Studying The Characteristic Parameters and External Freedom Degree of Polyethylene in different Molecular Weights By Hole Theory

T he external degree of freedom of linear and branched molecules of polymers played an important role in new polymer synthesis. The theory Simha-Somcynsky (SS) which established for the first time the statistical thermodynamics and polymers state of equation, using the mean field theory )that does not depend on the structural of materials). Expressed through the zeroth-order mean field theory through the hole vaccume cells as a measure of irregularity in the lattice. The (SS) theory has achieved quantitative success in statistical thermodynamics of polymers. The SS theory has been applied to extract the external degrees of freedom for polyethylene at different molecular weights. The external degree of freedom is very important parameter to specify the thermodynaics characteristic parameters. Applying SS hole theory, the thermodynamic properties of polyethylene was studied for temperature from 353K up to 473 and pressure from o.1Mpa up to150 Mpa. The minimum and maximum deviations


Introduction
Simha-Somcynsky (SS) Hole Theory: Simha-Somcynsky (SS) developed an equation of state (EOS) based on the lattice-hole model [1] introducing the temperature and volume dependent occupied site fraction, ( , ) y V T .The Simha and   Utraki [2] has differentiated from cell theories [3], the state an equation obtained is applied to low molecular weights [4], macromolecules and mixtures [5][6] and quantitative success has been achieved.Expressed with polymer filled and hole theory [7].The occupied site fraction, ( , ) y V T , and the complementary hole fraction ( , ) h V T , are given by the following equation: where N and h N are the number of molecules and holes respectively and s is the number of segments in a molecule.The configuration partition function [8], conf Z for N polymer molecules consisting of s segments and the proportion of occupied and empty cells, is given as in below.
Where   , g N y the combinatorial factor by Poser and Sanches [9]., that is the total number of distinguishable degenerate arrangements of the holes and molecules is expressed as The free volume f by Carri and Simha [10].expressing the amount of void in the lattice is given as in below; Here, the first part of the free volume an equation represent voids in filled cells, (free volume fraction) intracellular spaces, and the second part is the spaces in the lattice sites, that is represent empty cells.The total lattice energy 0 E for s-mer is given by; The total external degree of freedom   3c that determines the flexiblity for an ideal chain as given by; 33 cs  .... (7) By substituting all above equations in configuration partition function then substitutes in configurable free energy equation conf F we get configurable free energy of the system.
is obtained as.In the case of balance, the occupancy rate of the system does not change; In this case, the derivative of the system free energy should be equal to zero according to the occupancy rate: , , / 0 the derivative of free energy by volume gives the expression of pressure.

 
, , ; , Thus, the state equation to be used in polymers is obtained.
Explicit expression of reduced parameters that establish the link between theoretical values and experimental values is.
...( 13) The characteristic reducing parameters that establish the relationship between theoretical and experimental parameters are given below.
, , It is expressed.Then the charateristic parameters relalation equation as in below.
M O is the mass of the unit segment of the molecule and R gas constant.The Lennard-Jones-Devonshire, Prigogine cell model and fluid phase [12][13][14][15][16], and these cell models, a modified with the hexagonal package structure was developed by Dee and Walsh [17].In order to obtain the average interaction parameter and repulsive volume characteristics from these parameters.These parameters are separated into different molecular units that may be on the lattice.For this, the necessary decomposition equations are solved and interaction energy * and repulsive volume  * parameters are obtained.The average error and maximum error in the specific volume, are given as in below.max 100 max( ) The Results and Discussion  The values of charateristic parameters for different molecular weights of polyethylene are calculted intable2 to extract the chains of polyethylene is flexible or not as showed in graph6 .The figure 1 shows that the characteristic temperature of the polymer chain is highly dependent on the number of segments, in this study we have identified the characteristic temperatures increasing as the segment lentgh increases, the main reason belong to the number of contact pairs of segment increases and identified the segment length more 150, characteristic temperature remains constant.This graph shows that as the segment length increases, the degree of external freedom has been found to increase logarithmically.The most important thing that focuses us on this study is to determine the value of external degree of freedom and to learn about its behavior.The data of this graph have been extracted by using the SS theorey their themodynamic properties especially the hole fraction in the range of 353-473 K temperatures as in table3.Shows the relationship of different molecular weights to the temperature equivalent .There are two important determinations in this graph, the first is shown less hole fraction as the molecular weightincreases, the second identified the temperature increases leads to increase in hole fraction.The characteristic presure of low molecular weight in polyethylene (PE156) İ it is an expected value that will have high, characteristic pressure has decreased when molecular weight increased, but for macromolecularstructures (PE1100, PE2100) the characteristic pressure increase as the molecular weight increases, because cross-links are more in macromolecular structures.This graphic shows two independent equations.The first event characteristic volume and characteristic pressure decrease when increasing molecular weights (PE156, PE226 and PE338), the second event when molecular weghts increases (PE338, PE1100 and PE2100) leads to decreases in characteristic volume and relative increases in characteristic pressure, the reason for thisis that as the molecular weight increases the cross-links increases.

Fig. 6: Shows the relation between the characteristic parameters versus the inverse of molecular weights
The figure6 give for us the important information properties about chains behavior the characteristic parameters (P*, V* and T*) from the table2, with the inverse of the molecular weight.It gives us information about the structure of the chain molecule, it is found that the slope is proportion between externel degree of freedom to number of segment and found the value of this proportion around one and a half, this mean the polyethylene acts as if the average chain molecule acts as a rod and acts with less order of freedom than a monomer, obtaining this information is one of the aim of this study, because this information will help us synthesize new polymers.İn this study, Newton Raphson Method is used as one of the impotant tools, NRM methodis used in the solution of equations 10 and 12 nonlinear equations and an iteration is needed to start for most appropriate external degree of freedom solution by chracteristic parameter correlation (equa.15).İterasion is made up to the most appropriate value under the conditions.On the other hand the matrix is used to solve the state of equations for reduced temperature and reduced volume formed as a result of iteration.Since the matrix isnot a diagonal matrix, the Pseudo İnverse Matrix Technique (PİMT) were used so that the most appropriate parameter values were found forsolved equations.

Conclusions
This study has reached very good results that can be decided in accordance with its purpose, One of the most important results was determinedthat the external degree of freedom varies according to the weight of the polyethylene molecole.It has been lagarithmically elevated in macromolecular structure and the ratio of the external degree of freedom to the segment length indicates that there is relative flexibility in the chain structure of polyethlene for differet molecular weight s have been studied.Other parameters were to be compatible with the expermental especially the hole fraction and specific volume depends on temperature.As it is known, the polymertypes are few in nature, the industry world prefer polymer synthsis.The aim of this study is the ability of polyethylene to be synthesized.Finally, according to the results have been obtained by using the SS theory as in three tables, new polymers could be obtained for the polymerization process and brought to the world industry.

Fig. 1 :
Fig. 1: The values T* for different molecular weights of polyetylene as a function of segmental length.

Fig. 2 :
Fig. 2: showing the degree of external freedom with segment lentgh

Fig. 3 :
Fig. 3: The values hole and occupied site fractios for different molecular weights of polyetylene.

Fig. 4 :
Fig. 4: The values P* for different molecular weights of polyetylene as a function of segmental length.

Fig. 5 :
Fig. 5: Shows the relation between the characteristic V* versus the characteristic pressure P* .

Table 1 shows the characteristic parameters of polyethylene at different molecular weights Polymers
By applied the SS theory have been extracted some of important thermodynamic parameters as in table1, external degree of freedom, characteristic PVT, repulsive volume between segments and characteristic energy parameters, obtaining the values of a parameters is the purpose of this study.