1. Ricardo Faúndez-Carrasco Población
e-mail: ricardo.faundez-ca@estudiante.uam.es
Tutor: Giorgio Cinacchi. Departamento de Física Teórica de la Materia
Condensada, Universidad Autónoma de Madrid, Madrid, Spain.

2. 1. Definition of Liquid Crystals (LC’s)
2. Lebwohl – Lasher Model for LC’s
3. Characterization of the phase transition
4. Computational strategy
5. Results
6. Outlook
7. Bibliography
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3. • LC’s are materials that cannot be classified entirely as solid crystals
nor conventional liquids.
– May flow like a liquid
– Molecules with orientational and/or (partial) positional order
• Thermotropic mesophases: depend on the temperature.
• Aim: Find T at which the transition from an ordered nematic phase
to a disordered, isotropic liquid-like phase takes place.
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Nematics Cholesterics Smectics Columnar Phase

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• LC’s are materials that cannot be classified entirely as solid crystals
nor conventional liquids.
– May flow like a liquid
– Molecules with orientational and/or (partial) positional order
• Thermotropic mesophases: depend on the temperature.
• Aim: Find T at which the transition from an ordered nematic phase
to a disordered, isotropic liquid-like phase takes place.
Nematics Cholesterics Smectics Columnar Phase

5. • Maier-Saupe Theory: Interactions considered through a
mean-field model of these interactions, entirely due to van der Waals
forces. Proportional to second Legendre polinomial.
• Predictions:
– Threshold temperature Tc :
• Nematic phase if T < Tc
• The higher the temperature, the more disordered the system
– Universal value for the order parameter at the phase transition
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6. • Lattice model for the Maier- Saupe theory.
– Molecules arranged on the sites of a SC lattice
– Every molecule is described by an unit orientational vector.
– Pair potential for the interactions restricted to nearest
neighbours
– Normalised units (T*, U*, kB):
– Periodical Boundary Conditions
• Avoid superficial effects
• Create building blocks
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7. • Equilibration simulation: One has to reach equilibrium and only at that
point averages of quantities can be accumulated.
• Energy vs. Temperature: Determines the range within which the phase
transition takes place.
• Heat Capacity: Interpolation – Differentation. Gives precise T* of
transition.
• Order Parameter: Calculate order tensor – Diagonalisation – Biggest
eigenvalue
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8. • Due to Stanislaw Ulam & John von Neumann & Enrico Fermi.
• In a MC simulation the aim is to generate N-particle configurations
appearing with the Boltzmann frequency: Metropolis algorithm
1. Choose randomly one molecule and change its orientational vector.
2. Recalculate the change in energy of the system after that modification.
3. Generate a random number z between [0,1].
4. If z ≤ P, change the orientational vector – New configuration accepted
5. If the move cannot be accepted, the system goes back to the previous state but
nonetheless it is counted as a new configuration for statistical purposes.
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• One has to reach equilibrium and only at
that point averages of quantities can be
accumulated.
• Given a temperature T*, the system tend
to reach always the same equilibrium, no
matter the initial configuration.
• Larger times for disordered systems.
• Same effect in case orientational jumps
are smaller.
T* = 1.0
T* = 1.0

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• Phase transition at T* = 1.129
• At low T, ordered system
(preferentially aligned along z-axis)
– Nematic phase
• At high T, becomes increasingly
disordered – Conventional isotropic
phase
• Difference with literature due to
the size of the lattice.

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• Order Parameter
•Transition from ordered to disordered
•Agreement with theoretical predicitions
•Difference with literature due to the lattice size.
•Non-zero value due to the finite size of the system.
•Heat capacity
• Sharp peak at T = 1.129
•Reference T = 1.1232
•Discontinuities due to the
numerical differentiation

12. • First and fundamental step towards the generalisation for
helix-like molecules:
– Shape of the molecules: possible segmentation that forms an helix.
– Pair potential changed to take into account the segments (limited to nearest
neighbours distance).
– Degrees of freedom: rotation in a plane and around its main axis.
– New Boundary Conditions – Spiraling Boundary Conditions
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14. • Main text of reference in LC’s:
– de Gennes, P.G. & Prost, J. (1993). The Physics of Liquid Crystals, Second Edition, Oxford University Press, Oxford, Great Britain.
• Original paper on the Maier-Saupe theory:
– Maier W. and Saupe A. (1959). Z. Naturforsch. A 14, 882.
• Original paper on the Lebwohl-Lasher model:
– Lebwohl P. A. and Lasher G. (1972). Phys. Rev. A 6, 426.
• Metropolis et al algorithm:
– Metropolis,N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A., Teller, E. (1953). J. Chem.Phys. 21, 1087.
• More on the power’s method, how to obtain the biggest eigenvalue of a matrix:
– Classnotes of the subject "Computación Avanzada", by Prof. Alejandro Gutiérrez during year 2014-2015 of the B.Sc. in Physics,
Facultad de Ciencias, Universidad Autonónoma de Madrid, Madrid, Spain.
• Information on helicoidal molecules and the mesophases related to them, altogether with possible
modelization and properties:
– H.B. Kolli, E. Frezza, G. Cinacchi, A. Ferrarini, G. Giacometti, T.S. Hudson. (2014). J.Chem. Phys. Communications 140, 081101;
Hima Bindu Kolli, Elisa Frezza, Giorgio Cinacchi, Alberta Ferrarini, Achille Giacometti, Toby Hudson, Cristiano De Michele,
Francesco, Sciortino. (2014). Soft Matter 16, 8171.
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• All the figures were obtained using free, open-available software (xmgrace, gnuplot, QMGA)

15. • It is also possible to calculate the heat
capacity by another method:
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Based on statistical physics