A dendritic growth model involving interface kinetics for supercooled

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A dendritic growth model involving interface kinetics for supercooled water Tianbao Wang, Yongjun Lue, Liqiang Ai, Yusi Zhou, and Min Chen Langmuir, Just Accepted Manuscript • Publication Date (Web): 25 Mar 2019 Downloaded from http://pubs.acs.org on March 25, 2019

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Dendritic growth model involving interface kinetics for supercooled water Tianbao Wanga, Yongjun Lüb, Liqiang Aia, Yusi Zhoua, Min Chena,* a Department

of Engineering Mechanics, Center for Nano and Micro Mechanics, Tsinghua University, Beijing100084, China

b School

of Physics, Beijing Institute of Technology, Beijing 100081, China

ABSTRACT

The dendritic growth of ice in supercooled water droplets is studied theoretically and experimentally. The measured dendritic growth velocity of ice shows a good agreement with the prediction of the Langer and Müller-Krumbhaar (LM-K) growth model at supercoolings less than 7 K, whereas an increasing overestimation in the latter is observed as the droplets are further supercooled. The LM-K dendritic growth model is modified by considering the influence of interface kinetics. In the modified model, a dendrite grows in the limit of marginal stability coupled with diffusion at the liquidsolid interface, and the interface kinetics supercooling is introduced to predict the

*

Corresponding author, Tel.:+86 10 6279 7062, Email: [email protected]. 1

ACS Paragon Plus Environment

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dendritic growth velocity. The interface kinetics factor is obtained by fitting the experimental dendritic growth velocity within the framework of the modified model. This modification to the LM-K model well describes the dendritic growth of ice in water supercooled up to 25 K. It provides a solution to the dendritic growth of ice in the high supercooling regime and can serve as a reliable input for studies on icing problems in engineering fields.  INTRODUCTION Icing process of supercooled water has attracted increasing attention in many engineering fields, such as aviation industries1-3, road traffics4 and wind turbines5, because of its potential hazards. The impact of supercooled water droplets can cause ice accretion on cold substrates

6-9.

Experiments shown that the freezing process of

supercooled water droplets on a cold substrate starts with nucleation, followed by dendritic ice growth and equilibrium freezing 10-13. The growth velocities of dendritic ice in the bulk of supercooled water have been extensively studied in both experiment and theory 11, 14-20. However, a dendritic growth model applicable to supercooled water in deep supercooling regime is still lacking. In principle, given the thermal diffusion equations in liquid and solid phases combining specific initial and boundary conditions, dendritic growth in pure supercooled liquids can be solved. However, a number of approximations have been applied to simplify the theoretical model and gain the final solution. First, the shape of the dendrite tip was approximated as a rotating paraboloid in a supercooled pure melt 2


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21.

Then, Ivantsov gave the relationship between growth velocity and dendrite tip radius

by solving the diffusion process in front of the paraboloid tip 22. Langer and MüllerKrumbhaar (LM-K) developed a constrained growth model which assumes that the tip radius of growing dendrites is approximately equal to the lowest wavelength of the tip perturbation, namely, the limit of stability. 14, 23-25. The LM-K model well predicts the slow dendritic growth at small supercoolings irrespective of the chemical nature of systems. For supercooled water, Shibkov et al.

20, 26

have reported a good agreement

between the measured two-dimensional growth velocity of an ice crystal and the prediction of the LM-K model at supercoolings less than 5 K. A similar result is observed in supercooled water confined in the Helle-Shaw cell 11. However, growing overestimations are universally found as supercooling increases. For slow growth at small supercoolings, liquid atoms in front of the interface have enough time to relax and rearrange into an ordered crystalline lattice, which is the primary mechanism of interface movement. The effect of the limit of atomic transport near interface to the interface movement is actually disregarded. In other words, the interface can be approximated to be in a thermodynamic equilibrium, but this approximation fails for the rapid growth of the interface at large supercoolings. To reasonably interpret the dendritic growth of ice, particularly in the high supercooling regime, one should consider interface kinetics 27-28. The effect of interface kinetics has been considered in the Boettinger-Coriell-Trivedi (BCT) model

29

of dendritic growth by introducing the kinetics supercooling term 3


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Tk  v  , where v is the dendritic growth velocity and  is the constant interface kinetics coefficient

27-30.

Lü et al.

31

used this model for water and found a better

prediction of the dendritic growth velocity than that of the LM-K model, but deviations still exist at large supercoolings. Molecular dynamics (MD) simulation results have also shown that the kinetics coefficient of the ice-water interface is not constant

32-34.

The

growth velocity of ice crystals does not monotonically increase as supercooling increases, but decreases after it exceeds the maximum

32-34.

Wilson

35

and Frenkel

36

argued that movement of a planar liquid-solid interface is jointly determined by the thermodynamic driving force and the interface kinetics constraint. Atomic diffusion through the liquid-solid interface plays a crucial role in interface kinetics representation. As supercooling increases, the interface kinetics tends to dominate the interface movement over the thermodynamic driving. As a result, the growth velocity slows down. The Wilson-Frenkel model provides a microscopic description of the interface kinetics. If the rapid dendritic growth is supposed to obey the limit of stability, it can be modeled by coupling the LM-K model with the Wilson-Frenkel model. In this work, a dendritic growth model is proposed based on the LM-K model. This model considers the effect of interface kinetics by introducing an interface kinetics factor as suggested by the Wilson-Frenkel model. This coupled model well describes the dendritic growth of ice at large supercoolings up to 25 K.

 THEORETICAL BASIS

4


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The present model is established on the basis of the LM-K model of dendritic growth. The Wilson-Frenkel model is introduced to consider the effect of interface kinetics during dendritic growth. The coupled model for dendritic growth involving interface kinetics is expressed as three simultaneous equations. LM-K model of dendritic growth. The classical approach to describe the dendritic growth of a crystal from a pure melt is formulated in terms of free boundary problem. The temperature fields in solid and liquid phases satisfy the heat diffusion equation with two boundary conditions. The first condition is the conservation of energy, and the second condition is the relationship between the interface temperature and the thermodynamic melting point considering the effect of capillarity and interface kinetics. The solution for this free boundary problem was first derived by Ivantsov 22, who disregarded the effects of capillarity and interface kinetics and assumed a paraboloid shape of a dendritic tip. The tip moves at a constant velocity v , which is inversely proportional to the tip radius R . In Ivantsov’s solution, the dimensionless supercooling   c p T L is related to the Peclet number p as

  pe p E1  p  , where E1 is the standard error function E1  p   

(2)

 p

exp   y  vR . dy , and p  2 y

Here T  Tm  T is the initial bulk supercooling of the melt, Tm is the equilibrium melting temperature, T is the liquid temperature far from the liquid-solid interface, L is the latent heat per volume, c p is the specific heat per unit volume, and  is the

5


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thermal diffusivity. Ivantsov’s solution provides a coupling of v and R , whereas their single-valued dependence on supercooling is unavailable. The dendritic tip has a size at marginal stability, which was introduced by Langer and Muller-Krumbhaar in their model of dendritic growth (LM-K model)

14, 23.

They

analyzed the stability of the Ivantsov’s paraboloidal dendrites by treating the effect of surface tension as a linear perturbation, and divided the continuum family of Ivantsov’s solutions into stable and unstable regions. They assumed that the selected dendrite corresponds to the point of marginal stability separating the stable and unstable regions. This conjecture leads to the prediction of a universal selection parameter  defined as  =  d 0 vR 2  0.025 , which provides an additional relation between the growth velocity and the tip radius. Here d 0 is a capillarity length given by d 0  Tm c p L2 , where  is the solid-liquid surface tension. The growth velocity as a single-valued function of bulk supercooling is calculated on the basis of the marginal stability hypothesis. The dendritic growth velocity related to dimensionless supercooling  and dimensionless velocity V ( V  vd 0 2 ) is given as follows: V   p2.

(3)

The LM-K model has been verified by several experiments in transparent model substances including water, succinonitrile, pivalicacid, and camphene

14.

However,

LM-K predictions for water are much larger than experimental results in high supercooling ranges of   0.06

20.

Shibkov et al.

20, 26

6


suggested that local thermal

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equilibrium at growing solid-liquid interfaces in the LM-K model may not occur at higher supercooling levels if the transfer of water molecules across the interface is slow. Wilson-Frenkel model of interface kinetics. To consider the effect of interface kinetics during dendritic growth of a supercooled melt, one can use a kinetics supercooling term Tk in the boundary condition at the interface, that is, the interface temperature is modified as Ti  Tm  Tk . Mostly, Tk is related to the growth velocity v by a linear model with a constant interface kinetics coefficient  as

Tk  v  . As mentioned in the introduction, this assumption may not be suitable for some substances. For instance, the MD simulations of the crystal growth of LennardJones liquid 37 and silicon 38 have shown that the interface kinetics coefficients are not constant and the growth velocities are in good agreement with Wilson-Frenkel model. Rozmanov et al.

32

performed a MD simulation of ice crystal growth and found that

temperature dependent growth velocities can be fitted by a functional form similar to the Wilson-Frenkel expression. The Wilson-Frenkel model is relatively fine to describe interface kinetics compared to the aforementioned linear model. Thus for the present dendritic growth model, we choose the Wilson-Frenkel model to account the effect of interface kinetics. The Wilson-Frenkel model proposes the growth velocity of a pure material as follows 39:

v

6aD Ti    L   L  exp     exp    f ， 2   kTm   kTi   

7


(4)

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where a is the cube root of the atomic volume, D is the mass diffusion coefficient in front of the interface,  is an average diffusion jump distance in the liquid, Ti is the interface temperature, k

is the Boltzmann constant, and

f

is the fraction of

repeatable growth sites at the interface. Assuming that    2 af , we can rewrite Eq.(4) as follows: v

6 D Ti    L   L  exp     exp    .    kTm   kTi  

(5)

The parameters for water are chosen as L  6.0 103 J  mol1 , Tm  273.15 K , and k  1.38  1023 J  K 1 . The mass diffusion coefficient D Ti  in front of interface is

approximated by the bulk mass diffusion coefficient. The value of the interface kinetics factor  is determined by fitting with the experimental results on the basis of Eq. (5). To the authors’ knowledge, there is no accepted value of  for water in literature. For instance, Xu et al.

40

reported an interface kinetics factor by fitting with the planar-

interface growth velocity of ice in a temperature range from 180 K to 262 K. However, the maximum growth velocity of ice they measured was approximately 10 cm/s, which is much less than the measured maximum dendritic ice growth velocity (60 cm/s) in other reports 20. We choose all the available experimental results of dendritic ice growth velocity in literature to determine the interface kinetics factor  by applying the present model. Model of dendritic growth involving interface kinetics. In the LM-K model, the interface temperature is assumed to be the melting point without considering the effect of interface kinetics. Here, we introduce the interface temperature as a supercooled 8


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temperature Ti instead of the melting point Tm in Eq. (2), and express the solution for dendrite growth problem as follows:

Ti  T T T    m i  pe p E1  p  L cp L cp

(6)

Figure 1. Schematic of the present dendritic growth model. Ti is the interface temperature , which is related to the dendritic growth velocity by Wilson-Frenkel model and T is the liquid temperature far away from the interface.

In the present dendritic growth model (Figure 1), the interface temperature Ti is below the equilibrium melting point, and it is related to the dendritic growth velocity by Wilson-Frenkel model. In the LM-K model

14,

it is assumed that the stead-state

movement of the dendritic tip can be accurately approximated by the Ivantsov’s solution 22 for an isothermal, cylindrically symmetric, paraboloidal needle-like crystal. However, in the present model, the temperature along the interface should vary with the normal growth velocity considering that the interface temperature depends on the 9


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growth velocity, which differs from the case in LM-K model. Unfortunately, analytic steady-state solution of this dendritic growth problem is known only in the case where the interface is always isothermal. As in the BCT model

29, 41

in which the interface

temperature also depends on the growth velocity, we assume that Ivantsov’s solution (Eq. 5) is still applicable in the present model. With three unknowns, namely, Ti , p and v , the growth velocity v can be calculated by simultaneously solving three independent Eqs. (3), (5) and (6). When solving these equations, the interface kinetics factor  is determined by fitting with the experimental dendritic growth velocity data with least square method. In detail, different values of  are initially set, and the growth velocities at different supercoolings are solved. Then, the solved results are compared with the experimental data, and the sum of the squares of deviations is calculated.  that minimizes the sum of the squares of deviations is determined. 

EXPERIMENTAL METHOD

Figure 2. Schematic of the experimental apparatus.

The dendritic growth of ice in a single supercooled water droplet on a cold plate was investigated through high-speed CCD. The schematic of the experimental apparatus is 10


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given in Figure 2. The main component of the apparatus was a thermally controlled PMMA chamber. The chamber was cooled by a cold nitrogen supply. The water droplet was dispensed from a glass syringe connected to a micropump and deposited on a superhydrophobic surface. The superhydrophobic surface could delay the nucleation time of the water droplet so that the water droplet could be cooled to a higher supercooling. The environmental humidity was kept at ≈ 0% by dry nitrogen flow, to avoid frost formation on the substrate. The substrate was cooled by a thermoelectric cooler and controlled by a temperature controller module to maintain a constant supercooling throughout the experiment. The superhydrophobic surface was fabricated by coating the silicon substrate by silanized silica nanobeads with a diameter 30 nm dispersed in isopropanol (Glaco, Soft99). The details of the coating method can be found in another study 42. Ultrapure water with a resistivity of 18.2 MΩ∙cm was used in the experiments. First, a droplet was placed on the superhydrophobic surface, and the surface was cooled to the target temperature. After the temperature was kept for 2-3 min, the water droplet was cooled to the target temperature. This method could ensure the active control of supercooling from 5 K to 13 K for the droplets with a volume of 125 L. The crystallization of ice was triggered by a needle attached an ice crystallite. To achieve larger supercoolings, we reduced the droplet volume to 5−15 µL, in which crystallization occured spontaneously from the liquid-substrate interface due to heterogeneous nucleation. The maximum supercooling in the present experiments 11


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reached 21 K. The growth process was recorded with a high-speed CCD at a rate of 2,000 fps.

 RESULTS AND DISCUSSION

Figure 3. Dendritic ice growth in a supercooled water droplet. (a) Dendritic ice growth triggered by an ice crystallite. The temperature is −5 °C, and the droplet volume is 125 µL. (b) Spontaneous dendritic ice growth after nucleation. The temperature is −21 °C, and the droplet volume is 15 µL.

Figure 3(a) shows the dendritic growth triggered by an ice crystallite at a supercooling of 5 K. At low supercoolings, to clearly obtain an image of dendritic growth, we chose dendrites whose primary trunks are parallel to the focal plane and maintain a steady growth for tens of microseconds, as shown in Figure 3(a). The dendrite length is measured by using ImageJ and then adjusted by considering the refraction effect based on the optical analysis in Tracepro. The growth velocity of the dendrite can be calculated as the ratio between the dendrite length and the growth time. Figure 3(b) 12


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shows the spontaneous dendritic growth of ice after heterogeneous nucleation at a supercooling of 21 K. Nucleation occurs at 0 ms, and supercooling drives rapid crystal growth, leading to the dendritic morphology of interface due to thermal instability 12. After crystals grows rapidly, a mushy zone representing the mixture of dendrite cloud and water forms. Such growth morphologies at high supercoolings lead to a difficulty in distinguishing a single dendrite. To estimate the dendritic growth velocity, we first select a start frame when nucleation occurs, and then an end frame when dendritic arms grow over the whole droplet. The dendritic growth velocity can be calculated by dividing the maximum growth length by the duration of dendritic growth. Errors are introduced by image resolution, the adjustment due to refraction effect on the droplet surface, the fitting of the ratio between the dendrite length and time, and the judgment of nucleation and the end of dendritic growth.

Figure 4. Dendritic growth velocity in supercooled water as a function of bulk supercooling. The solid symbols denote the present experimental results and open symbols are the experimental results reported in previous studies. The dash line is the prediction of the LM-

13


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K model and the solid line provides the prediction of our model. One measurement is shown per data point.

The dendritic growth velocity of ice as a function of bulk supercooling is presented in Figure 4. For comparison, the growth velocities of two-dimensional single dendrite reported by Shibkov et al. 20, older experimental results 14-17, 19, 43-44, and the prediction of the LM-K model are also provided. Our experimental data agree well with the previous experimental results in literatures, and the LM-K model at supercoolings less than 7 K. As the supercooling exceeds 7 K, the LM-K model shows an overestimation of the growth velocity, and this finding becomes more distinct as supercooling increases. The overestimation is largely attributed to the absence of the interface kinetics effect in the LM-K model, as mentioned above. The effect of interface kinetics in the present dendritic growth model depends on two parameters, namely, the mass diffusion coefficient D Ti  and the interface kinetics factor  . The bulk mass diffusion coefficients obtained from the NMR experiments by Price et al.

45

are used to approximate D Ti  . 

is related to microscopic

quantities, including the atomic jump distance from liquid to interface and the repeatable growth sites at the interface. This factor is hardly determined directly by experiments, and the orders of magnitude of the reported estimations of  vary. Therefore, accurately predicting the dendritic growth velocity without any priori knowledge is difficult. However, if the present model reflects the physical nature of dendritic growth, and a good fit of  to the experimental data is expected. As a 14


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consequence, a well-worked dendrite growth model of ice used in high-supercooling environment can be available. A fit to the available experimental results is provided in Figure 4 based on the present model. The good consistency between them is held for   6.2 1011 m . However, in contrast to experiments reported by Shibkov et al.

20,

our model predicts that the

supercooling dependence of the dendritic growth of ice decreases as supercooling further increases. This finding suggests that dendritic growth prematurely steps into the interface-kinetics-dominated regime in our model compared with that of the experimental results. This deviation may be induced by the knowledge of  and D Ti  . First, the definition of D Ti  refers to the mass diffusion coefficient in front

of the liquid-solid interface 46. Conversely, the bulk mass diffusion coefficient of water is used in our calculation instead. The underlying ordered clusters and the density fluctuation close to the interface may cause a difference between the two coefficients. Second, the assumption that  is independent on supercooling is considered in the calculation.  may vary with supercooling, considering that the fraction of repeatable growth sites at the interface may depend on temperature 39, 47. The clarification of these potential factors requires further investigations, such as MD simulations, on the dynamic process of dendritic growth at an atomic level. As such, the present model with its parameter set can describe the dendritic growth of ice up to T ~25 K. This supercooling range covers most icing processes in engineering applications. Thus the model is beneficial to relevant research on icing. For instance, in the ice-shape 15


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simulations of aircraft icing, the growth velocity of dendritic ice in supercooled water is an important parameter

48,

and the present model may provide some support with

accurate input parameters.

Figure 5. Interface supercooling versus the bulk supercooling as predicted by the present mode for water.

Figure 5 shows the interface supercoolings versus different bulk supercoolings in the present model for water. The interface supercooling is approximately 0.29 K at the bulk supercooling of 7 K. Therefore, disregarding the interface supercooling is acceptable when the bulk supercooling is very small, as described in the LMK model. This neglect causes an error of 7.3% in the dendritic velocity. However, the interface supercooling exponentially grows with increasing bulk supercooling, and great deviations will be caused if the interface supercooling is further absent in the dendrite growth model. The present model involves the interface supercooling, and thus improves the prediction of dendrite growth velocities at large supercoolings.

16


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Figure 6. Dendritic growth velocity in supercooled nickel as a function of supercooling. A comparison between the experimental results (open circles), the LM-K model (dash line) and the present model (solid line) is provided.

The model is applied to dendritic growth in supercooled nickel melts (Figure 6) to further validate the present model. The experimental data is obtained from the work of Colligan and Bayles

49.

The mass diffusion coefficients of supercooled nickel are

obtained from the experiments by Bakker

50

and Maier et al.

51.

Other physical

properties of nickel are the same as those used in Coriell and Turnbull’s work 41. The results shown in Figure 6 indicate that the present model is also suitable for nickel. And the estimated  of nickel is 4.7 1016 m .

 CONCLUSIONS The dendritic growth velocity of ice in water droplets is measured in a supercooling range of T  21 K by using high-speed camera. Compared with the experimental results, the classical LM-K model for dendritic growth accurately predicts the growth velocities of ice at low supercoolings ( T  7 K ), but yields a remarkable 17


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overestimation as supercooling further increases. This deviation is because of the disregard of the effect of interface kinetics on the interface temperature. We modify the LM-K model by introducing the effect of interface kinetics as suggested by the WilsonFrenkel model. As in the LM-K model, the steady-state movement of the dendritic tip is approximated by the Ivantsov’s solution for an isothermal, cylindrically symmetric, and paraboloidal needle-like crystal. The modified model is expressed as three simultaneous equations, namely, Ivantsov’s solution, the marginal stability hypothesis, and the Wilson-Frenkel expression. The interface kinetics factor in the Wilson-Frenkel expression is determined by fitting current model with the experimental data of dendritic growth velocity. The model is also verified by the experimental dendritic growth data of nickel and exhibits a good transferability. For the dendritic growth of ice, current model shows good consistency with the experimental results up to a supercooling of 25 K.

 AUTHOR INFORMATION Corresponding Author *Email: [email protected]

 ACKNOWLEDGEMENTS This work is supported by the National Key Basic Research Program of China (No. 2015CB755801) and the National Natural Science Foundation of China (No. 51621062).

 REFERENCES 18


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A dendritic growth model involving interface kinetics for supercooled

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