Fractal theory study on morphological
dependence of viscosity of semisolid slurries
R. S. Qin and Z. Fan
A theoretical model is developed which enables the effect of particle morphology on the viscosity of semisolid slurries
to be assessed. Hausdroff dimensionality is introduced as a means of characterising the morphology of a solid
particle. The effective solid volume fraction, defined as the volume fraction of actual solid plus the entrapped liquid,
is derived analytically. The viscosity of semisolid slurries is calculated by the adaptation of Thomas’s empirical
approximation. The viscosity of a semisolid slurry is found to be a non-linear function of the Hausdroff
dimensionality. Slurries with particle morphologies of lower Hausdroff dimensionality have a higher viscosity for a
given solid fraction. The external variables, such as shear rate and cooling rate, affect the flow behaviour of
semisolid slurries primarily by changing the Hausdroff dimensionality and the linear size of solid particles. The
effect of particle size on the viscosity is also discussed.
MST/4694
At the time the work was carried out the authors were in the Department of Materials Engineering, Brunel University, Uxbridge,
Middx UB8 3PH, UK. Dr Qin ([email protected]) is now in the Department of Materials Science and Metallurgy, University
of Cambridge, Pembroke Street, Cambridge, CB2 3QZ, UK. Manuscript received 9 May 2000; accepted 27 November 2000.
# 2001 IoM Communications Ltd.
Introduction
Semisolid slurry (at a solid fraction less than 0.6) is a special
kind of solid – liquid suspension. Its main characteristics are
the morphological complexity of the solid phase and a
changeable degree of agglomeration among the solid
particles. As a result of the environment sensitive morphology of the solid particles and a stirring dependent
agglomeration reaction among the particles, semisolid
slurry behaves as a history dependent and non-Newtonian
fluid.1 There has been quite a lot of theoretical work which
has addressed the viscosity of semisolid slurries during the
last two decades.2 – 4 However, the physical understanding
of experimental evidence is still a daunting challenge.
The difficulties in modelling the viscosity of semisolid
slurries are manifold. The bottleneck is how to characterise
the morphology of the solid particle. As is well known, the
shape of a solid particle may be in the form of a dendrite,
rosette, or spheroid, depending on environmental conditions.5 Furthermore, the subdivisional characteristics of
dendrites are also environment sensitive according to
pattern formation theory.6 Neglecting the morphological
detail of the solid phase makes it impossible to understand
the whole flow behaviour of the semisolid slurry.
The literature on this topic is comprised of two different
types of contribution. The first focuses only on the effect of
agglomeration, such as in the models proposed by Mada
et al.2 and Kumar et al.,1 where the effect of particle
morphology is neglected. The solid particles are assumed to
be dense spheroids in those models. The second class of
contribution does include some effects of particle structure,
but these works do not concern themselves with the method
of characterising the morphology of the solid particle. For
example, Quemada7 considered the contribution of the
morphology using the concept of the effective solid volume
fraction. However, the model introduced an immeasurable
structural parameter S and did not specify the method to
determine its value. The research in this aspect has stayed at
a stage of phenomenological and qualitative description.
On the other hand, the highly developed fractal geometry
has proved to be a powerful method by which to describe irregular patterns.8 The evolution of the particle
morphology in semisolid slurries is from dendrite via
rosette to more or less spheroid as the shear rate increases.5
ISSN 0267 – 0836
In most cases the geometry of the solid phase is irregular. By
noting that the dendritic form is a typical fractal structure6
and that the solid – liquid interface morphology in directional solidification can be described quantitatively by
the fractal method,8 one may think that the irregular geometry of the particle morphology in a semisolid slurry can
be quantitatively characterised by fractal theory.9 The
Hausdroff dimensionality of the solid particle can be
measured by many different methods, such as slit island
analysis,8 profile analysis,10 direct surface area analysis,11
and variation-correlation analysis.12 The aim of the present
work is to illustrate how the viscosity is affected by the
particle morphology by means of the fractal theory.
Theoretical considerations
For a general solid – liquid suspension, the viscosity is a
function of the fraction of solid. A formula which has been
obtained from regression analysis of the empirical data of
Thomas13 can be expressed in the following form
m ~m=m ~1z2:5Wz10:05W2 zA exp(BW) : : : (1)
0
where m* is the relative or dimensionless viscosity, m is the
viscosity of the solid – liquid suspension, and m0 is the viscosity of the pure liquid, W is the volume fraction of solid in
the suspension, and A and B are constants which are found
to be 0.00273 and 16.6 respectively. The last two terms in
equation (1) represent the effect of particle agglomeration
(doublet, triplet, and so on) due to particle collision.
Equation (1) is suitable for solid – liquid suspensions
where the morphology of the solid particle is spheroid. For
semisolid slurries, the solid particle morphology is not
always so. Figure 1a is a possible example of a semisolid
slurry where the morphology of the solid particle is
dendritic. However, when the suspension is sheared, the
non-spheroidal particle will rotate and the entrapped liquid
will be forced to move together with the solid particle. The
result is that the entrapped liquid plays the role of a solid.
The solid particles with trapped liquid contained in them
are approximately spherical in morphology as illustrated
in Fig. 1b. The solid plus the entrapped liquid is defined as
the effective solid. Replacing the solid fraction W by
the effective solid fraction Weff leads to a condition in
Materials Science and Technology
September 2001 Vol. 17 1149
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Qin and Fan Fractal theory study on viscosity of semisolid slurries
(3{DH )=DH
N
fs3=DH :
Q
Weff ~k(3)k(DH ){3=DH
:
: (5)
Equation (5) shows that the effective solid fraction is
affected by the dimensionless factor Q/N, the real solid
fraction fs, and the Hausdroff dimensionality DH. In the
extreme case of DH~3, which means no liquid is entrapped
in the solid, equation (5) reduces to
Weff ~fs
a real (dendritic) solid particle; b effective solid particle
1 Schematic diagram illustrating concept of effective
solid fraction
which equation (1) can be used to describe the viscosity of
semisolid slurries. The task now is to derive a expression for
Weff.
Assume that the considered system contains N atoms.
Neglecting volume shrinkage during solidification, the
solidified atom number M at a temperature T is given by
M~fs N :
:
:
:
:
:
:
:
:
:
:
:
:
:
: (2)
where fs is the solid fraction which is a function of
temperature and can be obtained from consulting the phase
diagram. The M atoms are assumed to form Q particles of
the same size l. The length unit in this work is defined as the
atomic diameter. The actual volume of a solid particle
equals the number of atoms in the considered particle. The
effective solid fraction in such a length unit is obviously
pQl 3
Weff ~
:
6N
:
:
:
:
:
:
:
:
:
:
:
:
: (3)
Here one uses the Hausdroff dimensionality DH to characterise the morphology of the solid particles.8 DH is
sensitive to the external variables, such as the shear rate and
the cooling rate. Numerical simulation shows that DH~2.5
for a particle formed by a diffusion controlled mechanism
which corresponds to a zero shear rate.14 Also, DH~3 for
spheroids which formed at a very high shear rate. The
analytical relationship between the Hausdroff dimensionality and the external variables is not available in the
literature at present, but it is known that the higher the
shear rate (or the lower the cooling rate) the larger the
Hausdroff dimensionality. For a particle of Hausdroff
dimension DH, the volume V(l) of the region bounded by
the solid/liquid interface scales with the increasing linear
size l of the particle in a non-trivial way15,16
V(l)~k(DH )l DH
where
k(DH )~pDH =2
V(l)~
fs N
Q
:
:
:
:
:
:
:
DH
!
2DH |
2
:
:
:
:
:
:
:
:
:
:
:
: (4)
:
:
:
:
:
: (4a)
:
:
:
:
:
: (4b)
The factorials in equation (4a) can be calculated approximately by using Stirling’s formula17
DH =2
DH
DH
!~exp(DH =2)
(pDH )1=2 : : : : (4c)
2
2
Bringing equation (4) and (4b) into (3) and noting that
k(3)~p/6, the effective solid fraction takes the following
expression
Materials Science and Technology
September 2001 Vol. 17
:
:
: :
:
:
:
:
:
:
:
:
: (5a)
:
The dimensionless factor Q/N has the indirect physical
meaning of the number of solid particles per unit ‘volume’.
This factor can be calculated approximately according to
the physical and mathematical assumptions in the treatment
of Avrami.18 The essence of the assumptions in Avrami’s
treatment arises from the following:19 (a) as time proceeds
during particle growth, some grains grown from a site
active at various times in the past, may impinge upon grains
grown from other sites, and (b) active nucleation sites are
randomly distributed throughout the volume. Therefore,
one gets the following formula to estimate Q/N
Q=N~v=d 3
:
:
:
:
:
:
:
:
:
:
:
:
: (6)
where v is the volume of an atom and d is the average grain
size when the solidification is completed. Q/N can also be
measured experimentally by counting the number of solid
particles using SEM or optical microscopy. Replacing the
solid fraction W in equation (1) with the effective volume
fraction Weff given in equation (5), the viscosity of the
semisolid slurry can be obtained.
Numerical study and discussion
The Hausdroff dimensionality of a solid particle in a
semisolid slurry is related to the distribution of constituents,
the solid – liquid interfacial energy, undercooling, cooling
rate, shear rate, and many other factors. However, the
Hausdroff dimensionality is bounded by two limits which
are named the upper limit and lower limit. This is due to the
limitation that
fs ¡Weff ¡1 :
:
:
:
:
:
:
:
:
:
:
:
:
: (7)
The upper limit is equal to 3 according to equation (5a).
The lower limit can be calculated by letting Weff~1 in
equation (5). The upper limit implies that no liquid is
trapped in the solid particle while the lower limit implies a
solid skeleton is formed at that solid fraction. A numerical
analysis of the relationship between the lower limit of
Hausdroff dimensionality and the actual solid fraction fs is
illustrated in Fig. 2, where it is assumed that N/Q~1016
corresponding to an average grain size in a completely
solidified cast of about 100 mm.
Figure 2 shows that a network of solid phase can be
formed even at a very small solid fraction. For example, the
pattern in Meakin’s14 simulation has only a solid fraction of
about 0.004. Of course the solid network formed is very
weak when the solid fraction is small, due to the very thin
solid arm. The strength of the solid skeleton will develop as
the solid fraction increases. That is the reason for the
observation that a solid – liquid suspension exhibits an
apparent yield point of 106 dyne cm22 (1 dyne cm22~
1021 Pa) at a small solid fraction of 0.15 which increases
with increasing fraction of solid.20 Figure 2 also shows that
the critical solid fraction for the formation of a solid
network depends on the particle morphology. For a
morphology of DH~2.808 a network is formed from a
critical solid fraction of 0.1. While DH~2.92 corresponds to
a critical solid fraction of 0.4. It is well known that shearing
the slurry from a temperature above the melting point can
Qin and Fan Fractal theory study on viscosity of semisolid slurries 1151
Figure 3 shows the results of numerical calculation. There
are two limiting curves in the figure. One limiting curve of
DH~2.95733 corresponds to the extreme external condition
that a solid skeleton will be formed at a solid fraction of 0.6.
In terms of shear rate it is the lowest shear rate which
enables the slurry to behave as a fluid at a solid fraction of
less than 0.6; In terms of the cooling rate it is the highest
cooling rate. The other limiting curve of DH~3 corresponds
to the opposite extreme case, e.g. a high enough shear rate
to form a spheroid. The curve of DH~2.95733 shows that
the relative viscosity begins to increase sharply from a solid
fraction of about 0.3, while the curve of DH~3 shows that
the relative viscosity is very low even for a solid fraction of
near to 0.6. At fs~0.4 the apparent viscosity of a slurry
with DH~2.95733 is 60 times larger than that of a slurry
with DH~3. At fs~0.5 the ratio is 180. Comparing the
results in Fig. 3 with the experimental results reviewed by
Flemmings,6 the main characteristics coincide fairly well.
The size of the solid particles may also affect the viscosity
of the solid – liquid suspension for a given solid fraction.
Thus the effect of the particle size on the viscosity requires
to be studied. For a certain solid fraction, the particle size is
inversely proportional to the cube root number of particles.
Figures 4 and 5 show the numerical results for the size
related relative viscosity. In these figures the particle size is
normalised. The normalised particle size is taken to mean
the ratio between the true particle size and a standard
particle size. The standard particle size is defined as the
particle size which enables a number density of 1 cm23 at
various solid fractions.
Figure 4 shows the results of relative viscosity v.
normalised particle size at a solid fraction of 0.43 and
Hausdroff dimensionality of 2.95841, 2.975, 2.99, and 3. It
can be seen that the size effect is more important for a lower
Hausdroff dimensionality. For DH~3 the size of the solid
particle has no effect, just as suggested by Einstein’s work.
For a given fractal morphology of the solid phase, a smaller
particle size will result in a smaller relative viscosity.
Another calculation demonstrates the effect of particle
size on the viscosity of slurry at different solid fractions as
shown in Fig. 5. The Hausdroff dimensionality is chosen to
be 2.975 and the solid fractions are 0.2, 0.4, and 0.6. It is
found that the smaller the solid fraction the smaller the
contribution of the size effect. The figure shows that the size
effect cannot be neglected where the solid fraction is
relatively high.
3 Relationship between relative viscosity and solid
fraction for different morphologies: illustrated for
Hausdroff dimensionalities of 2.95733, 2.975, 2.99, and 3
4 Effect of particle size on relative viscosity for different
values of Hausdroff dimensionality where solid fraction is assumed to be 0.43
2 Relationship between lower limit of Hausdroff dimensionality and solid fraction
increase the Hausdroff dimensionality. So a solid network
will begin to form at a higher solid fraction. Therefore, the
reported observation that the shear stress at a solid fraction
of 0.4 is quite low for non-dendritic materials is, according
to the present model, due to the absence of a solid skeleton
at a high Hausdroff dimensionality with that solid fraction.21
The viscosity of semisolid slurry can be calculated under
the present theory. From the literature, the semisolid slurry
behaves as a fluid at a solid fraction of less than 0.6.
Otherwise the yield stress of the slurry would be very strong
and the slurry would behave as a viscoplastic solid. This
experimental phenomenon enables the suggestion that the
critical solid fraction for the formation of a solid network is
0.6. According to the numerical result found in Fig. 2, the
Hausdroff dimensionality for this slurry satisfies
2:95733¡DH ¡3
Materials Science and Technology
September 2001 Vol. 17
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Qin and Fan Fractal theory study on viscosity of semisolid slurries
the flow behaviour of semisolid slurries primarily by
changing the Hausdroff dimensionality of solid particles.
2. A solid skeleton can be formed at a very small solid
fraction if the Hausdroff dimensionality of the solid particle
is low. The critical solid fraction to form a solid network
depends on the Hausdroff dimensionality.
3. The particle size may affect the viscosity of semisolid
slurry at a high solid fraction and a small Hausdroff
dimensionality.
Acknowledgements
The authors wish to thank Brunel University and Ford
Motor Co. for financial support to this and other related
research projects. Thanks also go to Professor M. J. Bevis at
Brunel University for his encouragement.
References
5 Effect of particle size on relative viscosity at different
solid fractions (0.2, 0.4, and 0.6) where Hausdroff
dimensionality is chosen to be 2.975
Concluding remarks
A theoretical model based on fractal theory has been
developed to assess the effect of particle morphology on the
viscosity of semisolid slurries. A solid particle with a
Hausdroff dimensionality of less than 3 will entrap liquid in
it. This will increase the effective volume fraction of the
solid. By applying a fractal analysis the effective solid
volume fraction, defined as the volume fraction of actual
solid plus the entrapped liquid, has been derived theoretically. Analysis of the viscosity is enabled by adaptation
of an empirical expression of viscosity as a function of the
volume fraction of solid. It is found that the viscosity is
affected by the number of solid particles per unit volume,
the actual solid fraction, and the Hausdroff dimensionality
of the solid particle. Numerical calculation using the
developed model allows the following conclusions to be
made.
1. The viscosity of a semisolid slurry is a non-linear
function of the Hausdroff dimensionality. A slurry with a
particle morphology of lower Hausdroff dimensionality
exhibits higher viscosity for a given solid fraction. The
external variables, such as shear rate and cooling rate, affect
Materials Science and Technology
September 2001 Vol. 17
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