Modeling the Adsorption of Rheology Modifiers onto Latex Particles

Nov 2, 2015 - The Dow Chemical Company, Midland, Michigan 48674, United States ... Jonathan R. Brown , Youngmi Seo , Tiara Ann D. Maula , Lisa M. Hall...
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Modeling the Adsorption of Rheology Modifiers onto Latex Particles Using Coarse-Grained Molecular Dynamics (CG-MD) and SelfConsistent Field Theory (SCFT) Valeriy V. Ginzburg,*,† Antony Keith Van Dyk,‡ Tirtha Chatterjee,† Alan Isamu Nakatani,‡ Shihu Wang,§ and Ronald G. Larson§ †

The Dow Chemical Company, Midland, Michigan 48674, United States The Dow Chemical Company, Collegeville, Pennsylvania 19477, United States § Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109, United States ‡

S Supporting Information *

ABSTRACT: We model the adsorption of hydrophobically ethoxylated urethane (HEUR) thickeners onto two hydrophobic surfaces separated by a 50 nm gallery using coarse-grained molecular dynamics (CG-MD) with implicit solvent and three-dimensional self-consistent field theory (SCFT) with explicit solvent. The CG-MD simulations can be readily extended to encompass very long HEUR chains (up to 540 EO groups) but cannot with current computer speed simulate adsorption of HEURs with hydrophobes longer than 12 carbons (C12). The SCFT method can readily simulate HEURs with longer, C16, hydrophobes but has a greater challenge simulating very long EO chains. For HEURs with 180 EO units and C8 and C12 hydrophobes, both methods can be applied, allowing a combination of the two methods to span much of the parameter space of interest to experimentalists. It is demonstrated that depending on the hydrophobe strength and the HEUR concentration, HEUR chains can adsorb to the surfaces directly or indirectly (as adsorbed micelles or admicelles). We show that for hydrophobes as large or larger than C12 micellization and subsequent adsorption of the micelles play an important role in accurate prediction of adsorption isotherms and the structure of adsorbed layers and that micelles in solution form nodes that allow two or more HEUR chains to bridge the gallery between the two surfaces. The study suggests the need to investigate the influence of admicelles on the effective steric interaction potential, which, in turn, will influence both colloidal stability and rheology of HEUR thickened latex paints.

1. INTRODUCTION Hydrophobically modified ethylene oxide urethane (HEUR) thickeners have been used in paints and coatings for more than 25 years.1 The HEUR molecules are basically telechelic molecules with a poly(ethylene oxide) backbone linked by urethane groups. The end groups are usually hydrophobic alkyl groups of various carbon numbers (often referred to as “phobes”). Numerous researchers have studied their rheology, both in solution2 and in paint formulations (in the presence of pigments and latex binders).3−5 Among the experimental methods utilized to characterize these materials are steadyshear2a,f,6 and oscillatory-shear2c,6,7 rheology, neutron scattering,8 light scattering, NMR, and other techniques. In a pure aqueous HEUR solution, the molecules form a transient network (TN) consisting of flower micelles (FM) bridged by poly(ethylene oxide) (poly-EO) chains.6,9−11 Clusters of flower micelles grow as the HEUR concentration increases to form 3-dimensional networks and breakup as the shear rate is increased.12 In general, transient networks exhibit a Newtonian plateau at low shear rates (Brookfield range), then © XXXX American Chemical Society

undergo shear thinning at intermediate shear rates (KU range), and end up with another Newtonian plateau at high shear rates (ICI range). Under some circumstances, a weak shear thickening could be also observed.13 As discussed by Winnik and Yekta,10 the rheology of HEUR solutions is reasonably well understood. The oscillatory-shear viscosity of HEUR solutions is consistent with a simple Maxwell model in which the relaxation time corresponds to the characteristic pull-out time of the hydrophobe from the flower micelles. The steady-shear rheology, on the other hand, is more complicated, although the theoretical approaches of Tanaka and Edwards11,14 seem to capture most of the salient features, with recent analysis by Watanabe and co-workers15 offering additional insights into shear thickening mechanisms. The role of HEURs in a paint formulation is even more complex. In a typical paint formulation, the volume fraction of Received: September 21, 2015 Revised: October 21, 2015

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stretched adsorbed chains).23 As the concentration increases further, the crowding at the surface forces the polymers to form a second adsorbed layer bound to the first one, often by way of adsorbed micelles (“admicelles”). To understand the complexity of adsorbed HEUR layers beyond the simple brush theory, we performed threedimensional simulations using both CG-MD and SCFT. Where possible, the results from the two methods are compared, and in general, good agreement is found. The use of these two methods allows a quantitative comparison between them, allowing a check on their realism, since they use different approximations and assumptions. In addition, a different range of parameter space is readily accessible to each method; CGMD can simulate very long HEURs, corresponding to experimentally realistic molecular weights, but is limited to hydrophobes no longer than 12 carbons (C12), while SCFT can simulate longer hydrophobes (C16) but has more difficulty simulating very long HEUR chains. By demonstrating that both methods are accurate in regions of parameter space that both can access (i.e., relatively short HEURs with short hydrophobes), the two methods, when combined, encompass much of the range of parameter space that is of commercial interest.

solids is fairly high (about 1030% binder or latex and about 10−20% pigment). It is presumed that interactions between the pigment particles and HEURs are minimal.16 Thus, it is necessary to probe the interplay between the latex binder particles and the HEUR molecules. But what are the HEUR conformations? Do they directly bridge different binders, or do they generally adopt “loop” configurations? How does direct and indirect bridging and looping change with HEUR concentration and shear rate, and how does this impact the rheology? Recently, we used several experimental methods (small-angle neutron scattering (SANS),16a,17 nuclear magnetic resonance (NMR),16b and rheological characterization16a) to demonstrate that HEURs and latex particles can often combine to form a hybrid network or transient network of bridged particles (TNBP). The properties of the hybrid network and the distribution of latex and pigment cluster sizes are substantially influenced by the shell of HEUR molecules adsorbed onto latex particles. Depending on the adsorption characteristics, volume solids, interparticle distance, etc., HEUR molecules could either destabilize the paint suspension (e.g., by creating direct bridges between particles or by causing particle aggregation via the depletion mechanism) or, alternatively, improve that stability (by means of additional steric repulsion between adsorbed brushes).18 Furthermore, rheology of a paint depends on the interplay between the bridge lifetime and the effective shear rate.16a No comprehensive theory of this dependence is currently available, even though several authors used Monte Carlo and molecular dynamics to describe the behavior of telechelic polymers attached to flat surfaces and subjected to shear forces.19 Thus, predicting or describing the adsorption isotherms in HEUR/binder/water mixtures is a crucial first step for understanding the paint rheology. The adsorption of HEURs onto acrylic and other latices as a function of HEUR molecular structure (including hydrophobe type and overall molecular weight), presence of surfactants, latex surface chemistry, and other factors has been extensively studied by many researchers.1d,20 Experimentally, it has been shown that, in general, HEUR thickeners exhibit a strong affinity to adsorb onto latex surfaces, often displacing surfactants and oligomers which originally occupied the latex surface. Adsorption isotherms (the adsorbed amount of HEUR as a function of its bulk concentration) have often been shown to be consistent with the Langmuir21 model, with the maximum adsorbed amount estimated to be between 0.5 and 2.0 mg/ m2.22 Since adsorption isotherms are good measures of the thermodynamic interaction between the latices and the HEURs, it is important to be able to model them. Once our models are able to successfully describe the adsorption isotherms, it should be possible to utilize them to consider other effects, such as the competition between HEURs, oligomers, and surfactants for latex surfaces, the effective steric repulsion or attraction between the latex particles due to adsorbed chains, and, ultimately, the impact of adsorbed chains on the paint rheology. The problem of polymer adsorption onto surfaces has been studied extensively over the past several decades. Theoretical methods used in those studies include scaling theories, selfconsistent field theory (SCFT), atomistic and coarse-grained molecular dynamics (CG-MD), and Monte Carlo (CG-MC) methods. It has been shown that for small polymer concentration the adsorbed layer is in a “mushroom” regime (separated unstretched adsorbed chains); as the adsorption density increases, it transitions to a “brush” regime (overlapping

2. THEORY 2.1. Coarse-Grained Molecular Dynamics. Coarsegrained molecular dynamics (CG-MD) simulations have been widely used to simulate the self-assembly of short CnEm type of surfactants, where n is the number of carbons and m is the number of ethylene oxide monomers. Direct simulation of HEUR self-assembly is rare because of the relatively long chain length of HEURs and slow dynamics of hydrophobe aggregation. Recently, the development of coarse-grained, implicit-solvent MD approach24 has made it possible to simulate long HEURs on relatively large length and time scales. In this study, we employ these force field (FF) parameters to study the adsorption of HEURs to model latex surfaces. The coarse-grained (CG) MD simulations with implicit water model are performed using the GROMACS25 simulation package 4.5.5. The coarse-graining and parametrization of HEUR are based on the Dry Martini FF.24 Briefly, two types of CG beads are used for HEUR in this study: C1 and EO. Each C1 bead corresponds to a string of four carbon atoms while each EO bead corresponds to one ethylene oxide. The nonbonded interactions between these beads are modeled using a 6−12 Lennard-Jones potential with a cutoff distance of rc = 1.2 nm. The standard shift function in GROMACS25 is used to shift the LJ potential gradually to zero from a distance of rshift = 0.9 nm to rc so that both the energy and force becomes zero at rc. The bonded interactions are described by weak harmonic stretching and bending potentials and cosine-type angle potentials. To model the dihedral interactions, we utilize piecewise linear/harmonic distance restraints (bond type 10 in GROMACS) on the separation of the ith and i + 3rd EO beads in a PEO chain; this is done to enable larger timesteps (40 fs) while maintaining numerical stability. In all the simulations, an NVT ensemble is employed and the temperature is controlled at 300 K by a Berendsen thermostat26 with a coupling constant of 4 ps. The secondorder stochastic dynamic integrator in GROMACS is used, and all simulations are run with a time step of 40 fs. The CG parameters are summarized in Table 1, and the detailed description of potentials used is given in the Supporting Information. We also note that we validated these parameters B

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same range as CnEO180Cn. Three-dimensional periodical boundary conditions are applied. Each simulation is run for 2 μs for C8 HEURs and 6 μs for C12 HEURs in nominal simulation time. The last 1 μs for C8 HEURs and 2 μs for C12 HEURs is averaged to obtain equilibrium quantities. 2.2. Self-Consistent Field Theory. We refer the reader to several papers and reviews describing SCFT and its various applications, including prediction of adsorption behavior and polymer-induced interactions between colloidal particles.29 The specific geometry and setup used in our modeling are described in Figure 1. Within SCFT, polymer or oligomer chains are

Table 1. Nonbonded (Lennard-Jones) and Bonded Interaction Parameters Used in the Simulations Interaction Potentials LJ C1 C1 EO EO C1 EO bond C1−C1 EO−EO C1−EO angle

σ (nm) 0.47 0.43 0.47 d0 (nm) 0.480 0.330 0.390 θ0 (deg)

ε (kJ/mol) 4.50 0.20 1.90 kb (kJ/mol/nm2) 1250.0 16000.0 3000.0 kθ (kJ/mol/rad2)

C1−C1−C1 EO−EO−EO C1−C1−EO C1−EO−EO distance restraints

180 150 180 180 cutoff (r1, r2)

35.0 80.0 20.0 20.0 kdr (kJ/mol/nm2)

EOi EOi+3

r1 = 0.00 r2 = 0.60

60

Figure 1. Schematic of SCFT simulations: two flat surfaces (binder particle surfaces) separated by an aqueous solution of HEUR polymer.

treated as Gaussian (in the continuum version) or freely joined (in the lattice version). This enables one to write the expression for the partition function of the overall ternary system as follows (here, we are using the continuum rather than the lattice description):

by simulating the conformations of PEO chains in water and comparing them to experimental data; we also applied the same parameters to study micellization of alkylethoxylate surfactants.27 The model linear telechelic HEURs simulated by CG-MD have a backbone consisting of 180, 360, and 540 EO units ((Mn = 7920, 15 840, and 23 760 g/mol) and two hydrophobic terminal groups, i.e., hydrophobes. The hydrophobes are represented based on equivalent number of CH2 groups and typically include effective contributions from isocyanate linker and the hydrophobe itself.28 For the CG-MD simulations, we consider two types of hydrophobes: C8 (8 carbon atoms or 2 C1 CG beads) and C12 (12 carbon atoms or 3 C1 beads). For simplicity, a HEUR is labeled as CnEOmCn or Cn HEURs in the following, where n = 8 or 12 is the number of carbon atoms in one of the hydrophobes and m = 180, 360, or 540 is the number of EO units in the backbone. The latex surface is assumed to be hydrophobic and is modeled using a flat slab consisting of 800 linear alkyl chains. Each chain is 36 carbon atoms in length and is represented by 9 C1 beads joined linearly with the default C1 parameters in the Dry Martini FF.24 The slab is equilibrated first in a box of dimensions 10 nm × 10 nm × 8 nm for 200 ns. Periodic boundary conditions are applied in all three directions (X, Y, and Z). The slab thickness at equilibrium is 7.6 nm. It is then placed in the center of a box with dimensions of 10 nm × 10 nm × 57.6 nm, again, with periodic boundary conditions. This arrangement is essentially equivalent to creating two hydrophobic surfaces with an effective separation distance H = 50 nm. (We note that assuming monodisperse spherical latex particles and random close packing at ϕ* = 0.63, using H ≃ D[(ϕ*/ϕ)1/3 − 1], the separation distance H = 50 nm is achieved for a latex particle diameter D = 180 nm, and latex volume fraction ϕ* = 0.3, typical of commercial paint formulations.16a) After setting up the slab in the periodic box, 10−80 copies of CnEO180Cn are placed randomly in the gallery space to build initial configurations for our simulations. This range of numbers of HEURs corresponds to a range of volume fractions of roughly 0.034−0.233. The number of CnEO360Cn (from 5 to 40) and CnEO540Cn (from 4 to 27) in the gallery are adjusted accordingly in order to keep the volume fraction HEURs in the

Z=

⎤

⎡

∫ exp⎢⎣− F({W k}, T{φ}, ξ) ⎥⎦D{W }D{φ}Dξ B

(1)

⎡Q ⎤ ⎡Q ⎤ F = − nHEUR ln⎢ HEUR ⎥ − n w ln⎢ w ⎥− ⎢⎣ VφHEUR ⎥⎦ ρ0 VkT ⎣⎢ Vφw ⎥⎦ 1 V 1 V

∫ dr[WC(r)ϕC(r) + WE(r)ϕE(r) + Ww(r)ϕw(r)]

∫ drξ(r)[1 − ϕw(r) − ϕC(r) − ϕE(r) − ϕB(r)]+

1 V 1 V

∫ dr[χCE ϕC(r)ϕE(r) + χCw ϕC(r)ϕw(r) + χwE ϕw(r)ϕE(r)]+ ∫ dr[χBE ϕB(r)ϕE(r) + χBw ϕB(r)ϕw(r) + χBC ϕB(r)ϕC(r)] (2)

In eq 2, ϕB(r), ϕE(r), ϕC(r), and ϕw(r) are local (spatially varying) volume fractions of binder (B), EO-block (E), hydrophobe (C), and water (W), respectively; φHEUR and φW are the overall volume fractions of HEUR and water; then number densities of HEUR and water molecules are nHEUR = φHEUR/NHEUR, nw = φw/Nw. The single-molecule partition functions, QHEUR and QW, are functionals of the chemical potential fields WE, WC (for HEUR), and WW (for water). More details are provided in the Supporting Information. To ensure consistency with the CG-MD calculations, we set the reference unit mass to equal 56 g/mol or (CH2)4. For the HEUR chains, the effective degree of polymerization, NHEUR, is equal to the total HEUR molecular weight divided by the mass of the chosen reference repeat unit, i.e., MW/56. For water, the situation is more complicated. While naively, one would assume that Nw = 1 (and the effective repeat unit mass should be chosen to be 18 g/mol), recent experience in particle-based coarse-grained simulations suggest that a more appropriate choice would be Nw = 3−4, depending on the mass of the chosen repeat unit.30 This is due to strong hydrogen-bonding and formation of water clusters, resulting in a substantial C

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done in 1D where micelles are disregarded. As shown in Figure 2, the total adsorbed HEUR layer often consists of directly

reduction in translational entropy compared to the assumed scenario in which water molecules were behaving as pure monomers. In this paper, we take Nw = 4, again in order to facilitate comparison with CG-MD simulations. The volume fraction profile of the binder, ϕB(r), is equal to 1 inside the slab, 1/2 at the surfaces, and 0 in the gallery. All other volume fraction (“density”) profiles are determined self-consistently. Flory−Huggins parameters χij are described in Table 2. In Table 2. Flory−Huggins Interaction Parameters C4H8 H2O EO

C4H8

H2O

EO

0.00 4.00 0.00

4.00 0.00 0.06

0.00 0.06 0.00

Figure 2. Sketch of various HEUR configurations near the latex surface (black rectangle). Red ovals represent hydrophobes, and curved green lines depict EO chains. The adsorbed shell consists of the directly adsorbed hydrophobes as well as adsorbed micelles (admicelles) that could be weakly bound to the surface as shown. Some hydrophobes could also be unattached (“dangling ends”).

selecting those parameters, we assumed that χCW = 4.0 (so that the effective enthalpy of the water/hydrocarbon interaction is about 1kBT/CH2 group) and χCE = 0.0 (EO and hydrocarbon have no enthalpy of mixing), while letting χEW vary between 0 and 0.1 (since water is a good solvent for PEO) and choosing the value that gives the best agreement with the MD simulation (χEW = 0.06). The functional integral (1) is generally intractable analytically or numerically; the mean-field approximation is therefore used to find the “saddle point” of the free energy functional (2) with respect to local densities of water (ϕw(r)), EO (ϕE(r)), and (CH2)4 (ϕC(r)), chemical potential fields WC(r), WE(r), and Ww(r), and the pressure field ξ(r). Setting all functional derivatives of F to zero, we obtain typical SCFT self-consistency equations which are then solved in an iterative fashion (Picard algorithm) until the solution has converged. At the start of the simulation, all chemical potential fields are initialized as Gaussian random variables with noise amplitude of 0.03 and average value of 0 (small perturbation to push the system away from the unstable uniform solution). We consider three model telechelic HEURs having the same backbone length of 180 EO units (Mn = 7920 g/mol) and hydrophobes with “equivalent carbon atom numbers” of 8, 12, and 16 (to be labeled as C8, C12, and C16 HEURs, respectively). Chains with more than 180 EO groups become difficult to simulate with SCFT because the product of the Flory−Huggins interaction parameter and chain length (often called effective segregation strength, χN) then becomes very high. (Generally, SCFT begins to fail when effective segregation strength, χN > 40−50; thus, for parameters used here, maximum EO length should be about 200 monomers for the C16 hydrophobe, about 350 monomers for the C12 hydrophobe, and about 480 monomers for the C8 hydrophobe.) The simulation box is depicted in Figure 1. Working in the spirit of the Derjaguin approximation, we replace latex particles with flat surfaces. The neighboring particles are placed sufficiently far away that their interparticle distance (H = 50 nm) is much greater than twice the equilibrium end-to-end distance for the HEUR chains in water (2Ree ∼ 1215 nm, based on the estimates of Lee et al.30b) The calculations are performed in the canonical ensemble, so that the HEUR volume fraction, Φ, is specified and varied between 0.01 and 0.11. Expressed in terms of number of chains per unit area of latex surfaces, this corresponds to between 0.018 and 0.2 chains/nm2, or from 0.036 to 0.4 nm−2 hydrophobes per unit area. One crucial, often-missing, consideration in SCFT simulations is the role of micelles, since SCFT simulations are often

adsorbed chains and adsorbed micelles (“admicelles”). To ensure that we correctly describe the micelles, our SCFT simulations were performed in two steps. First, the surfaces were made “hydrophilic”; i.e., their Flory−Huggins parameters with all the species were set equal to those of water. That procedurerun for 2000 iterations or until the density convergence criterion, ⟨|ϕC(r) + ϕEO(r) + ϕW(r) + ϕB(r) − 1|⟩ < 5 × 10−4, is satisfied“nucleated” micelles or other aggregates close to the surfaces. On the other hand, the hydrophilic nature of the surface prevented the uniform, quasi1d increase of the hydrophobe density at the surface layer. As shown in Figure 3, this procedure indeed results in the

Figure 3. Density color maps for (a) hydrophobes and (b) EO groups at the end of the preparatory SCFT run (cross-section in the XZ-plane, perpendicular to the surfaces). Hydrophobe is C12, and HEUR volume fraction is 0.11. Micellar core diameter is about 2 nm, and overall diameter is about 5 nm.

formation of spherical micelles with core diameter about 1.5−2 nm and EO−corona thickness of about 1.5 nm. Second, the chemical potential fields from this calculation were used as input to a longer (5000 iterations or until convergence of 3 × 10−4) simulation with “hydrophobic” surfacesFlory−Huggins parameters with all the species were now set equal to those of the hydrophobes. To calculate the adsorbed amount X, we use the following expression: 1 X= dz[⟨ϕC(z)⟩ − ϕCbulk ] (3) 2 where ⟨ϕC(z)⟩ is the XY-average of the hydrophobe volume fraction in the z layer, and ϕbulk is equal to ⟨ϕC(z)⟩ in the C middle of the gallery. The factor 1/2 reflects the presence of two surfaces. Here, X is expressed in lattice units; it is straightforward to convert it to standard units such as mg/m2 or nm−2.

∫

D

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beginning of our simulations, the total number adsorbed hydrophobes (including both directly and indirectly adsorbed hydrophobes) is close to 80 for C12 HEURs. This is because of the quick formation of C12 aggregates even from initially random configurations and the formation of linkages between these aggregates and the surfaces via dangling HEURs. Later, we observe abrupt decreases and increases in the total number adsorbed hydrophobes, which correspond to desorption and readsorption of the admicelles from the surfaces mostly due to the dynamics of dangling chains that link admicelles to surfaces. Unlike C8, the C12 hydrophobes can form micelles, and if these micelles are connected to surfaces indirectly, via admicelles shown in Figure 2, their detachment from or reattachment to the surfaces results in a large change in the total number of adsorbed hydrophobes. The structure of the micelles is described in our recent paper;27 in particular, it was found that the average aggregation number (number of hydrophobes per micelle) in C12 HEUR is approximately 20−25. To better visualize molecular arrangements of HEURs in the gallery, we show snapshots of CG-MD simulations for both C8 and C12 systems at HEUR volume fraction of 0.117 (Figure 5).

We now proceed to describe simulation results. Wherever possible, CG-MD and SCFT results will be shown side by side for comparison.

3. RESULTS AND DISCUSSION 3.1. Structure of Adsorbed Layer. In the presence of a hydrophobic slab, many HEUR hydrophobes are directly adsorbed onto the surfaces; if two hydrophobes belonging to the same HEUR molecule are adsorbed onto the same surface, the HEUR molecule forms a loop. The hydrophobes can also be indirectly attached to surfaces via dangling chains or via admicelle(s), as schematically shown in Figure 2. Within CGMD, we can calculate the number of adsorbed hydrophobes in different states as a function of time for C8EO180C8 and C12EO180C12. The results for a HEUR volume fraction of 0.117 (40 HEURs or 80 hydrophobes in total) as a function of time are shown in Figure 4. It can be seen that C8 HEURs

Figure 4. Number of adsorbed hydrophobes vs nominal simulation time for (a) C8 HEUR and (b) C12 HEUR. The HEUR volume fraction is 0.117 (or 40 molecules) in both cases. The HEURs can be directly adsorbed on the surface (many being in a loop configuration) or indirectly adsorbed via admicelles or as dangling chains (“tails”; see Figure 2).

adsorb to surfaces very quickly, and the adsorbed amount reaches equilibration in less than 300 ns simulation time. On the other hand, the adsorption of the C12 hydrophobe is somewhat slower and reaches equilibration after roughly 1.5 μs simulation time. We notice that the number of adsorbed C8 hydrophobes fluctuates much more quickly around its equilibrium value than C12 does due to the shorter length and weaker adsorption of the former and thus its faster dynamics. The average equilibrium number of adsorbed C8 hydrophobes is 44.0 (55% of the total), of which about 89% belong to “loops” and the rest belong to “tails”. The total number of C8 hydrophobes adsorbed to surfaces (including both directly and indirectly adsorbed hydrophobes) is only slightly larger than that of the directly adsorbed C8. In fact, because C8 HEURs barely form micelles due to the short hydrophobe length and relatively long hydrophilic backbone, most of these indirectly adsorbed hydrophobes are in the form of dangling hydrophobes or unstable aggregates of sizes 2−5. For the same reason, there is also a large fraction of free C8 HEURs not adsorbed to surfaces, as can be seen from Figure 4a (the total number of adsorbed C8 hydrophobes is significantly less than 80, which is the total number of hydrophobes present in the gallery). For C12 HEURs, the average number of hydrophobes directly adsorbed to the surface is 51.0 (63.7% of the total), and about 97.6% of them are in loops; both values are slightly larger than for C8 HEURs. A large difference is seen in the indirectly adsorbed hydrophobes. As seen in Figure 4b, even at the

Figure 5. Snapshots of CG-MD simulation box. Left: C8 HEUR; right: C12 HEUR. Red spheres are EO beads, blue spheres are hydrophobes, and light-green areas correspond to the latex particles. The HEUR volume fraction is 0.117.

Here, red dots correspond to EO beads, and blue dots represent hydrophobes. For the case of C8 HEUR, one can see that in addition to directly adsorbed hydrophobes there are also many hydrophobes floating in the gallery. For the case of the C12 HEUR, however, almost all the hydrophobes are either adsorbed or associated into micelles. Furthermore, the micelles themselves are weakly attached to the latex surfaces (one or more HEUR chains have one hydrophobe in the micelle and another directly adsorbed onto a latex surface). Similar behavior can be observed in SCFT simulations as well. Figure 6 depicts SCFT density profiles of the EO-block at a HEUR volume fraction of 0.13. For the C8 HEUR, one can clearly see the layer of directly adsorbed molecules (loops and dangling chains); further away from the surfaces, the liquid is nearly uniform. For the C12 HEUR, on the other hand, the first layer is almost the same, but one can also clearly see the admicelles (red spheres). The C16 HEUR system (not shown) exhibits behavior very similar to the C12 one. We can understand the molecular arrangements in more detail by averaging the local densities (or volume fractions) of E

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Interestingly, CG-MD shows some depletion of hydrophobes 2−3 nm away from the surface, while SCFT does not seem to capture this effect. Figure 7c,d shows similar plots for the EOblock density and volume fraction. Both CG-MD and SCFT describe the formation of a “wet” brush having thickness of about 5−6 nm. This value is in good agreement with theoretical models23a,30b,31 and qualitatively in line with the SANS measurements of loop-forming swollen HEUR shells on acrylic binders16a,17b (although in the latter measurements, the HEUR molecular weight was much larger than in the current theoretical study). Simulations clearly show that at these, relatively low, HEUR concentrations, HEURs do not form micelles, preferring to directly adsorb onto the surfaces. Only at much higher concentrations, with the surfaces fully saturated, do stable micelles or even micellar networks form. This range of high HEUR concentration, however, is out of the scope of this study and will be addressed in future publications. For the C12-HEURs (Figure 8a−d), the behavior is quite different. At low HEUR volume fractions, most hydrophobes are still directly adsorbed to surfaces (Figure 8a,b, blue curves). However, as the HEUR concentration is increased to 11 vol % (Figure 8a,b, red curves), the polymers begin to form micelles, and those micelles are attached to the initial brush. This effect can be seen in both CG-MD and SCFT simulations, though there are small but pronounced quantitative differences (CGMD shows wider peaks and locates them further away from the latex surfaces than does SCFT). Note that the two peaks in

Figure 6. SCFT density color map for EO-blocks. Left: C8 HEUR; right: C12 HEUR. Red corresponds to EO-rich regions (directly adsorbed brush or admicelles), while blue depicts EO-poor regions (bulk solution away from the surfaces). The HEUR volume fraction is 0.13.

various species in the XY-plane and plotting those averages as a function of Z (direction normal to the surfaces). Figure 7 shows the density profiles for the C8 HEUR system. In Figure 7a, we plot the hydrophobe density calculated using CG-MD, while Figure 7b shows the hydrophobe volume fraction calculated using SCFT. It can be clearly seen that most of the hydrophobes are directly adsorbed onto the latex surfaces, while those in the bulk are dispersed fairly uniformly.

Figure 7. Density profiles of hydrophobe and EO-blocks along the Z (normal) direction as calculated using CG-MD and SCFT for the C8 HEUR solution (volume fractions 0.07 and 0.11). (a) CG-MD density of hydrophobes, i.e., C1 beads; (b) SCFT volume fraction of hydrophobes; (c) CGMD density of EO beads; (d) SCFT volume fraction of EO-blocks. F

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Figure 8. Same as Figure 7, but for the C12 HEUR.

SCFT have different heightsthe left one is composed of three micelles, and the right one only of two. In CG-MD, these micelles are often able to jump from one surface to the other, and so the profile is symmetric. The EO profiles (Figure 8c,d) show the same trends. For lower concentrations, the adsorbed polymers form a typical wet brush, while for higher concentrations, there are admicelles weakly attached to the brush. SCFT results for C16 HEUR (not shown) shows behavior very similar to that of the C12 HEUR, with slightly higher direct adsorption. 3.2. Adsorption Isotherms. We can now compute the adsorption isotherm, i.e., the adsorbed amount of HEUR per unit area as a function of the total HEUR volume fraction. In Figure 9, such an isotherm (more precisely, the fraction of hydrophobes adsorbed onto the surface vs total HEUR volume fraction) is plotted for C8 and C12 HEURs. (For the C16 system, all hydrophobes are found to be adsorbed for all HEUR concentrations.) It can be seen that there is an excellent agreement between the CG-MD simulation results (symbols) and SCFT results (lines), provided that the Flory−Huggins parameters in the SCFT calculation are those given in Table 2. The agreement becomes worse at higher HEUR concentrations, but for practical applications, the limits considered in this study should be quite sufficient (since typical HEUR concentrations in commercial waterborne paints seldom exceed several volume percent). Thus, three-dimensional SCFT calculations can be used to accurately compute adsorption isotherms and, from these, effective particle−particle interactions.

Figure 9. Total (both direct and indirect) adsorbed fraction of hydrophobes as a function of HEUR volume fraction for C8 and C12 HEUR solutions. Lines correspond to SCFT simulations, while symbols are results of CG-MD simulations.

From the CG-MD simulations, we can extract detailed information about the chain configurations (Figure 10a,b). At low HEUR volume fraction below 0.05, nearly all C8 hydrophobes (Figure 10a) are adsorbed to the surfaces in the form of loops. The fraction of adsorbed C8 hydrophobes starts to decrease once HEUR volume fraction reaches 0.05 (about 0.067 chains/nm2 or 0.133 hydrophobes/nm2), a critical condition where the surfaces are more or less covered fully G

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fraction is reached, as shown by the minimum in total adsorbed HEURs in Figure 10b, and then increases back to 100% adsorption. One possible explanation for such a minimum could be that at the onset of micellization a micelle has a preference for the adsorbed state but also spends substantial time in the bulk of the gallery. These dynamic processes of desorption and readsorption reduce the average fraction slightly. As the HEUR volume fraction is increased further, the admicelle state would become more favorable energetically compared to the free micellar state and the adsorbed fraction would again increase to 100%. Figure 11a shows the SCFT adsorption isotherms plotted in terms of adsorbed amount (chains/nm2). It can be seen that for the C8 HEUR, at volume fractions close to 0.10 (0.14 chains/ nm2), the adsorbed amount approaches saturation, while for the C12 and C16 HEUR, the relationship between adsorbed amount and total amount still remains linearall HEURs adsorb either directly onto the surface or indirectly as admicelles. The adsorbed amount plateaus for the C8 system at around 0.1 chains/nm2 or, given the HEUR molecular weight close to 8000 g/mol, at about 1.3 mg/m2, which is the right order of magnitude for these types of systems (see, e.g., ref 20a or 20c). For the case of C12 and C16 HEURs, we can determine the amount of directly and indirectly adsorbed chains (Figure 11b). The C12 HEURs all adsorb directly until the coverage becomes approximately equal to 0.1 chains/nm2; after that, additional molecules begin to adsorb indirectly. The C16 HEURs, with stronger binding of hydrophobes to the latex, form a slightly denser layer with coverage close to 0.12 chains/nm2. Finally, we note that adsorption depends not just on the hydrophobe strength but also on the backbone molecular weight. We performed CG-MD simulations for C8 and C12 HEURs with PEO chain lengths of 360 and 540 repeat units. Interestingly, our attempts to use SCFT to model those systems were not successful as SCFT numerical scheme failed to converge. The CG-MD results are shown in Figure 12a (for C8 HEURs) and Figure 12b (for C12 HEURs). In general, the adsorbed HEUR fractions (as a function of HEUR overall volume fraction) seem to be relatively independent of the molecular weight. However, these results are preliminary and more detailed investigation will be undertaken in the future.

Figure 10. Average fraction of adsorbed hydrophobes, relative to the total number of hydrophobes, as a function of HEUR volume fraction for (a) C8 and (b) C12 HEURs.

by the HEURs. The number of hydrophobes in loops above the critical volume fraction of 0.05 is nearly constant (on average 19.7 C8 HEURs or 39.4 C8 hydrophobes) in the range of HEUR volume fractions considered. The increase of HEUR volume fraction is also accompanied by an increase in the fraction of indirectly adsorbed HEURs as well as free HEURs in solution (as evidenced by the monotonic decrease in both direct and indirect adsorption). Compared to C8 HEURs, for C12 HEUR (Figure 10b), the decrease in the fraction of adsorbed HEURs occurs at a slightly larger critical volume fraction of around 0.07 (0.08 chains/nm2 or 0.16 hydrophobes/nm2), indicating a higher surface coverage for C12 vs C8. Similar to C8, the number of C12 hydrophobes directly adsorbed to surfaces increases at a slower rate after this critical HEUR volume fraction is reached, and so the fraction of hydrophobes adsorbed then decreases. In addition, above the critical concentration, the number of adsorbed loops becomes nearly constant, and so the fraction of HEURs in loops decreases, as shown in Figure 10b. However, the absolute number of loops is larger than for C8 HEURs (on average 24.1 C12 HEURs or 48.2 C12 hydrophobes), and its fraction can be fitted by 0.72/(HEUR volume fraction). However, the fraction of indirectly adsorbed C12 hydrophobes increases with HEUR volume fraction much more strongly than for C8 due to the adsorbed “admicelles” discussed above. The total fraction of adsorbed C12 hydrophobes is thus much larger than for C8 hydrophobes, as seen in Figure 10. As a matter of fact, instead of a monotonic decrease with HEUR volume fraction as in the case of C8 HEURs, the total fraction of adsorbed C12 hydrophobes decreases only slightly when its critical volume

Figure 11. (a) Adsorbed amount (in chains/nm2) as a function of HEUR volume fraction. (b) Amount of indirectly adsorbed HEURs vs total adsorbed amount for C12 and C16 HEURs. H

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Figure 12. Fraction of adsorbed HEURs as a function of HEUR volume fraction (from CG-MD simulations) for molecules with backbone lengths of 180, 360, and 540 EO repeat units: (a) C8 hydrophobe; (b) C12 hydrophobe.

4. CONCLUSIONS We performed three-dimensional mesoscale simulations (coarse-grained molecular dynamics and self-consistent field theory) to study the behavior of hydrophobically ethoxylated urethane (HEUR) thickeners near hydrophobic latex surfaces. The CG-MD study was performed with the Dry Martini force field, in which water molecules are not modeled explicitly, so the simulation speed increases dramatically compared to the original Martini. The SCFT model, on the other hand, accounts for water explicitly, parametrizing it as a four-bead chain to account for strong hydrogen bonding and formation of water clusters. In general the two models agree well, in terms of both the structure of adsorbed layer and the overall adsorption isotherms. Our study investigated adsorption and surface interaction behavior of HEUR polymers with short to medium sized PEG backbone (180 to 540 EO monomers or 7920 to 23 760 g/ mol), capped by hydrophobe groups on both ends. The hydrophobes were modeled as simple hydrocarbon oligomers, Cn, where C denotes one methyl group and n is the number of methyl groups in one hydrophobe. We considered n = 8, 12, and 16, which spans the typical range for commercial HEURs. CG-MD simulations were carried out for C8 and C12 hydrophobes, but for C16, CG-MD simulations could not be fully equilibrated, since for C16 the lifetime of hydrophobe− surface or hydrophobe−hydrophobe association is much longer than a microsecond. However, SCFT simulations could be equilibrated and showed that C16 hydrophobes adsorb very strongly onto the latex surfaces, as expected. The role of PEG molecular weight is also important, with our simulations coming close to that of commercial HEURs, which usually have Mw of around 30 kg/mol. Here, we found that while CG-MD can successfully describe molecular weight dependence of adsorption for smaller (C8) and medium (C12) hydrophobes, SCFT generally fails if the molecular weight becomes too high. However, more studies are needed to better understand the applicability of either of the two methods in the broader parameter space. We also found that for shorter, weaker hydrophobes like C8 HEURs do not form micelles, and thus adsorption is limited to a monolayer. In contrast, stronger hydrophobes like C12 (and C16) can associate into micelles, and these micelles, in turn, can be weakly bound to a monolayer directly adsorbed to the hydrophobic surface. Thus, the adsorption capacity of the surface increases substantially as a result of micellization; also, it

seems likely that the presence of the indirectly adsorbed admicelles helps improve steric stabilization of the dispersion. Accounting for these effects, however, requires going beyond the classical theory of steric stabilization by brushes31b,32 and will be the subject of future work.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.5b02080. Description of coarse-grained molecular dynamics (CGMD) interaction potentials; description of self-consistent field theory (SCFT) formalism (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected] (V.V.G.). Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We thank Profs. Kristen Fichthorn, Scott Milner, Darrell Velegol (Penn State), John Kieffer (University of Michigan), and Drs. Susan Fitzwater, Alex Kalos, and Fang Yuan (Dow) for helpful discussions. This work was supported by The Dow Chemical Company through Grant 223278AE.

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REFERENCES

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