David C. Morse

Scaling Properties of Stretching Ridges in a Crumpled Elastic Sheet

Strong deformation of a sheet of solid material often leads to a crumpled state having sharp points of high curvature. A scaling property governing the crumpled state has been numerically demonstrated by an examination of the ridges joining pairs of sharp points in a range of simple geometries of variable size. As the linear size X increases sufficiently, the deformation energy grows as X1/3 and consists of similar amounts of bending and stretching energy. The deformation energy becomes concentrated in a fraction of the sheet that decreases as X1/3. Despite this concentration, the local strain in the ridge decreases as X2/3. Nearly all the deformation energy in thin, crumpled elastic sheets was found to be concentrated in ridges that obey these scaling laws.

Viscoelastic behavior of cubic phases in block copolymer melts

The viscoelastic behavior of block copolymer melts that exhibit a cubic phase has been examined by oscillatory shear experiments. A low frequency plateau in the measured storage modulus that is absent in the disordered phase is found for both gyroid and body-centered cubic sphere phases of diblock and triblock copolymer melts. The magnitude of the apparent plateau modulus is found to be insensitive to strain amplitude for strains of 0.05%–5%, and so is believed to be characteristic of the true linear response. The value of the apparent terminal relaxation frequency, beneath which G″(ω)>G′(ω) is, however, sensitive to strain amplitude in the same range of strains and decreases steadily with decreasing strain, indicating that the terminal regime is highly sensitive to nonlinear effects. The presence of one or more entangled blocks is found to decrease the terminal frequency, and thus extend the range of linear behavior. Experimental results for the dependence of the plateau modulus and the unit cell size upon molecular volume yield effective power law exponents that are close, but not identical, to those predicted by self-consistent field theory.The viscoelastic behavior of block copolymer melts that exhibit a cubic phase has been examined by oscillatory shear experiments. A low frequency plateau in the measured storage modulus that is absent in the disordered phase is found for both gyroid and body-centered cubic sphere phases of diblock and triblock copolymer melts. The magnitude of the apparent plateau modulus is found to be insensitive to strain amplitude for strains of 0.05%–5%, and so is believed to be characteristic of the true linear response. The value of the apparent terminal relaxation frequency, beneath which G″(ω)>G′(ω) is, however, sensitive to strain amplitude in the same range of strains and decreases steadily with decreasing strain, indicating that the terminal regime is highly sensitive to nonlinear effects. The presence of one or more entangled blocks is found to decrease the terminal frequency, and thus extend the range of linear behavior. Experimental results for the dependence of the plateau modulus and the unit cell size up...

Droplet elasticity in weakly compressed emulsions

We discuss the deformation of a fluid droplet in an emulsion under external forces, such as those exerted by contact with neighboring droplets. We find that the deformation energy associated with a small droplet-droplet contact scales as f2 ln (1/f) with the force f exerted between droplets. We consider the equation of state of an emulsion in which droplets are assumed to interact only via such contact forces, and obtain an osmotic compressibility which diverges logarithmically with the osmotic pressure in the limit of small pressure.

Polymer-polymer interfacial slip in multilayered films

Significant slip can occur in the flow of a blend of two immiscible polymers due to reduced entanglements at their interface. The slip is of practical importance because of its effect on morphology and adhesion in, for example, disordered two-phase blends or multilayer films. Interfacial slip was quantified using two polymer pairs each with closely matched viscosity and elasticity but different miscibility (χ): polypropylene (PP)/polystyrene (PS) χ=0.04 and polyethylene (PE)/fluoropolymer (FP) χ≅0.1. To control the amount of interfacial area, we prepared alternating layers by coextrusion. The number of layers of PP/PS ranged from 20 to 640 while that for PE/FP was 80. Nominal viscosity of the multilayer samples was measured with three types of rheometers: an in-line slit-die rheometer, rotational parallel-disks, and sliding plate. Good agreement was found between the three methods. The nominal viscosity as well as shear normal stresses of the multilayer samples decreased with the number of layers. From th...

Brownian dynamics algorithm for bead-rod semiflexible chain with anisotropic friction

A model of semiflexible bead-rod chain with anisotropic friction can mimic closely the hydrodynamics of a slender filament. We present an efficient algorithm for Brownian dynamics simulations of this model with configuration dependent anisotropic bead friction coefficients. The algorithm is an extension of that given previously for the case of configuration independent isotropic friction coefficients by Grassia and Hinch [J. Fluid Mech. 308, 255 (1996)]. We confirm that the algorithm yields predicted values for various equilibrium properties. We also present a stochastic algorithm for evaluation of the stress tensor, and we show that in the limit of stiff chains the algorithm recovers the results of Kirkwood and Plock [J. Chem. Phys. 24, 665 (1956)] for rigid rods with hydrodynamic interactions.

Viscoelasticity of dilute solutions of semiflexible polymers

We show using Brownian dynamics simulations and theory how the shear relaxation modulus G(t) of dilute solutions of relatively stiff semiflexible polymers differs qualitatively from that of rigid rods. For chains shorter than their persistence length, G(t) exhibits three time regimes: At very early times, when the longitudinal deformation is affine, G(t) approximately t(-3/4). Over a broad intermediate regime, during which the chain length relaxes, G(t) approximately t(-5/4). At long times, G(t) mimics that of rigid rods. A model of the polymer as an effectively extensible rod with a frequency dependent elastic modulus B(omega) approximately ((i)omega)(3/4) quantitatively describes G(t) throughout the first two regimes.

Linear Response and Stability of Ordered Phases of Block Copolymer Melts

An efficient pseudo-spectral numerical method is introduced for calculating a self-consistent field (SCF) approximation for the linear susceptibility of ordered phases in block copolymer melts (sometimes referred to as the random phase approximation). Our method is significantly more efficient than that used in the first calculations of this quantity by Shi and Laradji and co-workers, allowing for the study of more strongly segregated structures. We have re-examined the stability of several phases of diblock copolymer melts and find that some conclusions of Laradji et al. regarding the stability of the gyroid phase were the result of insufficient spatial resolution. We find that an epitaxial (k = 0) instability of the gyroid phase with respect to the hexagonal phase that was considered previously by Matsen competes extremely closely with an instability that occurs at a nonzero crystal wavevector k.

On the chain length dependence of local correlations in polymer melts and a perturbation theory of symmetric polymer blends

The self-consistent field (SCF) approach to the thermodynamics of dense polymer liquids is based on the idea that short-range correlations in a polymer liquid are almost independent of how monomers are connected into polymers over larger scales. Some limits of this idea are explored in the context of a perturbation theory for symmetric polymer blends. We consider mixtures of two structurally identical polymers, A and B, in which the AB monomer pair interaction differs slightly from the AA and BB interactions by an amount proportional to a parameter alpha. An expansion of the free energy to first order in alpha yields an excess free energy of mixing per monomer of the form alphaz(N)phi(A)phi(B) in both lattice and continuum models, where z(N) is a measure of the number of intermolecular near neighbors per monomer in a one-component (alpha=0) reference liquid with chains of length N. The quantity z(N) decreases slightly with increasing N because the concentration of intramolecular near neighbors is slightly higher for longer chains, creating a slightly deeper intermolecular correlation hole. We predict that z(N)=z(infinity)[1+betaN(-1/2)], where N is an invariant degree of polymerization and beta=(6/pi)(3/2) is a universal coefficient. This and related predictions about the slight N dependence of local correlations are confirmed by comparison to simulations of a continuum bead-spring model and to published lattice Monte Carlo simulations. We show that a renormalized one-loop theory for blends correctly describes this N dependence of local liquid structure. We also propose a way to estimate the effective interaction parameter appropriate for comparisons of simulation data to SCF theory and to coarse-grained theories of corrections to SCF theory, which is based on an extrapolation of perturbation theory to the limit N-->infinity.

Renormalized one-loop theory of correlations in polymer blends.

The renormalized one-loop theory is a coarse-grained theory of corrections to the random phase approximation (RPA) theory of composition fluctuations. We present predictions of corrections to the RPA for the structure function S(k) and to the random walk model of single-chain statics in binary homopolymer blends. We consider an apparent interaction parameter chi(a) that is defined by applying the RPA to the small k limit of S(k). The predicted deviation of chi(a) from its long chain limit is proportional to N(-1/2), where N is the chain length. This deviation is positive (i.e., destabilizing) for weakly nonideal mixtures, with chi(a)N less than or approximately 1, but negative (stabilizing) near the critical point. The positive correction to chi(a) for low values of chi(a)N is a result of the fact that monomers in mixtures of shorter chains are slightly less strongly shielded from intermolecular contacts. The predicted depression in chi(a) near the critical point is a result of long-wavelength composition fluctuations. The one-loop theory predicts a shift in the critical temperature of O(N(-1/2)), which is much greater than the predicted O(N(-1)) width of the Ginzburg region. Chain dimensions are found to deviate slightly from those of a random walk even in a one-component melt and contract slightly as thermodynamic repulsion is increased. Predictions for S(k) and single-chain properties are compared to published lattice Monte Carlo simulations.

Entropy and fluctuations of monolayers, membranes, and microemulsions

Conformational entropy of the interfaces in microemulsions and surfactant solutions has been predicted, under some circumstances, to have significant effect upon their thermodynamic behavior. Continuum theories have led to a consistent phenomenological description of the effects of undulations, but conclusive experimental evidence for or against the predicted effects remains elusive in all but the lamellar phase. Work on the bicontinuous phases has led to an increasing focus on the role of topology and Gaussian rigidity, and on the role of anharmonic (in curvature) contributions to the bending energy.