b Instrucciones de Diseño para la Ordenanza 7.

It is well known that at small dipole strength the ST fluid does exhibit a GL phase sepa- ration, mainly due to the isotropic dispersion interaction from the LJ potential. The first computer simulations on GL equilibria for the ST fluid were performed by Smit, Williams, Hendriks, de Leeuw and van Leeuwen [1–3]. They employed the Gibbs ensemble Monte Carlo method (GEMC), a technique proposed by Panagiotopoulos [4], to simulate GL equilibrium without explicit knowledge of the chemical potential. Two phases are sim- ulated in two separate simulation boxes. The method employs MC steps with particle displacements and volume changes for each box including particle exchange between the two boxes, to impose internal and mutual equilibrium on the two phases. The long- range corrections were calculated by Ewald summation with ’tinfoil’ boundary conditions (cf. equation (4.3.5)). They got GL phase coexistence curves for the ST fluid for dipole strengths up to µ2 = 6 and compared them with curves obtained from first order pertur- bation theory [5] and first order perturbation theory with Pad´e approximation [6]. They found good agreement with first order perturbation theory only for GL coexistence curves with µ2 < 1, with Pad´e approximation up to µ2 < 4. Nevertheless in the critical region both methods are not satisfactory and overestimate the critical point.

In perturbation theory the Helmholtz free energy of the considered system is expressed in terms of a well known reference system and a perturbation of the reference system. In case of the ST fluid, the LJ fluid is an adequate reference system and an expansion in terms of the dipole moment µ yields the perturbation. The Pad´e approximation includes an estimate for the higher order terms of the expansion. First order perturbation theory

58 4 The Stockmayer fluid in literature

for dipolar systems refers to expansions of order µ4_{. The next order estimated by Stell et}

al. via Pad´e approximation is O(µ6). As equation of state for the LJ system, the modified Benedict-Webb-Rubin (MBWR) equation [7] was used. Perturbation theory calculations using Pad´e approximation applied to the pST fluid were also carried out by Vesely [8]. He calculated the internal energy, the compressibility factor and the mean dipole moment for dipole strength µ = 1 and polarizabilities up to α = 0.1. The results were also compared with values obtained from computer simulation [9]. He found substantial deviations for large polarizabilities, but his simulated systems were small (N = 108).

In reference [3], van Leeuwen compared the critical properties obtained from GEMC with properties obtained from the virial equation of state [10–12], including the third virial co- efficient, and density functional theory (DFT) [13], originally developed for higher dipole strengths. The former one does not even reproduce the critical behaviour qualitatively, because it is a low density theory and the critical densities for the investigated range of dipole strengths are close to 0.3. DFT shows an overly strong µ dependence of the critical properties. Van Leeuwen showed the increasing deviation from the principle of corresponding states [14] for the ST fluid with increasing dipole strength, especially in the liquid phase.

DFT is based on the fact that the thermodynamic potential of an ensemble, for example the Helmholtz free energy, can be written as a functional of the microscopic density. This functional is minimal for the physical realized density. From the thermodynamic potential all other thermodynamic properties can be derived.

Garz´on, Lago and Vega showed in reference [15] the equivalence of reaction field and Ewald summation simulations for GEMC simulations, by comparing their results for µ2 = 2 and 4 obtained using the reaction field method with the ones of Smit et al. [1]. They found differences neither in the location of the phase coexistence region, nor in the structure of the gas and liquid by observing the pair correlation functions.

Van Leeuwen and Smit investigated the GL phase coexistence region for the vLS fluid (cf. equation (2.3.1)) with the same method employed in [16]. They performed simulations with dipole strength µ2 = 4 and varied λ between 0 and 1. Most noticeably they found no coexistence region for λ . 0.3 (especially for the DSS fluid (λ = 0)). They claimed chain formation, observed for small values of λ, prohibits the phase transition. Since the vLS fluid can be mapped onto the ST fluid according to (2.2.2)-(2.2.7), the phase transition should disappear for the ST fluid for dipole strengths µ2

& 25. This seemed to be in line with a publication of Caillol [17], who performed GEMC and isothermal-isobaric MC sim- ulations of the DHS fluid and was not able to observe a GL transition. This conclusion is quoted frequently in the subsequent literature (e.g, [18–24]), although, van Leeuwen and Smit reported problems with the GEMC method due to formation of reversible dipole chains resulting in too low acceptance rates for the MC steps (cf. reference [25]). They tried to solve these problems employing the configurational bias MC technique.

Tavares, Telo da Gama and Osipov [21] calculated the free energy of a strongly polar fluid as a mixture of self-assembled chains applying several approximations to calculate the phase coexistence region and critical properties. They compared their results to the simulation data for the vLS fluid of van Leeuwen and Smit and found qualitatively good

4.1 Gas-liquid phase coexistence 59

agreement for λ ≥ 0.35. If λ is further decreased, the theory predicts coexistence for a fluid of chains, interacting via the isotropic dispersion interaction between the monomers. They suggest finite size effects as possible reason for the prevention of the phase transition in simulation, since the average chain length grows exponentially with decreasing λ. Their theory predicts the critical temperature and density to become zero for λ = 0.

Stevens and Grest performed GEMC simulations on the ST fluid with and without an applied electric field [18]. They provided GL coexistence curves without applied field up to dipole strength µ2 = 16, together with the values for the critical temperature and crit- ical density. Additionally, they obtained the field dependence of the coexistence curves for µ2 _{= 1 and 6.25. They provided the mapping laws of the vLS potential [16] onto the}

ordinary ST potential. To our knowledge they were the first who combined simulation data for GL coexistence with ferroelectric ordering. They found a ferroelectric phase in the dense liquid phase for µ2_{/T & 4, but the isotropic gas-ferroelectric liquid coexistence,}

as predicted by a theory of Zhang and Widom [26], seemed to be unlikely, since the order parameter decreases strongly entering the phase coexistence region. They relied on the disappearance of the GL coexistence region for the ST fluid above the dipole strength µ2 _{& 25, but it does not become clear if they checked this on their own or not.}

Stoll et al. determined vapour-liquid equilibria properties of the two-centre LJ plus axial point dipole fluid (2CLJD) [27] from MD simulations using the reaction field method. The 2CLJD potential model is composed of two identical LJ sites at fixed distance and a point dipole ~µ2CLJ D placed in the geometric center of the molecule, pointing along the molecular

axis. They employed the NPT plus test particle method introduced by M¨oller [28] to get GL coexistence curves. In the borderline case for vanishing LJ centre-centre distance, they provided GL phase coexistence and critical data for dipole strengths up to µ2

2CLJ D = 20,

corresponding to µ2 _{= 5 for the ST fluid.}

Kiyohara, Gubbins and Panagiotopoulos, to the best of our knowledge, published the only phase coexistence data from simulation on the pST fluid [29]. They performed grand canonical Monte Carlo (GCMC) simulations [30] and calculated the thermodynamic prop- erties from the histogram reweighting method [31]. They gave coexistence curves for dipole strengths µ2 _{= 1, 4 and polarizabilities α = 0, 0.03, 0.06 and compared their results with}

Wertheim’s perturbation theory [32], later improved by Kriebel and Winkelmann [33, 34]. Kiyohara et al. recognized an advantage of the GCMC method for phase equilibria in CPU time compared with the GEMC simulations.

As already mentioned Frodl, Groh and Dietrich provided various global phase diagrams for the ST fluid, obtained from DFT, including GL phase coexistence, ferroelectric order and the liquid-solid transition [13, 35–38]. The main focus of reference [13] was GL coex- istence with small dipole strengths µ2 _{≤ 4. In [35, 36] Groh and Dietrich investigated the}

phase coexistence and the isotropic liquid-ferroelectric liquid transition for samples of dif- ferent shapes. In [37, 38] they deal with the freezing of the ST system and the coexistence of ferroelectric liquid and ferroelectric solid. They found that for small dipole strengths µ2 < 1, the ferroelectric liquid is preempted completely by the freezing transition.

Russier and Douzi compared the GL phase transition of the ST fluid as a model for the dilute-dense phase transition of a colloidal ferrofluid [39]. They did not take the solvent

60 4 The Stockmayer fluid in literature

into account explicitly, thus they considered a one-component system whose interaction potential indirectly includes the influence of the solvent. They determined GL coexistence curves from a second order virial expansion via the Maxwell construction in the range of dipole strengths µ2 _{= 1 − 4 and compare their results with MC simulation [1–3] and}

DFT [13]. With this simple method they got surprisingly good results.

Dudowicz, Freed and Douglas applied a Flory-Huggins mean field lattice model for re- versibly associating polymers to the ST fluid to determine the GL phase coexistence region and the critical points for high dipole strengths [40]. In their model they found no evidence for the coexistence region to disappear for µ2 _{& 25. The critical temperatures and densi-}

ties they found were in good agreement with the existing data from GEMC [1–3, 16, 18].