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Self-assembly of polymer-grafted nanoparticles in thin films Thomas Lafitte,a Sanat K. Kumarb and Athanassios Z. Panagiotopoulos*a We use large-scale molecular dynamics simulations with a coarse-grained model to investigate the selfassembly of solvent-free grafted nanoparticles into thin free-standing films. Two important findings are highlighted. First, for appropriately chosen values of system parameters the nanoparticles spontaneously assemble into monolayer thick films. Further, the nanoparticles self-assemble into a variety of morphologies ranging from dispersed particles, finite stripes, long strings, to percolating networks. The main driving force for these morphologies is the competition between strong short-range attractions of the particle cores and long-range entropic repulsions of the grafted chains. The grafted nanoparticle
Received 3rd September 2013 Accepted 21st November 2013
systems provide practical means to realize two-length-scale systems that have been previously seen,
DOI: 10.1039/c3sm52328d
using a simple two-dimensional model [G. Malescio and G. Pellicane, Nat. Mater., 2003, 2, 97], to generate a variety of morphologies. However, there are only relatively narrow ranges of interaction
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strengths and chain lengths for which anisotropic self-assembly is possible.
1
Introduction
Self-assembly has allowed for the production of nanostructured materials with diverse properties, potentially useful for electronics, photonics and biomedical devices. Polymer-graed nanoparticles are promising building blocks for creating advanced materials; a large body of literature is devoted to understanding the parameters which control their morphologies.1–7 In many cases, the main driving force leading to the self-assembly of nanoparticles lies in the presence of some anisotropy at the level of their bare interactions. However, recent ndings on the structural behavior of spherical polymergraed nanoparticles8–10 have opened up new strategies to achieve anisotropic assemblies without the need for synthesizing anisotropically-shaped particles. While uniformly graing polymer chains onto the surface of inorganic particles was originally thought as a way to provide “steric stabilization” of the nanoparticles when dispersed into polymer matrices, this process is now known to be able to generate several anisotropic morphologies. This phenomenon which occurs under specic conditions for the graing density and gra chain length, is now understood as being akin to amphiphile molecule selfassembly.11 In particular, unfavorable interactions between the particle core and graed polymer chains cause them to phase separate. However, since they are constrained by chain connectivity, they can only microphase separate. The different
a
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA. E-mail: [email protected]
b
Department of Chemical Engineering, Columbia University, New York, New York 10026, USA
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morphologies obtained (e.g. strings, two dimensional sheets) depending on the architecture of the corona and nanoparticle core size result from a subtle balance between the enthalpic gain upon the core–core aggregation and the non-isotropic distribution of polymer segments around the particle.10,12 Most studies to date have been carried out for the case of selfassembled structures of graed nanoparticles suspended in a solvent. Much less is known on the limit of solvent-free conditions where the graed polymers effectively become the solvent.13–15 These novel materials are found to be able to exhibit liquid-like behavior under ambient conditions,14,16 and offer practical advantages compared to conventional nanocomposites due to their better processability. A recent simulation study,17 using a simple coarse-grained model, suggested the possibility of obtaining striped structures for graed nanoparticle systems with sufficiently low graing density, and investigated the effects of particle core size, graing density and temperature on the equilibrium morphologies. Other studies18–21 have focused on the dynamics of such systems. There has been a growing effort devoted in nding experimental techniques which allow the creation of robust thin lms22 and there are recent experimental observations for thin lms of solvent-free graed nanoparticles.23,24 An example of a simple model system that self-assembles in two-dimensions is the square-shoulder potential of Malescio and Pellicane.25 This model belongs to the class of soened core potentials which are characterized by a repulsive component consisting of an exclusion region plus a nite so repulsive shoulder. The existence of two well-dened length scales is sufficient to generate two-dimensional spatial modulations.25–27 However, an experimental realization of this model is only achieved under some
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very special conditions.28 The primary focus of the present study is to investigate structure formation for thin lms of solventfree graed nanoparticles. Our goal here is not to provide comparisons with the corresponding bulk systems, but rather it is to focus on the structural behavior of monolayers of nanoparticles and their phenomenological similarity with twodimensional core-soened model systems. The structure of this paper is as follows. Aer describing the model and simulation methodologies employed, we investigate the possibility of obtaining stable monolayer formation (lm thickness corresponding to only one inorganic core). We subsequently study the inuence of both particle–particle and particle–polymer interactions on the morphology of the suspended lm. We nd that for specic values of these parameters, our systems self-assemble into dened structures ranging from dispersed particles, nite stripes to percolating networks in two dimensions. Following this structural analysis, we employ the inverse-Boltzmann iterative procedure to derive effective nanoparticle–nanoparticle interaction potentials which accurately reproduce the pair correlation functions between the inorganic cores of the full system. We show that the resulting potentials are also able to reproduce the detailed morphologies of the graed nanoparticle thin lms, suggesting a strong relationship between anisotropic structure formation and the existence of two competing length scales.
2 Model and simulation methodology 2.1
Nanoparticle coarse-grained model
In this work, polymer-graed nanoparticles are modeled as spheres with f linear oligomer attached chains. A schematic illustration of the model is given in Fig. 1. Each chain is composed of m Lennard-Jones (LJ) beads of size s, bonded to form chains via harmonic potentials V(r) ¼ k(r s)2, where k denotes the spring constant. The spring constant k was set to a high value, k ¼ 50003/s2, to ensure tangential bonding between pairs of catenated beads. To dene the energy and length scale for the model, we follow the coarse-graining scheme used by Hong et al.,29 where each bead represents an ethoxy repeat unit (–CH2OCH2O–) and interacts with other beads through a cutand-shied Lennard-Jones potential U(r): 8 > < 43 s 12 s 6 U r # 2:5s cut r r (1) UðrÞ ¼ > : 0 r . 2:5s where Ucut denotes the value of the unshied interactions at the cutoff distance of 2.5s. Parameters s ¼ 0.4 nm and 3/kB ¼ 377 K were obtained from the critical properties of the ethoxy repeat unit, estimated through the group contribution method of Constantinou and Gani.30 The LJ bead mass is set to m ¼ 44 g mol1 appropriate for the ethoxy group. The particle core diameter was set to sc ¼ 3.5s in this study, corresponding to 1.4 nm, signicantly below the diameter range of silica particles commonly used in experiments, typically between 8 and 20 nm.8,24,31 On the other hand, sc ¼ 1.4 nm matches commonly used gold particle diameters.24,32 It was not
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Grafted nanoparticle coarse-grained model representations. (a) Two-dimensional schematic view. (b) Example of three-dimensional configuration, with f ¼ 12 and sc ¼ 3.5s.
Fig. 1
possible to investigate larger core diameters due to computational limitations, since many particles with attached chains need to be simulated to identify patterns and long-range order in the systems of interest. Prior attempts to simulate brush coated silica nanoparticles with large cores have been restricted to the study of one or two particles.33–36 In this work, we consider the particular case of f ¼ 12 chains tethered isotropically to the surface of the inorganic cores, corresponding to a graing density of 2 chains per s2. Three values for the polymer chain length are selected, m ¼ 20, 30 and 40. To allow fast computation, interactions between core particles are represented by a spherically symmetric, pair-wise additive potential. We used the procedure of Lee and Hua37 to determine this potential. A spherical packing unit (Si6O12) was rst chosen and assigned LJ parameters so as to accurately reproduce the known bulk properties of amorphous silica. Core–core interactions were obtained by summing up all the interacting beads which are placed to form two spherical shells of increasing diameter, in an approach similar to that used to obtain Hamaker constants. Using a numerical angle-averaging at various interparticle distances, we extract a sphericallysymmetric potential which can be used within our coarsegrained model. To perform the angle averaging, we xed the position of one nanoparticle and placed another spherical shell at a random location dened by a vector r which is subsequently rotated multiple times to random congurations. By repeating this procedure for different interparticle distances, we obtained a numerical estimate of the pair interaction between the core particles. The resulting shape of this potential depends strongly on core size, exhibiting an attractive part with shorter range and stronger attraction well depth than the underlying elementary LJ bead.37 In order to derive an analytical form for the angleaveraged pair interaction between spherical silica particles Uc(r) Lee and Hua37 advocated the use of a generalized LJ potential, sc 2l sc l (2) Uc ðrÞ ¼ 43c r r where sc approximately coincides with the particle diameter, and 3c and l are tted to the previously obtained tabulated pair potential.
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We applied this strategy for the case of small silica nanoparticles of diameter sc ¼ 1.4 nm and obtained l ¼ 7, 3c/kB ¼ 10 179K x 93/kB. For the case of gold nanoparticles of the same diameter, using the LJ potential of Kyrychenko,38 we obtained l ¼ 18 and 3c/kB ¼ 26 173K x 693/kB. Clearly, the overall shape of the coarse-grained pair potential representing the interactions between nanoparticles strongly depends on their precise chemistry. In order to maintain computational feasibility, we assumed a single shape, LJ 14-7, for the coarse-grained potential. However, we studied the effect of varying the attractive energy 3c. We considered three different values for core–core interaction: 3c ¼ 93, 3c ¼ 253 and 3c ¼ 353. In each case, the potential is cut and shied to zero at the cutoff distance of 2.5sc. The shape of the cross-interaction potential Ucp between the nanoparticle core and the polymer beads is dened by taking an arithmetic mean between attractive and repulsive exponents of the LJ 12-6 (polymer–polymer) and LJ 14-7 (core– core) potentials: 8 scp 13 scp 6:5 > < 43 Ucp r # 2:5scp cp r r Ucp ðrÞ ¼ (3) > : 0 r . 2:5scp where scp ¼ (sc + s)/2, corresponding to the widely used Lorentz–Berthelot combining rule39 and Ucp denotes the value of the unshied interactions at the cutoff distance of 2.5scp. In order to mimic the effect of graing polymer chains with different chemical compositions, the cross-interaction energy 3cp is considered as a variable parameter through the use of the pffiffiffiffiffiffi expression 3cp ¼ ð1 aÞ 3c 3: We investigated several different values of the core–polymer interaction 3cp by taking a ¼ 0, a ¼ 0.2, a ¼ 0.4, a ¼ 0.6 and a ¼ 0.8. This allowed us to span a broad region of parameter space from attractive to nearly purely repulsive cross-interactions between particle cores and polymer beads. 2.2
Simulation details
The large scale atomic/molecular massively parallel simulator (LAMMPS) was used to compute the structural behavior of the free-standing lms. All simulations were performed in orthorhombic boxes containing N ¼ 400 graed particles, under periodic boundary conditions with xed area in the xy plane. A rigid-body treatment was used between the central particle core and the rst LJ bead of each graed chain. The time evolution of the rigid body part of each particle was performed via constant NVE integration. Temperature was maintained at T ¼ 300 K using Langevin dynamics, in which the polymer chains are coupled to a heat bath through the use of a friction coefficient g and random force. The main objective of this study was to explore the structural behavior of the system, so the precise choice of the thermostat is not crucial. The equations of motion were integrated using the velocity-Verlet algorithm with a time pffiffiffiffiffiffiffiffiffiffiffiffiffi step dt ¼ 0.008s, where s ¼ ms2 =3 is the basic unit of time. To create free-standing monolayers, we started from initial congurations where the cores were placed randomly (ensuring no overlap between them) in an expanded orthorhombic simulation box with dimensions parallel to the lm set to
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Lx ¼ Ly ¼ 2000s. The dimension perpendicular to the lm was chosen to be Lz ¼ 3sc to ensure appearance of a monolayer. Polymer chains were tethered to the surface of the core particles and grown using self-avoiding random walks, with the constraint that the centers of the LJ beads are located in the interval 1.5sc < z < 1.5sc. In order to maintain the free monolayer structure during the pre-equilibration period, two impenetrable walls (truncated-shied 9-3 Lennard-Jones) were added to conne the system in the z-direction. The x- and y-box dimensions were subsequently reduced at a rate of 0.001 nm per time step, until the desired area fraction was obtained. During this initialization step, all the interactions were described solely by purely repulsive WCA potentials for the sake of computational efficiency and in order that the chains adopt stretched conformations. Once the desired dimensions of the box were reached, we switched on the full intermolecular interactions (i.e., the coarse-grained model described in the previous section) and let the system relax over 5 105dt. At this point the walls are removed so that the system is then equilibrated as a free-standing lm for an additional period of 6 106dt, which removes any effects of the initial connement. Subsequently, the measured properties are averaged over typical production runs of 2 106dt. The free-standing lms are stable over this period of time.
3 Results and discussion 3.1
Free-standing monolayer formation
For the different chain lengths of the graed particles studied in this work (m ¼ 20, 30, and 40), it was rst necessary to determine the area fractions A ¼ N/(LxLy) that led to stable free standing monolayers. We display in Fig. 2 three different simulations performed at increasing values of percentage area fraction A ¼ 13%, 23% and 27% in the particular case of systems with a graed chain length m ¼ 30, an energy parameter 3c ¼ 93, and a ¼ 0. It can be seen that for low values of the area fraction A, the monolayer is not mechanically stable and eventually develops holes aer sufficient time has elapsed. For higher values of A, free standing monolayers can be obtained where the cores arrange themselves in nearly 2-dimensional layers at the center of the lm. For enthalpic reasons, the central cores of the particles stay away from the vacuum boundaries and are surrounded by polymer beads. This can be seen in Fig. 3, where we plot the area density along the z direction for both the central core particles and the polymer beads corresponding to the thin lm displayed in Fig. 2(b). This behavior is related to the existence of a core–polymer attraction. Upon further increase of the area fraction, and hence, increase in geometric frustration, the nanoparticle cores gradually start to be displaced from the center plane of the lms. In the present study, we focus on area fractions for which the lms correspond to a stable monolayer with the particles preferentially near the center plane. We note that it is not possible to know in advance what precise threshold value of the area fraction A yields stable freestanding lms. These values depend on the graing density and polymer chain length. Film stability was found to be insensitive
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Fig. 2 Example of configurations obtained for systems with m ¼ 30, 3c ¼ 253 and a ¼ 0 at different area fractions. (a) A ¼ 13%: unstable film (b) A ¼ 23%: stable monolayer (c) A ¼ 27%: buckled monolayer. Top and bottom pictures show the top view and the side view of the films respectively. All three snapshots contain N ¼ 400 grafted nanoparticles; the linear dimensions of the three simulation boxes are different.
Fig. 3 Area density profiles of core particles, rc (red curve), and polymer beads, rp (blue curve), in the z direction for m ¼ 30, 3c ¼ 253 and a ¼ 0 with imposed area fraction A ¼ 23%.
to the value of the energy parameters. Typically, systems with short polymer chains are stable at higher values of A than particles graed with long chains. It was then necessary to perform multiple test runs for each graing chain length of interest. The goal here was not to determine a precise boundary of stability, but rather to select appropriate values of A to perform a structural study of a free standing monolayer. Based on these preliminary calculations, we selected the values A ¼ 17%, 23%, and 30% for graed chain lengths of m ¼ 20, 30, and 40, respectively.
3.2
present work. Fig. 4–6 show morphology diagrams of equilibrium congurations taken from simulations performed at different values of 3c and 3cp for each graed chain length studied. Several clearly dened structures are observed; dispersed particles, strings of width equal to one particle, stripes two or three particles wide, and clusters of particles. There seems to be a threshold value around a ¼ 0.6, at which a structural transition from dispersed particles to strings and stripes occurs. In addition, a minimum amount of core–core attraction is necessary to observe anisotropic structures. This supports the idea that the morphology of the graed particle systems results from a delicate balance between the enthalpic gain upon aggregation of the inorganic cores and the entropic gain associated with particle dispersion, mediated by the steric stabilization provided by the chains. The morphology diagrams also suggest that the main driving force for the formation of anisotropic structures lies in the inherent dislike between the central core and the polymer chains (modeled through the variation of the parameter a) and to a lesser extent to the core–core attraction. The precise architecture of the particles also affects their equilibrium spatial distribution. For the systems with chain length m ¼ 20, highly anisotropic objects are observed for a > 0.6 and typically ll the entire simulation box. In the case of longer chains, the particles form shorter strings of nite lengths. This is attributed to the increased entropic stabilization provided by the longer chains.
Anisotropic structure formation
To gain insight into the parameters that control the spatial distribution of the graed particles, we rst investigated the effects of changing both the core–core and the core–polymer attraction for different values of the polymer chain length. Other factors such as the particle core size and graing density may also play important roles, but are not considered in the
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3.3
Local structure
To further investigate the microscopic mechanisms at the origin of the 2-dimensional anisotropic aggregation of the graed particles, we study the microstructure of chains surrounding the inorganic core. The formation of “nanowires” is generally attributed to the equator region of the dimer
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Top view of equilibrium configuration of grafted particles (chain length m ¼ 20) in thin films for different values of core–core attraction 3c pffiffiffiffiffiffiffi and core–polymer attraction 3cp ¼ ð1 aÞ 3c 3: The polymer beads are not represented in other to better visualize the structures formed by the inorganic cores. Fig. 4
Top view of equilibrium configuration of grafted particles (chain length m ¼ 30) in thin films for different values of core–core attraction 3c pffiffiffiffiffiffiffi and core–polymer attraction 3cp ¼ ð1 aÞ 3c 3: The polymer beads are not represented in other to better visualize the structures formed by the inorganic cores. Fig. 5
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Top view of equilibrium configuration of grafted particles (chain length m ¼ 40) in thin films for different values of core–core attraction 3c pffiffiffiffiffiffiffi and core–polymer attraction 3cp ¼ ð1 aÞ 3c 3: The polymer beads are not represented in other to better visualize the structures formed by the inorganic cores. Fig. 6
nanoparticle surface being less shielded by the graed chains leading to an effective dipolar interaction.10,40 This phenomenon can be assessed by calculating the averaged 2-dimensional asphericity, Axy, of the graed nanoparticles. This quantity, initially proposed by Rudnick and Gaspari is dened as follows:41 * 2 + ly lx (4) Axy ¼ 2 lx þ ly where lx and ly are the squares of the main component of the 2-dimensional radius of gyration tensor Rgxy obtained by projection of the polymer bead coordinates onto the plane dened by the equation z ¼ 0. The quantity Axy has an upper bound of 1 in the limit case of a one dimensional rod-like particle and a lower bound of 0 for a molecule forming a disk shape. From the results reported in Fig. 7, it is clear that the formation of string morphologies is strongly correlated with an increase of the sphericity Axy. The effect is clear for systems with chain length m ¼ 20. The minimum value for Axy is found at low values of the core–core attraction 3c and high values of the core–polymer interaction 3cp (corresponding to low values of the parameter a). In this regime, the systems are well dispersed with negligible distortion of the chains, as one expects for systems dominated by entropic contributions to their free energy. The favorable cross-interaction leads to adsorption of the polymer chains on the ller surface creating an overall disk shape for the molecule, and hence, low asphericity. Increasing a, the degree of “dislike” between central core particles and polymer chains, leads to only
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minor increases in the asphericity for these systems. A different regime arises for 3c ¼ 253 and 3c ¼ 353. For these values of core– core attractions, both enthalpic and entropic contributions become relevant and the microscopic structure of the particles undergoes a transition around a ¼ 0.5 where a gradual increase of the asphericity is observed. This increase is over a similar range of a values to that corresponding to string and stripe patterns formation. This local rearrangement of the graed chains creating an ellipsoidal shaped particle is illustrated in Fig. 8, where we plot the projection of polymer–core distribution function on the axis dened by the eigenvectors of the gyration tensor Rgxy. It is clear from this gure that upon increasing a, the polymer beads position themselves farther away from the nanoparticle cores and the polymer chains are found to be stretched along a preferential direction leaving space for the core to aggregate. The asphericity Axy for longer chain lengths, m ¼ 30 and m ¼ 40 is also shown in Fig. 7. We note that the change in the chain microstructure with energetic parameters is lower in this case, due to larger contribution of the entropic effects. 3.4
Effective potentials
It has been shown in Section 3.3 that the self-assembly of isotropically graed nanoparticles into strings results from a balance of enthalpic and entropic effects. String formation can be thought of as being caused by a competition between shortrange attractive forces and long-range repulsion arising from dipole interactions.42,43 However, in a recent paper10 a different mechanism related to graing inhomogeneities has been
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proposed; this is likely to be relevant at lower graing densities than those considered here. Prior simulations have demonstrated that spherical particles interacting isotropically through a repulsive interparticle potential with two different length scales (the so-called core-soened potentials) are sufficient to yield spontaneous self-assembly into anisotropic structures.25,44–47 To gain insight into the link between the graed nanoparticle effective interactions and the physics of coresoened potentials, we developed effective two-dimensional coarse-grained potentials. This is done through the use of an inverse-Boltzmann procedure48–50 in which we iteratively update the effective pair interaction u(r) to match the pair correlation functions of the target system according to the following formula:
gi uiþ1 ðrÞ ¼ ui ðrÞ þ kT ln (5) gtarget
Asphericity Axy at various values of the core–core attraction 3c and core–polymer interaction a in the case of (a) m ¼ 20, (b) m ¼ 30 and (c) m ¼ 40. Fig. 7
where gi(r) is the 2-dimensional radial distribution obtained during the ith step and gtarget is the 2-dimensional distribution between the cores of the full system. The 2-dimensional MD simulations were performed at the same number density rc as the full systems. We found that 50 iterations were enough to achieve satisfactory convergence for each system. We present results of the inverse-Boltzmann procedure in Fig. 9 for systems with graed chain lengths m ¼ 20 and m ¼ 30. Since our main goal is to reproduce the anisotropic selfassembly with the derived spherical potentials, we focus our attention on the systems with strong core–core attraction, 3c ¼ 253, and study the inuence of varying the core–polymer interaction through the parameter a. For the sake of clarity, in Fig. 9 we show the effective potentials related to two different regimes: dispersed state (a ¼ 0.2) and aggregated state (a ¼ 0.6). For favorable core–polymer interactions, a ¼ 0.2, we obtain an
Fig. 8 Projection of the polymer–core distribution function on the axis defined by the eigenvectors of the gyration tensor Rgxy for the grafted nanoparticle with 3c ¼ 253 at different values of the core–polymer attraction a. Darker shades indicate lower values of the distribution function. Representative equilibrium configurations of the nanoparticle cores corresponding to each distribution functions are also displayed for the sake of clarity.
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30. A remarkable similarity between the full and spherically coarse-grained systems is obtained in each case. These results suggest that the formation of nite stripes is mainly an effect of the reduction of the area fraction of exposed cores with increasing chain length. Further work will be needed in this line to possibly develop a rigorous mapping between the parameters linked to the architecture of graed nanoparticle systems and the shape of the isotropic effective potential, and also to relate the results of the current work (with relatively high gra densities) with the previous work by Bozorgui et al.10 Effective potentials representing the interactions between particles as a function of the core–polymer attraction a, and chain length, (a) m ¼ 20 and (b) m ¼ 30.
Fig. 9
effective diameter of around 5s which conrms our previous nding that the polymer chains wrap around the nanoparticle and allow for a well-dispersed, sterically stabilized state. A more interesting scenario occurs for a ¼ 0.6, i.e., for graed nanoparticles exhibiting self-assembly. From an inspection of the effective potential, we notice the appearance of a well-dened so-core shoulder and a short ranged attractive part. In addition, it can be seen that the effective potentials derived for m ¼ 20 and m ¼ 30 are of the same general shape. Fig. 10 shows two snapshots representing equilibrium congurations calculated from the effective potentials corresponding to m ¼ 20 and m ¼
Fig. 10 Comparison of equilibrium configurations obtained with the full model and the spherically symmetric coarse-grained potentials obtained from the inverse-Boltzmann procedure. Two examples are displayed: (a) system with m ¼ 20, 3c ¼ 253, a ¼ 0.6 and (b) system with m ¼ 30, 3c ¼ 253, a ¼ 0.6. The corresponding effective potentials are reported in Fig. 9.
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4 Conclusions We have demonstrated that a three-dimensional coarse-grained model for solvent-free nanoparticles graed with polymer chains spontaneously assembles into two-dimensional lms for specic ranges of system parameters. Further, these nanoparticle monolayers show self-assembly behavior. This nding complements recent theoretical and experimental observations of structure formation for spherically-symmetric graed nanoparticles in solution and in solvent-free melts. In spite of the isotropic nature of their bare interactions, particles selfassemble into a variety of anisotropic structures, driven by the microphase separation between the inorganic core and the organic (graed) chains. The simple coarse-grained model for graed nanoparticle systems developed in this work allows sufficiently fast computation to consider the interactions of few hundreds of these graed nanoparticles, while keeping essential details related to the chain architecture and the intermolecular forces between the inorganic core and the graed polymer chains. We have examined in detail the behavior of graed nanoparticle systems with respect to variations in the relative strength of particle–particle and particle–polymer attractions, and for different graed polymer chain lengths. We found that for weak core–core attractions, entropic contributions dominate the phase behavior of the lms, and the nanoparticles stay in a dispersed state for any value of the core–polymer cross-interaction. For higher values of the core–core attractive energy, we found that a competition between enthalpic and entropic interactions arises. In this case, if there is a sufficient “dislike” between the central core and the polymer chains, the graed nanoparticles can self-assemble into stripes for which the morphology depends on the polymer chain lengths. Two different regimes were observed: percolating stripes for sufficiently short chains and string-like structures of nite size for longer chain lengths. There are therefore signicant similarities between the complex behavior of realistically modeled graed nanoparticles and the physics of radially symmetric pair potential with intermediate-range repulsive interactions, which are able to form anisotropic structures in two dimensions.25 Nevertheless, some of the morphologies observed for the earlier two-step model potentials, in particular hexagonal closedpacked ordered phases at relatively low area densities, are not observed in the systems studied in this work, even though it is likely that they would appear for larger particles and sufficiently short graed chains.
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Acknowledgements
Published on 26 November 2013. Downloaded by University of Sydney on 31/01/2014 13:23:10.
This publication is based on work supported by collaborative research grant CBET-1033155 (Princeton) and CBET-1033168 (Columbia) from the U.S. National Science Foundation. We thank Dr. Alexandros Chremos for help with setting up the simulations.
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20 S. Goyal and F. A. Escobedo, J. Chem. Phys., 2011, 135, 184902. 21 B. Hong, A. Chremos and A. Z. Panagiotopoulos, J. Chem. Phys., 2012, 136, 204904. 22 D. Lee, M. F. Rubner and R. E. Cohen, Nano Lett., 2006, 6, 2305–2312. 23 N. J. Fernandes, J. Akbarzadeh, H. Peterlik and E. P. Giannelis, ACS Nano, 2013, 7, 1265–1271. 24 P. Green, So Matter, 2011, 7, 7914–7926. 25 G. Malescio and G. Pellicane, Nat. Mater., 2003, 2, 97–100. 26 Y. Norizoe and T. Kawakatsu, J. Chem. Phys., 2012, 137, 024904. 27 Y. Norizoe and T. Kawakatsu, Europhys. Lett., 2005, 72, 583. 28 N. Osterman, D. Babic, I. Poberaj, J. Dobnikar and P. Ziherl, Phys. Rev. Lett., 2007, 99, 248301. 29 B. Hong, A. Chremos and A. Z. Panagiotopoulos, Faraday Discuss., 2012, 154, 29–40. 30 L. Constantinou and R. Gani, AIChE J., 1994, 40, 1697–1710. 31 Q. Zhang and L. A. Archer, Langmuir, 2002, 18, 10435–10442. 32 X. Chen and P. Green, So Matter, 2011, 7, 1192–1198. 33 D. Meng, S. K. Kumar, J. M. D. Lane and G. S. Grest, So Matter, 2012, 8, 5002–5010. 34 J. Kalb, D. Dukes, S. K. Kumar, R. S. Hoy and G. Grest, So Matter, 2011, 7, 1418–1425. 35 T. Ndoro, E. Voyiatzis, A. Ghanbari, D. Theodorou, M. B¨ ohm and F. M¨ uller-Plathe, Macromolecules, 2011, 44, 2316. 36 G. D. Smith and D. Bedrov, Langmuir, 2009, 25, 11239–11243. 37 C. K. Lee and C. C. Hua, J. Chem. Phys., 2010, 132, 224904. 38 A. Kyrychenko, G. V. Karpushina, S. I. Bogatyrenko, A. P. Kryshtal and A. O. Doroshenko, Comput. Theor. Chem., 2011, 977, 34–39. 39 J. S. Rowlinson and F. L. Swinton, Liquids and Liquid Mixtures, Butterworth, London, 1982. 40 L. Cheng and D. Cao, J. Chem. Phys., 2011, 135, 124703. 41 J. Rudnick and G. Gaspari, Science, 1987, 237, 384–389. 42 M. Seul and D. Andelman, Science, 1995, 267, 476–483. 43 A. Stoycheva and S. Singer, Phys. Rev. Lett., 2000, 84, 4657– 4660. 44 G. Malescio and G. Pellicane, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2004, 70, 021202. 45 G. Malescio, J. Phys.: Condens. Matter, 2007, 19, 073101. 46 F. Sciortino, S. Mossa, E. Zaccarelli and P. Tartaglia, Phys. Rev. Lett., 2004, 93, 055701. 47 S. Mossa, F. Sciortino, P. Tartaglia and E. Zaccarelli, Langmuir, 2004, 20, 10756–10763. 48 S. Jain, S. Garde and S. K. Kumar, Ind. Eng. Chem. Res., 2006, 45, 5614–5618. 49 N. G. Almarza, E. Lomba and D. Molina, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2004, 70, 021203. 50 A. P. Lyubartsev and S. Marcelja, Phys. Rev. E: Stat., Nonlinear, So Matter Phys., 2002, 65, 041202.
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Self-assembly of polymer-grafted nanoparticles in thin films Thomas Lafitte,a Sanat K. Kumarb and Athanassios Z. Panagiotopoulos*a We use large-scale molecular dynamics simulations with a coarse-grained model to investigate the selfassembly of solvent-free grafted nanoparticles into thin free-standing films. Two important findings are highlighted. First, for appropriately chosen values of system parameters the nanoparticles spontaneously assemble into monolayer thick films. Further, the nanoparticles self-assemble into a variety of morphologies ranging from dispersed particles, finite stripes, long strings, to percolating networks. The main driving force for these morphologies is the competition between strong short-range attractions of the particle cores and long-range entropic repulsions of the grafted chains. The grafted nanoparticle
Received 3rd September 2013 Accepted 21st November 2013
systems provide practical means to realize two-length-scale systems that have been previously seen,
DOI: 10.1039/c3sm52328d
using a simple two-dimensional model [G. Malescio and G. Pellicane, Nat. Mater., 2003, 2, 97], to generate a variety of morphologies. However, there are only relatively narrow ranges of interaction
www.rsc.org/softmatter
strengths and chain lengths for which anisotropic self-assembly is possible.
1
Introduction
Self-assembly has allowed for the production of nanostructured materials with diverse properties, potentially useful for electronics, photonics and biomedical devices. Polymer-graed nanoparticles are promising building blocks for creating advanced materials; a large body of literature is devoted to understanding the parameters which control their morphologies.1–7 In many cases, the main driving force leading to the self-assembly of nanoparticles lies in the presence of some anisotropy at the level of their bare interactions. However, recent ndings on the structural behavior of spherical polymergraed nanoparticles8–10 have opened up new strategies to achieve anisotropic assemblies without the need for synthesizing anisotropically-shaped particles. While uniformly graing polymer chains onto the surface of inorganic particles was originally thought as a way to provide “steric stabilization” of the nanoparticles when dispersed into polymer matrices, this process is now known to be able to generate several anisotropic morphologies. This phenomenon which occurs under specic conditions for the graing density and gra chain length, is now understood as being akin to amphiphile molecule selfassembly.11 In particular, unfavorable interactions between the particle core and graed polymer chains cause them to phase separate. However, since they are constrained by chain connectivity, they can only microphase separate. The different
a
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, USA. E-mail: [email protected]
b
Department of Chemical Engineering, Columbia University, New York, New York 10026, USA
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morphologies obtained (e.g. strings, two dimensional sheets) depending on the architecture of the corona and nanoparticle core size result from a subtle balance between the enthalpic gain upon the core–core aggregation and the non-isotropic distribution of polymer segments around the particle.10,12 Most studies to date have been carried out for the case of selfassembled structures of graed nanoparticles suspended in a solvent. Much less is known on the limit of solvent-free conditions where the graed polymers effectively become the solvent.13–15 These novel materials are found to be able to exhibit liquid-like behavior under ambient conditions,14,16 and offer practical advantages compared to conventional nanocomposites due to their better processability. A recent simulation study,17 using a simple coarse-grained model, suggested the possibility of obtaining striped structures for graed nanoparticle systems with sufficiently low graing density, and investigated the effects of particle core size, graing density and temperature on the equilibrium morphologies. Other studies18–21 have focused on the dynamics of such systems. There has been a growing effort devoted in nding experimental techniques which allow the creation of robust thin lms22 and there are recent experimental observations for thin lms of solvent-free graed nanoparticles.23,24 An example of a simple model system that self-assembles in two-dimensions is the square-shoulder potential of Malescio and Pellicane.25 This model belongs to the class of soened core potentials which are characterized by a repulsive component consisting of an exclusion region plus a nite so repulsive shoulder. The existence of two well-dened length scales is sufficient to generate two-dimensional spatial modulations.25–27 However, an experimental realization of this model is only achieved under some
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very special conditions.28 The primary focus of the present study is to investigate structure formation for thin lms of solventfree graed nanoparticles. Our goal here is not to provide comparisons with the corresponding bulk systems, but rather it is to focus on the structural behavior of monolayers of nanoparticles and their phenomenological similarity with twodimensional core-soened model systems. The structure of this paper is as follows. Aer describing the model and simulation methodologies employed, we investigate the possibility of obtaining stable monolayer formation (lm thickness corresponding to only one inorganic core). We subsequently study the inuence of both particle–particle and particle–polymer interactions on the morphology of the suspended lm. We nd that for specic values of these parameters, our systems self-assemble into dened structures ranging from dispersed particles, nite stripes to percolating networks in two dimensions. Following this structural analysis, we employ the inverse-Boltzmann iterative procedure to derive effective nanoparticle–nanoparticle interaction potentials which accurately reproduce the pair correlation functions between the inorganic cores of the full system. We show that the resulting potentials are also able to reproduce the detailed morphologies of the graed nanoparticle thin lms, suggesting a strong relationship between anisotropic structure formation and the existence of two competing length scales.
2 Model and simulation methodology 2.1
Nanoparticle coarse-grained model
In this work, polymer-graed nanoparticles are modeled as spheres with f linear oligomer attached chains. A schematic illustration of the model is given in Fig. 1. Each chain is composed of m Lennard-Jones (LJ) beads of size s, bonded to form chains via harmonic potentials V(r) ¼ k(r s)2, where k denotes the spring constant. The spring constant k was set to a high value, k ¼ 50003/s2, to ensure tangential bonding between pairs of catenated beads. To dene the energy and length scale for the model, we follow the coarse-graining scheme used by Hong et al.,29 where each bead represents an ethoxy repeat unit (–CH2OCH2O–) and interacts with other beads through a cutand-shied Lennard-Jones potential U(r): 8 > < 43 s 12 s 6 U r # 2:5s cut r r (1) UðrÞ ¼ > : 0 r . 2:5s where Ucut denotes the value of the unshied interactions at the cutoff distance of 2.5s. Parameters s ¼ 0.4 nm and 3/kB ¼ 377 K were obtained from the critical properties of the ethoxy repeat unit, estimated through the group contribution method of Constantinou and Gani.30 The LJ bead mass is set to m ¼ 44 g mol1 appropriate for the ethoxy group. The particle core diameter was set to sc ¼ 3.5s in this study, corresponding to 1.4 nm, signicantly below the diameter range of silica particles commonly used in experiments, typically between 8 and 20 nm.8,24,31 On the other hand, sc ¼ 1.4 nm matches commonly used gold particle diameters.24,32 It was not
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Soft Matter
Grafted nanoparticle coarse-grained model representations. (a) Two-dimensional schematic view. (b) Example of three-dimensional configuration, with f ¼ 12 and sc ¼ 3.5s.
Fig. 1
possible to investigate larger core diameters due to computational limitations, since many particles with attached chains need to be simulated to identify patterns and long-range order in the systems of interest. Prior attempts to simulate brush coated silica nanoparticles with large cores have been restricted to the study of one or two particles.33–36 In this work, we consider the particular case of f ¼ 12 chains tethered isotropically to the surface of the inorganic cores, corresponding to a graing density of 2 chains per s2. Three values for the polymer chain length are selected, m ¼ 20, 30 and 40. To allow fast computation, interactions between core particles are represented by a spherically symmetric, pair-wise additive potential. We used the procedure of Lee and Hua37 to determine this potential. A spherical packing unit (Si6O12) was rst chosen and assigned LJ parameters so as to accurately reproduce the known bulk properties of amorphous silica. Core–core interactions were obtained by summing up all the interacting beads which are placed to form two spherical shells of increasing diameter, in an approach similar to that used to obtain Hamaker constants. Using a numerical angle-averaging at various interparticle distances, we extract a sphericallysymmetric potential which can be used within our coarsegrained model. To perform the angle averaging, we xed the position of one nanoparticle and placed another spherical shell at a random location dened by a vector r which is subsequently rotated multiple times to random congurations. By repeating this procedure for different interparticle distances, we obtained a numerical estimate of the pair interaction between the core particles. The resulting shape of this potential depends strongly on core size, exhibiting an attractive part with shorter range and stronger attraction well depth than the underlying elementary LJ bead.37 In order to derive an analytical form for the angleaveraged pair interaction between spherical silica particles Uc(r) Lee and Hua37 advocated the use of a generalized LJ potential, sc 2l sc l (2) Uc ðrÞ ¼ 43c r r where sc approximately coincides with the particle diameter, and 3c and l are tted to the previously obtained tabulated pair potential.
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We applied this strategy for the case of small silica nanoparticles of diameter sc ¼ 1.4 nm and obtained l ¼ 7, 3c/kB ¼ 10 179K x 93/kB. For the case of gold nanoparticles of the same diameter, using the LJ potential of Kyrychenko,38 we obtained l ¼ 18 and 3c/kB ¼ 26 173K x 693/kB. Clearly, the overall shape of the coarse-grained pair potential representing the interactions between nanoparticles strongly depends on their precise chemistry. In order to maintain computational feasibility, we assumed a single shape, LJ 14-7, for the coarse-grained potential. However, we studied the effect of varying the attractive energy 3c. We considered three different values for core–core interaction: 3c ¼ 93, 3c ¼ 253 and 3c ¼ 353. In each case, the potential is cut and shied to zero at the cutoff distance of 2.5sc. The shape of the cross-interaction potential Ucp between the nanoparticle core and the polymer beads is dened by taking an arithmetic mean between attractive and repulsive exponents of the LJ 12-6 (polymer–polymer) and LJ 14-7 (core– core) potentials: 8 scp 13 scp 6:5 > < 43 Ucp r # 2:5scp cp r r Ucp ðrÞ ¼ (3) > : 0 r . 2:5scp where scp ¼ (sc + s)/2, corresponding to the widely used Lorentz–Berthelot combining rule39 and Ucp denotes the value of the unshied interactions at the cutoff distance of 2.5scp. In order to mimic the effect of graing polymer chains with different chemical compositions, the cross-interaction energy 3cp is considered as a variable parameter through the use of the pffiffiffiffiffiffi expression 3cp ¼ ð1 aÞ 3c 3: We investigated several different values of the core–polymer interaction 3cp by taking a ¼ 0, a ¼ 0.2, a ¼ 0.4, a ¼ 0.6 and a ¼ 0.8. This allowed us to span a broad region of parameter space from attractive to nearly purely repulsive cross-interactions between particle cores and polymer beads. 2.2
Simulation details
The large scale atomic/molecular massively parallel simulator (LAMMPS) was used to compute the structural behavior of the free-standing lms. All simulations were performed in orthorhombic boxes containing N ¼ 400 graed particles, under periodic boundary conditions with xed area in the xy plane. A rigid-body treatment was used between the central particle core and the rst LJ bead of each graed chain. The time evolution of the rigid body part of each particle was performed via constant NVE integration. Temperature was maintained at T ¼ 300 K using Langevin dynamics, in which the polymer chains are coupled to a heat bath through the use of a friction coefficient g and random force. The main objective of this study was to explore the structural behavior of the system, so the precise choice of the thermostat is not crucial. The equations of motion were integrated using the velocity-Verlet algorithm with a time pffiffiffiffiffiffiffiffiffiffiffiffiffi step dt ¼ 0.008s, where s ¼ ms2 =3 is the basic unit of time. To create free-standing monolayers, we started from initial congurations where the cores were placed randomly (ensuring no overlap between them) in an expanded orthorhombic simulation box with dimensions parallel to the lm set to
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Lx ¼ Ly ¼ 2000s. The dimension perpendicular to the lm was chosen to be Lz ¼ 3sc to ensure appearance of a monolayer. Polymer chains were tethered to the surface of the core particles and grown using self-avoiding random walks, with the constraint that the centers of the LJ beads are located in the interval 1.5sc < z < 1.5sc. In order to maintain the free monolayer structure during the pre-equilibration period, two impenetrable walls (truncated-shied 9-3 Lennard-Jones) were added to conne the system in the z-direction. The x- and y-box dimensions were subsequently reduced at a rate of 0.001 nm per time step, until the desired area fraction was obtained. During this initialization step, all the interactions were described solely by purely repulsive WCA potentials for the sake of computational efficiency and in order that the chains adopt stretched conformations. Once the desired dimensions of the box were reached, we switched on the full intermolecular interactions (i.e., the coarse-grained model described in the previous section) and let the system relax over 5 105dt. At this point the walls are removed so that the system is then equilibrated as a free-standing lm for an additional period of 6 106dt, which removes any effects of the initial connement. Subsequently, the measured properties are averaged over typical production runs of 2 106dt. The free-standing lms are stable over this period of time.
3 Results and discussion 3.1
Free-standing monolayer formation
For the different chain lengths of the graed particles studied in this work (m ¼ 20, 30, and 40), it was rst necessary to determine the area fractions A ¼ N/(LxLy) that led to stable free standing monolayers. We display in Fig. 2 three different simulations performed at increasing values of percentage area fraction A ¼ 13%, 23% and 27% in the particular case of systems with a graed chain length m ¼ 30, an energy parameter 3c ¼ 93, and a ¼ 0. It can be seen that for low values of the area fraction A, the monolayer is not mechanically stable and eventually develops holes aer sufficient time has elapsed. For higher values of A, free standing monolayers can be obtained where the cores arrange themselves in nearly 2-dimensional layers at the center of the lm. For enthalpic reasons, the central cores of the particles stay away from the vacuum boundaries and are surrounded by polymer beads. This can be seen in Fig. 3, where we plot the area density along the z direction for both the central core particles and the polymer beads corresponding to the thin lm displayed in Fig. 2(b). This behavior is related to the existence of a core–polymer attraction. Upon further increase of the area fraction, and hence, increase in geometric frustration, the nanoparticle cores gradually start to be displaced from the center plane of the lms. In the present study, we focus on area fractions for which the lms correspond to a stable monolayer with the particles preferentially near the center plane. We note that it is not possible to know in advance what precise threshold value of the area fraction A yields stable freestanding lms. These values depend on the graing density and polymer chain length. Film stability was found to be insensitive
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Fig. 2 Example of configurations obtained for systems with m ¼ 30, 3c ¼ 253 and a ¼ 0 at different area fractions. (a) A ¼ 13%: unstable film (b) A ¼ 23%: stable monolayer (c) A ¼ 27%: buckled monolayer. Top and bottom pictures show the top view and the side view of the films respectively. All three snapshots contain N ¼ 400 grafted nanoparticles; the linear dimensions of the three simulation boxes are different.
Fig. 3 Area density profiles of core particles, rc (red curve), and polymer beads, rp (blue curve), in the z direction for m ¼ 30, 3c ¼ 253 and a ¼ 0 with imposed area fraction A ¼ 23%.
to the value of the energy parameters. Typically, systems with short polymer chains are stable at higher values of A than particles graed with long chains. It was then necessary to perform multiple test runs for each graing chain length of interest. The goal here was not to determine a precise boundary of stability, but rather to select appropriate values of A to perform a structural study of a free standing monolayer. Based on these preliminary calculations, we selected the values A ¼ 17%, 23%, and 30% for graed chain lengths of m ¼ 20, 30, and 40, respectively.
3.2
present work. Fig. 4–6 show morphology diagrams of equilibrium congurations taken from simulations performed at different values of 3c and 3cp for each graed chain length studied. Several clearly dened structures are observed; dispersed particles, strings of width equal to one particle, stripes two or three particles wide, and clusters of particles. There seems to be a threshold value around a ¼ 0.6, at which a structural transition from dispersed particles to strings and stripes occurs. In addition, a minimum amount of core–core attraction is necessary to observe anisotropic structures. This supports the idea that the morphology of the graed particle systems results from a delicate balance between the enthalpic gain upon aggregation of the inorganic cores and the entropic gain associated with particle dispersion, mediated by the steric stabilization provided by the chains. The morphology diagrams also suggest that the main driving force for the formation of anisotropic structures lies in the inherent dislike between the central core and the polymer chains (modeled through the variation of the parameter a) and to a lesser extent to the core–core attraction. The precise architecture of the particles also affects their equilibrium spatial distribution. For the systems with chain length m ¼ 20, highly anisotropic objects are observed for a > 0.6 and typically ll the entire simulation box. In the case of longer chains, the particles form shorter strings of nite lengths. This is attributed to the increased entropic stabilization provided by the longer chains.
Anisotropic structure formation
To gain insight into the parameters that control the spatial distribution of the graed particles, we rst investigated the effects of changing both the core–core and the core–polymer attraction for different values of the polymer chain length. Other factors such as the particle core size and graing density may also play important roles, but are not considered in the
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3.3
Local structure
To further investigate the microscopic mechanisms at the origin of the 2-dimensional anisotropic aggregation of the graed particles, we study the microstructure of chains surrounding the inorganic core. The formation of “nanowires” is generally attributed to the equator region of the dimer
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Top view of equilibrium configuration of grafted particles (chain length m ¼ 20) in thin films for different values of core–core attraction 3c pffiffiffiffiffiffiffi and core–polymer attraction 3cp ¼ ð1 aÞ 3c 3: The polymer beads are not represented in other to better visualize the structures formed by the inorganic cores. Fig. 4
Top view of equilibrium configuration of grafted particles (chain length m ¼ 30) in thin films for different values of core–core attraction 3c pffiffiffiffiffiffiffi and core–polymer attraction 3cp ¼ ð1 aÞ 3c 3: The polymer beads are not represented in other to better visualize the structures formed by the inorganic cores. Fig. 5
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Top view of equilibrium configuration of grafted particles (chain length m ¼ 40) in thin films for different values of core–core attraction 3c pffiffiffiffiffiffiffi and core–polymer attraction 3cp ¼ ð1 aÞ 3c 3: The polymer beads are not represented in other to better visualize the structures formed by the inorganic cores. Fig. 6
nanoparticle surface being less shielded by the graed chains leading to an effective dipolar interaction.10,40 This phenomenon can be assessed by calculating the averaged 2-dimensional asphericity, Axy, of the graed nanoparticles. This quantity, initially proposed by Rudnick and Gaspari is dened as follows:41 * 2 + ly lx (4) Axy ¼ 2 lx þ ly where lx and ly are the squares of the main component of the 2-dimensional radius of gyration tensor Rgxy obtained by projection of the polymer bead coordinates onto the plane dened by the equation z ¼ 0. The quantity Axy has an upper bound of 1 in the limit case of a one dimensional rod-like particle and a lower bound of 0 for a molecule forming a disk shape. From the results reported in Fig. 7, it is clear that the formation of string morphologies is strongly correlated with an increase of the sphericity Axy. The effect is clear for systems with chain length m ¼ 20. The minimum value for Axy is found at low values of the core–core attraction 3c and high values of the core–polymer interaction 3cp (corresponding to low values of the parameter a). In this regime, the systems are well dispersed with negligible distortion of the chains, as one expects for systems dominated by entropic contributions to their free energy. The favorable cross-interaction leads to adsorption of the polymer chains on the ller surface creating an overall disk shape for the molecule, and hence, low asphericity. Increasing a, the degree of “dislike” between central core particles and polymer chains, leads to only
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minor increases in the asphericity for these systems. A different regime arises for 3c ¼ 253 and 3c ¼ 353. For these values of core– core attractions, both enthalpic and entropic contributions become relevant and the microscopic structure of the particles undergoes a transition around a ¼ 0.5 where a gradual increase of the asphericity is observed. This increase is over a similar range of a values to that corresponding to string and stripe patterns formation. This local rearrangement of the graed chains creating an ellipsoidal shaped particle is illustrated in Fig. 8, where we plot the projection of polymer–core distribution function on the axis dened by the eigenvectors of the gyration tensor Rgxy. It is clear from this gure that upon increasing a, the polymer beads position themselves farther away from the nanoparticle cores and the polymer chains are found to be stretched along a preferential direction leaving space for the core to aggregate. The asphericity Axy for longer chain lengths, m ¼ 30 and m ¼ 40 is also shown in Fig. 7. We note that the change in the chain microstructure with energetic parameters is lower in this case, due to larger contribution of the entropic effects. 3.4
Effective potentials
It has been shown in Section 3.3 that the self-assembly of isotropically graed nanoparticles into strings results from a balance of enthalpic and entropic effects. String formation can be thought of as being caused by a competition between shortrange attractive forces and long-range repulsion arising from dipole interactions.42,43 However, in a recent paper10 a different mechanism related to graing inhomogeneities has been
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proposed; this is likely to be relevant at lower graing densities than those considered here. Prior simulations have demonstrated that spherical particles interacting isotropically through a repulsive interparticle potential with two different length scales (the so-called core-soened potentials) are sufficient to yield spontaneous self-assembly into anisotropic structures.25,44–47 To gain insight into the link between the graed nanoparticle effective interactions and the physics of coresoened potentials, we developed effective two-dimensional coarse-grained potentials. This is done through the use of an inverse-Boltzmann procedure48–50 in which we iteratively update the effective pair interaction u(r) to match the pair correlation functions of the target system according to the following formula:
gi uiþ1 ðrÞ ¼ ui ðrÞ þ kT ln (5) gtarget
Asphericity Axy at various values of the core–core attraction 3c and core–polymer interaction a in the case of (a) m ¼ 20, (b) m ¼ 30 and (c) m ¼ 40. Fig. 7
where gi(r) is the 2-dimensional radial distribution obtained during the ith step and gtarget is the 2-dimensional distribution between the cores of the full system. The 2-dimensional MD simulations were performed at the same number density rc as the full systems. We found that 50 iterations were enough to achieve satisfactory convergence for each system. We present results of the inverse-Boltzmann procedure in Fig. 9 for systems with graed chain lengths m ¼ 20 and m ¼ 30. Since our main goal is to reproduce the anisotropic selfassembly with the derived spherical potentials, we focus our attention on the systems with strong core–core attraction, 3c ¼ 253, and study the inuence of varying the core–polymer interaction through the parameter a. For the sake of clarity, in Fig. 9 we show the effective potentials related to two different regimes: dispersed state (a ¼ 0.2) and aggregated state (a ¼ 0.6). For favorable core–polymer interactions, a ¼ 0.2, we obtain an
Fig. 8 Projection of the polymer–core distribution function on the axis defined by the eigenvectors of the gyration tensor Rgxy for the grafted nanoparticle with 3c ¼ 253 at different values of the core–polymer attraction a. Darker shades indicate lower values of the distribution function. Representative equilibrium configurations of the nanoparticle cores corresponding to each distribution functions are also displayed for the sake of clarity.
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30. A remarkable similarity between the full and spherically coarse-grained systems is obtained in each case. These results suggest that the formation of nite stripes is mainly an effect of the reduction of the area fraction of exposed cores with increasing chain length. Further work will be needed in this line to possibly develop a rigorous mapping between the parameters linked to the architecture of graed nanoparticle systems and the shape of the isotropic effective potential, and also to relate the results of the current work (with relatively high gra densities) with the previous work by Bozorgui et al.10 Effective potentials representing the interactions between particles as a function of the core–polymer attraction a, and chain length, (a) m ¼ 20 and (b) m ¼ 30.
Fig. 9
effective diameter of around 5s which conrms our previous nding that the polymer chains wrap around the nanoparticle and allow for a well-dispersed, sterically stabilized state. A more interesting scenario occurs for a ¼ 0.6, i.e., for graed nanoparticles exhibiting self-assembly. From an inspection of the effective potential, we notice the appearance of a well-dened so-core shoulder and a short ranged attractive part. In addition, it can be seen that the effective potentials derived for m ¼ 20 and m ¼ 30 are of the same general shape. Fig. 10 shows two snapshots representing equilibrium congurations calculated from the effective potentials corresponding to m ¼ 20 and m ¼
Fig. 10 Comparison of equilibrium configurations obtained with the full model and the spherically symmetric coarse-grained potentials obtained from the inverse-Boltzmann procedure. Two examples are displayed: (a) system with m ¼ 20, 3c ¼ 253, a ¼ 0.6 and (b) system with m ¼ 30, 3c ¼ 253, a ¼ 0.6. The corresponding effective potentials are reported in Fig. 9.
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4 Conclusions We have demonstrated that a three-dimensional coarse-grained model for solvent-free nanoparticles graed with polymer chains spontaneously assembles into two-dimensional lms for specic ranges of system parameters. Further, these nanoparticle monolayers show self-assembly behavior. This nding complements recent theoretical and experimental observations of structure formation for spherically-symmetric graed nanoparticles in solution and in solvent-free melts. In spite of the isotropic nature of their bare interactions, particles selfassemble into a variety of anisotropic structures, driven by the microphase separation between the inorganic core and the organic (graed) chains. The simple coarse-grained model for graed nanoparticle systems developed in this work allows sufficiently fast computation to consider the interactions of few hundreds of these graed nanoparticles, while keeping essential details related to the chain architecture and the intermolecular forces between the inorganic core and the graed polymer chains. We have examined in detail the behavior of graed nanoparticle systems with respect to variations in the relative strength of particle–particle and particle–polymer attractions, and for different graed polymer chain lengths. We found that for weak core–core attractions, entropic contributions dominate the phase behavior of the lms, and the nanoparticles stay in a dispersed state for any value of the core–polymer cross-interaction. For higher values of the core–core attractive energy, we found that a competition between enthalpic and entropic interactions arises. In this case, if there is a sufficient “dislike” between the central core and the polymer chains, the graed nanoparticles can self-assemble into stripes for which the morphology depends on the polymer chain lengths. Two different regimes were observed: percolating stripes for sufficiently short chains and string-like structures of nite size for longer chain lengths. There are therefore signicant similarities between the complex behavior of realistically modeled graed nanoparticles and the physics of radially symmetric pair potential with intermediate-range repulsive interactions, which are able to form anisotropic structures in two dimensions.25 Nevertheless, some of the morphologies observed for the earlier two-step model potentials, in particular hexagonal closedpacked ordered phases at relatively low area densities, are not observed in the systems studied in this work, even though it is likely that they would appear for larger particles and sufficiently short graed chains.
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Soft Matter
Acknowledgements
Published on 26 November 2013. Downloaded by University of Sydney on 31/01/2014 13:23:10.
This publication is based on work supported by collaborative research grant CBET-1033155 (Princeton) and CBET-1033168 (Columbia) from the U.S. National Science Foundation. We thank Dr. Alexandros Chremos for help with setting up the simulations.
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