Heterogeneous ice nucleation controlled by the coupling of surface crystallinity and surface hydrophilicity
HHeterogeneous ice nucleation controlled by thecoupling of surface crystallinity and surfacehydrophilicity
Yuanfei Bi, Raffaela Cabriolu, and Tianshu Li ∗ Department of Civil and Environmental Engineering, George Washington University, Washington,DC 20052
E-mail: [email protected]
Abstract
The microscopic mechanisms controlling heterogeneous ice nucleation are complex andremain poorly understood. Although good ice nucleators are generally believed to match icelattice and to bind water, counter examples are often identified. Here we show, by advancedmolecular simulations, that the heterogeneous nucleation of ice on graphitic surface is con-trolled by the coupling of surface crystallinity and surface hydrophilicity. Molecular levelanalysis reveals that the crystalline graphitic lattice with an appropriate hydrophilicity may in-deed template ice basal plane by forming a strained ice layer, thus significantly enhancing itsice nucleation efficiency. Remarkably, the templating effect is found to transit from within thefirst contact layer of water to the second as the hydrophilicity increases, yielding an oscillatingdistinction between the crystalline and amorphous graphitic surfaces in their ice nucleation ef-ficiencies. Our study sheds new light on the long-standing question of what constitutes a goodice nucleator. ∗ To whom correspondence should be addressed a r X i v : . [ phy s i c s . c h e m - ph ] O c t ntroduction In understanding and controlling the formation of ice, perhaps the most crucial question to an-swer is: What makes an effective ice nucleation center? The extensive studies on ice formation insupercooled cloud droplets may provide some useful insight to this question. In clouds, the lead-ing candidates for heterogeneous ice nucleation centers have been identified to be bacteria, pollengrains, mineral dust (e.g., clay), emission from aircraft (e.g., soot), and high-molecular-weight or-ganic compounds (long-chain alcohols). While general distinction exists in their chemical nature,these ice-nucleating agents share common structural features. For example, the surfaces of thesesubstances may contain chemical groups capable of forming hydrogen bond with water. As a con-sequence, layers of water molecules can be hydrogen-bonded to the surfaces. The surfaces mayalso exhibit the ordering patterns that resemble the structure of ice. Therefore water layers boundto surfaces may be ice-like, providing template for ice to nucleate.On the basis of laboratory experiments, general criteria for a surface to be a good ice nucleator(IN) were summarized, and these are: (1) IN should be highly water-insoluble; (2) IN shouldhave similar hydrogen bonds available at its surface; (3) IN should have an arrangement of atomsor molecules on its surface as close as possible to that of water molecules in some low indexplane of ice. However it should be stressed that these criteria may only be suggestive but notpredictive, and no unique correlation can be established between ice nucleation threshold andthese surface characteristics. For example, lattice match can be a good indicator, but it cannotaccount for the fact that different materials with the same lattice mismatch with ice, e.g., long-chain alcohol and lead iodide, exhibit different freezing temperatures. Amorphous or poorlyordered materials such as soot are also known to be effective ice nucleator, despite their non-crystalline nature. Crystalline soluble salts such as ammonium sulphate was also found to nucleateice. In addition, recent laboratory experiment using droplet freezing technique further showedthat non-clay mineral feldspar dominate ice nucleation in mix-phase clouds.Here we report direct computational evidence that the crystallinity and the hydrophilicity of thecarbon surface are strongly coupled to dominate the kinetics of ice nucleation. By systematically2arying the water-carbon interaction strength over a wide range and computing the ice nucleationrates explicitly, we observe a rich spectrum of heterogeneous ice nucleation behaviors on bothcrystalline and amorphous carbon surfaces. Remarkably, we find that only within certain rangesof hydrophilicity does the crystallinity of the carbon surface play an active role in nucleating ice,and within these ranges, crystalline carbon surface is found to be significantly more efficient thanthe amorphous. In other hydrophilicity, the role of crystalline ordering becomes negligible, andno appreciable difference is observed in the directly computed ice nucleation rates between thecrystalline and amorphous carbon surfaces. Our study demonstrates that the commonly adoptedindividual criterion for good IN alone may not be sufficient for predicting the ice nucleation ca-pacity of a surface, and points at ways of engineering surfaces for controlling ice formation. Methods
Calculation of ice nucleation rate.
The direct calculation of ice nucleation rates was conducted by employing the forward flux sam-pling (FFS) method. In FFS, the nucleation rate R was obtained through the product of initial fluxrate ˙ Φ λ and the growth probability P ( λ B | λ ) (namely, R = ˙ Φ λ P ( λ B | λ ) ), both of which can becalculated directly by sampling the nucleation trajectories in the parameter space defined by the or-der parameter λ . The p B histogram analysis in our previous study has demonstrated that a goodorder parameter λ is the number of water molecules contained in the largest ice nucleus for bothhomogeneous and heterogeneous ice nucleation on graphitic surface. The ice-like water moleculeis numerically identified by the local bond-order parameter q with a q > . The initial fluxrate ˙ Φ λ is obtained by N / t V , where N is the number of successful crossings to the interface λ from basin A , i.e. , the spontaneous formation of ice nucleus containing λ water molecules, t isthe total time of initial sampling, and V is the volume. The growth probability P ( λ B | λ ) is com-puted through P ( λ B | λ ) = ∏ ni = P ( λ i | λ i − ) , where P ( λ i | λ i − ) is the crossing probability for whicha trajectory starts from interface λ i − and ends on interface λ i . By firing a large number M i − of3rial shootings from the interface λ i − and collecting N i successful crossings at the interface λ i ,one estimates P ( λ i | λ i − ) = N i / M i − . The statistical uncertainty of P ( λ B | λ ) consists of both thevariance of binomial distributions of N i and the landscape variance of the configurations collectedat the previous interface λ i − . More details of ice nucleation rate calculations by FFS can befound in Ref.
The computed heterogeneous ice nucleation rates for all the conditions are listedin Table S1 in Supporting Information.
Systems.
Our molecular dynamics (MD) simulations were carried out using the monotonic-water (mW)model. The carbon-water interaction is described based on the two-body term of the mW model.To mimic a wide range of hydrophilicity of carbon surface, the water-carbon interaction strength ε is varied between ε and 10 ε , where ε = .
13 kcal/mol is the original water-carbon inter-action strength that reproduces the experimental contact angle (86 o ) of liquid water on graphite.The simulations include 4096 water molecules and 1008 carbon atoms, in a nearly cubic cell com-bined with a periodic boundary condition (PBC). The No´se-Hoover thermostat was employed tosimulate the isobaric-isothermal canonical ensemble (NPT), with a relaxation time of 1 ps and15 ps for the temperature and pressure, respectively. A time step of 5 fs is used throughout thesimulations. Model of amorphous graphene.
To study the role of crystallinity of the carbon surface, we amorphize the graphene surface by intro-ducing the Stone-Wales (SW) defects through the Wooten-Weaire-Winer (WWW) bond-switchingMonte Carlo (MC) method. In this approach, the carbon-carbon interaction is described by theKeating-like potential: V = ∑ i , j k (cid:2) r i j − r (cid:3) + ∑ i , j , k h (cid:20) (cid:126) r ki · (cid:126) r k j + r (cid:21) , (1)4here r = .
42 Å which denotes the equilibrium C-C bond length. The first and second terms ofthe potential describe the energies for bond stretching and bond bending, with k =7.5 kcal/mol and h ( h / k = .
2) being the corresponding spring constants, respectively. The MC simulation includestwo types of moves: a single particle displacement and a WWW bond-switching move. The lattermove allows introducing five- and seven-fold rings in the graphene sheet while preserving the localthree fold coordination of carbon atoms. The schematic diagram for the bond-switching move isshown in Fig. 1. By progressively introducing the bond-switching move, one may disrupt the trans-lational symmetry of the graphene while minimizing the coordinational defect. In our procedure ofamorphizing graphene, each MC step consists of one attempt of single particle displacement andone bond-switching move, both performed randomly throughout the graphene sheet. The standardMetropolis MC procedure was employed. To allow the structure to relax sufficiently, particularlyonce the bond-switching move is accepted, the single particle move and the bond-switching moveare performed at k B T = .
33 eV and 39 eV, respectively. After 10 MC steps, the structure wasquenched at k B T = .
033 eV for obtaining realistic atomic structure for amorphous graphene atroom temperature. Fig. 1 (b)&(c) show the distributions of the C-C bond length and C-C-C bondangle of the amorphous graphene. The topology analysis shows the amorphous graphene networkis composed of 35% pentagons, 37% hexagons, 22% heptagons, and 4.9% octagons, in agreementwith the ring statistics (34.5%, 38% 24%, and 4.5%, respectively) obtained by Kapko et. al. Results and Discussion
We first carried out study of heterogeneous ice nucleation on carbon surface with the originalwater-carbon interaction strength ( ε = ε ). Figure 2a shows the computed heterogeneous ice nu-cleation rates on both crystalline and amorphous graphene. At both temperatures investigated,no distinguishable difference is found on the computed ice nucleation rate between the two sur-faces. The result may appear surprising in its first glimpse but can be understood by the fact thatcrystalline graphene neither binds water nor templates ice. In fact the heterogeneous ice nucle-5 D i s t r i bu t i o n ( % ) C-C bond C-C-C angle (a)(b) (c)
Figure 1: ( a ) Bond interchange using the WWW algorithm to introduce the Stone-Wales defect ingraphene. The SW defect is generated by rotating a carbon-carbon bond by 90 degrees. Withinthe WWW framework, this is achieved by first randomly selecting a bond, e.g. , AC, followed bya random selection of two carbon atoms connecting to A and C, respectively, but on the oppositesides of bond AC, e.g. , atom B and D. Then the bond AB and CD are replaced by new bondsAD and BC (dashed lines). The final amorphous graphene structure yields a distribution of thecarbon-carbon bond length ( b ) and the carbon-carbon-carbon bond angle ( c ).6tion of graphene was recently attributed to the induced layering of water density perpendicularto water-carbon interface: Since the oscillation of water density near the water-carbon in-terface matches well the density profile of ice, and the motion of water molecules in the contactlayers is well restricted within the plane, the space that these water molecules can explore, hencethe entropic barrier of ice nucleation, is effectively reduced. It has been further suggested thatwater molecules in the first layer experience a nearly uniform potential from graphene. In otherwords, water molecules in contact layers see carbon surface as an atom-less flat surface. Indeed,our calculated density of water (Fig. 2b ) shows that the absence of crystallinity in graphene doesnot alter the profiles of water density normal to the surface. This is consistent with the observedinsensitivity of ice nucleation rate on surface crystallinity under the original surface hydrophilicity.
226 228 230 232 234 236 238
Temperature (K) N u c l e a t i o n r a t e ( m - s - ) CrystallineAmorphous
CrystallineAmorhous D e n s i t y o f w a t er ( n m - ) Distance to surface (Å) ε ba Figure 2: ( a ) Calculated heterogeneous ice nucleation rates on crystalline and amorphous grapheneat 230 K and 235 K. Inset shows the atomic structures of both crystalline and amorphous graphene.( b ) Calculated density profiles of water along the direction normal to the water-graphene interface,for different water-carbon interaction strengths.The interesting results are obtained when water-carbon interaction strength ε is allowed tovary. Increasing ε makes carbon atoms bind water more strongly, and correspondingly the sur-face becomes more hydrophilic. As shown in Figure 3, the variation of ε (hence hydrophilicity)7 water-carbon strength ( ε ) N u c l e a t i o n r a t e ( m - s - ) crystallineamorphous Water-carbon strength ( ε ) × -2 × × × R c / R a Hydrophilicity
Figure 3: Variation of the calculated ice nucleation rates with water-carbon interaction strength ε (in the unit of the original strength ε ) at 230 K, for both crystalline and amorphous graphene. Theblue diamonds indicate the calculated nucleation rate of ice forming on crystalline graphene witha stretched carbon-carbon bond length of 1.46 Å . Inset shows the ratio of the ice nucleation ratesbetween the crystalline and amorphous graphene R c / R a , as a function of water-carbon strength.The red horizontal line indicates a ratio of one. 8ields a rich spectrum of heterogeneous ice nucleation behaviors on both crystalline and amor-phous graphene. First, increasing surface hydrophilicity initially enhances the ice nucleation rateson both surfaces, which maximizes the ice nucleation efficiencies for the crystalline and amorphousat the water-carbon strength of 4 ε and 2 ε , respectively. A further increase of hydrophilicity, nev-ertheless, starts weakening the ice nucleation capacities of both surfaces, until the minimum nu-cleation rates are reached at 6 ε for both surfaces. Upon additional enhancement of hydrophilicity,both surfaces are found to re-gain their abilities of nucleating ice.Remarkably, in contrast to the behaviors observed at the original water-carbon strength ε ,the crystallinity of the graphene surface was found to play a significant role in facilitating icenucleation when surface hydrophilicity is varied. As shown in Figure 3, there exist ranges ofhydrophilicity where the crystalline graphene yields significantly higher ice nucleation rate thanthe amorphous graphene. This can be better illustrated by the inset of Fig. 3 which shows theratio of the ice nucleation rate resulted from the crystalline graphene to that of the amorphoussurface, R c / R a , as a function of water-carbon interaction strength ε . It is evident that within thelow ( ε ≤ ε ) and mid-high (6 ε ≤ ε ≤ . ε ) ranges of water-carbon interaction strength, thecrystalline graphene shows no appreciable difference from the amorphous in its ice nucleationability, whereas it becomes much more efficient within other ranges. In particular, at ε = ε the crystalline graphene yields an ice nucleation rate almost 10 times higher than the amorphous,strongly suggesting that the crystallinity is a key factor for ice nucleation under such hydrophilicity.The results immediately suggest the existence of a coupling resulted from the two surface char-acteristics that dictates ice nucleation, and that the coupling strength varies non-monotonically withthe surface hydrophilicity. To understand the origin of this coupling behavior, we first examine therecently proposed layering mechanism. Fig. 2b shows, however, that the density profile of waternormal to the carbon-water surface is essentially insensitive to the change of surface crystallinity,within the entire range of hydrophilicity investigated in this work. Clearly, the observed differenceof ice nucleation rates induced by crystallinity may not be explained by the layering mechanism.To shed light on its origin, we examine the structure of the critical ice nucleus forming on car-9 ε st layer 1 ε st layer 3 ε st layer 5 ε st layer 6 ε st layer 6 ε nd layer 7.5 ε nd layer 9 ε nd layer a b c d e f g h Figure 4: Evolution of ice layer (blue) in the critical nucleus forming on graphene (cyan) withwater-carbon interaction strength ε . ( a ) and ( b ) show the ice layer forming in the first contact layerof water on the crystalline and the amorphous graphene, respectively, at the original strength ε .With an increasing ε , the ice layer gains lattice registry with respect to the crystalline graphene,as in ( c ) and ( d ). Further hydrophilicity increase disfavors ice formation in the first contact layer,as in ( e ) where only dangling water molecules appear, and instead, facilitates ice nucleation inthe second contact layer, as in ( f ). The ice layer gradually aligns registered with the crystallinegraphene with the continuous increase of water-carbon strength, as in ( h ).10on surface. As heterogeneous ice nucleation aligns the basal plane of ice parallel to graphene, theice structure in the contact layers are expected to play an active role in ice nucleation. Fig. 4a &b show the structure of the first ice layer of a critical nucleus formed on the underlying substratewith the original ε = ε , for both crystalline and amorphous graphene. As expected, the basalplane of ice does not appear to match the underlying crystalline graphene lattice. Correspondinglythe computed ice nucleation rates do not exhibit fundamental difference when the crystallinity ofgraphene changes. As the water-carbon strength increases to 3 ε , the first layer of ice and thegraphene lattice are found to form a nearly commensurate structure, i.e. , each six member ring ofwater encloses a six member ring of carbon, as shown in Fig. 4c. On the basis of the underlyingcrystalline graphene lattice, the pseudo commensurate structure can be considered as a 3 × ε , but interestingly, disappears upon a further increase of water-carbon strengthto 6 ε . Under this hydrophilicity (6 ε ), it is found that the water molecules in the first contact icelayer no longer arrange themselves into an ice like structure (Fig. 4e). Instead, the basal plane ofice forms in the second layer of water, and it is further observed that the second ice layer, simi-lar to the first ice layer forming at the original water-carbon strength ε , is incommensurate withthe underlying graphene lattice (Fig. 4f). We also note that at this water-carbon strength (6 ε ),the computed ice nucleation rates for both crystalline and amorphous graphene become equiva-lent again, and reach their minima (see Fig. 3). More interestingly, upon a further increase inhydrophilicity, e.g. , ε = ε , the basal plane of ice in the second layer appears to form the 3 × ε ∼ ε . Through the formation of the pseudo commensurate structure, the crystalline graphenenow regains its efficiency for promoting ice nucleation.The molecular analysis of ice nucleus suggests that the higher ice nucleation efficiency of thecrystalline graphene within 2 ε ≤ ε ≤ ε and ε ≥ . ε is closely related to its ability of forming11 ε st layer 3 ε st layer 6 ε st layer 7.5 ε st layer 6 ε nd layer 9 ε nd layer a c d e f g h i b Figure 5: The in-plane distribution of water molecules in the contact layers of graphene at 230 K.For clarity, the distribution is defined as − Ln [ P ( x , y )] , where P ( x , y ) is the probability density offinding a water molecule located at ( x , y ) in the plane. ( a ) and ( c ) show the contour of − Ln [ P ( x , y )] computed for the first water layers in contact with crystalline graphene and amorphous graphene,respectively, with a carbon-water strength of ε = ε . As indicated by the contour legend, a darkercolor represents a higher probability of distribution. ( b ) is the zoom-in of a local region in ( a ),which shows water tends to be adsorbed above the center of carbon hexagonal ring. To demonstratethe influence of water-carbon interaction strength ε on the distribution of water in contact layers,( d ) ∼ ( i ) show the distribution − Ln [ P ( x , y )] as a function of both x and y for the same region as in( b ), with different hydrophilicity. In general, the increasing hydrophilicity of crystalline graphenepatterns the first layer of water, which resembles the underlying graphene lattice. When water-carbon strength is high enough (as in ( i )), the second layer of water is also patterned, as a resultof the strong localization of water in the adsorption sites in the first layer and the local tetrahedralordering of water. 12n ice layer nearly commensurate with the underlying graphene lattice. Essentially this can beunderstood on the basis of the templating effect, albeit that the templating of ice occurs over thesupercell of graphene lattice. The question now is: why does this templating effect only occur atcertain hydrophilicity?To answer this question, we examine the in-plane density of water in the contact layers ofgraphene. At a low water-carbon strength, i.e. , when carbon atom binds water weakly, watermolecules in the first contact layer only experiences a nearly uniform potential from the underlyinggraphene. This can be illustrated by the in-plane distribution of water density in Fig. 5d. Whencarbon binds water more strongly ( ε ≥ ε ), the in-plane density distribution of water in the firstlayer begins affected by the underlying atomic structures. In the case of crystalline graphene, watermolecules in the contact layer are found to preferentially locate themselves above the center of thecarbon hexagonal ring (see Fig. 5b), making those weak adsorption sites of water. Consequently,the density of water clearly displays a pattern reminiscent to graphene lattice, as shown in Fig.5a. The ordering of water density in the first layer, which partially matches the ice basal planeof ice, thus may effectively increase the possibility of forming an ice-like fragment. In the caseof amorphous graphene, the disordered, less uniform water density in the first layer (Fig. 5c) infact frustrates the crystalline ordering required for ice nucleation, which leads to a decrease ofice nucleation rate. As a consequence, the difference in the ice nucleation efficiency between acrystalline and amorphous graphene starts increasing with hydrophilicity.As water-carbon interaction becomes even stronger ( e.g. , 6 ε ), the water molecules in the firstlayer are further constrained around the adsorption sites (Fig.5f). This leads to two effects thatboth suppress ice nucleation on crystalline graphene surface. First, because the formation of thehexagonal ice patchworks on crystalline graphene requires that only two thirds of the adsorptionsites are filled while the rest one third are empty (see Fig. 4c & d), a full coverage of water withstrong binding strength in fact structurally hinders ice nucleation. Second, since the underlyinggraphene structure is rigid with a fixed carbon-carbon bond length d CC = i.e. , three times of d CC . This implies that the ice layerin such structure must be strained (compressed in this case), given that the equilibrium in-planelattice constant of ice is 4.5 Å. Therefore the strain energy cost of imposing such perfect matcheventually makes the formation of the strained ice layer energetically unfavorable in the contactlayer. To further demonstrate the role of strain, we increase the carbon-carbon bond length incrystalline graphene to 1.46 Å , in order to better match ice lattice. The calculated ice nucleationrate at both 6 ε and 7 . ε indeed show that the elongated crystalline graphene yields significantenhancement on ice nucleation rate (Fig. 3) relative to those of the unstrained graphene.As the first water layer becomes inactive, the nucleation of ice consequently starts occurring inthe second layer of water when ε ≥ ε . In this case, as the in-plane water density in the secondlayer is nearly uniform (Fig. 5h), the crystallinity of the underlying graphene becomes inactiveagain in ice nucleation, as evidenced by the synchronization of the computed ice nucleation rateswithin 6 ε ≤ ε ≤ . ε . Interestingly, a further increase of hydrophilicity ( ε ≥ ε ) is found tonearly immobilize water molecules within the first water layer in their adsorption sites. The stronglocalization of water in the first water layer essentially turns it into an image of the underlyingcrystalline graphene sheet. The second layer of water, now under the influence of the patterningfrom the first layer and the water-water interaction, becomes also structured (Fig. 5i) and tendsto facilitate the formation of the commensurate ice-like structure. In other words, the first and thesecond layers of water at the high hydrophilicity act almost as the graphene substrate and the firstlayer of water at the low hydrophilicity, respectively. In this way, the crystalline graphene sheetgains renewed ice nucleation capability, making it again superior than amorphous graphene fornucleating ice.The findings in this work have a few important implications regarding our understanding ofheterogeneous ice nucleation. Perhaps the most immediate implication is that neither surface crys-tallinity nor surface hydrophilicity alone may be a good indicator for the ice nucleation efficiencyof a surface. Instead, the combined surface characteristics may yield a complex coupling that dom-inates ice nucleation behaviors. This may potentially explain why materials that have similar lattice14ismatch with ice exhibit drastically different ice nucleation behaviors, and that no correlation hasbeen established between ice nucleation threshold and any of the crystallographic characteristics. Therefore a thorough understanding of the ice nucleation capacity for an IN should be achievedthrough a comprehensive study by explicitly considering all the necessary molecular details at thesurface. Second, it is envisioned that the surface chemistry and surface crystallinity may also becoupled with the elasticity of the substrate. In our modeling the graphene surface is consideredrigid. This is a good approximation as carbon-carbon bond is much stronger than the hydrogenbond of ice. When ice nucleates on a soft substrate that has a shear modulus lower than or com-parable to that of ice, the possible local deformation of the substrate should play an active roleif there exists a lattice mismatch between ice and the surface. Since a soft substrate may bettertemplate ice for a lower strain energy cost, the range of its lattice mismatch can be wider than astiff substrate for achieving the comparable ice nucleation efficiency. This also may explain whysome soft materials, e.g. , self-assembled monolayers of amphiphilic alcohols, are known as themost effective INs. Acknowledgment
The work is supported by NSF through award CMMI-1537286. T. L. also thanks the Sloan Foun-dation through the Deep Carbon Observatory for supporting this work.
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