COMPOSITES SCIENCE AND TECHNOLOGY ELSEⅤIER Composites Science and Technology 62(2002 www.elsevier.com/locate/compscitech The transverse thermal conductivity of 2D-SiCr/sic composites G.E. Youngblood a ,*. David J Senora, R.H. Jones. Samuel Graham Pacific Northwest National Laboratory, PO Box 999, MSIN K2-44, Richland, WA99352, US.A Sandia National Laboratories. Livermore. CA 94550. USA Received 21 June 2001; received in revised form 7 February 2002: accepted 7 February 2002 Abstract The Hasselman-Johnson(H-)model for predicting the effective transverse thermal conductivity(Kefr) of a 2D-SiCr/Sic com- posite with a fiber-matrix thermal barrier was assessed experimentally and by comparison to numerical FEM predictions. Agree- ment within 5% was predicted for composites with simple unidirectional or cross-ply architectures with fiber volume fractions of 0.5 or less and with fiber-to-matrix conductivity ratios less than 10. For a woven 2D-SiCdSic composite, inhomogeneous fiber packing d numerous direct fiber-fiber contacts would introduce deviations from model predictions. However, the analytic model should be very appropriate to examine the degradation in Kefr in 2D-woven composites due to neutron irradiation or due to other mechanical or environmental treatments. To test this possibility, expected effects of irradiation on Ker were predicted by the H-J model for a hypothetical 2D-SiCrSic composite made with a high conductivity fiber and a cvi-Sic matrix. Before irradiation, redicted Kefr for this composite would range from 34 down to 26 W/m K)at 200 and 1000C, respectively. After irradiation to saturation doses at 200 or 1000C, the respective Kefr-values are predicted to decrease to 6 or 10 w/(m-K).C 2002 Elsevier Science Ltd. All rights reserved Keywords: A. Ceramic-matrix composites(CMCs); B Modeling: B. Thermal conductivity; C Computational simulatio 1. ntroduction for possible applications in advanced nuclear fission Silicon carbide(SiC) exhibits favorable mechanical A major issue to be considered when using SiC/ SiC in ld chemical properties at high temperatures for many a high-temperature neutron radiation environment, or applications. Desirable properties of SiC that also make in other non-radiation environments where components it attractive for use in fusion reactor applications in a or structures are subjected to a high heat flux, is the neutron radiation environment are its dimensional sta- expected in-service behavior of its effective transverse bility in the 800-1000oC temperature range, low thermal conductivity, Keff. Knowledge about the expec induced radioactivity and low afterheat [1]. However, ted range of Kefr is necessary to optimize SiC/SiC con the brittle nature of SiC limits its use as a structural figurations for their intended uses. Several modeling material. As compared to monolithic SiC, continuous studies have shown how Kefr depends upon constituent fiber-reinforced SiC-matrix composites (SiCr/SiC) exhi- fiber and matrix thermal conductivity values, and their bit improved toughness with a high, non-catastrophic volume fractions and distributions [6-8]. However strain-to-failure [2]. For these reasons, SiC in the form many experimental measurements have indicated that of SiCr/SiC is being considered as a structural material interfaces between fibers and matrices in a composite for first wall or breeder blanket applications in introduce a thermal barrier that may affect Keff [9-12] advanced fusion power plant concepts in the US [3] Furthermore, Kefr may be affected by physical changes and in international programs [4, 5]. In the US, the of the interface and even the surrounding atmosphere DOE-sponsored Nuclear Energy Research Initiative As with mechanical behavior, to attain desired thermal (NERD) program also is examining SiC composites behavior of SiC SiC proper attention needs to be given to the design of the interphase and to the control of Corresponding author. Tel:+1-509. interfacial thermal effects Classical composite models recently have been up- dated to include the effect of interfacial thermal barrier 0266-3538/02/S. see front matter C 2002 Elsevier Science Ltd. All rights reserved. PII:S0266-3538(02)00069-6
The transverse thermal conductivity of 2D-SiCf/SiC composites G.E. Youngblooda,*,David J. Senora ,R.H. Jonesa ,Samuel Grahamb a Pacific Northwest National Laboratory, PO Box 999, MSIN K2-44, Richland, WA 99352, USA bSandia National Laboratories, Livermore, CA 94550, USA Received 21 June 2001; received in revised form 7 February 2002; accepted 7 February 2002 Abstract The Hasselman–Johnson (H–J) model for predicting the effective transverse thermal conductivity (Keff) of a 2D-SiCf/SiC composite with a fiber-matrix thermal barrier was assessed experimentally and by comparison to numerical FEM predictions. Agreement within 5% was predicted for composites with simple unidirectional or cross-ply architectures with fiber volume fractions of 0.5 or less and with fiber-to-matrix conductivity ratios less than 10. For a woven 2D-SiCf/SiC composite,inhomogeneous fiber packing and numerous direct fiber–fiber contacts would introduce deviations from model predictions. However,the analytic model should be very appropriate to examine the degradation in Keff in 2D-woven composites due to neutron irradiation or due to other mechanical or environmental treatments. To test this possibility,expected effects of irradiation on Keff were predicted by the H–J model for a hypothetical 2D-SiCf/SiC composite made with a high conductivity fiber and a CVI-SiC matrix. Before irradiation, predicted Keff for this composite would range from 34 down to 26 W/(m K) at 200 and 1000 C,respectively. After irradiation to saturation doses at 200 or 1000 C,the respective Keff-values are predicted to decrease to 6 or 10 W/(m–K). # 2002 Elsevier Science Ltd. All rights reserved. Keywords: A. Ceramic-matrix composites (CMCs); B. Modeling; B. Thermal conductivity; C. Computational simulation 1. Introduction Silicon carbide (SiC) exhibits favorable mechanical and chemical properties at high temperatures for many applications. Desirable properties of SiC that also make it attractive for use in fusion reactor applications in a neutron radiation environment are its dimensional stability in the 800–1000 C temperature range,low induced radioactivity and low afterheat [1]. However, the brittle nature of SiC limits its use as a structural material. As compared to monolithic SiC,continuous fiber-reinforced SiC-matrix composites (SiCf/SiC) exhibit improved toughness with a high,non-catastrophic strain-to-failure [2]. For these reasons,SiC in the form of SiCf/SiC is being considered as a structural material for first wall or breeder blanket applications in advanced fusion power plant concepts in the US [3] and in international programs [4,5]. In the US, the DOE-sponsored Nuclear Energy Research Initiative (NERI) program also is examining SiC composites for possible applications in advanced nuclear fission reactors. A major issue to be considered when using SiCf/SiC in a high-temperature neutron radiation environment,or in other non-radiation environments where components or structures are subjected to a high heat flux,is the expected in-service behavior of its effective transverse thermal conductivity, Keff. Knowledge about the expected range of Keff is necessary to optimize SiCf/SiC con- figurations for their intended uses. Several modeling studies have shown how Keff depends upon constituent fiber and matrix thermal conductivity values,and their volume fractions and distributions [6–8]. However, many experimental measurements have indicated that interfaces between fibers and matrices in a composite introduce a thermal barrier that may affect Keff [9–12]. Furthermore, Keff may be affected by physical changes of the interface and even the surrounding atmosphere. As with mechanical behavior,to attain desired thermal behavior of SiCf/SiC proper attention needs to be given to the design of the interphase and to the control of interfacial thermal effects. Classical composite models recently have been updated to include the effect of interfacial thermal barriers 0266-3538/02/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S0266-3538(02)00069-6 Composites Science and Technology 62 (2002) 1127–1139 www.elsevier.com/locate/compscitech * Corresponding author. Tel.: +1-509-375-2314; fax: +1-509-375- 2186. E-mail address: ge.youngblood@pnl.gov (G.E. Youngblood)
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 conductance,which includes the effect of imperfect matching 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 coating [13]. In particular,interfacial fiber-matrix debonding may develop in service due to thermal expansion mismatch 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 provide protection of SiC-type fibers during fabrication of SiCf/SiC and to provide a compliant layer for composite toughening. Thus,the interfacial conductance,and therefore Keff,will depend upon the fiber coating characteristics and the thermal,mechanical and neutron radiation exposure 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 conductance on Keff in the transverse direction are examined,although interfacial conductance effects on inplane 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,thermal 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,insulatingtype 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. Examination of Eq. (1) indicates that the value of the nondimensional parameter, Kf/ah,relative to the fiber-tomatrix conductivity ratio, Kf/Km,controls the overall effect of interfacial barrier resistances on Keff. For analysis,Eq. (1) can be written in a simpler nondimensional 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 reciprocal 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,respectively. To easily compare the effects of r and h on Keff, the same size fiber (a=5 mm) and the same matrix thermal conductivity [Km=20 W/(m K)] were assigned for this example. In these figures the units for h were selected 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 decoupling), R attains its minimum possible value independently of r and is given by Rmin=(1f)/(1 + f). 1128 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139
G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 5) For h-oo(perfect fiber-matrix thermal cou- fibers become thermally decoupled from the matrix and pling), R approaches a maximum value which is Rmin effectively represents the relative thermal con given by ductivity for the limiting case of dispersed cylindrical pores with a volume fraction /. This latter point has Rmax=(1-f)+r(1+O)/(1+f)+r(1-/) important consequences for SiC/Sic designed to have a high thermal conductivity value, i.e. for a composite (6) For r>l (rl and more so for higher fiber volume fractions K,=100W(mK) K=20 2. Validity of the Hasselman-Johnson model The preceding observations about the effect of inter facial conductance on Kefr are all based on the H-J model. The model, a modification of the rayleigh Maxwell equations [18, 19], is considered a dilute con- centration model where a single fiber is surrounded by matrix material and is subjected to periodic boundary 05 conditions. The effects of adjacent fibers on the local field response are not considered (non-interacting fibers). Ho microstructural constituents are expected for real com- Interfacial thermal conductance, h [w/(cm. K) posite materials. Therefore, the conditions for making reliable predictions of Kef for actual composites using the h-j model need to be assessed In this section, for composites containing a random 08 distribution of fibers having an interfacial thermal resistance the predictions based on the H-J model are red to numerical pi derived from a fi element model(FEM). Briefly, a square region of material containing a random distribution of uniaxial fibers was modeled (pseudo-representative volume ele ment or pseudo-RVE). Based on extensive analysis [20], the scale of the region was chosen such that the length of the side of the window(Lw) was seven times the dia- meter of the fibers within the window To simulate a representative volume element of material, pseudo- random-periodic boundary conditions were employed nterfacial thermal conductance, h [wr(cm .. K)] using the commercial finite element code ABAQUS Fig. 1.(a, b) Comparison of analytical solutions of the Hasselman- This was accomplished by creating another square Johnson Eq.(1)as a function of h for fiber volume fractions up to region whose sides were 1.5 L and concentric with the f=0.6 and for two different K/ Km ratios(r=5 and 0.2, respectively). pseudo-RVE(Fig. 2). Doubly periodic reflections of the For each case, the fiber radius a=5 um and Km=20 w/(m K) random geometry were made in the space between the
(5) For h!1 (perfect fiber-matrix thermal coupling), R approaches a maximum value which is given by Rmax ¼ ½ ð Þþ 1f rð Þ 1 þ f =½ ð Þþ 1 þ f rð Þ 1f : (6) For r>1 (r1,and more so for higher fiber volume fractions. 2. Validity of the Hasselman–Johnson model The preceding observations about the effect of interfacial conductance on Keff are all based on the H–J model. The model,a modification of the Rayleigh– Maxwell equations [18,19], is considered a dilute concentration model where a single fiber is surrounded by matrix material and is subjected to periodic boundary conditions. The effects of adjacent fibers on the local field response are not considered (non-interacting fibers). However,interactions between neighboring microstructural constituents are expected for real composite materials. Therefore,the conditions for making reliable predictions of Keff for actual composites using the H–J model need to be assessed. In this section,for composites containing a random distribution of fibers having an interfacial thermal resistance the predictions based on the H–J model are compared to numerical predictions derived from a finite element model (FEM). Briefly,a square region of material containing a random distribution of uniaxial fibers was modeled (pseudo-representative volume element or pseudo-RVE). Based on extensive analysis [20], the scale of the region was chosen such that the length of the side of the window (Lw) was seven times the diameter of the fibers within the window. To simulate a representative volume element of material,pseudorandom-periodic boundary conditions were employed using the commercial finite element code ABAQUS. This was accomplished by creating another square region whose sides were 1.5 Lw and concentric with the pseudo-RVE (Fig. 2). Doubly periodic reflections of the random geometry were made in the space between the Fig. 1. (a,b) Comparison of analytical solutions of the Hasselman– Johnson Eq. (1) as a function of h for fiber volume fractions up to f=0.6 and for two different Kf/Km ratios (r=5 and 0.2,respectively). For each case,the fiber radius a=5 mm and Km=20 W/(m K). G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139 1129
G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 Pseudo-RVE 25F K,/Km 1.5 c 0.5 想d h [w/(cmK) 3.5 K/Km=100 T1 1.5L RVE of dimension Lw and imposed periodic structural boundary 1.5 conditions nested within an outer window of dimension 1.5 Lw outer boundary and the pseudo-RVE. Periodic bound 0.5 ary conditions were then imposed on the outer region that induced random-periodic boundary conditions on he inner pseudo-RVE. Random-periodic boundary h [w/(cm2.K conditions have been shown to approximate the effec- tive response of an RVE even in cases where statistical- matrix composite with a normalized effective thermal conductivity homogeneity of microstructural quantities is not satis- R(=Ken/ Km)calculated by Eq(1). The results are presented as a fied [20-22]. The effective thermal conductivity(Ke) function of h for Vr values up to 0.5 and fiber-to-matrix conductivity then was determined by dividing the average heat flux ratios of(a)r=10, and(b)r=100 by the average temperature gradient determined between the boundaries of the pseudo-RVE. Further interface. The fiber perturbation is eventually cancelled details of the analysis are presented elsewhere [23] out as the Biot number (ah/K=1/ x)decreases, and the In Fig 3(a-c), the predictions of R(=Kcfr/Km)and the material behaves as a homogeneous continuum. This is numerical FEM results for fiber packing fractions he reason for the common crossover point for all fiber Ve=0.1-0.5 are compared for two Ko/Km ratios, r=10 volume fractions predicted by the H-J model as well as and 100, respectively. It is quite clear that differences the FEM method when r>1. This point, called the between the H-J model and the numerical results occur homogenization point, results in the local values of heat for Ve>0.3, but only for conditions approaching perfect flux being equal in the fiber and matrix for temperature- thermal coupling or decoupling(hoo or 0, respec- induced boundary conditions. Thus, rather than just a tively). Also, the differences between the analytical H-J global averaged response local homogeneity is achieved model and numerical FEM results increase with Further decreasing of the Biot number past the homo- increasing volume fraction genization point continues to decouple the conductivity The differences between the analytical model and the contribution of the fibers. The differences between Eq numerical FEM predictions are due to the inhomo- 1)and the fem predictions again increase as the per- geneity created in the microstructure accounted for by turbation due to the thermal decoupling of the fibers he FEM net, but not by the H-J model. When r>l, from the matrix becomes stronger. fiber interaction cannot be ignored as larger fiber In Fig. 4(a, b), a non-dimensional plot of the differ volume fractions introduce larger perturbations in local ence(error) between the H-J model and the numerical material response. However, as an interfacial thermal FEM predictions are presented for Vr=0.4 and 0.5. resistance is introduced, this perturbation in material respectively. For the range of parameters studied, the response is dampened and fiber interaction effects are H-J model predictions deviate from the numerical reduced. Deviations from the H-J model are then results by a maximum of 9% for the perfect thermal reduced as a result of introducing an imperfect thermal coupling case and a15% for the condition of thermally
outer boundary and the pseudo-RVE. Periodic boundary conditions were then imposed on the outer region that induced random-periodic boundary conditions on the inner pseudo-RVE. Random-periodic boundary conditions have been shown to approximate the effective response of an RVE even in cases where statisticalhomogeneity of microstructural quantities is not satis- fied [20–22]. The effective thermal conductivity (Keff) then was determined by dividing the average heat flux by the average temperature gradient determined between the boundaries of the pseudo-RVE. Further details of the analysis are presented elsewhere [23]. In Fig. 3(a–c),the predictions of R(=Keff/Km) and the numerical FEM results for fiber packing fractions Vf=0.1–0.5 are compared for two Kf/Km ratios, r=10 and 100,respectively. It is quite clear that differences between the H–J model and the numerical results occur for Vf>0.3,but only for conditions approaching perfect thermal coupling or decoupling (h!1 or 0,respectively). Also,the differences between the analytical H–J model and numerical FEM results increase with increasing volume fraction. The differences between the analytical model and the numerical FEM predictions are due to the inhomogeneity created in the microstructure accounted for by the FEM net,but not by the H–J model. When r>1, fiber interaction cannot be ignored as larger fiber volume fractions introduce larger perturbations in local material response. However,as an interfacial thermal resistance is introduced,this perturbation in material response is dampened and fiber interaction effects are reduced. Deviations from the H–J model are then reduced as a result of introducing an imperfect thermal interface. The fiber perturbation is eventually cancelled out as the Biot number (ah/Kf=1/x) decreases,and the material behaves as a homogeneous continuum. This is the reason for the common crossover point for all fiber volume fractions predicted by the H–J model as well as the FEM method when r>1. This point,called the homogenization point,results in the local values of heat flux being equal in the fiber and matrix for temperatureinduced boundary conditions. Thus,rather than just a global averaged response local homogeneity is achieved. Further decreasing of the Biot number past the homogenization point continues to decouple the conductivity contribution of the fibers. The differences between Eq. (1) and the FEM predictions again increase as the perturbation due to the thermal decoupling of the fibers from the matrix becomes stronger. In Fig. 4(a,b),a non-dimensional plot of the difference (error) between the H–J model and the numerical FEM predictions are presented for Vf=0.4 and 0.5, respectively. For the range of parameters studied,the H–J model predictions deviate from the numerical results by a maximum of 9% for the perfect thermal coupling case and 15% for the condition of thermally Fig. 2. Example finite element mesh for f=0.5 shown with a pseudoRVE of dimension Lw and imposed periodic structural boundary conditions nested within an outer window of dimension 1.5 Lw. Fig. 3. Comparison of the finite element results for a uniaxial fibermatrix composite with a normalized effective thermal conductivity R (=Keff/Km) calculated by Eq. (1). The results are presented as a function of h for Vf values up to 0.5 and fiber-to-matrix conductivity ratios of (a) r=10,and (b) r=100. 1130 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139
G E. Youngblood et al. /Composites Science and Technology 62(2002)1127-1139 vf=0.4 口 Kf/Km= 100 o KI/Km=1 1000 Biot number b) 山 O Ki/Km= 0.001 10 1000 Biot number Fig 4. Nondimensional plot of the error between the finite element results and Eq (1)for Kd Km=1, 10 or 100 and (a)Vr=0. 4 and (b)vr=0.5 decoupled fibers. As expected, the differences for the Based on values reported in the literature, h-values Ve=0.5 case are somewhat larger than for the Vf=0.4 between 0. 1 and 105 W/(cm2K)are possible [12, 24, 25 case. However, for r< 10 the differences are less than The H-J model should satisfactorily describe Keff of 5%. Thus, for composite systems with V<0.5 and many ceramic matrix composites, including SiC/SiC r-values between I and 10, the H-J model is applicable with simple unidirectional or cross-ply laminate fiber over a wide range of h-values. This criterion is often met architectures. More complicated theories that take into by typical commercial ceramic matrix composites. account fiber interaction may only be necessary when
decoupled fibers. As expected,the differences for the Vf=0.5 case are somewhat larger than for the Vf=0.4 case. However,for r410 the differences are less than 5%. Thus,for composite systems with Vf40.5 and r-values between 1 and 10,the H–J model is applicable over a wide range of h-values. This criterion is often met by typical commercial ceramic matrix composites. Based on values reported in the literature, h-values between 0.1 and 105 W/(cm2 K) are possible [12,24,25]. The H–J model should satisfactorily describe Keff of many ceramic matrix composites,including SiCf/SiC with simple unidirectional or cross-ply laminate fiber architectures. More complicated theories that take into account fiber interaction may only be necessary when Fig. 4. Nondimensional plot of the error between the finite element results and Eq. (1) for Kf/Km=1,10 or 100 and (a) Vf=0.4 and (b) Vf=0.5. G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139 1131
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 of
the 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 distortions in the theory due to fiber packing non-uniformity or percolation effects. Therefore,the analytic solutions described by Eq. (1) should be very appropriate 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 independently across a thin fiber-matrix interface with an average 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 contact 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 conductance, 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 Knudsen 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 transverse direction is expected to be small and T across the gap will be small. In this case,the radiative conductance, hr,is approximately given by [27] hr "Bð Þ Ti 3 ð7Þ where sB=5.67 108 W/(m2 K4 ) is the Stefan-Boltzmann 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 contributions of the three interfacial conductances, hd, hg and hr,will now be estimated and compared over the temperature range 300–1300 K. First,assume that hd results from asperities between a debonded and mismatched fiber-matrix interface with a mean roughness approximately equal to 0.1 mm. Then for a highly crystalline SiC asperity at the fiber surface with a thermal conductivity 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 decreases,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 maximum value of hr in the temperature range of interest is only about 0.01 W/(cm2 K) at 1300 K. The relative ranking for the separate interfacial conductances appears to be something like hc>hg> >hr. In particular,the radiative component fghr for transverse conduction across narrow 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
G E. Youngblood et al. /Composites Science and Technology 62(2002)1127-1139 the direct and gaseous conduction components might be 烈小点 comparable. The product components depend on the magnitudes of fa and fg, and should be very dependent upon the surface roughness, gap dimensions, tempera- ture and environmental conditions. Separately deter- mInin ng fa and fg would be dificult because these quantities likely vary with changing environmental con- ditions in a non-reproducible manner. However, hg 0 改 nder vacuum conditions. Therefore, the product terms fahd and ghg, can at least be estimated by performing experiments on the same sample in both vacuum and a known atmosphere. To a good approximation, the interfacial conductance measured in vacuum should represent only fahd term. Then to determine the component ghg, the total interfacial conductance h is Fig. 5. A transverse view by SEM of a polished surface for a uniaxial measured in the gaseous phase of interest, and fahd is Hi-Nicalon M/PIP-SiC (amorphous Ceraset M) composite after two subtracted from it crystallization temperature for Ceraset (a1600C) 4. The effective transverse thermal conductivity of However, the hTT conditions (30 min at 1100oC in example SiC/SiC composites argon) were sufficient to form a fairly stable, Si-C-O matrix structure and to expel most of the gaseous com 4.1. Uniaxial Hi-Nicalon/ PIP-SiC composite ponents. The infiltration and HTT cycle was repeated one time to increase the composite density. After the To experimentally assess the analytical solutions for second HTT, a rod was core-drilled from the center of the H-J model, the transverse Kefr for a particular uni- the plug, and several disks for thermal diffusivity mea- axial SiCe/ Sic composite was measured in three diffe surements were cut from the rod with fiber alignment ent atmospheres at a fixed temperature. Using Keff, Kf, either parallel or perpendicular to the flat disk surfaces Km, a and Ve values determined experimentally, the The individual disks were 62-mm diameter by 2-mm overall interfacial conductance, h, was calculated from thick. Normally, for laser flash thermal diffusivity mea- estimated by the method described above and compared however later tests requiring packaging in small y Eq.() for each atmosphere. Then fghg-values were surements one prefers a diameter-to-thickness ratio >5, to calculated values based on known values of gas ther- tion capsules limited the size of our disks to a relatively mal conductivity and a gap thickness estimated from small diameter microstructural analysis Representative fiber diameter and packing fraction The particular composite selected for this analysis was values were determined by SEM examination of made with Hi-Nicalon(Hi-Nicalon is a trademark used polished disc surface with fiber ends normal to the sur- by the Nippon Carbon Co., Yokohana 221, Japan) Sic face. Five randomly selected areas were examined using fiber contained within a polymer impregnated and pyr- a commercial image analysis routine(Prism View olyzed(PIP)-SiC matrix. The procedure used to fabri- Prism View is a trademark used by Analytical Vision, cate uniaxial Hi-Nicalon/PIP-SiC composites for this Inc, Rayleigh, NC, USA). The average diameter of the study follows. First, several Hi-Nicalon fiber tows(2- Hi-Nicalon fiber was 13.8+1.5 um, and a representative cm lengths) were aligned inside a piece of shrink-fit fiber packing fraction for the uniaxial composite was tubing that was then filled with a liquid pre-ceramic 0.566+0.040 polymer( CerasetTM, Ceraset is a trademark used by An SEM micrograph showing a transverse view of the DuPont Lanxide, Newark, DE, USA). The polymer was typical packing and alignment of the Hi-Nicalon fibers cured at 190C in a vacuum for 30 min so that the is given in Fig. 5. The space between the individual tubing simultaneously shrank and compressed the fibers fibers appears uniformly infiltrated by the PIP-SiC into a relatively high packing fraction (>0.6). After matrix. However, the matrix contains numerous curing, the now-rigid plug (8-mm diameter) was shrinkage cracks running both parallel and perpend removed from the tubing and wrapped in graphfoil cular to the fiber lengths. The parallel cracks obviously prior to a high-temperature treatment (HTT). The PIP- would interrupt the transverse heat conduction paths matrix component was purposely kept amorphous so between most of the filaments. Also, numerous direct that fiber contributions would dominate the overall fiber-to-fiber contacts are observed in Fig. 5, although composite thermal conductivity. For this reason, the the actual contact area between touching cylindrical HTT temperature(1100C)was selected below the Sic fibers should be limited
the direct and gaseous conduction components might be comparable. The product components depend on the magnitudes of fd and fg,and should be very dependent upon the surface roughness,gap dimensions,temperature and environmental conditions. Separately determining fd and fg would be difficult because these quantities likely vary with changing environmental conditions in a non-reproducible manner. However, hg 0 under vacuum conditions. Therefore,the product terms fdhd and fghg,can at least be estimated by performing experiments on the same sample in both vacuum and a known atmosphere. To a good approximation,the interfacial conductance measured in vacuum should represent only the fdhd term. Then to determine the component fghg,the total interfacial conductance h is measured in the gaseous phase of interest,and fdhd is subtracted from it. 4. The effective transverse thermal conductivity of example SiCf/SiC composites 4.1. Uniaxial Hi-NicalonTM/PIP-SiC composite To experimentally assess the analytical solutions for the H–J model,the transverse Keff for a particular uniaxial SiCf/SiC composite was measured in three different atmospheres at a fixed temperature. Using Keff, Kf, Km, a and Vf values determined experimentally,the overall interfacial conductance, h,was calculated from Eq. (4) for each atmosphere. Then fghg-values were estimated by the method described above and compared to calculated values based on known values of gas thermal conductivity and a gap thickness estimated from microstructural analysis. The particular composite selected for this analysis was made with Hi-Nicalon (Hi-Nicalon is a trademark used by the Nippon Carbon Co.,Yokohana 221,Japan) SiC fiber contained within a polymer impregnated and pyrolyzed (PIP)-SiC matrix. The procedure used to fabricate uniaxial Hi-Nicalon/PIP-SiC composites for this study follows. First,several Hi-Nicalon fiber tows ( 2- cm lengths) were aligned inside a piece of shrink-fit tubing that was then filled with a liquid pre-ceramic polymer (CerasetTM; Ceraset is a trademark used by DuPont Lanxide,Newark,DE,USA). The polymer was cured at 190 C in a vacuum for 30 min so that the tubing simultaneously shrank and compressed the fibers into a relatively high packing fraction ( f50.6). After curing,the now-rigid plug ( 8-mm diameter) was removed from the tubing and wrapped in graphfoil prior to a high-temperature treatment (HTT). The PIPmatrix component was purposely kept amorphous so that fiber contributions would dominate the overall composite thermal conductivity. For this reason,the HTT temperature (1100 C) was selected below the SiC crystallization temperature for Ceraset ( 1600 C). However,the HTT conditions (30 min at 1100 C in argon) were sufficient to form a fairly stable,Si–C–O matrix structure and to expel most of the gaseous components. The infiltration and HTT cycle was repeated one time to increase the composite density. After the second HTT,a rod was core-drilled from the center of the plug,and several disks for thermal diffusivity measurements were cut from the rod with fiber alignment either parallel or perpendicular to the flat disk surfaces. The individual disks were 6.2-mm diameter by 2-mm thick. Normally,for laser flash thermal diffusivity measurements one prefers a diameter-to-thickness ratio 55, however later tests requiring packaging in small radiation capsules limited the size of our disks to a relatively small diameter. Representative fiber diameter and packing fraction values were determined by SEM examination of a polished disc surface with fiber ends normal to the surface. Five randomly selected areas were examined using a commercial image analysis routine (Prism ViewTM; Prism View is a trademark used by Analytical Vision, Inc.,Rayleigh,NC,USA). The average diameter of the Hi-Nicalon fiber was 13.81.5 mm,and a representative fiber packing fraction for the uniaxial composite was 0.5660.040. An SEM micrograph showing a transverse view of the typical packing and alignment of the Hi-Nicalon fibers is given in Fig. 5. The space between the individual fibers appears uniformly infiltrated by the PIP-SiC matrix. However,the matrix contains numerous shrinkage cracks running both parallel and perpendicular to the fiber lengths. The parallel cracks obviously would interrupt the transverse heat conduction paths between most of the filaments. Also,numerous direct fiber-to-fiber contacts are observed in Fig. 5,although the actual contact area between touching cylindrical fibers should be limited. Fig. 5. A transverse view by SEM of a polished surface for a uniaxial Hi-NicalonTM/PIP-SiC (amorphous CerasetTM) composite after two infiltration and HTT-1100 C cycles with f=0.566. G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139 1133
G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 either parallel or perpendicular to the fiber alignment direction. The error bars represent the range of mea- sured values at each temperature. As expected, the thermal diffusivity values measured for the parallel alignment were considerably higher than those for the perpendicular alignment because the interfacial gap resistances significantly interfered with heat conduction for the latter case Sic plug derived from Ceraset TM-Sic matrix, a PIP- To simulate the amorphous PIP pre-ceramic polymer 059…n was prepared using the composite fabrication procedure described earlier except without fibers. The plug con- tained numerous small isolated bubbles. but was crack free. Three thermal diffusivity samples, cut from the amorphous PIP-SiC plug, had an average bulk density Fig. 6. Thermal diffusivity values for the uniaxial Hi-NicalonMPI 1.95+0.04 g/cm. In Fig. 7, the calculated thermal con- the fiber direction (in air for T400C). The PIP-SiC matrix material are shown(lower curve).The error bars indicate the range of several measurements made at each thermal conductivity was only 0.720+0.04 W/(m K)at 27oC, and increased gradually up to 1. 10+0.03 W/(m K)at 400C. Since the PIP-SiC material was purposely The effective thermal conductivity values(Kefr)were made amorphous, the measured K-values were lower determined Iction of temperature(T) by calculating than K-values expected for fully crystallized PIP-SIC Km(T)=a(D)·p(D)·Cp(7 (8) material Also shown in Fig. 7 are the calculated thermal con ductivity values() for Hi-NicalonTMfiber where a(n) and p()are the measured thermal diffusiv- curve). To attain Kf, a series model was ity and bulk density values for each disk, respectively. where The bulk density at room temperature for each disk was determined from its mass and dimensions, where the Kerr= Vr(K+(1-VrKm (9) dimensions were adjusted for temperature dependence by using the thermal expansion of B-SiC. The specific a method suggested by Brennan et al. [8]. For the pur heat,Cp, was calculated using the rule-of-mixtures poses of this calculation, Vr=0.566, the Km-data were based on an equivalent composition of Hi-Nicalon in extrapolated up to 1000C(shown dashed in Fig. 7). terms of SiC, C and SiO2 phases and literature values of and Kefrvalues were calculated by Eq.( 8)using the Cp(T) for each[29]. At 323 K, the calculated Cp-value measured a(T)-data for the parallel alignment case for Hi-Nicalon was 689 J/(kg K), which compares well shown in Fig. 6. Because the Km - values were relatively with a measured Cp-value of 670 J/(kg K) listed by small, any uncertainty in their extrapolated values ippon Carbon Co [30] would make little difference in the Kr-calculation by Eq Two laser flash systems were used to measure the ( 9) thermal diffusivity by a standard method [31]. The The thermal conductivity of this batch of Hi-Nic thermal diffusivity tests were conducted between room lon fiber increased from about 4.6 W/(m K)at 27C temperature and 400C using a modified Anter Flash- up to 5.7 W/(m K)at 1000C. The average bulk density lineTM 5000(Flashline 5000 is a trademark used by determined for several representative Hi-Nicalon fiber Anter Corporation, Pittsburgh, PA, USA) laser bundles by the liquid gradient method was 2.69+0.02 g/ measurement system. From 400 to 1000 oC, a custom- cm. Nominal density and thermal conductivity values built system with a tungsten mesh tube furnace con- for Hi-Nicalon fiber are 2.74 g/cm and 7.8 W/(m K)at tained in a steel bell jar was used. The Anter system 25C and 10. 1 W/(m K) at 500C, respectively [30] operated in room air, while the custom-built system The lower bulk density and thermal conductivity contained a vacuum or inert gas. The temperature was measured for our particular batch of Hi-Nicalon varied typically by 100C steps in each system. To likely are due to batch fiber fabrication difference determine a(D), at least six measurements were made at also suggested by other work in our laboratory each temperature step. Further experimental details are To determine the effect of changing the atmosphere given elsewhere [32 on the transverse Keff, one Hi-Nicalon/PIP-SiC sample In Fig. 6, a(T)-data are plotted for the uniaxial Hi- with fibers aligned parallel to the flat specimen faces was Nicalon/PIP-SiC composite with the heat conduction mounted in the custom-built diffusivity system. The
The effective thermal conductivity values (Keff) were determined as a function of temperature (T) by calculating Keff ð Þ¼ T ð Þ T ð Þ T CpðÞ ð T 8Þ where (T) and (T) are the measured thermal diffusivity and bulk density values for each disk,respectively. The bulk density at room temperature for each disk was determined from its mass and dimensions,where the dimensions were adjusted for temperature dependence by using the thermal expansion of b-SiC. The specific heat, Cp,was calculated using the rule-of-mixtures based on an equivalent composition of Hi-Nicalon in terms of SiC,C and SiO2 phases and literature values of Cp(T) for each [29]. At 323 K,the calculated Cp-value for Hi-Nicalon was 689 J/(kg K),which compares well with a measured Cp-value of 670 J/(kg K) listed by Nippon Carbon Co. [30]. Two laser flash systems were used to measure the thermal diffusivity by a standard method [31]. The thermal diffusivity tests were conducted between room temperature and 400 C using a modified Anter FlashlineTM 5000 (Flashline 5000 is a trademark used by Anter Corporation,Pittsburgh,PA,USA) laser measurement system. From 400 to 1000 C,a custombuilt system with a tungsten mesh tube furnace contained in a steel bell jar was used. The Anter system operated in room air,while the custom-built system contained a vacuum or inert gas. The temperature was varied typically by 100 C steps in each system. To determine a(T),at least six measurements were made at each temperature step. Further experimental details are given elsewhere [32]. In Fig. 6, a(T)-data are plotted for the uniaxial HiNicalon/PIP-SiC composite with the heat conduction either parallel or perpendicular to the fiber alignment direction. The error bars represent the range of measured values at each temperature. As expected,the thermal diffusivity values measured for the parallel alignment were considerably higher than those for the perpendicular alignment because the interfacial gap resistances significantly interfered with heat conduction for the latter case. To simulate the amorphous PIP-SiC matrix,a PIPSiC plug derived from CerasetTM pre-ceramic polymer was prepared using the composite fabrication procedure described earlier except without fibers. The plug contained numerous small,isolated bubbles,but was crackfree. Three thermal diffusivity samples,cut from the amorphous PIP-SiC plug,had an average bulk density 1.950.04 g/cm3 . In Fig. 7,the calculated thermal conductivity values for the simulated amorphous (1100 C) PIP-SiC matrix material are shown (lower curve). The thermal conductivity was only 0.7200.04 W/(m K) at 27 C,and increased gradually up to 1.100.03 W/(m K) at 400 C. Since the PIP-SiC material was purposely made amorphous,the measured K-values were lower than K-values expected for fully crystallized PIP-SiC material. Also shown in Fig. 7 are the calculated thermal conductivity values (Kf) for Hi-NicalonTM fiber (upper curve). To attain Kf,a series model was assumed where: Keff ¼ Vf Kf þ ð Þ 1Vf Km: ð9Þ a method suggested by Brennan et al. [8]. For the purposes of this calculation, Vf=0.566,the Km-data were extrapolated up to 1000 C (shown dashed in Fig. 7), and Keff-values were calculated by Eq. (8) using the measured a(T)-data for the parallel alignment case shown in Fig. 6. Because the Km-values were relatively small,any uncertainty in their extrapolated values would make little difference in the Kf -calculation by Eq. (9). The thermal conductivity of this batch of Hi-NicalonTM fiber increased from about 4.6 W/(m K) at 27 C up to 5.7 W/(m K) at 1000 C. The average bulk density determined for several representative Hi-Nicalon fiber bundles by the liquid gradient method was 2.690.02 g/ cm3 . Nominal density and thermal conductivity values for Hi-Nicalon fiber are 2.74 g/cm3 and 7.8 W/(m K) at 25 C and 10.1 W/(m K) at 500 C,respectively [30]. The lower bulk density and thermal conductivity values measured for our particular batch of Hi-Nicalon fiber likely are due to batch fiber fabrication differences,as also suggested by other work in our laboratory. To determine the effect of changing the atmosphere on the transverse Keff,one Hi-Nicalon/PIP-SiC sample with fibers aligned parallel to the flat specimen faces was mounted in the custom-built diffusivity system. The Fig. 6. Thermal diffusivity values for the uniaxial Hi-NicalonTM/PIPSiC composite shown in Fig. 5 measured parallel and perpendicular to the fiber direction (in air for T400 C). The error bars indicate the range of several measurements made at each temperature. 1134 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139
G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 Table I The interfacial conductance, h, at 400C determined by Eqs. (2)and (4) from measured values of thermal diffusivity for a Hi-NicalonTM/PIP-SiC (amorphous Ceraset)composite in different atmospheres. Data used in Eq (4): Kr=5.72 w/(m K), Km=1.05 w/(m k),f=0.566 and a=6.9 um Atmosphere a(400°C Kr(400°C) h(400°C) ×10-6m2/s (%) by Eq (8) ( Keff/Km) Eqs. (2)and (4 1024±0.050 0.975 17.7 Helium(≈105Pa) 1435±0.049 Vacuum(≈10Pa) 1062±0.034 a lg for 10-individual measurements or each condition sample was heated to 400C and held at that tempera- 0.040 for the composite; a=6.9+0. 8 um and Kr=5.72 ture while evacuating to better than 10 Pa for 17 h using W/(m K)for Hi-Nicalon fiber; and Km-1.05 W/(m K) a mechanical roughing-pump After the diffusivity mea- for the amorphous PIP-SiC. Thus, in Eq. (4)r=K urements in vacuum were completed, the furnace was Km=5.45 at 400C. back-filled with helium and evacuated six times to a The results are presented in Table I for each atmo- pressure slightly above atmospheric while holding the spheric condition. The h-values calculated for the three temperature constant. After the measurements in separated vacuum conditions were approximately the helium were completed, the system was evacuated again same, indicating that changing the atmosphere while for 17 h and the measurements were repeated in holding the temperature constant did not affect fa and vacuum. Then the backfill/evacuation procedure was fg, the nominal fractional area coverage for the direct repeated with argon and diffusivity measurements were contact and gas conduction mechanisms, respectively. It made again. Finally, the system was evacuated once also demonstrated that the introduced gases were com- more overnight and diffusivity measurements were pletely eliminated during each 17-h evacuation proce- repeated in vacuum for the third time. Ten diffusivity dure. The average h-value for the vacuum condition was measurements were made at 400+2C for each atmos- 18.9+1.0 W/(cm- K), which represents the product fdhd sheri condition in Eq.(6). The overall h-values determined with the The Kem-values were calculated by Eq.( 8)from the argon and the helium atmospheres were 29.7 and 39.6 measured a( T)-values for each atmosphere. Then with W/(cm- K), respectively. Therefore, the difference R=Kef/Km, h-values were calculated using Eq (4). For between these overall h-values and fahd represents ghg these calculations the following data at 400oC were i. e 10.8 and 20.7 W/(cm2 K)for argon ellum gas used: p=2.08 g/cm, Cp=1. 12 J/g K) and f=0.566+ conduction in the composite, respectively In Fig 8, for the given Kr and Km-values at 400oC the analytical solutions given by Eq (I)are plotted for fiber packing fractions Vr=0. 1, 0.4, 0.5 and 0.6. Super imposed on this plot are the measured R-values at each atmospheric condition forf=0.566. The data points fall in the transition region near the homogenization point where the R-values are most sensitive to the changing h- values, but are not so sensitive to the fiber volume frac tion. Importantly, in this region Eq. (1)most accurately describes the behavior of Keff and its dependence on h Since the minimum hd value is expected to be about 100 W/(cm-K), an fahd-value of 18.9 W/(cm K)implies that f a <0. 2, so fg.8. Examination of the extensive crack structure between the fibers and matrix, as depic ted in Fig. 5, suggests that these values for fa and fg are reasonable estimates for this particular uniaxial compo Temperature (C) site structure. Then, estimating that tal um from Fig. 5 Fig. 7. The calculated thermal conductivity values from 27 to 1000 C and using Kg=0. 269 W/(m K) for helium gas at one for a Hi- fiber(upper curve)and for amorphous Ceraset TM atmosphere and 400C [28], fghgfgKg/t=21 W/(cm that simulates the PIP-Sic matrix material (lower curve) K), which closely matches the 20.7 W/(cm K) value
sample was heated to 400 C and held at that temperature while evacuating to better than 10 Pa for 17 h using a mechanical roughing-pump. After the diffusivity measurements in vacuum were completed,the furnace was back-filled with helium and evacuated six times to a pressure slightly above atmospheric while holding the temperature constant. After the measurements in helium were completed,the system was evacuated again for 17 h and the measurements were repeated in vacuum. Then the backfill/evacuation procedure was repeated with argon and diffusivity measurements were made again. Finally,the system was evacuated once more overnight and diffusivity measurements were repeated in vacuum for the third time. Ten diffusivity measurements were made at 4002 C for each atmospheric condition. The Keff-values were calculated by Eq. (8) from the measured a(T)-values for each atmosphere. Then with R=Keff/Km, h-values were calculated using Eq. (4). For these calculations the following data at 400 C were used: =2.08 g/cm3 , Cp=1.12 J/(g K) and f=0.566 0.040 for the composite; a=6.90.8 mm and Kf=5.72 W/(m K) for Hi-Nicalon fiber; and Km=1.05 W/(m K) for the amorphous PIP-SiC. Thus,in Eq. (4) r=Kf/ Km=5.45 at 400 C. The results are presented in Table 1 for each atmospheric condition. The h-values calculated for the three separated vacuum conditions were approximately the same,indicating that changing the atmosphere while holding the temperature constant did not affect fd and fg,the nominal fractional area coverage for the direct contact and gas conduction mechanisms,respectively. It also demonstrated that the introduced gases were completely eliminated during each 17-h evacuation procedure. The average h-value for the vacuum condition was 18.91.0 W/(cm2 K),which represents the product fdhd in Eq. (6). The overall h-values determined with the argon and the helium atmospheres were 29.7 and 39.6 W/(cm2 K),respectively. Therefore,the difference between these overall h-values and fdhd represents fghg, i.e. 10.8 and 20.7 W/(cm2 K) for argon and helium gas conduction in the composite,respectively. In Fig. 8,for the given Kf- and Km-values at 400 C the analytical solutions given by Eq. (1) are plotted for fiber packing fractions Vf=0.1,0.4,0.5 and 0.6. Superimposed on this plot are the measured R-values at each atmospheric condition for f=0.566. The data points fall in the transition region near the homogenization point where the R-values are most sensitive to the changing hvalues,but are not so sensitive to the fiber volume fraction. Importantly,in this region Eq. (1) most accurately describes the behavior of Keff and its dependence on h. Since the minimum hd-value is expected to be about 100 W/(cm2 K),an fdhd-value of 18.9 W/(cm2 K) implies that fd40.2,so fg 0.8. Examination of the extensive crack structure between the fibers and matrix,as depicted in Fig. 5,suggests that these values for fd and fg are reasonable estimates for this particular uniaxial composite structure. Then,estimating that t 1 mm from Fig. 5 and using Kg=0.269 W/(m K) for helium gas at one atmosphere and 400 C [28], fghg fgKg/t=21 W/(cm2 K),which closely matches the 20.7 W/(cm2 K) value Table 1 The interfacial conductance, h,at 400 C determined by Eqs. (2) and (4) from measured values of thermal diffusivity for a Hi-NicalonTM/PIP-SiC (amorphous CerasetTM) composite in different atmospheres. Data used in Eq. (4): Kf=5.72 W/(m K), Km=1.05 W/(m K), f=0.566 and a=6.9 mm Atmosphere (400 C) [ 106 m2 /s] 1sa (%) Keff (400 C) by Eq. (8) [W/(m K)] R (400 C) (=Keff/Km) h (400 C) Eqs. (2) and (4) [W/(cm2 K)] Vacuum ( 10 Pa) 0.442 4.9 1.0240.050 0.975 17.7 Helium ( 105 Pa) 0.619 3.4 1.4350.049 1.367 39.6 Vacuum ( 10 Pa) 0.464 4.0 1.0750.043 1.023 19.7 Argon ( 105 Pa) 0.554 4.6 1.2840.059 1.223 29.7 Vacuum ( 10 Pa) 0.458 2.6 1.0620.034 1.011 19.2 a 1s For 10-individual measurements or each condition. Fig. 7. The calculated thermal conductivity values from 27 to 1000 C for a Hi-NicalonTM fiber (upper curve) and for amorphous CerasetTM that simulates the PIP-SiC matrix material (lower curve). G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139 1135
36 G E. Youngblood et al./Composites Science and Technology 62(2002)1127-1139 K,=572WmK) 2"r3a38E奖 K=1.05W/(mK "r 20 W/(m K) Fiber Interfacial thermal conductance, h [W/(cm K) Fig 8. Analytical solution of the Hasselman-Johnson Eq (1) for fiber Fig 9. Analytical solutions of the Hasselman-Johnson Eq (1)at 27C volume fractions up to 0.6. The selected Kr and Km-valt lues corres. for fiber volume fractions up to 0.6. The selected values for a. Kr, and pond to constituent values at 400C for the Hi-Nicalon/PIP-Sic Km correspond to constituent values at 27C for a hypothetical composite shown in Fig. 5. The three superimposed data points were 2D- Tyranno SATM/CVI-SiC composite. The determined for this composite(with f=0.566)in vacuum, argon or analytical solution at 1000C for f=0.4 only calculated using Eq.(6). However, using the same tion environment since each component consists mostly method for the argon gas case with Kg=0.0478 W/m of crystalline p-SiC. However, at the time of this study K),fgKgt 3.8 W/(cm- K)rather than 10.8 W/(cm=K) such a composite was not available for actual testing calculated using Eq (6) from data given in Table 1 predicted analytic solutions for such a The reasonable agreement and the correct hypothetical Tyranno SA/CVI-Sic composite are shown by these rough estimates suggest that the shown as a function of h for f-values up to 0.6 when thermal barrier model together with the H-J model using representative data: Kr=65 w/(m k)and Km=20 describe the behavior of Keff and its dependence on h W/(m K)at 27C and a=5 um. The dependence of R for the uniaxial Hi-Nicalon/PIP-SiC composite fairly onf and h is similar to the case examined in Fig. 1(a) and has a crossover at R=l for h=578 W/(cmK).At 1000C, for f=0.4 the h-value for crossover actually 4.2. Hypothetical 2D-Tyranno M/CVI-SiC composite increases to 910 W/(cmK), as indicated in Fig 9 by the (unirradiated and irradiated) bold dashed line. For hx104 W/(cm2 K)or greater, R becomes relatively independent of further increases in h For 2D-woven configurations, an increasing number Importantly, any degradation of an interface with such of fiber-fiber interactions within tightly packed tows are a high value of h would drop Keff into the transition expected to cause an under-estimate of Keff by Eq (1). zone where Kef would significantly decrease as h Furthermore, the degree of the under-estimate will be decreases further larger the further the fiber-matrix thermal conductivity The temperature dependence as well as the irradiation ratio Kr/Km departs from unity and for larger f-values, effects can be included in the analysis by expressing Ke as discussed in Section 3. With these limitations in and Km as a function of temperature and dose in the mind, the H-J model will be used in this section to pre- following manner. For many ceramics, the reciprocal of dict Kefr and assess degradation effects for a hypothe- the thermal conductivity can be fit to a linear function tical 2D-SiCeSiC composite designed to have a desired of temperature for the temperature range near and high Keff as well as dimensional stability in a radiation above the Debye temperature [34]. However, for SiC environment where the Debye temperature is relatively high(807C) The selected fiber, Tyranno SA(Tyranno SA is a [35], a good linear fit is not obtained immediately above trademark used by the Ube Industries, Ltd, Ube City 27C because of the steep temperature dependence of 755, Japan), has a crystalline, near-stoichiometric Sic the Sic heat capacity. However, by using reciprocal microstructure and a relatively high thermal con- thermal diffusivity data the influence of the heat capa- ductivity(65 W/(m K) at RT)[33]. Both the Tyranno city temperature dependence below the Debye tempera SA fiber and a chemical vapor infiltrated(CVI)-Sic ture is removed from Eq.( 8), and a good linear fit matrix should exhibit dimensional stability in a radia- generally i
calculated using Eq. (6). However,using the same method for the argon gas case with Kg=0.0478 W/(m K), fgKg/t 3.8 W/(cm2 K) rather than 10.8 W/(cm2 K) calculated using Eq. (6) from data given in Table 1. The reasonable agreement and the correct trends shown by these rough estimates suggest that the simple thermal barrier model together with the H–J model describe the behavior of Keff and its dependence on h for the uniaxial Hi-Nicalon/PIP-SiC composite fairly well. 4.2. Hypothetical 2D-Tyranno TM/CVI-SiC composite (unirradiated and irradiated) For 2D-woven configurations,an increasing number of fiber–fiber interactions within tightly packed tows are expected to cause an under-estimate of Keff by Eq. (1). Furthermore,the degree of the under-estimate will be larger the further the fiber-matrix thermal conductivity ratio Kf/Km departs from unity and for larger f-values, as discussed in Section 3. With these limitations in mind,the H–J model will be used in this section to predict Keff and assess degradation effects for a hypothetical 2D-SiCf/SiC composite designed to have a desired high Keff as well as dimensional stability in a radiation environment. The selected fiber,Tyranno SATM (Tyranno SA is a trademark used by the Ube Industries,Ltd.,Ube City 755,Japan),has a crystalline,near-stoichiometric SiC microstructure and a relatively high thermal conductivity (65 W/(m K) at RT) [33]. Both the Tyranno SA fiber and a chemical vapor infiltrated (CVI)-SiC matrix should exhibit dimensional stability in a radiation environment since each component consists mostly of crystalline b-SiC. However,at the time of this study such a composite was not available for actual testing. In Fig. 9,predicted analytic solutions for such a hypothetical Tyranno SA/CVI-SiC composite are shown as a function of h for f-values up to 0.6 when using representative data: Kf=65 W/(m K) and Km=20 W/(m K) at 27 C and a=5 mm. The dependence of R on f and h is similar to the case examined in Fig. 1(a) and has a crossover at R=1 for h=578 W/(cm2 K). At 1000 C,for f=0.4 the h-value for crossover actually increases to 910 W/(cm2 K),as indicated in Fig. 9 by the bold dashed line. For h 104 W/(cm2 K) or greater, R becomes relatively independent of further increases in h. Importantly,any degradation of an interface with such a high value of h would drop Keff into the transition zone where Keff would significantly decrease as h decreases further. The temperature dependence as well as the irradiation effects can be included in the analysis by expressing Kf and Km as a function of temperature and dose in the following manner. For many ceramics,the reciprocal of the thermal conductivity can be fit to a linear function of temperature for the temperature range near and above the Debye temperature [34]. However,for SiC where the Debye temperature is relatively high (807 C) [35],a good linear fit is not obtained immediately above 27 C because of the steep temperature dependence of the SiC heat capacity. However,by using reciprocal thermal diffusivity data the influence of the heat capacity temperature dependence below the Debye temperature is removed from Eq. (8),and a good linear fit generally is obtained. Fig. 9. Analytical solutions of the Hasselman–Johnson Eq. (1) at 27 C for fiber volume fractions up to 0.6. The selected values for a, Kf,and Km correspond to constituent values at 27 C for a hypothetical 2D-Tyranno SATM/CVI-SiC composite. The solid line represents the analytical solution at 1000 C for f=0.4 only. Fig. 8. Analytical solution of the Hasselman–Johnson Eq. (1) for fiber volume fractions up to 0.6. The selected Kf- and Kmvalues correspond to constituent values at 400 C for the Hi-NicalonTM/PIP-SiC composite shown in Fig. 5. The three superimposed data points were determined for this composite (with f=0.566) in vacuum,argon or helium atmospheres. 1136 G.E. Youngbloodet al. / Composites Science andTechnology 62 (2002) 1127–1139