Influence of pulsed laser processing conditions on dendrite tip radius

Applied Surface Science 109r110 Ž1997. 113–123

Influence of pulsed laser processing conditions on dendrite tip radius S. Konstantinov a

c

a,)

, I. Smurov b, G. Flamant

c

A.A. BaikoÕ Institute of Metallurgy, Russian Academy of Sciences, 49 Leninsky Prospekt, 117911 Moscow, Russia b Ecole Nationale d’Ingenieurs, 58 Rue Jean Parot, 42023 Saint-Etienne Cedex 2, France ´ Institut de Science et de Genie et Procedes, ´ des Materiaux ´ ´ ´ CNRS, BP 5 Odeillo, 66125 Font-Romeu Cedex, France Received 4 June 1996; accepted 28 October 1996

Abstract The correlation between laser processing conditions and microstructure of a solidified alloy is studied, namely, the dendrite growth during directional solidification of alloys induced by pulsed laser irradiation. A model including the thermal and solute diffusion processes as well as capillary effects at the solidrliquid interface is proposed. The dependence of dendrite tip radius on power density and pulse duration is obtained for relatively ‘soft’ regimes of laser processing, when evaporation and melt removal are negligible. The parameters of process conditions are found, when non-equilibrium conditions at the solidrliquid interface influence significantly the dendrite formation. The conditions when the dendrite growth is replaced by the planar one are determined.

1. Introduction Laser processing of alloys induces significant modification in the microstructure, which defines the material properties w1,2x. Thereby, one of the important problems of laser machining is the prediction of the correlation between process parameters and the resulting microstructure. Solidification of laser produced melt may be considered as directional, when the temperature gradient in liquid and solid has the same direction and the latent heat of fusion is dissipated through the solid. As the growth rate of solid phase increases with time during solidification, the morphology of liquidrsolid interface changes from planar to cellular and from cellular to dendritic. In particular, for directional solidification caused by laser melting, dendrite structure is formed upon resolidification of the alloy. Dendrite structures Žor dendrites. are characterized by the dendrite tip radius R, the primary spacing l1 Ždistance between neighbour dendrite needles., and secondary arm spacing l2 Ždistance between secondary branches of dendrites.. For directional solidification, these parameters are controlled by alloy composition as well as by the growth rate V and temperature gradient G imposed by processing conditions. Thus if the growth rate and temperature

)

Corresponding author.

0169-4332r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved. PII S 0 1 6 9 - 4 3 3 2 Ž 9 6 . 0 0 9 1 3 - 0

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gradient together with alloy composition are known, it is possible to determine the dendrite dimensions and, vice versa, if the dendrite structure is given, one could determine the dendrite growth rate. The physical processes determining the behaviour of a solidrliquid interface are heat transfer, phase transformation, diffusive mass transfer and capillary effect w3,4x. The self-consistent description of these processes can provide a full pattern of morphology of phase boundary during solidification. The macro-parameters of pulsed laser processing which determine the thermal course are the energy and temporal profile of the laser pulse and the spot size. A comprehensive analysis of dendrite tip growth has been done by R. Trivedi and W. Kurz, where the dendrite tip radius and tip composition are found as functions of melt undercooling or imposed velocity of solidrliquid interface determined from experiments w5x. In the present paper the comparatively ‘soft’ regimes of pulsed laser processing Žpower density of 10 8 –10 11 Wrm2 and pulse duration of 10y2 –10y9 s. of Al with 4 at% Cu alloy are examined. We neglect surface evaporation Žthat is discussed in Section 2.2.1., and the diameter of laser beam is assumed to be much larger than the melt pool depth, that permits to one dimensional treatment and to exclude the influence of convective heat transfer on solidification w6x. Thus, the thermal problem is reduced to the one dimensional transient meltingrsolidification task from which the temperature gradient at phase boundary and the growth rate of solid phase are obtained. Due to the large difference between thermal diffusivity Ž a. and solute diffusion coefficient Ž D . of metal alloys Ž a 4 D . the thermal and diffusion mass transfer problems may be separated for the processing conditions considered here. The concentration field near the dendrite needle, that may be considered as the quasi-steady state one, can be taken from the model of parabolic dendrite growth proposed by Ivantsov w7x. The aim of this paper is to determine the correlation between the parameters of pulsed laser processing and alloy micro-structure, namely, dendrite tip radius Ž R ..

2. Physical and mathematical model The simplest scheme of dendrite growth is shown in Fig. 1a. For directional solidification, the growth rate V is imposed by the processing conditions and it is equal to the velocity Ž Viso . of the isotherm corresponding to equilibrium liquidus temperature of the alloy, T l s TM y mC0 Žthis will be discussed in details below. For given V, the dendrite tip composition Ct should be determined by the solution of the diffusion problem Žtaking into account the capillary effect as well.. Dendrite tip temperature Tt is correlated with dendrite tip composition Ct through phase diagram Žsee, Fig. 1b., then undercooling DTt of the melt ahead of dendrite tip interface may be considered as a temperature difference between the liquidus temperature T l and the dendrite tip temperature Tt . 2.1. Characteristic scales of the solidification process after laser melting The prediction of the correlation between processing conditions and microscopic features of solidification rather complicated. It is necessary to analyse together a number of physical processes that control microstructure, i.e. thermal diffusion, solute diffusion and capillary effect. For directional solidification of alloys induced by pulsed laser action, the characteristic lengths of the thermal and solute diffusion processes can be estimated as follows: The heat affected depth l a l a ; Ž at .

1r2

.

Ž 1.

The freezing length l T l T ; DT0rG.

Ž 2.

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Fig. 1. Scheme of physical processes Ža., phase diagram Žb..

The solute diffusion length l D is determined by the steady-state diffusion problem in the moving coordinate system associated with solidrliquid interface, i.e. C s C0 Ž1 q w1rk y 1x expŽyVd xrD .. w8x, where d x represents the distance measured into the liquid from the dendrite tip. Hence, diffusion length can be estimated as: l D ; DrV .

Ž 3.

For typical laser irradiation conditions Žwith power density of about 10 8 y 10 11 Wrm2 ., the l D rl T ratio is much less than unity, i.e. the process has significantly different characteristic scales that allows for the simplification of the problem. The growth rate of solid phase is determined by the dissipation of the latent heat of fusion and by the temperature gradients. In a general case, this dissipation takes place on the interface having rather complicated shape. The characteristic size of the region, where the growth of solid phase occurs, equals to the freezing length l T , which is much narrower than the heat affected zone: l T - l a . That is why, from the macroscopic point

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of view, the latent heat of fusion is released at a plane, which coincides with the current position of equilibrium liquidus isotherm. It is assumed that the dendrite structure reproduces itself growing with a macroscopic velocity equal to the velocity of the liquidus isotherm ŽFig. 1a., i.e. the dendrite growth rate is stable. Thus, for directional solidification the growth rate of solid phase can be found from the macroscopic heat transfer problem. In the thermal problem of pulsed laser processing, the characteristic time of temperature variation obviously equals to the duration of heating Žlaser pulse duration.. Note, that the time of melt existence could be sufficiently different from the pulse duration, when melting starts close to the end of the laser action, i.e. the moment of the appearance of liquid phase t m is comparable to the pulse duration t : t m s 0.25p ŽT l y w x T0 . l2 ay1qy2 0 ; t . On the contrary, if t 4 t m the lifetime of liquid phase is less than 2–3 t 9 . In the present problem, the characteristic time is the solidification period, t T ; HrV where H is the maximum of melt depth and V is the average value of solidification rate. The problem can be considered as a steady-state one if its transient period is much less than the characteristic duration of the process. The transient period of the diffusive mass transfer is t D ; l D rV ; DrV 2 that is much shorter than the duration of heat transfer process Žestimated close to the solidification period t T . i.e. t D - t T . Consequently, the temperature field can be treated in the diffusion problem as a slowly varying function. Finally, the diffusion problem is reduced to a steady-state one with solidrliquid interface velocity as a parameter taken from the solution of the transient heat transfer problem. The interfacial energy effect Žcapillary effect. plays an important role in dendrite tip radius selection, when the balance between the destabilising effect of concentration and temperature gradients from the one side, and stabilising effect of the interfacial energy from the other side on morphology of solidrliquid interface is considered. In the present approach, the dendrite tip structure is determined on the basis of both heat transfer and diffusion problem. The diffusion problem is governed by the heat transfer phenomena but no counter influence is taken into account. At macroscopic level, the velocity of the solidrliquid interface and the temperature gradient are determined from the transient heat transfer problem. At microscopic level, the steady-state concentration field Žfollowing the movement of solidrliquid interface. is calculated Žnote that the diffusion phenomena are rapid as compared to the heat transfer.. Capillary effects also influence the dendrite tip radius. 2.2. Heat transfer problem 2.2.1. Mathematical model For directional solidification, the solid phase growth rate V and the temperature fields are determined by the processing conditions. The 1D thermal model includes heating, melting, cooling and solidification due to the laser pulse of rectangular time-profile impinging onto the metal surface. The energy flow is assumed to be absorbed on the irradiated surface, melting Žsolidification. is determined by the classical Stefan boundary condition, the thermal conductivity and thermal diffusivity in liquid and solid phase are assumed to be the same. As a result the heat transfer model may be written as follows w9x: The heat transfer equation is

ET

E 2T

, 0 - x - s Ž t . Ž liquid phase . , s Ž t . - x Et E x2 The boundary and initial conditions are ET q Ž t . s yl , x s 0, T s T0 , xs Ex sa

q Ž t . s q0

½

1, 0,

tFt t)t

T Ž x , 0 . s T0

Ž solid phase. , t ) 0.

Ž 4.

Ž 5. Ž 6.

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Stefan’s boundary condition at the solidrliquid interface

ET

y

Ex

xs sq0

ET Ex

Lr s

l

xssy0

T < xs sy0 s T < xssq0 s Tm y mC0 ,

Viso ,

x s sŽ t . ,

Ž 7.

where the velocity of equilibrium melting isotherm Viso is associated with the growth rate of dendrite tip V, since, for solidification Viso - 0, then V s yViso . In dimensionless terms, the system of Eqs. Ž4. – Ž7. can be rewritten as follows:

ET

s

E tX

E 2TX Ex

E TX E xX ET Ex

X

X

0 - tX -

X

x ss q0

ET

X

Ex

X

sX Ž t . - xX - , X

X

0-t -t ,

x s 0,

y X

0 - xX - sX Ž t . ,

,

s y1,

xs , X

X2

E TX E xX

s 0,

tX ) 0 xX s 0,

Ž 8. tX ) t X ,

T X s T0X ,

T X Ž x , 0 . s T0X s LXn , X

X

T X < x XssXy0 s T X < x XssXq0 s 1 y mX C0 ,

Ž 9. xX s sX Ž t . ,

Ž 10 .

x ss y0

where xX s xqrlTm , tX s tq 2 arl2 Tm2 , LX s LrcTm , n s Viso lTm rqa, mX s mrTm , T X s TrTm , t X s t q 2 arl2 Tm2 . It can be seen that the dimensionless solution of Eqs. Ž8. – Ž10. depends only on the dimensionless pulse duration t X and material properties. The dimensionless pulse duration t X is determined by the power density q and pulse duration t as well as by the material properties. When choosing q and t in such a way as to keep t q 2 s constant, the dimensionless solution of Eqs. Ž8. – Ž10. is invariant with respect to pulsed laser processing conditions. The physical meaning of this variation of the pulse duration and power density is that the surface temperature at the end of the laser pulse is the same. If, for a given couple of q and t , the resulting T Ž x s 0, t s t . s Tmax is low enough to neglect evaporation, for any q and t values satisfying the condition q 2t s constant, evaporation remains negligible. Note that the value of the constant should be always the same, and by varying this value Tmax will change. As a result for any chosen Tmax and q 2t s constant it is enough to solve the meltingrsolidification problem only once. 2.2.2. Numerical procedure The system of equations is solved numerically by the finite difference method. The position of equilibrium melting isotherm is continuously tracked and the latent heat release is treated as a moving boundary condition. In both regions of liquid and solid phases the moving nonuniform grids consist of a fixed number of nodes. Each grid node moves with a different velocity. The Crank–Nicolson technique of various derivatives is used. 2.2.3. Dimensionless solution of the heat transfer problem Fig. 2 presents the dimensionless velocity of the solidrliquid interface n versus coordinate xX , for dimensionless pulse duration t X s 10. Positive values of n correspond to melting, and negative ones to solidification. The absolute value of Õ during solidification process increases from 0 to maximum Ž nmax s 7.52 = 10y2 . and then monotonically decreases up to 4.64 = 10y2 . The average value of n during solidification is 6.03 = 10y2 . The difference between the mean value and the maximum or minimum at the end of solidification is less than 25%. This should be noted, because to obtain the value of dendrite tip radius the mean value of n Žduring solidification. and the mean value of the effective temperature gradient g will be used. The behaviour of the effective temperature gradient at solidrliquid interface is shown in Fig. 3. During solidification the absolute value of gradient decreases monotonically and its mean value is equal to y2.3 = 10y2 .

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X

Fig. 2. Dimensionless solidrliquid interface velocity Õ as a function of the dimensionless coordinate x .

2.3. Diffusion problem A comparison of the characteristic time scales of heat transfer Žsolidification period t T ; l TrV . and diffusion mass transfer Žtransient period t D ; DrV 2 . processes shows that t D rt T - 1. It means that the solution of the heat transfer problem may be considered as a slowly varying function Ži.e. parameter. in the solution of the diffusion problem. On the other hand, in the coordinate system associated with the phase boundary the diffusion process is a steady-state one, because the duration of diffusion process Žwhich is about solidification period t T . is much more than its transient period t D . It allows one to determine the dendrite tip composition Ž Ct . and concentration gradient Ž Gc . ahead of the dendrite growth front from the steady-state

X

Fig. 3. Dimensionless effective temperature gradient g at solidrliquid interface as a function of the dimensionless coordinate x .

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diffusion mass transfer problem, when the dendrite growth rate V is imposed by the transient heat transfer model. Assuming parabolic needle shape the solution can be obtained from Ivantsov’s model, which gives: Cl y C0 s Cl Ž 1 y k . In Ž P . Ct s Cl k GC s Ž 1 y k . VClrD, where C l is the solute concentration in liquid phase at the dendrite tip interface, the function In Ž P . s P expŽ P . E1Ž P . w10x, in which E1Ž P . is the exponential integral function, and P is the Peclet number of the solute of the dendrite tip, given by P s VRr2 D. The value GC can be therefore expressed as: GC s

Ž 1 y k . C0 V . D 1 y Ž 1 y k . In Ž P .

Ž 11 .

2.4. Selection criteria for dendrite tip radius The capillary effect has a stabilising influence on the solidrliquid interface. In order to take it into account and to select a unique radius from Ivantsov’s solution, another relationship, called the selection criteria is required. In terms of the corresponding temperature gradients which express driving forces for stabilisationrdestabilisation, the following expression can thus be written for the dendrite tip w11–13x mGC j C q G s

1

s

)

G R2

,

Ž 12 .

where the term mGC j C represents the effective liquidus temperature gradient, j C being a function of solute Peclet number, G is the effective temperature gradients at the solidrliquid interface Ž G s 0.5Ž Gs q G l ., Gs , G l are the temperature gradients in solid and liquid phase correspondingly., GrR 2 is the equilibrium temperature gradient due to curvature of the tip, s ) is the stability constant which usually equals to 0.02 w14,15x. Thus the Eq. Ž12. links the results obtained from thermal and diffusion problems together with capillary effect. The function j C is important for high Peclet number and is defined by w16x

jC s 1 y

2k 1 q Ž 1rs ) P 2 .

1r2

,

Ž 13 .

y1q2kV

where k V is the non-equilibrium distribution coefficient which depends on V. The deviation from equilibrium becomes important, when the interface Peclet number, Pi s a 0 VrDi , for solute redistribution Žin which D irV is the interface diffusion length, and a 0 is of the order of interatomic distance. is larger than the equilibrium distribution coefficient k, i.e. Pi ) k w17,18x. The expression which links the non-equilibrium distribution coefficient k V with the equilibrium one is obtained in w17,18x and is written as: k V s Ž k q Pi .rŽ1 q Pi .. For the considered situation, G is negative and mGC j C is positive. If the left side of Eq. Ž12. is negative, the condition of planar front growth takes place w19,20x. When it becomes positive, the transition from planar front to cellular and from cellular to dendrite growth occurs with increasing V. Substituting R s 2 DPrV and the value of GC from Eq. Ž11. in Eq. Ž12., and replacing k with k v , the expression for dendrite tip selection is written in terms of Peclet number, where the non-equilibrium effect is taken into account P2

m V Ž 1 y k V . C0 1 y Ž 1 y k V . In Ž P .

jC Ž P . q P 2

1 GV

GD V

s

s ) 4D

,

Ž 14 .

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where the velocity dependent liquidus slope

ž

mV s m 1 q

k y k V q k V ln Ž k Vrk . 1yk

/

is given by w5x.

3. Results and discussion In this chapter the variation of the laser processing parameters satisfying q 2t s constant are considered. The value of this constant is 2.12 = 10 13 W 2 P srm4 , which is determined by the properties of Al–4at%Cu binary alloy Žchosen as a model material.. The velocity of solidrliquid interface V and effective temperature gradient G, which are taken from the solution of the heat transfer problem are as follows: a q Vsn q, Gsg . Ž 15 . lTm l Varying the process parameters, for example, q from 10 8 to 10 11 Wrm2 and t from 10y2 to 10y8 s results in an increase in the solidrliquid interface velocity from 2.65 = 10y3 mrs up to 2.65 mrs. Note that it is enough to solve the dimensionless meltingrsolidification problem only once, and then one can vary q, t while keeping q 2 t s constant. 3.1. Dendrite tip selection criteria in terms of processing conditions Substituting Eq. Ž15. in Eq. Ž14., the dendrite tip selection criterion in terms of Peclet number and laser power density is Ps

(

G an

q

4s) DlTm

Ž mn Ž 1 y kn . C0r1 y Ž 1 y kn . In Ž P . . j c Ž P . q Ž gDrn a . Tm

.

Ž 16 .

Eq. Ž16. gives the dependence of the Peclet number on the laser power density. Let us analyse the Eq. Ž14. for the special cases of small and large Peclet numbers, i.e. slow and rapid solidification, respectively. For small q, the Peclet number as well as the Ivantsov’s function also turn to 0, while j c Ž P . s 1. Thus, the asymptotic of the Eq. Ž16. for small Peclet numbers is Ps

(

G an )

q

4s DlTm m Ž 1 y k . C0 q Ž gDrn a . Tm

,

for P ™ 0.

Ž 17 .

Peclet number turns to infinity for large q, the function j c Ž P . turns to 0. For this condition, the argument of the square root function in Eq. Ž16. becomes negative and Eq. Ž16. cannot be interpreted. Physically, it means that the condition of dendrite growth is cancelled and planar growth of solid phase takes place. 3.2. Non-equilibrium effect The non-equilibrium effect at the solidrliquid interface is taken into account in Eq. Ž14. for dendrite tip radius selection as a velocity dependent solute distribution coefficient kn and liquidus slope mn . The exact solutions of Eq. Ž17. without Ždashed line. and with non-equilibrium effect Žsolid line. are presented in Fig. 4. The calculations are made for Al–4at%Cu alloy with the following properties: a s 4.1 = 10y5 m2rs, L s 396 kJrkg, r s 2700 kgrm3 , l s 100 Wrm K, D s 4 = 10y9 m2rs, G s 10y7 m K. The

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Fig. 4. Peclet number versus energy density flux and laser pulse duration. Dashed line corresponds to the equilibrium conditions at solidrliquid interface, solid line corresponds to the non-equilibrium ones.

equilibrium phase diagram is defined by the following parameters: Tm s 933 K, m s 3.17 Krat%, k s 0.15 w21,22x. The mean dimensionless values of solidrliquid interface velocity V and effective temperature gradient G were chosen as n s 6.03 = 10y2 and g s y2.3 = 10y2 in Eq. Ž16.. The comparison of solid and dashed lines shows that the non-equilibrium effect plays an important role for laser power densities higher than 10 10 Wrm2 and, correspondingly, for pulse durations less than 1 m s. For laser

Fig. 5. Dendrite tip radius versus energy density flux and laser pulse duration. Dashed line corresponds to the equilibrium conditions at solidrliquid interface, solid line corresponds to the non-equilibrium ones.

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power densities greater than 2 = 10 11 Wrm2 , taking into account the non-equilibrium effect, the Eq. Ž16. has no solution Žsolid line is broken.. Thus, for these process parameters planar growth of solid phase takes place. Taking P s n aqRrlTm D, it is easy to obtain the dendrite tip radius dependence on q. It is necessary only to substitute P s n aqRrlTm D in Eq. Ž16.. If, we substitute P expressed in terms of q in Eq. Ž17., it can be seen that the dendrite tip radius is inversely proportional to the square root of q for small q values. The dendrite tip radius versus q is presented in Fig. 5. The solid line is broken at 2 = 10 11 Wrm2 , which means that this power density corresponds to a limit of dendrite growth, while the dashed line corresponding to equilibrium conditions gives values of dendrite tip radius with no physical meaning for high energy fluxes Žfor example, the limit when ˚ at q s 3 = 10 12 Wrm2 .. In w5x it is the Eq. Ž16. has no solution corresponds to dendrite tip radius R s 5 A called as an absolute stability limit. It should be noted, that the non-equilibrium effect increases the dendrite tip radius.

4. Conclusion An analytical formula ŽEq. Ž16. together with P s n aqRrlTm D . which characterizes the correlation between laser processing conditions Žlaser power density and pulse duration. and dendrite tip radius is given. This equation is valid for q 2t s constant. For such q and t values, the surface temperature at the end of the laser pulse is always the same. The dendrite tip radius is inversely proportional to the square root of the power density for 10 8 –10 9 Wrm2 . Non-equilibrium effects at solidrliquid interface become important above 10 10 Wrm2 . The absolute stability limit, i.e. when the dendrite growth is replaced by the planar one, is found at 2 = 10 11 Wrm2 . The dendrite tip radius corresponding to non-equilibrium conditions at solidrliquid interface is always larger than that for equilibrium ones.

5. Nomenclature a C0 Cl Ct cp D G GC H k L la lD lT m t tD tT DT 0 P q, q0

thermal diffusivity Žm2rs. initial composition Žat% Cu in A liquid composition ahead of dendrite tip Žat% Cu in Al. dendrite tip composition Žat% Cu in Al. heat capacity ŽJrkg P K. diffusion coefficient in liquid Žm2rs. effective temperature gradient at solidrliquid interface ŽKrm. interface concentration gradient Žat%rm. maximum of melting depth Žmm. equilibrium distribution coefficient fusion latent heat ŽJrkg. heat affected depth Žm. diffusion length Žm. freezing length Žm. liquidus slope ŽKrat%. pulse duration Žms. transient period of solute diffusion Žs. solidification period Žs. liquidus–solidus range ŽK. Peclet number density of absorbed heat flux ŽWrm2 .

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R s T Tl Tm t V Viso x

dendrite tip radius Žm. coordinate of solidrliquid interface Žm. temperature ŽK. liquidus temperature ŽK. melting point of pure material ŽK. time Žs. dendrite growth rate Žmrs. velocity of melting isotherm Žmrs. spatial coordinate Žm.

Greek G l r s) jC

symbols. Gibbs–Thomson coefficient ŽK m. thermal conductivity ŽWrm P K. material density Žkgrm3 . stability constant function of Peclet number

123

Acknowledgements The authors would like to thank Professor W. Kurz, Swiss Federal Institute of Technology, for fruitful discussion.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x w17x w18x w19x w20x w21x w22x

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