Article pubs.acs.org/Langmuir
Micellar Structures in Nanoparticle-Multiblock Copolymer Complexes Houyang Chen* and Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260-4200, United States S Supporting Information *
ABSTRACT: Brownian dynamics simulation is employed to examine the structure changes of complexes composed of a hydrophobic nanoparticle and a multiblock copolymer molecule (MCP). The dependence of the structure transitions on the radius of the nanoparticle, on the interactions between the hydrophobic segments of the MCP, and on the interactions between the hydrophobic segments and hydrophobic nanoparticle is examined. It is shown that the multiblock copolymer adsorbed on a nanoparticle can acquire the structure of a micelle.To better characterize the micelle generated and the structure changes in the nanoparticle−MCP complex, the mass dipole moment of the complex [the distance between the center of mass of MCP and the center of the nanoparticle minus the radius of the nanoparticle (DCC)], the density profiles of MCP segments around its center of mass and around the nanoparticle, the radius of gyration of the MCP, and the thickness of the micelle around the nanoparticle are determined. It was found that, when structural transition of the complex occurs, the above quantities change dramatically. The present simulation may provide new insights regarding the drug-loaded micelle interacting with a virus represented by a nanoparticle. nanoparticle (Lugo et al22). The dependence of surfactant aggregation on surface curvature was also investigated.22 However, the micelle generation in a polymer−nanoparticle complex and the transition from a nanoparticle inside a micelle to a micelle adsorbed on only a part of a nanoparticle were not examined. In this paper, we investigate complexes containing a nanoparticle and a multiblock copolymer molecule (MCP), in which the nanoparticle is either completely surrounded by the adsorbed MCP forming a micelle or only partially covered by a micelle or by adsorbed MCP. In addition, the transitions between these structures are examined. To better characterize the transitions and the morphologies of the complexes, the mass dipole moment of the complex [the distance between the center of mass of MCP and the center of the nanoparticle minus the radius of the nanoparticle (DCC)], the density profiles of the MCP segments around their centers of mass and around the nanoparticle, the thickness of the micelle and its radius of gyration, were calculated. It should be mentioned that in the current paper, we consider a “hard” nanoparticle interacting with a micelle, and not a “soft” one (e.g., a liquid droplet) interacting with the polymer molecule.23,24 The present simulations may provide insights regarding the drug
1. INTRODUCTION The adsorption of surfactants or polymers on nanoparticles has numerous applications and was widely investigated.1−15 A recent review is concerned with the critical adsorption of weak polyelectrolytes on curved surfaces and with the stable complexes formed by strong polyelectrolytes with oppositely charged spherical and cylindrical particles.16 In these papers, the dependence of the structure of complexes containing polymers (or surfactants) and nanoparticles on various factors, such as the natures of the solvent and polymer and the surface properties of the nanoparticle, were examined. In such complexes, the critical conditions for adsorption and the structure of the polymer are important. By employing Edward’s equation and the ground-state dominance approximation, the adsorption of polyelectrolyte chains from solutions on oppositely charged surfaces was examined by Cherstvy and Winkler.17,18 They found that, for low-dielectric constants, the charge density of the surface plays a crucial role in inducing the adsorption of polyelectrolyte chains. If the surfactants or polymers aggregate and generate micelles, they can be used as drug carriers19 and as nanoreactors.20 Using Monte Carlo simulations, Stoll and co-workers21 examined the surfactant aggregates adsorbed on soft hydrophobic particles. They found that, by increasing the concentration of surfactant and the interaction between the nanoparticles, elongated surfactant aggregates are generated. The adsorption of surfactants (pentaethylene glycol monododecyl ether or/and n-dodecylβ-maltoside) formed aggregates on the surface of silica © 2014 American Chemical Society
Received: July 29, 2013 Revised: February 22, 2014 Published: March 16, 2014 3723
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where r0 is the equilibrium bond distance, selected to be 1.2σ0, and Kb is the spring coefficient selected to be 1500ε0/σ20. Dilute solution simulations were carried out for a system of size (240σ0)3, which contains both a nanoparticle and a multiblock copolymer (MCP) (O10H10)12, with MCP composed of both hydrophilic (H) and hydrophobic (O) segments. The diameter of the segments of the multiblock copolymer is kept constant (σ0), and the diameter of the nanoparticle is varied. For simplicity, the nanoparticle is assumed frozen in the center of the simulation box. Using the Langevin thermostat25 for nanoparticle and segments of MCP, an implicit solvent is included. All molecular dynamics simulations were carried out by employing the large atomic/molecular massively parallel simulator (LAMMPS) package.27 As the initial configuration, the end of the MCP is considered to be a hydrophobic segment O which is generated at a distance σ0 from the surface of the nanoparticle; the other segments were generated randomly. After an initial configuration was generated, three processes were performed to optimize the initial configuration: (1) minimization using the conjugate gradient method, (2) simulation at high temperatures, and (3) annealing simulation. For simulation at high temperatures, the temperature T* = kBT/ε0 = 2.5 was selected. The simulation involved 105 BD steps, and for each step, a time step between 0.001τ and 0.003τ was employed. For the annealing process, a simulation with decreasing temperature from T* = 2.5 to T* = 1.0, involving 3 × 105 BD steps, was carried out; for each BD step, a time step between 0.001τ and 0.003τ was employed. After the initial configuration was generated and optimized, a simulation at T* = 1.0 involving 107 BD steps was performed, and for each BD step, a time step of 0.005τ was used. The configurations of the complexes were collected every 103 BD steps of the last 3 × 106 steps. It should be emphasized that (i) three independent initial conditions were employed, and the results were reproducible and (ii) 107BD steps are large enough to reach equilibrium (See Figure S1 of the Supporting Information).
loaded micelles interacting with a virus represented by the nanoparticle.
2. SIMULATION DETAILS AND MODEL The Brownian dynamics (BD) simulation25−28 employed is based on a coarse-grained model. The Langevin equation of motion of each segment i of mass mi is provided by r mi r( ï t ) = −∇ ∑ [Eij(rij) + Eb(rij)] − mi ζ r( i̇ t ) + Fi (t ) j
(1)
where mi, ri, and ζ are the mass, position vector, and friction coefficient of segment i, respectively. The coefficient ζ = 1.0 τ−1 [where τ = (m0σ20/ε0)1/2], mi = m0 for the segments of the copolymer and mi =50m0 for the nanoparticle. The quantities σ0, ε0, and m0 are the units of length, energy, and mass, respectively. The random force Fri (t) is considered to be Gaussian with a zero mean that satisfies the equation ⟨Fir(t ) ·F jr(t ′)⟩ = 6kBTmζδijδ(t − t ′)
(2)
where kB is the Boltzmann constant, T is the temperature, and δ is the Dirac delta function. The interactions between segments and between nanoparticle and segments are expressed in terms of the Lennard− Jones (LJ) potential: ⎧ ⎡⎛ ⎞12 ⎪ ⎢⎜ σij ⎟ ⎪ 4εij ⎜ ⎟ − Eij = ⎨ ⎢⎣⎝ rij ⎠ ⎪ ⎪ 0 ⎩
⎛ σij ⎞6 ⎤ ⎜⎜ ⎟⎟ ⎥ rij ≤ rc , ij ⎥ ⎝ rij ⎠ ⎦ rij > rc , ij
(3)
where rc,ij and rij are the cutoff distance and the distance between the centers of two segments or between the centers of nanoparticle and the segments of MCP, respectively. The length σij is the distance between the centers of beads (nanoparticle or segments) i and j when they are in contact, and εij is the interaction strength between beads i and j. The LJ potential parameters involved in the pair interactions between hydrophobic segments, hydrophilic segments, and nanoparticle are listed in Table 1. The LJ potential is shifted to zero at the cutoff distance. The interactions between two successive segments of the polymer chain are provided by the harmonic potential Eb =
1 Kb(rij − r0)2 2
3. RESULTS AND DISCUSSIONS When a multiblock copolymer molecule (MCP) is adsorbed on a nanoparticle, the structure of the MCP−nanoparticle complex can become very complicated. In this paper, we emphasize the structures in which micelles are generated. Two micellar structures have been identified: (i) a nanoparticle inside a micelle (see Figure 1e) and (ii) a micelle adsorbed on a part of the nanoparticle (see Figure 1f). A third structure consists of a MCP (which does not generate a micelle) adsorbed on a part of the nanoparticle (see Figure 1g). Figure 1 presents the structures generated when a nanoparticle interacts with a MCP molecule as functions of the radius (RP) of the nanoparticle, the interactions εOO between the hydrophobic segments of the MCP, and the interactions εOP between the hydrophobic segments of MCP and the hydrophobic nanoparticle. The other interaction parameters are fixed (see Table 1). For a low attractive interaction εOP between the hydrophobic segments of MCP and nanoparticle, MCP cannot overcome the entropy loss involved in its adsorption and prefers to stay away from the nanoparticle. For a moderate interaction εOP between the hydrophobic segments of MCP and the nanoparticle (εOP = 1.0, see Figure 1a) and small εOO, εOP can overcome the entropy loss and the MCP molecule is adsorbed on a part of the nanoparticle, generating a complex of MCP adsorbed on the nanoparticle. For εOP =1.0 and large εOO,
(4)
Table 1. Lennard−Jones Potential Parameters in the Simulations
interaction pairs O−O (hydrophobic segments− hydrophobic segments) O−H (hydrophobic segments− hydrophilic segments) O−P (hydrophobic segments− hydrophobic nanoparticle) H−H (hydrophilic segments− hydrophilic segments) H−P (hydrophilic segments− hydrophobic nanoparticle)
energy (ε0)
size (σ0)
cutoff distance (σ0)
εOO
σOO = σ0
2.5 σOO
εOH = 1.0 εOP
σOH = σ0
21/6 σOH
σOP = (σ0 + 2RP)/2 σHH = σ0
2.5 σOP
σHP = (σ0 + 2RP)/2
21/6 σHP
εHH = 1.0 εHP = 1.0
21/6 σHH
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Consequently, a complex of a micelle adsorbed on a part of the nanoparticle is formed. As shown in Figure 1 (panels a−d), by increasing εOP, the range of formation of a complex containing a nanoparticle inside a micelle increases. The morphologies of the above three complexes are presented in Figure 1 (panels e−g). Regarding the complex of a polymer adsorbed on a nanoparticle (Figure 1g), some results were reported previously.7,14 Using Monte Carlo simulations, Stoll and co-workers, who considered a polyelectrolyte and nanoparticles, found that the conformation of the polyelectrolyte is dependent on its rigidity and charge7 as well as on the salt valency.13 Feng and Ruckenstein14 identified complexes containing a polyampholyte and a charged nanoparticle and examined the effect of particle charge on the conformation of the polymer (not micelle) in the particle− polyampholyte complex. To better characterize the transitions between different complexes, a mass dipole moment of the complex [the distance between the center of mass (COM) of MCP and the center of the nanoparticle minus the radius of the nanoparticle (DCC)] was calculated. DCC is negative when the nanoparticle is inside the micelle and positive when the micelle is located only on a part of the surface of the nanoparticle. It should be mentioned that a negative value of DCC indicates a weak mass dipole moment of the complex, whereas a positive one means a strong mass dipole moment of the complex. A larger value of DCC means a stronger mass dipole moment. The mass dipole moment becomes zero when the absolute value of the negative DCC equals the radius of the nanoparticle (i.e., the center of mass of the polymer is located at the center of the nanoparticle). Figure 2 plots DCC as a function of the
Figure 1. State diagram (a−d) of complexes containing a nanoparticle and a micelle as functions of the radius of the nanoparticle (RP), the interaction between the hydrophobic segments εOO and between the hydrophobic segments and the hydrophobic nanoparticle and their snapshots (e−g). (a) εOP = 1.0 ε0; (b) εOP = 2.0 ε0; (c) εOP = 3.0 ε0; (d) εOP = 4.0 ε0. (e) Snapshot of a nanoparticle inside a micelle. (f) Snapshot of a micelle adsorbed on a nanoparticle. (g) Snapshot of a multiblock copolymer (MCP) adsorbed on a nanoparticle. For (a−d): ●, a nanoparticle inside a micelle; □, a micelle adsorbed on a part of nanoparticle; and △, a MCP adsorbed on a part of the nanoparticle. For e−g: hydrophobic segments, orange (gray in the white/black version); hydrophilic segments, yellow (white in the white/black version); and nanoparticle, black (black in the white/black version).
the hydrophobic segments of MCP self-aggregate and are adsorbed on a part of the nanoparticle, resulting in a kind of micelle adsorbed on the nanoparticle (see Figure 1a). For a strong interaction εOP between the hydrophobic segments of MCP and the hydrophobic nanoparticle (εOP = 2.0 − 4.0ε0, see Figure 1, panels b−d), MCP is adsorbed on the nanoparticle because εOP is large enough to overcome the entropy loss caused by MCP adsorption. For low εOO and small RP, the hydrophobic segments cover the surface of the nanoparticle, and because of weak interaction with the nanoparticle, the hydrophilic segments become located in the outer layer of the complex as much as the chain connectivity allows, generating a complex of a nanoparticle inside a kind of micelle. For low εOO and large RP, the number of hydrophobic segments may not be large enough to cover the entire surface of the nanoparticle. Then the hydrophobic segments together with the connected hydrophilic ones are adsorbed on a part of the surface of the nanoparticle, generating a complex of MCP adsorbed on a part of the nanoparticle. For moderate and large εOO (and any RP), the hydrophobic segments aggregate generating the core of a kind of micelle, and because the interaction with the nanoparticle of the hydrophilic segments is weaker than that between the hydrophobic segments, the former tend to be located on the outer layer of the structure.
Figure 2. The distance between the center of mass of MCP and the center of nanoparticle minus the radius of the nanoparticle (DCC) as functions of εOO when εOP = 2.0 ε0. RP (radius of a nanoparticle) = 2σ0 (○); RP = 3σ0 (□); RP = 4σ0 (△); RP = 5σ0 (▽); RP = 6σ0 (◇); RP = 7σ0 (×); RP = 8σ0 (⊕); RP = 9σ0 (⊞); RP = 10σ0 (short dashed line).
interaction εOO between hydrophobic segments and shows that it is negative when εOO is smaller than 0.8ε0 and positive when εOO is larger than 1.0ε0. In the former case, the nanoparticle is located inside the micelle, and the COM of MCP is inside the nanoparticle, leading to DCC < 0. In the latter case, MCP selfaggregates becoming a kind of micelle adsorbed on a part of the nanoparticle, and the COM of MCP is located outside the nanoparticle, leading to DCC > 0. DCC can provide the transition from a nanoparticle inside a micelle to a micelle adsorbed on a part of the nanoparticle. The density profiles offer additional information on the morphologies generated. The density profiles of MCP 3725
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the micelle is located close to the nanoparticle. In a comparison with the DCC results (previous section), one can conclude that a complex containing a micelle adsorbed on a part of a nanoparticle is generated. In the case of nonaggregated MCP (Figure 3c), the peak in the density profile of the hydrophobic segments around the nanoparticle (solid line in Figure 3c) is located close to the nanoparticle and the peak in the density profile of the hydrophilic segments around the nanoparticle is located somewhat further (dashed lines in Figure 3c). This can be explained as follows. The size of the nanoparticle in Figure 3c is large (much larger than the nanoparticle of Figure 3a) and the hydrophobic segments too few to cover the entire surface of the nanoparticle. The hydrophilic segments cover the surface of the nanoparticle because of chain connectivity between the hydrophobic and hydrophilic segments. However, because the interaction between the nanoparticle and the hydrophilic segments is weaker than that between the hydrophobic nanoparticle and hydrophobic segments, the former have the tendency to be located further from the nanoparticle. Figure 3 reveals that a complex containing a nanoparticle inside a micelle can be obtained for a particular size of the nanoparticle. A too large particle cannot be completely covered by the polymer. To investigate the micelle in the complex in more detail, its thickness is defined by
segments around its COM and around the nanoparticle are plotted for the morphologies: a nanoparticle inside a micelle, a micelle adsorbed on a part of the nanoparticle, and MCP adsorbed on a part of the nanoparticle (Figure 3). Figure 3a
Figure 3. Density profiles of MCP around its center of mass (COM) and around the nanoparticle. The solid line and the short dashed line are density profiles of hydrophobic and hydrophilic segments around the nanoparticle, respectively; circle and square are density profiles of hydrophobic and hydrophilic segments around the COM of MCP, respectively. d is the distance from the center of the nanoparticle or the COM of MCP. (a) RP = 3σ0, εOP = 2.0 ε0, εOO = 0.4 ε0; (b) RP = 3σ0, εOP = 2.0 ε0, εOO = 2.0 ε0; (c) RP = 7σ0, εOP = 2.0 ε0, εOO = 0.4 ε0; and (d) RP = 7σ0, εOP = 2.0 ε0, εOO = 2.0 ε0.
∞
θ=
∫R sρ(s) ds P
∞
∫R ρ(s) ds
− RP (5)
P
was also calculated. In the above expression, s is the distance between the centers of the nanoparticle and segment, and ρ(s) is the number density of MCP segments as a function of the distance from the center of the nanoparticle. Another useful quantity is the radius of gyration (Rg), which is defined as
presents the case of a nanoparticle inside a micelle. The peak of the density profiles of the hydrophobic segments around the nanoparticle (solid line in Figure 3a) is located close to the nanoparticle; the peak of the density profile of the hydrophilic segments around the nanoparticle (dashed line in Figure 3a) is somewhat further located, in a region where the density of the hydrophobic segments is negligible. In other words, the structure of the MCP is that of a micelle possessing a hydrophilic shell and a hydrophobic core. The peak in the density profile of the hydrophobic segments of MCP around its COM (○ in Figure 3a) is close to its COM. The density profile of the hydrophilic segments around the COM of MCP (□ in Figure 3a) is located in a region where the density of hydrophobic segments is negligible. Consequently, a kind of micelle around the COM of MCP is formed. The density profiles of the MCP segments around its COM and around the nanoparticle almost coincide, indicating that the complex consists of a nanoparticle inside a micelle. The case of a micelle adsorbed on a part of the nanoparticle provides quite different density profiles (Figure 3, panels b and d). The peaks of the hydrophobic segment density around the COM of the polymer (○ in Figure 3, panels b and d) are located close to its COM and have large values, indicating that the COM of the polymer is outside the nanoparticle. The peaks of the hydrophilic segments (□ in Figure 3, panels b and d) are located further from the COM of MCP. Thus a kind of micelle is formed. The density profiles of segments around the nanoparticle (the solid lines and dashed ones in Figure 3, panels b and d) show that
Rg =
1 M
∑ mi(ri − rcom)2 i
(6)
where ⟨...⟩ denotes the ensemble average, M is the mass of the MCP and rcom is the location of its center of mass. The thickness of the micelle is plotted in Figure 4a for a number of radii of the nanoparticle and a number of values of εOO. For a particular radius (RP) of the nanoparticle, the thickness remains the same when εOO increases from 0.2 to 0.6ε0 but starts to increase at εOO = 0.8ε0; for a particular εOO, the thickness increases with increasing RP. The radius of gyration (Rg) is
Figure 4. (a) Thickness of adsorbed MCP on the nanoparticle for εOP = 2.0 ε0, and (b) radius of gyration (Rg) of MCP for εOP = 2.0 ε0. RP = 2σ0 (○); RP = 3σ0 (□); RP = 4σ0 (△); RP = 5σ0 (▽); RP = 6σ0 (◇); RP = 7σ0 (×); RP = 8σ0 (⊕); RP = 9σ0 (⊞); and RP = 10σ0 (solid line). 3726
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Langmuir plotted in Figure 4b. For a selected RP and an εOO smaller than or equal to 0.8ε0, Rg remains constant when εOO increases; for a selected εOO and εOO lower than or equal to 0.8ε0, Rg increases with increasing RP. This occurs because, when RP increases, the polymer becomes more extended. In addition, with increasing RP, the structure of a nanoparticle inside a micelle changes to that of a polymer adsorbed on the nanoparticle (because the numbers of hydrophobic and hydrophilic segments are not large enough to cover the entire surface of the nanoparticle). When εOO becomes larger than or equal to 1.0 ε0, a kind of micelle is adsorbed on the nanoparticle possessing an Rg, which has the value of 3.6 ± 0.3σ0 regardless of the values of RP and εOO. This occurs because a stable micelle is formed and adsorbed on the surface of the nanoparticle. Figure 4b shows that a dramatic decrease in Rg occurs when the complex changes from a nanoparticle inside a micelle to a micelle adsorbed on a part of the nanoparticle, as well as from a polymer adsorbed on a part of the nanoparticle to a micelle adsorbed on a part of the nanoparticle.
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REFERENCES
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ASSOCIATED CONTENT
S Supporting Information *
Temperature and pairwise energy as well as radius of gyration of the multiblock copolymer as functions of the BD step. This material is available free of charge via the Internet at http:// pubs.acs.org.
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ACKNOWLEDGMENTS
We wish to thank the Center for Computational Research in the State University of New York at Buffalo for the computer time provided. The authors are grateful for the financial support from the State University of New York at Buffalo. H.C. is also grateful for the financial support from the National Natural Science Foundation of China (Grant 21206049).
4. CONCLUSIONS By employing Brownian dynamics simulation, the transitions from a nanoparticle inside a micelle to a micelle adsorbed on a part of the nanoparticle, from a copolymer adsorbed on a part of the nanoparticle to a micelle adsorbed on a part of the nanoparticle, as well as from a copolymer adsorbed on a part of the nanoparticle to a nanoparticle inside a micelle were determined. Additional information was obtained by calculating the mass dipole moment of the complex, the density profiles of the polymer segments around its COM and around the nanoparticle, the thickness of the micelle, and the radius of gyration of the copolymer. The above physical quantities change dramatically when structural transitions occur. It was found that when the nanoparticle is inside the micelle, the DCC of the complex is negative (which indicates that the mass dipole moment is weak), the density profiles of MCP segments around the nanoparticle coincide with the density profile of MCP segments around its COM, and the radius of gyration increases as the radius of nanoparticle increases. When the micelle is adsorbed on a part of the nanoparticle, DCC is positive (which indicates that the mass dipole moment is strong), and the radius of gyration remains the same when both the radius of the nanoparticle (surface curvature) and the interaction between hydrophobic segments are increased.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail: hchen23@buffalo.edu. *E-mail: feaeliru@buffalo.edu. Tel: +1 716 645 1179. Fax: +1 716 645 3822. Notes
The authors declare no competing financial interest. 3727
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(19) Miller, T.; Hill, A.; Uezguen, S.; Weigandt, M.; Goepferich, A. Analysis of Immediate Stress Mechanisms upon Injection of Polymeric Micelles and Related Colloidal Drug Carriers: Implications on Drug Targeting. Biomacromolecules 2012, 13, 1707−1718. (20) Menezes, W. G.; Zielasek, V.; Dzhardimalieva, G. I.; Pomogailo, S. I.; Thiel, K.; Wohrle, D.; Hartwig, A.; Baumer, M. Synthesis of stable AuAg bimetallic nanoparticles encapsulated by diblock copolymer micelles. Nanoscale 2012, 4, 1658−1664. (21) Arnold, C.; Ulrich, S.; Stoll, S.; Marie, P.; Holl, Y. Monte Carlo simulations of surfactant aggregation and adsorption on soft hydrophobic particles. J. Colloid Interface Sci. 2011, 353, 188−195. (22) Lugo, D.; Oberdisse, J.; Karg, M.; Schweins, R.; Findenegg, G. H. Surface aggregate structure of nonionic surfactants on silica nanoparticles. Soft Matter 2009, 5, 2928−2936. (23) Prestidge, C. A.; Barnes, T.; Simovic, S. Polymer and particle adsorption at the PDMS droplet-water interface. Adv. Colloid Interface Sci. 2004, 108−109, 105−118. (24) Philip, J.; Prakash, G. G.; Jaykumar, T.; Kalyanasundaram, P.; Mondain-Monval, O.; Raj, B. Interaction between emulsion droplets in the presence of polymer−surfactant complexes. Langmuir 2002, 18, 4625−4631. (25) Schneider, T.; Stoll, E. Molecular-dynamics study of a threedimensional one-component model for distortive phase transitions. Phys. Rev. B 1978, 17, 1302−1322. (26) Dünweg, B.; Paul, W. Brownian dynamics simulations without gaussian random numbers. Int. J. Mod. Phys. C 1991, 02, 817−827. (27) Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. J. Comput. Phys. 1995, 117, 1−19. (28) Feng, J.; Liu, H.; Hu, Y. Molecular dynamics simulations of polyampholytes inside a slit. Mol. Simul. 2005, 31, 731−738.
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Micellar Structures in Nanoparticle-Multiblock Copolymer Complexes Houyang Chen* and Eli Ruckenstein* Department of Chemical and Biological Engineering, State University of New York at Buffalo, Buffalo, New York 14260-4200, United States S Supporting Information *
ABSTRACT: Brownian dynamics simulation is employed to examine the structure changes of complexes composed of a hydrophobic nanoparticle and a multiblock copolymer molecule (MCP). The dependence of the structure transitions on the radius of the nanoparticle, on the interactions between the hydrophobic segments of the MCP, and on the interactions between the hydrophobic segments and hydrophobic nanoparticle is examined. It is shown that the multiblock copolymer adsorbed on a nanoparticle can acquire the structure of a micelle.To better characterize the micelle generated and the structure changes in the nanoparticle−MCP complex, the mass dipole moment of the complex [the distance between the center of mass of MCP and the center of the nanoparticle minus the radius of the nanoparticle (DCC)], the density profiles of MCP segments around its center of mass and around the nanoparticle, the radius of gyration of the MCP, and the thickness of the micelle around the nanoparticle are determined. It was found that, when structural transition of the complex occurs, the above quantities change dramatically. The present simulation may provide new insights regarding the drug-loaded micelle interacting with a virus represented by a nanoparticle. nanoparticle (Lugo et al22). The dependence of surfactant aggregation on surface curvature was also investigated.22 However, the micelle generation in a polymer−nanoparticle complex and the transition from a nanoparticle inside a micelle to a micelle adsorbed on only a part of a nanoparticle were not examined. In this paper, we investigate complexes containing a nanoparticle and a multiblock copolymer molecule (MCP), in which the nanoparticle is either completely surrounded by the adsorbed MCP forming a micelle or only partially covered by a micelle or by adsorbed MCP. In addition, the transitions between these structures are examined. To better characterize the transitions and the morphologies of the complexes, the mass dipole moment of the complex [the distance between the center of mass of MCP and the center of the nanoparticle minus the radius of the nanoparticle (DCC)], the density profiles of the MCP segments around their centers of mass and around the nanoparticle, the thickness of the micelle and its radius of gyration, were calculated. It should be mentioned that in the current paper, we consider a “hard” nanoparticle interacting with a micelle, and not a “soft” one (e.g., a liquid droplet) interacting with the polymer molecule.23,24 The present simulations may provide insights regarding the drug
1. INTRODUCTION The adsorption of surfactants or polymers on nanoparticles has numerous applications and was widely investigated.1−15 A recent review is concerned with the critical adsorption of weak polyelectrolytes on curved surfaces and with the stable complexes formed by strong polyelectrolytes with oppositely charged spherical and cylindrical particles.16 In these papers, the dependence of the structure of complexes containing polymers (or surfactants) and nanoparticles on various factors, such as the natures of the solvent and polymer and the surface properties of the nanoparticle, were examined. In such complexes, the critical conditions for adsorption and the structure of the polymer are important. By employing Edward’s equation and the ground-state dominance approximation, the adsorption of polyelectrolyte chains from solutions on oppositely charged surfaces was examined by Cherstvy and Winkler.17,18 They found that, for low-dielectric constants, the charge density of the surface plays a crucial role in inducing the adsorption of polyelectrolyte chains. If the surfactants or polymers aggregate and generate micelles, they can be used as drug carriers19 and as nanoreactors.20 Using Monte Carlo simulations, Stoll and co-workers21 examined the surfactant aggregates adsorbed on soft hydrophobic particles. They found that, by increasing the concentration of surfactant and the interaction between the nanoparticles, elongated surfactant aggregates are generated. The adsorption of surfactants (pentaethylene glycol monododecyl ether or/and n-dodecylβ-maltoside) formed aggregates on the surface of silica © 2014 American Chemical Society
Received: July 29, 2013 Revised: February 22, 2014 Published: March 16, 2014 3723
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where r0 is the equilibrium bond distance, selected to be 1.2σ0, and Kb is the spring coefficient selected to be 1500ε0/σ20. Dilute solution simulations were carried out for a system of size (240σ0)3, which contains both a nanoparticle and a multiblock copolymer (MCP) (O10H10)12, with MCP composed of both hydrophilic (H) and hydrophobic (O) segments. The diameter of the segments of the multiblock copolymer is kept constant (σ0), and the diameter of the nanoparticle is varied. For simplicity, the nanoparticle is assumed frozen in the center of the simulation box. Using the Langevin thermostat25 for nanoparticle and segments of MCP, an implicit solvent is included. All molecular dynamics simulations were carried out by employing the large atomic/molecular massively parallel simulator (LAMMPS) package.27 As the initial configuration, the end of the MCP is considered to be a hydrophobic segment O which is generated at a distance σ0 from the surface of the nanoparticle; the other segments were generated randomly. After an initial configuration was generated, three processes were performed to optimize the initial configuration: (1) minimization using the conjugate gradient method, (2) simulation at high temperatures, and (3) annealing simulation. For simulation at high temperatures, the temperature T* = kBT/ε0 = 2.5 was selected. The simulation involved 105 BD steps, and for each step, a time step between 0.001τ and 0.003τ was employed. For the annealing process, a simulation with decreasing temperature from T* = 2.5 to T* = 1.0, involving 3 × 105 BD steps, was carried out; for each BD step, a time step between 0.001τ and 0.003τ was employed. After the initial configuration was generated and optimized, a simulation at T* = 1.0 involving 107 BD steps was performed, and for each BD step, a time step of 0.005τ was used. The configurations of the complexes were collected every 103 BD steps of the last 3 × 106 steps. It should be emphasized that (i) three independent initial conditions were employed, and the results were reproducible and (ii) 107BD steps are large enough to reach equilibrium (See Figure S1 of the Supporting Information).
loaded micelles interacting with a virus represented by the nanoparticle.
2. SIMULATION DETAILS AND MODEL The Brownian dynamics (BD) simulation25−28 employed is based on a coarse-grained model. The Langevin equation of motion of each segment i of mass mi is provided by r mi r( ï t ) = −∇ ∑ [Eij(rij) + Eb(rij)] − mi ζ r( i̇ t ) + Fi (t ) j
(1)
where mi, ri, and ζ are the mass, position vector, and friction coefficient of segment i, respectively. The coefficient ζ = 1.0 τ−1 [where τ = (m0σ20/ε0)1/2], mi = m0 for the segments of the copolymer and mi =50m0 for the nanoparticle. The quantities σ0, ε0, and m0 are the units of length, energy, and mass, respectively. The random force Fri (t) is considered to be Gaussian with a zero mean that satisfies the equation ⟨Fir(t ) ·F jr(t ′)⟩ = 6kBTmζδijδ(t − t ′)
(2)
where kB is the Boltzmann constant, T is the temperature, and δ is the Dirac delta function. The interactions between segments and between nanoparticle and segments are expressed in terms of the Lennard− Jones (LJ) potential: ⎧ ⎡⎛ ⎞12 ⎪ ⎢⎜ σij ⎟ ⎪ 4εij ⎜ ⎟ − Eij = ⎨ ⎢⎣⎝ rij ⎠ ⎪ ⎪ 0 ⎩
⎛ σij ⎞6 ⎤ ⎜⎜ ⎟⎟ ⎥ rij ≤ rc , ij ⎥ ⎝ rij ⎠ ⎦ rij > rc , ij
(3)
where rc,ij and rij are the cutoff distance and the distance between the centers of two segments or between the centers of nanoparticle and the segments of MCP, respectively. The length σij is the distance between the centers of beads (nanoparticle or segments) i and j when they are in contact, and εij is the interaction strength between beads i and j. The LJ potential parameters involved in the pair interactions between hydrophobic segments, hydrophilic segments, and nanoparticle are listed in Table 1. The LJ potential is shifted to zero at the cutoff distance. The interactions between two successive segments of the polymer chain are provided by the harmonic potential Eb =
1 Kb(rij − r0)2 2
3. RESULTS AND DISCUSSIONS When a multiblock copolymer molecule (MCP) is adsorbed on a nanoparticle, the structure of the MCP−nanoparticle complex can become very complicated. In this paper, we emphasize the structures in which micelles are generated. Two micellar structures have been identified: (i) a nanoparticle inside a micelle (see Figure 1e) and (ii) a micelle adsorbed on a part of the nanoparticle (see Figure 1f). A third structure consists of a MCP (which does not generate a micelle) adsorbed on a part of the nanoparticle (see Figure 1g). Figure 1 presents the structures generated when a nanoparticle interacts with a MCP molecule as functions of the radius (RP) of the nanoparticle, the interactions εOO between the hydrophobic segments of the MCP, and the interactions εOP between the hydrophobic segments of MCP and the hydrophobic nanoparticle. The other interaction parameters are fixed (see Table 1). For a low attractive interaction εOP between the hydrophobic segments of MCP and nanoparticle, MCP cannot overcome the entropy loss involved in its adsorption and prefers to stay away from the nanoparticle. For a moderate interaction εOP between the hydrophobic segments of MCP and the nanoparticle (εOP = 1.0, see Figure 1a) and small εOO, εOP can overcome the entropy loss and the MCP molecule is adsorbed on a part of the nanoparticle, generating a complex of MCP adsorbed on the nanoparticle. For εOP =1.0 and large εOO,
(4)
Table 1. Lennard−Jones Potential Parameters in the Simulations
interaction pairs O−O (hydrophobic segments− hydrophobic segments) O−H (hydrophobic segments− hydrophilic segments) O−P (hydrophobic segments− hydrophobic nanoparticle) H−H (hydrophilic segments− hydrophilic segments) H−P (hydrophilic segments− hydrophobic nanoparticle)
energy (ε0)
size (σ0)
cutoff distance (σ0)
εOO
σOO = σ0
2.5 σOO
εOH = 1.0 εOP
σOH = σ0
21/6 σOH
σOP = (σ0 + 2RP)/2 σHH = σ0
2.5 σOP
σHP = (σ0 + 2RP)/2
21/6 σHP
εHH = 1.0 εHP = 1.0
21/6 σHH
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Consequently, a complex of a micelle adsorbed on a part of the nanoparticle is formed. As shown in Figure 1 (panels a−d), by increasing εOP, the range of formation of a complex containing a nanoparticle inside a micelle increases. The morphologies of the above three complexes are presented in Figure 1 (panels e−g). Regarding the complex of a polymer adsorbed on a nanoparticle (Figure 1g), some results were reported previously.7,14 Using Monte Carlo simulations, Stoll and co-workers, who considered a polyelectrolyte and nanoparticles, found that the conformation of the polyelectrolyte is dependent on its rigidity and charge7 as well as on the salt valency.13 Feng and Ruckenstein14 identified complexes containing a polyampholyte and a charged nanoparticle and examined the effect of particle charge on the conformation of the polymer (not micelle) in the particle− polyampholyte complex. To better characterize the transitions between different complexes, a mass dipole moment of the complex [the distance between the center of mass (COM) of MCP and the center of the nanoparticle minus the radius of the nanoparticle (DCC)] was calculated. DCC is negative when the nanoparticle is inside the micelle and positive when the micelle is located only on a part of the surface of the nanoparticle. It should be mentioned that a negative value of DCC indicates a weak mass dipole moment of the complex, whereas a positive one means a strong mass dipole moment of the complex. A larger value of DCC means a stronger mass dipole moment. The mass dipole moment becomes zero when the absolute value of the negative DCC equals the radius of the nanoparticle (i.e., the center of mass of the polymer is located at the center of the nanoparticle). Figure 2 plots DCC as a function of the
Figure 1. State diagram (a−d) of complexes containing a nanoparticle and a micelle as functions of the radius of the nanoparticle (RP), the interaction between the hydrophobic segments εOO and between the hydrophobic segments and the hydrophobic nanoparticle and their snapshots (e−g). (a) εOP = 1.0 ε0; (b) εOP = 2.0 ε0; (c) εOP = 3.0 ε0; (d) εOP = 4.0 ε0. (e) Snapshot of a nanoparticle inside a micelle. (f) Snapshot of a micelle adsorbed on a nanoparticle. (g) Snapshot of a multiblock copolymer (MCP) adsorbed on a nanoparticle. For (a−d): ●, a nanoparticle inside a micelle; □, a micelle adsorbed on a part of nanoparticle; and △, a MCP adsorbed on a part of the nanoparticle. For e−g: hydrophobic segments, orange (gray in the white/black version); hydrophilic segments, yellow (white in the white/black version); and nanoparticle, black (black in the white/black version).
the hydrophobic segments of MCP self-aggregate and are adsorbed on a part of the nanoparticle, resulting in a kind of micelle adsorbed on the nanoparticle (see Figure 1a). For a strong interaction εOP between the hydrophobic segments of MCP and the hydrophobic nanoparticle (εOP = 2.0 − 4.0ε0, see Figure 1, panels b−d), MCP is adsorbed on the nanoparticle because εOP is large enough to overcome the entropy loss caused by MCP adsorption. For low εOO and small RP, the hydrophobic segments cover the surface of the nanoparticle, and because of weak interaction with the nanoparticle, the hydrophilic segments become located in the outer layer of the complex as much as the chain connectivity allows, generating a complex of a nanoparticle inside a kind of micelle. For low εOO and large RP, the number of hydrophobic segments may not be large enough to cover the entire surface of the nanoparticle. Then the hydrophobic segments together with the connected hydrophilic ones are adsorbed on a part of the surface of the nanoparticle, generating a complex of MCP adsorbed on a part of the nanoparticle. For moderate and large εOO (and any RP), the hydrophobic segments aggregate generating the core of a kind of micelle, and because the interaction with the nanoparticle of the hydrophilic segments is weaker than that between the hydrophobic segments, the former tend to be located on the outer layer of the structure.
Figure 2. The distance between the center of mass of MCP and the center of nanoparticle minus the radius of the nanoparticle (DCC) as functions of εOO when εOP = 2.0 ε0. RP (radius of a nanoparticle) = 2σ0 (○); RP = 3σ0 (□); RP = 4σ0 (△); RP = 5σ0 (▽); RP = 6σ0 (◇); RP = 7σ0 (×); RP = 8σ0 (⊕); RP = 9σ0 (⊞); RP = 10σ0 (short dashed line).
interaction εOO between hydrophobic segments and shows that it is negative when εOO is smaller than 0.8ε0 and positive when εOO is larger than 1.0ε0. In the former case, the nanoparticle is located inside the micelle, and the COM of MCP is inside the nanoparticle, leading to DCC < 0. In the latter case, MCP selfaggregates becoming a kind of micelle adsorbed on a part of the nanoparticle, and the COM of MCP is located outside the nanoparticle, leading to DCC > 0. DCC can provide the transition from a nanoparticle inside a micelle to a micelle adsorbed on a part of the nanoparticle. The density profiles offer additional information on the morphologies generated. The density profiles of MCP 3725
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the micelle is located close to the nanoparticle. In a comparison with the DCC results (previous section), one can conclude that a complex containing a micelle adsorbed on a part of a nanoparticle is generated. In the case of nonaggregated MCP (Figure 3c), the peak in the density profile of the hydrophobic segments around the nanoparticle (solid line in Figure 3c) is located close to the nanoparticle and the peak in the density profile of the hydrophilic segments around the nanoparticle is located somewhat further (dashed lines in Figure 3c). This can be explained as follows. The size of the nanoparticle in Figure 3c is large (much larger than the nanoparticle of Figure 3a) and the hydrophobic segments too few to cover the entire surface of the nanoparticle. The hydrophilic segments cover the surface of the nanoparticle because of chain connectivity between the hydrophobic and hydrophilic segments. However, because the interaction between the nanoparticle and the hydrophilic segments is weaker than that between the hydrophobic nanoparticle and hydrophobic segments, the former have the tendency to be located further from the nanoparticle. Figure 3 reveals that a complex containing a nanoparticle inside a micelle can be obtained for a particular size of the nanoparticle. A too large particle cannot be completely covered by the polymer. To investigate the micelle in the complex in more detail, its thickness is defined by
segments around its COM and around the nanoparticle are plotted for the morphologies: a nanoparticle inside a micelle, a micelle adsorbed on a part of the nanoparticle, and MCP adsorbed on a part of the nanoparticle (Figure 3). Figure 3a
Figure 3. Density profiles of MCP around its center of mass (COM) and around the nanoparticle. The solid line and the short dashed line are density profiles of hydrophobic and hydrophilic segments around the nanoparticle, respectively; circle and square are density profiles of hydrophobic and hydrophilic segments around the COM of MCP, respectively. d is the distance from the center of the nanoparticle or the COM of MCP. (a) RP = 3σ0, εOP = 2.0 ε0, εOO = 0.4 ε0; (b) RP = 3σ0, εOP = 2.0 ε0, εOO = 2.0 ε0; (c) RP = 7σ0, εOP = 2.0 ε0, εOO = 0.4 ε0; and (d) RP = 7σ0, εOP = 2.0 ε0, εOO = 2.0 ε0.
∞
θ=
∫R sρ(s) ds P
∞
∫R ρ(s) ds
− RP (5)
P
was also calculated. In the above expression, s is the distance between the centers of the nanoparticle and segment, and ρ(s) is the number density of MCP segments as a function of the distance from the center of the nanoparticle. Another useful quantity is the radius of gyration (Rg), which is defined as
presents the case of a nanoparticle inside a micelle. The peak of the density profiles of the hydrophobic segments around the nanoparticle (solid line in Figure 3a) is located close to the nanoparticle; the peak of the density profile of the hydrophilic segments around the nanoparticle (dashed line in Figure 3a) is somewhat further located, in a region where the density of the hydrophobic segments is negligible. In other words, the structure of the MCP is that of a micelle possessing a hydrophilic shell and a hydrophobic core. The peak in the density profile of the hydrophobic segments of MCP around its COM (○ in Figure 3a) is close to its COM. The density profile of the hydrophilic segments around the COM of MCP (□ in Figure 3a) is located in a region where the density of hydrophobic segments is negligible. Consequently, a kind of micelle around the COM of MCP is formed. The density profiles of the MCP segments around its COM and around the nanoparticle almost coincide, indicating that the complex consists of a nanoparticle inside a micelle. The case of a micelle adsorbed on a part of the nanoparticle provides quite different density profiles (Figure 3, panels b and d). The peaks of the hydrophobic segment density around the COM of the polymer (○ in Figure 3, panels b and d) are located close to its COM and have large values, indicating that the COM of the polymer is outside the nanoparticle. The peaks of the hydrophilic segments (□ in Figure 3, panels b and d) are located further from the COM of MCP. Thus a kind of micelle is formed. The density profiles of segments around the nanoparticle (the solid lines and dashed ones in Figure 3, panels b and d) show that
Rg =
1 M
∑ mi(ri − rcom)2 i
(6)
where ⟨...⟩ denotes the ensemble average, M is the mass of the MCP and rcom is the location of its center of mass. The thickness of the micelle is plotted in Figure 4a for a number of radii of the nanoparticle and a number of values of εOO. For a particular radius (RP) of the nanoparticle, the thickness remains the same when εOO increases from 0.2 to 0.6ε0 but starts to increase at εOO = 0.8ε0; for a particular εOO, the thickness increases with increasing RP. The radius of gyration (Rg) is
Figure 4. (a) Thickness of adsorbed MCP on the nanoparticle for εOP = 2.0 ε0, and (b) radius of gyration (Rg) of MCP for εOP = 2.0 ε0. RP = 2σ0 (○); RP = 3σ0 (□); RP = 4σ0 (△); RP = 5σ0 (▽); RP = 6σ0 (◇); RP = 7σ0 (×); RP = 8σ0 (⊕); RP = 9σ0 (⊞); and RP = 10σ0 (solid line). 3726
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Langmuir plotted in Figure 4b. For a selected RP and an εOO smaller than or equal to 0.8ε0, Rg remains constant when εOO increases; for a selected εOO and εOO lower than or equal to 0.8ε0, Rg increases with increasing RP. This occurs because, when RP increases, the polymer becomes more extended. In addition, with increasing RP, the structure of a nanoparticle inside a micelle changes to that of a polymer adsorbed on the nanoparticle (because the numbers of hydrophobic and hydrophilic segments are not large enough to cover the entire surface of the nanoparticle). When εOO becomes larger than or equal to 1.0 ε0, a kind of micelle is adsorbed on the nanoparticle possessing an Rg, which has the value of 3.6 ± 0.3σ0 regardless of the values of RP and εOO. This occurs because a stable micelle is formed and adsorbed on the surface of the nanoparticle. Figure 4b shows that a dramatic decrease in Rg occurs when the complex changes from a nanoparticle inside a micelle to a micelle adsorbed on a part of the nanoparticle, as well as from a polymer adsorbed on a part of the nanoparticle to a micelle adsorbed on a part of the nanoparticle.
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ASSOCIATED CONTENT
S Supporting Information *
Temperature and pairwise energy as well as radius of gyration of the multiblock copolymer as functions of the BD step. This material is available free of charge via the Internet at http:// pubs.acs.org.
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ACKNOWLEDGMENTS
We wish to thank the Center for Computational Research in the State University of New York at Buffalo for the computer time provided. The authors are grateful for the financial support from the State University of New York at Buffalo. H.C. is also grateful for the financial support from the National Natural Science Foundation of China (Grant 21206049).
4. CONCLUSIONS By employing Brownian dynamics simulation, the transitions from a nanoparticle inside a micelle to a micelle adsorbed on a part of the nanoparticle, from a copolymer adsorbed on a part of the nanoparticle to a micelle adsorbed on a part of the nanoparticle, as well as from a copolymer adsorbed on a part of the nanoparticle to a nanoparticle inside a micelle were determined. Additional information was obtained by calculating the mass dipole moment of the complex, the density profiles of the polymer segments around its COM and around the nanoparticle, the thickness of the micelle, and the radius of gyration of the copolymer. The above physical quantities change dramatically when structural transitions occur. It was found that when the nanoparticle is inside the micelle, the DCC of the complex is negative (which indicates that the mass dipole moment is weak), the density profiles of MCP segments around the nanoparticle coincide with the density profile of MCP segments around its COM, and the radius of gyration increases as the radius of nanoparticle increases. When the micelle is adsorbed on a part of the nanoparticle, DCC is positive (which indicates that the mass dipole moment is strong), and the radius of gyration remains the same when both the radius of the nanoparticle (surface curvature) and the interaction between hydrophobic segments are increased.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail: hchen23@buffalo.edu. *E-mail: feaeliru@buffalo.edu. Tel: +1 716 645 1179. Fax: +1 716 645 3822. Notes
The authors declare no competing financial interest. 3727
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