Particle Size Distribution II - American Chemical Society

Chapter 17

Efficiency of Particle Separation in Capillary Hydrodynamic Fractionation (CHDF) 1

2

2

Downloaded by NORTH CAROLINA STATE UNIV on August 2, 2012 | http://pubs.acs.org Publication Date: September 24, 1991 | doi: 10.1021/bk-1991-0472.ch017

J. G. DosRamos, R. D. Jenkins, and C. A. Silebi 1

Matec Applied Sciences, Hopkinton, MA 01748 Department of Chemical Engineering and Emulsion Polymers Institute, Lehigh University, Bethlehem, PA 18015-3590

2

This paper presents a fundamental analysis from which we can determine the minimum residence time required to fully develop the concentration profile of a colloidal species in laminar flow through a capillary, and the specific resolution of the separation. Our analysis is based on the full development of the radial concentration profile of particles in the capillary. To estimate the minimum residence time required to fully develop the radial concentration profile, we evaluated the smallest eigenvalue of the particle diffusion equation from the Rayleigh quotient while taking into account the radial migration of particles caused by fluid inertial forces and the colloidal potential of interaction between the dispersed particles and the capillary wall. Measurements of the variance of the fractogram and of the average elutiontimeof the colloidal species determined the specific resolution. The ionic strength of the eluant and the lift forces exerted on the colloidal particles by the inertia of thefluidcan either increase or decrease the specific resolution, depending on the value of the product of the particle Reynolds number and the Peclet number. Fundamental theoretical calculations for the specific resolution agree well with experimentally measured values. When submicron colloidal particles of different sizes are transported by a fluid through an open capillary tube under laminar flow conditions, theyfractionateand emerge in order of decreasing diameter. The rate at which the particles are transported downstream (as measured by their elution times) can be quantified in terms of the eluant velocity and ionic strength, particle diameter, and capillary diameter. This difference in the rate of transport for particles of different sizes can be used to obtain the particle size distribution (PSD) of a broad distribution of particle sizes. Unfortunately, a sample composed of monodisperse particles elutes within a range of elution times rather that at a single elution time, this effect is referred to as axial dispersion or "instrumental band broadening". Axial dispersion is a serious obstacle to obtaining the PSD, and is the main cause of imperfect resolution in CHDF and in other fractionation methods. 0097-6156y91AM72-0264$06.00/0 © 1991 American Chemical Society

In Particle Size Distribution II; Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991.


17. D o s R A M O S E T A L .

Capillary Hydrodynamic Fractionation

265

Taylor [1] developed the theoretical framework from which the longitudinal dispersion of molecular species in laminar flow through capillary tubes could be evaluated in terms of fundamental parameters. Shortly afterwards, Aris [2] used the method of moments to generalize Taylor's analysis to include non-circular cross sections. In 1970 DiMarzio and Guttman [3] applied Taylor's analysis to evaluate the longitudinal dispersion coefficient of finite size spherical particles. DiMarzio and Guttman assumed that a Brownian particle suspended in a viscous fluid undergoing Poiseuille flow within a capillary tube samples all radial positions accessible to it with equal probability, provided that the residence time of the particle in the flow field is long. However, the finite size of the particle limits the approach of the particle center to a distance equal to its radius from the inner surface of the capillary wall. As a consequence of this, the particle is excluded from the slowest moving streamlines; this implies that the longitudinal dispersion of particles decreases somewhat with increasing particle size. Polymer colloids are useful as a model system because they have a broad range of well characterized, uniform particle sizes that can be used to investigate dispersion phenomena of particles flowing through capillary tubes under laminar flow conditions. Several experimental studies have been reported which demonstrate that, contrary to Taylor and Aris* results for solutes, when colloidal particles with diameters greater than l-|im are pumped through a capillary tube, their axial dispersion decreased with increasing eluant velocities. Noel et al. [4], using capillary tubes of 250- and 500 p,m inside diameter, reported such behavior for lO^im silica particles . In a similar study of particle fractionation by flow through capillaries, Brough et al. [5] found the same behavior for 2 |im diameter particles. According to McHugh [6], this phenomena results from the tubular pinch effect, where fluid inertial forces induce a radial migration of the particles toward a noncentral radial position [7]. Moreover, based on Taylor's minimum residence time criteria, McHugh also indicated that, for the experimental conditions of Noel et al., the steady state radial concentration profile did not fully develop because of the small diffusion coefficient of the particles and the relatively large eluant velocity and capillary diameters. He concluded that a more detailed analysis was needed to explain the reduction in axial dispersion with increasing particle size and eluant velocity. Recently, Silebi and DosRamos [8] used C H D F to obtain analytical separations of submicron sized particles using capillaries with diameters as small as 7 microns. In contrast to previous studies, these investigators found that, in addition to the effect of inertial forces on the particles, ionic strength influences the fractionation and the axial dispersion of particles in microcapillaries. Because of the well defined geometry of the microcapillary, C H D F is amenable to rather exacting theoretical analysis [9]. For laminar flow, the parabolic velocity profile provides an exact description of the flow distribution; the inertial and colloidal forces, which apply uniformly throughout the capillary tube, are equally welldefined in terms of particle size, capillary radius, eluant velocity and composition. In the present work, we apply Taylor's method of analysis of the dispersion phenomena in laminar flow to capillary hydrodynamic fractionation of submicron colloidal particles, and incorporate size exclusion, wall effects, and colloidal and inertial forces into the theory because all of these phenomena affect the particle displacement through the capillary and its radial distribution. The most important feature of the theory of dispersion as introduced by Taylor is that it describes the average concentration distribution in a complex three-dimensional system by the solution of the one-dimensional convective diffusion equation. As a result, the primary problem is to determine from first principles the dispersion coefficient associated with the one-dimensional dispersion equation.


266

P A R T I C L E S I Z E D I S T R I B U T I O N II

Theory Including fluid inertial forces and electrostatic repulsion between the particles and the capillary wall, the diffusion equation that governs the development of the particle concentration profile for the separation process described above is [9,10]:


3t

r

A L

r

ft

D

^c+rcfv

dr

1

T

1

*

-BL r

+

e

A r

d

r

J

(8) R

o "

e

rdr


268


where H(r*) = J{ v (r') - < v pz

p z

>}

f dr'

(9)

O

The solution of Equation 6 for N particles that are concentrated over a small length dZ at the entrance of the capillary (Z = 0) at time t = 0 is given by:


-A

C™ =

2

e *
40 for the larger particles in the mixture and RepPe < 40 for the smaller particles in the mixture. As the eluant velocity increases further, R e p P e for the smaller sized particle reaches a value of 40. Then the separation factors for both species have reached their limiting values, and further increases in average eluant velocity decreases the axial dispersion of both particle sizes to increase the specific resolution. For submicron particles, the efficiency of separation decreases with increasing average eluant velocity, since very large eluant velocities are required before R p P e exceeds 3. This is especially significant for small inner diameter capillaries, where the large pressure drops encountered in the microcapillaries places an upper limit on the average eluant velocity. Thus, under this limiting condition, smaller velocities will give a better separation for mixtures of the smaller particles.


s

s

e

e

MEAN ELUANT VELOCITY

(cm/s)

Figure 6: Theoretical calculation of the influence of the eluant velocity on the specific resolution for various pairs of particles. Capillary radius: 3 pm; Capillary length: 20 m; Ionic strength: 1 x 10~ M . 3


274


0.3 DC

Z

-.794-1.1

o


o CO LU £C

O

,357-. 109 fim

EE o LU CL CO

0.0

MEAN ELUANT VELOCITY (cm/s) Figure 7: Comparison between theoretical and experimental results of the specific resolution as a function of eluant velocity. Capillary radius: 17.4 Aim; Capillary length: 17.4 m; Ionic strength: 1 x 10~ M . 3

to DC

Z

3-

o J— ZD -J

o

CO LU

o o LU CL CO

2-

L 1

0

1

y r — D p : 0.357-0.109 pm _ _ ^ / ^ - D p : 0.176-0.088 Jim

* 2

—

3

4

5

6

MEAN ELUANT VELOCITY (cm/s) Figure 8: Comparison between theoretical and experimental results of the specific resolution as a function of eluant velocity. Capillary radius: 3.5 /xm; Capillary length: 8.5 m; Ionic strength: 1 x 10~ M . 3



275


Conclusions The appropriate criteria for the development of the steady state radial concentration profile has been established based on the evaluation of the smallest eigenvalue of the convective diffusion equation for the particles. The results show good agreement between the calculated and experimental values of Rf and R . For mixtures of particles in which RepPe is smaller than 3 for all species in the mixture, the specific resolution decreases as the eluant velocity increases. At ionic strengths greater than 1 0 " M , the experimental specific resolution of several pairs of particles was essentially constant as predicted by the theoretical analysis. The analysis provides a natural explanation for universal calibration: Rf becomes independent of particle type (as indicated by the particle's surface potential and Hamaker's constant) at very low eluant ionic strength because the double layer repulsive force dominates the van der Waals attractive force. s


3

Appendix

(Al)

Rewriting Equation 1 in dimensionless form yields:

(A2) where C

Z

1

=

°

R

2

- ^ { z - < v v R m

0

'

D~'

kT' V

>t } , E = 0 - | R 8

e p

P

^

2

f^df

e

o

J

D

c is the particle concentration at the entrance of the capillary, Z is the axial distance from the entrance of the capillary, Rep is the particle Reynolds number, and P is the Peclet number. The boundary conditions which accompany the diffusion equation correspond to symmetry around the capillary centerline and to no mass flux across the capillary wall: 0

e

gl

=0

(A3)

Equations A l , A 2 , and A3 constitute a Sturm-Liouville system. To find the conditions under which radial variations of concentration are reduced to a small fraction of their initial value through the action of molecular diffusion, it is necessary to calculate how rapidly the radial concentration profile approaches a time independent form. The solution of Equation A1 that gives the dependence of c upon r and T can be obtained by assuming a functionality dependence of the form


276


c(f ,t) = p(r) x(t). Substitution of this product solution into Equation A l after separating variables and integrating yields for the time and radial functions: x(t)=Ae

(A5)

x i


[•fD p' - p r D E ] ' - X p r = 0

(A6)

where the primes indicate ordinary differentiation with respect to radial position, and X is the separation constant or eigenvalue. As seen in Equation A 5 , the eigenvalue X governs approach to full development of the particle radial concentration profile. It is difficult to solve Equations A 6 analytically for the eigenvalue, so we instead form the Rayleigh quotient to quantify the upper bound of the smallest eigenvalue, which is used to examine the influence of particle size and flow conditions on entrance length. The following development presents the mathematical details for forming the Rayleigh ratio for Equation A6. To form the Rayleigh quotient, we first put Equation A 6 into Sturm-Liouville form: [p(f) P' ] ' + l> q(r) + w(r) ] p = 0

(A7)

where

p(r) = e x p { f ( i i ^ I + E ) d r }

(A8)

q(r) = 2®D

(A9)

1

J

(- )

W

r

=

p

r

r

f D

)[2!L±^IX] rD

(AIO)

Following Silebi and McHugh [15], we multiply Equation A 7 by an arbitrary estimating function F for p, integrate by parts, and rearrange to form the Rayleigh quotient: 1-K

J t p O ^ - w F ^ d ? - (pFF)l

1-K (

0

X =

(All) 1-K

Jql^dr As long as F satisfies the symmetry and no-flux boundary conditions, and is twice continuously differentiable, numerical integration of Equation A l l calculates the upper bound of the smallest eigenvalue. An appropriate estimating function for the radial concentration profile of the particles is: F(r) = [ 1 +ar + br + c r ] e 2

3

E

(A12)

The exponential factor is the fully developed concentration profile, obtained by solving the radial component of the diffusion equation when the total radial flux is



277


zero. The constants in Equation A12 are found by applying the boundary conditions and the orthogonality condition: 1-K

(A13)

jFrdf = 0 o


Thus result Equations A14 through A16: (A14)

a = O'l

r=0

3(1 -

r=0 u

K)

o

_

e' df E

b =

1-K

1-K

2(1

-

3(1 -

K)

o

0

j^ '

Jf e* df 3

E

e

1-K

E

&

1-K

2(1-K)

-O'\.

0 Jf

(A15)

K)"

=

0

0 3

e"

E

df

-Jf(l

1-K

_ + t{W)\.

M 0

e"

E

df

1-K

2(1 -

K)

JfV df E

1-K

(A16) 3(1 -

K )

2

Jf e- df 4

E

1-K

Direct numerical integration of Equation A l l calculates the smallest eigenvalue of the diffusion equation, and hence, the minimum residence time required to fully develop the radial particle concentration profile. In the absence of colloidal and inertial forces, and assuming constant diffusivity, the analytical solution of the above equations results in X= 14.7 D/(l - K ) . For zero particle size, this is the same result obtained from Taylor's analysis of the dispersion of a molecular species in laminar flow. 2


278

PARTICLE SIZE DISTRIBUTION II

Literature Cited


1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Taylor,G.I.Proc. R. Soc. Lond. 1954, A225, 473. Aris, R. Proc. Roy. Soc. A 1956, 235, 67. DiMarzio, E. A.; Guttman, C. M., Macromolecules 1970,3,131. Noel, R.J.; Gooding, K. M.; Regnier, F.E.; Ball, D.M.; Orr, C.; Mullins, M.E. J.Chromat. 1978,166,373. Brough, A.W.J.; Hillman, D.E.; Perry, R.W. J. Chromat. 1981, 208, 175. McHugh, A.J. CRC Critical Reviews in Analytical Chemistry 1984, 15, 63. Segre, G.; Silberberg, A. J. Fluid Mech. 1962, 14, 136. Silebi, C. A.; DosRamos, J. G. J. Coll. and Interf. Sci. 1989, 130, 14. DosRamos, J. G.; Silebi, C. A. J. Coll. and Interf. Sci. 1989, 133, 302. Silebi, C. A.; DosRamos, J. G. AIChE J. 1989, 35, 1351. Brenner, H. Progress in Heat and Mass Transfer 1972, 6, 509. Ananthakristan, V.; Gill, W. N.; Barduhn, A. J. AIChE J. 1965, 11, 1063. Brenner, H.; Gaydos, L.J. J. Colloid Interf. Sci. 1977, 58, 312. Nagy, D. J., Silebi, C.A.; McHugh, A.J.J.Colloid Interf. Sci. 1981, 79, 264. Silebi, C. A.; McHugh, A. J. J. Polymer Sci: Physics 1979, 17, 1469.

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January 14, 1991


Particle Size Distribution II - American Chemical Society

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