Vane rheometry of bentonite gels

291

Journal of Non-Newtonian Fluid Mechanics, 39 (1991) 291-310 Elsevier Science Publishers B.V., Amsterdam

Vane rheometry of bentonite gels N.J. Alderman, G.H. Meeten and J.D. Sherwood Schlumberger Cambridge Research, P.O. Box 153, Cambridge CB3 OHG (U.K.) (Received

April 26, 1990; in revised form November

22, 1990)

Abstract

We describe the use of a four-bladed vane to measure the yield stress 70 of a series of aqueous bentonite clay suspensions. An extension of the vane technique is given, showing how the shear modulus G and the yield strain y, can also be obtained. These techniques are validated by comparing the results with those obtained from conventional concentric cylinder rheometry at low clay concentrations where wall-slip is absent. A study of the clay concentration dependence is made for volume fractions ($) up to 0.1. The magnitudes of rO, G and y, are found to scale closely as a@ over the volume fraction range 0.01-0.1, with x approximating to 3 for rO, to 4 for G and to -1 for y,,. Whereas x is largely time-independent, a increases with time for both 70 and G as the dispersions spontaneously gel. Keywords: bentonite;

elasticity;

gels; rheology;

yield

1. Introduction

Oilfield fluids such as drilling muds, cements and fracturing fluids, particularly at high concentrations, are prone to slip against a smooth solid wall of a rheometer rotor or stator [1,2]. Rheological characterisation of these fluids is therefore made difficult. Although methods exist to allow for slippage by correcting the rheological data, it is preferable in practice to use measurement techniques for which slip is absent. Yield stress measurements with conventional measuring geometries are particularly prone to the effects of wall-slip and this has prompted research into other measuring geometries. The vane method, widely used in soil mechanics as a simple technique for in situ measurements of the shear strength of cohesive soils, was adapted by Dzuy and Boger [3] for the yield stress measurement of concentrated 0377-0257/91/$03.50

0 1991 - Elsevier Science Publishers

B.V.

292

Fig. 1. Vane geometry.

See Section 2.2 for reference

to the dimensions

D and H.

suspensions. The vanes used by Dzuy and Boger, and in the work reported here, consisted of four thin blades at equal angles around a small cylindrical shaft (Fig. 1). The use of the vane instead of the more conventional measuring geometries has two principal advantages. Firstly, any wall slippage is absent since the yield surface is within the material itself. Secondly, inserting the vane causes much less structural disruption to a sample than introducing the fluid into conventional measuring geometries. This is important for fluids, such as ours, having a fragile gel structure which can be destroyed by large strains. In their comparative study, Dzuy and Boger [3] found, for bauxite residue slurries, that yield stresses obtained from the vane method were in good agreement with those obtained from extrapolation of shear stress-shear rate data to zero shear rate, and also with those obtained from the stress relaxation method. The vane method has been successfully applied to yield stress measurements of concentrated suspensions of titanium oxide, uranium oxide and brown coal [4]; various commercial greases [5,6]; suspensions of illite [7]; oil-in-water emulsions [S] and cements [9]. Here we investigate the suitability of the vane method for determining the yield stress rO, the yield strain y,, and the shear elasticity G of gelled bentonite suspensions. The vane method has been used for measurement of these quantities in clay soils [lO,ll], where the clay volume fraction is

293

typically 0.5. We are unaware of its prior use for measuring yield strain and shear elasticity of weak mud gels, such as we describe, with a clay volume fraction range 0.01-0.1. Filtration at the mud-rock interface in a well-bore produces filtercakes consisting of mud with a clay volume fraction in the 0.1-0.5 range. Such filtercakes are typically a few mm thick and the vane technique is therefore unsuitable. The measurement of 70 for filtercakes using a squeeze-film method is described in the following paper [12]. Thus the vane and the squeeze-film technique together cover yield stress measurement over the clay volume fraction range 0.01-0.5. The closest previous work to ours is that of Dzuy and Boger [4] and James et al. [7]. In particular, the workers in ref. 7 studied illite suspensions, not dissimilar in concentration and structure to the montmorillonite clays used by us. They also studied effects of varying the vane geometry, and compared their T,, data with results from a conventional (Weissenberg) rheometer. However, neither of the above obtained G and y, from their vane results. 2. Principle

of the method

The vane was gently immersed into the suspension, and was rotated at constant speed. The resulting torque was measured as a function of time. We show below that the torque is independent of the speed if this is low enough to make the viscous part of the torque negligible compared to that arising from the yield stress. A typical torque-time curve is given in Fig. 2. As the vane rotates from rest, region A shows an initial transient response, which the rheometer manufacturers (Haake) attribute to gear train play, and which we ignore. The linear behaviour at region B originates from the Hookean elastic response of the sample. From the slope of the curve at B the elastic modulus

*r

torque/mN m

Fig. 2. A torque-time plot for an FB5 suspension of 11.4% by weight left to gel for 16 h after rolling. The vane speed was 0.98 rpm. Refer to Section 2 for the meaning of the symbols A-G.

294 of the suspension G can be evaluated. As the vane continues to rotate, the response becomes increasingly non-Hookean, as shown by region C of the torque-time curve. The maximum torque given by region D of the torquetime curve allows the yield stress of the suspension, TV,to be evaluated once the geometry of the yield surface and the shear stress distribution on this surface is known. Details of the method of calculating TV, G and the yield strain y, are given in Sections 2.1, 2.2 and 2.3. In Section 5 we discuss how errors caused by material non-linearity and singularities at the blade tips may affect the interpretation of the vane results. The rapid decline as shown by region E of the torque-time curve marks the transition of the sheared sample from a gel to a fluid. 2.1 Yield stress For calculating the yield stress, TV, from the measured maximum torque T,, Dzuy and Boger [3] assumed the material to yield over the whole of the cylindrical surface area, including the ends, circumscribing the blades of the vane. They obtained

(1) for a vane of height H and diameter D immersed in the sample fluid. The assumption of uniform shear stress distribution over the surface of the whole cylindrical area is an approximation. Using finite element analysis for a four-bladed vane rotating in a Bingham fluid, Keentok et al. [6] have shown that the stress is highest near the outer edges of the vane blades, being uniform along the cylindrical wall and a function of radial position r over the circular end surfaces. Dzuy and Boger [3,4] also considered the case when the stress was non-uniform over the ends of the cylinder. They assumed the stress at a radius r to follow TO(r/R)” for r _
Assuming m to be H-independent be written T, = 2aHR2~, t 2T,,

and to depend only on D, then (2) may

(3)

where 2T, is that part of the total torque originating from the two end surfaces. By using vanes of various H but of the same D, (3) enables 7. to

295 be measured without knowing m. We later compare our experimental results with eqns. (l), (2) and (3). The assumption that the yield surface diameter is equal to the diameter of the vane may be suspect, as there is experimental evidence [5,6,11] to suggest that the material may yield at a diameter 0, that is larger than the diameter of the vane. Studies of soils by Wilson [13], and Arman et al. [14], showed the increase in D to be small, causing insignificant errors in TV.Keentok et al. [5,6] found the ratio D,/D for greases to be 1.00-1.05 and highly dependent on rheology, whereas their computer simulations of a four-bladed vane rotating in a Bingham fluid produced a value of Q/D = 1.025. It was noted by Dzuy and Boger [4] that these results were based on observations long after yielding has taken place. Clearly, the actual yield area remains a matter of speculation but D = 0, can be taken within the experimental error of measuring the torque, i.e. a few percent. 2.2 Elastic modulus G To obtain G from the maximum slope of the torque-time curve, the vane is regarded as a cylinder which circumscribes the blades. Below its yield stress the material is assumed to behave as a Hookean elastic solid, i.e. 7=yG,

(4)

where y is the elastic strain imposed on the material. The total elastic torque T experienced by the material as the vane rotates from rest we write as T = T, + T2,

(5)

where Tl is the elastic torque between the walls of the cylinder and the container (defined by region A in Fig. 3) and T2 is an end correction due to the elastic torque between the end surface of the cylinder and the base of the container (defined by region B in Fig. 3). This allocation of torque into regions A and B cannot be exact but we show below that for our vanes Tz -==xTl. For calculating T,, the material is assumed to be sheared between two concentric cylinders of height H, with inner radius R and outer radius R,, the container radius. The shear stress 7(r) at the radius Y is then

r(r) = ~

T

2mr2H.

If +(r) is the rotation of the sample relative to its fixed outer boundary then the shear strain is y(r)

=rg.

(7)

296

-__-_____ ______.

A

me___________-

____________ .

I. : :, A

._-----___.

I

I

6

I :

j

I

0

: I I

Fig. 3. Dimensions the symbols.

I

, , *

required

for the calculation

of G. Refer to Section 2.1 for the meaning

of

Using (6) for T(Y), it follows that the rotation of the vane, relative to the sample boundary defined by the container, is

For estimating T2, the material in region B is assumed to be in torsion between two parallel plates of radius R. A standard result for a cylinder in torsion gives T

=

2

mGR4+(R) 2h

.

(9)

297 The relative magnitudes of Tl and T2 can be estimated from Tl _=_ Tz

8Hh

00)

R2’

As it is usual to have Hh X- R2 (e.g. for vane D in this work, Hh = 12R2), it can be seen from (10) that we may neglect T2. Differentiating (8) with respect to time, with T2 +c T,, gives

(11) where w = d+( R)/d t is the angular velocity of the vane. 2.3 Yield strain y, In Fig. 2 the maximum torque (from which we obtain rO), is well-defined, even when the maximum itself is not particularly pronounced, as was the case for the low concentration gels. The yield strain corresponding to r0 is, however, not very well-defined, partly because of the uncertainty of locating point D accurately within the broad maximum and partly because of the curvature at A. Thus we defined a yield strain yY corresponding to the experimentally well-defined strain difference between G and F in Fig. 2. By this definition

3. Experimental

validation of the vane method

3. I Materials Measurements were made on suspensions of l-22% by weight Clarsol FB5 bentonite (supplied by British CECA) in deionised water. The 22% suspension was prepared by adding 414.4 g bentonite to 1500 ml of deionised water in a 2 1 bottle over a period of 5 min during agitation with a Silverson high-shear mixer. After ensuring that no material remained on the mixer head, two stainless steel cylinders, 40 mm in diameter and 180 mm long, were inserted into the bottle containing the suspension. This bottle was then placed onto a Luckham Multimix roller and the suspension was homogenised by rolling at 100 r-pm for 3 h. It was then left to stand for 16 h for hydration and dispersion of the bentonite platelets. The other suspensions were prepared by diluting down the 22% suspension to the required concentration with deionised water. Using the roller,

298 these suspensions were then cold-rolled at 100 r-pm for 10 min to ensure homogeneity. These suspensions were then left to stand for 16 h to allow for further hydration and dispersion. To ensure that all the suspensions had the same initial shear history prior to measurement, they were cold-rolled at 100 r-pm for 10 mm. The bentonite volume fraction + was obtained from the weight fraction, assuming a bentonite crystalline density of 2600 kg mP3. The viscoelastic properties of the muds were found to depend on the gelation time t, elapsed after cold-rolling. In Section 4 we report the t, dependence of r0 and G. 3.2 Apparatus

and procedure

The vanes used in this study consisted of four 0.6 mm thick blades at right angles around a cylindrical shaft of diameter 3.0 mm. Five vanes were used, which we label A-E. Their other dimensions are given in Table 1. Vanes were directly attached to a Haake Ml50 drive and torque measuring head which was controlled by a Haake RVlOO unit for chart recorder output. The container holding the suspension was raised with a laboratory jack until the vane was immersed to the required position. To minimise the effect of wall boundaries, care was taken to ensure the following criteria [4] were met: (1) about (2) height (3)

The vane was placed near the centre of the container, which was three times larger in diameter than the vane. The suspension depth beneath the blades was greater than the blade H. The suspension height above the vane was greater than H/2.

The vane was then rotated with constant speed N and the resulting torque T was measured as a function of time. Using the 7.81% by weight bentonite

TABLE 1 Vane dimensions Vane

D (mm)

H (mm)

A B C D E

12.7 12.7 12.7 19.0 25.4

12.7 19.0 25.4 28.5 25.4

299 0.0005

0.0004

E ? “1 0.0003 ET ,o

El 3

o’ooo2

g

0.0001

0.0000 0.01

1.00

0.10

Rotational

speed/rpm

Fig. 4. Effect of vane rotational speed on the maximum torque measured with vane E fully immersed in the suspension.

suspension as the test fluid, the effect of N on the maximum torque T, was investigated using vane E immersed in the suspension. The results given in Fig. 4 show T, to be essentially constant when N I 1.0 t-pm whereas T, increases with N when N > 1.0 rpm. This behaviour is in good agreement with that obtained with other rheometers having stiff spring systems such as the Weissenberg rheogoniometer for bauxite slurries [3,4] and cements [9]. For the Haake Ml50 measuring head used here, the measuring spring deflects by 0.5” when subject to a torque of 14.7 mN m. The increase of T, with N in Fig. 4 presumably originates from the viscosity of the sample fluid. All subsequent vane measurements were carried out at 0.1 rpm. 3.3 Interpretation

of the T,,, data

For a range of bentonite concentrations the maximum torque T, was measured using vanes A, B and C. The results, shown in Fig. 5, confirm the linear dependence of T on H in (I), (2) and (3). However, apart from the two highest yield stress muds, the extrapolated lines in Fig. 5 pass through the origin within experimental error. This suggests that there is no end-contribution from the blades for the lower yield stress muds. A similar result was found by James et al. [7] for illite suspensions (their Fig. 8), but not by Dzuy and Boger [4] for bauxite suspensions.

300 0.0005 a

7.81% cl 6.99%

0.0004 E

* 10.0% 0 11.90% + 14.00% x 16.00%

? 2 0.0003 z E $ 0.0002

zi 0.0001

g

-a A-.

0.0000 0.000

I

I

0.005

0.010

I

I

0.015

0.020

I

0.025

0.030

H/m Fig. 5. Plots of T, vs. H to test eqn. (2).

Our results are compared in Table 2. Column 2 gives the results for vanes A, B and C, analysing the data using eqn. (1). The tabulated r0 corresponds to the mean value for the three vanes and the uncertainty was obtained from the spread of values. Column 3 gives r0 obtained using eqn. (2), giving m in column 4. Where values of m are not shown, this corresponds to the T,-axis

TABLE 2 Comparison of 70 values Solids concentration (W; by weight)

7.

7.81 8.99 10.00 11.96

3.2+0.2 7.2+0.5 9.8 + 0.6 18.5 f 1.7

3.8 8.9 12.1 22.7

_ _ _ _

14.00 16.00

31.3kl.O 61.6kl.O

33.9 63.8

2.0 0.7

Eqn. (1)

(Pa)

Eqn. (2) 7.

(Pa)

m

301

intercept in Fig. 5 being zero within the experimental uncertainty, with the implication that m + 00. Such values of m imply a very small end-contribution from the vane blades, while the values of = 2 and = 0.7 found for the two highest concentrations suggest that end-effects cannot be neglected. This result, and that of James et al. [7], suggests that for higher concentrations the stress around the vane is in line with the assumptions of Dzuy and Boger, while for lower concentrations there is a negligible stress over the circular ends of the vane. Interpretation via eqn. (1) gives values of r0 which are slightly smaller than those given by eqn. (2). However, the discrepancy is commensurate with experimental uncertainty. Hence, in Sections 3.4 and 4, where we study the concentration dependence of TV, we used the simplest method, i.e. one vane at one position in the sample and interpretation via eqn. (1). 3.4 Comparison

of vane and Carrimed

results for 7O

For each of the bentonite suspensions, the determination of r0 from the vane method was carried out using vane A immersed in the suspension. From the torque-time curve, the measured maximum torque T, was obtained, and r0 calculated, using eqn. (1). These r0 values are given in Table 3 together with the r0 values obtained from measurements using a Carrimed CSlOO controlled stress rheometer. This had a stainless steel recessed concentric cylinder geometry, where the radii of the bob and cup were respectively 8.63 mm and 9.33 mm and the height of the bob was 32 mm. The 7,, values obtained from the vane method and the Carrimed rheometer as a function of solids volume fraction are compared in Fig. 6. For the

TABLE 3 Comparison suspensions

of methods

used

to determine

Solids concentration

q, values

for

3-12%

by weight

Vane method

Carrimed

(W; by weight)

(% ; by volume)

7. (pa)

7. (pa)

2.97 3.99 4.99 5.83 7.00 8.00 8.99 9.99 11.4

1.23 1.67 2.12 2.47 3.03 3.46 3.92 4.38 5.06

0.4 1.4 2.1 2.8 6.0 8.6 13.0 20.7 28.9

0.1 0.9 2.1 2.6 _ 8.6 11.5 17.4 21.9

rheometer

bentonite

302 35.0

30.0

25.0 al ? ; 20.0 E 2; a 15.0 ;ii F 10.0

5.0

0.00

0.01

0.02

0.03

0.04

0.05

0.06

Solid volume fraction Fig. 6. Comparison of the yield stress-solid volume (squares) and the Carrimed rheometer (octagons).

fraction

plot

for the vane

method

2-3.5s by volume bentonite suspensions good agreement was found between the 7,, values determined from the two methods. At solids concentrations below 2% by volume, 7, values obtained from the vane method were slightly but systematically larger than those obtained from the Carrimed rheometer (see Table 3). This suggests that the vane technique may be suitable only for yield stresses greater than a minimum value, probably dependent in magnitude on the number of vanes. At solids concentrations above 3.5% by volume, the 70 values obtained from the Carrimed rheometer were found to be lower than those obtained from the vane method, possibly owing to wall slippage in the Carrimed rheometer. Figure 6 shows that either method can be used to characterise the rheology of bentonite suspensions provided that the yield stress is in the range 2-10 Pa (2-3.5% by volume), the vane method being preferable when 70 > 10 Pa and the concentric cylinder geometry being preferable when 70 < 2 Pa.

303 3.5 Comparison

of the vane and Bohlin results for G

Taking the 9.99% by weight bentonite suspension as the test fluid, the determination of G was carried out as a function of gelation time using vane D fully immersed in the suspension. Care was taken in these experiments to ensure the suspension did not approach yield, as measurements of G were then found to be influenced by the previous measurement. From the torque-time curves, the measured maximum slope required for the calculation of G using (11) was obtained as a function of gelation time. These results were compared with those obtained from low strain oscillation measurements using a Bohlin VOR rheometer with the C25 concentric cylinder geometry where the radii of the bob and cup were, respectively, 12.5 mm and 13.75 mm and the height of the bob was 37.5 mm. Figure 7 is a typical plot of Bohlin G results against the strain y used to measure the stress in the sample fluid, showing G to be strain-independent up to strains of about 0.03, thus indicating the presence of a Hookean or

100.0

(F

(d t

0

10.0

1.0

I-

1.0

% Strain Fig. 7. Strain-sweep plot obtained from the Bohlin rheometer for G at 0.002 Hz for the location of the linear viscoelastic region for a 9.99% by weight suspension.

304

250 5 200

-

150

-

100

I! -

50

-

al k 0

0

’

I

0

1000

I

1

2000

3000

I

4000

5000

TIME/S Fig. 8. Comparison of the G-time plot obtained from the vane method at 0.01 S-I (squares) and the Bohlin rheometer at 1% strain and 0.002 Hz (octagons) for 9.99% by weight bentonite suspension. linear viscoelastic region. A non-linear response is evident for y > 0.03. Subsequent measurements of G were made in the Hookean region as a function of the gelation time t,. Figure 8 shows a good agreement between G( tJ measured by the vane and by the Bohlin rheometer. In Section 4.4 we show that the yield strain y, as defined by (12) is not the same as the strain at which non-linearity begins to occur, e.g. = 0.03 in Fig. 7. 4. Vane investigation of gelation 4.1 Apparatus

and procedure

Having shown that the vane method gave reliable results, we used it to study the gelation time ( tg) and concentration (+) dependence of 70 and G in bentonite gels. Vane A was used in all measurements.

305 A 25% by weight suspension was prepared by adding 505.3 g of Clarsol FB5 bentonite to 1500 ml of deionised water and proceeding as previously described. The other suspensions were prepared by dilution with deionised water. Vane measurements were carried out on each suspension at gelation times (t,) of 0, 16, 69 and 120 h, using a fresh sample for each test. The gelation time was the time elapsed after suspensions had been cold-rolled for 10 min prior to the start of each experiment. From the maximum torque T,

10’

10”

10'

10'

loo

1*10-’

’

I

I

1’1P

III

I

1

10-l

1o-z

Solid

volume

III

fraction

Fig. 9. Variation of q, with solids volume fraction as a function of gelation time. Legend: 0 and solid line, t = 0 h; o and dotted line, t =16 h; A and dashed line, t = 69 h; 0 and dot-dash line, t = 120 h.

306 TABLE 4 A and B values in the yield stress relation (13)

te (h)

A (MPa)

B

0

0.2kO.l 1.3kO.5 5.5 * 2.5 3.0+ 1.5

3.0+0.1 2.9+0.1 3.2 f 0.1 3.0+0.1

16 69 120

and the maximum slope of the torque-time curve, the yield stress r0 and the elastic modulus G were evaluated using eqns. (1) and (ll), respectively. 4.2 Measurements

of r.

The dependence of r0 on solids concentration as a function of gelation time t, is shown in Fig. 9. Least squares fits were made to the relation 70 = A@

(13)

and A and B for each t, are given in Table 4. Within the experimental uncertainty we conclude that B is essentially constant and that A is time-dependent. More data are required to establish the proper dependency of A on t,. 4.3 Measurements

of G

The dependence of G on solids concentration as a function of gelation time t, is shown in Fig. 10. Least-squares fits were made to the relation G = C+“,

04)

where C and D for each time t, are given in Table 5. Within the experimental uncertainty, we conclude that D is essentially constant and that C is dependent on t,. More data are required to establish

TABLE 5 C and D values in the elasticity relation (14)

f, 03

C (MPa)

D

0

ll+_ 5 230 + 100 170+ 80 8Ok 30

4.2 f 0.1 4.3 + 0.1 4.2kO.l 4.0 & 0.1

16 69 120

307

A

I

A

1*10-’

1

I

I

III

I

I

to-’

10-l

1*10”

Solid

volume

J

III

fraction

Fig. 10. Variation of G with solids volume fraction as a function of gelation time. Legend: 0 and solid line, t = 0 h; o and dotted line, t = 16 h; A and dashed line, t = 69 h; 0 and dot-dash line, t =120 h.

the detailed dependency of C on t,. Values of D given in Table 5 are consistent with the range of 4-5 obtained by Rhandal and Tadros [15], who made viscoelastic measurements of sodium montmorillonite suspensions in the pH range of 2.5-8.8 as a function of gelation time, using a Bohlin VOR rheometer with concentric cylinder geometry.

308 4.4 Measurements

Using yY as a Within gelation

of y,

the above data for 70 and G in eqn. (12), we obtain the yield strain function of time and clay concentration. This is plotted in Fig. 11. the experimental uncertainty y, is independent of the time of and varies as (15)

4*10-l 1*10-

10-*

10-l

Solid volume fraction Fig. 11. Variation of yield strain yY with solid volume fraction of clay. Refer to Fig. 9 for the meaning of the symbols.

309 where E = 0.04 f 0.01 and F = - 0.95 &-0.05. For a 9.99% by weight suspension the Bohlin G data in Fig. 7 suggest that non-Hookean behaviour occurs at a strain of about 0.03. This we note to be about an order of magnitude smaller than y, obtained from the vane rheometer (Fig. 11) at the same clay concentration, presumably as the yield strain as defined by (12) includes some non-linear strain of the sample. 5. Discussion and summary We now consider briefly some aspects of the vane’s operation. After the artefact region A in Fig. 2, the torque at first grows linearly with time, but there is eventually a non-linear region C-D. An analysis of the deformation of a linear elastic solid leads to the appearance of singularities at the extremities of the vane. Any real material will become non-linear near the tips of the vane, and if the region of non-linearity is sufficiently large, the total torque will be affected. Thus we must ascertain the existence of a Hookean region (B in Fig. 2) and then obtain G from the slope measured at B. For the muds used here, separate measurements of G (using a Bohlin rheometer) at strains well below the yield strain, showed (Fig. 8) good agreement with the vane measurements. The good agreement found for the yield stress between the vane method and the Carrimed concentric cylinder rheometer (Fig. 6) for suspensions where smooth-surface slip is thought not to occur, suggests that the same definition of yield is being applied in the two geometries, and that the local yield around the vane tips is relatively unimportant. For yield stress measurements a good agreement was found between the vane and the concentric cylinder methods applied to bentonite suspensions of volume fractions between about 0.02 and 0.035. Below 0.02 we propose a failure of the simple vane assumptions. Above 0.035 we attribute the divergence to slip between the sample and the walls of the concentric cylinders. For measurements of shear elasticity the vane and the concentric cylinder methods agree well. However, the yield strain as obtained from the vane method using eqn. (12) appears to be about an order of magnitude greater than the strain at which the sample becomes non-linear. We think this is because the yield stress rO, as defined by the maximum torque on the vane (Fig. 2) is the stress to failure rather than the stress at which the material becomes non-Hookean. The magnitudes of rO, G and y, scale closely as a@ over the volume fraction range 0.008-0.1, with x being approximately 3 for rO, 4 for G and - 1 for y,. Whereas x is largely time-independent, a increases with gelation

310 time for q, and G, presumably owing to the spontaneous incorporation of bentonite platelets into an elastic network.

time-dependent

References 1 D.C-H. 2

3 4 5 6

7 8 9 10

11

12 13 14

15

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