Osmotic pressure measurements on strongly interacting polymer colloid dispersions

Colloids and Surfaces

A: Physicochemical and Engineering Aspects 161 (2000) 231–242

Osmotic pressure measurements on strongly interactingpolymer colloid dispersions

R.H. Ottewill *, A. Parentich 1, R.A. RichardsonSchool of Chemistry, Uni6ersity of Bristol, Cantocks Close, Bristol BS8 1TS, UK

Abstract

Equipment has been developed for the determination of the osmotic pressure of small-sized particulate dispersionsof polystyrene at low volume fractions. The specific conductance of the dispersion and that of a reservoir of salt, inequilibrium with the dispersion separated by a membrane, was measured simultaneously with osmotic pressure. Theresults are analysed in terms of a low volume fraction theory of osmotic pressure equilibrium between a colloidcompartment and salt solution compartment separated by a membrane. © 2000 Elsevier Science B.V. All rightsreserved.

Keywords: Osmotic pressure; Polymer–colloid dispersions; Electrostatic interactions

www.elsevier.nl/locate/colsurfa

1. Introduction

Fifty years ago the book by Verwey and Over-beek [1] was published. This work together withthat of Derjaguin and Landau [2], provided abasis for an understanding of the behaviour ofcolloidal dispersions in terms of a ‘soft’ pair-po-tential for particle–particle interactions. Althoughin many ways simplistic, the theory, which wasinitially developed to understand the coagulationof colloidal systems by electrolyte, has provedrobust and has been applied to understand a widerange of problems. It has also withstood the test

of direct experimental studies on measurements ofthe forces between two surfaces [3,4]. In addition,it has provided an inroad into understanding thebehaviour of concentrated dispersions where theinteractions are multibody rather than pairwise.These studies demanded systems of spherical par-ticles with a very narrow distribution of particlediameters and known surface charge. Thus, thedevelopment of synthetic methods such as thoseof Goodwin et al. [5,6] for the preparation ofpolymer colloids enabled detailed rheological andscattering studies to be initiated on monodisperseconcentrated dispersions [7–9]. Moreover, withthese systems, after cleaning by dialysis andmixed-bed ion-exchange resins the importantparameter surface charge density could be deter-mined by either conductimetric or potentiometrictitration.

* Corresponding author. Tel.: +44-117-928-7647; fax: +44-117-925-1295.

E-mail address: [email protected] (R.H. Ottewill)1 Present address: Curtin University of Technology, Perth,

Western Australia.

0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved.

PII: S0927 -7757 (99 )00373 -8

R.H. Ottewill et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 161 (2000) 231–242232

A basic experimental measurement which canalso be used to obtain fundamental informationabout colloidal interactions and changes of statein concentrated dispersions is that of osmoticpressure studied as a function of the volumefraction of the dispersion and electrolyte concen-tration. Moreover, osmotic pressure can be di-rectly related to the thermodynamic andstatistical mechanical properties of the system[10].

In early work, an apparatus designed to studythe interaction between assemblies of plate-likeclay particles [4,11,12] was adapted to studypolymer colloid particles in both aqueous [13,14]and nonaqueous dispersions [15]. These studiesclearly showed the formation of crystalline ar-rays of spherical colloidal particles [13–15].Adaptation of the equipment to enable opticaldiffraction measurements to be made simulta-neously with pressure [16,17] clearly showed thetransitions from a ‘gas-like’ structure to a ‘liq-uid-like’ structure and then to a single phase ofcolloidal crystals at a volume fraction of ca.0.55; the latter was in near agreement with thepredictions of Hoover and Ree [18] for hard-spheres. The co-existence of colloidal liquid andcrystalline states was also recognised in these ex-periments [17], as was the indication that elec-trolyte concentration changes caused similareffects to temperature changes in molecular sys-tems.

The early equipment was not found to besufficiently sensitive for studies of dispersions atrelatively low-volume fractions. It was, more-over, of considerable interest to study changesof electrolyte in the dispersion and in the reser-voir of electrolyte on the external side of theseparating osmotic membrane. Both the designof the new equipment and its use for studies ofwell-characterised small-size polystyrene colloidalparticles are described in this contribution. Thechoice of small particles was essential in orderto study interactions by osmotic pressure and inorder to correlate the results with previous workusing photon correlation spectroscopy, time-av-erage light scattering and small-angle neutronscattering [9,19–21].

2. Experimental

2.1. Materials

All the water used was doubly distilled in anall-Pyrex apparatus. The sodium chloride usedwas BDH Analar material.

The anion and cation exchange resins weresupplied by Bio-Rad in a specially precleanedform suitable for analytical use. The resins werefurther cleaned using the procedure of van denHul et al. [22] in order to remove any residualsoluble polyelectrolyte species. Particular care wastaken to ensure the criteria of purity were met, i.e.low absorption of ultra-violet radiation at 280 nmin the final wash water from the resins. Inaddition, the supernatant of the water basedmixed-bed resin stock was always checked at thiswavelength before the resin was used.

2.2. Polymer latices

Two different polystyrene latices were usedwhich were designated SLRR1 and KA7. Thepreparation of SLRR1 has been previouslydescribed [19]; KA7 was prepared by a seededgrowth process [23]. Both latices were cleaned byextensive dialysis against distilled water, dialysateto latex ratio 10:1, and then treated with mixed-bedion-exchange resin just before preparation of theexperimental samples.

The size of the particles in the latices wasdetermined by several methods, namely,transmission electron microscopy, photon-correlat-ion spectroscopy and small-angle neutron scatter-ing. The results are given in Table 1. The electronmicroscopy examination was carried out using aJeol-100CX electron microscope at low beamcurrents; calibration of the magnification wascarried out at the time of the examination using acarbon replica of a diffraction grating. The particlesize histogram was determined from the micro-graphs, using a Carl Zeiss TGZ3 particle sizeanalyser, and counting at least 1000 particles.

2.3. Determination of surface charge density

Following dialysis and ion-exchange treatment

R.H. Ottewill et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 161 (2000) 231–242 233

samples of the latices were titrated conducti-metrically with sodium hydroxide solutions [6,19].SLRR1 had a surface charge density of 4.2 mCcm−2 and KA7 a value of 1.4 mC cm−2.

2.4. Preparation of samples for osmotic studies

The stock dialysed latex was diluted with dou-bly-distilled water to the approximate volumefraction required for compression studies andfreshly dialysed. Cleaned ion-exchange resin, ca. 5g to 100 cm3 of latex, was then added and left toequilibrate for several days. The sample for usewas removed from the resin and either used di-rectly or dialysed against a sodium chloride solu-tion of the appropriate concentration with severalchanges of the salt solution.

2.5. Equipment for medium 6olume fractioncompression experiments

The compression cell and the hydraulic systemsupplying the pressure was adapted from a previ-ous design [17] and is illustrated in cross-sectionin Fig. 1. The cell was made of Perspex andpositioned between a stainless steel top and base.It was connected through a stainless steelthreaded connection in the base to an externalhydraulic system which was controlled manually.By pushing hydraulic fluid into the system themercury piston moved upwards and into the cell.

Fig. 1. Medium-volume fraction compression cell: (A) salt-so-lution capillary; (B) mercury capillary; (C) O-ring; (D) steelbolt; (E) positioning screw; (F) connection to mercury reser-voir and hydraulic system; (G) screw-threaded nut; (H) stain-less-steel top; (I) Nucleopore filter; (J) Perspex cell; (L) latex.

The applied pressure was read from the mercurycolumn in the glass tube attached to the hydraulicsystem.

The cell assembly was contained in a Perspexbox maintained at 25.090.2°C by means of aheating bulb regulated by means of a contactthermometer; a fan was used to maintain aircirculation. The water and mercury levels and themercury–water interface level were read using acathetometer.

A known weight of latex was introduced intothe cell and a Nucleopore filter was fixed intoposition. Filters of pore diameter 0.03 mm wereused for SLRR1 and 0.05 mm for KA7. The restof the apparatus was then assembled. The pres-sure in the cell was increased slowly to force airout of the labyrinth into the stainless steel top andto force the filtrate into the capillary. A tight-fitting glass bottle containing cotton wool satu-rated in the aqueous salt solution was placed over

Table 1Characterisation of polystyrene latices

Particle radius (A, )

SLRR1 KA7

Electron microscopyNumber average particle radius 156 240

28Standard deviation on the mean 4018% 18%Coefficient of variation on the mean

Small angle neutron scatteringParticle radius 158 225

15%Standard deviation 15%

Photon correlation spectroscopyZ-average particle radius 178 270


the end of the capillary to reduce evaporationlosses.

At equilibrium the levels of the water–latexinterface and the mercury were read to determinethe volume of contained latex. The pressure wasthen increased and the system allowed to attainequilibrium. For the dilute latex equilibrium wasreached within 24–48 h. With increasing volumefraction the time taken to reach equilibrium in-creased up to several days. During a compressionrun the volume fraction of the latex was typicallyincreased from B0.1 to ca. 0.35.

2.6. Equipment for the low 6olume fractioncompression experiments

The principle of the glass cell was identical tothat described previously [17] but the externalpressure application, using a manometer, wasmore accurate at low volume fractions. Thepresent cell was designed to incorporate electrodesinto the upper and lower chambers (see Fig. 2) inorder that changes in the salt concentration inthese compartments could be followed duringcompression.

The cell was constructed from a Pyrex filtrationapparatus (Gallenkamp). The upper part of thecell was made by attaching a glass tube of internaldiameter �8 mm to the filter part and putting agraduation mark on the tube just above the glass-blown joint. The lower part of the cell was madeby attaching a ‘J’ shaped glass tube of internaldiameter 4 mm to the filter part. Two small holes,180° apart, were drilled in the lower part of thecell just below the flange for the insertion of a pairof curved platinum electrodes. The electrodeswere sealed in from the outside with slow settingAraldite and the protruding platinum wires wereprotected with plastic wire sheathing. A pair offlat electrodes were inserted into the upper half ofthe cell in a similar manner.

The two halves of the cell were held togetherwith a metal clamp with a thick rubber ringplaced between the metal and the glass. An ‘O’ring between the two glass flanges was used toseal the cell. The complete cell was clamped insidea Perspex thermostat box at 25.090.2°C. Thetwo capillaries, the main body of the cell and the

dead space in the upper part of the cell werecalibrated using aliquots of mercury of knownweight at 25°C.

The cell constants of the two conductivity cellswere determined using a 1.0% w/v solution ofpotassium chloride of specific conductivity0.18804 V−1 m−1 at 25°C.

The cell was assembled as shown in Fig. 2.Before use, the cell was allowed to stand in dou-ble-distilled water for 1 day to leach out anyimpurities from the Araldite. It was then rinsedand dried. Mercury was introduced into the baseof the cell until the level was within the uniformpart of the body. A known weight of thermallyequilibrated latex was added carefully until it justprotruded from the top of the chamber. For theion-exchanged samples a fresh portion of ion-ex-change resin with the excess water removed wasadded along with the latex. The clean nucleopore

Fig. 2. Low-volume fraction compression cell: (A) salt solu-tion; (B) graduation mark; (C) platinum electrodes; (D) sin-tered-glass filter; (E) stainless-steel clamp; (F) O-ring; (G)Nucleopore filter; (H) latex; (I) mercury manometer.


filter was then slid into position and the twohalf-cells were clamped together.

Mercury was added down the side arm to dis-place the air in the filter support with dispersionmedium. When the liquid level was clear of thefilter a known weight of dialysate was added tothe upper part of the cell to bring the liquid levelabove the graduation mark. The cell was put inthe thermostat box with a plastic bottle, saturatedin dialysate, and placed over the end of the watercapillary in an attempt to reduce evaporation.

At equilibrium the conductivities in the upperpart of the cell and that of the latex were mea-sured. The levels of the liquid, mercury and latex-mercury interface were read, with respect to theposition of the Nucleopore filter, using acathetometer. The volume fraction of the latexand the applied pressure were then calculated.

A fresh aliquot of mercury (ca. 0.5 cm3) wasthen added down the side arm and a similarvolume of dialysate added to the evaporationcontrol mechanism; the system was then left toattain equilibrium.

The range of volume fractions which could beutilised in this apparatus was approximately0.01–0.2. Equilibrium was usually reached within24 h at the lower volume fractions, for the highervalues 2–3 days were required.

2.7. Calculation of osmotic pressure

The osmotic pressure and volume fraction werecalculated from the liquid levels measured relativeto some reference point, as shown in Fig. 2. Atequilibrium the pressure in the two sidearms canbe equated so that:

p= (H1−H2)rHg g− (H4−H3)rH2O g

+ (H3−H2)rlatex g

where p is the osmotic pressure generated in thelatex, in N m−2 and rHg, rH2O and rlatex thedensities of mercury, water and latex, respectively.rlatex was calculated using the equation:

rlatex=f rPS+ (1−f) rH2O

where f is the volume fraction of the dispersionand rPS the density of polystyrene which wastaken as 1.054 g cm−3 [24]. The volume fraction

Fig. 3. Schematic diagram of osmotic cell: (A) salt solution;(B) membrane; (C) colloidal dispersion.

of the latex in the cell was calculated from thevolume of liquid squeezed out of the latex, theinitial volume fraction and the total volume oflatex added to the cell. The final volume fractionwas checked at the end of the run using a dryweight method.

3. Osmotic pressure of charged polymer colloiddispersions

3.1. Introduction

Fig. 3 gives a representation of the experimentalconditions where compartment I is separatedfrom compartment II by a semipermeable mem-brane. Compartment I contains water and saltonly, e.g. a 1:1 electrolyte, and acts as a reservoir.Compartment II contains latex, water and salt,the latex constituting a non-diffusable colloidcomponent. The equilibrium condition in the cellis controlled by the equality of the electrochemicalpotential of the diffusable species in the two com-partments. Thus, for cations, anions and waterthe three conditions are given by [25]:

(p II−p I)V+ +F(c II−c I)=RT ln(x I+/x II

+) (1)

(p II−p I)V− +F(c II−c I)=RT ln(x I−/x II

−) (2)

(p II−p I)Vw=RT ln(x Iw/x II

w) (3)

where p is the pressure, V is the partial molarvolume, F the Faraday, c electrostatic potential


and x mole fraction, R the gas constant and Tabsolute temperature.In the absence of electrolytethen x I

w=1 and x IIw =1−xc where xc is the mole

fraction of the colloid component. Hence, for thissituation

(p II−p I)Vw= −RT ln(1−xc) (4)

If the effective charge on the colloid particle isdesignated as Z, then for the conditions of elec-troneutrality to be maintained it follows that

x I+ =x I

− =x (5)

and

x II+ =x II

− +Zxc (6)

where x is the mole fraction of the electrolyteions. This also gives

x Iw=1−x I

+ −x I− =1−2x (7)

x IIw =1−x II

+ −x II− −xc (8)

Using these relationships and Eq. (4) it followsthat Eq. (3) becomes

(p II−p I)Vw

=RT ln[(1−2x)/(1−x II+ −x I

− −xc)] (9)

At small xc the pressure term can be neglectedgiving

x I+x I

− =x II+x II

− =x2 (10)

As a result of the colloid component in compart-ment II an electrolyte difference will exist betweenthe two components which can be expressed as

x−x II+ =Dx+ for the cations and

x−x II− =Dx− for the anions (11)

From Eqs. (6) and (8) the following equation isobtained

x II− = −Zxc/2+ [(Zxc)2/4+x2]1/2 (12)

and thus,

1−x II+ −x II

− −xc=1−xc−2x [1+ (Zxc)/2x)2]1/2

(13)

There are two interesting cases of Eq. (13) whichcan be examined in the following manner.

3.2. Case 1 Zxc/2xB1

The last term in Eq. (13) can be rearranged andexpanded as a binomial series hence taking onlythe square term gives the RHS as:

1−xc−2x− (Zxc)2/8x2

so that

(p II−p I)Vw

=RT ln[(1−2x)/(1−xc−2x− (Zxc)2/4x)](14)

so that on expanding the natural logarithm weobtain

(p II−p I)Vw=RTxc[1+Z2xc/4x ]…

=RTxc+ (Z2x2c/4x) (15)

This equation, valid when 2x\Zxc therefore ap-plies to high electrolyte concentrations or to parti-cles with a small effective charge Z at low volumefractions. At low xc values Eq. (15) reduces to

(p II−p I)Vw=RTxc (16)

which is the classical osmotic pressure equationsince xc=cc/M with cc the colloid concentrationin g cm−3 and M the molecular mass of thecolloid component. It also follows that

x II− =x−Zxc/2 and xII+ =x+Zxc/2

In the case of low colloid concentrations thereforehalf of the charge of the non-diffusable colloidparticle is compensated by counter-ions and theother by a deficit of co-ions.

3.3. Case 2 Zxc/2x\1

For this situation Eq. (13) can be rewritten inthe form

1−x II+ −x II

− −xc=1−xc

− (Zxc)[1+ (2x/Zxc)2]1/2

(17)

Since 2x/Zxc is less than 1, using a binomialexpansion for the square root term and substitut-ing into Eqs. (6) and (8), followed by expansion ofthe natural logarithm term it follows that


(p II−p I)Vw/RT+2x=xc(Z+1) (18)

and if x is small then

(p II−p I)Vw=RTxc(1+Z)=RTcc(1+Z)/M(19)

For a negatively charged particle, Dx− will essen-tially represent the negative adsorption of co-ionsfrom the colloid compartment II as expressed byEq. (11). Expressing the square root term of Eq.(12) as a binomial expansion, combining it withEq. (11) and neglecting higher terms it followsthat

Dx− =x−x2/Zxc (20)

also

x II− =x2/Zxc and from (6) x II

+ =Zxc+x2/Zxc

For this set of conditions, appropriate for highcolloid concentrations and low electrolyte concen-trations, the charge on the particle is nearly com-pensated by an excess of counter ions. It is alsoclear from Eq. (18) that under these circumstancesthe counter ions are osmotically active as well asthe charged particles.

4. Results

4.1. Compression studies

The experimental results obtained for SLRR1at volume fractions up to 0.2 at nominal elec-trolyte concentrations of 10−5 (ion-exchanged)and 10−4 mol dm−3 sodium chloride are shownin Fig. 4 in the form of osmotic pressure againstvolume fraction of the latex, f. In both cases theosmotic pressures shown are of quadratic formwith respect to volume fraction, as expected fromEq. (15), and show, as anticipated, that increasingthe salt concentration from 10−5 to 10−4 moldm−3 decreases the osmotic pressure observed.

The results obtained with KA7 at the samenominal electrolyte concentrations are shown inFig. 5. The pressures developed at the same vol-ume fractions are lower than from SLRR1, asexpected from the larger diameter and lower sur-face charge density.

Fig. 4. Osmotic pressure against volume fraction for SLRR1at nominal salt concentrations of: �, 10−5 mol dm−3; ,10−4 mol dm−3. D, results obtained in low pressure glass cell.

4.2. Conductance studies

The specific conductance results obtained fromthe latex compartment and the external reservoirwhich were obtained simultaneously with the os-motic pressure measurements are shown in Figs. 6and 7 for SLRR1 at nominal sodium chlorideconcentrations of 10−5 and 10−4 mol dm−3 andin Fig. 8 for KA7 at a nominal concentration of10−4 mol dm−3.

Fig. 5. Osmotic pressure against volume fraction for KA7 atnominal salt concentrations of: �, 10−5 mol dm−3; , 10−4

mol dm−3. D, results obtained in low pressure glass cell.


Fig. 6. Specific conductance against volume fraction forSLRR1 in nominal 10−5 mol dm−3 salt: , reservoir com-partment; �, latex compartment.

Fig. 8. Specific conductance against volume fraction for KA7in nominal 10−4 mol dm−3 salt: , reservoir compartment;�, latex compartment.In each case it is clear that the specific conduc-

tance in the latex compartment increases withincrease in pressure and hence also volume frac-tion, whereas the specific conductance of thereservoir reaches a constant value beyond a cer-tain volume fraction. As also anticipated the spe-cific conductance values in the 10−4 mol dm−3

salt samples are much higher than those of thenominal 10−5 mol dm−3 samples.

5. Discussion

5.1. Properties of the particles

The surface charge density of the particles inthe latex SLRR1 was found to be 0.042 C m−2

which corresponds to an area per charge of 372A, 2 and hence a total number of charges perparticle of 822. Previously, the molecular mass ofthe particles was found to be 1.07×107 g mol−1

by light scattering [19]; this figure is in goodagreement with that calculated directly using theradius of 156 A, , estimated from electron mi-croscopy, and a particle density of 1.054 g cm−3.Similarly, for latex KA7, the surface charge den-sity of 0.014 C m−2 gave an area per charge onthe surface as 1143 A, 2 and hence the total numberof charges per particle as 633; the molecular massobtained by calculation was 3.7×107 g mol−1.The molecular masses of SLRR1 and KA7 were

Fig. 7. Specific conductance against volume fraction forSLRR1 in nominal 10−4 mol dm−3 salt: , reservoir com-partment; �, latex compartment.


used for the calculation of xc, the mol ratio of thelatex particles.

The surface potential, f0, of a charged sphericalunit is related to the surface charge density, s, bythe equation:

s=ere0f0(1+kR)/R (21)

where er is the relative permittivity of the medium(78.5), e0 is the permittivity of free space, R is theradius of the particle and k is the Debye–Huckelelectrolyte parameter. Eq. (21) gives an estimateof f0 in 10−5 and 10−4 mol dm−3 1:1 electrolyteof 815 and 623 mV for SLRR1 and KA7,respectively.

5.2. Estimation of the effecti6e charge, Z

For the conditions used in the present work oflow electrolyte concentrations and low volumefractions of the dispersions, in the region 0–0.3, itis clear that Zxc/2x is greater than unity andhence Eq. (18) was used to evaluate Z. However,some approximations were made in the derivationof this equation and the RHS is the leading termin a series expansion; hence it is only strictly validfor the lower volume fractions. The values for theeffective charge, Z, obtained are listed in Table 2and examples are given in Figs. 9 and 10 for latex

Fig. 9. Effective charge Z against volume fraction for SLRR1in a nominal salt concentration of 10−5 mol dm−3.

SLRR1 in 10−4 mol dm−3 salt. For both laticesthe value of Z was found to increase with volumefraction; also, the values were higher in the lowerelectrolyte concentrations.

An alternative representation of the data is toconsider the quantity, Z/N, with N as the totalnumber of charges per particle. This quantitygives the fractional charge on the particle and isfrequently used as a measure of the fraction ofsites without bound counter-ions for ionic surfac-tants [26]. This gave a normalised figure for com-parison of particles of different size and basic

Table 2Estimates of Z and Z/N for SLRR1 and KA7

Nominal 10−5 molVolume frac- Nominal 10−4 moltion dm−3 dm−3

Z Z/N Z Z/N

SLRR10.05 0.08660.15120

160 0.190.10 78 0.101200.15 0.15195 0.24

0.20 0.27 161 0.202230.25 208245 0.250.30

KA7153 0.28a0.05 1770.24

192 0.300.10 204 0.32269 0.420.15 248 0.39330 0.490.20 3100.52

0.593750.664180.25

a Very low pressure.Fig. 10. Effective charge Z against volume fraction for SLRR1in a nominal salt concentration of 10−4 mol dm−3.


charge density. The results given in Table 2 ap-pear to indicate that the effective charge alsoincreases as the volume fraction increases. Thismay arise from at least two effects, the increasein electrolyte concentration with increase in par-ticle concentration and also the effect of increas-ing particle–particle interactions.

5.3. Conductance effects

The conductance results shown in Figs. 6–8show a number of interesting effects. In all casesthe conductivity of the electrolyte in the externalreservoir levels off to an essentially constantvalue which then appears to be independent ofvolume fraction. For example, with SLRR1 innominally 10−4 mol dm−3 sodium chloride so-lution the conductance is constant over the vol-ume fraction range, ca. 0.075–0.2. The specificconductance in the ‘plateau’ region is 4.3×10−3

S m−1. From the tables of conductance datalisted by Harned and Owen [27,28], it can bededuced that the specific conductance of 10−4

mol dm−3 sodium chloride at 25°C is 1.26×10−3 S m−1 and that a specific conductancevalue of 4.3×10−3 S m−1 corresponds, in theabsence of H+ ions, to a sodium chloride con-centration of 3.3×10−4 mol dm−3. As expectedon the basis of the Donnan effect, dialysis ofthe latex does not produce the same electrolyteconcentration within the latex as in the di-alysate; an increase in salt concentration has oc-curred in the reservoir. At low volume fractionsof latex both the reservoir and the latex com-partment appear to be tending towards the spe-cific conductance value expected for 10−4 moldm−3 salt.

For the ion-exchanged SLRR1 latex (Fig. 6)the plateau occurs at a specific conductancevalue of 6.1×10−4 S m−1 which indicates anelectrolyte concentration close to 5×10−5 moldm−3 salt in the reservoir compartment on thebasis of sodium chloride concentration in theabsence of H+ ions. This parallels the effect ofnegative adsorption of anions as shown in Fig.11 which shows a plot of Dx− versus latex vol-ume fraction as calculated from Eq. (20) and acomparison with the experimental results.

Fig. 11. The difference in the mole fraction of anions incompartments I and II Dx− against volume fraction forSLRR1, nominally in 10−4 mol dm−3 salt. ----, theory fromEq. (20).

The results for latex KA7 behave in a similarfashion. For example, in a nominal salt concen-tration of 10−4 mol dm−3 salt the specific con-ductance is 3.5×10−3 S m−1, whichcorresponds to a salt concentration of 2.5×10−3 mol dm−3. Thus, the increase in salt con-centration in the reservoir appears to be smallerthan observed with SLRR1, which has a valueof 4.3×10−3 S m−1 (Fig. 7). This is a reason-able result as the latex has a smaller numberconcentration of particles with a lower surfacecharge density; again as the volume fraction de-creases the specific conductance tends towardsthe value expected for 10−4 mol dm−3 salt.

The results obtained for SLRR1 in nominally10−4 mol dm−3 electrolyte suggest a limitingvalue of Z of 6095 at low volume fractions.Hence, using this value the diffuse layer surfacecharge density can be calculated from:

sd=Ze/4pR2 (22)

where e is the fundamental charge on the elec-tron. This gives a value of 3.14×10−3 C m−2

and hence from Eq. (21) the diffuse layer poten-tial fd can be calculated. A value of 47 mV wasobtained which is in reasonable agreement withthe zeta-potential of 62 mV obtained from dy-namic light scattering measurements of elec-trophoretic mobility [20].


5.4. Particle–particle interactions

The results obtained in the compression experi-ments indicate that although the initial disper-sions were adjusted to a particular saltconcentration during compression, salt was trans-ferred from the dispersion compartment to thereservoir. However, as the volume fraction in-creased the conductance measurements indicatedthat the salt concentration in the reservoir becameconstant. Over the same range of volume fractionsthe effective charge on the particles showed anapparently continuous increase.

In the simplest case of two charged surfacesapproaching each other, in order to interact atconstant potential of the diffuse double layer thediffuse layer surface charge must decrease as theinterparticle distance decreases [1]. For constantcharge interaction in order to maintain the gradi-ent of potential/distance constant the diffuse layerpotential must increase.

In principle, polystyrene latex particles chargedby chemical groups on the surface, e.g. carboxylgroups, would be expected to behave as constantcharge systems at low volume fractions. However,as a consequence of the electrolyte changing in thelatex compartment with addition of an increasingamount of counter-ions from the particles othereffects may occur. As pointed out by Beresford-Smith et al. [29], the counter-ion effect is animportant one. For example, with particles ofSLRR1 at f=0.1 there are 6.29×1018 particlesper 1000 cm3 and with a charge per particle, iffully ionised, of 822 this corresponds to a saltconcentration of 8.6×10−3 mol dm−3 in addi-tion to the basic 10−4 mol dm−3 salt. If theeffective charge Z, at this volume fraction, istaken as 78 (see Table 2) then the effective in-crease in salt from unbound counter-ions amountsto 8.2×10−4 mol dm−3. In the present experi-ments there is qualitative agreement between theshape of the curve of specific conductance of thelatex compartment against volume fraction (Fig.7) and the increase in salt concentration arisingfrom the counter-ions. However, there will also bea contribution from the particles and from anyH+ ions present; the pH was not measured in thelatex compartment. Despite this it should be

noted that both calculated and measured conduc-tance values are of the same order 10−2 S m−1.

Also, with the increase in salt in the latexcompartment increased ionisation of the surfacegroups would be expected to occur and the in-crease in Z observed with increasing volume frac-tion could be interpreted in this way. Hence, arealistic model to explain the interaction resultscould well be by using a mechanism of chargeregulation [30,31].

The connection between the structure factorS(Q) as obtained, for example, by small-angleneutron scattering and osmotic pressure is a directone and will be discussed in more detail in a laterpublication [32]. It is clear from the present workthat considerable care has to be taken in adjustingthe elctrolyte concentration to obtain a reliablevalue for the analysis of results. This point isparticularly relevant to any latices treated withmixed-bed ion-exchange resins [33].

Acknowledgements

We wish to thank ESPRC (RAR) and CurtinUniversity (AP) for support of this work.

References

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Osmotic pressure measurements on strongly interacting polymer colloid dispersions