Anal. Chem. 1888, 60, 1109-1119
1109
Digital Simulation of Sedimentation Field-Flow Fractionation Mark R. Schure Laboratory Data Products Group, Digital Equipment Corporation, 1 Iron Way, MRO2-41E33, Marlborough, Massachusetts 01752
Dlgltal slmulatlon of the sedlmentatlon fleld-How fractlonation process Is accomplkhed by uslng a slmple method based on a random-walk Monte Carlo slmulatlon. Thls slmulatlon Is used to verlfy the present analyHcal theory at b w and medlun retention values. The model used here Includes the features of a rectangular c h a d and wall effects. The resutts -ate a very close correspondence between the analytical theory and the dmulatlon at medlum retentlon levels where most experbnents are conducted. The results of slmulation, however, suggest that large dlscrepancles exlst between theory and slmulatlon In the reglon of low retentlon due to translent behavlor whlch produces non-Gausslan zone shapes.
Field-flow fractionation (FFF) (1,2) is a term used to describe a class of techniques used for the separation and characterization of polymers, macromolecules,and particulate media in solution. Two of the subtechniques of FFF, thermal FFF and sedimentation FFF, are now available commercially and are being used for a wide variety of tasks in the laboratory. One of the outstanding features of these techniques is that physical characterization of a sample may be performed at the same time that separation is taking place. Toward this end, theoretical relationships are used to convert the resulting elution profile into a molecular weight or particle size distribution (3-7). Of all the different subtechniques of FFF, sedimentation FFF has been recognized to be the simplest because of the lack of secondary effeds which appear in the other techniques. These secondary effects include nonuniform viscosity in the flow profile of thermal FFF, an electrokinetic effect in the velocity profile of electrical FFF, and the effect of nonuniform cross-flow in flow FFF. Because it appears to be the simplest case of FFF, we will concentrate our efforts on the simulation of sedimentation FFF. The accuracy of the molecular weight or particle size distribution is directly related to the accuracy of the theory used for analysis of the elution profile. This situation is critical when deconvolution techniques (8,9) are used to remove the zone broadening that is inherent in the FFF experiment because the broadening function peak shape has a large effect on the resulting deconvolved fractogram. The theory of FFF has been developed in a number of different ways. Giddings and co-workers (10-14) have developed the theory of FFF using the nonequilibrium approach whereby the eluting zone is considered to be near equilibrium with respect to the spatial distribution of solute. Both retention (first statistical moment) and zone broadening (second statistical moment) are described. The assumptions carried out in this theoretical development suggest that the particle zone will sample all relevant flow velocities an infinite number of times, forming a zone near equilibrium with the surrounding flow streamlines. A number of other assumptions are made including viewing the particle as a point mass and with no interactions between walls, fluid, and other particles. Additional work has been carried out to examine the perturbations due to the steric behavior of large particles (15,16), the 0003-2700/88/0360-1109$01.50/0
effect of the infinite parallel plate assumption (17),channel end piece perturbations (18, 19),wall effects (20))and secondary relaxation effects in field programming (21). Verification of the theory has been performed for a number of cases; however, the properties of the sample itself (size, molecular weight, polydispersity, density) are always in question in these studies (4-6) because standards are often not well characterized, thus leading to an uncertainty which must be resolved by analysis from a host of other techniques. In addition, the theory has not been verified over the entire elution range. Additional contributions to FFF theory have been made by Gajdos and Brenner (22), who considered the effects of f i i t e particle size, wall potential, and nonspherical particles. Dynamics of the sample zone during elution have been considered by Subramanian et al. (23,24) in one of the few studies that considered the evolution of a zone as a function of time rather than as a steady-state process. This work did not consider a relaxation step to occur before the flow was applied and flow was considered to occur between two infinite parallel plates. Numerical methods were used to solve the partial differential equations of convection and diffusion. In the theoretical study of sedimentation FFF conducted by Berg and Purcell(25), a stochastic approach was used to define the zone-broadening parameters. The particle trajectory was considered to occur along discrete levels in the channel width as a convenience in formulating the equations. The theory was used to compare experimental results with satisfactory agreement. Again, flow was considered to occur between infinite parallel plates and only the long time, steady-state solution was considered. In an attempt to view the FFF process in three dimensions, Takahashi and Gill (26) used a moments-based generalized dispersion theory to solve the convection-diffusion equations which describe the FFF phenomenon. The steady-state solutions of Giddings et d. were compared with their results and it was determined that their results differed in the zonebroadening calculations by a factor of 6 for zones of medium retention. This discrepancy has not been seen in sedimentation FFF experiments. Because diffusion in the breadth direction will undoubtedly lead to incomplete sampling of the flow velocities in the breadth direction, the use of the steady-state assumption in the breadth direction largely overestimates zone broadening. Using a similar approach to Takahashi and Gill, Kim and Chung (27) have presented a theory that does include dispersion in all three dimensions and is totally free from the steady-state assumption. This theory is exceedingly complex due to the use of triple infinite series but does contain the essential features needed to evaluate the effect of the small walls. A review of the different approaches used in FFF theory is given by Lightfoot and co-workers (28) for work carried out prior to 1981. In this paper we discuss the evolution of the zone as well as the consequences of the rectangular channel, where flow retardation occurs along the edges of the channel. Toward this end, simulations are performed that use some of the original concepts of Berg and Purcell; however, the full rectangular channel is considered and no steady-state assump0 1988 American Chemical Soclety
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
The mean layer thickness, 1, can be expressed in the dimensionless form X which for sedimentation FFF has been shown to have the simple form (12)
where k is Boltzmann’s constant, T i s the absolute temperature, d is the particle diameter, G is the acceleration due to spinning the channel, and Ap is the difference between carrier fluid and particle density. The retention ratio R can be explicitly expressed solely as a function of X such that
R = 6X coth ( 1 / 2 X ) - 12X2 FLOV,
Flgure 1. Internal dimensions of an FFF channel.
tions are made. Wall effects are an optional feature of the simulation allowing for the particle diffusion coefficient to be a function of the position from the wall. This will allow the FFF process to be viewed as a dynamically evolving separation process with very few assumptions and limitations.
ANALYTICAL THEORY Since the results of simulation will be compared to the standard FFF theory of Giddings et al. ( I I , 1 2 ) ,the analytical theory will now be presented in brief. The channel used in most all FFF studies is shown in Figure 1 with the appropriate symbology describing the channel width w,breadth b, and length L, along with the corresponding vector directions x, y, and z. Missing from this diagram are the end pieces which bring fluid into and out of the channel. The effect of the endpieces will not be considered in this study. The distribution of solute in the x direction is the result of a simple interplay between the field-induced particle velocity, U , and the restoring force of diffusion. It can be shown that at equilibrium the distribution of solute with respect to the x position, c(x), is given by a simple equation (10) c ( x ) = coe-IUx/D = coe-x/l (1) where c,, is the concentration of solute at the x = 0 position, 1 is the mean layer thickness, which for well retained zones is where half of the zone is located above and half the zone is located below, and D is the diffusion coefficient for a particle in infinite solution. In this paper we will only consider particles that are more dense than the surrounding fluid so that particles will migrate under the influence of the field toward the lower (outer) wall. This is to be contrasted with the situation where particles are less dense than the fluid and hence migrate toward the higher (inner) wall. The retention ratio R, which is simply the ratio of the unretained void peak time, to,to the peak retention time, tr, is given as
The above equation may be modified to include the steric effect, which occurs for large particles because the particle center of mass is sufficiently far from walls as to exclude convection in the slow flow wall regions of the channel. In this case the retention ratio R may be written as (15)
R = ~ Y ( L-Ya2) + 6 X ( 1 -
u(x) = 6 ( u ) [ : -
51
where ( u ) is the average fluid velocity.
(3)
~ L Y )
1
1 - 2a coth -- 2x 1 -2x2LY (6)
where y is an empirical factor for lift forces and inertial effects (usually set to unity) and LY is the ratio r / w where r is the particle radius. Zone broadening is also described by theory where we use the plate height, H, as a measure of the zone dispersion. The plate height for sedimentation FFF zones is
(7) where a is the standard deviation of a zone in time units and x is the dimensionless nonequilibrium parameter (11).
COMPUTATIONAL PROCEDURE The random-walk Monte Carlo procedure used here is quite easy to understand and will be presented here in detail. The procedure consists largely of isolating the particle diffusion step and particle convection step. The particle is diffused in the x direction with a bias proportional to the strength of the applied field and is then allowed to diffuse randomly in the y and z directions for a time t,. The particle is then convected with the velocity of flow that is contained in the channel where the particle’s center of mass lies. The diffusion step is then applied again. This convection-diffusion cycle is performed until the particle passes a certain position or time in the channel. A t these points the particle positions are written to disk. A flow chart of this procedure is shown in Figure 2. Discrete positions are identified along the channel width where the particle’s center of mass is to be located. The spacing s between positions is picked so that the relationships s OS pm) particles which are typically separated and characterized bv sedimentation FFF. the eftect is verv small. This is also seen from Figure 3, 4,'and 6 where there is very little discernible tailing in the retained zone simulation results. As shown in Figure 14, the overall amount of solute in the edge region for the zone of medium retention is quite small; this suggests that the use of a theory which completely neglects the small walls and treats the channel as if it were two infinite parallel plates can he justified for sedimentation FFF. Caution, must he used, however, in justifying this assumption for thermal, electrical,and flow FFF where much smaller particles and macromolecules are likely to be separated and characterized. Although this edge effect is significant for the case highlighted in Figure 5, most all solutes used in sedimentation FFF have diffusion coefficients in excess of 400 times larger than that used to produce Figure 5. Diffusion in channel breadth will for all practical flow rates encountered in sedimentation FFF (or any other form of FFF) never achieve a steady-state distribution of solute in the breadth direction for channels which have large aspect ratios, such as those used in this study and in routine experimental fractionators. Theoretical treatments which attempt to induce the steadystate assumption in breadth diffusion, as was mentioned earlier in this paper, have been shown to be erroneous (26,
bo
(2Dt)'I2
+w
(19)
For the case where an unretained low molecular weight tracer substance is eluted (D= 8 X 10" cm2/s), i t is shown in Figure 13 that the zone forms a tail against the edge wall. The tail breadth, as calculated from the equation above is 0.0565 cm (bo/w = 4.45), which is in reasonable agreement with
lo Oo0 particles are used here.
23. The results of simulation suzgest that the analvtical theorv presently used in sedimentationFFF io not adequate over the entire elution ranne. Most serious is the error found in zones of low retention where the R value is approximately between 0.25 and 1. This suggests that quantitation in this area of retention will contain large errors in the resulting particle size distribution, especially when measuring the dispersity of the particle size distribution, On the other hand, the results of simulation suggest that for zones with R values less than 0.25, good agreement between the actual particle size and those measured by sedimentation FFF should he had when the flow rate is in the range studied bere. In fact, quantitative experiments should be designed by varying the field strength so that elution is had in the retention range of R < 0.25. The power of this trpe of simulation lies in the ability to model the complex interaction of convection and diffusion in the presence of a force field with very few assumptions. No
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ANALYTICAL CHEMISTRY, VOL. 60, NO. 11, JUNE 1, 1988
steady-state assumption need be made in any coordinate direction, and since the driving mechanisms leading to separation can be effectively decoupled, although occurring simultaneously, the power for expansion of detail is readily apparent. Although not given in this paper, the results of simulation have been shown to give accurate results for field programming in sedimentation FFF (42),with no restriction as to how fast the field is changed. In addition, the simulation of a simultaneous field and flow programmed experiment has recently been demonstrated in this laboratory. Very little extra computational complexity arises due to the incorporation of these extra features; this is not the case for analytical solutions to the coupled equations of convection and diffusion when field and flow programming are present in the timedependent formulation. It is possible to extend the simulation to higher retention levels than those given here; however, the present model lacks detail which may be important at these higher levels of retention. These include the complete lack of particle-particle interactions, a suitable electrostatic wall interaction potential, a friction factor for particle friction near the wall, and local viscosity effects which will be predominant in the middle of the zone and at the lower wall. When further data is available on some of the effects, it will be possible to extend the simulation to model some of these details because the Monte Carlo method is easily extendible for modeling the case where the probability of diffusing in some direction is position dependent. Inertial effects may be programmed so that the particle is given memory as to its momentum and local trajectory for the case of particles whose size and density is in the region where both diffusion and inertial effects are important. The concentration-dependent viscosity effect may not be included with this type of simulation because the model itself considers only one particle in the channel at any time; hence the model is only good for the situations where infinite dilution is valid. Since most FFF experiments appear to be reasonably free of injection concentration effects, this may not be important. Finally, it must be mentioned that no simulation can replace the mathematical theories of FFF, but rather simulation is extremely helpful in verifying whether mathematical assumptions made in the theory are valid and in developing empirical relationships for complex systems. This puts simulation in a complementary role to theory and experiment. As faster computers become routinely available (especially those with parallel computational capabilities), it is anticipated that simulation methods, such as the one presented here, will find increased use as an aid to the development of FFF theory. GLOSSARY a,,, a,, u2 velocity interpolation coefficients channel breadth edge influence distance concentration of solute with respect to x concentration of solute at the wall diffusion coefficient in bulk solution diffusion coefficient near wall in the x direction particle diameter acceleration due to spinning the channel plate height Boltzmann’s constant channel length infinite series index number of excursions over one mean layer thickness number of theoretical plates probability of particle diffusing to higher x probability of particle diffusing to lower x probability of particle diffusing to higher y probability of particle diffusing to lower y probability of particle diffusing to higher z probability of particle diffusing to lower z
retention ratio retention ratio estimated from the peak maximum particle radius spacing for discrete levels in x diffusion distance for t,’ retention time void time switching time for wall-free region generalized switching time absolute temperature field induced velocity of a particle volumetric flow rate fluid velocity in the infinite parallel plate model fluid velocity in the rectangular channel model average fluid velocity channel width width direction vector initial x position average distance traveled by a particle breadth direction vector length direction vector mean layer thickness mean layer thickness (dimensionless) wall effect constant density difference between particle and fluid standard deviation in time units viscosity nonequilibrium parameter average time for diffusion through one 1 the ratio r/w coordinate for wall effect uniform random number between 0 and 1 empirical steric correction factor
ACKNOWLEDGMENT The author wishes to thank Professor Karin D. Caldwell, of the University of Utah Center for Biopolymers at Interfaces, and Professor Howard Brenner, Department of Chemical Engineering, Massachusetts Institute of Technology, for many enlightening discussions during the course of this work. LITERATURE CITED Giddings, J. C.; Myers, M. N.; Caldwell, K. D. Sep. Sci. Technoi. 1981, 16, 549-575. Giddings, J. C. Sep. Sci. Technol. 1985, 19, 531-547. Kirkland, J. J.; Rementer, S. W.; Yau, W. W. Anal. Chem. 1981, 53, 1730-1736. Giddings, J. C.; Karalskakis, G.; Caldwell, K. D.; Myers, M. N. J. ColbM Interface Sci. 1983. 92, 66-80. Yang, F.-S.; Caldwell, K D.; Glddings, J. C. J. Colloid Interface Sci. 1983, 92, 81-91. Yang, F A . ; Caldwell, K. D.; Myers, M. N.; Giddings, J. C.; J. ColloM Interface Sci. 1983, 93, 115-125. Yau, W. W.; Kirkland, J. J. S e p . Sci. Technoi. 1981, 16, 577-605. Jansson, P. A. Deconvoiut/on with Applications to Spectroscopy;Academic: New York, 1964. Schure, M. R.; Giddings, J. C. unpublished work. Giddlngs, J. C. J. Chem. Phys. 1988, 4 9 , 81-85. Giddings, J. C.; Yoon, Y. H.; Caldwell, K. D.; Myers, M. N.; Hovingh, M. E.; S e p . Sci. Technol. 1975, IO, 447-460. Giddings, J. C.; Yang, F. J.; Myers, M. N. Anal. Chem. 1976, 4 8 , 1126-1132. Hovingh, M. E.; Thompson, G. H.; Giddings, J. C.Anal. Chem. 1970, 4 2 , 195-203. Giddings, J. C.; Caldwell, K. D.; Moellmer, J. H.; Dickinson, T. H.; Myers, M. N.; Martin, M. Anal. Chem. 1979, 57, 30-33. Giddings, J. C. S e p . Sci. Technoi. 1978, 13. 241-254. Peterson, R. E.; Myers, M. N.; Glddings, J. C. Sep. Sci. Technol. 1984, 19, 307-315. Giddings, J. C.; Schure, M. R. Chem. Eng. Sci. 1977, 4 2 , 1471-1477. Giddings, J. C.; Schure, M. R.; Myers, M. N.; Velez, G. R. Anal. Chem. 1984, 5 6 , 2099-2104. Williams, P. S.; Gddings, S. B.;Giddings, J. C. Anal. Chem. 1988, 5 8 , 2397-2403. Davis, J. M.; Giddings, J. C. S e p . Sci. Technol. 1985, 2 0 . 699-724. Giddings, J. C. Anal. Chem. 1986, 58, 735-740. Gajdos, L. J.; Brenner, H. S e p . Sci. Technol. 1978, 13, 215-240. Krishnamurthy, S.; Subramanian, R. S. Sep. Sci. Technol. 1977, 72, 347-379. Jayaraj. K.; Subramanian, R. S.; Sep. Sci. Technol. 1978, 13, 791-617. Berg, H. c.; Purcell, E. M.; Proc. Natl. Acad. Sci. U . S . A . 1967, 5 8 , 862-869.
Anal. Chem. 1988, 60, 1119-1124
(31) (32) (33) (34) (35) (36)
Takahashi, T.; Gill, W. N. Chem. Eng. Commun. 1980, 5, 367-385. Kim, E.-K.; Chung, I . J. Chem. Eng. Commun. 1986, 42, 349-385. Lightfoot, E. N.; Chlang, A. S.; Noble, P. T. Annu. Rev. FluM Mech. 1981. 73, 351-378. Cornish, R. J. Roc. R . Soc. London 1928, 120, 891-700. Press, W. H.; Flannery, B. P.; Teukolsky, S. A,; Vetterling, W. T. NumertCal Recipes; Cambrklge University Press: New York, 1986; Chapter 3. Happel, J.; Brenner. H. Low ReynoM’s Number ~drodymmlcs;Sljthoff and Noordhoff Aphen van den Rijn, The Netherlands, 1973. Brenner, H. Chem. €ng. Scl. 1981, 16, 2242-2251. Brenner, H.; Gaydos, L. J. J . ColloM Interface Scl. 1977, 58. 312-358. MacKay, G. D. M.; Suzukl, M.; Mason, S. G. J. ColbM Interface Scl. 1963, 78. 103-104. Atwood, J. G.; Golay, M. J. E. J. Chromatogr. 1981, 278, 97-122. Vandersiice, J. T.; Rosenfeld, A. 0.; Beecher. G. R. Anal. Chlm. Acta 1986. 779, 119-129.
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(37) Claesson, “Forces Between Surfaces Immersed in Aqueous Solutions”; Ph.D. Dissertation, 1988, Department of Physical Chemistry, The Royal Institute of Technology, Stockholm, Sweden. (38) Bike, S. G.; Prleve, D. C.; Knapp. T. L. Electrokinetic Lift of Colloidal Particles in Slow Flow Near a wall, 8lst Colloid and Surface Science Symposium, University of Michigan, June 21-24. 1987. (39) Saffman, P. G. J. FluMMech. 1985. 22, 385-400. (40) Ho, B. P.; Leal. L. G. J . FluMMech. 1974, 65, 385-400. (41) Caldweii. K. D., University of Utah, personal communication. (42) Hansen, M. E.; GkMlngs, J. C.; Schure, M. R.; Beckett. R., submitted for publication in anal. Chem.
RECEIVED for review November 5,1987. Accepted February 9,1988. Portions of this paper were presented at the 193rd Meeting of the American Chemical Society, Denver, CO, April 1987.
Neat and Admixed Mesomorphic Polysiloxane Stationary Phases for Open-Tubular Column Gas Chromatography G. M. Janini Department of Chemistry, Kuwait University, Kuwait 13060,Kuwait
G. M. Muschik and H. J. Issaq PRI, Frederick Cancer Research Facility, National Cancer Institute, P.O. Box B, Frederick, Maryland 21 701
R. J. Laub* Department of Chemistry, San Diego State University, San Diego, California 92182
A mesomorphic polydioxane (MEPSIL) solvent, employed elther neat or in admixture wlth SE-30 pdy(cRmethylsibxane) (PDMS) dlluent, is shown to be a hlghly selective statlonary phase for open-tubular column gas-liquid chromatography. Mixtures of cls and trans fatty add methyl esters (FAME) are resolved isothermally In under 10 min with neat nematic MEPSIL phase, where trans Isomers are retalned longer than the& cls counterparts. Programmed-temperature separations of more complex mixtures of FAME solutes also demonstrate that elution order is by increasing carbon number and that within a group of the same carbon number, retentions increase with increasing saturatlon, which Is completely o p p &e to the order of selectlvlty given by conventional phases. Shape seiectivlty of the nematic MEPSIL phase is further illustrated by the separation of samples Containing polychkrlnated Mphenyi (PCB) solutes, where Inter ala congener 198 elutes well before 128. Blending the MEPSIL solvent with SE-30 Is then shown not l o depress significantly the nematic/isotropic transltion of the pure mesomorph, and blends of up to 3:l SE30/MEPSIL are virtually as selective as neat MEPSIL at temperatures in excess of 300 O C . However, the crystal/nematlc trandtlon cannot thereby be lowered substantially, nor can the lower ihnrtlng chromatographic efficiency be improved l o much beyond that of the neat phase. Even so, blended-phase columns exhibit in excess of 3200 N m-‘ at elevated temperatures, which is sufficient to provide good resolutlon, isothermally, of 8 of 10 methylbenz[a]anthracene isomers at 240 OC, and of 9 of 10 methylbenZO[8]pyrene solutes at 275 O C .
The gas chromatographic (GC) analysis of samples comprised of isomeric solutes of near-identical vapor pressure 0003-2700/88/0380-1119$01.50/0
remains a challenging task in analytical separations. Moreover, the problem is often compounded by the sheer complexity of mixtures containing classes of materials such as polycyclic aromatic hydrocarbons (PAH), polychlorinated biphenyls (PCB), and fatty acid methyl esters (FAME) (1-3).Thus,even with columns of the highest available efficiency, it is fair to say even today that the resolution of isomers such as five-ring PAH (including, e.g., the notoriously carcinogenic compound benzo[a]pyrene) is not entirely satisfactory. Even more problematic is the separation of methyl-substituted PAH, such as methylbenz[a]anthracenes, methylchrysenes, and methylbenzo[a]pyrenes, with conventional GC phases. In contrast, it has been recognized for some time that liquid-crystalline (mesomorphic)stationary phases (the nematic state in particular) provide enhanced GLC separations of many isomeric compounds, including PAH, on the basis of solute geometry. Thus, to a first approximation, more rodlike solutes are retained longer due to their relative compatibility with such ordered solvents. As a result, a number of liquidcrystalline phases have been introduced over the last decade or so (4-8), the best-known being the commercially available materials (e.g., Alltech Associates) N,N’-bis(p-methoxybenzylidene)-a,a’-bi-p-toluidine (BMBT), N,N’-bis(p-butoxybenzy1idene)-a,a’-bi-p-toluidine (BBBT), and N,N’-bis(p-pheny1benzylidene)-a,&-bi-p-toluidine(BPhBT) developed by Janini and his co-workers (9-12). An example of the remarkable shape selectivity of nematic mesophases is that they yield base-line separation of most parent PAH, which was achieved for the first time with packed columns containing BMBT stationary phase (9).However, the utility of such low molecular weight liquid-crystalline phases is limited by excessive column bleed at elevated temperatures, moderate column efficiency, and restricted useful mesomorphic temperature ranges. The fabrication of open-tubular columns with 0 1988 American Chemical Society
