Collisional damping of spherical ice particles

This paper presents experimental values for the coefficient of restitution (en) for millimeter-sized ice particles colliding with massive walls at different temperatures. Three different wall materials are tested: hardened glass, ice and Acrylonitrile butadiene styrene (ABS) polymer. The results show a high sensitivity to impact velocity Vi, where en decreases rapidly with increasing Vi. The results also show a decrease in en with increasing temperature T. A novel model that predicts en based on the assumption of collisional melting and viscous damping caused by an increased premelted liquid-layer, is proposed. The model predicts both the velocity and the temperature trends seen in the experiments. The difference obtained in experiments between wall materials is also captured by the new model. A generalized regime map for ice particle collisions is proposed to combine the new model with previous work.


Introduction
Collisions of ice particles are important in many physical systems. A deep understanding of these collisions is important to model and predict how systems of ice particles will develop in time. Examples of such systems are ice particles in space that agglomerate, which is an important part of planet formation [1]; and ice particles in the atmosphere that transfer charge through collisions, thereby triggering thunderstorms [2] or grow and precipitate from the sky as snow and/or rain [3]. Snow and ice tend to accumulate on surfaces, such as buildings [4], infrastructure [5], and vehicles [6]. Significant research focuses on preventing snow sticking and accumulating. A fundamental understanding of the energy dissipation in collisions between ice particles and massive walls, is thus an important part of understanding how snow or ice accumulates on surfaces.
The energy dissipation for a particle colliding with a wall is an important subject in the field of particle technology [7][8][9][10]. A wellestablished measure to characterize the energy dissipation for particle collisions is the coefficient of restitution e n , which is the ratio between normal rebound velocity V r and normal impact velocity V i , where by definition it is expected that e n ∈ [0, 1]. A variety of different damping mechanisms affect the dependencies and trends of e n , and consequently, there are many different models for e n [11]. Examples of mechanisms that have commonly been studied are viscoelastic damping [12,13], adhesion (typically due to van der Waals forces) [14,15], plastic dissipation [16], and liquid layer damping [17][18][19][20][21]. The e n has also been extensively studied for ice particles and a wide range of studies examine ice particle impacts [22][23][24][25], and Güttler et al. [26] have compiled an excellent literature survey. They summarize experimental data of particles in general and for ice particles in particular. One of the most extensive works that examines the e n specifically for ice particles is by Higa et al. [27]. Those authors studied spherical ice particles with diameters from 2.8 to 72 mm for a variety of temperatures and impact velocities. A key finding from that study is that below a certain critical velocity, e n is constant and independent of V i , while above this velocity, e n rapidly decreases with increasing V i . Although many studies have examined ice particle collisions, to the best of our knowledge, no data has been published on the e n for ice particles smaller than 2.8 mm. Snow grains on the ground typically, transform from snowflakes into spherical-shaped particles through metamorphism [28], and this grains range from hundreds of microns to a few millimeters in diameter [29].
In this paper, we present experimental results that extend the knowledge of ice particle collisions by examining collisions of millimeter-sized ice particles with an ice wall, a hardened glass wall and an ABS polymer wall. These walls are interesting because they represent large variations in Young's modulus, and they are common surfaces on vehicles in the automotive industry. Icing of external automobile surfaces is becoming a key obstacle for automated cold-weather driving. We relate observations to previous experimental work by Higa et al. [27], and utilize a collisional melting model based on a framework proposed by Wettlaufer [30] to model collisional melting for collisions between ice particles. We arrive at a novel model for e n that predicts a viscous loss caused by a premelted liquid layer, and we show that the model captures experimental observations. Finally, we present a regime map of the e n for ice particles where we combine our previous work on damping due to van der Waals forces [31] with the expanded collisional model.

Measurements
Experiments were performed to measure the e n for ice particles colliding with massive walls of different materials. To validate the setup, collisions of glass particles against a glass wall were also measured at the same temperatures used in the ice particle experiments and at room temperature.

Experimental setup
Ice particles were created using a previously established method whereby a mixture of distilled water and food colorant is dropped into liquid nitrogen using a syringe [32]. The syringe allowed the size of the droplets to be uniform, and the smallest available syringe needle was used to create the smallest ice particles possible. Previous studies of millimeter-sized spherical glass particles colliding with glass walls [19,33], led us to use spherical glass particles to validate the experimental setup. Glass particles with 0.5 mm, 1 mm, and 2 mm in diameter were used. Particles were picked up using a paintbrush and released into a vertical metal pipe. The particles were then accelerated to a certain impact speed V i by gravity and collided with the massive wall placed under the pipe. The impact speed was varied by changing the length of the pipe. Ice, hardened glass, and ABS polymer were used as the massive wall materials where the ice wall was created by freezing distilled water in a silicone mold. A high-speed camera recorded each particle before and after impact and a laser connected to the camera triggered image acquisition when a particle crossed the laser beam prior to impact (Fig. 1). The experiments were recorded at 4000 frames per second. In order to focus this work on normal collisions between particles and walls, massive walls with low surface roughness were chosen to avoid tangential velocity components after impact and experimental samples with a clear tangential velocity component were removed from the data set. The experiments were conducted in a large cooling chamber with temperature control. A temperature probe used to monitor the temperature was located 20 cm from the massive wall and was used to monitor the temperature T with the mean and 1σ standard deviation to indicate uncertainty (Table 1). Recordings from the high-speed camera were analyzed to extract particle trajectories, velocities, and particle size. The image analysis was done using Fiji and the single-particle tracking tool TrackMate [34]. Before using TrackMate, images were pre-processed to improve particle detection by removing the background and masking the images. For each recorded frame, the particle diameter was calculated from the sum of the pixel area. The median of all calculated diameters were then taken to estimate the particle sizes, where median were used to eliminate outliers caused by edge effects. The trajectories resulting from the image analysis were post-processed to determine the velocity before and after each collision. The uncertainty in position was quantified by manually measuring the center positions of particles and averaging the deviation between these measurements and the output of the tracking tool. The obtained average positional uncertainty δx = 2.613 ⋅ 10 −5 m results in an velocity uncertainty δV = 0.105 m/s. The uncertainty in the coefficient of restitution, δe n , is then calculated by further propagation of uncertainties as Fig. 2 illustrates an ice particle colliding with a glass wall with the particle's center point trajectory extracted from the image analysis overlayed as circles.
For particle impacts that occur close to the bulk material melting temperature there is the potential that proximity to the melting temperature T m plays an important role in the collisions. Thus the homologous temperature T H = T/T m , where T is the experimental temperature, is often used to depict how close to the melting point the experiments come.

Experimental observations
The e n obtained for glass particles colliding with glass walls is shown in Fig. 3 as a function of V i . A melting temperature of T m = 1300K is used for glass particles to calculate T H and these measurements detected no difference for the e n at different temperatures, or for varying V i . The mean e n was found to be 0.94 ± 0.05 at the 2σ level.
An empirical function for e n proposed by Higa et al. [27] for ice particles impacting an ice wall is, where e qe is the coefficient of restitution in the quasi-elastic regime, and V c is the critical velocity and suggests e n varies only above V c . Measurements of ice particles colliding with an ice wall were conducted at three temperatures: 267 K, 262 K, and 254 K, which correspond to T H of 0.977, 0.959 and 0.930 respectively. The ice particles had an average diameter of 1.64 ± 0.15 mm. Fig. 4 shows the measured e n values with a non-linear least square fit of Eq. (3) superimposed. The uncertainties of the fitted V c and e qe are quantified as the matrix square roots of the covariance matrix from the non-linear fit.
Collisions of ice particles with a glass wall were measured at two temperatures: 254 K and 262 K. The resulting measurements of e n are shown in Fig. 5.
Similar to the experiments with a glass wall, measurements of ice particles colliding with an ABS polymer wall were conducted at two temperatures: 254 K and 262 K. The extracted e n are shown in Fig. 6, where it appears that for all observations that V i < V c .   A summary of the fitted parameters e qe and V c are shown in Table 1 in addition to the uncertainties estimated from the coviarance matrix.
The confidence interval for both e qe and V c in our experimental observations for ice particles colliding with an ice wall at T = 262 K are either very close to or overlap with the confidence intervals for the two parameters at T = 254 K and T = 267 K. A comparison of results for the experiments involving massive walls of ice and glass shows that the e n is lower for the glass wall than the ice wall. This is also reflected in the obtained parameters V c and e qe . For all experiments except those with the ABS polymer, a clear trend was obtained where e n decreased with increasing T H . This is maybe not surprising given the fact that T H is greater than 0.9, which are temperatures that are very close to the melting point of water.

Comparison with previous experimental results
The values of e n obtained for glass particles colliding with a glass wall shown in Fig. 3 are similar to previous findings [19,33] that found e n to range from 0.92 to 0.98. Since the experimental conditions were far from the melting temperature of the glass particles, with T H ≈ 0.2, we did not expect to see a difference between temperatures, and the values obtained imply that the experimental equipment was not affected operating in cold temperatures. Ice particles colliding with both glass and ice surfaces behave differently, which is also expected given that T H > 0.9 in our observations. Fig. 4 shows that Eq. (3) captures the rapid trend of a decrease in e n with increasing V i above the critical velocity V c . An empirical equation for the critical velocity V c was derived in Higa et al. [27] as a function of temperature and particle radius R. If we simplify this equation it can be re-expressed as, where k b is the Boltzmann constant, and K 1 and K 2 are constants extracted from the Higa et al. [27] results (K 1 = 7.0055 ⋅ 10 −7 and K 2 = 0.25). Fig. 7 shows the V c predicted by Eq. (4) for the three temperatures used in the ice particle collision experiments with an ice wall. The figure also shows the values obtained for V c from Table 1 and the values from Higa et al. [27] obtained at T = 261 K.

Collisional melting -Premelting due to an impact
The present work and the previous Higa et al. [27] experiments show that the e n , for ice particles colliding with massive walls, significantly decreases with increasing V i above certain velocities. For obtained values of V c ≈ 1 m/s, the velocity dependency of e n is close to , which is significantly more rapid than plastic dissipation models predict for e n , which are typically proportional to V i −1/4 [35]. Interestingly, this strong dependency has been seen before. A study of micrometer-sized metal particles colliding with a metal surface [36] found that the metal particles follow the plastic dissipation dependency however, e n begins to follow V i −1 , the same dependency found in this work for ice particles. However, those experiments were performed at room temperature meaning at low T H for the metal. Given that the T H in these ice particle experiments is close to unity it is not surprising that e n for ice particles begins to fall at lower velocities since the material will be more sensitive to melting. Sensitivity to T H has been observed for other materials, for example, in a recent study where steel balls collide with heated metal walls, Hashemnia [37] shows that e n begins to decrease rapidly for T H > 0.6. Another study of glass particles colliding with a PEG (Poly Ethylene Glycol) substrate reported that e n fell to zero as the substrate approached its melting temperature [38].
As shown in Fig. 7 the empirical equation for V c (Eq. (4)) predicts that V c decreases with increasing T H . However, physical reasoning suggest that V c should approach 0 m/s as T H → 1, which implies that something fundamental is missing in Eq. (4). We propose that the damping observed for ice particles can be explained by collisional melting and specifically an increase in premelting of a liquid layer on the ice surface due to the collision. The existence of equilibrium liquid-like layers on ice surfaces is well documented [39][40][41][42][43][44], and here we expect the collision to enhance the existence of that layer. Such films are often referred to as premelted layers, quasi-liquid layers (QLL) or distorted liquid-like materials, and the material properties of the liquid layers are not necessarily expected to be those of the bulk liquid.
Models have been proposed for ice particle collisions that predict an increased premelting due to a collision [2]. Wettlaufer [30] proposed a theoretical model for collisional fusion as an explanation for the rapid damping of ice particles observed by Higa et al. [27]. That model assumes that a fraction ξ of the kinetic energy will damage the interface, and the resulting disorder causes an enhanced premelted liquid layer. This liquid layer can subsequently freeze during impact, causing the particle to fuse with the colliding object at certain velocities. The model is intended to predict fusion between ice grains at astrophysically relevant temperatures and pressures.
While fusion is relevant for astrophysical conditions with short freezing times, for our operating conditions with temperatures significantly closer to the melting point of water, fusion cannot explain the observed temperature trend. Specifically, with increasing T H , the time scale for freezing by heat conduction is decreasing and at the same time the predicted d is increasing. This means that with increasing T H the rate of freezing is reduced while the amount of liquid to freeze is increasing. The experimental measurements show that the damping consistently increases with increasing temperature for both the ice walls and the glass walls. We therefore argue that the associated energy loss instead can be explained by the presence of an increased liquidlayer and the associated energy loss this layer causes. An equation for the liquid layer thickness d caused by ice particles colliding with other ice particles was derived by Wettlaufer [30]. By simplifying Eq. (3) in Wettlaufer [30] for ice particles colliding with walls we obtain, Here ρ s and ρ f are the density of solid and liquid water. T m and P m are the bulk melting temperature and pressure for water. E * is the effective Young's modulus in the collision defined as, where E i and ν i are the Young's modulus and Poisson's ratio for material i. The damage term ξ is used to describe how microscopic and mesoscopic changes lead to the deterioration of macroscopic material properties [45]. With this new formulation for d we make two interesting observations: 1) d is predicted to be linearly proportional to R, which means that the ratio, d/R, is independent of particle size. 2) d is proportional to E * 2/5 , which means that increased melting should be observed with increasing Young's modulus of the material walls. To illustrate how the liquid layer changes with V i , Fig. 8 shows the predicted d for ice particles upon impact with an ice wall as a function of V i for the temperatures 267 K, 262 K, and 254 K. The critical velocities found in this work are highlighted in the figure. The figure shows a rapid increase in d for velocities above V c and d is predicted to be a few microns. The secondary y-axis in the figure shows d/R, and we can see that the liquid layer thickness ranged from 0.2% to 0.4% of particle radius R around V c . In these predictions ξ = 0.15 is used based on the work by Wettlaufer [30]. The uncertainty of ξ will be discussed later in this section. The energy loss for a particle-wall collision with the presence of a liquid layer has been the subject of many studies. In Crüger et al. [33], the authors observed that the e n decreases for glass particles colliding with a wetted glass wall with increased liquid layer thickness. Matthewson [46] studied the adhesion of spheres due to a thin   liquid film, and show that the adhesive force can be divided into two components: a meniscus force and a viscous damping force. While the resulting energy loss for the meniscus force is expected to be constant for a specific particle size and independent of V i , the viscous damping loss depends on d, V i , and the viscosity η of the fluid. We, therefore, argue that the viscous damping force will dominate for the case of damping caused by an increased premelted liquid layer. Based on Reynolds lubrication theory it has been shown by Chan and Horn [47] and Matthewson [46] that the viscous force F v can be expressed as a function of the particles approaching velocity V and the separation distance x between the particle and the wall as, where h(x) is a function that depends on the wetting conditions in the collision. For a particle colliding with an infinite wetted wall with constant d, Chan and Horn [47] showed that, which is valid except for very thin films or when x tends to zero (when the two solid surfaces come closer to contact). The resulting energy loss E v , for a particle colliding with an infinite wetted wall was estimated by Sutkar et al. [20] to be where the path integral is defined as the collision trajectory and ε is the particle roughness. In our situation the wetting area is finite and the thickness d is increasing during the course of collision (due to melting). This means that we expect h(x) to be more complex than for an infinite wetted wall with constant d. It is however reasonable to expect that the resulting E v would take a similar form as Eq. (9) and therefore we propose where the constant a can be empirically determined. The total energy loss E t is then where E i is the initial kinetic energy and thus ξE i is the amount of energy that causes damage to the ice particle surface. The E t can also be expressed as a function of e n as We determine a using nonlinear curve fitting of the observed energy loss represented by Eq. (3). For our experimental values we find an average best fit of a = 1.46 for ice particles colliding with an ice wall, and therefore propose that a = 1.5 (or a = 3/2).
The premelted liquid layer is a distorted region with properties between solid and bulk liquid water [41], therefore, η is unknown but likely temperature dependent. The temperature dependent viscosity of a liquid can be expressed as, where G 0 + is a potential energy barrier empirically determined by fitting experimental data [48]. Using this we can express the viscous energy loss E v for an ice particle colliding with a massive wall as, where C and G 0 + are coefficients with units Pa⋅s and eV, respectively.
Here C represents the combined value of all constants in Eq. (10). The resulting coefficient of restitution can then be expressed as, Fitting the coefficients C and G 0 + to the experimental data from ice particles colliding with an ice wall at T = 267 K and T = 254 K we obtained average values for C = 0.5 ⋅ 10 −10 Pa⋅s and G 0 + = 0.5 eV.
The best fit gives remarkably similar results for the experiments at T = 267 K and T = 254 K with only a difference of 0.6% and 0.5% for C and G 0 + , respectively. Fig. 9 shows the predicted coefficient of restitution for ice particles colliding with an ice wall together with the experimental measurements from the present work. As shown in the figure, the new formulation for the energy loss captures the rapid increase of energy dissipation with increasing V i , and also captures the difference between temperatures. The new model predicts that e n approaches zero for a finite value of V i which is different from Eq. (3) where e n only approaches zero as V i → ∞.
In Fig. 10 the model predictions of e n for different wall materials and the experimental measurements for these wall materials are presented. The different wall materials are represented by different values of E * where the material properties for the different walls are the same as used in a previous study [31] except that the Young modulus for ice is calculated as the function of temperature as in Hobbs [49]. Notably the new model predicts the trend between wall materials that is observed in the experiments.
The Eq. (15) solutions are plotted in Figs. 9-10 with ξ = 0.15. However, the value of ξ is not well defined and Wettlaufer [30] bounded ξ between 0.1 and 0.2. The dependency of ξ on d in Eq. (5) is almost linear for the range of velocities examined herein. Therefore, changes in ξ will only alter the constant K 1 in Eq. (14), and thus, as long as ξ is a constant. it will not matter what non-zero value is used for the prediction of E v . Fig. 11 shows d/R as a function of ξ together with a linear dashed line to illustrate the linear dependence between d/R and ξ. The figure assumes V i = 5 m/s, which is a high V i in this work. The linearity further increases with decreasing velocity because the contribution of the contact pressure in Eq. (5) decreases.
As shown in Fig. 10, Eq. (14) provides a good fit to the experimental measurements for the ice particles with a diameter of 1.64 mm (R = 0.82 mm). To address how e n will scale with a change in particle size we extracted the values of e n published by Higa et al. [27] for ice particles with radii of 1.4 mm and 4.0 mm at T = 261 K. These results together with the e n predicted by the new model are shown in Fig. 12. Again the model captures the trend that is observed in the experiments.

General regime map for the coefficient of restitution of ice particle collisions
The measurements we have presented complement existing measurements and lead us to propose a regime map for ice collisions. We identify three e n regimes for ice particle collisions: an intermolecular adhesive regime, a collisional melting regime, and a fragmentation regime. The intermolecular adhesive regime is characterized by damping due to intermolecular forces. In previous work [31], we studied when an ice particle sticks or bounces depending on intermolecular forces. Based on the Johnson-Kendall-Roberts (JKR) model, at low V i , e n can be defined as, Fig. 10. The e n for ice particles colliding with different massive walls as a function of V i . Solid, dashed and dotted lines show predicted e n caused by collisional melting for ice particles colliding with an ice wall, a polymer wall and a glass wall respectively. Marker symbols (Circle, square and triangle) show experimental measurements for ice particles colliding with an ice wall, a polymer wall and a glass wall respectively.
where V L is the lower sticking velocity and for collisional velocities below V L , particles stick due to intermolecular adhesion. This lower sticking velocity is defined as, where K 1 ≈ 0.9355 is an integration constant [31]. From Eq. (17) it can be seen that the V L is mainly governed by the interaction properties E * and W (work of adhesion), but will also increase with decreasing R. The collisional melting regime is defined by Eq. (15). In this regime an ice particle colliding with a massive wall will have a rapid increase in energy loss with increasing V i until the particle begins to adhere to the wall. This will occur when the incoming kinetic energy is equal to the sum of energy causing damage at the interface and the energy lost due to viscous damping. This condition results in a separate, higher sticking velocity, V H , that can be defined as, where d H is the melted liquid layer thickness when this sticking occurs. However, d H is a nonlinear function of V i , as seen in Eq. (5) and therefore it is not possible to derive an explicit equation for the V H and Eq. (18) can only be solved implicitly. In Fig. 13 the sticking velocities V L and V H are plotted as a function of R for ice particles colliding with an ice wall, a polymer wall and a glass wall at T = 261 K. As concluded by Eidevåg et al. [31], a polymer wall will have a higher V L than a glass wall which means that a polymer wall will tend to accumulate more ice particles than a glass wall for V i in the intermolecular adhesive regime. However, for high velocity collisions, where collisional melting dominates, a glass wall have a lower V H than a polymer wall and therefore a glass wall is predicted to accumulate more ice particles than a polymer wall for V i in the collisional melting regime. For even greater velocities ice particles will continue to adhere until fragmentation begins to influence the impacts at a certain fragmentation velocity V F . Particles at these velocities, in excess of collisional fusion, will rebound but simultaneously fragment. Previous experiments have shown that for high velocity impacts, particles rebound and lose energy due to fragmentation but partial fragments may adhere to the wall [50]. Experimental data in this range is lacking and the e n for such impacts is ill-defined given that the impacting particle will not remain intact.
The outlined regimes can be convoluted to yield a map for ice particle impacts (Fig. 14). In the figure, e n is the convoluted coefficient of restitution that emerges from considering, e JKR due to intermolecular forces and e cm due to collisional melting. A fragmentation regime is indicated for velocities above an unknown V F to illustrate that this is an experimentally unexplored region. The sticking velocities V L and V H are also indicated in the figure. The combined model predicts a horizontal plateau around the point e n ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 1−ξ 2 q and this demostrates why previous studies have proposed a quasi-static regime for the e n of ice particles. However, the new combined model also predicts that e n will decrease for V i below this point.

Conclusions
In this study we have investigated the damping mechanisms of ice particles that collide with walls, specifically in terms of the coefficient of restitution e n . Experimental measurements of e n for millimetersized particles at different temperatures and for different wall materials were presented. We obtain a velocity dependent e n , similar to what has previously been observed by Higa et al. [27]. Different magnitudes of e n for ice particles colliding with different wall materials were also obtained. Collisions with a polymer wall had the highest e n and collisions with a hardened glass wall had the lowest e n . Fig. 13. The sticking velocities V L (a) and V H (b) for ice particles colliding with different massive walls as a function of R at T = 261 K. Solid, dashed and dotted lines shows the calculated sticking velocities for ice particles colliding with an ice wall, a polymer wall and a glass wall respectively. In (a) below each respective line particles sticking will occur. In (b) above each respective line particles sticking will occur. The red dotted line (e JKR ) shows the coefficient of restitution due to intermolecular forces. The dashed green line (e cm ) shows the coefficient of restitution due to collisional melting, and a fragmentation regime is also indicated where the e n becomes undefined. The combined e n is shown by the black solid line.
We propose that the rapid energy dissipation for ice particles colliding with walls can be explained by collisional melting and the associated viscous damping the increased premelted liquid-layer causes in a collision. A novel model has been derived that captures obtained temperature dependencies and observed differences attributed to wall material properties. We combined the new model with previous work [31] and proposed a regime map for the e n of ice particles.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.