- Email: [email protected]
Thixotropy of Highly Viscous Sodium (Carboxymethyl)cellulose Hydrogels
Thixotropy of Highly Viscous Sodium (Carboxymethyl)cellulose Hydrogels M. DOLZ†X, J. BUGAJ‡, J. PELLICER†, M. J. HERNAÄ NDEZ†,
AND
M. GOÄ RECKI‡
Received February 19, 1997, from the †Department of Thermodynamics, Faculty of Pharmacy and Physics, University of Valencia, 46100 Burjassot, Valencia, Spain, and ‡Department of Inorganic and Analytical Chemistry, Faculty of Pharmacy, K. Marcinkowski University of Medical Sciences, Grunwaldzka str., 6. 60−780 Poznan, PolandX. Accepted for publication September 5, 1997 Abstract 0 A general method to quantify the thixotropic behavior of systems with very low thixotropy is proposed. The areas enclosed by the rheograms τ ) f(γ˙ ) must be fitted to functions with well-determined boundary conditions. From these equations the corresponding thixotropic areas are obtained, together with the theoretical area enclosed by the rheogram corresponding to the maximum rheodestruction. The proposed method is applied to high viscosity sodium (carboxymethyl)cellulose gels.
shear rates (γ˘ min and γ˘ max, respectively) applied to the system and corresponding to agitation time, t. The mathematical functions, S ) S(t), cannot be arbitrary, for they must satisfy a series of clearly defined limiting conditions. In effect, examination of the experimental rheograms of a thixotropic system shows that, for t ) 0, the area S attains its maximum value, S ) Smax. In contrast, for very long agitation times, S tends toward a minimum limiting value, S ) Smin. Therefore, a simple expression for the function S(t) is
S(t) ) A+ Be-f(t)
(1)
Introduction The thixotropic behavior of certain systems, fluids, or semisolids that contain macromolecules such as stabilizers or thickeners is of particular interest in the technological processes of formulation and packaging in the pharmaceutical and food industries.1-3 Thixotropy involves a progressive decrease in viscosity, and thus of shear stress, due to the agitation produced on subjecting the system to continuous shear deformation, followed by the recovery of the rheological properties after a more-or-less prolonged rest period. The study of thixotropic systems is typically complicated, both when made on an empirical basis and in the context of theoretical models.4-7 Hysteresis experiments may also be used to investigate the rheological properties of thixotropic systems.8-12 In general, the approach involves the determination of the areas enclosed between one or more up-curve rheograms, corresponding to increasing shear rates, and different down-curve rheograms obtained with decreasing shear rates, following different agitation times. These areas are in turn referred to as thixotropic areas. It is normally accepted that greater thixotropic areas imply increased thixotropy, though this affirmation may be placed in doubt in certain cases. In fact, with the aim of comparatively studying the thixotropy of different gels, the use of a parameter known as the relative thixotropic area13 has been proposed that relates thixotropic areas to the total areas enclosed by the up curves corresponding to different gels. An additional difficulty in the experimental investigation of such systems arises when their thixotropic properties are limited, i.e., when the areas enclosed by the hysteresis cycles (ST) are very small compared to the area enclosed by the up curve. This problem becomes manifest when attempts are made to correlate the thixotropic areas obtained with the agitation times to which the system has been subjected. In fact, ST may vary on a random basis, following no logical law capable of accounting for its gradual increase over time. This randomness may be due to different causes, but particularly to the sensitivity of the viscometer employed. The present study proposes a method for solving the above problem on the basis of the prior determination, by means of a regression curve, of the form of the function S ) S(t), where S is the area determined for each of the curves obtained in a flow diagram, τ-γ˘ , between the minimum and maximum X
Abstract published in Advance ACS Abstracts, October 15, 1997.
© 1997, American Chemical Society and American Pharmaceutical Association
In the above equation, A and B are constants, and the concrete form of f(t) is dependent upon the rheological behavior of the system involved. However, the following two limiting conditions always apply
lim f(t) ) 0 tf0
(2)
lim f(t) ) ∞ tf∞
(3)
It is easy to verify that
A ) Smin and B ) Smax - Smin
(4)
Thus, equation [1] may also be expressed as follows
S(t) ) Smin + (Smax - Smin)e-f(t)
(5)
Once the function S(t) has been determined, it becomes evident that the thixotropic area (ST) is expressed by
ST ) Smax - S(t)
(6)
or, on considering eq 5,
ST ) (Smax - Smin)[1 - e-f(t)]
(7)
This expression provides the thixotropic areas as a function of the corresponding agitation times. However, knowledge of the absolute thixotropic areas does not suffice to comparatively investigate systems with highly different viscosities. In fact, thixotropic systems with very high viscosities may exhibit down curves that differ relatively little from the corresponding up curves, yet the enclosed areas may be large due to the high shear stresses reached in such systems. In the same way, in less viscous systems, the enclosed areas can be very small, though the rheogram up and down curves may differ considerably. The structural breakdown of more viscous systems may be relatively minor compared to less viscous systems, despite the fact that the absolute thixotropic area is much greater in the former.
S0022-3549(97)00075-0 CCC: $14.00
Journal of Pharmaceutical Sciences / 1283 Vol. 86, No. 11, November 1997
The relativizing of thixotropic areas has been carried out13 by means of the concept of relative thixotropic area:
ST SR ) 100 Smax
(8)
This equation allows us to calculate the percentage of rheodestroyed area caused by agitation, with respect to the total area enclosed by the rheogram up curve. Therefore, a system becomes more thixotropic as the value SR increases. By replacing (7) in (8), we see that
SR ) [100 - (SR)min][1 - e-f(t)]
(9)
Smin (SR)min ) 100 Smax
(10)
where
Equations 7 and 9 are formally identical and, because of that, the first one is always used. Obviously, to obtain the relative thixotropic area, Smax must be replaced by 100 and Smin by (SR)min, as defined in eq 10. The above concepts have been applied in the present study to the investigation of the thixotropic behavior of hydrogels containing different concentrations of high-viscosity sodium (carboxymethyl)cellulose (NaCMC); the small thixotropy of this substance requires the application of a mathematical treatment such as that described in the present paper.
Material and Methods The hydrogels of NaCMC (Aldrich Chemical Co., Milwaukee, WI) were studied at concentrations of 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, and 2.4% (w/w). The preparation of hydrogels was carried out by dispersing the amounts of NaCMC required for a total of 250 g at the desired concentration in isopropyl alcohol, which was used as dispersing agent. The dispersions were placed in a water bath at 85 °C and water at this temperature was added to each dispersion in a proportion of 2/3 of the whole amount required for every hydrogel. The gels were stirred at this temperature for 10 min in order to homogenize them, facilitate gel formation, and evaporate the alcohol. Care was taken to avoid the introduction of bubbles. Water was added after cooling the gel to room temperature (25 °C) to make up lost weight. The hydrogels were maintained at room temperature for 24 h, after which the experimental measurements were made. A Brookfield Digital DVII+ rotary viscometer was used to carry out the experimental measurement of apparent viscosity, η. The viscosity was determined for the eight rotational speeds provided by the viscometer. Taking into account the experimental characteristics involved, these correspond to shear rates,14 γ˘ , of 0.063, 0.126, 0.314, 0.628, 1.26, 2.51, 6.28, and 12.57 s-1. The measurements of apparent viscosity were initially made with the unagitated gels in increasing order of shear rate and in time intervals of 10 s, which enabled the corresponding up-curve rheograms to be obtained. Then, after agitation times of 1, 2, 3, 4, 5, 10, 15, and 20 min at the maximum rotational speed (60 rpm), the apparent viscosity values corresponding to the shear rates in decreasing order were measured in order to obtain the down-curve rheograms.
Results and Discussion On the basis of the experimental measurement of the viscosity of NaCMC dispersions as a function of shear rate (γ˘ ) and of agitation time (t), the corresponding shear stresses (τ) generated have been obtained for all the concentrations evaluated. As an example, Figure 1 shows the rheograms τ ) f(γ˘ ,t) corresponding to the hydrogel formulated with a 2% w/w concentration of NaCMC. These rheograms are qualitatively similar to those obtained for the rest of the concentra1284 / Journal of Pharmaceutical Sciences Vol. 86, No. 11, November 1997
Figure 1sShear stress as a function of shear rate for the 2% NaCMC hydrogels at different agitation times: filled circle, up curve (0 min); open box, 1 min; filled triangles, 2 min; open circles, 3 min; filled diamond, 4 min; plus sign, 5 min; plus in a box, 10 min; asterisk, 15 min; filled box, 20 min.
tions studied, and their convex shape confirms the pseudoplastic behavior of such systems. Attention should also be drawn to the progressive decrease in shear stress as agitation time is prolonged, this being typical of thixotropic systems. The decrease is most apparent during the first 2-3 min and becomes much less noticeable as agitation time is extended. This fact, and the limited sensitivity of the measuring viscometer, could account for the irregular distribution that the thixotropic areas may present when correlated to agitation time. In such cases thixotropy must be studied by the method described in the Introduction, i.e., on the basis of prior fits of the total areas, S, which determine both the up-curve rheogram and each of the down-curve rheograms obtained for different agitation times. The calculation of the areas has been made by numerical integration of the function, τ ) f(γ˘ ,t), between the values γ˘ o ) 0.063 and γ˘ 1 ) 12.56 s-1, these being the extreme shear rates of the viscometer employed. The values of S calculated for each NaCMC concentration were fitted by the least-squares method using equations of type (1) taking for f(t) the following function:
f(t) ) Kt1/2
(11)
The above function satisfies the limiting conditions required by eqs 2 and 3, and has been chosen in view of its simplicity and because it provides sufficiently acceptable correlation coefficients (r > 0.995) in all cases. The substitution of (11) in (5) in turn provides the following empirical model 1/2
S(t) ) Smin + (Smax - Smin)e-Kt
(12)
The parameter K characterizes the time variation of the area S. If the system in question is thixotropic, then K > 0, since S must decrease on extending agitation time. Moreover, K must be finite, for only in this case does the area depend upon agitation time. Finally, the influence of K upon the values of S(t) is clearly conditioned by the value of (Smax Smin), which in turn depends on the polymer concentration of the gel, as is demonstrated below. The graphic representations of the functions S(t) obtained for the different NaCMC concentrations employed are shown
Figure 2sS(t) values versus agitation times for the NaCMC hydrogels at various concentrations: filled circle, 2.4%; open box, 2.2%; filled triangle, 2.1%; o, 2.0%; filled diamond, 1.9%; plus sign, 1.8%; plus in a box, 1.7%; asterisk, 1.6%; filled box, 1.5%.
in Figure 2. It should also be pointed out that systems containing NaCMC concentrations of under 1.5% and over 2.4% have been evaluated, though in all cases the area obtained remained practically constant after the first minute of agitation. In view of the information provided by Figure 2, two fundamental aspects should be pointed out. First, the area determined by the different rheograms increases with the polymer concentration of the gels, this being a logical consequence of their rising viscosity. Second, for a given concentration, the area decreases as the agitation time is prolonged, this in turn being a result of the thixotropy of the system. The decrease becomes more accentuated as the concentration of NaCMC increases. According to our empirical model (12), the intersections of the different curves with the ordinate axis determine the maximum areas, Smax, enclosed by the up-curve rheograms corresponding to each concentration. These areas are in turn calculated experimentally by numerical integration of the different up curves, τup ) f(γ˘ ). On the other hand, and based on the empirical model (12), calculations were made of the minimum areas, Smin, determined by infinite agitation times, for each polymer concentration studied. Both Smax and Smin were fitted by the leastsquares method as a function of polymer concentration c, providing the following equations:
Smax ) (3.13c - 4.07) × 103 Pa‚s-1
(13)
Smin ) (2.01c - 2.39) × 103 Pa‚s-1
(14)
The correlation coefficients are greater than 0.994 in both cases. Clearly, these areas must be positive, which limits the application of eqs 13 and 14 to concentration values of 1.5% w/w, or higher. This may be attributable to the fact that the tridimensional networks of the NaCMC dispersions of lower concentrations are not totally structured, and hence their rheological behavior is more similar to that of a macromolecular suspension than of an actual gel. The graphic representations of eqs 13 and 14 are shown in Figure 3. In
Figure 3sSmax, Smin, and (ST)max as a function of the concentration of NaCMC hydrogels (filled circles correspond to experimental Smax values; Smin and (ST)max are theoretical curves).
Figure 4sK values of eq 12, versus NaCMC hydrogels concentration.
this figure we have only included the fitted values of Smax (experimental values), but not of Smin (not experimental values), for clarity. Finally, the fits performed also provide the values of the parameter K corresponding to the different concentrations. Figure 4 shows the graphic representation of K ) f(c), where K is seen to increase progressively beyond c ) 1.5% to a maximum value of 1.2 min-1/2 for a 2.0% w/w concentration of NaCMC. In principle, this indicates that a gel prepared with that concentration exhibits the greatest dependency of S upon agitation time. This gel may thus be considered to possess the greatest thixotropy of all the preparations studied. In the case of lower concentration gels, this dependency is less, a fact that, as before, may be attributed to their behavior as macromolecular suspensions. However, for concentrations of over 2%, K is found to decrease rapidly with rising concentration, thus reflecting less dependency of S on agitation time in the case of the lower polymer concentrations. This fact, in contrast to the previous case, may be explained by considering that the tridimensional
Journal of Pharmaceutical Sciences / 1285 Vol. 86, No. 11, November 1997
Figure 5sThixotropic areas as a function of agitation time for the NaCMC hydrogels at various concentrations: filled circle, 2.4%; open box, 2.2%; filled triangle, 2.1%; open circle, 2.0%; filled diamond, 1.9%; plus sign, 1.8%; plus in a box, 1.7%; asterisk, 1.6%; filled box, 1.5%.
networks of these more concentrated gels are highly structured and thus more difficult to deform by agitation. Clearly, the study of the thixotropic behavior of the gels analyzed is still incomplete, for the combined influence of all the parameters, i.e., K, Smax, and Smin, should also be analyzed. This influence has been studied on the basis of the thixotropic areas, ST, obtained as a function of agitation time, for each concentration. These areas have been calculated by means of eqs 7 and 11. The graphic representations of the functions ST ) f(t,c) are in turn reflected in Figure 5, where the thixotropic area is seen to increase on prolonging the agitation time, regardless of polymer concentration. Likewise, it should be pointed out that ST also increases with NaCMC concentration, when comparing the values corresponding to the same agitation times. All the functions represented in Figure 5 exhibit a tendency toward different maximum values with each concentration, when t tends toward infinity. These maximum values are clearly reflected by (ST)max ) (Smax Smin). Thus, from eqs 13 and 14, the maximum thixotropic area as a function of increasing polymer concentration is provided by the equation
(ST)max ) (1.12c - 1.68) × 103 Pa‚s-1
(15)
This indicates the linear increase of the maximum thixotropic areas, (ST)max, with NaCMC concentration, in the interval c ) [1.5, 2.4]% (see Figure 3). A similar result is obtained from the mean thixotropic areas, S h T, in the entire agitation time interval [t1, t2] studied, in this case [0, 20] min, calculated by the following expression13
S hT )
∫
1 (t2 - t1)
t2
t1
ST(t) dt
(16)
In fact, on representing the S h T values obtained for all the gels as a function of concentration (Figure 6), and fitting the points by the least-squares method using a linear function, we obtain the equation
S h T ) (0.762c - 1.095) × 103 Pa‚s-1 (r ) 0.990) (17) 1286 / Journal of Pharmaceutical Sciences Vol. 86, No. 11, November 1997
Figure 6sMean thixotropic areas versus NaCMC hydrogels concentration.
This indicates that an increase in polymer concentration produces a linear increment in the mean thixotropic area of the NaCMC hydrogels, within the concentration interval and agitation times considered in the present study. The fact that the gel with the greatest thixotropic area is precisely the preparation possessing the highest polymer concentrationsas deduced from eq 17sapparently contradicts the results regarding thixotropic behavior obtained from the study of the values of the parameters in eq 12 and from its justification based upon the tridimensional structure of the NaCMC gels. In this sense it was pointed out that the gel containing 2% w/w of polymer exhibited the greatest dependency of S on agitation time, and therefore should be the system with the greatest thixotropy. This problems is solved by considering that a greater thixotropic area does not necessarily imply greater thixotropy, since comparisons of the thixotropic behavior of different systems should be made on the basis of studies of the relative thixotropic areas, SR, defined by eq 9. In effect, on substituting the values of K, Smax, and Smin in eq 9, and taking as f(t) the function defined by eq 11, we calculated the relative thixotropic areas for all the concentrations and agitation times studied. The graphic representations of SR ) f(c,t) are shown in Figure 7, where it is seen that the lowest relative thixotropic area (lowest thixotropy) corresponds to gels with concentrations of 1.5, 1.6, and 1.7% w/w NaCMC, regardless of agitation time. From a 1.8% w/w NaCMC concentration onward, the differences between the relative thixotropic areas were so small (similar thixotropy) that they could be considered as due to experimental errors. However, there is a differential fact between high concentration gels (2.1, 2.2, and 2.4% w/w NaCMC) and those of intermediate concentrations (1.8, 1.9, and 2% w/w NaCMC). The graphics corresponding to the former ones present a different curvature, which implies a lower relative thixotropic area in the early minutes of agitation. On the other hand, when the agitation time grows, the relative thixotropic areas become higher than the areas obtained for the rest of the concentrations. These facts seem to indicate that the total breakdown of the high concentration gels structure also requires longer agitation times, although in low concentration gels the total breakdown attained is smaller. The quantitative analysis of the differences between the relative thixotropic areas may be made on the basis of the relative values of the maximum, (SR)max, and mean thixotropic areas, S h T, calculated from eqs 13, 15, and 17. The resulting expressions obtained (as percentages) are
Figure 7sRelative thixotropic areas as a function of agitation time for all NaCMC hydrogels concentration: filled circle, 2.4%; open box, 2.2%; filled triangle, 2.1%; open circle, 2.0%; filled diamond, 1.9%; plus sign, 1.8%; plus in a box, 1.7%; asterisk, 1.6%; filled box,1.5%.
(SR)max ) 35.8 S h R ) 24.4 -
22.4 3.13c - 4.07
10.4 3.13c - 4.07
(18)
Figure 8sRelative maximum and mean thixotropic areas as a function of the concentration of NaCMC hydrogels.
Acknowledgments J.B. thanks the Spanish Ministry of Foreign Affairs for their support in the form of a grant which made 3 months research possible in the Department of Thermodynamics, University of Valencia.
(19)
These allow us to calculate both relative thixotropic areas, and each is seen to tend toward a different limiting value. From eqs 18 and 19, it may be shown that the increase in the maximum and mean relative thixotropic areas is very slow, i.e., on the order of 3.7% and 1.7% respectively, upon increasing polymer concentration from 2.0 to 2.4% w/w NaCMC (Figure 8). The variations in thixotropic areas on this order of magnitude are those that may become masked in most cases as a result of experimental errors due to limitations in viscometer sensitivity, the lack of sample homogeneity, very small temperature variations in the course of the measurement process, and so on. However, gels that apparently lack thixotropy, such as those prepared with highly viscous NaCMC or many other systems (including hydrogels formulated with certain acrylic acid polymers that we are currently investigating), in reality exhibit weak thixotropic behavior that may easily be demonstrated and quantified by the procedure described in the present study.
References and Notes 1. Ford, E. W.; Steffe, J. F. J. Texture Stud. 1986, 17, 71-85. 2. Garza, S.; Faus, S.; Giner, J.; Martin, O.; Ibarz, A. Progress and Trends in Rheology IV: Proc. Fourth European Rheology Conference. 1994, Sevilla, Spain. 3. Dolz, M.; Gonza´lez, F.; Herra´ez, M.; Dı´ez-Sales, O. J. Disper. Sci Technol. 1994, 15 (2), 189-205. 4. Tiu, C.; Boger, D. V. J. Texture Stud. 1974, 5, 329-338. 5. Martı´nez-Padilla, L. P.; Hardy, J. J. Texture Stud. 1989, 20, 7185. 6. Amemiya, J. I.; Shoemaker, C. F. J. Food Eng. 1992, 16, 1724. 7. Go´recki M.; Bugaj J. Acta Pol. Pharm. 1994, 51-2, 191-194. 8. Harris, J. Nature 1967, 214, 797-797. 9. Cheng, D. C.-H. Nature 1967, 216, 1099-1100. 10. Mewis, J. J. Non-Newton Fluid Mech. 1979, 6, 1-20. 11. Dolz-Planas, M.; Roldan-Garcı´a, C.; Herra´ez-Domı´nguez, J.; Belda-Maximino, R. J. Pharm. Sci. 1991, 80 (1), 75-79. 12. Fresno, M. J.; Jime´nez, M. M.; Selle´s, E. Cienc. Pharm. 1993, 3 (2), 81-87. 13. Dolz, M.; Herna´ndez, M. J.; Pellicer, J.; Delegido, J. J. Pharm. Sci. 1995, 84 (6), 728-732. 14. More solutions to sticky problems, Brookfield Engineering Laboratories, Inc.: Stougton, MA, 1983.
JS970075+
Journal of Pharmaceutical Sciences / 1287 Vol. 86, No. 11, November 1997
AND
M. GOÄ RECKI‡
Received February 19, 1997, from the †Department of Thermodynamics, Faculty of Pharmacy and Physics, University of Valencia, 46100 Burjassot, Valencia, Spain, and ‡Department of Inorganic and Analytical Chemistry, Faculty of Pharmacy, K. Marcinkowski University of Medical Sciences, Grunwaldzka str., 6. 60−780 Poznan, PolandX. Accepted for publication September 5, 1997 Abstract 0 A general method to quantify the thixotropic behavior of systems with very low thixotropy is proposed. The areas enclosed by the rheograms τ ) f(γ˙ ) must be fitted to functions with well-determined boundary conditions. From these equations the corresponding thixotropic areas are obtained, together with the theoretical area enclosed by the rheogram corresponding to the maximum rheodestruction. The proposed method is applied to high viscosity sodium (carboxymethyl)cellulose gels.
shear rates (γ˘ min and γ˘ max, respectively) applied to the system and corresponding to agitation time, t. The mathematical functions, S ) S(t), cannot be arbitrary, for they must satisfy a series of clearly defined limiting conditions. In effect, examination of the experimental rheograms of a thixotropic system shows that, for t ) 0, the area S attains its maximum value, S ) Smax. In contrast, for very long agitation times, S tends toward a minimum limiting value, S ) Smin. Therefore, a simple expression for the function S(t) is
S(t) ) A+ Be-f(t)
(1)
Introduction The thixotropic behavior of certain systems, fluids, or semisolids that contain macromolecules such as stabilizers or thickeners is of particular interest in the technological processes of formulation and packaging in the pharmaceutical and food industries.1-3 Thixotropy involves a progressive decrease in viscosity, and thus of shear stress, due to the agitation produced on subjecting the system to continuous shear deformation, followed by the recovery of the rheological properties after a more-or-less prolonged rest period. The study of thixotropic systems is typically complicated, both when made on an empirical basis and in the context of theoretical models.4-7 Hysteresis experiments may also be used to investigate the rheological properties of thixotropic systems.8-12 In general, the approach involves the determination of the areas enclosed between one or more up-curve rheograms, corresponding to increasing shear rates, and different down-curve rheograms obtained with decreasing shear rates, following different agitation times. These areas are in turn referred to as thixotropic areas. It is normally accepted that greater thixotropic areas imply increased thixotropy, though this affirmation may be placed in doubt in certain cases. In fact, with the aim of comparatively studying the thixotropy of different gels, the use of a parameter known as the relative thixotropic area13 has been proposed that relates thixotropic areas to the total areas enclosed by the up curves corresponding to different gels. An additional difficulty in the experimental investigation of such systems arises when their thixotropic properties are limited, i.e., when the areas enclosed by the hysteresis cycles (ST) are very small compared to the area enclosed by the up curve. This problem becomes manifest when attempts are made to correlate the thixotropic areas obtained with the agitation times to which the system has been subjected. In fact, ST may vary on a random basis, following no logical law capable of accounting for its gradual increase over time. This randomness may be due to different causes, but particularly to the sensitivity of the viscometer employed. The present study proposes a method for solving the above problem on the basis of the prior determination, by means of a regression curve, of the form of the function S ) S(t), where S is the area determined for each of the curves obtained in a flow diagram, τ-γ˘ , between the minimum and maximum X
Abstract published in Advance ACS Abstracts, October 15, 1997.
© 1997, American Chemical Society and American Pharmaceutical Association
In the above equation, A and B are constants, and the concrete form of f(t) is dependent upon the rheological behavior of the system involved. However, the following two limiting conditions always apply
lim f(t) ) 0 tf0
(2)
lim f(t) ) ∞ tf∞
(3)
It is easy to verify that
A ) Smin and B ) Smax - Smin
(4)
Thus, equation [1] may also be expressed as follows
S(t) ) Smin + (Smax - Smin)e-f(t)
(5)
Once the function S(t) has been determined, it becomes evident that the thixotropic area (ST) is expressed by
ST ) Smax - S(t)
(6)
or, on considering eq 5,
ST ) (Smax - Smin)[1 - e-f(t)]
(7)
This expression provides the thixotropic areas as a function of the corresponding agitation times. However, knowledge of the absolute thixotropic areas does not suffice to comparatively investigate systems with highly different viscosities. In fact, thixotropic systems with very high viscosities may exhibit down curves that differ relatively little from the corresponding up curves, yet the enclosed areas may be large due to the high shear stresses reached in such systems. In the same way, in less viscous systems, the enclosed areas can be very small, though the rheogram up and down curves may differ considerably. The structural breakdown of more viscous systems may be relatively minor compared to less viscous systems, despite the fact that the absolute thixotropic area is much greater in the former.
S0022-3549(97)00075-0 CCC: $14.00
Journal of Pharmaceutical Sciences / 1283 Vol. 86, No. 11, November 1997
The relativizing of thixotropic areas has been carried out13 by means of the concept of relative thixotropic area:
ST SR ) 100 Smax
(8)
This equation allows us to calculate the percentage of rheodestroyed area caused by agitation, with respect to the total area enclosed by the rheogram up curve. Therefore, a system becomes more thixotropic as the value SR increases. By replacing (7) in (8), we see that
SR ) [100 - (SR)min][1 - e-f(t)]
(9)
Smin (SR)min ) 100 Smax
(10)
where
Equations 7 and 9 are formally identical and, because of that, the first one is always used. Obviously, to obtain the relative thixotropic area, Smax must be replaced by 100 and Smin by (SR)min, as defined in eq 10. The above concepts have been applied in the present study to the investigation of the thixotropic behavior of hydrogels containing different concentrations of high-viscosity sodium (carboxymethyl)cellulose (NaCMC); the small thixotropy of this substance requires the application of a mathematical treatment such as that described in the present paper.
Material and Methods The hydrogels of NaCMC (Aldrich Chemical Co., Milwaukee, WI) were studied at concentrations of 1.5, 1.6, 1.7, 1.8, 1.9, 2.0, 2.1, 2.2, and 2.4% (w/w). The preparation of hydrogels was carried out by dispersing the amounts of NaCMC required for a total of 250 g at the desired concentration in isopropyl alcohol, which was used as dispersing agent. The dispersions were placed in a water bath at 85 °C and water at this temperature was added to each dispersion in a proportion of 2/3 of the whole amount required for every hydrogel. The gels were stirred at this temperature for 10 min in order to homogenize them, facilitate gel formation, and evaporate the alcohol. Care was taken to avoid the introduction of bubbles. Water was added after cooling the gel to room temperature (25 °C) to make up lost weight. The hydrogels were maintained at room temperature for 24 h, after which the experimental measurements were made. A Brookfield Digital DVII+ rotary viscometer was used to carry out the experimental measurement of apparent viscosity, η. The viscosity was determined for the eight rotational speeds provided by the viscometer. Taking into account the experimental characteristics involved, these correspond to shear rates,14 γ˘ , of 0.063, 0.126, 0.314, 0.628, 1.26, 2.51, 6.28, and 12.57 s-1. The measurements of apparent viscosity were initially made with the unagitated gels in increasing order of shear rate and in time intervals of 10 s, which enabled the corresponding up-curve rheograms to be obtained. Then, after agitation times of 1, 2, 3, 4, 5, 10, 15, and 20 min at the maximum rotational speed (60 rpm), the apparent viscosity values corresponding to the shear rates in decreasing order were measured in order to obtain the down-curve rheograms.
Results and Discussion On the basis of the experimental measurement of the viscosity of NaCMC dispersions as a function of shear rate (γ˘ ) and of agitation time (t), the corresponding shear stresses (τ) generated have been obtained for all the concentrations evaluated. As an example, Figure 1 shows the rheograms τ ) f(γ˘ ,t) corresponding to the hydrogel formulated with a 2% w/w concentration of NaCMC. These rheograms are qualitatively similar to those obtained for the rest of the concentra1284 / Journal of Pharmaceutical Sciences Vol. 86, No. 11, November 1997
Figure 1sShear stress as a function of shear rate for the 2% NaCMC hydrogels at different agitation times: filled circle, up curve (0 min); open box, 1 min; filled triangles, 2 min; open circles, 3 min; filled diamond, 4 min; plus sign, 5 min; plus in a box, 10 min; asterisk, 15 min; filled box, 20 min.
tions studied, and their convex shape confirms the pseudoplastic behavior of such systems. Attention should also be drawn to the progressive decrease in shear stress as agitation time is prolonged, this being typical of thixotropic systems. The decrease is most apparent during the first 2-3 min and becomes much less noticeable as agitation time is extended. This fact, and the limited sensitivity of the measuring viscometer, could account for the irregular distribution that the thixotropic areas may present when correlated to agitation time. In such cases thixotropy must be studied by the method described in the Introduction, i.e., on the basis of prior fits of the total areas, S, which determine both the up-curve rheogram and each of the down-curve rheograms obtained for different agitation times. The calculation of the areas has been made by numerical integration of the function, τ ) f(γ˘ ,t), between the values γ˘ o ) 0.063 and γ˘ 1 ) 12.56 s-1, these being the extreme shear rates of the viscometer employed. The values of S calculated for each NaCMC concentration were fitted by the least-squares method using equations of type (1) taking for f(t) the following function:
f(t) ) Kt1/2
(11)
The above function satisfies the limiting conditions required by eqs 2 and 3, and has been chosen in view of its simplicity and because it provides sufficiently acceptable correlation coefficients (r > 0.995) in all cases. The substitution of (11) in (5) in turn provides the following empirical model 1/2
S(t) ) Smin + (Smax - Smin)e-Kt
(12)
The parameter K characterizes the time variation of the area S. If the system in question is thixotropic, then K > 0, since S must decrease on extending agitation time. Moreover, K must be finite, for only in this case does the area depend upon agitation time. Finally, the influence of K upon the values of S(t) is clearly conditioned by the value of (Smax Smin), which in turn depends on the polymer concentration of the gel, as is demonstrated below. The graphic representations of the functions S(t) obtained for the different NaCMC concentrations employed are shown
Figure 2sS(t) values versus agitation times for the NaCMC hydrogels at various concentrations: filled circle, 2.4%; open box, 2.2%; filled triangle, 2.1%; o, 2.0%; filled diamond, 1.9%; plus sign, 1.8%; plus in a box, 1.7%; asterisk, 1.6%; filled box, 1.5%.
in Figure 2. It should also be pointed out that systems containing NaCMC concentrations of under 1.5% and over 2.4% have been evaluated, though in all cases the area obtained remained practically constant after the first minute of agitation. In view of the information provided by Figure 2, two fundamental aspects should be pointed out. First, the area determined by the different rheograms increases with the polymer concentration of the gels, this being a logical consequence of their rising viscosity. Second, for a given concentration, the area decreases as the agitation time is prolonged, this in turn being a result of the thixotropy of the system. The decrease becomes more accentuated as the concentration of NaCMC increases. According to our empirical model (12), the intersections of the different curves with the ordinate axis determine the maximum areas, Smax, enclosed by the up-curve rheograms corresponding to each concentration. These areas are in turn calculated experimentally by numerical integration of the different up curves, τup ) f(γ˘ ). On the other hand, and based on the empirical model (12), calculations were made of the minimum areas, Smin, determined by infinite agitation times, for each polymer concentration studied. Both Smax and Smin were fitted by the leastsquares method as a function of polymer concentration c, providing the following equations:
Smax ) (3.13c - 4.07) × 103 Pa‚s-1
(13)
Smin ) (2.01c - 2.39) × 103 Pa‚s-1
(14)
The correlation coefficients are greater than 0.994 in both cases. Clearly, these areas must be positive, which limits the application of eqs 13 and 14 to concentration values of 1.5% w/w, or higher. This may be attributable to the fact that the tridimensional networks of the NaCMC dispersions of lower concentrations are not totally structured, and hence their rheological behavior is more similar to that of a macromolecular suspension than of an actual gel. The graphic representations of eqs 13 and 14 are shown in Figure 3. In
Figure 3sSmax, Smin, and (ST)max as a function of the concentration of NaCMC hydrogels (filled circles correspond to experimental Smax values; Smin and (ST)max are theoretical curves).
Figure 4sK values of eq 12, versus NaCMC hydrogels concentration.
this figure we have only included the fitted values of Smax (experimental values), but not of Smin (not experimental values), for clarity. Finally, the fits performed also provide the values of the parameter K corresponding to the different concentrations. Figure 4 shows the graphic representation of K ) f(c), where K is seen to increase progressively beyond c ) 1.5% to a maximum value of 1.2 min-1/2 for a 2.0% w/w concentration of NaCMC. In principle, this indicates that a gel prepared with that concentration exhibits the greatest dependency of S upon agitation time. This gel may thus be considered to possess the greatest thixotropy of all the preparations studied. In the case of lower concentration gels, this dependency is less, a fact that, as before, may be attributed to their behavior as macromolecular suspensions. However, for concentrations of over 2%, K is found to decrease rapidly with rising concentration, thus reflecting less dependency of S on agitation time in the case of the lower polymer concentrations. This fact, in contrast to the previous case, may be explained by considering that the tridimensional
Journal of Pharmaceutical Sciences / 1285 Vol. 86, No. 11, November 1997
Figure 5sThixotropic areas as a function of agitation time for the NaCMC hydrogels at various concentrations: filled circle, 2.4%; open box, 2.2%; filled triangle, 2.1%; open circle, 2.0%; filled diamond, 1.9%; plus sign, 1.8%; plus in a box, 1.7%; asterisk, 1.6%; filled box, 1.5%.
networks of these more concentrated gels are highly structured and thus more difficult to deform by agitation. Clearly, the study of the thixotropic behavior of the gels analyzed is still incomplete, for the combined influence of all the parameters, i.e., K, Smax, and Smin, should also be analyzed. This influence has been studied on the basis of the thixotropic areas, ST, obtained as a function of agitation time, for each concentration. These areas have been calculated by means of eqs 7 and 11. The graphic representations of the functions ST ) f(t,c) are in turn reflected in Figure 5, where the thixotropic area is seen to increase on prolonging the agitation time, regardless of polymer concentration. Likewise, it should be pointed out that ST also increases with NaCMC concentration, when comparing the values corresponding to the same agitation times. All the functions represented in Figure 5 exhibit a tendency toward different maximum values with each concentration, when t tends toward infinity. These maximum values are clearly reflected by (ST)max ) (Smax Smin). Thus, from eqs 13 and 14, the maximum thixotropic area as a function of increasing polymer concentration is provided by the equation
(ST)max ) (1.12c - 1.68) × 103 Pa‚s-1
(15)
This indicates the linear increase of the maximum thixotropic areas, (ST)max, with NaCMC concentration, in the interval c ) [1.5, 2.4]% (see Figure 3). A similar result is obtained from the mean thixotropic areas, S h T, in the entire agitation time interval [t1, t2] studied, in this case [0, 20] min, calculated by the following expression13
S hT )
∫
1 (t2 - t1)
t2
t1
ST(t) dt
(16)
In fact, on representing the S h T values obtained for all the gels as a function of concentration (Figure 6), and fitting the points by the least-squares method using a linear function, we obtain the equation
S h T ) (0.762c - 1.095) × 103 Pa‚s-1 (r ) 0.990) (17) 1286 / Journal of Pharmaceutical Sciences Vol. 86, No. 11, November 1997
Figure 6sMean thixotropic areas versus NaCMC hydrogels concentration.
This indicates that an increase in polymer concentration produces a linear increment in the mean thixotropic area of the NaCMC hydrogels, within the concentration interval and agitation times considered in the present study. The fact that the gel with the greatest thixotropic area is precisely the preparation possessing the highest polymer concentrationsas deduced from eq 17sapparently contradicts the results regarding thixotropic behavior obtained from the study of the values of the parameters in eq 12 and from its justification based upon the tridimensional structure of the NaCMC gels. In this sense it was pointed out that the gel containing 2% w/w of polymer exhibited the greatest dependency of S on agitation time, and therefore should be the system with the greatest thixotropy. This problems is solved by considering that a greater thixotropic area does not necessarily imply greater thixotropy, since comparisons of the thixotropic behavior of different systems should be made on the basis of studies of the relative thixotropic areas, SR, defined by eq 9. In effect, on substituting the values of K, Smax, and Smin in eq 9, and taking as f(t) the function defined by eq 11, we calculated the relative thixotropic areas for all the concentrations and agitation times studied. The graphic representations of SR ) f(c,t) are shown in Figure 7, where it is seen that the lowest relative thixotropic area (lowest thixotropy) corresponds to gels with concentrations of 1.5, 1.6, and 1.7% w/w NaCMC, regardless of agitation time. From a 1.8% w/w NaCMC concentration onward, the differences between the relative thixotropic areas were so small (similar thixotropy) that they could be considered as due to experimental errors. However, there is a differential fact between high concentration gels (2.1, 2.2, and 2.4% w/w NaCMC) and those of intermediate concentrations (1.8, 1.9, and 2% w/w NaCMC). The graphics corresponding to the former ones present a different curvature, which implies a lower relative thixotropic area in the early minutes of agitation. On the other hand, when the agitation time grows, the relative thixotropic areas become higher than the areas obtained for the rest of the concentrations. These facts seem to indicate that the total breakdown of the high concentration gels structure also requires longer agitation times, although in low concentration gels the total breakdown attained is smaller. The quantitative analysis of the differences between the relative thixotropic areas may be made on the basis of the relative values of the maximum, (SR)max, and mean thixotropic areas, S h T, calculated from eqs 13, 15, and 17. The resulting expressions obtained (as percentages) are
Figure 7sRelative thixotropic areas as a function of agitation time for all NaCMC hydrogels concentration: filled circle, 2.4%; open box, 2.2%; filled triangle, 2.1%; open circle, 2.0%; filled diamond, 1.9%; plus sign, 1.8%; plus in a box, 1.7%; asterisk, 1.6%; filled box,1.5%.
(SR)max ) 35.8 S h R ) 24.4 -
22.4 3.13c - 4.07
10.4 3.13c - 4.07
(18)
Figure 8sRelative maximum and mean thixotropic areas as a function of the concentration of NaCMC hydrogels.
Acknowledgments J.B. thanks the Spanish Ministry of Foreign Affairs for their support in the form of a grant which made 3 months research possible in the Department of Thermodynamics, University of Valencia.
(19)
These allow us to calculate both relative thixotropic areas, and each is seen to tend toward a different limiting value. From eqs 18 and 19, it may be shown that the increase in the maximum and mean relative thixotropic areas is very slow, i.e., on the order of 3.7% and 1.7% respectively, upon increasing polymer concentration from 2.0 to 2.4% w/w NaCMC (Figure 8). The variations in thixotropic areas on this order of magnitude are those that may become masked in most cases as a result of experimental errors due to limitations in viscometer sensitivity, the lack of sample homogeneity, very small temperature variations in the course of the measurement process, and so on. However, gels that apparently lack thixotropy, such as those prepared with highly viscous NaCMC or many other systems (including hydrogels formulated with certain acrylic acid polymers that we are currently investigating), in reality exhibit weak thixotropic behavior that may easily be demonstrated and quantified by the procedure described in the present study.
References and Notes 1. Ford, E. W.; Steffe, J. F. J. Texture Stud. 1986, 17, 71-85. 2. Garza, S.; Faus, S.; Giner, J.; Martin, O.; Ibarz, A. Progress and Trends in Rheology IV: Proc. Fourth European Rheology Conference. 1994, Sevilla, Spain. 3. Dolz, M.; Gonza´lez, F.; Herra´ez, M.; Dı´ez-Sales, O. J. Disper. Sci Technol. 1994, 15 (2), 189-205. 4. Tiu, C.; Boger, D. V. J. Texture Stud. 1974, 5, 329-338. 5. Martı´nez-Padilla, L. P.; Hardy, J. J. Texture Stud. 1989, 20, 7185. 6. Amemiya, J. I.; Shoemaker, C. F. J. Food Eng. 1992, 16, 1724. 7. Go´recki M.; Bugaj J. Acta Pol. Pharm. 1994, 51-2, 191-194. 8. Harris, J. Nature 1967, 214, 797-797. 9. Cheng, D. C.-H. Nature 1967, 216, 1099-1100. 10. Mewis, J. J. Non-Newton Fluid Mech. 1979, 6, 1-20. 11. Dolz-Planas, M.; Roldan-Garcı´a, C.; Herra´ez-Domı´nguez, J.; Belda-Maximino, R. J. Pharm. Sci. 1991, 80 (1), 75-79. 12. Fresno, M. J.; Jime´nez, M. M.; Selle´s, E. Cienc. Pharm. 1993, 3 (2), 81-87. 13. Dolz, M.; Herna´ndez, M. J.; Pellicer, J.; Delegido, J. J. Pharm. Sci. 1995, 84 (6), 728-732. 14. More solutions to sticky problems, Brookfield Engineering Laboratories, Inc.: Stougton, MA, 1983.
JS970075+
Journal of Pharmaceutical Sciences / 1287 Vol. 86, No. 11, November 1997