E3S Web of Conferences 124, 05013 (2019) https://doi.org/10.1051/e3sconf/201912405013 SES-2019 The problem of creating an automated system to control growth of single crystal sapphires from melt as a problem of control and monitoring of a complex nonlinear and dynamic system V A Petrosyan1,*, A V Belousov2, A G Grebenik2 and Yu A Koshlich2 1LLC "Techsapphire", Belgorod, Russia 2BSTU named after V.G. Shukhov, Belgorod, Russia Abstract. The paper mentions some problems of automated control system development for growth of large (150 kg and above) single crystal sapphires. We obtain an analytical equation for the temperature distribution and thermal stresses along the crystal axis during the growth phase. An analysis was carried out and numerical estimates were obtained for the axial distribution of components of thermoelastic stresses depending on the physical, optical, and geometric parameters of the crystal. It is shown that the cause of thermal stresses and blocks during crystal growth is the nonlinear temperature dependence of thermal conductivity and thermal expansion coefficients. 1 Introduction a forced convention and with different density of melt saturation with gas inclusions in these zones. As a result, Based on the general concepts of the theory of normal in the places of melting, more favorable conditions arise directional crystallization from the melt, it is known that for the capture of micron-sized bubbles and their further the crystallization rate 푣푐푟 is proportional to the growth into a single crystal. supercooling of the melt ∆푇 or the temperature gradient It follows from the above that an important at the phase interface. That is, 푣푐푟 ~ ∆푇 or 푣푐푟 ~ 푔푟푎푑푇. conclusion is that the magnitude of the supercooling at In the harmonic approximation for the growth rate, the crystallization front determines not only the the expression: conditions for the growth of a single crystal, but also its perfection. vcr = v01 + v sin( t + ) (1) We need to note one more important fact for the controlling systems of the growth rate and maintaining stability at the crystallization front. The fact is that for where 푣0 is an average crystal growth rate, 푣1 is an amplitude of change in growth rate due to temperature sapphire the melting point and the crystallization fluctuations in the melt near the phase transition temperature do not coincide. The melting point is well boundary, 휔 is a temperature oscillation frequency, 푡 is reproduced at different heating rates, and the time, 휑 is some phase. Based on equation (1), we can crystallization temperature depends on the cooling rate determine the magnitude of the dynamic supercooling or of the melt [1]. Therefore, in formula (2), ∆푇0 should be overheating of the melt at the crystallization front: understood as the average value of "physical supercooling". T = T + Bsin( t + ) (2) It follows from the above that fluctuations in the 0 magnitude of dynamic supercooling affect the change in the rate of crystallization and lead to stricter where ∆푇 is a fixed value of "physical supercooling" 0 requirements for the system for stabilizing gradients at (the difference between the melting point and the the crystallization front of 퐴푙 푂 , than, for example, for crystallization temperature), 퐵 is a supercooling change 2 3 silicon. amplitude. We will now define two cases of dynamic It can be seen from equation (2) that, depending on supercooling. The first is small fluctuations in the amplitude and phase of the change in the supercooling, within the zone of "physical supercooling, periodic localized melting of the crystal at supercooling", which are stochastic in nature and depend the crystallization front can occur with re-crystallization on many factors. These processes are described by of the melted layer. methods of nonlinear dynamics, in particular, the theory It should be noted that temperature fluctuation in of dynamic or deterministic chaos. individual local areas on the crystallization front may The second case is high-amplitude supercooling occur due to mixing of hotter and less hot melt layers at oscillations that go beyond the limits of "physical * Corresponding author: [email protected] © The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 124, 05013 (2019) https://doi.org/10.1051/e3sconf/201912405013 SES-2019 supercooling". These fluctuations can occur during This model is known from studies of plasma physics growth control using an automated system that operates and the theory of controlled thermonuclear fusion and is on a feedback signal from a weight sensor and regulates called the model of complex thermal structures. The the power at the heater. complex structures in this model are unstable and can Since dynamic weighing is an indirect method of controlling crystal growth, it becomes necessary to use exist only with the correct specification of the initial the so-called observation equation, which relates the data. In [5], we have investigated a similar model when force acting on the weight sensor with the required calculating the temperature distribution along the crystal crystal mass and the rate of its crystallization. axis during growth and optimized some process parameters based on identification with experimental 2 The problem of controlling the crystal data. growth using the feedback signal from In this case, we were guided by the fact that the the weight sensor parameters 푄(푇) and 퐾(푇) in the equation (3) have a different physical interpretation depending on the optical When crystals are grown by the modified Kyropoulos thickness of the layer 휏 = 훽0퐿 (where 훽0 is the method (GOI method), a significant part of the crystal is attenuation coefficient, 퐿 is the thickness of the layer), in the melt and a pushing force acts on the crystal. The which spreads the flow of heat. magnitude of this force depends on the fluctuation of the melt density depending on the temperature, variations in 3 The role of sapphire crystal the crystallization rate, changes in the angle of growth. It is also necessary to add the dependence of the weight transparency and the effect of nonlinear sensor readings on fluctuations in the temperature of its temperature parameters on the quality environment, on various kinds of vibrations, etc. Of all of crystals these parameters, we distinguish two main ones—the linear crystallization rate (the vertical component of the In the general case, the nonstationary problem of joint rate of front advancement into the melt during growth) radiation-conductive heat transfer in the direction of the and the shape of the crystallization front, depending on X axis in a layer of a semi-transparent medium in terms the angle of the growth cone. The weight sensor signal is of dimensionless parameters can be written as [6]: a function of these two parameters and the cause of the ( ,,, ) 22 ( ) 1 Q ( ) change in weight gain can be either a change in the = −r + H (, ) (4) linear crystallization rate or a change in the angle at the 2N * 2 apex of the growth cone. In view of this, the control object is not observable in a fairly wide "dead zone" [2- where 휃(휏, 휉) is a temperature, 푄푟(휃, 휉) is a radiation 4]. flux density, 퐻(휏, 휉) is a bulk power of internal sources, With this in mind, as well as the inertia of heat 푁∗ is a conductive-radiation parameter. transfer from the heater to the melt, there is a loss of In this equation, the parameter 푁∗ characterizes the controllability and the control system can enter the self- relative contribution of thermal conductivity compared oscillatory mode. During the growth process, turbulent to radiation. This parameter has large values when flows in the melt, vortex flows at the crystallization thermal conductivity prevails, and small ones when front, etc. can occur. radiation predominates. That is, when the medium is There are two main ways to stabilize a chaotic only scattering (푁∗ → ∞) and the radiation term in system: without feedback and with feedback. The first is equation (4) disappears, then it turns into the usual heat called chaos suppression, and the second is chaos conduction equation. control. The second method which includes active For a medium with a high optical thickness, the control will be described in future research. attenuation coefficient 훽0 is replaced by the absorption The task of searching for controlling external coefficient. That is, for a sapphire crystal, 퐾(푇) is the influences, which make it possible to minimize radiative thermal conductivity coefficient in the diffusion temperature fluctuations at the crystallization front, can approximation (Rosseland approximation [6]). The be formulated in a general form. Suppose that in the expression for this coefficient is [6-8]: system under consideration there are two competing processes: a nonlinear source 푄(푇), which reflects 16NT23 K = SL (5) feedback, and a dissipative source, nonlinearity of which SL 3 is determined by the coefficient 퐾(푇). If these functions have a power form, then the system can be described by where 푇 is a crystallization temperature, 휎 is a Stefan- the equation: 푆퐿 Boltzmann constant, 푁 is a refraction coefficient. TT If we calculate the conducted-radiation parameter ∗ 퐾퐹훽0 푊 (3) =+KTQT( ) ( ) 푁 = 2 3 for sapphire, then with 퐾퐹 = 3 , 훽0 = t x x 4푁 휏푇푆퐿 푚∙퐾 1 0.35 , 푁 = 1.79, 푇 = 2308퐾° and 휎 = 5.67 ∙ 푐푚 푆퐿 푊 10−8 will get 푁∗ = 0.0118. 푚2퐾4 2 E3S Web of Conferences 124, 05013 (2019) https://doi.org/10.1051/e3sconf/201912405013 SES-2019 Thus, in solving the problem of temperature where 훼푇 is a thermal expansion coefficient, 휇 is a distribution along the axis of a growing crystal, the Poisson's ratio, 퐸 is Young's modulus.