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Simulation of laser quenching to assess thermophysical properties of TiC
R~f/?actory Metals ,:~ Hard Materials 11 (1992) 113 119
Simulation of Laser Quenching to Assess Thermophysical Properties of TiC A. B ichli & A. Blatter Institute o f A p p l i e d Physics, U n i v e r s i t y o f Berne, CH-3012 Berne, Switzerland (Received 10 M a r c h 1992; accepted 6 July 1992)
Abstract: Pulsed laser quenching of vapor deposited TiC is simulated by a
numerical model which combines macroscopic heat flow and microscopic solidification kinetics. The model permits examination of laser induced melting as well as crystallization during cooling. Experimentally accessible aspects including the threshold laser fluence for melting, the melt depth, and the grain size can be calculated. The simulation shows the sensitivity of these aspects to various material parameters. By comparing model predictions with experimental measurements, it is therefore possible to estimate unknown thermophysical properties of TiC. Specifically, a high-temperature (2200-3340 K) thermal conductivity of the solid of 45 W / K m , a solid-liquid interracial energy of around 0.25 J/m '2, and an upper limit for the glass transition temperature of 1650 K have been evaluated.
INTRODUCTION
the simulation is to detail the thermal and microstructural evolution during laser melting and following rapid solidification of a high melting point material. The simulation permits isolation and identification of the effects of changes in individual parameters on experimentally observable aspects, such as laser fluence for incipient melting, melt depth, and grain size. Insisting on consistency between simulation and experiments confines the free variability of the model parameters and thereby allows the evaluation of thermophysical properties of TiC which are otherwise inaccessible. The interfacial energy between liquid and solid, the glass transition temperature, and high-temperature thermal conductivities are assessed in this way.
Nanosecond laser pulses permit selective melting and subsequent quenching of the near-surface region without affecting the underlying bulk. 1'2 The enormous quench rates achieved (typically 10 TM K/s) 3 lead to a high undercooling of the melt, governing solidification kinetics and microstructural features. The product is a homogeneous, extremely fine-grained or even amorphous surface microstructure having significant impact on the material's performance. We have recently demonstrated the potential of laser quenching to improve the tribological properties of high melting point TiC coatings prepared by Chemical Vapor Deposition ( C V D ) . 4'5 The originally columnar morphology was thereby transformed to a refined equiaxed microstructure and the surface was simultaneously polished. This was even achieved for TiC on graphite where mechanical attempts to polish resulted in cohesive failure within the substrate? This paper describes the computer simulation of pulsed laser quenching of TiC using a model which combines macroscopic heat flow and microscopic isothermal crystallization kinetics. The purpose of
EXPERIMENTAL BACKGROUND
This section is a brief summary of the experimental conditions, procedures and results as described in detail elsewhere. 4'5 TiC coatings of thickness 2-4 ~na were grown on various substrates by CVD. The deposits exhibited a columnar growth morphology and a pronounced micro-roughness. A consequence of this roughness, 113
Refractor)' Metals & Hard Materials 02634368/92 $05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain.
A. Biichli, A. Blatter
114
with dimensions in the sub-/~m range, was a greatly enhanced absorptance for the 1-06/tm radiation of the N d : Y A G laser. A reflectance R of 10% equivalent to an absorptance ( l - R ) of 90% was measured. The coatings were irradiated with 10 and 50 ns pulses from Q-switched N d : Y A G lasers at various fluences. The pulses had a Gaussian temporal profile. The threshold fluences for melting were 0-5 J / c m 2 (10 ns) and 1"2 J / c m 2 (50 ns). The melt depths attained at higher fluences were determined by cross-sectional SEM. The resolidified surface was microscopically flat and the absorptance accordingly decreased to 50 %. A thin film X-ray diffractometer (CuK~I) was employed to characterize the resolidified microstructure. Analysis of the broadened X-ray diffraction profiles revealed a refined microstructure with randomly orientated crystallites. A lower limit for the volume-weighted crystallite size of 25 nm was deduced.
THE S I M U L A T I O N M O D E L Macroscopic heat flow model The thermal evolution of the TiC coating during laser melting and resolidification is examined by numerically solving the one-dimensional heat flow equation using a finite-differences (6z, ~t), 1-domain enthalpy scheme which ensures energy conservation during phase changes. 68 The restriction to one dimension is justified as the short laser pulses give shallow (of the order of 1/zm) and wide (5 ram) melted zones. As the melted zone is also shallow compared to the coating thickness, the heat flux at the TiC-substrate interface is too small (until solidification is completed) to affect the evolving microstructure. The coating is therefore considered as semi-infinite for the calculations. Initially, all spatial elements 5z are assigned the enthalpy equivalent of room temperature. Heat flow through the sample-atmosphere interface is neglected. The macroscopic continuity equation to be solved then reads
I( t) (1 -- R) F(z) + V [K(T) V T(z,t)] - dH(dtT,z) (1) The first term describes the heat flux produced by the absorption of the laser light. A Gaussian shaped laser intensity I(t) is assumed, with its F W H M equivalent to the pulse duration. The reflectance R is assigned the measured value 0.1 for the original,
solid surface and 0"5 for the liquid. The distribution function F(z) expresses the penetration of the laser light into the material. 2 Thermal conductivity is denoted by K. The enthalpy is given by H(T)
(2)
=
H~(T) =
cp(®) d®
(3)
The enthalpy of the solid Hs(T) and the enthalpy of fusion, L = 5-83 G J / m ~ are tabulated) The solid fraction f8 must be calculated for all individual spatial elements at each time step. This is easily done for the melting process where no superheating of the solid is assumed. However, during the subsequent solidification period undercooling of the melt must not be neglected. In this case, nucleation and growth kinetics must be used to obtain an appropriate solid fraction model. 7 10
Microscopic solidification model For solidifying systems, f~ is readily obtained using Johnson-Mehl-Avrami isothermal transformation kinetics. 11 This requires calculation of the homogeneous nucleation of crystals (at high quench rates heterogeneous nucleation is not normally important) within a small isothermal control cell (~z) ~, and their three-dimensional growth. Isothermal steady state nucleation assumes that the equilibrium cluster population evolves rapidly enough to follow changes in temperature. Otherwise, the classical theory would overestimate the nucleation rate. The rate at which nuclei form in a control cell is given by 12
li(T) = (dz)aD(l ~-~s)Nv exp(-(A~-~) )
(4)
Here, ( 1 - f s ) N v is the number of nucleation sites with Nv = 5"1028 m -3 the molecular concentration. D is the diffusivity, a 0 = 3 nm an effective molecular diameter, k the Boltzmann constant, and AG* the activation energy for homogeneous nucleation. AG* is related to the solid-liquid interfacial energy O"by 12 16~ a 3 AG* - ~ - AGv~ (5) The volume free-energy difference between liquid and solid, AG v, is approximated by the DubeyRamachandrarao expression 13
rm
) (6)
Laser quenching to assess thermophysical properties of TiC
where T,, = 3340 K is the melting point. The excess specific heat Ac~)between liquid and solid is taken to be 0.8 times the volume entropy of fusion? 4 The precise value of Ac, is not important for the present situation. Crystallization restricts the undercooling to a few 100 K. This limits the second term in eqn. (6) to less than 5 % of the first term. Diffusion-limited crystal growth as found in m a n y cases of c o m p o u n d formation 1'~ can be expressed by the relationship TM
v(T) =J-D--(1--exp( - - ~ ) )
(7)
I
]
I
/.
610
5000
4000
% ~-
3000
2000
1000 300
20
O -
kT 3zca,,tl( T)
(8)
By definition, a viscosity of l0 le Ns m 2 is assumed at the glass transition temperature Tg. Based on the free volume theory, the viscosity between T,,, and T:, can be estimated by a Doolittle expression 17 which yields
(
,)))
810
I 100
120
t [ns]
a0 \
where the structural constant f is normally taken as unity, R is the gas constant, and the driving force for crystallization AG,, is related to AG,, by the molar volume. Note that the experimentally observed microstructure cannot be simulated on the assumption of collision-limited growth. Assuming that the diffusion coefficients for nucleation and growth are the same, and identical to the liquid diffusivity, they can be related to the viscosity t / b y the Stokes-Einstein equation
115
Fig. 1. The calculated course of temperature T of TiC upon irradiation with a 10 ns laser pulse (LP) of fluence I J/cm 2. Curves 1, 2, and 3 correspond respectively to the temperature in a depth of 10, 190, and 350 nm.
temperature falls below T,,. As soon as the integral becomes greater than one, nuclei of the critical volume have formed which subsequently grow according to eqn. (7). The volume of newly formed nuclei must not be neglected here for the evaluation of the solid fraction, since high cooling rates require such small numerical time steps that growth may be less than 1 ,~ per cycle. This must be compared to a critical radius of the order of nanometers. The crystallized volume at time t and hence the solid fraction to be introduced into the macroscopic eqn. (1), is derived from 47~
(Sz)3L(t) = n ( t ) ~ ( r ( t - S t ) +
v(T)St) ~
(12)
(9)
where r ( t - d t ) denotes the average grain radius. This increases by v(T) 5t during the present cycle fit to give the new average radius
The hole formation energy E . is calculated from T~17
(13)
r/(T) = 0"0033 exp 33-34 exp ~ - ~ - - ~
E . = (6.96 T,/-705)R
(10)
For a reasonable range of T~ this returns a viscosity at T,, of the order of 10 ,2 Ns m -2, a value typically encountered in metallic melts. The solid fraction j,~ and an average grain radius r are then derived as follows. First, the number of nuclei n(t) within each control cell is calculated:
n(t) =
h dr
(11)
o
The integration starts at t o, the time at which the II
This f ~ - r :~relationship neglects grain impingement during growth. A sophisticated fi'c impingement model reveals that deviations occur only for .f, > 0.7, and that the final average grain radius is underestimated by about 20 %.7
RESULTS AND DISCUSSION Figure 1 displays the simulated thermal evolution of three spatial TiC elements upon laser quenching, with a 10 ns pulse of fluence 1 J / c m 2. The material parameters used are those listed in Table 1. The first step in the simulation is the absorption of the laser ERM
II
116
A. Bgichli, A. Blatter
Table 1. TiC material parameters used in this work unless stated otherwise.
Parameter
Value
Reference
1.6
E ~0
1.1,
~ so
~"
Melting point Glass transition temperature Solid liquid interfacial energy Molar mass of TiC Molecular diameter Density of TiC at 3000 K Latent heat Original solid absorptance Liquid absorptance Thermal conductivity 300-2300 K Thermal conductivity 2300 K - T m Thermal conductivity of the liquid Enthalpy
T~ Tg
3340 K 1600 K
[19] *
~
0-25 J/cm 2
a0 p
60 amu 0"3 nm 4700 kg/m 3
[20] [21]
L (l-R)
5-83 G J / m 3 0-9
[9] [4]
(l-R) K
0-5 30-45 W / K m
K
45 W / K m
*
K,i
20 W / K m
*
u.
*
•
l ~000
•-,E 1.2
H(T)
[9]
* Parameters evaluated in this work.
energy. Absorption is in a surface layer (50-70 nm) and decreases exponentially over several spatial elements. The deposited energy heats and melts the surface element (curve 1). Towards the end of the laser pulse, the heat flow out of the element overbalances the energy supply by the laser. Consequently, the temperature goes through a maximum and plunges. The high quench rate at T m of the order of 1011 K/s causes the melt to undercool by a few 100 K before the first nuclei are formed. The latent heat released during their growth gives rise to recalescence as reflected by the second bump in curve 1. The heat diffusion from the surface into the material drives the melt front to a depth of 190 nm, as can be seen from the temperature curve of the corresponding element (curve 2). Deeper lying elements are heated without undergoing a phase transition (curve 3). The laser irradiation period involving material heating and melting, and the cooling period involving crystallization will be examined and discussed separately. The material parameters relevant for the laser irradiation period are the absorptance, the enthalpy, the latent heat, and the thermal conductivity (see Table 1). In the case of heating without melting the only unknown parameter left is the thermal conductivity between 2300 K and T m. As this controls the heat loss--and hence the enthalpy and temperature of the solid surface element, it can be deduced from a comparison of the simulated
I
1 1 ~'/
//
...,...
/
0.8
06
-
.,."
,.,-
/
.-$4I
s"
TIK] 3000/ / . y
1
/
* [22, 23]
I
2000
.,'"
...."
.•.•
0.4 5'
1
2's
35 '
45 '
55 '
1: [nsl
Fig. 2. Calculated threshold laser fluence F for melting as a function of pulse duration z for three different high-temperature extrapolations of the thermal conductivity K as indicated in the inset• Agreement with experimental values (closed circles) is satisfactory for a constant high-temperature conductivity of 45 W/Km.
minimum laser fluence required to induce melting with the experimentally determined threshold. The variation of the threshold fluence with laser pulse duration is shown in Fig. 2. The closed circles are the experimental values for the 10 ns and 50 ns laser pulses respectively. The simulated curves correspond to the three different extrapolations of the conductivity to temperatures above 2300 K, as indicated in the inset. Coincidence is found on the assumption of a constant high-temperature thermal conductivity of 45 W / K m . With this parameter fixed, the liquid properties determine the melt depth attained at a higher fluence. Liquid absorptance was found to be relatively unimportant compared to the liquid thermal conductivity for 10 ns laser pulses. A liquid absorption of 50 % was assumed, the value measured for the resolidified flat surface. The observed melt depths of 0.2 #m (at 1 J/cm 2) and 0"4 #m (at 2 J/cm ~) are fairly well reproduced by the simulation using a liquid thermal conductivity of 20 W / K m (Fig. 3). These parameters also correctly reproduce the melt depths observed with 50 ns laser pulses. The estimate of the liquid thermal conductivity is, however, less reliable than that of the high-temperature solid conductivity. The liquid conductivity could not be evaluated independently of an assumption of the liquid absorptance, and the melt depths could not be determined as accurately as the threshold laser fluences in the experiment. The temperature, nucleation rate, growth velocity, and solid fraction of the surface element during the solidification period are displayed in Fig. 4. Notice that the growth velocity of individual
Laser quenching to assess thermophysical properties of TiC 0.5
,
,
,
,
,
117 6
10000
0.4
8 -C
8000
4 _
E "
0.3
~
~'--
6000 o2~
0.2
_"!, :... .
01
4000
'" .." " .. I
30
............
60
90
2000
120
150
170
0
t [nsl
Fig. 3. Calculated position z of the solid-liquid interface in TiC irradiated with 10 ns laser pulses of fluences 1 J/cm 2 (dotted line) and 2 J/cm 2 (solid line). The maximum z reflects the melt depth. The respective surface temperatures T are also shown.
I
!
i
0.8
12
9
3&oo
oooF/ 3
..-=os
25
\ ,, ~ = . , J ~ _~ ' ~ , ~ " ~
0 . 4 ~
18oo 1700
1600
Tg[KI
A
.,~_ ..
0 0
O'[j/m
0 3 ~ 2]
30
1 35
,.i,..,..y": ...... 40
02
45
tins]
Fig. 4. Details of the solidification of the surface element after irradiation with a 10 ns, 1 J/cm 2 laser pulse: T, temperature, n/(fz) 3, nucleation rate; v, grain growth velocity; ~ , solid fraction.
grains (less than I r a / s ) is much lower than the resulting velocity of the solidification front which is about 25 m / s as inferred from Fig. 3. Nucleation becomes significant only when the liquid is undercooled, for the present situation to about 2940 K. As soon as the first nuclei have formed, they start to grow and the solid fraction increases. The associated liberation of latent heat causes the temperature to rise. Accordingly, the nucleation rate tails off rapidly. In fact, nuclei are only formed during a short time interval of 3 ns. This leads to a narrow grain size distribution of 1 6 + 2 n m within the surface element. There is, however, a decrease in grain size with depth, as the development of temperature, and consequently crystallization kinetics, is different in each spatial element (see Fig. 1).
Fig. 5. Calculated dependence of average grain radii r calculated for the surface element on interfacial energy a and glass transition temperature ~ . Laser pulse parameters were 10 ns and 1 J/cm 2.
The average grain size near the melt depth is approximately 8 nm. The simulated grain size depends heavily on a and Tg, parameters for which only estimates but no experimental values are available. As liquid TiC can be considered like a Ti-C alloy, a o- in the range 0.1-0-5 Jm 2 can be assumed (N. Eustathopoulos, pers. comm.). A thermodynamic model for binary systems indicates a value of 0-32 Jm .~.lS In glass forming systems, Tg has been found to vary between 0.44. T m and 0.66. T,~. This corresponds to the temperature range 1470-2200 K for TiC. This latter value has been taken from the literature. 9 Figure 5 illustrates the dependence of the average grain size in the surface element on a and Yg. For a ~> 0.4 the nucleation barrier is high, and nucleation is suppressed for all T~, i.e. a glass would form in clear contradiction to experiment. With decreasing a a sharp transition to complete crystallization occurs. The grain size first increases; at the lowest values of a, however, the grain size decreases again. Here, the low nucleation barrier produces a high number of nuclei which limits the grain size (eqn. 13). The actual number of nuclei which form is also responsible for the decrease of grain size with increasing T,. A higher T~ means a lower diffusivity and hence a lower r~ and v. Accordingly, f~ increases less rapidly (eqn. 12) and reheating by recalescence is reduced. This delays the drop of r~ in Fig. 4. As a consequence of this extended nucleation interval, more nuclei form. At even higher Tg diffusionlimited kinetics are so slow that crystallization is completely suppressed. As discussed in the previous section, the simulation underestimates the true grain radius by about 20%. Therefore, a calculated radius of at least 10 nm is required to be consistent with the experimental crystallite size of approximately 25 nm. Such large radii are only reproduced by the simulation within a confined area of the a-T~, plane, 11-2
118
A. Biichli, A. Blatter I
1
I
I
I
2000
1900 r<5nm
1800
...''"
'...
1700
:.:""'~.!
1600 1500 i I
0.1
I
02
I
0.3
However, the corresponding intensity ( > 10 TM W / c m ~) would be above the threshold normally observed for air-breakdown. Furthermore, the resulting melt depth would be of the order of 10 nm which is significantly less than the surface roughness of the TiC deposits. This makes the application of the present model questionable (assuming planar heat flow). CONCLUSION
I
0.4
I
0.5
O"[JIm 21 Fig. 6. Projection of Fig. 5 onto the a-Tg plane. Grain radii exceeding 10 nm, and hence comparable with experimental values, lie in the area enclosed by the solid line. For comparison, parameters outside the dotted line return radii which are far too small ( < 5 nm) to match with experimental grain size. The shaded area thus contains the most likely values of cr and Tg for TiC.
as illustrated in Fig. 6. Grain radii larger than 10 nm are obtained only within the shaded area, which therefore embodies the most likely combinations. An upper bound of 1650 K is inferred for Tg which is definitely lower than the value of 2200 K quoted in Ref. 9. Assuming a lower bound of 0.44. T,,, acceptable values for a are found in the range 0"2-0"35 Jm -~. The mean value of 0-25 Jm -2 is close to (though somewhat lower than) the estimated value of 0-32 Jm -2. The evaluated limits for a and Tg are rather stringent, even though they rely on the somewhat arbitrary criterion r > 10 nm. This is visualized in Fig. 6 by the dotted line. Although near the shaded area, it is still in the range of highly improbable parameters, grain radii on this line are 5 nm and are far too small to be comparable with experimental data. The major uncertainty in the evaluation of a and Tg is related to the approximation of viscosity to diffusivity. It may be noted, however, that the use of a Fulcher-Vogel interpolation scheme--which has also been applied successfully to undercooled metallic melts17--instead of eqn. (9), will leave the evaluated limits unchanged. Given o- = 0"25 J / c m 2 and Tg = 1650 K, the simulation can be used to determine whether chemically vapor-deposited TiC could be vitrified by melt quenching at all. To keep the fraction solid below 10_8 (the criterion for a glass), a quenching rate in excess of 1013 K / s would be required. This could in principle be accomplished by a 2 ps laser pulse with a fluence of about 0"05 J / c m 2 to ensure melting.
a-Tg
Nanosecond laser quenching of TiC has been examined by computer simulation and experiments. U n k n o w n material parameters used in the model were varied in such a way as to reconcile simulated predictions with experimental data. Thermophysical properties of TiC were evaluated. A constant thermal conductivity of 45 W / K m , between 2200 K and the melting point was derived. A value of the order of 20 W / K m was inferred for the liquid. Acceptable values for the solid-liquid interfacial energy were found in a range around 0.25 J / c m 2. The glass transition temperature is most likely to be in the range 1470-1650 K, significantly lower than previously assumed. Metastable phase formation was not observed in spite of extreme quench rates of 1011 K/s. The simulation made glass formation conceivable only under conditions of sub-picosecond laser quenching. This work indicates how the method of pulsed laser quenching can be used to predict the thermophysical properties of high melting point, reactive materials where direct experimental access may be difficult. ACKNOWLEDGEMENTS We are indebted to Professor Dr. H. P. Weber for helpful discussions. This work was funded by the Swiss Foundation for the Research in Microtechnics and by the Swiss Commission for the Encouragement of Scientific Research. REFERENCES 1. Picraux, S. T. & Pope, L. E., Science, 226 (1984) 615. 2. Von Allmen, M., Laser-Beam Interaction with Materials. Springer, Berlin, 1987. 3. von Allmen, M., Huber, E., Blatter, A. & Affolter, K., J. Rapid Solidif., 1 (1984-85) 15. 4. B~ichli, A. & Blatter, A., Surface & Coatings Techn., 45 (1991) 393-7. 5. B/ichli, A. & Blatter, A., in Proc. 5th Intern. Conf. in Surface Modification Techn., Sept. 1991, Birmingham, UK., 1992, 821 33.
Laser quenching to assess thermophysical properties o f TiC 6. Clyne, T. W., Metall. Trans. B, 15B (1984) 369 81. 7. Rappaz, M., Int. Materials Rev., 34 (1989) 93 123. 8. Kurz, W. & Fisher, D. J., Fundamentals o f S o l i d , cation, 3rd Edn. Trans Tech Publications, Switzerland, 1989, 2143. 9. Stull, D . R . & Prophet, H., J A N A F Thermochemical Tables, J. Phys. Chem. Re[i Data. 14 Suppl. 1 (1985) 640-2. 10. Morris, D. G. Acta Metall. 31 (1983) 1479 89. I 1. Christian, J. W., Trans:[~)rmation in Metals and Alloys. 2nd Edn, Pergamon Press, 1975. 12. Uhlmann. D. R., Hays, J. F. & Turnbull, D., Phys. Chem. Glasses. 7 (1966) 159. 13. Dubey, K.S. & Ramachandrarao, P., Aeta Metall. 32 (1984) 91. 14. Battezzatti, L. & Garrone, E. Z. Metallkd. 305 (1984) 75.
119
15. Vitta, S., Greet, A. L. & Somekh, R. E., Mater. Sci. Eng. 98 (1988) 105. 16. Turnbull, D. J. Phys. Chem. 64 (1962) 609. 17. Ramachandrarao, P., Cantor, B. & Cahn, R. W.. J. Mat. Sci. 12 (1977) 2488 502. 18. Warren, R. J. Mat. Sci. 15 (1980) 2489. 19. Ohtani, H., Tanaka, T., Hasebe, M. & Nishiyawa, T., Calphad 12 (1988) 235. 20. We have used the diameter of the metal atom as justified for metal metalloid alloys : Turnbull, D. Metall. Trans. A, 12A (1981) 695. 21. Touloukian, Y. S. lEd.], Thermophysical Properties ~/High Temperature Solid Material, vol. 5. 1967. 22. Morelli, D. T., Phys. Rev. B. 44 (1991) 5453 8. 23. Taylor, R. E., J. Am. Ceram. Soe. 44 (1961) 525.
Simulation of Laser Quenching to Assess Thermophysical Properties of TiC A. B ichli & A. Blatter Institute o f A p p l i e d Physics, U n i v e r s i t y o f Berne, CH-3012 Berne, Switzerland (Received 10 M a r c h 1992; accepted 6 July 1992)
Abstract: Pulsed laser quenching of vapor deposited TiC is simulated by a
numerical model which combines macroscopic heat flow and microscopic solidification kinetics. The model permits examination of laser induced melting as well as crystallization during cooling. Experimentally accessible aspects including the threshold laser fluence for melting, the melt depth, and the grain size can be calculated. The simulation shows the sensitivity of these aspects to various material parameters. By comparing model predictions with experimental measurements, it is therefore possible to estimate unknown thermophysical properties of TiC. Specifically, a high-temperature (2200-3340 K) thermal conductivity of the solid of 45 W / K m , a solid-liquid interracial energy of around 0.25 J/m '2, and an upper limit for the glass transition temperature of 1650 K have been evaluated.
INTRODUCTION
the simulation is to detail the thermal and microstructural evolution during laser melting and following rapid solidification of a high melting point material. The simulation permits isolation and identification of the effects of changes in individual parameters on experimentally observable aspects, such as laser fluence for incipient melting, melt depth, and grain size. Insisting on consistency between simulation and experiments confines the free variability of the model parameters and thereby allows the evaluation of thermophysical properties of TiC which are otherwise inaccessible. The interfacial energy between liquid and solid, the glass transition temperature, and high-temperature thermal conductivities are assessed in this way.
Nanosecond laser pulses permit selective melting and subsequent quenching of the near-surface region without affecting the underlying bulk. 1'2 The enormous quench rates achieved (typically 10 TM K/s) 3 lead to a high undercooling of the melt, governing solidification kinetics and microstructural features. The product is a homogeneous, extremely fine-grained or even amorphous surface microstructure having significant impact on the material's performance. We have recently demonstrated the potential of laser quenching to improve the tribological properties of high melting point TiC coatings prepared by Chemical Vapor Deposition ( C V D ) . 4'5 The originally columnar morphology was thereby transformed to a refined equiaxed microstructure and the surface was simultaneously polished. This was even achieved for TiC on graphite where mechanical attempts to polish resulted in cohesive failure within the substrate? This paper describes the computer simulation of pulsed laser quenching of TiC using a model which combines macroscopic heat flow and microscopic isothermal crystallization kinetics. The purpose of
EXPERIMENTAL BACKGROUND
This section is a brief summary of the experimental conditions, procedures and results as described in detail elsewhere. 4'5 TiC coatings of thickness 2-4 ~na were grown on various substrates by CVD. The deposits exhibited a columnar growth morphology and a pronounced micro-roughness. A consequence of this roughness, 113
Refractor)' Metals & Hard Materials 02634368/92 $05.00 © 1992 Elsevier Science Publishers Ltd, England. Printed in Great Britain.
A. Biichli, A. Blatter
114
with dimensions in the sub-/~m range, was a greatly enhanced absorptance for the 1-06/tm radiation of the N d : Y A G laser. A reflectance R of 10% equivalent to an absorptance ( l - R ) of 90% was measured. The coatings were irradiated with 10 and 50 ns pulses from Q-switched N d : Y A G lasers at various fluences. The pulses had a Gaussian temporal profile. The threshold fluences for melting were 0-5 J / c m 2 (10 ns) and 1"2 J / c m 2 (50 ns). The melt depths attained at higher fluences were determined by cross-sectional SEM. The resolidified surface was microscopically flat and the absorptance accordingly decreased to 50 %. A thin film X-ray diffractometer (CuK~I) was employed to characterize the resolidified microstructure. Analysis of the broadened X-ray diffraction profiles revealed a refined microstructure with randomly orientated crystallites. A lower limit for the volume-weighted crystallite size of 25 nm was deduced.
THE S I M U L A T I O N M O D E L Macroscopic heat flow model The thermal evolution of the TiC coating during laser melting and resolidification is examined by numerically solving the one-dimensional heat flow equation using a finite-differences (6z, ~t), 1-domain enthalpy scheme which ensures energy conservation during phase changes. 68 The restriction to one dimension is justified as the short laser pulses give shallow (of the order of 1/zm) and wide (5 ram) melted zones. As the melted zone is also shallow compared to the coating thickness, the heat flux at the TiC-substrate interface is too small (until solidification is completed) to affect the evolving microstructure. The coating is therefore considered as semi-infinite for the calculations. Initially, all spatial elements 5z are assigned the enthalpy equivalent of room temperature. Heat flow through the sample-atmosphere interface is neglected. The macroscopic continuity equation to be solved then reads
I( t) (1 -- R) F(z) + V [K(T) V T(z,t)] - dH(dtT,z) (1) The first term describes the heat flux produced by the absorption of the laser light. A Gaussian shaped laser intensity I(t) is assumed, with its F W H M equivalent to the pulse duration. The reflectance R is assigned the measured value 0.1 for the original,
solid surface and 0"5 for the liquid. The distribution function F(z) expresses the penetration of the laser light into the material. 2 Thermal conductivity is denoted by K. The enthalpy is given by H(T)
(2)
=
H~(T) =
cp(®) d®
(3)
The enthalpy of the solid Hs(T) and the enthalpy of fusion, L = 5-83 G J / m ~ are tabulated) The solid fraction f8 must be calculated for all individual spatial elements at each time step. This is easily done for the melting process where no superheating of the solid is assumed. However, during the subsequent solidification period undercooling of the melt must not be neglected. In this case, nucleation and growth kinetics must be used to obtain an appropriate solid fraction model. 7 10
Microscopic solidification model For solidifying systems, f~ is readily obtained using Johnson-Mehl-Avrami isothermal transformation kinetics. 11 This requires calculation of the homogeneous nucleation of crystals (at high quench rates heterogeneous nucleation is not normally important) within a small isothermal control cell (~z) ~, and their three-dimensional growth. Isothermal steady state nucleation assumes that the equilibrium cluster population evolves rapidly enough to follow changes in temperature. Otherwise, the classical theory would overestimate the nucleation rate. The rate at which nuclei form in a control cell is given by 12
li(T) = (dz)aD(l ~-~s)Nv exp(-(A~-~) )
(4)
Here, ( 1 - f s ) N v is the number of nucleation sites with Nv = 5"1028 m -3 the molecular concentration. D is the diffusivity, a 0 = 3 nm an effective molecular diameter, k the Boltzmann constant, and AG* the activation energy for homogeneous nucleation. AG* is related to the solid-liquid interfacial energy O"by 12 16~ a 3 AG* - ~ - AGv~ (5) The volume free-energy difference between liquid and solid, AG v, is approximated by the DubeyRamachandrarao expression 13
rm
) (6)
Laser quenching to assess thermophysical properties of TiC
where T,, = 3340 K is the melting point. The excess specific heat Ac~)between liquid and solid is taken to be 0.8 times the volume entropy of fusion? 4 The precise value of Ac, is not important for the present situation. Crystallization restricts the undercooling to a few 100 K. This limits the second term in eqn. (6) to less than 5 % of the first term. Diffusion-limited crystal growth as found in m a n y cases of c o m p o u n d formation 1'~ can be expressed by the relationship TM
v(T) =J-D--(1--exp( - - ~ ) )
(7)
I
]
I
/.
610
5000
4000
% ~-
3000
2000
1000 300
20
O -
kT 3zca,,tl( T)
(8)
By definition, a viscosity of l0 le Ns m 2 is assumed at the glass transition temperature Tg. Based on the free volume theory, the viscosity between T,,, and T:, can be estimated by a Doolittle expression 17 which yields
(
,)))
810
I 100
120
t [ns]
a0 \
where the structural constant f is normally taken as unity, R is the gas constant, and the driving force for crystallization AG,, is related to AG,, by the molar volume. Note that the experimentally observed microstructure cannot be simulated on the assumption of collision-limited growth. Assuming that the diffusion coefficients for nucleation and growth are the same, and identical to the liquid diffusivity, they can be related to the viscosity t / b y the Stokes-Einstein equation
115
Fig. 1. The calculated course of temperature T of TiC upon irradiation with a 10 ns laser pulse (LP) of fluence I J/cm 2. Curves 1, 2, and 3 correspond respectively to the temperature in a depth of 10, 190, and 350 nm.
temperature falls below T,,. As soon as the integral becomes greater than one, nuclei of the critical volume have formed which subsequently grow according to eqn. (7). The volume of newly formed nuclei must not be neglected here for the evaluation of the solid fraction, since high cooling rates require such small numerical time steps that growth may be less than 1 ,~ per cycle. This must be compared to a critical radius of the order of nanometers. The crystallized volume at time t and hence the solid fraction to be introduced into the macroscopic eqn. (1), is derived from 47~
(Sz)3L(t) = n ( t ) ~ ( r ( t - S t ) +
v(T)St) ~
(12)
(9)
where r ( t - d t ) denotes the average grain radius. This increases by v(T) 5t during the present cycle fit to give the new average radius
The hole formation energy E . is calculated from T~17
(13)
r/(T) = 0"0033 exp 33-34 exp ~ - ~ - - ~
E . = (6.96 T,/-705)R
(10)
For a reasonable range of T~ this returns a viscosity at T,, of the order of 10 ,2 Ns m -2, a value typically encountered in metallic melts. The solid fraction j,~ and an average grain radius r are then derived as follows. First, the number of nuclei n(t) within each control cell is calculated:
n(t) =
h dr
(11)
o
The integration starts at t o, the time at which the II
This f ~ - r :~relationship neglects grain impingement during growth. A sophisticated fi'c impingement model reveals that deviations occur only for .f, > 0.7, and that the final average grain radius is underestimated by about 20 %.7
RESULTS AND DISCUSSION Figure 1 displays the simulated thermal evolution of three spatial TiC elements upon laser quenching, with a 10 ns pulse of fluence 1 J / c m 2. The material parameters used are those listed in Table 1. The first step in the simulation is the absorption of the laser ERM
II
116
A. Bgichli, A. Blatter
Table 1. TiC material parameters used in this work unless stated otherwise.
Parameter
Value
Reference
1.6
E ~0
1.1,
~ so
~"
Melting point Glass transition temperature Solid liquid interfacial energy Molar mass of TiC Molecular diameter Density of TiC at 3000 K Latent heat Original solid absorptance Liquid absorptance Thermal conductivity 300-2300 K Thermal conductivity 2300 K - T m Thermal conductivity of the liquid Enthalpy
T~ Tg
3340 K 1600 K
[19] *
~
0-25 J/cm 2
a0 p
60 amu 0"3 nm 4700 kg/m 3
[20] [21]
L (l-R)
5-83 G J / m 3 0-9
[9] [4]
(l-R) K
0-5 30-45 W / K m
K
45 W / K m
*
K,i
20 W / K m
*
u.
*
•
l ~000
•-,E 1.2
H(T)
[9]
* Parameters evaluated in this work.
energy. Absorption is in a surface layer (50-70 nm) and decreases exponentially over several spatial elements. The deposited energy heats and melts the surface element (curve 1). Towards the end of the laser pulse, the heat flow out of the element overbalances the energy supply by the laser. Consequently, the temperature goes through a maximum and plunges. The high quench rate at T m of the order of 1011 K/s causes the melt to undercool by a few 100 K before the first nuclei are formed. The latent heat released during their growth gives rise to recalescence as reflected by the second bump in curve 1. The heat diffusion from the surface into the material drives the melt front to a depth of 190 nm, as can be seen from the temperature curve of the corresponding element (curve 2). Deeper lying elements are heated without undergoing a phase transition (curve 3). The laser irradiation period involving material heating and melting, and the cooling period involving crystallization will be examined and discussed separately. The material parameters relevant for the laser irradiation period are the absorptance, the enthalpy, the latent heat, and the thermal conductivity (see Table 1). In the case of heating without melting the only unknown parameter left is the thermal conductivity between 2300 K and T m. As this controls the heat loss--and hence the enthalpy and temperature of the solid surface element, it can be deduced from a comparison of the simulated
I
1 1 ~'/
//
...,...
/
0.8
06
-
.,."
,.,-
/
.-$4I
s"
TIK] 3000/ / . y
1
/
* [22, 23]
I
2000
.,'"
...."
.•.•
0.4 5'
1
2's
35 '
45 '
55 '
1: [nsl
Fig. 2. Calculated threshold laser fluence F for melting as a function of pulse duration z for three different high-temperature extrapolations of the thermal conductivity K as indicated in the inset• Agreement with experimental values (closed circles) is satisfactory for a constant high-temperature conductivity of 45 W/Km.
minimum laser fluence required to induce melting with the experimentally determined threshold. The variation of the threshold fluence with laser pulse duration is shown in Fig. 2. The closed circles are the experimental values for the 10 ns and 50 ns laser pulses respectively. The simulated curves correspond to the three different extrapolations of the conductivity to temperatures above 2300 K, as indicated in the inset. Coincidence is found on the assumption of a constant high-temperature thermal conductivity of 45 W / K m . With this parameter fixed, the liquid properties determine the melt depth attained at a higher fluence. Liquid absorptance was found to be relatively unimportant compared to the liquid thermal conductivity for 10 ns laser pulses. A liquid absorption of 50 % was assumed, the value measured for the resolidified flat surface. The observed melt depths of 0.2 #m (at 1 J/cm 2) and 0"4 #m (at 2 J/cm ~) are fairly well reproduced by the simulation using a liquid thermal conductivity of 20 W / K m (Fig. 3). These parameters also correctly reproduce the melt depths observed with 50 ns laser pulses. The estimate of the liquid thermal conductivity is, however, less reliable than that of the high-temperature solid conductivity. The liquid conductivity could not be evaluated independently of an assumption of the liquid absorptance, and the melt depths could not be determined as accurately as the threshold laser fluences in the experiment. The temperature, nucleation rate, growth velocity, and solid fraction of the surface element during the solidification period are displayed in Fig. 4. Notice that the growth velocity of individual
Laser quenching to assess thermophysical properties of TiC 0.5
,
,
,
,
,
117 6
10000
0.4
8 -C
8000
4 _
E "
0.3
~
~'--
6000 o2~
0.2
_"!, :... .
01
4000
'" .." " .. I
30
............
60
90
2000
120
150
170
0
t [nsl
Fig. 3. Calculated position z of the solid-liquid interface in TiC irradiated with 10 ns laser pulses of fluences 1 J/cm 2 (dotted line) and 2 J/cm 2 (solid line). The maximum z reflects the melt depth. The respective surface temperatures T are also shown.
I
!
i
0.8
12
9
3&oo
oooF/ 3
..-=os
25
\ ,, ~ = . , J ~ _~ ' ~ , ~ " ~
0 . 4 ~
18oo 1700
1600
Tg[KI
A
.,~_ ..
0 0
O'[j/m
0 3 ~ 2]
30
1 35
,.i,..,..y": ...... 40
02
45
tins]
Fig. 4. Details of the solidification of the surface element after irradiation with a 10 ns, 1 J/cm 2 laser pulse: T, temperature, n/(fz) 3, nucleation rate; v, grain growth velocity; ~ , solid fraction.
grains (less than I r a / s ) is much lower than the resulting velocity of the solidification front which is about 25 m / s as inferred from Fig. 3. Nucleation becomes significant only when the liquid is undercooled, for the present situation to about 2940 K. As soon as the first nuclei have formed, they start to grow and the solid fraction increases. The associated liberation of latent heat causes the temperature to rise. Accordingly, the nucleation rate tails off rapidly. In fact, nuclei are only formed during a short time interval of 3 ns. This leads to a narrow grain size distribution of 1 6 + 2 n m within the surface element. There is, however, a decrease in grain size with depth, as the development of temperature, and consequently crystallization kinetics, is different in each spatial element (see Fig. 1).
Fig. 5. Calculated dependence of average grain radii r calculated for the surface element on interfacial energy a and glass transition temperature ~ . Laser pulse parameters were 10 ns and 1 J/cm 2.
The average grain size near the melt depth is approximately 8 nm. The simulated grain size depends heavily on a and Tg, parameters for which only estimates but no experimental values are available. As liquid TiC can be considered like a Ti-C alloy, a o- in the range 0.1-0-5 Jm 2 can be assumed (N. Eustathopoulos, pers. comm.). A thermodynamic model for binary systems indicates a value of 0-32 Jm .~.lS In glass forming systems, Tg has been found to vary between 0.44. T m and 0.66. T,~. This corresponds to the temperature range 1470-2200 K for TiC. This latter value has been taken from the literature. 9 Figure 5 illustrates the dependence of the average grain size in the surface element on a and Yg. For a ~> 0.4 the nucleation barrier is high, and nucleation is suppressed for all T~, i.e. a glass would form in clear contradiction to experiment. With decreasing a a sharp transition to complete crystallization occurs. The grain size first increases; at the lowest values of a, however, the grain size decreases again. Here, the low nucleation barrier produces a high number of nuclei which limits the grain size (eqn. 13). The actual number of nuclei which form is also responsible for the decrease of grain size with increasing T,. A higher T~ means a lower diffusivity and hence a lower r~ and v. Accordingly, f~ increases less rapidly (eqn. 12) and reheating by recalescence is reduced. This delays the drop of r~ in Fig. 4. As a consequence of this extended nucleation interval, more nuclei form. At even higher Tg diffusionlimited kinetics are so slow that crystallization is completely suppressed. As discussed in the previous section, the simulation underestimates the true grain radius by about 20%. Therefore, a calculated radius of at least 10 nm is required to be consistent with the experimental crystallite size of approximately 25 nm. Such large radii are only reproduced by the simulation within a confined area of the a-T~, plane, 11-2
118
A. Biichli, A. Blatter I
1
I
I
I
2000
1900 r<5nm
1800
...''"
'...
1700
:.:""'~.!
1600 1500 i I
0.1
I
02
I
0.3
However, the corresponding intensity ( > 10 TM W / c m ~) would be above the threshold normally observed for air-breakdown. Furthermore, the resulting melt depth would be of the order of 10 nm which is significantly less than the surface roughness of the TiC deposits. This makes the application of the present model questionable (assuming planar heat flow). CONCLUSION
I
0.4
I
0.5
O"[JIm 21 Fig. 6. Projection of Fig. 5 onto the a-Tg plane. Grain radii exceeding 10 nm, and hence comparable with experimental values, lie in the area enclosed by the solid line. For comparison, parameters outside the dotted line return radii which are far too small ( < 5 nm) to match with experimental grain size. The shaded area thus contains the most likely values of cr and Tg for TiC.
as illustrated in Fig. 6. Grain radii larger than 10 nm are obtained only within the shaded area, which therefore embodies the most likely combinations. An upper bound of 1650 K is inferred for Tg which is definitely lower than the value of 2200 K quoted in Ref. 9. Assuming a lower bound of 0.44. T,,, acceptable values for a are found in the range 0"2-0"35 Jm -~. The mean value of 0-25 Jm -2 is close to (though somewhat lower than) the estimated value of 0-32 Jm -2. The evaluated limits for a and Tg are rather stringent, even though they rely on the somewhat arbitrary criterion r > 10 nm. This is visualized in Fig. 6 by the dotted line. Although near the shaded area, it is still in the range of highly improbable parameters, grain radii on this line are 5 nm and are far too small to be comparable with experimental data. The major uncertainty in the evaluation of a and Tg is related to the approximation of viscosity to diffusivity. It may be noted, however, that the use of a Fulcher-Vogel interpolation scheme--which has also been applied successfully to undercooled metallic melts17--instead of eqn. (9), will leave the evaluated limits unchanged. Given o- = 0"25 J / c m 2 and Tg = 1650 K, the simulation can be used to determine whether chemically vapor-deposited TiC could be vitrified by melt quenching at all. To keep the fraction solid below 10_8 (the criterion for a glass), a quenching rate in excess of 1013 K / s would be required. This could in principle be accomplished by a 2 ps laser pulse with a fluence of about 0"05 J / c m 2 to ensure melting.
a-Tg
Nanosecond laser quenching of TiC has been examined by computer simulation and experiments. U n k n o w n material parameters used in the model were varied in such a way as to reconcile simulated predictions with experimental data. Thermophysical properties of TiC were evaluated. A constant thermal conductivity of 45 W / K m , between 2200 K and the melting point was derived. A value of the order of 20 W / K m was inferred for the liquid. Acceptable values for the solid-liquid interfacial energy were found in a range around 0.25 J / c m 2. The glass transition temperature is most likely to be in the range 1470-1650 K, significantly lower than previously assumed. Metastable phase formation was not observed in spite of extreme quench rates of 1011 K/s. The simulation made glass formation conceivable only under conditions of sub-picosecond laser quenching. This work indicates how the method of pulsed laser quenching can be used to predict the thermophysical properties of high melting point, reactive materials where direct experimental access may be difficult. ACKNOWLEDGEMENTS We are indebted to Professor Dr. H. P. Weber for helpful discussions. This work was funded by the Swiss Foundation for the Research in Microtechnics and by the Swiss Commission for the Encouragement of Scientific Research. REFERENCES 1. Picraux, S. T. & Pope, L. E., Science, 226 (1984) 615. 2. Von Allmen, M., Laser-Beam Interaction with Materials. Springer, Berlin, 1987. 3. von Allmen, M., Huber, E., Blatter, A. & Affolter, K., J. Rapid Solidif., 1 (1984-85) 15. 4. B~ichli, A. & Blatter, A., Surface & Coatings Techn., 45 (1991) 393-7. 5. B/ichli, A. & Blatter, A., in Proc. 5th Intern. Conf. in Surface Modification Techn., Sept. 1991, Birmingham, UK., 1992, 821 33.
Laser quenching to assess thermophysical properties o f TiC 6. Clyne, T. W., Metall. Trans. B, 15B (1984) 369 81. 7. Rappaz, M., Int. Materials Rev., 34 (1989) 93 123. 8. Kurz, W. & Fisher, D. J., Fundamentals o f S o l i d , cation, 3rd Edn. Trans Tech Publications, Switzerland, 1989, 2143. 9. Stull, D . R . & Prophet, H., J A N A F Thermochemical Tables, J. Phys. Chem. Re[i Data. 14 Suppl. 1 (1985) 640-2. 10. Morris, D. G. Acta Metall. 31 (1983) 1479 89. I 1. Christian, J. W., Trans:[~)rmation in Metals and Alloys. 2nd Edn, Pergamon Press, 1975. 12. Uhlmann. D. R., Hays, J. F. & Turnbull, D., Phys. Chem. Glasses. 7 (1966) 159. 13. Dubey, K.S. & Ramachandrarao, P., Aeta Metall. 32 (1984) 91. 14. Battezzatti, L. & Garrone, E. Z. Metallkd. 305 (1984) 75.
119
15. Vitta, S., Greet, A. L. & Somekh, R. E., Mater. Sci. Eng. 98 (1988) 105. 16. Turnbull, D. J. Phys. Chem. 64 (1962) 609. 17. Ramachandrarao, P., Cantor, B. & Cahn, R. W.. J. Mat. Sci. 12 (1977) 2488 502. 18. Warren, R. J. Mat. Sci. 15 (1980) 2489. 19. Ohtani, H., Tanaka, T., Hasebe, M. & Nishiyawa, T., Calphad 12 (1988) 235. 20. We have used the diameter of the metal atom as justified for metal metalloid alloys : Turnbull, D. Metall. Trans. A, 12A (1981) 695. 21. Touloukian, Y. S. lEd.], Thermophysical Properties ~/High Temperature Solid Material, vol. 5. 1967. 22. Morelli, D. T., Phys. Rev. B. 44 (1991) 5453 8. 23. Taylor, R. E., J. Am. Ceram. Soe. 44 (1961) 525.