Activation Energy for Mobility of Dyes and Proteins in Polymer Solutions: From Diffusion of Single Particles to Macroscale Flow

We measure the activation energy $E_a$ for the diffusion of molecular probes (dyes and proteins of radii from 0.52 to 6.9 nm) and for macroscopic flow in a model complex liquid—aqueous solutions of polyethylene glycol. We cover a broad range of polymer molecular weights, concentrations, and temperatures. Fluorescence correlation spectroscopy and rheometry experiments reveal a relationship between the excess of the activation energy in polymer solutions over the one in pure solvent $\Delta E_a$ and simple parameters describing the structure of the system: probe radius, polymer hydrodynamic radius, and correlation length. $\Delta E_a$ varies by more than an order of magnitude in the investigated systems (in the range of ca. $1–15\mathrm{kJ}/\mathrm{mol}$) and for probes significantly larger than the polymer hydrodynamic radius approaches the value measured for macroscopic flow. We develop an explicit formula describing the smooth transition of $\Delta E_a$ from the diffusion of molecular probes to macroscopic flow. This formula is a reference for the quantitative analysis of specific interactions of moving nano-objects with their environment as well as active transport. For instance, the power developed by a molecular motor moving at constant velocity $u$ is proportional to $u^2\exp (E_a/RT)$.

Figure 1

Relative diffusivity at the macroscale for different PEG solutions plotted according to the proposed scaling equation. The slope of the concentrate linear fit corresponds to the coefficient $\gamma$ [see Eq. (5)], equal to ca. $3.96\mathrm{kJ}/\mathrm{mol}$.

Figure 2

Relative diffusivity experienced by molecular probes in PEG solutions plotted according to the proposed scaling equation. Probe descriptions: rho, rhodamine ($r_p=0.52\mathrm{nm}$); lys, lysozyme ($r_p=1.9\mathrm{nm}$); apo, apoferritin ($r_p=6.9\mathrm{nm}$).

Figure 3

Dependence of the $\gamma$ coefficient [Eqs. (7, 8)] on the probe/polymer size ratio. Relatively large molecular probes reproduce well the results of macroviscosity measurements. The shaded range denotes the maximal error of rheometry results.