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On the glass transition of the one-component metallic melts
Accepted Manuscript On the glass transition of THE ONE-component metallic melts A.I. Fedorchenko PII: DOI: Reference:
S0022-0248(17)30428-1 http://dx.doi.org/10.1016/j.jcrysgro.2017.06.011 CRYS 24214
To appear in:
Journal of Crystal Growth
Received Date: Revised Date: Accepted Date:
6 March 2017 13 April 2017 12 June 2017
Please cite this article as: A.I. Fedorchenko, On the glass transition of THE ONE-component metallic melts, Journal of Crystal Growth (2017), doi: http://dx.doi.org/10.1016/j.jcrysgro.2017.06.011
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ON THE GLASS TRANSITION OF THE ONE-COMPONENT METALLIC MELTS A. I. Fedorchenko Institute of Thermomechanics, Academy of Sciences of the Czech Republic, v. v. i., Dolejskova 5, 182 00, Prague 8, Czech Republic S. S. Kutateladze Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia ABSTRACT In this paper, the conditions for one-component metallic melts vitrification by quenching from a liquid state were formulated. It is shown that the tendency to the glass formation drastically increases with the temperature of melting. The maximum glass layer thickness and the associated cooling rates along with the vitrification temperatures was determined for Al, Cu, and Ni melts deposited on the Cu substrate. The results are in agreement with the available experimental data. Based on analytical solution of the impinging droplet solidification, the numerical value of the early-introduced asymptotic criterion, which separates equilibrium and non-equilibrium phase transitions, was determined. Good agreement between the calculated and experimental values of the thickness of the splats shows that criterion indeed predicts a priori a scenario of solidification. Keywords: A1. Equilibrium and non-equilibrium solidification A1. Criterion of the phase transition scenario A2. One-component metal melts B2. Glass transition B2. Temperature of vitrification
Corresponding author. E-mail address: [email protected]
1
Nomenclature a
thermal diffusivity (m2 s–1)
c
specific heat (J kg–1K–1)
d
droplet diameter (m)
da
atom diameter (m)
Fo
Fourier number,
h
height of the splat (m)
hP
Planck constant (J s)
Js
stationary nucleation rate (s–1m–3)
k
Boltzmann constant (J K–1)
K
kinetic coefficient (m s–1K–1)
Ku
Kutateladze number, L /( cd(s)Tm )
K
(s) (s) 1/ 2 relative thermal activity, [(s) in Eq. (7) d d cd /( s s cs )]
K
relative thermal activity, [(dl ) d(l)cd(l) /(s s cs )]1/ 2 in Eq. (24)
l
linear scale (m)
L
latent heat of melting (J kg–1)
Lv
volumetric latent heat of melting, L (J m–3)
Na
number of atoms per unit volume (m–3)
Nu
Nusselt number, d / (dl )
Pe
Peclet number, Ud / a d(l )
Pr
Prandtl number, /a
q
cooling rate (K s–1)
[Q]f
ad(s)t/d2
jump of heat fluxes at the phase transition front (J m–2s–1) 2
R
ratio of diffusive and kinetic speeds, ad(s) /( dKTm )
Tg
glass transition temperature (K)
Tm
melting temperature (K)
T
undercooling of the melt, (Tm – T) (K)
uf
speed of the phase transition front (m s–1)
U
droplet impact velocity (m s–1)
U0
activation energy (J)
zd
instantaneous coordinate of the droplet apex (m)
Greek symbols
heat transfer coefficient (W m–2 K–1)
dimensionless energy of activation, 2.3U 0 /( kTm )
root of transcendental Eq. (7)
thermal conductivity (W m–1 K–1)
kinematic viscosity, (m2 s–1)
density (kg m–3)
R
time of molecular relaxation (s)
dimensionless temperature, T/Tm
d
droplet dimensionless temperature, (Td Ts 0 ) /(Td 0 Ts 0 )
dimension coordinate of the front of phase transition (m)
dimensionless coordinate of the front of phase transition, /d
criterion of the phase transition, /( Lv K )
3
Subscripts d
refers to droplet
f
refers to the front of phase transition
g
refers to glass
m
refers to melting (solidification)
s
refers to substrate
0
refers to initial state
Superscripts l
refers to liquid state
s
refers to solid state
1. Introduction Although amorphous substance, such as “silicate glasses” was known from time immemorial, a breakthrough in obtaining metallic glasses occurred as recently as in 60-es of the 20th century. Klement et al. [1] in 1960 were the first who produced the metallic glass from an alloy (Au 75Si25). They developed a splat quenching technique and the cooling rate q reached about 106 K/s. This stipulated the emergences of the multiple technologies of quenching from a liquid state, such as thermal spray, melt spinning, gun technique, etc. [2, 3]. Investigations of the metallic glasses have shown that they possess a unique set of mechanical, electrical, magnetic, elastic, anti-corrosion, etc. properties [4]. For a long time it was believed that getting an amorphous state from one-component metal melt is not possible until 1985 when Bajkov et al. [5] produced nickel and molybdenum glasses. They developed a new technique based on the explosion of a thin wire by passing electric current and the deposition of the explosion products on the inner surface of
4
the copper drum; the cooling rate reached 1013 K/s. Increasing the rate of cooling results in a rich diversity of the phase transition scenarios from equilibrium solidification to non-equilibrium crystallisation and metallic glass formation. It should be noted that at high cooling rates the melt can solidify at significant undercooling, which may reach hundreds of degrees. In this case, the model of equilibrium crystallization (Stefan problem) fails to describe properly the dynamics of the phase transition and more complicated models, such as the non-equilibrium crystallization [6-10] or nucleation-controlled solidification [11 - 16] have to be used. All methods of the metallic glasses production by quenching from a liquid state possess one serious limitation - metallic glasses can be produced only from the samples having the small linear sizes (typically ribbons, foils, wires, or splats). This is due to the necessity to provide a very high cooling rate (q > 106 K/s) in order to achieve so-called glass-transition temperature Tg and simultaneously to avoid crystallization. The glass-transition or vitrification temperature Tg of a material characterizes the range of temperatures over which this glass transition occurs. It is always lower than the melting temperature, Tm, of the crystalline state of the material. For polymers, ionic liquids and even inorganic glass Tg/Tm is about 2/3 (the Boyer-Beaman rule) [17]. For metallic glass Tg/Tm 1/3 is essentially lower. Let us estimate the relationship between the linear size of the melt l and the
2 cooling rate: time scale of cooling l2/a, q (Tm – Tg)/ = Tm a / l 2 . Taking, for example, the 3 properties of the Ag melt [7]: (l) = 9320 kg/m3, (l) = 180 W/(mK), c(l) = 283 J/(kgK), a(l) = 0.710–4 m2/s we obtain the following rates of cooling for different l: q = 6104 K/s for l = 1 mm and q = 61010 K/s for l = 1 m. As we can see, indeed, the highest cooling rates can be achieved only for the tiny samples. Henceforth under the metallic glasses we will have in mind the amorphous metals produced by ultra rapid quenching (q 106 K/s) from a liquid state. 5
As mentioned above, the following scenarios of phase transitions are observed in increasing the rate of cooling: 1) equilibrium (Stefan problem)- Tf Tm Tf 0 and the energy balance
Luf [Q]f is maintained at the solid-liquid interface, where the change of the state occurs; here [Q]f is the jump of heat fluxes at the front of phase transition, L [J/kg] is the latent heat of melting (solidification), uf is the velocity of the front of phase transition; 2) non-equilibrium Tf 0, uf = KTf, where K is the kinetic coefficient, which can be considered as a constant [7, 10]; 3) vitrification when the melt temperature reaches Tg. From a practical point of view it is important to have some criterion allowing a priory predict the scenarios of solidification. As shown in Fedorchenko [7], the dimensionless combination Ω /( Lv K ) can play this role, where is the heat transfer coefficient, Lv = L is the volumetric latent heat of solidification. It is important to keep in mind that the inequality O(1) is a necessary but not sufficient condition for the melt vitrification. Despite the success in obtaining metallic glasses, the fundamental aspects of the physics of the amorphous state is not entirely understood. These include, for example, the problem of determining the fundamental difference between the atomic structures of melts and metal glasses, the atomic processes by which the metal melts acquire amorphous rigidity upon cooling, etc. These issues are the subject of intensive researches [17 – 22].
In this work we confine ourselves to the conditions of production of one-component metallic glasses obtained by quenching from the liquid state and formulate the sufficient conditions for the vitrification of metallic melts. Another issue we consider is the determination of ranges of
6
numerical values of the criterion , which correspond to equilibrium and non-equilibrium phase transitions. 2. Equilibrium and non-equilibrium phase transitions based on criterion The criterion [7] was derived based on the dimensional analysis therefore the numerical values of which separate equilibrium and non-equilibrium phase transitions are only known to within an order of magnitude. In order to obtain more accurate values of the criterion, we use experimental data on the heights of the solidified droplets (splats) [23, 24]. Experiments were carried out for the wide droplet/substrate combinations and in the conditions of the full control of parameters of the droplet and substrate. Comparison of experimental data with the results of calculations on the equilibrium model showed that along with the coincidence there were the substantial divergences for some combinations droplet/substrate [25]. These differences could be due to the deviation of the crystallization process from the equilibrium model. To verify this, let us consider the non-equilibrium model of solidification. Figure 1 shows the temperature distribution in the system "impinging droplet-substrate" which corresponds to the experimental conditions in [24, 25].
7
Fig. 1. The temperature distribution in the impinging droplet/substrate under the condition of non-equilibrium solidification. 2.1. Model of the non-equilibrium droplet crystallization Following the assumptions and results of [7], the problem statement takes the form:
t Ts a s zzTs at – < z < 0,
(1)
t Td( s ) ad( s ) zzTd( s ) at – 0 < z < (t).
(2)
At z = 0 the fourth kind boundary conditions are imposed:
Ts (t , 0) Td( s ) (t , 0), s ( z Ts ) (ds ) ( z Td ) .
(3)
At the front of the phase transition z = (t) the heat balance reads
(ds ) ( z Td ) d( s ) Ld (t ) / dt (Tm T f ) ,
(4)
where is the heat transfer coefficient. The law of the front motion
d (t ) / dt K (Tm Tf ) ,
(5)
the boundary and initial conditions Td(0, z) = Tm, Ts(0, z) = Ts(t, – ) = Ts0, close the problem statement. An exact solution of the problem stated by Eqs. (1)-(6) was found by Lyubov [26].
8
(6)
Following [26] and omitting cumbersome intermediate mathematics, the instantaneous dimensionless coordinate of the front of phase transition ( = /d) reads: Σ 2Fo1/ 2 , where Fo is the Fourier number, is the root of the equation:
1 s0 Ku (1 (dl,s ) NuR /
Ku )
1 / 2 [ K erf ( )] exp( 2 ) .
(7)
s) (l,s) Let us inspect the term (l, d NuR / Ku by substituting instead of d , Nu, R, and Ku their
s) (s) (l) (s) (s) (l) expressions: (l, d d / d , Nu d / d , R ad /( dKTm ) , Ku L /( cd Tm ) , s0 Ts0 / Tm ,
(ds ) cd( s ) d( s ) . As a result we arrive at the important conclusion: K s cs s
(dl, s) NuR / Ku /( Lv K ) Ω ,
(8)
that is the above-introduced dimensionless number does characterize the impact of kinetics itself on the crystallization process. At << 1 Eq. (7) becomes 1 s0 1 / 2 [ K erf ( )] exp( 2 ) . Ku
(9)
In this case the dynamics of solidification is determined solely by the criterion of the equilibrium crystallization Ku. At finite , the phase transition depends essentially on the kinetics. Now the height of the splat can be determined as follows. For the metal melts the Pr number is very small (10–2 – 10–3) therefore, the model of the ideal droplet spreading suits well. In
this
case the apex of the spreading droplet keeps the initial impact velocity U. The front of the phase transition moves toward to the apex of the spreading droplet as shown in Fig. 1. The time of their meeting determines time of the complete solidification. This means that Fo* is the root of the following equation:
1 PeFo 2Fo1/ 2 , ad(l,s) , and it reads
9
(10)
Fo * [2 2 Pe 2 ( 2 Pe)1/ 2 ](Pe) 2 .
(11)
Substituting Eq. (11) into Eq. (10) yields the dimensionless height of the splat H = h/d H 1 PeFo * 2Fo *1/ 2 .
(12)
Figure 2 displays comparison of the experimental splats heights for different combinations molten droplet/substrate [23, 24] and the results of calculations according to equilibrium and non-equilibrium models. In calculating the numerical values of the kinetic coefficient K and formula for the heat transfer coefficient
Nu d / (dl) (5.5Pe / )1/ 2 ,
(13)
are taken from [27].
Fig. 2. Comparison of experimental splats heights and theoretical predictions according to equilibrium and non-equilibrium models. As can be seen from Fig. 2, the numerical values of the splat heights Al/Al and Al/Ag, calculated in accordance with the equilibrium and non-equilibrium models are practically identical and equal to the experimental values. At 0.57 (cases Ni/Ag, Cu/Ag, Ag/Ag) the 10
equilibrium model stops working properly. Hence, for the applicability of the equilibrium model the inequality 0.1 should hold. 3. Sufficient conditions of the metallic melt vitrification As noted above, for the vitrification of molten metal by quenching from a liquid state the certain conditions must be fulfilled, which include the following. The high cooling rate should be realized by an appropriate choice of the droplet size and velocity, material of substrate and its initial temperature. During cooling the crystalline nucleus waiting time has to be bigger than the characteristic time scale tg of the melt cooling to vitrification temperature Tg. 3.1. Vitrification temperature and the rate of cooling On the very high cooling rates Tg already cannot be considered as a constant and a dependence of Tg on cooling rate has to be taken into consideration. A fundamental criterion for glass formation can be formulated in the form that at the temperature of vitrification the time of molecular relaxation R has to be of the same order as the characteristic observation or stay time t for the process considered [28], i.e.,
R t at T = Tg.
(14)
Differentiating Eq. (14) with respect to temperature results in
d R / dT 1 / q ,
(15)
where the cooling rate q = – dT/dt > 0. A quantitative measure for the ability of a system to flow is the shear viscosity . According to Frenkel [29] the shear viscosity and the average stay time R are strongly connected. This connection can be expressed by the following expressions
R R 0 exp[U 0 /( kT )] and R R 0 exp[U 0 /( kT )] . 11
(16)
By U0 the activation energy for the viscous flow is denoted, k is Boltzmann’s constant, R0 is the preexponential factor, R0 is the period of eigenvibrations of the building units of the melt. The latter is a practically temperature independent quantity with values of the order 10 –12 s to 10–13 s. Combination of Eqs. (15) and (16) yields q R kT 2 /U 0 . Accounting for condition R = t for T = Tg, T should be replaced by Tg resulting in
q R (Tg ) kTg2 / U 0 .
(17)
Taking the decimal logarithm from both sides of Eq. (17) gives 2 2 1 2.3k kTg 2.3k kTg 2.3k lg lg lg q = Tg U 0 U 0 R 0 q U 0 U 0 R 0 U 0
(18)
Introducing the dimensionless vitrification temperature g = Tg/Tm, the equation reads
g /[lg( C0 g2 / R 0 ) lg q] ,
(19)
where C0 kTm2 / U 0 , U 0 (2.3kTm ) . The very important conclusion follows from the equation: g depends strongly on the cooling rate q and the value of which reflects the properties of the considered substance. Figure 3 shows the dependence of the dimensionless glass transition temperature on the cooling rate for a different metal melts. Note that a similar formula given by the Eq. (2) in [18] differs from Eq. (18) by the first term C = constant on the RHS. Calculations show that although the first term on the RHS of Eq. (18) varies slightly for the selected material in the temperature range from Tm/3 to Tm, the values of Tg depend drastically on the C values. Equations (18) and (19) are devoid of this shortcoming therefore they are preferable for the calculation of the glass transition temperature.
12
1
R0=10-13s R0=10-12s
Ni
0.8
Cu Al Ni
0.6
Cu
g g=1/3
0.4
Al
0.2
0 6
8
10
12
lg q Fig. 3. Dependences of the dimensionless vitrification temperature g on the cooling rate q for two values of R0 = 10–12s, 10–13s and for different molten metals: Al (Tm = 933 K, = 1.4), Cu (Tm = 1356 K, = 1.5) and Ni (Tm = 1728 K, = 1.7). 3.2. Determination of the cooling rate from solution of the conjugate heat transfer droplet/substrate To estimate the cooling rate let us consider a simplified model of the conjugate heat transfer of a droplet with the substrate. The melt layer of thickness h and initial temperature Td0 brought into contact with semi-infinite substrate of the initial temperature Ts0. Then the mathematical statement of the problem is as follows:
t Td ad(l) zzTd , t Ts as zzTs ,
(20)
Td ( z,0) Td 0 , Ts ( z,0) Ts 0 ,
(21)
( z Td ) z 0 0, (Ts ) z Ts0 ,
(22)
Td (h, t ) Ts (h, t ), (l) d ( z Td ) z h s ( z Ts ) z h .
(23)
The z-axis is directed into the interior of the substrate with the beginning on the surface of the 13
melt. Applying the Laplace transform to Eqs. (20) - (23) and following Lykov approach [30] in solving such problems, the temperature distribution over the thickness of the melt layer takes a look: d ( , ) 1
1 1 K
n1
(2n 1) (2n 1) erfc 2 2
( ) n1 erfc
(24)
Averaging Eq. (24) over the layer thickness reads:
d ( ) 1
where erfc (x) =
2
2 Fo 1 K
e x
2
n1
n 1 n ierfc , Fo Fo
( ) n1 ierfc
d , ierfc (x) =
erfc( )d ;
(25)
K [(dl ) d(l) cd(l) /(s s cs )]1/ 2 ,
x
1
T T (1 K ) /(1 K ) , Θd d s0 , Θd Θd d ; Fo ta d(l ) / h 2 . Td0 Ts0 0 The series Eq. (25) converges rapidly, so we can confine ourselves to the first term of the series:
Θd 1
1 2 Fo . ierfc( 0) ierfc 1 K Fo
(26)
Equation (26) allows one to calculate the time of melt cooling Fo* from the initial temperature
d = 1 to the temperature of the substrate d = 0. Then the cooling rate q becomes q (Td0 Ts0 ) / t * ,
(27)
where t * h 2 Fo * / ad(l) . 3.3. The glass transition condition On the base of the above-obtained results the vitrification conditions can be formulated in the following manner. Firstly, in the process of cooling the melt temperature has to drop lower than the vitrification 14
temperature, that is,
Td (t * ) Tg (q) .
(28)
Secondly, the crystalline nucleus waiting time has to be bigger than the characteristic time scale tg of the melt cooling to vitrification temperature Tg(q): tg
h
3
Tg
J s dt h 3
0
Tm
Js dT 1, q
16 3Tm2 U J s (T ) N a C exp 0 exp 3 kL2 (T ) 2 T , kT v
(29)
(30)
where Js is the stationary nucleation rate [16], T = Tm – T is the melt supercooling, is the surface tension on the interface crystal-melt, C 2d a (kT )1/ 2 / hp , hP is Planck’s constant, da is the atom diameter, Na is the number of the atoms (molecules) per unit volume. The obtained results allow us to determine the maximum glass layer thickness hmax. If the thickness of the melt layer is greater than hmax, the melt crystallizes. The initial melts (Al, Cu, Ni) and substrate (Cu) temperatures were assumed to be equal to the melting temperature and 300 K, respectively. The values of hmax along with the associated cooling rates and vitrification temperatures are shown in Table. The computed results show good agreement with the experimental data [2]. 4. Conclusion Based on analytical solutions of the impinging droplet solidification in approximation of equilibrium and non-equilibrium crystallization, the finite numerical value of the early-introduced asymptotic criterion was determined. Namely, at 0.57 the equilibrium model of solidification stops working properly and for the applicability of the equilibrium model an 15
inequality 0.1 should hold. The sufficient conditions given by Eqs. (19), (27) and (28) for the one-component metallic melts vitrification allow one to determine the maximum glass layer thickness. The results shown in the Table demonstrate clearly that the tendency to the glass formation drastically increases with the temperature of melting. References [1] W. Klement, R. Willens, P. Duwez, "Non-crystalline structure in solidified gold-silicon alloys," Nature 187 (4740), 869–870 (1960). [2] Volume 20 of Treatise on Materials Science and Technology: Ultrarapid Quenching of Liquid Alloys /Ed. by H. Herman, Academic Press, 1981. [3] H. Miura, S. Isa, K. Omuro, N. Tanigami, Production of amorphous Fe-Ni based alloys by flame-spray quenching, Trans. of the Japan Inst. of Metals, 22 (9) (1981) 597-606. [4] K. Suzuki, H. Fujimori, K. Hashimoto, Ed. C. Masumoto, Amorphous Metals, Moscow, Metallurgiya, 1987. [5] A. P. Baikov, V. A. Ivanchenko, V. I. Motorin, S. L. Musher and A. F. Shestak, “The one-component metallic glasses from nickel and molybdenum,” Physics Letters 113A, No. 1, 38-40 (1985). [6] M. Chung, R. H. Rangel, Parametric study of metal droplet deposition and solidification process including contact resistance and undercooling effects, Int. J. Heat Mass Transfer 44 (2001) 605 - 622. [7] A. I. Fedorchenko, A.- B. Wang, Non-equilibrium solidification of the molten metal droplets impacting on a solid surface, Int. J. Heat Mass Transfer 50 (2007) 2463 – 2468. [8] H. Zhang, X. Y. Wang, L. L. Zheng, S. Sampath, Numerical simulation of nucleation, solidification, and microstructure formation in thermal spraying, Int. J. Heat Mass Transfer 47
16
(2004) 2191 – 2203. [9] H. Zhang, X. Y. Wang, L. L. Zheng, X. Y. Jiang, Studies of splat morphology and rapid solidification during thermal spraying, Int. J. Heat Mass Transfer 44 (2001) 4579 – 4592. [10] J. Fukai, T. Ando, Microstructure development in alloy splats during rapid solidification, Mater. Sci. Eng. A 383 (2004) 175–183. [11] V.P. Skripov, Metastable Liquids [in Russian], Nauka, Moscow (1972); [in English] Wiley, New York (1974). [12] T.W. Clyne, Numerical treatment of rapid solidification, Metall. Trans. B 15 (1984) 369–381. [13] W.T. Kim, D.L. Zhang, B. Cantor, Nucleation of solidification in liquid droplets, Metall. Trans. A 22 (1991) 2487–2501. [14] D.M. Herlach, Non-equilibrium solidification of undercooled metallic melts, Mater. Sci. Eng. R 12 (1994) 179–272. [15] K.- C. Chang, C.- M. Chen, Revisiting heat transfer analysis for rapid solidification of metal droplets, Int. J. Heat Mass Transfer 44 (2001) 1573–1583. [16] A. I. Fedorchenko, A. A. Chernov, Simulation of the microstructure of a thin metal layer quenched from a liquid state, Int. J. Heat Mass Transfer 46 (2003) 921 - 929. [17] P. G. Debenedetti, F. H. Stillinger, Supercooled liquids and the glass transition, Nature 410 (2001) 259 – 267. [18] L. N. Kolotova, G. E. Norman, V. V. Pisarev, Glass transition of an overcooled aluminum melt: a study in molecular dynamics, Russ. J. of Phys. Chem. A 89 (5) (2015) 802-806. [19] L. N. Kolotova, G. E. Norman, V. V. Pisarev, Glass transition of aluminum melt. Molecular dynamics study, J. of Non-Crystalline Solids, 429 (2015) 98-103. [20] C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, S. W. Martin, Relaxation in glassforming liquids and amorphous solids, J. Appl. Phys. 88 (2000) 3113-3157. 17
[21] I. Gutzow, J. Schmelzer, The Vitreous State Thermodynamics, Structure, Rheology and Crystallization, Springer, Berlin (1995). [22] P. G. Debenedetti, Metastable Liquids: Concepts and Principles, Princeton University Press (1996). [23] V. V. Kudinov, Plasma Coatings (in Russian), Nauka, Moscow (1977). [24] V. V. Kudinov, V. M. Ivanov, Plasma deposition of refractory coatings (in Russian), Mashinostroenie, Moscow (1981). [25] A.I. Fedorchenko, O.P. Solonenko, Dynamics of crystallization process of molten particles at their interaction with surface, in: O. P. Solonenko, A.I. Fedorchenko (Eds.), Plasma Jets in the Development of New Materials Technology, VSP, Utrecht, The Netherlands; Tokyo, Japan, 1990, pp. 283–297. [26] B.Ya. Lyubov, Theory of Crystallization in Large Volumes [in Russian], Nauka, Moscow (1975). [27] A. I. Fedorchenko, A.- B. Wang, Non-equilibrium solidification of the molten metal droplets impacting on a solid surface, Int. J. Heat Mass Transfer 50 (2007) 2463 - 2468. [28] I. Gutzow, J. Schmelzer, The Vitreous State. Thermodynamics, Structure, Rheology and Crystallization, Berlin, Springer (1995). [29] Ya. I. Frenkel, The Kinetic Theory of Liquids, Oxford University Press, Oxford (1946). [30] A. V. Lykov, Theory of Heat Conduction [in Russian], Vysshaya shkola, Moscow (1967); Analytical Heat Diffusion Theory (http://www.sciencedirect.com/science/book/9780124597563) [in English].
18
Table The computed maximum glass layer thickness hmax, the associated cooling rates q, vitrification temperatures Tg, and temperature of melting Tm along with experimental data [2] Melt/Substrate
hmax, nm
Tg, K
q, 1013 K/s
Tm, K
Al/Cu
15
535
8.4
933
Cu/Cu
27
670
3.4
1356
Ni/Cu
30/(30-40)*
825
1.5
1728
*
Experimental data [2].
19
Captions Fig. 1. The temperature distribution in the impinging droplet/substrate under the condition of non-equilibrium solidification. Fig. 2. Comparison of the experimental splats heights and theoretical predictions according to equilibrium and non-equilibrium models. Fig. 3. Dependences of the dimensionless vitrification temperature g on the cooling rate q for two values of R0 = 10–12s, 10–13s and for different molten metals: Al (Tm = 933 K, = 1.4), Cu (Tm = 1356 K, = 1.5) and Ni (Tm = 1728 K, = 1.7).
20
Conditions for metallic melts vitrification were formulated. Tendency of metal melt vitrification drastically increases with melting point. Proven that W criterion indeed predicts a priori a scenario of crystallization.
21
S0022-0248(17)30428-1 http://dx.doi.org/10.1016/j.jcrysgro.2017.06.011 CRYS 24214
To appear in:
Journal of Crystal Growth
Received Date: Revised Date: Accepted Date:
6 March 2017 13 April 2017 12 June 2017
Please cite this article as: A.I. Fedorchenko, On the glass transition of THE ONE-component metallic melts, Journal of Crystal Growth (2017), doi: http://dx.doi.org/10.1016/j.jcrysgro.2017.06.011
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
ON THE GLASS TRANSITION OF THE ONE-COMPONENT METALLIC MELTS A. I. Fedorchenko Institute of Thermomechanics, Academy of Sciences of the Czech Republic, v. v. i., Dolejskova 5, 182 00, Prague 8, Czech Republic S. S. Kutateladze Institute of Thermophysics, Siberian Branch of Russian Academy of Sciences, Novosibirsk 630090, Russia ABSTRACT In this paper, the conditions for one-component metallic melts vitrification by quenching from a liquid state were formulated. It is shown that the tendency to the glass formation drastically increases with the temperature of melting. The maximum glass layer thickness and the associated cooling rates along with the vitrification temperatures was determined for Al, Cu, and Ni melts deposited on the Cu substrate. The results are in agreement with the available experimental data. Based on analytical solution of the impinging droplet solidification, the numerical value of the early-introduced asymptotic criterion, which separates equilibrium and non-equilibrium phase transitions, was determined. Good agreement between the calculated and experimental values of the thickness of the splats shows that criterion indeed predicts a priori a scenario of solidification. Keywords: A1. Equilibrium and non-equilibrium solidification A1. Criterion of the phase transition scenario A2. One-component metal melts B2. Glass transition B2. Temperature of vitrification
Corresponding author. E-mail address: [email protected]
1
Nomenclature a
thermal diffusivity (m2 s–1)
c
specific heat (J kg–1K–1)
d
droplet diameter (m)
da
atom diameter (m)
Fo
Fourier number,
h
height of the splat (m)
hP
Planck constant (J s)
Js
stationary nucleation rate (s–1m–3)
k
Boltzmann constant (J K–1)
K
kinetic coefficient (m s–1K–1)
Ku
Kutateladze number, L /( cd(s)Tm )
K
(s) (s) 1/ 2 relative thermal activity, [(s) in Eq. (7) d d cd /( s s cs )]
K
relative thermal activity, [(dl ) d(l)cd(l) /(s s cs )]1/ 2 in Eq. (24)
l
linear scale (m)
L
latent heat of melting (J kg–1)
Lv
volumetric latent heat of melting, L (J m–3)
Na
number of atoms per unit volume (m–3)
Nu
Nusselt number, d / (dl )
Pe
Peclet number, Ud / a d(l )
Pr
Prandtl number, /a
q
cooling rate (K s–1)
[Q]f
ad(s)t/d2
jump of heat fluxes at the phase transition front (J m–2s–1) 2
R
ratio of diffusive and kinetic speeds, ad(s) /( dKTm )
Tg
glass transition temperature (K)
Tm
melting temperature (K)
T
undercooling of the melt, (Tm – T) (K)
uf
speed of the phase transition front (m s–1)
U
droplet impact velocity (m s–1)
U0
activation energy (J)
zd
instantaneous coordinate of the droplet apex (m)
Greek symbols
heat transfer coefficient (W m–2 K–1)
dimensionless energy of activation, 2.3U 0 /( kTm )
root of transcendental Eq. (7)
thermal conductivity (W m–1 K–1)
kinematic viscosity, (m2 s–1)
density (kg m–3)
R
time of molecular relaxation (s)
dimensionless temperature, T/Tm
d
droplet dimensionless temperature, (Td Ts 0 ) /(Td 0 Ts 0 )
dimension coordinate of the front of phase transition (m)
dimensionless coordinate of the front of phase transition, /d
criterion of the phase transition, /( Lv K )
3
Subscripts d
refers to droplet
f
refers to the front of phase transition
g
refers to glass
m
refers to melting (solidification)
s
refers to substrate
0
refers to initial state
Superscripts l
refers to liquid state
s
refers to solid state
1. Introduction Although amorphous substance, such as “silicate glasses” was known from time immemorial, a breakthrough in obtaining metallic glasses occurred as recently as in 60-es of the 20th century. Klement et al. [1] in 1960 were the first who produced the metallic glass from an alloy (Au 75Si25). They developed a splat quenching technique and the cooling rate q reached about 106 K/s. This stipulated the emergences of the multiple technologies of quenching from a liquid state, such as thermal spray, melt spinning, gun technique, etc. [2, 3]. Investigations of the metallic glasses have shown that they possess a unique set of mechanical, electrical, magnetic, elastic, anti-corrosion, etc. properties [4]. For a long time it was believed that getting an amorphous state from one-component metal melt is not possible until 1985 when Bajkov et al. [5] produced nickel and molybdenum glasses. They developed a new technique based on the explosion of a thin wire by passing electric current and the deposition of the explosion products on the inner surface of
4
the copper drum; the cooling rate reached 1013 K/s. Increasing the rate of cooling results in a rich diversity of the phase transition scenarios from equilibrium solidification to non-equilibrium crystallisation and metallic glass formation. It should be noted that at high cooling rates the melt can solidify at significant undercooling, which may reach hundreds of degrees. In this case, the model of equilibrium crystallization (Stefan problem) fails to describe properly the dynamics of the phase transition and more complicated models, such as the non-equilibrium crystallization [6-10] or nucleation-controlled solidification [11 - 16] have to be used. All methods of the metallic glasses production by quenching from a liquid state possess one serious limitation - metallic glasses can be produced only from the samples having the small linear sizes (typically ribbons, foils, wires, or splats). This is due to the necessity to provide a very high cooling rate (q > 106 K/s) in order to achieve so-called glass-transition temperature Tg and simultaneously to avoid crystallization. The glass-transition or vitrification temperature Tg of a material characterizes the range of temperatures over which this glass transition occurs. It is always lower than the melting temperature, Tm, of the crystalline state of the material. For polymers, ionic liquids and even inorganic glass Tg/Tm is about 2/3 (the Boyer-Beaman rule) [17]. For metallic glass Tg/Tm 1/3 is essentially lower. Let us estimate the relationship between the linear size of the melt l and the
2 cooling rate: time scale of cooling l2/a, q (Tm – Tg)/ = Tm a / l 2 . Taking, for example, the 3 properties of the Ag melt [7]: (l) = 9320 kg/m3, (l) = 180 W/(mK), c(l) = 283 J/(kgK), a(l) = 0.710–4 m2/s we obtain the following rates of cooling for different l: q = 6104 K/s for l = 1 mm and q = 61010 K/s for l = 1 m. As we can see, indeed, the highest cooling rates can be achieved only for the tiny samples. Henceforth under the metallic glasses we will have in mind the amorphous metals produced by ultra rapid quenching (q 106 K/s) from a liquid state. 5
As mentioned above, the following scenarios of phase transitions are observed in increasing the rate of cooling: 1) equilibrium (Stefan problem)- Tf Tm Tf 0 and the energy balance
Luf [Q]f is maintained at the solid-liquid interface, where the change of the state occurs; here [Q]f is the jump of heat fluxes at the front of phase transition, L [J/kg] is the latent heat of melting (solidification), uf is the velocity of the front of phase transition; 2) non-equilibrium Tf 0, uf = KTf, where K is the kinetic coefficient, which can be considered as a constant [7, 10]; 3) vitrification when the melt temperature reaches Tg. From a practical point of view it is important to have some criterion allowing a priory predict the scenarios of solidification. As shown in Fedorchenko [7], the dimensionless combination Ω /( Lv K ) can play this role, where is the heat transfer coefficient, Lv = L is the volumetric latent heat of solidification. It is important to keep in mind that the inequality O(1) is a necessary but not sufficient condition for the melt vitrification. Despite the success in obtaining metallic glasses, the fundamental aspects of the physics of the amorphous state is not entirely understood. These include, for example, the problem of determining the fundamental difference between the atomic structures of melts and metal glasses, the atomic processes by which the metal melts acquire amorphous rigidity upon cooling, etc. These issues are the subject of intensive researches [17 – 22].
In this work we confine ourselves to the conditions of production of one-component metallic glasses obtained by quenching from the liquid state and formulate the sufficient conditions for the vitrification of metallic melts. Another issue we consider is the determination of ranges of
6
numerical values of the criterion , which correspond to equilibrium and non-equilibrium phase transitions. 2. Equilibrium and non-equilibrium phase transitions based on criterion The criterion [7] was derived based on the dimensional analysis therefore the numerical values of which separate equilibrium and non-equilibrium phase transitions are only known to within an order of magnitude. In order to obtain more accurate values of the criterion, we use experimental data on the heights of the solidified droplets (splats) [23, 24]. Experiments were carried out for the wide droplet/substrate combinations and in the conditions of the full control of parameters of the droplet and substrate. Comparison of experimental data with the results of calculations on the equilibrium model showed that along with the coincidence there were the substantial divergences for some combinations droplet/substrate [25]. These differences could be due to the deviation of the crystallization process from the equilibrium model. To verify this, let us consider the non-equilibrium model of solidification. Figure 1 shows the temperature distribution in the system "impinging droplet-substrate" which corresponds to the experimental conditions in [24, 25].
7
Fig. 1. The temperature distribution in the impinging droplet/substrate under the condition of non-equilibrium solidification. 2.1. Model of the non-equilibrium droplet crystallization Following the assumptions and results of [7], the problem statement takes the form:
t Ts a s zzTs at – < z < 0,
(1)
t Td( s ) ad( s ) zzTd( s ) at – 0 < z < (t).
(2)
At z = 0 the fourth kind boundary conditions are imposed:
Ts (t , 0) Td( s ) (t , 0), s ( z Ts ) (ds ) ( z Td ) .
(3)
At the front of the phase transition z = (t) the heat balance reads
(ds ) ( z Td ) d( s ) Ld (t ) / dt (Tm T f ) ,
(4)
where is the heat transfer coefficient. The law of the front motion
d (t ) / dt K (Tm Tf ) ,
(5)
the boundary and initial conditions Td(0, z) = Tm, Ts(0, z) = Ts(t, – ) = Ts0, close the problem statement. An exact solution of the problem stated by Eqs. (1)-(6) was found by Lyubov [26].
8
(6)
Following [26] and omitting cumbersome intermediate mathematics, the instantaneous dimensionless coordinate of the front of phase transition ( = /d) reads: Σ 2Fo1/ 2 , where Fo is the Fourier number, is the root of the equation:
1 s0 Ku (1 (dl,s ) NuR /
Ku )
1 / 2 [ K erf ( )] exp( 2 ) .
(7)
s) (l,s) Let us inspect the term (l, d NuR / Ku by substituting instead of d , Nu, R, and Ku their
s) (s) (l) (s) (s) (l) expressions: (l, d d / d , Nu d / d , R ad /( dKTm ) , Ku L /( cd Tm ) , s0 Ts0 / Tm ,
(ds ) cd( s ) d( s ) . As a result we arrive at the important conclusion: K s cs s
(dl, s) NuR / Ku /( Lv K ) Ω ,
(8)
that is the above-introduced dimensionless number does characterize the impact of kinetics itself on the crystallization process. At << 1 Eq. (7) becomes 1 s0 1 / 2 [ K erf ( )] exp( 2 ) . Ku
(9)
In this case the dynamics of solidification is determined solely by the criterion of the equilibrium crystallization Ku. At finite , the phase transition depends essentially on the kinetics. Now the height of the splat can be determined as follows. For the metal melts the Pr number is very small (10–2 – 10–3) therefore, the model of the ideal droplet spreading suits well. In
this
case the apex of the spreading droplet keeps the initial impact velocity U. The front of the phase transition moves toward to the apex of the spreading droplet as shown in Fig. 1. The time of their meeting determines time of the complete solidification. This means that Fo* is the root of the following equation:
1 PeFo 2Fo1/ 2 , ad(l,s) , and it reads
9
(10)
Fo * [2 2 Pe 2 ( 2 Pe)1/ 2 ](Pe) 2 .
(11)
Substituting Eq. (11) into Eq. (10) yields the dimensionless height of the splat H = h/d H 1 PeFo * 2Fo *1/ 2 .
(12)
Figure 2 displays comparison of the experimental splats heights for different combinations molten droplet/substrate [23, 24] and the results of calculations according to equilibrium and non-equilibrium models. In calculating the numerical values of the kinetic coefficient K and formula for the heat transfer coefficient
Nu d / (dl) (5.5Pe / )1/ 2 ,
(13)
are taken from [27].
Fig. 2. Comparison of experimental splats heights and theoretical predictions according to equilibrium and non-equilibrium models. As can be seen from Fig. 2, the numerical values of the splat heights Al/Al and Al/Ag, calculated in accordance with the equilibrium and non-equilibrium models are practically identical and equal to the experimental values. At 0.57 (cases Ni/Ag, Cu/Ag, Ag/Ag) the 10
equilibrium model stops working properly. Hence, for the applicability of the equilibrium model the inequality 0.1 should hold. 3. Sufficient conditions of the metallic melt vitrification As noted above, for the vitrification of molten metal by quenching from a liquid state the certain conditions must be fulfilled, which include the following. The high cooling rate should be realized by an appropriate choice of the droplet size and velocity, material of substrate and its initial temperature. During cooling the crystalline nucleus waiting time has to be bigger than the characteristic time scale tg of the melt cooling to vitrification temperature Tg. 3.1. Vitrification temperature and the rate of cooling On the very high cooling rates Tg already cannot be considered as a constant and a dependence of Tg on cooling rate has to be taken into consideration. A fundamental criterion for glass formation can be formulated in the form that at the temperature of vitrification the time of molecular relaxation R has to be of the same order as the characteristic observation or stay time t for the process considered [28], i.e.,
R t at T = Tg.
(14)
Differentiating Eq. (14) with respect to temperature results in
d R / dT 1 / q ,
(15)
where the cooling rate q = – dT/dt > 0. A quantitative measure for the ability of a system to flow is the shear viscosity . According to Frenkel [29] the shear viscosity and the average stay time R are strongly connected. This connection can be expressed by the following expressions
R R 0 exp[U 0 /( kT )] and R R 0 exp[U 0 /( kT )] . 11
(16)
By U0 the activation energy for the viscous flow is denoted, k is Boltzmann’s constant, R0 is the preexponential factor, R0 is the period of eigenvibrations of the building units of the melt. The latter is a practically temperature independent quantity with values of the order 10 –12 s to 10–13 s. Combination of Eqs. (15) and (16) yields q R kT 2 /U 0 . Accounting for condition R = t for T = Tg, T should be replaced by Tg resulting in
q R (Tg ) kTg2 / U 0 .
(17)
Taking the decimal logarithm from both sides of Eq. (17) gives 2 2 1 2.3k kTg 2.3k kTg 2.3k lg lg lg q = Tg U 0 U 0 R 0 q U 0 U 0 R 0 U 0
(18)
Introducing the dimensionless vitrification temperature g = Tg/Tm, the equation reads
g /[lg( C0 g2 / R 0 ) lg q] ,
(19)
where C0 kTm2 / U 0 , U 0 (2.3kTm ) . The very important conclusion follows from the equation: g depends strongly on the cooling rate q and the value of which reflects the properties of the considered substance. Figure 3 shows the dependence of the dimensionless glass transition temperature on the cooling rate for a different metal melts. Note that a similar formula given by the Eq. (2) in [18] differs from Eq. (18) by the first term C = constant on the RHS. Calculations show that although the first term on the RHS of Eq. (18) varies slightly for the selected material in the temperature range from Tm/3 to Tm, the values of Tg depend drastically on the C values. Equations (18) and (19) are devoid of this shortcoming therefore they are preferable for the calculation of the glass transition temperature.
12
1
R0=10-13s R0=10-12s
Ni
0.8
Cu Al Ni
0.6
Cu
g g=1/3
0.4
Al
0.2
0 6
8
10
12
lg q Fig. 3. Dependences of the dimensionless vitrification temperature g on the cooling rate q for two values of R0 = 10–12s, 10–13s and for different molten metals: Al (Tm = 933 K, = 1.4), Cu (Tm = 1356 K, = 1.5) and Ni (Tm = 1728 K, = 1.7). 3.2. Determination of the cooling rate from solution of the conjugate heat transfer droplet/substrate To estimate the cooling rate let us consider a simplified model of the conjugate heat transfer of a droplet with the substrate. The melt layer of thickness h and initial temperature Td0 brought into contact with semi-infinite substrate of the initial temperature Ts0. Then the mathematical statement of the problem is as follows:
t Td ad(l) zzTd , t Ts as zzTs ,
(20)
Td ( z,0) Td 0 , Ts ( z,0) Ts 0 ,
(21)
( z Td ) z 0 0, (Ts ) z Ts0 ,
(22)
Td (h, t ) Ts (h, t ), (l) d ( z Td ) z h s ( z Ts ) z h .
(23)
The z-axis is directed into the interior of the substrate with the beginning on the surface of the 13
melt. Applying the Laplace transform to Eqs. (20) - (23) and following Lykov approach [30] in solving such problems, the temperature distribution over the thickness of the melt layer takes a look: d ( , ) 1
1 1 K
n1
(2n 1) (2n 1) erfc 2 2
( ) n1 erfc
(24)
Averaging Eq. (24) over the layer thickness reads:
d ( ) 1
where erfc (x) =
2
2 Fo 1 K
e x
2
n1
n 1 n ierfc , Fo Fo
( ) n1 ierfc
d , ierfc (x) =
erfc( )d ;
(25)
K [(dl ) d(l) cd(l) /(s s cs )]1/ 2 ,
x
1
T T (1 K ) /(1 K ) , Θd d s0 , Θd Θd d ; Fo ta d(l ) / h 2 . Td0 Ts0 0 The series Eq. (25) converges rapidly, so we can confine ourselves to the first term of the series:
Θd 1
1 2 Fo . ierfc( 0) ierfc 1 K Fo
(26)
Equation (26) allows one to calculate the time of melt cooling Fo* from the initial temperature
d = 1 to the temperature of the substrate d = 0. Then the cooling rate q becomes q (Td0 Ts0 ) / t * ,
(27)
where t * h 2 Fo * / ad(l) . 3.3. The glass transition condition On the base of the above-obtained results the vitrification conditions can be formulated in the following manner. Firstly, in the process of cooling the melt temperature has to drop lower than the vitrification 14
temperature, that is,
Td (t * ) Tg (q) .
(28)
Secondly, the crystalline nucleus waiting time has to be bigger than the characteristic time scale tg of the melt cooling to vitrification temperature Tg(q): tg
h
3
Tg
J s dt h 3
0
Tm
Js dT 1, q
16 3Tm2 U J s (T ) N a C exp 0 exp 3 kL2 (T ) 2 T , kT v
(29)
(30)
where Js is the stationary nucleation rate [16], T = Tm – T is the melt supercooling, is the surface tension on the interface crystal-melt, C 2d a (kT )1/ 2 / hp , hP is Planck’s constant, da is the atom diameter, Na is the number of the atoms (molecules) per unit volume. The obtained results allow us to determine the maximum glass layer thickness hmax. If the thickness of the melt layer is greater than hmax, the melt crystallizes. The initial melts (Al, Cu, Ni) and substrate (Cu) temperatures were assumed to be equal to the melting temperature and 300 K, respectively. The values of hmax along with the associated cooling rates and vitrification temperatures are shown in Table. The computed results show good agreement with the experimental data [2]. 4. Conclusion Based on analytical solutions of the impinging droplet solidification in approximation of equilibrium and non-equilibrium crystallization, the finite numerical value of the early-introduced asymptotic criterion was determined. Namely, at 0.57 the equilibrium model of solidification stops working properly and for the applicability of the equilibrium model an 15
inequality 0.1 should hold. The sufficient conditions given by Eqs. (19), (27) and (28) for the one-component metallic melts vitrification allow one to determine the maximum glass layer thickness. The results shown in the Table demonstrate clearly that the tendency to the glass formation drastically increases with the temperature of melting. References [1] W. Klement, R. Willens, P. Duwez, "Non-crystalline structure in solidified gold-silicon alloys," Nature 187 (4740), 869–870 (1960). [2] Volume 20 of Treatise on Materials Science and Technology: Ultrarapid Quenching of Liquid Alloys /Ed. by H. Herman, Academic Press, 1981. [3] H. Miura, S. Isa, K. Omuro, N. Tanigami, Production of amorphous Fe-Ni based alloys by flame-spray quenching, Trans. of the Japan Inst. of Metals, 22 (9) (1981) 597-606. [4] K. Suzuki, H. Fujimori, K. Hashimoto, Ed. C. Masumoto, Amorphous Metals, Moscow, Metallurgiya, 1987. [5] A. P. Baikov, V. A. Ivanchenko, V. I. Motorin, S. L. Musher and A. F. Shestak, “The one-component metallic glasses from nickel and molybdenum,” Physics Letters 113A, No. 1, 38-40 (1985). [6] M. Chung, R. H. Rangel, Parametric study of metal droplet deposition and solidification process including contact resistance and undercooling effects, Int. J. Heat Mass Transfer 44 (2001) 605 - 622. [7] A. I. Fedorchenko, A.- B. Wang, Non-equilibrium solidification of the molten metal droplets impacting on a solid surface, Int. J. Heat Mass Transfer 50 (2007) 2463 – 2468. [8] H. Zhang, X. Y. Wang, L. L. Zheng, S. Sampath, Numerical simulation of nucleation, solidification, and microstructure formation in thermal spraying, Int. J. Heat Mass Transfer 47
16
(2004) 2191 – 2203. [9] H. Zhang, X. Y. Wang, L. L. Zheng, X. Y. Jiang, Studies of splat morphology and rapid solidification during thermal spraying, Int. J. Heat Mass Transfer 44 (2001) 4579 – 4592. [10] J. Fukai, T. Ando, Microstructure development in alloy splats during rapid solidification, Mater. Sci. Eng. A 383 (2004) 175–183. [11] V.P. Skripov, Metastable Liquids [in Russian], Nauka, Moscow (1972); [in English] Wiley, New York (1974). [12] T.W. Clyne, Numerical treatment of rapid solidification, Metall. Trans. B 15 (1984) 369–381. [13] W.T. Kim, D.L. Zhang, B. Cantor, Nucleation of solidification in liquid droplets, Metall. Trans. A 22 (1991) 2487–2501. [14] D.M. Herlach, Non-equilibrium solidification of undercooled metallic melts, Mater. Sci. Eng. R 12 (1994) 179–272. [15] K.- C. Chang, C.- M. Chen, Revisiting heat transfer analysis for rapid solidification of metal droplets, Int. J. Heat Mass Transfer 44 (2001) 1573–1583. [16] A. I. Fedorchenko, A. A. Chernov, Simulation of the microstructure of a thin metal layer quenched from a liquid state, Int. J. Heat Mass Transfer 46 (2003) 921 - 929. [17] P. G. Debenedetti, F. H. Stillinger, Supercooled liquids and the glass transition, Nature 410 (2001) 259 – 267. [18] L. N. Kolotova, G. E. Norman, V. V. Pisarev, Glass transition of an overcooled aluminum melt: a study in molecular dynamics, Russ. J. of Phys. Chem. A 89 (5) (2015) 802-806. [19] L. N. Kolotova, G. E. Norman, V. V. Pisarev, Glass transition of aluminum melt. Molecular dynamics study, J. of Non-Crystalline Solids, 429 (2015) 98-103. [20] C. A. Angell, K. L. Ngai, G. B. McKenna, P. F. McMillan, S. W. Martin, Relaxation in glassforming liquids and amorphous solids, J. Appl. Phys. 88 (2000) 3113-3157. 17
[21] I. Gutzow, J. Schmelzer, The Vitreous State Thermodynamics, Structure, Rheology and Crystallization, Springer, Berlin (1995). [22] P. G. Debenedetti, Metastable Liquids: Concepts and Principles, Princeton University Press (1996). [23] V. V. Kudinov, Plasma Coatings (in Russian), Nauka, Moscow (1977). [24] V. V. Kudinov, V. M. Ivanov, Plasma deposition of refractory coatings (in Russian), Mashinostroenie, Moscow (1981). [25] A.I. Fedorchenko, O.P. Solonenko, Dynamics of crystallization process of molten particles at their interaction with surface, in: O. P. Solonenko, A.I. Fedorchenko (Eds.), Plasma Jets in the Development of New Materials Technology, VSP, Utrecht, The Netherlands; Tokyo, Japan, 1990, pp. 283–297. [26] B.Ya. Lyubov, Theory of Crystallization in Large Volumes [in Russian], Nauka, Moscow (1975). [27] A. I. Fedorchenko, A.- B. Wang, Non-equilibrium solidification of the molten metal droplets impacting on a solid surface, Int. J. Heat Mass Transfer 50 (2007) 2463 - 2468. [28] I. Gutzow, J. Schmelzer, The Vitreous State. Thermodynamics, Structure, Rheology and Crystallization, Berlin, Springer (1995). [29] Ya. I. Frenkel, The Kinetic Theory of Liquids, Oxford University Press, Oxford (1946). [30] A. V. Lykov, Theory of Heat Conduction [in Russian], Vysshaya shkola, Moscow (1967); Analytical Heat Diffusion Theory (http://www.sciencedirect.com/science/book/9780124597563) [in English].
18
Table The computed maximum glass layer thickness hmax, the associated cooling rates q, vitrification temperatures Tg, and temperature of melting Tm along with experimental data [2] Melt/Substrate
hmax, nm
Tg, K
q, 1013 K/s
Tm, K
Al/Cu
15
535
8.4
933
Cu/Cu
27
670
3.4
1356
Ni/Cu
30/(30-40)*
825
1.5
1728
*
Experimental data [2].
19
Captions Fig. 1. The temperature distribution in the impinging droplet/substrate under the condition of non-equilibrium solidification. Fig. 2. Comparison of the experimental splats heights and theoretical predictions according to equilibrium and non-equilibrium models. Fig. 3. Dependences of the dimensionless vitrification temperature g on the cooling rate q for two values of R0 = 10–12s, 10–13s and for different molten metals: Al (Tm = 933 K, = 1.4), Cu (Tm = 1356 K, = 1.5) and Ni (Tm = 1728 K, = 1.7).
20
Conditions for metallic melts vitrification were formulated. Tendency of metal melt vitrification drastically increases with melting point. Proven that W criterion indeed predicts a priori a scenario of crystallization.
21