上海交通大学：《材料加工原理 Principles of Materials Processing》教学资源（课件讲稿）凝固和铸造 Solidification and casting_Solidification of single phase alloys（3）

先进材料疑固实验室 Laboratory of Advanced Materials Solidification 1896 1920 1987 2006 Solidification of single phase alloys(3) Dr.Mingxu Xia anced Mat 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

1896 1920 1987 2006 Solidification of single phase alloys (3) Dr. Mingxu Xia

先进材料疑固实验室 References Laboratory of Advanced Materials Solidification M.C.Flemings,凝固过程 W.Kuz,凝固原理 介万奇，凝固技术 JA Dantzig and M Rappaz,Solidification ory of anced Materials Solid 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

References M.C.Flemings, 凝固过程 W.Kurz,凝固原理 介万奇，凝固技术 JA Dantzig and M Rappaz, Solidification

先进材料疑固实验室 OUTLINE Laboratory of Advanced Materials Solidification Solute Redistribution Solute distribution coefficient Solute redistribution in equilibrium state Solute redistribution in non-equilibrium state Solute distribution in front of S/L interface Application:Zone melting Morphology Instability of a S/L interface Constitutional undercooling Morphological instability of a S/L interface Solidification microstructure:Cells and dendrites ·Constrained growth Preferred growth direction ·Dendrite arm spacing Formation of equiaxed grains 。Cast microstructure 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

OUTLINE Solute Redistribution • Solute distribution coefficient • Solute redistribution in equilibrium state • Solute redistribution in non-equilibrium state • Solute distribution in front of S/L interface • Application: Zone melting Morphology Instability of a S/L interface • Constitutional undercooling • Morphological instability of a S/L interface • Solidification microstructure: Cells and dendrites • Constrained growth • Preferred growth direction • Dendrite arm spacing • Formation of equiaxed grains • Cast microstructure

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification Dendrite morphology in usual casting and ingot making processes remains largely unchanged over wide range of cooling rates.It simply becomes finer as heat is extracted at greater rate. PRIMARY PRIMARY ARMS ARMS (a) FIGURE 5-13 Schematic diagram of dendrite structure in Al-4.5%Cu alloy at (a)50 percent solid and (b)90 percent solid.(From Singh et al.) 熟 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing Dendrite morphology in usual casting and ingot making processes remains largely unchanged over wide range of cooling rates. It simply becomes finer as heat is extracted at greater rate

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification A convenient and widely used measure of the effects of solidification conditions on dendrite structure is dendrite arm spacing,i.e.,the spacing between primary,secondary,or higher- order branches. 生尘 (a) (b) FIGURE 5-14 Formation of new primary arms by branching from secondaries. 上浒充通大学 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing A convenient and widely used measure of the effects of solidification conditions on dendrite structure is dendrite arm spacing, i.e., the spacing between primary, secondary, or higher￾order branches

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification ⊕ The primary arm spacing is an important characteristic of columnar dendrites and has a marked effect on the mechanical properties.It is assumed that the cell or the dendrite envelope,representing the mean cross-section of the trunk and branches,can be described approximately by an ellipse.The radius of curvature of the ellipse R is given by: b2 R- a ATo b is the semi-axis,is proportional △T toλ1，the primary arm spacing, a G where the proportionality constant Te depends on the geometrical arrangement of the dendrites. Co b=0.58λ1 上游文通大学 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing The primary arm spacing is an important characteristic of columnar dendrites and has a marked effect on the mechanical properties. It is assumed that the cell or the dendrite envelope, representing the mean cross-section of the trunk and branches, can be described approximately by an ellipse. The radius of curvature of the ellipse R is given by: b is the semi-axis, is proportional to λ1, the primary arm spacing, where the proportionality constant depends on the geometrical arrangement of the dendrites

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification The semi-axis length,a,is given by the difference between the tip temperature and the root temperature,divided by the mean temperature gradient in the mushy zone: AT'T-Ts' a= G G ct a is the semi-axis length,AT is To given by the difference between △T the tip temperature T*and the root a s G temperature T's,divided by the Te mean temperature gradient in the mushy zone.T's is the solidus Co b=0.58λ1 temperature of the solid part at the root part. 上游文通大学 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing The semi-axis length, a, is given by the difference between the tip temperature and the root temperature, divided by the mean temperature gradient in the mushy zone: a is the semi-axis length, ΔT is given by the difference between the tip temperature T* and the root temperature T’s, divided by the mean temperature gradient in the mushy zone. T’s is the solidus temperature of the solid part at the root part

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification Therefore,it can be assumed that the tip radius and the length of the interdendritic liquid zone together determine the primary spacing(given a fixed temperature gradient),due to purely geometrical requirements: 1 AT'R 3AT'R 14= 0.58、 G G The radius of the dendrite tip can be calculated using the following equation: Dr R=2π VkATo W.Kurz,D Fisher,Fundamentals of solidification where D is the diffusion coefficient,I is the Gibbs-Thomson coefficient (S )6 is the surface energy,S,is the fusion entropy, V is the growth speed,k the partition coefficient,47 is the liquidus- solidus range at Co. 上泽充通大学 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing Therefore, it can be assumed that the tip radius and the length of the interdendritic liquid zone together determine the primary spacing (given a fixed temperature gradient), due to purely geometrical requirements: The radius of the dendrite tip can be calculated using the following equation: where D is the diffusion coefficient, Г is the Gibbs-Thomson coefficient (δ/ΔSf ), δ is the surface energy, ΔSf is the fusion entropy, V is the growth speed, k the partition coefficient, ΔT0 is the liquidus￾solidus range at C0. W. Kurz, D Fisher, Fundamentals of solidification

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification Due to the radius of the dendritic tips highly relies on the growth speed,the above equation can be written as: 4.3(△ToDT0.25 k0.25v0.25VG ⊕ The above equation shows temperature gradient G is more important than growth speed v. where D is the diffusion coefficient,I is the Gibbs-Thomson coefficient (/S),6 is the surface energy,S is the fusion entropy,I is the growth speed,k the partition coefficient,47 is the liquidus-solidus range at Co. 上游文通大学 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing Due to the radius of the dendritic tips highly relies on the growth speed, the above equation can be written as: The above equation shows temperature gradient G is more important than growth speed v. where D is the diffusion coefficient, Г is the Gibbs-Thomson coefficient (δ/ΔSf ), δ is the surface energy, ΔSf is the fusion entropy, V is the growth speed, k the partition coefficient, ΔT0 is the liquidus-solidus range at C0

先进材料疑固实验室 Dendrite arm spacing Laboratory of Advanced Materials Solidification 103 m 44 (ww)g 103 R 104 Al-2wt%Cu Ve G:10 K/mm 104 102 1 102 v(mm/s) anced Maw 上浒充通大 SHANGHAI JIAO TONG UNIVERSITY

Dendrite arm spacing