Shear-Dependent Interactions in Hydrophobically Modified Ethylene

Article pubs.acs.org/Macromolecules

Shear-Dependent Interactions in Hydrophobically Modified Ethylene Oxide Urethane (HEUR) Based Rheology Modifier−Latex Suspensions: Part 1. Molecular Microstructure Tirtha Chatterjee,*,† Alan I. Nakatani,∥ and Antony K. Van Dyk§ †

Analytical Sciences, The Dow Chemical Company, Midland, Michigan 48667, United States Analytical Sciences, The Dow Chemical Company, Collegeville, Pennsylvania 19426, United States § Dow Coatings Materials, The Dow Chemical Company, Spring House, Pennsylvania 19477, United States ∥

S Supporting Information *

ABSTRACT: We have studied the microstructure of latex suspensions formulated with hydrophobically modified ethylene (oxide) urethane (HEUR) thickener (or rheology modifier, RM) using small-angle neutron scattering under shear (rheo-SANS). Within the shear rate range studied (0− 1000 s−1), the neutron scattering profiles are consistent with a polydisperse core−shell model, with the latex particles comprising the core and an adsorbed layer of water-swollen RM on the latex surface forming the shell. The core−shell structure is isotropic under quiescent conditions but becomes anisotropic under shear (with the major axis along the vorticity direction). During shear, the solvent (D2O/H2O) is expelled (hydrodynamic squeezing) from the swollen polymer chains, and the shell structure becomes denser. The anisotropic shell is a result of differing degrees of compression along the flow and vorticity directions. With increasing shear rate, the shell thickness (in both the flow and vorticity direction) tends toward asymptotic values (with the shell thickness in the vorticity direction greater than the shell thickness in the flow direction) independent of the RM hydrophobe density (defined as the average number of hydrophobes per polymer chain). The RM concentration (w/w) in the adsorbed layer varies from ∼0.05−0.1 (at low shear) to ∼0.25−0.4 (high shear, ∼1000 s−1) with higher values for the RM polymer with higher hydrophobe density. The swollen RM-water shell substantially increases the effective volume fraction of the dispersed latex particles. We find, however, that accounting for this increase within the conventional effective hard-sphere (Krieger−Dougherty) dispersion rheology model does not fully explain the higher viscosity of the formulated mixture. We hypothesize the existence of latex−latex interactions mediated by RM polymer bridges even at high shear. The large-scale structure of the particle assembly will be reported in a subsequent manuscript.

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INTRODUCTION Hydrophobically modified ethylene oxide urethane (HEUR) thickeners are widely used as rheology modifiers in the coatings industry.1 These nonionic associative thickeners consist of a water-soluble poly(ethylene oxide) (PEO) backbone with various combinations of internal, terminal, and in some specific cases, pendant hydrophobes.2 PEO and other components are linked together with urethane (carbamate) groups based on isocyanate chemistry;3,4 acetal−polyether or ketal−polyether groups with base catalyzed gem dihalide chemistry;5 basecatalyzed epichlorohydrin or xylene dichloride chemistry;6 or aminoplast-ether (polyuril ether) chemistry.7 Generally, the terminal hydrophobes are straight chain alkyl groups that typically vary in size from C10 to C24 producing nonspecific hydrophobic interactions, similar to surfactants. The intermediate diurethane linkers are generally smaller, where they behave as weak hydrophobes, but may be larger and include additional hydrophobic groups to produce stronger hydrophobic interactions. Structurally, these molecules are similar to triblock copolymers where the hydrophilic middle block is © 2014 American Chemical Society

much larger than the hydrophobic end blocks. In most cases, the structural and rheological behavior of HEUR polymers in solution are a cross between surfactant solutions and typical triblock copolymer solutions.8 In aqueous solutions, HEUR polymers, above a threshold aggregation concentration, form flower-like micelles, where the hydrophobic groups assemble in a core.9 Further, those flowerlike-structures are dynamic in nature and are bridged through individual HEUR molecules to form a transient network. The steady shear behavior of this system is complex and has been studied extensively, both experimentally10 and theoretically.11 At low shear rate, the aqueous solution rheology of HEUR polymers is Newtonian followed by a shear-thickening regime, then a shear thinning regime at moderate and high shear rates, respectively. The shear thickening phenomenon was attributed to incomplete relaxation of a dissociating chain. On the basis of Received: July 25, 2013 Revised: January 10, 2014 Published: January 27, 2014 1155

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this, Marrucci and co-workers11d developed a nonlinear model, which considered both the bridging chain and dangling chain contributions to the viscosity. Later, Tripathi and co-workers11c reported a nonlinear two-species network model which quantitatively captured the nonlinear rheology of a HEUR aqueous solution. Recently, Watanabe and co-workers proposed two concentration dependent regimes for HEUR solution behavior.12 At low concentration, a sparsely connected network is formed where rheological properties are controlled by the spatial correlation of the micellar cores and the network functionality. This network exhibits a single Maxwellian shear relaxation. In contrast, at high concentration, a dense network is formed where multiple Maxwellian type relaxations are observed. At low shear rate, hydrophobe pullout from micelles does not occur at a significant rate. At intermediate shear, multiple dissociations of superbridges and their reassociation leads to a structural anisotropy responsible for shear thickening at intermediate shear rates. At high shear rates, the nonlinear shear thinning is observed due to flow-induced disruption of the network connectivity through conversion of superbridges into superloops, both having the same core structure. In addition, some shear-induced orientation of the fragmented network takes place under these conditions. By comparison, under oscillatory shear, these systems exhibit a Newtonian plateau at low frequency, followed by a shear-thinning region and subsequently a second Newtonian plateau at high frequency.13 A simple Maxwell model is capable of explaining this behavior with a single relaxation time corresponding to the hydrophobe relaxation time from the flower-like micelles.13,14 In contrast to HEUR solutions in water, the underlying structure responsible for the rheology of fully formulated paints (HEUR polymers with latex, pigment particles, and surfactant) is more complex and less understood. For sufficiently large hydrophobic end groups and an apolar latex surface, it has been proposed that the HEUR polymers adsorb on the latex surface via their hydrophobic end groups leaving a thin PEO shell extending into the solvent (H2O).15 Even when the lifetime of hydrophobe−latex association is short (≪1 s), a continuous adsorption−desorption mechanism, in concert with polymer topology of two or more hydrophobes per molecule, leads to an equilibrium core−shell architecture. Early research in this field proposed that in the presence of latex particles, a transient network of HEUR polymers coexists with HEUR adsorbed to the latex particle surfaces, with HEUR mediated bridges between the particles and the transient micelle network.16 Jenkins et al.15b summarized the HEUR−latex interactions as follows: given that one end of a HEUR molecule is adsorbed via a hydrophobe to a particular latex particle, the other hydrophobe may be connected to (a) the same latex particle (making a loop) or (b) another latex particle (creating a direct bridge) or (c) adsorbed to the end-hydrophobe of another HEUR polymer (creating a train) at higher HEUR and surfactant concentrations. However, Hemker et al.,17 Richey et al.,18 Jenkins et al.,15b Uemura and Macdonald,19 and Beshah et al.,20 reported that in the presence of latex particles, and below the HEUR-surfactant saturation of the latex surface, a HEUR transient network (through flower micelles) was not detected. Using HEUR with pyrene hydrophobes in a fluorescence spectroscopy study, Richey et al.18 as well as Hemker et al.17 reported that at 1% (w/w) concentration, all hydrophobes are attached to the latex surface. Macdonald and co-workers19,21 employed a pulsed-gradient spin echo nuclear magnetic resonance (PGSENMR) technique to quantitatively measure

the fraction of HEUR bound to latex particles. Uemura and Macdonald19 reported that at 1% (w/w) HEUR and 4% polystyrene latex loading, the fractional population of bound HEUR was 0.53. However, they started with a dilute solution (8.1% w/w latex in D2O), the aliquot (mixing of latex solution and HEUR solution) was centrifuged, and the pellet fraction was resuspended to its original volume. The centrifugation and resuspension may have introduced artifacts in adsorption quantification. A recent pulsed field gradient (PFG) NMR spectroscopy study by Beshah et al. reported that only at high concentrations of HEUR (>2% w/w) were hydrophobes and HEUR molecules detected in the medium.20 In contrast with the above studies, Pham et al.22 and Glass et al.23 hypothesized that some amount of the HEUR remains free or forms flower micelles of a transient network. It should be noted that even if HEUR were not adsorbed to latex in these experiments, the concentration in the medium would be far too low to generate the observed viscosity via a transient network of flower micelles mechanism and furthermore cannot account for the observed shear thinning rheology of HEUR-thickened latex dispersions. To date, it has been unclear what the relative contributions of transient HEUR network, hydrodynamic volume, and largescale structure are to viscosity, especially at the high shear rate limit, in commercially relevant coatings formulations. There are several proposed mechanisms suggesting dominant contributions from (a) a HEUR transient network in solution, (b) the hydrodynamic volume of HEUR-latex particles, or (c) hybrids of a and b. Most researchers assume that HEUR molecules form a transient network in solution in concert with some adsorption to latex where the latter provides hydrodynamic drag and enhanced network connectivity.16 Latex dispersions have been studied extensively18,24 with experimental, theoretical, and computational25 approaches. Krieger and co-workers26 proposed an equation for the dependence of the low-shear-rate dispersion viscosity on the hard-sphere colloid volume fraction (Krieger−Dougherty equation), and also investigated the flowcurves (viscosity vs shear stress). Pishvaei,27 Glass,28 and others have further investigated this scenario. Bicerano et al.29 analyzed the influence of hard particle shape on the viscosity, as well as the role of clustering and percolation. Structural measurements on HEUR-latex systems have been performed under dilute quiescent conditions, where photon correlation spectroscopy (PCS) measurements have measured the adsorbed layer thickness in dilute dispersions of latex.15b,22,30 At low latex particle concentrations, assuming a saturated adsorption condition, the diffusion coefficient of the particle is measured (using PCS) which is related to particle hydrodynamic radius through the Stokes−Einstein equation. The adsorbed layer thickness (Δ) is the difference in hydrodynamic radius between the bare particle and the particle with an adsorbed layer. The effective volume fraction is calculated assuming an isotropic shell layer thickness: ϕeff = ϕ*[1 + (Δ/r)3], where ϕ and r are the volume fraction and radius of the bare latex particle, respectively.22,31 While this “hard” adsorbed layer thickness assumption has been experimentally validated for zero and low shear regimes,22,31b this may not be applicable for the high shear regime or for formulations with a higher latex particle volume fraction that are of commercial interest. In this work, we concentrate on understanding the shearinduced changes in the microstructure of adsorbed HEUR polymers on latex surfaces as a function of hydrophobe density (number of hydrophobes per HEUR polymer chain) as the 1156

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Table 1. Descriptions of rheology modifiers used in this study RM

chemistry

RM-1 RM-2 RM-3

HEUR HEUR HEUR

average MW (kg/mol)a end hydrophobe size (no. of C atoms) 28−32 (L) 40−50 (H) 32−38 (I)

10−12 10−12 10−12

hydrophobe densityb

neutron SLDc (×10‑7 Å−2)

ICI viscosityd,e (Pa-s)

2 2−3 >3

7.29 ± 0.1 7.24 ± 0.1 7.12 ± 0.1

0.045 ± 0.005 0.105 ± 0.005 0.075 ± 0.005

a

From aqueous gel permeation chromatography (GPC) measurements. bDefined as the number of hydrophobes/chain. cCalculated data. Formulation with experimental latex (ϕ = 0.28). eImperial Chemical Industries (ICI) viscosity is a measure prominent in the coatings industry using a cone-and-plate fixture at 25 °C, producing a nominal shear rate of 10 000 s−1. d

sodium lauryl sulfate (SLS, based on monomer) and the batch was neutralized with ammonia (28% in water).32 HEUR Synthesis. The HEUR’s used in this study (hereafter, also referred to as rheology modifiers (RM’s)) were synthesized from the condensation of a hexamethylene diisocyanate (HDI) with PEO diol and a capping alkyl alcohol. The isocyanate condensation reaction was conducted around 80−110 °C and used either a tin (dibutyltin dilaurate) or bismuth (bismuth octoate) catalyst. Generally, the polymers were synthesized in a two-step process where the PEO diol, excess diisocyanate, and catalyst are first reacted to produce chain extension. After this reaction was complete, the capping alcohol was added. A mole excess of 10% capping alcohol was used to ensure that all isocyanate was reacted.33 Thus, some residual alcohol may be present in the samples, along with low molecular weight (MW) species such as diadduct (2 mol of capping alcohol with 1 mol of diisocyanate). Three different RM polymers were used which varied in their PEO backbone molecular weight and no. of hydrophobes per polymer chain. Cm denotes the hydrophobe size for the RM polymers, where m is the effective/equivalent number of methylene groups representative of the combined hydrophobic contributions of the isocyanate linker and alcohol capping agent moieties.34 For the RMs used in this study, m is approximately 10 to 12. The weight-average MWs (in kg/mol) are 28−32 (low), 40−50 (high), and 32−38 (intermediate) for RM-1, RM-2, and RM-3, respectively. For RM-1, the hydrophobe density (no. of hydrophobes/polymer chain) is ∼2 (low). Note that a single hydrophobe per chain is equivalent to a surfactant molecule. For RM-2 and RM-3, the hydrophobe densities are ∼2−3 (intermediate) and >3 (high), respectively. A summary of the RM polymer properties including their neutron scattering length densities (SLDs) and ICI viscosity values in the formulations are presented in Table 1. ICI (Imperial Chemical Industries) viscosity measurements are used prominently in the coatings industry. It is a cone and plate measurement (plate diameter 24.0 mm and a cone angle of 0.5°) conducted at 25 °C at a nominal shear rate of 10 000 s−1. Sample Preparation. As noted previously, since the RM polymers are presumed to be noninteractive with pigment, the focus of this study is on pigment-free formulations of the RMs and latex. The pigment-free formulations used in this study were prepared by combining the latex emulsions synthesized in H2O and D2O, along with pure H2O and D2O, RM, and neutralizing agent, a 95% solution of 2-amino-2-methyl-1-propanol, to maintain pH = 9.0 for all samples. From previous SANS measurements, the SLD of the latex was found to be 9.8 × 10−7 Å−2. The SLDs of different RM polymers are tabulated in Table 1. In earlier work, pure D2O-based samples were found to produce multiple scattering effects. In order to obtain sufficient scattering contrast between the latex particle and solvent, yet minimize multiple scattering effects, the Δρ value between the solvent medium and latex was selected to be 1.0 × 10−6 Å−2. A combination of D2O/H2O with a D2O wt. fraction of 0.367 (w/w) was therefore used as solvent medium with a SLD of 1.98 × 10−6 Å−2. The samples were prepared in 25 g batches and mixed on a Hauschild Flacktek SpeedMixer for 2 min. The liquids (H2O and D2O) were first premixed with the RM, after which the latex components were added and mixed. The pH adjustment was done on the final sample by titrating with a 95% solution of 2-amino-2-methyl-1-propanol, typically of around 20 mg. ICI measurements were made on the samples after equilibrating for an hour. The latex volume fraction (ϕ) in all

hydrophobe size (equivalent number of straight chain alkyl groups) is held constant. The formulations used in this work are all at moderately concentrated constant latex volume fraction (ϕ = 0.28) and fixed HEUR polymer concentration (1% (w/w)) which are typical concentrations used in commercial coatings. This HEUR concentration is below the concentration found by Beshah et al.20 where the HEUR molecules saturate the surface of the latex. While the ultimate goal of this line of work is to better understand fully formulated coatings in the wet state, for simplicity, pigment (TiO2) particles were not used in this study and the sodium lauryl sulfate surfactant concentration was fixed at 0.2% (w/w). We use small-angle neutron scattering (SANS) collected under Couette flow conditions (rheo-SANS) to study the latex-HEUR microstructure as a function of shear rate. Shear rate dependent anisotropy in the scattering patterns is observed, which we interpret as changes in the structure of both the adsorbed layer thickness and the latex-HEUR particle spatial arrangements. Furthermore, we estimate the effective hydrodynamic particle (core and shell) volume fraction considering the structural anisotropy using the HEUR layer thickness measured by SANS. After substituting this effective volume fraction into the Krieger−Dougherty dispersion rheology model, we find that the calculated viscosities are significantly lower than the measured viscosities. In some previous studies, the thickening of complex colloidal suspensions with HEUR molecules has been attributed to a contribution from the hydrodynamic volume of latex particles with adsorbed HEUR molecules, especially in the high shear rate region. This report presents for the first time the precise measurement of the in situ hydrodynamic volume of latex−HEUR particles as a function of shear rate and demonstrates conclusively that hydrodynamic volume alone cannot explain the experimentally measured viscosity even in the high shear rate regime.

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MATERIALS AND METHODS

Latex Synthesis. Latex representative of commercial grade latex was used for all the formulations. However, for the SANS study, synthesis was performed using 99% D2O instead of water in a semi batch “gradual addition” polymerization following a process based on commercial products. The nominal latex particle diameter used for this study was 120 nm measured using a centrifugation method (capillary hydrodynamic fractionation, CHDF, data not shown). The synthesis process employed D2O for reactor and monomer emulsion water as well as most water dilutions where practical. Surfactants were delivered as a solution in water and were not transferred to D2O. Small charges of materials which are normally stored as solutions in water, such as FeSO4 solution, hard bases, ammonia and biocides also did not include D2O. Unlike analytical tools such as NMR, absolute replacement of protonated solvent was not required for this investigation. The latex is a single stage copolymer of 46 butyl acrylate/52 methyl methacrylate/ 1.0 ureido monomer (CAS No. 86261-90-7)/1.0 methacrylic acid (% mass), which was thermally initiated with ammonium persulfate using a preformed polymer seed. This material was stabilized with 0.6% 1157

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formulations was maintained at 0.28. At ϕ = 0.28, the concentration of latex spheres is much lower than the concentration for entropy driven ordering/random jamming of hard spheres.35 The final RM loading was 1.0% (w/w) and final surfactant (SLS) loading was 0.2% (w/w). The typical formulation density was ∼1.05 g/cm3. Surfactants are ubiquitous and key components of latex-based coatings formulations, and although we do not explicitly study their role in this study, it is pertinent to briefly elaborate on their interactions with latex and HEUR RMs.36 We have constrained the SLS surfactant concentration in this study to 0.2% (below the critical micelle concentration, CMC, but within the lower quartile used in commercial coating formulations). Below the CMC (of mixed species including HEUR and surfactant), increasing the surfactant concentration raises the free, unassociated surfactant concentration in the aqueous phase which is in thermodynamic equilibrium with the concentration of surfactant adsorbed to latex surfaces. By definition, the latex surface approaches saturation with HEUR and surfactant as the medium surfactant concentration approaches the CMC. Adding more surfactant in excess of the CMC populates an additional phase (micelles), but does not significantly increase the concentration of the free surfactant in the medium, and hence does not directly increase the thermodynamic potential for increasing surfactant adsorption to latex. A significant indirect effect arises from the surfactant micelle formation where the micelle hydrophobic cores provide additional surface for HEUR end-hydrophobe adsorption. Thus, increasing micelle concentration decreases the fraction of latex surface relative to the total available apolar surface (latex surface plus micelle cores) in thermodynamic equilibrium with the aqueous medium monomer concentration. As equilibrium is maintained between species adsorbed to latex surfaces and micelle cores, some HEUR molecules are exchanged from the latex surface and migrate to micelles. It has been reported that such displacement of the HEUR polymers from the latex surface is most effective in presence of nonionic surfactants due to their synergy with latex particles.36 However, such redistribution of the HEUR molecules arises at surfactant concentrations greater than about 1% and at HEUR concentrations greater than about 2%, much greater than the amounts used in this study. We assume that at 1% (w/w) HEUR concentration and 0.28 volume fraction latex, HEUR polymer is not present in solution (either free or in micellar form) under quiescent conditions in line with Richey et al.18 and Beshah et al.20 Therefore, in this study, we do not consider any displacement of HEUR polymers from the latex surface by surfactant molecules. Rheo-SANS Measurements. The rheo-SANS measurements were performed on the NG7 30m SANS37 beamline at the National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR) in Gaithersburg, MD. An incident neutron wavelength (λ) of 8.403 Å and wavelength spread (full width at half-maximum) of Δλ/λ = 11% was used for all measurements. The sample to detector distance (SDD) was 15.3 m, which covers a scattering vector range of q = 0.001 Å−1 to 0.14 Å−1 [q = (4π/ λ)*sin(θ/2), where θ is the angle between the incident and scattered beam]. A MgF2 biconcave lens configuration was used for beam collimation. The instrument was equipped with a 640 mm × 640 mm 3 He position-sensitive 2D detector with a 5 mm × 5 mm resolution. The NIST Boulder shear cell38 consisting of an inner quartz cylindrical stator (outside diameter 60 mm) and an outer quartz cylindrical rotor (inside diameter 61 mm) with a 0.5 mm gap between the cup and bob was used for all measurements. Approximately 11.75 mL of sample was required to fill the shear cell. The outer cylinder was rotated to generate a simple shear field between the cylinders. A schematic of the shear cell geometry and position of the SANS detector relative to the laboratory coordinates is presented in Figure 1. The shear cell is equipped with a vapor trap that was filled with H2O to prevent sample drying during the course of the experiment. It is assumed that exchange between the H2O in the vapor trap and the H2O/D2O mixture in the sample is minimal so the composition of the solvent in the sample remains constant during the experiment. All experiments were performed at 25 °C. The incoming neutron beam is along the Y or velocity gradient direction and parallel to the shear plane (X−Y). The detector position is in the X−Z plane and is orthogonal to the

Figure 1. Schematic of the sample holder geometry and SANS detector position relative to the laboratory coordinates (X−Y−Z). shear plane. Hence, the horizontal axis of the SANS detector is parallel to the flow direction (X) and vertical axis of the detector is parallel to the vorticity direction (Z). Scattering from the empty cell and blocked beam were recorded for corrections to the sample scattering. For each sample, transmission measurements were made for the open beam and each sample under quiescent conditions only. We assume that sample transmissions do not change as a function of shear rate. Data correction and reduction were performed using IGOR-based routines available from the NCNR.39 The Jacobian corrections due to the curvature of the detector were performed on all data sets based on the isotropic scattering from a sheet of poly(methyl methacrylate). All data were placed on an absolute intensity scale using an attenuated empty beam. Each sample was measured at rest and at 4 different shear rates (10, 100, 316, and 1000 s−1). Additionally, for the latex−RM-1 formulation, rheo-SANS data were collected at 2500 s−1. Typically, the quiescent measurement was performed first, followed by the shear measurements, where the shear rate was increased in a stepwise fashion between measurements. For each measurement (quiescent or under shear), the counting time was sufficiently long (2700 s) to obtain a total detector count of 106 neutrons over the area of the detector. Rheology. The NIST Boulder shear cell is not equipped with transducers for simultaneous viscosity measurements during scattering. Therefore, viscosity measurements were conducted in-house. A stress controlled rheometer (TA Instruments AR-G2) equipped with a 40 mm diameter stainless steel, upper parallel plate and a Peltier plate temperature controlled lower plate was used for all the measurements. The test temperature was maintained at 25 °C, and the fixtures were zeroed before each sample loading. Samples were tested in a Controlled Stress Sweep mode where the applied stress was varied between 0.1 and 1000 Pa in eight equally spaced logarithmic increments per decade of applied shear stress. Stress sweeps were performed first by increasing stress then decreasing stress to check for flow hysteresis. The viscosity was recorded as a function of the applied stress. Dynamic frequency sweeps were performed on a Rheometrics Fluid Spectrometer (RFS-2) using 50 mm diameter parallel plate fixtures at 25 °C with a 0.5 mm gap. The frequency sweeps were conducted from 100 rad/s to 0.1 rad/s in eight equally spaced logarithmic increments per decade of frequency. In dynamic frequency sweep measurements, a frequency (ω) dependent small-amplitude oscillatory strain is applied. The strain amplitude was kept low (5% for these measurements) to ensure measurements were performed in the linear viscoelastic regime. The storage and loss moduli, G′(ω) and G″(ω), respectively, and complex viscosity (η*) were measured as a function of the frequency. The magnitude of the complex viscosity can be expressed as: |η*|= (G′2 + G″2)(1/2)/ω.

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RESULTS AND DISCUSSION Rheology of Latex−RM Formulations. The flow curves (viscosity as a function of shear rate) for the different latex− RM formulations are presented in Figure 2. It is difficult to directly interpret any specific information about the detailed latex−RM structure exclusively from the viscosity curves. The 1158

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Figure 2. Flow curves for the latex−RM formulations for which rheo-SANS measurements were performed. Absence of hysteresis in the flow curve suggests that structural changes under shear are completely reversible. In part a, four different regimes are shown for clarity: (1) first or low-shear plateau (2) first shear thinning regime; (3) second or intermediate-shear plateau; (4) second shear thinning regime.

Figure 3. Frequency (ω) dependent storage modulus (G′), loss modulus (G″), and magnitude component of the complex viscosity (|η*|) for different latex−RM formulations for which rheo-SANS measurements were performed.

(marked “1” in Figure 2a) can be interpreted as the regime where the large scale network structure of bridged particles is continuously deformed and rearranged, resulting in an equilibrium state under steady shear,40 followed by a shear thinning region (marked “2” in Figure 2a) around 1 s−1. Alternatively, the first plateau region may arise from wall slip/ lubrication between plates which has been discussed in a

shear-thinning behavior suggests that significant structural changes are occurring as the shear rate increases but does not convey what the scale of these structural changes might be. The shear dependent viscosity demonstrates two plateau regions with a third terminal Newtonian plateau region expected at very high shear rates beyond the range of shear rates accessible in this study. The first, low shear rate (γ̇ < 0.5 s−1) plateau region 1159

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Figure 4. (a) Representative 2D-SANS data collected on a formulation with 0.28 latex particle volume fraction (ϕ) and 1.0% (w/w) RM-1 associative thickener illustrate the evolution of structural anisotropy as a function of shear rate. The intensity scale bar is the same for all the three 2D-SANS data. (b) Sector-averaged (Δψ ± 10°) scattering intensities in the flow (ψ = 0°) and vorticity (ψ = 90°) direction as a function of the scattering vector (q). ψ is the azimuthal angle measured from the equator. The error (±1 standard deviation) in measurements of SANS intensity is less than the size of the markers used unless explicitly shown using error bars.

previous publication.41 In the first shear-thinning regime, the large-scale network is broken into fractal aggregates. The interaggregate bridging is disrupted and fractal aggregates form, which decrease in size with increasing shear rate,42,43 until an approach to an intermediate Newtonian plateau44 is observed at shear rates around 100 s−1 (Region “3” in Figure 2a). A second shear-thinning regime (Region “4” in Figure 2a) is observed around 1000 s−1 where fractal aggregates rupture and break up into smaller aggregates. At higher shear rates, above 10 000 s−1 and beyond the range accessible in this study, we expect a terminal Newtonian plateau corresponding to single particles. Additionally, there is no hysteresis observed for the reverse measurements. Therefore, the structural changes observed under shear are completely reversible. The linear frequency sweep data for the different latex−RM formulations are presented in Figure 3. As discussed previously, well above the critical aggregation concentration (cAC), the pure HEUR solution oscillatory shear rheology can be explained through a simple Maxwell model.9,13b The characteristic time scale (τ) corresponds to the hydrophobe relaxation time (both intra- and intermicellar). In the high frequency region (ω > τ−1), the elastic modulus is higher than viscous modulus (G′ > G″) and at low frequency (ω < τ−1), terminal relaxation behavior (i.e., G′ ∼ ω2, G″ ∼ ω and G″ > G′) is observed.9,13b In contrast, for the latex−RM formulations used in this study, G″ > G′ at high frequencies (ω > 10 rad/s), and both quantities show almost identical power-law scaling with frequency (G′ ∼ G″ ∼ ωn). Such behavior is often typical of nearly gelled systems8,45 with the presence of multiple characteristic sizes and cluster size distribution. Therefore, instead of a single relaxation time constant, the system demonstrates a broad distribution of multiple relaxation times consistent with a distribution of the number of HEUR molecular bridges between latex particles. In our case, the average value of the high-frequency scaling exponent, n, was found to be 0.65 ± 0.07. The exact values of n, for each case, are reported in Figure 3. Adolf and Martin demonstrated that the power-law scaling

exponent for a system with cluster molecular weight distribution should be 0.67 (where cluster fractal dimension is 2.5)46 which is in good agreement with the experimentally observed values for latex−RM formulations here. At intermediate frequencies (10 rad/s < ω < 1 rad/s), these formulations displayed some deviation from the terminal relaxation behavior. We find that in this frequency range G″ ∼ ω0.85±0.06 and G′ ∼ ω1.2±0.2. Further, at low frequency (ω < 1 rad/s), a nonterminal plateau like behavior in G′ was observed which hints at the presence of a network structure with long relaxation time. In this frequency zone, a crossover point was observed for RM-2 where ωcrossover = 0.133 rad/s. From the observed low-frequency dependence of G′ and G″ a similar crossover for RM-1 and RM-3 formulations is expected to appear at ω < 0.1 rad/s. This large-scale structure is also evident from the complex viscosity, where an upturn in |η*| at low frequency was observed. Therefore, the latex-HEUR formulation viscoelastic behavior under oscillatory shear is significantly different from pure HEUR solution viscoelastic response. Further investigation of the large-scale structure under quiescent and shear conditions will be the subject of a future report. Rheo-SANS on Latex−RM Formulations. Figure 4a illustrates representative 2D-SANS data collected at different shear rates for the RM-1 containing sample. The 2D-SANS pattern in the absence of shear (left, Figure 4a) is isotropic azimuthally with a maximum in scattering intensity shown by the yellow in the false color representation. With application of a moderate shear rate (100 s−1) (center, Figure 4a, flow direction is horizontal), the scattering pattern becomes anisotropic with the higher scattering intensity observed along the flow direction compared to the vorticity direction. At even higher shear rates (1000 s−1) (right, Figure 4a), the difference in intensity between the flow and vorticity direction is further enhanced. It should be noted that the color scale bar for all the 2D-SANS images is the same. The polydispersity of the latex spheres ensures that there will not be any shear-driven 1160

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Figure 5. Scattering vector dependent intensity profiles in (left column) flow and (right column) vorticity directions as a function of shear rate. Scattering intensity profiles obtained from (a and b) latex−RM-1 formulation, (c and d) latex−RM-2 formulation, and (e and f) latex−RM-3 formulations at different shear rates are presented. The error (±1 standard deviation) in measurements of SANS intensity is less than the size of the markers used unless explicitly shown using error bars. The solid lines represent the model fit to the data using a spherical core−shell form factor (with polydisperse core radius) with hard sphere structure factor accounting for the interparticle repulsion.

vorticity direction are identical as expected for an isotropic scattering pattern. The peak appearing at ∼0.009−0.01 Å−1 is the form factor peak representing the length scale of a single latex core − HEUR shell composite particle. The more intense low q peak at ∼0.005 Å −1 is the structure factor peak representing interparticle interactions. At moderate shear rate (γ̇ = 100 s−1) (center, Figure 4b), the form factor peak change is relatively small. In contrast, the structure factor peak broadens in the flow direction. Additionally, in the low-q region (q < 0.004 Å−1), the scattering intensity in the flow direction becomes greater than the vorticity direction. At the highest shear rate (γ̇ = 1000 s−1) (right, Figure 4b), the difference between the flow and vorticity direction intensity profiles becomes more pronounced. An increase in scattering intensity at low-q appears in the vorticity direction as well which is less obvious than in the flow direction. Consistent with our viscosity data, under quiescent conditions and in the low shear rate regime (regime 1 and 2 in Figure 2a), the large scale

ordering even when the effective hydrodynamic volume increases due to RM adsorption.47 Data analysis of anisotropic scattering patterns is challenging and nontrivial. Some common analysis techniques are (a) annular averaging at fixed q, (b) sector averaging, (c) spherical harmonic analysis, and (d) full 2D fitting. Sector averaging is most commonly practiced by researchers to study anisotropic structures. In a recent review article, Eberle and Porcar stated that 1D structural information with respect to a given direction can be extracted from sector averaging.48 To quantify the shear-induced anisotropy, representative qdependent scattering intensities (sector-averaged for Δψ = ± 10°) in the flow (ψ = 0°) and vorticity (ψ = 90°) directions are presented in Figure 4b. Here ψ is the azimuthal angle measured from the equator. Hereaf ter, the sector averaged intensities in the f low and vorticity directions will be referred to as the f low and vorticity direction scattering intensities, respectively. In the absence of shear (left, Figure 4b), the intensities in the flow and 1161

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Figure 6. (a) Spherical form factor fit to experimental data for a dilute (ϕ = 0.003) Latex solution. The best-fitted parameters are shown in the inset box. (b) Shear dependent scattering data from dilute latex solution (ϕ = 0.003).

form factor and, S(q) is the structure factor. The q-independent incoherent scattering background is represented by the bgd term. In eq 1, the P(q) accounts for scattering from a single particle and depends on the particle size and shape. On the other hand, S(q) accounts for the scattering arising from spatial correlations between the scattering particles over a large correlation volume. Latex Particle Form Factor, P(q). The latex particle form factor was obtained from a SANS measurement on a dilute solution of the pure latex in D2O (volume fraction, ϕ = 0.003). For sufficiently low volume fractions, S(q) = 1.0 for all q. The scattering intensity for a dilute solution can be written as:

structure of particle assemblies is larger than the minimum q range which can be accessed by the SANS experimental set up. We presume that with increasing shear rate, the large assemblies break into smaller clusters49 (regime 3 and 4 in Figure 2a) which become observable in the q range covered by the SANS experiment. Consequently, in the high shear regime, an increase in low-q scattering intensity is observed. Rheo-ultrasmall-angle neutron scattering (rheo-USANS) measurements are indicated to study this larger length scale as a function of shear rate. The shear rate dependence of the scattering intensities are presented in Figure 5a−f. Parts a and b of Figure 5 represent the scattering intensities in flow and vorticity directions, respectively, from the RM-1-containing formulation. Similarly, parts c (flow) and d (vorticity) of Figure 5 represent the scattering from the RM-2-containing formulation and parts e (flow) and f (vorticity) of Figure 5 represent the scattering from the RM-3-containing formulation, respectively. The different formulations containing the different RMs show the same general systematic variations in the scattering behavior as a function of shear rate. The detailed differences between the different RM−latex formulations are more apparent in the fits to the data, which are described in the next section. The shear rate dependent changes in the overall scattering intensity profile are stronger in the flow direction compared to the vorticity direction. In general, small changes are observed in the form factor peak as a function of γ̇, in both the flow and vorticity directions. With increasing γ̇, the structure factor peak becomes broader in the flow direction with an upturn in scattering intensity at low-q. In the vorticity direction, the structure factor peak intensity decreases with increasing shear rate, with little broadening (much less compared to the flow direction). In addition, a weaker upturn in vorticity direction scattering intensity is also observed at γ̇ = 1000 s−1. Model Fitting of the rheo-SANS Data. For the case of neutron scattering from a dense suspension of latex particles, the scattering intensity, I(q), can be expressed as: I(q) = ϕVp(Δρ)2 P(q)S(q) + bgd

lim I(q) = ϕVP(Δρ)2 P(q) + bgd

(2)

ϕ→ 0

The form factor for spherical latex particles is given as ⎡ 3(sin(qr ) − qr cos(qr )) ⎤2 ⎥ P(q) = F (qr ) = ⎢ (qr )3 ⎦ ⎣ 2

⎡ 3J (qr ) ⎤2 ⎥ =⎢ 1 ⎣ qr ⎦

(3)

where J1 is the spherical Bessel function. The polydispersity (p) of the sphere radius is incorporated by means of a Schulz distribution given by z = (1/p2)-1, where z is the width parameter of the distribution and p = σ/ravg where σ2 is the variance of the distribution. The details of the Schulz distribution can be found elsewhere.50 Incorporating polydispersity effects, eq 2 can be rewritten as: lim I(q) =

ϕ→ 0

⎛ 4π ⎞2 ϕ ⎜ ⎟ (Δρ)2 ⎝ 3 ⎠ ⟨VP⟩

∫0

∞

f (r )r 6F 2(qr ) dr + bgd (4)

Here f(r) is the polydispersity function; ⟨VP⟩ is the average particle volume that is related to the third moment of the Schulz distribution. SANS data from the dilute latex dispersion without RM (vol. fraction, ϕ = 0.003) and the model fit are presented in Figure 6a. The fitting variables in eq 4 were volume fraction (ϕ), mean particle radius (ravg), and the polydispersity factor (p) while the SLDs of the latex particles and solvent medium (pure D2O) and the background term (0.1 cm−1) were held constant. The returned fit value for ϕ was

(1)

where q is the scattering vector, ϕ is the volume fraction of the scattering particles, Vp is the volume of a single scatterer, Δρ is the scattering contrast (the difference between neutron scattering length densities between components), P(q) is the 1162

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sphere potential.55 This model introduces two more unknown parameters: a perturbation parameter (a function of well width and hard sphere diameter) and a stickiness parameter (a function of well depth and a perturbation parameter) which are strongly coupled. The main challenge of using this structure factor is the absence of any prior reported or experimental data to estimate either of these two parameters independently. Without the ability to constrain one of these two parameters, there are more adjustable fit parameters than are desirable to fit our data. Therefore, the HS structure factor was used exclusively in this study. The Combined Form Factor and Structure Factor. To extract the structural parameters from the latex−RM mixtures, we assume all of the RMs adsorb to the latex surface and form a “shell” around the spherical latex “core”. A form factor/ structure factor model, as described in eq 1, was fitted to the scattering data using a homogeneous spherical core−shell form factor (P(q)). The core−shell form factor has been derived for concentric spheres consisting of an inner sphere with polydisperse core radius surrounded by a uniform density outer layer or shell. For our system, assuming the shell layer is of uniform density is imperfect. It is believed that the PEO midblocks in the RMs are swollen in the presence of good solvent (H2O/D2O molecules) and the PEO areal density will vary in the radial direction from the latex surface to the outer shell boundary. The transfer energy from water to latex or self is on the order of 50 kJ/mol per hydrophobe and about 100 kJ/ mol for a HEUR molecule with two adsorbed hydrophobes.56 Kushare et al. report that the free energy change due to mixing, and the excess free energy change, for PEO solutions is 10−100 J/mol.57 Therefore, we conclude that adsorption of swollen HEUR PEO chains to latex is physically reasonable. It is also noted here that the RM concentration (1% (w/w)) used in these formulations is below their saturation condition (>2% (w/w)) experimentally determined by PFGNMR spectroscopy20 (for the same latex−RM formulations and at a similar latex volume fraction). However, this nonuniformity in areal coverage can be neglected here since the neutron experiment time scale is orders of magnitude larger than the characteristic latex−RM interaction time scale. A uniformly dense polydisperse core with a fuzzy shell layer would be a closer approximation to this picture. However, such a form factor model is not readily available and would add extra fitting parameters; therefore, the spherical core−shell form factor model with a homogeneous shell was used as a first approximation. This homogeneous spherical core−shell form factor model has been utilized by other investigators to describe SANS data from latex-associative polymer systems.15a,54,58 The spherical core−shell form factor, and the intensity data scaled to particle volume fraction can be expressed as:

0.0028, which is consistent with the expected volume fraction (0.003). The mean radius (ravg) of the latex particles is 535.2 ± 3.6 Å with a polydispersity (p) value of 0.135 ± 0.01. Before measuring the shear rate dependent scattering from the latex−RM mixtures, rheo-SANS experiments were also performed on the dilute latex dispersion without RM (ϕ = 0.003, in pure D2O) to determine whether shear-induced deformation of single latex particles occurs. The 1D scattering intensity profiles were obtained from azimuthally averaged sectors along the flow and vorticity directions. In Figure 6b, the scattering in the flow and vorticity directions for the dilute latex at different shear rates (10, 100, 316, and 1000 s−1) and at rest are presented. All the intensity profiles are identical within the error of the measurements, indicating that for a dilute suspension of the latex particles, the latex particle form factor is independent of shear rate and particle deformation does not occur over the range of shear rates studied. On the basis of this result, the average latex particle diameter and polydispersity values were held constant at 535.2 Å and 0.135, respectively, for the remainder of the SANS fitting. Latex Particle Structure Factor, S(q). We model the interparticle interaction using a hard sphere (HS) structure factor (S(q)) with Percus−Yevick closure.51 The Percus− Yevick approximation is quite accurate up to an (effective) volume fraction ∼0.45 for monodisperse hard spheres and found to be accurate for a larger range of volume fractions for polydisperse hard spheres.52 The hard sphere potential presumes that spheres interact through excluded volume only. Therefore, two such spheres are impenetrable when they are in contact with each other. Mathematically, the interparticle potential, U(a), is expressed as U (a) = ∞ ,

a ≤ 2r

U (a) = 0,

a > 2r

(5)

where a is the radial distance from the center of a sphere with diameter = 2r. The interparticle potential determines the equilibrium arrangement of particles or pair correlation function, g(r) from which, the structure factor, S(q), can be calculated through Fourier transformation. Given that the shell layers are predominantly made of soft PEO polymer chains, such a hard-sphere model may not be fully realistic. However, for colloidal particles with a steric-stabilization layer grafted onto the particle surface, Mewis and co-workers have shown that the hard sphere approximation is valid unless the stabilizer layer thickness/particle radius ratio is too small.31b Ottewill and co-workers treated SANS data from colloidal particles composed of a poly(methyl methacrylate) (PMMA) core with a shell of poly-12-hydroxy stearic acid (PHS), using “hard sphere” and “soft repulsive sphere” structure factors (with a homogeneous spherical core−shell form factor in both cases). They concluded, based on the model fit, that it was reasonable to treat the PMMA−PHS system as particles having a “nearly hard-sphere” behavior.15a,53 Recently, Zackrisson and coworkers54 introduced a single parameter, α, in their form factor-structure factor model to represent the deviation from hard sphere behavior. For PEG chains grafted to a PS latex, they found α varies between 1.0 and 0.96 for the ϕ range ∼0.004 to 0.4 g/mL. Therefore, the latex particles with grafted polymers on their surface may also be approximated closely by hard sphere structure factor. One alternative form for the structure factor is the sticky hard-sphere (SHS) model based on Baxter’s adhesive hard

P(q) =

ϕ ⎡⎢ 3V1(ρlatex − ρRM )J1(q⟨ravg ⟩) Vshell ⎢⎣ (q⟨ravg ⟩) 3V2(ρRM − ρsolvent )J1(q⟨ravg + Δ⟩) ⎤ ⎥ + bgd + (q⟨ravg + Δ⟩) ⎥⎦ 2

(6)

where ravg is the mean core (latex) radius calculated using the Schulz distribution, Δ is the shell (RM layer) thickness and, J1 is the spherical Bessel function. V1 is the inner sphere volume [4π/3⟨ravg⟩3] and V2 is the outer sphere volume [4π/3(⟨ravg⟩ + Δ)3]. Vshell is given as V2 − V1. In the absence of any known 1163

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Figure 7. Shear rate dependent adsorbed RM layer thickness in (a) flow and (b) vorticity directions for different RM-based formulations. The error (±1 standard deviation) in model fitted Δ values is less than the size of the markers used unless explicitly shown using error bars. The solid lines are a guide to the eye.

experimental findings about shell layer thickness distribution, no polydispersity function was assigned to this parameter. Combining eq 6 with the HS structure factor (eq 5) according to eq 1, the sector averaged SANS data were fit as a function of shear rate for the three formulations. The scattering intensity expressions presented through eq 1−6 are defined for azimuthally integrated I(q) as a function of q. However, these expressions are applied to fit the sector-averaged intensity data. In this treatment, completely uniform orientation of the long axis of revolution in the vorticity direction was assumed. Under these circumstances, the anisotropic particle form factor (e.g., perfectly oriented ellipsoid) may be approximated by an isotropic (sphere) form factor to fit the sector averaged data as demonstrated in the Appendix. Fitting to two orthogonal sector averaged data sets for each shear rate extracts two discrete shell thickness values and shell SLD values in the flow (ψ = 0° and Δψ = ± 10°) and vorticity (ψ = 90° and Δψ = ± 10°) directions as a function of shear rate (a schematic of this concept is presented in the Appendix, Figure 14). These two discrete thickness and SLD values must be connected with a thickness and SLD which gradually changes as a function of azimuthal angle. More sophisticated models are possible, but at the expense of more assumptions, and we believe the present analysis captures the essential features of the anisotropic structure. As noted previously, sector averaging of the anisotropic scattering data and subsequent fitting using an isotropic scattering model to extract structure parameters is commonly reported in the literature.59 One potential issue with using a spherical core−shell form factor to fit the sector averaged scattering from an elliptical shell-spherical core particle is the difference in intraparticle interference from an ellipsoid versus a sphere, which is not considered in the above treatment. The difference in intraparticle interference would be most pronounced for highly anisotropic particles and this effect decreases with decreasing anisotropy. It is assumed that for the shear rate window studied here, the core radius ≫ shell thickness. Since the core (latex particle) constituting the major component remains spherical as a function of γ̇ (Figure 6b), the overall anisotropy of the core− shell particle is presumed to be low. This assumption is shown to be valid with the shell thickness data extracted from the SANS model fit reported later. Therefore, the difference in intraparticle interference between a spherical core−shell and

the slightly elliptical core−shells in this study can be neglected. To further support this assumption, for a representative data set, the structural parameters extracted from the sector average fit were compared with full 2D fitting using an elliptical core− shell model and reported in the Supporting Information. It was found that both model fits returned comparable shell thickness values in the flow and vorticity directions. However, the present 2D fit capabilities do not allow a variable shell scattering length density relative to the particle coordinate axes. We will demonstrate below that variable shell density under shear is a critical consideration of this work. Finally, during the model fit to the sector-averaged SANS data, we assume the anisotropy in scattering patterns is due to both variations in the adsorbed layer thickness and the spatial arrangement of the core−shell particles in the flow and vorticity directions. For fitting to the combined core−shell scattering function, the average latex radius (core), core polydispersity, latex scattering length density (SLD), and scattering background were all held constant at 535.2 Å, 0.135, 9.8 × 10−7 Å−2, and 0.1 cm−1, respectively. Therefore, the extracted fit parameters were the core−shell volume fraction, adsorbed RM layer thickness (shell), shell SLD, and solvent SLD. The fit values for the shell layer SLD were constrained between the solvent SLD (i.e., no HEUR molecules adsorbed on the latex surface) and the pure RM SLD (i.e., an unswollen shell). Shear Dependent Structural Parameters. Fits to sector averaged scattering data for the different latex−RM formulations according to the previous section are presented in Figure 5a−f. The agreement between the model fit and experimental data is better for the vorticity direction scattering profiles than the flow direction profiles. While the model successfully captures the broadening of the structure factor peak for the flow direction, especially at high shear rates, it fails to account for the low-q upturn of the scattering data. It is presumed that this deviation in the flow direction indicates the presence of anisotropic fractal-like aggregates of the particles.41 Aggregation of the particles in the system is not considered in the present model fits. All the extracted fit parameters and error estimates are reported in the Supporting Information. The adsorbed RM layer thicknesses (Δ) extracted from the different formulations in the flow and vorticity directions are presented in Figure 7. On the basis of the extracted fit parameters, the following observations can be made: (1) The 1164

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Table 2. Theoretical Estimation of the Adsorbed RM Layer Thickness and Comparison with Measured Shell Thickness Values under Quiescent Conditionsa contour lengthc (L) (Å)

radius of gyration (Rg)d (Å)

Flory radiuse (RF) (Å)

Alexander−de Gennes calculation Δf (Å)

rheology modifier (RM) (MWeb (g/mol))

no loop

forming loop

no loop

forming loop

no loop

forming loop

no loop

forming loop

measured Δg (Å)

RM-1 (30K) RM-2 (22.5K) RM-3 (9K)

2409 1807 723

1204 903 361

67 58 36

47 41 26

139 117 68

92 77 45

172 −240 142 −198 77 −108

86−120 71−99 38−54

105 ± 3 81 ± 1 37 ± 2

Errors reported to ±1 standard deviation for the SANS model fit data; all the calculated parameters are reported to their nearest integer value. MWe = (average MW)/((average no. of hydrophobes/RM polymer) − 1). cL = Nb, where N is no. of Kuhn unit (=MWe/137) and b is Kuhn monomer length (11 Å for PEO). For loop formation: L = (N/2)b dRg = √[⟨R2⟩/6], for loop formation Rg =√[⟨R2⟩/12], where ⟨R2⟩ = Nb2 eRF = n3/5a, where n is the number of monomer and a is monomer size (2.78 Å for PEO). fCalculated using Alexander−de Gennes theory for ν/b3 values ranging from 0.12 to 0.33. gSANS model fit. a b

molecular weight between two adjacent hydrophobes, MWe (g/ mol), the number (N) of Kuhn segments present is (N = MWe/137). Considering ideal chains with N statistically independent Kuhn segments, the contour lengths (L = Nb) are calculated to be 2409, 1807, and 723 Å for the RM-1, RM-2, and RM-3 polymers, respectively. The maximum adsorbed layer thickness should therefore be half of the contour length (L) for the case where adjacent hydrophobe groups of the HEUR are adsorbed to the same latex particle (“loop”) yielding shell thicknesses of 1204, 903, and 361 Å, for RM-1, RM-2, and RM3, respectively. In the absence of any polymer−solvent interaction, the radius of gyration (⟨Rg⟩) is a more reasonable characteristic parameter to represent untethered polymer chains. For ideal flexible linear chains, ⟨Rg2⟩ = ⟨R2⟩/6, where ⟨R2⟩ = Nb2 (the mean-square end-to-end distance).60 Using this expression, the ⟨Rg⟩ values for RM-1, RM-2, and RM-3 are 67, 58, and 36 Å, respectively. Further, for looping conformations of these RMs, the radius of gyration becomes60 √[⟨R2⟩/12] and the values are 47, 41, and 26 Å, respectively. However, for polymers in a thermodynamic good solvent, such as PEO in water, repulsion between monomer segments leads to swelling and expansion of polymer chains. In that case, the effective size of the polymer coil is larger than the unperturbed dimension (Rg) and referred to as the Flory radius, RF = (n3/5)a, where n is the number of monomers and a is monomer size (2.78 Å for PEO). 61 For HEUR RMs in a looping conformation, the RF values are 92, 77 and 45 Å for RM-1, RM-2 and RM-3 polymers, respectively. The Flory values for loops in good solvent are consistent with the experimentally determined shell thicknesses. The values of the various dimensions calculated above are summarized in Table 2 for ease of comparison. Alternatively, for dense brushes in good solvent, the theory of Alexander and de Gennes62 predicts that the brush layer thickness (denoted as Δ) on a spherical particle surface (r ≫Δ) is: Δ ∼ L(σν/6b)1/3, where L is the contour length of the polymer, b is the Kuhn length, and ν is the excluded volume parameter. The surface coverage parameter is σ = (crρNA)/ (3ϕMW) (equivalent to the adsorbed amount (w/w), Γ = (cρ)/(ϕρp)), where c is the RM concentration (0.01), r is the nominal particle hydrodynamic radius (600 Å), NA is Avogadro’s number, ρ is the formulation density (∼1.05 × 106 g/m3), ϕ is the particle volume fraction, ρp is the latex particle density (∼1.2 × 106 g/m3), and MW is the polymer (HEUR) molecular weight (MWe in our case). Comparison with the area per adsorbed chain suggests a crowded layer consisting of stretched chains at the concentration c considered here, indicating that a brush, rather than mushroom,

shell thickness varies monotonically with RM hydrophobe density, but does not show a monotonic relationship with RM molecular weight. (2) At zero shear, the adsorbed RM layer shell is isotropic, demonstrated by the same fit shell thickness in both directions. (3) As γ̇ increases, the adsorbed RM layer thickness decreases. The extent of the decrease is anisotropic with the decrease in thickness being greater in the flow direction. (4) At high shear rate (1000 s−1), a finite thickness of the adsorbed RM layer remains on the surface. The measured Δ values at zero shear are 105 ± 3, 81 ± 1, and 37 ± 2 Å for the RM-1-, RM-2-, and RM-3-based formulations, respectively. The extracted values of the shell thickness are a strong function of the hydrophobe density and follow the trend: RM-1 > RM-2 > RM-3. In general, one might expect that a HEUR polymer with a higher MW should form a larger loop, thus producing a thicker adsorbed layer. On the other hand, the higher hydrophobe density per polymer chain constrains the loops to smaller sizes. Therefore, the RM that has the smallest distance between two adjacent hydrophobes is expected to form the thinnest shell and have the lowest single particle hydrodynamic volume. This suggests that the shell thickness depends on an effective MW (MWe, kg/mol) between two adjacent hydrophobes in RM polymers. For RM-1, MWe is 30 kg/mol (assuming 2.0 hydrophobes per RM polymer). For RM-2 and RM-3 polymers, the hydrophobe densities are higher (∼2−3, and >3, respectively), therefore, the effective molecular weights are lower. The MWe values are 22.5 kg/mol for RM-2 (assuming 3 equally spaced hydrophobes per RM polymer) and 9 kg/mol for RM-3 (assuming 5 equally spaced hydrophobes per RM polymer), respectively. The higher number of hydrophobes per chain act as topological constraints to create more contacts with the latex surface. Such multiple contacts per molecule with the latex surface restrict the degree of freedom of the backbone PEO chain and therefore we observe a decrease in measured adsorbed layer thickness with increasing hydrophobe density. Multilayer adsorption (hemimicellar structure)23 and multipoint adsorption (via hydrophobes and backbone)15b,20 have been postulated previously in the literature. It is important to compare the extracted shell thickness values with the expected chain dimensions of each of the RM polymers using standard polymer statistical mechanical arguments. For the simplest case of a linear PEO backbone with hydrophobe units attached to each end, the fully extended contour length can be calculated. The molecular weight and length (b) of the Kuhn segment for PEO is 137 g/mol (ca. 3 monomers/Kuhn segment) and 11 Å, respectively.60 For HEUR polymers (idealized as PEO chains) of given effective 1165

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Figure 8. Adsorbed RM layer neutron scattering length densities (model-extracted parameters) as a function of shear rate in flow and vorticity directions for (a) RM-1-, (b) RM-2-, and (c) RM-3-based formulations. The dotted lines denote the limiting SLD values, i.e., the solvent medium SLD and pure RM layer SLD. The error (±1 standard deviation) in model fitted SLD values is less than the size of the markers used unless explicitly shown using error bars. The solid lines are a guide to the eye.

conformation is applicable.15b,18,22,63 According to this definition, the surface coverage parameter is inversely proportional to the adsorbed area per chain. Considering that ν/b3 values range from 0.12 to 0.33 (calculated on the basis of the molecular dimension of PEO in dilute solution using Flory theory),64 the Δ values calculated for the different RM polymers using the Alexander−de Gennes approach are also reported in Table 2. The extracted Δ values from the SANS data are in close proximity to both the Flory calculation and the Alexander−de Gennes calculation, if one assumes that the RM polymers form loops. Therefore, we conclude that under quiescent conditions, the measured thickness of the adsorbed RM polymer layer on the latex surface is predominantly in the form of loops (swollen with solvent molecules) and the adsorbed layer thickness decreases with increasing RM polymer hydrophobe density due to multiple (>2) contact points with the latex particle surface. Under shear, the adsorbed RM layer thickness parameter decreases more in the flow direction than in the vorticity direction. A potential mechanism for the decrease in the adsorbed RM layer thickness is revealed in the extracted shell (RM layer) neutron scattering length densities (SLDs) in flow and vorticity directions. We have assumed that the shell consists of the swollen EO block of the RM with the hydrophobic groups tethered to the latex surface. A logical upper limit for the shell SLD would be that of the solvent medium SLD. Such a situation may arise when the number density of the adsorbed RM chains per unit latex surface area is negligible and the (neutron) contrast between shell and solvent medium is nearly zero. On the other hand, the lower limit of shell SLD should be the pure RM SLD, where the latex surface is fully covered with RM layers without any penetration of the solvent medium into the shell. An intermediate situation indicates the extent to which the RM shell is swollen. In Figure 8, the shear rate dependence of the shell SLDs in both the flow and vorticity directions for the different formulations is presented. Under quiescent conditions and at low shear rate (10 s−1), the RM-1 layer SLD is close to the solvent medium SLD, which supports our previous assumption of highly swollen adsorbed layers. For RM-2 and RM-3, the shell SLDs are intermediate between the solvent and pure RM SLD. Also at rest and at low shear rate, the flow and vorticity

direction shell SLD values are almost equal with exception of the RM-2 shell. There is no apparent reason for this discrepancy. With increasing γ̇, the RM layer SLD systematically decreases toward the pure RM SLD (lower limit). This suggests that with increasing γ̇, progressively more solvent molecules are expelled from the adsorbed layer. This compression and concomitant deswelling results in a decrease in the RM layer thickness with increasing shear rate. Since the compression is more pronounced in the flow direction compared to the vorticity direction, one would expect that the SLD of the shell to be lower in the flow direction compared to the vorticity direction. This is consistent with our observations for all RM’s in this study. Furthermore, we note that the approach of the adsorbed RM layer SLD to the pure RM SLD with increasing shear rate demonstrates that HEUR molecules remain adsorbed to the latex surface and do not detach with shear. A different compressive effect in the flow and vorticity directions is instrumental to the development of structural anisotropy. RM polymers with higher hydrophobe density having smaller adsorbed layer thicknesses that decrease with shear can be explained from a statistical thermodynamics point of view. Multiple latex−RM contacts (≥2) result in longer residence times of the RM polymers on the latex surface. Consequently, the polymers will explore more conformational space on the latex surface, including those states where more of the molecule is adsorbed (at the loss of higher conformational entropy), contributing to even longer residence times. There is some effect of the longer residence time on PEO chain conformation where the multiple hydrophobe−latex contacts per RM polymer chain lead to less extended loop conformations. Previously Richey and co-workers,18 based on their adsorption isotherm data, concluded that even when the polymer chains lose some of their conformational energy due to loop formation, the combined system (latex with adsorbed layer) gains more free energy to compensate for this loss. In other words, under static conditions, the enthalpy gain due to adsorption of hydrophobes on the latex surface compensates for the loss of conformational entropy of the backbone PEO chains. However, in this condition there is an entropy penalty for solvent molecules bound in the swollen shell. Under shear, the solvent molecule entropy increases when expelled from the 1166

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adsorbed layer and returned to the dispersion medium, outweighing the f urther loss of conformational entropy due to flattening of the HEUR loops and reduction of the shell thickness.15a The anisotropic deswelling presumably arises from a nonuniform stress distribution which leads to a higher compression along the flow direction. Under this condition, the local-scale microstructure may be at a non-equilibrium steady state, resulting from the coupling between the stress field and the adsorbed RM layer concentration field, analogous to shearinduced phase separation of polymer solutions.65 Anisotropy of the adsorbed HEUR layer could also be associated with relatively greater HEUR bridge density in the vorticity direction. Following the width/spread of the structure factor peak (q ∼0.005 Å) as a function of γ̇ in the flow and vorticity directions (Figure 5a−f), the interparticle ordering is more uniform in the vorticity direction compared to the flow direction. We postulate that the long-range spatial arrangement is akin to vorticity elongation and the log rolling process, which has been observed for many deformable soft materials including anisotropic particles in solution and polymer matrices, attractive emulsions near their colloidal glass transition, liquid crystals, and polymer blends.8,66 A detailed study of the shearinduced long-range order and the consequent effects on rheological properties is beyond the scope of the present work and is the subject of a future communication. The finite adsorbed layer thickness at high shear rate (γ̇ ≥ 1000 s−1) is consistent with the results reported by Beshah et al.20 and is unexpected based on transient network theory (TNT).11g−l According to the TNT, most of the HEUR molecules were assumed to be associated via hydrophobes with flower-like micelles and bridges between micelles, forming the transient network, with only a small number of HEUR hydrophobes associating with the latex. Recently, atomic force microscopy measurements were conducted67 on PEO chains capped with alkyl hydrophobes tethered on one end to a silica substrate and to a silicon nitride AFM probe tip on the other end. The measurements revealed that for a HEUR polymer chain with C12 hydrophobes, the average detachment force was about 60 pN. The Reynolds number, Re [=(ρ(ravg)2γ̇/η)(4π/ 3ϕ)2/3], for these formulations ranges from a maximum of about 6 × 10−5 at the highest shear rate accessible (2500 s−1) to less than 3 × 10−9 at the lowest shear rate employed of 0.1 s−1. For low Re, the drag force (FD) can be calculated using Stokes’ law: FD = −6πηravgv, where v is the velocity in the flow direction. Even at 2,500 s−1 shear rate, the calculated drag force on a HEUR PEO coil for these systems is 0.2 pN, which is much smaller than the pullout force. Therefore, we conclude that the finite adsorbed layer thickness we extract at high shear rate is not a fitting artifact and is representative of a persistent layer of adsorbed RM. The shear-induced structure of the latex−RM combination can be represented schematically as shown in Figure 9. The schematic is not exactly to scale but an effort has been made to scale the RM layer thickness with respect to the latex particle size. At zero shear, there is an isotropic RM layer adsorbed on the latex surface. The PEO segments of the HEUR RMs are swollen in the solvent medium and the layer thickness is of the same order of magnitude as predicted by both the Flory model and the Alexander−de Gennes dense brush model (on a spherical surface) for looping polymer chains. With the application of shear, solvent molecules are progressively expelled from the RM layer. This polymer compression and solvent expulsion leads to a decrease of the shell thickness with

Figure 9. Schematic representing the shear-induced structure from a single latex−RM pair point of view. These structures are time averaged where the time scale is orders of magnitude higher than RM adsorption−desorption time scale. An effort has been made to scale the RM layer thickness (based on SANS model fit data for formulation with RM-1 polymers) with respect to the latex particle size.

increasing shear. The thinning effect is different in the flow and vorticity directions, and a shear-induced anisotropic structure evolves. For reference, the gallery spacing, h (latex interparticle distance) can be expressed as: h = 2r[((π)/(2√6ϕ))1/3 − 1], assuming a uniform distribution of particles. For ϕ = 0.28 and nominal particle hydrodynamic radius 600 Å, h = 460 Å which indicates that for a uniform distribution (neglecting aggregation), a Gaussian RM random coil polymer conformation is too small to form a bridge between neighboring latex particles. However, h is smaller than the fully stretched HEUR polymer contour length (L) reported in Table 2. In the presence of shear, Brownian motion, mixing, and collisions, the polymer chains may adopt partially or fully stretched conformations (up to the limiting contour length) and at the same time, the local gallery spacing may decrease due to particle association/ aggregation driven by the attractive potential between stickyshell particles and shear. Under these conditions, RM polymers can form bridges between the neighboring latex particles.68,69 However, the rheo-SANS data collection time-scale (∼2700 s) is orders of magnitude longer than the HEUR polymer characteristic hydrophobe adsorption−desorption time scale (∼ms). Therefore, the structural parameters obtained from the rheo-SANS measurements represent the time-averaged structure. Since the time scale for reattachment of the hydrophobe groups is much faster than the experimental time scale, information concerning dynamic interparticle bridging is not possible with these experiments. The Extent of Swelling and Effective Volume Fraction Calculation. We have demonstrated that a finite shell thickness of RM molecules on the latex surface exists at shear rates as great as 1000 s−1. The decrease in shell thickness with increasing shear rate is explained through shear driven solvent expulsion from the shell layer. We can estimate the mean RM concentration in the adsorbed layer by deriving the following equations. The rheology modifier concentration (c) in the formulation is given as: c = mRM/(ρV), where mRM is the total RM mass, ρ and V are formulation density and volume, respectively. Assuming that the volume fraction of RM in the adsorbed layer (shell) is θRM and the shell volume fraction is ϕs then: θRM = cρ/(ϕsρRM) assuming that all RM polymers are in the adsorbed layer. Recently, this assumption has been experimentally verified through pulsed field gradient (PFG) NMR spectroscopy studies for RM concentrations 100 s−1), the fractional change in the effective volume with respect to their quiescent state volume follows an inverse relationship with hydrophobe density. For example, at 1000 s−1, the normalized effective volume fraction values are 0.66, 0.72, and 0.88 for RM1-, RM-2-, and RM-3-based formulations, respectively. This observation suggests that under the same shearing condition, compared to the quiescent state, the extent of change/ deformation in the shell volume is lower for the RM polymer with multiple hydrophobe-latex contact points. On the basis of the calculated ef fective volume f ractions, the RM concentration (w/w) in the adsorbed layer can be determined using Figure 10. For different RM-based formulations, the RM concentrations in the shell are presented in Figure 12 as a function of shear rate. Alternatively, these data

ρmedium (SLDmedium − SLDshell ) + ρRM (SLDshell − SLDRM ) (11)

For any adsorbed layer (shell) volume fraction, ϕs, the SLDshell can be calculated using eq 8, as all other variables are either known or measurable quantities. By using the SLDshell value, the concentration of RM in the adsorbed layer can be obtained using eq 11. The concentration of RM in the adsorbed layer (w/w), cRM, is theoretically calculated and plotted against the effective volume f raction (=ϕ + ϕs, derived below) in Figure 10. The following values are used to generate Figure 10: c = 0.010 (w/w); ρ = 1.050 g/cm3, ρRM = 1.257 g/cm3; ρmedium = 1.034 g/cm3; SLDRM =7.29 × 10−7 Å−2; SLDmedium = 1.98 × 10−6 Å−2. The largest adsorbed layer thicknesses have the highest swelling and effective volume f raction and the lowest RM backbone concentration. While Figure 10 is prepared using the value of the RM-1 SLD as SLDRM, negligible changes are expected if either the RM-2 or RM-3 SLD values are used to prepare this plot. In order to obtain the effective hydrodynamic volume of a latex particle with an adsorbed RM layer, the volume of an ellipsoid of revolution is calculated based on the structural parameters extracted from the rheo-SANS fittings. From the symmetry of the flow tensor, we assume that the shear dependent shell thickness in the gradient direction is the same as in the flow direction. Considering this, the volume of a single latex particle with an adsorbed RM shell layer can be calculated as follows: Vellipsoid = (4/3)π(ravg + ΔRM, flow)2(ravg + ΔRM,vorticity), where ravg 1168

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Figure 11. (a) Effective volume fraction as a function of shear rate for different RM-based formulations. (b) Normalized effective volume fraction as a function of shear rate for different RM-based formulations. To generate this plot the effective volume fraction data are normalized by their corresponding quiescent state effective volume fraction.

shell layer increases to ∼0.25−0.38 with the higher values for the RM-3 formulation (high hydrophobe density). This suggests that even when the solvent molecules are expressed from the adsorbed layer during shear, the RM polymer chains are still swollen to some extent and for the shear rate window studied here; a completely dry shell condition is never reached. Consequently, the lubrication approximation between adsorbed RM polymer chains and the latex surface remains valid, and the system can be considered further with appropriate fluid mechanics modeling. Estimates of Dispersion Viscosity as a Function of Hydrodynamic Volume and Shear Rate. The viscosity of non-dilute hard particle suspensions is typically described by the Krieger−Dougherty (K−D) equation:26b −2.5ϕ * ⎡ ϕsolid ⎤ η(ϕ) = ηs⎢1 − ⎥ ⎣ ϕ∗ ⎦

Figure 12. RM layer concentration (w/w) in the shell layer as obtained from Figure 10 as a function of shear rate for different RMbased formulations.

(13)

where ϕsolid is the volume fraction of solids, ϕ* is the volume fraction for maximum packing (for spherical particles, ϕ* = 0.64), η and ηs are the sample and solvent viscosity, respectively. While the K−D equation was developed for noninteracting hard sphere dispersions, Choi and Krieger demonstrated that the model was applicable to polymer stabilized colloidal dispersions as well when the thickness of the stabilizing layer was included in calculating volume fractions.26a In a similar approach, we assume that the shear

provide us an estimation of swelling (solvent concentration in shell = 1.0 − RM concentration in shell) of the shell layers as a function of shear rate. The RM concentration in the shell layer increases with increasing hydrophobe density and increases with decreasing MWe (Table 2). At low shear, the RM polymers are swollen and their concentration (w/w) in the shell layer varies from ∼0.05−0.1. At high shear, hydrodynamic squeezing takes place and the RM concentration (w/w) in the

Figure 13. Comparison of Krieger−Dougherty equation predicted viscosity to experimental data for representative formulations. The ICI viscosities are 0.045 ± 0.005 and 0.105 ± 0.005 Pa s for RM-1- and RM-2-based formulations, respectively. 1169

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trend is RM-2 (0.105 ± 0.005 Pa s) > RM-3 (0.075 ± 0.005 Pa s) > RM-1 (0.045 ± 0.005 Pa s). The ICI viscosity is believed to originate from a combination of single particle hydrodynamic volume and transient multiparticle drag (which is not considered by the transient network theory). Therefore, while the higher hydrophobe density (or lower MWe for a fixed HEUR MW) results in lower single particle hydrodynamic volume, it should increase the number of hydrophobe exchanges (adsorption/desorption process) during a second particle collision resulting in an increase in the probability and number of bridges between particles. On the other hand, the probability of bridging is expected to decrease as the interhydrophobe contour length (Table 2) approaches the gallery spacing. Therefore, although RM-3 has higher hydrophobe density than RM-2, the significantly lower MWe of RM-3 should result in a reduction in the probability of bridges between particles, in concordance with the observed ICI viscosity trend. In a previous publication, we demonstrated that the formulation viscosity as a function of shear rate can be predicted fairly well using structural parameters associated with fractal aggregates in a K−D type relation.41 However, in that exercise, the hydrodynamic volume of the fractal structure building block (i.e., individual particle) was calculated using a shear rate independent particle diameter (without considering the contribution of an adsorbed polymer shell). Pishvaei et al.27 found that small variations in the volume fraction of aggregated particles (i.e., actual volume fraction at the shear rate including effects of adsorbed species and shear-induced desorption) could produce large effects on viscosity. The deviation was greatest close to ϕ*, the volume fraction for maximum packing, where a 0.5% change in volume fraction changed the viscosity by a factor “close to 10”. Therefore, a small difference in effective hydrodynamic volume could cause a substantial difference in formulation viscosity. The Krieger−Dougherty rheological model, based on the exact shear-dependent effective hydrodynamic volume using the rheo-SANS data, and our formulation ϕ significantly less than ϕ*, under-predicts the experimentally measured viscosity. Therefore, failure of the Krieger− Dougherty model indicates that even at the ICI shear rate, single particle hydrodynamic volume alone cannot predict the formulation viscosity. Consequently, the formulation viscosity must evolve from the RM polymer mediated (direct HEUR bridges) particle−particle residual microstructure at high shear rate in concert with some contribution from the individual particle hydrodynamic volume. The shear rate dependence of the effective hydrodynamic volume of f ractal aggregates consisting of the core−shell RM−latex particles should be used in a K−D type equation to improve the formulation viscosity prediction and is the subject of a future manuscript.

rate dependence of the viscosity can be accounted for by knowing the rate dependence of the effective volume fraction of the latex particles with adsorbed RM shell. Using the effective (hydrodynamic) volume fraction calculated from the structural analysis of the rheo-SANS data (Figure 11a), and ηs = 0.004 Pa s, the viscosity is calculated using eq 13. The Peclet number (Pe = (6πη(ravg)3γ̇)/(kBT)) for these samples ranges from about 20 to about 8 × 10−4, so we conclude that both diffusion and viscous effects are important. Therefore, the particle motion can be characterized by simple planar viscous flow and we can ignore turbulence and tumbling effects implying the K−D approach for our samples is appropriate provided the calculated effective volume f raction is a valid estimate for ϕsolid. For two selected representative formulations, the viscosity calculated using the Krieger−Dougherty equation is compared with the experimental results in Figure 13. Use of the K−D equation with ϕsolid = ϕ = 0.28 should give a Newtonian viscosity profile (η(ϕ = 0.28) = 0.01 Pa s) over the entire shear rate range as presented by the solid black line in Figure 13. The viscosity prediction based on ϕsolid = ef fective volume f raction gives a shear rate dependent viscosity profile where the predicted viscosity values are orders of magnitude lower than the experimental data. However, the predicted viscosity data follow the same shear thinning nature as observed experimentally and at very high shear rate (∼1000 s−1) tend to converge toward the viscosity value calculated based on bare latex particles. It is expected that the K−D equation underpredicts the viscosity at lower shear rates where interparticle interactions dominate rheological behavior. However, even in the high shear rate region, the predicted and measured viscosities do not match. The ICI viscosities of these formulations are 0.045 ± 0.005 and 0.105 ± 0.005 Pa s for the RM-1- and RM-2-based formulations, respectively. While the viscosity prediction is limited to the highest γ̇ applied during the rheo-SANS experiment (2500 s−1 or 1000 s−1 depending on the formulation, which is lower than the ICI shear rate of 10000 s−1), it is clear that the prediction will fail to predict the ICI viscosity. However, it is apparent from Figure 13 that for different latex−RM formulations, the K−D equation prediction tends to converge toward the measured viscosity at high shear rates. We expect agreement in the terminal Newtonian plateau regime, beyond the range accessible in this study, at extremely high shear rates perhaps above about 105 s−1 where aggregates and structure are completely disrupted and only single particles remain. As mentioned earlier, the flow curves and viscoelastic response imply a larger scale, interparticle aggregate structure. With increasing shear rate, the large assemblies equilibrate into smaller clusters with length-scales accessible by SANS. Therefore, in the high shear regime, an increase in low-q scattering intensity is observed. With increasing shear rate, shell thickness decreases, while RM concentration in the shell increases. Anisotropy of the adsorbed HEUR layer presumably arises from a nonuniform stress distribution and could also be associated with relatively greater HEUR bridge density in the vorticity direction. The adsorbed layer thickness, as a function of hydrophobe density (number of hydrophobes/polymer chain), follows the trend: high (RM-3) < intermediate (RM2) < low (RM-1). The RM polymer with the lowest MWe is expected to form the thinnest shell and the lowest single particle hydrodynamic volume, which agrees with the rheoSANS result (Figure 11a). However, the ICI viscosities do not follow the same trend. For these formulations, the ICI viscosity

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SUMMARY In this work, the microstructure of a rheology modifier (RM) layer adsorbed onto the surface of an acrylic latex particle under shear has been studied. The formulation mimics architectural and industrial coatings, which are subjected to a wide range of application shear rates. The latex−RM formulation microstructure and corresponding rheological behavior are significantly different from that of aqueous HEUR solutions. For the formulation composition studied here (1% RM (w/w) and latex volume fraction 0.28), essentially all HEUR RMs are adsorbed on the particle surface. The focus of this study was to understand the local microstructure (adsorbed layer thickness) 1170

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for RM polymers with different hydrophobe density. It is assumed that the RM adsorbs to the surface of the latex forming a core−shell structure. Using rheo-SANS, the adsorbed RM layer thickness on the latex surface was extracted as a function of shear rate by fitting a spherical core−shell form factor along with a hard sphere structure factor model to the data. Under quiescent conditions, the RM shell thickness was found to be isotropic and inversely related to the RM polymer hydrophobe density. In the shell layer, the adsorbed RM polymers are highly swollen. As the hydrophobe density increases, multiple hydrophobe−latex contacts produce topological constraints to the backbone PEO chain conformation and the adsorbed layer thickness decreases. The extracted shell layer thickness from the SANS data agrees with the theoretical predictions for loop forming chains according to both the Flory calculations for swollen polymers and the Alexander−de Gennes dense brush theory. Under shear, the adsorbed shell layer becomes anisotropic with respect to the flow and the rate of decrease of the shell thickness in the flow direction is greater than in the vorticity direction. Further, the adsorbed layer thickness becomes almost independent of the hydrophobe density at high shear rates (∼1000 s−1 and above). As the RM layer thickness decreases, the scattering length density (SLD) of the adsorbed layer (shell) decreases as well. The RM layer SLD is close to the solvent medium SLD under quiescent conditions (for RM-1), indicating a highly swollen state of the RM polymers. With increasing shear, the RM layer SLD tends toward the pure RM SLD in both the flow and vorticity directions. This suggests that under shear, solvent molecules are expelled from the adsorbed layer leading to a thinning of the shell. The thinning effect is different in the flow and vorticity directions; therefore, a shearinduced anisotropic structure evolves. On the basis of materials balance calculations and the shell SLD data, the mean concentration (w/w) of RM molecules in the shell (adsorption layer) was calculated. At low shear rate, the RM concentration in the adsorbed layer is ∼0.05−0.1 which increases to ∼0.25− 0.38 at high shear rates with higher values for the RM polymer with higher hydrophobe density. Hence, even when shear driven solvent expulsion from the shell occurs, the RM molecules remain attached to the latex particles, and the adsorbed layer HEUR PEO backbone remains in a partially swollen state. Finally, the shear rate dependent ef fective hydrodynamic volume f ractions were calculated for the core−shell structure based on the rate dependence of the extracted ellipsoid of revolution. The calculated volume fraction was used to predict the viscosity of the formulation applying the Krieger− Dougherty equation. The predicted viscosity shows a similar shear thinning profile as observed from the experimental flow curve data; while the values are orders of magnitude lower, they suggest convergence at much higher shear rates than the highest shear rate studied here. This observation indicates that hydrodynamic volume alone cannot explain the viscosity of these systems. Consequently, we conclude that the high shear formulation viscosity must evolve from the microstructure (RM polymer mediated particle−particle interaction) in concert with some contribution from the individual particle (latex with adsorbed RM shell) hydrodynamic volume. The large-scale structure of the formulation under shear and at rest will be reported in a future communication.

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APPENDIX

A form factor/structure factor model, as described in eq 1, was fitted to the scattering data using a homogeneous spherical core−shell form factor, P(q), despite the anisotropy in the scattering patterns observed under shear. We illustrate the method with an example using oriented ellipsoidal particles. According to our coordinate system, X−Y−Z are defined as the flow, gradient (incident beam) and vorticity directions, respectively (Figure 1). Following the procedure described by Shibayama and coworkers59a we define two spherical coordinate systems, lab OXYZ and sample O-xyz, an azimuthal angle ψ from the X−Y plane to Z along the vorticity axis, and a polar angle μ from the X axis in the X-Y shear plane, and calculate spherical coordinates as follows: x = q cos ψ cos μ, y = q cos ψ sin μ, z = q sin ψ, and then rotate the sample coordinate system clockwise by an angle α about the X axis ⎡ qx ⎤ ⎡ 0 0 ⎤⎡ q cos ψ cos μ ⎤ ⎢ ⎥ ⎢1 ⎥ ⎥⎢ q ̂ = ⎢ qy ⎥ = ⎢ 0 cos α −sin α ⎥⎢ q cos ψ sin μ ⎥ ⎢ q ⎥ ⎣ 0 sin α cos α ⎦⎢ ⎣ q sin ψ ⎥⎦ ⎣ z⎦ ⎡ ⎤ q(cos ψ cos μ) ⎢ ⎥ = ⎢ q(cos α cos ψ sin μ − sin α sin ψ )⎥ ⎢ ⎥ ⎢⎣ q(sin α cos ψ sin μ + cos α sin ψ )⎥⎦

The scattering amplitude function for ellipsoids can be written as: fellipsoid (q ̂, R̂ ε) =

3J1(q ·̂ R̂ ε) q ·̂ R̂ ε

where Rε is the effective radius defined by Rε = ∥R̂ ε∥ and where R̂ ε = R(cos ψ cos μ, cos ψ sin μ, (1 + ε) sin ψ).70 For perfectly oriented ellipsoids, the long axis (1+ε)R is oriented in the vorticity direction Z, with α = 0°. The scattering intensity functions in the flow and vorticity directions can be obtained by setting ψ = 0°, where qx = q cos μ, qy = q sin μ, qz = 0; and ψ = 90° where qx = 0, qy = 0, qz = q respectively. We observe that (q̂ · R̂ ε)∥ = qR and (q̂ · R̂ ε)⊥ = q (1 + ε) R so one can write P (q) = Pellipsoid(q; α = 0, ψ = 0) = || f (q)̂ ||2 ⎡ 3J (qR ) ⎤2 ⎥ =⎢ 1 ⎣ qR ⎦

and P⊥(q) = Pellipsoid(q; α = 0, ψ = 90) = || f⊥ (q)̂ ||2 ⎡ 3J (q(1 + ε)R ) ⎤2 ⎥ =⎢ 1 ⎣ q(1 + ε)R ⎦

Hence, for oriented ellipsoids, the sector averaged 1D scattering profiles can be appropriately modeled using a spherical form factor where the sphere radius represents the particle dimension along those axes.59a,c,d A schematic representation of the above concept is presented in Figure 14. 1171

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(9) Winnik, M. A.; Yekta, A. Associative polymers in aqueous solution. Curr. Opin. Colloid Interface Sci. 1997, 2 (4), 424−436. (10) (a) Berret, J. F.; Sereo, Y.; Winkelman, B.; Calvet, D.; Collet, A.; Viguier, M. Nonlinear rheology of telechelic polymer networks. J. Rheol. 2001, 45 (2), 477−492. (b) Kaffashi, B.; Barmar, M.; Eyvani, J. The steady state and dynamic rheological properties of telechelic associative polymer solutions. Colloids Surf. A: Physicochem. Eng. Aspects 2005, 254 (1−3), 125−130. (c) Barmar, M.; Barikani, M.; Kaffashi, B. Steady shear viscosity study of various HEUR models with different hydrophilic and hydrophobic sizes. Colloids Surf. A: Physicochem. Eng. Aspects 2005, 253 (1−3), 77−82. (d) Barmar, M.; Kaffashi, B.; Barikani, M. Investigating the effect of hydrophobic structural parameters on the thickening properties of HEUR associative copolymers. Eur. Polym. J. 2005, 41 (3), 619−626. (e) Tam, K. C.; Jenkins, R. D.; Winnik, M. A.; Bassett, D. R. A structural model of hydrophobically modified urethane-ethoxylate (HEUR) associative polymers in shear flows. Macromolecules 1998, 31 (13), 4149−4159. (11) (a) Ahn, K. H.; Osaki, K. Mechanism of Shear Thickening Investigated by a Network Model. J Non-Newton Fluid 1995, 56 (3), 267−288. (b) Cifre, J. G. H.; Barenbrug, T. M. A. O. M.; Schieber, J. D.; van den Brule, B. H. A. A. Brownian dynamics simulation of reversible polymer networks under shear using a non-interacting dumbbell model. J Non-Newton Fluid 2003, 113 (2−3), 73−96. (c) Tripathi, A.; Tam, K. C.; McKinley, G. H. Rheology and dynamics of associative polymers in shear and extension: Theory and experiments. Macromolecules 2006, 39 (5), 1981−1999. (d) Vaccaro, A.; Marrucci, G. A model for the nonlinear rheology of associating polymers. J Non-Newton Fluid 2000, 92 (2−3), 261−273. (e) Vandenbrule, B. H. A. A.; Hoogerbrugge, P. J. Brownian Dynamics Simulation of Reversible Polymeric Networks. J. Non-Newton Fluid 1995, 60 (2− 3), 303−334. (f) Wang, S. Q. Transient Network Theory for ShearThickening Fluids and Physically Cross-Linked Systems. Macromolecules 1992, 25 (25), 7003−7010. (g) Tanaka, F. Thermodynamic Theory of Network-Forming Polymer-Solutions 0.1. Macromolecules 1990, 23 (16), 3784−3789. (h) Tanaka, F. Thermodynamic Theory of Network-Forming Polymer-Solutions 0.2. Equilibrium Gelation by Conterminous Cross-Linking. Macromolecules 1990, 23 (16), 3790− 3795. (i) Tanaka, F.; Edwards, S. F. Viscoelastic Properties of Physically Cross-Linked Networks 0.1. Nonlinear Stationary Viscoelasticity. J. Non-Newton Fluid 1992, 43 (2−3), 247−271. (j) Tanaka, F.; Edwards, S. F. Viscoelastic Properties of Physically Cross-Linked Networks 0.2. Dynamic Mechanical Moduli. J. Non-Newton Fluid 1992, 43 (2−3), 273−288. (k) Tanaka, F.; Edwards, S. F. Viscoelastic Properties of Physically Cross-Linked Networks 0.3. Time-Dependent Phenomena. J. Non-Newton Fluid 1992, 43 (2−3), 289−309. (l) Tanaka, F.; Edwards, S. F. Viscoelastic Properties of Physically Cross-Linked Networks - Transient Network Theory. Macromolecules 1992, 25 (5), 1516−1523. (12) (a) Suzuki, S.; Uneyama, T.; Inoue, T.; Watanabe, H. Rheology of Aqueous Solution of Hydrophobically Modified Ethoxylated Urethane (HEUR) with Fluorescent Probes at Chain Ends: Thinning Mechanism. Nihon Reoroji Gakkaishi 2012, 40 (1), 31−36. (b) Suzuki, S.; Uneyama, T.; Inoue, T.; Watanabe, H. Nonlinear Rheology of Telechelic Associative Polymer Networks: Shear Thickening and Thinning Behavior of Hydrophobically Modified Ethoxylated Urethane (HEUR) in Aqueous Solution. Macromolecules 2012, 45 (2), 888−898. (c) Suzuki, S.; Uneyama, T.; Watanabe, H. Concentration Dependence of Nonlinear Rheological Properties of Hydrophobically Modified Ethoxylated Urethane Aqueous Solutions. Macromolecules 2013, 46 (9), 3497−3504. (d) Uneyama, T.; Suzuki, S.; Watanabe, H. Concentration dependence of rheological properties of telechelic associative polymer solutions. Phys. Rev. E 2012, 86 (3), 031802. (13) (a) Annable, T.; Buscall, R.; Ettelaie, R. Network formation and its consequences for the physical behaviour of associating polymers in solution. Colloids Surf. A: Physicochem. Eng. Aspects 1996, 112 (2−3), 97−116. (b) Annable, T.; Buscall, R.; Ettelaie, R.; Whittlestone, D. The Rheology of Solutions of Associating Polymers - Comparison of

Figure 14. Schematic of using spherical core−shell form factor to extract structural information on anisotropic latex−RM pair.

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ASSOCIATED CONTENT

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AUTHOR INFORMATION

S Supporting Information *

Extracted model fit parameters and error estimates from rheoSANS data and a comparison of sector-average fitting to full 2D SANS fitting for a representative data set. This material is available free of charge via the Internet at http://pubs.acs.org/. Corresponding Author

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

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Drs. James Bohling and John Rabasco for latex and rheology modifier synthesis, respectively. We acknowledge Dr. David Mildner and Dr. Paul Butler for their help at the NG7 30m SANS beamline at the NCNR. We also thank Drs. Valeriy V. Ginzburg, Melissa M. Johnson, Mike Bender, Arthur Leman, Barrett Bobsein, Susan J. Fitzwater, John Rabasco, Aslin Izmitli, Kebede Beshah, and many other Dow colleagues for helpful discussions. We thank Thomas Suder for paint formulation preparation. Dow Coatings Materials, a business unit of The Dow Chemical Company, supported this research.

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REFERENCES

(1) Schaller, E. J.; Glass, J. E. Polymers as Rheology Modifiers, 1st ed.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991. (2) Howard, P. R.; Leasure, E. L.; Rosier, S. T.; Schaller, E. J. System Approach to Rheology Control. In Polymers as Rheology Modifiers, 1st ed.; Schulz, D. N., Glass, J. E., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 1991; pp 207−221. (3) Emmons, W. D.; Stevens, T. E. Polyurethane Thickeners in Latex Compositions. US Patent 4,079,028, 1978. (4) Kruse, U.; Crowley, B. C.; Mardis, W. S. Water dispersible, modified polyurethane thickener with improved high shear viscosity in aqueous systems. US 5,023,309, 1991. (5) Sau, A. C. Hydrophobically modified poly(acetal-polyethers). US 5,574,127, 1996. (6) Broadbent, R. W.; Breindel, K. Polymeric thickeners for aqueous compositions. US 6,107,394, 2000. (7) Glancy, C. W.; Steinmetz, A. L. Water based coating composition containing an aminoplast-ether copolymer. US 5,629,373, 1997. (8) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999. 1172

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NOTE ADDED AFTER ASAP PUBLICATION This article posted ASAP on January 27, 2014. The abstract graphic has been revised. The correct version posted on January 29, 2014.

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