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Effect of potential attraction term on surface tension of ionic liquids
Accepted Manuscript Effect of potential attraction term on surface tension of ionic liquids N. Vaziri, R. Khordad, G. Rezaei PII:
S0921-4526(17)31063-3
DOI:
10.1016/j.physb.2017.12.068
Reference:
PHYSB 310645
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 29 July 2017 Revised Date:
27 December 2017
Accepted Date: 28 December 2017
Please cite this article as: N. Vaziri, R. Khordad, G. Rezaei, Effect of potential attraction term on surface tension of ionic liquids, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/ j.physb.2017.12.068. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Effect of potential attraction term on surface tension of ionic liquids N. Vaziri, R. Khordad* and G. Rezaei
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Department of Physics, College of Sciences, Yasouj University, Yasouj, Iran
Abstract
In this work, we have studied the effect of attraction term of molecular potential on surface
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tension of ionic liquids (ILs). For this purpose, we have introduced two different potential models to obtain analytical expressions for the surface tension of ILs. The introduced
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potential models have different attraction terms. The obtained surface tensions in this work have been compared with other theoretical methods and also experimental data. Using the calculated surface tension, the sound velocity is also estimated. We have studied the structural effects on the surface tensions of imidazolium-based ionic liquids. It is found that the cation alkyl chain length and the anion size play important roles to the surface tension of the selected ionic liquids. The calculated surface tensions show a good harmony with
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experimental data. It is clear that the attraction term of molecular potential has an important role on surface tension and sound velocity of our system.
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Keywords: Surface tension, Ionic liquids, Potential model
*Corresponding author, E-mail: [email protected]
1
ACCEPTED MANUSCRIPT 1. Introduction Accurate knowledge of the thermodynamic properties of ionic liquids is one of main challenges in the scientific purposes like electrochemical stability, synthetic, separation processes, and industrial applications [1]. Among the thermodynamic properties, the surface tension is an important and useful quantity in heat transfer, mass transfer, flow, secondary
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and tertiary crude oil recovery, and gas condensate recovery [2, 3]. Surface tension is defined as the difference between the energy in the bulk and surface per unit area of interface. This parameter is a basic physical property that has a key role in development, design and simulation of many chemical engineering processes such as
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biochemical engineering, environmental protection, gas absorption, distillation, extraction and residual liquid saturation [4-7].
Hitherto, several works have been performed theoretically and experimentally to predict the
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surface tension of simple fluids and fluid mixtures [8-11]. The performed theoretical works to estimate the surface tension is based on statistical thermodynamics. Many authors have used different approaches including the corresponding-states theory [12], the parachor method and its derivatives [8, 13-15], the gradient theory of inhomogeneous fluids [16, 17], Monte Carlo simulation [18], density functional theory [19-24], molecular dynamic simulation [25],
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perturbation theory [26, 27], Gibbs method [28], and quantitative structure-property relationship (QSPR) [29-31].
Ionic liquids (ILs) are organic salts comprised of an anion and an organic cation. They are liquids at ambient temperature and have an appreciable liquid range. Ionic liquids were
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discovered in 1914 but their systematic study began with the present century [32]. Very interestingly, the physicochemical properties of ILs can be finely tuned by slight structural
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changes of the corresponding cations and anions. In recent years, an increasing interest has been devoted on the topic of ionic liquids due to their wide applications in the chemical industry such as such as homogeneous and biphasic transfer catalysts, multiphase homogeneous catalytic reaction and solvents in organic synthesis [33-41]. In order to use the ionic liquids in potential applications, we should know their physical properties such as viscosity, density and surface tension. As we know there are few accurate theoretical results for surface tension of ionic liquids compared with their density data. Hitherto, several theoretical methods have been employed to estimate the surface tension of ionic liquids. Therefore, the prediction of theoretical surface tension of ionic liquids accurately is useful to establish the surface tension data in the design specifications. Examples of the methods are the group contribution method [42], the corresponding-states 2
ACCEPTED MANUSCRIPT group-contribution method [43], and corresponding state theory [44]. To obtain more information about theoretical results of the ionic liquids, the reader can refer to [45-51]. In the previous works, authors have considered the columbic interactions, the hard-sphere repulsions and Lennard-Jones (LJ) dispersion forces in calculating surface tension of ILs. In this work, we have intended to answer the following question. What is the effect of potential
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attraction term of the LJ dispersion forces on the surface tension of ILs. For this purpose, we have selected two potential models which have the same behaviors as the LJ potential but with different attraction terms. It is to be noted that we have also used these attraction terms in nanostructures [52]. The new idea of the present work is considering contributions arising
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from the attraction term of the potential models on the surface tension of ILs. Since the surface tension of the ionic liquids is an important property, we intend to study this property theoretically. We have used two new potential models and molecular thermodynamic model
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to prediction of the surface tensions of 13 ionic liquids including 12 imidazolium based and 1
2. Theory and model
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phosphonium-based ones.
2.1. A new potential model
It should note that from Fowler model and KB (Krikwood-Buff) model [53, 54], the surface
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tension of multicomponent systems depends on intermolecular pair potential and pair correlation function. The accurate theoretically prediction of the surface tension will help to
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improve engineering applications. To this end, knowledge of accurate pair potential of a system is an important problem. We have tried to combine the previous potential models for obtaining better results. For this goal, we have selected two potentials which have the repulsion and attraction terms and the potentials have same behaviors as the LJ potential. We have considered the repulsion term of the LJ potential with two different attraction terms. The two potential models to show the interaction between two molecules with the separation of :
attraction terms of the potentials are as attraction well. In the following, we have introduced
i)
Potential model (1)
3
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= β β β , 1 ii)
Potential model (2)
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= β β β , 2
where and are the energy scale and the length scale, respectively. In above equation,
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and are constants equal to 2.3 and 2.0, respectively (see Fig. 1).
2.1. Surface tension
We have used the perturbation theory [25] to obtain the surface tension of ionic liquids. According to the theory, one can write the following expression
!
!
+ #$% + &'$( , 3
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where
=
and &'$( are the contributions from hard-sphere repulsion and the electrostatic
interactions to surface tension, respectively. The second term in Eq. (3) relates to the contribution of the new potentials to the surface tension. In the previous works [45], authors
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have considered *+ instead of the second term of Eq. (3). They have assumed *+ contributes from the Lennard-Jones (LJ) potential. !
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In order to calculate considered by
Here 0,- =
56 758
, the hard-sphere pair potential for fluid mixtures has been
,- =
β β€ 0,. 4 0 > 0,-
is the distance between the center of mass of two hard spheres and 0, , 0- are
the sphere diameters. The symbols 9 and : are used for each ions.
The surface tension can be calculated in terms of radial distribution function. It is expressed for non-simple molecules mixtures at low vapour pressures as [57, 58]
4
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0,- ; = = >, >- ? @ A B,- C 0 , 5 8 0 F
!
,,-G
E
where >, and >- are the molecular number density of components 9 and :, respectively, is
the distance between the molecules, B,- is the radial distribution function and ,- is the relation [18, 52]
0,-! 0
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pair potential function. With respect to the potential form of Eq. (4), we can use the following = βIJ KLM β 0,- N, 6
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where L is the Dirac delta function, IJ is the Boltzmann's constant and K is the system
temperature. Inserting Eq. (6) into Eq. (5) and taking into account integration of delta function and >, >- = , - > , we obtain F
D
,G -G
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,- !
F
; = = = >, >- ? C PβIJ KLM β 0,- NQB,- 0 = 8 E
F
;IJ K C β > = , - 0,B,-! 0 7 . 7 8 ,,-G
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Here, B,-! 0 7 is the pair radial distribution function (RDF) of hard-sphere mixtures at contact distance (0 7 ) and it is given by [55, 56]
T,T,1 + + , 8 1 β S 21 β S 21 β SU
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B,-! 0 7 =
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where S is the packing fraction and it is expressed by F
> S = = , V, 4 ,G
Where >, , and V, are the molecular number density, the mole fraction and the van der
Waals co-volume, respectively. The T,- is expressed as > V, VT,- = @ A 4 V,-
/U F
= , V, ,GE
/U
9
5
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In the case of hard-sphere mixture, the V,- parameter is given by
1 U V,- = PV, K + V- KQ 10 8
in terms of the reduced temperature KY = IJ K/. This parameter is given by [56] VK = β0.01054Z1 β 0.7613KY expβ0.7613KY ^ KYC
a exp `β
1.3227 KYC
ab 11
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0.3306
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+ 2.9387 _1 β `1 +
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This parameter represents the van der Waals co-volume which has been previously adopted
The last term of Eq. (3) corresponds to the contribution from the ionic electrostatic interaction. To obtain this term from Eq. (5), the radial distribution function requires. Blum and Hoye [59] have solved the mean spherical approximation and obtained the pair correlation function. They have used the following electrostatic potential
c, c, 12 E
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,- =
where c, and c- are the valences of components. E and are the permittivity of free space
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and the unit electric charge, respectively. Blum and Hoye [59] have obtained the following radial distribution function
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d c, c- e fY B,- = β , 13 E
where d = 1/IJ K and Ξ is a single shielding parameter. Wu et al. [60, 61] have employed
Eq. (13) and obtained &'$( as &'$(
,-
d;> c, c= @ A 14 32Ξ E
6
ACCEPTED MANUSCRIPT In above equations, Ξ is single shielding parameter which has the dimensions of an inverse length. This parameter reflects the range of long-ranged electrostatic forces operating between ions. To other word, this parameter reflects the mean distance among the ion pairs of electrolyte fluids. This parameter depends on temperature and it can be obtained by simple /
#
2; Ξ= h= >, c, i IJ K ,G
15
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iteration using the following iteration value
In order to obtain the second term of surface tension using Eq. (5), we have employed the
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radial distribution function which was previously developed by Xu and Hu [62] as U B,- = jM β Μ ,- N + Μ ,- β 0,Ξ΄ β Μ ,- /3,-β 16
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m
where j β E is the Heaviside step function and it is given by j β E = 1 > E < E n j β E = 0 j β E = 1/2 = E
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Also, L β E is Dirac delta-like function and it is given by
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L β E = 0 β E L β E = β = E
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In Eq. (16), ,-β = 1.1500,- and Μ ,- = 1.5750,- are the position of molecules and the outer
radius of the first coordination shell [63, 64], respectively. The two parameters ,-β and Μ ,- are
obtained using the molecular simulation [65].
Inserting Eqs. (1) and (16) into Eq. (5), after some difficult integration, we have obtained the following analytical expression for #$% for the potential model (1) as follows:
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#
%$s t ,-t ,U Cr = 3; = >, >- ,- ,- r@ A β MΜ U β 0,N βuu ,16Μ ,16 ,,,r q
4.6,-β ,-β ,-β U +` wxyβ @ @ A β A β @ @ A β Aa MΜ ,- U β 0,N , ,72 ,-v
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m
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} U {1 Β Μ ,1 Β + YΜ 68 YΜ 68 Β Β`40000 @ β A |12 ,CΒ eΒ Β eΒ { @279841 Β68 + 559682 Β68 + 279841A z ~ Β
68 Μ ,Μ ,CΒ eΒ Β68 + 276000 @ β A + 634800 @ β Aa , ,-
YΜ 68 eΒ Β68
+ 30000
Β
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+ 2 @15000
CΒ
YΜ
YΜ 68 eΒ Β68
+ 15000A Β9U
Β
YΜ 68 eΒ Β68
68 Μ ,Μ ,CΒ eΒ β 2 Β30000 @ β A + @30000 @ β A + 69000A Β68 , ,-
68 68 Μ , Β eΒ Β eΒ β A + 138000A Β68 + 69000Β Β9 Β68 ,-
YΜ
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+ @60000 @
YΜ
Μ ,Μ ,β 2 Β30000 @ β A + 138000 @ β A , ,-
YΜ
68 Μ ,Μ ,CΒ eΒ Β68 + h30000 @ β A + 138000 @ β A + 158700i , ,-
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YΜ
68 Μ ,Μ , Β eΒ Β68 + h60000 @ β A + 276000 @ β A + 314700i , ,-
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+ 158700Β Βy @Β
Β
YΜ 68 eΒ Β68
YΜ
+ 1ΒA
Μ ,Μ ,Μ ,β Β20000 @ β A + 144000 @ β A + 358800 @ β A , , ,C
U
8
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ΒΒ YΜ 68 {Β Μ , Β eΒ + 338560 @ β A + 1046362Β Β68 β 486680Β Β 17 ,ΒΒ { ΒΒ D
Β9Β c = =
ΒG
cΒ c cU = c + + + β― 18 IΒ 2Β 3Β
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Where
,
-
+
β
Where erf =
Mβ8Μ ,- β C
,-t
16Μ ,-
4Μ ,- ,-
9 ,-v
,-t
6,-
βuu
U MΜ ,- U β 0,N+
β 4Μ ,- ,-U β 5 ,-C N
192 ,-C
U ,-β MΜ ,- U β 0,N m
tβ
Μ ,- β ,11β; erf h2 @ Ai 192 ,-
exp @β
4Μ ,- 8Μ ,+ β 4A , ,-
,-β ,-β @1 β A exp β Β2 @ A β 2Β Β 19 , ,
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,- = 3; = = >, >- ,- ,-C Β
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Analytical expression for #$% for the potential model (2) is obtained as
Β eΒ 0w is the error function. βΒ E
Β
m
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It is to be noted that there are several parameters in Eqs. (17) and (19) such as ,-β , Μ ,- , >, , >- , ,- , ,- . Among these parameters, ,-β and Μ ,- depend on 0,- which is an important parameter.
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When 0,- is large, the pair correlation function approaches zero. Therefore, in our expansion,
the higher orders of 0,- are small and we have neglected the orders. The most important order
of 0,- is the first order. This parameter can also depend on temperature. In Eqs. (17) and (19), we have kept the terms which have same orders. The dependence of 0, is given by 1 + 0.2977IJ K/, 0, = 20 , 1 + 0.3316IJ K/, + 0.001047IJ K/,
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ACCEPTED MANUSCRIPT Also, other parameters can be depended on temperature. But, we have considered these parameters as constant values. In order to obtain Eq. (17), we have expanded the function Ββ to obtain the surface tension.
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3. Results and discussion In this paper, we have carried out the numerical results for the surface tension of ionic liquids using the perturbation method and introduction of two new potential models. The used parameters in this work have been presented in tables 1 and 2. These parameters have been
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taken from Ref. [66].
Fig. 1 shows the variation of new potential models (1) and (2) as a function of molecular distances. To compare, we have also plotted the (12-6) Lennard-Jones potential model. It is
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seen from the figure that two potentials have the same behaviors as the LJ potential.
The accuracy of our model has been further assessed by comparing our surface tension calculations with those obtained from the method of Gardas and Coutinho [67] and the corresponding-states group-contribution (CS-GC) method of Wu et al [43]. Our results were
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summarized in table 3 as the AAD (in %) of the calculated surface tensions of the some selected ILs, using the proposed molecular model (this work), the QSPR strategy of G & C and those obtained from the CSβGC method of Wu et al [43], from those measured Almeida et al [68]; Carvalho et al [69], Freire et al [70], Souckova et al [71], Kilaru et al [72], Pereiro
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et al [73]. It is observed from table 3 that our new model outperforms the QSPR strategy and CSβGC method. From 69 examined data points for some studied ILs, the AAD (in %) of the
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correlated surface tensions from the proposed model was found to be 2.21% (potential model 1) and 2.33% (potential model 2) which is lower than those obtained from the QSPR strategy and CS-GC method with AADβs equal to 10.31% and 4.85%, respectively. It should be added that the uncertainty of the experimental surface tension data used for comparisons was of the order of Β±0.0012 N m-1.
We have calculated the surface tension (STcal) of the ionic liquids using the perturbation
method Eq. (5) using the new potential models Eqs. (1) and (2). In order to show graphically the accuracy of our method, we have plotted the theoretical surface tension of the ionic liquids versus the experimental surface tension (STexp) [68, 74-77] in Figs. 2 and 3 for the potential models (1) and (2), respectively. The results show the good harmony between the experimental data and the correlated surface tension obtained from our proposed models. 10
ACCEPTED MANUSCRIPT From the figures, it is seen that the potential model 2 gives better results than the potential model 1. This means that the attraction term of the potential model can be influenced on the surface tension of the selected ILs. In Figs. 4 and 5, the surface tension (ST) of imidazolium-based ionic liquids has been plotted as a function of temperature for the two potential models (1) and (2). Our results in the two
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figures have been compared with experimental data [72, 78-81]. Two figures have been plotted in order to show the influence of structural effects on the surface tensions of the selected ionic liquids.
The effect of alky chain-length of the cation on the surface tension of the selected ionic
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liquids has been shown in Figs. 4(a) and 5(a). Also, the influence of anion size on the surface tension has been investigated in Figs. 4(b) and 5(b). It is seen from the Figs. 4(a) and 5(a) that the surface tension decreases with increasing the cation alkyl chain length of the selected ILs
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for the two potential models. It should be noted that the enhancement of cation alkyl chain length causes an increase of the van der Waals forces and aliphatic character ionic liquids. The factors lead to the dispersion of the ion charge and thereby the reduction of electrostatic forces.
It is clear from Figs. 4(b) and 5(b) that the surface tension is reduced with increasing the
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anion size of the selected ILs. Our results are in agreement with Ref. [82]. With increasing the anion size, the hydrogen bonding strength and the electrostatic forces reduce and the delocalization charge enhances. We can see an important behavior in Figs. 4(b) and 5(b). From the figures, it is clear that the surface tension obtained from the potential model (1) has
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better agreement with experimental data. It means that the surface tension depends on potential attraction term.
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One can theoretically calculate the sound velocity of the ionic liquids with knowledge of the surface tension. The relationship between the sound velocity and surface tension is as [83] Β£ Β‘= , 21 6.33 Γ 10e E >
where > and are the liquid density and the surface tension, respectively. We have selected
Β€=0.6714 from Ref. [67]. This parameter has been obtained by the correlation of experimental sound velocities of imidazolium-based ionic liquids and the least-squares
method [67]. Using the calculated surface tension in this work for two potential models and by using Eq. (21), we can compute the sound velocity. Figs. 6 and 7 show the relative 11
ACCEPTED MANUSCRIPT deviation of the calculated sound velocity Eq. (21) of imidazolium-based ionic liquids from the experimental data [84-90] over the temperature range within 278 K to 343 K. Our theoretical results in Figs. 6(a)-6(c) and 7(a)-7(c) have been compared with different references. The comparison has been made for two potential models (1) and (2) in Figs. 6 and 7, respectively. As shown in Figs. 6(a) and 7(a), the plots represent the relative deviations (in
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%) of our predictions from Refs. [84-86] measurements for which their uncertainties were of the order of 1.2 m/s, 2.0 m/s and 1.6 m/s for the potential model (1). Also, the uncertainties for the potential model (2) are of the order of 1.5 m/s, 2.3 m/s and 1.9. The figures 6(c) and 7(c) represent the relative deviations (in %) of the estimated sound velocities of
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[C2mim][NTf2] and [C4mim][BF4] from Refs. [89] and [90] measurements for which their uncertainties were equal to 1.2 m/s and 2.2 m/s, respectively, for the potential model (1). The uncertainties for the potential model (2) are of the order of 1.3 m/s and 2.5, respectively, for
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the potential model (2). It is observed from the figures that the deviation for the sound velocity from the potential model (2) is larger than that the potential model (1). This means
4. Conclusions
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that the attraction term of the potential model can be influenced on the sound velocity.
It is fully known that the tension at the surface of a liquid is one of the most important demonstrations of the intermolecular forces. The intermolecular forces composed of both
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repulsion and attraction terms. We have answered the following question in this work. What is the effect of potential attraction term of dispersion forces on the surface tension of ILs. We
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have calculated the surface tension of selected ILs using two potential models which have different attraction terms. The main advantages of the present model in comparison with the previous works are: (i) the statistical thermodynamic approaches expression (5) proposed in this work provide us good approach to predict the surface tension of ILs. (ii) Using the expression (5), we could show the effect of attraction term dispersion forces on the surface tension of ILs. (iii) The proposed combination model can perform in a predictive manner with no adjustable parameters. (iv) With considering temperature-dependent of the used parameters [Eqs. (11), (15) and (20)], we could obtain the acceptable results for the surface tension of ILs. (v) The surface tension of the selected ILs can be predicted using only four parameters , , Ξ and 0. (vi) In comparison with available data, we can say that the better
results for the surface tension of the selected ILs can be obtained by finding the accurate 12
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dispersion forces. (vii) The constant parameters ( and ) in new potential models can be
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influenced on our results.
Table I. The cation and anion parameters [66].
[C2mim]+
Ο/IJ Β₯ 869.78
Ξ΅ Β¦E
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Cation
Anion [BF4]-
4.981
Ξ΅ Β¦E
318.61
4.9704
1210.8
6.596
330.38
5.633
[NTf2]
[C6mim]+
252.18
6.313
[Triflate]-
320.61
6.441
[C8mim]+
268.38
6.720
[PF6]-
349.27
5.505
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+
[C4mim]
-
Ο/IJ Β₯
Table II. The ionic liquids parameters [66].
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Ξ΅ Β¦E [C2mim][NTf2] [C4mim][NTf2] [C6mim][NTf2] [C2mim][BF4] [C4mim][BF4] [C8mim][BF4] [C4mim][PF6] [C6mim][PF6] [C8mim][PF6] [C4mim][Triflate]
Ο/IJ Β₯ 1040.29 770.59 731.49 594.19 324.49 293.49 339.82 300.72 308.82 325.49
13
Ionic liquids 5.732 6.095 6.453 4.976 5.291 5.779 5.569 5.895 6.082 6.023
ACCEPTED MANUSCRIPT Table 3. The AAD (in %) of the calculated surface tensions of the some selected ILs, using the proposed molecular model (this work), the QSPR strategy of Gardas and Coutinho [67] and those obtained from the CS-GC method of Wu et al [43], all were compared with the measurements [68-73].
NP
[C4mim][NTf2]
293-343
[C6mim][NTf2]
This
This
worka
workb
06
1.02
1.03
293-343
06
0.24
0.26
[C2mim][NTf2]
293-343
06
0.45
0.49
[C2mim][Triflate]
293-343
06
0.60
0.63
[C2mim][BF4]
288-355.91
14
5.16
[C4mim][BF4]
298-343
06
[C4mim][PF6]
288-313
[C8mim][BF4]
QSPR
CS-GC
Data
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βK (K)
Ionic Liquid
source
3.23
[69]
6.59
5.20
[69]
5.33
2.00
[69]
10.46
1.24
[70]
5.34
19.0
7.54
[71]
2.31
2.58
9.48
--
[68]
06
2.16
2.43
2.45
--
[73]
288-343
07
4.01
4.14
15.96
6.14
[70]
[C6mim][PF6]
293-353
07
0.29
0.36
6.38
--
[70]
[C8mim][PF6]
293-313
05
1.98
2.03
11.93
--
[73]
69
2.21
2.33
10.31
4.85
----
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Overall
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3.98
14
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0
2
4
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0
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u(x)
(12-6) LJ potential Potential model (1) Potential model (2)
x
Fig. 1.
40
EP
45
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STcalc(mN.m-1 )
50
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55
35
30
25 25
30
35
40
45 -
STexp(mN.m 1 )
Fig. 2. 15
50
55
ACCEPTED MANUSCRIPT 55
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45
40
35
SC
STcalc(mN.m-1 )
50
25 25
30
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30
35
40
45
50
55
-
STexp(mN.m 1 )
50
ST(mN.m -1)
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45
(a)
EP
55
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Fig. 3
40
35
30
25 260
[C2mim][BF4] [C2mim][BF4] [C4mim][BF4] [C4mim][BF4] [C6mim][BF4] [C6mim][BF4] [C8mim][BF4] [C8mim][BF4]
280
300
320
T/K
16
340
360
380
ACCEPTED MANUSCRIPT (b)
48 46 44
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40 38
34 32 30 28 260
[C4mim][BF4] [C4mim][BF4] [C4mim][PF6] [C4mim][PF6] [C6mim][BF4] [C6mim][BF4] [C8mim][BF4] [C8mim][BF4]
280
SC
36
300
M AN U
ST(mN.m -1)
42
320
T/K
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EP
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Fig. 4.
17
340
360
380
ACCEPTED MANUSCRIPT (a)
50
[C2mim][BF4] [C2mim][BF4] [C4mim][BF4] [C4mim][BF4] [C6mim][BF4] [C6mim][BF4] [C8mim][BF4] [C8mim][BF4]
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40
35
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ST(mN.m -1)
45
25 260
280
300
M AN U
30
320
340
360
380
T/K (b)
44
40
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ST(mN.m -1)
38 36
[C4mim][BF4] [C4mim][BF4] [C4mim][PF6] [C4mim][PF6] [C6mim][BF4] [C6mim][BF4] [C6mim][PF6] [C6mim][PF6] [C8mim][BF4] [C8mim][BF4]
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Fig. 7
Figures Captions:
Fig. 1. Variation of potential function = / versus intermolecular distance
= / . Two new potentials have been compared with (12, 6) Lennard-Jones potential.
Fig. 2. The calculated surface tension (STcal) as a function of experimental surface tension (STexp) [68, 74-77] for the potential model (1). Fig. 3. The same as Fig. 2, but for the potential model (2). 22
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Fig. 4. The surface tension of imidazolium-based ionic liquids as a function of temperature. The solid lines and markers correspond to theoretical and experimental results, respectively
Fig. 5. The same as Fig. 4, but for the potential model (2).
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[72, 78-81].
Fig. 6. The relative deviation of the calculated sound velocity Eq. (15). Our results have been
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compared with experimental data. Three figures correspond to (a) [C4mim][Triflate] [84-86], (b) [C2mim][Triflate] [87, 88], and (c) [C2mim][NTf2]/[C4mim][BF4] [89, 90].
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Fig. 7. The same as Fig. 6, but for the potential model (2).
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References
[1] H. T. Davis, J. Stat. Phys. 24, 43 (1981).
[2] J. J. Bikerman, Physical Surfaces, Academic Press, New York (1970). [3] L. I. Rolo, A. I. Caco, A. J. Queimada, I. M. Marrucho, and J.A.P. Coutinho, J. Chem.
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Eng. Data 47, 1442 (2002).
[4] B. E. Poling, J. M. Prausnitz, J. P. O'Connell, The Properties of Gases and Liquids fifth
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ed., McGraw-Hill, New York (2001). [5] A. W. Neumann, R. David, Y. Zuo, Applied Surface Thermodynamics, second ed., Taylor and Francis Group, CRC Press, Boca Raton (2011). [6] H. Lin and Y. Y. Duan, Int. J. Thermophys. 24, 1495 (2003). [7] G. Zhao, S. Bi, J. Wu, and Z. Liu, J. Chem. Eng. Data 55, 3077 (2010). [8] D. B. MacLeod, Trans. Faraday Soc. 19, 38 (1923). [9] B. S. Carey, L. E. Scriven, and H. T. Davis, AIChE J. 26, 705 (1980). [10] L. A. Girifalco and R. J. Good, J. Phys. Chem. 61, 904 (1957). [11] F. M. Fowkes, J. Phys. Chem. 66, 1863 (1962). [12] E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945). [13] S. Sugden, J. Chem. Soc. Trans. 125, 32 (1924). 23
ACCEPTED MANUSCRIPT [14] S. Sugden, J. Chem. Soc. Trans. 125, 1167 (1924). [15] C. F. Weinaug and D. L. Katz, Ind. Eng. Chem. 35, 239 (1943). [16] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). [17] J. S. Rowlinson, J. Stat. Phys. 20, 197 (1979). [18] P. Attard and G.A. Moule, Mol. Phys. 42, 943 (1993).
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[19] M. Mecke, J. Winkelmann, and J. Fischer, J. Chem. Phys. 107, 9264 (1997). [20] D. Fu, J. F. Lu, J. C. Liu, and Y. G. Li, Chem. Eng. Sci. 56, 6989 (2001). [21] R. Evans, Adv. Phys. 28, 143 (1979).
[22] P. I. Teixeira and M. M. Telo da Gama, J. Phys. Condens. Matter 3, 111 (1991).
[24] J. Winkelmann, Phys. Chem. 98, 1308 (1994).
SC
[23] J. Winkelmann, U. Brodrecht, and I. Kreft, Phys. Chem. 98, 912 (1994).
[25] J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 (1967).
M AN U
[26] J. M. Haile and C. G. Gray, J. Chem. Phys. 64, 2569 (1976). [27] S. Toxvaerd, J. Chem. Phys. 57, 4092 (1972).
[28] Z. B. Li, Y. G. Li, and J. F. Lu, Ind. Eng. Chem. Res. 38, 1133 (1999). [29] A. Khajeh and H. Modarress, J. Hazard. Mater. 179, 715 (2010). [30] A. Khajeh and H. Modarress, Expet. Syst. Appl. 37, 3070 (2010).
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[31] A. Khajeh and H. Modarress, J. Chemometr 25, 333 (2011).
[32] P. Wasserscheid and T. Welton (Eds.), Ionic Liquids in Synthesis, Wiley-Verlag, Weinheim (2003).
[33] W. Liu, L. Cheng, Y. Zhang, H. Wang, and M. Yu, J. Mol. Liq. 140, 38 (2008).
EP
[34] W. Liu, T. Zhao, Y. Zhang, H.Wang, and M. Yu, J. Sol. Chem. 35, 1337 (2006). [35] Q. G. Zhang, F. Xue, J. Tong,W. Guan, and B. Wang, J. Sol. Chem. 35, 297 (2006).
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[36] L. A. S. Ries, F. A. do Amaral, K. Matos, E. M. A.Martini,M. O. de Souza, and R. F. de Souza, Polyhedron 27, 3287 (2008). [37] R. M. Lau, F. Rantwijk, K. R. Seddon, and R. A. Sheldon, Org. Lett. 2, 4189 (2000). [38] M. Koel, Crit. Rev. Anal. Chem. 35, 177 (2005). [39] M. Rantwijk and R.A. Sheldon, Chem. Rev. 107, 2757 (2005). [40] M. Haumann and A. Riisager, Chem. Rev. 108, 1474 (2008). [41] M. A. P.Martins, C. P. Frizzo, D. N.Moreira, N. Zantta, and H. G. Bonacorso, Chem. Rev. 108, 2015 (2008). [42] F. Gharagheizi, P. IIani, A. Kashkouli, and A. H. Mohammadi, Chem. Eng. Sci. 78, 204 (2012). [43] K. J. Wu, C. X. Zhao, and C. H. He, Fluid Phase Equilib. 328, 42 (2012). 24
ACCEPTED MANUSCRIPT [44] M. Mousazadeh and E. Faramarzi, Ionics 17, 217 (2011). [45] M. M. Alavianmehr, S. M. Hosseini, B. Haghighi, and J. Moghadasi, Chem. Eng. Sci. 122, 622 (2015). [46] R. Khordad, Int. J. Refrigeration (2017) in press. [47] E. Ghasemian and R. Zobeydi, Fluid Phase Equilib. 358, 40 (2013).
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[48] S. A. Mirkhani, F. Gharagheizi, N. Farahani, and K. Tumba, J. Mol. Liq. 179, 78 (2013). [49] Y. Nagata, T. Ohto, M. Bonn, and T. D. KΓΌhne, J. Chem. Phys. 144, 204705 (2016). [50] A. Ghoufi and P. Malfreyt, J. Chem. Phys. 136, 024104 (2012). [51] G. Galliero, J. Chem. Phys. 133, 074705 (2010).
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[52] A. Gharaati, R. Khordad, Superlatt. Microstruct. 48, 276 (2010). [53] R. H. Fowler, Proc. Royal SOC. (London), A159, 229 ( 1937). [54] J. G. Kirkwood and F. P. Buff, Chem. Phys. 17, 338 (1949).
M AN U
[55] T. Boublik, J. Chem. Phys. 53, 471 (1970).
[56] G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, J. Chem. Phys. 54, 1523 (1971).
[57] H. T. Davis, J. Chem. Phys. 62, 3412 (1975).
[58] P. H. Winterfeld, L. E. Scriven, and H. T. Davis, AIChE J 24, 1010 (1978).
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[59] L. Blum and J. S. Hoye, J. Phys. Chem. 81, 1311 (1977).
[60] J. Wu, J. Lu, and Y. Li, Fluid Phase Equilib. 101, 121 (1994). [61] J. Wu, J. Lu, and Y. Li, Fluid Phase Equilib. 107, 45 (1995). [62] Y. N. Xu and Y. Hu, Fluid Phase Equilib. 30, 221 (1986).
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[63] E. Chicurel-Uziel, J. Sound Vib. 305, 134 (2007). [64] W. R. Smith and D. Henderson, Mol. Phys. 19, 411 (1970).
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[65] Y. Hu, Fluid Phase Equilib. 17, 217 (1984). [66] M. M. Alavianmehr, S. M. Hosseini, and J. Moghadasi, J. Mol. Liq. 197, 287 (2014). [67] R. L. Gardas and J. A. P. Coutinho, Fluid Phase Equilib. 265, 57 (2008). [68] H. F. D. Almeida, J. A. Lopes-da-Silva, M. G. Freire, and J. A. P. Coutinho, J. Chem. Thermodyn. 57, 372 (2013). [69] P. J. Carvalho, M. G. Freire, I. M. Marrucho, A. J. Queimada, J. A. P. Coutinho, J. Chem. Eng. Data 53, 1346 (2008). [70] M. G. Freire, P. J. Carvalho, A. M. Fernandes, I. M. Marrucho, A. J. Queimada, J. A. P. Coutinho, J. Colloid Interface Sci. 314, 621 (2007). [71] M. Souckova, J. Klomfar, J. Patek, Fluid Phase Equilib 303, 184 (2011). [72] P. Kilaru, G. A. Baker, and P. Scovazzo, J. Chem. Eng. Data 52, 2306 (2007). 25
ACCEPTED MANUSCRIPT [73] A. B. Pereiro, J. L. Legido, A. Rodriguez, J. Chem. Thermodyn. 39, 1168 (2007). [74] H. F. D. Almeida, A. R. R. Telesa, M. G. Freire, and J. A. P. Coutinho, J. Chem. Thermodyn. 54, 49 (2012). [75] M. A. A. Beigi, M. Abdouss, M. Yousefi, S. M. Pourmortazavi, and A. Vahid, J. Mol. Liq. 177, 361 (2013).
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[76] M. Souckova, J. Klomfar, and J. Patek, Fluid Phase Equilib. 333, 38 (2012). [77] M. Souckova, J. Klomfar, and J. Patek, J. Chem. Thermodyn. 48, 267 (2012).
[78] J. Klomfar, M. Souckova, and J. Patek, J. Chem. Thermodyn. 42, 323 (2010). [79] J. Klomfar, M. Souckova, and J. Patek, J. Chem. Eng. Data 54, 1389 (2009).
SC
[80] M. Shamsipur, A. A. M. Beigi, M. Teymouri, S. M. Pourmortazavi, and M. Irandoust, J. Mol. Liq. 157, 43 (2010).
Data 54, 2803 (2009).
M AN U
[81] L. G. Sanchez, J. R. Espel, F. Onink, G. W. Meindersma, A. B. D. Haan, J. Chem. Eng.
[82] M. Deetlefs, C. Hardacre, M. Nieuwenhuyzen, A. Padua, O. Sheppard, and A. Soper, J. Phys. Chem. B 110, 12055 (2006).
[83] N. Auerbach, Experientia 4, 473 (1948).
[84] G. Garcia-Miaja, J. Troncoso, and L. Romani, Fluid Phase Equilib. 274, 59 (2008).
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[85] E. Vercher, P. J. Miguel, and F. J. Llopis, J. Chem. Eng. Data 57, 1953 (2012). [86] E. J. Gonzalez, N. Calvar, A. Dominguez, and E. A. Macedo, J. Chem. Thermodyn. 61, 64 (2013).
[87] G. Garcia-Miaja, J. Troncoso, and L. Romani, J. Chem. Thermodyn. 41, 334 (2009).
52, 1468 (2007).
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[88] E. Vercher, A. V. Orchilles, P. J. Miguel, and A. Martinez-Andreu, J. Chem. Eng. Data
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[89] M. Dzida, M. Chorazewski, M. Geppert-Rybczynska, E. Zorebski, M. Zorebski, M. Zarska, and B. Czech, J. Chem. Eng. Data 58, 1571 (2013). [90] A. Kumar, J. Solution Chem. 37, 203 (2008).
26
S0921-4526(17)31063-3
DOI:
10.1016/j.physb.2017.12.068
Reference:
PHYSB 310645
To appear in:
Physica B: Physics of Condensed Matter
Received Date: 29 July 2017 Revised Date:
27 December 2017
Accepted Date: 28 December 2017
Please cite this article as: N. Vaziri, R. Khordad, G. Rezaei, Effect of potential attraction term on surface tension of ionic liquids, Physica B: Physics of Condensed Matter (2018), doi: 10.1016/ j.physb.2017.12.068. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
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Effect of potential attraction term on surface tension of ionic liquids N. Vaziri, R. Khordad* and G. Rezaei
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Department of Physics, College of Sciences, Yasouj University, Yasouj, Iran
Abstract
In this work, we have studied the effect of attraction term of molecular potential on surface
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tension of ionic liquids (ILs). For this purpose, we have introduced two different potential models to obtain analytical expressions for the surface tension of ILs. The introduced
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potential models have different attraction terms. The obtained surface tensions in this work have been compared with other theoretical methods and also experimental data. Using the calculated surface tension, the sound velocity is also estimated. We have studied the structural effects on the surface tensions of imidazolium-based ionic liquids. It is found that the cation alkyl chain length and the anion size play important roles to the surface tension of the selected ionic liquids. The calculated surface tensions show a good harmony with
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experimental data. It is clear that the attraction term of molecular potential has an important role on surface tension and sound velocity of our system.
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Keywords: Surface tension, Ionic liquids, Potential model
*Corresponding author, E-mail: [email protected]
1
ACCEPTED MANUSCRIPT 1. Introduction Accurate knowledge of the thermodynamic properties of ionic liquids is one of main challenges in the scientific purposes like electrochemical stability, synthetic, separation processes, and industrial applications [1]. Among the thermodynamic properties, the surface tension is an important and useful quantity in heat transfer, mass transfer, flow, secondary
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and tertiary crude oil recovery, and gas condensate recovery [2, 3]. Surface tension is defined as the difference between the energy in the bulk and surface per unit area of interface. This parameter is a basic physical property that has a key role in development, design and simulation of many chemical engineering processes such as
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biochemical engineering, environmental protection, gas absorption, distillation, extraction and residual liquid saturation [4-7].
Hitherto, several works have been performed theoretically and experimentally to predict the
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surface tension of simple fluids and fluid mixtures [8-11]. The performed theoretical works to estimate the surface tension is based on statistical thermodynamics. Many authors have used different approaches including the corresponding-states theory [12], the parachor method and its derivatives [8, 13-15], the gradient theory of inhomogeneous fluids [16, 17], Monte Carlo simulation [18], density functional theory [19-24], molecular dynamic simulation [25],
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perturbation theory [26, 27], Gibbs method [28], and quantitative structure-property relationship (QSPR) [29-31].
Ionic liquids (ILs) are organic salts comprised of an anion and an organic cation. They are liquids at ambient temperature and have an appreciable liquid range. Ionic liquids were
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discovered in 1914 but their systematic study began with the present century [32]. Very interestingly, the physicochemical properties of ILs can be finely tuned by slight structural
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changes of the corresponding cations and anions. In recent years, an increasing interest has been devoted on the topic of ionic liquids due to their wide applications in the chemical industry such as such as homogeneous and biphasic transfer catalysts, multiphase homogeneous catalytic reaction and solvents in organic synthesis [33-41]. In order to use the ionic liquids in potential applications, we should know their physical properties such as viscosity, density and surface tension. As we know there are few accurate theoretical results for surface tension of ionic liquids compared with their density data. Hitherto, several theoretical methods have been employed to estimate the surface tension of ionic liquids. Therefore, the prediction of theoretical surface tension of ionic liquids accurately is useful to establish the surface tension data in the design specifications. Examples of the methods are the group contribution method [42], the corresponding-states 2
ACCEPTED MANUSCRIPT group-contribution method [43], and corresponding state theory [44]. To obtain more information about theoretical results of the ionic liquids, the reader can refer to [45-51]. In the previous works, authors have considered the columbic interactions, the hard-sphere repulsions and Lennard-Jones (LJ) dispersion forces in calculating surface tension of ILs. In this work, we have intended to answer the following question. What is the effect of potential
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attraction term of the LJ dispersion forces on the surface tension of ILs. For this purpose, we have selected two potential models which have the same behaviors as the LJ potential but with different attraction terms. It is to be noted that we have also used these attraction terms in nanostructures [52]. The new idea of the present work is considering contributions arising
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from the attraction term of the potential models on the surface tension of ILs. Since the surface tension of the ionic liquids is an important property, we intend to study this property theoretically. We have used two new potential models and molecular thermodynamic model
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to prediction of the surface tensions of 13 ionic liquids including 12 imidazolium based and 1
2. Theory and model
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phosphonium-based ones.
2.1. A new potential model
It should note that from Fowler model and KB (Krikwood-Buff) model [53, 54], the surface
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tension of multicomponent systems depends on intermolecular pair potential and pair correlation function. The accurate theoretically prediction of the surface tension will help to
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improve engineering applications. To this end, knowledge of accurate pair potential of a system is an important problem. We have tried to combine the previous potential models for obtaining better results. For this goal, we have selected two potentials which have the repulsion and attraction terms and the potentials have same behaviors as the LJ potential. We have considered the repulsion term of the LJ potential with two different attraction terms. The two potential models to show the interaction between two molecules with the separation of :
attraction terms of the potentials are as attraction well. In the following, we have introduced
i)
Potential model (1)
3
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= β β β , 1 ii)
Potential model (2)
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= β β β , 2
where and are the energy scale and the length scale, respectively. In above equation,
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and are constants equal to 2.3 and 2.0, respectively (see Fig. 1).
2.1. Surface tension
We have used the perturbation theory [25] to obtain the surface tension of ionic liquids. According to the theory, one can write the following expression
!
!
+ #$% + &'$( , 3
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where
=
and &'$( are the contributions from hard-sphere repulsion and the electrostatic
interactions to surface tension, respectively. The second term in Eq. (3) relates to the contribution of the new potentials to the surface tension. In the previous works [45], authors
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have considered *+ instead of the second term of Eq. (3). They have assumed *+ contributes from the Lennard-Jones (LJ) potential. !
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In order to calculate considered by
Here 0,- =
56 758
, the hard-sphere pair potential for fluid mixtures has been
,- =
β β€ 0,. 4 0 > 0,-
is the distance between the center of mass of two hard spheres and 0, , 0- are
the sphere diameters. The symbols 9 and : are used for each ions.
The surface tension can be calculated in terms of radial distribution function. It is expressed for non-simple molecules mixtures at low vapour pressures as [57, 58]
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0,- ; = = >, >- ? @ A B,- C 0 , 5 8 0 F
!
,,-G
E
where >, and >- are the molecular number density of components 9 and :, respectively, is
the distance between the molecules, B,- is the radial distribution function and ,- is the relation [18, 52]
0,-! 0
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pair potential function. With respect to the potential form of Eq. (4), we can use the following = βIJ KLM β 0,- N, 6
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where L is the Dirac delta function, IJ is the Boltzmann's constant and K is the system
temperature. Inserting Eq. (6) into Eq. (5) and taking into account integration of delta function and >, >- = , - > , we obtain F
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,G -G
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,- !
F
; = = = >, >- ? C PβIJ KLM β 0,- NQB,- 0 = 8 E
F
;IJ K C β > = , - 0,B,-! 0 7 . 7 8 ,,-G
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Here, B,-! 0 7 is the pair radial distribution function (RDF) of hard-sphere mixtures at contact distance (0 7 ) and it is given by [55, 56]
T,T,1 + + , 8 1 β S 21 β S 21 β SU
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B,-! 0 7 =
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where S is the packing fraction and it is expressed by F
> S = = , V, 4 ,G
Where >, , and V, are the molecular number density, the mole fraction and the van der
Waals co-volume, respectively. The T,- is expressed as > V, VT,- = @ A 4 V,-
/U F
= , V, ,GE
/U
9
5
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In the case of hard-sphere mixture, the V,- parameter is given by
1 U V,- = PV, K + V- KQ 10 8
in terms of the reduced temperature KY = IJ K/. This parameter is given by [56] VK = β0.01054Z1 β 0.7613KY expβ0.7613KY ^ KYC
a exp `β
1.3227 KYC
ab 11
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0.3306
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+ 2.9387 _1 β `1 +
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This parameter represents the van der Waals co-volume which has been previously adopted
The last term of Eq. (3) corresponds to the contribution from the ionic electrostatic interaction. To obtain this term from Eq. (5), the radial distribution function requires. Blum and Hoye [59] have solved the mean spherical approximation and obtained the pair correlation function. They have used the following electrostatic potential
c, c, 12 E
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,- =
where c, and c- are the valences of components. E and are the permittivity of free space
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and the unit electric charge, respectively. Blum and Hoye [59] have obtained the following radial distribution function
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d c, c- e fY B,- = β , 13 E
where d = 1/IJ K and Ξ is a single shielding parameter. Wu et al. [60, 61] have employed
Eq. (13) and obtained &'$( as &'$(
,-
d;> c, c= @ A 14 32Ξ E
6
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#
2; Ξ= h= >, c, i IJ K ,G
15
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iteration using the following iteration value
In order to obtain the second term of surface tension using Eq. (5), we have employed the
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radial distribution function which was previously developed by Xu and Hu [62] as U B,- = jM β Μ ,- N + Μ ,- β 0,Ξ΄ β Μ ,- /3,-β 16
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m
where j β E is the Heaviside step function and it is given by j β E = 1 > E < E n j β E = 0 j β E = 1/2 = E
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Also, L β E is Dirac delta-like function and it is given by
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L β E = 0 β E L β E = β = E
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In Eq. (16), ,-β = 1.1500,- and Μ ,- = 1.5750,- are the position of molecules and the outer
radius of the first coordination shell [63, 64], respectively. The two parameters ,-β and Μ ,- are
obtained using the molecular simulation [65].
Inserting Eqs. (1) and (16) into Eq. (5), after some difficult integration, we have obtained the following analytical expression for #$% for the potential model (1) as follows:
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#
%$s t ,-t ,U Cr = 3; = >, >- ,- ,- r@ A β MΜ U β 0,N βuu ,16Μ ,16 ,,,r q
4.6,-β ,-β ,-β U +` wxyβ @ @ A β A β @ @ A β Aa MΜ ,- U β 0,N , ,72 ,-v
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m
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} U {1 Β Μ ,1 Β + YΜ 68 YΜ 68 Β Β`40000 @ β A |12 ,CΒ eΒ Β eΒ { @279841 Β68 + 559682 Β68 + 279841A z ~ Β
68 Μ ,Μ ,CΒ eΒ Β68 + 276000 @ β A + 634800 @ β Aa , ,-
YΜ 68 eΒ Β68
+ 30000
Β
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+ 2 @15000
CΒ
YΜ
YΜ 68 eΒ Β68
+ 15000A Β9U
Β
YΜ 68 eΒ Β68
68 Μ ,Μ ,CΒ eΒ β 2 Β30000 @ β A + @30000 @ β A + 69000A Β68 , ,-
68 68 Μ , Β eΒ Β eΒ β A + 138000A Β68 + 69000Β Β9 Β68 ,-
YΜ
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+ @60000 @
YΜ
Μ ,Μ ,β 2 Β30000 @ β A + 138000 @ β A , ,-
YΜ
68 Μ ,Μ ,CΒ eΒ Β68 + h30000 @ β A + 138000 @ β A + 158700i , ,-
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YΜ
68 Μ ,Μ , Β eΒ Β68 + h60000 @ β A + 276000 @ β A + 314700i , ,-
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+ 158700Β Βy @Β
Β
YΜ 68 eΒ Β68
YΜ
+ 1ΒA
Μ ,Μ ,Μ ,β Β20000 @ β A + 144000 @ β A + 358800 @ β A , , ,C
U
8
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ΒΒ YΜ 68 {Β Μ , Β eΒ + 338560 @ β A + 1046362Β Β68 β 486680Β Β 17 ,ΒΒ { ΒΒ D
Β9Β c = =
ΒG
cΒ c cU = c + + + β― 18 IΒ 2Β 3Β
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Where
,
-
+
β
Where erf =
Mβ8Μ ,- β C
,-t
16Μ ,-
4Μ ,- ,-
9 ,-v
,-t
6,-
βuu
U MΜ ,- U β 0,N+
β 4Μ ,- ,-U β 5 ,-C N
192 ,-C
U ,-β MΜ ,- U β 0,N m
tβ
Μ ,- β ,11β; erf h2 @ Ai 192 ,-
exp @β
4Μ ,- 8Μ ,+ β 4A , ,-
,-β ,-β @1 β A exp β Β2 @ A β 2Β Β 19 , ,
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,- = 3; = = >, >- ,- ,-C Β
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Analytical expression for #$% for the potential model (2) is obtained as
Β eΒ 0w is the error function. βΒ E
Β
m
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It is to be noted that there are several parameters in Eqs. (17) and (19) such as ,-β , Μ ,- , >, , >- , ,- , ,- . Among these parameters, ,-β and Μ ,- depend on 0,- which is an important parameter.
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When 0,- is large, the pair correlation function approaches zero. Therefore, in our expansion,
the higher orders of 0,- are small and we have neglected the orders. The most important order
of 0,- is the first order. This parameter can also depend on temperature. In Eqs. (17) and (19), we have kept the terms which have same orders. The dependence of 0, is given by 1 + 0.2977IJ K/, 0, = 20 , 1 + 0.3316IJ K/, + 0.001047IJ K/,
9
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3. Results and discussion In this paper, we have carried out the numerical results for the surface tension of ionic liquids using the perturbation method and introduction of two new potential models. The used parameters in this work have been presented in tables 1 and 2. These parameters have been
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taken from Ref. [66].
Fig. 1 shows the variation of new potential models (1) and (2) as a function of molecular distances. To compare, we have also plotted the (12-6) Lennard-Jones potential model. It is
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seen from the figure that two potentials have the same behaviors as the LJ potential.
The accuracy of our model has been further assessed by comparing our surface tension calculations with those obtained from the method of Gardas and Coutinho [67] and the corresponding-states group-contribution (CS-GC) method of Wu et al [43]. Our results were
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summarized in table 3 as the AAD (in %) of the calculated surface tensions of the some selected ILs, using the proposed molecular model (this work), the QSPR strategy of G & C and those obtained from the CSβGC method of Wu et al [43], from those measured Almeida et al [68]; Carvalho et al [69], Freire et al [70], Souckova et al [71], Kilaru et al [72], Pereiro
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et al [73]. It is observed from table 3 that our new model outperforms the QSPR strategy and CSβGC method. From 69 examined data points for some studied ILs, the AAD (in %) of the
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correlated surface tensions from the proposed model was found to be 2.21% (potential model 1) and 2.33% (potential model 2) which is lower than those obtained from the QSPR strategy and CS-GC method with AADβs equal to 10.31% and 4.85%, respectively. It should be added that the uncertainty of the experimental surface tension data used for comparisons was of the order of Β±0.0012 N m-1.
We have calculated the surface tension (STcal) of the ionic liquids using the perturbation
method Eq. (5) using the new potential models Eqs. (1) and (2). In order to show graphically the accuracy of our method, we have plotted the theoretical surface tension of the ionic liquids versus the experimental surface tension (STexp) [68, 74-77] in Figs. 2 and 3 for the potential models (1) and (2), respectively. The results show the good harmony between the experimental data and the correlated surface tension obtained from our proposed models. 10
ACCEPTED MANUSCRIPT From the figures, it is seen that the potential model 2 gives better results than the potential model 1. This means that the attraction term of the potential model can be influenced on the surface tension of the selected ILs. In Figs. 4 and 5, the surface tension (ST) of imidazolium-based ionic liquids has been plotted as a function of temperature for the two potential models (1) and (2). Our results in the two
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figures have been compared with experimental data [72, 78-81]. Two figures have been plotted in order to show the influence of structural effects on the surface tensions of the selected ionic liquids.
The effect of alky chain-length of the cation on the surface tension of the selected ionic
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liquids has been shown in Figs. 4(a) and 5(a). Also, the influence of anion size on the surface tension has been investigated in Figs. 4(b) and 5(b). It is seen from the Figs. 4(a) and 5(a) that the surface tension decreases with increasing the cation alkyl chain length of the selected ILs
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for the two potential models. It should be noted that the enhancement of cation alkyl chain length causes an increase of the van der Waals forces and aliphatic character ionic liquids. The factors lead to the dispersion of the ion charge and thereby the reduction of electrostatic forces.
It is clear from Figs. 4(b) and 5(b) that the surface tension is reduced with increasing the
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anion size of the selected ILs. Our results are in agreement with Ref. [82]. With increasing the anion size, the hydrogen bonding strength and the electrostatic forces reduce and the delocalization charge enhances. We can see an important behavior in Figs. 4(b) and 5(b). From the figures, it is clear that the surface tension obtained from the potential model (1) has
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better agreement with experimental data. It means that the surface tension depends on potential attraction term.
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One can theoretically calculate the sound velocity of the ionic liquids with knowledge of the surface tension. The relationship between the sound velocity and surface tension is as [83] Β£ Β‘= , 21 6.33 Γ 10e E >
where > and are the liquid density and the surface tension, respectively. We have selected
Β€=0.6714 from Ref. [67]. This parameter has been obtained by the correlation of experimental sound velocities of imidazolium-based ionic liquids and the least-squares
method [67]. Using the calculated surface tension in this work for two potential models and by using Eq. (21), we can compute the sound velocity. Figs. 6 and 7 show the relative 11
ACCEPTED MANUSCRIPT deviation of the calculated sound velocity Eq. (21) of imidazolium-based ionic liquids from the experimental data [84-90] over the temperature range within 278 K to 343 K. Our theoretical results in Figs. 6(a)-6(c) and 7(a)-7(c) have been compared with different references. The comparison has been made for two potential models (1) and (2) in Figs. 6 and 7, respectively. As shown in Figs. 6(a) and 7(a), the plots represent the relative deviations (in
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%) of our predictions from Refs. [84-86] measurements for which their uncertainties were of the order of 1.2 m/s, 2.0 m/s and 1.6 m/s for the potential model (1). Also, the uncertainties for the potential model (2) are of the order of 1.5 m/s, 2.3 m/s and 1.9. The figures 6(c) and 7(c) represent the relative deviations (in %) of the estimated sound velocities of
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[C2mim][NTf2] and [C4mim][BF4] from Refs. [89] and [90] measurements for which their uncertainties were equal to 1.2 m/s and 2.2 m/s, respectively, for the potential model (1). The uncertainties for the potential model (2) are of the order of 1.3 m/s and 2.5, respectively, for
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the potential model (2). It is observed from the figures that the deviation for the sound velocity from the potential model (2) is larger than that the potential model (1). This means
4. Conclusions
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that the attraction term of the potential model can be influenced on the sound velocity.
It is fully known that the tension at the surface of a liquid is one of the most important demonstrations of the intermolecular forces. The intermolecular forces composed of both
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repulsion and attraction terms. We have answered the following question in this work. What is the effect of potential attraction term of dispersion forces on the surface tension of ILs. We
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have calculated the surface tension of selected ILs using two potential models which have different attraction terms. The main advantages of the present model in comparison with the previous works are: (i) the statistical thermodynamic approaches expression (5) proposed in this work provide us good approach to predict the surface tension of ILs. (ii) Using the expression (5), we could show the effect of attraction term dispersion forces on the surface tension of ILs. (iii) The proposed combination model can perform in a predictive manner with no adjustable parameters. (iv) With considering temperature-dependent of the used parameters [Eqs. (11), (15) and (20)], we could obtain the acceptable results for the surface tension of ILs. (v) The surface tension of the selected ILs can be predicted using only four parameters , , Ξ and 0. (vi) In comparison with available data, we can say that the better
results for the surface tension of the selected ILs can be obtained by finding the accurate 12
ACCEPTED MANUSCRIPT
dispersion forces. (vii) The constant parameters ( and ) in new potential models can be
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influenced on our results.
Table I. The cation and anion parameters [66].
[C2mim]+
Ο/IJ Β₯ 869.78
Ξ΅ Β¦E
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Cation
Anion [BF4]-
4.981
Ξ΅ Β¦E
318.61
4.9704
1210.8
6.596
330.38
5.633
[NTf2]
[C6mim]+
252.18
6.313
[Triflate]-
320.61
6.441
[C8mim]+
268.38
6.720
[PF6]-
349.27
5.505
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+
[C4mim]
-
Ο/IJ Β₯
Table II. The ionic liquids parameters [66].
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Ξ΅ Β¦E [C2mim][NTf2] [C4mim][NTf2] [C6mim][NTf2] [C2mim][BF4] [C4mim][BF4] [C8mim][BF4] [C4mim][PF6] [C6mim][PF6] [C8mim][PF6] [C4mim][Triflate]
Ο/IJ Β₯ 1040.29 770.59 731.49 594.19 324.49 293.49 339.82 300.72 308.82 325.49
13
Ionic liquids 5.732 6.095 6.453 4.976 5.291 5.779 5.569 5.895 6.082 6.023
ACCEPTED MANUSCRIPT Table 3. The AAD (in %) of the calculated surface tensions of the some selected ILs, using the proposed molecular model (this work), the QSPR strategy of Gardas and Coutinho [67] and those obtained from the CS-GC method of Wu et al [43], all were compared with the measurements [68-73].
NP
[C4mim][NTf2]
293-343
[C6mim][NTf2]
This
This
worka
workb
06
1.02
1.03
293-343
06
0.24
0.26
[C2mim][NTf2]
293-343
06
0.45
0.49
[C2mim][Triflate]
293-343
06
0.60
0.63
[C2mim][BF4]
288-355.91
14
5.16
[C4mim][BF4]
298-343
06
[C4mim][PF6]
288-313
[C8mim][BF4]
QSPR
CS-GC
Data
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βK (K)
Ionic Liquid
source
3.23
[69]
6.59
5.20
[69]
5.33
2.00
[69]
10.46
1.24
[70]
5.34
19.0
7.54
[71]
2.31
2.58
9.48
--
[68]
06
2.16
2.43
2.45
--
[73]
288-343
07
4.01
4.14
15.96
6.14
[70]
[C6mim][PF6]
293-353
07
0.29
0.36
6.38
--
[70]
[C8mim][PF6]
293-313
05
1.98
2.03
11.93
--
[73]
69
2.21
2.33
10.31
4.85
----
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Overall
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3.98
14
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0
2
4
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0
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u(x)
(12-6) LJ potential Potential model (1) Potential model (2)
x
Fig. 1.
40
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45
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STcalc(mN.m-1 )
50
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55
35
30
25 25
30
35
40
45 -
STexp(mN.m 1 )
Fig. 2. 15
50
55
ACCEPTED MANUSCRIPT 55
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45
40
35
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STcalc(mN.m-1 )
50
25 25
30
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30
35
40
45
50
55
-
STexp(mN.m 1 )
50
ST(mN.m -1)
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45
(a)
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55
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Fig. 3
40
35
30
25 260
[C2mim][BF4] [C2mim][BF4] [C4mim][BF4] [C4mim][BF4] [C6mim][BF4] [C6mim][BF4] [C8mim][BF4] [C8mim][BF4]
280
300
320
T/K
16
340
360
380
ACCEPTED MANUSCRIPT (b)
48 46 44
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40 38
34 32 30 28 260
[C4mim][BF4] [C4mim][BF4] [C4mim][PF6] [C4mim][PF6] [C6mim][BF4] [C6mim][BF4] [C8mim][BF4] [C8mim][BF4]
280
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36
300
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ST(mN.m -1)
42
320
T/K
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Fig. 4.
17
340
360
380
ACCEPTED MANUSCRIPT (a)
50
[C2mim][BF4] [C2mim][BF4] [C4mim][BF4] [C4mim][BF4] [C6mim][BF4] [C6mim][BF4] [C8mim][BF4] [C8mim][BF4]
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40
35
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ST(mN.m -1)
45
25 260
280
300
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30
320
340
360
380
T/K (b)
44
40
EP
34
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ST(mN.m -1)
38 36
[C4mim][BF4] [C4mim][BF4] [C4mim][PF6] [C4mim][PF6] [C6mim][BF4] [C6mim][BF4] [C6mim][PF6] [C6mim][PF6] [C8mim][BF4] [C8mim][BF4]
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42
32 30 28
26 260
280
300
320
340
Fig. 5
18
360
380
400
ACCEPTED MANUSCRIPT (a)
7 6 5
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Deviation%
4 3 2
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1 0
-2 280
290
300
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Garcia-Miaja(2008) Vercher,(2012)) Gonzalez,(2013)
-1
310
320
330
340
T/K
(b)
3
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2 1
-2 -3
EP
-1
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Deviation%
0
-4 -5 -6
-7 270
Garcia-Miaja(2009) Vercher,(2007))
280
290
300
310
T/K
19
320
330
340
ACCEPTED MANUSCRIPT (c)
8 6 4
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Deviation%
2 0 -2
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-4 -6 -8
290
300
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-10 280
Dzida et al(2013 kumar,(2008)
310
320
T/K
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Fig. 6.
20
330
340
350
ACCEPTED MANUSCRIPT (a)
1 0 -1
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Deviation%
-2 -3 -4
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-5 -6
-8 280
290
300
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Garcia-Miaja(2008) Vercher,(2012)) Gonzalez,(2013)
-7
310
320
330
340
T/K (b)
-7
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-8 -9
-12
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-11
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Deviation%
-10
-13 -14 -15
-16 270
280
Garcia-Miaja(2009) Vercher,(2007))
290
300
310
T/K
21
320
330
340
ACCEPTED MANUSCRIPT (c)
4 3 2
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Deviation%
1 0 -1
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-2 -3 -4 -5 280
290
300
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Dzida et al(2013 kumar,(2008)
310
320
330
340
350
T/K
.
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Fig. 7
Figures Captions:
Fig. 1. Variation of potential function = / versus intermolecular distance
= / . Two new potentials have been compared with (12, 6) Lennard-Jones potential.
Fig. 2. The calculated surface tension (STcal) as a function of experimental surface tension (STexp) [68, 74-77] for the potential model (1). Fig. 3. The same as Fig. 2, but for the potential model (2). 22
ACCEPTED MANUSCRIPT
Fig. 4. The surface tension of imidazolium-based ionic liquids as a function of temperature. The solid lines and markers correspond to theoretical and experimental results, respectively
Fig. 5. The same as Fig. 4, but for the potential model (2).
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[72, 78-81].
Fig. 6. The relative deviation of the calculated sound velocity Eq. (15). Our results have been
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compared with experimental data. Three figures correspond to (a) [C4mim][Triflate] [84-86], (b) [C2mim][Triflate] [87, 88], and (c) [C2mim][NTf2]/[C4mim][BF4] [89, 90].
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Fig. 7. The same as Fig. 6, but for the potential model (2).
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References
[1] H. T. Davis, J. Stat. Phys. 24, 43 (1981).
[2] J. J. Bikerman, Physical Surfaces, Academic Press, New York (1970). [3] L. I. Rolo, A. I. Caco, A. J. Queimada, I. M. Marrucho, and J.A.P. Coutinho, J. Chem.
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Eng. Data 47, 1442 (2002).
[4] B. E. Poling, J. M. Prausnitz, J. P. O'Connell, The Properties of Gases and Liquids fifth
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ed., McGraw-Hill, New York (2001). [5] A. W. Neumann, R. David, Y. Zuo, Applied Surface Thermodynamics, second ed., Taylor and Francis Group, CRC Press, Boca Raton (2011). [6] H. Lin and Y. Y. Duan, Int. J. Thermophys. 24, 1495 (2003). [7] G. Zhao, S. Bi, J. Wu, and Z. Liu, J. Chem. Eng. Data 55, 3077 (2010). [8] D. B. MacLeod, Trans. Faraday Soc. 19, 38 (1923). [9] B. S. Carey, L. E. Scriven, and H. T. Davis, AIChE J. 26, 705 (1980). [10] L. A. Girifalco and R. J. Good, J. Phys. Chem. 61, 904 (1957). [11] F. M. Fowkes, J. Phys. Chem. 66, 1863 (1962). [12] E. A. Guggenheim, J. Chem. Phys. 13, 253 (1945). [13] S. Sugden, J. Chem. Soc. Trans. 125, 32 (1924). 23
ACCEPTED MANUSCRIPT [14] S. Sugden, J. Chem. Soc. Trans. 125, 1167 (1924). [15] C. F. Weinaug and D. L. Katz, Ind. Eng. Chem. 35, 239 (1943). [16] J. W. Cahn and J. E. Hilliard, J. Chem. Phys. 28, 258 (1958). [17] J. S. Rowlinson, J. Stat. Phys. 20, 197 (1979). [18] P. Attard and G.A. Moule, Mol. Phys. 42, 943 (1993).
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[19] M. Mecke, J. Winkelmann, and J. Fischer, J. Chem. Phys. 107, 9264 (1997). [20] D. Fu, J. F. Lu, J. C. Liu, and Y. G. Li, Chem. Eng. Sci. 56, 6989 (2001). [21] R. Evans, Adv. Phys. 28, 143 (1979).
[22] P. I. Teixeira and M. M. Telo da Gama, J. Phys. Condens. Matter 3, 111 (1991).
[24] J. Winkelmann, Phys. Chem. 98, 1308 (1994).
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[23] J. Winkelmann, U. Brodrecht, and I. Kreft, Phys. Chem. 98, 912 (1994).
[25] J. A. Barker and D. Henderson, J. Chem. Phys. 47, 4714 (1967).
M AN U
[26] J. M. Haile and C. G. Gray, J. Chem. Phys. 64, 2569 (1976). [27] S. Toxvaerd, J. Chem. Phys. 57, 4092 (1972).
[28] Z. B. Li, Y. G. Li, and J. F. Lu, Ind. Eng. Chem. Res. 38, 1133 (1999). [29] A. Khajeh and H. Modarress, J. Hazard. Mater. 179, 715 (2010). [30] A. Khajeh and H. Modarress, Expet. Syst. Appl. 37, 3070 (2010).
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[31] A. Khajeh and H. Modarress, J. Chemometr 25, 333 (2011).
[32] P. Wasserscheid and T. Welton (Eds.), Ionic Liquids in Synthesis, Wiley-Verlag, Weinheim (2003).
[33] W. Liu, L. Cheng, Y. Zhang, H. Wang, and M. Yu, J. Mol. Liq. 140, 38 (2008).
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[34] W. Liu, T. Zhao, Y. Zhang, H.Wang, and M. Yu, J. Sol. Chem. 35, 1337 (2006). [35] Q. G. Zhang, F. Xue, J. Tong,W. Guan, and B. Wang, J. Sol. Chem. 35, 297 (2006).
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[36] L. A. S. Ries, F. A. do Amaral, K. Matos, E. M. A.Martini,M. O. de Souza, and R. F. de Souza, Polyhedron 27, 3287 (2008). [37] R. M. Lau, F. Rantwijk, K. R. Seddon, and R. A. Sheldon, Org. Lett. 2, 4189 (2000). [38] M. Koel, Crit. Rev. Anal. Chem. 35, 177 (2005). [39] M. Rantwijk and R.A. Sheldon, Chem. Rev. 107, 2757 (2005). [40] M. Haumann and A. Riisager, Chem. Rev. 108, 1474 (2008). [41] M. A. P.Martins, C. P. Frizzo, D. N.Moreira, N. Zantta, and H. G. Bonacorso, Chem. Rev. 108, 2015 (2008). [42] F. Gharagheizi, P. IIani, A. Kashkouli, and A. H. Mohammadi, Chem. Eng. Sci. 78, 204 (2012). [43] K. J. Wu, C. X. Zhao, and C. H. He, Fluid Phase Equilib. 328, 42 (2012). 24
ACCEPTED MANUSCRIPT [44] M. Mousazadeh and E. Faramarzi, Ionics 17, 217 (2011). [45] M. M. Alavianmehr, S. M. Hosseini, B. Haghighi, and J. Moghadasi, Chem. Eng. Sci. 122, 622 (2015). [46] R. Khordad, Int. J. Refrigeration (2017) in press. [47] E. Ghasemian and R. Zobeydi, Fluid Phase Equilib. 358, 40 (2013).
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[48] S. A. Mirkhani, F. Gharagheizi, N. Farahani, and K. Tumba, J. Mol. Liq. 179, 78 (2013). [49] Y. Nagata, T. Ohto, M. Bonn, and T. D. KΓΌhne, J. Chem. Phys. 144, 204705 (2016). [50] A. Ghoufi and P. Malfreyt, J. Chem. Phys. 136, 024104 (2012). [51] G. Galliero, J. Chem. Phys. 133, 074705 (2010).
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[52] A. Gharaati, R. Khordad, Superlatt. Microstruct. 48, 276 (2010). [53] R. H. Fowler, Proc. Royal SOC. (London), A159, 229 ( 1937). [54] J. G. Kirkwood and F. P. Buff, Chem. Phys. 17, 338 (1949).
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[55] T. Boublik, J. Chem. Phys. 53, 471 (1970).
[56] G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, J. Chem. Phys. 54, 1523 (1971).
[57] H. T. Davis, J. Chem. Phys. 62, 3412 (1975).
[58] P. H. Winterfeld, L. E. Scriven, and H. T. Davis, AIChE J 24, 1010 (1978).
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[59] L. Blum and J. S. Hoye, J. Phys. Chem. 81, 1311 (1977).
[60] J. Wu, J. Lu, and Y. Li, Fluid Phase Equilib. 101, 121 (1994). [61] J. Wu, J. Lu, and Y. Li, Fluid Phase Equilib. 107, 45 (1995). [62] Y. N. Xu and Y. Hu, Fluid Phase Equilib. 30, 221 (1986).
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[63] E. Chicurel-Uziel, J. Sound Vib. 305, 134 (2007). [64] W. R. Smith and D. Henderson, Mol. Phys. 19, 411 (1970).
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[65] Y. Hu, Fluid Phase Equilib. 17, 217 (1984). [66] M. M. Alavianmehr, S. M. Hosseini, and J. Moghadasi, J. Mol. Liq. 197, 287 (2014). [67] R. L. Gardas and J. A. P. Coutinho, Fluid Phase Equilib. 265, 57 (2008). [68] H. F. D. Almeida, J. A. Lopes-da-Silva, M. G. Freire, and J. A. P. Coutinho, J. Chem. Thermodyn. 57, 372 (2013). [69] P. J. Carvalho, M. G. Freire, I. M. Marrucho, A. J. Queimada, J. A. P. Coutinho, J. Chem. Eng. Data 53, 1346 (2008). [70] M. G. Freire, P. J. Carvalho, A. M. Fernandes, I. M. Marrucho, A. J. Queimada, J. A. P. Coutinho, J. Colloid Interface Sci. 314, 621 (2007). [71] M. Souckova, J. Klomfar, J. Patek, Fluid Phase Equilib 303, 184 (2011). [72] P. Kilaru, G. A. Baker, and P. Scovazzo, J. Chem. Eng. Data 52, 2306 (2007). 25
ACCEPTED MANUSCRIPT [73] A. B. Pereiro, J. L. Legido, A. Rodriguez, J. Chem. Thermodyn. 39, 1168 (2007). [74] H. F. D. Almeida, A. R. R. Telesa, M. G. Freire, and J. A. P. Coutinho, J. Chem. Thermodyn. 54, 49 (2012). [75] M. A. A. Beigi, M. Abdouss, M. Yousefi, S. M. Pourmortazavi, and A. Vahid, J. Mol. Liq. 177, 361 (2013).
RI PT
[76] M. Souckova, J. Klomfar, and J. Patek, Fluid Phase Equilib. 333, 38 (2012). [77] M. Souckova, J. Klomfar, and J. Patek, J. Chem. Thermodyn. 48, 267 (2012).
[78] J. Klomfar, M. Souckova, and J. Patek, J. Chem. Thermodyn. 42, 323 (2010). [79] J. Klomfar, M. Souckova, and J. Patek, J. Chem. Eng. Data 54, 1389 (2009).
SC
[80] M. Shamsipur, A. A. M. Beigi, M. Teymouri, S. M. Pourmortazavi, and M. Irandoust, J. Mol. Liq. 157, 43 (2010).
Data 54, 2803 (2009).
M AN U
[81] L. G. Sanchez, J. R. Espel, F. Onink, G. W. Meindersma, A. B. D. Haan, J. Chem. Eng.
[82] M. Deetlefs, C. Hardacre, M. Nieuwenhuyzen, A. Padua, O. Sheppard, and A. Soper, J. Phys. Chem. B 110, 12055 (2006).
[83] N. Auerbach, Experientia 4, 473 (1948).
[84] G. Garcia-Miaja, J. Troncoso, and L. Romani, Fluid Phase Equilib. 274, 59 (2008).
TE D
[85] E. Vercher, P. J. Miguel, and F. J. Llopis, J. Chem. Eng. Data 57, 1953 (2012). [86] E. J. Gonzalez, N. Calvar, A. Dominguez, and E. A. Macedo, J. Chem. Thermodyn. 61, 64 (2013).
[87] G. Garcia-Miaja, J. Troncoso, and L. Romani, J. Chem. Thermodyn. 41, 334 (2009).
52, 1468 (2007).
EP
[88] E. Vercher, A. V. Orchilles, P. J. Miguel, and A. Martinez-Andreu, J. Chem. Eng. Data
AC C
[89] M. Dzida, M. Chorazewski, M. Geppert-Rybczynska, E. Zorebski, M. Zorebski, M. Zarska, and B. Czech, J. Chem. Eng. Data 58, 1571 (2013). [90] A. Kumar, J. Solution Chem. 37, 203 (2008).
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