Yield Stress Bentonite Kelessidis

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Colloids and Surfaces A: Physicochem. Eng. Aspects 318 (2008) 217–226

Yield stress of water–bentonite dispersions V.C. Kelessidis a,∗ , R. Maglione b a

Department of Mineral Resources Engineering, Technical University of Crete, Polytechnic City, 73100 Chania, Crete, Greece b Vercelli, Italy

Received 9 August 2007; received in revised form 1 November 2007; accepted 20 December 2007 Available online 6 January 2008

Abstract Yield stress of aqueous bentonite dispersions was determined at two concentrations, with two bentonites, over a range of pH values, with the vane technique and by extrapolation of the full rheograms, derived with concentric cylinder viscometer, fitted to Herschel–Bulkley and to Casson models. All samples exhibited a yield stress and gave very similar yield stress values determined by the three techniques and hence, any of the techniques can be used for measurement of the yield stress. Data extrapolation using either the Herschel–Bulkley or the Casson model would be favoured, though, because it gives, in addition to the yield stress, the rheological model parameters. The close matching observed for all three techniques is attributed to preparation and intensive preshearing procedures, similar to ones experienced by fluids in flow situations. pH of dispersions affected their yield stress but the effect was different for the two bentonites and the two concentrations tested. Measurement time at each rotational speed should be kept at a minimum of 60 s. Bentonite dispersions build continuously structure over time and the yield stress evolution with time could be well described by power law. A model to predict yield stress, previously suggested for suspensions at the isoelectric point, could be a good starting point for yield stress prediction of bentonite dispersions. © 2008 Elsevier B.V. All rights reserved. Keywords: Yield stress; Bentonite; Vane; Herschel–Bulkely; Casson

1. Introduction Water–bentonite dispersions at concentrations more than 1% exhibit a yield stress, which is defined as ‘the stress above which the material flows like a viscous fluid’ [1]. The true existence of the yield stress of various dispersions and whether it is a material property has been debated over many years [2–7] but many researchers consider it a true material property [8–13]. This controversy is in essence, a controversy about the shear rate range that shearing is observed for relatively short times. If the very long-term stability of suspensions at extremely low shear rates is of interest, the existence or not of the yield stress could be questioned [7], but if, on the other hand, flow situations of such suspensions are of interest, as is the case for the current article, then the yield stress is indeed a reality. In the latter cases, the successful match of pressure loss predictions in flow of such dispersions through various conduits with lab-

∗

Corresponding author . Tel.: +30 2821037621. E-mail address: [email protected] (V.C. Kelessidis).

0927-7757/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2007.12.050

oratory and field measurements using rheological models with a yield stress [13–17] may be considered a direct proof that the yield stress is indeed an engineering reality. Furthermore, drilling industry relies heavily on the existence of a yield stress of the drilling fluids for suspending weighting solids (such as barite) and for transferring drill cuttings to surface [12,18,19]. Extensive reviews on yield stress materials have been presented previously [20–22]. Various techniques to determine the yield stress have been reported in the past and are used in research and industry and these have been reviewed by Nguyen and Boger [23]. No single method, however, has been universally accepted as the standard method for measuring the yield stress. Determination of the yield stress as a true material property is very difficult because not only it depends on the measurement technique, but also on the model used to evaluate rheological data [7,23,24]. Furthermore, the yield stress has been characterized as a timedependent property [9,24] although the intensive preshearing occurring prior to measurement according to drilling fluid industry practice [25,26] should minimize thixotropic effects [27]. Never-the-less, yield stress can be determined either by using

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Nomenclature A AH B d D h0 H K n r SQ2 t Tm

term in Eq. (9) Hamaker constant (J) term in Eq. (9) particle diameter (m) vane diameter (m) inter-particle distance (m) vane height (m) fluid consistency index (Pa sn ) fluid behavior index radius (m) sum of square errors (Pa2 ) time (s) torque (N/m)

Greek letters γ˙ shear rate (s−1 ) ζ zeta potential (V) μC Casson viscosity (Pa s) μp plastic viscosity (Pa s) τ yield stress (Pa) τB Bingham yield stress (Pa) τ CA Casson yield stress (Pa) τe yield stress on bottom and top surface of vane (Pa) τ HB Herschel–Bulkley yield stress (Pa) τ HSRB high shear rate Bingham yield stress (Pa) vane yield stress (Pa) τvn τLSRB linear shear rate Bingham yield stress τ y,max maximum yield stress (Pa) φ solid volumetric concentration

direct measuring devices or by implementing indirect measuring techniques. Direct measurement techniques rely on an independent assessment of the yield stress, normally carried out with the rotating vane method [8,11,24,28,29], a technique used widely in soil mechanics and adapted for use in fluids. Numerical simulation [30,31] has provided further support for the reliability of the technique. The vane has usually four thin blades and is rotated at very slow speed while immersed in the material. The resulting torque is measured continuously as a function of time and analysis of the curve provides the yield stress, which is often called the static yield stress [32]. The technique is not prone to errors attributed when use is made of the indirect techniques, because, firstly, no wall slippage occurs, with the material yielding in itself and secondly, it causes less structural disruption, which is particularly important for fluids having fragile gel structure like water–bentonite suspensions. The vane technique, although an established method used with many suspensions, it has not been used widely in the drilling fluids industry [19,33]. Various efforts have been reported in the past aiming at developing models which relate microstructure (the interaction among molecules and suspending particles) to macrostructure (rheology and in particular the yield stress) but none had significant

success and this could be the result of not knowing which is the real property that is measured. Hence, one must assess which of the yield stress values is the real engineering property, not dependent on the measuring methodology and furthermore a procedure should be standardized on how to measure it. It is the scope of this work to provide further evidence that the yield stress of aqueous bentonite dispersions is an engineering reality. The yield stress of bentonite dispersions will be determined at different conditions such as, different raw material, different preparation procedures, different concentrations and different pH values, by the vane technique and also estimated by extrapolation of concentric cylinder viscometric data using Herschel–Bulkley and Casson rheological models. The values will be compared so that a best approach for obtaining the engineering property of bentonite dispersions known as yield stress will be proposed. An attempt will also be made to predict yield stress of such dispersions from proposed relationships between microstructure and macroscopic properties, which, if successful, could provide evidence that yield stress is a material property. 2. Background theory The yield stress from the vane measurements can be computed, following the approaches of Nguyen and Boger [8], James et al. [10] and Alderman et al. [28]. If the stress is non-uniform over the circumscribed cylinder by the vane, then    D/2 τy D Tm = πDH τe (r)r dr (1) +2 π 2 0 where Tm is the maximum measured torque with the vane instrument, τ y is the yield stress, H is the height of the vane and D is the diameter of the vane. τ e is the shear stress developed on the bottom and top surfaces of the vane, which is assumed to vary from the center to the outer circumference by  m τe 2r = (2) τy D Substitution into Eq. (1) and integration yields,   πD3 H 1 Tm = τy + 2 D m+3

(3)

If m is zero or very small, it may be concluded that the fluid ‘yields’ and the shear stress along the radius of the bottom or top of the vane is constant and equals the yield stress, τvn [23,34]. James et al. [10] using vanes of different height and diameter in illite suspensions of various concentrations found that (m) ranged between 0.01 and 0.05, hence, it may be concluded that the stress is uniform over the cylindrical surface and equal to the yield stress. Setting thus m = 0 in Eq. (3) results in   1 πD3 H Tm = τvn + (4) 2 D 3 A typical stress (torque) versus time diagram is shown in Fig. 1, with data of sample 6, to be analyzed later. The value of the vane yield stress, τvn , is estimated then by Eq. (4), taking the maximum shear stress (torque) value from the diagram.

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In most cases, however, water–bentonite dispersions exhibit shear thinning behavior with a yield stress [14,27,41–44]. To take into account the non-linearity of rheological data of these bentonite suspensions, non-linear models such as Herschel–Bulkley model [45], Casson [46] or Robertson–Stiff [47] models have been used. The latter model uses three rheological parameters with none related to a true yield stress [48]. Hence, in this work, only the first two mentioned models have been utilized. The Herschel–Bulkley model, given by ˙ n τ = τHB + K(γ)

Fig. 1. Yield stress measurement with vane. Actual data of sample 6.

In the indirect techniques, the yield stress is estimated from the rheograms obtained with a viscometer after extrapolating the τ–γ˙ (shear stress–shear rate) curve to zero shear rate, with the rheogram obtained with a variety of instruments like rotating cylinders, cone and plate or parallel plates [35]. The parameter thus estimated is often called the dynamic yield stress [33,36]. This technique has been criticized and the estimated yield stress has not been considered by some researchers a true fluid yield stress because its accuracy depends on several factors such as, the assumed model, the consistency of the data or the instrument used [23]. Furthermore, at very low shear rates, slippage of the fluid near the wall may occur giving false readings. It is for these reasons that researchers should be aware, not only of these problems [23,28,37], but also of the methods used to identify them [38]. A number of models have been proposed in the past to estimate the yield stress of bentonite dispersions. The linear Bingham plastic model is very often used, τ = τB + μp γ˙

(5)

where τ is the measured shear stress, γ˙ is the imposed shear rate and μp is the plastic viscosity. The shear rate range over which Eq. (5) is fitted to derive the so-called Bingham yield stress differs among different investigators. Some researchers may utilize only the two high shear rate readings, as is done by drilling fluid industry [18,25,26], which are taken as the 600 rpm and 300 rpm (giving Newtonian shear rates on the fixed inner cylinder of 1021 s−1 and 511 s−1 , respectively for the rotating viscometer used in oil-field, with inner cylinder diameter of 1.7245 cm and an outer cylinder diameter of 1.8415 cm), thus obtaining the high shear rate Bingham yield stress, τ HSRB . Conversely, only the linear portion of the curve may be used, normally at high, and often more than two, shear rates [39,40] thus giving the linear shear rate Bingham yield stress τ LSRB . Many times, though, the full rheological data set obtained at all shear rates may be used thus giving the Bingham yield stress τ B . The use of various Bingham yield stresses, without many times having a reference on how exactly it was determined, could be one of the reasons that there are great discrepancies among yield stress values reported by different investigators, even for solid suspensions in under otherwise similar conditions.

(6)

uses three rheological parameters, the Herschel–Bulkley yield stress (τ HB ), the flow consistency index (K) and the flow behavior index (n). The Casson model is given by  √ √ τ = τC + μC γ˙ (7) where τ C is the Casson yield stress and μC is the Casson viscosity. Considering the yield stress an engineering reality, the question would arise as to which might be the yield stress of a particular suspension [7]. The answer would certainly depend on the application [4]. Barnes [7] claimed that for yield stresses determined for shear rates greater than 0.001 s−1 , the Sisko model [49] would be the best, but he finally suggested to use the Bingham model, because it has more worked out solutions to fluid problems, especially when compared to Casson or Herschel–Bulkley models. But, while this is true, there are several reports showing the disparity between theoretical and experimental results using the Bingham plastic model for the flow of many non-Newtonian fluids [16,17,41] and a reason for the discrepancies could be the different Bingham yield stresses used. On the other hand, the use of Herschel–Bulkley model has proven to be a good choice for many cases [27,42–44], while in many other cases, the Casson model has proven to work the best [41,50,51]. Of course, modeling can be attempted to describe the flow behavior of such suspensions with different flow equations, applicable only over a certain (and even narrow) range of shear rates. Another method, also, to determine the true yield stress could be through measurements of pressure loss − flow rate (p − Q) under various flow conditions, matching predictions with measurements in order to single out the true yield stress. For this approach to work, however, good theoretical models, covering laminar, transition and turbulent flows of such fluids in various conduit shapes is essential. 3. Materials, preparation and equipment 3.1. Materials Various bentonite-water dispersions of 5.0% (w/w) and 6.42% (w/w) were prepared using two commercial bentonite products, Zenith© (S&B Industrial Minerals SA, Greece), a Naactivated bentonite containing more than 90% montmorillonite and Wyoming bentonite (Haliburton-Cebo, Holland), a natural sodium montmorillonite, both used in oil-well drilling with par-

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Table 1 Sample information and measured and estimated values of yield stresses Sample #

Sample information (wt%)

pH

τvn (Pa)

τ HB (Pa)

R2c

SQ2 (Pa2 )

τ CA (Pa)

R2c

SQ2 (Pa2 )

1 2 3 4 5 6 7 8 9 10 11 12 13 14

Zenith 5% Zenith 5% Zenith 5% Zenith 5% Zenith 6.42% Zenith 6.42% Wyoming 5% Wyoming 5% Wyoming 5% Wyoming 5% Wyoming 6.42% Wyoming 6.42% Wyoming 6.42% Wyoming 6.42%

7.31 8.02 9.47 11.06 7.92 10.77 7.6 7.69 8.87 10.56 6.52 5.29 8.98 10.41

5.66 7.54 8.34 23.83 11.45 82.98 24.06 23.95 14.71 17.31 43.12 22.59 37.25 30.95

5.51 7.06 8.23 19.15 7.91 77.53 20.80 20.92 10.36 21.76 40.85 17.09 32.95 21.34

0.9993 0.9993 0.9908 0.9991 0.9985 0.9670 0.9984 0.9977 0.9987 0.9984 0.9984 0.9993 0.9991 0.9981

0.02 0.02 0.30 1.23 0.11 21.51 0.53 0.77 0.14 2.12 1.73 0.56 1.32 6.86

5.92 6.55 8.38 28.84 10.39 86.62 19.80 20.61 10.20 32.11 39.87 15.58 33.38 25.06

0.9928 0.9988 0.9882 0.9655 0.9675 0.9283 0.9962 0.9972 0.9992 0.9622 0.9980 0.9978 0.9976 0.9900

0.21 0.03 0.38 46.80 2.52 46.70 1.23 0.94 0.09 50.90 2.28 1.77 3.13 35.25

ticle sizes for both bentonites smaller than 70 ␮m, with most of them around 10–20 ␮m [27,44]. The samples were prepared according to American Petroleum Institute (API) procedures [25,26] with deionized water, using a Hamilton Beach high speed mixer to stir the samples at 11,000 rpm for 20 min when preparing the dispersion. In order to have dispersions covering an extended yield stress range, the pH of the dispersions was varied. The sample pH was adjusted to the desired value with 1 M NaOH or 5 M HCl, while the ‘natural’ pH of the dispersions, without any additive, was around 8.8. The samples were then poured in a covered container and left undisturbed for 16 h for full hydration at room temperature. Prior to testing, the samples were stirred for 5 min at 11,000 rpm and the final pH value was recorded and it is this value that is reported in Table 1, as there is a shift in pH of bentonite dispersions [27]. After pH measurement, the sample was poured into the viscometer container to get the rheograms and it was then poured into the container to measure the vane yield stress. 3.2. Equipment Rotational viscometric data was obtained with a variable speed rotational viscometer (Grace Instruments, USA) which offers electronically controlled and continuously varied speeds from 0.01 rpm to 600 rpm, connected to a PC for data storing and analysis. The inner fixed cylinder diameter is 1.7245 cm and the outer rotating cylinder diameter is 1.8415 cm thus giving a gap with a diameter ratio of δ = 1.06785. Viscometric data were obtained at fixed speeds of 600 rpm, 300 rpm, 200 rpm, 100 rpm, 60 rpm, 6 rpm and 3 rpm, which give Newtonian shear rates on the inner fixed cylinder of 1021.38 s−1 , 510.67 s−1 , 340.46 s−1 , 170.23 s−1 , 102.14 s−1 , 10.21 s−1 and 5.11 s−1 , respectively. The readings were taken from high to lower speeds, while rotation lasted for 60 s at each rotational speed, with readings recorded every 10 s, thus giving six measurements for each rotational speed. These six values were then averaged and recorded for rheological parameter estimation according to the two chosen models. Direct yield stress measurements were performed with a Brookfield yield stress vane measuring device using two dif-

ferent four bladed vane spindles, in order to cover the extended yield stress range, one with length of 4.333 cm, and diameter of 2.167 cm while the second had length of 2.535 cm and diameter of 1.267 cm. Vane was rotated at 0.1 rpm. 3.3. Methodology All measurements were done at 25 ◦ C. pH was measured with Inolab pH-meter and ranged from acidic (pH of 5.3) to alkaline (pH of 11.1). The conditions of testing the 14 samples are shown in Table 1. The effects of length of measuring time while at a particular rotational speed on the measured rheological properties was also determined. This was done by using measuring times of 60 s, 120 s and 180 s. An extensive literature search did not return any particular standard but the usual practice [25,26] recommends rotation until ‘reading stabilizes’. Such testing should be performed under otherwise similar conditions of samples. For this reason, three separate batches of Zenith dispersions at 6.42 wt.% were prepared exactly the same way, following API preparation procedures, with the samples left overnight for 16 h for full hydration and agitated for 5 min at 11,000 rpm prior to testing, thus experiencing the same preshearing history. The build up of the structure of these dispersions over time was measured by measuring the vane yield stress, using a 6.42% Zenith bentonite dispersion prepared as per API specifications and left to hydrate overnight for 16 h. Prior to first measurement (t = 0), the dispersion was agitated at 11,000 rpm for 5 min and then the vane yield stress was determined after 60 min, 120 min, 180 min and 240 min. At 300 min, the dispersion was presheared according to the normal practice (i.e. 11,000 rpm for 5 min) before final measurement. 4. Results and discussion 4.1. Comparison of yield stress measurement techniques For each of the samples, the rheograms were fitted to the Herschel–Bulkley and Casson models, computing the relevant rheological parameters together with the estimation of the good-

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Fig. 2. Rheograms of measured data, of Herschel–Bulkley and of Casson models, estimated yield stresses from Herschel–Bulkley and Casson models and measurement with vane for sample 6 (Zenith, 6.42%, pH 10.8).

ness of fit, using a standard non-linear regression package. A typical comparison for one sample (sample 6) is shown in Fig. 2, where the full rheograms, measured, predicted with the Herschel–Bulkley model and with the Casson model, are shown together with the Casson yield stress, the Herschel–Bulkley yield stress and the vane yield stress. For the particular case it is seen that τ HB is estimated at a lower value than τ CA , which is very close to the 6-rpm and the 3-rpm readings, while τvn , is between the two estimated yield stresses. The values of the yield stresses from the two rheological models are plotted versus the vane yield stresses in Fig. 3, with the error bars corresponding to the standard error, as estimated from the non-linear regression routine. Yield stresses cover an extended range between 5 Pa and 80 Pa, with all but one being less than 45 Pa. The results show that both τ HB and τ CA plot around the 1:1 (perfect matching) line with τvn and both values are close to each other. Thus, for these dispersions, any of the proposed techniques is good for measuring the yield stress. Concentric cylinder data would then be preferable because they also give the full rheograms from which the rheological model, either Herschel–Bulkley or Casson model, could be derived. The sum of square errors, SQ2 , an indication of the goodness of fit, defined as the sum of the square of errors between measured shear stress values and predicted shear stress values

221

Fig. 4. Sum of square errors in the estimating models by Herschel–Bulkley and Casson vs. the measured vane yield stress.

according to either of the rheological model, have been computed for all samples and are reported in Table 1 and plotted in Fig. 4 for both models. They vary between a very low 0.02 Pa2 and a maximum of 50.9 Pa2 , with the majority of them being less than 10 Pa2 , thus indicating a very good fit of either model with experimental data, which was also evident from the regression coefficient which for all samples was higher than 0. 99 but in one case, where it was 0.9283 (Table 1). For some samples (samples 1, 2, 3, 5, 9, and 13), SQ2 values are small for both models, for some samples (samples 7, 8, 11, and 12), SQ2 values are low for the Casson and high for the Herschel–Bulkley models and for samples 4, 6, 10 and 14, SQ2 values are high for Casson and low for Herschel–Bulkley models. However, these results do not seem to correlate with the small differences obtained between the Casson and the Herschel–Bulkley yield stress values. The Herschel–Bulkley model has also been found to perform good representation of rheograms of similar dispersions [27,36,43,44,52] while it has been reported [53,54] that solid particle suspension rheological data that were explained by Casson model, could be well described also by the Herschel–Bulkley model. The close match of the vane yield stress with either τ HB or τ CA can be further quantified if an index of the degree of deviation, DD , defined by Tsamantaki et al. [33] as DD =

2 N   τy,k −1 τvane

(8)

i=1

Fig. 3. Comparison of yield stress measured by vane to yield stress estimated from Herschel–Bulkley and Casson models for all 14 samples.

is computed, where τ y,k can be either τ HB or τ CA . A value of DD close to zero would indicate that τ y,k is very close to vane yield stress. Computation of the degree of deviation gave DD = 0.5 for the Herschel–Bulkley model and DD = 1.1 for the Casson model, indicating that not only the values are close to each other but also close to the vane yield stress. The close matching of vane yield stress with yield stresses estimated by extrapolation using either Herschel–Bulkley or Casson models observed for all samples in this work has been reported previously by only few investigators [19,55], while the majority of prior work shows much higher vane yield stress values than values obtained by extrapolation.

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Large differences between static and dynamic yield stresses of various yield-pseudoplastic fluids, including bentonite dispersions, have been reported by previous investigators. Saak et al. [56] referenced prior work on yield stress of oil well cement slurries, with Haimoni and Hannant [57] reporting static yield stresses twice as high as the dynamic ones determined with concentric cylinder viscometer and with Banfill and Kiching [58] reporting large differences between static yield stresses and dynamic yield stresses obtained with parallel plate viscometer, with all attributing these differences to wall slip in the concentric cylinder or parallel plate instruments. Ulherr et al. [59] reported static and dynamic yield stresses of Carbopol solutions and of TiO2 suspensions. Static yield stresses were determined by vane and slotted plate technique, while dynamic yield stresses were determined by rotating viscometer and extrapolation using fourth order polynomial, Herschel–Bulkley or Casson models. Static and dynamic yield stresses of Carbopol solutions were similar, but static yield stresses of the TiO2 suspensions were three times as much as the yield stresses determined by extrapolation. They have attributed the huge differences to wall slip and to prior shear history, expected to play significant role for TiO2 suspensions but not for Carbopol gels. Nguyen et al. [60] have compared yield stresses of 50% and 60% TiO2 dispersions employing static methods (vane and slotted plate) and from extrapolation of Herschel–Bulkley curves and found that vane yield stresses were 1.5 and 2.5 times more than the yield stresses obtained by extrapolation. The values reported were averages among different laboratories but it should be noted that preparation and preshearing procedures were not similar among the different laboratories. If the estimated yield stresses by extrapolation are considerably smaller than the vane yield stresses, then the fluid may ‘slip’. Most of the dynamic yield stress values obtained in this work using either the Herschel–Bulkley model or the Casson model were close to the static yield stresses measured by the vane and no huge differences were observed of the magnitude reported by previous investigators. Thus no slipping should be taking place within the concentric cylinder viscometer, and this should be attributed to the extensive and strong preshearing prior to measurement [27]. Zhu et al. [55] reported close matching of vane yield stresses with extrapolated yield stresses using Herschel–Bulkley model on parallel plate rheometric data, for 40–70% TiO2 water suspensions, with τ HB , though, being constantly slightly higher than τvn , however, they used sand paper to cover the plates avoiding thus wall slip while they applied prior shearing by stirring the samples with spatula before testing. Power and Zamora [19] found that τ HB was much closer to the vane yield stress for a series of drilling fluids, with yield stress values between 0 Pa and 15 Pa, particularly when compared to the high shear rate Bingham yield stress, τ HSRB or to the Bingham yield stress, τ B . But there was greater variation of τ HB with τvn compared to the variation reported in this work, giving a much higher degree of deviation of DD = 18. The similarities of the results of the work of Power and Zamora [19] with the results of this work are attributed to prior intensive

Fig. 5. Variation of vane yield stress with pH for two different bentonites (Z = Zenith, W = Wyoming) at the two stated concentrations.

shearing of the samples because they have also followed API specifications. 4.2. Effect of pH on the yield stress The variation of the vane yield stress with pH, grouped for the different bentonite types and concentrations is shown in Fig. 5. There are various trends for the different conditions. For the 5% concentration of bentonites and pH range of 7.0–10.5, very little variation with pH is observed of the vane yield stress. Zenith bentonite, however, at pH of 11.1, gives a vane yield stress which is twice as great compared to yield stresses at the other pH values. This behavior is different from the results of 6.42% Wyoming bentonite concentration, which showed a maximum in the vane yield stress at pH range of 6.5–8.0, similar to results from prior work [27], where a maximum of τ HB was obtained at pH of 8.8. Zenith bentonite results, although with only two points, show that vane yield stress drastically increased when pH was strongly alkaline. 4.3. Effect of measuring time and rest period In Fig. 6 the results of the tests to estimate the effect of measuring time at each rotational speed are shown, where data points (τ–t) are plotted for three rotational speeds (600 rpm, 200 rpm and 100 rpm) for the three measuring times of 60 s, 120 s and 180 s. It can be seen, Fig. 6a, that for the highest shear rate employed (600 rpm), shear stress values taken with 120 s rotational time almost coincide with the values taken with 180 s rotational time, while the values taken with 60 s rotational time fluctuate between the values of the 120 s time and values slightly lower, with differences less than 10%. For the lower rotational speeds (300 rpm and 200 rpm, Fig. 6b and c, respectively) the measured values essentially coincide for all three measuring times. Similar results were observed for the lowest rotational speeds, not shown here for brevity. Hence, this time series analysis shows that there exist no significant differences for rotation times of 60 s, 120 s and 180 s. Computation of τ HB and τ CA for each of the three samples gave the values depicted in Fig. 7. Interestingly enough, the yield stress estimated by

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Fig. 7. Variation of estimated yield stress by Herschel–Bulkley model and by Casson model with length of measurement time at each rotating speed. Zenith bentonite dispersion at 6.42% (w/w).

with a fairly high regression coefficient of 0.902. Interesting to note that the last measurement, taken after preshearing the sample, shows that preshearing after 300 min for 5 min did not affect the fluid structure since the point falls well within the curve that describes prior rest history of the dispersion. 4.4. Prediction of yield stress Models have been proposed to relate microstructure of charged particle dispersions to macroscopic parameters such as rheology and yield stress through phenomenological modeling using DLVO theory [61,62]. Proposed models [63–66] can be cast in a form of τy ∝ φ2 (A − Bζ 2 )

Fig. 6. Comparison of variation of shear stress with time allotted for measurement, for 180 s, 120 s, and 60 s for the three rotational speeds: (a) 600 rpm, (b) 300 rpm and (c) 200 rpm. Zenith bentonite, 6.42% (w/w).

(10)

where the term (A) represents van der Waals attractive forces, the term Bζ 2 represents electrostatic forces, ζ is the zeta potential and φ is solid volumetric concentration. Eq. (10) predicts a maximum yield stress at the isoelectric point (iep), i.e. the value of pH at which zeta potential is zero (ζ = 0), which is proportional to the square of volumetric concentration. The proposed equation described well experimental results as reported by Leong et al. [67] with ZrO2 particles at fairly high concentrations, with

the Casson model is almost the same with that estimated from the Herschel–Bulkley model, if data from 180 s measurement time are used, while it is larger at 120 s measurement time (ratio of τ CA /τ HB = 1.16) and much larger at 60 s (τ CA /τ HB = 1.25). Unfortunately, no vane yield stress measurements were made for these samples. The results of the time evolution tests of the vane yield stress while the dispersion is at rest are shown in Fig. 8. It is seen that the fluid builds continuously structure over time and the yield stress data follow a power type of growth, given by τvn = 22.7 + 0.076(t)0.893

(9)

Fig. 8. Evolution of vane yield stress with time. Zenith bentonite at 6.42% (w/w).

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yield stress measured by the vane technique, by Scales et al. [64] with alumina slurries and yield stress measured by vane, while Prestige [68] found Eq. (10) could describe Bingham yield stress data of galena suspensions at concentrations less than 10% but only for pH values greater than 6.0, while an additional parameter had to be included for the yield stresses taken at pH less than 6.0. Sakairi et al. [69] proposed also a model for bentonite dispersions in electrolyte solutions but the predictions have been recently questioned [70] because it grossly under predicted measured values and furthermore, it was found to work only in a very narrow range. Zeta potential of bentonite dispersions is negative over the entire pH range [39,71–73] with the iep, considered the pH of the medium at which the edges of bentonite platelets have no charge, covering a range of values of pH between 5.0 and 8.0 [27]. Eq. (9) could thus be applicable if there was a minimum of zeta potential over this pH range, but Missana and Adellm [71] have reported no variation of ζ with pH for sodium–bentonite dispersions, while Niriella and Carnahan [74] have reported a local maximum. In the absence of any predictive models, it would be interesting to compare maximum measured vane yield stress values, within the reported (iep) range of bentonite dispersions, with values predicted by the equation proposed by Zhou et al. [75] for concentrated alumina slurries, given by τy,max =

36AH φ2 24π2 h20 d

(11)

where AH is the Hamaker constant, φ is the solid volumetric function, h0 is the inter-particle distance and d is the diameter of the particle. To use Eq. (11), the Hamaker constant must be known but it has been reported as an extremely difficult to measure quantity and may be completely different from values predicted from theory [71,76]. For bentonite dispersions, a value of 2.25 × 10−20 J has been predicted [77], Missana and Adellm [71] had to use a higher value of 6.0 × 10−20 J and reported even higher values used by other investigators of 1 × 10−19 J, in order to match theoretical coagulation predictions with their experimental data. Missana and Adellm [71] attributed the need to use higher AH values in order to explain experimental results, to additional attractive forces, other than the van der Waals forces not considered in classical DLVO theory, such as hydration and swelling energies. Maximum yield stress values could be predicted using Eq. (11), taking values for particle diameter, d = 20 nm, for inter-particle distance, h0 = 1.0 nm and different values for the Hamaker constant, for the two concentrations tested. The computed values have been compared with the maximum measured values from both bentonite dispersions within the pH range of the (iep), with the results shown in Fig. 9. Measured maximum yield stresses for Zenith bentonite closely matched predicted maximum values, for the value of the Hamaker constant of AH = 0.04 × 10−20 J, while a good match was also evident for Wyoming bentonite dispersions but for higher Hamaker constant value, AH = 0.15 × 10−20 J. A range, of course, of degrees of matching could have been obtained taking different combina-

Fig. 9. Measured and predicted maximum yield stress values of Wyoming and Zenith bentonite dispersions for two different concentrations at various assumed values of Hamaker constant.

tions of inter-particle distances, particle diameters and Hamaker constants. Interestingly enough, the values of AH that gave very similar maximum yield stresses to the measured ones, are one to two orders of magnitude lower than the values utilized by previous investigators reported above. Zhou et al. [75] suggested that their proposed model should be accurate enough to a first order of magnitude. Of course, a difference of one or two orders of magnitude can result in complete different conclusions in stability predictions, with several authors using different values without stating the reasons for doing so [71]. These results then indicate that Eq. (11) could be a very good starting point for estimation of yield stress of bentonite dispersions provided that more accurate values of Hamaker constants, of particle diameter and of particle thickness become available. 5. Conclusions Aqueous dispersions of two different bentonites, used in oil well drilling, at 5 wt.% and 6.42 wt.% concentrations and over a range of pH values, exhibit non-linear rheograms that can be equally well described by Herschel–Bulkley and Casson models. The yield stress predicted by both models and the yield stress measured by the vane are very close to each other. This close match is attributed to the good preparation and preshearing procedures which are similar to the ones that the dispersions may experience under various flow conditions. pH affects yield stress of these dispersions and the variations observed is the same for the yield stresses measured by the three techniques. Different effects have been observed for the two different bentonites and for the two concentrations. Measuring time of shear stress at each rotational speed should be at a minimum of 60 s as no variation has been observed for the longer measurement times of 120 s and 180 s for all rotational speeds. The vane yield stress increases with rest time after intensive preshearing indicating that the fluid continuously builds structure. The evolution could be well described with a power

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V.C. Kelessidis, R. Maglione / Colloids and Surfaces A: Physicochem. Eng. Aspects 318 (2008) 217–226

law over a period of 5 h. Intensive preshearing after 5 h was not sufficient to destroy the structure of the dispersion. A theoretical model proposed for the maximum yield stress of charged particle suspensions at the isoelectric point could be a good starting point to predict yield stress of bentonite dispersions because it gives a close match with the maximum measured yield stress of the dispersions within the range of the iep, after using values of Hamaker constants which were one or two orders of magnitude lower than the ones reported previously.

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Acknowledgement

[25]

The authors would like to acknowledge Mrs. C. Tsamantaki for the data collection and analysis.

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