Self-Assembled Morphologies and Percolation Probability of Mixed

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Self-Assembled Morphologies and Percolation Probability of Mixed Carbon Fillers in the Diblock Copolymer Template: Hybrid Particle-Field Molecular Dynamics Simulation Ying Zhao, Giuseppe Milano, Yue Cong, Naisen Yu, Yangyang He, Yan Cong, Qing Yuan, and Bin Dong J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.5b04491 • Publication Date (Web): 13 Oct 2015 Downloaded from http://pubs.acs.org on October 20, 2015

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The Journal of Physical Chemistry

Self-Assembled Morphologies and Percolation Probability of Mixed Carbon Fillers in the Diblock Copolymer Template: Hybrid Particle-Field Molecular Dynamics Simulation 1*

Ying Zhao , Giuseppe Milano2,3, Yue Cong1, Naisen Yu1,Yangyang He1, Yan Cong1, Qing Yuan1, and Bin Dong1* 1

Institute of Nano-Photonics, School of Physics and Materials Engineering, Dalian Nationalities University, No. 18, Liaohe West Road, Dalian Economic & Technical Development Zone, Dalian 116600, China

2

Dipartimento di Chimica e Biologia and NANOMATES, Research Centre for NANOMAterials and nanoTEchnology at Università di Salerno, I-84084 via Ponte don Melillo Fisciano (SA), Italy 3

IMAST Scarl-Technological District in Polymer and Composite Engineering, P.le Fermi 1, 80055 Portici (NA), Italy

1*

Corresponding authors:

Tel.: +86-411-87658872; Fax: +86-411-87656331. Email address: [email protected] (Ying Zhao) ; [email protected] ( Bin Dong) .

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ABSTRACT The self-assembly of polymer composites of mixed carbon fillers including single walled carbon nanotube (SWCNT) and carbon black nanoparticles (CB NPs) in diblock copolymer (BCP) template are investigated using hybrid particle-field molecular dynamics simulation in this work. Simulations show, in agreement with experiments, the composites of BCP template with SWCNT has lower percolation threshold than that of BCP template with CB NPs. Moreover, the ratio between SWCNT and CB NPs has a strong influence on the percolation threshold of composites. The results of probability of percolation show that adding more SWCNT (compared with CB NPs) to the BCP template could decrease the percolation threshold. However, a synergistic effect of percolation of the mixed carbon fillers in BCP template has been found. In particular, a nonlinear relation following Boltzman function has been found and lowest percolation threshold exists with the volume ratio 4:1 (SWCNT: CB NPs) compared with the volume ratio of 1:1, 2:1, and 8:1 (SWCNT: CB NPs). The mixed carbon fillers also affect the morphologies of the BCP template, and the calculated radius of gyration of BCP shows that in a higher concentration of the mixed fillers, the stretching of BCP is stronger which results in the deformation of BCP template.


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1. Introduction Composites of block copolymers (BCP) with carbon-based fillers (e.g., carbon nanotubes (CNTs), carbon black nanoparticles (CB NPs), and graphite) have received tremendous attention in both academe and industry due to the applications in modern devices such as sensors, organic electronics, and nanolithography, etc.1-4 In comparison to homopolymer or polymer blends, BCP are more favorable due to their ability to self-assemble into various equilibrium structures and allow a better control of the distribution of nanofillers.5-8 Basically, the distribution of nanofillers in a polymer matrix significantly affects desirable properties in order that a more homogenous dispersion accompanying with strong interfacial interactions between the fillers and polymer matrix results in higher mechanical properties. On the contrary, an increase in the percolation probability causes the aggregation of nanofillers in the matrix, which forms a path for electron flow through the filler/polymer composite and enhances the electrical conductivity9-10 and dielectric permittivity. 11-14 Until now, the reported work primarily involves a single type of carbon filler in BCP matrix, especially CNTs-BCP composites15-18 due to excellent electronic and mechanical properties of single walled carbon nanotubes (SWCNT)19. For example, Meier et al. applied two different processing strategies namely direct melt mixing and solution-precipitation to prepare thermoplastic elastomeric composites by using styrene-ethylene butylene-styrene(SEBS) triblock copolymer and multiwall CNTs, and observed that the mechanical and electrical properties such as storage module and electrical percolation threshold are clearly affected by the processing approach.20 Kim and coworkers successfully and serially investigated hierarchically organized CNTs arrays on the selfassembled BCP nanotemplates including polystyrene-block-poly(methyl methacrylate) (PS-bPMMA) and poly(styrene-block-4-vinylpyridine) (PS-b-P4VP).15-16 Liu et al. demonstrated that PSfunctionalized CNTs could be accommodated in the cylindrical PS phase of a microphase-separated



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styrene-butadiene-styrene(SBS) triblock copolymer templates regardless of the molecular weight of the PS ligand.17 On the other hand, using mixed carbon fillers to prepare polymer composites has attracted considerable attention in the recent years.21-23 For example, however, in order to improve the homogeneity of CNTs in the polymer matrix, complicated processes are usually involved in the experiments, which results in costly fabrication of the two-phase CNT-polymer composites. Recently, the incorporation of NPs into CNT-polymer composites without complex experimental procedure has been found to achieve outstanding properties of the hybrid composites.21-23 Moreover, other multiphase composites, such as CNTs/nanosheets/polymer and CNTs/fibers/polymer composites have also been investigated because of their advanced combined properties.24-25 Correspondingly, the use of mixed carbon fillers can provide a balance of the resulting properties and economic cost. Therefore, it will be a good strategy to obtain the composites by combining BCP and mixed carbon fillers together, which composites will capitalize the advantages of both BCP and mixed carbon fillers and have outstanding properties. Recently, computer modeling and simulation are playing an ever-increasing role in exploring the reinforcement mechanism of electrical properties of polymer composites.26-30 For example, Feng et al.30 employed coarse grained models of hexagonal and square patterns of vertical posts on flat substrates and simulated the assembly of melt-cast block copolymers using Monte Carlo method. Their simulated results for high area density circuit boards are successfully consistent with experimental observations. There are numerous researches in modeling and simulation of polymer composites with a single type of carbon filler, however, only few reports involve mixed carbon fillers in the single polymer matrix or polymer blend.31-33. Rahatekar et al.

31

have used dissipative

particle dynamics method to investigate the mixtures of perfectly conducting fibers and spheres, as well as the mixtures of fibers of different aspect ratios. Their theoretical results suggest that 4 ACS Paragon Plus Environment

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mixtures of fibers of different aspect ratio may be helpful in reducing the volume fraction of high aspect ratio filler particles required to achieve significant electrical conductivity in composite materials. Chen et al.

32

have used Monte Carlo simulation to estimate the electrical percolation of

polymer-matrix composites containing hybrid fillers of CNTs and CB. Xiong et al. 33 have proposed a model based on excluded volume theory to describe the electrical percolation of mixed carbon fillers in polymer blends. To the best of our knowledge, modeling the self-assembly of mixed carbon fillers in BCP template is still lacking. In this paper, we focus on the investigation of CNT and CB NPs filled in the block coplymer template by using the hybrid particle-field molecular dynamics simulation. This paper aims to better understand process of the self-assemblies and morphologies of composites produced by mixed carbon fillers (SWCNTs and CB NPs) in the BCP template. In this paper, we use hybrid particlefield molecular dynamics simulation method and implement the coarse-grained (CG) model to describe the self-assemblies of the composites including mixed carbon fillers and BCP template. Furthermore, influence of dispersion of mixed carbon fillers on the probability of electrical percolation of the hybrid composites is also discussed including different ratios between SWCNTs and CB NPs and different concentrations of mixed carbon fillers. Finally, the phenomenon of BCP templates deformed by mixed carbon fillers is obviously observed and the radius of gyration of BCP is also calculated. 2. Simulation Methods and CG Models Hybrid particle filed combination method, including combined self consistent filed theory (SCF) and density functional theory (DFT)

29, 34-38

, and SCMF method which combines SCF and Monte

Carlo (MC) method39-40, have successfully been used in the investigation of polymer composites. In 2009, an approach that combines molecular dynamics (MD) and SCF has been reported, which is suitable for the treatment of atomistic force fields and/or specific coarse-grained models41-42. The 5 ACS Paragon Plus Environment


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difference between hybrid MD-SCF and MC based SCMF method is in force calculation part. In particular, to generate MD trajectories forces acting on particles are needed. This calculation is achieved through the calculation of spatial derivatives of density fields. In MC the statistical ensembles are generated using probabilistic schemes using changes in energy before and after a given move, such as Metroplis acceptance criterion43. From the point of view of Hamiltonian they are equivalent. The only difference is about time evolution of the system that in MD and in MDSCF simulation also is intrinsically related to the method formulation. Furthermore, as for specific application of this methodology to systems related to the ones considered here, the hybrid particlefield MD-SCF has been successfully applied for polymer and carbon nanotube systems44-48. The CG models considered in this paper have been developed in a hybrid particle-field (PF) scheme that combines particles with a field representation for non-bonded interactions. The main feature of the hybrid PF approach is that the evaluation of the non bonded force and its potential between atoms of different molecules, i.e. the most computationally expensive part of MD simulations can be replaced by the evaluation of an external potential dependent on the local density at position r in PF simulations. According to the spirit of SCF theory, a many body problem like molecular motion in many molecule systems is reduced into the problem of deriving the partition function of a single molecule in an external potential V(r). Then non bonded force between atoms of different molecules can be obtained from a suitable expression of the V(r) and its derivatives. In the frame of SCF theory, a molecule is regarded to interact with the surrounding molecules not directly but through a mean field. Assuming that the density dependent interaction potential W, where each component species is specified by the index K, takes the following form:

k T r  = ∫ dr  B   2 

{ ( )}

W  φK 

∑χ

φ

KK ' K

KK '

2 1    r + ∑ φ r − φ0   , 2κ  K K 

( r) φ ( ) K'

()

(1)


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()

Where φK r is the coarse-grained density of the species K at position r and χKK ' are the mean field parameters for the interaction of a particle of type K with density fields due to particles of type K’, it can be shown that, using the so called saddle point approximation, the external potential is given by:

()

VK r =

{

} = k T

δW  φK ( r )  

δφK ( r )

B

∑χ K'

φ

KK ' K '

( r) + κ1  ∑ φ ( r) − φ  . K

0

(2)

K

The main advantage of hybrid MD-SCF scheme is that the most computationally expensive part of the MD simulations, i.e. the evaluation of the non-bonded force between atoms of different molecules, is replaced by the evaluation of forces between single molecules and an external potential. In order to connect particles and field models, for the proposed hybrid MD-SCF scheme, it is necessary to obtain a smooth coarse-grained density function directly from the particle positions. This function is to obtain implementing a mesh based approach suitable to obtain also the density derivatives needed to calculate forces. The details of the implementation of this approach and a complete derivation of eq. 2 are reported elsewhere.41-42 The simulations reported here have been performed by using the parallelized version of the OCCAM code.49 The main advantage of hybrid particle-field approach is the lower computation cost. The ratio between the computational times for the particle-field approach, where pair forces between non bonded particles are not calculated and the standard particle-particle approach (such as MD or Dissipative Particle Dynamics, DPD) would be function of the system size and the number of processors employed. In particular, due to the very low amount and less frequent data exchange the larger the number of processors. In particular, in the case of field based approach such as MD-SCF, only information about density field (much smaller data size) is exchanged between processors (particle positions are not shared) and much less frequently. To have a more quantitative idea field 7 ACS Paragon Plus Environment


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based simulations and DPD methods, a recent paper of Sevink and Fraaije50 shows comparison between classical DPD models and its hybrid particle field version using an approach employed here. They found by comparison between single processor simulations of the same system a factor of about 4 between DPD and hybrid approach. Comparison for parallel applications has been not reported, but we can reasonably assume they will be similar to those reported in our previous study about MD code scaling.49 In particular, the particle-field simulations can take from 40% (using 16 processors) to 12% (using 64 processors) of the CPU time needed for state-of-the art particleparticle simulations. Due to SWCNTs as the ultimate polymer51, CG model for SWCNT (10,10) with the diameter σ of 1.4 nm is composed of 10 particles which corresponds to the length of SWCNT 11.48nm and represents for 1880 C atoms, shown in Figure 1(a). In this model, bonds between two successive particles are described by a harmonic potential Vbond(R) type, Vbond (R) =

1 2 K bond (R − Rbond ) , where 2

Rbond is the equilibrium distance which equals to 0.8×σ =1.12 nm. Moreover, the stiffness of the SWCNT is taken into account by a harmonic potential Vangle (θ ) depending on the cosine of angle between Vangle (θ ) =

atoms,

where

θ

is

the

angle

between

two

successive

bonds.

1 2 K angle {cos(θ ) − cos(θ 0 )} , θ0 is the equilibrium bond angle 180º for SWCNT. The bond 2

and angle force constants are 10000 kJ·mol-1 and 8000 kJ·mol-1 respectively. Additionally, the CG model of CB NPs composed by two connected particles is shown in Figure 1(b), which model has the diameter of 2.8 nm. Correspondingly, the CG model of BCP(A12B13) shown in Figure 1(c) is generic, but can well represent amphiphilic diblock copolymers such as PMMA-block-PDMS However, there is no bond angle for the flexible BCP, and the distance between two particles is 1.4 nm. This diblock copolymer is noncharged. (see Supporting Information for a full description of CG models and parameters). 8 ACS Paragon Plus Environment

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As the box size needs to be at least several lengths of CNTs in each dimension27, the length of each side of a three dimensional cubic box is set at four times of the length of SWCNT in order to investigate the properties of percolation threshold reasonably. Hybrid-Particle Field MD-SCF method could take the advantage of low computational cost in these simulations. Finally, the description of the simulation systems is reported in Table 1 and the interaction parameters between particles are shown in Table 3. To investigate the composites including SWCNT, CB NPs, and BCP template, the parallel hybrid particle-field MD-SCF program OCCAM is performed in the simulations. NVT ensemble has been conducted to keep temperature constant at 500K using Andersen thermostat with a collision frequency of 20ps-1. The time step used for the integration of the equations of motion is 0.03 ps. The density grid size is set as 2.1 nm and update frequency is set as 300 time steps to obtain reasonable results. The site percolation method as a specific mathematical model is used on a box lattice. This uses an L×L×L box matrix of one and zero, which is called site matrix. One represents an “occupied” site and zero means “empty” site. A cluster composed of CNTs and CB NPs is a set of occupied sites. When all of the occupied sites are connected to an occupied site by neighboring occupied sites. It should be noted that the neighbors are only in the x, y, and z directions, not along the diagonals. If one cluster could go through from above to below, from left to right, or from front to back, the cluster is called spanning cluster. The spanning cluster has an element on both two corresponding surfaces of the site matrix. A simple algorithm for labeling clusters on a grid was found by J. Hoshen and R. Kopelamen in 1976.52 The Hoshen- Kopelamen algorithm is used to investigate the percolation configurations. They give each site label a another index b(a). As long as a is a “good” label, b(a) is a. But once the cluster labeled with a turns out to be a sub-cluster of a cluster labeled with ci



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Where, N is the number of particles that composed of SWCNT and θij is the angle between the vector bi and bj. The vector bi = (ri − ri −1 ) / 2 is the one formed by the midpoint of two adjacent particles i and i-1. Note that θij = 0 corresponds to Q = 1 , whereas a random orientation of SWCNT

results in Q = 0 . The time evolution of global SWCNT order parameter Q is calculated for different ratios of SWCNT and CB NPs and shown in Figure 8. Green line reprensents the C fillers concentration of 1.0 vol% and red line reprensents the C fillers concentration of 6.0 vol%. Comparing these two different C fillers concentration at different ratios of SWCNT and CB NPs, we could find the the order parameter of SWCNT is different. Firstly, when the ratio of SWCNT and CB NPs less than 1 (Figure 8 (a) and (b)), the order parameter Q corresponding to C fillers concentration of 1.0 vol% (green line) is a little larger than of C fillers concentration of 6.0 vol% (red line). Moreover, at C fillers concentration of 6.0 vol%, order parameter Q of SWCNT increases with the increasing ratios from SWCNT : CB NPs = 1: 8 to SWCNT : CB NPs = 1: 4. Furthermore, when the ratios of SWCNT and CB NPs increases into 1:1 (Figure 8(c)) and 2:1 (Figure 8 (d)), order parameter Q of SWCNT corresponds to C fillers concentration of 6.0 vol% is larger than 0.7, which is a little larger than that of 1.0 vol% due to more components of SWCNT. Finally, increasing the ratios of SWCNT: CB NPs into 4:1 (Figure 8 (e) ) and 8:1 (Figure 8 (f)), we could find that much difference of order parameter Q between C fillers concentration of 1.0 vol% and 6.0 vol%. At lower concentration (1.0 vol%) with much more SWCNT components (4:1 and 8:1), SWCNT would like to distribute more randomly which results in the value of Q is about 0.5. When the C fillers concentration increases into 6.0 vol%, SWCNT distributes more orientationally and Q is obtained at about 0.75. Therefore, the global order parameter of SWCNT Q is increasing during the increasing ratios of SWCNT and CB NPs. 18 ACS Paragon Plus Environment

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3.3 Effect of different concentration of SWCNT and CB NPs on the morphology of the BCP template Mixtures of the BCP and NPs have been studied by Huh and co-workers using both MC simulations and theory based on a scaling model for BCP in the strong segregation limit

60

. They

observed various morphological transitions by varying the concentration and size of NPs. In the meanwhile, such morphological transitions induced by NPs filler are proved by the Jang et al. the experiment. Moreover, Shagolsem and Sommer

6

61

in

obtained the various structures of the

copolymer composites from BCP and NPs mixtures confined between two identical walls in slit geometry and summarized in a phase diagram constructed in diblock composition and nanoparticle concentration space by using molecular dynamics simulations. As discussed in earlier studies, NPs can influence/modify the preferred of the lamellae self-assembled by BCP. Here, using hybrid MDSCF simulation method, we explore a much wider parameter space of the concentrations and ratios of mixed fillers (SWCNT and CB NPs), obtain different structures formed by BCP. In the following, we briefly summarize the typical morphologies at the ratio of SWCNT: CB NPs 4:1 in Fig. 9. In Fig. 9 (a), (b), and (c), we display the snapshots of nanocomposite including block A (green particles), block B (blue particles), SWCNT (red particles), CB NPs (yellow particles) and the template shown by block B (blue particles), respectively, corresponding to the volume concentration of mixed fillers of 1.0 vol %. It indicates that SWCNT (red particles) and CB NPs (yellow particles) are distributed in the block A (green particles) phase, and the lamellar template is not changed due to adding lower concentration of SWCNT and CB NPs. Furthermore, increasing the concentration of mixed fillers into 4.0 vol%, the SWCNT (red particles) form into bundles and distribute in the phase of block A (green particles) in Fig. 9 (d). It is very interesting that the lamellar template is changed and perforated by the segregated SWCNTs bundles shown in Fig. 9 (e). And the excessive NPs would like to surround the SWCNTs and decrease the interfacial energy in the A block. This situation is also observed at the volume concentration of 8.0 vol%, the segregated SWCNT bundles 19 ACS Paragon Plus Environment


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and the holes in the template self-assembled by block copolymers in Fig. 9 (g) are larger than those with lower volume concentration of 4.0 vol%, which is the result of lower enthalpy energy. Such morphologies are referred to as self-assembled structures, and it is an exclusive characteristic of SWCNT/ CB NPs/ BCP mixtures. To characterize the overall size of BCP chains, the square of the radius of gyration ( Rg 2 ) is calculated which is defined as follows:

Rg 2 =

N 1 < ∑ (ri − rcm ) 2 > N i =1

(7)

Where ri and rcm denote the position vector of each particle in BCP chain and the vector of center-ofmass for the whole BCP chain, respectively. If the Rg 2 of BCP chain increases, it means the expansion of BCP chain. The time evolution of the square of the radius of gyration ( Rg 2 ) of total BCP chain is calculated and shown in Fig. 10. At the ratio between SWCNT and CB NPs of 1:1, Fig 10 (a) shows that after the equilibrium, Rg 2 (12.4 nm2) of total BCP chain at the concentration of 8.0 vol% of mixed carbon fillers is larger than that (12.2 nm2) at 1.0 vol% of mixed carbon fillers. It means that the BCP chain prefers to stretch at higher concentration of mixed carbon fillers, which minimizes the total free energy of the system and determines the conformation. When the ratio between SWCNT and CB NPs increases into 8:1 in Fig. 10 (b), at the higher ratio (8.0 vol%) Rg 2 of BCP chain (12.6 nm2) is larger than that (12.4 nm2) at the ratio of 1:1 (SWCNT: CB NPs).

This shows that the expansion of BCP chains increases with the increasing ratio between SWCNTs and CB NPs. The excess SWCNT would like to form the bundles in the block A6 and the enthalpic energy is gained due to the contact between block A and SWCNT bundles. The segregated SWCNTs could not move translational, thus the balance of stretching of BCP chains and SWCNTs’ 20 ACS Paragon Plus Environment

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concentration determines the equilibrium conformation of the composites. This could make the total energy of the system minimum. Therefore, the BCP chains would like to stretch in a larger degree at the higher concentration (8.0 vol%) and at larger ratio of of SWCNTs and CB NPs (8:1). Furthermore, the square of radius of gyration ( Rg 2 ) of whole BCP chain (shown in Fig.11(a)) and block B (shown in Fig. 11(b)) at the ratio of 1:1 and 8:1 (SWCNT: CB NPs) are calculated with the different volume concentration corresponding to the simulation time of 1.5 µ s which is the same to the one of Fig.10. From Fig. 11(a), we could find that Rg 2 of total BCP chain increases with the volume concentration more obviously at the ratio of 8:1 (SWCNT: CB NPs) than that at the ratio of 1:1 (SWCNT: CB NPs). However, this contribution of the rising Rg 2 of total BCP is mostly from the increased Rg 2 of block B shown in Fig. 11 (b), which indicates that the block B is more stretched and is consistent with the morphology of Fig. 9.

4. Conclusions The development and validation of coarse-grained models of composites including block copolymers (BCP), single walled carbon nanotube (SWCNT) and carbon black nanoparticles (CB NPs) have been reported. Hybrid particle-filed MD-SCF simulations of large-scale coarse-grained models of SWCNT and CB NPs interacting with BCP templates suitable to reach large time and length scales for several systems have been reported. Simulations show, in agreement with experiments, the composites of SWCNT/BCP template have lower percolation threshold than those of CB NPs/BCP template. Moreover, we focus on the effect of the ratios and concentrations on the morphologies and probability of percolation of composites. It is interesting that while the ratio of SWCNT to CB NPs is larger than 1.0, adding more SWCNT could decrease the percolation threshold of the composites. However, there is still a limitation that the electrical probability of percolation will reach the highest point at the ratio of 4:1 (SWCNT: NPs). Meanwhile, the mixed 21 ACS Paragon Plus Environment


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fillers also have a strong influence on the morphologies of BCP templates. At higher concentration of mixed fillers and higher ratio of SWCNT, larger percolated structures of BCP templates are obtained. The results in this work primarily complement the experiments with the possibilities on the preparation of the composites by adding mixed fillers of SWCNT and CB NPs into the BCP template.

Acknowledgements The project was supported by the National Natural Science Foundation of China (21203179, 11274057, 51102036, 11204024, 61205178), National Key Basic Research Program of China (973) (2012CB626801),

Fundamental

Research

Funds

for

the

Central

Universities,

China

(DC13010219.B), and Science and Technology Project of Liaoning Province, China (2012222009). The work was carried out at National Supercomputer Center in Tianjin, and the calculations were performed on TianHe-1 （ A ） .We thank Zhan-Wei Li from Changchun Institute of Applied Chemistry, Chinese Academy of Sciences, Yong-Lei Wang from Department of Materials and Environmental Chemistry, Stockholm University, and Zhen-Yi Zhang from Dalian Nationalities University for their useful discussions. Associated content The supporting information describes the details of the coarse-grained models and parameters. Furthermore, the CG models are validated by the morphologies and energy calculations.


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References

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TOC graphic: The morphologies of composites corresponding to the volume concentration. The red line represents the probability of percolation corresponding to different ratio of SWCNT: CB NPs. Above the red line, the systems are percolated.



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Table 1 – Simulated Systems Composition(no. of atoms) Systems

Vol %

SWCNT and CB NPs

Block copolymer

Particle no.

Box size [nm]

Simulated time[µs]

I

0.3

164

54750

54914

56.62

1.5

II

0.6

330

54750

55080

56.68

1.5

III

1

554

54750

55304

56.76

1.5

IV

2

1118

54750

55868

56.95

1.5

V

3

1694

54750

56444

57.14

1.5

VI

4

2282

54750

57032

57.34

1.5

VII

5

2882

54750

57632

57.54

1.5

VIII

6

3494

54750

58244

57.75

1.5

IX

7

4120

54750

58870

57.95

1.5

X

8

4760

54750

59510

58.16

1.5

XI

9

5415

54750

60165

58.37

1.8

XII

10

6084

54750

60834

58.59

1.8

XIII

11

6767

54750

61517

58.81

1.8

XIV

12

7466

54750

62216

59.03

1.8

XV

13

8181

54750

62931

59.26

1.8

XVI

14

8913

54750

63663

59.48

1.8

XVII

15

9662

54750

64412

59.72

1.8

XVIII

16

10429

54750

65179

59.95

1.8

XIX

17

11214

54750

65964

60.19

1.8

XX

18

12018

54750

66768

60.44

1.8

XXI

19

12843

54750

67593

60.68

1.8

XXII

20

13688

54750

68438

60.94

1.8


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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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Table 2 The number of molecules of mixed carbon fillers at different ratio (SWCNT: CB NPs). Vol%

1:8

1:4

1:1

2:1

4:1

8:1

0.3

2

72

3

67

8

42

11

27

13

17

15

9

0.6

4

145

7

130

17

80

22

55

26

33

29

18

1

6

247

11

222

28

137

37

92

44

57

49

31

2

12

499

23

444

56

279

75

184

89

114

99

64

3

19

752

34

677

85

422

113

282

136

167

151

92

4

26

1011

46

911

115

566

152

381

183

226

203

126

5

32

1281

58

1151

145

716

192

481

231

286

257

156

6

39

1552

70

1397

175

872

233

582

280

347

311

192

7

46

1830

83

1645

206

1030

275

685

330

410

367

225

8

53

2115

96

1900

238

1190

317

795

381

475

424

260


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Table 3 – Particle field interaction parameters(kJ/mol) SWCNT

CB NPs

A

B

SWCNT

0.0

0.0

10.0

40.0

CB NPs

0.0

0.0

10.0

40.0

A

10.0

10.0

0.0

40.0

B

40.0

40.0

40.0

0.0



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Figure 1. The Coarse-Grained (CG) model of (a) SWCNT, (b) CB NPs, and (c) BCP AmBn.


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Figure 2. Comparison of the self-assembly process of SWCNT/BCP system (a) and CB NPs/BCP system (b) at single filler 10.0 vol% concentration. In the figure, the total simulation time is 1.8 µs. The red particles represent for SWCNT, the yellow particles represent CB NPs, and blue particles represent block A of BCP template.



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Figure 3. (a) The probability of percolation correpsponding to the C concentration (vol%) SWCNT/BCP and CB NPs/BCP are shown. In the figure, the volume concentration of single filler is 2.0%, 6.0%, 10.0%, 15.0%, and 20.0%, respectively. The red particles represent for SWCNT, the yellow particles represent CB NPs, and blue particles represent block A of BCP template. (b) The morphologies of SWCNT/BCP composites at 8.0 vol% in the z direction and x, y plane respectively.


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Figure 4 The self-assembly process of composites at SWCNT: CB NPs = 4:1 and the volume concentration of 8.0 vol%. Red particles represent SWCNT, yellow particles represent CB NPs, and blue particles represent block B. The morphologies correspond to the self-assembly time of 0 µs , 0.06 µs , 0.6 µs , 1.5 µs ,respectively.



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Figure 5 The probability of percolation correpsponding to the C concentration (vol%) at the ratio of SWCNT:CB NPs = 1:8, 1:4, and 1:1, respectively(a), and the ratio of SWCNT and CB NPs 1:1, 2:1, 4:1, and 8:1, respectively (b).


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Figure 6 The morphologies of composites with the ratio of SWCNT: CB NPs = 1:8, 1:4, 1:1, 2:1, 4:1, and 8:1, corresponding to the volume concentration of 1.0 vol%, 2.0 vol%, 4.0 vol%, and 6.0 vol%, respectively. The red line represents the probability of percolation corresponding to different ratio of SWCNT: CB NPs. Above the red line, the systems are percolated.



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Figure 7 Denstiy profiles in z direction of SWCNT and CB NPs. The density profiles of z direction are calculated by dividing the length of the simulation box in z direction into r layers according to the grid size about 2.1 nm. Thus, r of X axis represents the number of layers in the z direction. In the figure, the red line reprensents SWCNT, the yellow line represents CB NPs, the green line represents block A, and the blue line reprensents block B. 38 ACS Paragon Plus Environment

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Figure 8 Time evolution of global SWCNT Order paramter is calculated corresponding to different ratios of SWCNT and CB NPs and different C filler concentions. In this figure, the green line represents the C filler concentration of 1.0 vol% and the red line reprensents the C filler concentration of 6.0 vol%. (a) SWCNT : CB NPs = 1:1; (b) SWCNT : CB NPs = 1:4; (c) SWCNT: CB NPs = 1:8; (d) SWCNT : CB NPs = 2:1; (e) SWCNT: CB NPs = 4:1; (f) SWCNT : CB NPs = 8:1.



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Figure 9 The morphologies of composites at the ratio of SWCNT: CB NPs = 4:1 corresponding to 1.0 vol% (a), (b), and (c); 4.0 vol% (d), (e), and (f); 8.0 vol% (g), (h) and (i), respectively. In the figure, red particles represent SWCNT, yellow particles represent CB NPs, blue particles represent the B block of BCP template, and green particles represent the A block of BCP template.


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Figure 10 The Rg2 (the square of radius of gyration) of BCP vs the simulation time at the volume concentration of the mixed fillers of 1.0 vol%(green line) and 8.0 vol%(red line), at the ratio of SWCNT: CB NPs =1:1 (a) and SWCNT: CB NPs = 8:1 (b).



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Figure 11 (a) Rg2 (the square of radius of gyration) of BCP (b) Rg2 of block B vs different volume concentration at the ratio of SWCNT: CB NPs = 1:1 (green line) and 8:1 (red line).


Self-Assembled Morphologies and Percolation Probability of Mixed

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