Numerical study of polymer tumbling in linear shear flows
Introduction
Thanks to recent improvements in experimental techniques it is nowadays possible to follow the motion of individual molecules in solvents [1], [2], [3], [4], [5], [6], [7], [8], [9]. The characterization of polymer dynamics at the level of a single molecule is a first step towards the understanding of mechanical interactions between biomolecules (see, e.g. [10], [11], [12], [13], [14], [15], [16], [17]), of the fundamental rheology of polymer solutions, and of the viscoelastic properties of more complex flows (see, for example [18], and references therein for elastic turbulence).
Recently measurements of elongation and orientation with respect to simple external flows have been performed [7], [9] in order to analyze how the conformation of a single molecule can be modified by an external field.
As the number of degrees of freedom needed to fully describe a macromolecule is extremely high it is necessary to formulate theoretical models, to verify them with numerical simulations, to simplify the problem , and to understand which observables play a key role.
Large numbers of papers have been written on the subject of single polymer dynamics in shear flows from experimental, numerical and theoretical viewpoint by Chu, Larson, Shaqfeh and their respective colla- borators.
A polymer molecule in a plane, linear, steady shear flow is oriented in the flow direction by the velocity field. As the rotational motion of the polymer is determined only by the velocity difference in the space points, when it is aligned along the shear direction the external flow effect becomes negligible, and the thermal noise is the most relevant external force. Thermal fluctuations can push the polymer to regions where the external flow is relevant again. In these cases the shear flow can induce a fast rotation and align the polymer again along the (reverse) flow direction, i.e. the polymer tumbles [21], [3], [4], [8], [9], [7].
This phenomenon can happen via several conformational pathways due to the complexity of the motion of a polymer molecule (see, for example [8], [9]), and can be fully described only by taking into account all the degrees of freedom of the polymer. Unfortunately in the framework of these complex polymer models it is very difficult to obtain analytical results.
The main goal of our paper is to test numerically the predictions of recent theoretical and experimental studies [20], [7] in a framework in which analytical results can be obtained [23], and to analyze the statistics of the tumbling times, i.e. the time between two subsequent flips of the polymer [21], [3], [7]. The simplest model that reproduces qualitatively the behavior of polymers is the dumbbell model [22]. This model allows one to carry out some analytical calculations in the case which we analyze [19], [20], [24], [23], it is very easy and fast to simulate numerically (see Setion 2 and [25]) and reproduces qualitatively recent experimental results [3], [7].
The paper is organized as follows: in Section 2, the evolution equation of the polymer and the numerical methods are briefly explained. Section 3 is devoted to the analysis of the statistics of thermal fluctuations of a flexible polymer placed into a linear shear flow. In our work we present the analysis of the stationary distribution of the polymer end-to-end vector and we study the distribution function of the polymer tumbling time, which can be measured experimentally. In Section 4, we study the angular dynamics of strongly elongated polymers, for which the size fluctuations are negligible. Finally, in Section 5, we study the elongation statistical properties of the end-to-end vector in a random flow with a large mean shear.
Section snippets
Basic relations and numerical analysis
We wish to analyze the behavior of a polymer in a generic simple shear flow experiencing the Langevin force [19]. In general there are two effects of the velocity field on a polymer molecule: the Lagrangian advection of the polymer and the elongation/relaxation dynamics due to velocity gradients. In all the cases discussed in this paper we disregard the Lagrangian dynamics by staying in the reference frame of the polymer center of mass. For the internal degrees of freedom of the polymer we use
Thermally driven polymers
In this section we will examine the case of a linear steady shear flow in the plane , where s is the shear rate. Eq. (1) has an explicit solution of the following form:where . At large times the initial polymer elongation is forgotten and after averaging over the thermal fluctuations one can easily obtain the following distribution function:
Strongly elongated and rigid molecules
Another physical situation we are interested in is the dynamics of strongly elongated polymers in random flows. In this model, described in detail in [20], [24], [23], the polymer is placed into a random flow above the coil-stretch transition, where the effect of the thermal fluctuations can be neglected. In this case the orientational dynamics of the polymer are decoupled from the evolution of the elongation, so that we can introduce the unit vector , obeying the following evolution
Polymer elongation in chaotic flows
In this final section we will study elongation statistics of the polymer in the case of random velocity plus mean shear. The polymer is not strongly elongated and can be treated as a linear dumbbell, as in Section 3. Such a situation corresponds to a flexible polymer in a chaotic flow below the coil-stretch transition[31]. Formally, this is the case when the maximum Lyapunov exponent is smaller than the inverse relaxation time , where the Lyapunov exponent is the rate of divergence or
Conclusions
The tumbling phenomenon [21], [3], [20], [7] has been studied in the framework of the linear dumbbell model, and some universal features of this motion are derived and numerically verified.
Three different examples of polymers in a linear, steady, plane, shear flow have been studied: (i) a flexible polymer experiencing thermal noise, (ii) a rigid polymer in a smooth random velocity gradient above the coil-stretch transition [31], [32], [6]), and (iii) a flexible polymer in a smooth random
Acknowledgements
We would like to thank A. Celani, M. Chertkov, I. Kolokolov and V. Lebedev for many fruitful and inspiring discussions and S. Geraschenko and V. Steinberg for many useful comments and interest in our work. We are grateful to Chris Carr and Colm Connaughton who helped us to develop the final form of this paper. We acknowledge hospitality of CNLS at Los Alamos National Laboratory where this work was partially done. KT was supported by RFBR grant 04-02-16520a and INTAS grant Nr 04-83-2922.
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