We present a scaling formula for size-dependent viscosity coefficients for proteins, polymers, and fluorescent dyes diffusing in complex liquids. The formula was used to analyze the mobilities of probes of different sizes in HeLa and Swiss 3T3 mammalian cells. This analysis unveils in the cytoplasm two length scales: (i) the correlation length ξ (approximately 5 nm in HeLa and 7 nm in Swiss 3T3 cells) and (ii) the limiting length scale that marks the crossover between nano- and macroscale viscosity (approximately 86 nm in HeLa and 30 nm in Swiss 3T3 cells). During motion, probes smaller than ξ experienced matrix viscosity: η(matrix) ≈ 2.0 mPa·s for HeLa and 0.88 mPa·s for Swiss 3T3 cells. Probes much larger than the limiting length scale experienced macroscopic viscosity, η(macro) ≈ 4.4 × 10(-2) and 2.4 × 10(-2) Pa·s for HeLa and Swiss 3T3 cells, respectively. Our results are persistent for the lengths scales from 0.14 nm to a few hundred nanometers.

We measured the viscosity of poly(ethylene glycol) (PEG 6000, 12,000, 20,000) in water using capillary electrophoresis and fluorescence correlation spectroscopy with nanoscopic probes of different diameters (from 1.7 to 114 nm). For a probe of diameter smaller than the radius of gyration of PEG (e.g. rhodamine B or lyzozyme) the measured nanoviscosity was orders of magnitude smaller than the macroviscosity. For sizes equal to (or larger than) the polymer radius of gyration, macroscopic value of viscosity was measured. A mathematical relation for macro and nanoviscosity was found as a function of PEG radius of gyration, R(g), correlation length in semi-dilute solution, xi, and probe size, R. For R < R(g), the nanoviscosity (normalized by water viscosity) is given by exp(b(R/xi)a), and for R > R(g), both nano and macroviscosity follow the same curve, exp(b(R/xi)a), where a and b are two constants close to unity. This mathematical relation was shown to equally well describe rhodamine (of size 1.7 nm) in PEG 20,000 and the macroviscosity of PEG 8,000,000, whose radius of gyration exceeds 200 nm. Additionally, for the smallest probes (rhodamine B and lysozyme) we have verified, using capillary electrophoresis and fluorescence correlation spectroscopy, that the Stokes-Einstein (SE) relation holds, providing that we use a size-dependent viscosity in the formula. The SE relation is correct even in PEG solutions of very high viscosity (three orders of magnitude larger than that of water).

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We study the phase diagram and scattering functions of a ternary mixture of A and B homopolymers, in equal concentrations, and AB diblock copolymer. We locate the disorder line, which marks the appearance of damped oscillatory decaying correlation functions in the system, and also the Lifshitz and equimaxima lines of the structure functions. At these lines, the oscillatory component becomes dominant in a given correlation so that the peak in scattering intensity is no longer at the zero wave vector. We find that while the Lifshitz line of the structure function of all A monomers is quite close to the disorder line, that of A monomers in the homopolymer only is far from it. This shows that the copolymer orders the homopolymers inefficiently, a behavior which is also reflected in the copolymer’s weak ability to solubilize them, and which contrasts with the amphiphilic solubilizers of oil and water. The disorder line and all Lifshitz lines meet at a Lifshitz tricritical point. As these ternary mixtures provide a rare opportunity to study this unusual point, we discuss its effects on the structure functions and surface tensions.