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Theoretical correlation between viscosities at dynamic and steady flow states in Aureobasidium pullulans culture fluids
JOURNAL OF BIOSCIENCE AND BIOENGINEERING Vol. 98, No. 6, 497–499. 2004
Theoretical Correlation between Viscosities at Dynamic and Steady Flow States in Aureobasidium pullulans Culture Fluids HISAMOTO FURUSE1* AND KIYOSHI TODA2 Institute of Molecular and Cellular Biosciences, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan1 and Department of Applied Chemistry and Biotechnology, Niigata Institute of Technology, 1719 Fujihashi, Kashiwazaki, Niigata 945-1195, Japan2 Received 19 May 2004/Accepted 22 September 2004
For the culture fluid and exopolysaccharide solution of the fungus Aureobasidium pullulans in a previous study, the angular frequency dependence of dynamic viscosity is well superimposed on the shear rate dependence of steady flow viscosity. For various polymeric fluids, the superimposition of the dependencies of steady flow viscosity and dynamic viscosity has also been realized. In this study, it is derived that, by assuming the sinusoidal oscillating flow of a viscoelastic fluid, complex viscosity is equivalent to viscosity in a steady flow measurement. A similar relation may also hold for the dynamic viscosity of an ideal viscous fluid. Generally, the measurement of the steady flow viscosity deprives a viscoelastic fluid of its elastic nature due to the disruption of the network structure causing the viscoelasticity. For the culture fluid and exopolysaccharide solution of A. pullulans, the correlation between the dynamic viscosity and the network structure was low. The dynamic viscosity in this case is consistent with the steady flow viscosity in the superimposing correlation. [Key words: Aureobasidium pullulans culture fluid, complex viscosity, dynamic viscosity, steady flow viscosity, viscosity correlation]
observed in various polymeric fluids is made in the argument that follows. The steady flow viscosity is given by
For the culture fluid and exopolysaccharide solution of the fungus Aureobasidium pullulans, while complex viscosity considerably shifted upwards from the steady flow viscosity, the angular frequency dependence of dynamic viscosity was well superimposed on the shear rate dependence of steady flow viscosity (1). In the culture fluid, the influence of strain amplitude on the angular frequency dependence of the dynamic viscosity was small, although the strain amplitude affected the network structure of the viscoelastic fluid considerably. That is, the correlation between the dynamic viscosity and the network structure was comparatively low. Hitherto, it has been experimentally found for various polymeric fluids that in most cases dynamic viscosity (complex viscosity in some cases of measurement) against angular frequency and steady flow viscosity against shear rate have a superimposing correlation (2, 3). In the present paper, it is theoretically shown, for approximate sinusoidal oscillating flow in the nonlinear behavior of a viscoelastic fluid, that the complex viscosity agrees with the viscosity obtained by measuring the shear stress and shear rate in a steady flow measurement. A similar argument can also be made in the case of an ideal viscous fluid, in which the dynamic viscosity is correlated with the measured viscosity. A simple explanation for this correlation
th = ----(1) g· where h, t, and g· represent the steady flow viscosity, shear stress, and shear rate, respectively. The Qouette flow, which is used for the viscosity measurement, approximates the above laminar shearing flow. On the other hand, the following relation defines the complex dynamic viscosity: h* = s*/(dg*/dq)
(2)
where s* and g* are the complex variables of stress and strain, respectively, and q represents the elapsed time (4). The sinusoidal flow of a viscoelastic fluid is expressed by g* = g0eiwq s* = s0ei(wq +d)
(3) (4)
where g0 and s0 are the maximum amplitudes of strain and stress, and w and d represent the angular frequency and phase-shift angle, respectively. By substituting g* and s* in Eqs. 3 and 4 into Eq. 2, h* is expressed as s0 s0 h* = ---------cos d - sin d - i ----------g0 w g0 w
* Corresponding author. e-mail: [email protected] phone/fax: +81-(0)467-52-6267 497
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498
J. BIOSCI. BIOENG.,
FURUSE AND TODA
This complex number has the form h* = h¢ - ih¢¢
(6)
where the first term at the right-hand side is called the dynamic viscosity. By comparing Eqs. 5 and 6, the following expression is obtained. s0 h¢ = ---------(7) - sind g0 w s0 h¢¢ = ---------(8) - cosd g0 w The absolute magnitude of the complex dynamic viscosity, which is simply called complex viscosity, is given by |h*| =
2
h¢ + h²
2
(9)
By using Eqs. 7 and 8, the complex viscosity is expressed as
s0 |h*| = ---------(10) g0 w in which the expression at the right-hand side is consistent with the dynamic viscosity of an ideal viscous fluid (at d = p/2 in Eq. 5). In the sinusoidal flow of a viscoelastic fluid, the shear stress and shear rate are measured with a phase shift (p/2 - d), under an external action creating the dynamic state. They are proportional to the applied shear strain amplitude, the direction of which is periodically changed. Accordingly, the ratios of those measurement quantities to the shear strain amplitude of the same phase, which are free from the dynamic amplitude, define the shear stress (t¢) and shear rate ( g· ¢) in a steady flow measurement. The shear stress in this case is composed of parts generated from shearing and oscillating states. The ratios are, for the shear stress given by the real part of Eq. 4, s0 cos(wq + d) s0 = -------g0 cos(wq + d) g0 and, for the shear rate expressed as g· = -wg0 sinwq, t¢ =
(11)
-wg0 sinwq =w (12) g0 cos(wq + p/2) The viscosity obtained by providing the values of the shear stress and shear rate in a dynamic state is correlated with the complex viscosity in the same state using the relation in Eq. 10 as g· ¢ =
t¢- = s0/g0 = |h*| h = -----(13) w g· ¢ Thus, the steady flow viscosity h obtained by measuring the shear stress (t) and shear rate ( g· ) agrees with the complex viscosity |h*| in which t¢ (= s0/g0) and g· ¢ (= w) in a corresponding dynamic state are respectively equal to the t and g· measurements. Conversely, the complex viscosity |h*| obtained by measuring the shear stress s0/g0 (= t¢) and angular frequency w (= g· ¢) agrees with the steady flow viscosity h obtained by using the t¢ and g· ¢ measurements as the shear stress (t) and shear rate ( g· ), respectively.
A treatment analogous to the above argument on the complex viscosity can be employed for the dynamic viscosity h¢ of an ideal viscous fluid, in which h¢ takes the same form as |h*| of a viscoelastic fluid (cf. Eqs. 5 and 10). From the presented consideration, if measurements of the steady flow viscosity and the complex viscosity result from the same state of a viscoelastic fluid, a superimposing correlation should be realized between these viscosity factors. However, since the measurement of the steady flow viscosity is always accompanied by the disruption of the network structure due to the shear stress generated on the flow plane even if the structure is created during the shearing flow, the fluid looses its elastic nature. Consequently, the viscoelastic fluid has a tendency to behave as an ideal viscous fluid. The small normal stress effect appearing in the Qouette flow, in the measurement of the viscoelasticity, is disregarded as the effect on the shearing flow in the present study. When the correlation between the dynamic viscosity and the network structure is low, as in the polymeric fluids of the microbial culture fluid in this study, the dynamic viscosity is in a state independent of the network structure. In this case, the fluid can also be approximately regarded as an ideal viscous fluid with the same viscosity as that given by the first term at the right-hand side of Eq. 5. Hence, in this case, the relation of h' against w in the dynamic state is superimposed on that of h against g· in the steady flow state (1). In this viscoelastic measurement of the A. pullulans culture fluid, viscosities between dynamic and steady flow states were somewhat different in a sample of low viscosity and in a high shear and angular frequency region (Fig. 1 in Ref. 1). The discrepancy in both viscosities was caused by the difference between fluid structures concerned with dispersing polymers and particles, i.e., partially gelling polymers and microbes, under steadily shearing and dynamic oscillating states, particularly at the concentration used in this study. In the shearing flow, the constituent of the fluid is ground into the dispersing polymers and particles that produce the fluid viscosity. On the other hand, in the oscillating state, there is at any time, on a small scale, the possibility that a network structure consisting of the polymers including the particles is created. The creation of the network causes a decrease in the dispersion of the polymers and particles, resulting in somewhat lower values of viscosity in the oscillating state than in the steadily shearing state. In the measurement of dynamic viscoelasticity, because an effect analogous to the network disruption mentioned above for the steady flow viscosity is also brought about, the nonlinear behavior of a viscoelastic fluid ordinarily emerges to a greater or lesser extent (for example, see Ref. 5). Thus, we must take note that the measurement in the dynamic state generally gives an approximate viscoelasticity value. REFERENCES 1. Furuse, H., Amari, T., Miyawaki, O., Asakura, T., and Toda, K.: Characteristic behavior of viscosity and viscoelasticity of Aureobasidium pullulans culture fluid. J. Biosci. Bioeng., 93, 411–415 (2002).
VOL. 98, 2004
2. Cox, W. P. and Merz, E. H.: Correlation of dynamic and steady flow viscosities. J. Polym. Sci., 28, 619–622 (1958). 3. Morris, E. R., Cutler, A. N., Ross-Murphy, S. B., Rees, D. A., and Price, J.: Concentration and shear rate dependence of viscosity in random coil polysaccharide solutions. Carbohydr. Polym., 1, 5–21 (1981).
NOTES
499
4. van Wazer, J. R., Lyons, J. W., Kim, K. Y., and Colwell, R. F.: Viscosity and flow measurement, p. 329–351. Interscience Publishers, John Wiley & Sons, New York, London (1963). 5. Amari, T.: Non-linear viscoelastic properties of concentrated suspensions. Prog. Org. Coat., 31, 11–19 (1997).
Theoretical Correlation between Viscosities at Dynamic and Steady Flow States in Aureobasidium pullulans Culture Fluids HISAMOTO FURUSE1* AND KIYOSHI TODA2 Institute of Molecular and Cellular Biosciences, The University of Tokyo, 1-1-1 Yayoi, Bunkyo-ku, Tokyo 113-0032, Japan1 and Department of Applied Chemistry and Biotechnology, Niigata Institute of Technology, 1719 Fujihashi, Kashiwazaki, Niigata 945-1195, Japan2 Received 19 May 2004/Accepted 22 September 2004
For the culture fluid and exopolysaccharide solution of the fungus Aureobasidium pullulans in a previous study, the angular frequency dependence of dynamic viscosity is well superimposed on the shear rate dependence of steady flow viscosity. For various polymeric fluids, the superimposition of the dependencies of steady flow viscosity and dynamic viscosity has also been realized. In this study, it is derived that, by assuming the sinusoidal oscillating flow of a viscoelastic fluid, complex viscosity is equivalent to viscosity in a steady flow measurement. A similar relation may also hold for the dynamic viscosity of an ideal viscous fluid. Generally, the measurement of the steady flow viscosity deprives a viscoelastic fluid of its elastic nature due to the disruption of the network structure causing the viscoelasticity. For the culture fluid and exopolysaccharide solution of A. pullulans, the correlation between the dynamic viscosity and the network structure was low. The dynamic viscosity in this case is consistent with the steady flow viscosity in the superimposing correlation. [Key words: Aureobasidium pullulans culture fluid, complex viscosity, dynamic viscosity, steady flow viscosity, viscosity correlation]
observed in various polymeric fluids is made in the argument that follows. The steady flow viscosity is given by
For the culture fluid and exopolysaccharide solution of the fungus Aureobasidium pullulans, while complex viscosity considerably shifted upwards from the steady flow viscosity, the angular frequency dependence of dynamic viscosity was well superimposed on the shear rate dependence of steady flow viscosity (1). In the culture fluid, the influence of strain amplitude on the angular frequency dependence of the dynamic viscosity was small, although the strain amplitude affected the network structure of the viscoelastic fluid considerably. That is, the correlation between the dynamic viscosity and the network structure was comparatively low. Hitherto, it has been experimentally found for various polymeric fluids that in most cases dynamic viscosity (complex viscosity in some cases of measurement) against angular frequency and steady flow viscosity against shear rate have a superimposing correlation (2, 3). In the present paper, it is theoretically shown, for approximate sinusoidal oscillating flow in the nonlinear behavior of a viscoelastic fluid, that the complex viscosity agrees with the viscosity obtained by measuring the shear stress and shear rate in a steady flow measurement. A similar argument can also be made in the case of an ideal viscous fluid, in which the dynamic viscosity is correlated with the measured viscosity. A simple explanation for this correlation
th = ----(1) g· where h, t, and g· represent the steady flow viscosity, shear stress, and shear rate, respectively. The Qouette flow, which is used for the viscosity measurement, approximates the above laminar shearing flow. On the other hand, the following relation defines the complex dynamic viscosity: h* = s*/(dg*/dq)
(2)
where s* and g* are the complex variables of stress and strain, respectively, and q represents the elapsed time (4). The sinusoidal flow of a viscoelastic fluid is expressed by g* = g0eiwq s* = s0ei(wq +d)
(3) (4)
where g0 and s0 are the maximum amplitudes of strain and stress, and w and d represent the angular frequency and phase-shift angle, respectively. By substituting g* and s* in Eqs. 3 and 4 into Eq. 2, h* is expressed as s0 s0 h* = ---------cos d - sin d - i ----------g0 w g0 w
* Corresponding author. e-mail: [email protected] phone/fax: +81-(0)467-52-6267 497
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498
J. BIOSCI. BIOENG.,
FURUSE AND TODA
This complex number has the form h* = h¢ - ih¢¢
(6)
where the first term at the right-hand side is called the dynamic viscosity. By comparing Eqs. 5 and 6, the following expression is obtained. s0 h¢ = ---------(7) - sind g0 w s0 h¢¢ = ---------(8) - cosd g0 w The absolute magnitude of the complex dynamic viscosity, which is simply called complex viscosity, is given by |h*| =
2
h¢ + h²
2
(9)
By using Eqs. 7 and 8, the complex viscosity is expressed as
s0 |h*| = ---------(10) g0 w in which the expression at the right-hand side is consistent with the dynamic viscosity of an ideal viscous fluid (at d = p/2 in Eq. 5). In the sinusoidal flow of a viscoelastic fluid, the shear stress and shear rate are measured with a phase shift (p/2 - d), under an external action creating the dynamic state. They are proportional to the applied shear strain amplitude, the direction of which is periodically changed. Accordingly, the ratios of those measurement quantities to the shear strain amplitude of the same phase, which are free from the dynamic amplitude, define the shear stress (t¢) and shear rate ( g· ¢) in a steady flow measurement. The shear stress in this case is composed of parts generated from shearing and oscillating states. The ratios are, for the shear stress given by the real part of Eq. 4, s0 cos(wq + d) s0 = -------g0 cos(wq + d) g0 and, for the shear rate expressed as g· = -wg0 sinwq, t¢ =
(11)
-wg0 sinwq =w (12) g0 cos(wq + p/2) The viscosity obtained by providing the values of the shear stress and shear rate in a dynamic state is correlated with the complex viscosity in the same state using the relation in Eq. 10 as g· ¢ =
t¢- = s0/g0 = |h*| h = -----(13) w g· ¢ Thus, the steady flow viscosity h obtained by measuring the shear stress (t) and shear rate ( g· ) agrees with the complex viscosity |h*| in which t¢ (= s0/g0) and g· ¢ (= w) in a corresponding dynamic state are respectively equal to the t and g· measurements. Conversely, the complex viscosity |h*| obtained by measuring the shear stress s0/g0 (= t¢) and angular frequency w (= g· ¢) agrees with the steady flow viscosity h obtained by using the t¢ and g· ¢ measurements as the shear stress (t) and shear rate ( g· ), respectively.
A treatment analogous to the above argument on the complex viscosity can be employed for the dynamic viscosity h¢ of an ideal viscous fluid, in which h¢ takes the same form as |h*| of a viscoelastic fluid (cf. Eqs. 5 and 10). From the presented consideration, if measurements of the steady flow viscosity and the complex viscosity result from the same state of a viscoelastic fluid, a superimposing correlation should be realized between these viscosity factors. However, since the measurement of the steady flow viscosity is always accompanied by the disruption of the network structure due to the shear stress generated on the flow plane even if the structure is created during the shearing flow, the fluid looses its elastic nature. Consequently, the viscoelastic fluid has a tendency to behave as an ideal viscous fluid. The small normal stress effect appearing in the Qouette flow, in the measurement of the viscoelasticity, is disregarded as the effect on the shearing flow in the present study. When the correlation between the dynamic viscosity and the network structure is low, as in the polymeric fluids of the microbial culture fluid in this study, the dynamic viscosity is in a state independent of the network structure. In this case, the fluid can also be approximately regarded as an ideal viscous fluid with the same viscosity as that given by the first term at the right-hand side of Eq. 5. Hence, in this case, the relation of h' against w in the dynamic state is superimposed on that of h against g· in the steady flow state (1). In this viscoelastic measurement of the A. pullulans culture fluid, viscosities between dynamic and steady flow states were somewhat different in a sample of low viscosity and in a high shear and angular frequency region (Fig. 1 in Ref. 1). The discrepancy in both viscosities was caused by the difference between fluid structures concerned with dispersing polymers and particles, i.e., partially gelling polymers and microbes, under steadily shearing and dynamic oscillating states, particularly at the concentration used in this study. In the shearing flow, the constituent of the fluid is ground into the dispersing polymers and particles that produce the fluid viscosity. On the other hand, in the oscillating state, there is at any time, on a small scale, the possibility that a network structure consisting of the polymers including the particles is created. The creation of the network causes a decrease in the dispersion of the polymers and particles, resulting in somewhat lower values of viscosity in the oscillating state than in the steadily shearing state. In the measurement of dynamic viscoelasticity, because an effect analogous to the network disruption mentioned above for the steady flow viscosity is also brought about, the nonlinear behavior of a viscoelastic fluid ordinarily emerges to a greater or lesser extent (for example, see Ref. 5). Thus, we must take note that the measurement in the dynamic state generally gives an approximate viscoelasticity value. REFERENCES 1. Furuse, H., Amari, T., Miyawaki, O., Asakura, T., and Toda, K.: Characteristic behavior of viscosity and viscoelasticity of Aureobasidium pullulans culture fluid. J. Biosci. Bioeng., 93, 411–415 (2002).
VOL. 98, 2004
2. Cox, W. P. and Merz, E. H.: Correlation of dynamic and steady flow viscosities. J. Polym. Sci., 28, 619–622 (1958). 3. Morris, E. R., Cutler, A. N., Ross-Murphy, S. B., Rees, D. A., and Price, J.: Concentration and shear rate dependence of viscosity in random coil polysaccharide solutions. Carbohydr. Polym., 1, 5–21 (1981).
NOTES
499
4. van Wazer, J. R., Lyons, J. W., Kim, K. Y., and Colwell, R. F.: Viscosity and flow measurement, p. 329–351. Interscience Publishers, John Wiley & Sons, New York, London (1963). 5. Amari, T.: Non-linear viscoelastic properties of concentrated suspensions. Prog. Org. Coat., 31, 11–19 (1997).