張相雄 陳 民

(清華大學(xué)工程力學(xué)系，北京100084)

1 Introduction

Liquid water can stay unfrozen as low as-40°C in a clean environment.1However，the crystallization can be remarkably catalyzed by external conditions，such as heterogeneous contact，confinement or static homogeneous electric field.The influence of electric field on the properties and the phase transition behavior of water is of fundamental importance in many fields such as food preservation，2-4climate change，5and power transmission in cold region.6As a kind of polar molecule，the structure of liquid water can be affected by external electric field.For example，as the field strength progressively increased，the first and the second peaks of the oxygen-oxygen radial distribution function(RDF)increase and the first minimum decreases with the position of the second peak shifting closer to the first.7Svishchev and Kusalik8-11produced ice crystal successfully within hundreds of picoseconds with the help of electric field(~5.0 V·nm-1)and found that the structure of the field induced crystal was sensitive to external conditions.They found that the induced ice phase was cubic ice(Icice)dominant at a proper density.9If the density is a little lower or higher，only low-density or high-density amorphous ice can be obtained.9However，the response of crystalline structure to the field strength has not been fully understood.One purpose of this work is to examine the influence of field strength on the structure of the field induced ice crystal.

A crystallization phenomenon is generally composed of two stages:nucleation and crystal growth processes.The field can induce the ice nucleation by reducing the entropic contribution to the liquid-solid phase transition free energy by restricting the orientational degrees of freedom perpendicular to the external field.12Intuitively，the field can enhance the hydrogen bond7，13and induce ice-like structure7，14in liquid water and then nucleation.Once nucleation starts，the nucleus will expand rapidly into the bulk.One question naturally suggested is how the field affects the ice crystal growth kinetics.Bartlett et al.15reported monotonously increase of the growth rate of ice crystal from supercooled vapor with field strength up to 5.0×104V·m-1.Latham and Saunders16carried out experiments with field strength up to 2.0×105V·m-1and showed that the aggregation rate of ice from water vapor raised monotonously as the field strength changed from 1.0×104to 1.5×105V·m-1and decreased at higher fields owing to the ejection of agglomerates of small ice crystal.Libbrecht and Tanusheva17applied different electric potentials to induce vaporice transition in a cloud chamber and found that the velocity of the dendrite tip increased as a function of the potential firstly，but once the potential exceeded a threshold value around 1400 V，the dendrite could not grow stably anymore and transformed into a needle morphology growing pattern.Libbrecht et al.18confirmed that the growth velocity of the dendrite tip reached a plateau as long as the potential exceeded 1000 V，roughly independent on the saturation.Compared with vapor-ice transition，it needs much stronger electric field to observe apparent influence of electric field on ice growth from liquid.Sun et al.2stated that the phase transition time was unaffected by the application of field between 1.0×103and 1.0×105V·m-1and depended only on the supercooling temperature by freezing a distilled liquid water column between two electrodes.Similar results were reported by Orlowska et al.4with field strength ranging from 0.0 to 6.0×106V·m-1.With combination of electric field and magnetic field，Hu et al.19studied the growth of ice from pre-existed hexagonal ice(Ih)crystal and showed that the critical strength at which the growth can be affected obviously depended on the solid/liquid interface with smallest values 106V·m-1and 0.01 T for{1010}surface.Investigation of the relationship between the field strength and the growth kinetics of ice from liquid water at the molecular level is relatively insufficient and the other motivation of this work is to examine the dependence of growth rate on the field strength.Using the molecular dynamics simulation technique，we examined the influence of field strength on the crystalline structure and the crystallization growth kinetics.

2 Simulation methods

2.1 Identification of ice components

To monitor the phase transition process，we used the CHILL algorithm proposed by Moore et al.20to identify the ice components from liquid water.They classify the water molecules into four groups(Icice，Ihice，defective ice(I)，and liquid/amorphous ice)based on parametera(i，j)which is defined as

Yl，m(r→ij)is the spherical harmonic function，r→ijis the unit vector connecting i and j.i is the index of the central water molecule and j is one of its four nearest neighboring water molecules.qlm(i)is the local orientational bond order parameter vector for each water i.(i)is the corresponding complex conjugate ofl is adopted to be 3 in this algorithm，m is an integer between-3 and 3.The hydrogen bonds between water molecules in ice can be identified as staggered bond witha(i，j)=-1 or eclipsed bond witha(i，j)=-0.11.The local structure of a water molecule can be identified through the hydrogen type connecting the molecule and its four nearest neighbors.For a Icmolecule，it is connected to all its four nearest neighbors with staggered bonds.For a Ihmolecule，it has three staggered bonds and one eclipsed bond.Besides，one molecule which does not completely satisfy the criterion for Icand Ihwill be identified as defective ice.Fig.1 shows three successive snapshots of a typical crystal growth process under field strength(E)=30.0 V·nm-1with different ice components shown in different colors.The growing ice nucleus mainly consists of cubic ice and is surrounded by defective ice all the time.Hexagonal ice molecules emerge and dissolve occasionally and the number is no more than six.

2.2 Simulation process

Fig.1 Three successive configurations of a typical crystal growth process

One of the prominent effects of electric field is the alignment preference of the water molecules along the field direction.As a consequence，the density of the dipolar water system can be affected by the external electric field.21-23Toney et al.24indicated more than two-fold increase of the density near a charged surface.With this consideration，we constructed a water system with two liquid-vapor interfaces in x-axis direction，which allowed the system to change its density without resistance during the phase transition.The dimensions of the simulation cell were Lx=6.0 nm，Ly=Lz=2.0 nm，and periodic boundary conditions were used in all three directions.The water slab containing 508 water molecules lied between-2.0 and 2.0 nm in x-axis direction.The waterwater interactions were described by the rigid and non-polarizable extended simple point charge model(SPC/E)potential25(melting temperature Tm~214 K).26This intermolecular potential has been widely used in icing/melting investigations.10，27，28The SPC/E and TIP4P29potentials exhibit quite similar results in structural predictions for most of the ice phases except ice II.30The other two models designed for ice are TIP4P/ice31model and Six-Site32model.We can see that they give very similar structural predictions for Icunder electric field compared with SPC/E model in the following discussion.Besides，under such high electric field，the influence of the flexibility and polarizability of the water model on the structural prediction needs to be concerned.This issue will be addressed in further investigations.The systems were firstly equilibrated 0.30 ns at 300 K in canonical ensemble(NVT)with the time step of 1.5 fs and then simply quenched to 200 K.Simultaneously，homogeneous electric field was applied along zaxis，with strengths of 0.5，1.0，4.0，5.0，10.0，20.0，30.0，or 40.0 V·nm-1.The water molecules in the amino acid cracks33or near the atomic force microscope tip34experience electric field with similar strength.Moreover，the water molecules could stay unionized according to ab initio calculations35.Ten simulations with different initial configurations for each strength were run 18 ns for crystallization.The time is long enough for phase transition as E≥4.0 V·nm-1.To monitor the structural evolution，the configurations were stored every 200 steps.The temperature was controlled through the Nose-Hoover thermostat and the equations of motion were integrated by the Velert algorithm.We carried out all simulations using LAMMPS code package.36

3 Results and discussion

3.1 Structure analysis

Nucleation events do not occur in the systems with E=0.5，1.0 V·nm-1after 18 ns simulations.As E=4.0-40.0 V·nm-1，the systems transform rapidly into crystalline phase as long as the field turned on.The change in structural order caused by increasing field strength can be reflected by the radial distribution function.When determining RDF，densities of Icand liquid water，only center atoms locating in interval(-1.0 nm，1.0 nm)were selected to eliminate the influence of surface molecules.The RDF and density of Icare calculated when the crystallization process has finished.To confirm the reliability of the RDF calculations，we carried out simulations using TIP4P/ice model at 200 and 270 K，respectively under E=5.0 V·nm-1.The other settings for the system are the same with the simulations for SPC/E model.The RDF profiles for the three models shown in Fig.2(a)are qualitatively consistent.The height of the first and the second peaks are higher at 200 K than those at 270 K for TIP4P/ice model.This kind of variation caused by temperature change is also observed by Murdachaew et al..37Based on these results，we suppose that the SPC/E model can describe the right structural evolution as a function of the strength of electric field to a certain extent.The long-range intermolecular correlation shown in Fig.2(b)suggests the appearance of crystalline structure.38The crystal is Icdominant according to ice component analysis.As shown in Fig.2(b)，compared with liquid state，the first shell is enhanced.Al-tering the field strength，the height and position of the first peak and the first minimum of goo(r)are not affected as obviously as that in liquid phase.7The second shell is strongly affected by increasing the field strength.Similar with liquid，under stronger field，the height of the second peak becomes higher and the position approaches closer to the first.At the same time，the second minimums get larger.These changes indicate that the Iccrystal becomes denser under higher electric field.

Fig.2 (a)Oxygen-oxygen radial distribution function gOO(r)for different water models under E=5.0 V·nm-1;(b)gOO(r)of Ic crystal under different field strengths

According to the Bernal-Fowler rules，39including Icand Ih，in all kinds of common ice crystals，each oxygen atom locates at the center of a tetrahedron with its four closest neighboring oxygen atoms at the vertices.The relative local tetrahedral order can be measured by order parameter q(i)，40which is defined as

where i is the index of the central oxygen，j and k are the indexes of two oxygen atoms at the vertices，and?jikis the angle between them.The tetrahedral order parameter equals to unit for perfect tetrahedron.Compared with RDF，the tetrahedral order parameter can provide some intuitive structural information.To evaluate the change of Icstructural，when calculating the q，only oxygen atom itself with all of its four nearest neighbor atoms belonging to Icis considered.The average q，the bond length and the density are listed in Table 1.As can be seen from Table 1，q decreases a little with increasing the field strength which reflects the gradual deviation of the Icice structure from perfect tetrahedron.This evolution suggests that under external electric field the Icstructure is distorted.The average bond length connecting the central oxygen atom and its four nearest neighbors becomes slightly shorter as the field becomes stronger，ranging from 0.2770 to 0.2753 nm which is not reflected obviously in RDF.This tendency is consistent with the ab initio results.35This evolution means that the ice crystal becomes denser under higher electric field.The densities of Icshown in Table 1 reveal this tendency.The resulting densities are much larger than the proper densities in Svishchev and Kusalik's work.9The simulation time scale might be contributed to the difference and the conclusion of density dependent phase transition phenomenon might need to be verified.As shown in Table 1，the density of the liquid water increases obviously as a function of the strength of the electric field.21，22However，the density of the Icis similar(increased about 0.094 g·cm-3)and is a bit larger than that without electric field(0.964 g·cm-3).30It is questionable to set the density of the system as a constant value to simulate liquid-icephase transition under electric field with magnitude of 10.0V·nm-1.

Table 1 Structural parameters under electric field with different strengths

3.2 Crystal growth

When cooling a liquid sample exposed to a higher electrostatic field，it will freeze at a higher temperature but the corresponding phase transition time is longer.2，4However，it is difficult to isolate the net influence of electric field on crystal growth in experiments.To obtain some details of the crystallization kinetics of supercooled liquid water under electric field，the Avrami equation41is used:

where f(t)is the fraction of the number of water molecules belonging to the cubic ice in the system，f1andf2are fractions at initial and final states，rate coefficient k and Avrami exponent n are fitting parameters.The value n is determined by the shape of the growing nucleus and whether the phase transformation is controlled by volume diffusion(n is a fraction or n=1 for growth on cylindrical nucleus)or interface transfer(n=1，2，3，4).42The evolution of the averagef(t)is presented in Fig.3.The solid lines in Fig.3 are the best fits of equation(4).To observe the crystalline growth process clearly，only the initial 4 ns results are plotted.It shows that there is no obvious induction period and the phase transformation can be recognized as instantaneous nucleation.The total fraction of Icis merely determined by the crystal growth velocity.Using fitting parameters k and n，the characteristic time scale(τ)of the crystallization can be expressed as τ=k-1/n.The fitting results and the time scales are listed in Table 2.The kinetic exponent is less than 1.0 and the phase transformation is controlled by diffusion mechanism.Additionally，the crystal growth rate correlates positively with the electric field strength.It increases obviously(4.0-10.0 V·nm-1)at first and then increases slower.The characteristic time for E=10.0 V·nm-1is a little larger than that for E=20.0 V·nm-1.This fluctuation could be rationalized that when the field intensity is larger than 10.0 V·nm-1the influence of the field on the kinetics of water molecules approaches the limit.The growth velocity reaches a plateau gradually when the field is larger than 10.0 V·nm-1.Similar phenomenon is also observed by Libbrecht et al.18when studying the influnce of the electric potential on the growth velocity of the dendrite tip of ice.As compared with the growth rate of SPC/E ice without electric field，43the crystal growth of supercooled water with about-14 K supercooling26under external electric field as high as 4.0-40 V·nm-1is an extremely rapid process.Although the field strengths used in our simulations are much higher than that used in previous experiments，our results can also provide some clues to understand the influence of the field on the crystal growth.It is reasonable to conclude that the external electric field can not only induce the nucleation process but also accelerate the growth process.

Fig.3 Evolution of the average fraction of Icice molecules in the systems under different field strengths

Table 2 Kinetic coefficients as a function of field strength

To understand the acceleration for a bit，we examined the influence of the electric field on the rotational dynamics during the crystal growth process by calculating the autocorrelation function Ct=＜P2[u(0)·u(t)]＞44and Cd=＜P2[u1(0)·u1(t)]＞，where u(t)is the unit cross product of the two oxygen-hydrogen vectors in the same water molecule，u1(t)is the unit vector of water dipole，and P2(x)is the second Legendre polynomial.Because of the abrupt change of the orientations as long as the field turns on，Ctand Cddiminish rapidly down to very small values fluctuating around zero.In order to observe the influence of electric field strength on the rotational dynamics，we chose the coordinates at t=0.15 ns as the initial configuration.Accordingly，the time variable t shown in Fig.4 is t0

Fig.4 Rotational autocorrelation functions Ctand Cdat T=200 K with different field strengths

-0.15 ns，where t0is the real time.As shown in Fig.4，both Ctand Cddecay exponentially as a function of time45and approach a platform very rapidly with values bigger than zero.The orientations of the water molecules are highly correlated after 0.15 ns simulation.The evolutions of Ctand Cdsuggest that during the growth process the dipole direction of the water molecules rotates to the proper direction much earlier than the rotation around the dipole vector.The molecules need much longer time to adjust their orientations along u(t)vector to proper directions for forming ice crystal.Larger Ctreflects stronger correlation between the initial and the final configurations.As shown in Fig.4，at higher electric field，the system needs shorter time to approach the steady platform.This means that the field can accelerate the rotation of water molecules to proper orientations for crystallization and this kind of acceleration is enhanced at higher field.Besides，under such strong electric field，the system containing only 508 molecules forms polymorphs in some cases，due to the extremely low energy barrier reduced by the strong electric field and the very high growth velocity.Another possible source of the decrease of Ctis the rearrangement of water molecules in neighboring nuclei during the consolidation process.However，such kind of mechanism has not been observed in all simulations.It might need much longer time to capture such kind of picture.46

Because of the preferential alignment of water molecules along the field direction，only very small amount of Ihmolecules(no more than 6 molecules)can form in the system.However，as shown in Fig.1，during the whole crystallization process，the Icnucleus is always surrounded by defective ice molecules.Although defective ice is not a good indicator of the crystallization advance，41it will be very interesting to find out what role it plays during the Icproduction process.To understand this，we trace the original state(Ih，defective ice or liquid)of new formed Icmolecules every 200 time steps.We defined two types of Icmolecules，i.e.，type A:Icmolecules transform from defective ice molecules and Type B:Icmolecules transform directly from liquid molecules.As can be seen from Fig.5，at the very beginning，about half amount of new formed Icmolecules transforms from defective ice and the rest from liquid water.The percent of typeAmolecules increases rapidly to as high as 80%at the fast growth region and keeps at a platform at the following slow growth region，while the percent of molecules transforming directly from liquid water decreases down to approximate 20%.Increasing the electric field can enhance the fraction of type Amolecules.Such enhancement might accelerate the crystal growth rate.Although the states used to trace the new formed Icice production trajectory is not consecutive(Δt=0.0003 ns)，the results can also reveal the primary characteristics.The results suggest that the defective crystalline phase plays an important role in the crystallization transition.

Fig.5 Average fraction of Icmolecules transforming from ice I(A)and liquid(B)molecules

4 Conclusions

In this work，we examined the influence of the strength of the electric field on the crystalline structure and the crystal growth process.The results suggest that the first neighbor of a ice molecule is not affected as obviously as that in liquid state while the second solvation shell is changed remarkably under electric field as high as 4.0-40.0 V·nm-1.Besides，the density of the produced Icis higher than that without field and the tetrahedral structure is distorted.The density becomes larger at higher electric field，increasing from 0.98 to 1.08 g·cm-3in our simulations.The simulations also reveal that the field can not only induce the nucleation process but also accelerate the crystal growth process.This kind of acceleration is partially attributed to the enhancement of rotational dynamics under higher electric field.Additionally，we find that only small amount of Icmolecules transform directly from liquid water and most of them transform from defective ice molecules based on component analysis.The important role of the defective I ice in phase transition is due to the interface transfer mechanism.Our results provide some insights into understanding the influence of electric field on the crystallization of supercooled water.The influence of the electric field with much smaller magnitude on the growth of ice crystal will be concerned in further research.One possible solution is to introduce ice nucleus into the system at the very beginning.Besides，the influence of the flexibility and polarizability of the water model on the growth kinetics under electric field will be considered.

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