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Kinetic model of mass transfer through gas–liquid interface in laser surface alloying
Applied Surface Science 109r110 Ž1997. 143–149
Kinetic model of mass transfer through gas–liquid interface in laser surface alloying A.G. Gnedovets
a,)
, O.M. Portnov b, I. Smurov c , G. Flamant
d
a
d
BaikoÕ Institute of Metallurgy, Russian Academy of Sciences, Leninsky pr., 49, 117911 Moscow, Russia b Paton Welding Institute, Bozhenko, 11, KieÕ 252650, Ukraine c Ecole Nationale d’Ingenieurs de Saint-Etienne, 58 rue Jean Parot, 42023 Saint-Etienne Cedex, France ´ Institut de Science et de Genie et Procedes, ´ des Materiaux ´ ´ ´ C.N.R.S., B.P. No. 5, 66125 Font-Romeu Cedex, France Received 4 June 1996; accepted 14 October 1996
Abstract In laser surface alloying from gas atmosphere neither surface concentration nor the flux of the alloying elements are known beforehand. They should be determined from the combined solution of heat and mass transfer equations with an account for the kinetics of interaction of a gas with a melt. Kinetic theory description of mass transfer through the gas–liquid interface is applied to the problem of laser surface alloying of iron from the atmosphere of molecular nitrogen. The activation nature of gas molecules dissociation at the surface is considered. It is shown that under pulsed-periodic laser action the concentration profiles of the alloying element have maxima situated close to the surface of the metal. The efficiency of surface alloying increases steeply under laser-plasma conditions which results in the formation of highly supersaturated gas solutions in the metal.
1. Introduction Laser surface alloying represents a promising direction in the technology of materials processing because it makes it possible to produce local zones with increased service characteristics: microhardness, wear resistance, etc., on the surface of components and tools. The alloying elements can be predeposited on the materials surface as a coating or in a powder form, or can be introduced into the melt from liquid and gas phase w1–3x. To describe mass transfer in the metal it is necessary to determine the concentration of the alloying
)
Corresponding author.
solute on its surface at each specific period. The surface concentration can be specified in alloying from a layer of a coating deposited in advance on the treated material w4,5x. In laser surface alloying from the gas atmosphere w3,6x neither surface concentration nor the flux are known in advance and should be determined from the combined solution of heat and mass transfer problems in the metal and the gas phase. It is important to note that not all collisions of gas molecules with the metal surface are accompanied by a chemical reaction Ždissolution or formation of a compound.. In the metals only gases in the atomic state can be dissolved efficiently w7x and, consequently, the rate of the alloying from the atmosphere of multiatomic gases is determined by the dissocia-
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 7 4 6 - 5
144
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
tion which can take place either in the volume of the gas or on the heated surface. At moderate densities of the energy flux Ž q - 10 6 Wrcm2 ., the gas atmosphere remains transparent for radiation, hence plasma formation and dissociation of molecules do not occur. As a result of the decrease of the binding energy of the atoms of the absorbed gas molecules, dissociation is most likely to take place on the heated surface of the metal. In this work, the results of mathematical modeling of admixture absorption from the atmosphere of a molecular gas by a metal during pulsed-periodic irradiation are presented. The iron–nitrogen system is considered as an example. The model is based on the kinetic theory description of interaction of a gas with a melt. For the sake of simplicity and in an effort to test the proposed approach, only the diffusion transfer of the alloying element is assumed. As a result, the proposed solution of mass transfer through the gas–liquid interface may be considered as a kind of boundary condition for the general problem of admixture redistribution in laser alloying from gas phase. The relative importance of the mechanisms of the alloying element transfer in the metal, i.e. convection and diffusion, are determined by a number of factors, among which are the energy input per pulse, diameter of the zone of action, duration of laser action, spatial distribution of the beam, etc. w4,5x. It was shown w4x that in several cases the formation of a comparatively large molten pool Ždepth about ; 50– 100 m m. does not accompanied by an intensive convective mass transfer. In this case the proposed approach may be used for the estimations of concentration fields in pulsed laser alloying from gas phase.
Fig. 1. Schematic diagram of the main molecular processes and structure of the flow at the gas–liquid interface.
4. Intensity of gas generation from the metal depends only on the surface concentration of the dissolved gas and surface temperature. 5. Diffusion is the main mechanism of transfer of the alloying element in the metal. The main microscopic molecular processes taking place at the gas–metal interface, the structure of the gas flow and the selected coordinate system are represented schematically in Fig. 1. Since the efficiency of dissociation of the molecules on the surface, solubility, and diffusion of the solute atoms of the gas in the metal depend greatly on its temperature, to describe mass transfer in the melt it is necessary to solve jointly the diffusion problem
EC Et
2. Mathematical model
s
E Ex
ž
DŽ T .
EC Ex
/
,
C Ž x , 0 . s C0 ,
C Ž y , t . s C0 , yD Ž T . E CrE x
xs 0 s j
Ž 1.
and the heat conduction problem The basic assumptions for the present study are as follows: 1. In the metal only gas in the atomic state can be dissolved efficiently. 2. The gas atmosphere is transparent for laser radiation and plasma formation and dissociation of molecules do not take place in the gas phase. 3. Gas dissociation takes place on the heated surface of the metal.
r Ž c p q Lm d Ž T y Tm . .
ET Et
s
E Ex
žŽ
l T.
ET Ex
/
,
T Ž x , 0 . s T0 , T Ž y , t . s T0 ylŽ T . E TrE x
xs 0 s q
Ž 2.
The density of the total mass flow of the gas into the melt j involved in the boundary condition for the diffusion Eq. Ž1. is not known beforehand. In terms
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
of the velocity distribution function of the gas molecules, it can be specified as j s y mÕX f dz
Ž 3.
H
and should be determined from the solution of the kinetic problem for the gas phase. The following nomenclature is used in Eqs. Ž1. – Ž3.: C s C Ž x, t . is the alloying element mass concentration; D is the diffusion coefficient; T s T Ž x, t . is the metal temperature; x is spatial coordinate directed inside the material; t is time; r , c p and l are, respectively, the metal density, specific heat capacity and thermal conductivity; Lm is latent heat of melting; Tm is the temperature of melting; d ŽT y Tm . is delta function which is used for the treatment of latent heat evolution; q is the density of the absorbed energy flux of laser radiation; f s f Ž X, z . is the velocity distribution function; X is spatial coordinate directed into the gas phase; z is the velocity and m is the mass of the gas molecule. The gas content in metal wNx Žin wt%. is related to its concentration C as wNx s Ž Crr . P 100%. In the present approach, the heat transfer and diffusion problems are linked by the temperature distribution inside the metal, although the heat conduction is considered independent on the solute concentration field. The frequency of the collisions of the gas molecules with the surface is determined by solving the system of gas-kinetic equations in the Knudsen layer adjacent to the melt. The method of solution can be based on the bimodal approximation of the velocity distribution function w8,9x q aq Ž X . fq Ž z . q ay Ž X . fy Ž z .
Ž 4.
Ž X . of the spatial coorwith unknown functions a " i dinate X and half-ranged Maxwellians f i" Ž z . s n i
ž
m 2p kTi
surface and at the boundary of the Knudsen layer, q respectively; subscript ) denotes fq s y fs collisions; superscripts qŽy. correspond to molecules with the velocity ÕX G 0 Ž- 0.; parameters n i , Ti and u i specify the number density, temperature and mass flow velocity, in particular, u s s 0, u ) s Ž2 kTsrp m.1r2 , n ) s n sr2, T) s Ž1 y 2r3p .Ts . The Ž X . satisfy the conditions at unknown functions a " i q y infinity Ž X s . aq s 0, ay s ) s 0, a s 1, a s 1, so that the parameters n , T and u correspond, respectively, to the macroscopic density, temperature and mass flow velocity of the gas at the boundary between the Knudsen layer and hydrodynamic region adjacent to it. At the metal surface Ž X s 0. aqs 0 is determined from the condition that the and aq s outgoing flow is formed of the gas molecules diffusely reflected by the surface or liberated from the melt. Ž X . in the velocity disUnknown coefficients a " i tribution function can be determined from the set of the moment equations
Ef
HQ z E X dz s D Ž Q L
3r2
/
L
.
1 2
1 y py1 r2 ty1r2 ps q 12 a t )1r2 F) ay s s Ž 0. q 2 a F a Ž 0.
sS
Ž 6.
ps q 14 2 t ) G) y Ž 1 y a . p 1r2 ts1r2 t )1r2 F) ay s Ž 0. q 14 2G y Ž 1 y a . p 1r2 ts1r2 F ay Ž 0 . s S 2 q 1r2
Ž 7.
py1 r2 ts1r2 ps q t )1r2 H) y Ž 1 y a . ts t )1r2 F) ay s Ž 0. q H y Ž 1 y a . ts F ay Ž 0 . s S Ž S 2 q 5r2 .
Ž 8.
2
=exp y
Ž 5.
For the molecular characteristics Q L s m, mÕ and mÕ 2r2 the collision terms DŽ Q L . s 0, and as it follows from Eq. Ž5., corresponding mass, momentum and energy conservation equations which connect the parameters of the gas flow at the metal surface Ž X s 0. and at the outward boundary of the Knudsen layer Ž X s . are written as
1 4
q y y f Ž X , z . s aq s Ž X . fs Ž z . q a ) Ž X . f ) Ž z .
145
m Ž Ž Õ X y u i . q Õ Y2 q ÕZ2 . 2 kTi
Here subscripts i assume the values i s s, ) , ; subscripts s and denote the conditions at the metal
y where ay s s a ) n ) rn , t i s TirT , ps s PsrP 1r2 Ž . u ir 2 kTirm , Pi s n i kTi and the functions
and Hi are defined as Fi s Si erfc Si y py1 r2 exp Ž ySi2 .
, Si s Fi , Gi
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
146
Gi s Ž Si2 q 1r2 . erfc Si y py1 r2 Si exp Ž ySi2 . Hi s 12 Si Ž Si2 q 5r2 . erfc Si y py1 r2 Ž 1 q 21 Si2 . exp Ž ySi2 . Here it is taken into account that the melt absorbs only molecules which undergo dissociation Žthe probability of this process is a s a 0 expŽyTd) rTs . with the effective temperature of dissociation Td) s Ž Ed y 2 Eg,Me .rk, where Ed is the energy of molecule dissociation in the gas, Eg,Me is the binding energy between the gas atom and the metal., undissociated molecules are diffusely reflected by the surface, and the gas liberated from the melt is characterized by the temperature Ts and partial pressure Ps . Ž . Eqs. Ž6. – Ž8. contain three unknowns: ay s 0 , yŽ . a 0 and S . The desired mass flow of the gas into the melt j s yHmÕX f dz ' ymn Ž2 kT rm.1r2 S is expressed in terms of the speed ratio S s u rŽ2 kT rm.1r2 , which is determined from the solution of this system of equations. The gas temperatures and pressures at the boundary of the Knudsen layer Žsubscript . and in hydrodynamic region Žg. are related as w9x TgrT s Ž PgrP
.
Ž g y1 .r g
,
PgrP s 1 q 12 Ž g y 1 . M
2 g r Ž g y1 .
Ž 9.
where M s Ž2rg .1r2 S is Mach number, g s c prc v is specific heat ratio, Tg ' TN 2 and Pg ' P N 2 are the temperature and pressure of the surrounding gas, respectively. One more parameter to be determined is the dimensionless pressure of the gas generated from the melt ps s PsrP . In the particular case of thermodynamical equilibrium Ž ts s 1, S s 0., one can obtain ps s a , as it follows from Eqs. Ž6. – Ž8.. Since the intensity of gas generation from the melt is determined only by the surface temperature Ts and subsurface gas concentration wNxs , the partial pressure of the liberated gas is Ps s a Ž w N x srK N .
2
iron over different temperature ranges can be found in w7x. It is important to take into account the exponential temperature dependence of the diffusion coefficient of the alloying element in the melt D s D 0 expŽyTD rT ., the activation nature of dissociation of gas molecules on the surface with intensity ; expŽyTd) rTs . Žfor nitrogen in iron D 0 s 4.1 P 10y7 m2rs, T D s 7.6 P 10 3 K, Td) s 2.3 P 10 4 K., the dynamics of the metal melting, and asymmetry of its heating and cooling with respect to the end of the laser pulse.
3. Results and discussion 3.1. Metal alloying under conditions close to thermodynamic equilibrium The most simple situation is mass transfer through gas–surface interface under the conditions close to thermodynamic equilibrium when the melt is heated uniformly and its temperature coincides with the temperature of the surrounding gas. This problem is well studied both theoretically and experimentally as applied to a number of metallurgical processes Žremelting or refining of metals and alloys., which enables to use this case to test the proposed model. Fig. 2 shows the variation of nitrogen concentration depending on the holding time of iron in nitrogen atmosphere for two different situations: Ž1. metal
Ž 10 .
no matter whether or not the alloying conditions are equilibrium. The equilibrium constant K N s wNxrŽ PN .1r2 depends on the temperature as log K N 2 s ArT q B. The data on the solubility of nitrogen in
Fig. 2. Temporal variations of gas content in iron melt depending on the holding time in nitrogen atmosphere. Remelting regimes: PN 2 s 0.01 MPa, T s1953 K, wNx 0 s 0.001% Ž1. and PN 2 s 0.008 MPa, T s1873 K, wNx 0 s 0.01% Ž2.. Points – experimental data w7x, solid lines – calculated results.
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
147
remelting Žinitial nitrogen content wNx 0 is less than the equilibrium value wNxe at a given temperature T . with gas absorption and Ž2. metal refining ŽwNx 0 ) wNxe . with gas generation from the melt. It is vital to note that the characteristic period of the melt saturation is in the range of tens of seconds and is several orders of magnitude greater than laser pulse duration. 3.2. Metal alloying under the action of single laser pulse As an example, the effect of a single laser pulse Ž q s 6.5 P 10 4 Wrcm2 , t s 4 ms. on a semi-infinite metallic solid Žiron. in a gas atmosphere Žnitrogen at normal pressure. is examined. The typical stages of formation of the alloyed layer in the metal under the effect of the single pulse are shown in Fig. 3 which gives the spatial variations of the concentration profiles of dissolved nitrogen wNx Žin wt%. in the iron melt at different times t and temporal behavior of subsurface Žat x s 0. gas content wNxs . The distribution of the alloying element in the surface layer of the metal, heated by laser beam, is determined by the competition of the following processes: Ž1. adsorption and dissociation of gas molecules colliding with the surface and their dissolution in the melt; Ž2. gas generation from the melt; Ž3. diffusion redistribution of the dissolved gas in the melt. Heating of the metal which absorbs laser radiation leads to an increase of the efficiency of the dissociation of nitrogen on the surface and its solubility in the melt, and the flow of the alloying element into the metal and the gas content rapidly increase ŽFig. 3a, curves 1–4.. As a result of a high heating rate the actual concentration of nitrogen in the iron greatly lags behind the equilibrium concentration which corresponds to the pressure of the surrounding gas and the metal temperature Žat the end of the pulse, the surface temperature Ts s 2.6 P 10 3 K, subsurface concentration wNx s 0.035%; at PN 2 s 0.1 MPa at the melting point of iron Tm s 1.8 P 10 3 K the equilibrium gas content wNxe s 0.044%.. After completing the pulse, the gas continues to penetrate into the heated metal for a certain period of time Ž; 2 ms. but due to the diffusion flow of the dissolved solute into the volume of the melt, the concentration profiles ‘fade away’ and the subsurface concentration of nitrogen decreases ŽFig. 3a, curves 5–7.. As a result of rapid
Fig. 3. Profiles Ža. and temporal behavior of subsurface content Žb. of nitrogen in pulsed laser irradiation of iron. The numbers of the curves Ža. denote time in ms. Alloying regime: q s6.5P10 4 Wrcm2 , t s 4 ms, P N 2 s 0.1 MPa.
cooling of the melt caused by heat conductivity, and an exponential decrease of the diffusion coefficient of nitrogen in iron, the process of redistribution of the solute ‘freezes up’ ŽFig. 3b. and a thin nitrated layer is ‘fixed’ under the surface ŽFig. 3a, curve 8.. The nitrogen concentration in the layer monotonically decreases with increasing distance from the surface and the thickness of the layer equals ; 15 m m. 3.3. Metal alloying under pulsed-periodic laser action The main relationships of the alloying dynamics under the action of a single laser pulse remain valid
148
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
Fig. 4. Nitrogen flow Ža. and concentration profiles Žb. in pulsedperiodic laser irradiation of iron. The numbers of the curves denote the number of the pulse. Alloying regime: q s6.5P10 4 Wrcm2 , t s 4 ms, f s10 Hz, P N 2 s 0.1 MPa.
also for the subsequent pulses, although repeated irradiation of the metal is accompanied by important special features, as it will be discussed later. Fig. 4 shows the time dependencies of the dimensionless flow S and ‘frozen’ spatial profiles of the concentration of nitrogen wNx during the action of a series of laser pulses Ž q s 6.5 P 10 4 Wrcm2 , t s 4 ms, f s 10 Hz.. Influence of the dissolved gas on thermophysical and optical properties of the metal is not taken into account in the present study. In pulsed-periodic laser radiation, as a result of ‘freezing’ of the alloying element in the cooling stage in the period between two adjacent pulses, each consecutive pulse already acts on the alloyed metal. This is accompa-
nied by a large increase of the importance of the process of gas generation from the melt because in low-temperature regions of the thermal cycle the solution of the gas in the metal becomes supersaturated. The total mass flow of the gas j s ymn Ž2 kT rm.1r2 S is determined as the difference between the flows of the gas molecules moving into the melt and generated from it. Starting approximately at the eighth or ninth pulse, the mean nitrating rate greatly decreases ŽFig. 4a. and failure radiation no longer causes any large increase of the gas content ŽFig. 4b.. The highest rate of gas generation is observed in the cooling section after the start of solidification of the melt. This is associated with jump-like reduction of the solubility of nitrogen in delta iron. Gas generation from the melt leads to the depletion of the subsurface zone in the alloyed layer and to the appearance of maxima on the concentration profiles of nitrogen in the multiple irradiated metal. The concentration maxima ŽFig. 4b. are placed at a depth of the order of the diffusion length ; Ž Dtg .1r2 ; 5 m m from the surface, where tg is the characteristic time of gas generation. The calculations show that the thickness of the nitrated layer, formed during pulsed-periodic laser radiation in the millisecond range, may exceed ; 30–50 m m. The order of magnitude of this value corresponds to the size of the heat affected zone because diffusion into the unheated volume of the metal is extremely small. The nitrogen concentration reaches the values close to the solubility limit at the solidification point. 3.4. Metal alloying under plasma conditions The efficiency of surface alloying increases steeply under the laser action with plasma formation near the irradiated metal. Although the description of the dynamics of laser-produced plasma is rather complicated w10x, qualitative analysis can be performed in the framework of the present approach. In the situation being considered, gas dissociation on the metal surface is of minor importance. At a characteristic plasma temperatures T ; 10 4 K even at a low degrees of ionization, practically all nitrogen in plasma region is dissociated and can be easily absorbed by the melt. Concentration profiles and temporal behavior of subsurface nitrogen content in iron calculated on the basis of the proposed model
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
149
4. Conclusions Ž1. Kinetic theory analysis is applied to the problem of laser surface alloying of a metal from the atmosphere of a molecular gas. The proposed solution of mass transfer through the gas–liquid interface may be considered as a kind of boundary condition for the general problem of admixture redistribution in laser alloying from gaseous phase. Ž2. The concentration profiles of the alloying element in pulsed-periodic laser alloying from the gas atmosphere have local maxima close to the surface of the metal. Ž3. The efficiency of surface alloying increases steeply under laser-plasma conditions which results in the formation of highly supersaturated gas solutions in the metal.
References
Fig. 5. Concentration profiles Ža. at different times t s1 m s Žcurve 1., 5 m s Ž2., 25 m s Ž3., 125 m s Ž4. and temporal behavior of subsurface content Žb. of nitrogen in surface alloying of iron under plasma conditions. Alloying regime: PN 2 s 0.1 MPa, TN 2 s 10 4 K, Ts s 2.1P10 3 K.
for the dissociated Žmonatomic. gas are shown in Fig. 5. It must be emphasized that the alloying rate under laser-plasma conditions increases to an extent that in a short time period Žof the order of few m s. the nitrogen solution becomes supersaturated.
w1x C.W. Draper and J.M. Poate, Int. Med. Rev. 30 Ž1985. 85–108. w2x G. Herziger and E.W. Kreutz, Phys. Scripta T 13 Ž1986. 139–150. w3x A. Uglov, M. Ignatev, A. Gnedovets and I. Smurov, J. Phys. Colloq. 50 Ž1989. C5-727–735. w4x I.Yu. Smurov and L. Covelli, in: Pulse laser alloying: Theory, experiment, ICALEO 92 Proc. Orlando, Florida 25–29 October 1992, eds. D. Farson, W. Steen and I. Miyamoto ŽLaser Mater. Process, Vol. 75, Laser Institute of America, 1993. pp. 265–277. w5x I. Smurov, L. Covelli, K. Tagirov and L. Aksenov, J. Appl. Phys. 71 Ž1992. 3147–3158. w6x R.V. Arutyunyan, V.Yu. Baranov, L.A. Bolshov, V.V. Vityukov, E.V. Evstratov, V.P. Kiselev, A.M. Kovalevich, B.S. Podolskii and Yu.Yu. Stepanov, Poverkhn. Fiz. Khim. Mekh. 5 Ž1988. 129–137. w7x V.V. Averin, A.V. Revyakin, V.I. Fedorchenko and L.N. Kozina, Nitrogen in Metals ŽMetallurgiya, Moscow, 1976.. w8x T. Ytrehus, Phys. Fluids 26 Ž1983. 939–949. w9x T. Bergstrom and T. Ytrehus, Phys. Fluids 27 Ž1984. 583– 588. w10x V. Mazhukin, I. Smurov and G. Flamant, J. Comp. Phys. 112 Ž1994. 78–90.
Kinetic model of mass transfer through gas–liquid interface in laser surface alloying A.G. Gnedovets
a,)
, O.M. Portnov b, I. Smurov c , G. Flamant
d
a
d
BaikoÕ Institute of Metallurgy, Russian Academy of Sciences, Leninsky pr., 49, 117911 Moscow, Russia b Paton Welding Institute, Bozhenko, 11, KieÕ 252650, Ukraine c Ecole Nationale d’Ingenieurs de Saint-Etienne, 58 rue Jean Parot, 42023 Saint-Etienne Cedex, France ´ Institut de Science et de Genie et Procedes, ´ des Materiaux ´ ´ ´ C.N.R.S., B.P. No. 5, 66125 Font-Romeu Cedex, France Received 4 June 1996; accepted 14 October 1996
Abstract In laser surface alloying from gas atmosphere neither surface concentration nor the flux of the alloying elements are known beforehand. They should be determined from the combined solution of heat and mass transfer equations with an account for the kinetics of interaction of a gas with a melt. Kinetic theory description of mass transfer through the gas–liquid interface is applied to the problem of laser surface alloying of iron from the atmosphere of molecular nitrogen. The activation nature of gas molecules dissociation at the surface is considered. It is shown that under pulsed-periodic laser action the concentration profiles of the alloying element have maxima situated close to the surface of the metal. The efficiency of surface alloying increases steeply under laser-plasma conditions which results in the formation of highly supersaturated gas solutions in the metal.
1. Introduction Laser surface alloying represents a promising direction in the technology of materials processing because it makes it possible to produce local zones with increased service characteristics: microhardness, wear resistance, etc., on the surface of components and tools. The alloying elements can be predeposited on the materials surface as a coating or in a powder form, or can be introduced into the melt from liquid and gas phase w1–3x. To describe mass transfer in the metal it is necessary to determine the concentration of the alloying
)
Corresponding author.
solute on its surface at each specific period. The surface concentration can be specified in alloying from a layer of a coating deposited in advance on the treated material w4,5x. In laser surface alloying from the gas atmosphere w3,6x neither surface concentration nor the flux are known in advance and should be determined from the combined solution of heat and mass transfer problems in the metal and the gas phase. It is important to note that not all collisions of gas molecules with the metal surface are accompanied by a chemical reaction Ždissolution or formation of a compound.. In the metals only gases in the atomic state can be dissolved efficiently w7x and, consequently, the rate of the alloying from the atmosphere of multiatomic gases is determined by the dissocia-
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 7 4 6 - 5
144
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
tion which can take place either in the volume of the gas or on the heated surface. At moderate densities of the energy flux Ž q - 10 6 Wrcm2 ., the gas atmosphere remains transparent for radiation, hence plasma formation and dissociation of molecules do not occur. As a result of the decrease of the binding energy of the atoms of the absorbed gas molecules, dissociation is most likely to take place on the heated surface of the metal. In this work, the results of mathematical modeling of admixture absorption from the atmosphere of a molecular gas by a metal during pulsed-periodic irradiation are presented. The iron–nitrogen system is considered as an example. The model is based on the kinetic theory description of interaction of a gas with a melt. For the sake of simplicity and in an effort to test the proposed approach, only the diffusion transfer of the alloying element is assumed. As a result, the proposed solution of mass transfer through the gas–liquid interface may be considered as a kind of boundary condition for the general problem of admixture redistribution in laser alloying from gas phase. The relative importance of the mechanisms of the alloying element transfer in the metal, i.e. convection and diffusion, are determined by a number of factors, among which are the energy input per pulse, diameter of the zone of action, duration of laser action, spatial distribution of the beam, etc. w4,5x. It was shown w4x that in several cases the formation of a comparatively large molten pool Ždepth about ; 50– 100 m m. does not accompanied by an intensive convective mass transfer. In this case the proposed approach may be used for the estimations of concentration fields in pulsed laser alloying from gas phase.
Fig. 1. Schematic diagram of the main molecular processes and structure of the flow at the gas–liquid interface.
4. Intensity of gas generation from the metal depends only on the surface concentration of the dissolved gas and surface temperature. 5. Diffusion is the main mechanism of transfer of the alloying element in the metal. The main microscopic molecular processes taking place at the gas–metal interface, the structure of the gas flow and the selected coordinate system are represented schematically in Fig. 1. Since the efficiency of dissociation of the molecules on the surface, solubility, and diffusion of the solute atoms of the gas in the metal depend greatly on its temperature, to describe mass transfer in the melt it is necessary to solve jointly the diffusion problem
EC Et
2. Mathematical model
s
E Ex
ž
DŽ T .
EC Ex
/
,
C Ž x , 0 . s C0 ,
C Ž y , t . s C0 , yD Ž T . E CrE x
xs 0 s j
Ž 1.
and the heat conduction problem The basic assumptions for the present study are as follows: 1. In the metal only gas in the atomic state can be dissolved efficiently. 2. The gas atmosphere is transparent for laser radiation and plasma formation and dissociation of molecules do not take place in the gas phase. 3. Gas dissociation takes place on the heated surface of the metal.
r Ž c p q Lm d Ž T y Tm . .
ET Et
s
E Ex
žŽ
l T.
ET Ex
/
,
T Ž x , 0 . s T0 , T Ž y , t . s T0 ylŽ T . E TrE x
xs 0 s q
Ž 2.
The density of the total mass flow of the gas into the melt j involved in the boundary condition for the diffusion Eq. Ž1. is not known beforehand. In terms
A.G. GnedoÕets et al.r Applied Surface Science 109 r 110 (1997) 143–149
of the velocity distribution function of the gas molecules, it can be specified as j s y mÕX f dz
Ž 3.
H
and should be determined from the solution of the kinetic problem for the gas phase. The following nomenclature is used in Eqs. Ž1. – Ž3.: C s C Ž x, t . is the alloying element mass concentration; D is the diffusion coefficient; T s T Ž x, t . is the metal temperature; x is spatial coordinate directed inside the material; t is time; r , c p and l are, respectively, the metal density, specific heat capacity and thermal conductivity; Lm is latent heat of melting; Tm is the temperature of melting; d ŽT y Tm . is delta function which is used for the treatment of latent heat evolution; q is the density of the absorbed energy flux of laser radiation; f s f Ž X, z . is the velocity distribution function; X is spatial coordinate directed into the gas phase; z is the velocity and m is the mass of the gas molecule. The gas content in metal wNx Žin wt%. is related to its concentration C as wNx s Ž Crr . P 100%. In the present approach, the heat transfer and diffusion problems are linked by the temperature distribution inside the metal, although the heat conduction is considered independent on the solute concentration field. The frequency of the collisions of the gas molecules with the surface is determined by solving the system of gas-kinetic equations in the Knudsen layer adjacent to the melt. The method of solution can be based on the bimodal approximation of the velocity distribution function w8,9x q aq Ž X . fq Ž z . q ay Ž X . fy Ž z .
Ž 4.
Ž X . of the spatial coorwith unknown functions a " i dinate X and half-ranged Maxwellians f i" Ž z . s n i
ž
m 2p kTi
surface and at the boundary of the Knudsen layer, q respectively; subscript ) denotes fq s y fs collisions; superscripts qŽy. correspond to molecules with the velocity ÕX G 0 Ž- 0.; parameters n i , Ti and u i specify the number density, temperature and mass flow velocity, in particular, u s s 0, u ) s Ž2 kTsrp m.1r2 , n ) s n sr2, T) s Ž1 y 2r3p .Ts . The Ž X . satisfy the conditions at unknown functions a " i q y infinity Ž X s . aq s 0, ay s ) s 0, a s 1, a s 1, so that the parameters n , T and u correspond, respectively, to the macroscopic density, temperature and mass flow velocity of the gas at the boundary between the Knudsen layer and hydrodynamic region adjacent to it. At the metal surface Ž X s 0. aqs 0 is determined from the condition that the and aq s outgoing flow is formed of the gas molecules diffusely reflected by the surface or liberated from the melt. Ž X . in the velocity disUnknown coefficients a " i tribution function can be determined from the set of the moment equations
Ef
HQ z E X dz s D Ž Q L
3r2
/
L
.
1 2
1 y py1 r2 ty1r2 ps q 12 a t )1r2 F) ay s s Ž 0. q 2 a F a Ž 0.
sS
Ž 6.
ps q 14 2 t ) G) y Ž 1 y a . p 1r2 ts1r2 t )1r2 F) ay s Ž 0. q 14 2G y Ž 1 y a . p 1r2 ts1r2 F ay Ž 0 . s S 2 q 1r2
Ž 7.
py1 r2 ts1r2 ps q t )1r2 H) y Ž 1 y a . ts t )1r2 F) ay s Ž 0. q H y Ž 1 y a . ts F ay Ž 0 . s S Ž S 2 q 5r2 .
Ž 8.
2
=exp y
Ž 5.
For the molecular characteristics Q L s m, mÕ and mÕ 2r2 the collision terms DŽ Q L . s 0, and as it follows from Eq. Ž5., corresponding mass, momentum and energy conservation equations which connect the parameters of the gas flow at the metal surface Ž X s 0. and at the outward boundary of the Knudsen layer Ž X s . are written as
1 4
q y y f Ž X , z . s aq s Ž X . fs Ž z . q a ) Ž X . f ) Ž z .
145
m Ž Ž Õ X y u i . q Õ Y2 q ÕZ2 . 2 kTi
Here subscripts i assume the values i s s, ) , ; subscripts s and denote the conditions at the metal
y where ay s s a ) n ) rn , t i s TirT , ps s PsrP 1r2 Ž . u ir 2 kTirm , Pi s n i kTi and the functions
and Hi are defined as Fi s Si erfc Si y py1 r2 exp Ž ySi2 .
, Si s Fi , Gi
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Gi s Ž Si2 q 1r2 . erfc Si y py1 r2 Si exp Ž ySi2 . Hi s 12 Si Ž Si2 q 5r2 . erfc Si y py1 r2 Ž 1 q 21 Si2 . exp Ž ySi2 . Here it is taken into account that the melt absorbs only molecules which undergo dissociation Žthe probability of this process is a s a 0 expŽyTd) rTs . with the effective temperature of dissociation Td) s Ž Ed y 2 Eg,Me .rk, where Ed is the energy of molecule dissociation in the gas, Eg,Me is the binding energy between the gas atom and the metal., undissociated molecules are diffusely reflected by the surface, and the gas liberated from the melt is characterized by the temperature Ts and partial pressure Ps . Ž . Eqs. Ž6. – Ž8. contain three unknowns: ay s 0 , yŽ . a 0 and S . The desired mass flow of the gas into the melt j s yHmÕX f dz ' ymn Ž2 kT rm.1r2 S is expressed in terms of the speed ratio S s u rŽ2 kT rm.1r2 , which is determined from the solution of this system of equations. The gas temperatures and pressures at the boundary of the Knudsen layer Žsubscript . and in hydrodynamic region Žg. are related as w9x TgrT s Ž PgrP
.
Ž g y1 .r g
,
PgrP s 1 q 12 Ž g y 1 . M
2 g r Ž g y1 .
Ž 9.
where M s Ž2rg .1r2 S is Mach number, g s c prc v is specific heat ratio, Tg ' TN 2 and Pg ' P N 2 are the temperature and pressure of the surrounding gas, respectively. One more parameter to be determined is the dimensionless pressure of the gas generated from the melt ps s PsrP . In the particular case of thermodynamical equilibrium Ž ts s 1, S s 0., one can obtain ps s a , as it follows from Eqs. Ž6. – Ž8.. Since the intensity of gas generation from the melt is determined only by the surface temperature Ts and subsurface gas concentration wNxs , the partial pressure of the liberated gas is Ps s a Ž w N x srK N .
2
iron over different temperature ranges can be found in w7x. It is important to take into account the exponential temperature dependence of the diffusion coefficient of the alloying element in the melt D s D 0 expŽyTD rT ., the activation nature of dissociation of gas molecules on the surface with intensity ; expŽyTd) rTs . Žfor nitrogen in iron D 0 s 4.1 P 10y7 m2rs, T D s 7.6 P 10 3 K, Td) s 2.3 P 10 4 K., the dynamics of the metal melting, and asymmetry of its heating and cooling with respect to the end of the laser pulse.
3. Results and discussion 3.1. Metal alloying under conditions close to thermodynamic equilibrium The most simple situation is mass transfer through gas–surface interface under the conditions close to thermodynamic equilibrium when the melt is heated uniformly and its temperature coincides with the temperature of the surrounding gas. This problem is well studied both theoretically and experimentally as applied to a number of metallurgical processes Žremelting or refining of metals and alloys., which enables to use this case to test the proposed model. Fig. 2 shows the variation of nitrogen concentration depending on the holding time of iron in nitrogen atmosphere for two different situations: Ž1. metal
Ž 10 .
no matter whether or not the alloying conditions are equilibrium. The equilibrium constant K N s wNxrŽ PN .1r2 depends on the temperature as log K N 2 s ArT q B. The data on the solubility of nitrogen in
Fig. 2. Temporal variations of gas content in iron melt depending on the holding time in nitrogen atmosphere. Remelting regimes: PN 2 s 0.01 MPa, T s1953 K, wNx 0 s 0.001% Ž1. and PN 2 s 0.008 MPa, T s1873 K, wNx 0 s 0.01% Ž2.. Points – experimental data w7x, solid lines – calculated results.
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remelting Žinitial nitrogen content wNx 0 is less than the equilibrium value wNxe at a given temperature T . with gas absorption and Ž2. metal refining ŽwNx 0 ) wNxe . with gas generation from the melt. It is vital to note that the characteristic period of the melt saturation is in the range of tens of seconds and is several orders of magnitude greater than laser pulse duration. 3.2. Metal alloying under the action of single laser pulse As an example, the effect of a single laser pulse Ž q s 6.5 P 10 4 Wrcm2 , t s 4 ms. on a semi-infinite metallic solid Žiron. in a gas atmosphere Žnitrogen at normal pressure. is examined. The typical stages of formation of the alloyed layer in the metal under the effect of the single pulse are shown in Fig. 3 which gives the spatial variations of the concentration profiles of dissolved nitrogen wNx Žin wt%. in the iron melt at different times t and temporal behavior of subsurface Žat x s 0. gas content wNxs . The distribution of the alloying element in the surface layer of the metal, heated by laser beam, is determined by the competition of the following processes: Ž1. adsorption and dissociation of gas molecules colliding with the surface and their dissolution in the melt; Ž2. gas generation from the melt; Ž3. diffusion redistribution of the dissolved gas in the melt. Heating of the metal which absorbs laser radiation leads to an increase of the efficiency of the dissociation of nitrogen on the surface and its solubility in the melt, and the flow of the alloying element into the metal and the gas content rapidly increase ŽFig. 3a, curves 1–4.. As a result of a high heating rate the actual concentration of nitrogen in the iron greatly lags behind the equilibrium concentration which corresponds to the pressure of the surrounding gas and the metal temperature Žat the end of the pulse, the surface temperature Ts s 2.6 P 10 3 K, subsurface concentration wNx s 0.035%; at PN 2 s 0.1 MPa at the melting point of iron Tm s 1.8 P 10 3 K the equilibrium gas content wNxe s 0.044%.. After completing the pulse, the gas continues to penetrate into the heated metal for a certain period of time Ž; 2 ms. but due to the diffusion flow of the dissolved solute into the volume of the melt, the concentration profiles ‘fade away’ and the subsurface concentration of nitrogen decreases ŽFig. 3a, curves 5–7.. As a result of rapid
Fig. 3. Profiles Ža. and temporal behavior of subsurface content Žb. of nitrogen in pulsed laser irradiation of iron. The numbers of the curves Ža. denote time in ms. Alloying regime: q s6.5P10 4 Wrcm2 , t s 4 ms, P N 2 s 0.1 MPa.
cooling of the melt caused by heat conductivity, and an exponential decrease of the diffusion coefficient of nitrogen in iron, the process of redistribution of the solute ‘freezes up’ ŽFig. 3b. and a thin nitrated layer is ‘fixed’ under the surface ŽFig. 3a, curve 8.. The nitrogen concentration in the layer monotonically decreases with increasing distance from the surface and the thickness of the layer equals ; 15 m m. 3.3. Metal alloying under pulsed-periodic laser action The main relationships of the alloying dynamics under the action of a single laser pulse remain valid
148
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Fig. 4. Nitrogen flow Ža. and concentration profiles Žb. in pulsedperiodic laser irradiation of iron. The numbers of the curves denote the number of the pulse. Alloying regime: q s6.5P10 4 Wrcm2 , t s 4 ms, f s10 Hz, P N 2 s 0.1 MPa.
also for the subsequent pulses, although repeated irradiation of the metal is accompanied by important special features, as it will be discussed later. Fig. 4 shows the time dependencies of the dimensionless flow S and ‘frozen’ spatial profiles of the concentration of nitrogen wNx during the action of a series of laser pulses Ž q s 6.5 P 10 4 Wrcm2 , t s 4 ms, f s 10 Hz.. Influence of the dissolved gas on thermophysical and optical properties of the metal is not taken into account in the present study. In pulsed-periodic laser radiation, as a result of ‘freezing’ of the alloying element in the cooling stage in the period between two adjacent pulses, each consecutive pulse already acts on the alloyed metal. This is accompa-
nied by a large increase of the importance of the process of gas generation from the melt because in low-temperature regions of the thermal cycle the solution of the gas in the metal becomes supersaturated. The total mass flow of the gas j s ymn Ž2 kT rm.1r2 S is determined as the difference between the flows of the gas molecules moving into the melt and generated from it. Starting approximately at the eighth or ninth pulse, the mean nitrating rate greatly decreases ŽFig. 4a. and failure radiation no longer causes any large increase of the gas content ŽFig. 4b.. The highest rate of gas generation is observed in the cooling section after the start of solidification of the melt. This is associated with jump-like reduction of the solubility of nitrogen in delta iron. Gas generation from the melt leads to the depletion of the subsurface zone in the alloyed layer and to the appearance of maxima on the concentration profiles of nitrogen in the multiple irradiated metal. The concentration maxima ŽFig. 4b. are placed at a depth of the order of the diffusion length ; Ž Dtg .1r2 ; 5 m m from the surface, where tg is the characteristic time of gas generation. The calculations show that the thickness of the nitrated layer, formed during pulsed-periodic laser radiation in the millisecond range, may exceed ; 30–50 m m. The order of magnitude of this value corresponds to the size of the heat affected zone because diffusion into the unheated volume of the metal is extremely small. The nitrogen concentration reaches the values close to the solubility limit at the solidification point. 3.4. Metal alloying under plasma conditions The efficiency of surface alloying increases steeply under the laser action with plasma formation near the irradiated metal. Although the description of the dynamics of laser-produced plasma is rather complicated w10x, qualitative analysis can be performed in the framework of the present approach. In the situation being considered, gas dissociation on the metal surface is of minor importance. At a characteristic plasma temperatures T ; 10 4 K even at a low degrees of ionization, practically all nitrogen in plasma region is dissociated and can be easily absorbed by the melt. Concentration profiles and temporal behavior of subsurface nitrogen content in iron calculated on the basis of the proposed model
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4. Conclusions Ž1. Kinetic theory analysis is applied to the problem of laser surface alloying of a metal from the atmosphere of a molecular gas. The proposed solution of mass transfer through the gas–liquid interface may be considered as a kind of boundary condition for the general problem of admixture redistribution in laser alloying from gaseous phase. Ž2. The concentration profiles of the alloying element in pulsed-periodic laser alloying from the gas atmosphere have local maxima close to the surface of the metal. Ž3. The efficiency of surface alloying increases steeply under laser-plasma conditions which results in the formation of highly supersaturated gas solutions in the metal.
References
Fig. 5. Concentration profiles Ža. at different times t s1 m s Žcurve 1., 5 m s Ž2., 25 m s Ž3., 125 m s Ž4. and temporal behavior of subsurface content Žb. of nitrogen in surface alloying of iron under plasma conditions. Alloying regime: PN 2 s 0.1 MPa, TN 2 s 10 4 K, Ts s 2.1P10 3 K.
for the dissociated Žmonatomic. gas are shown in Fig. 5. It must be emphasized that the alloying rate under laser-plasma conditions increases to an extent that in a short time period Žof the order of few m s. the nitrogen solution becomes supersaturated.
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