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Retention behavior of colloidal dispersions in sedimentation field-flow fractionation
Advances
in Colloidand ZnterfaceScience,53 (1994) 129-140
129
Elsevier Science B.V. 00220 A
RETENTION BEHAVIOR OF COLLOIDAL DISPERSIONS SEDIMENTATION FIELD-FLOW FRACTIONATION YASUSHIGE
IN
MORI
Department of Chemical Engineering and Materials Science, Doshisha University, Tanabe, Kyoto 610-03, Japan
CONTENTS Abstract ........................ 1. Introduction ..................... 2. Theory ........................ 2.1 General principle ................ 2.2 Transport equation of particles in FFF .... 2.3 Retention theory of FFF ............ 2.4 Retention equation presented by Giddings . . 2.5 Particle velocity in z-direction ......... 2.6 Particle-wall interaction ............ 3. Comparison with experimental data ........ 3.1 Using aqueous solutions as the carrier .... 3.2 Using methanol solutions as the carrier .... 4. Conclusions ...................... References ........................
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . , . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
129 130 130 130 131 133 133 134 135 137 137 138 140 140
ABSTRACT Sedimentation field-flow fractionation (SdFFF) has a high resolution over a wide range of particle size compared with other methods of size determination of colloidal dispersions. However, the retention behavior in SdFFF depends strongly on the experimental parameters, especially the ionic strength of the carrier solution. The sizes calculated from the results of the experiment by using the theory of Giddings are underestimated if the carrier solution with low surfactant concentration is used, compared with those determined by other techniques. To explain this phenomenon, this review discusses theoretically the perturbation behavior in the retention by the particlewall interaction due to electrostatic repulsive and van der Waals’ attractive forces, and by the contribution of particle motion in shear flow. The influence of particle-wall interaction was not significant at low sample concentrations below 0.1% and at high ionic strengths of the carrier solution above 10m3 M. However, under low ionic strength conditions of the carrier solution, the concentration profiles in the channel spread widely, and the estimated retention time decreases from the Giddings’ theory. The effect of particle motion in shear flow appears OOOl-8686/94/$26.00
0 1994 - Elsevier Science B.V. All rights reserved.
SSDZ OOOl-8686(94)00220-7
130 at the measurement of large particle with high ionic concentration solution, and the retention ratio will become smaller. But this effect is not so large compared with particle-wall interaction.
I. INTRODUCTION
Field-flow fractionation (FFF) is classified as a one-phase chromatography technique in which an externally adjusted force field is applied to the suspended particles under motion in a channel [1-3]. Sedimentation FFF (SdFFF) is one of the sub-techniques in FFF that uses centrifugal force field as the external field. It is suitable for the characterization and fractionation of colloidal particles in the submicron range. The theoretical basis of SdFFF was originally presented by Giddings and his co-workers [4-6], and the established equation requires the densities of particles and the carrier solution only in addition to the experimental operation conditions, such as flow rate, external field strength, temperature, and channel dimensions. Recently, perturbations of the retention time of particles from Giddings' theory were reported, in which the particle-wall interaction, particle-particle interaction, and the overloading of the sample were taken into consideration [7-11]. In this review, the experimental results on the retention behavior are reported using various carrier solutions. The effect of particle-wail interaction including electrostatic and van der Waals' forces and the contribution of particle motion in shear flow are discussed and compared with the experimental results.
2. THEORY
2.1 General Principle Figure 1 shows the outline of the separation principle of F FF and gives an overview of the experimental equipment. The separation occurs inside a narrow channel, in which the carrier solution flows as a laminar flow between the smooth and parallel walls. When an external field is applied in the direction perpendicular to the flow, i.e. x axis, the particles are concentrated against the accumulation wall. The formed concentration
131
carrier solution flow I stop external field
)
I
time
flow
analysis period
heavy particles
0 light particles high velocity
accumulation wall Fig. 1. Schematic diagram of field-flow fractionation
and the principle of separation.
gradient induces diffusion in the reverse direction. After a certain time, the steady state concentration profile is reached which depends on the particle mass. Light particles are distributed near the center of the channel compared with heavy particles. The average transport speed of light particles, therefore, is higher than that of heavy particles, because of the parabolic flow profile in the channel. The size analysis by FFF is normally going as the three stages. The first stage is the injection period, in which the sample is brought into the inlet of the channel from the injector under the external field. The relaxation period starts to form the steady-state concentration profile, after stopping the flow of the carrier solution. At the end of the relaxation period, the carrier solution flows again. Particles move towards the channel by each velocities under the external field, in the analysis period. 2.2 Transport Equation of Particles in FFF The motion of particles in the channel of FFF is determined by the combined action of the flow, the applied field, and the Brownian migration. The flux of particles can be written as a vector sum of the x axis component, J,, in the direction of the induced external field, which is
I
(
132
perpendicular to the z axis, the direction of fluid flow, and the z axis component, J,.
(1)
Jx=-Dp$+Uc
(2)
Here, c is the particle concentration at the local point in the channel. U and vP is the particle velocity in the LX-and z-direction in the channel, respectively. Dp is the diffusion coefficient of particles whose diameters are dp, and is expressed as follows by the use of the Stokes-Einstein relationship Dp=& P
(4)
where k is the Boltzmann constant and T is the absolute temperature. The expression of U is derived by the balance of the viscous resistance force of Stokes’ law and the external force, F,,, F l-J=---- ext 3w$
(5)
In the operation of FFF, the concentration gradient in the x direction can safely be assumed to be at steady state or quasi-equilibrium condition in Eq. (2). Also, the first term in Eq. (3) is usually much smaller than the second one. We then obtain
J, = vpc
(7)
133
2.3 Retention
Theory of FFF
The retention ratio R of particles is defined as the ratio of the elution time for the particles, tP, to that for a non-retained peak, t,. R = t&,
(8)
Since the average velocity of the particle, VP,and that of fluid, Um, are inversely proportional to tP and to, respectively, the ratio R can also be written as
(9)
2.4 Retention Equation Presented by Giddings Giddings assumed that a particle moves with the same velocity as the fluid at the center of the particle. Then vP in Eq. (9) is replaced by the flow velocity of the carrier solution, u. vp=u=
(10)
Furthermore, the centrifugal force, F,, was only considered as the external force in SdFFF. Fext = F, = - xApd; G/ 6
(11)
Here,Gis the field strength and Apisthedensitydifferencebetween particleandcarriersolutionlnthiscase,theconcentrationprofileC(x), andtheretentionratio,R,wereobtainedasfollows[5].
R = 6a(l-
a) + 6h(l-
2a) coth [ [%)-i%]
(13)
134
where co is the particle concentration at the distance radius, a, from the accumulation wall (X = 0).
of the particle
a = dJ(2w) = alw
(14)
h = (6k T, I (7cApd; Gw)
(15)
2.5 Particle Velocity in z-Direction First assumption of Giddings’ will be checked by using the reports about the motion of particles in a shear flow near a planar wall. The plotted points in Fig. 2 show the ratio of the particle velocity and the flow velocity at the center of the particle if the flow velocity is not disturbed by the particle, which were exactly calculated by Goldman et al. WI. They also presented approximate formula near the wall, as follows
(16)
0.8
-
10-6
0 calculated data by Goldman et al. [12]
0.01
10-4
1
10
(x-alla Fig. 2. The ratio of particle velocity and fluid velocity as a function of the distance of the particle from accumulation wall. The curve is calculated by Eqs. (16) and (17).
135
When the gap width between the particle surface and the wall is much smaller than the particle radius, the estimated equation is
s= U
P2
0.6376 - 0.2 ln((x -a) la)
(17)
The two approximate equations should be switched somewhere in order to calculate the particle velocity in the whole channel. The switched point was chosen at x/u = 1.14, and the values of p1 and p2 were taken as 0.32 and 0.809, respectively, instead of 0.3125 and 0.7431, which were reported by Goldman et al. [12]. The solid line in Fig. 2 is the calculated one with the above values, which is in good agreement with the exact values. We can, therefore, use these approximate equations, that is the slip factor of the particle velocity, to calculate the particle velocity in the channel. The lift force on a particle, which acts in the shear flow towards the perpendicular direction of the flow, i.e. x-direction, should also be considered, but the centrifugal force always works in the channel of SdFFF, and is usually much larger than the lift force. This lift force could safely be neglected. 2.6 Particle-Wall
Interaction
Interaction among materials would be significant when the distance between them approaches micrometer order. In the case of SdFFF, the particles are concentrated near the accumulation wall. The force due to this interaction should be considered in addition to the external force. The total potential energy of interaction between a particle sphere and a planar wall, VPw,is obtained as the sum of that of London-van der Waals attraction, V&lu, and of electrostatic repulsion, VRpw 113,141.The potential energy due to London-van der Waals attractive force for particle-wall interaction is
(18)
where A1s3 is the effective Hamaker constant between the particle and the wall material in the carrier solution.
136
The expression of electrostatic interaction is known only as approximate equations at the limited value of KU where UK refers to the DebyeHiickel electrical double layer thickness for Ku >> 1 VR,pw =
for
64rc&q,k27%$‘S’w
KCI <<
VR,pw =
a
Z2 e20
In [ 1 + exp(-lc(X - a))]
(19)
1
1287~E E. k2 p ‘up yw ~
(x +a)
Z2 ei
where ‘yi = tanh
ax
=p
ln 1 + : exp(-lc(x - a))
1 (20)
(particle), w (wall), and
e. is the charge of an electron, ~~ is the permittivity of the vacuum, E is the dielectric constant of the carrier solution, pi is the surface potential, NA is Avogadro’s constant, 2 is the valence of the ion in the carrier solution, and I is the ionic strength of the carrier solution. The particle-wall interaction force, Fpw, can be calculated by differentiating Vpw with respect to x.
Fpw=-
avpw &
(21)
Figure 3 illustrates the concentration profiles obtained from Eq. (6) considering the particle-wall interaction as an extra force, for polystyrene latex (dp = 506 nm, pp = 1054 kg/m3). The profile at high ionic strength of the carrier solution is approximately similar to that calculated by Eq. 12 (line S). But as the ionic strength decreases, the profile spreads widely, and the position of average concentration along the x axis deviates from that calculated by Eq. (12). The slip factor of the particle velocity is also plotted in Fig. 3, which affects largely when the distance from the
137
60 dp = 506 nm I = 10-3 M
\
10
cc> =O.lwt. %, A= 2.05 x 10-3
/
\I=10-6M
-I
0
0.7
0.6 0
0.5
1
1.5
2
2.5
3
x / pm Fig. 3. Local concentration profile of 506 nm polystyrene latex particles by Eq. (12) (line S) and with particle-wall interaction, and the ratio of particle velocity and fluid velocity. $, = -80 mV, up, = -10 mV, A,,, = 5.6 x 1O-21J, 2 =l, T = 293 K.
accumulation wall is smaller than the particle diameter. The effect of slower particle velocity compared with fluid velocity will increase at higher ionic strength condition. In the case of larger particle diameter, or higher external field strength, this effect must be considered. 3. COMPARISON
WITH
EXPERIMENTAL
DATA
3.1 Using Aqueous Solutions as the Carrier Figure 4 shows the comparison of the observed retention ratios and the calculated ones. The experiments were carried out using polystyrene latex whose diameter ranged from 200 to 1000 nm. Two different solutions were used as the carrier solution, that is distilled water and 0.1 mM sodium dodecylbenzene sulfonate (SDS) anionic surfactant solution. The ionic strength of 0.1 mM SDS solution was determined to be 1.05 x 1OA M from the concentration of SDS and from the measurement of the hydrogen ion concentration. The ionic strength of distilled water, 4 x lo4 M, was the value of the measured hydrogen ion concentration, although the accurate measurement of ion concentration in this region is difficult due to the absorption of the carbon dioxide into water.
138
0.1
1/ h Fig. 4. Comparison of the observed and the calculated retention ratios with aqueous solution as the carrier. $, = -54 mV, (I, = -24 mV, Alz3 = 2.5 x 1O-2oJ, 2 = 1, T = 295 K.
In the case of SDS solution, fair agreement between the experimental results and the calculated ones by Eq. (13) (line S) is recognized, at low values of l/h. But the observed data with distilled water deviate significantly from line S. However, the calculated retention ratios including all interactions (the solid lines in Fig. 4) were in good agreement with the experimental values. At low ionic strength, the effect of the slip factor is negligible because particles can never be concentrated near the accumulation wall due to the high repulsive electrostatic force. In the case of high ionic strength and large particle diameter, this effect appears. The calculated line considering this effect agrees better with the experimental data, although this effect is not so large. 3.2 Using Methanol
Solutions
as the Carrier
Figure 5 shows the experimental results when methanol solutions were used as the carrier solution. Equation (13) does not predict the observed data using pure methanol as the carrier solution. When metha-
139
10
100
1000 I/
10000
h
Fig. 5. Comparison of the observed and the calculated retention ratios with methanol solution as the carrier. $, = -77 mV, Q, = -24 mV, AIz3 = 4.0 x 10-20 J, Z =l, T = 295 K.
no1 with Aerosol OT (AOT) surfactant was used, the retention ratios approached the values predicted by Eq. (13), in a similar fashion to the case of the water system using SDS solution. These discrepancies were tried again to be explained with the particle-wall interaction, and the slip factor of the particle velocity. In the case of pure methanol, the ionic strength should be taken to be about low5 M, to fit the observed data of smaller size latex with the measured 5 potential of polystyrene latex in methanol. The measured value of water content in methanol was about 50 mM. As this value is 5000 times higher than the ionic strength, the ionic strength used in the calculation might be reasonable if water is assumed to be dissociated slightly into methanol. The distribution ofwater in the particle-methanol system may also be one possibility to account for this result. As the surface of the particle is hydrophilic, water tends to be absorbed on the particle surface, and the ion concentration near the particle surface, therefore, might become higher compared with that in the bulk of methanol.
140
For the AOT solution, the retention ratios agreed with the calculated values with all interactions, when the ionic strength of AOT solution was taken to be the same value of AOT concentration. The calculated values by Eq. (13) were also in good agreement with the experimental data. It is interesting to note that Eq. (13) could be used to predict the retention ratio at some conditions of the carrier solution. 4.
CONCLUSIONS
At low ionic strength in the carrier solution, the retention of SdFFF deviates from that predicted by the theory presented by Giddings, resulting in the underestimation of the particle size. In this case, the retention ratio can better be estimated by considering particle-wall interaction. The particle velocity is smaller than the fluid velocity near the wall. This effect appears at measurement of large particle with high ionic concentration solution, and the retention ratio will become smaller. But this effect is not so large compared with particle-wall interaction. When methanol solution was used as the carrier, the behavior of the retention ratio is apparently the same as when the aqueous solution was used. The observed retention ratio could be estimated by the particle-wall interaction, when the proper ionic strength of the carrier solution is chosen. REFERENCES 1
10 11 12 13 14
K.D. Caldwell, in: H.G. Barth (Ed.), Modern Methods of Particle Size Analysis. John Wiley & Sons Inc., 1984,211 pp. J.C. Giddings, Sep. Sci., 1 (1966) 123. J. Janca, Field-Flow Fractionation. Marcel Dekker, New York, 1987. J.C. Giddings, J. Chem. Phys., 49 (1968) 81. J.C. Giddings, Sep. Sci. Technol., 13 (1978) 241. M.E. Hovingh, G.H. Thompson and J.C. Giddings, Anal. Chem., 42 (1970) 195. Y. Mori, K. Kimura and M. Tanigaki, Anal. Chem., 62 (1990) 2668. Y. Mori, B. Scarlett and H.G. Merkus, J. Chromatogr., 515 (1990) 27. T. Hoshino, M. Suzuki, K. Ysukawa and M. Takeuchi, J. Chromatogr., 400 (1987) 361. M.E. Hansen and J.C. Giddings, Anal. Chem., 61(1989) 811. M.E. Hansen, J.C. Giddings and R. Beckett, J. Colloid Interface Sci., 132 (1989) 300. A.J. Goldman, R.G. Cox and H. Brenner, Chem. Eng. Sci., 22 (1967) 653. Y. Kitahara and A. Watanabe, Kaimen Denki Gensyou, Kyouritu Shuppan, Tokyo, 1972. E.J.W. Verwey and J.T.G. Overbeek, Theory of the Stability of Lyophobic Colloid. Elsevier, Amsterdam, 1948.
in Colloidand ZnterfaceScience,53 (1994) 129-140
129
Elsevier Science B.V. 00220 A
RETENTION BEHAVIOR OF COLLOIDAL DISPERSIONS SEDIMENTATION FIELD-FLOW FRACTIONATION YASUSHIGE
IN
MORI
Department of Chemical Engineering and Materials Science, Doshisha University, Tanabe, Kyoto 610-03, Japan
CONTENTS Abstract ........................ 1. Introduction ..................... 2. Theory ........................ 2.1 General principle ................ 2.2 Transport equation of particles in FFF .... 2.3 Retention theory of FFF ............ 2.4 Retention equation presented by Giddings . . 2.5 Particle velocity in z-direction ......... 2.6 Particle-wall interaction ............ 3. Comparison with experimental data ........ 3.1 Using aqueous solutions as the carrier .... 3.2 Using methanol solutions as the carrier .... 4. Conclusions ...................... References ........................
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . , . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
129 130 130 130 131 133 133 134 135 137 137 138 140 140
ABSTRACT Sedimentation field-flow fractionation (SdFFF) has a high resolution over a wide range of particle size compared with other methods of size determination of colloidal dispersions. However, the retention behavior in SdFFF depends strongly on the experimental parameters, especially the ionic strength of the carrier solution. The sizes calculated from the results of the experiment by using the theory of Giddings are underestimated if the carrier solution with low surfactant concentration is used, compared with those determined by other techniques. To explain this phenomenon, this review discusses theoretically the perturbation behavior in the retention by the particlewall interaction due to electrostatic repulsive and van der Waals’ attractive forces, and by the contribution of particle motion in shear flow. The influence of particle-wall interaction was not significant at low sample concentrations below 0.1% and at high ionic strengths of the carrier solution above 10m3 M. However, under low ionic strength conditions of the carrier solution, the concentration profiles in the channel spread widely, and the estimated retention time decreases from the Giddings’ theory. The effect of particle motion in shear flow appears OOOl-8686/94/$26.00
0 1994 - Elsevier Science B.V. All rights reserved.
SSDZ OOOl-8686(94)00220-7
130 at the measurement of large particle with high ionic concentration solution, and the retention ratio will become smaller. But this effect is not so large compared with particle-wall interaction.
I. INTRODUCTION
Field-flow fractionation (FFF) is classified as a one-phase chromatography technique in which an externally adjusted force field is applied to the suspended particles under motion in a channel [1-3]. Sedimentation FFF (SdFFF) is one of the sub-techniques in FFF that uses centrifugal force field as the external field. It is suitable for the characterization and fractionation of colloidal particles in the submicron range. The theoretical basis of SdFFF was originally presented by Giddings and his co-workers [4-6], and the established equation requires the densities of particles and the carrier solution only in addition to the experimental operation conditions, such as flow rate, external field strength, temperature, and channel dimensions. Recently, perturbations of the retention time of particles from Giddings' theory were reported, in which the particle-wall interaction, particle-particle interaction, and the overloading of the sample were taken into consideration [7-11]. In this review, the experimental results on the retention behavior are reported using various carrier solutions. The effect of particle-wail interaction including electrostatic and van der Waals' forces and the contribution of particle motion in shear flow are discussed and compared with the experimental results.
2. THEORY
2.1 General Principle Figure 1 shows the outline of the separation principle of F FF and gives an overview of the experimental equipment. The separation occurs inside a narrow channel, in which the carrier solution flows as a laminar flow between the smooth and parallel walls. When an external field is applied in the direction perpendicular to the flow, i.e. x axis, the particles are concentrated against the accumulation wall. The formed concentration
131
carrier solution flow I stop external field
)
I
time
flow
analysis period
heavy particles
0 light particles high velocity
accumulation wall Fig. 1. Schematic diagram of field-flow fractionation
and the principle of separation.
gradient induces diffusion in the reverse direction. After a certain time, the steady state concentration profile is reached which depends on the particle mass. Light particles are distributed near the center of the channel compared with heavy particles. The average transport speed of light particles, therefore, is higher than that of heavy particles, because of the parabolic flow profile in the channel. The size analysis by FFF is normally going as the three stages. The first stage is the injection period, in which the sample is brought into the inlet of the channel from the injector under the external field. The relaxation period starts to form the steady-state concentration profile, after stopping the flow of the carrier solution. At the end of the relaxation period, the carrier solution flows again. Particles move towards the channel by each velocities under the external field, in the analysis period. 2.2 Transport Equation of Particles in FFF The motion of particles in the channel of FFF is determined by the combined action of the flow, the applied field, and the Brownian migration. The flux of particles can be written as a vector sum of the x axis component, J,, in the direction of the induced external field, which is
I
(
132
perpendicular to the z axis, the direction of fluid flow, and the z axis component, J,.
(1)
Jx=-Dp$+Uc
(2)
Here, c is the particle concentration at the local point in the channel. U and vP is the particle velocity in the LX-and z-direction in the channel, respectively. Dp is the diffusion coefficient of particles whose diameters are dp, and is expressed as follows by the use of the Stokes-Einstein relationship Dp=& P
(4)
where k is the Boltzmann constant and T is the absolute temperature. The expression of U is derived by the balance of the viscous resistance force of Stokes’ law and the external force, F,,, F l-J=---- ext 3w$
(5)
In the operation of FFF, the concentration gradient in the x direction can safely be assumed to be at steady state or quasi-equilibrium condition in Eq. (2). Also, the first term in Eq. (3) is usually much smaller than the second one. We then obtain
J, = vpc
(7)
133
2.3 Retention
Theory of FFF
The retention ratio R of particles is defined as the ratio of the elution time for the particles, tP, to that for a non-retained peak, t,. R = t&,
(8)
Since the average velocity of the particle, VP,and that of fluid, Um, are inversely proportional to tP and to, respectively, the ratio R can also be written as
(9)
2.4 Retention Equation Presented by Giddings Giddings assumed that a particle moves with the same velocity as the fluid at the center of the particle. Then vP in Eq. (9) is replaced by the flow velocity of the carrier solution, u. vp=u=
(10)
Furthermore, the centrifugal force, F,, was only considered as the external force in SdFFF. Fext = F, = - xApd; G/ 6
(11)
Here,Gis the field strength and Apisthedensitydifferencebetween particleandcarriersolutionlnthiscase,theconcentrationprofileC(x), andtheretentionratio,R,wereobtainedasfollows[5].
R = 6a(l-
a) + 6h(l-
2a) coth [ [%)-i%]
(13)
134
where co is the particle concentration at the distance radius, a, from the accumulation wall (X = 0).
of the particle
a = dJ(2w) = alw
(14)
h = (6k T, I (7cApd; Gw)
(15)
2.5 Particle Velocity in z-Direction First assumption of Giddings’ will be checked by using the reports about the motion of particles in a shear flow near a planar wall. The plotted points in Fig. 2 show the ratio of the particle velocity and the flow velocity at the center of the particle if the flow velocity is not disturbed by the particle, which were exactly calculated by Goldman et al. WI. They also presented approximate formula near the wall, as follows
(16)
0.8
-
10-6
0 calculated data by Goldman et al. [12]
0.01
10-4
1
10
(x-alla Fig. 2. The ratio of particle velocity and fluid velocity as a function of the distance of the particle from accumulation wall. The curve is calculated by Eqs. (16) and (17).
135
When the gap width between the particle surface and the wall is much smaller than the particle radius, the estimated equation is
s= U
P2
0.6376 - 0.2 ln((x -a) la)
(17)
The two approximate equations should be switched somewhere in order to calculate the particle velocity in the whole channel. The switched point was chosen at x/u = 1.14, and the values of p1 and p2 were taken as 0.32 and 0.809, respectively, instead of 0.3125 and 0.7431, which were reported by Goldman et al. [12]. The solid line in Fig. 2 is the calculated one with the above values, which is in good agreement with the exact values. We can, therefore, use these approximate equations, that is the slip factor of the particle velocity, to calculate the particle velocity in the channel. The lift force on a particle, which acts in the shear flow towards the perpendicular direction of the flow, i.e. x-direction, should also be considered, but the centrifugal force always works in the channel of SdFFF, and is usually much larger than the lift force. This lift force could safely be neglected. 2.6 Particle-Wall
Interaction
Interaction among materials would be significant when the distance between them approaches micrometer order. In the case of SdFFF, the particles are concentrated near the accumulation wall. The force due to this interaction should be considered in addition to the external force. The total potential energy of interaction between a particle sphere and a planar wall, VPw,is obtained as the sum of that of London-van der Waals attraction, V&lu, and of electrostatic repulsion, VRpw 113,141.The potential energy due to London-van der Waals attractive force for particle-wall interaction is
(18)
where A1s3 is the effective Hamaker constant between the particle and the wall material in the carrier solution.
136
The expression of electrostatic interaction is known only as approximate equations at the limited value of KU where UK refers to the DebyeHiickel electrical double layer thickness for Ku >> 1 VR,pw =
for
64rc&q,k27%$‘S’w
KCI <<
VR,pw =
a
Z2 e20
In [ 1 + exp(-lc(X - a))]
(19)
1
1287~E E. k2 p ‘up yw ~
(x +a)
Z2 ei
where ‘yi = tanh
ax
=p
ln 1 + : exp(-lc(x - a))
1 (20)
(particle), w (wall), and
e. is the charge of an electron, ~~ is the permittivity of the vacuum, E is the dielectric constant of the carrier solution, pi is the surface potential, NA is Avogadro’s constant, 2 is the valence of the ion in the carrier solution, and I is the ionic strength of the carrier solution. The particle-wall interaction force, Fpw, can be calculated by differentiating Vpw with respect to x.
Fpw=-
avpw &
(21)
Figure 3 illustrates the concentration profiles obtained from Eq. (6) considering the particle-wall interaction as an extra force, for polystyrene latex (dp = 506 nm, pp = 1054 kg/m3). The profile at high ionic strength of the carrier solution is approximately similar to that calculated by Eq. 12 (line S). But as the ionic strength decreases, the profile spreads widely, and the position of average concentration along the x axis deviates from that calculated by Eq. (12). The slip factor of the particle velocity is also plotted in Fig. 3, which affects largely when the distance from the
137
60 dp = 506 nm I = 10-3 M
\
10
cc> =O.lwt. %, A= 2.05 x 10-3
/
\I=10-6M
-I
0
0.7
0.6 0
0.5
1
1.5
2
2.5
3
x / pm Fig. 3. Local concentration profile of 506 nm polystyrene latex particles by Eq. (12) (line S) and with particle-wall interaction, and the ratio of particle velocity and fluid velocity. $, = -80 mV, up, = -10 mV, A,,, = 5.6 x 1O-21J, 2 =l, T = 293 K.
accumulation wall is smaller than the particle diameter. The effect of slower particle velocity compared with fluid velocity will increase at higher ionic strength condition. In the case of larger particle diameter, or higher external field strength, this effect must be considered. 3. COMPARISON
WITH
EXPERIMENTAL
DATA
3.1 Using Aqueous Solutions as the Carrier Figure 4 shows the comparison of the observed retention ratios and the calculated ones. The experiments were carried out using polystyrene latex whose diameter ranged from 200 to 1000 nm. Two different solutions were used as the carrier solution, that is distilled water and 0.1 mM sodium dodecylbenzene sulfonate (SDS) anionic surfactant solution. The ionic strength of 0.1 mM SDS solution was determined to be 1.05 x 1OA M from the concentration of SDS and from the measurement of the hydrogen ion concentration. The ionic strength of distilled water, 4 x lo4 M, was the value of the measured hydrogen ion concentration, although the accurate measurement of ion concentration in this region is difficult due to the absorption of the carbon dioxide into water.
138
0.1
1/ h Fig. 4. Comparison of the observed and the calculated retention ratios with aqueous solution as the carrier. $, = -54 mV, (I, = -24 mV, Alz3 = 2.5 x 1O-2oJ, 2 = 1, T = 295 K.
In the case of SDS solution, fair agreement between the experimental results and the calculated ones by Eq. (13) (line S) is recognized, at low values of l/h. But the observed data with distilled water deviate significantly from line S. However, the calculated retention ratios including all interactions (the solid lines in Fig. 4) were in good agreement with the experimental values. At low ionic strength, the effect of the slip factor is negligible because particles can never be concentrated near the accumulation wall due to the high repulsive electrostatic force. In the case of high ionic strength and large particle diameter, this effect appears. The calculated line considering this effect agrees better with the experimental data, although this effect is not so large. 3.2 Using Methanol
Solutions
as the Carrier
Figure 5 shows the experimental results when methanol solutions were used as the carrier solution. Equation (13) does not predict the observed data using pure methanol as the carrier solution. When metha-
139
10
100
1000 I/
10000
h
Fig. 5. Comparison of the observed and the calculated retention ratios with methanol solution as the carrier. $, = -77 mV, Q, = -24 mV, AIz3 = 4.0 x 10-20 J, Z =l, T = 295 K.
no1 with Aerosol OT (AOT) surfactant was used, the retention ratios approached the values predicted by Eq. (13), in a similar fashion to the case of the water system using SDS solution. These discrepancies were tried again to be explained with the particle-wall interaction, and the slip factor of the particle velocity. In the case of pure methanol, the ionic strength should be taken to be about low5 M, to fit the observed data of smaller size latex with the measured 5 potential of polystyrene latex in methanol. The measured value of water content in methanol was about 50 mM. As this value is 5000 times higher than the ionic strength, the ionic strength used in the calculation might be reasonable if water is assumed to be dissociated slightly into methanol. The distribution ofwater in the particle-methanol system may also be one possibility to account for this result. As the surface of the particle is hydrophilic, water tends to be absorbed on the particle surface, and the ion concentration near the particle surface, therefore, might become higher compared with that in the bulk of methanol.
140
For the AOT solution, the retention ratios agreed with the calculated values with all interactions, when the ionic strength of AOT solution was taken to be the same value of AOT concentration. The calculated values by Eq. (13) were also in good agreement with the experimental data. It is interesting to note that Eq. (13) could be used to predict the retention ratio at some conditions of the carrier solution. 4.
CONCLUSIONS
At low ionic strength in the carrier solution, the retention of SdFFF deviates from that predicted by the theory presented by Giddings, resulting in the underestimation of the particle size. In this case, the retention ratio can better be estimated by considering particle-wall interaction. The particle velocity is smaller than the fluid velocity near the wall. This effect appears at measurement of large particle with high ionic concentration solution, and the retention ratio will become smaller. But this effect is not so large compared with particle-wall interaction. When methanol solution was used as the carrier, the behavior of the retention ratio is apparently the same as when the aqueous solution was used. The observed retention ratio could be estimated by the particle-wall interaction, when the proper ionic strength of the carrier solution is chosen. REFERENCES 1
10 11 12 13 14
K.D. Caldwell, in: H.G. Barth (Ed.), Modern Methods of Particle Size Analysis. John Wiley & Sons Inc., 1984,211 pp. J.C. Giddings, Sep. Sci., 1 (1966) 123. J. Janca, Field-Flow Fractionation. Marcel Dekker, New York, 1987. J.C. Giddings, J. Chem. Phys., 49 (1968) 81. J.C. Giddings, Sep. Sci. Technol., 13 (1978) 241. M.E. Hovingh, G.H. Thompson and J.C. Giddings, Anal. Chem., 42 (1970) 195. Y. Mori, K. Kimura and M. Tanigaki, Anal. Chem., 62 (1990) 2668. Y. Mori, B. Scarlett and H.G. Merkus, J. Chromatogr., 515 (1990) 27. T. Hoshino, M. Suzuki, K. Ysukawa and M. Takeuchi, J. Chromatogr., 400 (1987) 361. M.E. Hansen and J.C. Giddings, Anal. Chem., 61(1989) 811. M.E. Hansen, J.C. Giddings and R. Beckett, J. Colloid Interface Sci., 132 (1989) 300. A.J. Goldman, R.G. Cox and H. Brenner, Chem. Eng. Sci., 22 (1967) 653. Y. Kitahara and A. Watanabe, Kaimen Denki Gensyou, Kyouritu Shuppan, Tokyo, 1972. E.J.W. Verwey and J.T.G. Overbeek, Theory of the Stability of Lyophobic Colloid. Elsevier, Amsterdam, 1948.