Casting (1998) Part 3 pot

where D S is the diffusion coefficient in the solid phase, t f is the local solidification time, and λ is the dendrite arm spacing. A more accurate or exact solution for this model has been obtained (Ref 5). Equation 4a and a similar equation given in Ref 6 approximate the exact solution below f S < 0.9. Equation 4a is applicable not only to platelike dendrites but also to columnar dendrites if 2B in Eq 4b is doubled. It also agrees with Eq 1 for D S or B ? 1 and with Eq 3 for D S or B = 1, respectively. The Brody-Flemings equation (Ref 7) is not applicable for B > 0.5. Third, there is a solid-state diffusion and solute buildup ahead of the solid-liquid interface. An analytical solution for this actual case has not been obtained. However, in the case where D S = 0 and a solute boundary layer controls the solute transfer in the liquid, the effective partition coefficient k ef has been derived for semi-infinite volume-element and steady-state conditions (Ref 8): * (1)exp(/) s ef ocL C k k CkkRD δ ≡= +−− (Eq 5) where C* S is the solid composition at the interface, R is the growth rate, and δ c is the solute boundary layer ahead of the interface. The coefficient k ef can be used for k in Eq 3: (1) (1) ef k L s o C f C − =− (Eq 6) and C S = k ef · C L (Eq 7) Although Eq 5 cannot be directly applied to dendritic solidification, it is useful for an understanding of the formation of microsegregation. Further, it is applied to evaluate macrosegregation in single-crystal growth. References cited in this section 2. G.H. Gulliver, J. Inst. Met., Vol 9, 1913, p 120 3. E. Scheil, Z. Metallkd., Vol 34, 1942, p 70 4. I. Ohnaka, Trans. ISIJ, Vol 26, 1986, p 1045 5. S. Kobayashi, Tetsu-to-Hagané (J. Iron Steel Inst. Jpn.), Vol 71, 1985, p S199, S1066 6. T.W. Clyne and W. Kurz, Trans. AIME, Vol 12A, 1981, p 965 7. H.D. Brody and M.C. Flemings, Trans. TMS-AIME, Vol 236, 1966, p 615 8. G.F. Bolling and W.A. Tiller, J. Appl. Phys., Vol 32, 1961, p 2587 Microsegregation In practice, microsegregation is usually evaluated by the Microsegregation Ratio, which is the ratio of the maximum solute composition to the minimum solute composition after solidification, and by the amount of nonequilibrium second phase in the case of alloys that form eutectic compounds. Some data and an isoconcentration profile for an Fe-25Cr-19Ni columnar dendrite (Ref 9) are given in Table 1 and Fig. 3. Table 1 Microsegregation ratio (numbers without dimension) and amount of nonequilibrium second phase (mass%) Microsegregation ratio Alloys, mass% Mo (1.4-2.0) Carbon steel (0.3-0.4C) Cr (1-5, increases with C up to 1.4%) Fe-(1-3)Cr-C Mo (2.7-3.8), Cr (1.4-1.5) 1.2Cr-0.25Mo steel Ni (1.2-1.4), Cr (1.3-1.5), Mo (2.6-3.8) 2.8Ni-0.8Cr-0.5Mo steel Mn (1.3-1.8) 1.5Mn steel Ni (1.06-1.07), Cr (1.3) 18Cr-8.6Ni stainless steel Ni (1.1), Cr (1.1-1.3) 25Cr-19Ni stainless steel Si (1.8-3.1), Mn (1.3-1.8) 19Cr-15Ni stainless steel P (36 for cooling rate T • = 0.083 K/s) P (30 for cooling rate T • = 0.167 K/s) P (15 for cooling rate T • = 0.833 K/s) 22Cr-20Ni stainless steel Al (1.9-2.0), Ti (2.1-2.2) Ni-5Al-13Ti V (1.3) Ti-(2-10)V Sn (1.6-3.7, decreases with growth rate) Cu-8Sn Cu (1-2 vol%) Al-2Cu Cu (3.9 area% for equiaxed structure; 1.5-3 area% for columnar structure) Al-4.5Cu Cu (2.8 vol%) Al-6.5Cu Cu (4.1 vol%) Al-6.5Cu-0.26V Cu (4 vol%) Al-6.5Cu-0.1Ti Mg (4-7 area% for equiaxed structure; 1-4 area% for columnar structure) Al-10.4Mg P (25-26.5 mass%) Fe-4P Fig. 3 Isoconcentration profile in an Fe-25Cr-20Ni columnar dendrite. Equation 3 is often used in the case of the lower back diffusion parameter B (for example, for aluminum alloy castings), and Eq 1 and 4a are used in the case of higher B (for example, for steel castings). However, it is not easy to estimate the real microsegregation as listed in Table 1, because the real phenomena are very complicated and the solid composition after solidification cannot be calculated by Eq 4a if the finite solid diffusion is not negligible. The following points should be considered. Solidification Mode and Structure. Microsegregation varies considerably with the history of the growth of the solid. For example, microsegregation often increases with cooling rate in the case of equiaxed dendritic solidification, but it decreases in the case of unidirectional dendritic solidification. This is because, in the former case, the liquid composition is rather uniform in the interdendritic liquid, and Eq 4a is applicable. In the latter case, the solute buildup on the dendrite tip cannot be neglected, and equations such as Eq 5 or the Solari-Biloni equation (Ref 10), which considers the solute buildup ahead of the dendrite tip and dendrite curvature, should be used. Alternatively, a numerical calculation is necessary. Estimating the microsegregation in an equiaxed globular grain structure requires information on its formation mechanism and the history of the grain (that is, dendrite melt-off and settling in the liquid). Morphology of the Dendrite and Diffusion Path. In the case where solid-state diffusion is not negligible, the diffusion path or the morphology of the solid is very important. Although Eq 4a can be applied to the volume element in a primary or secondary dendrite array, the real diffusion occurs three dimensionally in both dendrites. Therefore, careful attention is required to determine the dendrite spacing λ. One method is to employ the mean value of the primary and secondary dendrite arm spacing, λ = (λ 1 + λ 2 )/2 (Ref 6). Equation 8 is also recommended (Ref 4): 22 11 [(1)/] [(1)/] 2 11 11 1 (1) 1 k k s L s os f C f Cf ψ ψ ψ ψ ψ − −  − =−  −  (Eq 8) where the subscripts 1 and 2 are used for the state f S ≤ f S1 , and f S > f S1 , respectively. Thus, if the diffusion path changes from the primary dendrite to the secondary dendrite at the fraction solid f S1 , then λ 1 and λ 2 , are used for ψ 1 and ψ 2 , respectively. Phase Transformation. If a phase transformation occurs during solidification, the microsegregation can change considerably because the equilibrium partition coefficient varies with phase. For example, the k of phosphorus in steel castings is 0.13 in ferrite and 0.06 in austenite. Therefore, as shown in Fig. 4, in steel castings the peritectic reaction, which is affected by carbon composition, greatly affects microsegregation (Ref 11). If it is assumed that the phase change occurs at f S = f S1 , then Eq 8 is also applicable, but it may result in a large error. A more accurate estimation of microsegregation requires numerical calculations that take into consideration the amount of change in each phase (Ref 11, 12). Fig. 4 Effect of carbon concentration and cooling rate on phosphorus concentration in an Fe-C- 0.016P dendrite upon cooling to 1537 K. Effect of Third Solute Element. Figure 5 shows that the partition coefficient is affected by the third solute element (Ref 13). In aluminum alloys, chromium decreases the partition coefficient of magnesium. Further, it should be noted that the dendrite morphology varies with solute elements resulting in a different diffusion effect. Fig. 5 Variation of equilibrium partition coefficient with third solute elements in an iron-carbon alloy. Dendrite Coarsening. Because coarsening or remelting of the dendrites occurs during solidification, the dendrite spacing is not constant, and the resolved solid dilutes the liquid composition. Although a numerical analysis has been performed, such effects have not been made clear (Ref 14). Movement of the Liquid Phase. In many cases, the interdendritic liquid does not remain stationary but moves by solidification contraction or by thermal and solutal convections, resulting in varying degrees of microsegregation. Temperature and Concentration Dependency of Diffusion Coefficient. Physical properties such as D S and D L are temperature and concentration dependent. Because the diffusion coefficient may vary by an order of magnitude in the case of a large solidification interval, both values must be closely monitored. This can be determined by numerical calculation. Undercooling. In actual use, undercooling at the dendrite tip does exist. However, it is not a factor that affects microsegregation in typical solidification processes, with the exception of welding and unidirectional solidification (Ref 13). Other Effects. When a very high temperature gradient exists (for example, over 40 K/mm), the Soret effect, which considers solute transport to be a function of a temperature gradient, becomes a factor (Ref 15). References cited in this section 4. I. Ohnaka, Trans. ISIJ, Vol 26, 1986, p 1045 6. T.W. Clyne and W. Kurz, Trans. AIME, Vol 12A, 1981, p 965 9. M. Sugiyama, T. Umeda, and J. Matsuyama, Tetsu-to-Hagané (J. Iron Steel Inst. Jpn.), Vol 63, 1977, p 441 10. M. Solari and M. Biloni, J. Cryst. Growth, Vol 49, 1980, p 451 11. Y. Ueshima, S. Mizoguchi, T. Matsumiya, and H. Kajioka, Metall. Trans. B, Vol 17B, 1986, p 845 12. H. Fredriksson, Solidification and Casting of Metals, The Metals Society, 1979, p 131 13. Z. Morita and T. Tanaka, Trans. ISIJ, Vol 23, 1983, p 824; Vol 24, 1984, p 206; and private communication 14. D.H. Kirkwood, Mater. Sci. Eng., Vol 65, 1984, p 101 15. J.D. Verhoeven, J.C. Warner, and E.D. Gibson, Metall. Trans., Vol 3, 1972, p 1437 Microsegregation in Rapid Solidification Processing In rapid solidification processing, the solid growth rate can be very high, resulting in a completely different solute distribution. If the atomic motions responsible for interface advancement are much more rapid than those necessary for the solute element to escape at the interface, microsegregation-free or diffusion-free solidification can occur (Ref 16). The nonequilibrium partition coefficient k N is considered to increase monotonically with velocity (Ref 17): * * [(.)/] 1[(.)/] i N i kRD K RD +Λ = +Λ (Eq 9) where D* i is the interface interdiffusivity (Ref 17) and Λ is the interatomic spacing of the solid. However, the solute trapping is also dependent on solute concentration, and the complete equation has yet to be formulated. References cited in this section 16. J.C. Baker and J.W. Chan, Solidification, American Society for Metals, 1970, p 23 17. M.J. Aziz, J. Appl. Phys., Vol 53, 1982, p 1158; Appl. Phys. Lett., Vol 43, 1983, p 552 Macrosegregation Macrosegregation is caused by the movement of liquid or solid, the chemical composition of which is different from the mean composition. The driving forces of the movement are: • Solidification contraction • Effect of gravity on density differences caused by phase or compositional variations • External centrifugal or electromagnetic forces • Formation of gas bubbles • Deformation of solid phase due to thermal stress and static pressure • Capillary force Macrosegregation is evaluated by: • Amount of segregation (∆C): ∆C = C S - C o • Segregation ratio or index: C max /C min or (C max - C min )/C o • Segregation degree (in percent): 100 C S /C o ) where C o is the initial alloy composition, C S is the mean solid composition at the location measured, and C max and C min are the maximum and minimum compositions, respectively. For example, the following carbon segregation index has been empirically obtained for steel ingots (except hot top) (Ref 18): (C max - C min )/(C o D) (%) = 2.81 + 4.31 H/D + 28.9 (%Si) + 805.8 (%S) + 235.2 (%P) - 9.2 (%Mo) - 38.2 (%V) (Eq 10) where D and H are ingot diameter and height in meters, respectively. Macrosegregation is especially important in large castings and ingots, and it is also a factor in some aluminum or copper alloy castings of small and medium size. Various types of macrosegregation and their formation mechanisms are described below, mainly for the case of k < 1. In the case of k > 1, similar but converse results are obtained. Plane Front Solidification. When plane front solidification occurs, as in single-crystal growth, the formation mechanism of macrosegregation is rather simple, and Eq 6 can be applied. A schematic of the typical solute distribution is shown in Fig. 6. The solid composition of the initially solidified portion is low and has an approximate value of kC o , which gradually increases with time because of diffusion as the solute is pushed ahead, resulting in a higher concentration at the finally solidified portion or the ingot center. This segregation is termed normal segregation. As seen from Eq 5, the degree of normal segregation increases with decreasing growth rate (R) or the solute boundary layer thickness (δ c ), which decreases with increasing intensity of the liquid flow. Fig. 6 Typical solute distribution in plane front solidification where R is the growth rate. Further, changes in the growth rate during solidification results in a segregation as shown in Fig. 6. If the growth rate increases suddenly from the steady state to a higher rate, then a larger effective partition coefficient is realized, resulting in a concentration higher than the mean composition, which is termed positive segregation. (The normal segregation is a type of positive segregation.) Conversely, a sudden decrease in the growth rate R results in a solute-poor band or negative segregation. If R or δ c varies periodically, then periodical composition change, which is termed banding or solidification contour, occurs. In either case, it is essential to consider the fluid flow and the solute boundary layer δ c , which typically ranges from 0.1 to 1 mm (0.004 to 0.04 in.) for the single-crystal growth of metals. Gravity segregation is caused by the settling or floating up of solid and liquid phases having a chemical composition different from the mean value. For example, the initially solidified phase or melted-off dendrites settle in the bottom of the casting because they are of higher density than the liquid. This phenomenon can be the source of negative cone, which often occurs in steel ingots, as shown in Fig. 7. If lighter solids such as nonmetallic inclusions and kish or spheroidal graphites are formed, they can float up to the upper part of the casting. Fig. 7 Typical macrosegregation observed in steel ingots. A-segregation and V- segregation are discussed later in this article. In steel castings, the interdendritic liquid is often lighter than the bulk liquid and floats up, resulting in positive segregation in the upper part of the casting. Although various types of macrosegregation are caused by the gravity effect, the compositional change between the upper and lower parts of a casting due to the simple gravity effect is called gravity segregation. In centrifugal casting, centrifugal force simulates gravity and can cause compositional changes between the internal and external parts of the casting. Liquid Flow Induced Segregations in the Mushy Region. If only the liquid flows * in the mushy region, where a concentration gradient exists, macrosegregation occurs. Equation 12 can be derived by assuming the local equilibrium condition and the constant liquidus slope and by neglecting the dendrite curvature effect (Ref 20): 11 (1)1. 1 n L s L u C f fA tkUCt β ∂ ∂−   =−+−   ∂−∂   (Eq 11) where sL s ρρ β ρ − = and ρ S and ρ L represent the density of the solid and the liquid, respectively; u n is the flow velocity normal to the isotherms; and U is the velocity of isotherms. In this equation, A = 0 corresponds to zero diffusion in a solid and: (1)(1) s s kf A f β = −− (Eq 12) corresponds to complete diffusion in a solid. * If it is assumed that there is no shrinkage (β = 0) and no fluid flow (u n = 0), Eq 12 is integrated to give either the equilibrium or Scheil equations (Eq 1 and 3) depending on the choice of A. For example, in the case of no diffusion in the solid (A = 0), Eq 12 is integrated to: [(1)/] (1) k L s o C f C ξ − =− (Eq 13) where ξ (1 - β) (1 - u n /U) and k are assumed to be constant. The following results can be seen from Eq 13: • In the case of ξ= 1 or u n /U = -β/(1 - β), Eq 13 is identical to Eq 3 and the average composition of the solid is C o , which means that no macrosegregation occurs • In the case of ξ> 1 or u n /U < -β/(1 - β), C L becomes lower than the value calculated by Eq 3 , which indicates that negative segregation may occur • In the case of 0 < ξ< 1 or 1 > u n /U > -β/(1 - β ), positive macrosegregation occurs. For example, because at the mold wall u n = 0, it results in positive segregation described below The segregation shown in Fig. 8, which is called inverse segregation, is where solute concentration is higher in the earlier freezing portion (Ref 21). This is caused by the solute-enriched interdendritic flow due to solidification contraction, which is the main driving force, plus the liquid density increase during cooling. ** If a gap is formed between the mold and the solidifying casting surface, then the interdendritic liquid is often pushed into the gap by static pressure or by the expansion due to the formation of gas bubbles or graphite in the liquid. This results in severe surface segregation or in exudation, a condition in which a solute-rich liquid covers the casting surface and forms solute-rich beads. Fig. 8 Inverse segregation in an Al-4.1Cu ingot with unidirectional solidification. Changes in liquid velocity may cause segregation. For example, the change in the shape of the casting shown in Fig. 9 can change the velocity, resulting in a segregation, as shown in Fig. 10 (Ref 21). Fig. 9 Simulated fluid flow at 50 s after cooling and macrosegregation in an Fe-0.25C specimen. [...]... p 131 13 Z Morita and T Tanaka, Trans ISIJ, Vol 23, 19 83, p 824; Vol 24, 1984, p 206; and private communication 14 D.H Kirkwood, Mater Sci Eng., Vol 65, 1984, p 101 15 J.D Verhoeven, J.C Warner, and E.D Gibson, Metall Trans., Vol 3, 1972, p 1 437 16 J.C Baker and J.W Chan, Solidification, American Society for Metals, 1970, p 23 17 M.J Aziz, J Appl Phys., Vol 53, 1982, p 1158; Appl Phys Lett., Vol 43, ... Jackson, Interaction Between Particles and a Solid-Liquid Interface, J Appl Phys., Vol 35 (No 10), 1964, p 2986 2 P.K Rohatgi, R Asthana, and S Das, Solidification, Structures and Properties of Cast Metal-Ceramic Particle Composites, Int Met Rev., Vol 31 (No 3) , 1986, p 115 3 K.C Russell, J.A Cornie, and S.Y Oh, Particulate Wetting and Particle: Solid Interface Phenomena in Casting Metal Matrix Composites,... as having r < λ/l, while large particles have r > λ/l Equations 16 and 17 were derived for small and large particles, respectively: 1/ 3 0.14 B3  σ SL  Rcr =   η r  B3 r  (Eq 16) 1/ 4 0.15 B3  ∆SG  Rcr =   η r  B3VB  (Eq 17) All the above approaches assume a planar liquid/solid interface, only one particle at the interface, and thermal conductivity of the particle KP equal to that of the... Jackson, Interaction Between Particles and a Solid-Liquid Interface, J Appl Phys., Vol 35 (No 10), 1964, p 2986 2 P.K Rohatgi, R Asthana, and S Das, Solidification, Structures and Properties of Cast Metal-Ceramic Particle Composites, Int Met Rev., Vol 31 (No 3) , 1986, p 115 3 K.C Russell, J.A Cornie, and S.Y Oh, Particulate Wetting and Particle: Solid Interface Phenomena in Casting Metal Matrix Composites,... between the particle and the solid, the particle will not be engulfed In other words, for a particle to be pushed, mass transport in the liquid layer is required between the particle and the solid The concept of a critical interface rate Rcr, below which particles are pushed and above which particles are engulfed, was postulated For a particle to be pushed, a repulsive force must exist between the particle... force is again described by Eq 5 (Ref 8) They derived Eq 11, 12, 13, and 14 For small, smooth particles (r < rb): η ² R²r 3 = N 4ψ (α ) kT σ SL ao 9π (Eq 11) For bigger particles with bumps (r ≥ rb): η ² R ² rb3 + 2ψ (α ) 4ψ (α ) g ∆ρ aoη Rr ³rb = kT σ SL ao 9α 9π (Eq 12) For very large particles (r ? rb): η Rr 3 = 2α NkT σ SL π rb g ∆ρ (Eq 13) In the absence of bumps, Eq 14 can be written: η Rr 4 = 2α... in calculations The second group consists of: • • • • Particle shape Particle aggregation, that is, gas, liquid, or solid Convection level in the liquid Density of liquid (ρL) and particle (ρP) Equations 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, and 19 have been derived on the assumption of spherical particles As expected, particles of nonspherical shape will tend to position themselves... for thermodynamic calculations of particle entrapment Source: Ref 4 Similarly, when moving from 3 to 4, the change in free energy is: ∆F34 = 1 1 (σPS - σPL) + σSL 2 4 (Eq 2) where σ is the interface energy between particle (P), liquid (L), and solid (S), in various combinations The net change in free energy during engulfment is: ∆Fnet = ∆F 23 + ∆F34 = σPS - σPL (Eq 3) If ∆Fnet < 0, engulfment is to... carbide alloys, and particulate metal-matrix composites are examples in which solid particles interact with the solid/liquid interface during solidification (Ref 2, 3) Basically, when a moving solidification front intercepts an insoluble particle, it can either push it or engulf it Engulfment occurs through the growth of the solid over the particle, followed by enclosure of the particle in the solid... materials (Fig 3a) If the interface is cellular, particles can be pushed at the tips of the cells while they are entrapped at cell boundaries, as observed for particles in the ice-water (Ref 1), Pb-1Sn-iron (Ref 15), and waternylon systems (Ref 15), resulting in an aligned particulate composite (Fig 3b) When the interface breaks down into dendrites, a more complex situation occurs, with small particles . Metal- Ceramic Particle Composites, Int. Met. Rev., Vol 31 (No. 3) , 1986, p 115 3. K.C. Russell, J.A. Cornie, and S.Y. Oh, Particulate Wetting and Particle: Solid Interface Phenomena in Casting. steel Ni (1.1), Cr (1.1-1 .3) 25Cr-19Ni stainless steel Si (1.8 -3. 1), Mn (1 .3- 1.8) 19Cr-15Ni stainless steel P (36 for cooling rate T • = 0.0 83 K/s) P (30 for cooling rate T • = 0.167. T • = 0. 833 K/s) 22Cr-20Ni stainless steel Al (1.9-2.0), Ti (2.1-2.2) Ni-5Al-13Ti V (1 .3) Ti-(2-10)V Sn (1.6 -3. 7, decreases with growth rate) Cu-8Sn Cu (1-2 vol%) Al-2Cu Cu (3. 9 area%