G E. Youngblood et al. /Composites Science and Technology 62(2002)1127-1139 [13]. Interfacial thermal barriers are quantitatively where h is the effective interfacial conductance; Km and characterized by a value called the interfacial con- Kr are the thermal conductivity values of the matrix and ductance, which includes the effect of imperfect match- fiber constituents, respectively; and Vr and a are the ing of surfaces at an interface as well as the effect of fiber volume fraction and radius, respectively. Exami interfacial gaps brought about by debonding of the fiber nation of Eq. (1)indicates that the value of the non from the matrix or microcracking within the fiber coat- dimensional paramete 0, K p fah, relative to the fiber-to- ing [13]. In particular, interfacial fiber-matrix debonding matrix conductivity rat f Km, controls the overall may develop in service due to thermal expansion mis- effect of interfacial barrier resistances on Keff match or thermomechanical fatigue: or in a radiation For analysis, Eq(1) can be written in a simpler non- environment, due to differential swelling/shrinkage dimensional form by making the substitutions: characteristics of the irradiated fiber-matrix constituents The overall interfacial conductance in this paper will be Ker/Km=R interpreted in the broadest sense to also include the Kr/Km =r effective heat transfer coefficient of a thin fiber coating, K/ah=x which may or may not act as a thermal barrier [14, 15]. Vr=f For instance, thin (100-500 nm)pyrocarbon(PyC) or boron-nitride fiber coatings are commonly used to pro- then, using the algebraic substitutions A=(1 x+ r) vide protection of Sic-type fibers during fabrication of and B=(1+ x-r), Eq (1)becomes toughening. Thus, the interfacial conductance, and there- R=[l-(B/A)//[+(B/A)/T (3) fore Keff, will depend upon the fib and the thermal, mechanical and neutron radiation expo- In Eq. 3),lAl is always greater than B), A is always sure histories, and perhaps the surrounding atmosphere positive and b can be positive or negative. Therefore, that can permeate any interfacial gaps [16] he thermal conductivity ratio R is less than or greater The purposes of this study are: first, to assess the than 1 for b being positive or negative, respectively validity limitations for using a particularly simple, but Also, Keff=Km (i.e. R=1) for B=0, or equivalently for very useful model derived by Hasselman and Johnson x=r-l. An explicit solution for x, which is the reci- [13] to predict Kefr for a two-dimensional (2D)SiCr/Sic procal of the Biot number for heat transfer at the fiber composite; and second, to examine the expected effects surface, is given in terms of measurable quantities R, r of temperature and atmosphere on Keff for an example and f by combination of currently available SiC fiber and matrix types. Only the effects of changes in the interfacial con x={(R+1)r-1)+(r+1)(1-R){R+1)+(R-1) ductance on Kefr in the transverse direction are exam- ined. although interfacial conductance effects on in- lane Kefr also are important and expected For In Fig. 1(a, b), the relative thermal conductivity R is instance, the in-plane Keff can be significantly affected plotted as a function of h for fiber volume fractions by microcracking within the composite matrix and by f=0. 1, 0.4, 0.5 and 0.6 for two different fiber /matrix debonding of the fiber from the matrix, as shown by Lu thermal conductivity ratios, r=5 and r=0.2, respec- and Hutchinson [17]. However, the thickness and the tively. To easily compare the effects of r and h on Keff, transverse Kefr of a structural wall govern, in part, ther- the same size fiber (a=5 um) and the same matrix ther mal management in a system, which is the primary focus mal conductivity [Km=20 W/(m K)] were assigned for here. Finally, the expected effects of radiation exposure this example. In these figures the units for h were selec on the interfacial conductance and ultimately on Kefr are ted to be 104 w/(m'K). The numerical labels on the considered for a hypothetical SiC/SiC composite that plot cover a range 0. 1-105 W/( For composites was designed to have a high thermal conductivity containing Sic-type fibers, h-values ranging from I to 400 W/cmK) have been reported [12] L. The hasselman-Johnson model The following observations are made: In 1987, Hasselman and Johnson [13] derived an 1)Asf→0(e.g.f=0.1),R→ I for all values of h. expression for the transverse Keff of dispersed uniaxial (2) For r>l, there is a common crossover point at fibers in a matrix with thermal barriers(thin, insulating R=l for all values of f when x=r-1 type fiber coatings or fiber/matrix debonds) given by (3)For r<l, there is no crossover point and R <I for all values off and h Kerr=Km[(Kr/Km-1-kr/ah)Vr+(1+Kr/Km+kr/ah) (4)For h-0(complete fiber-matrix thermal decou pling), R attains its minimum possible value inde- (1-Kr/Km+ Krahvr+(1+Kr/Km+ krah (1) ndently of r and is given by Rmin=(1-f/(1+f[13]. Interfacial thermal barriers are quantitatively characterized by a value called the interfacial con￾ductance,which includes the effect of imperfect match￾ing of surfaces at an interface as well as the effect of interfacial gaps brought about by debonding of the fiber from the matrix or microcracking within the fiber coat￾ing [13]. In particular,interfacial fiber-matrix debonding may develop in service due to thermal expansion mis￾match or thermomechanical fatigue; or in a radiation environment,due to differential swelling/shrinkage characteristics of the irradiated fiber-matrix constituents. The overall interfacial conductance in this paper will be interpreted in the broadest sense to also include the effective heat transfer coefficient of a thin fiber coating, which may or may not act as a thermal barrier [14,15]. For instance,thin (100–500 nm) pyrocarbon (PyC) or boron-nitride fiber coatings are commonly used to pro￾vide protection of SiC-type fibers during fabrication of SiCf/SiC and to provide a compliant layer for composite toughening. Thus,the interfacial conductance,and there￾fore Keff,will depend upon the fiber coating characteristics and the thermal,mechanical and neutron radiation expo￾sure histories,and perhaps the surrounding atmosphere that can permeate any interfacial gaps [16]. The purposes of this study are: first,to assess the validity limitations for using a particularly simple,but very useful model derived by Hasselman and Johnson [13] to predict Keff for a two-dimensional (2D) SiCf/SiC composite; and second,to examine the expected effects of temperature and atmosphere on Keff for an example combination of currently available SiC fiber and matrix types. Only the effects of changes in the interfacial con￾ductance on Keff in the transverse direction are exam￾ined,although interfacial conductance effects on in￾plane Keff also are important and expected. For instance,the in-plane Keff can be significantly affected by microcracking within the composite matrix and by debonding of the fiber from the matrix,as shown by Lu and Hutchinson [17]. However,the thickness and the transverse Keff of a structural wall govern,in part,ther￾mal management in a system,which is the primary focus here. Finally,the expected effects of radiation exposure on the interfacial conductance and ultimately on Keff are considered for a hypothetical SiCf/SiC composite that was designed to have a high thermal conductivity. 1.1. The Hasselman–Johnson model In 1987,Hasselman and Johnson [13] derived an expression for the transverse Keff of dispersed uniaxial fibers in a matrix with thermal barriers (thin,insulating￾type fiber coatings or fiber/matrix debonds) given by: Keff ¼Km½  ð Þ Kf=Km1Kf =ah Vf þ ð Þ 1þKf =KmþKf =ah ½  ð Þ 1Kf =Km þ Kf =ah Vf þ ð Þ 1 þ Kf =Km þ Kf =ah 1 ð1Þ where h is the effective interfacial conductance; Km and Kf are the thermal conductivity values of the matrix and fiber constituents,respectively; and Vf and a are the fiber volume fraction and radius,respectively. Exami￾nation of Eq. (1) indicates that the value of the non￾dimensional parameter, Kf/ah,relative to the fiber-to￾matrix conductivity ratio, Kf/Km,controls the overall effect of interfacial barrier resistances on Keff. For analysis,Eq. (1) can be written in a simpler non￾dimensional form by making the substitutions: Keff=Km ¼ R Kf=Km ¼ r Kf=ah ¼ x Vf ¼ f ð2Þ then,using the algebraic substitutions A=(1 + x + r) and B=(1 + xr),Eq. (1) becomes: R ¼ ½  1ð Þ B=A f =½  1 þ ð Þ B=A f : ð3Þ In Eq. (3),|A| is always greater than |B|, A is always positive and B can be positive or negative. Therefore, the thermal conductivity ratio R is less than or greater than 1 for B being positive or negative,respectively. Also, Keff=Km (i.e. R=1) for B=0,or equivalently for x=r1. An explicit solution for x,which is the reci￾procal of the Biot number for heat transfer at the fiber surface,is given in terms of measurable quantities R, r and f by: x¼ f R½ þ ð Þ þ1 ð Þ r1 ð Þ rþ1 ð Þ 1R
= f Rð Þþ þ 1 ð Þ R1
: ð4Þ In Fig. 1(a,b),the relative thermal conductivity R is plotted as a function of h for fiber volume fractions f=0.1,0.4,0.5 and 0.6 for two different fiber/matrix thermal conductivity ratios, r=5 and r=0.2,respec￾tively. To easily compare the effects of r and h on Keff, the same size fiber (a=5 mm) and the same matrix ther￾mal conductivity [Km=20 W/(m K)] were assigned for this example. In these figures the units for h were selec￾ted to be 104 W/(m2 K). The numerical labels on the plot cover a range 0.1–105 W/(cm2 K). For composites containing SiC-type fibers, h-values ranging from 1 to 400 W/(cm2 K) have been reported [12]. The following observations are made: (1) As f ! 0 (e.g. f=0.1), R ! 1 for all values of h. (2) For r >1,there is a common crossover point at R=1 for all values of f when x=r 1. (3) For r <1,there is no crossover point and R <1 for all values of f and h. (4) For h!0 (complete fiber-matrix thermal decou￾pling), R attains its minimum possible value inde￾pendently of r and is given by Rmin=(1f)/(1 + f). 1128 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139