Two-phase thermodynamic (2PT) method and the DoSPT code

From the standpoint of MD simulation methods, the determination of ∆𝐺_𝑚𝑖𝑥 is complicated by the fact that the determination of the entropy of mixing (∆𝑆_𝑚𝑖𝑥) is not straightforward.

Methods such as umbrella sampling,³⁴ Widom particle insertion,³⁵ and thermodynamic integration,³⁶ can effectively probe the entropy of an ensemble, though each has potential complications depending on the system under investigation. Another means to determine entropy is to evaluate the vibrations of the system. Here, the system thermodynamic characteristics can be evaluated within the context that the ensemble density of states (DoS) is comprised of harmonic oscillators. While such an assumption is generally valid for solids, the anharmonic nature of low-frequency and diffuse vibrational modes in liquids and gases renders severely limits the assumption.³⁷ To overcome this constraint for liquids, Lin,

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Blanco, and Goddard proposed the two-phase thermodynamic (2PT) model.³⁸ Here, the thermodynamics characteristics of a liquid are determined from the DoS obtained from the Fourier transform of the velocity autocorrelation function,³⁹ with the total DoS (𝐷𝑜𝑆(𝑣)) being the sum of two components – a gas component that contains the anharmonic diffusive vibrations (𝐷𝑜𝑆_{𝑑𝑖𝑓𝑓𝑢𝑠𝑒}(𝑣)) and a solid component that contains the harmonic vibrations (𝐷𝑜𝑆_{𝑠𝑜𝑙𝑖𝑑}(𝑣)) – as shown below:

𝐷𝑜𝑆(𝑣) = 𝑓 × 𝐷𝑜𝑆_{𝑑𝑖𝑓𝑓𝑢𝑠𝑒}(𝑣) + (1 − 𝑓) × 𝐷𝑜𝑆_{𝑠𝑜𝑙𝑖𝑑}(𝑣) Equation 2.25

A key feature of 2PT theory is the fluidicity parameter (𝑓), which is a function of the system properties (e.g. self-diffusion, density, and temperature) that is solved self-consistently from the MD simulation.⁴⁰ The system thermodynamics are recovered by applying statistical weighting functions to each respective component. Since the inception of the 2PT method, it has been shown to provide accurate determinations of the absolute entropies and free energies for a variety of liquid systems, including those with considerable chemical and physical complexity.37, 38, 41-56

In the work presented in this thesis, the free energy of different mixtures was calculated through the 2PT method. Each system was equilibrated using the general simulation protocol introduced in Section 2.5, followed by a 20 ps MD simulation with the same NVT ensembles using the previous equilibrium system. The DoSPT code developed by Caro and co-workers^{57, 58} was used for the 2PT calculations; DoSPT is Fortran implementation that

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allows the computation of entropies from MD simulation within the framework of the 2PT method. More detailed explanation of DoSPT can be found in http://dospt.org/.

Free energy of solvation

Solvation describes the interactions among the solvent and the dissolved solute molecules, and provides insight into how solvent and solute behave in different environments. The solvation of a component is only favorable when the overall Gibbs energy of the solution is decreased in comparison to the separated two components, (∆𝐺_{𝑠𝑜𝑙𝑣} is negative). Here, we used a free energy perturbation method to determine the free energy of solvation of several solvents.

If we consider a process of solute molecule A solvated into a solvent environment S, the imaginary start and end states can be expressed as the Figure 2.1:

Figure 2.1. Pictorial representation of the starting (State A) and ending (State B) of solvation.

Here, state A can be described as the molecule A is isolated from solvent S and state B is the when molecule A is dissolved in solvent S. The free energy difference between these two states determines the relative probability 𝑝_𝐴 and 𝑝_𝐵,

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𝑝_𝐴

𝑝_𝐵 = 𝑒𝑥𝑝^{𝐹𝐵−𝐹𝐴}

𝑘_𝐵𝑇 Equation 2.26

where 𝑘_𝐵 is Boltzmann’s constant and 𝑇 is the temperature. Principally, the free energy can be calculated by forcing the system into a situation that the system does not want to be, and then calculate how much it does not want to be there. In the free energy perturbation method, this is expressed by coupling the two components in the system with an interaction variable 𝜆, such that:

𝐸_{𝑡𝑜𝑡𝑎𝑙} = 𝐸_𝐴−𝐴 + 𝐸_𝐵−𝐵+ 𝜆 × 𝐸_𝐴−𝐵 Equation 2.27

The variable 𝜆 is varied from 0 to 1; as a point of reference, the molecule A is effectively turned off from the system (when 𝜆 = 0).

In this study, the solvation energy were calculated using the Bennett acceptance ratio (BAR)⁵⁹ perturbation method as implemented in the GROMACS software suite. According to BAR, the free energy difference can be calculated directly if the two states A and B are close enough. In this case, the switching between two states becomes possible in both directions, as demonstrated in Figure 2.2 by the double arrows.

Figure 2.2. Pictorial representation of solving the particle into the solvent by varying interaction variable λ.

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By creating many 𝜆 values between the initial and end states, we can effectively turn on and off the interaction between the solute and solvent molecules. For each 𝜆 value, a series of MD simulations is conducted, and we use the same workflow as described above. The simulation will calculate the energy difference between two neighboring λ values, while all the energy difference combined will provide the total energy of solvate the molecule A into the solvent environment (∆𝐺_{𝑠𝑜𝑙𝑣}).

Natural order parameter

Order parameters were determined using the method described by Dewar and Camp⁶⁰. In this method, the global ordering of the model molecule is assumed to involve preferential ordering of a set of molecular axes. The unit vectors and ê₂ describe the orientations of the two bending “arms” of the molecule, with ê₁× ê₂ = −𝑐𝑜𝑠𝛾. The molecular frame is defined by three orthonormal vectors given by:

𝑎̂ = _|𝑒̂^{𝑒̂1−𝑒̂2}

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where (𝜃, 𝜙, 𝜓) are the Euler angles in the rotation matrix mapping the molecular frame defined by Equation 3.14 - 3.16 to the laboratory frame⁶¹. The laboratory frame is assigned in each molecular model by taking the director of the most aligned molecular axis to define the laboratory Z axis, while the second-most aligned molecular axis is taken to be the laboratory Y axis, and the X axis is orthogonal to Y and Z. Therefore, 𝑄₀₀² = 1 and 𝑄₂₂² = 0 denote a phase with perfect uniaxial orientational ordering. In their simulations, the directors and order parameters were obtained by diagonalizing the order tensors⁶⁰, shown in Equation 3.19, where I is the second-rank unit tensor.

𝑄_𝑎𝑎 = ¹

2𝑁_𝑚∑^𝑁_𝑖=1^𝑚(3𝑎̂_𝑖𝑎̂_𝑗− 𝐼 ) Equation 2.33

Radial distribution function

Radial distribution functions (RDF), also called pair distribution functions or pair correlation functions 𝑔(𝑟), define the probability of finding a particle in a shell 𝑑𝑟 at the distance 𝑟 of another particle chosen as a reference point, such as:

𝑑𝑛(𝑟) =^𝑁

𝑉𝑔(𝑟) 4𝜋𝑟²𝑑𝑟 Equation 2.34

where 𝑉 is the model volume and 𝑁 is the total number of atoms. If we consider dividing the physical space volume 𝑉 into shell 𝑑𝑟, then it is possible to compute the number of atoms 𝑑_𝑛(𝑟) at a distance between 𝑟 and 𝑟 + 𝑑𝑟 from a given atom. The RDF is strongly dependent on the type of matter, therefore the profile of solids, liquids and gases are significantly different.

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In the solid state, the materials structure is specific over a long range, therefore the profile of RDF for solid state materials appears as discrete peaks. Each peak has a broadened shape which is caused by the vibration of the particle around its lattice site. In liquids, due to the more dynamic nature of the molecules, the liquid does not maintain a constant structure and loses the long-range structure. Therefore, the first coordination spheres will occur as a sharp peak that indicates the first sphere. At longer ranges, the molecules become more independent of each other and the distribution returns to the bulk density. Because of the loosely packed nature, the RDF for liquids does not have exact intervals. In gases, because of the non-existence of regular structure, the RDF only has a single coordination sphere then rapidly decays to the normal bulk density of gas.

Simulation of x-ray diffraction patterns

X-ray diffraction (XRD) is a powerful nondestructive technique used to obtain the structural information of crystalline materials. The intensities shown on XRD patterns are produced by constructive interference of a monochromatic beam of x-rays that scattered by atoms in a periodic lattice. The diffraction of x-rays by crystals is described by Bragg’s Law as shown below:

𝑛𝜆 = 2𝑑 𝑠𝑖𝑛 𝜃 Equation 2.35

where 𝑛 in an integer, 𝜆 is the characteristic wavelength of the X-rays impinging on the crystallize sample, 𝑑 is the interplanar spacing between rows of atoms, and 𝜃 is the angle of the x-ray beam with respect to these planes. While the Equation 3.35 is satisfied, x-rays scattered by the atoms in the plane of a periodic structure are in phase and diffraction occurs

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in the direction defined by the angle 𝜃. From the experimental measurement, the x-ray diffraction patterns consist of a set of diffracted peaks with the angles at which they are observed. These patterns are the fingerprint of periodic atomic arrangements in a given material, and can be used to identify the chemicals by comparing this diffraction pattern to a database of known patterns.