trixotropia barnes (1996)

ELSEVIER J. Non-Newtonian Fluid Mech., 70 (1997) 1-33

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Thixotropy a review

Howard A. Barnes

Unilever Research Laboratory, Bebington, Merseyside L63 3JW, UK

Received 16 November 1996; received in revised form 16 December 1996; accepted 6 January 1997

Abstract

The ensuing mechanical response to stressing or straining a structured liquid results in various viscoelastic phenomena, either in the linear region where the microstructure responds linearly with respect to the stress and strain but does not itself change, or in the nonlinear region where the microstructure does change in response to the imposed stresses and strains, but does so reversibly, The complication of thixotropy arises because this reversible, microstructural change itself takes time to come about due to local spatial rearrangement of the components. This frequently found time-response of a microstructure that is itself changing with time makes thixotropic, viscoelastic behaviour one of the greatest challenges facing rheologists today, in terms of its accurate experimental characterisation and its adequate theoretical description. Here a history of thixotropy is given, together with a description of how it is understood today in various parts of the scientific community. Then a mechanistic description of thixotropy is presented, together with a series of applications where thixotropy is important. A list of different examples of thixotropic systems is then given. Finally the various kinds of theories that have been put forward to describe the phenomenon mathematically are listed. 1997 Elsevier Science B.V.

Keywords: Microstructure; Stress; Strain; Viscoelastic

1. Introduction

The growing use of clay-based structurants together with the increasing presence of flocculated structures in home, personal and chemical products and precursors has led to the appearance o f th ixotropy in a widening range of situations, quite apart f rom its presence in systems long known to display the phenomenon. Difficulties then arise in mixing and handl ing these materials because thixotropic structures progressively break down on shearing and slowly rebuild at rest. The time-scales involved can range from many minutes in the case of breakdown to many hours in rebuilding.

0377-0257/97/$17.00 1997 Elsevier Science B.V. All rights reserved. PII S0377-0257(97)00004-9

2 H.A. Barnes/J. Non-Newtonian Fluid Mech. 70 (1997) 1-33

Thixotropy has been deliberately built into products to make them usable by non-experts-- with the best-known example being thixotropic paints--however, as will be shown here, what is usually wanted in these cases is extreme shear-thinning. However the way in which this is brought about usually introduces thixotropy as well, which is then almost always an unwanted nuisance. However the phenomena still has to be understood, and hence the need for an up-to-date review.

Major post-war reviews of thixotropy have been produced by Bauer and Collins, 1967 [1], Mewis, 1979 [2], Cheng, 1982 [3], and Godfrey, 1983 [4]. While the general areas they cover are also dealt with here, they are well worth consulting for interesting examples of thixotropic systems not cited here.

2. A history of thixotropy

2.1. Origins

In 1923, Schalek and Szegvari found that aqueous iron oxide "gels have the remarkable property of becoming completely liquid through gentle shaking alone, to such an extent that the liquified gel is hardly distinguishable from the original sol. These sols were liquified by shaking, solidified again after a period of time ... the change of state process could be repeated a number of times without any visible change in the system" [5]. The term thixotropy was then coined by Peterfi in 1927 [6], in the first paper that properly described the phenomenon. The work combines the Greek words thixis (stirring or shaking) and trepo (turning or changing).

Although no mention of the phenomenon appeared in the seminal rheology text of the day 'The Viscosity of Liquids', by Emil Hatschek [7], (especially the chapter on colloidal solutions), by 1935 Freundlich had published a book called 'Thixotropie' [8] devoted to the subject, having been the first to introduce it into the title of a paper when he described the flow properties of aluminium hydroxide gels. Freundlich and co-workers soon found thixotropic effects manifested by a whole variety of systems including vanadium pentoxide sols, starch pastes, gelatin gels, pectin gels and many more.

Thixotropy originally therefore referred to the reversible changes from a flowable fluid to a solid-like elastic gel. Previously these kinds of physical changes had only been known to occur by changing the temperature, when such gels would melt on heating and then re-solidify on cooling. It was believed that a new kind of phase change had been found.

2.2. Progress

Early work in this area in the USA is exemplified by a series of three papers by McMillen in 1932 [9], reporting the results of his doctoral investigations into the thixotropy of a large number of flocculated paints. He showed that the fluidity (the inverse of viscosity) as a function of rest time decreased in some cases by four orders of magnitude, showing almost a quadratic dependence on rest time.

Writing in the UK in 1942, Scott-Blair [10] stated that 'the whole subject [of thixotropy] is so very new'. But then went on to list over 80 papers on the subject (see pp. 61-64). (In the second

H.A. Barnes/J. Non-Newtonian Fluid Mech. 70 (1997) 1-33 3

edition of this book published in 1949, nearly 120 papers on thixotropy are cited.) Among the examples of thixotropic materials he gives are clays and soil suspensions, creams, drilling muds, flour doughs, flour suspensions, fibre greases, jellies, paints, carbon black suspensions and starch pastes. He also lists a number of papers on so-called thixotrometers, instruments specially devised to characterise the phenomenon. In this respect he raised some interesting points, among them whether thixotropy ought to be studied at constant rate of shear or at constant stress? This is still a most controversial question.

Scott-Blair quotes Hamaker's explanation of thixotropy as being due to the secondary minimum so that 'particles can form a loose association which is easily destroyed by shaking but re-establishes itself on standing'. This explanation still stands. With our present knowledge of microstructural changes, it is probably safe to say that all materials that are shear thinning are thixotropic, in that they will always take a finite time to bring about the rearrangements needed in the microstructural elements that result in shear thinning. As Scott-Blair concluded all those years ago "If this recovery is very rapid, the phenomenon is observed as structural viscosity [shear thinning]; if slow, it is observed as thixotropy". However even Scott-Blair sometimes confused thixotropy with shear thinning, as in his example of the importance of thixotropy for drilling muds that must be runny [sic] when lubricating the drill, but "of a high enough consistency at rest to avoid settling of suspended matter".

An important point he made concerned a suggestion that certain results of flow in capillary tubes of suspensions--that we now believe showed migration of particles away from the wall and thus, have an easier flow in small rather than large tubes--was due to thixotropy. He refuted this by showing that doubling the tube length halved the flow rate for a given driving pressure.

Pryce-Jones [11] (the first well-known Welsh rheologist) studied about 250 paints all in a state of light flocculation, using his own thixotrometer [12]. He noted that "It is a well-established fact that thixotropy is more pronounced in systems containing non-spherical particles", this is obviously so because they have to find themselves in the best 3D structure by rotation as well as movement, and progress from a solid gel to a freely flowing liquid due to complete microstructural breakdown, see Fig. 1.

Thixotropy is one of the few original technical terms used in pre-war, European rheology circles that has survived, unlike 'structural viscosity' (Strukturviskositaet, which we now understand as shear thinning) and 'false body' (now understood as extreme shear thinning with thixotropy) which have fallen by the wayside ~.

However, as late as 1953, Roscoe [13] still referred to 'false body' as different from thixotropy. The 'false body' had an apparent yield stress [stress at low shear rate following shearing at a high shear rate] that recovered quickly, while the thixotropic material takes some times before relatively fast recovery takes place. Today we understand that they are both manifestations of thixotropy. False bodies were taking a long time to die.

Jobling and Roberts in 1957 [14] commented that "thixotropy now has an even less distinct connotation. Electronic methods of measurement have shown that the time-lag required before

Readers with an interest in the historical derivation of scientific expressions are directed to Scott-Blair [10], p. 52. All Scott-Blair's books were written as personal memoirs and are very evocative of the man himself for those who knew him.


the original structure is regained may be very short indeed and it then becomes difficult to distinguish between a thixotropic material with a very short recovery time and a material whose viscosity falls with increasing rate of shear and depends for all practical purposes only on the instantaneous rate of shear. The latter effect is frequently called 'structural viscosity'". They went on to say "We endorse Pryce-Jones's plea that in the absence of authoritative definitions, terms such as ... thixotropy should not be used unless the intended meaning is made clear". In the Discussion section of this paper, Marcus Reiner notes that 'structural viscosity' and 'thixotropy' are seen as the same thing by some, with structural viscosity seen as a material with "nearly zero time of recovery".

The full extent of thixotropy was maintained by Bauer and Collins in their 1967 review [1]: "When a reduction in magnitude of rheological properties of a system, such as elastic modulus, yield stress, and viscosity, for example, occurs reversibly and isothermally with a distinct time dependence on application of shear strain, the system is described as thixotropic". They went on to say that thixotropy was "usually conceived as an unusual property of very special materials, sol-gel systems such as aqueous iron oxide dispersions, thixotropy in the sense described above has been found to be exhibited by a great many and a large variety of systems. Along with the breakdown in structure, other non-rheological features change, such as conductivity and dielectric constant". Lastly they noted that "The terms used by Freundlich are now seen to be archaic, viz liquefaction, re-solidification, sol. These had some obvious meaning for the qualitative changes brought about in low concentration dispersions of highly insoluble oxides of needle-like crystals such as iron oxide and vanadium pentoxide in low-viscosity aqueous media".

Nowadays thixotropy is sometimes used to include all time effects in a movement to non-linear behaviour, see for instance Cheng [15], but especially Lapasin and Pricl [16], who illustrate thixotropic behaviour by the transient response of viscosity and normal force of

S~KING / Ski,,. . . . . / e

Completely structured- giving ~)~4?>'ft'~,/~--T:~--~.~_2LI ~'~-'~'A ~"~//~\~,~,,V~f ' elastic, so,d-,ke response ." I I ."/.

Partly structured - giving '-....._~" ~-~'~- yt~f ~/ t /~-" ; I "-k viscoelastic response i/,~'~/-"~/',.-,~-.,~#~ ~" ~ ~,~ ~i

Completely unstructured ~ '~/4~ ~ - giving viscous, shear

-thinning response

Fig. 1. Breakdown of a 3D thixotropic structure.

H.A. Barnes / J. NonNewtonian Fluid Mech. 70 (1997) 1-33 5

polymer solutions. They then noted that the stress overshoot on start-up increases with increasing rest time. This is an interesting point--build up in polymer solutions is usually considered to be rapid, and rest times are rarely considered necessary. Weakly cross-linked gels would give the same thixotropic effect as a flocculated system. 2.3. How is" thixotropy generally understood today7

One of the first definitions of thixotropy was given by Freundlich and Rawitzer [17] who stated that "By thixotropy is meant the phenomena of concentrated gels ... which solidify to gels which may again be liquified to sols. The resolidification occurs repeatedly, at constant temperature with a constant speed". (This would be very far from the kind of definition offered today.) However, Pryce-Jones [11] soon afterwards stated that the true meaning of thixotropy was "an increase of viscosity in a state of rest and a decrease of viscosity when submitted to a constant shearing stress".

It is clear that people using the word thixotropy today fall into two camps: first those who understand it in the latter (Pryce-Jones) sense as the time response of the microstructure brought about by shearing or resting, respectively, and the rheological effects arising therefrom. In these circles it is often used in a very narrow sense of viscosity changes only, with no reference to the reversible transition from gel-like to fluid-like behaviour.

Secondly there are those--often in industrial circles--who understand thixotropy in its original Freundlich and Rawitzer sense, as stated above, of conferring gel-like properties to a liquid which disappear on shaking but reappear on standing. This particular property adds considerable advantage to the practical use of materials such as paints, adhesives and coatings. Unless this different use is borne in mind, severe misunderstand will arise on reading the general literature. The former group would understand the use of thixotropy by the latter group as conferring extreme shear thinning to a liquid by, for instance, the addition of so-called thixopropes or by inducing flocculation. On the other hand the latter group only see the temporal properties of thixotropy as an irritation, because the desired reversible gelled state they want takes some time to disappear on shearing or reappear on standing. It is not surprising therefore that thixotropy has, on occasions, been confused with shear thinning.

There are various definitions of thixotropy offered in the current general scientific literature, e.g., scientific dictionaries and encyclopedias, that reflect these two points of view. Some are misleading, with even the best being incomplete. The following are a selection that illustrate the situation:

Oxford Encyclopedic Dictionary of Physics, [18]--"Thixotropy: Certain materials behave as solids under very small applied stresses but under greater stresses become liquids. When the stresses are removed the material settles back into its original consistency. This property is particularly associated with certain colloids which form gels when left to stand but which become sols when stirred or shaken, due to a redistribution of the solid phase."

Chambers Dictionary of Science and Technology [19]--"Rheological property of fluids and plastic solids characterised by a high viscosity at low stress, but a decreased viscosity when an increased stress is applied. A useful property of paints, because it makes for a thick film which is nevertheless easily worked."


McGraw-Hill Dictionary of Scientific and Technical Terms [20]--"Property of certain gels which liquify when subjected to vibratory forces, such as ultrasonic waves or even shaking, and then solidify again when left standing. Thixotropic clay: a clay which weakens when disturbed and increases in strength upon standing."

Van Nostrand's Scientific Encyclopedia [21]--"A thixotropic fluid is a fluid whose viscosity is a function not only of the shearing stress, but also of the previous history of motion within the fluid. The viscosity usually decreases with the length of time the fluid has been in motion. Such systems commonly are concentrated solutions of substances of high molecular weight, or colloidal suspensions."

Oxford Concise Science Dictionary [22]--"More common, however, is the opposite effect in which the viscosity depends not only on the viscosity gradient but also on the time for which it has been applied. These liquids are said to exhibit thixotropy. The faster a thixotropic liquid moves the less viscous it becomes. This property is used in nondrip paints (which are more viscous on the brush than on the wall) and lubricating oils (which become thinner when the parts they are lubricating start to move)."

Chambers 20th Century Dictionary [23]--"Thixotropy: the property of gels of showing a temporary reduction in viscosity when shaken or stirred."

Definitions given in more specialised dictionaries emphasise the time aspect of thixotropy:

Polymer Technology Dictionary [24]--"Thixotropy. A term used in rheology which means that the viscosity of a material decreases significantly with the time of shearing and then, increases significantly when the force inducing the flow is removed."

Polymer Science Dictionary [25]--"Time-dependent fluid behaviour in which the apparent viscosity decreases with the time of shearing and in which the viscosity recovers to, or close to, its original value when shearing ceases. The recovery may take place over a considerable time. This may sometimes occur with polymer systems, when molecular disentanglement increases with time of shearing."

The definition of thixotropy in the rheological literature has changed over the years. The (American) Society of Rheology was quoted by Reiner and Scott-Blair in 1949 [26] as having defined thixotropy as "that property of a body by virtue of which the ratio of shear stress to rate of deformation [viscosity] is temporarily reduced by previous deformation". Some time later thixotropy was defined as "a comparatively slow recovery, on standing, of the consistency lost through shearing". However, thixotropy has recently been defined by Barnes, Hutton and Walters [27] (following the 1975 British Standards Institution definition) as the "decrease [in time] of ... viscosity under constant shear stress or shear rate, followed by a gradual recovery when the stress or shear rate is removed". (As we shall see, this is not a comprehensive description of the phenomenon--a better description would cover the temporal rheological response of a microstructure to changes in imposed stress or strain rate.)


It is obvious then that while most rheologists take a general view of thixotropy which covers all time effects resulting from microstructure changes, in the popular image, the older idea of a gel-sol transition on shearing-resting is still often held. Strictly speaking, what is usually meant by thixotropy in that case would now be termed 'extremely shear thinning', to give near solid-like properties at rest but flow under stress, as for instance in thixo-forming using semi-molten metals or the addition of so-called thixotropes to paints.

A better and extended definition of thixotropy is clearly needed, and it should contain the idea of both considerable shear thinning (i.e., gel-fluid transition) and also time changes over and above those encountered when in its structured state the thixotropic material might be viscoelastic with its attendant time effects.

3. Description of the phenomenon

3. I. General considerations

All liquids with microstructure can show thixotropy, because thixotropy only reflects the finite time taken to move from any one state of microstructure to another and back again, whether from different states of flow or to or from rest. The driving force for microstructural change in flow is the result of the competition between break-down due to flow stresses, build-up due to in-flow collisions and Brownian motion. Brownian motion is the random thermal agitation of atoms and molecules that results in elements of the microstructure being constantly bombarded, which causes them to move to a favourable position where they can--given the necessary attractive force--attach themselves to other parts of the microstructure. Very occasionally situations arise where existing weakly attached microstructural elements--brought together by collision during shear--are slowly torn apart by the constant action of the random Brownian motion. In that case, the opposite to thixotropy is seen, i.e., antithixotropy, where flow structures and rest destructures the material.

The term microstructure, as used here, while usually associated in thixotropic systems with flocculated particles, can also mean alignment of fibres; favourable spatial distribution of particles or drops, or entanglement density or molecular associations in polymer solutions. All these determine the level of viscosity and elasticity, and they all take time to change from one state to another under the action of shear and/or Brownian forces. In these cases, the maximum microstructure is seen when alignment and spatial distribution are random (in three dimensions) and entanglement density is at a maximum. Both these conditions result in the greatest viscous (and usually elastic) response. On the other hand, minimum microstructure is when there is maximum alignment with the flow of fibres; the drop or particle spatial distribution is asymmetrical in the flow direction, or there are a minimum number of entanglements or associations--all these leading to minimum viscous and elastic response.

When the timescales displayed in these changes become long compared with either the response time of a viscometer (or rheometer), or the flow-time in a particular flow geometry, we can speak sensibly about thixotropy. These timescales often range from seconds to hours, with rebuilding usually taking much longer than breakdown.

8 H.A. Barnes / J . Non-Newton ian Fluid Mech. 70 (1997) 1 -33

+,a

0

....................... equ i l ibr ium .............................................

0 shearing time

Fig. 2. Shearing a thixotropic liquid after short and long rest times.

3.2. Typical behaviour

If we place a thixotropic material into a viscometer (see Fig. 2) and apply a constant shear rate, the measured viscosity will decrease with time, but it will eventually steady out to a constant value. If we then switch off the shear and allow the material to rest for a long time (without drying or any other artifacts such as sedimentation or separation occurring), and switch the shear on again, the measured viscosity will be initially higher, but it will then again decrease and end up at the same value as that which was seen after the original long-term shearing. However, the level for the original value will not necessarily be the same, because that will depend on how carefully or vigorously the material was initially loaded into the viscometer and how long it was left to rest before shearing.

If on the other hand a third experiment is performed where the material is allowed to come to equilibrium and then allowed to rest for the same time as before, the results will be identical. If now, after equilibrium is achieved, the shear rate is instantaneously decreased to a lower value, the measured shear stress drops instantly, but thereafter it will slowly increase towards a new equilibrium.

(If instead of applying a given shear rate we applied a particular shear stress, then the inverse applies--the shear rate would increase as the structure breaks down and the change to another (lower) shear stress will result in a sudden decrease in shear rate followed by a further drop, see Fig. 3.)

If we now enquire what is happening on a microscale, we can imagine the picture presented in Fig. 4, where the viscosity/shear-rate behaviour of a typical thixotropic material (which for the sake of argument could be a flocculated suspension) is presented. We start from point a where the microstructure at rest is a series of large flocs. Then if the applied shear rate is increased progressively and sufficient time allowed, the floc size decreases until at a high enough shear rate, the floc has disintegrated completely into its constituent primary particles.

We now imagine another experiment where the shear rate is increased step-wise from a to end up at point b. Instantaneously, the floc size will be that appropriate to the shear stress conditions at point a, but as it experiences the higher shear rate at position a', it begins to erode, until it reaches an equilibrium size appropriate to the higher rate. This process can take some time. If now the opposite happens--the shear rate is instantaneously decreased--the individual particles

H.A. Barnes//J. Non-Newtonian Fluid Mech. 70 (1997) 1-33 9

J~

,T i

STEP EXPERIMENTS

. . . . . . . . . . . . . . . . . va lues -~

> Time

> Time

. . . . . . . . . . . . -4_ . . . . . . . 7 ~ ~ . ~ equilibrium

Time

I Time

>

Fig. 3. Two kinds of step experiment.

(which gave the low viscosity) begin to collide and flocculate until the size of the floc so-formed is appropriate to the new, lower shear rate. This process also takes time, but the build-up proceeds at a different rate than does breakdown.

Any concentrated suspension of particles is shear thinning, thus when we consider flocs, they too will show the phenomena as illustrated in Fig. 4. If we imagine that the particles in a floc are permanently glued together and thus, the floc size is fixed, the flow curve of such a suspension of fixed-size flocs would follow the lines shown in the figure according to the floc size. If the floc size is now decreased (and the overall concentration maintained) but again the floc size remains unchanged when sheared, the flow curve will be lower and (for a given concentration) the degree of shear thinning decreases.

m

o m

~a

,t=

shear rate (log scale)

Fig. 4. Microstructure and flow curves of a flocculated suspension.

10 H.A. Barnes /J. Non-Newtonian Fluid Mech. 70 (1997) 1-33

However because we are interested in systems flocculated in a secondary minimum, the floc size is not constant but decreases with increasing shear rate (or shear stress). Because the viscosity decreases with floc size (see above), we now have a double shear-thinning effect. This means that flocculated systems are very shear thinning, see the equilibrium curve in Fig. 4. This extreme shear-thinning also results in such flocculated, thixotropic systems appearing to have a yield stress, since the stress only decreases very slowly as the shear rate is decreased.

The true steady-state behaviour of a thixotropic liquid is seen both after an infinite shearing time at any shear rate or shear stress of interest or infinite rest time. Of course, as all these equilibrium states are approached asymptotically, one comes close to this state after a reasonable rather than an infinite time, but even then, breakdown times of hours and rebuilding times of days might be necessary to fully describe a very thixotropic system.

In a flocculated system, breakdown is towards an equilibrium situation that is governed by a balance of hydrodynamic shear stresses pulling structures apart by erosion, and a combination of Brownian and shear forces building the structure up by collision and accretion of particles which agglomerate into flocs. The forces holding the structure together are colloidal in nature, acting over short distances ( ~ 10 nm) within the composite particle. At rest, only the Brownian rebuilding forces are present, and these are quite small compared to the shearing forces, being of order 1 kT. This means that the rebuilding time can be very long, since this small, random force takes a long time to rearrange large particles that as flocs are getting larger and larger as they move into a most favourable structure, which is then manifested as a higher and higher viscosity.

The typical response to a step-wise change from one steady state condition to another is, in terms of the viscosity, often characterised by the so-called stretched exponential model:

F/ = Y]e,oQ -~- (g /e ,~ - - r /~ ,o ) (1 - - e-(t/~)~), (1)

where r/e,0 is the viscosity at the commencement of shearing, ~/e.~ the viscosity after shearing for an infinite time, z is a time constant and r is a dimensionless constant (which in the simplest case is unity). This equation can cope with build-up and break-down in step-up or step-down tests, with the values of ~ and r depending on both the level and the direction, i.e., going from 10 to 1 s - 1 will not have the same value of f found in going from 1 to 10 s- 1. The values of r also vary with the conditions of the test, as well as the particular system being tested, and they decrease linearly with the log of the shear rate, typically from 0.7-0.9 at l0 -2 s-1 to 0.3-0.4 at 1 s- 1 in Mewis' experiments [28]. He also showed that for a typical flocculated system (fumed silica in paraffin oil), z is a decreasing function of r/e.O ~/~/oont, where qcont is the viscosity of the continuous phase, which when taken into account, properly describes the effect of temperature. (He also reports optical, dielectric and conductivity results on these systems).

Heymann et al. [29] investigated the build-up after shearing of the yield stress of five newsprint inks, with the formulation containing carbon black. A period of pre-shearing was carried out at 1500 Pa, and then the flow curves were measured as a function of time. For those inks, judged to have a yield stress ~y, an equation similar to Eq. (1) was used to describe the rebuilding:

ra t ) = o Oy + [try -- or](1 -- e-~t/~)) (2)


For these materials so described, the values of v were found to be well over 100 s. For this reason, it was judged that the recovery behaviour had no relevance to the printing process, because these process times are of the order of a few seconds at most.

Not only can shear break down or build up flocs, but it can also change their internal morphology. For instance, the work of Mills et al. [30] showed that shearing freshly prepared flocculated suspensions can densify individual floes, causing reorganisation within the flocs. This shows how important it is to 'condition' such systems before shearing them in experiments to elucidate their thixotropic behaviour. They found that following prolonged shearing, loose- packed flocs became tightly packed and more monodispersed. This can be interpreted as a permanent loss of structure--rheomalaxis or rheo-destruction.

The experiments of Wolthers et al. [31] show us how important thixotropy can be in a typical flocculated systems. They examined the behaviour of a depletion-flocculated polymer latex suspension and showed that the shear stress dropped significantly with time, and was still decreasing after 1 h. The values of the initial and final viscosities they found were at least an order of magnitude different. They also found that the unsheared samples were more open flocs while shearing made to fresh flocs more dense.

3.3. Antithixotropy

As we have already seen, given the right kind of attraction between particles, shearing can promote temporary aggregation rather than breakdown, due to the collision of these attractive particles--this results in anti-thixotropy. Doraiswamy et al. [32] have shown that the average agglomerate size of a suspension in shear was given by

r -- = exp[(8~b~Et)/3Tc], (3) F0

where E was the probability that a collision of particles would result in agglomeration, and r and r0 were the radius of the sheared and unsheared flocs, respectively. They then described an experiment where a 12% phase volume suspension of inorganic pigment in a low molecular weight polymeric fluid grows from 3.4 to 88.1 jam after flowing through a 50 m long tube of diameter 0.038 m. This meant that the value of E was as low as 0.006. They pointed out that this value will tend to zero above a critical stress when the agglomerates begin to disintegrate. In separate experiments on a 4 gm glass beads in corn syrup suspension in a cone-and-plate viscometer, they showed that the agglomerate size tended to a steady value after shearing for more than about 2 h. Chang and Smith [33] showed that this kind of behaviour is seen in certain nuclear waste simulate slurries, especially if silica (SiO4) is present.

Antithixotropy can also arise because certain floes can become looser and more open under the action of shear and thus the viscosity increases. One good example of this negative (or anti-) thixotropy is ferric-oxide suspensions in mineral oil, see [34]. Suspensions of acicular, fine maghemite are known to show a strong tendency to flocculate due to the magnetic properties of the particles. Like other such suspensions, there is a range of flow conditions under which shear-enhanced collisions make structure rather than break it. In this case the onset of antithixotropy is around 50 s 1. Below that range the suspensions are thixotropic in the normal way, while above this critical value, strong anti-thixotropy sets in. They conclude that the shear loosens the ftocs, which then have a higher effective phase volume.


viscoelastic response

~ 4

~ equ~

shearing time, t

inelastic, viscous response

shearing time, t

Fig. 5. Different kinds of thixotropic behaviour on start up of shear after various degrees of rest (neglecting instrument inertia).

4. Viscoelasticity and thixotropy

Viscoelastic systems sheared in their linear region show time-dependency because the microstructure takes time to respond to the flow/stress. At short times (high frequencies) the structures cannot respond quickly, and we see an elastic response, while given time (low frequencies), the system can adjust itself continuously, i.e., it can flow, showing viscous effects. Thus, when observed over all time (and frequency) scales, the system is viscoelastic.

For non-linear viscoelastic--or simply inelastic but shear-thinning systems--not only does the microstructure take time to respond to the flow, but it is also changed by the flow and this change will itself take time. This is the essential difference between linear viscoelasticity and thixotropy--that while both are time effects, the former is in the linear region, where the structure responds but remains unchanged and the latter takes place in the non-linear region where the structure is broken down by deformation as well as responding to it.

Shear thinning can occur for many reasons, e.g.,: alignment of rod-like particles in the flow direction; loss of junctions in polymer solutions; rearrangement of microstructure in suspension and emulsion flow and breakdown of flocs.

Since changes in any of these states take some time to come about (either from rest or from some other configuration), thixotropy is always (in principle) to be expected from any shear- thinning mechanism. However thixotropy becomes significant when the time-scale over which it is seen becomes significantly longer than the response times of instruments used to measure rheology or longer than some flow time in a process, e.g., the average time a liquid takes to flow through a pipe. Thixotropic timescales can be longer than viscoelastic timescales and are practically important when these timescales become minutes and longer for breakdown. This will inevitably mean many more minutes or even hours for recovery of the structure. Given the nature of most microstructural features which produce thixotropic effects (see above), it is obvious that at conditions near the fully structured rest state, viscoelastic effects will also be seen. The typical response to a startup experiment from a rested state shows this, see Fig. 5. Only a few theories have sought to account for this effect, which shows concurrently a breakdown/rebuilding of both viscous and elastic responses.

H.A. Barnes /J. Non-Newtonian Fluid Mech. 70 (1997) 1-33 13

The picture presented becomes even more complicated when we consider the linear viscoelastic response of a rebuilding structure, where the typical storage and loss modul i - -G' and G"--curves evolve with time. This can be used as a measure of the rebuilding mode since the test is conducted at low-enough stresses/deformations that the evolving structure is unaffected by the deformation. The growth of G' particularly is very sensitive to structure rebuilding. Barut et al. [35] studied an acrylic polymer solution in a mixture of solvents, with TiO2 or a mixture of TiO2 and alumino-silicate particles. These were pre-sheared in a controlled-stress rheometer at 200 Pa for 2 min and after this ceased, the linear oscillatory properties were monitored for 10 min at strains less than 0.1 over a frequency range 0.03-62.8 rad s 1. The form of the rebuilding curve of the storage modulus was of a stretched exponential form:

G' = Gk- (G" - G;)exp( - ktP). (4)

Williams and Ren [36] used the Virtual Gap Rheometer operating over the range 250-800 Hz to examine the rebuilding of 0.045 g ml-1 aqueous Laponite RD dispersions (synthetic hectorite-type clay, circular discs ,-- 300 x 10 A), again measured by G' but now derived from the phase velocity. At these high frequencies, G' is a particularly sensitive measure of structure as the solid-like response of the clay. They showed that restructuring was significant over the first 10 min or so, but was still going on after 20 min. Viewed on a log scale there is a rise from an initial value at small times and a fast build up through a power-law region, eventually (as it must) flattening out at times greater than 20 min.

Bouda and Mikegovfi [37] simultaneously monitored the AC conductivity and the storage modulus G' to establish the buildup of a carbon black network in a polyethylene melt. After an initiation period where the conductivity was constant, it then increased rapidly. The time a 7.4 weight % sample took to show the rapid increase was nearly 200 rain. Similar behaviour was also observed for G'. They explained the observations as resulting from the setting up of a continuous network, and using percolation theory showed that the observed behaviour was explainable on the basis of diffusion-limited aggregation of small clusters of primary carbon- black particles under the action of Brownian motion. As the flocs collide and stick together they eventually form an interconnecting network as the percolation threshold is reached, at which point both the electrical conductivity and the storage modulus G' rose rapidly, see also [38].

On the other hand, Greener and Connelly [39] point out how easy it is to misinterpret thixotropic loops, particularly if there is viscoelasticity present in the sample being tested. They compare the supposed thixotropic behaviour of an aqueous polyacrylamide solution with the predicted behaviour of the Wagner model and show that all the effects are accounted for by the non-linear viscoelastic behaviour, and not by true thixotropy!

5. Typical thixotropic experiments

5.1. Thixotropic /hysteresis loops

One of the favourite ways of measuring thixotropy is to perform a loop test; that is to say, to linearly increase the shear rate (or sometimes shear stress) from zero to a maximum value, and then to return at the same rate to zero. This test could then be repeated again and again,

14 H.A. Barnes / J . Non-Newtonian Fluid Mech. 70 (1997) 1-33

until eventually, a constant loop behaviour is seen, see Fig. 6. The area between the up and down curve is automatically measured in some computer-based rheometers as a measure of thixotropy.

This kind of test is to be depreciated, for although useful as a quick, qualitative procedure it has a number of bad points. First, the loop test is often carried out too quickly, and inertia effects due to the measuring head are introduced but not always recognised. (However, inertial effects can now be accounted for by some rheometer software packages.) Secondly, a test where both shear rate and time are changed simultaneously, on a material where the response is itself a function of both shear rate and t ime--as thixotropy obviously is-- is bad experimentation, because the response cannot then be resolved into the separate effects arising from both variables. However the problem in interpreting the loop is even more difficult when we realise that the first part of the behaviour on start-up is essentially elastic. As the strain becomes large, this moves to non-linear elastic response. If the behaviour were just linear elasticity, then the strain in a simple loop test increases parabolically, and the curve is concave to start with, but soon turns over. At the same time the viscous behaviour can become apparent in the first tendency to flatten out. The viscous behaviour then itself becomes nonlinear as the structure begins to break down at large strains. As the strain rate increases further, the liquid would like to shear-thin, but this takes time since the structure cannot adjust itself fast enough to respond to the increasing shear rate. When the time is long enough and the structure has broken down, the down curve will be under the up curve. Rebuilding will then begin to take place slowly. Even with an apparatus that responds perfectly to the stress and strain, interpreting the data to obtain the parameters corresponding to a model is very difficult if not impossible.

A simpler and more sensible test for a thixotropic liquid is performing and deriving results from step-wise experiments where the shear rate or stress is changed from one constant value to another with a carefully controlled prehistory. Even so, it is impossible to eliminate an elastic response and instrument inertia.

5.2. Start-up experiments

Any experiment that starts from rest is another kind of thixotropic test. The typical behaviour of strain- or stress-controlled experiments is shown in Fig. 7. Most, if not all, thixotropic

. . . . . . . . . . second loop n'th loop

shear rate, 1/s

Fig. 6. A typical thixotropic-loop test.


time of shearing

Fig. 7. Various flow regimes after start up.

materials that have been at rest for some time show viscoelastic behaviour, so the immediate response in such tests is elastic, then depending on the conditions, a thixotropic response will be seen either as an overshoot in the stress in strain-controlled experiments or an increase in the slope of the strain-time curve in creep tests. This sometimes happens after a critical strain has been achieved. This initial elastic response giving way to a thixotropic viscous response makes the behaviour quite complicated.

5.3. Artifacts involved in measuring thixotropy

The greatest difficulty in understanding and modelling thixotropic materials is knowing the effect of the--often unknown--deformation prehistory on the liquid of interest. This is particularly true in situations where a thixotropic liquid is subject to mixing and pumping, or even the seemingly simple task of filling a viscometer or rheometer. In both cases it would be interesting to be able to predict the subsequent flow even if only the initial response was known, say the initial torque on a viscometer operating at a certain shear rate, or the torque on a mixing vessel prior to the liquid being pumped out.

A number of methods have been devised to establish a consistent initial condition: fixed rest time after sample loading; pre-shear at a prescribed shear rate for a prescribed time followed by a set rest period; pre-shearing to equilibrium at a low shear rate followed by testing at a higher shear rate, etc. These eliminate the problem in characterisation, but they can never completely eliminate the problem in practise, since the effects of prehistory on a previously untested sample is always unknown.

The mechanical inertia of the rotating members in viscometers and rheometers means that in experiments where the shear rate or stress is changed quickly or instantaneously, the instrument response is delayed, and this is often mistaken for thixotropy, and, even if thixotropy is present, it can complicate its measurement. Sometimes the presence of a low compliance spring complicates the measurement because the output of the experiment relates to the spring winding up.

For thixotropic materials, as we have seen, a typical source of the phenomenon is breakdown of flocs. However the appearance of apparent slip at the wall, e.g., depletion, can also arise from flocs because they are large, see Barnes [41]. This is a case where observing the pressure gradient

16 H.A. Barnes / J . Non-Newtonian Fluid Mech. 70 (1997) 1-33

alone will not differentiate or account for both effects. A detailed investigation of the internal flow profile will alone show what is happening. As well as pipe flows, there are many apparently simple flows where the shear rate or shear stress is not constant spatially, for instance wide-gap concentric cylinders and parallel plates. These geometries are often used in the rheological characterisation of thixotropic liquids, and unwittingly many workers are unaware of the complications.

6. Engineering consequences of thixotropy

6.1. Flow in mixers

Edwards et al. [40] found that the behaviour of a range of thixotropic materials in a series of mixers was quite easy to characterise if one assumed that the mixer behaved in the same way as a viscometer running at the same shear rate as the average shear rate in the mixer. They used an average shear rate for the flow in a cylindrical vessel with anchor, helical ribbon and helical screw impellers, given by the impeller rotational speed N (rev s- l) times a constant depending on the impeller geometry, k, where values of k ranged from about 12 for the helical screws, around 20 for the anchor to 30 for the helical ribbon. They compared the torque produced in the mixer with the signal from a viscometer running at the same shear rate, both of which could halve over the course of the experiment. For salad cream, tomato ketchup, yoghurt, paint and 3 and 4% aqueous Laponite dispersions, they found that the average viscosity measured as a function of time in the mixer at a given impeller speed compared well with that from a viscometer running at the same shear rate. For the salad cream, tomato ketchup, yoghurt and paint, the viscosities agreed to within 10%. The predicted values for 3 and 4% Laponite agreed reasonably well for the anchor and helical ribbon, but were between 20 and 40% too low for the helical screws. This latter fact was probably due to the very non-Newtonian and thixotropic nature of the Laponite dispersions.

6.2. Flow in pipes

When a thixotropic liquid enters a long pipe from a large vessel where it has been allowed to rest, the development of the velocity and pressure fields in the pipe is very complicated. The large pressure involved in the start-up of flow of a thixotropic liquid can cause significant problems in terms of the necessary pump performance. Often cavitation can be caused because, although a pump could cope with sheared material, it might be unable to initiate flow of material that has been at rest for some time. Cavitation in the liquid within the pump can then ensue.

Once flow has started, the liquid near the pipe wall is subjected to the highest shear rate but the lowest velocity, hence it is subjected to the shear for longer than the fluid flowing at the middle of the pipe. This results in a very fast and prolonged breakdown near the wall, giving a low viscosity layer that effectively lubricates the inner, more-viscous layers. If the pipe is long enough, the flow profile will evolve such that eventually the steady state profile is established. However, for short pipes, characterising the flow can be quite complicated, with a non-linear


pressure profile down the pipe being a distinct possibility. Distinguishing thixotropy from a developing slip layer caused by particle depletion can be very difficult (see Barnes [41]).

Simple approaches have been made to analyze this problem mathematically, see Godfrey's summary [4], but the ones worth mentioning are due to Kemblowski and Petera [42] and Cheng and co-workers [43]. The kinds of important situations analyzed include start-up of fully structured thixotropic fluids and then steady-state flow, in both cases with either a constant flow rate or a given pressure drop. The analysis was complex and required extensive numerical computation.

7. Examples of systems and studies from the literature

There is a very large number of systems that have been found to be thixotropic; previous reviews have listed many examples (see Bauer and Collins [1], Mewis [2], Cheng [3], and Godfrey [4]). Here a set of largely new examples are given, with special emphasis on hitherto unreported eastern European and oriental studies. In order to save space, a large number of examples are presented in tabular form in Tables 1 and 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8 Table 9 Table 10, grouped appropriately, with each paper summarised in one phase. Reference can be made to them at the reader's convenience. Then follows a more detailed discussion of some recent examples.

7.1. Thixotropic paints, inks and coatings (see Table 1)

When coatings are applied to vertical or inclined surfaces, the time taken for rebuilding to occur can cause the material to drain. This is obviously undesirable. The thixotropic breakdown of paints is important when such paints are being put onto the brush or brushed out. The desirable properties of non-drip might appear quite quickly, but the paint has to be worked to make it thin enough to apply evenly.

7.2. Thixotropic detergent systems (see Table 2)

Thixotropy in commercial detergent liquids can give rise to problems when they have to be poured from containers or poured into machines. Then dispersion can be a problem. If rebuilding is very slow, physical instability can result due to sedimentation or creaming.

7.3. Thixotropic clay suspensions (see Table 3)

Clays are probably the best known examples of thixotropy, because of the extreme changes brought about under shear. A clay suspension can be shaken in a bottle, and the sound generated is almost water-like, but on standing the clay can become completely gelled and manifests a ringing sound if tapped in a glass container. Clays such as the natural Bentonites and the manufactured and modified Laponites, because of the extensive nature of the very thin sheets from which they are made up, give a very good thickening effect at relatively low concentrations, without giving unwanted viscoelastic effects found with some kinds of polymeric

18

Table 1 Paints, inks and coatings

H.A. Barnes/J. Non-Newtonian Fluid Mech. 70 (1997) 1-33

Author(s) Ref. Details

Overdiep Kuwano et al. Iizuka and Na-

ganuma Cornelius Galli Kutik Kerle Pila Papo and Sturzi Parmentier Dubyaga et al. Alessandrini Walton Craw ford Staus and Edser Berry Iizuka and Na-

ganuma Kobs and Voigt Tsuritani Toussaint

[44] Effect of thixotropy on film formation [45] Thixotropy in automobile paints [46] Review of thixotropes for paints

[47] Thixotropy in urethane modified alkyd resins [48] Fumed and precipitated silica, and clays as thixotropes for plastic coatings [49] Amorphous silica gel as thixotropes [50] Thixotropy in polyester-based thixotropic resins [51] Thixotropes for vinyl resin thick layer systems [52] Bentone as a thixotrope for oxidised bitumen protective coatings [53] Hydrogenated castor oil as thixotrope for wood treatment coating [54] Effect of thixotropy on film properties of polyether polyurethanes [55] Montmorillonite thixoptrope in oxidised bitumen paint [56] Production of thixotropic paints [57] Review of advantages of thixotropy for all coatings [58] Thixotropic vinyl-based polymer household emulsion paint [59] Thixotropic alkyd-resin and polyamide-based household paints [60] Thixotropes for high-solids coatings

[61] Effect of thixotropy on thick film screen printing [62] Thixotropy of printing suspension and paste inks [63] Thixotropy in organic coatings

thickeners. However because of the size of the clay plates the rate of structuring is very slow and also the difference between the fully formed structure at rest and the flowing dispersion is very large.

7.4. Thixotropic oils and lubricants (see Table 4)

Greases are thixotropic because of the flocculation of the dispersed material suspended in the oil phase. This is very important in lubricating situations where the grease has to break down on shearing so that no unwanted extra drag is experienced in bearings.

Table 2 Detergent systems


Wolff [64] Rounds [65] Moreno et al. [66]

Moreno et al. [67]

Thixotropy in micellar surfactant solutions containing aromatic solubilisers Thixotropic in rod-like micellar surfactant solutions Thixotropy in systems containing anionic surfactant, ethoxylated alcohol, and sodium or triethanol-ammonium sulfate Effect of ethanol, urea, or sodium silicate on thixotropic properties of commercial heavy duty liquid detergents


Table 3 Clay suspensions


Aripov Legrand and

Costa Haret and Zarea Shevchuk and

Shishkevich Orazmuradov et

al. Ryabchenko and

Agabal'yants Schulz Egashira Rosenqvist Nakaishi and Ya-

sutomi Bendukidze et al. Txoperena

[68] Thixotropy of montmorillonite dispersions [69] Effect of shearing on thixotropic bentonite muds

[70] Effect of vibrations, electrolytes and temperature on thixotropy of clays [71] Effect of oil-water emulsions on thixotropy of montmorillonite-based muds

[72] Effect of electrolyte and various colloids on thixotropy of bentonite clay suspensions

[73] Orientation-thixotropic effect in aqueous dispersions of montmorillonite clays

[74] Review of modified Bentonite (organoclay) organophilic thixotropes for coatings [75] Effect of colloid chemistry on thixotropy of clays [76] Thixotropy in quick clay [77] Effect of time scale of measurement on flow of clay suspensions

[78] Effect of Bentonite on storage properties of chlorinated rubber enamels [79] Thixotropy in coatings for wood paints and varnishes

7.5. Thixotropic coal suspensions (see Table 5)

Coal-oil and coal-water suspensions show considerable thixotropy, and problems with start up of pumps after stopping the flow can cause problems. The pump duties required for flow of sheared suspensions are very different from start up of a rested suspension. This can result in pump failure, since start up torques can be very high.

Table 4 Oils and lubricants


Wislicki [80] Thixotropic properties ester-based synthetic lubricating oils Czarny [81] Thixotropy in lithium and calcium-based greases Szilas [82] Thixotropy in crude oils flowing in pipelines

20

Table 5 Coal suspensions

H.A. Barnes / J . Non-Newtonian Fluid Mech. 70 (1997) 1-33


Usui [83] De Jong [84] Chakraborty [85] Sabadell et al. [86]

Thixotropy and physical stability of coal slurries Thixotropy of coal-water mixtures Thixotropic coal-water suspensions Thixotropy of coal-oil suspensions

Table 6 Metal slushes


Erz Vives et al. Flemings et al. Mada and

Ajersch

[87] Thixocasting--a casting process for near-net-shape manufacture aluminium alloys [88] Thixotropy of two-phase metallic slurries [89] Review of thixo-casting [90] Thixotropic of semi-solid aluminum-6% silicon carbide alloy reinforced with silicon car-

bide particles

7.6. Thixotropic metal slushes (see Table 6)

If metals are sheared just below their melting point, they take on the appearance of a shear-thinning liquid. However, they have been described as thixotropic. While they might show some thixotropy, the title is probably a misnomer because shear thinning is perhaps even more important than thixotropy. In fact, as stated above, thixotropy is probably a nuisance, because what is needed is a fast-responding, very shear-thinning, liquid-like material for casting.

7. 7. Thixotropic rubbers solutions (see Table 7)

The manufacture of black rubber tyres etc., uses carbon black. When dispersed, the particles are attracted to one another, and form a network throughout the rubber solution. The rubber solution itself is shear thinning, and the carbon black network also renders a degree of shear thinning, but it is also thixotropic.

Table 7 Rubber systems


Gent and [91] Alan

Lomov et al. [92] Katishonok et [93]

al. White [94]

Thixotropy in rubbers containing particulate fillers

Thixotropy of nitrile rubber system with combined steady-state flow and vibration Thixotropic for solutions of chlorinated rubber

Review of thixotropy in carbon-black rubbers

H.A. Barnes/J. Non-Newtonian Fluid Mech. 70 (1997) 1-33

Table 8 Food and biological systems

21


Benezech and Main- [95] Thixotropy in set, stirred and drinking yoghurts gonnat

Noik et al. [96] Battistoni et al. [97] Morris et al. [98] Baxter [99] Divjak [100]

Thixotropy of xanthan protein complexes obtained by fermentation Thixotropy of digested municipal solid-wastes Thixotropy of acetan and related xanthan gum polysaccharide solutions Thixotropy of sludge in anaerobic digestors Thixotropy in xanathan-gum-based low-calorie mayonnaises and salad dressings

7.8. Thixotropic food and biological systems (see Table 8)

Many food and biological systems are well known examples of thixotropy. For instance, stirring yoghurts makes them thinner, but leaving them to rest thereafter thickens them again. Thixotropy in food thickeners such as xanthan gums can cause problems in that the suspending properties they give to liquids may take time to appear after shearing and this could cause some initial sedimentation or creaming of suspended material.

7.9. Thixotropic creams and pharmaceuticals products (see Table 9)

Creams and other personal-product and pharmaceutical materials are given 'body' by using materials--so-called thixotropes--that happen to be thixotropic. Here the original meaning of thixotropy as conferring gel-like properties is still very often the controlling idea. The time effects seen in using these materials are therefore again only of nuisance value.

7.10. Other thixotropic suspensions (see Table 10)

Some other examples are cited where thixotropy is seen in structured liquids of different kinds, most of which are flocculated in some way, ranging from flocculation induced by decreasing droplet size in emulsions through to the deliberate addition of thixotropes.

Table 9 Creams and pharmaceuticals


Okor [101] Junginger [102] Matsumoto and Nakata [103] Assmus [104] Yamamura [105]

Effect of phenol on thixotropy of pharmaceutical gels Thixotropy of cosmetic and dermatological cream bases Effect of natural waxes on thixotropy of cosmetics Thixotropy in some gelling cosmetic oils Natural wax as a thixotrope for cosmetic pharmaceuticals

22 H.A. Barnes /J. Non-Newtonian Fluid Mech. 70 (1997) 1 33

Table 10 Other thixotropic suspensions


Ueda Faitel'son Nemets et al. Shul'man et

al. Veliev and

Abdinov Stepita Ito Gao et al. Arima and

Eguchi Kruger and

Hulsenberg Fedotova Sakka and

Kozuka Tretinnik

[106] Thixotropy in fine emulsions produced using a high-pressure homogenizer [107] Thixotropic properties of epoxy resin polymers [108] Thixotropy in aluminum and zirconium oxides suspensions [109] Thixotropy of electrorheological suspensions

[110] Thixotropy in magnetically active (ferromagnetic dust) liquids

[111] Alkali metal silicates and montmorillonite as thixotropes in limestone-based plaster [112] Thixotropy in silica(Aerosil)-based adhesives [113] Thixotropes (silicas, calcium carbonate or PVC) in unsaturated polyester resin [114] Aerosil as a thixotrope

[115] Effect of electrolyte on thixotropy in ceramic slips

[116] Effect of additives on aqueous poly(vinyl alcohol) thixotropy [117] Development of thixotropy in metal alkoxide solutions undergoing hydrolysis and con-

densation reactions. [118] Thixotropic in silicate suspensions

7.11. More examples in detail

It is interesting to select a group of recent papers to study their different approaches. Ramsay and Linder [119] studied montmori l lonite c lay - -a naturally occurring clay from Crook County, Wyoming USA. The size of these clay plates is considerably bigger than Laponite and their shape is less well defined. Using small-angle neutron scattering (SANS), they showed that under shear, the short-range interactions disappeared. They showed that the anisotropic structure brought about by shearing remained for many hours, even days, after the shear was removed. This was in accord with their G' measurements. These showed that the structure built up quickly while the particles were still aligned. It took a long time before complete isotropy was achieved. They suggest that montmori l lonite is well described by this explanation, but the breakup of flocs is a more likely explanation in Laponite with its smaller plates. Thus, the montmori l lonite samples are more thixotropic. The rebuilding is driven by Brownian forces which take longer to align larger particles like montmoril lonite. They also note in passing that the linear elastic limit moves to higher shear.

Zhang and Nguyen [120] used a technique to measure the rebuilding of the yield stress for mayonnaise that relied on the position of apparent cessation of flow in a wide-gap concentric cylinder apparatus. Up to 1000 s was needed before the yield-line came to a constant position.

Takeeshi, Wei and Sato [121] studied the thixotropic response of fumed silicas (Aerosil 200 and OX 50) suspended in polybutadientes of various viscosities (1, 11, 33 Pa s). After shearing, the normal network structure rebuilds slowly, with considerable differences still occurring at 8 h, but interestingly if a 15% strain was imposed for even 2 min, the structure became stiffer still from rest. This is an example of a shear-enhanced flocculation if the shear is limited.


Breton-Dollet et al. [122] measured the thixotropic behaviour of maize starch pastes using a loop-test method, noting the area under the curve as a measure of thixotropy. It remained fairly constant as a function of temperature for the first loop for maize, but was lower and thereafter decreased significantly as a function of temperature for the second and third loops. Unfortu- nately in this case, no attempt seems to have been made to establish any equilibrium data. For waxy starch pastes, the thixotropy was lower and a decreasing function of temperature and the number of loops. The difference in behaviour was attributed to the presence of amylose and amylopectin in the maize starch.

Berland and Launay [123] examined the thixotropic behaviour of wheat flour doughs and showed that thixotropic effects are seen in flows other than steady shear. They measured the thixotropic recovery of the storage modulus measured in the linear region (0.2% strain) after straining at higher strain amplitudes in the nonlinear region. Recovery is complete--although taking many minutes--up to large amplitude stains of around 20%. Above that value, irreversible changes seemed to occur.

7.12. Blood

Thixotropy in blood has been recognised for a long time, but a major problem encountered in investigating restructuring of individual blood cells into aggregates called rouleau is that sedimentation takes place quite quickly as the aggregates form. Hence the full potential of restructuring is not always seen. Of course the circulation takes blood through many flow regimes, and the breakdown and recovery of structure is important especially in blood sensitive to aggregation. A general review on the experimental work on human blood viscoelasticity and thixotropy was presented by Stoltz and Lucius [124], who showed that some models used for polymer solutions could also be used to account for the phenomena observed with blood (transient and periodic flow). Being more specific, Huang et al. [125] studied the effect of haematocrit (red cell concentration) on thixotropic parameters. They found that the rate constant of thixotropy--in this case arising from the breakdown of rouleuax into individual red cells--was independent of the haematocrit.

8. Mathematical theories

8.1. Requirements of useful models

The ideal model to describe thixotropic behaviour would start from the fact that some fundamental, rheology-determining physical entity takes time to change when the flow field around it has changed or is changing. As we have already said, this might be for example the size of a floc; the orientation angle of an alignable particle or the density of transient entanglements. In the simplest models, all such fundamental parameters change instantly with, for instance, shear, shear rate or shear stress. First we have to know at what rate these changes take place, and then if we know how the microstructure relates to the stress, we can predict the overall behaviour. Most workers in this field have used theories to describe viscous thixotropic phenomena, and only a few have attempted to describe viscoelastic effects. Representative examples of classes of theories are described below and others in that field are noted.


8.2. Viscous theories

Current theories for thixotropy fall into three groups: first those that use a very general description of microstructure described by a numerical value of a scalar parameter, typically 2, and then use d2/dt as the working parameter; second those who attempt some direct description of the temporal change of microstructure as for instance, the number of bonds, or an attempt at describing real floc architecture using fractal analysis, and third those that use the viscosity- time data itself on which to base a theory.

8.2.1. Indirect microstructural theories Most workers in this area have developed mathematical theories of thixotropy based on the

numerical scalar measure of structure, often designated by 2. Using this simplistic cypher, completely built structure is represented by 2 = 1 and completely broken-down structure as 2 = 0. In the simplest case of a typical, inelastic, non-Newtonian liquid with upper and lower Newtonian viscosity plateaus, 2 = 1 corresponds to t/o and 2 = 0 corresponds to t/a, with points between taking intermediate values.

Then thixotropy is usually introduced via the time derivative of the structure parameter, d2/dt, which is given by the sum of the buildup and breakdown terms, which, in the simplest theories, are only controlled by the shear rate and the current level of structure 2. The most general description of the rate of breakdown due to shearing is given by the product of the current level of structure and the shear rate raised to some power, and the driving force for buildup as controlled by the distance the structure is from its maximum value, i.e., (1 - 2), raised to another power. Then

d2 = g(~, 2) = a(1 - 2) h - c2~ d, (5)

where a, b, c and d are constants for any one system. If the value of g(~, 2) is negative, the system is breaking down towards equilibrium; if it is positive, it is building up towards equilibrium. At equilibrium, for every value of ~ there is a particular value of 2 which in this equation is found by setting d,~/dt to zero.

Godfrey [4] has summarised the development of these two-process models beginning with Goodeve and Whitfield in 1938 [126] which led to an earlier version of the above equation due to Moore [127] with both b and d as unity, and progressing through Cheng and Evans [128] who made b unity, but allowed d to be non-unitary, through to the very general 'structural kinetics' model of Mewis, with both powers taking non-unitary values. This latter formulation was also used by Lapasin's group [129]. The next step in this kind of approach is to relate the structure 2--as calculated using the equations above--to the stress tr or viscosity ~/ in some flow equation. As we shall see this has been done in a variety of ways which ranged from a simple Bingham equation, through the Cross model to a Cross-like model containing a yield stress. Most of the differences between the theories in this area are accounted for in the various combinations of these structure change and structure-viscosity formulations.

Baravian, Quemada and Parker [130] recently proposed a modification of the Cheng and Evans approach. Their study is worth looking at in detail for two reasons. First it follows the tradition route to describing thixotropy, and second, it shows how long and seemingly involved the procedure to describe thixotropy becomes.


They postulate that the relationship between viscosity and structure is given by

q(0-, t) = q(2) = (1 K2)2 , K= 1 - (6) - \r io /

where % and ~/~ are the usual limiting values of viscosity at very low and very high shear rate/stress, respectively. This assumes that the effect of stress on the viscosity is also accounted for within the current value of the structural parameter 2, which can itself be written as

where q is the current value of viscosity. Then for any system one has to find the values of % and r/~ (and thus, K), and from these, all values of r/can be converted into values of R. They found that the relationship between the equilibrium value of the structural parameter 2 and the shear stress 0- was given by

Requit(0- ) ~--- [1 -I- (0-/0-c) p] --1 (8)

From any one particular equilibrium position of stress, 1 Pa jumps were made up and down and plots of dR/dt versus R were made. These were of the form

dR 1 ~1 (9)

This kind of curve was obtained for many values of equilibrium stress. The values of the constants were then described by the equation:

R(t) = Requi~ + (2ini- 2eq,i0exp - t/ta2equi~ (10)

Thus, for all values of stress, they could now define 2in~, Requil and 6. Using the stress-up as well as stress-down data, the values of 6 were the same as a function of stress. This theory was able to describe a loop test very well, once instrument inertia had been accounted for.

Other variations on this theme include that of De Kee et al. [131], (following Tiu and Boger [132]), who described the break-down behaviour of various food systems by

dR - - c~ 'd(R - - Requil) n. (1 1)

dt

The viscosity was described by a multiple exponential-type flow law

0(9) = R(0-o + 9Zr/i,e- 'P~). (12)

The theory was used to characterise viscosity decay curves for yogurt and mayonnaise.

8.2.2. Direct structure theories Denny and Brodkey's applied reaction kinetics to thixotropy via a simple scheme that looked

at the distribution of broken and unbroken bonds [133]. The number of these bonds was later linked to viscosity. The forward and reverse rate constants k'~ and k2 represented the breakdown kinetics in flow, and the buildup kinetics, see for instance van den Tempel [134] who related bonds to flocculated system of fat globules. Denny and Brodkey wrote down the rate of structure breakdown as (cp. Ruckenstein and Mewis [135]):

26 H.A. Barnes/ J . Non-Newtonian Fluid Mech. 70 (1997) 1 -33

d(unbroken) - kl(unbroken) n - k2(broken) m, (13)

dt

and solved to give the viscosity by assuming it is linearly proportional to the amount of unbroken structure, with a maximum value when completely structured of r/o and a minimum value when completely destructured of r/~. The rate constant k2 is assumed to be independent of shear rate, being merely a description of Brownian collisions leading to restructuring, while the rate of breakdown constant is related to shear rate by a power-law expression.

The way the well-known Cross model was derived is instructive [136]. Assuming that a structured liquid was made up of flocs of randomly linked chains of particles, Cross derived a rate of equation of the form:

dN = k2P - (ko + kl~)m)N, (14)

dt

where N was the average number of links per chain, k2 was a rate constant describing Brownian collision, ko and kl were rate constants for the Brownian and shear contributions to breakdown, P was the number of single particles per unit volume, and m was a constant less than unity. At equilibrium dN/dt is zero, so

k2P N~= ( k, )" (15)

ko 1 +~o~) m

Then assuming that the viscosity was given by a constant, r/~ plus a viscous contribution proportional to the number of bonds Are, he derived his well-known relationship:

F/e -- r/a, _ 1 (16) - - k l "m r/o r/~ 1 +~7

Cross could of course have used the non-equilibrium data to derive a thixotropic model as later others did using his model.

Lapasin et al. [137] used a fractal approach to describe flocculated suspensions. The analysis of fractal flocs using computer simulations based on the ballistic, cluster-cluster aggregation mechanism suggested that Cfp in Quemada's model, i.e., 2/[q]Om in the KD equation, is given by

B + A (N~+x I - 1) (17) Cfp= B + Nmax - 1

where A and B are constants, Nmax is the number of primary particles in a floc, and fl is given by 3/D- 1, where D is the fractal dimension of the floc. Then Lapasin et al. argue that (cf. Cheng and Parker)

dNm,x - - - a (N - Nmax) - b~rP(Nmax - - Nmax~) (18)

dt

where Nmax.~ is the lower limit to which Nmax tends as the shear stress o- becomes infinite; a, b and p are material constants. This can be solved for N >> Nmax to give


aN Nmax = Nmax,~ + _~_o- 1, (19)

which, once we have replaced the lumped parameter (aN/b) L'p by a critical stress oc and substituted into the Krieger-Doughery equation, gives

q -(1/2) qb B+A Nm,~x~ + a /s

- - = 1 (20)

Lr/~J ~bmax BA-Nmax,~ +(~--~-~ " - -1 \o-,./

This equation describes the breakdown of flocs under flow, and predicts a yield stress and a finite high-shear-rate viscosity. It described very well the behaviour of both TiO2 and mica dispersions in steady state. (A much more complicated theory of this kind had been proposed by Eyring much earlier [see [138]]).

8.2.3. Simple viscosity theories Frederickson [139] formulated an expression for the rate of change of fluidity (the inverse

of viscosity) of a non-Newtonian system as

and used this expression for steady state flow (dO/dt = 0), recovery at rest (?', = 0), the sudden application of stress and then a loop test. Like the Cross model, this model also has higher and lower Newtonian plateaus at equilibrium, and if the quadratic dependence on shear-rate is replaced by a power-law relationship, they become similar.

Mewis and Schryvers [28] have recently devised a theory that also circumvents the use of any structural parameter such as 2, and instead used the viscosity as a direct measure of structure. They proposed that the rate of change of viscosity rather than rate of change of structure be related to the viscosity difference between the steady state and current values of viscosity (not the structure difference), respectively, viz:

dq d t - K[~/s(~") - r/]" (22)

This integrates to give

q = q ..... - [r/e,~ - r/e,O][(q - 1)Kt(r/e -s_ - qe,0) n-1 + 1] ''(l --") (23)

Mewis and Schryvers then made the (Newtonian) assumption that r/e,0 = r/i,~ which makes the use of the equation simpler since it is much easier to measure the eventual viscosity of the initial steady state condition. This assumes that the viscosity at the end of the initial steady-state period is the same as that at the beginning of the new shear-rate test, i.e., that the system is essentially Newtonian between those conditions. This is reasonable under some conditions but as Mewis's previous work has shown, it is only strictly true for most systems at higher shear rates. Using a value of n of 5/3, they fitted experimental data very well for carbon black in mineral oil.

Kristensen et al. [140] modelled the thixotropic breakdown behaviour of maize starch pastes using the equation:

28 H.A. Barnes,/J. Non-Newtonian Fluid Mech. 70 (1997) 1-33

( J / - qoo)' -m = [(m -- 1)kt + 1]0/0 - qoo)' -m (24)

where r/0 and qoo are the asymptotic values of viscosity r/ (representing the fully structured and fully destructured states, respectively) measured at time t any particular shear rate, and k and m are material constants. This approach had been used with previous success for minerals and industrial suspensions (see [141]). For the starch paste studied they found that the data was satisfactorily described by m = 3, thus suggesting a third-order kinetic model. Over the shear rate range tested, the steady-state flow curves could be described by a power-law model with an approximate half-power.

8.3. Viscoelastic theories

Almost any viscoelastic theory can have thixotropy introduced if the particular parameters that give the viscous and elastic responses are made to change in the way we have described for purely viscous behaviour. Probably the best example of this approach is given by the model due to Acierno et al., [142] who considered a model based on a series on Maxwell elements. These papers have had a large influence in the field, and have already been cited 49 and 104 times respectively at the time of writing. The number of elements in their model was such that they could be represented by a continuous spectrum. Simplifying the model to a discrete series, it is possible to write down the behaviour as

a = Z,a,, ~ + O, d t \O J = 0;i (25)

where a~ is the stress, G~ the modulus, and 0 the relaxation time (= ~Ii/G~ and labelled 2g in their paper) of the ith element in the discrete spectrum.

Thixotropy is introduced via the well-used structure parameter 2 (labelled as x; in their paper):

G~ = Go,2;, 0~ = 00,2] '4 (26)

The rate equation is then given by

a ,FE, (27) dt 0i 0i L6;]

i / ~ /

where Ei is the instantaneous elastic energy in the ith element. This is the same as the Moore linear model, except that now ~ is replaced by the generalised expression (E/Gi)~/2/Oi that accounts for the elasticity as well as the viscosity. The theory gives an excellent description of most of the rheological behaviour of a low-density polyethylene melt in shear and exten- sional transient and steady-state flow. The model is equivalent to the Moore model if reduced to the viscous case. Shoong and Shen [143] introduced a power-law dependence of the breakdown term which then compares with the Cheng and Evans' inelastic model. An obvious extension is to raise both terms to non-unitary powers.


9. The break-down and build-up of isolated flocs

The two key mechanisms of thixotropy for typical systems are floc erosion and Brownian collisions. Work has been published on these topics for diluted flocs and it is instructive to relate these studies to thixotropic build-up and breakdown of suspensions of flocs.

The diffusion rate of isolated flocs decreases significantly as their size grows. Reynolds and Goodwin [144] measured the diffusion coefficients of isolated flocs and found a rapid decrease of diffusion rate with floc size, with the particular value depending on the floc geometry which they studied as linear or clustered flocs. As these are quite moderate floc sizes, it is obvious that very large flocs of hundreds or thousands of primary particles will move very sluggishly. The effect of primary particle size on translational diffusion coefficient was worked out by Einstein who showed a simple inverse dependence on size. However the rotational diffusion of particles scales as the inverse cube of particle size (see [145]). This behaviour explains why exponential- type expressions for rebuilding contain a driving force such as (1 - 2), because rebuilding starts at a given floc size that grows and then the diffusion coefficient decreases. This means that collisions become less frequent, and as rebuilding progresses it gets slower and slower, but theoretically never stops.

The breakdown of isolated flocs in imposed shear fields has been studied by a number of workers. Sonntag [145] has summarised the results as dr = c~ ~ where g has been measured as 0.2 or 0.5. The surface shear stress experienced by an isolated floc is given by dfdqp, where dr is the floc size, d the size of the primary particles, t/the viscosity of the continuous phase and ~ is the shear rate. It is this stress which produces surface erosion of primary particles if it is greater than some bond shear strength between the primary particles [see Mtihle [146]]. This expression shows that floc breakdown in a given shear field is fastest for the largest sized flocs, i.e., at the shortest times, as well as being proportional to the shear rate raised to a power. These facts are reflected in the structure breakdown criteria normally used, d2/dt Qc -2 .~ ",n.

I0. Overall conclusions

Thixotropy comes about first because of the finite time taken for any shear-induced change in microstructure to take place. The microstructure is brought to a new equilibrium by competition between the processes of tearing apart by stress and flow-induced collision, in a time that can be minutes. Then, when the flow ceases, Brownian motion is able to move the elements of the microstructure around slowly to more favourable positions and rebuild the structure. This can take many hours to complete. The whole process is completely reversible.

The position of various areas of thixotropy today is well summarised by three quotations: (1) As to the proper understanding of the word itself, we can do no better than quote from

Lewis Carroll's Alice in Wonderland, "When I use a word," Humpty Dumpty said ... "it means just what I choose it to mean--neither more nor less". "The question is", said Alice, "whether you can make so many words mean so many different things".

(2) As to the state of experimental investigation in thixotropy, we quote Godfrey [4], "Thixotropy is one of the more complex characteristics associated with the behaviour of non-Newtonian liquids ... most of the available [experimental] data leave something to be desired".


(3) Then to assess the state of proper theoretical understanding of thixotropy, particularly in viscoelastic systems, "The results obtained so far do not allow [sic] to consider the modell ing of the thixotropic viscoelastic behaviour as a solved problem", Lapasin and Pricl [16], but see [147].

Thus, we conclude that there is plenty of scope for more work in these three areas; viz, a fuller definition, a better collection of data on well-characterised model systems and then the need for a comprehensive theory that describes both viscous and viscoelastic thixotropic effects: compu- tational efforts such as molecular modell ing will be very important in such studies because of the large number of variables that would ensure, as will the introduction of thixotropy into CFD codes.

References

[1] W.H. Bauer and E.A. Collins, in F.R. Eirich (Ed.), Rheology: Theory and Applications, Vol. 4, Academic Press, New York, 1967, ch. 8.

[2] J. Mewis, J. Non-Newtonian Fluid Mech., 6 (1979) 1. [3] D.C.-H. Cheng, Report Number CR 2367 (MH), Warren Spring Laboratory, UK, October 1982. [4] J.C. Godfrey, Ph.D. Thesis, University of Bradford, UK, June 1983. [5] E. Schalek and A. Szegvari, Kolloid Z, 32 (1923) 318; 33 (1923) 326, English translation given in Bauer and

Collins [1]. [6] T. Peterfi, Arch. Entwicklungsmech. Organ. 112 (1927) 680, Verhanitlungen 3rd. Intern. Zellforschung-Kongr.,

Arch. Exp. Zellf, 15 (1934) 373. [7] E. Hatschek, The Viscosity of Liquids, G. Bell, London, 1928. [8] H. Freunlich, Thixotropie, Hermann, Paris, 1935. [9] E.L. McMillen, J. Rheol. 3 (1932) 75, 164&179.

[10] G.W. Scott-Blair, A Survey of General and Applied Rheology, Pitman, London, 1943. [11] J. Pryce-Jones, JOCCA, 17 (1934) 305; 19 (1936) 395; 26 (1943) 3. [12] J. Pryce-Jones, J. Sci. Instr., 18 (1941) 39. [13] R. Roscoe, in J.J. Hermans (Ed.), Suspensions in Flow Properties of Disperse Systems, North-Holland,

Amsterdam, 1953, Ch. 1. [14] A. Jobling and J.E. Roberts, in C.C. Mill (Ed.), Some observations on dilatancy and thixotropy, Rheology of

Disperse Systems, Pergamon Press, London, 1959, Ch. 7. BSR Conf., Swansea, Sept. 1957. [15] D.C.-H. Cheng, Rheol. Acta 25 (5) (1986) 542-554. [16] R. Lapasin and S. Pricl, Rheology of Industrial Polysaccharides, Blackie, London, 1995, pp. 196. [17] H. Freundlicb and W. Rawitzer, Kolloid-Z., 41 (1927) 102. [18] In J. Thewlis (Ed.), Oxford Encyclopedia Dictionary of Physics, Pergamon Press, Oxford, 1962. [19] In T.C. Collocott (Ed.), Chambers Dictionary of Science and Technology, W&R Chambers, Edinburgh, 1971. [20] In S.P. Parker (Ed.), McGraw-Hill Dictionary of Scientific and Technical Terms, 4th edn., McGraw-Hill Book

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New York, 1976. [22] Oxford Concise Science Dictionary, 2nd edn., Oxford University Press, Oxford, 1991. [23] Chambers 20th Century Dictionary, Chambers, London, 1993. [24] In T. Whelan (Ed.), Polymer Technology Dictionary, Chapman & Hall, London, 1994. [25] In M.S.M. Alger (Ed.), Polymer Science Dictionary, Elsevier Applied Science, London, 1990. [26] M. Reiner and G.W. Scott-Blair, Rheological terminology in Rheology, Theory and Applications, Vol. 4,

Academic Press, New York, 1967, Ch. 9. [27] H.A. Barnes, J.F. Hutton and K. Walters, An Introduction to Rheology, Elsevier, Amsterdam, 1989. [28] J. Mewis and J. Schryvers, unpublished International Fine Particle Research Institute report, 1996.

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Quebec City Canada, pp. 526 527. [32] D. Doraiswamy, R.K. Gupta and S. Chimmili, in A. Ai't-Kadi et al. (Eds.), Proc. XIIth Int. Congress on

Rheology, Laval University, 1996, Quebec City, Canada, pp. 579. [33] C. Chang and P.A. Smith, Rheol. Acta, 35 (4) (1996) 382. [34] H. Kanai and T. Amari, Rheol. Acta, 34 (3) (1995) 303. [35] M. Barut, R. Lapasin, A. Zupanic and M. Zumer, in A. A'it-Kadi et al. (Eds.), Proc. Xllth Int. Congress on

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trixotropia barnes (1996)

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