Liquid-state and soft-matter physics

A.Y. 2020/2021
Overall hours
Learning objectives
The course aims at teaching the fundamentals of the theory of equilibrium properties of classical liquids such as fluids of neutral and charged particles, polymer solutions, and colloidal dispersions. Starting from a basic knowledge of statistical mechanics, we will first consider the mean-field approximation for interacting particles, and subsequently the density-density correlation function together with some approximate methods for determining it. We will then introduce the concept of effective interaction, which will be applied in several cases involving polymer solutions, stabilization of colloidal dispersions, and osmotic depletion forces.
Expected learning outcomes
Students who attend the course will develop the following skills:

1) They will be able to establish the link between the microscopic interaction and the macroscopic properties of the fluid in some paradigmatic cases, and to grasp the similarities as well as the differences between them.

2) They will be able to use the mean-field approximation and to understand its limitations.

3) They will learn some approximate methods for determining the correlation functions, and those for determining the thermodynamics from the correlations.

4) They will get familiar with the simplest equations of state for uncharged and charged fluids and polymer solutions and with the qualitative features of their phase diagrams.

5) They will learn the basic features of colloidal dispersions, including electrostatic and steric stabilization.

6) They will acquire knowledge about the origin and the description of osmotic depletion forces by examining in detail a specific model of a colloid-polymer mixture.
Course syllabus and organization

Single session

Lesson period
Second semester
In case of mobility restrictions ensuing from COVID-19 pandemics, lectures will be delivered remotely in pre-recorded format by uploading audio-video lecture files to ARIEL official University repository. Real-time virtual meetings may be scheduled upon students' request to answer their questions, and will be handled by Zoom.
Course syllabus
- A short reminder of basic thermodynamics and statistical mechanics; phase diagrams.

- Elements of liquid-state theory: dispersion forces; mean-field approximation; hard-sphere gas; van der Waals gas; fluid-fluid and fluid-solid transitions; correlation functions; structure factor; perturbative methods; random-phase approximation; Coulomb gas; Debye-Hückel and Poisson-Boltzmann equations.

- Polymer solutions: random-coil model; Flory-Huggins model; soft repulsive interactions between polymer chains; correlation length; scaling.

- The colloidal state: size and density of particles; Brownian motion; Hamaker constant; Derjaguin approximation; coagulation; steric stabilization; electrostatic stabilization; DLVO potential.

- Depletion forces: Asakura-Oosawa model and effective potential.
Prerequisites for admission
1) Fundamentals of thermodynamics: thermodynamic potentials and expression of observables in terms of said potentials.

2) Fundamentals of statistical physics:
a) Canonical and grand canonical ensembles, partition function, link with thermodynamics.
b) Energy flucutations in the canonical ensemble. Particle number fluctuations in the grand canonical ensemble.
c) Classical ideal gas in the canonical and grand canonical ensembles. Thermal wavelength.
d) Configurational part of the partition function. Excess free energy.
e) Ideal gas in an external field: barometric law.
Teaching methods
During lectures, both slides and blackboard are used. In order to see theory at work, special emphasis is given to its applications to specific models. These applications are described rather in detail, so as to allow students to master the methods being used and to understand their advantages and limits.
Teaching Resources
P.N. Pusey, "Colloidal Suspensions", Les Houches, session LI, 1989: Liquids, Freezing, and Glass Transition, Elsevier Science, 1991.

P.G. De Gennes, "Scaling Concepts in Polymer Physics", Cornell University Press, Ithaca, 1988.

M. Doi, "Introduction to Polymer Physics", Oxford University Press, New York, 2006.

C. N. Likos, "Effective Interactions in Soft Condensed Matter Physics", Physics Reports 348, 267 (2001).

J. N. Israelachvili, "Intermolecular and Surface Forces", Academic Press, London, 1998.

J.-L. Barrat and J.-P. Hansen, "Basic Concepts for Simple and Complex Liquids", Cambridge University Press, Cambridge, 2003.

J.-P. Hansen and I. R. McDonald, "Theory of simple liquids", Academic Press, London, 1986.

Xerox copies of the slides used in the course.
Assessment methods and Criteria
The examination consists in an oral test, during which students will be asked questions about topics dealt with in the course. Since the answers require describing mathematical equations or diagrams, students will need to use pen and paper. The questions may concern both the quantitative treatment of specific models by the methods which were applied to them in the course (e.g., obtain the equation for the probability density of the configurations of a polymer modelized as a random walk), and a more qualitative, phenomenological discussion on general aspects (e.g., describe the phase diagram of a simple fluid on the temperature-density plane). Both kind of abilities will be evaluated, and special attention will be made to make sure that students have grasped the logical connection between the points of an argument. The duration of the examination is typically between 1h and 1h 30'. This allows the students to answer questions fully, and the teacher to evaluate students' knowledge at depth instead of having to rely on how they answered just a single question.
FIS/03 - PHYSICS OF MATTER - University credits: 6
Lessons: 42 hours
Professor: Pini Davide Enrico