The Anomalously High Rate of Crystallization, Controlled by Crystal

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The anomalously high rate of crystallization, controlled by crystal forms under the conditions of a limited liquid volume S. Y. Misyura Cryst. Growth Des., Just Accepted Manuscript • DOI: 10.1021/acs.cgd.7b00980 • Publication Date (Web): 30 Jan 2018 Downloaded from http://pubs.acs.org on January 31, 2018

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Crystal Growth & Design

The anomalously high rate of crystallization, controlled by crystal forms under the conditions of a limited liquid volume Misyura S.Y.* 1. Kutateladze Institute of Thermophysics Siberian Branch, Russian Academy of Sciences, 1 Akad. Lavrentyev Ave., Novosibirsk, 630090 Russia, 2. National Research Tomsk Polytechnic University, 30 Lenin Ave., Tomsk 634050, Russia Abstract Non-isothermal evaporation and crystallization in a thin layer of an aqueous salt solution have been studied experimentally. The results of these studies are important for a wide range of modern technologies, associated with obtaining thin coatings and crystallization in thin layers and films. The crystallization patterns are shown to differ for different salts and change both over the crystalline hydrate surface and over time. The growth kinetics of crystal hydrates has been examined. A qualitative behavior of crystallization curves may both coincide and differ from the behavior of the curves, obtained using statistical approach. It is shown that this difference is determined by a changing ratio of rates of evaporation to crystallization. Changing evaporation velocity it is possible to change crystal forms and crystallization velocity. It is well known that, the crystallization rate increases as a square root of time. However, according to our results the crystallization rate may both rise and fall over time and have an extremum. For all studied crystalline hydrates of salts it was possible to find dendrites. For the NaCl crystals, the dendrites were not found. A neglect of the crystal form (“crystal habit”) may result in both overestimating and underestimating the crystallization rate by three orders of magnitude. The morphology of the crystal forms is shown to control the kinetics and leads to anomalously high crystallization rates. The kinetic constant derived in the experiment is by three orders higher than the constant, calculated by the known kinetic expressions, which do not take into account “crystal habit”. Ahead of the crystallization front there is a moving thermal front, arising in a metastable solution. Consideration of heat transfer and crystal forms allows quantitatively and qualitatively describing the difference in the behavior of crystallization. Key words: evaporation rate, crystal habit, crystallization rate, crystal growth kinetic ____________ *E-mail address: [email protected]

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1. Introduction At solution evaporation, a large number of different crystalline phases and mixtures of sulfates, nitrates, chlorides, and carbonates can form. An increased attention in the literature is focused on such brines as sodium sulfate and magnesium sulfate, which are widely distributed in nature and are of interest for geochemistry [1, 2]. Studying the crystallization of an aqueous solutions of salts has wide practical application, e.g., for the purpose of purification and for the production of crystalline compounds [3], and for the development of porous building materials. The formation of crystals and crystal hydrates in a porous medium was considered in [2, 4-6]. Crystalline structures in the double salts were studied by Lindsrom et al. [6]. Heterogeneous formation of the crystal hydrates and crystals of salt in a small space is significantly complicated and controlled by the free surface energy of the phase boundary. Thus, in the porous space of rocks, crystallization can occur at the surface of droplets and at the free surface of a thin layer of a solution, containing a large amount of solid impurities. A wide range of problems is associated with studies of evaporation and crystallization in droplets and thin layers of aqueous solutions. In recent years, numerous studies of the behavior of droplets of salt solutions on the hydrophobic surfaces have been conducted. Applying a thin hydrophobic film can effectively combat metal corrosion [7, 8]. Studies on salt crystallization in capillary tubes and in small droplets indicate that the peripheral regions with interphase boundaries are extremely favorable for nucleation and growth of salt crystals. The strong convective effect within a film of a capillary tube during water evaporation leads to salt accumulation and crystallization [9]. When water evaporates from the surface of droplets of Na2SO4 solution, anhydrous crystals (thenardite) are formed and grow close to contact lines. In the final stage of evaporation, dendritic anhydrous crystals (thenardite dentritic growth) prevail [10]. This behavior of crystal growth depends on the properties of the free surface of the droplet. Droplet evaporation and crystallization behavior in the double salt solution (NaNO3-Na2SO4-H2O solution droplet) are presented by Linnow et al. [11] using Raman microscopy and phase equilibrium diagrams. The formation of crystals and their subsequent behavior in droplets of aqueous solutions of NaCl and CaSO4 are controlled by wettability [12]. In this article, SEM micrographs were used to consider crystallization in droplets, evaporating on various surfaces with contact angles varying in the range 5-112°. Growing crystals of NaCl on the droplet free surface move along the surface from the droplet edge towards its center due to capillary forces [13]. Crystals of NaCl move in the direction of the droplet center, and crystals of CaSO4·2H2O (gypsum crystals) remain attached to the contact line (not moving to the center of the droplet) [12]. Crystallization in the droplet always starts at the edges, near the contact line, and further moves to the center of the 2 ACS Paragon Plus Environment

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droplet. At that, crystals differ for different salts and change with increasing temperature of the free surface [6, 11, 14, 15]. Crystallization in the solution layer differs from that in the droplets by the fact that the crystallization nuclei arise randomly on the free surface and usually do not correspond to the point of the lowest temperature. Three important approaches for describing crystallization may be mentioned: 1) the statistical approach based on a probabilistic description of the generation and development of a new phase (Kolmogorov); 2) the dynamics of crystallization is mainly determined by heat fluxes and thermal inertia of the medium, and at that, the kinetic equation is usually preset; 3) the problem of kinetics is solved in isothermal formulation. Kolmogorov was one of the first to build a statistical theory of isothermal growth of a new crystalline phase in an unlimited volume and at constant rates of crystal growth and emergence of crystallization centers over time [16]. Further, this theory was developed taking into account the dimensions, and the crystallization curves were shown to differ for the volume, layer and at a rod growth [17]. Significantly complicated is the simulation of crystallization in solutions at large supersaturation. The crystallization kinetics has been well investigated for quasistationary and quasi-isothermal conditions with a low degree of the solution supersaturation. In real natural conditions and in technological devices, the behavior of complex solutions may deviate from the equilibrium, and the solution turns out to be in the supersaturated condition. In the case of fast cooling or heating speeds, the crystallization (melting) starts in the metastable region, and the effective activation energy (Eeff) depends on the degree of the solution metastability. Then, the value of Eeff may be different from the Ee when the pressure, temperature and concentration of the components correspond to the equilibrium curve for isothermal case. To describe the kinetics of non-isothermal crystallization it is possible to apply the effective activation energy Eeff [18]. Crystallization with a strong degree of supercooling in the metal melts is significantly different from the quasi-equilibrium kinetics [19]. Unusually high rates of both heterogeneous crystallization and dissociation are observed in non-equilibrium systems at high external heat fluxes [20-22]. Significantly complicated is simulation of crystallization in a confined volume, e.g. in droplets, thin layers, on the surface of small particles and in minichannels [23]. High rates of surface crystallization of ice at dissociation of gas hydrates are associated with the formation of dendrites, which are clearly visible with an electron microscope [24]. At high crystallization rates in metal alloys, the kinetics of phase transition is driven by the structures of various morphology, including rod-shaped forms [25]. High crystallization rates are usually supposed to require a high degree of solution supersaturation. However, in reality this is not always true. There are processes in which the dominant role is played by the morphology of crystals, determining high rates of phase transitions. Moreover, these solutions may be very close to the 3 ACS Paragon Plus Environment


equilibrium curve. The aim of this paper is to demonstrate that in thin layers of aqueous salt solutions at high heat fluxes there is heterogeneous and anisotropic crystallization with abnormally high crystallization rates at extremely low deviation of the system from the equilibrium curve. The behavior of the system may be described by taking into account kinetics and thermal inertia of the medium, when the thermal inertia limits the crystal growth. Self-organization of dendrites allows achieving highrates of reactions at the phase transition. 2. Experimental results 2.1. Experimental technique A schematic drawing of the measurement setup is shown in Fig. 1(a1), where 1 – electronic balance; 2 – heater; 3 – metal working section; 4 – thermocouples; 5 – liquid; 6 – thermal imager (video camera). Fig. 1(a2) shows the growth of a crystal (crystal hydrate of salt) on a liquid-gas surface. The growth occurs in the longitudinal direction x(t). In the transverse direction the crystal growth is observed only in the beginning of the crystallization, and further the thickness of the crystal remains quasiconstant. Ambient air temperature was 21 °C, ambient pressure was 1 bar, and the relative humidity of ambient air was 35-40 %. The work area was a metal cylinder with a diameter of 75 mm and a height of 70 mm, which provided a uniform heat flux across the surface of the cylinder.

Fig. 1. (a1) The scheme of experimental setup: 1 – electronic balance; 2 – heater; 3 – metal working section; 4 – thermocouple; 5 – liquid; 6 – thermal imager (videocamera); (a2) Crystal growth on a free liquid surface The whole working unit was located in a closed housing to maintain the desired 4 ACS Paragon Plus Environment

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ambient temperature and humidity. The working material of the setup was titanium. The initial height of the solution layer for all experiments was 3 mm. Deviation of the initial layer height from 3 mm did not exceed 0.2 mm for the entire surface. The initial mass salt concentration С01 was determined by densimeters. The salt concentration continuously increased over time due to water evaporation from the solution surface. Current mass salt concentration Сi was determined using the weight methods by determining the reduction of the solution mass at evaporation (Сi = (msalt/msol)·100%, where msalt is mass of salt and msol is mass of solution). This method serves to determine the average concentration for the entire layer. The relative measurement error for the liquid mass did not exceed 5 %. The temperature of the solution surface (Ts) was measured by thermal imager (6) (NEC-San Instruments). Thermal measurements were carried out using ten-fold magnification of close-up lens and a high-precision measurement cell. The thermograhic image resolution was 100 µm/pixel. The resolution of video camera (Kodac EktaPro Digital High Speed Camera) was 4 µm (800 x 600 pixels with speeds up to 1000 full frames per second). The surface areas of a crystal were determined using the high-speed camera. These areas were distinguished and calculated using the software. Test experiments have shown that the change in salt concentration has no influence of thermal imager measurements, i.e. differs within its measurement error. The wall temperature (Tw) was kept constant and was determined by the readings of graduated thermocouples, located under the wall surface. The maximal measurement error α was within ± 12 %. On the basis of experimental data the values of heat transfer coefficient α for liquid were determined as αi =

k ∆mi , S si ∆ti (Twi − Tsi )

(1)

where k is the specific heat of evaporation, mi is the current value of droplet mass; S is the area of liquid free surface (liquid-gas), ti is the current time, Twi is the current wall temperature under the solution layer (average over the surface), Tsi is the current temperature of the interface (liquid-gas) (average temperature over the entire surface). Subscript i relates to local values. The experiments were carried out for aqueous solutions of salts of CaCl2, LiCl, LiBr, and NaCl. The initial concentration of salts was the same and increased with time growth, since as a result of a heat flux from the heated wall and water evaporation, the amount of water in the layer decreased. Test experiments have showed that at slow evaporation (at low wall temperatures of 40-60 °C) the crystallization starts at concentrations that are consistent with the equilibrium crystallization curves for the above salts. The difference in concentration was in the range of 0.5-1 %. Despite that the absolute error of the thermal imager was about 5 ACS Paragon Plus Environment


0.5 ºC, for determining the solution supersaturation it was important to know the relative temperature difference ∆T1 = Tecr – Te. Here, Te is the temperature of the free surface of liquid, corresponding to the supersaturation curve, and the temperature Tecr corresponds to the dimensions of the thermal imager at the beginning of crystallization. Relative measurement errors of ∆T1 were in the range of 20-30 %. In the event of crystallization, for a period of approximately 0.1-1 s, the surface temperature of the emerging crystal increased from Te to Tecr. Equilibrium curves and data for ∆T1 served to determine the value of ∆С1; ∆С1 = Сe – Сecr, where Сe is the concentration of the supersaturated solution at the beginning of crystallization, and Сecr is the concentration in the point for the equilibrium crystallization curve. 2.2. Different kinetics of the crystallization curves Two approaches to describing the crystallization developed in parallel for a long time: a statistical approach disregarding the local structures and the thermodynamic one taking into account the system deviation from equilibrium and different crystal forms. The first approach formulated by Kolmogorov and Avrami considered an infinitely large volume of a medium, a random law of crystal nucleation, no effect of crystals on each other (which is possible at their large distance from each other), a constant rate of crystal nucleation, and a constant growth rate of crystals. Given these assumptions, the expression (2) was obtained in Refs. [16, 17] and used to plot curves in Fig. 2, Vi/V0 = 1-exp(-k1wck2tk3) (2) where Vi is the current volume of crystals, V0 is the volume of the whole region, k1, k2, k3 are the constants, c is the constant velocity of crystal growth, w is the crystal nucleation rate, c is the growth rate of the crystal nucleus.

Fig. 2. Crystallization curves for different dimensions: 1 – volumetric crystal growth with continuous emergence of new centers of crystallization with nucleation rate w 6 ACS Paragon Plus Environment

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(w = const) and continuous growth of each crystal with the speed c (c = const); 2 – growth of plates (w = const, c = const); 3 – growth of rods (w = const, c = const); 4 – growth of plates (w = 0, c = const) Curve 1 corresponds to the volumetric crystal growth with continuous emergence of new centers of crystallization with nucleation rate w (w = const) and a continuous growth of each crystal with the speed c (c = const); Curve 2 is plotted for the growth of plates (w = const, c = const); Curve 3 – for the growth of rods (w = const, c = const) and Curve 4 for the growth of plates (w = 0, c = const). For curve 1 k2 = k3 = 3; for curve 2 k2 = k3 = 2 and for curve 3 k2 = k3 = 1. For curve 4, the growth of the new phase volume occurs from one or more initial crystal nuclei, and then new crystals are not formed. In real conditions, the crystal growth usually takes place under varying external conditions and at the presence of space limitations. Most modern technologies are based on thin layers, films, and mini- and micro-channels, when at phase transitions there are rapid changes of temperatures, pressures and heat fluxes over time. Further it will be shown that the change of thermal boundary conditions leads to the transition from one fundamental curve to another (in accordance with curves 1-4 in Fig. 2). Fig. 3 presents experimental data for crystallization of three aqueous salt solutions: 1 – CaCl2; 2 – LiCl; 3 and 4 – LiBr; 5 – NaCl. 1.0 0.8 1 0.6

2

F/F0

3 0.4

4 5

0.2

1

4

3

2

0.0 0

100

200

300

400

500

t, s Fig. 3 Relative change in the area of crystallization (F) over time for aqueous solutions of salts: 1 – CaCl2; 2 – LiCl; 3, 4 – LiBr, 5 – NaCl (C01 = 10 % for curves 1-3; 5 and C01 = 30 % for curve 4)



The initial concentration of salt (at the initial time of evaporation) C01 = 10 % for curves 1-3; 5 and C01 = 30% for curve 4. The point of origin corresponds to the beginning of crystallization. In accordance with the crystallization curves [26-28] for Ts = 75-78 °C, formulas of crystal hydrates and crystals have the form: curve 1 is for crystal hydrate of the salt CaCl2·2H2O; curve 2 corresponds to the crystal hydrate of the salt LiCL·H2O; curves 3 and 4 are for crystal hydrate LiBr·H2O; and curve 5 is for the crystals of NaCl salt (not salt hydrate). For curve 4 (C01 = 30 %) the initial rate of crystallization was significantly lower. And in 20% of cases, two centers grew and merged. However, the first center quickly interacted with the second one, and therefore, two centers practically did not affect the curve, which almost coincided with the kinetics of growth from one center. The qualitative behavior of curves 1-5 (Fig. 3) differs from the behavior of crystallization in a large unlimited volume (Fig. 2). In the final stage of crystallization, the growth rate of the area drops significantly, and all curves have a close tangent of the inclination angle. This is because the conditions of heat transfer change with the growth of the crystalline film. The surface covered with crystals practically excludes evaporation, and as a result, the temperature of the solution below the crust starts growing. When the crust approaches the edges of the cylindrical working section (the crust area is about 80-90 % of the entire area of the surface), the solution temperature before the film rises and approaches the temperature of the equilibrium crystallization curve. As a result, the thermodynamic state of the solution near the edges of the heated section is weakly dependent on the initial conditions. In the analysis of crystallization kinetics it is important to know the local values of temperature and concentration of the solution as well as the degree of deviation from the equilibrium crystallization curve. Fig. 4 shows a diagram in coordinates T, C. T1 is the temperature of the solution (averaged over the height of the layer) prior to crystallization. This temperature corresponds to the equilibrium value for the salt concentration Ce1 on the free surface. The salt concentration in the solution (average in the layer) before the crystallization equals to . This concentration corresponds to the equilibrium temperature of the interfacial surface Te1. Since temperature decreases at evaporation, then T1 > Te1. Since water evaporates from the surface of the solution, there will be the transverse gradient of concentration and < Ce1, i.e., the salt concentration will always be higher near the interface. A necessary condition for evaporation is the system deviation from thermodynamic equilibrium, i.e., Ce1 ≠ 0. Before the beginning of crystallization this condition was always fulfilled in the experiments that led to metastable states (supersaturation of solutions). It is important to note that the points (Te1, ) and (T1, Ce1) are located on the equilibrium curve, plotted for constant values of the equilibrium partial vapor pressure 8 ACS Paragon Plus Environment

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ps1 = const. Then Ce1 k1 + k2Te1 (with increasing temperature Te1 the equilibrium value of salt concentration Ce1 increases), where constants k1 and k2 depend on the pressure ps1. Low evaporation rate leads to a very low deviation of the solution from the equilibrium curve of crystallization and to the formation of only one center of crystallization (or two centers which happens very rarely). This suggests that the incubation period of the first crystal centre is much less than that of other centers. Technical tasks are often associated with high degree of supercooling. In this case, many centers of crystallization appear at the same time, and at their growth there is a continuous interaction of the crystallization fronts that significantly changes the kinetics of crystallization. When the solution passes the equilibrium point (T1, Ce1), it is in a metastable state with salt concentration of ( > ). By analogy, this metastable region corresponds to a pair of values (Te2, ) and (T2, Ce2). Moreover, the points in this metastable region lie on a different equilibrium curve with ps2, ps2 < ps1, as with increasing salt concentration, the equilibrium partial vapor pressure decreases (ps1 corresponds to curve (a), and ps2 corresponds to curve (b)). Values Te1, Te2 and Tecr were measured by thermal imager. Te2 corresponded to the temperature of interface of the solution before the beginning of crystallization. Tecr corresponded to the temperature at the beginning of crystallization.

Fig. 4. Deviation of parameters of the aqueous salt solution (curves (a, b)) from the crystallization curve (c); ps1 (curve (a)) > ps2 (curve (b)) At crystallization there is the temperature shock ∆T1, which is kept constant for a certain period of time. The crystallization curve (c) differs from the curves (a) and (b). For all experiments the values of ∆T1 = Tecr- Te2 ranged within 0.2-1.0 K. In this case, 9 ACS Paragon Plus Environment


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the jump of the salt concentration due to the phase transition С1 (∆С1 = Ce2 - Cecr), in accordance with the equilibrium crystallization curves, corresponded to the interval 0.1-0.3 % for different salts. For these values of ∆С1 the supersaturation for solutions S = Ce2/Cecr = 1.001-1.005. It is surprising that such extremely low supersaturation corresponded to the anomalously high crystallization rates, which will be discussed below. All the studied salt solutions had one center of crystallization. Crystallization of the aqueous solution of NaCl occurred by continuous growth in the number of crystallization centers (Fig. 5). There are various ways to calculate the crystallization rate vcr. The following describes the method for determining the vcr. To simplify the calculation of the current volume, the thickness of the crystal film δcr (Fig. 14(b2)) may be taken as the time average value, i.e.

, where t1 is the total time

of crystallization. Then, the current value of the volume Vi = Fiδcr, where Fi is the current surface area of the film.

t=2s

t=90s

t=250s

Fig. 5. Growth of NaCl crystals and their merger over time Curve 5 for the salt of NaCl (Fig. 3) corresponds to curve 1 in Fig. 2. This curve has 4 characteristic regions (Fig. 3): 1) the emergence and growth of several crystals; 2) spontaneous generation of a large number of new crystals (increase in the rate of crystal nucleation (increasing w)); 3) constant rate of new crystals nucleation (w = const); 4) new crystals interact and merge but are not formed. Fig. 6 gives experimental data on the change of the number of crystallization centers N over time. As it is apparent from the graph, only for time 2 s < t < 20 s, N const. For 20 s < t < 200 s, there is a sharp rise for N. After 200 s, the number of crystals drops sharply as the crystals merge. However, the merger of crystals does not lead to a drop in the growth rate of F/F0 in Fig. 3 (Curve 5). The inclination of curve 5 remains constant up to t = 370-400 s. Crystal merger results in the reduction of the overall length of the crystallization front, but at that the growth rate of single crystals increases 10 ACS Paragon Plus Environment

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due to the rise in supersaturation (water continuously evaporates from the solution). Probably, two factors with opposing action cancel each other out. Thus, for nonisothermal solutions it is easy to make a mistake about the influence of various factors on the kinetics of crystallization. 1000.00

100.00

N 10.00

1.00 1

10

100

1000

t, s Fig. 6. Change in the number of crystals of NaCl salt over time For salts of CaCl, LiCl, and LiBr, the crystal growth took place in one center, so the characteristic radius of the crystal surface Ri was determined as Ri = (Fi/π)1/2, where Fi is the current surface area of the crystal, which increases with time. The area F and the number of crystals N (for NaCl salt) are determined using software in the photographic measurements. If we analyze the behavior of each crystal separately, the difference in the growth rates will amount to tens of percent due to the random nature of the process, and also due to the mutual influence of the crystals on each other. It is more correct to determine the characteristic average size of crystal for each t using expression = (Fi/Ni)1/2, where a is the length of the crystal on the surface of the solution. If crystal growth (plate) is realized from a single center (w = 0), the curve corresponds to expression (3). Fi/F0 Vi/V0 = 1-exp(-kc2t2), (δi δ0). (3) The expression (3) describes the growth of the area (Fi), volume (Vi), where Vi = Fiδcr, δcr const) for curves 1-4 (Fig. 3) and for curve 4 (Fig. 2)). These curves are characterized by continuous decrease of the growth rate of the dimensionless volume with time. Therefore, it is important to note the difference between the two curves: 3 (LiBr, C01 = 10 %) and 4 (LiBr, C01 = 30 %) in Fig. 3. Curve 3 (Fig. 3) corresponds to curve 4 in Fig. 2 in case of the growth of one center of crystallization. And curve 4 (Fig. 3) corresponds to curve 2 in Fig. 2, as if new centers of crystallization 11 ACS Paragon Plus Environment


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continuously appear and grow. In fact, curves 3 and 4 (Fig. 3) had only one crystal center. This contradiction is easy to explain if one measures the local rates of solution evaporation and the crystallization rate at the same time. 2.3. Heat and mass transfer and evaporation rate In the experiments, great attention was paid to the calculation of heat balance. The relative error of the heat balance did not exceed 5-7 % at a maximum change in heat flux with time. Since the nature of crystallization curves depends on the heat transfer, the heat transfer coefficients for the liquid and gas phases and the ratios of crystallization rates to evaporation speed were determined, which helped in the analysis of crystallization kinetics. Experimental data on heat transfer and crystallization are introduced in Table 1, where δl is the average thickness of the solution layer at the beginning of crystallization; δcr is the average thickness of the crystalline film; Vf is the velocity of liquid-gas interface motion and characterizes the solution evaporation rate (since water evaporates from the free liquid surface, the height of the liquid layer decreases, i.e. the interface continuously moves towards a solid wall) Vf = Je/(ρ·F), Je = dm/dt - is the rate of the solution mass loss (kg/s); F is the area of the horizontal surface of the liquid layer, and ρ is the density of the solution. Values Je, ρ, F are taken before the beginning of crystallization; the values Vf, δl, α, are measured before crystallization; the crystallization rate V0cr is measured at the initial moment of crystallization, α is the heat transfer coefficient of the solution, C01 is the mass concentration of salt at the initial moment of evaporation (t = 0), and the wall temperature before the crystallization Tw0 = 348-351К. Changes of ∆T1 and δcr are given at an increase in time (t0→tf, t0 is the initial crystallization time, and tf is the final crystallization stage). Table 1. Experimental data for aqueous solutions of salts CaCl2

LiCl

LiBr

LiBr

NaCl

10

10

10

30

10

Vf ·10 , mm/s

6.6

3

2.3

0.75

15

Vcr0, for t=1s,

10.1

2.2

1.2

0.03

0.3

(Vcr0/Vf) ·10-4

1.5

0.73

0.5

0.04

0.02

∆T1, K

0.9→0.2

0.8→0.3

-

-

1.0→0.2

α, Wm-2K

370

310

170

105

505

δl, mm

0.58

0.54

0.28

0.96

1.2

C01, % 4

mm/s


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Type of crystallization δcr, mm

surface

surface

surface

surface

volumetric

0.02→0.015

0.017→0.015

0.01→0.022

0.011→0.021

-

∆T1 = Tecr-Te2 (Fig. 4), ∆T1, α, δcr, Vcr, Vf and S = Ce2/Cecr were determined as an average value for 5-10 experiments. The temperature difference ∆T1 for LiBr salt was not recorded by the thermal imager, as it was very low (∆T1 < 0.1K) and was within the measurement error of the devices. As can be seen from the table, the ratios Vcr0/Vf correlate with the rate of surface crystallization. So, for the LiBr salt for curve 4 (С01 = 30 %, Fig. 3) the ratio Vcr0/Vf is approximately an order smaller than for curve 3 (С01 = 10 %). Fig. 7(a) shows typical variants of the beginning of crystallization, realized in two different ways A–B1–B2 and A–C1-C2. Crystallization of LiBr with C01 = 10 % (curve 3 in Fig. 3) is implemented according to A–B1, when the temperature of the solution in the vicinity of the “crystal film” (metastable state) grows slowly, and the supersaturation grows rapidly. In this case, the crystallization rate is high (see Table. 1). The option A–C1 corresponds to LiBr with C01 = 30 % (curve 4 in Fig. 3). In this case, the crystallization rate at the initial moment is much slower.

Fig. 7. (a) Phase diagram of an aqueous salt solution; (b, c) - a one-dimensional growth of crystallization (for (b) ∆Tcr = 0, for (c) ∆Tcr 0) As a result, the solution (A–C1) falls within an area with a lower supersaturation than the solution for the case (A–B1). By definition of supersaturation obtain the condition for the ratio of salt concentrations (CeB/CeA) > (CeC/CeA), where CeA corresponds to the equilibrium crystallization curve, and values CeB and CeC correspond to salt concentrations in the metastable region. Point D2 corresponds to the unsaturated 13 ACS Paragon Plus Environment


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region, when the solution is under a crystalline film. Temperature increase under the film and slow diffusion in the liquid lead to the unsaturated state and the crystal growth in the transverse direction stops. As a result, there are two states at the same time: supersaturation before the crystallization front and undersaturation under the crystalline film. The rate of nucleation w is expressed through supersaturation using expression (4) [3] −

∆)* ! "#$%& + ! +

,

(4)

where T is the absolute temperature; k is the Boltzmann constant (1.3805·10-23 JK-1; γ is the interfacial tension; v is the molecular volume, and supersaturation S = Ce2/Cecr (Fig. 4). The so-called classical theory of homogeneous nucleation utilizes the concept of a clustering mechanism of reacting molecules. The assembly of molecules is a “critical nucleus” (“a critical cluster”) with a nearly perfect form. The free energy changes are associated with the process of homogeneous nucleation. The overall excess free energy ∆G is equal to the sum of the surface excess free energy ∆Gs and the volume excess free energy ∆Gv (∆GV + ∆GS = 4/3πr3∆Gv+ 4πr2γ). A “critical nucleus” (rc) is obtained by maximizing ∆G (d∆G/dr = 0). The nucleation rate w may be expressed in the form of Arrhenius reaction velocity equation. Supersaturation is determined using Gibbs-Thomson relationship for a non-electrolyte and is related to ∆Gv. The size of the “critical nucleus” is related to supersaturation S. Let us suppose that in the case of a single growing crystallization center, new centers can also appear, and the site of origin of these centers relates to the crystallization front. These new centers are not formed randomly over the entire volume (surface) of the solution, but they are very close to the surface of the crystallization front. Thus, the rate of crystal (“plate”) growth vcr will be proportional to w (vc ~ w), and further this will enable us to associate vcr with solution supersaturation S in accordance with (4). It is important to note the reasons for which the crystallization curves for one of the growing center in one case correspond to curve 4 in Fig. 2 (w = 0), and in the other case, to curve 2 (w = const). Curve 3 in Fig. 3 (LiBr, C01 = 10 %) has a high initial rate of crystallization. The form of this curve corresponds to curve 4 in Fig. 2. Due to the high crystallization rate, the solution supersaturation (before crystallization front) falls. The temperature of the solution increases, and the point B1 shifts to point B2. The decrease of supersaturation S (expression (4)) with time leads to a drop in crystallization rate vcr. If the supersaturation S is not reduced, the vcr wouldn't decrease with time, as the transverse thickness of the crystal does not change with time. Fundamentally different is the nature of curve 4 in Fig. 3 (LiBr, C01 = 30 %). Low evaporation rate leads to a lower initial supersaturation (point C1). Due to a low 14 ACS Paragon Plus Environment

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crystallization rate, the salt concentration before the crystallization front significantly increases due to water evaporation. Point C1 is shifted to point C2. The increase of supersaturation with time leads to increased crystallization rate, and curve 4 (Fig. 3) corresponds to the type of curve 2 (Fig. 2). Thus, the transition from curve 4 to curve 2 (Fig. 2) is not associated with an increase in the number of crystallization centers. This transition (change of the fundamental curve) is associated with a change in thermal boundary conditions, which control the nature of crystallization. 2.4. Various types of crystal forms at crystallization of aqueous salt solutions To identify specific crystalline forms (“crystal habits”) the photos of crystallization of salt solutions of NaCl, CaCl2, LiBr and LiCl were taken (top view (6), Fig. 1(a1)). In these solutions, the formation of dendrite forms is specific for the whole period of crystal growth. For the salt of NaCl the dendrites were absent, and NaCl had only the form of a cube. The cube height increased at a much slower rate than the size of the cube faces in the horizontal plane. The lack of dendrites led to the fact that the growth rate of the cube was two to three orders lower than for other salts, which formed crystal hydrates. Further it will be shown that the growth kinetics of the cube was somewhat higher than the curve, calculated according to the standard approach. The overestimation of the growth rate is easily explained and predicted, considering different sizes of the cube faces. The lack of growth in the transverse direction (over the liquid height) is compensated by a faster growth in the direction of the horizontal plane. Fig. 8 shows photographs of crystals of NaCl salt (top view).

t=160s

10mm

Fig. 8. Forming crystals of NaCl from an aqueous solution of NaCl salt (C01 = 10 %, Ts = 75-78K) The crystal forms are different for different locations, taken on the surface of the crystal hydrate. The photo (top view) of the crystal hydrate CaCl2·2H2O shows a different “crystal habit” for locations A, B, C (Fig. 9). However, for all cases, dendrites 15 ACS Paragon Plus Environment


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having different spatial orientation and different places of occurrence are clearly distinguished. A B C B A C

t=82s (a)

15 mm (b)

13 mm

11 mm

(c)

d)

Fig. 9. Photos of crystal hydrates CaCl2·2H2O (C01 = 10 %, Ts = 75-78K) Fig. 10 shows photographs of the growth of crystal hydrate LiBr·H2O at different moments of time t.

A B (a) t=2s

C (b) t=12s

A

15.2 mm (d)

(c) t=52s B

10.1 mm

t=12s (e)

(f)

C

12.5 mm (m)

Fig. 10. Photos of crystal hydrates LiBr·H2O (C01 = 10 %, Ts = 75-78K) 16 ACS Paragon Plus Environment

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It is evident that the shape of the perimeter of crystal film changes over time. First, there is the growth of square in the horizontal plane (t = 2s), but already in the 12th second the square loses its shape due to a more isotropic growth of crystallohydrate in different directions. At that, there is an anisotropic growth, much faster growth in one direction Fig. 10(f). The photo (f) clearly shows the formation of main stem and primary and secondary branches. Everywhere at the edges of the growing crystal hydrate there are dendrites whose stems are placed parallel to each other (Fig. 10(m)). This suggests that their growth is not directly associated with the place of origin of the initial cristal nucleus. When approaching the edges of the heater (3) (Fig. 1(a1)) the crystal faces are smoothed Fig. 10(c, m), and their form becomes more rounded. This is due to the fact that with time supersaturation decreases, conditions approach the equilibrium, and the anisotropy of the growth along certain directions decreases. Markedly differing patterns, in contrast to the previously discussed crystal hydrates, are observed for LiCl·H2O (Fig. 11).

D B A

C (a) t=2s

(b) t=7s

A

B

(e)

(f)

4.5mm

(c) t=22s

(d) t=220s

C

(m)

13.5mm

D

(n)

7mm

Fig. 11. Photos of crystal hydrates LiCl·H2O (C0 = 10 %, Ts = 75-78K) This figure shows patterns at different points in time. It is seen that the “wavy character” of the surface of the salt of crystal hydrate increases over time. The surface becomes “rippled”. At the same time, the dendrites are located more chaotically (not parallel as in the previous cases). It seems that dendrites’ stems do not grow 17 ACS Paragon Plus Environment


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straightforward but are distorted and become twisted. Such distortions from the straight line are clearly visible at the edges of the growing crystal Fig. 11(f). On the edges, there are also clearly visible growing dendrites (Fig. 11(n)), whose stems, unlike the previous case (Fig. 10(m)), are not quite parallel to each other. It is important to note that the “crystal habit” is dependent on the heat flux. With the wall temperature increase the heat flux and the evaporation rate increase. At that, the solution near the free liquid surface falls in the range of higher supersaturation. An experiment was conducted for an aqueous solution of the salt of CaCl2 at a wall temperature Tw = 125-130 °C (Fig. 12).

3 2 (a)

(b)

1

c)

Fig. 12. a) Photograph of the formation of dendrite forms during the growth of crystal hydrate CaCl2·2H2O (Tw = 125-130 °C); b) Enhanced thermal imaging for the development of “rod” forms; c) Thermal imaging for the crystallization front motion This pattern (Fig. 12(a)) differs substantially from those previously discussed. Dendrites mainly grow from one center of crystallization, and stems have a straight form. The growth of primary and secondary branches is somewhat depressed. The crystal growth on the surface is much more uniform. Thus, with increasing supersaturation and growth rate of the crystal at the free surface of the solution, the morphology of crystalline forms somewhat changes, which may simplify the modeling of the growth kinetics. Note several important features of the crystal growth, obtained by magnifying thermal images. Fig. 12(b) shows strong temperature inhomogeneity (anisotropy of the thermal field) along and across the direction of the dendrites’ stems. The largest heat release is in places of the larger mass of the crystal (“rod”), and less heat will be observed in the thinner plate, located between the two “rods”. Thus, for a combination “rods+plates” (Fig. 14(a4)) there will be a nonuniform thermal field, which it is important to take into account for different technologies, when thermal inhomogeneity affects the quality of the coatings. In Fig. 12(c) with thermal image, there are three regions. The first region (1) corresponds to the temperature of the 18 ACS Paragon Plus Environment

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supersaturated solution, where crystallization has not yet occurred. In this region, active centers of crystallization have not formed. The region 3 corresponds to the growth of the crystal film. Due to crystallization the surface temperature increases by ∆Tcr (Tcr=Ts+∆Tcr, where Ts is the temperature of the solution before crystallization). The temperature value for the region 2 is between areas 1 and 3. Thus, ahead of the crystallization front there is a moving thermal front, the precursor of crystallization. Estimate the order of magnitude of the thermal front velocity due to thermal diffusion. vT=a/lT =0.01-0.03 mm/s (where a is the thermal diffusivity of the fluid, lT is the width of the thermal front, and lT=3-7 mm). The experimental value of the crystallization front velocity at small times for CaCl2 corresponds to 10 mm/s, which exceeds the thermal diffusion velocity by three orders of magnitude, and only for large times they are of the same order. This feature may be easily explained, assuming that the thermal front is due to the fact that not all but only part of critical nuclei start growing. These active nuclei are in the metastable supersaturated region of aqueous salt solution. Most probable occurrence of active crystallization centers is in the area with the maximum degree of supersaturation, i.e. on the free surface of the liquid. On the free surface, there is the maximum salt concentration due to water evaporation. If the critical nuclei are assumed to have an elongated rather than spherical shape, i.e. the form of “rods” or “plates”, then a weakly discernible “heat” front will outrun the “main” front of crystallization, having a clear visible form. It is obvious that the critical nuclei in the form of “rods” will be less resistant to external hydrodynamic disturbances. Therefore, only a small part of clusters in the form of critical nuclei will not break up and begin to grow. As a result, the observed thermal shock is many times lower than the temperature growth at the crystallization front of the growing crystal. The mechanism of sustainable growth of critical nucleus is discussed later in more detail. In conclusion of this paragraph let us consider the characteristic crystalline patterns for different crystal hydrates of the salt Fig. 13. The growth of dendrites from a single center Fig. 13(a) corresponds to Fig. 10(d), Fig. 11(e), and Fig. 12(a). The growth of the main stem and the subsequent increase of the secondary branch Fig. 13(b) corresponds to Fig. 10(m) and 10(f). For Fig. 10(f) the secondary branches are as strong as the main stem, and for the case of Fig. 10(m) the secondary stems are substantially less than the main one. The growth of dendrites on the edges of the crystal Fig. 13(c), not associated with the primary crystal center, is observed in almost all photographs and is more apparent in Fig. 9(b), Fig. 10(m), and Fig. 11(n). The predominant crystalline growth in one direction in the form of main stem and branch (Fig. 13(d)) corresponds to Fig. 10(f). The simultaneous growth of crystals from several centers (Fig. 13(e)) is presented in Fig. 9(d). Rough and distorted surface of the crystal hydrate (“wavy”), when dendrites are randomly directed with respect to each other (Fig. 13(f)) is shown in Fig. 11(d, f, m). 19 ACS Paragon Plus Environment


Fig. 13. Different patterns at growing crystal hydrates of salts 2.5. Crystals forms controlling the kinetics of crystallization When modeling the crystallization, the reaction coefficient is usually constant, moreover, ∆T =Tw - Ts and supersaturation S are constant as well. These conditions may only be fulfilled in unlimited volume, and if the influence of local crystal forms (“crystal habit”) does not change the reaction rate. In reality, one often has to face the crystallization, limited in space, and the multicomponent solutions, alloys. In addition, solutions are always filled with surface-active substances which greatly affect the surface properties of phase boundaries. As mentioned above, the considered problem is a typical example, when the boundary conditions and the rate of reaction significantly change during crystallization. The Stefan approximation implies the saturation condition at the interface (Fig. 7(b)). Temperature rise at the interphase boundary (Fig. 7(c)) is associated with the crystallization rate using the expression (5) [29], where the degree n is often taken equal to one, and K is the kinetic constant. Vcr = K(∆T)n (5) For simplicity of further calculations, a complex variety of crystalline forms may be simplified, considering that more complex shapes can be obtained by superposition of simple ones. In the considered problem we are dealing with “crystal habit” (Fig. 14). It is assumed that on the free surface of a thin liquid layer “rods” (a1), “plates” (“films”) (a2), and “cube” (a3) are formed. The growth of crystal hydrates for (a1) occurs on one axis, and crystal hydrates (a2) grow in the horizontal plane. For volumetric cube of NaCl crystals (a3) the growth occurs on five faces, except for the axis perpendicular to the top surface. These forms are conditional and do not reflect the changes of the crystal lattice. It is further assumed that at the level of the crystal lattice (crystal 20 ACS Paragon Plus Environment

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structure) there are no changes that affect the crystallization kinetics, i.e. the kinetics is determined by a conventionally selected “crystal habit”, such as a dendrite form (a1).

Fig. 14. “crystal habit (forms)”: “rod” (a1, b1); “plate” (film) (a2, b2); “cube” (a3); rods and plates (a4) The selected crystalline forms will simplify the calculation of heat transfer. The growth rate of salt crystals and crystal hydrates of salts will be determined by the form, heat transfer and diffusion. It is therefore important to determine the selected forms, for which the heat transfer can provide the crystallization rates, obtained in the experiment. The most frequently the problems of heat and mass transfer consider the motion of a plane crystallization front in an unlimited volume (neglecting the “crystal habit”) or the growth of the crystal nucleus with a spherical shape. Evaporation and crystallization in a thin layer are fundamentally different from abovementioned approaches; and additional approaches and hypotheses are necessary to more accurately predict the kinetics of crystallization. At crystal growth, the transition of one form to another is possible. Changes in “crystal habit” over time and space imply a continuous self-organization in a complex non-equilibrium thermodynamic system and a superposition of entropy changes ∑∆S, which must be considered when modeling the kinetics of crystallization. Emergence and development of various types of “crystal habit” in these problems is the subject of further research. Experimental data of this article provides interesting and important material on the development of crystalline forms for different solutions. Fig. 15 presents experimental data on the change of the crystallization rate with time for the growth of a film of crystal hydrates (C0 = 10 %, Ts = 75-78K): ( CaCl2 – 21 ACS Paragon Plus Environment


curve 1, LiCl – curve 2, LiBr – curve 3, and for the growth of crystals of NaCl salt (curve 4). As can be seen from Fig. 15, the rate of crystallization varies significantly over time for all curves. And in the final stage of crystallization, the rates for curves 13 are close to each other. As mentioned before, this is due to the limited surface of the work area and the increasing temperature of the solution due to the growth of the crystalline film. 100.00

1

2

3

4

5

6

7

8

2

3

10.00

vcr·103, m/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.00 0.10 0.01

1 0.00 1

10

100

1000

t, s Fig. 15. Changes in crystallization rate over time for C01 = 10 %, Ts = 75-78K: 1 – CaCl2; 2 – LiCl; 3 – LiBr; 4 – NaCl; 5-7 approximation curves; 5 – Vcr = 1.2(t)-0.5; 6 – Vcr = 1.9(t)-0.5; 7 – Vcr = 8.5(t)-0.9; 8 – Vcr = 0.3(t)-0.65 Regardless of how many crystal centers appear and grow (curve 1-3 for a single crystal, and curve 4 for continuously emerging new crystals), three distinct modes are realized (Fig. 15) at generalizing the crystallization rate vcr in a logarithmic scale in the form of power dependence vcr = a/(t)n: 1) for the first region the crystallization rate is dramatically reduced by power law due to the change of boundary conditions of heat transfer and reduction of the degree of the solution supersaturation S = Ce/Cecr; S = f (∆T1, ∆C1), where Ce is the value of salt concentration on the free liquid surface in a metastable area, and Cecr is the salt concentration corresponding to the equilibrium crystallization curve, ∆T1, ∆C1 in accordance with Fig. 4. However, for this time field the degree n=const, which may be related to self-similar power law of decreasing ∆Tcr with time growth (∆Tcr = b/tn1, where n1=const). 2) For the second region, the exponent n changes drastically with the increase of time n=var; and n increases substantially and exceeds 1. The growth of crystals leads to a multifold exceedance of the longitudinal size over the transverse one, which virtually eliminates the diffusion in the longitudinal 22 ACS Paragon Plus Environment

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direction because of its inertia. As a result, under the crystal there is a balance or a low degree of undersaturation. During crystallization the thickness of the crystalline film may both grow and decrease with time. 3) Due to the reduction of supersaturation at the end of the second region the value ∆Tcr approaches zero, while n1=0. Thus, the third region is characterized by quasi-equilibrium crystallization under identical thermal conditions. It is no coincidence that for all the curves 1, 2 and 3, the value of vcr is the same. The velocity value for crystallization (curves 1-3 in Fig. 15) for the third section vcr=0.02-0.03 mm/s, which is close to the velocity value for the calculated curve 3 in Fig. 18 (vcr=0.01 mm/s for t=100s). There may be an erroneous idea that the kinetics for the third section is not associated with “rod-plate” crystal habit, but has the character of a “volume-cubic” crystal growth. However, this curve 3 (Fig. 18) is obtained considering only the kinetics of volume growth (further, we will consider expression 11 for the kinetics of growth) and neglecting the heat transfer. Accounting of heat transfer will certainly lead to a slower growth of crystal hydrates. Therefore, high experimental values of vcr prove that the third section is also controlled by the “rod” forms. Curves 5-8 for Fig. 15 are generalized by the following relationships: curve 5 – Vcr = 1.2(t)-0.5; curve 6 – Vcr = 1.9(t)-0.5; 7 – Vcr = 8.5(t)-0.9; and 8 – Vcr = 0.3(t)-0.65. Fig. 16 demonstrates that different evaporation rates of the solution Vf (Table 1) can lead to both increase and decrease of crystallization rate over time. Curve 1 corresponds to C01 = 10 %, and for curve 2 C01 = 30 %. Curves 3, 4 in Fig. 16 are generalized by the dependences: curve 3 – Vcr = 1.2(t)-0.5 and curve 4 – Vcr = 0.032(t)0.5. As can be seen, both curves have a long time interval with constant degree n. The reason of changes in the character of the crystallization curve with changes in initial salt concentration was discussed in the previous paragraphs in more detail. 100.00 10.00

vcr·103, m/s

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1

3

2

4

1.00 0.10 0.01 0.00 1

10

100

1000

t, s Fig. 16. Changes in crystallization rate over time for LiBr (Ts = 75-78K): 1 – C01 = 23 ACS Paragon Plus Environment


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10 %; 2 – C01 = 30 %; 3, 4 – approximation curves; 3 – Vcr = 1.2(t)-0.5; 4 – Vcr = 0.03(t)0.5 In Fig. 17 experimental data on crystallization rate are generalized in dimensionless coordinates z = hx (where h is the ratio of heat transfer coefficient of the solution α to the coefficient of thermal conductivity λ, and x is the distance) and τ = ah2t (where a is the thermal diffusivity, and t is the time). Curves 6, 7 (Fig. 17 are built on the expression (6) [29]. Curve 6 corresponds to CaCl2·2H2O salt hydrate; k=0.08; ∆T=0.5°C, curves 7 corresponds to NaCl salt crystal; k=0.5; ∆T=1.0°C z = k√. (6) Curves 1-4 (Fig. 17) for the growth of crystal hydrates of salts show a multifold overestimation of the growth rate in generalized coordinates, which take into account both temperature gradients and physical and chemical parameters of the media (heat conductivity, the heat transfer coefficient, heat flux, etc.). The overestimation of the experimental data (curve 5, NaCl) for z results from deflection of the crystal form from the cube due to the thin layer of the aqueous salt solution NaCl. 100.00 1 10.00

2 3

z

1.00

4 5

0.10

6 7

0.01 0.01

0.1

1

10

100

τ Fig. 17. Changes in crystallization rate in the generalized coordinates z and τ (C01 = 10 %, Ts = 75-78 K): 1 – CaCl2; 2 – LiCl; 3 – LiBr (С01 = 10 %); 4 – LiBr (С01 = 30 %); 5 – NaCl; 6 – curve is built on expression (6) (for CaCl2·2H2O; k = 0.08; ∆T = 0.5 °C); and 7 – the calculation by (6) (for NaCl; k = 0.5; ∆T = 1.0 °C) To calculate the crystallization rate there is a need in equations both for the kinetics of crystal growth and for thermal balance. It is known that at very small times and large temperature gradients, the kinetics determines the growth rate of the crystal. Over times of the order of 1-100 ms and above the fair is the Stefan problem, when it is possible to neglect the inertia of the kinetics, and the growth process is determined by 24 ACS Paragon Plus Environment

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the thermal task; i.e., the thermal relaxation time by many orders exceeds the relaxation, associated with kinetics [30]. The equations of heat balance for a “rod” (7), “plate” (8) and “cube” (9) are presented in a simplified form to identify the effect of structural parameters on thermal inertia (7-9), and the kinetic equation is written in the form (10), rcrπr2ρl(dx/dt) + ρlCp∆Tcrπr2(dx/dt) = – qe + (qw – qα – qr)F1, (7) 2Rrcrπδcrρl(dR/dt) + 2RρlCp∆Tcrπδcr(dR/dt) = – qe + (qw – qα – qr) F1, (8) 2 2 3rcra ρl(da/dt) + 3ρlCp∆Tcra (da/dt) = – qe + (qw – qα – qr) F1, (9) Vcr = dx/dt (or dR/dt) = K∆T, (10) 2 ρl is the liquid density; F1 = πD /4 (D = 75 mm, Fig. 1(a1)); Cp is the fluid heat capacity, ∆Tcr is the temperature difference during crystallization, r is the radius of the cylinder of a “rod” (r = const, Fig. 14(b1)), R is the radius of the “plate” Fig. 14(b2), δcr is the plate thickness; a is the cube edge length Fig. 14(a3), qw is the density of the heat flux from the wall to liquid (measured with thermocouples located on the wall), qe is the heat flux from the solution evaporation, qe = reJv (re - specific heat of evaporation, Jv is the mass flow of vapor), rcr is the specific heat of crystallization; qα is the density of heat flux due to free convection in the vapor-gas medium, qα = α(Ts – T0), α is the heat transfer coefficient of the solution, T0 = 294 K; and qr is the density of the radiation heat flux The growth of crystal hydrates takes place on the surface of a thin fluid layer (Fig. 1(a2)). The growing crystal is heated by heat flux from the wall and as a result of phase transition (solution-crystal), and is cooled from the heat of evaporation, free convection and radiation flux. At the same time, the following simplifications are introduced: the radius of the rod r (Fig. 14(b1)) and the plate thickness δcr (b2) are constant in the process of crystallization, and the rod’s length l and the plate radius R change. With the growth of the crystalline film, of course, the stationary thermal regime is violated. However, due to the rapid growth of the crystal film, the temperature of the wall and fluid increases quite slowly. When the crystal surface covers 60-70 % of the entire surface, the temperature rises less than by 2-3K. Thus, it is possible to consider the problem in quasi-isothermal statement. In addition, due to the small thickness of the plate and the small rod radius, one may neglect spatial gradients within the crystal (small numbers Bi). It is important to note that the evaporation rate is influenced by all the key parameters: diffusion in gas, free convection in the solution, the Marangoni flow on the liquid-gas surface, and diffusion in solution and in the gas. At large times, i.e. approaching the crystallization point, the evaporation rates become very low. Therefore, in the first approximation, the flows (qw, qe, qr, qa) in the right part of equations (7-9) may be set constant. The values of these flows are taken based on experimental data. It is possible to qualitatively examine the simultaneous impact of “crystal habit” 25 ACS Paragon Plus Environment


and thermal factor (thermal inertia of the system) on the growth rate of crystals (crystal hydrates). Assume that during crystallization, three crystal shapes are formed: “rods”, “plates” and “cubes”. At that, let the growth rate of the crystal mass (crystal hydrate) is constant with increasing time. The experimental values of the crystal thickness δcr at large times are of the order of 0.01-0.03 mm. Estimates show that if plates have constant thickness of 0.01 mm, the crystallization rate will only 6 times exceed the one for the cubic crystal. For the thickness of 0.1 mm, the excess in the rate will only be two times. For the case of the rod growth and the rod diameter of 0.01-0.03 mm, the rate of crystallization (“rods” grow only on the free surface of the solution) will exceed that of bulk crystallization of NaCl by 2-3 orders of magnitude, which corresponds to the experimental data of Fig. 15. Theoretical value of the thickness δcr = 0.01 mm corresponds to the experimental value δcr = 0.01-0.022 mm from Table 1. The article does not consider how a diffusion transfer in an aqueous salt solution affects the dynamics of crystallization. However, it is possible to qualitatively assess the role of diffusion. To simplify the analysis, we may consider the case of propagation of a thermal and diffusion layer in one-dimensional formulation for an infinitely thick layer (or small times), and in the absence of convective flows and the smallness of thermal and diffusion changes at the interface. Then the ratio of the heat flux q to the diffusion flux is proportional to (a/D)1/2, where a is the thermal diffusivity of the fluid, and D is the diffusion coefficient in the solution. Since a is about two orders of magnitude greater than D, the diffusion transfer will be much slower than heat transfer. In this case, the characteristic time and the characteristic crystallization rate will be controlled by diffusion. In this case, the calculated curve 4 in Fig. 18 will have a 10 times lower velocity value compared with experiment (curve 1 for CaCl2). Thus, simultaneous consideration of the kinetics of the rods’ growth, the thermal inertia and the diffusion again leads to a significant underestimation of the calculated data compared with the experiment. This contradiction may be eliminated by assuming that the active crystal nucleus in the form of “rods” are formed in metastable solution before the crystallization. The time period of the solution evaporation, of the order of 100-500s, is quite sufficient for the formation of the crystal nucleus before the crystallization. In connection with the foregoing, one can reasonably assume that the crystallization velocity is controlled by the combined influence of the kinetics of “crystal habits” and the heat inertia, and the diffusion may not be considered in the first approximation. Thus, to provide the crystallization rate in the experiment, the anisotropic crystalline forms have to be generated at the initial moment of crystallization. In addition, there is a significant thermal anisotropy in the cross section of the crystalline film, which follows from the infrared images. Kinetics of the crystal surface growth may be represented by several stages. In a metastable supersaturated solution appear 26 ACS Paragon Plus Environment

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unstable dendrites, which become arranged orderly on the surface at the occurrence of the first active center of crystallization. Further, the dendrites (“rods”) are connected with a thin “film” (“plate”). This two-stage growth is a restriction neither for kinetics, nor for heat transfer, and at the same time it allows for the thickening of both “rods” and “plates”. The kinetic constant K in the expression (10) is defined in accordance with (11) [3], (d4ρcrrcr/hTcr)exp(-∆E/RTcr) (11) where d is the characteristic size of the atom, ρcr is the density of the crystal, rcr is the heat of crystallization, h is the Planck constant, ∆E is the activation energy, and Tcr is the crystallization temperature. For salt hydrate CaCl2·2H2O the kinetic constant K = 0.1 mm/sK. Then the estimated growth rate of the crystal hydrate at ∆T = 0.5 K will be equal to Vcr1 = K∆T = 0.1 mm/sK·0.5 K = 0.05 mm/s. Then, the experimental value of velocity Vcr2 for the initial moment of crystallization for CaCl2·2H2O will exceed the theoretical value by approximately three orders of magnitude, Vcr2/Vcr1 = 10/0.05 = 200 (for t = 1s); Vcr2/Vcr1 = 50/0.05 = 1000 (for t = 0.1s). Calculation of kinetic constant for a crystal of NaCl salt (11) gives a value of K = 15mm/sK, and a value Vcr1 = K∆T = 15 mm/sK·1K= 15 mm/s. Vcr1/Vcr2 = 15/0.04 = 375 (for t = 1s). Thus, the estimated velocity value taking into account kinetics is almost three orders of magnitude higher than the experimental value of volumetric (cubic) crystal growth. Fig. 18 shows experimental data and calculated results. Curve 3 for the growth of crystal hydrate CaCl2·2H2O is obtained without taking into account the “rods”: the crystal growth is in the form of a cube (9) and in accordance with (11). Curve 4 for the growth of CaCl2·2H2O is obtained with the growth of “rods” (expression (7)) and heat transfer. In this crystal growth, both the morphology and the heat transfer govern the rate of longitudinal crystallization. Curve 5 for the crystal growth of NaCl is obtained without taking into account the heat transfer and considering only the kinetics of the process (11). Curve 6 is obtained with account for the heat transfer and the growth of a cubic crystal by the expression (9). The calculated data of film thickness give values of approximately 0.01 mm. The experimental values for film thickness give δcr = 0.01-0.022 mm. Thus, it is justified to switch from Fig. 3 to Fig. 2, as in calculations δcr = const. The growth of the film significantly reduces evaporation flux and leads to an increase of temperature under the crust. As a result, the lower surface of crystal hydrate is at the point D2 (Fig. 7(a)) (solution in undersaturated condition). However, melting of the lower surface of the crystalline crust is extremely slow because of the strong inertia of diffusion. In addition, the rapid longitudinal growth of the crust virtually eliminates the diffusion flow of water and salt molecules in the solution in the longitudinal direction. 27 ACS Paragon Plus Environment


100.00 1 10.00

vcr·103, m/s

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2 3

1.00

4 0.10 5 0.01

6

0.00 1

10

100

1000

t, s Fig. 18. Changes in crystallization rate versus time (C01 = 10 %, T s = 75-78 K): 1 – experimental data for CaCl2; 2 – experimental data for NaCl; 3 – calculation for CaCl2 kinetics for the cubic growth without heat transfer; 4 – calculation for CaCl2 and for the “rods” taking into account kinetics and heat transfer; 5 – calculation for NaCl neglecting heat transfer; 6 – calculation for NaCl taking into account kinetics and heat transfer In conclusion, once again, let us get back to the question of emergence of a critical nucleus. In the experiments, the anomalously high crystallization rates were realized at extremely low supersaturation S = 1.001-1.005. At such low supersaturation ∆Gv (the volume excess free energy) becomes much less than ∆Gs (the surface excess free energy) in expression 4. Thus, no matter what the critical radius and supersaturation are, kinetics is determined only by ∆Gs. In this case, all experimental curves for crystallization are to be the same and to be generalized by expression (6). To understand this contradiction, assumptions for expression (4) have to be considered. It is assumed that the critical cluster of molecules forms a perfect sphere. In addition, new centers of crystallization arise continuously (spontaneously). In our case (except for NaCl), there is only one center. However, in this case as well, it may be assumed that at the crystallization front motion, new centers spontaneously arise on its boundary, i.e. the crystallization front is the initiator of the spontaneous growth of new centers. This assumption is partially confirmed by the thermal images, described previously. Thus, there is one basic assumption – spherical critical nucleus. It is obvious that from the point of view of resistance to hydrodynamic disturbances, the sphere is the optimal shape. However, considering the probability of formation of critical nucleus, which will grow steadily, the “rod” and “plate” are more preferred. Let us consider Fig. 19. 28 ACS Paragon Plus Environment

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Fig. 19. Critical nucleus in a thin layer of a liquid solution Spheres with radius r1 are inscribed in the cylinder (“rod”) and “plate”. The surface of all the spheres significantly exceeds the surface of the cylinder (“rod”) and the “plate”. The volume of the cylinder and the plate is slightly higher than the volume of the inscribed spheres. The ratio |∆G1 |/|∆G2 | > 1, where ∆Gc is the overall excess free energy of the cylinder (plate) (∆Gc = ∆Gvc(the volume excess free energy of cylinder) + ∆Gsc (the surface excess free energy of cylinder)), ∆Gs is the overall excess free energy of the spheres (∆Gs = ∆Gvs + ∆Gss). In large volume, the formation of such critical forms of clusters is unlikely, however, for thin layers it is quite possible. It may be assumed that a thin layer of solution with a high evaporation rate behaves in a metastable region similar to liquid crystals. At that, the properties of both fluidity and crystal anisotropy show up. Fig. 19 shows 3 characteristic radii r1, r2 and r3. Both in the solution and in gas there is an intense circulation due to thermo-capillary convection and Marangoni flow. As a result, “rod” and “plate” will be subjected to external disturbances from the external and surface flows. The increase in length L for “rod” and “plate” will cause the loss of stability and rupture (the breakup) of the “rod”(“plate”). These forms will be stable under some limit (critical) radius r2cr and r3cr. Assume that r2(r3) >> r1, then L >> r1. In this case, there will be a faster growth in the direction of the axis of the “rod”. The most favorable conditions for the formation of such forms are observed on the free liquid surface, characterized by the maximum supersaturation and the minimum perturbations that partially attenuate when convective vortices approach the free surface. It is important to note that a thin layer of liquid is not necessary. A thin thermal boundary layer is sufficient. This layer is always available, if the hot melt is poured on the cold surface. At high thermal conductivity of the wall and the melt there will be a substantial supercooling of the melt; and on small times there will be a very thin thermal boundary layer with high temperature gradient. 29 ACS Paragon Plus Environment


Not surprisingly, in this case, there will also be favorable conditions for the formation of non-spherical crystal nucleus. 3. Conclusion Experimental studies of the growth of crystals and crystal hydrates of salt in a thin layer of aqueous solutions of salts CaCl2, LiBr, LiCl and NaCl have been carried out. The growth kinetics of crystals and crystal hydrates has been considered. It is shown that the qualitative behavior of crystallization curves can both coincide and differ from the behavior of the curves, obtained using statistical approach and taking into account the form of the crystals (“crystal habit”). The form of the crystallization curve depends on the ratio of evaporation velocity Vd to the crystal growth rate (crystal hydrate) Vcr. Therefore, at the initial time of crystallization, Vcr may be both maximum and minimum, and the derivative dVcr/dt can be both positive and negative. Calculations for the crystallization rate have been made to take into account the heat balance and the “crystal habit”: crystal in the form of a “cube”, “plate” and “rod”. It is shown that the maximum velocity of the crystallization front is achievable only for “rods” (dendrites structures). In this case, the crystal growth can be considered as a superposition of two motions – the initial growth of the “rods” and the subsequent growth of the “film” between the “rods”, which implies a highly anisotropic behavior of salt hydrates. Experimental data also indicate the presence of thermal anisotropy in the cross section of crystal hydrates. Ahead of the crystallization front there is a moving thermal front, arising in a metastable solution. The authors offer a hypothesis on the formation of critical clusters (in the form of critical dendrites) in a metastable liquid state. For all investigated crystal hydrates of salts, the dendrites have been identified, different patterns have been described, and different scenarios of the dendrites appearance on the free liquid surface and their growth have been discussed. A neglect of the form of crystal structures can give both overestimation and underestimation of crystallization rate by three orders of magnitude. So, the kinetic calculation for the NaCl crystal gives an overestimation of velocity Vcr by almost three orders of magnitude. For the crystal hydrates of CaCl2 salt, LiBr salt and LiCl salt, on the contrary, the kinetic calculation gives an underestimation of Vcr by three orders of magnitude for t = 0.1s. Taking morphology of the crystal forms into account allows quantitatively and qualitatively describing the kinetics of crystal growth. In many technical problems, the growth of thin films and their crystallization depend on highly non-stationary heat transfer and non-stationary evaporation rate, as well as on the morphology of crystal forms. 30 ACS Paragon Plus Environment

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There are three possible favorable conditions for the formation of dendrites: the presence of high gradients of concentration or temperature in a thin boundary layer; |∆G1 |/|∆G2 | > 1; r2 and r3 < rcr. These conditions are formed in many technical fields. Modern technology has been dealing with ever decreasing spatial volumes: flows in mini- and micro-channels, followed by phase transitions and high energy density; the formation of nano-coatings by plasma spraying; chemical reactions in thin layers; creating microstructured surfaces; optical technology; high velocity of liquid cooling on the substrates in cryobiology and medicine; the creation of high-strength microstructured shells of ice on the surface of the particles of gas hydrates, etc. Further development of these technologies largely depends on the accuracy of simulation and adequate reflection of physicochemical processes at the micro level, taking into account self-organization of different crystal forms. Proper control of heat fluxes and evaporation rate of volatile substances will improve the quality of these technologies. Acknowledgment Experiments were carried out at the National Research Tomsk Polytechnic University. The methodology of calculation and using the software and library programs with respect to research of the processes of heat exchange intensification at the interphase under the conditions of phase transformations and chemical reactions was carried out within the framework of the development program of the National Research Tomsk Polytechnic University in the project of the leading universities of the world 5-100. References 1.

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TOC-synopsis page

The anomalously high rate of crystallization, controlled by crystal forms under the conditions of a limited liquid volume Misyura S.Y. 1. Kutateladze Institute of Thermophysics Siberian Branch, Russian Academy of Sciences, 1 Akad. Lavrentyev Ave., Novosibirsk, 630090 Russia, 2. National Research Tomsk Polytechnic University, 30 Lenin Ave., Tomsk 634050, Russia TOC graphic

Synopsis The morphology of the crystal forms is shown to control the kinetics and leads to anomalously high crystallization rates. The kinetic constant derived in the experiment is by three orders higher than the constant, calculated by the known kinetic expressions


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The Anomalously High Rate of Crystallization, Controlled by Crystal

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