Brownian dynamics simulations of bead-rod-chain in simple shear flow and elongational flow
Introduction
Polymer liquids in flows exhibit several interesting phenomena. Such behavior is typically due to the non-Newtonian nature of the macromolecular fluid. For example, polymer solutions and melts in shear flows undergo a decrease in viscosity with increase in shear rate. This shear-thinning behavior is observed in most polymer solutions that have a shear rate dependent viscosity, although there are a few polymer solutions that are dilatant (i.e. that exhibit shear-thickening). The normal stresses are also non-zero and shear rate dependent. There are many experimental investigations of shear-thinning (see, for example, [1], [2], [3], [4]).
Another interesting phenomenon is the existence of the coil-stretch transition when the polymeric liquid is under an elongational flow. The nature of this transition is still unclear, although De Gennes has predicted the presence of a first order phase transition for two-dimensional elongational flows [5].
Experiments performed by Odell et al. and by others have shown the critical strain rate εc to be related to the molecular weight of the polymer by a power law, εc∼N−β, β=1.5 independent of solvent quality [6], [7], [8], [9], [10], [11]. Different values of β have also been reported by other authors for good solvents [12], [13].
To calculate the rheological properties of the complex fluid, the exact approach would be to include the effect of solvent molecules as well as macromolecules. Although such calculations are extant, the overwhelming computational burdens involved make their application still very limited.
The solvent is therefore usually neglected and the system is approximated by theoretical models. The simplest model is the Rouse model [18], where the solvent is approximated by pure random noises. However, as it completely ignores the effect of excluded-volume and hydrodynamic interactions (HI), a simulation of a dilute polymer solution based on the Rouse model would not take into account the long-ranged hydrodynamic coupling, and would not predict shear-thinning behavior, nor yield correct scaling laws for material and transport functions.
The Kirkwood–Riseman–Zimm model [19], [20] takes HI into account with a pre-averaging approximation and obtains correct scaling laws, but the material functions are independent of shear rate. Several authors have contributed other models, improving over the pre-averaging HI approximation and including excluded volume effects [21], [22], [23], [24], [25], [26], [27], [28], [35], [37], [38], [39], [40]. In particular, Fixman made a correction to the Zimm pre-averaging approximation by his perturbative calculation of the effect of HI [21], [22], [23], [24], [25]. Öttinger developed a consistently averaging model and obtained the correct flow-rate dependence for the material functions [36]. Further details of theory, simulation models and experimental data, can be found in, for example, Bird et al. [31], [32], Yamakawa [33], Öttinger [34] and in Petera and Muthukumar [38] (here-after referred to as P–M) and the references cited therein. Öttinger has explicitly reformulated the diffusion equation for polymer chains under constraints and with HI into a set of stochastic differential equations; he has demonstrated the construction of Brownian dynamics simulation algorithms and the technique of calculation therein of stresses [35]. P–M implemented Öttinger's bead-rod-chain model in a Brownian dynamics simulation with excluded volume effects and obtained correct shear-thinning behavior [38]. Here in this paper, following the same algorithm as in P–M, we have simulated a single freely jointed chain, with number of units N up to 60, under shear and elongational flows. We calculated both configurational and rheological quantities mainly in the mid and high strain rate regimes. We also paid attention to the coil-stretch transition in the low flow rate regime. As mentioned earlier, it is still debatable whether the coil-stretch transition is a first order transition; our simulations for short chains indicate that the transition approaches discontinuity as the chain length increases.
This paper proceeds as follows. We present our simulation model in some detail in Section 2. Section 3 gives the results of our simulations, and Section 4 contains the discussion of results and conclusions.
Section snippets
Algorithm
We modeled the polymer with N+1 beads connected by N freely rotating rigid rods of length l. The system is described by a diffusion equation derived by P–M from the work of Öttinger by using connector vectors instead of positional vectors.
If denotes the position vector of the ith bead, the connector vector between beads i and i−1 will be given by The stochastic differential equation for the connector vectors is [38]
Results for shear flow
In order to verify our implementation of the algorithm described above, we simulated the same system as in P–M, but with a much longer chain, N=60. A very good agreement was found on comparing the results of our long chain simulation to their simulation with shorter chains. Results are summarized in 3.1 Configurational quantities, 3.2 Rheological quantities.
Results for elongational flow
For elongational flow, we are principally interested in the critical strain rate εc of the coil-stretch transition. This has been observed by experiments [6], [11], [15] and in some simulations [29], with and without including HI. Our results are summarized in Fig. 7, Fig. 8, Fig. 9, Fig. 10. We have plotted Rg2 as a function of strain rate in Fig. 7. The orientational order parameter, S, for the system, is defined as the ensemble-averaged value of the single bond order parameter Si for bond i,
Discussion and conclusion
We implemented a Brownian dynamics algorithm for the simulation of a bead-rod-chain model for a dilute polymer solution under pure shear and elongational flows. The bead-rod-chain model constrains the bond length, thereby preserving the crucial property of inextensibility of the chemical bond, while models with bond extensibility might fail to display shear-thinning at high shear rates. We showed that our simulation produced the correct shear-thinning behavior even at very high shear rates. The
Acknowledgements
We would like to thank D. Petera and Prof. H. C. Öttinger for helpful discussions. Acknowledgment is made to the Materials Science Research and Engineering Center, University of Massachusetts, Amherst.
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