Iron carbide (Fe3C) particles are dispersed in iron. The surface energy of Fe3C is 2J/m2, the shear modulus of iron is 64GPa and the Burgers vector of iron is 0.25nm. What minimum surface energy of the obstacle is required to induce dislocations to be pinned at the obstacles and bow out around them?
8 J/m^2
the answer has bben verifieded
To find the minimum surface energy of the obstacle required to induce dislocations to be pinned at the obstacles and bow out around them, we need to consider the equilibrium conditions for the interaction between dislocations and the obstacle.
When a dislocation encounters an obstacle, it can either bow out and move around the obstacle, or get pinned at the obstacle. The critical condition for the dislocation to be pinned and for bowing out to occur is when the line tension force acting on the dislocation is balanced by the pinning force exerted by the obstacle.
The line tension force acting on the dislocation can be calculated using the formula:
F = μbL
where:
F is the line tension force
μ is the shear modulus of iron (64 GPa or 64 x 10^9 Pa)
b is the Burgers vector of iron (0.25 nm or 0.25 x 10^-9 m)
L is the length of the dislocation line interacting with the obstacle
Now, the pinning force exerted by the obstacle can be derived from the surface energy. In this case, the surface energy of Fe3C particles is given as 2 J/m^2.
The pinning force can be estimated using the formula:
F_p = γ √(aL)
where:
F_p is the pinning force
γ is the surface energy of the obstacle (2 J/m^2)
a is the atomic spacing (~ interplanar spacing)
Now, since the line tension force and pinning force should balance each other, we can equate them:
μbL = γ √(aL)
Simplifying the equation:
L/√L = γ/μb√a
Squaring both sides:
L = (γ^2a)/(μ^2b^2)
Now, we need to determine the interplanar spacing or atomic spacing (a) since it is not given in the question. The interplanar spacing can be estimated using the formula:
a = d / sqrt(h^2 + k^2 + l^2)
where:
d is the interplanar spacing of the crystal planes (unknown)
h, k, l are the Miller indices (unknowns)
Without additional information or specific crystallographic details, it's not possible to calculate the interplanar spacing (d). The given information does not provide the Miller indices (h, k, l) or any indications about the crystal structure of the obstacle.
Therefore, we cannot determine the minimum surface energy of the obstacle required to induce dislocations to be pinned at the obstacles and bow out around them without the crucial information about the crystal structure and interplanar spacing.