Rheological characterisation of shear thickening TiO2 suspensions in low molecular polymer solution

PROGRESS IN ORGANIC COATINGS ELSEVIER

Progress in Organic Coatings 30 (19971 67-78

Rheological characterisation of shear thickening Ti02 suspensions in low molecular polymer solution Andreja Zupani-it”, hDeptrrtmrnt

“Fflcult~ of Chemistry und Chrmicol Technolog!, of Chemical, En~~ironmental cmd Raw Materials

Roman0 Lapasinb, Miha Turnera University Engineering.

qf Ljubljuntr. L’niversit~

A.fker?ew
5, SI-1001, Ljubljnna. Siovenia Piuzde Europtr 1. I 34127,

Trieste,

Italy

Received 16 July 1996: accepted 29 July 1996

Abstract

The rheological behavior of titanium dioxide (rutile) suspensions has been investigated under steady shear conditions in order to study the influence of disperse phase concentration. The solid particles were dispersed in a Newtonian vehicle, made of epoxy and acrylic resins dissolved in a solvent mixture at different solid volume fractions a, ranging from 0.16 to 0.40 (the values do not take account of resin components). The suspensions examined exhibit a Newtonian plateau at very low shear stresses and shear thinning behavior in low and moderately high shear stress regions. At a solids volume fraction @ higher than 0.20, the onset of shear thickening behavior is observed in the highest shear stress region. For these suspensions, time-dependent effects can be assumed negligible even at the highest particle concentration examined and the shear-dependent behavior is almost reversible as is usually found for stabilised suspensions. A satisfactory fitting of both shear-thinning and shear-thickening behavior is given for the suspensions examined by modified versions of the KriegerDougherty and Cross equations. Accordingly. the critical shear stresses and shear rates for the relevant conditions are defined. A thorough analysis of the whole data set clearly shows that the modified Eilers equation provides the best correlation for the concentration dependence of relative viscosity, on condition that shear stress is used as a proper rcferencc quantity instead of shear rate. 0 1997 Elsevier Science S.A. Keywords:

Suspension rheology; Shear thickening;

Critical shear rate or shear stress: Maximum

1. Introduction

Colloidal suspensionsare widely used in a number of industrial applications and are usually formulated to display well-defined flow behavior. In all stagesof paint technology, for example, the properties of pigment suspensions and ready mixed paints depend significantly on flow behavior [ 11.This applies equally to paint manufacture, paint application and the achievement of specific coating effects. Mill basepreparation is one of the most important processesof paint manufacture. It must ensurean appropriatebreakdown of pigment particle agglomeratesand aggregatesand complete wetting of the pigment surfaceby the suspendingmedium. According to Patton [2], mill bases,intended for high speeddissolver. are usually formulated to provide borderline ‘dilatancy’ to obtain maximum effectiveness of particle deagglomeration.For efficient formulation, the basic rules which govern the rheological behavior of pigment suspensions should, therefore. be known.

0300-9440/97/S I7.00 % 1997 Elscvier Scicncc S.A. 411 rights rcscrwd PI/ SO300-‘)J-10~96)0067~-4

packing fraction

The rheology of concentrated suspensionshas been the subject of continuing researchand in the last 20 years more attention hasbeenpaid to the description and understanding of shearthickening in concentrated suspensions.The phenomenonof shearthickening of suspensionsin general has no useful applications in industrial production. In several mechanical operations, the increasein viscosity with shear motion causes an increasein power consumption and overloading can occur. However, there are alsosomeadvantages in industrial situations (mill bases). 1.1. Flow

behavior

of concentrated

suspensions

Above a certain particle concentration, suspensionsare generally non-Newtonian fluids, whose flow properties are influenced by a large number of factors [3]. The increasein particle concentrationsoften causesa very complex rheological behavior

of suspensions

[4]. Rheological

properties

and especially their concentration dependencecan provide

68

A. ZupanCiE

et al. / Progrrss

in Organic

detailed information about the microstructure of the suspension. Concentrated suspensions of non-aggregated solid particles usually have high, but finite viscosity at very low shear rates. When the shear rate increases, the viscosity begins to decrease (shear thinning region) and at higher shear rates, it reaches an apparently constant value. With a further increase in shear rate, suspension viscosity begins to increase. After this shear thickening region, the viscosity can either become so great that fracture of the sample occurs, the viscosity can level out to a new plateau value or even decrease. If these suspensions are measured in the appropriate high shear rate range, they always show shear thickening behavior. It is important to note that for these suspensions, shear thickening behavior is almost immediately reversible, which means that as soon as the shear rate (or stress) decreases, the viscosity also immediately decreases [5]. Shear thickening generally takes place when suspensions are deflocculated, that is in the absence of overall attraction between particles. The stability of suspensions against flocculation and the degree of flocculation are governed by the interactions between particles in the suspension. In concentrated stabilized suspensions, the average particle separation is very small and is strongly affected by the relative positions of particles. When the shear rate increases above a certain critical point, the viscosity of the suspension begins to increase due to the transition from a two-dimensional layered arrangement of particles to a random three-dimensional form [5]. The original two-dimensional particle arrangement is caused by the flow itself rearranging the particles into closely placed sheets flowing over each other. This layered arrangement gives the lowest possible viscosity for a suspension. At higher shear rates, the hydrodynamic action that brought about this layering gradually disrupts. The existence of particle ordering in shear flows has been demonstrated using light scattering [6] and by small angle neutron scattering [7-91. The actual nature of shear thickening behavior depends on the characteristics of the disperse phase, the nature of the dispersing medium, the type of the imposed flow field and particle interactions. Boersma et al. [lo] suggested scaling analysis, which indicated that the critical shear rate for the onset of shear thickening of stabilized suspensions is determined by solid volume fraction particle size, shape and size distribution, viscosity of the dispersing medium, and magnitude of repulsive stabilising forces. They assumed that interparticle forces (electrostatic or steric) keep the particles in a layered structure at relatively low shear rates. At a certain higher shear rate, the shear forces that push the particles together become larger in magnitude than interparticle forces. As a result, particles are moved from their equilibrium positions, leading to transition from a layered to a disordered structure. The shear thickening behavior of stabilized concentrated suspensions therefore begins when hydrodynamic forces become sufficient to overcome the colloidal repulsive forces

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which promote ordered flow. The critical shear rate of the onset of shear thickening is very high for moderately concentrated suspensions. It decreases with increasing particle concentration and approaches zero when the concentration is close to the maximum packing fraction, 9,. A reduction in particle size would move shear thickening to much higher shear rates. Barnes [5] concluded from all the data he collected on shear thickening stabilized dispersions that the critical shear rate has an inverse quadratic dependence on particle size. Jansma and Qutubuddin [l l] showed that broader particle size distribution of stabilized dispersions significantly reduces the viscosity, leads to higher shear rates where shear thickening sets in, and increase in viscosity may not be so extensive. Also in the case of particle size, size distribution and shape, the critical parameter is ap,, which suggests that anything that changes +,,, modifies the form of shear thickening (including the influence of shape and particle interactions). The severity of shear thickening depends on the particle concentration in proportion to the maximum packing fraction. 1.2. Models for describing shear dependent behavior The suspension microstructure depends on the balance between the rate of formation of links between particles via Brownian motion and its rate of breakdown by both shear stress effects and Brownian motion. The shear viscosity at a certain particle concentration can be related to the applied shear stress. T, using the equation proposed by Krieger and Dougherty for describing pseudoplastic behavior [12]:

where no and n- are zero- and infinite-shear viscosity, respectively, and rc is the characteristic shear stress at which n = (qO + n-)/2. Many experimental works have been done on the rheology of well-defined colloidal suspensions of spherical particles. Often systematic deviations between the experimental data and Eq. (1) have been observed, especially at higher particle concentrations [ 131. Better agreement with experimental data can be obtained by using the modified equation: (2) It has been pointed out by Russel et al. [14] that Eq. (2) correlates well the viscosity of hard sphere suspensions at intermediate stresses, with 1 I n I 2. To describe pseudoplastic behavior of suspensions in the form of dependence of viscosity on shear rate, a suitable four-parameter equation was proposed by Cross [ 151:

Parameter T, can be considered as a characteristic

shear

A. ZupanEii

et al. /Progress

in Organtc

rate which corresponds to a balance between destruction and formation of interparticle links. The value of exponent n was originally set equal to 2/3. A similar form of the model was derived by Liu et al. [16] and was successfully applied to describe shear- and time-dependent behavior of different structured fluids. The authors used a simple kinetic model in which structural disruption and reformation processes were approximated by a first order kinetics. De Kee et al. [ 171 later modified the model, making it suitable to describe also shear thickening behavior and rheopexy. They approximated the various functions which determine the structural kinetics by power law relationships, making in this way a model with a limited range of applicabilities. Many applications of Eqs. (2) and (3) have been made to evaluate the pseudoplastic behavior of various suspensions, stabilized and structured, in a wide shear rate interval for a large particle concentration range. For highly concentrated suspensions, shear thickening behavior at higher shear rates can be expected, the phenomenon being more pronounced in stabilized suspensions. However, the critical shear condition of the onset of shear thickening and the severity of the phenomenon cannot be easily predicted and this is probably the reason why much less effort has been made to derive a model suitable for describing such a complex shear-dependent behavior. Some relations for describing shear-thinning/shear-thickening behavior of specific systems can be found in the literature. Hanks [ 181 studied the flow properties of coal slurries and applied the following equation to evaluate shear-dependent behavior: 7=(vo-

bm’ w-m))j/+ -pJ

b(m’-m) m j;;l’i/(1-,II,+b~“-‘“-“” (4)

Parameter To indicates the critical shear rate of the onset of shear thickening and 90 the minimum viscosity at the point of qo. Eq. (4) allows correlation of the critical shear rate. By applying this 6-parameter (polynomial) relation, it is difficult to determine correct parameter values, since they are strongly correlated. A model for describing shear dependent behavior of well-defined stabilized acrylic dispersions in a wide range of particle concentrations was proposed by Wagstaff and Chaffey [ 191. At lower particle concentrations, the authors applied the Cross model and used it also to determine the parameters introduced in their empirical model for describing the flow behavior of highly concentrated suspensions where shear thickening was observed at high shear rates. Another modification of the Cross model was proposed by Roper III and Attal [20] for describing flow behavior of shear thickening paper coatings. The authors combined the Cross model (low shear rates) with the Gillespie model (high shear rates) and obtained the empirical equation with 6 parameters which allows correlation of the experimental data in a wide range of shear rates. The attention of their modelling was focused on finding the mathematical form to describe the end of the

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30 (IW7)

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69

shear thickening regime and the condition when the viscosity of such suspensions starts to decrease again at very high shear rates. 1.3. Concentration

dependence of relative viscosity

For moderately and highly concentrated suspensions, the increase in particle concentration is reflected in asymptotic increase of the relative viscosity (nr = n/n,J. When particle concentration approaches a certain maximum packing fraction a,, the relative viscosity approaches a very high (infinite) value. This corresponds to the state of dispersion in which the amount of the dispersing medium is just sufficient to fill the existing voids between the dispersed particles in their state of maximum packing allowed by the boundary conditions, i.e. spatial packing geometry and degree of dispersion. Under such conditions the particle interlocking prevents the suspension from flowing. To describe the dependence of relative viscosity on the solid volume fraction of concentrated suspensions, asymptotic equations should be taken into consideration. To date, no rigorous hydrodynamic analysis has been developed for such concentrated suspensions (real systems of practical interest), because of the complexity of particle-particle and particles-suspending medium interactions. Often semi-empirical and empirical equations are used to describe the relation nr(@), Many equations can be found in the literature which assume the existence of maximum limit concentration through the parameter +,,,. Krieger and Dougherty [12] were the first to define the volume fraction dependence of ndl and nr-. In both limits the data can be correlated by the following equation: qr = (1 - @/$J’@m

(5)

Parameters a, and [n] represent the maximum packing fraction and the intrinsic viscosity, respectively. Q, depends on the type of particle packing and ranges between 0.53 and 0.74 for non-interactive monodisperse spherical particles and it is very sensitive to particle size distribution and shape. For spherical particles, the values of [q] are often between 2.5 and 5 and therefore, [n] loses any theoretical significance [21]. The value of 2.5 was developed theoretically by Einstein. The upper bound is found experimentally for real systems and depends on the degree of particle aggregation and the size of electroviscous effects. For real systems, [v] represents the effective shape of particles imposed by their surface condition and is usually higher than 2.5 due to particle aggregation. Thus, for uncharged spherical particles there is no shear dependence in [VI. Barnes et al. [22] used Eq. (5) to evaluate the concentration dependenceof relative viscosity for a numberof experimental investigations on suspensionsof asymmetric particles from the literature. They showedthat increasein particle asymmetry in the suspensionleads to higher value of [q] and to lower values of a,,,, but the product of the two

70

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terms changes a little. The exponent ([TJ]~,,,) in Eq. (5) is often found to be very close to 2. This value often gives the best fitting results for disordered hard sphere suspensions [22]. If the exponent is set equal to 2, Eq. (5) can be reduced to a relation with only one parameter, @,,,. This empirical simplification of Eq. (5) is known as the Maron and Pierce equation and many successful applications of this simple empirical relation can be found in the literature (e.g. [23251). Tsutsumi et al. [26] examined rheological properties of flocculated coal-oil suspensions. They found similarities between their floes-aggregate structural relation and the Robinson’s equation [27]:

where [n], according to Tsutsumi et al.. represents morphological properties of aggregates, thus accounting for the geometrical irregularity. Another very popular empirical equation, proposed by Eilers [28], has been found very reliable for Newtonian viscosity of dispersed spheres. Wildemuth and Williams [29] modified the equation to be applicable also for describing shear-dependent effects of concentrated suspensions:

The application of Eqs. (5)-(7) to the suspensions with interacting particles implies that parameters +‘m and [q] are associated with the shape. size and size distribution of particles, floes and aggregates present at certain circumstances in the suspension. The parameters, therefore, depend not only on the structural conditions of suspensions, but also on shear rate (or shear stress), since relative viscosity is also influenced by the applied shear action in the same way as suspension viscosity. Some relations which connect the particle concentration dependencies and the influence of shear action of shear thinning suspensions can be found in the literature [2932]. The common basic concept of all these works is that the maximum packing fraction depends on the shear stress or the shear rate. In the low shear stress limit, lower values of (P, are a consequence of a random initial orientation of particles and possible aggregated microstructure. As shear stress increases, aggregates can break down into smaller units. The size, shape and size distribution of the aggregates change, leading to higher a,,, values. At a higher shear stress limit, the flow permits orientation of particles and particle migration (in polydisperse systems) to fit particles into voids optimally. For shear-thinning suspensions, a,,, exhibits two limiting values: +‘main the low shear stress limit and a Ill-> in the high shear stress limit. The authors proposed equations in which the values of Q, continuously increase between the two limits. In this manner, the shear dependent behavior of +,,, excludes the transition from shear thinning to shear thickening behavior. Above a critical shear rate,

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30 (1997)

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when shear thickening begins, the data of relative viscosity may be also fitted to one of the many available asymptotic expressions discussed. However, important features in shear thinning and thickening regimes would still be unaccounted for Ill]. Parameters am and [v], evaluated from the equations for describing relation q,(a), are often calculated only for two limiting situations of shear-dependent behavior (70 and n _ ). Many experimental investigations, especially for welldefined stabilized suspensions, have been made in this way [13,33,34]. The application of ~~(9) and qIcc(@) for @-correlation purposes avoids the question of whether the full non-Newtonian curve should be correlated as a function of r or of r, or whether (P, is physically dependent on shear rate or shear stress. In order to isolate the influence of shear effects, relative viscosity can be plotted against the reduced solid volume fraction (a/+,) instead of + [32,34]. Wildemuth and Williams [30] collected several sets of non-Newtonian data for suspensions of non-spherical particles from various investigations found in the literature. The analysis, performed by extrapolating rr (+) at constant + to obtain a,,,(i) showed deviations in relative viscosity dependence on the reduced solid volume fraction, +/9m(i/). The influence of the selected reference quantity (shear rate or shear stress) on dependence ~,((a) was confirmed with their investigations on coal suspensions in various suspending media. They showed that the determination of parameters a,,, and [n] according to a suitable model for describing ?I,(%), depends on whether selected reference quantity is shear stress or shear rate. Also, the variations of parameters with shear rate differ significantly from those with shear stress. To check the influence of shear rate and shear stress on the shape of the nl- - (P/+, curve, the data of relative viscosity were plotted as a function of +/9,,,(+) and @#B’,(r). In the first case a single curve of nr - a/@,,,(+) was observed for each set of T, whereas in the second case the uniform curve of ~7~- 9/@,(r) of the same 7, data was found for a wide range of shear stresses. They suggested that shear stress is the proper reference quantity when the q,.(a) dependence of full non-Newtonian behavior of suspensions with non-spherical particles is studied. In this work, the rheological behavior of pigment (TiOl) suspensions in a polymer solution, intended for high solid coatings, is studied. The characterization of such a complex industrial suspension is usually not interesting from a theoretical point of view, but very important from the viewpoint of practical interest in paint manufacture. Rheological investigation of such industrial systems may contribute to the determination of relations between shear-dependent properties, structural conditions of disperse phase and concentration dependence of relative viscosity. The objectives of this work are: (i) to check the possibilities of characterizing the rheological behavior of stabilized suspensions by different measuring procedures; (ii) to define a suitable rheological model for describing the shear-dependent beha-

A. Zupani-i?

et d. /Progress

in Orgunk

vior of examined suspensions; (iii) to analyze the dependence of relative viscosity on particle concentration; and (iv) to correlate the rheological behavior with the structural condition of the disperse phase. All these items represent a fundamental basis for establishing correct procedures to be used in industrial labs for product control and for defining suitable criteria to be followed in the formulation and optimization of the products.

2. Experimental 2. I. Formulation

of suspensions

The suspensions examined are representative of the formulation used in the paint industry for the preparation of high solid baking enamels. The raw materials as well as the preparation procedure were selected by taking into account a final commercial product. As a disperse phase, suspensions contained titanium dioxide (rutile), KRONOS 2310. The pigment, with an average primary particle size of 200 nm and density of 4.1 g/cm’, was prepared by the chloride process and the particle surface was treated with aluminum, silicon and zirconium compounds. The suspensions were prepared at various solid volume fractions, ranging from 0.16 to 0.40. The disperse medium was the same for all suspensions. It was made of epoxy and acrylic resins dissolved in a solvent mixture and exhibited Newtonian behavior (qm = 0.338 Pa s). The formulation of suspension at the highest solid volume fraction is shown in Table 1. The preparation procedure was defined, starting from a theoretical background for paint manufacturing, and taking into account the specific characteristics of the components. Acrylic resin, solvents and the solid phase were first homogenized inside a high speed dissolver by gently mixing, and then the stirrer speed was set to 2000-2500 rev./min for 15 min. In the next step, epoxy resin was added and mixed for 5 min at 1000 rev./min. In the last step, a pearl mill (Eigher) was used for 12 min to ensure an adequate dispersion quality. Suspensions at lower concentrations were prepared from the concentrated mother batch by diluting with a dispersing medium (resins and solvents) and re-mixing in the high speed dissolver at 1000 rev./min for 5 min. The experiments were performed 2 weeks after the preparation. 2.2. Rheometer and experimentLr1 procedures

Coatings

71

low to moderate viscosities. The usual method of dealing with this problem is to lower the speed change sufficiently to eliminate errors caused by inertia [35]. In linear ramp programming, the inertial discrepancy is the most pronounced at low shear rates. It may be advantageous to utilize a logarithmic ramp where the inertial acceleration is greatly reduced and ensures sufficient information in the low shear stress range. The experiments were performed in a logarithmic increase and decrease of shear stress. Times of 5 min for stress acceleration, as well as for stress deceleration were long enough to obtain a regular flow curve even for a low viscosity suspending medium. To set a uniform starting point for experimental tests connected to sample loading and, hence, to obtain good repeatability of experimental data under ‘unsteady’ shear conditions, preshear and equilibration were applied and afterwards a logarithmic stress ramp was carried out. A preshear of 50 Pa for 30 s was considered sufficient to approach constant viscosity values. After the first 30 s of equilibration, the repeatability of experimental data was beyond expectation. The flow behavior of the examined suspensions in the low shear rate range was examined using creep experiments: a given shear stress, ranging from 0.03 to 5 Pa, was applied to the sample, and deformation (strain) was measured as a function of time. A shear rate value can be determined from the slope of the strain-time curve when linear behavior is attained, after a certain measuring time. 2.3. Visualization

of the microstructure

of suspensions

In order to obtain a complete picture of the microstructure of suspensions, scanning electron microscopy (JEOL JSM T220) was used. The samples have to be completely dry and carefully prepared before the microscopy is performed. To keep the particle structure intact, i.e. the same as in the suspension at rest, the drying should be very slow. The specific characteristics of the disperse medium, the combination of resins and solvents used for paint hardening at elevated temperatures, require drying at high temperatures. The best visualization was obtained for highly concentrated suspensions. The suspension at + = 0.4 was spread on the support in a thin film. The sample was allowed to dry very slowly. first at room temperature for 2 h, then at 60°C for Table

I

Composition

Rheological investigations were carried out at 20 f 0.1 “C by using a controlled stress rheometer Cart?-Med CSLlOO, equipped with cone and plate geometry of 2” cone and of 6 cm diameter. A solvent trap was necessary to reduce sample drying out. Whenever the imposed torque changes with time as in the ramp function of programming of the torque, the inertial effects due to the controlled stress rheometer cannot be neglected. Errors are especially serious with samples of

30 (19971 67-78

of suspension

Ingredient Carboxylated polyacrylic Epoxy resin Solvent Solvent Solvent Solvent Pigment

at the highest

solid volume

fraction

+ = 0.40

Company

Component

Vol.%

BASF

Luprenal

DOW Chemicals BASF Shell DOW Chemicals EXXON Chemicals KRONOS

DER 331 8.18 3.48 Butyl glycol 2.65 Butyl diglycol acetate Propyl glycol diacetate I .35 6.42 Solvesso 150 Titanium dioxide 23 10 40.0

LRXX41

37.9

resin

A. ZupanCi?

et al. /Progress

in Organic

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30 (1997)

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region, Such reversible shear-dependent behavior is often found for suspensions with the highest degree of dispersion

[51. 3.2. Models for describing shear dependent behavior

Fig. 1. Scanning electron micrograph volume fraction * = 0.40.

of suspension

at the highest

solid

1 h. The process was completed by drying at 180°C for 30 min. The drying process allows the complete solvent evaporation and hardening of resins. Hence, the micrograph in Fig. 1 demonstrates the conditions of maximum particle packing, available for the suspension at rest. The suspension forms a very closely packed structure of spherical particles where smaller particles fill the voids of closely spaced bigger ones (Fig. 1). The particle size distribution ranges from primary particles of 0.2 pm in diameter to aggregates of around 0.8 pm. Also aggregates have a quasi-spherical shape. Some clusters of aspect ratio of about 2, which are composed of two or more spherical particles, overlapped by dried resin, can also be visualized in the picture.

3. Results and discussion 3.1. Shear dependent behavior The influence of particle concentration on shear-dependent behavior of examined suspensions is demonstrated in Fig. 2. Time-dependent effects on the flow behavior are almost negligible, so that it can be assumed that the flow curves exhibit quasi-steady-state viscosity values. At very low shear stresses, viscosity-dependence was also measured with creep tests. In a broad stress region, the shear-dependence is examined by applying logarithmic stress ramp tests, As an example, a suspension at the highest solid volume fraction exhibited a Newtonian plateau in a very low shear stress region, shear thinning in an intermediate shear stress range and shear thickening at higher shear stresses. Evidently, the experimental results, evaluated with both experimental procedures, coincide in the same shear stress range. If the suspensions at the highest volume fraction exhibit a Newtonian plateau at very low shear stresses, this can also be expected for suspensions at lower particle concentrations. At a solid volume fraction higher than 0.2, shear thickening is observed in the highest shear stress

When the suspensions examined exhibit shear thickening, various processes have to be taken into account, which can, under certain circumstances, lead to another Newtonian plateau, or to a downturn of the flow curve; discontinuities of flow curves may also be expected [5]. Our experimental results cover the shear conditions up to the onset of shear thickening only. The investigated suspensions comprise particles of quasi-spherical shape. The actual particle size and particle size distribution cannot be determined. If the microstructure of the suspensions is not precisely defined, a rheological model cannot be developed solely on the basis of the theoretical background. Some relations proposed for describing shear thickening systems [ 18,191 were taken into account to correlate the experimental data but they do not provide an acceptable fitting quality. The combined Cross-Gillespie model, proposed by Roper III and Attal [20], was not suitable due to the absence of experimental data in the highest shear rate range. For describing the shear-dependent behavior of the suspensions investigated, the expression is derived by a modification of available rheological models describing pseudoplastic behavior in an empirical way. The shear viscosity is assumed to be the sum of two contributions (shear thinning and shear thickening). At low flow conditions, particle ordering and aggregate breakdown are the predominant mechanisms. and both are promoted by shear action and govern the shear-dependent behavior of stabilized and flocculated suspensions, respectively. The resulting viscosity decrease can be suitably described by the KriegerDougherty equation, Eq. (2), or the Cross equation, Eq. (3). depending on the independent variable (shear stress or

P

Fig. 2. Flow curves obtained with loparithmlc stress ramp experiments (0) for suspensions at various solid volume fractions a, ranging from 0.24 to 0.40 and comparison with experimental data evaluated by creep experiments (m) for suspension at + = 0.4.

73

q(thickening)

= Kf"

or q(thickening)

= C’v

(8)

(9)

(10)

0.14

"

0.0001

,,,I 0.001

"',,! 001

,I

.,""!

0.1 shear

" ,,,,~ 1

"""'1

10

"4 100

low

rate q (l/s)

Fig. 3. Shear rate dependences of suspensions at various solid volume fractions +, ranging from 0.16 to 0.40. Solid curves passing through experimental data illustrate the calculated values according to Eq. (IO).

shear rate) adopted. When the shear action is sufficiently strong, particle order-disorder transition may occur and becomes the prevailing mechanism which leads to a viscosity increase. In the absence of any established criteria for the shear thinning-shear thickening transition and because of the limited range of shear stresses examined where shear thickening was observed, we adopted a simple power law relation for the shear thickening contribution, valid for the whole shear range. For the correlation of shear-dependent behavior in the whole shear stress (or shear rate) range the following empirical expressions were obtained: q = q(thinning) + q(thickening)

The experimental data for the evaluation of the parameters of Eqs. (9) and (10) are taken from the flow curves when the shear stress in the experimental procedure logarithmically increases. As an objective function in the minimization procedure, the mean squared relative deviation (MSRD) is used. The viscosity dependence on shear rate for suspensions at various solid volume fractions ranging from 0.16 to 0.40 are shown in Fig. 3. The solid curves passing the experimental data represent the values evaluated by applying Eq. (10). Apparently, the equation fits the experimental data very well in the whole shearrate range examined, aswell asthe particle concentration range. Table 2 reports the resultsobtained from the fitting of the experimental data for suspensionsat various solid volume fractions. The comparison of MSRD values shows that both Eqs. (9) and (10) correlate well with the experimental data of all suspensionsexamined. The estimated values of q. and q- obtained from Eqs. (9) and (10) are nearly the same at a given solid concentration. For the suspensions which exhibit shear thickening at a higher shearrate than the range explored (Q > 0.2), q- is close to the minimum value attained by the experimental viscosity in the intermediate shearregion (central plateau region). Fig. 4 shows

Table 2 Parameter Q

values of Eqs. (9) and (IO)

for examined

VII

‘I-

T.

(Pa s)

(Pa s)

(Pal

(a) Eq. /9J (shetrr ~tw.u depen&nceJ 0.40 302.2 7.78 0.36 40.7 2.94 0.32 18.9 2.11 0.28 10.7 1.67 0.24 5.00 1.27 0.20 1.56 1.10 0.16 1.19 0.94

0

suspensions,

0.3 1 0.27 0.25 0.22 0.20 0.1’) 0.10

mean squared relative I,

K

1.71 I .33 1.27 1.16 1.01 1.00 1.00

0.033 0.033 0.033 0.033 0.033 -

T-

C

(Pas)

(Pa s)

(Pa”’

1.9 14.2 24.9 35.9 67.3 339 562

(MSRD)

and critical

shear conditions

,fl

MSRD

0.83 0.65 0.57 0.45 0.46 -

3.53 8.00 -1.30 2.50 3.61 4.15 2.25

1’

MSRD

1.41 0.85 0.6.5 0.50 0.46 -

4.28 3.67 3.05 3.54 3.77 2.84 1.31

x IO’

(Pa’-“’ a)

m

(b) Eq. (10~ (sheur rote depmdencr~ 0.40 311.3 7.20 0.36 39.9 2.94 0.32 18.5 2.04 0.28 10.7 1.62 0.24 5.02 1.24 0.20 1.50 1.07 0.16 1.12 0.92

deviattons

0.80 0.84 0.80 0.79 0.78 0.79 0.8 1

0.04 0.04 0.04 0.04 0.04 -

a)

~C,,f (Pal 22.8 21.1 17.9 17.3 14.3 -

x IO’

Ycr,, (S-9 4.6 6.6 9.3 11.3 12.6 -

critical shear stress (rccrit.uanks) and shear rate (jctit.nanks) values of the onset of shear thickening evaluated from Eq. (4). Even if the fitting quality was rather poor when Eq. (4) was applied, both parameters coincide with the values obtained by applying Eqs. (9) and (10). A slight but regular increase in the parameters 7, and T,,~ is noticed with increasing particle concentration (Fig. 5a), whereas the values of parameters qc and +oit decrease (Fig. 5b). 3.3. Concentration

014

0.18

0.22 solid

0.26

0.30

0.34

0.38

0.42

volume fraction, 0, (i)

Fig. 4. Parameters to and TJ- evaluated volume fraction +.

by Eqs. (9) and (10) versus solid

that both parameters, q0 and TV-, asymptotically increase with increase in particle concentration. The values of exponent n, found in the literature for the evaluation of sheardependent behavior with Eq. (2), are often close to unity and can deviate with increasing solid volume fraction in the range 1 I n I: 2, especially for concentrated suspensions. When Eq. (9) is used to fit the experimental data of suspensions examined, the values of exponent n also increases with increasing particle concentration in the same range. When the Cross model is applied to describe the pseudoplastic behavior of various suspensions, the values of exponent n are usually lower than 1. and are close to 2/3, which was originally proposed by Cross [ 151. The values of exponent R from Eq. (10) are also less than 1 and do not change much with variation in particle concentration (n = 0.8). The shear thickening behavior at higher shear stresses (or shear rates) is described by the power law terms in Eqs. (9) and (10). A preliminary fitting showed that the parameters K and C for the examined suspensions are not strongly influenced by particle concentration. In order to avoid the problem deriving from the correlation between these parameters and the relevant exponents (m and p. respectively), the parameters in the final curve fitting are fixed to a constant value (Table 2). Hence, m and p values provide a direct measure of the severity for shear thickening contribution in the examined shear stress range. Both parameters regularly increase with increasing G. All suspensions examined exhibited shear thickening behavior at relatively low solid volume fractions. A slight increase in viscosity in the higher shear rate range was first observed at @ = 0.24. The r,,, and qcn, values reported in Table 2 were calculated from Eqs. (9) and (10) and correspond to the critical shear conditions for the onset of shear thickening. Their dependence on solid concentration @ is in accordance with data reported in the literature [5]. Fig. 5 shows the influence of the solid volume fraction on the critical shear thinning and shear thickening conditions. The values T,,,~and yo,, were compared with the correlated

dependence

The analysis of the concentration dependence of suspensions starts with the application of the Krieger-Dougherty equation, Eq. (5). Relative viscosity is calculated from the ratio of the suspension viscosity and the viscosity of the suspension medium (7, = 0.338 Pa s). In the first step, the influence of particle concentration on relative viscosity was analyzed by examining data subsets at a constant shear rate. The variation of the model parameters @‘mand [r] with shear rate shows an increase only in the very low shear rate range and displays a maximum around 0.5 s-’ (Fig. 6). The values of parameter a,,, range between 0.44 and 0.79, which is normally found for different packing geometries of spherical particles. For shear thinning suspensions a,,, should increase with increasing shear rate (or shear stress) [30,31.36]. The shear thickening behavior begins at shear rates higher than 0.5 s-’ (see Table 2), so it cannot be a correct explanation for @,,,decrease above 0.5 s-‘. The influ100 (a) Glit.Hmk* f . .._...

*- . . .._._

L

. ..j

. . . . .

l

‘iii

0 c

10

Gfit -o-eq.

I E03

-eq --*-

1

9 9 eq.4

---

r.

0

0 0.1 0.14

~,~~:"":~"'I"~'~""~"":"' 0.18 0.22 solid

0.26 volume

0.3 fraCiiOn

0.34

0.36

1 0.42

0 (0

o.ool~,“‘:“‘,:,,“I”“I”‘,I”“I”,” 0.14

0.18

0.22 solid

0.26 volume

0.3 fraction

0.34 *

0.38

0.42

(0

Fig. 5. Particle concentration dependencies of critical shear conditions expressed by parameters (a) 7,. r,,,, from Eq. (9). compared with T,,,, HAnl, from Eq. (4) and (b) yL. y,,,, from Eq. (IO), compared with qCr,, H~,,L\ from Eq. (4).

a08 f

(a)

0.7 4

0.1 shear

1 rate -j (l/s)

10

100

6.0

00

(b)

F 0.01

’

“““1

I

“l’s”l

0.1 shear

’ “““‘1 1 rate, $ (l/s)

’ 10

“‘“‘q 100

Fig. 6. Shear rate dependence of parameter5 from the Krwger-Dougherty equation (Eq. (5)). (a) dependence of maximum packing fraction +,,,. (h) dependence of intrinsic viscosity [TJ] and product of [?].a,,,.

ence of shear rate on parameter [r] and on the product [q] x +,,, is shown in Fig. 6b. When shear rate increases. the variation of both quantities also displays a maximum and unexpectedly strong shear rate dependence. From the variation of [r] values between 3.6 and 6.1, which is. however, in accordance with existing literature data, it may be concluded that the shape of particles and aggregates strongly depends on shear rate. This hypothesis is somewhat uncertain. since the micrograph in Fig. I shows that the particles and aggregates have quasi-spherical shape at rest conditions. According to Barnes et al. 151. the values of the product [n] x +,, should be more or less independent of shear rate (or shear stress), even if the influence of shear is reflected in the variation of both parameters. The variation of the product [q] x @,,,with shear rate seems to be higher than expected and also the values of the product are much higher than 2, which is often found in real systems.According to Eq. (5) the fitting quality of vr(+) is rather poor, especially at lower shearrates. The values of MSRD. used as minimization criteria for fitting the experimental data. range between 0.37 at lower shear rates and 0.09. In order to isolate the sheareffects, the relative viscosity was plotted against the reduced solid volume fraction (CV a,,,) instead of Cp.Fig. 7 showsthat, for the examined subpensions,noticeable differences are observed between the qr - @/+J;i) curves at the various shearrates, Quite similar considerationswere madeby Wildemuth and Williams [30]. In the next step, the analysis of q,(%) was performed at various constant shearstresses.Among the several relations from the literature, the best results were obtained with the Krieger-Dougherty (Eq. (5)). Robinson (Eq. (6)) and the modified Eilers (Eq. (7)) equations. A relevant model for

describingrelation q,(+) and evaluating correct values of the parameters [v] and 9, from the equations applied was selected. starting from the statistics of fitting quality in terms of the values of the mean squaredrelative deviation (MSRD). The best fitting resultswere obtained by applying the Robinson equation, Eq. (5). The MSRD values ranged between 0.09 and 0.04, except in the very low shear stress range where the MSRD values are around 0.17. The equation also makesit possibleto find somesimilarities with the theoretical approachof the Aocs-aggregate structural model describedby Tsutsumi et al. 1261. To check the influence of shear-dependentbehavior on the concentration dependenceof the relative viscosity, the experimental data, in terms of relative viscosity, are plotted againstthe reducedsolid volume fraction G./@,,,(T),asshown in Fig. 8. In contrast to Fig. 6, where a different curve of vr@+$,,(i/) is observed for each set of data at a certain shear rate, the experimental data lie on the samecurve V&J’/@,,,(~) in the whole shearstressrange examined, when parameter a,,, is derived from Eqs. (5) (6) and (7) at the referenceshear stresses.Fig. 8a demonstratesthat values of qr in the low shear stress range systematically deviate from the ql--@/ G),,,(T)general curve pattern, if @‘mis evaluated from the Robinson equation, Eq. (6). In Fig. 8b and c, the same data of relative viscosity are plotted against the reduced solid volume fraction @‘/@Jr), the parametera,,, being evaluated from the Kriegher-Dougherty equation, Eq. (3, and the modified Eilers equation, Eq. (7) respectively. In both figures. no systematic deviation is observed from the general pattern of the curve. and the scattering of relative viscosity data is sensibly reduced in the case of the modified Eilers equation. Accordingly, we can argue that the Robinsonequation is not suitablefor describing the dependenceof qr(+), whereas the modified Eilers equation provides the best data correlation. The shear stressdependenceof maximum packing fraction, evaluated from Eqs. (5) (6) and (7) is shown in Fig. 9. Even if a.,,,dependson the equation used, the shapeof a’,(r) curves derived from Eqs. (5) (6) and (7) are very similar. In the low shear stressregion. a,,, increaseswith increasing

100 T

Fig. 7. Relative viscosity versus reduced solid volume correlated accordmg to Eq. (5) for pr data at wnstant

L

fraction Q/a,,,. a,,, is shear rates.

(a)

8,

- from Robinson

equation .

0.62 e’ E 0.58 .o 5 g 0.54

. 0 low shear .

stresses .

l

+m.

Elks

+

Krieger-Dougherty

-A-

Robinson

@ P 0.50 ii E 0.46 2 f

0.42 0.38

I 0.2

0.3

0.4

0.5

0.6 aJ/a,

0.7

0.8

0.9

1000 (b)

Q,

horn

Krieger-Dougherty

equation

0.3

0.4

0.5

0.6 am,

0.7

0.8

(Eq.

0.9

1

loo0

(c) am - from m. Eilers S Cl00

equation

. A .

i

i’.,,,‘.,..~,,,.,‘...l..,.l.“‘,‘.”,.”” 0.2

Fig. 8. Relative correlated Dougherty ~7~data

0.3

viscosity

by using equation at constant

0.4

versus

0.5

(5))

and

dependence equation

modified

Etlers

10 stress,

100

of maximum packing fraction (Eq. (6)). the Krieger-Dougherty equation

1000

T (Pa)

+,,,. evaluequation

(Eq. (7)).

about systematic data deviation in the low shear stress region from the general curve pattern in Fig. 8a. Thus, Eq. (6) can be excluded from the evaluation of the correct values of parameters @‘mand [q]. The parameter [v], correlated from Eqs. (5) and (7), displays almost the same values in the whole shear stress range examined. The variation of [q] with shear stress is small and may lead to a conclusion that deviations of primary particles and aggregates from spherical shape are rather insensitive to the variation of the shear stress. The influence of shear stress on the values of the product [q] x Cp, determined from Eqs. (5), (6) and (7) is shown in Fig. 11. The values of [v] x a,,,, obtained from Eq. (6). exhibit strong shear stress dependence in the low shear stress range. When the values of [r] x G,,, are evaluated by using Eqs. (5) and (7), the variation of the product with shear stress is relatively low and the mean value of 2.4 is found in both cases.

4. Conclusions 0.6 @I~,

reduced

solid

(a) the Robinson equatton (Eq. (5)) and (c) modified shear

Shear

Ftg. 9. Shear stress ated by the Robinson

0 .

02

1

0.1

1

0.7

0.8

volume

fraction

0.9

1

a/@,,,. @‘m is

(Eq. (6)). (b) the Krieger\tqEilers equation (Eq. (7)) for

Rheological investigations of TiOl suspensions were carried out using a controlled stress rheometer. Test procedures, logarithmic stress ramp and creep were determined

stresses.

shear stress and reaches its maximum value in the shear stress range where the onset of shear thickening occurs for highly concentrated suspensions. The observed fp, decrease at higher shear stresses can be ascribed to the appearance of shear thickening behavior. All these considerations on the shear dependence of 9, further confirm the validity of using shear stress as the reference quantity in the analysis of the concentration dependence of relative viscosity [35]. The shear stress-dependence of the parameter [TJ]in Eqs. (5), (6j and (7) is illustrated in Fig. 10. When Eq. (6) is applied to correlate experimental data. the parameter [q] exhibits high values and a strong decrease in the low shear stress range. This is not in accordance with an existing microstructure in suspensions and confirms the observations

.2 8:: 6 ‘5 .o 4 ‘5E

em.

.6

+

Krieger

+

Robinson

2

Eilars Dougherty

1 shear

Fig.

IO. Shear

Robinson modified

stress

dependence

10 stress.

1000

100 r (Pa)

of intrinsic

equation (Eq. (6)). the Krieger-Dougherty Eilera equation (Eq. (7)).

viscosity

[n]. evaluated equation

by the

(Eq. (5)) and

A. Zupnn?if

-=-m.

et al. /Progress

in Orgunk

Eilers

+

Kriager

-A-

Robinson

Dougherty

p r 2.5 2 1.5 I

1 0.1

1 shear

10 stress

100

looil

T (Pa)

Fig. I I. Shear stress dependence of [n]+,. Parameters are evaluated by the Robinson equation (Eq. (6)). the Krieger-Dougherty equation (Eq. (5)) and modified Eilers equation (Eq. (7)).

with regard to specific characteristics of the investigated pigment suspensions and provided a suitable evaluation of shear-dependence in a wide shear stress range. Examined suspensions, containing TiO? particles only, exhibited a shear-dependent behavior which is usually found for stabilized suspensions. In particular, time-dependent effects on shear-dependent behavior could be considered as negligible. For suspensions at a higher particle concentration, a Newtonian plateau in the low shear stress region could be detected. The suspensions exhibited shear thinning behavior in the intermediate shear stress range and, at a solids volume fraction @ higher than 0.20, shear thickening behavior was observed in the highest shear stress region. In describing the shear-dependent behavior of these suspensions, the relations proposed in the literature were applied to correlate the experimental data. but did not provide an acceptable quality of fitting. A better correlation throughout the whole shear rate (or shear stress) range was obtained by two empirical expressions, Eqs. (9) and (10). Both equations were derived by the modification of pseudoplastic Krieger-Dougherty and Cross models. It was assumed that shear viscosity could be the sum of two contributions (shear thinning and shear thickening). In the absence of established criteria for the shear thinningshear thickening transition, a simple power law relation for the shear thickening contribution was adopted, valid in the whole shear range. Accordingly, the critical shear rates and shear stresses for the relevant transition conditions are defined. When shear stress is adopted as a reference quantity, the interval between the two transition conditions shifts slightly towards higher shear stress values with increasing solids content. The influence of particle concentration on relative viscosity was analyzed by examining data subsets at a constant shear rate and subsets at a constant shear stress. The best results were obtained by the Krieger-Dougherty equation (Eq. (5)) and the modified Eilers equation (Eq. (7)). When shear rate was used as a reference quantity, the variation of the model parameters a,,, and [n] with shear rate showed an increase only in the very low shear rate range and displayed

Coatings

30 (1997)

67-7X

17

a maximum around 0.5 s-‘. In order to isolate the shear effects, the relative viscosity was plotted against the reduced solid volume fraction (G+@,) instead of %. For the examined suspensions, noticeable differences were observed between n1 - a/@,,,(T) curves at various shear rates. When the parameter @‘mwas derived from the equations examined at reference shear stresses, the experimental data lay on the same curve n, - a/@‘,(r) in the whole shear stress range examined. Even if a,,, is dependent on the chosen equation, the shape of a&r) curves was very similar. In the low shear stress region, G,,, increased with increased shear stress and reached its maximum value in the shear stress range where the onset of shear thickening occurs for highly concentrated suspensions. The subsequent +,,, decrease at higher shear stresses could be ascribed to the existence of shear thickening behavior. All these considerations on the shear dependence of 9, confirm the validity of using shear stress as a reference quantity in the analysis of the concentration dependence of the relative viscosity.

References [ I ] R. Lambourne, Paints rtnd .Sur$~ce Cocrtinqs. Ellis Horwood. Chichester. 1987. [2] Patton. Point Flout rend Pigment Dispemrons. 2 edn., Wiley, New York, 1979. [3] D.J. Jeffrey and A. Acrivos. AIChE J.. 22 (1976) 417-432. [3] J. Mewis and A.J.B. Spaul. .I. Cdoid Intrtfbce Sci.. 6 (1976) 173200. 151 H.A. Barnes, J. Rhed., 33 ( 1989) 329-366. [6] R.L. Hoffman, Tmns. Sot. Rhrol.. /6 (1972) 155-173. [7] B.J. Ackerson, J.B. Hayfter, W.A. Clark and L. Cotter, J. Chem. Phys.. 84 (1986) 234-2349. [8] H.M. Law, R. Bung, E. Hadicke and R. Hingmann. Proc. Xth Int. Con~r. on Rhrolog~, Brussels. Elsevier. Amsterdam, 1992, pp. 616618. [9] M.K. Chow and F.Z. Zukoski, J. Rhaol.. 39 (1995) 33-59. [IO] W.H. Boersma, J. Laven and H.N. Stein, AIChE J., 36 (1990) 321332. [I I] J.B. Jansma and S. Qutubuddin. ./. Rhed., 39 (1995) 161-178. 121 1.M. Krieger and T.J. Dougherty. Trcms. SM. Rheol., 3 (1959) 137152. I?] B.E. Rodriguez. E.W. Kaler and M.S. Wolfe. I~~ngrnuir, 8 (1992) 2382-2389. II] W.B. Russel, D.A. Saville and W.R. Schowalter. Cdoidal Dispersiom. Cambridge University Press. New York, 1989. Chapter 14. 1S] M.M. Cross, J. Colloid Sci., 20 (1965) 417-437. 1161 T.Y. Liu. D.S. Soong and D. de Kee, Chrm. Eng. Commun., 22 (1983) 273-285. [ 171 D. de Kee and C.F. Chan Man Fong, PoI!m. Eng. Sci., 34 (1994) 438-445. [ 181 R. Hanks. Proc. fXth fnt. Gong. Rheol. Mu.ricn, 2 (1984) 355-360. [ 191 I. Wagstaff and C.E. Chaffey. J. Colloid fnterfirce Sci., 59 (1977) 53-G. [20] J.A. Roper III and J.F. Attal. Toppi Joumtrl, 76 (1993) 56-61. [?I] W.B. Russel and P.R. Sperry. frog. Org. Coat., 23 (1994) 305-324. [22] H.A. Barnes. J.F. Hutton and K. Walters, AH introduction to Rheolo,ey. Elsevier, Amsterdam, 1989. [23] D. Quemada, Rhed. A&r. 17 (1989) 632-642. [24] A.B. Metzner, J. Rheol.. 29 (1985) 739-775. [25] R. Lapasin and S. Pricl. Cwt. J. Chem. Eng., 70 (1992) 20-27.

[26]

A. Tsutsumi. K. Yoshida, M. Yui. S. Kanamori and K. Shibata, Pmder Technol., 78 (1994) 165-172. [27] J.V. Robinson, Trans. kc. Rheol.. / (1957) 15. 1281 H. Eilers, Kolloid Z., 97 (1941) 313-321. [29] C.R. Wildemuth and M.C. Williams, Rheol. Actu. 23 (1984) 627635. [30] C.R. Wildemuth and M.C. Williams. Rhro/. Acttr. 24 (1985) 75-91. [31] R. Lapasin. Chem. Biochem. Eug. Q.. I (1987) 143-150.

[32] M.L. Wang and T.C. Cheau. Rheol. Acta, 27 (1988) 596-607. 1331 J. Mewis, W.J. Frith, T.A. Strivens and W.B. Russel, AIChE J., 35 (1989) 415-422. [34] D.A.R. Jones, B. Leary and D.V. Boger. J. Colkoid Int