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We compare the results of the approximations involved in the SAFT-D theory proposed by us [10,11] and that of the Ghonasgi and Chapman [7].On first line of approach, Dickman and Hall [12] and Honnell and Hall [13] derived the Generalized Flory (GF), the Generalized Flory-Huggins (GPH) and Generalized-Flory-dimer (GFD) equation of state for the hard sphere chain fluids, Thus two important theories are the Generalized-Flory-dimer theory (GFD) [13] and statistical associating fluid theory-dimer (SAFT-D) [7]. GFD theory is developed on the assumption that the probability of inserting a test chain into chain fluids can be related to the probabilities of inserting a monomer into a monomer fluid and dimer into dimer fluid. The GFD theory underestimates the probability of inserting a monomer into a chain fluid by ≤40% at liquid density. The errors associated with the probability of inserting the second bead next to the first bead in dimer fluid are much lower ≤10%. Inspite of these errors, GFD theory does well in predicting equation of state of chain fluids due to cancellation of errors.

Recently Kumar et al. [4] have tested the GFD theory by predicting chemical potential over a wide range of densities. The figure of merit that they have used in the test is the residual chemical potential μ^r minus zero density chemical potential {μ^r(ρ)=0}, is related to the insertion probability P_n(ƞ) through the expression μ^r-μ^r (ƞ=0)= -kT In P_n(ƞ). SAFT-D theory [7] also suffers from inconsistency with Monte Carlo (MC) values at packing fractions higher than 0.35 and with increasing chain length. The equation of state in SAFT-D theory requires only the contact values of the hard sphere correlation function and hard sphere site-site correlation function g(σ). Ghonasgi and Chapman [7] have employed the expression of g(σ) of the dimer with an assumption that the correlation function at contact of the dimer, tetramer, octamer and so on will remain the same. The theory can be applied readily to molecules having longer chain lengths. We have re examined the equation of state of the SAFT-D theory [10]. In this work we have employed the contact value of the correlation function g(σ) proposed by Chiew [14] who has derived analytical expression for the average correlation function at contact as function of chain length m and hard sphere volume function ƞ. We have employed the expression of g(σ) for m component separately without making any assumption. We have found that our equation of state predicts better results than those obtained Ghonasgi and Chapman [7]. However, the equation of state is not very sensitive to the errors associated with the approximations assumed in theory. Thus, it is important to evaluate the accuracy of the approximations. In the present work, we have examined the accuracy of our proposed equation of state by finding the residual chemical potential and compared with SAFT-D (now referred as SAFT-D1) proposed by Ghonasgi and Chapman. Thus the aim of the present work is to show that the residual chemical potential {μ^r- μ^r (ƞ=0)} associated with modified SAFT-D (now referred to SAFT-D2) equation of state predict better values than SAFT-D1 model and in good agreement with the results obtained through configurational bias Monte Carlo method (CBMC) by kumar et al. [4].

Recently, we have also proposed a new equation of state of chain molecules using statistical associating fluid theory (SAFT) [11]. The formalism derived is based on the assumption that chain is formed by the pair of trimers. Let us consider a long chain of m tangent hard spheres composed of pairs of trimers. The equation of state for trimer in SAFT can be written as

------------(Eq.- 1)

with
        -------------(Eq.-2)
             
Where Z^HSis the equation of state of hard spheres described by Carnahan and Starling [15] and ᶯ is the packing fraction or volume fraction and is defined as ᶯ=πρmσ³/6 pair of trimers can form a chain which has six segments i.e. a hexamer. The equation of state for a hexamer in SAFT-D can be written as
  -------(Eq-3)                             
Similar, the equation of state of 12-mers, 24-mers, and 48-mers, and so on can be written similarly.
A general expression can be written as
   -(Eq-4)      

this equation can be solved further by using the site-site correlation function at contact g_m(σ) proposed by Chiew [14]
 -----(Eq-5)                        
In this model, we have further applied the site-site correlation function at contact as a function of chain length instead of assuming an approximation that correlation function is independent of chain length. Thus, it is worthwhile to include this model for comparison with SAFT-D1 model.
Methods for the Determination of  Chemical Potential
From the knowledge of the equation of state (Z) of hard sphere chain fluid, one can obtain the residual Helmholtz free energy as
 ----(Eq-6)  
where ƞ is the packing fraction

From the residual Helmholtz free energy, one can obtain the residual Chemical Potential µ^res ,as follows
          ----- (Eq-7)                                    
 
In the present work, we have employed three model based on the statistical associating fluids theory known as SAFT-D1 model proposed by Ghonasgi and Chapman and two models SAFT-D2 and SAFT-T, proposed by us [10,11].

SAFT-D1 model

The equation of SAFT-D1 model [7], is written as
--(Eq.-8)
The expression for residual Helmholtz free energy and Chemical potential can be derived from equation (6) and (7) as
---(Eq.-9)
and
--(Eq.-10)
SAFT-D2 model

SAFT-D1 theory is reformulated to yield an improved equation of state for the hard sphere chain fluid, is given by [10].
 -(Eq.11)
Helmholtz free energy and residual chemical potential of equation [11] can be derived as 
--(12)

----(Eq.-13)

SAFT- T Model (Trimer model)
The equation of state for m-mers in trimer model can be written as [11]
--(Eq.-14)
The Helmholtz free energy and residual chemical potential  for trimer model can be derived as:
--(Eq.- 15)

and

---(Eq.-16)
The compressibility of hard spheres (Z^HS ) can be accurately determined from the Carnhan-Starling (CS) equation (15). and Bμ^res (η = 0) for equation (10), (13) and (16)

The accuracy of the SAFT-D theory is dependent on the accurate representation of the site-site correlation function for disphere and the group of segments. The figure of merit that we have used in the present work, is the numerical determination of residual chemical potential (μ^r- μ^r (η=0)) for hard chain fluids containing chains of length m-2, 4, 6, 8, 12, 16 and 24. The value of μ^r(η=0) is zero in all three equation (10), (13) and (16), we now consider how the equation (11) and equation (14) are different from equation (8) firstly, we examine the accuracy of dimer. Figures 1.shows the results of the residual chemical potential of dimer of the SAFT-D1 or SAFT-D2 (both are same) with that of Tildesley- street expression [16] that is given by

    

We find an excellent between two results which means discrepancy arises 1-2 for higher mers. Figure 2. shows the residual chemical potential for 4, 8 and 16 mers employing SAFT-D2 model. Comparison with SAFT-D1 model shows that the discrepancy between SAFT-D1 and SAFT-D2 models increases with the increase of chain length. The result of residual chemical potential obtained for SAFT-D2 model shows a good agreement with results obtained from configurational bias Monte Carlo method. The numerical values of CBMC method are determined using equation (32) and table 1 of the reference [4]. Similarly, figure 3.shows the residual chemical potential for 6, 12 and 24 mers employing SAFT-T model and Monte Carlo values. We find the discrepancy between SAFT D1 model and SAFT-T model increases with the increase of chain length. Monte Carlo values for 6, 12 and 24 mers are obtained by considering mean values of the respective mers given in the table 1 of the reference [4]. For example for 6-mers.we have considered the mean values of 4 and 8-mers.
Table - 1. Comparison of the chemical potential from SAFT-D1, SAFT- D2 and SAFT-T models with Monte-Carlo data [17].

 

Fig 1- Comparison between the Tildesley and Streett values (circles) and SAFT -D2 predictions (solid line) of the chemical potential (μ^r/KT) for dimer.

Fig-2 Comparison of the chemical potential (μ^r/KT) between SAFT-D1 and SAFT-D2 predictions with simulation values [10] for 4-mers, 8-mers and 16-mers. [ρ*=6η/π].

Fig-3 Comparison of the chemical potential (μ^r/KT)  between SAFT-D1 and SAFT-T predictions with simulation values [10] for 6-mers, 12-mers &24-mers. [ρ*=6η/π].

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