34
A. Hu et al.
method. The energy balance in control volume of each nodal point (m) is used as the
following governing equation [119, 120]:
Q laser + Q cond − Q conv − Q rad = E st ,
(1.3.23)
where Q laser is the heat addition by laser and considered for the first nodal point (m
= 1 or x = 0) only, Q cond is the net heat conduction flow to the neighboring nodes,
Q conv and Q rad are the convection and radiation heat loss through the CuNW surface,
and E st is the change of stored heat in the control volume of each node. Here,
for simplification, this analysis excludes heat dissipation through the substrate. This
omission of the CuNW/substrate conduction heat transfer is because its inclusion
can impose excessive complexities (from contact shape, roughness, and interfacial
transport) although their influence is not significant (due to roughness and smaller
actual contact area) and can be included in other surface heat dissipation terms (Q conv
and Q rad ). Q laser is given as the product of laser heat flux q in , absorptance (ratio of the
absorbed to the incident radiant power) β, and the first node surface area, i.e., Q laser
= q in βA, and our simulations employ q in = 4.46 × 10
10 W/m
2 for CW laser and β
= 0.5 at the wavelength of CW laser (532 nm) [121]. Q cond is calculated using nodal
temperatures (T m ) and Cu thermal conductivity, k = 400 W/m K with 1-D Fourier
heat conduction equation [119]. The convection and radiation heat transfer are calculated as Q conv = hA(T m – T ∞ ) and Q rad = εσ A(T
4
m − T
4
surr ) [119]. Here, σ represents
the Stefan-Boltzmann constant, and we use room temperature for the surrounding
and ambient air temperatures (T surr = T ∞ = 303 K), convection coefficient of h =
2000 W/m
2 K, and Cu emissivity of ε = 0.07 [122]. Lastly, specific heat capacity,
c = 385 J/kg K and density ρ = 8960 kg/m
−3 of Cu are used for the calculation of
the stored thermal energy change, i.e., E st = ρVc(∂T m /∂t), where V is the control
volume of nodal point m.
Two-Temperature Model for 1-D Nanowire With a short pulse of high-intensity
laser irradiation, nonequilibrium between electron and lattice systems dynamically
changes, and to address this nonequilibrium dynamics, electron and lattice temperatures of CuNW (T e and T l ) are separately calculated. As in the analysis of CW laser,
1-D model and FDM are employed, but for each nodal point, the thermal energy
balances for electron and lattice are considered, given by [13]:
for electron T e , Q laser − Q el→latt + Q el,cond = E el , and
(1.3.24)
for lattice T l , Q el→latt + Q latt,cond − Q conv − Q rad = E latt .
(1.3.25)
Here, the laser heat addition (Q laser = q in βA) is applied to the first node as in the
CW laser simulation but considered only in T e analysis [13, 123]. At the wavelength
of FS laser (1030 nm), β has been reported as 0.06 [121]. During a pulse with 300 fs
of duration, we assume a constant laser heat flux, q in = 1.77 × 10
17 W/m
2 , which
ensures the employed average power of FS laser (5 mW) considering 1 μm of laser
A. Hu et al.
method. The energy balance in control volume of each nodal point (m) is used as the
following governing equation [119, 120]:
Q laser + Q cond − Q conv − Q rad = E st ,
(1.3.23)
where Q laser is the heat addition by laser and considered for the first nodal point (m
= 1 or x = 0) only, Q cond is the net heat conduction flow to the neighboring nodes,
Q conv and Q rad are the convection and radiation heat loss through the CuNW surface,
and E st is the change of stored heat in the control volume of each node. Here,
for simplification, this analysis excludes heat dissipation through the substrate. This
omission of the CuNW/substrate conduction heat transfer is because its inclusion
can impose excessive complexities (from contact shape, roughness, and interfacial
transport) although their influence is not significant (due to roughness and smaller
actual contact area) and can be included in other surface heat dissipation terms (Q conv
and Q rad ). Q laser is given as the product of laser heat flux q in , absorptance (ratio of the
absorbed to the incident radiant power) β, and the first node surface area, i.e., Q laser
= q in βA, and our simulations employ q in = 4.46 × 10
10 W/m
2 for CW laser and β
= 0.5 at the wavelength of CW laser (532 nm) [121]. Q cond is calculated using nodal
temperatures (T m ) and Cu thermal conductivity, k = 400 W/m K with 1-D Fourier
heat conduction equation [119]. The convection and radiation heat transfer are calculated as Q conv = hA(T m – T ∞ ) and Q rad = εσ A(T
4
m − T
4
surr ) [119]. Here, σ represents
the Stefan-Boltzmann constant, and we use room temperature for the surrounding
and ambient air temperatures (T surr = T ∞ = 303 K), convection coefficient of h =
2000 W/m
2 K, and Cu emissivity of ε = 0.07 [122]. Lastly, specific heat capacity,
c = 385 J/kg K and density ρ = 8960 kg/m
−3 of Cu are used for the calculation of
the stored thermal energy change, i.e., E st = ρVc(∂T m /∂t), where V is the control
volume of nodal point m.
Two-Temperature Model for 1-D Nanowire With a short pulse of high-intensity
laser irradiation, nonequilibrium between electron and lattice systems dynamically
changes, and to address this nonequilibrium dynamics, electron and lattice temperatures of CuNW (T e and T l ) are separately calculated. As in the analysis of CW laser,
1-D model and FDM are employed, but for each nodal point, the thermal energy
balances for electron and lattice are considered, given by [13]:
for electron T e , Q laser − Q el→latt + Q el,cond = E el , and
(1.3.24)
for lattice T l , Q el→latt + Q latt,cond − Q conv − Q rad = E latt .
(1.3.25)
Here, the laser heat addition (Q laser = q in βA) is applied to the first node as in the
CW laser simulation but considered only in T e analysis [13, 123]. At the wavelength
of FS laser (1030 nm), β has been reported as 0.06 [121]. During a pulse with 300 fs
of duration, we assume a constant laser heat flux, q in = 1.77 × 10
17 W/m
2 , which
ensures the employed average power of FS laser (5 mW) considering 1 μm of laser