A Tracker is an object that can be attached to a Molecule when it first enters the Reactor as part of the 'food supply'. When the Molecule takes place in an Interaction, the Tracker is passed on to one of the Interaction's Product Molecules chosen according to an arbitrary rule (if the Interaction is a Fission), or to the single Product Molecule (if the Interaction is a Transformation or Construction).
This process goes on so that as Interactions take place the Tracker is continually passed from Reactant to Product and traces a path from Molecule Type to Molecule Type. Eventually the Tracker will either stop at a particular Molecule Type that takes place in no further Interactions, or will return to a Molecule Type it has previously visited. When this happens, a Cycle has occurred. SimSoup detects Cycle Occurrences and identifies each as an instance of a particular Cycle Type.
The plot below shows initial output from SimSoup using Trackers. The scenario setup was as follows:-
The plot shows the following:-
I have been thinking for some while about how to represent these factors in SimSoup, and this article outlines some initial thoughts on this topic.
Potential energy is energy associated with a system's configuration, and is measured by reference to some alternative configuration of the system. For example, if a spring starts from its unstretched state and is then stretched, the work done in stretching it is converted to potential energy. This potential energy can be released if the spring is allowed to return to its unstretched state. Similarly, if two oppositely charged particles are moved apart from an initial position, the system acquires potential energy that can be released if the particles move back to their initial configuration.
A molecule or ion of a particular species has a defined potential energy. When a reaction event takes place, the energy released or absorbed is the difference between the potential energy of the reactants of the products.
The figure above shows an example reaction profile. The reaction coordinate represents the configuration and motion of the participants in the reaction. On the left hand side of the figure, the reaction coordinate represents a situation in which Reactants are about to interact. On the right hand side of the diagram, the reaction coordinate represents a situation in which Products have just been formed by a reaction event.
The figure shows that the Reactants initially have a potential energy Vr, and that as the reaction progresses the potential energy increases until it reaches a maximum value, after which the potential energy decreases until it reaches a value Vp associated with the products.
The configuration that corresponds to the region close to the maximum is called the activated complex. The difference between the initial potential energy of the reactants (Vr), and the maximum in potential energy associated with the activated complex is known as the activation energy, Ea. The activation energy is the minimum kinetic energy that the reactants must have in order to form products.
In a bimolecular reaction in which there are two reactant species, B and C, the total rate of the reaction is:-
Entropy is not disorder. Entropy is not a measure of disorder or chaos. Entropy is not a driving force. Energy's diffusion, dissipation, or dispersion in a final state compared to an initial state is the driving force in chemistry. Entropy is the index of that dispersal within a system and between the system and its surroundings
where T is temperature, and R is the gas constant.
The relationship holds for both unimolecular reactions and bimolecular reactions, and for reactions taking place both in gases and in solutions.
We can interpret the exponential factor e-Ea/RT as the fraction of potential reaction events in which sufficient kinetic energy is available to overcome the activation energy, Ea.
In the case of a reaction taking place between two molecules in the gas phase, e-Ea/RT is given by the Boltzmann distribution as the fraction of collisions with a kinetic energy in excess of Ea. This kinetic energy would be derived from the thermal motion of the two reactant molecules. In the case of a liquid phase reaction taking place in a solution, some or all of the kinetic energy would be derived from kinetic energy of the solvent molecules.
The 'pre-exponential' factor, A, can be interpreted as the frequency of potential reaction events. In the case of a gas phase reaction between two colliding molecules, two factors affect this frequency:-
Suppose that the activation energy for the reverse reaction is greater than for the forward reaction (as is the case in the reaction profile diagram above, in which the potential energy of the products is less than that of the reactants).
In this situation, the exponential factor e-Ea/RT will be larger for the forward reaction than for the reverse reaction. If the conditions determining the pre-exponential factor, A, are such that A has the same value for both the forward and reverse reactions, then the rate constant for the forward reaction will be greater than that for the reverse reaction.
This reflects the fact that it is easier for a molecule of X to acquire the activation energy for the forward reaction than for a molecule of Y to acquire the higher activation energy required for the reverse reaction.
It is not necessarily the case that a reversible reaction always prefers the direction that results in lower potential energy. For example, consider the following bimolecular reaction:-
Suppose that there is a very stringent steric requirement for the forward reaction such that the C and D molecules must approach each other with a very specific orientation, but that there is no similar requirement for the reverse reaction. This would lead to a situation in which the pre-exponential factor for the forward reaction is very low. As a result of this, the rate constant for the reverse reaction can be higher than that for the forward reaction, even if the activation energy for the reverse reaction is higher.
This can be understood in terms of an activation entropy component in the pre-exponential part of the Arrenhuis equation. In the bimolecular example above, the requirement for two previously separated molecules to combine (with a stringent steric requirement) into a single molecule means that the single E molecule has a lower entropy than the two separated C and D molecules. This can outweigh the exponential factor e-Ea/RT that would favour the forward reaction.
We can conclude therefore that the rate constants of reactions between molecular species in chemical networks are determined by thermodynamic factors including the activation energy and activation entropy, and that the Arrenhius equation is a reflection of the operation of these factors.
To incorporate thermodynamic principles into SimSoup, the following enhancements are planned:-
The Manhattan Plot below shows the variability in the Reactor Composition at different times. The Reactor Composition is defined by the number of Molecules of each type present in the Reactor. In other words, the Reactor Composition represents the 'mix' of Molecule Types present.
The most straightforward way to read the Manhattan Plot is horizontally from left to right. The point at the far right of each 'row' corresponds to the Reactor Composition at a particular time (the 'Base Time'). Points to the left correspond to the Reactor Composition at earlier times, with the darkness of each point indicating the degree of difference from the Reactor Composition at the Base Time. A light coloured point therefore indicates a time at which the Reactor Composition was close to that at the (later) Base Time.
The striking diagonal pattern indicates that the Reactor Composition is behaving cyclically. Following any horizontal line across the plot, we see that most of the path is black, but that it turns white for a short period at intervals of approximately 200,000 timesteps. The white sections along the line indicate periods during which the system had (almost) the same Reactor Composition.
In short, the Reactor Composition is constantly changing, but the system returns periodically to the same Reactor Composition. In other words, the system has an oscillating behaviour.
The two plots below show details of the behaviour in the Reactor. The first of the two shows rates of specific Interaction Types, and the second shows the numbers of specific Molecule Types in the Reactor as the simulation proceeds.
It can be seen that there are large variations, with 'spikes' occurring in the Interation Rates.