Gwenn Boedec

Julien Deschamps
email: julien.deschamps@univ-amu.fr

Disk wrinkling under gravity

Keywords: Elasticity, Wrinkled thin plate, Instability, Föppl-von Karman equation

We study the deflection by gravity of a circular elastic disk deposited on a rigid support. The axisymmetric deflection induces a compressive orthoradial stresses which leads to a wrinkling instability above a critical threshold of the dimensionless gravity force. We study this instability by a combination of experiments, numerical simulations and analytical tools, with a particular focus on the role of geometry. We show that aspect ratio is a crucial parameter that controls both the threshold of instability and the most unstable mode. The influence of this parameter on the threshold can be catched by introducing a new nondimensionalization of the transverse load.

Introduction

Depositing a circular napkin on a circular table with a smaller radius, one may observe that the outer boundary undulates, with a wavelength depending on the ratio of the napkin radius and the table radius. This is a manifestation of an elastic instability called wrinkling, where the system releases some stretching energy by developing out-of-plane undulations. Many different systems may develop wrinkles, like stretched [START_REF] Cerda | Wrinkling of an elastic sheet under tension[END_REF] or sheared [START_REF] Wong | Wrinkled membranes i: experiments[END_REF] rectangular elastic plates, tensed circular elastic films under indentation [START_REF] Huang | Capillary wrinkling of floating thin polymer films[END_REF][START_REF] Holmes | Draping films: A wrinkle to fold transition[END_REF][START_REF] Paulsen | Curvature-induced stiffness and the spatial variation of wavelength in wrinkled sheets[END_REF], or deposited on a droplet [START_REF] King | Elastic sheet on a liquid drop reveals wrinkling and crumpling 30 as distinct symmetry-breaking instabilities[END_REF]. One prototypical situation to study wrinkling is the Lamé setup, where an annulus of elastic material is submitted to radial tension on its edges [START_REF] Davidovitch | Prototypical model for tensional wrinkling in thin sheets[END_REF]. Depending on the differential tension, a zone of compressive hoop stress may develop, which leads ultimately to the formation of wrinkles. Role of geometry is intricate : theoretical analysis [START_REF] Coman | On the wrinkling of a prestressed annular thin film in tension[END_REF] shows that aspect ratio has an importance on the wavenumber selection, and that pockets of instability overlap, which may lead to multistable systems. While the Lamé setup is well-defined theoretically, its experimental realization is trickier since it requires to induce a differential stresses at the edges while still keeping the sheet planar (before wrinkling) : this has been re-cently realized by using surface tension variations with surfactants [START_REF] Piñeirua | Capillary buckling of a floating annulus[END_REF].

In other situations, the sheet is generally deformed out of plane, and this deformation might also generate compressive hoop stress : in a setup such as the napkin on the table, the gravity-induced deflection of the sheet is in fact the driving mechanism for the development of a region under compression. A similar situation may be encountered in the context of lightweight deployable space structures, like solar sails or solar power satellites [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF][START_REF] Sader | Shear-induced buckling of a thin elastic disk undergoing spin-up[END_REF]. In this case, the weight is replaced by the load of the solar light so that maintaining an (almost) flat shape is important for efficiency, and may be attained by using centrifugal forces.

In this paper, we study gravity-induced wrinkling of the overhang part of an elastic disk deposited on a rigid circular support of a smaller size. A theoretical study by [START_REF] Cerda | The elements of draping[END_REF] has shown that in the limit of a punctual support, there is a bifurcation between a one-folded shape similar to a d-cone, and a two-folded shape. This study was mostly concerned with asymptotic behavior in the fully non-linear regime (large deflection). On the other hand, [START_REF] Chen | Non-axisymmetric warping of a heavy circular plate on a flexible ring support[END_REF] study theoretically (numerically) the deformation under gravity of a circular plate supported by an inner ring, in the small to moderate deflection regime.

They show that first an axisymmetric solution exists (no wrinkles), and that, for further loading, this solution destabilizes, with a critical mode number and a critical load depending on the aspect ratio. [START_REF] Chen | Warping of stationary and rotating heavy disks[END_REF] then extended the theoretical study to include the effect of rotation and conducted some experiments showing the coexistence of cos 2θ and cos 3θ deformations. Recently, [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF] conducted a detailed analysis of spinning transversely loaded membranes, coupling experiments and numerical analysis with a buckling analysis of Föppl-von Karman equations to study the effect of rotation on the wrinkling patterns. In the same manner, [START_REF] Sader | Shear-induced buckling of a thin elastic disk undergoing spin-up[END_REF] considered the impact of the spin-up of the disk on the buckled modes. Despite the strong effect of the aspect ratio on the critical load and on the observed wavenumber, both [START_REF] Chen | Warping of stationary and rotating heavy disks[END_REF] and [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF] conducted experiments with a unique fixed aspect ratio (0.1 for [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF] and 0.3 for [START_REF] Chen | Warping of stationary and rotating heavy disks[END_REF]), focusing mostly on the importance of rotation : No systematic experimental diagram study was conducted. As a first step the present study is concerned with the influence of aspect ratio for a non-rotating disk, deflected only by gravity. We systematically vary the relevant parameters to obtain an experimental phase diagram of the system, and show that the two dimensionless parameters can be collapsed in a single one controlling the transition between wrinkled and unwrinkled state. We rationalize this finding by developing an original asymptotic analysis which predicts analytically the instability criteria. These results are systematically validated by comparison with a buckling analysis close to the threshold and full numerical simulations of thin-shell equations for the nonlinear regime.

System description

We consider a disk of radius b made of an elastic material. The thickness of the disk h is such that h/b 1, thus the disk may appropriately be described as a thin plate whose non-deformed shape is planar. This disk is deposited on a circular rigid support of radius a < b and we let it hang freely under the action of gravity. Our system is thus equivalent to an annulus of elastic material, whose inner edge is clamped and whose outer edge is free (see figure 1 a). While this clamped boundary condition is not strictly enforced experimentally we checked that no detachment of the inner part occurs in the data reported in this paper.

Due to gravity the annulus deflects downwards. For very small deflections, the shape remains axisymmetric as the annulus adopts a conical shape, but as the deflection, i.e. the effect of gravity, increases, the outer edge starts to undulate orthoradially as a wrinkling instability develops on the annulus (see figure 1 b). This instability can be simply understood as it is not possible for the outer edge to be deflected without stretching, either radially or orthoradially, the sheet. A part of the resulting stretching energy due to compressive hoop stress can be relaxed by developing out-ofplanes undulations at the expense of some bending energy that is energetically favorable since the sheet thickness is small. Following [START_REF] Chen | Non-axisymmetric warping of a heavy circular plate on a flexible ring support[END_REF][START_REF] Chen | Warping of stationary and rotating heavy disks[END_REF][START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF], two dimensionless parameters control this instability.

First, a dimensionless measure of gravity G is defined as :

G = 12(1 -ν 2 )ρgb 4 Eh 3
where ρ is the volume mass of the disk, g is the acceleration due to gravity, E, ν are the Young modulus and Poisson coefficient of the material. Second, a geometrical parameter is involved : the aspect ratio of the annulus :

α = a b
3. Methods

Experiments

We use disks cut into elastic sheets made of four different mate- 1.

Once the disk is characterized, it is gently deposited on cylindrical support of radius a. We simulate our setup using an in-house code which solves thin plate equations based on an isogeometric framework. The code is built upon a method using subdivision elements [START_REF] Cirak | Subdivision surfaces: a new paradigm for thin-shell finite-element analysis[END_REF][START_REF] Cirak | Fully c1-conforming subdivision elements for finite deformation thin-shell analysis[END_REF], with the improvement of [START_REF] Green | Second-order accurate constraint formulation for subdivision finite element simulation of thin shells[END_REF] for dealing with boundary conditions: we use a clamped boundary for the inner edge and a free boundary condition for the outer edge. The plate kinematics are based on Kirchoff-Love theory, assuming that in the deformed configuration, the plate director remains normal to the mid surface. Within this framework, plate deformation is completely given in terms of membrane strains and bending strains defined on the mid surface. Note that we do not restrict the description to linearized kinematics and resolve the non-linear problem with a modified Newton-Raphson solver. For the material behavior, a generalized Hooke law is chosen, meaning that we have a linear elastic material behavior, but with all geometrical nonlinearities. The plate is discretized using Catmull-Clark elements.

Material E (MPa) h (µm) b (mm) PDMS 0.2 (±0.02)-2.3 (±0.1) 220 (±8)-1800 (±60) 15-120 (±1) Silicone 0.5-1 (± 0.1) 150 (±6)-3200 (±50) 25-135 (±1) Latex 2.6 (±0.1) 150-230 (±20) 15-70 (±1) Mylar 4000 (±200) 125 (±20) 55-110 (±1)

Stability analysis

We use the method of [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF] to compute the critical load. We briefly summarize the main ideas here, and refer to their work for a complete description of the method. The principle is as follows :

complete Föppl-von Karman equations 1-2 are solved via a boundary value problem solver (e.g. the solve bvp function in Python) to first obtain a stationary axisymmetric solution for a given couple (G, α):

D∇ 4 z =∂ rr z 1 r ∂ r φ + 1 r 2 ∂ θθ φ + ∂ rr φ 1 r ∂ r z + 1 r 2 ∂ θθ z -2 1 r ∂ rθ φ - 1 r 2 ∂ θ φ 1 r ∂ rθ z - 1 r 2 ∂ θ z + ρgh (1)
∇ 4 φ = Eh 1 r ∂ rθ z - 1 r 2 ∂ θ z 2 -∂ rr z 1 r ∂ r z + 1 r 2 ∂ θθ z (2)
where z(r, θ) is the vertical deflection of the deformed disk (figure 1 a) and φ(r, θ) is the Airy function related to the stress components:

σ rr = 1 h 1 r ∂ r φ + 1 r 2 ∂ θθ φ σ θθ = 1 h ∂ rr φ σ rθ = - 1 h ∂ r 1 r ∂ θ φ
The solution satisfies clamped boundary conditions (no displacement, no rotation) at the inner edge and free boundary (no stresses, no moments) at the outer edge. Then, the stability of this axisymmetric solution with respect to buckling is computed by introducing a perturbation with an assumed form z(r, θ) = Z(r) exp(inθ). After 130 linearization of the Föppl-von Karman axisymmetric equations, the resulting eigenvalue problem is solved, using again a boundary value solver. Depending on the sign of the eigenvalue, a small perturbation would either be damped (stable) or amplified (unstable). By coupling this with a root-finding algorithm, one can find for a given 135 α and mode number n the critical load G crit for which the eigenvalue sign changes.

Scaling of instability

To gain some insight in the behavior of the system, we develop a simplified theoretical analysis based on different scalings of the 140 energies.

We assume the position of a point initially at (r, θ, 0) on the flat disk can be described as (r + u r , θ, z(r, θ)) (figure 1

rr ∼ ∂ r u r + 1 2 (∂ r z) 2 θθ ∼ u r r + 1 2r 2 (∂ θ z) 2 κ rr ∼ ∂ rr z κ θθ ∼ ∂ r z r + ∂ θθ z r 2
There are three energies densities involved in the problem :

• stretching energy

E s ∼ Eh (1 -ν 2 ) S 2 rr + 2
θθ + 2ν rr θθ dS

• bending energy

E b ∼ Eh 3 12(1 -ν 2 ) S κ 2 rr + κ 2 θθ + 2νκ rr κ θθ dS
• gravity energy

E g ∼ S hρgzdS
For a given set of parameter, the solution to the problem is given 150 by a minimization of the total energy of the system, which results

in Föppl-von Karman equations. However, as these equations are notoriously difficult to solve in a general case, we resort to scaling analysis under simplifying assumptions to try to gain some insight in the behavior of the system. Poisson coefficient effect (ν = 0). With these variables, and using a bar over dimensionless quantities, we estimate for instance:

rr ∼ U l ∂ r u + 1 2 δ l 2 (∂ r z) 2
where the scaled quantities ∂ r u or ∂ r z are expected to be of order one if the scaling is correctly chosen. For the sake of clarity of the presentation, we will omit the scaled terms in the following, and keep only the scalings. Thus, the previous equation reads :

rr ∼ U l + 1 2 δ l 2
Likewise, the orthoradial stretching is estimated as

θθ ∼ U b + 1 2b 2 δ 2 A 2 n 2 sin(nθ) 2
Note that there is a difference in scaling of

∂ r u ∼ U l ∂ r u and u r ∼ U b u r
because for derivation the relevant length scale is the width of the annulus and not its radius.

With these two scalings for the stretching we can estimate for instance the unwrinkled (A = 0) stretching energy as

E s ∼ Eh S [ 2 rr + 2 θθ ]dS ∼ SEh U l + 1 2
δ 2 l 2 2 + U b 2 ( 3
)
where S is the surface of the annulus. We proceed to analyse the system in the two asymptotic limits of α → 0 (point like support) and α → 1 (quasi 2D system). Details of calculation are provided in Appendix A and Appendix B.

Results

For small deflection of the outer edge, we observe experimentally that the plate adopts an axisymmetric conical shape (figure 2 a).

For a given plate, the maximal deflection increases when the aspect ratio decreases as shown in figure 2 b. For a fixed aspect ratio, increasing the dimensionless gravity G also increases the maximal deflection. We observe that when the deflection reaches a critical value, the outer edge develops an undulated shape, revealing a wrinkling instability.

We plot in figure 3 all experimental points classified in three categories: self contact, axisymmetric and wrinkled. We focus particularly on the last two and observe that there is a well-defined boundary between these two regions. This frontier appears even clearer as we plot in figure 4 that, for a given mode number n, the neutral curve (zero growth rate) G crit (α) is weakly decreasing for small aspect ratios, reaches 190 a minimum and then increases rapidly. Close to threshold, full numerical simulations are in excellent agreement with linear stability analysis, and also show that G = cst is a relevant criteria for almost all the range of aspect ratio. However both predict a slightly higher threshold (G ≈ 25). In the limit α → 1 we notice a weak increase 195 of the threshold. This result is in good agreement with the scaling analysis developed in Appendix B. In the limit α → 0 the asymptotic analysis presented in Appendix A shows that the wrinkling instability develops when the outer edge deflection reaches a critical value.

In the bending regime, this deflection scales as G , which gives the 200 scaling G ≈ cst. In the limit α → 1 it is shown that G slightly diverges as (1 -α) -1/2 . While these results come from asymptotic analysis and are strictly speaking only valid in their respective limit, we notice that they hold beyond their region of derivation. Note also

G ∼ (1 -α) -1/2 (Appendix B)
. the fit is performed over the linear stability envelope for α > 0.8 and gives G = 0.7 + 15.9(1 -α) -1/2 . that the difference between the two scalings is weak: turning back to 205 non-dimensionalization, the α → 0 limit indicates that one should rescale the length with (1 -α) 4 while the α → 1 limit indicates that one should rescale the length with (1 -α) 4.5 . This is an effect too much subtle to be of practical impact or even experimentally accessible. Thus for simplicity we propose a criterion G = cst over the full range of α to be the most adequate.

It is interesting to look at the evolution of the most unstable mode as a function of α : we show in Appendix B that the optimal wavenumber diverges as n ∼ (1 -α) -1 in the limit α → 1. This is clearly confirmed both by linear stability analysis and experiments in figure 6. The interpolation of the experimental data gives n = 1.67(1 -α) -0.98 in very good agreement with [START_REF] Sader | Shear-induced buckling of a thin elastic disk undergoing spin-up[END_REF]. Note that this scaling, derived from energy minimization, can also be interpreted by a simple argument : the wavy pattern amplitude decreases from a maximal amplitude at the outer boundary to zero amplitude at the inner boundary. Thus, the pattern can extend radially at most on a length scale given by the width of the annulus. On the other hand, it is known that persistence length L of a wavy pattern depends on the wavelength λ and the amplitude A [START_REF] Vandeparre | Wrinkling hierarchy in constrained thin sheets from suspended graphene to curtains[END_REF] :

L/λ = (A/h) 1/2 .
Thus, assuming that the wavelength of the pattern is of the same size that the width of the annulus gives

λ = 2πb n ∼ (b -a), which upon inversion yields n ∼ 1 1-α .
Finally, one can also notice that regions where axisymmetric state is linearly unstable to only one mode are the exceptions rather than in very good agreement with the scaling predicted in the α → 1 limit (Appendix B) and with [START_REF] Sader | Shear-induced buckling of a thin elastic disk undergoing spin-up[END_REF].

the rule : for the most of the diagram, several modes can grow 230 and interact nonlinearly. Thus, we analyze the modes selection ex-perimentally by perturbing by hand the outer edge. For small aspect ratios and weak gravity, mode 2 (figure 2 a) is the only mode observed: even if other modes are linearly unstable, they always destabilize to form a mode 2. For α → 0, increasing gravitational effects, a symmetric mode 2 can destabilize into an asymmetric one, consistent with [START_REF] Cerda | The elements of draping[END_REF] predictions, but is often associated with selfcontact, a regime we do not explore in this paper (see figure 2 b).

Increasing the aspect ratio, modes higher than 2 can be observed.

We construct the experimental zone of existence of different modes, The deflection of a circular annulus by gravity leads to a wrinkling instability of the outer edge, where geometry plays a crucial role to determine both the threshold and the optimal wavenumber. We have shown that this influence can be caught in a modified dimensionless parameter G which express the ratio between normal load and bending forces. Using this parameter, the instability threshold is well described over the whole range of aspect ratio by a simple criterion G ≈ 20. To interpret this criteria, we develop a scaling analysis showing that wrinkling is associated with a critical deflection of the outer edge (which generates circumferential stresses). This deflection is governed by bending both in the α → 0 limit (where deflection are small enough for stretching effects to be negligible) and in the α → 1 limit (where radial displacement cancels the stretching energy to leading order in (1 -α), meaning that δ/h ∼ G over the whole range of α as shown in Appendix A and Appendix B.

One possible attractive extension of this study would be to include other external forces, like centrifugal ones, as these forces are relevant in the deployment and stability of spacecraft struc- in the α → 0 limit and Ω = Ω(1 -α) 2 in the α → 1 limit. We test the former scaling by using the results of [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF] where the critical 275 curves G(Ω) obtained by theoretical analysis were reported for four different values of α : extracting the data from figure 12a of [START_REF] Delapierre | Wrinkling of transversely loaded spinning membranes[END_REF] and replotting it (figure 8) in terms of G and Ω indicates that the scaling collapses the data onto a single curve. Even if it is mathe-α as high as 0.7. It would be interesting to test this scaling for more values of α and test it experimentally , especially close to α ≈ 1 to see whether or not there is an universal critical curve valid for all aspect ratio, a fact that might be useful for the design of such structures.

From a more fundamental point of view, the analysis developed here may also extend to the classical Lamé setup, where a similar blowup of eigenvalues as a function of aspect ratio is observed [START_REF] Coman | On the wrinkling of a prestressed annular thin film in tension[END_REF].

Including the aspect ratio into the control parameter may help to delineate the respective influence of geometry and physics.

295

In the limit of a point-like support, there is only one length scale b = l. The radial displacement is determined by minimizing the total energy with respect to U , which is equivalent to minimizing the stretching energy with respect to U because at first order the radial displacement U appears only in the stretching energy. Thus, differentiating equation (3) gives :

U ∼ - δ 2 4b
The bending energy of an unwrinkled sheet can be estimated as

E b ∼ 2SEh 3 δ b 2
2 . Thus, the typical vertical deflection solves the minimization of the total (stretching + bending + gravitational) energy, which reads number is diverging as a function of 1-α, which is clearly confirmed by figure 6 where we represent for each mode the range of α where it is the most dangerous. There is a very good agreement between this scaling and experimental data, as well as values coming from the stability analysis (which keeps all the complexity of the Föpplvon Karman equations). Note that the scaling (B.2) predicts that

δ h ∼ (1 -α) -1 2
, which means that the critical G should be a weakly diverging function of (1-α). This is consistent with what we observe in linear stability analysis and full numerical simulations.

Figure 1 :

 1 Figure 1: a. Definition of the system. b. Mechanism of wrinkling

  rials : PDMS, Silicone, Latex and Mylar. Latex disks are cut from commercial dental dam and Mylar disks are obtained from standard stencil sheets. PDMS and Silicone sheets are formed via the same process : we mix two liquid components, the polymer base and the curing agent. (Silgard 184 purchased from Merck for PDMS and RTV 181 from Esprit Composite for Silicone). We put the mixture into a vacuum chamber to remove air bubbles and then mold the mixture between two planar parallel plates made of aluminium with a controlled spacing. We then let the mixture cure for 24h-72h at room temperature. The elastic sheet is then gently detached from the mold. Whatever the material, the thickness of the disk is measured in different locations with a Palmer micrometer and av-eraged. The Young modulus is measured with a homemade tensile test machine. For PDMS, different proportion of the mixture allows to change the Young modulus by an order of magnitude. For each material, diameters, thicknesses and Young moduli ranges are reported in table

  a) with u r the radial displacement and z(r, θ) = w(r)(1 + A cos(nθ)) the deflection. With this assumption, and keeping the leading non-145 linearities due to geometry (only quadratic terms in the deflection, not in displacement), we have to dominant order to express strains and curvatures:

155

  Assume a planar ring of elastic material of inner radius a i.e. αb and outer radius b. We denote the typical deflection at the outer boundary by δ (= w(b)), the typical size of radial displacement by U and the width of the annulus by l = b(1 -α). We neglect the

bFigure 2 :

 2 Figure 2: Comparison of experimental and numerical shapes of

Figure 3 :

 3 Figure 3: Experimental stability boundary : filled symbols are wrinkled

Figure 4 :

 4 Figure 4: Experimental stability boundary : filled symbols are wrinkled states, open symbols are axisymetric ones. Grey solid line is G * =

Figure 5 :

 5 Figure 5: Stability boundaries of different modes computed by a lin-

Figure 6 :

 6 Figure 6: Evolution of most unstable mode number as a function of

from n = 2 Figure 7 :

 27 Figure 7: Experimental diagram of the zone of existence of different

Figure 8 : 3 b 2

 832 Figure 8: Critical curves G, Ω of [10] rescaled using the parameters

Ehδ 3 2b 4 + 3 3b 4 - 1 , bending is dominant and δ h ∼ G , while for δ h 1 , stretching is dominant and δ h ∼ G 1 3 2 θθ dS ∼ Eh S - δ 2 b 2 + δ 2 A 2 n 2 2b 2 2 dSδ 4 n 4 16b 4 -A 2 δ 4 n 2 4b 4 + h 2 12 A 2 δ 2 n 4 b 4 n 2 - 1 3 h δ 2 (A. 1 )

 43411322222412421 Eδh ρgh = 0 which gives two different regimes. For δ h Wrinkling threshold We now estimate the threshold for wrinkling by inserting a non zero amplitude modulation : z(r, θ) = δ(1 + A cos nθ) 300 and in κ θθ ∼ δ b 2 -δAn 2 b 2 cos(nθ). With this in mind, and recalling that U ∼ -δ 2 b , we clearly see the physical mechanism responsible for wrinkling : elastic energy due to the orthoradial stretching E s θθ ∼ Eh S sin(nθ) 2 2 dS can be diminished by allowing wrinkles to grow. However, this is balanced by a cost in bending energyExpanding and then integrating over θ, simple algebra permits to obtain the A-dependent terms of the energy asE A ∼ 3A 4 EhSThus, amplitude can be found by minimization of the energy with respect to A which leads toA = 0 or A ∼1Positivity of the square root requires that δ h > n √ 3 , meaning that there is a threshold value of the deflection below which axisymmetric state (A = 0) is stable.

Table 1 :

 1 Range

of parameters for each material. Typical uncertainties are indicated in parenthesis 3.2. Numerics

Acknowledgements

Centre de Calcul Intensif d'Aix-Marseille is acknowledged for granting access to its high performance computing resources.

Unwrinkled state 305

In the limit of an elastic disk slightly larger than its support, the orthoradial stretching θθ ∼ U b is subdominant compared to radial

, which is the dominant term in the stretching energy. Thus, radial displacement is found by minimization of radial stretching

and cancels to leading order the stretching energy. Deflection under gravity is then governed by bending only and leads to δ h ∼ G

Wrinkling threshold

Inserting the expression for radial displacement (B.1) in the stretching energy, one can compute the next order of the stretching energy in powers of (1 -α). Minimizing the total energy with respect to amplitude and mode number yields the following scalings :

Again, for a fixed aspect ratio, there is a critical deflection for which wrinkling is energetically favorable. The most favorable mode 310 106.224301


Laure Lecacheux

Abdelkrim Sadoudi

Agnès Duri

Véronique Planchot

Thierry Ruiz
email: .thierry.ruiz@umontpellier.fr

The role of Laplace pressure in the maximal weight of pendant drops

Keywords: Pendant drop, Maximal drop weight, Capillary force, Laplace pressure, Curvature, Image analysis. Nomenclature

Hypothesis

The value of the maximal weight of a pendant drop formed at the end of a syringe needle is lower than the intensity of the corresponding capillary force. The balance of the external forces applied to the maximal pendant drop must be completed by the overpressure generated by the piston of the syringe. Inside the drop, the Laplace pressure corresponds to this overpressure.

Experiments

Pendant drops are made with three liquids and five different needle diameters. The influence of Laplace pressure on the maximal weight is experimentally highlighted by modulating the

Introduction

Pendant drops at the end of a syringe capillary, are easily made objects that have a regularity of size and shape, offering the possibility to use them in multiple applications. They can be used as micromanipulators to capture small objects by capillary interaction thanks to the formation of a liquid-bridge at their apex [START_REF] Obata | A scheme for micro-manipulation based on capillary force[END_REF][START_REF] Lambert | A study of capillary forces as a gripping principle[END_REF]. These same capillary interactions are also used to structure assemblies of colloidal or granular particles. Liquid marbles are other applications that implement these surface interactions with colloids [START_REF] Aussillous | Liquid marbles[END_REF][START_REF] Bormashenko | New insights into liquid marbles[END_REF][START_REF] Mchale | Liquid marbles: topical context within soft matter and recent progress[END_REF]. Drops are also used for agglomeration or granulation unit operations [START_REF] Emady | Granule formation mechanisms and morphology from single drop impact on powder beds[END_REF][START_REF] Rossetti | Rupture energy and wetting behavior of pendular liquid bridges in relation to the spherical agglomeration process[END_REF]. The drop properties (in particular their size and volume), which interact with powder to associate them into agglomerated structures [START_REF] Hapgood | Drop penetration into porous powder beds[END_REF][START_REF] Bellocq | Impact of fluidized bed granulation on structure and functional properties of the agglomerates based on the durum wheat semolina[END_REF], require a precise control in order to best master the associative processes [START_REF] Iveson | Nucleation, growth and breakage phenomena in agitated wet granulation processes: a review[END_REF][START_REF] Mandato | Liquids' atomization with two different nozzles: Modelling of the effects of some processing and formulation conditions by dimensional analysis[END_REF]. Because the small drop volume is circumscribed by surface tension, it can also be directly used as a millireactor where various chemical reactions can be performed [START_REF] Abdelaziz | Green chemistry and nanofabrication in a levitated Leidenfrost drop[END_REF][START_REF] El-Nagar | Platinum nanostructure tailoring for fuel cell applications using levitated water droplets as green chemical reactors[END_REF].

The pendant drop of a colloidal suspension or a macromolecular solution constitutes a reactional volume in which a "stoichiometric forcing" can occur as the volume decreases by evaporation [START_REF] Boissiere | Aerosol Route to Functional Nanostructured Inorganic and Hybrid Porous Materials[END_REF]. In spray-drying, specific reactions are achievable and the control of the structural mechanisms to which they lead, requires the mastering of the evolution of the drop volume which can be studied in a pendant configuration [START_REF] Sadek | Shape, shell, and vacuole formation during the drying of a single concentrated whey protein droplet[END_REF]. Pendant drops are also employed to determine the surface tension of a liquid [START_REF] Gunde | Measurement of the surface and interfacial tension from maximum volume of a pendant drop[END_REF][START_REF] Morita | Influence of drop volume on surface tension evaluated using the pendant drop method[END_REF][START_REF] Berry | Measurement of surface and interfacial tension using pendant drop tensiometry[END_REF], from the analysis of the shape of a pendant drop in equilibrium and matching its contouring curve to a mathematical solution of the Young-Laplace equation. This inverse method allows to identify the surface tension value which represents the fitting factor.

There is a second method, called the drop weight method [START_REF] Yildirim | Analysis of the drop weight method[END_REF], which is based on the comparison between the weight of a falling drop (measured) and its theoretical value estimated from Tate's law [START_REF] Tate | On the magnitude of a drop of liquid formed under different circumstances[END_REF]. The surface tension is the adjustment parameter that allows to fit these two values. Tate's law relates the maximum weight of a pendant drop to the capillary force that holds the drop in equilibrium at the end of a syringe needle, just before it falls off.

Since only part of the pendant drop falls off, leaving a residual fraction on the needle, it is necessary to correct the weight of an "ideal drop" given by Tate's law by a correction factor.

Among the most widely used works for this purpose, that of Harkins and Brown [START_REF] Harkins | The determination of surface tension and the weight of falling drops: The surface tension of water and benzene by the capillary height method[END_REF] proposes values of the correction factor tabulated on a chart. But the exact calculation of the maximum weight just before its detachment would ensure even greater precision.

The aim of this work is to determine the maximal weight of a pendant drop at the end of a syringe needle by experimental investigations and theorical developments. We propose to define a consistent expression of the maximum weight of a pendant drop. No particular application of this result will be favored. Experimental results related to the measurement of the maximal weight of drops performed with three liquids of different known surface tensions and five different needle sizes, are presented. These experiments are carried out at small Bond numbers of the needle in order to achieve the same drop shape (piriform). We observe that the weight of the pendant drop is not equal to the capillary force which ensures the drop/needle connection. This gap is attributed to the lack of a force in the balance of external forces applied to the drop. In order to reveal the importance of this missing force, mechanical stress tests are performed by adding a calibrated glass bead at the apex of the drop, without changing the intensity of the capillary force. Beads of increasing mass induce a variation of the mean curvature directly associated with a variation of the maximal weight of the drops following a master curve. The repercussion of these variations on the Laplace pressure generates a force which completes the force balance. To demonstrate mathematically this important point, a physical reasoning, based on the fluid mechanics, is developed. These results allow us to introduce the Laplace pressure in the mathematical model which predicts the maximal weight of a pendant drop.

Experimental procedure

Prototype description

A prototype, presented in Figure 1, was designed to study the formation of a pendant drop at the end of a capillary until its maximal volume, just before its detachment. It consists of a PPT syringe composed of a body of an internal section Ω & = 61 mm 2 , a piston and a tapered tip allowing the adaptation of a straight cut steel needle (Doseurope) of section Ω ( . Five different needle sizes were used. Their external diameter is: 0.23, 0.5, 0.8, 1.26 and 1.80 mm and their internal diameter ( is respectively: 0.11, 0.26, 0.51, 0.84 and 1.37 mm. In order to ensure a constant flow rate of liquid ( ~ 0.097 to 0.387 mm 3 .s -1 ), a syringe pump driven by a stepper motor (MicroLYNX, M-1410-0.75D) is used to provide a vertical displacement according to . This flow rate gives a velocity intensity in the needle comprised between 0.26 and 5 mm.s -1 . For each test, the laminar flow inside the needle is achieved with a Reynolds number comprised between 18×10 -3 (Triacetin with the largest needle) and 5.6×10 -1 (water with the smallest needle). The evolution of the pendant drop formation until its maximal volume is filmed using a USB microscope (DinoLite®) placed in front of the needle with an acquisition rate of 25 images.sec -1 . A LED backlight system is also used to enhance the contrast.

The prototype is placed on an insulated support to avoid any vibration and is isolated in a plexiglass chamber to minimize any contamination. All the tests are performed in quintuples, at a controlled temperature of 25.0 ± 0.2 °C, and a relative humidity of 40 ± 16 %. The total duration of formation of a drop is between 1 and 2 minutes which allows to neglect the evaporation phenomena [START_REF] Portuguez | Evolution of a Water Pendant Droplet: Effect of Temperature and Relative Humidity[END_REF].

Characteristics of liquids

Pure water, Tween® 80 (Sigma-Aldrich) diluted at 2×10 Bond number of the drop: = -! " ⁄ . < 1 (Fig. 2) and the Bond number of the needle:

= - 2! " ⁄ . <
, ensure a piriform geometry of the drops.

Description of the mechanical test

This mechanical test, adapted from the works of Li et al. [START_REF] Li | Direct visualization of particle attachment to a pendant drop[END_REF] and Neeson et al. [START_REF] Neeson | Compound pendant drop tensiometry for interfacial tension measurement at zero bond number[END_REF], consists in loading with a calibrated particle of known mass a pendant drop obtained from a needle of a given diameter. Then the maximal weight of the loaded drop just before its detachment is measured. In order to highlight the role of Laplace pressure on the weight of a pendant drop, this test allows to modulate the main curvatures of the drop (see video).

The aim of this test is to achieve conditions that ensure invariance of the capillary force at the drop/needle wetted perimeter while modulating Laplace pressure within the drop. When the drop reaches a stage of development corresponding to about 30% of its maximum volume, the glass bead is then approached to its apex. At its contact, the drop exerts a capillary traction that captures the bead and keeps it in suspension in a position that respects the global axisymmetry (Fig. 2). The hydrodynamic flow conditions in the syringe allow a quasistationary evolution of the growth of the loaded drop. The shape of the drop, and in particular its main curvatures, adapt to the resultant external forces applied to it [START_REF] Neeson | Compound pendant drop tensiometry for interfacial tension measurement at zero bond number[END_REF]. Six spherical glass beads (density of 2.501 g.cm -3 ) of different masses (0.62, 1.48, 2.55, 4.52, 8.38 and 12.03 mg) are used. The Bond numbers of the glass beads are less than 1, which ensures their capillary capture by the pendant drop.

The operating conditions are identical to those used during the formation of a single pendant drop and the evolution of the loaded drops is also captured by the USB microscope (Fig. 1).

Determination of drop characteristics

An image analysis method was developed to determine the maximum pendant drop volume and the two radii of curvature. Technical details are given in the Supplementary Materials file.

Maximal volume. Each drop is filmed (from 1 to 2 min at 25 fps) during its entire development and all the images are extracted (from 1500 to 3000) then analyzed to precisely determine the maximum volume of each pendant drop. The linear progression of the volume of the drop, due to the constant flow, makes it possible to locate the maximal volume reached with an error less than 0.01 %. The maximum weight is then calculated from the determination of the volume.

Radius of curvature, main curvature and Laplace pressure. The image, corresponding to the limit stage of drop development, is used to determine the curvature radii. The curvature radius

, is determined at the level of the maximum diameter in the horizontal plane. The curvature radius , is located in the vertical plane, and corresponds to the radius of the circle being superimposed with the drop curve [START_REF] Berry | Measurement of surface and interfacial tension using pendant drop tensiometry[END_REF]. The main curvature of the drop is then deduced from these measurements according to its standard definition in 3D space, 2 = 2 3 + 2 5 . The Laplace pressure in the drop is then calculated according to its definition [START_REF] De Gennes | Capillarity and wetting phenomena: drops, bubbles, pearls, waves[END_REF]:

∆ = 2 .
This procedure is used for single and loaded drops.

Results and discussion

Maximal weight of a pendant drop

Pendant drops are formed at the end of needle of a syringe using a stationary flow rate of the liquid, Q, ensured by a force applied on the piston. The maximal mass reached by the drop, , is calculated from the measurement of the volume. The laminar regime ensures the stability of the drop formation and we verify that neither the value of the flow nor the height of liquid ( ) between the surface of the piston and the lowest part of needle, does not change the maximal mass of the drop (see Supplementary Materials file). This point implies that the hydrostatic pressure does not affect the maximal weight of a pendant drop generated by a syringe.

In these conditions, the balance of the external forces applied to a pendant drop takes into account only three actions (the weight, the capillary force exerted at the triple line and inertia [START_REF] De Gennes | Capillarity and wetting phenomena: drops, bubbles, pearls, waves[END_REF]), and can be written as follow:

+ = 7 8 9 : ≈ 0 8 (1)
where the inertial force is negligible when a steady state is imposed (a constant flow rate implies that the barycentric velocity of the drop inertia center, , remains constant). The projection of this vectorial equation on the horizontal axis cancels out, while on the vertical axis we find that the weight must be compensated by the projection of the capillary force on . Assuming that the intensity of the capillary force is equal to [START_REF] De Gennes | Capillarity and wetting phenomena: drops, bubbles, pearls, waves[END_REF]: = = γcos -., for each tested condition, the wetting angle at the drop connection on the outer perimeter of the needle is checked. It is always close to 0° and does not impact the value of the capillary force, as already observed by Nazari et al. [START_REF] Nazari | Drop formation from a capillary tube: Comparison of different bulk fluid on Newtonian drops and formation of Newtonian and non-Newtonian drops in air using image processing[END_REF]. With this definition, the maximal weight of the pendant drop is given by:

= = (2)
This expression is well known as the Tate's law [START_REF] Tate | On the magnitude of a drop of liquid formed under different circumstances[END_REF][START_REF] Wilkinson | Extended use of, and comments on, the drop-weight (drop-volume) technique for the determination of surface and interfacial tensions[END_REF], and the maximal weight defined by Equation ( 2) is classically called the weight of the "ideal drop". We remind that this maximum weight is the weight of a pendant drop still attached to its needle and not the weight of a detached drop. By plotting the measurements of the maximum drop weight as a function of the corresponding capillary force for each tested case, it can be clearly observed in Figure 3a that Equation ( 2) is not verified. The capillary force is systematically higher than the weight, and this gap is about 35.0% ±1.2% on average. Surprising as it may seem, we are not aware of any experimental data in the scientific literature on the weight of pendant drops. All the numerous works deal with detached drops with all the precautions on the validity of Tate's law. Lee et al. [START_REF] Lee | A critical review: Surface and interfacial tension measurement by the drop weight method[END_REF] pointed out that the force balance used to define the weight of the "ideal drop" does not include a compressive force in the upper plane of the drop. This balance needs to be redefined [START_REF] Garandet | Considerations on the pendant drop method: a new look at Tate's law and Harkins' correction factor[END_REF][START_REF] Kumar | On the validity of force balance models for predicting gravity-induced detachment of pendant drops and bubbles[END_REF]. Therefore, such a gap implies to consider again the balance of external efforts applied to a pendant drop [START_REF] Kumar | On the validity of force balance models for predicting gravity-induced detachment of pendant drops and bubbles[END_REF] taking into account the repercussion of the force increment applied on the piston, , called the Laplace force, on the pendant drop. A drop generated by a syringe is only developed by the action of a force increment on the piston. This force is oriented in the same direction as the weight and is transmitted until the top surface of the drop: = ( /4, via an overpressure throughout the liquid column. We can show that this overpressure corresponds to the Laplace pressure in the pendant drop. The expression of the external force balance becomes:

+ + ≈ 0 8 (3)
Under all these hypotheses and by introducing the definition of the Laplace pressure in the external force balance exerted on the pendant drop, Equation (3) can be written as:

= = -Δ = ( /4 (4)
This equation involves the internal ( , and external diameters of the needle. From

Laplace's pressure measurements, we verify in Figure 3b that the maximal weight of a pendant drop is then well described by Equation (3).

We propose in the following section an experiment at the drop level in order to highlight the role of the Laplace pressure and the trend with which it acts on the maximal weight drop. The implication of Laplace pressure in the balance of external forces applied to a pendant drop has already been suggested by Garandet et al. [START_REF] Garandet | Considerations on the pendant drop method: a new look at Tate's law and Harkins' correction factor[END_REF]. However, our approach highlights the noncontribution of hydrostatic pressure, contrary to the equation presented by Garandet et al. [START_REF] Garandet | Considerations on the pendant drop method: a new look at Tate's law and Harkins' correction factor[END_REF]. We also provide an explanation of the origin of this involved Laplace force and a justification by establishing a model under well-identified hypotheses.

Drop loading to highlight the Laplace pressure

The drop keeps held to the needle by the capillary force at its triple line which equilibrates the weight added to the force applied on the piston transmitted to the drop via the pressure in the liquid column. The test consists of keeping the capillary force constant (constant needle diameter and constant surface tension of the liquid) and varying the drop weight by distributing it between the liquid and a glass bead placed at the apex. Whatever its weight, the presence of a glass bead does not modify the intensity of the capillary force exerted on the pendant drop due to the fact that: (i) the wetted perimeter is unchanged, (ii) the contact angle value just before drop detachment has been systematically measured by image analysis (Fig. 4) and remains unchanged: ≈ 0°, and (iii) the liquid surface tension is not modified (the beads are carefully cleaned beforehand). Thus, the force balance (Eq. 4) is satisfied for the maximal total weight of the loaded drop: * = + * , where * is the liquid weight and is the glass bead weight. The weight of the bead has an impact on the shape of the drop (Fig. 4) and therefore on the Laplace pressure inside the drop. The drop loading with a glass bead constitutes therefore a reliable mechanical test with a fixed capillary force.

For each liquid, increasing loads with calibrated glass beads were realised for all pendant drops obtained with the five needles. Figures 5 presents an illustration of the maximal weight of the loaded pendant drops, as a function of the bead mass. Whatever the needle diameter and the liquid, the maximal weight of the loaded drops is not constant but decreases with the increase of the bead mass. In all cases, > * and so, the hypothesis of the capillary force, only opposed to the weight of the maximal pendant drop in the external force balance, is invalidated by the imposed conditions in these experiments.

This result indicates that at constant capillary force, if the maximum weight of a pendant drop decreases with increasing loads, then the Laplace pressure should increase to ensure the external force balance (Eq. 4).

In order to determine the Laplace pressure in the drop, the mean curvature is measured. For a given needle, as the particle mass increases, a loaded drop adopts an increasingly elongated shape (Fig. 4). The axisymmetry being satisfied, the quantitative analysis of the main curvatures (Fig. 2) shows that the increasing mechanical solicitation induces an elongation of the axial radius of curvature (vertical plane), , and simultaneously a narrowing of the radial radius of curvature (horizontal plane), (Fig. 6a). For each liquid and each needle, the radii of curvature of the loaded drops are resized by those of the unloaded drops made with the same needle: * / and * / . The antagonistic variation of the two radii of curvature systematically leads to an increase of the mean curvature defined by,: 2 * = 2 3 * + 2 5 * , for all loaded pendant drops.

Figure 6b shows that 2 * /2 ratio increases linearly with the solid mass fraction and is superimposed on a single curve for all experiments.

For each tested liquid and needle, experiments show that loading induces a decrease of the maximal weight of the pendant drops (Fig. 5) and an increase of the Laplace pressure (Fig. 7).

This point represents an experimental validation of the influence of the Laplace pressure in the weight of a pendant drop. The expression of the external force balance given by Equation

(3) seems more correct than Equation (1).

Because of the glass and liquid incompressibility, the volumes of unloaded drops and loaded drops are respectively given by: = /# , and * = * /# + /# . For all maximal drops (loaded and unloaded), a well-defined relationship is shown between the mean curvature and the volume of a maximal pendant drop (Fig. 8). This power law dependence between these geometrical quantities constitutes a master curve for the maximal pendant drops. When unloaded maximal pendant drops are quasi-spherical, which is the case for very small drops, ≈ ≈ , the curvature follows a power law with the volume:

2 ~ " 3 G .
Such a trend is experimentally obtained as shown in Figure 8a.

Regardless of the mechanical loading, for each liquid and according to the variation of the mean curvature, Laplace pressure decreases with the maximal weight of the pendant drops (Fig. 8b). The surface tension value discriminates Laplace pressure levels within drops made of different liquids. At equal density, the mixture of water and Tween 80 has a lower surface tension than water, so the Laplace pressure is lower.

Demonstration of maximal pendant drop weight expression

For a pendant drop generated at the end of a syringe needle, the external force balance (Eq. 3)

includes (Fig. 9): (i) the weight of the drop: , (ii) the force increment applied on the piston that propagates until the drop via the internal section of the needle (= ( /4): , and (iii) the capillary force at the wetted perimeter (here the outer perimeter of the needle): . The rigorous demonstration that the intensity of the force is related to the Laplace pressure is not so obvious. The intensity exerted on a pendant drop can be written as a function of the overpressure at the piston (Δ ) within the fluid at the end of the needle: = Δ = ( /4

(the force exerted on the piston equals: Δ Ω % ). The overpressure Δ can be calculated from the momentum balance of the flowing fluid between the piston and the end of the needle.

Assuming that the liquid is incompressible and the flow in a steady-state, Bernouilli's equation gives:

+ # --Δ . + # H 5 I J 5 = + # H 5 I KL 5 + Δ → (5)
with Δ , the displacement corresponding to the piston lowering, and respectively the pressures in the liquid at the piston interface and at the end of the needle, and Δ → the drop pressure between the piston and the end of the needle. During all the pendant drop formation, the ratio Δ / remains below 1 %. The term contributions related to the kinetic energy:

# /Ω % and: and ~10 Pa for all other conditions. Therefore, the viscous dissipation can be neglected compared to the other pressure values (~10 5 Pa). Under these considerations, the balance (Eq. 5) can be reduced to the hydrostatic part:

≈ + # (6)
As the liquid was initially drawn into the syringe by the application of a vacuum, the condition of hydrostatic equilibrium that precedes the application of the overpressure to the piston can be written as: T = T + # , where T and T are the pressures at equilibrium after the entrance of the liquid. When the pendant drop is generated, the overpressure applied to the piston is given by: Δ = -T , and is transmitted to the end of the needle by: Δ = -T . The momentum balance (Eq. 6) leads to the conclusion that Δ ≈ Δ . In quasi-static conditions, the overpressure applied to the piston is transmitted to the drop. We find that hydrostatic pressure no longer plays a role on the drop weight, as experimentally verified (see Supplementary materials file).

For the studied configuration and the implemented process conditions, we verified that the stationary kinetics of the drop growth can be assimilated to a succession of equilibrium states (1.8×10 -2 ≤ ≤ 5.6×10 -1 ). Under this hypothesis of local quasi-static state, we consider that the expression of the overpressure within the drop can be given by the Laplace-Young relation. Consequently, Δ = Δ and so the "Laplace force" is given by: = Δ = ( /4.

Under all these hypotheses, the expression of the maximal weight of a pendant drop at the end of a syringe needle is consistent with Equation (4). By noting the "ideal drop" weight: V = = , equation (Eq. 7) provides a correction to Tate's law in the case of a pendant drop generated by a syringe, that is expressed as follows:

= V -Δ = ( /4 (7)
Figure 3b shows the set of experimental results that validates Eq. 7. Figure 9 illustrates more intuitively the meaning of Eq. 7: the maximum weight of the pendant drop is equal to the capillary force exerted at the end of the needle, from which must be subtracted the force coming from the pressure increment applied to the piston and which imposes the value of the Laplace pressure within the drop. We provide a justification of Eq. 7 by establishing this model under well-identified hypotheses and mathematical developments and it differs from that one suggested by Garandet et al. [START_REF] Garandet | Considerations on the pendant drop method: a new look at Tate's law and Harkins' correction factor[END_REF].

Conclusion

This work contributes to the characterization of the weight of pendant drops. It experimentally and mathematically demonstrates the influence of the Laplace pressure on the maximum weight of a pendant drop. The involved physical phenomena in the equilibrium of a pendant drop are well known. However, the writing of the balance of the applied external forces to the pendant drop associated with the rigorous demonstrations of (i) the non-influence of the hydrostatic pressure (which appears in the equation of Garandet et al. [START_REF] Garandet | Considerations on the pendant drop method: a new look at Tate's law and Harkins' correction factor[END_REF]) and (ii) the equality of the overpressure exerted on the piston with the Laplace pressure, lead to an original equation (Eq. 7) providing a correction to the well-known Tate's law. Tate's law [START_REF] Tate | On the magnitude of a drop of liquid formed under different circumstances[END_REF],

based on the theoretical balance of applied external forces to a drop, is usually used to theoretically calculate this weight. Despite numerous criticisms [START_REF] Lee | A critical review: Surface and interfacial tension measurement by the drop weight method[END_REF][START_REF] Garandet | Considerations on the pendant drop method: a new look at Tate's law and Harkins' correction factor[END_REF][START_REF] Kumar | On the validity of force balance models for predicting gravity-induced detachment of pendant drops and bubbles[END_REF], to the best of our knowledge, no experimental measurements relating to the validation of Tate's law are reported in the scientific literature. Our work thus constitutes a new basis of reflection associated with experimental data in order to elucidate the factors that determine the weight of a pendant drop.

We have carried out the weight measurement of maximal pendant drops at the end of a syringe needle of different sizes for three liquids of contrasting capillary lengths. The results show a gap with the capillary force alone. These experimental results invalidate Tate's law under the employed conditions. This difference, highlighted with three liquids, is corrected by taking into account the exerted force on the piston in the applied external force balance to a pendant drop. This force can be related to the Laplace pressure within the drop. A mechanical loading experiment, with imposed capillary force adapted from the works of Li et al. [START_REF] Li | Direct visualization of particle attachment to a pendant drop[END_REF] and Neeson et al. [START_REF] Neeson | Compound pendant drop tensiometry for interfacial tension measurement at zero bond number[END_REF], is proposed in order to highlight the role of Laplace pressure in the mathematical expression of the maximal weight.

The principle of the mechanical loading with a bead attached to the drop apex, allows to modulate its main curvature and therefore the Laplace pressure. The results show that the mean curvature increases with the mechanical loading according to a master curve which is independent of the liquid characteristics and the needle size. If they are extended to a larger number of liquids, they would provide the basis for a remarkable universality. For each unloaded pendant drop, the curvature is a decreasing function of the drop volume. The relationship between Laplace pressure and the applied force on the piston is mathematically

established. An analytical expression of the maximal weight of a pendant drop is deduced.

The model, which considers the weight equal to the capillary force, is then revisited and replaced by a new model that explicitly incorporates the Laplace pressure. It is experimentally validated for each tested case.

Our approach can be extended to determine the liquid-solid adhesion work using centrifugal adhesion balance [START_REF] Tadmor | Solid-Liquid Work of Adhesion[END_REF][START_REF] Yadav | A Novel Technique Enables Quantifying the Molecular Interaction of Solvents with Biological Tissues[END_REF]. For this, one needs to replace the piston motion with the increase of the effective gravity performed with centrifugal device. It could be also used to define with high precision the volume that constitutes a pendant drop, especially in the context of downsizing technologies [START_REF] Basaran | Small-scale free surface flows with breakup: drop formation and emerging applications[END_REF]. The better knowledge of the maximum weight of a pendant drop could participate to the improvement of particle coating techniques as used for marbles [START_REF] Bormashenko | New insights into liquid marbles[END_REF][START_REF] Mchale | Liquid marbles: topical context within soft matter and recent progress[END_REF]. In the pharmaceutical field, this work could have repercussions on the precision of active drugs formulation distributed as drop (eye drops, perfusion, oral solution in drops…).

1 ]

 1 Diameter of the needle [m] force increment applied by the piston on the pendant drop [N] capillary force at the wetted perimeter [N] gravity acceleration [m.s -2 ] height of liquid colum in the syringe [m] needle length [m] mass of a glass bead [kg] maximal mass of a pendant drop [kg] * maximal mass of a loaded pendant drop [kg] * mass of the liquid in a loaded pendant drop [kg] pressure at the end of the needle [Pa] pressures at the piston [Pa] feeding volume flow rate [m 3 .s -1 ] Reynolds number [-] radial curvature radius [m] axial curvature radius [m] barycentric velocity of the drop [m.s -1 ] maximal volume of a single pendant drop [m 3 ]

5 ,

 5 take values of the order of magnitude of 10 -4 Pa at most and are therefore negligible. Due to the large gap between the internal sections of the syringe and the needle (Ω ( /Ω % ≈ 0.025 at most), the pressure drop in the body of the syringe is negligible compared to the one in the needle. The flow occurs in laminar regime so, we assume that the pressure drop is given by Hagen-Poiseuille formula:Δ → = OP K H QR KL S, with the viscosity of the liquid and the length of the needle. With these value ranges of the parameters, this pressure drop does not exceed ~10 2 Pa in the worst case (Triacetin with the smallest needle)

Figure 1 .

 1 Figure 1. Scheme of the experimental prototype for pendant drop generation and contouring acquisition of the maximal pendant drop.

Figure 2 .

 2 Figure 2. Illustration of the curvature radii measurement for a single pendant drop (left) and a loaded pendant drop with a glass bead (right).

Figure 3 .

 3 Figure 3. (a) Capillary force as a function of the maximal drop weight (Eq. 2) for all experiments, (b) capillary force minored by the "Laplace force" versus maximal drop weight (Eq. 4) for all experiments. The order of the error magnitude is about 10 -4 mN. The dashed line is a guideline.

Figure 4 .

 4 Figure 4. Illustration of the drop loading with a glass bead at constant capillary force. The images correspond to pendant water drops just before their detachment and formed with a same needle of diameter ( = 1.80 mm) and an increasing load.

Figure 5 .

 5 Figure 5. Maximal mass of loaded pendant drops as a function of bead mass for Water (circle), Tween 80 (square) and Triacetin (triangle). For each liquid only results, obtained for needles of external diameters equal to 1.26 (empty symbols) and 1.80 mm (blanck symbols), are shown. The dashed lines are a guideline. Full results are available in the Supplementary Materials file.

Figure 6 .

 6 Figure 6. Variation of the loaded pendant drop curvatures as a function of the solid mass fraction for all experiments: (a) dimensionless curvature radii, (b) dimensionless mean curvature. The dashed lines are a guideline.

Figure 7 .

 7 Figure 7. Laplace pressure within the maximal pendant drops as a function of the particle mass for (a) Water, (b) Tween 80, (c) Triacetin and different external diameters of the needle ( = 0.23, 0.50, 0.80, 1.26 and 1.80 mm). The order of the diameters is respected for each figure. The dashed lines are a guideline.

Figure 8 .

 8 Figure 8. (a) Variation of the mean curvature as a function of the maximum pendant drop volume (R² = 0.9736) for all liquids and needles used (Log-Log representation), (b) Laplace pressure within the maximal pendant drops as a function of its weight.

Figure 9 .

 9 Figure 9. Scheme of the external forces applied to a maximal pendant drop generated by a syringe.

Figure 1 .

 1 Figure 1. Scheme of the experimental prototype for pendant drop generation and contouring acquisition of the maximal pendant drop.

Figure 2 .

 2 Figure 2. Illustration of the curvature radii measurement for a single pendant drop (left) and a loaded pendant drop with a glass bead (right).

Figure 3 .

 3 Figure 3. (a) Capillary force as a function of the maximal drop weight (Eq. 2) for all experiments, (b) capillary force minored by the "Laplace force" versus maximal drop weight (Eq. 4) for all experiments. The order of the error magnitude is about 10 -4 mN. The dashed line is a guideline.

Figure 4 .Figure 5 .

 45 Figure 4. Illustration of the drop loading with a glass bead at constant capillary force. The images correspond to pendant water drops just before their detachment and formed with a same needle of diameter ( = 1.80 mm) and an increasing load.

Figure 6 .Figure 7 .Figure 8 .

 678 Figure 6. Variation of the loaded pendant drop curvatures as a function of the solid mass fraction for all experiments: (a) dimensionless curvature radii, (b) dimensionless mean curvature. The dashed lines are a guideline.

Figure 9 .

 9 Figure 9. Scheme of the external forces applied to a maximal pendant drop generated by a syringe.

Acknowledgments

The authors thank the project DEFI Blé Dur (2016-2022), that has received funding from Programme d'investissement d'avenir (PIA), and the French Région Occitanie for their financial supports.

the volume predicted by Tate's law, varies according to a unique characteristic curve as a function of the capillary radius scaled by the cube root of the detached drop volume. What happens to this relationship if we now consider the real maximum weight of a pendant drop and not the one given by Tate's law? This last point, which constitutes our future work, will bring a new representation of the weight of the drops that fall from a capillary.

CRediT authorship contribution statement