BOOKCOMP, Inc. — John Wiley & Sons / Page 966 / 2nd Proofs / Heat Transfer Handbook / Bejan 966 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 [966], (20) Lines: 634 to 676 ——— 8.44412pt PgVar ——— Normal Page PgEnds: T E X [966], (20) suggested that when the grouping, NT/φ, where N is the number of atomic layers in the thin film, T the temperature, and φ the Debye temperature, is less than unity, the heat capacity can be expected to display sensitivity to the characteristic dimension. Their results show that such size effects on the thermodynamic properties are more important at cryogenic temperatures. 13.3.1 Spreading Resistance In chip packages that provide for lateral spreading of the heat generated in the chip, the increasing cross-sectional area for heat flow in the “layers” adjacent to the chip reduces the heat flux in successive layers and hence the internal thermal resistance. Unfortunately, however, there is an additional resistance associated with this lateral flow of heat, which must be taken into account in determination of the overall chip package temperature difference. The temperature difference across each layer of such a structure can be expressed as ∆T = qR T (13.19) where R T = R 1D + R sp (13.20) or R T = ∆x kA + R sp (13.21) For the circular and square geometries common in microelectronic applications, Negus et al. (1989) provide an engineering approximation for the spreading resistance R sp of a small heat source on a thick substrate or heat spreader, insulated on the sides and held at a fixed temperature along the base as R sp = (0.475 − 0.626 + 0.13ζ) 3 k √ a (13.22) where ζ is the square root of the heat source area divided by the substrate area, k the thermal conductivity of the substrate, and a the area of the heat source. The spreading resistance R sp from eq. (13.22) can now be added to the one- dimensional conduction resistance to yield the overall thermal resistance of that layer. It is to be noted that the use of eq. (13.22) requires that the substrate be three to five times thicker than the square root of the heat source area. Consequently, for relatively thin layers on thicker substrates, such as thin lead frames or heat spreaders interposed between the chip and the substrate, eq. (13.22) cannot be expected to provide an acceptable prediction of R sp . Instead, use can be made of the numerical results plotted in Fig. 13.6 to obtain the appropriate value of the spreading resistance. BOOKCOMP, Inc. — John Wiley & Sons / Page 967 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 967 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 [967], (21) Lines: 676 to 699 ——— 0.96115pt PgVar ——— Normal Page PgEnds: T E X [967], (21) Figure 13.6 Spreading resistance on thin layers. Kennedy (1959) analyzed heat spreading from a circular uniform heat flux source to a cylindrical substrate with isothermal temperature boundaries at the edges, the bottom, or both, and presented the results in graphic form. Spreading resistance charts prepared originally by Kennedy (1959) were reproduced by Sergent and Krum (1994). Although the boundary conditions and geometry assumed by Kennedy (1959) do not match the mixed boundary conditions and rectangular shapes found in most electronic packages, the spreading resistance results can be used with acceptable accuracy in many design situations (Simons et al., 1997). Using the spreading re- sistance factor H from the appropriate Kennedy graph, the spreading resistance is calculated using R sp = H kπ √ a (13.23) where k is the thermal conductivity and a is the heat source area (Figs. 13.7 to 13.9). Song et al. (1994) developed an analytical model to estimate the constriction– spreading thermal resistance R sp from a circular or rectangular heat source to a sim- ilarly shaped convectively cooled substrate. Lee et al. (1995) extended the solutions provided by Song et al. (1994) to present closed-form expressions for dimensionless constriction–spreading thermal resistance based on average and maximum tempera- ture rise through the substrate. Their set of equations is R sp,avg = 0.5(1 − ζ) 3/2 φ c k √ A source (13.24) BOOKCOMP, Inc. — John Wiley & Sons / Page 968 / 2nd Proofs / Heat Transfer Handbook / Bejan 968 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 [968], (22) Lines: 699 to 709 ——— * 15.91125pt PgVar ——— Normal Page * PgEnds: Eject [968], (22) R sp,max = 1/ √ π (1 − ζ) 3/2 φ c k √ A source (13.25) φ c = tanhλ c τ + (λ c · Bi) 1 + (λ c · Bi) tanhλ c τ (13.26) λ c = π + 1 e √ π (13.27) τ = δ b (13.28) Bi = hr k (13.29) Figure 13.7 Spreading resistance factor H 1 for surface z = w at zero temperature. (From Simons et al., 1997.) BOOKCOMP, Inc. — John Wiley & Sons / Page 969 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 969 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 [969], (23) Lines: 709 to 725 ——— 0.25099pt PgVar ——— Normal Page PgEnds: T E X [969], (23) 0.01 0.1 1.0 0.01 0.1 1.0 10 ab/ Spreading Resistance Factor H 1 w b a z T =0 0.01 0.015 0.02 0.025 0.03 0.04 0.05 0.06 0.08 0.10 0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.50 2.00 2.50 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 wb/ r Figure 13.8 Spreading resistance factor H 2 for surface z = w at zero temperature. (From Simons et al., 1997.) In eqs. (13.24)–(13.29), R sp,ave is the constriction resistance based on the average source temperature, R sp,max the constriction resistance based on the maximum source temperature, δ the fin thickness, r the outer radius of the substrate, h the convective heat transfer coefficient, and k the thermal conductivity. The authors claim the ex- pressions to be accurate to within 10% for a range of source and substrate shapes and for source and substrate rectangularity aspect ratios less than 2.5. Use of the convective boundary condition on the substrate base makes these relations especially BOOKCOMP, Inc. — John Wiley & Sons / Page 970 / 2nd Proofs / Heat Transfer Handbook / Bejan 970 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 [970], (24) Lines: 725 to 738 ——— -1.57901pt PgVar ——— Normal Page PgEnds: T E X [970], (24) Figure 13.9 Spreading resistance factor H 3 for surface z = w at zero temperature. (From Simons et al., 1997.) well suited to the analytical determination of the spreading resistance of a chip-to- heat sink assembly as well as a chip encapsulated in a convectively cooled plastic package. 13.3.2 Heat Flow across Solid Interfaces Heat transfer across an interface formed by the joining of two solids is accompanied by a temperature difference caused by imperfect contact between the two solids. Even when perfect adhesion is achieved between the solids, the transfer of heat is impeded by the acoustic mismatch in the properties of the phonons on either side of the interface. Traditionally, the thermal resistance arising due to imperfect contact has been called the thermal contact resistance. The resistance due to the mismatch in the acoustic properties is usually termed the thermal boundary resistance. The thermal contact resistance is a macroscopic phenomenon, whereas thermal boundary resistance is a microscopic phenomenon. Thermal Contact Resistance When two surfaces are joined, as shown in Fig. 13.10, asperities on each of the surfaces limit the actual contact between the two BOOKCOMP, Inc. — John Wiley & Sons / Page 971 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 971 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 [971], (25) Lines: 738 to 763 ——— -2.1539pt PgVar ——— Normal Page * PgEnds: Eject [971], (25) Figure 13.10 Contact and heat flow at a solid–solid interface. solids to a very small fraction, perhaps just 1 to 2% for lightly loaded interfaces, of the apparent area. As a consequence, the flow of heat across such an interface involves solid-to-solid conduction in the area of actual contact, A co , and conduction through the fluid occupying the noncontact area, A nc , of the interface. At elevated tempera- tures or in vacuum, radiation heat transfer across the open spaces may also play an important role. The pressure imposed across the interface, along with microhardness of the softer surface and the surface roughness characteristics of both solids, deter- mine the interfacial gap δ and the contact area A co . Assuming plastic deformation of the asperities and a Gaussian distribution of the asperities over the apparent area, for the contact resistance R co , Cooper et al. (1969) proposed R co = 1.45 k s (P/H ) 0.985 σ | tan θ| −1 (13.30) where k s is the harmonic mean thermal conductivity, defined as k s = 2k 1 k 2 /(k 1 +k 2 ); P the apparent contact pressure; H the hardness of the softer material; and σ the root- mean-square (rms) roughness, given by σ 1 = σ 2 1 + σ 2 2 (13.31) where σ 1 and σ 2 are the roughness of surface 1 and 2, respectively. The term |tan θ| in eq. (13.30) is the average asperity angle: | tan θ| 2 =|tan θ 1 | 2 +|tan θ 2 | 2 (13.32) This relation neglects the heat transfer contribution of any trapped fluid in the inter- facial gap. In the pursuit of a more rigorous determination of the contact resistance, Yovano- vich and Antonetti (1988) found it possible to predict the area-weighted interfacial gap, Y , in the form BOOKCOMP, Inc. — John Wiley & Sons / Page 972 / 2nd Proofs / Heat Transfer Handbook / Bejan 972 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 [972], (26) Lines: 763 to 790 ——— -0.7157pt PgVar ——— Short Page PgEnds: T E X [972], (26) Y = 1.185σ −ln 3.132P H 0.547 (13.33) where σ is the effective rms as given by eq. (13.31), P the contact pressure (Pa), and H the surface microhardness (Pa) of the softer material, to a depth of the order of the penetration of the harder material. Using Y as the characteristic gap dimension and incorporating the solid–solid and fluid gap parallel heat flow paths, Yovanovich (1990) derived for the total interfacial thermal resistance, R co = 1.25k s |tan θ| σ P H 0.95 + k g Y −1 (13.34) where k g is the interstitial fluid thermal conductivity. In the absence of detailed information, σ/| tan θ| can be expected to range from 5 to 9 µm for relatively smooth surfaces. Thermal Boundary Resistance When dealing with heat removal from a chip and thermal transport in various packaging structures at room temperature and above, the thermal boundary resistance R b is generally negligible compared to the contact re- sistance. However, at the transistor level, where interfaces—often formed by epitaxial thin film deposition, through atomistic processes such as physical vapor deposition— may be nearly perfect, the thermal boundary resistance should be included. Two theoretical models are widely used to predict the thermal boundary resistance R b : the acoustic mismatch model (AMM) and the diffuse mismatch model (DMM). The former is based on the specular reflection of sound waves at the interface and the latter is based on the diffuse scattering of phonons at the interface. Swartz and Pohl (1989) provide a comprehensive discussion of both AMM and DMM models for the thermal boundary resistance and have shown that the microscopic thermal boundary resis- tance resulting from the mismatch in the acoustic properties in the low-temperature limit can be obtained by the following DMM equation: R b = π 2 30 k 4 b ¯ h 3 jj c −2 1,jj × j c −2 2,jj jj c −2 1,jj + jj c −2 2,jj T −3 (13.35) where k b is the Boltzmann constant, ¯ h the Planck constant divided by 2π, c the speed of sound, jj the mode of sound (jj = 1 for the longitudinal mode, and jj = 2 for the transverse mode), and the subscripts 1 and 2 refer to the two solids in contact. Note that this relation is strictly valid only at very low temperatures. Similar equations to estimate R b using DMM at high temperatures or AMM can be obtained from Swartz and Pohl (1989). Although AMM and DMM are based on very different physical arguments, they appear to yield identical results for most material pairs (Swartz and Pohl, 1989). Both these models are very good in predicting R b at very low temperatures but fail miserably at high temperatures for various reasons, such BOOKCOMP, Inc. — John Wiley & Sons / Page 973 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 973 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 [973], (27) Lines: 790 to 815 ——— 0.33711pt PgVar ——— Short Page PgEnds: T E X [973], (27) as increased scattering of phonons and deviation from the Debye density of states (Swartz and Pohl, 1989). Interstitial Materials In describing heat flow across an interface, eq. (13.34) assumed the existence of a fluid gap, which provides a parallel heat flow path to that of the solid–solid contact. Because the noncontact area may occupy in excess of 90% of the projected area, heat flow through the interstitial spaces can be of great importance. Consequently, the use of high-thermal-conductivity interstitial materials, such as soft metallic foils and fiber disks, conductive epoxies, thermal greases, and polymeric phase-change materials, can substantially reduce the contact resistance. The enhanced thermal capability of many of high-performance epoxies, thermal greases, and phase-change materials commonly in use in the electronic industry is achieved through the use of large concentrations of thermally conductive particles. Successful design and development of thermal packaging strategies thus requires the determination of the effective thermal conductivity of such particle-laden interstitial materials and their effect on the overall interfacial thermal resistance. Comprehensive reviews of the general role of interstitial materials in controlling contact resistance have been published by several authors, including Sauer (1992). When interstitial materials are used for control of the contact resistance, it is desirable to have some means of comparing their effectiveness. Fletcher (1972) proposed two parameters for this purpose. The first of these parameters is simply the ratio of the logarithms of the conductances, which is the inverse of the contact resistance, with and without the filler: χ = ln κ cm ln κ bj (13.36) in which κ is the contact conductance, and cm and bj refer to control material and bare junctions respectively. The second parameter takes the thickness of the filler material into account and is defined as η = (κδ filler ) cm (κδ gap ) bj (13.37) in which δ is the equivalent thickness. The performance of an interstitial interface material as decided by the parameter defined by Fletcher (1972), in eqs. (13.36) and (13.37) includes the bulk as well as the contact resistance contribution. It is for this reason that in certain cases the thermal resistance of these thermal interface materials is higher than that for a bare metallic contact because the bulk resistance is the dominant factor in the thermal resistance (Madhusudan, 1995). To make a clear comparison of only the contact resistance arising from the interface of the substrate and various thermal interface materials, it is important to measure it exclusively. Separation of the contact resistance and bulk resistance will also help researchers to model the contact resistance and the bulk resistance separately. BOOKCOMP, Inc. — John Wiley & Sons / Page 974 / 2nd Proofs / Heat Transfer Handbook / Bejan 974 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 [974], (28) Lines: 815 to 837 ——— 4.61006pt PgVar ——— Normal Page PgEnds: T E X [974], (28) Equations (13.36) and (13.37) by Fletcher (1972), show that the thermal resistance of any interface material depends on both the bond line thickness and thermal con- ductivity of the material. As a consequence, for materials with relatively low bulk conductivity, the resistance of the added interstitial layer may dominate the thermal behavior of theinterface and may result in an overall interfacial thermal resistance that is higher than that of the bare solid–solid contact (Madhusudan, 1995). Thus, both the conductivity and the achievable thickness of the interstitial layer must be considered in the selection of an interfacial material. Indeed, while the popular phase-change materials have a lower bulk thermal conductivity (at a typical value of 0.7 W/m · K) than that of the silicone-based greases (with a typical value of 3.1 W/m · K), due to thinner phase-change interstitial layers, the thermal resistance of these two categories of interface materials is comparable. To aid in understanding the thermal behavior of suchinterface materials, it is useful to separate the contribution of the bulk conductivity from the interfacial resistance, which occurs where the interstitial material contacts one of the mating solids. Fol- lowing Prasher (2001), who studied the contact resistance of phase-change materials (PCMs) and silicone-based thermal greases, the thermal resistance associated with the addition of an interfacial material, R TIM can be expressed as R TIM = R bulk + R co 1 + R co 2 (13.38) where R bulk is the bulk resistance of the thermal interface material, and R co the contact resistance with the substrate, and subscripts 1 and 2 refer to substrates 1 and 2. Prasher (2001) rewrote eq. (13.38) as R TIM = δ κ TIM + σ 1 2κ TIM A nom A real + σ 2 2k TIM A nom A real (13.39) where R TIM is the total thermal resistance of the thermal interface material, δ the bond-line thickness, κ TIM the thermal conductivity of the interface material, σ 1 and σ 2 the roughness of surfaces 1 and 2, respectively, A nom the nominal area, and A real the real area of contact of the interface material with the two surfaces. Equation (13.39) assumes that the thermal conductivity of the substrate is much higher than that of the thermal interface material. The first term on the right-hand side of eq. (13.39) is the bulk resistance, and other terms are the contact resistances. Figure 13.11 shows the temperature variation at the interface between two solids in the presence of a thermal interface material associated with eq. (13.39). Unlike the situation with the more conventional interface materials, the actual contact area between a polymeric material and a solid is determined by capillary forces rather than surface hardness, and an alternative approach is required to determine A real in eq. (13.39). Modeling each of the relevant surfaces as a series of notches, and including the effects of surface roughness, the slope of the asperities, the contact angle of the polymer with each the substrates, the surface energy of the polymer, and the externally applied pressure, a surface chemistry model was found to match very well with the experimental data for PCM and greases at low pressures (Prasher, 2001), as shown in Fig. 13.12 for BOOKCOMP, Inc. — John Wiley & Sons / Page 975 / 2nd Proofs / Heat Transfer Handbook / Bejan LENGTH-SCALE EFFECTS ON THERMOPHYSICAL PROPERTIES 975 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 [975], (29) Lines: 837 to 848 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [975], (29) Figure 13.11 Temperature drops across an interface. PCM. Unfortunately, it has not yet been found possible to determine the contact area with a closed-form expression. It is also to be noted that eq. (13.39) underpredicts the interface thermal resistance data at high pressures. Thermal Conductivity of Particle-Laden Systems Equation (13.39) shows that the bulk and contact resistance of the thermal interface material are dependent on the thermal conductivity of the interface material. The thermal conductivity of a particle-laden polymer increases nonlinearly with increasing volume fraction of the conducting particle, as suggested in Fig. 13.13. One of the most commonly used models for predicting the thermal conductivity of a particle-laden, two-phase system is the Lewis and Nielsen (1970) model. This model calculates the thermal conductivity of two-phase system using . BOOKCOMP, Inc. — John Wiley & Sons / Page 966 / 2nd Proofs / Heat Transfer Handbook / Bejan 966 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 [966],. ζ) 3/2 φ c k √ A source (13.24) BOOKCOMP, Inc. — John Wiley & Sons / Page 968 / 2nd Proofs / Heat Transfer Handbook / Bejan 968 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 [968],. especially BOOKCOMP, Inc. — John Wiley & Sons / Page 970 / 2nd Proofs / Heat Transfer Handbook / Bejan 970 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 [970],