BOOKCOMP, Inc. — John Wiley & Sons / Page 361 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 361 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 [361], (101) Lines: 4013 to 4018 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [361], (101) 10 0 10 2 10 3 10 4 10 5 10 2 10 1 10 3 P g (torr) h g (W/m . K) 2 Experiment Theory He He N 2 N 2 Nickel 200 = 2.32 m P=0.52 MPa ( / ) = 3.6Y YDH PH/ = 1.7 10 e ϫ Ϫ4 Figure 4.28 Gap conductance model and data for conforming rough Ni 200 surfaces. (From Song, 1988.) 200 surfaces. The plastic deformation model was used to calculate Y. The points for M ∗ < 0.01 correspond to the high-gas-pressure tests (near 1 atm), and the points for M ∗ > 2 correspond to the low-gas-pressure tests. 4.17 JOINT CONDUCTANCE ENHANCEMENT METHODS In many electronics packages the thermal joint conductance across a particular joint must be improved for the thermal design to meet its performance objectives. If the joint cannot be made permanent because of servicing or other considerations, the joint BOOKCOMP, Inc. — John Wiley & Sons / Page 362 / 2nd Proofs / Heat Transfer Handbook / Bejan 362 THERMAL SPREADING AND CONTACT RESISTANCES 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 [362], (102) Lines: 4018 to 4029 ——— 4.097pt PgVar ——— Long Page PgEnds: T E X [362], (102) Figure 4.29 Dimensionless gap conductance model and data for conforming rough Ni 200 surfaces. (From Song, 1988.) conductance must be “enhanced”; that is, it must be improved above the bare joint situation utilizing one of several known techniques, such as application of thermal interface materials (TIMs): for example, thermal grease, grease filled with particles (also called paste), oils, and phase-change materials (PCMs). Enhancement of the joint conductance has also been achieved by the insertion of soft metallic foils into the joint, or by the use of a relatively soft metallic coating on one or both surfaces. More recently, soft nonmetallic materials such as polymers and rubber have been used. One may consult review articles by Fletcher (1972, 1990), Madhusudana and Fletcher (1986), Madhusudana (1996), Marotta and Fletcher (1996), Prasher (2001), Savija et al. (2002a, b), and other pertinent references may be found in these reviews. This section is limited to a few examples where models and data are available. BOOKCOMP, Inc. — John Wiley & Sons / Page 363 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 363 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 [363], (103) Lines: 4029 to 4060 ——— 2.33809pt PgVar ——— Long Page PgEnds: T E X [363], (103) 4.17.1 Metallic Coatings and Foils An effective method for enhancement of joint conductance consists of vapor depo- sition of a very thin soft metallic layer on the surface of the substrate. The layer thickness is often less than 100 µm; it is in “perfect” thermal and mechanical contact with the substrate, and its bulk resistance is negligibly small relative to the contact resistance. The thermal resistance at the layer–substrate interface is also negligible. A comprehensive treatment of the theoretical development and experimental ver- ification of the thermomechanical model can be found in Antonetti (1983) and An- tonetti and Yovanovich (1983, 1985). In the following discussion, therefore, only those portions of the theory needed to apply the model to a thermal design problem are presented. The general expression for the contact conductance of the coated joint operating in a vacuum is h c = h c H S H 0.93 k 1 + k 2 Ck 1 + k 2 (W/m 2 · K) (4.296) where h c is the uncoated contact conductance, H S the microhardness of the softer substrate, H the effective microhardness of the layer–substrate combination, C a spreading–constriction parameter correction factor that accounts for the heat spread- ing in the coated substrate, and k 1 and k 2 the thermal conductivities of the two sub- strates, respectively. The coated contact conductance relationship consists of the product of three quan- tities: the uncoated contact conductance h c , the mechanical modification factor (H S / H ) 0.93 , and the thermal modification factor. The uncoated (bare) contact conductance may be determined by means of the conforming, rough surface correlation equation based on plastic deformation: h c = 1.25 m σ 2k 1 k 2 k 1 + k 2 P H S 0.95 (W/m 2 · K) (4.297) where H S is the flow pressure (microhardness) of the softer substrate, m the combined average absolute asperity slope, and σ the combined rms surface roughness of the joint. For a given joint, the only unknowns are the effective microhardness H and the spreading–constriction parameter correction factor C. Thus, the key to solving coated contact problems is the determination of these two quantities. Mechanical Model The substrate microhardness can be obtained from the fol- lowing approximate relationship (Hegazy, 1985): H S = (12.2 −3.54H B ) σ m −0.26 (GPa) (4.298) which requires the combined surface roughness parameters σ and m and the bulk hardness of the substrate H B . In the correlation equation the units of the joint rough- ness parameter σ/m are micrometers. For Ni 200 substrates, H B = 1.67 GPa. BOOKCOMP, Inc. — John Wiley & Sons / Page 364 / 2nd Proofs / Heat Transfer Handbook / Bejan 364 THERMAL SPREADING AND CONTACT RESISTANCES 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 [364], (104) Lines: 4060 to 4103 ——— -1.19986pt PgVar ——— Normal Page * PgEnds: Eject [364], (104) The effective microhardness must be obtained empirically for the particular layer (coating)–substrate combination under consideration. This requires a series of Vick- ers microhardness measurements which will result in an effective microhardness plot similar to that shown in Fig. 4.30 (e.g., a silver layer on a Ni 200 substrate). The effective Vickers microhardness measurements, denoted H , are plotted against the relative indentation depth t/d, where t is the layer thickness and d is the indentation depth. The three microhardness regions were correlated as H = H S 1 − t d + 1.81H L t d for 0 ≤ t d < 1.0 (4.299) 1.81H L − 0.21H L t d − 1 for 1.0 ≤ t d ≤ 4.90 (4.300) H L for t d > 4.90 (4.301) where H S and H L are the substrate and layer microhardness, respectively. The Ni 200 substrate microhardness is found to be H S = 2.97 GPa for the joint roughness parameter values: σ = 4.27 µm and m = 0.236 rad. The Vickers microhardness of the silver layer is approximately H L = 40 kg/mm 2 = 0.394 GPa. The relative indentation depth is obtained from the following approximate corre- lation equation (Antonetti and Yovanovich, 1983, 1985) t d = 1.04 t d P H −0.097 (4.302) To implement the procedure (Antonetti and Yovanovich, 1983, 1985) for finding H from the three correlation equations requires an iterative method. To initiate the iterative method, the first guess is based on the arithmetic average of the substrate and layer microhardness values: H 1 = H S + H L 2 (GPa) For a given value of t and P , the first value of t/d can be computed. From the three correlation equations, one can find a new value for H : say, H 2 . The new microhardness value, H 2 , is used to find another value for t/d, which leads to another value, H 3 . The procedure is continued until convergence occurs. This usually occurs within three or four iterations (Antonetti and Yovanovich, 1983, 1985). Thermal Model The spreading–constriction resistance parameter correction fac- tor C is defined as the ratio of the spreading–constriction resistance parameter for a substrate with a layer to a bare substrate, for the same value of the relative contact spot radius : C = Ψ( , φ n ) Ψ( ) (4.303) BOOKCOMP, Inc. — John Wiley & Sons / Page 365 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 365 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 [365], (105) Lines: 4103 to 4109 ——— * 22.25099pt PgVar ——— Normal Page PgEnds: T E X [365], (105) Figure 4.30 Vickers microhardness of a silver layer on a nickel substrate. (From Antonetti and Yovanovich, 1985.) BOOKCOMP, Inc. — John Wiley & Sons / Page 366 / 2nd Proofs / Heat Transfer Handbook / Bejan 366 THERMAL SPREADING AND CONTACT RESISTANCES 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 [366], (106) Lines: 4109 to 4155 ——— 4.24411pt PgVar ——— Normal Page PgEnds: T E X [366], (106) The dimensionless spreading–constriction resistance parameter is defined as Ψ( , φ n ) = 4k 2 a R c (4.304) where k 2 is the thermal conductivity of the substrate that is coated, a is the con- tact spot radius for the layer on the substrate, and R c is the spreading–constriction resistance of the contact spot. The spreading–constriction resistance parameter with a layer on the substrate is (Antonetti and Yovanovich, 1983, 1985) Ψ( , φ n ) = 16 π ∞ n=1 J 2 1 (δ n ) (δ n ) 3 J 2 0 (δ n ) φ n γ n ρ n (4.305) The first of these, φ n , accounts for the effect of the layer though its thickness and thermal conductivity; the second, γ n , accounts for the contact temperature basis used to determine the spreading–constriction resistance; and the third, ρ n , accounts for the contact spot heat flux distribution. For contacting surfaces it is usual to assume that the contact spots are isothermal. The modification factors in this case are γ n = 1.0 and φ n = K (1 + K) +(1 −K)e −2δ n τ (1 + K) −(1 −K)e −2δ n τ (4.306) where K is the ratio of the substrate-to-layer thermal conductivity, τ = t/a is the layer thickness-to-contact spot radius ratio, and ρ n = sin δ n 2J 1 (δ n ) (4.307) The parameter δ n are the eigenvalues, which are roots of J 1 (δ n ) = 0. Tabulated values of C were reported by Antonetti (1983) for a wide range of the parameters K and τ . Details of the thermomechanical model development are given in Antonetti (1983) and Antonetti and Yovanovich (1983, 1985). The thermomechanical model of Antonetti and Yovanovich (1983, 1985) has been verified by extensive tests. First the bare joint was tested to validate that part of the model. Figure 4.31 shows the dimensionlessjointconductancedata and theory plotted versus the relative contact pressure for three joints having three levels of surface roughness. The two surfaces were flat; one was lapped and the other was glass bead blasted. All tests were conducted in a vacuum. The agreement between the model given by the correlation equation and all data is very good over the entire range of relative contact pressure. The bare surface tests were followed by three sets of tests for joints having three levels of surface roughness. Figure 4.32 shows the effect of the vapor-deposited silver layer thickness on the measured joint conductance plotted against the contact pres- sure. For these tests the average values of the combined surface roughness parameters were σ = 4.27 µm and m = 0.236 rad. For the contact pressure range the substrate BOOKCOMP, Inc. — John Wiley & Sons / Page 367 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 367 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 [367], (107) Lines: 4155 to 4157 ——— 3.42099pt PgVar ——— Normal Page PgEnds: T E X [367], (107) 10 0 10 1 10 0 10 1 Relative Pressure 10ϫ 4 P H Dimensionless Conductance /10hmkϫ 4 h mk P H = 1.25 0.95 ( ( Specimens 08/09 Specimens 10/11 Specimens 26/27 Specimens 34/35 Figure 4.31 Dimensionless contact conductance versus relative contact pressure for bare Ni 200 surfaces in a vacuum. (From Antonetti and Yovanovich, 1985.) microhardness was estimated to be H S = 2.97 GPa. The layer thickness was between 0.81 and 39.5 µm. The lowest set of data and the theoretical curve correspond to the bare surface tests. Agreement between data and model is very good. The highest set of data for layer thickness of t = 39.5 µm corresponds to the infinitely thick layer where thermal spreading occurs in the layer only and the layer microhardness con- trols the formation of the microcontacts. Again, the agreement between experiment and theory is good. BOOKCOMP, Inc. — John Wiley & Sons / Page 368 / 2nd Proofs / Heat Transfer Handbook / Bejan 368 THERMAL SPREADING AND CONTACT RESISTANCES 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 [368], (108) Lines: 4157 to 4180 ——— 0.097pt PgVar ——— Normal Page PgEnds: T E X [368], (108) 10 0 10 3 10 0 10 1 10 2 Pressure (kN/m ) 2 Contact Conductance (kW/m . K) 2 Specimens Coating 08/09 None 0.81 m 39.5 m 5.1 m 1.4 m 1.2 m 0.81 m 1.2 m 1.4 m 39.5 m 5.1 m ” 22/23 16/17 10/11 12/13 18/19 14/15 Upper bound Infinite coating Lower bound No coating Figure 4.32 Effect of layer thickness and contact pressureon joint conductance: vacuum data and theory. (From Antonetti and Yovanovich, 1985.) The difference between the highest and lowest joint conductance values is ap- proximately a factor of 10. The enhancement is clearly significant. The agreement between the measured values of joint conductance and the theoretical curves for the layer thicknesses: t = 0.81, 1.2, 1.4, and 5.1 µm is also very good, as shown in Fig. 4.32. All the test points for bare and coated surfaces are plotted in Fig. 4.33 as dimensionless joint conductance versus relative contact pressure. The agreement between experiment and theory is very good for all points. BOOKCOMP, Inc. — John Wiley & Sons / Page 369 / 2nd Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 369 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 [369], (109) Lines: 4180 to 4180 ——— 4.097pt PgVar ——— Normal Page PgEnds: T E X [369], (109) 10 0 10 1 10 2 10 0 10 1 10 2 Relative Pressure 10ϫ 4 Dimensionless Conductance 10ϫ 4 h mk P H Ј ЈЈ = 1.25 0.95 h mk Ј Ј h mk P H Ј ЈЈ = 1.25 0.95 h mk Ј Ј P HЈ ( ( Series A = 4.27 m Series B = 1.28 m Series C = 8.32 m Figure 4.33 Dimensionless joint conductance for a bare and silver layer on Ni 200 substrates versus relative contact pressure. (From Antonetti and Yovanovich, 1985.) A parametric study was conducted to calculate the enhancement that can be achieved when different metal types are used. The theory outlined earlier will now be applied to a common problem in electronics packaging: heat transfer across an aluminum joint. What is required is a parametric study showing the variation in joint conductance as a function of metallic coating type and thickness for fixed surface BOOKCOMP, Inc. — John Wiley & Sons / Page 370 / 2nd Proofs / Heat Transfer Handbook / Bejan 370 THERMAL SPREADING AND CONTACT RESISTANCES 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 [370], (110) Lines: 4180 to 4188 ——— -0.83296pt PgVar ——— Normal Page PgEnds: T E X [370], (110) TABLE 4.20 Assumed Nominal Property Values of Four Coatings k(W/m · K)H(kg/mm 2 ) Lead 32.4 3.0 Tin 58.4 8.5 Silver 406.0 40.0 Aluminum 190.0 85.0 roughness and contact pressure. The thermophysical properties of the coatings and the aluminum substrate material are presented in Table 4.20. Figure 4.34 shows the effect of the metallic layers on joint conductance. As shown in this figure, except for a very thin layer (about 1 µm), the performance curves are arranged according to layer microhardness. Lead with the lowest microhardness has Figure 4.34 Effect of layer thickness for four metallic layers. (From Antonetti and Yovano- vich, 1983.) . Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 361 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 [361],. Proofs / Heat Transfer Handbook / Bejan 362 THERMAL SPREADING AND CONTACT RESISTANCES 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 [362],. Proofs / Heat Transfer Handbook / Bejan JOINT CONDUCTANCE ENHANCEMENT METHODS 363 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 [363],