Dynamics of Semiflexible Polymer Solutions in the Highly Entangled Regime

We present experimental evidence that the effective medium approximation (EMA), [D. C. Morse, Phys. Rev. E 63, 031502 (2001)], provides the correct scaling law of the plateau modulus $G^0\propto {\rho }^{4/3}{L}_p^{-1/3}$ (with $\rho$ the contour length per unit volume and $L_p$ the persistence length) of semiflexible polymer solutions, in the highly entangled regime. Competing theories, including a binary collision approximation (BCA), instead predict $G^0\propto {\rho }^{7/5}{L}_p^{-1/5}$. We have tested both predictions using F-actin solutions which permit experimental control of $L_p$ independently of other parameters. A combination of video particle tracking microrheology and dynamic light scattering yields independent measurements of $G^0$ and $L_p$, respectively. Thus we can distinguish between the two proposed laws, in contrast to previous experimental studies focused on the (less discriminating) concentration dependence.

Figure 1

Mean-square displacement (MSD) versus lag time, for $0.489\mu \mathrm{m}$-diameter beads in F-actin solutions at different concentrations (mg/ml) (system 2). The solid line represents the MSD of the beads suspended in water at $25°\mathrm{C}$.

Figure 2

Storage and loss moduli versus frequency, for F-actin solution at a concentration $1\mathrm{mg}/\mathrm{ml}$ (system 2). The lines are guides for the eye. The same features are also exhibited by system 1.

Figure 3

Plateau modulus $G^0$ versus F-actin concentration, for Systems 1 (circle) and 2 (square). The open triangles are bulk rheology results taken from Ref. 13. The short dashed and dash-dotted lines are linear fits to the data on the log-log plots (power laws); the dotted and dashed lines show the BCA ($G^0\propto c^{7/5}$) and EMA ($G^0\propto c^{4/3}$) scaling predictions, respectively. The inset (system 2) shows the actual existence of a different power-law regime at concentrations lower than the tightly entangled regime (cf. Ref. 18).

Figure 4

Double logarithm of the normalized dynamic structure factor $g^{(1)}(q,t)/g^{(1)}(q,0)$ versus logarithm of time, at a scattering angle of 90°, for a solution at F-actin concentration $c=0.4\mathrm{mg}/\mathrm{ml}$ (system-1). Similar data are obtained for system 2. The lines are guides for the eye.

Figure 5

Normalized initial decay rate ${\Gamma }^0$ versus scattering wave-vector $q$, and the best fit of Eq. (9) using $a_h$ as a free parameter, for a solution at F-actin concentration $c=0.1\mathrm{mg}/\mathrm{ml}$ (system-2). Inset: Diameter distribution from TEM image analysis.

Figure 6

Persistence length $L_p$ versus wave vector $q$ for a solution at F-actin concentration $c=0.2\mathrm{mg}/\mathrm{ml}$ (system-1) (derived using $a_h=9.2\mathrm{nm}$).