Simulations of adsorption from flowing solutions in a slit or capillary with a finite adsorption constant at the walls

COLLOIDS AND SURFACES ELSEVIER

B

Colloids and Surfaces B: Biointerfaces 4(1995) 111 120

Simulations of adsorption from flowing solutions in a slit or capillary with a finite adsorption constant at the walls P h i l i p p e D 6 j a r d i n *, I s a b e l l e C o t t i n Institut Charles Sadron, 6 rue Boussingault, 67083 Strasbourg Cedex, France Received 12 January 1994; accepted 10 August 1994

Abstract

Simulations of adsorption from flowing solutions of a solute with diffusion coefficient 2.0 x 10 -7 cm2 S -1, typical for fibrinogen, were performed for a slit or capillary of dimension 0.2-10 mm and computational results compared to the approximate relation k - I = k a l + kLe1, where k is the apparent kinetic constant, ka is the kinetic adsorption constant at the interface and kLevis the kinetic constant relative to a purely transport controlled process (L6vSque model). The variations of k -1 with distance from the slit or capillary entrance were examined. When approaching "chemical" control, a good estimation of k, could be obtained whereas the diffusion coefficient was overestimated by about 30%, while near the L6v6que regime we obtained, as expected, a good evaluation of the diffusion coefficient.

Keywords: Adsorption; Capillary; Flowing solution; Slit 1. Introduction

To describe the adsorption kinetics of (bio)polymers at interfaces, a detailed mathematical analysis is necessary for several reasons. Firstly, results may be compared to those from optical I-1-3] and radiolabeling [4] techniques which enable continuous recording of the adsorbance of solute. Secondly, the processes at the interface and in solution may be complex: conformational changes of adsorbed molecules, stabilization of the adsorbed layer and the presence of complexes in solution and at the interface [5]. As in our laboratory, we are mainly concerned with experiments using solutions flowing through a tube [6] or capillaries [7]; we were interested in simulating adsorption processes with a non-zero shear rate at the interface. An initial study was performed for

stagnant solutions [8] similar to an earlier study [9]. In a following work, an approximate analytical solution for a system with a finite adsorption constant k, and a fixed thickness of diffusion was proposed [10], on the basis of simulations which maintained the bulk concentration throughout the kinetic process at a chosen distance from the wall. Application to flowing dilute solutions in the first steps of the adsorption process led to an expression for the apparent kinetic constant k in terms of the finite "chemical" adsorption constant ka at the interface and the constant kLev of the L6vaque model which assumes an infinite adsorption constant and a semi-infinite medium k l=k~a +kc~lv where kLev = 0.538(02~,/X) 1/3

* Corresponding author. 0927-7765/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0927-7765(94)01161-3

(la)

(lb)

where D is the diffusion coefficient of the solute, Y

112

P. Ddjardin, L Cottin/Colloids Surfaces B: Biointerfaces, 4 (1995) 111-120

is the shear rate at the wall and x is the distance from the entrance of the slit. The basis of Eq. (la) is the following. The Nernst depletion layer thickness Yr~ normal to the wall under steady-state conditions for C(x, y) was assumed to be the same as in the L6v6que model, while the stationary concentration at the wall C(x, 0) was estimated from the equality of the rate of transport and the rate of adsorption at the interface with the finite kinetic constant ka OF(x, t)/Ot = kaC(x , 0 ) = D[C b -- C(x, 0)]/y N

(2)

where C b is the constant bulk concentration far from the wall and F is the interfacial concentration. Following this reasoning and since kLev is proportional to x -1/3, it was proposed that k -1 should be a linear function of x 1/3. The simulations performed in the present work permit a more realistic verification of relation Eq. (la) than in previous papers. Rather than choosing an arbitrary value of the Nernst thickness layer for one computation, calculations are carried out for a slit or capillary 10 cm long with a width or radius varying between 0.2 and 10 mm. Numerous cases of similar type were studied by Adamczyk [ 11] where analytical solutions were provided as in the textbook of Levich [12].

y = b/2

OC/Oy = 0

(4c)

y = 0

D(OC/@) = kaC(x , O, t) x ( 1 - F(x, t)/F~,t )

(4d) Although chosen to be of the Langmuir type, the excluded surface factor may be taken as unity for calculations performed at the low concentration C u = 0.1 tag ml -I, where estimation of the slope t3F/Ot is limited to very early times after establishment of the steady-state concentration profile. This factor plays a role at high concentrations when the interracial concentration becomes comparable to its saturation value F~,t = 1 ~tg cm -2. The Poiseuille velocity profile for a slit is v(y) = ?y( 1 - y/b)

(5)

where ? is the shear rate at the wall. Similar equations hold for the cylindrical geometry of a capillary of radius R, r being the distance to the capillary axis OC ~C O ~ /- - / O / rC \ Ot - - v ( r ) ~ x + - r ~r \ O r ) with a velocity profile v(r) = 0.57R [ 1 - (r - R) z ]

2.1. Fundamental equations in a continuum

For a solute of diffusion coefficient D and concentration C(x, y, t), where x is the distance from the entrance of the slit, y is the distance from the wall and t is the time, the differential equation for a mechanism taking into account convection and diffusion with a velocity profile v(y) is c)t --

c~C

c~2C

v ( y ) ~ x + D--oy 2

(7)

Substituting b = 2R in Eq. (5) gives the profile of Eq. (7). We neglected for both geometries the diffusion term D(OUC/~x 2) along the flow direction.

2. Computational model

c~C

(6)

(3)

The boundary conditions are as follows, where b is the slit width and Cb is the bulk solution concentration

2.2. Discretization of space

We used uniform mesh sizes of 2 and 0.5 mm along the flow direction and 1 gm in the direction normal to the wall, which corresponds to a grid of 50 × 100 and 200 × 100 for a slit of length 10 cm and width 200 tam. Whatever the size of the slit or capillary, a grid normal to the wall was always created to a depth of 100 tam. The explicit expression for a discretized form of Eq. (3) for a cell (i, j) in the volume is (Fig. 1) 3C(i, j ) / A t = D[C(i j + 1, t) - 2C(i, j, t)

t= 0

x > 0

C(x, y, 0) = 0

(4a)

+ C(i, j -

1, t)]/(Ay) 2

Vt

x <<_0

C(x, y, t) = Cb

(4b)

- v ( j ) [ C ( i , j , t) - C ( i - 1,j, t)]/Ax (8)

P. Dkjardin, I. Cottin/Colloids Surfaces B: Biointerfaces, 4 (1995) 111 120

j=

1

2

3

4

5

6

113

7

1

1

C=CI: C=Cb C=Cb C:Cb C---Cb C=Cb C=Cb C=Cl i=I

~

i=3

•

i=4

Flow

i=5

7(

..

t

AX

" <

> Ay

Non adsorbing surface B Adsorbing surface

Fig. 1. Discretization of space for computation. The first slab (i = 1) is in contact with a non-adsorbing surface and is permanently filled with solution at concentration Cb, while the concentration elsewhere is zero at time zero. Arrows indicate the linear velocity field near the wall.

where C(i, j, t) represents the average concentration in the cell (i, j) at time t and v ( j ) the velocity in its center. The concentration gradient is zero at the center of the slit (boundary condition 4c), which gives the specific formulation AC(i, Jmax)/At = - O[C(i, j . . . . t) - C(i, Jmax -- 1, t)]/(Ay) 2 - V(Jmax)[C(i, J . . . . t) -- C(i -- 1, j . . . . t ) ] / A x

calculated A F ( i ) / A t using E q . ( l l ) and finally AC(i, 1)/At using Eq. (10). Such a procedure avoids introducing an imaginary concentration behind the wall [ 13]. The condition (4b) was incorporated in the model by creating a first slab of thickness A x containing Jmax cells, where the concentration was maintained at the bulk concentration Cb throughout the computation, the concentration elsewhere being zero at time zero (condition 4a). Similar equations with corrections for curvature were applied to the capillary geometry (see Appendix).

(9) For cells (i, 1) in contact with the wall (i >/2), the boundary condition (4d) is applicable

3.1. Analysis at low concentration ( Cb = 0.11~g ml 1)

AC(i, 1)~At = O[C(i, 2, t) - C(i, t, t)]/(Ay) 2 - v(1)[C(i, 1, t) - C ( i -- [ A F ( i ) / A t ] / A y

1, 1, t ) ] / A x

(10)

where

~Jr(i)/At =

ka C(i , O, r)(1 - F(i, t)/F~,t)

3. Results and discussion

(11)

In practice, we first estimated the volume concentration C(i, O, t) at the wall from the linear extrapolation C(i, 0, t) = 1.5C(i, 1, t) - 0.5C(i, 2, t) and then

Computational results were analyzed for a shear rate of 500 s -~ and a diffusion coefficient D of 2.0 x 10 - 7 c m 2 s -1, which is typical for fibrinogen [-14], according to Eq.(1). Three dimensions of the slit or capillary were considered: b (or diameter) = 0.2, 2 and 10 mm, conditions under which the thickness of the Nernst layer at the end of the slit or capillary was always much smaller than the

P. Dkjardin, I. Cottin/Colloids Surfaces B." Biointerfaces 4 (1995) 111-120

114

half width of the slit or the capillary radius. This situation is visualized in Fig. 2 which shows the steady-state concentration profiles for the two goemetries and smallest dimensions. The depletion layer has a total maximal thickness of 60 gm. A plot of the velocity profile shows it to be far from linear in this domain, while a comparison of steadystate profiles for slit and capillary geometry is in accordance with less transport of matter through the center of the capillary than through the center of the slit due to curvature. Let us recall that in the L6v~que solution, the velocity profile is assumed to be strictly linear. According to Eq.(1), the plot of Cb/(OF/&) should be a linear function of x 1/3, the ordinate at the origin being 1/k, and the slope (s) leading to the diffusion coefficient through the relation D = 2.53s- 3/27-1/2

(12)

Examples of such plots are presented in Figs. 3a 1.0 r

~ /

/

J

J

/ / 0.8 I-

/

1/ /

-t 2

/

/ / / / /

06]

/ /

.Q

/

L)

/

/

o O

/ /

0.4

/ / / / /

/

0.2

0.0 ~ 0

~ 20

0 40

60

80

100

Distance to the w a l l [pro]

Fig. 2. Scale on the left: steady-state concentration profiles normalized to the bulk concentration C b = 0 . 1 gg m1-1 for a slit of width 2 m m (solid line) and a capillary of radius 0.1 m m (dotted line) at 10 cm from the entrance (from b o t t o m to top, ka = 10 -3, 2.0 x 10 -4, 10 -5 cm s-a). Shear rate at the wall is 500 s 1. Scale on the right: velocity profile (dashed line).

and 3b for a slit width or capillary diameter of 0.2 mm and in Fig. 4 for a situation similar to a semi-infinite medium. The linearity predicted by Eq. (1) is indeed observed for a half width or radius of 5 mm (Fig. 4), while very small positive curvature appears for a slit width of 0.2 mm. However for a capillary of the same dimension there are strong effects of curvature. Linear fits are relative to x between 2 and 10 cm, a reasonable domain of experimental measurements. We expect these curvature effects to be minimized near the entrance as the depletion layer becomes thinner. Using Eq. (1), a linear fit of these computational results led to estimations ka(app ) and D(app) of the kinetic constant and diffusion coefficient, respectively, which are compared in Figs. 5 and 6 to the real values ka and D introduced as parameters in the calculations by plotting the ratios r(k,) = ka(app)/k~ and r(D) = D(app)/D against k,. Results for the determination of ka are shown in Fig. 5a for a slit and in Fig. 5b for a capillary, while Figs. 6a and 6b give the corresponding results relative to the determination of D. It is apparent from these plots that Eq. (1) is approximate as even for the largest dimension of 10 mm, where both conditions of linear velocity gradient and negligible curvature (capillary case) in the depletion layer are met, we cannot recover the values of ka and D introduced in the calculations. In the region of high ka, corresponding to conditions approaching those of the L6v6que regime, the slope leads to a good value of the diffusion coefficient with an overestimation of k,, whereas for the slower interfacial adsorption process the determination of k~ is possible, but there is an overestimation of the diffusion coefficient of up to 31% for ka= 10 S cms-1. Moreover the influence of curvature is still detectable for a capillary diameter of 2 mm while for a slit of the same width the limit of an infinite width is almost attained. These results correspond qualitatively to what would be expected from a physical point of view. At low k~, the process is controlled by the interface and k ~ ka, while for high ka values the process is controlled by the transport and k ~ kL,v; hence a good estimate of the diffusion coefficient D. There is, however, no hope in this range for an

P. DEjardin, I. Cottin/Colloids Surfaces B." Biointerfaces, 4 (1995) 111-120 4e+4[ '

4e+4

115

-i

(a)

i (b)

/ ///

i

/// 3e+4

3e+4 . / / "/y

/ /

//,,/

[

./.// ///

~" 2e+4

/

:

g

//o /o/°

/-/" / / "

/// ~" 2e+4 f /

///

/~

// ///

.t..,

/

o"

//

//

~'~

/o//c o

z/.

/~

/// le+4 / /

le+4

Oe+O

0.0

i

05

1.0 x 1/3

1.5

2.0

2.5

[cm v3]

Oe+O/ / / / O0

/6

~ 0.5

//

10 x 1/3

15

2.0

2.5

[cm 1/3]

Fig. 3. Inverse of the apparent kinetic constant Cb/(gF/~t) plotted against distance from the entrance to the power 1/3: (a) for a slit of width 0.2 ram; (b) for a capillary of radius 0.1 mm. The solid line gives the linear fit of the computation results (2 cm < x < 10 cm) and the dashed line Eq. (la). From bottom to top, k a = 10 -3, 2 x 10 4, 10-4, 5 x 10 5 cm s 1. The empty symbols refer to distances from the entrance smaller than 2 cm.

accurate determination of k, from experimental data through extrapolation to x 1/3 = 0 (Fig. 4). One condition of the Lrv~que model is the invariant zero value of the volume concentration at the wall, a property which disappears with a finite adsorption constant. In Fig. 7, the steadystate concentration at the wall C(x, 0) normalized to the bulk concentration Cb is plotted against x and compared to the approximate theoretical value (1 +k,/kLev) -~ deduced from Eq.(2). It can be seen that this approximation underestimates the concentration at the wall and, especially near the entrance, the variation of C(x, 0) with x cannot be neglected. Artefacts due to the choice of mesh size in the x and y directions were also verified. Calculations performed with a mesh size of Ay reduced from the usual 1 gm to 0.2 gm showed no difference in results. As shown in Fig. 7, the Ax step should be smaller as the concentration gradi-

ent near the entrance is steeper. In Table 1, we compared results from computations performed with mesh sizes (Ax, Ay)=(0.5 mm, 1 gin) and (2 mm, 1 gm) for a large slit. Similar variations were obtained for smaller slit dimensions. A few percentages of difference in k,(app) at high ka values are the highest variations, therefore convergence is probably attained in practice in the present calculations. A numerical diffusion effect [ 15], if any, is small. Moreover, for higher values of k,, it seems likely that the experimental precision would not permit also a confident determination of k~ near the Lrvrque regime.

3.2. Analysis at high concentration (Cb > 10 Itg m1-1 ) Fig. 8 illustrates the variation of the interfacial concentration with time at different distances from

P. D~jardin, 1. Cottin/Colloids Surfaces B." Biointerfaces, 4 (1995) 111-120

116

4e+4

jjJJ

J

2.0

.1

jJJ 3e+4

A

jjjJJJJ /"J

T

//"

///

.//"

.f/S

//./

2e+4 "0

1.5

///

/-/

..../j

.'// j//

1.0

~/ 10 -s

10 4

10-3

log10 (k=[cm/s])

7/"

///

/.~

2.0

le+4

i

i

a Slit

/

.4zz z 1.5

Oe+O 0.0

0.5

1.0

x 1/3

1.5

2.0

2.5

[cm 1/3]

Fig. 4. Inverse of the apparent kinetic constant Cb/(QF/~t) plotted against distance from the entrance to the power 1/3 for a capillary of radius 5 m m (the results for a slit of width 10 m m were identical). Solid line gives the linear fit of the computation results (2 cm < x < 10 cm) and the dashed line Eq. (la). From bottom to top, ka = 10 3, 2 x 10 -4, 10 -4, 5 x 10 5 cm s -1. The empty symbols refer to distances from the entrance smaller than 2 cm.

the entrance for a high kinetic constant ka = 10 -3 cm s -1. A slight acceleration of the adsorption process (positive curvature O2F/Ot2) occurs after a certain delay which depends on the position examined, and this phenomenon appears from an almost complete coverage starting from the entrance. Relative to downstream positions, the entrance would appear to move during the adsorption process, thus reducing the Nernst layer thickness and increasing the rate of adsorption. When the kinetic constant ka decreases, this difference between the functions F(x, t) at differrent x vanishes (Fig. 9) with a constant negative curvature c32F/~t 2. It is well known that the Langmuir model does

1.0 10-s

10-4

10-3

Ioglo (ka[Cm/s]) Fig. 5. Ratio of the kinetic constant deduced from computation results to the value k a given as an entry parameter for calculations plotted against ka. (a) Slit; (b) capillary. Width or diameter, 0.2 m m (©); 2.0 m m ([Z); 10.0 m m (+).

not accurately describe the kinetics of protein adsorption due, for example, to an inappropriate model of sites, surface diffusion and/or interaction between adsorbed molecules. A simple access to an experimental excluded surface factor needs a constant concentration near the wall. For high ka, this occurs when full coverage is almost attained, thus decreasing sharply the rate of adsorption and leading to a flat concentration profile C(x, y) = Cb. This decrease is especially strong in models of random sequential adsorption (RSA) in a continuum [16] or on randomly distributed sites [17]. We examined the model of RSA using the excluded

P. Dkjardin, Z. Cottin/Colloids Surfaces B: Biointerfaces, 4 (1995) 111 120 1.50

117

1.0 r , _ _

- -

b capillary 1.25 08

A

¢'~ ,~- 1.00 0.6 .Q

0.75

¢J O

04

0.50 10-5

104

10.3

10glo (ka[cm/s]) 1.50

0.2

a Slit 1.25 0.0

L

-

-

. . . . .

~

0

2

A

a

a .

.

.

.

.

4

~

_ _

6

•

__

._J

8

10

x [cm]

10 0

Fig. 7. Normalized volume concentration at the wall under steady-state conditions for kinetic constants (from bottom to top) of 10 -3 , 2 × 1 0 -4 and 1 0 - S c m s i and a slit of 10mm. The continuous line gives the approximate theoretical value (1 + ka/kLev) 1.

!

0.75

050 __L

.

.

.

10 "s

.

.

10 .4

10 .3

Table 1 Comparison of k,(app) and D(app), determined according to Eq. (1), from computations with different Ax mesh sizes

1O810 ( k , [ c m / s ] )

Fig. 6. Ratio of the diffusion coefficient deduced from computation results (Eq. (la)) to the value 2.0 × 10 7 cm 2 s-1 given as an entry parameter for calculations plotted against k~ entry parameter. (a) Slit; (b) capillary. Width or diameter, 0.2 mm (O); 2.0 mm (D); 10.0 mm (+).

surface factor ~ proposed

by Schaaf and

[-16] t o fit t h e i r n u m e r i c a l

simulations

~b = (1 - x')3/( 1 --F ClX' 4- c2 X'e 4- c3 X'3)

Talbot

(12)

w h e r e Cl = 0.8120, c2 = 0 . 2 3 3 6 , c3 = 0 . 0 8 4 5 a n d x ' =

ka

Ax

(cm s -1)

(mm)

k.(app) (cm s -1)

D(app) (cm 2 s- 1)

10 4 10 4

0.5 2.0

1.051x10 4 1.049 x 10 4

2.224x 10 7 2.229 x 10 -7

2 x 10 -4 2 x 10 -4

0.5 2.0

2.237 x 10 -4 2.236 X 1 0 - 4

2.106 x 10 -7 2.108 × 10 -7

5 × 10 -4 5 × 10 4

0.5 2.0

6.246 X l0 4 6.398 X 10 4

2.027 X 10 7 2.021 x 10 7

10 -3 10 -3

0.5 2.0

1.365 × 10 -3 1.455 x 10 3

2.005 X 10 7 1.998 × 10 -7

0/0o0 = F/Foo, 0 b e i n g t h e f r a c t i o n o f s u r f a c e c o v ered with protein molecules. As the jamming is 0 ~ = 0 . 5 4 7 f o r c i r c u l a r d i s k s , s i m u l a t i o n s performed

with an arbitrary

Variations

of the

interfacial

limit were

~ , t = 0 . 5 4 7 p,g c m -2. concentration

with

time exhibited very slow evolution

t o its l i m i t i n g

value. In experiments where the precision of the results would allow us to plot the slope dF/dt

P. Dkjardin, I. Cottin/ Colloids Surfaces B." Biointerfaces, 4 (1995) 111-120

118 1.0

'

i

T

-

~

r

_

~

-

//" /'. /

0.8 /

/

/

/

/

/

/

/

/

~

/

.."

/

/

/ /

./ /

.

......./;..

10

20

>>>~_

f..

///--'>J" ././////"

_

0.2

==

'/3//.././>" /,.

i

0.3

c-,l

0.6

I

~

0.1

,~-/

bl) =L

0.4

//

0.0

/ / <. .>'/~U./~/7

////~;~/'

y

0.2

1.0 0.8

0.0 0

2

4

i

i

6

8

i

r

10 0.6

1.0

m

v

'

0.4 0.8

0.2 0.0

0.6

30

0

I

I

10

20

30

0.4 0.2

&--, ~

0.8 ~

//.

/ ~ ~ - - - -

0.6 r

O0 0

/jj:7~:~:.S~I~

"

~.~K ./~.~..,~. ~

2

4

6

8

. . . .

10

time [min]

Fig. 8. Interracial concentration plotted against time at different distances from the entrance of a slit of width 10 m m (equivalent to a semi-infinite medium) for k a = 1 0 - a c m s - L Average distance x = 0.5, 0.9, 1.9, 2.9, 3.9, 4.9, 5.9, 6.9, 7.9, 8.9, 9.9 cm; F~t = 1.0 ~tg cm 2. U p p e r graph, Cb = 20 Ixg ml ~; lower graph, C b = 10 i.tg m1-1.

against ( 1 - F/F~,t), an interfacial concentration limit could be derived as well as an exponent ~ of the power law dF/dt ~ (1 - F / F ~ t ) ~. Such a verification (Fig. 10, top) requires the interfacial concentration to be within about 15% of its saturation value with k~= 1 0 - 3 c m s -~. Under these conditions, the final adsorption process is sufficiently slow to ensure a constant bulk concentration at the interface along the entire length of the slit or capillary.

4. Conclusion

In a simulation study of the early steps of an adsorption process, verification of the approximate formula Eq. (1) which was designed to take into

L.

~ _

0.2

o

0

2

4

6

8

10

time [rain]

Fig. 9. Interracial concentration plotted against time at different distances from the entrance of a slit of width 10 m m (equivalent to a semi-infinite medium) for different values of ka. Average distance x = 0.5, 0.9, 1.9, 2.9, 3.9, 4.9, 5.9, 6.9, 7.9, 8.9, 9.9 cm; Cb = 20 ixg ml 1; F,a t = 1.0 ~g cm 2 U p p e r graph, k a = 10 - s cm s - l ; middle graph, ka = 10 -4 cm s - ~; lower graph, k a = 1 0 3 c m s -1.

account coupling between the interracial reaction and transport to the interface by diffusion and convection demonstrated the limits of this approximation. Far from the L6v6que limit, accurate values of the kinetic constant could be obtained whereas the diffusion coefficient was overestimated up to about 30%, while for large values of the kinetic constant it was possible to determine only the diffusion coefficient, as it occurs in the L6v6que model. The discrepancies between the approximate relation Eq.(1) and these computational results indicate the need to derive an expression giving a more accurate description of the transition from a

P. Dkjardin, I. Cottin/ Colloids Surfaces B." Biointerfaces, 4 (1995) 111-120

calculations with k~ = 10 -3 c m s - 1 allowed verification of the introduced power law above 85% of the jamming limit, when conditions of negligible depletion near the wall are met.

10 -2

~r.~ m

119

10-3

E r,.) ::L

Acknowledgements .,~

104

10-5 0.01

J

1.00

0.10

(1-F/Fsa t) ,

i

i

This work was supported by the " G D R 960 Biochromatographie" from CNRS (B. S6bille). The author also thanks Professor P. Schaaf for discussions concerning the RSA model and for providing Ref. [16].

i

Appendix In a capillary model, the equations analogous to Eqs. (8)-(10) of the main text for a slit may be defined as follows

0.4

)

(A1)

r ( j ) = R -- j J r

0.2 ~-~

0.0 0

. 4

.

.

. 8

.

. 12

.

. 16

20

time [rain] Fig. 10. Simulations with an excluded factor describing a random sequential adsorption process (see Eq.(12) in text) (Cb = 20 gg ml-1; k , = 1 0 3 cm s-l; Fsa~ = 0.547 gg cm-Z). Lower graph {from left to right): average distance x = 0.5, 1.9, 5.9, 9.9cm. Upper graph (from top to bottom): x = 0 . 5 , 1.9, 9.9 cm.

where R is the radius of the capillary and r ( j ) is the distance of the cell (i, j) to the capillary axis. The average variation of concentration in a cell (i, j) over time At after time t is AC(i, j ) / A t -- f ( j ) D { [ C ( i ,

j + 1, t ) -

C(i, j, t ) ] r ( j ) / A r

- [C(i, j, t) - C(i, j - 1, t)] Jr(j) + A r ] / A r } - v ( j ) [ C ( i , j, t ) -

C(i-

1, j, t ) ] / A x

(A2)

where transport controlled reaction to a chemical one, work which is currently in progress in our laboratory [ 18 ]. From a computational point of view, it could be useful, as suggested by one referee, to use a non-uniform grid to describe properly the entrance region [13]. However, these results suggest that the linear variation of k-1 with x ~/3 still is a good approximation in the transition from one regime to another, at least on a scale of 10 cm, but with variations in slope and ordinate at the origin. Simulation of the final steps of the adsorption process was performed within a RSA model. The

f ( j ) = 2/{ [r(j) + Ar] 2 -- r(j) 2 }

(A3)

Special formulations hold for a cell at j = j m a x and for a cell near the wall at j = 1 A Cfi, Jmax)/Z] t = f ( J m a x ) D [ C ( i , Jmax -- 1, t) -- C(i, Jm~x, t)]/(Ar) 2 -- V(Jma,,)[C(i, Jmax, t) -- C(i - 1, Jmax, t)]/zlx

(A4) where f ( J ~ x ) = 2/{ [r(Jmax) + Ar] 2 -- r(jm.x) 2 }

(A5)

P. DOjardin, I. Cottin/ Colloids Surfaces B: Biointerfaces, 4 (1995) 111-120

120

When computation is performed over the entire section of the capillary, then r(jmax) = 0. AC(i, 1)~At = T ( 1 ) D [ C ( i , 2, t ) -

C(i, 1, t ) ] r ( 1 ) / A r

-- v(j)[C(i, 1, t) - C(i - 1, 1, t ) ] / A x -- [ A F ( i ) / A t ] / A r

(A6)

where f(1) = Z/JR 2 - r( 1 )2]

(A7)

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