On the Calculation of
Surface Thickness from New Approaches
Mohsen Najafi¹ and Ali Maghari²
1-Department of Chemistry, Faculty of Science ,Tarbiat Modares University,Tehran,Iran [email protected]
2-Department of Physical Chemistry , University College of Science ,University of Tehran,Tehran,Iran [email protected]
Abstract
In this paper, we focus on interfacial thickness and develop two new methods to predict it for non-polar molecules like n-Alkanes. At first, by using of experimental surface tension data and SAFT-DFT approach, interfacial thickness is calculated. This is a typical inversion method. Later, by using a physical analysis of interfacial region, we obtain a new general equation for interfacial thickness.
Keywords: Surface Thickness; Density Functional Theory; SAFT; n-Alkanes
Introduction
Fluid interfaces are important in several phenomena concerning phases at equilibrium.
Because of this, there are many studies on interfacial properties especially surface thickness and surface tension experimentally and theoretically. But despite their importance, there is still a considerable lack of experimental data and accurate models for their prediction.
What is very important indeed is that, the surface region of fluids is an inhomogeneous one and there is the density gradient in it. Inhomogeneity in the surface region causes to exist density gradient d ρ(z) / d z ≠ 0 in the vertical direction, that is, ρ(z) is changed
with z . The density profile ρ(z) is expressed as the following equation [1-2]:
ρ(z,T ) = A(T ) − B(T ) tanh Y (z,T ) (1)
in which A = 1/ 2(ρ l + ρ v ) , B = 1/ 2(ρ l − ρ v ) and Y = 2(z − z0 ) / δ (T ) = 2∆z / δ (T ) .
The superscript v and l denote liquid and vapor saturated phases, respectively, δ stands
2
for the surface thickness and z0 is the position of the Gibbs dividing surface and is
1 l v
given by ρ(z0 ) = (ρ + ρ ) . The surface thickness δ plays an important role in the calculation of density profile. In this paper, we focus on interfacial thickness and develop two new methods to predict it.
Inversion Method
The Density Functional Theory (DFT) is the most popular approach to the statistical mechanics of inhomogeneous fluids. The density gradient and the heterogeneity are taken into account in this theory. In this theory, the surface tension γ can be expressed
as follows [3]
∞
γ =
∫
−∞ [ p − p[ρeq (z)]]dz (2)in which p = Z l ρ l kT = Z v ρ v kT
and p[ρeq (z)] = Zρeq (z)kT are normal and
tangential pressure respectively. p[ρeq (z)] is functional of density profile ρeq (z) that
is expressed as Eq. (1) and Z is the compressibility factor which may be obtained from an equation of state. It is clear that the surface thickness δ plays an important role in the calculation of density profile and then the surface tension. In fact, in this theory, the surface tension γ is a function of surface thickness, i.e. γ = ƒ (δ ) . In the other word, having the surface thickness and using Eqs. (1) and (2), the surface tension of fluids can be calculated theoretically. Inversely, if it is considered the thickness as a adjustable parameter and the surface tension is given experimentally, the surface thickness of fluids can be calculated. In fact, the surface thickness is obtained as the equality of theoretical with experimental surface tension i.e. the surface thickness is changed until γ = γ exp . This is a typical inversion method. In this method, we use the original
SAFT EOS as a compressibility factor Z for n-alkanes as a follows [4]
Z = 1 + Z hs + Z dis + Z chain (3)
Common Intersection Method
The density profile of methane in different temperatures is shown in Fig.(1) using original SAFT-DFT method. It is clear that all curves pass the common point and this
c c
r r
matter has been shown in the other literatures too. In other word, there is a very small region in the interfacial region that its density does not change with changing temperature.
The density of the region is the critical density and has mathematically
(
dρ (z) / dT)
z or ρ = 0Using Eq.(1) and the above condition, we obtain
d ln δ
=
1 dBr
tanh 2Y
− dAr (4)
dT W dT c dT
r r r r
in which Tr = T / Tc ,
A = A / ρ c , B = B / ρ c , Yc is the value of Y at the intersection
point and
W = 2B Y
[
1 − tanh 2 2Y]
r r c c
Finally, the interfacial thickness can be numerically calculated by integrating the following expression:
ln δ (Tr ,2 )
=
Tr , 2 Λ(T ) dT (5)
δ (Tr ,1 ) Tr ,1 r r
in which Λ(Tr ) = d ln δ (Tr )
dTr . This is the Common Intersection Method (CIM). Eqs (4)
and (5) are the key result that allows obtaining the surface thickness as a function of reduced temperature from the common intersection method (CIM).
Results
The widths of vapor-liquid interface of n-alkanes as a function of temperature from inversion method and the curve slopes of the surface thickness of n-alkanes versus temperature from a theoretical approach (Eq.(4)) are shown in Figs.(2) and (3)
∫
respectively. It is clear that firstly, the surface thickness and slopes are increased with increasing temperature and goes to the infinity when the temperature is approached to the critical value. Secondly, the interfacial width and slopes at a given temperature are found to decrease with increasing the chain length. Thirdly, the slopes of vapor-liquid interface of molecules goes to constant value when temperature decreases.
Once the surface thickness of a fluid is known at temperature Tr,1 , it can be found at other temperatures Tr,2 by integration of Eq. (5). For this purpose, we selected the triple
point as the lower interval of the integral, that is Tr,1 = Tr,tp and obtained the vapor-liquid surface thickness of chain fluids at the triple point from the surface tension data by the inversion method.
The surface thicknesses of n-alkanes as a function of temperature, obtained from the Eq. (5) with using its data point at triple point temperature are shown in Fig. 4. It can be seen that as the temperature approaches the critical temperature, the interfacial thickness diverges. Moreover, it can be shown that, if one uses surface thickness δ as a function of reduced temperature Tr , then, to a pretty good approximation, all chain fluids show a nearly linear behavior of ln δ (Tr ) versus ln(1 − Tr ) or may be scaled as a power law
upon approaching the critical point with critical exponent. The curve behavior of surface thickness for n-alkanes obtained from two mentioned above methods are rather the same and have a good agreement with the other results. The experimental data has taken from NIST Chemistry WebBook [5]
30000
25000
20000
15000
10000
5000
0
-20 -15 -10 -5 0 5 10 15 20 z l o
Figure 1. Density profile of methane
8
7 Inversion
6
5
methane ethane
4 propane
butane
3 pentane
hexane heptane
2 octane
nonane decane 1
0 100 200 300 400 500 600
T / K
Figure 2. The interfacial thickness of n-Alkanes from inversion method
95 130 170 185 Tc
ðlo Density profile (mol m-3)
100 90 80 70 60 50
methane ethane propane butane pentane hexane heptane octane nonane decane
*dln6/dT
40 30 20 10
0
0 100 200 300 400 500 600 700
T (K)
Figure 3. The curve slopes of the surface thickness of n-Alkanes versus temperature from CIM
50
45
40
35
30
25
m e t h a n e e t h a n e p r o p a n e b u t a n e
15 pentane
hexane heptane
10 octane
nonane decane 5
0 100 200 300 400 500 600 700
T (K)
Figure 4. The interfacial thickness of n-Alkanes versus reduced temperature from CIM
References
1. G. A. Chapela, G. Saville, S. M. Thompson and J. S. Rowlinson, J. Chem. Soc. Faraday Trans. II, 8(1977)133
2. M.M. Telo da Gama and R. Evans, Mol. Phys., 38(1979)367
3. J.F. Lu, D. Fu, J.C. Liu, Y.G. Li, Fluid Phase Equilibria, 194-197(2002)755 4. S. H. Huang and M. Radosz, Ind. Eng. Chem. Res., 29(1990)2284
5. NIST Chemistry Webbook, NIST Standard Reference Database Number 69, National .Institute of Standards and Technology, http://www.webbook.nist.gov.
ð(Å)
20
