Dynamics of Strongly Deformed Polymers in Solution

– Bead spring models for polymers in solution are nonlinear if either the finite extensibility of the polymer, excluded volume effects or hydrodynamic interactions between polymer segments are taken into account. For such models we use a powerful method for the determination of the complete relaxation spectrum of fluctuations at steady state. In general, the spectrum and modes differ significantly from those of the linear Rouse model. For a tethered polymer in uniform flow the differences are mainly caused by an inhomogeneous distribution of tension along the chain and are most pronounced due to the finite chain extensibility. Beyond the dynamics of steady state fluctuations we also investigate the nonlinear response of the polymer to a large sudden change in the flow. This response exhibits several distinct regimes with characteristic decay laws and shows features which are beyond the scope of single mode theories such as the dumbbell model. Introduction. – The slow internal dynamics of polymers makes polymer solutions viscoelastic which is manifest in such spectacular effects as turbulent drag reduction. In spite of active research for more than 60 years already, the understanding of these phenomena is still incomplete [1, 2, 3, 4, 5, 6, 7, 8]. Progress towards a microscopically founded rather than empirical description of flowing polymer solutions is possible only by a thorough analysis of the interplay between single polymers and the flow. However, only recent experiments on individual DNA molecules allow to observe the molecular conformations [9, 10, 11, 12] and computer simulations facilitate a treatment of the nonlinear interactions within the polymer-solvent system [13, 14, 15, 16, 17]. The dynamics of fluctuations of a polymer in flow is commonly characterized by relaxation times and modes, which also determine the linear viscoelastic properties. These quantities are known analytically only for the linear Rouse and Zimm models [5, 6]. Questions arising are: Is there an effective method for determining the complete relaxation spectrum and the corresponding modes for fluctuations around an arbitrary stationary state also for more realistic nonlinear polymer models? What is their dependence on the polymer deformation and which decay laws describe the whole relaxation process of a strongly stretched polymer to Typeset using EURO-TEX 2 EUROPHYSICS LETTERS equilibrium? Here we give answers for the model problem of a tethered polymer in uniform flow which has been much investigated recently both experimentally [9, 10] and theoretically using blob models [18, 15] and bead spring models [13, 14, 15, 16, 17]. By simulation of bead spring chains we determine the relative importance for the polymer dynamics of various interactions such as finite extensibility, excluded volume interactions (EVI), and hydrodynamic interactions (HI). Due to Brownian motion, even in a steady state the beads fluctuate around some timeindependent average positions. In the equilibrium state of a tethered polymer these average positions all coincide with the tether-point, while if the polymer is stretched by a flow or force they assume non-trivial values. Our analysis of the conformational fluctuations in such a statistically steady state is based on the covariance matrix of bead positions. This approach yields the full relaxation spectrum rather than only the longest relaxation time considered in previous works and it provides numeric values for the relaxation times which cannot be obtained from common scaling arguments. Furthermore it gives a corresponding complete set of uncorrelated relaxation modes and thus constitutes a generalization of the normal mode analysis used for the linear models of Rouse and Zimm which is applicable to nonlinear polymer models. Theories of polymer deformation in simple shear or elongational flows rely on the so-called time criterion [7]: the polymer will be significantly deformed when the inverse shear or elongation rate κ is shorter than the longest polymer relaxation time. Since the latter is conformationdependent because of hydrodynamic backflow, it was argued that a hysteretic transition between a coiled and a stretched polymer conformation may occur upon varying κ [7, 8]. Implicit in the use of the time criterion is the assumption that the polymer is in the steady state corresponding to the local flow field at any time. This requires variations in the flow field experienced by the polymer to be slow – an assumption which is probably not valid in flows as they occur e.g. in turbulent drag reduction. An interesting model for such rapidly varying flows is provided by the immediate start–up and cessation of simple flows like the uniform flow acting on a tethered polymer considered here. For this case we identify several regimes governed by different decay laws which have not been described before. This complex behavior cannot be captured by a simple dumbbell model and it coincides with the relaxation of fluctuations in the steady state only in the linear response regime of polymer dynamics. Bead Spring Polymer Model. – In our simulations we follow the Brownian dynamics of a bead spring model as described in more detail in Refs. [16, 17]. The equation of motion for the vector R = (R1, ...,RN) of all bead positions may be written in the form