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A heterogeneous chemical reaction involves substances in different states, such as gaseous and solid in this case. The reaction shown here can be summarized by the equation “A B → C“, meaning that there are three types of molecules: type A, shown here in red, type B, shown in blue, and type C, shown in green. Whenever a molecule of type A encounters a molecule of type B, they have a certain probability of forming a molecule of type C. Unlike in most previous simulations of chemical reactions on this channel, the A molecules are kept initially in a gaseous phase, and the B molecules in a solid phase. In this simulation, there are initially 3035 molecules of type A, 1666 molecules of type B, and no molecules of type C. The molecules interact via a Lennard-Jones potential. Reactions between A and B occur with a given probability per time step when they are closer than a fixed distance. B molecules are 50% heavier than A molecules, and C molecules weigh as much as an A molecule and a B molecule together. The total momentum is conserved during a reaction. The separation between solid and gaseous phase is achieved by coupling all molecules above a certain height to a thermostat, while the molecules below are subject to a large viscous friction, which acts like a thermostat at zero temperature. For that reason, the A molecules approaching the lower region tend to slow down and pile up, a little bit like a gas condensing on a cold surface. This also slows down the chemical reaction, as the C molecules tend to act like an insulating layer, similar to corrosion on a metal. Reactions are highlighted by a small disc that quickly fades. To guide the eye, whenever three molecules of the same type are close enough, the triangular region between them is filled with the same color. In addition, a line is drawn between molecules when they are closer than a given distance (the reaction distance is smaller, however). The graph at the top right shows the number of molecules of each type as a function of time. This simulation has two parts, showing the same evolution at two different speeds: Time lapse: 0:00 Slow motion: 0:34 In the first part, time has been accelerated by a factor 3. 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. 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 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 Render time: 42 minutes 15 seconds Color scheme: Turbo, by Anton Mikhailov Music: “Close My Mouth“ by Silent Partner Reference: Leimkuhler, B., Noorizadeh, E. & Theil, F. A Gentle Stochastic Thermostat for Molecular Dynamics. J Stat Phys 135, 261–277 (2009). ~theil/ Current version of the C code used to make these animations: Some outreach articles on mathematics: (in French, some with a Spanish translation)
