Alexandrov, D. V.

Makoveeva, E. V.

Malygin, A. P.

IOP Conference Series Materials Science and Engineering

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin

1Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012034 doi:10.1088/1757-899X/192/1/012034Mathematical modeling of the stationary nucleationand crystallization process in supersaturated systemswith a crystallizerE V Makoveeva1, A P Malygin1 and D V Alexandrov11 Ural Federal University, Department of Theoretical and Mathematical Physics, Laboratoryof Multi-Scale Mathematical Modeling, 620000 Ekaterinburg, Russian FederationE-mail: eugenya1m@gmail.comAbstract. Motivated by important industrial applications we consider the growth prosess ofsolid particles in a supersaturated (supercooled) system with allowance for a crystallizer. Theparticle-radius distribution function satisfies the second order kinetic equation supplimented bydifferent boundary conditions. An exact steady-state analytical solution is found. We show thattwo different types of analytical solutions for the kinetic equation exist.1. IntroductionThe process of growth of a new phase in metastable systems can be divided into three stages:nucleation, growth of critical nuclei and relaxation of a new phase to equilibrium. The divisionis conditional, because all these processes occur simultaneously. Modeling the processes offormation of a new phase is important both for fundamental research in the field of phasetransitions and for industrial applications. The particle size distribution can have a decisiveinfluence on the properties of the systems in the production of powder materials from melts,precipitation from solutions, synthesis of colloidal solutions and so on.In this study, the process of formation and growth of particles of a new phase under stationaryconditions is considered. Such conditions can be ensured when the particles of the solid phaseare continuously removed from the liquid and the supersaturation (supercooling) is artificiallymaintained. The removal of particles can be realized both as a result of processes of precipitationof large particles due to sedimentation, fragmentation and sedimentation of large volumes of thenew phase, and also due to artificial deposition, for example, during centrifugation.The problem of stationary distribution was previously considered by Buevichi and Mansurov[1]. However, they considered the determenistic growth of solid crystals without theirfluctuations. The present paper takes into account random processes in the Fokker-Planckequation for the particle-radius distribution function.2. ModelIt is assumed that the physical properties of both the solution and the crystals are independent ofsupersaturation (hereinafter, supersaturation will also be understood as supercooling) and time.Supersaturation is homogeneous throughout the reservoir volume. The nucleation kinetics isstationary. This is permissible as the time of nucleation process, as a rule, is several orders of21234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012034 doi:10.1088/1757-899X/192/1/012034magnitude smaller than the time of continuous crystallization for any real supersaturation. Thesize of spherical crystals is determined by using one parameter, their radius.The crystal radius is a random variable satisfying the stochastic equation [2]dr = g(t, r)dt+√2DW, (1)where t is the growth time, g(t, r) is the growth rate of a particle that has no oscillations, D is afunction that expresses the speed of free oscillations (the coefficient of mutual Brownian diffusionof a particle) and W is the Wiener process. The distribution function f of particles of radius rfor the stochastic equation (1) considering the crystallizer (crystal derivation) is described by akinetic equation of the Fokker-Planck type [3].∂f∂t+∂∂r(gf) + h(r)f =∂∂r(D∂f∂r), r > r∗. (2)The function h(r) describes the rate of crystal extraction, r∗ is a critical radius. The exactdefinition of D is a difficult problem in statistical physics. However, in the first approximationone can assume that D is proportional to the nucleation rate of growth [3, 4]D = d1g(t, r), (3)where d1 is a coefficient.The rate of particle growth g can be found from a problem of the Stefan type in a metastablesystem [4, 5]. The result can be expressed in the formg(t, r) =β∗ 4 θ1 + β∗qr, q =LVλl(4)if the system is a supercooled melt and in the formg(t, r) =β∗ 4 C1 + β∗qr, q =CpDl(5)if the system is a supersaturated solution. Here, β∗ is the kinetic coefficient, λl and Dlare the temperature conductivity and diffusion coefficients, 4C is the supersaturation, 4θis the supercooling. The heat (mass) balance equation for the dimensionless supercoolingW = 4θ/4 θ0 (supersaturation W = 4C/4 C0) can be written as [4].w(τ) = 1− b∫ ∞r∗r3f(τ, t)dr, τ > 0, (6)where b = 4piLV /(3ρmCm 4 θ0) if the system under consideration is a supercooled melt, andb = 4piCp/(34 C0) if the system is represented by a supersaturated solution. Here LV is thelatent heat parameter, Cp is the concentration at saturation, ρm and Cm are the density andspecific heat of the mixture and subscript 0 designates the initial state.For a complete investigation of the problem, it is necessary to set the initial and boundaryconditions. As the initial condition for the distribution function, it is usually assumed that thereare not nuclei in the system:f = 0, τ = 0. (7)The initial condition for w becomesw = 1, τ = 0. (8)The first boundary condition for the density distribution function takes the form31234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012034 doi:10.1088/1757-899X/192/1/012034gf −D∂f∂r= I(w), r = r∗, τ > 0. (9)The empirical expressions for the nucleation rate which are frequently used in analyzing manyindustrial processes can be expressed asI = I∗(4θ)p = (I∗(w4 θ0)p (10)for supercooled melts andI = I∗(4C)p = (I∗(w4 C0)p (11)for supersaturated solutions. Expressions (10) and (11) are frequently called the Meirs nucleationkinetics. It follows from the normalization condition that the distribution function vanishes forsufficiently large r.Generally speaking, two boundary conditions are possible. The first of them reflecting thedistribution function decaying isf = 0, r →∞. (12)The second one describes the case when the density distribution function vanishes for a finite rf = 0, r = r0. (13)3. Steady-state regimeWe are dealing with two processes: the formation and growth of crystals and their removal.These processes can balance each other. Thus, if the supersaturation is artificially maintained,it is possible to achieve the existence of a solution in the form of a stationary (time-independent)distribution-density function. The equation becomes an ordinary differential equation, and theinitial conditions are no longer required. I this paper, we assume that the rate of the crystalremoval is independent of their radii. The experiment shows that the crystal growth rate g isfrequently constant. In this case, the kinetic equation takes the formDddr(dfdr)− g ddr(f)− hf = 0, (14)ddr(dfdr)−A ddr(f)−Bf = 0, A = gD, B =hD. (15)The solution of this equation has the formf = C1eλ1r + C2eλ2r, (16)λ1 =A+√(A2 + 4B)2, λ2 =A−√(A2 + 4B)2, (17)where C1 and C2 are determined from the boundary conditions, consequently two cases arepossible.In the first case, (12), we haveC1 = 0, (18)C2 =Ig(1− d1λ2)eλ2r∗ . (19)41234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012034 doi:10.1088/1757-899X/192/1/012034In the second case, (13), we getC1 = − Ieλ2r0g((d1λ1 − 1)eλ2r0eλ1r∗ − (d1λ2 − 1)eλ1r0eλ2r∗) , (20)C2 =Ieλ1r0g((d1λ1 − 1)eλ2r0eλ1r∗ − (d1λ2 − 1)eλ1r0eλ2r∗) . (21)The analytical solutions (16)-(21) can be rewritten in the dimensionless form asF (x) =fC2= Ceµ1x + eµ2x, (22)C =C1C2, x =rd1, Λ =4BA2, (23)µ1 =12(1 +√1 +4BA2), µ2 =12(1−√1 +4BA2). (24)Here the reduced constant C is introduced asC = 0 (25)in the case of boundary conditions (18), (19) andC = −e(µ2−µ1)x0 (26)in the case of boundary conditions (20), (21).4. ConclusionsFigures 1 and 2 show graphs of the distribution function for different boundary conditions. Ofspecial interest is the case when the distribution function vanishes for finite r = x0d1. As wecan see, the function F in Figures 1 and 2 does not have extremal points and is a monotonicallydecreasing function. Thus, we come to the conclusion that under stationary conditions withincreasing radius of the nuclei the probability of finding such nuclei in the crystallizer decreases.The difference between two boundary conditions (12) and (13) shown in Figures 1 and 2 consistsin the fact that F = 0 at x = x0 in Figure 2 while F → 0 at x→∞ in Figure 1. This is connectedwith different boundary conditions (12) and (13) initially used in our model. Generally speaking,both of them are possible if the system approaches to its steady-state. This, in turn, depends,first of all, on system equilibria and how it tends to them. It depends on the level of systemsupercooling (supersaturation) as well. In addition, Figure 2 demonstrates the presence ofinflection points appearing as a result of the boundary condition (13).Note that increasing Λ (withdrawal rate h) decreases the distribution function and the numberof crystals as is seen from comparison of the left and right panels in Figures 1 and 2.The distribution funstion behavior illustrated in Figures 1 and 2 is a direct consequence ofthe second spatial order of the kinetic equation (2) and corresponding boundary conditions.Indeed, if we are dealing with the first order kinetic equation the distribution function breaksdown at a finite crystal radius [6, 7]. An important point is that the distribution function (22)is always decreasing because its second derivative is negative at any x.51234567890STPM2017  IOP PublishingIOP Conf. Series: Materials Science and Engineering 192 (2017) 012034 doi:10.1088/1757-899X/192/1/012034a) b)Figure 1. Dimensionless distribution function versus dimensionless cristal radius at a) Λ = 0.5and b) Λ = 0.3 plotted for the case of boundary condition (12).a) b)Figure 2. Dimensionless distribution function versus dimensionless cristal radius at a) Λ = 0.5and b) Λ = 0.3 plotted for the case of boundary condition (13), x0 = 30.5. References[1] Buyevich Yu A and Mansurov V V 1990 Kinetics of the intermediate stage of phase transition in batchcrystallization J. Cryst. Growth 104 861-867[2] Gardiner C W 1983 Handbook on stochastic methods: for physics, chemistry and the natural sciences (Berlin,Germany: Springer)[3] Lifshitz E M and Pitaevskii L P 1981 Physical kinetics(New York, NY: Pergamon Press)[4] Buyevich Yu A, Alexandrov D V and Mansurov V V 2001 Macrokinetics of crystallization (NewYork, NY:Begell House)[5] Kirkaldy J S and Young D J 1987 Diffusion in the condensed state (London, UK: Institute of Metals)[6] Alexandrov D V and Malygin A P 2013 Transient nucleation kinetics of crystal growth at the intermediatestade of bulk phase transitions J. Phys. A: Math. Theor. 46 455101[7] Alexandrov D V 2014 Nucleation and crystal growth kinetics during solidification: Tye role of crystallitewithdrawal rate external heat and mass sources Chem. Eng. Sci. 117 156-160AcknowledgmentsThis work was supported by the Russian Foundation for Basic Research (grant no. 16-08-00932)and the Ministry of Education and Science of the Russian Federation (project no. 1.9527.2017).

Mathematical modeling of the stationary nucleation and crystallization process in super saturated systems with a crystallizer

https://elar.urfu.ru/bitstream/10995/73799/1/10.1088-1757-899X-192-1-012034.pdf

Mathematical modeling of the stationary nucleation and crystallization process in super saturated systems with a crystallizer

Abstract

Similar works

Full text

Available Versions

Institutional repository of Ural Federal University named after the first President of Russia B.N.Yeltsin