56 5.8 Liquid-liquid coexistence in the diagram of the packing fractions of the two species for. 57 5.9 Liquid-liquid coexistence in the diagram of the packing fractions of the two species for.
Phase transitions and metastable thermodynamic states
- First Order Phase Transition
- First Order Verses Second Order Phase Transitions
- Importance Sampling and the Metropolis Method
- Monte Carlo Algorithm
It is important to consider how the Gibbs potential changes when the temperature of the system changes. Then the discontinuities in the slope of the Gibbs free energy corresponding to the thermodynamically stable phase are only discontinuities in the entropy of the system (entropy is the partial derivative of the Gibbs potential with respect to temperature).
Gibbs Ensemble Monte Carlo Method
- Acceptance Rules
- Extension of the Acceptance Rules to a Binary Mixture
- Case Study: GEMC applied to a Particles in a Square Well Potential
- Symmetric mixture with non-additive parameter ∆ = 0.2
- Asymmetric mixture with non-additive parameter ∆ = 0.0
Here we find that the ratio between probability densities for the new and old states is given by 3.20). Applying the same method to calculate the ratio of the probability densities for the case where a particle of speciesBis displaced within box 1 gives the acceptance rule:. By imposing the condition of detailed balance, we determine that the ratio of the probability densities of the two states n and o is given by: .
At the beginning of the simulation, we randomly place the same number of particles in field 1 and field 2. This can be seen both from the compressibility factor (βP/ρ) and from the contact values of the radial distribution functions. In fact, in most cases the statistical errors of our data overlapped with those reported in the literature.
However, it should be noted that larger deviations were observed in the contact values of the radial distribution function of the larger sphere (BB) interactions.
Particle Insertion: The Constant-NPT Ensemble Test
The Chemical Potential µ and the Widom method
When sampling the phase space of a system, the free energy cannot be written as a simple average of microscopic functions of the coordinates and momenta of the system's particles. The chemical potential of a species in a mixture is defined as the change in free energy of a system when a particle of that species is added to or removed from the mixture while the number of other species remains constant. U =U(sN+1)−U(sN) and the excess part of the chemical potential µex can be written as
The above method is known as the Widom insertion method and as explained above it consists of adding a "ghost" particle with a randomly generated position inside the simulation box, and calculating the corresponding change in the energy of the system in order to calculate exp(−β∆U). The Widom insertion method described so far has been considered in the canonical ensemble (NVT), but it can be easily adapted to other ensembles [103]. Since we compare our estimate of the chemical potential with several others reported in the NPT ensemble, it is worth noting the differences with the formula reported in Eq.
Using the procedure previously applied in the NPT ensemble, it is in fact possible to show that the same separation between an ideal and an excess contribution to the chemical potential holds (even though the contribution of the ideal will part are different [103]), and that the fluctuating quantity to be averaged out is no longer exp(−β∆U), but rather V exp(−β∆U).
Relating the Constant-NPT Ensemble Monte Carlo Results to a GEMC with-
As reported in Table (4.4), our results are comparable to those reported in the literature and provided us with further evidence for the reliability of the implemented GEMC code for NAHSMs.
Symmetric mixture with non-additive parameter ∆ = 0.2
In the rest of the chapter, we will use reduced units of pressureP σ3/ and chemical potential μi/, unless otherwise specified, where =kBT = 1. In the rest of the thesis, when writing chemical potentials we will always neglect the termlnΛ3i , which is identical in both coexisting phases. Thermodynamic properties are invariant with respect to the change of compositions xi of two types for symmetric mixtures (σA = σB), i.e.
Thus the following conditions apply (for symmetrical mixtures):. and the coexisting compositions xiα of the α species in phase are symmetric with respect to the equimolar composition xA=xB = 0.5:. In the MIX1 [44] approximation, the excess Helmholtz free energy Aex of the mixture is written as:. Also in this case, the knowledge of EOS (Eq. 5.8)) is sufficient to calculate the Helmholtz free energy of the system by integration [6].
Within DFT calculations, the bulk potential Ω[ρA, ρB] is written as a function of the one-body density profilesρi as.
Results and Discussion
It is also worth noting that the smaller error we estimated with our simulations in the unmixing region allows a more accurate understanding of the theoretical predictions than has been achieved in the past. S provides a quantitative measure of the weight of spatial correlations involving more than two particles in the configurational entropy of the system. Even in this case, there is no estimate of the critical point since the construction of the spinodal curve is not based on a calculation of the free energies of the two coexisting phases.
We see that the two IETs provide a fair description of the demixing region, even if they are not quantitative, especially in the region with higher packing fractions of the smaller species. However, the critical point prediction of the two theories (located in the empty region of the two coexisting branches) seems reasonable compared to the GEMC estimate (diamond symbol in figure). The reason for this is that theoretical calculations tend to overestimate the magnitude of the prediction. the two-phase region, and we had to select a state point that was still in the stable region of the thermodynamic plane, such as.
DFT tends to overestimate the extent of the two-phase region compared to GEMC data, with the prediction of the critical point becoming less accurate as the non-additivity parameter is increased from ∆ = 0.2 (Fig.
Conclusion
In this section we explicitly consider the equation of state (EOS) of a binary hard disk mixture. The particles interact via hard disk pair potentials as already reported for hard spheres in Eq. 4.1) with the difference that we only consider interactions in two dimensions (xandy) since we are dealing with mixtures of hard disks. In the remaining part of the thesis we will refer to the virial expansion arrested after the third (D=E= 0 in Eq.
By knowing the virial coefficients, some improved equations of state can be constructed, which should in principle improve the accuracy of the virial expansion. Therefore, we will consider the modified virial expansion (RVE) proposed by Baus and Colot [93] to obtain an (approximate) equation of state for a symmetric NAHD mixture. Similarly to the virial expansions associated with Eq. 6.1), in the rest of the paper we will refer to the modified viral expansions, held to a third (c3=c4= 0 in Eq.
For a symmetric mixture, the condition required for the stability of a material (at equilibrium, the chemical potentials of the two distinct phases are equal) provides us with the identity.
Monte Carlo simulation
The coefficients c0, c1, c2, c3 and c4 depend only on the mole fraction and accurately reproduce the first virial coefficients. The critical point falls atxc1=xc2= 1/2 and after determining the value of ∆, one can find a solutionxi(with i=1,2) of Eq. We can then trace the coexistence lines using a method such as the double tangent method (see Ref for details).
The acceptance rate for particle displacements was set at 40% and cubic periodic boundary conditions were applied to the simulation box [103]. The total number of discs in the two boxes was fixed at 2000, and the achievement of conditions for phase coexistence was checked by verifying the equality of the pressures in the two boxes according to Eq. 5.2), and the similarity of the chemical potentials of the two species, as calculated according to the formula Eq. Each GEMC cycle consists of a number of attempts to displace particles in the two boxes equal to the total number of particles, an attempt to change the total area of the two boxes, and a number of particle exchanges between the two boxes between 1% and 5 % of the total number of particles.
The critical densityρcr of the systems considered was estimated by the critical power law Eq. 5.6) with reference to βbeta which is the critical exponent related to the order parameter, whose exact value for the two-dimensional case is β enxcr1 = 0.5 for symmetric mixtures.
Results and Discussion
6.2-6.6, the compressibility factor was plotted as a function of the packing fraction according to different values of the mole fraction and. In fact, as a general rule we note that the critical packing fraction of the binary mixture tends to become smaller as the non-additivity parameter increases. Now it is interesting to assess the performance of the two schemes compared to NVT MC computer simulation.
The observed difficulty in finding a general trend arises because the convergence of the virial expansion is not well understood for binary mixtures. The solid red line and dashed blue line are the fifth-order virial expansion and rescaled truncated virial expansion predictions [93], respectively;. 0.2 (see second panel from top), the theories generally tend to follow the GEMC data at mole fractions < 0.05, except for the virial binodal curve.
Thus, we clearly see that all theories tend to underestimate the GEMC critical density for ∆ ≥ 0.2.
Conclusion
Where (Ns) corresponds to the number of successful configurations and (Nt) is the total number of trial configurations. The accuracy of the MC depends on the number of route configurations and for the group integralJ we have that. The number of moves in our MC runs is comparable to the period of the standard Fortran pseudorandom number generator (PRNG), namely 232.
Some of the most interesting properties of MT19937 are: i) a very long period ii) a very high even distribution of points in space up to 623 dimensions and iii) successful results in several tests for statistical randomness. If we write the Boltzmann factor in terms of Mayer functionsf(i, j), where. 7.2) This is the kind of integral that appears in the definition of the particle density distributions. The value of a labeled diagram is the value of the integral that the diagram represents.
Thus, an unlabeled diagram makes use of a combinatorial factor related to the topological structure of the diagram.
Fluid-fluid coexistence in the diagram of the packing fractions of the two species for
The fifth virial coefficient plotted as a function of the mole fraction for positive
Compressibility factors as a function of the total packing fraction at different values
Phase coexistence boundary of a symmetric NAHD mixture for different values of ∆
