Viscous Fingering in Size Exclusion Chromatography - American

Jun 1, 1996 - Department of Chemical Engineering, Thornton Hall, University of Virginia,. Charlottesville, Virginia 22903-2442. A two-dimensional hybr...
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Viscous Fingering in Size Exclusion Chromatography: Insights from Numerical Simulation T. Tucker Norton and Erik J. Fernandez* Department of Chemical Engineering, Thornton Hall, University of Virginia, Charlottesville, Virginia 22903-2442

A two-dimensional hybrid finite difference/particle-tracking simulation has been used to study the flow nonuniformities induced by the viscous fingering instability during size exclusion chromatography of proteins. The numerical simulation faithfully reproduces the morphology and dynamics of viscous fingering observed in magnetic resonance images of sample migration through operating chromatography columns. In addition to sample intrinsic viscosity, solute molecular diffusivity and packing heterogeneities are found to have a significant impact on finger dynamics. More so than in chromatography of solutes which do not enhance viscosity, packing heterogeneities lead to considerable increases in HETP for viscous samples. Where intraparticle mass transfer resistances are small, the effect of fingering on HETP is almost independent of velocity. Introduction Displacement of a miscible viscous liquid by a less viscous one typically leads to formation of “fingers” of the less viscous fluid which rapidly invade the displaced one. This flow instability yields a diverse set of behavior encompassed by the term viscous fingering (Homsy, 1987). Viscous fingering poses difficulties in a variety of engineering contexts, including secondary and tertiary oil recovery, groundwater transport, and packed bed separations. In these situations, the flow nonuniformity produced by viscous fingering is undesired and reduces efficiency. In chromatography, viscous fingering causes peak tailing which elevates column HETP for viscous samples such as proteins, synthetic polymers, and sugars (Flodin, 1961; Moore, 1970; Yamamoto et al., 1986; Byers et al., 1990). In adsorption separations it reduces elution efficiency of viscous samples during elution with less viscous solvents (Hill, 1952). Solutions to the problem have been proposed, including batch methods and the use of eluents with viscosities matching that of the sample (Czok et al., 1991), although adding solutes to accomplish this is typically undesirable. Batch methods have also been proposed, although this concedes the resolution provided by chromatography (Emne´us, 1968). All of these studies have shown that loss of efficiency is strongly dependent on viscosity of the sample; however, the quantitative relationships between important factors such as reduced velocity, sample size, packing heterogeneity, and solute physical properties to column performance parameters are not available. Because of the importance of viscous fingering in enhanced oil recovery, a number of experimental and theoretical studies of viscous fingering have been performed. Since the first detailed observations of viscous fingering by Hill (1952), most studies of miscible viscous fingering have focused on the contribution of dispersion and permeability heterogeneity to the morphology and growth rates of the developing fingers. Several investigators have performed linear stability analyses of the instability, incorporating aspects such as nonstep initial profiles, different viscosity-concentration profiles, and * To whom correspondence should be addressed. Phone: (804)-924-1351. Fax: (804)-982-2658. Email: [email protected].

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anisotropic dispersion (Chuoke, 1982; Tan and Homsy, 1986; Zimmerman and Homsy, 1991). These analyses have shown that dispersion is a stabilizing force, restricting finger wavelengths and growth rates by smoothing concentration gradients. Most simulation efforts to examine nonlinear fingering behavior have been directed toward petroleum-engineering applications, under conditions of high viscosity ratios (µsample/ µsolvent), stratification of permeability heterogeneity, and essentially infinite domains (Christie et al., 1990; Tchelepi et al., 1993). Little work, however, has been done on the aspects particular to chromatography, where finite injection volumes with dissimilar sample properties are passed through a domain of small width and finite length. Capturing viscous fingering experimentally in threedimensional porous media has proven to be a nontrivial task. Many investigators have used Hele-Shaw cells and thin glass bead packs which approximate a threedimensional domain with a pseudo-two-dimensional one. Optical or acoustic methods have allowed threedimensional samples to be examined although the details of fingering are masked by cross-sectional averaging (Christie et al., 1990; Bacri et al., 1992). Recently, magnetic resonance imaging (MRI) has provided direct analysis of the three-dimensional movement of a liquid through a porous medium over time (Gummerson et al., 1979; Rothwell and Vinegar, 1985; Guillot, 1988; Bayer et al., 1989). Recently MRI has been applied successfully to examination of viscous fingering in chromatography and porous media (Athalye et al., 1991; Davies et al., 1992; Plante et al., 1994). The ability to visualize viscous fingering in operating chromatography columns now allows comparison with theoretical descriptions of finger behavior. In this work, we have used numerical simulation to study the effect of sample properties on the growth and development of the viscous fingers in a chromatography bed. Specifically, the effects of sample molecular weight, sample size, packing heterogeneities, and reduced velocity have been examined. The results show that a disparity between glycerol and albumin in fingering behavior can be attributed to differences in molecular diffusivity. Further, the sample band leading edge is resistant to finger breakthrough when µsample > µsolvent, leading to substantial finger cross flow. Finally, the impact of © 1996 American Chemical Society

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reduced velocity and packing heterogeneities on HETP have been evaluated.

manufacturer. A linear velocity of 0.12-0.18 cm/min was used for all experiments. Glycerol was used as a model viscosity-enhancing solute.

Methods Numerical Simulation. A two-dimensional hybrid finite difference/particle-tracking numerical simulation of Orr and co-workers (Araktingi and Orr, 1990) was adopted to model miscible viscous fingering. The mathematical representation of the flow is given by the convection-dispersion equation,

∇‚(D‚∇c) - ∇(cv) )

∂c ∂t

(1)

Darcy’s law,

k v ) - (∇P + Fg) µ

(2)

and the continuity equation,

∇‚v ) 0

(3)

where D is the dispersion tensor, c is solute concentration, v is the velocity vector, k is the column permeability, µ is viscosity, P is pressure, F is the density, and g is the gravitational vector. The simulation was modified to incorporate finite width sample volume injections and actual viscosity-concentration data instead of viscosity approximation by the quarter-power blending rule. Intraparticle mass transfer and solute adsorption are not included in this model. Additional details of the simulation algorithm are given elsewhere (Araktingi and Orr, 1990). The simulation parameters were selected to correspond with a size exclusion chromatography column employed in our laboratory, measuring 14 cm in length by 3.2 cm in diameter. The simulation flow was horizontal with flowrates on the order of milliliters per minute. The average particle size of the porous media was 30 µm, with Peclet numbers with respect to particle diameter on the order of unity. No adjustable parameters were used to improve the quality of the simulation results. Simulations were run on IBM RS6000 machines and typically consumed 2-6 h of shared computational time. While the simulation performed here were only twodimensional for computational economy, previous simulation studies incorporating all three spatial dimensions showed little difference from their two-dimensional counterparts in most situations. Only in cases of strong gravity effects did the two- and three-dimensional cases diverge (Tchelepi and Orr, 1993). Magnetic Resonance Imaging. Magnetic resonance images were collected at 74.57 MHz for 1H using a 1.75 T magnet (Nalorac) and console (Tecmag, Houston, TX) using the established spin-warp method as described previously (Fernandez et al., 1995). While the sample moves during the acquisition time, the effective “blurring” of the image is less than might be expected since most of the large signal intensity data is collected during a short period in the middle of the acquisition time. Size Exclusion Chromatography. A 3.2 cm diameter glass column (Amicon, Natick, MA) was used for all experiments. The column was packed with Toyopearl HW 65S media with a 30 µm average particle diameter (TosoHaas, Montgomeryville, PA). The column was packed under flow as recommended by the

Results Validation. To show that the simulations reproduce the physical situation accurately, we have performed a comparison between MRI measurements of viscous fingering in a chromatography column and numerical simulation with no adjustable parameters. One important parameter particular to a given column is the packing heterogeneity. Our previous studies of fingering by MRI have revealed that packing heterogeneity near the inlet of the column is common and is a critical factor in fingering (Fernandez et al., 1995). In fact, in our experience this region of the column appears to harbor most of the packing nonuniformities, with the body of the column being quite homogeneous. To obtain the distribution and size of these important packing variations to be used in the simulation, we have used MRI analysis of matched viscosity sample for which viscous fingering effects are eliminated. By monitoring the sample band shape over the length of the column, an approximate description of the permeability heterogeneity near the inlet may be obtained. Figure 1a shows the MRI results used to approximate the inlet permeability field, with flow from left to right and the field of view of 3.2 cm width and 14 cm length. Rows a, b, and c represent runs of 0% glycerol, 20% glycerol, and 40% glycerol samples, respectively, on the same column. The viscosity ratios (M) are 1, 1.73, and 3.65, respectively. Permeability variations are present in four regions across the width of the column, with two at the walls of the column and the remaining two located approximately quarter- and halfway from the bottom wall. The evolution of the sample band shape during elution reveals differences in permeability encountered along the flow direction. In this case, most of the flow nonuniformity occurs at the column inlet. For the purposes of simulation, the permeability field present in the column was approximated as a region of lower permeability near the inlet with the magnitude of the permeability change dictated by the nonuniform displacement of the sample band in Figure 1a. In all cases, the maximum variation of permeability was 30% of the original value. This permeability field was used along with experimental parameters to produce the simulated displacement shown in Figure 2. The values of average column permeability, sample volume, sample viscosity, and flowrate were taken from the experimental conditions of Figure 1 and dispersion calculated according to available correlation (Perkins and Johnston, 1963). Figure 2 shows representative images throughout the displacement. The image reveals dispersive effects that spread the contours on the sample band boundaries, as well as nonuniform flow caused by viscous fingering at the unstable rear edge of the sample band. Note that like the experimental results, the leading edge is flat and self-stabilizing. The trailing edge is convoluted by the invasion of solvent fingers due to the viscous gradient between sample and eluent, delaying portions of the sample band and altering the structure of the resultant chromatogram. A visual comparison of the simulation results and MRI measurements shows that both results agree at least qualitatively. Using the permeability field as detailed above, the simulated displacement of a 40% glycerol sample injection produces a distorted sample

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Figure 1. Magnetic resonance visualization of viscous fingering in chromatographic media. Flow is from left to right, with the column inlet on the left. Rows a, b, and c are 0% v/v (M ) 1), 20% (M ) 1.73), and 40% (M ) 3.65) glycerol sample concentrations, respectively. From Fernandez et al., 1995. Reproduced with permission from the American Institute of Physics.

Figure 2. Numerical simulation results under the same conditions as Figure 1. The permeability field was estimated from the distortion of the sample band in Figure 1a, as detailed in the text. Dimensions of the domain are 14 cm length and 3.2 cm width.

band shape which resembles the fingering observed in MRI images. The simulated finger wavelengths are approximately 5-8 mm in agreement with experimental observations in Figure 1. The simulated finger locations along the cross-section and their relative magnitude also resemble their experimental counterparts. This comparison confirms that inlet packing quality influences the location, magnitude, and wavelength of the developing fingers. Slight differences in finger magnitude are likely due to incomplete characterization of the original permeability field. Since the MRI signal is nonlinear with respect to concentration, measurement of exact displacement from the sample band is difficult. Anatomy of a Viscous Finger. Simulation analysis of the local velocity and pressure fields around a finger provides a mechanistic understanding which can lead to the implementation of solutions to the viscous fingering problem. In order to focus on the mechanisms involved for a simple case of an isolated finger, a region of high permeability was inserted in the middle of the column cross-section, creating a dominant finger. This technique enables a single finger to grow rapidly in the

center of the column, thereby reducing competition from other much smaller fingers. Figure 3 displays the velocity field around the emerging finger at a relatively early time within the column. To highlight the nonuniform flow effects due to viscous fingering and permeability variations, the average superficial velocity has been subtracted from the image to yield a Lagrangian velocity field. Since the permeability field is homogeneous except near the inlet, all nonzero velocity vectors in this figure may be attributed to the flow nonuniformities provided by flow instability alone. The most striking feature in the flow field are two regions of counter-rotating flow located on either side of the finger. These vortices are a result of the development and propagation of the solvent finger into the sample band. At the finger tip, the vortices act to draw solvent laterally away from the finger tip, causing finger broadening. In cases of multiple fingers, the presence of such vortices may contribute to the often described tip spreading and coalescence mechanisms observed in previous simulation studies (Tan and Homsy, 1988).

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Figure 3. Anatomy of a viscous finger. A region of high permeability at the center of the column inlet was used to seed the initial finger. The Lagrangian velocity field is superimposed on the 0.5 concentration contour to show the presence of flow nonuniformities.

The vortices also play a role specific to finite sample volumes not described elsewhere to our knowledge. As a finger approaches the leading edge of the sample band, it fails to penetrate the sample band but rather shifts its primary growth direction to a more lateral one. This effect has been documented in our experiments and confirmed in simulation results. The Lagrangian velocity field explains these observations by the presence of significant cross flow near the leading edge and the finger tip. The vorticity generated by the fingers combined with the stability of the leading edge yields a stronger cross flow than is observed without the leading edge present. For large samples, the fingers create a measure of cross flow which results in finger spreading. However, for smaller samples where the finger encounters the leading edge of the sample band, the size of the transverse velocity component is larger and further tip spreading is observed. In fact, in some cases such as Figure 4, a substantial cross flow is observed, such that the finger appears to “turn” and grow laterally in one transverse direction only. We have also observed this behavior in some experiments (data not shown). These interactions of fingers with a leading edge are particular to chromatography since in the petroleumengineering examples the leading edge is usually not encountered. Consequently, the effect of finite sample volume with regard to finger development has not been addressed. Although this behavior is peculiar to the chromatographic case, in practice one would hope to avoid such an extreme case of finger invasion of the sample band. Diffusive Contribution to Viscous Fingering. MRI visualization shows altered fingering profiles for different samples as they pass through the same chromatography column. In many cases, the differences may be ascribed to changes in viscosity ratio, known to alter both finger wavelength and growth rate. Nevertheless, other factors contribute to the growth rates of fingers. Visualization of the viscous-fingering patterns in samples of glycerol versus samples of bovine serum albumin (BSA) reveals differences in finger wavelength (Fernandez et al., 1995). The viscosity ratio of 1.3 for BSA is much lower than that for glycerol (1.73) under the conditions of those experiments (µ ) 1.3 × 10-3 Pa s at 43 mg/mL of BSA versus 1.73 × 10-3 Pa s at 20% glycerol), and yet finger wavelengths are smaller for BSA.

Figure 4. Illustration of effect of cross flow on the direction of finger growth. The primary direction of finger growth has shifted to the transverse direction, parallel to the leading edge. Size of the domain is 14 cm in length by 1.6 cm in width, with flow from left to right. The simulation images were acquired at 0.2, 0.6, and 0.8 fraction of progress through column.

A potentially important difference in physical properties of the two solutes is molecular diffusivity. Chuoke performed a basic linear stability analysis over twenty years ago to calculate the fastest growing finger wavelength (Gardner and Ypma, 1984),

λmax )

(

)( )

µs + µo Dt µ x5 - 2 s - µo v 2π

(4)

where µs and µo are the viscosity of the sample and solvent, respectively, Dt is the transverse dispersion coefficient, and v is the local interstitial velocity. Despite the assumptions of a linear viscosity-concentration profile, nonadsorptive porous medium, and isotropic dispersion independent of velocity, certain conclusions may still be drawn. The importance of relative viscosity is shown in the second factor, which decreases rapidly with increasing viscosity ratio at small ratios (M < 3) before asymptotically approaching the limit of unity at higher ratios. For chromatographic purposes, most separations are performed in this small ratio regime where changes in viscosity ratio can significantly alter the wavelength. Transverse dispersion likewise can be affected by the choice of sample. Employing the Perkins-Johnston correlation for transverse dispersion (Perkins and Johnston, 1963) enables an estimation of the importance of molecular diffusivity

Dt )

Do + 0.055vdp Fφ

(5)

where Do is the molecular diffusivity, v is the local velocity, Fφ is a resistivity parameter of the packed bed, and dp is the particle diameter. Given that the molecular diffusion coefficients of BSA and glycerol differ by

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Figure 5. Effect of diffusivity on numerical simulation results. Four runs were performed under identical viscosity ratios but with varying diffusion coefficients. Parts a-d use Deff ) 8.3 × 10-6 (glycerol), 5.7 × 10-6, 3.2 × 10-6, and 6.0 × 10-7 cm2/s (BSA), respectively.

over an order of magnitude (approximately 6 × 10-7 versus approximately 1 × 10-5 cm2/s, respectively), this diffusional change may suffice in controlling finger wavelengths even with minor viscosity ratio changes. Equations 4 and 5 therefore dictate a decreasing most dangerous wavelength from glycerol to BSA despite a concomitant decrease in viscosity ratio which agrees with the expectation of decreased wavelengths with a reduction in dispersion. In addition, the use of numerical simulation affords the possibility of isolating the effect of differing molecular diffusivity while holding constant the viscosity ratio for displacements of glycerol and BSA samples. Figure 5 demonstrates the magnitude of the molecular weight effect for both glycerol and BSA at identical viscosity ratios. As before, flow is to the right with gravity acting in the downward direction. Two other cases are also shown with diffusivities intermediate to those of glycerol and BSA for comparison. The glycerol case shows the broadest fingers which appear slightly less invasive than the fingers invading the BSA sample, though this distinction is sufficiently small to be attributed to noise in the data. The finger wavelengths are approximately 2 mm for BSA and 4-5 mm for glycerol. These wave-

lengths agree well with experimental observations (Fernandez et al., 1995). To make a more quantitative comparison between simulation and experiment, we have employed two metrics of the effects of viscous fingering. Fitting the chromatogram to an error-function solution of the convective-diffusion equation with incorporation of finite sample width effects permits calculation of the HETP

C(z,t) )

{[ ] [

]}

(z - z0 - a) (z - z0) C0 erf - erf 2 x2Hz0 x2Hz0

(6)

where C0 is the initial concentration, H is the HETP, a is the sample band width in the flow direction, z is position in column, and z0 is the average position of the sample band at time t. The presence of viscous fingering progressively increases the HETP as the fingers delay portions of the sample, leading to “shouldering” and broadening of the chromatogram. Figure 6 demonstrates the procedure for a unit viscosity case, as well as a 40% glycerol case. While the unit viscosity case has a typical Gaussian profile, the 40% glycerol case demonstrates the effect of fingering on the shouldering

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Figure 7. The dimensionless line integral (DLI) as a function of time for the BSA and glycerol cases. The DLI was calculated from Figure 5.

Figure 6. Fitting the 1-D concentration profile to calculate HETP. Figure 6a,b shows the chromatograms for the M ) 1 and M ) 3.65 viscosity ratios, respectively.

of the chromatogram, impacting the goodness of fit of equation 6 and increasing the HETP. The second measurement is the dimensionless finger line integral (DLI), defined as

DLI )

∫

1 C(x,y) dL W L(c))cj

(7)

where W is the domain width, C(x,y) is the concentration field, and cj is the contour level of interest. The DLI gives values of unity for a perfectly stable, noiseless rear interface perpendicular to flow, with increasing values proportional to the amplitude and wavenumber of fingers along the front. This measure of finger growth was found to have advantages over the standard mixing length definitions of Lδ and Lσ (Zimmerman and Homsy, 1991; Zimmerman and Homsy, 1992), which are not easily adapted to pulse injections. Whereas the peak maximum of pulse injections decreases with time due to dispersive effects, the maximum of a step injection remains constant. Lδ, for example, requires this maximum be constant and also that the concentration profile be single-valued for concentration which does not occur in pulse injections. Both the DLI and HETP measurements may be used here to contrast the differences between BSA and glycerol. Use of the DLI enables the construction of Figure 7. In effect, the DLI may be considered a measure of the extent of fingering through penetration depth, with higher values being interpreted as more deleterious to separation efficiency; however, caution

must be used in cases of widely varying wavelength over a run due to the competing effects of finger wavelength and growth rates. At the first point in the column, both cases have DLIs near unity as the disturbance has had little time to grow and noise in the data predominates. The DLI values then separate and show monotonically increasing rates of growth with decreasing diffusivity. The simulations show that the smallest diffusivity exhibits the most narrow fingers which tend to have slightly greater amplitudes. By the end of the study, the sample band has passed through 80% of the column and the two cases are easily distinguished from one another by DLI and visual inspection of the sample band. The second metric of band broadening we have employed is HETP. HETP is a useful parameter for analysis of simulation and experimental data since it is used widely as a measure of chromatographic performance and allows ready comparison between simulation and subsequent experiments. Curve-fitting eq 4 to the cross-sectionally averaged concentration profiles throughout the displacement allows determination of the HETP for each of the simulation cases. Unlike matched viscosity samples which only exhibit peak broadening due to dispersive effects, the samples under study here do not remain constant but show an increase in HETP with time due to the progressive solvent penetration of the sample. By the time the samples have progressed halfway through the column, the reduced HETP values are 7.6 and 7.4 for glycerol and BSA, respectively. At 80% progression through the column, the values are 9.2 and 9.5, respectively. These values are sufficiently close to one another to be considered identical. Because the effect of wavelength on the chromatogram is negligible in comparison to penetration depth, the deterioration of separation quality may be attributed to the solvent penetration alone. Importantly, the HETP is approximately identical between the two samples, showing that the HETP measurement lacks the sensitivity to detect changes of this magnitude of finger wavelength. The similarity of the two cases indicates the relative proximity in growth rate of the fingers, confirming the finger penetration depths previously shown in Figure 5. The shapes of the µ(c) curves have been shown to have measurable, sometimes dramatic, impact on finger behavior. Examination of the concentration-viscosity relationship for both glycerol and BSA reveals differences in the slopes at the endpoints, which have consequences for the growth of the viscous finger

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Figure 8. Velocity and permeability heterogeneity effect on the reduced HETP. Open symbols represent the cases with the nonuniform permeability used in Figure 2.

instability. Manickam and Homsy derived a stability relation based on these slopes,

-

(

)

(dµ/dc)c)0 + (dµ/dc)c)1 R+1

(8)

where R is the viscosity ratio and c ) 0 and c ) 1 are the endpoints of the viscosity-concentration curve (1993). Comparison of BSA and glycerol viscosityconcentration profiles for the same viscosity range (R ) 1.15) gives values for glycerol approximately 33% greater than those for BSA, indicating that the glycerol case is more unstable. At higher concentrations, the viscosity of glycerol solutions increases much more rapidly than does the viscosity of BSA. Given the instability dependence on the derivative of viscosityconcentration curves, the disparity between glycerol and BSA only increases with concentration. Viscous Fingering, Velocity, and HETP. In addition to the above basic mechanistic simulation studies of sample band geometry and solute molecular diffusivity, we endeavor to examine the effect of velocity on the amplitude of viscous fingering. While chromatographers using SEC must remain near the minimum of the HETP curve to achieve optimal resolution, a compromise is often struck between operation time and optimum HETP. Consequently, the highest flowrate consistent with the separation requirements is typically used. The numerical simulation was used to develop a reduced HETP versus reduced velocity plot for viscosity ratios of 1 and 3.65 (Figure 8), corresponding to water and 40% glycerol, respectively. The two cases therefore serve to demonstrate the effects of viscous fingering over a range of velocities. These curves were generated using homogeneous permeability fields, with exceptions to be noted later. Both cases display the expected rapidly decreasing HETP in the diffusive regime but show divergent behavior in the convective regime. The unit viscosity ratio exhibits a flat curve at higher reduced velocities whereas the higher viscosity ratio curve does not reach the same minimum and increases gradually with reduced velocity. Because the simulation does not yet account for intraparticle mass transfer, the expected behavior for the unit viscosity case in the convective regime is the flat line shown here. Interestingly, the increase in HETP with velocity in the viscous fingering case underscores the separation penalty suffered by increasing the flowrate, even in the absence of intraparticle mass transfer.

The two additional points on the figure were calculated for the two viscosity ratios using identical but heterogeneous permeability fields, similar to the one used earlier in this work. Noting the displacements of the respective points from their homogeneous permeability curves, it is clear that the separation efficiency of the unit viscosity ratio case is not significantly affected. By contrast, the heterogeneous 3.65 viscosity ratio case shows an HETP almost twice that of the homogeneous permeability case. The consequences of poor packing are clear: as the viscosity ratio increases, the penalty paid for permeability heterogeneity correspondingly increases dramatically. Examination of the fingering patterns in Figure 2, which used heterogeneous permeability fields, also shows this behavior. Therefore, good column packing is critically important for viscous sample separations, more so than separations near viscosity ratios of unity. Discussion In this work, two-dimensional numerical simulation results reproduce, at least semiquantitatively, the viscous fingering behavior observed in chromatography columns by MRI. Without the use of adjustable parameters, growth rates, wavelengths, and nonlinear finger behavior such as spreading and tip-splitting are all predicted. Any discrepancies between simulation and experiment are likely due to errors in the estimation of the permeability field and values of axial and transverse dispersion which are not accurately known in this regime. We plan to obtain more accurate permeability fields which may be available through direct MRI velocity measurements in an operating column. The comparisons between experiment and simulation also indicate that the two-dimensional simulations are sufficient to capture the behavior of fingering in threedimensional chromatography columns. The simulation proved particularly valuable in explaining features of viscous fingering particular to chromatography. To our knowledge, this is the first simulation study of interactions between viscous fingers in a finite length viscous band. The simulations establish that lateral flow at the tips of fingers is enhanced when they near the leading edge, leading to substantial cross flow which we have observed experimentally (Fernandez et al., 1995). This stability of the leading edge perhaps explains why previous studies have noted that peaks from viscous samples continue to begin eluting at the same time regardless of sample size (Czok et al., 1991) and why larger sample volumes suffer greater loss of resolution from fingering (Moore, 1970). The effects of solute molecular weight on fingering can be significant according to the simulations. The markedly lower molecular diffusivities of proteins compared with small molecules such as glycerol result in corresponding reduced finger wavelengths, and possibly elevated finger growth rates. This is consistent with the important contribution molecular diffusivity makes to dispersion, particularly the transverse component important to fingering, at the low flowrates used in SEC. Although only one protein was studied numerically in this work, these observations may be universal for globular proteins because they have consistent intrinsic viscosities and molecular weight dependence of molecular diffusivity (Tanford, 1961). Rodlike and random coil proteins, on the other hand, will be sensitive to the viscous-fingering phenomenon because of their high intrinsic viscosities and lower molecular diffusivities (Tanford, 1961).

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In these studies of molecular weight and diffusivity, homogeneous permeability fields allowed for accurate comparisons between the two systems without complicating the picture with other variables. However, this also means the extent of fingering observed in the simulations will be less than what would be observed experimentally. It should also be noted that employing a particular nonuniform permeability field to compare BSA and glycerol finger morphology will tend to favor finger growth into one of the samples over the other due to “resonance” effects. As has been shown in other simulations of miscible fingering (Tan and Homsy, 1992) and other simulations we have performed (data not shown), disturbances with correlation length scales comparable to the natural finger wavelength will be most effective in accelerating finger growth. Thus, BSA will be more sensitive to smaller length scale packing heterogeneities than glycerol. Applications of these results to column design and packing procedures will thus require the consideration of the magnitude and correlation length of the permeability fluctuations. Our simulation studies of the effect of reduced velocity and permeability heterogeneity reinforce the importance of packing heterogeneities on the magnitude of fingering. In fact, solutes prone to viscous fingering are even more sensitive to flow nonuniformities caused by poor packing than solutes that do not affect sample viscosity. There are questions left open by the work, however. While this investigation has some application to other types of isocratic chromatography, the effects of adsorption and intraparticle mass transfer have not been incorporated and will have a strong impact on the results at some point. For example, as Hill noted some years ago (1952), viscous fingering can cause significant flow nonuniformities in adsorptive sugar purification. While the separation could still be performed, the efficiency of elution was severely compromised. In a similar fashion, protein samples desorbed during step gradient elution may suffer from viscous fingering, resulting in undesired product dilution. Conclusions The results presented in this work have provided us with a much better mechanistic understanding in viscous fingering in size exclusion chromatography. In addition to viscosity ratio, sample physical properties, sample size, and column packing heterogeneities have been shown to have a strong effect on chromatographic performance for viscous samples. In the absence of intraparticle mass transfer effects, reduced velocity has a weak effect on the degree of fingering observed. In the near future, we plan to apply these simulation techniques to study the effects of adsorption and mass transfer limitations on fingering. This will allow us to evaluate potential solutions to the viscous fingering problem under a more realistic set of conditions. Acknowledgment We thank Chad Grotegut for assistance with the magnetic resonance imaging experiments. The authors would also like to thank Hamdi Tchelepi for helpful discussions on implementation of the numerical simulation. This work has been supported by the National Science Foundation, the donors of The Petroleum Research Fund, administered by the American Chemical Society, and DuPont. Computational support for this research provided partially by a grant from the IBM

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Received for review December 27, 1995 Accepted March 25, 1996X IE950777E

X Abstract published in Advance ACS Abstracts, June 1, 1996.