Viscoelastic Response of a Complex Fluid at Intermediate Distances

Abstract

The viscoelastic response of complex fluids is length-and time-scale dependent, encoding information on intrinsic dynamic correlations and mesoscopic structure. We study the length scale above which bulk viscoelasticity sets in, and the material response that precedes it at shorter distances. We show that the crossover between these two regimes may appear at a surprisingly large distance. We generalize the framework of microrheology to include both regimes and apply it to F-actin networks, thereby extracting their dynamic correlation length from their bulk and local viscoelastic properties. PACS numbers: 47.57.Qk, 87.16.dm, 87.16.dj, 87.16.Ln Most fluids in nature and industry are complex, or structured [1], in the sense that they include mesoscopic elements in between the molecular and macroscopic scales. For example, in suspensions, micron-scale solid particles are dispersed in a molecular fluid, and in polymer gels the polymer chains form a network embedded within a molecular solvent. Consequently, the response of complex fluids to stress is characterized by intermediate length and time scales.The bulk viscoelastic response of such materials is commonly measured using macrorheology [2]. Similar information, for a wider frequency range and smaller material quantity, can be extracted from microrheology by following the motions of embedded tracer particles [3][4][5][6][7][8]. In one-point (1P) microrheology [3-5] the thermal fluctuations of a single particle are used to infer the viscoelastic properties of the medium via a generalized Stokes-Einstein relation (GSER). It has been found that this measurement is affected by the local environment of the tracer particle [9,10], and thus may fail to reproduce the material's bulk response. Two-point (2P) microrheology [6] overcomes this obstacle by tracking the correlated motions of particle pairs as a function of their separation. 2P measurements have focused on asymptotically large separations, where the pair correlation has a universal form due to momentum conservation.The current work addresses two questions: i) Beyond what length scale does the bulk viscoelastic behavior emerge? ii) What is the material response at smaller length scales? We find that the leading correction to the asymptotic behavior at large distances, referred to hereafter as the subdominant response, may be unexpectedly large, causing the bulk response to set in at surprisingly large distances. The physical origin of the subdominant response, which is unique to complex fluids, is different from that of the asymptotic one. It is related as well to a conservation law (of fluid mass rather than momentum), resulting in a generic system-independent form. The study of this distinctive regime leads to a more com-plete description of the complex-fluid response.We first derive the generic form of the subdominant response and, subsequently, confirm the general predictions in a specific theoretical example, the two-fluid model of polymer gels [8,11,12]. Extending the framework of microrheology to include...

Response of a polymer network to the motion of a rigid sphere

“…(18), depending on the fluid viscosity η and characteristic pore size ξ. Hence, ξ is identified with the network's mesh size [10], up to a proportionality factor close to unity [12,13]. It also characterizes the spatial decay of transverse (shear) stresses due to the friction between the two components and, therefore, decreases with increasing Γ .…”

“…Theoretical arguments given in ref. [12] showed that the intermediate response was fundamentally different from the macroscopic one, in that it was related to mass transport, rather than momentum transport, of the fluid. As we shall see below, the intermediate region corresponds to the relative motion of the material's two components (network and solvent), whereas in the asymptotic region the material moves collectively, as a whole.…”

“…A recent study, applying two-point microrheology to entangled F-actin networks [12,13], revealed a wide range of distances, intermediate between the microscopic scale (the network's mesh size ξ) and the macroscopic (asymptotically large) one, in which the dynamic pair correlations were qualitatively different from those at larger distances. This intermediate behavior ended at a distinct dynamic length, ℓ c , much larger than ξ, which marked the crossover to the macroscopic response.…”

“…The boundary conditions at infinity are, therefore, u(r → ∞, θ) = v(r → ∞, θ) = ∇p(r → ∞, θ) = 0. (12) As to the boundary conditions at the surface of the sphere, we will examine various possibilities in sect. 4. 3 General solution…”

“…As we shall see below, the intermediate region corresponds to the relative motion of the material's two components (network and solvent), whereas in the asymptotic region the material moves collectively, as a whole. The application of these ideas to a combination of one-and two-point measurements gives rise to an extension of microrheology, allowing for better characterization of complex fluids (e.g., extracting their correlation length) [12,13].…”