Colloid dynamics in semiflexible polymer solutions

We investigate the dynamics of monodisperse colloidal polystyrene particles suspended in solutions of the semiflexible polymer filamentous actin, over a range of filament lengths that either exceed or are substantially less than the particle radius. The filament length is controlled by the capping protein gelsolin, and particle surface chemistries that minimize the adsorption of filaments are used. The particle dynamics are measured on short time scales using diffusing wave spectroscopy. A sharp transition in the initial particle diffusivity marks the expected shift from a dilute to a tightly entangled polymer network as the filament average length increases. In both the dilute and entangled regimes, the measured particle dynamics are compared with the theories of rodlike and semiflexible polymer solution rheology using the generalized Stokes-Einstein relationship. In the dilute limit, the particle dynamics are in good agreement with theory. However, in the tightly entangled regime, the particle response is consistent with polymer depleted near the surfaces of the particles. The magnitude of the depletion layer thickness depends strongly on particle size and weakly on filament length. This behavior is in agreement with nonlocal entropic repulsions and the loss of conformational entropy associated with rodlike molecules near impenetrable particles. These results illustrate the use of microrheology as a method to investigate local structure and dynamics in colloid-polymer solutions.

Figure 1

The multicomponent correlations of filled semiflexible networks are dependent on the relative ratios of the particle radius $a$, polymer contour length $L$, persistence length $L_p$, polymer concentration $c$, and particle volume fraction $\varphi$. $D$, $\mathit{LE}$, and $\mathit{TE}$ indicate dilute, loosely entangled, and tightly entangled regimes, respectively. In this work, we investigate the response of particles as the filament length is changed, illustrated by the line labeled 1. The effect of particle size was investigated previously (line labeled 2) [30].

Figure 2

(a) Mean-squared displacements of particles in $0.63\mathrm{mg}∕\mathrm{ml}$ F-actin with various average filament lengths, $L$, controlled by the addition of gelsolin. Filled square: unshortened F-actin $(L\approx 20\mu \mathrm{m})$; open square: $L\approx 5\mu \mathrm{m}$; filled circle: $L\approx 3\mu \mathrm{m}$; open circle: $L\approx 1.5\mu \mathrm{m}$; filled triangle: $L\approx 1.2\mu \mathrm{m}$; open up triangle: $L\approx 0.5\mu \mathrm{m}$; cross: $L\approx 0.15\mu \mathrm{m}$; and open down triangle: $L\approx 0.053\mu \mathrm{m}$). The solid line has a slope 1 for comparison. (b) Linear plots demonstrate the decrease in mean-squared displacement as the filament length increases.

Figure 3

The particle short-time diffusion coefficients as a function of actin filament length. The actin concentration is $0.63\mathrm{mg}∕\mathrm{ml}$. The bold open circle corresponds to the point closest to the expected transition to the tightly entangled fluid regime. Lines are shown to guide the eye.

Figure 4

The overlap concentration $\overline{c}$ and entanglement concentration $c^{*}$ marking the boundaries between the dilute $\left(D\right)$, loosely entangled (LE), and tightly entangled (TE) regimes are plotted as a function of filament length. The transition point of the initial diffusivity with filament length (Fig. 3) is in excellent agreement with the calculated transition to the tightly entangled regime, as shown by the bold open symbol.

Figure 5

Mean-squared displacements of particles in $0.63\mathrm{mg}∕\mathrm{ml}$ F-actin comparing three particle surface chemistries: untreated PS (open triangle); PS with adsorbed BSA (open square); PS with covalently attached PEG (filled triangle). The subdiffusive response of untreated PS particles, $⟨\Delta r^2\left(t\right)⟩\sim t^{0.75}$, is in good agreement with previous microrheology studies. Both PEG and BSA-treated particles exhibit nearly identical responses with significantly less subdiffusion.

Figure 6

(Color online) DWS data using $2a=1.053\mu \mathrm{m}$ particles (open down triangle, black: PS with adsorbed BSA; gray: untreated PS) [30] and $2a=1.072\mu \mathrm{m}$ (open up triangle, green: unshortened F-actin $(L\approx 20\mu \mathrm{m})$; dark yellow: $L\approx 5\mu \mathrm{m}$; red: $L\approx 3\mu \mathrm{m}$; violet: $L\approx 1.5\mu \mathrm{m}$; orange: $L\approx 1.2\mu \mathrm{m}$; blue: $L\approx 0.5\mu \mathrm{m}$; pink: $L\approx 0.15\mu \mathrm{m}$; black: $L\approx 0.053\mu \mathrm{m}$) in $0.63\mathrm{mg}∕\mathrm{ml}$ F-actin is in good agreement with measurements using laser tracking microrheology for $2a=0.914\mu \mathrm{m}$ (filled diamond, black: PS adsorbed with BSA; gray: untreated PS) in $1.0\mathrm{mg}∕\mathrm{ml}$ unshortened F-actin [33] and multiple particle tracking for $2a=0.84\mu \mathrm{m}$ PS coated with PEG (filled circle, green:$L\approx 17\mu \mathrm{m}$; dark yellow:$L\approx 5\mu \mathrm{m}$; red: $L\approx 2\mu \mathrm{m}$; blue: $L\approx 0.5\mu \mathrm{m}$) [45] in $1.0\mathrm{mg}∕\mathrm{ml}$ F-actin. Differences in the particle size and F-actin concentration are adjusted by the concentration dependence of the modulus and particle radius $c^{1.8}a$ [46]. $⟨\Delta r^2\left(t\right)⟩$ and $a$ have units of length in meters and $c$ is in mg/ml. Two solid lines have slopes 1 and $3∕4$, respectively, for comparison.

Figure 7

The experimental $G^{\prime }\left(\omega \right)$ (filled triangle) and $G^{″}\left(\omega \right)$ (open triangle) calculated from the particle dynamics using the generalized Stokes-Einstein relationship compared to the dilute solution theories of rods [$G^{\prime }\left(\omega \right)$, solid gray line; $G^{″}\left(\omega \right)$, dashed gray line] and semiflexible [$G^{\prime }\left(\omega \right)$, solid black line; $G^{″}\left(\omega \right)$, dashed black line] polymer rheology calculated using Eqs. (10, 11, 12). The plots show four average filament lengths in the dilute regime: (a) $L\approx 0.053\mu \mathrm{m}$. The black dotted lines are the moduli calculated using Eqs. (6, 7) with the fitted time scale $\tau =2.65\mu \mathrm{s}$; (b) $L\approx 0.15\mu \mathrm{m}$; (c) $L\approx 0.5\mu \mathrm{m}$; and (d) $L\approx 1.2\mu \mathrm{m}$.

Figure 8

(a) $G^{″}\left(\omega \right)$ and (b) $G^{\prime }\left(\omega \right)$ calculated using the generalized Stokes-Einstein relationship in the tightly entangled regime for four filament lengths (filled square: $L\approx 1.5\mu \mathrm{m}$; open triangle: $L\approx 3\mu \mathrm{m}$; filled circle: $L\approx 5\mu \mathrm{m}$; and open square: $L\approx 20\mu \mathrm{m}$) compared to the theory of entangled solution rheology of semiflexible polymers (solid lines) using Eq. (13) [12].

Figure 9

Illustration of the viscoelastic shell model used to calculate the depletion layer thickness $\Delta$.

Figure 10

The depletion shell thickness $\Delta$ (symbols) calculated using the viscoelastic shell model and nonlocality length, $\lambda$ (lines) from PRISM theory as a function of particle size [(a) shell thickness data obtained from Ref. 30 and filament length for two particle sizes, (b) square: $3.21\mu \mathrm{m}$ diameter; and triangle: $1.07\mu \mathrm{m}$ diameter.] In (b), the upper and lower solid lines show the calculated nonlocality length for the larger and smaller particles, respectively. They have been shifted (dashed lines) to compare the trends.

Figure 11

The dependence of the depletion shell thickness $\Delta$ on both particle size and filament length is collapsed onto a single curve by the nonlocality length $\lambda$ [circle: data from Fig. 10a; square: $3.21\mu \mathrm{m}$ diameter particles shown in Fig. 10b; triangle: $1.07\mu \mathrm{m}$ diameter shown in Fig. 10b]. Note that the depletion shell thickness is 1.5–1.9 times larger than $\lambda$. The line is a least-squares fit to the data.