Tracer Microrheology Study of a Hydrophobically Modified Comblike

Mar 16, 2015 - The viscoelastic properties of associative polymers are important not only for their use as rheology modifiers but also to understand t...
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Tracer Microrheology Study of a Hydrophobically Modified Comblike Associative Polymer Ahmed A. Abdala,*,† Samiul Amin,‡ John H. van Zanten,§ and Saad A. Khan∥ †

Qatar Environmental and Energy Research Institute (QEERI), Qatar Foundation, Doha, Qatar Malvern Instruments, Columbia, Maryland 21046, United States § Biomanufacturing Training and Education Center, North Carolina State University, Raleigh, North Carolina 27695-7928, United States ∥ Department of Chemical and Biomolecular Engineering, North Carolina State University, Raleigh, North Carolina 27695-7905, United States ‡

ABSTRACT: The viscoelastic properties of associative polymers are important not only for their use as rheology modifiers but also to understand their complex structure in aqueous media. In this study, the dynamics of comblike hydrophobically modified alkali swellable associative (HASE) polymers are probed using diffusing wave spectroscopy (DWS) based tracer microrheology. DWS-based tracer microrheology accurately probes the dynamics of HASE polymers, and the extracted microrheological moduli versus frequency profile obtained from this technique closely matches that obtained from rotational rheometry measurements. Quantitatively, however, the moduli extracted from DWS-based tracer microrheology measurements are slightly higher than those obtained using rotational rheometry. The creep compliance, elastic modulus, and relaxation time concentration scaling behavior exhibits a power-law dependence. The length scale associated with the elastic to glassy behavior change is obtained from the time-dependent diffusion coefficient. The Zimm−Rouse type scaling is recovered at high frequencies but shows a concentration effect switching from Zimm to more Rouse-like behavior at higher concentrations.

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INTRODUCTION Hydrophobically modified alkali soluble emulsion (HASE) polymers are water-soluble associative polymers with a comblike structure composed of pendant hydrophobic groups randomly grafted to a polyelectrolyte backbone. Because of their advantages over other associative polymers in terms of cost and wide formulation latitude,1 they are used as rheology modifiers in a wide range of applications, including paint formulations,2,3 paper coatings,4 personal and home care products,5 UV-photoprotective and aerated emulsions,6 fabric softeners,7 and glycol-based aircraft anti-icing fluids.8 Similar to surfactants, HASE polymers are capable of nonspecific inter- and intramolecular hydrophobic associations forming transient network structures and exhibiting a range of rheological behavior, which depends on the structure and flexibility of the backbone, the structure and concentration of the macromonomer, and the polymer concentration. The effects of the polymer architecture on microstructure and solution rheological behavior have been the subject of previous studies.9−13 The polymer concentration plays a critical role in the nature of the network structure and the rheology of HASE polymers. At low polymer concentration (0.5−1 wt %), the intermolecular © 2015 American Chemical Society

hydrophobic association dominates and dictates the solution rheology. At intermediate concentrations, both intramolecular and intermolecular associations exist and contribute to the solution rheology.9 The dynamics of HASE polymers, and associative polymers in general, is usually characterized using rotational rheometry. However, rotational rheometry measurements are associated with a number of limitations: First, rotational rheometry cannot access high frequency data, which are important not only to probe short time/length scale behavior but also to simulate processing and application conditions. Second, dynamic measurements require application of oscillatory strain low enough to remain in the viscoelastic region but high enough to yield a measurable torque. This applied strain may affect the microstructure resulting in an inability to probe the intrinsic microstructure. Finally, rotational rheology measurements require several milliliters of sample which may not be available for very expensive, hazardous, and/or small-scale samples. Therefore, an alternative, complementary technique for measuring polymer solution dynamics is desirable. Although conventional light scattering in principle can probe polymer solution dynamics, its application is limited to sufficiently low Received: December 18, 2014 Revised: March 12, 2015 Published: March 16, 2015 3944

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polymer solution dynamics are characterized at frequencies not accessible by traditional mechanical rheometry.

concentration to ensure that the single scattering limit is being considered in order to meaningfully analyze the scattering data. To overcome this limitation, a few recent techniques have been developed such as low coherence dynamic light scattering (LCDLS),14 noninvasive backscatter (NIBS),15 3D dynamic light scattering (3DDLS),16 and diffusing wave spectroscopy (DWS).17 All these techniques allow probing of dynamics in concentrated and turbid solutions and also allow tracer microrheology to be carried out.18 The DWS technique however has a unique advantage as it allows the probing of very short time dynamics or high frequency viscoelastic response. This very short time dynamics cannot be probed either in traditional DLS based microrheology or in the newer DLS techniques mentioned. High frequency rheology of complex fluids does not only contain information about the fluid relaxation mechanisms but also provides insight into their microstructure, interactions, and local dynamics.19,20 In addition to optical techniques, few recent mechanical rheometry techniques such as torsional resonators21 and piezo rotary vibrator (PRV)22 are also capable of probing the high frequency dynamics. Willenbacher and Oelschlaeger provide a good and concise review of these techniques.19 DWS was introduced by Pine et al. to study dynamic processes in turbid media.17 Mason and Weitz introduced the DWS-based tracer microrheology method a few years later.23 Since then, DWS-based tracer microrheology has been used to study collective dynamic properties for a variety of systems, including polymer solutions and gels,17,18,23−25 associative polymers,26,27 surfactants,28,29 solution of actin filaments,30−35 colloidal suspensions and gels,36,37 concentrated emulsions,38−40 foams,41,42 magnetorheological suspensions,43 and nematic liquid crystals.44 Recently, particle tracking microrheology has been extended to monitoring tissue and living cells,45−48 online monitoring of polymerization reaction,49 nanoparticle imaging,50 and measurement of glass transition in colloids.51 Compared to rotational rheology measurements, DWS has several advantages. First, dynamic properties are measured by applying an extremely small strain which does not affect the system structure. Second, it requires very small sample volumes, approximately hundreds of microliters. Third, while rotational rheometry measurements cannot probe the very short time behavior, DWS can probe both the long and short time behavior by using cells of variable width, thereby providing dynamic data over a very wide frequency range, up to megahertz range, which is not accessible by rotational mechanical rheometry. Finally, and most importantly, owing to its multiple scattering nature, DWS is capable of resolving angstrom-scale particle motions and therefore very short time (high frequency) dynamics. The use of DWS to probe polymer solution dynamics requires the addition of optical probes to render the medium highly scattering; polystyrene spheres are usually used as the optical probes. This resembles well a practical application of HASE associative polymers in paints and coatings where the associative polymer exits in conjunction with colloidal particles. In this study, DWS-based tracer microrheology is utilized to study the dynamics of associative polymers in aqueous media by tracking the Brownian motion of polystyrene spheres embedded in the polymer solution. The microrheological data obtained using DWS are compared to those obtained using conventional rotational rheometry. In addition, the HASE

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EXPERIMENTAL SECTION

The associative polymer considered in this study is a hydrophobically modified alkali-soluble (HASE) polymer synthesized by Dow Chemicals, via emulsion polymerization of methacrylic acid (MAA), ethyl acrylate (EA), and a hydrophobic macromonomer. Figure 1

Figure 1. Schematic representation of the architecture of a typical HASE polymer and its molecular structure. Here, p = 40 and R ≡ C22H44; x/y/z = 43.57/56.21/0.22 by mole. shows a schematic representation of the polymer structure composed of a random copolymer backbone of methacrylic acid and ethyl acrylate and pendant hydrophobic macromonomers. The macromonomers are end-capped with C22H45 alkyl hydrophobes that are separated from the polymer backbone by 40 units of poly(ethylene oxide) (PEO). Preparation method details can be found in a previous publication.52 Prior to use, polymer lattices were dialyzed against deionized water using cellulosic membranes for at least 3 weeks with daily change of water and then freeze-dried. A 5.0 wt % stock solution was prepared and pH adjusted to 9.0 ± 0.1 using 1 N NaOH with an ionic strength of 10−4 M KCl. Solutions containing 1.0, 2.0, and 3.0 wt % polymer were prepared from the stock solution by dilution. The dynamic properties of the HASE polymer solutions were measured using a stress-controlled rheometer (TA AR2000, TA Instruments) equipped with a 40 mm cone and plate with 0.04 rad cone angle. Both creep and frequency sweep experiments were carried out using a strain within the linear viscoelastic (LVE) regime. A homemade transmission-mode DWS setup was utilized, a schematic of which is shown in Figure 2. In this setup, a beam from a diode pumped solid state (DPSS) Nd:YAG laser (wavelength of 532 nm in vacuo) was incident upon a flat cell that contained the polymer solution with embedded polystyrene spheres as optical probes. The multiply scattered light was collected using an ALV SI/SIPD photon detector via a single mode optical fiber. In order to ensure point-topoint geometry, the single mode optical fiber has an attached gradient refractive index (GRIN) lens with a very narrow angle of acceptance. The output from the ALV SI/SIPD photon detector was fed into a cross-correlator. The measured intensity autocorrelation function was converted into an electric field autocorrelation function using the Siegert relationship. Mathematical Treatment of DWS Data. The electric field autocorrelation function g1(t) obtained from DWS measurements is related to the mean-square displacement of the probe particles, ⟨Δr2(t)⟩, through53

g1(t ) =

∫0

∞

⎡ 1 s⎤ P(s) exp⎢− k 0 2⟨Δr 2(t )⟩ ⎥ ds ⎣ 3 l* ⎦

(1)

where P(s) is the path length distribution function, k0 is the wave vector, ⟨Δr2(t)⟩ is the particle mean-squared displacement, and l* is 3945

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Figure 3. Evolution of the mean-square displacement, ⟨Δr2(t)⟩, of 0.996 μm PS spheres in HASE polymer solution of varying concentrations.

Figure 2. Transmission mode DWS setup. The beam is focused and incident upon a flat cell containing the HASE polymer solution and embedded spherical optical probes. The light is multiply scattered and collected by two cross-correlated PMTs.

temporal evolution of ⟨Δr2(t)⟩ for different polymer concentrations (Figure 3), the PS sphere displacement becomes slower with increasing polymer concentration. This behavior is expected as the medium viscosity and viscoelastic properties are enhanced by increasing polymer concentration. Moreover, the observed ⟨Δr2(t)⟩ temporal behavior exhibits initially subdiffusive behavior at short times followed by the onset of a plateau at intermediate times and finally exhibits subdiffusive behavior at the longest times considered here. The observed temporal behavior is independent of polymer concentration within the investigated concentration range, implying that the HASE polymer solution dynamics influencing the probe motion are of the same origin. A similar ⟨Δr2(t)⟩− time profile has also been observed for (PEO−PPO−PEO) triblock copolymer (Pluronic F108) micellar soft crystals.57 However, there are two differences between the current systems and the pluronic soft crystals. First, the plateau onset and the onset of the final subdiffusive region shift to shorter time as the HASE polymer solution concentration increases. Second, the short time behavior of the probe particles’ meansquared displacement becomes increasingly more subdiffusive with increasing polymer concentration. These differences are attributed to relaxation time changes with increasing polymer concentration because the higher concentration solutions exhibit slower relaxation processes owing to viscoelastic network enhancement. The ⟨Δr2(t)⟩−time profile can be explained as follows: at short times, the spheres diffuse in the viscoelastic media (initial subdiffusive region) before they are entrapped by the polymer network (the plateau region) and finally escape after network relaxation (final subdiffusive region). It should be noted that although the ⟨Δr2(t)⟩−temporal behavior observed here is similar to that previously observed for Pluronic copolymer soft crystals, the relaxation mechanisms for the two systems are very different. While the Pluronic copolymer molecular crystals relax via micellar rearrangements,48 HASE polymer solutions relax by disengagement of associated hydrophobes. The short time scaling behavior is interesting to note. It becomes increasingly subdiffusive with increasing polymer concentration, with scaling exponents decreasing from 0.54 to 0.31. This behavior is different than what was observed by Oppong and de Bruyn for hydrophobically modified hydroxyethyl cellulose solutions.18 In that system, the behavior is almost diffusive at short times for the lowest concentration

the distance over which light becomes completely randomized. ⟨Δr2(t)⟩ of the spherical probes was extracted pointwise from the electric field autocorrelation function through a bisection rootsearching program. It is worth noting that in contrast to dynamic light scattering (DLS), in which the length scale over which particle motion is probed can be adjusted by varying the scattering angle and/ or the wave vector, the length scale over which the motion is probed in DWS is adjusted by varying the cell thickness L. In this study, cells with 2 mm and 10 mm thickness are used. The viscoelastic properties of the medium are extracted from the ⟨Δr2(t)⟩ data through either direct conversion of the complex viscoelastic modulus (G*) or conversion of the creep compliance (J(t)). G* is obtained from tracer microrheology experiments through application of the generalized Stokes−Einstein relation:38 G̃(s) ≈

kBT 6πas⟨Δr 2̃ (s)⟩

(2)

This expression requires transformation from time domain to frequency domain. Therefore, conversion of the ⟨Δr2(t)⟩ to creep compliance is a more direct method, which yields better results as J(t) is directly proportional to ⟨Δr2(t)⟩:34,54 πa J(t ) = ⟨Δr 2(t )⟩ kBT (3) The frequency-dependent storage (G′) and loss moduli (G″) can be obtained from J(t) using various methods.55 However, it is difficult to implement some of these methods, in particular the direct transformation methods, due to the very large temporal dynamic range and logarithmic spacing of the measured ⟨Δr2(t)⟩. Moreover, the accuracy of these methods can be reduced by the noise in ⟨Δr2(t)⟩ and the rapid changes in the logarithmic slope of ⟨Δr2(t)⟩.54 To avoid these potential problems, G′ and G″ are calculated using the retardation spectrum L(τ) by a regularized fit of the creep compliance using a set of impartial basis function as described by Mason et al.54

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RESULTS AND DISCUSSION The microstructure, and thus the dynamics, of HASE polymer solutions are highly dependent on polymer concentration as the observed behavior goes through many transitions within a limited concentration range.56 In this regard, the effect of polymer concentration on the HASE polymer solution dynamics was investigated with DWS-based tracer microrheology. Figure 3 shows the temporal behavior of the ⟨Δr2(t)⟩ of 0.966 mm PS sphere embedded in HASE polymer aqueous solutions of varying concentration. As indicated by the 3946

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dynamics, the creep compliance was converted to the corresponding dynamic moduli. Figure 5 shows G′ and G″ as functions of angular frequency (ω) for HASE polymer solutions of varying concentration. As seen in Figure 5, and as expected from the J(t) comparison, there is excellent qualitative agreement between the moduli extracted from the DWS-based tracer microrheology and mechanical rheometry, regardless of the polymer concentration. Validity of the DWS Data. Figures 4 and 5 considered together indicate that although there is good qualitative agreement between DWS-based tracer microrheology and mechanical rheometry, the quantitative agreement is somewhat lacking. This variation between DWS-based tracer microrheology and mechanical rheometry could arise from several factors, including the range of validity of the generalized Stokes−Einstein relation (GSER), the presence of structural inhomogeneity, and possible interactions between the probe particles and the HASE polymer, which may include absorption or depletion effects. The range of validity of the GSER has been the focus of several publications.59−62 Levine and Lubensky51 found that there is a large frequency range over which the GSER is valid in many systems. The upper frequency range is bounded by the inertial effects that typically become significant at frequencies higher than 1 MHz and thus may be safely ignored. On the other hand, the low frequency range is bounded by a characteristic frequency, ωc, under which the effective decoupling of the network and fluid dynamics becomes significant. An order of magnitude estimate of ωc can be determined from the relation59

samples exhibiting a value of about 1 and then decreasing only to a value of 0.8 at 0.5 wt %. This behavior could be attributed to both the lower concentration range investigated in that study and also limitations in the high frequency data range utilized for scaling due to having used DLS rather than DWS. The other aspect which could be attributed to the strong concentration dependence of the short time behavior our HASE system exhibits is the strength of the intermolecular interactions and the strong increase of these with concentration. In the hydroxyethyl cellulose study, the long-time scaling is also closer to diffusive having many of the investigated concentrations exhibiting a value close to 0.8. This sort of behavior is usually reported in the case of slightly interconnected polymer networks.58 The measurements presented here did not indicate the presence of purely diffusive behavior at long times. Figure 4 compares DWS-based tracer microrheology creep compliance and mechanical rheometry data. Excellent qual-

ωc = Figure 4. Creep compliance, J(t), obtained from mechanical rheometry (symbols) and tracer microrheology (lines) for different concentrations of HASE polymer at 25 °C.

⎛ G ⎞⎛ ξ ⎞ 2 Gξ 2 = ⎜ ⎟⎜ ⎟ ⎝ η ⎠⎝ R ⎠ ηsR2

(4)

where G is the shear modulus that can be taken as G*(ω∼0), ηs is solvent viscosity, ξ is the network mesh size, and R is the radius of the probing particles. The mesh size of the polymer network is of the same order as the persistence length and can be estimated as63

itative and good quantitative agreement between DWS-based tracer microrheology and mechanical rheometry is observed at all concentrations. In addition, the presented data reveal a major advantage of combining microrheology and standard rheometry measurements; i.e., data over a very wide range of time scale (8 decades) are obtained. To further examine the ability of DWS-based tracer microrheology to correctly probe the HASE polymer solution

⎛ k T ⎞1/3 ξ=⎜ B ⎟ ⎝ G0 ⎠

(5)

where kB is the Boltzmann constant, T is the temperature, and G0 is the plateau modulus. Therefore, for the HASE polymer

Figure 5. Elastic modulus (G′) (a) and viscous modulus (G″) obtained from mechanical rheometry (symbols) and DWS-based tracer microrheology (lines) for different concentrations of HASE polymer. 3947

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Langmuir solutions considered here, G0 ranges between 25 and 400 Pa and ξ changes from 55 nm at 1 wt % to 22 nm for at 5 wt %. To approximate the lower frequency limit, the G/η ratio is assumed to be on the order of 100−1000 s−1, G*(ω=0.001) ∼ 0.1−1.0 Pa, and ηs = 1 mPa·s. A sphere radius that is 20−50 times larger than the mesh size yields an estimate for ωc of 0.3− 0.5 Hz (0.03−0.08 rad/s). Therefore, the majority of the DWSbased microrheology data are valid with only a very small fraction falling below this predicted lower frequency limit. Nevertheless, the discrepancy between DWS-based microrheology and shear rheometry data exists for frequencies larger than the lower critical frequency limit, which cannot be fully attributed to the GSER validity range. Another factor that may be responsible for the deviation between the DWS-based microrheology and mechanical rheology measurements is the presence of local inhomogeneity in the HASE polymer solutions. This can be examined by comparing the measured rheological properties probe size dependence. In addition, changing the sphere size can provide information on the frequency range over which the GSER is valid. Figure 6 shows the creep compliance obtained by DWSbased microrheology measurements of a 0.9 wt % HASE

polymer solution using different size PS spheres. The data extracted using intermediate size spheres (0.511−0.966 μm) collapse perfectly onto a single curve in acceptable agreement with the shear rheology data, thereby illustrating both the validity of the GSER and the absence of local inhomogeneity effects at these length scales. However, the data extracted using the smallest and the largest spheres show significant deviation. The deviation observed for the largest spheres could be caused by the interparticle interactions or aggregation. Aggregation would result in an underestimate of the probe size leading to a concurrent underestimation of the creep compliance as shown in eq 3. On the other hand, for the case of the smallest sphere, d = 0.195 μm, their size is close to the mesh size or correlation length of the HASE polymer solutions. As such, these smaller spheres may not be fully entrapped by the HASE polymer solution microstructure. In general, the deviation between the data measured using DWS-based tracer microrheology and mechanical rheometry could be attributed to the perturbation of the polymer matrix by the probe spheres or to measurement errors. For example, DWS requires independent measurement of l* (the distance at which light becomes completely randomized); any error in the value of l* will affect the calculated ⟨Δr2(t)⟩. On the basis of this discussion, we can summarize that the GSER is valid when spheres of suitable sizes are used, the microstructure of the polymer solution is insensitive to the probe size, and the decoupling between the solvent and the network does not play a significant role on the DWS-based tracer microrheology measurements. It is most likely the discrepancy arises from differences between the micro- and macrorheological properties. Time-Dependent Diffusion Coefficient. The timedependent diffusion coefficient, D(t), can be extracted from the mean-square displacement:

D(t ) =

⟨Δr 2(t )⟩ 6t

(6)

Figure 7a shows the evolution of D(t) for different HASE polymer solution concentrations. As expected from the observed mean-squared displacement behavior of the probe particles, regardless of the HASE polymer solution concentration, the diffusion coefficient decreases continuously with time, and the motion of the probe spheres cannot be described by a single diffusion coefficient. Higher polymer concentrations

Figure 6. Creep compliance, J(t), obtained from DWS-based tracer microrheology using different sphere sizes embedded in 0.9% polymer solution. The line represents the creep compliance obtained from mechanical rheometry measurement.

Figure 7. Time-dependent diffusion coefficient, D(t), of 0.966 μm spheres embedded in HASE polymer solutions of varying concentration as a function of (a) time and (b) the average sphere displacement. 3948

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Langmuir result in a lower diffusion coefficient due to the increase in the viscoelasticity of the physical network. At very short times and small length scales, the spheres are subjected to small amplitude, high frequency lateral fluctuations of the polymer network.48 In this limit, the medium behavior is dominated by the glassy nature of the polymer network, and the diffusion coefficient approaches that of the spheres in pure water (D0), 0.46 μm2/s. At long times or long length scales, the medium behavior is dominated by its elastic nature, and the probe spheres become elastically trapped by the network structures. Therefore, a very low diffusion coefficient is expected at long times and higher concentrations. The very small diffusion coefficient, up to 5 orders of magnitude lower than the diffusion coefficient of the 0.966 μm spheres in water (D0), corresponds to near arrest of the sphere by the network. Moreover, from the values of the diffusion coefficient presented in Figure 6, one would expect the sphere size to be much larger than the network mesh size. Hence, if the sphere size were much smaller than the network mesh, a diffusion coefficient similar to D0 would be expected. Information about the HASE polymer solution dynamics at different length scales can be extracted from the behavior of the diffusion coefficient as a function of the sphere displacement, as shown in Figure 7b. Three distinct dynamical regions are observed for all concentrations considered. In the first region at short displacements, the diffusion coefficient shows a nearplateau behavior. This region is followed by a sharp downward transition, and finally a near plateau region is established at longer displacements. The length scale for the sharp transition onset, which is related to the relaxation time, decreases from 10 to 2.5 nm as the concentration is increased from 1.0 to 5.0 wt %. Although the relaxation time increases with polymer concentration, the average displacement at the corresponding relaxation time decreases as concentration increases due to the concomitant increase in solution viscosity and viscoelastic moduli. Similar transition behavior as observed in Figure 7b has been reported for actin filament networks.34 The dynamics in the initial near-plateau region corresponds to a behavior dominated by the hydrodynamic interactions. As the spheres approach the polymer network, they experience the dynamics of the elastic medium, and they finally become entrapped in the elastic cage formed by the viscoelastic network. Scaling Behavior. The concentration dependence of material functions often reveals valuable information about a fluid’s complex microstructure. In particular, the rheology of associative polymer solutions is strongly dependent on their concentration as they can undergo several structural transitions in a small concentration range.56 At low concentrations, intramolecular associations are the dominant hydrophobic interaction mode; as the concentration increases, some of the intramolecular associations are converted to intermolecular associations. At sufficiently high concentrations, the intermolecular associations are the dominant mode of interaction. In this regard, the high frequency G′ and J(t) are examined as a function of HASE polymer solution concentration in Figure 8. Here G′ is considered at a fixed frequency of 10 rad/s; this is bounded by the highest frequency accessible by the mechanical rheometry measurements for the lowest concentration. J(t) was taken at t = 50 s, bounded by longest time accessible by DWS-based tracer microrheology. Both G′ and J(t) exhibit power-law concentration dependence within the entire studied concentration range. As observed from Figure 8, G′ and J(t) scale as G′ ∝ c1.7 and J(t) ∝ c−1.8. It is

Figure 8. Scaling of elastic modulus (G′), creep compliance (J(t)), and longest relaxation time (τL) with concentration. G′ is taken at a fixed frequency of 10 rad/s and J(t) at a fixed time of 50 s.

worth mentioning that while the concentration scaling of the G′ at various frequencies yields the same scaling exponent, J(t) concentration scaling at different times yields scaling exponents that vary from −1.4 at short times to −1.9 at long times. The observed G′ concentration dependence is somewhat weaker than the theoretical prediction of the sticky reptation model, which predicts a scaling exponent of 2.2 for good solvent increasing slightly to 2.3 in θ-solvent.47 Nevertheless, different exponents were also reported for similar associative polymers. English et al. reported G′ scaling exponents of 1.4, 2.8, and 6.5 for HASE polymers with C8, C16, and C20 hydrophobes, respectively.64 In fact, we previously obtained a G′ concentration scaling exponent of 1.8 for several HASE polymers in full agreement with the 1.7 obtained here.65 The J(t) concentration scaling exponent of 1.8 is similar to that observed by Ng et al. for hydrophobically modified ethoxylate urethane (HEUR) polymers.66 Figure 8 also shows the concentration scaling of the longest relaxation time (τlong), defined as the reciprocal of the crossover frequency (G′ = G″). τlong values in the plot are the average of τlong obtained from DWS-based tracer microrheology and mechanical rheometry, and the error bars represent the standard deviation. The very small error bars demonstrate the ability of DWS-based microrheology to accurately interrogate HASE polymer solution dynamics. The scaling shows that τlong ∝ c0.80, in excellent agreement with the sticky-reptation theory prediction, τ long ∝ c5/6 .56 The observed τlong and G′ concentration scaling suggests a scaling exponent of 2.6 for the steady shear viscosity (η), based on the transient network theory prediction, η = G′τlong.62 This exponent is in full agreement with that obtained in our laboratory for HASE polymers with different backbone composition and hydrophobic contents.65 High frequency rheology provides information about the internal dynamics of the short polymer segments. The Rouse model predicts power-law dependence of the high frequency G′ and G″ − ηsω on frequency with exponent of 0.5. This model is good for accurately predicting short time dynamics in polymer melts or concentrated systems, where hydrodynamics do not play a very dominant role. Hydrodynamic interactions are taken into account in the Zimm model. The Zimm model predicts a power law dependence of 2/3 and crossover to G″ − ηsω = 0 (i.e., G″ ∝ ω) at frequencies higher than the reciprocal of the shortest relaxation time.67 Our high frequency data (ω = 104− 105) show power-law dependence of G′, G″, and G″ − ηsω with 3949

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exponent that varies slightly with polymer concentration as shown in Table 1. For low concentration, 1 wt %, we recover

power-law exponent, n G′

G″

G″ − ηsω

1.0 2.0 3.0 5.0 avg

0.44 0.40 0.48 0.34 0.42 ± 0.06

0.66 0.59 0.49 0.38 0.53 ± 0.06

0.55 0.54 0.46 0.38 0.48 ± 0.08

the 2/3 or 0.66 exponent consistent with the Zimm model. However, we do not observe the crossover to a constant dynamic viscosity (η′ = G″/ω = ηs), suggesting the polymer network is still able to relax fast enough to contribute to the solution viscosity at frequencies as high as 105 rad/s. At higher concentrations, we see a behavior which is closer to the Rouse model exhibiting values which are close to 0.5 (e.g., at 3 wt %). Moreover, the internal bending mode scaling of 0.75 was not observed in any of the concentrations.

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CONCLUSIONS

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AUTHOR INFORMATION

REFERENCES

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Table 1. Exponent of the Power-Law Scaling of High Frequency G′, G″, and G″ − ηsω versus ω conc (wt %)

Article

DWS-based tracer microrheology was utilized to characterize the HASE polymer solution dynamics as a function of polymer concentration. DWS-based tracer microrheology provided information on the HASE polymer solution dynamics over a very wide frequency range, including high frequencies that are not accessible by mechanical rheometry. To assess the quality of the DWS-based tracer microrheology data, creep compliance and dynamic moduli data obtained from DWS-based microrheology were compared to those obtained using small strain rotational rheometry. Regardless of the excellent qualitative agreement between the two techniques, there exist quantitative discrepancies that cannot be explained by the validity range of DWS-based tracer microrheology. These discrepancies may result from differences in the system dynamics at the macroscopic and the micro/nanoscopic level. Finally, the creep compliance, high frequency elastic modulus, and relaxation time concentration dependence determined from DWS-based tracer microrheology and mechanical rheometry measurements reveal concentration scaling behavior with exponents consistent with reported theoretical predictions and experimental data. The high frequency scaling and short time dynamics information obtained provide interesting new insights into the effect of concentration on Rouse−Zimm type behavior and short time subdiffusive type response. These results taken together suggest that DWS-based tracer microrheology is a viable technique to probe the associative polymer solution dynamics over a very wide range of time scales.

Corresponding Author

*(A.A.) E-mail: [email protected]. Notes

The authors declare no competing financial interest. 3950

DOI: 10.1021/la504904n Langmuir 2015, 31, 3944−3951

Article

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