Mathematical modeling of laser induced heating and melting in solids

Optics & Laser Technology 33 (2001) 533–537

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Mathematical modeling of laser induced heating and melting in solids Z.H. Shena; ∗ , S.Y. Zhanga , J. Lub , X.W. Nib b Department

a Institute of Modern Acoustics, Nanjing University, Nanjing 210093, People’s Republic of China of Applied Physics, Nanjing University of Science & Technology, Nanjing 210094, People’s Republic of China

Received 18 November 2000; accepted 20 December 2000

Abstract An analytical method for treating the problem of laser heating and melting is developed in this paper. The analytical method has been applied to aluminum, titanium, copper, silver and fused quartz and the time needed to melt and vaporize and the e6ects of laser power density on the melt depth for four metals are also obtained. In addition, the depth pro8le and time evolution of the temperature of aluminum before melting and after melting are given, in which a discontinuity in the temperature gradient is obviously observed due to the latent heat of fusion and the increment in thermal conductivity in solid phase. Additionally, the calculated melt depth evolution of fused quartz c 2001 Published by Elsevier Science Ltd. induced by 10:6
m laser irradiation is in good agreement with the experimental results.  Keywords: Laser; Heating; Melting

1. Introduction When a high power laser irradiates a material surface, a part of the laser energy is absorbed and conducted into the interior of the material. If the absorbed energy is high enough, the material surface will melt and even vaporize. On basis of this phenomenon, many practical applications including, for example, welding, cutting of metals, drilling of holes, laser shock hardening, laser glazing and information recording have been developed. The study of laser induced heating and melting has attracted great interest [1–5] and the results obtained are of great importance for achieving high quality materials processing with lasers. To simplify the problem, it is necessary to assume that the process of laser heating and melting is a linear process, that is to say, the physical parameters of the material, including density, thermal conductivity, thermal capacity, optical absorptivity, etc., are independent of the temperature. In this study, a one-dimensional heat conduction problem is solved approximately in the solid and liquid regions by assuming ∗

Corresponding author. Present address: Department of Optics, Teaching Group No. 902, East China Institute of Technology, Nanjing 210014, People’s Republic of China. E-mail address: [email protected] (Z.H. Shen). c 2001 Published by Elsevier Science Ltd. 0030-3992/01/$ - see front matter
PII: S 0 0 3 0 - 3 9 9 2 ( 0 1 ) 0 0 0 0 5 - 6

that the thermophysical properties of the material are independent of the temperature. The computations of the depth pro8le and time evolution of the temperature before melting as well as after melting are carried out for many materials including metals and non-metals. By this method, variation of the melt depth with time is obtained, in good agreement with the experimental results. The e6ects of the laser power density on the melt depth and the irradiation time on melting and vaporization are also calculated. 2. Mathematical model The geometry of laser irradiation and the resulting liquid and solid regions are shown in Fig. 1. The diameter of the laser beam is broad enough compared to the molten region and the thickness of the material is much greater than the thermal penetration depth, so that the problem can be solved in one dimension and a semi-in8nite model may be accepted. Accordingly the surface of the material reaches the fusion point or not, the whole process of laser induced heating and melting in the material is divided into two steps: before melting and after melting. Step A. Before melting. The thermal conduction model can be described by the following equations before the

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Z.H. Shen et al. / Optics & Laser Technology 33 (2001) 533–537

1 @Ts (x; t) @2 Ts (x; t) − = 0; @2 x s @t

S(t) 6 x ¡ ∞

(12)

with boundary conditions −kl

@Tl (x; t) = Al I; @x

x = 0;

Ts (x; t) = Tl (x; t) = Tm ; s L

−ks

@Ts (x; t) = As I; @x

0 6 x ¡ ∞;

x = 0;

S(t) = 0; (1) (2)

Ts (x; t) = T0 ;

x → ∞;

(3)

Ts (x; t) = T0 ;

t = 0;

(4)

where Ts ; ks ; s and As are the temperature, thermal conductivity, thermal di6usivity and absorptivity of the solid phase respectively. T0 is the ambient temperature and I is the power density of laser beam. We 8rst assume a temperature pro8le, which satis8es the boundary condition (3): Ts (x; t) = Tw (t)e−x= (t) ;

(5)

where Tw (t) represents the temperature of the surface and

(t) is a temporal function representing the temperature penetration depth in the solid. Substituting expression (5) into Eqs. (2) and (1), we get the following relations: dTw (t) s = 2 Tw (t); dt

(t)

(6)

As I

(t): Tw (t) = ks

(7)

x = S(t);

x→∞

(15) (16)

t = tm ;

(17)

where Ti ; ki and i = ki =ci i are temperature, thermal conductivity and thermal di6usivity of the ith phase respectively (i = s (solid phase) or l (liquid phase)), i and ci are the density and heat capacity of the ith phase. Al is the absorptivity of liquid phase, Tm is the melting point and L is the fusion latent heat of the irradiated material, S(t) is the position of the interface between solid phase and liquid phase. The temperature pro8les in liquid region and solid region are assumed as Tl (x; t) = Tw (t)e−x= l (t)

(18)

Ts (x; t) = Tm e−x−S(t)= s (t)

(19)

which satisfy Eqs. (14) and (16) and l (t) and s (t) are two temporal functions representing the temperature penetration depth in liquid and solid regions. Satisfying Eq. (11) at x =0 by using expression (18) and Eq. (12) at x = S(t) by using expression (14), we obtain dTw (t) l Tw (t) = 0; − 2 dt

l (t)

(20)

s dS(t) = : dt

s (t)

(21)

Substituting expression (18) into Eq. (13), we get

According to Eqs. (6) and (7), one gets As I (2t)1=2 ; Tw (t) = T0 + √ cs  s k s


(t) = 2s t:

(9) (10)

Step B. After melting. The thermal conduction equations in liquid and solid regions can be described as @2 Tl (x; t) 1 @Tl (x; t) − = 0; l @t @2 x

(14)

and initial condition

temperature of the surface reaches the fusion point: @2 Ts (x; t) 1 @Ts (x; t) − = 0; 2 @ x s @t

x = S(t);

@Ts (x; t) @Tl (x; t) dS(t) − kl = ks ; dt @x @x

Ts (x; t) = T0 ; Fig. 1. The geometry of laser irradiation.

(13)

0 6 x ¡ S(t);

(11)

k‘ Tw (t) = A‘ I:

‘ (t)

(22)

Substituting expressions (18) and (19) into Eq. (15) and by using Eq. (14), we obtain   Tm k‘ ks dS(t) = − : (23) dt s L ‘ (t) s (t) According to Eqs. (20) – (23), the following relations are obtained: 1=2  2‘ A2‘ I 2 Tw (t) = t + C ; (24) 0 k‘2

Z.H. Shen et al. / Optics & Laser Technology 33 (2001) 533–537

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Table 1 Thermophysical parameters of some materials [6,9,10]

s (kg=m3 ) ‘ (kg=m3 ) cs (J=kg K) c‘ (J=kg K) ks (W=m K) k‘ (W=m K) Tm (K) Tv (K) Lv (105 kg−1 )

Aluminum

Silver

Copper

Titanium

Fused quartz

2700 2385 917 1080 238 100 933 2793 3.88

10500 9300 235 292 429 361 1234 2485 1.112

8960 8000 386 480 401 342 1358 2836 2.047

4500 4110 528 700 21.6 20.28 1940 3558 3.65

2650 2350 863 1150 1.67 2.87 1883 2270 1.46

Fig. 2. Melt depth evolution of fused quartz. Table 2 Absorptivity of some materials [10] Absorptivity

Aluminum

Silver

Copper

Titanium

Solid phase Liquid phase

0.0588 0.064

0.02 0.043

0.02 0.058

0.257 0.433



1=2 2‘ A2‘ I 2 t + C ; 0 k‘2   1=2 2‘ A2‘ I 2 s L T m ks

s (t) = s + t + C0 ; A‘ ITm L k‘2  1=2 k‘ 2‘ A2‘ I 2 S(t) = t + C 0 A‘ I k‘2

‘ (t) =

k‘ A‘ I

ln

[2‘ A2‘ I 2 t=k‘2 + C0 ]1=2 ; Tm

where C0 = Tm2 −

(25) (26) Fig. 3. Melt depths evolution of four di6erent metals.

(27)

‘ ks2 A2‘ (Tm − T0 )2 . s k‘2 A2s

3. Numerical results and discussions We apply the above analytical solutions to aluminum, titanium, copper, silver and fused quartz. Table 1 shows their thermophysical parameters in solid and liquid phases and Table 2 shows their absorptivity at 1:06
m. 3.1. Melt depth The propagation of the solid–liquid interface expressed by Eq. (27) shows that the melt depth increases rapidly at the beginning of laser irradiation and then slowly after a certain time. Such a trend is also observed in experimental studies for the laser drilling of fused quartz [7] and aluminum [8]. The calculated melt depth evolution of fused quartz induced by 10:6
m laser irradiation compared to the experimental data [7] is shown in Fig. 2. The curve is obtained by assuming the absorptivity of the fused quartz to be 50%. This 8gure shows a good agreement between the theory and the experiment for irradiation times less than about 3 s, but

some deviations appear as the time increases. In fact, vapor and plasma will occur at large irradiation times; which block the incident laser light and absorb a portion of laser energy, but such e6ects are ignored in this work. As a result, the theoretical results seem to overestimate the melt depth for large irradiation times. Therefore, the model expressed by Eq. (27) is accurate when the vapor or plasma is not very strong, which is usually the case at the beginning of the irradiation process. The variations of melt depth with 1:06
m laser irradiation time for four di6erent metals are shown in Fig. 3. At the beginning, the melting velocity is high and then decreases to a low value. The trend is the same as that of fused quartz and is also observed in experimental studies. 3.2. Temperature pro;le and evolution The temperature 8elds as the functions of depth for aluminum at di6erent irradiation times are plotted in Fig. 4. Curves 1 and 2 represent the temperature distribution in the solid before melting, while curves 3 and 4 represent that after melting. The temperature within the liquid phase decreases rapidly from the surface temperature to the melting temperature. Beyond the solid–liquid interface in the solid phase region, the temperature decreases to the ambient temperature with a relatively gradual gradient. Such a discontinuity in the temperature gradient is obviously observed due

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Z.H. Shen et al. / Optics & Laser Technology 33 (2001) 533–537 Table 3 The time needed to melt and vaporize for some materials at di6erent power densities

Melting time (s) Vaporizing time (s)

Al Ti Cu Ag Al Ti Cu Ag

I (W=m2 ) 108

109

1010

3.414 0.1045 194 115 5.517 0.243 315 238

3:4 × 10−2 1:045 × 10−3 1.94 1.15 5:517 × 10−2 2:43 × 10−3 3.15 2.38

3:4 × 10−4 1:045 × 10−5 1:94 × 10−2 1:15 × 10−2 5:52 × 10−4 2:43 × 10−5 3:15 × 10−2 2:38 × 10−2

Fig. 4. Temperature distribution of aluminum at di6erent irradiation time.

depths of copper and silver will exceed those of aluminum and titanium at higher power density. 3.4. The time for surface to melt and vaporization Let Tw (t) in Eq. (9) equal the fusion temperature Tm . The time for the surface reaching the melting point tm is tm =

(Tm − T0 )2 cs s ks : 2(As I )2

(28)

Let Tw in Eq. (24) equal the vaporization temperature Tv . The time for the surface reaching the vaporization point tv is Fig. 5. InPuence of power density on melt depth.

to the latent heat of fusion and the increment in thermal conductivity in the solid phase. It is also seen from the 8gure that the evolution of the surface temperature after melting is much faster than before melting, which results from the higher absorptivity and lower thermal conductivity in liquid phase. 3.3. The e
tv =

(Tv2 − C0 )k‘2 : 2(A‘ I )2 ‘

(29)

The results of four metals are shown in Table 3. It is shown that titanium needs least time to reach fusion temperature and vaporization temperature among these four metals due to its low thermal conductivity although it has relatively high fusion and vaporization temperatures. 4. Conclusions An analytical method for treating the problem of the laser heating and melting is developed in this paper by suggesting a simple and reasonable temperature pro8le. We apply the analytical method to aluminum, titanium, copper, silver and fused quartz. The temperature pro8le and evolution of aluminum before melting as well as after melting is described. A discontinuity in the temperature gradient is obviously observed due to the latent heat of fusion and the increment in thermal conductivity in solid phase. The calculated melt depth evolution of fused quartz is in good agreement with the experimental results. The e6ects of laser power density on the melt depths for four metals are also obtained. It can also be concluded that titanium needs least time to reach fusion temperature and vaporization temperature among these four metals due to its low thermal conductivity although it has relatively high fusion and vaporization temperatures.

Z.H. Shen et al. / Optics & Laser Technology 33 (2001) 533–537

Acknowledgments This work was supported by the Trans-Century Talent Training Program Foundation of the State Education Ministry of China and the Nature Science Foundation of Jiangsu Province (China), as well as the Postdoctoral Foundation of Nanjing University (China). References [1] El-Adawi MK, El-Shehawey EF. Heating a slab induced by a time-dependent laser irradiation—An exact solution. J Appl Phys 1986;60(7):2250–5. [2] Hassan AF, El-Nicklawy MM, El-Adawi MK. A general problem of pulse laser heating of a slab. Opt Laser Technol 1993;25(3): 155–62.

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