Numerical calculations of temperature distribution of double layer metallic surface treated by laser beams

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Optics & Laser Technology 36 (2004) 523 – 528 www.elsevier.com/locate/optlastec

Numerical calculations of temperature distribution of double layer metallic surface treated by laser beams M. Soukieh∗ , B. Abdul Ghani, M. Hammadi Atomic Energy Commission (AECS), P.O. Box 6091, Damascus, Syria Received 8 April 2003; received in revised form 14 December 2003; accepted 15 December 2003

Abstract A mathematical model of laser beam treatment of double layer alloys (Ni/Fe, Al/Fe and Cr/Fe systems) describing the e5ect of laser beam on di5erent physical and geometrical parameters of coated layer system has been adapted. The numerical solutions of the non-homogeneous heat-transport di5erential equation could estimate the temperature of the treated region. The suggested model allows investigation of the temperature distribution as a function of treated surface and laser parameters. The physical parameters of the treated materials were taken as functions of temperature due to the change in the temperature of the treated double layer materials. ? 2003 Elsevier Ltd. All rights reserved. Keywords: Temperature distribution; Metallic surface; Laser beam

1. Introduction The improvement of the physical and mechanical properties of metallic alloy surface are still important scienti9c and technical problems. There are di5erent methods for surface modi9cation of the metallic alloys based on the increase of the heating and cooling rates, such as plasma, ion beam, laser, electron beam, and others [1,2]. Laser has unique properties for surface heating, when the electromagnetic radiation 9eld of a laser beam is absorbed within the surface layers of the treated materials (Fig. 1a). The metallic laser processing of metals can be used for cases, which cannot be treated by classical methods. The advantages of laser surface processing compared with alternative processes are chemically clean, minimum distortion, controlled thermal pro9le and therefore shape and location of heat-a5ected region and it is relatively fast and easy to automate. Practically, the used pulsed lasers have power densities in the range of 108 –109 W=cm2 with duration in the range of 10−3 –10−9 s and the laser heating spot is in the range of 10−2 –100 cm. Laser beams a5ecting the metal surfaces could lead to phase transformation and ∗ Corresponding author. Tel.: +963-1161-11926/7; fax: +963-116112289. E-mail address: [email protected] (M. Soukieh).

0030-3992/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlastec.2003.12.008

structural changes as a result of physical and chemical processes. Recently, new techniques called “laser implantation”, have been developed, by treating double-layer systems to form new phases between the substrate and the coating layer. Metal surface processing depends on laser beam parameters (such as power density, intensity, wavelength, beam divergence, beam diameter, incident angle and processing time) on material parameters (such as structure, chemical composition, phase, surface absorption of the laser beam and thermal conductivity) [3]. The heating Fux has a high energy density inside the limited processing region, which could lead to locally heated surface (∼ mm2 ) at the rate of 1010 K=s in comparison with the other heating methods (104 K=s). The temperature of locally heated materials could be higher than the phase transition temperature AC1 (in case of iron-based alloys), but less than the melting point temperature Tm [4]. The cooling rate of the treated surface depends on its thermal conductivity. The direct measurement of the heating temperature and its distribution throughout the processed laser materials is very diHcult, so usually it is numerically estimated. Therefore, this work will concentrate on the mathematical model describing the physical processes, which occur during the laser processing of double layer metallic surfaces and estimate the temperature of the treated materials and its distribution.

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M. Soukieh et al. / Optics & Laser Technology 36 (2004) 523 – 528

where T (x; t) is the spatial and temporal distribution of the temperature, q(x; t) = ((1 − r)=‘)q0 (x; t) is the energy density of the absorbed laser beam, r is the reFection coeHcient of the processed surface, q0 (x; t) is the spatial distribution of the laser power density in the exposed region, ‘ is the thickness of the treated material, (T ) is the thermal conductivity coeHcient, c∗ (T ) = c(T ) + LiF; V (T − TiF; V )=(T − TiF; V ) is the speci9c heat [7], LiF; V is the latent heat, (T − TiF; V ) is the Dirac delta function, TiF; V is the temperature of phase transition (fusion or vaporization) and (T ) is the material density. The solution of Eq. (1) can be obtained using the concept of the regular mesh in Cartesian system, where the step is equal to Lx. The following relation gives the approximation of the 9rst derivative:

Laser beam

Coated material

l1

Substrate

l2

x

(a)

λi− λi+

Laser beam

λi−1

@T (xi ; tj ) ∼ 1 [T (xi+1 ; tj ) − T (xi−1 ; tj )]: = @x 2Lxi

i−1

λi

i

λi+1

The second derivative can be approximated as

i+1

@2 T (xi ; tj ) ∼ 1 [T (xi+1 ; tj ) − 2T (xi ; tj ) − T (xi−1 ; tj )]; = @x2 Lxi2

(b) Fig. 1. (a) The scheme of the laser beam e5ect on the coated material surface, (b) the energy balance at the boundary of any sequential two sub-layers.

i = 2; 3; 4; : : : ; (m − 1); j = 2; 3; 4; : : : ; (n − 1):

When a laser beam incidents on a metal surface, a part of it reFects and another part is absorbed within a layer thickness 10−5 –10−6 cm (Fig. 1a). Consequently, the absorbed laser radiation could lead to an increase in the electron gas temperature Te inside the sub-surface layer faster than the crystal lattice atoms by several orders due to the di5erence between electron and atom masses. The change of the crystal lattice atoms temperature Tc can be expressed as dTc T e − Tc ; = dt r

(3)

The time derivative can be approximated as follows: @T (xi ; tj ) ∼ 1 [T (xi ; tj+1 ) − T (xi ; tj−1 )]: = @t 2Lt

2. Mathematical model

(4)

The numerical solution of Eq. (1) can be given by the following relation: T (xi ; tj+1 ) = T (xi ; tj ) +

n−1


RT (xi−1 ; tj ) − 2T (xi ; tj )

j=1

+T (xi+1 ; tj ) +

qabs p(xi ; tj )Lt ; c∗ (T (xi ; tj )) (T (xi ; tj ))Lx (5)

where −9

−11

s) [1]. The duwhere r is the relaxation time (10 –10 ration of the output laser pulse d (in case of free running regime) could reach the order of 10−3 –10−9 s. The laser pulse consists of several pulses (in this case); the duration of each can be about 10−6 s, which is three to 9ve orders greater than the relaxation time. After the relaxation time the temperature di5erence LT = Te − Tc will be a minimum. Therefore, the heating gradient could be described according to the classical laws of physics. The heating gradient in homogeneous and isotropic media for one-dimensional space can be expressed by the following second-order partial di5erential equation [5,6]: (T )

(2)

@2 T (x; t) @T (x; t) ; + q(x; t) = (T )c∗ (T ) 2 @x @t

(1)

R=

(T (xi ; tj ))Lt ; i ; tj )) (T (xi ; tj ))

c∗ (T (x

where p(xi ; tj ) is the power ratio released within the time period Lt and qabs is the absorbed power. The initial condition is: T (x; 0) = T0 = Tambient , 0 6 x 6 ‘ = ‘1 + ‘2 and the boundary conditions are:     @T @T −(T ) = q0 (x; t); −(T ) = 0: @x x=0 @x x=‘ The variation of the material density as a function of temperature can be given by the linear function: (T ) = 0 + 1 T . The variation of the thermal conductivity coeHcient is given as: (T ) = 0 + 1 T + 2 T 2 . The variation of the speci9c heat as a function of temperature can be written in good approximation as a second-degree polynom

M. Soukieh et al. / Optics & Laser Technology 36 (2004) 523 – 528

525

Table 1 The physical parameters as functions of temperature for some metals

Fe

20◦ C ¡ T ¡ 770◦ C 910◦ C ¡ T ¡ 1557◦ C T ¿ 1557◦ C 20◦ C ¡ T ¡ 770◦ C 770◦ C ¡ T ¡ 910◦ C 900◦ C ¡ T ¡ 1400◦ C 1350◦ C ¡ T ¡ 1550◦ C

(T ) = 83:337 − 0:12205T + 1:2720e−4 T 2 − 7:5463e−8 T 3 (T ) = −24:318 + 0:11298T − 7:9225e−5 T 2 + 1:9954e−8 T 3 (T ) = 11:423 + 2:8497e−2 T − 6:681e−6 T 2 + 3:2186e−10 T 3 C(T ) = 43:29 + 1:6166T − 5:3833e−3 T 2 + 6:3042e−6 T 3 C(T ) = 3:1729e5 − 1088:4T + 1:2473T 2 − 4:7643e−4 T 3 C(T ) = 484:40 + 0:24771T − 1:1202e−4 T 2 + 3:2115e−8 T 3 C(T ) = 4699:3 + 11:15T − 7:6589e−3 T 2 + 1:7738e−6 T 3

(T ) = 7977 − 0:62T

Ni

0◦ C ¡ T ¡ 400◦ C 400◦ C ¡ T ¡ 1400◦ C 100◦ C ¡ T ¡ 358◦ C 358◦ C ¡ T ¡ 1400◦ C

(T ) = 93:947 − 0:12933T + 1:7864e−4 T 2 − 1:4809e−7 T 3 (T ) = 56:230 + 2:0794e−2 T + 1:1830e−6 T 2 − 5:5316e−10 T 3 C(T ) = 557:11 − 2:0988T + 1:3060e−2 T 2 − 2:0857e−5 T 3 C(T ) = 422:12 + 0:32191T − 3:0126e−4 T 2 + 1:5894e−7 T 3

(T ) = 8900 − 0:45T

Cr

0◦ C ¡ T ¡ 1200◦ C 0◦ C ¡ T ¡ 2000◦ C

(T ) = 86:497 + 3:8834e−2 T − 2:8969e−4 T 2 + 4:2469e−7 T 3 C(T ) = 501:41 − 0:10168T + 2:8776e−4 T 2 + 1:654e−9 T 3

(T ) = 8807:5 − 1:5271:T

Al

0◦ C ¡ T ¡ 660◦ C 660◦ C ¡ T 660◦ C ¡ T

(T ) = 236:4 + 5:377e−2 T − 2:47e−4 T 2 + 1:94e−7 T 3 (T ) = 58:525 + 6:08e−2 T − 1:948e−5 T 2 + 1:638e−9 T 3 C(T ) = (T )=(T )= (T )

(T ) = 2699 − 0:178T

of the form: c(T ) = c0 + c1 T + c2 T 2 . The above-mentioned physical parameters for some metals [8,9] are presented in Table 1.

3. Numerical solution of the heat transport equation A computer program based on 9nite di5erence method was used to solve Eq. (1) as a one-dimensional problem for di5erent coated metals. The adapted model assumed that the coated material is divided into small slabs with a thickness of Lx and the processing time is also divided into N periods with a step of Lt = d=v, where d is the diameter of the laser spot and v is the scanning speed of the laser spot. The laser power can be estimated by the relation q0 (x; t)|x=0 =P=(0:5d)2 , where P is the laser beam power. The energy di5uses from layer i into the two considered directions as shown in Fig. 1b; The 9rst direction is toward the sub-layer i − 1 and the second one is towards the sublayer i + 1. The following relations give the thermal conductivity ± (T )i between the (i − 1), (i) and, (i + 1) layers: i± (T ) =

i i±1 (Lxi ± Lxi±1 ) : i Lxi±1 + i−1 Lxi

The following relations give the (absorbed, di5used) power (energy) and temperature at a depth x of the treated material: qabs = q0 (1 − r)Lt; ± qdi5 = i± [Ti (x ± 1; t − 1) − Ti (x; t − 1)]Lt;

− + + qdi5 ; qidi5 = qdi5

Ti (x; t) = Ti (x; t − 1) +

qabs + qidi5 :

i ci∗ Lxi

In the fusion (vaporization) case, the temperature of treated material at a depth x is calculated as follows: Ti (x; t) = TiF; V (x) + fLiF; V =ci∗ ;
f= i

×

qabs + qidi3 − i · ci∗ · Lxi · [TiF; V (x; t)−Ti (x; t−1)] ; V

i · Lxi · LF; i

where qabs , qidi5 are the inlet and outlet to sublayers absorption and di5usion energy, f is the fraction of the fused (vaporized) material at the i layer. The criteria for stability of the numerical calculations is given by the relation (i Lti = i ci∗ Lxi ) 6 12 . The used program allows the investigation of temperature distribution in iron coated with different metallic layers treated with laser beams, and estimates the e5ect of laser input parameters (such as: (P: 1000 –4000 W), (d: 0 –2 mm), (v: 5 –10 mm=s)) on the treated surface under heating temperature. The reFectivity of the coated layers was chosen in the range of 83–97%. 4. Results and discussion The temperature distribution of laser processing of iron coated by di5erent metals (iron, nickel, chromium,

M. Soukieh et al. / Optics & Laser Technology 36 (2004) 523 – 528

Temperature (C)

1200

Surface z = 100 µm

Fe Fe

800

z = 500 µm

400

z = 1000 µm z = 2000 µm

z = 500 µm

400

z = 1000 µm z = 2000 µm

0 50

100

0

150

Time (ns)

(a)

Surface 1200

z = 100 µm

Cr Fe

800

z = 500 µm

400

50

z = 1000 µm z = 2000 µm

150

Surface

1200

0

100

Time (ns)

(b)

Temperature (C )

0

Temperature (C )

Al Fe

800

0

Ni Fe

z = 100 µm 800

z = 500 µm 400

z = 1000 µm z = 2000 µm

0 0

(c)

Surface z = 100 µm

1200

Temperature (C )

526

50

100

150

Time (ns)

0

50

100

150

Time (ns)

(d)

Fig. 2. Temperature distribution on iron surface coated by di5erent metals with a thickness of 100 m treated by laser beam (diameter: 1 mm, scanning speed of laser spot: 10 cm=s, power: 1000 W, reFectivity: 90%).

1500 Temperature (C)

aluminum) for di5erent parameters of the laser beam (power, diameter, scanning speed of laser spot, : : : etc.) is studied. The amount of heat, which is required to vaporize surface layers of the materials, is in the order of 10 eV=atom. Therefore, the absorbed energy by one atom as a result of a5ecting one peak of free running laser pulses is about one order greater than the required energy to vaporize the processing region. The maximum heating temperature is determined not only by the power density, but also by the absorption and roughness of the treated surfaces. The numerical solutions of the non-homogeneous partial di5erential heat transport of Eq. (5) can estimate the temperature distribution during the laser processing for di5erent coated material parameters (reFectivity, thickness). Due to the change in the treated double layer material temperature, the physical parameters (T ), (T ), (T ), c(T ) were taken as functions of temperature (Table 1). The considered thickness of the iron substrate is about 10 mm coated by di5erent metal layers with thickness of 100 m. Fig. 2 shows the calculated temperature change of the iron coated by aluminum, chromium, and nickel. It can be noticed from Fig. 2 that (in case of Al/Fe, Cr/Fe, Ni/Fe systems) the maximum temperature is in the range of (1100 –1300◦ C), and it decreases by increasing the distance from the spot center of the laser beam. The calculated results show that, the temperature in the spot center of the laser beam could not melt the coated materials, while the heating

1200 900 600

Ni Fe

300 0 83

85

87

89

91

93

95

Reflectivity (%) Fig. 3. The surface temperature as a function of the reFectivity.

process could lead to atomic mixing between the atoms of the two considered layers, and therefore, new phases can be formed. It can also be seen from Fig. 2, in the case of the coated iron with aluminum layers, that the treated material temperature can be greater than the melting point or vaporizing temperature of the aluminum layer. Therefore, the di5usive process between substrate and coated materials cannot occur. Finally, the heated region is in the range of approximately 500 m. Figs. 3–6 give the temperature changes in the sub-surface layers of coated iron by nickel (thickness 100 m) as a function of laser beam parameters and the parameters of coating materials (reFectivity and thickness of coated

M. Soukieh et al. / Optics & Laser Technology 36 (2004) 523 – 528

1240

2500 2000

Ni Fe

1500 1000 500 0

Temperature (C)

Temperature (C)

3000

527

1220 1200

Ni Fe

1180 1160 1140

0

0.4

0.8

1.2

1.6

2

0

Diameter (mm)

20

40

60

80

100

Thickness (µm)

Fig. 4. The surface temperature as a function of the laser beam diameter.

Fig. 7. The surface maximum temperature as a function of coated material thickness.

Temperature (C)

3000 2500 2000 1500

Ni Fe

1000 500 0 0

1000

2000

3000

4000

Power (W) Fig. 5. The surface temperature as a function of the laser beam power.

Temperature (C)

1500 1200

Finally it is important to estimate the damaged laser threshold intensity of the material structure using the relation Ith ≈ [Tm ( 1 c1 ‘1 + 2 c2 ‘2 )]=Ad (This approximate formula is derived from the analytical solution of Eq. (1) [10]), Tm is the melting point of the coating materials, A is the laser absorption coeHcient of the treated surface. The calculated damage threshold intensity of the laser beam (duration of about 10−6 s and absorption coeHcient A = 0:05) for double layer alloys Ni/Fe, Al/Fe systems with substrate thickness of 10 mm and coated material thickness of 100 m will be approximately 1:3 × 1014 (W=cm2 ) and 6:8 × 1012 (W=cm2 ), respectively, while the applied intensity of the laser beam is 1:3 × 105 (W=cm2 ), which means that using laser intensity in our calculation could not lead to damage the coated substrate.

900

Ni Fe

600

5. Conclusion

300 0 4

6

8

10

Speed (mm/s) Fig. 6. The surface temperature as a function of the scanning speed of the laser spot.

material). From Figs. 3 and 4 it can be seen that the temperature of the treated surface increases with increasing power density and decreasing surface reFectivity. In addition, it can be noticed from Fig. 5 that the temperature of the Ni/Fe system increases non-linearly with increasing power of the laser beam. Fig. 6 shows that the temperature decreases by increasing the scanning rate of the laser source on the coated iron by nickel. Fig. 7 shows the surface maximum temperature as a function of coated material thickness. It can be noticed from this 9gure that the maximum temperature increases by increasing the thickness of the coated material.

The surface treatment is a subject of consideration up to now, because it seems to o5er the chance to save strategic materials to improve the surface properties for discrete areas; the laser has competitors and can give a wide variety of treatment such as hardening, annealing, and surface alloying of coated materials. A computer program based on 9nite di5erence method was used to solve Eq. (1) in case of di5erent coated metals for estimating the heating temperature of double layer alloys (M/Fe, M=Al, Cr, Ni systems). The numerical solutions of the heat transport equation predict that, the heating temperature could reach (1100 –1300◦ C) of the processing metal surface by laser beams. This model could estimate the e5ect of laser parameters on the heating temperature of the coated system. The physical parameters of the coated materials were taken as functions of temperature (for more accuracy) due to the change in the treated double layer material temperature. Also this work estimates the laser threshold energy density for material structure damage. The used method of calculation in this work can be applied for any coated metallic system.

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M. Soukieh et al. / Optics & Laser Technology 36 (2004) 523 – 528

Acknowledgements The authors would like to express their thanks to the Director General of AECS Prof. I. Othman for his continuous encouragement, guidance and support. References [1] Mirkin LI. Physical basic of laser beam material processing. Moscow, Moscow University, 1975. p. 124 [in Russian]. [2] Vykovski YuA. Ion and laser implantation of metallic materials. Moscow, Energy-Atom, 1991. p. 57 [in Russian]. [3] Ashby MF, Easterling KE. The transformation hardening of steel surfaces by laser beam. Acta Metall 1984;32:1935–48.

[4] Jain AK, Kulkani VN, Sood DK. Pulsed laser heating calculation incorporating vaporization. J Appl Phys 1981;25:127–33. [5] Grigorjants AG, Safonov AN. Methods of material surface processing using laser beams. Moscow, Vishaya Shkola, 1987 [in Russian]. [6] Wddershoven FP. Simple analytical model for continuous wave-laser melting alloying for reFectivity change at the solid–liquid boundary. Jpn J Appl Phy 1989;28:1842–4. [7] Steen WM. Laser material processing. Berlin: Springer, 1991. p. 164. [8] Touloukian YS. Thermo-physical properties of matter. In: TPRC Data Series, vol. 13. New York, NY: Plenum press; 1972. [9] Prigojac AG. Basic laser treatment of material. Moscow; 1989 [in Russian]. [10] Grigorov DZ. Heating of double layer structural by laser radiation. Phys Chem Process Mater 1977;4:14. [in Russian].