Main Page
During my postdoctoral study at Caltech, I collaborated with Professors John Brady and Zhen-Gang Wang, in the area of Polymer Rheology. In particular, I studied the static and dynamic properties of semiflexible polymers near equilibrium, with main focus on the scaling behavior in the stress relaxation based on the relation between the polymer configuration and stress. Common examples of semiflexible polymers include biopolymers such as DNA, actin filaments, microtubules and rod-like viruses as well as a host of stiff synthetic polymers such as Kevlar and polyesters. In the case of biopolymers, the properties of interest are often on length scales that are comparable to the persistent length of the molecules. For stiff synthetic polymers, stiffness is responsible for the macroscopic alignment of the chains in the system which imparts unique mechanical properties for these materials. Although I explicitly considered the relaxation of a single polymer chain in a viscous solvent, my results are valid even for concentrated polymer solutions and networks as long as the relaxation of interest occurs on length scales shorter than that characterizing the entanglements or crosslinks.In order to identify the effects of local stiffness on the properties of semiflexible polymers, I employed the Brownian Dynamics method developed recently by Grassia and Hinch (1996). This method is based on a (flexible) bead-rod model with fixed bond-lengths. Using this model, the total contour length of the polymer is always fixed; thus, we avoid unrealistic behavior that would arise from large fluctuations of the polymer length, if the links were extensible. To account for polymer stiffness, I used the Kratky-Porod wormlike chain model. This model allows a continuous crossover from a freely-jointed flexible chain to a rigid rod as the bending energy increases. A Monte-Carlo Metropolis method, based on the chain bending energy, was employed to generate equilibrium configurations. To be able to perform large-scale dynamic simulations (i.e. study very long chains), the algorithm was parallelized using Message Passing Interface (MPI). All computations were performed on the multiprocessor computers SGI Origin 2000 and NT Supercluster provided by the National Center for Supercomputing Applications (NCSA) at Urbana, and on the Compaq Alpha Beowulf Cluster provided by the Center for Advanced Computing Research (CACR) at Caltech.
We calculated two kinds of properties: conformational and material ones. These properties were expressed as a function of the two parameters of the model, the number of rods N and the bending parameter E or equivalently, the chain length L and the persistent length Lp. (Note that L = N b while Lp = E b , where b is the fixed bond length.) An extensive study was conducted, covering a broad range of chain stiffness. I summarize briefly our main results. The influence of the bending energy on the long-time behavior of the polymer stress can be divided into two regimes: the coil-like regime for small stiffness E (or more precisely for E << N), and the rod-like regime for large E (or E > N). The longest relaxation time for both the conformational and material properties was found to be an increasing function of the number of rods and the bending parameter. This behavior results from the rotation (or translation) of the whole polymer chain. A scaling law was provided based on the average length of the chain and the diffusivity of the center of mass. The longest relaxation time for the stress autocorrelation in the coil-like regime is twice the one for the rotational correlation, in agreement with the analytical predictions for the Rouse model and zero bending potential. On the other hand, the longest relaxation time for the stress autocorrelation in the rod-like regime is three times the one for the rotational correlation, in agreement with the analytical predictions of the rigid-rod model.
At very early times, the stress relaxation for a given polymer chain exhibits a plateau, followed by fast decay towards the rotation regime. This change is caused by the transverse relaxation of the chain ends. The relevant time scale, for very stiff chains, is inversely proportional to the polymer stiffness E. An accurate estimation for this time scale (as well as a scaling law) for any bending parameter E, may be found from the relaxation of the chain ends. Over a wide intermediate-time window spanning several decades the stress relaxation is described by a single power law t-a, with the exponent a apparently varying continuously from 1/2 for flexible chains, to 5/4 for very stiff ones. At intermediate times, both transverse and longitudinal relaxations contribute to the polymer stress decay. Our study identifies the limits of validity of the t-3/4 power law at short times predicted by recent theories. A new regime is identified, the ultra-stiff chains, where this behavior disappears. In the absence of Brownian motion, the purely mechanical stress relaxation produces a t-3/4 power law for both short and intermediate times. (In this case the relaxation is governed by transverse motion only as in the short-time t-3/4 relaxation.) In contrast, for ultra-stiff chains over the same time interval a different power law (t-5/4) was observed; this points to the essential role of thermal motion at any finite temperature.
My results are in good agreement with current experimental findings. It is well known that flexible polymers show high-frequency t-1/2 scaling of the shear modulus, in agreement with our E=0 results. Concentrated solutions and networks of actin filaments were found to exhibit a high-frequency viscoelastic modulus t-0.75--0.78. In addition, my results may explain recent experimental findings that suggest a high-frequency viscoelastic modulus varying from t-3/4 to t-1. Finally, I hope that my study motivates experiments that cover all the stiffness regimes that I have identified.
This study is now complete and a manuscript [ 1 ] has been published in the Phys. Rev. E, while two additional manuscripts [ 2 , 3 ] are being prepared. Our results were presented at the AIChE Annual Meeting, in November 2000, in Los Angeles, California (see abstract), the 72nd Annual Meeting of the Society of Rheology (abstract), as well as the AIChE 2001 Annual Meeting in Reno, Nevada (abstract).
This work was supported by the National Science Foundation under the grant DMR-9970589, 1999-2002. The computations were performed on multiprocessor computers provided by the National Center for Supercomputing Applications at Urbana and the Center for Advanced Computing Research at Caltech.