January 2007 Mechanical Design for Accommodating Thermal ex 17 for P into Eq. (20) for AK an 1.1 sion into(21)gives the following final result for the degradation in the fracture strength due to thermal stresses induced by the L EBC 07 ErcπA3Kc A special case of Eq.(24)can be reduced to a physically nportant result. For this purpose assume tha = tates that the elastic modulus for the topcoat and the Fig8. A plot of fc), as given by Eq(28)as a function of c, for materials is the same and that the periodic spacing of the estimating the degradation in the fracture strength is about equal to the faw size(which is approximately 10 um). In this case Eq. (24)reduces to the follow expression slurry of zirconia which is spun coated on to the substrate. This procedure results in a type of structure shown schematically in to the dia (26) latex spheres, while width of the columns is varied by changing geometrical analysis gives the following relationship Note that =s(ny) as it follows from Eq(19)that y n+l with j Equation(26)can be expressed in another form by substituting for A, from Eq. (14). Assuming again that EBcR Erc and that (Lhro)≈1, we obtain where vr is the volume fraction of the latex spheres. The result btained with latex spheres having a diameter of 25 um and olume fraction v=0. 5 is shown in the micrograph in Fig9. As on (30) this constitution gives an aspect ratio A NI and wa25 um In this example the columnar structure is made from cles of zirconia while the continuou made from electron-beam physical-vapor deposited HfO (=u According to the design map in Fig. 4, the aspect ratio of one falls to the left hand side of the minimum. Since in the present case pa0.5, we note that for"safe"operation of the coating, it radation in strength, which is given by Eq(1 aso'the deg- is necessary that (hrd/W<0.4. Since Wx25 um, this condition Note that since f(c), in Eq(28), depends only on means that hrc should be less than about 10 primarily on c, except for the additional effect of Kic. Therefore, The results from two EBCs, one with hrc 40 um and the the degradation in strength can be estimated by plotting f(e), as other with hrc 10 um after exposing to streaming humid en- a function of c. This plot is given in Fig. 8. The result is that f(c) vironment at 1250C for 30 h are shown in Fig. 10. while the is of the order of unity, varying from approximately 0.3 to 1.1 itely 0.3 to l 1, thick topcoat delaminates, the thin topcoat, which meets the safe over a wide range of values for c. Taking the near-highest value of this range that is c= l, and assuming that Kic a 2 as dis- cussed earlier (for silicon nitride), we have that the degradation As Prepared After Exposure in strength is approximately equal to af 24r (29) that is. less than 2%. The conclusion to be drawn from the above analysis is that the columnar structure of the bond coat Si3N4 will reduce the influence of the topcoat on the fracture strength of the substrate to just a few percent 50 Detailed experiments for the survivability EBCs prepared ac- ording to the guidelines developed in this paper are being re- o ported separately, where the processing of the multilayer coatings on silicon-nitride is also described. Here we report on the concept for creating the columnar structure, and study whether or not the design-map in Fig. 6 gives a credible predic tion. Note that the safe design space depends on the thickness of the topcoat relative to the width of the columns, and upon the pect ratio of the columns. The method for controlling the above design parameters illustrated in Fig 9. The effective width of the columns and the Fig9. An experimental strategy for creating a columnar structure for pect ratio is controlled by introducing latex spheres into a accommodating the thermal strain in the topcoat.tion for Pj into Eq. (20) for DK and again inserting this expres￾sion into (21) gives the following final result for the degradation in the fracture strength due to thermal stresses induced by the coating: s0 f sf ¼ 1
L a EBC ETC 1 pA3 rK
IC Xn j¼0 u
jOj (24) A special case of Eq. (24) can be reduced to a physically important result. For this purpose assume that EBC ETC 1; and; L a 1 (25) which states that the elastic modulus for the topcoat and the column materials is the same and that the periodic spacing of the columns is about equal to the flaw size (which is approximately equal to 10 mm). In this case Eq. (24) reduces to the following expression: s0 f sf ¼ 1
1 pA3 rK
IC Xn j¼0 u
jOðnjÞ (26) Note that Oj 5 O(nj) as it follows from Eq. (19) that Yj 5 nj where nj 5 1,2, y, n11 with j 5 0, 1,y, n when La. Equation (26) can be expressed in another form by substituting for A3 r from Eq. (14). Assuming again that EBCETC and that (L/hTC)1, we obtain Dsf sf ¼ sf
s0 f sf ¼ fðcÞ 12pK
IC (27) where fðcÞ ¼ c Xn j¼0 m
jOðnjÞ (28) Note that since f(c), in Eq. (28), depends only on c, the deg￾radation in strength, which is given by Eq. (27), also depends primarily on c, except for the additional effect of K
IC. Therefore, the degradation in strength can be estimated by plotting f(c), as a function of c. This plot is given in Fig. 8. The result is that f(c) is of the order of unity, varying from approximately 0.3 to 1.1, over a wide range of values for c. Taking the near-highest value of this range, that is c 5 1, and assuming that K
IC 2 as dis￾cussed earlier (for silicon nitride), we have that the degradation in strength is approximately equal to: Dsf sf : 1 24p (29) that is, less than 2%. The conclusion to be drawn from the above analysis is that the columnar structure of the bond coat will reduce the influence of the topcoat on the fracture strength of the substrate to just a few percent. V. Experiments Detailed experiments for the survivability EBCs prepared ac￾cording to the guidelines developed in this paper are being re￾ported separately,10 where the processing of the multilayer coatings on silicon-nitride is also described. Here we report on the concept for creating the columnar structure, and study whether or not the design-map in Fig. 6 gives a credible predic￾tion. Note that the safe design space depends on the thickness of the topcoat relative to the width of the columns, and upon the aspect ratio of the columns. The method for controlling the above design parameters is illustrated in Fig. 9. The effective width of the columns and the aspect ratio is controlled by introducing latex spheres into a slurry of zirconia which is spun coated on to the substrate. This procedure results in a type of structure shown schematically in Fig. 9. The height of the columns is equal to the diameter of the latex spheres, while width of the columns is varied by changing the volume fraction of the latex spheres in the slurry. Simplified geometrical analysis gives the following relationship: Ar ¼ hBC W ¼ ffiffiffiffiffiffiffi 6vf p r (30) where vf is the volume fraction of the latex spheres. The result obtained with latex spheres having a diameter of 25 mm and volume fraction vf 5 0.5 is shown in the micrograph in Fig. 9. As expected from Eq. (30) this constitution gives an aspect ratio Ar1 and W25 mm. In this example the columnar structure is made from particles of zirconia while the continuous topcoat is made from electron-beam physical-vapor deposited HfO2. According to the design map in Fig. 4, the aspect ratio of one falls to the left hand side of the minimum. Since in the present case r0.5, we note that for ‘‘safe’’ operation of the coating, it is necessary that (hTC/W)r0.4. Since W25 mm, this condition means that hTC should be less than about 10 mm. The results from two EBCs, one with hTC40 mm and the other with hTC10 mm after exposing to streaming humid en￾vironment at 12501C for 30 h are shown in Fig. 10. While the thick topcoat delaminates, the thin topcoat, which meets the safe Fig. 8. A plot of f(c), as given by Eq. (28) as a function of c, for estimating the degradation in the fracture strength. Fig. 9. An experimental strategy for creating a columnar structure for accommodating the thermal strain in the topcoat. January 2007 Mechanical Design for Accommodating Thermal Expansion Mismatch 175