1 s2.0-0377025779870019-main
Upcoming SlideShare
Loading in...5
×
 

1 s2.0-0377025779870019-main

on

  • 180 views

research

research

Statistics

Views

Total Views
180
Views on SlideShare
180
Embed Views
0

Actions

Likes
0
Downloads
0
Comments
0

0 Embeds 0

No embeds

Accessibility

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

1 s2.0-0377025779870019-main 1 s2.0-0377025779870019-main Document Transcript

  • 1 Journal of Non-Newtonian Fluid Mechanics, 6 (1979) l-20 0 Elsevier Scienkc Publishing Company, Amsterdam - Printed in The Netherlands THIXOTROPY - A GENERAL REVIEW J. MEWIS Department (Belgium) of Chemical Engineering, Katholieke (Received October 9, 1978, accepted October Uniuereiteit Leuven, B-3030 Leuven 10,1978) The time-dependent behaviour associated with thixotropy rather than with viscoelasticity is discussed. The evolution of the concept is traced back and a generalized definition is accepted. Subsequently, the various experimental methods are considered with which meaningful measurements of thixotropy can be performed. The specific experimental difficulties encountered with the systems under consideration are treated separately. It is impossible to enumerate all materials that show thixotropy or antithixotropy. Instead, they are classified in groups according to their origin or application. The resulting table is used to deduce the characteristics which accompany thixotropic phenomena. This leads to a discussion of the time effects in terms of the molecular or microscopic structure. An attempt is made to provide a systematic outline of the published models and possible constitutive equations for thixotropic materials. Both inelastic and viscoelastic descriptions are included. 1. Introduction The interest in thixotropic phenomena is nearly as old as modem rheology. An increasing number of real materials has been found to show these effects. Also, they have been applied in various industrial applications. Nevertheless, thixotropy has never appeared in the mainstream of rheological research. Together with the related effect of yield stress, it constitutes probably the major remaining problem in theoretical rheology. By the same token, about a thousand papers in the field have not resulted in a satisfactory picture of the mechanisms governing thixotropy.
  • 2 The present paper attempts to review the available information, to emphasize the relevance of the topic and to indicate areas in which progress can and should be made. 2. Concept The term thixotropy was originally coined to describe an isothermal, reversible, gel-sol (solidiiquid) transition due to mechanical agitation [l-3]. From the beginning it was realized that sol-gel transitions occurred not as frequently as other and less drastic changes in consistency. Gradually, the latter have been also considered as thixotropic phenomena. The early history has been well reviewed by Bauer and Collins [4]. At present, there exists a rather general agreement to call thixotropy the continuous decrease of apparent viscosity with time under shear and the subsequent recovery of viscosity when the flow is discontinued [ 4-61. The material is assumed to be at rest for a sufficiently long time before the said experiment is performed. The opposite effect also exists and is known as antithixotrophy [6,7]. Although the word rheopexy has been used as a synonym for antithixotropy, its recommended use is for systems, the rate of structural recovery of which can be accelerated by vibrations of well defined characteristics [ 6,8]. From the definition, it is clear that thixotropy is associated with timedependent effects. Time effects are reflected in shear rate effects but the relationship might be complicated. The suggestion by Goodeve and Whitfield [ 91 to describe shear thinning by a thixotropic coefficient has been followed by a few authors but has now been abolished. As viscoelasticity deals with similar effects, the relative position of both approaches comes into question. First it should be pointed out that the definition of thixotropy does not exclude elastic effects. This statement is supported by experimental evidence (e.g. [lO,ll]). On the other hand, the time scale of thixotropy seems not necessarily associated with the time scale for viscoelastic relaxation. A relation exists, both physically and mathematically, with the nonlinear effects in viscoelasticity. The correspondence has been used to develop thixotropic theories of nonlinear viscoelasticity [12,13]. There, thixotropy refers to an approach rather than to a phenomenon. The phenomenon itself becomes noticeable in viscoelastic materials when the stress or viscosity recovery after cessation of flow takes more time than the stress relaxation. 3. Experimental methods Starting from the definition of thixotropy one can investigate experimental techniques to detect and evaluate thixotropy in real materials. As one is looking for time effects, it can be expected that transient measurements provide useful information. This turns out to be correct but it should be kept in mind that various parameters will affect the measured time function.
  • 3 Numerous transients have been published in the literature. The earlier work has been reviewed by Bauer and Collins [4]. Basically, three types of suitable transient experiments have been proposed: (1) a step function in shear rate [14] or shear stress [ 153 ; (2) a consecutive linear increase and decrease in shear rate [ 161; (3) a sinusoidal change in shear rate [ 171. To be meaningful, the material should be in a well defined condition at the beginning of a transient. This requirement is not met in the earlier work. Normally the sample is sheared homogeneously until no further changes occur. At that stage, a sudden change in stress or rate of strain is applied. It is generally assumed that an equilibrium is reached prior to the application of the jump. This is not a trivial condition: if the changes are not completely reversible, no real equilibrium exists. A simple procedure including alternately higher and lower shear rates [ 181, allows verification of reversibility and equilibrium. Nevertheless, this step is still often neglected. In most viscometers the kinematics are controlled rather than the stresses. Hence experiments with sudden changes in shear rate appear more frequently than stress jumps (e.g. [ 19-211). Modifications to apply a constant stress have been described (e.g. 122-241). Despite the large number of experimental studies, very few have varied the parameters in a systematic way. One particular kind of step function is of special interest. It consists of a sudden drop in shear rate from a finite value to zero. In this manner, the rate at which the material regains its equilibrium rest condition can be analyzed. As such, it measures the basic thixotropic phenomenon. Besides, the reversibility of the thixotropic decay under shear can be verified. The problem is that the system has to be disturbed in order to measure the extent of recovery. The standard procedure has been to start the flow again after a given time of rest and measure the overshoot stress. A diagram of this stress plotted against time of rest provides the curve for thixotropic recovery (e.g. [ 25-27 3). The time scale for recovery can be too small to be accurately measured but can also reach days or weeks. The characteristic thixotropic recovery can also be encountered in viscpelastic systems [ 10,111, although it has not attracted much attention from polymer rheologists. Instead of using overshoot, recovery can be followed by applying oscillatory flow [11,28,29]. If the amplitude is kept small enough, the technique can be nondestructive 1111. For viscoelastic thixotropic systems, one can follow the recovery by measuring the overshoot in normal stress differences [lo] or even the change in the viscoelastic relaxation spectrum [28,30]. Clearly, the later method, whenever applicable, provides more detailed information about the changing system than any other of the previous ones. The complex effect of the measuring conditions on the results implies that step functions are not very suitable for a fast screening or comparison of thixotropic materials. Green and Weltmann [ 161 suggested measuring the
  • 4 stress under consecutively increasing and decreasing shear rates. If the sample is thixotropic, the two curves do not coincide, causing a hysteresis loop. Repetition of the experiments shifts the hysteresis until finally an equilibrium loop is described. The surface area of a loop can be used as a measure for the degree of thixotropy. This technique is useful in industrial work and has been applied in a large number of papers (e.g. [ 31-331). The initial resulting hysteresis loop depends on the previous shear history and both the rate of change and the maximum value of the shear rate. The effect of shear history can be eliminated by using the equilibrium loop. The latter is often very small. Containing a more complex shear history than the step function, the relation between loop area and material parameters is complicated [ 34,351. In addition, ‘viscoelastic relaxation can cause hysteresis in the absence of any thixotropy [ 36,371. The triangular shear rate history of the hysteresis experiment can be replaced by other time functions. The only one used is the sine wave [ 17,381. The amplitude is taken sufficiently large to extend into the nonlinear region. Again the stress-rate of strain relation describes a time dependent hysteresis. Eventually a Fourier analysis can be applied to the varying response [ 381. Under oscillatory flow, the viscosity decrease can sometimes be delayed [ 381. The delay can facilitate the study of time phenomena. As will be seen later, there are other reasons which make such experiments attractive. The responses to step functions and the hysteresis loops describe thixotropy in part. The measurements should be supplemented with the equilibrium rheogram as in the case of other non-Newtonian liquids. The same procedure holds for antithixotropic systems. All available experimental evidence seems to have been obtained under simple shear flow. From the beginning, attempts have been made to supplement or to replace mechanical measurements on thixotropic systems by other, nondestructive, techniques [ 391. No generally applicable method seem to be available to date. For sufficiently transparent materials, optical measurements can be used. Weymann et al. applied light scattering [40]. The system discussed in ref. 11 shows some peculiar transient flow birefringence. On the whole the optical methods have not been very successful in thixotropy. They are complicated to use and are only applicable on a small percentage of the materials discussed in this paper. Dielectric techniques have been used more frequently but are limited mainly to materials containing conductive elements in a nonconductive medium. Since the effect was first noticed [41], different applications have been published. Changes in d.c.-conductivity and in static dielectric constant as a function of time or shear rate have been reported [42,43]. Other workers investigated the dielectric spectrum as a function of shear rate [ 441. Dielectric measurements have been compared with the corresponding rheological data both for thixotropic [ 45,461 and antithixotropic [ 471 samples.
  • 5 Qualitatively, the dielectric and the mechanical behaviour of structured systems can be correlated but more systematic comparisons are needed before a general evaluation can be made. Recently transient dielectric spectra have been obtained [48] on systems for which transient mechanical spectra are also available [ 30,381. In this case a common model has been suggested for both [49]. It can be concluded that dielectric methods, whenever applicable, provide a powerful tool in studying thixotropy. In most cases the shape of the dielectric spectrum does not change but the height can vary considerably. Up to now it has been assumed that the data had been obtained in properly designed experiments. For a substantial number of published data, it is not sure whether this condition is satisfied. The more important problems are enumerated below. As in viscoelastic materials, the stress changes with time and shear rate. Hence some o.f the instrumental requirements are related to those of nonlinear viscoelastic fluids [ 501. As the time effects are normally very slow, instruments with a shear rate gradient in the sample, as in capillaries, are not recommended. Indeed, large entrance effects can be expected and only the equilibrium behaviour can be adequately derived from the data. An occasional correlation between capillaries and rotational instruments [ 511 can only be explained by the absence of time effects or by a peculiar time-shear rate correspondence for the sample at hand. For the analysis of time effects, a constant shear rate throughout the sample is essential. Even in the normal rotary instruments which are used for non-Newtonian liquids, problems may arise with thixotropic samples. The reason being that the latter are most often strongly nonlinear at low shear rates and even show a yield stress. For a coaxial viscometer, this can lead to severe restrictions on the radii ratio [ 521 as illustrated by Helsen et al. [48]. A plate-and-cone geometry turns out to pose less problems [ 531. Some authors have considered “superanomalous” systems where three different equilibrium viscosities should exist for each stress within a certain stress range [ 20,541. This means that there exists a region where the equilibrium shear stress decreases with increasing shear rate. It can be considered as a region with a negative power-law index, which would clearly give rise to stability problems in any measuring device. With heterogeneous materials, slip might occur which can be reduced by means of corrugated walls. However this will not solve the stability problem. Finally, most thixotropic materials have a tendency to show shear fracture [ 551. At higher shear rates they creep out of the gap thereby reducing the cross-section of sheared material. The effect is most pronounced in plate-andcone viscometers and in parallel-plate instruments but it can also occur in coaxial cylinders. From the literature, it is not always clear whether the shear rate limits dictated by shear fracture have been respected. Hutton [ 551 has applied an elastic fracture condition on the appearance of shear fracture in silicone oils. An attempt to correlate the critical shear rate for different
  • 6 viscoelastic and thixotropic materials did not give any result [ 561. The previous discussion refers mainly to the measurement of the equilibrium viscosity. For the determination of transients, additional problems arise. If the variable structure, which causes thixotropy, is not deformable enough, the structural changes can be inhomogeneous throughout the sample. Further, the rigidity and inertia of the instrument can interfere with the overshoot measurement [ 57,581. It can be concluded that the absolute values of most overshoot recovery-time curves are open to question but this does not affect the accuracy of the thixotropic time scale derived from the experiments. If oscillatory measurements are used instead of overshoot, it must be verified that the vibrations do not interfere with the recovery process. If a molecular aggregation causes the thixotropic effects, the deformability is usually sufficiently large to avoid interference [ll] With . particulate structures this is not the case [ 281. Except for a correct measurement of stress, the step function method requires a sudden change in shear rate. The jump has always to be faster than the thixotropic changes in the sample. In some cases a jump time of the order of 30 seconds has been reported to be adequate [ 591. More often, much faster changes are required. A necessary modification for available viscometers has been suggested [ 351. Sinusoidal and linear changes in shear rate require a variable drive system [ 60,611. Also in hysteresis loops, the inertia effects must be considered [ 62, 631. In addition, the problem of a constant shear rate throughout the sample must be envisaged as in the other transients. This holds in particular for the transients under oscillatory flow with large amplitudes. It remains to be established under which conditions a given material deforms homogeneously during the said experiment. 4. Occurrence and explanation Before the quantitative description is looked into, the physical phenomena leading to thixotropy will be discussed. The latter property is by no means unusual. Over the years it has been attributed to hundreds of systems. In some cases the data are doubtful, owing to either inadequate experimentation or to a deviating use of the term thixotropy. The cases where shear thinning or shear thickening are meant [ 91 should be disregarded, as well as irreversible effects of shear. In some systems the changes are partially reversible [ 65,661. Lists of known thixotropic substances have been compiled by Green [ 671, Fisher [68], Weltman [69] and Bauer and Collins [4]. Lists of individual products are becoming continuously larger and do not help to deduce the essential features. Instead, the thixotropic materials are grouped here in classes according to origin or application. The classification (Table 1) is not unique and completeness cannot be claimed. This survey suffices to demonstrate the ubiquitous nature of such materials. However, the listing as such does not provide any clue as to the underlying
  • TABLE 1 Classes of materials in which thixotropy can he encountered Class Reference Crude oil and oil products Greases and waxes Pigment dispersions (paints, printing inks, paper coatings) Fibre dispersions Liquid crystals and micellar structures Food products (natural and synthetic) Resinous and polymer systems (filled and unfilled) Biological systems Dispersions of clay minerals and similar products (ceramics, cement, concrete, drilling muds) Emulsions and latices Molten salts Rubber and elastomers ____ 66,70 45,71,72 27, 29, 69,73, 28, 65, 75 79,80 46,81--85 74 10,11,83 86-89 17, 21, 23, 31, 33, 54, 90, 91 22,92 93 94,91 processes. If one considers the nature of the materials involved some similarities between various groups become noticeable. In the first place, there are hardly any materials consisting of only a single component. Most of them are clearly heterogeneous systems containing a finely dispersed phase. As such they are colloidal in nature. Others are solutions but then the solute is often at the edge of solubility. It is obvious that thixotropy always presupposes some molecular or microscopic process for changing the consistency. In most cases, some kind of association process occurs. In the absence of flow, either molecules or particles link together. The linking forces are rather weak and under shear the hydrodynamic forces are large enough to break the links. The reduction in size of the associated structural units corresponds to a smaller resistance against flow. In colloidal dispersions, the suggested process is determined by the interaction forces between particles. Whenever there is suitable balance between the attraction forces (van der Waals forces) and the electrostatic and steric repulsion forces, a weak coagulation takes place. The result is the formation of isolated large floes or of a single particulate structure throughout the whole material. In the first case, one would expect a liquid-like behaviour at low shear rates. The second gives rise to a yield stress. It is the stress required to rupture the particulate network (e.g. [96,97]). The available evidence for the structural changes as well as the relationship between rheological properties and stability parameters has been reviewed already [4,98-100 1. Only the results will be discussed here. Shear thinning and thixotropy are often pronounced in dispersions of platelets and other anisometric particles [ 19,13,28,31,32,83]. Under suitable conditions, the edges and the plates can carry opposite electric charges. The result is heteroflocculation (edge-side contacts) and an open card-house
  • 8 structure [ 101,102]. Also spherical particles produce shear thinning and thixotropy [ 11,22,26,46,92,101]. The gel-like behaviour at low concentrations can be understood in terms of a network consisting of chains of spheres. In systems without dispersed particles, weak intermolecular interactions can cause a similar behaviour. Chemical bonds can eventually be ruptured reversibly by mechanical action, as in rubber. In most cases they will be too strong but all lower energy mechanisms for bonding seem to be possible, going from entanglements [lo] to hydrogen bonds [ 11,231. With organometal compounds like aluminium and titanium esters, chelation can occur. The evidence for the bonding mechanism is generally derived from the chemical composition rather than from a direct measurement. The discussion given above pertains to the effect of shear. In order to give rise to a noticeable thixotropy, the recovery of structure must be sufficiently slow. Various phenomena can slow down the structural changes. Diffusion becomes effective if the medium is sufficiently viscous [lo] or if the particle shape reduces mobility, as in the rotation of long rods and fibres. Also mobility is reduced quite drastically if the structure is already partially developed. In such a case, one would expect the rate of change to slow down more with the degree of structure than expected from models with constant mobility. This latter conclusion seems to be confirmed by a number of experimental results as will be discussed in the next section. This does not mean that the experiments prove the presence of a diffusional effect. Other mechanisms can lead to similar deviations in the kinetics. In disperse systems, the presence of repulsion forces and hydrodynamic interaction between the particles provide additional sources of slowed down aggregation [ 1031. In a few systems [ 80,951, order-disorder transitions are responsible for the thixotropic behaviour. Antithixotropic phenomena are less clearly understood and also occur less frequently. The literature is often confusing due to lack of agreement on definitions and also due to inadequate experimentation. Quite often antithixotropy is concluded to exist from an inverse hysteresis loop, without any verification of reversibility. With a few exceptions, antithixotropy is found in dispersions [ 69,104,105] and polymer solutions near the edge of solubility [ 106,107 1. Savins has reviewed the earlier work [ 1071. Antithixotropy is the contrary of thixotropy in the sense that shear induces aggregation of particles or molecules. This effect is well known but is usually irreversible. Antithixotropy requires that, upon releasing the stress or strain rate from the aggregates, the change in configuration, orientation or interaction field suffices to restore the less structured rest condition. The time scale seems to be generally much smaller than in thixotropic systems. Therefore the difference with shear thickening is less well defined [ 4,105b,107]. A single material can show both shear thinning or thixotropy and antithixotropy. In that case, antithixotropy will appear at higher shear rates than the other effects.
  • 9 From the previous discussion it can be concluded that the general principles of thixotropic and antithixotropic behaviour are understood. Although these phenomena occur in a large number of important materials, it remains extremely difficult to make any predictions. The specific mechanisms which determine the response of individual systems or classes of systems have not been analysed in a systematic manner. 5. Theoretical description Various approaches can be followed to develop a theoretical description for thixotropic materials. A possible choice would be to start from the general axiomatic theory of continuum mechanics and see how the present materials can be fitted in. Following this route, difficulties are encountered very soon. Some of the obvious simplifications which seem valid in normal polymer rheology cannot be applied here without further verification. The fact can be mentioned that some thixotropic materials behave solid-like at low stress levels and liquid-like at high stress levels. As a result the consecutive order approximations must fail. Also the structural recovery is based on interactions which can have a non-negligible range of action. Whether the presence of such interactions eventually invalidates the principle of local action is not known. As a large number of thixotropic materials show hardly any elastic effects, there is an interest in inelastic theories of the said phenomenon. Existing theories generalize the Reinex-Rivlin constitutive equation by making the relation between stress rij and strain rate +ij dependent on time. Slibar and Paslay [ 1081 started from a Bingham body, where the critical value of the stress rc,it is made a function of the shear history: Yijtt) = 211Yij(t) O 0X5< 9 =yGit Tij(f) 9 Jm Tcrit(t) > 9 Tcrit(t) , (1) 7 where t’ C-m Tcrit(t) dwj exp[-a(t - t’)] dt’ = 71 - (71 p +] ,/@8 t cc-m - 70) . (2) exp[--dt - t’)l dt’ The model constants are q, ro, r,, a! and 0. The analysis assumes that the material behaves as a Bingham body, the critical stress of which depends on shear history through a simple exponential memory function. Clearly, a large number of similar models can be generated by modifying various assumptions. Very few comparisons with experiments have been performed [ 1081.
  • 10 As with most other thixotropic models, it describes qualitatively the typical features of thixotropy. However, it relies on a constant plastic viscosity and Bingham behaviour for each structural condition, assumptions which have not been substantiated [ 19,109]. The analysis suggested by Harris [ 1101 avoids the explicit use of the Bingham model. It does not specify the shape of the rheogram, which makes it more flexible but also less directly useful. A distinction is made between time-dependent viscous forces, eqn. (3), and a timedependent yield condition, eqn. (4). The latter is based on a critical strain energy density Ti which is taken to be proportional to the second invariant of the stress [ 1111: Tij(t) = 2rl(11+, (3) Ilr,)*j(t) if Ti(t) > r,,it(t) - (4) For the difference between q(t) and an assumed rest viscosity q1 and similarly for the corresponding term T(t) - T, integral memory functions are suggested that describe the changes with time and shear rate: q1 -v(t) =j Tl - T(t) = j fi [II&‘), f2[II&‘), Irl,(t’)]M(t - t’) dt’ , (5) III&)]N(t - t’) dt’ . (6) The memory functions M and N can be continuous functions or discrete relaxation distributions can be introduced. In this manner, eqns. (5) and (6) generalize eqns. (1) and (2). There remains the problem of how the yield criterion will be incorporated into the rheological constitutive equation. Harris suggests a possible form presented earlier by Oldroyd [ 1111. It consists of a kind of superposition, but its physical meaning is not very clear. Quite generally, the yielding process is insufficiently known. Additionally, there is some confusion in the literature between the real yield stress and the apparent yield stress [ 1001. The latter is only a model parameter for describing the rheogram. Harris did not apply his general analysis to experimental data. The usefulness of the approach has to be verified. Nothwithstanding the artifical combination of viscous and yield stresses, the model is illustrative for the use of more than one time function to describe the thixotropic changes. This point will be reconsidered below. The generalized continuum theory approach is not the only possible one. A large number of theoretical analyses have been based on the argument of structural kinetics. This means that it is understood that the change in time of rheological parameters is caused by changes in the internal structure of
  • 11 the material. The nonlinear, time-dependent behaviour can then be described by a set of two equations. The first gives the instantaneous stress as a function of the instantaneous kinematics for every possible degree of structure S. The second is a kinetic equation, which describes the rate of change for the degree of structure as a function of the instantaneous value of S and the instantaneous kinematics: F(t) = 6 [G (0, S(01 , (7) dWt)ldt = f4 [r’ (t), S(t)1 . (8) For the time being, the algebraic nature of S(t) is not further specified. This alternative approach has been successfully applied to the rheology of dilute dispersions, where the microstructure is well understood (e.g. [112, 1131). Its use is now advocated for larger classes of materials [ 1141. The points to be discussed in relation to eqns. (7) and (8) include: the type of the equations and the meaning of S(t). In most of the earlier work, the degree of structure was characterized by the number n of bonds or links that contribute to thixotropy [ 21,22,26,46,115]. Each change in n is normally supposed to cause a proportional change in Newtonian viscosity. This determines the shape of eqn. (7). For the rate equation (8), single-order kinetics have been suggested for recovery and breakdown, with variable effects of shear rate. The resulting equation is comparable to an equilibrium reaction in chemical kinetics. A rather general form compatible with the previous arguments is given by eqns. (9) and (10): 712(t) = [en(t) + b1?12(U 3 dn(Wdt = -4~n~912(t)~ + k2[no - r~(t)]~~~~(t)~ , (9) (10) where a, b, c, d, ki, a and fi are constants and n, is the limiting number of thixotropic links. Equations (7)-(8) and (9)-(10) can describe qualitatively known thixotropic effects such as stress decay, stress recovery and overshoot. Although the second set has already nine constants, it still does not accurately describe the published data [ 19-211. There are also some fundamental problems. Equation (9) assumes that each degree of structure shows a Newtonian response. The available evidence is open to debate but suggests a more complicated relationship [ 19,20,109]. Clearly, no yield stresses can be described in this manner. In addition, eqns. (7)-(10) are baaed on the assumption that n is an adequate measure for the degree of structure. A number of authors have used the apparent viscosity or fluidity as a measure of structure. This is equivalent to the use of n assuming that 9 is proportional to n [ 116-1181. Hence the same restrictions apply. Considering a thixotropic dispersion it seems reasonable to assume that not all the structural units or aggregates are identical [ 21,22,118,120]. Therefore a structural parameter X has been introduced which does not refer any-
  • 12 more to n [ 20,109]. Cheng [ 1091 has developed a general framework for these theories starting from eqns. (11) and (12): F(t) = ;I [A(t), II&) 7(t)] . (11) WOldt = gzLW% ~~#)I . (12) In a Tij*ij plot, the behaviour C= be represented by a series of constant A curves and by a series of constant dX/dt lines. Eventually the struct,ural parameter can be eliminated [ 1211: dTn/dt = 47129 912) d+lz/dt + P(712,%2) . (13) Equations (11)-(13) as well as the other thixotropic theories have only been applied in shear flow. Cheng’s theory does not provide a specific model description but it suggests the use of two material functions as well as experimental techniques to determine them [109,121,122]. There has been some discussion about the adequacy of a single, scalar, parameter for the complete description of structure [ 123,124]. Mewis [ 1241 has reviewed the available experimental evidence against a one-to-one correspondence between a scalar structural parameter and the rheological characteristics. He also showed the existence of a structural hysteresis phenomenon which cannot be described by a single X [ 1251. It should be pointed out that dilute dispersions already required a second-order tensor for their material characterization. Cheng has extended his theory to incorporate two scalar parameters Xi [ 1231. Rather dilute dispersions with a yield stress seem to constitute a class of materials where a more complex structural characterization is required [ 30,125]. Comparing the kinetic equation (10) and the experimental evidence, the general conclusion is that the shear rate has usually only a weak direct influence. In the models fl is most often put equal to zero. Further the values of c and d, used either with n or q as structural parameters are low, typically c = 0, d = 2. It is assumed in eqn. (10) that the mechanisms of breakdown and recovery of structure are each characterized by a single rate constant. Considering the complex structure which often must be built up, together with the relative weakness of thixotropic bonds, a certain distribution of rate constants seems possible and even probable. If one looks at some of the available data, a more gradual change than that predicted from a single rate constant can be noticed [11,19,21]. In particular, the data of ref. 11 can be understood as resulting from a first-order reaction with a distribution of rate constants. The spreading can be compared with memory functions of the continuum theory models that have more than a single relaxation time. Of course, the number of constants in the model will increase even more if the said distribution is taken into account. Besides the continuum theories and the structural kinetics, there exists a third approach. As in other branches of rheology, one can start from the microstructure to calculate the rheological behaviour. Only for the simplest
  • 13 structures has the microstructural approach been successful [112,113]. For thixotropic materials, a few, albeit crude, attempts have been made. They refer to dispersions and try to give a more detailed picture of the microstructure than the structural kinetics models. Mooney [22] calculated the size distribution of aggregates us&g van Smoluchowski’s theory of slow coagulation during shear. Immobilization is considered the cause of viscosity increase due to aggregation. The idea was further pursued by van den Tempel [ 1191. The analysis considers the shear effects but not the time effects. Casson [ 1261 has investigated the particular case of rodlike aggregates. The aggregates am assumed to have equal length. This length is calculated as a function of shear rate, using various simplifications. Time effects have been calculated for the aggregate size distribution. Weymar-mhas considered spherical aggregates [21,127] whereas Ruckenstein and Mewis [ 1281 worked on chain-like structures. The latter two analyses calculate the thixotropic behaviour but the structural picture remains elementary. They do not deal with a yield stress. It could pointed out that the Ruckenstein-Mewis model gives a distribution of structures which can be described by a single structural parameter [124]. A modified version of the Ruckenstein-Mewis model was used on experimental data by Hudson et al. [29]. Chaffey [ 1291 has made a comparative study of the microstructural models. For the time being, the main difficulty with these models has been in the very incomplete picture we have of the detailed structure of thixotropic materials. Also different subclasses are known to have completely different microstructures: entangled and gelled macromolecules, particulate chains and networks, spherical aggregates, and card-house structures. It is not quite clear how a single explicit model could cover the detailed behaviour of the various subclasses. Except for the three major groups of thixotropic models, some more isolated analyses have been suggested. The only ones which have been applied to a certain extent and with some success are derived from Eyring’s theory of rate processes [31,33,130]. The analysis is based on the presence of various flow mechanisms. Each causes a Newtonian or a non-Newtonian contribution to the shear stress r12 in proportion to the number Xi of flow units of its kind [33] : 712 = (xlPll~lkil2 + (X2/~2b~h-92%2 + W3/~3b~h-1P3%2. (14) In a thixotropic material, flow units of one kind can change to another kind. The theory of rate processes predicts a kinetic equation of the type: -dX,/dt = X2& exp(c+q2/kT) - X1kb exp(-ef2/kT) , (15) where k, and k, are the rate constants for the forward and backward reactions. The results predict reasonably well the measured transients but usually the available experimental evidence is limited. The parameters are obtained through curve fitting. Hence, any conclusion about the prevailing flow
  • 14 mechanisms from the goodness of fitting is questionable [ 31,331. Other criticisms refer to the calculation of the activation free energies. Russian workers have modified Eyring’s theory to derive rheological constitutive equations with stress as the independent parameter [131-1331. Again a limiting Newtonian viscosity is assumed and the viscosity anomaly discussed above is treated in detail, They have concentrated on the equilibrium behaviour. Finally, it might seem attractive to apply thermodynamic arguments to develop thixotropic theories. The theory of rate processes is only based in part on thermodynamic arguments. Huang [ 1341 claims a thermodynamic approach but mainly follows a structural kinetics route. He uses a tensorial parameter for molecular arrangement, its rate of change is assumed to be proportional to shear stress. Several explicit and implicit assumptions are made. The time functions are unusual and are not discussed. The general thermomechanical theory has not yet produced results in the area of thixotropy. Sarti and Astarita [ 1351 have made a preliminary discussion of the problem of introducing structure in thermomechanics. Spaull [ 136 ] has pointed out the similarity between the structural parameter and the de Donder parameter from nonequilibrium thermodynamics. This correspondence places the thixotropic theories in a larger perspective but, up to now, this approach has not been pursued any further. 6. Viscoelastic thixotropic theories Evidence for the existence of viscoelastic thixotropic materials has been discussed above. From this follows the need for a corresponding theory. There is also a second reason for developing such theories. Some investigators have realized that the mechanism of thixotropy or variable structure can be applied to generate nonlinear models from linear constitutive equations. Thus structural kinetics equations have been introduced in viscoelasticity without reference to thixotropy [ 1371. They will be included in the discussion as far as they are of interest for thixotropy. Available models can be divided into three classes, which bear some resemblance to the classes of models for inelastic materials as discussed above. The first could be linked with the usual phenomenological theories of nonlinear viscoelasticity. There, nonlinearity is introduced in a way that differs from the one used in thixotropic continuum theories. In principle, the two could be combined into an overall theory but no such analysis has been found in the literature. The fact that the material response can still change even after the stresses are completely relaxed, is essential in the present problem. Apparently both viscoelastic and structural memory functions are required. The second class of models corresponds to the structural kinetics approach in the purely viscous case but the instantaneous response must now be viscoelastic. The entanglement density is a possible structural parameter for polymer systems which can and has been used in the kinetic equation [ 1371. As in the purely viscous case the structural parameter could be
  • 15 replaced by a rheological characteristic if they are linked by a one-to-one correspondence. The kinetic equation must then be written in terms of the relaxation spectrum. Different constitutive equations are generated depending on the spectral characteristics and the nature of the rate determining parameter: stress, kinematics or a combination of both. Brodkey and co-workers [ 24,138 ] used the viscous thixotropic equation and superimposed on it a shear rate dependent retardation time. Instead, one can write kinetic expressions for the various relaxation mechanisms that constitute the linear relaxation spectrum. The destruction of the Maxwell mechanisms has been associated with a critical elastic strain by Trapeznikov [ 13, 139,140]. Others have applied a critical elastic strain energy for a single relaxation mechanism [ 171, for a discrete [7] or a continuous spectrum [12]. Experimental evidence is restricted to simple shear flow. The correspondence with theory is reasonable. Oddly enough the thixotropic recovery has apparently not been studied with these models. Regarding the results on inelastic systems, it is not quite certain if a linear behaviour can be associated with each degree of structure. As with other nonlinear viscoelastic theories, it is not clear yet whether the changes in spectrum are determined by the kinematics, the stress, an energy term or another parameter. A comparison of transients under constant stress and under constant strain rate could help to clarify the question but very few experiments of this kind are available [ 241. If an energy term is chosen, specific differences between stationary and oscillatory flow can be predicted [ 121. Again systematic experiments are lacking although some thixotropic resonance under vibrations has been described [ 1411. An additional difficulty arises when the viscoelastic material shows also a yield stress. Even without thixotropy the problem has not been satisfactory solved. Vinogradov has suggested the use of rupturing Maxwell--Schwedoff elements [ 1421. The third class of viscoelastic thixotropic models would comprise the microstructural models. From those presented above, only the RuckensteinMewis model can be readily generalized to include elasticity. The potential forces between the particles in the chain correspond to spring-like elastic forces [ 100,143 1. The particles themselves can cause viscous dissipation. Hence, under suitable conditions the chains can react as bead-spring necklaces. The difference with the behaviour of the models used for macromolecules lies in the smaller number of beads in each chain and in the rupture that can take place under shear. Bchoukens and Mewis have generalized the Ruckenstein-Mewis model and have obtained spectra that qualitatively agree with this picture [ 1441. The previous discussion refers to reversible phenomena. The kinetic approach can also be followed to describe the viscoelastic response during irreversible changes in ageing systems [ 1451 but this aspect is not considered here.
  • 16 7. Conclusion Thixotropy, like viscoelasticity, is a phenomenon which can appear in a large number of systems. It only requires the presence of a reversibly variable structure. The time scale for thixotropic changes is measurable in various materials, including important commercial and biological products. The general theory could be fitted into general nonlinear continuum mechanics, but very little work has been done to study the validity here of the various simplifications usually applied in viscoelasticity. Thixotropy suggests the eventual usefulness of a different theoretical approach based on structural kinetics. The basic structural changes that cause thixotropy are known but the information has not yet been quantified. This is not due to the lack of experimental tools. The necessary experimental methods to study thixotropy are available, although quite often they have not been properly used. In most cases the data are too fragmentary for a systematic study. Very few attempts have been made to express the data in a consistent set of parameters, covering the total behaviour. Thixotropy seems to be one of the most neglected areas of rheology, notwithstanding the industrial and scientific importance of materials exhibiting this property. Notation a, b, c, d tvgi ki M,N : T Ti t, t’ xi II, III SY 0Li9 Pi 09 T, ++ij Z,Xi 7, Tij model constants characteristic functions Boltzmann constant rate constants memory functions for thixotropic changes number of reversible links degree of structure absolute temperature critical strain energy (Harris model) time number of flow units of kind i second and third invariant of a tensor material characteristics rate of strain tensor and components viscosity structural parameter stress tensor and components References 1 E. Schalek and A. Szegvary, Kolloid 2 T. Peterfi, Arch. Entwicklungsmech. Z., 32 (1923) 318. Org., 112 (1927) 680.
  • .17 3 H. Freundlich and W. Rawitzer, KoIIoid Z., 41 (1927) 102. 4 W.H. Bauer and E.A. Collins, in F.R. Eirich (Ed.), Rheology Theory and Applications, Academic Press New York, 4 (1967) 423. 5 M. Reiner and G.W. Scott Blair, in F.R. Eirich ed., Rheology Theory and Applications, Academic Press N.Y., vol. 4 (1967) p. 461. 6 Selected Definitions, Terminology and Symbols for Rheologicai Properties, IUPAC, Oxford, 1976. 7 A.I. Leonov and G.V. Vinogradov, Dokl. Akad. Nauk SSSR, 155 (1964) 406. 8 H. Freundhch and F. Juhusberger, Trans. Faraday Sot., 31(1935) 920. 9 C.F. Goodeve and G.W. WhitfieId, Trans. Faraday Sot., 34 (1938) 511. 10 A.A. Trapeznikov and GH. Lesina, Russ. J. Phys. Chem., 45 (1971) 868. 11 J. Mewis and R. de Bleyser, J. CoiIoid Interface Sci., 40 (1972) 360. 12 A.I. Leonov, in W.R. Showalter et ai. (Eds.), Progress in Heat and Mass Transfer, Vol. V, Pergamon Press, Oxford, 1972, p.151. 13 O.V. Voinov and A.A. Trapeznikov, Dokl. Akad. Nauk SSSR, 202 (1972) 1049. 14 J. Pryce-Jones, J. Sci. Instrum., 18 (1941) 39. 15 E.L. McMillen, J. Rheol., 3 (1932) 164, 179. 16 H. Green and R.N. Weltmann, Ind. Eng. Chem. Anal. Ed., 15 (1943) 424. 17 N.F. Astbury and F. Moore, Rheol. Acta 9 (1970) 124. 18 J. Pawlowski, Chem. Ing. Techn., 28 (1956) 791. 19 D.D. Joye and G.W. Poehlein, Trans. Sot. Rheol., 15 (1971) 51. 20 E. MyIius and E.O. Reher, Plasm Kautsch., 19 (1972) 420. 21 H.A. Mercer and H.D. Weymann, Trans. Sot. Rheol., 18 (1974) 199. 22 M. Mooney, J. CoIloid Sci., 1 (1946) 195. 23 P.A. Rehbinder, in S. Onogi (Ed.), Proc. 5th Int., Congr. Rheol., Vol. 2, Univ. Tokyo Press, Tokyo, 1970, p. 375. 24 K.H. Lee and R.S. Brodkey, Trans. Sot. Rheol., 15 (1971) 627. 25 S.Y. Weiier and P.A. Rehbinder, DokI. Akad. Nauk SSSR, 49 (1945) 354. 26 S. Peter, KoiIoid Z., 113 (1949) 29. 27 D.J. Doherty and R. Hurd, J. Oil Colour Chem. Assoc., 41(1958) 42. 28 J. Mewis and L. Heginckx, Rheol. Acta, 11 (1972) 203. 29 N.E. Hudson, M.D. Bay&s and J. Ferguson, Rheol. Acta, 17 (1978) 274. 30 J. Mewis and G. Schoukens, J. Chem. Sot., Faraday Disc., to be published, 1978. 31 A.F. Gahysh, T. Ree, H. Eyring and I. Cutler, Trans. Sot. Rheol., 5 (1961) 67. 32 W. Hoffman, Rheol. Acta, 3 (1964) 172. 33 I. Park and T. Ree, J. Korean Chem. Sot., 15 (1971) 293. 34 J. Harris, Nature, 214 (1967) 796. 35 D.C-H. Cheng, Nature, 216 (1967) 1099. 36 A.G. Frederickson, Principles and Applications of Rheology, Prentice Hall. 37 B.D. Marsh, Trans. Sot. Rheol., 12 (1968) 478. 38 G. Schoukens, A.J.B. SpauIi and J. Mewis, in C. Kiason and J. Kubat (Ed&), Proc. VIIth Int. Congr. Rheol., Chalmers Univ., Gothenburg, 1976, p. 498. 39 S.S. Kistler, J. Phys. Chem., 35 (1931) 815. 40 R.A. Ross, H.D. Weymann and M.C. Huang, Phys. Fluids, 16 (1973) 784. 41 A. Parts, Nature, 155 (1945) 236. 42 M.J. Forster and DJ. Mead, J. Appl. Phys., 22 (1951) 705. 43 (a) A. Voet, Am. Inkmaker, 35 (1957) 34. (b) A. Voet and P. Aboytee, Rubb. Chem. Technol., 43 (1970) 1359. 44 (a) A. Bondi and C.J. Penther, J. Phys. Chem., 57 (1953) 74. (b) T. Hanai, N. Koizumi and R. Gotoh, Bull. Inst. Chem. Res. Kyoto Univ., 40 (1963) 240. 45 Yu.F. Deinega, A.U. Dumanski, G.V. Vinogradov and V.P. Pavlov, Kolloidn. Zh.. 22 (1960) 16.
  • 18 46 J.M.P. Papenhuijzen, in S. Onogi (Ed.), hoc. 5th Int. Congr Rheol.,Vol. 2, Univ. Tokyo Press, Tokyo, 1970, p. 353. 47 A-A. Trapeznikov, G.G. Petrzhik and T.I. Korotina, Dokl. Akad. Nauk SSSR, 176 (1967) 378. 48 J.A. Heisen, R. Govaerts, G. Schoukens, J. de Graeuwe and J. Mewis, J. Phys. E., 11 (1978) 139. 49 G. Schoukens, Ph.D dissertation, Katholieke Univ. Leuven, 1978. 50 K. Walters, Rheometry, Chapman and Hall, London, 1975. 51 W.H. Bauer, D.O. Shuster and S.E. Wiberley, Trans. Sot. Rheol., 4 (1960) 315. 52 J.R. van Wazer, J.W. Lyons, K.Y. Kim and R.E. Colwell, Viscosity and Flow Measurements, Interscience, New York, 1963. 53 D.C.-H. Cheng, Brit. J. Appl. Phys., 17 (1966) 253. 54 G.M. Bartenev and N.V. Ermilova, Kolloid Z., 29 (1967) 771. 55 J.F. Hutton, Nature, 200 (1963) 646. 56 E. Husson, dissertation, Katholieke Univ., Leuven, 1976. 57 K. Lederer and J. Schurz, Monatsh. Chem., 103 (1972) 840. 58 M.A. Lockyer and K. Walters, Rheol. Acta, 15 (1976) 179. 59 V.V. Chavan, A.K. Deysarkar and J. Ulbrecht, Chem. Eng. J., 10 (1975) 205. 60 R.B. Bird and B.D. Marsh, Trans. Sot. Rheol., 12 (1968) 478. 61 P. Oroscz, N. Siskovic, C.R. Huang and R.G. Griskey, J. Phys. E., 6 (1973) 389. 62 D.C.-H. Cheng, Rheol. Acta, 4 (1965) 257. 63 C.C. Mill, J. Oil Colour Chem. Assoc., 51 (1968) 861. 64 W.H. Bauer, A.P. Finkelstein and S.E. Wiberley, A.S.L.E. Trans., 3 (1960) 215. 65 V.A. Fedotova, P.I. Paraska and P.A. Rehbinder, Kolloidn. Zh., 33 (1971) 277. 66 N. Petrellis and R.W. Flumerfelt, Can. J. Chem. Eng., 51 (1973) 291. 67 H. Green, Industrial Rheology and Rheological Structures J. Wiley, New York 1949. 68 E.K. Fischer, Colloidal Dispersions, J. Wiley, New York 1950. 69 R.N. Weltmann, in F.R. Eirich (Ed.), Rheology Theory and Applications, Vol. 3, Academic Press, New York, 1960, p. 189. 70 G.W. Govier and M. Fogsrasi, J. Can. Petrol. Techn., (1972) 42. 71 T.W. Martinek and D.L. Kiass, N.G.L.I. Spokesman, 29 (1965) 219. 72 M.A. Kassem, A.A. Kassem and H.A. Salama, Fette Seifen Anstrichm., 72 (1970) 366. 73 P.F.S. Cartwright, Brit. Inkmaker, (1966) 83. 74 W. Schempp and H.T. Tran., Das Papier, 31 (1977) V34. 75 P. Lepoutre and A.A. Robertson, TAPPI, 57 (1974) 87. 76 J.E. Bujake, J. Pharm. Sci., 54 (1965) 1599. 77 M. Woodman and A. Marsden, J. Pharm. Pharmacol., Suppl., 18 (1966) 198s. 78 H.C. Mital and J. Adotey, Pharm. Acta Helv., 47 (1972) 508. 79 W.H. Bauer, N. Weber and S.E. Wiberley, J. Phys. Chem., 62 (1958) 106. 80 M.J. Groves and A.B. Ahmad, Rheol. Acta, 156 (1976) 501. 81 S.J. Higgs and R.T. Norrington, Process Biochem., 6 (1971) 52. 82 J.R. Smith, T.L. Smith and N.W. Tschoegl, Rheol. Acta, 9 (1970) 239. 83 0-F. Ferraro, Mod. Plast., 46 (1968) 98. 84 R.J. Lippe, S.P.E. Ann, Conf., 15 (1969) 66. 85 S.A. Nikitna and L.I. Peregudova, Kolloidn. Zh., 35 (1973) 977. 86 T. Ohnushi, J. Polym. Sci., 62 (1962) 842. 87 B. Bloch and L. Dintenfass, Aust. N.Z.J. Surg., 33 (1963) 108. 88 C.E. Miller and H. Goldfarb, Trans. Sot. Rheol., 9 (1965) 135. 89 CR. Hunag, N. Siskovic, R.W. Robertson, W. Fabrisiak, E.H. Smitherberg and A.L. Copley, Biorheology, 12 (1975) 279. 90 U. Hoffman, Kolloid Z., 216 (1967) 370. 91 M. Schijn, Rheol. Acta, 9 (1970) 1.
  • 19 92 93 94 95 96 F.L. Saunders, J. Colloid Sci., 23 (1967) 230. G.K. Moiseev and A.N. Sevorov, Kolloidn. Zh., 35 (1973) 580. H. Freundlich, Trans. Inst. Rubber Ind., 11 (1935) 55. F.H. Miiller and H. Martin, Kolloid Z.Z. Polym., 119 (1961) 119. H. Sonntag and K. Strenge, Koagulation und Stabilitiit disperser Systeme, VEB Deutscher Verlag Wiss., Berlin, 1970, 97 G.D. Parfitt (Ed.), Dispersion of Powders in Liquids, Elsevier, London 1969. 98 H. Freundlich, Thixotropy, Hermann and Cie, Paris, 1935. 99 U. KSnig, Chem. Ztg., 92 (1968) 343. 100 J. Mewis and A.J.B. Spaull, Adv. Colloid Interface Sci., 6 (1976) 173. 101 A. Weiss, R. Fahn and U. Hofmann, Naturwisschenschaften, 39 (1952) 351. 102 H. van Olphen, Proc. 4th Nat. Conf. on Clays and Clay Minerals, Nat. Res. Council, Wash., 1956, p. 204. 103 J.Th.G. Overbeek, J. Colloid Interface Sci., 58 (1977) 408. 104 J.B. Yannas and R.N. Gonzalez, Nature, 191 (1961) 1384. 105 D.C.-H. Cheng, (a) Nature 245 (1973) 93. (h) Warren Spring Lab. Report, 1973. 106 J. Crane and D. Schiffer, J. Polym. Sci., 23 (1957) 93. 107 J.G. Savins, Rheol. Acta, 7 (1968) 87. 108 A. Slibar and P.R. Paslay, in M. Reiner and D. Abir (Eds.), Proc. Int. Symp. on Second-order Effects in Elasticity Plasticity and Fluid Dynamics, Pergamon Press, Oxford, 1964, p. 314. 109 D.C.-H. Cbeng and F. Evans, Brit. J. Appl. Pbys., 16 (1965) 1599. 110 J. Harris, Rheol. Acta, 6 (1967) 6. 111 J.G. Oldroyd, Proc. Cambridge. Philos. Sot., 100 (1947) 396 and 521. 112 H. Brenner in W.R. Schowalter et al. (Eds.), Progress in Heat and Mass Transfer, Vol., 5, Pergamon Press, 1972, p. 89. 113 G.K. Batchelor, Ann. Rev. Fluid Mech., 6 (1974) 227. 114 G. Astariia, IUTAM Meeting, Louvain-la-Neuve 1978, published in J. Non-Newtonian, Fluid Mecb. 5 (1979) 125-140. 115 B.T. Storey and E.W. Merrill, J. Polym. Sci., 33 (1958) 361. 116 D.A. Denny and R.S. Brodkey, J. Appl. Phys., 33 (1962) 226. 117 K.L. Pinder, Can. J. Chem. Eng., 42 (1964) 132. 118 A.G. Frederickson, AIChEJ., 16 (1970) 436. 119 M. van den Tempel, in P. Sherman (Ed.), Emulsion Rheology, Pergamon Press, Oxford, 1963, p.1. 120 R.A. Ritter and G.W. Govier, Can. J. Chem. Eng., 48 (1970) 505. 121 D.C.-H. Cheng, Rheol. Acta, 12 (1973) 228. 122 D.C.-H. Cheng, Nature, 245 (1973) 93. 123 D.C.-H. Cheng, J. Phys. D., 7 (1974) L155. 124 J. Mewis, J. Phys. D., 8 (1975) L148. 125 J. Mewis, A.J.B. Spaull and J. Helsen, Nature, 253 (1975) 618. 126 N. Casson, in C.C. Mill (Ed.), Rheology of Disperse Systems, Pergamon Press, Oxford, 1959, p. 84. 127 H.D. Weymann, M.C. Hunag and R.A. Ross, Phys. Fluids, 16 (1973) 775. 128 E. Ruckenstein and J. Mewis, J. Colloid Interface Sci., 44 (1973) 532. 129 C.E. Chaffey, J. Coiioid Interface Sci., 56 (1976) 495. 130 S.J. Hahn, T. Ree and H. Eyring, Ind. Eng. Chem., 51 (1959) 856. 131 N.V. Mikhailov and A.M. Lichtheim, Kolloidn. Zh., 17 (1955) 364. 132 G.M. Bartenev and N.V. Ermilova, Kolloidn. Zh., 31 (1969) 169. 133 P.F. Ovchinnikov, N.N. Kruglitskii and N.V. Mikhailov, Reologiyz Tiksotropnykh Sistem, Nauka Dimka, Kiev 197 2. 134 C-R. Huang, Chem. Eng. J., 3 (1972) 100. 135 G.C. Sarti and G. Astarita, Trans. Sot. Rheol., 19 (1975) 215.
  • 20 136 A.J.B. Spauii, J. Chem. Sot. Faraday Disc. I., 73 (1977) 128. 137 D. Acierno, F.P. La Mantia, G. Marrucci and G. Titomanlio, J. Non-Newtonian Fluid Mech., 1(1976) 125. 138 W.E. Lewis and R.S. Brodkey, in S. Onogi (Ed.), Proc. 5th Int. Congr. Rheol., Vol. 4, Tokyo Univ. Press, Tokyo, 1970, p. 141. 139 AA. Trapeznikov, Dokl. Akad. Nauk SSSR, 102 (1955) 1177. 140 A.A. Trapeznikov, Russ. J. Phys. Chem., 41(1967) 664. 141 I.N. Achverdov, Dokl. Akad. Nauk B. SSR, 16 (1971) 992. 142 I.A. Glushkov, G.V. Vinogradov and V.A. Rozhkov, Polym. Mech. 10 (1976) 779. 143 M. van den Tempel, J. Colloid Sci., 16 (1961) 284. 144 G. Schoukens and J. Mewis, J. Rheol., 22 (1978) 381. 145 A. Siiberberg and G. Tzur, in C. KIason and J. Kubat (Eds.), Proc. 7th Int. Congr. Rheol., Gothenburg, 1976, p. 129.