the rheology of strongly-flocculated suspensions

Journal of Non-Newtonian Fluid Mechanics, 24 (1987) 183-202 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

183

THE RHEOLOGY OF STRONGLY-FLOCCULATED SUSPENSIONS

R. BUSCALL *l, I.J. MCGOWAN ‘, P.D.A. MILLS ‘, R.F. STEWART ‘, D. SUTTON ‘, L.R. WHITE * and G.E. YATES ’

I ICI PLC, Corporate Colloid Science Group, PO Box 11, The Heath, Runcorn, Cheshire, WA7 4QE (U.K.) 2 Department of Mathematics, University of Melbourne, Parkville, Victoria 2060 (Australia)

(Received June 30,1986; in revised form September 4,1986)

The rheology of strongly-flocculated dispersions of colloidal particles has been investigated at particle concentrations where a continuous network is formed rather than a collection of discrete floes. Such networks are shown to possess a true yield stress in both shear and in uniaxial compression (as realised in a centrifuge). Properties measured as a function of particle concentration and particle size include the yield stresses in shear ( uY) and compression (P,,); the limiting and strain-dependent, instantaneous shear moduli Go and G(y); the elastic recovery at finite strains, and the rate of centrifugally-driven compaction. The yield stresses and moduli appear to show a power-law dependence on particle concentration with G, and P,,, having the same power-law index and u,, a somewhat lower one. The data are in part consistent with predictions based on the idea that the networks have a heterogeneous structure comprising a collection of interconnected fractal aggregates. The behaviour as a function of particle size and concentration is however not completely scaleable as might be expected on this basis. Thus, whereas the shear yield stress could be scaled to remove its dependence on particle radius Q and volume fraction 9 (over the measured range 0.25 pm < a < 3.4 pm; 0.05 < C#J < 0.25) as could the strain dependent modulus (0.25 < a < 1.3 pm; 0.08 < C#I < 0.25), the particle-size and concentration dependence of Py and G, could only be scaled for particles with radii between 0.16 and 0.5 pm, smaller and larger particles having different and much higher power-law index in respect of their concentration dependencies. In the case of the smaller particles the failure of the scaling is thought

* To whom correspondence should be addressed.

0377-0257/87/$03.50 8 1987 Elsevier Science Publishers B.V.

184

to be due to an anomaly since these particles distort significantly under the influence of the strong van der Waals forces and this causes the aggregates to be more compact then they otherwise would be. The reasons for the failure at larger sizes is not clear.

1. lntroductioIl

Sufficiently dilute suspensions of strongly-flocculated particles (flocculated in the primary minirnum) consist of a collection of discrete floccules which recent experiments show [1,2] to have a fractal structure, such a structure is in accord with what is seen in computer simulations [3,4] of diffusion-limited aggregation (DLA). In more concentrated suspensions the floccules cease to be discrete and above some critical volume fraction ?s a network forms. The critical volume fraction es depends strongly on particle size and shape: For spherical particles it can be anywhere between 0.05 and 0.3 depending on particle size; in the case of highly anisometric particles it can be as low as 0.01. The rheological and transport properties of strongly flocculated suspensions change abruptly at +g. For example, in shear, strongly-flocculated suspensions develop a true yield stress and show instantaneous and delayed elasticity. In the case of sedimentation the rate of fall of the solid-liquid boundary slows markedly at 4 as the mechanism of sedimentation changes from one involving the fall of discrete floes to a collapse of a network under its own weight. Likewise cohesive filter-cakes are formed above & and t ese show a compressibility which decreases

I rapidly with increasing conce tration. This paper summarises the results and conclusions of an extensive study

of the shear rheology, sedimentation behaviour and compressibility of strongly-flocculated networks. The data presented refers to experiments on well-characterised polymer latices (spherical particles) and on natural clay minerals (plates and rods). A much wider range of materials have, however, been studied and it is believed that the data displayed below are qualitatively representative of strongly-flocculated materials in general.

2. Experimental

Experimental procedures have been described and only a brief description will be given here.

(i) Shear rheology

in detail elsewhere [5-71

Shear moduli at small strains were measured using a Pulse Shearometer (Rank Brothers Scientific Instruments, Bottisham, Cambridge). This instru-

185

ment measures the wave-propagation modulus G” at a frequency of ca. 200 Hz and at peak strains L 10m4. The measurements were made only when damping was negligible. Under such conditions and given the very small strains involved, it can be assumed that G” approximates to G,,, the instantaneous, infinitesimal modulus of the network.

Creep, yield stress and continuous shear measurements were performed using a controlled stress rheometer, the Sangamo Viscoelastic Analyser, fitted with Couette rotors.

(ii) Sedimentation and network compression

Measurements were made both under normal gravity and in batch centrifuges fitted with swing-out rotors and stroboscopic illumination. A range of centrifuges was used so as to cover a wide range of speeds (100-1.5 X lo4 rpm) and accelerations (- 2-5 x lo4 g).

Initial boundary sedimentation rates were obtained from plots of the height of the sediment-supernatant boundary against time. The compressive strength of the networks (compressive yield stress PJ +)-see below) was obtained from curves of equilibrium sediment height versus acceleration using the method described by Buscall and White [7], according to this the compressional yield stress is given parametrically by

p,(+) = Apg&,H,(l - H,,/2@, (I)

G%f4)[1- (Heq + WR] += [(H,,+S)(l-H,/R)+~~/2R]’

(2)

where ‘pO = initial volume fraction of solid, Ho = initial height of the column of suspended material, H,(g) = the equilibrium sediment height, g = centrifugal acceleration (g = u2R), R = radius of rotation at the bottom of the sediment, S = d Heq/d In g. A quantity related to P,,( $) is the so-called network modulus [6]

K(9) = d J’,(+)/d ln 9.

3. Materials

The polystyrene lattices used were surfactant-free, monodisperse polystyrene lattices prepared by the usual methods [5]. The attapulgite clay was a sample of Attagel 50 (Lawrence Chemical Co., Mitcham, Surrey). The bentonite used was a sample of sodium bentonite (ex BDH). The latex was purified by dialysis and then flocculated by the addition of barium chloride.

186

The bentonite was similarly flocculated by adding an excess of strong electrolyte. The attapulgite as received was, whilst suspendable in water, colloidally unstable and required no additives to cause flocculation.

4. Results and discussion

(i) Shear rheology

The shear rheology of strongly flocculated monodisperse polystyrene lattices will be illustrated using data for lattices with the following particle diameters: 0.49, 1.0, 2.6 and 3.4 pm. Creep studies using the controlled-stress rheometer showed that the dispersions exhibited solid-like viscoelastic behaviour at sufficiently small stresses, that is, the low shear viscosity &as immeasurably large ( % lo6 Pa s). On increasing the stress there was a change from solid-like to. fluid-like behairiour over a very narrow range of applied stress. The networks thus, to all intents and purposes, displayed a true yield stress. Attention here will be focussed on the yield stress and the elastic behaviour below the yield stress. The viscosity above the yield stress has been discussed elsewhere [5].

As explained elsewhere the yield stress can be strongly influenced by the mechanical history of the sample [5]. In particular, prolonged shearing prior to measurement can, under certain conditions of solids concentration and shear stress, cause a large decrease in yield stress and rearrangement of structure [5,8]. Such effects will not however be considered here and the results displayed refer to freshly flocculated samples subjected to minimal mechanical pretreatment. Repeated measurements showed that the yield stress was subject to a significant but non-systematic variation about a mean, the standard deviation being typically between 15 and 30% of the mean. This variation, which would be unacceptable in many areas of rheology, appears to be an inherent feature of strongly flocculated networks. One possibility is that it may be due to structural inhomogeneity of the network, the annular gap in the couette being large compared to the particle size but perhaps not so large compared to the length scale associated with density variations in the network.

The effect of particle size and concentration on the mean yield stress (average of - 10 measurements) is illustrated in Fig. 1. Both variables have a strong effect on the yield stress as might be expected. The range of both particle size and concentration covered was insufficiently wide and the data insufficiently detailed to determine the form of the relationship between uY and +, or a, very precisely or unambiguously. The data are, however, not inconsistent with power-law behaviour; that is

q-V; an, (4)

187

1 .Opm

0.5pm t

i-7 I

wy/Pa

I i

6- I / 5- I I

4- ,(! /

3- // //

/ 2- /

l- /

/

AA /' 2.6 pm

/’ ----I3 3.5flm --- d S-s-J==_ -- 0 I r I , I

0.05 0.10 0.15 0.20 0.25 0

Fig. 1. Plot of yield stress in shear versus volume fraction for coagulated polystyrene lattices of various diameters.

with m = 2.5-3 and n = -2. This can be seen from Fig. 2 which shows a log-log plot of the quantity

t

s = +v%e,)‘9 (5)

where aref is the radius of the smallest latex used, against $J_ The reference line has a slope of 3.

Fig. 2. Log-log plot of S = IJ,u*/~$~ against volume fraction for the four different size latices.

188

Viscoelastic behaviour could be observed below the yield stress. The instantaneous modulus was measured by two independent methods, wave- propagation and creep. These two methods allowed a wide range of strain amplitudes to be studied. In the case of the wave propagation method the absolute peak strain was not known precisely (it was however very small). The relative peak strain could however be varied over a range of about five by varying the potential applied to the input transducer. By such means it was possible to show that the technique measured a strain-independent modulus; this device thus probed the linear regime.

The smallest strain amplitude that could be resolved with acceptable accuracy in creep was - 3 x 10m4. Instantaneous moduli measured at strains of this order are compared with moduli obtained by wave-propagation in Fig. 3. The agreement is very satisfactory given the difficulty of working at such small strains. At strain amplitudes of ca. 10h3 and above the instantaneous modulus showed a strong strain-dependence as can be seen from a typical curve plotted in Fig. 4. Also shown is the elastic recovery observed on the removal of the applied stress. The latter is included to show that the instantaneous strain is substantially but not fully recovered. This implies that the strained structure is not quite the same as the unstrained structure, repeated measurements however showed no evidence of progressive degradation, as did not repeated measurements of the yield stress. The origin of the loss is thus not entirely clear. One possibility would be a

102 1

lo-

GdkPa

l-

t,, , I I I

0.05 0.10 0.15 0.20 0.25

0

Fig. 3. Comparison of limiting shear moduli obtained by wave propagation (0) and creep (O), for the 0.49 pm diameter latex.

189

1, Elastic Recovery %

Fig. 4. Effect of shear strain (y) on the instantaneous shear modulus and elastic recovery for the 0.49 pm diameter latex at $3 = 0.08.

tendency for slip to occur at the wall of the couette and this might also offer an alternative or additional explanation of the dispersion of the yield stress. However, the loss increases only slowly with strain and a stronger effect might be expected were slip to be the explanation. It is thus believed that the interesting strain-dependence of the modulus and the elastic loss are intrin- sic features of the network. In particular they are believed to reflect in part the nature of the force-distance curve of the bonds between the particles and in part the heterogeneous nature of the network for reasons which will now be discussed.

The particles are presumably held in a deep potential minimum arising from a balance between van der Waals attraction and some short-range but significantly distance-dependent repulsion. An anharmonic potential of this type implies a force coefficient for an interparticle bond that decreases with increasing displacement for finite displacements as is argued in Appendix 1. This may, in part, explain the behaviour of G(y), however it is considered that the network is too compliant for this to be the sole explanation. The elastic strain prior to yield was typically 0.03. Even if the network deformed affinely this would imply elastic displacements in the interparticle bond distance of order I

s max - @a + 4JY9 (6) where 8, is the equilibrium bond-length; non-affine deformation would imply even larger displacements in certain bonds. Interparticle bonds would only be expected to be stable to stretching up to some critical displacement 8, where the bond potential shows a point of inflection. It is argued in the Appendix that this distance, whilst clearly larger than S,, is only of the order

190

l-

0.9-

WW30)

0.8-

0.7-

0.6 -

0.5--

OA-

0.3-

0.2--

0.01

Fig. 5. Normalised plot of shear modulus against log strain for 2.6 gm latex at various volume fractions between 0.1 and 0.25 (0). Also shown are some data for the 0.49 pm latex

(6).

of S,,, and 8, itself should be small, of the order of nanometres at most. S,, calculated from eqn. (6) on the other hand, is of the order of tens of nanometres. This discrepancy can be resolved by postulating that a significant fraction of bonds do break below the yield point, thereby increasing the compliance of the remaining network. This idea has the attraction that it explains the incomplete elastic recovery observed at finite strains. If the fraction of elastic loss is used as a measure of the bonds broken, then from Fig. 4 it might be inferred by extrapolation that the network yields (that is it loses its large-scale, three-dimensional connectivity), when perhaps 30-50% of the elastically effective bonds are broken. A figure of this order would seem plausible when it is realised that for the network to be space-filling the mean coordination number of the particles has to be of order 3. The implication is thus that the network yields when its functionality is reduced to the point when it ceases to be cross-linked.

G(y) curves for a larger latex (2~ = 2.6 pm) were rather similar in form except that there was a less pronounced tail at large y. These curves could be normalised by plotting G(y)/GO against y/yref as is shown in Fig. 5. yref in this case tended to decrease with increasing solids content from 0.073 at C#B = 0.1 to 0.015 at + = 0.25. Surprisingly, there was no strong or systematic effect particle size on yref (a measure of the maximum strain the network can withstand), the maximum strain achievable being typically a few per cent.

191

The above discussion refers to dispersions of spherical particles. Some- what similar non-linear elasticity in creep has been observed by Rehbinder et al. [12,13] for plate-like particles (bentonite gels). These authors postulate an alternative mechanism for the observed dependence of modulus on strain in which they invoke reorientation and slip of the platelets forming the network. Their mechanism and ours are not however mutually exclusive and clearly there are new possibilities when the particles are highly anisometric. Rehbinder et al. consider the elasticity to be in part entropic in origin whereas any entropic contribution can probably be discounted for dispersions of spherical particles with diameters of the order of 100 nm or more (the particle number concentration being much smaller than it is in the case of the very thin clay plates).

(ii) Sedimentation and network compression

That strongly flocculated networks show a yield stress in respect of uniaxial loading can be seen in Fig. 6 which shows the effect of increasing the compressive stress in the network on the equilibrium sediment height for a flocculated sodium montmorillonite suspension, the quantity plotted on the abscissa being the compressive stress at the bottom of the sediment.

P max = Apw2R(1 - H,,/2R)H&, (7)

It can be seen that no consolidation of the network occurs until a critical stress PC = PJ #By) is exceeded. The yield stress function P,( C#J) can be measured either by determining PC for a series of initial, starting concentra-

Equilibrium sediment volume

I I I I I I

0 100 200 300 400 500 600 Centrifugal acceleration /ms2

Fig. 6. Plot of equilibrium sediment volume against centrifugal acceleration for flocculated bentonite dispersions. O-44; w/v; O-3.58 w/v.

192

Py/kPa

50-

20-

10-

5-

2-

l-

0.5-

0.1 0.15 0.2 0.3 0

Fig. 7. Log-log plot of compressional yield stress against volume fraction for 0.49 pm diameter latex. O-l.2 cm diameter tube, initial volume fraction, $,, = 0.1; O-l.2 cm tube, e0 = 0.05; r-1.45 cm tube, &, = 0.05; r-2.3 cm tube, & = 0.05. Reference line has a slope of 4.3.

tions $J@, or by analysing the curve of He4 versus P,, for Pm, > P,( &,) as described in the experimental section.

Data for a strongly flocculated polystyrene latex with a particle diameter of 0.5 pm are shown in Fig. 7. The measurements were made using two different centrifuges; three different tube diameters and two starting concentrations. The good agreement between the various sets of data shows that there is, no wall effect, also that the structure of the network is independent of the initial concentration. The data fall onto a power-law curve of form

Py ((p) = 1.02 X 107+4.31 Pa. (8)

This curve is compared with data for other particle sizes in Fig. 8. The curves for 0.33 pm and 0.96 I_cm diameter particles show a similar power-law dependence, the effect of increasing particle size being to decrease the front factor. Data for small and large particles show a stronger concentration dependence. In the case of the large particles this is believed to be because the floccules making up the network may be too small (compared to the

193

I 0.06(-’ O) 0.33(4.4) : of 0.5(4.31)

Py/kPa I v

Fig. 8. Log-log plot of compressional yield stress against volume fraction for various particle diameters. Figures shown are particle diameter in pm, and in brackets the power-law index.

particle diameter) to behave as fractal objects (see Appendix 2). In the cases of very small particles, electron microscopy showed that distortion and partial fusion of the particles had occurred reducing the openness of the aggregate structure, this is believed to be a consequence of Hertzian deformation of the particles under the influence of the very strong van der Waals forces. This would also explain the reversal of the effect of particle size on ~$s seen for small particles. Data for more rigid particles with diameter c 0.1 pm would be of interest in order to confirm that this explanation is correct but such data have yet to be generated.

A theory of consolidation of flocculated networks having a compressive yield stress has been developed by Buscall and White [7]. According to this, initial rate of fall of the sediment depends upon P ,,,ax = Apg*d+,Ho, through

0; L%x < q&J bw#dl -+iM,

9, [1-f#$]: Pm,>Py(~o,),

(9)

194

1 dH

- i*z I

X10%

Fig. 9. Plot of initial boundary sedimentation-rate coefficient against reciprocal, reduced, normal stress for a strongly flocculated network of attapulgite clay particles.

where g* = w2R(1 - H,/2R) for a centrifuge (for normal gravity, g* is the acceleration due to gravity) and P,, = Apg*&,H,; qS is the viscosity of the medium and K, is the permeability of the network to fluid. From eqn. (9) a plot of (l/g) dH/dt 1 o against g* should be a straight line, that this appears to be so for materials with a compressive yield stress can be judged from preliminary data shown in Fig. 9. The intercept can be used to obtain the permeability K,($). From experimental work done so far it has been found that measured values of K,(G) are not inconsistent with predictions based on the Kozeny-Carman equation which has

K

P

where Z is a characteristic particle dimension. However, too few data have been generated to test this relationship in detail and so no definite conclusions can be made as to its validity at this stage. I’,,(+) and K,(+) are key parameters in the theory of the separation of flocculated solids from liquids. P,(G) shows a very strong dependence on $ (Fig. 8) and because of this our prejudice is that the kinetics of separation are probably very sensitive to the precise form of I’,( +) but not so much to KP( +).

195

(iii) Comparisons between shear and compressional behaviour

Figure 10 compares the shear modulus G, with the “network modulus” defined as

The correspondence between the two quantities is remarkable (this has been observed to be the case for a fairly wide range of materials, although in some cases the shift on the modulus axis is larger than shown by these data). Such data show

A possible meaning of this relationship can be established as follows. The uniaxial modulus of the network just below the yield point, MY(+) say, is given by

M,(9) = p,(+)/%W~ (13)

where cc(+) is the (unknown) elastic, strain just prior to yield. One might anticipate a relationship between M,, and G,,, e.g.

G(0); K(0)/kPa

1001

o’0’$02 ’ ,O.d6 1 0:l , ‘DA

0.1 0.16 0L

Fig. 10. Plot of limiting shear modulus and the network modulus defined as K = d P,, /d ln $I against volume fraction for attapulgite clay (A) and 0.5 pm diameter latex (L).

196

0

Fig. 11. Comparison of compressional (m) and shear (A) yield stresses for 0.49 gm diameter latex.

or in view of the experimental observation K( +) and G( (p)

p,W/e,(+) - d P,(+)/d ln 9. (19

Figure 7 shows that 2Y( $I) typically shows power-law behaviour and so this becomes

+“A(+) - 9 d(V)/d+ 06)

or cc #f( c#J). The observed relationship between K( $I) and G,( t#~) thus appears to imply that yield in uniaxial compression occurs at a critical level of network elastic strain which is independent of the density and thus the structure of the network. This presumably means that yield occurs when a critical level of strain is reached in the interparticle bonds.

Other quantities that might fruitfully be compared are the yield stresses P,,( +) and uY( +). This is done in Fig. 11. The data show that flocculated networks are much stronger in uniaxial compression than they are in shear. This is not too surprising particularly if it is supposed that only those bonds in tension break and not those in compression. Thus, in shear approximately equal numbers of bonds will be stretched and compressed, whereas in compression the majority of bonds will be in a state of compression; bonds will only be stretched in regions of the network where heterogeneity of the structure gives rise to net torques.

5. Concluding remarks

It has been shown that strongly flocculated networks show a yield stress both in shear and in uniaxial compression. (Note that the results shown here

197

pertain to only three materials; polystyrene latex, attapulgite clay and bentonite; however we observe qualitatively similar behaviour with other materials.)

All of the measured properties show a stronger dependence upon particle concentration than might be expected were the networks to have a uniform structure [9]. The observed behaviour is more consistent with a structure comprising interpenetrating fractal aggregates. Recent theoretical calculations [lo] predict a dependence of modulus on concentration of something like G - e4 for such structures. A similar dependency is seen experimentally for spherical particles with diameters between 0.3 and 1 pm. For smaller and larger particles a much stronger dependence is seen, implying an even more heterogeneous structure. The reasons for this are not entirely clear, however in the case of the smaller particles, microscopic evidence suggests that the flocculated particles are grossly distorted and partially fused and this may have an influence by making the aggregates more dense than they would otherwise be.

Fruitful comparisons can be made between various properties. Thus it is found that the shear modulus and the yield stress in uniaxial compression are proportional and nearly numerically equal. This is taken to indicate that yield in uniaxial compression takes place above a critical strain independent of volume fraction. Comparison of shear and compressional yield stress shows that the networks are much weaker in shear than in compression. The shear yield stress and the shear modulus are not proportional implying that yield in shear does not occur at a constant level of critical strain; surprisingly perhaps, the critical strain appears to decrease with increasing particle concentration. However the strain dependence of the instantaneous modulus and the elastic recovery below the yield stress suggest that a significant fraction of bonds may be broken before yield occurs and the additional compliance this allows might well be expected to be inversely related to concentration. It is also perhaps pertinent to note that calculations made using model potentials imply that were no bonds broken, G(y,)/G(O) should not perhaps be less than - 0.45. The amount of elastic loss just prior to yield is estimated to be about 40% and so if this fraction of bonds are broken prior to yield, the modulus just before yield might be expected to be G(y,) = 0.45 x 0.4 G(0) = 0.2 G(0) and this is what is observed. If this interpretation is correct then it would appear that the networks yield when roughly half the elastically effective bonds are broken.

One important feature of the shear viscoelastic behaviour is that very small strain amplitudes need to be employed in order to obtain a linear response, typically < 5 x 10m4. This means that it would be impossible to obtain linear data using most available oscillatory rheometers having strain as the input. However the existence (sometimes) of a second near plateau

198

(cf. Fig. 4) could mislead the user of such a machine into believing that the modulus was independent of strain at much higher strain amplitudes. It has been demonstrated that a combination of wave propagation at a single frequency, together with creep, yield good results, a controlled-stress oscillatory rheometer would perhaps also prove useful.

The idea of a compressional yield stress can be developed into a theory of sedimentation and consolidation. Preliminary results suggest that this may correctly describe the sedimentation of flocculated networks.

Finally, whilst strongly flocculated networks do not show universal behaviour, certain properties and/or certain properties in certain particle-size regimes appear to be scaleable. Thus the shear yield stress data appear to scale like oY - G3a- *; the strain-dependent modulus appears to scale like WYW, - -In Y/Y,~, where yref = yref( $, a), and in the intermediate particle size regime, 0.3-l pm, the compressional yield stress and the shear modulus scale like G, Py - (t#~/&~~)-~ with IYI = 4.3 + 0.1, where C#Q = &.r( a). The existence of such scalings might be taken as further support for the idea that such networks comprise interacting fractal aggregates. Further work is however required in order to establish this. Likewise, further work is required in order to understand the very strong (- @lo) dependencies for Py and G for small and large particles and to elucidate the mechanism of the finite-strain elasticity and yield. In the latter context it would, with hind- sight, have no doubt been fruitful to examine the retarded elasticity in creep in detail, however this aspect awaits a separate detailed study.

References

1 D.A. Weitz and M. Oliveria, Phys. Rev. Lett., 52 (1984) 1433. 2 M. Matsushita, K. Sumida and Y. Sawada, J. Phys. Sot. Jpn., 54 (1985) 2786. 3 P. Meakin, J. Colloid Interface Sci., 102 (1984) 491. 4 P. Meakin, Phys. Rev., A27 (1983) 604; A27 (1983) 1495. 5 R. Buscall, P.D.A. Mills and G.E. Yates, Colloids & Surfaces, 18 (1986) 341. 6 R. Buscall, Colloids & Surfaces, 5 (1982) 269. 7 R. Buscall and L.R. White, J. Chem. Sot., Farad. Trans., I (in press). 8 B.A. Firth and R.J. Hunter, J. Colloid Interface Sci., 57 (1976) 266. 9 M. van den Tempel. J. Colloid Sci., 16 (1961) 284; J. Colloid Interface Sci., 71 (1979) 18.

10 R.C. Ball and W.D. Brown, Cambridge University (paper in prep.). 11 J.N. Israelachvili, Adv. Colloid Interface Sci., 16 (1982) 31. 12 P.A. Rehbinder, in S. Onogi (ed.), 5th Int. Congr. Rheol., Vol. 2, Tokyo Univ. Press, 1970,

p. 375. 13 E.D. Scukin and P.A. Rehbinder, Kolloidn. Zh. USSR, 33 (1971) 450.

199

Appendix 1

Form of the force constant for finite strains suggested by model interparticle potentials

The particles are assumed to be held in a potential energy minimum formed as a net result of the van der Waals potential and some short-range repulsion. The orign of short-range forces between colloidal particles re- mains obscure but the existence of a strong short-range repulsion with non-negligible range (1-2 nm) has been demonstrated by means of direct force measurements between mica plates immersed in electrolyte solutions [ll]. The existence of a yield stress also implies a distance-dependent repulsion since in order for interparticle bonds to withstand stretching forces below some critical level without breaking there has to be a point of inflection in the potential energy curve. It is thus supposed that the particles are held by a highly anharmonic potential comprising a (roughly) ( R - 2~2) -l van der Waals attraction and a much steeper repulsion. Such a potential curve exhibits a force coefficient that decreases with strain and so the shape of G(y) could in part reflect this. In order to test this, calculations were made using a force-law of the type

fb = S-2 - k 6-P 07) where 6 = R - 2a = surface to surface separation and where the first term represents the van der Waals force and the second the short-range potential, the coefficient k being chosen so as to place the potential minimum at any desired value of 8, = ( R - 2 a) O, the equilibrium particle separation.

It has to be recognised that in a sheared network some bonds will be compressed and some stretched. The apparent force coefficient should thus reflect his. It would seem reasonable to assume that, say, half are stretched and half compressed, and so at finite displacements the force coefficient has to be suitably averaged. Clearly if the potential is anharmonic the deformation cannot be affine since only limited displacement can occur on the compressive side. It would seem sensible to assume rather that the force developed in compressed and stretched bonds is the same, in which case the average spring rate is

E(8) = 2fbA 16, I + I a- I), (18)

where a+(fb) and Ufd are the compressive and stretching displacements elicited by a force fb. It is also supposed that stretched bonds are only stable to breakage up to the point of inflection on the stretching side, this being at

8, = S,( p/2)1’(P-2). (1%

From this 8,/S, is rather insensitive to j3 as can be seen from Table 1.

200

TABLE 1

B WJO 3 1.5 5 1.36 7 1.29 9 1.24

11 1.21

The critical displacement is thus of order 8,. 8, itself has to be small in order to obtain large van der Waals forces, of order 1 nm at most, and so 8, has likewise to be of order 1 nm. Choosing other forms for the repulsive potential does not change this conclusion. The contribution of a critically stretched bond to the macroscopic strain has thus to be of order SJ(2a + &,) and thus of order S,/2a which for 6, = 1 nm and a = 0.25 pm is 0.002, and at a = 1.3 pm, 0.004. The real networks, however, withstood macroscopic strains of between 0.01 and 0.07. It thus appears possible, if not likely, that bonds are broken prior to yield, causing the network to become more compliant. The value of the calculated reduced force constant I(fi”‘)/5(0), where firit = f,( S,), t urns out to be independent of the values assigned to 8, and /3 and thus the shape of the potential, having a value of - 0.45. It is thus tempting to conclude that the effect of finite displacement on the force coefficient of the bonds alone cannot account for total drop in G(y) prior to yield, this being typically 0.8.

An objection to the above argument is that the process implied appears to be unstable. That is, if the application of a stress causes some bonds to break then the stress is concentrated in fewer bonds, which in turn should break, and so on. However, the heterogeneity of the structure guarantees that there should be paths through the system of varying tortuousity. Thus, this criticism can be negated by arguing that initially only the most direct pathways support the stress so that when bonds break in these pathways the stress is taken up in more tortuous pathways which did not initially contribute. On this basis, yield occurs when the number of bonds potentially available become too few to support the stress. In order words the deformation is progressive and highly non-affine.

Appendix 2

Fractal nature of strongly flocculated networks

The data presented show that G and Py show an apparent power-law dependence on volume-fraction, the power-law index being about 4 for

201

intermediate particle. sizes (0.15 < a < 0.5 pm). A strong dependence of this type cannot be modelled by assuming that the network structure is uniform, that is by assuming that the coordination number of each particle is equal to the average coordination number in the network [9]. Such an approach gives something like G - $L The implication is thus that the networks are highly non-uniform. The elastic behaviour of such networks has recently been modelled by Ball and Brown [lo] who assumed that the network comprises a set of bonded fractal aggregates. Such aggregates have the property that their density decreases as a power-law function of distance from their centre of mass. Thus, as a collection of such objects is concentrated it can be imagined that the number of bonds between the aggregates will rise with concentration in a highly non-linear fashion. Ball and Brown estimate the concentration dependence of, say, G by deducing the elastic and geometric properties of individual aggregates from computer simulations and then using a scaling argument to calculate the elasticity of an interpenetrated and bonded collection of such objects. It has recently been demonstrated experimentally [2] that polystyrene latex does form fractal aggregates in dilute dispersions. The approach would thus appear to be valid for the case where flocculation is carried out in the dilute state ( C$ +K +g) and a network is then formed by concentrating the system so that I$ > +g. It is however less clear that it is valid when the flocculation is carried out in the concentrated regime. However, we have found that the properties of flocculated networks tend (provided they are subjected to a minimum of mechanical pretreatment) to be independent of the volume fraction at which the flocculation is performed, and this arguably justifies general application of their approach.

Ball and Brown predict power-law behaviour with an index which depends upon the mechanism of aggregation. For diffusion-limited (i.e. fast) aggregation they predict a power of 3.4 and for chemically-limited (slow) aggregation they predict 4.4. A value of 3.4 is thus anticipated for the present systems, as compared to the experimental value of 4.3 for the intermediate size regime. It would thus appear that the real aggregates have a somewhat higher Hausdorfer dimensionality than do theoretical aggregates. The calculation of Ball and Brown can probably be regarded as successful nevertheless as it (a) predicts power-law behaviour and (b) predicts a power significantly greater than the value of - 1 predicted by models that assume a uniform structure.

Appendix 3

Nature of the yield stress

The use of the idea of a yield stress in describing the behaviour of strongly flocculated networks probably deserves some further comment

202

since the term is used in a number of different ways in the literature. In shear, we have used the term to describe systems that are so strongly shear-thinning that a small increase in shear stress in a critical range causes the viscosity to decrease from some immeasurably large value to a value of order 1 Pa s or less. The’rate of charge of viscosity in the region of the yield stress (da/da) being at least lo6 s, and probably much larger. Problems of base-line drift, evaporation, and so on, make it difficult to make even order of magnitude estimates of the actual low-shear viscosity. We suspect however that it can be very large indeed. The reason for stating this is as follows. In the compression experiments (sedimentation experiments) it is possible to observe the behaviour of the networks over very long periods of time without ambiguity. Thus, sediments judged to have reached equilibrium have occasionally been kept for periods in excess of one year without any further creeping consolidation being observed. If a compressional creep viscosity qC is defined as the ratio of the mean compressive stress to the rate of volume-strain it can be deduced from such observations that vC > 10” Pa s. (For example, no perceptible volume-change in 1 year means in practice that any change in volume is < l%, so AV/Vc 0.01, the time scale is - 3 x lo7 s and the mean compressive stress is typically of the order 10 Pa under normal gravity and so qC > 3 X 10’ X lo/O.01 Pa s.)

the rheology of strongly-flocculated suspensions

Documents