J. CHEM. SOC. FARADAY TRANS., 1993, 89(16), 3091-3097 3091

Chronoamperometric Mobilities in the Electrodynamics of a Non-aqueous Sol : Depletion, Turbulence and Electron Transfer in Sol Electrodeposit ion

David R. Rosseinsky* and John S. Graham Department of Chemistry, The University, Exeter, UK EX4 4QD Malcolm T. Connah Malvern Instruments Ltd., Spring Lane South, Malvern, Worcester, UK WR14 1AT

The validity of a simple chronoamperometric technique for measuring the electrophoretic mobilities of colloid particles has been established by comparison of the results with those obtained from photon correlation spec- troscopy, a well established method. Resolution of several flaws in the theory underlying chronoamperometric electrophoresis has led to a rationale of electrophoretic deposition based on turbulent liquid flow. The new theory also accounts for observations regarding the charge-transfer processes necessarily accompanying parti- cle deposition.

Colloid mobilities may be measured by means of inter alia a simple chronoamperometric electrophoresis experiment,' which is particularly suited to non-aqueous sols. The disper- sion is contained between two closely spaced cylindrical elec- trodes, and on application of a potential, particles migrate across the cell gap to deposit on the inner electrode. The elec- trophoretic mobility is obtained from logarithmic plots of the observed current i us. time t.

This theory, however, is flawed on several counts. The stated diffusion requirements remain unsatisfied. Further- more, the theoretically linear semi-logarithmic plots are in fact initially curved. There are also unexplained (and unremarked ') two-fold factors between electrophoretic mobi- lities independently calculated from the initial experimental currents and those from In i us. t plots.

We aim here to remedy these deficiencies and to present a mechanism for the electrochemical processes accompanying deposition. To this end, chronoamperometric experiments were performed on carbon-black sols in hydrocarbon- surfactant and the mobilities checked by photon correlation spectroscopy (PCS), which additionally yielded particle sizes. In order to establish the associated physical properties of the sol, the adsorption isotherm of the stabilising surfactant on the particles was established to achieve a quantitatively con- sistent description of the system.

Original Cbronoamperometry (CA) Equation' for Electrophoresis

A uniform distribution of particles throughout the cell is assumed' (unjustifiably: see Discussion and Appendix 11) to pertain over the entire course of particle depletion, requiring the particle diffusion velocity to be substantially greater than the electrophoretic velocity. The rate of particle depletion is' then :

dn n dt z

- -=-

where n is the number density of particles at time t and z is the average time for a typical particle to cross the cell gap, the 'transit time'. On integration,

n = no exp( - :)

where no is the initial number density of particles. The current i between electrodes of area A arising from the migra- tion of particles with velocity u and bearing a charge 4 is

i = qunA

and for the initial current i, , i, = qunO A

We thus obtain:

i = qunoA exp( - :) = i, exp( - :) Hence z, from a plot of In i against t ; the mobility p, as an average, follows from

u L2 p = - = -

E Ez

where E is the field strength and L is the length of inter- electrode gap. (The geometry and properties of the system disallow application of ordinary electrolyte theory for i.)

Experimental In the apparatus for chronoamperometric electrophoresis (schematic in Fig. 1) two concentric cylindrical aluminium electrodes separated by 4.5 mm are mounted in a cell with a Perspex base. As particles migrate across the cell gap under a high field, 50-1500 V cm-', the current measured on a Keithley 61OC analogue electrometer (input impedance 1014 R), was plotted on a high-input impedance (10 MR) chart recorder.

The instrument employed for PCS was a Malvern Zetasi- zer 3 in conjunction with an AZ26 low-conductivity cell.

X-Ray fluorescence spectroscopy, on a Philips PW 1400 fully automated sequential X-ray spectrometer, was used to determine the amount of zirconium surfactant adsorbed at the solidfliquid interface, for constructing an adsorption iso- therm.

The colloidal dispersions were akin to liquid toners used in reprographics. They consisted of four components, Raven 1020 carbon black, Neocryl B-702, a thermoplastic acrylic polymer from Shell, used here as a steric stabiliser, Isopar G, a hydrocarbon solvent consisting of C,-C straight chains, and zirconium versetate, a metal soap surfactant from Shell,

J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

4 cathode

3092

ii

high tension source

electrometer 4 cathode

insulating base Fig. 1 The chronoamperometry system

used as a dispersing agent and to confer charge on the par- ticles. (The substitution of nonane as the dispersion medium led to dispersions which flocculated in a few days: nonane is noticeably less viscous than Isopar G which could account for its inefficacy.)

Typically carbon black (2 g) was added to an Isopar G solution (100 ml) of polymer (10 g) and surfactant (0.2 g) in a ceramic plot. After stainless-steel balls were inserted, the sealed pot was rotated on its horizontal axis for 24 h. The resulting concentrate was diluted 100-fold to give a disper- sion necessarily aged for 4 days to permit adsorption pro- cesses to reach equilibrium.

For PCS2 the dispersions were diluted 25 times with a clear isotonic fluid obtained from centrifuged dispersion, undiluted dispersions absorbing too much of the incident laser beam. Dilution had no noticeable effect on the mea- sured mobilities (Table l). In each PCS measurement, three readings were taken and the average mobility calculated.2 A similar procedure was adopted for the complementary chro- noamperometric measurements.

Two sets of experiments established whether mobility from chronoamperometry was independent of the initial number density of particles. First, gross dilutions of the colloid con- centrate were carried out (series I). Secondly, a series of graded dispersions were produced in which only the number of particles varied, the bulk concentrations of polymer and surfactant remaining constant throughout (series 11). The range of dilution employed in both cases was limited by the need of a large enough deposition current to give reliable data and at the upper extreme by the necessity to avoid incip- ient flocculation of highly concentrated dispersions.

The X-ray fluorescence spectrometer for determining the amount of zirconium per unit mass of particles was cali- brated using aqueous solutions of ZrC1, of known concentra-

Table 1 Variation of mobility (PCS) with dilution of sol

electrophoretic mobility/lO-’ m2 V-’ s- ’ dispersion diluted by

6 x 12 x 25 x 50 x

1.16 0.98 1.11 1.05

tion. Particles were extracted by centrifugation from various dispersions differing only in surfactant content. The centri- fuged deposit was ground to a uniform particle size, mixed with cellulose powder in a 1 : 10 ratio and the mixture pressed into tablets for fluorescence spectroscopy.

For each dispersion, the specific surface area exposed to solution was calculated from the mean particle size (ca. 0.2 pm) as determined by PCS, and from the mass of one particle and the total mass of deposit was calculated a value for the surface concentration of adsorbed zirconium.

Results and Discussion Characterisation of the Dispersions

Measurements of mobility (expressed as zeta potential) in four different dispersions by CA and by PCS were in close agreement providing a sufficiently low field strength (< 100 V cm-’) was used for the CA measurements (Table 2). A field strength of 380 V cm-’ was used for the PCS measurements.

Owing to the substantial difference between the field strengths employed in the comparison of mobilities from CA and PCS, the dependence of particle mobilities obtained from chronoamperometry on field strength was investigated. The results are shown in Fig. 2. They agree with similar observa- tions made by S t ~ t z , ~ who interpreted the rise as being due to partial separation of the particle from its double layer.

Electrophoretic mobility was found to be independent of the initial number density of particles, varied in the two ways described in ‘Experimental’. The results are shown in Table 3 (series I) and Table 4 (series 11).

The dependence of mobility on polymer concentration was determined (Fig. 3), falling initially steeply and then approaching a constant value of 8.6 x lo-’’ m2 V-’ s-’ . A sophisticated theory4,’ accounts for an increase in mobility with increasing polymer concentration as observed for pre- served erythrocytes. Occasionally, however, the a ~ t h o r s ~ , ~

Table 2 Zeta potentials from PCS and CA

70 105 77 95

77 104 80 93

2.2 2 ~

1.2

2 4 6 8 10 12 14 16 18 20 applied potential/lO-’ V

Chronoamperometric mobility us. applied field strength

1

Fig. 2

J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89 3093

Table 3 Dependence of mobility on particle number density (series I)

relative initial particle concentration

electrophoretic mobility mz V-' S-I

0.25 0.5 1 .o 1.5 2.0 2.5 3 .O 3.5 4.0 4.5 5.0

average mobility

1.30 1.15 1.30 1.21 1.36 1.39 1.28 1.33 1.12 1.22 1.25

1.27 k 0.03

found that for other types of particles (aqueous dispersions of bentonite, rutile and quartz were studied) the mobility was reduced by increasing the polymer content. Such sols obtain at least part of their charge from specific ion adsorption (unlike the biological cells) and there thus exists the possi- bility that polymer chains could compete with ions for adsorption sites on the particles, thereby reducing the surface potential. There is evidence6v7 for such competitive adsorp- tion, hence for our dispersions, competitive adsorption prob- ably accounts for the observed decrease in mobility. Since mobility is independent of particle number density, it is clear that the mobility here is dependent on the ratio of the con- centration of surfactant to that of polymer.

Table 4 Dependence of mobility on particle number density (series 11)

relative initial particle electrophoretic mobility concentration m2 V-' s-

0.5 1 .o 2.0 3.0 5.0 6.0

average mobility

1.28 1.27 1.55 1.41 1.37 1.29

1.36 0.03

r

I v)

c I > E ri

0 I

1.4

1.2

1

Depletion and Turbulence

A typical diffusion coefficient for the 0.2 pm particles used here may be estimated from the Stokes-Einstein equation:

D = kT/67cqa x 1 x m2 s - '

where T = 298 K and q, the solvent viscosity, is 1 x kg ,-1 s-l . During the course of a typical CA experiment (of say 100 s duration) a particle will have moved a distance 1 x , /(Dt) = 1 x mm, whereas the halfway distance between the electrodes is 2.25 mm. Thus for the high field conditions employed here, the original authors' assumption,' cited in our introduction, is invalid, and local particle- depleted zones are bound to arise, such non-uniform distribu- tions thus apparently invalidating the theory used to calculate mobilities from chronoamperometry. However, the theory appears to be supported by the close agreement between mobilities measured by CA and by PCS. This paradox can be resolved only if turbulent liquid motion? replenishes depleted zones, thus restoring a uniform distribu- tion of particles (Fig. 4).

A value of 1.05 x lo-'' m2 V-' s-' was measured, by chronoamperometry for the mobility of the carboxylate anions. The surfactant dissociates as:

Zr'"(RCO,), -+ Z r ' " ( R C 0 ~ ) ~ + RCO,.

The frictional drag on the cation will be substantially greater than that experienced by the anion due to its size, thus causing it to be the slower moving of the two. A computer- generated space-filling representation of the versetate anion [Fig. 5(a)] shows it to be cylindrical; this probably accounts in part for its low mobility which is 10 times less than that of the particles,

Thus following complete particle deposition, 9/10 of the

Fig. 4 Schematic depiction of depletion and (curved arrow) of effec- tive distance over which replenishing particles are back-transferred. The extent of depletion is the ratio of distances x/L.

Fig. 3 sion

0 2 4 6 ' 8 '10 '12 '14

polymer content of concentrate ( o/o w/v)

Chronoamperometric mobility us. polymer content of disper-

t Joule heating may be discounted as a vehicle for liquid turbu- lence. The heat h generated over duration t (here ca. 1 min) by current i (here 1 FA maximum) through a resistance R is given by h = i2Rt = 0.6 J. The consequent temperature rise AT = hM,/c,pV = 0.01 K is far too small to induce significant convection. Here M, = molar mass (142 g mol-I), c, = heat capacity (233 J K-' mol-I), p = density of liquid (0.73 g cm-3), V = volume of fluid (40 an3); the values of molar mass, molar heat capacity and density, unavailable for Isopar G, are for the physically similar n-decane of comparable chain length.

3094 J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

Fig. 5 (a) Space-filling representation of the cylindrical versetate anion RCO,. (b) Likely square-planar configuration for an adsorbed neutral surfactant molecule where X is CO, OH or COY.

fluid volume would be occupied by uncompensated anions comprising an enormous negative charge density, this being matched by the comparably large positive charge in the metal of the anode. This is clearly an unstable situation and in reality the field so developing within the fluid will be suffi- cient to force the anions to migrate to the anode with a mobility exceeding their viscosity-limited mobilities. For charge carriers under such conditions, the Reynolds number is high and the flow becomes turbulent.' Turbulence often involves the formation of vortices or small localised regions of turbulent flow.' In such circumstances, sol particles required to move towards the anode will do so accompanied by counter-ions. There is thus a net neutral particle moving backwards. Such a mechanism would not invoke counter- current contributions, arising from back-transfer of positive particles, to the observed net current. Furthermore, nil- current motion of the anions towards the anode is thereby effected by vortical turbulence, as depicted schematically in Fig. 6. Here only two particles are shown; in reality many particles will comprise one vortex with approximately equal numbers in pro- and anti-field motion.

The number of back-transferred particles required to restore uniformity after a period of depletion is obtained as follows. In Fig. 4, x/L is the extent of depletion. The popu- lated region contains a number of particles N = No(l - x/L) ,

+ r

/ /

I L I

I

I I

/$ I

I I I I I I

5 \ \

\ \

__+

\ \ , ,

\ \ \ \ \

\ \

I I

I I I

I f

/ I

Fig. 6 Simplified diagram of the mechanism of vortex-controlled motion in turbulent liquid flow

where N o is the initial number of particles. Following replen- ishment of the depleted zone by back transfer, the number density of particles in both the bulk (a) and newly replenished (b) zones is n, = nb = no(l - x/L) . The volumes of the two zones are V, = A ( L - x ) and V, = Ax, where A is the recei- ving electrode surface area. Hence multiplying n, and nb by the appropriate volumes gives the particle numbers in each region,

N , = No(l - x /L) , and N b = No(x/L)(l - x/L) .

Thus the number of back-transferred particles required to restore uniformity is N(Nb/N, + N b ) = N(x/L) .

The effect of turbulence on the distance travelled by the particles over an entire experiment is readily deduced. In a simplified depiction of chronoamperometric electrophoresis, a proportion of the particles is abstracted by deposition, causing a zone depleted of particles to appear. Then follows restoration of the system to uniform particle number density by back-transfer of a fraction of the remaining dispersed par- ticles. This cycle is repeated continuously until all the dis- persed particles have been deposited. Expressions for the numbers of deposited particles ( N , , N , , . . . etc.) and of back- transferred particles ( N l , x , N 2 , x , ... etc.) in terms of the initial number of particles and the extent of the depleted zone x / L (assumed constant throughout) then follow, from the pre- ceding paragraph, as:

N , = No(l - x /L) ; Nl, x = N o ( x / W - x / L )

N , = No( l - x / Q 2 ; N 2 , = No(x/L)(l - x/L)2

N3 = No(1 - x / L ) ~ ; N3 ,x = NO(x/L)(l - x / L ) ~

The total distance travelled, weighted according to particle number, is given by:

L x - - ( N o + N , + N , + N , + - -) forward 2 L

L + 2 (Nl,x + N 2 , x + N 3 , x + - . .) reverse ( 1 )

The simple theory' gives eqn. ( 1 ) as merely (L/2)N0. There- fore the ratio of the effective distance travelled to that without turbulence can be expressed as :

co co

x = ( x / L ) C(1 - x/L)' + ( x / L ) ( 1 - X / L y (2) j = O k = 1

Eqn. (2 ) may be written as:

00

x = 2(x/L) c (1 - x/L)' - x / L j = O

1 - ( 1 - x / L ) (3)

Hence the ratio of turbulence to non-turbulence driven dis- tances is 2 - (x /L) . In the limit of small x / L (as will be the case experimentally), turbulence causes the particles to travel twice the L / 2 distance anticipated in a continuous depletion regime over an entire CA experiment (see Appendix 11). Thus for the same net motion over distance L / 2 in the two cases, clearly the net (observed) velocities will be u for the former and 2u for the latter. Since turbulence in our model is always preceded by depletion, the t = 0 velocities will be the same, uo , for both cases (only with persistent depletion does deposi- tion continue at velocity uo). Hence uo = 2u, and comparably Po = 2P.

J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89 3w5

-16*8 r 7 = In 2

-17.6 n i

0 20 40 60 80 100 120 140 160 180 time/s

Fig. 7 Semi-logarithmic plot of a typical CA current transient (i , is a small background current) showing two-fold ratio between observed and backextrapolated initial currents. 0, Experimental decay; (-) theoretical decay.

In semi-logarithmic plots of CA current transients (Fig. 7), strong deviation from linearity is notable during the initial stages of deposition; for the case shown, linearity only sets in after 80 s. An approximate In 2 difference between the experi- mental and the back-extrapolated initial current is also evident. The mobility p,, calculated from the experimental initial current i, by means of p, = io L/EQo, is expected, from the preceding argument, to be twice that obtained from the slope of the semi-logarithmic straight line. Both of these two- fold relationships are demonstrated graphically in Fig. 8, where p, and i, are plotted against p and ib, respectively, from a wide variety of separate experiments. The best-fit straight line passing through the data has a slope of just 2, as predicted by the depletion/turbulence model.

Electron Transfer and Charge Accumulation during Deposition It is necessary in particle electrodeposition for the charge- carrying entity to lose its charge (or part of it) at the depos- iting aluminium electrode by a net electron-transfer process, in order for the liquid-phase current and that in the external circuit, carried by electrons, to be identical. There are two

p/1 o - ~ m2 V-' s-l 0 2 4

f 0 6l

1 0

0 2 4 id1 0-* A

Fig. 8 Observed i, us. back-extrapolated ib and po (calculated from io) us. pz (calculated from transit time t). Best-fit straight line has a slope of just 2, as required by depletion/turbulencc model. 0, Initial currents; 0, mobilities.

possible redox reactions at the cathode:

ZP + k,, + Zro

ZP + 2e,, Zf The reduction of Z P to Z p is possible in view of the exis- tence of Zm in c~mplexes.~ At the anode, the requisite elec- tron loss is undoubtedly effected by oxidation of the carboxylate anions, loss of CO, and radical combination :

RCO, RCO', + eAl

RCO; -+ R + CO,

2R' -+ R,

There is support for this mechanism" from studies on the oxidation of carboxylic acids at platinum electrodes. No bubbles are seen during sol deposition, but C 0 2 is appre- ciably soluble in hydrocarbon solvents (0.0133 mol dm-j in n-decane or 1500 cm3 of C02 per litre of n-decane at STP").

Following complete particle deposition, a slow release of charge was observed over a 14 h period [Fig. 9(a)]. This process follows a first order rate law Q = Q,[l - exd-kt)] where the rate constant k = 0.22 s-l and the limiting value Q, = lo-' x (17.35 & 0.05) C, which is ca. 20% of the total deposition charge, implying that the deposition current is partially attributable to non-faradaic processes. The circuit used to measure the stored charge is shown in Fig. 9(b).

18

16

0 l4

0 12 m

1 6 c

4

2w 0 2 4 6 8 10 12 14 16 18

time/h

cell

source

I 1 Fig. 9 The release of hitherto stored charge. (a) Accumulated charge us. time; (b) circuit used for measurements in (a).

3096 J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89

0 Q,

6 8 L ‘i,

I I I I

i 0 20 40 60

time/s

Fig. 10 Calculated excess current us. time. 0, Data points; (-) calculated fit.

The ‘excess current’ (i - i’) evident in Fig. 7 is plotted against time in Fig. 10. The data fit the equation, a single- exponential decay, (i - i’) = A exp(-&), where A = 25.17 nA and B = 0.1024 s - ’ for the example shown. Graphical integration of this plot yields an ‘excess’ charge. A compari- son of the values of the stored charge, excess charge and the ratio of stored to deposition charge is shown in Table 5 for four separate experiments. The depletion/turbulence model can account for these observations. During the initial depletion-governed stage of deposition, the anions remain effectively stationary within the fluid. There is thus no charge transfer at the electrodes and the observed current flow is non-faradaic. Once turbulence commences, reduction of Zr”’ occurs and thus, following complete particle deposition, the cell may be regarded as a combination of two capacitors (Fig. 11). At the cathode unreduced ZrtV from the depletion period is present. Electrons residing within the cathode counter- balance the positive charge of the deposit. Concomitant holes are present within the anode, these counterbalancing the remaining anions. When the two electrodes are connected via a coulometer, the electrons within the cathode can move towards the positively charged anode. This is accompanied by the observed repeptisation of those particles still bearing Zr”, during which the diffuse double layer reforms around these particles. This mechanism accounts for the discharge current being opposite in sign to that accompanying deposi- tion, and it also explains why the release of the stored charge is slow to reach completion as observed, the rate-limiting process being the repeptisation of hitherto deposited par- ticles. Finally, Qstored = QexceSS (Table 5) is rationalised.

Particle Charge and Adsorption at the Solid/Liquid Interface

A value for the specific charge (charge per unit mass of deposited particles) was calculated from the amount of zir-

Table 5 Comparison of measured stored charge and calculated excess charge for four separate CA experiments on same dispersion

Qstored Qstorcd Qexcess lo2 x -

run 110-7 c /lo-7 c Qdeposition

1.74 1.67 21.2 1.84 1.78 21.3 1.75 1.74 24.6 1.83 1.78 22.3

average 1.79 & 0.03 1.74 f 0.03 22.4 & 0.8

short circuit cathode 4 >anode

fluid I

Fig. 11 Schematic depiction of the distribution of ions within the CA cell during initial stages of release of stored charge. The 1.h.s. arrows show the repeptisation and the r.h.s. arrows indicate accom- panying anion diffusion.

conium present within the samples of electrodeposited and centrifuged particles, as determined by X-ray fluorescence spectroscopy. The centrifuged particles were analysed to exclude the possibility of free ions discharging at the cathode during chronoamperometry, and thereby contaminating the deposit with excessive amounts of zirconium. Both types of deposit were found to contain 5.05 mg of Zr per gram of particles. Because of the low permittivity of the medium, ionic zirconium will be present in the liquid as [Zr(RCO,),]+, implying a specific charge due to adsorbed Zr species as 5.35 C g-’. This is ca. 1.5 x lo3 greater than the value measured from chronoamperometry (3.61 x C g-l). Thus, of all the zirconium moieties adsorbed at the solid/liquid interface, only ca. 1 in 1500 are present as cations resulting from the dissociation of [Zr(RCO,),] to [Zr(RCO,),]+. This is in approximate accord with a similar conclusion drawn by Novotny,’2 ‘When the fraction of ionised molecules on the particle surface is compared with the number of adsorbed molecules on the surface, it is found that approximately 1 out of lo4 are ionised.’

An isotherm for the adsorption of zirconium versetate at 298 K is shown in Fig. 12. The data for the dependence of

9

2 3l 0 0 20 40 60 80 100

C/10-4 mol-’ dm-3

Fig. 12 Isotherm for the adsorption of surfactant. 0, Experimental; (-) Langmuir fit.

J. CHEM. SOC. FARADAY TRANS., 1993, VOL. 89 3097

surface concentration r on solution concentration c show a good fit to the Langmuir equation:

rm bc 1

1 + bc TmL r=- and o=-

where r, = saturation concentration (found to be 3.31 x mol m-2), bccexp(-A,,,G/RT) (found to be 9.52 mol-’ dm3), whence CJ = 0.50 nm2. A computer- calculated representation of the versetate anion [Fig. 5(a)] gives an approximate value of 0.126 nm2 for the cross- sectional area of the versetate ion. An adsorbed neutral molecule will probably possess distorted square-planar c~ordination’~ [Fig. 5(b)], giving a contact area of 4 x 0.126 nm2. An empirical calculation thus gives the minimum likely area per adsorbed molecule as 0.5 nm2. This is in excellent agreement with the corresponding value obtained from the isotherm.

Conclusion Chronoamperometry has been shown to be a valid technique for measuring the parameters associated with colloid disper- sions. A new mechanism of chronoamperometry based on turbulent liquid motion has been shown to be capable of accounting for the previously unremarked two-fold differ- ences between mobilities calculated from the initial current and from the straight-line slope of the semi-logarithmic current transients. The new model also accounts for the observed electrochemical phenomena accompanying deposi- tion. Characterization of the dispersions used was supported by means of an isotherm for adsorption at the solid/liquid interface.

Electrodeposition in a cell must involve electron transfer at an electrode. This may not be a necessity in liquid-toner elec- trographics, where the fixed-phase attracting charge is an electrostatic one pinned at sites on an essentially insulating material. Nevertheless the occurrence of turbulence can arise from local charge density imbalance at temporarily depleted regions, during sol deposition in the image forming process.

A preliminary a c c o ~ n t ’ ~ has been published.

We thank Drs. D. E. Wilson and J. P. N. Haxwell of Coates Lorilleux for continuous support and advice, Mr. D. McGee of Gestetner Manufacturing Ltd. for discussions, the SERC and Coates Lorilleux Ltd. for a CASE studentship for J.S.G., and Dr. David Williams of Imperial College, London, for his assistance with the molecular modelling.

Appendix I. Values of mobility, deposition charge and particle diameter can be shown to be mutually consistent by calculating via Stokes’ law a value for the mobility from the charge per par- ticle, its radius and the viscosity of the dispersion medium and comparing the result with the corresponding PCS mea- surement. The properties of the sol required for this compari- son are given in Table 6. If the double-layer thickness is much greater than the particle diameter (ica 4 l), which is the case for dispersions in apolar media, the charge counter to the particle charge is too far away to impinge on the hydro- dynamic flow around the particle.” From Stokes’ law the drag force on the particle is f = 6ntfau, where a is the particle diameter and tf is the fluid viscosity (here 1.014 x kg m-’ s-’). Hence from balancing the electrical and drag forces,16*17 4 E = 6ntfau and q = 6ntfap. The charge per parti-

Table 6 Summary of properties of sol required for comparison of calculated mobility (from deposition charge) with that measured from PCS

experimental property method value

1.22 x 10-9 electrophoretic PCS

mass of deposit, M gravimetry 1.79 x 10-3 g

,2 v-1 s-l mobility 1.69 x C deposition charge, Q CA

density of deposit, p free flotation 0.960 g particle diameter PCS 210 nm

cle is given by q = m(Q/M) = 4.40 x 10- l8 C, where Q is the total deposition charge, M is the total mass of deposit and m is the mass of one particle = 4/3n(~/2)~p = 4.66 x lo-’’ The mobility is obtained hence as p = 4/6nqa = 1.10 x 10- m2 v-1 s-l , cf. PCS value of 1.22 x m2 V-’ s - l . Some slight discrepancy might have arisen from different intrinsic weightings associated with each of the averaged parameters used for the ‘typical’ particle in the dispersion, and from applying the macroscopic Stokes’ law to such small particles (athough Walden’s rule implicitly based on Stokes’ law works well for much smaller ions in aprotic solvents).

11. It is necessary to show that for an ever-uniform com- position regime, the ‘transit time’ 7 is indeed the time taken to traverse the distance L. Substitution of no = N , A L and Q = 4 N , , where Q is the total charge carried by all the par- ticles, into i, = qun, A gives i, = Qu/L (Al). Since the deposi- tion (‘electrochemical’) charge is necessarily equal to the sum total charge borne by all the particles prior to deposition,

gs

Q = ri dt = i, [exp( - t / 7 ) dt = i, z (A2)

Substitution for Q of eqn. (A2) into eqn. (Al) gives, for the ever-uniform composition regime maintained by turbulence, u = L/7, hence 7 is indeed the time to traverse L.

References 1 2

3 4 5

6

7

8

9

10

11 12

13 14

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Paper 3/00651D; Received 2nd February, 1993