Nucleation on Strongly Curved Surfaces of Nanofibers
It is well known that the existence of the energy barrier of nucleation is a result of the interplay of two antagonistic tendencies: an endeavor of the system to go from initial metastable phase to a more favorable one, and a general trend to minimize the
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Nucleation on Strongly Curved Surfaces of Nanofibers Pavel Demo, Alexey Sveshnikov, and Zdenˇek Koˇz´ısˇek
It is well known that the existence of the energy barrier of nucleation is a result of the interplay of two antagonistic tendencies: an endeavor of the system to go from initial metastable phase to a more favorable one, and a general trend to minimize the area of interfaces between different phases in the system. The former leads to a negative volume contribution GV to the total Gibbs free energy of the cluster formation G, whereas the latter corresponds to the positive surface contribution GS : G D GV C GS D n C n2=3 ;
(19.1)
where is the difference of the chemical potentials of the initial metastable and the newly growing phases, n is the number of building units in the cluster, is the excess surface energy, and stands for the shape factor, which describes the ratio of the surface area of the cluster to its volume. Because the shape factor appears in the expression (19.1) for the cluster’s Gibbs free energy, it influences all basic parameters of the nucleation process, including the critical size nc and the stationary nucleation rate I: nc D
2 3
3 ;
(19.2)
P. Demo • A. Sveshnikov • Z. Koˇz´ısˇek Institute of Physics, Academy of Sciences of the Czech Republic, Cukrovarnick´a 10, 162 53 Praha 6, Czech Republic P. Demo () • A. Sveshnikov Faculty of Civil Engineering, Czech Technical University in Prague, Th´akurova 7, 166 29 Praha 6, Czech Republic e-mail: [email protected]; [email protected] ˇ ak and P. Simon ˇ J. Sest´ (eds.), Thermal Analysis of Micro, Nano- and Non-Crystalline Materials, Hot Topics in Thermal Analysis and Calorimetry 9, DOI 10.1007/978-90-481-3150-1 19, © Springer ScienceCBusiness Media Dordrecht 2013
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4 3 3 : I / exp 272 kT
(19.3)
Consequently, the knowledge of the precise value of the shape factor is no less important for the correct calculation of these parameters than the chemical potential or the surface tension . For example, the shape factors for the spherical and the cubic nucleus are, correspondingly, sphere D .36/1=3 v2=3 ;
(19.4)
cube D 6v2=3 ;
(19.5)
and
where v represents the volume of the building unit in the nucleus. Thus, the ratio of the critical size for these two relatively similar shapes is ncube c sphere
nc
D 1:9:
(19.6)
If the shape of the growing cluster deviates from the spherical cluster much more, the corresponding change of the critical size (19.2) and especially of the nucleation rate (19.3) might be several orders of magnitude. In general, the determination of the exact shape of the growing cluster is an extremely difficult task. Usually the assumption of the local thermodynamic equilibrium is utilized to simplify the problem. According to Frenkel [1] and Moran et al. [2], the shape of the nucleus minimizes its surface energy under the condition of the constant volume of the nucleus. In other words, we assume that the characteristic time of relaxation processes in the nucleus is m
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