A Dissipative Particle Dynamics Study - American Chemical Society

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Modeling of Polyethylene and Functionalized CNT Composites: A Dissipative Particle Dynamics Study Yao-Chun Wang,† Shin-Pon Ju,*,† Huy-Zu Cheng,‡ Jian-Ming Lu,§ and Hung-Hsiang Wang† Department of Mechanical and Electro-Mechanical Engineering, Center for Nanoscience and Nanotechnology, National Sun-Yat-Sen UniVersity, Kaohsiung, Taiwan 80424, Republic of China, Department of Materials Science and Engineering, I-Shou UniVersity Kaphsiung Taiwan 840, Republic of China, National Center for High-Performance Computing, Tainan, Taiwan 74147, Republic of China ReceiVed: October 8, 2009; ReVised Manuscript ReceiVed: January 3, 2010

Dissipative particle dynamics (DPD), a mesoscopic simulation approach, has been used to investigate the effect of volume fraction, the different degree of functionalization, and the effect of PE length on the structural property of the immiscible polyethylene (PE)/carbon nanotube (CNT) in a system. In this work, the interaction parameter in DPD simulation, related to the Flory-Huggins interaction parameter, χ, is estimated by the calculation of mixing energy for each pair of components in molecular dynamics (MD) simulation. The immiscibility property of CNT and PE polymer induces the phase separation and exhibits different architectures at different volume fractions. In order to observe the effects of different degrees of functionalization, we change the repulsive interaction parameter to simulate the different degrees of functionalization. In order to observe the effect of volume fraction and different degrees of functionalization, the radius of gyration and order parameter are used to observe the arrangement of the polymer chains and CNT, respectively. We find that different degrees of functionalization will affect the final equilibrium structure significantly when the volume fraction of CNT is lower than 50%. We also find that the microstructure arrangement of PE is dependent on the equilibrium phase. Finally, for a shorter PE chain system, we only need to decrease the repulsive interaction parameter slightly in order to distribute the CNT and the PE in the system. In addition, it should be noted that different interaction parameters do not relate to any real functional group, but only present different interaction degree in the simulation, which can reflect the different chemical functionalizations. 1. Introduction Theoretical calculations and experimental measurements on carbon nanotubes have shown lots of exceptional properties in recent years. Because of these properties, CNTs have been proposed for several applications, such as sensors,1,2 their ability to store gas,3 polymer-nanotube composites,4,5 and as surfactants.6 The polymer-nanotube composites have especially attracted broad attention in the industrial and research communities.7-10 In order to use CNTs efficiently and satisfy the demand requirements, the functionalization of CNTs is an essential area of study. There are two methods to reach functionlization. The first method is noncovalent functionalization. This method utilizes different polymers adsorbed on the surface, and it can ensure that the surface structure of the CNT cannot be destroyed and can act as a solute in an organic solvent.11,12 The second method is chemical functionalization. This uses the covalent bond to link the functional group with the surface atom, improving the solubility and enhancing the interfacial adhesion.13-15 Therefore, the functionalization of CNTs in order to control the CNT-polymer composite characteristics is a growing area of research.16-19 Polyethylene (PE) is a widely used polymer material in several areas, and it comprises 20% of the plastic production in the world.20 PE has several good properties, such as excellent chemical resistance, good impact and drop * Corresponding author. E-mail: [email protected]. Fax: 886-7-5252132. Tel: 886-7-5252000Ext. 4231. † National Sun-Yat-Sen University. ‡ I-Shou University. § National Center for High-Performance Computing.

resistance properties, and high durability. The structural behavior of CNT-polymer composites has attracted broad interest in research communities,21 because they have useful mechanical properties. Consequently, PE-functionalized CNT composites should have a more complex structural behavior and display different structures. Computer simulations have been used to study polymer blend,22-26 diblock copolymer,27-29 and triblock copolymer properties.30 On a different scale, there are series of well developed simulation techniques such as the molecular dynamics (MD) and Monte Carlo (MC) methods on an atomistic level, as well as dissipative particle dynamics (DPD), lattice Boltzmann methods (LBM), and dynamic mean field theory (MF) on the mesoscopic scale. Using atomistic simulation tools, we can analyze the molecular structure and dynamic behavior of molecules. Because they are limited in the time- and lengthscale in simulation and cannot effectively prevent a configuration becoming trapped at a local minimum energy, it is difficult to observe the phase transformation process of a polymer blend and diblock copolymer system. Therefore, atom-based simulations cannot predict more realistic structures on a mesoscopic scale. For structural predictions on this scale, mesoscopic simulations such as DPD, LBM, and MF are effective methods to reflect the mixture process between two or more polymers. Because the simulation methods mentioned above are all focused on a specific scale, it is necessary to use a method to bridge the gap between atomistic and mesoscopic simulation to compensate for the insufficiency of the time- and length-scale and gather sufficient data.

10.1021/jp909644b  2010 American Chemical Society Published on Web 02/04/2010

Dissipative Particle Dynamics Study of CNT Composites Recently, the DPD method has found broad use in many areas, such as the investigation of the composites of CNT-polymer,31 the formation of micelle in the solvent,32 the shear force that induces the structural transform of the lamellar phase,33 and the viscosity property of polymer.34 For bridging the gap between atomistic and mesoscopic simulations, Groot and Warren established a link between DPD and the FloryHuggins theory for a polymer system, where the Flory-Huggins (FH) parameter (χ) of the polymer can be calculated by a microscopic simulation method. The FH parameter (χ) of PHB/ PE, PHB/PEO, PEO/PE,35 and PHB/PEO/PE25 has been calculated by the Monte Carlo simulation, and PS/PE, PE/PS, PP/ PS,36 and HDPE/LLDPE37 has been calculated by MD simulation. Based on the FH parameter, the DPD method has been used to investigate the morphology of polymer blends,36 polymer-CNT composities,16 and surfactants.17,28,38 In this study, the hierarchical procedure for bridging DPD and MD methods is used to study the effects of volume fraction, degree of functionalization, and chain length on the phase and the structural arrangement. In order to explain the effect of the functionalization, calculations of the gyration radius and order parameter are used to observe the detailed arrangement of the polymer chain and the CNT, respectively, in the PE-CNT composite system. 2. Simulation Detail DPD simulations were carried out in order to investigate the phase behavior of a PE-CNT composite. Before starting to use DPD simulations, two parameters, the compressibility parameter and the mixing energy, should be obtained from the MD simulation. These parameters cannot be used directly in the DPD simulation before they are transferred by a coarse-grain mapping procedure. After the transferring process, the repulsive interaction parameter can be obtained, which is used in the DPD simulation. Therefore, we separate the simulation detail section into three parts. The first section is the MD simulation, the second section is the coarse grain mapping, and the third section is DPD simulation. 2.1. Molecular Dynamics Simulation. Molecular dynamics simulation was carried out using the Discover and Amorphous Cell module of Material Studio 4.3 which was developed by Accelrys Software Inc. The compass potential and Andersen thermostat were used in our simulation. The time step of 1 fs was set for the time integration. To calculate the compressibility, the mixing energy, and the Flory-Huggins parameter, χ, the equilibrium structure of the CNT, PE, and the CNT-PE composite should be obtained from MD. For the pure PE system, 60 PE chains were confined to a volume of 21.6 × 21.6 × 21.6 Å3 (0.85 g/cm3), and each chain has three monomers. The system performed the NVT ensemble and equilibrated for 400 ps at 300 K. For pure CNT system, 9 (5,5)SWCNTs were placed in a volume of 38.1 × 38.1 × 24.6 Å3. The NPT ensemble was performed at 1 atm and equilibrated at 300 K for 400 ps. Thus, the density of 2.11 g/cm3 was obtained for the pure CNT system. Then, the density of 2.11 g/cm3 was set in the NVT ensemble and equilibrated for 200 ps at 300 K. For the CNT-PE composite system, the method to obtain the equilibrium structure was the same as the pure CNT system. The number of molecules and final density at each volume fraction were listed in the Table 1. The Flory-Huggins parameter χ can describe the mixing effect. The relationship between Flory-Huggins parameter χ and mixing energy shows below

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( )

χ ) Vseg

∆Emix RT

(1)

where R is the gas constant, ∆Emix is the cohesive energy density which is obtained from the MD simulation as mention in section 2.1, and Vseg is the volume of the polymer segment corresponding to the bead size in the MD simulation. Based on the Flory-Huggins theory, every bead has the same volume, and the polymer is assumed to be a chain that consists of several coarse-grain beads. In our MD simulation, the volume of PE with PE monomers is 168.35 Å3, and that of CNT is 177.63 Å3 Therefore, the volume of each bead is roughly set at 177.63 Å3 which is close to the volume of CNT and that of PE. The method of dimensionless compressibility was obtained from the slope of line form the ref 39 Hence, to obtain the according number density at different target pressure, the equilibrium structure of PE by the NVT MD simulation continues to perform the NPT simulation at different target pressure at 300K. The 200 ps of NPT MD simulation are performed to equilibrate structure of PE polymer system, and then to obtain the corresponding number density. 2.2. Coarse Grain Mapping. In the DPD simulation, the total force acting on a DPD bead i is expressed as summation over all the other bead, j, of the conservative force, a dissipative force, a random force, and a spring force. The conservative force is a soft repulsive force, where the interaction strength of this repulsive force is decided by the repulsive interaction parameter (aij). When bead i and j are the same substance, the repulsive interaction parameter is obtained from the compressibility parameter. In section 2.1, MD is used to calculate the compressibility parameter from the PE polymer system, which we then match to the DPD system’s dimensionless compressibility40

κ-1

|

) DPD

[ ]

∂pDPD 1 kBTDPD ∂FDPD

) T

[ ][ ]

∂pMD ∂pMD 1 kBTMD ∂FDPD ∂FMD Nmk-1

|

)

T

(2)

MD

where F is the number density, Nm is the coarse-graining parameter, kB is the Boltzmann constant, and T is the system temperature. In this study, the Nm is set 1 in MD section. Then the repulsive parameter (aii) of the same kind of polymers can be determined from the relationship between the aii and the dimensionless compressibility parameter which is found in a reference from Groot and Warren.41

(k-1 - 1)kBT aii ) (R ) 0.101 ( 0.001 F > 2) 2RF

(3)

It should be noticed that the equation 3 only establishes when the number density (F) is larger than 2. In order to simulate more efficiency, we choose the minima value of 3. TABLE 1: Number of Molecules and Final Density at Each Volume Fraction volume fraction PE/CNT 1/1 1/4 1/6 1/14 1/20 density (g/cm3) 1.3077 0.992 0.9276 0.8568 0.8318 number of molecules 1:14 1:30 1:48 1:70 1:100 (CNT:PE)

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Groot and Warren’s study shows that they can insert the mixing effect as ∆a into the repulsive interaction parameter aij for different kind of beads by the Flory-Huggins parameter, χ. For the case in which the reduced density F is 3, this relationship is as follows:

χ ≈ 0.286(∆a)

(4)

aij ) aii + ∆a

(5)

From ref 42, they use the repulsive interaction parameter in the DPD simulation and obtain the surface tension. However, this is not accurate because the surface tension of experimental data is a constant. Therefore, they assume that a range of ∆a has a linear variation between 15 to 115, with a χ value of 0.3 at ∆a ) 15 and a value of 0.2 at ∆a ) 115. After modifying the ∆a, the surface tension is a constant and is close to the experimental data. 2.3. Dissipative Particle Dynamics Simulation Method. In the present research, the DPD simulation method was adopted to investigate the effect of volume fraction of a PE-CNT composite on the structural property; moreover, it also compare the difference between symmetrical arrangement and asymmetrical arrangement of the triblock copolymer on the microstructure. Equations 6 and 7 only describe the condition that the DPD simulation follows Newton’s equation of motion.

db r )b Vi dt

(6)

dV b ) bf i dt

(7)

However, in a DPD simulation, all of the beads in the system are of the same volume regardless of the number of and kinds of different molecules comprising the bead. This assumption is required, because the system must conform to the Flory-Huggins χ-parameter theory.42 For simplicity, the masses of all particles in the system are normalized to 1. Equation 8 represents the fact that the total force consists of four forces. The interaction force on bead i is given by the sum of a conservative force, FCij , a dissipative force,, FDij , a random force, FRij , and a spring force, FSij.41

fi )

∑ (FijC + FijD + FijR + FijS)

(8)

j*i

where conservative force represents a purely repulsive force, dissipative force represents the friction between DPD beads that reduces velocity differences between the particles, random force works to conserve the system temperature, and the spring force is used to binding the intrapolymer beads. The second and third forces are responsible for the conservation of total momentum in the system. All of the forces act within a sphere of cut off radius rC, which also defines the system’s length scale. The conservative force with a linear approximation is given by

FijC )

{

aij(1 - rij /rc) (rij < rc) (rij > rc) 0

(9)

Figure 1. Different repulsive interaction parameters and the corresponding mixing energies.

where rij is the distance between bead i and bead j and aij is the repulsive interaction parameter, where aij is the repulsive parameter describing the interaction strength between beads. When the i material is the same as the j material, the repulsive interaction parameter is obtained from the compressibility parameter. In our DPD simulation, the cell volume is 10 × 10 × 10, and the number density of system is 3 (F ) 3). The system contains 3000 beads. It consists of 250 chains and every chain consists of 12 beads. The chain length is fixed at 12 beads at every volume fraction (including 1/1, 1/4, 1/6, 1/14, and 1/20). We can adjust the bead ratio to reach the different volume fractions. In order to describe currently the structure of CNT, the potential of bond extension and angle was performed for CNT and shown as follows:

US )

∑ 21 Cb(rb - r0b)2

(10)

∑ 21 ka(θa - θ0a)2

(11)

b

UA )

a

where Cb and ka are force constants representing the bond stretch and bond bending, respectively, and θa, rb, θ0a , and rb0 are the bending angle, the length, the equilibrium angle of the bending angle and the equilibrium length of the bond. It should be note that every PE bead is composed of 3 ethylene monomers and every CNT bead is composed of 2 unit. 3. Result and Discussion Since there are currently many methods to functionalize CNT with different functional groups,18,19 the interaction strength between CNT and polymer materials can effectively be adjusted. In order to investigate the effect of different interaction strengths between PE and functionalized CNT on the microstructure by DPD, different repulsive interaction parameters are adopted to reflect different interaction strengths between PE and the CNT which has been functionalized by different functional groups. Figure 1 shows different repulsive interaction parameters in DPD simulations and the corresponding mixing energies between PE and the functionalized CNT for fraction 1/1. The mixing energy is defined as eq 1. It can be seen that there is a linear relationship

Dissipative Particle Dynamics Study of CNT Composites


TABLE 2: Repulsive Interaction Parameter and Equilibrium Phase at Each Volume Fraction volume fraction PE/CNT aij equilibrium phase

1/1 27.7 L

1/4 26.79 Cy

1/6 24.781 Cy

1/14 26.723 Cy

1/20 26.471 Cy

between repulsive interaction parameter and the mixing energy when CNTs modified by different functional groups are mixed with the PE material. First, the mixtures of pristine CNTs and PE at different volume fractions are considered. II lists the repulsive interaction parameters and the equilibrium phases at five CNT/PE volume fractions (1/1, 1/4, 1/6, 1/10, and 1/14). For these cases, the initial repulsive parameters between polymer beads and between CNT beads are both 19.66, values lower than those PE and CNT interaction strengths given in Table 2, indicating stronger interaction between PE chains or CNTs than for the PE-CNT. The two capital letters, L and Cy, stand for the lamellae and cylinder phases. For the CNT/PE mixture at 1/1 volume fraction, both CNT and PE parts are in the lamella arrangement, and for the other four volume fractions, the aggregations of CNTs display cylindrical arrangements surrounded by the PE polymer. The snapshots of lamellae and cylindrical phases are shown in

Figure 3. CNT order parameter profiles against different repulsive parameters at five different volume fractions.

Figure 2, panels a and b, respectively. The interaction strength between PE polymer chains is stronger than that between the CNT and PE polymer, so the PE part and CNT part tend to be separated, which has been reported in previous experimental studies.43 To reach the most stable CNT arrangement, all CNT principle axes should be almost parallel to one another, which can be revealed by high order parameters over 0.95 for the pristine CNTs mixed with PE polymer at different volume fractions, as shown in Figure 3. When the fraction of CNT is higher than 50%, the PE polymer and CNT will form a lamellae phase. Otherwise, the CNTs will form the cylindrical arrangement surrounded by the PE polymers. To analyze the arrangements of CNTs and the PE chain conformations when CNT and PE are mixed at different volume fractions, the order parameter and mean square radius of gyration are used. The order parameter P2 was determined by44

P2 )

3〈cos2 θ〉 - 1 2

(12)

where P2 is the second-rank order parameter common used in the analysis of liquid crystals, and the value for cos θ is written as

cos θ ) u · n

Figure 2. Equilibrium phase of (a) lamella and (b) cylinder, respectively.

(13)

where u is a unit vector representing the long axis direction of a certain molecule in the molecular system and can be calculated from the eigenvector corresponding to the smallest eigenvalue of the moment of the inertia tensor for the specific molecule. The unit vector n stands for the director of the molecular system and can be found by diagonalizing a second-rank ordering tensor Q of the system. The angled bracket pair used in eq 12 denotes the ensemble and time average. From definition of the order parameter, eq 13, the molecular system tends to isotropic (in a disordered conformation) as P2 approaches zero, whereas behaves like a crystal (in an ordered-arranged conformation) as P2 approaches unity. Means square radius of gyration (〈Rg2〉) provides information on the mass distribution of the chain in the system, which also

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plays a central role in interpreting light scattering and viscosity measurements. If all beads have the same mass n

〈Rg2〉

∑

1 ) 〈(r - rc)2〉 n i)1 i

(14)

where ri denotes the coordinate of the particle, rc denotes the coordinate of center of mass of the polymer chain, and n is the bead number in a chain. Additionally, it can be represented as the tensor in different directions as follows: n

〈Gxy〉 )

∑

1 〈(r - rcx)(riy - rcy)〉 n i)1 ix

(15)

where rix and riy denote the position vector of the particle i and rcx and rcy denote the position vector of the center of mass of polymer chain. The three eigenvalues of G are denoted by Rg12, (major-axial, which is the largest eigenvalue) Rg22, and Rg32. The summation of Rg12, Rg22, and Rg32 is 〈Rg2〉, which can be used to roughly determine the structural arrangement of a chain in the system. Figure 3 presents the CNT order parameter profiles against different repulsive parameters at five different volume fractions. At a specific volume fraction, different DPD repulsive parameters indicate that the CNTs have been modified by different functional groups and the interaction strength between the pristine CNTs and PE polymer has therefore been adjusted. In Figure 3, the largest repulsive parameter for each profile represents the mixture of pristine CNTs and PE polymer, with the CNTs ordered by lamellae or cylindrical phase arrangement, as listed in Table 2. For the case of CNT/PE volume fraction of 1/1, different functional groups seem to have no effect on the arrangements of CNTs, because the order parameters are higher than 0.8 for all repulsive parameters and all equilibrium structures are in lamellae phases. For the other four volume fractions, three characteristic repulsive parameter ranges are seen to correspond to different CNT arrangements when CNT is functionalized by different groups. The region where the CNT order parameter is smaller than 0.3 is designated as region I, and that larger than 0.8 is regarded as region III. The region with an order parameter value between 0.3 and 0.8 is designated as region II. For convenience in comparing the results between different volume fractions, two repulsive parameters, 20.36 and 23, are used to indicate the region boundaries between the three repulsive parameter regions, designated by dashed lines. In region I, CNTs are in a disordered arrangement at the 1/4, 1/6, 1/14, and 1/20 volume fractions. Moreover, the CNT order parameters do not exhibit significant change with the increase of repulsive parameter, which can be attributed to the repulsive force between the CNT and PE at these volume fractions that is too weak to form the cylindrical phase. However, at 1/1 volume fraction, because the most stable arrangement of CNTs is aligned in parallel along the axial direction, it is easier to form the lamellae phase. In Region II, except for the 1/1 volume fraction, the order parameters of CNT generally increase significantly when the repulsive parameter increases. Further, from 1/4 to 1/20 (that is, with a relative decrease of CNT to PE), the point at which the order parameter shows significant increase (i.e., its slope rapidly increases), follows an increasing trend. For example, the point at which this occurs is 20.36, 20.56, 20.66, and 21.66 for the volume fractions of 1/4, 1/6, 1/14, and 1/20, respectively.

We name these values critical-repulsive interaction parameters, signifying where the repulsive interaction strength is large enough to allow both the PE chains and the CNTs to congregate with like material. When the repulsive parameter reaches a value of 23, the order parameter of CNT maintains a value of 0.95. At a repulsive parameter value higher than 23, the equilibrium phase remains in the cylinder phase. Figure 4 shows the congregating process of the CNT at 1/4 volume fraction. It can be clearly seen that the CNT remains distributed relatively evenly throughout the DPD system when the repulsive parameter is smaller than the critical-repulsive interaction parameter as shown in Figure 4(a). The CNTs start to congregate when the repulsive parameter becomes larger than the critical-repulsive interaction parameter, as shown in Figure 4b-d. Note that the proportion of CNTs affects the adjustment range of the repulsive parameter necessary to reach a 0.95 order parameter that forms the cylinder phase. For example, at a 1/20 volume fraction, the system has the fewest CNTs, and the repulsive parameter must be increased from 20.36 to 23 to reach the value of 0.95. However, at the 1/4 volume fraction, only an increase from 20.36 to 21.56 is required. This suggests that it is difficult to adjust the CNT order parameter by adjusting the repulsive parameter when the proportion of CNT is larger than 5% (1/ 20), and that the proportion of CNT added to reinforce composite material should then be lower than 5%. Such an addition has not only the benefits of improve the material properties through reinforcement, but also ease in controlling the structure during the process of reinforcement. This has been confirmed in several studies.8,45 The reason for this is that the distance between CNT and CNT is longer at a smaller volume fraction, and therefore the required repulsive parameter necessary for CNT to congregate is larger. In region III, all of the CNT order parameters are about 0.95, even at a higher repulsive parameter. Figures 5 and 6 show the profiles of 〈Rg2〉and the corresponding projected values of 〈Rg2〉in three major axial directions for the different repulsive interaction parameters at different volume fractions. Two dashed lines are again to indicate three different characteristic repulsive parameter regions. The circle in Figure 5 denotes the 〈Rg2〉 of the pure PE system. The three majoraxials shown in Figure 6b-d has values close to 0.62, indicating that the microstructure is spherical at Region I. Only the results for four volume fractions (1/1, 1/4, 1/6, and 1/14) are shown in Figure 6 because the results of the volume fraction of 1/20 are very similar to those at 1/14. For repulsive parameters within Region I, except for the case at 1/1 volume fraction, the values of 〈Rg2〉 are about 1.86 to 1.88, and the CNTs are in disordered arrangements, as can be seen the snapshots shown in Figure 3c. Figure 6b-d shosw that the projected values of the three major-axes are almost the same at Region I, which indicates that the microstructure of a PE chain tends to be spherical in that region, the disorder phase. This is why CNTs can move easily in the environment, as shown as Figure 7a, where the long gray rectangles represent the CNTs, and the blue spheres represent the PE chains forming spherical microstructures. Further confirmation of this comes from the small CNT order parameter. We also observe that 〈Rg2〉 of PE at 1/20 volume fraction and 〈Rg2〉 in a pure PE system are almost the same. At the 1/1 volume fraction, the equilibrium phase is the lamellae. According to the three profiles in Figure 6a, the microstructures of the PE chain are disk-like because there are two longer major-axes and one shorter major-axis. The disklike microstructure of the PE chain is preferable in forming the lamellae phase and contacting with other disk-like PE molecules.



Figure 4. Congregating process of the CNT at 1/4 volume fraction.

Figure 5. Radius of gyration for the different repulsive interaction parameters at different volume fractions.

At the interface of the PE polymer and CNTs, the disk-like microstructure of the PE chain is more stable because it will

form layers with the CNTs, as shown in Figure 7b, where the long gray rectangles represent the CNTs, and the blue disks represent the PE chains forming disk-like microstructures. Because the equilibrium phase is lamellae, and the microstructure is disk-like, the fact is that there is one shorter major-axial causes a much smaller 〈Rg2〉 than other volume fractions. The reason for the significant difference in results for 〈Rg2〉 of PE at 1/4, 1/6, 1/14, and 1/20 volume fractions is the different equilibrium phase. In region II of Figure 5, the repulsive interaction parameter is strong enough to affect the PE molecules and CNTs. Except for the 1/1 volume fraction, the equilibrium phase of other volume fractions transfer from disorder to the cylindrical phase. Moreover, the 〈Rg2〉 of PE molecules also change in this region because of the change of microstructure. From Figure 6b-d, the microstructures transfer from spherical to the ellipsoid with the increasing repulsive interaction parameter because there are two shorter major-axials and one longer major-axial. At the 1/1 volume fraction, the microstructure of PE molecules still maintains a disk-like structure and becomes thinner with an increasing repulsive interaction parameter. Comparing these two microstructures, the shorter major-axial of the disk-like one is much shorter than that of the ellipsoid. Therefore, the 〈Rg2〉 of PE at the 1/1 volume fraction is smaller than that of the others.

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Figure 6. Radius of gyration in three major axial directions for the different repulsive interaction parameters at (a) 1/1, (b) 1/4, (c) 1/6, and (d) 1/14 volume fractions.

At these volume fractions which have the cylindrical phase, 〈Rg2〉 decreases with the increasing repulsive interaction parameter, with their respective order increasing from 1/4 to 1/20. The largest 〈Rg2〉 is at the 1/20 volume fraction; as the volume fraction becomes larger, the 〈Rg2〉 of PE molecule becomes smaller. The reason for this is the number of CNTs in the system. As the number of CNTs increases, the cylinder becomes larger. The PE molecules which are affected by the repulsive interaction parameter on the interface increase, and the number of microstructures which transfer their structure from spherical to ellipsoid increases as well. From Figure 6b-d, we can observe that, for each volume fraction, the repulsive interaction parameters for microstructures which start to transfer are the same as the critical-repulsive interaction parameters. Moreover, the near-constant values of the repulsive interaction parameter of the three major-axials in region II are the same as repulsive interaction parameter for an order parameter of CNT of 0.95 at each volume fraction (from Figure 3). This indicates that the change of microstructure will be affected by the CNT arrangement. When the CNT change their arrangement, the microstructures of PE molecules also change. That is why the PE molecules at 1/4 volume fraction changes microstructure during

Figure 7. phases.

Snapshots of forming (a)disorder and (b)lamellae


Figure 8. CNT order parameter against the different repulsive interaction parameter for 4 PE molecules of different lengths at 1/14 volume fraction.

the increase of repulsive interaction parameter from 20.36 to 21. In region III, all of the 〈Rg2〉 maintain constant values and microstructure even if the repulsive interaction parameter is increased because the equilibrium phase does not change. Comparing these three regions in Figures 5 and 6, we can find that the equilibrium phase corresponds to the microstructure. For example, the microstructure at the disorder phase is spherical, and that at the cylindrical phase is ellipsoid. This phenomenon was also observed in our previous study ref 39. Although the preceding results are from mixtures of modified CNTs and PE polymers with 12 beads per chain, PE chain length will have an effect on the CNT arrangement at different repulsive interaction parameters, and will be similar for each volume fraction. There are four different lengths discussed here: 3 beads, 12 beads, 48 beads, and 96 beads per PE chain. The order parameter of CNT is calculated for different PE chain lengths at each volume fraction. Figure 8 presents the CNT order parameter against the different repulsive interaction parameter for four PE molecules of different lengths. We show only the 1/14 volume fraction because the result is similar to other volume fractions. When the length of PE is shorter, the system requires a larger repulsive interaction parameter to allow the CNT order parameter to reach the value of 0.95. In other words, we only need to decrease repulsive interaction between functional CNT and shorter PE molecules slightly more than for functional CNT and longer PE molecules, because longer polymers become intertwined easier than shorter polymers. Since the different repulsive interaction parameter represents the different degree of functionalization between functional CNT and PE, we only need to functionalize the CNT by decreasing the repulsive interaction parameter to a value of 23 and the shorter polymers will not maintain the cylindrical phase, and PE molecules and CNTs will distribute. This is why the value of the order parameter of the CNT decreases when the repulsive interaction parameter is 23 in the 3 beads PE molecule system. 4. Conclusion The DPD simulation method was carried out to investigate the phase behavior, the different degree of the functionalization, and the chain length effect of PE-CNT composite on different volume fraction. The χ-parameter, which bridges

J. Phys. Chem. C, Vol. 114, No. 8, 2010 3383 the gap between atomistic and mesoscopic simulation, was obtain by molecular dynamics simulation, and was found to be in good agreement with experimental results. For different degrees of functionalization, we observe that when the volume fraction of CNT is larger than 50% (5/5 volume fraction), the effect of functionalization is not significant. When the volume fraction of CNT is smaller than 50%, the repulsive interaction parameter is larger than critical-interaction repulsive parameter, and the CNTs will start to congregate together. If the repulsive interaction parameter is smaller than that, the CNT will distribute in the PE system. We also find that as the volume fraction of CNT is lower, it is easier to control the change of structure at different degrees of the functionalization. At each volume fraction and repulsive interaction parameter examined, the microstructure of PE depends on the equilibrium phase. For example, the lamellae phase has a disk-like microstructure of PE. In terms of PE chain length, we find that it is easier for the shorter PE chain system than for longer chain system to allow CNT and PE to distribute by decreasing the repulsive interaction parameter. Acknowledgment. The authors would like to thank the National Science Council of Taiwan for support this study, under Grant Nos. NSC98-2221-E-110-022-MY3. References and Notes (1) Zhao, Q.; Frogley, M. D.; Wagner, H. D. Polym. AdV. Technol. 2002, 13, 759. (2) Li, J.; Lu, Y.; Ye, Q.; Cinke, M.; Han, J.; Meyyappan, M. Nano Lett. 2003, 3, 929. (3) Dillon, A. C.; Jones, K. M.; Bekkedahl, T. A.; Kiang, C. H.; Bethune, D. S.; Heben, M. J. Nature 1997, 386, 377. (4) Lioudakis, E.; Kanari, C.; Othonos, A.; Alexandrou, I. Diamond Relat. Mater. 2008, 17, 1600. (5) Al-Ostaz, A.; Pal, G.; Mantena, P. R.; Cheng, A. J. Mater. Sci. 2008, 43, 164. (6) Angelikopoulos, P.; Bock, H. J. Phys. Chem. B 2008, 112, 13793. (7) Dintcheva, N. T.; La Mantia, F. P.; Malatesta, V. Polym. Degrad. Stab. 2009, 94, 162. (8) Sato, Y.; Hasegawa, K.; Nodasaka, Y.; Motomiya, K.; Namura, M.; Ito, N.; Jeyadevan, B.; Tohji, K. Carbon 2008, 46, 1509. (9) Qi, D.; Hinkley, J.; He, G. W. Model. Simul. Mater. Sci. Eng. 2005, 13, 493. (10) Kang, Y. K.; Lee, O. S.; Deria, P.; Kim, S. H.; Park, T. H.; Bonnell, D. A.; Saven, J. G.; Therien, M. J. Nano Lett 2009, 9, 1414. (11) Yang, M. J.; Koutsos, V.; Zaiser, M. J. Phys. Chem. B 2005, 109, 10009. (12) Steuerman, D. W.; Star, A.; Narizzano, R.; Choi, H.; Ries, R. S.; Nicolini, C.; Stoddart, J. F.; Heath, J. R. J. Phys. Chem. B 2002, 106, 3124. (13) Zhao, W.; Song, C. H.; Zheng, B.; Liu, J.; Viswanathan, T. J. Phys. Chem. B 2002, 106, 293. (14) Coleman, K. S.; Bailey, S. R.; Fogden, S.; Green, M. L. H. J. Am. Chem. Soc. 2003, 125, 8722. (15) Hu, H.; Zhao, B.; Hamon, M. A.; Kamaras, K.; Itkis, M. E.; Haddon, R. C. J. Am. Chem. Soc. 2003, 125, 14893. (16) Chakraborty, A. K.; Coleman, K. S.; Dhanak, V. R. Nanotechnology 2009, 20, 6. (17) Ju, S. Y.; Utz, M.; Papadimitrakopoulos, F. J. Am. Chem. Soc. 2009, 131, 6775. (18) Gao, Y.; Shi, M. M.; Zhou, R. J.; Xue, C. H.; Wang, M.; Chen, H. Z. Nanotechnology 2009, 20, 9. (19) Hu, H.; Zhao, B.; Hamon, M. A.; Kamaras, K.; Itkis, M. E.; Haddon, R. C. J. Am. Chem. Soc. 2003, 125, 14893. (20) Maraschin, N. J.; Miller, R. C. Mod. Plast. 1997, 74, 33. (21) Kyu, T.; Hu, S. R.; Stein, R. S. J. Polym. Sci. Pt. B-Polym. Phys. 1987, 25, 89. (22) Tiller, A. R.; Gorella, B. Polymer 1994, 35, 3251. (23) Fan, C. F.; Olafson, B. D.; Blanco, M.; Hsu, S. L. Macromolecules 1992, 25, 3667. (24) Lee, S.; Goo Lee, J.; Lee, H.; Mumby, S. J. Polymer 1999, 40, 5137. (25) Yang, H.; Li, Z. S.; Lu, Z. Y.; Sun, C. C. Eur. Polym. J. 2005, 41, 2956. (26) Groot, R. D.; Madden, T. J. J. Chem. Phys. 1998, 108, 8713.

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