The deposition of colloidal particles on smooth solids

J O U R N A L O F COLLOID AND I N T E R F A C E SCIENCE 9.9. 342--351

(1966)

The Deposition of Colloidal Particles on Smooth Solids J. K. M A R S H A L L 1 AND J. A. K I T C H E N E R

Department of Mining and Mineral Technology, Imperial College, London, S.W.7., England Received October 27, 1965 A novel method based on the rotating disc principle has been developed for investigating the interaction between small particles and smooth planar solids. The deposition of carbon black particles from dilute aqueous suspensions on to glass, polystyrene, polyvinylformaldehyde, regenerated cellulose, and a styrene-vinylpyridine copolymer has been studied with various concentrations of anionic or cationic surfactant and additions of KC1. Deposition of single particles occurred only from slightly unstable dispersions. Deposition was greatest when particles and solids had zeta potentials of opposite sign, though the maximum deposition possible according to the theory of the rotating disc was never reached, presumably owing to slow coagulation in the suspensions. Deposition was reduced by increased zeta potentials of like sign or by decreased salt concentration, but the results cannot be explained quantitatively by the Derjaguin-Landau-Verwey-Overbeek theory except by postulating that the surfaces make contact at asperities of the order of 50 A--a possibility that cannot be entirely excluded.

INTRODUCTION Deposition of particles from colloidal suspensions on to much larger particles is of technical importance in detergency (where liberated dirt particles may cause "redeposition" on to cleaned textile fibers), in the removal of colloidal particles from water by sand filters, and in the "slime-coating" of mineral grains in the froth flotation process. Basically, the interaction of small particles with planar solids should be represented by an extension of the classical theory of stability of colloids; in addition, it may include the case of interaction of materials of different nature, corresponding to "heterocoagulation" of colloids (i, 2). Amongst the few published investigations of this type of system are the work of Hunter and Alexander (3) on deposition of kaolin on sand and various technical studies of redeposition on textiles (summarized by i Present address: Unilever Research Laboratories, Port Sunlight, Cheshire.

D u r h a m (4)). The removal of particles already attached is a quite distinct problem which has also recently received some preliminary study (5). Because of the complicated h y d r o d y n a m ics involved in granular filters and in washing of textiles it is not possible to use such systems for more t h a n qualitative checks of the theory of colloid stability. Such studies have been made b y D u r h a m (6) and yon Lange (7), and their results suggest t h a t the classic Derjaguin-Landau-¥erwey-Overbeek theory should provide a sound basis for representation of the phenomenon of deposition from suspensions. However, it is clear t h a t more fundamental investigation is needed with effectively spherical particles and smooth surfaces. The convective transport must be controlled and the surfacechemical nature of both materials must be studied. Furthermore, the amount and condition of both the dispersed and the deposited material must be determined; over-all 342

DEPOSITION OF COLLOIDAL PARTICLES ON SMOOTH SOLIDS measurements of the mass deposited are inadequate because aggregation may be present to an unknown extent. The present paper reports measurements of the deposition of graphitized carbon black particles from dilute detergent solutions on to discs of glass, regenerated cellulose, polystyrene, and other plastics. The hydrodynamics of the mass transport was controlled by use of the rotating disc winciple. The electrostatic interaction between the particles and deposition "substrate" was varied by changing the concentration or type of surfactant and by varying the concentration of indifferent electrolyte

(i~Cl). EXPERIMENTAL

Materials The carbon black was graphitized "Sterling M T " , kindly provided by Dr. W. R. Smith of Cabot Carbon Inc. The particles, which are partially developed polyhedra (8) rather than perfect spheres, have an average diameter of 0.45~ and some are cemented in small aggregates that are difficult to break up. For comparison, some measurements were made with graphitized "Sterling F T , " 0.18t, diam. Dispersions (0.01 wt. %) were prepared in dilute solutions of "Aerosol O T " (sodium di-ethylhexylsulfosuccinate) or in eetyltrimethylammonium bromide (CTAB) with the aid of ultrasonics. After 30 rain. of treatment the optical density approached a constant value; however, in order:to standardize the degree of dispersion, the ultrasonic treatment was continued until the percentage transmission (of white light.) through 10 cm. of a twenty times diluted sample of suspension was 31% 4- 1%. The dispersion then contained about 6 X 10s pai'tides per milliliter, as estimated from mass concentration and, independently, by counting under the ultramicrosfope.: The majority of the particles were monomeric, but a proportion consisted of double or triple aggregates (as seen by electron microscopy) and a small residue remained nondispersible. The dispersions of carbon in most of the pure surfactant solutions (i.e,, without added

343

KC1) remained practically unchanged in optical density (therefore degree of dispersion) for the duration of the deposition exp e r i m e n t s - u s u a l l y 30 min. ; but ill certain experiments in which KC1 was added slow aggregation oeeurred, as shown by decreasing optical density. Therefore it was necessary to relate deposition to the average state of dispersion. The surfactants used were sufficiently pure to show no minimum in s.t.--concentration curves. Their adsorption isotherms on the same graphitized carbons have been previously reported from this laboratory (9).

Rotating Disc Apparatus The apparatus, based on considerations recommended by Riddiford (10), is shown schematically in Fig. 1. The drive and shaft were of heavy, precision, construction to avoid vibration that would disturb the desired laminar flow of solution across the face of the disc. The body of the disc (4 cm.

t

II II

,I]

B

II

i~

D

± _

_

E

_ -

F

J

Fla. 1. Diagram of the rotating disc apparatus for controlled deposition. (A)--eonstant speed motor with reduction gears; (B)--roller bearing; (C)--eollet chuck; (D)--eover; (E)--glass dish containing dispersion; (F)--plastie disc; (G)-deposition surface.

344

MARSHALL AND KITCHENER

solution without carbon. The disc was then removed and placed to drain and dry in desiccator. The cover slip was detached and the wax cleaned off the back with solvent. Finally the surface was examined in a microscope at 480X magnification with dark-ground illumination and the superficial density of particles determined by counting over at least 30 different areas. Where the deposition was uniform the mean deviation of counts was about 4-3 %. Only single particles or small, unresolvable aggregates Deposition Surfaces were counted; the presence of larger aggreThe glass surfaces consisted of 22 mm. gates, if any, was noted. Results were exdiam. circular microscope cover glasses. pressed conveniently by the number of (refr. index 1.542). These were cleaned' particles deposited Nd per 104 ~2 of surface; with hot chromic acid, thoroughly washed, typically, N'd ranged from 5 to 500. and then drained in an upright position in a Zeta-Potentials desiccator. Plastic surfaces were prepared Zeta-potentials of particles and substrates by coating such cover slips from solutions of the plastics, using the falling level method were determined by microelectrophoresis in employed for preparing support films for a novel flat, demountable cell described electron microscopy. Such films are known elsewhere (13). Mobilities were converted to be very smooth, as shown by metal to nominal zeta-potentials by the Helmholtzshadowing. The plastics studied were poly- Smoluchowski equation without refinements. styrene (PS) (a sample from Dow Chemical RESULTS Co., as used for electron microscopy), polyConditions for Regular Deposition vinylformaldehyde (PVF) (from Shawinigan Ltd.), and a eopolymer of 80 % 2-vinylpyriCharacteristic regular deposition of indine (PVP) with 20 % styrene, prepared by dividual particles (Fig. 2A) was found only the method of Birbeck and Stacey (12) and with dispersions that were very slowly kindly given by Mr. M. S. C. Birbeek. Re- aggregating. Highly stabilized dispersions generated cellulose was in the form of "Visk- gave negligible deposit. On the other hand, ing" dialysis sheet; it was washed for several rather rapidly aggregating suspensions gave days before use and was attached to a rota- irregular, weakly adhering deposits containtion disc by a special tapered outer ring to ing a wide range of aggregate size and hold the sheet taut and flat. showing spiral flow-lines across the disc The deposition surfaces studied were se- (Fig. 2B), presumably caused by "avalanchleered for the variety of chemical natures ing" of aggregates detached by the flow of they provide; thus, glass is hydrophilic and liquid. Quantitative measurements were anionic; polystyrene is strongly hydrophobic therefore restricted to moderately stable and practically nonionic; polyvinylformalde- dispersions--namely, those containing very hyde is less hydrophobic and weakly anionic; dilute detergents with various additions of polyvinylpyridine is hydrophilic and cati- KC1. onic; cellulese is hydrophilie and weakly The number of individual particles deanionic. posited in given time (at constant r.p.m.) was found to be accurately proportional to Determination of Deposition the number in solution, at least up to a The deposition substrates were rotated in concentration of 0.0018 wt. % of Sterling the dispersions for the required time and FT (= 3 X 10s paiticles per milliliter). then the bath was lowered and repla~ed for For a given solution the number Ne in30 sec. by a rinse bath containing the same creased with time of exposure of the disc to diam.) was turned from polymethacrylate plastic and was thinly coated with paraffin wax, which was also used to attach the experimental surface to the center of the face. The speed of rotation was generally 200 r.p.m. The suspension (250 ml.) was contained in a dish cf 10 cm. diam. The dimensions are sufficiently near to theoretical requirements of the rotating disc system so that Levich's theory (11) can be employed.

DEPOSITION OF COLLOIDAL PARTICLES ON SMOOTH SOLIDS

345

FIG. 2. Photomicrographs of carbon black deposits. (A)--Uniform deposit of individual particles (dark-ground illumination). (B)--Deposit produced by a rapidly coagulating suspension (transmitted light). the suspension, though the increase was less than proportional to the time. For example, from a suspension of 0.01g. carbon in 0.24 millimolar Aerosol + 0.02 M KC1 the deposition rate was initially 6 per minute (per 104 ~2), and after 30 min. it had fallen to about 1 per minute. This was not due to any potential saturation of the surface, as the superficial density had reached only Nd = 84, corresponding to < 0 . 2 % of the number that could be accommodated in cubic packing. Rather is it related to the disappearance of individual particles from the suspension through slow aggregation; the optical transmission (after 20X dilution), initially 30 % for 10 cm., rose to 47 % after 30 rain. (aggregates obstructing less light than their constituent particles separately). A few small aggregates were deposited but not included in the count. Because of this complication, experiments had to be conducted under strictly comparable conditions and Nd values, must be interpreted with regard to the rate of aggregation of the various suspensions. The deposition was found to increase with increasing rate of revolution of the disc.

Over the range 100 to 400 r.p.m., 2Va increased roughly proportionally to the square root of the r.p.m.; above 500 r.p.m, onset of turbulence enhanced deposition.

Effect of Su~actants on Deposition Relatively high concentrations of either agent (e.g., 2.5 n ~ l . without KC1) reduced deposition to low values (N'd < 5) on all substrates--as is, of course, to be expected for detergents. But with low concentratio.ns some striking differences were observed with different substrates; in certain systems a trace of surfactant greatly reduced deposition whereas in others it enhanced it (Fig. 3). The explanation of these results is bound up with the sign and magnitude of the charges on the particles and substrates.

Nominal Zeta-Potentials (~) Figure 4A and B, respectively, show nominal zeta-potentials (~') for carbon and various substrates in the pure surfactant solutions. In pure water graphitized carbon particles are weakly negatively charged; their potential is made increasingly negative by

346

MAI~SHALL AND KITCHENER Concentrcition -0025

~°°~ . . --

200 ~

1

~,,,,,,,

-

of

SCMC (wt.percent)

10of

.0075

-

T ~

~

A

Aerosol

0

CTAB

•

SCMC

*50

OT

I CTAB

0

100

o l~' 0

",O...-r..,~

•

, c,_.;._ "10

"05 Conceotrafiorl of Aeroso[ or C TA B

a~_

"l 5

(raiUirnotelUter)

-5 tog

-4 -3 -2 C (C in millirnole][ for 1& 2 ; wt,'[o for SCMC)

-1

500 + 100

0 Celtophane 400 -

//"

~ 0 0

/

• gloss +50-

~

5 B

o

10o

k

I PVP + Aerosol 2PVP + SCMC 3 gimss + Aerosol 4 gloss +CTAB 5 Cellophane + CTAB

-50

""-0f

-100

I

-5

.01

"02

-03

.0z,

-05

"06

-4

-t_

-3

-2

-1

log C ( Cos above)

Concentration ( mil[imole/liter )

]?me. 3. Effect of low concentrations of sur-

Fie. 4. Zeta-potentials (nominal) of carbon dispersions (A) and substrates (B).

SOiTIe

fact~nts on deposition on to (A)--polyvinylpyridine eopolymer: (B)--regenerated cellulose the ionic surfactant is enhanced by presence and glass (with Aerosol). of indifferent electrolyte. physical adsorption of Aerosol and is neuEffect of KCl on Deposition tralized and reversed by chemisorption Generally the addition of KC1 to a carbon and then physical adsorption of CTAB (cf. reference 9). Glass is strongly negative and + substrate system containing a stabilizing insensitive to Aerosol but reversed by concentration of surfactan% increased the CTAB. PVP is weakly positive in water deposition (Fig. 5). The effects became and is reversed by Aerosol. Regenerated marked with 0.01 M KC1 or more and corcellulose behaves somewhat similarly to responding reductions in the stability of glass, but with numerically smaller poten- the carbon dispersions alone were readily tials. Polystyrene and polyvinylformalde- detected by changes of optical density. This hyde were not studied in detail, but both is clearly due to the reduction cf range of action of the diffuse electrical double layers adsorbed Aerosol markedly. The effect of added KC1 on the zeta- between mutually repelling bodies. The sharp minimum in the curve for PVP potentials was complicated. Above 0.01 M there was the usual reduction of potential (Fig. 5) is problematical. Significantly, it due to reduction of the thickness (increase does not occur with any of the other subof capacity) of the diffuse double layer. Up strates. It may possibly be due to the great to this region most of the systems contain- effect of very small concentrations of KC1 ing detergent showed f increasing with addi- in enhancing the adsorption of Aerosol on tions of I(C1, indicating that adsorption of the originally positive surface; the Aerosol

DEPOSITION OF COLLOIDAL

PARTICLES ON SMOOTH SOLIDS

347

AEROSOL(raM ) SUBSTRATE

120

0-2~

-

/

100

80

.o

,0

0

/

/

O

_/

//

0.2

2-4

0

•

~

,t,

RVP

Celtophane

k

0,4

0,6

ID

~

PVF

El

II

gross

O.g

Concentration of IKCl (mote]i)

FIG. 5. Influence of KC1 on deposRion on various substrates in presence of Aerosol OT. (0.24 and 2.4 millimoles per liter). alone is enough to give a negative sign, but the f-potential is increased markedly by KC1 of around 10-3 M. It may be that below this concentration local positive regions still exist. Alternatively, it may be that adsorbed Aerosol is displaced by the arrival of carbon particles.

the first data for deposition that can be compared with theoretical expectations from the theory of sphere-plate interaction, and when this comparison is carried out certain discrepancies appear concerning the absolute deposition rates. These will now be considered.

DISCUSSION

Deposition without Electrostatic Barrier

The qualitative features of the results are readily understandable. The rate of deposition is controlled by (a) mass transfer through the solution up to the surface, fop lowed by (b) adhesion or nonadhesion, determined by surface-chemical forces. The major forces being long-range London-vart der Waals forces of attraction and electrical double-layer interaction, which is repulsive for like sign and attractive for uNike sign, it follows that all particles capable of reaching the surface should adhere if the surfaces are uncharged or oppositely charged, whereas when they have like signs the proportion of particles adhering will be reduced by a factor that may be large or small, depending on the magnitudes of the potentials and of the Debye parameter K of the solution. This is broadly what has been observed. However, the present results provide

When no barrier is present (surfaces uncharged or of opposite sign) the rate of deposition should be controlled only by diffusion and convection up to the surface. For the rotating disc system, the transport can be calculated by the theory of Levich. The maximum flux (j) of particles up to the surface is given by [11

j = D(Co/A),

where D is the Brownian diffusion coefficient of the particles; for spherical particles, the Stokes-Einstein equation gives D = k T / 6 ~ a (a being the particle radius) ; A is the Nernst equivalent thickness of the diffusion layer which, according to the theory of Levich (14), is given by A = 1.61

,

I21

348

MAI~SHALL AND KITCHENER TABLE I DEPOSITION RATES FOR SYSTEMS WITH PARTICLES AND SUBSTRATES I-IAvING

OPPOSITE Solution

A

Distilled water /" =

B

c. 10-6 M eetavlona

SIGN OF POTENTIAL.

Substrate

Carbon

Jexpt.b

Jtheor.

PVP:

Sterling MT (gr)

1.8 )< 103

1.8 X 104

c. +

~- =

2.0 X 108

1.8 X 104

10 Inv.

glass f = -12 Inv.

-29

mv.

Sterling MT (gr) ~ = +37 Inv.

Initially 10.8 M: depleted by adsorption. ~ Average for 30 min. period. where ~ is the kinetic viscosity and o~ the rate of rotation of the disc (in radians per second). Hence the maximum deposition rate is obtained from j

=

0.621 CoD2/3~-ll6w1/2.

[3]

Putting in numerical values for the present system, namely, a = 2.25 X 10-s era., , = 8.97 × 10-3 Stokes (hence, D = 1.08 × 10-s cm. s sec.-i), ~ = 20.5 radians see. -I (hence A = 3.58 X 10-4 era.), Co ~ 6 X 10s; leads to j ~ 1.8 X 104 (particles per era? per sec.) In Table I this theoretical value is compared with the highest rates observed in two systerns having opposite charges. A third systern, namely, cellophane in dilute cetavlon, gave a maximum similar to that for case B, with j~xpt. = 2.2 X 103 presumably corresponding to opposite signs of zeta-potential in the equilibrium solution (depleted by adsorption). It is seen that the maximum experimental deposition rates were an order of magnitude smaller than the theoretical maximum rate; the discrepancy is beyond experimental uncertainty, despite the approximate nature of the data. The explanation is almost certainly that the dispersions showing maximum deposition inevitably had little or no detergent in solution (pure water or 10-6 M eetavlon) and hence were undergoing some aggregation during the period of the experiment. Typically, the optical transmission rose from 31% to about 38 %. According to Eq. [3], aggregation would reduce j by its effect on both Co and D. It is not possible to establish the quantitative relation between changing optical

transmission and j; but in the one system examined (PVP in 0.24 raM. Aerosol + 0.02 M KC1), the deposition rate after 30 minutes was ~ t h of the initial rate, during which time the optical transmission had risen from 30 to 40 %. The suspensions coneerned in Table I being distinctly less stable than in this example, it seems likely that the low average deposition rate in these experiments is due to slowing of deposition with time. This does not exclude the possibility that the retardation might be due in part to gradual neutralization of the average charge of the plane surface as oppositely charged particles deposit on it. The results therefore suggest that under ideal conditions (a stable dispersion and a bare solid of opposite charge) the theoretical equation [3] might well be valid. The effective range of the Nernst boundary layer being about 3.5~, whereas the surface forces are significant only out to <0.1~, maximum deposition is controlled solely by diffusion; any acceleration due to van der Waals forces or double-layer attraction cannot significantly increase the deposition beyond this limit.

Deposition against an Electrostatic Barrier Double-layer repulsion can, however, reduce the deposition rate by interposing an energy barrier and thereby inducing a finite concentration (C8) of particles in the liquid very close to the solid. Consider a system in which the carbon dispersion is virtually stable (showing constant optical density), the ionic strength is low, and carbon and substrate are both negatively charged; for example, carbon (~" = --43 my.) depositing on glass (~ =

DEPOSITION OF COLLOIDAL PARTICLES ON SMOOTH SOLIDS --108 my.) from suspension in 0124 mM. Aerosol. The number of particles deposited per 104~2 was 2.0, giving je,~t. = 11. This is so small compared with jth .... possible by diffusion transport (18,000) that the diffusion step is unimportant, the surface deposition step being rate-controlling. The concentration just outside the energy barrier must be practically equal to Co. According to the current theory of colloid stability (Derj aguin-Landau-Verwey-Overbeck), the energy barrier extends over hundreds of Angstrom units; therefore, transport across this barrier must be treated as diffusion in a force field (rather than by the rate theory of chemical kinetics). An expression for linear diffusion in a potential energy gradient is derived in the Appendix. The retardation factor resulting from the field E = E(x) can be evaluated if E can be expressed in simple analytical f o r m - - o r otherwise by numerical integration of f~ exp (E/t~T) dx. The impossibility of interpreting the slow deposition data obtained experimentally on the basis of the straightforward theory is proved by even the most approximate calculations. Consider, for example, the deposition from dispersions of carbon in 0.24 mM. of Aerosol containing 0.02 N KC1. These dispersions were slightly unstable and produced slow, but readily measurable, deposition on the various negatively charged substrates, N'e being of the order of 20 (Fig. 5). In this medium both the carbon and the deposition substrates had zetapotentials of about 50 my. Figure 6 shows an approximate potential energy curve for the theoretical interaction of a sphere and a plane with surface potentials of 50 my, in 0.01 N KC1, with an assumed value of the London-Hamaker constant A = 5 X 10-~3 erg (arrived at from London theory by use of optical data (13b). The interaction has been taken as equal to twice that for interaction of equal spheres, the formulas for repulsion and attraction being, respectively: VR

-= ea¢02{ln [1 -}- exp (--Kh0)]};

VA=-F

~oo "

349

io

cN@; 5

>

2oo ~.(Z) o

r

-i

FIG. 6. Approximate theoretical potential energy curve for interaction of a sphere (radius 0.225~) with a semi-infinite plate: ¢0 = 50 mv; A = 5 X 10-~3 erg; ~ = 3.3 X 106 era.-I (0.01 N KC1). (Here e is the dielectric constant, a is the radius of the spheres, h0 is the shortest distance between sphere and plate, and F is a factor, less than unity, correcting for the retardation effect. It has been shown elsewhere (13b) that the factor for sphere-plate is approximately the same as for spheresphere interaction and the formula given by Schenkel and Kitchener (15) has therefore been used.) For a rough estimate, the energy barrier may be replaced by a linear rise of about 10-11 era ( = 250 kT) over a distance of about 60 A. Applying the formula [3] (Appendix) gives j = 10-99, signifying that deposition should be completely impossible! Similarly, a barrier of }4 X 250 k T should mean perfect stability of the carbon dispersions. This highly anomalous result definitely cannot be ascribed to the (recognized) inaccuracies in the expressions for Vs or VA, or to the choice of value of A or the identification of ~0, the surface potential, with ~'. In order to fit the experimental data, the energy barrier would have to be very much smaller--about 15 kT. (The corresponding barrier for sphere-sphere interaction would be 7.5 kT, and this would be consistent with the slow coagulation observed in the carbon dispersions themselves.) One is therefore forced to the conclusion

350

MARSHALL AND KITCHENEIR

that either the D-L-V-O theory is numerically inaccurate by more than an order of magnitude or, for some reason, the theory cannot be applied directly to either the deposition system or the coagulation of carbon black dispersions. One possible explanation for the deposition results might be that minute mechanical disturbances in the liquid, arising from the necessarily imperfect rotating disc, provide activation energy for the particles to cross the repulsion barrier. But this cannot apply to the carbon dispersions. When a similar conclusion was arrived at from earlier work on suspensions of microscopic spheres of sulfonated polystyrene (15), it was suggested that floeculation was occurring into the secondary minimum (which must always be present if a very high potential energy barrier is encountered). This explanation might apply to the present carbon dispersions, but alone it cannot explain the observed deposition data for two reasons: first, deposition into the secondary minimum should be equal to the maximum possible diffnsional flux, whereas much smaller depositions were observed, indicating the surmounting of a barrier; secondly, the deposition was always measured after a period of washing which, presumably, would have removed particles in a shallow secondary minimum. In fact, even prolonged washing did not remove them. Strictly, the deposition problem should be treated by the full theory of "heterocoagulation" for interaction of dissimilar surfaces (of. Derjag~in (i), Bierman (2)). This treatment leads to somewhat smaller energy barriers for interaction of surfaces of potentials @t and ~2 than use of the geometrical mean potential; but it seems that the order of magnitude would still be too great. In any case, this complication does not arise with the carbon-carbon interaction in the dispersions. A more likely explanation of the results is that the particles and the surfaces were not smooth enough to be treated as ideal spheres and planar substrates. The doublelayer "thickness" parameter, I/K, is typically only about 30 A. If aspirities of about 50 A were present, the particles would first move into the secondary minimum (at about i00 A) and spend some time there, undergoing

Brownian motion and radial flow over the surface of the disc, until a chance contact caused seizure on a projection. The local repulsion between such small projections would be insignificant. Electron micrographs of shadowed replicas from glass and plastic surfaces (16) suggest that no projections larger than about 100 A are present, but the possibility of smaller ones cannot be excluded. In support of the roughness theory is the observation that, although all the systems studied responded qualitatively in the same manner to reduction of zeta potentials and increase of ionic strength, the absolute depositions at comparable values of the semiempirical stability parameter 0 = h f ~/KA varied widely from one substrate to another (Fig. 7), suggesting that deposition was dependent only partly on double-layer repulsion and London-van der Waals attraction, but was also influenced by some other specific factor--conceivably surface roughness. Significantly, points for a given substrate, but obtained from various concentrations of surfactant and electrolyte, fell on a smooth curve. 100 0

AEROSOL (+KCI) PVP

z~ Ce[tophane ~7 PS 80 -

\

©

PVF

[]

gldss

•

AEROSOL ( + SCMC) PVP

~& C e t t o p h e n e

6C

PVF Nd :

40

20

0 ~

~

2 e(.lO

4

~

6

.

8

n)

FIG. 7. Partial correlation of deposition with the parameter 0 (=¢1¢~/d). (Solid points: with added sodium earboxymethyl cellulose.)

DEPOSITION-OF COLLOIDAL PARTICLES ON SMOOTH SOLIDS APPENDIX

Di:~usion of Charged Colloidal Particle8 go a Wall Bearing an Electrical Double Layer of Like Sign The general theory of diffusion in a field of force was developed by Smoluchowski (17). The stationary linear diffusion current, j, in the direction x, from a plane A to a plane B against a potential (per particle) E = E(x), is given b y

C ( k T / m ) exp (E/kT)I~ j ~---

B

[1] "*"

~ ¢ ~ exp ( E / k T ) dx where C is the local concentration, m the mass of the particles diffusing and ~ the frictional coefficient, which, for a sphere of radius a, is given by ¢ = 67ran~m, v being the viscosity of the medium. The numerator means the difference between the value of the function at planes A and B. For the case of diffusion up to a plate at which all particles adhere, CB = 0, while if plane A is t a k e n at the outer edge of the electrical double layer, for slow deposition C~ ~ Co, the bulk concentration, and EA = 0. Equation [1] therefore reduces to

CokT j ~

I"B

[1'] """

67ra~ J~ exp ( E / k T ) dx

represents the retardation of the diffusion across the region of the potential gradient. This t e r m tends to unity for Em << kT, whereas for systems where E,~ > 5kT, the 1 in the denominator becomes negligible compared with the exponential term. In deposition kinetics, E,, represents approximately the height of the total potential energy barrier. ACKNOWLEDGMENTS The authors thank Unilever Research (Port Sunlight) for award of a research studentship to J. K. M., during the tenure of which the present work was carried out. REFERENCES

1. DERAGUIN, B. V., Discussions Faraday Soc. 18, 85 (1954). 2. BIERMAN,A., J. Colloid Sci. 10,231 (1955). 3. HUNTER, R. J., AND ALEXANDER, A. E., or.

Colloid Sci. 18, 820 (1963). 4. DUnHAM, K., ed., "Surface Activity and Detergency." Macmillan, London, 1961. 5. (a) ZIMAN, A. D. AND DERJAGUIN, t~. V.,

Kolloidn. Zh. 25, 135 (1963). (b) BOHME, G., AND KRUPP, H., et al., Proc.

Intern. Congr. Surface Activity 4th Brussels 1964, in the press. 6. DUnHA~, I~., J. Appl. Chem. 6,153 (1956). 7. VON LANGE, ]:I., Kolloid Z. 154, 103 (1957); ibid. 156, 108 (1958). 8. GRahAm, D., AND KAy, W. S., J. Colloid Sci. 16, 182 (1961). 9. SALEEB, F. Z. AND KITCttENER,

I f E(x) were known (e.g., for a calculated Verwey-Overbeek energy curve) a solution of [1'] could be obtained by numerical integration. However, for the present purpose it will be sufficient to consider a simple case for which the integral can be performed, namely, a linear potential gradient. Suppose the potential increases from zero at A to a m a x i m u m value E~ at B, and the diffusion p a t h A B has a length ~; therefore E = (E,~/~)x. Hence,

Cok T / 67rav J = (6kT/Em) exp (E,~x/SkT)]~ "'" "

[2]

= D ( C ° / ~ ) [ e x p (E~,/kT)(E~/kT)--1] " " . [3] where D is the diffusion coefficient according to the Stokes-Einstein formula ( = k T /

6~na). I t is seen t h a t the t e r m in square brackets

351

J. A., (a)

Proc. Intern. Congr. Surface Activity 4th Brussels, 1964), in the press. (b) J. Chem. Soc. 167, 911 (1965). 10. RIDDIFORD, A. C., Advan. Electrochem. Electro-

chem. Eng. 4 (1966) in press. 11. LEVIeH, V. G., "Physicochemical Hydrodynamics" (Trans.), p. 60 Prentice-Hall, New York, 1962. 12. BIRBECK, M. S. C., AND STACE¥, K. A., or. Biophys. Biochem. Cyto~. 5, 167 (1959). 13. (a) MAaSHALI,,J. K., To be published. (b) MARSHALL, J. K., P h . D . thesis, U n i v e r s i t y

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