Determination of linear viscoelastic properties of an entangled polymer melt by probe rheology simulations

Particle rheology is used to extract the linear viscoelastic properties of an entangled polymer melt from molecular dynamics simulations. The motion of a stiff, approximately spherical particle is tracked in both passive and active modes. We demonstrate that the dynamic modulus of the melt can be extracted under certain limitations using this technique. As shown before for unentangled chains [Karim et al., Phys. Rev. E 86, 051501 (2012)], the frequency range of applicability is substantially expanded when both particle and medium inertia are properly accounted for by using our inertial version of the generalized Stokes-Einstein relation (IGSER). The system used here introduces an entanglement length $d_T$, in addition to those length scales already relevant: monomer bead size $d$, probe size $R$, polymer radius of gyration $R_g$, simulation box size $L$, shear wave penetration length $\mathrm{\Delta }$, and wave period $\mathrm{\Lambda }$. Previously, we demonstrated a number of restrictions necessary to obtain the relevant fluid properties: continuum approximation breaks down when $d\gtrsim \mathrm{\Lambda }$; medium inertia is important and IGSER is required when $R\gtrsim \mathrm{\Lambda }$; and the probe should not experience hydrodynamic interaction with its periodic images, $L\gtrsim \mathrm{\Delta }$. These restrictions are also observed here. A simple scaling argument for entangled polymers shows that the simulation box size must scale with polymer molecular weight as $M_w^3$. Continuum analysis requires the existence of an added mass to the probe particle from the entrained medium but was not observed in the earlier work for unentangled chains. We confirm here that this added mass is necessary only when the thickness $L_S$ of the shell around the particle that contains the added mass, $L_S>d$. We also demonstrate that the IGSER can be used to predict particle displacement over a given timescale from knowledge of medium viscoelasticity; such ability will be of interest for designing nanoparticle-based drug delivery.

Figure 1

Values of the linear storage, $G^{\prime }\left(\omega \right)$ (open symbols) and loss, $G^{\prime \prime }\left(\omega \right)$ (filled symbols) moduli of entangled polymer melt of chain length $N=80$ obtained from NEMD (squares) simulations. Values are compared with the literature data (circles) obtained using Green-Kubo simulations [49].

Figure 2

Comparison of frequency dependence of viscoelastic moduli as determined by active rheology, NEMD, and Green-Kubo techniques. The upper four (solid-star, dashed-plus, filled circles, and filled squares) data sets are for $G^{\prime \prime }\left(\omega \right)$ and the lower four data sets (solid-star, dashed-plus, empty circles, and empty squares) are for $G^{\prime }\left(\omega \right)$. Solid-star lines differ from corresponding dashed-plus lines in that the former represent results obtained by accounting for medium and particle inertia while the latter represent results obtained by ignoring the medium and particle inertia in the analysis procedure. Uncertainties are shown on the results that were obtained by accounting for inertia; these were estimated by using the blocking method [50].

Figure 3

Averaged MSD (black dots) of a probe particle from 10 independent replicas. Solid red line is the fitting curve obtained by Eq. (6), with ${\alpha }_1=2.807,{\alpha }_2=1.877,{\alpha }_3=0.2111,{\alpha }_4=0.7933,{\tau }_0=0.08,{\tau }_1=1.6,{\tau }_2=37,{\tau }_3=600,{\tau }_4=150000$ $\left({\chi }^2=0.0086\right)$. Inset shows the logarithmic slope of MSD with respect to time (calculated using Euler differencing approach), where a transient caging effect due to the entangled mesh is evident from the minimum in the slope.

Figure 4

Comparison of the $G^{*}\left(\omega \right)$ values obtained by passive probe rheology, NEMD, and Green-Kubo techniques. The upper red solid star and red dashed plus lines represent $G^{\prime \prime }\left(\omega \right)$ and the lower blue solid star and blue dashed plus lines represent $G^{\prime }\left(\omega \right)$ values obtained probe rheology. For the NEMD and Green-Kubo values, the open symbols denote $G^{\prime }\left(\omega \right)$ while the solid symbols denote $G^{\prime \prime }\left(\omega \right)$.

Figure 5

Comparison of active probe rheology results for two different particle sizes, $R=2.5$ and 12.0. The upper two curves along with the filled symbols are for $G^{\prime \prime }\left(\omega \right)$ and the lower two curves along with the open symbols are for $G^{\prime }\left(\omega \right)$.

Figure 6

Frequency dependence of the penetration depth (solid black line) and the wavelength (dashed red line) of the propagating shear wave calculated using the literature Green-Kubo values. The minimum distance between the moving probe particle and its images (upper) and the probe particle size (lower) are shown by horizontal (dotted blue) lines.

Figure 7

Comparison of the probe particle MSD measured in the passive rheology simulations (dotted) and that predicted from literature $G^{*}$ values [49] (shown in inset). Solid line is derived from the IGSER, whereas dashed line is from the conventional GSER. In inset, solid lines are fitting curves by a superposition of the Zimm model (with the relaxation time 20 and the amplitude 0.12) and the two-mode Maxwell model (with the relaxation times $\{1100,65\}$ and the amplitudes $\{0.038,0.065\})$.

Figure 8

Comparison of active rheology results obtained by considering the probe particle mass to be $m_{\mathrm{observed}}=10500$ and $m_{\mathrm{eff}}=13500$. The results from NEMD simulations and literature Green-Kubo calculations are also shown.