Double layer calculations for the attachment of a colloidal particle with a charged surface patch onto a substrate

Rajasekar Vaidyanathan

Bird Machine Company, So. Walpole, MA, USA

The attachment of a particle onto a substrate with a like surface charge may be greatly facilitated in the presence of an oppositely charged surface patch on the particle. Calculations are presentedfor the effect of variousparameters on particle attachment. These include the patch charge, patch size, particle size and the ionic strength of the medium.

Keywords: colloid; particle attachment; patchy surface charge; double layer

Introduction

The electrostatic behavior of colloids or surfaces in aqueous systems possessing heterogeneous charge characteristics has drawn increasing interest in many recent studies.‘-6 Classically, the surface charge de- termining the electric double layer is assumed to be uniformly smeared on the surface, and this picture is often adequate for explaining various observations re- lating to processes such as deposition, coagulation, and electrophoresis. However, charge heterogeneities on the length scale of the surface do occur in many situa- tions where it is essential to account for them in order to understand the physics of the observed phenomena.

For example, many types of clays are known to possess a heterogeneous surface charge distribution. These may affect important engineering properties such as the rheology of clay suspensions7 or the electri- cal resistivity of oil reservoir formations in the presence of clay fines.2 Anderson and coworkers3,4 examined theoretically the electrophoresis of a nonuniformly charged particle and showed that the electrophoretic mobility depends on certain moments of the zeta poten- tial distribution on the particle. Vaidyanathan and Tien5 examined the deposition of uniformly charged particles onto patchy-charged substrates. The charge on the patches was of the same sign as the particles, while the region between patches was assumed to be uncharged. Lesins and Ruckenstein6 studied the adsorption of pro- teins in chromatography columns and concluded that,

Address reprint requests to Dr. Vaidyanathan at Bird Machine Com- pany, 100 Neponset St., So. Walpole, MA 02071, USA. Received 16 October 1991; accepted 23 December 1991

98 Sep. Technol., 1992, vol. 2, April

in some cases, proteins may adsorb onto similarly charged adsorbents due to an oppositely charged patch on the protein. This paper pertains to this last problem.

Here, the objective is to obtain some general under- standing of the role played by various parameters af- fecting the attachment of patchy-charged particles us- ing simple double layer calculations. These parameters include the size of a patch, its zeta potential, the parti- cle size and ionic concentration of the medium. The calculations made here follow closely the approach used in Ref. 5 and may be considered an extension of that work.

Model

The interaction forces between a sphere (particle) and plane (substrate) are examined below. Unless noted otherwise, lengths are scaled with the sphere radius a, potential energies with kT, and forces with kT/a. The plane has a zeta potential I&, while the sphere has a zeta potential 5, on most of its surface, except in the region of a single charged circular patch where the zeta poten- tial is I;p. Here, the zeta potential refers to the local zeta potential that may be related to the local charge density using the Guoy-Chapman theory.3 The patch, of radius p, may in principle be located anywhere on the particle surface. However, if 4, and 5, are of the same sign and opposite to that of &,, then the preferred orientation will be with the patch region facing the plane as shown in Figure I, in order to minimize the potential energy of the system. This situation is examined below. (When the particle has more than one oppositely charged patch, there may be several preferred orientations of the particle surface relative to the plane. However, if

0 1992 Butterworth-Heinemann

Attachment of a patchy-charged particle onto a substrate: R. Vaidyanathan

about the double layer interaction for each of the possi- ble orientations.)

-.-

d h

The double layer interaction energy for the above system, V,,, may be calculated using Derjaguin’s ap- proximation based on the energy of flat, parallel double layers interacting at constant surface potential. (In principle, the same approximation may be employed to determine V,, for double layers interacting at constant surface charge or via linear superposition.) Dejaguin’s method has been found in the past to provide accurate results for ~a > 5, where K is the reciprocal double layer thickness. In this well-known procedure, the in- teracting surfaces are approximated by a set of flat, parallel rings that interact according to the flat double layer expression. Details of the method may be found in most past studies on double layer interactions.5’8 Straightforward application of the method to the pres- ent situation leads to the following expression for V,,:

I { [

1 - exp( - 2NDLd)

vD’ = N’ log 1 - exp( - 2NDLh) 1 Y + N2 log

1 - exp( -NDLh) Figure 1 Interaction of a particle with a charged surface patch and a substrate. The patch is oppositely charged relative to the

K 1 + exp(- N,,h) 1

rest of the particle surface and the substrate. x 1 + exp(-N,,4

( 1 - exp( - ND& )I the size of the patches are much smaller than the dis- + Ns log[l - exp( - 2N,,h)]

tance separating them, the preferred orientations would likely correspond to Figure I for each patch. In + N4 log

[

1 + exp( - N,,h)

that case, the procedure described below for a single- 1 - exp( - N,,h) II (1) patch particle may still provide useful information where d is the separation distance between the particle

Notation

: particle radius T absolute temperature Hamaker constant V

d dimensionless distance between particle and dimensionless total interaction energy (Equation 12)

substrate (Figure 2) V DL dimensionless double layer interaction F dimensionless total interaction force energy (Equation 1)

(Equation 13) FDL dimensionless double layer interaction force

(Equation 8) FL0 dimensionless London interaction force

(Equation 10) h dimensionless distance between periphery of

patch and substrate (Figure 1) Z ionic concentration of univalent electrolyte k Boltzmann’s constant, 1.38 x 1O-23 J/“K Ni dimensionless groups defined in Equations 2,

. . .) 5; i = 1, 2, 3, 4 N,, dimensionless double layer thickness

(Equation 7) N,, dimensionless London group (Equation 11) P dimensionless patch radius PC minimum value of p necessary to produce

particle attachment

V,, dimensionless London interaction energy (Equation 9)

Greek letters

co permittivity of vacuum, 8.8542 x 1O-‘2 C’IJ - m

% relative permittivity of aqueous medium, 80 at 20°C

Fll reciprocal double layer thickness local zeta potential on most of the particle surface

&We local zeta potential on patch value of &, at base conditions (Base conditions section) zeta potential of the substrate

&ote: Dimensionless lengths are scaled with a potential energies with kT and forces with kT/a:)

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Attachment of a patchy-charged particle onto a substrate:

and the plane, and h is the distance from the periphery of the patch to the plane (Figure 1). h may be expressed in terms of the patch radius as:

h = 1 + d - (1 - p*)“* (2)

and the dimensionless groups, N,, . . . , N4, ND, are defined as follows:

(3)

(5)

(6)

(7)

In Equation 3, co is the permittivity of free space and E, the relative permittivity of the medium. The double layer force, FDL, corresponding to Equation 1 is given by the following expression:

FDL = - NDL N,{cosh(N,,d) - cosh(l\r,,h)

+ N2[cosh(NDLh) - cosh(N&)] (8)

+ N,[l - cosh(N,,h)] - N4 cosh(l\r,,h)}

A number of expressions are available in the litera- ture’ for the London potential, VLo, and force, FL,; here, only the simplest possible unretarded expressions are used for purposes of illustration:

d log-

d+2 1 - (2/3)N,,

FLo = d*(d + 2)* (10)

(9)

where

N LO = AIkT (11)

and A is the Hamaker constant. The total surface inter- action energy, V, and interaction force, F, are given by:

v = VDL + v,, (12)

F = F,, + FL0 (13)

Results and discussion

The comparison of the above model with experiments is complicated by the fact that data on the values of &, and p are usually not available.‘O Estimates of these parameters are perhaps possible from a knowledge of the surface chemistry of the particle, but the author is unaware of any such effort in the literature. On the other hand, it is possible to obtain some general under- standing of the critical parameters in the problem with

R. Vaidyanathan

6 lc-

k 5-

0

-5-

-w- I I I Illll~ I I IllIll

0.01 0.10 100

d

Figure 2 Dimensionless interaction force (taller curve) and en- ergy (shorter curve) in the absence of a patch, as a function of the particle-substrate separation distance. Physical variables are at the base values defined in Base conditions section.

sample calculations. This is the approach adopted below.

Base conditions

The following values for the various dimensionless groups were chosen as base conditions:

N LO = 0.1, N,, = 10, N, = 4,

N2 = -1, N3 = 1, N4 = 1.

The above were obtained using the following values for the physical variables:

a = IOnm, -6 = 4, = 4, = 20mV,

A== lo-*’ J, l/K = lnm, T = 300K.

The size of the patch is variable. Plots of F vs. d and V vs. d shown in Figure 2 were obtained at the above conditions using Equations 12 and 13 with p = 0, i.e., in the absence of a patch. The F vs. d plot has a maxi- mum of F = 24 at d = 0.05 which is a repulsive force barrier against particle attachment. We shall first see how the presence of a patch changes this scenario. Next, the effect of choosing other values of the physical variables will be observed.

100 Sep. Technol., 1992, vol. 2, April

Attachment of a patchy-charged particle onto a substrate: R. Vaidyanathan

Effect of patch size on interaction force profile and force barrier

Let us now assume a patch is present on the particle, opposite in charge to the rest of the particle surface/ substrate. The values of the physical variables are as- sumed to correspond to the base conditions. Plots of the interaction force profiles for different values of the patch size are shown in Figure 3. As the patch radius p increases from 0 to 0.34, the interaction force barrier drops from 24 to 0. The smallest patch size that elimi- nates the force barrier (i.e., permits particle attach- ment) will be called pC, i.e., in this case pC = 0.34. The extent to which different sets of conditions favor particle attachment may be gauged by the relative val- ues of pC under those conditions. Thus a small value of pC indicates a more favorable condition than does a larger value.

Another important point to note in Figure 3 is that as p increases from 0, not only is the force barrier diminished, it also occurs at a larger separation dis- tance. The outward movement of the force barrier oc- curs in all sample calculations reported here. This phe- nomenon has important consequences in situations where fluid convection not only plays a role in particle transport but also determines particle attachment. One such example is in the filtration of relatively large parti-

0.10

d 100

Figure 3 Change in the interaction force profile as a function of the patch size, p. From top to bottom the force profiles corre- spond to p = 0, 0.1, 0.2 and 0.34.

0.6

0.5

0.4

S

0.3

0.2

0.1

0.0

Figure 4 Dependence of pc on the zeta potential of the patch. Other variables remain at their base values.

cles in stagnation point flow~.~ The drag component pushing the particle towards the substrate in the trajec- tory equation increases with the distance of the particle center from the plane surface.’ Therefore, the attach- ment of very large particles is influenced by convec- tion. The above results indicate that the presence of even a small charged patch could exert a major influ- ence on the attachment of large particles by magnifying the role of convection.

Effect of the magnitude of patch surface potential on particle attachment

As &, increases in magnitude, one expects p, to de- crease due to increased electrostatic attraction be- tween the patch and the substrate. In other words, a smaller patch size will be sufficient to bring about parti- cle attachment. To illustrate this effect calculations were made by varying $,/{,b”“’ (where {p is the value of $, under base conditions) and keeping other physical vartables at their base values. However, the dimen- sionless groups N,, . . . , IV4 do change from their base values due to changes in &,.

The results shown in Figure 4 indicate that pC is not a very strong function of <,,. For a 30-fold increase in [,,, pC is reduced only by about a factor of three. Therefore, as far as attachment is concerned, the exis-

Sep. Technol., 1992, vol. 2, April 101

Attachment of a patchy-charged particle onto a substrate: R. Vaidyanathan

100

&

0.10

0.01

b

1 10 100 1000

ND,

Figure5 Dependence of&on reciprocal double layerthickness, K, (open triangles) and on particle radius, a, (solid triangles). The abscissa NDL changes in proportion to a change in K, or to a change in a, when other variables are fixed at the base values. The base value of NDL is 10.

tence of the oppositely charged patch seems more criti- cal than the actual charge or zeta potential on the patch.

Effect of ionic concentration on particle attachment

The reciprocal double layer thickness, K, changes with ionic strength, I, according to

K c( Z”* (14)

for univalent electrolytes. A change in K then results in a change in ND, via Equation 7. With other parameters fixed, increasing Z leads to better screening of the sur- face charge and hence one expects the interaction force barrier to drop, followed by a decrease in p,. The de- pendence of pc on K (NDL) is shown in Figure 5 using open triangles with other dimensionless groups re- maining at their baseline values. The value of p, drops at an increasing rate as NDL increases from its base value of 10. For NDL slightly in excess of 200, the surface charge is so effectively screened that pc is virtu- ally zero, i.e., the particle will deposit onto a similarly charged substrate even in the absence of a patch.

Effect of particle size on attachment at fixed ionic concentration

This is a particularly interesting case where both NDL and N, change in proportion to the change in a (Equa- tions 3 and 7), while all other dimensionless groups and physical variables stay at their baseline values. Results are shown as solid triangles in Figure 5. An increase in a by a factor of 100 (rise in ND, from 10 to 1000) leads to a reduction inp, by a factor of 10 (0.34 to 0.035). In fact, the results on the log-log scale indicate that pC a NE;‘*. Since pC is scaled using a, one concludes that the physical patch size corresponding to pC changes only as al/Z. In other words, a large particle requires a proportionally smaller sized patch to enable deposition compared to that required for depositing a smaller par- title .

The weak dependence of pC on the particle radius arises because the major contribution to the double layer interaction is from those regions on the inter- acting surfaces that are at the closest separation from each other.” Therefore, in the present case the domi- nant contribution to F,, is from the patch. Hence a small increase in patch size offsets a much larger in- crease in particle size with regard to the effect on FDL.

The effect of changing ionic strength or K vs. chang- ing the particle size may be compared by looking at the two curves in Figure 5. It is clear that an increase in K by a certain factor generally has a larger effect on particle attachment than an increase in particle radius by the same factor. However, it should be remembered that a given change in K results from a much larger change in ionic strength due to Equation 14; hence the relative effects of K and Z on p, are different.

Conclusions

Sample calculations have been made to understand the important parameters in the attachment of a particle with a charged surface patch onto a substrate. The patch is oppositely charged relative to the rest of the particle surface and to the substrate. Constant-poten- tial double layer interactions and attractive London interactions are assumed.

It is found that the presence of a patch not only reduces the interaction force barrier, it also moves it to a larger separation distance. This phenomenon fa- vors the adsorption of relatively large particles whose attachment may be assisted by fluid convection, since convection is stronger at larger distances from the (sta- tionary) substrate. At high ionic strengths, the attrac- tive London component of the surface forces is domi- nant, while the double layer component is much smaller due to charge screening. Therefore, the patch size nec- essary to produce attachment, p,, decreases with an increase in ionic strength. At fixed ionic concentration, when the particle radius a is increased, the physical patch size necessary for attachment is found to increase as a’/*. In other words, a large particle size requires a proportionally smaller patch to deposit it compared to that required by a small particle. Finally, p, is found to depend only weakly on the patch zeta potential, with

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Attachment of a patchy-charged particle onto a substrate: R. Vaidyanathan

a 30-fold increase in potential producing a mere 3-fold decrease in pC. Thus, with regard to particle attach- ment, the existence of an oppositely charged patch seems more critical than the magnitude of the patch charge.

5.

6.

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forces due to surface inhomogeneities in the narticle denosition

7.

_ process. Colloids and Surfaces 1987, 22, 2b7 . 8.

2. Cohen, R.R. and Radke, C.J. A patchy-charge model for the surface conductivity of clays. SPE 15968. Presented at the SPE International Symposium on Oilfield Chemistry, San Antonio, 9. TX, February 4-6, 1987

3. Anderson, J.L. Effect of nonuniform zeta potential on particle movement in electric fields. J. Colloid Interface Sci. 1985,105, 10. 45 11.

4. Fair, M.C. and Anderson, J.L. Electrophoresis of nonuni-

formly charged ellipsoidal particles. .I. Colloid Interface Sci. 1989, 127, 388 Vaidyanathan, R. and Tien, C. Hydrosol deposition in granular media under unfavorable surface conditions. Chem. Eng. Sci. 1991, 46,967 Lesins, V. and Ruckenstein, E. Chromatographic probing of protein-sorbent interactions. J. Colloid Interface Sci. 1989, l32,566 Tadros, Th.F. Settling of suspensions and prevention of forma- tion of dilatant sediments. In Solid/Liquid Dispersions. Th.F. Tadros, ed. London: Academic Press, 1987, pp. 225-274 Hogg, R., Healy, T.W. and Fuerstenau, D.W. Mutual coagula- tion of colloidal dispersions. Trans. Faraday Sot. 1966, 66, 1638 Gregory, J. Approximate expressions for retarded van der Waals interaction. J. Colloid Interface Sci. 1981, 83, 138 Ruckenstein, E. Personal communication, 1991 Verwey, E.J.W. and Overbeek, J.T.G. Theory of the Stability of Lyophobic Colloids. Amsterdam: Elsevier, 1948

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