Límites de Atterberg (ASTM D 4318; AASHTO T 89-90 Y T 90-87)

This methodology was pioneered by Frenkel and Ladd [38] and is based upon the construction of an unphysical, reversible path from a state of known free energy, an Einstein crystal with the same structure as the solid being investigated. The solid under consideration is coupled to its lattice sites by gradually turning on harmonic springs, and when the coupling is sufficiently strong, the solid is indistinguishable

from an Einstein crystal. In an Einstein crystal, each atom is an independent harmonic oscillator, all oscillating with the same frequency and with a potential energy function given by

U_Ein= 1 2 N X i=1 αi(ri−r0,i)2 (3.4)

where r_0,i is the associated equilibrium lattice position of particle i at position r_i and αi is the corresponding coupling parameter. The Helmholtz free energy can easily be calculated, as needed for a reference state [39, p.253–254].

The Einstein crystal has the same structure as the solid being investigated, hence the path is very likely to be reversible, i.e. not pass through any phase transitions.

The path is constructed by adding to the unperturbed hamiltonianH_Model the hamiltonian corresponding to the Einstein crystal,HEin

H(λ) =λHEin+ (1−λ)HModel. (3.5) The free energy of a real crystal is related to that of the Einstein crystal with spring constant λ. For a value of λ = 0 we retain the original solid hamiltonian and for λ= 1 the path goes exactly the hamiltonian of the Einstein crystal.

The derivative of the free energy with respect to the spring constant λis ∂F ∂λ =−kBT ∂ ∂λ ln Z ... Z exp[−β(H(λ)] (3.6) =hVi_λ (3.7)

this leads to the free energy of the crystal (λ= 0) as

F(λ= 0) =F(λ)− Z 1

0

hViλ0dλ0. (3.8)

The implementation requires simulations to find the optimal spring constant and then further simulations to sample the free energy derivative at varyingλ-points. Quantifying the error from these calculations can be very expensive. The obvious, and most common, way is to repeat the calculation many times and look at the variance in the results. This will take up a lot of computer time and will also give no handle on systematic errors. There is no easy or smart way of calculating the error from these simulations or how the errors get propagated through simulations to the final result.

Absolute free energies are generally very large quantities, but the difference in free energy between crystal polymorphs close to a phase transition can be extremely small∼10−5kBT. This is often smaller than the typical error in an Einstein crystal calculation.

This method also depends on a lot of parameters that need to be chosen and optimised in order to get an accurate result. Hence, it is not the most reliable way to find small free energy differences.

This method was first applied to the hard sphere system to calculate the free energy difference between face centred cubic (fcc) and hexagonal close packed (hcp) phases. Calculations were made at a reduced densityρ/ρcp = 0.7360, which is the density of solid-fluid coexistence, and at a density ofρ/ρcp= 0.7778.

For each simulation Frenkel and Ladd performed ten runs for different values ofλ. The chosen values ofλ were different for each density. The free energy of the Einstein crystal was approximated by a cluster expansion. Eq. (3.8) was adapted so that the integrand is a slowly varying function of the integration variable.

∆F =− Z λmax 0 hr2iλ(λ+c) dλ (λ+c) (3.9) =− Z ln(λmax+c) ln(c) hr2iλ(λ+c) dln(λ+c). (3.10)

wherehr2_{i is the mean square particle displacement, λ}

max is the maximum spring constant used, decided for each system separately. c is a constant, chosen to be c = exp(3.5) ≈ kBT σ2/hr2i where σ is the hard sphere diameter. The centre of mass needs to be constrained to ensure that the mean square displacement does not diverge. Without this constraint, the mean square displacement would become of order L2, where L is the box length, as λ → 0. The integrand in eq. (3.10) would sharply peak aboutλ= 0 and greatly affect the accuracy of the result. This constraint of the centre of mass involves difficult computation, and especially for MC simulations it is not trivial.

To implement this constraint Frenkel and Ladd expressed all coordinates rel- ative to the centre of the periodic box. One needs to keep track of the displacement of the centre of mass of the system by updating and changing the entire box coor- dinates after every move. This makes it a costly undertaking, yet necessary for this method. One must also take into account this constraint in the centre of mass with a (lnV)/N term to the free energy.

The system sizes studied were chosen so that perfect stacking of the crystals was ensured. It was found that the excess free energy depended on the shape of

the system box and so there is significant difference between the free energy of a 54 = (3×3×6) system and a cubic 216 = (6×6×6) system. One also must compare with results from studies that have taken this into account in the same way. When looking at finite size effects this extrapolation can have a huge effect on the results, as seen in the work of Frenkel and Ladd at a density of ρ/ρcp= 0.7360. The value calculated for the free difference between fcc and hcp hard sphere crystals put the free energy difference in the interval−0.001<∆F∞ <0.002, where ∆F =Fhcp−Ff cc for both system sizes.

This shows that the method is not accurate enough to fully resolve the free energy difference. The error is of the same order as the difference between the absolute values, for example at a density ofρ/ρcp= 0.7360 the free energy difference is ∆F = 0.0009 and the error on the absolute free energy for the fcc phase is 0.0010. In summary, the errors are very large and uncertain, and the final result doesn’t even definitively determine the most stable phase. This shows that the method is not best used for such small free energy differences.

Polson et al. [82] corrected the formulas in this method, showing that the true free energy of a crystal is (2/N) ln(N) lower than that in the original paper. Even though this tends to zero asN → ∞, for the finite sizes studied this will have a non-negligible effect. Polson et al. [82] only calculated results for the free energy of soft spheres, which cannot be directly compared with the hard sphere results. Nevertheless, the error in their calculations was given as an order of magnitude lower than that quoted in the original Frenkel and Ladd paper. No explanation of where this reduction in error came from was given. Their final value for the free energy difference is ∆Fhcp-fcc/(N kBT) = 0.0028(8), showing that the fcc crystal phase is more stable.