Environmental Engineering Reference
In-Depth Information
simultaneous relaxation of a large number of layers [118]. In our formula, the relaxation
effect is simply considered by adding Eq. (4.2) into Eq. (4.1). According to tables 9~11, this
measure leads to satisfactory results.
It should be noted that as a simple model without free parameter, the above formula
supplies a new insight and another way for a general estimation of surface energy of
elements. This success is difficult to achieve for present first principles calculation.
Moreover, our model also supplies a basis of comparison and supplement for further
theoretical and experimental considerations on γ sv0 values of elements.
Recently, Lodziana et al . have proposed that surface energy for θ-alumina is negative
[152]. Their use of the term ″negative surface energy″ can and has caused confusion in the
scientific community [153]. Before long, a clarification is present [153]: In a single-
component system, the Gibbs dividing surface can be located arbitrarily so that there is no
surface excess quantity, which is to say that because the solid is surrounded by its own
vapor, there are no chemical effects to consider. For all stable solids then, energy input is
required for creating new surface area and hence all clean solids have positive surface
energies .
The situation is slightly more complicated in a multicomponent system (such as θ-
alumina + water), where chemical effects must be considered [153]. In such a system, the
Gibbs dividing surface can be located such that there is no excess term for one component,
but this only leads to non-zone excess quantities for the other components, which alters the
solid′s surface energy. Physically, this is a result of the interaction energies between the solid
surface and the other components. Thus, in addition to reversible work for creation of new
(clean) surface area, surface energy here also includes chemical interactions between the
newly formed surface and the surroundings. Adsorption on solid surfaces is typically an
exothermic process and it reduces the solid′s surface energy. Others have indeed shown that
chemical effects can lead to negative surface energies [154].
The Size-Dependent Surface Energy γ sv ( D )
The thermodynamic behavior of nanocrystals differs from that of the corresponding bulk
materials mainly due to the additional energetic term of γ sv ( D ) A —the product of the surface
(or interfacial) excess free energy and the surface (or interfacial) area. This term becomes
significant to change the thermal stability of the nanocrystals due to the large surface/volume
ratio of nanocrystals or A / V ∝ 1/ D [155-158]. When the surfaces of polymorphs of the same
material possess different interfacial free energies, a change in phase stability can occur with
decreasing D [159]. Despite of the fundamental thermodynamic importance of γ sv ( D ), few
reliable experimental or theoretical values are available [25,116]. The effects of size and
surrounding of nanocrystals on γ sv ( D ) are hardly studied [77,159-160].
However, in mesoscopic size range, the size-dependence of the liquid-vapor interface
energy γ lv ( D ) was thermodynamically considered fifty years ago by Tolman and Buff,
respectively [113,161]. The final form of the analytical equation is as follows [113],
γ lv ( D )/γ lv0 = 1-4δ/ D +…
(4.9)
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