Lattice Site Temperature Research Articles

Non-equilibrium energy transport in a thin metallic film: Analytical solution for radiative transport equation

Non-equilibrium energy transfer in thin aluminum film is examined and analytical solution for the radiative transport equation is presented for the electron and lattice subsystems. Electron–phonon coupling is incorporated in the radiative transport equation to account for the energy exchange between the electron and the lattice sub-systems during their thermal communications. The radiative transport equations are reduced to a system of integral equations in the form of Fredholm integral equation of the second kind and they are solved analytically through the integral transformations technique. The analytical solution for temperature distribution in lattice and electron sub-systems is simulated for various film thicknesses. The analytical solution of two-equation model is also presented to compare the findings of the radiative transport in the film. The analytical solution of the radiative transport equation is validated through the numerical predictions. It is found that numerical predictions agree well with the findings of the analytical solution. Lattice site and electron temperatures obtained from the analytical solution differ from those resulted from the two-equation model for the aluminum film thickness of 0.1µm because of the ballistic behavior of the phonons emitted at high temperature edge of the film.

A generalized heat conduction solution for ultrafast laser heating in metallic films

A generalized analytical solution for the electron and lattice temperature profiles in a metallic film exposed to an ultrafast laser source is obtained using the superposition and Fourier methods in conjunction with solution structure theorems. The generalized temperature profile in algebraic form is applicable for the solution of three thermal models, namely, phonon–electron two-temperature interaction model, electron kinetic theory model, and the improved electron kinetic theory model. By selecting the appropriate coefficients as appear in the original governing partial differential equation of a particular model, temperature profiles for both the electron and lattice can readily be obtained. A comparison of the aforementioned models in graphical form is made. Results from the phonon–electron two-temperature interaction model agree well with the improved electron kinetic theory model for electron and lattice temperatures. However, due to the omission of the time rate change of heat diffusion and laser source, temperature predictions from the electron kinetic theory model are slightly higher than temperatures from the other two models; particularly, the effect is more profound at the location of the energy incident surface. Otherwise all three models compare extremely well for the lattice site temperatures.

Short-pulse heating and analytical solution to non-equilibrium heating process

Analytical solution for lattice and electron temperatures is presented for non-equilibrium energy transfer in gold substrate subjected to a time exponentially decaying short-pulse heating. The finite Lie point symmetries and similarity solution are incorporated in the analysis. Electron temperature distribution obtained from the analytical solutions is compared to its counterpart predicted from the numerical simulations. It is found that the rate of electron temperature decay in the surface region is high, which in turn, results in high rate of lattice site temperature increase due to the collisional energy transfer from the electron sub-system to the lattice-sub system.

Laser Short-Pulse Heating: Influence of Spatial Distribution of Absorption Coefficient on Temperature Field in Silicon Film

Laser short-pulse heating of silicon film is considered and effect of the spatial distribution of absorption coefficient on electron and lattice site temperatures is examined. The high absorption region in the silicon film resembles the presence of absorbed particles in this region, since the absorption coefficient of silicon at the wavelength of the laser irradiation is significantly low. Electron and lattice temperatures are predicted using the electron kinetic theory approach. Three different spatial distributions of absorption coefficient are considered in the simulations. It is found that electron temperature attains the highest for the case of high absorption coefficient located in the surface region of the silicon film. As the high absorption region moves inside the film, electron and lattice temperatures become low.

Exact solution for temerature field due to non-equilibrium heating of solid substrate

Non-equilibrium energy transfer between electron and lattice sub-systems due to short-pulse heating is formulated and the closed form solution for electron and lattice site temperatures is presented. The electron kinetic theory approach is incorporated to formulate non-equilibrium energy transfer in the electron and lattice sub-systems. The method of Lie point symmetries is used in the exact solution of governing energy equation. In the analysis, the volumetric heat source, representing the laser heating pulse, and surface heat source, corresponding to short thermal contact of the surface, are incorporated and the analytical solutions for each heating source are presented. Electron temperature distribution obtained from the closed form solution is compared with its counterpart predicted from the numerical simulation. It is found that the results obtained from the closed form agree well with electron temperature predictions obtained from numerical simulation.

Radiative phonon transport in silicon and collisional energy transfer in aluminum films due to laser short-pulse heating: Influence of laser pulse intensity on temperature distribution

Energy transfer across aluminum and silicon films through phonon transport is examined in line with the laser short-pulse interaction with the aluminum film. The modified two-equation model is incorporated to compute electron and lattice site temperatures in the aluminum film while phonon radiative transport is used to predict equilibrium temperature in the silicon film. The thermal boundary resistance is considered at the interface of the films in the analysis. The numerical scheme using the finite difference method is adopted to solve the governing equations of energy. It is found that lattice site temperature rise is gradual in the aluminum film in the late heating period. However, equilibrium temperature decay is sharp in the region of silicon interface during this period. The thermal boundary resistance lowers lattice site temperature considerably in the region of the aluminum interface.

Analytical solution for electron and lattice site temperatures due to laser-induced non-equilibrium energy transport in metals

The present study describes the modeling of short-pulse laser heating of gold and copper materials. The energy transport due to a laser short pulse is formulated using the electron kinetic theory approach. Electron and lattice temperature distributions inside the substrate material during the laser short pulse are formulated analytically. In the analysis, the Laplace transformation technique is used. The results obtained from the analytical solutions are compared with numerical predictions and experimental data. It is found that the results obtained from the numerical and analytical solutions for temperature distributions are in good agreement with the experimental data.

Laser short-pulse heating of silicon film with the presence of metallic substrate

Laser interaction of silicon film located at he top of metallic substrate is examined and energy transport in electron and lattice sub-systems are formulated using the electron kinetic theory approach. The simulations are repeated for different substrate materials, namely gold, silver, and copper. It is found that electron temperature in the silicon film rises in the vicinity of the silicon–metallic substrate interface, despite the fact that energy absorption from the irradiated filed is significantly low in the silicon film. Lattice site temperature rises rapidly in the early heating period at the interface. In addition, lattice site temperature increase is higher in the silicon film than that corresponding to the metallic substrate.

Study into laser short-pulse heating of a layered structure

Laser short-pulse heating of a lead—silicon—gold-layered structure is considered and non-equilibrium equation in the lattice and electron subsystems is formulated using the electron kinetic theory approach. The Seebeck coefficient in the metallic and silicon layers is also formulated. Electron and lattice site temperature rise in the subsystems and the Seebeck coefficients are computed for time exponentially decaying pulse. The study is extended to include the influence of the first layer (lead layer) thickness on temperature rise and the Seebeck coefficients. It is found that the lattice site temperature across the interface of the lead and silicon layers increases sharply. The Seebeck coefficient predicted in the silicon layer is higher than in the metallic layers in the structure.

Non-equilibrium energy transport in nano-layer gold coating onto silica

The non-equilibrium energy transport in a metallic material with two layered structure having different material properties results in thermal separation of electrons and lattice sub-systems during a laser short-pulse irradiation. In this case, electron and lattice site temperature distributions become material properties and thickness of each layer dependent. In the present study, energy transport in a nanometer scale gold coating onto silica due to laser short-pulse irradiation is examined. The Seebeck coefficient in the gold coating is predicted and compared to the results obtained from the previous formulation. It is found that the maximum value of electron temperature in the gold coating increases with increasing coating thickness and lattice site temperature results in a peak in the silica in the region close to the coating silica interface. The Seebeck coefficient predicted agrees well with the results of the previous formulation.

Analytical solution for non-equilibrium energy transfer in gold: Influence of ballistic contribution of electrons on energy transfer

The analytical solution for non-equilibrium energy transfers in gold substrate and ballistic contribution of electrons to the energy transfer mechanism is examined. The non-equilibrium energy equation including the ballistic contribution of electrons is obtained using the electron kinetic theory approach. The analytical solution using the perturbation method is introduced to formulate lattice and electron temperature distributions in the film. A numerical method using the finite difference scheme is employed to predict and compare electron and lattice site temperatures to those obtained from the analytical solution. It is found that temperature remains high in the surface region of the gold film due to the cases: (i) accounting for the ballistic contribution of electrons to non-equilibrium energy transfer, and (ii) excluding the ballistic contribution to the non-equilibrium energy transfer. This is true for electron and lattice temperatures.

Laser short-pulse heating of metallic surface: Consideration of Seebeck effect

In the present study, laser short-pulse heating is formulated using an electron kinetic theory approach. Temperature predictions are compared with that obtained from two-equation model. Seebeck effect is considered during the heating process. The predicted Seebeck coefficients are compared with the results based on the early formulation. Electron excess energy loss due to Seebeck effect is compared with electron mean energy. It is found that Seebeck coefficient decays sharply in the surface region due to sharp decay of electron temperature in this region. Seebeck coefficient obtained from the present study is in agreement with the predictions based on the early formulation, provided electron temperature is used in the previous formulation. However, Seebeck coefficient differs significantly once the lattice site temperature is used in the previous formulation. Electron excess energy loss due to Seebeck effect is considerably less than electron mean energy, i.e. the ratio is in the order of 10−5.

Symmetries and approximate solution of energy transfer equations in short pulse laser heating

Energy transfer equations in laser heating are considered. Symmetries of the equations are calculated by Lie Group theory. Using the symmetries, the equations are transferred to an ordinary differential system. Equations are approximately solved by strained parameter method, a perturbation technique. A boundary value problem in which the laser beam transfers heat to the surface is treated. Closed form approximate solutions for electron and lattice site temperature rise are obtained for a solid layer heated at the surface with a time decaying intensity pulse. Analytical solutions are verified by numerical simulations.

Laser Short-Pulse Heating of a Gold Surface: Comparison of Absorption and Surface Heat Flux Heating Situations

Laser irradiation of a gold surface via volumetric absorption and surface heat flux heating are examined. Electron and lattice site temperature increases are predicted and compared for two heating situations. Time-dependent pulse intensity is used to simulate the temporal variation of an actual laser pulse. Nonequilibrium energy transfer in the substrate material is considered, and the electron kinetic theory approach is accommodated when modeling the heating process. It is found that electron and lattice site temperatures attain higher values in the surface region with absorption heating than with surface heat flux heating.

Laser Short-Pulse Heating of Gold–Silver Assembly: Entropy Generation Due to Heat and Electricity Flows in Electron Subsystem

ABSTRACT Laser short-pulse heating of a two-layer assembly consisting of gold–silver is considered. In the assembly, a gold layer of 3.24 × 10−8 m thickness is situated on top of a silver layer. Electron kinetic theory approach is introduced when predicting electron and lattice site temperature distributions. Seebeck coefficient is predicted and compared with values obtained from the previous formulation. Entropy generation in the electron subsystem is formulated and the Seebeck coefficient is accommodated to account for the thermoelectric effect. It is found that lattice site and electron temperature profiles decay smoothly across the gold–silver interface. The Seebeck coefficient follows electron temperature profiles and its predicted values agree well with the results obtained from the previous formulation. Entropy generation attains high values in the surface region because of the high flux term (J Q ) in this region. Entropy contribution due to Seebeck effect is smaller than that corresponding to energy flow in the electron subsystem.

Heating of Gold Due to Surface Heat Flux and Entropy Generation

A non-equilibrium heating situation occurs when the intensity of heat flux is high and the heating duration is shorter than the thermalization time of the substrate material. In the present study, the heating of gold due to time exponentially decaying surface heat flux is considered. Electron and lattice site temperatures are predicted from the electron kinetic theory approach. To quantify the thermodynamic irreversibility in the thermal system, entropy analysis in the electron and lattice subsystems is performed and thermal communication between both systems is formulated. It is found that electron and lattice subsystems separate thermally. Electron temperature rise is rapid while lattice temperature rise is gradual during the early heating period. Volumetric entropy generation in the electron subsystem is higher than it is in the lattice subsystem.

Analytical solution for temperature field in electron and lattice sub-systems during heating of solid film

The analytical solution for non-equilibrium temperature field in solid substrate is presented. Closed form solutions for electron and lattice site temperature rise are obtained for a solid layer heated at the surface with a time-decaying intensity pulse. In the analytical solutions, a perturbation method of strained parameters is introduced. Temperature simulations are carried out for a gold layer with different thicknesses. It is found that increasing layer thickness lowers electron and lattice site temperatures at the surface. Electron temperature at the surface decays sharply with progressing heating period, which is more pronounced for thin layer. Moreover, lattice site temperature continues to rise despite reducing electron temperature in the surface region. The results obtained from the analytical solution for the lattice site temperature agrees well with the numerical predictions.

Improved formulation of electron kinetic theory approach for laser ultra-short-pulse heating

The formulation of energy transport in electron and lattice sub-systems due to the laser ultra-short-pulse heating of metallic surface is carried out using an electron kinetic theory model. The rate of electron energy gain from the irradiated field and its dissipation through the collisional process are taken into account in the analysis. The constant and variable physical properties are introduced in the numerical simulations. The two-equation and electron kinetic theory models formulated previously are also employed for the comparison purposes. It is found that the improved electron kinetic theory formulation predicts relatively lower lattice site temperatures as compared to the two-equation and electron kinetic theory models formulated previously. The results of the improved formulation is similar to that obtained for the hyperbolic heating model.

Laser short pulse heating: Influence of pulse intensity on temperature and stress fields

Laser short-pulse heating of gold surface is considered and the influence of laser pulse intensity on the temperature and stress fields is investigated. Laser step input pulses with different pulse lengths and the same energy content are employed in the simulations. The electron kinetic theory approach employing thermomechanical coupling is introduced to model the non-equilibrium energy transport in the electron and lattice sub-systems. Thermal stress development in the lattice sub-system and temperature rise in the lattice and electron sub-systems are computed. It is found that electron temperature rises rapidly while lattice site temperature rise is gradual in the early heating period, which is more pronounced for high intensity pulses. Thermal stress component in the axial direction is compressive and its magnitude is considerably less than the yielding limit of the substrate material.

An approach for analytical solution pertinent to lattice temperature variation due to laser short-pulse heating

Non-equilibrium energy transport in the surface region of the metallic substrate occurs due to a laser irradiation, which in turn results in thermal separation of electron and lattice sub-systems. As the heating period exceeds the thermal relaxation time, both sub-systems become thermally in equilibrium having the identical temperatures. When electron and lattice temperatures become identical the corresponding instant can be called thermal equilibrium time. In the present study, an analytical formulation of lattice site temperature distribution in the domain of thermal equilibrium time is obtained. Temperature differences and temperature distributions in electron and lattice sub-systems are computed for gold. It is found that electron and lattice temperatures become identical for the heating period beyond the thermal equilibrium time. Temperature distribution obtained from the analytical solution and numerical predictions agree well.