3
$\begingroup$

I'm an applied math graduate student studying the cytoskeleton. I wanted to know of any reference(s) providing the most general mathematical theory of polymer dynamics, think an updated version of Doi & Edwards. This could be a textbook or some thorough review papers.

For example, Doi and Edwards tends to treat systems as having uniform concentration without the interplay of how concentration gradients can interact with contour length distribution functions for flexible polymers or orientational distribution functions for rigid rods. It also doesn't examine semiflexible polymers in depth.

Don't get me wrong - it's an invaluable reference. I'm wondering if anyone has further expanded/generalized the theory since then and what mathematics they've used in the process.

$\endgroup$
4
$\begingroup$

Apart from Doi and Edwards, these are the key books on polymer theory:

  1. Bird, Hassager, Armstrong - Dynamics of Polymeric Liquids (Two volume set)
  2. Ronald Larson - Constitutive equations for polymer melts and solutions
  3. Yamakawa - Helical Wormlike Chains in Polymer Solutions
  4. Yamakawa - Modern Theory of Polymer Solution
  5. Prof. Dr. Hans Christian Öttinger - Stochastic Processes in Polymeric Fluids - Tools and Examples for Developing Simulation Algorithms
  6. Karl Freed - Renormalization Group Theory of Macromolecules

For semiflexible polymers see papers by Roland G. Winkler, B. J. Cherayil.

If you want to understand Polymer Field Theory and Path Integral Techniques then these should do:

  1. M.Chaichian, A. Demichev - Path integrals in physics (2 volume set)
  2. H. Kleinert - Path Integrals in Quantum Mechanics, Statistics, Polymer Physics & Financial Markets

More info:

The two volume set by Bird, Hassager is one of the most consulted reference for Polymer theory and Rheology (i.e. behavior of polymers under flows). The helical worm like chains book by Yamakawa is one of the latest additions to the subject and treats the topics you are interested in i.e. Semiflexible and Stiff chains. Öttinger's book includes many numerical and Simulation schemes to study polymer dynamics. He is a renowned polymer physicist and gets heavily cited in polymer literature. He has a very unique style of writing and his papers are the most lucid ones for beginning graduate students.

Freed's book is a great resource for those who want to dive deep into the functional integral techniques routinely employed to study many path dependent observables and continues to be the best resource to understand critical phenomena in polymer, polymer field theory and renormalization. But it is quite heavy. He was one of the first few who gave polymer physics a new direction in the 80's by showing how RG techniques can be applied to polymer problems. He has authored numerous seminal papers on the treatment of Semiflexible Polymers as well.

Semiflexible Polymers are not that hard to treat compared to flexible ones. One introduces bending rigidity into the model by defining a tangent vector along the chain contour which serves as measure of persistence length and that is longer compared to the flexible case. One needs to use subtle boundary conditions at the two ends of the chain to suppress unphysical fluctuations that then leads to a 4th order pde which can be decomposed in terms of normal modes just like the Rouse and Zimm models. Winkler's papers are the best source to learn about the subtle intricacies in modelling Semiflexible Polymers.

Moving from single chain systems you may want to study entangled polymer solutions and melts since you want to model viscoelastic systems which carries the memory of deformation since all living systems are nothing but memory fluids. For that I suggest you to get accustomed with the Generalized Langevin Equation formalism and later move onto the Fractional calculus approach to model such memory systems. Fractional calculus approach is currently a hot topic to study non-Gaussian statistics and anomalous diffusion problems in polymers and related fields. Here we leave the regime of Markovian into Non-markovian dynamics. This will give an overview of modelling aspects from a materials science/engineering perspective and introduce you to the phenomena of ergocity breaking in statistical mechanics and provide you with the microscopic roots to viscoelasticity and Fractional Kinetics. In addition, you may want to study dynamics not only in the presence of white noise but colored noise with long-range correlations for e. g. Fractional Gaussian Noise, Mittag-leffler type noise.

Moreover, an exploration into Nonequilibrium Stochastic thermodynamics and Fluctuation theorems based on polymer or protein manipulation (pulling experiments) where one calculates the distribution of work done and heat dissipated during such processes, excess entropy production and Kullback-Leibler divergence as a measure of nonequilibrium behavior, can push you into a very intellectually fruitful trajectory.

Moving on, I must stress you to learn path integral techniques if you really want to build Intuition into chain dynamics. It's a beautiful technique to explore the field theory (quantum and Statistical) aspects of stringy objects. From your post, it seems like you are a fresher. Pick up Feynman, Hibbs's path integral book and then move onto Chaichian, Demichev. It will open a whole new world to you and I promise you will start looking for problems you can solve using this technique to study contour length fluctuations and two-point correlation functions. One can study multipoint correlations as well. So invest at least a year learning it and reproducing key papers. Avoid Kleinert's book initially but keep it aside for more technical stuff. After you have mastered the key techniques, you want to explore Doi-Peliti and Martin-Siggia-Rose-Janssen–De Dominics formalism. For that, Alex Kamenev's Field Theory of Non-Equilibrium Systems is the best resource.

Apart from the book suggestions, I have laid out a plan for discovering intriguing stuff related to polymer physics. I hope that helps you in your future endeavors.

$\endgroup$
  • $\begingroup$ The banner at the top of the page says, Please write substantial answers that detail the style, content, and prerequisites of the book, paper or other resource. Would you be able to add these details in? Otherwise, your answer risks deletion. $\endgroup$ – Kyle Kanos Dec 28 '18 at 11:18
  • $\begingroup$ @KyleKanos added more info as suggested. $\endgroup$ – noisyoscillator Dec 28 '18 at 16:14

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.