A Dissipative Particle Dynamics Study - ACS Publications - American

Article pubs.acs.org/JPCB

Encapsulation of Nanoparticles During Polymer Micelle Formation: A Dissipative Particle Dynamics Study Kyaw Hpone Myint,†,‡,§ Jonathan R. Brown,§ Anne R. Shim,§ Barbara E. Wyslouzil,§,⊥ and Lisa M. Hall*,§ †

Department of Chemistry, Berea College, Berea, Kentucky 40404, United States Department of Physics, Berea College, Berea, Kentucky 40404, United States § William G. Lowrie Department of Chemical and Biomolecular Engineering, The Ohio State University, Columbus, Ohio 43210, United States ⊥ Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210, United States ‡

S Supporting Information *

ABSTRACT: The formation of block copolymer micelles with and without hydrophobic nanoparticles is simulated using dissipative particle dynamics. We use the model developed by Spaeth et al. [Spaeth, J. R.; Kevrekidis, I. G.; Panagiotopoulos, A. Z. J. Chem. Phys. 2011, 134 (16), 164902], and drive micelle formation by adjusting the interaction parameters linearly over time to represent a rapid change from organic solvent to water. For different concentrations of added nanoparticles, we determine characteristic times for micelle formation and coagulation, and characterize micelles with respect to size, polydispersity, and nanoparticle loading. Four block copolymers with different numbers of hydrophobic and hydrophilic polymer beads, are examined. We find that increasing the number of hydrophobic beads on the polymer decreases the micelle formation time and lowers polydispersity in the final micelle distribution. Adding more nanoparticles to the simulation has a negligible effect on micelle formation and coagulation times, and monotonically increases the polydispersity of the micelles for a given polymer system. The presence of relatively stable free polymer in one system decreases the amount of polymer encapsulating the nanoparticles, and results in an increase in polydispersity and the number of nanoparticles per micelle for that system, especially at high nanoparticle concentration. Longer polymers lead to micelles with a more uniform nanoparticle loading. precipitation (FNP).22−24 In these techniques, the polymer and particles are first dispersed in a good solvent. The solvent− solute interactions are then manipulated−by adding a cosolvent or dropping the temperature−to induce micelle formation. The time scales over which micelle formation occurs can vary from milliseconds in FNP 25−27 to about 15 min for the coprecipitation method.20 In FNP experiments, Prud’homme et al.23 found that slower mixing times (τmix > ∼ 100 ms) yield bigger micelles as slower changes in solvent quality permit unimer exchange to take place for a longer time. In contrast, rapid mixing (τmix < ∼ 40 ms) at high polymer and solute concentrations produced micelles that were kinetically frozen on the time scale of the experiment. When small hydrophobic molecules (solutes) were present, they found that both polymer-only micelles and solute-containing micelles were formed in all their experiments.24 Although FNP can produce micelles of controllable size using different polymers and solutes,28−37 micelle formation using this technique is not fully understood at the molecular level.38 This is largely because

1. INTRODUCTION Nanoparticles encapsulated in block copolymer micelles have great potential for use in the biomedical arena with applications in medical imaging,1−8 drug delivery,5−9 and molecular biomarker-based separation and manipulation.4,10−12 For example, Gao and colleagues6 devised a technique to encapsulate doxorubicin (DOXO) and superparamagnetic iron oxide nanoparticles (SPIONs) into polymeric micelles, and attach cRGD ligands to them. The multifunctional nanoparticles were then used to target specific cancer cells, providing an efficient and MRI-ultrasensitive drug delivery system.6 Similarly, Winter et al.10 encapsulated fluorescent quantum dots (QDs) and SPIONs in polystyrene-b-polyethylene glycol (PS91-b-PEG409)4,13 copolymer micelles, and used these to quickly analyze, separate, and manipulate different biomarkers. An important advantage of using block copolymers with distinct hydrophobic and hydrophilic regions, is that they can encapsulate hydrophobic species as they self-assemble into micelles, and then transport, protect, and manipulate their contents in an aqueous environment.3,14−18 A number of techniques have been developed to encapsulate hydrophobic nanoparticles in block copolymer micelles19 including coprecipitation,20 heating−cooling,21 and flash nano© 2016 American Chemical Society

Received: July 21, 2016 Revised: September 12, 2016 Published: October 17, 2016 11582

DOI: 10.1021/acs.jpcb.6b07324 J. Phys. Chem. B 2016, 120, 11582−11594

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The Journal of Physical Chemistry B

formation process, as well as on final micelle properties. During the initial stages of micelle formation, free polymers and NPs come together to form small new aggregates that then merge to form larger, more stable micelles. In addition to estimating the characteristic times associated with new micelle formation and coagulation, we also characterize the effect of polymer composition and nanoparticle concentration on the average micelle size, composition, and polydispersity throughout the micelle formation process. We use the interaction parameters developed by Spaeth et al.,38,39 changing the solvent−polymer interaction parameters over time from those corresponding to a good solvent (THF) to those corresponding to a poor solvent (water). The hydrophobic NPs in this study are composed of beads with the same parameters as those of the hydrophobic polymer, but the nanoparticle beads are bonded together to form small spheres. This is a generic representation of a hydrophobic nanoparticle, such as the QDs or/and magnetic nanoparticles typically used in experimental work.1,3,11,12 The structure of this paper is as follows. In Section 2, we discuss the parametrization of our DPD model and the method used for the simulation. In Section 3, we present and analyze the results of the simulations. We begin with the dynamic evolution of the simple NP-only and polymer-only systems, while defining how we characterize different parameters such as micelle formation time, coagulation time, and polydispersity. We then discuss and characterize the behavior of the NPcontaining polymer systems. We conclude, in Section 4, with a summary of our results and comment on potential future work to improve understanding of the micelle formation process.

directly observing the micelle formation process during the mixing step is difficult due to the short time scales involved (about ns to ms).38 Thus, experiments are generally limited to characterizing the initial and final states of the system. Recently, Spaeth and colleagues38,39 used dissipative particle dynamics (DPD) to investigate micelle formation in the FNP process. DPD is a coarse-grained method in which one particle bead represents the center of mass of a group of atoms. The beads are very soft and highly overlapping, allowing efficient simulations that can start to approach the time and length scales of interest in micelle formation. DPD simulations of micellization have previously been performed by multiple groups, and this work was recently reviewed.40 Generally, the focus has been on short polymers or surfactants whose micelles can equilibrate within the time scale of the simulation. Long polymers present multiple challenges, though one can effectively simulate over longer times using a coarse-grained, implicit solvent model,39,41 and instead of brute force equilibration, advanced methods, such as umbrella sampling, can be used to determine the appropriate micelle size distribution.41 The experimental application of interest here uses relatively long block copolymers to encapsulate hydrophobic nanoparticles. These block copolymer micelles are not expected to fully reach thermodynamic equilibrium.42,43 Therefore, our simulations attempt to approximately reproduce the kinetics of the micelle formation process during FNP as inspired by refs 38 and 39, such that the micelles form similarly to those in relevant experimental systems (we expect both the simulated and experimental systems to be in a kinetically trapped state). The parameters in Spaeth et al.’s38,39 DPD model were chosen to match the experimental work of Kumar et al.28 that used polystyrene-b-polyethylene glycol (PS10-b-PEG68) as the diblock copolymer, itraconazole, an antifungal drug, as the hydrophobic solute, and THF and water as the solvent and antisolvent, respectively. Spaeth et al. explored the parameters affecting the micelle formation process including the mixing times, the solute solubility, the polymer block lengths, and solvent and polymer concentrations.38 In agreement with the experimental work,22−24 they found that micelle size increased with increasing mixing time. Furthermore, decreasing the solute and polymer concentrations, and increasing the hydrophilic length of the polymers, led to smaller nanoparticles due to the lower rate of micelle coagulation at low concentration and the enhanced stabilization of micelles by the longer hydrophilic region of the polymer.38 Spaeth et al. considered small molecule encapsulants that outnumbered the polymer by a ratio of ∼2.7:1. Our work focuses on a different limit, that of incorporating a small number of relatively large nanoparticles into micelles. More recently, Pi et al.44 used DPD to study encapsulation of cadmium selenide (CdSe) nanoparticles coated with trioctylphosphine oxide (TOPO) in polystyrene-b-poly(ethylene oxide) (PS-b-PEO). For a fixed number of nanoparticles, they focused on the effect of polymer block sizes on the micelle properties including the dimensions of the PS core and the PEO shell. They found that the diameter of the whole micelle, that of hydrophobic core, and the thickness of the hydrophilic outer shell are proportional to the MWs of the PS-b-PEO block copolymer, PS segment, and PEO segment, respectively.44 The goal of the current article is to use DPD to investigate the effects of nanoparticle loading, polymer length, and polymer hydrophobic fraction on the kinetics of the micelle

2. MODEL AND METHODS 2.1. The DPD Model. DPD simulations of the polymer− nanoparticle−solvent system, illustrated in Figure 1, were

Figure 1. System components include the block copolymer, nanoparticles, and water. In this case the block copolymer consists of 17 polystyrene (S) and 33 polyethylene glycol (G) beads, as indicated by the subscripts. The number following N is the ratio of (nanoparticle beads):(polymer beads) in the system. The number of water beads is adjusted to maintain a constant water:polymer bead ratio.

carried out using LAMMPS (large scale atomic-molecular massively parallel simulator). DPD employs a soft, conservative repulsive force FC, a dissipative/frictional force FR, and a random/impulsive force FD to create a system of soft overlapping particles, or beads, that each represent multiple atoms and that collectively conserves momentum. The system evolves under Newton’s laws of motion where the force on bead i due to bead j is given by

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Fi ⃗ = (F C + F D + F R )riĵ

(1)

F C = a ijω(r )

(2)

F D = −γω 2(r )(riĵ ·vij⃗ )

(3) DOI: 10.1021/acs.jpcb.6b07324 J. Phys. Chem. B 2016, 120, 11582−11594

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F R = βω(r )α(Δt )−1/2

(4)

ω(r ) = 1 − r /rc

(5)

and repulsion parameter of aij = 25ε). This produces nanoparticles with an amorphous internal structure containing 103 beads. All bead−bead distances in the sphere less than 1σ were found and bonds were created for these bead−bead interactions to hold the sphere together. The bond lengths between the beads were assigned to keep approximately the same length, but were rounded to the nearest of 0.1, 0.3, 0.5, 0.7, or 0.9σ (except one bead initially not within 1σ of other beads was bonded at bond length 0.9σ to its 2 nearest neighbors). The spring constant k = 30.5ε/σ2 of the internal particle bonds was the same as that for polymer bonding. The shapes of the nanoparticles vary slightly from each other due to the flexible nature of the harmonic bonding, and the bonds also cause the nanoparticle radius during the simulation to be smaller than that of the initial sphere cut out of the fluid. Specifically, the average extent of the nanoparticles (the distance from the lowest to highest x-coordinate of all bead centers of mass, but averaged over the three coordinates) is approximately 2.2 nm, corresponding to a total nanoparticle diameter of approximately 3.2 nm after considering the range of the bead−bead interactions from the center of mass. 2.2. Simulation Details and Interaction Parameters. The pairwise interactions between the beads are chosen to represent the chemical interactions of the atomistic system at a coarse-grained level; parameters are as in refs 38 and 39 and are listed in Table 1. Typically, the NPs that are encapsulated in

and the force is cut off after r = rc. Here, aij is the repulsion coefficient, rc is the diameter of the bead (for all beads types here, rc = 1σ; this is used as our unit of length), γ is the friction coefficient, r̂ij is the unit vector in the direction from beads j to bead i, vi⃗ j is the relative velocity of bead i versus bead j, α is a Gaussian random number with a mean of 0 and variance of 1, Δt is the time step size, and ω(r) is a weighting factor that varies linearly between 0 and 1. The random force is related to the frictional force according to a fluctuation−dissipation relation,45 β = (2kBTγ )1/2

(6)

where kB is the Boltzmann constant and T is the temperature. Further details on the DPD method can be found in the original paper of Groot and Warren.46 For all of the systems, the characteristic length and mapping to real units was σ = 1 nm and the characteristic energy was ε = kBT = 4.114 × 10−21J (the thermal energy at 298 K). Each bead had a unit characteristic mass, m = 1, which we map to 200 g/ mol as in ref 39. The bead number density of the system was ρ = 3.0σ as is typical in DPD simulations. The dimensionless time unit is τ = σ(m/ε)1/2 and a mapping to time can be calculated from the other units. However, the DPD method essentially accelerates the dynamics of the system. Therefore, we report times using a mapping to real time of τ = 250 ps as calculated by Spaeth et al. to match the polymer diffusion to experimental results.39 A time step of Δt = 0.03 τ and a periodic cubic box were used. Simulating micelle formation experiments is challenging because of the large number of molecules involved and the relatively long experimental time scales compared to typical simulation times, even when solvent/antisolvent mixing is rapid and a highly coarse-grained method such as DPD is used.38 We used the relatively high polymer concentration as in ref 38 (32 solvent beads per polymer bead), which was reported to be more than 20 times higher than in the experimental systems of Kumar et al.32,38 This promotes faster micelle formation, making the simulations tractable while maintaining enough solute and polymer in the system to form multiple large micelles. The polymer parameters were also kept the same as in ref 38, to represent polystyrene-b-polyethylene glycol block copolymers in a solvent that is changed from THF to water (see details below).38,39 One hydrophobic bead represents 2 PS monomers and one hydrophilic bead represents 3.4 PEG monomers. Though mapping a bead of 200 g/mol to PEG, based on mass, would give a slightly different number of monomers per bead, this mapping was proposed to approximately reproduce the proper experimental Rg for 20 bead chains in water.38 Mapping 11 water molecules onto a DPD solvent bead approximates the appropriate bead mass and density of water. Neighboring polymer beads were bonded by harmonic bonds using Fbond = k(rij − r0)

Table 1. Pairwise Repulsion Parameters (aij) for the Various Bead−Bead Interactions Given in Units of ε pair type

solvent

NP

PS

solvent

25

NP

initial: 25 final: 54

25

PS

initial: 25 final: 54

25

25

PEG

25

40

40

PEG

25

micelles are hydrophobic. As the simplest case, we consider NPs that are chemically identical to PS. There are 2 sets of parameters for solvent-NP and solvent-PS interactions: one for the favorable solvent (before mixing) that represents THF, and another for the unfavorable solvent (after mixing) that represents water. The friction coefficients of 18ετ/σ2 (for solvent-PEG and solvent-PS interactions) and 4.5ετ/σ2 (for all other interactions) are used as in refs 38 and 39 to match the ratio of the solute and the diblock diffusion coefficients. Initially, the polymers and nanoparticles were dissolved in the good solvent (THF) and equilibrated for 100 000 time steps or 0.75 μs. The THF was rapidly changed into a poor solvent for PS (water) in a stepwise fashion over 800 evenly spaced increments, referred to as the “mixing time”. Specifically, the solvent-NP and solvent-PS repulsion parameters were increased by 29/800 every 1000 time steps, 800 times total for an overall mixing time of 800 000 time steps or 6 μs. Changing parameters in this way was proposed in ref 39 to represent the very fast mixing of solvents in the FNP process.22−24 Finally, the system continued to evolve with pure water as the solvent for an additional 800 000 time steps, or 6 μs. During this third period the solvent-PS and solvent-NP bead repulsion

(7)

with r0 = 0.8σ and k = 30.5ε/σ2 (corresponding to 0.25 N/m), where rij is the distance between the neighboring bonded beads. The spherical nanoparticles (NPs) were created by cutting a sphere of radius 2σ from a configuration of a DPD simulation of pure solvent (unconnected DPD solvent beads with ρ = 3.0σ 11584


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3. RESULTS AND DISCUSSION 3.1. Identifying Aggregates and Micelles. Aggregates/ micelles are found by grouping any beads that are within 1 bead diameter of each other into the same aggregate; all beads in the same molecule are included in the same aggregate by default. Free polymers (or NPs) are those polymers (or NPs) that have no bead within the cutoff distance of beads on any other polymer or NP. Polymer-only micelles, Figure 2a, are defined as

parameters were constant at aij = 54ε. One of the differences between the work in refs 38 and 39 versus this study is that our systems include equilibration time before mixing: the free polymers and nanoparticles are started as though they were initially in THF at equilibrium. The effect of mixing time was studied in ref 38 as noted above. To limit the computational expense of the current study, we use a single mixing time of 6 μs; this is shorter than the shortest mixing time studied in ref 38, but longer than the instantaneous mixing considered in ref 39 and 44. The four types of polymers studied are detailed in Table 2. Each polymer contained either 25 beads or 50 beads, and the Table 2. Composition of the Four Polymers Simulated polymer length [bead]

PS [bead]

PEG [bead]

25 25 50 50

5 8 10 17

20 17 40 33

Figure 2. Cross-sectional view of typical micelles at the end of the simulation: (a) polymer-only micelle formed in the S17G33[N0] system; (b) NP-containing micelle formed in the S17G33[N0.32] system.

aggregates of two or more polymers. As expected, in a poor (aqueous) solvent the hydrophobic interior (pink beads) is protected by the hydrophilic exterior (teal beads). NPcontaining micelles, Figure 2b, are defined as aggregates comprised of at least one polymer and one nanoparticle. In these micelles, NPs (purple beads) are in the hydrophobic core, in contact with each other or the hydrophobic block of the polymer (pink beads). Micelles generally appear roughly spherical in shape. Aggregates containing only NPs are rarely found, and never found at the end of the simulation time (other than in the NP-solvent systems); we include such aggregates as micelles in the calculation of micelle size and polydispersity to allow for clearer comparison of aggregates between NP-solvent and NP-polymer−solvent systems. We characterize the aggregates by counting the number of polymers and NPs in each, and calculate their size and polydispersity. As a measure of size, we report the extent of the micelle LE, defined as the average distance across the micelle,

length of the PS block was chosen to create polymers that were 1/5 or approximately 1/3 hydrophobic. The first polymer listed has the same length and composition studied by Spaeth et al.38,39 All simulations involving polymer maintained the same ratio of polymer beads to water beads. The short (N = 25) polymer systems contained 450 diblock copolymers and 360 000 water beads, whereas the long (N = 50) polymer systems had twice as many polymer and water beads (720 000). The number of nanoparticles was set to 0, 18, 35, 53, or 70 to create NP bead:polymer bead ratios of ∼0, 0.16, 0.32, 0.49, and 0.64, respectively, for N = 25 polymers, or ∼0, 0.08, 0.16, 0.24, and 0.32 for N = 50 polymers. Note that for clear comparison we report bead ratios, analogous to weight ratios, however, the molecular NP:polymer ratio is smaller because a single NP contains more beads than a polymerthere are always more than 6 polymer molecules per NP. For comparison, simulations were also run with NPs and solvent only. These systems had 360 000 water beads and 18, 35, 53, or 70 NPs. To identify the systems more easily we use the nomenclature illustrated in the Figure 1. The number of polystyrene (S) and polyethylene glycol (G) beads in each polymer is indicated by the subscripts. The ratio of NP beads:polymer beads in the system is denoted by the subscripted number after N. For each system studied, we ran three independent simulations starting from different random initial configurations. The amount of free and aggregated material was quantified as discussed below, and the results for the three simulation runs were averaged together. The standard deviation in these amounts between the three runs was calculated to provide error bars in the figures. Quantities that characterize the distribution of aggregatesthe median number of polymers or nanoparticles per aggregate, the median aggregate size, and polydispersitywere calculated based on the entire set of aggregates across all 3 runs. We reported the median values rather than the mean values to avoid the influence of outliers on a given quantity. To give a measure of the width of distributions, the standard deviation about the median, σmedian was approximated by multiplying the standard deviation of the mean by 1.25. This approximation assumes the samples are normally distributed, and gives us rough estimate of σmedian.

LE =

(Δx + Δy + Δz) 3

(8)

where Δx, Δy, and Δz are the distances between the maximum and the minimum bead centers along each axis, considering all beads in the micelle. We also determined the radius of gyration Rg (see the Supporting Information). Polydispersity is calculated using ⎛ N ⎞2 Polydispersity = 1 + ⎜ std ⎟ ⎝ Nmean ⎠

(9)

In eq 9, Nmean is the mean number of beads per aggregate and Nstd is the standard deviation in the number of beads per aggregate. For polymer-free systems, the free nanoparticles are also included in the polydispersity calculation. Before addressing systems containing both polymers and nanoparticles (NPs), we first consider the simpler NP-solvent and polymer− solvent systems. 3.2. Nanoparticle−Solvent Systems. Figure 3 summarizes the quantitative data derived from three independent simulations of the NP-solvent system containing 70 nanoparticles. Initially, most of the NPs are free; at the end of the 12 μs simulation, essentially all of the nanoparticles are incorporated into a few large aggregates. This behavior is 11585


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coagulation likely for each NP/aggregate collision. At very long times, the system would presumably form a single aggregate whose size would depend on the number of NPs in the simulation. (The NP-solvent simulations are presented for comparison/context versus the NP-polymer−solvent results, rather than to reproduce a particular experimental system of interest.) As illustrated in Figure 3, the number of NP aggregates peaks near t = 2 μs when about 50% of NPs in the system are still free. The general shape of the aggregate number curve can be understood by considering the competition between aggregate formation and aggregate growth either by the addition of free NPs to an existing aggregate or via aggregate coagulation. Initially, most of the NPs are free, thus the probability that a NP encounters another NP rather than an aggregate is high and the number of aggregates increases. As more aggregates form and grow by single NP addition, the decrease in the concentration of free NPs reduces the new aggregate formation rate. Finally, coagulation of existing aggregates into larger ones decreases the total number of aggregates. We can characterize the kinetics of aggregate (micelle) formation and coagulation by estimating the times associated with these processes. Although there are many possibilities, here we define the aggregate (micelle) formation time τF in the

Figure 3. Number of free nanoparticles and nanoparticle aggregates versus time for the NP-solvent simulation with 70 NPs. The red curve is a fit of the coagulation model to the data points between t0 = 2 μs, the time corresponding to the maximum in the aggregate number, and t = 7.5 μs.

expected since the NPs are hydrophobic and, in the final aqueous solvent, the lack of a protective coating makes

Table 3. Characteristics of the Micelle Formation Process, Including the Time Corresponding to the Peak in the Number of Aggregates t0, the Micelle (Aggregate) Formation Time τF, and Coagulation Time τC for All Simulated Systemsa polymer

NP bead: polymer bead ratio

t0

τF [μs]

τC [μs]

polydispersity [at 12 μs]

median num poly/ micelle

median num NP/ micelle

total num micelles

S17G33

0 0.08 0.16 0.24 0.32

2.25 2.25 2.25 2.25 2.25

2.8 2.9 3.1 2.9 3.0

2.7 3.0 2.7 2.5 2.4

1.17 1.19 1.14 1.22 1.25

18 18 22 22 25

0 1 1 2 3

71 65 61 63 54

S10G40

0 0.08 0.16 0.24 0.32

3.00 3.75 3.00 3.00 3.00

4.4 4.1 3.9 4.2 4.2

5.2 5.0 5.3 5.6 3.9

1.13 1.22 1.20 1.29 1.31

12 13 15 14 16.5

0 0 1 1 2

105 101 92 89 82

S8G17

0 0.16 0.32 0.49 0.64

3.00 3.00 3.00 2.25 2.25

4.3 4.2 4.5 4.2 4.4

1.5 1.6 1.5 1.1 1.2

1.16 1.25 1.51 1.36 1.50

25 29 28 37 36

0 1 1 3 5

53 44 40 35 33

S5G20

0 0.16 0.32 0.49 0.64

5.25 4.50 3.75 3.00 3.00

4.5 5.1 5.7 5.2 5.9

2.5 2.9 3.3 2.2 3.4

1.52 1.77 1.70 1.77 1.91

12 17 28 33 26

0 0 1.5 2 3

105 64 44 39 35

Nonly

18 35 53 70

4.50 2.25 2.25 1.50

4.0 4.0 2.8 3.2

4.6 4.0 3.5 3.0

1.66 1.58 1.33 1.67

0 0 0 0

5 14 16.5 32

7 6 8 7

NPb NP NP NP

a The standard of deviation across the 3 equivalent simulations performed for each system is about 0.5 μs for micelle (aggregate) formation time, and about 0.3 μs for coagulation time. The median num poly/micelle, median num NPs/micelle, and total num micelles columns report the median number of polymer molecules per micelle, median number of NPs per micelle, and total number of micelles, respectively, across the 3 simulations for each system. bThe NP-only systems contain no polymer and 18, 35, 53, or 70 NP in the box corresponding to NP:water bead ratios of 0.0052, 0.010, 0.015, and 0.020.

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The Journal of Physical Chemistry B following way. If Ci and Cf are the initial and final free NP concentrations, τF corresponds to the time required to reduce the free NP concentration from Cf + 0.95(Ci − Cf) to Cf + 0.05(Ci − Cf). For systems that contain polymers, τF is based on the free polymer concentration rather than the free NP concentration. For coagulation, we obtain the characteristic time by fitting the aggregate number curve after the maximum to 1 1 − = 8πDr(t − t0) N (t ) N0

(10)

where t is time, t0 is the time of the peak in the number of aggregates, N(t) is the number of aggregates at time t, N0 is the maximum number of aggregates, D is the diffusivity, and r is the radius of the aggregates.47 We fit the data from t0 to t = 7.5 μs, to obtain values for N0 and Dr, from which we determined the coagulation time:47 τC =

1 8πN0Dr

(11)

The derivation of eq 10 assumes coagulation starts from a monodisperse size distribution and that Dr is constant. Given that most of coagulation occurs between relatively small aggregates whose sizes do not vary greatly, and that D ∝ r−1, eqs 10 and 11 should be a valid way to estimate the coagulation time. The good agreement between the fit and the simulation data, and the fact that the values of N0 are very close to the maximum values of N observed in the simulation, provides further support for this approach. Table 3 summarizes the characteristic times associated with the simulations. For the NP-solvent simulations, lower particle concentrations generally lead to longer characteristic times, consistent with the decreased probability of two particles or aggregates interacting when concentrations are reduced. 3.3. Polymer−Solvent Systems. Figure 4 shows snapshots of the formation of polymer micelles over time from one polymer−solvent simulation of S17G33, the polymer with the largest number of hydrophobic beads considered here. For the same polymer, Figure 5 summarizes the percent of free polymer and the number of micelles as a function of time. After equilibration in the good solvent, polymers appear randomly dispersed throughout the system (Figure 4a) with about 5% of this polymer in aggregates (Figure 5) that are primarily dimers. As the solvent-PS repulsion increases during mixing, polymers aggregate rapidly into small micelles (Figure 4b). This trend continues until there is little or no free polymer left in the system (Figure 4c). Even before mixing is complete (t = 6 μs), the smaller micelles evolve into larger, more stable micelles (Figure 4d,e) by merging with each other. The general steps of micelle formation mirror those observed for NP aggregation and also hold for the other polymer compositions. For most of the polymer systems, all of the polymer was incorporated into micelles before the mixing time was over. Only in the S5G20 system, was a small amount (10−15%) of free polymer present at the end of the simulation. For S5G20, the presence of free polymer in the final aqueous state reflects the fact that this polymer contains the smallest number of hydrophobic beads, and that these can be stabilized in the aqueous phase, and shielded from other polymers’ hydrophobic segments, by the relatively large hydrophilic part of the polymer. As in the NP-solvent systems, the general shape of the micelle number curve in Figure 5 can be understood by

Figure 4. Snapshots of polymers at various times during simulation of the S17G33[N0] system. Hydrophobic polymer beads are pink, and hydrophilic polymer beads are teal; solvent is not shown. As the solvent−polymer interactions become unfavorable, new micelles form, and these coagulate until the system is reasonably stable. Some beads are unwrapped across periodic boundaries such that each free polymer or aggregate appears contiguously.

Figure 5. Percentage of free polymers (polymers that have no beads within the cutoff distance of any other polymer’s beads) and the number of micelles versus time in the S17G33[N0] system. The red line is a fit to the coagulation model for times between t0 and 7.5 μs.

considering the competition between new micelle formation and micelle growth by single polymer incorporation or coagulation. Unlike the NP-solvent systems, however, the hydrophilic region of the polymer can stabilize the micelles in the final aqueous environment. Thus, at the end of the simulation, the number of micelles is much larger and coagulation has noticeably slowed relative to the fit of eq 10 that is based on the data up to 7.5 μs. The characteristic times 11587


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Collisions between these regions of the polymers drive aggregation, and the probability of two hydrophobic regions interacting will depend on their extent. Thus, longer PS regions will enhance micelle formation rates and, therefore, decrease micelle formation times. Although we do not have a direct comparison, the total length of the polymer also appears to play a role; for roughly the same number of PS beads, the micelle formation time for S10G40 is close to or even slightly higher than that for S8G17. This behavior is consistent with the decreased diffusivity of the longer polymer reducing the collision probability and/or the steric interference by the longer PEG region present in the longer polymer. In particular, shielding of the hydrophobic beads by a relatively long hydrophilic part can increase the stability of the free polymers and act as a barrier to aggregation for these polymer compositions. This means that small clusters will form less easily than for polymers that are more hydrophobic, and that these aggregates may break up more easily unless they reach a size beyond which micelles are significantly more stable than free polymers. More systematic studies would be required to determine which effect dominates. As illustrated in Figure 6, the coagulation time is largely controlled by the number of hydrophilic beads on the polymer. A bigger hydrophilic region increases the steric hindrance around the micelles, reducing the ability of two micelles to merge. The polymer length also seems to play a role since the S5G20 coagulation time is quite comparable to that of S17G33. In part this may reflect the increased solubility of the S5G20 polymer in the aqueous environment. In particular, for S5G20, over the entire simulation time, free polymers persist and continue to form new aggregates, and conversely, small aggregates continue to break apart to generate free polymers. Figure 7 illustrates the time evolution of the median size (extent) and polydispersity of micelles for the four polymers considered here. As expected, the median extent of the micelles increases with time and with polymer length (Figure 7a, analogous R g results are available in the Supporting Information). For the long polymers, the slight decrease in particle size during the early stages of mixing (t < 2 μs) reflects the fact that during this time period the aggregates are small

derived from the polymer−solvent simulations are included in Table 3 and are also presented in Figure 6.

Figure 6. Micelle (aggregate) formation (left axis, blue squares/ diamonds) and coagulation (right axis, red triangles) times as functions of the number of PS (bottom) and PEG beads (top) in the polymer, respectively. The data are the values derived from polymer−solvent simulations only (open symbols for short polymer, open symbols with cross for long polymer) or averaged across all NP concentrations for a given polymer (closed symbols). The standard deviation across the three runs in micelle (aggregate) formation time for the NP-polymer data is about 0.5 μs and in coagulation time is about 0.3 μs.

In general, micelle formation times decrease with an increase in the number of PS beads whereas coagulation times increase with the number of PEG beads, and these trends do not change significantly with the presence of nanoparticles. Both trends can be understood intuitively. In particular, as the repulsive forces between the hydrophobic PS beads and the solvent increase during mixing, interactions between the hydrophobic regions of the polymers are favored over interactions with the solvent.

Figure 7. Variation in the (a) median extent LE and (b) polydispersity of micelles as a function of time for the 4 polymer−solvent systems. Near the beginning (at 1.5 μs) of the simulation, 1.25 times the standard deviation in size, σmedian (a measure of the width of the distribution) is ∼1.0 nm whereas at the end of the simulation (at 12 μs) is σmedian ∼2.2 nm. 11588


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The Journal of Physical Chemistry B primarily dimers. As the solvent−polymer interactions become less favorable, the polymers adopt a less open configuration thereby initially decreasing the median size of the micelle. For these polymers, LE increases systematically with the length of the hydrophobic region and the effect is more pronounced for the short polymers. The polydispersity of each system (Figure 7b) reaches a maximum at times very close to t0, the time corresponding to the peak in the number of aggregates, and the longer polymers have a lower maximum polydispersity. At the end of the simulation time, the S5G20 polymer maintains a higher polydispersity relative to the other 3 systems. In part the trends reflect the shorter micelle formation times for the longer polymers that generates a relatively uniform pool of small aggregates when coagulation begins to control particle growth. If the aggregates coagulate uniformly as well, polydispersity will be low. In contrast, if aggregate formation is spread out in time, and small aggregates can break apart, then some of the aggregates that form early can grow to larger sizes while those formed later stay small since much of the free polymer has been consumed. This can lead to higher polydispersity until coagulation dominates aggregate growth and reduces the polydispersity. Thus, it is not surprising that the system with the highest free polymer solubility also exhibits the highest polydispersity. 3.4. NP-Polymer−Solvent Systems. Snapshots from one of the NP-polymer−solvent systems, S17G33[N0.32], are presented in Figure 8 to show qualitatively how micelles evolve with time in these systems. The corresponding quantitative results summarized in Figure 9. At the end of every NP-polymer system simulation, all of the NPs are encapsulated within micelles similar to those shown in Figure 8e and Figure 2b. The trends illustrated here are representative of those observed for the other NP-containing systems. In particular, as solvent quality decreases, the free NPs and polymers quickly form aggregates until both are largely consumed (Figure 9a,b). Polymers outnumber the NPs by a factor at least 6.4 (or as much as 25, depending on the system) and though NPs have many beads, they are compact relative to polymers. Thus, it is intuitive that the NPs are more likely to encounter a polymer than another NP and to form NPcontaining micelles rather than NP-only aggregates. Similarly, free polymers are initially more likely to encounter other polymers than NPs, and, thus, there are more polymer-only micelles during the early stages of micelle formation (from ∼0 to 2 μs) than NP-containing micelles. However, once the concentrations of free NPs and polymer are reduced, the polymer-only micelles coagulate with each other and with the NP-containing micelles. Only 1 NP is required to shift a micelle from the polymer-only to NP-containing class, thus the number of polymer-only micelles decreases more rapidly than the number of NP-containing micelles. Coagulation continues until the micelles reach a relatively stable size and the number of NPcontaining micelles, Figure 9c, stays approximately constant on the time scale of the simulation. It is interesting to note that within any one micelle, there is not always a compact region of NPs in the center, but rather the NPs can be spread out throughout the hydrophobic core. This is not surprising given that the NP bead−bead interaction parameters are the same as the bead-PS interaction parameters (Figure 10). Table 3 summarizes the characteristic micelle formation and coagulation times for all systems. As is also illustrated in Figure 6, the averages of the micelle formation and coagulation times

Figure 8. Snapshots of NPs and polymers at various times during one simulation of the S17G33[N0.32] system. The NP-polymer−solvent systems evolved in a manner similar to the NP-solvent and polymer− solvent systems. NP-NP aggregation is much rarer than NP-polymer or polymer−polymer aggregation. Some beads are unwrapped across periodic boundaries such that each free polymer, free NP, or aggregate appears contiguously. NP beads are purple, hydrophobic polymer beads are pink, and hydrophilic polymer beads are teal; solvent is not shown. At bottom right, the polymer has been rendered transparent so that the distribution of NPs can be seen more clearly.

for the NP-polymer−solvent systems follow essentially the same trend established for the polymer−solvent systems, with only slight increases in micelle formation time and slight decreases in coagulation time for most of the polymer compositions. Thus, under our conditions polymer composition has a far greater effect on these two parameters than does the NP concentration. This behavior reflects the following facts. First, the polymer greatly outnumbers the NPs, and, thus, the micelle formation kinetics will be dominated by the polymer− polymer interactions, and second, the exterior of the micelle and its ability to merge is dominated by the hydrophobic portion of the polymer and the NPs do not change this significantly. Figure 11a illustrates the change in LE with time for selected systems and at the end of the simulation for all systems as a function of the number of NPs. The size of NP-containing micelles evolves in time similarly to that of polymer-only micelles in the polymer−solvent simulations. Intuitively, the median LE (Figure 11b) at the end of the simulation increases as the number of NPs in the simulation increases, and the effect is stronger for the short polymers than the long polymers. The latter reflects the relative sizes of the NPs, LE ∼ 2.22 nm, and the short LE ∼ 2.53 nm versus long LE ∼ 4.01 nm polymers. Qualitatively similar trends are observed for the median Rg of the micelles (see Supporting Information). 11589


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Figure 9. Percentages of (a) NPs and (b) polymers that are in aggregates or free, and (c) the change in the number of aggregates with and without NPs as a function of time in the S17G33[N0.32] system. The large number of polymers in the system ensures that NPs quickly form NP-containing micelles while polymers can initially form polymer-only micelles. Coagulation decreases the number of micelles as well as the ratio of pure-polymer to NP-containing micelles in the system. The red line in (c) is a fit to the coagulation model for times between t0 and 7.5 μs.

Figure 10. Snapshots of NP-polymer−solvent systems with the lowest (top row) and the highest (bottom row) NP concentration, at 12 μs. NP beads are purple, hydrophobic polymer beads are transparent pink, and hydrophilic polymer beads are transparent teal; solvent is not shown.

Figure 11. (a) The median extent LE versus time for the NP-polymer micelles for systems with the largest number of nanoparticles; the results are very similar to those observed for the polymer-only micelles in Figure 7a. At 1.5 μs σmedian is ∼0.5 nm and at 12 μs σmedian is ∼2.2 nm. (b) Median extent LE at the end of the simulation for all NP-polymer−solvent systems; for a given polymer composition, LE generally increases with the number of NPs in the system. Average of σmedian across all data points is ∼2.7 nm. 11590


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Figure 12. Polydispersity of micelles (a) versus time for systems with the largest number of nanoparticles and (b) at the end of the simulation for all polymer−solvent and NP-polymer−solvent systems. Adding NPs generally increases the polydispersity, and this effect is greater for the shorter polymers. The polymer with the shortest hydrophobic section, S5G20, shows the most variation in polydispersity.

Figure 13. (a) The median number of polymers per polymer-only or NP-containing micelle versus time for selected S17G33 systems. At 1.5 μs σmedian is ∼1.2 and at 12 μs it is ∼10. (b) The average number of polymer-only and NP-containing micelles at the end of the simulation as a function of the number of NPs in the system for S17G33 systems. The error bars correspond to the standard deviation across 3 runs.

simulation (middle curve) as well as for both NP-containing and pure polymer micelles of NP-polymer−solvent systems with two different levels of NPs. In both cases, the NPcontaining micelles have significantly more polymers, and are therefore bigger, than the pure polymer micelles. Furthermore, this difference is more pronounced for the higher NP concentrations. As illustrated in Figure 9c, between 2 and 4 μs, the populations of NP-containing and pure polymer micelles are comparable, but the sizes (Figure 13a) are not. Thus, polydispersity increases. As coagulation proceeds, the size difference between these populations becomes less important because the number of polymer-only micelles decreases rapidly relative to the NP-containing micelles. Figure 13b illustrates the distribution between the two types of micelles at the end of the simulation as a function of the number of NPs. The increased

The polydispersity of the micelles versus time for the systems reported in Figure 11a is shown in Figure 12a. The increase in the peak polydispersity in the presence of the NPs is striking especially for the shortest, most hydrophilic polymerand the time where this occurs is again close to t0. In general, all polymer compositions show an increase in polydispersity with increasing nanoparticle concentration (Table 3, Figure 12b), with the shorter polymers more affected by the presence of nanoparticles than the longer polymers. The increased polydispersity for NP-containing systems can be better understood by examining the evolution of the two subpopulations of micelles present in NP-polymer−solvent simulations, i.e., those that contain NPs and those that do not. For the S17G33 polymer, Figure 13a shows the median number of polymers per micelle versus time for the polymer−solvent 11591


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Figure 14. (a) Median number of NPs per NP-containing micelle and (b) polydispersity in the number of NPs per NP-containing micelle, at the end of the simulation, as a function of the number of NPs in the system. In part (a), the average of σmedian across all data points is ∼2.9.

more hydrophilic polymer (S10G40) that can shield the hydrophobic cores most easily displays the most uniformity (lowest polydispersity) in NP loading.

number of polymers associated with the NP-containing micelles is consistent with NPs increasing the size of the hydrophobic core of the micelles and, therefore, increasing the amount of hydrophilic polymers needed to stabilize the micelle. The trends observed for the other polymer compositions are comparable to those observed for the S17G33 polymer discussed here. Figure 14a,b present the median number of NPs and polydispersity in the number of NPs per NP-containing micelle at the end of the simulation as a function of the number of NPs present in the system. As expected, the NP loading typically increases with an increase in NP concentration. However, for S5G20 systems, the median number of NPs drops for the highest NP loading. This is the polymer composition with the shortest hydrophobic region, whose free polymers are relatively stable, and this is the only polymer composition for which a significant amount of free polymer is present at the end of the simulation. The presence of free polymers in this system means there is a smaller amount of polymers in micelles shielding the cores of NP-containing micelles compared to the otherwise similar S8G17 systems. With a small amount of NPs, the S5G20 polymers participating in micelles might be enough to shield the NPs from the solvent. However, with the largest number of NPs, insufficient shielding of NPs means that, especially for large aggregates, there is a greater chance that the hydrophobic cores will be exposed and these large NP-containing aggregates can continue to grow. This causes the S5G20 system at high loading to have a few very large NP-containing micelles, along with some smaller NP-containing micelles, which explains the smaller median number of NPs per NP-containing micelle for S5G20[N0.64] system in Figure 14a. Moreover, it also explains the significant jump in NP polydispersity from 53 to 70 NP containing systems for the same polymer. Similarly, one would expect that making polymers shorter will decrease the steric shielding and increase the possibility of hydrophobic cores of NP-containing micelles to be exposed and to coagulate into larger micelles of higher loading. Figure 14 does show that shorter (comparing red and blue versus green and brown lines) polymers typically have higher NP loading and polydispersity. Especially with a large number of NPs present, the longer and

4. CONCLUSIONS We used a DPD model with changing polymer−solvent interactions39 to simulate encapsulation of spherical NPs in block copolymer micelles during and after a rapid change from a good to a poor solvent. We focused on how polymer composition and nanoparticle concentration affect micelle formation as quantified by micelle formation and coagulation times. Furthermore, we characterized different micellar properties including micelle size, polydispersity, and the number of polymers/nanoparticles per micelle. We found that for the polymer−solvent simulations, micelle formation time decreases as the number of hydrophobic beads on the polymer increases, whereas coagulation time increases with the number of hydrophilic beads on the polymer. Shorter micelle formation times were also associated with lower polydispersity in the final micelle distribution. The micelle formation and coagulation times were relatively insensitive to the addition of nanoparticles, most likely because the ratio of polymer to NPs was always large (greater than ∼6). Addition of more NPs to a given system results in an increase in the polydispersity of the system. Furthermore, an increase in NP concentration results in increase in the number of polymers and NPs per micelle. This can be attributed to the fact that NPcontaining micelles can accommodate more polymers since their hydrophobic cores are larger than those of polymer-only micelles. Our results provide the correlation between PS/PEG block length and important parameters, such as micelle formation time, coagulation time, and polydispersity. Further work of interest includes exploring the effect of longer polymers with several PS/PEG ratios, variable polymer concentration, and longer mixing times on the micelle formation process. 11592


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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.jpcb.6b07324. Radius of gyration, the number of micelles, overall polydispersity, and polydispersity in NP loading for all systems (PDF)

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AUTHOR INFORMATION

Corresponding Author

*Email: [email protected]; Phone: 616-688-1017. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This material is based upon work supported by the National Science Foundation under grant CMMI-1344567. This work was also supported by an allocation in computing time from the Ohio Supercomputer Center. We thank Jessica Winter, Gauri Nabar, Kil Ho Lee, and Matthew Souva for helpful discussions throughout the course of this work.

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REFERENCES

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