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Surface tension of silanes: A new equation
Fluid Phase Equilibria xxx (2015) 1e6
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Surface tension of silanes: A new equation* Giovanni Di Nicola*, Gianluca Coccia, Mariano Pierantozzi Politecnica delle Marche, Ancona, Italy DIISM, Universita
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2015 Received in revised form 27 August 2015 Accepted 18 September 2015 Available online xxx
This work presents a new formula to calculate the surface tension of silicon based molecules. As a first step, the raw surface tension data of silanes (experimental, smoothed and predicted) accepted by DIPPR database were collected. After the data analysis, only the experimental and smoothed data were considered for the calculations. The selected data were regressed with a scaled equation based on four input parameters: radius of gyration, critical density, critical temperature and acentric factor as fluid constants. Particularly relevant is the introduction of the radius of gyration as fluid parameter, that is a noticeable difference with respect to other previous literature methods. The selected data also were analyzed with the most reliable semi-empirical correlating methods in literature based on the corresponding states theory. The proposed equation is very simple and gives noticeable improvement with respect to existing equations (AAD ¼ 3.7%). © 2015 Elsevier B.V. All rights reserved.
Keywords: Factor analysis Radius of gyration Silanes Surface tension
Another easy and empirical formula was proposed by Macleod [4]:
1. Introduction Silanes are versatile materials used in a wide range of applications including adhesion promoters, crosslinking agents, dispersing agents, coupling agents, and surface modifiers. Surface tension of silanes is particularly important, because of a growing number of applications that involves silanes in controlling the interaction of water with a surface, such as hydrophobicity and hydrophylicity. Several models that have been proposed to date can be applied to describe the surface tension, but none of them was specifically oriented to silanes. The surface tension s (mN m1) of a fluid can be expressed as a function of temperature using an equation similar to that of van der Waals [1].
T n s ¼ s0 1 TC
(1)
where Tc (K) is the critical temperature, s0 (mN m1) and n are empirical constants obtained with the least square method in a fit to the experimental data available on surface tension [2,3].
* Paper presented at the 19th symposium on thermophysical properties, June 21e26, 2015, Boulder, CO, USA. * Corresponding author. E-mail address: [email protected] (G. Di Nicola).
s ¼ Kðrl rv Þ4
(2)
The formula suggests the existence of a simple relationship between surface tension and the difference between the densities of the liquid and of its vapor; K is a temperature independent constant characteristic of each substance. One year later, Equation (2) was modified by Sudgen [5]:
s ¼ ½Pðrl rv Þ3
(3)
where K ¼ P4, suggesting that the parameter P was temperature dependent; he called this parameter parachor. Parachor values of various compounds were calculated by Quayle [6] using experimental surface-tension data. Parachor property can be related to the critical properties of compounds including critical temperature and molar volume as follows [7,8]: 7=8
P ¼ 0:324$Tc4 $vc
(4)
where T is the temperature (K), v is the molar volume (m3 kmol1) and the subscript c denotes the critical value. Equation (3) may be considered as an equation of state [9] of the interface because it has been shown to be a good predictor of surface tension if equilibrium densities and experimental data for the parachor are used. As Escobedo and Mansoori suggested [10],
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its good performance and extremely easy analytical form have made Equation (3) a very popular method for calculating surface tension, but it has some shortcomings, as the percent deviation in predicting surface tension increases with increasing complexity of the molecular structure. Ferguson and Kennedy [11] and Guggenheim [12] used the corresponding states principle to correlate the surface tension in the low temperature range; this decision was justified under the assumption that, since the corresponding states principle was derived from an equation of state, surface tension is also expected to follow the same principle. Brock and Bird [13] developed this concept for non-polar liquids and proposed the following formula applicable to the range of temperature far from the critical point: 2
1
11
=
=
s ¼ Pc 3 Tc 3 Q ð1 Tr Þ 9
(5)
where Pc (bar) is the critical pressure, Tr is the reduced temperature, T/Tc. Q is the Riedel parameter developed in terms of Pc and Tbr according to the Miller suggestion,
Tbr lnðPc 1:01325Þ 0:279 1 Tbr =
Q ¼ 0:1196½1 þ
(6)
Tbr is the reduced normal boiling temperature, Tb/Tc, and Tb is the normal boiling temperature (K). In some cases, the surface tension can be represented in terms of an additional parameter. One example is the so-called acentric factor, u, initially introduced as an empirical parameter by Pitzer to explain the deviation from the corresponding states principle, as defined for noble gases, when applied to larger molecules. Pitzer's [14] relation in terms of Tc (K), Pc (bar), and u leads to the following corresponding state relationship for s: 2
1
s ¼ Pc3 Tc3
2 11 1:86 þ 1:18u 3:75 þ 0:91u 3 ð1 Tr Þ 9 19:05 0:291 0:08u
(7)
For compounds that exhibit hydrogen bonds, Sastri and Rao [15] proposed a modified expression for the Brock and Bird correlation: y
s ¼ KPcx Tb Tcz
1 Tr 1 Tbr
m (8)
where Pc is in bar. Coefficients were regressed for different families separately. To underline the work done in this research field, other recent equations have been proposed. Miqueu et al. [16] proposed the following:
s ¼ kTc
Na Vc
2 3
ð4:35 þ 4:14uÞt 1:26 1 þ 0:19t 0:5 0:487t
(9)
where t ¼ 1 T=Tc , and k; NA ; Vc ; u are Boltzmann's constant, Avogadro's number, the critical volume, and the acentric factor, respectively. This equation represents the surface tension for 31 substances up to the critical point with an average absolute deviation of less than 3.5%. Other relevant equations, again based on the corresponding states principle, were recently proposed by Xiang [17]. These equations describe the surface tension for non-polar, polar hydrogen-bonding, and associating substances. The equations represent the surface tension well, but they are also slightly more complex compared with previous equations, since they have a large number of parameters, including the aspherical factor. Recently, Di Nicola et al. [18,19] proposed two equations
specifically aimed at refrigerants. Furthermore, Di Nicola and Pierantozzi [20] issued an equation to predict the surface tension of binary refrigerant systems. A new scaled formula to calculate the surface tension of ketones [21] and alcohols [22] was also presented by the same authors. The scaled formula for alcohols was as following:
D s ¼ A$ð1 Tr ÞB $ 1 þ FC s0
(10)
where s is the surface tension (N/m), s0 is a scaled factor ¼ (k·Tc)/G2r , Tr is the reduced temperature, k is Boltzmann's constant (J/K), Gr is the radius of gyration (m), F is an adimensional term ¼ Na·rc·G3r , Na is Avogadro's number (mol1), rc is the critical density (mol m3). In this paper, to predict the surface tension of silanes, a modified version of equation (10) is presented. 2. Data analysis In this paper, the raw surface tension data (experimental, smoothed and predicted) accepted by DIPPR database [23] were collected and analyzed. The main advantage of the DIPPR database is that it collects data from a wide range of sources and evaluates them critically. References, notes, and quality codes for all data points are given [23]. During the data collection, a fluid by fluid analysis was performed. In the DIPPR database in its more recent version, the surface tension of 36 silanes are reported. The silanes are divided in the following subfamilies: 3 silanes, 13 chlorosilanes, 5 cyclosiloxanes, 6 linear siloxanes and 9 are classified by DIPPR database under the voice “other”, that can be considered as 8 Alkyl alkoxysilanes and 1 silazane. In Table 1, a summary of the range of data collected for surface tension and temperatures together with the number of points (experimental, smoothed and predicted) and the four fluid parameters adopted during the regressions are reported. The surface tension of silanes are generally predicted by DIPPR database with the Sudgen equation [5] and smoothed by Jasper [24]. In Fig. 1, all the collected from DIPPR data and calculated surface tension data are reported, split in experimental, predicted and smoothed. For the calculations, the more recent equation and better performing literature equation is considered [16]. Deviations produced by equation (9) are also reported in Table 1. From the figure and the Table analysis, it is evident that the data showing higher deviations values are generally the predicted ones. In a fact, deviations much higher than 10% were achieved for the following silanes: Methyl Chlorosilane (AAD ¼ 18.5%), Hexadecamethylcyclooctasiloxane (AAD ¼ 18.3%), [3-(2,3-epoxyproxy) propyl] trimethoxy Silane (AAD ¼ 35.3%), Bis[3-(trimethoxysilyl) propyl]disulfide (AAD ¼ 23.9%), Dimethyl-dimethoxy Silane (AAD ¼ 22.5%), Gammaaminopropyltriethoxy Silane (AAD ¼ 28.6%) and Methyl Silane (AAD ¼ 46.6%). Going into deeper, the histogram of the predicted, experimental and smoothed data are reported in Fig. 2. From the histogram, it is evident that the predicted data contribution to databank is particularly relevant at extreme surface tension values. However, from 5 to 40 mN m1 the surface tensions are well represented by the experimental data. In addition, always considering Fig. 1, the surface tension data smoothed by Jasper [24] generally gave quite low deviations. For these reasons, in this paper only the experimental and smoothed data were considered for the further calculations. Among numerous physical parameters, the fluid parameters most likely to influence experimental surface tension of silanes were identified using factor analysis [25,26]. The factor analysis
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Table 1 Summary of the collected data and calculated surface tension of silanes. P: predicted; E: experimental; S: smoothed. Name
3-chloropropyltrichlorosilane (3-methylacryloxypropyl) trichlorosilane Dichlorosilane Dimethyldichlorosilane Diphenyldichlorosilane Ethyltrichlorosilane Methyl Chlorosilane Methyl Dichlorosilane Methyl Trichlorosilane Phenylmethyldichlorosilane Tetrachlorosilane Trichlorosilane Trimethylchlorosilane Decamethylcyclopentasiloxane Dodecamethylcyclohexasiloxane Hexadecamethylcyclooctasiloxane Hexamethylcyclotrisiloxane Octamethylcyclotetrasiloxane Decamethyltetrasiloxane Dodecamethylpentasiloxane Eicosamethylnonasiloxane Octadecamethyloctasiloxane Octamethyltrisiloxane Tetradecamethylhexasiloxane [3-(2,3-epoxyproxy)propyl] trimethoxysilane [3-(mercapto)propyl]triethoxysilane Bis[3-(triethoxysilyl)propyl]disulfide Bis[3-(trimethoxysilyl)propyl]disulfide Dimethyldimethoxysilane Gamma-aminopropyltriethoxysilane Tetraethoxysilane Trimethoxysilane Hexamethyldisilazane Methyl Silane Silane Tetraethyl Silane a
Family
s range
Number of points E
S
T Range (K)
rc
u
2049.18 1506.02
0.381 0.627
2.9 9.6
451.5 2.714 520.35 3.535 814 5.857 559.95 4.014 442 2.5754 483 3.2563 517 3.84 689 4.827 507 3.77 479 3.439 497.75 3.4378 617.4 6.372 645.8 7.243 689.2 8.595 554.2 4.919 586.5 5.828 599.4 6.832 628.4 7.314 707.2 12.3 688.9 11.41 564.4 5.735 653.2 8.701 701.4 6.642
4219.41 2793.30 1483.68 2481.39 4065.04 3460.21 2941.18 1968.50 3067.48 3731.34 2732.24 813.01 675.68 505.05 1445.09 1016.26 909.09 714.29 409.84 405.52 1152.07 595.24 1366.12
0.163 0.267 0.428 0.270 0.225 0.276 0.263 0.402 0.232 0.203 0.270 0.622 0.710 0.883 0.496 0.584 0.660 0.720 1.050 0.984 0.538 0.777 0.760
3.5 2.5 3.5 7.6 18.5 0.9 1.8 10.0 1.7 1.8 2.3 9.2 12.8 18.3 4.1 8.6 3.1 8.4 11.5 16.4 4.6 11.1 35.3
647.5 900.3 736.2 524 634.6 592.2 525 544 352.5 269.7 606
1358.70 666.67 900.90 2267.57 1251.56 1631.32 1550.39 2450.98 4878.05 7535.80 1703.58
0.837 0.889 1.155 0.263 0.890 0.510 0.633 0.322 0.131 0.094 0.400
14.7 8.6 24.0 22.5 28.6 2.8 6.5 6.1 46.6 10.5 2.1
Tc (K)
(N$m1).103
Total
P
Chlorosilane Chlorosilane
16 15
16 15
0 0
0 0
3.54318e38.9293 4.5931e43.808
Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Cyclosilane Cyclosilane Cyclosilane Cyclosilane Cyclosilane Linear siloxane Linear siloxane Linear siloxane Linear siloxane Linear siloxane Linear siloxane Alkyl alkoxysilane
10 19 7 16 10 9 23 11 22 27 12 39 24 16 25 28 15 12 19 8 12 6 12
10 10 0 16 10 0 11 11 0 0 1 16 16 16 16 16 0 0 16 0 0 0 12
0 9 7 0 0 9 12 0 14 23 11 23 8 0 9 12 15 12 3 8 12 6 0
0 0 0 0 0 0 0 0 8 4 0 0 0 0 0 0 0 0 0 0 0 0 0
2.7424e30.317 2.8358e32.113 26.66e37.24 2.76892e41.6079 2.9034e30.174 16.65e22.12 4.201e31.14 2.9438e40.412 15.73e21.26 15.52e21.09 14.54e19.61 2.3232e23. 2.4136e21. 2.55016e20.42 2.67498e14.351 2.5118e18.4 14.7e18. 15.84e18.1 2.296e25.514 15.85e18.78 14.26e17.3 16.1e18.5 7.-75.
173e413. 197.05e468.31 303.15e413.15 167.55e503.96 139.05e397.8 268.15e308.15 195.35e455.35 229.65e620.1 273.15e328.15 263.15e313.15 273.15e323.15 229.15e555.66 270.15e581.22 304.7e620.28 337.15e498.78 290.73e527.85 293.15e333. 293.15e328.15 202.15e636.48 298.15e337.15 293.15e328.15 293.15e328.15 160.75e618.22
Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Silazane Silane Silane Silane
15 16 16 12 16 13 11 10 11 9 14
15 16 16 11 15 0 11 0 11 0 0
0 0 0 1 1 13 0 10 0 0 8
0 0 0 0 0 0 0 0 0 9 6
4.8173e42.685 4.0143e36.952 5.3898e47.099 3.5609e34.531 4.735e64.966 16.75e22.21 3.1765e36.48 11.5e18.24 1.6253e21.274 14.06e26.22 13.68e23.69
210e582.75 270e810.27 250e662.58 193e471.6 140e571.14 289.65e353.15 159.6e472.5 298.15e380. 116.34e317.25 88.5e161.3 285.15e398.65
224.3e594.99 661.1 220e637.29 708.1
Gr (Å)
(mol m3) 5.297 6.474
6.16 9.732 9.223 3.791 5.944 5.4604 3.954 5.420a 1.909 1.0735 5.251a
AAD Eq. (9) [16]
Regressed values.
involves analyzing relationships among items of an instrument or large numbers of variables in order to obtain a smaller number of variables or factors. In this way, having a smaller number of variables facilitates theory development and testing. Thus, factor analysis offers not only the possibility of gaining a clear view of the data, but also the possibility of using the output in subsequent analyses. ub Let x be a vector of independent variables with xi 2½xlb i¼ i xi ; 1; …; N and y be the dependent variable. We define
Fig. 1. Calculated surface tension (equation (9)) versus DIPPR surface tension.
)
( Dþ ¼ j
x2Djxj >
xub þxlb i i 2
)
( and D j ¼
x2Djxj >
xub þxlb i i 2
; the effect of
xj is then:
Eff xj ¼
P
yðxÞ x2Dþ j
P
yðxÞ x2D j card Dþ card D j j
(11)
where cardðDþ Þ denotes the cardinality of the set Dþ . j j The factor analysis was performed for the experimental and the smoothed data only. For the factor analysis, the following physical parameters were taken into account: experimental temperature, critical temperature, critical density, critical pressure, acentric factor, boiling point temperature, melting point temperature, triple point temperature and radius of gyration. The acentric factor and the state properties are typical corresponding states principle parameters, while the radius of gyration, already adopted by us in recent papers [21,22], plays an important role in chemistry [27,28]. In a fact, it is usually a better estimate of the chain dimensions than the root-mean-squared end-to-end distance, but the end-to-end distance is difficult to measure, while the radius of gyration can be measured by a light scattering technique. Going more in detail with the radius of gyration analysis, Fig. 3 shows the relationship between the radius of gyration and the surface tension at boiling point for the silanes for which the
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Fig. 2. Histogram of experimental, predicted and smoothed data on the surface tension of silanes.
experimental and smoothed data were collected. From Fig. 3, a general trend is clearly evident. In particular, surface tension at boiling point was found to be decreasing with radius of gyration with an evident common trend, excluding three fluids, for almost all the silanes, witnessing the validity of the choice of the radius of gyration as fluid parameter. All the selected fluid physical parameters considered for the factor analysis were collected from the DIPPR database [23]. For our calculations, only values accepted by the database were considered. For two fluids, namely hexamethyldisilazane and tetraethyl silane, the radius of gyration was not found. Since a noticeable crosscorrelation between radius of gyration and critical density for the fluids is generally present, in Fig. 4 the behavior of the radius of gyration with the critical density was analyzed also for all the silanes listed in Table 1. Finally, the obtained correlation between critical density and radius of gyration (reported in Fig. 4 as dashed line) was considered for the calculation of the two missing radii of gyration:
Gr ¼ 6:184$e4:72$10
8
0:000578rc þ0:572r2c
(12)
According to equation (12), the following values were derived for hexamethyldisilazane (Gr ¼ 5.420 Å) and tetraethyl silane
Fig. 4. Radius of gyration versus critical density for the analyzed silanes.
(Gr ¼ 5.251 Å). The factor analysis showed the following statistical results: a strong effect was found, as expected, for the experimental temperature (Eff ¼ 0.0110 N/m). Between the previously adopted
Fig. 3. Surface tension at boiling point versus radius of gyration.
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3. A new equation Starting from Equation (10), the following modified equation was finally proposed:
D s ¼ A$ð1 Tr ÞB $ 1 þ FC $uE s0
Fig. 5. Scatter plot of surface tension data of silanes versus (1Tr)B. Notation as in Fig. 3.
physical constants of the fluids [21,22], the following effects were obtained for the critical temperature (Eff ¼ 0.0023 N/m), critical density (Eff ¼ 0.0029 N/m) and radius of gyration (Eff ¼ 0.020 N/ m). Between the additional parameters to be selected, the following effects were obtained: acentric factor (Eff ¼ 0.0033 N/m), molecular mass (Eff ¼ 0.0018 N/m), boiling point temperature (Eff ¼ 0.0016 N/m), reduced dipole moment (Eff ¼ 0.0014 N/m), critical pressure (Eff ¼ 0.0003 N/m) and critical compressibility factor (0.0001 N/m). Since the farer is the Eff value from zero, the higher is the statistical relevance of the parameter under analysis, the acentric factor was selected as additional parameter. In addition, when the Eff value is positive, the property (the surface tension in this case) is expected to be increasing with the selected parameter. This means that since the Eff value of the acentric factor is positive, the surface tension should increase with the increasing of the term containing the acentric factor. From the Eff values analysis, it is also expected that the surface tension should decrease with the critical density and the radius of gyration, while it should increase with the critical temperature.
(13)
where s0 and F are defined as for equation (10). To optimize the coefficients, the LevenbergeMarquardt curve-fitting method was adopted [29]. This method is actually a combination of two minimization methods: the gradient descent method and the GaussNewton method. The following coefficients were found after regression: A ¼ 0.00031, B ¼ 1.22975, C ¼ 0.10397, D ¼ 14.93220 and E ¼ 0.33689. Comparing equation (13) with equation (10), it is evident that the term with acentric factor, u, was added. Since the coefficient E and the acentric factors are positive and smaller than unity (excluding the acentric factors of two fluids), the new term has a direct effect on the surface tension calculation, as required by the factor analysis. In Fig. 5, a scatter plot of surface tension data of silanes divided by family versus (1Tr)B is presented; it shows a rather positive correlation and a common trend for all fluids, which underlines the importance of this term of equation (13) in calculating the surface tension. This is also confirmed by NIST's REFPROP program [30], where the same term in Equation (1) is adopted as a reference [3,4]. As shown in a previous paper [22] and from Fig. 4, the crosscorrelations between the critical density and the radius of gyration would suggest that one of the parameters could be substituted, giving the possibility to remove a parameter from equation (13), simplifying the final version of the formula. However, substituting the critical density by a correlation containing the radius of gyration in equation (13), the AAD significantly increased, unlike what happened for alcohols. Finally, equation (13) was left in its originally proposed version. In order to compare the proposed equation with the literature ones, deviations were calculated as follows:
Table 2 Summary of deviations of surface tension experimental data for the different equations. Notation as in Table 1. Name
N. of points (E þ S)
AAD Eqs. 5 þ 6 [13]
AAD Eq. (7) [14]
AAD Eq. (8) [15]
AAD Eq. (9) [16]
AAD Eq. 13
Dimethyldichlorosilane Diphenyldichlorosilane Methyl Dichlorosilane Methyl Trichlorosilane Tetrachlorosilane Trichlorosilane Trimethylchlorosilane Decamethylcyclopentasiloxane Dodecamethylcyclohexasiloxane Hexamethylcyclotrisiloxane Octamethylcyclotetrasiloxane Decamethyltetrasiloxane Dodecamethylpentasiloxane Eicosamethylnonasiloxane Octadecamethyloctasiloxane Octamethyltrisiloxane Tetradecamethylhexasiloxane Dimethyldimethoxysilane Gamma-aminopropyltriethoxysilane Hexamethyldisilazane Tetraethoxysilane Silane Tetraethyl Silane Mean
9 7 9 12 22 27 11 23 8 9 12 15 12 3 8 12 6 1 1 10 13 9 14 e
2.5 11.3 2.3 1.9 1.7 2.2 1.1 13.6 16.1 2.6 8.0 2.3 10.2 16.3 14.7 2.3 14.2 22.4 18.3 2.4 10.8 5.0 1.5 6.0
6.7 6.9 7.6 7.1 5.1 7.1 4.3 5.3 6.9 3.5 0.6 6.0 0.8 1.0 1.4 3.5 3.5 19.1 11.5 2.3 17.3 17.4 6.1 6.0
4.7 6.9 1.5 4.9 4.8 5.2 4.2 13.6 14.7 7.7 10.3 18.6 19.3 37.8 34.8 13.5 22.1 14.5 24.5 3.6 9.5 19.5 6.4 10.8
1.5 3.5 0.9 1.3 1.7 1.8 2.4 9.3 12.8 3.1 8.2 3.1 8.4 12.1 16.4 4.6 11.1 21.7 28.5 2.8 6.5 10.5 2.1 5.3
1.4 3.7 1.0 1.4 1.9 2.4 2.2 4.0 7.4 2.2 2.8 3.5 2.7 4.8 10.3 1.5 4.6 21.8 25.5 3.7 13.3 1.1 3.6 3.7
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specifically oriented to silanes. Amongst the literature equations, the better results in terms of deviations were achieved by equations adopting the acentric factor as parameter, confirming the validity of the choice of the additional parameter.
References
Fig. 6. Residuals of surface tension data of silanes versus reduced temperature. Notation as in Fig. 3.
AAD ¼
n
sexp scalc
1X $100 sexp n i¼1
(14)
The respective AAD values were reported in Table 2. The proposed equation is generally giving the lowest deviations, of course considering that none of the considered equations was specifically designed for silanes. The residuals produced by Equation (13) are reported in Fig. 6. The prediction of surface tension of silanes is generally good, both at the low and the high reduced temperatures. Excluding one silane that generally shows higher residuals (namely tetraethoxysilane) and two silanes that present only one experimental datum (namely Dimethyldimethoxysilane and Gammaaminopropyltriethoxysilane), residuals are generally well below ±2 mN m1. As additional comment, it can be noticed that the validity of the choice of the acentric factor as additional parameter is confirmed also from the fact that the equations (7) and (9) give the better results in terms of deviations (AAD ¼ 6% and AAD ¼ 5.3%, respectively) between the literature equations. 4. Conclusions Starting from recently proposed equations for ketones and alcohols, a new equation to predict the surface tension of silanes was developed. After a factor analysis, the acentric factor was selected as additional parameter. The final equation adopts four fluid parameters, namely the critical temperature, the radius of gyration, the acentric factor and the critical density. The proposed equation was regressed with experimental and smoothed data collected from the DIPPR database. The data predicted by the database were found to be not reliable and consequently not considered during the calculations. The obtained formula is scaled and rather simple. No literature equation adopts the radius of gyration as fluid parameter for the surface tension description. When compared to the equations existing in the literature, the proposed correlation gives the lowest deviations in terms of AAD for almost all the silanes under study. It has to be pointed out that none of the literature equations was
[1] J.D. van der Waals, The thermodynamic theory of capillarity under the hypothesis of a continuous variation of density, J. Stat. Phys. 20 (1879) 200e244. ~ a, M.I. Parra, Recommended Correlations for the Surface [2] A. Mulero, I. Cachadin Tension of Common Fluids, J. Phys. Chem. Ref. Data 41 (2012), 043105 1:13. ~ a, Recommended Correlations for the Surface Tension of [3] A. Mulero, I. Cachadin Several Fluids Included in the REFPROP Program, J. Phys. Chem. Ref. Data (2014), http://dx.doi.org/10.1063/1.4878755. [4] D.B. Macleod, On a relation between surface tension and density, Trans. Farad. Soc. 19 (1923) 38e41. [5] S. Sugden, The variation of surface tension with temperature and some related functions, J. Chem. Soc. 125 (1924) 32. [6] O.R. Quayle, The parachors of organic compounds. An interpretation and catalogue, Chem. Rev. 53 (1953) 439e589. [7] J.R. Fanchi, Calculation of parachors for compositional simulation, J. Pet. Technol. 85 (1985) 2049e2050. [8] J.R. Fanchi, Calculation of parachors for compositional simulation, an update, SPE Res. Eng. 5 (1990) 433e436. [9] M.E. Boudh-Hir, G.A. Mansoori, Statistical mechanics basis of Macleod's formula, J. Phys. Chem. 94 (1990) 8362e8364. [10] J. Escobedo, G.A. Mansoori, Surface tension prediction for pure fluids, AIChE J. 42 (1996) 1425e1433. [11] A. Ferguson, S.J. Kennedy, Free and total surface energies and related quantities, Trans. Farad. Soc. 32 (1936) 1474e1481. [12] E.A. Guggenheim, The principle of corresponding states, J. Chem. Phys. 13 (1945) 253e261. [13] J.R. Brock, R.B. Bird, Surface tension and the principle of corresponding states, AIChE J. 1 (1955) 174e177. [14] K.S. Pitzer, Thermodynamics, McGraw Hill, New York, 1995. [15] S.R.S. Sastri, K.K. Rao, A simple method to predict surface tension of organic liquids, Chem. Eng. J. Biochem. Eng. J. 59 (1995) 181e186. [16] C. Miqueu, D. Broseta, J. Satherley, B. Mendiboure, J. Lachaise, A. Graciaa, An extended scaled equation for the temperature dependence of the surface tension of pure compounds inferred from an analysis of experimental data, Fluid Phase Equilib. 172 (2000) 169e182. [17] H.W. Xiang, The Corresponding-states Principle and its Practice, Elsevier, Amsterdam, 2005. [18] G. Di Nicola, C. Di Nicola, M. Moglie, A new surface tension equation for refrigerants, Int. J. Thermophys. 34 (2011) 1e18. [19] G. Di Nicola, M. Moglie, A generalized equation for the surface tension of refrigerants, Int. J. Refrig. 34 (2011) 1098e1108. [20] G. Di Nicola, M. Pierantozzi, Surface tension prediction for refrigerant binary systems, J. Refrig. 36 (2013) 562e566. [21] G. Di Nicola, M. Pierantozzi, A new scaled equation to calculate the surface tension of ketones, J. Therm. Anal. Calorim. 116 (2014) 129e134. [22] G. Di Nicola, M. Pierantozzi, Surface tension of alcohols: a scaled equation and an artificial neural network, Fluid Phase Equilib. 389 (2015) 16e27. [23] DIPPR-801, Project 801, Evaluated Process Design Data, Public Release Documentation, Design Institute for Physical Properties (DIPPR), American Institute of Chemical Engineers, AIChE, 2013. [24] J.J. Jasper, The surface tension of pure liquid compounds, J. Phys. Chem. Ref. Data 1 (1972) 841e1009. [25] NIST/SEMATECH e-Handbook of Statistical Methods, 2012. http://www.itl. nist.gov/div898/handbook/. [26] A. Turco, Factor Analysis Usage and Interpretation, Technical Report 2011009, Esteco, 2011. [27] C. Vega, L.G. MacDowell, P. Padilla, Equation of state for hard n-alkane models: long chain, J. Chem. Phys. 104 (1996) 701e713. [28] M.Yu Lobanov, N.S. Bogatyreva, O.V. Galzitskaya, Radius of gyration as an indicator of proteine structure compactness, Mol. Biol. 42 (2008) 623e628. [29] D.W. Marquardt, An algorithm for least-squares estimation of nonlinear parameters, J. Soc. Ind. Appl. Math. 11 (1963) 431e441. [30] E.W. Lemmon, M.L. Huber, M.O. McLinden, REFPROP: Reference Fluid Thermodynamic and Transport Properties, NIST Standard Reference Database 23, Version 9.12, National Institute of Standards and Technology, Gaithersburg, MD, 2013.
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Fluid Phase Equilibria j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / fl u i d
Surface tension of silanes: A new equation* Giovanni Di Nicola*, Gianluca Coccia, Mariano Pierantozzi Politecnica delle Marche, Ancona, Italy DIISM, Universita
a r t i c l e i n f o
a b s t r a c t
Article history: Received 15 July 2015 Received in revised form 27 August 2015 Accepted 18 September 2015 Available online xxx
This work presents a new formula to calculate the surface tension of silicon based molecules. As a first step, the raw surface tension data of silanes (experimental, smoothed and predicted) accepted by DIPPR database were collected. After the data analysis, only the experimental and smoothed data were considered for the calculations. The selected data were regressed with a scaled equation based on four input parameters: radius of gyration, critical density, critical temperature and acentric factor as fluid constants. Particularly relevant is the introduction of the radius of gyration as fluid parameter, that is a noticeable difference with respect to other previous literature methods. The selected data also were analyzed with the most reliable semi-empirical correlating methods in literature based on the corresponding states theory. The proposed equation is very simple and gives noticeable improvement with respect to existing equations (AAD ¼ 3.7%). © 2015 Elsevier B.V. All rights reserved.
Keywords: Factor analysis Radius of gyration Silanes Surface tension
Another easy and empirical formula was proposed by Macleod [4]:
1. Introduction Silanes are versatile materials used in a wide range of applications including adhesion promoters, crosslinking agents, dispersing agents, coupling agents, and surface modifiers. Surface tension of silanes is particularly important, because of a growing number of applications that involves silanes in controlling the interaction of water with a surface, such as hydrophobicity and hydrophylicity. Several models that have been proposed to date can be applied to describe the surface tension, but none of them was specifically oriented to silanes. The surface tension s (mN m1) of a fluid can be expressed as a function of temperature using an equation similar to that of van der Waals [1].
T n s ¼ s0 1 TC
(1)
where Tc (K) is the critical temperature, s0 (mN m1) and n are empirical constants obtained with the least square method in a fit to the experimental data available on surface tension [2,3].
* Paper presented at the 19th symposium on thermophysical properties, June 21e26, 2015, Boulder, CO, USA. * Corresponding author. E-mail address: [email protected] (G. Di Nicola).
s ¼ Kðrl rv Þ4
(2)
The formula suggests the existence of a simple relationship between surface tension and the difference between the densities of the liquid and of its vapor; K is a temperature independent constant characteristic of each substance. One year later, Equation (2) was modified by Sudgen [5]:
s ¼ ½Pðrl rv Þ3
(3)
where K ¼ P4, suggesting that the parameter P was temperature dependent; he called this parameter parachor. Parachor values of various compounds were calculated by Quayle [6] using experimental surface-tension data. Parachor property can be related to the critical properties of compounds including critical temperature and molar volume as follows [7,8]: 7=8
P ¼ 0:324$Tc4 $vc
(4)
where T is the temperature (K), v is the molar volume (m3 kmol1) and the subscript c denotes the critical value. Equation (3) may be considered as an equation of state [9] of the interface because it has been shown to be a good predictor of surface tension if equilibrium densities and experimental data for the parachor are used. As Escobedo and Mansoori suggested [10],
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its good performance and extremely easy analytical form have made Equation (3) a very popular method for calculating surface tension, but it has some shortcomings, as the percent deviation in predicting surface tension increases with increasing complexity of the molecular structure. Ferguson and Kennedy [11] and Guggenheim [12] used the corresponding states principle to correlate the surface tension in the low temperature range; this decision was justified under the assumption that, since the corresponding states principle was derived from an equation of state, surface tension is also expected to follow the same principle. Brock and Bird [13] developed this concept for non-polar liquids and proposed the following formula applicable to the range of temperature far from the critical point: 2
1
11
=
=
s ¼ Pc 3 Tc 3 Q ð1 Tr Þ 9
(5)
where Pc (bar) is the critical pressure, Tr is the reduced temperature, T/Tc. Q is the Riedel parameter developed in terms of Pc and Tbr according to the Miller suggestion,
Tbr lnðPc 1:01325Þ 0:279 1 Tbr =
Q ¼ 0:1196½1 þ
(6)
Tbr is the reduced normal boiling temperature, Tb/Tc, and Tb is the normal boiling temperature (K). In some cases, the surface tension can be represented in terms of an additional parameter. One example is the so-called acentric factor, u, initially introduced as an empirical parameter by Pitzer to explain the deviation from the corresponding states principle, as defined for noble gases, when applied to larger molecules. Pitzer's [14] relation in terms of Tc (K), Pc (bar), and u leads to the following corresponding state relationship for s: 2
1
s ¼ Pc3 Tc3
2 11 1:86 þ 1:18u 3:75 þ 0:91u 3 ð1 Tr Þ 9 19:05 0:291 0:08u
(7)
For compounds that exhibit hydrogen bonds, Sastri and Rao [15] proposed a modified expression for the Brock and Bird correlation: y
s ¼ KPcx Tb Tcz
1 Tr 1 Tbr
m (8)
where Pc is in bar. Coefficients were regressed for different families separately. To underline the work done in this research field, other recent equations have been proposed. Miqueu et al. [16] proposed the following:
s ¼ kTc
Na Vc
2 3
ð4:35 þ 4:14uÞt 1:26 1 þ 0:19t 0:5 0:487t
(9)
where t ¼ 1 T=Tc , and k; NA ; Vc ; u are Boltzmann's constant, Avogadro's number, the critical volume, and the acentric factor, respectively. This equation represents the surface tension for 31 substances up to the critical point with an average absolute deviation of less than 3.5%. Other relevant equations, again based on the corresponding states principle, were recently proposed by Xiang [17]. These equations describe the surface tension for non-polar, polar hydrogen-bonding, and associating substances. The equations represent the surface tension well, but they are also slightly more complex compared with previous equations, since they have a large number of parameters, including the aspherical factor. Recently, Di Nicola et al. [18,19] proposed two equations
specifically aimed at refrigerants. Furthermore, Di Nicola and Pierantozzi [20] issued an equation to predict the surface tension of binary refrigerant systems. A new scaled formula to calculate the surface tension of ketones [21] and alcohols [22] was also presented by the same authors. The scaled formula for alcohols was as following:
D s ¼ A$ð1 Tr ÞB $ 1 þ FC s0
(10)
where s is the surface tension (N/m), s0 is a scaled factor ¼ (k·Tc)/G2r , Tr is the reduced temperature, k is Boltzmann's constant (J/K), Gr is the radius of gyration (m), F is an adimensional term ¼ Na·rc·G3r , Na is Avogadro's number (mol1), rc is the critical density (mol m3). In this paper, to predict the surface tension of silanes, a modified version of equation (10) is presented. 2. Data analysis In this paper, the raw surface tension data (experimental, smoothed and predicted) accepted by DIPPR database [23] were collected and analyzed. The main advantage of the DIPPR database is that it collects data from a wide range of sources and evaluates them critically. References, notes, and quality codes for all data points are given [23]. During the data collection, a fluid by fluid analysis was performed. In the DIPPR database in its more recent version, the surface tension of 36 silanes are reported. The silanes are divided in the following subfamilies: 3 silanes, 13 chlorosilanes, 5 cyclosiloxanes, 6 linear siloxanes and 9 are classified by DIPPR database under the voice “other”, that can be considered as 8 Alkyl alkoxysilanes and 1 silazane. In Table 1, a summary of the range of data collected for surface tension and temperatures together with the number of points (experimental, smoothed and predicted) and the four fluid parameters adopted during the regressions are reported. The surface tension of silanes are generally predicted by DIPPR database with the Sudgen equation [5] and smoothed by Jasper [24]. In Fig. 1, all the collected from DIPPR data and calculated surface tension data are reported, split in experimental, predicted and smoothed. For the calculations, the more recent equation and better performing literature equation is considered [16]. Deviations produced by equation (9) are also reported in Table 1. From the figure and the Table analysis, it is evident that the data showing higher deviations values are generally the predicted ones. In a fact, deviations much higher than 10% were achieved for the following silanes: Methyl Chlorosilane (AAD ¼ 18.5%), Hexadecamethylcyclooctasiloxane (AAD ¼ 18.3%), [3-(2,3-epoxyproxy) propyl] trimethoxy Silane (AAD ¼ 35.3%), Bis[3-(trimethoxysilyl) propyl]disulfide (AAD ¼ 23.9%), Dimethyl-dimethoxy Silane (AAD ¼ 22.5%), Gammaaminopropyltriethoxy Silane (AAD ¼ 28.6%) and Methyl Silane (AAD ¼ 46.6%). Going into deeper, the histogram of the predicted, experimental and smoothed data are reported in Fig. 2. From the histogram, it is evident that the predicted data contribution to databank is particularly relevant at extreme surface tension values. However, from 5 to 40 mN m1 the surface tensions are well represented by the experimental data. In addition, always considering Fig. 1, the surface tension data smoothed by Jasper [24] generally gave quite low deviations. For these reasons, in this paper only the experimental and smoothed data were considered for the further calculations. Among numerous physical parameters, the fluid parameters most likely to influence experimental surface tension of silanes were identified using factor analysis [25,26]. The factor analysis
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G. Di Nicola et al. / Fluid Phase Equilibria xxx (2015) 1e6
3
Table 1 Summary of the collected data and calculated surface tension of silanes. P: predicted; E: experimental; S: smoothed. Name
3-chloropropyltrichlorosilane (3-methylacryloxypropyl) trichlorosilane Dichlorosilane Dimethyldichlorosilane Diphenyldichlorosilane Ethyltrichlorosilane Methyl Chlorosilane Methyl Dichlorosilane Methyl Trichlorosilane Phenylmethyldichlorosilane Tetrachlorosilane Trichlorosilane Trimethylchlorosilane Decamethylcyclopentasiloxane Dodecamethylcyclohexasiloxane Hexadecamethylcyclooctasiloxane Hexamethylcyclotrisiloxane Octamethylcyclotetrasiloxane Decamethyltetrasiloxane Dodecamethylpentasiloxane Eicosamethylnonasiloxane Octadecamethyloctasiloxane Octamethyltrisiloxane Tetradecamethylhexasiloxane [3-(2,3-epoxyproxy)propyl] trimethoxysilane [3-(mercapto)propyl]triethoxysilane Bis[3-(triethoxysilyl)propyl]disulfide Bis[3-(trimethoxysilyl)propyl]disulfide Dimethyldimethoxysilane Gamma-aminopropyltriethoxysilane Tetraethoxysilane Trimethoxysilane Hexamethyldisilazane Methyl Silane Silane Tetraethyl Silane a
Family
s range
Number of points E
S
T Range (K)
rc
u
2049.18 1506.02
0.381 0.627
2.9 9.6
451.5 2.714 520.35 3.535 814 5.857 559.95 4.014 442 2.5754 483 3.2563 517 3.84 689 4.827 507 3.77 479 3.439 497.75 3.4378 617.4 6.372 645.8 7.243 689.2 8.595 554.2 4.919 586.5 5.828 599.4 6.832 628.4 7.314 707.2 12.3 688.9 11.41 564.4 5.735 653.2 8.701 701.4 6.642
4219.41 2793.30 1483.68 2481.39 4065.04 3460.21 2941.18 1968.50 3067.48 3731.34 2732.24 813.01 675.68 505.05 1445.09 1016.26 909.09 714.29 409.84 405.52 1152.07 595.24 1366.12
0.163 0.267 0.428 0.270 0.225 0.276 0.263 0.402 0.232 0.203 0.270 0.622 0.710 0.883 0.496 0.584 0.660 0.720 1.050 0.984 0.538 0.777 0.760
3.5 2.5 3.5 7.6 18.5 0.9 1.8 10.0 1.7 1.8 2.3 9.2 12.8 18.3 4.1 8.6 3.1 8.4 11.5 16.4 4.6 11.1 35.3
647.5 900.3 736.2 524 634.6 592.2 525 544 352.5 269.7 606
1358.70 666.67 900.90 2267.57 1251.56 1631.32 1550.39 2450.98 4878.05 7535.80 1703.58
0.837 0.889 1.155 0.263 0.890 0.510 0.633 0.322 0.131 0.094 0.400
14.7 8.6 24.0 22.5 28.6 2.8 6.5 6.1 46.6 10.5 2.1
Tc (K)
(N$m1).103
Total
P
Chlorosilane Chlorosilane
16 15
16 15
0 0
0 0
3.54318e38.9293 4.5931e43.808
Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Chlorosilane Cyclosilane Cyclosilane Cyclosilane Cyclosilane Cyclosilane Linear siloxane Linear siloxane Linear siloxane Linear siloxane Linear siloxane Linear siloxane Alkyl alkoxysilane
10 19 7 16 10 9 23 11 22 27 12 39 24 16 25 28 15 12 19 8 12 6 12
10 10 0 16 10 0 11 11 0 0 1 16 16 16 16 16 0 0 16 0 0 0 12
0 9 7 0 0 9 12 0 14 23 11 23 8 0 9 12 15 12 3 8 12 6 0
0 0 0 0 0 0 0 0 8 4 0 0 0 0 0 0 0 0 0 0 0 0 0
2.7424e30.317 2.8358e32.113 26.66e37.24 2.76892e41.6079 2.9034e30.174 16.65e22.12 4.201e31.14 2.9438e40.412 15.73e21.26 15.52e21.09 14.54e19.61 2.3232e23. 2.4136e21. 2.55016e20.42 2.67498e14.351 2.5118e18.4 14.7e18. 15.84e18.1 2.296e25.514 15.85e18.78 14.26e17.3 16.1e18.5 7.-75.
173e413. 197.05e468.31 303.15e413.15 167.55e503.96 139.05e397.8 268.15e308.15 195.35e455.35 229.65e620.1 273.15e328.15 263.15e313.15 273.15e323.15 229.15e555.66 270.15e581.22 304.7e620.28 337.15e498.78 290.73e527.85 293.15e333. 293.15e328.15 202.15e636.48 298.15e337.15 293.15e328.15 293.15e328.15 160.75e618.22
Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Alkyl alkoxysilane Silazane Silane Silane Silane
15 16 16 12 16 13 11 10 11 9 14
15 16 16 11 15 0 11 0 11 0 0
0 0 0 1 1 13 0 10 0 0 8
0 0 0 0 0 0 0 0 0 9 6
4.8173e42.685 4.0143e36.952 5.3898e47.099 3.5609e34.531 4.735e64.966 16.75e22.21 3.1765e36.48 11.5e18.24 1.6253e21.274 14.06e26.22 13.68e23.69
210e582.75 270e810.27 250e662.58 193e471.6 140e571.14 289.65e353.15 159.6e472.5 298.15e380. 116.34e317.25 88.5e161.3 285.15e398.65
224.3e594.99 661.1 220e637.29 708.1
Gr (Å)
(mol m3) 5.297 6.474
6.16 9.732 9.223 3.791 5.944 5.4604 3.954 5.420a 1.909 1.0735 5.251a
AAD Eq. (9) [16]
Regressed values.
involves analyzing relationships among items of an instrument or large numbers of variables in order to obtain a smaller number of variables or factors. In this way, having a smaller number of variables facilitates theory development and testing. Thus, factor analysis offers not only the possibility of gaining a clear view of the data, but also the possibility of using the output in subsequent analyses. ub Let x be a vector of independent variables with xi 2½xlb i¼ i xi ; 1; …; N and y be the dependent variable. We define
Fig. 1. Calculated surface tension (equation (9)) versus DIPPR surface tension.
)
( Dþ ¼ j
x2Djxj >
xub þxlb i i 2
)
( and D j ¼
x2Djxj >
xub þxlb i i 2
; the effect of
xj is then:
Eff xj ¼
P
yðxÞ x2Dþ j
P
yðxÞ x2D j card Dþ card D j j
(11)
where cardðDþ Þ denotes the cardinality of the set Dþ . j j The factor analysis was performed for the experimental and the smoothed data only. For the factor analysis, the following physical parameters were taken into account: experimental temperature, critical temperature, critical density, critical pressure, acentric factor, boiling point temperature, melting point temperature, triple point temperature and radius of gyration. The acentric factor and the state properties are typical corresponding states principle parameters, while the radius of gyration, already adopted by us in recent papers [21,22], plays an important role in chemistry [27,28]. In a fact, it is usually a better estimate of the chain dimensions than the root-mean-squared end-to-end distance, but the end-to-end distance is difficult to measure, while the radius of gyration can be measured by a light scattering technique. Going more in detail with the radius of gyration analysis, Fig. 3 shows the relationship between the radius of gyration and the surface tension at boiling point for the silanes for which the
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Fig. 2. Histogram of experimental, predicted and smoothed data on the surface tension of silanes.
experimental and smoothed data were collected. From Fig. 3, a general trend is clearly evident. In particular, surface tension at boiling point was found to be decreasing with radius of gyration with an evident common trend, excluding three fluids, for almost all the silanes, witnessing the validity of the choice of the radius of gyration as fluid parameter. All the selected fluid physical parameters considered for the factor analysis were collected from the DIPPR database [23]. For our calculations, only values accepted by the database were considered. For two fluids, namely hexamethyldisilazane and tetraethyl silane, the radius of gyration was not found. Since a noticeable crosscorrelation between radius of gyration and critical density for the fluids is generally present, in Fig. 4 the behavior of the radius of gyration with the critical density was analyzed also for all the silanes listed in Table 1. Finally, the obtained correlation between critical density and radius of gyration (reported in Fig. 4 as dashed line) was considered for the calculation of the two missing radii of gyration:
Gr ¼ 6:184$e4:72$10
8
0:000578rc þ0:572r2c
(12)
According to equation (12), the following values were derived for hexamethyldisilazane (Gr ¼ 5.420 Å) and tetraethyl silane
Fig. 4. Radius of gyration versus critical density for the analyzed silanes.
(Gr ¼ 5.251 Å). The factor analysis showed the following statistical results: a strong effect was found, as expected, for the experimental temperature (Eff ¼ 0.0110 N/m). Between the previously adopted
Fig. 3. Surface tension at boiling point versus radius of gyration.
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5
3. A new equation Starting from Equation (10), the following modified equation was finally proposed:
D s ¼ A$ð1 Tr ÞB $ 1 þ FC $uE s0
Fig. 5. Scatter plot of surface tension data of silanes versus (1Tr)B. Notation as in Fig. 3.
physical constants of the fluids [21,22], the following effects were obtained for the critical temperature (Eff ¼ 0.0023 N/m), critical density (Eff ¼ 0.0029 N/m) and radius of gyration (Eff ¼ 0.020 N/ m). Between the additional parameters to be selected, the following effects were obtained: acentric factor (Eff ¼ 0.0033 N/m), molecular mass (Eff ¼ 0.0018 N/m), boiling point temperature (Eff ¼ 0.0016 N/m), reduced dipole moment (Eff ¼ 0.0014 N/m), critical pressure (Eff ¼ 0.0003 N/m) and critical compressibility factor (0.0001 N/m). Since the farer is the Eff value from zero, the higher is the statistical relevance of the parameter under analysis, the acentric factor was selected as additional parameter. In addition, when the Eff value is positive, the property (the surface tension in this case) is expected to be increasing with the selected parameter. This means that since the Eff value of the acentric factor is positive, the surface tension should increase with the increasing of the term containing the acentric factor. From the Eff values analysis, it is also expected that the surface tension should decrease with the critical density and the radius of gyration, while it should increase with the critical temperature.
(13)
where s0 and F are defined as for equation (10). To optimize the coefficients, the LevenbergeMarquardt curve-fitting method was adopted [29]. This method is actually a combination of two minimization methods: the gradient descent method and the GaussNewton method. The following coefficients were found after regression: A ¼ 0.00031, B ¼ 1.22975, C ¼ 0.10397, D ¼ 14.93220 and E ¼ 0.33689. Comparing equation (13) with equation (10), it is evident that the term with acentric factor, u, was added. Since the coefficient E and the acentric factors are positive and smaller than unity (excluding the acentric factors of two fluids), the new term has a direct effect on the surface tension calculation, as required by the factor analysis. In Fig. 5, a scatter plot of surface tension data of silanes divided by family versus (1Tr)B is presented; it shows a rather positive correlation and a common trend for all fluids, which underlines the importance of this term of equation (13) in calculating the surface tension. This is also confirmed by NIST's REFPROP program [30], where the same term in Equation (1) is adopted as a reference [3,4]. As shown in a previous paper [22] and from Fig. 4, the crosscorrelations between the critical density and the radius of gyration would suggest that one of the parameters could be substituted, giving the possibility to remove a parameter from equation (13), simplifying the final version of the formula. However, substituting the critical density by a correlation containing the radius of gyration in equation (13), the AAD significantly increased, unlike what happened for alcohols. Finally, equation (13) was left in its originally proposed version. In order to compare the proposed equation with the literature ones, deviations were calculated as follows:
Table 2 Summary of deviations of surface tension experimental data for the different equations. Notation as in Table 1. Name
N. of points (E þ S)
AAD Eqs. 5 þ 6 [13]
AAD Eq. (7) [14]
AAD Eq. (8) [15]
AAD Eq. (9) [16]
AAD Eq. 13
Dimethyldichlorosilane Diphenyldichlorosilane Methyl Dichlorosilane Methyl Trichlorosilane Tetrachlorosilane Trichlorosilane Trimethylchlorosilane Decamethylcyclopentasiloxane Dodecamethylcyclohexasiloxane Hexamethylcyclotrisiloxane Octamethylcyclotetrasiloxane Decamethyltetrasiloxane Dodecamethylpentasiloxane Eicosamethylnonasiloxane Octadecamethyloctasiloxane Octamethyltrisiloxane Tetradecamethylhexasiloxane Dimethyldimethoxysilane Gamma-aminopropyltriethoxysilane Hexamethyldisilazane Tetraethoxysilane Silane Tetraethyl Silane Mean
9 7 9 12 22 27 11 23 8 9 12 15 12 3 8 12 6 1 1 10 13 9 14 e
2.5 11.3 2.3 1.9 1.7 2.2 1.1 13.6 16.1 2.6 8.0 2.3 10.2 16.3 14.7 2.3 14.2 22.4 18.3 2.4 10.8 5.0 1.5 6.0
6.7 6.9 7.6 7.1 5.1 7.1 4.3 5.3 6.9 3.5 0.6 6.0 0.8 1.0 1.4 3.5 3.5 19.1 11.5 2.3 17.3 17.4 6.1 6.0
4.7 6.9 1.5 4.9 4.8 5.2 4.2 13.6 14.7 7.7 10.3 18.6 19.3 37.8 34.8 13.5 22.1 14.5 24.5 3.6 9.5 19.5 6.4 10.8
1.5 3.5 0.9 1.3 1.7 1.8 2.4 9.3 12.8 3.1 8.2 3.1 8.4 12.1 16.4 4.6 11.1 21.7 28.5 2.8 6.5 10.5 2.1 5.3
1.4 3.7 1.0 1.4 1.9 2.4 2.2 4.0 7.4 2.2 2.8 3.5 2.7 4.8 10.3 1.5 4.6 21.8 25.5 3.7 13.3 1.1 3.6 3.7
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specifically oriented to silanes. Amongst the literature equations, the better results in terms of deviations were achieved by equations adopting the acentric factor as parameter, confirming the validity of the choice of the additional parameter.
References
Fig. 6. Residuals of surface tension data of silanes versus reduced temperature. Notation as in Fig. 3.
AAD ¼
n
sexp scalc
1X $100 sexp n i¼1
(14)
The respective AAD values were reported in Table 2. The proposed equation is generally giving the lowest deviations, of course considering that none of the considered equations was specifically designed for silanes. The residuals produced by Equation (13) are reported in Fig. 6. The prediction of surface tension of silanes is generally good, both at the low and the high reduced temperatures. Excluding one silane that generally shows higher residuals (namely tetraethoxysilane) and two silanes that present only one experimental datum (namely Dimethyldimethoxysilane and Gammaaminopropyltriethoxysilane), residuals are generally well below ±2 mN m1. As additional comment, it can be noticed that the validity of the choice of the acentric factor as additional parameter is confirmed also from the fact that the equations (7) and (9) give the better results in terms of deviations (AAD ¼ 6% and AAD ¼ 5.3%, respectively) between the literature equations. 4. Conclusions Starting from recently proposed equations for ketones and alcohols, a new equation to predict the surface tension of silanes was developed. After a factor analysis, the acentric factor was selected as additional parameter. The final equation adopts four fluid parameters, namely the critical temperature, the radius of gyration, the acentric factor and the critical density. The proposed equation was regressed with experimental and smoothed data collected from the DIPPR database. The data predicted by the database were found to be not reliable and consequently not considered during the calculations. The obtained formula is scaled and rather simple. No literature equation adopts the radius of gyration as fluid parameter for the surface tension description. When compared to the equations existing in the literature, the proposed correlation gives the lowest deviations in terms of AAD for almost all the silanes under study. It has to be pointed out that none of the literature equations was
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Please cite this article in press as: G. Di Nicola, et al., Surface tension of silanes: A new equation, Fluid Phase Equilibria (2015), http://dx.doi.org/ 10.1016/j.fluid.2015.09.037