Phase Stability in the Nanocrystalline Tio 2 System
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2 a
(1)
in addition to the environment pressure (P,) (assuming surface tension y=surface stress). The change of the Gibbs free energy (G) of the system dG = -SdT + (V - V,,ot)dP + V,,vdPx, + IdA
(2)
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Mat. Res. Soc. Symp. Proc. Vol. 481 ©1998 Materials Research Society
where S is the entropy of the system, T the temperature, V the volume of the system, V,,,o the total volume of the volume phase of all the particles involved, and A the total area of all the particles involved. At a constant temperature and an environment pressure, Equ.2 becomes dG = Vo.dP,., + IdA
(3)
Figure 1. A free spherical particle with a radius a. For the system, the total number of particles (n), A and V,,,o are respectively,
A
(4)
/ -a3 3V
n
= 4ra2
5)
Va
(6) - o53 a 3 where V, is the molar volume of the solid. Applying Maxwell relation in Equ.3, and inserting Equs. 1, 5 and 6 into the relation, we obtain .
V,,0 =-4,r(a -V-
diny
2(1-
a) (aa
1+2(1---
(7)
-
Integration of above equation yields ' 2(1 x)2 -- l+2(l_'2x In{(2§j -
(8)
where y, represents the surface tension of macroscopic particles (a--• oc). Figure 2 shows that the surface tension decreases markedly as the particle size approaches the dimension of the thickness of the surface phase. Now consider the free energy change (AG) of the system when the particle size varies from infinite dimension to nanometer-scale while fixing other conditions. Inserting y from Equ.8 into Equs.1 and 3, and inserting Equs.5 and 6 and the resulted Equ. 1 into
620
Equ.3, a differential equation relating AG with a is generated. Equ.9 is based on a regression of the numerical integration of the generated equation. AG = G(size a) - G(macro - size) =
a
.Vf()) a
(9)
where
f (x) = 5.00- 7.52x + 8.11x 2 - 3.29X3
(O30 nm, AG--5VyJa ,,l/d, within a short temperature range, Hgb can be represented as Hgb=(A+B.T)/d (J/mol)
621
(11)
Fitting DSC scanning data in [2] to Equ. 11, we obtain A=3268.4 and B=65.9 (1073-1163 K). In [3], the heat capacity (Cp) of 35 nm-grained rutile was determined within ca. 600950 K. Cp of this sample has excess value relative to that calculated from thermodynamic data [4]. This excess in Cp also arises from grain-boundaries, and is called grainboundary heat capacity (Cgb). Theoretical study [5] shows that, at T=0 K, the excess heat capacity (Cexe) of a nanocrystalline substance=0; with increase in T, Cexe increases, and it reaches a maximum at a certain T; continue to increase T, Cexe decreases rapidly. In accordance with the theoretical study, we use the following empirical equation to represent the Cgb of 35-nm grained rutile (35NMGR) in [3] and that derived from Equ. 11: Cgb
= aT3e-bT
(12)
(J / mol. K)
where a=4.93 x10-8 and b=3.22x10 3 . With Equ. 12, Hgb-Hgb(O K) (and the grain-boundary free energy Ggb-Ggb(O K) ) can be calculated from 0 K to T for 35NMGR. Hgb(O K)=Ggb(O K)=850 J/mol • K is such determined that the calculated Hgb-Hgb(O K) within 1073-1163 K coincide with those given by Equ. 11. Using Equ.4 of [2], the grain-boundary area (Agb)
