1.
1
Journal of NonNewtonian
Fluid Mechanics, 6 (1979) l20
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, B3030
Leuven
10,1978)
The timedependent 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.
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, gelsol (solidiiquid)
transition due to mechanical agitation
[l3].
From the beginning it was realized that solgel 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 [ 461. 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.
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. [ 19211). Modifications to apply a constant
stress have been described (e.g. 122241). 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. [ 2527 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.
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. [ 31331).
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 stressrate 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 nonNewtonian 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.
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 timeshear
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
nonNewtonian 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 plateandcone 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 powerlaw 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
crosssection of sheared material. The effect is most pronounced in plateandcone viscometers and in parallelplate 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.
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 recoverytime 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
7.
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
46,8185
74
10,11,83
8689
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 liquidlike 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 [4,98100 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 (edgeside contacts) and an open cardhouse
8.
8
structure [ 101,102]. Also spherical particles produce shear thinning and
thixotropy [ 11,22,26,46,92,101].
The gellike 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, orderdisorder 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.
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 solidlike at
low stress levels and liquidlike 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 nonnegligible 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 ReinexRivlin 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’ Cm
Tcrit(t)
dwj
exp[a(t
 t’)] dt’
= 71 
(71
p +]
,/@8
t ccm

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.
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
timedependent 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.
11
the material. The nonlinear, timedependent 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), singleorder 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 [ 19211. 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 [ 1161181. 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.
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 onetoone 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
secondorder 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 firstorder 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.
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. Weymarmhas considered spherical aggregates [21,127] whereas Ruckenstein and
Mewis [ 1281 worked on chainlike 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 RuckensteinMewis model gives a distribution of structures which can be described
by a single structural parameter [124]. A modified version of the RuckensteinMewis 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 cardhouse 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 nonNewtonian contribution to the shear stress r12 in proportion to the number Xi of flow units
of its kind [33] :
712 = (xlPll~lkil2
+ (X2/~2b~h92%2
+ W3/~3b~h1P3%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.
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 [1311331. 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.
15
replaced by a rheological characteristic if they are linked by a onetoone
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 coworkers [ 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 MaxwellSchwedoff 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 springlike elastic
forces [ 100,143 1. The particles themselves can cause viscous dissipation.
Hence, under suitable conditions the chains can react as beadspring 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
RuckensteinMewis 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.
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
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