THE continued growth in performance and functionality

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1 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. Therml Contct Resistnce: Effect of Elstic Deformtion MjidBhrmi,M.MichelYovnovich,ndJ.RichrdCulhm Abstrct Existing models over-predict the therml contct resistnce of conforming rough joints t low contct pressures. However, the pplicble pressure rnge in the microelectronics industry is low due to lod constrints. In this pper new model is presented which is more suitble for low pressures. The ssumes plstic deformtion t microcontcts. The effect of elstic deformtions beneth the microcontcts is determined by superimposing norml deformtions in n elstic hlf-spce due to djcent microcontcts. The model lso ccounts for the vrition of the effective microhrdness. A prmetric study is conducted to investigte the effects of min contct input prmeters on the elstic effect. The study revels tht the elstic deformtion effect is n importnt phenomenon especilly in low contct pressures. The is compred with experimentl dt nd good greement is observed t low contct pressures. Keywords Therml Contct Resistnce, Plstic Deformtion, Elstic Deformtion, ow Contct Pressure, Microhrdness, Hlf-spce, Roughness, Modeling, Experimentl Dt. Nomenclture A = re, m = rdius of microcontcts, m = specimen rdius, m = Vickers microhrdness coefficient, P c = Vickers microhrdness coefficient, [ ] E = Young s modulus, P E = effective elstic modulus, P F = pplied lod, N H mic = microhrdness, P H = non-dimensionl microhrdness H mic /E k = therml conductivity, W/mK = distnce between microcontcts, m m = combined men bsolute surfce slope, [ ] n = number of microcontcts P = pprent contct pressure, P = therml joint resistnce, K/W r = rdil position, m x = non-dimensionl position r/ Y = men surfce plne seprtion, m Greek γ = plsticity index H mic /E m ε = reltive rdius p A r /A η = density of microcontcts, m Λ = non-dimensionl length b / (σ/m) λ = non-dimensionl seprtion Y/ σ σ = combined RMS surfce roughness, m υ = Poisson s rtio ω = norml elstic deformtion, m Subscripts = pure plstic model vlue, = solid, = pprent r = rel s = solid, micro I. Introduction THE continued growth in performnce nd functionlity of microelectronic nd vionic systems hs resulted in significnt increse in het dissiption rtes nd presents gret chllenge to therml engineers. The het generted must pss through complex network of therml resistnces to dissipte from the junction to the surroundings. A significnt resistnce in the network is the therml constriction/spreding resistnce through microcontcts t the interfce between the pckge nd its het sink. Therefore, n ccurte knowledge of mechnics of the contct, i.e. number nd size of microcontcts, is essentil for the therml resistnce nlysis. When rndom rough surfces re plced in mechnicl contct, rel contct occurs t the summit of surfce sperities which re clled microcontcts. The rel contct re A r, the summtion of the microcontcts, forms smll portion of the nominl contct re, typiclly few percent of the nominl contct re. The contct between two Gussin rough surfces is modeled by the contct between single Gussin surfce, tht hs the combined chrcteristics of the two surfces, with perfectly smooth surfce. The combined roughness σ nd surfce slope m cn be found from q q σ = σ + σ nd m = m + m () M. Bhrmi is Post-Doctorl Fellow t the Microelectronics Het Trnsfer bortory, University of Wterloo (MHT), Wterloo, Cnd. E-mil: mjid@mhtlb.uwterloo.c. M. M. Yovnovich is Distinguished Professor Emiritus of Mechnicl Engineering nd the founder of the MHT, University of Wterloo, E-mil: mmyov@mhtlb.uwterloo.c. J. R. Culhm is n Associte Professor of Mechnicl Engineering nd director of MHT, t the University of Wterloo Cnd. E-mil: rix@mhtlb.uwterloo.c. To study the constriction/spreding resistnce of microcontcts, the joint is usully studied in vcuum where the het trnsfer between contcting bodies occurs only vi conduction through microcontcts. Therml Contct Resistnce (TCR) of conforming rough surfces in vcuum is proportionl to the rel contct re []. TCR cn be decresed by reducing the roughness nd out-of-fltness

2 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. of the surfces before the interfce is formed or incresing contct pressure. However, mnufcturing highly finished surfces is not prcticl due to cost constrints. Also, lod constrints on electronic components mke it unfesible to use high contct pressure. Very little hs been done for light pressures <. MP, which is the pplicble rnge for microelectronics devices. Existing models such s [], [3] cn predict TCR for moderte to high contct pressures ccurtely. Milnez et l. [4] experimentlly studied low contct pressure joints in vcuum nd showed tht the models [], [3] overestimte the TCR t low pressures. They clled this phenomenon the trunction effect nd ttributed this trend to the Gussin ssumption of the surfce sperities which implies tht sperities with infinite heights exist. Milnez et l. [4] proposed correltions for mximum sperities heights s functions of surfce roughness. Existing plstic models [], [3] do not consider the effect of elstic deformtions beneth the microcontcts. These effects would be negligible if the elstic modulus of contcting bodies were infinity nd/or the distnce between the neighboring microcontcts ws smll enough so the elstic deformtion ws the sme for ll microcontcts. In relity, none of the bove is true nd the elstic deflection underneth microcontct is lwys lrger thn the deformtion outside the microcontct re (men plne). Mikic [5] ws the first one to point out this problem nd proposed model. However, his model did not consider the effect of the elstic deformtion of neighboring microcontcts nd vrition in the effective microhrdness which ws reported lter by Hegzy [6]. Also Mikic did not compre his model ginst experimentl dt. II. Problem Sttement Figure shows the cross-section of the contct schemticlly; note tht the surfce slopes of sperities, m, is exggerted. In relity, the surfce sperities cn be visulized s shllow hills nd vlleys. In this study the microcontcts re ssumed to deform plsticlly, resons supporting this ssumption re discussed in the next section. Consider plsticlly deformed microcontcts s loded res on n elstic hlf-spce. The pressure pplied on the microcontcts is the effective microhrdness of the joint. As elstic deformtions occur beneth the plstic microcontcts, men seprtion between the two contcting surfces decreses, compred with the vlue predicted by the pure plstic model where the elstic deformtion is neglected, i.e. Y < Y. As result of smller seprtion, more microcontcts re formed which in turn leds to higher rel contctrethtisequivlenttolowertcr. The gol of this study is to investigte the effect of elstic deformtions beneth the plsticlly deformed microcontcts on TCR. A new model is proposed tht ccounts for the elstic deformtion of microcontcts nd vrition of effective microhrdness with men rdius of microcontcts. A novel numericl lgorithm is presented tht stisfies the force blnce. The is compred with experimentl dt. Y, m Y Fig.. ) geometry of contct, pure plstic model, m b) plticlly deformed sperities with elstic deformtion men plne men plne Effect of elstic deformtion on men seprtion III. Why Plstic Microcontcts? Different pproches hve been tken to nlyze the deformtion of sperities by ssuming plstic [], elstic [7], or elstoplstic [8], [9] regimes t microcontcts. It hs been observed through experiments tht the rel contct re is proportionl to the pplied lod []. However, if simple elstic deformtion, following the Hertzin theory, is ssumed for sperities, the rel contct re will not be linerly proportionl to the lod, insted one obtins A r F /3. Greenwood nd Willimson (GW) [7] developed n elstic contct model. They proposed tht s the lod increses new microcontcts re nucleted while the men size of microcontcts remins constnt; the GW model stisfied the observed proportionlity A r F. As result,neffective elstic microhrdness cn be defined for elstic models which shows tht the ssumption of elstic nd/or plstic deformtion of sperities leds to similr results [7], []. Greenwood nd Willimson [7] introduced plsticity index s criterion for plstic flow of microcontcts. They reported tht the lod hs little effect on the deformtion regime. Bsed on plsticity index, Greenwood nd Willimson concluded tht except for especilly smooth surfces, the sperities will flow plsticlly under the lightest lods. Considering n indenttion hrdness for sperities, Persson [] lso concluded tht except for polished surfces ll microcontcts deform plsticlly. Mikic [5] proposed plsticity index γ = H mic /E m to determine the deformtion mode of sperities ( γ.33 sperities deform plsticlly () γ 3. sperities deform elsticlly The effective elstic modulus E is defined s E = υ E + υ E where H mic is the effective microhrdness. Mikic [5] lso concluded tht for most engineering surfces the sperity deformtion mode is plstic nd the verge sperity pressure is the joint effective microhrdness. According to [5], the deformtion mode of sperities depends on mteril

3 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. 3 Hrdness H (GP) Vickers micro-hrdness t d V d V /t=7 mcro-hrdness SS 34 Ni Zr-.5% wt Nb Zr-4 3 Indenttion depth t (µm) Fig.. Mesured hrdness nd microhrdness, Hegzy [6] properties E,H mic nd the shpe of sperities m; lso it is not function of the pplied lod. IV. Microhrdness Microhrdness cn vry throughout the mteril s the indenttion depth is incresed [3]. Microhrdness depends on severl prmeters, men surfce roughness, men slope of sperities, method of surfce preprtion, nd pplied pressure. Depending on the surfce preprtion, microhrdness cn be much greter thn the bulk hrdness [], [6]. As shown in Fig., microhrdness decreses with incresing depth of the indenter until bulk hrdness is obtined. Hegzy [6] concluded tht this increse in the plstic yield stress (microhrdness) of the metls ner the free surfce is result of locl extreme work hrdening or some surfce strengthening mechnism. He proposed empiricl correltions to ccount for the decrese in microhrdness with incresing depth of penetrtion H mic = (d v/7) c (3) where H mic is the Vickers microhrdness in GP, d v = d v /d nd d = µm, d v = 7t is the Vickers indenttion digonl in µm. The correltion coefficients nd c re determined from Vickers microhrdness mesurements. Eqution (3) is generl nd cn lso be used for surfces tht hve constnt microhrdness H mic,e by substituting = H mic,e nd c =. Song nd Yovnovich [4] proposed correltion to clculte H mic s follows: P P = H mic (.6σ /m) c ( +.7c ) V. Pure Plstic Models Cooper et l. (CMY) [], bsed on the level-crossing theory nd using the sum surfce pproximtion, derived reltionships for men microcontct size nd density of microcontcts η by ssuming hemisphericl sperities whose (4) heights nd slopes hve Gussin distributions r 8 ³ σ = exp π m λ erfc (λ) η = ³ m exp λ 6 σ erfc (λ) where λ = Y/ σ is the dimensionless seprtion. They lso showed tht the rtio of the rel contct re to the pprent re is relted to the men seprtion (5) A r = P = erfc (λ) (6) A H mic where H mic is the effective microhrdness of the softer mteril in contct nd P = F/A is the nominl contct pressure. Pullen nd Willimson [5] experimentlly investigted plstic flow under lrge lods. They ssumed tht mteril displced from the contcting regions must repper by rising some prts of the non-contcting surfce. They ssumed tht the volume of mteril remined constnt nd tht the mteril tht is plsticlly displced ppers s uniform rise over the entire surfce. Since the uniform rise will not ffect the shpe of the surfce outside the contct re, they showed tht the contct re due to the interction of microcontcts is not proportionl to the norml lod t reltively high lods; nd proposed s good pproximtion; A r /A = P p / ( + P p ),wherep p = P m /H mic. VI. Present Model The modeled geometry of the contct is shown in Fig. 3. Nine microcontcts, nmed A to I, re shown in Fig. 3 s htched identicl circles of rdius. The microcontcts re ssumed to be rrnged in squre rry where the shortest distnce between neighboring microcontcts is. From Fig. 3, one cn clculte the reltive rdius ε s ε = r Ar A = π The norml elstic displcement of hlf-spce produced s result of pplying uniform pressure distribution H mic over circulr re of rdius cn be determined from [6] 4H mic ³ r πe E ω = 4H mic πe r E ³ r µ r ³ K r r< r (8) where r is the rdil loction mesured from the center of the loded re; nd E ( ) nd K ( ) re the complete elliptic integrls of the second nd the first kind, respectively. The men elstic deformtion of the loded circulr re is ω =6H mic /3πE. Eqution (8) cn be non-dimensionlized nd re- (7)

4 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. 4 F C A D M x H origin ω totl - ω A ω mp ω totl ω mp self nd neighboring microcontcts ω totl self, due to microcontct "A" only ω A I E B G ω - -3 ε = (A r /A ) ω = π E' ω /4H mic men plne ω mp net displcement ω totl ω mp Fig. 3. Modeled geometry of contct, pln view ε - rrnged in the following form: µ ε πx E π ε ω = µ µ ε x E µ ε ε πx πx K πx where x< ε π x ε π (9) ω = 4H mic πe ω = 4H mic πe ε ω () nd x = r/. Reltionship for the deformtion outside the contct re, i.e., x>ε/ π is complex s given in Eq. (9). The following simpler reltionship cn be used to clculte the deformtion of hlf-spce outside the loded re over wide rnge of x nd ε with resonble ccurcy: ω =.6 ε x x> ε π () The elstic deformtion of hlf-spce is liner, thus superposition cn be pplied to clculte the totl elstic deformtion due to the microcontct A nd neighboring microcontcts. The line AM, Fig. 3,ischosensrepresenttive(or men plne) of the hlf-spce to estimte the totl elstic deformtions due to microcontcts A to I. Using superposition the elstic deformtion of the men plne AM is ω AM = ω A + ω D + ω E +ω B +ω H +ω G () where due to symmetry ω B = ω C, ω G = ω I, nd ω F = ω H. The non-dimensionl men elstic deformtion underneth the microcontct A, <x<ε/ π, due to its neighbors nd itself is ω = 4ε 3 π {z } due to A Z +.5π ε π X x i x + b i dx {z } due to neighbors x< ε π (3) where i nd b i re the constnts due to chnging the loction of the origin from ech neighboring microcontct to the Fig. 4. Elstic deformtion beneth microcontct A, for <x< ε/ π center of the microcontct A. Using the sme method, the men deformtion for the rest of the men plne AM, i.e., ε/ π <x</ cn be found: ω =.5ε 4 ε π Z / X ε π x i x + b i dx ε π <x< (4) Finlly the men elstic deformtion of the men plne AM cn be clculted using ω nd ω. Figure 4 shows the men deformtion due to microcontct A only, totl men deformtion considering the effects of neighbors, nd the net elstic displcement for the microcontct A s the reltive rdius of microcontcts ε vries. As expected, for smll vlues of ε, i.e. <. reltively low contct pressure, the effects of neighboring microcontcts is smll nd cn be ignored. As ε increses, the effect of neighboring microcontcts become more significnt, lso the displcement of the men plne increses. As result of these two competing trends, the net elstic deformtion beneth the microcontct A becomes smller nd eventully the net displcement pproches zero t reltively lrge lods. A. Numericl Algorithm A new numericl model is developed to ccount for the elstic deformtion beneth microcontcts nd vritions in microhrdness. The numericl lgorithm used in the is described below nd is shown in Fig. 5. Cooper et l. s (CMY) [] reltionships, Eqs. (5), (6) re used to clculte the men rdius,densityη,reltive rdius ε, nd men seprtion of the joint λ. Song nd Yovnovich s correltion [4], Eq. (4), is employed to estimte n effective microhrdness H for the pure plstic model. The subscript indictes the pure plstic model vlues. The net men elstic deformtion, ω net = ω ω AM,is then clculted using reltive rdius ε sdescribedinthe

5 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. 5 previous section. A new men seprtion between contcting surfces cn be found from λ = λ ω net / σ (5) With new seprtion λ, one cn determine new density of microcontcts η from Eq. (5). Applying force blnce, new men rdius of microcontcts is determined from s = F π H mic A η (6) Since the men rdius of the microcontcts chnges s the pplied lod vries, the microhrdness will lso chnge ccording to level-crossing theory, Eq. (3). If the print re in Vickers test is ssumed to be equl to the microcontct re, reltion between the Vicker s digonl nd the men size of microcontcts cn be found s START from CMY s = / E, ', F, σ, m, c, c A = πb s - P λ = erfc H' 8 σ = exp π m m exp η = 6 σ erfc ε = σ.6 m σ ( +.7c ) πη H ' = c P = F / A s ( λ ) erfc( λ ) ( λ ) ( λ ) d v = π (7) clculte ω net Therefore, new effective microhrdness cn be computed from the new rdius of microcontcts. This procedure continues until the difference between the new men rdius of microcontcts nd the old one becomes negligible. The results of the bove procedure, i.e. the men rdius nd the density of microcontcts, re used to clculte the TCR of the joint. The therml resistnce nlysis is bsed on the premise tht there re n (= ηa ) identicl circulr microcontcts of rdius tht provide n prllel pths for therml energy to be trnsferred in the contct plne. The constriction/spreding resistnce of the joint then cn be determined by employing the flux tube solution [3] s: H mic = c c π 7σ m = λ = λ ω / σ η = 6 = net m σ exp ( λ ) ( λ ) erfc F π H mic Aη : cross-level theory force blnce cceptble END ( ε).5 = (8) η A where =k k / (k + k ) isthehrmonicmenoftherml conductivities of the contcting bodies. B. Trends of Contct Prmeters The is run for joint s the pplied lod is vried over wide rnge to study the trends of the contct prmeters; see Figs. 6 nd 7 for the contct input prmeters. The contct prmeters clculted by both the present nd the pure plstic models re listed in Tble I. Figures 6 nd 7 show the rtio of the contct prmeters s the non-dimensionl pressure P/H mic is incresed. As shown in these plots, the rtio of seprtions λ /λ is greter thn one for the entire comprison due to elstic deformtions effect. As result of smller seprtion, more microcontcts re formed i.e. n/n >, the rel contct re is incresed A r /A r, thus therml resistnce is decresed / >. It cn lso be seen tht these rtios A r /A r,n/n,nd / decrese s the pplied lod increses which indictes tht the elstic deformtion effect becomes less importnt t higher lods. This is consistent with the trend seen in Fig. 4. Figure 6 shows tht the rtio of microcontcts rdius / < throughout the comprison. However, it should Fig. 5. Numericl lgorithm used in / λ / λ H' = (.6 σ' /m) c H mic /H' contct prmeters σ =.9µm m=.49 =.7GP c =-.37 E' =.9 GP H = P/H mic Fig. 6. Rtio of clculted vlues by over pure plstic model for: men seprtion, men rdius of microcontcts, nd effective microhrdness

6 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. 6 TABE I Contct prmeters clculted by present nd pure plstic models pure plstic model F (N) λ ε (µm) η λ ε (µm) η E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E contct prmeters σ =.9µm m =.49 =.7 GP c = -.37 E' =.9 GP H =.4 n/n A r /A r / n/n / A r /A r P/H mic Fig. 7. Rtio of clculted vlues by over pure plstic model for: density of microcontcts, rel contct re, nd TCR R j = Λ 9 7 H =.7 (E' = 6 GP) pure plstic model E' = 6 GP E' = 6 GP E' = GP 5 3 H =.5 (E' = 6 GP) H =.7 (E' = GP) contct dt σ =.99µm m =.83 = GP c =-.67 pure plstic model (E' ) H =H mic /E' Λ =b /(σ /m) =5mm P/H mic Fig. 8. Effect of elstic modulus E / on be noted tht the bsolute rdius of microcontcts,, increses by incresing the lod, see Tble I. Therefore, the effective microhrdness H mic decresessthelodincreses, see Eqs. (7) nd (3). The effective microhrdness shown in Fig. 6 is non-dimensionlized with respect to H = (.6σ/m) c. Note tht H remins constnt throughout the comprison. C. Effect of Elstic Modulus E Non-dimensionl joint resistnces of typicl joint is shown in Fig. 8 over wide rnge of the non-dimensionl pressure P/H mic. Four vlues of E =, 6, 6, nd (pure plstic model) hs been selected to investigte the effect of elstic modulus, E, on TCR. The contct input prmeters re shown in the figure nd re kept constnt s the effective elstic modulus E is chnged. The rtio of the effective microhrdness over the effective elstic modulus H = H mic /E is used s to lbel these four curves. It cn be seen tht s H pproches zero, i.e. E, the pproches the pure plstic model nd the elstic effect vnishes. Therefore, it my be concluded tht the non-dimensionl prmeter H is mesure of how importnt is the elstic deformtion effect. Also note tht the difference between the nd the pure elstic model decreses s P/H mic increses. Beyond certin pressure the difference between the pure plstic model nd the (three vlues of E ) becomes negligible. This is in greement with the observed trend in Fig. 4 which indictes tht the effect of elstic deformtion is more importnt t lighter lods.

7 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, σ =.5µm σ =µm σ =µm σ =5µm 4 pure plstic model Milnez T dt 3 /.3.. contct prmeters =.3GP σ [µm] m c =.5.53 =5mm.76.9 E' =.9 GP.9 =9.9W/mK P/H mic Fig. 9. Effect of surfce roughness σ on - Milnez [4] SS T mteril: SS 34 σ =.7µm m =.4 = 8.87 W/mK E' =.9 GP =.7 GP c = -.37 = 5 mm H =.7 ( )/ rnge is 5% D. Effect of Roughness σ To study the effect of roughness on the elstic deformtion effect, the is run over rnge of pplied lod for typicl contct. Four levels of roughness, i.e..5,,, nd 5 µm re chosen. The predicted joint resistnces by the re normlized with respect to the pure plstic model resistnces / nd plotted versus the non-dimensionl pressure P/H mic in Fig. 9. The contct input prmeters re shown in the figure nd re kept constnt s roughness level is chnged. The effective microhrdness is ssumed to be constnt, i.e. it is not function of the penetrtion depth or microcontcts rdius by setting c =, to investigte the effect of roughness only. As shown in Fig. 9, t fixed non-dimensionl pressure P/H mic, the elstic effect is lrger t smller roughness levels. This phenomenon cn be explined s follows. The verged non-dimensionl net elstic deformtion beneth microcontcts ω net is function of reltive rdius ε only which is function of the non-dimensionl pressure P/H mic, see Eq. (6). Therefore, t fixed ε or P/H mic the net deformtion ω net is the sme for ll roughness levels. However, from Eqs. () nd (5), one cn conclude tht the reduction in the men seprtion λ, due to the elstic effect, is proportionl to the rtio of /σ, where reltive rdius ε nd microhrdness H mic re constnt. For smller roughness levels, the rtio /σ is lrger, therefore the net reduction in the non-dimensionl seprtion λ is lrger. This leds to more new microcontcts, lrger rel contct re nd therefore less TCR or equivlently more elstic effect. E. Comprison With Dt The is compred with experimentl dt of Milnez et l. [4] in Figs. to. They collected three sets of dt for SS 34, the surfce prmeters of ech test re listed in the corresponding plot. The pure plstic model is included in the comprisons to better show the effect of elstic deformtion. The verge difference Fig.. Comprison of with Milnez et l. [4] dt, test T 4 pure plstic model Milnez T dt 3 Milnez [4] SS T mteril: SS 34 σ =.9µm m =.49 = 9.9 W/mK E' =.9 GP =.7GP c =-.37 = 5 mm H =.4 ( )/ rnge is 4% Fig.. Comprison of with Milnez et l. [4] dt, test T between the nd the pure plstic model over the pplied lod rnge is lso reported in the comprisons. The non-dimensionl prmeter H is lso shown for ech set of dt where H mic is n verge vlue of the effective microhrdness over the comprison rnge. As shown in the plots, the dt of [4] show better greement with the t reltively low lods nd move towrd the pure plstic model t higher lods. The three sets of [4] dt differ only in roughness levels which re.7,.9, nd 3.7 µm. From Figs. to cn be seen tht the difference between the nd the pure plstic model becomes smller t dt set T3 where surfce roughness is the lrgest of the three sets of dt. This is consistent with the trend seen in Fig. 9, see lso section VI-D. The is lso compred with experimentl dt of Hegzy [6] in Figs. 3 to 5. Hegzy conducted severl experiments with four different lloys. Here, three

8 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, pure plstic model Milnez T3 dt 3 pure plstic model PZ456 - Milnez [4] T3 mteril: SS 34 σ =3.7µm m =.7 =8.78W/mK E' =.9 GP =.7 GP c =-.37 = 5 mm H =. ( )/ rnge is 5% Hegzy [6] PZN56 mteril: Zircloy 4 σ =3.4µm m =.9 = 8.6 W/mK E' = 57.6 GP =3.3GP c = -.45 = 5 mm H =.38 ( )/ rnge is 6%. 3 4 Fig.. Comprison of with Milnez et l. [4] dt, test T3 Fig. 4. Comprison of with Hegzy [6] dt, test PZ456 pure plstic model PZN - Hegzy [6] PZN mteril: Zr-.5%wt. σ =.99µm m =.83 =.3 W/mK E' = 57.6 GP =5.88GP c = -.67 =5mm H =.53 ( )/ rnge is 3%. 3 4 Fig. 3. Comprison of with Hegzy [6] dt, test PZN different mterils re chosen, i.e. Zr-.5%wt. Zircloy 4 nd nickel ; which cover reltively wide rnge of contct input prmeters elstic modulus E, roughness σ, therml conductivity, nd microhrdness coefficients nd c. The mteril properties nd surfce prmeters relistedinthefigures. The dt of [6] re closer to the t smller lods nd move towrd the pure plstic model t lrger lods, i.e. the sme s Milnez et l. [4] dt. VII. Conclusion The effect of elstic deformtion of the substrte underneth the microcontcts is studied on the TCR of rough conforming joints in vcuum. The microcontcts re ssumed to deform plsticlly. A new model is developed tht ccounts for elstic deformtions of the substrtes due to self nd neighboring microcontcts using superposition of elstic deformtions in hlf-spce. The pure plstic model PNI - - ( )/ rnge is 7%. Hegzy [6] PNI mteril: nickle σ =.9µm m=. = 75.8 W/mK E' =.9 GP =6.3GP c = -.64 = 5 mm H = Fig. 5. Comprison of with Hegzy [6] dt, test PNI lso ccounts for the vrition in the effective microhrdness nd stisfies the force blnce. The microcontcts re ssumed s identicl circles which re rrnged in squre rry. Reltionships re derived for the verge norml elstic deformtion of the elstic substrte beneth microcontcts. The net elstic deformtion of the microcontcts is clculted. Using the levelcrossing theory, the men seprtion between two contctingbodiesismodified for the net elstic deformtion. An itertive numericl lgorithm is developed to compute the modified men size, density, nd the contct resistnce of the joint. The force blnce principle is used in the numericl solution. The trends of the re studied nd compred with the pure plstic model where the elstic deformtion is completely ignored. It is observed tht s result of the elstic deformtion the men seprtions be-

9 SEMI-THERM, SEMICONDUCTOR THERMA MEASUREMENT AND MANAGMENT SYMPOSIUM, SAN JOSE. MAR 5-7, 5. 9 tween two contcting surfces becomes smller; thus more microcontcts re nucleted, the rel contct re is incresed, nd therml contct resistnce is decresed It is lso shown tht the elstic deformtion effect becomes less importnt t higher lods. This is direct result of tht the net elstic deformtion pproches zero t reltively high lods, i.e. the elstic deformtion is uniform for the whole men plne. A prmetric study is performed to investigte the effects of min input contct prmeters on the joint resistnce. A non-dimensionl prmeter H = H mic /E is introduced s mesure of importnce of the elstic deformtion. As H pproches zero, i.e. E, the pproches the pure plstic model. It is lso shown tht for fixed contct, the elstic effect is more significnt t smller roughness levels. The is compred ginst experimentl dt of [4] nd [6]. The experimentl dt cover reltively wide rnge of input contct prmeters. The dt show good greement with the t low contct lods. The dt however move towrd the pure plstic model t high contct lods. As result of this trend, one cn conclude tht the present nd the pure plstic models present lower nd higher bounds for the therml joint resistnce of the conforming rough surfces in vcuum, respectively. [] D. Tbor, The Hrdness of Metls, Oxford University Press, Amen House, ondon E.C.4, UK, 95. [] J. A. Greenwood nd J. H. Tripp, The elstic contct of rough spheres, Trnsctions of the ASME: Journl of Applied Mechnics, vol. 89, no., pp , 967. [] B. N. J. Persson, Sliding Friction Physicl Principles nd Applictions, Springer, Berlin, Germny,. [3] M. M. Yovnovich nd E. E. Mrott, Therml Spreding nd Contct Resistnces, chpter 4, Het Trnsfer Hndbook, Editors: A. Bejn nd D. Krus, John Wiley nd Sons Inc., Hoboken, New York, USA, 3. [4] S. Song nd M. M. Yovnovich, Reltive contct pressure: Dependence on surfce roughness nd vickers microhrdness, AIAA Journl of Thermophysics nd Het Trnsfer, vol.,no., pp , 988. [5] J. Pullen nd B. P. Willimson, On the plstic contct of rough surfces, Proc., Roy. Soc., ondon, A37, pp , 97. [6] K.. Johnson, Contct Mechnics, Cmbridge University Press, Cmbridge, UK., 985. Acknowledgments The uthors grtefully cknowledge the finncil support of the Centre for Microelectronics Assembly nd Pckging, CMAP nd the Nturl Sciences nd Engineering Reserch Council of Cnd, (NSERC). References [] M.Bhrmi, J.R. Culhm, nd M. M. Yovnovich, Therml contct resistnce: A scle nlysis pproch, To be ppered in Journl of Het Trnsfer, ASME, Also pper No. IMECE3-4497, 5. [] M.G.Cooper,B.B.Mikic,ndM.M.Yovnovich, Therml contct conductnce, Interntionl Journl of Het nd Mss Trnsfer, vol., pp. 79 3, 969. [3] M. M. Yovnovich, Therml contct correltions, AIAA Pper No. 8-64, lso Progress in Aeronutics nd Aerodynmics: Spcecrft Rditive Trnsfer nd Temperture Control,edited by T. E., Horton, vol. 83, pp , 98. [4] F.H.Milnez,M.M.Yovnovich,ndM.B.H.Mntelli, Therml contct conductnce t low contct pressures, Journl of Thermophysics nd Het Trnsfer, vol. 8, pp , 3. [5] B. B. Mikic, Therml contct conductnce; theoriticl considertions, Interntionl Journl of Het nd Mss Trnsfer, vol. 7, pp. 5 4, 974. [6] A. A. Hegzy, Therml Joint Conductnce of Conforming Rough Surfces: Effect of Surfce Micro-Hrdness Vrition, Ph.D. thesis, University of Wterloo, Dept. of Mech. Eng., Wterloo, Cnd, 985. [7] J. A. Greenwood nd B. P. Willimson, Contct of nominlly flt surfces, Proc., Roy. Soc., ondon, A95, pp. 3 39, 966. [8] W.R.Chng,I.Etison,ndD.B.Bogy, Anelstic-plstic model for the contct of rough surfces, JournlofTribology, vol. 9, pp , 987. [9] Y. Zho, D. M. Miett, nd. Chng, An sperity model incorporting the trnsition from elstic deformtion to fully plstic flow, JournlofTribology,Trns.ofASME,vol., pp ,.