G E. Youngblood et al. /Composites Science and Technology 62(2002)1127-1139 the extreme conditions of perfect fiber-matrix thermal effect, fa+fg=1(>1). Heat transfer through gas filling coupling or decoupling exist, or in the case of unusually any interfacial gaps, described by the ghg term in Eq high -or r-values (6), depends upon the ratio of the mean free path of the Most woven 2D-SiCr/SiC composites have a fiber gas molecules to a characteristic gap width(the Knud volume fraction of about 0.4; however, the localized sen number). Therefore, hg itself depends upon gas type, fiber packing within the individual tows likely exceeds temperature and pressure as well as the actual gap his value. In fact, observations of several SiCr/Sic width. The third term on the right in Eq (6) is usually composite cross-sections by SEM reveal that fiber small because an interfacial debonded gap in the trans- g tractions typically range from 0.6 to 0.65 within verse direction is expected to be small and AT across the individual fiber bundles. Furthermore, many direct gap will be small. In this case, the radiative con fiber-fiber contacts occur especially at the numerous ductance, hr, is approximately given by [27] fiber bundle crossover points. For r>l, Kefr may be enhanced by percolation effects that become more h1≈Eo(T)3 important as the number of fiber-fiber contacts increase, perhaps forming significant conductive chain where o=5.67x10-8 W/(m2 K)is the Stefan-Boltz formations. Nevertheless, the relative changes in the mann constant of radiation, s is a mean emissivity for predicted Kefr should be little affected by moderate dis- the interface surfaces and Ti is the average temperature tortions in the theory due to fiber packing non-uni- in the gap. formity or percolation effects. Therefore, the analytic For a typical SiCr/ SiC composite, the relative contri- solutions described by Eq.(1)should be very appro- butions of the three interfacial conductances, hd, hg and priate to analyze degradation mechanisms induced even hr, will now be estimated and compared over the tem- in woven 2D-SiCr/Sic by neutron radiation exposure or perature range 300-1300 K. First, assume that hd results by other mechanical or environmental treatments from asperities between a debonded and mismatched fiber-matrix interface with a mean roughness approxi- mately equal to 0.1 um. Then for a highly crystalline 3. A simple thermal barrier model SiC asperity at the fiber surface with a thermal con ductivity value of Kd 100 W/(m K), ha might be as high If the mechanisms of heat transfer operate indepen- as Ka/105 W/(cm2 K). If the fiber surface actually dently across a thin fiber-matrix interface with an aver- consists of a continuous, low conductivity interphase of age thickness, t, and temperature difference, AT, the thickness al um(e.g. an SiO2 oxidation layer or a fiber heat flux can be described by a simple heat flow model coating with a low thermal conductivity value of Kd l for parallel conduction paths [16] W/(m K), then hd might attain a value as low as 100 W/(cm- K). Thus, for a SiCr/SiC composite ha-values Tht=hAT=l(aKa/0)(g Kg/t)+(rKr/D AT, (5) may cover a relatively wide range from 102 to 105 WI (cm=K). Second, consider that hg might result when a here K; is the effective interfacial thermal conductivity. relatively large interfacial gap is produced by oxidizing The component thermal conductivities, Kd, Kg and Kr away a PyC fiber coating of thickness 0.2 um, which is describe the heat transfer through points in direct con- then filled with dry air at atmospheric pressure. Since act between the fiber and matrix, through any gap Kg=0.026 W/(m K) for ambient air [28], hgsKg/t al between the fiber and matrix, and across these gaps by W/(cm- K). At 1300 K, hg would increase to 35 W/ radiation, respectively. Also, the direct contact and the(cm- K)because of the temperature dependence of K, gas gap fractional area coverage at the fiber-matrix Since hg increases inversely as the gap thickness decrea- interface are given by fd and fg, respectively. It is ses, somewhat higher hg values are possible until the gap assumed that the fractional area coverage for heat dimensions limit hg, i.e. when the gas mean free path transfer by gaseous conduction and radiation are approximately equals t. For air at ambient conditions, equivalent so that fg=fr. The overall interfacial con- this occurs for t0. I um. Thus, the expected hg values ductance. h is then defined as should lie within a range of 10-100 w/(cm K)or less for lower gas pressures. Finally, by Eq.(7)the max h=fahd +fhg +fghr (6) imum value of h in the temperature range of interest is only about 0.01 W/(cmK)at 1300 K. The relative rank In Eq.(6), the net interfacial conductance due to ing for the separate interfacial conductances appears to fiber-matrix contact (aha) depends in a complicated be something like he>hg>>hr. In particular, the radia- manner on both the surface roughness and a thermal tive componentfghr for transverse conduction across nar- resistance mismatch at the interface as well as a possible row gaps is expected to be negligible up to 1300 K. constriction"phenomenon near the direct contact points While the product component fahd usually will be along the interface [26]. Without(with) the constriction larger than ghg, if fd is small enough the magnitudes ofthe extreme conditions of perfect fiber-matrix thermal coupling or decoupling exist,or in the case of unusually high f- or r-values. Most woven 2D-SiCf/SiC composites have a fiber volume fraction of about 0.4; however,the localized fiber packing within the individual tows likely exceeds this value. In fact,observations of several SiCf/SiC composite cross-sections by SEM reveal that fiber packing fractions typically range from 0.6 to 0.65 within individual fiber bundles. Furthermore,many direct fiber–fiber contacts occur especially at the numerous fiber bundle crossover points. For r>1, Keff may be enhanced by percolation effects that become more important as the number of fiber–fiber contacts increase,perhaps forming significant conductive chain formations. Nevertheless,the relative changes in the predicted Keff should be little affected by moderate dis￾tortions in the theory due to fiber packing non-uni￾formity or percolation effects. Therefore,the analytic solutions described by Eq. (1) should be very appro￾priate to analyze degradation mechanisms induced even in woven 2D-SiCf/SiC by neutron radiation exposure or by other mechanical or environmental treatments. 3. A simple thermal barrier model If the mechanisms of heat transfer operate indepen￾dently across a thin fiber-matrix interface with an aver￾age thickness, t,and temperature difference, T,the heat flux can be described by a simple heat flow model for parallel conduction paths [16]: Ki T=t¼h T¼ ð Þþ fdKd=t fgKg=t þð Þ frKr=t
T; ð5Þ where Ki is the effective interfacial thermal conductivity. The component thermal conductivities, Kd, Kg and Kr, describe the heat transfer through points in direct con￾tact between the fiber and matrix,through any gap between the fiber and matrix,and across these gaps by radiation,respectively. Also,the direct contact and the gas gap fractional area coverage at the fiber-matrix interface are given by fd and fg,respectively. It is assumed that the fractional area coverage for heat transfer by gaseous conduction and radiation are equivalent so that fg=fr. The overall interfacial con￾ductance, h,is then defined as h ¼ fdhd þ fghg þ fghr: ð6Þ In Eq. (6),the net interfacial conductance due to fiber-matrix contact ( fdhd) depends in a complicated manner on both the surface roughness and a thermal resistance mismatch at the interface as well as a possible ‘‘constriction’’ phenomenon near the direct contact points along the interface [26]. Without (with) the constriction effect, fd+fg=1 (>1). Heat transfer through gas filling any interfacial gaps,described by the fghg term in Eq. (6),depends upon the ratio of the mean free path of the gas molecules to a characteristic gap width (the Knud￾sen number). Therefore, hg itself depends upon gas type, temperature and pressure as well as the actual gap width. The third term on the right in Eq. (6) is usually small because an interfacial debonded gap in the trans￾verse direction is expected to be small and T across the gap will be small. In this case,the radiative con￾ductance, hr,is approximately given by [27] hr "Bð Þ Ti 3 ð7Þ where sB=5.67 108 W/(m2 K4 ) is the Stefan-Boltz￾mann constant of radiation, e is a mean emissivity for the interface surfaces and Ti is the average temperature in the gap. For a typical SiCf/SiC composite,the relative contri￾butions of the three interfacial conductances, hd, hg and hr,will now be estimated and compared over the tem￾perature range 300–1300 K. First,assume that hd results from asperities between a debonded and mismatched fiber-matrix interface with a mean roughness approxi￾mately equal to 0.1 mm. Then for a highly crystalline SiC asperity at the fiber surface with a thermal con￾ductivity value of Kd 100 W/(m K), hd might be as high as Kd/t 105 W/(cm2 K). If the fiber surface actually consists of a continuous,low conductivity interphase of thickness 1 mm (e.g. an SiO2 oxidation layer or a fiber coating with a low thermal conductivity value of Kd 1 W/(m K),then hd might attain a value as low as 100 W/(cm2 K). Thus,for a SiCf/SiC composite hd-values may cover a relatively wide range from 102 to 105 W/ (cm2 K). Second,consider that hg might result when a relatively large interfacial gap is produced by oxidizing away a PyC fiber coating of thickness 0.2 mm,which is then filled with dry air at atmospheric pressure. Since Kg=0.026 W/(m K) for ambient air [28], hg Kg/t 13 W/(cm2 K). At 1300 K, hg would increase to 35 W/ (cm2 K) because of the temperature dependence of Kg. Since hg increases inversely as the gap thickness decrea￾ses,somewhat higher hg-values are possible until the gap dimensions limit hg,i.e. when the gas mean free path approximately equals t. For air at ambient conditions, this occurs for t 0.1 mm. Thus,the expected hgvalues should lie within a range of 10–100 W/(cm2 K) or less for lower gas pressures. Finally,by Eq. (7) the max￾imum value of hr in the temperature range of interest is only about 0.01 W/(cm2 K) at 1300 K. The relative rank￾ing for the separate interfacial conductances appears to be something like hc>hg> >hr. In particular,the radia￾tive component fghr for transverse conduction across nar￾row gaps is expected to be negligible up to 1300 K. While the product component fdhd usually will be larger than fghg,if fd is small enough the magnitudes of 1132 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139