Model of Primary Recrystallization in Pure Copper

Abstract

A new model that describes primary recrystallization in metals has been proposed. This model determines the rate of recrystallization nucleus growth in terms of the kinetics of the decreasing power of disclination dipoles distributed within nucleus boundaries. The equation relating the volume fraction of recrystallized material to the temperature and annealing time has been derived. This equation has the form of the Avrami equation at n = 1 and an activation energy of diffusion along nonequilibrium grain boundaries.

APPENDIX

Numerical values of parameters

Label	Parameter	Typical numerical value	Source
Ab, Aω, Aρ	Numerical coefficients	10	[19]
b	Burgers vector	2.56 × 10–10 m	[23]
k	Coefficient in the Avrami equation	10–1–10–5 s–n	[4]–[11]
d *	The mass transfer scale at which the total charge of disclination dipoles is equal to zero	0.3 μm	[19]
D b	Diffusion coefficient at equilibrium boundaries	3.8 × 10–13 cm2/s	[19]
$D_{b}^{*}$	Diffusion coefficient at nonequilibrium boundaries	5 × 10–13–5 × 10–11 cm2/s	[19]
$D_{{b0}}^{*}$	Pre-exponential multiplier of the diffusion coefficient along nonequilibrium grain boundaries	9.8 × 10–2 cm2/s	[19]
D L0	Pre-exponential multiplier of the diffusion coefficient in the melt	1.5 × 10–3 cm2/s	[19]
G	Shear modulus	42 GPa	[23]
k	Boltzmann constant	1.38 × 10–23 J/К	[23]
M	Mobility of the defective boundary	10–12 cm3 N–1 s–1	[19]
M b	Mobility of the defect-free boundary	10–8 cm3 N–1 s–1	[19]
M ω	Mobility of the disclination dipole	10–12 cm3 N–1 s–1	[19]
M ρ	Mobility of OMDs	10–9 cm3 N–1 s–1	[19]
n	Coefficient in the Avrami equation	0.32–4.8	[4]–[11]
P	Driving forces of the growth of recrystallization nuclei	104 N/cm2	[20]–[22]
Q	Activation energy in the Avrami equation	7.6–15.3 kTm	[4]–[11]
$Q_{b}^{*}$	Diffusion activation energy along nonequilibrium boundaries	6–8.5 kTm	[19]
Q L	Diffusion activation energy in the melt	3.6 kTm	[23]
R	Universal gas constant	8.3	[23]
t	Incubation period of recrystallization	3600 s	[19]
t 3	Typical time of the decreasing power of a disclination dipole	103 s	[19]
T m	Melting temperature of copper	1357 К	[23]
W 1	Values used to calculate the diffusion coefficient along nonequilibrium boundaries and dependent on the thermodynamic characteristics of the material	6.75 kTm	[19]
W 2		11.3 kTm	[19]
Z 1		12 k	[19]
Z 2		9.6 k	[19]
α	Relative free volume of grain boundaries	0.35–0.4	[19]
α*	Threshold free volume of boundaries	0.5	[19]
Δα	Defect-induced change in the free volume of boundaries	0.01–0.1	[19]
αB	Coefficient in Eq. (6)	0.02 (at T = 0.5Tm)	[19]
δ	Grain boundary width	5 × 10–8 cm	[19]
φ	Numerical coefficient in Eq. (4)	0.1	[19]
ρbΔb	Density of OMDs	10–3	[19]
ω	Power of a disclination dipole	10–2–10–1	[19]
ω0	Initial power of a disclination dipole	10–1	[19]
Ω	Atomic volume of copper	1.18 × 10–29 m3	[23]
GΩ/kTm	–	26.5	–