THERMODYNAMIC PROPERTIES OF A SIMPLE FLUID MULTI Yukawa USING A VARIATIONAL METHOD

Jose Noe Pacheco Felipe Herrera1*, Alfonso Vera Cruz1,Alma Y. Govea Salazar2,Eduardo González JiménezJanuary

1Universidad Autónoma de Puebla, Faculty of Physics and Mathematics, Av San Claudio CU and Rio Verde S / N, Bldg 111 th, Col. San Manuel, Puebla, Mexico, CP 72570. Faculty of Physics and Mathematics.

2Technological University of Mixteca, Agribusiness Institute, Carretera a Acatlima KM 2.5, Huajuapan de León, Oaxaca, Mexico, CP 69000. E-mail: nherrera@fcfm.buap.mx

Abstract

The perturbation theory derived from statistical mechanics of dense systems allows for the state equation analytically for when the intermolecular interaction is represented as a linear combination of Yukawa type potential and taking as the reference potential of hard sphere interaction. This equation of state depends on the ratio of the characteristic molecular diameters. In this work we determine this ratio using a variational method on the results obtained by Guerin, our approach will call VMSA, it means minimizing the Helmholtz free energy with respect to a parameter, where  is the distance at which the interaction potential is zero. The effective diameter R depends on the density, temperature and potential parameters. Thus, we can achieve results consistent with the thermodynamics and are comparable with those reported by the method of computer simulation using the Monte Carlo method, particularly for the compressibility factor.

Keywords: perturbation theory, typical diameters, effective potential.

Abstract

The perturbation theory derived from statistical mechanics of dense systems Allows for the state equation analytically for Such a system s, which depends on the ratio of the characteristic molecular Diameters. In this study Determines That ratio by using a variational method, will call Our approach VMSA, Which Involves Minimizing the Helmholtz free energy by,  is the distance Where the energy with Respect to a Potential interaction WHERE parameter is equal to zero. The Effective diameter R depends on the density, temperature and Potential parameters. Thus, we Obtain results close to Those Reported by the Monte Carlo method for the compressibility factor Obtained from a perturbation theory for Which Is an Effective Potential interaction Formed by the interaction of hard sphere plus a Multi-Yukawa contribution.

Keywords: perturbation theory, characteristic Diameters, Effective potential.

PACS: 01.40.-d; 61.20.Gy

1. Introduction

From the results of the theory of statistical mechanics ensembles constructed the classical theory of liquids, which allows up to now have theories able to describe the thermodynamic behavior of various systems [1.2]. A common feature of these theories is that in calculating the thermodynamic properties requires knowledge of an analytical potential interaction between the components of the system, which is generally very difficult to obtain, and the correlation functions. One approach that has yielded very good results is to propose models of molecular interaction usually consisting of two terms, one that takes into account short-range interaction and one for long-range interaction [1,3,4,5]. The first term can be of the hard sphere and soft sphere, but in general is a repulsive term that takes into account the excluded volume effect of particles and has been shown experimentally that the effect is predominant peaks main scattering electromagnetic radiation when you affect on a fluid [2]. The second term of the effective potential may include both attractive and a repulsive part, because the experimental results show that the potential of average force is generally an oscillatory function, which is impossible to represent by a single interaction term. On this basis, in the second half of last century theories of disturbance proposed that consider the effective interaction potential consists of a reference part represented by the hard sphere potential plus a contribution due to the disturbance. Within these theories are the theory of Barker and Henderson (BH) [6], Weeks-Chandler-Andersen [5], among others. Recently Guérin [8] obtained the thermodynamic properties for fluid interaction potential perturbation represented by two Yukawa terms, one attractive and one repulsive, and the hard sphere interaction potential as dominant, he used the results of perturbation theory first-order developed by Tang and Lu [7]. The equation of state obtained by Guerin is analytical, but it depends on a parameter involving the ratio of two diameters, R is the equivalent diameter or a pseudo-hard sphere diameter, which is a necessary element in perturbation theory, while  is the distance at which the potential vanishes. According to the theory of BH [4], R can be obtained by integrating the Mayer function and depends mainly on temperature. For its part Guérin thermodynamic properties obtained for constant values of the parameter  obtaining results very close to those predicted by the MC method, however, the value used for this parameter has no physical justification. In this work we Guérin analytical results and we use the Gibbs condition Bogolioubov (GB) [8] to find an effective diameter that minimizes the Helmholtz free energy of the system. When applied to the condition GB Guerin expressions leads to a nonlinear equation for the parameter .The results show that the values of this parameter depends on the density, potential parameters and temperature and have a satisfactory agreement with those reported by MC at low and intermediate densities, but at high densities are significant deviations. The results show that the perturbation theory of Tang and Lu predicted similar results and anfiend Monsoori theory, but with the advantage that the calculation of the parameter  is very simple.

2. Methodology

Consider a monodisperse fluid whose effective potential is given by the HS potential plus a combination of interactions represented as a linear combination of Yukawa type terms (MYHS)

ur = ∞, for r<σ (1a)

and

ur = εφrσ = εφx, for r>σ, (1b)

where x = r / σ and φ (x) is given by:

φx = cx im-1iexp-αi (x-1) (1c)

In equation (3), m is the number of Yukawa terms, i   and  are the parameters of the inverse of the scale and intensity of the interaction respectively, and the same λ = Rσ, where R is the effective diameter hard sphere of particles and is a condition that u()= 0, [8]. The constant c is obtained by determining the minimum of φ (x) equal to -1.

Recently Tang et al., [7] have proposed a new theory of disturbances in which expands the radial distribution function to first order, ie, where the first contribution is the radial distribution function of the reference system and second term is associated with the disturbance. With this we can write the Helmholtz free energy as a second order as [8]:

a = a0 + a1 + a2 (2)

With

                                  (3)

In the above equations  is the packing fraction (η = πρσ3λ3 / 6 = ρ * λ3),0 is the Helmholtz free energy Carnahan Starling (CS),1 is the term to first order perturbation theory while2 is written in terms of g1(r).

When we use the potential MYHS in equations above the resulting expressions are close to the Laplace transforms of rg0(r) and rg1(r) given by:

                                              (4)

Wherteim [9] obtained an analytical expression for G0(s)within the PY approximation for a fluid of hard spheres of diameter R given by

                                               (5)                

with

                                               (6)

For its part Tang et al. [10] derived an expression for G1(s)for a two-term Yukawa potential with a hard shell diameter R and consider a monodisperse fluid with a single extent ofinteraction. However, the inclusion of different amplitudes in the interaction and consider M = 2 Yukawa terms in a monodisperse system is straightforward, so that for systems with different amplitude and of equal diameter. We have:

                                     (7)

By using the above expressions we can obtain explicitly the contributions to the reduced free energy as follows [6]:

12η a11 = (1-η) 2cλt i (-1) i exp ⁡ (αi 1-λ) αi Lαi λQαi λ2 λ (8.1a)

a12 =- 12η g0Rcλ you (-1) iαi λ2 Ki (8.1b)

a21 =- 6η cλ t2 i, j-1i + + αjλ jαiλ αj1-λQ2αiλ expαi + Q2 αjλ (8.1c)

6ηcλ T2i a22 = (-1) i expαi (1-λ)-1jαj Q2αiλj λ2 (8.1d)

where t = KBT / ε, kB is the Boltzmann constant and T the absolute temperature, L (z) and Q (z) are given by equations (6) and Ki is given by,

Ki = 1 + αi λexpαi (1-λ) - (1-αi) (9)

The excess internal energy is obtained from the following thermodynamic expression

                                                  (10)

Differentiating the Helmholtz free energy with respect to the density allows us to obtain the compressibility factor as:

                                                (11)

Where Z0 is the Carnahan-Starling expression for the compressibility factor and

and after taking the derivatives we obtain:

                              (12)       

Where in the above equations denotes the derivative of Q with respect.Equation (11) is the reduced pressure or compressibility factor as can be seen has an explicit analytical expression in terms of the characteristic parameters of the interaction is, in terms of the amplitude or depth of interaction, the inverse length potential range of the ratio of characteristic diameters of the particles and their density. But it remains to determine the parameterreduced diameter,there are at least three ways to find this parameter [11]: i. Using the equation of Barker and Henderson [4], ii. Using an expression that was achieved by adjusting simulation values and extending the theory of BH, which was obtained by Verlet and Weis [12] and iii. Using a variational method using the Gibbs criterion Bogolioubov [8].

The first method calculates the parameter using the expression:

                                        (13)

As shown above expression to calculate the ratio of the diameters in terms of the thermal bath temperature and the parameters that define the effective potential, where the result is an explicit dependence largely a function of temperature. The general integral must be numerically [6].

While the second approach is determined from an expression that was derived from the analysis of the results by computer simulation and using the idea of the BH theory, the best fit is expressed as:

                                                 (14)

As is evident from the reduced temperature dependence in this approach, see Figure 1, however there is a difference regarding the calculation of BH since the slope of the curve is convex in VW [12] and calls in BH [4], although in both approaches the diameter decreases with increasing temperature, note that if VW again only preserves amplitude information of the interaction of the effective potential. In both approaches we have for the ratio of the diameters do not have the density dependence and no way of how calculated.

The third method is usually used to calculate the pseudo-diameter hard sphere is known as a variational method, which involves applying a condition of minimizing the Helmholtz free energy with respect to R. This procedure was applied to the results of the BH perturbation theory, and other authors, obtaining reasonably good results in the prediction of thermodynamic properties [11]. Patiend Guerin's work [8], we have the energy and compressibility factor in terms of parameter ,however in the work of this author found that not calculated the ratio of the diameter but uses some empirically known values that reproduce with good accuracy the simulation results [6]. In this work we took on the task of applying the variational method, assuming that the parameter  has the properties to minimize the Helmholtz free energy, since we know that it is a functional, with this information we can establish that the stability condition requires condition is satisfied:

The third method is a variational (VMSA)

∂ a ∂ λρ *, T *= 0 (16)

Which involves solving the equation:

∂ a ∂ λη, T * + ∂ a ∂ ηρ *, T * ∂ η ∂ λρ *, T *= ∂ a ∂ λη, T * + 3Zλ = 0 (17)

This expression is a nonlinear equation for the parameter,explicit expressions of equation (17) are widespread but is obtained directly from equations (2) and (11). Equation (17) is a nonlinear equation for the parameter,the selection criterion of the physical root is that if the reference system is the hard sphere is then taken root close to unity. Note that equations (16) and (17) and the general expressions for free energy and compressibility factor contain no restrictions as those imposed on the work of Guerin, [8], which calls, for example, that a21 = a22 = 0, for λ> 1. In this work, the unknown parameter we must minimize the free energy only.

3. Results

Expressions are obtained for m = 2 terms of Yukawa, but its extension to more terms is straightforward, this paper presents its application to double Yukawa potential only, as we have reported results from the literature for some of them.

To apply our results and in order to compare our results with those reported in reference [6], we rewrite the potential in the form used by Guerin, so we have that for when the system has a perturbation potential written as a sum of two terms Yukawa type and scope is unique, we have:

                    (17)

Figure 2 shows the dependence of the parameter  based on the reduced temperature T *, for a fluid with an effective potential perturbation with parameter values ,a= 2.6509 2= 14.8105 c = 2.0053, [14]. Note that as the temperature increases reduced the value of the parameter decreases as expected, showing the same qualitative behavior presented in Figure 1. But at very low densities the slope is similar to the result of Barker and Henderson, however, with increasing density of such behavior is predicted by the Verlet-Weis

formula.Figure 3 shows the dependence of the parameter for different values of the reduced temperature as a function of reduced density. Note that the qualitative behavior is similar to that obtained by Mansoori and Canfiel [13], who used a variational method similar to this job, but the expression of free energy was different and the acquisition of such a procedure parameter requires extensive computation. However, it may be noted that since the Tang perturbation theory is in terms of the expansion in the radial distribution function, we find that this approach predicts a density dependence much stronger than in the variational theory of Mansoori and Canfield [13].

Figure 4 shows the isotherm at T *= 1.35 obtained by equation (11) for the compressibility factor usingparameter values  obtained by solving the equation (17), depending on the number density *, compared with the results of Monte Carlo [11].

From the results shown we can see that the variational method agrees qualitatively with the results of Monte Carlo, but there are deviations that need to be analyzed, either by changing the hard sphere contribution in compresibililidad factor or by using a theory disturbance to the highest order. Figure 2 shows the dependence of small-diameter parameter as a function of temperature, the behavior is expected, on the other hand Figure 3 shows the dependence of this parameter as a function of reduced density and we can see by inspection that at low density parameter is almost constant at fixed temperature, density effects are felt at high densities can also appreciate that at high densities(*> .85) perturbation theory of Tang et al. (TPMSA) no longer correctly predicts the thermodynamic properties and that may be useful in establishing the limits of validity of the approach.

4. Conclusions

Predicting the behavior of the parameter  VMSA obtained with the theory qualitatively agrees with what is expected using the expression of Barker and Henderson, and a fixed density is qualitatively comparable with the form of Verlet and Weis, however to have a VMSA important difference from the two approaches mentioned above, since in our case the mentioned parameter depends on temperature, density and the parameters that define the interaction potential, the main difference is the dependence on density, this makes our approach in principle is more complete than the other two, because physically it should be, ie the effective diameter representative of the system should depend on the thermodynamic system defined by temperature and density, the temperature dependence one would be restricted. Now having deviations from the Monte Carlo values for certain densities allows us to grasp that this is because our approach to second-order perturbation is still incomplete and that the same reference system of hard spheres is relatively good but not quite correct, there are proposals to improve the prediction of them is to use different expressions for the reference system see reference [11].

This work complements the results of Guerin for the thermodynamic properties of a system with interaction potential dominant field disturbance lasts longer written as a sum of Yukawa type waves.

Dedication

The authors wish to dedicate this work to Professor Lesser Blum.

References

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Figure captions

    FIGURE 1:Reduced diameter  by Verlet and Weis for a LJ fluid type [11].

FIGURE 2: Dependence of the parameter  with the reduced temperature for a fluid hard sphere Yukawa plus a double shock.

FIGURE 3: Dependence of the diameters with the density at different temperatures for the same system in Figure 2.

Figure 4. Comparison of isotherms, solid line represents the isotherm obtained by Eq. (17) and points were obtained by MC simulation [11].

Figure 1

Figure 2

Figure 3

Figure 4