Ind. Eng. Chem. Res. 1996, 35, 3399-3402
3399
A Continuum Approach to Microscopic Surface Tension for the n-Alkanes Yong-Zhuo Su and Raymond W. Flumerfelt* Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843
Microscopic surface tensions of cavities in liquids or of droplets with radii down to about 0.5 nm have been predicted. Surface energies of small clusters are calculated from the unretarded dispersion forces at a spherical interface. The resulting expression for surface tension is a function of basic intermolecular properties and cluster size, with predictions for macroscopic surface tensions being in good agreement with experimental data (flat surface). The surface tension is found to decrease with decreasing radius, and for n-alkanes, such size effects occur below 2 nm; above this size, the predicted results closely approximate the planar values. Introduction Recently, there has been growing interest in sizedependent properties of small clusters. In particular, the effect of cluster size on surface tension is of fundamental and practical importance in a number of fields. Surface tension of very fine droplets containing 100 or fewer molecules is crucially involved in theories of nucleation (Zettlemoyer, 1969; Derjaguin, 1972; Ruckenstein and Nowakowski, 1990). Knowledge of microcluster surface properties is also of great significance in understanding of microemulsion systems (Bourrel and Schechter, 1988) and in industrial applications of microemulsions such as tertiary oil recovery, pharmaceuticals, and food processing, in nanotechnology, and in many other areas. Ever since Gibbs (1961) discussed the size effects and surface tension, many efforts have been reported to examine this subject. For example, variations of surface tension with droplet size have been argued on the grounds of thermodynamics (Tolman, 1949; Melrose, 1968) and statistical mechanics (Kirkwood and Buff, 1949). Sinanoglu (1981) derived expressions that relate microcluster surface tension to the macroscopic entropies and enthalpies. To date, most studies depend on a knowledge of macroscopic quantities and cannot predict values a priori. Predictions using statistical mechanics analysis are rather complicated and limited in practical application. Therefore, a simple continuum approach to self-consistent and quantitative prediction is appealing. From a molecular point of view, the surface tension of nonpolar systems arises entirely from the unbalanced dispersion forces at an interface. In many systems, the dispersion forces constitute the most important contribution to the total van der Waals force. They are unique in that these forces interact between all atoms and molecules, even completely neutral ones such as nalkanes. Dispersion interactions are long-range forces and can have effects at interaction distances on the order of tens of nanometers. A knowledge of the interaction potential function between molecules allows a calculation of the surface tension (Fowkes, 1964). Padday and Uffindell (1968) calculated macrosurface tension for n-alkanes by considering that two semiinfinite and parallel surfaces of a liquid, separated by a large or infinite distance, are brought together to form * Author to whom correspondence should be sent. Current address: College of Engineering, Box 870200, University of Alabama, Tuscaloosa, AL 35487-0200.
S0888-5885(96)00104-2 CCC: $12.00
a single phase. The change in total interaction energy of the system arising from only dispersion forces was obtained by summing the change of every pair interaction potential according to the pairwise additivity. Israelachvili (1973) tackled the same problem and calculated Hamaker constants on the basis of the Lifshitz theory of dispersion forces. Recently, the calculation of Hamaker constants has been refined using better UV absorption spectra data (Israelachvili, 1991). Pairwise summation (de Boer, 1936; Hamaker, 1937) is based on the assumption that interactions between pairs of particles can be simply added together to obtain the total interaction energy. This assumption is not correct for condensed media because it ignores the influence of surrounding particles on the pair interaction. The problem of additivity is avoided in the Lifshitz theory (Lifshitz, 1956; Dzyaloshinskii et al., 1961) where, by a treating bodies as continua, the interaction forces are derived in terms of bulk dielectric constants and refractive indices. The pairwise additivity, however, remains valid even within the framework of continuum theories, so long as the Hamaker constant is calculated using the Lifshitz theory (Israelchvili, 1991). In this paper, we explore the feasibility of using continuum theory of long-range intermolecular forces to investigate the effect of cluster size on surface tension. This is done by calculating the change in interaction energy required to create a spherical void inside a liquid. The resulting expression for surface tension is a function of the Hamaker constant characterizing liquid-liquid molecular interactions, the intermolecular distance, and the cluster size. Predictions from the latter for macroscopic surface tensions are in good agreement with previous measured results for n-alkanes. Theory Consider the hypothetical process (see Figure 1) in which a continuous liquid state is transferred to a state with a spherical cavity under isothermal and reversible conditions. It is analogous to the cohesion process in which two block materials are separated from contact. The free energy change associated with this process can be expressed in terms of
∆Φ ) (Φf - Φi) ) 4πR2γ
(1)
where Φ is the total potential energy in the system © 1996 American Chemical Society
3400 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996
Figure 2. Configurations of interacting particles.
when 0 < r < d0, Figure 1. A continuous liquid state is transferred to a state with a spherical cavity.
attributable to dispersion forces, R is the cavity radius, and γ is the surface tension. To estimate the surface tension from long-range attractive energy, two important assumptions are involved. (1) The intermolecular forces are limited to unretarded dispersion forces. Contributions of gas-gas and gas-liquid interactions are neglected compared to those associated with liquid-liquid interactions. The Sutherland model (Hirschfelder et al., 1954) is employed for the pair interaction potential:
{
φ1 ) -
(∫
d0
0
∫θπ r16r′2 sin θ dr′ dθ + 0
∫d∞ ∫0π r16r′2 sin θ dr′ dθ
φ2 ) -
(∫
2A π
r-d0
0
d0 ) [6M/(πFnuNA)]1/3
(3)
where M is the molecular weight, nu the number of repeating units in the molecule, NA Avogadro’s constant, and F the mass density of the liquid. Israelachvili (1973) treated atoms of hydrocarbons in a close packed arrangement to calculate the interfacial distance between the adjacent planes, and other investigators assumed that spherical -CH2- segments are arranged in close packing (Padday and Uffindell, 1968; Slattery, 1990). Because only unretarded dispersion forces are considered, the interaction potential energy per unit volume at a point (r,ϑ,φ) with any other point (r′,ϑ′,φ′) in the system is
A π2
∫V rdv6
(4)
ij
where dv is the volume element around any point (r′,ϑ′,φ′) in the body with volume V, and A is the Hamaker constant
A ) π2q2C
(5)
with q being the number of molecules per unit volume. For the interaction potential in the initial state, the integration with respect to r can be divided into two regions:
(6)
∫0π r16r′2 sin θ dr′ dθ + ij
∫r-d ∫θ r+d0
π 0
1 2 r′ sin θ dr′ dθ + rij6 ∞ π 1 r′2 sin θ dr′ dθ (7) 6 r+d0 0 rij
)
∫ ∫
(2)
This potential represents two identical rigid spheres of diameter d0, which attract each other according to an inverse power law, where C is the constant for pair potential energy, and rij is the distance between the centers of the particles. (2) Simple liquids like n-alkanes can be viewed as amorphous, continuously uniform matter up to the equilibrium spacing between constituent components such as the repeat unit -CH2-. The intermolecular distance or hard sphere diameter d0 is estimated assuming the spherical symmetry of the repeat unit -CH2-, i.e.,
ij
when d0 < r < Q0,
0
rij < d0 C φij ) r > d0 rij6 ij
)
ij
0
∞
φ)-
2A π
where
rij ) (r2 + r′2 - 2rr′ cos θ)1/2
(8)
and θ0 is determined by
d02 ) r2 + r′2 - 2rr′ cos θ0
(9)
as shown in Figure 2. To find the total energy of a system enclosed by a spherical surface of radius Q0, we sum point pairwise interaction energies and divide by 2. Thus the total attractive energy for initial state Φi is given by
∫0d
2Φi ) 4π(
0
φ1r2 dr +
∫dQ
0
0
φ2r2 dr)
(10)
Similarly, Φf, the total interaction energy for the final state, can be obtained by summing every pair interaction potential over the regions of R < r < R + d0 and R + d0 < r < Q. Using eq 1, we finally obtain
γ)
[
]
2R2 ln(2R/d0) 5 A + 2 2 18 d 3 8πR 0
(11)
which gives the surface tension in terms of the Hamaker constant A, the cluster radius R, and d0, the intermolecular distance. This result provides the basis for determining cluster size effects and the effects of the key intermolecular parameters. Finally, it should be noted that macroscopic surface tensions are measured at interfaces between two unbounded phases or in the limit as R f ∞. Equation 11 then gives
γ∞ )
A 4πd02
(12)
It is the prediction of eq 12 that can be compard to experimental data. Results and Discussion In Table 1, we compare the surface tension calculated using eq 12 with measurements of γ∞ (Jasper, 1972) for
Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 3401 Table 1. Macroscopic Surface Tensions of the n-Alkanes at 20 °C
g
n-alkane carbon no.
Fa (g/cm3)
Ma (g/mol)
db (nm)
nc
a
νec (1015 s-1)
Ac (10-20 J)
Ad (10-20 J)
γ∞(expt)e (mN/m)
γ∞(calc)f (mN/m)
γ∞(calc)g (mN/m)
5 6 7 8 9 10 11 12 13 14 15 16
0.6262 0.6603 0.6837 0.7025 0.7176 0.7300 0.7402 0.7487 0.7564 0.7628 0.7685 0.7733
72.15 86.18 100.21 114.23 128.26 142.29 156.31 170.34 184.37 198.39 212.42 226.45
0.418 0.410 0.405 0.401 0.398 0.395 0.393 0.392 0.390 0.389 0.388 0.387
1.349 1.365 1.378 1.387 1.395 1.402 1.407 1.411 1.415 1.418 1.421 1.423
1.837 1.887 1.921 1.948 1.972 1.985 1.997 2.012 2.021 2.034 2.039 2.046
2.99 2.98 2.98 2.97 2.97 2.98 2.95 2.99 2.95 2.94 2.94 2.94
3.75 4.07 4.32 4.50 4.66 4.82 4.88 5.04 5.05 5.10 5.16 5.23
3.75 4.05 4.31 4.48 4.65 4.81 4.86 5.02 5.03 5.08 5.15 5.19
16.05 18.40 20.14 21.62 22.85 23.83 24.66 25.35 25.99 26.56 27.07 27.47
17.08 19.17 20.93 22.18 23.36 24.48 25.02 26.01 26.30 26.71 27.19 27.54
18.27 19.74 21.01 21.83 22.63 23.42 23.70 24.44 24.52 24.75 25.07 25.27
a Lide (1995). b Calculated using eq 3. c Hough and White (1980). Calculated using eq 14.
d
Calculated using eq 13. e Jasper (1972). f Calculated using eq 12.
n-alkanes at 20.0 °C. In arriving at these results, the nonretarded Hamaker constants were calculated using an approximate expression, which is valid for media whose absorption spectra follow a single characteristic frequency. For two identical phases 1 interacting across a medium 3 (Israelachvili, 1991),
(
)
1 - 3 3 A ) kT 4 1 + 3
3hνe (n12 - n32)2
2
+
2 2 3/2 16x2 (n1 + n3 )
(13)
where k is the Boltzmann constant, T the absolute temperature, the dielectric constant, h the Planck constant, νe the characteristic absorption frequency, and n the refractive index. Note the good agreement between values calculated from eq 13 for the Hamaker constant and those computed more rigorously by solving the full Lifshitz equation over the complete dispersion function for the dielectric constant (Hough and White, 1980). As shown in Table 1, our predicted results agree within 5% of measured values. Table 1 also compares eq 12 with
γ∞ )
A 24πD02
(14)
using D0 ) 0.165 nm recommended by Israelachvili (1991). This “cut-off” distance is estimated very skillfully by calculating the surface energy from the binding energy between surface atoms for an idealized planar close packed solid. Equation 12 predicts macroscopic surface tensions more accurately than does eq 14. Temperature Dependence. The temperature variation of macroscopic surface tension may also be predicted. The Hankinson-Brobst-Thomson (HBT) techniques (Reid et al., 1987) are used to obtain the saturated density of liquids at various temperatures. The temperature dependence of the dielectric constant of liquids can be found in Lide (1995), and the refractive index can be estimated by Maxwell’s relationship:
n2 )
Figure 3. Comparisons between experimental and predicted macrosurface tensions for n-alkanes; lines are calculated values.
(15)
There are no data available for absorption frequency dependence on temperature for the n-alkanes, hence, νe is assumed to be a constant. Figure 3 compares the predictions from eq 12 for macroscopic surface tensions of several n-alkanes with experimental values at different temperatures. In general, the predicted values are slightly higher than the measured values at lower temperatures ( 2 nm. It suggest that the bulk surface tension is a valid concept for small clusters down to molecular dimensions, at least for nonpolar liquids. Experimental studies on limits of applicability of the Kelvin equation also imply similar results (Fisher and Israelachvili, 1981). Variations of surface tension with temperature and cluster size for n-octane and n-dodecane are shown in Figures 5 and 6. It should be noted that we have assumed the cluster sizes to be time invariant. Actually, because of molec-
3402 Ind. Eng. Chem. Res., Vol. 35, No. 10, 1996 q ) number density rij ) distance between two points R ) cluster radius T ) temperature (K) Greek Letters γ ) surface tension (mN m-1) ) dielectric constant νe ) characteristic absorption frequency (s-1) F ) mass density of bulk phase (kg m-3) φij ) pair potential energy between two points φ ) total potential energy per unit volume at a point Φ ) total potential energy of the system (J)
Literature Cited Figure 5. Microscopic surface tension as a function of the cluster size for n-octane at different temperatures.
Figure 6. Microscopic surface tension as a function of the cluster size for n-dodecane at different temperatures.
ular fluctuations, the cluster sizes will also fluctuate. For very small clusters, such effects would produce time variations in the effective surface tension. Although neglected here, such variations could be important in phase change nucleation and other short-term processes. Conclusions A simple theory applicable to nonpolar liquids has been developed to predict surface tension dependence on cluster size and temperature. Predictions for macroscopic surface tension from eq 12 are in good agreement with the experimental values for n-alkanes. The surface tension is found to decrease with decreasing radius, and for n-alkanes, such cluster size effects occur below 2 nm; above this size, the predicted results closely approximate the planar values. This report also demonstrates that continuum theory can be applied to very small clusters with proper recognition given to intermolecular forces and to the range of rigid core repulsion. The information obtained here can provide a useful starting point for future work. Acknowledgment This work has been supported by the Advanced Technology Program of the State of Texas (Project No. 999903-274). Nomenclature A ) Hamaker constant (J) C ) constant for pair potential energy d0 ) hard sphere diameter h ) Planck’s constant (6.62 × 10-34 J s-1) M ) molecular weight (g mol-1) n ) refractive index NA ) Avogadro’s number (6.023 × 1023 mol-1) nu ) number of repeating units in a molecule
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Received for review February 22, 1996 Accepted July 11, 1996X IE960104B X Abstract published in Advance ACS Abstracts, September 1, 1996.
