Colloidal Particles at Solid–Liquid Interfaces: Mechanisms of Desorption Kinetics

JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.

206, 322–331 (1998)

CS985636

Colloidal Particles at Solid–Liquid Interfaces: Mechanisms of Desorption Kinetics Michael Weiss,*,1 Yves Lu¨thi,* Jaro Ricˇka,* Thomas Jo¨rg,† and Hans Bebie† *Institute of Applied Physics, Sidlerstr. 5, University of Bern, CH-3012 Bern, Switzerland; and †Institute of Theoretical Physics, Sidlerstr. 5, University of Bern, CH-3012 Bern, Switzerland Received February 17, 1998; accepted May 1, 1998

We study the sorption of colloids on equally charged surfaces. Our focus is on the time scale from hours to weeks, where adsorption is not an irreversible process but interplays with (spontaneous) desorption. Using model calculations, we show how the desorption kinetics is influenced by readsorption, a potential barrier, a secondary potential minimum, local variation of the potential, and bond aging. In the experimental part we present results of in situ observation of the sorption kinetics of polystyrene latex particles onto a glass surface. Combining the evanescent field method with video microscopy, we were able to identify the particle arrival and departure times individually and therefrom determine the adhesion time distribution function. The nonexponentiality of this function can be explained by a gamma distribution of the potential depth at the binding sites as well as by logarithmic bond aging. © 1998 Academic Press Key Words: colloids; readsorption; nonexponential desorption kinetics.

1. INTRODUCTION

The deposition of colloidal particles on a collecting surface and their release from it are of great interest to a large scientific community. Ecologists, for example, consider the colloidal transport of pollutants in groundwater (1). Medical research deals with several major diseases caused by colloidal transport and deposition such as cardiac infarctions, strokes, and metastases, the latter originating from malignant cells released from a primary tumour (2). There are also important industrial applications such as emulsion paints and water-soluble glues. For almost 50 years, DLVO theory (3, 4) has been accepted as the standard theory to describe adsorption and desorption phenomena. Some words about why we cannot use it for our experiment may therefore be appropriate: DLVO theory has been quite successful in predicting adsorption rates when the charges of particles and surface are oppositely signed (5–9). However, its failure in the case of equally signed charges, where adsorption is controlled by the height of the electrostatic barrier that an adsorbing particle has to overcome, is dramatic (5, 10, 11). Indeed, for the parameters of our experiment, DLVO theory predicts a barrier height of several hundred kBT, 1

To whom correspondence should be addressed.

0021-9797/98 $25.00 Copyright © 1998 by Academic Press All rights of reproduction in any form reserved.

which would prevent any adsorption or desorption, in strong contradiction to our experimental findings. The difficulty in establishing a correct theory originates in the exponential dependence between barrier height and transition rate: Only slight changes of the barrier height lead to huge differences in the rate at which the barrier is passed. Hence, seemingly secondary effects such as surface charge fluctuations and mobility or surface heterogeneities, which are difficult to control and to predict, can increase the sorption rates by orders of magnitude and are usually referred to as possible explanations of the deviations between experiment and theory (10, 12, 13). It is possible that DLVO theory is in general correct even for equally signed charges on particles and surfaces, in predicting that on most parts of the surface, there is no adsorption at all taking place. However, surfaces may contain oppositely charged spots which will then serve as binding sites. This was also found in our experiment (for details, see (14)). Binding sites and other models of surface heterogeneity help a lot to explain adsorption under unfavorable conditions (12, 15, 16). More details about the interaction between binding sites and particles need to be known, however, to predict desorption kinetics. Desorption takes place on a time scale of hours to years, and it is difficult to keep experimental conditions constant during such a long time (it is, e.g., a considerable problem to keep the particles from coagulating) and to obtain reproducible results. This may be one of the reasons experimental data on desorption in colloid–interface systems are rare. Probably the most systematic study comes from Kallay et al. (17). The study most similar to ours is from Meinders et al. (18 –20); concerning desorption kinetics, their focus is on the role of collisions. The investigation of freshly contaminated and 30-year-old soil samples by Connaughton et al. (21) encompasses the largest time scale on which desorption experiments have been performed. If desorption were a stationary single-rate process, an exponential adhesion time distribution would be expected. However, in all experiments cited above, and also in ours, an increasing resistance to desorption with increasing adhesion time has been observed. This is, the adhesion time distribution decreases slower than an exponential function. In various papers, it has been noted that a distribution of desorption rates

322

COLLOIDAL DESORPTION

can explain this effect quantitatively (17, 21), but an interpretation of the effect is either missing or given in terms of nonidealities. We believe that deviations from exponential release kinetics are inherent to the interaction of colloids and surfaces and deserve more attention than they are usually paid for. In this paper, we will reduce the explanation by a rate distribution to the more fundamental concept of a potential distribution, and we will consider alternatives to it. Our list of possible reasons for the phenomenon is of course only a selection based on subjective estimations of plausibility and our actual knowledge. Nevertheless, we think that our selection elucidates the role of some phenomena which are important for a broad variety of colloidal solid–liquid interface systems. The list contains the following: Readsorption. Particles that have left the surface can return to it due to diffusion. This may introduce non-rate-like processes. Secondary minimum. After escape from the primary binding site, a particle may be trapped in a secondary minimum from which it has an increased probability to return to the primary minimum. Distribution of potential depth. As a distribution of the potential depth implies a desorption rate distribution, this is essentially the standard explanation: For long binding times, the slower rates dominate, hence the increasing resistance to desorption. However, this explanation makes sense only when a typical ‘‘wavelength’’ l of a fluctuation is much larger than the contact area of one binding site and much smaller than the total observed area. This is very likely for our experiment, because the binding sites are isolated, so that l is about the average distance of two binding sites. Aging of binding sites. Plastic deformation of particles may increase the binding force with increasing binding time. This also reduces the desorption speed with increasing adhesion time. In order to see which of these effects can quantitatively explain our experimental findings, we have investigated four corresponding model situations. This investigation requires that we carefully discuss under which conditions we can give sound definitions of our observable quantities, what these definitions look like, and what they depend on. This is done in Section 2, together with the discussion of the readsorption problem. In Section 3, models for the secondary minimum, a potential depth distribution, and bond aging are presented and calculated. Both the readsorption and the secondary minimum can be ruled out as explanations of our nonexponentialities. However, we will show in Section 4 that both the distributed potential and the aging model are in good agreement with the experimental data. We conclude with Section 5. 2. DEFINITIONS

Probably the main reason for the lack of reliable results on desorption is the subtleties in defining the relevant observables:

323

they depend to a great extent on details of the experiment. In the present discussion we concentrate on experiments that are performed in situ, in the presence of the liquid. Even the most simple definition, namely that of an adsorbed particle, requires some caution. It is natural to regard a particle as adsorbed when it is found in the primary or in a secondary minimum of the interaction potential U. Theoretically, the limit b of the interaction range can be taken as the outermost local maximum of 2­U 2 /­ z 2 , where z is the distance between the particle and the surface. However, the potential is never known with an accuracy that would permit a reasonable estimate of b. Furthermore, the depth information about the particles is very limited; essentially, all we can expect is that there is an observation depth d which permits us to distinguish with some accuracy particles with z , d from particles with z . d (of which we treat the latter as invisible). However, d is considerably larger than b; even with the near-field technique of evanescent field, d amounts to a few tens of nanometers. Therefore, in order to distinguish particles in a potential minimum from other visible particles, we have to use the difference in the time scales: We regard those particles as adsorbed which are visible and (almost) immobilized for a time t which is much larger than the timescale (d 2 b)2/(2D) of diffusion within the boundary between b and d (D is the diffusion coefficient). To test for immobilization, we need a sufficient lateral resolution. In our experiment, we consider a particle as adsorbed when it is visible in at least two subsequent images, which were taken in a time interval of at least 100 s, and when its position change during this interval does not exceed a small value Dr. In order to account for possible migration of binding sites, it can be reasonable to choose Dr greater than the lateral resolution. The definition of adsorption is insensitive to d as long as the probability of finding a particle in the range [b, d] is much smaller than the probability of finding it in the range [0, b]. How large d can be chosen depends, inter alia, on the concentration of particles in the bulk, but as soon as d is smaller than a particle radius (which can easily be accomplished with the evanescent field method), its influence on the definition of adsorption can safely be neglected. In order to define the observables of desorption kinetics, we guide ourselves by a gedankenexperiment: Suppose that the investigated surface is brought into contact with the particle suspension, where particles are deposited and released during a time t. Then we replace the suspension with a pure liquid and measure the decrease of the density of adsorbed particles s(t, t) from t to t 1 t. (The switch to the pure liquid is realized in a real experiment simply by not counting all particles that adsorb later than t.) The measurement can be done in two ways which lead to completely different results: First, we can simply count all particles which are found to be adsorbed on the observed surface at some time t 1 t. Thereby, we also include particles which leave the surface but get readsorbed again. We denote the density obtained in this way as sr(t, t). Secondly, we can exclude readsorbing particles from our counting. In this

324

WEISS ET AL.

way, we obtain another density sd(t, t). In our real experiment, we exclude all particles from the counting whose lateral position changes more than Dr between two subsequent images. This method is not perfect, but it can be improved considerably by flow parallel to the surface which carries readsorbing particles further away than pure diffusion would. In our gedankenexperiment, we can simply remove all particles as soon as they pass some distance c from the surface for the first time. The parameter c here accounts for Dr and for the flow strength, and also for the geometry of the setup, which has a strong influence on whether a particle gets readsorbed on the observed surface or elsewhere. Because of the last factor, c is also finite for sr. In fact, in our gedankenexperiment, sd and sr can be regarded as limiting cases of one single function s(t, t; c), corresponding to small and large values of c, resp. The limit c* between small and large values is set so that for sd, the timescale of diffusion in the range [b, c] is much smaller than the time scale of escape from the surface, and vice versa for sr. A complicating effect occurs when there is a variation in the depth V of the potential well (V1 1 V2 in Fig. 1) or when the particle–wall bond is subject to aging. For an illustration, let us consider the case of distributed depth, where p(V)dV be the probability to find a particle in a well with depth V. In our gedankenexperiment, this distribution depends on the history of the surface before it was inserted into the pure liquid: Since the particles stay longer in the deeper sites, p(V) will be progressively skewed toward large V with increasing t. A similar effect occurs with aging. Indeed, in our experiment, we observe a slow down of the decay of sd with increasing t(sd (t, t ) 5 D(t, t )/S in Ref. (22); S is the observed surface area). To account for this in our gedankenexperiment, we only select those particles for our desorption measurements which adsorb right before switching to the pure liquid, i.e., during a small time interval [t2Dt, t]. We denote the resulting quantities as dsr(t, t) and dsd(t, t), thereby indicating their differential character. In order to get rid of the proportionality to the initial coverages, it is suitable to consider the normalized quantities ds˜ r~t, t) [ dsr(t, t)/dsr(t, 0) and ds˜ d~t, t) [ dsd(t, t)/dsd(t, 0) only. Just as sd may depend on t because of the variation in the depth of the potential, ds˜ d, too, may explicitly depend on t if the height V2 of the barrier varies. For example, if V2 is correlated with the depth of the well, the decay of ds˜ d~t, t) will slow with increasing t. If the barrier height is much smaller than the depth of the well, this slowing will end as soon as most adsorption sites are occupied, but there may also be permanent effects, originating, e.g., from nonstationarity of the state of the surface. As for ds˜ r~t, t), slow processes in the liquid may preserve the influence of the initial condition for very long times, thereby causing an additional explicit t-dependence. Our experimental finding that ds˜ d does not depend on t greatly simplifies the analysis of the measurements. The evolution of ds˜ d~t, t) is an experimental estimate of the distribution of adhesion times Pd(t|t). This distribution is the probability that a particle which has adsorbed within an infin-

itesimal time interval [t2dt, t ] remains permanently adsorbed at least until the time t 1 t. (In view of our experimental finding, we will drop the explicit t dependence of Pd in the following discussion.) The counterpart to Pd is the recurrence probability Pr(t|t), i.e., the probability that a particle which has adsorbed within an infinitesimal time interval [t2dt, t ] is still or again adsorbed somewhere on the observed surface at the time t 1 t. Pr is estimated by ds˜ r~t, t). Pr is the relevant and only accessible quantity in many complex situations like filtration and colloidal transport in the subsurface of an aqueous environment. However, Pd is independent of specific features of the experiment such as the geometry of the flow and of the surface and thus is the more fundamental quantity. Also, it is much more accessible to theory. In our experiment, where we can measure Pd, we will therefore focus on it. In order to elucidate the dependence of Pr and Pd on those details of the experiment which influence the readsorption, we investigate the model situation of one-dimensional Brownian motion in a piecewise constant potential sketched in Fig. 1. We will refer to this model as the readsorption model. Within its framework, we define

P ~t ! 5

E

d

W ~ z, t !dz

[1]

0

where W ( z, t )dz d t is the probability of finding a particle in the space–time interval ( z, z 1 dz ) 3 ( t , t 1 d t ) , and the initial condition is that at t 5 0, the particle is at a random position within the potential well. As in our gedankenexperiment, c accounts for all factors which influence readsorption: P can be tuned from Pd to Pr simply by moving the absorbing boundary c from a small value to infinity. Following our former discussion, we do not choose an observation depth d larger than a particle radius. Let us now consider the formal aspects of the model. It is described by the one-dimensional Smoluchowski equation

F

G

­W ~ z, t ! D ­ ­ ­ 5 W ~ z, t ! V~z! 1 D ­t ­ z kB T ­ z ­z

[2]

for the probability density W, with

V~z! 5

H

2V 1 0 # z , a V2 a # z , b 0 b#z,c .

[3]

As this potential is piecewise constant, the Smoluchowski equation reduces to a simple diffusion equation, ­W ~ z, t ! ­ 2 W ~ z, t ! 5D ­t ­ z2

[4]

325

COLLOIDAL DESORPTION

heat conduction and has been verified in the complete analytical solution. Under these conditions, W(z, t) obeys P˙ d ~ t ! 5 2j ~ t ! 5 2D

W ~c, t ! 2 W ~a 1 0, t ! , c2a

[6]

where the dot means differentiation in respect to t, and f (x 6 0) is defined as lime30 f (x 6 e) with e . 0. Using W(c, t ) 5 0, W (a10, t ) 5 W(a20, t )e2V1 /(kBT), and Pd (t ) 5 aW(a 2 0, t ), we find P˙ d ~ t ! 5 2e 2V1 /~kB T! FIG. 1. The potential of the readsorption model with primary minimum V1 and potential barrier V2. The absorbing boundary c is a measure of our ability to detect readsorptions. The electrostatic barrier can be switched off by setting V2 5 0.

with additional constraints at z 5 a and z 5 b. We take the diffusion tensor D as a constant because its dependence on the distance from the wall can be included in the form of the potential and in a coordinate transformation. Our initial condition is

W ~ z, 0! 5

H

1 0,z#a a 0 a , z # c.

[5]

Any initial distribution which is nonzero only within the well will converge to this distribution on a time scale given by t0 5 a2/(2D). This is usually the fastest time scale of the system. The boundary at 0 is reflecting; the boundary at c is absorbing. V1 and a describe the attractive part of the interaction potential; V2 and b characterize the electrostatic repulsion. This part may be omitted by setting V2 5 0 and b 5 a. For the calculations we assume V1 $ 5kBT. Equation [4] can be solved analytically by means of Laplace transformation (23). We restrict ourselves to the solution in the four cases of small and large c, both with and without an electrostatic potential barrier. With ‘‘small’’ and ‘‘large’’ c, we mean c ! c* and c @ c*, where c* is obtained in the model by equating the timescale of escape from the potential well, which is found to be given by (23) te 5 (a2 /D)e2(V11V2 )/(kBT) , with the time scale tD 5 c2/(2D) of diffusion within the boundary [0, c], yielding c* 5 =2ae(V11V2 )/(kBT) . We consider the case of large c only in the limit c 3 `. 1. Small c and V2 5 0 (P 5 Pd). Recalling that within certain limits, P is insensitive to the optical resolution d, we consider only the case d 5 c. As tD ! te, the probability current j(z, t) for any fixed time t is almost independent of z on the interval (a, c). This is intuitively clear from the analogy to

D P ~t ! ; a ~c 2 a ! d

[7]

hence, we have desorption at a rate of k1 5

De 2V1 /~kB T! . a ~c 2 a !

[8]

The assumption Pd (t ) 5 aW(a 2 0, t ) is valid only when thermal equilibration within the well is much faster than diffusion from a to c, i.e., when a ! c 2 a. This assumption is not restrictive because smaller values of c would in any case be unrealistic. Thus, in the range of interest, the desorption rate is inversely proportional to c. This suggests that the observed desorption rate depends considerably on our ability to detect readsorptions. However, the way c affects Pd is simple: it just rescales the time unit. This is true even when there is a distribution of the potential depth, because c is independent of V1. 2. Small c and V2 . 0 (P 5 P d ). A similar reasoning as in case 1 again leads us to a rate equation for Pd. The rate is now given by k2 5

De 2V1 /~kB T! . a @~c 2 b ! 1 ~b 2 a !e V2 /~kB T!#

[9]

For large values of V2 (i.e., (b2a)eV2 /(kBT) @ c2b at constant c), the second term in the denominator dominates, and the rate is almost independent of the choice of c. Pd depends only weakly on the ability to detect readsorptions. For larger values of c (opposite case with constant V2, but still in the range c ! c*), the first term in the denominator dominates, and the influence of the barrier diminishes. 3. c 3 ` and V2 5 0 (P 5 P r ). This case is exactly solvable (23) if we set d 5 a, again making use of our freedom in choosing d within certain limits. The solution is an infinite series, which for t @ a2/(2D) is well approximated by P r ~ t ! 5 e t /teErfc Î t / t e ,

[10]

where t e 5 (a 2 /D )e 2V 1 /(k B T ) and Erfc( x ) 5 (2/ = p ) * ` x 2 e 2j d j is the complementary error function. Clearly, the time

326

WEISS ET AL.

evolution is nonexponential, which is due to the fact that diffusional processes on all time scales contribute to Pr. 4. c 3 ` and V2 . 0 (P 5 P r ). The same definitions, restrictions, and arguments as in case 3 apply. We have calculated that the solution for d 5 a is now given by

2K1 K2 Pr ~t ! 5 p

E

`

0

S

D

D 2 tz a2 dz z2 @~ A~z!!2 1 K22 ~B~z!!2 # sin2 z exp 2

[11]

where A ~ z ! 5 K 1 sin z cos k z 1 cos z sin k z B ~ z ! 5 K 1 sin z sin k z 2 cos z cos k z

k5

b2a a

S D S D

K 1 5 exp

V1 1 V2 kB T

K 2 5 exp

V2 . kB T

A derivation of this result will be published in a separate paper. A comparison of Eq. [10] and Eq. [11] is shown in Fig. 2. Obviously, the barrier V2 has no influence on the long-time evolution of Pr. This is because processes on this time scale are controlled by diffusion over long distances rather than by the comparably fast process of crossing the barrier (whose time scale is e 2V 2 /(k B T ) (b2a ) 2 /D ) . In our experiment, either the first or the second case is realized. We cannot specify this more precisely, because we do not know whether the electrostatic barrier, which should exist due to the

FIG. 2. Pr(t) in the readsorption model for various values of the potential depth V1 (units of kBT). The continuous lines are the solutions in the absence of an electrostatic barrier (V2 5 0); the corresponding dashed lines are for V2 5 1. The parameters are a 5 1, b 5 10, D 5 1 (dimensionless units).

FIG. 3. A sketch of the two-state model. k 1 , k 2 , and k 3 indicate the rates at which transitions between the primary potential minimum, the secondary minimum, and bulk (represented as a sink) occur.

equally signed charge of particles and surface, is also present at the sorption sites. We therefore have to keep in mind that the time scale of our measurements may depend on our choice of Dr and of the flow strength. Also, from this information, the readsorption model shows that readsorptions do not lead to nonexponentialities of Pd. Even if there are small variations in the spatial dimensions of the potential, i.e., in a and b, the deviations of Pd from a single exponential will be small, because their influence on the rates in Eqs. [8] and [9] is only linear. The nonexponentialities found in our experiment must have other reasons. 3. MODELS

3.1. The Secondary Minimum or Two State Model Let us now assume that we have a potential with a deep primary minimum, an electrostatic barrier, and a secondary minimum (less deep than the primary one). Assume further that a particle from the primary minimum jumps over the barrier. Without the secondary minimum, it would quickly be removed, but instead, it is trapped and can perform attempts to jump back. How does this influence Pd? In a potential with a barrier, the relevant diffusional processes happen on a time scale which is much faster than the time scale of adsorption and desorption processes. It is therefore possible to neglect the details of these processes and to assume rates for the jumps over the barrier. Thus, an adsorbed particle can essentially assume one of two states (cf. Fig. 3): State 1 is in the potential well, while state 2 is in the secondary minimum on the other side of the potential barrier. Let n1 and n2 be the probability tho the particle is in state 1 or 2, resp. The adhesion time distribution Pd can most easily be calculated in the situation of our gedanken experiment where no particles are supplied for t . 0. It is then simply given by Pd ~t ! 5 n1 ~t ! 1 n2 ~t ! .

[12]

327

COLLOIDAL DESORPTION

We assume that particles will jump from state 1 to state 2 at a rate k1 and from state 2 to state 1 at a rate k2, and that particles in state 2 will be removed at a rate k3. The rate equations of the system are n˙ 1 ~ t ! 5 2k 1 n 1 ~ t ! 1 k 2 n 2 ~ t !

[13a]

n˙ 2 ~ t ! 5 k 1 n 1 ~ t ! 2 ~k 2 1 k 3 !n 2 ~ t ! .

[13b]

The initial condition is that a particle has just been adsorbed; i.e., n1(0) 5 0, n2(0) 5 1. As long as all three rates are positive, the exact solution for Pd is given by P ~ t ! 5 Ae

2l1 t

1 ~1 2 A !e

2l2 t

.

[14]

In the presence of a deep well, i.e., for k 1 ! k 2 , k 3 , the coefficients are well approximated by l 1 5 k 1 k 3 /(k 2 1 k 3 ), l 2 5 k 2 1k 3 , and A 5 k 2 /(k 2 1k 3 ) . 2 The second term is important at the beginning, indicating fast emptying of the secondary minimum. After a time t2 5 1/(k2 1 k3), most of the remaining particles (their fraction is given by A) are in the primary minimum. After the initial emptying of the secondary minimum, the desorption rate drops to l1. Hence, apart from the different behavior for times of order t2 (typically seconds), desorption still remains a rate process, although the rate, l1, is decreased by a factor of k 3 /(k 2 1k 3 ) , compared to the situation without a secondary minimum. This behavior is very similar to that in the readsorption model: The rate k3 takes over the role of the absorbing boundary c, while k2 is the counterpart to V2. Also, for a high barrier (k 2 ! k 3 ) , where the secondary minimum becomes more and more unimportant, the desorption rate is almost independent of the removal rate k3.

A variation in the depth of the potential well is described by a two-parameter distribution function for real numbers that vanishes for nonpositive values. We choose a gamma distribution with mean ^U& and variance (DU ) 2 5 ^U 2 & 2 ^U & 2 , i.e.,

SD

1 U G ~a ! b

a

e 2U/b

dU , U

[15]

where

a 5 ~^U &/DU ! 2 and b 5 ~DU ! 2 /^U & .

[16]

Neglecting variations in the curvature of the potential at the bottom and at the barrier, we can, in the strongly damped limit, calculate the corresponding rate distribution using Kramer’s result (24, 25), With the initial condition n1(0) 5 1, n2(0) 5 0, the same values for l6 result, but A512k 1 k 2 /(k 2 1k 3 ). 2

[17]

where

k5

D kB T

Î U0 ~ z min !|U0 ~ z max !| 2p

[18]

and zmin and zmax are given by the global minimum and maximum of 2­ 2 U ( z )/­ z 2 (for simplicity, we do not consider secondary minima here). We obtain g ~k !dk 5

S

k B T ln~k / k ! 1 G ~a ! b

S

D

a

D

dk k B T ln~k / k ! . b k ln~k / k !

3 exp 2

[19]

As the escape at a given rate k is described by e2kt, we get

Pd ~t ! 5

E

`

e 2kt g ~k !dk ,

[20]

0

which is the Laplace transform of the rate distribution function. It turns out that the adhesion time distribution calculated in this way can well be fitted to our experimental data (see Fig. 4). While the details of the fitting procedure are the subject of Section 4, we want to point out here that the distribution, given by ^U &'20k B T, DU'1.3k B T is remarkably narrow: This makes the distributed potential model a very likely, though not the only possible explanation of our data. 3.3. The Aging Model

3.2. The Distributed Potential Model

p ~U !dU 5

k ~U ! 5 k e 2U/~kB T! ,

One of the classical paradigms of adsorption and desorption is the hardness of adsorbate and adsorbent: Neither the surface nor the adsorbed particle is assumed to undergo changes after the adsorption, and hence, the interaction potential is regarded as independent of time. Only recently have attempts been made to go beyond this paradigm (26). The aging model, which also assumes a time-dependent interaction, is mainly motivated by a result from measurements of static friction. There is a certain overlap between colloidal science and tribology which is due to surface interaction: As is well known among tribologists, the effective area of contact between two surfaces is usually much smaller than it appears and is proportional to the load (27–29). The radius of an effective contact is comparable to that of a colloidal particle, and therefore, the surface interaction is similar. Static friction is defined as the minimal lateral force that must be applied to two bodies in contact to make them slip relative to each other. This force is proportional to the normal load, the proportionality factor being the static friction coefficient ms. It is well known that the static friction coefficient ms increases slowly with the contact time t (30, 31). Various

328

WEISS ET AL.

phenomenological expressions have been suggested to fit the experimental data (32), among which is a dependence of the form (33)

m s ~ t ! 5 a s 1 b s ln~ t / t s 1 1! ,

[21]

for which a theoretical derivation is presented in (34). As the proportionality between static friction and effective contact area Aeff is well confirmed, the time dependence of ms must be attributed to changes of Aeff due to plastic processes. Hence, the total interaction potential of a single surface contact has the same time dependence of ms. Applying this concept to the surface potential of a colloidal particle, we make the ansatz U ~ t ! 5 U 0 1 U 1 ln~ t / t 0 1 1! .

[22]

In analogy to Eq. [17], this leads to a time-dependent escape rate coefficient, k ~ t ! 5 k e 2U0 /~kB T!

S

t 11 t0

D

2U1 /kB T

.

[23]

The adhesion time distribution obeys P˙ d ~ t ! 5 2k ~ t ! P d ~ t !,

P d ~0! 5 1

[24]

and hence

E

H

t

Pd ~t ! 5 exp@ 2

k~t!dt# 5 exp 2

0

FS D GJ

n k0 t0 t 1 1 2 1 , [25] n t0

where k 0 5 k e 2U0 /~kB T! and n 5 1 2

U1 . kB T

[26]

U 1 ,k BT must be assumed to obtain reasonable results. In the long-time limit, t @ t0, Eq. [25] simplifies to P d ~t ! 5 exp@ 2 ~ mt ! n #

[27]

with

m5

S D

1 k0 t0 t0 n

1/n

.

[28]

Equation [27] represents a stretched exponential relaxation law which is found in a broad variety of condensed matter systems (35–37). A fit of Eq. [25] or [27] to the data again works very well, which is shown in the following.

4. EXPERIMENT AND THEORY

4.1. Setup As the details of the experimental setup are described elsewhere (14, 22), we only give the most essential information here: We used fluorescently labeled surfactant-free carboxyl polystyrene latex particles with radius 155 nm. The density of the particles (1.055 g/cm3) is sufficiently close to that of water to neglect sedimentation. The surface density of the carboxyl charge groups is 5.85 3 1013 cm22. The latex particles were suspended at a number concentration of cB 5 4.7 3 1024 mm23 in a carbonate buffer solution of pH 9.5. This high pH value compared with the pK value of 5 of the carboxyl groups ensures their almost complete dissociation, thus yielding a surface charge density of 29.36 mC/ cm2. Sodium chloride was used to adjust the ionic strength to 0.023 mol/l (conductivity 0.17 m21 V21), corresponding to a Debye screening length of 2 nm. No emulsifier was added. The experiments were carried out at 23.0°C, and a temperature stability of 60.05°C was maintained. Various tests for constancy of the experimental conditions were performed (14). We used a parallel plate channel with dimensions lx 5 34.3 mm (parallel to the flow), ly 5 9.1 mm (width), and lz 5 1 mm (height). The observed area was located in the middle of the upper x, y-plane. In this area, the flow velocity profile close to the surface is given by the linear expression vx (z ) 5 az with a ' 100 s21. These conditions guarantee laminar flow, but the Pe´clet number is large enough so that the adsorption is convection- rather than diffusion-dominated. Also, the flow is fast enough to restrict undetected readsorptions to a time scale considerably smaller than a typical adhesion time. The material of the flow cell (including the observed area) is optical glass K5. The adsorption surface is thus of optical smoothness. The surface charge density is estimated to be 211 mC/cm2. For the observation of the particles, the evanescent field method (38) combined with video image microscopy (39) has been used. The penetration depth ze of the evanescent field (intensity profile I ( z ) 5 I 0 e 2z /z e ) was adjusted to 86 nm. The lateral optical resolution was 1.0 mm. The effective area of a particle in the pixel image encompassed 3 3 3 pixels. The images were taken at time intervals of approximately 100 s during the first 1000 s after the beginning of the experiment and later every 1200 s during a measuring time of 4 days. This resulted in approximately 300 digital images. The particle position determination has been performed using the freely distributed STARMAN software (40). The limited optical resolution of 1 mm requires corrections relating the number of detected particles to the number of real particles. The corresponding formulas can be found in (22).

329

COLLOIDAL DESORPTION

FIG. 4. Main graphic. The number of residents R(t) as a function of the time t after insertion of the colloidal particles into the flow channel. The dashed line is the best fit of the distributed potential model to the data, corresponding to ^U& 5 19.9kBT and DU 5 1.3kBT. The continuous line is the best fit to the aging model and corresponds to m 5 3.55 3 1025 s21 and n 5 0.45. Inset. The uncorrected direct measurements of the adhesion time distribution 3(t|t) are shown for the indicated values of t. It can be seen that 3 is independent of t. The stepwise character of the experimental data is due to the discreteness of the particle numbers. Also shown is Pd according to the models, with the same fit parameters as before.

4.2. Determination of Pd(t) Performing the corrections given in (22), we can determine the number of residents R(t) and the influx J(t). J(t) is defined as N A ~t, Dt ! J ~t ! 5 lim Dt Dt30

[29]

where NA(t, Dt) is the number of particles adsorbing within the time interval (t 2 Dt, t). A direct estimation of the adhesion time distribution is possible from the measurement of 3 ~ t |t ! 5 lim

Dt30

N AD ~t, t , Dt ! N A ~t, Dt !

[30]

where NAD(t, t, Dt) is the number of particles which arrive on the observed surface in the time interval (t 2 Dt, t). It can be seen from the inset of Fig. 4 that 3 appears to be independent of t. Unfortunately, there is no explicit correction formula for 3, as there is for J and R, to account for unresolved multiplets in the images (22). Therefore, the comparison of 3 with the

model functions for Pd (inset of Fig. 4) mainly serves illustrative purposes. In order to fit the model functions [20] and [25] to the data, we have to use the relation

R ~t ! 5

E

t

P d ~t 2 q !J ~ q !d q .

[31]

0

Here, Pd is one of the model functions [20] and [25] and J is an interpolation of the (corrected) experimental data Ji(ti). We actually fit the function [31] to the (corrected) data Ri(ti). Our fitting parameters are a and b or k0, n, and t0. As it is impossible to fit k independent of the other parameters, we give an estimate for the unknown second derivatives in Eq. [18]: Assuming a typical potential of V 5 ^U&, or V 5 U0 and a typical length scale of l0 5 10 nm, a typical second derivative is of order ^U&/l20. Assuming D 5 10212 m2/s, we arrive at an estimate of k ' V /(k B T ) z 1000 s21. In Fig. 4 the corrected values Ri(ti) are plotted together with the fitted functions R(t) based on the two models. The distributed potential model fits best with a ' 230 and b ' 0.085kBT, while for the aging model, the best fit is achieved with k0 '

330

WEISS ET AL.

0.02 3 1025 s21, n ' 0.45, and t0 ' 0.06 s. The small value of t0 compared to the time scale of the experiment justifies the use of Eq. [27], and a fit to it indeed results in the same values for n and m as in the three-parameter fit (for which m can be calculated from Eq. [28]). One finds m ' 3.55 3 1025 s21. From a and b, it can readily be calculated via Eq. [16] that ^U& ' 19.9kBT and DU ' 1.3kBT. A calculation of U0 from Eq. [26] is more delicate because we are very close to the long time limit, where k0 and t0 no longer have any individual significance, but enter only via the combination [28]. Thus, the error in k0 is very large (of the order of k0 itself), but on the other hand, U0 depends only very weakly on k0 (Eq. [26]). Therefore, we can at least give a rough estimate of U0, which amounts to 13 6 3kBT. Although the ageing model fits the data slightly better than the distributed potential model, we do not prefer it to the latter. In fact, one could proceed the inverse way and construct a potential distribution from a smoothed interpolation of the data, which would fit the data ‘‘perfectly’’ just by construction. Similarly, one could find a ‘‘perfect’’ aging law. But the interesting point is that a quite common distribution and a quite common aging law can explain our data rather than that we can construct some distribution or some aging law that does so. 5. CONCLUSIONS

In studies of colloidal desorption, the interesting observables are the recurrence probability, Pr(t|t), and the adhesion time distribution, Pd(t|t). In many experiments, Pr is the only relevant and accessible quantity. However, Pd is independent of the flow and surface geometry, and therefore more fundamental. In any measurement of Pd, it must be possible to distinguish permanent stays on the surface from readsorptions. This requires a sufficient lateral resolution and is greatly facilitated when desorbing particles are carried away by flow. However, when the interaction between the particles and the surface lacks a repulsive electrostatic barrier, Pd depends strongly on the ability to detect readsorptions. On the other hand, when a barrier of only a few kBT exists, this dependence vanishes rapidly. With our experimental technique which combines evanescent wave illumination of fluorescently labeled particles with digital video microscopy, we are able to give estimates 3(t|t) of the adhesion time distribution which are only limited by the pair resolution (22). An explicit dependence on the adsorption time t was not found. Via Eq. [31], model functions for Pd(t) can be tested without the restriction of limited pair resolution. This has been done with the distributed potential model and with the ageing model. From the four reasons we could imagine for a nonexponential Pd, only two approved to be applicable to our data. Nonexponentialities due to readsorption are important for measurements of Pr, but not of Pd. A secondary minimum may lower the rate of desorption but does not qualitatively

affect the rate behavior. A gamma distribution of the potential depths and a logarithmic increase of the interaction strength are, however, suitable to explain our experimental findings. The distributed potential model directs towards the same direction as the widely accepted distribution of desorption rates. Starting from the potential rather than from the rate is more natural because it is the potential that determines the rate. Also, it facilitates the interpretation of the results: A mean value of about 20kBT and a standard deviation of 1.3kBT, which characterize the potential distribution, can easily be considered realistic values. The aging model is motivated by similar findings on measurements of static friction coefficients. It results approximately in a stretched exponential for the adhesion time distribution function and can be fitted to our experimental data even slightly better than the distributed potential model. However, we do not prefer either of the models to the other one. The wide presence of nonexponential desorption kinetics and the plausibility of its explanations have led us to consider it not as a nonideality of the system but as an interesting and inherent phenomenon that deserves further attention. As we have already stated in the Introduction, we do not claim that the explanations presented here are the only possible ones. In (14), we show that the mean lifetime of a binding site correlates excellently with the surface dissolution rate. This observation is not incompatible with a stretched exponential adhesion time distribution law and could be another explanation of the nonexponential behaviour. Also, the ion concentration in the layer between the surface and a particle need not remain constant while the particle is adsorbed, which could be another reason for a time-dependent interaction potential. It is most probable that various effects—including variations in the potential depth among the binding sites and bond aging— do exist and contribute to the nonexponential form of Pd(t), but it is certainly desirable to confirm the existence of each phenomenon individually. More detailed observations of single particles at adsorption sites could help to do so. Deformations of the particles could be detected, e.g., with AFM technique; a positive finding would support the aging model. Optical tweezers could be used to desorb single particles, thereby precisely measuring the force between a particle and a binding site. AFM investigations of the binding sites might also help us to understand their properties better and open up even completely new aspects of adsorption and desorption kinetics. ACKNOWLEDGMENTS We thank M. Borkovec, M. Elimelech, E. Mann, H. Behrens, F. Seddighi, M. Semmler, and Th. Binkert for many stimulating discussions and continuing interest. This work has been supported by the Swiss National Science Foundation.

COLLOIDAL DESORPTION

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9.

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Ryan, J. N., and Elimelech, M., Colloids Surfaces A 107, 1 (1996). Ruckenstein, E., and Chen, J. H., J. Colloid Interface Sci. 128, 592 (1989). Derjaguin, B. V., and Landau, L. D., Acta Physicochim. 14, 633 (1941). Verwey, E. J. W., and Overbeek, J. T. G., ‘‘Theory of the Stability of Lyophobic Colloids.’’ Elsevier, Amsterdam, 1948. Van de Ven, T. G. M., ‘‘Colloidal Hydrodynamics,’’ Colloid Science Series. Academic Press, San Diego, 1989. Israelachvili, J. N., ‘‘Intermolecular and Surface Forces,’’ 2nd ed. Academic Press, London, 1991. Adamczyk, Z., Siwek, B., Zembala, M., and Belouschek, P., Adv. Colloid Interface Sci. 48, 151 (1994). Adamczyk, Z., and Warszyn´ski, P., Adv. Colloid Interface Sci. 63, 41 (1996). Elimelech, M., Gregory, J., Jia, X., and Williams, R., ‘‘Particle Deposition and Aggregation: Measurement, Modelling and Simulation,’’ Colloid and Surface Engineering Series. Butterworth Heinemann, Oxford, 1995. Hull, M., and Kitchener, J. A., Trans. Faraday Soc. 65, 3093 (1969). Adamczyk, Z., Dabros, T., Czarnecki, J., and Van de Ven, T. G. M., Adv. Colloid Interface Sci. 19, 183 (1983). Bowen, B. D., and Epstein, N., J. Colloid Interface Sci. 72, 81 (1979). Sjollema, J., and Busscher, H. J., J. Colloid Interface Sci. 116, 382 (1989). Lu¨thi, Y., Ricˇka, J., and Borkovec, M., J. Colloid Interface Sci. 206, 314 (1998). Vaidyanathan, R., and Tien, C., Chem. Eng. Sci. 46, 967 (1991). Sjollema, J., and Busscher, H. J., Colloids Surfaces 47, 337 (1990). Kallay, N., Barouch, E., and Matijevic´, E., Adv. Colloid Interface Sci. 27, 1 (1987). Meinders, J. M., Noordmans, J., and Busscher, H. J., J. Colloid Interface Sci. 152, 265 (1992). Meinders, J. M., and Busscher, H. J., Colloid Polym. Sci. 272, 478 (1994). Meinders, J. M., and Busscher, H. J., Langmuir 11, 327 (1995). Connaughton, D. F., Stedinger, J. R., and Lion, L. W., Environ. Sci. Technol. 27, 2397 (1993).

22. 23. 24. 25. 26.

27. 28.

29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40.

331

Lu¨thi, Y., and Ricˇka, J., J. Colloid Interface Sci. 206, 302 (1998). Berdichevsky, V., and Gitterman, M., J. Phys. A 29, 1567 (1996). Kramers, H. A., Physica 8, 284 (1940). Ha¨nggi, P., Talkner, P., and Borkovec, M., Rev. Mod. Phys. 62, 251 (1990). Cerofolini, G. F., Meda, L., and Bandosz, T. J., in ‘‘Adsorption and its Application in Industry and Environmental Protection’’ (A. Dabrowski, Ed.). Elsevier, Amsterdam, 1998. (In press) Bowden, P., and Tabor, D., ‘‘The Friction and Lubrication of Solids.’’ Clarendon, Oxford, 1950. ‘‘Fundamentals of Friction: Macroscopic and Microscopic Processes,’’ NATO ASI Series E (I. L. Singer and H. M. Pollock, Eds.), Vol. 220. Kluwer Academic, Dordrecht, 1992. ‘‘Physics of Sliding Friction,’’ NATO ASI Series E (B. N. J. Persson and E. Tosatti, Eds.), Vol. 311. Kluwer Academic, Dordrecht, 1996. Rabinowicz, E., ‘‘Friction and Wear of Materials.’’ Wiley, New York, 1965. Heslot, F., Baumberger, T., Perrin, B., Caroli, B., and Caroli, C., Phys. Rev. E 49, 4973 (1994). Oden, J. T., and Martins, J. A. C., Comput. Methods Appl. Mech. Eng. 52, 527 (1985). Scholz, C. H., ‘‘The Mechanics of Earthquakes and Faulting,’’ Chap. 2. Cambridge Univ. Press, Cambridge, UK, 1990, and references therein. Persson, B. N. J., ‘‘Sliding Friction. Physical Principles and Applications.’’ Springer-Verlag, Berlin, New York, 1998. Klafter, J., and Shlesinger, M., Proc. Natl. Acad. Sci. USA 83, 848 (1986). Shlesinger, J., Annu. Rev. Phys. Chem. 39, 269 (1988). Scher, H., Shlesinger, M., and Bendler, J., Physics Today 44, 26 (1991). Lok, B. K., Cheng, Y.-L., and Robertson, Ch. R., J. Colloid Interface Sci. 91, 87 (1983). Dabros, T., and Van de Ven, T. G. M., Colloid Polym. Sci. 261, 694 (1983). Penny, A. J., and Morris, P. W., ‘‘Starman—Crowded-Field Stellar Photometry.’’ Documentation and program code available http://ast.star. rl.ac.uk/dev/starman/main.html, 1995.