V Thiévenaz

T Séon

C Josserand

Solidification dynamic of an impacted drop

Keywords:

This paper is dedicated to the solidification of a water drop impacting a cold solid surface. In a first part, we establish a 1D solidification model, derived from the Stefan problem, that aims at predicting the freezing dynamic of a liquid on a cold substrate, taking into account the thermal properties of this substrate. This model is then experimentally validated through a 1D solidification setup, using different liquids and substrates. In a second part, we show that during the actual drop spreading, a thin layer of ice develops between the water and the substrate, and pins the contact line at its edge when the drop reaches its maximal diameter. The liquid film then remains still on its ice and keeps freezing. This configuration lasts until the contact line eventually depins and the liquid film retracts on the ice. We measure and interpret this crucial time of freezing during which the main ice layer is built. Finally, we compare our 1D model prediction to the thickness of this ice pancake and we find a very good agreement. This allows us to provide a general expression for the frozen drop main thickness, using the drop impact and liquid parameters.

Introduction

When a liquid drop is put in contact with a cold substrate, either by impact or deposition, the freezing of the liquid can lead to unexpected final shapes. Understanding the coupling between drop impact hydrodynamics and solidification, that builds the frozen structure, is crucial in many different contexts : airplane icing [START_REF] Baumert | Experimental and numerical investigations on aircraft icing at mixed phase conditions[END_REF], ice accretion on wires or roadways due to freezing rain [START_REF] Jones | A simple model for freezing rain ice loads[END_REF], 3D printing [START_REF] Lipson | Fabricated: The new world of 3D printing[END_REF], surface metal coating technology [START_REF] Pasandideh-Fard | Splat shapes in a thermal spray coating process: simulations and experiments[END_REF][START_REF] Fauchais | Knowledge concerning splat formation: an invited review[END_REF], etc. Indeed, although most of the airplane icing configurations concern impact of ice crystal [START_REF] Vidaurre | Particle impact and breakup in aircraft measurement[END_REF][START_REF] Hauk | Ice crystal impact onto a dry solid wall. particle fragmentation[END_REF], the impact of water droplets on subfreezing substrates can be of great interest in the dynamics of icing formation [START_REF] Schremb | Normal impact of supercooled water drops onto a smooth ice surface: experiments and modelling[END_REF]. Today, the optimal design and coating of the surfaces to avoid icing remains an important open problem [START_REF] Cao | Anti-icing superhydrophobic coatings[END_REF][START_REF] Kreder | Design of anti-icing surfaces: smooth, textured or slippery?[END_REF]. Freezing rain may cause hazardous conditions for pedestrians and cars or break tree limbs and power lines, and thus may cause immeasurable economic losses [START_REF] Jones | Ice accretion in freezing rain[END_REF]. In thermal spray deposition, the thickness and geometry of the solidified splat is important for the quality of the coating and depends on the melting of the spray with the substrate [START_REF] Chandra | Formation of solid splats during thermal spray deposition[END_REF]. Similarly, multiple droplet impacts, coupling the fluid dynamics with the solidification thermal processes, produce complex splat patterns that determines the coating quality [START_REF] Dhiman | Freezing-induced splashing during impact of molten metal droplets with high weber numbers[END_REF].

In that context, it is important to have a precise characterization of the thickness of the residual solid layer. It could for instance help to improve the existing ice accretion models, crucial for aircraft icing or ice load on transmission lines [START_REF] Schremb | Normal impact of supercooled water drops onto a smooth ice surface: experiments and modelling[END_REF], or increase coating efficiency formed by the impact and solidification of molten thermal spray particles [START_REF] Dhiman | Predicting splat morphology in a thermal spray process[END_REF]. Moreover, because of thermal contraction when the solid layer becomes cooler, and depending on the thickness of the frozen impacted drop and the substrate temperature, the frozen structure can either remain stuck on the substrate or detach through a self-peeling process (de [START_REF] De Ruiter | Self-peeling of impacting droplets[END_REF] or even fragment into a myriad of small ice pieces [START_REF] Ghabache | Frozen impacted drop: From fragmentation to hierarchical crack patterns[END_REF].

When the drop is simply deposited on the substrate, the frozen drop shape and thickness depend on the contact line solidification dynamics [START_REF] Ruiter | Contact line arrest in solidifying spreading drops[END_REF][START_REF] Tavakoli | Spreading and arrest of a molten liquid on cold substrates[END_REF][START_REF] Schiaffino | Motion and arrest of a molten contact line on a cold surface: an experimental study[END_REF]. In this paper, we aim at obtaining a prediction of the final ice layer thickness resulting from the impact and solidification of a drop on cold surfaces. This imposes to consider the complex coupling between freezing and hydrodynamics. Indeed, right after impact, while the drop spreads on the substrate [START_REF] Josserand | Drop impact on a solid surface[END_REF], a thin solid crust layer forms between the liquid and the substrate (de [START_REF] De Ruiter | Self-peeling of impacting droplets[END_REF]. Afterwards, as the solid layer keeps growing from the substrate [START_REF] Marin | Universality of tip singularity formation in freezing water drops[END_REF][START_REF] Gao | Precise deposition of molten microdrops: the physics of digital microfabrication[END_REF], the remaining liquid can retract on its solid layer [START_REF] Bartolo | Retraction dynamics of aqueous drops upon impact on non-wetting surfaces[END_REF], preventing the final solid layer to reach a uniform thickness. For the sake of clarity, this paper will be therefore divided into two parts: in the first part, the solidification dynamics of a liquid suddenly put in contact with a cold substrate will be tackled from a general point of view and, in the second part, the results obtained will be applied to the particular case of a water drop impacting such a cold substrate.

The paper is organized as follows: in the first part, after introducing the problem of a melt freezing on a substrate (section 2.1), we develop and interpret the associated theoretical model (section 2.2), and we compare it to a dedicated model experiment (section 2.3). In the second part, the drop impact experimental setup and the measurement techniques involved are first described (section 3.1), then, we depict and interpret the hydrodynamic behaviour of the water layer pinned at the edge of its growing ice (section 3.2), and finally, we present our experimental results relating to the formation of the residual ice layer, in the light of the thermal and hydrodynamic behaviours presented before (section 3.3).

Solidification dynamics of a liquid on a substrate

In this section, we introduce a simplified 1-D solidification model that aims at describing the temperature distribution and the solidification front dynamics in the problem of a liquid film on a cold solid substrate.

Presentation of the problem

We consider a material existing in two phase states : liquid and solid. Then, an infinite flat solid substrate, filling the halfspace z < 0, with a temperature T s below the melting temperature T m , is in contact with the solid layer of the material (for 0 < z < h(t)), that is growing in its melt (z > h(t)). h(t) is the position of the solidification front (see schematic Fig. 1). This configuration belongs to the large class of Stefan problems, a particular kind of boundary value problem where a phase boundary can move with time. It is named after Josef Stefan who first, in 1889, solved the original configuration in which a solidification front propagates between two phases (liquid and solid) of the same material, without substrate [START_REF] Stefan | Über die theorie der eisbildung, insbesondere über die eisbildung im polarmeere[END_REF][START_REF] Brillouin | Sur quelques problèmes non résolus de la physique mathématique classique propagation de la fusion[END_REF]. In figure 1, this original Stefan problem amounts to replace the substrate by the solid phase.

Configurations belonging to Stefan problems, even restricted to the solidificationmelting phase change, are abundant, in an impressively wide range of application. We can cite as examples, the solidification of the earth, supposed to be molten at the origin, which triggered the first work of interest in this area, by Lamé and Clapeyron (1831) (Lamé & Clapeyron 1831). The formation of ice crystals, like snowflakes, that grow out from seeds in an environment of supersaturated water vapour. In this problem the ingredients of the original Stefan problem are not sufficient, Gibbs-Thomson equation and 3D effects need to be added [START_REF] Langer | Instabilities and pattern formation in crystal growth[END_REF]. The modelling of the dynamics of sea ice on the surface of the polar oceans [START_REF] Worster | Solidifcation of fluids[END_REF] or the shape of an icicle [START_REF] Neufeld | On the mechanisms of icicle evolution[END_REF], where the liquid motions and the associated convective heat transfert have to be considered. Idem for various lava flows that cool and gradually solidify until they come to rest [START_REF] Griffiths | The dynamics of lava flows[END_REF]. Cooling can occur from the surrounding atmosphere (or water) or from the underlying solid [START_REF] Huppert | Phase changes following the initiation of a hot turbulent flow over a cold solid surface[END_REF], and in ancient time, some of the lavas were hotter than today and were even capable of melting the underlying rock and shaping their own thermal erosion bed [START_REF] Huppert | The intrusion of fluid mechanics into geology[END_REF]. In industry, Stefan problems are a lot studied in the context of melting or solidification of metals or metal alloys [START_REF] Viskanta | Heat transfer during melting and solidification of metals[END_REF], where the properties of the solid (in particular, its mechanical and thermal properties) are functions of the kinetics of solidification, such that Czochralski crystal growth, used for example in the fabrication of semi-conductor wafers (Nishinaga 2014), laser welding [START_REF] Cline | Heat treating and melting material with a scanning laser or electron beam[END_REF][START_REF] Allmen | Laser-beam interactions with materials: physical principles and applications[END_REF], or synthesis of nanoparticles from metal films melting [START_REF] Font | Substrate melting during laser heating of nanoscale metal films[END_REF]. Finally, this class of problems is also a fantastic playground for mathematicians [START_REF] Rubinstein | The Stefan Problem[END_REF][START_REF] Gupta | The Classical Stefan Problem -Basic Concepts, Modelling and Analysis[END_REF]. But to the best of our knowledge, we are not aware of a system of equations modeling solid growth by a sudden contact between a liquid and a cold solid substrate.

In the following, we assume the melt stays at rest, at constant temperature everywhere (T m ). We neglect the variation of heat capacity and of thermal conductivity with temperature. We also neglect the thermal expansion, and more generally the variation of density. In other words, the thermal parameters of the media are the following: the latent heat of solidification L, the heat capacity C pk , the thermal conductivity λ k , the heat diffusion coefficient D k = λ k /(ρ k C pk ) and therefore the density ρ k , are taken as constant. The subscript k stands for l in the liquid phase, i in the solid phase and s for the substrate (see Fig. 1).

The choice of a one-dimensional model will be discussed in §2.3 and is supported by the strong aspect ratio of the impacted drop that will be considered later. We assume that the liquid is at the melting temperature T m : this approximation will also be discussed in §2.3. It can be justified a priori by both, the small thickness of the liquid layer and the little energy needed to cool the water down to its melting temperature :

C pl (T d -T m ) ∼ 4000×20 ∼ 8•10 4 J•kg -1
, where T d is the initial temperature of the liquid drop, compared to the latent heat for solidification (L ∼ 3 • 10 5 J•kg -1 ).

Under these assumptions, the mathematical problem that need to be solved is the following (see Fig. 1): a constant temperature T m in the liquid, two heat equations for the temperature field T (z, t) :

∂T ∂t = D s ∂ 2 T ∂z 2 for z 0 ; ∂T ∂t = D i ∂ 2 T ∂z 2 for 0 z h(t), (2.1) Cold substrate z Solid Liquid h(t) 0 ∞ T T m T 0 T S T 1 T 0 0 Figure 1.
Summary of the model hypotheses: A finite layer of solid lies between the semi-infinite melt (z > h(t)) and the semi-infinite substrate (z < 0). The temperature of the whole melt is set constant at the melting point (T = Tm), while the temperature of the substrate tends to TS when z goes to -∞. The temperature in the solid phases is given by a set of two heat equations, with a specific diffusion coefficient D k for each phase, coupled by the temperature and heat flux continuity at z = 0. At the solidification front (z = h(t)), the Stefan condition imposes the downward thermal flux be equal to the latent heat liberated by the freezing.

and four boundary conditions at the two interfaces :

T (0 -, t) = T (0 + , t) ; λ s ∂T ∂z (0 -, t) = λ i ∂T ∂z (0 + , t), (2.2)
which impose both the continuity of the temperature and of the heat flux at the substrate/solid interface (z = 0), and

T (h(t) -, t) = T m ; λ i ∂T ∂z (h(t) -, t) = ρ i L dh dt , (2.3)
which impose both the continuity of the temperature, and the law of motion of the solidification front (z = h(t)). This energy conservation law under liquid-solid phase change is called the Stefan condition. Here, it imposes that the solidification front velocity is proportional to the rate at which latent heat can be transported in the solid phase. The solidification front is thus controlled by the diffusion in the solid and the substrate through the Stefan condition. Finally, we impose a constant temperature T s far in the substrate :

lim z→-∞ T (z, t) = T s , (2.4)
and we complement this set of equations by the initial conditions taken at t = 0:

T (z, 0) = T s for z 0 and h(0) = 0, (2.5)

indicating that at t = 0 the liquid is suddenly put in contact with the substrate.

Solution of our unidimensional solidification model

Similarity analysis shows that this diffusive problem exhibits a self-similar solution, with the usual self-similar variable involved in diffusion problems η = z √ t , even in the presence of the moving solidification front. In this case, the solidification follows the usual diffusive front growth law:

h(t) = D eff t (2.6)
where D eff is the effective diffusion coefficient that determines the growth of the solid layer. It is really the quantity of interest that we need to compute and thus relate to the different thermal properties of our problem.

Introducing the self-similar variable in the set of Equations (2.1)-(2.5), we obtain the following solutions for the temperature field:

T (z, t) = T 0 + (T 0 -T s ) • Erf z 2 √ D s t for z 0 and
(2.7)

T (z, t) = T 0 + e s e i (T 0 -T s ) • Erf z 2 √ D i t for 0 z h(t) (2.8)
where e s,i = λ s,i ρ s,i C ps,i are the effusivities of the substrate and the solid, and T 0 the contact temperature at the solid/substrate interface (a constant in time in this self-similar framework). The effusivity of a material is the physical quantity that witnesses both its heat capacity and its ability to diffuse it. T 0 is an integration constant to be determined by the boundary conditions. It corresponds to the temperature at the solid/substrate interface and the self-similar behaviour indicates that this temperature is a constant in time! The error function, Erf is defined here by:

Erf(x) = 2 √ π x 0 e -ξ 2 dξ.
(2.9)

Then, by imposing the Stefan condition at z = h(t) (eqs. 2.3), we obtain the following transcendental equation:

St = √ πβ 2 e β 4 e i e s + Erf √ β 2 (2.10)
that links St, the Stefan number, with the ratio of the diffusion coefficients β, defined respectively by:

St = C pi (T m -T s ) L and β = D eff D i (2.11)
From the relation (2.10) it is easy to deduce the asymptotic behaviours for small and large Stefan numbers:

β ∼ 4e 2 s πe 2 i St 2 for St 1,
and β ∼ 4 ln(St) for St 1.

(2.12)

An interesting physical quantity to compute is the dimensionless solid/substrate interface temperature T0 yielding:

T 0 = T 0 -T s T m -T s = 1 1 + es ei Erf √ β 2
(2.13) such that T 0 varies between 0 and 1 with T 0 = 0 for T 0 = T s and T 0 = 1 for T 0 = T m (see Fig. 1). Then from the asymptotic relations between β and St (Eqs. 2.12), we obtain that T 0 → T m for St 1, while for St 1 it converges to an intermediate temperature:

1 0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 1 0 2 1 0 3 St = cP( Tm -Ts ) L 0.2 0.4 0.6 0.8 1.0 T 0 = T0 - TS Tm - TS Ice Marble Steel Copper (b) St = 250 St = 0.25 1 0 -2 -1 0 1 0 1 1 0 -4 1 0 -3 1 0 -2 1 0 -1 1 0 0 1 0 1 Copper Steel Marble Ice (d)
lim St→∞ T 0 = T s + (T m -T s ) 1 1 + es ei (2.14)
which corresponds to the interface temperature when two infinite media are suddenly put in contact (de [START_REF] De Ruiter | Self-peeling of impacting droplets[END_REF].

In order to understand the thermal fields in the substrate and the solid layer, we plot the vertical position normalized by the thickness of the growing ice layer (z/ √ D eff t) as a function of the normalized temperature profiles: The melt is water as it is the most common in our experiments and four substrates have been chosen : copper, steel and marble in order to have a large range of thermal conductivity, and ice in order to compare our result with the classical Stefan problem with no substrate. We observe that the temperature profiles evolve from the substrate temperature T s (T (z→-∞, t) = 0) up to the solidification temperature T m (T (h(t), t) = 1). In the high Stefan numbers regime (Fig. 2 (a)), the temperature profiles exhibit the shape of the error function within both the solid and the substrate domains. The temperature gradient is discontinuous at the contact between the ice and the substrate (z=0), because of the discontinuity of the thermal conductivities. For St = 0.25 (Fig. 2 (b)), the temperature profile tends to be linear in the ice while it still exhibits an error function profile in the substrate.

T (z, t) = T (z, t) -T s T m -
Figures 2 (c) and (d) present respectively the normalized temperature of the solid/substrate interface, T 0 (Eq. 2.13), and the normalized effective diffusion coefficient β = D eff /D i (Eq. 2.6), both plotted as functions of the Stefan number, for the same four couples melt/substrate as before. Remember that increasing the Stefan number corresponds to decrease the substrate temperature.

To better understand the different regimes observed in figure 2, we can investigate the two asymptotic behaviours of our model: the solidification-dominated regime (St 1) and the solid-cooling dominated (St 1). Let us first consider the latter case of large Stefan numbers (St 1) that corresponds to C pi ∆T L, meaning that the latent heat L released by the solidification at the solid/liquid interface z = h(t), is negligible compared to the heat energy released by cooling the solid (C pi ∆T ) from the melting temperature T m to the substrate one T s . In this case, the substrate is almost not warmed up by the solidification and does not influence the front propagation. As a consequence, the dynamics is dominated by the self-similar variation of the temperature profiles with time; they exhibit two Erf function that join at the solid/substrate interface (Fig. 2 (a)). If the diffusive coefficients in the solid and the substrate are the same, which is here the case when the substrate is ice and almost the case when the substrate is marble, the Erf shape of the temperature profile are symmetrical with respect to z = 0. In other cases (steel and copper), this symmetry of the error function is broken. In this limit, the temperature of the solid/substrate interface is the asymptotic limit given by the equation (2.14) (Fig. 2 (c)). Finally, figure 2 (d) confirms that in this limit (St 1), the front propagation dynamics is not much influenced by the substrate thermal conductivity as all the substrates gather on the classical Stefan problem (dark blue curve) with no substrate.

In the opposite limit (St 1), the latent heat released by the solidification is much larger than the heat released by cooling the solid down to T s , indicating that the heat reaching the substrate comes mainly from the solid/liquid interface. The solid layer is inert in the sense that it only transfers the heat flux coming from the solidification to the substrate: the heat flux stays constant through the solid layer. This explains why the temperature field is linear in the solid (Fig. 2 (c)). The slope is selected by the front dynamics and evolves slowly with time, it can be estimated using Eq. (2.12) for St 1 (β ∼ St 2 ) as follows :

λ i ∂T ∂z = ρ i L dh dt = 1 2 ρ i L D eff t ∼ Stρ i L D i t .
This linear profile in the solid joins an Erf profile in the substrate which goes deeper (relatively to h(t)) compared to the large Stefan regime. The freezing process is efficient enough to warm the substrate up. This explains why the interface temperature T 0 increases (Fig. 2(d)) and tends toward T m (T 0 → T m for St → 0 (Eq. 2.13)). Finally, the effective diffusion coefficient is smaller than in the large Stefan regime (Fig. 2 (d)). This can be understood easily since decreasing the Stefan number corresponds to warming the substrate up, so the ice grows slower. In this regime we observe that all the curves follow D eff ∼ St 2 D i as shown on Eq. (2.12). What is very interesting is that D eff with metal substrates differs by several orders of magnitude from the one with no substrate (dark blue curve). Indeed, as the Stefan number decreases, the solid gets more and more inert, and the solidification front propagation becomes more and more influenced by the substrate thermal parameters. In this case, the higher the substrate thermal conductivity, the faster the solidification front propagation (see Eq. (2.12)). This property demonstrates how important it is to take into account the heat transfert within the substrate.

Experimental comparison

In the aim of validating our model and its assumptions, we carry out a model experiment which consists in putting in contact a liquid with a substrate at a temperature below the liquid solidification temperature (see Fig. 3 (a)). The liquid is initially poured in a transparent PMMA tube of 3.4 cm diameter. We use two different liquids (water and hexadecane) and two different substrates (copper and steel). We also varied the initial temperature of the water (20 and 0 • C) and the thickness of the initial water layer (between 4 millimeters and a few centimeters). Finally, the propagation of the solidification front (h(t)) is recorded using a camera, between one and ten frames per second depending on the experiment.

Figure 3 (b) presents the growth of the ice layer (h(t)) with respect to time for water solidifying on a copper substrate. The blue and red squares are for water at 20 • C and two different temperatures of the substrate, respectively -38 • C and -26 • C as indicated on the graph. As expected the freezing front propagates faster when the substrate is colder. On the same graph, the empty triangles show the front propagation with water initially at 0 • C. Their variations coincide with the preceding ones on the whole range of time, indicating that solidifying water at 20 • C or at 0 • C gives rise to the same solidification dynamics. This important observation justifies a posteriori the approximation to consider the liquid layer at T m in the model and to neglect the heat flux in the liquid. Superimposed on these four experimental curves, the lines are the best fit using a square root function of the form :

h(t) = D exp t,
(2.15)

with D exp an experimental diffusion coefficient that quantifies the solidification dynamics. We observe that, as shown in the model, the solidification front follows a diffusive law. Note that in the experiments shown here, the same diffusive law is followed up to 7mm, suggesting that the 1D approximation seems valid at least up to a one fifth aspect ratio. Finally, we plot on the same graph with dashed line the variation of h(t) given by the classical Stefan problem associated to the two substrate temperatures. In these plots, the substrate is treated as if it was ice and the thermal parameters of the substrate are consequently note taken into account. It appears that, by promoting the heat transfert, the substrate highly increases the front propagation velocity. By fitting the solidification front evolution by Eq. (2.15) on each experiment we obtain D exp , the experimental diffusion coefficients. On Fig. 3 (c) we compare them to the effective diffusion coefficient given by our 1D model D eff , for all of our experiments : water (0, 20 • C and different liquid thicknesses) and hexadecane, on copper and steel (see the different markers on the graph), and for various substrate temperatures between 0 and -50 • C, indicated by the colour of the markers. All the data gather along the black dashed line that has a slope of 1, indicating that for each experiment D exp D eff : our 1D model provides therefore an excellent estimation of the dynamics of the upwards solidification when a liquid is placed on the top of a cold substrate.

Solidification dynamics of an impacted drop of water

With the relevant model for the solidification dynamics of a liquid put suddenly in contact with a cold substrate, let us now consider our experiment of a water drop impacting a cold surface and see whether it can predict the final thickness for the frozen impacted drop.

Experimental setup and qualitative description

The classical drop impact setup consists of a syringe pump pushing a liquid through a capillary tube from which the drop falls. As the pumping is slow enough, the size of the drop is entirely controlled by the radius of the capillary tube. We used two different tubes of inner diameter 1600 µm and 250 µm leading to two drop radii: R = 1.9 mm and 1.2 mm, yielding respective volumes 30 µL and 7 µL . The impact velocity U 0 is controlled by the height of fall H, which in our case ranges from 15 cm to 45 cm, so that U 0 ranges from 1.7 m.s -1 to 3 m.s -1 (following roughly U 0 = √ 2gH). Our substrates, made of blocks (100×100×30 mm) of different materials (steel, copper and marble), are placed into a bowl and cooled down by pouring a certain amount of liquid nitrogen. The minimal temperature reached in this work is around -80 • C. Due to the substrate heat capacity and to the bowl thermal isolation, it takes several hours for the system to warm up to room temperature. The change in the substrate temperature is thus much less than 1 • C during the time of an experiment which is roughly 1 second.

In order to minimize frost formation, the whole system is placed inside a regulated atmosphere chamber which allows us to drastically reduce the humidity (less than 1% humidity inside the chamber). The substrate temperature T s is measured before each experiment using a surface thermometer. The dynamic of the impact is studied using a high-speed camera, and the height profile of the frozen drop is extracted with a polychromatic confocal sensor (CCS OptimaPlus from STIL Optics) moving along a translation platform.

At room temperature, as a water drop impacts a solid substrate, it spreads, reaches a maximal radius [START_REF] Laan | Maximum diameter of impacting liquid droplets[END_REF] and immediately starts to retract back to an equilibrium radius [START_REF] Bartolo | Retraction dynamics of aqueous drops upon impact on non-wetting surfaces[END_REF][START_REF] Josserand | Drop impact on a solid surface[END_REF]. Figure 4 (a)-(h) shows a sequence of snapshots of a water drop impacting a sub-zero substrate. Here again, the drop spreads rapidly and reaches a maximal spreading diameter (a)-(b). But then the drop does not retract, it is stuck at its maximal diameter (c). This pinning is due to the formation of a thin layer of ice between the substrate and the liquid during the spreading. On images (c)-(e) the drop is thus made of a thin layer of ice, attached to the substrate and growing vertically, beneath a thicker layer of water where capillary waves are damped. This remaining water layer is not stable due to its high aspect ratio, so that it still needs to retract in order to reach its equilibrium, and we observe on (e) where the liquid layer actually just started to depin from the edge. Then, the liquid retracts on the ice layer (f), until it reaches its typical equilibrium contact angle on ice [START_REF] Knight | Experiments on the contact angle of water on ice[END_REF] and forms a spherical-cap-shaped drop (g). Eventually, it completely freezes, yielding a pointy ice drop [START_REF] Anderson | The case for a dynamic contact angle in containerless solidification[END_REF][START_REF] Snoeijer | Pointy ice-drops: How water freezes into a singular shape[END_REF][START_REF] Marin | Universality of tip singularity formation in freezing water drops[END_REF], on top of an ice pancake (h).

Figure 4 (i) shows the height profile of the frozen drop (h) obtained by scanning the drop diameter with our optical profilometer. This profile shows clearly two different parts: a quasi-cylindrical plate (the so-called ice pancake), whose thickness is denominated as h p , on top of which we find an ice pattern, consequence of the water retraction on ice. We thus define clearly h p as the ice layer thickness of the edge of the structure when it depins. This structure with an ice pancake in contact with the impacted solid is found anytime a drop impacts a sub-zero cold surface. The ice pattern that is eventually formed on the top of the ice pancake can exhibit different shapes that will be the subject of future works.

Here we focus on the ice pancake that is, in fact, crucial in all relevant applications, in particular for the stability of the frozen drop. Indeed, as the temperature of the ice varies, thermal expansion/contraction generates elastic stress that can lead to the delamination of the pancake (de [START_REF] De Ruiter | Self-peeling of impacting droplets[END_REF] or the formation of cracks [START_REF] Ghabache | Frozen impacted drop: From fragmentation to hierarchical crack patterns[END_REF], influenced by the thickness h p . The aim of the present paper is thus dedicated to the quantification of this thickness h p

The stationary contact line (SCL) regime

As we observe on figure 4, after spreading, the contact line seems steady (b-e), before a dewetting transition occurs leading to the retraction of the water film on the ice layer. It is thus expected that the thickness of the ice pancake h p is built by thermal conduction during this time. Consequently, we need first to characterize the duration of this regime, that we call stationary contact line (SCL) regime [START_REF] Rivetti | Universal contact-line dynamics at the nanoscale[END_REF]. Figure 5 (a) presents the variation of the liquid film radius during an impact, plotted as a function of time, for three different substrate temperatures. The dashed line shows the liquid drop radius evolution at room temperature and the two solid lines at freezing temprature : -14 and -30 • C. Without freezing (dashed line), the drop spreads rapidly, reaches its maximal diameter and almost instantaneously retracts. The behaviour is different when the drop freezes. Indeed, after having reached its maximal diameter (illustrated on the curve with the inserts), the stationary contact line regime is observed : the liquid remains attached to the ice layer close to its maximum radius. There, the liquid radius barely varies with time during approximately 60 -70 ms before the retraction of the liquid film starts. This show a sequence at different times (indicated on each image) of a drop of water of radius 1.9 mm impacting at velocity 2.6m.s -1 on an aluminium substrate at temperature -9 • C. (i) is a height profile of the final frozen splat, measured using the optical profilometry. The aspect ratio is 20, so the real splat is much flatter.

time between the spreading and the retraction regimes defines the SCL time : τ SCL , as shown on the curves. We note that the two SCL times, for the two different temperatures, seem to be quite close.

Figure 5(b) shows the dependence of the time τ SCL with the temperature, for different heights of fall and same drop size (1.9 mm) and substrate material (steel). Instead of the substrate temperature T s , we use ∆T = T m -T s , where T m is the freezing temperature of the liquid (here for water T m = 0 • C so that ∆T = -T s with the Celsius scale), which is the relevant temperature here, because we saw that the energy needed to cool the drop down to T m can be neglected. The first observation is that τ SCL always reaches a plateau where it is independent of ∆T . The value of this plateau strongly depends on the drop impact velocity and thus on the initial spreading of the liquid film : the larger the maximal diameter at impact, the shorter the liquid layer stays stationary atop the ice pancake before retracting. These results suggest therefore that τ SCL might be independent of the heat transfert and due to the sole liquid dynamics.

Figure 5(c) presents the variation of τ SCL with ∆T for our three different substrates: marble (triangle markers), steel (square), copper (circle), and two different drop radii on steel substrate (full and empty squares). In each case, the plateau regime is quickly reached confirming this independence of SCL time with ∆T . As expected, the drop size, as the drop impact velocity, seems to play a role in the selection of the plateau value of τ SCL . What is more surprising here is the variation of the plateau value with the substrate material : τ SCL decreases as the substrate thermal conductivity increases. This appears as a contradiction with the previous result that τ SCL does not depend on the substrate temperature and, thus, seems to be independent of the thermal parameters! We will come back on these observations in the following.

To explain the existence of this delay τ SCL between spreading and retraction when water freezes, we will use an argument of a previous work [START_REF] Rivetti | Universal contact-line dynamics at the nanoscale[END_REF] on the relaxation of a contact line pinned at the edge of a polymer film. Similarly, we assume here that the time τ SCL is due to the relaxation dynamics of the contact angle θ formed by the liquid film and the growing ice pancake, and pinned at its edge. The contact line relaxation starts at an angle given by the spreading dynamics (influenced by the drop impact velocity, the substrate wetting angle, and the drop size), and ends when θ reaches a critical value θ * enabling its depinning, required prelude to its receding motion. In the experiment of [START_REF] Rivetti | Universal contact-line dynamics at the nanoscale[END_REF], this angle is θ * = 4.5 • ± 0.5 • and surprisingly appears to be independent of the liquid and of the film thickness.

Because the liquid film is very thin, its dynamics can be taken in the lubrication regime so that the contact line relaxation is expected to follow a capillaro-viscous relaxation time t w ∝ (h 0 η)/γ, with the proportional coefficient being a function of θ * , being about 10 5 in their experiments. Note that, as soon as the contact line retracts, they showed that θ increases to a receding contact angle which stays roughly constant during dewetting.

In order to show that the same relaxation dynamics is at play in our experiments, we compare on figure 5 (d) the characteristic thickness h w of the liquid film that corresponds to the relaxation time τ SCL following this capillaro-viscous dynamics, i.e. h w = γτ SCL /(10 5 η), and the characteristic thickness of the water film h exp tot formed at the end of the spreading, neglecting ice formation : h exp tot = V tot /πR 2 max with V tot the total volume of the drop and R max the spreading radius. Remarkably, the experimental data gather along a line of slope 1, suggesting that this relaxation scenario of the contact line is correct. Note that, since h exp tot corresponds to the total height of the spread drop, it is the sum of the ice pancake and the water film, so that the intersection of the dotted line with the x-axis gives a consistent estimation of the ice pancake thickness. We point out the fact that the thicknesses of the ice and liquid layer are variying in time, so that h exp tot is just a rough approximation of the liquid film thickness. This might explain the data scattering around the linear prediction.

We have used for comparison the same value of θ * than for the polymer films (leading to the 10 5 prefactor for h w ), while there is no reason that this value relevant for water films. In fact, when trying to fit the best line for the data of figure 5(d), we found an angle slightly different but with no significant improvement when comparing with the θ * correlation, so that we have kept this value for the sake of simplicity. The fact that this dewetting angle θ * appears to be almost independent on the substrates, the liquids and the film thicknesses, while our experiments and those of [START_REF] Rivetti | Universal contact-line dynamics at the nanoscale[END_REF] concern film with totally different thickness is a very interesting and intriguing result. It might shed light on an universal dewetting mechanism that should deserve specific investigations in the future.

Finally, following these results, the stationary contact line time (τ SCL ) varies linearly with the thickness of the liquid pancake at impact (h exp tot ). As the spread drop diameter increases with the impact velocity, the liquid pancake is thinner and we understand the decrease of τ SCL when the impact velocity increases, observed on figure 5 (b). These results resolve also the contradiction that we draw concerning the dependance of τ SCL with the different substrate. The contact line relaxation dynamics suggests indeed that the substrate intervenes through its wetting properties that influences the spreading radius, and therefore liquid film thickness pinned on the thin ice layer. In this scenario, the thermal properties of the substrate plays no direct role, explaining thus the independence of τ SCL with the temperature.

The ice pancake

At this point, we know the time during which the ice pancake is building (τ SCL ) and we have a validated model for the ice growth dynamics (D eff ); we now need the experimental measurements of the pancake thickness: h p . Figure 6 (a)-(c) show the ice thickness h p , deduced from the height profiles of the frozen drops (see Fig. 4 (i)), as a function of ∆T , respectively for four different drop impact velocities (a), three different substrates (b) and two different drop sizes (c). These control parameters, and their corresponding markers, are the same as those used in figure 5.

These graphs show that the underlying ice layer becomes thicker when: (i) the substrate is colder (Fig. 6 (a), (b) and (c)); which is expected since more liquid can be frozen during the solidification time that appeared to be mostly constant with ∆T (Fig. 5 (b) and (c)). (ii) The substrate heat conductivity increases (Fig. 6 (b)): the heat is indeed transferred to the substrate with higher efficiency, so that the freezing front can propagate faster during a solidification time that, again, does not vary much (Fig. 5 (d)). (iii) The drop ∆T for different impact velocities, on the same substrate (steel) and with the same drop radius (1.9mm). The colder the substrate is, the thicker the ice layer is. For a given temperature the ice layer is thinner for the highest impact velocities. (b) Evolution of hp along ∆T , for different substrates, with the same impact velocity (2.6 m/s) and drop radius (1.9 mm). For a given temperature, the more conductive the substrate, the thicker the ice layer. (c) Evolution of hp with ∆T for different drop radius, on the same substrate (steel) and at the same velocity (2.6 m/s). The smaller the drop the less it freezes. (d) Rescaling of the measurements against the model: the measured diffusivity of the solidification front, h 2 p /τSCL is plotted against the effective diffusion coefficient obtained through the previously described model (Eq. 2.6 and 2.10). The dashed line has a slope 1.

impact velocity is slower (Fig. 6(a)), or the drop size is smaller (Fig. 6(c)), because the spread drop stays freezing longer before retracting (Fig. 5 (b) and (c)).

We can now compare our experimental results to our theoretical model. Figure 6 (d) presents the variation of h 2 p /τ SCL , which represents the experimental diffusion coefficient of the solidification front, as a function of the theoretical diffusion coefficient D eff , for all the substrate materials, substrate temperatures, drop sizes and impact velocities investigated experimentally. The substrate temperature ranges from -5 • C to -80 • C, which yields the Stefan number ranging from approximately 10 -2 to 1. We observe a nice collapse of all the data into a straight line of slope 1. We can therefore conclude that the ice pancake thickness is well described by the expression :

h 2 p D eff τ SCL
This indicates clearly that our simplified 1D model accounts well for this problem and that the ice pancake formation is controlled by the thermal diffusion during the depinning time of the remaining liquid film.

We observe that some experiments are not collapsing on the line with the others. They appear particularly for an effective diffusive coefficient greater than 3.10 -7 m 2 /s. This limit of our model probably comes from the hypothesis of semi-infinite water in the freezing model. Indeed, if we estimate the thickness of the water film at impact h exp tot with the drop volume V tot and the spreading radius R max , we find h exp tot ∼ V tot /πR 2 max ∼ 200µm, with a drop of volume V tot = 30µL spreading to a radius of R max = 7mm. This thickness is smaller than the highest values of h p , meaning that almost the whole water drop freezes during τ SCL and the hypothesis of semi-infinite water does not hold anymore.

Conclusion

By studying precisely the freezing of a drop impacting a cold substrate, we have shown that the dynamics could be broken down into four phases. (1) A rapid spreading of the drop that ends with a very thin ice layer on top of which a liquid water layer is pinned.

(2) A stillness period during which the water layer is almost at rest, the ice layer growing by thermal exchange, and the contact angle of the liquid layer relaxing. (3) Then, after the time τ SCL , when the contact angle has reached a depinning threshold, the water film retracts on the ice layer until it reaches an equilibrium state. (4) The last phase consists in the freezing of the remaining liquid, forming the final ice pattern. The paper is devoted to the characterization of the thermal dynamics during the second phase time τ SCL when the contact line is pinned. Our goal is to determine the subsequent thickness of the ice pancake, h p .

We have firstly developed a unidimensional solidification model that considers the heat diffusion in the solid and in the ice coupled with the Stefan condition for the solidification front. A dedicated experiment has been run in order to validate this model, showing its relevance and checking its hypotheses, in particular the one-dimensional geometry, the liquid initially taken at T m , and the semi-infinite liquid phase. This leads to the characterization of the diffusion front dynamics, that involves the effective diffusion coefficient D eff .

We have observed that, over a large range of temperature, τ SCL is independent of the temperature and of the substrate thermal parameters, proving that this phase is controlled by the contact angle relaxation. Therefore, we have shown that τ SCL can be roughly expressed by the unique expression :

τ SCL = 10 5 η γ 70 + V tot πR 2 max
Using the model, we provide a general expression for the ice pancake thickness :

h 2 p D eff τ SCL = 10 5 D eff η γ 70 + V tot πR 2 max
that accounts quantitatively for almost all of our experiments. This quantity h p has many practical interests since it provides an estimate of the splat thickness formed by the impact. Controlling this thickness is thus crucial for coating and 3D printing technology [START_REF] Lipson | Fabricated: The new world of 3D printing[END_REF] and for predicting the further mechanical behaviour of the splat [START_REF] Ghabache | Frozen impacted drop: From fragmentation to hierarchical crack patterns[END_REF][START_REF] De Ruiter | Self-peeling of impacting droplets[END_REF]. Our work provides therefore a general framework to model and study more complex configuration such as multiple drops impacts for airplane icing or spray coating [START_REF] Chandra | Formation of solid splats during thermal spray deposition[END_REF] for instance.

Figure 2 .

 2 Figure 2. Results of the model for the different substrates (copper: red, steel: orange, marble: light blue) and ice (dark blue) that is for comparison with the Stefan problem with infinite ice and infinite liquid water. (a) & (b): Dimensionless temperature profiles T (z, t) = T (z,t)-Ts Tm-Ts as function of the self-similar variable z/ √ D eff t, obtained from Eq. 2.8 and 2.7 for two characteristic Stefan number St = 250 (a) and St = 0.25 (b). (c) Dimensionless contact temperature, obtained from Eq. 2.13. Regardless of the substrate, it goes from 1 for St → 0 to the asymptotical values (1 + es/ei) -1 (indicated on the figure) when St → ∞, reminiscent of the contact temperature of two infinite bodies initially at different temperatures. (d) The effective diffusion coefficient of the solidification front normalized by the thermal diffusion coefficient of the ice, β = D eff /Di, as a function of the Stefan number. The differences between substrates are more important at low Stefan number, and the asymptotic regime at low Stefan number β ∝ St 2 is indicated (dashed line).

  T s for two specific values of the Stefan number : a large value St = 250 1 (figure 2(a)) and a value of order one St = 0.25 ∼ 1 (figure 2(b)) ; and for four couples melt/substrat.

Figure 3 .

 3 Figure 3. (a) Schematic of the experimental setup: A liquid-filled cylinder is set upon a cold substrate. The liquid starts freezing into a solid layer of thickness h(t), which is monitored using a camera. (b) Growth of the solid phase as a function of time. h(t) is plotted in the case of water freezing on a copper plate at two different temperatures (blue and red squares), the liquid being initially at room temperature. The same experiments are repeated with water slightly warmer than 0 • C (triangles), showing no differences. The dynamic of each experiment is fitted by a square root function (full lines), in order to get its diffusion coefficient Dexp. Dashed lines represents the dynamic predicted by the classical Stefan problem (without the substrate), for the two different temperatures. (c) Comparison of the experimental freezing dynamic and the theory:The diffusion coefficients Dexp and D eff , respectively experimental and theoretical, are compared for a wide range of parameters. The colour refers to the substrate cooling ∆T = Ts -Tm. Circles (respectively squares) represents water at room temperature freezing on copper (respectively steel). Stars refer to thin thicknesses of water freezing on copper, triangles to 0 • C water freezing on copper. Finally, pentagons are for hexadecane freezing on both metals. The y = x guideline is shown as a dashed line.

Figure 4 .

 4 Figure 4. Snapshots (a)-(h)show a sequence at different times (indicated on each image) of a drop of water of radius 1.9 mm impacting at velocity 2.6m.s -1 on an aluminium substrate at temperature -9 • C. (i) is a height profile of the final frozen splat, measured using the optical profilometry. The aspect ratio is 20, so the real splat is much flatter.

Figure 5 .

 5 Figure5. (a) The dynamic of spreading and retraction of the drop. The radius of the liquid film is plotted as function of time. Inset pictures show the aspect of the film on the different stages. On all graphs, impact velocities are represented by colours (light blue for the slowest to dark purple for the fastest), substrates by symbols (circles for copper, squares for steel and triangles for marble) and drop radii by the filling (full symbols for R = 1.9mm, empty symbols for R = 1.2mm). (b)τSCL as a function of the temperature ∆T for different impact velocities ). The drop radius is 1.9mm and the substrate is steel. (c)τSCLas a function of the temperature ∆T for different substrates. The impact velocity is 2.6 m/s and the drop radius is 1.9mm (full symbols). Results for a drop of R = 1.2mm impacting on steel at the same velocity are shown with empty squares. (d) Theoretical prediction for the water film thickness hw as a function of its experimental estimation h exp tot . Each point represents a series experiments with given impact velocity, substrate and drop radius. The error bars contain the variations within each series. The dashed line is a linear fit with the slope set to one.

Figure 6 .

 6 Figure6. (a) Evolution of the underlying ice plate thickness hp as a function of the temperature ∆T for different impact velocities, on the same substrate (steel) and with the same drop radius (1.9mm). The colder the substrate is, the thicker the ice layer is. For a given temperature the ice layer is thinner for the highest impact velocities. (b) Evolution of hp along ∆T , for different substrates, with the same impact velocity (2.6 m/s) and drop radius (1.9 mm). For a given temperature, the more conductive the substrate, the thicker the ice layer. (c) Evolution of hp with ∆T for different drop radius, on the same substrate (steel) and at the same velocity (2.6 m/s). The smaller the drop the less it freezes. (d) Rescaling of the measurements against the model: the measured diffusivity of the solidification front, h 2 p /τSCL is plotted against the effective diffusion coefficient obtained through the previously described model (Eq. 2.6 and 2.10). The dashed line has a slope 1.


Flavien Lécuyer

Valérie Gouranton

Adrien Reuzeau

Ronan Gaugne

Bruno Arnaldi

Authoring AR Interaction by AR

Figure 1: From left to right, the user (a) adds an interactive behaviour on a bottle of glue, (b) imports the interactions in the environment, and (c) uses the interaction to pour the virtual glue from the real bottle into a virtual pot

Introduction

Augmented reality usage is rapidly growing in multiple domains [BCL15]. As the general demand for augmented reality experiences grows fast, the question of making the applications more interactive becomes more important too. While the design of interactive objects in virtual reality is now quite democratized, it is still difficult to provide such interaction in augmented reality. In the creation of applications, one of the main time-consuming tasks is to take into account the needs of the application. This requires many discussions between the client -who is often an expert in the concerned domain (e.g. an educational expert for training systems), and is not familiar with computer science -and the developer. The developer to adapt easily to different contexts. To make the designer capable of designing interactive augmented reality applications, the tool is based on the use of extensible libraries. The interactions are created with a novel paradigm, based on existing work from virtual reality, which we adapted to take into account the difference between real and virtual objects, which is crucial for augmented reality. Those libraries allow a developer to create new interactions with an ensured reusability thanks to the object-relation paradigm [LC07]. Since the tracking is another difficulty for a novice, our tool integrates the management for the tracking through off-the-shelf solutions, making the overlay of virtual 3D objects easy to edit for the author of an application. The architecture of the tool allows the integration of other tracking solutions, so that a developer can enrich it and customize the tool to adapt it to the user's needs.

In this paper, after the analysis of related works in Section 2, we describe the problematic of authoring interaction for augmented reality in Section 3. We then give a brief overview of our solution in Section 4, to give more details in Section 5. We also illustrate the use of this solution with an example in Section 6. A video of this example os also available at https://vimeo. com/349891514.

Related works

In this section, we give an overview of works related to the authoring of interactive reality applications. For the creation of augmented reality applications, the designers need easy to use authoring tools, that allow to manage the placement of virtual objects according to the real world. For this, there are authoring tools designed to let a designer create an application with no code, albeit with little to no interaction. We start this section by presenting those works. To make interactive augmented reality applications, it is important to have interactions that are easy to implement, instead of having to code them anew for each application. To improve the creation of augmented reality applications, some works focused on easing the design of interaction for augmented reality. We describe them in the second part of this section. Another way of easing the creation of interactive applications is to improve the reusability of the code. This has been done for virtual reality, with the objectrelation concept, which we focus on in the last part of this section.

As augmented reality is a strong tool for the presentation of augmented information, some tools aim at providing the means for an author to create augmented presentations. The first of them, Pow-erSpace, is based on the use of the slideshow editor PowerPoint, which is well-known by the general user, and allows them to be more efficient during the authoring [HR02]. However, the content authored with PowerSpace is limited to the capacity of a slideshow. This slideshow format can also be authored directly in-situ, as done with ASMIM [NMTS18], where the author can create steps based on the environment, augmented with some layers of information. A similar effort can be found with AMIRE-ES [HSZ05], where the author can create an augmented reality guide that can be used with different devices. In ACARS, the authors also propose a limited customization for some of the properties of the objects according to the user's preferences [ZON13]. The combination of on-site and off-site use allows a more refined authoring. This in-situ authoring can also be adapted for the augmentation of outdoor environments, as with Sketching up the world [LMZ * 12]. A more immersive presentation of the content can be found by using outdoor AR, where the augmentation is provided at given points in space. To this extent, Guven et al propose to augment a 3D space by selecting specific points, and associating content to it [GF03]. Similarly, AR Game Builder associates content to GPS positions [START_REF] Klopfer | Augmenting your own reality: Student authoring of science-based augmented reality games[END_REF]. This content takes the form of text and images to narrate a story. Finally, AREDA proposes to record directly the user to generate an application [START_REF] Bhattacharya | Augmented reality via expert demonstration authoring (AREDA)[END_REF]. The application generated is similar to a slideshow, with each step displayed when the object placement from the previous step is done by the user.

Augmented reality can be used with more complex interactions. In Virtual Museums, the authors propose to associate 3D models with fiducial markers [LSB * 04]. The augmentation is interactive and reacts to the state of the markers, and can be customized through a desktop interface to modify the visualisation of the object (e.g. color, position, ...). The authoring can also be done in immersion in the environment. By providing a tangible interface for the authoring, Lee et al. allow the user to act directly on the environment with simple metaphors [LNBK04], with some simple interactions for the augmented objects. For more customizable interactions, Broll et al. propose to add behaviours on the objects [START_REF] Broll | Interactive Bits: Prototyping of Mixed Reality Applications and Interaction Techniques through Visual Programming[END_REF]. The behaviours come from scripts and are attached automatically on certain objects, making it difficultly usable for a domain expert. The same observation can be made for AR-Room, where the behaviours must be created with scripts by a computer scientist [START_REF]AR-Room: a rapid prototyping framework for augmented reality applications[END_REF]. This system of behaviours is also present in APRIL to make the objects interact, although the interactions are still strongly oriented towards augmented presentations [LS05]. Those behaviours can be triggered by certain conditions, such as the contexts presented in k-MART [CKL * 10], in which the user can select when a given behaviour is to be used. It is to note, however, that the behaviours proposed are quite limited. In DART, the behaviours function as a combination between cues and actions, which make an object react to certain conditions (e.g. occlusion of a marker) with a given action (e.g. an animation) [MGDB04]. The user can combine both as they want to generate more complex behaviours, but the available cues are limited to what a marker can do. A similar proposition is made with BEAR, where a given set of states can be associated with actions on the objects [LC10]. However, the creation of the behaviours require some knowledge in computer science, making the tool less adapted to a designer.

In virtual reality, more attention have been put on the efficient design of interactions. Particularly, the object-relation paradigm aims at improving the reusability of the interactions that have been defined, through the separation of the action (represented by the relation) and the objects involved. Although any number of object can be involved in a relation, this is particularly interesting for actions that require more than two objects, as the complexity remains low on the developer's end. To ensure an easy reuse of the interactions, as well as a good efficiency during the design of those interactions, we designed a new methodology for the design and implementation of the interactions, based on the object-relation paradigm. More details about this are given in Section 5.1. In the same way, the tracking is managed as the integration of third party software. The integration is managed in a modular way, so that any third party software can be used, according to what is needed for the application.

On the other side, the designer is given an authoring tool, which is enriched by those libraries, to create interactive augmented reality applications. The main interface of this tool is illustrated in the Figure 3. As illustrated in Figure 2, those libraries are then used to enrich the authoring tool, which can in turn be used by the designer to create an interactive augmented reality application.

To help create interactive applications with this novel paradigm for augmented reality, we provide an authoring tool based on it, to easily create interactive augmented reality applications. In the rest of this paper, we present more details in Section 5, which we illustrate with an example in Section 6.

Authoring AR by AR

In this section, we give more details on the methodology and authoring tool to create interactive augmented reality applications. which proposes a framework independent from any platform, to make it more versatile.

To conclude this discussion about the existing authoring tools for AR, an important observation can be made: for now, the authoring tools have mostly followed the tendency of using AR mainly as a presentation means. Although some other tools tried to bring more interaction into AR, their lack of affordability prevented them from truly attaining this goal. In parallel, some models have emerged for virtual reality to manage this kind of interaction, but they only manage virtual objects, and cannot be used as is for AR.

Needs

Few augmented reality applications have put a focus on the interaction. In order to democratize interaction for augmented reality applications, we need to provide efficient ways to create it. In most works, the position and occlusion state of the real objects are observed, and used to trigger some behaviours -usually, some animation -once certain conditions on those properties are verified. Because of this, the interactions between the real and virtual objects are often limited (mostly occlusion and collisions). However, a higher level of interaction could be proposed in augmented reality, similar to the one provided in virtual reality, where the objects can have complex behaviours and evolve according to the state of the world and the user's actions.

Indeed, the manipulation of real objects is a powerful means to make the user get a feeling akin to the reality, while the overlay of virtual elements gives more information and improves the experience. Thanks to this, and although augmented reality is more constrained than virtual reality, it has a great potential in terms of fully interactive experiences. Thanks to many works on it, the tracking solutions for augmented reality can now be considered sufficient for the average needs, at least for indoor applications [MUS16]. However, interactions for augmented reality have been far less studied. Because of this, it is still usually the role of the developer to create such interactions, according to the needs of a designer.

Solution overview

To support the creation of such interactive augmented experiences, we propose an authoring tool, using a novel methodology for the implementation of the interactions, based on the object-relation paradigm already used in virtual reality. To palliate the lack of interactivity in augmented reality applications, we propose a new framework for the creation of interactions in augmented reality. This framework is divided in two parts: on the side of the developer, we provide a new methodology to design and implement the interactions. For the designer -whom can be a domain expert, an artist, or a graphist for instance -we provide an authoring tool based on this novel methodology, to make an easy use of the Both parts are designed in a scalable way, in order to make it easy to adapt to any domain of application.

Usually, the work of the developer consists in fulfilling the demands of the designer, with the difficulty induced by the difference in their vocabularies. With our framework, the role of the developer is deported upstream. Indeed, the developer becomes in charge of Figure 4: To extend the object-relation to AR, we make the distinction between real and virtual objects, with links to the tracking and 3D models libraries respectively Figure 5: An editor is provided to ease the creation of new relations First, we describe how the interactions are designed in Section 5.1, and how they are used in the authoring in Section 5.2. We then present how the tracking and virtual objects are managed in the authoring, in Sections 5.3 and 5.4 respectively. Finally, we discuss how the authoring data is managed in the tool in Section 5.5.

Creation of the interactions

The interactions are created by the developer, with the help of the object-relation paradigm. The object-relation paradigm is a paradigm created to manage interactions for virtual reality. Its goal is to design the interactions in such a way that encourages the reuse, instead of the reimplementation.

Usually, the object-relation models are based on two main concepts: the types objects and the relations (see Figure 4). The objects in the environment are made interactive by receiving types (e.g. Screwer), which gives them behaviours (e.g. to turn when used), and bear properties that can be used by the relations to interact with the objects (e.g. a screw type). The types can also be defined parametric types, using values to modify their behaviour. For instance, with a single type Screwable and a value screw_type, multiple screw types can be easily defined. The types can also be enriched with variables representing the current status of the object.

Those variables can be manipulated by the relations to modify the behaviour and status of the objects. In complement, the relations define the needed types for the objects, a condition for the execution (e.g. compatible screw types), and an action to execute (e.g. screw and/or display some text). The object-relation paradigm also allows to create relations involving more than two objects, such as soldering. For this, a relation with four objects (a soldering iron, two pieces to connect, and a metal source) could be created. This separation between the types, representing the capacities of the objects, and the relations representing the actions, makes the definition of the interactions more flexible.

For the developer, the interest of this approach is twofold. First, the use of the object-relation paradigm allows to gain time during the coding of interactions, whether they are reused or implemented using the interface. Secondly, the work of the developer is pushed upstream in the development process. This is particularly useful, as the work of the developer can be retargeted on the creation of code libraries, instead of being constantly going and coming with the designer to adapt the code to their needs.

For the use with augmented reality, we add a particular type on the objects, depending on whether it is virtual or physically real. This allows to define different behaviours for a same interaction, depending on the nature of the objects interacting. For instance, assembling a virtual tip on a real screwdriver will not be managed the same way as if the tip was real and the screwdriver virtual.

To make the creation of new interactions easier and faster, an C# interface is provided to create new relations. A graphical editor is also integrated in Unity, for both the types and relations, and can be used to ensure that the relation is properly built. An example, for the relation RelationPick, is given in Figure 5.

Integration of the interactions in the authoring tool

In the authoring tool, the types and relations defined in the libraries by the developer can be used as add-ons by the designer.

As can be seen in Figure 3, when an object is detected by the tracking system, a panel with its current types will be automatically displayed on the left of the screen, and a second panel with the available types is displayed on the right. This allows the designer to add and customize as many types as needed for a single object. For the screwdriver on Figure 3, we can see the type Alias that is already attached on the object, as well as the InteractionArea one, that can be modified to customize the bounding box that will be taken into account for the interactions. The MainComponent one is used to let the screwdriver interact with the tips.

The types that are mandatory for the object, such as the Alias or Virtual one (depending on the physical/virtual nature of the object), are placed on the object from the start and cannot be removed by the designer. As such, they are not proposed on the available types, and are locked on the inspection of current types. Similarly to the list of available types, a list of available relations can be accessed from the authoring tool. On the example in Figure 6, no relation are used in the application. However, some relations on the DIY theme are proposed and can be selected by the designer. For the screwing action, for instance, the designer would need the

virtual objects as it appears in the application

To prevent this, the virtual representation and interactive areas are automatically simplified to rectangular bounding boxes, as shown in Figure 7. As an example, we already integrated Vuforia using this architecture. For the case of Vuforia, the virtual representations are set as Vuforia targets, which are then automatically updated by the tracking API. Those representations also bear the Alias type, which will be used by the relations to get the necessary information about the real object. For the developer, this just means that the third party API needs to be integrated normally, with the addition of the Alias type on the virtual representation used to translate the tracking information in the virtual environment. The real objects are then registered in a library, which is used by the authoring tool to automatically recognize them.

Integrate the 3D models

Since both the developer and the designer can bring their own 3D models, it is important to let them both have a way to integrate them in the application. The 3D models provided for the creation of virtual objects in the authoring tool are retrieved from a specific folder on the device on which the authoring tool is installed. To provide the designer some basic virtual objects to use in the authoring tool, some simple 3D models are provided by default. Others 3D models can be added by placing the files in the specific folder. However, to ease the process, the designer can import new 3d models directly from the authoring tool. This lets the designer modify by themselves the contents of the library, with their own 3D models if needed. Thanks to this, a developer is no longer required to integrate the 3D models in the application. As the authoring tool is implemented using the Unity game engine, the format for the 3D models are the ones supported by Unity (OBJ, FBX, etc.). Once those objects are added to the authoring tool, they will then appear in a list, as displayed in Figure 8. This list can then be used by the designer to add objects in the scene.

Export of the content

To use the application after the authoring is done, the designer needs a way to export the created content. Inside of the authoring tool, the data is structured in three parts, as shown in Figure 9: the scene, the interactions, and the tracking information. Those three parts are integrated in a Unity project that is used in real time by This integration of the types and relations is especially appreciable for the designer, as it allows them to experiment with the available interactions by themselves, without having to request the help of a developer to do so. Since the integration of the types and relations in the authoring tool is automatic, the work of the developer is also eased here.

Integration of the tracking libraries

To prevent any constraint from the choice of a single tracking API that becomes integrated in the authoring tool, we decided to integrate the tracking third parties as modules that can be added to the application. Thanks to this, the choice in the tracking API can be made according to the needs of the designer. We defined a minimal common API to communicate with any tracking system, in order to ensure its genericity. To make the rest of the application independent from the tracking software, each real object is represented virtually in the environment. This virtual representation has the Alias type affixed to it. The Alias type is in charge of updating the virtual representation's pose according to the tracking information, and can be used by the relations to get information on the object, or for occlusion for instance. Although any kind of object can be tracked, given that the tracking software in use is able to detect it correctly, the use of a complex model makes it rapidly quite heavy to compute, and difficult to manipulate for the designer.

Figure 9: The export from the authoring tool generates the code from the application the authoring tool, and that can also be exported for further use, such as refining the animations for a better experience. The choice of those three parts is motivated by the wish to keep the libraries as well separated as possible from the content that has been authored, which in turn makes it simpler for a developer to reuse the generated code. To do so, the content is integrated in the scene, and references the different libraries. As illustrated in Figure 9, the authoring generates the scene, in which the objects are enhanced with types. Some specific Unity objects are also added for the management of the tracking and the relations. First, the scene created from the authoring tool is exported. Since the virtual objects are managed by a scene graph, in which each object is placed relatively to another one that already exists, the scene in the Unity project is constructed with this same scene graph.

The code for the interactions is distributed between the two main entities of the object-relation paradigm: the relations, and the types. The types are added as Unity components on the objects, that can easily be removed or added from the Unity editor after the authoring. The relations are added in the Unity project as separate objects, with a behaviour coming from the relation itself. The types and relations are created using the framework #FIVE [BGB * 15], which we extended to manage augmented reality. This choice is motivated by the strong independance of #FIVE regarding the target device, which makes it easier to adapt for augmented reality. The engine that manages the test and launch of the relations is also using #FIVE, and is automatically included in the project. Since the relations and types are taken from libraries, there are not supposed to be modified by a developer. However, this is still possible.

Thirdly, the tracking information is integrated in the Unity project, using the format preferred by the tracking API in use. Usually, the tracking information is put on each virtual representation, so that it can be tracked when the project is launched.

The use of multiple third parties, such as #FIVE to manage the relations, and the use of third party tracking, is motivated by the need for the tool to be as scalable as possible. To this end, the tool is also designed to be simple to adapt to other third parties.

Example of use

In this section, we detail how the authoring tool can be used, through an example. First, we present a use case in Section 6.1. We chose the domain of DIY for this use case as the actions for this domain are quite well-known, but the approach could be, of course, scaled to other domains. Then, we illustrate how a designer would add a real object for such a use case.

Use case

In order to demonstrate the use of our authoring tool, we designed a short use case for illustration sake. The goal of this use case is to build an application in which a user will build a piece of furniture with different tools. The application, designed specifically to demonstrate the authoring, uses several real props, as well as a tablet to augment the environment, as shown in Figure 10. The realization of the piece of furniture is divided in two tasks: an assembly and a screwing task. The user can do those two tasks in any order.

The first task consists in placing a virtual logo on the real plank. To do it, the user must first take some virtual glue with a real brush and spread it on the plank. Then, the user must prepare the logo by cutting it out from a virtual piece of paper, and place it on the plank, where the glue is.Finally, the task is completed by using a real pallet to flatten the logo on the plank.

The screwing task consists in screwing some virtual screws and bolts on a real plank. In this task, there are three virtual screws: two cross-head ones and a slot-head one, as well as a virtual bolt to tighten. To help in the task, the user can use an electric screwdriver -a real one -with different virtual tips, and a real wrench. The use of the screwdriver is illustrated in Figure 11. in Figure 12, the types that can be added are displayed on the right panel. The ones already present on an object are displayed on the left panel, from which the designer can customize their properties. In this inspector panel, the designer can get the information about the object and its types. If the type bears some properties -as, in this case, a list of possible contents -the value can be modified from the inspector too.

For the objects to interact with each other, the designer chooses the relations to use as being possible relations for the environment. To do this, a list of available relations is proposed to the author, as shown in Figure 6. In this list, the designer can select the relation they want and add it to the scene. Once the relation is added in the scene, it can be use by the object-relation engine, which will try to match objects to launch the relations.

Conclusion

In this paper, we presented a new methodology for the creation of interactions in augmented reality, supported by an authoring tool. Our goal with this methodology and tool is to give to the designers of all domains the opportunity to create interactive experiences with augmented reality, without feeling limited by the technology barrier. Giving the designers this opportunity is particularly important, as the democratisation of augmented reality can difficultly be reached without an important contribution from the designers.

To make the interactions easily accessible for those designers, we designed a specific paradigm for the coding of said interactions. This paradigm is based on the object-relation paradigm, which is quite useful for the creation of interactions in virtual reality. We adapted this paradigm for augmented reality, by letting it take into account whether an object is real and virtual. With this paradigm, the interactions created by a developer are by nature easy to reuse, and will enrich libraries for the authoring tool.

Thanks to this novel paradigm for the design of interactions, and to the authoring tool supporting it, the creation of interactive augmented reality experiences is made easier for people without particular knowledge in computer science. This paradigm and this tool With these objects and interactions, a complete interactive application can be generated without the need for the author to know anything about coding. For someone who is not yet familiar with the tool, the creation of this application can be done in around an hour an a half, considering that the developer has already configured the tracking system. In this time, five minutes are used to manage the relations, and approximately five minutes are spent for each object, to select the types and configure them. The work by the developer prior to the authoring is difficult to estimate, as it depends highly on the tracking system to integrate.

In order to illustrate how the designer would use the tool for the creation of an application, let us study a short example. In this example, and unless specified, the designer is the one doing the authoring. In many cases during the use of an augmented reality application, a real object is used to affect the virtual environment.

To create an interactive environment, the designer can add virtual objects, real objects and interactions, in any order.

Setting of a real object

For the addition of a real object, the designer needs to initialize the tracking and configure the real object in the environment. It is to note that the setting of a virtual object is mostly the same, except that the first three steps are replaced by the selection of a 3D model in the tool. This is done through the following steps:

Tracking initialization First, the designer must get tracking information about the object. For this, the designer can scan the object beforehand, to get the tracking features that will serve for the recognition afterwards. With the tracking system we used for this example, Vuforia, the designer can place the object on a reference pattern, use the provided application (Scanner) to get the tracking data and use it in the authoring tool. This is done easily, by adding a Vuforia object recognition manager for the real object, directly in the game engine by using a prefab and selecting the file containing the features for the intended object. This prefab is integrated in the authoring tool, and uses the scripts from the tracking API to make them easier to use. Although this can be done by a designer, this is still easier for a developer. Of course, this tracking initialization process is dependent of the tracking system in use, so this step can differ with another system.

Creation of the virtual representation

Once the object is recognized in the environment, a virtual representation of it is automatically created and co-localized with the real object. Once the virtual representation is added, it is automatically used to generate occlusion with the virtual objects. To make it simpler for the author, the virtual representation is not displayed. This lets the designer act on the real object in a more transparent manner.

Addition of the Alias type Once the object is recognized and its appearance is set; the Alias type is automatically added on it to state that the object with this type is in fact the virtual representation of an object tracked in the real world. This type modifies the activity of the object according to this fact.

Addition of the types for the object When the object is integrated in the environment, it is time for the designer to define what types it should bear. This can be done using the object inspector. As can also be used directly by computer scientists, to help them design their applications faster. The sources of the authored application can also be retrieved, to be modified externally by a developer who would want it.

To be able to evaluate further our solution, we would like to use it on other domains. This would allow to create new libraries for other domains, to enrich the list of libraries to include in the authoring tool. For instance, we think about using this authoring methodology for the medical domain, which presents many different objects, each with a precise use. Because of this, it would allow us to work on a consequent set of new relations, types, and 3D models.

In the same kind of idea, it would be interesting to integrate new tracking libraries. For instance, since many applications use fiducial markers as their tracking targets, ARToolkit would be a good addition to the authoring tool [BPKM00].

  It has first been proposed with the model Cause & Effects [LC07]. It has also been used by the models Storm [MGA07], and Mascaret [CTB * 12], which integrated a graphic model to simplify the use of their models. Lanquepin et al. also proposed Domain-DL, integrated in the platform HUMANS [LCL * 13], to benefit from the other models in the platform, such as a scenario model. The most recent model is #FIVE [BGB * 15],

Figure 2 :

 2 Figure 2: The main concepts of the framework

Figure 3 :

 3 Figure 3: The inspector for the types in the authoring tool

Figure 6 :

 6 Figure 6: The list of relations in the tool Figure 8: List of virtual objects as it appears in the application

Figure 7 :

 7 Figure 7: A real screwdriver, with its interactive area in red. The virtual representation is voluntarily modified to change the interactible zone

Figure 10 :

 10 Figure 10: Setting of the application, with the tablet and the real objects

Figure 11 :

 11 Figure 11: Example of an interaction from the use case. The user is using the screwdriver to drive the top left screw

Figure 12 :

 12 Figure 12: Inspector for an object. The left panel indicates the current types for the object, and the right one proposes the available types to add on the object.

Acknowledgements

This work is part of the ANR-16-FRQC-0004 INTROSPECT project, and the SUNSET project funded by the ANR-10-LABX-07-01 "Investing for the Future" program.