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Rheological model to predict the thixotropic behaviour of colloidal dispersions
Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
Rheological model to predict the thixotropic behaviour of colloidal dispersions J. Labanda∗ , P. Marco, J. Llorens Chemical Engineering Department, University of Barcelona, c/Mart´ı i Franqu`es 1, 08028 Barcelona, Spain
Abstract We propose a rheological structural model to describe the thixotropic behaviour of colloidal dispersions. The model shows a certain connection between the structural changes of colloidal dispersions with viscosity. Viscosity depends on the size, shape and orientation of colloidal aggregates. Instantaneous shear rate changes cause instantaneous changes in aggregate shape and orientation, but non-instantaneous changes in aggregate size. The kinetic process in changing aggregate size determines the thixotropic behaviour of colloidal dispersions. The model differentiates two characteristic thixotropic times: one for structure formation and other for structure destruction. The elastic effects due to non-instantaneous changes of aggregate shape and orientation are also included in the model. The model has the following parameters: (i) three parameters to determine equilibrium or steady-state rheology; (ii) two parameters to set the viscosity for a given aggregate size at a given shear rate; (iii) two characteristic thixotropic times, which depend on shear rate; (iv) a characteristic elastic modulus. Each group of parameters are fitted independently with appropriate rheological tests and coupling among them is prevented. The model has been tested with consecutive lineal increases and decreases of the shear rates with time levels of maximum shear rates. Carbopol solutions of different molecular weights have been analysed with good agreement between experimental and predicted viscosity. © 2004 Elsevier B.V. All rights reserved. Keywords: Rheology; Thixotropy; Colloidal dispersions; Carbopol
1. Introduction Thixotropy is caused by time-consuming structural changes in colloidal dispersions at a given shear rate. The literature describes theoretical structural models [1–4], which use a structural parameter, and two fundamental equations: steady state and kinetic. The structural parameter, λ, is a scalar that accounts for the structural level of the colloidal system. For a completely broken down structure λ = 0 and for a completely built-up structure λ = 1. Recently, Bautista et al. [5] introduced viscoelastic aspects in the models and Quemada [6] used the concept of effective volume fraction to describe the structural parameter. All these structural models describe certain aspects of thixotropic behaviour, but complex behaviour is difficult to explain. The aim of this study is to develop a rheological structural model to describe the complex rheology behaviour of ∗
Corresponding author. E-mail address: [email protected] (J. Labanda).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.08.062
Carbopol solutions when the shear rate changes. Carbopol is a commercially polyacrylic acid found in many cosmetic and pharmaceutical products. The formation of networked microgel structures above a given polymer concentration, ionic strength and pH, is its principal feature [7]. Rheological studies of Carbopol gels have shown weak viscoelasticity at rest [8–9], yield stress [10] and shear-thinning at steady state [11–13]. In non-steady-state conditions, Carbopol gels show complex thixotropy caused by break down and built up of micro-gels aggregates.
2. Theory The proposed structural model is based on the distinction between structural level of aggregates, quantified by a structural parameter, S, and the shape and orientation of these aggregates, quantified by a shape factor, F. The viscosity of the colloidal system depends on the structural parameter and the shape factor. The viscosity of the colloidal suspension,
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J. Labanda et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
which mat vary over time, t, is associated with the structural level and shape factor as follow:
level is only connected with the viscosity through Eq. (1). Therefore, the relation between viscosity and shear rate is:
˙ t) = F (γ, ˙ S(γ, ˙ t )) S(γ, ˙ t) η(γ,
˙ t) γ(t) ˙ − σ(t) = η(γ,
(1)
˙ ) is the structural parameter at the instant before where S(γ,t t, t = t – t. Changes of shape and orientation are elastic deformations with very short relaxation times and, therefore, almost non-time dependant. However, changes of structural level are clearly time dependant. The shape factor depends on shear rate and structural level as follows:
β
˙ ),t ) ˙ S) = γ˙ α·S(γ(t F (γ,
(2)
where α and β are parameters that can be determined from experimental data. For a given shear rate, the non-equilibrium structural parameter varies with time following a first order kinetic process due to structure break down and build up because of shear forces and particle interactions, respectively. This kinetic process has just two kinetic parameters: thixotropic time for construction of the structure, tSF , and thixotropic time for destruction of the structure, tSD . The thixotropic time is the time needed to reach the equilibrium structural level after shear rate changes. The evolution of the structural parameter with time is obtained by integrating the kinetic equation. Then, t ˙ ˙ ), t) − Se (γ(t))] ˙ ˙ S(γ(t), t) = [S(γ(t exp − + Se (γ(t)) tS (3) where t = t – t and tS is tSF when the structure level increases with time or tSD when the structure level decreases with time. ˙ Se (γ(t)) is the equilibrium structural parameter for a given ˙ shear rate, whose evolution with time is given by γ(t). The viscosity of most colloidal suspensions, at steady state, follows pseudo-plastic behaviour with power law dependence on shear rate, even at very low shear rates. Obviously, at rest the system shows the maximum viscosity or zero shear rate viscosity, η0 . Then, the equilibrium viscosity, ηe , can be calculated from the following equations: η0 , for γ˙ = 0 ˙ = (4) ηe (γ) a γ˙ n−1 , for γ˙ = 0 where a is consistency index and n the pseudo-plasticity index. On the other hand, from Eq. (1), the equilibrium viscosity can be calculated as: β
˙ ˙ = γ˙ αSe (γ) ˙ ηe (γ) Se (γ)
(5)
˙ can be obtained The equilibrium structural parameter, Se (γ), by solving Eq. (5). Finally, the shear stress, σ(t), for a viscoelastic system can be obtained from a mechanical Maxwell model, which has the elastic parameter, G, which is related with the strength of the structure but not with the structural level. The structural
˙ t) ∂σ(t) η(γ, G ∂t
(6)
The elastic relaxation time of the structure, λ, depends on ˙ time and shear rate and its value is η(γ,t)/G. Some colloidal suspensions show a finite and measurable value for the elastic relaxation time. When colloidal suspensions present null values for the elastic relaxation time, they only exhibit viscous behaviour and Eq. (6) becomes the typical Newton relation: ˙ t) γ(t). ˙ σ(t) = η(γ, In principle, the proposed structural model has been developed to determine the rheological behaviour of colloidal suspensions of any nature and concentration with shear rate changing with time. The model has a total of eight parameters that characterise the complete rheological behaviour of a colloidal system.
3. Experimental Colloidal dispersions of two polymers (Carbopol) have been analysed: Carbopol-940 of molecular weight = 4 × 106 g/mol and Carbopol-941 of molecular weight = 1.25 × 106 g/mol. Carbopol is the name of commercial polymers composed of long chains of polyacrylic acid cross-linked with allyl sucrose. The powdered form of these polymers was obtained from B.F. Goodrich. Polymer dispersions were prepared with de-ionised water. Dispersions were stirred at 500 rpm for the period needed to reach transparent gels. pH was fixed to 8.0 with NaOH. The analysed polymer concentrations were: 0.153 wt.% for Carbopol-940 and 0.186 wt.% for Carbopol-941. The rheological experiments were carried out using a rotational rheometer with controlled shear rate (Haake Rotovisco RV100). A concentric cylinder sensor was used. Shear rates ranged from 0.5 to 2000 s−1 and shear stress from 0.1 to 150 Pa. Measurements were made at 25 ± 0.1 ◦ C. After keeping the sample undisturbed for 7 h, the shear rate was linearly increased in 5 h from zero to the maximum shear rate. The maximum shear rate was maintained for 3 h. Finally, the shear rate was returned to zero in 5 h by lineal decrease over time. Equilibrium was obtained independently by maintaining interjacent shear rates until constant values were achieved for the shear stress.
4. Results and discussion Equilibrium values for viscosity were found by applying different shear rates and waiting for equilibrium. Eq. (4) fit the equilibrium values for viscosity. Simultaneous resolution of model equations permitted the fitting of experimental data to values calculated from the model. The values for the parameters were found with following considerations: (1) The
J. Labanda et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
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Table 1 Calculate parameters of proposed model that permit to fit the rheological behaviour of both Carbopol solutions η0 (Pa s) a (Pa sn ) n αa βa tSD (s) tSF (s) G (Pa)
Carbopol-940
Carbopol-941
7.00 5.28 0.424 −0.463 0.158 0.121γ˙ 0.105 1.263γ˙ 1.362 32.0
4.00 4.55 0.384 −0.383 0.165 0.351γ˙ 0.445 0.378γ˙ 0.383 32.0
a Values of α and β calculate directly the dimensionless shape factor following Eq. (2), where shear rate is used (s−1 ) and structural parameter is used in (Pa s).
elastic relaxation time is not zero, and therefore Eq. (6) was used. The value for the elastic parameter, G, was the same for both polymers because they both had the same monomeric composition. (2) The parameters of the kinetic processes of formation and destruction of structure were initially assumed to be independent of shear rate. Later, for a more precise estimate, these parameters were considered shear rate dependent as follow: tSF = ϕF × γ˙ χF ;
tSD = ϕD × γ˙ χD
(7)
where ϕF , χF , ϕD and χD are kinetic parameters. (3) The calculated shear stress at low shear rate was sensitive to theoretical zero shear rate, η0 . Therefore, viscosity at zero shear rate was deduced by fitting calculated data to experimental data at low shear rates. Table 1 shows the calculated parameters for both polymer solutions. Calculated shape factors, F, are much lower than 1 indicating that changes in structure shape and orientation are important. Therefore, the elastic deformations with very low relaxation time are significant. Fig. 1 and Fig. 2 show the evolution of experimental and calculated shear stress over shear rate of two different tests
Fig. 2. Experimental (triangles) and predicted (solid line) data of Carbopol940 solution at the maximum shear rate of 1080 s−1 . Filled circles correspond to equilibrium values.
for Carbopol-940 and 941, respectively. Fig. 1 shows the calculated and experimental data using thixotropic parameters with and without dependence on shear rate. Better fits were found with thixotropic parameters dependent on shear rates. The rheological behaviour of both solutions is well described by the proposed model. Carbopol-941 (low molecular weight) shows classic positive thixotropy and Carbopol-940 (high molecular weight) shows negative thixotropy.
5. Conclusions The proposed rheological structural model predicted positive and negative thixotropy of two colloidal dispersions of different polymers. The model takes into account the equilibrium viscosity, structural level, thixotropic time, and elasticity. Two aspects characterise the rheological behaviour of these polymers: (1) Shape factors are low, indicating that their structure is easily deformed and oriented in the shear rate. Chains of polymer Carbopol-940 are more easily deformed than those of Carbopol-941. (2) Calculated thixotropic time for destruction of the structure is short, indicating that structure break down is fast. On the other hand, thixotropic time for construction of the structure is significant, indicating that the process of structure built up is low.
Acknowledgement This study received financial support from the Project CICYT (PPQ2002-04115-C02-02). Fig. 1. Experimental (triangles) and predicted (solid line) data of Carbopol941 polymer solution at the maximum shear rate of 1620 s−1 . Discontinuous line corresponds to predicted data with thixotropic time independent of the shear rates and filled circles correspond to equilibrium values.
References [1] D.C.-H. Cheng, F. Evans, Br. J. Appl. Phys. 16 (1965) 1599.
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J. Labanda et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
[2] C. Tiu, D.V. Boger, J. Texture Stud. 5 (1974) 329. [3] A. Alessandrini, R. Lapasin, F. Sturzi, Chem. Eng. Commun. 17 (1982) 13. [4] J. Llorens, E. Rud´e, C. Mans, Prog. Colloid Polym. Sci. 100 (1996) 252. [5] F. Bautista, J.M. de Santos, J.E. Puig, O. Manero, J. Non-Newton. Fluid Mech. 80 (1999) 93. [6] D. Quemada, Eur. Phys. J.: Appl. Phys. 5 (1999) 191. [7] J.O. Carnali, M.S. Naser, Colloid Polym. Sci. 270 (1992) 183.
[8] Y.I. Cho, J.P. Hartnett, W.Y. Lee, J. Non-Newton. Fluid Mech. 15 (1984) 61. [9] N. Unlu, A. Ludwig, M. van Ooteghem, A.A. Hincal, Pharmazie 46 (1991) 784. [10] R. Pal, Chem. Eng. Commun. 98 (1990) 211. [11] W.H. Bauer, W.H. Fischer, S.E. Wiberley, Trans. Soc. Rheol. 5 (1965) 221. [12] B.W. Barry, M.C. Meyer, Int. J. Pharm. 2 (1979) 1. [13] B.W. Barry, M.C. Meyer, Int. J. Pharm. 2 (1979) 27.
Rheological model to predict the thixotropic behaviour of colloidal dispersions J. Labanda∗ , P. Marco, J. Llorens Chemical Engineering Department, University of Barcelona, c/Mart´ı i Franqu`es 1, 08028 Barcelona, Spain
Abstract We propose a rheological structural model to describe the thixotropic behaviour of colloidal dispersions. The model shows a certain connection between the structural changes of colloidal dispersions with viscosity. Viscosity depends on the size, shape and orientation of colloidal aggregates. Instantaneous shear rate changes cause instantaneous changes in aggregate shape and orientation, but non-instantaneous changes in aggregate size. The kinetic process in changing aggregate size determines the thixotropic behaviour of colloidal dispersions. The model differentiates two characteristic thixotropic times: one for structure formation and other for structure destruction. The elastic effects due to non-instantaneous changes of aggregate shape and orientation are also included in the model. The model has the following parameters: (i) three parameters to determine equilibrium or steady-state rheology; (ii) two parameters to set the viscosity for a given aggregate size at a given shear rate; (iii) two characteristic thixotropic times, which depend on shear rate; (iv) a characteristic elastic modulus. Each group of parameters are fitted independently with appropriate rheological tests and coupling among them is prevented. The model has been tested with consecutive lineal increases and decreases of the shear rates with time levels of maximum shear rates. Carbopol solutions of different molecular weights have been analysed with good agreement between experimental and predicted viscosity. © 2004 Elsevier B.V. All rights reserved. Keywords: Rheology; Thixotropy; Colloidal dispersions; Carbopol
1. Introduction Thixotropy is caused by time-consuming structural changes in colloidal dispersions at a given shear rate. The literature describes theoretical structural models [1–4], which use a structural parameter, and two fundamental equations: steady state and kinetic. The structural parameter, λ, is a scalar that accounts for the structural level of the colloidal system. For a completely broken down structure λ = 0 and for a completely built-up structure λ = 1. Recently, Bautista et al. [5] introduced viscoelastic aspects in the models and Quemada [6] used the concept of effective volume fraction to describe the structural parameter. All these structural models describe certain aspects of thixotropic behaviour, but complex behaviour is difficult to explain. The aim of this study is to develop a rheological structural model to describe the complex rheology behaviour of ∗
Corresponding author. E-mail address: [email protected] (J. Labanda).
0927-7757/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2004.08.062
Carbopol solutions when the shear rate changes. Carbopol is a commercially polyacrylic acid found in many cosmetic and pharmaceutical products. The formation of networked microgel structures above a given polymer concentration, ionic strength and pH, is its principal feature [7]. Rheological studies of Carbopol gels have shown weak viscoelasticity at rest [8–9], yield stress [10] and shear-thinning at steady state [11–13]. In non-steady-state conditions, Carbopol gels show complex thixotropy caused by break down and built up of micro-gels aggregates.
2. Theory The proposed structural model is based on the distinction between structural level of aggregates, quantified by a structural parameter, S, and the shape and orientation of these aggregates, quantified by a shape factor, F. The viscosity of the colloidal system depends on the structural parameter and the shape factor. The viscosity of the colloidal suspension,
124
J. Labanda et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
which mat vary over time, t, is associated with the structural level and shape factor as follow:
level is only connected with the viscosity through Eq. (1). Therefore, the relation between viscosity and shear rate is:
˙ t) = F (γ, ˙ S(γ, ˙ t )) S(γ, ˙ t) η(γ,
˙ t) γ(t) ˙ − σ(t) = η(γ,
(1)
˙ ) is the structural parameter at the instant before where S(γ,t t, t = t – t. Changes of shape and orientation are elastic deformations with very short relaxation times and, therefore, almost non-time dependant. However, changes of structural level are clearly time dependant. The shape factor depends on shear rate and structural level as follows:
β
˙ ),t ) ˙ S) = γ˙ α·S(γ(t F (γ,
(2)
where α and β are parameters that can be determined from experimental data. For a given shear rate, the non-equilibrium structural parameter varies with time following a first order kinetic process due to structure break down and build up because of shear forces and particle interactions, respectively. This kinetic process has just two kinetic parameters: thixotropic time for construction of the structure, tSF , and thixotropic time for destruction of the structure, tSD . The thixotropic time is the time needed to reach the equilibrium structural level after shear rate changes. The evolution of the structural parameter with time is obtained by integrating the kinetic equation. Then, t ˙ ˙ ), t) − Se (γ(t))] ˙ ˙ S(γ(t), t) = [S(γ(t exp − + Se (γ(t)) tS (3) where t = t – t and tS is tSF when the structure level increases with time or tSD when the structure level decreases with time. ˙ Se (γ(t)) is the equilibrium structural parameter for a given ˙ shear rate, whose evolution with time is given by γ(t). The viscosity of most colloidal suspensions, at steady state, follows pseudo-plastic behaviour with power law dependence on shear rate, even at very low shear rates. Obviously, at rest the system shows the maximum viscosity or zero shear rate viscosity, η0 . Then, the equilibrium viscosity, ηe , can be calculated from the following equations: η0 , for γ˙ = 0 ˙ = (4) ηe (γ) a γ˙ n−1 , for γ˙ = 0 where a is consistency index and n the pseudo-plasticity index. On the other hand, from Eq. (1), the equilibrium viscosity can be calculated as: β
˙ ˙ = γ˙ αSe (γ) ˙ ηe (γ) Se (γ)
(5)
˙ can be obtained The equilibrium structural parameter, Se (γ), by solving Eq. (5). Finally, the shear stress, σ(t), for a viscoelastic system can be obtained from a mechanical Maxwell model, which has the elastic parameter, G, which is related with the strength of the structure but not with the structural level. The structural
˙ t) ∂σ(t) η(γ, G ∂t
(6)
The elastic relaxation time of the structure, λ, depends on ˙ time and shear rate and its value is η(γ,t)/G. Some colloidal suspensions show a finite and measurable value for the elastic relaxation time. When colloidal suspensions present null values for the elastic relaxation time, they only exhibit viscous behaviour and Eq. (6) becomes the typical Newton relation: ˙ t) γ(t). ˙ σ(t) = η(γ, In principle, the proposed structural model has been developed to determine the rheological behaviour of colloidal suspensions of any nature and concentration with shear rate changing with time. The model has a total of eight parameters that characterise the complete rheological behaviour of a colloidal system.
3. Experimental Colloidal dispersions of two polymers (Carbopol) have been analysed: Carbopol-940 of molecular weight = 4 × 106 g/mol and Carbopol-941 of molecular weight = 1.25 × 106 g/mol. Carbopol is the name of commercial polymers composed of long chains of polyacrylic acid cross-linked with allyl sucrose. The powdered form of these polymers was obtained from B.F. Goodrich. Polymer dispersions were prepared with de-ionised water. Dispersions were stirred at 500 rpm for the period needed to reach transparent gels. pH was fixed to 8.0 with NaOH. The analysed polymer concentrations were: 0.153 wt.% for Carbopol-940 and 0.186 wt.% for Carbopol-941. The rheological experiments were carried out using a rotational rheometer with controlled shear rate (Haake Rotovisco RV100). A concentric cylinder sensor was used. Shear rates ranged from 0.5 to 2000 s−1 and shear stress from 0.1 to 150 Pa. Measurements were made at 25 ± 0.1 ◦ C. After keeping the sample undisturbed for 7 h, the shear rate was linearly increased in 5 h from zero to the maximum shear rate. The maximum shear rate was maintained for 3 h. Finally, the shear rate was returned to zero in 5 h by lineal decrease over time. Equilibrium was obtained independently by maintaining interjacent shear rates until constant values were achieved for the shear stress.
4. Results and discussion Equilibrium values for viscosity were found by applying different shear rates and waiting for equilibrium. Eq. (4) fit the equilibrium values for viscosity. Simultaneous resolution of model equations permitted the fitting of experimental data to values calculated from the model. The values for the parameters were found with following considerations: (1) The
J. Labanda et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
125
Table 1 Calculate parameters of proposed model that permit to fit the rheological behaviour of both Carbopol solutions η0 (Pa s) a (Pa sn ) n αa βa tSD (s) tSF (s) G (Pa)
Carbopol-940
Carbopol-941
7.00 5.28 0.424 −0.463 0.158 0.121γ˙ 0.105 1.263γ˙ 1.362 32.0
4.00 4.55 0.384 −0.383 0.165 0.351γ˙ 0.445 0.378γ˙ 0.383 32.0
a Values of α and β calculate directly the dimensionless shape factor following Eq. (2), where shear rate is used (s−1 ) and structural parameter is used in (Pa s).
elastic relaxation time is not zero, and therefore Eq. (6) was used. The value for the elastic parameter, G, was the same for both polymers because they both had the same monomeric composition. (2) The parameters of the kinetic processes of formation and destruction of structure were initially assumed to be independent of shear rate. Later, for a more precise estimate, these parameters were considered shear rate dependent as follow: tSF = ϕF × γ˙ χF ;
tSD = ϕD × γ˙ χD
(7)
where ϕF , χF , ϕD and χD are kinetic parameters. (3) The calculated shear stress at low shear rate was sensitive to theoretical zero shear rate, η0 . Therefore, viscosity at zero shear rate was deduced by fitting calculated data to experimental data at low shear rates. Table 1 shows the calculated parameters for both polymer solutions. Calculated shape factors, F, are much lower than 1 indicating that changes in structure shape and orientation are important. Therefore, the elastic deformations with very low relaxation time are significant. Fig. 1 and Fig. 2 show the evolution of experimental and calculated shear stress over shear rate of two different tests
Fig. 2. Experimental (triangles) and predicted (solid line) data of Carbopol940 solution at the maximum shear rate of 1080 s−1 . Filled circles correspond to equilibrium values.
for Carbopol-940 and 941, respectively. Fig. 1 shows the calculated and experimental data using thixotropic parameters with and without dependence on shear rate. Better fits were found with thixotropic parameters dependent on shear rates. The rheological behaviour of both solutions is well described by the proposed model. Carbopol-941 (low molecular weight) shows classic positive thixotropy and Carbopol-940 (high molecular weight) shows negative thixotropy.
5. Conclusions The proposed rheological structural model predicted positive and negative thixotropy of two colloidal dispersions of different polymers. The model takes into account the equilibrium viscosity, structural level, thixotropic time, and elasticity. Two aspects characterise the rheological behaviour of these polymers: (1) Shape factors are low, indicating that their structure is easily deformed and oriented in the shear rate. Chains of polymer Carbopol-940 are more easily deformed than those of Carbopol-941. (2) Calculated thixotropic time for destruction of the structure is short, indicating that structure break down is fast. On the other hand, thixotropic time for construction of the structure is significant, indicating that the process of structure built up is low.
Acknowledgement This study received financial support from the Project CICYT (PPQ2002-04115-C02-02). Fig. 1. Experimental (triangles) and predicted (solid line) data of Carbopol941 polymer solution at the maximum shear rate of 1620 s−1 . Discontinuous line corresponds to predicted data with thixotropic time independent of the shear rates and filled circles correspond to equilibrium values.
References [1] D.C.-H. Cheng, F. Evans, Br. J. Appl. Phys. 16 (1965) 1599.
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J. Labanda et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 249 (2004) 123–126
[2] C. Tiu, D.V. Boger, J. Texture Stud. 5 (1974) 329. [3] A. Alessandrini, R. Lapasin, F. Sturzi, Chem. Eng. Commun. 17 (1982) 13. [4] J. Llorens, E. Rud´e, C. Mans, Prog. Colloid Polym. Sci. 100 (1996) 252. [5] F. Bautista, J.M. de Santos, J.E. Puig, O. Manero, J. Non-Newton. Fluid Mech. 80 (1999) 93. [6] D. Quemada, Eur. Phys. J.: Appl. Phys. 5 (1999) 191. [7] J.O. Carnali, M.S. Naser, Colloid Polym. Sci. 270 (1992) 183.
[8] Y.I. Cho, J.P. Hartnett, W.Y. Lee, J. Non-Newton. Fluid Mech. 15 (1984) 61. [9] N. Unlu, A. Ludwig, M. van Ooteghem, A.A. Hincal, Pharmazie 46 (1991) 784. [10] R. Pal, Chem. Eng. Commun. 98 (1990) 211. [11] W.H. Bauer, W.H. Fischer, S.E. Wiberley, Trans. Soc. Rheol. 5 (1965) 221. [12] B.W. Barry, M.C. Meyer, Int. J. Pharm. 2 (1979) 1. [13] B.W. Barry, M.C. Meyer, Int. J. Pharm. 2 (1979) 27.