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Polymer dynamics


Polymer dynamics > How does stress relax in melts of unlinked rings?

Polymer physicists have had great success in understanding the way that entangled polymers move to explore new random-walk conformations, under the action of random thermal forces. This understanding allows us to predict how polymer melts and solutions will flow, for a great variety of polymers (linear, branched, mixtures) and kinds of flow (shear, extensional, mixed).

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This modern understanding of polymer motion is based on the idea that the constraints on the motion of a given polymer due to entanglement with other polymers in the system, can be represented by a confining tube. The tube constrains the ways in which a polymer can explore new conformations. A linear polymer so confined can slip randomly back and forth in the tube (reptation); a star polymer can withdraw one of its “arms” temporarily down the tube, and then re-extend it along a new path (arm retraction).

How, then, do a melt of long unlinked ring polymers move? This vexing question is not of practical importance (melts of unlinked rings are very hard to make, and unlikely to ever be a commercial product), but is a challenging test of our understanding of polymer dynamics.

Most attempts to understand how rings move begin with the notion that unlinked rings in a melt adopt conformations similar to a single ring brought into a lattice of fixed obstacles. Such a ring must constantly "double back" on itself, leading to a so-called "lattice animal" conformation. The ring then moves by withdrawing small loops and re-extending them nearby.

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Recently, we have found a way of describing this process quantitatively, which makes analogies to successful theories for motion of star polymers. The goal is to evaluate the distribution of relaxation times for different portions of a ring configuration. Any given bond on a lattice animal divides the animal into two "subtrees". A bond can only relax when one of its two subtrees "evaporates" - moves completely across the bond into the other subtree. This motion happens as a result of the random moves of small peripheral loops - "leaves" on the tree.

Most segments in an animal are "peripheral"; they divide the animal into one small and one large subtree. A few segments are "central"; they divide the animal more equally. These segments take a long time to relax. We have carried out Monte Carlo simulations to find the distribution P(τ) of segment relaxation times τ, and constructed an analytical theory for P(τ) adapting methods used to describe star polymers. We ultimately find power-law stress relaxation in entangled melts of unlinked rings, in agreement with recent careful experiments.

(For more information: S. T. Milner and J. D. Newhall, "Stress relaxation in entangled melts of unlinked ring polymers", Physical Review Letters 105, 208302 (2010).)

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Master plot of average segment lifetime τ(c) versus "centrality" c (centrality = size of smaller subtree) for animals with number of segments N=50 (blue circles), 100 (red squares), 200 (green stars), compared to analytical theory (black curve).

Polymer dynamics

Polymer dynamics > How can we simulate the dynamics of entangled polymers most efficiently?

Analytical tube-based models for the dynamics of entangled polymers have scored many successes, in detailed comparison with rheology data for linear and branched polymers and their mixtures. But as structures become more complicated, or the system more compositionally varied, these models become too cumbersome to use.

A discretized version of such models, called "sliplink" models, replaces the tube with a sequence of discrete pairwise entanglement points, called sliplinks. This formulation is amenable to brownian dynamics simulations; the sliplinks move along with the mean flow, while the chain that "threads" the sliplinks undergoes a form of Rouse dynamics. The sliplinks have stochastic dynamics of their own; when the end of one chain passes through one of its sliplinks, some other sliplink on some other chain is likewise relaxed.

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In a sliplink model, the tube is replaced with a sequence of "links" through which the chain passes.

This approach works well, but is inefficient because most of the computer time is spent describing the short-time local wiggling of the Rouse chain. That wiggling ultimately gives rise to contour length fluctuations, reptation, and all the slow relaxations of interest; but typically we are not interested in the details of the short-time motion.

It turns out there are much more efficient ways of generating valid future configurations of a fluctuating Rouse chain, than by just numerically integrating the Rouse equation with noise, as has commonly been done. In particular, because the underlying Rouse equation is linear, it can be formally solved in terms of the noise history.

This allows valid configurations of the chain to be generated at arbitrary future times with no more work than a Fourier transform and generation of new gaussian mode amplitudes. This useful result holds even in shear or extensional flow.

In practical applications, not all of the time evolution operator can be solved exactly. For example, in a sliplink simulation, the affine motion of the sliplinks under the action of the flow cannot be incorporated into the exact solution. This motion is most easily described in real space, while the Rouse solution is carried out in Rouse modes (Fourier modes of the arclength variable).

To deal with this situation, we have developed a technique of stochastic operator splitting. It is conceptually related to methods for deterministic PDEs such as the alternating direction implicit (ADI) method for solving time-dependent diffusion equations in multiple space dimensions. The basic idea is to split the full time evolution into a all the parts (noise, linear operators) that can be solved exactly in Rouse modes, and another part best handled in real space. Then we alternate timesteps with the two operators.

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Weak convergence of stochastic Euler (red) and stochastic splitting (blue), error versus step size.

We can analyze how the approximate splitting method solution approaches the exact solution as the timestep is made smaller. There are two measures of accuracy relevant for stochastic problems: strong convergence, meaning fidelity of the solution for a given noise history; and weak convergence, meaning accurate calculation of quantities when averaged over the noise. Typically we are chiefly interested in weak convergence.

We have tested our splitting method for the simple example of a Rouse chain in shear flow with a nonlinear spring between beads on the chain. The nonlinear spring constant spoils the exact solution; we treat the nonlinear force in real space, the rest in Rouse modes. We have compared our new method to the usual stochastic Euler method (see figure 1); the Euler method converges with weak order 1, while our new method is weak order 2 (as we have shown analytically).

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Figure 2. Example snapshots of a Rouse chain in shear flow, with exact solution (black) versus increasingly short timesteps (red, blue).

Even more promising than the superior accuracy of stochastic splitting is its stability: even with large timesteps, its deviations from the exact trajectory tend to be bounded, and are confined to shorter wavelength modes of the chain (see figure 2). This behavior is again similar to that of the deterministic Crank-Nicholson or ADI methods, and results from the exact solution for high-derivative operators in the splitting method.

Finally, our stochastic splitting method is generalizable to a wide variety of spatially varying order parameters in one or more spatial dimensions, evolving under combined action of noise, linear and nonlinear forcing, and flow.

(For more information: M. P. Howard and S. T. Milner, Numerical simulation methods for the Rouse model in flow. Physical Review E 84, 051804 (2011).)

Polymer dynamics

Polymer dynamics > How do spreading or sintering polymer droplets flow?

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What do hitting a golf ball, drying paint, and big plastic children’s toys have in common? A golf ball, at the moment the driver strikes it, flattens against the face of the driver, forming a "Hertzian contact". Droplets of viscous fluid deposited on a wettable flat surface adopt similar shapes as they spread. The velocity field of the spreading particle at early times turns out to be strongly analogous to the displacement field of the elastically deformed golf ball.

Films of latex paint, when viewed with a powerful microscope, are revealed to consist initially of separate spherical latex particles, which merge together into a continuous film as the paint dries. This merging process is known as "sintering", and is a widespread physical phenomenon. It occurs for example in the manufacture of large plastic objects (including children’s toys) by a process called rotomolding, in which a large mold filled with small plastic granules is heated until the granules sinter together.



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Sintering of two spherical plastic particles, heated until the plastic is a molten viscous fluid, is essentially just the spreading of two droplets against each other, driven by the reduction in free energy associated with the decrease in area of the droplet-vapor interface. So the early stages of sintering and spreading are closely related phenomena.

Recent experiments by Laura Ramirez in Prof. Darrell Velegol’s group have focused on the time-dependence of the contact area of a viscous droplet (a molten but viscous micron-sized polystyrene particle) spreading on a flat surface (a silicon wafer). Remarkably, a power-law relation between spreading time t and contact radius a is found at "early times" (contact small compared to droplet), of the form a/R ~ (tγ/ηR)1/3 for a droplet of radius R, viscosity η, and surface tension γ.

We have exploited the analogies between sintering and spreading, and between liquid droplets spreading and elastic spheres deforming into a Hertzian contact, to construct scaling arguments and analytical calculations for the growth of contact radius versus time. We find results in very good agreement with the power-law relation found in spreading experiments. Extending our approach to viscoelastic droplet spreading, we find a general relation between contact radius and creep compliance.

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Contact radius 'a' as a function of spreading time t [from Ramírez et al., "Controlled flats on spherical polymer colloids", Langmuir 26, 7644 (2010).]

Polymer dynamics

Polymer dynamics > How does the chiral structure of DNA affect its mechanical properties?

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DNA is a semiflexible polymer; we typically characterize it by its "bending stiffness" (as for any bendable rod) or "persistence length" (the length scale on which it would bend under thermal fluctuations). But what aspects of its bending and thermal fluctuations are affected by the "handedness" of DNA?

It turns out that DNA reveals its handedness in a tendency, once bent into an arc, to "coil" preferentially in one sense over the opposite sense, rather like a length of hemp rope "prefers" to be coiled in one sense rather than the opposite sense. This "tendency" can be expressed as a nonlinear elastic coefficient, which couples the square of the local curvature of the DNA path to the local torsion ("torsion" is the rate at which the normal to the path rotates around the tangent).

The magnitude of this chiral coupling can be determined by performing atomistic molecular dynamics simulations on a short length of DNA. (Basically, on a computer you take a piece of DNA, "grab hold" of it, bend it, and then try to twist it first one way, then the other. Or, you can sit back and watch as thermal fluctuations explore those motions for you, and note which occur more frequently.)

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In recent work, undergraduate Pat Hrisko and postdoc Hujun Shen from Prof. Coray Colina’s group in Materials Science and Engineering have carried out atomistic MD simulations on DNA strands in explicit solvent (water). The strands studied were three helical repeats long - just enough to define one torsional angle. When the strands were made to bend by imposing external forces, the chiral elasticity revealed itself by a definite bias in the torsional angle. We find that for bends with radius shorter than about 5nm (i.e., at about the persistence length), chiral elasticity will strongly influence DNA conformations.


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Probability distribution for torsional angle in two different DNA strands, each three helical repeats long, both "encouraged" to bend by imposed external forces. Open squares, repeating ATAT... strand; filled circles, repeating CGCG... strand. Both show torsional bias (of opposite sign).

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