Figure 1

(a) Persistence forces. The forces ${\mathbf{F}}_1$ and ${\mathbf{F}}_2$ act on the monomers 1 and 2 and are normal to the bonds 1-3 and 2-3, respectively. The force ${\mathbf{F}}_3$ acting on the monomer 3 is opposite to the vector sum ${\mathbf{F}}_1+{\mathbf{F}}_2$. All forces lie in the plane defined by the monomers 1, 2, and 3. (b) Thin nonslip layers near the walls parallel to the direction of the application of the external force on the bead. The water spheres in the layers are only able to move perpendicular to them. (i) the bead, (ii) the boundary layer in which the nonslip conditions are imposed, and (iii) the water sphere at the boundary.Reuse & Permissions

Figure 2

(Color online) Magnetic bead (large, brown) moving in the polymer network. The polymers are shown as small colored spheres, with beads of the same color belonging to the same polymer. For the sake of visibility (i) we show only each tenth monomer sphere, and (ii) polymers are not shown if they are located between the observer and the sphere or on the other side of the sphere from the observer. (a) The bead in a rest state (before the force is applied) surrounded by the polymers. (b) The bead during its motion along the $Ox$ axis. The concentration of the polymers is increased in front of the bead and decreased behind it.Reuse & Permissions

Figure 3

(a) Typical displacements of the bead with time. Curves 2–5 correspond to the forces $f_2=400\mathrm{FU}$, $f_3=800\mathrm{FU}$, $f_4=1000\mathrm{FU}$, and $f_5=1300\mathrm{FU}$ applied to the bead. All curves were obtained for the system containing 4500 polymers. The dashed lines show the best fit to each curve by Eq. (1). (b) The bead displacement versus time shown on a double-logarithmic plot reveals two distinct regimes of motion: the initial regime characterized by the exponent ${\alpha }_1\approx 0.75$ followed by the regime with the exponent ${\alpha }_2\approx 0.5$. Thick solid lines indicate slopes with the exponents ${\alpha }_1=3∕4$ and ${\alpha }_2=1∕2$. Responses of the four systems are shown with various numbers of polymers: 2900 (dot-dashed), 3500 (dotted), 4000 (dashed), and 4500 polymers (thin solid line). All cases shown here correspond to the force of 1000 FU applied to the bead.Reuse & Permissions

Figure 4

(a) The coefficient $K$ characterizing the bead displacement in the square-root regime [Eq. (1)] plotted versus force in a double-logarithmic scale exhibits a linear force dependence for the analyzed systems with 2900 (open circles), 3500 (open squares), 4000 (open triangles) and 4500 (filled spheres) polymers. The solid line indicates the slope equal to 1. (b) Dependence of the coefficient $K$ on the concentration of polymers for forces $f=200$ (open circles), 400 (open squares), 800 (triangles), 1000 (filled circles), and 1300 FU (filled squares). The solid line shows the slope equal to ${\gamma }_2=-1.4$. (c) The bead displacements, normalized by $D_{\Vert }^{1∕2}$ with four different dissipative constants, ${\gamma }_{\mathrm{mw}}$, of monomer-water interaction, collapse onto the same line, indicating the dependence $x\sim D_{\Vert }^{1∕2}$.Reuse & Permissions

Figure 5

(a) Number of monomers neighboring the bead at the front (i) and at the rear (ii) hemispheres versus time during the bead motion. The external force is applied at $t=0$. (b),(c) The distribution of the concentration of monomers (arbitrary units) in the vicinity of the bead as a function of distance $r$ from the bead center and the azimuthal angle $\theta$. The concentration is shown in the rest state (b) and during the bead motion (c). The moment of time for which the image (c) has been constructed corresponds to the square-root regime of the bead motion.Reuse & Permissions

Figure 6

The forces exerted on the moving bead by water $F_{\mathrm{w}}$ (a) and (b), and by polymers $F_{\mathrm{pol}}$ (c). (a) The force due to water acting on the bead at the beginning of bead motion gives the main contribution to the resistance force. (b) At later times the water contribution to the resistance gradually decreases. (c) The force exerted by polymers on the bead gradually increases with time until its absolute value reaches that of the external force (which is in the shown case $f=1000\mathrm{FU}$).Reuse & Permissions

Figure 7

Three contributions to the force exerted on the bead by the polymers: (a) the viscous, (b) the random, and (c) the conservative force (in force units used here). The time interval $6.5⩽t⩽6.9\mathrm{TU}$ corresponds to the square-root regime. The dashed line in (c) indicates the force equal to $-1000\mathrm{FU}$.Reuse & Permissions

Figure 8

(a) The longitudinal $⟨R_{\Vert }^2⟩$ (left axis) and transverse $⟨R_{\perp }^2⟩$ (right axis) MSDs of polymers with constant length $(L=34.5\mathrm{LU})$ in networks with various mesh sizes. The solid arrows indicate the axis corresponding to the data. The dashed arrow shows the direction of increasing mesh size $\xi$. (b) The radius of the fluctuation tube as a function of mesh size. Dots show the simulation data, while the solid line is a linear fit. (c) The longitudinal (left axis) and transverse (right axis) MSDs of polymers with various contour lengths in a solution with a constant mesh size $\xi \approx 1.57\mathrm{LU}$. The solid arrows indicate the correspondence between the data and the axes. The dashed arrow shows the direction of increasing polymer contour length $L$. The dash-dotted line shows the slope with $2D_{\Vert }\approx 0.077{\mathrm{LU}}^2∕\mathrm{TU}$.Reuse & Permissions

Figure 9

Motion of polymers located at time $t=6.5\mathrm{TU}$ in a cylindrical region, coaxial with the direction of bead motion, the $Ox$ axis, in front of the moving bead (a). (b) Displacement of the monomers. The solid line shows the average displacement $\delta x=\delta x\left(t\right)$ of the monomers while the dashed line shows the fit to the data with $\delta x\approx C{t}^{1∕2}$ with $C=0.167\mathrm{LU}∕{\mathrm{TU}}^{1∕2}$. (c) The longitudinal (j) and transverse (jj) MSDs of segments of polymers formed by the monomers in front of the bead. For comparison, the longitudinal (i) and transverse (ii) MSDs of polymers in the bulk are shown.Reuse & Permissions

Figure 10

Bead motion in pure water. (a) Velocity of the bead moving under a constant applied external force versus time. The inset shows the dependence of the velocity on the force in the regime of steady motion. (b) $Ox$ projection of the mean square displacement of the bead during applied-force-free Brownian motion. The linear relation $⟨x^2\left(t\right)⟩=0.0289t$ is shown by the dashed line. (c) The radial component of the water velocity $\mathbf{U}$ in front of and behind the bead $(\theta =0,\theta =\pi )$ normalized by the bead velocity $v_{\mathrm{st}}$ in the steady regime. The results of our simulation are shown by the solid line and the analytical results [60] by the dashed line. (d) The radial and angular functions ${\Delta }_r(r,\theta )$ (solid line) and ${\Delta }_{\theta }(r,\theta )$ (dashed line) represent measures of deviations of the simulated velocities from the analytical solution of the Stokes equation [60]. They are shown at the section $\theta =\pi ∕4$ where they are maximum.Reuse & Permissions