Laser processing of materials. Temperature distributions

Transcription

1 Laser processing of materials Temperature distributions Prof. Dr. Frank Mücklich Dr. Andrés Lasagni Lehrstuhl für Funktionswerkstoffe Sommersemester 7 Contents: Temperature distributions 1. Definitions. Diffusion length 3. Temperature distribution in Bulk materials i. Heat diffusion equation ii. Constrain conditions iii. The error function iv. General solution v. Formulae for different conditions vi. Eamples of laser heating 4. Temperature distribution in Thin Films i. Heat diffusion equation ii. Constrain conditions iii. Formulae for different conditions iv. Cooling of thin-films v. Heat transfer to Substrate vi. Lateral Heat-transfer in the film

2 Temperature distributions in bulk materials LASER BEAM δ : light penetration depth hp : heat affected zone κ t (diffusion length) heat heat hp κ t k: thermal diffusivity t: interaction time The temperature rise is basically controlled by δ, hp and r. Temperature distributions in bulk materials EXAMPLE light penetration depth (δ) for metals: mts Thermal diffusivity (k) for metals: m²/s LASER TYPE: For t = 1 ns X hp ~.1-1µm For t = 1ms X hp ~ 1-3µm

3 Temperature distributions in bulk materials How to calculate the temperature as function of the time and depth? q T K +. (. q = ρ c q ) = α q (1 R) e p T t α α =1/δ : absorption coefficient R: reflectivity; K: thermal conductivity q : Power density (laser-light flu density [W/cm²]) c p : specific heat T K (, y, z, t) = (no heat lost (no radiation)) q= q= T (,y,z,t) = T Temporal condition: T (,y,z,) = T Temperature distributions in bulk materials In one-dimensional case, for uniform surface irradiation, and constant K, c p, α : T (, t) = T (1 R) q + K (1 R) q + α K (1 R) q + α K kt ierfc kt ep( α kt α ) erfc α ep( α kt + α ) erfc α (1 R) q α K kt kt kt + kt e α Where: erfc(u) is the complementary error function and ierfc(u) its integral

4 Temperature distributions in bulk materials Complementary error function: Integral of the complementary error function: Temperature distributions in bulk materials However, different simplified equations can be used in different situations (δ, hp and r ): (q = q (1-R)) metals polymers

5 Temperature distributions in bulk materials Heat depth of penetration Temperature distributions in bulk materials General solution for: Most useful condition: metals, and normal laser beams diameters (mm-cm) (1 R) q T (, t) = T + kt ierfc (during irradiation) K kt (1 R) q T (, t) = T + t t k ierfc kt ierfc ( p ) K kt ( t t p ) k (after the laser pulse with duration t p )

6 Temperature distributions in bulk materials Temperature distributions in bulk materials Eample: Laser heating of stainless steel AISI 34: t p = 1-3 s t = t p / Melting point t = t p Melting point Temperature at different intensities before laser is turned off

7 Temperature distributions in bulk materials Eample: Laser heating of stainless steel AISI 34: t p = 1-3 s t = t p Melting point Melting point t = 1 t p Temperature at different intensities after laser is turned off Temperature distributions in bulk materials Eample: Laser heating of stainless steel AISI 34: t p = 1-3 s Melting point

8 Temperature distributions in thin films. α1 q( ) = q ( A1 ) e q α =1/δ : absorption coefficient for h>δ => we can neglect the wave reflected from the film substrate However, heat release does not follow light-absorption law given that the diffusion length >> δ (up to h ~ 5 µm) In case of thicker films it is necessary to consider that heat release is not uniform in depth h K T / (, y, z, t) = (1) () Temperature distributions in thin films. α1 q( ) = q ( A1 ) e q Considering that r >>(κ1τ) 1/ => K T + q T. 1, 1, 1, (, t) = ρ1, c p1,. A1 q1 = q h. (this means that T 1 =C along ) t q = α q ( 1 R1 A1 )ep[ α ( h)] K T / (, y, z, t) = h T 1 ()=Cte (1) (1-R 1 -A 1 ) = D 1 ( = h) () Observation: R 1 +A 1 +D 1 = 1

9 Temperature distributions in thin films Considering:. α1 q( ) = q ( A1 ) e q T 1 ( ) = T ( ) = T at t = T 1 ( t) / = at = T t) = T ( t) at = h 1 ( K T / (, y, z, t) = h (1) () T t T t k = k ) 1( ) ( 1 at = h T ( t) = T at = Temperature distributions in thin films τ = laser pulse duration Thin-film temperature-time dependence R 1 +A 1 +D 1 = 1 D 1 = (Transmission) D 1 > ; A 1 > (A 1 +D 1 = 1 R 1 ) D = D 1 > ; A 1 > (A 1 +D 1 = 1 R 1 ) D > D 1 > ; A 1 ~ (D 1 = 1 R 1 ) D = D 1 > ; A 1 ~ (D 1 = 1 R 1 ) D >

10 Temperature distributions in thin films Numerical calculations q = 1 1 W/m² Film thickness = 5µm Substrate: Glass Film s temperature Film s temperature Temperature distributions in thin films Cooling of thin-films Cooling of the film depends on the heating regime For the case of an opaque film on any substrate, the temperature at the film after the laser interaction time (τ) is given by: h T ( t) = T1 ( τ ) erf τ < 1ns k( t ) τ [ t / τ ( t τ ) / τ ] µs T ( t) = T ( τ ) τ > 1 Eample: considering: τ = 1ns, h=1nm, k = 6E-3 cm²/s (glass), the time t to cool the film up to.1t 1 is 1 ns! q A1 τ T1 ( τ ) = ρ c h 1 p1 q A1 k T τ τ ( ) = K π Question: why are such cooling-times etremely short?

11 Temperature distributions in thin films Heat transfer to Substrate 1 - During laser irradiation, a more or less significant part of the thermal energy is drained through the substrate - Only a part of the substrate with depth l p is heated up: l p ~ (κ τ) 1/ 3 - The energy efficiency of the treatment can be described as: η= 1 Q d /Q a Q a = energy absorbed Q d = energy dissipated 4- The energy efficiency can be written in terms of thermal properties of the substrate and the film: ρ1c p1h η = 1+ with: ψ = ψ π ρ c k t p Temperature distributions in thin films Heat transfer to Substrate With the rise of τ (lower ψ), the film-heating efficiency drops Q d = Q d = Q a quickly! π ρ1c p1 h If: τ 4 c ρ p k then η > ½ E.g.: 1 nm Cu-film on quartz substrate : τ < 36 ns Consequently: laser thin-film treatment should be carried out in a pulsed regime at short times. This provides lower energy loss and lower risk of substrate damage!

12 Temperature distributions in thin films h Lateral Heat-transfer in the film π r π r h Generally, the lateral heat flow weakly affects the film heating due to: π r q > π r h From 3D analysis it can be proved that if: r > k 1 τ Then the lateral heating is negligible => q Strong lateral heat flow T c: temperature at the center without considering lateral heat flow T r c: temperature at the center considering lateral heat flow Temperature distributions in thin films Lateral Heat-transfer in the film Eample: Cu-Film k1 τ [ µm] Without lateral heating r > k 1 τ With lateral heating r < k 1 τ Laser interaction time [ns]

13 Temperature distributions in thin films Film-to-substrate adhesion In reality, film to substrate adhesion is not uniform over the contact area (defects, impurities, islands, etc). In case of ideal adhesion, the thermal resistance at the film-substrate interfece is zero. In case of absent of adhesion, the thermal resistance is infinite (adiabatic system). Then, the heat flow through the substrate can be written as: S1 j = S + S where S 1 is the overall area with ideal adehsion S the area with zero adhesion 1 Temperature distributions in thin films Film-to-substrate adhesion S1 j = S + S 1 The temperature of a film is lower than an adiabatically heated one but higher than a film in ideal thermal contact! The lower the adhesion, the closer the heating conditions of being adiabatic. Adiabatic heating Ideal thermal contact Question: why is the difference between curves 1 and smaller for short laser interaction-time?

14 Laser ablation Ablation: loss of material => liquid epulsion, vaporization Vapor consist on: clusters, molecules, atoms, ions, and electrons Vapor/plasma plume Liquid epulsion The higher the laser-light intensity the higher the density of species The energy required to remove an atom from a solid can be estimated from: H a [J/atom] = H V [J/g] / N s H V : enthalpy of vaporization N s =L/M: atom number density (L=Avogadro Number; M=atomic weight) P P S Plume Laser ablation - vaporization However the plume will absorb and scatter the incident laser radiation! Considering only evaporation: Energy input: E = τ [( P P )( 1 R ) P ] input s L P: laser power P s : power absorbed by the vapor plume P L : energy looses (heat conduction, radiation, convection, reaction enthalpies, etc.) R: reflectivity τ: dwell time of the laser beam P L h Energy to required to vaporize a volume A. h: E [ L + L + c ( T T ) + c ( T T )] vap = A h ρ v m ps m L v : latent heat of vaporization; L m : latent heat of melting C ps, C pl : specific heat of solid and liquid, respectively T m : melting point; T v : vaporization point; T : initial temperature ρ : density h: ablated depth; A: ablated area pl v m

15 Laser ablation - vaporization When plasma is produced, an important part of the energy is absorbed by the plume and the calculation is more complicated P P S Plume P L h A ρ Laser ablation - vaporization Combining both equations: h ( P Ps )( 1 R) Pl + c ( T T ) + c ( T T ) [ L + L ] τ v m ( φ φ )( 1 R) s ρ H v ps m φl τ Φ : laser fluence (J/cm²) This equation is only valid if: l α (optical penetration depth) << For laser powers similar to the vaporization energy: P s ~ if h ~ diff. length => bulk heating is minimized => P L ~ ( φ)( R) pl h 1 Ablation rate is proportional to Laser fluence (relative low energy densities, without considering τ ρ H v liquid epulsion) v m κ t

16 Laser ablation - vaporization One-dimensional model (surface temperature ): Considerations: We ignore any liquid layer T vs is the temperature at the solid-vapor interface and v vs is its velocity Any attenuation of incident laser light by the plume is ignored ( 1 (1 R) P ξ 1) T = + ( + ) s T Lv L m cp ρvvs = v ( ) vvs v ep T s E ξv = k B v H k B a v v o is in the order of the sound velocity within the solid ξ v is the activation energy for vaporization and can be replaced by the enthalpy of vaporization per atom/molecule we need to solve equations (1) and () simultaneously Laser ablation Influence of liquid layer: Liquid epulsion Vapor Recoil force The recoil pressure (p rec ) is originated because of the momentum conservation of the evaporated species. This is in the order of the saturated vapor pressure at T s (p sat ) and increases nonlinearly with P (W/cm²) p ~ p rec sat ( T ) s The melt-ejection flu (J) is given by: (w = laser-beam radius) J 1 1/4 m ~ prec w