Lifshitz interaction can promote ice growth at water-silica interfaces
Mathias Boström, Oleksandr I. Malyi, Prachi Parashar, K. V. Shajesh, Priyadarshini Thiyam, Kimball A. Milton, Clas Persson, Drew F. Parsons, Iver Brevik
LLifshitz interaction can promote ice growth at water-silica interfaces
Mathias Boström,
1, 2, ∗ Oleksandr I. Malyi, Prachi Parashar,
1, 4, † K. V. Shajesh,
1, 4
PriyadarshiniThiyam, Kimball A. Milton, Clas Persson,
2, 5, 7
Drew F. Parsons, and Iver Brevik ‡ Department of Energy and Process Engineering,Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Centre for Materials Science and Nanotechnology,University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway School of Materials Science and Engineering, Nanyang Technological University,50 Nanyang Avenue, Singapore 639798, Singapore Department of Physics, Southern Illinois University-Carbondale, Carbondale, Illinois 62901 USA Department of Materials Science and Engineering,Royal Institute of Technology, SE-100 44 Stockholm, Sweden Homer L. Dodge Department of Physics and Astronomy,University of Oklahoma, Norman, Oklahoma 73019, USA Department of Physics, University of Oslo, P. O. Box 1048 Blindern, NO-0316 Oslo, Norway School of Engineering and Information Technology,Murdoch University, 90 South Street, Murdoch, WA 6150, Australia
At air-water interfaces, the Lifshitz interaction by itself does not promote ice growth. On thecontrary, we find that the Lifshitz force promotes the growth of an ice film, up to 1–8 nm thickness,near silica-water interfaces at the triple point of water. This is achieved in a system where the com-bined effect of the retardation and the zero frequency mode influences the short-range interactionsat low temperatures, contrary to common understanding. Cancellation between the positive andnegative contributions in the Lifshitz spectral function is reversed in silica with high porosity. Ourresults provide a model for how water freezes on glass and other surfaces.
Although water in its different forms has been studiedfor a very long time, several properties of water and iceremain uncertain and are currently under intense inves-tigation [1–4]. The question we want to address in thepresent paper is to what extent the fluctuation-inducedLifshitz interaction can promote the growth of ice films atwater-solid interfaces, at the triple point of water. Parti-cles and surfaces, e.g., quartz, soot, or bacteria, in super-cooled water are known experimentally to nucleate iceformation [5–7]. Here, we focus on interfaces betweenwater and silica-based materials and examine the rolesof several intervening factors in the sum over frequencymodes (Matsubara terms) contributing to the Lifshitzfree energy.Quantum fluctuations in the electromagnetic field re-sult in van der Waals interactions, which in their un-retarded form were explained by London in terms offrequency-dependent responses to the fluctuations in thepolarizable atoms constituting the material medium [8].The understanding of these interactions was revolution-ized when Casimir introduced retardation effects [9]. Thetheory was later generalized by Lifshitz to include dielec-tric materials [10, 11]. The Lifshitz formula in Eq. (1),derived for three-layer planar geometries [11], gives theinteraction energy between two semi-infinite dielectricmedia described by their frequency-dependent dielectricpermittivities as well as the dielectric permittivity of themedium separating them (see Fig. 1).The purpose of the present work is twofold. First, wewant to show that a finite size ice film, nucleated by asolid-water interface, can be energetically favorable even Lε water ε ice ε silica Figure 1. Ice ( ε ) of thickness L at the interface of water( ε ) and silica ( ε ), illustrated here as three planar regions ofinfinite extent. when only the Lifshitz interaction is accounted for. Sec-ond, we want to highlight a relevant contribution fromthe zero frequency term in the expression for the Lifshitzenergy in a region where it is not expected to be impor-tant. The temperature dependence of the Casimir forcebetween metal surfaces [11–14] relies strongly on the ex-act behavior of the low-frequency dielectric function ofmetals. These and many other investigations have pro-vided support for the notion that the zero frequency termwould only be relevant at high temperatures or large sur-face separations at a moderate temperature. In biologi-cal systems that involve water, the zero frequency termcontributes substantially to the total Lifshitz interactionenergy because of the high static dielectric permittivityof the water compared to the interacting media [15, 16].In this paper, we will show that for three-layer planargeometries, where an attractive-repulsive force transitioncan occur, it is possible to find systems in which the com- a r X i v : . [ c ond - m a t . o t h e r] A p r bined effect of retardation and the zero frequency termdetermines what happens with the interaction across ex-tremely thin sheets. The mechanism behind this is a can-cellation between the positive (repulsive) and negative(attractive) contributions from the different frequency re-gions, which leads to a diminished contribution from thenonzero Matsubara terms and thus renders the zero fre-quency Matsubara term dominant.We emphasize that the system, in spite of its appar-ent simplicity, is far from trivial. The resulting value ofthe Lifshitz energy is dependent on an interplay betweendifferent factors:(i) The crossing in the curves for the permittivities ε as functions of the imaginary frequency ζ , where thecrossing occurs in the optical region, results in a switchfrom attractive to repulsive contributions to the Lifshitzforce.(ii) The need to include retardation effects in the for-malism: This may appear surprising, as retardation ef-fects due to the finite speed of light c are usually relatedto cases where the gap widths are large.(iii) The dominant role played by the zero frequencyMatsubara term n = 0 , which is a direct consequence ofthe aforementioned two factors: This may also be some-what unexpected, in view of the circumstance that the n = 0 term is usually taken to be important only inthe limits of large separation distance L at a moderatetemperature T , or high temperature at moderate separa-tion. (Observe that in the special case of a nondispersivemedium the single nondimensional parameter of impor-tance in the Lifshitz sum-integral is Lk B T / (cid:126) c .)The need to include all these effects stems of coursefrom the complicated Lifshitz sum-integral, when the dis-persive properties of the material components are ac-counted for accurately. In a three-layer planar system,where medium is interacting with medium acrossmedium , the system tries to minimize the interac-tion energy, which manifests as a force of attraction if [ ε ( iζ ) − ε ( iζ )][ ε ( iζ ) − ε ( iζ )] > and a force of repul-sion for [ ε ( iζ ) − ε ( iζ )][ ε ( iζ ) − ε ( iζ )] < . These condi-tions for attraction and repulsion must hold over a widefrequency range because they occur within the Lifshitzsum-integral. The plausibility of the repulsive Lifshitzforce between two dielectric objects with an interveningmedium of suitable dielectric permittivity was first dis-cussed by Dzyaloshinskii et al. [11] and has been observedexperimentally [17–21]. Earlier experimental and theo-retical studies are comprehensively discussed in Ref. [22].Elbaum and Schick observed that the difference betweenthe dielectric permittivities of ice and water changes signat the transition frequency ( ζ a ≈ . × rad/s), asshown in Fig. 2 [23]. Thus, the contribution to the Lif-shitz force, above and below the transition frequency ζ a ,is attractive and repulsive in nature, respectively. Fur-thermore, the difference between the dielectric permit-tivities of ice and water changes sign again at frequencies . . ζ a ζ (rad/s) ε ( i ζ ) van Zwol et al. , data set 1van Zwol et al. , data set 2GrabbeMalyi et al. , V=44.53 ˚A Malyi et al. , V=68.82 ˚A Malyi et al. , V=141.87 ˚A WaterIce
Figure 2. (Color online) Permittivity as a function of fre-quency for ice, water, and different silica materials. The staticvalues ε (0) for ice and water are . and . , respectively,using data from Elbaum and Schick [23]. For different SiO materials, the static values are . , . , and . using datafrom Malyi et al. [24] for volumes . , . , and . SÅ , respectively (here extended to include phonon contribu-tions), . from Grabbe [25], and . from data set 1 anddata set 2 of van Zwol and Palasantzas [26]. The transitionfrequency, ζ a ≈ . × rad/s, is where the permittivitiesof ice and water cross in the optical frequency region. lower than the first Matsubara frequency, thus affectingthe overall behavior of the Lifshitz force. Elbaum andSchick showed that these attractive and repulsive con-tributions for the ice-water-vapor system, at the triplepoint, lead to the formation of a thin layer of water atthe interface of ice and vapor [23]. The scale for the thick-ness of the layer of water is set by the transition distance c/ζ a . Most often it is argued that the retardation effectscan be neglected if the distance is less than a few tens ofnanometers. However, several studies [23, 26–29] high-light the importance of including the retardation effecteven at the separation distances of less than nm.We investigate if a thin layer of ice at the interface ofsilica and water will grow (freeze) or vanish (melt), nearthe triple point of water, assisted exclusively by the Lif-shitz interaction. In Ref. [30], Elbaum and Schick findthat a thin sheet of ice does not grow at the water-vaporinterface. In contrast, we report that the Lifshitz forcedoes assist ice growth at the silica-water interface. Thethickness of the ice layer formed at the silica-water inter-face varies with the permittivities of the silica substrate.(In Ref. [31], Dash et al. thoroughly reviewed a relatedphenomenon of the premelting of ice, which was also con-sidered by some of us in Ref. [32] where we showed thatit is essential to have a vapor layer between ice and asilica surface to have premelting of the ice.)To study ice growth at the silica-water interface, weconsider a model system with a planar silica surface inter-acting with water across a thin planar ice film of thickness L , as illustrated in Fig. 1. The ice sheet thicknesses thatwe discuss are typically in the range 1–8 nm. Recently,Schlaich et al. [33] showed that the dielectric functionsfor films thicker than 1 nm approached their bulk values.Thus, to predict trends, it should be sufficient to usebulk dielectric functions for the thin ice layer. The Lif-shitz interaction free energy per unit area F is expressedas a sum of Matsubara frequencies, ζ n = 2 πn/ (cid:126) β [11], F ( L ) = ∞ (cid:88) n =0 (cid:48) g ( L, iζ n ) , β = 1 k B T , (1)where g ( L, iζ n ) obtains contributions from the transverseelectric (TE) and the transverse magnetic (TM) modes, g ( L, iζ n ) = 1 β (cid:90) d k (2 π ) (cid:8) ln (cid:2) − e − γ L r TE r TE (cid:3) + ln (cid:2) − e − γ L r TM r TM (cid:3) (cid:9) . (2)Here, γ i = (cid:113) k + ( ζ n /c ) ε i , k is the magnitude of thewave vector parallel to the surface, and the prime on thesummation sign indicates that the n = 0 term should bedivided by 2. We have used the notations r TE ij = γ i − γ j γ i + γ j and r TM ij = ε j γ i − ε i γ j ε j γ i + ε i γ j (3)for the TE and TM mode reflection coefficients.We use dielectric functions for different silica, eachwith a specific nanoporosity, or average volume ( V ) perSiO unit, computed directly from first-principles calcu-lations, as reported in our previous work [24]. However,since the phonon contribution to the dielectric functionat imaginary frequencies can have a noticeable impacton the Lifshitz forces, we model the phonon parts ofthe dielectric functions using the single-phonon Lorentzmodel and the Kramers-Heisenberg equation [34]. Here,the longitudinal frequency ( ζ LO = 0 . eV), taken tobe the same for all considered systems, is determinedfrom the fitting of the multiphonon contribution to thedielectric function of quartz. The longitudinal and trans-verse optical frequencies for quartz are taken directlyfrom the experimental data [35]. At the same time,the single-phonon transverse frequency ζ TO is computedfrom the Lyddane-Sachs-Teller equation [36] using thefitted ζ LO and dielectric constants reported in our previ-ous work [24]. We also use parametrized model dielectricfunctions for different silica materials based on the opti-cal data and the Kramers-Kronig relationship given byGrabbe [25] and two separate data sets by van Zwol andPalasantzas [26] for comparison. We take the dielectricfunctions of ice and water at T = 273 . K from Elbaumand Schick [23]. Figure 2 shows the plots of dielectricfunctions for ice, water, and different silica materials. ζ n ( × rad/s) g ( L , i ζ n )( n J / m ) GrabbeMalyi et al. , V=44.53 ˚ A Malyi et al. , V=68.82 ˚ A Malyi et al. , V=141.87 ˚ A van Zwol et al. , data set 1van Zwol et al. , data set 2 ε water ε ice L ε silica ζ a Figure 3. (Color online) Spectral function g ( L, iζ n ) as a func-tion of Matsubara frequency ( ζ n ) for a silica-ice-water sys-tem with L = 2 . nm thick ice film. We compare the resultusing different silica dielectric functions presented in Fig. 2.The zero frequency contributions for various silica materials(not shown in the figure) are at least one order of magnitudehigher than the contributions from other Matsubara frequen-cies ( ≈ − nJ/m ). All the curves vanish at the same pointcorresponding to the transition frequency ζ a ≈ . × rad/s. In Fig. 3 we plot the spectral function g ( L, iζ n ) inEq. (2), for different silica-ice-water systems, at L = 2 . nm. The total Lifshitz energy is the area under thecurve(s), getting positive contributions from the posi-tive area and negative contributions from the negativearea. The cancellation between these contributions re-sults in a dominant role for the n = 0 Matsubara term.In the symmetric systems involving water ( ε = ε ) thelarge static dielectric permittivity of water compared tothe interacting media causes an increase in the factor r TM (0) r TM (0) ≈ . . This enhances significantly thecontribution of the n = 0 term to the total interactionenergy [15]. By contrast, in our asymmetric silica-ice-water system the above factor is approximately . dueto very similar values of the static dielectric permittivi-ties of ice and water. The contribution of the n = 0 termis therefore not enhanced here.We nevertheless find that the n = 0 Matsubara termis crucial for all separation distances, as shown in Fig. 4.It is evident from the plot that if we ignore the retar-dation effect, then there will be a complete freezing ofthe water, which, however, is not a natural phenomenon.The contribution to the Lifshitz energy from the n = 0 term is always attractive and considerably influences theequilibrium thickness as well as the stability of the icesheet as compared to the contributions from the n > .This conclusion is true for most materials with a low di-electric constant that can serve as nucleation sites for ice Ice fi lm thickness (nm) E n e r g y ( n J / m ) total non-retarded n > n = 0 Figure 4. (Color online) Contributions to the Lifshitz freeenergy per unit area for a V = 68 . Å ) system as a functionof ice film thickness. Four different curves are shown: thetotal nonretarded energy, the contributions from the n > term to the retarded energy, the total retarded energy, andthe contribution from the n = 0 term alone. formation but not for metals, where the n = 0 term givesa repulsive contribution.An estimate for the stable thickness of ice formed atthe interface of the silica-water system is obtained [29]by replacing the two exponentials in Eq. (2) with a stepfunction, e x ∼ θ ( x ) . This corresponds to γ L ≈ ,which leads to L ≈ c/ ζ a (cid:112) ε ( ζ a ) = 7 . nm. This esti-mate is similar to the equilibrium thicknesses of the icesheets for the broad range of the silica-water interfacescalculated using the complete Lifshitz energy of Eq. (1),shown in Table I. This stable thickness corresponds to anextremum in the plots of the total Lifshitz energy versusthe separation distance L in Fig. 5. The last columnin Table I shows the relative contribution of the n = 0 term with respect to the total energy at the equilibriumthickness. It is clear that the contribution from the n = 0 term is dominant in most cases, even at the small separa-tion distances, and even exceeds the contribution comingfrom the n > terms in some cases.Typically for the Casimir interaction between twoatoms, retardation effects become relevant for distanceregimes set by the cube root of the polarizability of theatoms, which serves as the scale for the retardation ef-fects. In our system, the characteristic frequency is thetransition frequency ζ a , which sets the scale for retarda-tion to be nm. This includes the speed of light in theintermediate medium.We summarize our results for ice formation near silicasurfaces in Fig. 5 and Table I. We find that the systemshows the behavior of the vapor-ice-water interface ofRef. [30], i.e., the intermediate layer vanishes, for very Ice fi lm thickness (nm) E n e r g y ( n J / m ) GrabbeMalyi et al. , V=44.53 ˚ A Malyi et al. , V=68.82 ˚ A Malyi et al. , V=141.87 ˚ A van Zwol et al. , data set 1van Zwol et al. , data set 2 Figure 5. (Color online) Lifshitz free energy per unit area forsilica-ice-water systems as a function of the ice film thicknessusing different silica dielectric functions presented in Fig. 2.Volume (Å ) Ice film thickness (nm) F eqn=0 /F eq n = 0 term and the total retarded Lif-shitz energy at the equilibrium ice film thickness. The plotsfor volumes 35.68 and 106.39 Å are not shown in Figs. 3 and5. high nanoporosity (large V for the SiO material). Thespectral function g ( L, iζ n ) in this case is reversed (seeFig. 3). In this limit when the substrate behaves moreas a vapor, there is no ice growth due to Lifshitz forces,as predicted by Elbaum and Schick [30]. For these cases,due to the attractive n = 0 contribution, there is a globalenergy maximum around L =4-5 nm and a local very weakenergy minimum around L =1–2 µ m. However, for a largerange of different silica materials, we predict a surfacespecific ice growth near the silica-water interface. Thetransition point between a stabilized thin ice layer anddestabilized ice growth is apparent from the dielectricspectrum of nanoporous silica, seen in Fig. 2. The stablethin layer is lost when the silica porosity is high enoughto cause its dielectric function to remain below that ofice and water.The study of ice formation at a silica interface has sig-nificant applied value as the model system for how waterfreezes on glass, rocks, and soil surfaces. Quasiliquid lay-ers are observed to form on solid-ice interfaces, dependingon the surface density and roughness [37–41]. Optical re-flection measurements have demonstrated the existenceof up to a few tens of nanometer thick premelted watersheets on ice crystal surfaces [42–45]. Ice in contact withsilica has been found to have a 5–6 nm thick quasiliquidlayer on the surface with a density similar to high-densityamorphous ice [37]. Several measurements have been car-ried out aiming at an understanding of the structure ofthe ice surface [46–49]. For a thorough review on the pre-melting of the ice, see Dash et al. in Ref. [31]. From ourstudy, we find that the Lifshitz force promotes freezingin the limit of low porosity, analogous to the reductionin the premelting layer observed with decreasing temper-ature [41]. In another experimental study, Bluhm andSalmeron [50] observe a 0.7 nm thin sheet of ice formedat the mica-water interface. We obtain a thickness 2.7nm for ice formation on mica using the above techniqueswith the dielectric permittivity of mica from Ref. [51].In real systems, optical properties, surface charges,surface roughness [3], the density of the material, grav-ity [28, 52], ions [40, 53, 54], the presence of gas layers onice premelting in pores [32] and so on influence the totalenergy of the system. It is an advantage of the theorythat different properties can be analyzed separately.In summary, the investigations of ice growth, due tothe Lifshitz interaction, near different materials requirea detailed knowledge of the dielectric functions for a largerange of frequencies. The zero frequency term, althoughof fundamental interest in its own right, can under spe-cific circumstances also play a major role in determiningthe stability and thickness of a thin layer near surfaces atmuch shorter distances than one would normally expect.Elbaum and Schick observed that the Lifshitz interactionis not sufficient, by itself, to promote ice growth at thewater-vapor surface [30]. In contrast, we predict a growthof nanosized ice films driven by the Lifshitz interactionat certain silica interfaces in ice-cold water. We suggestthat it should be possible to measure them, perhaps withthe use of already available experimental techniques [55].We acknowledge support from the Research Council ofNorway (Projects 221469 and 250346). We also acknowl-edge access to high-performance computing resources viaSNIC and NOTUR. ∗ [email protected] † [email protected] ‡ [email protected][1] A. Arbe, P. Malo de Molina, F. Alvarez, B. Frick, and J.Colmenero, Phys. Rev. Lett. , 185501 (2016).[2] M. J. Gillan, D. Alfe, and A. Michaelides, J. Chem. Phys. , 130901 (2016).[3] J. Benet, P. Llombart, E. Sanz, and L. G. MacDowell,Phys. Rev. Lett. , 096101 (2016).[4] A. Lintunen, T. Hölttä, and M. Kulmala, Sci. Rep. ,2031 (2013).[5] R. Pandey, et al. , Sci. Adv. , e1501630 (2016).[6] B. J. Murray, D. O’Sullivan, J. D. Atkinson, and M. E.Webb, Chem. Soc. Rev. , 6519 (2012).[7] C. Hoose and O. 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