^{1,a)}, Ioannis G. Kevrekidis

^{1,2}and Athanassios Z. Panagiotopoulos

^{1,3}

### Abstract

We have developed explicit- and implicit-solvent models for the flash nanoprecipitation process, which involves rapid coprecipitation of block copolymers and solutes by changing solvent quality. The explicit-solvent model uses the dissipative particle dynamics (DPD) method and the implicit-solvent model uses the Brownian dynamics (BD) method. Each of the two models was parameterized to match key properties of the diblock copolymer (specifically, critical micelle concentration, diffusion coefficient, polystyrene melt density, and polyethylene glycol radius of gyration) and the hydrophobicsolute (aqueous solubility, diffusion coefficient, and solid density). The models were simulated in the limit of instantaneous mixing of solvent with antisolvent. Despite the significant differences in the potentials employed in the implicit- and explicit-solvent models, the polymer-stabilized nanoparticles formed in both sets of simulations are similar in size and structure; however, the dynamic evolution of the two simulations is quite different. Nanoparticles in the BD simulations have diffusion coefficients that follow Rouse behavior (*D* ∝ *M* ^{−1}), whereas those in the DPD simulations have diffusion coefficients that are close to the values predicted by the Stokes–Einstein relation (*D* ∝ *R* ^{−1}). As the nanoparticles become larger, the discrepancy between diffusion coefficients grows. As a consequence, BD simulations produce increasingly slower aggregation dynamics with respect to real time and result in an unphysical evolution of the nanoparticle size distribution. Surface area per polymer of the stable explicit-solvent nanoparticles agrees well with experimental values, whereas the implicit-solvent nanoparticles are stable when the surface area per particle is roughly two to four times larger. We conclude that implicit-solvent models may produce questionable results when simulating nonequilibrium processes in which hydrodynamics play a critical role.

Financial support for this work was provided by a NIRT award from the National Science Foundation (NSF) (CBET-0506966), with additional support from the Princeton Center for Complex Materials, a National Science Foundation (NSF) MRSEC (Award DMR-0819860), from National Science Foundation (NSF) grant (CMMI-1002469), and from the US DOE (DE-SC 0002097). Additionally, the authors are grateful for discussions with Professor Robert K. Prud'homme, Varun Kumar, Stephanie J. Budijono, and Suzanne M. D'Addio.

I. INTRODUCTION

II. MODELS AND METHODS

A. Implicit-solvent model

1. Characteristic scales

2. Interaction parameters

3. Simulation details

B. Explicit-solvent model

1. Characteristic scales

2. Coarse-grained mapping

3. Interaction parameter determination

4. Friction coefficients

5. Simulation details

C. Nanoparticle identification

D. Rescaling of time

III. RESULTS AND DISCUSSION

A. Dynamics of aggregation

B. Nanoparticle structure and size

C. Nanoparticlediffusion coefficients

D. Nanoparticle stability

E. Relative efficiency of BD versus DPD

IV. CONCLUSIONS

### Key Topics

- Nanoparticles
- 116.0
- Diffusion
- 45.0
- Polymers
- 43.0
- Solvents
- 35.0
- Solution processes
- 32.0

## Figures

Number of nanoparticles, weight-averaged number of solute molecules per nanoparticle, and weight-averaged number of diblock molecules per nanoparticle vs time. Results shown are an average of three randomly initialized simulations, with the error bars showing standard deviations at a few points during the simulation. Solid symbols are for DPD simulations and open symbols are for BD simulations. Triangles are for favorable solute–PS interaction and circles for unfavorable solute–PS interaction.

Number of nanoparticles, weight-averaged number of solute molecules per nanoparticle, and weight-averaged number of diblock molecules per nanoparticle vs time. Results shown are an average of three randomly initialized simulations, with the error bars showing standard deviations at a few points during the simulation. Solid symbols are for DPD simulations and open symbols are for BD simulations. Triangles are for favorable solute–PS interaction and circles for unfavorable solute–PS interaction.

Distribution of solutes in clusters of different sizes after 30 ns (top) and 70 ns (bottom). Clusters containing 20 or more solutes were included in the calculation but are not shown in this figure. The solid line represents a theoretical prediction based upon a diffusion-limited Smoluchowski approach (Ref. 56). The dashed line represents the same theoretical prediction, but with the time rescaled by a factor of 0.75. Symbols are as in Fig. 1.

Distribution of solutes in clusters of different sizes after 30 ns (top) and 70 ns (bottom). Clusters containing 20 or more solutes were included in the calculation but are not shown in this figure. The solid line represents a theoretical prediction based upon a diffusion-limited Smoluchowski approach (Ref. 56). The dashed line represents the same theoretical prediction, but with the time rescaled by a factor of 0.75. Symbols are as in Fig. 1.

Number of diblocks per nanoparticle vs number of solutes per nanoparticle (both weight-averaged) at various points in time during in the simulations. The solid line represents the ratio between the total number of polymers and solutes present in the simulations. Symbols are as in Fig. 1.

Number of diblocks per nanoparticle vs number of solutes per nanoparticle (both weight-averaged) at various points in time during in the simulations. The solid line represents the ratio between the total number of polymers and solutes present in the simulations. Symbols are as in Fig. 1.

Snapshot of nanoparticles formed in a DPD simulation (top) and BD simulation (bottom) with favorable solute–PS interactions. The solute beads are shown in blue, PS beads in red, and PEG beads as transparent gray. Note the uniform coverage of PS beads on the nanoparticle surface and the uniform distribution of PEG chains around the outside of the nanoparticles. Movie files of a DPD simulation (movie 1) and a BD simulation (movie 2) are available (Ref. 58).

Snapshot of nanoparticles formed in a DPD simulation (top) and BD simulation (bottom) with favorable solute–PS interactions. The solute beads are shown in blue, PS beads in red, and PEG beads as transparent gray. Note the uniform coverage of PS beads on the nanoparticle surface and the uniform distribution of PEG chains around the outside of the nanoparticles. Movie files of a DPD simulation (movie 1) and a BD simulation (movie 2) are available (Ref. 58).

Snapshot of nanoparticles formed in a DPD simulation (top) and BD simulation (bottom) with unfavorable solute–PS interactions. The color scheme for the beads is the same as in Fig. 4. Note the clustering of PS beads on the nanoparticle surface, which leaves large areas of the core exposed. Movie files of a DPD simulation (movie 3) and a BD simulation (movie 4) are available (Ref. 58).

Snapshot of nanoparticles formed in a DPD simulation (top) and BD simulation (bottom) with unfavorable solute–PS interactions. The color scheme for the beads is the same as in Fig. 4. Note the clustering of PS beads on the nanoparticle surface, which leaves large areas of the core exposed. Movie files of a DPD simulation (movie 3) and a BD simulation (movie 4) are available (Ref. 58).

Radius of gyration of solute beads vs the number of solute molecules in a nanoparticle. The statistical uncertainty of each data point is no larger than the symbol size. Symbols are as in Fig. 1.

Radius of gyration of solute beads vs the number of solute molecules in a nanoparticle. The statistical uncertainty of each data point is no larger than the symbol size. Symbols are as in Fig. 1.

Mean distance between all PS beads in a nanoparticle and the nanoparticle center of mass vs the maximum solute radius. The statistical uncertainty of each data point is no larger than the symbol size. Symbols are as in Fig. 1. The solid black line of unit slope is a guide to the eye.

Mean distance between all PS beads in a nanoparticle and the nanoparticle center of mass vs the maximum solute radius. The statistical uncertainty of each data point is no larger than the symbol size. Symbols are as in Fig. 1. The solid black line of unit slope is a guide to the eye.

Overall nanoparticle radius vs number of solutes in the nanoparticle. The overall nanoparticle radius is the distance between the terminal bead in a PEG block and the nanoparticle center of mass, averaged over all diblocks in the nanoparticle. The statistical uncertainty of each data point is no larger than the symbol size. Symbols are as in Fig. 1.

Overall nanoparticle radius vs number of solutes in the nanoparticle. The overall nanoparticle radius is the distance between the terminal bead in a PEG block and the nanoparticle center of mass, averaged over all diblocks in the nanoparticle. The statistical uncertainty of each data point is no larger than the symbol size. Symbols are as in Fig. 1.

Diffusion coefficient vs overall nanoparticle radius for the nanoparticles in the BD and DPD simulations. The solid line shows the Stokes–Einstein relation. Error bars show statistical uncertainties for a few representative data points. Symbols are as in Fig. 1.

Diffusion coefficient vs overall nanoparticle radius for the nanoparticles in the BD and DPD simulations. The solid line shows the Stokes–Einstein relation. Error bars show statistical uncertainties for a few representative data points. Symbols are as in Fig. 1.

Diffusion coefficients for nanoparticles from BD simulations, along with the curve *D* = *kM* ^{−1}. The constant *k* was chosen so that the curve passes through the infinite dilution diffusion coefficient of the solute for the implicit-solvent model. Error bars show statistical uncertainties for a few representative data points. Symbols are as in Fig. 1.

Diffusion coefficients for nanoparticles from BD simulations, along with the curve *D* = *kM* ^{−1}. The constant *k* was chosen so that the curve passes through the infinite dilution diffusion coefficient of the solute for the implicit-solvent model. Error bars show statistical uncertainties for a few representative data points. Symbols are as in Fig. 1.

Surface area per polymer vs number of solutes in nanoparticle for simulations with favorable solute–PS interactions. Symbols are as in Fig. 1. The solid line represents the surface area occupied by a randomly coiled diblock (Ref. 16).

Surface area per polymer vs number of solutes in nanoparticle for simulations with favorable solute–PS interactions. Symbols are as in Fig. 1. The solid line represents the surface area occupied by a randomly coiled diblock (Ref. 16).

## Tables

Pairwise interaction parameters for the implicit-solvent model studied with BD simulations. Energies are given as multiples of ε_{ BD }, where ε_{ BD }/*k* _{ B } = 372.5 K.

Pairwise interaction parameters for the implicit-solvent model studied with BD simulations. Energies are given as multiples of ε_{ BD }, where ε_{ BD }/*k* _{ B } = 372.5 K.

Repulsion parameters used in the explicit-solvent DPD simulations, in units of ε_{ DPD }, where ε_{ DPD }/*k* _{ B } = 298 K.

Repulsion parameters used in the explicit-solvent DPD simulations, in units of ε_{ DPD }, where ε_{ DPD }/*k* _{ B } = 298 K.

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