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 liquidussolidus 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