Concentration Fluctuations In Polymer Solutions Research Articles

Concentration fluctuations in polymer solutions under mixed flow

In this work, we extend the classical analysis of concentration fluctuations in polymer solutions under shear flow to consider the same phenomenology under mixed (shear + extensional) flows. To investigate this phenomenon, we couple stress and concentration using a two-fluid model with fluctuations driven by thermal noise incorporated through a canonical Langevin approach. The polymer stress is governed by the Rolie-Poly model augmented with finite extensibility to account for large stretching of chains at high Weissenberg numbers. Perturbing the equations about homogeneous flow for weak amplitude inhomogeneities, but arbitrary flow strength, we solve for the steady state structure factor (Fourier transformed pair correlation function) under general linear flows using a unique method of characteristics solver. Under shear flow, the model predicts butterfly patterns in accord with previous experimental and theoretical work, including a full rotation of peaks past the flow axis. In addition, the magnitude of the structure factor initially grows with the Weissenberg number until reaching a maximum at intermediate shear rates and decaying thereafter. Under mixed flow, the butterfly patterns as well as the location and magnitude of the peak structure factor are strongly tied to both the flow type parameter and the Weissenberg number (the characteristic strain rate). As expected, for flows characterized as strong, the scattering patterns typically appear like a rotated version of pure extension. However, as the flow type approaches the pure shear limit, the influence of shear flow on the butterfly patterns becomes more pronounced. In particular, for large Weissenberg numbers, contrary to expectations, the flow type need not be very near shear flow in order for the scattering patterns to no longer be simply rotated versions of extensional flow.

Concentration fluctuations in polymer solutions under extensional flow

In this work, we extend the classical analysis of concentration fluctuations in polymer solutions under shear flow to consider the same phenomenology under extensional flow. Experimental work by van Egmond and Fuller [Macromolecules 26, 7182–7188 (1993)] revealed a four-lobe scattering pattern for a polystyrene solution in a planar extensional flow field. Similar to earlier results found in shear, they find the existence of finite-wavelength peak intensity locations. To investigate this phenomenon, we couple stress and concentration using a two-fluid model with fluctuations driven by thermal noise incorporated through a canonical Langevin approach. The polymer stress is governed by the Rolie–Poly model augmented with finite extensibility to account for large stretching of chains at high Weissenberg numbers. Perturbing the equations about homogeneous planar extensional flow for weak amplitude inhomogeneities, but arbitrary flow strength, we solve for the steady correlations. The resulting structure factor ...

Effect of oscillatory shear on polymer solutions.

We consider the effects of oscillatory shear flow on small concentration fluctuations in polymer solutions, which we model as highly asymmetric entangled polymer blends. A simple model that enables us to predict the scattering patterns in the flow-vorticity plane is presented. It is shown that peaks in the patterns occur in the flow direction at a finite wave vector, their position strongly depends on the angular frequency of oscillatory shear flow, whereas the intensity of these peaks is rather controlled by the amplitude of the flow. The model reproduces characteristic double-winged anisotropic scattering patterns called "butterfly," which are stable and thus represent concentration fluctuations rather than phase separation.

Slip, Concentration Fluctuations, and Flow Instability in Sheared Polymer Solutions

Recent experiments suggest that shear-enhanced growth of concentration fluctuations in polymer solutions is strongly influenced by solid boundaries. We analyze the dynamics of a model of a sheared polymer solution, accounting for the effect of stress variations on mass flux and for wall-slip. If (and only if) these effects are present, the flow exhibits a boundary-localized instability consistent with experimental observations. Even in the absence of flow instability, thermally-driven concentration fluctuations are significantly enhanced near the boundary.

Dynamics of concentration fluctuations in polymer solutions with spatiotemporal correlated noise.

We study the problem of concentration fluctuations in polymer solutions under the influence of spatiotemporal correlated noise. We find that in contrast to the case of white noise, where the dynamic structure function is characterized by a single decay rate ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{k}}$ which is given in terms of the static structure function g(k) in the colored noise case, the decay rate takes a different form depending on a characteristic wave vector \ensuremath{\kappa}, given by the solution of the equation 1+${\ensuremath{\xi}}^{2}$${\mathrm{\ensuremath{\kappa}}}^{2}$-\ensuremath{\tau}${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{\ensuremath{\kappa}}}$=0, with \ensuremath{\xi} being the correlation length of the polymers and \ensuremath{\tau} being the correlation time of the colored noise. For the wave vector k\ensuremath{\ll}\ensuremath{\kappa}, the decay rate is ${\mathrm{\ensuremath{\Gamma}}}_{\mathit{k}}$, just as in the white noise case. For k=\ensuremath{\kappa}, the decay rate is ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{\ensuremath{\kappa}}}$, but the decay is modified by an extra factor (1+${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{\ensuremath{\kappa}}}$t) where t is the time. For k\ensuremath{\gg}\ensuremath{\kappa}, the decay rate is (1+${\ensuremath{\xi}}^{2}$${\mathit{k}}^{2}$)/\ensuremath{\tau}. Since our result should hold as long as the correlation time \ensuremath{\tau} is not exactly zero, it should lead to experimentally verifiable consequences in dynamic light scattering in polymer solutions in short time scales.

Dynamical theory of concentration fluctuations in polymer solutions under shear.

Dynamical theory of concentration fluctuations in polymer solutions under shear.

Flow-induced concentration fluctuations in polymer solutions: Structure/property relationships

Mechanical and optical rheometric measurements are reported on solutions of polystyrene dissolved in dioctyl phthalate, a solution that can undergo an apparent phase separation upon the application of shear. Solutions prepared using three molecular weights ranging from one to four million were studied. Time-temperature superposition was observed to apply for these solutions up to and including the onset of an apparent shear thickening of the steady shear and first normal stresses. Optical measurements employing turbidity and scattering dichroism determined that concentration fluctuations were enhanced by flow and grew parallel to the vorticity axis below the critical velocity gradient for the onset of the apparent shear thickening effect. Prior to the onset of thickening, the fluctuations were observed to rearrange and orient parallel to the flow direction. Second normal stress difference measurements indicate these solutions have a negative ratio of the second to the first normal stress differences. It is interesting to point out that the ratio tends to zero in the vicinity of the shear rate range at which shear thickening occurs.

“Butterfly” Light Scattering Pattern in Shear-Enhanced Concentration Fluctuations in Polymer Solutions and Anomaly at High Shear Rates

The effects of simple shear flow on concentration fluctuations in semidilute solutions of high molecular weight polystyrene are investigated by the flow light scattering method. The shear-rate (\(\dot{\gamma}\) ) dependence of scattering was found to fall into four regimes: Regime I at \(\dot{\gamma} \dot{\gamma}_{c}\), \(\dot{\gamma}_{c}\) being the critical shear rate for the shear-enhanced concentration fluctuations. In Regime II, a unique butterfly-type scattering pattern with a dark streak perpendicular to the flow direction appears. The intensity parallel to the flow increases by about 100 times with increasing \(\dot{\gamma}\) . In Regime III at higher shear rates, the intensity increase with \(\dot{\gamma}\) tends to saturate, and finally in Regime IV at even higher shear rates, the intensity further increases and the butterfly pattern changes into a two-point pattern elong...

Time-dependent small-angle light scattering of shear-induced concentration fluctuations in polymer solutions

A small-angle light-scattering experiment probing the time-dependent behavior of a semidilute polymer solution under simple shear is presented. Specifically, this paper investigates time-dependent anisotropic growth and orientation of concentration fluctuations in a semidilute solution of polystyrene in dioctyl phthalate above the quiescent cloud-point temperature. Two flow cells, which permitted investigation of the structure and dynamics of concentration fluctuations in both the flow-shear and the flow-vorticity planes, were used. Concentration fluctuations were found to be enhanced anisotropically by the shear flow in both planes. In the flow-vorticity plane, shear-enhanced concentration fluctuations orient along the vorticity direction, while in the flow-shear plane, the axis of average orientation is at an angle between 135° and 45° to the direction of flow. Qualitative trends of this anisotropic growth of concentration fluctuations are in agreement with the predictions of the Helfand–Fredrickson model in the flow-shear plane. Furthermore, the Onuki–Doi theory relating the scattering dichroism to the structure factor has been verified experimentally by comparing scattering dichroism calculated from scattering patterns with previous experimentally measured dichroism. Scattering dichroism, both calculated from structure factors and experimentally determined over a range of shear rates and temperatures, is demonstrated to depend only on the Weissenberg number. Relaxation of shear-enhanced fluctuations on cessation of flow allowed for the calculation of a diffusion coefficient.

Enhanced concentration fluctuations in polymer solutions under shear flow.

We report an elastic-light-scattering experiment for a semidilute polymer solution under a uniform laminar shear flow. The concentration fluctuations are greatly enhanced by shear flow and their structure is highly anisotropic in both the weak (\ensuremath{\gamma}\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\tau}}}_{\mathit{d}}$\ensuremath{\ll}1) and strong (\ensuremath{\gamma}\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\tau}}}_{\mathit{d}}$\ensuremath{\gg}1) shear regimes, where \ensuremath{\gamma}\ifmmode \dot{}\else \.{}\fi{} is the shear rate and ${\mathrm{\ensuremath{\tau}}}_{\mathit{d}}$ is the longest relaxation time of the system. The anisotropy is qualitatively different from binary liquid mixtures under a shear flow as a result of the strong coupling between the concentration fluctuations and the hydrodynamic flow.

Shear-Enhanced Concentration Fluctuations in Polymer Solutions as Observed by Flow Light Scattering

The effects of simple shear flow on concentration fluctuations in polymer solutions are investigated as a function of S , T , c , and N by the flow light scattering method. Here, S is the shear rate, T the temperature, c the polymer concentration, and N the degree of polymerization of the polymer. Over the range of the concentration studied in this experiment (a semidilute regime covering c / c * =4 to 20, where c * is the overlap concentration), strong shear-enhanced concentration fluctuations were observed above a certain critical shear rate S c . For S > S c , the integrated scattered intensity I ⊥ in the direction perpendicular to the flow increases at least by two orders of magnitude compared to that of the homogeneous solution. The scattering patterns observed for S > S c have two main chracteristics: (i) they are very diffuse, extending up to high scattering vectors and (ii) they have a sharp and strong destructive interference in the direction perpendicular to the flow.