COMPUTATIONAL MODELLING OF HARD AND SOFT MATERIALS

In modern materials science research, numerical simulations have become essential for complementing experiments and theory, as they reveal how atomic and electronic structures influence the properties and macroscopic behaviour of materials.
This course aims to introduce the key theoretical and computational methods used to study materials across different length scales. Topics will range from a quantum mechanical descriptions of electronic structure based on density functional theory to molecular dynamics and coarse-grained approaches for modelling large systems. The course will cover both the theoretical foundations of these methods and their practical applications, featuring hands-on tutorials with open-source software.
This course is suitable for both computational and experimental scientists whose work may require insight into materials properties through numerical calculations. By the end of the course, students will be able to independently conduct simulations, analyse the results, and critically assess findings from the scientific literature.

1. Knowledge and understanding
• Basic knowledge of the computational methods commonly used in research in condensed and soft matter physics.
• Familiarity with the open-source software implementing those methods.
2. Ability to apply knowledge and understanding
• Independently set-up simulations for a broad range of problems and systems in material science.
• Select appropriate numerical techniques, libraries, and data visualization tools for each specific problem.
• Critically analyze the results obtained from numerical simulations.
3. Autonomy of judgment
• Identify and correct errors through a critical analysis of the methods applied and the results obtained.
4. Communication skills
• Clearly and precisely communicate the acquired knowledge using appropriate terminology, both in written and oral forms.
• Present simulation results using graphs and figures that comply with the standards of scientific publications.
5. Learning skills
• Take notes by selecting and organizing information based on its relevance and priority.
• Achieve sufficient autonomy in identifying and using computational tools and open-source software to solve problems in materials science.

To enroll in this course, students are required to have a solid knowledge of Classical Physics, Statistical Mechanics and Quantum Mechanics at the level covered in a M.Sc. scientific degree program, such as Physics or Engineering Physics. A prior knowledge of condensed matter physics at the level of a M.Sc. course (see for example https://www.unive.it/data/course/441360 ) and computer programming may be beneficial.

Part 1: (Electronic Structure Theory)
1.1 Review of solid-state physics
Schrödinger equation; Adiabatic approximation; Bloch Theorem; Electron’s band structure; Phonon’s band structure
1.2 Introduction to Density Functional Theory, fundamentals
Foundational theorems of Density Functional Theory; Kohn-Sham equations; Exchange-correlation functionals
1.3 Introduction to DFT, implementation
Localized basis functions and plane waves; Pseudopotentials; Atomic structures (visualization), calculation of total energy, cohesive energy, and numerical convergence tests
1.4 DFT calculations in practice (hands-on tutorial)
Ionization potential and electron affinity of molecules; Electronic bands of solids; Ionic force ad structural optimization

Part 2: Soft Matter
2.1 Statistical Thermodynamics
Review of Thermodynamic potentials and Legendre transformation; Gibbs ensembles (NVE, NVT, NPT, μVT); Universality and Scaling.
2.2 Liquid Theory
Virial expansion; Perturbation theory; Mean Field theory; Phase transitions; Maxwell construction and van der Waals gas; Integral equation theory; Electrostatic theory, Debye Huckel theory; Polar and non-polar solvents.
2.3 Colloidal systems
Energy and length scales; Packing problems; Entropically driven transitions, Glasses and gels; Depletion interactions; Patchy particles; Janus fluids.
2.4 Liquid Crystals
Historical perspectives; Liquid crystals phases; Technological applications; Theoretical approaches (Onsager theory, Density Functional Theory).
2.5 Polymers: Equilibrium properties
Linear polymers; Connection with Diffusion Equation; Phase diagram; Flory Theory; Solvent effects; Polymer solutions; Flory-Huggins for solutions; Experimental probes.

Part 3: Practical Cases Studies in Soft Matter
3.1 Theoretica bases
Theoretical basis for Numerical Simulations; Monte Carlo Methods and Metropolis algorithm; Molecular Dynamics; GROMACS and LAMMPS software packages;
3.2 Numerical simulations:
Generic polymer chains, lipid membranes, DNA and proteins.

General part: Fluids and Colloidal Systems
• 530.13 KARDM M. Kardar, Statistical Physics of Particles (Cambridge Univ. Press 2007)
• 547.7 HAMLI Hampley, Introduction of Soft Matter (Wiley 2002)
• 541.3 FUNIC1 Lyklema, Fundamental of interfaces and colloidal science Vol 1-5 (Academic 1991)
• 530.413 FOFTM Gompper, Schick, Soft Matter Vol 1-3 (Wiley 2006)
• 530.4 HANSJP Hansen, McDonald, Theory of Simple Liquids (Academic 2006)
• 530.42 BARRJ Barrat, Hansen, Basic Concepts for Simple and Complex Liquids (Cambridge 2003)

Electronic Structure
• Robert G. Parr and Weitao Yang, Density-Functional Theory of Atoms and Molecules (Oxford University Press)
• Richard M. Martin, Electronic structure: Basic theory and practical methods (Cambridge University Press)

Numerical and Computational Techniques
• 532.01 ALLEN MP Allen, Tildesley, Computer Simulations of Liquids (Clarendon 1987)
• 539.6 FREND Frenkel, Smit, Molecular Simulations (Academic 2002)

Polymers
• LT547.7 DOIM Doi, Edwards, Theory of Polymer Dynamics (Oxford, 1986)
• 530.41 RUBIC Rubinstein, Colby, Polymer Physics (Oxford 2003)

The exam consists of an individual project and an oral exam.

Project: Some possibilities include
a. Calculation of electronic and vibrational properties of 2d materials.
b Magnetism of a selected material.
c. Properties of defect and impurities in a selected semiconductor.
d. Equation of state ad a function of temperature of a simple liquid
e. gas-liquid phase coexistence of a simple liquid

Oral Exam (30-40 minutes): During the oral exam, the student will present the project results and answer in-depth questions on the fundamental concepts of the course. This session will also serve to verify the student's understanding of the models and techniques applied in the project.

oral

• Excellent (27-30/30): The exam will be considered fully successful if the project contains mostly correct results, and the student can clearly explain their conclusions, respond effectively to most of the teacher’s questions, and demonstrate a comprehensive understanding of the subject.

• Good (22-26/30): An average evaluation will result from a project with at least half of the results correct. During the oral exam, with the assistance of the teacher's questions, the student will demonstrate a good understanding of the key concepts and how to address errors in the project.

• Sufficient (18-21/30): The exam will be considered sufficient if the project contains errors in approximately two-thirds of the results, but the student demonstrates, during the oral exam, an adequate knowledge of the most important concepts covered in the course.

The problems and fundamental equations relevant to each part of the course will be introduced using a traditional approach, with all calculations done on a blackboard or whiteboard. Examples of applications of the methods for electronic structure calculations will be carried out using the open-source Quantum Espresso package.