Effect of Carrier Fluid Viscosity on Retention Time and Resolution in

ARTICLE pubs.acs.org/ac

Effect of Carrier Fluid Viscosity on Retention Time and Resolution in Gravitational Field-Flow Fractionation Seungho Lee,† Da Young Kang,† Miri Park,† and P. Stephen Williams*,‡ † ‡

Department of Chemistry, Hannam University, Daejeon 305-811, South Korea Department of Biomedical Engineering, Lerner Research Institute, Cleveland Clinic, 9500 Euclid Avenue, Cleveland, Ohio 44195, United States

bS Supporting Information ABSTRACT: Gravitational field-flow fractionation (GrFFF) is a useful technique for fast separation of micrometer-sized particles. Different sized particles are carried at different velocities by a flow of fluid along an unobstructed thin channel, resulting in a size-based separation. They are confined to thin focused layers in the channel thickness where force due to gravity is exactly opposed by hydrodynamic lift forces (HLF). It has been reported that the HLF are a function of various parameters including the flow rate (or shear rate), the size of the particles, and the density and viscosity of the liquid. The dependence of HLF on these parameters offers a means of altering the equilibrium transverse positions of the particles in GrFFF, and hence their elution times. In this study, the effect of the viscosity of the carrier fluid on the elution behavior (retention, zone broadening, and resolution) of micrometer-sized particles in GrFFF was investigated using polystyrene (PS) latex beads as model particles. In order to change the carrier liquid viscosity without affecting its density, various amounts of (hydroxypropyl) methyl cellulose (HPMC) were added to the aqueous carrier liquid. It was found that particles migrate at faster rates as the carrier viscosity is increased, which confirms the dependence of HLF on viscosity. At the same time, particle size selectivity decreased but peak shape and symmetry for the more strongly retained particles improved. As a result, separation was improved in terms of both the separation time and resolution with increase of carrier viscosity. A theoretical model for plate height in GrFFF is also presented, and its predictions are compared to experimentally measured values.

F

ield-flow fractionation (FFF) is a family of elution-based separation techniques that employ external fields applied perpendicularly to the fluid flow direction in thin channels. The fluid has a parabolic flow velocity profile across the thickness of the channel with the fastest flow velocity at the center and zero velocity at the walls. The field acts on the sample components, driving them toward one of the walls. Particle diffusion or hydrodynamic forces tend to oppose this transport toward the wall resulting in steady-state distributions of the components across the fluid velocity profile. Different sample components have different steady-state distributions and are therefore carried by different fluid velocities resulting in their separation. FFF is applicable to separation and characterization of various types of macromolecules and particles in the size range of a few nanometers up to several tens of micrometers.15 Gravitational field-flow fractionation (GrFFF) uses the Earth’s gravity as the external field and is relatively simple in principle and operation.6 GrFFF is usually applied to separation of particles ranging from a few micrometers up to about 50 μm in diameter. The particles are separated based on their size with larger ones eluting earlier than the smaller ones. GrFFF has been shown to be useful for separation of various types of micrometer-sized particles including coal particles,7 silica particles,815 clay particles,16 starch granules,1719 yeast,2024 and pollens.25 It r 2011 American Chemical Society

has also been shown that GrFFF is sufficiently gentle to allow separation of samples such as lymphocytes,26 red blood cells,2729 parasites,29,30 stem cells,31,32 and bacteria.33 The optimization of experimental conditions for particle analysis in the various types of FFF generally involves the adjustment of the applied field strength. Although the gravitational field itself cannot be varied, indirect means of changing the forces acting on particles in GrFFF have been used, such as changing the angle between the gravitational field and the flow axis of the channel34,35 and changing the density of the carrier liquid.36 At relatively low flow rates, particles stay in close proximity to the bottom wall of the channel while they are driven along the channel by the flow (steric mode). As flow rate is increased, particles are subject to hydrodynamic lift forces (HLF) which raise them some distance above the wall to form thin equilibrium layers where gravitational and lift forces are balanced (hyperlayer or focusing mode).8,12,37,38 The HLF have been studied using the technique of sedimentation FFF which uses a centrifuge to generate the transverse force across the channel thickness. In these experiments, the flow rate and the field Received: December 3, 2010 Accepted: March 19, 2011 Published: April 05, 2011 3343

dx.doi.org/10.1021/ac103002g | Anal. Chem. 2011, 83, 3343–3351

Analytical Chemistry

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strength may be varied independently of one another.3941 Later the effect of carrier fluid viscosity was also included.42 The HLF were found to increase rapidly as a particle approaches the wall, and this strong distance dependence means that particles may find different equilibrium positions for a wide range of applied field strengths. Conversely, the equilibrium positions may be altered by changing the strength of the HLF at some fixed field strength. It was found that HLF are a function of various parameters including the flow rate (or fluid shear rate), the size of the particles, and the density and viscosity of the carrier liquid. The flow rate or the fluid viscosity may therefore be altered to change the strength of the HLF acting on particles of any given size. The particle equilibrium positions and hence their retention ratios could thereby be influenced, where a particle retention ratio is the ratio of particle elution velocity to mean fluid velocity. The purpose of this study was to investigate the effect of the carrier viscosity on retention, zone broadening, and resolution in GrFFF, while other parameters were kept constant.

’ THEORY Retention in GrFFF. The fluid velocity in the thin FFF channel follows a parabolic profile across the channel thickness described by x x 1 ð1Þ vðxÞ ¼ 6Ævæ w w

where x is the distance from the accumulation wall (the wall toward which the particles are driven by the applied field), w is the channel thickness, and Ævæ is the mean fluid velocity. In GrFFF, retention of a sample is measured by the retention ratio, R, which corresponds to the ratio of particle migration velocity along the channel to the mean fluid velocity. It is determined experimentally as the ratio of channel void time t0 to the particle retention time tr. In the ideal model of steric FFF, the particle migrates in close proximity to the accumulation wall and its velocity vp corresponds to that of the fluid at the position of its center, at a distance of one radius from the wall. The retention ratio R would then be related to the particle radius a by vp t0 a a 1 ¼ ¼6 R ¼ 6R ð2Þ w w Ævæ tr in which R is the ratio of particle radius a to channel thickness w. In practice, the retention ratio is given by R ¼ 6γR

ð3Þ

in which γ is a dimensionless factor that is included to take into account the effect of the lift forces and hydrodynamic effects that tend to retard the motion of particles relative to that of the fluid at the position of the particle centers when they are close to the channel wall.35,37,43 In the steric mode, where particles are carried along the length of the channel close to the accumulation wall, they move with a velocity close to that of the fluid at a distance of one particle radius from the wall, and γ is therefore close to unity. The magnitude of γ may be even smaller than unity due to the hydrodynamic effects mentioned above. In the hyperlayer mode, where particles are lifted away from the accumulation wall during migration, γ can be much larger than unity. In the hyperlayer mode of GrFFF, γ can be used as a measure of the distance of the

particles from the bottom wall of the channel. For example, if γ = 2 then the particles are raised to a point where the particle centers are at a distance of approximately twice the particle radius from the wall. Because γ varies with experimental conditions, a calibration plot of log(tr) versus log(d) (where d is the particle diameter) for a set of particle standards of known size and density is usually employed for size determination by GrFFF. Such plots tend to be fairly linear over reasonable size ranges of, say, a 10-fold diameter range. The absolute value of the slope of this calibration plot is defined as the size-based selectivity, Sd. Hydrodynamic Lift Forces. Fluid inertia has been shown to result in hydrodynamic lift on particles entrained in shear flow.44 For plane Poiseuille flow, as found in the parallel plate FFF channels, theory predicts that, in the absence of any transverse field, freely rotating particles will be driven toward two stable equilibrium positions at x/w corresponding to approximately 0.19 and 0.81.45,46 The fluid inertial force FLi is described by the equation Fhvi2 a 4 x FLi ¼ 13:5π g ð4Þ w w2 in which F is the fluid density, and g(x/w) is given approximately by40,41 x x x x g ¼ 19:85 0:19 0:5 0:81 ð5Þ w w w w The fluid inertial lift force is linearly dependent on fluid density and is not a function of the fluid viscosity. The theory predicting the inertial lift force is known to break down for particles in close proximity to a bounding wall (due to approximations made in its derivation), and it is this near-wall region that is of most relevance to steric FFF. Experiments have demonstrated the existence of a viscosity-dependent lift force that dominates in the region close to the wall and is significant even for particles relatively far from the wall—the experiments reported below show that particles may be driven to transverse position x of more than 4.5a in the case of the higher viscosity carrier solutions. Experiments involving measurement of polystyrene standard particle retention in sedimentation FFF under a wide range of field strength, carrier flow rate conditions, and carrier viscosity allowed the extraction of an empirical equation describing the near-wall dependence of hydrodynamic lift force on particle size, fluid shear rate, fluid viscosity, and the distance between the particle and the wall. For fluid viscosities that do not greatly exceed that of water, the empirical dependence was found to be given by3942 FLw ¼ C

a3 ηs0 a3 ηÆvæ ¼ 6C wδ δ

ð6Þ

where C is an empirical dimensionless coefficient, η is the dynamic viscosity of the carrier liquid, s0 is the fluid shear rate at the channel wall, and δ is the distance between the particle surface and the accumulation wall of the channel so that δ = x a when the center of the particle is at distance x from the wall. For higher fluid viscosities, experiments suggested a stronger dependence on viscosity.42 Equilibrium Particle Position and Retention Ratio. It is simple to calculate, via eqs 4 and 5, the maximum predicted inertial lift force on a 12 μm diameter particle entrained in a fluid flow rate of 2 mL/min (corresponding to Ævæ = 0.833 cm/s for a 3344

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channel of 2 cm breadth and 200 μm thickness, as used in the experiments reported below) to be 1.46 1013 N. The gravitational force on a 12 μm diameter polystyrene particle (of density 1.05 g/mL) immersed in an aqueous solution of density 1.00 g/mL is 4.44 1013 N. A maximum inclination of 60 to the horizontal was used for the experiments reported, and at this angle the cross-channel component falls to 2.22 1013 N. The inertial lift force is therefore predicted to be insufficient to prevent the 12 μm particles from contacting the wall under any of the experimental conditions reported below. The empirical lift force corresponding to eq 6, being inversely dependent on the distance δ between the particle surface and the wall, will always be sufficient at some point to counter gravitational force and result in a particle equilibrium position above the wall, no matter what the value of C. If we assume for the moment that the inertial contribution to lift force is relatively insignificant, then the net force Fnet on a particle in the direction away from the wall is given by Fnet ¼

6Ca3 ηÆvæ FG cos θ wδ

ð7Þ

where FG is the force on the particle due to gravity (equal to (4/3)a3ΔFG, where ΔF is the difference between the particle density and fluid density and G is the acceleration due to gravity or 9.81 m/s2) and θ is the angle of the channel to the horizontal. The particle equilibrium position is given when the net force becomes zero. Rearranging eq 7 with Fnet = 0 gives the solution: δeq ¼

9 CÆvæη 2 πwΔFG cos θ

ð8Þ

and xeq = a þ δeq. Note that because of the cubic dependence of the empirical force FLw on a, the value of δeq is predicted to be independent of particle size. It was mentioned above that hydrodynamic effects influence particle velocity when the particle is entrained in a bounded fluid flow. Goldman et al.47 showed that a spherical particle entrained in fluid flow bounded by a plane wall moves at a velocity that is smaller than the undisturbed fluid velocity at the position of the particle center. The relative retardation is a function of the ratio of δ/a so that vp ¼ f ðδ=aÞvf

ð9Þ

where vp is the particle velocity and vf is the undisturbed fluid velocity at the position of the particle center. The limiting behavior for f(δ/a) as δ/a f 0 was shown39 to be given by f ðδ=aÞ ¼ 1=ð0:6600 0:2693 lnðδ=aÞÞ

ð10Þ

which is better than 0.15% accurate for δ/a < 0.005. For δ/a g 0.5, the following equation is accurate to better than 1%: 5 a 3 ð11Þ f ðδ=aÞ ¼ 1 16 x Taking into account the equilibrium position of the particles in the parabolic fluid velocity profile (which may correspond to eq 8) the particle retention ratio is then given by vp xeq xeq xeq ¼ 6f ðδeq =aÞ 1 ð12Þ R ¼ 6f ðδeq =aÞ Ævæ w w w

that is greater than a and the particle retardation function f(δeq/a). When particles are forced close to the wall, and xeq is only slightly greater than a, then the value of f(δeq/a) principally determines the value of γ. In this case, γ can be less than unity. When particles are driven away from the wall by lift forces, then f(δeq/a) approaches unity, and the value of γ approaches the ratio of xeq/a. In this case, γ can be considerably greater than unity. Influence of Fluid Viscosity on Plate Height. Here we shall again consider the effective lift force to be described by the empirical eq 6. The particles approach their equilibrium position where the net force becomes zero, and the position is assumed to correspond to that given by eq 8. We can expect the particles to undergo random fluctuations around this equilibrium position as they are carried along the channel, being subjected to the restoring force Fnet given by eq 7. If we assume that only thermal energy causes these random fluctuations, we can derive an expression for the concentration profile around the equilibrium position by taking a mathematical approach similar to that taken for hyperlayer FFF.48,49 First, we expand Fnet as a Taylor series around the equilibrium position as follows: 0

Fnet ðxÞ ¼ Fnet ðxeq Þ þ ðx xeq ÞFnet ðxeq Þ þ

ðx xeq Þ2 00 Fnet ðxeq Þ 2!

ðx xeq Þ3 000 Fnet ðxeq Þ þ 3 3 3 ð13Þ 3! Steric and hyperlayer FFF exhibit very high efficiency, and the concentration distributions must therefore be very narrow. Consequently, we may consider only the first two terms in the series and expect sufficient accuracy for predicting the concentration profile. The first differential of Fnet with respect to x, at the equilibrium position is given by þ

0

Fnet ðxeq Þ ¼

6Ca3 ηÆvæ FG cos θ ¼ δeq wδeq 2

ð14Þ

Realizing that Fnet(xeq) = 0, we obtain Fnet ðxÞ

ðx xeq Þ FG cos θ δeq

ð15Þ

The potential energy U for a particle as a function of position x is then given by Z x FG cos θ UðxÞ ¼ Fnet ðxÞ dx ¼ ðx xeq Þ2 ð16Þ 2δeq xeq The Boltzmann equation then provides the concentration profile: ! UðxÞ 2 FG cos θ cðxÞ ¼ c0 exp ¼ c0 exp ðx xeq Þ kT 2δeq kT ð17Þ in which c0 is the concentration at the equilibrium position. This equation describes a Gaussian distribution around the equilibrium position with standard deviation σx given by

The dimensionless factor γ of eq 3 is seen to be due to the influence of lift forces that result in an equilibrium position xeq

σx ¼ 3345

δeq kT FG cos θ

!1=2

¼ δeq

1=2 wkT 6Ca3 ηÆvæ

ð18Þ



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An expression for theoretical plate height H as a function of σx has been derived for hyperlayer FFF48,49 based on a random walk model. The plate height is given by 2 2m4 σx 4 dv ðxeq Þ ð19Þ H ¼ vðxeq ÞD dx

Resolution. Resolution between two neighboring Gaussian peaks is given by the standard chromatographic measure:

Rs ¼

H ¼

0

12m σ x Ævæ ½Γð1 ΓÞwf ðδeq =aÞ þ ð1 2ΓÞf ðδeq =aÞ Dw2 Γð1 ΓÞf ðδeq =aÞ 4

ð23Þ

where tr1 and tr2 are the retention times of the earlier and the following peak, respectively, and w1 and w2 are the widths of the two peaks (also in units of time) measured between the intercepts with the baseline of the tangents drawn on each side of the peaks at the points of inflection.

where m is a parameter of order 2 and D is the particle diffusion coefficient. (The random walk is assumed to carry the particle over the distance of (mσx around the equilibrium position xeq.) In the case of steric/hyperlayer FFF, the particle retardation effect close to the channel wall must be taken into account. It follows that 4

2ðtr2 tr1 Þ w2 þ w1

’ EXPERIMENTAL SECTION

2

Polystyrene Latex Beads. The samples were polystyrene (PS) latex beads having nominal diameters of 8, 12, and 20 μm obtained from Fluka AG (Steinheim, Germany) and 40 μm obtained from Duke Scientific Corporation (Palo Alto, CA, U.S.A.). They had very narrow size distributions. The batches used were calibrated by the suppliers and mean diameters and standard deviations of 8.020 ( 0.098 μm, 11.85 ( 0.123 μm, 20.28 ( 0.02 μm, and 40.25 ( 0.60 μm, respectively, were reported. The beads had a density of 1.05 g/mL. They were supplied as 1% suspensions and were diluted with the carrier solution at 1 drop/mL before injection. Gravitational FFF. The GrFFF channel used in this study was assembled in the same manner as described in a previous report.35 The channel was 200 μm thick, 2 cm wide, and 51 cm long and is shown in Figure 1. Particle suspensions were injected directly into the channel using a 50 μL syringe (Hamilton Co., Reno, NV, U.S.A.) through a rubber septum. The carrier liquid was supplied by a Young-Lin SP930D HPLC pump (Seoul, Korea). The particles in the FFF effluent were monitored by a Young-Lin M720 UV detector at the operating wavelength of 254 nm. For channel angle variation, the GrFFF channel was tilted in either up-flow or down-flow position as in a previous report.35 The channel angle was varied from 0 to 60 at 10 intervals. In all experiments, the channel flow was interrupted for 3 min following sample introduction to allow the particles to relax to the accumulation wall. The sample would be eluted on resumption of the flow. A channel flow rate of 2 mL/min was used for all of the experiments. Adjusting Viscosity of the Carrier Solutions without Changing Their Density. The viscosity of the GrFFF carrier liquid was varied by adding different amounts of (hydroxypropyl) methyl cellulose (HPMC-7509, Sigma Aldrich, St. Louis, MO, U.S.A.) to either pure water or to water with 0.1% FL-70 and 0.02% NaN3. The latter additives have been widely used in GrFFF for separation of various types of particles. Carrier liquids of six different compositions (denoted A1, A2, A3, B1, B2, B3) were prepared for this study, and their compositions

ð20Þ in which Γ represents xeq/w, and f0 (δeq/a) is the differential of the retardation function with respect to x at the position xeq. When the particles are forced close to the wall, where f(δ/a) is given by eq 10, then f0 (δeq/a) = (0.2693/δeq)(f(δeq/a))2 and eq 20 reduces to H ¼ 2 2 12m4 σ x 4 Ævæ ½0:2693Γð1 ΓÞwðf ðδeq =aÞÞ =δeq þ ð1 2ΓÞf ðδeq =aÞ Dw2 Γð1 ΓÞf ðδeq =aÞ

ð21Þ When particles are driven far from the wall, where f(δ/a) is given by eq 11, eq 20 reduces to " #2 5 R3 1 2Γ þ ð2 ΓÞ 16 Γ3 12m4 σ x 4 Ævæ ! ð22Þ H ¼ Dw2 5 R3 Γð1 ΓÞ 1 16 Γ3

Figure 1. GrFFF channel used in this study. The channel outline is cut from a Mylar spacer that is then sandwiched between two glass plates that are in turn sandwiched between two acrylic blocks. The assembly is bolted together. Holes through one of the glass plates allow for inlet and outlet tubing. The septum injector at the channel inlet is on the right.

Table 1. Solvent Systems Used in This Study for GrFFF Carrier Liquid solvent system water þ HPMC-7509

water with 0.1% FL-70 0.02% NaN3 þ HPMC-7509

carrier

% HPMC-7509

viscosity η (cp)

A1 A2

0 0.05

0.96 1.11

A3

0.07

1.33

B1 B2

0 0.05

0.70 1.23

B3

0.07

1.51

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Table 2. Retention Ratio Rexp and Plate Height Hexp Data Measured at 2 mL/min for 12 μm PS Beads in Various Carrier Liquidsa upward flow angle (deg) 0

10

20

30

40

50

60

a

carrier

Rexp

δexp (μm)

HB (cm)

downward flow Hexp (cm)

A1

0.25

3.5

0.0029

0.06

A2

0.35

7.0

0.0074

0.03

A3

0.43

10.0

0.013

0.05

B1

0.13

0.41

0.0003

0.05

B2

0.20

2.0

0.0020

0.09

B3

0.23

2.8

0.0036

0.07

Rexp

δexp (μm)

HB (cm)

Hexp (cm)

A1

0.26

3.9

0.0033

0.05

0.27

4.5

0.0035

0.05

A2

0.39

8.2

0.0090

0.04

0.35

7.4

0.0076

0.02

A3

0.46

11.0

0.015

0.11

0.44

10.7

0.014

0.05

B1

0.13

0.43

0.0003

0.05

0.13

0.46

0.0003

0.04

B2 B3

0.22 0.25

2.7 3.4

0.0028 0.0044

0.03 0.10

0.21 0.23

2.3 3.3

0.0023 0.0038

0.10 0.05

A1

0.27

4.2

0.0040

0.06

0.28

4.7

0.0043

0.06

A2 A3

0.40 0.46

8.8 11.0

0.011 0.016

0.04 0.08

0.36 0.45

8.2 11.2

0.0088 0.016

0.02 0.06

B1

0.13

0.49

0.0003

0.06

0.13

0.56

0.0003

0.05

B2

0.22

2.7

0.0031

0.04

0.21

2.5

0.0026

0.09

B3

0.26

3.9

0.0057

0.07

0.25

3.5

0.0048

0.07

A1

0.28

4.5

0.0051

0.06

0.29

5.2

0.0054

0.08

A2

0.41

9.0

0.013

0.04

0.39

8.8

0.012

0.01

A3

0.49

12.2

0.021

0.04

0.47

12.6

0.019

0.06

B1

0.14

0.56

0.0005

0.06

0.14

0.68

0.0005

0.05

B2

0.23

3.0

0.0041

0.06

0.22

2.9

0.0034

0.08

B3

0.27

4.2

0.0073

0.08

0.25

4.2

0.0061

0.04

A1

0.30

5.0

0.0074

0.07

0.30

6.0

0.0078

0.08

A2

0.46

11.0

0.020

0.03

0.40

10.4

0.016

0.03

A3

0.53

13.6

0.030

0.08

0.50

14.0

0.028

0.04

B1 B2

0.14 0.24

0.64 3.1

0.0007 0.0054

0.07 0.05

0.14 0.23

0.90 3.3

0.0007 0.0052

0.05 0.06

B3

0.27

4.2

0.0094

0.06

0.27

4.5

0.0094

0.06

A1 A2

0.32 0.52

5.8 13.2

0.012 0.035

0.08 0.03

0.33 0.44

7.8 12.6

0.013 0.027

0.10 0.02

A3

0.57

15.3

0.048

0.09

0.54

17.9

0.044

0.05

B1

0.16

0.90

0.0012

0.07

0.15

1.2

0.0012

0.07

B2

0.30

5.1

0.014

0.04

0.25

4.2

0.0085

0.06

B3

0.30

5.0

0.016

0.05

0.28

6.0

0.015

0.05

A1

0.37

7.5

0.028

0.12

0.38

4.5

0.029

0.18

A2

0.54

14.3

0.062

0.03

0.50

7.4

0.055

0.04

A3

0.64

18.5

0.093

0.04

0.63

10.7

0.090

0.06

B1

0.17

1.3

0.0028

0.07

0.17

0.46

0.0026

0.08

B2

0.31

5.3

0.024

0.06

0.27

2.3

0.018

0.03

B3

0.33

6.2

0.035

0.04

0.33

3.3

0.034

0.04

δexp are calculated from Rexp and HB via eq 24 with m set to 2.

and measured viscosities are listed in Table 1. The carrier viscosities were measured using a VIBRO SV-10 viscometer (Daegoo, Korea). The addition of up to 0.0007 g/mL of HPMC-7509 to the carrier solutions has a negligible effect on

the carrier density and even on the difference in density between the PS beads and carrier density. This is important because of the dependence of δeq on ΔF as well as η, as shown by eq 8. 3347



’ RESULTS AND DISCUSSION The experimentally determined retention ratio Rexp, corresponding δexp, and the plate height Hexp for 12 μm PS beads in

Figure 2. Variation in retention ratio R of 12 μm PS latex beads with channel angle using the different carrier solutions. The upper figure shows data for upward channel tilt, and the lower shows data for downward tilt.

ARTICLE

the six different carrier liquids at various channel angles are summarized in Table 2, and retention ratios and plate heights are plotted in Figures 2 and 3. Also listed in Table 2 are the predicted contributions to plate height due to Brownian motion HB. It can be seen in Figure 2 that, for both up- and down-flow positions, the retention ratio gradually increased with the channel angle for each of the carrier liquids. This is because the component of the gravitational force exerted on the particles across the channel thickness decreases with increasing channel angle for both up- and down-flow.35 The equilibrium distance from the wall for the particles, where the component of gravitational force across the channel thickness is balanced by HLF, therefore increases with increasing angle of the channel from the horizontal. The retention ratio predicted for ideal steric FFF is equal to 6R, or 0.18. It can be seen that only in the case of carrier solution B1, the lowest viscosity solution, do the observed Rexp fall significantly below 0.18 and γ values fall below unity. There does not appear to be a significant difference in retention ratios for upward and downward flow at any given angular departure from the horizontal for the range of angles examined. The observed retention ratios plotted against the carrier viscosity for the two solvent systems are shown in Figure S1 in the Supporting Information. In each case, the retention ratio increases with increasing carrier viscosity, indicating a viscosity dependence of HLF in agreement with eq 6. In the case of carrier solutions A, the retention ratios approach very high values as particles are driven far from the wall. The fact that the two solvent systems A and B result in very different retention ratios indicates that viscosity is not the only factor influencing lift force. The lower ionic strength carrier solutions A1, A2, and A3 give the highest retention ratios, corresponding to the highest δeq. The influence of ionic strength on retention ratios was reported for previous experiments involving sedimentation FFF.50 Neither

Figure 3. Variation of plate height H of 12 μm PS latex beads with viscosity using the different carrier solutions. The panels at the top and bottom are for solvent systems A and B, respectively. The panels on the left are for upward channel tilt and those on the right are for downward tilt. 3348



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electrostatic repulsion between the particles and the channel wall nor electrokinetic effects could provide a satisfactory explanation. Equation 20 predicts the contribution to plate height due to random Brownian fluctuations of the particles around their equilibrium position. If the diffusion coefficient is replaced by the StokesEinstein equation kT/6πηa, and σx by eq 18, then eq 20 becomes HB ¼ 2 72πam4 kTÆvæ ηδeq 2 ½Γð1 ΓÞwf 0 ðδeq =aÞ þ ð1 2ΓÞf ðδeq =aÞ w2 Γð1 ΓÞf ðδeq =aÞ ðFG cos θÞ2

ð24Þ The plate heights predicted by this equation are listed in Table 2 along with the experimentally measured values. The predicted plate heights correspond to values determined by eq 24 with m assumed to be equal to 2 and with δeq and Γ corresponding to the experimentally observed retention ratios. The predicted plate heights do not therefore depend on some assumed coefficient C in eq 6; only the dependence of lift force on the reciprocal of δ is assumed. Also, the values for f(δeq/a) and f0 (δeq/a) were determined numerically and the limiting expressions of eqs 10 and 11 were not used. Predicted plate heights calculated using eq 24 and listed in Table 2 do not account for contributions due to sample polydispersity, sample volume, extrachannel volumes, channel end and edge effects, channel nonuniformities, etc., and are not surprisingly far smaller than the observed plate heights in most cases. Many of these additional contributions to plate height are expected to be small in comparison to observed plate heights. For example, the plate height contribution due to sample injection volume (Hinj = (L/12)(Vinj/V0)2, where Vinj is the injection volume, and plug injection is assumed) should be of the order of 0.0026 cm. The polydispersity contributions are similarly small as we shall see following the discussion of selectivity below. These predicted plate heights in Table 2 are also based on the assumption that only Brownian motion disturbs the particles from their equilibrium position across the channel thickness. Brownian motion may therefore be discounted as the dominant contribution to band spreading. Other phenomena may also disturb the particles, however. Viscous interactions between the particles, of the type giving rise to shear-induced diffusion,5154 may contribute to particle self-diffusion and to enhanced particle diffusion across concentration gradients. This enhanced gradient diffusion is related to the phenomenon of viscous resuspension.55,56 These effects have been predicted to scale with sa2 where s is the local shear rate,51 and with φ2 where φ is the particle volume fraction.53,54 The enhanced self-diffusion of particles within the thin hyperlayer will tend to reduce the plate height contribution due to finite hyperlayer thickness because the rate of exchange of particle positions will be higher than predicted by Brownian motion. A mechanism involving particle trajectory swapping in wall-bounded shear flow has been proposed to contribute to selfdiffusion, even at low concentrations.57 The enhanced gradient diffusion will tend to increase the hyperlayer thickness and add to plate height, however. It is apparent that the contribution to σx2 due to Brownian motion is proportional to δeq2 when lift force is dependent on the reciprocal of δ (see eq 18). This is because as δeq decreases the walls of the potential energy well become steeper and the particles are more sharply focused. However, shear-induced gradient diffusion will tend to oppose the focusing

Figure 4. Calibration curves obtained at the flow rate of 2 mL/min with four PS latex beads (8, 12, 20, and 40 μm) using the different carrier solutions. Linear regression yielded the following selectivities Sd and squares of correlation coefficients (R2): A1, 0.60 (0.98); A2, 0.44 (0.97); A3, 0.40 (0.97); B1, 0.85 (0.99); B2, 0.71 (0.99); B3, 0.71 (0.99).

of the particles into a very thin hyperlayer. It will tend to broaden the foot of the potential energy well, which will result in a contribution to σx2. Not only will the relative effects of shearinduced gradient diffusion increase with decrease of δeq (because of the tendency for narrowing of the hyperlayer thickness), the effects are predicted to increase with the increase of shear rate as particles are driven closer to the wall. On the other hand, the disturbances due to shear-induced gradient diffusion would be predicted to decrease along the length of the channel as the particles become more dispersed, although this effect would not be apparent from measurements of overall plate height for an eluting sample zone. From the plots of experimentally measured plate heights for 12 μm PS particles versus viscosity shown in Figure 3, we see that in the solvent system A, the plate height apparently decreases and then increases as the carrier viscosity is increased, and this is observed for both upward and downward flow. For solvent A1, there is a gradual decrease in measured plate height as the channel is lowered from 60 to 30, followed by little change to 0, for both upward and downward flow. This is what we might expect with the increasing effective cross-channel field strength and the resulting decrease in δeq and reduction in σx2. However, for upward flow with solvent A3 there is no clear trend. In the solvent system B, there was no apparent minimum in plate height at intermediate viscosity. Figure 4 shows GrFFF calibration curves obtained with four PS beads (8, 12, 20, and 40 μm) at a flow rate of 2 mL/min in the different solvents with the channel placed horizontally. In the solvent system A, the slope of the calibration curve (or the sizebased selectivity, Sd) decreases as the viscosity increases. This is also true of the solvent system B although there is little difference between selectivities for solvents B2 and B3. This is as expected. If the HLF has a dependence on particle size and δ as described by eq 6, then δeq should be independent of particle size as shown by eq 8. In this case, the selectivity has been shown to correspond to the slope of a plot of log(f(δeq/a)(δeq þ a)) versus log(d) for the particular fixed value of δeq.40 This is apparent by consideration of the approximate dependence of R on δeq as given on the right-hand side of eq 12, remembering that xeq = δeq þ a. Replacing R with t0/tr, we obtain ðδeq þ aÞ t0 ¼ 6f ðδeq =aÞ tr w 3349

ð25Þ



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Table 3. Retention Times and Resolutions Measured from Fractograms Shown in Figure 5

Figure 5. GrFFF separation of a PS mixture (8, 20, and 40 μm) obtained using the different carrier solutions. The channel was horizontal, and the flow rate was 2 mL/min in each case.

Rearranging eq 25 and taking logarithms we obtain the equation ! t0w ð26Þ log tr ¼ log logðf ðδeq =aÞðδeq þ aÞÞ 6 which explains the aforementioned correspondence of the selectivity to the slope of a plot of log(f(δeq/a)(δeq þ a)) versus log(d). Selectivity is the absolute slope of a plot of log(tr) versus log(d), and since the quantity (t0w/6) is constant, this must be equal to the absolute slope of the plot of log(f(δeq/a)(δeq þ a)) versus log(d). These plots of log(f(δeq/a)(δeq þ a)) versus log(d) are slightly curved, as are the calibration plots, but the slope tends to decrease with increase of δeq. The value of δeq increases with viscosity, and therefore Sd is expected to decrease with increase of carrier viscosity. The trend continues with solvent system B. The δeq are lower than for system A, and the selectivities are higher. The selectivity is apparently related to δeq rather than viscosity; the relationship with viscosity is indirect. Therefore, making the assumption that δeq for 12 μm particles, as listed in Table 2 for each carrier solution at the channel angle of 0, applies to the other particle sizes under the same conditions, we can determine predicted selectivities from plots of log(f(δeq/a)(δeq þ a)) versus log(d) over the particle size range of 840 μm for each δeq. For the δeq of 0.41, 2.0, 2.8, 3.5, 7.0, and 10.0 μm, corresponding to carrier solutions B1, B2, B3, A1, A2, and A3, respectively, we calculate selectivities of 0.80, 0.68, 0.64, 0.60, 0.49, and 0.42. These agree very well with the observed selectivities of 0.85, 0.71, 0.71, 0.60, 0.44, and 0.40, respectively. The contribution to apparent plate height due to the polydispersity of the PS standards may be calculated using the measured selectivities. This contribution Hp is given by Hp = LSd2(σd/d)2, where σd is the standard deviation in particle diameter d. For example, the 12 μm particles should contribute between 0.0009 and 0.0039 cm to plate height, depending on the carrier solution, for a horizontal channel. Other particle sizes should have very similar contributions as they have very similar polydispersities. This is small in comparison to the measured plate heights listed in Table 2, but of the same order as the predicted Brownian contributions. Also, as the channel is tilted with respect to the horizontal, δeq tend to increase and

carrier

tr, 8 min

tr, 20 min

tr, 40 min

Rs 20, 40

Rs 8, 20

A1

4.50

2.86

1.73

3.6

1.9

A2 A3

3.03 2.53

2.26 1.93

1.50 1.33

4.9 5.0

3.1 2.9

B1

8.51

4.44

2.31

6.0

3.1

B2

5.83

3.36

1.90

6.1

4.4

B3

5.56

3.16

1.76

4.6

4.8

selectivities would be expected to decrease. This would reduce the polydispersity contributions to plate height. Figure 5 shows separations of a mixture of three PS beads (8, 20, and 40 μm) obtained using the different carrier liquids with the channel in the horizontal position. The resolutions for consecutive pairs of peaks determined for the fractograms are summarized in Table 3. It can be seen that, as the carrier viscosity increases, the retention times of course decrease for both solvent systems A and B, consistent with increased HLF. The important observation is that analysis times decrease without loss of resolution (see Table 3). In fact, the resolution of the latereluting peaks increases with the addition of HPMC because the peak symmetry improves. This improvement in symmetry occurs with the lifting of the particles away from the channel wall. We would expect the Brownian contribution to plate height to increase with viscosity (see values of HB in Table 2), but the Brownian contribution is apparently not the major contribution to plate height. The polydispersity contribution to plate height, of course, has no influence on resolution as selectivity changes. The increase of viscosity by 40% for system A and 120% for system B results in a 40% and 45% reduction in analysis time, respectively, with improved resolution.

’ CONCLUSIONS Results indicate that carrier viscosity is an important variable that may be exploited in GrFFF to reduce analysis times. The carrier viscosity strongly affects particle retention through its influence on HLF. Increase of carrier viscosity increases hydrodynamic lift, causing particles to elute more quickly. The zone broadening is apparently not strongly influenced by carrier viscosity so that the analysis times may be reduced without loss of resolution. Resolution of more strongly retained particles may even improve with increase of carrier viscosity as they are driven further from the channel wall. For the experiments reported, the contributions to plate height due to sample volume, sample polydispersity, and Brownian motion have been shown to be small compared to the measured plate heights. The overall plate height in steric/ hyperlayer FFF is the result of many different contributions. The work reported here suggests that efficiencies may be improved by reducing some of these contributions. Extrachannel volumes must be minimized, and the channel must be fabricated as carefully as possible to reduce variations in flow velocity across channel breadth. If shear-induced diffusion contributes to band spreading then these effects may be reduced by reducing sample concentration, although this is limited by detector sensitivity. This work shows that we can expect further improvements in separation for this simple technique of GrFFF in the future. 3350



’ ASSOCIATED CONTENT

bS

Supporting Information. Additional information as noted in text. This material is available free of charge via the Internet at http://pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*Phone: þ12164441217. Fax: þ12164449198. E-mail: willias3@ ccf.org.

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Effect of Carrier Fluid Viscosity on Retention Time and Resolution in

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