Fluctuation spectroscopy of surface melting of ice without, and with impurities
aa r X i v : . [ c ond - m a t . o t h e r] N ov Fluctuation spectroscopy of surface melting of ice without, and with impurities
Takahisa Mitsui ∗ and Kenichiro Aoki † Research and Education Center for Natural Sciences and Dept. of Physics,Hiyoshi, Keio University, Yokohama 223–8521, Japan
Water is ubiquitous, and surface properties of ice has been studied for some time, due to itsimportance. Liquid-like layer (LLL) is known to exist on ice, below the melting point. We usesurface thermal fluctuation spectroscopy to study LLL, including its thickness, for pure ice, and forice with impurities. We find that the properties of LLL are experimentally those of liquid water,with thickness much smaller than previous results. We also find that impurities cause LLL to bethicker, and be quite inhomogeneous, with properties depending on the dopant.
The importance of understanding the surface proper-ties of ice in air can not be understated, which is crucialfor the clarification of the melting, freezing process it-self, as well being important to a broad range fields inscience [1–3]. Theoretically, the existence of LLL (alsocalled “quasi-liquid layer”, or QLL) has been discussedsince 19th century, and its properties, including the ex-istence, has been studied from various points of view— thermodynamic arguments[4–6], and more recently,state of the art molecular dynamics simulations havebeen employed[7–15]. Due to its importance, and itsexperimental difficulty, surface melting of ice has beenanalyzed experimentally using a multitude of methods,such as ellipsometry[16–18], X-ray diffraction[19], pro-ton backscattering[20], photoelectron spectroscopy[21],atomic force microscopy[22–24], and surface sum-frequency generation (SFG) spectroscopy[25–27]. For anumber of reasons, including the thinness of LLL, whichcan be nm order or smaller, the study of LLL remainsto be an experimentally challenging problem. This diffi-culty is evidenced by the thickness measurements of LLLof ice, which vary by orders of magnitude[3, 18, 28].In this work, we optically measure the surface ther-mal fluctuations of LLL on ice, in air. By measuring thethermal motions of the molecules directly, in additionto clearly differentiating the liquid and solid phases, theproperties of the material become apparent[29–34]. Sucha method, while not previously applied to surface melt-ing, has proven to be effective in understanding the prop-erties of various liquids, complex fluids, and viscoelasticmaterials[35–40]. This experimental method for measur-ing the properties of LLL, distinct from previous meth-ods, enables us to obtain a different perspective on LLL.The surface thermal fluctuations, which are spontaneous,reveal the properties of the material underneath, provid-ing information whether LLL has the properties of waterin the bulk, in addition to the behavior of its thickness.The importance of thermal fluctuations of the LLL wererecognized, and their properties were recently analyzed ∗ E–mail: [email protected] . † E–mail: [email protected] . using simulations[41]. There, it was found that the sur-face fluctuations are similar to those of water–vapor in-terfaces, with the fluctuation spectra strongly affected bythe thickness of LLL, consistent with our results. Also, ithas been found using SFG spectroscopy[25–27], that thesurface of LLL behaves similarly to bulk supercooled wa-ter, seemingly consistent with our results. We find thatLLL is much thinner than most of the previous exper-imental results, and that the additions of impurities atppm levels thicken LLL. Furthermore, by scanning thesurface at µ m level resolution, we directly observe thatimpurities cause inhomogeneities, with their propertiesdepending on the dopant. ADCADC MultiplierAveragerFFTFFTComputerMicroscopeObjective DEPD1DEPD2
Laser 1532 nmLaser 2491 nm
FR PBS DM1DM2 x-y ScannerPeltierIce
FIG. 1: Experimental setup: Linearly polarized laser lightwith wavelengths 532, 491 nm are shone on the sample ice sur-face, with powers 340, 270 µ W at the surface. The reflectedlight is directed to two dual-element photodiodes (DEPD1,2)corresponding to Laser 1,2. The differences in the light beampowers in the DEPDs are digitized using analog-to-digitalconverters (ADC), Fourier transformed (FFT), and the av-eraged correlation is computed (averager). Faraday rotator(FR) is used to rotate the polarization of the light by π/ The experimental setup is shown in Fig. 1: Light isshone on the surface of ice, with LLL expected to be onit. The reflected light is detected by the dual-elementphotodiode (DEPD). The surface acts as a partial mir-ror, and the two elements in DEPD produce the samephotocurrent, if the surface is not fluctuating. The sur-face fluctuates thermally, and produces fluctuations inthe photocurrent difference, whose power spectrum isthe inclination fluctuation spectrum of the surface, upto a constant. The measurement is performed using twolight sources with different wavelengths, in order to usecorrelation analysis to statistically reduce the extrane-ous noise[37], to orders of magnitude below the shot-noise level, often referred to as the “Standard QuantumLimit”[42]. The beam radius (waist) at the sample, w ,is 1 . µ m, and the sample can be moved horizontally intwo dimensions by 15 µ m in each direction, allowing usto scan the surface. Recently, LCM-DIM (laser confo-cal microscopy combined with differential contrast mi-croscopy) has been developed to optically observe icesurfaces[43, 44]. Our method measures the thickness ofLLL, which is not measurable in LCM-DIM. LCM-DIMhas been used to measure transient properties of surfacestructures, in contrast to our thermal equilibrium mea-surements, and while both methods are optical measure-ments, we find these methods complementary. Due tothe breadth of interest, importance of the surface melt-ing phenomena, and the difficulties in measuring theirproperties, various experimental approaches have beenbrought to bear on this problem. We believe that itis important to investigate the phenomena from differ-ent perspectives, both experimental and theoretical, toclarify the inner workings underlying them. One impor-tant character of our approach is that it makes opticalmeasurements of only the spontaneous surface fluctua-tions, with no external stimulation, and is hence min-imally invasive. The time-scales for our measurementsat relatively longer time scales, at least order of tensof seconds. The approach is suited for measurementsof thermal equilibrium properties, and not for observingtransient effects. The measurements made in this workare stable on the order of hours.The liquid sample to be frozen was put into the stain-less steel container (diameter 3.5 mm, depth 1.5 mm), andthermoelectrically cooled. Due to supercooling, the liq-uid froze at around − ◦ C, without, and with impurities.For pure water, and NaCl solution cases, the temperatureof the liquid was then raised to 0.07 ◦ C, and kept at thattemperature till the water layer on ice was about 0.5 mm,which took around 30 minutes. Next, the sample liquidtemperature was lowered gradually to − . ◦ C, takingover an hour. After this, the measurements were thentaken, at various temperatures. For the Volvic sample,the same procedure led to too much precipitation of min-erals, leading to unwanted noise in the measurement dueto the unevenness of the surface. Therefore, to reduce theprecipitation, the temperature of the sample was raisedto to − . ◦ C after freezing the supercooled Volvic water,and kept constant for over an hour, after which measure-ments at various temperatures were taken. Water samplewas purified using the EMD Millipore Purification Sys-tem (Merck Millipore, Germany).Our experimental setup measures the thermal fluctua- -18 -17 -16 -15 -14 -13 -12 -11
100 1000 10000 100000 1e+06 1e+07 S ( f ) [r a d / H z ] f [Hz] FIG. 2: Surface thermal fluctuation spectra of waterlayer of various thicknesses: Fluctuation spectra for su-percooled water (∆ T = 17 .
89 K, red), and LLL with(∆ T [K] , h [m]) = (0 . , . × − ) (green), (0 . , . × − ) (blue), (0 . , . × − ) (magenta), (0 . , . × − )(grey), (0 . , . × − ) (orange), (1 . , . × − ) (cyan),(24 . , . × − ) (yellow). Corresponding theoretical spec-tra for water with finite depth are also shown(black), andagree well with the measured spectra. tion spectra of the averaged inclination within the beamspot on the sample[37], S ( f ) = Z ∞ dk k e − w k / F ( k, f, h ) . (1)Here, f is the frequency, and F ( k, f, h ) is the spectralfunction of the thermal fluctuations of the fluid surface,and depends on LLL thickness, h [32]. The spectrumis uniquely determined by the bulk properties of water(density, surface tension, viscosity), beam size, h , andthe sample temperature. Gravitational effects are negli-gible at our sample size. Some examples of spectra forLLL, and water at various h, ∆ T = T m − T are shown inFig. 2. T is the temperature of the sample surface, and T m is the bulk melting temperature of ice. It can be seenthat the spectral shape depends strongly on h , and theexperimental spectra agree quite well with the theoreticalspectra of surface thermal fluctuations of water with fi-nite thickness. The known bulk properties of supercooledwater[45–48] were used to compute the theoretical spec-tra. We note that surface thermal fluctuations of solidice without LLL should not only be much smaller, butbehave as 1 /f [36, 37], which is incompatible with themeasurements. For LLL with h &
10 nm, the spectralshape and the magnitude are sensitive to the fluid prop-erties of LLL, density, surface tension, and viscosity, aswell as its thickness, in our experimental setting. In par-ticular, in all the spectra analyzed, the viscosity inferredfrom the spectrum is consistent with the known physi-cal properties of supercooled water. For h .
10 nm, thetheoretical spectrum behaves as S ( f ) ≃ π k B Tηw h f , (2)where η is the viscosity of (supercooled) water, k B is theBoltzmann constant, and T is the temperature. This1 /f behavior is indeed observed for thinner LLL inFig. 2. While the spectral magnitude is quite sensitive tothe thickness, the spectral shape is rather insensitive tothe properties of LLL. h [ µ m ] ∆ T [K] FIG. 3: The temperature dependence of the thickness of LLLfor pure ice: The three types of points ( (cid:13) , △ , ✷ ) correspondto three data sets taken with three different samples, on threedifferent days, which are seen to be consistent within error. The dependence of h for LLL of pure ice on the tem-perature is shown in Fig. 3. h was estimated fromthe surface thermal fluctuation spectra (Fig. 2): For h &
10 nm, theoretical spectral shape was fit to themeasurements, and for smaller h , the measured spec-trum was normalized using the shot-noise level. The re-lation between h and ∆ T has been studied previouslyby a number of authors, using various other methods,and h differs by orders of magnitude, depending on themethod used[3, 18, 28]. Our results extend over a muchwider range of temperatures than that previously coveredwith any one method. The results in Fig. 3 are consis-tent between measurements of different samples takenon different days, and also between measurements takenwith the temperatures in ascending, and descending se-quences. This strongly suggests that the results are equi-librium properties. Our results for h are smaller thanthose previously measured for ∆ T <
T >
10 K, except for photoelectronemission spectroscopy study of LLL of ice in pure wa-ter vapor[21]. Compared with this, our results for h areof the same order but slightly larger in this temperaturerange. On the theoretical side, some thermodynamic con-siderations predict much thicker[5], and also thinner[6]LLL. Recent molecular dynamics simulation results exist for ∆ T & ∼ h / ( ηf ) for thin LLL, larger vis-cosities lead to larger h values in our results (Fig. 3), sothat a dramatic rise in the viscosity for smaller h seemsunlikely. ∆ T =0.033 K, 0.010 K, 0.008 K, 0.007 K, 0.006 K x [ µ m ] y [ µ m ] h [ µ m ] FIG. 4: Spatial distribution of LLL thicknessof pure ice: h , at four temperatures, ∆ T =0 . , . , . , . , .
006 K. h is larger at lowertemperatures (larger ∆ T ). h distributions are seen to berelatively uniform, with slight variations. An important questions is whether LLL is homoge-neous: Our measurement system allows for scanning at µm level, since the light beam is focused, with w =1 . µ m. In Fig. 4, the dependence of the h on the surfacelocation is shown, for the surface of pure ice, at differ-ent temperatures. h values are seen to be reasonablyuniform in the scanned region, for LLL of pure ice. Forsmall ∆ T , h is strongly dependent on the temperature(Fig. 3), so some variation is visible. Molecular dynamicssimulations exist that suggest the transition from liquidto solid is not sharp for LLL[15, 41]. The properties wefound through the surface thermal fluctuations are con-sistent with a thin layer having the properties of bulkwater, as seen in Fig. 2. We expect this not to precludea gradual transition from the liquid-like to the solid-likestructure at the bottom, as long as waves with ampli-tudes larger than those at the liquid-gas interface do notpropagate in this transition layer. It would be interest-ing to investigate if more precise measurements, and fur-ther information can be extracted from surface thermalfluctuation measurements, especially when LLL is thin,around 1 nm or less. Thermodynamic arguments com-bined with microscopic simulation suggest, interestingly,that the ice-liquid interface has a roughness, at atomicscales[52]. However, since our measurements are aver-aged within the beam at the µ m order, such roughnessis not observable. ∆ T =0.95 K(a) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =7.9 K(b) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =14.9 K(c) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =19.9 K(d) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =27.9 K(e) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =2.4 K(f) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =3.9 K(g) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =7.9 K(h) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =9.9 K(i) x [ µ m ] y [ µ m ] h [ µ m ] ∆ T =14.9 K(j) x [ µ m ] y [ µ m ] h [ µ m ] FIG. 5: Spatial distribution of h for ice with NaCl ((a)—(e)),and Volvic ((f)—(j)), as the temperature is lowered, for thesame area. Theoretically, impurities can greatly affect the over-all thickness of LLL[53, 54], and are perhaps the majorcause of their large observed disparities[2]. To study theeffect of impurities, in Fig. 5, the spatial dependences of h are shown for frozen NaCl solution (10 ppm by weightbefore freezing), and Volvic (water with minerals roughly60 ppm by weight)[55] at various temperatures, as theyare being cooled. Such direct observations of the inhomo-geneities have been observed for the first time. We choseVolvic, which contains various minerals, as a model ofwater in a natural setting. There are clear qualitative dif-ferences from the properties of LLL of pure ice, and alsobetween LLL with different impurities. First, we see thatin both cases, LLL with h > . µ m exists for ∆ T > h distributions are quite in-homogeneous. There is also a distinct difference betweenthe effect of two impurities: A reasonably thick LLL ex-ists for frozen NaCl solution down to temperatures much lower than that for Volvic, which is not as inhomogeneousas the latter. For Volvic, channels of LLL form, whichhave the bulk properties of water, that grow narrower andshallower as the temperature lowers. Interestingly, veinlike structures have been observed in glaciers[2, 56]. Thecause for the distinct difference between ice with NaCland Volvic can perhaps be attributed to the difference inthe solubility of the impurities. Minerals within Volvicare not as soluble in water as NaCl, in general, probablyleading to more inhomogeneities and precipitation. Wehave scanned the surface of ice with glucose, and havefound that the behavior is similar to that of NaCl solu-tion. Results for ice with NaCl at various concentrations,and frozen Evian water[57], and are also consistent withthe above considerations. The concentration per unitarea of the NaCl solution in Fig. 5 can be estimated tobe 90 µ · mol/m . 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