Impact of an ice particle onto a dry rigid substrate: Dynamic sintering of a residual ice cone

Ice particle impact onto a cold dry rigid substrate leads to particle deformation and breakup. If the impact velocity is high enough, the deformation is governed mainly by inertial and plastic stresses. Particle deformation may lead to the development of multiple cracks and the formation of a fragmented particle zone in the vicinity of the target surface. Moreover, a small solid residual ice cone, formed from fine particle fragments, remains attached to the substrate. In the present study the normal impact of nearly spherical ice particles, their deformation and fragmentation are observed using a high-speed video system. The size and mass of the residual ice cone are measured for impact velocities ranging from 11.2 ms -1 to 73.2 ms -1 and initial particle diameters ranging from 1.89 mm to 4.44 mm. A theoretical model for the ice particle collision and deformation is used to estimate the residual ice cone size. The model is based on a hydrodynamic approach describing particle deformation and is able to predict well the maximum radius of impression and the collision duration. The radius of the impression is used as the main length scale for an empirical model for the geometry of the residual ice cone.


Introduction
The physical processes involved in an ice particle impact event are of interest in different areas of research. In the solar system, impact, fragmentation and sticking of icy bodies are relevant mechanisms for asteroid and planet formation. For instance, bonding of ice particles at low velocity impacts has been investigated (Hatzes et al., 1991;Bridges et al., 1996;Heißelmann et al., 2010;Shimaki and Arakawa, 2012), concluding that one of the necessary conditions for the particle sticking behavior is a sufficient duration of contact, since sticking has only been observed below a critical relative velocity. Hypervelocity collisions of ice particles are studied in order to better understand and model the kinetic processes in the solar system, including formation of planets or Saturn rings. The studies include the characterization of the crater geometry in ice targets after particle impact (O'Keefe and Ahrens, 1985;Lange and Ahrens, 1987;Kato et al., 1995;Shrine, 2002;Grey, 2003;Kraus et al., 2011;Koschny et al., 2001;Burchell, 2013), ejection velocity of the fragments  or restitution coefficients after particle impact (Higa et al., 1998).
In aeronautics, ice particle impacts are investigated to better understand and model hail impact damage to aircraft structures. Impact velocities in the order of 100 ms -1 are studied, which are comparable with the speed of flight of an aircraft. For instance, ice projectile deformation, development and propagation of cracks and the formation of crushed fragments have been studied in detail (Pan and Render, 1996;Combescure et al., 2011). The residual displacement of cylindrical ice projectiles are measured for different impact velocities. The force produced during an ice particle impact and the resulting damage to composite structures is investigated in Juntikka andOlsson (2009), Rhymer (2012), Tang et al. (2019) and Tippmann et al. (2013). Further research on ice particle impact is motivated by the problem of ice crystal ice accretion in the compressor stages of aircraft gas turbine engines (Mason et al., 2006(Mason et al., , 2011Currie et al., 2012Currie et al., , 2014Veres et al., 2012;Veres and Jorgenson, 2013;Ayan et al., 2015;Bucknell et al., 2018). When an aircraft flies through clouds containing ice crystals, these crystals are ingested into the engines, where they impact on the rotating and stationary engine parts. In this context, single particle impact experiments have been conducted in order to investigate the fragment size and their dynamics after impact (Guégan et al., 2011;Vargas et al., 2015). The study of Hauk et al. (2015) provides experimental data and a scaling for the size-dependent threshold velocity for ice particle breakup. While passing through the gas turbine engine, the ice fragments start to melt due to a rising air temperature. The walls in this region are heated and may lead to partial crystal melting during impact. If melt water generated by these mechanisms wets the compressor surfaces, adhesion of impacting ice particles is promoted, which may result in the accretion of ice layers. Shedding of these ice layers (Mason et al., 2011;Mazzawy, 2007;Kintea et al., 2016) can cause mechanical damage, loss of engine controlability or even combustor flameout, making ice crystal icing a major concern for safe aircraft operation.
Ice particle impact experiments have shown that some of the particle fragments stick to the impact surface even with the particle and target being at subfreezing temperatures (Palacios et al., 2014;Hauk 2016). Thus, this sticking mechanism does not depend on a macroscopic liquid layer either on the ice particle or the target. If the impact surface is at a temperature above the freezing temperature, this sticking residual mass partially melts and acts as an additional source for liquid water, promoting adhesion of subsequent impacting ice particles.
In the present study, the formation of a residual ice mass after normal impact of a nearly spherical ice particle onto a dry solid substrate is studied experimentally and modeled theoretically. The impact, deformation and fragmentation of the ice particle are observed using a highspeed video system but used only for visualization of the process and for measuring the particle diameter and velocity prior to impact. The mass of the nearly conical residual ice cone and its area on the target surface are measured for different particle diameters and impact velocities. A theoretical description of particle deformation during an impact, based on a simplified hydrodynamic model (Roisman, 2021) is used to describe the residual ice cone size. The velocity field in the deforming particle is approximated by a kinematically admissible inviscid flow. The pressure field in the particle is integrated, accounting for the plastic rheology of the material, including hardening effects, namely the dependence of the yield strength of ice on the local strain rate of the flow. Finally, the equations of motion of the particle are solved, which allow prediction of the maximum dimensionless particle dislodging size. The dimensionless particle dislodging parameter is then used for the development of an appropriate scale for the typical size of the residual ice cone. An empirical model for the cone height is proposed for engineering applications.

Experimental method
The experiments are designed for the measurement of the residual mass after a normal impact of an ice particle onto a solid substrate. The experimental method includes a technique for controlled and repeatable generation of spherical ice particles of a given diameter and a technique for the measurement of the residual mass.

Ice particle generation
Ice particles are generated from purified de-ionized water (Millipore, Milli-Q®). The water is degassed in a glass bottle attached to a vacuum pump (Adixen AMD4), in order to minimize the inclusion of gas bubbles in the ice particles. The pump provides a reduced pressure below 0.2 kPa. Half an hour before the ice particle formation, the glass bottle is evacuated and manually shaken. Shaking of the bottle initiates intensive cavitation in the system which enhances the degassing process (Juan Gallego-Juarez, 2015). If no more air bubbles form after the shaking, the water is considered to be ready for ice particle generation.
The ice particles are generated by gently dripping water drops onto a reservoir of liquid nitrogen, as reported in Vargas et al. (2015). Upon impact onto the liquid nitrogen surface, the drop surface freezes almost

Nomenclature
Greek letters Mass of the impacting ice particle (kg) M res Measured mass of the residual ice cone (kg) N Pixel number in an ice particle image (− ) p Pressure (kg m − 1 s − 2 ) R Initial ice particle radius (m) r j Distance of a pixel to the particle symmetry axis (m) U(ζ) Characteristic particle velocity (m s − 2 ) U* Characteristic velocity at the maximum impact force (m s − 2 ) U 0 Ice particle impact velocity (m s − 2 ) V water Meltwater volume (m 3 ) Axial and radial coordinates (− ) instantly, maintaining its nearly spherical shape. A few seconds after the impact onto the nitrogen, while the ice particle is still floating on top of the liquid nitrogen surface due to the inverse Leidenfrost effect (Hall et al., 1969), it is picked up by a trowel and stored inside a freezer at an ambient temperature of − 20 • C. Using syringe needles of different sizes, particles were generated having diameters ranging from 1.89 mm to 4.44 mm. Due to the degassing procedure, all ice particles appear transparent in the experiments. This method is a feasible way of generating a sufficient number of almost spherical ice particles with a minimum of gaseous inclusions, and results in highly repeatable experiments. However, due to the cryogenic conditions the drop is exposed to upon deposition on the liquid nitrogen surface, the chance for freezing of a drop from the outside inward also makes the method prone to result in initial cracks of the ice particles. These pre-cracked particles cannot be identified during or after particle generation. However, they can be easily observed in the high-speed videos and are not considered further in the analysis.

Impact experiments
The setup for the impact experiments, schematically shown in Fig. 1, consists of an ice particle gun, a transparent solid substrate and a highspeed video system.
The ice particle gun is driven by pressurized gaseous nitrogen which is stored inside a small tank connected to a solenoid valve and a magazine made of aluminum. The magazine is used to load a sabot with an ice particle prior to the experiment. When the solenoid valve is opened, the ice particle shielded by the sabot is accelerated inside of an aluminum tube. At the end of the aluminum tube, the sabot is stopped and the ice particle is ejected from the gun.
All impact experiments are conducted inside a cooling chamber providing an ambient air temperature fluctuating in time and space between − 7 • C and − 15 • C. This arrangement ensures that all parts of the experimental setup are at subfreezing temperatures prior to the experiments and during handling of the ice particles. Since the ambient temperature fluctuations are large, the temperature of the target is directly measured using a thermocouple attached to the surface at a distance of 20 mm from the impact point. Additionally, a thermocouple is attached to the surface of the aluminum magazine and insulated from the ambient. In preliminary tests, a thermocouple was placed inside a particle before placing this test particle inside the magazine. This test confirmed that after approximately 30 seconds, the particle core temperature reached the magazine temperature measured at its surface. Therefore, the magazine temperature has been subsequently used to measure the ice particle temperature. In the experiments, the target temperature varied between − 6.7 • C and − 10.6 • C and the particle temperature varied between − 9.4 • C and − 10.9 • C. An influence of these temperature fluctuations on the experimental results is possible but was not observed in the present study. Since the experimental results are repeatable and show a clear trend as it can be seen in Fig. 8, the influence of the temperature fluctuations is not expected to change the significance of the experimental results. If the target and particle temperature are controlled more precisely, it is possible that the scatter in the data can be reduced.
The impact of the ice particle onto a glass target is captured using a high-speed video system, which includes a high-speed video camera (Photron MC2.1) and a LED backlight (Veritas Constellation 120E). The recording speed is set to 8000 frames per second and the spatial resolution is 77.52 μm/pixel. The visualization allows evaluation of the initial particle size and the impact velocity obtained using in-house image processing algorithms realized in Matlab. After a background subtraction and binarization of the images of the ice particle prior to impact, the position of the particle centroid and the area of the particle is obtained. In the next step, the motion of the particle centroid is plotted against time for all relevant images. Since the ice particle travels with a constant velocity, the plotted data exhibits a linear increase with a slope that corresponds to the velocity. This slope is quantified using a linear least squares fit to the data.
Ice particles having a non-spherical shape were discarded for the experiments. Given that the particle is accurately depicted as a body of revolution, the particle geometry can be defined entirely by a single video frame in which the particle symmetry axis is aligned with the focal plane. The frame most closely capturing a view of the symmetry axis is selected by comparing the ratio between the maximum and minimum feret diameter, d f,max /d f,min , for all frames before the impact and choosing the frame with the largest ratio. The symmetry axis through the particle centroid is found and the distance of each pixel inside the particle to this axis in metres, r j , is determined. The particle volume V 0 is then computed as where x px is the width of one pixel in metres and N is the number of all pixels in the ice particle image. The particle diameter D 0 is then defined as the volume equivalent sphere diameter using V 0 as the particle volume The uncertainty of the particle velocity is estimated as follows. Prior to impact, the particle is observed to travel at least half the distance of the image field of view, which is equal to a distance of 256 pixels using  Science and Technology 194 (2022) 103416 the present optical setup. If the particle deviates from a perfect spherical shape and rotates slightly during its flight, the measured centroid position deviates from the true centroid position. As a conservative estimate, the measured centroid of the particle is assumed to be shifted a distance of 5 pixels from its true centroid during the particle flight path. As the time between the video frames is known precisely, the uncertainty of the measured velocity is directly proportional to the uncertainty of the position of the particle centroid, leading to an uncertainty of 5 pixels/256 pixels ≈2.0%.

Cold Regions
For the uncertainty estimation of the particle diameter D 0 , the measurement error is assumed to be in the order of one pixel. Since the pixel size is a fixed quantity in the measurements, the uncertainty is highest for the smallest measured particle (4.1%, D 0 = 1.89 mm) and lowest for the largest measured particle (1.7%, D 0 = 4.44 mm).
In Fig. 2, an example image sequence of an impacting ice particle is shown. In the experiments the ice particle gun is oriented vertically to eject the particles upward, which can be seen in the left images. In the time sequence at t = 0 s, an ejection of fine fragments near the impact surfaces can be identified, analogous to a prompt splash during liquid drop impact. The impact leads to particle fragmentation and formation of a cloud of large fragments.
As shown in Fig. 2, some of the ice particle fragments remain adhered to the target long after the impact, despite the downward directed acceleration of gravity. This residual ice mass exhibits approximately a conical shape and is observed to strongly adhere to the target surface, although the particle and target temperature were below − 6.7 • C and − 9.4 • C, respectively.

Measurement of the residual ice cone mass and base diameter
In the present experiments, it is difficult to directly measure the residual mass using a laboratory precision balance. The strongly adhering residual mass would have to be removed from the target surface and transferred to a balance while making sure no mass is lost. Detaching the solid residual mass mechanically is very difficult due to its strong bonding to the surface and parts of the ice could either remain on the surface, or on the tool used for removing the ice. Melting the residual mass prior to its transfer is disadvantageous, since liquid water wets the target made of sapphire glass very well, making it difficult to remove the residual mass. Furthermore, it must be ensured that an insignificant amount of water evaporates during this process. In order to measure the mass of the residual cone sticking to the target precisely, a novel indirect measurement technique was developed, which is illustrated in Fig. 3. After the impact experiment, a heated glass plate with a thin tissue wipe (Kimtech Science) on top is pressed against the target glass plate. As a result, the residual ice cone melts and wets the tissue leading to a higher transparency of the wetted tissue area. This process is recorded in an view orthogonal to the target surface with a spatial resolution of 46.08 μm/pixel. Using image processing, the wetted tissue area is measured and can be related to the melted water volume. This wetted area -melted water volume relationship is calibrated for the tissue paper by placing drops of known volume between 0.2 μl and 1.4 μl onto the glass target using a 5 μl Hamilton syringe. The wetted tissue area is measured using the same procedure as for the residual ice cone mass measurement. In Fig. 4, the results of the calibration measurements are shown.
The expression for the linear calibration curve is where V water is the imbibed meltwater volume and A wetted is the wetted tissue area. The error of the dispensed drop volume in the calibration measurements cannot be directly quantified. Therefore, the error is assumed to be in the order of one graduation step of 0.05 mm 3 of the syringe, which is shown as vertical error bars in Fig. 4. The horizontal error bars correspond to one standard deviation of 6 experimental repetitions. The calibration curve is found by fitting a linear function to the measurements. In the experiments, the residual mass was measured while varying the particle diameter between 1.89 mm and 4.44 mm and the impact velocities between 11.18 ms -1 and 73.19 ms -1 . The minimum measured water volume of the residual ice mass is 0.170 mm 3 , therefore being in the range of the calibration measurements. In addition to the residual mass measurement, the area of the residual mass A res visible in the view normal to the impact surface is measured using image processing. For this purpose, the same camera as for the measurement of the wetted tissue is used, having a spatial resolution of 46.08 μm/pixel.
Using the measured area A res , a circle equivalent diameter of the residual mass can be computed as d max = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ ̅ 4A res /π √ .

Fig. 2.
Example image sequence of an ice particle impacting onto the glass target, showing the particle before impact (t = − 0.13 ms), the ejection of fine fragments (t = − 0 ms) and ejected fragments of the particle after breakup. After collision of the ice particle, a residual ice cone is formed which adheres to the target surface. In this example the particle diameter is 2.30 mm and the impact velocity is 21.63 ms -1 .

Dynamics of an impacting ice particle
Predicting the amount of the observed residual ice mass adhering to the target can be highly relevant to engineering applications. In the following section, a physics based model predicting the residual mass geometry is presented. Experimental results are then compared to these model predictions. The particle fragmentation process is governed by the particle deformation caused by the stress field in the impacting particle. The flow field in the impacting particle is governed by inertia, yield strength in the plastic region and elastic stresses in the elastic region. The flow is influenced also by fragmentation of the particle and by elastic waves in the solid substrate. The modeling of such a flow is usually rather complicated. Such models are based on finite element computations (Rhymer, 2012;Juntikka and Olsson, 2009), discrete element models (Carmona et al., 2008), implementation of the dynamic material behaviour of ice in commercial codes (Miljkovi et al., 2010) or hybridized numerical codes (Leinhardt and Stewart, 2009), which include various numerical techniques for different stages of particle impact, deformation and penetration into the target material.

Hydrodynamic model of particle deformation
A simplified hydrodynamic model for crater formation in ice due to a particle impact has been developed in (Sherburn and Horstemeyer, 2010). The hydrodynamic modeling of problems related to penetration mechanics and collisions is a well-known approach (Ben-Dor et al., 2013;Yarin et al., 2017;Schremb et al., 2019), which leads to significantly simplified methods of solution and to the prediction of characteristic integral quantities, like penetration depths of a rigid projectile into a metal target (Roisman et al., 2001(Roisman et al., , 1999 or the final shape of the projectile after the impact onto a rigid substrate (Taylor, 1948;Jones et al., 1998). Such kind of models also allow to predict the distribution of fragment sizes after penetration of metal projectiles into a metal target of finite width (Yarin et al., 2000).
The hydrodynamic approach to problems in penetration mechanics is often based on the assumption of similarity of flows with analogous geometry, but with materials of different rheological properties. For instance, in Fig. 5(a) the time sequence of a single water drop onto a dry substrate is shown, which illustrates the main features of drop deformation and ejection of a radially spreading thin lamella. An ice particle impact, shown in Fig. 5(b), exhibits a similar deformation during impact at comparable dimensionless time t = 2 U 0 /D 0 obtained by normalizing the physical time t with the particle radius D 0 and impact velocity U 0 . During particle impact a material element is compressed mainly in the axial direction and is stretched in the radial direction. This deformation leads to the formation of the cracks propagating in the particle material. The relative volume of the voids continuously grows in time. Intersection of the voids leads to the formation of void clusters. The behavior of such a granular flow is indeed comparable to the plastic deformation of a body penetrating into a plastic target. This process is analysed in detail in Yarin et al. (2000).
In Fig. 5(b) at t = 80 μs, longitudinal cracks can be observed, which have already reached the rear end of the particle. Nevertheless, the effect of the cracks on the evolution of the particle shape remains only minor and it continues to deform as a continuous body. The behavior of the fractured body is characterized by the probability of the lacunae openings in the particle material, which is equal to the relative volume of these voids. Only if the probability reaches the percolation threshold the material can be considered as fractured, consisting of separate grains (Staufer, 1985). The flow in the fractured region can be then considered as a dense granular flow. The fractured zone first reaches the spreading front of the particle at the substrate. This leads to the generation of flow of fine particle fragments moving in the radial direction. At the later stages of particle impact the deformation rate reduces and the flow is governed by the elastic energy collected in the deforming fragments of the particle. This elastic energy leads to the rebound of the particle fragments from the substrate. The hydrodynamic model holds only up to the moment when elastic energy release of the fragments starts to play a role.
Consider a normal impact of a spherical particle onto a perfectly rigid, perfectly smooth target. Denote D 0 and U 0 the particle initial diameter and impact velocity, respectively. The expressions for the velocity and the stress field in the ice particle are obtained in Roisman (2021). Because of the importance of the hydrodynamic model reported in Roisman (2021) for the discussion in the present study, its main elements are repeated here. The model predicts the evolution of the force produced by the particle impact and the maximum particle deformation. In this study, the model is used to estimate the size of the residual ice cone. A kinematically admissible flow in a deforming particle is assumed, which is determined by the instantaneous dimensionless drop dislodging parameter ζ(t) and the characteristic velocity U(t) of the particle,  where R = D 0 /2 is the initial radius of the particle. Starting from ζ(t = 0) = 0 at the instance of contact of the particle with the target, the dimensionless drop dislodging parameter ζ(t) can be interpreted as dimensionless displacement of a material point in the far field of a velocity field describing the particle deformation. The velocity field in the particle is approximated by a potential flow field around a flat disc having the size of the impression radius a(t) in an uniform stream with velocity U(ζ), which satisfies the continuity equation and the condition of the target impenetrability over a spot of the impression radius. The impression radius, shown in Fig. 5(c), is a function only of the drop dislodging parameter and for small times can be roughly estimated by the impression radius of a truncated sphere, a = R ̅̅̅̅̅ 2ζ √ . Correspondingly, the velocity field u is given in the form (Batchelor, 1967) where φ is the velocity potential, ξ, η are dimensionless elliptic coordinates defined through where z and r are the axial and radial coordinates, respectively. The rateof-strain tensor E of the flow field (Eq. (6)) is the symmetric part of the velocity gradient in the vicinity of the impact axis where the symbol ⊗ denotes the usual tensor product, and γ ≡ ̅̅̅̅̅̅̅̅ 2/3 √ ̅̅̅̅̅̅̅̅̅̅ ̅ E : E √ is the equivalent rate of strain. In the perfectly plastic flow the stress tensor in the particle is determined by the velocity field and by the yield strength Y, where p is the pressure and I is the unit tensor. The elastic stresses are assumed to be negligibly small in comparison to the plastic stresses and are neglected in this study. The expression Eq. (10) for the deviatoric part of the stress tensor (σ ′ ) in the vicinity of the impact axis (linearized for small r) is obtained in the form The yield stress is a material property of ice. It is known that it depends on the instantaneous strain rate and can therefore be expressed as where y(γ) is a dimensionless function of the equivalent rate of strain γ, determined in Eq. (9) and Y 0 is the static yield strength at γ→0. The solution of the momentum balance equation in the particle flow, with ρ denoting the ice density, is obtained in Roisman (2021) using the expressions for the velocity field Eq. (5) and for the stress field Eq. (10). Integration of the differential Eq. (14) yields the expression of the stress tensor in the particle, which allows estimation of the total force of the particle interaction with the wall. Finally, the momentum balance equation of the deforming particle in the integral form yields the following set of differential equations in dimensionless form (Roisman, 2021) where while the dimensionless values are determined as The function C is determined by the form of the dependence of the yield strength on the local strain rate. We use the expression obtained based on the assumed dependence where χ and τ are empirical constants. The values of Y 0 = 5.6 MPa, τ = 7 ×10 − 4 and χ = 5.0 are obtained in Roisman (2021) using experimental data for the peak force produced during ice particle impact. The system of the ordinary differential Eqs. (15) has to be solved subject to the initial conditions Here, h is the dimensionless particle height, scaled with the particle radius R. For the present analysis, the system of equations is solved using the variable-step, variable-order solver "ode15s" implemented in the commercial software package Matlab, which is based on the numerical differentiation formulas of orders 1 to 5 (Shampine and Reichelt, 1997).
As the solution of the system of equations is singular at t = 0, a finite value of t = 0.001 was used as initial condition instead. A smaller initial value of t did not significantly change the obtained results.

The total force
The total force produced by the particle impact on the target is expressed in the form (Roisman, 2021) The value of the force initially increases, since the impression radius increases during particle deformation. At some instant, corresponding to ζ = ζ*, the force F z (ζ) reaches a maximum value and then starts to reduce, since the characteristic particle instantaneous velocity U(ζ) decreases. The instant ζ* can be considered as an important characteristic deformation point, where the velocity U(ζ) has reached the characteristic velocity U*. The force F z (ζ) applied to the plastic particle is always positive. The magnitude of the velocity U(ζ) decreases in time. At some instant the velocity magnitude reaches zero. This instant corresponds also to the maximum value of the dimensionless particle dislodging ζ = ζ max . In Fig. 6, values of ζ* and the corresponding dimensionless particle velocity U */U 0 obtained from numerical evaluation of the present model are shown as functions of the characteristic final particle dislodging parameter ζ max . The data points in Fig. 6 are obtained by solving the differential equation system Eq. (15) for the same values of D 0 and U 0 as measured in the experiments. The value U */U 0 is close to 0.5 for a wide range of parameters and only slightly increases with ζ max . The value of ζ* increases almost linearly with the maximum drop dislodging, ζ * ≈0.72 ζ max .

Experimental results and discussion
The impacting particle breaks up as soon as the impact velocity is higher than a characteristic threshold velocity, which depends on the particle size (Hauk et al., 2015). The impact velocities in our experiments are much higher than the threshold velocities. The larger fragments rebound from the target while fine particles remain on the surface forming a solid, nearly conical shape.

Diameter of the residual ice cone
High-speed drop impact onto a solid substrate leads to the generation of a radially spreading thin lamella. Such a lamella can be observed during inertia dominated liquid drop impact, as shown in Fig. 5(a) and (c). The spreading diameter d of the lamella is defined in Fig. 5(c). In analogy to a drop impact, for an ice particle impact the formation of the flow in the lamella is indicated by the observation of a radial stream of fine particle fragments ejected parallel to the target substrate.
These fine fragments are not considered in the current theory. On the other hand, for inertia dominated particle impact, the particle deformation can be compared to the deformation during single drop impact with high Reynolds and Weber numbers (Yarin et al., 2017) as shown in Fig. 5(a) and (b). It is assumed that during ice particle impact, fine fragments generated near the impact point radially spread on the target surface up to a characteristic instance in time, where the spreading comes to rest and the fragments form the residual ice cone. In the present experiments, the base diameter of the residual ice cone d max is measured and compared to the lamella spreading diameter d for a liquid drop impact. The kinematics of the drop spreading diameter has been investigated in detail in Rioboo et al. (2002). It has been shown that for relatively small drop deformations, ζ < 0.2, the scaled spreading diameter does not depend on the drop material properties like viscosity or surface tension. The dependence of d/D 0 on the drop dislodging parameter ζ is determined as where the factor b = 2.05 was determined in Rioboo et al. (2002). In Fig. 7, the measured base diameter of the residual ice cone d max , scaled by the initial particle diameter D 0 is plotted as a function of the characteristic final particle dislodging parameter ζ max obtained by evaluation of the present model. Additionally, the evolution of a liquid drop spreading diameter as a function of the drop dislodging, determined in Eq. (23), is plotted as black dashed line in the same graph. The agreement between the measurements and the predictions for liquid drop impact is rather good. It supports the hypothesis that the residual ice cone is formed after a spreading motion of fine fragments near the impact point, which comes to rest at the final dislodging ζ max of the ice particle. It can be considered as validation of the theory and of the assumed dynamic material properties of the ice particles.
We can subdivide all the particle fragments created after particle breakup into three main groups: (i) relatively large particle fragments of size comparable with the particle diameter, (ii) fine fragments formed from the spreading lamella and (iii) sintered particles, forming the residual particle cone remaining on the substrate.

Estimation of the residual mass
In Fig. 8, the dependence of the dimensionless residual mass scaled by the particle mass, M res /M 0 , on the impact velocity U 0 is shown for several diameter classes of the impacting particle. The value of M res /M 0 increases almost linearly with U 0 and seems not to depend on the particle diameter.
As shown in Fig. 7, the diameter of the residual ice cone is determined by the maximum spreading diameter of the lamella d max . In order to predict the total mass of the residual ice cone, its height h res has to be modelled. In this study the characteristic height of the residual ice cone is estimated by approximating its shape by a cone which with the help of Eq. (23) yields where M res is the measured residual mass and ζ max is the theoretically predicted dimensionless particle dislodging. In our experiments the Fig. 6. Point of maximum force during ice particle deformation. The values of the dimensionless dislodging parameter ζ* and the particle instantaneous velocity U* at this point are shown as a function of the maximum particle dislodging parameter ζ max . Fig. 7. The dimensionless spreading diameter of the residual ice cone d max scaled by the initial particle diameter D 0 as a function of the characteristic final particle dislodging parameter ζ max . The dashed line corresponds to the spreading diameter of a liquid drop (Rioboo et al., 2002), defined in Eq. (23).

Fig. 8.
Experimental data for the mass of the residual ice cone as a function of the impact velocity for different particle diameters. maximum radius of the impression, a max = D 0 ζ 1/2 max / ̅̅̅ 2 √ , is comparable with the initial particle diameter, but the cone height is much smaller than D 0 . Therefore, it is reasonable to assume that the geometry of the residual ice cone is determined by a and does not depend on the particle diameter D 0 . Using dimensional reasoning under the assumption that the particle yield strength Y 0 and the surface energy of ice γ = 0.19 Nm -1 , obtained in Gundlach et al. (2011), are key influencing factors, the maximum impression radius a max can be made dimensionless resulting in Using the expression for h res in Eq. (24), the dimensionless height ℋ of the residual ice cone is defined as The dimensionless height ℋ is related to the cone angle where small values of ℋ correspond to a flat cone. If ℋ is not dependent on ℛ, all cone shapes are the same for different cone sizes. The experimentally determined dependence of ℋ on ℛ is shown in Fig. 9. It is interesting, that for the relatively high values of the dimensionless radius ℛ > 2 × 10 4 , the dimensionless cone height is nearly constant ℋ ≈ 0.12. Some increase of the values of ℋ for smaller values of ℛ can be explained by the influence of additional factors, like interaction with the elastic waves in the substrate and in the particle, interaction of large particle fragments, etc. This result indicates that some additional study is probably required for better modeling of the cone shape at smaller impact velocities.
By inserting the predicted value of ζ max from the hydrodynamic model for ζ in Eq. (23), the ice cone base diameter can be estimated and by using the dimensionless ice cone height ℋ = 0.12 and b = 2.05 (see Eq. (23)), an approximation of M res /M 0 can be obtained as In Fig. 10, the predicted values of M res /M 0 are compared to the experimental results. Although a deviation of the predicted from the measured residual mass exist, the general trend of the data is predicted well and Eq. (27) can be used for the engineering modeling of the processes involved in ice accretion.

Impact induced melting at subfreezing temperatures
One of the possible mechanisms of formation of the residual cone of sintered ice fragments is partial melting of fine fragments due to dissipation. The dissipation function Φ ≡ σ ′ ij E ij near the particle axis, which is obtained using the expressions Eq. (8) and Eq. (12) for the rate of strain tensor E and of the deviatoric stress σ ′ , is used to estimate the possible temperature change of the ice particle fragments during impact The heat produced through dissipation is therefore The integration of the heating must follow a corresponding material point. Integration in the Lagrangian system requires to follow the evolution of the z-coordinate of the material point where z 0 is the initial z-coordinate of the material point at t = 0. The temperature distribution can be estimated assuming adiabatic heating The predicted temperature elevation due to heat dissipation is shown in Fig. 11 as a function of the axial coordinate z for various values of ζ max .
Since for the small particles in our experiments the value of the effective rate-of-strain is relatively high, the constant maximum yield strength Y = 6Y 0 is used for the analysis. The values shown in Fig. 11 can be considered as an upper bound for the temperature increment due to particle impact, since no energy spent on the fragmentation is taken into account. Also, the effect of heat conduction in the particle and in the target is not accounted for in the model and it has to be noted that friction among disrupted ice fragments may be an additional phenomenon relevant for heat generation during impact. Nevertheless, the temperature estimations in Fig. 11 clearly demonstrate that the temperature rise estimated using the present model can be significant and should lead to the partial melting of the particles. Moreover, the melting of the particles occurs first at the fragment surfaces due to the premelted liquid water layer (Wettlaufer and Worster, 2006;Dash et al., 2006).  This process can potentially enhance the process of fragment sintering.

Conclusions
In this study the normal impact of spherical ice particles onto a dry rigid substrate has been studied experimentally and modelled theoretically. The focus of the study has been on the creation of a solid residual mass after ice particle impact and fragmentation. In the performed experiments the mass and surface covering area remaining at the impact surface after collision of the residual ice were measured for various particle diameters and impact velocities. The residual ice mass after impact resembled a conical shape. A theoretical model for particle deformation based on a hydrodynamic approach often used in the description of problems in penetration mechanics, is used to describe the residual ice cone size. The plastic flow in the particle is approximated by a kinematically admissible inviscid velocity field which satisfies the continuity equations and the impenetrability of the solid wall on the impression spot. The flow allows estimation of the stress field in the plastic flow and the total force produced by particle collision. The model is able to predict the maximum impression radius which agrees very well with the experimental data.
The dimensionless height ℋ of the residual ice cone has been shown to correlate well with the dimensionless impression radius ℛ obtained from the theoretical model. An empirical relation has been proposed for the further engineering modeling of the processes involved in ice accretion. The physics behind the formation of the residual ice cone is not completely clear. It can be a result of the adhesion of fine ice fragments pressed together during particle impact. Dissipation in the deforming particle can lead to local melting of the fragments and subsequent fast solidification after impact. The model for the mass of the residual ice cone can be potentially used for the description of the heat flux during the collision of an ice particle with a hot solid substrate, associated with the ice crystal accretion problem.

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.