Crystallization --> General

Estimation of surface energy.

by Leonid Sakharov

A phenomenon of surface energy plays a central role in analyses of most aspects of phase transitions in general and crystallization specifically. If for liquid surface energy (tension) can be measured directly for crystals such measurements practically impossible to carry out with any level of reliability. Most pronounce phenomena where surface energy of crystalline phase could be estimated on the base of experimental data are kinetic of crystal nucleation and growth. There two major complications for direct applying such kind of experimental data, for example temperature dependence of crystal growth rate, to calculate value of surface energy: few such experiments made with accuracy better than 3%; absence of simple, linear like, formula defined or included as coefficient a surface energy.

A numerical simulation of crystal growth phenomena explicitly demands to set value of surface energy as one of the most influential input parameter. At very least one could want to get reasonable estimation as starting point for defining a range of surface energy values to move on.

A commonly accepted explanation for phenomena of surface energy and its other manifestation, surface tension, is distortion of central like symmetry of chemical bond forces for atoms on surface comparable to atoms inside solid or liquid phases. In publication [1] the calculation of surface energy are coming from thought experiment of splitting a crystal by half creating two new surfaces and estimating energy it should take for breaking all chemical bonds along. The bond energy there is associated with heat of sublimation.

Embracing in general approach described above one could point out on obvious limitations for its direct application any other cases except crystal with known topology of chemical bonds and measured sublimation energy in vacuum. In this article more general approach will be evaluated and compared with data well established data for surface energy of several liquids.

Let presume that phase, Phase 1, with lower level of energy (crystal or liquid) is in direct contact with other phase, Phase 2, (crystal, liquid or gas). A difference between energy levels in these two phases is directly measurable parameter called heat of phase transformation or change of enthalpy - ΔHv, specific volume enthalpy of phase transition. Letís also presume that there are border layer between there two phases with effective thickness, Ls, where energy level changing linearly as it is shown at Fig. 1. It is worth to note that linear function of changing energy level is obviously an rough approximation of reality that is necessary to be able to characterize the border layer by single parameter -  Ls.  

  

A value of surface energy in such model can be defined as average energy of surface layer that will be assigned to Phase 1 and calculated by formula:


σ =0.5* ΔH * Ls                                                                        (1)

A coefficient 0.5 reflects linear slope of changing energy along surface layer.

It is reasonable to assume that value of effective thickness of surface layer should be close enough to the length of chemical bonds that is for purposes of most rough estimation could be considered as 0.1 nm.

Table 1 presents calculation of effective thickness of surface layer for several liquids these have well defined in handbooks necessary values to relay on:

Liquid

ΔH, J/cm3,

 (vaporization)

σ, erg/cm2

Ls, nm

Acetic acid

414 27.6 0.133

Acetone

426 23.7 0.111

Diethyl ether

264 17 0.128

Ethanol

660 22.27 0.067

Glycerol

1255 63 0.100

n-Hexane

219 18.4 0.167

Isopropanol

575 21.7 0.075

Mercury

10000 487 0.097

Methanol

871 22.6 0.051

Water

2443 75.64 0.0619

 

As Table 1 shows calculated values of Ls are intact with our assumption. Average value Ls for these ten liquids is 0.099 nm (0.028) that is in very good agreement with initial assessment.
Thus usage of formula (1) with actual value specific change of enthalpy at phase transition and effective value of thickness of surface layer taken as 0.1 nm should result estimation of surface energy with accuracy no worse than 50%.

References:

1.
Surface Tension of Crystals, Safa Kasap, Department of Electrical Engineering. University of Saskatchewan. Canada,


Submitted at Nov. 23, 2011; 19:52

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