# Polymer Physics (PHYS*7330)

Code and section: PHYS*7330*01

Term: Winter 2021

Instructor: Robert Wickham

## Course Information

### Course Objectives

This course will introduce students to the central concepts, current topics and themes of polymer physics. The objective is to provide students from a variety of backgrounds with the basic knowledge and mathematical tools that will be foundational as they advance through the field. Students should enter with a practical knowledge of undergraduate calculus, differential equations, chemistry and physics. A solid background in thermodynamics and statistical mechanics will be helpful, since this course will develop and apply concepts from these subjects in the context of polymeric systems. In this sense, PHYS*7330 is one cap-stone course for the thermodynamics/statistical mechanics stream in the undergraduate and graduate curriculum.

We will closely follow the topic selection and structure in the textbook Polymer Physics by Rubinstein and Colby. Topics we aim to cover include: properties of polymers, ideal chains, statistical description of chain conformations, scattering methods to measure chain size, real chains with excluded volume and solvent effects, thermodynamics of polymer blends and solutions, experimental investigation of polymeric phase behaviour, unentangled and entangled polymer dynamics, viscoelasticity of melts and solutions.

Students will refine their analytical and problem-solving skills through regular written assignments, involving problems drawn largely from the end-of-chapter problems in Polymer Physics. Basic information about polymer physics and theoretical/experimental methods from the first half of the course, that students should internalize, will be tested during a closed-book midterm. An end-of-term written paper, in lieu of a final exam, is intended to improve a students' written communication skills. Writing this paper will allow the student to explore a topic of their choice that is beyond the discussion in class. In consultation with the instructor, early on, a student will choose a current topic in polymer physics to be the subject of their paper. The topic should not be directly related to the student's area of research, and should not overlap extensively with material covered in class. To identify current topical areas, browsing abstracts for the DPOLY talks at the 2021 APS March Meeting (and possibly seeing these presentations during the meeting!) may be helpful.

### Class Schedule and Location

Tuesdays and Thursdays, 8:30 am - 9:50 am, synchronous (live) via Zoom. Unless there are objections, I will be recording these lectures and posting them on CourseLink.

First Lecture: Tuesday, January 12th
Last Lecture: Thursday, April 15th

The course runs for 24 lectures; however, there will be gaps in the lecture schedule (no lectures) during the week of February 15th (Winter Break) and (perhaps) the week of March 15th (APS March meeting). We can discuss this.

### Course Instructor

Name: Rob Wickham
Email: rwickham@uoguelph.ca

### Office Hours

Notionally, Tuesday, Wednesday 2:30 pm - 3:30 pm. Please send me an email if you wish to schedule a Zoom meeting at these times.

### Course Website

I will post lecture notes, problem sets, and other course-related material on CourseLink.

### Course Materials

#### Required Textbook

• M. Rubinstein and R. H. Colby, Polymer Physics (Oxford University Press, 2003).

#### Other References

• M. Doi, Introduction to Polymer Physics
• P. G. de Gennes, Scaling Concepts in Polymer Physics
• Y. Grosberg and A. R. Khokhlov, Giant Molecules: Here, There, and Everywhere
• H. Morawetz, Polymers: The Origins and Growth of a Science
• Y. Grosberg and A. R. Khokhlov, Statistical Physics of Macromolecules
• T. Kawakatsu, Statistical Physics of Polymers: An Introduction
• G. H. Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers
• G. Strobl, The Physics of Polymers
• M. Doi and S. F. Edwards, The Theory of Polymer Dynamics

### Evaluation

Assessment Weight Due Date
Assignments (5) 40%

every other Thursday:

• Feb. 4th
• Feb. 25th,
• Mar. 11th
• Apr. 1st,
• Apr. 15th (last day)
Midterm Exam 20% Friday March 5th, 1 - 4 pm
Term Paper  40% Monday, April 19th

Physics is not done in a vacuum. (OK, sometimes it is...) Students may discuss assignments amongst themselves but their written solutions must not be shared with
anyone (this would be an example of plagiarism).

Plagiarism is the act of appropriating the ...composition of another, or parts or passages of his [or her] writings, or the ideas or language of the same, and passing them off as the product of one's own mind...'' (Black's Law Dictionary). A student found to have plagiarized will receive zero for the work concerned. Collaborators shown to be culpable will be subject to the same penalties.

### Course Outline

This course follows closely the topic sequence, selection and presentation in Rubinstein and Colby, Polymer Physics. We will cover material in Chapters 1-3, on single chain physics of ideal, and real, chains; Chapters 4 and 5 on the thermodynamics of polymer blends and solutions, and Chapters 8 and 9 on unentangled, and entangled, polymer dynamics. Below is an (aspirational) outline of what I will discuss in the lectures.

#### Week 1

• 1. Chapter 1: Overview and physical context, polymer architectures, chain composition, chain conformations, length scales, conformation of ideal coil and self-similarity, solution concentration regimes
• 2. Molar mass distribution, polydispersity, examples of polymerization methods: condensation and addition, results for molar mass distributions, averages, polydispersity index

#### Week 2

• 3. Two methods to measure molar mass: osmotic pressure, thermodynamics of the problem, Mn; light scattering, Rayleigh scattering, polarizability scales as N, Nw, ideal chains
• 4. Chapter 2: Ideal chains, origin of chain flexibility, chain conformations, freelyjointed chain, universal description of ideal chains, freely-rotating chain, worm-like chain limit

#### Week 3

• 5. Radius of gyration, introduction to distribution of end-to-end vector, random walk and diffusion
• 6. Single-chain thermodynamics, force to stretch a chain, scaling argument with tension and thermal blobs (our first scaling argument)

#### Week 4

• 7. Exact calculation of force versus extension for FJC, measuring conformations: scattering (intro)
• 8. Scattering: Debye function, neutron scattering data shows chain is ideal in melt state, small- and large-q limits, pair correlation function.

#### Week 5

• 9. Chapter 3: Real chains, effective interaction, excluded volume, solvent quality
• 10. Size of a real chain (self-avoiding walk), Flory mean-field theory, scaling theory for force-extension of a real chain, example of cylindrical confinement, confinement free energy

#### Week 6 (after Winter Break)

• 11. More examples of scaling arguments: parallel plates, cylindrical pore, adsorption of a polymer on a surface, temperature effects, thermal blob
• 12. Solvent quality, distribution of the end-to-end vector of a real chain

**** END OF (SINGLE-CHAIN) MATERIAL FOR THE MIDTERM ****

#### Week 7

• 13. Chapter 4: Motivation to study thermodynamics of mixing and phase separation in a polymer blend, Flory-Huggins mean-field lattice model: what is mean-field theory? entropy of mixing, counting calculation, discussion of the result for the entropy
• 14. Nature of the interaction energy, energy of mixing (mean-field argument), Flory- Huggins interaction parameter ($x$), properties of the energy, Flory-Huggins free energy, relation to solubility parameters, limitations of the theory. Empirical formula for chi (relation to $T$) and typical values

#### Week 8

• 15. Thermodynamics of the binary polymer blend: free energy of mixed vs. two phase states, double-tangent construction, binodal (numerical solution), chemical potential, behaviour for highly-asymmetric blend compositions
• 16. Phase behaviour of the blend II, derivation of chemical potentials, linear stability analysis, metastability and instability, spinodal, detailed blend phase diagram, critical point, composition fluctuations

#### Week 9 (after March Meeting)

• 17. Size of composition fluctuations in the stable mixed phase of the blend, relation to/interpretation of small-angle scattering, extension to finite-q, Ornstein-Zernike formula, mention random-phase approximation
• 18. Osmotic pressure of polymer blend, dilute limit, virial expansion: excluded volume and chi, screening of excluded volume in the polymer melt: Flory's (mean-field) argument for ideal chain conformations in a melt, scaling arguments, two-dimensional case.

#### Week 10

• 19. Chapter 5: Polymer solutions: temperature-composition phase diagram, solvent quality, correlation length: behaviour with volume fraction, chain conformational behaviour, xi for semi-dilute, good solvents, (de Gennes) scaling theory, length scales: random walk of correlation blobs, crossover to concentrated regime
• 20. Osmotic pressure in semi-dilute good solvents, data and scaling theory, Alexanderde Gennes polymer brush

#### Week 11

• 21. Chapter 8: Unentangled polymer dynamics, overview of regimes, mathematics of Brownian motion, velocity autocorrelation and time scales, Brownian motion: noise characteristics (thermostat), mean-square displacement, Einstein relation
• 22. Rouse bead-spring model for chain dynamics, chain center of mass diffusion Rouse time, Rouse modes of relaxation (calculation of normal modes), interpretation in terms of coherent motion of sections of the chain, animation (importance of noise), Self-similarity and scaling: sub-diffusive monomer motion.

#### Week 12

• 23. Chapter 9: Experimental evidence for breakdown of Rouse theory at large N, what is an entanglement? Edwards tube model predictions (tube and entanglement blob/strand), plateau modulus and entanglement strand length.
• 24. Chain dynamics in the tube: de Gennes reptation in polymer melts, diffusion along 1D tube, scaling predictions: reptation time, viscosity, chain CoM diffusion in 3D, comparison with data, reptation time is now the longest chain stress relaxation time, hierarchy of timescales in oscillatory shear experiment, Doi-Edwards model for stress relaxation and viscosity in entangled polymer melts

### Course Evaluation

The Department of Physics requires student assessment of all courses taught by the Department. These assessments provide essential feedback to faculty on their teaching by identifying both strengths and possible areas of improvement. In addition, annual student assessment of teaching provides part of the information used by the Department’s Tenure and Promotion Committee in evaluating the faculty member's contribution in the area of teaching.

The Department's teaching evaluation questionnaire invites student response both through numerically quantifiable data, and written student comments. In conformity with University of Guelph Faculty Policy, the Department’s Tenure and Promotion Committee only considers comments signed by students (choosing "I agree" in question 14). Your instructor will see all signed and unsigned comments after final grades are submitted. Written student comments may also be used in support of a nomination for internal and external teaching awards.

Note: No information will be passed on to the instructor until after the final grades have been submitted.

## University Statements (applicable to Guelph)

### E-mail Communication

As per university regulations, all students are required to check their e-mail account regularly: e-mail is the official route of communication between the University and its students.

### When You Cannot Meet a Course Requirement

When you find yourself unable to meet an in-course requirement because of illness or compassionate reasons, please email the course instructor to make arrangements.

### Drop Date

At Guelph, the last date to drop one-semester courses, without academic penalty, is Monday, April 12th. For regulations and procedures for Dropping Courses, see the Graduate Calendar.

### Copies of out-of-class assignments

Keep paper and/or other reliable back-up copies of all out-of-class assignments: you may be asked to resubmit work at any time.

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For more information, contact SAS at 519-824-4120 ext. 56208 or email sas@uoguelph.ca or see the SAS website.