fraction is large. The classical curve given by eq 1 and corresponding to the infinite dilution situation is also shown on this figure. On Figure 8, the relative deviation, ( R - Ro)/Ro,of R from its limiting value at infinite dilution, Ro, is plotted versus (4), for various A0 values. ROis, for a given ho, given by eq 1. This relative deviation increases steadily with (4) but this increase is larger for small XO. This arises from the fact that the influence of the excluded volume interactions is more
+
0.0
0
2
4
6
8
10
a>% Flgure 8. Relative deviation, (R - Ro)/Ro,of R from its infinite dilution limit, Ro, versus ( 4 ) for varlous Xo values: Xo = 0.01, 0.015, 0.02, 0.03,0.04,0.05,0.06, 0.07, 0.08,0.09,0.1 from the upper to the lower curve.
significant at the relatively large volume fractions reached for low XOnear the accumulation wall. On Figure 9, the relative error on R is plotted versus A0 for different values of (4). This relative error is larger a t low XO where particles are more highly compressed near the wall than at large ho. The influence of the excluded volume interactions on retention is seen to be quite significant. For instance, for a 1% suspension of a particulate species with a ho value equal to 0.03, the deviation from the high dilution retention factor (Ro= 0.17) reaches 35%. Clearly, these deviations due to excluded volume interactions between particles have to be taken into account for proper characterization of a particulate material from its experimentally measured retention factor. Indeed, if the classical retention equation 1 is used to extract an apparent X parameter, Xapp, from the experimentally determined R value (i.e. solving eq 1 for X with R known, where Xapp now replaces XO in this equation), one obtains a value which may significantly differ AnalyticalChemistry, Vol. 66,No. 10, May 15, 1994
1725
(R-R,)IR
J
I
\
". 0.00
0.08
0.04
0.12
L o
Figure 9. Relative deviation, (R - Ro)/Ro,of R from Its lnfinlte dilution iimlt, Ro, versus Xo for various (4) values: (4) = 0.5 % , 1 %, 2 % , 3%, 4 % , 5 % , 6%, 7 % , 8%, from the lower to the upper curve.
0.0
0.2
0.4
-
0.6
0.8
1.0
R
Flgure 10. Relativedeviation, (Aw ho)/&,of the basic FFF parameter obtainedfrom the experimentallydetermined retentionfactor by means of the classical retention equation (1) from the true parameter, Xo, versus R for various average volume fractionsof the suspension: (4) = 0.5%, 1 % , 2 % , 3%, 4 % , 5 % , 6 % , 7 % , 8 % , from the lower to the upper curve.
from the true XOparameter which characterizes the particulate species. The corresponding relative error on A, (Xapp - ho)/ho, is plotted versus R on Figure 10 for different values of the average volume fraction. The relative error on X is seen to decrease with increasing R, as expected since, for a given ( $), the effect of the interactions between particles is larger at low R where the particle zone is more strongly compressed near the accumulation wall than at large R. In the case of the above example of a 1% suspension of particles characterized by a XO value equal to 0.03, R is equal to 0.228 and it can be seen from Figure 10 that the error made on the determination of A, and hence on the determination of the effective mass of the particles, reaches then 38%. Since the retention factor at finite concentration is larger than at infinite dilution, the particles will then be considered as smaller (or less dense) than they actually are. The relative error on the effective mass may be quite significant and reach much larger values if the suspension is highly concentrated. For instance, the analysis of a 5% suspension of particles with A0 = 0.03 will lead to an error of 57% by default on their effective mass (Le. a 25%error on the particle diameter). The small undulations 1720
Analytical Chemistty, Vol. 66,No. 10, May 15, 1994
observed on some curves have no particular meaning, they are most likely due to numerical artifacts. Retention Time and Peak Shape. In practice, the error on X made when neglecting the effect of the excluded volume interactions for characterizing, for instance, the particle diameter, is not as bad as indicated by the above figures. Indeed, these figures, relating R to (4) and Ao, have to be regarded as providing local values since the average volume fraction in the center of the sample zone, ($), is decreasing when this zone is traveling along the FFF channel due to the dispersion process. Accordingly, an apparent retention factor, Rapp,is experimentally determined from the retention time, t,, of the particulate sample in the channel as Rapp= P/t,, where to is the channel void time, which is the elution time of a species insensitive to the field and moving at the carrier average velocity. The calculation of the retention time and of Rappby proper integration of the local R values is a problem of extreme complexity which requires the calculation of the coefficient x of the nonequilibrium term of the plate height expression2 for such complicated concentration profiles and velocity profiles as those given by eqs 19 and 23 in order to compute the dependence of the average volume fraction at the center of gravity of the zone with the distance along the channel. However, Rappis related by means of Figures 6 or 7 to ho and to some apparent average volume fraction, ( 4)app, the value of which is intermediary between the volume fraction of the injected sample, ( @ ) i n , and the average volume fraction of the effluent at time t,, ( 4)OUt. HOWclose ($)app is to ( @ ) i n or depends on the difference between the two extreme volume fractions and on the rate of variation of the average volume fraction in the center of the zone with the distance along the channel. One can get an estimate of the significance of the concentration influence on retention in practical operation by looking at the fractional distance, z / L , on which the center of the zone of the particulate species has to move along the channel in order for its average volume fraction to be diluted by some factor m. Let p = ( $)in/ ( @)out be the dilution factor at the channel outlet. A very rough estimate of z / L can be obtained by assuming that the zone has a Gaussian shape and that the plate height does not depend on z (which, of course, is not rigorously correct if concentration effects affect the migration velocity of the zone). One then gets
Here z and L should, for correctness, be measured from the position of center of the zone at the end of the injection process rather than from the channel inlet. Consider that (@)in is of the order of a few percent. One can consider that particle interactions affect significantly the rate of migration of the zone when ( 4 ) is larger than about OS'%, i.e. m is smaller than about 10 (this number depends of course on the precise value of (4)inand on XO). The influence of the concentration effect on the retention time will be observed if the fractional distance z / L is not too small, say larger than 5%, i.e. ( p 2 1) smaller than 2000 and p smaller than about 45. The factor (p2 - 1) represents the ratio of the increase of the variance in thechannel length (Le. HLif the plate height, H , isconstant) to the initial length-based variance of the zone at the end of
the injection process, o20. This initial variance is equal to qV,Z/b2w2, where q is a numerical factor which accounts for the influence of the injection device on the shape of the zone, V, is the the injected sample volume, and b and w are the breadth and thickness of the channel. The lowest value of q (q = 1/12) is obtained for an ideal piston injection without any mixing of the sample with the carrier liquid during the injection process. In practice, depending on the design of the injection device, q may be several times larger than this lowest value. With typicalvalues of the parameters in sedimentation FFF (L = 75 cm, b = 2 cm, w = 0.25 mm, H = 2 ~ m , 4V,~ = 20 pL, q = l ) , one gets p = 30. This dilution factor is smaller than the critical value estimated above. One therefore expects that concentration effects are likely to measurably influence the retention time in sedimentation FFF. Still, in the above example, as in practice, the injection volume is not optimized with respect to dilution and detectability as it could be. Such an optimization leads, in liquid chromatography, to a dilution factor as low as 3 or 450which, if this result also applies in FFF, should lead to a significant influence of the excluded volume interactions between particles on their retention time. Besides its influence on the elution time of the center-ofgravity of the particle zone, the dependence of the retention factor on the average volume fraction must affect the shape of the residence time distribution. Since, according to Figure 6, R increases with ( #), the high concentration parts of the zone tend to move faster and elute earlier than the low concentration parts on the edges of the particle zone. The elution profile of the monodisperse particle zone should no longer be symmetrical but should have an asymmetrical shape with a rather sharp ascending front and a tailing descending front. This phenomenon is very similar to that observed in nonlinear chromatography with Langmuir-type isotherms. Comparison with a Previously Published TheoreticalModel. Inagaki and Tanaka have previously developed a model for retention in FFF with solute interaction^.^' Their starting differential equation of the concentration gradient in the channel thickness is identical to eq 9 with a virial expression of X limited to the first order in concentration, i.e. X = Xo(1 BMc), where B is the second coefficient of the virial expansion for the sedimentation equilibrium formula (this is twice the osmotic second virial coefficient) and M the molecular weight of the solute. They found that the relative variation of the retention factor from its high dilution limit, ( R - Ro)/Ro, is proportional to the cross-sectional average concentration. This corresponds to the behavior observed on Figure 8 at low average volume fractions (or concentrations). Developing the retention equation for a parabolic velocity profile (this corresponds to the hypothesis of Inagaki and Tanaka) and a general X expression of the type A = Xo(1 k’4), one gets to the first order in (4)
+
+
with, in that case, Ro = 6 ho( 1 - 2x0). One notes in passing that, since our starting X expression is the same as that of Inagaki and Tanaka if k‘4 is replaced by BMc, one should obtain their result by replacing k ’ ( 4 ) by B M ( c ) in eq 6 of their paper. However, the comparison indicates that the factor
1 / 4 is missing in their equation. More interestingly, if, according to the virial development of X for hard spheres given in eq 14, one puts k’ = 8 in eq 29, this equation should give the slope of R vs (4) in Figure 6 at vanishing average volume fractions. Close inspection of Figure 6 (or of Figure 7) shows that the slope is actually slightly smaller than that given by eq 29. This is what one expects. Indeed, the derivation of this equation was made on the assumption of a parabolic flow profile while, as discussed above, the concentration effect on the viscosity profile in the zone tends to reduce the retention factor. The difference between the slope given by eq 29 and the actual slope reflects the contribution of the concentrationinduced distortion of the flow profile to the retention. Comparison with Published Experimental Data and Effect of van der Waals and ElectrostaticInteractions. The influence of the concentration effect on the profile of the elution peak has been studied by means of the Edgeworth-Crambr leastsquares fitting procedure by Reschiglian et al.52 It was shown that the deviation from the symmetrical shape, measured for instance by the peak skew, increases with increasing sample amounts. The peaks at high concentration have a relatively sharp ascending front and a tailing descending front, as expected from the analysis of the dependence of the retention with the concentration. Overloading effects have been experimentally studied in sedimentation FFF and discussed by Hansen et ale9who found that the retention factor increases as the sample concentration increases, in agreement with our theoretical expectation. Furthermore, their curves of R versus injected amount (i.e. volume fraction in injected sample since the injection volume is kept constant) in Figure 6 of ref 9 have a shape very similar to that of our Figure 6 , especially with the curve corresponding to XO = 0.01, a value very close to the experimental value for 0.46 pm polystyrene latex particles analyzed at 64g. In addition, an increase in the field strength was found to lead to a decrease in the apparent particle diameter, which implies that the retention factor does not decrease as fast with decreasing XOas it should according to the classical retention theory. This again is in agreement with the expected effect of the excluded volume interactions. The data of Hansen et al. show that the ionic strength of the suspension medium has a significant influence on the deviation of R from Ro. For a given injected amount of particulate sample, the deviation increases with decreasing ionic strengths. This result is interpreted as arising from the electrostatic interactions (repulsion) between particles stabilized by electrical charges. As the ionic strength decreases, the thickness of the electrical double layer associated with each particle increases, thus enhancing the range of the repulsive interactions and the minimum distance of approach of the particles. The effect of the ionic strength on the retention is therefore similar to that of the average volume fraction. One can consider that due to the electrostatic interactions, particles have an effective hard sphere diameter, deff,larger than the geometrical diameter, dp This repulsion effect can be moderated by the attractive van der Waals interactions (49) Karaiskakis. G.; Myers, M. N.; Caldwell, K. D.; Giddings, J. C. Anal. Chem. 1981, 53, 1314-1317. (50) Karger, B. L.; Martin, M.; Guiochon, 0.AMI. Chem. 1974,46,1640-1647. (51) Inagaki, H.; Tanaka, T. Anal. Chem. 1980,52, 203-205. (52) Reschiglian, P.; Blo, G.; Dondi, F. Anal. Chem. 1991, 63, 12C-130.
AnalyticalChemistry, Vol. 66,No. 10, May 15, 1994
1727
between particle^.^^ It can be shown18 that the hard sphere model can be applied to real particles provided that dcffis defined as
where @ is the interparticle interaction potential (taking into account both van der Waals and electrostatic interactions), r the center-to-center interparticle distance, and u that distance for which @(a) = 0. Then, the theoretical retention model presented here can be applied to real suspensions by using the effective average volume fraction, ( 4)cff, based on the effective particle diameter instead of (4), with (4Ieff/(4)= (deff/ dp)3. Because of the cubic dependence of ( 4 ) e n on deff, a slight deviation of dcfffrom dp may result in a significant increase of the effective average volume fraction. As (dcffdp) can be considered as equal to some small multiple of the double layer thickness, 1 / ~ which , itself is proportional to 1/Ill2,wherelis the ionic strength of theelectrolytic solution? one expects that, for relatively small values of ~ - l / d the ~, relative deviation of ( 4)efffrom (4) will also be proportional to l/I1P, A theoretical model of retention accounting for Londonvan der Waals (attractive) and electrostatic (repulsive) particle-particle interactions has been developed by Mori et a1.I0 This model is simplified as it does not take into account the distribution of the interparticle distances at a given volume fraction and, apparently, neglects the viscosity effect. Nevertheless, it is instructive as it demonstrates that the injected sample amount significantly influences the retention factor through the particle-particle interactions. As in the case of excluded volume interactions, the retention factor is then found to increase with increasing concentrations. Influence of the Finite Size of the Particles on Retention. In the model developed above, excluded volume interactions obviously arise from the finite size of the particles. The influence of a specific particle diameter is reflected through the basic parameter XO, but not explicitly in the expression of thedeviation of R from Ro. That is, for two samples of different particles, the local retention factor should be the same if XO is kept constant (i.e. if the product dp3 a2,where s2 is the angular velocity of the sedimentation FFF channel, is constant), and if the averagevolume fraction of the two samples is the same. Then, obviously, the number concentration of the sample of largest particles is lower than that of the smallest particles. However, the model will not correctly describe experimental results if particle-wall interactions, which depend on the size of the particles but not on their concentration, are significant. A retention model of sedimentation FFF at finite concentrations with particle-wall interactions can be built by means of the same approach as that used by Giddings for modeling retention in steric FFF.54 That is, the above model is applied to a fictitious channel corresponding to the core accessible to the particle’s centers. This fictituous channel is made of the actual channel minus an exclusion layer near both the accumulation and depletion walls. This channel moves, with (53) Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (54) Giddings, J. C. Sep. Sci. Techno/. 1978, 13, 241-254.
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Analytical Chemistry, Vol. 66,No. 10, May 15, 1994
respect to the actual laboratory channel, with a velocity equal to the carrier velocity at a distance from the walls equal to the exclusion layer thickness. All migration quantities (flow velocity, particle average velocity, retention factor) associated with the fictituous channel are then converted to quantities related to the actual channel by an appropriate change in the frameof reference. For a hard sphere suspension, the thickness of the exclusion layer should be equal to the particle radius. In the presence of London-van der Waals and electrostatic interactions between the particles and the walls, the effective distance can be calculated from the form of the particle-wall interaction potential, at least when the repulsive interactions dominate the attractive interactions. The influence of these particle-wall interactions on retention has been previously discussed.8-10 Application of the Model to Polymeric Solutions. Experimental investigations of the influence of the sample amount on the retention of polymeric solutions have generally led to a decrease of the retention factor with increasing concentrations, both in thermal FFF55-57 and in flow FFF.56 This trend is opposite to that predicted and observed in sedimentation FFF. However, in contradiction with these experimental results, one should note that a decrease in the calculated weightaverage molecular weight determined from thermal FFF measurements with increasing sample amount has been observed in temperature programming condition^,^^ which indicates an earlier elution of the more concentrated samples. It has been already noted that the above retention model of concentrated hard sphere suspensions strictly applies to sedimentation FFF as the starting force balance equation (eq 6 ) may not have the same expression in other forms of FFF. This means that the dependence of X on 4 may not be exactly given by eq 14 for fields other than centrifugal. However, this dependence, even limited to the first order in 4, is so strong (see Figure 1) that one does not expect that an exact derivation of the X expression for other field types should revert the sign of the second virial coefficient of X or, even, change significantly its value. Although the structure of a polymeric chain is different from that of a hard sphere, one can attribute to the chain a hydrodynamic diameter, dh, such that its hydrodynamic behavior is similar to the behavior of a rigid sphere of diameter dh. Also, although the viscosity of a macromolecular solution increases rapidly with increasing concentrations of the polymer, especially for high molecular weight polymers, the viscosity behavior of the solution is not expected to be much different from that of a particle suspension at the same volume fraction. Indeed, the successful universal calibration concept in size-exclusion~hromatography5~ is based on the application of the Einstein equation for the viscosity of hard spheres to relate the intrinsic viscosity of the polymer to its hydrodynamic volume. Accordingly, one expects that the retention model for concentrated hard sphere suspension should apply to polymeric solutions, at least as far as one is concerned by the sign of the deviation from the infinite dilution retention. ~~
~~~~~~
~
( 5 5 ) Myers, M. N.; Caldwell, K. D.; Giddings, J. C. Sep. Sci. 1974, 9, 47-70. (56) Caldwell, K. D.;Brimhall, S.L.; Gao, Y.;Giddings, J. C. J . Appl. Polym. Sci. 1988, 36, 703-719. (57) Jan&, J.; Martin, M. Chromarographia 1992, 34, 125-131. ( 5 8 ) Kirkland, J. J.; Rementer, S.W. Anal. Chem. 1992, 64, 904-913.
The fact that a generally negative deviation of the retention factor is observed at high concentrations both in thermal FFF and flow FFF indicates that this behavior is likely to be specific of the polymeric nature of the solution rather than of the type of field. A distinct feature of polymers, when compared to rigid particles, is the possibility of entanglement of the chains of two or more macromolecules when they come in close contact to each other. This arises for some critical transition concentration, c*, between the dilute and so-called “semidilute” regimes. One can easily imagine that the transport properties (viscosity, diffusivity) of an assembly of entangled macromolecules are considerably different from those of the individual molecules and that this assembly will behave as a molecule of much higher molecular weight and thus will be more retained than the freely behaving macromolecules. Caldwell et al. computed the critical concentration and found that, depending on the field intensity and on the molecular weight, the polymer concentration at the accumulation wall near the channel inlet can be much larger than c* in conditions encountered in FFF experiment^.^^ Another indication that the behavior noted above is well associated to the polymeric nature of the sample rather than to the type of field used can be given by the comparison of sample amount effects on retention of polymers56and of rigid species with the same field type. The superimposed fractograms of a polystyrene latex sample analyzed by thermal FFF at various sample loads reveal that the retention time slightly, but perceptibly, decreases with increasing sample amount.60 Besides, two recent studies performed with latex particles in flow FFF with a hollow fiber arrangemenF and with proteins and viruses in an asymmetrical flow FFF channeP2 both indicate that, with relatively large sample concentrations, the retention factor is generally larger than expected on the basis of the infinite dilution retention model, when the ionic strength is relatively low, and that this deviation increases when the cross flow increases or when the ionic strength decreases. These behaviors are opposite to those of the polymeric samples but are well in agreement with our model for hard spheres. This illustrates that proteins and viruses, which have a more or less globular and compact configuration, behave more as rigid particles than as random coil molecules. However, it is interesting to note that, when the ionic strength is increased, an increase in the sample load and/or the cross flow can lead to a retention factor smaller than expected. This reversal in retention behavior at high ionic strengths can be interpreted as arising from the domination of attractive interactions over screened repulsive interactions between the particles. There should therefore be a behavioral similarity between associated colloidal particles and entangled polymers. Clearly, further studies are needed to better understand the influence of concentration effects on the retention of polymeric solutions. If the value of the sample concentration required to observe a significant entanglement effect is relatively large, it may happen that the retention factor goes to a maximum for some injection concentration, with R first increasing at relatively low concentration, as for hard sphere (59) Grubisic, Z.; Rempp, P.; Benok H. J. Polym. Sci. E 1967, 5, 753-759. (60) Liu, G.; Giddings, J. C. Chromatographia 1992, 34, 483-492. (61) Carlshaf, A.; Jbnsson, J. A. Sep. Sci. Techno/. 1993, 28, 1191-1201. (62) Litztn. A,; Wahlund, K.-G. J . Chromatogr. 1991, 548, 393-406.
suspensions, then decreasing with increasing concentrations, due to the onset of the semidiluted behavior. Application of the Model to Other FFF Modes of Retention. As discussed above, the retention factor obtained from eq 26 as a function of (6)and corresponds to a local value since the averagevolume fraction depends on the mean axial position of the zone in the Brownian mode of retention, so that an appropriate averaging is needed to obtain the apparent retention. There are, however, other operating modes of retention in FFF where such an averaging process is not required because the average volume concentration is constant along the channel. This happens when a more or less concentrated suspension is continuously introduced in the channel so that a steady-state concentration profile is eventually achieved along the channel. This situation arises in FFF with secondary chemical equilibria (SCE-FFF), an operating mode in which the carrier liquid contains a dispersed phase such as an emulsion. Sample components are then allowed to distribute themselves between the continuous mobile phase and the dispersed phase which, because it is more or less compressed near one channel wall when an external field is applied, moves more slowly than the continuous phase.63 The retention time of sample components then depends on their distribution coefficient. This method allows the separation of low molecular weight species.64 The initial approximate theory of this retention mode63has been extended to any value of the distribution coefficient and of the retention factor of the dispersed mobile phase component.65 However, because this latter model assumes an exponential concentration distribution of the dispersed phase as well as a parabolic flow profile, it is limited to relatively low volume fractions of the dispersed phase. The present retention model of concentrated suspensions can be applied for predicting the retention in SCE-FFF with relatively large volume fractions of the dispersed phase. Work in this direction is in progress. In another FFF operating mode, called focusing (or hyperlayer) FFF, a suspension of fine particles having a density different from that of the suspending liquid is introduced continuously in the channel to form a transversal density gradient when these particles relax to their nonuniform equilibrium concentration profile under the action of the centrifugal force.66 Then, sample species injected separately move transversally toward some focusing position where their density matches the density of the carrier. They are then transported axially at this focusing distance from the wall and separation occurs, in principle, according to the particle density rather than to the particle size. The theory of this focusing mode has been developed by taking into account the viscosity effect using a semiempirical expression of the viscosity dependence of the volume fraction and by assuming that the concentration distribution of the fine carrier particles is e ~ p o n e n t i a l .It~ ~is clear that this theory is only approximate because it does not take into account the deviation of the carrier particle concentration profile from its infinite dilution exponential shape so that the density gradient is less steep and, consequently, the distortion of the velocity profile from (63) Berthod, A.; Armstrong, D. W. Anal. Chem. 1987, 59, 241C-2413. (64) Berthod, A.; Armstrong, D. W.; Myers, M. N.; Giddings, J. C. Anal. Chem. 1988,60, 2138-2141. (65) Martin, M. Chromatogrophia 1992, 34, 325-330. (66) Giddings, J. C. Am. Lab. 1992, November, 2OD-2OM.
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the parabolic shape less important than assumed. This correction can be made by means of the model developed above. Nevertheless it is unlikely that it will significantly modify the conclusions about the potential of this FFF mode drawn by Schure et Finally, one should emphasize that the retention model of concentrated suspensions presented here cannot be applied in the lift mode of operation, Le. when the concentration profiles are significantly affected by hydrodynamic lift forces of inertial origin exerted on the particles. In practice this limits the application of the model to colloidal particles with a diameter smaller than about 1 pm and/or to relatively low flow rates of the carrier liquid. GLOSSARY
channel breadth second virial coefficent for sedimentation number concentration of the particles number concentration of the particles a t the accumulation wall critical transition concentration between dilute and semidilute regimes constant defined by equation 24 effective particle diameter hydrodynamic diameter particle diameter diffusion coefficient of the particles diffusion coefficient of the particles a t infinite dilution force exerted by the field on one individual particle gravitational acceleration radial distribution function plate height ionic strength Boltzmann constant virial coefficient in the expression of h as a function of 4 limited to the first order in 4 dilution factor molecular weight channel length pressure numerical factor for the shape of the zone at the end of the injection process center-to-center interparticle distance retention factor retention factor a t infinite dilution apparent retention factor defined as ratio of channel void time to retention time reduced transversal coordinate = ratio of distance from accumulation wall to channel thickness channel void time retention time of the particles absolute temperature
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field-induced velocity of the particles field-induced velocity of the particles at infinite dilution velocity of the suspension averagevelocity ofthe suspension in thechannel thickness particle volume injected sample volume channel thickness distance from accumulation wall axial channel coordinate compressibility factor defined by eq 13 mean distance (from surface to surface) of nearest neighboring particles viscosity of the suspension viscosity of the suspension at infinite dilution reciprocal double layer thickness dimensionless parameter defined by eq 10 ratio of characteristic constant of the exponential concentration profile to channel thickness = infinite dilution limit of h apparent h value deduced from retention R by means of the classical retention eq 1 osmotic pressure dilution factor a t the channel outlet center-to-center interparticle distance for which the interaction potential is zero initial length-based variance of the zone at the end of the injection process volume fraction of the particles volume fraction of the particles at the accumulation wall volume fraction of the particles at the depletion wall volume fraction a t the inflection point in the particle concentration profile average volume fraction of the particles in the channel thickness apparent average volume fraction effective average volume fraction of the particles volume fraction of the particles in the injected sample average volume fraction of the particles in the channel thickness a t the channel outlet interparticle interaction potential dimensionless coefficient in the nonequilibrium term of plate height angular velocity of the sedimentation FFF channel
ACKNOWLEDGMENT We are very grateful to F. Dondi of the University of Ferrara for pointing our attention to the work of Einstein on the diffusion coefficient of diluted suspensions. Recelved for review October 15, 1993. Accepted February 24, 1994.'
* Abstract published in Advance ACS Abstracts, April
1, 1994.
