A R C H I V E S O F M E T A L L U R G Y A N M A T E R I A L S Volume 56 20 Issue 2 OI: 0.2478/v072-0-0035-7 W. WOŁCZYŃSKI, T. OKANE, C. SENEROWSKI, B. KANIA,. ZASAA, J. JANCZAK-RUSCH META-STABLE CONITIONS OF IFFUSION BRAZING METASTABILNE WARUNKI SPAJANIA YFUZYJNEGO A thermodynamc justfcaton for the jont formaton s developed on the bass of the dffuson brazng of the N/Al/N system. The phenomena of dssoluton and soldfcaton were ncluded nto the descrpton. The frst sold/sold transformaton s also dscussed. Manly, a descrpton for the sothermal brazng occurrence n the meta-stable condtons s developed. It nvolves the applcaton of the crteron of hgher temperature of the sold / lqud (s/l) nterface. The dssoluton of the fller metal n the substrate s descrbed by the N 0 solute concentraton wthn the dssoluton zone (lqud flm) dstngushed at the substrate surface. The selecton of the N 0 parameter by the dssoluton s justfed by the Thermocalc calculaton of the N-Al phase dagram for meta-stable equlbrum. Accordng to the model assumptons, the soldfcaton s accompaned by parttonng or by undercooled pertectc reacton resultng n formaton of the ntermetallc phase. The average Al solute concentraton measured across some Al 3 N 2 /Al 3 N/Al 3 N 2 jonts confrms that the N 0 - solute concentraton s conserved wthn the analyzed jont sub-layers. The N-Al phase dagram for meta-stable equlbrum referred to the soldfcaton s also calculated by means of the Thermocalc Software. It allows locatng the soldfcaton path, s/l nterface path and redstrbuton path onto the mentoned dagram. Superposton of both calculated phase dagrams s also gven to show that the jont formaton occurs cyclcally under the meta-stable condtons. Keywords: meta-stable equlbrum, jont formaton model, dssoluton, soldfcaton Zaproponowano termodynamczne uzasadnene dla kształtowana sę złącza na przykładze spajana dyfuzyjnego systemu N/Al/N. o proponowanego opsu włączono zjawska rozpuszczana krystalzacj. Perwsza przemana w stane stałym równeż poddana jest dyskusj. Główne, ops dotyczy przebegu spajana dyfuzyjnego w warunkach metastablnych. Wymaga on zastosowana kryterum wyższej temperatury frontu krystalzacj. Rozpuszczane składnka łączącego (Al) w podłożu (N) jest opsane stężenem N 0 w strefe rozpuszczana (w cekłej błonce wyróżnonej tuż przy powerzchn podłoża). Selekcja parametru N 0 jest uzasadnona oblczenam z użycem program Thermocalc Software Programem tym wyznaczono dagram fazowy N-Al dla równowag metastablnej odnesony do zjawska rozpuszczana. Zgodne z założenam proponowanego opsu krystalzacj towarzyszą rozdzał składnka lub przechłodzona reakcja perytektyczna skutkujące formowanem sę fazy mędzymetalcznej. Średne stężene Al zmerzone w złączach Al 3 N 2 /Al 3 N/Al 3 N 2 potwerdza, że stężene N 0 jest zachowane w podwarstwach analzowanych złącz. Równeż, wyznaczono dagram fazowy N-Al dla równowag metastablnej odnesony do krystalzacj. Wykorzystano ponowne program Thermocalc Software. Pozwolło to na zlokalzowane śceżk krystalzacj, śceżk frontów krystalzacj, śceżk redystrybucj na dagrame fazowym równowag metastablnej. Superpozycja obydwu oblczonych dagramów jest także pokazana celem zlustrowana, ż formowane sę złącza zachodz cyklczne w obserwowanych warunkach metastablnych. Nomenclature dx k nfntesmal amount of the substrate melted due to the dssoluton of the lqud Al solute, [mole fr.], ndex of the range of soldfcaton; = for the range N 0 N ; = 2 for the range N N 2 ; = 3 for the range N 2 N F, (phase dagram for stable equlbrum), Fgure 3, partton rato varyng wth the concentraton of solute n the lqud, =,..., n, [mole fr./mole fr.], INSTITUTE OF METALLURGY AN MATERIALS SCIENCE, POLISH ACAEMY OF SCIENCES, 30-059 KRAKÓW, 25 REYMONTA STR., POLAN AIST NATIONAL INSTITUTE OF AVANCE INUSTRIAL SCIENCE & TECHNOLOGY, UMEZONO --, 305 8568 TSUKUBA, JAPAN MILITARY UNIVERSITY OF TECHNOLOGY, KALISKIEGO 2, 00-908 WARSZAWA, POLAN AGH UNIVERSITY OF SCIENCE & TECHNOLOGY, FACULTY OF PHYSICS AN AVANCE INFORMATICS, 30-059 KRAKÓW, 9 REYMONTA STR., POLAN EMPA SWISS FEERAL LABORATORIES FOR MATERIALS SCIENCE & TECHNOLOGY, UEBERLAN 29, 86 000 UEBENORF, SWITZERLAN
32 k 0 k L L 0 l 0 N B N F N L N S N 0 ˆN 0 N N 2 N M 2 N N ( ) N s frst component of the partton rato ndependent of any changes n the solute concentraton along lqudus lne, =,..., n, [mole fr./mole fr.], second component of the partton rato ndependent of any changes n the solute concentraton along the lqudus lne, =,..., n, [mole fr./mole fr.], amount of the lqud at the start of soldfcaton, usually L 0 =, [dmensonless], amount of the lqud at the begnnng of a gven range of soldfcaton,, =,..., n, Eq. (A), [mole fr.], solute redstrbuton after both parttonng and back-dffuson, =,..., n, Eq. (5P), Eq. (5C), [mole fr.], concentraton of the fller metal n ts lqud soluton at the T T temperature, [mole fr.], solute concentraton n the lqud, observed along the lqudus lne for a gven range of soldfcaton, (local soldfcaton path), =,..., n, Eq. (), Eq. (P-3P), Eq. (C-3C), [mole fr.], solute concentraton at the sold / lqud nterface, observed along the soldus lne for a gven range of soldfcaton, (local sold/lqud nterface path), =,..., n, Eq. (4P), Eq. (4C), [mole fr.], solute content at the begnnng of soldfcaton path (on lqudus lne), [mole fr.], average solute concentraton measured across the formed mult-layer, [mole fr.], solute concentraton n the lqud at the frst pertectc reacton (phase dagram for stable equlbrum), Fgure 3, [mole fr.], solute concentraton n the lqud at the second pertectc reacton (phase dagram for stable equlbrum), Fgure 3, [mole fr.], solute concentraton n the lqud at the pertectc reacton (phase dagram for the meta-stable equlbrum), Fgure 9, [mole fr.], solute concentraton n the lqud at a gven pertectc reacton (phase dagram for stable equlbrum), Fgure 3, Eq. (2), =, 2, [mole fr.], solute concentraton n the lqud at a gven pertectc reacton, (for = 2, 3) or solute concentraton at the begnnng of soldfcaton path (for = ), (phase dagram for stable equlbrum), Eq. (), Fgure 3, [mole fr.], solute concentraton wthn the zone s, (Fgure 9), [mole fr.], N ss solute concentraton wthn the zone ss, (Fgure 9), [mole fr.], N mn solute concentraton at the mnmum of lqudus lne, phase dagram for meta-stable equlbrum, Fgure 8, Fgure 9, [mole fr.], n number of ranges of soldfcaton for a gven soldfcaton path, n = 3, s ss T f T T 32 T T R T T t K t F t t M t 3 B zone of saturaton created n the substrate, zone of super-saturaton created n the substrate, temperature at the sold / lqud nterface of a gven phase / compound, f, f AlN, Al 3 N 2, Al 3 N (n short, f, 32, 3)) as t results from the ntersecton of a gven meta-stable soldus lne wth the lne whch denotes N 0 concentraton, [K], temperature of the AlN phase sold /lqud nterface, Fgure 7, [K], temperature of the Al 3 N 2 phase sold /lqud nterface, Fgure 7, [K], temperature at whch the mnmum of lqudus lne appears n the phase dagram for meta-stable equlbrum, Fgure 8, [K], real temperature of bondng (temperature of the nterconnecton formaton), [K], meltng temperature of the Al fller metal, [K], tme of the soldfcaton progress, Fgure 5a, [s], tme at whch soldfcaton s completed, Fgure 5b, Fgure 3, [s], tme, [s], tme of the begnnng of the frst sold / sold transformaton, Fgure 3, [s], tme of the brth of the Al 3 N phase wthn the N/Al/N nterconnecton, Fgure 3, [s],
33 X 0 x x x 0 x max x mn x max α β ex β n λ F λ F 2 λ K λ K 2 x mn amount of the sold at a gven stage (progress) of soldfcaton when the process s arrested and morphology frozen, (calculated for a total system along a global soldfcaton path) [dmensonless], current amount of the sold, [dmensonless], amount of the prmary phase whch takes part n a gven pertectc reacton at the th range of soldfcaton, =,..., n, [dmensonless], amount of the sold at a gven stage (progress) of soldfcaton when the process s arrested and morphology frozen, (calculated for a gven th range of soldfcaton along a local soldfcaton path), [dmensonless], sum of amounts of both prmary phase remanng after pertectc reacton and pertectc phase resultng from the pertectc reacton at a gven soldfcaton range, =,..., n, [dmensonless], amount of the prmary phase remanng after the pertectc reacton at a gven range of soldfcaton =,..., n, [dmensonless]; denotes the amount of pertectc reacton product obtaned at a gven range of soldfcaton, =,..., n,, [dmensonless]; a gven pertectc reacton occurrng at the N solute concentraton (as observed on lqudus lne) s treated as belongng to the th range of soldfcaton, (stable equlbrum phase dagram), back-dffuson parameter for both parttonng and redstrbuton, =,..., n, [dmensonless], redstrbuton extent coeffcent, =,..., n, defned n detals by Wołczyńsk et al. [3], [dmensonless], redstrbuton ntensty coeffcent, =,..., n, defned n detals by Wołczyńsk et al. [3], [dmensonless], thckness of the Al 3 N 2 phase sub-layer wthn the nterconnecton formed at tme t F, Fgure 5, [µm], half the thckness of the Al 3 N phase sub-layer wthn the nterconnecton formed at tme t F, Fgure 5, [µm], thckness of the Al 3 N 2 phase sub-layer wthn the nterconnecton formed at tme t K, Fgure 5, [µm], half the thckness of the Al 3 N phase sub-layer wthn the nterconnecton formed at tme t K, Fgure 5, [µm].. Introducton A soldfcaton, a dssoluton and some sold/sold transformatons are predcted by Tuah-Poku, et al., [] as phenomena to be observed durng dffuson brazng. The soldfcaton and dssoluton are subjected to consderaton n the current model. However, exceptonally the frst sold / sold transformaton s strctly connected wth soldfcaton and dssoluton, through the mass balance, Kloch et al., [2]. A dssoluton of substrate by the lqud fller metal occurs contnuously wthn a zone dx Fgure. The dssoluton prepares the lqud flm wthn the zone dx. Once the zone becomes lqud the mass of ths lqud flm dffuses towards the jont symmetry axs and soldfcaton occurs at the s/l nterfaces, Wołczyńsk et al., [3]. Soldfcaton s accompaned by the undercooled pertectc reactons, Fgure 2. Accordng to the concept of Chuang et al., [4], undercooled pertectc reactons take place at the sold/lqud (s/l) nterface. The occurrng of pertectc reactons at the s/l nterface of cells, formed wthn a gven sub-layer, are marked schematcally wth some arrows n Fgure 2. Manly, the nternal channels wthn cells are employed for the flow of dssolved substrate towards the s/l nterfaces (bulk dffuson). The dssolved substrate contaned wthn zone dx, has the solute concentraton equal to N 0, Fgure. Under descrbed condtons, a bondng process can result n a jont that reproduces mcro-structurally a sequence of nter-metallc compounds or phases as t s vsble n an adequate phase dagram for stable equlbrum, Jacobson and Humpston [5], Jangwe et al., [6] and Wołczyńsk et al., [7].
34 Fg.. N-substrate dssoluton wthn ts nfntesmally small amount, dx (the so-called zone dx stuated just at the substrate surface) due to dffuson of lqud soluton of the fller metal, N F, through the channels stuated among some cells of Al 3 N 2 or Al 3 N phases formed n a gven sub-layer, respectvely; a/ zone dx s just formed, b/ begnnng of the zone dx formaton; a reacton wthn the zone dx s: lqud(n F ) + N undercooledlqud(n 0 ) Fg. 2. The undercooled pertectc reactons at the front of cells (as t results from the phase dagram for stable equlbrum): N solute concentraton wthn undercooled lqud for the frst pertectc reacton resultng n formaton of the sub-layer that contans cells of the Al 3 N 2 ntermetallc phase, N 2 solute concentraton wthn undercooled lqud for the second pertectc reacton resultng n formaton of the sub-layer that contans cells of the Al 3 N ntermetallc compound; a/ the channels predcted n the model for bulk dffuson are shown schematcally and completed wth some arrows that sgnfy occurrence of a gven pertectc reacton; b/ channels observed at the cross-secton of the sub-layer of the δ phase wthn the Fe/Zn/Fe system, [8] The analyss s focused on the formaton of the N/Al/N jonts n relaton to the dffuson brazng. It makes t possble to develop a descrpton for the soldfcaton preceded by the dssoluton and a new explanaton for the role of the soldfcaton path for the creaton of the jont sub-layers. Addtonally, a thermodynamc justfcaton for the begnnng of soldfcaton path, N 0 and the end of soldfcaton path, N F, both are gven. The Thermocalc Software calculaton of the N-Al phase dagram for meta-stable equlbrum wll gve the grounds for ths justfcaton. 2. escrpton of the NAl 3 N 2 /Al 3 N/Al 3 N 2 -N jont formaton The system selects one and only one nomnal concentraton of the solute, N 0 at a gven temperature, T R, Kloch et al., [2]. The selecton of the N 0 parameter by the system occurs n such a way to ensure the flm, dx, (zone dx), to be lqud. The zone dx s created at the surface of a substrate, Fgure. It s evdent that the lqud layer, dx, s strongly undercooled from ts lqudus temperature. Thus, dssoluton occurs at a gven temperature, T R, as long as t s necessary to transform sold substrate
35 zone, dx, nto the lqud flm, dx. Once the dx zone becomes lqud and ts concentraton acheves selected N 0, the undercooled lqud flm s ready to be subjected to soldfcaton nvolvng undercooled meta-stable pertectc reactons, Fgure 2. The average Al-solute content measured wthn the N/Al/N jont obtaned after dfferent perods of tme s always equal to the N 0 concentraton. Thus, the N 0 parameter s stuated on the lqudus lne as the begnnng of soldfcaton path, Fgure 3. Accordng to the current model assumpton, the constant parttonng s justfed for the formaton of an ntermetallc phase. In the case of an nter-metallc compound formaton a varyng partton rato s to be appled. Thus, the unversal defnton of the k partton rato was ntroduced: k (x) = k 0 +kl N /N L (x) =,..., n () As [ the assumed ] pertectc reacton x +lqud(n ) x max x mn occurs durng soldfcaton, therefore, the amount of the prmary phase s to be defned accordngly. x (α, l0, N, N, k ) = [( = l 0 ( ) N ( α / k ) / ) / N (k ) ( α k ) ] (2) Fg. 3. N-Al phase dagram for stable equlbrum wth the N 0 N N 2 N F soldfcaton path and wth the k N 0 k N k 2 N k 2 N 2 k 3 (N 2 )N 2 k 3 (N F )N F hstorcal, sold / lqud nterface path. T R = 700 C, and N 0 = 0.64 [mole fr.] Eq. () s unversal one and can be appled to the descrptons of pertectc reacton when: a/ an ntermetallc phase s formed, then k L = 0, b/ an ntermetallc compound s formed, then k 0 = 0. The x max and x mn parameters are defned n detal by Wołczyńsk et al. [3]. Fnally, the model allows determnng of the rato of sub-layers thckness: λ K /λk 2 = xmax /x max 2 +x 3, Fgure 4, wth some explanatons shown n Fgure 5. Fg. 4. Reproducton of the sub-layers thckness rato together wth the measured solute redstrbuton profle resultng from the undercooled pertectc reactons. Theoretcal profles are reproduced across the half a jont N / Al 3 N 2 / Al 3 N / Al 3 N 2 / N; the rato of sub-layers s reproduced for the soldfcaton path N 0 N N 2 N F shown n Fgure 3; the measurement ponts are taken from the EAX analyss; the soldfcaton was carred out at 700 0 C for 2 s
36 Fg. 5. The meanng of some parameters n soldfcaton of the Al 3 N 2 /Al 3 N multlayer on the N-substrate shown at: a/ a gven step of the Al 3 N 2 /Al 3 N mult-layer formaton, b/ completon of the Al 3 N 2 /Al 3 N multlayer formaton The smulaton shown n Fgure 4 s obtaned as f the formaton of jont occurs accordng to the phase dagram for the stable equlbrum, Fgure 3. Some equatons appled to the smulaton are shown n detals n the Appendx, Eqs. (P-5P) and Eqs. (G-5G). Addtonally, Eq. (2) was used to calculate x, x 2 and x 3 parameters. Accordng to the appled phase dagram for stable equlbrum two pertectc reactons were calculated durng smulaton: x +lqud(n ) [ ] x max and x 2 +lqud(n 2 ) [ ] x2 max x mn 2. The result of smulaton shown n Fgure 4 should be the same as the result of smulaton based on the phase dagram for meta-stable equlbrum. However, the phase dagram for meta-stable equlbrum should be much more realstc to descrbe perfectly the phenomena whch occur durng the formaton of jont. Accordng to the current model, the solute concentraton wthn a gven sub-layer s the same as the solute concentraton at the pertectc pont n the phase dagram. Thus, the Al solute concentraton equals 60 at.%, for the Al 3 N 2 ntermetallc phase and 75 at.% for the Al 3 N ntermetallc compound. However, t s to be verfed whether the rato of sub-layers thckness depends on the real temperature, T R, at whch the experment was performed. 3. Experment x mn A set of experments was carred out n order to study a sequence of the ntermetallc sub-layers formaton. The experment of the N/Al/N dffuson jont formaton was performed at dfferent temperatures: 700, 750, 800, 850, 900, 950 and 000 0 C. An Al-fol 25 µm thck was located between two peces of nckel then tghtened. The N/Al/N system was put nto a vacuum furnace, then heated untl a gven temperature. At each of the mposed temperature the Al-fol was melted and the system subjected to dssoluton / soldfcaton. The soldfcaton was halted after a gven perod of tme and morphology of the nterconnectons was frozen, Fgure 6. The Al-solute segregaton profles were analyzed across the frozen nterconnectons by means of the EAX mcro-analyzer. Next, an average content of the Al-solute, ˆN 0, was determned for each of the studed jont. The estmated results are gathered n Table. TABLE Estmated values of the Al solute content, ˆN 0 wthn the N/Al/N jonts formed at dfferent temperature T R, due to soldfcaton arrested at tme, t, wth t < t F. T R [ 0 C] ˆN 0 [at. %] 700 63.5 750 63.6 800 65.7 850 66.5 900 64.5 950 63.7 000 62.5 Snce the values of the ˆN 0 parameter (Table ) are smlar to each other, the average Al solute content, denoted as N 0, and representatve for all nvestgated jonts was calculated. It s equal to N 0 0.64 [mole fr.], ( N 0 (/7) ˆN 0 ). The revealed nterconnectons morphology proved that the rato of sub-layers thckness does not change wth the T R temperature mposed n a gven experment. It should be emphaszed that accordng to the mass balance the average solute content, N 0 revealed n the frozen nterconnectons should be equal to the ntal solute content, N 0, from whch the nterconnectons were formed.
37 The experments allowed the dentfcaton of the ntermetallc compound / phase sub-layers, Fgure 6. The areas whch contans the N F soluton are vsble n Fgure 6a. The pure Al - areas resultng from the so-called mants phenomenon revealed by Wołczyńsk et al. [7], (full consumpton of the Al 3 N compound by the Al 3 N 2 domnant phase) and descrbed by the adequate reacton: 2Al 3 N Al 3 N 2 +3Al, are also vsble, Fgure 6b. 4. Meta-stable condton of the jonng technology N 0 parameter localzaton on the lqudus lne, as shown n Fgure 3, ndcates that the AlN phase should be formed as the frst sub-layer just at the substrate surface. Meanwhle, expermental observatons confrm that the Al 3 N 2 phase sub-layer s formed, and AlN phase does not appear durng experment, Fgure 6. It s evdent that the soldfcaton occurred under meta-stable condtons snce the AlN phase was not formed. The transton of the system from stable condtons nto meta-stable condtons can be justfed by means of the Umeda-Okane-Kurz crteron, (UOK crteron), Umeda et al. [9]. The schematcally drawn N-Al phase dagram for stable equlbrum s shown n Fgure 7. Fg. 7. Applcaton of the UOK crteron to the N-Al phase dagram for stable equlbrum; two meta-stable soldus lnes are added: frst for AlN phase and second for Al 3 N 2 phase to explan a competton between the phases appearance Fg. 6. a.) The N / Al 3 N 2 / Al 3 N / Al 3 N 2 / N N/Al/N jont morphology, (T R = 800 C), as observed at tme t < t F ; the ˆN 0 parameter estmated; b.) The N / Al 3 N 2 / Al 3 N / Al 3 N 2 / N N/Al/N jont morphology, (T R = 950 C), as observed at tme t > t M ; the ˆN 0 parameter estmated The characterstc N 0 = N 0 solute content s also stuated n the phase dagram. The ntersectons of the N 0 content wth both meta-stable soldus lnes defnes T j temperatures of the s/l nterface for both dscussed phases ( j = AlN, Al 3 N 2 ). Accordng to the mentoned crteron ths phase s formed whch manfests a hgher temperature of ts s/l nterface durng soldfcaton. In the case of the N-Al system T32 > T as vsble and wrtten n Fgure 7. Therefore, the Al 3 N 2 phase sub-layer s selected by the system nstead of expected AlN phase sub-layer formaton.
38 Addtonally, some calculatons made by means of the Thermocalc Software allowed some N-Al phase dagrams for meta-stable equlbrum to be reproduced, respectvely: a/ for a dssoluton, Fgure 8a, b/ for a soldfcaton resultng n formaton of both observed phases, Fgure 8b. The mnmum shown n Fgure 8, defnes the mnmal temperature, T. At the T temperature the dssoluton (descrbed n Fgure ) occurs as easly as possble. In other words, the appearance of mnmum suggests that even at the T temperature the lqud soluton can exst wthn the zone dx, Fgure, under mposed meta-stable condton of the N/Al/N nterconnecton formaton. The phase dagram for dssoluton s superposed onto the phase dagram for soldfcaton n order to emphasze that both dscussed phenomena occur cyclcally, dssoluton: N F N 0, and soldfcaton: N 0 N F, Fgure 9. The dssoluton path, N F N 0 begns at the lqudus lne of the meta-stable equlbrum phase dagram calculated for soldfcaton and s completed at the lqudus lne mnmum of the meta-stable equlbrum phase dagram calculated for dssoluton (dashed green lne), Fgure 9. Fg. 9. Superposton of the N - Al phase dagrams for meta-stable equlbrum calculated: a/ for the dssoluton phenomenon dscussed n Fgure 8a, and b/ for soldfcaton resultng n the N/Al 3 N 2 /Al 3 N/Al 3 N 2 /N jont formaton, Fgure 8b Fg. 8. N-Al phase dagram for meta-stable equlbrum calculated for: a/ the solublty of Al (fller metal appled n the technology) n the N substrate, b/ for soldfcaton under meta-stable condton The calculated phase dagram (meta-stable equlbrum) for the dssoluton, Fgure 8, manfests the mnmum at the Al-solute concentraton N m n 0.64 [mole fr.]. It s evdent that the N m n parameter value has the same meanng as the N 0 parameter value that represents dssoluton phenomenon n the model descrbed by the scheme shown n Fgure. The soldfcaton path N 0 N F, s stuated on the lqudus lne of the meta-stable equlbrum phase dagram calculated for the soldfcaton, Fgure 9. The N F parameter shown n Fgure, Fgure 3 and Fgure 6a was measured by means of the EAX technque, Fgure 0a, (Nexp F.) and next, t was determned theoretcally, Fgure 0b, (N F theor. ). Addtonally, solute concentraton n the zones ss and s s determned for a gven T R sotherm. The N F theor. value was determned as the ntersecton of the T T sotherm and lqudus lne of the phase dagram for meta-stable equlbrum, Fgure 0b.
39 The ss pont results from the ntersecton of the T R sotherm and soldus lne whch s drawn n the meta-stable equlbrum phase dagram calculated for dssoluton, whle s pont s determned by soldus lne of the meta-stable equlbrum phase dagram for soldfcaton. Fg.. ES observaton of the full consumpton of the Al 3 N coupled phase by the Al 3 N 2 domnant phase accordng to reacton: 2Al 3 N Al 3 N 2 + 3Al; the pure Al precptates dffuse towards the jont axs of symmetry Fg. 0. Method of determnaton of the N F parameter: a/ expermentally wth N/Al/N-700 nterconnecton for whch soldfcaton was arrested and morphology frozen, b/ due to the Al-N meta-stable equlbrum phase dagram, (T T s the Al meltng pont, N F s the soluton of the N n the Al fller metal, as shown n Fgure ) 5. rft of the system towards the thermodynamc equlbrum The analyzed N/Al 3 N 2 /Al 3 N/Al 3 N 2 /N jont was obtaned under meta-stable condton. Ths s a natural tendency of ths knd of systems to acheve a thermodynamc equlbrum. The equlbrum would be reached when the analyzed N/Al 3 N 2 /Al 3 N/Al 3 N 2 /N jont s transformed nto N/(N)/N jont. Therefore, some sold / sold transformatons are needed. The frst transformaton n sequence was observed n the studed jont, Fgure. Fg. 2. Half the dffuson jont N/Al 3 N 2 /Al 3 N/Al 3 N 2 /N; a/ just at the completon of soldfcaton, t S t F ; zone dx dsappeared; b/ durng the frst sold / sold transformaton; N 0 parameter s the same for both analyzed phenomena
320 The mass balance measured at dfferent soldfcaton tmes confrms that the N 0 solute concentraton s conserved durng all soldfcaton tme, Fgure 6. The characterstc N 0 solute concentraton s shown schematcally for the completed soldfcaton, Fgure 2a. However, t should be emphaszed that the mentoned N 0 solute concentraton s also conserved durng the so-called mants phenomenon, Fgure 2b. It s evdent that the system became closed and no exchange of the mass between substrate and jont s possble durng ths frst sold / sold transformaton. The cells of both Al 3 N 2 and Al 3 N phases are always vsble nsde the proper sub-layers. The cellular morphology of cells shown schematcally n Fgure 2 were confrmed expermentally due to the SEM analyss appled to the jont frozen durng arrested soldfcaton [0]. the phenomenon of parttonng along the soldfcaton path N 0 N2 M, (t s marked n Fgure 3), and dffuson nto the sold; a smulaton of the Al 3 N 2 ntermetallc phase sub-layer s shown n Fgure 4, 6. Concludng remarks / regardless of the real temperature, T R, mposed n the technology, (Table ), the begnnng of soldfcaton path s always the same, and s equal to N 0, 2/ regardless of the real temperature, T R, the end of soldfcaton path s always the same and s equal to N F, whch was confrmed expermentally, Fgure 0a, and predcted theoretcally, Fgure 0b; a good agreement was obtaned between both N F (exp.) and N F (theor.) parameters, 3/ the N 0 solute concentraton s created wthn the zone dx as a result of dssoluton, Fgure -2, 4/ the N 0 solute concentraton s reproduced subsequently wthn the jont as an average solute concentraton, Table, 5/ the N 0 solute concentraton created wthn the zone dx corresponds well to the N mn solute concentraton whch s manfested by the meta-stable equlbrum phase dagram for dssoluton, Fgure 8, N 0 N mn, 6/ the N 0 solute concentraton plays essental role n the competton between stable and meta-stable formaton of the nterconnecton, Fgure 7, 7/ the N 0 solute concentraton located on the lqudus lne, Fgure 3, s treated as an ntal condton for the dfferental equaton whch descrbes the soldfcaton, 8/ accordng to the current model assumpton, creaton of the zone dx, follows the reacton: N F + N s(n s ) + ss(n s s) + dx(n 0 ), as drawn n Fgure 3, 9/ accordng to the meta-stable equlbrum phase dagram, Fgure 9, the frst Al 3 N 2 ntermetallc phase sub-layer (n sequence of appearance) s formed due to Fg. 3. Localzaton of the zone dx, zone ss and zone s wthn the N substrate 0/ accordng to the same phase dagram, Fgure 9, the second Al 3 N ntermetallc compound sub-layer s formed due to the reacton between the N2 M lqud and Al 3 N 2 prmary phase, (x max x mn n Fgure 4) next due to the parttonng along the soldfcaton path N2 M N F, (x 2 x max n Fgure 4) wth dffuson nto the sold, (t s marked n Fgure 3), / localzatons of the N2 M lqud and Al 3 N 2 prmary phase n the growng jont suggest that pertectc reacton (not pertectc transformaton) takes place to form Al 3 N compound at the N2 M pont, Fgure 9, (accordng to the descrpton by Phelan et al. []), 2/ the Gbbs Phase Rule appled to the current model and expermental observatons yelds the Number of egrees of Freedom l = 0, as requred for the sothermal soldfcaton mposed to the jont technology: l = c p + ; thus l = c p + = 0, wth c N, Al, therefore, p = 3, 3/ n fact, p lqud(n 0 ) d x, Al 3 N 2, Al 3 N = 3, as t s justfed by the current model and adequate experments, Fgure 6, Fgure 0a, 4/ the model s unversal one and can be appled to other stuatons where the brazng s expected; t s suggested that brazng occurs n the case of massve rolls soldfcaton, when the sold core nteracts on the surroundng lqud steel added n the second step of the technology.
32 Fg. 4. Smulaton of both sub-layers formaton; x amount of the Al 3 N 2 prmary phase formed due to soldfcaton path N 0 N M 2 shown n Fgure 9; x max x mn amount of the Al 3 N compound formed due to the undercooled pertectc reacton at N M 2 pont on soldfcaton path: (x x mn )[k N M 2 ] + (xmax x )[N M 2 ] (xmax x mn )[k 2 N M 2 ]; x 2 x max amount of Al 3 N compound formed due to the parttonng along the soldfcaton path N M 2 N F shown n Fgure 9, Eq. (5C); N F remanng last part of the lqud whch returns back to the process of dssoluton; N theoretcal solute redstrbuton wthn the Al 3 N 2 prmary phase, Eq. (6P); k 2 N M 2 solute redstrbuton wthn the Al 3 N compound; measurement pont are marked by red squares Acknowledgements The fnancal support from the Polsh Mnstry of Scence and Hgher Educaton (MNSW Poland) under the contract: N R5 006 004 s gratefully acknowledged. Appendx The followng mathematcal formalsm was appled to the current model: / formaton of the ntermetallc phase, k L = 0 The solute concentraton n the lqud, changes durng precptaton of prmary phases along a gven local soldfcaton path,, n accordance wth the relatonshp: N L ( x;α, l 0, N ), k 0 = N l0 +α k0 x x l 0 k 0 α k 0 (P) The Eq. (P) s the soluton to dfferental equaton for soldfcaton/mcro-segregaton: dn L /dx = ( k0 )N L (x)/(l0 + α k 0 x x) wth the ntal condton: N L (0; α, l 0, N ( ), k 0 ) = N ( ) (2P) (3P) The phenomenon of parttonng results n the formaton of the sold / lqud nterfaces, and hstory of the solute concentraton at the exsted nterfaces are gven as: N S (x; α, l 0, N ( ), k 0 ) = k0 N L (x; α, l 0, N ( ), k 0 ) (4P) The phenomenon of back-dffuson superposes solute parttonng. The superposton of both phenomena results n the solute redstrbuton, whch s the functon of the solute concentraton at formerly (hstorcally) exstng sold / lqud nterfaces. N B (x; x 0, α, l 0, N ( ), k 0) = [ + β ex (x; x 0, l0, k0 )βn (x 0; α, l 0, k )] N S (x; α, l 0, N ( ), k 0) The pertectc reacton requres the so-called shfted redstrbuton formed wthn the prmary phase: N (x; x 0, α, l 0, N ( ), k 0) = [N B (x + x x mn ; α, l 0, N ( ), k 0)] (6P) 2/ general descrpton of the ntermetallc phase / compound formaton, wth k 0 0; k L 0
322 The solute concentraton n the lqud, changes durng precptaton of prmary phases along the gven local soldfcaton path,, n accordance wth the relatonshp: B = ( N L (x) N M 2 kl ) α (bc) N L(x;α, l0, N, k ) =[N /( k 0 )] k0 kl +( k0 kl )[(l0 +α k0 x x)/l0 ] α k 0 (G) C = N L (x) [ ( α ( 2 F l.l; α + 2; N L (x) ( N M 2 kl ) )) ] + α + l (c3c) Eq. (C) s the soluton to dfferental equaton for soldfcaton/mcro-segregaton: dn L /dx = [( k0 )N L (x) kl N ( )]/(l 0 + α k 0 x x) (2G) wth the ntal condton: N L (0; α, l 0, N, k 0, kl ) = N (3G) The hstory of solute concentraton, at the appearng nterfaces s descrbed as follows: N S (x; α, l 0, N, k ) = k 0 N L (x; α, l 0, N, k ) + k L N (4G) Consequently, the solute redstrbuton after parttonng and back-dffuson s gven by: N B(x; x0,α, l0, N, k ) = [ +β ex (x; x 0, l0, k )β n (x 0,α, l0, k ) ] N S(x;α, l0, N, k ) (5G) 3/ formaton of the ntermetallc compound, k 0 = 0 The general formulaton of the dfferental equaton s: dn L/dx =[( k0 )N L k LN ] /(l 0 + α (k 0 + k L[N (0C) /N L (x)]x x) Its smplfed form approprate to the formaton of the ntermetallc phase (wth k L = 0) s presented by Eq. (P). However, ts smplfed form adequate to the formaton of the ntermetallc compound can be obtaned by ntroducton of the k 0 = 0 nto the Eq. (0C). Then, the followng dfferental formula s to be obtaned: dn L /dx = [N L k L N ]/(l 0 + α (k L [N /N L (x)]x x) (C) whch s to be solved wth the ntal condton: N L (0; α, l 0, N, k L ) = N (2C) to obtan the functon N L (x) whch can be calculated numercally for a gven value of the x: where x = l 0 {AB + C} [M] (3C) M = ( N L (x) N M 2 kl ) ( α + l ) (d3c) The hstory of solute concentraton, at the appearng nterfaces s descrbed as follows: N S ( x; α, l 0, N, k L ) = k L N L ( x; α, l 0, N, k L ) (4C) The solute redstrbuton of the formed ntermetallc compound s equal to solute concentraton at the appearng sold / lqud nterfaces: ( N B x; x 0, α, l 0, N ) (, k L N S x; α, l 0, N ), k L (5C) There s no need to calculate the amount of pertectc reacton because the ntermetallc compound s formed due to the parttonng phenomenon, only. All the above relatonshps Eq. (P-6P), Eq. (G-5G) and Eq. (C-5C) are formulated wth the use of: l 0 = x 0 = L 0, = ; L 0 = j= xmax j, = 2,..., n; X 0, = ; X 0 = j= xmax j, = 2,..., n; (A) (2A) Addtonally, the followng defnton of the coeffcent of the redstrbuton extent was formulated; β ex (x; x 0, l0, k ) s the soluton to the followng equaton: ( + β ex )N S (x;, l0, N, k ) = N S (x0 ;, l0, N, k ) (3A) Moreover, the defnton of the coeffcent of the redstrbuton ntensty was also formulated; β n (x 0; α, l 0, k ) s the soluton to the followng equaton: ) α A = N α + [ 0 N L (x) ] α ( N0 N ( 2 MkL ( ) ) ] (a3c) [α 2F l, l; α + 2; N 0 N M 2 k L α x 0 [ 0 +β n β ex (x; x 0, l0, k ) ] N S(x;α, l0, k )dx = l 0 N (l 0 x0 )NL (x0 ;α, l0, N, k ) (4A)
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