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Self-Assembly of Organophilic Nanoparticles in a Polymer Matrix: Depletion Interactions Ssu-Wei Hu,† Yu-Jane Sheng,*,† and Heng-Kwong Tsao*,‡ † ‡
Department of Chemical Engineering, National Taiwan University, Taipei, Taiwan 106, Republic of China Department of Chemical and Materials Engineering, Department of Physics, National Central University, Jhongli, Taiwan 320, Republic of China ABSTRACT: The dispersion of nanoparticles in a polymer matrix has been proven a challenge. A recent experiment reveals that it can be controlled by the relative size of nanoparticle/matrix polymer instead of their compatibility. Dissipative particle dynamics simulations are thus employed to investigate self-assembly of organophilic nanoparticles and dispersion of organophobic nanoparticles. The degree of aggregation in terms of the mean aggregation number is evaluated to explore the aggregation kinetics of nanocubes and nanoplatelets. The influence of the length of the matrix polymer on the aggregation behavior is studied as well. It is found that the depletion attraction plays the major role for the aggregation of organophilic nanoparticles. Large nanoplatelets prefer clustering, while small nanoplatelets tend to disperse because the depletion attraction grows with the face area of nanoparticles. On the other hand, the slow aggregation kinetics, hindered by the energy barrier associated with depletion interactions and low nanoparticle diffusivity, is responsible for the low degree of aggregation for organophobic nanoparticles.
I. INTRODUCTION Polymer systems are extensively used due to their unique characteristics, being lightweight and their ease of production. As compared to metal and ceramic materials, however, polymer materials have lower mechanical strength and thermal stability. To improve the properties of polymer materials, low volume fraction of nanofillers is embedded into a polymer matrix to form nanocomposite materials. As a result, the mechanical or thermal properties of polymer matrices, such as impact strength or thermal conductivity of epoxy resin,1,2 are significantly enhanced by blending inorganic nanoparticles into polymer materials. Moreover, nanoscale dispersion of filler in the matrix may introduce new physical properties and novel behaviors that are absent in the unfilled matrices, effectively changing the nature of the original matrix. For instance, properties like high gas barrier and antiflammability are presented in polymer materials by incorporation of high aspect ratio nanoplatelets. Superior barrier properties against gas form an absolutely necessary requirement for the use of materials in packaging and storage applications. In general, thick material is required to be used to provide a barrier to various gases. However, polymer/clay nanocomposites reduce the thickness of the commercial packaging laminates and make the polymer materials more light and also transparent as the nanoscale dispersed filler would not scatter light.3,4 The performance of nanomaterials fabricated by dispersing nanoparticles in polymer melts is able to far exceed that of traditional composites, and property improvements have been achieved in a variety of nanocomposites.5,6 However, the dispersion of particles in polymeric materials has proven difficult and r 2011 American Chemical Society
frequently results in phase separation and agglomeration. Because inorganic nanofillers typically dislike an organic matrix, the nanoparticles tend to aggregate into spherical clusters instead of dispersing in polymer matrix. As a consequent, nanofillers do not bring about improvement to material properties. In the process of nanoparticle dispersion, both thermodynamic and kinetic processes play significant roles. To control the dispersion or aggregate structure of nanoparticles in a polymer matrix, various methods were adopted, such as adding dispersing agent into the nanocomposites7 and grafting polymer chains on nanoparticle surface.8 In addition, a method such as using ultramicrotome to shorten carbon nanotubes was found to improve dispersion without reducing their thermal conductivity.9 In general, two different approaches of dispersion strategies are employed to improve the degree of dispersion: surface modification and size control. On the basis of the concept of nanoparticle/polymer compatibility, organophobic nanoparticles are liable to aggregate because of the enthalpy gain associated with the reduction of contact area between nanoparticles and polymer matrix. A useful strategy for nanoparticle dispersion is therefore the surface modification of nanoparticles. Note that coating a thin film of material on the surface of nanoparticles is a general approach to tailor the surface for specific physical, optical, electronic, chemical, and biochemical properties. As the surface of the Received: November 1, 2011 Revised: December 10, 2011 Published: December 13, 2011 1789
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organophobic nanoparticle is covered with a organophilic, protected layer, it is anticipated that nanoparticles tend to disperse in the polymer matrix rather than cluster together due to the entropy gain of nanoparticles. The typical approach of surface modification is to graft the nanoparticle with polymers. Because grafting chains are usually the same as the matrix polymer, they are able to shield the nanoparticle surface from the immiscible polymer matrix.10,11 As a result, the nanoparticle becomes organophilic because of grafting of organophilic chains. Recently, an approach based on the control of the relative size of the nanoparticle and matrix polymer has been proposed.12 Three kinds of nanoparticles made of polyethylene, fullerene, and polystyrene are blended with linear polystyrene of different chain length. The polyethylene nanoparticles and C60-fullerene nanoparticles are classified as organophobic nanoparticles because both of them are known to have very limited miscibility with linear polystyrene. On the contrary, the cross-linked polystyrene nanoparticle can be regarded as organophilic nanoparticles because they are made of the same materials. The experimental observations demonstrate that the mixture of nanoparticles and polymer melt becomes phase-separated as the radius of gyration of matrix polymer is smaller than the size of nanoparticles. On the contrary, the dispersion of nanoparticles into a linear polymer melt is enhanced if the polymer radius of gyration is greater than the nanoparticle size. In other words, the degree of nanoparticle dispersion can be controlled by the nanoparticle-to-polymer size ratio regardless of the nanoparticle/polymer compatibility. This consequence reveals that under some circumstances, organophilic nanoparticles can cluster together while organophobic nanoparticles are able to disperse in a polymer matrix. Evidently, there exist discrepancies between the dispersion strategies based on nanoparticle/polymer compatibility and relative size control. In this Article, dissipative particle dynamics (DPD) simulations are adopted to investigate the aggregation of organophilic nanoparticles and the dispersion of organophobic nanoparticles. After the brief description of the system and simulation method, the results are organized as follows. First, the degree of dispersion in terms of the mean aggregation number is employed to examine the aggregation of nanoparticles in a matrix polymer. The nanoparticles are uniformly dispersed initially. The variation of the aggregation number with the time is examined, and the influence of the length of the matrix polymer on the aggregation number is studied as well. The structure of the aggregate is analyzed through the radial distribution function. Second, to examine the mechanism responsible for aggregation or dispersion of nanoparticles, the effective interparticle interaction is evaluated via a direct calculation of the force acting on two nanoparticles at a fixed distance. To show the effect of the shape of nanoparticles, both nanocubes and nanoplatelets are considered. Finally, a mechanism based on the depletion force and slow aggregation kinetics is proposed to explain the dispersion strategy of relative size control.
II. MODEL AND SIMULATION METHOD The dissipative particle dynamics (DPD) method, introduced by Hoogerbrugge and Koelman in 1992,13 is a particle-based simulation technique. This method combines some of the detailed description of Molecular Dynamics simulations, and the DPD beads obey Newton’s equation of motion:1416 dri ¼ vi , dt
dv i fi ¼ dt mi
ð1Þ
where a bead with mass mi represents a block or cluster of atoms or molecules moving together in a coherent fashion. These DPD beads are subject to soft potentials and governed by predefined collision rules. As a result, this mesoscale method allows the simulation of hydrodynamic behavior in much larger, complex systems, up to the microsecond range. The interparticle force Fij exerted on bead i by bead j is made of a conservative term (FCij ), a dissipative term (FD ij ), and a random term (FRij ). Thus, the total force acting on bead i is given by fi ¼
∑ ðFCij þ FDij þ FRij Þ
ð2Þ
j6¼ i
The sum acts over all beads within a cutoff radius rc = 1 beyond which the forces are neglected. Typically, the conservative force is represented by a soft-repulsive interaction: FCij ¼ aij ðrc r ij Þrij , for rij e rc ; 0, for rij > rc
ð3Þ
where rij is the distance between the two beads, and rij is the unit vector in the direction of the separation. The interaction parameter aij represents the maximum repulsion between bead i and j, and the dissipative force is proportional to the relative velocity, D FD ij = γω (rij 3 vij)rij, where γ is the friction coefficient, and vij is bead i with respect to bead j. The random the velocity vector of force is related to the temperature, FRij = σωRθijrij, where σ = (2γkBT)1/2 represents the noise amplitude, andθij is the random number whose average number is zero. The weight functions ωD and ωR are rij-dependent, and we choose ωD = (ωR)2 = (1 rij/rc)2 for rij e rc and 0 otherwise to satisfy the fluctuationdissipation theorem. In DPD simulations, all of the units are scaled by the particle mass m, cutoff distance rc, and thermal energy kBT. As a result, the length, time, the interaction parameter, and the number density of the system are dimensionalized by r = r0 /rc, t = t0 /(mr2c / kBT)1/2, aij = aij0 /(kBT/r2c ), and F = Ntot/(L/rc)3, where x0 denotes unit before scaled and Ntot is the total number of DPD beads within the simulation box with the length L. In our system, there are two different species of DPD beads, including beads for forming polymer matrices (S) and organophilic (or organophobic) beads for constructing nanoparticles (N). The box size in our simulations was 65 65 65 with periodic boundary conditions in the x, y, and z directions. The system density was set as F = 3 and thus Ntot = 823 875. Note that generally the box size effect is prominent for small systems. Our system is with box length = 65, which can be considered large enough to neglect the box size effect. The repulsive interactions between the same species are set as aSS = aNN = 25. We have chosen aNS = 25 for organophilic nanoparticles but aNS = 45 for organophobic nanoparticles. Note that if species i and j are fairly compatible, one has aij ≈ 25. As the incompatibility between i and j rises, the value of aij increases. A nanoparticle is modeled by connecting beads with springs, and the arrangement of DPD beads in a nanoparticle is illustrated in Figure 1. In this study, three shapes of nanoparticles are considered, including cube, sphere, and platelet. Examples of nanocube include Co3O417 and silsesquioxane,18 while examples of nanoplatelets are graphene,19 silver,20 zirconium phosphate,21 and laponite.22A nanocube is constructed from the basic unit, in which the DPD beads are arranged in a body-centered cubic (bcc) structure. For instance, a 4 4 4 nanocube is formed by repeating bcc units three times in x, y, and z directions and thereby consists of 91 beads. As long as a nanocube is built, a 1790
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Figure 1. Schematic diagrams of nanocube, nanosphere, and nanoplatelet.
nanosphere can be simply created by removing beads outside a sphere with its center coinciding with that of the nanocube. For instance, a nanosphere with its radius 3 can be obtained from a nanocube of edge length 4, as shown in Figure 1b. A nanoplatelet is constructed with 2 layers of hexagon, which consists of beads in a hexagonal arrangement.23 The size of the nanoplatelet is characterized by the diagonal span on the hexagonal face. For example, Dn denotes a nanoplatelet with the diagonal distance being n DPD beads. The nanoplatelet D5 is demonstrated in Figure 1c. The volume fraction of nanoparticles in the system is defined as jnp = NnpMnp/Ntot, where Nnp and Mnp represent the total number of nanoparticles in the system and total number of beads per nanoparticle, respectively. In this work, we have jnp = 0.022. The adjacent beads in a nanoparticle are connected via harmonic spring forces FSij represented by FSij = C(rij req)rij, where the spring constant is C = 100 and req denotes the equilibrium length with a typical value 0.7. The spring force is used to impose connectness among beads in a nanoparticle, and the choice of C and req will not influence the qualitative behavior of the system studied in our work. To further stabilize the structure of a nanoparticle, additional harmonic spring forces2426 are also imposed for nonadjacent beads in our simulations. In general, face diagonal and space diagonal forces are adopted. For example, in a bcc lattice, corner-to-corner forces and face diagonal forces are used in addition to the center-to-corner force. Similarly, for a nanocube, face diagonal forces on the six planes and space diagonal forces are introduced. The difference between these additional forces is the equilibrium length. For instance, √one has req = 0.7 for the center-to-corner force, req = 0.7(2/ 3) for the cornerto-corner force, and req = 0.7(2(6)1/2/3) for the face diagonal force. Once all forces are specified, Newton’s equation of motion was integrated using the modified velocity-Verlet algorithm with λ = 0.65. The DPD time step was set at a relatively small value Δt = 0.04 to avoid divergence of the simulation, and the total DPD steps are 1.2 106. The nanoparticles may self-assemble into a variety of superstructures, which can be characterized by the radial distribution function g(r). It describes how the number density of nanoparticles varies as a function of distance from one particular nanoparticle and is defined as g(r) = [dn(r)/4πr2Δr]/(Nnp/ V), where dn(r) represents the number of nanoparticles at a distance between r and r + Δr from a specified particle and V is the system volume. Note that the interparticle distance r is defined as the distance between the centers of mass of the two
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Figure 2. The variation of the mean aggregation number of organophilic nanoparticles with time steps in polymer matrices with different chain lengths. The result for nanospheres is shown in the inset.
nanoparticles. Moreover, to quantify the size of the self-assembly, we adopt the weight-average aggregation number, ÆPæ:
∑i Ni Pi 2 hP i ¼ ∑i Ni Pi
ð4Þ
where Pi depicts the aggregation number of ith cluster and Ni is the number of clusters with aggregation number, Pi. To understand why the organophilic nanoparticles aggregate in a polymer matrix, the mean force or the potential of mean force between two nanoparticles is calculated. As the separation between two nanoparticles are given, the average forces acting on the two particles, which keep a set of constituent DPD beads fixed at {x1...xM}, are evaluated over all of the configurations of all of the remaining DPD beads in the system with {xM+1...xN}. The definition is given by27 Z Fðx1 :::xM ÞeβUðx1 :::xN Þ dxMþ1 :::dxN Z ÆFæfx1 :::xM g ¼ ð5Þ eβUðx1 :::xN Þ dxMþ1 :::dxN The mean force exerted on a nanoparticle by the surrounding polymers is acquired by summing all of the forces acting on fixed DPD beads. For a symmetrical arrangement, the total force acting on the first particle is essentially the same as that on the second one, but they are in opposite directions. The separation-dependent interparticle force is then obtained by varying the interparticle distance. When the separation is far enough, the mean force acting on a nanoparticle vanishes.
III. RESULTS AND DISCUSSION The aggregation of organophilic nanocubes and nanoplatelets in a polymer matrix of various chain lengths is investigated by DPD simulations. The dispersion of organophobic nanoparticles is explored as well. The degree of dispersion is expressed in terms of the mean aggregation number. We focus on the relation between the effective interaction between two nanoparticles and the self-assembly behavior. The effect of the chain length of the polymer matrix on the interparticle interaction is studied. In addition, the influences of the size and shape of the nanoparticle are discussed. A. Aggregation of Organophilic Nanocubes and SelfAssembly Structure. Figure 2 shows the variation of the mean 1791
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Figure 3. Snapshots of the aggregation of organophilic nanocubes in polymer matrices with different chain lengths (Ls) at 1.2 106Δt.
Figure 5. The structure of the self-assembly of organophilic nanocubes analyzed by the radial distribution function for Ls = 12 at the time steps 1.2 106.
Figure 4. The variation of the mean aggregation number with the polymer chain length at various time steps for the organophilic nanocubes.
aggregation number ÆPæ of nanoparticles with the time steps in polymer matrices with the chain length Ls = 1, 12, and 60. Note that the length of polymer (Ls) is proportional to the degree of polymerization, which quantifies the number of monomers incorporated into the chain. The snapshots of the nanocube distribution at 1.2 106Δt are given in Figure 3. When the polymer length is small, for example, Ls = 1, the organophilic nanocubes disperse quite well in the polymer matrix, and the mean aggregation number is essentially around unity all of the time. This result is consistent with our intuition that organophilic particles like a polymer matrix therefore tend to spread in the polymer matrix to increase the particle entropy. However, for Ls = 12 and 60, the mean aggregation number is significantly greater than unity and continues growing with time. This consequence indicates that the aggregation behavior does take place for organophilic nanoparticles as the polymer chain length is significantly large. This result is somewhat surprising. Moreover, ÆPæ of Ls = 60 is always less than that of Ls = 12. It is worth noting that nanoparticles with the spherical shape do not cluster together in a polymer matrix. As illustrated in the inset of Figure 2, the mean aggregation number remains around unity all of the time for Ls = 12 and 60. This result reveals that the shape of nanoparticles plays an important role for the aggregation of organophilic nanoparticles. The disappearance of the depletion attraction for the nanosphere can be attributed to the vanishing of the depletion zone as will be discussed more in Figures 7 and 9. Evidently, the chain lengths of polymer matrices affect the aggregation behavior. Because the aggregation of nanoparticles is a time-dependent process as well, that is, ÆPæ (t, Ls), the influence of the polymer length on ÆPæ is realized at a specified time.
Figure 4 illustrates the variation of the mean aggregation number with the polymer chain length at various time steps for the organophilic nanocubes. Although the degree of aggregation rises with increasing the polymer length for shorter Ls, it decays with increasing polymer length for longer Ls. That is, there exists a maximum ÆPæ, which occurs at Ls = 12. For 4 e Ls e 12, the growth of ÆPæ with Ls further confirms that the organophilic nanoparticles can cluster together and reveals the existence of effective attractions between organophilic nanoparticles. On the other hand, the simulation result that the nanoparticles tend to reduce the tendency of aggregation as Ls > 12 seems to be consistent with the experimental observation based on the dispersion strategy of relative size control. Note that a significant degree of aggregation is still observed for Ls > 12. The aforementioned results reveal that the concept of nanoparticle/ polymer compatibility cannot be applied to the analysis of selfassembly of nanoparticles directly. The structure of the nanocube self-assembly formed in a matrix polymer with Ls = 12 can be realized from the radial distribution function. As shown in Figure 5, a lot of peaks appear in g(r) at 1.2 106Δt, and the peak height declines generally with increasing r. This result clearly suggests that there exists a packing order within the aggregate. The first peak locates at r = 3, revealing that two nanocubes often stack together in a face-toface fashion. The second peak locates at about r = 4.2 and is the result of L-shape packing. The configurations corresponding to the rest of the peaks are demonstrated in Figure 5. The structural analysis reveals that the aggregate tends to have a cuboid shape and the size is about 33, which is consistent with the mean aggregation number 26. The formation of the aggregate with the face-to-face arrangement implies the existence of the effective attraction between organophilic nanocubes. B. Effective Interparticle Interactions between Nanocubes. The effect of solvent quality is often manifested through uniform distribution or phase separation of the solutes. It is wellknown that solutes are uniformly distributed in a good solvent and phase-separated in a poor solvent. Moreover, polymers swell in a good solvent and collapse in a poor solvent. In this study, the solvent quality and nanoparticle/polymer compatibility are 1792
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Figure 6. The mean force between organophilic nanocubes is plotted against the interparticle separation for the face-to-face configuration.
reflected through the interaction parameter between a DPD bead associated with nanoparticles (N) and that associated with matrix polymers (S). As aNS = 25, the simple solute that is made of single DPD bead spreads uniformly, and the polymer made of 250 DPD beads swells substantially in the solvent (Ls = 1). On the contrary, for aNS = 45, most of the simple solutes form a large, spherical cluster with a small amount of monomers still resided in the bulk due to a limited solubility. Similarly, all polymers collapse together and form a polymer-rich phase with a spherical shape in such a poor solvent. To explain the attraction responsible for the self-assembly of organophilic nanoparticles, the effective interaction between two nanoparticles is directly calculated by DPD simulations. For simplicity, the face-to-face configuration with the separation H is considered, and the mean force acting on the two nanocubes fixed in the space is obtained for various interparticle separations. The separation H is defined as the distance from the center of the beads on the surface of the first nanocube to that of the second one. As shown in Figure 6, the mean force (Fd) is plotted against the interparticle separation (H) for aNS = 25. Note that the mean force acting on an isolated nanocube is zero and the root-meansquare force is ÆF2dæ1/2 ≈ 60, which is insensitive to the polymer length. Several interesting features are observed. First, when the two nanoparticles are in close contact before strong hard-core repulsions, an evident attractive well is seen at 0.5 j H j 1.0. It should be emphasized again that all interactions among DPD beads are purely short-ranged, repulsive forces. Significant repulsions arise as H j 0.5, representing hard-particle interactions. Second, when the two nanocubes move away from the attractive region, the repulsive force appears at the distance 1.0 j H j 1.4. Third, as the two nanoparticles are moved further apart, the magnitude of the force decays rapidly, but its oscillatory characteristic can be seen from the inset for Ls = 60. Fourth, the interaction behavior of Ls = 120 is quite similar to that of Ls = 12, and the interaction range is only a few DPD beads. Our findings reveal repulsiveattractive oscillatory depletion forces between nanocubes. Moreover, the fact that the range of interaction is a few monomer diameters regardless of polymer chain length indicates that local monomer packing correlation is the key variable, not the radius of gyration.28 Also, as we can see from Figure 6 that the first attraction well locates for 0.5 j H j 1.0, and because the center-to-center distance for two closely packed nanocubes is about 2.5, the first peak of the g(r) should situate at
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3.0 j r j 3.5, which corresponds quite well to the result shown in Figure 5. The effective attraction between organophilic nanoparticles in polymer melts is essentially caused by the depletion of polymer beads in the near-contact region of nanoparticles and can be called the depletion interaction. It is generally referred to the force between two colloidal (hard) particles in the presence of nonadsorbing polymers.29 Around each nanoparticle, there is a depletion region unavailable to polymer beads because of the hard-core potential. The physical origin of the depletion force is that the overlap of the restricted volumes (depletion zone) associated with the two nanoparticles increases the volume accessible to polymer beads. As a consequence, the entropy of the polymer matrix is increased (ΔS > 0) and the internal energy change associated with the contact of two organophilic nanoparticles is negligible. Thus, their free energy decreases according to the contribution TΔS, resulting in an attractive interaction, which is essentially an entropic force. The behavior of depletion interaction for polymer melts or concentrated solutions is very different from that for semidilute or dilute polymer solutions. While the monotonic decaying behavior is observed in the latter case, the depletion interaction is generally oscillatory with attractive and repulsive components associated with local monomer packing correlations in the former case.28,29 Our simulation results of nanocubes are qualitatively consistent with the previous work of spherical nanoparticles by Hooper et al.28 The potential of mean force is the average work needed to bring the two particles from infinite separation to a distance H. It can be simply obtained by the integration of the mean force with respect to the interparticle separation: Ud ðHÞ ¼
Z H ∞
Fd ðHÞ dH
Similar to the well-known DLVO potential for colloid aggregation, the primary minimum (well) and primary maximum (barrier) exist in the depletion potential. The magnitudes of the energy well and barrier for Ls = 12 are estimated as 24.4 and 7.7 kBT, respectively. For charged colloids, the primary minimum originates from van der Waals attraction, while the primary maximum comes from electrostatic double layer repulsion. However, for organophilic nanoparticles, the primary minimum and primary maximum emerge all from the depletion interaction and local monomer packing correlations. Note that the energy well and barrier for Ls = 1 are calculated as 4.9 and 4.5 kBT, respectively. Such a weak depletion force explains why nanocubes can disperse well in a polymer matrix of small molecular size. Because the nanocube is not spherically symmetry, the depletion interaction depends on the relative orientation between the two nanocubes, in addition to the interparticle distance. To understand the effect of relative orientation on the interparticle interaction, the other two configurations besides the face-to-face configuration are considered, including the face-to-corner and corner-to-corner configurations. Following the same method for the face-to-face configuration, the mean forces are calculated for the other two configurations. As shown in Figure 7, the depletion interactions for these two configurations are very insignificant in comparison with that of the face-to-face configuration. The disappearance of the depletion attraction can be attributed to the vanishing of the depletion zone in the vicinity of the corner of the nanocube. This result explains why the secondary structure in 1793
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Figure 7. The depletion forces calculated for three different configurations of nanocubes: face-to-face, face-to-corner, and corner-to-corner, in a polymer matrix (Ls = 12).
Figure 8. The variation of the mean aggregation number with the time steps for nanoplatelets D5, D7, and D9 in a polymer matrix Ls = 12.
the aggregate prefers the face-to-face configuration if the nanocube is regarded as a primary structure. C. Self-Assembly of Organophilic Nanoplatelets and Depletion Attraction. While organophilic nanocubes self-assemble in a polymer matrix, the aggregation of nanospheres is not evident as illustrated in the inset of Figure 2. This result indicates that the depletion attraction, which is responsible for aggregation of organophilic nanoparticles, depends on the shape of the nanoparticle. To examine the effect of the shape on the depletion interaction, we consider nanoplatelets in a polymer matrix. Figure 8 shows the variation of the mean aggregation number with the time steps for different sizes of organophilic nanoplatelets with the same thickness. Note that the size of the nanoplatelet follows the order D9 > D7 > D5. For smaller nanoplatelets such as D5 in a polymer matrix with length Ls = 12, the mean aggregation number is around unity, and this indicates that the depletion attraction is too weak to cause aggregation. The snapshot of D5 distribution is shown in the lower part of Figure 8. On the other hand, for larger nanoplatelets like D7 and D9, the mean aggregation number grows with time. The snapshot of D7 aggregation is also illustrated in the upper part of Figure 8. Obviously, larger nanoplatelets cluster, while smaller nanoplatelets
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Figure 9. The variation of the depletion force with the separation between two organophilic nanoplatelets in a polymer matrix with Ls = 12 for the face-to-face configuration.
disperse. The rod-like self-assembly takes place as the face area of the nanoplatelet is large enough. Evidently, the formation of aggregates of D7 and D9 indicates that there must exist interparticle attractions between organophilic nanoplatelets. Figure 9 shows the variation of the interparticle force with the separation between two nanoplatelets in a polymer matrix with Ls = 12 for the face-to-face configuration. Three different sizes of nanoplatelets are considered, and the face area rises with increasing diagonal distance. Similar to the shortrange interparticle force associated with nanocubes, the attractive forces occur at 0.7 j H j 1.1, and the repulsive forces arise at H j 0.7 and 1.1 j H j 1.5 for nanoplatelets. At the same separation H, the effective attraction and repulsion grow with the size of the nanoplatelet, that is, Fd(D9) > Fd(D7) > Fd(D5). In terms of the potential of mean force, the energy wells are 31.8, 75.2, and 119.7 kBT for D5, D7, and D9, respectively. The energy barriers are 11.7, 31.5, and 53.7 kBT, respectively. The energy well of larger nanoplatelets, that is, D7 and D9, is deep enough to stabilize the stacked structure in a polymer matrix. Note that the interaction energy of depletion force is proportional to the product of the number density, thermal energy, and depletion volume change. As the face area of a nanoplatelet is increased, its depletion zone rises as well. Consequently, the overlap of the depletion zones associated with the nanoplatelet faces leads to the decrease of the depletion volume and thereby increases the polymer entropy. In a polymer matrix with Ls = 12, such an entropy force is weak for D5 but strong for D7 to form rod-like aggregates. However, for a polymer matrix with Ls = 1, the entropy force is too weak to drive the cluster formation even for D9. In addition to the snapshot, the self-assembly structure of organophilic nanoplatelets due to depletion attractions is further realized by the radial distribution function. As illustrated in Figure 8 for nanoplatelets D7 in polymers with Ls = 12, nanoplatelets tend to stack regularly in a face-to-face manner, like coin stacking. In other words, nanoplatelets aggregate into rod-like aggregates. In this case, the radial distribution function g(r) at 1.2 106Δt exhibits periodic peaks as shown in Figure 10. The peak located at r = 1.5 corresponds to the two contact nanoplatelets, and the peak at r = 3.0 corresponds to the two nanoplatelets separated by one nanoplatelet. Similarly, the peaks at r = 4.5 or 1.5m denote the two nanoplatelets separated by two 1794
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Figure 10. The structure of the self-assembly of nanoplatelets (D7) analyzed by the radial distribution function for Ls = 12 at the time steps 1.2 106.
or (m 1) nanoplatelets. The small peak at r ≈ 9 reveals that the nanorod consists of about 7 nanoplatelets on average. If the aggregate is rod-like, then the peak value is expected to decline monotonically with increasing separation. Nonetheless, it is interesting to note that the second peak is lower than the third peak as shown in Figure 10. The extra contribution to the third peak comes from the edge-to-edge configuration, of which the center-to-center distance is about 4.34. Because the edge-to-edge interaction simply between two neighboring nanoplatelets is weak, the large cluster of nanoplatelets is driven by the edgeto-edge depletion attraction between two nanorods. That is, as the time proceeds, nanorods form first, and the aggregate of nanorods appears later on. This is confirmed by following the dynamics of the simulation. As demonstrated in Figure 8, a large cluster contains about 4 nanorods and thereby about 28 nanoplatelets. This result is consistent with the mean aggregation number 27. D. Dispersion of Organophobic Nanoparticles. According to nanoparticle/polymer compatibility, organophobic nanoparticles tend to cluster together to reduce their contact with polymer matrix. In general, longer matrix polymers can lead to higher degree of aggregation due to depletion forces. However, it is somewhat counterintuitive that for the same time period, the mean aggregation number in the polymer matrix of Ls = 1 is significantly greater than that in the polymer matrix of Ls = 60. In fact, the mean aggregation number declines with increasing chain length of matrix polymers as illustrated in Figure 11 at 4 105Δt. The snapshots of the systems are shown in the inset of Figure 11. This simulation result agrees qualitatively with experimental observation and implies that the degree of nanoparticle dispersion may be enhanced by increasing the matrix polymer length. In other words, it is consistent with the dispersion strategy of relative size control, that is, dispersion by making the polymer radius of gyration greater than the nanoparticle size, which seems to be irrelevant to the nanoparticle/polymer compatibility. To explore the mechanism of aggregation, the effective interaction between two organophobic nanocubes is calculated. A strong interparticle attraction is observed due to both organophobicity and depletion. That is, the energy well associated with organophobic nanoparticles (aNS = 45) is substantially deeper than that associated with organophilic nanoparticles (aNS = 25). The former is about 111 kBT, while the latter is about
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Figure 11. The variation of the mean aggregation number of organophobic nanocubes with the polymer length at the time steps 4 105. In the inset, aggregative snapshots for polymer lengths Ls = 1, 12, and 60 are shown.
24 kBT for Ls = 12. On the basis of such strong interparticle attractions, organophobic nanoparticles will cluster together eventually. In contrast to entropy-driven aggregation for organophilic nanoparticles, one expects enthalpy-driven clustering for organophobic nanoparticles. Similar to the depletion force of organophilic nanoparticles, there exists an energy barrier due to local monomer packing correlation, in addition to the energy well. Such an energy barrier is enhanced by organophobicity and can be as large as 25.1 kBT for Ls = 60. Evidently, the interparticle attraction is unable to explain why the degree of dispersion grows with Ls. If the formation of aggregates of organophobic nanocubes is thermodynamically more favorable than dispersion, then their small degree of aggregation must be attributed to kinetic resistance to clustering. Our simulations show that the Brownian diffusivity of nanoparticle (D) declines with increasing particle size and polymer chain . length (Ls). According to our simulations, one has D ≈ L1/2 s Experimentally, the polymer melt viscosity dependence on the degree of polymerization is found to follow the power law with the exponent about 3.3 due to chain entanglement, which is absent in DPD simulations. This result indicates that the Brownian motion of nanoparticles in a polymer matrix decays rapidly as the polymer length is increased. Therefore, the collision rate is hindered. Moreover, the energy barrier also rises with increasing Ls. Because of these two factors, dispersion of organophobic nanoparticles observed by experiments may be explained by very slow aggregation kinetics controlled by a small diffusion constant and high energy barrier. The longer is the polymer length, the lower is the degree of aggregation. If the aforementioned scenario is correct, then the degree of nanoparticle dispersion should be dependent on the processing procedures associated with solidification (curing) of polymers. If the solidification procedure is fast enough, then there is no time for nanoparticles to cluster. Thus, nanoparticle dispersion can be obtained because of low degree of aggregation. On the contrary, as the solidification procedure is very slow, nanoparticles have a sufficient time for collision and crossing the energy barrier. Consequently, large clusters are observed, and a high degree of aggregation is resulted. In fact, it has been reported that the degree of aggregation of organophobic C60 fullerene nanoparticles 1795
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The Journal of Physical Chemistry C in linear polystyrene relies on the processing method.30 Initially, the uniform dispersion is prepared by blending fullerence nanoparticles and polystyrene in a common good solvent (toluene). When drops of the dispersion are added into methanol, coprecipitation of polystyrene and nanoparticles in a few seconds leads to homogeneous blends. On the other hand, as the dispersion is cast onto a glass slide, drying at room temperature for 3 days yields large phase-separated domains. Evidently, the difference between the two processing procedures is the time of solidification. Dispersion is observed for the rapid precipitation method, while aggregation is seen for the slow solvent evaporation method. This experimental result confirms our model of nanoparticle dispersion.
IV. CONCLUSION The properties of polymer materials can be greatly improved by the incorporation of nanoparticles into polymer matrices. Because inorganic nanofillers typically dislike an organic matrix, the nanoparticles tend to cluster together. Therefore, the dispersion of nanoparticles in a polymer matrix has been proven a challenge. Recently, an approach based on the control of the relative size of the nanoparticle and matrix polymer has been proposed to obtain nanoparticle dispersions. The experimental observation reveals that organophilic nanoparticles with their size greater than the radius of gyration of polymers can cluster together. On the contrary, organophobic nanoparticles with their size less than the radius of gyration of polymers are able to disperse. This consequence is contradictory to the general belief that organophilic particles tend to spread in a polymer matrix, while organophobic particles are liable to aggregate. That is, the nanoparticle/polymer compatibility is not relevant for nanoparticle dispersion. To explore the mechanism of the aggregation of organophilic nanoparticles and the dispersion of organophobic nanoparticles, DPD simulations are performed to study self-assembly of nanocubes and nanoplatelets. The degree of aggregation in terms of the mean aggregation number is evaluated to monitor the aggregation kinetics. The influence of the chain length of the polymer matrix on the aggregation behavior is investigated as well. The self-assembly of organophilic nanocubes is clearly observed. As the length of the matrix polymer is increased, the mean aggregation number rises for shorter polymers, but it decays for longer polymers. The interparticle interaction between two nanoparticles is directly calculated, and it is found that the depletion attraction plays the major role for the aggregation of organophilic nanoparticles. Polymer-induced depletion interactions exhibit the repulsiveattractive oscillatory characteristic, and the range of interaction is a few monomer diameters regardless of polymer chain length. This result reveals that local monomer packing correlation is the key variable, not the radius of gyration. To investigate the effect of the nanoparticle shape on the aggregation and depletion forces, nanoplatelets in a polymer matrix are considered. It is found that large nanoplatelets cluster, while small nanoplatelets disperse. This is because the depletion attraction grows with the face area of nanoplatelets. If organophilic nanoplatelets are regarded as a primary structure, the formation of nanorods induced by the depletion force associated with the face-to-face configuration can be considered as a secondary structure. The edge-to-edge depletion attraction between nanorods is further invoked to form the tertiary structure of the nanoplatelet self-assembly. In summary, the aggregation of
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organophilic nanoparticles is simply driven by the entropy gain of polymer matrices. Although organophobic nanoparticles are anticipated to aggregate based on nanoparticle/polymer compatibility, our simulations indicate that the mean aggregation number of organophobic nanocubes at any specified time period declines with increasing the length of matrix polymers. That is, the longer is the polymer length, the lower is the degree of aggregation. This result is consistent with the dispersion strategy of relative size control but cannot be elucidated by the strong interparticle attraction caused by organophobicity and depletion. The latter also results in a high energy barrier in addition to the deep energy well. Because the nanoparticle diffusivity grows about cubically with the polymer length, the experimental finding of nanoparticle dispersion by long matrix polymers may be explained by very slow aggregation kinetics controlled by small diffusion constant and high energy barrier. As a consequence, the degree of nanoparticle dispersion should be dependent on the processing procedures associated with solidification (curing) of polymers. In fact, the nanoparticle aggregation observed in the solvent evaporation method (slow procedure) but the dispersion seen in the rapid precipitation method (fast procedure) confirm our prediction.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected] (Y.-J.S.),
[email protected] (H.-K.T.).
’ ACKNOWLEDGMENT This research work is supported by the National Science Council of Taiwan. Computing time, provided by the National Taiwan University Computer and Information Networking Center for High-Performance computing and National Center for High-performance Computing (NCHC), is gratefully acknowledged. ’ REFERENCES (1) Allaoui, A.; Bai, S.; Cheng, H. M.; Bai, J. B. Compos. Sci. Technol. 2002, 62, 1993–1998. (2) Clancy, T. C.; Gate, T. S. Polymer 2006, 47, 5990–5996. (3) Koo, J. H. Polymer Nanocomposites: Processing, Characterization, and Applications; McGraw-Hill: New York, 2006. (4) Chung, D. D. L. Composites Materials: Functional Materials for Modern Technologies; Springer: New York, 2003. (5) Geng, H.; Rosen, R.; Zheng, B.; Shimoda, H.; Fleming, L.; Liu, J.; Zhou, O. Adv. Mater. 2002, 14, 1387–1390. (6) Park, S. H.; Poy, A.; Beaupre, S.; Cho, S.; Coates, N.; Moon, J. S.; Moses, D.; Leclerc, M.; Lee, K.; Heeger, A. J. Nat. Photonics 2009, 3, 297–303. (7) Vestberg, R.; Piekarski, A. M.; Pressly, E. D.; Vanberkel, K. Y.; Malkoch, M.; Gerbac, J.; Ueno, N.; Hawker, C. J. J. Polym. Sci., Part A: Polym. Chem. 2009, 47, 1237–1258. (8) Li, Q. F.; Xu, Y.; Yoon, J. S.; Chen, G. X. J. Mater. Sci. 2011, 46, 2324–2330. (9) Wang, S.; Liang, R.; Wang, B.; Zhang, C. Carbon 2009, 47, 53–57. (10) Yan, S.; Yin, J.; Yang, Y.; Dai, Z.; Ma, J.; Chen, X. Polymer 2007, 48, 1688–1694. (11) Akcora, P.; Liu, H.; Kumar, S. K.; Moll, J.; Li, Y.; Benicewicz, B. C.; Schadler, L. S.; Acehan, D.; Panagiotopoulos, A. Z.; Pryamitsyn, V.; 1796
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