J. Phys. Chem. 1996, 100, 5433-5439
5433
Dynamic Viscoelasticity of an Aqueous System of a Poly(ethylene oxide)-Poly(propylene oxide)-Poly(ethylene oxide) Triblock Copolymer during Gelation Bo Nystro1 m* and Harald Walderhaug Department of Chemistry, UniVersity of Oslo, P.O. Box 1033, Blindern, N-0315 Oslo, Norway ReceiVed: August 25, 1995; In Final Form: NoVember 13, 1995X
Viscoelastic properties during thermo-induced gelation of a binary poloxamer (Pluronic F68)/water system have been determined by oscillatory shear and shear stress relaxation measurements. The polymer concentration has been kept constant at 35 wt %. A sol-gel transition is observed in the range 34-37 °C. The oscillatory shear data at temperatures around the gel temperature reveal a complex pattern of behavior. The results cannot be described by a single Maxwell element behavior as demonstrated by converting the data to ColeCole plots. Shear stress relaxation experiments show that the stress relaxation at the lowest temperatures of measurement (34 and 35 °C) can be described initially by a single exponential followed at longer times by a stretched exponential profile. At 36 °C and higher temperatures, a new relaxation mode in the form of a power law enters at intermediate times, between the exponential and stretched exponential domains. The power law part of the relaxation function has its maximum time window at 37 °C (incipient gel) where it covers a time region of more than 3 orders of magnitude. The power law exponent is generally close to 0.5, except at the lowest temperature (36 °C) of power law behavior, where it is close to 0.6. A relaxation exponent of 0.5 can be rationalized within a framework of the fractal model for polymer networks. When the oscillatory shear data are transformed into equivalent shear stress relaxation data, they are shown to be compatible with the experimental shear stress relaxation data at all temperatures.
Introduction Polymeric surfactants of the type poly(ethylene oxide)-poly(propylene oxide)-poly(ethylene oxide) (PEO-PPO-PEO) triblock copolymers, generally called poloxamers (commercially available from BASF Corp., under the name Pluronics), have recently attracted great interest, both as a result of their commercial utility in cosmetic and pharmaceutical industries1,2 and because of their novel physical behavior.3-5 It is wellknown that aqueous solutions of certain Pluronic polymers exhibit interesting temperature-induced micellization6-14 and gelation8,12-15 phenomena. These systems usually show inverse temperature behavior; that is, with increasing temperature, the concentrated (typically > 20 wt %) aqueous solution is transformed into a rigid transparent gel with a high yield stress value.7 Recent small-angle neutron-scattering results by Wanka et al.7 and by Mortensen et al.16 on aqueous Pluronic systems suggest that the gels consist of close-packed arrays of micelles of roughly the same size as those in the micellar solution phase. A further increase in temperature leads to a breakdown of the gel structure, again resulting in a solution of low viscosity.17 The temperature range for the gel zone depends on the concentration and on the composition of the Pluronic copolymer.15 Although there are a number of rheological studies7,12,13,17,18 of Pluronic polymers, a detailed picture of the viscoelastic response during the temperature-induced gelation process is still lacking. The principal aim of this paper is to gain further insight into the complex mechanisms that govern the rheological properties of these systems. Here we report results from oscillatory shear and stress relaxation measurements at various temperatures on an aqueous gelling solution (35 wt %) of the commercial sample of Pluronic F68 (EO78-PO30-EO78). This work will reveal an intricate X
Abstract published in AdVance ACS Abstracts, March 1, 1996.
0022-3654/96/20100-5433$12.00/0
rheological behavior of this system as the gel evolves. Although the rheological features of this system are quite intriguing, the phase diagram of the Pluronic F68 systems is rather simple,19 with a single isotropic phase at concentrations below 50 wt % and at temperatures over an extended range. Theoretical Background For systems exhibiting linear viscoelasticity, it has been found20-22 that the relaxation modulus can be described by a simple power law at the gel point
G(t) ) St-n λ0 < t < ∞
(1)
where the gel strength parameter S (dimension Pa sn) and relaxation exponent n are the only material parameters. The parameter n may theoretically have values in the range 0 < n < 1. In the Rouse model, n ) 1/2. However, there are also other situations23 that may lead to the power law G(t) ∼ t-1/2. The shortest time of power law behavior λ0 marks the lower cut-off due to glass transition or entanglement effects for very long polymer chains. The linear viscoelastic behavior at the gel point can be expressed with the aid of the gel equation,20,21 which is a linear viscoelastic constitutive equation for the stress
σ(t) ) S∫-∞(t - t′)-nγ˘ (t′) dt′ t
(2)
This expression relates the instantaneous stress tensor σ(t) to the strain history given by the rate of deformation γ˘ (t). In spite of the great capability of stress relaxation measurements to extract important information about viscoelastic polymeric systems, it is a rather neglected technique in the studies of rheological properties. Oscillatory shear experiments are more frequently employed to explore the viscoelastic characteristics of gelling polymer systems in terms of the storage modulus G′(ω) (ω is the © 1996 American Chemical Society
5434 J. Phys. Chem., Vol. 100, No. 13, 1996
Nystro¨m and Walderhaug
frequency), the loss modulus G′′(ω), and the complex shear modulus G* ) G′ + iG′′. In this type of experiment, a strain γ ) γ0 sin ωt is applied. Then for a linear response, the resulting stress will be given by
σ ) γ0(G′(ω) sin ωt + G′′(ω) cos ωt)
(3)
where G′(ω) represents the stress contribution which is in phase with the strain and G′′(ω) gives the contribution from the stress which is π/2 out of phase with the strain. Because the complex shear modulus G*(ω) is a transform of G(t), the same scaling behavior also governs dynamic mechanical experiments where G′ and G′′ for the incipient gel are related as
G′ ) G′′/tan δ ) SωnΓ(1 - n) cos δ 0 < ω < 1/λ0
(4)
where Γ(1 - n) is the gamma function. The phase angle between stress and strain (δ) is independent of frequency but proportional to the relaxation exponent22
δ ) nπ/2 with tan δ ) G′′/G′
(5)
Early oscillatory shear measurements by Chambon and Winter20,21 on stoichiometrically balanced end-linking reactions on the silicone PPO-polyurethane systems yielded critical gels with relaxation exponents n ) 0.5. However, several studies24-29 on chemical and physical gelling systems have revealed that n assumes different values, depending on properties such as structure, cross-linking density, and polymer concentration. The storage and loss moduli for viscoelastic materials are obtainable from the relaxation modulus by the Fourier transforms
G′(ω) ) ω∫0 G(t) sin(ωt) dt ∞
G′′(ω) ) ω∫0 G(t) cos(ωt) dt ∞
(6)
The unique solution22 for eq 6 together with eq 5 (where the Kramers-Kronig30,31 relations have been used) yields the power law relationship of the stress relaxation modulus (see eq 1). The corresponding relations to calculate G(t) from G′(ω) and G′′(ω) are30
G(t) ) (2/π)∫0 (G′/ω) sin(ωt) dω ) ∞
(2/π)∫0 (G′′/ω) cos(ωt) dω (7) ∞
In this work, we will make a comparison of measured G(t) data with the stress relaxation data converted from the experimental storage and loss moduli data. We should note that in oscillatory shear experiments, there is a finite number of measuring points available at discrete times (frequencies) extending over a finite domain in the time axis (frequency axis). Due to this, we have to resort to approximation methods in the evaluation of G(t) from G′(ω) and G′′(ω). This problem has been addressed by Ferry and co-workers30 and by Schwarzl.32 The linear relaxation modulus can be derived from the dynamic moduli with an approximation formula. In this study, the numerical conversion formula, elaborated by Schwarzl32 for the calculation of G(t), has been adopted.
G(t) = G′(ω) - 0.496G′′(ω/2) + 0.065{G′(ω/2) G′(ω/4)} + 0.073{G′(ω) - G′(ω/2)} + 0.111{G′(2ω) G′(ω)} + 0.030{G′(16ω) - G′(8ω)} + 0.007{G′(64ω) G′(32ω)} + ... (8a)
where ω ) 1/t. The absolute error of this approximation is (0.014G′(ω). This formula will have limits for the relative error which tend to -∞ and +∞ for large values of tan δ, and therefore, it will fail in the high tan δ region. However, this domain is not encountered in the analysis of the present data. For small values of tan δ, the following simple formula will be sufficient:32
G(t) = G′(ω) - 0.566G′(ω/2) + 0.203G′′(ω)
(8b)
In the analysis of the viscoelastic response of ionic surfactant systems33-35 and so-called associative thickeners,36,37 the simplest model of a viscoelastic fluid, the Maxwell approach, consisting of an elastic component (spring) connected in series with a viscous component (dashpot), is frequently used
G′(ω) )
G∞ω2τ2 1+ω τ
2 2
G′′(ω) )
G∞ωτ 1 + ω2τ2
(9)
where G∞ is the plateau modulus and τ is the relaxation time. In order to illustrate the “goodness of fit” of this model, the dynamic moduli data are usually plotted in the form of a ColeCole plot,30 which, for a Maxwell fluid should be semicircular and be described by
G′′(ω) ) (G′(ω)G∞ - (G′)2(ω))m
(10)
where m ) 0.5. This model implies a pure exponential decay of the stress relaxation function (G(t) ∼ exp(-t/τ)). Experimental Section Materials. A commercial poloxamer 188 sample (Pluronic F68) obtained from Fluka was used in this study without further purification. According to the manufacturer, this polymer contains approximately 80% by weight oxyethylene units and the nominal molecular weight is ∼8350 g mol-1. Solutions of 35 wt % were prepared. The solutions were prepared by weighing the components. Great care was exercised in the preparation procedure to obtain homogeneous solutions. Oscillatory Shear and Stress Relaxation Experiments. Oscillatory shear and stress relaxation measurements were carried out in a Bohlin VOR rheometer system using a doublegap concentric cylinder at temperatures below 34 °C, an ordinary concentric cylinder geometry (C 25; inner radius 12.5 mm) at 34 and 35 °C, and a cone-and-plate geometry, with a cone angle of 5o and a diameter of 30 mm, at higher temperatures. The double-gap device is applicable for low viscous liquids. In this work, a maximum strain amplitude of 5% (strain less than 0.01) was employed. The values of the amplitude were checked in order to ensure that all measurements were conducted within the linear viscoelastic region, where the dynamic storage modulus (G′) and loss modulus (G′′) are independent of the strain amplitude. We observed that at strain amplitudes of 10% and larger, the nonlinear viscoelastic regime was entered. In a stress relaxation experiment, the material is subjected to a rapidly applied small strain which is held constant for the remainder of the experiment, and the decay of stress in the viscoelastic material is monitored as a function of time. From this type of measurement, the stress relaxation modulus G(t) ) σ(t)/γ0, the ratio of stress to the constant strain, is determined. A layer of silicone oil was added onto the top of the sample in the cell to avoid evaporation of solvent. The rheometer is equipped with a temperature control unit that was calibrated to give a temperature in the sample chamber within 0.1 °C of the set value at all temperatures considered. At each temperature, the sample was allowed to equilibrate for some time before the
Dynamic Viscoelasticity of Aqueous PEO-PPO-PEO
J. Phys. Chem., Vol. 100, No. 13, 1996 5435
Figure 2. Viscoelastic loss tangent as a function of temperature for the aqueous Pluronic system (35 wt %) at the frequencies indicated.
Figure 1. Temperature dependences of the storage modulus (G′) and the loss modulus (G′′) for the aqueous Pluronic system (35 wt %) at the frequencies indicated.
measurements were commenced. The reproducibility of an experimental run with a new sample solution was usually better than (5%. The same reproducibility was observed when the sample was allowed to stand in the cell at a given temperature (temperatures both in the pregel and the postgel domains were checked) for a day before measurements were repeated. Results and Discussion Oscillatory Shear Experiments. Measurements of G′ and G′′ at constant frequencies (0.1, 0.5, and 1.0 Hz) as a function of temperature are shown in Figure 1. It is generally observed that at low temperatures, G′′ > G′ (viscoelastic liquid) and at high temperatures, G′ is somewhat higher than G′′. It seems that the crossover G′ ) G′′ is located at about the same temperature (34 °C) for the different frequencies displayed. This crossover has been used38 as a hallmark of the gel point. However, recent studies28 have shown that this definition of the gel point is rather arbitrary and cannot be used for an exact determination of the gel point. In this context, it is interesting to note that with the test tube “tilting”39 method, where the gelation temperature was determined by tilting the test tube containing the solution, a higher gel point temperature of about 37 °C was found. In this approach, the temperature at which the solution no longer flows is taken as the temperature of gelation. This method served to define the sol-gel transition temperature to about (1 °C. When checked, the stiff gel was found to be immobile in the inverted tube over long times. The reason for this discrepancy in the determination of the incipient gel phase, as well as the other anomalous effects discussed below, is not clear at the present stage. Figure 2 shows the temperature dependence of the loss angle at four different frequencies for the Pluronic F68 system (35 wt %). The gel point of a thermoreversible gelling system may be located22 by observation of a frequency-independent value of tan δ when plotted vs temperature. It has been argued28,40 that the gel point is more accurately determined from this type of plot than by the plot procedure depicted in Figure 1. However, it is evident from Figure 2 that a rather complex picture appears. The temperature dependence of the loss angle in Figure 2 suggests that viscoelastic properties of the system through the gelation process can for each frequency be described by different stages along the temperature coordinate. An initial
Figure 3. Plot of the complex modulus (G*) and the dynamic viscosity (η′), at a frequency of 0.1 Hz, vs temperature for the aqueous Pluronic system (35 wt %).
strong drop of tan δ occurs between 33 and 34 °C, which probably can be associated with the rapid growth of clusters, making G′ measurable. At higher temperatures, the growing population of clusters results in a rise in tan δ, and this growth phase continues until the tan δ values reach maxima at a temperature near 37 °C. At this temperature, a gel network, consisting of interconnected micelles,41 has probably been formed. At still higher temperatures, a slight decrease of tan δ is observed. Usually24,28,29 a frequency-independent value of tan δ, which is considered to be the signature of the gel point, is observed for the incipient gel. However, for the present system, this type of behavior is not detected in the graph of Figure 2. The anomalous behavior observed in Figure 2 suggests that the gelation process of the present Pluronic system is intricate and exhibits unconventional features. A similar scenario as that displayed in Figure 2 has previously been reported42 from oscillatory shear measurements on a gelling system of tetraethoxysilane. However, for this system, the intersection point of gelation could easily be identified. The viscoelastic properties of the Pluronic system are further illustrated in Figure 3, where the temperature dependences of the complex modulus G* and the dynamic viscosity η′ ) G′′/ω are depicted at a constant frequency (0.1 Hz). We note that both G* and η′ show a significant rise at temperatures in the range 35-39 °C. This trend indicates that the viscoelastic response of the system increases, which is a typical phenomenon for a gelling polymer. Above 39 °C, a weak decrease is observed. This may suggest an incipient breaking down of the network formed by bridging between the micelles. In Figure 4, the dynamic storage and loss moduli are plotted against frequency for the Pluronic system at various temperatures. The curves have been purposely shifted vertically by a
5436 J. Phys. Chem., Vol. 100, No. 13, 1996
Figure 4. Frequency dependences of the storage modulus G′ (open symbols and stars) and the loss modulus G′′ (solid symbols and crosses) at different stages of the gel-forming process for the aqueous Pluronic system (35 wt %). The curves have been shifted vertically by a factor B of the value listed in the insert.
factor B (see insert in Figure 4) for easier comparison. At 33 °C, before the gel zone, the system tends to behave classically as a Newtonian liquid (G′ ∼ ω2 and G′′ ∼ ω1). The experimental scatter of the G′ data precludes concluding that power law behavior was followed, but the storage modulus data are consistent with the power law G′ ∼ ω2. The rather large scatter in the experimental points of G′ at 33 °C is probably due to the weak elastic response at this temperature (low viscous solution). As the gelation process proceeds, a more complex picture emerges. At 34 °C, G′ becomes much higher than G′′ at higher frequencies (solidlike state). At higher temperatures, the general trend is that at low frequencies, we observe a viscous behavior with G′′ > G′, while at higher frequencies, depending on the temperature, G′ increases to cross G′′, and above this frequency, G′ exceeds G′′, which suggests that the elastic response dominates. These results indicate that the Pluronic F68 (35 wt %) system becomes more viscoelastic at higher frequencies, which is typical for networks containing entangled or interconnected chains. Another notable feature in Figure 4 is that the frequency of intersection ω* (G′ ) G′′) or the equivalent quantity, i.e., the time of intersection τ* (τ* ∼ 1/ω*), exhibits a complex temperature dependence. The value of τ*, which is related to a characteristic stress relaxation time of the system, is expected to be shifted toward longer times as the network evolves and strengthened at higher temperatures. However, such a progressive trend is not observed. The reason for this irregularity can probably be traced to changes in the frequency dependences of the dynamic moduli with increasing temperature. An analogous effect is also observed in the stress relaxation modulus (cf. the discussion below concerning the stress relaxation function). As will be discussed below, the decay profile of the stress relaxation function changes considerably with increasing temperature. This type of effects may give rise to anomalous crossover effects of the type observed in Figure 4. For an incipient gel, theory predicts (see the theoretical section above) that the storage and loss moduli are parallel and scale with frequency as G′ ∼ G′′ ∼ ωn over an extended frequency domain. This type of behavior is not observed at any temperature (oscillatory shear measurements have been carried out at more temperatures close to the gel zone than those presented in Figure 4) for the present system and therefore stresses the unusual properties of this gelling Pluronic system. It may be
Nystro¨m and Walderhaug
Figure 5. Cole-Cole plots of loss modulus vs storage modulus for the aqueous Pluronic system (35 wt %) at temperatures close to the gelation zone. The solid curves are calculated from eq 10 with m ) 0.47. The dotted curves illustrate pure Maxwellian response with m ) 0.50.
argued that the gel structure of an incipient thermoreversible gel is fragile and that this circumstance may prevent us from observing a power law behavior. However, we may note that power laws of G′ and G′′ in frequency have been reported29 for thermoreversible weak gels. Furthermore, it will be shown below that the relaxation moduli, calculated from the storage and loss moduli (see Figure 4), are compatible with the corresponding ones obtained experimentally from stress relaxation measurements. These results reveal power law domains at intermediate times. In Figure 5, the dynamic moduli, at different temperatures in the gel zone, are plotted in the form of a Cole-Cole plot, which, for a Maxwell fluid, should be semicircular and is described by eq 10. By allowing m to float, the best fit is obtained at all temperatures when m ) 0.47. However, a comparison between the experimental data and the theoretical model (eq 10) (solid curves) reveals, at all temperatures, significant deviations. The dotted curves depict a pure Maxwellian response (m ) 0.50). These findings suggest that the decay of the stress relaxation function is not a single exponential (cf. the discussion below). In Figure 6a, tan δ is found to decrease with increasing frequency, as is typical for a viscoelastic liquid before the gel point. According to the conventional criterion for an incipient gel, tan δ is expected to be independent of frequency. However, such a trend is not observed at any temperature for the present system. The variations of the dynamic viscosity η′ over an extended frequency range at different temperatures are given in Figure 6b. Again the results indicate that the polymer solution becomes more viscoelastic with increased frequency dependence and higher values of η′ at lower frequencies, typical for polymer network systems containing entangled or interconnected chains, as the temperature is increased. At low temperatures (23 and 33 °C), η′ is virtually independent of frequency, and the Pluronic system displays rheological characteristics typical of unentangled polymer solutions. Figure 7 shows a double-logarithmic plot of the complex modulus G* vs ω (s-1) for the Pluronic system at different temperatures. At low temperatures (23 and 33 °C), in the liquidlike state, a power law G* ∼ ω1.0 is observed. At higher temperatures, a more solidlike behavior emerges and only a
Dynamic Viscoelasticity of Aqueous PEO-PPO-PEO
Figure 6. Frequency dependences of the viscoelastic loss tangent (a) and the dynamic viscosity (b) for the aqueous Pluronic system (35 wt %) at the temperatures indicated.
Figure 7. Frequency dependence of the complex modulus for the aqueous Pluronic system (35 wt %) at the temperatures indicated.
weak frequency dependence of G* can be registered. In recent oscillatory shear measurements12 on the Pluronic P-94 sample (∼25 wt %), a stiff gel was observed in the approximate range 30-45 °C. After the “melting” of this gel at higher temperatures, it was succeeded by formation of a weak gel at temperatures above 58 °C. This latter state was referred to as the “soft” gel, since its elastic modulus is more than 2 orders of magnitude smaller than that of the “hard” gel at lower temperature. The strong viscoelastic response observed for the present Pluronic sample suggests that the gel formed in this work can be characterized as a hard gel. Shear Stress Relaxation. In Figure 8a-d, the relative shear relaxation moduli G(t)/G(0), where G(0) is the initially measured relaxation modulus (plateau modulus), are depicted for the same system (Pluronic F68; 35 wt %) at different temperatures during the course of the gelation process. At the two lowest temperatures of measurement (34 and 35 °C), the decay of the stress
J. Phys. Chem., Vol. 100, No. 13, 1996 5437 relaxation function (see Figure 8a) can initially be described by a single exponential followed by a stretched exponential (the value of the stretched exponent β is 0.40) at longer times (the solid curve in Figure 8a represents the inset expression). It is interesting to note that the decay of the relative shear relaxation modulus, calculated from the data of G′ and G′′ at 34 °C (see Figure 4) with the aid of eq 8, exhibits a very similar profile (see the inset plot of Figure 8a) as that obtained from the shear stress relaxation experiments. At 36 °C (approaching the gel zone) and higher temperatures, a new relaxation mode in the form of a power law G(t)/G(0) ∼ t-n evolves at intermediate times (see Figure 8b-d). Figure 8d constitutes a summary of the decay profiles of the stress relaxation functions at various temperatures. The curves were shifted vertically by a factor B (see the inset) to avoid overlap. The stress relaxation results at higher temperatures indicate three relaxation processes: a shorttime initial decay followed by a power law at intermediate times and then a long-time tail. The general picture that emerges is that the decay at short times has an exponential profile (see the solid curves in Figure 8a and b) and the mode at long times can be described by a stretched exponential (see the inset plot of Figure 8c). At 37 °C (cf. Figure 8b), the incipient gel zone, the width of the power law domain is at a maximum. We may note that the calculated relaxation function (see the inset plot of Figure 8b) from G′ and G′′ data at 37 °C by means of eq 8 and G(t)/G(0) determined from experimental data exhibit the same pattern of behavior over a wide time range. The shape of this profile is quite different from that observed at 34 °C (cf. Figure 8a). A close inspection of the stress relaxation data reveals that the width of the power law domain increases in the pregel region, with a maximum (the data span about 3.5 decades of time) (see Figure 8b) around 37 °C (incipient gel), and then gradually contracts in the postgel regime (see Figure 9). We may note that a similar phenomenon has been inferred43 from dynamic rheological data, converted into data in the time domain, during the gelation of a radiation cross-linked polyethylene. In this case, the relaxation modulus exhibited a power law region, which gradually extended over a wider time window when approaching the gel point, and then decayed slowly beyond this point. The three relaxation modes observed from the stress relaxation results may be rationalized in the following way. As the temperature-induced gel evolves, the conjecture is that a temporal network of interconnected micelles41 is formed. The initial relaxation process at short times is probably governed by some kind of rearrangement of network topology.30 The second stage of the relaxation process at intermediate times, where the power law behavior prevails, is probably associated with disengagement of junctions or bridges which hold the network together. We expect that the number and strength of these junctions are related to the extension of the power law regime. The present results suggest that the strength of the network is highest at 37 °C, with a power law region extending over a wide time window. At longer times, where the decay is described by a stretched exponential, a stage is approached where the chains are completely disentangled. The absence of the power law regime of the stress relaxation function at low temperatures (34 and 35 °C) probably indicates that a network formed by bridging between the micelles has not yet developed. In this context, it should be noted that three relaxation modes of the same type as those detected here have also been observed44 in the time correlation functions obtained from dynamic light scattering measurements on the same system and at the same polymer concentration. However, the time scales
5438 J. Phys. Chem., Vol. 100, No. 13, 1996
Nystro¨m and Walderhaug
Figure 8. Plot of the relative shear relaxation modulus as a function of time for the aqueous Pluronic system (35 wt %) at the following temperatures (°C): (a) 34 (the solid curve is fitted with the aid of the inset equation; the inset plot has been constructed from dynamic moduli data (34 °C) which have been converted into stress relaxation data by means of eq 8); (b) 37 (incipient gel) (the initial dashed curve illustrates single-exponential behavior of the stress relaxation function; the inset plot depicts the stress relaxation function calculated from dynamic moduli data at 37 °C with the aid of eq 8); (c) 40 (the solid curve illustrates single-exponential behavior of the stress relaxation function; the inset plot demonstrates the stretched exponential character of the relative shear relaxation modulus at long time). (d) An illustration of the profile of the decay of the relative shear relaxation modulus during the gelation process. The curves have been shifted vertically by a factor B of the value listed in the insert.
Figure 9. Illustration of the width of the time window of power law behavior of the stress relaxation function for the aqueous Pluronic system (35 wt %) at various temperatures.
over which these relaxation modes operated were much shorter. The value of the power law exponent n is close to 0.5 at all temperatures, except at 36 °C (approaching the incipient gel), where the value is somewhat higher (see Figure 8d). It has been found45 for other gelling systems that a deficiency in the entanglement density may give rise to a slight augmentation in the value of the relaxation exponent. The reported value of 0.5 for the relaxation exponent can be rationalized in the framework of the fractal model for polymer networks. In terms of the fractal dimension df for a monodisperse system of polymers, a dynamic scaling analysis of flexible fractals in the Rouse limit46-48 yields a relaxation exponent of n ) df/(2 + df). Hess et al.23 have argued that flexible clusters may swell
from the percolation value (df ) 2.5) to df ) 2. In this case, n assumes a value of 0.5. We may also note49 that the Θ chain with df ) 2 gives n ) 0.50 and the good-solvent chain with df ) 1.67 yields n ) 0.45. This information may be relevant because light scattering measurements50 on dilute solutions of this Pluronic system have shown that the thermodynamic conditions become poorer at higher temperatures. In this context, we may also note that in a recent study,51 a mechanical ladder model was presented which mimics the rheological properties of systems of cross-linking polymers near the gel point. This model, which consists of an infinite number of springs and dashpots, yields a value of n ) 1/2 for ladderlike structures. Relaxation exponents of 0.5 have been observed from oscillatory shear measurements on critical gelation of biopolymer systems52-54 as well as on incipient gels from other types of systems.20,21,55,56 In Figure 10, the temperature dependence of G(0) is illustrated. The value of G(0) is virtually constant at low temperatures, but as the gel evolves, a strong rise of G(0) is the prominent feature. At the highest temperatures, the curve tends to level off. These findings indicate that the viscoelastic response increases strongly when the gel forms. Conclusions In this study, oscillatory shear and shear relaxation experiments have been carried out on a triblock copolymer (Pluronic F68) dissolved in water that undergoes a temperature-induced sol-gel transition. The measurements reveal unusual rheological properties of this Pluronic system. Although a macroscopic gelation temperature (37 °C) was established by the absence of
Dynamic Viscoelasticity of Aqueous PEO-PPO-PEO
Figure 10. Temperature dependence of G(0) for the aqueous Pluronic system (35 wt %).
flow in the solution-containing flask, we did not observe a frequency-independent value of the loss tangent when plotted vs temperature. Furthermore, the frequency dependence of the dynamic moduli (G′ and G′′) for the incipient gel could not be described by a simple power law over an extended frequency domain. The results from the stress relaxation measurements show the following essential features. At low temperatures (34 and 35 °C) in the pregel regime, the stress relaxation function can be described by two relaxation processes: an exponential at short time followed by a stretched exponential at long times. At higher temperatures, approaching the gel zone, a new relaxation mode in the form of a power law comes into play at intermediate times between the exponential and the stretched exponential decays. The power law exponent is close to 0.5 at all temperatures, except at the lowest temperature where a somewhat higher value (0.6) is observed. The width of the time window of power law behavior increases as the gel zone is approached, with a maximum for the incipient gel, and decreases in the postgel regime. When the dynamic moduli data are converted into equivalent stress relaxation data in the time domain, the calculated relaxation functions are compatible with the experimentally determined stress relaxation functions at all temperatures. The general picture that emerges from this rheological study is that a strong network, consisting of interconnecting micelles, is formed as the gel evolves. References and Notes (1) Tuzar, Z.; Kratochvil, P. In Surface and Colloid Science; Matijevic´, E., Ed.; Plenum: New York, 1993; Vol. 15, Chapter 1, pp 1-83. (2) Schmolka, I. R. In Polymers for Controlled Drug DeliVery; Tarcha, P. J., Ed.; CRC: Boca Raton, FL, Ann Arbor, MI, Boston, 1991; Chapter 10, pp 189-214. (3) Chu, B. Langmuir 1995, 11, 414. (4) Almgren, M.; Brown, W.; Hvidt, S. Colloid Polym. Sci. 1995, 273, 2. (5) Alexandridis, P.; Hatton, T. A. Colloids Surf., A: Physicochem. Eng. Aspects 1995, 96, 1. (6) Zhou, Z.; Chu, B. Macromolecules 1988, 21, 2548. (7) Wanka, G.; Hoffmann, H.; Ulbricht, W. Colloid Polym. Sci. 1990, 268, 101. (8) Deng, Y.; Ding, J.; Stubbersfield, R. B.; Heatley, F.; Attwood, D.; Price, C.; Booth, C. Polymer 1992, 33, 1963. (9) Bedells, A. D.; Arafeh, R. M.; Yang, Z.; Attwood, D.; Heatley, F.; Padget, J. C.; Price, C.; Booth, C. J. Chem. Soc., Faraday Trans. 1993, 89, 1235.
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