Lumped Heat Capacity For an internally heated solid of relatively high ther-mal conductivity which is experiencing no external cooling, solution of the energy equation reveals that the t
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Figure 13.12 Surface chemistry model of contact resistance
k = km1+ SBϕ
wherek mis the thermal conductivity of the continuous phase or base polymer,ϕ the
particle volume fraction, andS a shape parameter that increases with aspect ratio.
Table 13.6 provides the value of A for dispersed polymers The constant B in eq.
(13.40) can be estimated using the expression
B = kp /km− 1
wherekpis the thermal conductivity of the filler and
TABLE 13.6 Values of A for Several Dispersed Types
Aspect Ratio of Dispersed Phase (Length/Diameter) A
Source: Cross (1996).
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ψ = 1 +1− ϕm
ϕ2
whereϕmis the maximum packing fraction Table 13.7 lists values ofϕmforspheres and rods in different packing configurations
Anothercommonly used model forpredicting thermal conductivity of two-phase
systems is effective medium transport (EMT), discussed by Devpura et al (2000).
Both the EMT and Lewis–Nielsen models were developed for moderate filler density (up to around 40% by volume) and often fail to predict the thermal conductivity at higherpercentages of the filler The Nielsen model becomes unstable above a 40%
volume fraction, as shown by Devpura et al (2000), whereas the EMT model under-predicts the thermal conductivity above 40%
Devpura et al (2000) have proposed a new model, based on the formation of a percolation network of the filler, for calculating the thermal conductivity of high-volume-fraction particle-laden systems The change in the conductivity of the matrix from its value at the percolation threshold (percentage of filler particles at which percolation starts) is given by
∆k =kf (p − pc)0.95±0.5 (13.43) wherepc is the volume fraction at the percolation threshold andp is the volume
fraction The thermal conductivities calculated using the percolation, EMT, and Lewis-Nielsen models are shown in Fig 13.13, where the percolation model appears
to provide a useful upper bound on the thermal conductivity Unfortunately, how-ever, the threshold valuep c needed in eq (13.43) can only be determined from a full numerical simulation Devpura et al (2000) have provided an algorithm for the percolation modeling of particle-laden systems
The effective thermal conductivity of a particle-laden polymeric system is also dependent on the interfacial resistance between the particle and the matrix Devpura
TABLE 13.7 Maximum Packing Fraction φm
Uniaxial simple cubic 0.785
Three-dimensional random 0.52
Source: Cross (1996).
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0 2 4 6 8 10 12 14 16
K)
Percolation model Experimental data EMT (Maxwell-Eucken equation) Nielsen Model
Figure 13.13 Comparing the percolation model with experimental data and other existing models fora bimodal distribution of Al2O3filler(65µm : 9µm = 4 : 1) in polyethylene
matrix(k f = 42.34/W/m · K, k m = 0.36 W/m · K, y = 25, z = 40) (From Devpura et al.,
2000.)
et al (2000) and Davis and Artz (1995) have shown that below a critical dimension
of the particles, the thermal conductivity of a two-phase polymeric system may decrease relative to the thermal conductivity of the matrix, despite the use of highly conducting fillers, owing to high interfacial resistance at the particle–matrix interface
For spherical particles in low volume fractions, Davis and Artz (1995) have used the
effective medium theory to provide an expression for the thermal conductivity for a
particle-laden system:
k
km =
kp(1 + 2α) + 2km+ 2ϕkp (1 − α) − km
k p (1 + 2α) + 2k m− ϕk p (1 − α) − k m (13.44)
whereϕ is the volume fraction of the particles and α is given by
α = Rmckm
wherer is the radius of the particle and Rmcis the interface resistance between the particle and the matrix Higher values ofα, which can be due to eitherhighervalues
ofRmc, highervalues ofkm, or smaller radius of the particles, will lead to a decrease
in the thermal conductivity of the particle-laden systems Nan et al (1997) provide a comprehensive comparison of EMT with the interfacial term in it with experimental
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Figure 13.14 Measured thermal contact conductance of a sodium silicate–based thermal in-terface material, as a function of volume percentage of boron nitride (BN) particles, normalized with respect toh cfor no BN particles (From Xu et al., 2000.)
data available in the literature It is to be noted that eq (13.44) can be used for
a low-volume fraction, probably up 40% For a higher-volume fraction, numerical techniques such as the percolation model of Devpura et al (2000) must be used
Devpura et al (2000) have shown that increasing values ofα increases the percolation
threshold The thermal characteristics of other types of high-performance thermally enhanced interface materials are described in Madhusudan (1995)
Effect of Filler Concentration on Mechanical Strength Along with the beneficial effect on thermal conductivity, increasing the particle volume fraction results in an increase in the mechanical rigidity of the material, as reflected in the variation in the shear modulus of particle-laden systems shown by Lewis and Nielsen (1970) Consequently, at a given pressure, a higher particle volume fraction in the interface material may result in a higher bond-line thickness and smallerArealthen those for a material with a low particle volume fraction As a result of this trade-off between thermal conductivity and mechanical rigidity, the minimum thermal resistance may not occur at the maximum particle loading condition No analytical study concerning this phenomenon has been reported in the literature; however, Xu
et al (2000) did confirm that the minimum thermal resistance occurs at less than the maximum particle loading Their results, presented in terms of contact conductance, which is the inverse of resistance, are shown in Fig 13.14
13.3.3 First-Order Transient Effects
Variations in component power dissipation, including power-up and power-down protocols, power-line surges, and lightening strikes, as well as performance-driven fluctuations, can result in significant thermal transients at all the relevant packaging levels Common commercial practice in the early years of the twenty-first century
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generally involves design and analysis for the worst-case steady-state conditions
However, detailed design and development of avionics for terrestrial and space appli-cations, as well as equipment design forotherharsh environments, often includes de-tailed numerical modeling of the complete performance- and environmentally driven temporal temperature variations in critical devices Moreover, predictions of die at-tach, wire bond, solder bump, and encapsulant failure rates—even under more benign circumstances—often require knowledge of the history of the die-bond temperature gradient and the temperature difference between the chip and the encapsulant A de-tailed treatment of thermal transients in electronic equipment, on multiple length and time scales, is beyond the scope of the present discussion Nevertheless, some in-sight into these effects can be gained from the use of judiciously selected first-order equations
Lumped Heat Capacity For an internally heated solid of relatively high ther-mal conductivity which is experiencing no external cooling, solution of the energy equation reveals that the temperature will undergo a constant rise rate, according to
dT
dt =
q
whereq is the rate of internal heating (W), m the mass of the solid (kg), and C p the specific heat of the solid (J/kg · K) Values of Cp are given in Table 13.3 for a wide variety of packaging materials Equation (13.46) assumes that internal temperature variations are small enough to allow the entire solid to be represented by a single
temperature This relation, frequently called the lumped-capacity solution, can be
used with confidence when the thermal conductivity of the solid is high If the solid
of interest is subjected to convective heating or cooling by an adjacent fluid, the tem-perature can be expected to rise to an asymptotic, steady-state limit If the convective heat exchange is represented by a heat transfer coefficient boundary condition, the temperature of the solid is found to vary as
T (t) = T (0) + ∆Tss1− e −hAt/mC p
(13.47)
where∆Tssis the steady-state temperature determined by the convection relation of
eq (13.5) andmCp /hA is the thermal time constant of this solid.
Heat flow from such a convectively cooled solid to the surrounding fluid encoun-ters two resistances, a conduction resistance within the solid and a convection re-sistance at the external surface When the internal rere-sistance is far smaller than the external resistance, the temperature variations within the solid may be neglected and use made of the lumped capacity solution The Biot number Bi, representing the ratio
of the internal conduction resistance to the external convective resistance, can be used
to determine the suitability of this assumption:
Bi= internal conduction resistance
external surface convection resistance = L/kA
1/hA =
hL
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whereh is the heat transfer coefficient at the external surface, k the thermal
conduc-tivity of the solid, andL the characteristic dimension, best defined by the quotient of
the volume divided by the external surface area For Bi< 0.1 it is generally acceptable
to determine the solid temperature with the lumped capacity approximation
Thermal Wave Propagation Thermal diffusion into a previously unheated solid can be viewed as an ever-expanding wave whose propagation rate is determined by the thermal diffusivity of the material For one-dimensional heat flow into a solid with invariant properties, which experiences a step change in surface temperature, the penetration depthδ can be expressed approximately as (Eckert and Drake, 1987)
whereα is the thermal diffusivity, equal to k/ρC p Strictly speaking, this relation can only be used to determine the location of the thermal front in a homogeneous solid However, with a stepwise change in properties and/or a judicious choice of the effective thermal diffusivity, it can often provide insight into the thermal behavior of the chip, substrate, package, or module affected by the thermal transient of interest
Chip Package Transients In a typical IC package (see Fig 13.15), the flow of heat from the active layer of the silicon chip through the die bond and encapsulant to the external package surfaces provides inherent time intervals for the thermal model-ing and analysis of IC packages Followmodel-ing Mix and Bar-Cohen (1992), the temporal
behaviorof chip packages can be classified into fourtime intervals: an early or chip period, when effects are confined to the chip; an intermediate period, when the die bond and local encapsulant (orpackage case) are involved; the quasi-steady period,
when the entire package is responding to the dissipation and transfer of heat; and
fi-nally, the steady-state period, when the temperatures everywhere have stabilized In
Fig 13.16 a semilog plot is used to display the temporal temperature variation of the active surface of the silicon for a convectively cooled package subjected to a constant powerdissipation and a 1-ms duration pulse of energy followed by a constant power dissipation A comparison of these temperature variations reveals the pulse-constant
Figure 13.15 Typical plastic chip package
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Figure 13.16 Thermal response of a typical plastic package
condition to generate a far more complex response than the constant power condition
It is therefore this condition that will serve as the vehicle for this brief exploration of the transient thermal behavior of a plastic IC package
During the initial period, heat dissipated by the circuitry just below the surface of the chip propagates into the silicon and across the nearby interface into the encap-sulant During this chip period, of order 0.0 < t < 1 ms, the chip temperature can
be expected to follow eq (13.50), commonly used to determine the temperature of a semi-infinite body that is subjected to a uniform heat flux (Eckert and Drake, 1987):
T (x, t) = q k
4αt
π e −x
2/4αt − x
π
4αterfc
x
√
4αt
(13.50)
It must be noted, however, that the presence of encapsulant above the chip reduces the heat flowing through the chip, in proportion to the ratio of the thermal effusivities, defined as theρC p k product, of the joined materials Assuming perfect contact
be-tween the silicon and the encapsulant and uniform power dissipation at the interface, the effusivities can be used to define a partitioning coefficient for the silicon that can
be used to obtain the net heat flow into the silicon orencapsulant:
qs
qs + q e = (
ρC pk)s (ρC p k) s + (ρC p k) e (13.51)
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Fortypical encapsulants (e.g., polysulfides, polyurethane, epoxides), the effusivity
of silicon is found to dominate this relation, and as a result, 90% or more of the heat generated diffuses into the silicon In recognition of the contact resistance between the encapsulant and the chip and/orin the interest of an upper-bound estimate, the chip temperature in eq (13.50) can be evaluated at the full surface heat flux (i.e.,q e
is negligible)
Although the internal thermal response of a multilayered structure can be related
to the thermal time constants for diffusion across individual layers, it is often the convective time constant for the entire package that determines the gross thermal behavior A composite convective thermal time constant for a plastic IC package can
be expressed as
τ =
n
(ρC p V ) n
where
(ρCp V )nis the summed lumped heat capacity of all the materials consti-tuting the package andA is the surface area available for external heat transfer For
a typical plastic package in which the mass of the chip and metallization is nearly negligible,
(ρC p V ) nis approximately equal to the heat capacity of the plastic en-capsulant The IC package time constantτICP can be expected to apply when the thermal front has reached the external surfaces (or case) of the package and is found
to be valid forthe time intervalτ < t < tss Using the package time constant, the temperature during this time period can be expressed to a first approximation by a relation of the form
∆T r,z = (1 − e −t/τ )(∆Tr,z)ss (13.53) where(∆Tr,z)ssrepresents the steady-state temperature at the location of interest Fig-ures 13.17 through 13.20 display the results of finite-element simulations for a typical plastic package (Mix and Bar-Cohen, 1992) and lend credence to the existence of dis-tinct time domains in the thermal transient behavior of an IC package Comparison of these values with the temperatures obtained from the analytical relations, eqs (13.50) through (13.53), suggest that judiciously selected analytical relations can yield results that are within some 5% of the more detailed computational results
13.3.4 Heat Flow in Printed Circuit Boards
Anisotropic Conductivity Prediction of the temperature distribution in a con-ductively cooled printed circuit board (PCB) necessitates modeling of heat flow in a multilayer composite structure with two materials (electrical conductor and dielec-tric) of vastly different thermal properties The complex heat flow patterns that de-velop in the PCB as a result of heat diffusing from high-power components into the surrounding board and/or as heat flows toward the cooled edges of the board may
be analyzed with the aid of a resistive network The multidimensional nature of the heat flow requires that effective thermal resistances be determined for each of the pri-mary directions, although often a bi-directional description distinguishing between
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10 20 30 40 50
Time (s)
Top surface of Si
Top surface of d/a
Bottom surface of d/a
Half-thickness (bottom encaps)
Node 401 Node 161 Node 41 Node 201 Analytical Analytical
Figure 13.17 Temperature as a function of time (early time)
20 21
23 24
22
26 25
Time (s)
Top surface of Si Top surface of d/a
Bottom surface of d/a
Half-thickness (bottom encaps)
Node 401 Node 161 Node 201 Node 41
Figure 13.18 Temperature as a function of time (intermediate time)
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20 30 40
60
50
Time (s)
Top surface of Si Top surface of d/a
Bottom surface of d/a
Half-thickness (bottom encaps)
Node 401 Node 161 Node 201 Node 41
Figure 13.19 Temperature as a function of time (quasi-steady)
20 30 40
60
50
Time (s)
Pulse: 55.6 W; Steady: 0.862 W
Top surface of Silicon
Analytical
Figure 13.20 Comparison of model as a function of analytical predictions (quasi-steady)