Supporting Information: Molecular Insight into. the Slipperiness of Ice

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1 Supporting Information: Molecular Insight into the Slipperiness of Ice Bart Weber,,, Yuki Nagata,, Stefania Ketzetzi, Fujie Tang, Wilbert J. Smit, Huib J. Bakker, Ellen H. G. Backus, Mischa Bonn, and Daniel Bonn, Van der Waals-Zeeman Institute, IoP, University of Amsterdam, Science Park 94, 198XH Amsterdam, The Netherlands Current address: Advanced Research Center for Nanolithography (ARCNL), Science Park 11, 198 XG Amsterdam, Netherlands. Max Planck Institute for Polymer Research, Ackermannweg 1, Mainz, Germany AMOLF, Science Park 14, 198 XG Amsterdam, The Netherlands These authors contributed equally 1

2 .15 Steel on Glass 88 MPa.1.5 F (N) Friction Force F (N) A (1-9 m 2 ) Steel on Ice 1.2 MPa Contact Area A (m 2 ) 1-7 Supplementary Figure 1: Interfacial shear stress. Using the effective elastic modulus E obtained in the indentation experiments, the Hertzian contact area A between the sphere and the ice can be calculated: A = π ( 3RN 3. The friction force obtained at low temperatures ( 16 C data from Supplementary Figure 2) is proportional to A with proportionality constant 1.2 MPa. We replace the ice with float glass and obtain a proportionality constant of 88 MPa between friction force and Hertzian contact area using the glass modulus Eglass = 64 GPa, this corresponds to a friction coefficient of.6 at a normal force of.1 N. 4E ) 2 Friction on different ice surfaces The ice is cooled by a thermostat bath for temperatures down to 25 C and with liquid nitrogen for temperatures down to 1 C. The surface temperature of the ice is measured by a thermocouple wire that is inserted into the setup and in contact with the ice surface. During the sliding, the normal force is varied between and 5 N, the average friction coefficient obtained at these normal forces is plotted; we ensure that the sampling of normal forces in this range is the same for all measurements. The temperature dependence of the friction coefficient was measured on both polycrystalline and monocrystalline ice cut in the basal plane. The agreement between both curves (Supplementary Figure 5) confirms that grain boundary melting does not play a role in the frictional behavior. On heavy water ice, the temperature dependence of the friction coefficient is like that on normal water ice but 2

3 .7 T = -16 C T = -2 C Rescaled Friction Force F.6.5 F N2/3 F N F N3/ T = -1 C Rescaled Normal Force N Supplementary Figure 2: The relation between frictional and normal force. The normal force dependence of the friction force at different temperatures is shown. Inset: the ice surface after sliding at low (left top) and high (right bottom) temperatures..8 Glass on Ice Friction Coefficient (-) Temperature ( C) Supplementary Figure 3: Glass on ice friction. Experimental protocol is identical to that used for the steel sphere in the main text. The steel sphere is replaced with a glass precision sphere of the same diameter (4.8 mm). The Arrhenius fit from the main text (green line) matches the data. 3

4 (a) Im χ (2) (a.u.) DAA Exp. DA Exp. -28 C -88 C.2 DAA DA IR Wavenumber λ -1 (cm -1 ) (b) Amplitude A (a.u.) Experiment Simulation DAA DA Temperature T ( C) Supplementary Figure 4: Sum-Frequency Generation Spectroscopy. (a) Experimentally measured SFG spectra of the free O-H stretch mode at the ice-basal face for 88 C and 28 C. DAA and DA molecules (see text) are fitted to the experimental data using two Gaussians centered at 3695 cm 1 (red) and 3715 cm 1 (blue), respectively. (b) Amplitude of the DAA and DA contributions as a function of temperature for both experimental and simulated SFG spectra. 4

5 shifted by roughly 4 C, the melting point difference between normal and heavy water. Friction Coefficient (-) Ice Monocrystalline Ice Heavy Ice Temperature T ( C) Supplementary Figure 5: The temperature dependence of the friction coefficient on different ice surfaces. Steel sphere-on-ice sliding tests were performed, as in the main text, at a sliding speed of.38 mm s 1 using a 2.38 mm radius stainless steel sphere and normal forces in the range -5 N. Polycrystalline (black circles) and monocrystalline (red triangles) ice have a similar temperature dependence of the friction coefficient. Heavy water ice (blue squares) shows a temperature shift in the frictional behavior corresponding to the difference in melting point between normal and heavy water. Ploughing force The ploughing cross section A p, illustrated in Supplementary Figure 6 is given by 1 A p = 2 3 cd, with c the contact arc length and d the depth to which the sphere penetrates the ice, for c << R. We approximate c 2r and from the right-angled triangle with sides R, r and R d we obtain d r2 2R both for small contacts c << R. Plugging these approximations into the formula for the ploughing cross section we obtain A p = 2r3. The ploughing cross 3R section multiplied with the penetration hardness gives the ploughing force. Assuming that the contact area during sliding is a half circle and that the contact pressure is equal to the 5

6 penetration hardness we obtain a ploughing force: F p = 4N πR πp h N 2r R d Supplementary Figure 6: Ploughing geometry. The ploughing cross section A p for a sphere of radius R pushed into an ice surface with force N. Indentation experiments We mount the 2.38 mm steel sphere that was used in the sliding experiments into a tensile tester to measure force vs. deformation curves for the sphere on ice contact. The sphere is loaded onto the ice surface at a constant rate of 2 N min 1 up to a maximum force of 1 N after which the load is removed at the same rate. Meanwhile, the tensile tester measures the deformation with respect to the point of initial contact and plots this vs. the contact force (Supplementary Figure 7). We see that at low temperatures the contact is reversible and therefore elastic. The Hertzian relation between penetration depth d and normal force ( 3N 3 N: d =, with R = 2.38 mm the radius of the sphere is fitted to the data to obtain 4 RE ) 2 E =.84 GPa, the effective ice modulus (Supplementary Figure 7). As the temperature is increased, the plastic, non-reversible contribution to the deformation increases. At 1 C, the contact is fully plastic; upon retracting the sphere, none of the ice deformation is reversed. In plastic contact, the contact area is given by πr 2 = N p h. Since r 2 = 2Rd for d << R, we can write d = N 2πRp h and it follows that the slope d N in the plastic loading curve is inversely 6

7 proportional to the penetration hardness: d N = 1 2πRp h. The penetration hardness obtained from the plastic loading curves is shown in Supplementary Figure Penetration Depth d (m) T = -1 C T = -12 C E* =.84 GPa Normal Force N (N) Supplementary Figure 7: Indentation of the ice surface with a stainless steel sphere. At varying temperatures, the ice surface is indented with the stainless steel sphere that was used for the friction experiments. At temperatures below 5 C we observe elastic behavior; the ice surface is reversibly deformed according to the Hertzian relation between penetration ( ) 2 3N depth d and normal force N: d = 4 3, with R = 2.38 mm the radius of the sphere and RE E =.84 GPa the effective elastic modulus of the ice, in good agreement with its literature value. At temperatures above 5 C, the contact becomes plastic and the ice does not push back when the sphere is lifted. At 1 C, the penetration depth d is an order of magnitude larger than it was at temperatures below 5 C. Frictional heating We calculate the temperature at the sphere on ice interface using the theory of heat diffusion, assuming that all frictional heat is absorbed by the sphere, there is no lateral heat diffusion and that the sphere is a semi-infinite solid (Supplementary Figure 9). In reality, the ice also absorbs heat and there is significant lateral heat diffusion while the heat capacity of the sphere is so large compared to the frictional heat flux that at the experimental time scales the sphere can be considered semi-infinite. The calculated surface temperatures therefore must 7

8 (Pa) Penetration Hardness p h Surface Temperature T ( C) Supplementary Figure 8: Ice hardness. The ice penetration hardness obtained from the d relation between penetration depth and contact force: = 1 N 2πRp h. The red curve p h = 11.1 MPa T ( C) is used as input for the ploughing force model presented in the main text. be larger than the actual surface temperature in the experiment. Nonetheless, we see that the sphere surface temperature is hardly increased by the frictional heat, even at the highest sliding speeds (Supplementary Figure 9). The sphere follows a circular sliding path over the ice surface with radius 9 mm. Because the ice surface is not perfectly aligned with the sphere rotation plane, the sphere is not in contact with the ice during the full rotation (except for experiments conducted in the plastic regime where the sphere penetrates the ice deep enough to remain in contact during the entire rotation). As a result, the sliding distance for each rotation typically does not exceed 1 cm, strongly limiting the effect of frictional heating. Molecular dynamics simulation We conducted molecular dynamics (MD) at the basal face of ice (Ih) in contact with vacuum. The MD simulation of the ice-air interface has been conducted elsewhere. 2 Therefore, here we explain the MD simulation protocols briefly. We used the POLI2VS force field model 8

9 SurfaceTemperature T ( C ) e-6 m/s 3.8e-5 m/s 3.8e-4 m/s 3.8e-3 m/s 9.5e-2 m/s Sliding Distance s (m) Supplementary Figure 9: Frictional heating. The surface temperature of a semi-infinite steel solid warmed up by frictional heat. During sliding a heat flux F v is generated by contact area A A due to friction force F at sliding speed v. If this heat flux is absorbed by the steel sphere, which we model as a semi-infinite solid, the temperature at the sliding interface as a function of time t is given by: T (t) = T + 2F v t, with, T A πρcκ = 5 C the initial temperature, ρ = kg m 3 the density of stainless steel, c = 48 J kg 1 K 1 the specific heat of stainless steel, and κ = 27 W m 1 K 1 the heat conductivity of stainless steel. We plot the relation between this surface temperature and the total sliding distance for sliding speeds ranging from m/s, using a friction force F = µ 1 N =.3 N and contact area A = m 2, corresponding to the Hertzian contact area at 1 N normal force. 9

10 for water molecules. 3 This model predicts a melting temperature of 265 ± 5 K and a surface tension of 6.6 mn/m without the long-range correction. 4 These are in reasonable agreement with experiment. In the MD simulation of the ice-vacuum interface, the simulation cell contained 1344 water molecules. Since the density of ice changes with temperature, 5 we set the simulation cell size to Å Å 6 Å at 13 C, Å Å 6 Å at 88 C, Å Å 6 Å at 73 C, Å Å 6 Å at 58 C, Å Å 6 Å at 43 C and Å Å 6 Å at 28 C, where 112 water molecules formed a basal face bilayer of ice. The system thus consisted of 12 bilayers (Supplementary Figure 1). Periodic boundary conditions were used. The electrostatic forces were calculated with the Ewald method. The time step for the equations of motion was set to.4 fs. The reversible reference system propagator algorithm (RESPA) method was employed to integrate the equation of motion. 6 We obtained over 7 ns MD trajectory for each temperature, which was used for the analysis. SFG spectra calculation The resonant part of the SFG signal can be calculated via the truncating response function formalism: 7 τ χ res,(2) xxz (ω, r t ) = iq(ω) R xxz(t, (2) r t )f(t)e iωt dt (1) where the z-axis is the surface normal and the xy-plane is the ice surface. The χ res,(2) xxz component corresponds to the SFG signal at the SPP-polarization configuration of the sumfrequency, visible, and infrared beams. Q(ω) = β ω/(1 e β ω ) (2) is the harmonic quantum correction factor. 8 β = 1 k B T, with k B the Boltzmann constant and T the temperature in Kelvin, f(t) is the Hann window function, 1

11 Supplementary Figure 1: Ice bilayers. Density profile of the basal ice slab simulated at 17K, 2K, and 23K. 11

12 cos 2 (πt/2π) for < t < τ, f(t) = for t > τ (3) where we used τ = 1.24 ps to compute the SFG spectra. The time correlation function R (2) xxz(t, r t ) is given by R (2) xxz(t, r t ) = gsc(z 3 i ())µ z,i ()α xx,i (t), (4) i where µ a,i (t)(α ab,i (t)) is the a component of the molecular dipole moment (the ab component of the molecular polarizability) of water molecule i at time t. Here, we considered only the autocorrelation function and neglected the cross-correlation terms such as µ z,i ()α xx,j (t) for i j, because the free O-H stretch SFG signal at 37 cm 1 is dominated by the autocorrelation terms. Since the two ice surfaces are identical in the slab model, the dipole moments of the two interfaces cancel out. In order to avoid this cancellation, we multiplied the dipole moment by the screening function if z z c1 g sc (z) = sign(z) cos 2 ( π( z z c2) ) if z 2(z c1 z c2 ) c1 < z z c2, 1 if z c2 < z (5) where z is the z-coordinate of the oxygen atom of a water molecule. Here, we set z c1 = 7 Å and z c2 = 8 Å, where the origin point is set to the center of mass of the system. The range of z c1 < z < z c2 is located in-between the bulk ice bilayers. To compensate the nuclear quantum effects on the frequency shift of the O-H stretch mode, we multiplied the factor of Furthermore, since the classical correlation function cannot predict properly the temperature dependence of the spectral amplitude, we scaled the maximum height of the simulated 37 cm 1 peak to that of the experimental data. 12

13 Definition of free O-H group of water When a H-atom does not form any hydrogen bond, its O-H group is categorized as a free O-H group. To identify the water molecules that have free O-H group(s), we used a geometrybased definition. In this definition, we categorized the hydrogen atoms of water into the nonhydrogen bonded sub-ensemble and the hydrogen-bonded sub-ensemble. When the possible water dimer (i, j) conformation has an intermolecular O i... O j distance (R) less than 4 Å and the O i -H i... O j angles (φ) are larger than 135, we defined that the i-j water dimer forms the hydrogen bond through the H i atom (Supplementary Figure 11). These criteria for R and φ reproduce a positive 37 cm 1 SFG spectral feature of the water air and ice air interface. A more detailed discussion is given in reference. 9 Supplementary Figure 11: Schematic of water dimer. R and φ are used for defining the free O-H groups. References (1) Harris, J.; Stöcker, H. Handbook of Mathematics and Computational Science; Springer New York, (2) Smit, W. J.; Tang, F.; Sánchez, M. A.; Backus, E. H. G.; Xu, L.; Hasegawa, T.; Bonn, M.; 13

14 Bakker, H. J.; Nagata, Y. Excess hydrogen bond at the ice-vapor interface around 2 K. Phys. Rev. Lett. 217, 119, (3) Hasegawa, T.; Tanimura, Y. A polarizable water model for intramolecular and intermolecular vibrational spectroscopies. J. Phys. Chem. B 211, 115, (4) Nagata, Y.; Hasegawa, T.; Backus, E. H. G.; Usui, K.; Yoshimune, S.; Ohtoc, T.; Bonn, M. The surface roughness, but not the water molecular orientation varies with temperature at the water-air interface. Phys. Chem. Chem. Phys. 215, 17, (5) Feistel, R.; Wagner, W. A new equation of state for H 2 O ice Ih. J. Phys. Chem. Ref. Data 26, 35, 121. (6) Tuckerman, M.; Berne, B. J.; Martyna, G. J. Reversible multiple time scale molecular dynamics. J. Chem. Phys. 1992, 97, (7) Nagata, Y.; Mukamel, S. Vibrational sum-frequency generation spectroscopy at the water/lipid interface: Molecular dynamics simulation study. J. Am. Chem. Soc. 21, 132, (8) Berens, P. H.; Wilson, K. R. Molecular dynamics and spectra. I. Diatomic rotation and vibration. J. Chem. Phys. 1981, 74, (9) Tang, F.; Ohto, T.; Hasegawa, T.; Xie, W. J.; Xu, L.; Bonn, M.; Nagata, Y. Definition of free OH groups of water at the airwater interface. J. Chem. Theory Comput. 218, 14,