Applied Mathematical Modelling

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1 Applied Mathematical Modellig 37 (013) Cotets lists available at SciVerse ScieceDirect Applied Mathematical Modellig joural homepage: The effective permeability of the uderfill flow domai i flip-chip packagig Coie Yag a, We-Bi Youg b, a Departmet of Systems ad Naval Mechatroic Egieerig, Natioal Cheg Kug Uiversity, Taia, Taiwa, ROC b Departmet of Aeroautics ad Astroautics, Natioal Cheg Kug Uiversity, Taia, Taiwa, ROC article ifo abstract Article history: Received 0 October 010 Received i revised form 16 March 01 Accepted 0 March 01 Available olie 9 March 01 Keywords: Uderfill Flip-chip Capillary flow Power-law fluid I order to reach the goals of high electrical performace ad dese packagig withi the limited space, the flip-chip techology becomes popular i electroics packagig. I the flip-chip assembly, differece betwee thermal expasio coefficiets of the chip ad substrate may cause thermal fatigue at solder joits. To avoid this thermal fatigue, epoxy ecapsulat is filled ito the gap betwee the substrate ad chip by the capillary force. Because of the small space i the flow domai, the uderfillig flow ca be assumed as a flow i porous medium. Permeability is used to characterize the flow field of the space amog the substrate, chip, ad solder bumps. I this study, a umerical method is used to determie the effective permeability for the uderfillig flow domai. Aalysis of the three dimesioal flow i a uit cell of the uderfill flow domai is performed. The resultig average velocity ad pressure gradiet are used to calculate the apparet permeability. Compariso with the aalytical approximatio for the permeability i literature is also performed. The effective permeability calculated usig the proposed umerical method gives reasoable predictio of the uderfill flow as compared to the experimetal result. Ó 01 Elsevier Ic. All rights reserved. 1. Itroductio Flip-chip packagig is a itegrated circuit (IC) packagig techique that uses solder bumps to coect chip with substrate. The mismatch of thermal expasio coefficiets teds to cause fatigue at solder juctios. Epoxy ecapsulat is used to solve this problem ad improve the reliability of flip-chip packagig. The ecapsulat is filled ito the gap betwee the chip ad substrate by the capillary force so that the thermal stresses may disperse ito the uderfill materials to avoid crack geeratio. Plety of studies related to the uderfill process ca be foud i literatures. The Washbur model for flow i betwee parallel plates is the most usual assumptio to describe the fillig flow i uderfill ecapsulatio [1 9]. Ha ad Wag [5] used a Hele Shaw model to perform both theoretical ad experimetal studies. Sice they uderestimated the effect of solder bumps, the results were deviated from the actual flow. Nguye et al. [10] used quartz dies to perform uderfill experimets with differet bump arragemets. They foud that the velocity of the flow frot at boudary was larger tha those at the ceter due to the edge effect. Fie et al. [11] used differet bump desities at ceter ad outside regios. They reported that lowerig the bump desity at ceter regio ca get a more uiform flow frot, ad lower the boudary effects. Youg ad Yag [], Youg [4] treated the uderfill flow domai as a porous medium, ad described the flow field with a modified Hele Shaw model. The fluid was treated as a highly viscous fluid, ad the flow was drive by the capillary force Correspodig author. address: yougwb@mail.cku.edu.tw (W.-B. Youg) X/$ - see frot matter Ó 01 Elsevier Ic. All rights reserved.

2 1178 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) amog the chip, substrate, ad bumps. Together with Darcy s law, they predicted the flow frot evolutio i the uderfill. Youg ad Yag [1] studied the effect of cotact agle ad bump arragemet. They foud smaller cotact agles have larger capillary pressures. They also reported both experimetally ad theoretically that icreased the bump pitch did ot alter the fillig time much util it reached a critical value. Wa et al. [13] proposed a umerical model for the predictio of flipchip uderfill flow. I this model, the power-law costitutive equatio was used to describe the o-newtoia behavior of ecapsulat fluids ad a time-depedet velocity boudary coditio was applied. Epoxy moldig compoud with silica fillers is used as the uderfill material, ad it usually exhibits a o-newtoia behavior durig the fillig process. Thus, a o-newtoia flow effect must be cosidered i modelig the uderfill flow. Because the uderfill flow domai cotais may solder bumps, it is quite complex i geometry for the flow simulatio. A simpler way to avoid complex geometry is to treat the flow domai as a porous medium. Uder this assumptio, the correspodig permeability is the major characteristics of the flow domai. I this study, we aalyze the flow field of a threedimesioal uit cell i the flow domai. The uderfill material is treated as a o-newtoia fluid. The effective permeability ca be determied by the correspodig average velocity ad pressure gradiet derived from the flow field of the uit cell. Compariso with the aalytical approximatio for the permeability i literature is also performed. The effects of the bump stad height, pressure gradiet, ad bump size o the effective permeability are discussed.. Permeability of the o-newtoia flow I order to simplify the aalysis of the uderfill flow, the flow field is assumed as a flow through a space betwee two parallel plates with a cylider bak, as show i Fig. 1. The cyliders represet the bumps while the parallel plates represet the chip ad substrate. To further simplify the aalysis, the flow field is treated as a flow through porous medium. For a porous medium the Darcy s law reads: * k u ¼ rp; ð1þ l where u * is the velocity vector, k is the permeability tesor l is the viscosity, ad p is the pressure. The permeability is a value used to characterize the fluidity of the porous medium ad is usually has a uit of m. For a o-newtoia fluid, the power law ca be used to model the shear rate depedet viscosity, ad is writte as: l ¼ m _c 1 ; where m is the cosistecy idex, _c s the shear rate, ad is the power-law idex. For < 1 the fluid is called a shear thiig fluid, or pseudo plastic fluid. For > 1 the fluid is called a shear thickeig, or dilatat fluid. The power-law viscosity model has good accuracy at high shear rate, but it teds to overestimate the viscosity at low shear rate. To solve this problem, a limit value for the viscosity is give, ad the power-law is modified as: l ¼ m _c 1 for _c > _c c ; l ¼ m o ¼ m _c 1 c for _c 6 _c c ; where _c c is the critical shear rate for the uderfill ecapsulat. The power law will result i a ureal large value of viscosity at the low shear rate. I reality, the viscosity of a o-newtoia fluid usually reaches a costat value called the zero-shear-rate viscosity as the shear rate is gettig lower. Therefore, it ca be defied a critical shear rate to avoid the failure of the power-law model at the low shear rate. A costat viscosity is used whe the shear rate is below this value. For flow betwee two parallel plates, the average viscosity ca be derived as: ðþ ð3þ Fig. 1. The schematic diagram of the flow domai i a flip-chip uderfillig process.

3 l ¼ h Z h 0 m 1 1 z dp dx 1 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) ! dz ¼ d þ 1 m 1 dp dx 1 1 h ; ð4þ where h is the distace betwee the chip ad substrate or the bump height, ad the average velocity is: u ¼ Z h udz ¼ þ1 dp h h m þ 1 dx : ð5þ 0 Usig the above model of average viscosity ad velocity, we ca derive the aalytical permeability betwee two parallel ifiite flat plats based o Darcy s law. The permeability is give as: k ¼ d 1 1 4ð þ 1Þ 1 þ h ¼ kh ; ð6þ 1 where k ¼ 4ð þ 1Þ d ¼ m _c c hjdp=dxj : d þ ; ð7þ 1 Cosider the permeability for two parallel plates with a cylider bak, the permeability model should be modified as follow [14]: 0 R d R 1 d 1 dx 1 k kh þ 0 kc 3 0 cdx A ¼ k 1 h þ S d p Z d 1 c dx! 3 ; ð9þ 0 d where c ¼ S qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d 4x ; ð10þ S is the bump pitch, ad d is the bump diameter. I Eq. (9), we use a simple model to calculate the permeability of the uderfill flow domai by averagig the permeability betwee two parallel plates ad solder bumps. This aalytical model eglects the effects of three-dimesioal flow. I order to uderstad the three-dimesioal effects o the permeability, a three-dimesioal uit cell flow model is used to simulate the uderfill flow, as show i Fig.. The uit cell is defied betwee two solder bump ad represets the basic space for the flow domai i the uderfill. By aalyzig the steady-state flow through the uit cell, the fluidity of the uit cell ca be characterized by the effective permeability defied i the followig. ANSYS FLUENT is used to simulate the flow i the uit cell model. No-slip boudary coditios are applied at the bottom boudary ad cylider walls which represet the substrate ad bumps, respectively. The top boudary where we apply the symmetry boudary coditio is set at the mid-plae betwee the substrate ad the chip. Pressure differece is applied across the frot ad rear boudaries. ð8þ Fig.. A three-dimesioal uit cell model for the uderfill flow.

4 1180 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) Fig. 3. The fiite elemet mesh of the three-dimesioal uit cell model. Fig. 4. The viscosity distributio o the uit cell model. From the average velocity ad pressure gradiet obtaied by the simulatio, oe ca fid the effective three-dimesioal permeability. I the simulatios, the fluid desity is set at 1600 kg/m 3, the cosistecy is 0.94 N-s/m, the power-law idex is 0.76, the critical shear rate is s 1, the bump diameter is 100 lm, the bump pitch is 00 lm, ad the height is 50 lm. The effective permeability is defied as: k e ¼ u m 0 jdp=dxj ; ð11þ

5 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) Fig. 5. A typical velocity field at the mid-plae betwee the chip ad substrate. Fig. 6. Aalytical ad umerical effective permeability for differet power law idex. where u is the average velocity from the umerical simulatio. The aalysis is based o the steady-state pressure drive flow through the uit cell to determie the permeability of the geometric structure. The effective permeability also depeds o the pressure gradiet oliearly. The approximatio for the effective permeability i the uderfill simulatio will

6 118 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) deped o the pressure gradiet at differet locatios. Based o the argumet, the effective permeability with respect to pressure gradiet must be determied for a give uderfill geometric structure. After that, the simulatio ca be performed with assumed porous medium for the uderfill flow domai. Based o the geometric data ad the pressure differece at each locatio, the correspodig permeability ca be determied for the calculatio. I actual uderfill flow, the pressure differece is low ad its effect o the permeability is low ad ca be eglected for simplicity. Certaily, some error i associated with this assumptio may occur i the simulatio. The effective permeability ca be related to the permeability as: k e ¼ k l m 0: ð1þ Fig. 7. The effective permeability as a fuctio of pressure gradiet for a give geometry, ad the bump diameter is 100 lm, the bump pitch is 00 lm, ad the height is 50 lm. Fig. 8. The effective permeability as a fuctio of the bump pitch, ad the bump diameter is 100 lm, ad the height is 50 lm.

7 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) Results ad discussios I the umerical simulatio, the meshig umber should be large eough to reach a covergig flow rate. The umerical error may reduce to a acceptable rage as the flow rate coverges. I our studies, the flow rate coverges as the meshig umber reaches 500,000. The fiite elemet model of the three-dimesioal uit cell after meshig is show i Fig. 3. For a o-newtoia fluid, the viscosity i the flow domai varies with locatios, as show i Fig. 4. Sice the uderfill ecapsulat i this study is cosidered as a shear thiig material, larger viscosity value is obtaied at locatios with smaller shear rate, ad vice versa. The viscosity is quite low at the wall of the bump due to the high shear rate i this regio. A typical velocity distributio at the mid-plae betwee the chip ad substrate is show i Fig. 5. Higher velocity is observed at the regio betwee the bumps because of the arrow width. The effective permeability as a fuctio of power-law idex is show i Fig. 6 both aalytically ad umerically. The aalytical data are obtaied by Eqs. (9) ad (1), ad the umerical data are obtaied by applyig the umerical results to Eq. (11). For the Newtoia flow as = 1, the aalytical ad umerical effective permabilities are also quite differet. The umerical result is smaller, ad the differece comes from the three-dimesioal effect that is eglected i the aalytical model. The effect of the three dimesioal flow o the effective permeability is sigificat ad must be cosidered i order to have correct aalysis of the uderfill flow. It is obvious that the effective permeability determied by this umerical method will give the better approximatio for the uderfill flow. It must be oticed that defiitio of effective permeability i Eq. Fig. 9. The effective permeability as a fuctio of the bump height, ad the bump diameter is 100 lm, ad the bump pitch is 00 lm. Fig. 10. The effective permeability as a fuctio of the bump diameter, ad the bump pitch is 00 lm, ad the height is 50 lm.

8 1184 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) (11) usig the zero shear rate viscosity istead of the viscosity. Durig the simulatio of a uderfill flow usig the Darcy s law, the average velocity is calculated istead of the actual velocity. Therefore, the true shear rate caot be available to determie the viscosity i the simulatio. With the effective permeability, a costat zero shear rate viscosity is used i the simulatio ad the depedet of the flow o the shear rate is accouted for by the effective permeability. The effective permeability defied i this study is ot oly a fuctio of the geometry but also the velocity. Sice the velocity is related to the pressure gradiet, the effective permeability may chage with the pressure gradiet for a give geometry. Fig. 7 shows the depedace of effective permeability o the pressure gradiet, it icreases with the pressure gradiet. Bump pitch is the distace betwee two earby bump ceters. The effective permeability icreases as bump pitch icreases, ad approaches a costat value, as show i Fig. 8. For large legth of bump pitch, the viscous force caused by the bumps is much less that by the chip ad substrate. Thus, the effective permeability will cease to vary with the bump Fig. 11. Numerical effective permeability for differet power law idex ad bump pitch. Fig. 1. Numerical effective permeability for differet power law idex ad bump diameter.

9 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) pitch ad be domiated by the bump height. Figs. 9 ad 10 show the effects of bump height ad diameter o the effective permeability, respectively. Uder the same pressure gradiet, as the bump height icreases, the fluid has more space to flow, ad the effective permeability icreases. As the bump diameter icreases, the permeability cotiuously decreases ad reaches zero util the bump diameter is equal to bump pitch. Nearly liear relatio is obtaied for the bump diameter. Figs shows the k e diagrams with differet bump pitch, diameter, ad height, respectively. The effective permeability decreases as power-law idex icreases for all cases. The effective permeability slightly icreases as bump pitch icreases, diameter decreases, or height icreases. The results also show that the effective permeability is highly depeds o the power-law idex. Differet bump size or pitch has little effects o the effective permeability. Compared to the bump pitch ad size, the bump height has more effect o the effective permeability. To compare with the available experimetal result of the uderfill flow, a chip size of 5 5 mm with a full array of bump patter is selected from i the literature [15]. The uderfill process is performed uder a costat temperature, 90 C. The cotact agle betwee the ecapsulat ad glass substrate ad chip is The cotact agle betwee the ecapsulat ad solder bump is The surface tesio is N/m. A costat average viscosity, 0.48 Pa s, is used Fig. 13. Numerical effective permeability for differet power law idex ad bump height. Fig. 14. Calculated ad experimetal results of uderfill fillig rate with respect to time for a flip-chip o board.

10 1186 C. Yag, W.-B. Youg / Applied Mathematical Modellig 37 (013) i the simulatios for the ecapsulat assumig that the degree of reactio is low durig the uderfill process. The bump diameter, bump pitch, ad bump height are 150, 300, ad 100 lm, respectively. From the three-dimesioal uit cell flow simulatio, the effective permeability for this flip-chip desig is m. The pressure depedece is eglected for simplicity due to its mior effect at the low pressure differece i actual uderfill flow. The correspodig capillary pressure for the flip-chip desig is N/m based o the formula i the literature [15]. The uderfill flow of the flip-chip ca be derived from Eq. (1) for a Newtoia fluid as: l ¼ P ok e t m o ; ð13þ where l is the uderfill flow distace ad P o is the capillary pressure. With Eq. (13), the compariso of calculated ad experimetal fillig percetage of the flip-chip is show i Fig. 14. The predictio of the fillig flow gives a reasoable result based o the effective permeability. 4. Coclusios I the flip-chip uderfillig flow field, the flow domai is simplified as two parallel plates with cylider bumps, ad o- Newtoia power-law is used to model the viscosity of the ecapsulat. A three-dimesioal uit cell model is proposed to simulate the uderfill flow. The simulated average velocity ad pressure gradiet are used to determie the effective permeability. Usig the o-newtoia viscosity model to predict the permeability gives a result closer to the actual flow field, especially for small scale systems. The effect of the three dimesioal flow o the effective permeability is sigificat ad must be cosidered i order to have correct aalysis of the uderfill flow. To obtai a higher effective permeability, a smaller power-law idex is desired. For uderfill ecapsulat that has the power-law idex aroud , the correspodig o- Newtoia effects are ot sigificat. The most direct way to icrease the permeability is to chage the bump pitch ad size. Icreasig the bump pitch, icreasig the bump height, or decreasig the bump diameter ca all icrease the permeability. Refereces [1] Y. Guo, G. Lehma, T. Driscoll, E. Cotts, A model of the uderfill flow process: particle distributio effects, i: Electroic Compoets ad Techology Coferece, 1999, pp [] W.B. Youg, W.L. Yag, The effect of solder bump pitch o the uderfill flow, IEEE Tras. Adv. Packag. 5 (00) [3] W.B. Youg, W.L. Yag, Uderfill viscous flow betwee parallel plates ad solder bumps, IEEE Tras. Compo. Packag. Techol. 5 (00) [4] W.B. Youg, Aisotropic behavior of the capillary actio i flip chip uderfill, Microelectro. J. 34 (003) [5] S.J. Ha, K.K. Wag, Aalysis of the flow of ecapsulat durig uderfill ecapsulatio of flip-chips, IEEE Tras. Compo. Packag. Mauf. Techol. B. Adv. Packag. 0 (1997) [6] J.L. Wag, Uderfill of flip chip o orgaic substrate: viscosity, surface tesio, ad cotact agle, Microelectro. Reliab. 4 (00) [7] J.L. Wag, Flow time measuremets for uderfills i flip-chip packagig, IEEE Tras. Compo. Packag. Techol. 8 (005) [8] J.W. Wa, W.J. Zhag, D.J. Bergstrom, A aalytical model for predictig the uderfill flow characteristics i flip-chip ecapsulatio, IEEE Tras. Adv. Packag. 8 (005) [9] J.W. Wa, W.J. Mag, D.J. Bergstrom, A theoretical aalysis of the cocept of critical, clearace toward a desig methodology for the flip-chip package, J. Electro. Packag. 19 (007) [10] L. Nguye, C. Queti, P. Fie, B. Cobb, S. Bayyuk, H. Yag, S.A. Bidstrup-Alle, Uderfill of flip chip o lamiates: simulatio ad validatio, IEEE Tras. Compo. Packag. Techol. (1999) [11] P. Fie, B. Cobb, L. Nguye, Flip chip uderfill flow characteristics ad predictio, IEEE Tras. Compo. Packag. Techol. 3 (000) [1] W.B. Youg, W.L. Yag, Uderfill of flip-chip: The effect of cotact agle ad solder bump arragemet, IEEE Tras. Adv. Packag. 9 (006) [13] J.W. Wa, W.J. Zhag, D.J. Bergstrom, Numerical modelig for the uderfill flow i flip-chip packagig, IEEE Tras. Compo. Packag. Techol. 3 (009) [14] C.L. Lai, W.B. Youg, A model for uderfill viscous flow cosiderig the resistace iduced by solder bumps, J. Electro. Packag. 16 (004) 1 9. [15] S.W. Peg, W.B. Youg, Applicatio of the uderfill model to bump arragemet ad dispesig process desig, IEEE Tras. Electro. Packag. Mauf. 33 (010) 1 18.