34
1 Surface Thermodynamics of Solid Electrode
Fig. 1.12 Two distinct operations of stretching (W 0 ) followed by separating (W 2 ) or of separating
(W 3 ) followed by stretching (W 1 ) on a cube [10]. The reversible works of W 0 + W 2 are equivalent
to those of W 1 + W 3 . Reprinted from [10] with permission of ASM International
(2) Now let the cube be returned to its original dimensions and then be separated
(cleaved) along an xy plane into two halves. The work W 3 required to separate the
cube into two halves is given by
W 3 = 2γ.
(1.136)
Furthermore, stretch each half in the x-direction by dx, keeping the y edge but the z
edge not constant. Let this work denote by W 1 . The work involved on the route (1),
in which the cube is first stretched and then separated, is the same as that on the route
(2), in which the cube is first separated and then stretched. As a result, the following
relationship holds:
W 0 + W 2 = W 1 + W 3 .
(1.137)
The substitution of Eqs. (1.135) and (1.136) into Eq. (1.137) leads to
W 0 + 2(γ + γ )(1 + dx) = W 1 + 2γ.
(1.138)
The difference (W 1 − W 0 ) in the work required in the stretching stage of the two
routes is two times as much as the product of the component g xx of a force (surface
stress) in the newly formed surface and of the distance dx through which the force
acts, that is,
W 1 − W 0 = 2g xx dx.
(1.139)
1 Surface Thermodynamics of Solid Electrode
Fig. 1.12 Two distinct operations of stretching (W 0 ) followed by separating (W 2 ) or of separating
(W 3 ) followed by stretching (W 1 ) on a cube [10]. The reversible works of W 0 + W 2 are equivalent
to those of W 1 + W 3 . Reprinted from [10] with permission of ASM International
(2) Now let the cube be returned to its original dimensions and then be separated
(cleaved) along an xy plane into two halves. The work W 3 required to separate the
cube into two halves is given by
W 3 = 2γ.
(1.136)
Furthermore, stretch each half in the x-direction by dx, keeping the y edge but the z
edge not constant. Let this work denote by W 1 . The work involved on the route (1),
in which the cube is first stretched and then separated, is the same as that on the route
(2), in which the cube is first separated and then stretched. As a result, the following
relationship holds:
W 0 + W 2 = W 1 + W 3 .
(1.137)
The substitution of Eqs. (1.135) and (1.136) into Eq. (1.137) leads to
W 0 + 2(γ + γ )(1 + dx) = W 1 + 2γ.
(1.138)
The difference (W 1 − W 0 ) in the work required in the stretching stage of the two
routes is two times as much as the product of the component g xx of a force (surface
stress) in the newly formed surface and of the distance dx through which the force
acts, that is,
W 1 − W 0 = 2g xx dx.
(1.139)