1 Mathematical Modeling of Particle Chromatography D. C. FRANCIS and A. J. McHUGH Downloaded via 185.252.218.81 on August 7, 2018 at 13:06:52 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.
Department of Chemical Engineering, University of Illinois, Urbana, IL 61801
A discussion is given of the mathematical modelling of the separation mechanisms associated with the packed column chromatography of particulate systems. Primary emphasis is on the derivation of models for the HDC and pore partitioning processes which occur with porous packing systems. Comparison is made of predictions for the separation factor - particle size behavior for a purely flow-through model, published earlier, and models developed herein to account for simultaneous pore partitioning effects. Comparison to literature data indicates that accounting for pore partitioning leads to a more accurate f i t . The results of these calculations indicate the need for further experimental studies to characterize the model parameters associated with the possible separation mechanisms. A l a r g e and important c l a s s of c o l l o i d s are the polymer l a t e x e s which c o n s i s t of charged (by ionogenic surface groups and/or adsorbed s p e c i e s ) g e n e r a l l y s p h e r i c a l p a r t i c l e s w i t h diameters ranging from tens of nanometers to microns. The r o l e of p a r t i c l e s i z e a n a l y s i s i n c h a r a c t e r i z i n g such systems, f o r both fundamental s t u d i e s and t e c h n o l o g i c a l a p p l i c a t i o n s , i s e q u i v a l e n t i n scope t o that of molecular weight a n a l y s i s i n c h a r a c t e r i z i n g bulk polymers. Reviews of the v a r i o u s techniques and important areas of a p p l i c a t i o n of p a r t i c l e s i z e a n a l y s i s can be found i n s e v e r a l references (e.g. (_1,»_2)). I n the p a s t , analyses of submicron p a r t i c l e s have been l i m i t e d to time-consuming techniques, such as e l e c t r o n microscopy, o r , t o methods such as l i g h t s c a t t e r i n g , which r e q u i r e a f a i r l y narrow s i z e d i s t r i b u t i o n f o r accuracy. R e c e n t l y , r e p o r t s of a number of s t u d i e s of a new method have been published i n which m o d i f i c a t i o n s 0097-6156/ 84/0245-0003506.50/0 © 1984 American Chemical Society
Provder; Size Exclusion Chromatography ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
4
SIZE E X C L U S I O N
CHROMATOGRAPHY
of the chromatographic techniques used i n polymer molecular weight a n a l y s i s have been employed t o determine p a r t i c l e s i z e and p a r t i c l e s i z e d i s t r i b u t i o n of suspensions (see r e f e r e n c e Ç3) f o r a b r i e f overview). These papers stem from experimental s t u d i e s (4^5) which demonstrated that s t a b i l i z e d , d i l u t e suspensions of l a t e x p a r t i c l e s f r a c t i o n a t e by s i z e when pumped through beds of porous or nonporous packing. There i s now a c l e a r i n d i c a t i o n that the development of such techniques f o r s i z i n g submicron p a r t i c l e s w i l l have much the same impact on the science and technology of c o l l o i d a l systems as l i q u i d s i z e e x c l u s i o n chromatography a n a l y s i s has had on t h e f i e l d of bulk polymers. The purpose of t h i s paper i s t o present a b r i e f overview and d e s c r i p t i o n of a m o d e l l i n g approach we are t a k i n g which i s aimed at developing a q u a n t i t a t i v e understanding of the mechanisms and s e p a r a t i o n c a p a b i l i t i e s of p a r t i c l e column chromatography. The main emphasis has been on the a p p l i c a t i o n of fundamental treatments of the convected motion and porous phase p a r t i t i o n i n g behavior of charged Brownian p a r t i c l e s t o the development of a m e c h a n i s t i c r a t e theory which can account f o r the unique s i z e and e l e c t r o c h e m i c a l dependent s e p a r a t i o n behavior e x h i b i t e d by such systems. Background D e s c r i p t i o n and Review of S e p a r a t i o n Mechanisms The experimental methods reported f o r p a r t i c l e chromatography have employed g l a s s or s t a i n l e s s s t e e l columns packed w i t h nonporous copolymer o r g l a s s beads, porous g e l m a t r i c e s , o r v a r i o u s GPC porous g l a s s m a t e r i a l s . Most s t u d i e s have analyzed polymer l a t e x s o l u t e p a r t i c l e s suspended i n s t a b i l i z e d aqueous media w i t h the common mode of s i g n a l d e t e c t i o n being l i g h t s c a t t e r i n g . Small's work (h) w i t h v a r i o u s nonporous packing systems demonstrated that f o r a range of eluant i o n i c s t r e n g t h s , l a r g e r l a t e x p a r t i c l e s e l u t e from the column ahead of s m a l l e r ones and that the primary f a c t o r s a f f e c t i n g the e l u t i o n time were eluant i o n i c s t r e n g t h , packing diameter, and flow r a t e . The f r a c t i o n a t i o n process occurs s o l e l y i n the mobile phase and r e s u l t s from the f a c t that the l a t e x p a r t i c l e s a r e p r e f e r e n t i a l l y excluded from the slower moving s o l v e n t streamlines nearest the packing surfaces and thus o b t a i n average v e l o c i t i e s i n excess of the s o l v e n t and these v e l o c i t i e s i n c r e a s e w i t h s o l u t e s i z e . The name Hydrodynamic Chromatography or HDC has t h e r e f o r e been used t o d e s c r i b e the process. A number of p u b l i c a t i o n s (6-10) have demonstrated that the s i z e s e p a r a t i o n mechanism i n HDC can be d e s c r i b e d by the p a r a l l e l c a p i l l a r y model f o r the bed i n t e r s t i c e s . The r e l e v a n t e x p r e s s i o n f o r the s e p a r a t i o n f a c t o r , Rp, ( r a t i o of eluant t r a c e r t o p a r t i c l e mean r e s i d e n c e times) i s g i v e n by, %
= / p
m
Provder; Size Exclusion Chromatography ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
(1)
1.
FRANCIS AND McHUGH
Mathematical
Modeling
5
where the p a r t i c l e and marker average v e l o c i t i e s through the and , are given by
bed,
m
R R - ο - ρ v
W " m — v
+
< v
m
/~t\
JL s i si — l - * = - l
D
\
^
>
1
F i g u r e 8. Schematic i l l u s t r a t i o n o f bed cross s e c t i o n f o r the combination model. See t e x t f o r e x p l a n a t i o n o f nomen clature .
Provder; Size Exclusion Chromatography ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
1.
FRANCIS AND McHUGH
Mathematical
19
Modeling
flow-through. The p o r t i o n of the cross s e c t i o n a s s o c i a t e d w i t h the p a r t i t i o n i n g process i s denoted 1, and c o n s i s t s of l a r g e , flow-through i n t e r s t i t i a l c a p i l l a r i e s , i , t o which are attached stagnant pore volume c y l i n d e r s , p, w i t h which p a r t i c l e s p a r t i t i o n during passage through the i tubes. D e r i v a t i o n of the s e p a r a t i o n f a c t o r f o r t h i s model f o l l o w s the development g i v e n e a r l i e r f o r a purely flow-through system ( 1 3 ) . The p a r t i c l e e l u t i o n volume i s g i v e n by V
=
R,P
%
< l >
( 1 7 )
where i s the e l u a n t flow r a t e , and i s the mean r e s i d e n c e time. Since the average residence time i s the sum of the times the p a r t i c l e spends i n each c a p i l l a r y , « n
+ n
i
A
g
+ ti ±
s
(18)
l
where n^ i s the t o t a l number of c a p i l l a r i e s of type j the p a r t i c l e samples, and ^ i s the average r e t e n t i o n time i n a c a p i l l a r y of type j . Since the p r o b a b i l i t y that a p a r t i c l e w i l l sample a c a p i l l a r y of type j i s given by n
( 1 9 )
Pj - j V Q f
where Ν. i s the number of c aapi p i l l a r i e s of type j i n a bank, and i s the f l o w r a t e i n a c a p i l ll a ir y of type j , then the t o t a l number of c a p i l l a r i e s of type j a p a r t i c l e samples i s tVj » ηρ^ » where η i s the number of banks. c a p i l l a r y of type j i s g i v e n by
nNjqj/Q
(20)
F
The average marker v e l o c i t y
ina
where a j i s the c r o s s - s e c t i o n a l area of the c a p i l l a r y and the average time f o r a p a r t i c l e i n a g i v e n c a p i l l a r y i s
j
- -gy-
(22)
Ρ j Combining Equations 17 t o 22 y i e l d s 1_
\
R_
~~ V
¥
1_ m
R
Tl
0
f
,^s +
V
m
_1_ R
+
F,s
V
m
R
w
l
< V
>
m i
m
< v > ,
F ,1
m 1
K
. J
where V j i s the volume of a l l c a p i l l a r i e s of type j , and % j I s the s e p a r a t i o n f a c t o r i n a c a p i l l a r y of type j . * f
Provder; Size Exclusion Chromatography ACS Symposium Series; American Chemical Society: Washington, DC, 1984.
20
SIZE E X C L U S I O N C H R O M A T O G R A P H Y
The e x p r e s s i o n f o r the mean p a r t i c l e r e t e n t i o n time (Equation 11) may be w r i t t e n f o r the marker and manipulated t o y i e l d . m
i 1+ Κ σ m
mi
(24) K
J
S u b s t i t u t i n g Equations 13 and 24 i n t o Equation 23 y i e l d s
-\
k
t , y
