Langmuir 1994,10, 3898-3901
3898
Adsorption Rate in the Convection-Diffusion Model P. Dejardin,* M. T. Le, J. Wittmer, and A. Johner Institut Charles Sadron (CNRS-ULP), 6 rue Boussingault, 67083 Strasbourg Cedex, France Received March 18, 1994. In Final Form: July 11, 1994@ This study shows that adsorption experiments under flow allow determination of a phenomenological kinetic constant k,, provided the system corresponds to an appropriate regime. Measurements of the adsorption rate along a capillary enable calculation of both the kinetic constant k a and the diffusion coefficient D of the solute, and the latter can be compared with independent results from other techniques. Alternatively, measurements at a single position give an estimate of 12, when the diffusion coefficient D is known. The conditions under which these results may be derived are discussed in detail.
I. Introduction There has been constant interest in the adsorption of proteins and solid particles over the last three decades. Recently data have become available from optical’-3 and radiolabeling t e c h n i q ~ e s ~which -~ enable continuous recording of the adsorbance of a solute, and there is experimental evidence that the mechanism may be complex, as for example in exchange processes.’ Detailed quantitative analyses of such experiments are, however, lacking, and we propose in the present paper an analysis of adsorption under flow. Adsorption experiments are often performed under static conditions. In the case of a capillary system, conditions have been thoroughly analyzed in the litera t ~ r e ? -while ~ simulations with analysis of the sticking coefficient have also been performed using a Langmuir excluded surface factor.’O Although static experiments can in principle give information on the interfacial process, usually described by the kinetic constant k , for irreversible adsorption, this type of study is nevertheless difficult to perform. Early stages are sensitive to the filling process and later stages subject to large depletion layers. Moreover, it would appear difficult to avoid convection at the level of such layers. Alternatively, adsorption experiments can be performed under flow, and it would seem easier to impose the velocity field of the flow. However, interpretation of the results is less straightforward. As will be shown, adsorption experiments under flow should allow relatively simple determination of the kinetic constant k,, provided the system obeys an appropriate kinetic regime, but even for moderate shear rates at the solutiodsubstrate interface the phenomenological constant k , may become shear rate dependent, as when a protein molecule first binds weakly to the surface before it sticks irreversibly. Under these conditions, it should nevertheless be possible to calculate Abstract published in Advance A C S Abstracts, September 1, 1994. (1)Lok,B.K.; Cheng,Y.-L.;Robertson, C. R. J . Colloid Interface Sci. 1983,91,87. (2) Elgersma, A. V.; Zsom,R. L. J.;Lyklema, J.; Norde, W. Colloids Surf. 1992,66,17. (3) Hlady, V.;Reinecke, D. R.;Andrade,J. D. J . Colloid Interface Sci. 1988,111,555. (4) Voegel, J. C.;de Baillou, N.; Sturm,J.;Schmitt, A. Colloids Surf. 1984.10.9. (5j Boumaza,F.;Dbjardin,P.;Yan, F.;Bauduin, F.;Holl, Y. Biophys. Chem. 1992,42,87. (6) Yan, F.; Dbjardin, P. Langmuir 1991,7 , 2230. (7)Scott, C. J . Biomater. Sci. Polym. Ed. 1991,2,173. (8) Young, B. R.; Pitt, W. G.; Cooper, S.L. J . Colloid Interface Sci. 19M 125 2AR - __,---, --. (9) Wojcechowski, P.; Brash, J. L. J . Colloid Interface Sci. 1990,140, @
299
(10)Weaver, D. R.;Pitt, W. G. Biomaterials 1992,13,577.
k , a t zero shear rate by extrapolation. Adsorption under flow furthermore occurs commonly in nature, and its dependence on shear rate is in itself interesting. In previous work, we have already simulated adsorption processes with a non-zero shear rate at the interface. A n approximate analytical solution was proposed for a system with a finite adsorption constant k , and a fured thickness of the diffusion 1ayer,I1on the basis of simulations which maintained the same bulk concentration throughout the kinetic process a t 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 k , at the interface and the constant k h v of the LBveque model, which assumes an infinite adsorption constant and a semi-infinite medium:
(la) with
r2) 2
k,
= 0.538
1/3
X
(lb)
In relation lb, D is the diffusion coefficientof the solute, y the shear rate at the wall, and x the distance from the
entrance of the slit. The basis of eq l a is as follows: the Nernst depletion layer thickness YN normal to the wall under steady state conditions for C(xy)was assumed to be the same as in the L6vSque model, while the stationary concentration at the wall C(x,O)was estimated by equating the rates of transport and adsorption at the interface through the finite kinetic constant k,. Recently, results of further simulations12have shown some departure from relation l a , leading to overestimation of the diffusion coefficient by about 30%at low k, and to overestimation of k , at high values of this parameter. Therefore, in order to go beyond this approximation, we present here a more detailed analysis of the problem and also discuss finite medium and curvature effects. In the following section we treat the semi-infinite case. Special attention is paid to steady state situations which are likely to be exploited experimentally, and the simple relation l a will be considered in detail. Section I11briefly discusses finite size effects for slits and capillaries. 11. Crossover from Surface-Controlledto
Transport-ControlledAdsorption For a solute of diffusion coefficientD and concentration C(xy,t),where x is the distance from the entrance of the (11)Dbjardin, P. J . Colloid Interface Sci. 1989,133,418. (12)Dbjardin, P.; Cottin, I. To appear in Collids Surf. B: Bwinterfaces.
0743-7463/94/2410-3898$04.50/0 0 1994 American Chemical Society
Adsorption Rate in the Conuection-Diffusion Model
Langmuir, Vol. 10, No. 10, 1994 3899 X
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equation shows that the concentration becomes stationary, indicating a constant rate of surface coverage. At a distancex from the entrance, the characteristic relaxation time for the adsorption rate is found to be
(s) 2
=
Figure 1. Reduced adsorption rate klk. = g(X) as a function ofreduced distance to the entranceX=x/Lco. The approximate given by eq 9 and the asymptotic expansion given in eq 6b are also shown (dashed lines).
slit, y the distance from the wall, and t the time, we take into account convection and diffusion in the velocity profile u(y) by applying the differential equation
-a_C at
- u @ )a -c+ D -
ax
a2c
(4)
This relation states that the depletion hole of order covers by convection a distance x in the direction of the flow during a time z, and in the following discussion we focus on the stationary regime where reliable measurements can be performed. Close to the slit entrance adsorption is governed by interfacial reactions, whereas at greater distance the solution near the interface becomes dilute and adsorption is controlled by transport from the bulk solution. The crossover point between these two regimes is expected to occur when k, % khv, at a distance L,,% D2y/ka3from the entrance. More precisely, direct solution of eqs 3 and 4 yields the adsorption rate:
with (2)
ay2
X = X
Throughout this section we neglect finite size effects and consider the case of simple shear with shear rate y in a semi-infinite medium. The velocity profile then reduces to u(y) = yy, while the bulk concentration c b enters through the boundary conditions
t =o x
113
'0 C(xy,O) = 0
where r(2/3)
3 ~ 3 ,
L,,=3 -
(rm)
(34 (3b)
(5b)
=a,
k:
(54
the prefactor being close to 0.387. The functiong(;k?allows the following expansions:
1 1 - -x>>1,g(x,= -r(2/3) x i 1 3 r(i/3) p 3
As we are not interested in saturation effects,any excluded surface term is neglected. Solution of the time-dependent
(6b)
k,l k
1
1
2
3
4
5
6
Figure 2. Plot of the inverse reduced adsorption rate k,/k against the inverse reduced Lev6que constant k$kbv. The approximate
given by eq 10b and the asymptotic expansion given by eq 11 are also shown (dashed lines).
3900 Langmuir, Vol. 10, No. 10, 1994
Dkjardin et al.
Table 1. Crossover Length L,, (cm)and Critical Width b* (mm)for Various Values of the Kinetic Constant k. and Difftusion Coefficient D D (cm2s - l ) 2 10-7
10-5 8 x lo3 4 104
[Y = 200 8-11 [y
= 1000 8-11
[b* = 0.12mml 1 x 10-6
2
[ y = 200 8-11 [ y = 1000 9-11
+ G(2/3,X) - G(1/3,X)
(7a)
with
although this relation would enable a two-parameter fit. The function g(X) may also be interpreted in terms of the effective constant k:
8x 4 x 10-2
[b* = 0.0012mml
[b* = 0.012mml 2 x 102 103
[b* = 0.06 mml
Equation 6a describes the interface-limited regime and eq 6b the IRv6que transport-limited regime. Due to the l / 3 powers, there nevertheless exists a large crossover region which covers about four decades of position along the tube. Hence we do not use the explicit function g(X) plotted in Figure 1:
10-3
8 40
105 106
[b* = 0.6 mml
g(X, = e-x
10-4
0.2 1.0 [b* = 0.006 mml
measurements in the far khv-l regime do not allow accurate extrapolation a t vanishing khv-l. In Figure 2 we plot k$k as given byg(X) in eq 7a against k$kh,. This permits estimation of the accuracy of such a determination ofk,, and the two approximate equations (lob and 11)can be seen to give a satisfactory description of the kinetics over the entire range. Another parameter of interest when, for example, measurements of radioactivity are performed during adsorption of labeled particles is the amount of missing solute per unit adsorbing area in the depletion hole:
with q(X) = -G(l,X)
+ G(4/3,X) + G(2/3,X)
(12b)
and for X 1 where
111. Finite Size Effects Experiments are most often carried out in slits or capillaries where finite size effects are to be expected as the size of the depletion hole becomes comparable to the width of the slit or the radius of the capillary. Assuming Poiseuille flow characterized by a shear rate y at the interface, we now consider the validity of relation 10b in these devices of finite size. For a slit of half-width b or a capillary of radius b, the adsorption rate at lowest order in X and l/b is
where a is '/Zo for a slit and 3 / 1 for ~ a capillary. Finite size effects become more important far from the entrance, and at the crossover distance L, the size b must be compared with the length b* = [~1T(l/3)m(2/3)1 D/k., the prefactor being close to 0.1 for a slit or 0.6 for a capillary. For b larger than b*, finite size effects may be neglected, while for b of order b* eq 13 allows correction. Approximants of the type
for slits and capillaries compare reasonably well with earlier simulations.
IV. Discussion the prefactor being close to 0.684. Although it is theoretically possible t o derive k , from this asymptotic behavior,
In this paper we propose a method to derive the phenomenological constant k , governing the interfacial reaction in adsorption experiments under flow. The simplest procedure is to determine the apparent constant
Adsorption Rate in the Convection-Diffusion Model
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Time (mn) Figure 3. Experimental data obtained using the radiolabeling technique. A plot of measured radioactivity against time shows the steady state regime and an activity drop when the capillary is rinsed with solvent, which allows calibration. Fibrinogen concentration 6.8 pg mL"; pH 7.4; Tris buffer 0.05 M containing 0.10 M NaC1; silica capillary radius 265 pm. k from the measured steady state and the relation r'= kCb and to plot k-l as a function of the known quantity D2/3k~ev-1, according to eq 10. This gives both k, and the diffusion coefficientD of the solute. Measurements must be performed at x < L,,,as the region close to the entrance of the device (x < Leo) is most sensitive to k,. The crossover distance L,, is proportional to k,-3, D2,and the shear rate y at the interface. Only the shear rate itself is a free parameter, and one cannot expect the interface-controlled regime to be easily obtained for an arbitrary proteid surface system. Outside this regime, information may be obtained from the asymptotic expansion ofk-l (eq ll),but since extrapolation from the far kLev-l is of limited accuracy, in practice k , should not exceed about 20khv.A two-parameter fit (D, k,) using the fullg(X) function could be helpful, especially near kLev = ka. For experiments performed in slits or capillaries, finite size effects may be ignored over the entire region x 5 L,,, provided the transverse dimension is larger than a characteristic length b" = PD/k,, where p is 0.1 for slits and 0.6 for capillaries. Explicit numerical examples using diffusion coefficients typical of proteins like fibrinogen and lysozyme are given in Table 1. Recently, we studied the adsorption of fibrinogen on siliica under flow ( y = 200 s-l). Silica capillaries of diameter 530 pm were first treated with dilute sulfochromic acid a t 50 "C, followed by a mixture of 30%(w/w) aqueous HzO2, 25% (wlw) aqueous NH3, and water (respective volume ratios 25/5/70) at 80 "C under flow conditions for 1h. This cleaning procedure, completed by thorough rinsing with deionized water (SuperQ,Millipore), ensured a high negative charge density at the interface. Data are presented in Figure 3 for a first series of experiments in which the adsorption rate was measured at a single position during the successivepassage ofbuffer, labeled solution, and buffer through a bundle of capillarie~.A ~ ,large ~ steady state is clearly evident before saturation. The measured constant k is close to 4.6 x c d s , and from the diffusion coefficient D = 2.7 x 10-7 cm2/s at 37 "C and the distance to the entrance x = 7.5 cm, we calculate k h v = 6.7 x c d s . Thus the experimental point lies close to the crossing of the two
Time (mn)
Figure 4. Plot of interfacial over bulk fibrinogenconcentration againsttime for adsorption from diluted human plasma (0.1%). T = 37 "C; capillary radius 265 pm. Upper curve: y = 500 s-l;
cb = 2.8 pg/mL.
Lower curve: y = 200 s-l; c b = 3.6 pg/mL.
approximate laws of eqs 10b and 11, where these laws are cds least applicable. Relation 10b gives k, = 1.1x while relation 11gives k, = 1.0 x c d s . Since these estimates are not far from the value obtained using the full expressiong(X) (k, = 1.2 x c d s ) , inmost practical cases, eqs 10b and 11 should be sufficiently accurate in view of the experimental precision. The aim of this short section was mainly to show that the method we propose to determine k, can be applied even under less favorable conditions. Consistency and reproducibility were further verified with additional experiments where labeled fibrinogen was added to diluted human plasma. Results are given in Figure 4 as a plot of the interfacial concentrationn r over bulk concentration against time. At a shear rate of 500 s-l, the initial experimental kinetic constant of 5.5 x c d s led to k, = 1.1 x c d s from law 10b and to k, = 0.95 x cm/s from law 11. In another experiment, with a higher concentration of labeled fibrinogen to increase the sensitivity of detection, the same initial c d s at y = 200 s-l experimental constant of 4.6 x was obtained as in the case of a single protein solution.
