Mechanistic investigation of electric field-activated self-propagating reactions: experimental and...

*Correspondence address: Dipartimento di Ingegneria Chimicae Materiali, Centro Studi sulle Reazioni Autopropaganti (CESRA) andUnita di Ricerca del Consorzio Interuniversitario Nazionale per laScienza e Tecnologia dei Materiali, Universita degli Studi di Cagliari,Piazza d'Armi, 09123 Cagliari, Italy. Tel.: #39-070-675-5058; fax:#39-070-675-5067.

E-mail addresses: [email protected] (G. Cao), [email protected] (Z. A. Munir).

Chemical Engineering Science 56 (2001) 683}692

Mechanistic investigation of electric "eld-activated self-propagatingreactions: experimental and modeling studies

R. Orru!, A. Cincotti", G. Cao!,",*, Z. A. Munir#!Dipartimento di Ingegneria Chimica e Materiali, Centro Studi sulle Reazioni Autopropaganti (CESRA) and Unita% di Ricerca del Consorzio Interuniversitario

Nazionale per la Scienza e Tecnologia dei Materiali, Universita% degli Studi di Cagliari, Piazza d'Armi, 09123 Cagliari, Italy"CRS4 (Center for Advanced Studies, Research and Development in Sardinia), VI Strada OVEST Z.I. Macchiareddu 09010 UTA (Cagliari), Italy

#Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, USA

Abstract

The mechanism of electric "eld-activated self-propagating reactions is investigated using the combustion front quenchingtechnique. In particular, previously published experimental results obtained through the "eld-activated combustion synthesis (FACS)of b-SiC, TaC, Ti

3Al and B

4C}TiB

2are re-examined and compared. Pre-combustion and combustion stages involved during

synthesis wave propagation are postulated for all systems. Subsequently, modeling results aimed at simulating the process where anelectric "eld-activated, self-propagating reaction takes place are presented. In particular, a one-dimensional model of FACS techniqueis developed to simulate the rapid quenching of the reaction during its progress as the applied "eld is turned o!. A rate expressionwhich accounts for the in#uence of temperature, particle size, compaction density, reactant stoichiometry, and inert content isincluded in the model. ( 2001 Elsevier Science Ltd. All rights reserved.

Keywords: Field-activated combustion synthesis; Combustion front quenching; Reaction mechanism; Heterogeneous model; Theoretical simulation

1. Introduction

It is well known that relatively high exothermic reac-tions of either solid}solid or gas}solid type are able topropagate in the form of a combustion wave upon igni-tion. This area of reaction engineering is very importantand relatively poorly understood (Varma, Rogachev,Mukasyan, & Hwang, 1998). The process is complexinvolving a wide spectrum of physico-chemicalphenomena (reaction, di!usion, nucleation, graingrowth, etc.).

A promising technique known under the acronym ofSHS (self-propagating high-temperature synthesis)

for the preparation of advanced materials (Merzhanov& Borovinskaya, 1972; Munir & Anselmi-Tamburini,1989; Varma et al., 1998) as well as for other interestingapplications in the "eld of environmental chemical engin-eering (cf. Orru, Sannia, Cincotti, & Cao, 1999c; Caoet al., 1999) is based on the occurrence of self-propagatingreactions. An empirical criterion typically adopted fordetermining the feasibility of self-propagating reactions isan adiabatic temperature, ¹

!$, exceeding 2000 K. But

this is only one parameter a!ecting the occurrence ofSHS reactions (Munir, 1998).

Recently, a new method, based on the use of an electric"eld to activate self-propagating reactions in less-exothermic systems, was developed (cf. Munir, Lai& Ewald, 1995). Through this method, referred to in theliterature as "eld-activated combustion synthesis(FACS), the synthesis of materials heretofore not pos-sible by normal SHS (e.g., SiC, SiC}AlN, TaC,B4C}TiB

2, MoSi

2}SiC, and Ti

3Al) was successfully

demonstrated (cf. Feng & Munir, 1995a; Xue &Munir, 1996a,b, 1997; Gedevanishvili & Munir, 1998;Orru, Cao & Munir, 1999a). SHS and FACS processesare typically characterized by high temperatures (up to

0009-2509/01/$ - see front matter ( 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 2 7 6 - 1

Fig. 1. Schematic representation of the FACS apparatus.

about 3500 K), reaction waves which propagate atrelatively high speeds (up to 25 cm/s), and low-energyrequirements, etc.

Although these processing conditions make these tech-niques advantageous for industrial applications, processscale-up and control may not be straightforward. Inorder to overcome this problem a fundamental know-ledge of reaction and structure formation mechanism isnecessary.

Considerable e!ort has been made at the experimentallevel in order to understand the basic mechanisms occur-ring during self-propagating reactions. Di!erent andcomplementary approaches have been proposed in theliterature, i.e. investigation of the e!ect of processingvariables, the synchrotron radiation technique, the par-ticle-foil technique, the combustion front quenching tech-nique (Mukasyan & Borovinskaya, 1992; Varma et al.,1998; Gras, Charlot, Ga!et, Bernard, & Niepche, 1999).The latter one, which is based on the rapid extinction ofthe reaction front during its progress, is often preferreddue to its simplicity and since it provides a relativelyaccurate picture of the physico-chemical phenomena in-volved during reaction evolution. By investigating prod-uct microstructure and composition at di!erent areasrelative to the location of the frozen reaction wave, it ispossible to identify intermediate species and phase trans-formation taking place.

For those less-exothermic systems which are syn-thesized through the FACS technique, quenching of thereaction can be obtained by turning o! the electric "eldduring front evolution. This technique has an advantageover the wedge}quench technique. The FACS quenchingis due to a sudden and abrupt decrease in the energy ofthe wave while the other is due to a gradual increase inthe heat loss due to conduction.

In this paper we focus on aspects related to the mech-anistic investigation of electric "eld-activated self-propagating reactions. In the "rst part, we re-examineand compare previously published experimental resultsobtained during the FACS of b-SiC, TaC, Ti

3Al and

B4C}TiB

2from elemental powders (Feng & Munir,

1995a; Xue & Munir, 1996a,b; Orru, Cao & Munir,1999b). It should be noted that b-SiC and Ti

3Al are

characterized by an adiabatic temperature less than2000 K and thus the occurrence of a self-propagatingreaction is thermodynamically limited. On the contrary,although for the other two systems ¹

!$is signi"cantly

higher of the minimum value of 2000 K, they cannot besynthesized by SHS without activation. In Section 4 ofthis paper, we present modeling results aimed atsimulating the process where electric "eld-activatedself-propagating reactions take place. In particular, aone-dimensional model of the FACS technique isdeveloped which is able to simulate the rapid quenchingof the reaction during its progress as the applied "eld isturned o!.

2. Remarks on previous experimental observations anddiscussion

In order to provide a background for the results of theFACS method, a brief description of the experimentalsetup and the procedure is provided here. A schematicrepresentation of the principle of FACS and its use forthe quenching of the combustion front is shown inFig. 1 (Feng & Munir, 1995a).

Typically, powders of elemental reactants are mixed instoichiometric amounts and pressed into round-endedrhombohedral pellets. The pellets were placed betweentwo graphite electrodes across which a voltage was ap-plied. The sample was ignited by means of a tungsten coilplaced few millimeters away from one end. The electric"eld was applied simultaneously with the ignition source.The latter one was turned o! immediately after the reac-tion was initiated. The reaction front propagatedthrough the sample only in the presence of the electric"eld and in a direction perpendicular to it. Since, aspreviously observed, the systems investigated are notself-sustaining, when the "eld was turned-o! during wavepropagation, the combustion wave stopped immediately.The process evolution was recorded on a video recorderand the wave velocity was measured from video record-ings using a time-code generator. Real-time current (I)and voltage (<) data were also acquired during wavepropagation (Feng & Munir, 1995a).

It was reported that during the "eld-activated combus-tion synthesis of b-SiC, TaC and Ti

3Al, the propagation

of the wave is only possible when the applied "eld isabove a threshold value, E

t. For these systems, the re-

ported values of Etare 6.8, 8.0 and 3.8 V/cm, respectively.

However, for the case of the composite yB4C}TiB

2, the

threshold "eld was found to depend on the value of y, i.e.,the molar ratio B

4C/TiB

2. Above the threshold, the

reaction front propagates through the sample with a ve-locity which increases with an increase in "eld strength

684 R. Orru% et al. / Chemical Engineering Science 56 (2001) 683}692

Fig. 2. SEM micrograph of the quenched sample and corresponding XRD patterns for the FACS of b-SiC.

for all systems investigated. The removal of the "eld insuch samples causes the reaction front to stop and thecombustion process to extinguish.

For the case of b-SiC, Fig. 2 (Feng & Munir, 1995a)shows a SEM micrograph of the quenched front. Fourdistinct regions can be clearly identi"ed from this "gure.XRD analysis results of samples obtained from thesezones are also reported in Fig. 2. Region 1 represents theelemental reactants. In Region 2, in addition to Si andC along minor amounts of the product, SiC, are present.The latter compound is the major phase in Region 3 butSi and C are still present. Finally, only b-SiC was ob-served in Region 4. More detailed microstructural exam-ination of these zones by SEM were accomplished. Inparticular, it was observed that the transition from Re-gion 1 to Region 2 corresponds to the melting of Si (Feng& Munir, 1995a).

The same approach described above to investigate thequenched samples was adopted for the other systemsconsidered in this work, i.e. Ti

3Al (Orru et al., 1999b),

TaC (Xue & Munir, 1996b) and B4C}TiB

2(Xue

& Munir, 1996a).On the basis of these experimental results, it is possible

to postulate the sequence of transformation duringFACS of b-SiC, Ti

3Al, TaC, and B

4C}TiB

2, as sum-

marized in Tables 1(a)}(d), respectively.For the case of b-SiC, Table 1(a), it is seen that the

combustion is preceded by the melting of silicon par-ticles, a process that takes place in the leading edge of thereaction wave. This phenomenon is very important since

it gives rise to an increase in the interconnectivity be-tween particles. In addition, due to the relatively highvalue of the electrical conductivity of liquid silicon (rela-tive to the corresponding values of the solid reactantsand product), the current is mainly con"ned to this re-gion (Feng & Munir, 1995a). From the compositionaland microstructural investigation of the quenched fronta gradual conversion of reactants to b-SiC is seen.

A mechanism of formation of SiC by combustion syn-thesis is recently proposed in the literature (Narayan,Raghunathan, Chowdhury, & Jagannadham, 1994). Asobserved in the previously reported FACS case (Feng& Munir, 1995a), the combustion reaction is preceded bythe melting of silicon. Then, the formation of SiC involvesthree di!erent steps: the "rst consists of the dissolution ofcarbon into liquid silicon, the second stage relates to thedi!usion of carbon through liquid silicon, and "nally thethird stage relates to the formation of SiC by precipita-tion from a saturated liquid solution. In addition, thelatter step is followed by the growth of the carbide grains.

The sequence of transformations involved in the FACSof Ti

3Al is summarized in Table 1(b) (Orru et al., 1999b).

In order to obtain the desired phase, a pre-combustionstage, i.e. the melting of aluminum, and two subsequentcombustion steps are required. The "rst step is related tothe formation of secondary (undesirable) aluminides,with TiAl

3being the "rst phase formed. During the

second combustion step these intermediate phases inter-act with unreacted titanium and "nally forming Ti

3Al.

This conclusion was also supported by the results

R. Orru% et al. / Chemical Engineering Science 56 (2001) 683}692 685

Table 1Transformation zones and their characteristics during FACS

Zones Chemical species Nature of structural transformation Results of the chemical interaction

(a) SiCReactants Si, C No transformation No interactionPre-combustion Si, C Melting of silicon No interactionCombustion Si, C, b-SiC Chemical reaction Formation of b-SiCFinal product b-SiC * No chemical interaction

(b) Ti3Al

Reactants Ti, Al No transformation No interactionPre-combustion Ti, Al Melting and spreading of aluminum No interactionCombustion (1) Ti, Al, TiAl

3Chemical reaction Formation of TiAl

3Combustion (2) Ti, TiAl, Ti

3Al Chemical reaction Disappearance of TiAl

3Formation/disappearance of TiAlFormation of Ti

3Al

Product Ti3Al * No chemical interaction

(c) TacReactants Ta, C No transformation No interactionCombustion (1) Ta, C, Ta

2C Di!usion of C into Ta lattice Formation of Ta

2C

Combustion (2) Ta, C, Ta2C, TaC Di!usion of C into Ta

2C Formation of Ta

2C

Transformation of Ta2C into TaC Formation of TaC

Di!usion of C into TaCFinal product TaC * No chemical interaction

(d) B4C}TiB

2Reactants Ti, B, C No transformation No interactionCombustion (1) B, C, Ti, TiB

2Chemical reaction Formation of TiB

2Combustion (2) B, C, TiB

2, B

4C Chemical reaction Formation of B

4C

Final products TiB2, B

4C * No chemical interaction

obtained during the FACS of Ti3Al (Orru et al., 1999a),

where the "eld-activated combustion wave propagatesthroughout the entire sample. The corresponding tem-perature pro"les associated with the passage of the waveshowed two peaks. The "rst one, which is always presentindependent of the applied "eld level, corresponds to the"rst stage of combustion. On the other hand, the secondpeak, almost always absent at low "elds, increasedstrongly with increasing "eld strength. Correspondingly,the formation of a mixture of phases, i.e. TiAl

3, TiAl,

unreacted Ti, and Ti3Al, was found to take place at low

"elds, while the latter phase is the only one formed whenapplying high "elds. In particular, this is the case whenusing an electric "eld equal to 14.3 V/cm, correspondingto the conditions considered in the reported work formechanistic investigations (Orru et al., 1999b).

Since the adiabatic temperature of TaC is less than itsmelting point (4153 K) as well as the melting points of Ta(3289 K) and C (4100 K), a solid}solid mechanism is theonly likely process during the combustion synthesis ofthis carbide phase. The transformations taking placeduring FACS of TaC are summarized in Table 1(c). Theinitial stage consists of the di!usion of carbon into tanta-lum with the formation of Ta

2C. This step is followed by

the transformation of Ta2C to TaC, which occurs as

a consequence of the di!usion of C into the Ta2C lattice.

Since only trace amount of Ta2C is found in the quen-

ched wave, the latter step is assumed to occur relativelyrapidly and therefore is not considered rate controlling.The progress of the synthesis involves also the di!usionof carbon into TaC, which is the controlling step. Thefact that this slow step is limiting the process evolution,may be the reason for the relative di$culty of establish-ing and sustaining a combustion wave in such systems ascompared to the case of other carbides, i.e. TiC, wheremelting phenomena are also involved. In addition, it wasfound (Xue & Munir, 1996b) that resulting tantalumcarbide di!raction peaks shifted when moving from thereactants region to the "nal product one, indicatinga change in the lattice parameter corresponding to thespeci"c elemental ratio. In particular, the lattice para-meter value of stoichiometric TaC, 4.455 As , was obtainedonly at the end of the combustion process. This is pos-sible because tantalum carbide exists in a relatively widerange of composition, as it may be seen from the Ta}Cphase diagram (Thaddaeus, 1986). This aspect furthercon"rms that the stoichiometric TaC is obtained onlywhen carbon completely disappears as a consequence ofa gradual di!usion of C into the TaC

1~xphase.

The removal of the "eld, applied in a direction perpen-dicular to that of the combustion wave, provides anexcellent means for investigating the sequence of phaseformation during FACS of B

4C}TiB

2. The correspond-

ing postulated transformation zones are summarized in


LgLt

"Ga*1,

6

oB

1#2b

b

D0

d2p0

expA!E

R¹B(a!g)(1!g)1@3

(a!g)1/oA#(b#g)1/o

p

,

a(1,6

aoB

1#2bb

D0

d2p0

expA!E

R¹B(1!g)(1!ag)1@3

(1!g)1/oA#(b/a#g)1/o

p

,

(2)

Table 1(d). It is apparent that the synthesis of this com-posite occurs through two sequential reactive steps. Inparticular, the "rst stage, involving only the interactionof titanium and boron but not carbon, results in theformation of TiB

2. This combustion step provides heat

to activate the second one, which leads to the formationof B

4C. Both steps take place in the combustion wave but

the "rst one occurs near the leading edge of the wavewhile the second one close to the trailing edge.

3. FACS modeling

As indicated in the introduction section, FACS in-volves complex physico-chemical phenomena, such asmelting and di!usion of reactants, chemical reactionswith formation of intermediate phases, nucleation and

grain growth, all contributing to the formation of themicrostructure of the "nal product. Therefore, the under-standing of FACS fundamentals represents a scienti"cand technological challenge and, simultaneously, the keyapproach to develop a proper process model which canbe used to address suitable control strategies. From theexperimental results discussed above it has been demon-strated that by quenching the reaction front during itspropagation it is possible to identify the evolution ofreactants to products as well as of the structural andchemical transformations. Although this technique isable to provide detailed information of a macrokinetictype useful for process understanding, the estimate of thephysico-chemical parameters involved may be obtainedonly by direct comparison between experimental andtheoretical results. This goal cannot be achieved if thetypically heterogeneous reacting mixture is considered ashomogeneous, as is the case in many literature accounts.In this work, we develop a one-dimensional model of theFACS technique which takes into account the heterogen-eous nature of the system and is able to simulate therapid quenching of the reaction during its progress as theapplied "eld is turned o!. It is worth noting that, al-though a two-dimensional approach may be more ap-propriate to simulate FACS processes (Feng & Munir,1995b; Feng, Graeve, & Munir, 1998), in this work wefocus our attention on the use of heterogeneous kinetics.

By assuming that the starting powders behave like anisotropic mixture and mass di!usion of the solid react-ants and products does not occur, the model consists of

the following energy balance:

oCp

L¹Lt

"

LLxAk

L¹Lx B#(!*H) C0

B

LgLt

#pA<

¸yB

2

!2Ch(¹!¹

0)#l(¹4!¹4

0)

w D!2C

;(¹!¹0)

¸y

D. (1)

This energy balance takes into account the rate of heatconduction and heat generation by chemical reaction, therate of heat generation from the electric "eld by Jouleheating as well as the rate of the heat dissipation due toradiative cooling from the sample surface. This is analog-ous to the model proposed by Feng et al. (1998).The energy balance is coupled with the following massbalance:

where g"1!(dp/d

p0)3 for a'1, g"[1!(d

p/d

p0)3]/a

for a(1, CA"C0

B(a!g), C

B"C0

B(1!g) and C

P"

C0B(b#g), while the meaning of all symbols is reported in

the Notation section. Here, the reaction rate equation isbased on the model developed by Kanury (1992), inwhich the reaction between a low-melting-point compon-ent (A) and a high-melting (solid) one (B), is considered.In particular, the model assumes that A melts and #owsby capillary forces around B particles, and the reaction isproportional to the instantaneous surface area of thesolid component. It should be noted that the rate expres-sion accounts for the in#uence of temperature, particlesize, compaction density, reactants proportion and inertcontent.

The mass and energy balances above are also coupledwith the following electropotential equation:

0(t(t1<"

<*/

1#(w/¸y)R

0:Lx

0pdx

, (3)

t*t1, <"0, (4)

which is derived by assuming that the electromagneticphenomenon is static or quasi-static, i.e. by neglecting thetime-derivative terms in Maxwell's equations, and con-sidering an ohmic material. The dependencies of thethermophysical properties, such as thermal conductivity,electrical conductivity, and heat capacity on temperatureand composition are taken from the literature, as re-ported in Table 2. In order to solve the above equations,the following functional relationships have been


Table 2Values of the physico-chemical parameters used during numerical simulations and corresponding literature references

Parameter Value References

a 1 *

b 0 *

C0B

3.67 104 mol/m3 Feng et al. (1998)C

p,A(¹) * Kubaschewski, Alcock and Spencer (1993)

Cp,B

(¹) * Kubaschewski et al. (1993)C

p,P(¹) * Kubaschewski et al. (1993)

dp0

5]10~6 m Feng and Munir (1995a)D

01.9]10~5 m2/s *

E 4.5]105 J/mol *

h 45 W/m2 K Huque and Kanury (1993)¸x

0.0208 m Feng and Munir (1995a)¸y

0.0141 m Feng and Munir (1995a)kA(¹) * Madelung (1996)

kB(¹) * Touloukian (1965)

kP(¹) * CRC (1994) 75th Ed.

kG3!1)*5%

100 W/m K Touloukian (1965)M

A28.085 g/mol *

MB

12.01 g/mol *

MP

40.19 g/mol *

R 8.314 J/mol K *

R0

0.02 ) Feng et al. (1998)s 0.007 m *

¹0

298 K *

t1

2 s *

<*/

14 V *

w 0.0068 m Feng and Munir (1995b)b 0.0025 Kanury (1992)!*H 66.9]103 J/mol Kubaschewski et al. (1993)l 5.67]10~8 J/s m2 K4 *

o 1.47]106 g/m3 Feng et al. (1998)o5)

2.3]106 g/m3 Feng et al. (1998)oA

8.26]104 mol/m3 *

oB

1.88]105 mol/m3 *

oP

8.0]104 mol/m3 *

pA(¹) * Touloukian (1965)

pB(¹) * International Critical Tables (1926)

pP(¹) * International Critical Tables (1926)

accounted for:

oCp"C

AM

AC

p,A(¹)#C

BM

BC

p,B(¹)

#CPM

PC

p,P(¹) , (5)

ln(p1)"

CA

ln(pA(¹))#C

Bln(p

B(¹))#C

Pln(p

P(¹))

CA#C

B#C

p

,

p"Ao

o5)Bp1

, (6)

k"AoothB

CAkA(¹)#C

BkB(¹)#C

PkP(¹)

CA#C

B#C

p

, (7)

;"As

kG3!1)*5%

#

1

hB~1

. (8)

The following initial and boundary conditions areused:

t"0 ∀x ¹"¹0, g"0, (9)

x"0 ∀t'0 ¹"g(t), (10)

x"¸x

∀t'0 !kL¹Lx

"h(¹!¹0)#l (¹4!¹4

0).

(11)

It should be noted that g(t) appearing in Eq. (10) is anempirical function obtained by interpolating the temper-ature pro"le at the base where the reacting sample isignited. The equations described above are solved simul-taneously by adopting a "nite-di!erence scheme at theinternal node points and integrating the resulting systemof ordinary di!erential equations using standard rou-tines. An appropriate "nite-di!erence scheme also ap-plies to the boundary condition (11), where non-linearities are handled with the Newton}Raphsonmethod. During the computation, 300 discretizationpoints were generally kept. Higher values of the internalnode points did not cause any change in the results.


Fig. 3. Computed voltage and current outputs during FACS without (dotted lines) and with (solid lines) electric "eld interruption during wavepropagation.

4. Modeling results and discussion

As noted in Section 1, one of the aims of the presentinvestigation is to contribute to the development ofa model of the FACS technique which may be used toobtain macrokinetic parameters of the process by directcomparison between experimental data and model re-sults. The model parameters reported in Table 2, are usedfor the computations. The reaction Si#CPSiC is con-sidered as model system in this work. This choice isrelated to the results reported in Section 2 (cf. Table 1). Itwas seen that the reaction is preceded by the melting ofSi, and the system does not involve the formation ofintermediate phases, in accordance with the adoptedkinetic equation. Therefore, we took advantage of thecorresponding values reported by several studies avail-able in the literature, as indicated in Table 2. Unfortu-nately, to the best of our knowledge the values of thepre-exponential e!ective di!usivity and the activationenergy are not available in the literature and thereforethe corresponding values were chosen in the range oftypical self-propagating reactions.

In agreement with reported experimental results, thepresent model predicts that in the absence of an imposed"eld no self-sustaining combustion waves can be in-itiated, but that such waves can be activated in thepresence of a "eld above a minimum (threshold) value, i.e.7.8 V/cm. Modeling results of outputs of current andvoltage during FACS are shown in Fig. 3 (dotted lines)for the case when the electropotential equation (3) ismaintained throughout the process. It is seen that whenthe wave is initiated, the current increases rapidly andthen reaches a relatively #at level during wave propaga-tion before the voltage is turned o! when the wave

reaches the opposite side of the sample. The changes involtage correspond to those of the current, but of courseare in the opposite direction. It is worth noting that thetheoretical prediction of current and voltage outputsshown in Fig. 3 are consistent with the correspondingexperiment measurements (cf. Feng and Munir, 1995a).In addition, the calculated value of propagating frontvelocity, i.e. about 0.5 cm/s, are in a reasonable goodagreement with the corresponding experimental one, i.e.about 0.4 cm/s (cf. Feng & Munir, 1995a), obtained underthe same operating conditions summarized in Table 2.

On the other hand, when the "eld is turned o! duringwave propagation (cf. Fig. 3 ; solid line), i.e. both theelectropotential equations (3) and (4) are used witht1"2 s, the front becomes quenched during its progress

as it can be seen from Fig. 4 where the time}spaceconversion pro"les are depicted. It is clear from Fig. 4that the velocity of the wave rapidly decreases after thetime when the "eld is turned o!, until the front is stopped.

The variation of the calculated ratio dp/d

p0and the

corresponding extent of conversion as a function of thespatial distance along the pellet axis at t"2.2 s is shownin Fig. 5. This "gure is representative of the sampleportion where the combustion wave is frozen. In particu-lar, moving from the region on the right corresponding tothe reactants, the particle diameter of the solid compon-ent "rst sharply decreases in a relatively narrow region(about 1 mm thick) and then continues to decrease ata much slower rate, until the product region isapproached. It is seen that the predicted reactionfront width matches fairy well with the thickness ofRegion 3 depicted in Fig. 2, which is the sampleregion where the reaction product, i.e. b-SiC, has beenformed.


Fig. 4. Degree of conversion pro"les from modeling studies when the electric "eld is interrupted during wave propagation.

Fig. 5. Calculated degree of conversion and dp/d

p0ratio in a sample portion where the combustion wave is frozen.

Ideally, by taking advantage of the experimental in-formation obtainable through FACS quenching and re-lating it to the spatial evolution of the particle diameterof the solid reactant, it is possible to obtain reliablemodel kinetic parameters following standard "tting pro-cedures. The proposed model appears then to be able tocapture the essential features related to the microstructureevolution observed experimentally during FACS. Thus,it may represent a good starting point towards the devel-opment of a more accurate description of the phenomenataking place during electric "eld-activated self-propagat-ing reactions. It should be noted that for this type ofreactions the relevant physico-chemical phenomenacannot be uncoupled, as performed when evaluating, forexample, kinetic parameters of a classical heterogeneous

chemical reaction, and thus the development of a com-prehensive model of the type proposed here and in pre-vious studies available in the literature (cf. Feng &Munir, 1995b; Feng et al., 1998), where mass, energy, andelectropotential equations are solved simultaneously, ismandatory.

5. Concluding remarks

In the present work we devote our attention to themechanistic investigation of electric "eld-activated self-propagating reactions. These reactions have in generalreceived less attention in the chemical engineering litera-ture (cf. Luss, 1990) despite their dramatic level of


application in the "eld of inorganic and organic chem-istry as well as material science and technology (Munir& Anselmi-Tamburini, 1989; Varma et al., 1998). Afterthe re-examination of previously published experimentalresults obtained during the FACS of b-SiC, TaC, Ti

3Al

and B4C}TiB

2from elemental powders, we devote

our attention to provide a contribution towards thedevelopment of a reliable model of FACS quenchingtechnique which might be used to quantitatively describethe evolution of either the reaction front or the productmicrostructure. The main idea is to couple the proposedmodel with a physico-chemical description of appropri-ate mechanism of structure formation. The "nal goal willbe the possibility to predict the microstructure of "nalproducts so as to synthesize materials with tailored prop-erties using the FACS method, whose advantages overconventional techniques are reported in the literature(cf. Feng & Munir, 1995a; Xue & Munir, 1996a,b, 1997;Gedevanishvili & Munir, 1998; Orru et al., 1999a).

Notation

a moles of melting reactant, A per mole of non-melting reactant, B in the initial mixture

b moles of inert diluent, P per mole of nonmeltingreactant in the initial mixture

C concentration, mol/m3

Cp

heat capacity, J/mol/Kdp

average particle diameter of the solid reactant,B, m

D pre-exponential e!ective di!usivity of themelting reactant, A, m2 s~1

E activation energy, J/molg(t) function appearing in Eq. (10), Kh natural convection heat transfer coe$cient,

W/m2/Kk thermal conductivity, W/m/K¸x

sample length, m¸y

sample height, mM molecular weight, g mol~1

R universal gas constant, J/mol/KR

0external circuit resistance, )

s graphite electrode thickness, mt time, st1

time at which the "eld is interrupted, s¹ temperature, K¹

0initial temperature, K

< voltage, Vw sample width, mx longitudinal coordinate, m

Greek letters

b ratio between the thickness of product and theinitial particle diameter of the nonmelting reac-tant

!*H reaction heat, J/molg extent of conversion of the nonmelting reactanto density, g/m3

p electrical conductivity, )~1/ml Stefan}Boltzmann constant, J/s/m2/K4

Superscripts and subscripts

A melting reactantB nonmelting reactantin applied valueP reaction productth theoretical0 value at t"0

Acknowledgements

This work has been performed in the framework ofa cooperation agreement between the University of Cali-fornia, Davis, USA and the Universita degli Studi diCagliari, Italy. Support by the (US) National ScienceFoundation (ZAM) is gratefully acknowledged.

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