Water Interfaces in Latex Paint - American

Sep 8, 2014 - Department of Chemical Engineering, The Pennsylvania State University, 120 Fenske ... rheology modifiers are added to paint formulations...
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Molecular View of Polymer/Water Interfaces in Latex Paint Zifeng Li,† Fang Yuan,‡ Kristen A. Fichthorn,† Scott T. Milner,*,† and Ronald G. Larson*,‡ †

Department of Chemical Engineering, The Pennsylvania State University, 120 Fenske Laboratory, University Park, Pennsylvania 16802, United States ‡ Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States S Supporting Information *

ABSTRACT: To obtain a molecular level view of the interface of a latex particle with water solvent, atomistic simulations using two different force fields were performed for a slab of methyl methacrylate (MMA)/n-butyl acrylate (BA) random copolymer in water. The carbonyl groups at the polymer/water interface were found to orient significantly toward the water phase. We calculated the copolymer/water and the copolymer/vacuum interfacial tensions, and predict a contact angle for water on copolymer of 77°, consistent with experimental values. We also calculated bulk densities of MMA/BA copolymer, and PMMA and PBA homopolymers, which compare well with experiment, and the structure factor for PMMA, which agrees with neutron scattering results. The simulated Tg of PMMA is not far from its experimental value, despite the much shorter chains and faster cooling rates of simulations versus experiments, because these two effects tend to cancel each other. When we correct for these effects, the predicted Tg is within 5 K of the experimental value.

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INTRODUCTION Since environmental regulations have required manufacturers to reduce volatile organic compounds (VOCs) in paint, waterborne latex paints have taken an increasing market share over traditional, organic-solvent-based paint.1 Compared to solvent-based paints, current latex formulations have a longer shelf life, lower toxicity, lower flammability, and a decreased likelihood of reacting with the substrate. However, waterborne coatings have inferior leveling properties, toughness, resistance to dirt and fingerprints, and ability to withstand freeze−thaw cycles.2 As a result, solvent-based paints are still primarily used in the industrial sector. To strengthen the advantages of latex with regard to stability and overcome the rheology problems, a delicate balance is needed of some seemingly contradictory properties. Latex particles should maintain a stable suspension without aggregating when the paint is stored, yet they should be able to amalgamate into a continuous film as quickly as possible after the paint is applied to a surface. These separation and aggregation properties are mainly the result of a balance between interparticle van der Waals and electrostatic forces. Charged species near the particle surfaces, including negatively charged dissociated comonomers, surfactants and ions are crucial to achieving this balance. Ideally, paint formulations should exhibit shear-thinning behavior. During application, the paint must flow easily and form a smooth surface without dripping off the brush or running down a vertical surface. To achieve these properties, rheology modifiers are added to paint formulations. Rheology modifiers form a network in the aqueous phase, but they can © XXXX American Chemical Society

also bind to the surfaces of the latex particles, and thus influence interparticle forces and shelf life. In addition, surfactant competes for accessible surface area with the rheology modifier, which may lead to complicated interactions between these additives. Modern latex paints and coatings have remarkably complex formulations, which typically consist of polymer binders, surfactants, rheology-modifying polymers, and pigments. For such a complex system, it is the interfacial region between polymer binder particles (referred to as polymer or latex polymer below) and water that plays a key role in the stability and rheology of the paint. In this polymer/water interfacial region, the interactions between the surface and key solution species, such as surfactants, salts and rheology modifiers, affect the stability and rheology. To analyze these interactions near the polymer/water interface, understanding the interfacial structure and properties is essential. Here, we apply molecular dynamics (MD) simulations to the study of the surface of an acrylic latex particle, composed of methyl methacrylate (MMA)/n-butyl acrylate (BA) random copolymer. No simulation study of the interfacial structure of acrylic latex polymer in water has been reported heretofore. However, there are both experimental and simulation studies of the polymer/water interfacial structure for other polymers, including PMMA, which are particularly relevant to the work reported here.3−5 Received: April 25, 2014 Revised: July 25, 2014

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Recent studies of polymer/water interfacial structure have provided insight into the local restructuring of polymer in response to water. The density profile of the PMMA/water interface has been measured for supported films by specular neutron reflectivity3 and obtained from MD simulations.4 A PMMA film is thickened by water compared with the pristine film in air, indicating that water molecules penetrate into the polymer film. The orientation of PMMA groups at the polymer/water interface has been studied by sum frequency generation vibrational spectroscopy.5 It was found that the carbonyl groups in the side chains orient toward the water phase to form hydrogen bonds with water. Lee et al.4 observed in their atomistic MD simulations that among all the polar atoms of PMMA, the carboxyl oxygens formed the greatest number of hydrogen bonds with water. The surface tension at the polymer/water interface is fundamental to the blending, adhesion and wetting of paint. Dee and Sauer6 compiled an extensive review of theoretical and experimental studies of the surface tension of liquid polymer/ vacuum interfaces. The temperature dependence of the surface tension of a polymer is typically linear over a range of temperatures, with a negative slope that may be interpreted as a surface entropy. A number of theoretical models predict the surface tension from bulk thermodynamic properties, but are unable to accurately calculate the contribution from surface conformational entropy.6 This surface conformational entropy reflects how the polymer chain ends, side groups and chemical structure pack at the interface, where the chains are no longer random coils. However, the surface tension, including the surface conformational entropy, can be directly calculated from MD simulations based on a carefully tested interaction potential. Gloor et al.7 reviewed different approaches for calculating surface tension using MD simulations. The truncated potentials used in MD simulations artificially lower the surface tension; this deficit can be corrected analytically by applying a “tail correction”.8,9 To gain a molecular level understanding of the surface properties of MMA/BA copolymer in water, we built a chemically detailed model of the polymer/water interface. Below, we discuss the results of MD simulations based on this all-atom model, in which we describe several bulk and interfacial properties of latex copolymer. As we discuss below, our results are in good agreement with experiment. These results validate our simulation methods and force fields, and set the stage for studies of the interactions between the particle surface and surface charge groups, surfactants, counterions, rheology modifiers, and other components of latex paints.

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Figure 1. Structural units of MMA/BA copolymer. planned future simulations of complex waterborne coating materials, which contain sodium dodecyl sulfate, rheology modifiers made of poly(ethylene glycol) and urea linkers, and other molecules. All the CGenFF parameters used in this study are listed in the Supporting Information. To model commercial latex copolymers, we construct random atactic copolymer chains of MMA and BA monomers with a 1:1 molar ratio. Our simulations contain 36 short chains of 20 monomers, which are roughly 50 times shorter than commercial MMA/BA copolymers. The choice of 20 monomers represents a compromise between computational cost and fidelity to the local structure of commercial polymers, which have a very low concentration of chain ends. To mimic commercial latex chains, each chain in the simulation is constructed by randomly selecting the species and tacticity of each monomer. As a result, each chain is different, and can contain different numbers of MMA and BA monomers. To achieve a 1:1 overall molar ratio of MMA to BA, we randomly added MMA and BA monomers one at a time with equal probability. To control chain tacticity, we built mirror images of each monomer, for which the side groups pointed toward opposite directions. Each MMA or BA added had equal probability to be either the original monomer or its mirror image, resulting in atactic chains. Monomers in chain ends have similar structure to monomers in the middle, except for an extra hydrogen on the first or last backbone carbon atom. We also built PMMA and PBA homopolymers so that simulations of these could be compared to those for MMA/BA copolymers. The homopolymers also consist of 36 atactic chains of 20 monomers, the same as for the MMA/BA copolymer. To create a reasonable initial configuration, we built all-trans chains, rotated the chains to randomize their conformations, and subsequently packed them at a low density to avoid overlap. To build all-trans chains, we created each monomer with its first backbone carbon atom located at the origin. A translation vector between the first backbone carbon of adjacent monomers was computed. To polymerize a chain, we translated the Nth monomer by (N − 1) times this translation vector. In this way, an all-trans chain was obtained. From the all-trans configurations, we prepared random walk conformations by rotating each dihedral to the trans, gauche +, or gauche − angles with equal probability. The oligomeric chains had no self-overlap after these rotations. Having constructed the individual chains, we translated them to fill a cubic simulation box at a sufficiently low density that chains did not overlap. The chains were packed with a minimum interchain distance of 7 Å, which is twice the Lennard-Jones (LJ) diameter σ for an alkane carbon.10 The resulting box size is 11 nm on each side, with a corresponding low density. To equilibrate the initial configuration, we performed an energy minimization, followed by a constant number, volume, and temperature (NVT) simulation with periodic boundary conditions. For energy minimization, the steepest descent and the l-BFGS (limitedmemory Broyde−Fletcher−Goldfarb−Shannon) integrator14 were utilized successively, with a force tolerance of 10 kJ/(mol·nm). The NVT simulation was run for 10 ns at high temperature (500 K) to accelerate equilibration. The temperature was maintained by the velocity rescaling thermostat,15 which is known to correctly render the canonical ensemble. To reach the natural density of the bulk polymer, we used a constant number, pressure, and temperature (NPT) simulation with periodic boundary conditions to further equilibrate the initial

METHODS

Atomistic MD simulations can provide a molecular level view of MMA/BA latex copolymer structure and properties. In our simulations, we used both the OPLS (Optimized Potentials for Liquid Simulations)10 and the CHARMM General Force Field (CGenFF).11 As the name suggests, the OPLS potentials have been extensively tested by comparing simulation results to measured properties for a wide variety of small organic liquids.12 Also, OPLS contains all the parameters needed to simulate copolymers made from MMA and BA monomers (structural units shown in Figure 1). CGenFF covers a wide range of organic molecules,11 and it is fully compatible with other versions of CHARMM, for example CHARMM27. 13 Since CHARMM27 contains more accurate force field parameters for certain molecules than does CGenFF, while CGenFF covers more molecular structures than does CHARMM27, combining parameters from both CGenFF and CHARMM27 is a promising approach for B

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configurations. NPT simulations were conducted at 500 K and 1 bar pressure for 30 ns. We used the Berendsen barostat16 to reach the target pressure and then switched to the Parrinello−Rahman barostat17 for further equilibration. We did not start with the Parrinello−Rahman barostat, since we found it either reaches the target pressure very slowly (with a large time constant), or can cause numerical instabilities (with a small time constant) if the pressure is far from equilibrium. In our NPT equilibration runs at 500 K and melt density, a typical polymer chain diffuses about four times its radius of gyration within 20 ns, indicating the effectiveness of our equilibration. After NPT equilibration, the cubic box dimensions of MMA/BA copolymer, PMMA, and PBA homopolymers were 5.38, 4.89, and 5.67 nm, respectively. All MD simulations used the leapfrog algorithm with a 2 fs time step. For efficiency, we enforced constant bond lengths with the LINCS bond constraint algorithm.18 The cutoff radius of the nonbonded potentials was 11 Å. We updated the neighbor list every 10 time steps. The van der Waals potential was shifted, and smoothly switched to zero beginning at 10 Å. The long-range electrostatic forces were computed using the particle-mesh Ewald19 method. All simulations were performed using the GROMACS package.20 Typically, our simulation contains about 13 000 polymer atoms and 8400 water molecules. Our simulations run at a rate of about 10 ns/ day using eight processors on a typical Linux cluster.

Figure 2. Temperature dependence of the density for copolymer and homopolymers, from OPLS simulations. The dashed lines are linear fits to the data over high and low temperature ranges, and the intersection points represent the Tg values.

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RESULTS AND DISCUSSION We verified the OPLS and CGenFF force fields by simulating bulk properties of the polymers, including the melt density, the glass transition temperature (Tg) and the structure factor S(q). Bulk Density and Glass Transition. Polymer dynamics slow down dramatically near Tg, which makes it difficult to achieve consistent results in MD simulations. To conduct our simulation in an appropriate temperature range, knowing the simulated Tg is essential. However, the experimental Tg is not necessarily a good estimate of the simulated Tg. The cooling rates we can achieve in all-atom MD simulations are 10 orders of magnitude higher than in typical experiments, which is known to increase Tg.21−23 We also have about 50 times lower molecular weights, which decreases Tg.24 Although chemically specific MD simulation is a powerful tool, only a few studies25,26 have been reported to address the effect of cooling rate and molecular weight on Tg using all-atom or united-atom MD. To determine Tg in our simulations, we calculated the mass density as a function of temperature. The slope of the density− temperature curve represents the thermal expansion coefficient, which drops discontinuously when approaching Tg from above. This method has been previously applied to a coarse-grained model27 and recently to an atomistic simulation of cross-linked epoxy systems,28 and is similar to the experimental determination of the Tg of thin films by ellipsometry.29 To obtain the temperature dependence of the density, we simulated a staged cooling, in which we began with a temperature of 500 K and lowered the temperature in 20 K increments until we reached a temperature of 100 K. We performed 5 ns NPT simulations at P = 1 bar for each temperature to obtain the average density. This corresponds to an average cooling rate of 4 K/ns. Figure 2 shows the simulated bulk densities of MMA/BA copolymer, PMMA homopolymer, and PBA homopolymer as functions of temperature obtained from simulations using the OPLS potentials, along with the available experimental density for PMMA.30 Corresponding results for the CGenFF force fields are shown in Figure 3.

Figure 3. Temperature dependence of the density for copolymer and homopolymers, from CGenFF simulations.

The break point at which the thermal expansion coefficient (the slope of the density curve) changes, indicates the glass transition temperature. To extract a value for Tg, we fit the data above and below the visually identified break point by straight lines; the intersection of the two lines occurs at Tg. The value of Tg increases as we go from PBA to MMA/BA copolymer, and from copolymer to PMMA, following the same trend as the experimental Tg. Above Tg, the simulated thermal expansion coefficient of PMMA at 140 °C is 3.9 × 10−4 K−1, which is about 70% of the experimental value (5.7 × 10−4 K−1).30 Below Tg, the simulated thermal expansion coefficient of PMMA at 60 °C is 1.7 × 10−4 K−1, about 70% of the experimental value (2.5 × 10−4 K−1).30 No experimental data for density versus temperature are available for PBA or the copolymer. In Figure 2, we observe that the simulated density of our short PMMA chains (20 monomers) are about 5% lower than the experimental values for polymeric PMMA. Qualitatively, this difference is expected, because of the greater free volume associated with the chain ends, which constitute 10% of our chains, but are negligible for polymeric PMMA. No such difference is evident in Figure 3, which paradoxically may indicate that the density for this potential is slightly too high. For efficiency, we have used bond constraints in our simulations. These have a modest effect on the thermal expansion coefficients, as detailed in the Supporting InformaC

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⎛ M ⎞ Tg = Tg ∞⎜1 − 0 ⎟ Mn ⎠ ⎝

tion. Simulations without bond constraints give a somewhat larger thermal expansion coefficient for PMMA above Tg, with no change below Tg, in better agreement with experiment. Removing bond constraints for PBA and the copolymer has little effect above Tg, and gives somewhat lower thermal expansion coefficients below T g. As we do not have experimental results for PBA or the copolymer, we do not know whether this improves agreement with experiment or not. To test the performance of CGenFF, the glass transition temperatures of PMMA, PBA and their 50:50 molar copolymer have been calculated using a procedure similar to that applied with the OPLS force field (see Figure 3). In these CGenFF simulations, for each temperature, a 2 ns NPT simulation at P = 1 bar was performed, resulting in a cooling rate of 10K/ns, somewhat higher than was used for the OPLS simulations. Cooling from a high temperature, the slope of the density vs temperature changes at the glass transition temperature, from a high value in the rubbery state to a lower value in the glassy state. The resulting glass transition temperatures of the copolymer and homopolymers are listed in Table 1, for both the OPLS and CGenFF force fields, together with the experimental results.

where Tg∞ is the long-chain limiting value of the glass transition temperature. Mn is the number-averaged molecular weight, and M0 is a material constant of order the molecular weight of a monomer (M0 = 181 g/mol for PMMA.33) As the molecular weight increases, the Tg approaches the limiting value for long chains. In our PMMA simulations we use short chains with a molecular weight Mn = 2002 g/mol and find an uncorrected Tg value of 364 K (see Figure 3 and Table 1). To infer the corresponding value of Tg for long chains at the same cooling rate, we use eq 1 to obtain Tg,∞ = 400.2 K. The effect of simulating such short chains is to shift the simulated Tg downward by 36 K. The cooling-rate dependence of Tg can be inferred from the Vogel−Fulcher−Tammann (VFT) equation,34 which describes the temperature dependence of a microscopic relaxation time τ as ⎛ B ⎞ τ(T ) = τ∞ exp⎜ ⎟ ⎝ T − T∞ ⎠

Table 1. Simulated and Experimental Tg (K) for MMA/BA Copolymer and PMMA and PBA Homopolymersa polymer

PMMA

copolymer

PBA

experimental OPLS CGenFF

38131 364 ± 16 363 ± 25

28632 317 ± 6 327 ± 14

21931 272 ± 8 312 ± 15

(1)

(2)

where τ∞ is the high-temperature limit of the time scale, B is an empirical material constant that plays the role of an energy barrier, and T∞ is the Vogel temperature, at which the time scale would formally diverge. The temperature dependence of relaxation times is often expressed in terms of a shift factor a(T), such that34 ⎛ τ (T ) ⎞ c 0(T − T0) a(T ) = log10⎜ ⎟= 1 ⎝ τ(T0) ⎠ T − T0 + c 20

a

The error in Tg is based on the 68% (1σ ) confidence region of the fitting parameters.

(3)

in which T0 is an arbitrary reference temperature, often taken equal to the experimental Tg. Comparing eqs 2 and 3, we find c02 = T0 − T∞ and B = c01c02 log 10. When we cool a system at a rate Γ = −∂T/∂t, any temperature-dependent relaxation time τ(T) becomes an increasing function of time, which we write as τ(t). If the relaxation time τ(t) grows faster than time is passing, chains will not be able to fully equilibrate: the relaxation time will grow increasingly out of reach. On the basis of this, we may write a criterion for the temperature at which the system falls out of equilibrium at finite cooling rate:

As can be seen in Table 1, the glass transition temperatures predicated by CGenFF agree reasonably well with those from OPLS, although CGenFF leads to higher densities for all three polymers than obtained from the OPLS force field. For example, the density of PMMA given by CGenFF is 1158 kg/ m3 at 300 K, which is about 2% lower that the experimental value, but about 3% higher than that calculated with the OPLS force field. Simple theoretical models are available to describe the dependence of Tg on both molecular weight and cooling rate. We assume that the dependencies of Tg on chain length and cooling rate are independent, additive corrections. Hence we separately correct for the effects of short chains and fast cooling in our simulations to obtain a Tg value relevant to long chains at slow cooling rates, which may be compared to experiment. (Recent work by Khare et al.28,42 estimates the shift in simulated Tg due to fast cooling by a similar approach, but does not include corrections due to the short chains typically used in simulations.) Here we predict an experimental Tg for PMMA only, since the experimental parameters in the models we use to correct Tg for the effects of chain length and cooling rate are not presently available for PBA or the random copolymer. Flory and Fox24 predicted that Tg should decrease as the molecular weight decreases. This lowering of Tg is proportional to the number density of chain ends, since monomers at the chain ends have more free volume:

∂τ(T ) ∂τ =1 = −Γ ∂T ∂t

(4)

Assuming the VFT dependence of τ(T) in eq 4, and that the factor B/(T − T∞)2 changes very little for modest changes in temperature, we can compare the criterion (eq 4) for two different cooling rates. In our case, we can relate the simulated Tg (Tg,sim) measured at a simulation cooling rate Γsim to an experimental Tg (Tg,exp), measured at a much lower cooling rate Γexp: ⎛ ⎞ τ(Tg ,exp) Γsim B B ⎟ = = exp⎜⎜ − Tg , sim − T∞ ⎟⎠ Γexp τ(Tg , sim) ⎝ Tg ,exp − T∞ (5)

Taking the log and expanding to first order in ΔTg = Tg,sim − Tg,exp, we obtain D

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c 20 c10

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log10(Γsim/Γexp)

For comparison to neutron scattering, g(r) is defined using the partial radial distribution functions

(6)

To use eq 6, we need the following values for atactic PMMA: the experimental glass transition temperature (Tg = 381 K), the Vogel temperature (T∞ = 301 K), and the coefficients c01 = 34 and c02 = 80 K (with a reference temperature T0 = Tg, which gives an energy barrier B = 6263 K).34 With these values, the prefactor to the log in eq 6 equals 2.35; thus we may say that Tg for PMMA shifts by about 2.4 K downward per decade decrease in the cooling rate. Then, using the simulation cooling rate (Γsim = 4 K/ns) and typical experimental cooling rates in differential scanning calorimetry (Γexp = 10 K/min),29 we compute the expected difference between simulated and experimental Tg values due to the large disparity in cooling rates, as ΔTg = 24.4 K. The simulation value for Tg is thus 24 K higher than it would be if we were able to cool at experimental rates. Now we combine the two corrections we have computed, to predict from our simulation Tg for short chains and fast cooling, a value of Tg for long chains and experimental cooling rates: Tg ,exp = Tg , sim + 36 K − 24 K = 376 K

g (r ) =

1 N

j

α

(7)

Figure 4. Structure factor of PMMA from present work and literature.35

OPLS and CGenFF force fields show three peaks in the q region from 0 to 3.5 Å−1 with positions in agreement with previous simulation results.35 The peak between 0.5 to 1 Å−1 was attributed to interchain correlations, while the other two peaks result from interactions involving both the side groups and the main chain segments.35 In Figure 4, it is evident that all three simulations display three peaks, of roughly the same magnitude and location as the experimental scattering result. Clearly, the precise positions of the simulation peaks are shifted to somewhat lower q relative to the experimentsthe middle peak in all the simulations is particularly far off the experimental location. One possible reason for these discrepancies is that the simulations use short chains (20 monomers for our OPLS simulations), while the scattering was performed on polymer samples. The simulated density for these short chains is 5% lower than the experimental density for polymers, which is expected, since the prevalent chain ends contribute free volume and lower the density. In addition, we note that the simulations were performed on atactic chains (relevant to commercial materials) while the experiments were carried out using 80% syndiotactic material.

k

(8)

∫0

∞

dr r 2

sin qr (g (r ) − 1) qr

(11)

Here cα = Nα/N, where N is the total number of atoms, and bα represents the neutron scattering length of species α. All samples are fully deuterated in all the analysis in the current work. PMMA has been thoroughly investigated using WANS and MD simulations by Genix et al.,35 who break down the structure factor of PMMA into contributions from three molecular substructures: the main chain, the α -methyl group, and the side group. In the current work, the OPLS and CGenFF force fields are tested by computing the static structure factor of PMMA and comparing the result with experimental WANS data and with the simulation results from Genix et al.35 As can be seen in Figure 4, results from both

We calculate S(q) through the radial distribution function g(r) using the Fourier transform relation S(q) = 1 + 4πρ

(10)

⟨b2⟩ = (∑ cαbα)2

∑ bjbk⟨eiq|r − r |⟩ j,k

⟨b2⟩

with

which is reasonably close to the experimental value of 381 K. Broadly speaking, the effects of short chains and fast cooling rates in simulations tend to cancel, so that uncorrected Tg values from MD simulations of oligomers may be unexpectedly close to experimental values. Binder et al. have expressed concerns that the physical mechanisms relevant to the experimental glass transition may not be fully established on nanosecond time scales in all-atom MD simulations. 23 We know that the VFT equation successfully describes the temperature dependence of the cooling rate for typical experimental cooling rates (about 10 K/ min, corresponding to a relaxation time of about 100 s). We expect the VFT-based estimate will fail eventually at a very high cooling rate (of order 104 K/ns), for which the chain relaxation time at Tg would approach 1 ps, the time scale of collisions between atoms. However, we believe that VFT-based estimates can still be applied at the high cooling rates relevant to atomistic simulations. Although the corresponding relaxation times are quite short compared with experimental time scales, there is no obvious upper limit to the applicability of VFT other than the collision time scale itself. Structure Factor. Wide-angle neutron scattering (WANS) is commonly used to analyze the local structure of polymer melts. Here, we use the static structure factor S(q) obtained from WANS to test our MD simulations by converting the radial distribution functions from MD to S(q) via Fourier transform, which we then compare to the experimental S(q). The structure factor S(q) is defined as S(q) =

∑α , β cαbβ cβbβ gαβ (r )

(9)

Our simulation results for g(r) extend to large enough separations that g(r) smoothly approaches unity, so that extending the integration of eq 9 to large r is easily done. E

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Preparing Latex/Water Interfaces. To probe latex/water interfaces, we created copolymer/water interfaces by creating polymer surface slabs and introducing water molecules. We examine density profiles of polymer and water, the restructuring of the polymer surface when polymer forms hydrogen bonds with water, and the surface tensions of the polymer/water and polymer/vacuum interfaces. All these simulations used the OPLS force-field parameters. In these simulations, we model our polymer surface as a slab, since at the length scale of our simulation (a few nanometers), the polymer surface is essentially flat. The initial configuration of the slab was prepared by two different methodsthe “NPT method” and the “wall method”, to ensure that our results did not depend on the initial state. For the NPT method, the initial configuration was obtained from an equilibrated bulk NPT simulation using periodic boundary conditions, resulting in a cubic box 6 nm on a side. This configuration was then placed in the center of a rectangular box, with dimensions 6 nm (x) by 6 nm (y) by 12 nm (z), where chains that crossed the periodic boundaries in the z direction of the original box were assigned to either one side or the other of the slab, depending on which side the center of mass of the chain resided. This results in a 3 nm vacuum layer in the z direction on either side of the polymer slab. The configuration was then equilibrated with an NVT simulation at 500 K for 20 ns to allow the resulting rough surface to relax. In the second wall method, we obtained the initial configuration by sandwiching the polymer chains at low density between two rigid walls, and compressed the system to reach the target density. We placed the walls perpendicular to the z direction, while maintaining periodic boundary conditions in the x and y directions. The walls were composed of uncharged LJ alkane carbon with a density of 1000 kg/m3. The initial system was a cubic box 11 nm on a side. To compress the chains to a realistic melt density, we repeatedly shrank the system isotropically by a small factor (0.99) by rescaling the box dimensions and all atom coordinates, followed by a short 100 ps NVT simulation between rescalings to relax the system locally, until the desired density was reached. At this point, the system was a cubic box 6 nm on a side. We then increased the box size in the z direction to 12 nm, which created a region of vacuum on either side of the polymer slab, and equilibrated the system by running an NVT simulation for 20 ns. In both methods, the final polymer slab is 6 nm thick (in the z direction), with a square cross section of 6 by 6 nm. Recall that the system has periodic boundary conditions, so that the slab is in fact periodically continued in the x and y directions, with chains able to cross these periodic boundaries while remaining in the slab of melt. Our choice of a slab dimensions represents a compromise that results in a reasonably high surface area, while maintaining a slab thickness large enough that the density and local structure of the polymer in the interior of slab is representative of bulk material. The polymer/water interfaces were then created by introducing water molecules above and below the polymer slab (shown in Figure 7). We first inserted water molecules to fill the portion of the simulation box not occupied by polymer. To avoid atom overlap, water molecules were then removed from the box where the distance between any atom of water and any atom of the polymer was less than the sum of the van der Waals radii of both atoms. We used the extended simple

To check this, we have performed simulations with 80% syndiotactic chains of 20 monomers using the OPLS potentials, and find the position of the middle peak is indeed closer to the experimental value. We also calculated the structure factor of PBA using the CGenFF and OPLS force fields. As shown in Figure 5, the two

Figure 5. Structure factor of PBA.

force fields agree reasonably well in the positions of the three peaks located approximately at 0.33 Å −1 (I), 1.2 Å −1 (II), and 3.1 Å −1 (III). The peak at q = 0.33 Å−1 can be assigned to the average intermolecular interactions, i.e., the distance between chains. This peak is at a smaller q than the lowest-q peak of the PMMA structure factor in Figure 4, indicating a larger separation between chains in PBA. This could be due to the bulkier side group in PBA (COOCH2CH2CH2CH3) compared to that in PMMA (COOCH3). Finally, the structure factors of the 50/50 (by mole) copolymer of PMMA and PBA for CGenFF and OPLS force fields, shown in Figure 6, are very similar, indicating that the force fields predict similar local ordering of the polymers.

Figure 6. Structure factor of MMA/BA copolymer.

To summarize, CGenFF and OPLS force fields result in similar structure factors for PMMA, PBA and the copolymer. The first distinct peak at low q, which results from intermolecular interactions, occurs at the lowest q value for PBA, followed by the copolymer and then PMMA, indicating that PBA has the largest main chain average separation distance, while PMMA has the smallest. F

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Density Profile. A sharp interface forms between the immiscible copolymer slab and the adjacent water region. The simplest characterization of this interface is its density profile. To obtain the density profiles, we divided the simulation box into 100 bins along the z direction, normal to the interfaces, and calculated the mass density of polymer or water within each bin. The last 5 ns of data in the NPT annealing at 363 K and 1 bar, was used to obtain the time-averaged density. The mass density profiles of water and polymer in the slabs prepared using the NPT and wall methods are shown in Figure 8. Before adding the water, the surface of the slab prepared by

Figure 7. Snapshot of the polymer slab in water.

point charge (SPC-E) water model.36 Water molecules were constrained using the SETTLE algorithm.37 After adding the water, we first performed a brief NVT simulation (2 ns at 363 K) with the polymer molecules frozen in place, to eliminate any grossly distorted arrangements of water molecules resulting from the algorithm used to add the water to the system. Next, we released the constraints on the polymer molecules, and performed a short NPT simulation (24 ns at 1 bar pressure, 363 K) as a first step in equilibrating the system. We chose the temperature to be 363 K (=90 °C), to simulate (with a small margin of safety) on the liquid side of the vapor− liquid transition for water at standard pressure. In this way, we simulated at the highest possible temperature we could reach without increasing the pressure above 1 bar, so that the glassy polymer system would relax as quickly as possible. Even with this choice, the simulation temperature is not very far above the simulated glass transition temperature of our copolymer system (Tg = 317 K, see Table 1). We expect the surface of the polymer slab to be somewhat plasticized by water, which helps to speed the equilibration of conformations near the interface. However, to give the interior of the slab every opportunity to respond to the presence of the water interfaces, we carried out two cycles of alternating high- and low- temperature NPT simulations. In each cycle, we raised the temperature to 500 K (about 180 K above the simulated Tg), and performed a 24 ns NPT simulation at 30 bar pressure (sufficient to keep the water liquid). Then we lowered the temperature back to 363 K, and performed another 24 ns NPT simulation at 1 bar pressure. For each of the 24 ns NPT simulations (whether at 500 or 363 K), the first 4 ns of this simulation used the Berendsen barostat, and the remaining 20 ns used the Parrinello−Rahman barostat. The Berendsen barostat is more robust to abrupt changes in the system (such as those resulting from a large temperature change), while the Parrinello−Rahman barostat reproduces the correct ensemble of density fluctuations. After this process was complete, the system dimensions were 58.5, 58.5, and 116.9 Å along the x, y, and z directions respectively. The final slab thickness was 44.4 Å, which is about three times of radius of gyration of the polymer chain.

Figure 8. Density profiles for the MMA/BA copolymer and water (obtained using OPLS force field).

the wall method was much smoother than the surface of the slab prepared by the NPT method. However, after adding water and equilibrating, the surfaces prepared by the two methods had very similar density profiles, indicating a well equilibrated surface from both methods. The simulated bulk density of polymer fluctuates around 1046 kg/m3. The simulated bulk density of water is 947 kg/m3, 1.9% smaller than the experimental value of 965 kg/m3.31 A slightly smaller density than the experimental value is expected for the SPC-E water model.38 The density profiles of polymer and water overlap at the interface, indicating that water molecules penetrate into the polymer surface. To investigate whether the interface of the MMA/BA copolymer is broadened by water, as was found experimentally for PMMA,3 we calculated the interfacial widths of the copolymer/water and the copolymer/vacuum interfaces. To obtain copolymer/vacuum interfaces, we removed the water from an equilibrated system, and re-equilibrated with an NVT simulation at 363 K for 40 ns. The last 20 ns of the NVT data were used to calculate the density profiles of copolymer/water and copolymer/vacuum systems. To measure the interfacial widths, the polymer density profiles in water and vacuum were fitted by a hyperbolic tangent function: ρ (z ) =

⎛ z − z0 ⎞ 1 1 ⎟ (ρ + ρ2 ) + (ρ1 − ρ2 ) tanh⎜ ⎝ d ⎠ 2 1 2

(12)

where ρ1 and ρ2 are the bulk densities on either side of the planar interface, centered at z0 with a normal thickness d. Typically, the interfacial width t is reported as t = 2.1972d, i.e., the width between 10 and 90% of the density increase, which is the “10−90 thickness”.39 Since the density profiles include two interfaces (see Figure 8), to gain better statistics, we merged the data of two interfaces by reflecting the right-hand density profile to coincide with the G

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left-hand profile before fitting. The fitted copolymer/water interfacial width (t = 4.75 ± 0.22 Å) is comparable to (but slightly smaller than) the copolymer/vacuum interfacial width (t = 5.47 ± 0.21 Å) interface. The measured interfacial width could be affected by surface roughness or capillary waves. To test whether surface roughness influences the measurement of interfacial width, we checked the roughness of copolymer/water interface by dividing the simulation box into four equal quadrants in the x−y plane. The density profiles in each quadrant were fitted using eq 12 to obtain the center of the interface and the interfacial widths. The locations of these planar interfaces (z0 in eq 12) in each of the four quadrants are close to each other, with a 0.53 Å standard deviation. The interfacial widths are also found to be similar to each other, with a standard deviation of 1.21 Å. Therefore, the amplitude of the surface roughness is about 1 Å, which is not large enough to significantly affect the measured interfacial width. Hydrogen Bonding and Surface Structure. Hydrogen bonds form between polymer oxygens and water hydrogens at the polymer/water interface, which affects the structure of the polymer surface. In this section, we examine the fraction of polymer groups forming hydrogen bonds at the interface, the role of polymer surface relaxation in the formation of hydrogen bonds, and the orientation of polymer surface groups. Hydrogen bonds are identified in the simulation by a geometrical criterion: (1) the distance between water oxygen and polymer oxygen must be less than 3.5 Å, and (2) the hydrogen bond angle (formed by the water oxygen, water hydrogen, and polymer oxygen) should be greater than 150°. To determine how much polymer participates in hydrogen bonding, we calculated the fraction of carbonyl oxygens forming hydrogen bonds at the interface. To obtain the fraction of hydrogen-bonding carbonyls, we calculated the concentration profiles in the z direction of all the carbonyl oxygens, and those carbonyl oxygens that participate in hydrogen bonds, shown in Figure 9. These concentration profiles were averaged over 600 frames spanning 4.8 ns. The curves were smoothed using a Gaussian kernel with a 2 Å width.40 As shown in Figure 9, about 20% of the carbonyl oxygens at the interface form hydrogen bonds. The fraction of carbonyl

oxygens forming hydrogen bonds is simply the ratio of the total number of hydrogen-bonding carbonyls to the total carbonyls in the interfacial region (defined in the caption of Figure 9). Nearly all the interfacial hydrogen bonds with the polymer were formed by carbonyl oxygens, with only 2% formed by ester oxygens from MMA or BA monomers. About 6% of carbonyl oxygens formed two hydrogen bonds, both of which are counted when calculating the number density profile. To find out if polymer relaxation plays an important role in the formation of hydrogen bonds, we “freeze” the polymer slab by restraining the position of all polymer atoms when water is first introduced, and then “thaw” the slab. The surface density of hydrogen bonds increased from 1/nm2 to 1.6/nm2 after the polymer was allowed to reorganize in water. This shows that reorganization of polymer surface groups promotes the formation of hydrogen bonds. Since polymer surface groups reorganize to form hydrogen bonds, we may expect these surface groups to alter their orientation with respect to the same groups in the bulk. To quantify the orientation of carbonyl groups after equilibration, we calculated the angles between carbonyl dipole (from carbon to oxygen) and the interface normal (toward water/vacuum). The analysis was based on 1300 frames spanning 10 ns. The carbonyl groups at the water interface orient toward the interface significantly compared with the carbonyls in the bulk, as shown in Figure 10. In the figure, the orientation

Figure 10. Orientation distributions P(cos θ) of carbonyls on MMA or BA monomers at the polymer/water and the polymer/vacuum interfaces, and orientation of all carbonyls in the bulk of the slab. An isotropic distribution corresponds to a constant value of 0.5. Angle scale corresponding to cosine values is shown at the top.

distributions of interfacial MMA and BA carbonyls are displayed separately. The angles for interfacial carbonyls were all skewed toward small values, indicating orientation normal to the interface. By contrast, the carbonyls in the bulk are distributed isotropically, as indicated by their nearly constant distribution (the small deviations from a horizontal line are typical of the statistical noise in our results). Preferential orientation of interfacial carbonyls has been observed qualitatively for PMMA/water interfaces using sum frequency generation vibrational spectroscopy.5 Carbonyls at the polymer/vacuum interface also tend to orient normal to the interface. To clearly show the effect of water on carbonyl alignment, we compare the distribution of carbonyls at the polymer/water interface with those at the

Figure 9. Density profile of all carbonyl oxygens (dashed) and carbonyl oxygens forming hydrogen bonds (solid). We define the boundary between interface and bulk arbitrarily to be where the concentration of the bonded carbonyls drops below 5% of the peak value. H

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The sums in eq 16 run over each LJ atom type i; ρi,A and ρi,B are the number densities of atom type i in the bulk of phase A (the middle of the polymer slab) or phase B (the water), respectively. The LJ energy and distance parameters for atom type i are denoted εi and σi respectively. (A geometric average combination rule is assumed for both ε and σ, consistent with the conventions in GROMACS.) The LJ cutoff radius is denoted as rc , and d is the interfacial thickness. The tail correction term achieves an upper bound of

polymer/vacuum interface, also shown in Figure 10. The distributions were obtained at the same temperature (363 K) using the same polymer chains. The angles at the water interface were found to be more skewed toward small values, with an average of 45° compared with 65° at the vacuum interface. (These average angles were computed by cos θavg = ∫ 1−1dμ P(μ)μ, where μ = cos θ.) Furthermore, the distribution of carbonyls at the water interface is narrower than that at the vacuum interface, again indicating that carbonyls are more strongly oriented at a water interface than at a vacuum interface. In Figure 10, we see that BA carbonyls are somewhat more strongly aligned with the surface normal than those on MMA monomers. Surprisingly, MMA carbonyls have a slightly stronger tendency to form hydrogen bonds than BA. We find that MMA forms 63% of the hydrogen bonds, even though our chains are constructed from an equal number of MMA and BA monomers. Surface Tension. A key property of the interface is its free energy. Using MD, we calculated the free energies of the copolymer−water interface γsl and of the copolymer-vacuum interface γsg. Using these values, we can then predict the contact angle of water on the copolymer in vacuum using Young’s equation: γsg = γlg cos θ + γsl

γtail ,max = 3πrc −2(ρA − ρB )2

in the limit of a sharp interface. From this limit, we see clearly that the tail correction decreases with increasing cutoff radius as rc−2. We computed the interfacial energy as a function of temperature to extrapolate the value down to the room temperature, where the simulation is remarkably slow because it is near the glass transition. Our calculation of the surface tension of the copolymer/water and the copolymer/vacuum interfaces began at 500 K and progressed downward in temperature to 360 K in steps of 20 K. At each temperature, for the copolymer/water system, an NPT simulation was performed first to let the system explore its natural dimensions, followed by an NVT simulation to further equilibrate and collect the data. The target pressure of the NPT simulation was chosen to be slightly (1.13 times) higher than the vapor pressure of water at the corresponding temperature to maintain water as a liquid. For the copolymer/vacuum system, only an NVT simulation was performed. The total simulation time at each temperature varied from 75 to 260 ns, with longer simulation time necessary at lower temperatures because of the slow dynamics as Tg was approached. The surface tensions with tail corrections decrease monotonically with increasing temperature, as shown in Figure 11. The tail correction is about 7% of the total surface tension

(13)

The resulting contact angle can be compared with experiment for validation. To compute the interfacial energies, the simplest method is the mechanical approach,41 where the surface tension is calculated from the pressure anisotropy: γ=

Px + Py Lz Pz − 2 2

(14)

where Lz is the simulation box length in the z direction (normal to the interface), and the prefactor of 1/2 accounts for the presence of two interfaces in the system (see Figure 7). Here Pz and (Px + Py)/2 are the normal and lateral components of the pressure tensor. This method enables us to calculate the interfacial tension from an NVT simulation. The surface tension calculated using eq 14 does not take into account the contribution of long-range LJ interactions beyond the simulation cutoff radius. To restore the contribution of long-range LJ interactions, we perform a “tail correction”.8,9 To compute the tail correction, we first fitted the number density profile to a hyperbolic function, as suggested by eq 12, for all LJ atom types. Each atom with different LJ parameters is considered as a different LJ atom type. The tail correction to the interfacial tension is then calculated as γtail = 12π (ρA − ρB )2

∫r

c

∞

dr

∫0

1

ds

(17)

⎛ rs ⎞ 3s 3 − s coth⎜ ⎟ 3 ⎝d⎠ r

Figure 11. Temperature dependence of surface tension for copolymer against water or vacuum. The lines are linear least-squares fits of the data. The error bars are discussed in the main text.

(15)

in which the coefficients ρA and ρB are defined by ρA =

∑ εi1/2σi3ρi ,A

for copolymer/water and 30% for copolymer/vacuum. The surface entropy − ∂γ/∂T, given by the slope of the data, is 0.176 mN/(m·K) for copolymer/vacuum and 0.092 mN/(m· K) for copolymer/water. The surface entropy of PMMA/ vacuum has been reported as 0.076 mN/(m·K) experimentally.43 No experimental data are available for PBA and MMA/ BA copolymer.

u

ρB =

∑ εi1/2σi3ρi ,B

(16)

i

The quantities ρA and ρB have dimensions of (energy) below).

1/2

(see I

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To estimate the error bar for our surface tension values, we need to know the effective number of independent measurements of ΔP(t) obtained in a given simulation run. Therefore, we calculated the autocorrelation function of surface tension after equilibration at each temperature. We extracted a characteristic relaxation time by fitting the autocorrelation function to

The molecular weight dependence of the surface tension should not significantly affect the surface tension we calculate. Experimentally, the difference between the surface tension of a polymer with molecular weight 3000 g/mol (our polymer is 2280 g/mol) and one with infinite molecular weight is typically less than 1 mN/m.43 To estimate the contact angle of water on copolymer in air, we extrapolated γsl and γsg linearly to room temperature (the straight lines in Figure 11). The room temperature surface tensions obtained from these extrapolations are γsl = 41.4 ± 4.6 mN/m and γsg = 54.9 ± 10.5 mN/m. We then predict the contact angle θ for water on the copolymer surface with Young’s equation (eq 13), using our simulation value for the water-vacuum interfacial tension of γ = 62.58 mN/m. The simulation value for γlg is slightly lower than the experimental value of 72.75 mN/m.31 We use only simulation results in predicting the contact angle, in the hope that systematic errors in the interfacial tensions that enter the Young equation may tend to cancel. Our predicted contact angle is θ = 77.5 ± 10.8°, which is about 8% larger than the experimental contact angle for MMA/BA copolymer with water of 71.8 ± 1.5°. The predicted contact angle of copolymer also sits nicely between the experimental contact angles for the corresponding homopolymers, shown in Table 2.

⟨γ(0)γ(t )⟩ = At −ae−t/ τ

where τ is a relaxation time, and A and a are fitting parameters. The autocorrelation functions of the surface tension for copolymer/water and its fits based on eq 18 at different temperatures are shown in Figure 12. There is a fast relaxation occurring within the first 0.1 ps at all temperatures, followed by a slow relaxation with a relative mean-square fluctuation amplitude of about 1%. The surface tension error bar is obtained by adding the variances of the slow and fast processes: 1/2 ⎡ ⎛ τfast τ ⎞⎤ Δγ = ⎢σT 2⎜ + 0.01 ⎟⎥ ⎣ ⎝ T T ⎠⎦

polymer

PMMA

copolymer

PBA

54−74 NA

71.8 ± 1.5 77.5 ± 10.8

100−115 NA

(19)

σT2

where is the variance of the data set of time duration T. Here τ is the slow relaxation time from fitting, and τfast is taken as a constant value of 0.02 ps. Because τ grows much larger than 100 ps at low temperature, the slow relaxation ultimately dominates the error bar. The relaxation time increases dramatically as the temperature decreases toward the glass transition temperature. To measure the effective energy barrier, we fit the temperature dependence of relaxation time to Arrhenius equation (shown as inset in Figure 12)

Table 2. Contact Angle (deg) for Copolymer, PMMA, and PBA with Water experimental present work

(18)

⎛E ⎞ τ = τ0 exp⎜ a ⎟ ⎝ RT ⎠

Our surface tension results are obtained from the time average of the pressure difference ΔP(t), defined as ΔP(t) = Pz(t) − [Px(t) + Py(t)]/2, which becomes increasingly slow to relax and hence correlated in time as the temperature approaches Tg from above. As a result, a simulation time series for ΔP(t) of a given length contains effectively fewer independent measurements for simulations carried out at temperatures approaching Tg. This implies a larger uncertainty in simulation values for surface tensions near Tg.

(20)

where the effective energy barrier is Ea = 81.7 ± 6.8 kJ/mol. This energy barrier Ea is expected to be similar to the energy barrier B in VFT equation (eq 2) in the high temperature limit. Our energy barrier Ea of the copolymer is in fact of the same magnitude as the energy barrier B in the VFT equation (eq 2) for PMMA (6264 K).

Figure 12. Autocorrelation function (ACF) of surface tension (γ) from 500 to 363 K in 20K decrement (left to right). The inset shows the temperature dependence of the relaxation time. The lines are fits of the data based on eq 18 J

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CONCLUSIONS

In this paper, we studied the interfacial structure and associated energies of a 50/50 mol percent MMA/BA latex copolymer in water using chemically detailed MD simulations. We first calculated bulk properties to test the validity of the force field parameters, including the bulk density, glass transition temperature and structure factor of MMA/BA copolymer, as well as PMMA and PBA homopolymers. The simulated Tg of the copolymer lies between the Tg values of the corresponding homopolymers. We used the simulated Tg of oligomeric PMMA to predict the experimental Tg value by correcting for the fast cooling rate and short chains in our simulations. These two corrections tend to cancel each other. When both corrections are applied, our simulated Tg value of 364 K corresponds to a predicted value for long chains at slow cooling rates of 376 K, within 5K of the experimental value of 381 K. We are not able to correct Tg for PBA or copolymer because the WLF and Flory−Fox experimental parameters needed to computed the corrections are not available. The structure factor of PMMA is in reasonable agreement with literature data, and the structure factor of all three polymers calculated using two different force fields are reasonably consistent with each other. We examined the structure of the copolymer/water interface. We found that water molecules penetrate into the polymer slab at the interface, and form hydrogen bonds with polymer carbonyl groups. About 20% of polymer carbonyls at the interface form hydrogen bonds with water. The freedom of carbonyl groups at the interface of an initially dry slab to reorient when water is next to the slab was found to contribute significantly to the formation of hydrogen bonds. Carbonyl groups at the polymer/water interface orient more toward the water phase, resulting in an average angle of about 45° with the interface normal, compared to 65° for carbonyls at the polymer/vacuum interface, while carbonyls in the center of the polymer slab are isotropically oriented. Our simulations also provide interfacial tension values for the copolymer/water and the copolymer/vacuum interfaces. We use these values together with the experimental water/vacuum surface tension to predict a contact angle of a water droplet on a copolymer surface of 77.5 ± 10.8°, which is within 10 deg of the experimental value. To obtain the interfacial tensions at room temperature, we calculated the temperature dependence of copolymer/water and copolymer/vacuum interfacial energy over a range of temperatures from 500 to 363 K and extrapolated the results to room temperature, where direct measurements are impractical due to slow dynamics near the glass transition. Having established this interfacial model, we will in future work explore the colloidal stability of copolymer latex particles by adding surface charged acidic comonomer, surfactants and salts. We can also calculate the interactions of the surface with different surfactants and rheology modifiers, which may serve as a guide to tune the rheology of latex formulations.

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AUTHOR INFORMATION

Corresponding Authors

*(S.T.M.) E-mail: [email protected]. *(R.G.L.) E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS

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REFERENCES

We thank Antony Van Dyk, Susan Fitzwater, and Valeriy Ginzburg for helpful discussions and the Dow Chemical Company for financial support.

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ASSOCIATED CONTENT

S Supporting Information *

Effect of bond constraint and CGenFF parameters for polymers. This material is available free of charge via the Internet at http://pubs.acs.org. K

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L

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