Article pubs.acs.org/IECR
Rheology of Poly(acrylic acid): A Model Study Yuchen Wang,† Richard A. Pethrick,‡ Nicholas E. Hudson,‡ and Carl J. Schaschke*,† †
Department of Chemical and Process Engineering, University of Strathclyde, 75 Montrose Street, Glasgow, Scotland WestCHEM, Department of Pure and Applied Chemistry, University of Strathclyde, 295 Cathedral Street, Glasgow, Scotland
‡
ABSTRACT: In this study, we examine the rheological characteristics of poly(acrylic acid) (PAA) polymers in aqueous solutions at 20 °C using various techniques. We also develop a model to simulate the rheological behavior of the polymer solutions, because, in some applications, the polymer is required to be dispersed within a short period of time, during which highshear mixing is commonly used to achieve the necessary level of homogeneity. The model, which involves consideration of reptation, relaxation of the chains by slippage or loop relaxation, and normal Rouse dynamics, is appropriately adapted to reflect the structural changes of the polymer molecules and then used to identify possible relationships between molecular structure and viscoplastic behavior. The simple theory for flexible polymers is modified to include the possibility of complex formation by association between neighboring polymer molecules. We conclude that, although the model is sensitive to the average molar mass of the polymer, additional and more extensive high-shear-rate data are required to explore fully the validity of the Rouse description. However, the scaling of the viscosity by the addition of an increased friction factor provides a reasonable description of the short-range motions observed.
1. INTRODUCTION Poly(acrylic acid) (PAA) is a water-soluble polyelectrolyte system that has wide applications as a fluid thickener, as well as a dispersing, suspending, and emulsifying agent in pharmaceuticals and cosmetics.1−5 PAA is also widely used as a binder in the production of ceramics and in deicing fluids used in aeronautics applications.6 PAA is widely used in deicing fluids to achieve the desired viscoplastic characteristics at temperatures on the order of 0 °C. As a component of deicing fluids, the rheological characteristics of the polymers are critical. A typical deicing fluid consists of a water/glycol mixture, with the composition adjusted to achieve the desired depression of the freezing point to allow the heated fluid to remove ice from the critical surfaces. A low concentration of PAA is added to the fluid to create a stable thin film that suppresses ice formation during the period when the aircraft taxies before reaching the hold position prior to takeoff. During the short period between the aircraft’s release from the hold position at the end of the runway to the point at which it reaches rotational speed, when the aircraft begins to lift from the runway, the thin film must be cleanly removed from the wings and fuselage. The fluid layer has to be sufficiently stable to provide the necessary cover when the aircraft is taxiing, but it must be easily released as the air shear increases during the period up to rotation. PAA, as with many polyelectrolytes, is able to form gel structures at high concentration and has been associated with problems associated with jamming flight control systems when residues build up in aerodynamically quiet areas. In moist conditions, these deposited gels can reswell, expanding in volume by several orders of magnitude, with the possible result of a malfunction of critical control equipment and jamming of flight control systems.6 It is desirable to be able to understand the way in which the molecular architecture of PAA creates the desirable viscoplastic characteristics so as to be able to design systems that do not suffer from insoluble gel formation. In this work, we investigate the rheological characteristics of PAA and © 2012 American Chemical Society
attempt to model its shear response behavior using models that explore the possible relationships between molecular structure and viscoplastic behavior. Using the model created in this work, new formulations are being developed using water/glycol mixtures that attempt to address the gelation problem and will be the subject of future publications PAA is a classic example of a polyelectrolyte and has been extensively investigated.1−5,7−13 At very high dilution, a large fraction of the carboxyls are ionized and repel one another, so that the macromolecule can adopt a loose conformation. As a consequence of the negative charge repulsion on the carboxyl group, the end-to-end distance of and volume occupied by the polymer coil are large and depend on the pH of the medium. With increasing concentration, the degree of dissociation of the carboxyls decreases, with a consequent change in the size of the polymer. The size of the PAA, and consequently the viscosity of the solution, is influenced by the pH of the medium. In the case of nonpolyelectrolytes, increasing the concentration of the polymer in solution leads to a point where dramatic changes in the viscosity occur that can be associated with the onset of entanglement, and the rheology becomes controlled by reptation processes.14,15 In the case of polyelectrolyte systems, polymer−polymer interactions can lead to significant effects on the rheological characteristics of the materials.16,17 The behavior of flexible polymers in dilute solution depends on a number of parameters, including the average molecular weight and the molecular weight distribution, the rigidity of the polymer backbone, the chain topography (extent of chain branching), the concentration, and the interaction between the polymer and solvent. For an ideally flexible monodispersed polymer in an ideal solvent, the relaxation behavior can be Received: Revised: Accepted: Published: 16196
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Figure 1. FTIR spectra of carbomers A and B.
2. EXPERIMENTAL SECTION 2.1. Materials. Two poly(acrylic acid) (PAA) polymers (carbomers A and B) were supplied by Lubrizol (Brussels, Belgium) and were obtained as flocculated solid particulate materials of approximately 0.2-μm diameter and with nominal average molar masses of 4 × 106 and 5 × 106 g/mol, respectively. Carbomer B was designated as a copolymer. 2.2. Polymer Structure Characterization. PAA is commercially available in a range of average molar masses and is designated as a copolymer that can contain vinyl pentaerythritol. During the drying process, PAA can form anhydride cross-links that can influence the effective molar mass and hydrodynamic volume of the polymer. To probe the structure of these polymers, Fourier transform infrared (FTIR) spectroscopy, Fourier transform nuclear magnetic resonance (FT-NMR) spectroscopy, and intrinsic viscosity (IV) measurements were carried out on the carbomer polymers. 2.3. FTIR Spectroscopy. Transmittance FTIR spectra were obtained using a Perkin-Elmer 100 FTIR spectrometer, with 32 scans per spectrum, at a resolution of 4 cm−1. The solid PAA was dispersed as a mull in Nujol (low-molar-mass paraffin) and measured on KBr plates. The spectra were normalized using the C−H stretch at 2940 cm−1. Because the C−H stretch is also found in the mull, the intensities of the other peaks are not quantitative because of the variation of the concentration of PAA in the sample. 2.4. FT-NMR Spectroscopy. 1H FT-NMR spectra were obtained at a frequency of 400 MHz using a Buker AV 400 instrument. The signal was obtained using 30 scans at 25 °C
satisfactorily described by two parameters: the excluded-volume integral, β, and the statistical or Kuhn segment length, lk, which is related to the persistence length, lp.16,17 In dilute solution, the rheological behavior of low-molar-mass flexible polymers can be described by Rouse theory.14,18 Increasing the concentration and molar mass results in an increased occurrence of polymer− polymer interactions, and for high-molar-mass polymers, a significant contribution to the viscosity arises from reptation. In polyelectrolytes, electrostatic forces are dominant over the normal intermolecular interactions, and in dilute solution, these forces cause the chain to adopt a highly extended structure. Addition of inert electrolytes, such as NaCl, to the polymer solution produces strong electrostatic screening, and the highly expanded polyelectrolyte coil shrinks.19,20 At high salt concentrations, unperturbed dimensions are approached, but “salting out” of polyelectrolytes may occur. 21 Further complications involve defining whether the number of dissociated or “noncondensed” counterions remains constant with added salt. A study of the elongational viscosity of PAA showed that self-association was an important contribution to the rheology.7 In this work, we consider an extension of this model to explore the various contributions that lead to the observed shear-dependent behavior in PAA. The intrinsic viscosity of PAA has been studied, and when the effects of counterions were incorporated into the analysis, it was found that the Huggins constant can be calculated.22 The intrinsic viscosity is investigated as part of the characterization of the materials used in this study. 16197
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Figure 2. FT-NMR spectra of (a) carbomer A and (b) carbomer B in D2O after 8 days of dissolution.
with suppression of the solvent spectrum. The solutions were obtained by dissolving the PAA in D2O and leaving the solutions for a period of 8 days. 2.5. IV Measurements. To avoid the possibility of shear degradation of the polymer, aqueous solutions were produced by slow stirring using a magnetic stirrer at a concentration of 0.0891 g/dL for PAA-[A] and 0.0952 g/dL for PAA-[B]. Dissolution of the polymer in water was very slow and typically took 5 days to produce a homogeneous solution. These stock solutions were diluted with distilled water to obtain the desired concentrations for the measurements. Dilute-solution viscosity measurements of PAA in deionized water were carried out using an Ubbelohde 0C suspended level viscometer thermostatically controlled in a water bath at 20 °C. Flow times were determined electronically and in triplicate, and the average values were used. The relative viscosity, ηrel, was determined for the solution relative to the solvent (deionized water) as ηrel = tps/ts, where tps and ts are the flow times for the polymer solution and solvent, respectively. The specific viscosity is given by ηsp = (tps − ts)/ts. Plots of the natural logarithm of the relative viscosity divided by the concentration (ln η rel /c) and the specific viscosity divided by the concentration (ηs/c) versus concentration (c) allow for the determination of the intrinsic viscosity, [η]. 2.6. Rheology. For the rheological measurements, the solutions were prepared using a method that closely resembles that used in industry. This was done by dispersing the solid powder in demineralized water using a Silverson L4R laboratory mixer operating at approximately 4800 rpm for 5 min and then 3600 rpm for 55 min. The solutions obtained contained small air bubbles that were released over time. Typically, the solution would be kept at room temperature for
2−3 days before measurements were carried out; at this point, no bubbles were visible in the solution. Creep and steady shear measurements were carried out on a AR1000N rheometer (TA Instruments, Crawley, U.K.) using a 4-cm parallel-plate geometry fitted with a solvent trap. Temperature was controlled using the Peltier effect, with a water bath maintained at 20 °C as a heat source/sink. The steady-shear experiments were performed in the range of 0.5− 1000 s−1, and no flow instabilities were observed. The lowest shear rate measured in the creep experiments was ∼2 × 10−5 s−1, and this was maintained over the final 2 min of the 5 min of creep at the lowest stress obtainable. Oscillatory shear measurements were carried out on a shearstress-controlled Carri-Med CSL2500 rheometer (TA Instruments, Crawley, U.K.) using a 4-cm parallel plate fitted with a solvent trap and operating over a frequency range from 0.01 to 100 rad s−1. Temperature was controlled by using the Peltier effect, with a water bath maintained at 20 °C as a heat source/ sink.
3. RESULTS AND DISCUSSION 3.1. Polymer Characterization. This study represents an attempt to understand the relationship between the molecular structure of a polymer and its rheological characteristics. Carbomers are high-average-molar-mass poly(acrylic acids), but they contain elements in their structure as a consequence of their synthesis that may influence their rheological character.17,18 The molecular structure of the polymers was determined using a combination of FTIR, FT-NMR, and IV measurements. 3.1.1. FTIR Spectroscopy. The FTIR spectra of the two carbomer polymers investigated were obtained as Nujol mull dispersions (Figure 1). The spectra were normalized using the 16198
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C−H stretch at 2940 cm−1 as a reference. This vibration is, however, also characteristic of the Nujol mull, and as a consequence, differences in the intensities of the other peaks in the comparative spectra do not necessarily reflect differences in the constitution of the polymers. Both spectra have characteristic strong absorption at ∼1720 cm−1 associated with the C O stretch of the carboxylic acid. Absorption in the range below 3400 cm−1 is associated with the −OH stretch of the carboxylic acid, which is broadened, reflecting the mobility of the proton of the carboxylic acid and its ability to interact with several sites in the molecule. The absorption in this range changes with the concentration of Nujol in the mull and is sensitive to changes in the structure of the polymer in the mull. No real differences were seen between the spectra of the two polymers. 3.1.2. FT-NMR Spectroscopy. The solutions in D2O for the FT-NMR study were obtained by leaving the polymer to dissolve over a period of 7 days. Clear homogeneous solutions were obtained that could be used for NMR examination. The spectra (Figure 2) were obtained using a pulse sequence that suppresses the resonance due to solvent. The selection of the relaxation frequency in the case of carbomer A was less precise than that in the case of carbomer B; as a consequence, there are differences in the solvent breakthrough peaks observed at ∼5.6 ppm. In the case of carbomer B, there is a spinning sideband at ∼2.14 ppm. To aid assignment of the spectra, theoretical peak shifts for model structural elements of PAA were calculated using ChemDraw software. The calculations were performed for the acrylic acid repeat unit and predicted peaks at 1.75 ppm for the degenerate CH2 protons and 2.35 ppm for the C−H proton. The presence of anhydride groupings arising from the dehydration of neighboring acrylic acid groups or the formation of cross-links between different polymers would give peaks for the CH2 protons at 1.88 and 1.63 ppm, because the protons would cease to be degenerate as a consequence of ring formation. The CH peak was predicted to be at 2.321 ppm, which is essentially the same as in the acrylic acid repeat unit. Close inspection of the peaks in the CH2 region indicate that there are resonances at ∼1.26 ppm. The incorporation of methacrylic acid into the polymer would account for a peak in this region. Carbomer copolymers are known to be modified by the incorporation of vinyl pentaerythritol as a cross-linker. The CH2 group of this element of the structure occurs at 3.45 ppm. The shift for the OH group is difficult to predict and theoretically should occur at 11.0 ppm; however, in the spectra, it is observed at 8.35 ppm. Small differences are expected between prediction and experiment as a result of solvent shift effects. Analysis of the spectra indicates that carbomer A is predominantly poly(acrylic acid) but contains a small amount of anhydride and methacrylic acid units. Using the peak intensities in the C−H region, the compositions shown in Table 1 were predicted. The peaks overlap, and hence, there is a significant level in uncertainty in the fractions calculated. The very weak peaks in
the range of 3.5 ppm in carbomer B are indicative of the pentaerythritol unit and indicate that this polymer is very lightly cross-linked. The intensity of this peak indicates that the pentaerythritol units are present on the order of less than 3 wt %. Quantification of the peak intensities is only approximate, as the intensities can be influenced by differences in chain mobility. The anhydride units can create cross-links between PAA chains that effectively increase the molecular weight of the PAA but also convert it from a linear- to a branched-chain polymer structure. Branched chains are also created by the incorporation of pentaerythritol into the PAA. 3.1.3. IV Measurements. The rheology of a polymer is sensitive to its hydrodynamic volume, which, in turn, is reflected in its molar mass. For a high-molar-mass polymer, the hydrodynamic volume provides an estimate of the size of the polymer, assuming that the polymer behaves as a flexible chain.16,17 For polyelectrolytes, the hydrodynamic volume depends on the medium in which the polymer is dispersed.22 In media of high dielectric constant, polyelectrolytes are often ionized, and the molecules assume extended configurations as a result of electrostatic interactions between charged groups. The total electrostatic repulsion depends on the number of charged groups and the ions present in solution. The former is usually expressed in terms of the degree of ionization, i. The ions present in solution are the counterions created by the dissociation of the polymer and ions that can be added to the solution, usually inorganic or organic salts. The ionic strength, μ, of the solution can then be defined as μ = 1/2∑j(cm)jmj2, where (cm)j is the molarity and mj is the charge on the jth ion. Salts with low molar mass suppress the electrostatic repulsion of charged groups. At sufficiently high concentration, ce, the repulsion is reduced to such an extent that the configuration and the solution properties are similar to those of a nonionic polymer and the behavior of the chains becomes governed by excluded-volume factors. The Mark− Houwink−Kuhn−Sakurada equation can be used to describe the intrinsic viscosity, [η], as [η] = KMα, where K and α are constants that depend on the polymer, temperature, and solvent used. Chu et al.23 and Tsvetkov et al.24 reported studies of the measurement of the intrinsic viscosity of PAA with added salt. The values of K and α were shown to depend on the salt concentration. For PAA in solutions with NaBr at 25 °C, values of K × 103 and α have been reported, as follows: 124 and 0.50 at cNaBr = 1.5 M, 25.4 and 0.75 at cNaBr = 0.1 M, and 13.6 and 0.89 at cNaBr = 0.01 M.25−27 At high ionic strength, the value of α approaches the ideal random coil coefficient of 0.5; however, at these high values, the polymer is approaching conditions where precipitation can occur due to salting out. For this study, we followed the procedure outlined by Tsvetskov et al.,24 in which the PAA was neutralized by the addition of an appropriate quantity of sodium hydroxide and sodium chloride was used as the neutral electrolyte (concentration of 0.012 M). The data obtained are presented in Figure 3. The appropriate values of the coefficients of the Mark−Houwink−Kuhn− Sakurada equation are K = 11.9 × 10−3 mL g−1 and α = 0.8596.23 The high value for α is consistent with the chain being in an extended conformation under these conditions. The intrinsic viscosity values were obtained using Figure 3 as 54.7 ± 0.9 for PAA-[A] and 10.37 ± 0.22 for PAA-[B]. Using these values of the Mark−Houwink−Kuhn−Sakurada coefficient25 gives estimates of 3.86 × 106 g/mol for PAA-[A] and of 5.6 × 105 g/mol for PAA-[B]. As pointed out by Volk et al.,2 the
Table 1. Analysis of the 1H NMR Spectra fraction resonance (ppm)
carbomer A
carbomer B
assignment
1.84 1.64 1.26
0.21 ± 0.04 0.76 ± 0.05 0.03 ± 0.02
0.24 ± 0.04 0.72 ± 0.05 0.04 ± 0.03
anhydride acrylic acid methacrylic acid 16199
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Figure 3. Determination of intrinsic viscosity at 20 °C in deionized water for (a) PAA-[A] and (b) PAA-[B]. Symbols are (+) ln(ηrel)/c and (×) ηsp/ c.
associated with changes in the extent to which PAA formed clusters rather than degrading. 3.2. Rheological Measurements. PAA is used as a thickener at significantly higher concentrations than those used in the intrinsic viscosity measurements. Combining the intrinsic viscosity measurements with more conventional rotational shear observations of the viscosity, it is possible to identify the point at which polymer−polymer interactions can be assumed to start to modify the rheological characteristics of these materials. Solutions were created with concentrations in the ranges 0.2145−0.0134 g/dL for PAA-[A] and 0.4369− 0.0546 g/dL for PAA-[B]. They were neutralized to a pH of 7, and 0.012 M sodium chloride solid was added as an inert electrolyte. The zero-shear viscosities were estimated from creep measurements, by steadily reducing the applied stress and determining the resultant shear rate in the dissipative part of the response curve. The results were extrapolated back to a notional shear rate of 10−6 s−1 (simulating zero shear), which were the final values used (Figure 4). Whereas, for the most dilute solutions, the viscosity was almost independent of shear
effective molar mass can be influenced by association with the electrolyte and the pH. The pH values, measured using a glass electrode, of the original solutions before neutralization were 2.97 for PAA-[A] and 2.94 for PAA-[B]. Titration of the polymers yielded equivalent masses of 79.9 g for PAA-[A] and 81.6 g for PAA-[B]. These values indicate that the polymers were cross-linked through the formation of anhydride linkages or possibly hydrophobically modified by incorporation of methyl acrylic acid or a proportion of pentaerythritol units. The intrinsic viscosity and, hence, hydrodynamic volume of a branched polymer arising from the formation of anhydride cross-links is lower than that of the equivalent linear polymer. As a consequence, the value of Mw obtained above must be considered to represent lower limiting values, and that for PAA-[B] is artificially low. The intrinsic viscosities of solutions produced using the slow dissolution method and the higher-shear approach were compared and were found to be comparable, indicating that viscosity changes occurring during dissolution were probably 16200
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Figure 4. Viscosity/shear rate plots for (a) PAA-[A] and (b) PAA-[B]. Symbols are (a) (×) 0.2415, (○) 0.1608, (+) 0.1072, (□) 0.0804, (Δ) 0.0536, (◊) 0.0268, and (*) 0.0134 g/dL; (b) (×) 0.4369, (○) 0.3276, (+) 0.2184, (□) 0.1638, (Δ) 0.1092, and (◊) 0.0546 g/dL.
stages of takeoff. Deicing fluids use concentrations where polymer−polymer interactions are expected to occur, and hence, any theory developed to describe the shear thinning should incorporate these molecular features. For an ideal polymer chain, increasing the concentration changes the viscosity behavior as the isolated polymer chains start to interact, leading to the range in which reptation becomes an important process. Plotting the relative viscosity against concentration clearly indicates that there was a threshold concentration, ct, at which polymer−polymer interactions started to play a dominant role. For PAA-[A], this occurred at 2.1 × 10−2 g/dL, and for PAA-[B], it occurred at 8.0 × 10−2 g/dL. If the polymers had similar architectures, then the product ct[η] would be expected to be constant. The value for PAA-[A] was 1.167, and that for PAA-[B] was 0.832. For linear nonionic polymers, a value of between 1 and 2 for the product c[η] usually indicates the onset of polymer− polymer interactions. However, if the polymer is more branched, the hydrodynamic volume of the molecules does not scale simply with the molar mass. The differences in the
rate, at higher concentrations, significant shear thinning was observed. Combining this data with the viscosity measurements obtained using the Ubbelohde viscosity measurements produced a plot of the viscosity as a function of concentration (Figure 5). For PAA-[A], in the low concentration range, the viscosities measured from shear rate experiments were slightly higher than those measured from Ubbelohde viscosity experiments; the shear rate was less well-defined in the capillary viscometer, and a lower value would be consistent with shear thinning. For both systems, a clear concentration could be identified at which the rheological behavior changed from that associated with “isolated” polymer chains to one in which polymer−polymer interactions have to be considered. It is also clear from the data presented in Figure 4 that these solutions were able to exhibit marked shear thinning, and in the context of their application as thickeners, this shear rate dependence could be very important in understanding their actions in applications where changes are occurring in the shear stresses, such as an aircraft in the initial 16201
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Figure 5. Plots of the relative viscosity versus concentration for PAA-[A] (Δ,×) and PAA-[B] (+,○). Symbols are (Δ,+) extrapolated 10−6 s−1 shear rate values; (×,○) Ubbelohde-viscometer-measured values.
values of c[η] for PAA-[A] and PAA-[B] are a reflection of the more highly branched structure in PAA-[B] compared with PAA-[A] and support the conclusions from the equivalent mass analysis, as well as the spectroscopic observations. 3.3. Yield Stress. The existence of a yield stress in viscoplastic materials, such as concentrated suspensions, pastes, foams, and composites (i.e., fluids containing particles of colloidal size), has been well documented.28 Whether a polymer solution can exhibit a yield stress has been the subject of much discussion,29 but the conclusion that has emerged is as follows: At very low levels of shear, these fluids can exhibit very high levels of viscosity; then, over a limited range of shear stress, the viscosity can fall dramatically. This has been compared to yield in solids, and so a new concept of a “yield stress” has evolveda “static” one, whereby a finite stress has to be developed for the fluid to flow, and a “dynamic” one, in which the increasing stress in a slowly flowing fluid causes the viscosity to fall rapidly. The viscosity was measured over a range of shear stresses for PAA-[A] and PAA-[B] (Figure 6). It is difficult from the plots to determine precise points at which the viscosity/stress curves change slope; however, there is a clear progression for each polymer system. For PAA-[A], the 0.2415 g/dL curve changes slope sharply at a stress of approximately 1−2 Pa. Dropping the concentration to 0.1608 g/dL reduces the critical point to a value of stress of approximately 0.9−1.5 Pa. Decreasing the concentration further to 0.1072 g/dL further decreases the critical stress to 0.3−0.45 Pa. A further decrease of the polymer concentration to 0.0804 g/dL gives a critical stress level of 0.02−0.03 Pa. At a concentration of 0.0536 g/dL, the critical stress is about 0.015 Pa. The lower concentrations do not show a change of viscosity with increasing stress level. These shear thinning characteristics parallel the observations (Figure 4) of the shear rate behavior and are indicative of complex deformation of the polymers during shear. In the context of deicing fluids, as air flow increases over the aerofoil, the levels of stress will increase, and as a consequence, the ability for the fluid to be removed from the wing will be increased.
For PAA-[B], the points at which changes occur are more clearly defined. At a concentration of 0.4369 g/dL, the critical stress level is 2−3 Pa. Reducing the concentration to 0.3276 g/ dL reduces the break point to a stress of 0.2−0.4 Pa. Further reduction of the concentration to 0.2184 g/dL reduces the break point to a stress of 0.015−0.03 Pa . Lowering the concentration to 0.1638 g/dL reveals a break point at a stress level of 0.008−0.012 Pa. Comparison of the data for the two polymers indicates that there are significant differences between these materials in terms of their ability to form extended network structures that lead to enhanced viscosity. If the polymers had similar architectures, then it would be expected that the variation of the break points would scale as the molar mass. The intrinsic viscosity and titration data indicate that these polymers had different architectures and that PAA-[A] was more highly branched than PAA-[B]. Higher concentrations of PAA-[B] had to be used to achieve the same viscosity, and this is consistent with the concept of a lower molar mass but is inconsistent in terms of the equivalent mass of the polymers. 3.4. Modeling of the Viscosity Curves. By combining shear rate and creep data, one can investigate an extended time range, and a typical curve is shown in Figure 7 for PAA-[A] at a concentration of 0.1608 g/dL. At very low shear rates, the solutions exhibited very high viscosities, but they exhibited dramatic shear thinning as the rate was increased. The dynamic response of a flexible polymer in solution can be described by Rouse theory, 18 in which the relaxation times are dictated by the molar mass and the solvent medium. For nonpolymer electrolytes, in the range where polymer−polymer interactions become significant, reptation has to be included in the description of the motion.14,15 Studies of concentrated polymer solutions and melts of flexible polymers have shown that, in dilute solution and for low strain, stress relaxation occurs in two steps: the relaxation of chain segments between fixed entanglement points and the relaxation of the entanglement points through reptation motion.30,31 However, in the nonlinear region, at large strain, there appears to be an additional relaxation process that lies between the other two processes. 16202
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Figure 6. Viscosity/shear stress plots for (a) PAA-[A] and (b) PAA-[B]. Symbols are (a) (×) 0.2415, (○) 0.1608, (+) 0.1072, (□) 0.0804, (Δ) 0.0536, (◊) 0.0268, and (*) 0.0134 g/dL; (b) (×) 0.4369, (○) 0.3276, (+) 0.2184, (□) 0.1638, (Δ) 0.1092, and (◊) 0.0546 g/dL.
Hudson et al.7 have proposed a model in which a pair of polymer molecules form a ladder-type of structure in which loops are distributed along the ladder in a random manner. The loops are able to execute relaxation behavior that is rather like that of the Doi model31 in which the chains are constrained by the entanglements. In Hudson et al.'s theory,7 the molar mass is assumed to be that of the isolated chain. In the present model, we allow for the interactions to produce an effective molecule that is a multiple of an isolated polymer chain. It is also recognized that the molar mass distribution must be included in the calculation of the Rouse and reptation contributions for realistic modeling of the dynamic characteristics of the polymer. In the approach employed here, it was assumed that the contributions to the overall viscosity are additive and, therefore, the effects of the molar mass distribution can be simulated by the calculation of the contributions from the individual polymer species. 3.4.1. Rouse Theory with Molar Mass Distribution. From the intrinsic viscosity, the molar masses of the polymers as isolated species were determined using the Mark−Howick−
Kuhn−Sakurada equation. The effect of the molar mass distribution was included in the form of the distribution function f (x ) =
⎛ (ln x − μ)2 ⎞ exp⎜ − ⎟ 2σ 2 ⎝ ⎠ x 2πσ 2 1
(1)
where σ and μ are the standard deviation and the mean, respectively, of the distribution variable’s natural logarithm. The variable, x, is the number of repeat units in the chain. The Rouse first normal mode, τ1, can be defined as N
τ1 =
∑i − 1 6η0fi Mi N
∑i − 1 π 2ρRTfi
(2)
where ρ is the solution density, η0 is the zero-shear-rate viscosity of the solution, M is the molar mass, R is the universal gas constant, T is the absolute temperature, and N is the total number of modes within a polymer chain. 16203
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Figure 7. Shear rate dependence at 20 °C of the viscosity for PAA-[A] at 0.1608 g/dL. Symbols are (+) viscosity measured by creep tests; (×) viscosity measured by steady shear flow tests.
Following the approach of Doi and Edwards,15,31 the Rouse contribution to the viscosity describes the relaxation of the polymer chain between entanglements or, in the case of polyelectrolytes, between points of clustering and makes a frequency- and shear-rate-dependent contribution η1(ω) = ηs +
6(η0 − ηs) π
2
N
∑ p=1
units, and Ne is the number of Kuhn units between entanglement points, Ne = Mc/Mk, where Mc is the critical molar mass of entanglement and Mk is the molar mass of a Kuhn unit. This form of the equation refers to the process of retraction of a segment of the chain that has been affinely deformed when the system was subjected to a step shear event. It has previously been shown that this type of solution exhibits elongational flow and that relaxation behavior in this region can be envisaged as chains moving in and out of the dynamic clusters as flow occurs. A number of alternative models of this type of chain motion exist, all leading to equations for τb of similar form.32−35 Typically, the value for Mc is on the order of (20−30) × 103 and depends on the polymer type.36 In Rouse theory, it is assumed that the friction coefficient is equal to the solution viscosity. However, in the case of a polyelectrolyte, this might not necessarily be correct, as electrostatic interactions that expand the chain also have a significant effect on the local chain mobility; it is therefore appropriate to make ηs an adjustable variable. In a polyelectrolyte, the Kuhn length can be significantly increased as a consequence of electrostatic repulsion opening the chain dimensions. The viscosity then has the form
p2 4
P + ω 2τr 2
(3)
where ηs is the solvent viscosity, p is the mode number, ω is the frequency, and τr is the Rouse relaxation time. 3.4.2. Reptation and Entanglement Clustering. The reptation model is based on the concept that, for high-molarmass polymers, their motion is constrained by entanglements and these constraint points form a “tube”. The average number of entanglements stays constant, although the polymer chain moves so as to break and make new entanglements. The formation of the entanglements leads to a viscosity that exhibits a rapid increase with molar mass, similar to that observed in Figure 5. Entanglement depends on the hydrodynamic volume of the polymer and rises to approximately the third power of the size. The relaxation contribution to the viscosity has the form η1(ω) = ηs + +
6(η0 − ηs) π
2
8(ηe − η0) π2
N
∑ p=1
Ne
∑ p=1
p 4
p + p
η1(ω) = ηs +
2
ω 2τr 2 +
2
p4 + ω 2τd 2
∑
where Ne is the number of segments between entanglement points and τd is the terminal relaxation time, given by τd = (1/ π2)(ζa2Ne/kT)N3, where a is the tube diameter (a2 = Neb2), b is the monomer length (in this case, the Kuhn length), ζ is the bead friction coefficient of Rouse motion, and k is the Boltzmann constant. Although this equation describes the behavior of a polymer melt, it is not necessarily appropriate for a system with nonlinear characteristics.31 Doi has shown that, for concentrated solutions, an additional contribution associated with contour length relaxation must be added; it has the form τb = ζb2N2Ne2/3π2kT. In this case, N is the total number of Kuhn
π2
1 8(ηe − η0) 2 π2
Ne
(4)
6(η0 − ηs)
p=1
N
∑ p=1
p2 p4 + ω 2τr 2
Ne
∑ p=1
p2 4
2
p + ω τb
2
+
1 8(ηe − η0) 2 π2
p2 p4 + ω 2τd 2
(5)
This theory, however, assumes that the polymeric entity undergoing relaxation is a single polymer chain and does not allow for the dynamic (hydrogen-bonding) interactions that are found in polyelectrolytes. 3.4.3. Complexation. The problem of dynamic complexation was considered by Hudson et al.,7 who assumed that a fraction g (
