Influence of Morphology on the Mechanical Properties of Polymer

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Influence of morphology on the mechanical properties of polymer nanocomposites filled with uniform or patchy nanoparticles Lu Wang, Zijian Zheng, Theodoros Davris, Fanzhu Li, Jun Liu, Youping Wu, Liqun Zhang, and Alexey V Lyulin Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.6b01049 • Publication Date (Web): 26 Jul 2016 Downloaded from http://pubs.acs.org on July 29, 2016

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Influence of morphology on the mechanical properties of polymer nanocomposites filled with uniform or patchy nanoparticles Lu Wang1, 2, Zijian Zheng1,2,3, Theodoros Davris5, Fanzhu Li1,2,3, Jun Liu1, 2, 3*, Youping Wu1, 2, 3, Liqun Zhang1, 2,3,4*, Alexey V. Lyulin5* 1

Key Laboratory of Beijing City on Preparation and Processing of Novel Polymer Materials, People’s Republic of China 2 Beijing Engineering Research Center of Advanced Elastomers, People’s Republic of China 3 Engineering Research Center of Elastomer Materials on Energy Conservation and Resources, Ministry of Education, PRC 4 State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, 100029 Beijing, People’s Republic of China 5

Group Theory of Polymers and Soft Matter, Department of Applied Physics, Technische Universiteit Eindhoven,5600 MB, Eindhoven, The Netherlands *Corresponding author: [email protected] or [email protected] or [email protected]

1

ACS Paragon Plus Environment

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Abstract In this work we perform molecular-dynamics simulations, both on the coarse-grained and the chemistry-specific levels, to study the influence of the morphology on the mechanical properties of polymer nanocomposites (PNCs) filled with uniform spherical nanoparticles (which means without chemical modification) and patchy spherical nanoparticles (with discrete, attractive interaction sites at prescribed locations on the particle surface). Through the coarse-grained model, the non-linear decrease of the elastic modulus (G') and the maximum of the viscous modulus (G'') around the shear strain of 10% is clearly reproduced. By turning to the polybutadiene model, we examine the effect of the shear amplitude and the interaction strength among uniform NPs on the aggregation kinetics. Interestingly, the change of the G' as a function of the aggregation time exhibited a maximum value at intermediate time attributed to the formation of a polymer-bridged filler network in the case of strong interaction between NPs. By imposing a dynamic periodic shear, we probe the change of the G' as a function of the strain amplitude while varying the interaction strength between uniform NPs and its weight fraction. A continuous filler network is developed at a moderate shear amplitude, which is critically related to the interaction strength between NPs and the weight fraction of the fillers. In addition, we study the self-assembly of the patchy NPs, which form the typical chain-like and sheet-like structures. For the first time, the effect of these self-assembled structures on the visco-elastic and stress-strain behavior of PNCs is compared. In general, in the coarse-grained model we focus on the size effect of the rough NPs on the Payne effect, while some other parameters such as the dynamic shear flow, the interaction strength between NPs, the weight fraction and the chemically heterogeneous surface of the NPs are explored for the chemistry-specific model.

2


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I. Introduction Polymer nanocomposites (PNCs), prepared by introducing nanoparticles (NPs) into a polymer host, receive a considerable scientific and technological interest attributed to their excellent mechanical,1 thermal,2 electrical3 and gas-barrier properties.4 The microscopic structure of PNCs is critically dependent on the dispersion morphology of the NPs, the interfacial interactions between polymer and NPs, and the geometrical characteristics of the NPs, such as size and shape. For instance, for spherical NPs, Liu et al.5 studied the effect of the polymer-filler interaction and the filler size on the filler aggregation process through coarse-grained simulation, and found that the aggregation extent reaches the minimum in the case of moderate polymer-filler interaction. The dispersion mechanism has also been studied for the case of non-spherical or anisotropic NPs in the polymer matrix.6-8 Due to the development of the synthesis technique, novel NPs are now being fabricated, for instance, Janus particles, which are characterized by two regions of different surface chemical composition. They exhibit energetic interactions that are related not only to their separation but also to their orientation.9 Patchy particles with discrete, attractive interaction sites at prescribed locations on the particle surface have been simulated by Glotzer et al.10-11 Various structures such as chains, sheets, rings, icosahedra, square pyramids, tetrahedra, as well as twisted and staircase structures, were formed by self-assembly. Meanwhile, Walther et al.12 have shown that the Janus particles can be located exclusively at the interface of the two polymer phases, exhibiting better efficiency compared to compatibilizers based on a block copolymer, 3


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especially when the loading is high. Dissipative particle dynamics (DPD) simulations of immiscible polymer blends have shown that the Janus NPs can function as a more effective stabilizing agent than homogeneous NPs.13 By taking advantage of the directional interaction between patchy particles, it is important to study their self-assembly in the polymer matrix. To the best of our knowledge, the structure-property relations in this case received rather little attention. The visco-elastic properties are important aspects of PNCs, since they are used mostly under conditions of dynamic deformation (this is of special importance for elastomer-based nanocomposites such as automobile tires). The nanoparticle-polymer interface has been found to have a strong effect on the visco-elastic properties of PNCs.14 Hattemer et al.15 studied the effect of grafting the NPs on the polymer visco-elastic properties by calculating the frequency-dependent storage and loss modulus of coarse-grained models of PNCs via molecular-dynamics simulation , and found that the nanocomposite moduli increases with decreasing NP size and increasing NP loading, grafted chain length, and grafting density with varying frequency dependence. Raos et al.16-17 studied the effect of different dispersion states (including dispersed, moderately aggregated, and completely aggregated) on the visco-elastic behavior through coarse-grained DPD simulations, and observed a nonlinear viscoelastic behavior as a function of the shear amplitude. However, the characteristics of this nonlinearity are different from the experimentally observed. In addition, Gersappe et al.18 observed that above a critical concentration of filler particles, the network structure formed between the fillers and the polymers strongly 4


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affects the dynamics of the nanocomposite under shear. For instance, at low filler volume fraction φ = 2% , no evidence of a percolating network is observed. While for φ = 5% and 10% , the presence of a major cluster is found, spanning the entire simulation system. Shen et al.19 particularly examined the influence of the grafted chain length and density on the non-linear behavior, namely the Payne effect.20 It is shown that the non-linear relation of the storage modulus and loss modulus as a function of the strain amplitude decreases with the increase of grafted chain length and density. Chen et al.21 also studied the Payne effect through coarse-grained simulations and found that the magnitude of the Payne effect depends on the polymer-particle interaction and the filling loading. But in their studies the particles were taken ideally spherical and purely repulsive, which is a rather strong simplification. In the present study, we tried to include the shape roughness and different (also attractive) interactions between NPs into account, which are important but never studied before. Our goal is just to understand the potential of this model, and the importance of these modifications in reproducing the Payne effect. The comparisons with the reality, as well as the more detailed comparison with the united-atom simulations is the subject of the future publication. In the present study we focus on the coarse-grained and the united-atom dynamics simulations of polymer nanocomposites filled with uniform NPs. Our aim is, first of all, to see the ability of this kind of molecular modeling to reproduce the experimentally-observed drop of the elastic modulus upon increasing the strain magnitude (Payne effect) and provide possible molecular insights into the physical 5


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mechanisms. We also study the effect of the dynamic periodic shear flow on the aggregation process of the uniform NPs and the visco-elastic properties of the nanocomposite. In addition, we examine the structural evolution of the NPs under shear flow, by varying the filler weight fraction and the interaction strength between the NPs. Lastly, we consider the self-assembly of patchy NPs in the polymer matrix, and the effect of the self-assembled structures on the visco-elastic properties and the stress-strain behavior of the nanocomposite. II. Simulation models and methods In this paper we adopt the united-atom models, both on the very coarse-grained level, Fig. 1(a), and on the more chemistry-specific level, to study the cis-polybutadiene (cis-PB) polymer-based composite. In cis-PB model each carbon atom and its bonded hydrogen atoms are grouped to a single large atom, as shown in Fig. 1(b). In the simulated coarse-grained model the polymer matrix consisted of 100 linear polymer chains of 50 monomers each of mass m. Periodic boundary conditions were implemented for all three dimensions of the simulation box. Newton’s equations of motion were integrated using the velocity-Verlet algorithm with a time step of

δ t = 0.005τ where τ = σ m / ε , and σ and ε are the corresponding parameters of the Lennard-Jones interaction, see Eq. 1 below. The interactions between non-bonded beads were modeled with a modified Lennard-Jones (LJ) 12-6 potential, 12

6

σ  σ  U nb ( r ) = 4ε [  −   ] + S f ( r ) r r

out ( r < rcut )

(1)

where S f ( r ) is a switching function that smoothly ramped the energy to zero from 6


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in out rcut = 2.5σ to rcut = 3.5σ . The LJ parameters for the monomer units were σm= σ,

εmm = ε, mm = m, and for the filler beads σf=σ. The LJ parameter ε of the interaction among monomer and filler beads was equal to ε mf = 2ε . The value ε ff = 50ε was used for filler beads belonging to the same NP (the number of filler beads comprising a NP was between 125 and 1000 beads, depending on the desired diameter of the NP). For the interactions among filler beads belonging to different NPs we used the LJ potential, Eq. 1, with ε ff = ε . In the coarse-grained model the covalently-bonded beads

interacted

through

a

combination

of

an

attractive

Finite-Extensible-Nonlinear-Elastic (FENE) potential, and a repulsive and truncated LJ 12-6 potential,

U bn ( r ) = −0.5k FENE R

2 max

  r 2   σ 12  σ 6   ln 1 −  + 4 ε   −    + ε   r     Rmax    r 

(2)

where the parameters kFENE and Rmax denote the stiffness and the maximum elongation of the spring, respectively, which were set equal to kFENE = 30ε / σ 2 and

Rmax = 1.5σ , respectively. The LJ parameters ε and σ in Eq. 2 had the same values with their non-bonded counterparts. The coarse-grained simulations were performed under the NPT ensemble at T = 0.65 and P = 0 by employing the Nose-Hoover thermostat and barostat. During both the equilibration and production runs we used the values recommended by the large scale atomic/molecular massively parallel simulator (LAMMPS ） 22 for the thermostat and barostat parameters, i.e., the temperature and pressure were allowed to relax to the specified average value over a time interval of 0.5τ and 5.0τ, respectively. Systems containing NPs with different average size in the range 6 – 13 were simulated. The filler volume fraction was 7


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fixed at 28.5%. The force-field parameters for the coarse-grained systems are listed in the Table 2(c). When we map the coarse-grained model to real polymers, the interaction parameter is set to be about 2.5-4.0 kJ mol-1 for different polymers.23 It is noted that the persistence length is about 0.676 for this coarse-grained model of polymer chains, and for real polymers, the range of the persistence length is between 0.35nm and 0.76nm. Each bead with its diameter equal to 1 nm roughly corresponds to 5 repeating units of polyethylene. Thus, our simulation is considered to be within a realistic range.24-25 In the simulated united-atom model of cis-polybutadiene the total force-field interaction energy was given by:

Etotal = Ebond (r ) + Eangle (θ ) + Edihedral (ϕ ) + Enon−bodning (r )

(3)

The bond stretching (r), bond angle bending(θ) and dihedral angle torsion(φ) are given by:

Ebond ( r ) = kb (r − r0 )2 Eangle (θ ) = kθ (θ − θ 0 )

2

(4) (5)

6

Edihedral (ϕ ) = ∑ kn (1 − cos ( nϕ ) )

(6)

n =1

where kb and r0 represent the stiffness of the harmonic spring and the equilibrium bond length for the three different types of bonds in the cis-1,4-PB molecule, respectively. kθ and θ0 represent the bond angle potentials and the equilibrium bond angle.26-27 The dihedral potentials of different magnitudes kn are associated with the three dihedral angles, φ1, φ2, and φ3. The non-bonded interaction is modeled through 8


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the truncated and shifted Lennard-Jones (TSLJ ) equation as follows:

E non − bonding ( r ) = 4 ε [(

σ r

)12 − (

σ r

) 6 ] + C , r ≤ rc

(7)

where r denotes the distance between two atoms, σ is the distance at zero energy and ε is the energy well depth, and C is a constants to maintain the continuity of the interaction. The force-field parameters of cis-PB are listed in Table 1, following Tsolo.28 The simulation box contains 100 polymer chains and each chain contains 128 carbon atoms. Note that, although these chains are rather short, they already show the characteristic static and dynamic behavior of long polymer chains.29 Decorating the surfaces of particles with precise patterns of atoms with specific interactions that can promote an anisotropic association among the particles, can potentially offer a universal way to rationally self-assemble particles into predictable, precise, and ordered structures. Therefore, in the model of cis-PB-based nanocomposite we consider two types of nanoparticles: (i) uniform nanoparticles (C60), which have homogeneous surface characteristics as Fig. 1(c)-(1), and (ii) patchy nanoparticles, which surface has specific arrangements of attractive patches so as to induce the organization of particles from a disordered state into unique structures. We consider two cases of patchy NPs: patchy2 and patchy4. Patchy2 is referred to C60 with two diametrically opposed circular patches as displayed in Fig. 1(c)-(2) and (3). Similarly, patchy4 is referred to C60 with four circular patches arranged in an equatorial plane on the four vertices of a square as shown in Fig. 1(c)-(4). However, when we construct the patchy4 model, the molecular symmetry is 9


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lowered from Ih to C2h (the molecule retains a two-fold axis, a symmetry plane orthogonal to it, and an inversion center) when the structure is changed from homogeneous C60 to patchy4. The interaction parameters among the patchy nanoparticles are shown in Table 2(b). By compounding uniform C60 with cis-PB, we focus on the effect of the NP-NP interaction strength and the loading of NPs. The chemical modification of the particle surface is modeled by varying the C60-C60 interaction strength. In this respect we simulate three different systems (see Table 2(a)). Case I refers to systems with weakly attractive filler-filler interactions and in Case II the interaction strength between fillers is moderate. Finally, Case III represents the case with strongly attractive filler-filler interactions. In addition we study the effect of the filler loading on the visco-elastic properties of the nanocomposites, by varying the number of C60 from 10 to 160, thus effectively varying their weight fraction as well. By compounding patchy C60 with cis-BR, we focus on the effect of the self-assembly structure of fillers on the resulting properties. The fast development and progress on the synthesis of Janus NPs provide the opportunity to fabricate the new material with ordered structures. In this work we not only study the visco-elastic properties but also the mechanical properties of different self-assembly structures of fillers. In the PB-based simulations we adopt the NPT ensemble by employing the Nose-Hoover thermostat and barostat, by setting the pressure P = 1atm . There are two temperatures adopted in the PB-based simulations: 298K (the room temperature) 10


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and 373K (the common temperature adopted when testing the storage modulus in the experiment). Periodic boundary conditions are employed in all three directions during the simulations. For PB simulations the velocity-Verlet algorithm is used to integrate the equations of motion, with a timestep δ t = 1 fs . Two different deformation modes are simulated. The first is the oscillating simple shear mode as Fig. 2(a). Oscillating shear mode was used both in coarse-grained and PB simulations. Concerning the simple shear mode, the SLLOD30 equations of motion are used along with the Lees–Edwards “sliding brick” boundary conditions.20 The upper yz plane of the simulation box is shifted along the z direction such that each point in the simulation box can be considered as having a “streaming” velocity. This position-dependent streaming velocity is subtracted from each atom's actual velocity to yield a thermal velocity which is used for the calculation of the temperature and for thermostatting. The shear strain is defined as follows:

γ yz = dz / Lx(0)

(8)

where the offset dz is the transverse displacement distance in the shear direction from the unstrained orientation, and Lx (0) is the box length perpendicular to the shear direction.31 During the oscillatory shear deformation, the shear strain is characterized by a sinusoidal function:

γ yz (t ) = γ sin(2πν t )

(9)

where γ and ν are the oscillatory shear strain amplitude and shear frequency, respectively. 11


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For the coarse-grained model, the shear frequency is set to = 0.01 , i.e., imposing one cycle of constant-amplitude oscillation every 100τ , and the shear strain amplitude ranges from 0.01 to 0.15. In PB simulation the shear frequency is set to ν = 10ns −1 . The shear strain amplitude ranges from 0.05 to 1.0 ( γ =0.05，0.10， 0.15 ， 0.25 ， 0.30 ， 0.40 ， 0.50 ， 1.0). The average shear stress σ s is obtained from the deviatoric part of the stress tensor

σ s = Pxy = Pyx . Under oscillatory shear

deformation the shear stress can be expressed through a sinusoidal function,

σ xy (t) = σ 0 sin(2πν t + δ ) = σ ' sin(2πν t ) + σ '' cos(2πν t )

(10)

Where σ0 is the stress amplitude, and δ is the loss angle. The in-phase and out-of-phase components of the complex shear modulus ( G * = G '+ iG '' ), i.e., the storage modulus G ' and loss modulus G '' , can be derived from the σ ' and σ '' parameters, G'=

σ' σ '' G '' σ '' G '' = tan δ = = γ γ G' σ '

(11)

The second mode of deformation, uniaxial tension mode, are applied to study the mechanical properties of polymer nanocomposites filled with patchy NPs, where we follow previous approach.32 As illustrated in Fig. 2(b), the box length in the z direction is increased at a constant engineering strain rate while the box lengths in the

x and y directions are reduced simultaneously so as to maintain the box volume being

unchanged.

The

tensile

strain

rate

ε&

is

specified

as

ε& = ( L(t ) z − Lz (0)) / ( Lz (0) ∗∆t ) = 0.000001 fs−1 , meaning that the box length in the z direction is doubled after 1ns and the tensile strain rate is the same as the

simulation work carried out by Hossain et al.33 and Zheng et al.29. The average stress 12


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σ in the z direction is obtained from the deviatoric part of the stress tensor

σt = (1+ µ)(−Pzz + P) ≈ 3(−Pzz + P) / 2 , where P = ∑i (Pii / 3) is the hydrostatic pressure. The parameter µ stands for the Poisson’s ratio, which is equal to 0.5 in our simulation, because rubbery materials are often regarded as incompressible during the deformation process. All MD simulations are carried out using the LAMMPS software developed by Sandia National Laboratories.22 For all cases, we make sure that each chain has diffused a distance at least 2Rg , with Rg being the root mean square radius of gyration of the polymer chains, to obtain well-equilibrated systems. A more detailed description of the simulation techniques used in this study can be found in our previous publications.34-36 III. Results and discussion

3.1 Uniform nanoparticle We start the discussion with the results of the coarse-grained MD simulations. Our goal here is to check their ability to reproduce the Payne effect. The results are shown in Fig. 3, where the dependence of both the elastic G’ and viscous G’’ modulus on the shear amplitude are shown for NPs of different diameters. Few conclusions can already be drawn on this stage: (i) the mechanical reinforcement, i.e., the increase of the elastic modulus upon increasing the filler surface area (by using smaller filler particles at fixed filler volume fraction) is clearly observed at very small (< 0.01) shear strains; (ii) a pronounced drop in G’ for large (> 0.1) shear strain amplitudes (i.e. the Payne effect) was observed; (iii) a maximum in G’’ was also observed at a 13


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strain amplitude of 0.1. We may conclude that the MD simulations on this coarse-grained level are capable of reproducing the qualitative features of the mechanical reinforcement and the experimentally-observed Payne effect. In reality, as mentioned in the introduction, the important effects which only partly can be modeled on the coarse-grained level, could also contribute to the Payne effect. These are,

for

example,

specific

filler-surface

modification

and

structure,

chemically-specific intrachain and interchain interactions which define chain flexibility, specific polymer-filler interactions. In order to try to understand the importance of these effects, in the present study we also performed the simulations on a more detailed level. We simulated the united-atom model of cis-polybutadiene filled with uniform C60 nanoparticles and patchy nanoparticles with different attractive patches on their surfaces. However, we want to point out that through the coarse-grained simulation, the size effect of the NPs is solely probed, which is not accessible through the adopted united-atom simulation. In what follows we examine how the external shear force influences the aggregation process of the NPs and the mechanical properties in a chemistry-specific polybutadiene-based nanocomposite. To validate that our simulation model and method are accurate, we calculate the glass-transition temperature (Tg) of the simulated cis-PB system, which is compared with the work from Tsolou et al.28 It is known that the glass-transition temperature signifies the transition between the rubbery and glassy state. Fig. S1 (Supporting Information) shows the specific volume (reciprocal of the density) as a function of the temperature for cis-PB/C60 system. A distinct kink in the curve indicates the 14


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occurrence of the glass transition. The estimated value of Tg is 180K for pure cis-PB, which agrees with other MD simulation work (185K),37 and is only a little higher than the experimental value (170K).38 Besides, a small increase in Tg is observed with the increase of the weight fraction of C60 , attributed to the interfacial interaction to constrain more polymer chains. This finding is consistent with the experimental results.39-40

To obtain an initial state with a relatively uniform dispersion of NPs, we employ repulsive interactions between NPs. The repulsive NP interactions are subsequently changed into attractive interactions. To monitor the aggregation process, we trace the total interaction energy between C60-C60, as shown in Fig. 4. To check the effect of the filler-filler interaction strength on the aggregation process, we consider three values for the C60-C60 interaction strength: ε C60 −C60 = 0.1kal / mol (Fig. 4(a)),

εC

60 − C60

= 1.2kcal / mol (Fig. 4(b)), and ε C60 −C60 = 3.6kcal / mol (Fig. 4(c)). Evidently,

the results show that with the increase of the shear amplitude and the increase of the C60-C60 interaction strength, the rate and degree of aggregation are both greatly enhanced. This is attributed to the fact that a larger shear amplitude increases the contact probability between the NPs. Fig. 5 shows indicative snapshots of equilibrated,

aggregated

NPs

for

different

shear

amplitude

values

and

ε C60 −C60 = 3.6kcal / mol . The degree of the aggregation state is accelerated as the shear amplitude is increased ( γ = 0, 0.2, 0.6, 0.8, 1.0 ), as shown in Fig. 4(d). As we can see, the NPs aggregate with the simulation time under the shear deformation process. So we can call the time as the aggregation time when the 15


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system is under the oscillatory shear. We also probe the change of the composite mechanical properties as a function of the aggregation time. To this end we trace the change of the storage modulus as a function of the aggregation time. The obtained results are shown in Fig. 6, where we consider three different values of the shear amplitude: γ = 0 , γ = 0.2 and γ = 0.6 . The statistically averaged storage modulus displays a non-monotonic behavior for all values of γ . As illustrated in Fig. 6, a maximum peak occurs at intermediate time, which is attributed to the formation of a filler network during the aggregation process by the bridging of NPs via polymer chains: at initial time the fillers are well dispersed, whereas at larger time they form large clusters, with a polymer-bridged filler network forming at intermediate time. As for the Payne effect, we turn our focus to the change of the storage modulus as a function of the strain amplitude for three different values of the filler-filler interaction strength, Fig. 7(a). For weak filler interactions, ε C60 −C60 = 0.1kcal / mol , the fillers are uniformly dispersed, and an appreciable decrease of the storage modulus versus strain occurs. Based on the change of the total NP-NP and NP-BR interaction energy, as shown in Fig. S2(b) and (c), we conclude that the Payne effect is induced mainly by NP-NP direct contacts. A more pronounced decline can be observed to occur for εC60 −C60 = 1.2kcal / mol at a strain amplitude of 15% . The increase of the total interaction energy between NPs-NPs and NPs-BR leads to the stronger NPs aggregation during the shear deformation. For both values of EC60 −C60 and the storage modulus displayed a monotonic dependence on the strain amplitude. On the other hand, in the case of ε C60 −C60 = 3.6kcal / mol , a non-monotonic change 16


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of the storage modulus can be observed. At low strain amplitudes the NPs tend to form small isolated clusters which lead only to local contributions to the reinforcement. This explains the greater values of the storage modulus in systems with high NP-NP interaction strength compared to those of systems with low NP-NP interaction strength. However, under external shear deformation, the isolated small clusters will approach each other, merge, and form a continuous filler network running throughout the whole system (such as when the strain amplitude is equal to

γ = 0.25 ). By further increasing the shear amplitude, the continuous network structure gradually transfers into an isolated large cluster similarly to the phase separation situation. We consider two C60 as neighbors when their center-to-center distance is less than

12.06 Angstrom based on the following formula: o

R = RC60 + 2.5 ∗ σ atom = 12.06 A ,where RC60 represents the radius of C60 , σ atom denotes the diameter of carbon atoms on the surface of C60. According to the Lennard-Jones potential function, when the distance of carbon atoms on the surface between different C60 is larger than 2.5 ∗ σ atom , no any interaction exists between these two atoms. Therefore, the critical distance to calculate the number of neighboring fillers is RC60 + 2.5 ∗ σ atom . That is to say, only when the distance between different C60 is less than 12.06Å, then the two C60 is considered to become neighboring. By calculating the number of neighboring C60 and the total filler-filler and filler-polymer interaction energy, a similar non-monotonic trend is seen for

εC

60 − C60

= 3.6kcal / mol , as displayed in Fig. S2(a), (b) and (c). In order to see the

structural evolution of the filler particles under shear, Fig. 7(b) and (c) show the 17


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Page 18 of 45

representative snapshots of the particles’ morphology at several values of the strain amplitude,

εC

60 − C60

at

two

different

values

of

the

NP-NP interaction

strength,

= 3.6kcal / mol and ε C60 − C60 = 1.2kcal / mol . Apparently, a moderate filler

network occurs for ε C60 − C60 = 1.2kcal / mol , which becomes gradually dismantled while the shear amplitude is increased, leading to the decrease of the storage modulus, see Fig. 7(a). However, for ε C60 −C60 = 3.6kcal / mol , a more continuous network structure is developed at moderate, γ = 0.25 , values of the strain amplitude, as verified by the continuous and ordered network structure formed by the NPs, Fig. 7(d) (in order to see clearly the network structure we unfolded the morphology along the x, y and z directions). For γ = 0.5 , though, no such continuous network is found, Fig. 7(e). These observations rationalize the appearance of the peak value in the storage modulus versus the strain amplitude, which is observed for ε C60 −C60 = 3.6kcal / mol . We can see that the results of the united-atom simulations at intermediate strength of the NP-NP interaction are qualitatively reproduced also on the coarse grained level, as shown in Fig. 3(a), namely the drop of the modulus is clearly seen. In both cases weakly-interacting filler particles are losing the direct contacts under deformation, which finally leads to the Payne effect. The aggregation effect which is seen on the atomistic level at high NP-NP strength is not observed on the coarse-grained level: more filler particles, which are interacting stronger, are necessary to study this further. It is well known that the volume fraction of the NPs has a great influence on the mechanical performance of the nanocomposite. For that reason, we quantify the 18


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effect of NP loading on the visco-elastic properties of the systems, while keeping the NP-NP interaction strength constant and equal to ε C60 −C60 = 3.6kcal / mol , Fig. 8(a). When the filler weight fraction is small, ( w = 4.0% ), the storage modulus exhibits a monotonic dependence on the strain amplitude, whereas for w = 14.3% and

w = 25.0% , the dependence is non monotonic. We can explain this discrepancy as follows: at small strain amplitudes, the NPs exist as isolated clusters in the polymer matrix without inter-connecting with each other. An increase of the shear amplitude leads to the approaching of the NPs, their clustering, and the formation of a continuous and ordered network structure which results in the appearance of the maximum value of the storage modulus. However, by further increasing the strain amplitude to γ = 0.5 , the continuous network gradually dismantles, which is accompanied by a decrease of the storage modulus. The dependence of the number of neighboring C60 and of the total filler-filler and polymer-filler interaction strength on the strain amplitude is displayed in Fig. S3(a), (b) and (c), respectively, for the same values of the filler mass fraction. The structural evolution of the particle morphology for w = 4.0% and w = 14.3% is shown separately in Fig. 8(c) and (d). Here we infer that there could exist a critical weight fraction of NPs, above which the isolated NPs clusters become gradually connected to each other, thus forming a continuous network structure. However, if one further increases the weight fraction of the NPs to

w = 40% , the storage modulus displays a monotonic decrease, as shown in Fig. 8(b). A possible explanation is that at such a high weight fraction of NPs, a filler network that spans the whole system is already formed at initial time, and is gradually broken 19


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Page 20 of 45

as the strain amplitude is increased. The snapshots of the structural evolution of the filler network, Fig. 8(e), indicate the phase separation by the formation of isolated cylinder structures. 3.2 Patchy nanoparticle Due to directional interactions, patchy nanoparticles with specific arrangements of attractive sites at prescribed locations on the particle surface tend to form novel structures in the polymer matrix, compared to uniform NPs. As shown in Fig. 9(a), patchy2 particles tend to form a chain-like structure in the considered range of filler weight fraction, i.e., w = 4.0% , w = 14.3% and w = 25.0% ; a branch-like structure occurs at w = 25.0% . On the other hand, patchy4 NPs tend to form a sheet-like structure for w = 6% , w = 17% and w = 29% , as shown in Fig. 9(b). In this part, we mainly study the effect of the self-assembled structures on the storage modulus and the mechanical property. In the experimental test, the storage modulus is obtained by varying the shear amplitude and the temperature always ranges from 60 oC to 100 oC . In our simulations, the simple shear with different strain amplitude ( γ =0.05,

0.10, 0.15, 0.25, 0.30, 0.40, 0.50, 0.08, 1.0) is maintained for 20ns, and after that we collect the data from the last 1ns at 373 K and get the result as plotted in Fig. 9(c) and (d). The obtained results indicate that the self-assembled linear structure induces a better mechanical reinforcement and a more prominent non-linear behavior compared to that of the self-assembled sheet structure. Further, we examine the change in the number of neighboring C60, within a distance of 12.06Å, as a function of the strain amplitude. Although the storage modulus evidently changes at small strain 20


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Langmuir

amplitudes, the number of neighboring C60 does not change to a noticeable degree. An additional noteworthy observation is that at small strain amplitudes the change in the storage modulus of systems containing a sheet-like filler structure is more stable than that of systems comprised of linearly structured filler particles. Finally, for PB-based polymer nanocomposites we performed the uniaxial tensile deformation to study the effect of the self-assembled structures on the stress-strain behavior. To this end, we examine the cases of chain-like and sheet-like structures filled polymer nanocomposites. During the simulation the temperature is set to 298K as adopted in the experiment. Before the deformation, we cross-link the PB elastomer matrix following the approach as follows: After enough equilibration, the permanent cross-linking bonds are imposed in the system, by randomly selecting one pair of beads, which belong to two different chains. If the center-to-center distance between two atoms from different polymer chains is smaller than 5.20Å, then a bond modeled by the harmonic potential energy is produced between these two randomly chosen atoms and the chemical coupling density (defined as the ratio of the number of the cross-linked bonds to the total box volume) is ρ1 = 0.0011 . We want to point out that although the critical distance to form the chemical cross-linking bond is a little large, such as around 5.20 Å, leading to a little jump of the total energy in the simulation during the cross-linking process, this long distance can guarantee to effectively form many chemically cross-linked bonds. The results, along with a number of representative snapshots, are shown in Fig. 10. Apparently, the pure PB system exhibited the lowest tensile stress, while the chain-like structure leads to 21


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Page 22 of 45

more improved stress-strain behavior compared to the sheet-like structure. In order to explain the reason for this phenomenon, we calculate the total interaction energy between C60-C60 and C60-PB during the tensile deformation as shown in Fig. 10(b) and (c). The total interaction energy between C60-PB is much larger for cis-PB/patchy2 system than that of cis-PB/patchy4 system, indicating that the self-assembled chain-like structure has more interfacial area than that of the sheet-like structure. Meanwhile, we observe that the total interaction energy between C60-C60 decreases faster for the sheet-like structure than that of the chain-like structure. The Video S1 and S2, respectively for systems with patchy2 and patchy4 NPs, also show that during the tensile deformation the sheet-like structure has more probability to become broken-up than the chain-like structure. There are two reasons for this phenomenon: one is that the one dimensional linear structure is more flexible and smooth than the two dimensional sheet structure during the deformation process. The other reason is that the patchy4 can not self-assemble to form a perfect sheet-like structures. During the tensile process, the defects of the sheet-like structures will induce the broken-up of the sheet-like structures at the small strain. In conclusion, the chain-like structure is better than the sheet-like structure in improving the mechanical property. IV. Conclusions

By employing both the coarse-grained polymer composite model and the united-atom model of the chemistry-specific cis-polybutadiene-based composite, we 22


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Langmuir

performed molecular-dynamics simulations of polymer nanocomposites filled with uniform and patchy nanoparticles (NPs). We show the ability of the MD simulations to qualitatively reproduce the experimentally observable Payne effect. The effect of the dynamic periodic shear amplitude and the interaction strength between uniform NPs on the aggregation kinetics are monitored, and the change of the storage modulus as a function of the aggregation time is quantified. By imposing a dynamic periodic shear on the equilibrated state, and varying the interaction strength between uniform NPs and its weight fraction, we examine the change of the storage modulus as a function of the strain amplitude. In addition, we study the self-assembly of the patchy NPs in the polymer matrix, which form typical chain-like and sheet-like structures, and compare its effect on the visco-elastic and stress-strain behavior of the nanocomposite. Overall, this study aspires to provide with scientific guidelines for the fabrication of high performance polymer nanocomposites filled with uniform or patchy NPs.

23


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Page 24 of 45

Table 1. The force-field parameters for the cis-PB model. The united atom model of cis-PB is shown in Figure 1(b).

(a) Bond stretching

kb(kcal (mol Å2)-1)

r0( Å)

CH2-CH2

331.5

1.54

CH2-CH

384.5

1.50

CH=CH

516.5

1.34

(b) bond angle bending

kθ(kcal mol-1)

θ0(deg)

CH2-CH2-CH

57.5

111.65

CH2-CH-CH

44.7

125.89

(c) dihedral angle

K1

K2

K3

K4

K5

K6

CH2-CH=CH-CH

-

12.1

-

-

-

-

CH2-CH2-CH=C

0.5165

-0.236

0.2777

0.1315

0.173

0.082

CH-CH2-CH2-CH

-0.444

0.3095

-1.8195

-0.033

-0.1235

-0.095

(d) non-bonding

ε(kcal mol-1)

σ(Å)

rcutoff

CH2

CH2

0.0936

4.009

10.023

CH2

CH

0.1015

3.793

9.483

CH

CH

0.1000

3.385

8.463

24


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Langmuir

(a) The coarse-grained model of the nanocomposite

(b) The united atom model of cis-PB

(c) The filler NPs

Figure 1. Model building blocks studied in this work. (a)The coarse-grained model of the nanocomposite. Four filler particles of diameter 10 times the diameter of the monomer beads are displayed at T=0.65. For clarity, only a fraction of the polymer chains is shown. Periodic boundary conditions are used in all three directions, the filler volume fraction is 28.5%. (b) The united atom model for cis-1,4-polyisoprene. (c) The filler NPs used in this work: (1) Uniform C60 without chemical modification (2) Side view of a patchy sphere with two diametrically opposed circular patches, denoted by patchy2. (3) Top view of patchy2. (4) Patchy sphere with four circular patches arranged in an equatorial plane, denoted by 25


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Page 26 of 45

patchy4. Note that the lime and pink beads represent “atoms” in the patches. The blue beads represent the rest of the particle.

Figure. 2 (a) Schematic view of the shear deformation of the simulation box. The shear force is in the z direction with its gradient in the x direction. The boxes enclosed by the solid and dash lines are the deformed and undeformed boxes, respectively, and they have the same volume. (b) Schematic view of the uniaxial tension of the simulation box along the z-axis direction.

26


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Langmuir

Table 2. (a) The force-field parameters for the uniform C60 filled system model. (b) The force-field parameters for the patchy C60 filled system model. (c) The force-field parameters for the coarse-grained model. C60* represent the carbon atoms without patchy.

(a) uniform NP System

ε(kcal mol-1)

Equal

rcutoff

time

Number & weight fraction of C60

C60-CH2 C60-C60

C60-PB

C60-C60 C60-CH

9.483

Initial state

3.6

Case I

0.1

0.1

3.791

10ns 8.463 9.483

0.1

8.463

20ns

N=80 w=27%

20ns

N=80 w=27%

8.463 9.483 Case II

1.2

0.1

8.463 8.463

N=10 w=4.5%

9.483 Case III

3.6

0.1

8.463

20ns 8.463

N=40 w=16% N=80 w=27%

(b) patchy NP

*

C60 -

C60*

C60*- CH2

Patchy - CH2

C60*- CH

Patchy - CH

0.1

0.1

9.483

9.483

8.463

8.463

*

C60 -Patchy Patchy-Patchy

ε(kcal mol-1)

0.1

0.1

0.5

rcutoff

3.791

3.791

8.463

27


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Page 28 of 45

(c) coarse-grained model monomer-monomer

bead-bead of

bead-bead of

different NP

same NP

ε(MD units)

1

1

rcutoff

3.5

3.5

monomer -NP

50

2

3.5

3.5

28


Page 29 of 45

Storage modulus

(a) 25 20

Particle diameter (σ: diameter of a monomer unit) 13.0σ 10.0σ 8.0σ 6.5σ

15 10 5 0 0.01

0.1

Strain Amplitude γ

(b) loss modulus

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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20 18 16 14

Particle diameter (σ: diameter of a monomer unit) 13.0σ 10.0σ 8.0σ 6.5σ

12 10 8 6 0.01

0.1

Strain Amplitude γ Figure 3. Shear-strain dependence of (a) the storage G’ and (b) the loss G’’ modulus for the simulated coarse-grained model of a nanocomposite. The mechanical reinforcement upon filler loading and the Payne effect are clearly observed. T=0.65, filler volume fraction is 28.5 %.

29


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(a) ( Kcal/mol)

-200 -400 εC

=0.1kcal/mol 60-C60

-600

60-C60 EC

γ=0 γ=0.2 γ=0.6 γ=0.8 γ=1.0

0

-800

-1000 -1200 0

1

2

3

4

5

6

7

8

9

10

Time/ns

(b)

γ=0 γ=0.2 γ=0.6 γ=0.8 γ=1.0

0

( Kcal/mol) 60-C60

-4000

EC

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

Page 30 of 45

-8000 εC

-12000

=1.2kcal/mol 60-C60

-16000 -20000 -24000 -28000 0

1

2

3

4

5

6

7

8

9

10

Time/ns

30


Page 31 of 45

10000

(c)

γ=0 γ=0.2 γ=0.6 γ=0.8 γ=1.0

0

EC -C ( Kcal/mol) 60 60

-10000 -20000 -30000

εC

=3.6kcal/mol 60-C60

-40000 -50000 -60000 -70000 -80000 -90000 0

1

2

3

4

5

6

7

8

9

10

Time/ns

(d) -C ( Kcal/mol) 60

0 -15000 -30000 εC

=0.1kcal/mol 60-C60

-45000

εC

=1.2kcal/mol

εC

=3.6kcal/mol

60-C60

60 EC

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

Langmuir

-60000

60-C60

-75000 0.0

0.2

0.4

0.6

0.8

1.0

Strain Amplitude γ Figure 4. The strain-amplitude dependence of the total interaction energy between C60-C60 when

εC

60 − C60

(a)

εC

60 − C60

= 0.1kcal / mol

,

(b)

εC

60 − C60

= 1.2kcal / mol

and

(c)

= 3.6kcal / mol . (d) The strain-amplitude dependence of the total interaction

energy between C60-C60 at t=4ns for various values of

ε C60 − C60 . Note that the temperature is

fixed at T = 298 K and the dynamic periodic shear frequency is

f = 1010 Hz .The weight

fraction of the NPs is 25.0%.

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Page 32 of 45

Figure 5. Snapshots of C60 filled polymer systems. (a) Initial dispersion state of C60 in the polymer matrix. The final equilibrium dispersion state of C60 for various shear amplitudes : (b) γ = 0 (c) γ = 0.2 (d) γ = 0.6 (e) γ = 0.8 and (f) γ = 1.0 . Note that the blue beads represent C60 , and the polymer chains is fully transparent. The mass fraction of C60 is w=25.0% and the C60-C60 interaction strength is

εC

60 − C60

= 3.6kcal / mol .

32


Page 33 of 45

1000

Storage modulus/MPa

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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γ=0 γ=0.2 γ=0.6

800 εC

60-C60

=3.6kcal/mol

600

400

200

0 0

2

4

6

8

10

Time/ns

Figure 6. The change of the storage modulus as a function of the aggregation time for various dynamic shear amplitudes. The temperature is fixed at T = 298 K and the dynamic periodic shear frequency is

f = 1010 Hz . The mass fraction of C60 is w=25.0%

and the C60-C60 interaction strength is

εC

60 − C60

= 3.6kcal / mol .

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Langmuir

(a) 700

εC60-C60=0.1kcal/mol

600

Storage modulus/MPa

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


500 400

Page 34 of 45


300 200

(ii)

(i)

100

(iii)

(iv)

0 -100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Strain Amplitude γ

(b) ε C

(c) ε C

60 − C60

60 − C60

= 3.6kcal / mol

= 1.2kcal / mol

34


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(d) continuous NP network for ε C

(e) isolated NP cluster for ε C

60 − C60

60 − C60

= 3.6kcal / mol

= 3.6kcal / mol

Figure 7. The strain-amplitude dependence of (a) the storage modulus and snapshots of the C60

filled

εC

60 − C60

εC

60 − C60

polymer

= 3.6kcal / mol

system and

under (c)

εC

different 60 − C60

shear

amplitudes

= 1.2kcal / mol .

In

the

for case

(b) of

= 3.6kcal / mol the structure of the NPs aggregate is unfolded in the x, y and z

directions for the following values of the dynamic shear amplitude: (d) γ = 0.25 and (e)

γ = 0.5 . The temperature is fixed at T = 373 K and the dynamic periodic shear frequency is

f = 1010 Hz . The mass fraction of C60 is w=25.0%. Note that the blue beads

represent C60 , and the polymer chains is red.

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Langmuir

Storage modulus/MPa

(a) 700

w=4.0%

600

w=14.3%

500

w=25.0%

400 300 200

100 0

(iii)

(ii)

(i)

-100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Strain Amplitude γ

(b) 6000 Storage modulus/MPa

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

Page 36 of 45

w = 40.0%

5000 4000 3000 2000 1000

0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 Strain amplitude γ

(c) w = 4.0%

36


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(d) w = 14.3%

(e) w = 40.0% Figure 8. The strain-amplitude dependence of the storage modulus when the weight fraction of the filler is fixed at (a) 4.0%, 14.3%, 25.0% and (b) 40%. The snapshots of the dynamic evolution process of the filler network during the shear deformation for different values of the shear amplitude when (c) w = 4.0% , (d) w = 14.3% and (e) w = 40.0% . Note that the temperature is fixed at T = 373 K , the dynamic periodic shear frequency is

f = 1010 Hz , and ε C

60 − C60

= 3.6kcal / mol . The blue beads represent C60 , and the

polymer chains is red.

37


Langmuir

(a) chain-like structure

(b) sheet-like structure

(c)

70 60

Storage modulus/MPa

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

Page 38 of 45

50 40

w=25%-patchy2 w=14%-patchy2 w=4%-patchy2 w=29%-patchy4 w=17%-patchy4 w=6%-patchy4

30 20 10 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Strain amplitude γ

38


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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(d)

3.0

Number of neighboring C60

Page 39 of 45

2.5

w=25%-patchy2 w=14%-patchy2 w=4%-patchy2 w=29%-patchy4 w=17%-patchy4 w=6%-patchy4

2.0 1.5 1.0 0.5

0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1

Strain amplitude γ

Figure 9. (a) Snapshots of the linear structures formed by setting the number of patchy2 NPs equal to 10, 40 and 80. (b) Snapshots of the sheet structures formed by setting the number of patchy4 NPs equal to 16, 49 and 100. The strain-amplitude dependence of (c) the storage modulus, and (d) the average number of neighboring C60 within a distance of 12.06Å from each C60 for different ordered structures. Note that ε C

* * 60 − C 60

= 0.1kcal / mol ,

ε patchy − patchy = 0.5kcal / mol ,and ε patchy − C = 0.1kcal / mol . The temperature is fixed at * 60

T = 373 K . The blue beads represent the patchy C60 , and the polymer chains is fully transparent.

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Langmuir

(a) 8000 7000

Tensile stress(MPa)

6000

pure cis-PB cis-PB/patchy4 cis-PB/patchy2

5000 4000 3000 2000 1000 0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Tensile strain(%)

(b)

-200 -400

cis-PB/patchy2 cis-PB/patchy4

EC -C (Kcal/mol) 60 60

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

Page 40 of 45

-600 -800

-1000 -1200 -1400 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Tensile strain(%)

40


Page 41 of 45

(c)

-4200

EC -PB(Kcal/mol) 60

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-4400 -4600

cis-PB/patchy2 cis-PB/patchy4

-4800 -5000 -5200 0.0

0.5

1.0

1.5

2.0

2.5

3.0

Tensile strain(%) Figure 10. (a) The tension curves for the pure cis-PB and the cis-PB filled with patchy2 and patchy4 NPs. The mass fraction of the fillers is w = 29% . (b) The total C60-C60 interaction energy and (c) the total C60-PB interaction energy during tensile deformation. Note that

εC

* * 60 − C60

= 0.1kcal / mol , ε patchy − patchy = 0.5kcal / mol and ε patchy −C * = 0.1kcal / mol . 60

The temperature is fixed at T = 298 K . The blue beads represent the patchy C60 , and the polymer chains is fully transparent.

Acknowledgements

This work is supported by the National Basic Research Program of China 2015CB654700(2015CB654704), the Foundation for Innovative Research Groups of the NSF of China (51221002), the National Natural Science Foundation of China (51333004 and 51403015), the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase).The cloud calculation platform of BUCT is also greatly appreciated. AVL and TD acknowledge the FOM Foundation for the support of the research project #11VEC06. Coarse-grained MD simulations were sponsored by the Stichting Nationale Computer faciliteiten (National Computer Facilities Foundation, NCF) through the usage of its supercomputer facilities, with ﬁnancial support from the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (Netherlands Organization from Scientiﬁc 41


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Table of Content Graphics

Influence of morphology on the mechanical properties of polymer nanocomposites filled with uniform or patchy nanoparticles Lu Wang1, 2, Zijian Zheng1,2,3, Theodoros Davris5, Fanzhu Li1,2,3, Jun Liu1, 2, 3*, Youping Wu1, 2, 3, Liqun Zhang1, 2,3,4*, Alexey V. Lyulin5*

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Influence of Morphology on the Mechanical Properties of Polymer

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