Molecular Dynamics simulation methods
1.4 Computing physical properties using Molecular Dynamics simulationsDynamics simulations
where Q is the parameter of the thermostat and τ the one of the barostat.
Hoover has shown [65] that these equations allow recovering the NPT ensemble.
1.4 Computing physical properties using Molecular Dynamics simulations
The method used to simulate an ensemble of interacting particles was exposed in the previous section. Those ensembles can be considered as those treated in statistical physics. This section exposes the definitions of some condensed matter properties used in this thesis and how they can be expressed in the frame of statistical physics, mostly using the linear response theory. The method using these expressions to compute the physical quantities from a Molecular Dynamics model is then exposed. As a recall, Molecular Dynamics being a classical method, quantum effects cannot be represented with these simulations.
As a result, only the classical version of statistical physics is useful and the equipartition of energy is valid as explained in AnnexeB.
1.4.1 Reaching equilibrium
The first step when conducting a Molecular Dynamics simulation is to equili-brate the system. The first structure imposed via atomic positions has to be as physical as possible but it remains difficult to a priori find the equilibrium positions, especially in fluids. During the first steps, the system evolves signifi-cantly and rapidly towards a final stage. This latter can be reached after a few nanoseconds but this duration depends on the system and is usually about a few picoseconds. A relevant technique to verify that the equilibrium is reached is to check the potential and kinetic energies but other parameters can be of interest, especially when the volume or the temperature is also relaxing.
Investigations on the physical properties of the system can start only after equilibrium is reached. This section details how to perform these calculations for several examples of physical properties.
1.4.2 The radial distribution function
The radial distribution function is a pair correlation of the positions of the par-ticles present in the system. It carries an information about the structure of an ensemble of particles and thus, it is widely used in soft matter to caracterise the phase of a fluid. We present here its definition from the microscopic description and the method to compute this function using Molecular Dynamics.
1.4.2.1 n-particle density
Let a group of N particles be in canonical ensemble NVT. The partition function of the microscopic configurations {r1, ..., rN} is given by:
Z =
Let’s define the probability associated to a partial configuration where only the first nparticles are taken into account:
P(n)(r1, ..., rn) = 1 As all particles are identical, all their permutations have the same probability.
There are Nn
ways of selectednparticles in the system andn!combinations of them to occupy the positionsr1, ..., rn. This allow us to compute the probability that positions r1, ..., rn are occupied by any particle:
ρ(n)(r1, ...rn) = N!
(N −n)!P(n)(r1, ..., rn). (1.80) This is called the n-particle density.
Using eq. 1.80the average 1-particle density V1 R
So the average 1-particle density is equal to the average density NV , that is the origin of the name of this function, but it can also be interpreted as the number of particle in the position r, which is why its integral is N. Actually R R ρ(2)(r1, r2)d3r1d3r2 =N(N−1), as ρ(2) is the pair density:
Noticing that in the case of an homogeneous fluid,ρ(1) does not depend on the position as it is the probability to find a particle at the positionr, thus:
1 V
Z
ρ(1)(r)d3r =ρ(1) =ρ. (1.84) 1.4.2.2 The correlation function g(n)
By focusing on the correlation between atoms to extractρ(n), one defines:
g(2)(r1,...,rN)= ρ(2)(r1, r2)
ρ(1)(r1)ρ(1)(r2). (1.85) g(2)(r1, r2)is called the pair correlation function or radial distribution function.
In an homogeneous and isotropic fluid,g(2) depends only on the distance |r2− r1|, which allows us to write: g(2)(r1, r2) = ρ(2)(|rρ22−r1|)=g(r).b
The probability to find a particle in the position r2 knowing there is an atom atr1 is expressed by the conditional probability:
ρ(2)(r1, r2)
ρ(1)(r1) =ρ(1)(r2)g(2)(r1, r2) =ρg(r). (1.86)
g can also be written the same way as ρ(n), keeping in mind that the particles The pair correlation function that is here defined from the microscopic states can be measured in diffraction experiments (see Annexe C).
1.4.2.3 Computing the radial distribution function from Molecular Dynamics
Molecular Dynamics simulations allow the computation of the radial distri-bution function, g, by temporal average, which is equivalent to the ensemble average in ergodic conditions. g is computed as:
g(r) = 1 This average formula lies in the computation of an histogram of the distances r = |ri(t)−rj(t)| using all atoms i, j from 1 to N and all time steps from 1 to T. This histogram is then normalized by the surface of the sphere at the distancer. gis then called the pair correlation function (and sometimes radial distribution function). This calculation can be done on any system and reflects its structure. There are two possibilities to take into account the boundaries of the system here :
• without periodic boundary conditions in the calculation of the histogram, there are size effects as all atoms miss neighbours, especially the atoms close to the border of the system. For this reason, a large system is re-quired so that the number of atoms missing neighbours is small compared to the core atoms.
• with periodic conditions, the histogram is computed by taking into ac-count the images of the system due to the periodic boundary conditions.
This method induces errors such as an atom correlated to itself.
An example of the pair correlation function of the oxygen atoms of water us-ing TIP3P is provided in Figure 1.3 and is in good agreement with previous calculations in the literature [41].