Assessment of the physical mechanism
of the effect of mechanical activation
of a charge mixture on subsequent SHS
Boris B. Khina
Physico-Technical Institute of the NationalAcademy of Sciences of Belarus, Minsk, Belarus
The mechanical activation (MA) of powder mixtures is used as a stage of charge preparation for subsequent production of a number of powder materials by self-propagating high-temperature synthesis (SHS). It permits changing the SHS parameters and producing a target product in systems where “ordinary” SHS is hampered of impossible because of thermodynamic or kinetic limitations. The mechanically activated (or assisted) SHS (MASHS) is currently a subject of extensive investigation but up to now a mechanism of the influence of MA on subsequent SHS is not well understood. Current situation in this area restrains a wider use of this prospective synthesis method and the development of novel advanced materials on its basis. In this work, a system of relatively simple estimates is developed to analyse the physicochemical mechanism of the effect of preliminary MA on SHS, in particular, the role of the stored energy of cold plastic deformation and that of accumulated point defects. A brief analysis of the known theories of MASHS is presented. A qualitative explanation of the effect of MA on SHS is proposed, which is based on the modern theory of deformation-enhanced interdiffusion during MA and state-of-the-art conceptions about solid-state nucleation in the field of concentration gradient.
1. INTRODUCTION
Self-propagating high-temperature synthesis (SHS), or combustion synthesis (CS), is known as an effective, cost and energy efficient method for producing a wide range of refractory compounds such as carbides, nitrides, borides, silicides, intermetallics, etc. and composite materials [1]. A necessary condition for the CS to proceed is the existence of a strongly exothermal reaction which provides sufficient heat release to sustain the combustion. A main drawback of SHS is the impossibility of controlling the synthesis in situ after the charge mixture of reactive powders is ignited. Besides, in a number of practically important systems, the CS is difficult to implement because of either thermodynamic (insufficient heat release of the main reaction) or kinetic limitations (e.g., low value of the diffusion coefficient in a solid-product layer or short lifetime of a transient melt wherein crystallization of the product grains can occur). The efficiency of SHS in such systems can be improved by preheating the charge mixture to a certain temperature T0>298 K, applying external electric or magnetic field during combustion (the so-called field-activated SHS [2]), and adding subsidiary exothermic reactants to the mixture to increase the adiabatic temperature. The former two methods require special equipment and decrease the cost and energy efficiency of SHS while the latter entails additional operations to separate the target product after synthesis.
Because of the above reasons, mechanical activation or alloying (MA) of initial powder mixtures has acquired a certain use to exert an influence on the SHS process, and the so-called mechanically activated (or assisted) SHS (MASHS) is currently a subject of extensive experimental investigation [3-7]. It was observed experimentally that in a number of systems preliminary MA of the charge mixture brought about a decrease in both the ignition (Tig) and combustion (TSHS) temperatures along with changes in the SHS wave velocity uSHS and an increase of the conversion degree (01); in some cases MA changed the phase composition of the final products. In the Ni-Ti and Ni-Al systems, with increasing the time of MA (tMA) in a high-speed planetary mill, the values of uSHS and TSHS were found first to increase and then to decrease substantially, and the position of a maximum in curves TSHS(tMA) and uSHS(tMA) did not coincide: the former was observed at smaller tMA than the latter [7]. At certain regimes of MA, a phenomenon of purely solid state combustion (PSSC) was observed in the Ni-Al system with 13 wt.% Al during the synthesis of compound Ni3Al: the SHS temperature fell down below the melting point of aluminum, TSHS < Tm(Al)=933 K [6,7].
Despite extensive experimental investigation, intricate physicochemical mechanisms responsible for a strong influence of MA of a charge powder mixture on subsequent SHS are not clearly understood yet. It is known that during initial stages of MA of metal-metal or metal-nonmetal systems, composite (e.g., lamellar) particles are formed due to fracturing and cold welding processes in a comminuting device for MA such as a vibratory or planetary mill or attritor [8]. In regard to subsequent SHS, this increases the contact surface area between unlike substances and decreases the characteristic size of solid particles/layers of the reactants. However, the reasons for the experimentallyobserved effects cannot be reduced only to the aforesaid factors, the latter can only explain an increment of the conversion degree . An increase in the contact surface area of the reactants after MA, which reduces heat losses during combustion, along with the observed buildup of the conversion degree will eventually lead to an increase in the SHS temperature: the latter is supposed to approach the adiabatic value, TSHSTad. As a consequence, the combustion velocity should increase continuously according to the Zeldovich-Frank-Kamenetskiy formula, which is widely used in the SHS theory and practice [9,10]. But, as mentioned above, this is not the case: after prolonged milling, both the TSHS and uSHS values were observed to decrease substantially [3-7]. In experimental work [6,7], the observed PSSC phenomenon was attributed to the influence of non-equilibrium point defects (mainly vacancies), which were accumulated in metallic reactants during intensive periodic plastic deformation (IPPD) in the course of MA, on the formation the product phase (Ni3Al) in the SHS wave. In Refs. [6,7] it is considered that the relaxation of these defects to the equilibrium concentration did not occur during heating of the activated powder mixture in the so-called preheat zone of the SHS wave, and this effect is called “reverse quenching of the nonequilibrium defects” [7]. However, this explanation was presented merely on a conceptual level and was not supported by experimental investigation nor by numerical estimates [6,7].
In a series of theoretical papers [11-13], the effect of preliminary MA on SHS was attributed to a decrease of the apparent activation energy of combustion by the value of energy stored in solid reactants during MA. However, the numerical value of this parameter was not reported. Moreover, all the calculations were performed using dimensionless complexes [11,13], thus real parameter values and their intrinsic physical meaning seem vague.
In connection with the above, the objective of this research is (i) to perform numerical assessment of the main physicochemical factors, which may be responsible for the influence of MA on subsequent SHS, using relatively simple physical estimates, (ii) to carry out critical analysis of the existing theories of MASHS using realistic parameter values, and (iii) to develop a physical explanation (at the current stage, on a qualitative level only) for the observed effects basing on certain results of the state-of-the-art solid-state diffusion theory.
2. Assessment of the stored energy in reactants AFTER ma
It is known that during cold plastic deformation of metals only 0.5-5% of the spent energy is stored in the form of non-equilibrium defects while the main portion of the energy dissipates in the form of heat [14]. Let us estimate the energy stored in metallic reactants due to cold work during MA (ES), which could affect the apparent activation energy of solid-state reaction during subsequent SHS due to distortion of the crystal lattice of the parent phases (pure metals), using the Ni-Al and Fe-Al systems as examples. The stored energy is assessed as
ES = Eed + Epd +Ed + Egb,(1)
where Eed is the energy of residual elastic deformation, Epd is that of point defects generated by plastic deformation during MA, Ed is the elastic energy associated with dislocations, and Egb is the energy of grain boundaries. These parameters should be estimated in terms of melting enthalpy (Hm(Ni)=17.8 kJ/mol, Hm(Al)=10.5 kJ/mol, Hm(Fe)=13.8 kJ/mol) because the maximal energy that can be accumulated in a metal due to IPPD corresponds to solid-state amorphization, which is known to occur during prolonged MA [8]; the enthalpy of the latter is close to 0.8Hm. It should be outlined that the regimes of MA, which are used as a preliminary stage for SHS, are typically intended to obtain composite particles with small lamella thickness and high contact surface area of the solid reactants, and the processing is terminated long before the amorphization can occur.
Contrary to popular opinion that nanocrystals do not contain dislocations, recent experimental investigations have revealed the presence of dislocations inside nanograins (not only in the grain boundaries) whose density, on the example of nickel, was found to reach = 4.71011-1.31012 cm2[15]. Hence formula (1) is also valid for nanostructured powder materials obtained by MA.
The residual stress in materials cannot exceed the limit of elasticity 0.2. Hence an upper estimate for Eed (per one mole) can be obtained using a simple formula that corresponds to the case of an elastically strained rod [16]:
,(2)
where Y is the elastic (Young) modulus, is the molar mass and d is density. The values of 0.2 and Y for severely cold-worked Ni, -Fe and Al [17] are presented in Table 1. Then from Eq.(1) we obtain Eed=1 J/mol for Ni, 1.3 J/mol for -Fe and 0.7 J/mol for Al, which constitutes only 0.006-0.009% of Hm of the corresponding metals, i.e. the value of Ed is vanishingly small.
In metals after severe cold work, the dislocation density, , is of the order of 1012 cm–2. The stored energy associated with dislocations is estimated as [16]
Ed= fGb2/d,(3)
where f=0.5-1 is a coefficient accounting for the type of dislocation, G is the shear modulus and b is the Burgers vector. Using reference data [18,19] (see Table 1) we obtain: Ed=0.32 kJ/mol for Ni, 0.28 kJ/mol for -Fe and 0.21 kJ/mol for Al. This constitutes about 2% of Hm for the corresponding pure metals, thus the value of Ed is small.
Table 1. The parameter values used in calculations [17-19,23].
Metal / 0.2, MPa [17] (cold worked) / Y, GPa [17] (cold worked) / G, GPa [18] (at 300 K) / b, m [19] / Hfv, eV [23] / D0*, cm2/s [19] / E*, kJ/mol [19]Al / 98.1 / 70.8 / 25.4 / 2.861010 / 0.76 / 2.25 / 144.35
-Fe / 263 / 213 / 64 / 2.481010 / 1.08 / 2.0 / 251.0
Ni / 247 / 216 / 78.9 / 2.491010 / 1.40 / 1.27 / 281.16
The energy associated with boundaries of nanograins, which are formed in the course of IPPD during MA, is calculated as Egb = gbS, where S is the total interface area and gb is the energy per unit surface area. To roughly estimate the value of S, it can be assumed (only for the sake of simplicity) that cubical grains with size a are stacked into a simple cubic lattice. Then the total interface area in volume V is S3V/a, and per unit volume we have s=S/V=3/a. Hence the energy of grain boundaries per one mole of a metal is estimated as
Egb= 3gb/(ad).(4)
In literature, experimental data of the energy of nanograin boundaries, gb, are reported for -Fe with the grain size a=10 nm: gb=0.16 J/m2 [20], and for nanocrystalline Cu (a=11 nm): gb=0.29 J/m2 (after 10 h milling time) [21]. Then, using Eq.(4), we obtain Egb= 340 J/mol for -Fe, which constitutes 2.2% of Hm. For copper Egb= 617 J/mol, which equals to 4.7% of Hm (for Cu Hm=13.02 kJ/mol [19]). Thus, the energy stored in metallic reactants after MA, which is associated with nonograin boundaries, is also small.
It is known that the concentration of non-equilibrium vacancies generated by plastic deformation via the Hirsch-Mott mechanism is substantially higher than that of self-interstitials [22]; the role of the latter will be analyzed below. Then we assume that EpdEv where Ev is the stored energy associated with excess vacancies, which is expressed as
,(5)
where Hfv is the enthalpy of vacancy formation expressed in eV, NA is the Avogadro number and Cv is the vacancy concentration. The concentration of vacancies accumulated in metals in the course of MA is not reported in literature, but known are the values measured experimentally after equal channel angular pressing (ECAP) and repetitive cold rolling (RCR), where the deformation conditions are still more severe than in MA (the latter were estimated using the concept of Hertzian collision in Ref. [24]). For fine-grained copper after ECAP and RCR, Cv=5104 at the dislocation density of 41011 cm2 (Fig.2 in Ref. [25]). For copper Hvf=1.17 eV [23], which is larger than for Al and -Fe (Table 1). Then from Eq.(5) we obtain Ev = 56.5 J/mol, which constitutes 0.43% of Hm(Cu). Hence this value is also small.
The above numerical estimates have demonstrated that the stored energy associated with the defects accumulated in metals after MA is small: ES < 10% of Hm, and this energy is small in comparison with a typical heat release of the reaction during SHS even in weakly exothermic systems. This result agrees qualitatively with the data reported for MASHS in the Fe-Si system (ES15-20% of Hm) [26] wherein, however, this value was presented without any calculations.
In theoretical works [11-13], the kinetic of reaction during SHS was described by a traditional model /t ~ k0exp[−E/(RT)], and it was postulated that after MA the apparent activation energy of combustion decreased by the by the value of energy stored in solid reactants during MA: E = E0 – aEES, where E0 is the activation energy without MA and aE is a dimensionless coefficient accounting for the relaxation of the excess energy. However, all the calculations were performed using dimensionless complexes =RTSHS/E0 and 0=ES/(RTSHS), where R is the universal gas constant [11,13]. Also, it was assumed that the combustion temperature, TSHS, is not affected by MA (see Fig.3 in Ref. [11]). Let us link the model parameters to a particular MASHS process. After MA, in the system Ni-13 wt.% Al, the observed SHS temperature was TSHS=820 K < Tm(Al) [6,7]. Calculations were performed for the dimensionless following values: =0.08, 0=5 and aE=1 (see caption to Fig.3 in Ref. [11]). Therefore, without MA the activation energy of SHS was E0=85.2 kJ/mol. This is a rather low value for a solid-state reaction, which is typically controlled by diffusion across the product-phase layer separating the initial solid reactants, but it can be considered as almost admissible: for comparison, the activation energy for diffusion in non-stoichiometric compound Ni0.46Al0.54 is 115.9 kJ/mol [27]. But for 0=5, the stored energy of deformation appears to be strongly overestimated: ES=34.1 kJ/mol, which exceeds the melting enthalpy of Ni by the factor of 1.9 and that of aluminum by the factor of 3.2. Then the obtained activation energy for MASHS, E = E0 – aEES = 51.2 kJ/mol, is extraordinarily low and can correspond only to diffusion in a metallic melt: e.g., for liquid iron E=51.2-65.7 kJ/mol in different temperature ranges [19]. In modeling MASHS in the Ti-N system, the stored energy was estimated proportionally to the broadening of X-ray diffraction lines of Ti after MA, which led to strongly overestimated value ES=36.6 kJ/mol [12], which by the factor of 2.4 exceeded the titanium melting enthalpy, Hm(Ni)=15.5 kJ/mol. As is known, the broadening of X-ray lines increases substantially when the grain size diminishes down to nanoscale, while the associated stored energy is small, as demonstrated above. Thus, the parameter values used for modeling MASHS in works [11-13] appear to be physically meaningless, and therefore the approach itself seems incorrect.
2. Estimation of Relaxation of Non-equilibrium Vacancies
The excess vacancies distributed uniformly in the metal can cause the distortion of the crystal lattice that may be responsible for a change in the apparent activation energy of a solid-state heterogeneous reaction during SHS. This idea was used for interpreting (on a conceptual level only) the effects observed in MASHS [6,7]. However, severely cold-worked metals after MA contain a lot of dislocations and grain boundaries, which act as sinks for non-equilibrium point defects. Let us estimate the relaxation of non-equilibrium vacancies, which were accumulated in metals during MA, in non-isothermal conditions corresponding to heating of the powder charge in the preheat zone of the SHS wave, using the method developed in Ref. [28].
Density of the edge components of dislocation loops, which act as volume-distributed sinks for non-equilibrium point defects, is estimated as e=/2 [16,22]. For calculations we take =1010-1011 cm–2, which is one-two order of magnitude below a typical value attained in metals after MA. Average spacing between edge dislocations is L ~ e1/2, and hence the maximal distance traveled by a vacancy to the sink is Lv ~ L/2 = (2)1/2. In isothermal conditions, the characteristic diffusion length is estimated as ~ Dvt, where Dv is the diffusion coefficient of vacancies and t is time. For non-isothermal conditions T=T(t), this dependence is expressed in the integral form:
.(6)
To determine the value of Dv, the following simple expressions known in the diffusion theory can be used [23]:
, ,(7)
where D* is the self-diffusion coefficient in a metal, D0* and E* are the preexponent and activation energy for self-diffusion, Cv0 is the equilibrium vacancy concentration.
To examine the relaxation of vacancies in non-isothermal conditions, we assume linear heating with a certain rate vT; this corresponds to SHS in the thermal explosion mode. Then T(t) = T0 + vTt, where T0=298 K is the initial temperature. Combining Eqs.(6) and (7), we obtain:
,(8)
where Tr is the temperature at the attainment of which complete relaxation of vacancies occurs.
The dependence of Tr on vT obtained by numerical solution of Eq.(8) is presented in Fig. 1; the heating rate was varied from 101 to 106 K/s. The parameter values used in calculations (Hfv, D0* and E*) for Al, Ni and -Fe are listed in Table 1. It should be noted that the calculated Tr values are the upper-level estimates since we have taken the maximal vacancy diffusion length and lower-level estimates for .
Let’s compare the results of calculations with experimental data on MASHS. In the Ni-Al system, the maximal heating rate in the SHS wave was vT=6000 K/s for TSHS900 K (below the Al melting point) [6,7], then the heating time from T0 to TSHS was 0.1 s. From Fig. 1 it is seen that during this short time the non-equilibrium vacancies in both Al and Ni can relax completely, thus their effect on the solid-state interaction kinetics during SHS is insignificant. For MASHS in the Fe-Al system, TSHS=1173-1200 K (above Tm(Al)) and vT=800-1600 K/s (depending on the process conditions) [29]. From Fig. 1 it is seen that complete relaxation of non-equilibrium vacancies in both Al and Fe can occur in the preheat zone of the SHS wave: in this case Tr<TSHS.