Average position y (divided by channel height H) for the center of mass of unadsorbed moieties (ynadv) of interfacial chains as a function of time for linear and SCB PE melts at intermediate current strength. Right) Schematic description of the characteristic molecular mechanisms of interfacial ring chains for each flow regime. Probability distribution of the angle (θloop) of the loop cross-sectional plane with respect to.
In particular, the interfacial slip phenomena have been recognized as one of the most exciting research topics since Mooney's pioneering work because they deny the traditional no-slip boundary condition, the assumption that fluids at a fluid-wall interface move at the same speed as the wall, which is routinely have been applied to a number of fluid dynamics problems. However, these works necessitate further exploration of the nature of such regimes, as the studied regimes have certain limitations based on several hypothetical assumptions (e.g., weak excluded volume effect). In this study, we aim to elucidate the fundamental molecular properties of the stress relaxation process that occurs in polymer systems upon cessation of steady shear flow.
- Unentangled polymer dynamics
- Entangled polymer dynamics
The non-entangled polymer model, called the Rouse model (also known as bead-entropic spring model) describes the dynamics by sharing polymer chains with N beads and N-1 springs (size b) based on the Gaussian spring (Gaussian spring constant 3k TB2 k = b, where kB and b denote Boltzmann's constant and Kuhn's length, as shown in Figure 2.1.4). Note that the Rouse model neglects the excluded volume effect and the hydrodynamic interaction; therefore, it effectively elucidates the dynamics of non-entangled polymer melts. This indicates that the chain is spread longer than the Rouse time, while the chain has modes shorter than the Rouse time.
- Standard flows: shear and elongational flow
Note that the stress tensor predicted by Newton's constitutive equation can be written as Therefore, a fluid subjected to uniaxial extensional flow would undergo both expansion in the z direction and contraction in the x and y directions. The former has the same shape of the velocity profile with the uniaxial extensional flow, except for the negative extensional velocities, and the latter has no deformations in the y direction.
Nonequilibrium molecular dynamics simulation
- Steady shear flow
- Unsteady shear flow: start-up shear and cessation of steady shear flow
Note that at start-up shear (i.e. yux= ) or stopping steady shear flow (i.e. u=0), the flow velocity U q(i) along the velocity gradient direction gradually develops or decreases as a function of time, ultimately the steady state speed of the corresponding imposed strain rate is reached or. The flow velocity U q(i) at atomic position qi was calculated from a 5th order polynomial fit in each MD step throughout the system. The true idiosyncratic angular momentum of each atom was then calculated by subtracting the flow velocity at its position from its laboratory momentum.
Simulation model and system details
We have performed direct atomistic NEMD simulations of various entangled linear polyethylene (PE) melts with complicated chain architectures under unsteady and steady shear in a wide range of flow strengths, corresponding to the Weissenberg number Wi (defined as the product of the terminal relaxation time d of the system and the imposed strain rate.The NEMD simulations were performed with the well-known p-SLLOD algorithm6,7 implemented by a Nosé-Hoover thermostat8,9 and standard Lees-Edwards sliding brick boundary conditions13 using the reversible reference system propagator algorithm (r- RESPA)10 (see Table 3.2.1 for details of the systems).
RESULTS AND DISCUSSION
Confined polymer melts under steady-shear flow
- Interfacial slip
- Linear polymer melts
- Short-chain branched polymer melts
- Ring (cyclic) polymer melts
In the first (weak flow) regime, interfacial polymer chains undergo the z-to-x rotation in the vortex plane due to the applied shear flow (x direction). In the second (intermediate) regime, the flow strength becomes comparable to the wall friction, resulting in the repeated chain release-attachment (out of plane logging). As mentioned, the increasing trend of ds in the weak flow regime for the linear polymer is attributed to the z-to-x in-plane chain rotation, which increases slip via reduction of interfacial friction against the wall in the flow direction.
4.1.4, where it shows the xx component of the gyration tensor Ginf for the interfacial chains as a function of the Wi number. 4.1.5, where these structural and dynamic features of the SCB polymer underlie a very small variation of ds even in the intermediate flow regime (see middle image of the bottom panel in Figure 4.1.3). Top) Schematic illustration of the characteristic molecular mechanism of the interfacial chains at high flow fields and the probability distribution function (PDF) of the representative mesoscopic chain conformations [Stretched (S), Fold (F), Half-dumbbell (HD), Coil (C) ] obtained by the brightness analysis during a tumbling cycle for the linear and SCB PE systems.
This mechanism results in a movement of the center of mass of the chain in the direction of flow, which contributes to polymer slip at the wall. In contrast, in the case of the loop migration mechanism (right panel in Fig. 4.1.9), the loops move (i.e., the perturbed parts propagate) along the chain against the flow direction. The probability distribution function (PDF) result for these two loop mechanisms (calculated directly by tracking the individual loop motion) confirms this behavior (top panel of Figure 4.1.10).
Second, two different types of loop tumbles (parallel and vertical; see Figure 4.1.11) appear for the ring polymer, depending on the orientation of the loop cross-section plane with respect to the z-x interfacial (or vorticity) plane. Snapshots of a selected interfacial chain for describing the representative molecular mechanisms (parallel and vertical loop tumbling) exhibited by interfacial chains for the C400 ring system under strong flow fields, depending on the angle of the loop cross-sectional plane with respect to the x-z interfacial plane (i.e. say, the loop plane is aligned parallel to the x-z (vorticity) and the x-y (shear) planes for the parallel and vertical tumbling of the loop, respectively). Probability distribution of the angle (θloop) of the loop cross-sectional plane with respect to the x-z plane.
Polymer melts undergoing start-up shear flow
- Transient behaviors of properties
- Molecular mechanisms behind the overshoot
Note that the transient behavior of the orientation tensor of entangled strands is very similar to that of the stress and the birefringence tensor. Furthermore, the plot of the orientation tensorSxyee [which represents the largest length scale of chain structure (i.e. the chain end-to-end vector)], quantitatively fits the stress and birefringence curves. However, it is important to note that the overall evolution (i.e., upslope, maximum, and descent) of the birefringence as a function of strain exactly coincides with that of the strain.
While the transient behavior of the intermolecular LJ shear stress xyinter and normal stress difference N1inter. Schematic illustration of the general characteristic molecular mechanisms of polymer chains associated with the stress overshoot phenomena upon exposure to the flow field. Note that the transient behavior of the shear stress is mainly determined by the intermolecular LJ interaction (i.e. xyinter) in the corner region of Sec 2 for all Wi numbers studied here (Fig. 4.2.5); that is, the xy overshoot is consistent with that of interxy.
Thus, we can think that the transient nonlinear shear stress behaviors for polymer systems under shear are essentially related to the chains belonging to the Sec 2 angular region, regardless of the imposed current strength. Average angular velocity vector field for clockwise (+, black arrows) and counter-clockwise (−, red arrows) rotation of the chain at each orientation angle before overstressing. The consequence of this chain stretching is the growth of the normal voltage difference, which eventually leads to an overshoot of N1 (Figure 4.2.7).
Although seemingly coincidental, this result may support the physical aspect of the normal stress overshoot associated with the overall chain stretch.
Stress relaxation upon cessation of steady shear flow
- Stress relaxation
- Individual chain analysis
- Two relaxational processes: structural/orientational and thermally-driven relaxation
First, we plot transient behavior of the stress tensors and chain structure with orientation in Figure 4.3.1. 4.3.1, the transient behavior of stress is directly compared to that of the average chain orientation angle and chain end-to-end distance R. Meanwhile, in addition to the flow-associated structural and induced orientational relaxations, a thermally driven orientational relaxation independent of the flow field will make the system reach the original spatial homogeneity and isotropicity of the equilibrium state (i.e. xy =N1=0).
Because the number of chains for large R generally increases with increasing applied shear, the contribution of the stretched chains to the overall structural and induced orientational relaxations will become larger with increasing Wi number, facilitating the stress relaxation. Physically, this requires the homogeneity and isotropicity of the entire system with complete structural and orientational relaxations of all chains (i.e. a constant increase in the fold conformation for 0t 2R via the structural relaxation combined with the induced orientational relaxation also occurs.
These molecular features of the structural relaxation process are schematically summarized in the bottom panel in Fig. Top) Transient behavior of the probability distribution function (PDF) for the five representative mesoscopic chain configurations [end-kink (E-K), end-center-kink (E-C-K), coil, fold, stretch] calculated from the brightness analysis after stopping the steady shear at t = 0. Accordingly, the structural and induced orientation relaxations of a chain are with a given R value and its contribution to xy and N1 would vary with the Wi number due to the change of the surrounding environment with the flow strength.
In contrast to the essentially flow-independent nature of mechanism (ii), mechanism (i) is strongly dependent on the imposed flow type and strength due to a large variation of the probability distribution of R (and its mean) and the anisotropic hydrodynamic friction of the chain's movement as a function of the flow field.
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Molecular characteristics of stress relaxation of polymer melts upon cessation of uniform shear using molecular dynamics simulation.