Anal. Chem. 1989, 6 1 , 811-819
Registry No. PMMA, 9011-14-7; PEG, 25322-68-3;polystyrene, 9003-53-6.
LITERATURE CITED Clark, D. T. Crit. SdM State Meter. Scl. 1978, 8 , 1. Steffens, P.; Niehuls, E.; Frlese, T.; Qrelfendorf, D.; Bennlnghoven, A. J . Vac. Scl. T e c M . low, A3, 1322. Brlgga. D. Surf. Interface Anal. 1986. 9,391. Manern. D. E.; Lin, F. T.; Hercules, D. M. Anal. Chem. 1084, 56, 2762. Nuwaysh, L. D.; Wlkins, C. L. Anal. Chem. 1988, 6 0 , 279. Bktsos, I. V.; Hercules. D. M.; van Leyen, D.; Bennlnghoven, A. Macromolec&s 1987. 20, 407. Becker, C. H.; Glilen, K. T. Anal. Chem. 1084, 56. 1871. Pallix, J. B.; Becker, C. H.; Newman, N. Meter. Res. Soc. Bull. 1987, X I I , No. 8, 52. Wllllams, P.; Sundqvist. B. Fhys. Rev. Len. 1987, 58. 103. Tabst, J. C.; Coner, R. J. Anal. Chem. 1984, 56, 1662. Tembreull, R.; Lubman, D. M. Anal. Chem. 1987, 59, 1003. Berkowltz, J. Photoabsotptlon. Photobnkatbn, and Photoelectron Spectroscopy; Academic Press: New York, 1979. ReM, N. W. Int. J . Mess Spectrom. Ion Fhys. 1971, 6 , 1. Feldmenn, D.; Kutzner, J.; Laukemper, J.; MacRobert. S.; Welge, K. H. A w l . Phys. B 1987, 4 4 , 81. Schhle, U.; PaHlx. J. B.; Becker, C. H. J . Vac. Scl. Techno/. A 1088, 6,936. SchUhle, J.; Pallx, J. B.; Becker, C. H. In Ion Formation from Organic SoMs IFOS I V ; Benninghoven, A,, Ed.; Wiiey: New York, In press. SchUhle, U.; Pallix, J. B.; Becker, C. H. J . Am. Chem. Soc. 1988, 110, 2323. Briggs. D.; Woonon, A. 8. Swf. Interface Anal. 1982, 4 , 109.
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Briggs, D. Surf. Interface Anal. 1082, 4 , 151. Becker, C. H.; Glllen, K. T. J . Vac. Scl. Technol. A 1985, 3 , 1347. Zych. L. J.; Young, J. F. IEEE J . Quantum Electron. 1978, OE-14, 147. Kung, A. H. Opt. Len. 1983, 8 , 24. Brown, A.; Vickerman, J. C. Surf. Interface Anal. 1988, 8 , 75. Turner, D. W.; Baker, C.; Baker, A. D.; Brundle, C. R. Molecular Photoelectron Spectroscopy; Wlley: New York, 1970. Long, S. R.; Meek, J. T.; Reiily, J. P. J . Chem. phvs. 1989, 79, 3206. Bischel, W. K.; Juslnski, L. E.;Spencer, M. N.; Eckstrom, D. J. J . Opt. Soc.Am. 1985, 82, 877. Reiily, J. P.; Kompa, K. L. J . Chem. Phys. 1080, 73, 5488. Rossi, M.; Eckstrom, D. J. Chem. Phys. Len. 1085, 120, 118. Coison. S. D. Nwl. Instrvm. Methods Fhp. Res. Sect. B 1987, 827, 130 and references therein. Hager, J. W.; Wallace, S. C. Anal. Chem. 1988, 60. 5. Robinson, P. J.; Holbrook. K. A. Unhnokular Reactions; Wlley-Interscience: New York, 1972. Hunt. D. F.; Shabanowltz,J.; Yates, J. R., I11 J . Chem. Soc., Chem. Commun. 1087, 548.
RECEIVED for review December 14,1987.Resubmitted October 3,1988.Accepted January 23,1989. The authors thank NSF Division of Materials Research and Perkin-Elmer Corp., Physical Electronics Division, as well as SRI Internal Research and Development funding for financial support. U.S.also thanks Deutsche Forschungsgemeinschaft for a visiting fellowship.
Retention Perturbations Due to Particle-Wall Interactions in Sedimentation Field-Flow Fractionation Marcia E. Hansen’ and J. Calvin Giddings* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 I n thls paper theoretical and experlmental results are obtained relating to the perturbation In retentlon h sedimentation fleld-flow fractionation due to particle-wall electrostatk and van der Waals lnteractlons. These perturbatlons are described In relatlonshlp to standard retention theory, an Ideal theory whose basic assttmptkns are summarlzed. A general equation in Integral form Is given for retentlon ratlo R , and It Is shown how the standard retention theory and sterlcally corrected retentlon theory are thereby obtakred. Expregskns are given for the potential energy of a colloklal partlcle near a wall resulting from electrostatk and van der Waals Interactions; these Interactions alter the concentratlon profile In the fldd-flow fractlonatlon channel In a way that requires numerkal Integratlon to get R . By the use of estimated interactlon parameters, R Is plotted against fleld strength and lonlc strength for several wall materlals Including stainless steel and fluorocarbon resin. Experimental results are reported for five different carrler sdutlons lncludlng dlstllled water. The agreement between theory and experhnental results Is very good conrldering the approximate nature of the parameters used. Both the calculations and the measurements show that the retentive perturbations are smaller for fluorocarbon resin than for stainless steel, Hastdloy C, and polyhlde surfaces. An Intermodlate lonlc strength also appears to be opttmal. A new separatlon technlque based on the c o m b h t h Ot (Idbfiow frac#onatkn and potential barrier chromatography Is suggested, and b porslble advantages are dlscumed.
* Author to whom correspondence should be addressed.
Present address: T h e Prodor and Gamble Co., Ivorydale Technical Center, Cincinnati, OH 45217. 0003-2700/89/0361-0811$01.50/0
INTRODUCTION It was shown over 10 years ago that sedimentationfield-flow fractionation (SdFFF) was capable of separating colloidal particles a t high levels of resolution (1). It was also pointed out a t about that time that the theoretical basis of sedimentation FFF was sufficiently rigorous to make possible the calculation of particle mass, diameter, density, and related properties for uncharacterized colloidal particles based on experimentally measured retention parameters (2-4). The particle size calculated for polystyrene latex beads on the basis of retention measurements was found generally to agree quite well with values obtained from the suppliers and/or electron microscopy (4,5). To continue to improve the accuracy of particle size characterization by FFF, it is important to understand the theoretical basis of the FFF retention equations and to carefully examine any perturbations that may introduce errors into the calculations. The underlying assumptions of the standard FFF retention theory are as follows: (a) The channel volume can be considered as the space between infinite uniform parallel planes. (b) The flow between the planar walls is parabolic. (c) The sample is composed of noninteracting “point” particles. (d) The transverse concentration distribution of particles is close to equilibrium. (e) The field-induced forces are uniform across the channel. (0 No extraneousnonuniform forces are acting transversely between the channel walls. With these five assumptions, the retention ratio R can be described by the ‘standard” mathematical equation, first reported in 1970 (6),as follows: 0 1989 American Chemlcal Society
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ANALYTICAL CHEMISTRY, VOL. 61, NO. 8, APRIL 15, 1989
R = 6h [coth (1/2h)
- 2x1
(1)
where the retention parameter X can generally be expressed as the ratio of two energies (3)
= kT/(JFlw)
(2)
in which kT is the mean thermal energy of the system, w is the channel thickness, and Ir;lis the absolute value of the force exerted on the particle (whose retention ratio is R ) by the applied field. Clearly, the above assumptions represent an ideal model of system behavior that will, at best, only be approximated in the laboratory. In most cases involving normal FFF operation, it is desirable to seek laboratory systems of such construction that the departures from the ideal model are minimal. (For large particles, where steric FFF and related methods are used, eq 1is totally inappropriate.) The fact that this standard theory is so often found to be in good accord with experimental measurement is an indication that the primary features of the model can be translated successfully into laboratory practice. Nonetheless, there are an increasing number of cases in which departures from this model have been examined. In some cases these studies have been done for theoretical completeness, but in other cases they have been introduced to account for observable discrepancies between the standard theory and the corresponding experimental practice. Among the corrections considered, subdivided according to the categories listed above, are the following: (a) departures from infinite parallel plane geometry by virtue of edge effects (7),surface texture (8), and the introduction of annular geometry (9, 10); (b) the departure from parabolic flow caused generally by edge effects (7) and, specifically for thermal FFF, by temperature and viscosity gradients across the channel (11);(c) finite particle size effects due to steric exclusion at the walls (12);(d) nonequilibrium effects due to flow (13), primary relaxation (6), and secondary relaxation (14-16); (e) small deviations from a uniform field resulting from the radial symmetry of centrifugal forces (10); (f) the presence of extraneous forces arising because of adsorption (17,18) and, especially for large particles, hydrodynamic lift forces (19). In this paper, we will consider another kind of departure from model behavior. This departure, falling in category (e) above, is based on the existence of forces of both attractive and repulsive form between the sample particles and the channel walls. It has long been recognized that various electrostatic and van der Waals forces play an important part in colloid chemistry, particularly in relationship to aggregation and adhesion (20,211. These forces clearly exist between the particles in an FFF channel and the walls of the channel. However, they have generally been considered to be of such short range as to not significantly influence FFF retention. In this paper, we show first that special electrolyte conditions must be established in order to minimize the effects of these forces and, second, even with the forces reduced, there is still an observable, although not large, effect on particle retention. We will show how the effect of such forces can be estimated. This study arose out of experimental anomalies associated with the FFF retention of polystyrene latex beads run under different electrolyte conditions. It was found that the retention showed a stronger dependence upon ionic strength than anticipated. Accordmgly, we engaged in a more thorough study to better understand (and correct for) the ionic strength effects observed in these experimental studies. Almost all of the work done in evaluating SdFFF capabilities and theory has utilized monodisperse polymer latex particles. These samples, with their narrow diameter range and known density, are ideal probes for testing FFF system behavior. The standard retention-diameter relationship (a
special case of eq l),as well as the associated band broadening theory (221, has been generally confirmed by using such samples (23). As early as 1974, Giddings et al. found that the combination of standard retention theory and SdFFF yielded the size of five polystyrene beads to an accuracy of a few percent (5). This work was done with the particles suspended in an aqueous 0.1% by weight solution of FL-70, a mixed anionic/nonionic surfactant. Sodium azide was added (0.02% by weight) to the carrier solution as a bacteriocide. More recently we have run such model latex particles in carrier solutions of widely varying (but more controlled) composition. On one end of the spectrum, distilled water has been used. Also, surfactants of a more defined nature than FL-70, including sodium dodecyl sulfate and Triton X-100, have been employed at concentrations similar to those in the FL-70 experiments. In addition, solutions have been used with small concentrations of sodium azide alone. We have found generally that retention is measurably modulated by the type of solution used in the experiments. These results are generally consistent with the more limited experiments reported recently by Hoshino et al. on the effects of detergent additives on SdFFF retention (18). Colloids are by nature charged, q d they are thus subject to electrostatic interactions. The stability of a colloidal system in resistance to aggregation is provided by repulsive interactions, usually originating in the surface charge on the particles. The solution ionic strength is a factor affecting these forces and the resultant stability. Similarly, the electrical properties of the surface of the flow channel influence the particle-wall interaction in FFF. Depending on the respective signs of the surface charges, the adsorption of colloidal particles onto the channel wall will be suppressed or promoted due to charge-charge interaction between the particles and the wall. Usually the two have like charge and the interaction is repulsive. Opposing the electrostatic repulsive forces are the London dispersion forces or van der Waals attractions. These are independent of solution ionic strength and dependent most strongly on the dielectrical properties of the materials and of the carrier solution involved. The classical Derajaguin-Landau-Verwey-Overbeek (DLVO) theory characterizes the energy of interaction between surfaces by combining these repulsive and attractive potentials into a net potential energy profile (21). In solutions of low ionic strength the electrostatic repulsion is predicted to dominate the interaction between surfaces of like charge; the particles in an FFF channel would thus seek a position somewhat removed from the channel wall. In solutions of high ionic strength, the Coulombic screening reduces the repulsive potential and the net potential can become attractive. This eventually leads to the flocculation of colloids or, in the case of particlewall interactions, deposition of the colloid on the wall. The colloidal forces thus described have been shown to be significant in hydrodynamic chromatography (24), reversing the normal diameter-retention trend a t high ionic strength (where the larger particles enjoy a stronger van der Waals attraction to the chromatographic packing and so elute later than the smaller). This observation has led to a new separation technique, potential barrier chromatography (%), based solely on these electrostatic particle-wall interactions. While the magnitude of these forces in chromatography is evident, for FFF techniques the particle-wall interactions are not expected to exert such a dominating effect on retention behavior. In comparison to hydrodynamic chromatography, the surface area available for interaction is much reduced in the open FFF channels. Therefore these forces are expected to merely perturb the retention mechanism in FFF. Also, for
ANALYTICAL CHEMISTRY, VOL. 61, NO. 8, APRIL 15, 1989
effective separations in hydrodynamic chromatography, every sample particle must travel through the column in constant close proximity (a few particle radii) to the packing surface; in normal FFF processes (at modest retention), only a small fraction of the particle population will be within a few radii of the channel wall at any given time. Particle-wall interactions, therefore, should modulate the FFF retention behavior in the following fashion. At low ionic strength the particle zone will be measurably repelled out of the sluggish flow region near the wall and should elute earlier than predicted. The concentration profiles in solutions of high ionic strength will be enriched in the area of net attractive potential close to the wall and will be excessively retained. Several forces are active when a particle approaches a wall. The static forces include the van der Waals and electrostatic repulsive forces mentioned above. Also, Born repulsion and solvent restructuring forces may create some aberrant results at very short distances. Additionally, hydrodynamic forces due to shear near a wall (19) will be important for larger particles at high flow rates. However, the first two forces are generally the dominating attractive and repulsive forces present for colloidal species. Our approach in this paper will be to include these forces in FFF retention theory to account for the interaction of sample particles with the channel wall. Retention behavior found with diverse solution conditions and channel surfaces will be compared to this new model of FFF retention, which includes the electrostatic particle-wall interactions.
THEORY General Theory of Retention. The retention ratio R in field-flow fractionation is given by the ratio of the sample band velocity to the average fluid velocity in the channel; it is, equivalently, the ratio of channel void volume V‘ to retention volume VI. A general expression for R is (6)
R=
(C(X)V(X))
w x w c ( x ) u ( x ) dx
-
(3) x w c ( x ) dx ~ w u ( x dx )
(c(x))(u(x))
0
where c ( x ) and u ( x ) are the sample concentration and velocity profiles, respectively, expressed as a function of the channel thickness dimension x . The triangular brackets represent the cross sectional average of the enclosed quantity. Equation 3 shows that the retention characteristics of the system are specified uniquely by the forms of c ( x ) and u ( x ) . The velocity profile u(x) under laminar flow conditions is given by the parabolic expression U(X)
= ~ ( u ) [ ( x / w-) ( x / w ) ~ I
(4)
where ( u ) is the average flow velocity and w is the thickness of the channel. The other term, the concentration profile 4x1, may be assumed as that existing under steady-state or equilibrium conditions (6), which corresponds to postulate d noted earlier. For the case in which all forces on a particle derive from a potential energy function V(x),the equilibrium concentration profile is given by the Boltzmann expression c ( x ) = co exp[-V(x)/kTj
(5)
where V ( x )is the potential energy of a particle whose center of mass is at coordinate position x and co is the concentration at the arbitrary position (ideally at the wall) where V(x) = 0. For pointlike particles in sedimentation FFF having no interactions except with the applied field, V(x) can be assumed with little error to equal V(x) =
I+
= m’Gx
(6)
813
where m’is the effective particle mass (true mass less buoyant mass) and G is the sedimentation field strength expressed as acceleration. If we assume that this same potential energy function applies to spherical particles of diameter d and density pa, eq 6 assumes the form 0
) = (4~/3)(d/W(p,- p)Gx
(7)
where p is the carrier density. The substitution of eq 7 into eq 5 and the substitution of this and eq 4 into eq 3 followed by integration gives us the standard retention equation for FFF noted in eq 1. For sedimentation FFF, where V(x) is expressed by eq 7, the retention factor X is
For particles of finite diameter d where the particle center can approach no closer than distance d/2 to the wall, the potential function of eq 7 is applicable only between 0.5d and w - 0.5d;it is effectively infinity outside of this range. Consequently, the integrals used to evaluate the terms in eq 3 containing c ( x ) will be evaluated with a lower integration limit of x = 0.5d and an upper limit of x = w - 0.5d. When this is done, the retention ratio becomes
[
R = CY - a2)+ 6X(1 - 2a) coth
(F) A] -
(9) which is a sterically corrected retention equation (12). In this expression, a is a dimensionless ratio scaled to particle radius a: a = a / w . Under ordinary circumstances particles are well retained, and eq 9 can be replaced by the limiting form
R = 6a + 6X
(10)
However, because of complications due to frictional drag and lift forces, it is normally better to replace 6a in eq 10 by 6ya, where y is a dimensionless steric correction factor inserted to account for these hydrodynamic effects. Previous studies in the steric transition region where a X found y ranging from 0.5 to 3 for l-pm particles run a t 60 gravities and at flow rates ranging from 0.5 to 4 mL/min in a typical 0.025-cm-thick channel (26). Particle-Wall Interactions. Colloidal particles near a wall experience two important kinds of interactions. The f i t is an electrostatic interaction between the charged surfaces of the particle and of the wall. The surface charges are generally of like sign, and so the forces are repulsive. These forces can be represented by the following potential energy function expressed in terms of the distance h = x - a, the gap width between the planar wall and the closest surface element of a spherical particle (24) where B is given by
B = 1 6 ~ a ( k T / e )tanh ~ (e$l/4kT) tanh (e$,/lkT)
(12)
in which c is the dielectric constant of the medium, a is the radius of the sphere, e is the electronic charge, and GIand G2 are the surface potentials of the sphere and plane, respectively. The constant K in eq 11 is the reciprocal double-layer thickness. Gouy-Chapman theory is used for the structure of the double layer, which gives the expression (20,21)
in which ciis the bulk solution concentration of ionic species
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ANALYTICAL CHEMISTRY, VOL. 81, NO.
8,APRIL 15, 1989
i and zi is its charge. The quantity I is the ionic strength of the suspending medium. Equation 11is applicable only when the distance 1 / K is small compared to particle radius a and to distance h; that is, when KU >> 1 and K X >> 1. The second important category of particlewall interactions consists of attractive forces of the London-van der Waals type. These forces are generally assumed to equal the sum of the forces between all of the atomic centers in the two respective materials (sphere and wall) appropriately weighted for the distance between centers. Such a summation leads to the potential energy expression (20)
v, = --
-In
4"
-I"
a.
1=10C~
Concentration
20
Standard
10
(y)] (14)
0
which approaches -Aa/6h for h
