1. Introduction
A complex fluid may exhibit significant thixotropic behavior due to the presence
of microstructures, which gradually breakdown under a shear field, thus showing a
decrease in viscosity with time [1-3]. The time-dependent viscosity profile can be
simply described by a structural kinetics model, in which the instantaneous
microstructure is characterized by a dimensionless structure parameter , having a
value between 0 and 1, denoting the completely collapsed state and the initial
fully-structured state (t = 0) [3-6].
The structure parameter concept can also be
incorporated into a constitutive equation to describe the time-dependent rheological
behavior of complex fluids [7-9].
However, the resulting models are often
complicated and a non-linear regression method is required to determine the model
parameters. The structural kinetics model, on the other hand, presents a simple
framework for understanding the time-dependent viscosity data.
The structural kinetics model normally consists of two parts: the structure
build-up and breakdown.
Our past experience has shown that for a large number of
thixotropic polymer solutions, samples no longer exhibit thixotropic characteristics
after shear even after a considerable period of time. This fact suggests that the effect
of structure build-up is negligible during the viscosity measurement and only the
structure breakdown term is required in modeling the time-dependent viscosity profile.
Thus, with the assumption that the structure breakdown process follows an nth-order
reaction, the structural kinetics model can be simplified as a differential kinetics
equation [1]:
d
  k n
dt
(1)
where k is the shear rate dependent rate constant.
By defining the structure
parameter in terms of apparent viscosities:   (    ) /( 0    ) , where  0 and
 are the asymptotic values of the apparent viscosity at t = 0 and t → ∞,
respectively, one can integrate Eq (1) to obtain



 1


1 n


 (n  1)kt  1 0  1


1 n
(2)
Eq (2) has been successfully used to describe the time-dependent viscosity profile at a
fixed shear rate for thixotropic polymer solutions [1-3].
To best-fit the experimental data, there are three parameters required to be
determined in Eq (2): the kinetics order, n; the rate constant, k; and the ratio of the
initial to equilibrium viscosity,  0 / .
A typical approach is to assume an integer
kinetics order and calculate the other parameters through a linear plot of Eq (2).
Despite of simplicity of this approach, difficulties have been encountered in
evaluating the parameters. Martinez-Padilla and Hardy [10] have reported that the
structural kinetics model is valid only in the initial period of a time-dependent
viscosity profile for fluid foods when using n = 2.
The data fitting can be improved
by using non-integer kinetics orders along with non-linear regression methods [3,7].
However, the variation in the kinetics order makes a further analysis on the rate
constant impossible and the model parameters lose their physical significance.
The failure in fitting data using a single value kinetics order is possibly due to the
scattering of data, especially when the resolution of the viscometer is low.
We found
that the data fluctuations near the equilibrium viscosity are exaggerated in a plot
against time, thus resulting in significant errors in determining the model parameters.
In this work, different plot methods are proposed to circumvent this problem.
First,
the second-order structural kinetics data is used to evaluate the effectiveness of the
plot methods. Then the approach is extended to solve the first- and third-order
problems.
Finally, the effectiveness of different plot methods in finding the kinetics
order is compared.
2. Second-order structural kinetics model
Since the time-dependent viscosity profile was often modeled by the
second-order structural kinetics, we start with the second-order equation.
The
structural kinetics equation can be readily obtained by setting n = 2 in Eq (2):
1
1





 1   0  1 (1  kt)
 

 

(3a)
Eq (3a) is the direct form of the second-order structural kinetics equation, and are
used exclusively in the literature to estimate the other parameters in a plot against
time (referred to as Method I in this paper).
But as mentioned earlier, this method
may lead to serious errors in estimating the parameters. Thus two other different
linear transformations of the structural kinetics equation are proposed:
1
1




t 
 1   0  1 (k  t 1 )
  
  
(3b)
 


 

t   1  k 1   0  1    1 
  
       
(3c)
1
1


 1 against the reciprocal of time (referred
Eq (3b) suggests a linear plot of t 
 

1




to as Method II), while eq (3c) suggests a linear plot of t 
 1 against   1
  
  
(referred to as Method III). All the three equations can be used to estimate the
parameters k and  0 / ; however, the plot of Eq (3a) may show significant
1


deviation from linearity near the equilibrium when both values of t and   1 are
  
large. The other two methods circumvent this problem by using variables t 1 and


  1 instead, whose values near the equilibrium become so small that the data
  
fluctuation will have little effect on the evaluation of the model parameters.
To test the validity of these methods, a set of ‘experimental’ data with known
parameter values and standard deviation was used in the linear plots of Eqs (3). A


set of error-free data was first generated from Eq (3a) by setting  0  1 = 0.5 and k
  
= 0.1 s-1, and then the viscosity data were allowed to vary according to a normal
distribution with a preset standard deviation.
The latter was accomplished by using
the Box-Muller algorithm suggested by Numerical Recipes in Fortran [11].
Figure 1 shows the linear plots of Eqs (3) using a set of simulated data generated
by the second-order kinetics with a standard deviation of 0.02 with respect to


  1 ( = 0.02).
  
It can be seen that the plot becomes scattered near the


equilibrium extreme ( 
 1  0 ) when using Method I (Figure 1(a)).
  
In contrast,
this scattering is partially suppressed when using Method II (Figure 1(b)) or becomes
less significant when using Method III (Figure 1(c)), indicating that the two proposed
linear transformations are better than the conventional equation in fitting
error-containing data.
To further elucidate the effectiveness of these linear transformations, the means
and standard deviations of the model parameters were calculated over one hundred
different sets of ‘experiment’ data using the three methods.
Table 1 lists the ratios of
the estimated parameter value to the actual value (mean ± standard deviation) for


different degrees of error in 
 1 .
  
At low errors ( = 0.005), Method I
underestimates the parameters by approximately 20% while both Methods II and III
generate parameters with values almost identical to the actual values.
the deviation from the actual values becomes more pronounced.
As  increases,
At large errors ( =
0.02), though the estimated parameters obtained from the three equations all show
significant deviation from the actual values, Methods II and III still give fair results
and are superior to Method I.
These comparisons indicate that for error-containing
data, Methods II and III are far better than Method I in estimating the model
parameters.
3. Structural kinetics with other orders
A similar approach can be used to solve the first- and third-order structural
kinetics problems:
By setting the value of n equal to 1 or 3 in Eq (1) and integrating,
one can obtain the typical structural kinetics equations




ln   1  ln  0  1  kt
 

 

(4a)
and



 1


2
2


  0  1 (1  2kt)


(5a)
for the first- and third-order structure breakdown processes, respectively (Method I).
In order to enhance the accuracy in estimating model parameters, Eqs (4a) and (5a)
are rearranged and two types of linear transformation are introduced:




t 1 ln   1  k  t 1 ln  0  1
  
  
(4b)
 




 
t ln 1 
 1  k 1  ln  0  1 ln 1 
 1  1
  
   
   
(4c)


t 
 1
  
1
2
2


  0  1 (2k  t 1 )
  
(5b)
2
2
2


 
 
1    0
  1    1
t   1 



2
k


  
      
(5c)
Eqs (4b) and (5b) are analogous to Eq (3b) in using the variable t 1 instead of t
(Method II). Eqs (4c) and (5c), on the other hand, is similar to Eq (3c) in using the




variables ln   1 and   1
  
  
2
2




instead of ln   1 and   1 ,
  
  
1
respectively (Method III).
The difference between the fitting of these linear transformations is illustrated by
the linear plots of simulated ‘experimental’ data sets described previously.
Figures 2
and 3 compare the plots of first- and third-order structural kinetics viscosity data
using Eqs (4) and (5), respectively.
It can be seen that the linear plot using the
conventional method (Figure 2(a)) results in an erroneous slope in the fit because of
the significant data scattering near the equilibrium extreme,. The data scattering is
almost suppressed when using Method II, as shown in Figure 2(b), but still exists in
the linear plot of Method III (Figure 2(c)).
Figure 3, on the other hand, is similar to
Figure 1 in that the data scattering observed in the conventional plot (Figure 3(a)) is
partially suppressed and becomes less significant when using Methods II and III
(Figures 3(b) and 3(c)).
These figures all show the capability of Methods II and III
to reduce the effect of data errors.
The major advantage of Methods II and III is their ability to obtain more accurate
estimates of the model parameters.
This feature can be elucidated by comparing the
estimated parameters calculated from different methods.
One hundred different sets
of ‘experiment’ data at fixed  (= 0.005) for each case were used and the resulting
parameter values are listed in Table 2.
Together with Table 1, the parameter values
indicate that as the kinetics order increases, the error in parameter estimation becomes
more serious when using the conventional method.
As we can see, this error is
effectively suppressed by using the proposed linear transformations (Methods II and
III).
Here, the improvement by Methods II and III is similar to that found for the
second-order structural kinetics problem.
4. Determination of kinetics order
Another concern regarding the structural kinetics model is the determination of
the kinetics order. This is usually done by examining the linearity of plots of
experimental data using the conventional linear transformation for each kinetics order.
Though it has been shown that the linear transformations suggested by Methods II and
III are superior to the conventional method in estimating the rate constant and
viscosity ratio, their effectiveness in determining the kinetics order is uncertain and
requires further evaluation.
Arbitrarily chosen data sets with n = 1, 2, and 3 as described above were used to
evaluate the effectiveness of the three methods in determining the kinetics order.


The sum of squared errors (SSE) in 
 1 between data points and predictions
  
from the model was thus calculated for each linear transformation.
The difference
between the SSE value of the preset n and that obtained from different n is used as a
measure to evaluate the ability to correctly identify the kinetics order.
The condition
with more negative SSE value indicates a more effective method for kinetics order
determination.
The SSE values for each data set calculated from different methods are listed in
Table 3.
If the model yields a negative rate constant, it will be considered incorrect
and the corresponding SSE value will not be calculated.
set with n = 1.
We first consider the data
It is evident that Method I is the best choice since it yields only
negative k values when using the second- and third-order structural kinetics models.
For the data set with n = 2, Method I results in the most negative SSE value when
using the first-order structural kinetics model or leads to a negative k value when
using the third-order structural kinetics model, indicating that it performs better than
the other two methods.
Likewise, Method I also yields the most negative SSE
values for the data set with n = 3.
Therefore, from these results, we conclude that the
conventional method is superior to Methods II and III in identifying the correct
kinetics order.
5. Summary
The conventional method used to estimate the parameters of the structural
kinetics model may lead to serious errors due to the exaggeration of viscosity
fluctuation near the equilibrium.
Two linear transformations of the model were thus
proposed in order to circumvent this problem.
Simulated error-containing
time-dependent viscosity data were used to evaluate the effectiveness of these
methods for models with different kinetics orders.
The fitting result shows that the
proposed methods are capable of suppressing the error, leading to significant
improvement in the estimation of the rate constant and the initial to equilibrium
viscosity ratio.
In contrast, the conventional method exhibits a better performance
than the other two methods in the determination of the kinetics order.
According to these results, we propose that a two-step approach should be
adopted to determine the model parameters:
order by the conventional method.
Step 1 – determination of the kinetics
Step 2 – estimation of the rate constant and the
initial to equilibrium viscosity ratio by either of the two proposed methods.
With
this approach, it is possible to obtain reasonable estimates of the model parameters
from low-resolution time-dependent viscosity data.
References
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(2002) 157.
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Figure Captions
Figure 1. The linear plots of the second-order structural kinetics viscosity data using
(a) eq (3a), (b) eq (3b), and (c) eq (3c). The data points were generated based on
 0

  1 = 0.5, k = 0.1 s-1 with a standard deviation of 0.02 with respect to
  


  1 .
  
Figure 2. The linear plots of the first-order structural kinetics viscosity data using (a)
eq (4a), (b) eq (4b), and (c) eq (4c). The data points were generated based on
 0

  1 = 0.5, k = 0.01 s-1 with a standard deviation of 0.02 with respect to
  


  1 .
  
Figure 3. The linear plots of the third-order structural kinetics viscosity data using (a)
eq (5a), (b) eq (5b), and (c) eq (5c). The data points were generated based on
 0

  1 = 0.5, k = 0.25 s-1 with a standard deviation of 0.02 with respect to
  


  1 .
  