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Modelling stationary regime of the reaction of self-propagating high-temperature synthesis in nanolayered materials (phenomenological model): 1. single-stage reaction.

Recently, the processes of self-propagating high-temperature synthesis (SHS) have been studied quite extensively on the nanolevel not only in the powder systems [1] but also in multilayered structures. The interest in these structures is determined by the fact that they can be used as intermediate layers in the formation of permanent joints in low-weldability materials, for example, composites or intermetallics [2, 3].

As a result of the generation of a large amount of heat during the SHS reaction in multilayered systems based on intermetallic-forming components they can be used as local heat sources in brazing [4]. It is also promising to use these systems as the initial material in the production of thin intermetallic foils.

To ensure the required parameters of the solid-phase reactions in the multilayered structure, it is necessary to take into account a number of factors, such as the chemical and concentration compositions of the foils, the thickness of the foils and the period of the multilayer, the spraying conditions of the foil (the degree of non-equilibrium of defects, the presence of intermediate phases and the probability of formation of these phases), and the reaction initiation parameters (temperature and ignition time).

The experimental solution of this problem requires the formulation of a large number of experiments. One of the methods of simplifying the problem is the construction of theoretical models which make it possible to predict the characteristics of the combustion front in the SHS process in relation to the experimental conditions.

In a general case, the SHS reaction should be studied in the non-stationary conditions, because the systems in which they take place are usually characterised by the complex phase diagrams. In this case, the phases can grow both simultaneously or successively and, consequently, there is competition between exo- and endothermic processes with changing local temperature which in turn predetermines the course of evolution of phase formation. The behaviour of the system becomes difficult to predict and because of the large number of the varied parameters it is necessary to find the required SHS conditions.

According to the authors, this problem can be solved by the construction of a simulation model of the self-consistent solution of heat conductivity and diffusion equations taking into account the phase characteristics changing discretely in time and in space.

This is due to the fact that the diffusion parameters of the phases and their thermodynamic stimuli of transformation depend, on the one hand, on the temperature and concentration fields and, on the other hand, affect them as a result of exo- or endothermic reactions. A preparatory stage of the construction of such a model is the development of a series of simple phenomenological models for describing the partial cases of the structure of the combustion front of the SHS reaction.

Since the diffusion transfer of the combustion products in the direction of propagation of the front is almost completely negligible [1], during the diffusion process in the direction normal to the propagation of the combustion front, the front itself travels the distance [square root of (][a.sup.2] / D ~ [10.sup.3] greater ([a.sup.2] is the thermal diffusivity coefficient, D is the diffusion coefficient). Consequently, the profile of the combustion front can be divided into the required number of the intervals to ensure that in each interval the temperature is regarded as constant, the diffusion process is isothermal and the duration of passage of the front is sufficient for the formation of the final product of solid-phase combustion.

For example, the Al/Ni system is characterised in most cases by a sequence of the formation of equilibrium phases with the increase of the nickel content and the generation of heat in the condition of the presence of a sufficiently large number of reagents both for mutual diffusion in the bulk specimens [5] and in interaction in thin films [6-8]. If it is assumed that external heat removal and the losses of heat in melting of aluminium are small, the resultant final phase will be determined by the initial composition of the foil.

The aim of the present work is the phenomenological description of the stationary propagation of the flat front as a result of the formation of a single intermediate phase. The model should be capable of predicting the main parameters (the speed of propagation of the front and the temperature of the front) on the basis of the characteristics of the multilayer structure (the period of the multilayers, the ratio of the number of the components, the degree of nonequilibrium of the structure). The solution of the inverse problem makes it possible to determine the optimum parameters of the structure for obtaining the required rate, temperature of the front and (if necessary) products of the SHS reaction with the required properties.

The construction of the model of processes in the nanolayered structure enables us to use directly the laws of mutual and reaction diffusion with a correction for the spatial nonuniformity of the temperature field [9], whereas in the case of the powder systems it is necessary to use the general equations of chemical kinetics with some fitting coefficients (the interpretation of these coefficients is not always unambiguous).

We examine a nanolayered two-component foil in the form of alternating M layers of the components A and B (Fig. 1), the width and the period of the multilayer 4l, where l corresponds to half the thickness of the layer of the same component (the thicknesses of the layers A and B are identical).

In contrast to the mutual diffusion approach, used in [9], it is assumed that the heat does not generate in the entire volume and only on the moving interphase boundaries. The model of propagation of the front is constructed using the reaction diffusion equations. This is more efficient in the formation of intermediate compounds with narrow regions of homogeneity in the SHS process.

[FIGURE 1 OMITTED]

The following main assumptions of the phenomenological model can be defined:

--the propagation front of the reaction is flat and stationary;

--as regards the concentration and temperature, all the phase boundaries correspond to the equilibrium diagram;

--a single phase forms during the passage of the front;

-the diffusion flows in the front are directed mainly normal to the direction of front propagation (on the condition that the width of the combustion front is considerably greater than the period of the multilayer);

--the resultant phase is characterised by a narrow homogenising range with similar values of the concentrations [c.sub.left], [c.sub.right] ([DELTA]c = [c.sub.right] [c.sub.left] << 1) at the boundaries [x.sub.left], [x.sub.right]. In this case, we can use the constant flow equation [10], i.e., the densities of the flows [J.sub.left] and [J.sub.right] on the left and right boundaries are approximately the same and equal to the density of the flow inside the phase. This flow density is determined by the integration, i.e., from the mean value the concentration dependence of the diffusion coefficient [bar.D] (c):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[FIGURE 2 OMITTED]

where W is the atomic volume;

--there is no heat removal through the external surfaces of the multilayer.

The investigated the growth of the intermediate [delta]-phase between the [alpha]-phase (solid solution based on A) and the [zeta]-phase (solid solutions on the basis of B) with the thermodynamic stimulus per atom [DELTA][g.sub.[delta]([alpha],[zeta])] (Fig. 2) [11]. The thermodynamic stimulus depends on temperature but in calculations it is assumed to be constant because of the insufficient number of the actual data. Since the foil has a periodic structure, we select the minimum period with thickness 2l, where l is the 1/4 period of the multilayer; [DELTA][y.sub.0] is the initial thickness of the layer of the phase formed prior to the SHS reaction (Fig. 3).

In the thin section dx, normal to the direction of propagation of the front, the formation of the phase during time dt takes place in the interlayer d[DELTA][y.sub.[delta]](x), containing d[DELTA]y(x) dxW/[OMEGA]. atoms (Fig. 4). The generated heat [DELTA][g.sub.[delta]([alpha],[zeta])] d[DELTA][y.sub.[delta]](x)W/[OMEGA]/dt [DELTA][g.sub.[delta]([alpha],[zeta])]d[DELTA][y.sub.[delta]](x)dxW/[OMEGA]/dt is used for heating the interlayer dx throughout the entire thickness 2l. Thus, the variation of temperature in the section dx is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [c.sub.p] is the specific heat capacity, p is density.

[FIGURE 3 OMITTED]

In the stationary combustion regime each point of the front moves with constant speed v and is characterised by temperature T(x), the width of the phase formed [DELTA]y(x) between two layers, heat generation per atom during unit time [q.sub.[delta]](x). It is evident that the diffusion characteristics depend on temperature, determined by heat generation which in turn depends on the efficiency of the diffusion process. Thus, the solution of the problem is the determination of self-consistent profiles T(x), [DELTA]y(x) and q(x). For this purpose, it is necessary to use the iteration procedure which is stopped when these profiles are stabilised. To construct the self-consistent model, requiring numerical calculations, we determine the profile of the width of the resultant [delta]-phase. For this purpose we write the equation of the balance of matter for the moving interphase boundary [y.sub.[alpha][beta]] (component A/intermediate phase) and for the moving interphase boundary [y.sub.[alpha][zeta]] (intermediate phase/component B), using the approximation (1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

where [c.sub.[delta]] is the mean concentration in the [delta]-phase; [D.sub.[delta]] is the diffusion coefficient in the [delta]-phase; [DELTA][c.sub.[delta]] is the homogeneity range of the [delta]-phase.

[FIGURE 4 OMITTED]

After relatively simple mathematical transformations we obtain:

d[DELTA][y.sup.2.sub.[delta]](t, x)/dt = 2/[c.sub.[delta]](1-[c.sub.[delta]])[D.sup.W.sub.[delta]] (4)

here [y.sub.[delta][xi]] - [y.sub.[alpha][delta]] = [DELTA][y.sub.[delta]](t,x);[D.sup.W.sub.[delta]] = [D.sub.[delta]][DELTA][c.sub.[delta]] -| is the

diffusion permeability of the phase which depends on temperature (the function of time and coordinate) and is the combination of diffusion [D.sup.*.sub.[delta]](A), [D.sup.*.sub.[delta]] (B) of the marked atoms A, B and the thermodynamic stimulus of the transformation [DELTA][g.sub.[delta]([alpha][zeta])](T) [12]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the pre-exponential multiplier and the activation energy of diffusion, respectively; [k.sub.B] is the Boltzmann constant.

After integrating equation (4) it must be taken into account that the interlayer [DELTA]y[delta](t, x) contains the layer [DELTA][y.sub.0] with diffusion permeability [D.sub.0.sup.W] in which the reaction took place even prior to the start of the SHS:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where t is the time at which the interlayers of the new [delta]-phase come into contact. We transfer to new variables

[xi] = x - [upsilon]t',t' = x - [xi]/[upsilon], dt' = d[xi]/[upsilon] so that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Consequently, at [??] = 0 the interlayers come together [DELTA][y.sub.[delta]](t, 0) = 2l and the solution of (6) has the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

and the speed of propagation of the stationary front is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In the heat conductivity equation, we transfer to the introduced variable [??]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

Equation (2) is substituted into equation (9), followed by preliminary substitution dt = d[??]/[upsilon], and we obtain the equation describing the dissipation of heat in the front (0 < [??]):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

The formal solution of equation (9) leads to the following integral equation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

which can be solved using the iteration self-consistent procedure of the simultaneous determination of the profile of variation of temperature (11) and speed (8).

In order to avoid using the described iteration procedure, the proposed model can be simplified by obtaining simple analytical estimates of the maximum temperature in the front [T.sub.f] and the speed of the front u. This requires another assumption: when solving the heat conductivity equation (but not the diffusion equation!) it is necessary to ignore the heat generated as a result of the formation of the new phase. Consequently, in the quasi-stationary approximation [partial derivative]T/[partial derivative]t = [upsilon][partial derivative]T/ [partial derivative]x the heat conductivity equation has the form

-[upsilon][partial derivative]T/[partial derivative]t = [a.sup.2][[partial derivative].sup.2]T/[partial derivative][x.sup.2] = 0 with the solution of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the width of the front is

L = [a.sup.2]/[upsilon]. (12)

[FIGURE 5 OMITTED]

During the time [tau] ~ L/[upsilon] = [a.sup.2]/[[upsilon].sup.2] of passage of the front through the width L the [delta]-phase with the equilibrium concentration [c.sub.5] should close up as a result of reaction diffusion

2/[c.sub.5](1-[c.sub.5])[[integral.sup.[tau].sub.0]D(T(t))dt [approximately equal to] [(2l).sup.2] - [([DELTA][y.sub.0]).sup.2], (13)

where

D(T) = [D.sub.0[delta]] exp([Q.sub.[delta]]/[k.sub.B]T)[DELTA][g.sub.P][delta]([alpha][xi])/ [k.sub.B]T.

As a result of the integration of the equation (13) using quite simple but cumbersome mathematical transformations we obtain the expression for the speed of propagation of the stationary front:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

where [T.sub.f] is the maximum temperature at the front. It is estimated using equation (10). Since the maximum temperature is reached during the time of passage t of the reaction throughout the entire thickness of the effective interlayer 2l- [DELTA][y.sub.0], the variation of temperature is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

It is evident that as the thickness of the initial interlayer [y.sub.0] increases, the maximum temperature [T.sub.f] at the front decreases. At [y.sub.0] = 0 the maximum temperature [T.sub.max] = [T.sub.0] [DELTA][g.sub.[delta]([alpha][xi])]/[c.sub.p] [rho][OMEGA]. is reached.

The proposed analytical estimate does not take into account the effect of heat generation on the temperature profile (zero in the right-hand part of the heat conductivity equation), i.e., this model is not self-consistent.

[FIGURE 6 OMITTED]

The quantitative estimates of the proposed models with one intermediate phase were obtained using the parameters described in [9]: [c.sub.[delta]] = 0.5; [D.sup.*.sub.0[delta]] = 1.5 X [10.sup.-5] [m.sup.2]/s; [Q.sup.[delta]] = 2.7 x [10.sup.-19] J; [a.sup.2.sub.[delta]] = 7.451 x [10.sup.-5] [m.sup.2]/s; [T.sub.0] = 300 K; the thermodynamic stimulus of the formation of the 5-phase [DELTA] [g.sub.[delta]([alpha][zeta])] = 7.36549 x [10.sup.-20] J was determined from the equation (15) at [DELTA][y.sub.0] = 0 and [T.sub.f] = 1919 K.

The results can be efficiently analysed using the parameter f = (2l-[y.sub.0])/(2l), which determines the fraction of the interlayer which has not reacted in the process of production of the multilayer foil where a phase can form in the process of SHS. This means that f shows the efficiency of the SHS reaction (at f [congruent to] 1 the temperature of the front reaches the maximum value [T.sub.f] = [T.sub.max]) and may have values from zero (phase formation took place throughout the entire thickness of the interlayer) to 1 (the multilayer film consists of pure components without intermediate phases). Therefore, the introduced parameter f is referred to as the coefficient of efficiency of the interlayer. In computer calculations, the value of this coefficient is in the range 0.5 [less than or equal to] f < 1, because the multilayer foil with f < 0.5 is not used efficiently in the SHS processes.

The quantity l (a quarter of the period of the multilayer) varied from [DELTA][y.sub.0] (f = 0.5, half the interlayer has reacted) to 200 nm (the maximum value in spraying), and the thickness of the interlayer [DELTA][y.sub.0], which interacted prior to the passage of the combustion front, varied from 0.1 nm f [congruent to] 1, smaller than the interplanar spacing) to 100 nm (f = 0.5, half the maximum thickness of the interlayer in spraying). The graphs show the results for [DELTA][y.sub.0], having the values of 0.1; 0.4; 1.6; 6.4; 25.6 nm.

In the computer calculations the maximum temperature at the front (Fig. 5, I) was fixed; the speed of the front was calculated from equation (8) (Fig. 5, II). The results were compared with the estimates obtained from the simplified analytical model: for the speed of the front using equation (14), for the maximum temperature at the front from equation (15). The speeds of the front were in qualitative agreement. For quantitative agreement, it was sufficient to introduce an adjustable multiplier p in the equations (12) and (14):

L = p[a.sup.2]/[upsilon] (12*)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14*)

Where [T.sub.f] = [T.sub.0] + ([T.sub.max] -[T.sub.0])f. (15*)

The points in Fig. 5, II show the profiles of the speed of the front [upsilon] at the optimum adjustable multiplier p = 4.04 (determined for the maximum value of the dependence (4l) at [DELTA][y.sub.0] = 0.1 nm).

Since the results of the proposed models are in qualitative agreement, the resultant relationships will be analysed using the analytical expressions (14) and (15). Equation (15) shows that the maximum temperature at the front depends on the thermodynamic stimulus of the formation of the new phase.

[FIGURE 7 OMITTED]

The dependence of temperature on the period of the multilayer 4l and the thickness of the initial interlayer of the new phase [DELTA][y.sub.0] is determined by the coefficient of efficiency of the interlayer f and is linear (Fig. 5, I, b). At fixed [DELTA][y.sub.0], the asymptotic value of temperature at the front is reached when the unit of the multilayer become sufficiently long for heating taking into account [DELTA][y.sub.0]: as the thickness of the initial interlayer increases, the period of the multilayer 4l at which the maximum temperature in the front is reached, becomes greater (Fig. 5, I, a).

The non-monotonic form of the dependence of speed on the period of the multilayer correlates not only with the analytical estimates for solid-phase [9] and gas-phase [13] combustion but also with the experimental results obtained for the TiAl system [14]. In the range of low values of l the coefficient of efficiency tends to the value 0.5, which indicates slow heating of the multilayer is which, according to the authors of [14], is caused by the 'increase of the specific surface of the interlayer boundaries per unit volume'.

The reasons for the non-monotonic dependence of the speed of the front on the period of the multilayer will now be analysed from the mathematical viewpoint. If the temperature of the front in equation (14*) is assumed to be the function of the coefficient of efficiency (15*) and 1/4[l.sup.2] - [DELTA][y.sup.2.sub.0] is regarded as 1/4[l.sup.2]f(2 - f), the speed of the front can be described by a function of two variables:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

i.e., the speed is determined by two such competing factors (without taking into account the constant quantities);

--the diffusion factor, inversely proportional to the period of the multilayer [M.sub.dif] (l) = 1/l and associated with the parabolic growth of the new phase (as the value of l increases, the diffusion path becomes longer); decreasing with increasing f (Fig. 6a);

--the thermal factor, which depends on the temperature of the front as a function of the coefficient of efficiency

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and increases with increasing f (Fig. 6b).

In this case, the maximum speed is obtained for the same value of f, regardless of the period of the multilayer structure (Fig. 6, II, b). In turn, the position of the maximum is determined by the activation energy of diffusion for the given phase: at Q = 2.7 x [10.sup.-9] J the maximum speed corresponds to the value f = 0.85; at Q = 1.35 x [10.sup.-19] J the maximum is displaced to f = 0.675.

The decrease of the speed of the front at f close to 0.5 can be explained by the controlling effect of the thermal factor [M.sub.heat] which decreases at a considerably higher rate than the rate of increase of the diffusion multiplier [M.sub.dif,] i.e., an increase of the period of the multilayer is accompanied by an increase of the diffusion path of the atoms, extending the heating time and, correspondingly, making it less effective.

In addition, equation (16) shows that the product of the speed in the period of the multilayer is determined by the coefficient of efficiency l(f)[upsilon](f) = Func(f). To verify this relationship, determined on the basis of the simple analytical model, the dependence l(f)[upsilon](f) was constructed and the variation of the value of [DELTA][y.sub.0] at fixed values of l for the results obtained for the self-consistent model (Fig. 7). The curves coincide (small differences are caused by the errors of linear extrapolation in the determination of the speed for the fixed value of l from the dependences [upsilon](l) at the fixed value of [DELTA][y.sub.0] (Fig. 6a). The results against confirm the adequacy of the simple evaluation procedure.

Since the multilayer foils are non-equilibrium objects, it is not guaranteed that the diffusion and thermodynamic parameters of the model discussed here can be taken from the tables of physical quantities. This relates especially to the stimulus for the transformation and the activation energy of diffusion. It is proposed to interpret them as adjustable parameters in the equation (14*). It should be mentioned that the derivative ln l(f)[upsilon](f) with respect to f (taking into account (15*)) depends on only on the single parameter [Q.sub.5]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Comparison with the experimental curves can be used for the single-parameter adjustment of the value [Q.sub.5]. The adjustable value [Q.sub.5] may be used for determining the second free parameter [D.sup.05][DELTA][g.sub.5( [alpha],[xi])].

Conclusions

1. A phenomenological approach was used for describing the stationary regime of the process of SHS, controlled by reaction diffusion. The proposed models can be used for evaluating the effect of phase formation on the parameters of the combustion front.

2. Comparison of the results, obtained from the simple analytical evaluation, with the results of numerical calculations using the more complicated self-consistent scheme enables the adjustable parameter to the selected. The use of this parameter in equation (14*) can be used for a simple evaluation of the speed of the combustion front.

3. A self-similarity was detected in the behaviour of the multilayer system at the fixed value of the proposed efficiency parameter f.

4. The scaling relationships, determined using simple phenomenological models, can be used for evaluating the diffusion and thermodynamic parameters of the system, and also for adjustment (calibration) of the computer calculations of temperature and the speed of the combustion front to the experimental data.

5. The correlated model, adequately describing the actual experiments with SHS in thin films may be used for predicting the course of the SHS reactions in a wide range of the characteristics of the multilayer foils (the thickness and period of the multilayer, concentration composition, number of defects, etc). Consequently, it is possible to optimise the parameters of the course of the SHS reaction in thin foils without carrying out a large number of experiments.

References

[1.] Merzhanov A.G., Solid flame combustion, ISMAN, Chernogolovka, 2000.

[2.] Ishchenko A.Ya., et al., Avt, Svarka, 2007, No. 7, 5-9.

[3.] Ustinov A.I., et al., Intermetallics, 2008, volume 16, 1043-1045.

[4.] Shishkin A.E., et al., Metallofizika Noveish. Tekhnol., 2009, No. 9, 1179-1188.

[5.] Larikov L.N., et al., Diffusion in ordered alloys, Naukova Dumka, Kiev, 1975.

[6.] Colgan E.G., Mater. Sci. Rep., 1990, vol. 5, 1-44.

[7.] Ma E., et al., J. Appl. Phys., 1989, vol. 65, 2702-2712.

[8.] Ustinov A.I., et al., Sur. Coat. Technol., 2008, vol. 202, 3832-3838.

[9.] Mann B., et al., J.Appl. Phys., 1997, vol. 82, 1178-1188

[10.] Gusak M. and Yarmolenko M.V., Jibid, 1993, vol. 73, 4881-4884.

[11.] Gusak A.M., et al., Models of solid-phase reactions, ChNU, Cherkassy, 2004.

[12.] Gusak A.M. and Tu K., Acta Mater., 2009, vol. 57, 3367-3373.

[13.] Strunin V.A. and Manelis G.B., Fiz. Goreniya Vzryva, 2004, No. 3, 22-27.

[14.] Rogachev A.S., et al., ibid, 2004, No. 2, 45-51.

T.V. Zaporozhets, A.M. Gusak and A.I. Ustinov

B. Khmel'nitskii Cherkask National University

E.O. Paton Electric Welding Institute, Kiev
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Title Annotation:GENERAL PROBLEMS OF METALLURGY
Author:Zaporozhets, T.V.; Gusak, A.M.; Ustinov, A.I.
Publication:Advances in Electrometallurgy
Date:Jan 1, 2010
Words:4273
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