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BOOKCOMP, Inc. — John Wiley & Sons / Page 976 / 2nd Proofs / Heat Transfer Handbook / Bejan 976 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [976], (30) Lines: 848 to 881 ——— -0.36981pt PgVar ——— Normal Page * PgEnds: Eject [976], (30) Figure 13.12 Surface chemistry model of contact resistance. k = k m 1 + SBϕ 1 − Bψϕ (13.40) where k m is the thermal conductivity of the continuous phase or base polymer, ϕ the particle volume fraction, and S 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 = k p /k m − 1 k p /k m + S (13.41) where k p is 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 Spheres 1 1.5 Randomly oriented rods 2 1.58 4 2.08 6 2.8 10 4.93 15 8.38 Source: Cross (1996). BOOKCOMP, Inc. — John Wiley & Sons / Page 977 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 977 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [977], (31) Lines: 881 to 929 ——— 0.33461pt PgVar ——— Normal Page PgEnds: T E X [977], (31) ψ = 1 + 1 − ϕ m ϕ 2 m φ (13.42) where ϕ m is the maximum packing fraction. Table 13.7 lists values of ϕ m for spheres and rods in different packing configurations. Another commonly used model for predicting 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 higher percentages 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 =  k f (p − p c )  0.95±0.5 (13.43) where p c is the volume fraction at the percolation threshold and p 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 value p 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 φ m Shape of Particles Type of Packing φ m Spheres Face-centered cubic 0.7405 Hexagonal close 0.7405 Body-centered cubic 0.6 Simple cubic 0.524 Random close 0.637 Random loose 0.601 Rods or fibers Uniaxial hexagonal close 0.907 Uniaxial simple cubic 0.785 Uniaxial random 0.82 Three-dimensional random 0.52 Source: Cross (1996). BOOKCOMP, Inc. — John Wiley & Sons / Page 978 / 2nd Proofs / Heat Transfer Handbook / Bejan 978 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [978], (32) Lines: 929 to 949 ——— 1.63112pt PgVar ——— Normal Page PgEnds: T E X [978], (32) 010 3020 40 50 60 70 80 0 2 4 6 8 10 12 14 16 Conductivity (W/m . K) Ϫ11 Percentage filler (Al O ) 23 Percolation model Experimental data EMT (Maxwell- Eucken equation) Nielsen Model Figure 13.13 Comparing the percolation model with experimental data and other existing models for a bimodal distribution of Al 2 O 3 filler (65 µm: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 k m =  k p (1 + 2α) + 2k m  + 2ϕ  k p (1 − α) − k m   k p (1 + 2α) + 2k m  − ϕ  k p (1 − α) − k m  (13.44) where ϕ is the volume fraction of the particles and α is given by α = R mc k m r (13.45) where r is the radius of the particle and R mc is the interface resistance between the particle and the matrix. Higher values of α, which can be due to either higher values of R mc , higher values of k m , 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 BOOKCOMP, Inc. — John Wiley & Sons / Page 979 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 979 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [979], (33) Lines: 949 to 956 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [979], (33) Figure 13.14 Measured thermal contact conductance of a sodium silicate–based thermal in- terface material, as afunction of volume percentage of boron nitride (BN) particles, normalized with respect to h c for 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 of α 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 smaller A real then 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 BOOKCOMP, Inc. — John Wiley & Sons / Page 980 / 2nd Proofs / Heat Transfer Handbook / Bejan 980 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [980], (34) Lines: 956 to 980 ——— 3.3861pt PgVar ——— Normal Page * PgEnds: Eject [980], (34) 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 for other harsh 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 mC p (13.46) where q 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 C p 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) + ∆T ss  1 − e −hAt/mC p  (13.47) where ∆T ss is the steady-state temperature determined by the convection relation of eq. (13.5) and mC p /hAis 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 resistance 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 k (13.48) BOOKCOMP, Inc. — John Wiley & Sons / Page 981 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 981 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [981], (35) Lines: 980 to 1008 ——— -0.773pt PgVar ——— Normal Page PgEnds: T E X [981], (35) where h is the heat transfer coefficient at the external surface, k the thermal conduc- tivity of the solid, and L 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 depth δ can be expressed approximately as (Eckert and Drake, 1987) δ = √ 12αt (13.49) where α 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. Following Mix and Bar-Cohen (1992), the temporal behavior of chip packages can be classified into four time intervals: an early or chip period, when effects are confined to the chip; an intermediate period, when the die bond and local encapsulant (or package 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 power dissipation 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. BOOKCOMP, Inc. — John Wiley & Sons / Page 982 / 2nd Proofs / Heat Transfer Handbook / Bejan 982 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [982], (36) Lines: 1008 to 1028 ——— 0.2472pt PgVar ——— Long Page * PgEnds: Eject [982], (36) 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 4αt erfc  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 the ρC 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 or encapsulant: q s q s + q e = (  ρC p k) s (  ρC p k) s + (  ρC p k) e (13.51) BOOKCOMP, Inc. — John Wiley & Sons / Page 983 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 983 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [983], (37) Lines: 1028 to 1080 ——— -0.02003pt PgVar ——— Long Page PgEnds: T E X [983], (37) For typical 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/or in 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 hA (13.52) where  (ρC p V) n is the summed lumped heat capacity of all the materials consti- tuting the package and A 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) n is approximately equal to the heat capacity of the plastic en- capsulant. The IC package time constant τ ICP 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 for the time interval τ <t<t ss . 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 )(∆T r,z ) ss (13.53) where (∆T r,z ) ss represents 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 BOOKCOMP, Inc. — John Wiley & Sons / Page 984 / 2nd Proofs / Heat Transfer Handbook / Bejan 984 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [984], (38) Lines: 1080 to 1080 ——— 11.854pt PgVar ——— Normal Page PgEnds: T E X [984], (38) 0.0000 0.0010 10 20 30 40 50 Temperature (°C) 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). 0.00 0.02 0.04 0.06 0.08 0.10 20 21 23 24 22 26 25 Temperature (°C) 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). BOOKCOMP, Inc. — John Wiley & Sons / Page 985 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 985 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [985], (39) Lines: 1080 to 1080 ——— 10.854pt PgVar ——— Normal Page PgEnds: T E X [985], (39) 0 100 200 300 400 500 600 20 30 40 60 50 Temperature ( C)° 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). 0 100 200 300 400 500 600 20 30 40 60 50 Temperature ( C)° Time (s) Pulse: 55.6 W; Steady: 0.862 W Top surface of Silicon Analytical Node 401 Analytical Figure 13.20 Comparison of model as a function of analytical predictions (quasi-steady). . 0.82 Three-dimensional random 0.52 Source: Cross ( 1996 ). BOOKCOMP, Inc. — John Wiley & Sons / Page 978 / 2nd Proofs / Heat Transfer Handbook / Bejan 978 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [978],. BOOKCOMP, Inc. — John Wiley & Sons / Page 976 / 2nd Proofs / Heat Transfer Handbook / Bejan 976 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [976],. twenty-first century BOOKCOMP, Inc. — John Wiley & Sons / Page 980 / 2nd Proofs / Heat Transfer Handbook / Bejan 980 HEAT TRANSFER IN ELECTRONIC EQUIPMENT 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 [980],

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