Supplementary material
Molecular simulation study of role of polymer–particle interactions in the straindependent viscoelasticity of elastomers (Payne effect)
Yulong Chen,2,3 Ziwei Li,2,3 Shipeng Wen,1,3 Qingyuan Yang,1,5,a) Liqun Zhang,
Chongli Zhong,1,5 and Li Liu2,3,4,a)
1State
Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology,
Beijing 100029, China
2State
Key Laboratory of Chemical Resource Engineering, Beijing University of Chemical Technology,
Beijing 100029, China
3Beijing
Engineering Research Center of Advanced Elastomers, Beijing University of Chemical
Technology, Beijing 100029, China
4Key
Laboratory of Carbon Fiber and Functional Polymers, Ministry of Education, Beijing University of
Chemical Technology, Beijing 100029, China
5Laboratory
of Advanced Nanostructural Materials, Beijing University of Chemical Technology, Beijing
100029, China
a)Corresponding
author: qyyang@mail.buct.edu.cn, LiuL@mail.buct.edu.cn
This supplementary document contains the following material.
SI. Details of the initial simulation system construction
SII. Mean-square displacements of NPs across polymer network after crosslinking
SIII. Additional analyses of polymer dynamics and the calculations of mean relaxation times (MRT) of polymer
monomers.
SIV. Interfacial potential energy of polymers from NPs during oscillatory shear
SV. Comparison of the IIDS factors of polymer in unfilled systems at different temperatures and monomers
located in first layer in filled systems
SVI. Strain-dependent viscoelasticity of filled system at different temperatures
1
SI. Details of the initial simulation system construction
To generate the initial configuration for each simulation system, all the polymer chains, crosslinking agent
molecules (CLAs) and nanoparticles (NPs) were placed in a very large simulation box, so the densities of the
systems were low (the number densities of the polymer beads (PBs) were about * = 0.01). Such systems were
first relaxed using equilibrium molecular dynamics (EMD) simulation in the canonical (NVT) ensemble at the
fixed temperature T* = 1.0 without any attractive interactions for 2103 (2106 timesteps). Then, all the
attractions involved were switched on (the polymer–particle interaction strength (pn) was set at 2.0 so that the
NPs could be well dispersed in the matrix before the crosslinking procedure1) and the isothermal-isobaric (NPT)
ensemble was performed at temperature T* = 1.0 and pressure P* = 0 to reduce the system volume and to
increase the number density of PBs to the melt density (around * = 0.85). Following this, all structures of the
systems were equilibrated over time. To make sure whether the systems reached equilibrium, the mean-square
radius of gyration 𝑅𝑔2 of the polymer chains with time have been monitored (see Figure S1(a)). It was found that
4𝑅𝑔2 fluctuated around 29.4 (𝑅𝑔 =2.71) after 2.5103. This is the signature when system reached equilibrium. At
the same time, the mean-square displacement (MSD) of molecules exceeded 4𝑅𝑔2 at 2.5103 (see Figure S1(b)),
meaning that the polymer chains moved by at last 2𝑅𝑔 when the systems equilibrated. In addition, the dispersion
states of NPs, using snapshots and radial distribution function, were also checked to confirm the equilibrium of
the systems. In the current study, all the systems were implemented for at least 5103
to ensure fully
equilibrated.
(a)
MSD of molecules
32
4Rg
2
31
30
29
28
27
0
3
1x10
3
2x10
3
3x10
time/
3
4x10
3
5x10
60
(b)
50
40
30
20
10
0
0
3
1x10
3
2x10
3
3x10
time/
3
4x10
3
5x10
FIG. S1. (a) Running values of the mean-square radius of gyration 𝑅𝑔2 with simulation time, (b) mean-square
displacement (MSD) of polymer chains, i.e., MSD of center of mass of polymer chains, as a function of time, for
the system with NP volume fraction  = 0.152 and polymer–particle interaction εpn = 2.0.
2
Once reaching equilibrium, chemical bonds between PBs and CLAs were generated if their distances were
less than 1.25, where one CLA was allowed to link two PBs with only one CLA for each PB. These bonds were
also described using the FENE potential with parameters the same as those for the bonds in the polymer. Through
this procedure, all the polymer chains could be connected by the CLAs to form the crosslinked network. Then, pn
were tuned in a broad range (from 4.0 to 12.0) and NPT-MD simulation at T* = 1.0 and P* = 0 was further
conducted to equilibrate each system for about 104 until no evident change was achieved for the results with
longer time. Depending on the filler loading and polymer–particle interaction, the number densities of PBs
obtained were about * = 0.86–0.92, and the number densities of crosslinking sites were found to range from
about 0.04 to 0.05 –3, which were defined as ρcl = Ncl/V, where Ncl is the number of the CLAs (Ncl = 800) and V
is volume of simulation box (ranging from 15 000 to 20 000  3). It has been demonstrated that when mapping the
coarse-grained bead–spring model to real polymers, the diameter () of the PB corresponded to segmental
lengths of about 0.5 to 1.3 nm,2 depending on the specific polymers. As a result, the crosslinking densities of the
simulation systems were approximately 3  10 5 to 7  10 4 mol/cm3, which were within the ranges (10–5 to 10–4
mol/cm3) for real elastomers.3 It should be noted that there were a certain number of cross-links between the
monomers in the same chain (self-linkage). In fact, the formation of self-linkages is certainly possible in real
crosslinking or vulcanization processes.
3
SII. Mean-square displacements of NPs across polymer network after crosslinking
It is generally accepted that crosslinking can prevent the diffusion of NPs to some extent. In order to check
whether the NPs could truly diffuse across the polymer network, after crosslinking, we have measured the MSD
of NPs with time, as shown in Figure S2. The slope of the MSD versus time is proportional to the diffusion rate
of the NPs. It can be found that in the initial very short simulation time, the diffusion rate increased with
increasing polymer–particle interaction, which was due to the thermodynamically non-equilibrium states of the
systems when the polymer–particle interaction has just been tuned from 2.0 to larger value at the beginning of
this stage. In the following stage, as the systems gradually reached equilibrium, the diffusion rate of the NPs
decreased. Finally, the higher the εpn was, the lower the diffusion rate behaved, because the polymer segments
were tightly bound to the NPs at high polymer–particle attraction that can hinder the diffusion of the NPs. These
results demonstrated that the NPs can diffuse in the network of our model systems.
MSD of NPs
10
pn = 4.0
8
pn = 8.0
6
pn = 12.0
4
2
0
0
3
1x10
3
2x10
3
3x10
3
4x10
3
5x10
time/
FIG. S2. Mean-square displacement (MSD) of nanoparticles (NPs) as a function of time for systems ( = 0.152)
after cross-linked.
4
SIII. Additional analyses of polymer dynamics and the calculations of mean relaxation times
(MRT) of polymer monomers.
A dynamic parameter, i.e., the incoherent intermediate dynamic structure (IIDS) factor  qs (t ) , was adopted
to characterize the dynamics of polymers at the segmental scale in the initial systems, as defined by4-7
qs (t ) 
1 M
1 M sin(qrm (t ))
 exp[ iq  (rm (t )  rm (0))]  
M m1
M m1
qrm (t )
(1)
where M is the number of polymer monomers; q and q are the wave vector and number, respectively; (rm(t) –
rm(0)) and rm (t ) are the displacement vector and scalar of monomer m after time t, respectively. Similar to the
literature,7 the value of q was set to 6.9 in this study which also corresponds to the first peak location of the static
structure factor obtained from our unfilled system (Figure S3).
Figure S4 shows the IIDS factors calculated for polymer monomers located in the first layer that
surrounding the NPs (one bead thickness) and that of all the monomers in the systems with three different filler
volume fractions, where the result for the unfilled system is also included for comparison. It can be found that
within the broad range of pn examined, the IIDS factors for the monomers in the first layer almost coincide for
all the systems if t ≤ 0.1. This short-time regime corresponds to the ballistic motion of a monomer. 6 As time
increased, the decay of  qs (t ) becomes much slower with increasing polymer–particle interaction. This can be
explained by the fact that higher interfacial strength will result in a more significant reduction of the mobility of
chain segments near the NP surfaces.8,9 In addition, the increase of filler volume fraction can also slow down the
decay of  qs (t ) , since the monomer feel more constraints imposed by its neighboring NPs.
The decay of qs (t ) for all the monomers in each filled system also manifests a similar behavior to that of the
monomers in the first layer but with a faster rate, and both of them are slower than that for the neat system, as can
be reflected from the comparisons given in Figures S4. On the other hand, for the systems with weak polymer–
particle interactions and low filler loadings, the changes of  qs (t ) for all the monomers are very limited (dash
lines, Figure S4(a), while the decay rates become significantly slower with increasing εpn or , and even can be
close to those for the monomers located in the first layer (Figure S4(c)) because of more monomers subjected to
the constraints from NPs. Moreover, for the systems with  = 0.265 and εpn = 8.0–12.0, the decay curves of  qs (t )
5
occur a plateau after t > 0.1 and then drop quickly, as shown in Figure S4(c). The former phenomenon is
generally regarded as the β-relaxation with the latter as the α-relaxation.6 In the β-relaxation regime, the
monomers move within the cages composed by their nearest neighbors (monomers or NPs), while in the αrelaxation regime, the monomers start to leave these cages, and the temporarily frozen structures “melts”.
In addition, the mean relaxation time (MRT), i.e., the α-relaxation time τα, of all the polymer monomers and
the monomers present in the first layer (one bead thickness) in each system was calculated which was defined as
the time where qs (t ) decays to 1/e.7
6
2.5
S (q )
2.0
1.5
1.0
0.5
0.0
4
8
12
16
q
20
FIG. S3. Static structure factor S(q) for the neat elastomeric system, which was calculated according to the
equation given in ref. 7. The location of the first peak corresponds to q = 6.9.
1.0
1.0
(a)
0.8
0.6
s
s
q (t)
q (t)
pn
Increasing
0.6
0.4
0.2
0.0 -2
10
(b)
0.8
0.4
0.2
-1
10
0
10
t/
1
0.0 -2
10
2
10
10
-1
0
10
10
1
t/
10
2
10
3
10
1.0
(c)
-rela
0.8
xation
-
re l
ax
s
q (t)
0.6
ati
on
0.4
0.2
0.0 -2
10
-1
10
0
10
1
10
t/
2
10
3
10
4
10
FIG. S4. Incoherent intermediate dynamic structure functions (  qs (t ) ) for the polymer monomers located in the
first layer (solid lines) and for all the monomers (dash lines) in the systems with different polymer–particle
interactions (pn ranges from 4.0 to 12.0) at three filler volume fractions: (a)  = 0.066, (b)  = 0.152, and (c)  =
0.265. In each subfigure, the result for the unfilled system is also included for comparison (black dash line).
7
SIV. Interfacial potential energy of polymers from NPs during oscillatory shear
-24100
(a)
-24200
Eint
-24300
-24400
-24500
-24600
-24700
0
-24100
50
100
t/
150
200
50
100
150
200
(b)
-24200
Eint
-24300
-24400
-24500
-24600
-24700
0
t/
FIG. S5. Interfacial potential energy Eint (in units of ε) of polymers from NPs in the system with  = 0.066 and pn
= 10.0 during two periods of oscillatory shear at strain amplitudes of (a) γ0 = 0.1 and (b) γ0 = 0.4 with shear
frequency  = 0.01.
8
SV. Comparison of the IIDS factors of polymer in unfilled systems at different temperatures and
monomers located in first layer in filled systems
1.15
1.10
V(T)
1.05
1.00
Tg
0.95
0.90
0.0
0.2
0.4
0.6
0.8
1.0
T
FIG. S6. Dependence of the effective volume V(T) per monomer on temperature for the unfilled cross-linked
system. The abrupt change in the curve is generally considered as a signature of the glass transition for polymers.
The results indicate that the glass transition temperature Tg was about 0.4.
1.0
T=0.1
T=0.2
T=0.3
T=0.4
0.6
s
q(t)
0.8
0.4
T=0.5
T=0.6
T=0.8
0.2
First layer
T=1.0
0.0 -2
10
-1
10
0
10
1
10
t/
2
10
3
10
4
10
FIG. S7. Influence of temperature on the incoherent intermediate dynamic structure factor qs (t ) for all the
monomers in the unfilled system. The result for the monomers located in the first layer (black line) around
nanoparticles in the filled system is also shown for comparison, where the filler volume fraction was  = 0.265
and the polymer–particle interaction was pn = 12.0. It can be seen that the relaxation time for the monomers in
the first layer is faster than that for the unfilled system at temperatures below Tg, indicating that the polymer shell
in such system still not in the glassy state.
9
SVI. Strain-dependent viscoelasticity of filled system at different temperatures
20
5
G'
G"
12
6
(a)
T = 0.6
T = 0.8
T = 1.0
16
3
4
2
0.1
0.2
0.3

0
0.4
0.5
(b)
0.4
0.5
4
8
0
0.0
T = 0.6
T = 0.8
T = 1.0
1
0.0
0.1
0.2
0.3
0
FIG. S8. Dependence of (a) storage modulus G′ and (b) loss modulus G″ on the shear strain amplitude for the
filled system with  = 0.152 and with high polymer–particle interaction strength (pn = 10.0) at three different
temperatures, where the shear frequency  = 0.01.
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