Shear-banding and superdiffusivity in entangled polymer solutions

Using high-resolution confocal rheometry, we study the shear profiles of well-entangled DNA solutions under large-amplitude oscillatory shear in a rectilinear planar shear cell. With increasing Weissenberg number (Wi), we observe successive transitions from normal Newtonian linear shear profiles to wall-slip dominant shear profiles and, finally, to shear-banding profiles at high Wi. To investigate the microscopic origin of the observed shear banding, we study the dynamics of micron-sized tracers embedded in DNA solutions. Surprisingly, tracer particles in the shear frame exhibit transient superdiffusivity and strong dynamic heterogeneity. The probability distribution functions of particle displacements follow a power-law scaling at large displacements, indicating a Lévy-walk-type motion, reminiscent of tracer dynamics in entangled wormlike micelle solutions and sheared colloidal glasses. We further characterize the length and time scales associated with the abnormal dynamics of tracer particles. We hypothesize that the unusual particle dynamics arise from localized shear-induced chain disentanglement.

Figure 1

Linear viscoelasticity of a highly entangled DNA solution. The storage modulus ($G^{\prime }$), the loss modulus ($G^{\prime \prime }$), and the loss tangent ($tan\delta$) were measured in a frequency sweep under SAOS. Solid squares and diamonds are for $G^{\prime }$. Empty squares and diamonds are for $G^{\prime \prime }$. Solid triangles (up-pointing and down-pointing) are for $tan\delta$. The results were used to determine the relevant tube parameters ${\tau }_d$, ${\tau }_R$, $G_N^0$, and $Z$. Squares and up-pointing triangles were obtained at room temperature ${23}^{\circ }\mathrm{C}$, whereas diamonds and down-pointing triangles were taken at ${13}^{\circ }\mathrm{C}$ and shifted to room temperature using the principle of time-temperature superposition.

Figure 2

Schematic of the experimental setup. The area of the wafer is $5\times 5{\mathrm{mm}}^2$. The thickness of the sample is $100\mu \mathrm{m}$. While the wafer is fixed, the coverslip is moved by the piezoelectric stage, which induces a linear shear in the confined sample. The solvent trap creates a sealed chamber, preventing the evaporation of the solvent. An inverted confocal microscope (represented here by the objective) is used for imaging the shear flow within the sample.

Figure 3

Normalized velocity profiles of DNA solutions under shear. Black squares are for a semidilute DNA solution at the overlap concentration $c^{*}=0.05$ mg/ml. Others are for the high-concentration, entangled DNA solution. Solid lines indicate piecewise linear fits. A Cartesian coordinate system is defined, where $x$, $y$, and $z$ are the flow, shear gradient, and vorticity directions, respectively. Velocity amplitudes, $V\left(y\right)$, are normalized by the velocity amplitude of the moving plate, $V_0=2\pi f{A}_0$ with $A_0=150\mu \mathrm{m}$.

Figure 4

Phase diagram of highly entangled calf thymus DNA solutions under LAOS. The amplitude of shear strain is fixed at ${\gamma }_0=1.5$. (a) Contour map of the wall-slip parameter, $(b_h+b_l)/2H$, at different shear rates ${\dot{\gamma }}_0$ and gap thicknesses $H$. Solid dots correspond to our experimental points. (b) Contour map of the shear-banding parameters, $|{\dot{\gamma }}_h-{\dot{\gamma }}_l|/{\dot{\gamma }}_0$, from the same experiments. (c) Contour map of the shear-banding parameter in terms of Wi and the slip parameter, $b_{\mathrm{avg}}/H=(b_h+b_l)/2H$.

Figure 5

Dynamics of particles in the shear frame. (a) Mean-squared displacements (MSDs) of particles along the flow ($x$) and vorticity ($z$) directions in the two coexisting shear bands. The slopes indicate the superdiffusive motion of particles at intermediate times and the diffusive motion at long times. (b) The ratio of MSDs, $\langle x^2\rangle /\langle z^2\rangle$, at $\mathrm{\Delta }t=50$ cycles versus local shear strain, $\gamma$. The dashed line indicates the prediction of Taylor dispersion. Different $\gamma$ are achieved by varying $A_0$ while keeping $f=4$ Hz.

Figure 6

Lévy walk of tracer particles. (a) Probability distribution functions (PDFs) of particle displacements over 20 shearing cycles in the high- and low-shear-rate bands. PDF in the low-shear-rate band is fitted by a Gaussian distribution. PDF in the high-shear-rate band is fitted by Eq. (2). (b) The fraction of jumpers performing the Lévy walk, $w$, at different heights. The corresponding velocity profile is shown for comparison. The shaded area indicates the high-shear-rate band. The fraction of jumpers in the sample with a linear velocity profile at $A_0=37.5\mu \mathrm{m}$ (Wi $\approx 9.4$) is also shown.

Figure 7

Dynamic heterogeneity of tracer particles. (a) Four-point susceptibility, ${\chi }_4\left(t\right)$, at different heights in the shear-banding flow. (b) The peak value of ${\chi }_4\left(t\right)$, ${\chi }_{4,p}$, at different heights. The corresponding shear-banding velocity profile is shown for comparison. The shaded area indicates the high-shear-rate band. The ${\chi }_{4,p}$ for the sample with a linear shear profile at $A_0=50\mu \mathrm{m}$ (Wi $\approx 12.5$) is also shown.

Figure 8

Correlation of jumper dynamics. (a) The autocorrelation of particle velocities, $C\left(t\right)$, in the low- and high-shear-rate bands. Time is measured in the unit of the number of shear cycles. (b) The spatial correlation of particle velocities, $C\left(r\right)$, in the low- and high-shear-rate bands. The radial distance $r$ is normalized by the diameter of tracer particles $d=1.1\mu \mathrm{m}$. The solid line indicates a double-exponential fit.