Aggregation of "water" molecules
Channel: Nils Berglund
Date: 22/03/2024

This new aggregation video features a simple model for water molecules. I don't claim this is physically meaningful, but I wanted to find out what kind of cluster shapes are created in this situation. 
The water molecules consist of one oxygen and two hydrogen atoms, with a mass ratio of 16:1 and a charge ratio of -2:1. They are modeled in such a way that the H-O-H angle at equilibrium is 104.45 degrees. Atoms that belong to different "molecules" interact via a Coulomb potential, complemented by a Lennard-Jones between particles of opposite charge, to avoid their collapse on a single point. In addition, atoms that come within a certain distance can "react", thereby becoming members of a same "molecule". These reactions are highlighted by a quickly fading white disc. Particles within a molecule no longer interact via a Coulomb potential, but instead via a strong harmonic force, that tends to maintain their distance constant. Furthermore, particles that have more than two "partners" are decoupled from the thermostat controlling the system, to avoid its slowing down too much, due to all the energy being soaked up by small vibrations inside clusters. 
The video has two parts, showing the same simulation with two different color schemes:
Clusters: 0:00
Kinetic energy: 1:33
In the first part, particles in the same cluster are given the same color, which is updated when several clusters merge. In the second part, the particles' color hue depends on their kinetic energy.
To save on computation time, particles are placed into a "hash grid", each cell of which contains between 3 and 10 particles. Then only the influence of other particles in the same or neighboring cells is taken into account for each particle. 
The temperature is controlled by a thermostat, implemented here with the "Nosé-Hoover-Langevin" algorithm introduced by Ben Leimkuhler,  Emad Noorizadeh and Florian Theil, see reference below. The idea of the algorithm is to couple the momenta of the system to a single random process, which fluctuates around a temperature-dependent mean value. Lower temperatures lead to lower mean values. 
The Lennard-Jones potential is strongly repulsive at short distance, and mildly attracting at long distance. It is widely used as a simple yet realistic model for the motion of electrically neutral molecules. The force results from the repulsion between electrons due to Pauli's exclusion principle, while the attractive part is a more subtle effect appearing in a multipole expansion. For more details, see https://en.wikipedia.org/wiki/Lennard-Jones_potential

Render time: 19 minutes 22 seconds
Compression: crf 23
Color scheme: Turbo, by Anton Mikhailov
https://gist.github.com/mikhailov-work/6a308c20e494d9e0ccc29036b28faa7a

Music: "The Making of Thumbs Up" by Joe Bagale

Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). https://doi.org/10.1007/s10955-009-9734-0
http://www.maths.warwick.ac.uk/~theil/HL12-3-2009.pdf

Current version of the C code used to make these animations: 
https://github.com/nilsberglund-orleans/YouTube-simulations
https://www.idpoisson.fr/berglund/software.html
Some outreach articles on mathematics:
https://images.math.cnrs.fr/_Berglund-Nils-1343_.html
(in French, some with a Spanish translation)

#molecular_dynamics #coagulation