Structure and Dynamics Properties at Interphase ... - ACS Publications

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Structure and Dynamics Properties at Interphase Region in the Composite of Polystyrene and Cross-Linked Polystyrene Soft Nanoparticle Tao Chen,† Hu-Jun Qian,*,† You-Liang Zhu,‡ and Zhong-Yuan Lu† †

State Key Laboratory of Supramolecular Structure and Materials, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, China ‡ State Key Laboratory of Polymer Physics and Chemistry, Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Changchun 130022, China S Supporting Information *

ABSTRACT: It is crucial to understand how much and how far the structure and dynamics properties of the polymer melt can be affected by incorporating nanoparticles (NPs) in the system. Here, we show that in an athermal all-polymer nanocomposites prepared from mixtures of linear polystyrene chains and cross-linked polystyrene NPs, the influence range of the NP on the melt polymer properties, namely interphase thickness, depends on the properties investigated. The local segment conformations can be affected in the range of an NP radius. However, if the interphase thickness is defined by the influence range of the NP on the radius of gyration (Rg) of the melt chain, it can be 1.5Rg for short-chain melts where the chain has a Rg smaller than particle radius Ra. For intermediate chain length polymer melts (Rg > Ra), the influence range of the NP does not change with the chain length. With the chain length further increasing, melt chains cannot feel the existence of the NP; the influence of the NP on the Rg tensors of the melt chain can be negligible. For the dynamics properties, the effects caused by the NP can be differently at different length and time scales. The mobility of the monomers can be slowed down in the vicinity of the NP, but they can have a faster mobility in the tangential direction than in the radial direction of the NP surface. More importantly, the segmental relaxation of the melt chains can be accelerated by the fast thermal deformations of the loose NP surface structures. We demonstrate that the softness of the NP and its deformability are crucial to the dynamics properties of the nanocomposites.

I. INTRODUCTION Polymer/nanoparticle nanocomposites (PNCs) are known to be very promising material due to their optimized mechanical,1−9 optical,10−14 and electrical15,16 properties. In the past decades, though many investigations have been performed in experiments17−19 and computer simulations20−28 in this field, the structure−property relationship and the mechanism of dramatic enhancements in various properties are still subject to investigation. A well-established consensus is that the PNC material properties are very much related to the dispersion state of NPs,29 which is largely determined by the entropic effect and enthalpic interactions between NPs and polymers. As in the case of inorganic NP/polymer nanocomposites, the interactions between the NP and the melt polymer chains in the interphase region can be tuned by grafting identical polymers onto the NP surface. At the same time, many investigations have shown that the delicate balance between entropic and enthalpic interactions can be influenced by many factors, i.e., NP size,20−26 grafting density,21−25,30 and graft chain length.20,23−25,30 Recent molecular dynamics (MD) simulations by Jayaraman and coworkers31 even showed that the mixing entropy of wetting the © XXXX American Chemical Society

grafted chains by the melt chains can also be influenced by the chain flexibility of both grafted and free melt chains. All these factors make the system much complex yet far from to be fully explored. For instance, there are numerous experimental works6,7,32,33 have shown that the attractive polymer−NP interactions can result in a bound layer with reduced segmental mobility in the vicinity of NPs and consequently an obvious increase in the glass transition temperature (Tg) in PNCs. However, there are also certain evidence8,34,35 that such effects on the segmental mobility and Tg by the presence of the NPs are very weak. Being different from grafted inorganic NPs, single-molecule NPs prepared by an intramolecular cross-linking process36,37 from linear polymer chain precursors are another important NP type. Mackay and co-workers synthesized polystyrene (PS) NPs by cross-linking linear PS chains into spheres.29,38,39 They mixed these NPs into linear PS chain melt. The interactions in Received: November 26, 2014 Revised: March 20, 2015

A

DOI: 10.1021/ma502383n Macromolecules XXXX, XXX, XXX−XXX

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both melt polymer and nanoparticle, distribution of the melt polymer segmental radius of gyration, and their orientation along the NP surface as well as the mean-square displacement of the melt monomers and segmental relaxations in the vicinity of the particle surface are characterized. The results of the present work are expected to be useful for understanding the origin governing the alternation of the structure and the dynamics properties induced by the loading of soft cross-linked NPs into polymer melts.

this system are much simpler. This system serves as an ideal model system where the NP and the polymer have identical composition and only the entropic effect dominates; therefore, the nuisance enthalpic interactions and others can be reduced to a minimum. In such a system, some interesting phenomena have also been reported, such as a swelling effect39 of the free melt chains and reductions in both the viscosity and the glass transition temperature38 induced by dispersion of NPs in the polymer melt. Although an earlier computer simulation40 predicted that a solid filler NP interacting weakly with polymer matrix can speed up the polymer dynamics in the vicinity of the NP and hence can cause a reduction in Tg, the situations in PNCs with the cross-linked soft NPs can be expected to be much different, especially for the properties of the NP itself. An obvious difference is that the cross-linked NPs are soft in nature and are deformable in the shape, and there are no free chain ends at the outside surface of the NP if compared with grafted NPs. In addition, traditional inorganic NPs often result in serious density distortions at the solid−liquid contact in polymer melts. However, liquid−liquid like contact is expected for the case of the cross-linked NP in polymer melt. Such differences may result in different wetting states between NPs and the melt chains. Undoubtedly, the presence of the NPs in a polymer melt can induce alternations in both the structure and the dynamics of the melt polymers at their contact region. On the other hand, the large ratio of the particle surface area (NP/polymer contact region) to the polymer volume in the PNC system requires a fully understanding on how much and how far the structure and the dynamics alternations can happen in the interphase region. To explore the interphase properties in PNCs with a silica NP, Müller-Plathe and co-workers performed both extensive fully atomistic21 and coarse-grained23,30 molecular dynamics simulations and found that the melt polymers had a layering structure and a reduced dynamics around the NP surface. Depending on the properties investigated, the interphase thickness (i.e., the distance beyond which the polymer has bulklike behavior) varied from 1 to 3 radius of gyration (Rg) of the bulk polymer. Similarly, using a Monte Carlo methodology based on polymer mean field theory, Vogiatzis and Theodorou28 investigated the structure and the wetting state of polymer layers grafted to NPs in silica/PS nanocomposites. They demonstrated that the dimensions of the grafted brush chains and its wetting behavior were mainly determined by the grafting density and the grafted chain length. All of the knowledge we learn from these simulations will help us to better manipulate the interfacial interactions between NPs and polymers and eventually to achieve designable PNC properties. However, in the case of the cross-linked PS NP composite system, despite of a few simulation studies26,41 using coarsegrained models, a detailed study of the structure and dynamics properties and their relationships at interphase region using a realistic model is still missing. In this work, employing a previously developed coarsegrained (CG) PS model42 which is capable of preserving internal structures of the PS melt, cross-linked PS NPs with different cross-linking densities and particle sizes are constructed. The structure and dynamics properties at the PS/ cross-linked PS NPs interphase region are examined by large scale coarse-grained molecular dynamics simulations. The influences of particle size, cross-linking density, and the chain length of the melt polymers are clarified. The interphase properties including interpenetrated density distributions of

II. COARSE-GRAINED MODELS AND SIMULATION DETAILS 1. Coarse-Grained Models. As mentioned above, a CG PS model developed by one of the authors42 is employed for this study. In this CG model, each styrene unit is represented by one CG bead. To account for the stereochemistry of atactic PS chains, two bead types of R and S are defined according to the absolute configuration of the styrene monomer given by the direction of the phenyl ring against the backbone. Using this model, neat PS melts with different chain lengths (N = 10−500, where N is the number of styrene units in each chain) are simulated up to 1 μs with five parallel runs for each chain length. From these simulations, we calculate the radius of gyration of the PS chains; the results are shown in Figure 1.

Figure 1. Radius of gyration of PS melt chains with different chain lengths. They are consistent with the experimental data from ref 43 and Monte Carlo simulation results from ref 28.

They are in good agreement with available experimental data43 and Monte Carlo simulation results.28 We note that due to the smoother CG energy landscape and due to the fact that friction forces are not explicitly incorporated in the equations of motion, we have an acceleration of the dynamics in the system.44−47 Experimentally, cross-linked PS nanoparticles were synthesized by an internal cross-linking process between benzocyclobutene units attached on linear PS chain in dilute solution.36 To faithfully simulate this process, the following steps are adopted to fabricate the cross-linked NPs in our simulations: (1) A single chain configuration is picked up from above wellequilibrated melt state, and then it is swollen in a solvent of PS2 dimer, which has a similar molecular size as phenyl ether used in experiments.38 After a long time equilibration run, the polymer chains are well swollen. For instance, a PS chain with N = 500 monomers (denoted by PS500 in the following) has a radius of gyration of ⟨Rg⟩ = 7.34 nm after this process, which is much larger than the melt value of 5.66 nm. (2) Starting from B


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agreement with the experimental values of 2.0 and 3.6 nm,39 demonstrating a very good reliability of our model. In our simulation, cross-linked soft NPs prepared from different parallel runs usually also have one or two cross-linking bond differences. But the structure and the dynamics properties found at interphase region in PS/NP composite are almost identical; therefore, we believe that small difference between CG and UA models presented in Figure 2 can be neglected. 2. Simulation Details. In our simulations, the initial configurations are generated at approximately a density of 0.65 g/cm3. The generated cross-linked PS NP obtained from the above procedure is first placed at the center of the simulation box; the linear PS chains are then generated around the NP by a random walk model with the bond length fixed at an equilibrium length of 0.5 nm. A random series of R or S configurations are specified along the chain with a half/half ratio. A distance threshold of rmin = 0.5 nm is used to avoid unphysical monomer−monomer overlaps in the initial configuration. In order to efficiently equilibrate the system and to release the local stress especially in the long chain systems, a designed equilibration process is performed: (1) First, local packing and chemical bonds are relaxed using a softcore nonbonded interaction potential (shown in Supporting Information Figure S1) in a short NVT run of 2 ns with a small time step of 0.5 fs. During this process the full bond potentials are used to keep the connectivity of the polymer chain with the angle potentials switched off. (2) Second, angles are slowly relaxed by adding a soft-core angle potential (shown in Supporting Information Figure S2) in an NVT simulation of 0.2 ns, where the angle potential is replaced by a plateau when the potential energy is higher than 85.0 kJ/mol. (3) After the relaxation of local packing and local bonds/angles, we switch on all the full potentials and slowly enlarge the time step (δt) from 0.5 to 5 fs in sequentially performed NVT simulations: an NVT run for 0.1 ns with δt = 0.5 fs followed by another 0.1 ns with δt = 1 fs, then for 0.5 ns with δt = 2 fs, and for 25 ns with δt = 5 fs. Finally, the system is equilibrated under NPT condition for 25 ns with δt = 5 fs. All above equilibration runs are performed at a high temperature of T = 1000 K. After the relaxation at high temperature described above, the system is addressed to an annealing process being repeatedly performed between 1000 and 500 K. Namely, the system is first slowly cooled down from 1000 to 500 K and then slowly heated back up to 1000 K. Afterward, this cooling and heating process is repeated once again and finally the system is cooled down to 500 K. The step size of the temperature change in this process is always ±50 K. For instance, the cooling process is performed gradually from 1000 K to 950, 900, 850, ..., 550, and finally 500 K. The simulation on each temperature is composed of 1 ns NVT simulation followed by 2 ns NPT simulation with an atmospheric pressure. After this equilibration process, the final density of the system will be between 0.93 and 1.00 g/cm3 depending on the matrix chain length, which is in good agreement with either atomistic 42,50 or other CG 51,52 simulations. Detailed results can be found in the next section. Finally, unless otherwise stated, at least three parallel production runs are performed for each system at 500 K at an atmospheric pressure for at least 1 μs with a time step of δt = 5 fs. All the parallel runs have totally different initial configurations generated with different random number seeds. The trajectory frames are saved every 50 ps. All the structure properties are derived from these replica trajectories. In order to analyze some short time dynamics properties, each system is

this configuration, some styrene monomers along the PS chain are randomly picked up and specified as cross-linker units. Thereafter we simulate the cross-linking process in a 40 ns NVT simulation similar to the simulation work of Liu, Mackay, and Duxbury.48 We calculate the distance between every possible pair of cross-linkers after every 0.8 ps. If any pair has a distance in the range of 0.73 ± 0.1 nm, chemical bonds will be created between them, where 0.73 nm is the equilibrium distance between the centers of mass of two cross-linked benzocyclobutene units measured from a well-optimized atomistic model. In our simulations, such cross-linking process is only possible if the two cross-linker units are at least separated by 14 normal PS units (∼2 Kuhn segments) along the chain backbone. The cross-linkers used in experiments are benzocyclobutene, which has only two more carbon atoms attached on the benzene ring. Therefore, we simply use the same nonbonded CG potential as PS monomers for crosslinker units in our simulations. This assumption is also in line with the experimental ansatz38 that the composite of polystyrene/cross-linked polystyrene NP can be served as an ideal system in which to study and delineate nanoscopic effects without the nuisance of enthalpic forces. We use a Gaussian potential49 to describe the newly formed bonds between crosslinkers. An equilibrium bond length of 0.73 nm and a bond strength similar to the normal S−S (R−R) bonds are used in our simulations. Note that the PS chains and NPs with different number of styrene units are denoted by the notations of NP and PS followed by a number indicating the size of the PS chain or NP, respectively. For instance, PS250 indicates a PS chain with 250 styrene units. NP250 is an NP with 250 styrene units. Figure 2

Figure 2. Number of reacted cross-linkers in an NP started from a linear polystyrene chain with 250 styrene units. The black squares are the results from the current work. The red dots are the results of a united atom model taken from ref 48. The blue triangles are the difference between the two models.

presents the number of reacted cross-linkers in a PS250 chain as a function of the percent of the cross-linkers after the simulated cross-linking process. The results from our coarsegrained simulations are compared with that of the united-atom (UA) model from ref 48, represented by red circles in Figure 2. Although the results are different between two models, we believe such small difference is quite reasonable considering the softness of the CG model, which apparently has a better crosslinking efficiency as compared to the UA model. Particularly, the radius of NP250 and NP1300 with 20% cross-linkers are 1.96 and 3.38 nm, respectively, which are in very good C


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NP250 containing 20% or 40% cross-linkers. Color lines indicate the results for the systems with an NP with 20% crosslinkers (denoted as 20%-NP in the following) embedded in PS melts with different chain lengths. The black line shows the results of NP250 with 40% cross-linkers in PS250 melt. Figure 3b shows the results for the system with NP1300 in PS melts with different chain lengths. The density in the bulk region is around 0.93 g/cm3 for short PS10 melt and around 1.00 g/cm3 for long chains melt. It is in very good agreement with atomistic simulations42,50 of short chains and other CG simulations with long chains.51,52 We also note that the simulated density value for long chain melt is a little (∼4%) higher if compared with the value of 0.953 g/cm3 predicted from an experimental equation of state reported in ref 54. This is because our CG model is developed based on an atomistic simulation42 of short chain PS10 melt with a density of 0.944 g/cm3. Different from the largely oscillating density profiles in the case of hard inorganic NPs,30 our soft all-polystyrene nanocomposite system has very smooth density distribution profiles. Obviously this is mainly due to the softness of the NP. A snapshot of the NP250 and the variation of the size of its N radius (⟨Ra⟩ = (1/N)∑i=1 (ri − rcm)2) are illustrated in Supporting Information Figure S3. The outskirt of the particle is clearly composed of deformable loops. On the other hand, the NP itself has a similar density since NP and PS melt have the same chemical composition as the PS melt. Therefore, the two components are easy to penetrate with each other, leading to an interphase which has smooth density profiles, where the entropy penalty experienced by the melt chain and the enthalpic interaction at the interphase region are reduced to a minimum. The results in Figure 3a also indicate that the NP will be more compact and rigid when it contains more cross-linkers. The penetration of the melt chains is much less on the surface of the NP250 with 40% cross-linkers (denoted as 40%-NP250) than that with 20% cross-linkers (20%-NP250), and the 40%NP250 has a little bit higher density in the core of NP. The radius of 20%-NP250 has a value of Ra = 1.96 nm, which almost coincides with the position (density crossover point) where the melt chain monomers and NP monomers have the same density. As shown in Figure 3b, in the case of 20%-NP1300 system, particle radius Ra and density crossover point have values of 3.38 and 3.50 nm, respectively, which are also at nearly the same position. However, in the case of the 40%NP250 system, the density crossover point (2.07 nm) is obviously larger than the particle radius (Ra = 1.83 nm), implying that the melt chains are repelled away from the NP. In the case of 20%-NP1300 system, there is obvious penetration of the short PS10 chains even into the NP core (red lines in Figure 3b), while such penetration is much less in the case of 20%-NP250 system (red lines in Figure 3a). This phenomenon again verifies that the NPs in our system are soft. Schneider et al.55 have observed similar swelling effect in a composite of cross-linked PMMA particle and linear PMMA melt. Different performance of 20%-NP1300 and 20%-NP250 may come from the different reaction pathways in the crosslinking process in which the linear precursor chain gradually collapses into an NP. Since the contact possibility of two crosslinkers separated with a long distance along the chain contour is rather small, such cross-linking process often starts with the cross-linking reactions between nearby cross-linkers along the chain contour, which has also been reported by Pomposo and co-workers.56 In the case of the 20%-NP250 system, the linear

simulated for another 50 ns at the end under NVT condition, and frames are recorded every 5 ps. The Berendsen thermostat and barostat are used for all the simulations in the present study, with a thermostat coupling time of 0.5 ps and a barostat coupling time of 5 ps. The simulated systems are listed in Table 1, which records the melt polymer chain length, number of the Table 1. Simulated Systems of One-NP Nanocompositesa NP250

NP1300

N

nchains

nmonomers

L (nm)

nchains

nmonomers

L (nm)

10 30 50 100 150 200 250 300 350 500

975 325 195 97 265 250 439 300 314 220

10000 10000 10000 9950 40000 50250 110000 90250 110150 110250

12.3 12.1 12.0 12.0 19.0 20.5 26.7 25.0 26.6 26.6

1900 623 574 437 331 300 451 366 314 270

20300 19990 30000 45000 50950 61300 114050 111100 111200 136300

15.5 15.2 17.3 19.8 20.6 21.9 26.9 26.8 26.7 28.6

a N is the melt chain length, nchains is the number of melt chains, nmonomers is the total number of styrene monomers in the system, and L is the edge size of the simulation box.

melt polymer chains, the total number of the styrene units in the system, and the final equilibrated simulation box size under NPT condition. All the CG simulations are performed with the GALAMOST simulation package.53

III. RESULTS AND DISCUSSION 1. Structure Properties at the Interphase Region. The density distribution of the constituent components at the NP/ melt interphase region is a crucial issue since it is directly related to the wetting state of the NP. Figure 3 shows the density of the styrene units belonging to both the melt chains and the NP as a function of the distance to the NP center of mass. Figure 3a shows the results for the systems with an

Figure 3. Density distribution of styrene units along the radial direction of the nanoparticle in the systems with a nanoparticle of (a) NP250 and (b) NP1300. The dashed lines are the density profile of nanoparticle monomers, the dash-dot lines are for the melt monomers with different chain lengths, and the solid lines are the total. The x-axis is the distance from the center of mass of the nanoparticle. D


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From the results in Figure 4, the bulk chain length shows little influence on the segment Rg, which is consistent with the results observed by Ghanbari et al. in the system of silica/PS nanocomposites; i.e., the systems with different chain lengths have almost the same trend.30 In Figure 4a, the Rg of segment with 5 monomers in PS10 melt is obviously larger than that of the other chain lengths. It can be attributed to a large content of free chain ends in PS10 system, which results in a relatively larger free volume and lower density. As presented in Figures 4a and 4d, the segment tends to be compact when its length is rather small, while it tends to be swollen when its length is larger, as shown in Figures 4c and 4f. The increase of the Rg is intuitive and has been observed in other works;21,23,26,30 the decrease of the Rg of the PS20 chain in melt around a grafted NP also has been observed by Ndoro et al.21 This can be interpreted in terms of the confinement effect on the rough surface of the NP. Short segments are easily trapped in some local rugged structures on the NP surface. To verify this, a hard spherical NP built with 251 highly bonded monomers (∼900 bonds in the NP) is simulated in a PS250 melt. For comparison, a pit structure is made on the surface of this NP by cutting off a crescent structure (see Figure S5 in Supporting Information). The simulation results of these two NPs in PS250 melt showed that even very short segments with 5 monomers are swollen around the spherical NP. However, the Rg of the segments with less than 15 monomers are obviously suppressed around the NP with a crescent pit, as shown in Figure S6. Since the only difference between these two systems is the pit structure on the NP surface, the above observations clearly demonstrate that the decrease of the segment Rg comes from the confinement effect induced by the NP surface pit structure. As shown in the inset of Figure S3, the surface of the NP in our simulations is very rough, and it has many pits. Based on this scenario, the fact that short segments of 5 monomers on the 20%-NP250 surface (Figure 4a) and those of 10 monomers on the 20%-NP1300 surface (Figure 4d) are confined is easy to understanda larger NP has bigger surface pit structures and thus can confine longer segments. However, when the length of the segments increases further, the pit structure can no longer hold them and the chain segments can easily escape from these confinements. Instead of being confined in small pit structures, longer chain segments prefer to adopt elongated chain conformations orienting along the surface of the NP (see the discussion below). Therefore, the segments with intermediate number of monomers are partially confined on the NP surface, having Rg values increased or decreased occasionally, as shown in Figure 4b for the segments with Nseg = 10 monomers around NP250 and in Figure 4e for the segments with Nseg = 20 monomers around NP1300. When the segments are longer enough, they are always swollen on the surface of the NP, as shown in Figure 4c for segments with Nseg = 20 monomers around NP250 and with Nseg = 40 monomers around NP1300 (Figure 4f). As have been reported by Müller-Plathe and co-workers21 for the silica/PS nanocomposites, such a swollen effect caused by NP on the local chain conformations is mainly due to the orientation of the chain segments along the NP surface. Such orientation is characterized by an orientation angle defined between the longest principal axis of the gyration tensor of the segment and the vector pointing from the NP center of mass to the segment center of mass, as shown in Figure 5. The orientation angles of the segments whose Rg values are plotted in Figure 4 are shown in Figure 6 correspondingly.

precursor chain length is rather small and there are only 50 cross-linker units along the chain. During the cross-linking process we find two reaction cores growing simultaneously, which eventually grow into a spherical NP. However, in the case of 20%-NP1300 system, the precursor chain is long enough and has totally 260 cross-linkers. At the beginning of the cross-linking process, several collapsed cores are formed simultaneously, and they eventually collapse into a relatively looser NP. Figure S4 in the Supporting Information shows the representative snapshots of the key events during the crosslinking process of both 20%-NP250 and 20%-NP1300. Another important effect caused by the presence of the NP is the perturbations on the local chain conformations around the NP. We show in Figure 4 the radius of gyration of the segments

Figure 4. Radius of gyration of the segments with (a) 5 monomers, (b) 10 monomers, and (c) 20 monomers in the NP250 system as a function of the distance to the center of mass of the nanoparticle. The figures in the right column are for the segments with (d) 10 monomers, (e) 20 monomers, and (f) 40 monomers in the system with NP1300. Black vertical dashed lines indicate the particle radius Ra; red ones indicate the influence range of the NP. The distance between the black and red dashed lines can be defined as the interphase thickness. To estimate the accuracy of the sampling, representative error bars are shown. The data in this figure are averaged over ⟨Rg2⟩1/2 values from trajectory frames. The error bars are therefore calculated via σ = [(1/n)∑ni=1(x − x)̅ 2]1/2, where x is value of ⟨Rg2⟩1/2 averaged over segments located in a specific bin in each frame and ⟨...⟩ denotes the average over segments. Note that the calculations of average orientation angles, ⟨Rg⟩, for melt chains and their corresponding error bars in next figures are carried out in a similar way; i.e., averages and error bars are calculated over discrete values from trajectory frames.

with different lengths. To improve the sampling, all the possible segments with specified segment length are included in the analysis. For instance, along a PS10 chain, there are totally 6 segments that have 5 monomers; i.e., the first to the fifth monomer is the first segment, the second to the sixth monomer is the second segment, ..., and finally the sixth to the tenth monomer is the sixth segment. E


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segments in the current investigated range of segment size; the trend is in line with the results observed in ref 21. However, the absolute values of the orientation angles in our system are much smaller than those at various PS/solid interfaces.21,57,58 To verify the underlying reason for the weaker perturbation caused by the cross-linked NP, we calculate the segmental orientation angles of matrix chains surrounding an ideal spherical hard NP which is comparable to sold inorganic NP21 model. We also calculate the matrix chain segmental orientation angles when the cross-linked NP is set as a rigid body. The results for both systems are shown in Figure S7 of the Supporting Information. In both cases, we obtain very strong orientation effects as observed at PS/solid interfaces.21,57,58 We note that both NP models are composed of our CG PS monomers and have the same CG potential; therefore, the weak orientation effect observed at the interface of PS/ cross-linked NP is mainly due to the softness of the cross-linked NP surface. Based on the above results on the segment Rg and orientation angles, the interphase thickness, i.e., the influence range of the NPs on these local properties can be roughly equal to the radius of the NP, i.e., around 2.0 nm in the case of NP250 and 3.5 nm in the case of NP1300, as indicated by the distance between the two vertical dashed lines (black and red) in both Figures 4 and 6. The above discussion is focused on the local properties of the chain segments. Figure 7 shows the radius of gyration of the

Figure 5. A cross-linked NP250 with 20% cross-linkers (in red) surrounded by several linear PS melt chains (shown in gray lines). For clarity, the other chains in the system are not shown. To depict the definition of the orientation angle, a segment with 20 monomers on the top of the NP are shown in gray beads. The arrow on the top of it depicts the longest principal axis of its gyration tensor; the other arrow is the normal vector pointing from the NP center of mass to the segment center of mass. The angle θ between these two arrows defines the orientation angle.

Figure 7. Melt chain radius of gyration ⟨Rg⟩ normalized by the corresponding pure melt values ⟨Rg⟩melt for the systems with a (a) NP250 and (b) NP1300. The tangential (solid line) and radial (dashed line) components of ⟨Rg⟩ are plotted for melt chains in the system with (c) an NP250 and (d) an NP1300 as a function of the distance to the NP center of mass. Melt polymer chain length is varied from PS10 to PS500. The left two figures are for the melts with an NP250, and the right two are for the melts with an NP1300. The binning size used for the calculation of ⟨Rg⟩ in this figure is 0.1 nm.

melt PS chains with different chain lengths (ranging from PS10 to PS500) at different distances to the NP center along the radial direction of the NP surface. The reported ⟨Rg⟩ data in Figure 7 are averaged from the instantaneous ⟨Rg⟩ values in discrete trajectory frames. In each frame, the ⟨Rg⟩ values of matrix chains are specified into corresponding bins where the chains reside. The chains found in a specific bin might be different for different frames. Figure 7a shows the results for the composite with an NP250 particle, in which the ⟨Rg⟩ values of PS chain are normalized by their corresponding pure melt

Figure 6. Orientation angles of the segments mentioned in Figure 4.

Overall, for the segments far away from the NP surface, they adopt random coil configurations and have an average orientation angle of 57.3° (see eq S1 in Supporting Information for the deduction). Near to the NP surface, the orientation angles get larger and the chain segments are oriented parallel to the NP surface. Such orientation is stronger for longer F


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Macromolecules values (⟨Rg⟩melt). Except the PS10 chain which is confined locally in rugged NP surface showing weakly suppressed ⟨Rg⟩ profiles at the NP surface, all other chains are apparently swollen in the vicinity of the NP surface. Such a swollen effect can be attributed to the alignment of the chain segments along the NP surface as discussed above. In the case of the NP1300 (Figure 7b), similar swollen effects are found for systems with melt chain length larger than 100 (from PS100 to PS500). But for short-chain PS10, PS30, and PS50 melts, due to the looser surface structure of the NP1300, the melt chains even longer than PS10; i.e., PS30 and PS50 are also confined in rugged surface, resulting in decreasing chain ⟨Rg⟩. Considering the NP radius of Ra ∼ 1.96 and 3.38 nm for NP250 and NP1300, respectively, the interphase thickness where the melt chains are swollen is around 1.0 nm in both cases. Beyond this expansion region, there is a kind of contraction well where the melt chain ⟨Rg⟩ values decrease. After this contraction well, the ⟨Rg⟩ values of PS chain slowly converge to the pure melt values, i.e., ⟨Rg⟩/ ⟨Rg⟩melt = 1.0. In order to interpret these results in more detail, we have calculated the radius of gyration tensors along the radial (⟨Rg⟩⊥) and tangential (⟨Rg⟩∥) directions of the NP surface: ⟨R g⟩ = (

1 N2

⟨R g⟩ =

∑ ∑ ((ri − rj) j>i i=1

1/2 ⎛1 2 2 ⊥ ⎞ ⎜ (⟨R ⟩ − ⟨R ⟩ )⎟ g g ⎝2 ⎠

z z MSD⊥(t , rlayer ) = ⟨δ(ri⊥ − rlayer )|r ⊥i (t ) − r ⊥i (0)|2 ⟩

(5)

1 z z (MSD(t , rlayer ) − MSD⊥(t , rlayer )) 2 (6)

where indicates the distance of the layer to the NP center and r⊥i is the radial position of the PS monomer i relative to the NP center. The system is divided into 2 nm thickness shells around the NP, and the dynamics properties are characterized in these shells. MSD⊥ and MSD∥ indicate the mobility of the monomers along the radial (⊥) and the tangential (||) directions relative to the NP surface. To eliminate the influence of the NP mobility, all the calculations of MSDs are performed in the coordinate frame of the NP center of mass (CM), namely all the monomer coordinates in eqs 4−6 are reset relatively to that of the NP CM in each trajectory frame. The results are presented in Figures 8a and 8b for PS250 melts with an NP250 and an NP1300, respectively. From the figure we clearly see that the PS monomers have an obvious heterogeneity in the diffusion along the tangential and radial directions of the NP. Especially in the vicinity of the NP, due to the hindrance of the NP the mobility of the PS monomers along the radial direction is much smaller than that along the tangential direction of the NP surface, as shown in 2−4 nm shell around NP250 and 4−6 nm shell around NP1300. A comparison of the PS monomers mobility between different shells also shows that the monomers near the NP are less mobile than those in the bulk. We have also calculated the self-scattering function of the matrix PS monomers surrounding the NP in different shells:

(1)

j>i i=1

ri + rj

(4)

rzlayer

N−1

N−1

z z MSD(t , rlayer ) = ⟨δ(ri⊥ − rlayer )|ri(t ) − ri(0)|2 ⟩

z MSD (t , rlayer )=

∑ ∑ (ri − rj)2 )1/2

⎛ 1 ⟨R g⟩ = ⎜⎜ 2 ⎝N ⊥

property. Specifically, Mackay and co-workers have shown a non-Einstein-like decrease in viscosity for the current investigated all-polystyrene nanocomposites.38 Similar results were also observed in other systems.59−61 Such variations of the dynamical properties induced by the loading of the NPs are undoubtedly related to the interphase structure and dynamics at the nanoparticle/polymer contact region. In order to explore the underlying mechanism of the viscosity reduction in this system, the local dynamics properties at the interphase region are characterized in detail in our simulations. First, the layer-resolved mean-square displacements (MSDs) of linear PS chain monomers are calculated along the tangential (||) and the radial (⊥) directions of the NP surface using the following equations:

⎞1/2

2⎟

) |ri + rj| ⎟⎠

(2)

(3)

The results are shown in Figure 7c for NP250 and in Figure 7d for NP1300. Clearly, the radial components decrease due to the presence of the NP, while the tangential components of ⟨Rg⟩ increase correspondingly. Therefore, the interphase boundary can be defined as the point where the radial and the tangential components of ⟨Rg⟩ coincide. The arrows in Figure 7 mark the positions whose distances are Ra + 1.5Rg away from the NP center of mass. Apparently for short chains with Rg smaller than Ra, interphase thickness is about 1.5Rg, for example from PS10 to PS50 in systems with NP250 (Figure 7c) and from PS10 to PS150 in systems with NP1300 (Figure 7d). It is similar to the PS grafted silica systems23,30 even though they have an interphase size of 1.0Rg ∼ 3.0Rg. With increasing PS chain length, the interphase thickness does not change anymore, meaning that the influence range of the NP reaches a maximum, as indicated by the vertical dashed lines in Figure 7c,d for NP250 (Ra = 2.0 nm) with chains of PS100 (Rg = 2.5 nm, which is larger than Ra) to PS300 and for NP1300 (Ra = 3.5 nm) with chains of PS200 (Rg = 3.6 nm, which is also larger than Ra) to PS500. With the PS chain length further increased, as indicated by PS350 and PS500 in Figure 7c for NP250, the polymer coils are much larger than the NP (Rg > Ra); they can hardly “feel” the existence of the NP, and therefore their Rg values do not change much by the addition of NP. In comparison, due to the larger size of NP1300, even the PS500 chain can “feel” the existence of the NP; an obvious difference between tangential and radial components of Rg tensors still exists (Figure 7d). 2. Dynamics Properties at the Interphase Region. The dynamics of the NP/polymer nanocomposites is an intriguing

N

S(q, t ) =

n 1 ⟨∑ exp{−iq[rj(t ) − rj(0)]}⟩ Nn j = 1

(7)

where Nn indicates the number of PS monomers in a shell. The value of q = 0.78 nm−1 = (π/4) nm−1 is chosen according to ref 24, which leads to a phase difference of π when a monomer is displaced from its position by a distance equal to the NP diameter (∼4 nm). The results are shown in Figure 9, which also verify the conclusion that the presence of the NP can reduce the mobility of the surrounding monomers, in line with the results of MSD presented above. We also have calculated the relaxation of the chain segments in 4, 6, and 8 nm thick shells surrounding the NP. This calculation is performed on finally saved 50 ns NVT trajectory frames. A chain segment is partitioned into a specific shell if the G


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Figure 10. Autocorrelation function of the end-to-end vectors of the segments with (a) 20 monomers and (b) 70 monomers in 4, 6, and 8 nm thick shells around the nanoparticle. The melt chain has a chain length of 250 monomers, and the particle is NP250. The bulk results are provided for comparison.

Figure 8. Layer-resolved mean-square-displacement of the monomers along the tangential (T) and radial (R) directions of the nanoparticle surface in PS250 melt loaded with a nanoparticle (a) NP250 or (b) NP1300. The calculations are performed in the coordinate frame of the nanoparticle center of mass.

monomers of PS chains in the PS250/NP250 nanocomposite system. In both cases with segment lengths of 20 and 70, the relaxation is found accelerated in the vicinity of the NP surface. This result is consistent with the experimental observations of the viscosity reduction in linear PS melts caused by the loading of cross-linked PS NP.38 Such acceleration of the segmental relaxation can be attributed to the fast thermal deformations of the looped surface structures of the soft nanoparticle. The same result is also obtained for PS250/NP1300 nanocomposite, as presented in Figure S8 of the Supporting Information for the accelerated relaxation of the segments with 20 monomers. To support this conclusion, we set the NP250 as a rigid body in our simulations and find that the segmental relaxation slows down as the segment comes closer to the NP, as presented in Figure S9 of the Supporting Information. This comparison suggests that the acceleration of the segmental relaxation originates from the softness and deformability of the crosslinked PS NP surface. Note that the decrease of monomer mobility obtained from mean-square displacement and the acceleration of segmental relaxation obtained from end-to-end autocorrelation function are not contradictory. The mean-square displacement characterizes the translational motion while the end-to-end autocorrelation function characterizes the rotational motion. Moreover, they depict different motion modes at different length scales. A similar conclusion that the mobility of melt chains in nanocomposites can show different behavior at different length scales has also been reported in ref 62. Comparing with atomistic or other CG simulations of PS/ silica22,24 or PMMA/silica23 interfaces, our system shows both similarities and differences. For instance, the deceleration and the directional heterogeneity of the monomer MSD in the vicinity of the NP surface found in our system are in line with these references. Such phenomena can be attributed to the steric hindrance effect caused by the presence of the NP. We note that the differences in the dynamics properties for chains

Figure 9. Self-scattering function of the matrix styrene monomers in different layers surrounding the nanoparticle in PS250 melt with a nanoparticle (a) NP250 or (b) NP1300. The calculations are performed in the coordinate frame of the nanoparticle center of mass.

middle point of its end-to-end vector is located in the shell for more than 50 ps. Afterward, the autocorrelation function of the segmental end-to-end vector is calculated in these shells. Figure 10 shows the results for the segments with 20 and 70 H


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affected by the NP differently at different length and time scales. The comparison between relaxations of segments around a soft NP and a rigid body NP with the internal degrees of freedom artificially frozen demonstrates that the NP surface deformability is crucial to the dynamics properties of the nanocomposites. Finally, the influence on local structure properties by the cross-linked soft NP investigated in this study is very similar to that of the conventional spherical (like SiO2) hard NPs. The matrix chain segments are found to be oriented along the surface of the NP except some small segments confined in rugged NP surface pit structures. In terms of the dynamics properties, the influence of the cross-linked soft NP is found to be very special. The melt polymer monomers can be decelerated in the vicinity of the NP surface, while the relaxations of segments can be accelerated by large deformation of the cross-linked soft NP surface structure, which is on the contrary if compared to the hard NP case. The results of this study are expected to shed new light on better understanding and design of the polymer/nanoparticle composites properties.

belonging in different regions shown in Figures 8−10 are relatively small. It is due to the soft nature of the cross-linked NP and the fact that the NP has the same composition as matrix polymer chains. However, acceleration found in relaxation of matrix chains caused by the large surface deformation of the cross-linked NPs is unique in our system.

IV. CONCLUSIONS Using a well-developed coarse-grained PS model, cross-linked PS NPs with different particle sizes and cross-linking densities have been built up by a random reaction process in solution between cross-linkers randomly distributed on the PS chain backbone. The resulting cross-linked PS NPs have similar particle sizes and cross-linking efficiencies as those of an atomistic model reported by Liu, Mackay, and Duxbury.48 The sizes of the resulted NPs are also in very good agreement with experiments.39 On the basis of this model, we have performed large-scale coarse-grained molecular dynamics simulations to investigate the structure and dynamics properties at the interphase region in polystyrene/cross-linked polystyrene nanocomposites. CG simulations of such all-polystyrene nanocomposites with one NP loaded in PS melts with different chain lengths are performed up to 1 μs. The influences of particle size, cross-linking density, and the chain length of melt polymers have been investigated. Overall, due to the inherent softness of the NP, the NP/ polymer interphase has a smooth density interpenetration between NP and melt polymers. With the fixed particle size, NPs with more cross-linkers are more compact and less penetrable by the melt chains. On the other hand, with the volume fraction of the cross-linkers fixed, larger NPs will have looser internal structure which will be easier for the melt chains to penetrate. Moreover, short segments of melt chains can be trapped and therefore be compressed in rugged NP surface structures. In the case of the larger NP, longer segments can be trapped since larger NP has more rugged surface structure. On the contrary, longer segments are elongated along the tangential direction of the NP surface, which is in good agreement with others’ work.21,23,26,30 In the vicinity of the NP, the melt polymer chain segments tend to orient parallel to the surface of NP, which is in line with the simulation results of the PS/silica nanocomposite system.21−23,26,30 Both the results of the segmental orientation and the segment ⟨Rg⟩ show that the influence range of NP; i.e., interphase thickness can be roughly equal to the radius of the NP. However, based on the analysis of radial and tangential components of the melt chain Rg around the NP, the interphase thickness can be ∼1.5Rg for short chain melts where the chain has a Rg smaller than the particle radius Ra, while for the longer chain melts (Rg is larger than Ra), the interphase thickness reaches a maximum and does not change with the chain length. With the chain length further increasing, melt chains cannot feel the existence of the NP; the perturbation of the NP on the melt chain Rg can be negligible even in the vicinity of the NP. We have also investigated the influence of the NP on the dynamics of the melt monomers and segmental relaxations. We find that the mobility of the monomers is slowed down due to the steric hindrance effect in the vicinity of the NP. In the meantime, they have a faster mobility in the tangential direction than that in the radial direction of the NP surface. More importantly, the segment relaxations of the melt chains are accelerated by the thermal deformations of the loose NP surface structures. Therefore, the mobility of the matrix may be

■

ASSOCIATED CONTENT

* Supporting Information S

Equation S1 and Figures S1−S9. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (H.-J.Q.). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is supported by the National Science Foundation of China (21204029, 21374043) and subsidized by the National Basic Research Program of China (973 Program, 2012CB821500). H.J.Q. and Z.Y.L. are also thankful for the support of the Jilin Province Science and Technology Development Plan (20130101020JC, 20140519004JH).

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REFERENCES

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