Influence of external temperature field to period of eutectic pattern
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Influence of external temperature field to period of eutectic pattern.
Gus’kov A.P. Institute of Solid State Physics Academy of Sciences of Russia, Moscow District, Chernogolovka, 142432, Russia INTRODUCTION We used a mathematical model of crystal growth of [1] for the description of an eutectic pattern [2]. But the stationary problem gives physically unrealizable solutions (fig. 1) for values of system parameters corresponding to real experiments. Now known models [1,3-7] of crystal growth, used for a research of the interface stability of directed crystallization, not take into account of an external temperature field too. As examples of application of these models the regimes of crystal growth are usually used. The purpose of this work - to construct a model of a binary melt crystallization taking into account an external temperature field. Within the framework of this model we deduce analytical dependence of period of the eutectic structure on parameters of the system. We demonstrate, that for real parameters of the system, the parameters of the external temperature field weakly influence period of the eutectic pattern. This outcome is observed in experiments. We also explain the reason of joint emerging eutectic and dendrite of a structure for want of growth of eutectic crystals. We also explain the reason of joint both eutectic growth and dendrite growth under eutectic growth. THEORY Let T(y,z,τ) be the temperature normalized to the phase transition temperature Te0 and at initial impurity concentration C0; C(y,z,τ) is the impurity concentration normalized to the initial one; y,z,τ are the dimensionless coordinates and the time: y=αyr, z=αzr, τ=α2χ0τr; D is the dimensionless factor of diffusion in a melt, D = Dr/χ0; χ = χr/χ0 is the factor of thermal diffusivity, ε is the heat of phase transition normalized to the specific heat capacity and the temperature of phase transition. yr,zr,τr,Dr,χr,εr are dimensional quantities, χ0=10-5m2s-1, α=10 2 m-1. We take into account the heat conduction in solid and liquid phases and diffusion of impurity in the liquid phase. To reduce the calculations in the equations we do not write down the coordinate x. The values relating to the solid phase are designated by stroke. We describe the external temperature field Text(z) by function of internal sources as exponential function
φ 0 T ′ (∞) + T (∞) + Tx − T ′ (∞) ⋅ exp ext ext ext T (∞) + Tx − T ′ (∞) ext ext T = ext φ0 z ( ) exp T ∞ + T ⋅ x T ext x
(
)
M7.7.1
(
)
z . (1)
The parameters of function (1) satisfy to following conditions: At point at infinity the temperature values are T ′ (−∞) and T (∞) ; the sewing of function (1) is made at the point z ext
ext
= 0, at this point T ′ (0) = T (0) and temperature gradient are equal to specific values φ0. The ext
ext
constant Tx is searched from boundary conditions. We do not give here setting and solution of the stationary problem. The solution of one is similar to the solution of a problem of the heat conduction with external fie
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