FUNDAMENTALS OFTHE SQUEEZE-FLOWBETWEEN A HEAT · PDF fileFUNDAMENTALS OFTHE...

FUNDAMENTALS OF THE SQUEEZE-FLOW BETWEENA HEAT SINK AND A FLIP-CHIP

M.A. Marois and M. LacroixDepartment of Mechanical Engineering, Universite de Sherbrooke

Sherbrooke, Quebec, Canada IlK 2RlContact: [email protected]

Received May 2008, Accepted August 2008No. 08-CSME-16, E.LC. Accession 3053

ABSTRACTThe paper presents the fundamentals of the squeeze-flow of the thermal interface material (TIM) that

takes place during the pressing of a heat sink to the back side of a flip-chip is studied. A two-dimensionalstring model is developed for predicting the time-varying plate separation and squeeze-rate in terms of thesqueeze force. The predictions are compared to a one-dimensional string model and to a squeeze-dropflow model. Results indicate that the flow resulting from the squeezing of a string of TIM between tworigid plates is truly two-dimensional. The effect of surface tension and of the heat transfer is found to benegligible under the assembly conditions. The flow behaviour of the TIM with suspensions of highthermal conductivity particles is also investigated. It is shown that the fluid remains Newtonian forparticle volume fractions smaller than 30%. For volume fractions larger than 30%, the fluid becomesNon-Newtonian during the early stages of the squeezing process, i.e. fort:=:;; Is. In the later stageshowever (t > lOs), the fluid may be considered Newtonian.

ETUDE FONDAMENTALE DE L'ECOULEMENT D'ECRASEMENT ENTRE UNDISSIPATEUR THERMIQUE ET UNE PUCE ELECTRONIQUE

RESUMEL'article presente une etude fondamentale qui a ete realisee a. propos de l'ecoulement de la pate

thermique lors de l'assemblage d'un module electronique. Cette pate s'ecoule lorsque qu'elle est presseeentre les surfaces de la puce electronique et du dissipateur thermique. Un modele cartesienbidimensionnel est developpe pour predire la distance entre les surfaces et la vitesse de pressage de la pateen fonction de la pression exercee sur les surfaces. Les resultats obtenus sont compares a. un modelecartesien unidimensionnel et a. un modele radial de gouttelette. Le modele revele qu'un traitementbidimensionnel est necessaire pour caracteriser rigoureusement Ie phenomene d'ecoulement. On observeque I'effet de la tension de surface ainsi que de la temperature est negligeable lors du proceded'assemblage. Le modele prend aussi en compte la presence de particules metalliques en suspension dansla pate. Ces particules sont utilisees pour augmenter la conductivite thermique. On montre que pour unefraction volumique inferieure a. 30%, Ie fluide peut etre considere newtonien. Au-dela. de cette limite, Iecomportement du fluide devient non newtonien au debut du pressage c'est-a.-dire pour t:=:;; Is . Plus tard(t > lOs ), I'ecoulement devient similaire a. celui d'un ecoulement newtonien.

Abstract: Keywords: Squeeze-flow, Non Newtonian, rigid plates, flip-chip, thermal interface material.

Transactions ofthe CSME Ide fa SCGM Vol. 32, No. 3-4,2008 467

INTRODUCTION

In the squeeze-flow problem, a material is deformed between two approaching rigid bodies and thesqueeze force Fw and plate separation bet) (and the squeeze-rate vw(t) = db(t) / dt) are correlated. This

problem has been the subject of few investigations in the past. Studies were conducted bothexperimentally and computationally for Newtonian as well as Non-Newtonian fluids under different flowconditions [1-7].

Thel111al Interface Material(J1M) /' Thefmal heatiSink

Contacts $tructure"e-tween 'eh1,pInterconneXion~

Fig. 1 : Schematic of a flip-chip with a thermal sink attached to its back side.

The problem of squeeze-flow between rigid plates fmds an interesting application in the assembly ofelectronic components. Indeed, in order to improve heat dissipation from a high power flip-chip, a heatsink is often fixed to it (Fig. 1). This is achieved first by putting a string of thermal interface material(TIM) on the back side of the chip and, next, by pressing the heat sink against it. As a result, the materialis squeezed while it fills the space between the parallel plates.This problem is of interest to the electronic packaging industry for two reasons: first, it is a key step in theassembly process. Second, it affects the structural integrity and the thermal performance of the resultingelectronic module.

Surprisingly, no fundamental study has been reported in the open literature on this matter. The presentpaper remedies this situation.

A two-dimensional mathematical model for the squeeze-flow of a string of thermal interface materialis first developed and studied with regards to the assembly process. Next the effect of surface tension andof the heat transfer on the behaviour of the squeeze-flow is examined. Finally, a model is presented forthe Non-Newtonian behaviour of a TIM filled with suspensions of high thermal conductivity particles.

PROBLEM STATEMENT AND ASSUMPTIONS

A schematic representation of the problem is depicted in Fig. (2).

aFig. 2 : Schematic of a string of material squeezed between two rigid plates.

Transactions ofthe CSME Ide La SCGM Vol. 32, Nq, 3-4, 2008 468

A string of material of width a is placed between two rigid plates separated by an initial distance bo . At

time t = 0 , a constant force Fw is applied on the top plate. As a result, the top plate moves downward

with a velocity vw(t) and the material is squeezed and gradually fills the space between the plates.

Typical conditions for the assembly process of a heat sink to a flip-chip, hereupon called the standardconditions for the assembly process, are summarized in Table 1.

Table 1 : Standard conditions for the assemblv DrocessSymbol Parameter Mae:nitude

0'0 Maximum stress (N/m2) 9.87

/l viscosity (N*s/m2) 20

Ut Thermal diffusivity (m2/s) 9.4E-08

y Surface tension (N/m) 0.036

e Contact angle (radian) 0

kp Permeability (m2) 2E-13

Fw Pressing force (N) 100

L Dimension (m) 0.03

bo Initial plate separation (m) 0.003

bfinal Final plate separation (m) 0.00010

To TIM Temperature (K) 320

Tb Plate temperature (K) 300

llHB Non-Newtonian viscosity (N*sn/ m2) 161.41

n Exponent 0.66

p Density (kg/m3) 1700

tlot Squeezing time (s) 20

final volume3C;

initial volume

I:: factor 1.1712E-04

<lo Initial width of string (m) 0.003

V Volume (m3) 2.70E-07

k Thermal conductivity (W1m K) 8

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Seven assumptions are made regarding the modeling of the squeeze-flow:1. The thermal interface material (TIM) is incompressible and its flow is creeping.2. The flow velocity is time-independent.3. The distance separating the rigid plates (z -direction) is small with respect to their dimensions in

the x and y-directions, i.e., z «x,y.4. The pressure distribution inside the thermal interface material is two-dimensional P = P(x, y) .5. Surface tension is negligible.6. The flow is isothermal.7. The flow is Newtonian.

The assumption 3 neglects the influence of the terms that could be of importance at the beginning ofthe process when the width and the height of the string are similar, which should stand for a really smalltime. In the next section, a two-dimensional squeeze-flow model for a string of TIM is presented. Thismodel, called the basic two-dimensional string model, rests on the above assumptions. The model is nextimproved by removing assumptions 5, 6 and 7 in turn.

THE BASIC TWO-DIMENSIONAL STRING MODEL

Based on the foregoing assumptions, the mass and momentum conservation equations for the flow of asqueezed string of TIM may be sated as

(1)

(2)

(3)

According to assumption 1, the inertia terms in Eqs. (2) and (3) are negligible with respect to the viscous

a2vx a2vx a2vx a2vy a2vy a2vyterms. Moreover, from assumption 3, --2 '--2 «--2 and --2'--2 «--2 . As a result, Eqs.ax By az ax By az(2) and (3) reduce to

(4)

(5)

Both equations are integrated twice to yield the velocity distribution in the liquid layer

Transactions ofthe CSME Ide La SCGM Vol. 32, No. 3-4, 2008 470

(6)

(7)

1 ap ( 2 )vx(x,y,z) = 2Jt aX z -bz

1 ap ( 2 )vy(x,y,z) = 2Jt By z -bz

The velocity distributions of Eqs. (6) and (7) are then employed to determine the mass flow rates mx and

my, i.e.,

(8)

(9)

b L ap b3

mx = fLvx(z)pdz = -p---o 2Jt ax 6

b L OP b3

m = fLv (z)pdz = -p---y 0 y 2Jt By 6

However, from the mass conservation equation, Eq. (1),

(10)

or

(11)

bi8v 8v 8v JII a: + a; + a: dz = 0

Substituting Eqs. (8) and (9) into Eq. (11) and performing the integration results in a single differentialequation for P(x,y) ,

(12)a2 p a2 p--+--=ax2 By2

Eq. (12) is subjected to the symmetrical boundary conditions {y = ±~) = 0 and p(x = ±~ ) = O. Eq.

(12) is solved using the method of separation of variables. Its solution is

(13)

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where k = !!...-(2n +1). The initial pressure distribution (13) of the useful region is depicted in Fig. 3.n 2L

Fig. 3 : Pressure distribution inside the squeezed material.

It is seen that the pressure drops chiefly near the free surfaces, which mean all over the y-direction whileit remains nearly uniform in the direction perpendicular to it (x-direction). Substituting the pressuredistribution (13) back into Eqs. (6) and (7) yields the velocity components, i.e.,

(14)

(15)

( ) 3vw [ 2] 32~ (-It () . ( )vx x,y,z = -3 z -bz -8x+-L. ()cosh kny sm knx4b L n=O k 2 h knacos --

n 2

24 00 (-It+1

vy(x,y,z) = ~w [Z2 -bzl! ()Sinh(knY)COS(knX)b L n=O k 2 h knacos --

n 2

The applied force on the top plate Fw is now related to the downward velocity, i.e., the squeeze-rate,

vw (i.e., vw = -db / dt ) via the following relation

(16)~7i

Fw = f fp(x, y)dxdy

-~ -7i

Substituting the pressure distribution (13) back into Eq. (16) and carrying out the double integrationyields

(17)


a 24 ~ tanh(n-a(2n + 1)/(2L)) .for which £ = - - -5LJ ( )5 . Eq. (17) may be solved for the plate separatlOn

8L 7r n=O 2n + 1

bet) for two distinctive time periods of the squeezing process.

In the first case, it is assumed that the liquid layer has already reached the extremities of the rigidplates in the y-direction and it overflows. As a result, the width of the TIM string remains fixed, i.e.a*" a(t). In this case, the solution ofEq. (17) is

(18)[

F 1 1 ]-~b(t) = _W__t+_4J.u} £ b

o2

For convenience, Eq. (18) is cast in the following dimensionless form:

(19) [ ]-~

6= ~r+1 2

for which t5 =.!!-. and r =bo2 Fwt . A plot ofEq. (19) is shown in Fig. (4). For comparison purposes, the

bo

8,uA2

time-varying plate separation predicted with the simplified one-dimensional string model, i.e.,

_ [327r ]-~t5 = [2r+1] ~ and with the squeeze-drop flow model [1], i.e., t5 = -3-r+1 are also shown. The

one-dimensional string model was developed in a way similar to that of the two-dimensional string modelexcept for the fact that the pressure distribution is dependent on the y-direction only, i.e., P = P(y) . As

for the squeeze-drop flow model, it is a one-dimensional radial model, that is P = P(r) .

-Ic.o 10

-IDString-2DString-Drop

-210 oL----'------'---'---'----'---'-----'

20 40 60 80 100 120 140't

Fig. 4 : Time-varying dimensionless plate separation for a = constant.


This figure reveals that the squeeze-rate is very steep in the early stages of the squeezing process andthen it quickly levels off. The squeeze-rate is faster for the drop than for the string. The figure also showsthat the predictions of the one-dimensional string model are way off.

The second case seeks a solution for Eq. (17) in the early stages of the squeezing process that is beforethe free interface of the material reaches the y-extremities of the plates. In this case, the width of the stringis time-dependent, i.e., a = aCt) . Then,

(20)L/ b )da _ 2 72 a

-=2vy=- f fvy(x,-,z dzdxdt bL -7i 0 2

F h h · 1 fi d' . db db da drom t e c am ru e or envatlves, V w = -- = ---, andt da dt

(21)

Integration of Eq. (21) yields,

db = _ bL2 {f[tanh(k~a/2)]}-1

da 8 n=O kn

(22)

for which ; = a . Fig. (5) illustrates the predicted temporal variation of the dimensionless plateL

separation in the early stages of the squeezing process.

o10 .-----~--~--~--~-----,

-lDString--2D String--Drop

-210 '--__-'--__-'-__--'-__---'-_----.J

10 20 30 40't

Fig. 5: Time-varying dimensionless plate separation for the entire squeezing process.


Once again, it is seen that the predictions of the one-dimensional string model are inaccurate. Also, thepredicted squeeze-rate for the two-dimensional string model remains smaller than that for the drop model.Indeed, in order to yield the very same final layer thickness, the initial volume of TIM for the string must

be larger than that for the droplet (to compensate for the overflow). Since the applied force Fw is the

same for the three models, the resulting stress component (J'zz for the string model is smaller that that for

the drop model and so is its squeeze-rate.

THE EFFECT OF SURFACE TENSION

The basic two-dimensional string model developed in the previous section ignores the effect of surfacetension on the squeeze-flow. But is this effect influential during the assembly of the heat sink to the flipchip? This question is examined next by removing assumption no. 5 from the basic two-dimensionalstring model.

... .. 2rcoseSurface tensIOn IS taken mto account by Imposmg a pressure p,. = b(t) at both free surfaces of the

string. The contact angle e is shown in Fig. 6.

Fig. 6: Schematic of a string of material: Effect of surface tension.

Eq. (12) is then solved with this new pressure boundary condition and the resulting squeeze-rate becomes

(23) v = _ db = b3

F;ot (!)w dt 8L4 JL C

where F;ot =Fw + F p =Fw - P,aL . Eq. (23) is conveniently cast in the following dimensionless form

(24)

JlV Vwviscosity forces to capillary forces and defined as Ca = wo . Also, A = -- where vwo

reos e vwo

and A = L2. Fig.7 shows a plot of the dimensionless plate separation 8 in terms of Ca.

The dimensionless number characterizing the flow is the Capillary number representing the ratio of3bo Fw

8,uA2


o10 ,---~

I-Strmg~1-Drop 1imit

Region where effect ofsurface tension is important

·110

·310

·2CQIO

-410 L.

6,-----'--4------'--::-.2 --'-:c-"'---------"'----' 2

10 10 10 10Ca

Fig. 7 : Effect of surface tension on the plate separation.

This figure delineates the influence of surface tension on the flow. It is assumed here that the effect of

surface tension becomes negligible for Fp / Fw ::;; 0,1. Due to its larger free surface, the squeeze-flow of

the string appears to be more sensitive to the effect of surface tension than that of the droplet. However,under the standard conditions for the assembly process (Table 1), Ca = 11.57 for the string and thedroplet. It is seen that for this value of Ca, the effect of surface tension on the squeeze-flow is

insignificant for 0 ~ 10-4. Consequently, it is acceptable to ignore the effect of surface tension on thesqueeze-flow during the entire assembly process.

THE NON-ISOTHERMAL FLOW

Heating the TIM reduces its viscosity and, as a result, eases the squeeze-flow. This effect was alsoinvestigated. Assumption no. 6 from the basic two-dimensional string model was removed and a

temperature dependent relation for the viscosity of the form f-l = f-loeP(To-T) was suggested. The thermal

energy conservation equation for the squeeze-flow of a string of TIM may then be stated as

(25) (aT aT aT aTJ (aZT aZT aZTJpc -+v -+v -+v - =k --+--+-- +,wD

p at x ax Y By z az ax z Byz az z

in which k is the thermal conductivity and <I> is the viscous dissipation term, i.e.,

Eq. (25) is subjected to the very same assumptions stated in section 3 above. As a result,

a2T a2T aZT aT aT aT . ·····d h . fl b--2 + --Z <<--2 ' V - + v - «v - and VISCOUS dISSIpatIOn mSI e t e creepmg ow may eax By az -'ax YBy zaz


neglected. Eq. (25) reduces to

(27) (aT +v aT) = ka2TfX p at Z az az 2

Eq. (27) may be cast in the following dimensionless form

(28)aear

v bThe Peelet number of the flow is defined as Pe =~ where at is the thermal diffusivity. Also,

at

v z () v (77, r) T - Tr =t8

0~, 77 =-,8

0=8 50 ' Sz = z () and e= 0 . To is the initial temperature of the

bo b V w r Tb - ToTIM and Tb is the plate temperature.

Eq. (28) was solved with a finite-difference method. This solution gives the relation e(r) that reveals

the limit of the operating conditions for which heating the TIM is of interest, which has been fixed at

e> 0.01 after O.ls. This limit is shown in Fig. 8 in terms of the initial layer thickness of the TIM (bo )

versus the Peelet number of the flow, fixed by the operating conditions. The solution for the squeeze-dropflow problem is also reported in this figure. As expected, the thicker the initial layer of the TIM, the morebeneficial is the heating effect on the squeeze-flow. On the other hand, heating the TIM is increasinglyuseless as the Peelet number of the squeeze-flow increases. It is elear that for the standard assemblyconditions heating the TIM is of no interest.

- String limit-Drop limit._._. String

--Drop

Region where heating isuseful

3

3.5

2.5~'-' 2.Q<:I

1.5Region where heating is

useless.,....,..-~---~.

0.5 .---,...........--.-.----~.- '"----Squeezing conditions

PeFig. 8 : Effect of heating the TIM on the squeeze-flow.


SUSPENSIONS OF PARTICLES

In order to increase its thermal conductivity and consequently enhance heat dissipation, metallicparticles are sometimes added to the TIM. These spherical particles are usually made of silver and theirdiameters range from few microns to about 50 microns [8]. Experiments carried out by Prasher et al. [9]have shown that the behaviour of squeeze-flows with suspensions of particles may be approximated as aNon-Newtonian Herschel-Bulkley material at strains greater than the elastic limit. This limit is reachedwhen the flow velocity exceeds a critical velocity Dc given by the following relation [2]

(29)

n is the power law index, 77HB is the Herschel-Bulkley effective viscosity, Dc is the critical speed, k p is

the permeability of the flow and 0'0 is the yield stress. For typical TIMs, the magnitude of k p is of the

order of 2.10-13 m 2 [2]. Fig. 9 shows that for such fluids the squeeze-flow behaves as a Herschel-Bulkleymaterial. Therefore, the analysis presented in section 3 may be repeated here for Non-Newtonian fluids.

~ 10-5......'

;;;t

10-5 10-4

--k = 2*1O-13m2p

- k = 2*10-Um:2p

- k = 2*10-9m2p

region where HerschelBulk!ey model is vali

b(m)

Fig. 9 : Herschel-Bulkley flows for different permeabilities.

We recall the general momentum conservation equation, i.e.,

(30) ~ (pV) = -V· (pVv) - V.f> - - .(j + pgat

From the assumption that O'zy ,0'zx »0'zz' 0'xy' 0'xx' 0'yy' Eq. (30) yields the following stress components

in the x and y-directions respectively:

(31)

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0' = (~-z) apxz 2 ax

Vol. 32, No. 3-4, 2008 478

(32)

The shear stress is then (j = CTyzY + CTxz,x and its norm is given by

(33)

For CTyz 2 +CTxz 2 = I/CTI/2 2': CTo2, the squeeze-flow behaves as afor which p[(:r+(:JTHerschel-Bulkley material. For such materials, the stress/strain relation may be expressed as [3] :

(34) aij =('lHBY"-' +~ },

. tijrij d' Ovi 8v}The strain rate is defined as r = -- an rij = - +- .2 8x} 8x;

problem, f zy' f zx >> f zz , f xy , f xx' f yy and Eqs. (31) and (32) become

For the present squeeze-flow

(35)

(36)

Ov 8P/8x OvyHowever, from Eqs. (35) and (36), _x = --- and, as a result,8z 8P/8y 8z

(37) . _ (8PJ-1

Ovx _ (8P)-IOvyr-x - ---X - --8x 8z 8y 8z

With the help of equation (37), Eqs. (35) and (36) become:

(38)

(39)

Transactions ofthe CSME Ide la SCGM

Ovx = _ X (~Jl;;[(1- 2zJQ _I]l;;8z Q 77HB b

Ovy= _~(~Jl;;[(1- 2zJQ _I]l;;

8z Q 77HB b

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· ( \y, b b OP b OPfor whIch Q= X 2+y2 J2=--X, x=-- and y=--. On the other hand, for

2ao 2To ox 2'0 Oy

a yz 2 + a.tz

2 = Ilof < a 02

, the strain rate is null, i.e., r= 0 . Then, according to the Von Mises criterion

[4], the strain rate becomes null for (Jyz 2 + a xz 2 =Ilof = (J02

. At this point, the z-position is denoted

by Zy. Invoking this criterion and using Eqs. (31) and (32), the plate separation may then be expressed as

(40)

For Z < Zy' the Herschel-Bulkley material is accelerating. For Z ~ ZY , the flow is stagnant. Eqs. (38) and

(39) are solved for both theses flow regimes, i.e.,

(J1. { .~ }2 (J II b n 2z II 11+1

xn- ~~ "2 n+1 [(I-/;)0-1] -[0-1] 1. for z < Zy

(41) v =x y,;t' J" b n [ r1.-XO- _0_ --- 0-1 II for Z ~ ZyTJHB 2n+l

1. { "1. }2 (J II b n 2z II 11+1rn- kJ "2;;-;\ [(1-/;)0-1] -[0-1] 1. for Z < Zy

(42) v =y y,;-rn-'(~J·k_n-[O-lry,; for z ~ ZyTJHB 2 n + 1

The relation between the pressure distribution and the squeeze-ratevw is obtained by invoking Eq. (10),

i.e.,

(43)

Substituting Eqs. (41) and (42) into Eq. (43) and integrating yields


(44)

ax ay ax ay VW (77HBJ,Y,; .where If/x =X- +Y-, If/y = X- +Y- and Sn =-2-z -- .. It IS noted that, for theax ax 8y 8y b (T 0

. ay1D- model of the squeeze-flow of a Herschel-Bulkley matenal, n =-Y and If/y =Y 8y' and

Eq. (44) reduces to

(45) _2n_+_2 (_ Y -1 )zn+'y'; - 2(- Y -1r+,Y,; }2n+1

2

2.5

Eq. (45) is equivalent to the solution reported by Covey and Stanmore [5]. Furthermore, for a Newtonianfluid, (To = 0 and n = 1, and Eq. (44) reduces to Eq. (12) presented above.

As an example, Fig. 10 shows the temporal variation of the plate separation predicted with Eq. (44).The solution for the Newtonian fluid is also depicted.

X 10-3

3 ~::::..;;;;;:::;;:;;;;;:-'"-r===;;==;=~:::=:=:::;;::=:H--Herschel-Bulkley fluid--Newtouiau fluid

Herschel-Bulkley flui- with newtonian fluid

properties

=:;;1.5

0.5

0'--------'--------'-------'

time. (5)

Fig. 10 : Time-varying plate separation predicted for Newtonian and Non-Newtonian fluids.

This figure reveals that, in the very early stages of the squeezing process, the squeeze rate for the NonNewtonian fluid is two orders of magnitude larger than that of the Newtonian fluid. During this shortperiod of time, the string of Herschel-Bulkley material spreads quickly and, as a result, the magnitude ofits strain rate is significantly larger than the one prevailing in the Newtonian fluid. This behaviour is


illustrated in Fig. 11. It is seen that the apparent viscosity of the Herschel-Bulkley material diminishesexponentially with the strain rate and, for strain rates larger than 0.3 S-1 , it becomes smaller than that ofthe Newtonian fluid. On the other hand, for t ~ Is, Fig. 10 shows that the predictions for both models arenearly similar. Note that for t ~ 00 , the plate separation b becomes null for the Newtonian material. Forthe Herschel-Bulkley material however, the plates never touch (b *- 0) since the apparent viscosity Jiapp

becomes infinite as the strain rate r tends to zero. This is illustrated in Fig. 11.

250.-----.-------r------,..-----.-----,

-newtonian--Herschel-Bulkley1200

\...."~ 15010+---------------"'"'1Q~

fI:-~>

...,.Z 100;~c.-< 50

°0 2 4 6 8 10

strain rate (5-1)

Fig. 11 : Apparent viscosity versus strain rate.

Results reported thus far are valid for a particle volume fraction ¢ = 50%. The particle volume

fraction ¢ is defined as the ratio of the volume of particles to that of the particles and of the fluid.

However, as reported by Prasher et AI. [1] in Table 2, the particle volume fraction strongly influences thephysical properties of the Herschel-Bulkley material and, consequently, the behaviour of the NonNewtonian fluid.

t . If f th H h I B lidthf ff IT bl 2 En t ftha e : ec 0 e ar IC e vo ume rac Ion on e proper les 0 e ersc e - u ey rna ena.

Silicon based thermal interface material (5.16 N .s/m2)

¢ 1JHB 11 uy(N/m 2)

40% 74.097 0.7467 1.8350% 161.410 0.6600 9.8760% 444.210 0.4956 55.2

This is illustrated in Fig. 12. As ¢ increases, the initial squeeze-rate augments and the behaviour of the

fluid departs increasingly from that of a Newtonian fluid. Also, the fmal plate separation is larger formaterials with larger particle volume fractions. These findings are in agreement with the results reportedby Prasher et AI. [1].

Transactions ofthe CSME Ide la SCGM Vol. 32, No. 3-4,2008 482

X]Q';"3~'

2,5

'~.

.:t U

i

QL- ~-__,_=_-__,_-~~------',

';;''','',-,.,:5, ' ,..,'1;'6IV! :1,0';,

tim~;(:S) ,Fig. 12 : Time-varying plate separation for different particle volume fractions ¢.

CONCLUSION

The squeeze-flow problem of a thermal interface material that takes place during the pressing of a heatsink to the back side of a flip-chip was studied. The main conclusions that can be drawn from this studyare the following:

• The flow resulting from the squeezing of a string of material between two rigid plates is trulytwo-dimensional.

• The squeeze-flow is characterized by two regimes. The first regime prevails before the freeinterface reaches the extremities of the plates. In this case, the time-varying liquid layer thickness(and squeeze-rate) is determined with Eqs. (20) and (22). The second regime prevails once thefree interface has crossed the extremities, i.e., when the material overflows the plates. In thiscase, the time-varying liquid layer thickness (and squeeze-rate) is predicted by Eq. (18).

• The effect of surface tension on the squeeze-flow of the thermal interface material was found tobe negligible under the standard conditions.

• Heating the thermal interface material for reducing its viscosity and therefore enhancing itssqueeze-flow is of no interest for the actual assembly process.

• The thermal conductivity of the interface material may be enhanced with dispersed solid particles.For TIMs with particle volume fractions smaller than 30%, the behaviour of the squeeze-flowremains Newtonian. However, for particle volume fractions larger than 30%, its behaviourbecomes Non-Newtonian. In this case, the time-varying liquid layer thickness (and squeeze-rate)is determined with Eq. (44).

• Simulations have shown that the squeeze-flow of a TIM with suspensions of particles behave asa Non-Newtonian fluid in the early stages of the squcezing process, i.e. for t ~ Is. In the laterstages however (t > lOs ), the behaviour of the squ el'l'-flow is indistinguishable from that of aNewtonian fluid.

ACKNOWLEDGEMENTS

The authors are grateful to the Natural Sciences and Engineering Research Council of Canada and to theFonds Quebecois de la Recherche sur la Nature et les Technolos ics for their fmancial support.

Transactions ofthe CSME /de fa SCGM Vol. 32, No. 3-4, 2008 483

REFERENCES

[1] Prasher, RS., Shipley, J.e., Prstic, S., Koning, P., Wang, J., Rheological Study of Micro ParticleLaden Polymeric Thermal Interface Materials: Modeling Part 2, America Society of MechanicalEngineers, Proceeding of the Heat Transfer Division - 2002 Volume 7: Heat Transfer in ElectronicEquipment, vol. 7, 2002, p. 53-59.

[2] Chaouche, M., Chaari, F., Racineux, G., Poitou, A., Comportement de fluides pateux en ecoulementd'ecrasement, Rheologie, vol. 3,2003, p. 46-52.

[3] Zhang, J., Khayat, R, Noronha, P., Three-dimensional lubrication flow of a Herschel-Bulkley fluid,International Journal for Numerical Methods in Fluids, vol. 50, 2006, p.511-530.

[4] J.D.Sherwood, D.Durban, Squeeze-flow of a Herschel-Bulkley fluid, J. Non-Newtonian Fluid Mech.,vol. 77, pp. 115-121, 1998.

[5] Covey, G.H., Stanmore, B.R., Use ofthe parallel-plate plastometer for the characterisation ofviscousfluids with a yield stress, Journal ofNon-Newtonian fluid mechanics, vol. 8, 1981, pp. 249-260.

[6] Philip J. Leider et RByron Bird, Squeezing Flow between Parallel Disks. 1. Theoretical Analysis, Ind.Eng. Chern., Fundam., vol. 13, no. 4, 1974, pp. 336-341.

[7] Delhaye, N., Poitou, A., and Chaouche, M., Squeeze Flow of Highly Concentrated Suspensions ofSpheres, J. Non-Newtonian Fluid Mech., vol. 94, pp. 67-74, 2000.

[8] Huang, W., Stan, RM., Skadron, K. Parameterized Physical Compact Thermal Modeling, IEEETransactions on Components, Packaging, and Manufacturing Technology Part A, vol. 28, nO 4, 2005, p.615-621.

[9] Prasher, RS., Shipley, J., Prstic, S., Koning, P., Wang, J.-L. Thermal Resistance of Particle LadenPolymeric Thermal Interface Materials, Transactions of the ASME, vol. 125,2003, p. 1170-1175.

NOMENCLATURE

A Area (m2)

b Plate separation (m)

cp Heat capacity (J .kg -1 . K- 1)

Fw Pressing force (N)

g Gravitational acceleration (m .s -2 )

k Thermal conductivity (W . m-1K-1)

kp Permeability (m2)

L Flip-chip width (m)m Mass flow raten power law index

P Pressure (N . m-2)

Pr Pressure at the free surface ( N .m-2 )

t Time (s)T Temperature (K)Tb Plate temperature (K)

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TIM

Dcv

vx,y,z

Thermal Interface Material

Critical velocity ( m . s -1 )

Velocity (m· S-I)

Squeeze-rate (m . s -1 )

Mean velocity (m . s -1 )

Volume (m 3)

Coordinates (m)

Greek symbols

a Width of TIM string (m)

at Thermal diffusivity (m 2• s-1 )

TJHB Herschel-Bulkley consistency index (No sn .m -2 )

r Surface tension (N .m -I )

f Strain rate (S-I)

e Contact angle between the TIM and the plate (radian)

1.1. Viscosity (N· s· m-2)

fl app Apparent viscosity (N· s· m-2)

p Density (kg· m-3)

a Shear stress (N .m-2 )

a 0 yield stress (N .m-2 )

¢ Particle volume fraction

Subscripts

ototx,y,z

Relative to the initial conditionsTotalRelative to the coordinates


Transactions ofthe CSME /de fa SCGM Vol. 32, No. 3-4, 2008 486

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