Chapter 4 Temperature Dependence of Polymer Diffusion in MWCNT/PS
4.3.2 Temperature dependence of tracer diffusion coefficients at fixed MWCNT
loadings
The tracer diffusion of homopolymers in well-entangled melts can be described by
the reptation model, 36-37
π· = 4
15
πππ0ππ΅π
π2π
0(π) (4.5)
where T is temperature, ΞΎ0 is the monomeric friction coefficient, Meis the entanglement
molecular weight, M0 is the monomeric molecular weight, and M is the tracer molecular
weight. The M-2 dependence of D has been confirmed experimentally.38-49 If we assume
that Me is temperature independent, Equation 4.5 shows that D/T is proportional to ΞΎ0(T)-1
with the same functional dependence on temperature. Thus, the WLF equation, Equation
4.2, sufficiently describes the temperature dependence of both D/T and ΞΎ0(T). This is well
documented for the tracer diffusion coefficients in homopolymer melts.30-31, 50
In Figure 4.4 we plot the temperature dependence of the tracer diffusion
coefficients for three polymer nanocomposites (0.5, 2, and 6 wt% MWCNT) as log(D/T)
versus 1/(T-Tβ). The WLF equation (Equation 2) predicts a straight line with a slope of
-C2. For polystyrene, viscosity measurements found C2= 710 K and Tβ = 322 K.51 This Tβ
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glass transition temperature (Tg). TheTgβsof our polymer nanocomposites are independent
of the MWCNT loading (Figure 4.1), suggesting the absence of strong attraction between
the PS and MWCNTs.52 Thus, we fix Tβ (322 K) and use C1' and C2 as free parameters to
fit Equation 4.2. Figure 4.4 shows the fitting results and C2 values associated with three
MWNCT loadings, where C1' for all CNT concentrations is ~ -10.5. The fitting results of
C1 and C2 for other MWCNT concentrations and the error analysis are provided are
provided in Figure 4.5. The WLF equation accurately captures the temperature
dependence of the tracer diffusion coefficient in MWCNT/PS nanocomposites across three
decades of (D/T). While the variations in C2 for the three MWCNT loadings in Figure 4.4
is small (within 10 %), the middle composition (2wt% MWCNT) has the largest C2 value,
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Figure 4.4. Temperature dependence of tracer diffusion coefficients for 0.5, 2, and 6 wt% MWCNT/PS nanocomposites, plotted as D/T versus 1/(T-Tβ) with Tβ = 322 K. The red
lines are best fits of the WLF equation (Equation 4.2) to the data and the values of C2 are
given. These curves are vertically shifted for clarity.
Figure 4.5. Fitting results for C1 and C2 for all MWCNT concentrations, except 0.7 wt%. We include results from samples with 0.1 and 1 wt % MWCNT although the temperature range studied is narrower and, consequently, the error bars are larger. At 0.7 wt%, the temperature range is too narrow (only ~ 20ΒΊC) to have reliable fitting results. As can be seen in the figure, C1 is approximately constant across the whole concentration, and C2
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Previously, we developed a trap model22 to describe the diffusion minimum
(D/D0)min that occurs in our MWCNT/PS and SWCNT/PS nanocomposites systems. It is
a phenomenological model that simulates the center-of-mass diffusion of polymer chains
through a three-dimensional lattice filled with cylindrical fillers. The model permits
anisotropic polymer diffusion near the cylinders (Dper and Dpar), but does not consider the
effect of temperature. There are three parameters in the simplest variation of the trap
model: p0 describes the diffusion both inside and outside the traps, p1 is the probability
for polymer chains to diffuse into or out of the cylindrical traps, and r is the radius of the
cylindrical traps. The significant slowing down of the center-of-mass diffusion at low trap
concentration requires a lower jump probability for entering or escaping the cylindrical
trap around the cylindrical filler (p1 < p0), and the recovery rate at higher particle
concentration is due to the percolation of the traps. In our previous comparison of
experimental and simulation results, an increase in r correlates to an increase in Mw of the
matrix polymer chains.From Figure 4.3(b), we find that the diffusion minimum occurs at
2wt% at all temperatures, and this implies that the trap radius is independent of
temperature. We also observe here that the decrease of the diffusion coefficients is
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chains to enter or escape the trap (higher p1). Alternatively, (p0 β p1) appears to decrease
at higher temperature, which corresponds to less anisotropy between the center-of-mass
diffusion parallel and perpendicular to the cylindrical trap, MWCNT in our experiments.
Karatrantos et al. observed anisotropic diffusion coefficients near a SWCNT in
their simulation studies,25 although only when there is an attractive interaction between
the SWCNT and polymer chain. In their simulations, they also found that self-diffusivity
decreases with the addition of SWCNT. To date only one SWCNT concentration (0.4 v%)
has been simulated, which is below 2wt%, where we observe (D/D0)min in MWCNT/PS
nanocomposites. To evaluate the polymer-MWCNT interactions in our nanocomposites,
we fit the tracer diffusion coefficients to a modified WLF equation with an Arrhenius
term53 logπ· π = πΆ1β²β πΆ2 1 (π β πβ)β πΈπ π π (4.6)
where Ea is the activation energy. The parameters C2 and Tβ are fixed to 710 K-1 and 322
K, corresponding to the PS melt.51 This approach embodies the assumption that changes
in the tracer diffusion coefficients are dictated by the polymer-MWCNT interactions. For
this assumption to be valid, the activation energy should increase with polymer-MWCNT
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data provides activation energies of 3.6 kJ/mol, 6.8 kJ/mol, and 3.9 kJ/mol, for
nanocomposites with 0.5, 2, and 6 wt% MWCNT, respectively. The normalized
interaction energy for the 0.5 wt% MWCNT composite is Ea/R = 433 K, which suggests
that the activation energy is reasonable and comparable to temperatures used in our
diffusion studies (425 K to 487 K). However, the activation energies do not increase
monotonically with nanotube concentration and, thereby, invalidates the assumption of
Equation 4.6 that the polymer-MWCNT interfacial interactions alone impede polymer
diffusion. While polymer-MWCNT interactions might make important contributions to
polymer dynamics, attractive interactions are inconsistent with the (D/D0)min found as a
function of increasing MWCNT concentration for all temperatures studied. Moreover, our
Tgmeasurements found no substantial influence upon adding MWCNTs, which further
suggests the negligible effect of energetic interactions between these polymer chains and
MWCNTs in this temperature range. Thus, the trap model suggests that the deeper
minimum observed in D/D0 at lower temperatures corresponds to greater anisotropy in
the local diffusion coefficients parallel and perpendicular to the MWCNTs (Dpar > Dper),
and molecular dynamics simulations suggest that the origin of this anisotropy is not
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A salient finding of this paper is the good fit of the WLF equation (Equation 4.2,
Figure 4.4) for the tracer diffusion coefficient in MWCNT/PS nanocomposites. This
result suggests that the observed temperature dependence of the tracer diffusion arises
primarily from the temperature dependence of the monomeric friction coefficient and the
fractional free volume in the system. The increase of C2 at 2 wt% MWCNT implies that
near the rheological percolation threshold there is a minimum in Ξ±f, the thermal expansion
coefficient of free volume. This implies that the addition of small quantities of MWCNT
(< 2 wt%) to polystyrene significantly decreases the thermal expansivity of the free
volume in the nanocomposite relative to the neat homopolymer and at higher
concentrations the thermal expansivity gradually returns to that of the neat homopolymer.
Interestingly, tracer polymer diffusion is apparently dominated by the
temperature-dependent changes in the fractional free volume of the polymer matrix. In
contrast, linear viscoelastic measurements detect a combination of these changes in the
fractional free volume in polystyrene (reduce Gβ) and the presence of high aspect ratio,
semi-flexible rods (increase Gβ). It is well documented that the addition of carbon
nanotubes, or a host of other high aspect ratio nanoscale fillers, may produce a transition
110
nanoparticles and their eventual formation of a reinforcing network. Future studies of
polymer dynamics in nanocomposites should seek to address the free volume in the
system and its thermal expansivity as a function of nanoparticle concentration and
evaluate local heterogeneities in the free volume that might account for anisotropy in
polymer diffusion near high aspect ratio nanoparticles.
4.4 SUMMARY
In this chapter, we investigated the temperature-dependence of the tracer diffusion
coefficient of dPS in MWCNT/PS nanocomposites. Firstly, as a function of MWCNT, D
exhibits a minimum at 2wt% MWCNT, independent of temperature. Secondly, the
minimum in D/D0 becomes shallower at higher temperatures, which suggests that the
mechanism causing the diffusion to slow down at low MWCNT concentrations is less
effective at higher temperatures. Thirdly, at fixed nanotube concentrations, the WLF
equation describes the temperature-dependence of the tracer diffusion coefficients in
MWCNT/PS nanocomposites. When the WLF equation was modified to include
MWCNT-polymer interactions, the results for the activation energy were not reasonable.
Thus, we conclude that polymer diffusion in these polymer nanocomposites is associated
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is associated with a minimum in the thermal expansion coefficient of free volume.
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