Studying the Relationship between Gradient Energy Coefficient with Occupied Site Fraction , Temperature , Density and Surface Tension for Liquid Cyclohexane

The Gradient energy coefficient ( ) in polymers and oligomers (a few segments) play an important role in polymer blending, wetting, coating, adhesion process, foaming and a comprehensive role in the description as well as understanding of many processes especially in innovation of new polymer with classification of materials. As a result, the main reason in this study appear which has how the gradient energy coefficient changes from bulk to surface, has been establishing the new equation that helped us to extract the correlation between gradient energy coefficient and different parameters. The gradient energy coefficient has been related with hole fractions, temperature and density. The relationships between the above-mentioned parameters are then drawn according to our model. Both Simha-Somcynsky (SS) and CahnHilliard (CH) models are employed together to calculate the thermodynamic properties of cyclohexane, namely, the hole fraction in temperature range at 313-473 K various presures up to 150 Mpa. Our values for the average and maximum percentage deviation of the specific volume of cyclohexane are calculated as 0.0196% and 0.0678% respectively


Introduction
The Cahn-Hilliard (CH) theory that includes the free energy profile during the phase separation in an inhomogeneous mixture density for a homogeneous system and  is a positive materials constant called the gradient energy coefficient or interaction coefficient for the component and the second part from same equation is the composition gradient contribution to the free energy [1].The lattice fluid (LF) theory [2] is able to describe the thermodynamic properties of both low and high molecular weight.The LF model in conjunction with the Cahn-Hilliard theory (CH) is employed to develop a method for calculating the surface tension of nonpolar and slightly polar liquids of arbitrary molecular weight has worked by Poser and Sanchez [3] to extract the surface tension and surface density profile of polymers in broad range of temperature.Poser and Sanchez, Kahl and Enders [4], Dee-Sauer and C.Miqueu and co-workers they has worked extensively on the surface tension and surface density profile of polymers for broad ranges of temperatures and molecular weights using the Cahn-Hillard density gradient [5].theory in conjunction with the Flory, Orwoll, and Vrij (FOV) [6] and Sanchez and Lacombe [7] (SL) equation of state theories.B. Sauer and T. Dee they has worked to obtained the surface tension and the gradient energy coefficient (  ) for linear, branched n-alkanes.The surface tension increase and surface entropy [8] decrease with increasing molecular weight.
The surface thermodynamics properties of polymers are strongly correlated with the bulk properties, the bulk properties are inherently hole fraction dependent so we could constitute the correlate between surface tension and surface density profile with hole fraction or free volume of the bulk, Carri and Simha (CS) examined the relation between surface tension and hole fraction of the bulk properties by means of SS lattice-hole theory [9].The Simha and Somsynsky (SS) come up with an arrangement in the model increasing the disorder by employed hole fractions in the underlying quasi lattice model.An established equation of state (e.o.s) was intensively applied to low and high molar mass of liquid polymers [10 and 11] and mix of molecular weight of different polymer, with significant quantitative achievement [12], The quantitative success of hole theory (SS) encouraged us to employ in conjunction with the Cahn-Hillard density gradient theory to inspection how the hole fraction changes from the bulk to surface and effectives on surface tension and its correlation with the surface density profile

Theories 1. The Cahn-Hilliard Theory
The Cahn-Hilliard theory [13] correlates the thermodynamic characteristics of a system with an interface between two non-condensing phases.In the interface between the liquid and the gas phase of a pure polymer in equilibrium condition, the density of the composition discontinues or behaves as gradient, consider a binary alloy in a two-phase equilibrium state.For the free energy of inhomogeneous systems, the density gradients varies from the bulk or liquid density to the surface or the vapor density continuously.This means that the Helmholtz free energy density, α, of a system with an interface can be obtained by expanding the Helmholtz free energy in Taylor series around the equilibrium state: where 0 ()  is the local free energy density of homogeneous polymer system and the coefficients of Laplacian and gradient density terms are The subscript 0 in Eq. ( 2) indicates that the derivatives are to be evaluated in the limit of   and 2   going to zero.Here the density variation is assumed comparatively small to the reciprocal of the intermolecular distance.The Helmholtz free energy, A, of a system of volume where  is the gradient energy coefficient for the system.It is composed of two terms: the first is the local free energy of homogeneous system and the second is composition gradient contribution to the free energy.
We could write Eq. ( 3) in finally shape of surface tension for a planar interface tension is given where is the difference between the Helmholtz free energy density of a homogeneous fluid of density  and two-phase equilibrium mixtures with liquid and gas states.
where 0 l  and 0 v  are the equilibrium chemical potentials of liquid and vapor, The appropriate form of the Euler equation says: where I represents the integrand of Cahn-Hillard equation.If we apply the integrand of Eq. (5) in Eq. ( 4) [14], we obtain a differential equation whose solution is the composition profile corresponding to invariance value (maximum, minimum or saddle points) of the integral.The condition for invariance value is 2 () In this equation the constant value must be zero, and also both Simha-Somcynsky (SS) developed an equation of state (EOS) based on the lattice-hole model [15] introducing the temperature and volume dependent occupied site fraction, ( , ) y V T .The occupied site fraction, ( , ) y V T , and the complementary hole fraction, ( , ) h V T , are given by the following equation: where s is the number of segments in a molecule, and N and Nh are the number of molecules and holes respectively.The SS theory is formulated in terms of scaled volume, scaled temperature and scaled pressure, viz.: where the scaling parameters are as follows: V * is defined by molar volume   s of the molecule, as a balance between attraction and thermal energy contributed by the external degrees of freedom (where k is the Boltzmann's constant), and P * is then assigned by the ratio between chain attraction energy   z q and volume   s .Here 3c that appears explicitly in the equilibrium condition is the total degrees of freedom of molecule.
In this article, we have considered the ideal chain flexibility employing 33 cs .The configurational partition function for the ensembly can be written as where E0 is the total lattice energy of the system employed Lennard-Johns potential energy, f is the free volume, and ( , ) g N y is the combinatorial factor that is the total number of distinguishable degenerate arrangements of the holes and molecules by Boltzmann's equation.
It can be calculated from the mixing entropy of an assembly of molecules and holes as ln ( , ) Hence the combinatorial factor is expressed as Using the coupled Eqs. ( 14) -( 15), we can determine the scaling parameters, ,,  , and the structural parameter 3c/s, which can be obtained by superimposing experimental P-V-T data on the theoretical ,, P V T surface.Having these parameters at hand, we can compute the hole fraction, h(V,T)=1-y, of the lattice model (as a measure of the free volume [16]).
From temperature dependence of the hole fraction law we have the relation: where the right hand side is obtained by substituting Eq (12) with Eq (13).The chemical potential of the system is given by Vol where the right hand side of Eq(21).isobtained by substituting Eq(17).Substituting Eqs(21) into Eq (8)., the scaled surface tension can be written as   Where Vtheory is the specific density calculated from the SS theory and Vexp is the one calculated from the Tait equation, and N is the number of data.

Working Tools
Has been employed the Cahn-Hilliard (CH) theory in conjunction with Simha-Somcynsky (SS) theory, has been derived manually the new equation that helped us to extract the correlation between gradient energy coefficient with hole fractions, temperature, density and another parameters.These studies are written in a mathematica code program.At first, PVT data for the SS theory, are calculated from the modified cell model(MCL) starting atmospheric pressure to 150Mpa and for the temperature range from 313-473 K, then the relationships between the parameters are then drawn by using the origin program and Math Type program to write the equations.

Results and Discussion
The relationship between the parameters are extratcted based on theoritically caluculations are shawn in Table 2  T(K)     3) and ( 4) respectively.They show how the increasing values of temperature and hole fraction lead to decreasing in gradient energy coefficient because the increasing in temperature follows increasing in hole fraction.The scaled surface tension is plotted against scaled temperature, figure (5), to show how the increasing in temperature is leading to decreasing in gradient energy coefficient.The correlation between scaled energy gradient coefficient and scaled surface tension is plotted as in figure (6), that shows the pattern of increaasing surface tension with increasing gradient energy coefficient.
Initially, PVT data for SS theory, are calculated from the modified cell model (MCL) at various conditions, starting at atmospheric pressure upto 150 Mpa and at the temperature range from 313 to 473 K. Our average and maximum values of percentage deviation of specific volume for cyclohexane are calculated as 0.0196% and 0.0678% respectively.

Conclusion 1.
Has been certified that the hole fraction increase from bulk to surface or interface.

2.
Only a very small fraction of the cyclohexane chains are close enough to a surface in order for their physical state and behavior are different.

Vol: 14
No:2, April 2018 DOI : http://dx.doi.org/10.24237/djps.1402.403BP-ISSN: 2222-8373 E-ISSN: 2518-9255 of binary polymers system.We are employed properties of SS theory as a nested with CH in the range of about 473 K temperature and up to about 1500 bar pressure.We have obtained %0.0196 maximum deviation in volume.


tend to zero when x   .Hence a minimum value can be expressed as: ://dx.doi.org/10.24237/djps.1402.403BP-ISSN: 2222-8373 E-ISSN: 2518-9255 2. The Simha-Somcynsky (SS)-EOS Theory EOS equation, derived from the configurational Helmholtz energy, is site fraction can be obtained from the minimization of the Helmholtz energy of an ensemble, ()KT is quite slowly varying function.Therefore, the hole faction h or occupied site fractiony satisfies constant yV C  (17)For a binary system (occupied and unoppupied), Gibbs free energy of mixing

1 Cy
b and s refer to bulk and surface.On the other hand the scaled   is expressed in terms of chemical potential difference as scaled value of chemical potential in equation (22) we get the final value of surfac etension is cyclohexane material studied in this article in unique weight which was fitted for temperature range of (313-473) K and pressure range of 0.1-150MPa as written above.The SS Vol: 14 No:2, April 2018 DOI : http://dx.doi.org/10.24237/djps.1402.403BP-ISSN: 2222-8373 E-ISSN: 2518-9255theory employs these calculated specific density data to obtain the characteristic parameters viz. the scaling pressure, scaling temperature and scaling volume.These parameters are simultaneity fitting of the density data with the theory using the coupled Eqs.(13)-(14).Table1shows these computed parameters with the average and maximum relative percentage error in parameters of cyclohexane used in this work are shawn in Table1

Figure 1 :
Figure 1: Plot of reduced gradient energy coefficient as a function of occupied site fraction for C6H12.

Figure 2 :
Figure 2: Plot of reduced gradient energy coefficient as a function of density for C6H12.

Figure 3 :VolFigure 4 :Figure 5 :
Figure 3: Plot of reduced gradient energy coefficient as a function of temperature for C6H12.

Figure 6 :
Figure 6: Plot of reduced gradient energy coefficient as a function of reduced surface tension for C6H12

3 . 4 .
When increasing the molecular weight of the polymer that refer to increasing in the length of segments that leads to increasing in the surface intensity (equation 23) and decreasing the interface thickness.The most of the polymers component is immiscible to blending, So main reason belong to multiphase, our the best conclusion from this study the high degree of polymerization, İt will help us to innovation the new polymers.Because the high degree of polymerization reducing (eliminating) the different phases.

5 .
Our model can be applied to all polymers.

Table 1 :
The critical value of surface tension, temperature and characteristic parameters of6 12

Table 2 :
Reduced surface tension and gradient energy coefficient6 12