124
J. Liu et al.
F tri (φ, n 0 ) =
B
l
2
n
2
0 −
n
3
0
6
+
n
4
0
12
+
3
16
(B
l
− B
x
) − n 0 (1 − n 0 )
φ
2
−
1
32
−
n 0
16
φ
3
+
45
1536
φ
4
(5.39)
In addition, minimizing the free energy functional F tri with respect to φ and n 0 gives
[7]:
φ =
4
5
1
2
− n 0 +
1
4
− 5(B l − B x ) + 4n 0 (1 − n 0 )
n 0 =
1
2
− 3
185
4
− 185(B l − B x )
/37
(5.40)
As for liquid phase, φ = 0, therefore the free energy functional for liquid phase is:
F liq =
B
l
2
n
2
0 −
n
3
0
6
+
n
4
0
12
(5.41)
A phase diagram of the coexistence of triangle lattice and liquid phase is obtained by
comparing F tri and F liq . An example of such a phase diagram is shown in Fig. 5.9.
In addition, a triangle lattice of a rectangle TSV are shown in Fig. 5.10a.
Fig. 5.9 Sample phase diagram of the coexistence of triangle lattice and liquid phase. T&L denotes
the coexistence of two phases