Freezing of few nanometers water droplets

Water-ice transformation of few nm nanodroplets plays a critical role in nature including climate change, microphysics of clouds, survival mechanism of animals in cold environments, and a broad spectrum of technologies. In most of these scenarios, water-ice transformation occurs in a heterogenous mode where nanodroplets are in contact with another medium. Despite computational efforts, experimental probing of this transformation at few nm scales remains unresolved. Here, we report direct probing of water-ice transformation down to 2 nm scale and the length-scale dependence of transformation temperature through two independent metrologies. The transformation temperature shows a sharp length dependence in nanodroplets smaller than 10 nm and for 2 nm droplet, this temperature falls below the homogenous bulk nucleation limit. Contrary to nucleation on curved rigid solid surfaces, ice formation on soft interfaces (omnipresent in nature) can deform the interface leading to suppression of ice nucleation. For soft interfaces, ice nucleation temperature depends on surface modulus. Considering the interfacial deformation, the findings are in good agreement with predictions of classical nucleation theory. This understanding contributes to a greater knowledge of natural phenomena and rational design of anti-icing systems for aviation, wind energy and infrastructures and even cryopreservation systems.

ouds strongly depend on water rates, and on the distribution particles 1,2 . In particular, the from the liquid may affect n and hydrological fluxes in in nanoscopic ice particles tens of thousands of water ratory [5][6][7][8][9] , and the onset of ice ter clusters containing as few as and growth of ice within water exploring water behaviour in crystallization of ice can hardly s temperature range owing to es. However, reducing sample on events, and at the nanoscale water. Indeed, by suspending w thousand molecules in a er was supercooled to a This technique also allowed emperature range over which . molecular beam experiments, ssumed to be size independent. lid in bulk and micron size e justified in the nano metre ts showed that confinement at lk phase diagram and shift the ale molecular simulations to of ice nucleation rates within Figure 1 | Size dependence of ice nucleation rates in the mW water droplets at 230 K. The solid black squares denote the calculated ice nucleation rate within the mW droplets, the dash lines indicate the computed ice nucleation rate in the mW bulk water, the green circles denote the corrected ice nucleation rate by using the effective volume, and the purple diamonds represent the calculated nucleation rate in bulk liquid with the density matching that of the corresponding droplet. The effective volume of the droplet is defined as the total volume minus the surface-like volume (see Supplementary Note 3). The thick blue line represents the predicted nucleation rate based on our model (equation (3)). The statistical uncertainty of the computed nucleation rate is mainly due to the error in the calculated growth probability P(l n |l 0 ) (see Methods) that is attributed to both the variance of the binomial distribution of N, i.e., the number of configurations collected at each interface, and the landscape variance of

S8
Supplementary Figure 10. I-V curve across nanopores measured as a function of temperature for water nanodroplets in the pores of membranes in the sizes of (a) 80 nm (b) 40 nm and (c) 10 nm. The average of two temperatures between which the jump in resistance is observed is assigned as ice nucleation temperature (TN). Figure 11. I-V curves measured across the membrane when there is no water inside the pores in the membrane with pore diameter of 80 nm. There is no nonlinear jump in electrical resistance down to -32°C.      Figure 19. FTIR spectrum of nanodroplets in various pore dimensions when pores are filled with water without addition of oil around them. (a) For 150 nm nanodroplets, at the temperature of 7°C, the OH stretch peak is narrowed and red-shifted to wavenumber of ~3200 cm -1 . This red shift indicates the water-ice phase change and provides TN value at this length scale. (b) The red shift occurs at temperature of -7°C for 80 nm water droplets as well. (c) The red shift occurs at temperature of -7°C for 40 nm water droplets as well. (d) For 20 nm nanodroplets, at the temperature of -7°C, the OH stretch peak is narrowed and red-shifted to wavenumber of ~3200 cm -1 . This red shift indicates the water-ice phase change and provides TN value at this length scale. (e) The red shift occurs at temperature of -7 o C for 10 nm water droplets as well. (f) For 5 nm water droplets, the narrowing of OH stretch peak occurs at the temperature of -3 o C. (g) The TN value for 2 nm droplets becomes ~ 0 o C.  Transmittance (arb. units)

Supplementary
0°C 3400 3200 3000 Transmittance (arb. units)    The geometry and dimension of pores in the nanomembranes are probed with scanning probe microscopy (SPM) as shown in Supplementary Figure 1 ( 1) Where ∆ denotes the density difference between water and octane, is the gravitational acceleration and 4 is the radius of curvature at the drop apex, Supplementary Figure 3. The differential form of Young-Laplace equation in terms of arc length (s) is written as where the bar indicates dimensionless values scaled by 4 . The boundary conditions are S23 @ = 0: ̅ = 0; ̅ = 0 ; = 0 The input to this system of equations is or specifically 01 .
Once this value is given, one can solve this system of equations numerically. We determined numerically the shape of the droplet for a range of surface tension values and compared the calculated droplet shape with the one measured, Supplementary Figure 4. For each considered value of surface tension, we determined the coefficient of determination (R 2 ) between the calculated shape and measured shape. The surface tension with the highest value of R 2 provides gives us 01 through pendant droplet method, Supplementary Table 2. As a control experiment, we compared the measured interfacial tension for octane-water with the reported value in the literature which shows less than 5% error.

Supplementary Note 3. The existence of water nanodroplets in the pores and volume effect
The electrical conductance across nanopores of 80 nm is measured through the four-probe method.
If the pores are fully filled with octane, the electrical resistance is approximately 5.5 GΩ, while the introduction of nanodroplets in these pores drops the resistance to approximately 3 GΩ. Note that the existence of any blocked air bubble in the pore would result in the insulating characteristics of these pores.
We performed quartz crystal microbalance (QCM) measurement on the membranes before and after filling with water to determine the volume and length of water nanodroplets inside the pores. A piece of membrane was mounted on a gold chip before and after filling with water. This chip was placed in an electrochemical quartz crystal microbalance (Gamry's eQCM 10M). The weight of water inside the pores was measured based on the changes in the resonant frequency of the oscillating quartz crystal before and after filling with water. We also compared the measurements with the theoretical values calculated from the membrane properties provided by the vendor. The results are tabulated in Supplementary Table 3. The volume of water within the pores is 0.817 53 for 20 nm pores, 0.511 53 for 10 nm pores, and 1.312 53 for 5 nm pores.
To clarify the droplet volume effect, we should note that the droplets in these pores are elongated ellipsoid droplets with a smaller diameter in the range of few nm while the other diameter in tens of micrometer. The approximate volume of isolated water droplet in the pores are given in Supplementary Table 4.
We compared the volume of these droplets with the volume of water droplets reported by Li et al. 4 , Supplementary Figure 6. As shown, for water droplets higher than 6.1 nm, the ice nucleation rate is similar to bulk water. The volume of 6.1 nm droplet is 9.5 *10 -10 pL. All of the droplets studied in this work have volumes at least 2 orders of magnitude higher than this limit. Thus, the volume effect on ice nucleation rate is insignificant in this study.

Supplementary Note 4. The experimental setup for E-resistance metrology
The schematic of the experimental setup is shown in Supplementary Figure 7. In this metrology, the nanomembrane was placed between two reservoirs separated by a heat conducting wall. We measured the temperature of two reservoirs during the experiments to ensure isothermal

S24
conditions. The coordinates of the electrodes are fixed. Once the water nanodroplets are formed inside the pores, the temperature of the system in a quasi-static manner is reduced and the I-V curves across the pores are probed.
A 4-point probe consists of four electrical probes in a line, with equal spacing between each of the probes. Based on this method, a high impedance current source is used to supply current through the outer two probes; a voltmeter measures the voltage across the inner two probes (See Supplementary Figure 8). Due to the high impedance of voltmeter, no current flows through inner electrodes. In this case, the voltage drop is measured between two inner electrodes where wire resistances (Rw2 and Rw3) and contact resistances (Rc2 and Rc3) do not contribute in voltage measurement and it is just the resistance of sample (Rs2) which leads to the voltage drop (∆ ).
We explored one polarity where we applied current and measured the voltage. In fact, what is important here are non-linearities in the resistance and by this method, we can observe those nonlinearities and jump in the resistance due to solid-liquid phase change.
Based on the 4-point probing method, the voltage is measured between two inner electrodes and these two electrodes are pretty close to the membrane, almost attached to the membrane. As a result, almost all of the voltage drop here comes from the membrane and the liquid inside the membrane and the changes observed in the resistance are due to the characteristic changes in the liquid inside the membrane. The liquid phase contains water and octane. Octane conductivity is due to the existence of a tiny amount of Span80 and conductivity of water is due to the sodium and chloride ions. Addition of sodium and chloride ions increases the conductivity of water due to ion conductivity. The most important feature is that, as water freezes, ions cannot move, and this increases ice resistance compared to water by more than three orders of magnitude. On the other hand, when there is no ion inside the water, the resistance of ice and water is in the same order of magnitude. It should be noted that these ions are not soluble in octane, as octane is completely non-polar, and after adding NaCl to octane it precipitates and octane resistivity, which is in order of Giga ohm (GW), does not change. Thus, sodium and chloride ions cannot be responsible for the charge transport inside the octane. Also, HLB (Hydrophilic-lipophilic balance) value for span 80 is 4.3 which means that span 80 is oil soluble and water-insoluble 2 .
We took two independent approaches to show that the electrical resistance of water and ice could differ by more than three orders of magnitude.
Approach 1: As mobility of ions in ice approaches zero, the resistance of ice is close to its pure ice value as reported as 10 7 ohm.m. (C. Jaccard, Mechanism of the electrical conductivity in ice, Annals of the New York Academy of Sciences, 125, 390-400, 1965 5 ). The electrical resistivity of water with 50 ppm of salt is reported ~100 ohm.m (Steve Felber, Water Fundamentals Handbook, DRI-STEEM, 2017 6 ). This suggests that there are five orders of magnitude difference between resistance of water solution and pure ice. Please note that the membrane is made of many pores surrounded by water/octane and calculation of total resistance of the system will have high uncertainty.

Approach 2:
In the second approach, we measured the electrical resistance of water solution and ice filled in a plastic tube with inner diameter of 1 mm as shown in Supplementary Figure 9. The tube was filled with water solution with 50 ppm NaCl and four-probe electrodes were attached to the tube. The specific electrical resistivity of the water is measured at 1 o C as 395 ohm.m. The S25 temperature of the system is dropped to -10 o C allowing to ice form in the tube. The specific electrical resistivity of ice was measured as 2093333 ohm.m. That is, the specific resistivity of ice is ~5000 times higher than that of water solution.

Supplementary Note 5. Electrical resistance metrology
The water-ice phase transformation for different size nanodroplets is shown in Supplementary  Figure 10. The non-linear shift in resistance across the pores indicates the phase change temperature (TN).
We measured temperature-dependent I-V curves for octane-only filled membranes for 80 nm membrane and results are shown in Supplementary Figure 11. From this figure, we cannot observe any non-linearity down to -32°C. However, we observed non-linearity in resistance in water-filled 80 nm membranes around -12°C. This proves that non-linearities in resistance are not due to Octane.
We conducted the complete I-V experiments for pore dimension of 80 nm in two cases: (1) Pure Octane case at two temperatures of -10 and 25 o C and (2) for water droplets surrounded by the Octane. The results for case (1) are shown in Supplementary Figure 12 suggesting Ohmic behavior. The results for case (2) are shown in Supplementary Figure 13 that indicate the nonlinearity in the system resistance is observed in both positive and negative domains with Ohmic characteristics. Note that the minimum current value with the source meter system is 10 pA and we have high uncertainty in low current region.

Supplementary Note 6. FTIR metrology
The FTIR experimental setups are shown below, Supplementary Figure 14. We used peltier coolers to control the sample temperature. In Attenuated Total Reflectance (ATR) mode, once the sample was placed on the FTIR stage, the peltier coolers were placed on a part of the membrane covered by a coverslip. For probing the sample temperature, a thermocouple was attached to the edge of the membrane. Because the membrane is placed between coolers, FTIR stage and coverslip, the vapor condensation from the surrounding environment is minimal. Furthermore, to prevent any possible frost formation, nitrogen gas was purged in the setup continuously. In the transmission mode, the membrane was sandwiched between coverslips and mounted on a stand and the light passed through the sample. C-H peaks from oil are subtracted using background and the resolution of the instrument is 4 cm -1 .
The water-ice phase transformation probed by FTIR metrology is shown in Supplementary  Figure 15. The narrowing and shift in OH stretch peak indicate the phase change temperature (TN).

Supplementary Note 7. Heterogeneous nucleation of nanodroplets
Gibbs energy barrier for heterogeneous ice nucleation is written as 7,8 :

S26
where "# denotes interfacial tension between water and ice, ∆ is the chemical potential difference between ice and water phases and ( , ) denotes the surface function. Surface function depends on interfacial tensions, , and surface geometry, , defined as follows where 7# is oil-water interfacial tension, 7" is oil-ice interfacial tension, "# is ice-water interfacial tension and is the ice embryo contact angle with the oil-ice interface. depends on interfacial geometry and is written as where 8 is the critical nucleolus radius and is the radius of curvature of the oil-water interface.
8 is written as 7,9 Where ∆ 9 denotes the free energy difference per unit volume between water and ice, ∆ 9 denotes enthalpy of phase change per unit volume, / is the melting temperature of ice, and is the temperature of the system. Also, ice-water interfacial tension is estimated as 8 "# = 23.24 j 235.8 k 4.6; ( 10) The surface function, ( , ), for a concave surface is written as 8,10 ( , ) = 1 2 If one considers the effect of oil-water interface geometry, , upon decreasing pore diameter, we observe a decrease in ( , ) and as a result a reduction in ∆ * , as shown in Supplementary  Figure 16 for two different temperatures.
Here, we have distinguished heterogeneous and homogeneous modes of nucleation through Gibbs energy barrier (∆ * ) analysis. If the nucleation in these pores were homogenous, we should not S27 observe any effect of confinement size on the nucleation temperature. For example, for the case of pure water in the confinement, the nucleation temperature varies from -7 to 0 C, Supplementary  Figure 19.
To further confirm this, we performed another set of experiments with membrane with pore dimension of 10 nm where we switched the encapsulating Octane phase with PDMS with various modulus as shown in Supplementary Table 6. In these experiments, we initially filled the membranes with water. Mixtures of Sylgard (i.e. ratio of base to crosslinker) and octane (as solvent) with three concentrations are developed. Each mixture is applied to the water filled membranes and allowed to be cured for 24 hrs at room temperature. Note that Octane evaporates during this period. That is, the water droplet are encapsulated with PDMS with different modulus. After, we performed FTIR analysis in transmission mode and the results are shown in Supplementary Figure 18. The results show that as the modulus of interface increases, due to the effect of surface factor on Gibbs energy barrier, ice nucleation temperature increases. This is another proof that the ice nucleation is heterogenous and not within the bulk of the liquid. This additional experiment also confirms that the soft octane-water interface leads to suppression of freezing to extremely low temperatures.

Supplementary Note 8. Effect of pores wall active sites on ice nucleation
We conducted ice formation experiments in pores just filled with water and no oil through FTIR metrology with different pore sizes. The results are shown in Supplementary Figure 19. As shown, ice nucleation temperature increases as the size decreases. In this case, the water has contact with the inner surface of the pores and the pore's walls act as nucleation sites.
Ice nucleation temperature as a function of pore dimension is shown in Supplementary Figure  20. Note that this is completely in contrast to the case that the oil-water interface exists in which ice nucleation temperature drops at lower pore dimensions.
The observed nucleation temperature could be explained through the thermodynamics of ice nucleation. In this case, nanodroplets experience negative pressure inside the pores due to the curvature of the water meniscus. Surface factor, ( , ), and chemical potential difference, ∆ , are calculated for this scenario as a function of pore dimension and are plotted in Supplementary  Figure 21a. As shown, as the size decreases, the surface factor decreases as well, primarily due to the concave interface of pore walls. In addition, the negative Laplace pressure in the nanodroplet decreases, which in turn leads to an increase in the chemical potential difference. The decrease in ( , ) and the increase in ∆ both lead to drops in the Gibbs energy barrier for ice nucleation, Supplementary Figure 21b.
We assume that oil wets the surface and water is encompassed by oil, as the oil has lower surface tension than water and the system is stable when the oil wets the pore walls. Thus, water does not have contact with pore walls and active sites. Also, even if we consider that some pores have defects or water has contact with the pore wall and freeze in higher temperatures, we can observe the changes in both electrical resistance and FTIR method, when most of the pores are frozen. In other words, freeing of few pores does not show significant changes in results and significant changes occur when the majority of pores are frozen.

S28
To further demonstrate that pore walls (with their potential active nucleation sites) are not involved in ice nucleation, we calculated surface factor, ( , ), and chemical potential difference, ∆ , for a hypothetical case where water contained inside the pores is surrounded by the oil at the ends, but oil does not wet the interior pore walls. Therefore, water is in direct contact with the walls within the pores. In this case, due to the curvature of the oil-water interface, there is a positive pressure build-up inside the water. The surface factor for the inner concave surface of pore walls is also known, thus Gibbs free energy can be obtained. As shown in Supplementary Figure 22a, upon decreasing the size, and ∆ both decrease and the effect of is more significant. Thus, ∆ * must decrease (Supplementary Figure 22b) for smaller sizes which is in contrast with our experimental observations; hence this scenario fails i.e., the walls are wetted with the oil.
The concavity is defined with respect to viewing from inside the droplet. Due to the fact that ice nucleates inside the water not oil, the oil is a concave substrate for ice nucleation as shown in Supplementary Figure 23. Also, for a concave interface, ΔG * decreases as the size of droplet decreases. (i.e. f(m,x) becomes smaller for smaller droplets).

Supplementary Note 9. Theoretical ice nucleation temperature of nanodroplets based on the classical nucleation theory (CNT)
Ice nucleation rate which is a function of Gibbs energy barrier is defined in Eq. (S9) 9,12 . Where -9 is ice nucleation delay time and K is a kinetic constant and defined as follows 12 :

= ( 10)
Where is Zeldovich non-equilibrium factor, is the rate of addition of atoms or molecules to the critical nucleus and is the number of atomic nucleation sites per unit volume. By substituting Eq. (S1) in Eq. (S9), we have = u− 16 "# 6 ( ) 3 = ( ∆ ) 3 v ( 14)