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Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study
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10.1063/1.3493464
/content/aip/journal/jcp/133/15/10.1063/1.3493464
http://aip.metastore.ingenta.com/content/aip/journal/jcp/133/15/10.1063/1.3493464

Figures

Image of FIG. 1.
FIG. 1.

Phase diagram of the symmetric binary (A,B) Lennard-Jones mixture with , cf. Eqs. (7)–(10), in the plane of variables and relative concentration of A particles (; ) for fixed . The cross shows the critical point, as obtained previously (Ref. 48). The horizontal broken line means that phase coexistence is studied at .

Image of FIG. 2.
FIG. 2.

Effective free energy of finite-size boxes of linear dimension with plotted vs at for the model of Fig. 1. The estimation of the size-dependent interfacial tension is indicated.

Image of FIG. 3.
FIG. 3.

Extrapolation of as function of , cf. Eq. (12), in order to estimate .

Image of FIG. 4.
FIG. 4.

Effective free energy plotted vs at for , as indicated.

Image of FIG. 5.
FIG. 5.

(a) Snapshot of a spherical droplet configuration formed by A particles in the background of B-particles (not shown) for , , and . (b) Same as (a), but for a cylindrical droplet, choosing . Note that our method does not at all suppress statistical fluctuations in the size and shape of these droplets, which therefore have spherical or cylindrical symmetry on the average only.

Image of FIG. 6.
FIG. 6.

Plot of vs at for . Data refer to a single run at each size to illustrate the typical noise level (200 Monte Carlo steps per particles have been used for each window of the successive umbrella sampling). For the final analysis, five such runs were averaged over.

Image of FIG. 7.
FIG. 7.

Schematic explanation of how the estimation of the functions and together allows the estimation of the concentration difference and free energy difference due to a droplet.

Image of FIG. 8.
FIG. 8.

Plot of of spherical A-rich droplet at for the binary symmetric LJ mixture of Fig. 1. The description in terms of the capillarity approximation of CNT is shown as a broken curve, using as obtained in Fig. 3. The full curve is a superposition of independent simulation results for , 16, 18, 20, 22, and 24, where a running averaging was done using the combined data set.

Image of FIG. 9.
FIG. 9.

Plot of vs . Here is taken from Fig. 3, while is estimated using Eq. (14). Ideally the estimates obtained from different values of should superimpose on a single curve. The scatter between the curves for different values of is due to residual statistical errors. The thin straight line is a fit function giving .

Image of FIG. 10.
FIG. 10.

Effective free energy density of the single-component Lennard-Jones fluid at plotted vs density for three values of , as indicated in the figure.

Image of FIG. 11.
FIG. 11.

Plot of for the one-component Lennard-Jones fluid as a function of , at , for three values of , as indicated.

Image of FIG. 12.
FIG. 12.

Extrapolation of as a function of for the simple LJ fluid at , giving . Similar exercise at gives .

Image of FIG. 13.
FIG. 13.

Plots of of spherical droplets and bubbles for the one-component LJ fluid, at , as a function of sphere radius . The capillarity approximation (CNT), is included, using the estimate of from Fig. 12.

Image of FIG. 14.
FIG. 14.

Same as Fig. 13, but for cylindrical droplets and bubbles. Note that here the -axis corresponds to the surface free energy per unit height of the cylinder.

Image of FIG. 15.
FIG. 15.

Plots of vs for spherical droplets and bubbles for the LJ fluid at (a) and (b) . Fits to functional forms (20) are included.

Image of FIG. 16.
FIG. 16.

Same as Fig. 15, but for cylindrical droplets and bubbles.

Image of FIG. 17.
FIG. 17.

(a) Radius-dependent Tolman length for bubbles and droplets with a corresponding linear fit in the range . (b) Surface tension ratio as function of the equimolar radius for bubbles and droplets with a corresponding quadratic fit in the range . All results are for the temperature .

Image of FIG. 18.
FIG. 18.

(a) Radius-dependent Tolman length for bubbles and droplets with a corresponding linear fit in the range . (b) Surface tension ratio as function of the equimolar radius for bubbles and droplets with a corresponding quadratic fit in the range . All results are for the temperature .

Tables

Generic image for table
Table I.

Comparison between DFT and simulation results for the coexistence densities and liquid-vapor surface tension at the two investigated temperatures.

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/content/aip/journal/jcp/133/15/10.1063/1.3493464
2010-10-18
2014-04-16
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Curvature dependence of surface free energy of liquid drops and bubbles: A simulation study
http://aip.metastore.ingenta.com/content/aip/journal/jcp/133/15/10.1063/1.3493464
10.1063/1.3493464
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