Viscoelastic Properties in Water of Comb Associative Polymers Based

Nov 15, 1997 - Richard Jenkins* and David Bassett. Union Carbide Corporation, UCAR Emulsion Systems, Research and Development,. 410 Gregson Drive ...
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Langmuir 1997, 13, 6903-6911

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Viscoelastic Properties in Water of Comb Associative Polymers Based On Poly(ethylene oxide) Bai Xu, Ahmad Yekta, and Mitchell A. Winnik* Department of Chemistry and Erindale College, University of Toronto, Toronto, Ontario M5S 3H6, Canada

Kayvan Sadeghy-Dalivand and David F. James Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada

Richard Jenkins* and David Bassett Union Carbide Corporation, UCAR Emulsion Systems, Research and Development, 410 Gregson Drive, Cary, North Carolina 27511 Received June 21, 1996. In Final Form: July 28, 1997X We describe aspects of the rheological behavior in aqueous solution of three associative polymers with a comb architecture. These polymers of identical structure but different molecular weight are based on poly(ethylene glycol) of M ) 8400, joined by a coupling agent bearing a C14H29- group. In steady shear, these polymer solutions exhibit a strong increase in low shear viscosity for concentrations between 1 and 2 wt %, and a sharp shear thinning transition at a shear rate of ca. 100 Hz. Some differences are seen between the “as prepared” and recrystallized samples. Oscillatory shear and first normal stress difference (N1) measurements were carried out on the lowest molecular weight sample, with an average of three pendant C14H29- groups per chain, and compared to the behavior of a telechelic polymer of similar molecular weight with C16H33- end groups. N1 measurements on both systems show similar behavior: a sharp increase in N1 which persists well into the shear thinning domain, followed by a decrease in N1 as the shear rate is increased further. The decrease in N1 and the strong shear thinning together suggest that the networks break down at high shear rates. In oscillatory shear experiments, major differences between the two types of polymers are apparent. The comb polymer exhibits a broad distribution of relaxation times, with a longest relaxation time of ca. 7 s, 2 orders of magnitude longer than that found for the telechelic polymer. From independent information about micellar structures present in the system, we calculate the functionality of the networks formed. For a given concentration, we find a much higher fraction of bridging chains and a much lower fraction of looped chains for the comb polymer than for the corresponding telechelic polymer.

Introduction In the preceding paper,1 we described the synthesis, characterization, and micellization properties of water soluble polymers containing C14H29 groups separated by poly(ethylene oxide) (PEO) blocks of molecular weight 8400. The PEO segments have a very narrow molecular weight distribution (Mw/Mn ) 1.05), but because the polymers themselves are prepared by a condensation reaction, they have a rather broad molecular weight distribution. Here we report the steady shear viscosity profiles of all three polymers and take a deeper look at the oscillatory shear and first normal stress difference response of the lowest molecular weight sample, Comb-81. This is the sample for which we have characterized the structure of the micelles formed by the polymer over the range of concentrations of interest.1 This sample has nearly the same molecular weight as another polymer sample of similar structure, but with C16H33- groups attached to the chain ends. With these two samples, we are able to compare the behavior of comb and telechelic HEUR associative polymers.2-10 There is now a rich literature on the structure formed and properties of linear associative polymers with hydrophobic end groups in aqueous solution.2-14 * To whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, November 15, 1997. (1) Xu, B.; Li, L.; Winnik, M. A.; Jenkins, R. D.; Bassett, D.; Wolf, D.; Nuyken, O. Langmuir 1997, 13, 0000.

S0743-7463(96)00613-0 CCC: $14.00

1 H NMR analysis indicates1 that Comb-81 contains 47 mmol of C14H29 groups per gram of polymer. By gel permeation chromatography (GPC), this polymer has a

(2) (a) Jenkins, R. D. Ph.D. Thesis, Lehigh University: Bethlehem, PA, 1990. (b) Jenkins, R. D.; Silebi, C. A.; El-Aasser, M. S. Polym. Mater. Sci. Eng. 1989, 61, 629. (c) Jenkins, R. D.; Silebi, C. A.; El-Aasser, M. S. In Advances in Emulsion Polymerization and Latex Technology: 21st Annual Short Course; El-Aasser, M. S., Ed.; Chapter 17, Lehigh University, June 1990. (3) (a) Water Soluble Polymers; Glass, J. E., Ed.; Advances in Chemistry Series No. 213; American Chemical Society: Washington, DC, 1986. (b) Polymers in Aqueous Media; Glass, J. E., Ed.; Advances in Chemistry Series No. 213; American Chemical Society: Washington, DC, 1989. (c) Polymers as Rheology Modifiers; Schulz, D. N., Glass, J. E., Eds.; ACS Symposium Series 462; American Chemical Society: Washington, DC, 1991. (d) Hydrophilic Polymers: Performance with Environmental Acceptance; Glass, J. E., Ed.; ACS Adv. Chem. Ser. 248; American Chemical Society: Washington, DC, 1996. (e) Franc¸ ois, J. Prog. Org. Coatings 1994, 24, 67. (f) Maechling-Strasser, C.; Clouet, F.; Franc¸ ois, J. Polymer 1993, 33, 1021. (g) Hester, R. D.; Squire, D. R., Jr. J. Coatings Technol. 1997, 69, 109. (4) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. J. Rheol. 1993, 37, 695. (b) Annable, T.; Buscall, R.; Ettelaie, R. Colloids Surf., A 1996, 112, 97. (5) Annable, T.; Buscall, R.; Ettalaie, R.; Shepherd, P.; Whittlestone, D. Langmuir 1994, 10, 1060. (6) (a) Fonnum, G.; Bakke, J.; Hansen, F. K. Colloid Polym. Sci. 1993, 271, 380. (b) Hulde´n, M. Colloids Surf., A 1994, 82, 263. (7) (a) Wang, Y.; Winnik, M. A. Langmuir 1990, 6, 1437. (b) Yekta, A.; Duhamel, J.; Brochard, P.; Adiwidjaja, H.; Winnik, M. A. Macromolecules 1993, 26, 1829. (c) Yekta, A.; Duhamel, J.; Adiwidjaja, H.; Brochard, P.; Winnik, M. A. Langmuir 1993, 9, 881. (8) Yekta, A.; Xu, B.; Duhamel, J.; Adiwidjaja, H.; Winnik, M. A. Macromolecules 1995, 28, 956. (9) Rao, B. H.; Uemura, Y.; Dyke, L.; Macdonald, P. M. Macromolecules 1995, 28, 531.

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broad molecular weight distribution including a low molar mass tail, and we calculate Mn ) 38 400. The peak molecular weight, Mpeak ) 67 300, corresponds to an average of 3.0 hydrophobes per chain. This polymer has the structure

Comb-81, like the telechelic polymers of M ) 34 000 and 51 000 with C16H33- end groups, undergoes association in water at very low concentrations, below 50 ppm.6 It is not yet known if the transition is sharp, in the way that micellization occurs at the critical micelle concentration (cmc) for traditional surfactants, or whether association occurs over a broader range of concentrations. This distinction is difficult to make experimentally if the polymers have a significant molecular weight distribution. Within the range of concentrations that one can study by dynamic light scattering (DLS) and pulsed-gradient spinecho (PGSE) NMR, one finds that both the telechelic and Comb-81 polymers form individual micelle-like association structures, which have been referred to as “flower-like micelles”15 or “rosettes”;8-10 cf. Figure 1. These are relatively compact objects, with diameters of 40-50 nm for the polymers so far examined, in which the PEO chains connecting two hydrophobes loop back into the same micelle. In these micelles, the free chain ends which dangle out into the solution are those which do not contain a hydrophobic end group. For the telechelic polymers with C16H33- end groups, the flower-like micelles on average consist of about 10 polymer molecules, and the micellar cores contain 20 hydrophobes.8-10 In the case of Comb-81, which contains an average of 3.0 hydrophobes per chain, the micellar cores contain 15 C14H29- groups, and the micelles contain on average five polymer molecules.1 At higher concentrations, a secondary association process takes place, in which some chains rearrange to bridge neighboring micelles.4,5,7,16 These objects grow in size with increasing polymer concentration, leading to a sol-gel-like transition in the system, and eventually the associated polymers fill all the available space in the (10) (a) Xu, B.; Yekta, A.; Li, L.; Masoumi, Z.; Winnik, M. A. Colloids Surf., A 1996, 112, 239. (b) Yekta, A.; Xu, B.; Winnik, M. A. In Solvents and Self-Organization of Polymers; Webber, S. E., Munk, P., Tuzar, Z., Eds.; Dordrecht: Kluwer, 1996; pp 319-330. (11) (a) Alami, E.; Rawiso, M.; Isel, F.; Beinert, G.; Binana-Limbele, W.; Franc¸ ois, J. In Hydrophilic Polymers; Glass, J. E., Ed.; ACS Adv. Chem. Ser. 248; American Chemical Society: Washington, DC, 1996; p 343. (b) Abrahmsen-Alami, S.; Alami, E.; Franc¸ ois, J. J. Colloid Interface Sci. 1996, 179, 20. (c) Franc¸ ois, J.; Maitre, S.; Rawiso, M.; Sarazin, D.; Beinert, G.; Isel, F. Colloids Surf., A 1996, 112, 251. (12) Alami, E.; Almgren, M.; Brown, W. Macromolecules 1996, 29, 2229. (13) Persson, K.; Wang, G.; Olofsson, G. Faraday Trans. 1994, 90, 3555. (14) Persson, K.; Bales, B. L. Faraday Trans. 1995, 91, 2863. (15) Semenov, A. N.; Joanny, J.-F.; Khokhlov, A. R. Macromolecules 1995, 28, 1066. (16) Borisov, O. V.; Halperin, A. Macromolecules 1996, 29, 2612. (17) Sadeghy-Dalivand, K. Ph.D. Thesis, University of Toronto, 1996.

Figure 1. A model for the associating structure of a telechelic AT. At concentrations too low to measure, the polymers exist as dissociated entities. At higher concentrations they associate to give well-defined “flower-like” micelles, which undergo secondary association to form secondary aggregates held together by bridging chains. There is evidence that across this transition, the number of hydrophobic groups per micelle core remains constant. The associated structures become larger with increasing concentration and ultimately fill space. We envision shear thinning in the system to involve breakup of the structure, via loop-to-bridge transitions,4 to form smaller entities.

solution. The presence of these larger objects and the networks they form account for the remarkable increase in zero-shear viscosity found in these solutions with increasing polymer concentration. For the telechelic polymers, the association structure has been described in terms of a transient network that fragments or rearranges when the solutions are subjected to a sufficiently strong shear stress. A picture of this structure, for the telechelic polymers, is given in Figure 1. In this figure we present the view that from a structural point of view, shear-induced fragmentation operates in much the same way as dilutioninduced changes in the network structure, with localized rearrangement of bridging chains to looping chains, a bridge-to-loop transition.4,5,7,8 Experimental Section The preparation and characterization of the comb polymers we examine here are described in the preceding paper,1 and corresponding information on the end-capped polymer AT22-3 has been reported previously.7b,8 Some rheological measurements were carried out on the comb polymers in the “as prepared” state. For comparison between AT22-3 and Comb-81, both polymers were recrystallized from hot ethyl acetate to remove low molecular weight impurities. Viscosity Measurements. Steady shear and linear viscoelastic measurements at Union Carbide were carried out at 25 °C on non-recrystallized samples using a Bohlin VOR rheometer (Bohlin Instruments, Cranbury, NJ) and three different fixtures: a tapered plug for very high shear rates; a cone and plate (30 mm diameter, 2.5° cone angle) for medium and high shear rates; and a Mooney-Couette cup and bob for low shear rates. The data collected using the various fixtures were superimposed to form composite shear-viscosity profiles. Brookfield viscosity standards were run periodically to calibrate the instrument. In the steady shear viscosity measurements, a delay time of 10 s was used between successive shear rate increments to allow the imposed flow and the fluid’s response to it to reach equilibrium. The strain amplitude employed in the oscillatory mode was chosen to be large enough to produce a measurable torque (typically 90 mrad, 20% strain), yet low enough to be in the linear range. Experiments in Toronto employed a Rheometrics RAA analyzer with a cone and plate geometry (50 mm diameter, 0.04 rad cone angle). Stock solutions consisting of 3.0 wt % recrystallized

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Figure 2. Plots of steady shear viscosity vs shear rate for the three (unrecrystallized) comb polymers in water at 25 °C at 1.0, 1.5, and 2.0 wt % concentrations: Comb-81, Mv ) 97 000; Comb-82, Mv ) 184 000; Comb-83, Mv ) 248 000. polymer in deionized water were prepared. The various samples examined were prepared by dilution of this stock solution. The linear viscoelastic region was determined to be below 20% strain by strain sweep experiments at frequencies of 1, 10, and 100 Hz. First Normal Stress Difference (N1) Measurements. The experimental measurements18 of N1 were carried out with a strain-controlled Rheometrics RFS rheometer in cone and plate geometry (cone diameter 50 mm, cone angle 0.04 rad). A 100 g-force transducer was used to sense the normal force that tends to separate the cone and plate during shear. This transducer was calibrated on a daily basis using standard weights. Data were obtained over a wide range of shear rates, typically from 10 to 2 × 103 s-1. The minimum shear rate is dictated by the sensitivity of the transducer and the level of the noise at low force readings. The maximum shear rate is dictated by a flow instability often observed for AT solutions at high shear rates. The normal force measured in this way is then transformed to the first normal stress difference using the following equation (which is valid only for cone and plate geometry)

N1 ) 2F/πR2

(1)

where R is the radius of the cone and F is the normal force.

Results Steady Shear Measurements. We first present steady shear viscosity profiles measured at Union Carbide on the samples as synthesized. Profiles for the three polymers at 1.0, 1.5, and 2.0 wt % (10, 15, and 20 g/L) are presented in Figure 2. The viscosity profile consists of a low-shear plateau, followed by a shear-thinning region. The shear rate at the onset of shear-thinning decreases (18) Israelachvili, J. Intermolecular and Surface Forces; Academic Press: San Diego, CA, 1992.

Figure 3. Plots of steady shear viscosity vs shear rate for recrystallized samples of associating polymers in water at different concentrations: (top) Comb-81; (bottom) telechelic polymer AT22-3. Both polymers have molecular weights of ca. 50 000.

as the polymer concentration increases and as the polymer molecular weight increases. The shear-thinning nature of the viscosity profiles is not a strong function of the polymer molecular weight. Some data are missing in the shear-thinning regions of some of the viscosity profiles due to the highly elastic nature of the solutions. At moderate shear rates, the Weissenberg effects caused the solutions to swell out of the shearing fixture. These data were omitted. At low shear rates, the normal stresses (see below) were low enough that the sample was not displaced. At high shear rates, the samples were less elastic, possibly due to partial disruption of the associative network. All the viscoelastic material properties were nearly independent of strain up to a strain amplitude of 0.1. At a test frequency of 1 Hz, the storage modulus was larger than the loss modulus for all samples. These data show that these polymers form robust elastic networks in aqueous solution. In Toronto, further rheological measurements were carried out on Comb-81 on a sample purified by recrystallization. This treatment removes low molecular weight impurities, particularly excess hydrophobe not bound to the polymer. It is an essential step in the fluorescence experiments used to determine the end-group aggregation number. It has relatively little effect on the viscosity profile of telechelic polymers like AT22-3, but the comb polymers are much more sensitive to this purification step. Plots of the steady shear viscosity vs shear rate for solutions of purified Comb-81, at various polymer concentrations, are presented in Figure 3a. Comparison of Figures 2a and 3a for the solutions at 1.5 and 2.0 wt %

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indicate that at low shear rates the purified samples are more viscous than their precursors by a factor of 2 to 3. This is consistent with a viscosity-lowering effect of substances removed through the recrystallization procedure. The absolute value of the low shear viscosity is very sensitive to the purification procedure, and all experiments reported here are for a single batch of purified polymer. At relatively low concentrations (1.2 and 1.5 wt %), the viscosity behavior of Comb-81 resembles that of the telechelic polymer AT22-3 shown in Figure 3b. In this concentration range and for shear rates less than about 30 s-1, both polymers exhibit shear-rate independent (Newtonian) viscosities. At higher shear rates, there is slight shear-thickening followed by severe shear thinning. In the shear-thinning regime, the viscosity (η) follows a power-law relation with shear rate (γ˘ ). That is, the relation is described by the usual power-law formula (η ) mγ˘ n) where m and n are constants. For both materials, the exponent n is close to -1, which indicates that the shear stress is virtually independent of shear rate. The onset of shear-thinning shifts smoothly to lower shear rates with increasing polymer concentration and occurs over similar ranges of shear rates for both polymers. If the onset is associated with a relaxation time of the system,2,4,8,10 then this time becomes longer as the polymer concentration is increased. At all concentrations, the comb polymer has a higher viscosity than the telechelic. Both polymers have similar molecular weights. With a shorter spacing between hydrophobes in the comb polymer, more bridges, and therefore larger networks, are possible for this polymer. At concentrations of 2 wt % and above, one notices a significant difference in the viscosity behavior between the two polymers. While the end-capped polymer exhibits Newtonian viscosities prior to shear-thinning at all concentrations, the more concentrated Comb-81 solutions show a gradual decrease in viscosity with increasing γ˘ prior to the shear thinning. For these concentrations, with the Toronto instrument, there is no clearly identifiable zero-shear viscosity. At lower frequencies, cf. Figure 2, the zero-shear viscosities become apparent. Oscillatory Shear Measurements. The measured values of storage modulus (G′) vs frequency (ω) for Comb81 at different concentrations are plotted in Figure 4a, and corresponding values of G′′(ω) are plotted in Figure 4b. At high frequencies the storage moduli appear to approach constant values, but there are no well-defined plateaus. Here there are relaxation modes in the networks with time constants smaller than 0.001 s. At low frequencies, the slope of the log-log G′ plot is not -2, and G′′ does not vary with ω1, behavior expected for simple Maxwell fluids and also characteristic of telechelic HEUR samples.2,4 Thus the plateau modulus GN° and the longest relaxation time cannot be found directly from the G′ plot. Because the plateau modulus cannot be identified in Figure 4a, we adopt a different approach. The maximum in G′′(ω) is associated with the longest relaxation times of the relaxation spectrum of the network. We can use this spectrum to estimate GN° 2a

GN° )

∫-∞a′ G′′(ω) d ln(ω)

2 π

Figure 4. Plots of G′ (a) and G′′ (b) vs frequency for solutions of Comb-81 in water.

The viscoelastic response of Comb-81 solutions to oscillatory shear seen in Figure 4 is consistent with a distribution of relaxation times in the system. The response of a viscoelastic system to oscillatory shear can be expressed as an integral equation over the relaxation spectrum H(τ) and the relaxation time τ.

G(t) )

(3)

In the frequency domain, the storage modulus G′(ω) and the loss modulus G′′(ω) can be written as19

G′(ω) )

∫-∞∞H(τ) 1 +ω ωτ 2τ2

]

d(ln τ)

(4)

G′′(ω) )

∫-∞∞H(τ) 1 +ωτω2τ2

d(ln τ)

(5)

[

2 2

[

]

The zero-shear viscosity η0 can be calculated if the relaxation spectrum is known.

(G′′ω ) ) ∫

η0 ) limη′ ) lim ωf0

ωf0

∞

τH(τ) d(ln τ)

(6)

-∞

The relaxation spectrum H(τ) can be calculated from the approximation formulas proposed by Tschoegl:12

(2)

The upper limit, a′, is chosen so that G′′(ω) contains the terminal loss peak only. In other words, it is chosen to include the contributions of the longest times of the relaxation spectrum within the range of integration. It is straightforward to integrate the G′′(ω) curve to obtain GN°.

∫-∞∞H(τ) exp(- τt) d(ln τ)

H(τ) ) H(τ) )

| |

dG′ 1 d2G′ + d(ln ω) 2 d(ln ω)2

1/ω)x2τ

dG′ 1 d2G′ d(ln ω) 2 d(ln ω)2

1/ω)τ/x2

(19) Tschoegl, N. W. Rheol. Acta 1971, 10, 582.

(7) (8)

Viscoelastic Properties in Comb Associative Polymers

Figure 5. Plot of the relaxation spectrum τH(τ) vs τ for Comb81 at 2.0 wt % in water.

Equation 7 is used when the slope of H(τ) is positive, and eq 8 is used when the slope is negative. To calculate the relaxation spectrum from the above equations, we first fit an algebraic curve to the plot of G′ vs ln(ω). Two derivatives are obtained from this curve and used for the calculation of H(τ) from eqs 7 and 8. The relaxation spectrum is the plot of H(τ) vs τ. As an example, we plot the relaxation spectrum of Comb-81 at 20 g/L as τH(τ) vs ln(τ) in Figure 5. As a check on its meaningfulness, we used this spectrum to calculate the zero-shear viscosity of the sample according to eq 10. The integral of this curve is not inconsistent with the measured low-shear viscosity of this solution (see Figure 3a). Complex and steady-shear viscosities were compared for the three comb polymers using the unpurified samples. We find that in the shear thinning region, solutions of the two higher molecular weight comb polymers exhibit similar slopes in the log-log plots of complex viscosity and steady-shear viscosity vs shear rate, but the steady shear viscosity is significantly larger. Thus the CoxMertz rule is not satisfied, a common occurrence for associating polymers. This rule describes the case in which the magnitude of the complex viscosity and the steady shear viscosity as a function of shear rate superpose when plotted on the same graph. For Comb-81, these two viscosities have different magnitudes and a different dependence on shear rate. First Normal Stress Difference Measurements. Elastic properties of solutions under shear are often revealed through measurements of the first normal stress difference, N1.20 Normal stresses in steady shear arise from deformation of the microstructures present in the system. Hydrodynamic forces induce the deformation, and the counteracting forces due to Brownian motion try to restore the system to equilibrium and give rise to the normal stresses. When a viscoelastic material is sheared between two parallel surfaces, normal stresses are developed: T11 in the direction of shear, T22 transverse to the flow and in the plane of shear, and T33 in the third direction, also perpendicular to the flow but parallel to the plates. The normal stress differences are defined as N1 ) T11 - T22 and N2 ) T22 - T33. N1 is positive for almost all materials, which means that the cone and plate surfaces tend to be pushed apart. For the solvent and for other simple, Newtonian fluids, N1 is zero. For solutions of linear polymers, N1 increases with shear rate at all shear rates.20,21 (20) Graessley, W. W. Adv. Polym. Sci. 1982, 47, 68.

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Figure 6. Plots of the steady shear viscosity and the firstorder normal stress difference of polymer AT22-3 in water at 1.0 wt % and 27.6 °C.

Figure 7. Plots of the steady shear viscosity and the firstorder normal stress difference of polymer Comb-81 in water at 1.49 wt % and 27.6 °C.

First normal stress difference (N1) measurements have not previously been reported for solutions of associative polymers. In Figure 6 we plot both N1 and viscosity for AT22-3 at 1.0 wt % and 27.6 °C as functions of shear rate. The viscosity profile is similar to that shown in Figure 2a for samples at 25 °C. It shows a plateau at low shear rates, then a region of slight shear thickening, and finally a shear thinning region. N1 becomes significant only at shear rates in which the system is well into the shearthickening regime. Here N1 increases with shear rate, passing through a maximum when the system is well into the shear thinning region. When the shear rate is increased further, the value of N1 actually drops. This behavior indicates that the networks are breaking up. A solution of Comb-81 polymer at 1.49 wt % and 27.6 °C shows very similar behavior (Figure 7). Discussion The behavior of associative polymers in solution is commonly described in terms of the theory of transient networks.22 These networks are formed from flexible chains containing association sites often referred to as “stickers”. In the model of Tanaka and Edwards22 association is stabilized by a binding energy which must be overcome for dissociation of the transient junction to occur. Subjecting the network to a shear strain can lead to rearrangement of the network. The rate of this process (21) Odell, J. A.; Keller, A.; Miles, M. J. Polymer 1985, 26, 1219. (22) Tanaka, F.; Edwards, S. F. J. Non-Newtonian Fluid Mech. 1992, 43, 247-309.

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is determined by a prefactor related to the strain and the activation energy for dissociation. The Tanaka and Edwards theory treats fully formed networks in which all chains act as bridges. Annable et al.4,5 recognized the importance of loops in the association network. These loops are mechanically inactive but contribute to the structure of the network formed. As the concentration of polymer is increased, some looping chains rearrange to form bridges, and the fraction of chains involved in bridges increases. In this concentration range where macroscopic networks are present, we8,10 and Annable et al.4,5 have described the breakdown of the network in response to shear in terms of bridge-to-loop transitions in the system. At much lower concentrations, the solution can contain a number of different species, ranging from single chains in solution to individual micelles to higher aggregates of finite size. Association at Low Polymer Concentration. Traditional surfactants undergo a sharp onset of association at a concentration referred to as the critical micelle concentration (cmc). Less is known about the onset of association in associating polymers, but for telechelics, some information is available from fluorescent probe,7,11,12 light scattering,12,23 PG-NMR,1,9,24 and differential scanning calorimetry13 measurements. In many cases it appears that the association transition is broader than the cmc transition of low molecular weight surfactants, but this may be a consequence of the relatively broader molecular weight distribution in the polymeric systems. What is clear is that there is an entropy penalty for loop formation.23,25 For telechelics, the onset of association is much more complex in systems in which long soluble chains contain only weakly associating end groups.18 When the interaction between the hydrophobic substituents is strong, polymer association appears at concentrations well below that at which the sharp increase in solution viscosity occurs. Fluorescent probe measurements on AT22-3 indicate that association can be detected at cpol < 10 ppm. PG-NMR measurements on Comb-81 are consistent with association at the lowest concentrations (0.1 wt %) where signal can be detected. Larger aggregates are detected in the PG-NMR signal at higher polymer concentrations. The sudden strong increase in the solution viscosity which occurs in the region of cpol ) 10 g/L is likely due to the onset of network formation. Semenov et al.15 developed a theory of association for polymers containing strongly associating end groups. Because of the strong interaction between the end groups, association commences at low concentration to form flower micelles, individual micelles of looped chains. Because these micelles experience a weak attraction and no repulsion beyond their corona radius, the micelles are predicted to phase separate into a polymer-rich phase with micelles at the space-filling concentration φ* (dimensionless, c* in g/L), and a polymer-lean phase containing a dilute gas of micelles. It is important to realize that these are starlike micelles26 and that the chains in the corona are stretched. They have a radial dimension which is larger than the radius of gyration RG of the free chain in solution. In our previous experiments on AT22-3, we reported the detection of flower micelles through a combination of dynamic light scattering and PG-NMR measurements.8,9 (23) Raspaud, E.; Lairez, D.; Adam, M.; Carton, J. P. Macromolecules 1994, 27, 2956. (24) Xu, B.; Li, L.; Yekta, A.; Masoumi, Z.; Kanagalingam, S.; Winnik, M. A.; Zhang, K.; Macdonald, P. M.; Menchen, S. Langmuir 1997, 13, 2447. (25) ten Brinke, G.; Hadziioannou, G. Macromolecules 1987, 20, 486. (26) Halperin, A. Macromolecules 1987, 20, 2943.

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These micelles contain an average of 20 C16 chain ends and ca. 10 polymer molecules, very similar to the size determined from fluorescence quenching experiments. PGNMR experiments on Comb-81 are consistent with flower micelles comprised of 15 hydrophobes, representing on average five polymer chains with three hydrophobes each. Here, too, we find a similar size from fluorescence quenching experiments.1 AT22-3 contains only 1.7 end groups per chain, so that there are some free chain ends lacking a hydrophobic sticker. In Comb-81, the hydrophobes are in the chain interior and the ends are free. The chains with free ends are likely to extend into the solution beyond the corona of loops, giving the micelles a hydrodynamic radius and an intrinsic viscosity somewhat larger than that of micelles containing only looped chains. On the basis of the hydrodynamic radii of the flower micelles, we estimate that a random close-packed array of spheres for both polymers corresponds to 1.4 wt % polymer and that the micelles fill space (as Vornoi polyhedra) at 1.7 wt %. We do not observe phase separation at cpol < 1.4 wt %, but we do find higher aggregates present in the system by dynamic light scattering and PG-NMR measurements. In our systems, the chains with dangling ends may provide a weak steric barrier which might limit the extent of flower micelle association at low polymer concentration and suppress phase separation. Similar end-group aggregation numbers are found at higher concentrations for both AT22-3 and Comb-81 where the solution viscosity is much higher, indicating that over the range of concentrations examined, the core of the micelle-like junctions maintains their size through the transition from flower micelles to extended networks. Relaxation Times. The most remarkable feature of the dynamic viscoelastic behavior of telechelic AT’s is the observation that the data can be fitted very well to a single element Maxwell model and characterized in terms of a single plateau modulus and a single relaxation time. Annable et al.4 attributed this single relaxation time to the exit rate of the hydrophobic group from its micelle, and the onset of shear thinning in the steady shear experiment corresponds approximately to shear rates faster than the reciprocal relaxation rate of the system. These authors reported a remarkable set of experiments involving a mixture of telechelic polymers with different end groups. They found that the oscillatory stress-strain behavior was more complex but could be fitted to a sum of relaxation times corresponding to the relaxation times of the individual end groups. This result is consistent with what is known in the surfactant literature about the exit rates of surfactants from micelles.18 This rate depends very strongly on the length of the hydrophobic tail and is related to the free energy of transfer of the hydrophobe from the micelle to the water phase. Comb-81 has a more complex behavior, which cannot be characterized in terms of a single relaxation time. At a concentration of 20 g/L, for example, the longest relaxation time is about 7 s (i.e., ln τ ) 2), as indicated by the onset of the sharp decline in τH(τ) in Figure 5. This time is much longer than the times of order 0.1 s associated with the telechelic polymer solutions at similar concentrations. For a relaxation time of order 10 s, it should be possible to observe the decay in stresses when the solution comes to rest after cessation of steady shear. When such experiments were conducted with the Comb-81 solutions, the rheometer torque was observed to decay over a period of several seconds, consistent with a relaxation time on the order of 10 s. When the same tests were carried out on AT22-3 solutions, the torque relaxed to zero much faster, in fact faster than the detection capabilities of the rheometer.

Viscoelastic Properties in Comb Associative Polymers

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type associative polymers. They examine bead-spring chains with 10 “sticky” beads connected by nine Gaussian springs.28 Their model includes only binary associations, so that the beads can be free or paired, leading to the three types of association structures shown below. Association of two chain ends (1) gives an extended linear chain. Pairing of a chain end with an interior bead (2) gives a branch with a functionality fn of three, whereas pairing of two interior beads (3) gives a branch with a functionality of four. Even though the bead aggregation number is at most two, this is sufficient for network structures to form.

Figure 8. Plot of the relaxation times Tau, calculated from the frequency at which G′ ) G′′, as a function of the concentration of Comb-81.

Other analyses of the 20 g/L Comb-81 data reveal that the solution has short relaxation times as well. One analysis is that of finding the frequency ω at which G′ ) G′′ and identifying ω-1 as a relaxation time. For a Maxwell fluid, ω-1 is identical with the fluid’s relaxation time; for a fluid with a distribution of relaxation times, ω-1 can be interpreted as a mid-range value. For the 20 g/L solution in question, Figure 4 shows that G′ ) G′′ at a frequency of 15 s-1, and so the corresponding relaxation time is 0.067 s. In Figure 8 we plot the relaxation times associated with the frequencies at which G′ ) G′′ and note that these relaxation times increase strongly with increasing polymer concentration. Another technique of finding a characteristic relaxation time is to use N1 data at low shear rates. Almost all constitutive equations and most viscoelastic data show that N1 varies as γ˘ 2 as the shear rate goes to zero. For two-parameter constitutive equations, e.g., the Maxwell model, the relation between N1 and the fluid properties at low shear rates is

N1 ) 2ητγ˘ 2

(9)

where η is the low-shear viscosity. The variation of N1 with γ˘ 2 is seen in Figures 6 and 7, where the low shear N1 data have slopes close to 2 on these log-log plots. From Figure 7 and from eq 9, it is found that τ is about 0.008 s (the value of η was taken to be 3 Pa‚s, from Figure 3a). The data in Figure 7 are for a concentration of 14.9 g/L, and so the corresponding time would be about 0.01 s for 20 g/L, the concentration of interest. Hence for the 20 g/L Comb-81 solution investigated here, the relaxation times cover 3 decades, from 0.01 s based upon the N1 data through 0.07 s based on the Maxwell fluid behavior to 10 s based upon G′ data. The wide distribution contrasts sharply with the single times of order 0.1 s for telechelic polymers and reflects the more complex networks which can be formed from comb polymers. Recently Groot and Agteroff27,28 reported simulations of the viscoelastic properties at different concentrations of a polymer with several features in common with comb(27) (a) Groot, R. D.; Agterof, W. G. M. J. Chem. Phys. 1994, 100, 1649, 1657. (28) Groot, R. D.; Agterof, W. G. M. Macromolecules 1995, 28, 6284. (29) Graessley, W. W. Macromolecules 1975, 8, 186.

The first interesting property of these networks is that they are characterized by a broad distribution of viscoelastic relaxation times. These arise from the multiple bridges formed from individual bead-spring chains so that shear-induced dissociation of an individual association site does not break the mechanical connectivity of the network and stress is only partly relieved. The authors comment that stress is most likely transferred to the neighbors and only slowly diffuses away by the breaking of other bonds. This is the distinguishing feature between comb-type structures and telechelic chains. Since telechelics are linked only at the chain ends, dissociation of a single chain end from a network junction leads to rapid relaxation of the stress on that chain through its Rouse modes. Superbridges. In their simulation of associating beadand-spring chains, Groot and Agteroff27 investigated the distribution of structures present in the system as a function of chain density (concentration). Near the gel point they find a very large cluster size distribution. The system is a weak gel because some structures span the simulation volume. As the density is increased, the system shifts toward full gels (100% connections between the edges of the space) and stronger gels. This picture is consistent with the pictures drawn in Figure 1, which emphasize the structures or clusters of finite size. More important is the idea that the gel point coincides with the polymer concentration where the large increase in solution viscosity of associating polymers occurs. Experimentally it is known that the plateau modulus and the longest relaxation time in associating polymer solutions both increase as the polymer concentration is increased. Annable et al.4 found that relaxation times in their telechelic systems increased linearly with concentration and ascribed the increase to the formation of superbridges. Superbridges (see Figure 9) are extended structures in which several polymers are linked to form a single active chain. If one of the links in this chain breaks, the complete connection is lost. Long superbridges are susceptible to more rapid scission, since the probability per unit time that any of the links of a chain breaks is proportional to the number of links in the connection. As the polymer concentration is increased, the average length of the superbridges decreases, and the relaxation times become longer. Groot and Agteroff27 emphasize the important role played by superbridges in determining the properties of associated polymers in solution. Their simulations show that the network relaxation time varies as the chain

6910 Langmuir, Vol. 13, No. 26, 1997

Xu et al.

Figure 9. A representation of superbridges formed in an associating polymer system containing a significant fraction of looped chains. In this we emphasize that the junctions within the superbridge contain polymer loops because the micelle cores which serve as junctions in the superbridge must on average contain the same number of hydrophobic associating groups (15 for Comb-81, 20 for AT22-3) as in the association structure as a whole.

density F to the first power and that the plateau modulus increases as F3. In their view, their simple model works to explain many features of a variety of associating systems (even gelatin) because close to the sol-gel transition, the dominant effect determining the modulus is the formation of superbridges that are larger than the correlation length of individual polymer molecules. This process, they state, is independent of the precise chemical nature of the individual association sites, and hence a rather universal behavior of the complex modulus can be expected. Since the low shear viscosity is the product of plateau modulus and the long-time relaxation time, they predict that η0 should be proportional to F4. They argue that these changes come about because of structural effects of the gel, that the shorter relaxation at low concentration and the third power density dependence of Gn° have a common origin. The active connections in the system are superbridges. The Functionality of the Network and the Number of Bridging Chains. The number of bridging chains in a polymeric network can be calculated from the magnitude of the plateau modulus by assuming that elasticity is a consequence of these bridging chains, which are the mechanically active components of the system.20 By analogy with the classical theory of rubber elasticity, the pseudoequilibrium modulus for an ideal network is given by

GN° ) νRT

(10)

where ν is the molar density of elastically effective chains, R is the gas constant, T is the absolute temperature. This type of analysis has been applied previously to solutions of telechelic associative polymers.2,4,6 As Jenkins demonstrated,2 entanglements make very little contribution to the viscoelastic properties of this system and can be neglected in the analysis of rheological data in these types of systems. In these models, it was common to compare the magnitude of ν with the molar concentration of polymer chains in the solution, from which one calculated the curious result that the molecular weight between junctions Me was many times larger than the molecular weight of the polymer. In our approach, we chose a model in which the fundamental building block of the network is the micelle itself.10 We use eq 10 in conjunction with our knowledge of the micellar structure to calculate the functionality of the network, fn, defined as the average number of bridging chains per junction The number of elastically effective chains connected to each micelle is given by the expression:

fn ) 2ν/[micelle] ) 2νNR/qRcpol

(11)

where NR is the number of hydrophobic groups per micelle, qR is the hydrophobic group content of the polymer (in units of mol/g of polymer), and cpol is the polymer

Figure 10. Plot of the calculated functionality of the networks formed, as a function of the concentration of Comb-81.

concentration in g/L. Although comb polymers are in general more complicated than the telechelic polymers, we still expect that the molecular weight of the polymer between bridges, Me, cannot be higher than the Mn of the polymer.10 In our calculations of fn, we used the value of qR determined by 1H NMR measurements, which is not dependent on the molecular weight of Comb-81. By combining eqs 10 and 11, we obtain

fn ) 2Gn°NR/(RTqRcpol)

(12)

These fn values represent an average over the entire system and increase linearly with increasing polymer concentration as is shown in Figure 10. In Figure 10, we find values of the functionality less than 3 at polymer concentrations where strong enhancement of the solution viscosity occurs. At these concentrations near the sol-gel transition, not all micelles in the system make elastic contributions to the network. In analyzing our data in terms of eq 11, we divide 2ν, as a measure of the number of chain ends of mechanically active chains, by the total number of micelles in the system. As the Groot and Agteroff28 simulations make clear, in this concentration range, the system contains a broad distribution of species ranging from single micelles to larger aggregates of finite size in addition to the percolation structures which dominate the mechanical properties of the solution. Thus our analysis calculates a value of fn averaged over all the species in the system. As the polymer concentration is increased, the micellar phase fills a greater fraction of the total volume. At c* ≈ 1.4 wt % for both AT22-38 and Comb-81,1 spheres of undeformed micelles just fill space in the solution. Above this concentration, the micelle cores are forced closer together. This proximity promotes loop-to-bridge transitions and increases the fraction of bridging chains, leading to an increase in fn. We can use the calculated values of the functionality to estimate the fraction of chains involved in bridges in the AT solutions. Since each micelle shares fn/2 bridging polymer chains, this fraction can be described by the expression fbridge ) (fn/2)/(number of chains per micelle). Values for fbridge and Me are more difficult to estimate here than in the case of telechelic polymers, since the comb polymers are in fact mixtures with on average three hydrophobic groups per chain, based upon Mpeak ) 67 000. Unlike the telechelic polymers, the comb polymers have more than two hydrophobic groups per chain. Thus one

Viscoelastic Properties in Comb Associative Polymers

Figure 11. Plot of the fraction of bridging chains in networks formed by Comb-81, as a function of the concentration of Comb81.

comb polymer chain can serve to bridge more than two micelles, depending on where hydrophobic groups reside. This means that comb polymers are more efficient in creating bridging chains. As an example, the solution of Comb-81 at cpol ) 20 g/L has 25% of its chains involved in bridging micelles. For AT22-3, the C16H33- end-capped polymer of the same molecular weight, at 20 g/L only 10% of the polymer chains are involved as network bridges.10 A plot of the percentage of bridging chains vs polymer concentration is shown in Figure 11. The very high percentage of bridging chains present in solution implies that the extent of intramolecular association is rather low. Most of the polymer molecules provide bridging chains for the network. This also explains in a more quantitative way the enhanced viscosity for the Comb-81 polymer solutions compared to AT22-3 at similar concentrations. Summary From the experiments reported here and elsewhere, a picture emerges of the general features of the structures formed by telechelic and comb associative thickeners based upon PEO. At very low concentrations, the polymers associate through their hydrophobic substituents to form micelle-like entities. In the case of the C16H33- end-capped telechelics and the C14H29- substituted combs examined here, the initial association occurs at such low concentrations that there is a range of concentrations in which the solution, still in the low viscosity regime, is comprised of individual flower-like micelles, with a well-defined hydrophobic group aggregation number NR. At higher concentrations, secondary association takes place to form higher order structures and networks. These networks are responsible for the large increase in viscosity of the solutions with increasing polymer concentration and for the elasticity of the solutions. In these solutions, the sharp increase in viscosity occurs at concentrations just below c* (calculated from the hydrodynamic radius of the micelles found at low concentration). On this basis, one can infer that the sol-gel

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transition occurs at concentrations just below that at which the micelles themselves would fill space. In some telechelic systems of slightly different chemical structure, evidence for further ordering within the network has been obtained from scattering experiments, consistent with the formation of cubic phases. In the aqueous systems studied by Franc¸ ois and co-workers,11 the scattering experiments indicate an increase in core size at these high concentrations. In an analogous triblock copolymer system studied in a nonpolar solvent by Raspaud et al.,23 the micelle size appears to remain unchanged over the range of concentrations at which the cubic phase can be studied. In the telechelic polymers and in Comb-81, shear thickening is observed, but it appears for polymer concentrations above 1 wt %, i.e., at concentrations just above the sol-gel transition. It is noteworthy that shear thickening coincides with the pronounced increase in the first normal stress difference. In transient network models, such as that of Vrahopolou and McHugh,30 shear thickening occurs when the force necessary to stretch a chain increases faster than that required for Gaussian chains. For the shear thickening observed here, a different mechanism is more likely. Tam31 has recently shown, through superposition of oscillations on steady shear experiments for the linear polymer AT22-3 that the plateau modulus increases in the shear thickening regime. Thus shear thickening results from a shear-induced increase in the number of mechanically active chains. Shear thinning in transient networks is extremely subtle. As a reviewer has emphasized to us, tiny changes in the disengagement time with shear rate (due to chain elasticity) can give rise to strong non-Newtonian effects. Tanaka and Edwards have shown that substantial shear thinning can occur from the effects of chain stretching on the time constant, even in the case of fully formed networks. The reviewer comments that since shear thinning does not go away at high concentrations for real associating telechelics, the dominant effect must be that of the stored elastic energy on the time constant itself. We like to approach this issue from the perspective of the structural aspects of the response of the system of shear. In the case of fully formed networks, the response to shear may be one of network reorganization in which chain ends change junctions. In the concentrations range where superbridges play a dominant role in determining the magnitude of the modulus and the time constant, shear thinning in our view results in breaking of a connection, followed by reinsertion of the dangling chain end into the micelle to which it is attached, a bridge-to-loop transition. Acknowledgment. The authors thank NSERC Canada for their financial support and Dr. J. Cloyd at Eastman Chemical Co. for helpful discussions. We also thank a reviewer whose critical comments on our original submission prompted us to think more deeply about our results. LA960613I (30) Vrahopolou, E. P.; McHugh, A. J. J. Rheol. 1987, 31, 371. (31) Tam, K. C.; Winnik, M. A.; Jenkins, R.; Bassett, D. Manuscript in preparation.