Article pubs.acs.org/Macromolecules
Shear-Dependent Interactions in Hydrophobically Modified Ethylene Oxide Urethane (HEUR) Based Coatings: Mesoscale Structure and Viscosity Antony K. Van Dyk,*,§ Tirtha Chatterjee,† Valeriy V. Ginzburg,† and Alan I. Nakatani∥ §
Dow Coatings Materials, The Dow Chemical Company, Collegeville, Pennsylvania 19426, United States Materials Science and Engineering, The Dow Chemical Company, Midland, Michigan 48674, United States ∥ Analytical Sciences, The Dow Chemical Company, Collegeville, Pennsylvania 19426, United States †
S Supporting Information *
ABSTRACT: We have investigated the in situ mesoscale structure of paint formulations under shear using ultra small-angle neutron scattering (rheo-USANS). Contrast match conditions were utilized to independently probe the latex binder particle aggregates and the TiO2 pigment particle aggregates. Two different latex chemistries and two different hydrophobically modified ethylene oxide urethane (HEUR) rheology modifiers were studied. The rheo-USANS data reveal that both the latex particles and the TiO2 particles form transient aggregates which are fractal in nature. The structures depend on the chemistry of the binder particles, the type of rheology modifier present and the shear stress imposed upon the formulation. The aggregate size of both the latex and pigment generally decreases with increasing shear stress. In two of the formulations studied, the latex and TiO2 correlation lengths remain large even at high shear stress and are characteristic of TiO2 crowding. In a third formulation, shear induces string-like aggregate structures of TiO2, and a further increase in shear leads to pigment particles becoming more uniformly dispersed. The changes in the latex and pigment transient aggregate structures correlate with the changes observed in their viscosity flow curve profiles. We have used this correlation to develop an elementary viscosity prediction model based on the structural parameters extracted from the rheo-USANS data. Using a single fitting parameter and only the latex transient fractal aggregate structural parameters, good agreement between the measured and calculated viscosity is obtained. This implies that the structural parameters extracted from the scattering data are representative of the colloidal structure under shear and that energy dissipation from transient fractal aggregates of latex is the predominant mechanism of viscosity creation in HEUR thickened latex paints.
1. INTRODUCTION In colloidal suspensions, a wide range of particle organization develops through the interplay between Brownian, repulsive, and attractive forces.1 The macroscopic properties of the colloidal suspension are governed by the spatial organization of particles or collections of particles (colloidal aggregates). With the application of a flow field, these aggregates rearrange (internally through deformation, or externally through breaking up or by further aggregation) to accommodate hydrodynamic and colloidal forces.2 As a consequence, colloidal solutions and suspensions show various nonlinear rheological properties (e.g., shear thinning, thixotropy, dilatancy etc.), which are caused by a change in structure imposed by the external flow field. Significant progress has been made to understand the structural alteration of model colloids under shear.2a,c,3 However, © 2015 American Chemical Society
commercial applications of colloidal suspensions such as paints in the coatings industry are complex in nature and the relationship between the model colloidal systems to commercial materials is often difficult to assess.4 Paints are complex formulations of polymeric binders, inorganic pigments, dispersants, surfactants, colorants, rheology modifiers, and other additives. The preparation of these formulations is also a complex process and depends on the ingredient compositions, order of mixing, processing parameters (such as mixing speed, power per unit volume, and the formulation pH among others). In general, the composition and chemistry (molecular) govern Received: October 24, 2014 Revised: January 27, 2015 Published: March 3, 2015 1866
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interplay with shear is needed to predict the overall formulation viscosity via hydrodynamic drag. The transient fractal aggregates studied here are “soft” and comprise transient polymeric bridges between particles, rather than rigid colloidal aggregates. Considering that for both latex aggregates (via RM molecules8a) and pigment aggregates (via polymeric dispersant molecules11) these bonds are transient molecular bridges of relatively long flexible polymers in solution we suppose that direct mechanical coupling (bending moments, torque) between particles is negligible, in contrast to rigid fractal aggregates.12 However, it is instructive to review the behavior of more familiar rigid aggregates under shear. Colloidal aggregates are often described in terms of two limiting cases: diffusion limited type aggregation (DLA-type) and reaction limited type aggregation (RLA-type). For DLAtype aggregates, the growth of the aggregates occurs through a fast collision process (single collisions with the aggregate limited by the particle diffusion time scale), resulting in open, branched structures with low fractal dimension, typically between 1.5 and 1.9.13 On the other hand, RLA-type aggregation occurs slowly (low sticking probability resulting in multiple collisions with the aggregate) producing denser aggregates. Because of the slow rate of aggregation, the aggregating particles have the opportunity to explore a large number of mutual configurations. Therefore, RLA-type aggregates form more dense or compact structures with a high fractal dimension, typically ranging from 2.3 to 3.0.13 Lagrangian particle simulations for non-Brownian particle aggregates under a linear flow field conducted by Harada and co-workers14 and Higashitani and co-workers15 provide insights on the evolution of the characteristic length scale and fractal dimension of DLA and RLA type systems under flow. The simulation studies analyzed the effect of shear stress on colloidal aggregates with different initial fractal dimensions as a function of time. The simulation results report both types of aggregates decrease in size with increasing time. The DLA-type aggregates break at early time and the resultant smaller aggregates are denser (i.e., an increase in fractal dimension). The DLA-type structures initially have many branches and these branches readily break under stress. In contrast, below the yield stress, the RLA-type aggregates slowly disperse (characteristic length decreases) with little change in fractal dimension. When the hydrodynamic stress exceeds the yield stress, larger aggregates rupture into smaller aggregates but still largely preserve their density (i.e., little or no change in their fractal dimension values). Structural studies of commercially relevant paints in the wet state have been difficult to accomplish due to the opacity of the suspensions, relatively high volume fraction of solids, and relatively large particle sizes (100−1000 nm). Static light scattering is capable of providing size information but is limited to dilute systems in which multiple scattering has been eliminated. A flow cell can be coupled with a particle sizer in the dilute limit,16 however, a light scattering based technique is not suitable to probe a mixture of different types of particles (e.g., latex and TiO2) independently. Microscopy methods have been very successful on dried films and diluted suspensions deposited on grids, however, examination of higher concentration suspensions in the wet state involves very specialized techniques.17 We have previously reported small-angle neutron scattering under shear (rheo-SANS) data10 on latex-RM mixtures, which indicated that the range of SANS scattering vector, q (= 4π/λ sin(θ/2), where λ is the incident neutron wavelength and θ is the scattering angle), does not allow a rigorous determination of
the formulation microstructure (colloidal/mesoscale). In typical paint formulations, the volume fraction of solids ranges from about 15% to 45% v/v, with about 5% to 35% v/v being polymeric latex binders. The rest of the solids are primarily inorganic pigment particles (TiO2 and extender such as CaCO3, clay, silicates, talc, etc.). Latex particles for coatings applications range from fairly monodisperse, to bimodal and polydisperse in nature. The nominal diameters range from about 100 to 300 nm, a length scale where surface chemistry is important due to high solid−liquid interfacial area. Additionally, in this colloidal size range interparticle forces, Brownian forces and hydrodynamic effects are of comparable magnitude. On the other hand, the nominal pigment and extender particle diameters range from 300 to 10000 nm with significant size polydispersity. Predicting the paint viscosity as a function of shear rate is a challenging task due to the formulation complexity containing structures with different hierarchical length scales and the alteration of structures on these different length scales under the influence of an external flow field. A commercially successful paint exhibits a desired viscosity profile over shear rates from ∼10−5 s−1 for settling to >104 s−1 for brushing, rolling, and spray applications. To achieve the desired viscosity profile over a wide shear rate range, rheology modifiers (RMs) such as hydrophobically modified ethylene oxide urethane (HEUR) polymer are widely used in the coatings industry.5 HEUR polymers are nonionic, associative thickeners consisting of a hydrophilic poly(ethylene oxide) (PEO) backbone with various combinations of internal, terminal, and in some specific cases, pendant hydrophobes.6 Typical HEUR RM concentrations in coatings formulations range from about 0.02 to 5% (w/w) based on total formulation weight. In addition, various surfactant packages (latex stabilizing and formulation surfactants) are also added to the formulations which typically vary from 0.1 to 1.0% (w/w). For a pure HEUR RM in aqueous solution, the thickening mechanism is reasonably well understood.7 In contrast, due to formulation complexity, a full understanding of the fundamentals of paint thickening is still somewhat elusive, but has been attributed to changes in the hierarchical structure existing in the paint. It has been established in various experimental studies that the hydrophobic groups of the HEUR molecules preferentially adsorb onto latex surfaces.8 A proposed mechanism for the thickening derives from the synergy between the transient network of the HEUR RM molecules in solution with a degree of adsorption of HEUR molecules to latex surface, where the latter provides hydrodynamic drag and enhanced network connectivity.5c,9 This interpretation does not consider the contribution of particle−particle (mesoscale) interactions. In a previous publication it has been demonstrated that, below a threshold concentration, which is a function of the RM and surfactant concentrations and chemistry, latex particle size, and latex concentration, all the HEUR hydrophobes are preferentially adsorbed to surfaces of the latex particles,8a and furthermore are not displaced by shear.8c,10 Therefore, it appears that the concentration of HEUR molecules remaining in solution is insufficient to provide an effective HEUR transient network thickening mechanism. Precise measurement of the latexHEUR particle hydrodynamic volume as a function of shear rate conclusively demonstrated that a conventional effective hard-sphere dispersion rheology model (Krieger−Dougherty) cannot explain the experimentally measured viscosity even in the high shear rate regime.10 We hypothesize that a critical understanding of the aggregated particle structures and their 1867
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above for the synthesis in pure H2O. 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. 2.2. Rheology Modifiers (RM). Two different commercial hydrophobically modified ethoxylated urethane (HEUR)26 associative thickeners, referred to as RM-1 and RM-4, were used in this study. Their synthesis procedure can be found in a previous publication.10 The weight-average molecular weights (Mw in kg/mol) are ∼30 and ∼65 for RM-1 and RM-4, respectively. Cm denotes the effective/ equivalent number of methylene groups representative of the combined hydrophobic contributions of the isocyanate linker and alcohol capping agent moieties.27 The Cm values are ∼12 and ∼18 for RM-1 and RM-4, respectively. The hydrophobe density is defined as the average number of hydrophobes per polymer chain.10 The hydrophobe density is 2 for RM-1 and >3 for RM-4. 2.3. Paint Formulation. The formulations used in this work are all at moderately concentrated constant latex volume fraction (ϕlatex = 0.28) and TiO 2 pigment (DuPont Ti-Pure R-706 pigment dispersed with TAMOL (Trademark of the Dow Chemical Company (“Dow”) or an affiliated company of Dow) 1124 Dispersant 1% w/w based on pigment) volume fraction (ϕTiO2 = 0.07) which are typically used in commercial coatings. The sodium lauryl sulfate surfactant concentration was fixed at 0.2% (w/w), below the critical micelle concentration, CMC, but within the range used in commercial coating formulations. The HEUR RM concentration (0.2% w/w in paint-1, and 0.04% w/w in paint-2 and -3) was kept low to avoid substantial changes in the RM shell microstructure in the presence of shear as reported previously.10 The total volume solids (VS) in the formulation was 0.35, where the pigment volume concentration [PVC = 100*ϕTiO2/(ϕTiO2+ϕlatex)] was 20%. Three different pairs of paints based on latex-A1, latex-A2, or latex-B were studied. For each pair, mixtures of H2O and D2O were used as solvent. The relative amount of D2O was chosen to match the neutron scattering length density (SLD) of either the latex particles or the pigment particles (discussed below). At the TiO2 SLD, the TiO2 particles were contrast matched and the scattering from latex particles (latex-A1, latex-A2, and latex-B) was collected. At the respective latex SLDs, the latex particles (A1/A2/B) were contrast matched, and the scattering exclusively from the TiO2 particles was collected. It should be noted that under shear, the latex particles are nondeformable.10 Summaries of the paint compositions are presented in Table 1.
large scale structures (especially at zero and low shear). The q range of the ultra small-angle neutron scattering (USANS) instrument is more suitable, since the lower minimum q value allows larger length scales to be examined. In the past, the structure of different complex fluids including anisotropic nanoparticle networks in polymer under quiescent and shear conditions,18 hydrogels,19 and polymer stabilized pigment (TiO2) dispersion in slurries11 and in epoxy coatings,20 etc., have been studied using the USANS technique. In this work, we concentrate on understanding the shear induced changes on the mesoscale (larger than a single particle) structure of latex binder aggregates and pigment aggregates. Since the inorganic pigment (TiO2) and polymer latex binder constitute the two highest solid volume fraction components of the paints, understanding the relationship between the latex structure and pigment structure is key to advancing the commercial understanding of these materials. From a formulator’s perspective, the ability to understand and control the interplay between these various solid components (in the presence and absence of flow) is critical to pigment utilization and obtaining a high quality coating. Application of a contrast matching technique, where the neutron scattering length density of the suspending medium (water/D2O mixture) is matched to either the latex or pigment, permits probing the shear dependence of the TiO2 aggregates and latex binder particle aggregate structures independently. The characteristic length scales and fractal dimensions of the latex binder aggregates and TiO2 particle aggregates as a function of shear rate are extracted by fitting fractal based models to the rheo-USANS data. The shear dependence of the structural evolution for different latex binder aggregates and TiO2 aggregates are discussed in terms of their aggregation mechanism and reconciled with reported simulation studies. In order to achieve best fits to the scattering data, it was necessary to assume that the observed scattering arises from a linear combination of scattering from particle aggregates and free particles.21 We develop an elementary viscosity prediction expression based on the Rouse22 model, which gives a fundamentally different dependence on the fractal dimension, Df, compared to a hydrodynamic viscosity model based on the approach of Krieger and Dougherty.23 Good agreement between the model prediction and experimentally measured flow curves establishes that for the formulations studied here, the dissipation of energy from the transient fractal aggregates of latex particles is primarily responsible for the observed viscosities.
Table 1. Descriptions of the Paint Formulations Used in This Study
latex
total volume solids (VS)b (v/ v)
pigment volume concentration (PVC)c (%)
rheology modifier
RM concentration (%, w/w)
A1 A2 B
0.35 0.35 0.35
20 20 20
RM-1 RM-4 RM-4
0.20 0.04 0.04
2. EXPERIMENTAL SECTION 2.1. Latex Synthesis. Three different acrylic polymer latices were synthesized separately in pure H2O and in pure D2O by conventional emulsion polymerization techniques. Two different batches of latex-A were prepared, which are referred to as latex-A1 and latex-A2. These were single stage copolymers of butyl acrylate and methyl methacrylate with 1.0% methacrylic acid, which were thermally initiated with ammonium persulfate, stabilized with 0.6% sodium lauryl sulfate (SLS, based on monomer), and the batches were neutralized with ammonia (28% in water).24 Samples for this study were prepared using a preformed polymer seed. Latex-A1 and latex-A2 have nominal particle diameters 1780 ± 15 and 1220 ± 20 Å, respectively. The other latex is referred to as latex-B, which is also a butyl acrylate and methyl methacrylate copolymer binder but with 2-methacryloyloxyethyl phosphate stabilization.25 Latex B has a nominal particle diameter of 2000 ± 200 Å. All reported particle diameters were measured using a Brookhaven BI-90 particle sizer. Latex-A1 and latex-B are considered to be of comparable nominal particle size. Synthesis of the complementary latex binders in pure D2O was performed using 99% D2O following a process similar to that described
paint
a
paint-1 paint-2 paint-3
a All paints contained a fixed (0.2%, w/w) amount of surfactant (sodium lauryl sulfate). bVS = ϕlatex + ϕTiO2, ϕlatex = latex volume concentration, ϕTiO2 = pigment (TiO2) volume concentration. cPVC = 100 × ϕTiO2/(ϕTiO2 + ϕlatex).
2.4. Contrast Match Experiments. The contrast match points of latex-A1, latex-A2, and latex-B were determined using a series of diluted suspensions prepared by mixing the D2O and H2O based latices in various ratios. Measurements were performed on the NG-7 30 m SANS instrument, using an incident wavelength of 9 Å and standard collimation to obtain the total neutron scattering intensity from each latex sample at the different ratios of H2O to D2O. Similar experiments were conducted on dilute TiO2 suspensions to obtain the 1868
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Macromolecules Table 2. Contrast Match Points for Latex and TiO2 component
volume fraction D2O for matching
scattering length density (×1010 cm‑2)
latex-A1 latex-A2 latex-B TiO2
0.223 0.223 0.197 0.445
0.981 0.981 0.806 2.53
a
particle size (R0) (radius, Å)a 703 526 831 1476
± ± ± ±
0.1 0.1 15 6
polydispersitya 0.10 0.15 0.25 0.29
± ± ± ±
0.001 0.001 0.016 0.002
BI 90 particle size (RH) (radius, Å) 890 ± 8 610 ± 10 1000 ± 100
Particle sizes ⟨R0⟩, and polydispersity indices (p), were measured in previous experiments (unpublished data) and used in this study.
contrast match point for the TiO2. The scattering intensity, I, is directly proportional to the number of scattering particles (∼ volume fraction, ϕ), the particle volume (Vp), the square of the difference in neutron scattering length densities (SLD), (Δρ)2, the structure factor, S(q) (where for the dilute case S(q) = 1), and the particle shape (or the form factor), P(q), according to eq 1, where bgd is the incoherent background scattering contribution:
I(q) = ϕVp(Δρ)2 P(q)S(q) + bgd
vorticity (Z) direction; therefore the structural parameters obtained from the USANS data are more representative of the structure in the flow (X) direction. The in situ rheo-USANS measurements for each sample were performed in the following order of shear rates: (2500), 1000, 100, 10, 1, (0.1), 1, 10, 100, and 1000 s−1 (2500 s−1). The parentheses indicate that only some of the formulations were tested at 0.1 and 2500 s−1. Because of time limitations, only selected formulations were studied at 0.1 s−1. Some formulations with higher viscosity were more prone to expulsion from the shear cell at high shear rates. Therefore, the maximum shear rate imposed upon these samples was 1000 s−1. The measurements were done in this order to investigate potential hysteresis in the scattering behavior. The measurement time for each shear rate was approximately 3.4 h. Transmission measurements were obtained for the open beam and each sample under quiescent conditions. We assume that sample transmissions do not change as a function of shear rate. All data were corrected for the empty cell scattering. Empty cell scattering and transmission corrected intensity was placed on an absolute scale using an attenuated empty beam. Finally, the intensity data were desmeared to correct for the instrument slit smearing effect. Data correction, reduction, and desmearing were performed using code based on IGOR Pro software routines available from the NCNR.30 2.6. Rheology. All viscosity measurements were conducted using a stress controlled rheometer (TA Instruments, AR-G2) with a 40 mm diameter stainless steel, upper parallel plate and Peltier plate temperature controlled lower plate. The test temperature was maintained at 25 °C, and the fixtures were zeroed before each sample loading. The 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.
(1)
At the contrast match point, the mixed solvent SLD and the scattering particle SLD are equal, so that Δρ ∼ 0, and the total scattering intensity has a minimum value. The scattering intensity at the contrast match point is not zero due to incoherent scattering primarily from the hydrogen bearing components. The contrast match point can be obtained by fitting a plot of the total scattered intensity, I, as a function of the D2O concentration to a quadratic function. The value of the D2O concentration at the minimum of the parabola yields the contrast match point for the primary scattering particles. Alternatively, the square root of the total detector counts can be plotted as a function of the D2O concentration resulting in a V-shaped plot. Linear regression fits to each arm of the plot are performed and the intersection of the two lines provides the contrast match point for the particle. The experimentally determined contrast match points, determined from averaging the results of the two methods above, are given in Table 2. 2.5. Rheo-Ultra Small-Angle Neutron Scattering (Rheo-USANS). The USANS from the samples was measured as a function of shear rate on the BT-5 Perfect Crystal Diffractometer USANS instrument at the National Institute of Standards and Technology (NIST) Center for Neutron Research (NCNR) in Gaithersburg, MD.28 The perfect crystal diffractometer (BT5) instrument has a high angular resolution in the scattering plane and a relatively poor resolution in the perpendicular (vertical) direction (slit smearing). The covered scattering vector (q) range is 3 × 10−5 Å−1 to 2.66 × 10−3 Å−1. A neutron wavelength of 2.38 Å with a full-width at half-maximum (fwhm) of the scattering vector in vertical direction (Δqv) of 0.117 Å−1 was used. The in situ rheo-USANS measurements were performed using the NIST Boulder shear cell (Couette geometry)29 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. Approximately 11.75 mL of sample was required to fill the shear cell. The shear cell is equipped with a vapor trap which was filled with H2O to prevent solvent evaporation 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 ambient temperature. The outer cylinder was rotated, with the rotation axis vertical, to generate a simple shear field between the cylinders. The shear flow is horizontal and is defined as the X coordinate of the flow field. The incoming neutron beam (as well as the outgoing beam) is along the Y coordinate of the flow field or velocity gradient direction and parallel to the shear plane (X−Y). The data were collected on a detector with slit smearing. The resolution in X is 2 × 10−5 Å−1, while the resolution in Z is 0.2 Å−1. Because of the instrument collimation, intensity values are integrated in the Z direction and no conclusions on structural anisotropy information can be extracted from the USANS data, in contrast to the case for the two-dimensional SANS detector. Furthermore, due to slit smearing, the detector has limited resolution in the
3. RESULTS AND DISCUSSION 3.1. Rheo-USANS Intensity Profiles. Rheo-USANS data from all six samples (3 paint formulations, each at two contrast match conditions) as a function of shear rate are presented in Figure 1a−f. The scattering intensity, I(q), is plotted as a function of scattering vector, q, at different shear rates for each formulation. Changes in the shear rate were implemented in a stepwise fashion and no hysteresis in the results was observed. Therefore, only the data collected under the descending shear cycle have been presented. In parts a, c, and e of Figure 1, the scattering from the latex particles/aggregates present in paints1, -2, and -3, respectively, are shown as a function of shear rate. In these cases, a contrast match condition for the TiO2 particles was present. The scattering profiles in parts b, d, and f of Figure 1 show the scattering from the TiO2 particles/aggregates in the formulations where the latex-A1, latex-A2, and latex-B particles/aggregates, respectively, were contrast matched. From the USANS scattering profiles presented in Figure 1, the following observations can be made: (1) At low q, the scattering intensity, I(q), decreases with increasing shear rate. At higher q values, the scattering intensities are independent of shear rate. This behavior is observed for both the latex 1869
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Figure 1. Rheo-USANS intensity profile as a function of the scattering vector (q) for the paint-1 formulation (VS = ϕlatex + ϕTiO2 = 0.35, PVC = 20%) at (a) TiO2 contrast matched condition and (b) latex-A1 contrast matched condition for different shear rates. Similar plot for the paint-2 formulation at (c) TiO2 contrast matched condition and (d) latex-A2 contrast matched condition for different shear rates. Scattering intensity profiles for (e) TiO2 contrast matched and (f) latex-B contrast matched conditions in paint-3. Plots a, c, and e represent scattering from latex-A1, -A2, and -B particles/aggregates as a function of shear rate, respectively and plots b, d, and f represent scattering from the TiO2 particles/aggregates. The error (±one standard deviation) in measurements of the USANS intensity is less than the size of the markers used unless explicitly shown using error bars. The solid lines are fit (eq 8 or 9) to the experimental data.
scattering and the TiO2 scattering. (2) At high γ̇ (2500 s−1 or 1000 s−1 and in some cases even at 100 s−1), a q independent Guinier scattering regime was approached. This suggests that above a critical shear rate, the structure is nearly homogeneous
at large length scales. (3) At lower shear rates (≤100 s−1), the low-q intensity demonstrated a power law dependence, indicating the presence of large scale structures (aggregates of particles). 1870
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core−shell particles with polydisperse radius, the analytical expression of the form factor, P(q)spherePCS, is given by
It is well documented in the literature that colloidal particles (latex or TiO2) may form aggregates which are fractal in nature.11,21,31 These aggregates are characterized by their correlation length, ξ, which is a descriptor of the overall size, and their self-similarity dimension or fractal dimension, Df. Under shear conditions, these fractal structures are expected to evolve through changes in both the correlation length and fractal dimension. Therefore, the USANS scattering data were fitted with a fractal aggregate scattering model to extract structural parameters as a function of shear rate. 3.2. Modeling of the USANS Data. The scattering from aggregated colloidal particles can be treated as scattering from fractal like objects. In this case, an individual latex or TiO2 particle constitutes the building block of the aggregate.31a,32 The general expression for scattering was given in eq 1. The scattering function from fractal aggregates has been developed by Teixeira32b and can be used for the scattering from either the latex particles (TiO2 contrast matched) or the TiO2 particles (latex contrast matched). We have previously shown that, although the TiO2 particles are nonspherical, carry an inorganic passivation layer of Al2O3 and SiO2, and have a colloidal stabilization layer of dispersant in water, with some dispersant molecules comprising bridges between pairs of pigment particles, they are nevertheless adequately modeled with a uniformly dense spherical form factor.11 The particle form factor, P(q), is assumed to be different for the TiO2 (hard sphere) and the latex (spherical core−shell) due to the adsorbed layer of RM on the surface of the latex. The specific differences between the P(q) functions will be discussed following the discussion of the structure factor, S(q). 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. For fractal aggregates, Chen and Teixeira32a,b proposed the correlation length, ξ, as a cut off length scale to describe the pair correlation function, g(r), at large r to avoid divergence in the S(q) evaluation. This length scale (ξ) physically represents the characteristic distance above which the mass distribution does not follow the fractal law. From Teixeira, the structure factor of a fractal-like aggregate can be expressed as S(q)fractal = 1 +
sin[(Df − 1)tan−1(qξ)] (qR )Df
P(q)spherePCS =
+
1
⎡ 3J (q⟨R 0⟩) ⎤2 ⎥ P(q)sphere = Vp 2Δρ2 ⎢ 1 ⎣ q⟨R 0⟩ ⎦
(6)
where ⟨R0⟩ is the mean particle radius (equal to the building block dimension, ⟨R⟩), and Δρ is the contrast factor (neutron SLD difference) between the particle (ρTiO2) and the suspending medium (ρsolvent), Vp = (4/3)π⟨R0⟩3, and J1 is the first order spherical Bessel function. The polydispersity of the sphere (TiO2) radius is incorporated by means of a Schulz distribution. Contrast factors are calculated from neutron scattering length density differences (Δρ) between the TiO2 particles (ρTiO2) and the suspending medium (ρsolvent). Under shear, large fractal aggregates of latex and TiO2 are expected to be restructured and broken into smaller fractal aggregates and finally into single particles at sufficiently high shear rate. Therefore, for each of the particle types (latex or TiO2), we assume that the total scattering intensity is a linear combination of scattering from two independent structures/ sources: the fractal aggregates and individual particles.21 For each of the independent scattering structures/sources, the intensity can be expressed as a product of the corresponding form factor (P(q)) and structure factor (S(q)) weighted by the corresponding volume fractions, respectively:
⎤(Df − 1)/2
⎥ q 2ξ 2 ⎦
where Df is the fractal dimension of the aggregates, Γ(x) is the gamma function, ⟨R⟩ is the building block radius, and ξ is the correlation length of the fractal aggregates. The radius of gyration, RG, of the fractal aggregates is related to ξ and Df through the following relation:
I(q) = ϕfractalP(q)S(q)fractal + ϕsphereP(q)S(q)particle + bgd
(3)
(7)
The aggregation number, N, defines the number of particles per aggregate and is determined by N = Γ(Df + 1)(ξ /R )Df
2
(5)
(2)
R G = ξ Df (Df + 1)/2
3V2(ρRM − ρsolvent )J1(q⟨R 0 + Δ⟩) ⎤ ⎥ (q⟨R 0 + Δ⟩) ⎦
where ⟨R0⟩ is the mean core radius (latex particle radius) assuming a Schulz distribution of radius values, Δ is the shell (RM layer) thickness, and J1 is the first order 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.33 V1 is the inner sphere volume, 4π/3⟨R0⟩3, and V2 is the outer sphere volume, 4π/3(⟨R0⟩ + Δ)3. Vshell is given as V2-V1. In this case, the building block radius is ⟨R0⟩ = (⟨R0⟩ + Δ). Contrast factors are calculated from neutron scattering length density differences (Δρ) between the latex particles (ρlatex), the RM shell (ρRM) and the suspending medium (ρsolvent). We assume that the RM polymers do not adsorb on the TiO2 particle surfaces and that adsorbed dispersant molecules can be ignored.11 Therefore, the form factor for TiO2 scattering was assumed to be that for a polydisperse sphere (P(q)sphere). The spherical form factor, P(q) can be expressed as
Df Γ(Df − 1) ⎡ 1+ ⎣⎢
ϕ ⎡ 3V1(ρlatex − ρRM )J1(q⟨R 0⟩) ⎢ Vshell ⎣ (q⟨R 0⟩)
This applies to either the latex or TiO2. The P(q) are either the spherical core−shell form factor (for the latex scattering) or the uniformly dense sphere form factor (for the TiO2 scattering). S(q)fractal is the expression derived by Teixeira for the scattering from fractal aggregates discussed above (eq 2). For both the individual latex particles with an RM shell layer and individual TiO2 particles, it is assumed that the particles are impenetrable. Therefore, at long-range, the isolated individual spheres (core− shell or pigment) interact only through the excluded volume.
(4)
We have previously demonstrated that RM polymers adsorb on latex particle surfaces to form a spherical core−shell structure with the latex particles comprising the core.8b,10 Therefore, a core−shell spherical form factor with polydisperse core radius is assumed for latex/RM scattering. For spherical 1871
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Similarly, for the scattering from TiO2 particles (i.e., latex particles are in the contrast match condition), there are nine coefficients associated with the scattering function. Four of these coefficients are known, ϕTiO2, ρTiO2, ρsolvent, and bgd terms, and were held constant during fitting for all data sets. The values of ⟨R0⟩ and p were determined previously and were also held fixed. The three remaining coefficients, the volume fraction of TiO2 aggregates, ϕf ractal_TiO2, and ξ and Df of the TiO2 fractal aggregates, were the only parameters extracted from the fit. Finally, for scattering from latex and TiO2 aggregates, the values of Df were constrained between the limits of 1.0 and 3.0. The latex and TiO2 correlation lengths and fractal dimensions as a function of shear rate for all of the formulations and the volume fractions of aggregates present in each formulation as a function of shear rate are presented in Tables 3 and 4, respectively. There are two major assumptions associated with the structural parameters extracted from model fits to the rheo-USANS data. The first assumption arises from the measurement system. Under shear, it is possible for the aggregated particles to form anisotropic structures.2b,c,36 For these rheo-USANS measurements, the detector has limited q resolution in the vorticity direction. Further, the fractal model is isotropic in nature and does not consider any anisotropy of structures. Therefore, the structural parameters (ξ and Df) obtained from rheo-USANS fitting, especially at high shear rates, are not truly representative of the overall aggregated structure and rather are a closer representation of the structure in the flow direction. The second assumption arises from the approach used to fit the USANS data. In eqs 7, 8a and 9a, it is assumed that the fractal aggregates and single/individual particles scatter independently. However, according to eq 10, for a three-component system, there are three partial structure factors, Sij, to consider. In general
We model this interparticle interaction using a hard sphere (HS) structure factor (S(q)HS) with Percus−Yevick closure.34 The hard sphere potential presumes no interaction for radial distance (r) greater than the particle diameter (U(r) = 0 for r > 2⟨R0 + Δ⟩ and U(r) = ∞ for r ≤ 2⟨R0 + Δ⟩). Justification for using a HS structure factor even when the shell layer is comprised of swollen soft PEO polymer chains can be found in our previous publication.10 Therefore, the scattering intensity for the latex particles (i.e., with the TiO2 particles in the contrast match condition) is given as I(q)|latex = ϕfractal _latexP(q)spherePCS S(q)fractal + ϕsphere _latex × P(q)spherePCS S(q)HS + bgd
(8a)
and the relationship between the volume fraction of aggregated latex and individual latex particles is given as ϕfractal _Latex + ϕsphere _Latex = ϕLatex
(8b)
where ϕlatex is the total volume fraction of latex in the formulation (ϕlatex = 0.28). The analogous relations for the TiO2 scattering (i.e., with the latex particles in the contrast match condition) are given as I(q)|TiO2 = ϕfractal _TiO P(q)sphere S(q)fractal + ϕsphere _TiO 2
× P(q)sphere S(q)HS + bgd
2
(9a)
and ϕfractal _TiO2 + ϕsphere _TiO2 = ϕTiO2
(9b)
where ϕTiO2 is the total volume fraction of TiO2 particles in the formulation (ϕTiO2 = 0.07). The structural parameters were extracted based on the model described above by fitting the desmeared35 experimental data using a nonlinear regression routine based on the IGOR Pro software available from NCNR.30 For the scattering from the latex particles (i.e., TiO2 contrast match condition), the model function (eq 8) has 11 coefficients associated with it. These are the total volume fraction ϕlatex, the volume fraction of particles in aggregates, ϕf ractal_latex, the building block dimensions (core radius, ⟨R0⟩, radius polydispersity, p, and shell thickness, Δ), the fractal aggregate structural parameters (the correlation length, ξ and fractal dimension, Df), the three scattering length densities (SLDs) (ρlatex, ρRM, and ρsolvent), and the incoherent background scattering, bgd. Since the volume fractions of aggregates and free particles are related by eq 8b and the total volume fraction ϕlatex is independently fixed, only the volume fraction of aggregates is a fit parameter. Values of ⟨R0⟩, the polydispersity index (p), ρlatex, and ρRM, were measured in previous experiments (unpublished data) and these coefficients are fixed. The solvent SLD, ρsolvent, was fixed to match the TiO2 particle neutron SLD. The shell thickness, Δ, varies with the shear rate where the thickness progressively decreases with increasing shear rate.10 However, considering the small length scale of Δ compared to the particle and aggregate sizes, this minor variation in Δ with shear rate was assumed to be negligible and a constant value of Δ = 10 Å based on our prior measurements at 2500 or 1000 s−1 was used.10 The bgd term was arbitrarily chosen to be constant at 1 cm−1. Therefore, in all cases only three of the 11 coefficients are fit parameters which are extracted by the nonlinear regression routine: (1) the volume fraction of latex aggregates, ϕfractal_latex; (2) ξ; and (3) Df.
I(q) = |A(q)|2 = b12S11(q)P1(q) + b2 2S22(q)P2(q) + 2b1b2S12(q)P1(q)P2(q) + bgd
(10)
where A(q) is the scattering amplitude, the bi are the coherent scattering length differences between different scattering elements and solvent, Pi(q) is the form factor for primary particles, and bgd is the incoherent background scattering contribution. For simplicity, we have omitted the corresponding volume fraction weighting. In our case primary particle composition and shapes are the same for both scattering elements (fractal aggregates and single particles) and, therefore, b1 = b2 and P1(q) = P2(q). Our model does not consider the S12 cross term contribution. Analytical treatment of the interference cross terms can be found in the scattering literature.37 For three component systems, more rigorously, the cross term should be calculated by decomposing scattering intensities into partial scattering functions by contrast variation.38 However, this approach is not feasible here as the fractal aggregate building block and individual primary particle composition are the same (i.e., b1 = b2) and cannot be contrast matched independently. It should be noted that the cross term contribution is significant when the structural length scales of the two scattering components present in the system are comparable.21 In the case of colloidal aggregates, the correlation length is typically orders of magnitude greater than the single particle radius, therefore the cross term contribution is generally neglected.21 This is largely true for formulations under quiescent conditions and in the low shear regime. However, 1872
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Table 3. Correlation Lengths and Fractal Dimensions of Different Latices and TiO2 Aggregates as a Function of Shear Rates Extracted from Rheo-USANS Dataa paint-1, latex-A1/TiO2 RM-1 formulation latex-A1 shear rate (s‑1)
shear stress (Pa)
2500 1000 100 10 1.0
386 210 37.4 5.36 1.289
shear rate (s‑1)
shear stress (Pa)
1000.0 100.0 10.0 1.0 0.1
56.8 33.8 12.6 2.5 0.908
shear rate (s‑1)
shear stress (Pa)
1000.0 100.0 10.0 1.0 0.1
98.6 80.5 56.1 16.0 11.50
TiO2
ξ (Å) × 103 18 19 20 20 20
± ± ± ± ±
ξ (Å) × 103
Df
Df
1.45 ± 0.05 1.60 ± 0.02 12 ± 1.9 1.90 ± 0.02 12 ± 1.0 2.00 ± 0.01 12 ± 0.79 2.43 ± 0.01 12 ± 0.26 paint-2, latex-A2/TiO2 RM-4 formulation
0.40 2.4 2.0 1.2 0.64 latex-A2
± ± ± ± ±
ξ (Å) × 103
Df
latex-B ± ± ± ± ±
0.14 0.05 0.04 0.01
Df
2.56 ± 0.01 5.3 ± 0.38 2.69 ± 0.01 10 ± 0.44 2.66 ± 0.01 13 ± 0.48 2.69 ± 0.01 16 ± 0.47 2.71 ± 0.01 22 ± 0.94 paint-3, latex-B/TiO2 RM-4 formulation
0.16 0.56 1.2 1.0 1.4
2.58 2.56 2.46 2.61 2.35
± ± ± ± ±
0.09 0.04 0.03 0.02 0.02
TiO2
ξ (Å) × 103 1.2 5.2 6.2 9.9 80
± ± ± ±
TiO2
ξ (Å) × 103 10 21 28 36 34
2.80 2.10 2.04 2.54
ξ (Å) × 103
Df
0.12 0.21 0.23 0.44 30
2.63 1.99 1.99 1.80 1.63
± ± ± ± ±
0.17 0.03 0.02 0.02 0.02
2.6 15 100 100 180
± ± ± ± ±
Df
0.55 1.8 13 67 120
1.10 1.10 1.10 1.09 1.10
± ± ± ± ±
0.10 0.10 0.07 0.09 0.08
a Errors reported to ±1 standard deviation for the USANS model fit data; all the correlation length parameters are reported to their nearest integer value.
Table 4. Fractal Transient Aggregate Volume Fractions of Different Latices (ϕf ractal_latex) and TiO2 (ϕfractal_TiO2) as a Function of Shear Rates Extracted from Rheo-USANS Dataa paint-1 shear rate (s‑1) 2500 1000 100 10 1 0.1 a
paint-2
paint-3
latex-A1
TiO2
latex-A2
TiO2
latex-B
TiO2
ϕfractal_latex × 10‑3
ϕfractal_TiO2 × 10‑3
ϕfractal_latex × 10‑3
ϕfractal_TiO2 × 10‑3
ϕfractal_latex × 10‑3
ϕfractal_TiO2 × 10‑3
± ± ± ± ±
0.82 ± 0.2 13 ± 0.8 20 ± 0.9 42 ± 0.8
± ± ± ± ±
7.7 ± 3 7.7 ± 0.6 7.0 ± 0.5 7.0 ± 0.6 11 ± 0.7
31 39 39 34 42
1 1 1 1 1
28 36 36 28 28
± ± ± ± ±
1 1 1 1 1
4.9 5.6 7.7 7.7 8.4
± ± ± ± ±
0.4 0.3 0.3 0.2 0.3
56 42 42 34 39
2 1 1 1 1
Errors are reported to ±1 standard deviation for the USANS model fit data; volume fraction parameters are reported with two significant digits.
at high shear rate, the fractal aggregate RG (∼ξ) may be comparable with the single particle dimension (⟨R⟩) and omission of the cross term may produce systematic errors in the structural parameters. Further evaluation of the cross term contribution under high shear rate is reported in the Supporting Information where we demonstrate that omission of the cross term does not significantly change the fitted results. 3.3. Colloidal Aggregate Structural Parameters. The shear stress corresponding to the applied shear rates (for the rheo-USANS experiment) were calculated according to eq 11:
η=
σ γ̇
formulations in this study were selected to demonstrate a range of rheological behaviors from mildly shear thinning to steeply shear thinning with yielding behavior (Figure 2). The extracted fractal dimensions, Df, and correlation lengths, ξ, as a function of shear stress for these formulations are presented in parts a and b of Figure 3 and parts a and b of Figure 4, for the latex and TiO2 aggregates, respectively. It is important to note that the variable plotted on the abscissa is shear stress and not shear rate. In conventional colloidal aggregates, the aggregates are typically presumed to be rigid solids having internal cohesive strength.14,39 The breakup of such aggregates occurs if the hydrodynamic stress surpasses the cohesive strength. Therefore, the aggregate correlation lengths are presented as a function of shear stress rather than shear rate. While flow-induced aggregation of particles, which is controlled by the particle collision frequency and
(11)
We use this relationship to estimate the shear stress in the rheo-USANS experiments since the shear rate was controlled in these experiments (values are reported in Table 3). The paint 1873
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Figure 2. Viscosity results for (a) paint-1, (b) paint-2, and (c) paint-3.
Figure 3. USANS extracted structural parameters (a) fractal dimension, Df, and (b) correlation length, ξ, as a function of the shear stress for latex aggregates. The error (±1 standard deviation) in model fit parameters is less than the size of the markers used unless explicitly shown using error bars.
Figure 4. USANS extracted structural parameters (a) fractal dimension, Df, and (b) correlation length, ξ, as a function of the shear stress for TiO2 aggregates. The error (±1 standard deviation) in model fit parameters is less than the size of the markers used unless explicitly shown using error bars.
aggregates conserve their correlation lengths while gradually evolving from a dense (i.e., higher Df) to a more open (i.e., lower Df) structure. By comparison, the TiO2 aggregates in paint-1 exhibit a relatively dense structure and fractal dimension between ∼2.0 and ∼2.8 over the range of shear stress studied (Figure 4a). Additionally, the TiO2 aggregate correlation lengths exhibit negligible stress dependence over this stress range (Figure 4b). For paint-2, the latex-A2/TiO2 formulation with RM-4, the latex aggregates have a fractal dimension which remains approximately
aggregate structure, is also possible, it was not observed for our systems. The fractal dimension of the latex aggregates in paint-1 (the latex-A1/TiO2 formulation with RM-1) decreases with increasing shear stress from ∼2.4 to ∼1.4 (Figure 3a). The aggregate size for latex-A1 is about 2 × 104 Å at the low shear stress region (
