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Electrical resistivity of metal matrix composites
Acta Materialia 51 (2003) 6191–6302 www.actamat-journals.com
Electrical resistivity of metal matrix composites Shou-Yi Chang a, Chi-Fang Chen a, Su-Jien Lin a,∗, Theo Z. Kattamis b b
a Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 300, Taiwan Department of Metallurgy and Materials Engineering, University of Connecticut, Storrs, CT 06269-3136, USA
Received 19 November 2002; received in revised form 30 July 2003; accepted 31 July 2003
Abstract Theoretical models for predicting the electrical resistivity of metal matrix composites reinforced with continuous fibers, short fibers, and particulates were developed by integrating thin slices of composite cells. The experimental electrical resistivity of aluminum, copper, and silver matrix composites was measured and compared with theoretical values derived from the models. Experimental resistivity of composites followed the trend of theoretical prediction, increasing with increasing volume fraction and decreasing size of reinforcement. Deformation regions containing residual stresses and dislocations formed around the reinforcement and raised the resistivity of composites. The magnitude of residual stresses and the dislocation density were found to depend on the type, size and shape of reinforcement, as well as the matrix type. The effective size of the deformation regions varied due to their overlapping and better fitted the calculated curves through empirical modification. Theoretical prediction of resistivity that takes into account the effect of residual stresses and dislocations, and the overlapping of deformation regions agreed reasonably well with experimental results. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Metal matrix composites; Electrical properties; Theory & modeling; Dislocations; Deformation regions
1. Introduction Metal matrix composites (MMCs) have been developed and applied in aerospace, automobile, and electronic industries since the 1960s, because of their excellent mechanical, electrical, and thermal performances. In the electrical contact and electronic packaging industries these composites have especially attracted interest in recent years,
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because of their low electrical resistivity, good thermal conductivity, and high mechanical strength. One of the most important properties of metal matrix composites for electrical and electronic applications is electrical resistivity, which has been previously modeled [1–5]. However, the theoretical electrical resistivity predicted by these models does not closely match experimental results. Also, during fabrication of composite materials, large amounts of residual stresses and dislocations form and affect their electrical resistivity. Some models have been developed to calculate the dislocation density and distribution of residual stresses [6–11],
1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00462-2
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but very limited work has been reported on their effect on electrical resistivity of composites. The mechanism by which residual stresses and dislocations affect the electrical resistivity of composites needs to be elucidated. During processing of composites, the difference between the coefficients of thermal expansion of the reinforcement and the matrix leads to the formation of residual stresses and dislocations. Arsenault and Shi [8] calculated the dislocation density in composites by using a prismatic punch method, and found that the type of matrix, as well as the type, volume fraction, shape and size of the reinforcement affect the distribution and magnitude of residual stresses, and the dislocation density. Hence, the effect of residual stresses and dislocations on the electrical resistivity of composites may be investigated by varying the amounts of residual stresses and dislocations, which can be produced using different reinforcements, matrices, and heat treatments. The purpose of this research program was to investigate the electrical resistivity of composites and evaluate the effect of residual stresses and dislocations thereon. Theoretical models of the electrical resistivity of metal matrix composites are introduced in which integral summation of thin slices of composite cells is performed, taking into account the effects of residual thermal stresses and dislocations and the overlapping of deformation zones.
앫 The flow of current in both the reinforcement and the matrix is by free electron migration. Under these assumptions, the electrical resistivity of a composite specimen is equal to that of a composite cell, and can be calculated by integration over thin slices of the composite cell. The calculation of the electrical resistivity of composites reinforced with continuous fibers differs from that for whiskers or short fibers, and particulates, because of the difference in reinforcement distribution and geometry, as will be discussed below. 2.1. Continuous fiber-reinforced composites For a continuous fiber-reinforced metal matrix composite, the calculation of electrical resistivity that follows will assume that electrons flow in the longitudinal direction and fibers are aligned: (1) in the longitudinal direction, (2) in the transverse direction, and (3) the fibers are cross-stacked. Longitudinal and transverse composite cells are schematically illustrated in Fig. 1.
2. Theoretical models The calculation of the theoretical electrical resistivity of a metal matrix composite, will be based on four basic assumptions, as follows: 앫 The composite consists of two phases, the reinforcement and the matrix with no interfacial reaction layer. 앫 The reinforcement is uniformly distributed within the matrix, hence its spacing is constant. 앫 The composite cell represents the whole specimen and consists of a reinforcing element surrounded by the matrix at the same volume fractions as those in the entire specimen.
Fig. 1. Schematic representation of (a) longitudinal and (b) transverse composite cells in a continuous fiber-reinforced composite.
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2.1.1. Fibers aligned in the longitudinal direction Fig. 1(a) shows a longitudinal composite cell with a length of w, a width of 2Df, a height of 2Df, and a fiber radius of r. The electrical resistance of the longitudinal composite cell, RL, can be simply obtained by combining in parallel the electrical resistance of reinforcement and matrix, Rf and Rm, respectively as: RL ⫽
Rf·Rm wrfrm ⫽ 2 Rf ⫹ Rm 4Df [rf(1⫺Vf) ⫹ rmVf]
(1)
where rf and rm are the electrical resistivities of the reinforcement and the matrix, respectively, and Vf is the volume fraction of the fiber. Since RL = w rL 2, the electrical resistivity of the longitudinal 4Df composite cell is given by: rL ⫽
r fr m rf(1⫺Vf) ⫹ rmVf
rm 1⫺Vf
(3)
2.1.2. Fibers aligned in the transverse direction Fig. 1(b) shows a transverse composite cell with a length of w, a width of 2Df, a height of 2Df, and a fiber radius of r. The electrical resistance of this transverse composite cell can be obtained by calculating that of a 1/4 transverse composite cell, which is divided into two regions: region 1 is composed of both reinforcement and matrix with a combined electrical resistance of R1, and region 2 is only composed of matrix with a electrical resistance of R2. In region 1, the differential electrical resistances of reinforcement and matrix are dRf = dx dx and dRm = rm , respectively. Fig. 1(b) rf whf whm shows that hf = rsinq, hm = Df⫺rsinq, x = rcosq, and dx = ⫺rsinq·dq. Hence, the differential electrical resistance of region 1: dR1 ⫽
dRf·dRm dRf ⫹ dRm
rmrf(⫺rsinq) dq w[rmrsinq ⫹ rf(Df⫺rsinq)]
The average electrical resistance of region 1, R1, p can be obtained by integrating dR1 from q = to 2 0 r q = 0 as R1 = pdR1. Since R1 = r1 , the average wDf
冕
(4)
2
electrical resistivity of region 1, r1, may be expressed as: r1 ⫽
冕
Df 0 rmrfrsinq dq r p(rf⫺rm)rsinq⫺rfDf
(5)
2
Following a detailed calculation, r1 is then obtained as:
冦
冋冪
rfDf ⫹ (rf⫺rm)r rfDf⫺(rf⫺rm)r
2rf2Dftan⫺1
Dfrm r1 ⫽ r
(2)
If the reinforcement is an insulator, i.e. rf ⬎ ⬎ rm, the electrical resistivity can be simplified as: r⬁L ⫽
⫽
6293
冑
(rf⫺rm) r D ⫺(rf⫺rm) r 2 f
2 f
2 2
册
冧
prf ⫺ 2(rf⫺rm)
(6) The width of the 1/4 composite cell, Df, varies with the volume fraction of reinforcement and can be obtained from the following simple calculation: Vf ⫽
pr2 4D2f
⇒
Df ⫽
冪4V r p
(7)
f
冪
r 4Vf , the electrical resistivity = Df p of the 1/4 composite cell is: By setting b =
冦
rm r1 ⫽ b
冋冪
2r2ftan⫺1
册
rf ⫹ (rf⫺rm)b rf⫺(rf⫺rm)b
冧
prf ⫺ 2(r f⫺rm) (rf⫺rm)冑r ⫺(rf⫺rm) b 2 f
2 2
(8) Region 2 consists only of matrix, hence r2 = rm and the electrical resistivity of the 1/4 transverse composite cell, rT(1 / 4), may be expressed by: r Df⫺r ⫽ r1b ⫹ rm(1⫺b) rT(1/4) ⫽ r1 ⫹ rm Df Df
(9)
and the electrical resistance of this 1/4 composite cell, by:
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rT(1/4) Df ⫽ RT(1/4) ⫽ rT(1/4) Df·w w
(10)
The composite cell comprises four identical 1/4 composite cells. Hence, the electrical resistivity of a composite cell, rT, equals rT(1/4), and consequently the electrical resistance of the composite cell, RT, equals RT(1/4). Thus, the electrical resistivity of the transverse composite cell may be expressed by:
冦
rT ⫽ r1b ⫹ rm(1⫺b) ⫽ rm 1⫺b
冋冪
2r2ftan⫺1
rf ⫹ (rf⫺rm)b rf⫺(rf⫺rm)b
册
冧 (11)
When D fr, i.e., the volume fraction of reinforcement tends to zero, b⬇0 and rT⬇rm. If the reinforcement is an insulator, i.e. r fr m, the electrical resistivity of the transverse composite cell can be simplified as:
r ⫽ rm ⬁ T
冦
p 1⫺b⫺ ⫹ 2
2tan⫺1
冋冪 册 b⫹1 b⫺1
冑1⫺b
2
Basically, for a composite reinforced by twodimensionally and randomly distributed short fibers the expression for electrical resistivity is similar to that of a cross-stacked, continuous fiberreinforced composite. The electrical resistivity also combines longitudinal and transverse terms, but the volume fractions of longitudinal and transverse fibers are no longer exactly equal to 0.5. According to Halpin and Tsai [12], 3/8 of all fibers contribute to the transfer of electrons in the longitudinal direction and 5/8 to the transfer in the transverse direction. Hence, considering a given volume of composite: 5 3 N ⫽ NL ⫹ NT 8 8
prf ⫹ ⫺ 2 2 2 2(r f⫺rm) (rf⫺rm) rf ⫺(rf⫺rm) b
冑
2.2. Whisker or short fiber-reinforced composites
冧
(14)
where N, NL, and NT are the total number of fibers, and those aligned in the longitudinal and transverse directions, respectively. The electrical resistance of the composite with short fibers, rc(sf), can then be obtained by combining those of longitudinal and transverse composite cells and can be expressed by: rC(sf) ⫽
8rL·rT 5rL ⫹ 3rT
(15)
2.3. Particulate-reinforced composites (12)
Fig. 2 shows a particulate-reinforced composite cell with a length of w, a width of 2Dp, a height
2.1.3. Cross-stacked fibers For a cross-stacked continuous fiber reinforced metal matrix composite, the electrical resistance of the composite can be expressed as the combination of the electrical resistance of longitudinal and transverse composite cells. By averaging the inverse resistivities in the longitudinal and transverse directions the electrical resistivity of the fiber-reinforced composite, rc(f), may be expressed by: rC(f) ⫽
2rL·rT rL ⫹ rT
(13)
Fig. 2. Schematic representation of the composite cell in a particulate-reinforced composite.
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of 2Dp, and a particulate radius of r. The composite cell is composed of eight 1/8 composite cells, and each 1/8 composite cell is divided into two regions. In region 1, the differential electrical resistances of 4dx the reinforcement and the matrix are dRp = rp 2 px 4dx and dRm = rm , respectively, where x (2Dp)2⫺px2 = rcosq and dx = ⫺rsinq·dq, and rm and rp are the electrical resistivities of the matrix and particulate reinforcement, respectively. The differential electrical resistance of region 1 may be expressed by: dR1 ⫽ ⫽
dRp·dRm dRp ⫹ dRm
By integrating, the electrical resistance of region 1 is obtained as:
冕
R1 ⫽ dR1 ⫽
(17)
冤
4rmrp
tan⫺1 [4rpD ⫺(rp⫺rm)pr ](rp⫺rm)p
冑
2 p
r1 ⫽
2
(rp⫺rm)pr2
冪4r D ⫺(r ⫺r )pr p
2 p
p
冥
2
m
L r Because R1 = r1 = r1 2, the electrical resistivity A Dp of region 1 is:
(20)
冤
4rmrp
tan⫺1 h冑[4rp⫺(rp⫺rm)ph ](rp⫺rm)p 2
R(1/8) ⫽ R1 ⫹ R2 ⫽
4D2prmrp
r冑[4rpD2p⫺(rp⫺rm)pr2](rp⫺rm)p
冤冪
tan⫺1
冥
(rp⫺rm)pr2 4rpD2p⫺(rp⫺rm)pr2
The width of the 1/8 composite cell, Dp, varies with the volume fraction of reinforcement and can be obtained as follows: 4 (2Dp)3 ⫻ Vf ⫽ pr3 3
⇒
r By setting h = = Dp
冪
Dp ⫽
冪6V r
3
p
f
3
6Vp , it follows that: p
(19)
p
p
冥
2
m
⫽
r1r ⫹ rm(Dp⫺r) D2p
(21)
r1h ⫹ rm(1⫺h) Dp
Since a composite cell is composed of eight identical 1/8 composite cells, its electrical resistivity, rC(p), equals r(1/8), and consequently the electrical resistance of the composite cell, RC(p), equals 1/2 R(1/8) by calculation. The electrical resistivity of the composite cell may be expressed by: rC(p) = r1h + rm(1⫺h). Hence,
冦
rC(p) ⫽ rm 1⫺h
⫹
冤
4rp
tan⫺1 冑[4rp⫺(rp⫺rm)ph ](rp⫺rm)p 2
(18)
r1 ⫽
(rp⫺rm)ph2
冪4r ⫺(r ⫺r )ph
Region 2 consists only of the matrix phase, so r2 = rm. The electrical resistance of the 1/8 composite cell is then given by:
(16)
⫺4rmrprsinq dq rmpr2sin2q ⫹ rp(4D2p⫺pr2sin2q)
6295
(rp⫺rm)ph2
冪4r ⫺(r ⫺r )ph p
p
m
冥冧
2
(22) When D pr, i.e., the volume fraction of reinforcement tends to zero, then h⬇0 and rC(p)⬇rm. If the reinforcement is an insulator, i.e. rp ⬎ ⬎rm, the electrical resistivity of the composite cell may be simplified as:
冦
r⬁C(p) ⫽ rm 1⫺h ⫹
4tan⫺1
冋冪
ph2 4⫺ph2
冑p(4⫺ph ) 2
册
冧
(23)
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2.4. Effect of residual stresses and dislocations on electrical resistivity During cooling, following processing or heattreatment of composite specimens thermal stresses are generated in the matrix, in the vicinity of the reinforcement/matrix interface, due to the difference in coefficient of thermal expansion (CTE) of these two phases. Part of these stresses may be relieved through yielding of the matrix which is manifested by plastic strain and formation of dislocations. The other part of these stresses will remain in the composite as residual thermal stresses. Even for furnace-cooled specimens both residual thermal stresses and dislocations will coexist. Strain fields in the matrix adjacent to the reinforcement form energy barriers to the flow of electrons, raising the electrical resistivity of composites. Different matrices and reinforcements result in different residual stress distribution and dislocation densities, hence in different sizes of deformation regions, leading to different effects on electrical resistivity. Arsenault and Shi [8] calculated the dislocation density, rdisl, in a quenched composite specimen, using a model based on a prismatic punching method. They established that: rdisl ⫽
BVfe b(1⫺Vf)d
(24)
where b is the Burgers vector of the matrix, and Vf and d are the volume fraction and the shortest dimension of the reinforcement, respectively. The strain misfit between reinforcement and matrix, ⑀, can be obtained by ⌬CTE × ⌬T, where ⌬CTE is the elementary difference in CTE between reinforcement and matrix, and ⌬T is the difference between heat treatment temperature and room temperature. The geometric parameter of the reinforcement B is equal to 12 when the dimensions of the reinforcement in all directions are the same, as in equiaxed particulates, and equal to 4 if the dimension in one direction is much larger than in the other directions, as in whiskers and fibers [8]. From Eq. (24), it is clear that the dislocation density depends on the type, volume fraction and geometry of the reinforcement, as well as on the type of the matrix. A highly strained deformation region containing
residual stresses and dislocations affects the electrical resistivity of the composite. For a reinforcement of radius r, the radius of the deformation region around the reinforcement may be assumed to be ar (including the radius of the reinforcement), where a is an experimental constant. A large a value corresponds to a large deformation region. The a value is determined by the dislocation density, and depends on the type of matrix and the type, size and shape of the reinforcement, but not its volume fraction. It can be expressed [8] by: Be (a⫺1)⬀ bd
(25)
According to calculations by Flom and Arsenault [10], a = 2.3 for SiCp/Al composite waterquenched from 470 °C. For a particulate reinforcement the volume fraction of the deformation region surrounding the reinforcement, VIIp, may be expressed by: VIIp ⫽ (a3⫺1) ⫻ Vp
(26)
where Vp is the volume fraction of particulates. For a fibrous reinforcement with a large aspect ratio, the volume fraction of the deformation region, VIIf, may be expressed by: VIIf ⫽ (a2⫺1) ⫻ Vf
(27)
The size of the deformation region affects the electrical resistivity of the composite. A modification term C(Vf, a), or C(Vp, a) must be included in the calculated theoretical electrical resistivity of composites [4]. For a fiber-reinforced composite, this term is: C(Vf,a) ⫽
冉 冊
a2⫺1 Vf a
(28)
A similar expression is valid for whiskerreinforced composites, in which Vf is replaced by Vw. For a particulate-reinforced composite, the modification term is: C(Vp,a) ⫽
冉 冊
a3⫺1 Vp a2
(29)
Thus, the modified theoretical electrical resistivity of a fiber-reinforced composite, taking into account
S.-Y. Chang et al. / Acta Materialia 51 (2003) 6191–6302
the effect of residual stresses and dislocations, can be written as:
冋 冉 冊册
r’C(f) ⫽ rC(f) 1 ⫹
a2⫺1 Vf a
(30)
The theoretical electrical resistivity of the particulate-reinforced composite can be modified as:
冋 冉 冊册
a3⫺1 Vp r’C(p) ⫽ rC(p) 1 ⫹ a2
(31)
3. Experimental procedure 3.1. Preparation of metal matrix composite specimens Alternative layers of properly spaced continuous molybdenum fibers (Mof) and stainless steel fibers (SSf) with the same diameter of 150 µm and AA1100 aluminum foils 100 µm thick were stacked and vacuum diffusion-bonded for 10 min at 500 °C, under a pressure of 100 MPa to obtain Mof/Al and SSf/Al composites containing 5–45 vol% fibers [13]. The diffusion-bonded composite specimens were subsequently furnace-cooled. Aluminum matrix specimens containing no fibers were also prepared by diffusion bonding stacked pure aluminum foils following the same procedure. Copper and silver matrix composites containing 10–40 vol% ceramic reinforcements of different types, sizes, and shapes, including silicon carbide whiskers (SiCw, 0.4 µm in diameter), carbon short fibers (Carbonsf, 7 µm in diameter), Saffil short fibers (Saffilsf, 3.8% SiO2-96.2% Al2O3, 3 µm in diameter), silicon carbide particles (SiCp, 12.5, 5, and 1.5 µm), and alumina particles (Al2O3p, 12.5, 5, 3, 1, and 0.3 µm), were produced by electroless copper or silver plating and hot pressing at 600 °C, under a pressure of 300 MPa in air for 15 min, followed by furnace cooling. [14]. For comparison purposes, electrolessly-precipitated pure silver powders from the same silver plating solution were prepared and hot pressed to obtain unreinforced silver matrix specimens. However, because of the difficulty in electroless precipitation of copper powders, commercial oxygen-free, high conductivity
6297
copper powders were hot-pressed under the same conditions as for silver, to obtain unreinforced copper matrix specimens. These aluminum, copper, and silver matrix composite specimens were then cut to a size of 3 × 4 × 40 cm3 and polished for measurement of electrical resistivity. 3.2. Measurement of electrical resistivity Electrical resistivity of these metal matrix composite specimens was evaluated by a four-point probe method, using a constant input current (I = 0.1 Amp), measuring the voltage (V), and using the following relationship: V ⫽ I ⫻ R, and R ⫽ rE ⫻ (L / A)
(32)
where R is electrical resistance, E is electrical resistivity, and L and A are the length and cross-sectional area of specimens, respectively. Three to four specimens of each type were tested. Pure indium (In) metal was used for electrical contacts and platinum (Pt) wires of 0.1 mm in diameter were used for connection, in order to reduce contact and wiring resistances. The current was input through the electrical contacts at both ends of the specimens, and the voltage was measured between two other contacts on the top faces of the specimens with a spacing of 36 mm between them. Repeated measurements and reversed current inputs were conducted to exclude the effect of contact or wiring resistance and to obtain precise electrical resistivity measurements.
4. Results and discussion 4.1. Experimental electrical resistivity and effect of residual stresses and dislocations Figs. 3–7 show the experimental electrical resistivity of various metal matrix composite specimens reinforced with continuous fibers, whiskers, short fibers, and particulates. In these measurements the data scatter was less than 3%. The theoretical resistivity of these composites was calculated without consideration of residual stresses and dislocations, using Eqs. (13), (15) and (23). Results were plotted in these Figures with an a value of 1.0. Experi-
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Fig. 3. Electrical resistivity of Mof/Al and SSf/Al composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
Fig. 4. Electrical resistivity of SiCw/Cu and SiCw/Ag composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
mental results follow the trend of calculated curves which show that the resistivity increases with increasing volume fraction and decreasing size of reinforcement. However, the experimental data exhibited some positive deviation from the predicted curves, because deformation regions increased the electrical resistivity of composites. This increase may be related to the size of the deformation region. Hence, the theoretical resis-
Fig. 5. Electrical resistivity of Carbonsf/Ag and Saffilsf/Ag composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
Fig. 6. Electrical resistivity of SiCp/Ag composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
tivity for each type of specimen was calculated for different a values and plotted in the same Figs. 3– 7. The experimental resistivity of each composite fitted well one of the theoretical curves corresponding to a specific a value. These experimentally effective a values for different composites were listed in Table 1. By comparing the effective a values of these composites, the effects of the following characteristics of the reinforcement and of the matrix on electrical resistivity were established:
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Fig. 7. Electrical resistivity of Mof/Al and SSf/Al composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
1. Reinforcement volume fraction. According to the calculation by Arsenault and Shi [8] and the investigation by Flom and Arsenault [10], the size of deformation regions, within which the dislocation densities were the same, was basically independent of the volume fraction of reinforcement, whereas the total dislocation density depended on it. In the present study, the a values of same type of composites with differ-
6299
ent contents of reinforcements were found to be almost the same. 2. Reinforcement shape. According to the work of Zhao, Zhijian and Yingkuo [11] and Vogelsang, Arsenault and Fisher [6], a higher dislocation density was detected around the sharp corners of SiC particles and whiskers, whereas almost no dislocations formed near their flat surfaces. Comparing the electrical resistivity of 3 µm Al2O3p/Ag with that of Saffilsf/Ag with 3 µm short fiber diameter, Al2O3 particulates had more sharp corners and contributed to a larger increase in electrical resistivity than did the Saffil short fibers, due to the corresponding larger a value. Also, the effect of particulates was more significant, because the volume fraction of the deformation region in a particulate-reinforced composite is proportional to (a3⫺1), whereas it is proportional to (a2⫺1) in a fiber-reinforced composite. 3. Reinforcement size. For a reinforcement size below 10 µm the dislocation density increased drastically [8] and significantly affected properties. With decreasing reinforcement size, the total stress and dislocation density, as well as the radius of the deformation region increased, as did the a value. Hence, the composites reinforced with fine SiC whiskers, SiC particu-
Table 1 Experimental and calculated a values for furnace-cooled composites, using as a reference a 10 µm SiCp/Al composite that was waterquenched from 470 °C [10] Composite specimens
d (µm)
B
SiCp/Al Mof/Al SSf/Al SiCw/Cu SiCw/Ag Carbonsf/Ag Saffilsf/Ag SiCp/Ag
10.0 150 150 0.4 0.4 7.0 3.0 12.5 5.0 1.5 12.5 5.0 3.0 1.0 0.3
12 4
Al2O3p/Ag
⌬CTE (10⫺6/°C) ⌬T (°C)
⑀ (10⫺2)
a (exp.)
a (theo.)
Vc
2.3 1.0 1.0 2.0 1.4 1.2 1.2 1.3 1.6 1.8 1.2 1.3 1.4 1.7 2.0
– 1.01 1.01 6.20 5.32 1.25 1.44 1.42 2.04 4.46 1.35 1.87 2.45 5.34 15.48
– 0.77 0.77 0.02 0.03 0.50 0.38 0.18 0.06 0.01 0.20 0.08 0.04 0.003 0.0001
445 175
12
24.0 23.0 13.0 13.0 15.7 15.7 12.1 15.7
275
1.07 0.40 0.23 0.46 0.43 0.43 0.33 0.43
12
13.0
275
0.36
4 4
350 275
6300
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lates, and Al2O3 particulates had much larger a values than those reinforced with larger, 150 µm diameter molybdenum and stainless steel fibers, carbon and Saffil short fibers, and 12.5 µm average size SiC and Al2O3 particulates. 4. Type of reinforcement and matrix. Residual stresses were more significant for reinforcements with lower CTEs. Comparing the a value of SiCp/Ag to that of Al2O3p/Ag, and Carbonsf/Ag to Saffilsf/Ag, SiC particulates and carbon short fibers had lower CTEs than Al2O3 particulates and Saffil short fibers, resulting in larger residual stresses and dislocation densities and larger a values. The matrix type also affected the a values through the CTE value and that of the Burgers vector, which obviously is not the same for aluminum, copper or silver. 4.2. Theoretical electrical resistivity and effect of deformation zone overlapping To further clarify the effect of deformation regions on electrical resistivity of composites containing different reinforcements and matrices, the theoretical sizes of deformation regions, i.e. the a values, were calculated by using as a reference the value of a = 2.3 for 10 µm particulate SiCp/Al composite that water-quenched from 470 °C [10]. For the calculation of ⑀ in Eq. (25) the differences between room temperature and the annealing temperature of matrix metals were adopted as ⌬T values, instead of the actual processing temperatures, because most dislocations were eliminated above the annealing temperature of matrix metals during furnace cooling. Stresses and dislocations remained in the matrix metals only below their annealing temperatures. The annealing temperatures for commercial pure aluminum, copper, and silver are 300, 375, and 200 °C, respectively [15– 17]. Also, the Burgers vectors of these face-centered cubic metals are simply related to their atomic radii. Using the basic properties of reinforcements and matrices listed in Table 2 [17], and the reference a value [10], the theoretical a values in furnace-cooled composites containing different reinforcements and matrices were calculated and listed in Table 1 for comparison with experimentally derived a values.
For composites with small theoretical a values, as those reinforced by large-sized fibers or particulates (Mof/Al, SSf/Al, Carbonsf/Ag, Saffilsf/Ag, 12.5 µm SiCp/Ag, and 12.5 µm Al2O3p/Ag), experimental a values matched the theoretical values very closely. This matching would suggest that the models introduced predict with accuracy the electrical resistivity of metal matrix composites. Thus, in the 150 m continuous fiberreinforced aluminum matrix composites, in which there was presumably no deformation region, the experimental resistivity fits very well the theoretically predicted curves with an a value of 1.0, as shown in Fig. 3. However, in composites with small-sized reinforcements (SiCw/Cu, SiCw/Ag, 1.5 µm SiCp/Ag, and 1 µm, 0.3 µm Al2O3p/Ag), the theoretical a values were very large, and the deformation regions easily overlapped with each other, i.e. ar reached Df and Dp in Eqs. (7) and (19), respectively. The overlapping of deformation regions arrests further expansion, hence the experimental a values were much lower than the theoretical. The last column in Table 1 lists the critical volume fractions of different reinforcements, Vc, which were calculated from Eqs. (7) and (19). Once the volume fraction of the reinforcement exceeds Vc, a reaches a constant value, and Eq. (25) is no longer valid for the calculation of effective a values. For example, the experimental a values for SiCw/Cu and 1 m Al2O3p/Ag reached constant values of 2.0 and 1.8, respectively. These values are much lower than the theoretical values of 6.20 and 5.34, respectively, because the critical volume fractions of SiC whiskers and Al2O3 particulates for overlapping of the deformation zones were as small as 0.02 and 0.003. Therefore, the effective a values need to be further modified by empirical fitting. Fig. 8 exhibits on a logarithmic scale relationships between experimental and theoretical a values for particulate and short fiberreinforced silver matrix composites. Modification terms were obtained by curve-fitting these data with the following representative equation: loga(exp) ⫽ m ⫹ n ⫻ loga(theo)⇒
(33)
a(exp) ⫽ 10m ⫻ an(theo) The constants m and n represent the formation
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6301
Table 2 Basic properties of reinforcements and matrices used in this research [17] Material
Al
Cu
Ag
Mof
SSf
SiCw/p
Carbonsf
Saffilsf
Al2O3p
CTE (×10⫺6) Structure ˚) Atomic Radius (A
28.0 FCC 1.43
17.0 FCC 1.28
19.7 FCC 1.44
5.0 – –
15.0 – –
4.0 – –
4.0 – –
7.6 – –
6.7 – –
In summary, during the calculation of the effective sizes of deformation regions, the theoretical a values obtained by Eq. (25) are not valid once the deformation regions overlap. Modified Eqs. (34) and (35) need to be considered to calculate more precise α values. These modified a values applied to Eqs. (30) and (31) will then more precisely predict the theoretical electrical resistivity of metal matrix composites.
5. Summary
Fig. 8. Logarithmic scale relationship between experimental and theoretical a values of short fiber- and particulatereinforced silver matrix composites.
capability of matrix deformation regions. The larger n is, the closer become the experimental and theoretical values of a. For short fiber and particulate-reinforced silver matrix composites, the following respective modification Eqs. (34) and (35) were obtained: asf(exp) ⫽ 1.16 ⫻ a0.11 sf(theo)
(34)
ap(exp) ⫽ 1.22 ⫻ a0.20 p(theo)
(35)
Short fibers contributed smaller m and n values than particulate reinforcements, indicating more rapid and extensive overlapping of deformation regions. That seemed opposite to the proposed concept of critical volume fractions listed in Table 1, but it is easy to understand that the layers of inplane distributed short fibers were closely packed and easily contacted each other in the hot-pressing direction, leading to the small saturated sizes of deformation regions.
Theoretical models for predicting the electrical resistivity of metal matrix composites reinforced with continuous fibers, short fibers, and particulates were developed by integrating thin slices of composite cells. The experimental electrical resistivity of aluminum, copper, and silver matrix composites was measured and compared with theoretical values derived from the models. Experimentally measured electrical resistivity of composites follows the trend of theoretical prediction, increasing with increasing volume fraction and decreasing size of reinforcement. Deformation regions containing residual thermal stresses and dislocations formed around the reinforcement and contributed to the increase in electrical resistivity. The magnitude of residual stresses and the dislocation density were found to depend on the type, size and shape of reinforcement, as well as the type of matrix. The effective size of the deformation regions varied due to their overlapping and better fitted the calculated curves through empirical modification. Theoretical prediction of resistivity that takes into account the effect of residual stresses and dislocations and the overlapping of deformation regions agreed reasonably well with experimental results.
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Acknowledgements The authors are pleased to acknowledge the financial support for this research by The National Science Council, Taiwan, under Grant No. NSC91-2216-E-007-029. The authors also acknowledge the assistance to this research by Mr Gong-Jin Hong, Cher-Hao Hsue, and Min-Hsiang Chen.
References [1] Schoutens JE. Electricity conductivity in continuous-fiber composites. In: Everett RK, Arsenault RJ, editors. Metal matrix composites: Mechanisms and properties. Boston, MA: Academic Press Inc.; 1991. [2] Rayleigh. Phil Mag 1982;34:481. [3] Davis A. J Phys 1974;D7:120. [4] Schoutens JE, Roig FS. J Mater Sci 1987;22:181. [5] Maxwell JC. In: A treatise on electricity and magnetism. Vol. 1. NY: Dover Publications; 1954, p. 435–49.
[6] Vogelsang M, Arsenault RJ, Fisher RM. Metall Trans A 1986;17A:379. [7] Arsenault RJ, Fisher RM. Scripta Metall 1983;17:67. [8] Arsenault RJ, Shi N. Mater Sci Eng 1986;81:175. [9] Kim CT, Lee JK, Plichta MR. Metall Trans A 1990;21A:673. [10] Flom Y, Arsenault RJ. Mater Sci Eng 1985;75:151. [11] Zhao Z, Zhijian S, Yingkun X. Mater Sci Eng A 1991;A132:83. [12] Halpin JC, Tsai SW. AFML-TR 67-423 1969;June. [13] Hwang YH, Horng CF, Lin SJ, Liu KS, Jahn MT. Mater Sci Technol 1997;13:982. [14] Chang SY, Lin JH, Lin SJ, Kattamis TZ. Metall Trans A 1999;30:1119. [15] Dean WA. In: Lampman SR, Zorc TB, editors. Metals handbook. 10th ed. Vol. 2. Metals Park, OH: American Society for Metals; 1990, p. 209. [16] Robinson P. In: Lampman SR, Zorc TB, editors. Metals handbook. 10th ed. Vol. 2, Metals Park, OH: American Society for Metals; 1990, p. 265. [17] Coxe CD, McDonald AS, Sistare Jr, GH, In: Lampman SR, Zorc TB, editors. Metals handbook. 10th ed. Vol. 2, Metals Park, OH: American Society for Metals; 1990, p. 699.
Electrical resistivity of metal matrix composites Shou-Yi Chang a, Chi-Fang Chen a, Su-Jien Lin a,∗, Theo Z. Kattamis b b
a Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 300, Taiwan Department of Metallurgy and Materials Engineering, University of Connecticut, Storrs, CT 06269-3136, USA
Received 19 November 2002; received in revised form 30 July 2003; accepted 31 July 2003
Abstract Theoretical models for predicting the electrical resistivity of metal matrix composites reinforced with continuous fibers, short fibers, and particulates were developed by integrating thin slices of composite cells. The experimental electrical resistivity of aluminum, copper, and silver matrix composites was measured and compared with theoretical values derived from the models. Experimental resistivity of composites followed the trend of theoretical prediction, increasing with increasing volume fraction and decreasing size of reinforcement. Deformation regions containing residual stresses and dislocations formed around the reinforcement and raised the resistivity of composites. The magnitude of residual stresses and the dislocation density were found to depend on the type, size and shape of reinforcement, as well as the matrix type. The effective size of the deformation regions varied due to their overlapping and better fitted the calculated curves through empirical modification. Theoretical prediction of resistivity that takes into account the effect of residual stresses and dislocations, and the overlapping of deformation regions agreed reasonably well with experimental results. 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Metal matrix composites; Electrical properties; Theory & modeling; Dislocations; Deformation regions
1. Introduction Metal matrix composites (MMCs) have been developed and applied in aerospace, automobile, and electronic industries since the 1960s, because of their excellent mechanical, electrical, and thermal performances. In the electrical contact and electronic packaging industries these composites have especially attracted interest in recent years,
Corresponding author. Tel.: +886-3-5719543; fax: +8863-5722366. E-mail address: [email protected] (S.-J. Lin). ∗
because of their low electrical resistivity, good thermal conductivity, and high mechanical strength. One of the most important properties of metal matrix composites for electrical and electronic applications is electrical resistivity, which has been previously modeled [1–5]. However, the theoretical electrical resistivity predicted by these models does not closely match experimental results. Also, during fabrication of composite materials, large amounts of residual stresses and dislocations form and affect their electrical resistivity. Some models have been developed to calculate the dislocation density and distribution of residual stresses [6–11],
1359-6454/$30.00 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/S1359-6454(03)00462-2
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but very limited work has been reported on their effect on electrical resistivity of composites. The mechanism by which residual stresses and dislocations affect the electrical resistivity of composites needs to be elucidated. During processing of composites, the difference between the coefficients of thermal expansion of the reinforcement and the matrix leads to the formation of residual stresses and dislocations. Arsenault and Shi [8] calculated the dislocation density in composites by using a prismatic punch method, and found that the type of matrix, as well as the type, volume fraction, shape and size of the reinforcement affect the distribution and magnitude of residual stresses, and the dislocation density. Hence, the effect of residual stresses and dislocations on the electrical resistivity of composites may be investigated by varying the amounts of residual stresses and dislocations, which can be produced using different reinforcements, matrices, and heat treatments. The purpose of this research program was to investigate the electrical resistivity of composites and evaluate the effect of residual stresses and dislocations thereon. Theoretical models of the electrical resistivity of metal matrix composites are introduced in which integral summation of thin slices of composite cells is performed, taking into account the effects of residual thermal stresses and dislocations and the overlapping of deformation zones.
앫 The flow of current in both the reinforcement and the matrix is by free electron migration. Under these assumptions, the electrical resistivity of a composite specimen is equal to that of a composite cell, and can be calculated by integration over thin slices of the composite cell. The calculation of the electrical resistivity of composites reinforced with continuous fibers differs from that for whiskers or short fibers, and particulates, because of the difference in reinforcement distribution and geometry, as will be discussed below. 2.1. Continuous fiber-reinforced composites For a continuous fiber-reinforced metal matrix composite, the calculation of electrical resistivity that follows will assume that electrons flow in the longitudinal direction and fibers are aligned: (1) in the longitudinal direction, (2) in the transverse direction, and (3) the fibers are cross-stacked. Longitudinal and transverse composite cells are schematically illustrated in Fig. 1.
2. Theoretical models The calculation of the theoretical electrical resistivity of a metal matrix composite, will be based on four basic assumptions, as follows: 앫 The composite consists of two phases, the reinforcement and the matrix with no interfacial reaction layer. 앫 The reinforcement is uniformly distributed within the matrix, hence its spacing is constant. 앫 The composite cell represents the whole specimen and consists of a reinforcing element surrounded by the matrix at the same volume fractions as those in the entire specimen.
Fig. 1. Schematic representation of (a) longitudinal and (b) transverse composite cells in a continuous fiber-reinforced composite.
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2.1.1. Fibers aligned in the longitudinal direction Fig. 1(a) shows a longitudinal composite cell with a length of w, a width of 2Df, a height of 2Df, and a fiber radius of r. The electrical resistance of the longitudinal composite cell, RL, can be simply obtained by combining in parallel the electrical resistance of reinforcement and matrix, Rf and Rm, respectively as: RL ⫽
Rf·Rm wrfrm ⫽ 2 Rf ⫹ Rm 4Df [rf(1⫺Vf) ⫹ rmVf]
(1)
where rf and rm are the electrical resistivities of the reinforcement and the matrix, respectively, and Vf is the volume fraction of the fiber. Since RL = w rL 2, the electrical resistivity of the longitudinal 4Df composite cell is given by: rL ⫽
r fr m rf(1⫺Vf) ⫹ rmVf
rm 1⫺Vf
(3)
2.1.2. Fibers aligned in the transverse direction Fig. 1(b) shows a transverse composite cell with a length of w, a width of 2Df, a height of 2Df, and a fiber radius of r. The electrical resistance of this transverse composite cell can be obtained by calculating that of a 1/4 transverse composite cell, which is divided into two regions: region 1 is composed of both reinforcement and matrix with a combined electrical resistance of R1, and region 2 is only composed of matrix with a electrical resistance of R2. In region 1, the differential electrical resistances of reinforcement and matrix are dRf = dx dx and dRm = rm , respectively. Fig. 1(b) rf whf whm shows that hf = rsinq, hm = Df⫺rsinq, x = rcosq, and dx = ⫺rsinq·dq. Hence, the differential electrical resistance of region 1: dR1 ⫽
dRf·dRm dRf ⫹ dRm
rmrf(⫺rsinq) dq w[rmrsinq ⫹ rf(Df⫺rsinq)]
The average electrical resistance of region 1, R1, p can be obtained by integrating dR1 from q = to 2 0 r q = 0 as R1 = pdR1. Since R1 = r1 , the average wDf
冕
(4)
2
electrical resistivity of region 1, r1, may be expressed as: r1 ⫽
冕
Df 0 rmrfrsinq dq r p(rf⫺rm)rsinq⫺rfDf
(5)
2
Following a detailed calculation, r1 is then obtained as:
冦
冋冪
rfDf ⫹ (rf⫺rm)r rfDf⫺(rf⫺rm)r
2rf2Dftan⫺1
Dfrm r1 ⫽ r
(2)
If the reinforcement is an insulator, i.e. rf ⬎ ⬎ rm, the electrical resistivity can be simplified as: r⬁L ⫽
⫽
6293
冑
(rf⫺rm) r D ⫺(rf⫺rm) r 2 f
2 f
2 2
册
冧
prf ⫺ 2(rf⫺rm)
(6) The width of the 1/4 composite cell, Df, varies with the volume fraction of reinforcement and can be obtained from the following simple calculation: Vf ⫽
pr2 4D2f
⇒
Df ⫽
冪4V r p
(7)
f
冪
r 4Vf , the electrical resistivity = Df p of the 1/4 composite cell is: By setting b =
冦
rm r1 ⫽ b
冋冪
2r2ftan⫺1
册
rf ⫹ (rf⫺rm)b rf⫺(rf⫺rm)b
冧
prf ⫺ 2(r f⫺rm) (rf⫺rm)冑r ⫺(rf⫺rm) b 2 f
2 2
(8) Region 2 consists only of matrix, hence r2 = rm and the electrical resistivity of the 1/4 transverse composite cell, rT(1 / 4), may be expressed by: r Df⫺r ⫽ r1b ⫹ rm(1⫺b) rT(1/4) ⫽ r1 ⫹ rm Df Df
(9)
and the electrical resistance of this 1/4 composite cell, by:
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rT(1/4) Df ⫽ RT(1/4) ⫽ rT(1/4) Df·w w
(10)
The composite cell comprises four identical 1/4 composite cells. Hence, the electrical resistivity of a composite cell, rT, equals rT(1/4), and consequently the electrical resistance of the composite cell, RT, equals RT(1/4). Thus, the electrical resistivity of the transverse composite cell may be expressed by:
冦
rT ⫽ r1b ⫹ rm(1⫺b) ⫽ rm 1⫺b
冋冪
2r2ftan⫺1
rf ⫹ (rf⫺rm)b rf⫺(rf⫺rm)b
册
冧 (11)
When D fr, i.e., the volume fraction of reinforcement tends to zero, b⬇0 and rT⬇rm. If the reinforcement is an insulator, i.e. r fr m, the electrical resistivity of the transverse composite cell can be simplified as:
r ⫽ rm ⬁ T
冦
p 1⫺b⫺ ⫹ 2
2tan⫺1
冋冪 册 b⫹1 b⫺1
冑1⫺b
2
Basically, for a composite reinforced by twodimensionally and randomly distributed short fibers the expression for electrical resistivity is similar to that of a cross-stacked, continuous fiberreinforced composite. The electrical resistivity also combines longitudinal and transverse terms, but the volume fractions of longitudinal and transverse fibers are no longer exactly equal to 0.5. According to Halpin and Tsai [12], 3/8 of all fibers contribute to the transfer of electrons in the longitudinal direction and 5/8 to the transfer in the transverse direction. Hence, considering a given volume of composite: 5 3 N ⫽ NL ⫹ NT 8 8
prf ⫹ ⫺ 2 2 2 2(r f⫺rm) (rf⫺rm) rf ⫺(rf⫺rm) b
冑
2.2. Whisker or short fiber-reinforced composites
冧
(14)
where N, NL, and NT are the total number of fibers, and those aligned in the longitudinal and transverse directions, respectively. The electrical resistance of the composite with short fibers, rc(sf), can then be obtained by combining those of longitudinal and transverse composite cells and can be expressed by: rC(sf) ⫽
8rL·rT 5rL ⫹ 3rT
(15)
2.3. Particulate-reinforced composites (12)
Fig. 2 shows a particulate-reinforced composite cell with a length of w, a width of 2Dp, a height
2.1.3. Cross-stacked fibers For a cross-stacked continuous fiber reinforced metal matrix composite, the electrical resistance of the composite can be expressed as the combination of the electrical resistance of longitudinal and transverse composite cells. By averaging the inverse resistivities in the longitudinal and transverse directions the electrical resistivity of the fiber-reinforced composite, rc(f), may be expressed by: rC(f) ⫽
2rL·rT rL ⫹ rT
(13)
Fig. 2. Schematic representation of the composite cell in a particulate-reinforced composite.
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of 2Dp, and a particulate radius of r. The composite cell is composed of eight 1/8 composite cells, and each 1/8 composite cell is divided into two regions. In region 1, the differential electrical resistances of 4dx the reinforcement and the matrix are dRp = rp 2 px 4dx and dRm = rm , respectively, where x (2Dp)2⫺px2 = rcosq and dx = ⫺rsinq·dq, and rm and rp are the electrical resistivities of the matrix and particulate reinforcement, respectively. The differential electrical resistance of region 1 may be expressed by: dR1 ⫽ ⫽
dRp·dRm dRp ⫹ dRm
By integrating, the electrical resistance of region 1 is obtained as:
冕
R1 ⫽ dR1 ⫽
(17)
冤
4rmrp
tan⫺1 [4rpD ⫺(rp⫺rm)pr ](rp⫺rm)p
冑
2 p
r1 ⫽
2
(rp⫺rm)pr2
冪4r D ⫺(r ⫺r )pr p
2 p
p
冥
2
m
L r Because R1 = r1 = r1 2, the electrical resistivity A Dp of region 1 is:
(20)
冤
4rmrp
tan⫺1 h冑[4rp⫺(rp⫺rm)ph ](rp⫺rm)p 2
R(1/8) ⫽ R1 ⫹ R2 ⫽
4D2prmrp
r冑[4rpD2p⫺(rp⫺rm)pr2](rp⫺rm)p
冤冪
tan⫺1
冥
(rp⫺rm)pr2 4rpD2p⫺(rp⫺rm)pr2
The width of the 1/8 composite cell, Dp, varies with the volume fraction of reinforcement and can be obtained as follows: 4 (2Dp)3 ⫻ Vf ⫽ pr3 3
⇒
r By setting h = = Dp
冪
Dp ⫽
冪6V r
3
p
f
3
6Vp , it follows that: p
(19)
p
p
冥
2
m
⫽
r1r ⫹ rm(Dp⫺r) D2p
(21)
r1h ⫹ rm(1⫺h) Dp
Since a composite cell is composed of eight identical 1/8 composite cells, its electrical resistivity, rC(p), equals r(1/8), and consequently the electrical resistance of the composite cell, RC(p), equals 1/2 R(1/8) by calculation. The electrical resistivity of the composite cell may be expressed by: rC(p) = r1h + rm(1⫺h). Hence,
冦
rC(p) ⫽ rm 1⫺h
⫹
冤
4rp
tan⫺1 冑[4rp⫺(rp⫺rm)ph ](rp⫺rm)p 2
(18)
r1 ⫽
(rp⫺rm)ph2
冪4r ⫺(r ⫺r )ph
Region 2 consists only of the matrix phase, so r2 = rm. The electrical resistance of the 1/8 composite cell is then given by:
(16)
⫺4rmrprsinq dq rmpr2sin2q ⫹ rp(4D2p⫺pr2sin2q)
6295
(rp⫺rm)ph2
冪4r ⫺(r ⫺r )ph p
p
m
冥冧
2
(22) When D pr, i.e., the volume fraction of reinforcement tends to zero, then h⬇0 and rC(p)⬇rm. If the reinforcement is an insulator, i.e. rp ⬎ ⬎rm, the electrical resistivity of the composite cell may be simplified as:
冦
r⬁C(p) ⫽ rm 1⫺h ⫹
4tan⫺1
冋冪
ph2 4⫺ph2
冑p(4⫺ph ) 2
册
冧
(23)
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2.4. Effect of residual stresses and dislocations on electrical resistivity During cooling, following processing or heattreatment of composite specimens thermal stresses are generated in the matrix, in the vicinity of the reinforcement/matrix interface, due to the difference in coefficient of thermal expansion (CTE) of these two phases. Part of these stresses may be relieved through yielding of the matrix which is manifested by plastic strain and formation of dislocations. The other part of these stresses will remain in the composite as residual thermal stresses. Even for furnace-cooled specimens both residual thermal stresses and dislocations will coexist. Strain fields in the matrix adjacent to the reinforcement form energy barriers to the flow of electrons, raising the electrical resistivity of composites. Different matrices and reinforcements result in different residual stress distribution and dislocation densities, hence in different sizes of deformation regions, leading to different effects on electrical resistivity. Arsenault and Shi [8] calculated the dislocation density, rdisl, in a quenched composite specimen, using a model based on a prismatic punching method. They established that: rdisl ⫽
BVfe b(1⫺Vf)d
(24)
where b is the Burgers vector of the matrix, and Vf and d are the volume fraction and the shortest dimension of the reinforcement, respectively. The strain misfit between reinforcement and matrix, ⑀, can be obtained by ⌬CTE × ⌬T, where ⌬CTE is the elementary difference in CTE between reinforcement and matrix, and ⌬T is the difference between heat treatment temperature and room temperature. The geometric parameter of the reinforcement B is equal to 12 when the dimensions of the reinforcement in all directions are the same, as in equiaxed particulates, and equal to 4 if the dimension in one direction is much larger than in the other directions, as in whiskers and fibers [8]. From Eq. (24), it is clear that the dislocation density depends on the type, volume fraction and geometry of the reinforcement, as well as on the type of the matrix. A highly strained deformation region containing
residual stresses and dislocations affects the electrical resistivity of the composite. For a reinforcement of radius r, the radius of the deformation region around the reinforcement may be assumed to be ar (including the radius of the reinforcement), where a is an experimental constant. A large a value corresponds to a large deformation region. The a value is determined by the dislocation density, and depends on the type of matrix and the type, size and shape of the reinforcement, but not its volume fraction. It can be expressed [8] by: Be (a⫺1)⬀ bd
(25)
According to calculations by Flom and Arsenault [10], a = 2.3 for SiCp/Al composite waterquenched from 470 °C. For a particulate reinforcement the volume fraction of the deformation region surrounding the reinforcement, VIIp, may be expressed by: VIIp ⫽ (a3⫺1) ⫻ Vp
(26)
where Vp is the volume fraction of particulates. For a fibrous reinforcement with a large aspect ratio, the volume fraction of the deformation region, VIIf, may be expressed by: VIIf ⫽ (a2⫺1) ⫻ Vf
(27)
The size of the deformation region affects the electrical resistivity of the composite. A modification term C(Vf, a), or C(Vp, a) must be included in the calculated theoretical electrical resistivity of composites [4]. For a fiber-reinforced composite, this term is: C(Vf,a) ⫽
冉 冊
a2⫺1 Vf a
(28)
A similar expression is valid for whiskerreinforced composites, in which Vf is replaced by Vw. For a particulate-reinforced composite, the modification term is: C(Vp,a) ⫽
冉 冊
a3⫺1 Vp a2
(29)
Thus, the modified theoretical electrical resistivity of a fiber-reinforced composite, taking into account
S.-Y. Chang et al. / Acta Materialia 51 (2003) 6191–6302
the effect of residual stresses and dislocations, can be written as:
冋 冉 冊册
r’C(f) ⫽ rC(f) 1 ⫹
a2⫺1 Vf a
(30)
The theoretical electrical resistivity of the particulate-reinforced composite can be modified as:
冋 冉 冊册
a3⫺1 Vp r’C(p) ⫽ rC(p) 1 ⫹ a2
(31)
3. Experimental procedure 3.1. Preparation of metal matrix composite specimens Alternative layers of properly spaced continuous molybdenum fibers (Mof) and stainless steel fibers (SSf) with the same diameter of 150 µm and AA1100 aluminum foils 100 µm thick were stacked and vacuum diffusion-bonded for 10 min at 500 °C, under a pressure of 100 MPa to obtain Mof/Al and SSf/Al composites containing 5–45 vol% fibers [13]. The diffusion-bonded composite specimens were subsequently furnace-cooled. Aluminum matrix specimens containing no fibers were also prepared by diffusion bonding stacked pure aluminum foils following the same procedure. Copper and silver matrix composites containing 10–40 vol% ceramic reinforcements of different types, sizes, and shapes, including silicon carbide whiskers (SiCw, 0.4 µm in diameter), carbon short fibers (Carbonsf, 7 µm in diameter), Saffil short fibers (Saffilsf, 3.8% SiO2-96.2% Al2O3, 3 µm in diameter), silicon carbide particles (SiCp, 12.5, 5, and 1.5 µm), and alumina particles (Al2O3p, 12.5, 5, 3, 1, and 0.3 µm), were produced by electroless copper or silver plating and hot pressing at 600 °C, under a pressure of 300 MPa in air for 15 min, followed by furnace cooling. [14]. For comparison purposes, electrolessly-precipitated pure silver powders from the same silver plating solution were prepared and hot pressed to obtain unreinforced silver matrix specimens. However, because of the difficulty in electroless precipitation of copper powders, commercial oxygen-free, high conductivity
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copper powders were hot-pressed under the same conditions as for silver, to obtain unreinforced copper matrix specimens. These aluminum, copper, and silver matrix composite specimens were then cut to a size of 3 × 4 × 40 cm3 and polished for measurement of electrical resistivity. 3.2. Measurement of electrical resistivity Electrical resistivity of these metal matrix composite specimens was evaluated by a four-point probe method, using a constant input current (I = 0.1 Amp), measuring the voltage (V), and using the following relationship: V ⫽ I ⫻ R, and R ⫽ rE ⫻ (L / A)
(32)
where R is electrical resistance, E is electrical resistivity, and L and A are the length and cross-sectional area of specimens, respectively. Three to four specimens of each type were tested. Pure indium (In) metal was used for electrical contacts and platinum (Pt) wires of 0.1 mm in diameter were used for connection, in order to reduce contact and wiring resistances. The current was input through the electrical contacts at both ends of the specimens, and the voltage was measured between two other contacts on the top faces of the specimens with a spacing of 36 mm between them. Repeated measurements and reversed current inputs were conducted to exclude the effect of contact or wiring resistance and to obtain precise electrical resistivity measurements.
4. Results and discussion 4.1. Experimental electrical resistivity and effect of residual stresses and dislocations Figs. 3–7 show the experimental electrical resistivity of various metal matrix composite specimens reinforced with continuous fibers, whiskers, short fibers, and particulates. In these measurements the data scatter was less than 3%. The theoretical resistivity of these composites was calculated without consideration of residual stresses and dislocations, using Eqs. (13), (15) and (23). Results were plotted in these Figures with an a value of 1.0. Experi-
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Fig. 3. Electrical resistivity of Mof/Al and SSf/Al composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
Fig. 4. Electrical resistivity of SiCw/Cu and SiCw/Ag composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
mental results follow the trend of calculated curves which show that the resistivity increases with increasing volume fraction and decreasing size of reinforcement. However, the experimental data exhibited some positive deviation from the predicted curves, because deformation regions increased the electrical resistivity of composites. This increase may be related to the size of the deformation region. Hence, the theoretical resis-
Fig. 5. Electrical resistivity of Carbonsf/Ag and Saffilsf/Ag composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
Fig. 6. Electrical resistivity of SiCp/Ag composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
tivity for each type of specimen was calculated for different a values and plotted in the same Figs. 3– 7. The experimental resistivity of each composite fitted well one of the theoretical curves corresponding to a specific a value. These experimentally effective a values for different composites were listed in Table 1. By comparing the effective a values of these composites, the effects of the following characteristics of the reinforcement and of the matrix on electrical resistivity were established:
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Fig. 7. Electrical resistivity of Mof/Al and SSf/Al composites vs. volume fraction of reinforcement. Experimental points (E) and theoretical curves (T).
1. Reinforcement volume fraction. According to the calculation by Arsenault and Shi [8] and the investigation by Flom and Arsenault [10], the size of deformation regions, within which the dislocation densities were the same, was basically independent of the volume fraction of reinforcement, whereas the total dislocation density depended on it. In the present study, the a values of same type of composites with differ-
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ent contents of reinforcements were found to be almost the same. 2. Reinforcement shape. According to the work of Zhao, Zhijian and Yingkuo [11] and Vogelsang, Arsenault and Fisher [6], a higher dislocation density was detected around the sharp corners of SiC particles and whiskers, whereas almost no dislocations formed near their flat surfaces. Comparing the electrical resistivity of 3 µm Al2O3p/Ag with that of Saffilsf/Ag with 3 µm short fiber diameter, Al2O3 particulates had more sharp corners and contributed to a larger increase in electrical resistivity than did the Saffil short fibers, due to the corresponding larger a value. Also, the effect of particulates was more significant, because the volume fraction of the deformation region in a particulate-reinforced composite is proportional to (a3⫺1), whereas it is proportional to (a2⫺1) in a fiber-reinforced composite. 3. Reinforcement size. For a reinforcement size below 10 µm the dislocation density increased drastically [8] and significantly affected properties. With decreasing reinforcement size, the total stress and dislocation density, as well as the radius of the deformation region increased, as did the a value. Hence, the composites reinforced with fine SiC whiskers, SiC particu-
Table 1 Experimental and calculated a values for furnace-cooled composites, using as a reference a 10 µm SiCp/Al composite that was waterquenched from 470 °C [10] Composite specimens
d (µm)
B
SiCp/Al Mof/Al SSf/Al SiCw/Cu SiCw/Ag Carbonsf/Ag Saffilsf/Ag SiCp/Ag
10.0 150 150 0.4 0.4 7.0 3.0 12.5 5.0 1.5 12.5 5.0 3.0 1.0 0.3
12 4
Al2O3p/Ag
⌬CTE (10⫺6/°C) ⌬T (°C)
⑀ (10⫺2)
a (exp.)
a (theo.)
Vc
2.3 1.0 1.0 2.0 1.4 1.2 1.2 1.3 1.6 1.8 1.2 1.3 1.4 1.7 2.0
– 1.01 1.01 6.20 5.32 1.25 1.44 1.42 2.04 4.46 1.35 1.87 2.45 5.34 15.48
– 0.77 0.77 0.02 0.03 0.50 0.38 0.18 0.06 0.01 0.20 0.08 0.04 0.003 0.0001
445 175
12
24.0 23.0 13.0 13.0 15.7 15.7 12.1 15.7
275
1.07 0.40 0.23 0.46 0.43 0.43 0.33 0.43
12
13.0
275
0.36
4 4
350 275
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lates, and Al2O3 particulates had much larger a values than those reinforced with larger, 150 µm diameter molybdenum and stainless steel fibers, carbon and Saffil short fibers, and 12.5 µm average size SiC and Al2O3 particulates. 4. Type of reinforcement and matrix. Residual stresses were more significant for reinforcements with lower CTEs. Comparing the a value of SiCp/Ag to that of Al2O3p/Ag, and Carbonsf/Ag to Saffilsf/Ag, SiC particulates and carbon short fibers had lower CTEs than Al2O3 particulates and Saffil short fibers, resulting in larger residual stresses and dislocation densities and larger a values. The matrix type also affected the a values through the CTE value and that of the Burgers vector, which obviously is not the same for aluminum, copper or silver. 4.2. Theoretical electrical resistivity and effect of deformation zone overlapping To further clarify the effect of deformation regions on electrical resistivity of composites containing different reinforcements and matrices, the theoretical sizes of deformation regions, i.e. the a values, were calculated by using as a reference the value of a = 2.3 for 10 µm particulate SiCp/Al composite that water-quenched from 470 °C [10]. For the calculation of ⑀ in Eq. (25) the differences between room temperature and the annealing temperature of matrix metals were adopted as ⌬T values, instead of the actual processing temperatures, because most dislocations were eliminated above the annealing temperature of matrix metals during furnace cooling. Stresses and dislocations remained in the matrix metals only below their annealing temperatures. The annealing temperatures for commercial pure aluminum, copper, and silver are 300, 375, and 200 °C, respectively [15– 17]. Also, the Burgers vectors of these face-centered cubic metals are simply related to their atomic radii. Using the basic properties of reinforcements and matrices listed in Table 2 [17], and the reference a value [10], the theoretical a values in furnace-cooled composites containing different reinforcements and matrices were calculated and listed in Table 1 for comparison with experimentally derived a values.
For composites with small theoretical a values, as those reinforced by large-sized fibers or particulates (Mof/Al, SSf/Al, Carbonsf/Ag, Saffilsf/Ag, 12.5 µm SiCp/Ag, and 12.5 µm Al2O3p/Ag), experimental a values matched the theoretical values very closely. This matching would suggest that the models introduced predict with accuracy the electrical resistivity of metal matrix composites. Thus, in the 150 m continuous fiberreinforced aluminum matrix composites, in which there was presumably no deformation region, the experimental resistivity fits very well the theoretically predicted curves with an a value of 1.0, as shown in Fig. 3. However, in composites with small-sized reinforcements (SiCw/Cu, SiCw/Ag, 1.5 µm SiCp/Ag, and 1 µm, 0.3 µm Al2O3p/Ag), the theoretical a values were very large, and the deformation regions easily overlapped with each other, i.e. ar reached Df and Dp in Eqs. (7) and (19), respectively. The overlapping of deformation regions arrests further expansion, hence the experimental a values were much lower than the theoretical. The last column in Table 1 lists the critical volume fractions of different reinforcements, Vc, which were calculated from Eqs. (7) and (19). Once the volume fraction of the reinforcement exceeds Vc, a reaches a constant value, and Eq. (25) is no longer valid for the calculation of effective a values. For example, the experimental a values for SiCw/Cu and 1 m Al2O3p/Ag reached constant values of 2.0 and 1.8, respectively. These values are much lower than the theoretical values of 6.20 and 5.34, respectively, because the critical volume fractions of SiC whiskers and Al2O3 particulates for overlapping of the deformation zones were as small as 0.02 and 0.003. Therefore, the effective a values need to be further modified by empirical fitting. Fig. 8 exhibits on a logarithmic scale relationships between experimental and theoretical a values for particulate and short fiberreinforced silver matrix composites. Modification terms were obtained by curve-fitting these data with the following representative equation: loga(exp) ⫽ m ⫹ n ⫻ loga(theo)⇒
(33)
a(exp) ⫽ 10m ⫻ an(theo) The constants m and n represent the formation
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Table 2 Basic properties of reinforcements and matrices used in this research [17] Material
Al
Cu
Ag
Mof
SSf
SiCw/p
Carbonsf
Saffilsf
Al2O3p
CTE (×10⫺6) Structure ˚) Atomic Radius (A
28.0 FCC 1.43
17.0 FCC 1.28
19.7 FCC 1.44
5.0 – –
15.0 – –
4.0 – –
4.0 – –
7.6 – –
6.7 – –
In summary, during the calculation of the effective sizes of deformation regions, the theoretical a values obtained by Eq. (25) are not valid once the deformation regions overlap. Modified Eqs. (34) and (35) need to be considered to calculate more precise α values. These modified a values applied to Eqs. (30) and (31) will then more precisely predict the theoretical electrical resistivity of metal matrix composites.
5. Summary
Fig. 8. Logarithmic scale relationship between experimental and theoretical a values of short fiber- and particulatereinforced silver matrix composites.
capability of matrix deformation regions. The larger n is, the closer become the experimental and theoretical values of a. For short fiber and particulate-reinforced silver matrix composites, the following respective modification Eqs. (34) and (35) were obtained: asf(exp) ⫽ 1.16 ⫻ a0.11 sf(theo)
(34)
ap(exp) ⫽ 1.22 ⫻ a0.20 p(theo)
(35)
Short fibers contributed smaller m and n values than particulate reinforcements, indicating more rapid and extensive overlapping of deformation regions. That seemed opposite to the proposed concept of critical volume fractions listed in Table 1, but it is easy to understand that the layers of inplane distributed short fibers were closely packed and easily contacted each other in the hot-pressing direction, leading to the small saturated sizes of deformation regions.
Theoretical models for predicting the electrical resistivity of metal matrix composites reinforced with continuous fibers, short fibers, and particulates were developed by integrating thin slices of composite cells. The experimental electrical resistivity of aluminum, copper, and silver matrix composites was measured and compared with theoretical values derived from the models. Experimentally measured electrical resistivity of composites follows the trend of theoretical prediction, increasing with increasing volume fraction and decreasing size of reinforcement. Deformation regions containing residual thermal stresses and dislocations formed around the reinforcement and contributed to the increase in electrical resistivity. The magnitude of residual stresses and the dislocation density were found to depend on the type, size and shape of reinforcement, as well as the type of matrix. The effective size of the deformation regions varied due to their overlapping and better fitted the calculated curves through empirical modification. Theoretical prediction of resistivity that takes into account the effect of residual stresses and dislocations and the overlapping of deformation regions agreed reasonably well with experimental results.
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Acknowledgements The authors are pleased to acknowledge the financial support for this research by The National Science Council, Taiwan, under Grant No. NSC91-2216-E-007-029. The authors also acknowledge the assistance to this research by Mr Gong-Jin Hong, Cher-Hao Hsue, and Min-Hsiang Chen.
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