J O Ël De Coninck

Thierry Huillet

WETTING IN 1+1 DIMENSIONS WITH TWO-SCALE ROUGHNESS

Keywords: classification: 68.08.Bc, 68.03.Cd, 05.40.-a Wetting, Cassie-Baxter, Wenzel, hydrophobicity, roughness

We show that a two-scale model in 1+1 dimensions enhances superhydrophobicity. The two scales may differ by a factor of order two or three, or by a large factor in a scaling limit. In both cases, we compute explictly the macroscopic contact angles as function of the flat material contact angle and aspect ratios. In addition to the Cassie-Baxter states with air cushion below the droplet and to the Wenzel states, completely wet, there appear several mixed states with air trapped in corners.

Introduction

Nature has designed many different examples of superhydrophobic surfaces for plants or insects. When a water droplet is deposited on it, high values of both advancing and receding contact angles are observed. Superhydrophobicity will also provide special characteristics such as rolling motion of a deposited water drop with a very low tilting angle or rebound of the drop when impacting the surface. For such surfaces, roughness plays a key role. When deposited on such surfaces, it is expected that the drop can be in at least two different states: in contact everywhere with the solid surface, i.e. the Wenzel state [1], or in contact with the top elements of the surface, the Cassie-Baxter state, [START_REF] Cassie | Wettability of porous surfaces[END_REF]. The basic idea for such systems is that Nature will minimize the free energy leading to the conclusion that if the equilibrium contact angle θ 0 corresponding to the flat surface satisfies some inequality, the drop will be in the Cassie-Baxter state leading to superhydrophobicity. In 1+1 dimensions, or in three dimensions with grooves, we can build a regular surface obtained as a periodic substrate with unit cell of parameters (a, b, c) as shown in Fig. 4. When dealing with such simple geometry, this inequality is well known and can be written as cos θ 0 < -(1 -φ)/(r -φ), where r is the Wenzel roughness of the surface, defined as the total area of the surface divided by its projection, and φ is the covered fraction, defined as the total area at the top level divided by the total projected area. As can be easily seen, the larger r, the more the surface is likely to be in the Cassie-Baxter state. This is the reason why it is believed that a very rough hydrophobic surface can be superhydrophobic. These considerations are generally summarized in a single graph describing the cosine of the contact angle θ of a rough surface versus the cosine of the contact angle θ 0 of the corresponding flat surface, as represented in Fig. 1.

For nanoparticles on top of a solid flat surface (see [START_REF] De Coninck | Is superhydrophobicity robust with respect to disorder?[END_REF]), the same type of results are obtained with more curved lines instead of straight lines. One of the remarkable properties of these systems is that the slope at the origin (cos θ 0 = 0, cos θ = 0) for the Wenzel regime is always the roughness r, and that the slope at (cos θ 0 = -1, cos θ = -1) is always the surface fraction φ. This kind of simple reasoning is based on surfaces with a single scale topography. Nature has designed remarkable hydrophobic surfaces showing 2 scales of roughness (see Fig. 2). It is the purpose of this paper to analyze in details how a second topography scale can improve the possible superhydrophobicity of a surface. Indeed, from the previous considerations, one may expect that by adding a second scale of grooves or nanoparticles, we will increase (resp. decrease) inevitably r (resp. φ) leading to an improvement of the Cassie-Baxter regime, compared to a one-scale situation. Dealing often with biomimetics, several authors have considered this kind of problem and are usually going in this direction, see [START_REF] Sajadinia | Thermodynamic analysis of the wetting behavior of dual scale patterned hydrophobic surfaces[END_REF], [START_REF] Feng | Super-Hydrophobic Surfaces: From Natural to Artificial[END_REF], [START_REF] Jeong | Nanoengineered Multiscale Hierarchical Structures with Tailored Wetting Properties[END_REF], [START_REF] Hipp | Systematic control of hydrophobic and superhydrophobic properties using double-rough structures based on mixtures of metal oxide nanoparticles[END_REF], [START_REF] Patankar | Mimicking the Lotus Effect: Influence of Double Roughness Structures and Slender Pillars[END_REF], [START_REF] Shirtcliffe | Dual-scale roughness produces unusually water repellent surfaces[END_REF] just to quote a few. However, the rules to combine the different roughnesses or surface area are not particularly clear, [START_REF] Passoni | Multiscale Effect of Hierarchical Self-Assembled Nanostructures on Superhydrophobic Surface[END_REF]. In a few publications moreover, [START_REF] Teisala | Nanostructures Increase Water Droplet Adhesion on Hierarchically Rough Superhydrophobic Surfaces[END_REF], [START_REF] Ensikat | Superhydrophobicity in perfection: the outstanding properties of the lotus leaf[END_REF], it is shown that some double scale or hierarchical structures are not even particularly favorable for a Cassie-Baxter state to emerge.

There is thus a real need for a rigorous analysis of the effect of the second scale roughness in the problem of superhydrophobicity. We address this problem here systematically in the context of the simple groove geometry (1+1 dimensions). We show rigorously that the second scale roughness will induce a series of mixed state between Wenzel and Cassie-Baxter. These states will certainly affect the global slipperiness of the corresponding surface. We also study in detail the case of two independent scales with respective surface fractions φ 2 (small scale), φ 1 (large scale) and respective roughnesses r 2 , r 1 where we show a very rich variety of mixed states depending on the various geometrical parameters. In a scaling limit where the two scales differ by a very large factor (resulting in separation of scales), the global Cassie-Baxter state appears with a surface fraction φ = φ 2 φ 1 < min (φ 2 , φ 1 ) and the global Wenzel state with a roughness r = r 2 + r 1 -1 > max (r 2 , r 1 ).

The paper is organized as follows.

• In Section 2, we first address the following problem: consider a regular (groove) surface obtained as a periodic substrate with unit cell of parameters (a, b, c) , as displayed in Fig. 4. We show that, besides the pure Cassie-Baxter state and the pure Wenzel state, an unsuspected mixed state (called CW) is also present theoretically in the picture, for some values of the aspect ratios of the unit cell. The latter model is called a 1-scale model as compared to the 2-scale model of interest in this paper.

• In Section 3, a two-scale model is considered: a large-scale (a 1 , b 1 , c 1 ) pattern within which small-scale patterns (a 2 , b 2 , c 2 ) are nested additively on their flat horizontal parts. Here k 2 ≥ 1 small rectangular protrusions of height b 2 and width a 2 are inserted between two large rectangular protrusions of height b 1 > b 2 and width a 1 ; and k 1 ≥ 1 small rectangular protrusions of height b 2 and width a 2 are inserted on top of the large rectangular protrusion of height b 1 . The precise two-scale model is sketched in Fig. 5.

We work under the simplifying and realistic hypothesis that the mixed state CW at scale 2 does not contribute to the overall picture and we distinguish two cases: either the Cassie-Baxter state at small scale 2 wins over Wenzel state or Wenzel state at small scale 2 wins over the Cassie-Baxter state. We describe the variety of states (either pure or mixed) that can arise in such a 2-scale model, some stable, some unstable; we compute their free energies and the resulting contact angles. The subset of stable states will be met successively when the angle θ 0 is varied on its full range and we sketch the scenarios. We also show that:

-There exists a geometry (a value of the aspect ratio parameters together with a value of k := (k 1 , k 2 )) and a range of θ 0 for each of these states to dominate the other ones.

-In general, when the geometry is fixed, only a subset of the whole set of states (typically 2, 3 or 4) will contribute to the final 2-scale Young-Dupré curve giving the macroscopic contact angle cos θ as a function of cos θ 0 on the whole range cos θ 0 ∈ [-1, 0] . In any case, the final Young-Dupré curve giving the contact angle cos θ as a function of cos θ 0 is obtained while considering the maximum of the curves cos θ versus cos θ 0 for all contributing states to the wetting problem just discussed.

• In Section 4, we consider the fate of the latter configurations (states) in a scaling limit of the latter 2-scale model. We show that the final Young-Dupré curve is explicitly given by

cos θ = max cos θ CB1CB2 , cos θ CW1CB2 , cos θ W1CB2 , cos θ W1W2 , cos θ CB1W2 , cos θ CW1W2 ,
where only 6 states are present, with S 1 S 2 where S ∈ {CB, CW, W } standing for the various states expected at the two different scales. We determine which of these states is the winner and in which range of θ 0 .

Rectangular protrusions. One scale

Consider a periodic substrate with unit cell of parameters (a, b, c) as shown in Fig. 4. For this substrate, define

(1) r = 1 + 2b a + c
, its roughness and

φ = a a + c
, the rectangular protrusions covered fraction in the CB state.

Let γ SV , γ SL be the flat Solid-Vapor and Solid-Liquid surface free energies per unit length of substrate. The Young-Dupré relation giving the contact angle of a liquid droplet on a flat substrate is: γ LV cos θ 0 + γ SL = γ SV where γ LV is the Liquid-Vapor interface free energy per unit length. The Solid-Vapor surface free energy over one unit cell of rough pattern is: F r SV = (a + 2b + c) γ SV . The Liquid-Vapor interface free energy for a length (a + c) cos θ is:

F LV = γ LV (a + c) cos θ.
-Cassie-Baxter (CB) configuration:

The Solid-Liquid surface free energy over one unit cell of rough pattern is: F CB SL = aγ SL +(2b + c) γ SV +cγ LV . Thus, the CB macroscopic contact angle on the periodic rough substrate (a, b, c) is given by [START_REF] Cassie | Wettability of porous surfaces[END_REF] cos

θ CB = F r SV -F CB SL γ LV (a + c) = φ cos θ 0 -(1 -φ) .
-Wenzel (W) configuration: this configuration occurs when the liquid wets completely the valleys.

Surface free energy : F W SL = (a + 2b + c) γ SL . The W macroscopic contact angle of the liquid over the periodic rough substrate (a, b, c) is given by

(3) cos θ W = F r SV -F W SL γ LV (a + c) = r cos θ 0 .
The two lines of equation ( 2)-(3) intersect at point

(cos θ 0 , cos θ) = (-(1 -φ) / (r -φ) , -r (1 -φ) / (r -φ))
inside the square [-1, 0] 2 , as in Fig. 1. Fix θ 0 . Then the CB state wins over W if

cos θ CB ≥ cos θ W or else if b ≥ -c (1 + cos θ 0 ) / (2 cos θ 0 ).
-mixed Cassie-Baxter-Wenzel (CW) state. This configuration is obtained while attaching linearly the liquid interface ending on the flat side of size c of the rectangular protrusion (a, b, c) and starting from the top-right corner of the first rectangular protrusion of height b and symmetrically, while attaching linearly the liquid interface starting on the flat side and ending at the top-left corner of the second rectangular protrusion of height b of the pattern (see Fig. 4). This is of course possible only if c is large enough to avoid overlaps of the two parts of the symmetric linear interfaces.

Let -tan θ 0 =: b/x. The surface free energy of the mixed CW state over a unit cell reads:

F CW SL = aγ SL + (2b + x) γ SV + (c -2x) γ SL + 2 √ b 2 + x 2 γ LV .
The corresponding CW macroscopic contact angle on the periodic rough substrate (a, b, c) is

cos θ CW = F r SV -F CW SL γ LV (a + c) = 1 a + c (a + c -2x) cos θ 0 -2 b 2 + x 2 = cos θ 0 + 2b a + c cos 2 θ 0 sin θ 0 - 2b a + c 1 sin θ 0 . Else, ( 4
) cos θ CW = F r SV -F CW SL γ LV (a + c) = cos θ 0 - 2b a + c sin θ 0 = -(r -1) sin θ 0 + cos θ 0 .
The CW contact angle curve is a non-linear, convex function of cos θ 0 , starting from (-1, -1) with a slope -∞.

The macroscopic contact angle on the whole range of cos θ 0 is given by, (see Figure 3):

cos θ = max cos θ CB , cos θ CW , cos θ W .
Remarks:

(i) We observe that cos θ CW meets cos θ W at θ 0 = 3π/4 and at this value of cos θ 0 we have cos

θ CW = cos θ W > cos θ CB if and only if r < √ 2 + φ 1 - √ 2 or else c > 2b/ √ 2 -1 .
We conclude that the CW part of the curve cos θ will be seen if and only if c > 2b/ √ 2 -1 and this will then happen with cos θ 0 in the range

cos θ 0 ∈ -(c/(2b)) 2 -1 (c/(2b)) 2 +1 , -1/ √ 2
. This is illustrated in Fig. 3. The lower bound of the range is the value of cos θ 0 for which cos θ CW = cos θ CB . When c = 2b/ √ 2 -1 , the three curves cos θ CB , cos θ CW , cos θ W all meet at abscissa -1/ √ 2.

(ii) If we impose c < 2b/ √ 2 -1 , (the well is deep enough) the CW part of the curve cos θ will never be seen for any θ 0 and the W state will win over CB for all angles θ 0 such that b < -c (1 + cos θ 0 ) / (2 cos θ 0 ) , or else for all angles θ 0 such that cos θ 0 > -c c+2b > -1/ √ 2.

Rectangular protrusions: two scales

Consider a periodic substrate where k 2 ≥ 1 small rectangular protrusions of height b 2 and width a 2 are inserted between two large rectangular protrusions of height b 1 > b 2 and width a 1 (see Fig. 5). Assume that the consecutive protrusions are equally spaced by c 2 , with c 2 → 0 corresponding to close packing. The length of the large rough pattern in the valleys is thus:

C 1 := k 2 a 2 + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 ,
whereas the projected length of the valleys is

c 1 := k 2 a 2 + (k 2 + 1) c 2 , so C 1 = c 1 + 2 (k 2 + 1) b 2 .
We also assume that k 1 ≥ 1 small rectangular protrusions of height b 2 and width a 2 are inserted on top of the large rectangular protrusion of height b 1 . The length of the large rough pattern on top of the flat roofs is thus:

A 1 = k 1 a 2 + 2 (k 1 -1) b 2 + (k 1 -1) c 2 , whereas the projected length of the top roofs is a 1 := k 1 a 2 + (k 1 -1) c 2 so A 1 = a 1 + 2 (k 1 -1) b 2 .
The total projected length of the large scale pattern is thus

a 1 +c 1 = (k 2 + k 1 ) (a 2 + c 2 ), so A 1 + C 1 = (k 2 + k 1 ) (a 2 + c 2 ) + 2 (k 2 + k 1 ) b 2 .
For each 2 scales, we also have the covered fractions

φ 2 = a 2 / (a 2 + c 2 ) and φ 1 = a 1 / (a 1 + c 1 ) = 1 k 2 + k 1 a 1 a 2 φ 2 = (k 1 (a 2 + c 2 ) -c 2 ) (k 2 + k 1 ) a 2 φ 2 .
Concerning roughness, we have

r 2 = 1+ 2b 2 a 2 + c 2 and r 1 = 1+ 2b 1 a 1 + c 1 = 1+ 1 k 2 + k 1 2b 1 a 2 + c 2 = 1+ 1 k 2 + k 1 b 1 b 2 (r 2 -1)
and the roughness of the 2-scale substrate will be seen to be r

= r 2 + r 1 -1 = 1 + (r 2 -1) 1 + 1 k2+k1 b1 b2 = 1 + 2b2 a2+c2 1 + 1 k2+k1 b1 b2 .
We end up with a large-scale (a 1 , b 1 , c 1 ) pattern within which small-scale patterns (a 2 , b 2 , c 2 ) were nested additively on their flat horizontal parts. Because c 1 and a 1 are affine functions of k 2 and k 1 , the larger (k 2 , k 1 ) are, the more disparate the two scales are.

In the sequel, we assume

(H) : 2b 2 > c 2 √ 2 -1 or r 2 > 1 + (1 -φ 2 ) √ 2 -1
(the small scale well is deep enough) so that, at the small scale 2, the W state only competes with state CB. It will win over CB for those angles θ 0 such that cos

θ 0 > -c2 c2+2b2 > -1/ √ 2.
In the complementary large range cos θ 0 < -c2 c2+2b2 only the CB state should be observed. In between, we assume that the liquid interface joins linearly the two latter points, without visiting the valleys of the in-between small protrusions (valid under (H) in the range cos θ 0 < -c2 c2+2b2 ). On top of the flat roofs, with k 1 small protrusions inserted, we also assume that the liquid interface does not visit the valleys. These are mixed Cassie-Wenzel (CW) states that have to be taken into account in the wetting problem. We do not consider the mixed CW states, say CW k (l), l = 1, ..., (k 2 + 1) /2 , obtained while following the paths down the valleys of the in-between small protrusions because, in view of the condition (H), these configurations lose over the latter ones CW k (l)

θ 0 Figure 5. Unit cell of a 2-scale periodic substrate. k 1 = 3, k 2 = 4 with the state CW k (2). in the range cos θ 0 < -c2 c2+2b2 . The state CW k (2) is illustrated in Figure 5 when k = (k 1 = 3; k 2 = 4).
For a 2-scale pattern with k 2 rectangular protrusions of sizes (a 2 , b 2 ) inserted in the valleys and k 1 rectangular protrusions of sizes (a 2 , b 2 ) inserted on the top roofs of the large scale pattern, we easily get the surface free energy of the CW k (l) state over a unit cell as:

F CW k (l) SL = k 1 a 2 γ SL + (k 1 -1) c 2 γ LV + (k 1 -1) (2b 2 + c 2 ) γ SV + (k 2 + 2 -2l) a 2 γ SL + (2b 1 + 2 (l -1) a 2 + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 ) γ SV + 2 (lc 2 + (l -1) a 2 ) 2 + b 2 1 + (k 2 + 1 -2l) c 2 γ LV .
The corresponding Solid-Vapor surface free energy is:

F r SV = (A 1 + 2b 1 + C 1 ) γ SV .
The Liquid-Vapor interface free energy for a length (a 1 + c 1 ) cos θ is: γ LV (a 1 + c 1 ) cos θ.

Thus, with l = 1, ..., (k 2 + 1) /2 and k := (k 2 , k 1 ) and ∆F

CW k (l) SL = F r SV -F CW k (l) SL (5) cos θ CW k (l) = ∆F CW k (l) SL (a 1 + c 1 ) γ LV =: a k (l) cos θ 0 -b k (l) ,
where

a k (l) = (k 2 + k 1 + 2 -2l) a 2 (k 2 + k 1 ) (a 2 + c 2 ) ; b k (l) = 2 (lc 2 + (l -1) a 2 ) 2 + b 2 1 + (k 2 + k 1 -2l) c 2 (k 2 + k 1 ) (a 2 + c 2 )
.

To decide now which CW k (l) state will win on some range of cos θ 0 , we need to consider the free energy restricted to the well (which will then only depend on k 2 ):

The Solid-Vapor surface free energy over the 2-scale rough pattern in the well is:

F r SV = (2b 1 + C 1 ) γ SV .
The surface free energy of the CW k (l) state restricted to the well is:

F CW k (l) SL = (k 2 + 2 -2l) a 2 γ SL + (2b 1 + 2 (l -1) a 2 + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 ) γ SV + 2 (lc 2 + (l -1) a 2 ) 2 + b 2 1 + (k 2 + 1 -2l) c 2 γ LV . It is independent of k 1 . Thus ∆F CW k (l) SL : = F r SV -F CW k (l) SL = (2b 1 + C 1 ) γ SV -(k 2 + 2 -2l) a 2 γ SL -(2b 1 + 2 (l -1) a 2 + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 ) γ SV -2 (lc 2 + (l -1) a 2 ) 2 + b 2 1 + (k 2 + 1 -2l) c 2 γ LV = γ LV   (k 2 + 2 -2l) a 2 cos θ 0 - 2 (lc 2 + (l -1) a 2 ) 2 + b 2 1 + (k 2 + 1 -2l) c 2   .
When l varies, we have a pencil of possibly intersecting straight lines with positive slopes, as functions of cos θ 0 . For each θ 0 , there is a value of l such that ∆F

CW k (l) SL
is maximum (the surface free energy

F CW k (l) SL
minimum), so we have to find

l (cos θ 0 ) = arg max l=1,..., (k2+1)/2 ∆F CW k (l) SL .
It can be checked that the function l → ∆F

CW k (l) SL
is concave and that its maximizer is a positive real number, independent of k 2 , namely

l * (cos θ 0 ) = 1 a 2 + c 2 a 2 + b 1 (c 2 -a 2 cos θ 0 ) a 2 (1 + cos θ 0 ) (2c 2 + a 2 (1 -cos θ 0 )) .
The function l * is monotone decreasing with cos θ 0 varying from -1 to 0, with

l * (-1) → ∞ and l * (0) = a 2 + b 1 c 2 / a 2 (2c 2 + a 2 ) / (a 2 + c 2 ) .
The final integral value of l maximizing

F r SV -F CW k (l) SL is thus either l * = l * (cos θ 0 ) or l * = l * (cos θ 0 ) . Comparison of the values of F r SV -F CW k (l) SL
at both endpoints gives the value of l * to be chosen. This can partially be decided while observing that, as

l → ∞, F r SV -F CW k (l) SL ∼ -2la 2 (1 + cos θ 0 ) (negative slope) while as l → -∞, F r SV -F CW k (l) SL ∼ 2l (a 2 + 2c 2 -2a 2 cos θ 0 ) (positive slope), showing that F r SV -F CW k (l) SL is skewed around l * (cos θ 0 ) . Due to a 2 + 2c 2 -2a 2 cos θ 0 > a 2 (1 + cos θ 0 )
, it bends to the left and in most cases so l * = l * (cos θ 0 ) . To simplify notation, we shall formally work with l * (cos θ 0 ), meaning either l * (cos θ 0 ) or l * (cos θ 0 ) whichever is relevant.

Coming back to (5), we conclude that the Young-Dupré equation for the mixed CW k states is [START_REF] Jeong | Nanoengineered Multiscale Hierarchical Structures with Tailored Wetting Properties[END_REF] cos for each value of θ 0 for which there is a unique value l = l * (cos θ 0 ) corresponding to a CW k (l) state. Therefore, cos θ CW k , as a broken line, is made of superposing linear pieces in the range cos

θ CW k = ∆F CW k (l * (cos θ0)) SL (a 1 + c 1 ) γ LV =: a k (l * (cos θ 0 )) cos θ 0 -b k (l * (cos θ 0 )) ; θ 0 A 1 B 2 A 2 C 2
θ 0 ∈ [-1, 0].
In the range cos θ 0 > -c2 c2+2b2 , still under (H), one should consider similar mixed states CW k (l) except that the liquid interface now wets the solid outside its leaning part (in particular visits the valleys of the in-between small protrusions). Details of the corresponding configuration computations are left to the reader.

3.2.

Mixed state with interface hitting the well inside a valley. From a top right corner, point A 1 , draw a straight line with slope tan θ 0 (see Figure 6). This line meets the substrate either on a roof at scale 2, or in the bottom at scale 2, at point B 2 , or on the vertical boundary at scale 2. In this last case the liquid contact angle is less than π/2, therefore less than θ 0 , so that the contact point will move up to the corner, a left corner of protrusion at scale 2, point A 2 . The other two cases are local minima of energy, with liquid contact angle equal to θ 0 . We denote C 2 the corner opposite A 2 . Here we will show that the "bottom" case, with liquid vapor interface A 1 B 2 , has energy greater than the configuration with liquid vapor interface A 1 A 2 , or greater than the configuration with liquid vapor interface A 1 C 2 followed by a Wenzel configuration from C 2 to A 2 . The first case will occur when the Cassie-Baxter state at scale 2 wins over the Cassie-Wenzel and Wenzel states at scale 2; the second case will occur when the Wenzel state at scale 2 wins over the Cassie-Wenzel and Cassie-Baxter states at scale 2. Under hypothesis (H) the Cassie-Wenzel state never wins at scale 2. Therefore the "bottom" case, with liquid vapor interface A 1 B 2 , will never win.

The interfaces A 1 A 2 and A 1 C 2 have slope respectively larger and smaller than tan θ 0 , but do not have a well-defined liquid contact angle because they meet the substrate at two corners.

Let us first show that A 1 B 2 loses over A 1 A 2 , in the appropriate range of θ 0 . The argument goes in two steps. First, consider translations of the pattern at scale 2, keeping A 1 B 2 fixed. In the comparison between A 1 B 2 , held fixed, and A 1 A 2 , mobile, the most unfavorable case for A 1 A 2 is the rightmost position, when A 1 B 2 touches the corner C 2 . It is enough to consider this case. Second, vary point A 1 along the line of slope θ 0 going through C 2 . The most unfavorable case for A 1 A 2 θ0 θ0 Next let us show that A 1 B 2 loses over A 1 C 2 , in the complementary range of θ 0 . Denote f (C 2 ) the energy of the configuration with liquid-vapor interface along A 1 C 2 , and f (B 2 ) the energy of the configuration with liquid-vapor interface along A 1 B 2 . Again, consider translations of the pattern at scale 2, keeping A 1 B 2 fixed. In the comparison between A 1 B 2 , held fixed, and A 1 C 2 , mobile, the most unfavorable case for A 1 C 2 is now the leftmost position, when B 2 is in the bottom corner below A 2 . Indeed, when translating by dx,

f (C 2 ) -f (B 2 ) -→ f (C 2 ) -f (B 2 ) -dx γ LV (cos θ -cos θ 0 )
with tan θ the slope of A 1 C 2 . Second, vary point A 1 along the line of slope θ 0 going through B 2 in the bottom corner. The most unfavorable case for A 1 C 2 is now when

A 1 goes to infinity. Denote D 2 the orthogonal projection of C 2 onto A 1 B 2 . Then f (C 2 ) -f (B 2 ) = (b 2 + c 2 )(γ SL -γ SV ) -|D 2 B 2 | γ LV and |D 2 B 2 | = -c 2 cos θ 0 + b 2 sin θ 0 , so that f (C 2 ) -f (B 2 )
γ LV = -b 2 (cos θ 0 + sin θ 0 ) < 0 in the Wenzel range under hypothesis (H).

3.3.

Mixed state hitting the well on top of a roof. Another family of mixed states which deserves attention is the following: it can happen that the line of slope tan θ 0 hits the l-th small scale motif (a 2 , b 2 , c 2 ) at a point on top of its roof, starting from the point at top of the right vertical substrate interface of height b 1 and symmetrically, while appending linearly the liquid interface at a point on a flat top-roof of the (k 2 -l + 1) -th rectangular protrusion of height b 2 , ending at the top-left corner of the second rectangular protrusion of height b 1 of the pattern. See Fig. 7. As before, in between, the interface is a horizontal line encompassing the intermediate small-scale wells. We shall call these mixed states CW roof k (l) . Following the study on the states CW k (l) , we easily get the surface free energy of the CW roof k (l) state as: For some a ∈ (0, a 2 ) ,

F CW roof k (l) SL = k 1 a 2 γ SL + (k 1 -1) c 2 γ LV + (k 1 -1) (2b 2 + c 2 ) γ SV + [(k 2 + 2 -2l) a 2 -2a] γ SL + (2b 1 + 2 (l -1) a 2 + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 + 2a) γ SV + 2 (lc 2 + (l -1) a 2 + a) 2 + b 2 1 + (k 2 + 1 -2l) c 2 γ LV .
CW roof k (l) SL := F r SV - F CW roof k (l) SL
, we get [START_REF] Hipp | Systematic control of hydrophobic and superhydrophobic properties using double-rough structures based on mixtures of metal oxide nanoparticles[END_REF] cos

θ CW roof k (l) = ∆F CW k (l) SL (a 1 + c 1 ) γ LV =: a k (l) cos θ 0 -b k (l) ,
where

a k (l) = (k 2 + k 1 + 2 -2l) a 2 -2a (k 2 + k 1 ) (a 2 + c 2 ) ; b k (l) = 2 (lc 2 + (l -1) a 2 + a) 2 + b 2 1 + (k 2 + k 1 -2l) c 2 (k 2 + k 1 ) (a 2 + c 2 )
.

Observing lc 2 + (l -1) a 2 + a = b 1 / tan (π -θ 0 ) , we obtain a k (l) = [(k 2 + k 1 ) a 2 + 2lc 2 ] sin θ 0 + 2b 1 cos θ 0 (k 2 + k 1 ) (a 2 + c 2 ) sin θ 0 b k (l) = 2b 1 + (k 2 + k 1 -2l) c 2 sin θ 0 (k 2 + k 1 ) (a 2 + c 2 ) sin θ 0 ,
giving, from [START_REF] Hipp | Systematic control of hydrophobic and superhydrophobic properties using double-rough structures based on mixtures of metal oxide nanoparticles[END_REF], the final explicit relation

cos θ CW roof k (l) = (k 2 + k 1 ) a 2 + 2lc 2 (k 2 + k 1 ) (a 2 + c 2 ) cos θ 0 - 2b 1 sin θ 0 (k 2 + k 1 ) (a 2 + c 2 ) - (k 2 + k 1 -2l) c 2 (k 2 + k 1 ) (a 2 + c 2 )
.

For each l = 1, ..., (k 2 + 1) /2 , the range of θ 0 for which the latter equation makes sense clearly is defined by

b 1 / tan (π -θ 0 ) ∈ [lc 2 + (l -1) a 2 , lc 2 + la 2 ] .
Varying l = 1, ..., (k 2 + 1) /2 , we end up with a relationship cos θ CW roof k , as a function of θ 0 only.

Mixed state (W C k states). (i)

There is yet another family of mixed states that needs to be considered (see Fig. 8). Let W C k (l), l = 1, ..., (k 2 + 1) /2 be the Wenzel-Cassie states obtained while appending linearly the liquid interface on the top-left corner of the l-th rectangular protrusion of height b 2 inside the valley, starting from a point of the left vertical substrate interface of height b 1 (making a contact angle θ 0 with the liquid interface) and symmetrically, while appending linearly the liquid interface on the top-right corner of the (k 2 -l + 1) -th rectangular protrusion of height b 2 , ending at a symmetric point of the right rectangular protrusion of height b 1 of the pattern. In between, we assume that the liquid interface joins linearly the two latter points, without visiting the valleys of the in-between small protrusions. On top of the flat roofs, with k 1 small protrusions inserted, we also assume that the liquid interface does not visit the valleys. These are mixed Wenzel-Cassie (WC) states that have to be taken into account in the wetting problem.

In Fig. 8, where

k 1 = k 2 = 2, a 2 = b 2 = c 2 = 1 and 5 -2 √ 2 > b 1 ≥ 1,
the state WC 22 (1) minimizes the energy under the two conditions: arccos (-1/3) ≤ θ 0 ≤ (3π) /4 and sin θ 0 -(1 + b 1 ) cos θ 0 ≤ 2. The condition arccos (-1/3) ≤ θ 0 comes from the fact that the Cassie-Baxter state at scale 2 wins over Wenzel state at scale 2. The condition θ 0 ≤ 3π/4 comes from θ 1 > θ 0 , required for the state to be a minimizer. The last condition comes from WC k (1) to win over CB k (the pure CB state).

We now show that the only WC states that need to be considered are the extremal ones, corresponding either to l = 1 or to the maximal value l * that l can take. And then we compute the macroscopic contact angles of these two configurations.

Consider indeed the difference between the free energies between the states W C k (1) and W C k (2) inside the well. We have

F W C k (2) SL -F W C k (1) SL = b 1 -(a 2 + 2c 2 ) tan θ 0 -π 2 γ SL + (a 2 + 2c 2 ) tan θ 0 -π 2 γ SV + (a2+2c2)γ LV cos(θ0-π 2 ) + a 2 γ SV - b 1 -c 2 tan θ 0 -π 2 γ SL + c 2 tan θ 0 -π 2 γ SV + c2 cos(θ0-π 2 ) γ LV + a 2 γ SL + c 2 γ LV = (a 2 + c 2 ) tan θ 0 - π 2 + a 2 (γ SV -γ SL ) + a 2 + c 2 cos θ 0 -π 2 -c 2 γ LV . Thus F W C k (2) SL -F W C k (1) SL γ LV = a 2 + c 2 sin θ 0 1 -cos 2 θ 0 + a 2 cos θ 0 -c 2 = (a 2 + c 2 ) sin θ 0 + a 2 cos θ 0 -c 2 = (a 2 + c 2 ) (sin θ 0 + cos θ 2 )
if cos θ 2 = a2 a2+c2 cos θ 0 -c2 a2+c2 defines the CB contact angle over the pattern (a 2 , b 2 , c 2 ) , as from (2). We conclude that state W C k (2) wins (loses) over W C k (1) if and only if sin θ 0 +cos θ 2 < 0 (sin θ 0 +cos θ 2 > 0). But then also W C k (l + 1) wins (loses) over W C k (l) if and only if sin θ 0 +cos θ 2 < 0 (sin θ 0 +cos θ 2 > 0), so that the overall winning state is either W C k (l * ) or W C k (1), depending on sin θ 0 +cos θ 2 < 0 or sin θ 0 + cos θ 2 > 0.

Let us compute the relationship cos θ versus cos θ 0 corresponding to the state W C k (1) : it will exist if b 1 ≥ c 2 / tan (π -θ 0 ) . If this is the case, the surface free energy of the W C k (1) state reads:

F W C k (1) SL = k 1 a 2 γ SL + (k 1 -1) c 2 γ LV + (k 1 -1) (2b 2 + c 2 ) γ SV + (k 2 a 2 + 2 (b 1 -c 2 / tan (π -θ 0 ))) γ SL + (2c 2 / tan (π -θ 0 ) + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 ) γ SV + 2c 2 1/ tan 2 (π -θ 0 ) + 1 + (k 2 -1) c 2 γ LV . With F r SV = (A 1 + 2b 1 + C 1 ) γ SV = ((k 2 + k 1 ) (a 2 + 2b 2 + c 2 ) + 2b 1 ) γ SV and ∆F W C k (1) SL := F r SV -F W C k (1) SL , we get cos θ W C k (1) = ∆F W C k (1) SL (a 1 + c 1 ) γ LV = cos θ 0 ((k 2 + k 1 ) a 2 + 2b 1 + 2c 2 / tan θ 0 ) -(2c 2 / sin θ 0 + (k 2 -1) c 2 ) a 1 + c 1 = ((k 2 + k 1 ) a 2 + 2b 1 ) cos θ 0 -2c 2 sin θ 0 -(k 2 + k 1 -2) c 2 (k 2 + k 1 ) (a 2 + c 2 ) .
Let us now compute the relationship cos θ versus cos θ 0 corresponding to the state W C k (l * ) : it will exist and will be different from W C k (1) if:

l * > 1, b 1 ∈ ((l * c 2 + (l * -1) a 2 ) / tan (π -θ 0 ) , ((l * + 1) c 2 + l * a 2 ) / tan (π -θ 0 )] and k 2 ≥ 2l * -1.
If this is the case, the surface free energy of the W C k (l * ) state reads:

F W C k (l * ) SL = k 1 a 2 γ SL + (k 1 -1) c 2 γ LV + (k 1 -1) (2b 2 + c 2 ) γ SV + ((k 2 -2 (l * -1)) a 2 + 2 (b 1 -(l * c 2 + (l * -1) a 2 ) / tan (π -θ 0 ))) γ SL + (2 (l * c 2 + (l * -1) a 2 ) / tan (π -θ 0 ) + 2 (k 2 + 1) b 2 + (k 2 + 1) c 2 ) γ SV + 2 (l * c 2 + (l * -1) a 2 ) 1/ tan 2 (π -θ 0 ) + 1 + (k 2 -2 (l * -1) -1) c 2 γ LV .
We thus get

cos θ W C k (l * ) = ∆F W C k (l * ) SL (a 1 + c 1 ) γ LV = ((k 2 + k 1 ) a 2 -2 (l * -1) a 2 + 2b 1 ) cos θ 0 -2 (l * c 2 + (l * -1) a 2 ) sin θ 0 -(k 2 + k 1 -2l * ) c 2 (k 2 + k 1 ) (a 2 + c 2 ) .
(ii) Consider the same configurations as the ones just discussed except that now, the liquid interface joins linearly the two latter points (l-th rectangular protrusion and (k 2 -l + 1) -th rectangular protrusion on which the interface is based), while visiting the valleys of the in-between small protrusions. This case arises when the Wenzel state at scale 2 wins over the Cassie-Baxter state at scale 2, or else if cos θ 0 > -c 2 / (2b 2 + c 2 ) . Proceeding similarly, we get that the difference of the free energies between l and l + 1

F (l+1) SL -F (l) SL γ LV = (a 2 + c 2 ) (sin θ 0 + cos θ 0 ) + 2b 2 cos θ 0 .
Thus depending on -tan θ 0 > a2+2b2+c2 a2+c2 (or <), we conclude that the corresponding WC states that need only to be considered are the extremal ones, corresponding either to l = 1 or to the maximal value l * that l can take, respectively. 3.5. Pure Cassie-Baxter state. This case corresponds to an interface which is in CB state at both scales. The surface free energy of the CB k state of the 2-scale pattern with k = (k 2 , k 1 ) small protrusions inserted is

F CB k SL = k 1 a 2 γ SL + (k 1 -1) c 2 γ LV + (k 1 -1) (2b 2 + c 2 ) γ SV + (2b 1 + C 1 ) γ SV + c 1 γ LV = ((k 2 + k 1 ) c 2 + k 2 a 2 ) γ LV + k 1 a 2 γ SL + ((k 2 + k 1 ) (2b 2 + c 2 ) + k 2 a 2 + 2b 1 ) γ SV . Thus (8) cos θ CB k = F r SV -F CB k SL (a 1 + c 1 ) γ LV = k 1 a 2 cos θ 0 -((k 1 -1) c 2 + c 1 ) a 1 + c 1 = k 1 a 2 (cos θ 0 + 1) (k 2 + k 1 ) (a 2 + c 2 ) -1 =: φ CB k (1 + cos θ 0 ) -1.
where φ CB k is the equivalent 2-scale covered fraction. Note

φ CB k = k 1 a 2 (k 2 + k 1 ) (a 2 + c 2 ) > φ 2 φ 1 = a 2 a 2 + c 2 a 1 a 1 + c 1 = a 2 a 2 + c 2 k 1 a 2 + (k 1 -1) c 2 (k 2 + k 1 ) (a 2 + c 2 )
and φ CB k < min (φ 2 , φ 1 ) .

3.6. Pure Wenzel state. This case corresponds to an interface which is in W state at both scales. The contact angle of the pure-W k state of the 2-scale substrate with k = (k 2 , k 1 ) small protrusions inserted is given by

cos θ W k = F r SV -F W k SL (k 2 + k 1 ) (a 2 + c 2 ) γ LV , where
F W k SL = (A 1 + 2b 1 + C 1 ) γ SL .
This leads to Suppose hypothesis (H) holds so that the mixed state CW at scale 2 does not contribute to the overall picture.

(9) cos θ W k = 1 + 2 b 1 + (k 2 + k 1 ) b 2 (k 2 + k 1 ) (a 2 + c 2 ) cos θ 0 =: r W k cos θ 0 with equivalent roughness r W k = r 2 + r 1 -1 > max (r 2 , r 1 ) where r 2 = 1 + (2b 2 ) / (a 2 + c 2 ) and r 1 = 1 + (2b 1 ) / ((k 2 + k 1 ) (a 2 + c 2 )) . Note also r 2 = r 1 ⇔ b 1 = (k 2 + k 1 ) b 2 .
One can distinguish two cases:

• Either the Cassie-Baxter state at small scale 2 wins over the Wenzel state (the range cos θ 0 ≤ -c2 c2+2b2 ). Here then cos θ 0 increases from -1 to -c2 c2+2b2 . We exhibited the following scenario involving mixed states: when θ 0 decreases starting from π, the state corresponding to an interface in Cassie-Baxter states both at small scale 2 and large scale 1 (the pure Cassie-Baxter state) becomes unstable at some θ 0 < π. The system enters into a family of states remaining in Cassie-Baxter state on top of the large-scale structure but with bending symmetric interfaces inside the well trapping air in the corners and still in Cassie-Baxter states in between the well's protrusions. Inside the well, the endpoint of the interface can either launch on top of any roof or to the left corner of any small protrusion at the bottom of the well. These are the states CW roof k (l) and CW k (l) respectively. All other possible states are unstable: the case of the interface hitting the well inside a valley or the vertical well of a small bottom protrusion. When θ 0 decreases more, a new class of states emerges: the ones obtained while appending linearly the liquid interface on the top-left corner of any rectangular small protrusion inside the valley, starting from a point of the left vertical large-scale substrate interface (making a contact angle θ 0 with the liquid interface) and symmetrically.

• Or the Wenzel state at small scale 2 wins over the Cassie-Baxter state (the range cos θ 0 ≥ -c2 c2+2b2 ). Here cos θ 0 increases from -c2 c2+2b2 to 0. Although the details of the computations were left to the reader, we can state the following obvious facts: when θ 0 increases starting from 0, the state corresponding to an interface in Wenzel states both at small scale 2 and large scale 1 (the pure Wenzel state) becomes unstable at some θ 0 > 0. A new class of states emerges: the ones obtained while appending linearly the liquid interface on the top-left corner of any rectangular small protrusion inside the valley, starting from a point of the left vertical largescale substrate interface (making a contact angle θ 0 with the liquid interface) and symmetrically. Outside the bending interfaces, the states of interest remain in Wenzel state both on top of the large-scale structure and at the bottom of the well. Increasing again θ 0 , the system enters into a family of states remaining in Wenzel state on top of the large-scale structure but with bending symmetric interfaces inside the well trapping air in the corners and still in Wenzel states in between the well's protrusions. Inside the well, the endpoint of the interface can either launch on top of any roof or to the left corner of any small protrusion at the bottom of the well. This family of states is the Wenzel counterpart of the later states CW roof k (l) and CW k (l). Finally, the system enters a state which is of Wenzel type at scale 2 (on the top plateau of the large-scale structure) and Cassie-Baxter at scale 1, with the interface jumping directly over the well. Furthermore:

-There exists a geometry (a value of the aspect ratio parameters together with a value of k := (k 1 , k 2 )) and a range of θ 0 such that each of these states will dominate the other ones.

-In general, when the geometry is fixed, only a subset of the whole set of states (typically 2, 3 or 4) will contribute to the final Young-Dupré curve giving the rough contact angle cos θ as a function of cos θ 0 on the whole range cos θ 0 ∈ [-1, 0] . In any case, the final Young-Dupré curve giving the contact angle cos θ as a function of cos θ 0 is obtained while considering the maximum of the curves cos θ versus cos θ 0 for all contributing states to the wetting problem just discussed. It thus presents itself typically as a broken line.

Separation of scales: scaling limit

We now consider the scaling limit of the latter configurations. Assume one takes the limit min (k 2 , k 1 ) → ∞, k 1 /k 2 → ρ and b 1 = (k 2 + k 1 ) b, b > 0. In this limit, the two scales become separate.

With λ = k 1 + k 2 , with a := ρ 1+ρ (a 2 + c 2 ), b and c := 1 1+ρ (a 2 + c 2 )
, in the limit we have: a 1 ∼ λa, b 1 = λb and c 1 ∼ λc, so this scaling limit model is similar to a one scale model (a, b, c).

4.1.

Mixed or pure states in the scaling limit. Let us first consider the CB states.

• It follows from ( 8) that the Young-Dupré equation giving cos θ CB k as a function of cos θ 0 becomes in the limit:

cos θ CB∞ = ρ 1 + ρ φ 2 (cos θ 0 + 1) -1.
Because

φ 1 = a 1 / (a 1 + c 1 ) ∼ a/ (a + c) = ρ/ (1 + ρ), this is also cos θ CB∞ = φ 1 φ 2 (cos θ 0 + 1) -1.
Renaming CB ∞ as CB 12 to signify that we have a CB state at both scales 1 and 2, we get

cos θ CB12 = φ CB12 (cos θ 0 + 1) -1 with φ CB12 = φ 1 φ 2 .
Remark: The limiting covered fraction is thus exactly φ CB12 = φ 1 φ 2 .

• Concerning the CW states, in view of

l → l * (cos θ 0 ) ∼ k 2 + k 1 a 2 + c 2 b (c 2 -a 2 cos θ 0 ) a 2 (1 + cos θ 0 ) (2c 2 + a 2 (1 -cos θ 0 )) =: (k 2 + k 1 ) l, from (6)
a k (l) = (k 2 + k 1 + 2 -2l) a 2 (k 2 + k 1 ) (a 2 + c 2 ) ∼ a l = 1 -2l a 2 a 2 + c 2 = φ 2 1 -2l ; b k (l) = 2 (lc 2 + (l -1) a 2 ) 2 + b 2 1 + (k 2 + k 1 -2l) c 2 (k 2 + k 1 ) (a 2 + c 2 ) ∼ b l = 2 (c 2 + a 2 ) 2 l 2 + b 2 + 1 -2l c 2 a 2 + c 2 .
This corresponds to a mixed limiting Cassie-Baxter-Wenzel state of equation cos θ CW∞ = a l cos θ 0 -b l , which is nonlinear in cos θ 0 .

In the limit, we have 2 contact angles, one θ 0 at the smooth vertical walls of the limiting large structure, the other θ 2 at its rough horizontal walls, in the wells. The contact angle θ 2 is characterized by tan (π

-θ 2 ) = b/ (c 2 + a 2 ) l , therefore (c 2 + a 2 ) 2 l 2 + b 2 = b 1 + 1/ tan 2 (θ 2 ) = b/ sin (θ 2 ) . We also have 1 + tan 2 (π -θ 2 ) = (a2+c2) 2 (c2-a2 cos θ0) 2 = 1 cos 2 (θ2) so cos θ 2 = cos θ CB 2 = - c 2 -a 2 cos θ 0 a 2 + c 2 = φ 2 (1 + cos θ 0 ) -1;
it is the expected contact angle of CB states on a substrate with solid-liquid contact angle θ 0 .

The latter Young-Dupré equation of the CW ∞ states is thus finally amenable to the simple expression

cos θ CW1CB2 = -(r 1 -1) sin θ CB 2 + cos θ CB 2 , in terms of θ CB 2 .
Here the corresponding state has been renamed CW 1 CB 2 to signify that we have a CW state at scale 1 and a CB state at scale 2. In the latter equation,

r 1 = 1 + (2b) / (a 2 + c 2 ). Coming back to θ 0 cos θ CW1CB2 = -(r 1 -1) sin(arccos (φ 2 (1 + cos θ 0 ) -1)) + φ 2 (1 + cos θ 0 ) -1,
which is nonlinear in cos θ 0 . Note that the state CW 1 CB 2 only exists under the condition -tan θ CB 2 > (r 1 -1) / (1 -φ 1 ).

• Consider the WC states in the scaling limit. If sin θ 0 + cos θ CB 2 < 0 the overall winning state is W C k (l * ) which coincides in the scaling limit with CW ∞ . Suppose sin θ 0 +cos θ CB 2 > 0. Then we need to consider W C ∞ as the scaling limit of W C k (1).

With γ

(2)

SL and γ (2)
SV the surface tensions of substrate 2, we have

F W C∞ SL = (a 1 + c 1 ) γ CB2 SL + 2b 1 γ SL = (a 1 + c 1 ) γ (2) SV -cos θ CB 2 γ LV + 2b 1 γ SL F r SV = (a 1 + c 1 ) γ (2) SL + 2b 1 γ SV . Thus cos θ W C∞ = F r SV -F W C∞ SL (a 1 + c 1 ) γ LV = 2b 1 (γ SV -γ SL ) + (a 1 + c 1 ) cos θ CB 2 γ LV (a 1 + c 1 ) γ LV
or (renaming W C ∞ as W 1 CB 2 to signify a W state at large scale 1 and CB state at small scale 2): cos θ W1CB2 = (r 1 -1) cos θ 0 + cos θ CB 2 .

• Concerning the pure Wenzel states, it follows from (9) that cos θ W∞ = (r 1 + r 2 -1) cos θ 0 =: r W∞ cos θ 0 ,

where 1 + 2b a2+c2 = r 1 = r 2 = 1 + 2b2 a2+c2 .
The state W ∞ may be renamed as W 12 to signify a Wenzel state at both scales 1 and 2.

Remark: The limiting roughness is thus

r W∞ = r 1 + r 2 -1, with r 1 = r 2 if b = b 2 .
• Concerning the state which is CB at scale 1 and W at scale 2, we get similarly: cos θ CB1W2 = φ 1 (r 2 cos θ 0 + 1) -1.

• Concerning finally the state which is CW at scale 1 and W at scale 2, we would get proceeding similarly:

cos θ CW1W2 = -(r 1 -1) sin θ W 2 + cos θ W 2 .
(which corresponds to the equation of states of CW 1 CB 2 where θ W 2 was substituted to θ CB 2 ).

φ 1 = ρ 1 + ρ , φ 2 = a 2 a 2 + c 2 , r 1 = 1 + 2b a 2 + c 2 , r 2 = 1 + 2b 2 a 2 + c 2 ,
the curve giving the limiting contact angle cos θ as a function of cos θ 0 is potentially made of the 6 pieces

cos θ CB12 = φ 1 φ 2 (cos θ 0 + 1) -1, cos θ CW1CB2 = -(r 1 -1) sin θ CB 2 + cos θ CB 2 , cos θ W1CB2 = (r 1 -1) cos θ 0 + cos θ CB 2 , cos θ W12 = (r 1 + r 2 -1) cos θ 0 , cos θ CB1W2 = φ 1 cos θ W 2 + 1 -1, cos θ CW1W2 = -(r 1 -1) sin θ W 2 + cos θ W 2 .
Indeed, the final Young-Dupré curve to consider is cos θ = max cos θ CB12 , cos θ CW1CB2 , cos θ W1CB2 , cos θ W12 , cos θ CB1W2 , cos θ CW1W2 , where we recall cos θ CB 2 = φ 2 (1 + cos θ 0 ) -1 and cos θ W 2 = r 2 cos θ 0 .

To determine which of these states is the winner, one can also use the following formulae giving the differences of the pairwise free energy densities in each regime:

• Cassie-Baxter state CB 2 at small scale 2:

F CB12 -F W1CB2 (a 1 + c 1 ) γ LV = (1 -φ 1 ) 1 + cos θ CB 2 + (r 1 -1) cos θ 0 F CB12 -F CW1CB2 (a 1 + c 1 ) γ LV = (1 -φ 1 ) 1 + cos θ CB 2 -(r 1 -1) sin θ CB 2 F W1CB2 -F CW1CB2 (a 1 + c 1 ) γ LV = -(r 1 -1) cos θ 0 + sin θ CB 2 .
• Wenzel state W 2 at small scale 2:

F CB1W2 -F W12 (a 1 + c 1 ) γ LV = (1 -φ 1 ) 1 + cos θ W 2 + (r 1 -1) cos θ 0 F CW1W2 -F W12 (a 1 + c 1 ) γ LV = (r 1 -1) cos θ 0 + sin θ W 2 F CB1W2 -F CW1W2 (a 1 + c 1 ) γ LV = (1 -φ 1 ) 1 + cos θ W 2 -(r 1 -1) sin θ W 2 .
-1 -0.9 -0.8 -0. In Figs. 9 and 10 we only look at the range of θ 0 corresponding to a Cassie-Baxter state CB 2 at scale 2. The red line represents the state W 1 CB 2 , the two other lines represent the states CB 12 (starting from (-1, 1)) and CW 1 CB 2 .

In Fig. 9 we show a two-scale situation for which the state CW 1 CB 2 does not contribute to the wetting problem.

In Fig. 10, we show a two-scale situation for which CW 1 CB 2 contributes. Here the 3 states will coexist in the scaling limit.

For such values of the aspect ratios, the transition occurs at φ 1 = 1/7 and r 1 = 1 + 2/7 and the range of θ 0 corresponding to CB 2 is cos θ 0 ≤ -c2 c2+2b2 = -1/3. In Figs. 11 and12, we look at the range of θ 0 corresponding to a Wenzel state W 2 at scale 2, for the same values as before of the covered fractions and roughnesses at both scales. The red line represents the state CB 1 W 2 , the green line represents the state CW 1 W 2 and the blue line represents the state W 12 . In both cases, the pure Wenzel state W 12 dominates the states CB 1 W 2 and CW 1 W 2 . The range of θ 0 corresponding to W 2 is cos θ 0 ≥ -1/3.

In any case, we find a significant improvement of the 2-scale model compared to the strict one-scale model (a, b, c) corresponding to the latter model with φ 1 , r 1 → 1 (or φ 2 , r 2 → 1) where it is seen that only the three distinct states of the one scale regime can remain in the limit. This is shown in Fig. 13 where the 2 red lines correspond to a 1-scale model with φ 1 , r 1 → 1 and a 2 = b 2 = c 2 = 1 (φ 2 = 1/2 and r 2 = 2), compared to a 2-scale model with φ 2 = 1/2 and r 2 = 2 and φ 1 = 1/9 and r 1 = 1 + 2/9. The 6 blue curves correspond to the 6 states derived in the scaling limit and each can be identified from Figs. 10 and12; they are all below the red ones translating an increase of the contact angle in the 2-scale situation compared to the 1-scale one. Here the full range of cos θ 0 is considered. We note however in this particular example that the Cassie-Baxter state in the 2-scale model is screened by mixed states on a large range of cos θ 0 . This raises the following question: can one find a regime of the parameters (φ 1 , r 1 ) and (φ 2 , r 2 ) for which the cos θ 0 -range of the 2-scale CB state does not diminish as compared to the one of the 1-scale model? This question can be answered as follows. First consider the value of θ 0 for which θ CB The condition under which CB 12 wins over W 1 CB 2 and CW 1 CB 2 is thus cos θ CB12 ≥ max cos θ W1CB2 , cos θ CW1CB2 , at the value of θ 0 given by cos θ 0 = -1-φ 2 r2-φ 2

.

The condition cos θ CB12 ≥ cos θ W1CB2 occurs (by direct computation) if

1 -φ 1 r 1 -1 ≤ 1 -φ 2 φ 2 (r 2 -1)
.

Suppose the state CW 1 CB 2 contributes, requiring -tan θ 2 > r1-1

1-φ 1

.

The condition cos θ CB12 ≥ cos θ CW1CB2 then holds if

(1 -φ 1 ) 2 -(r 1 -1) 2 (1 -φ 1 ) 2 + (r 1 -1) 2 ≥ r 2 (1 -φ 2 ) r 2 -φ 2 .
To see why the last condition holds, recall F CB12 -F CW1CB2 (a 1 + c 1 ) γ LV = (1 -φ 1 ) 1 + cos θ CB Another scenario under which the cos θ 0 -range of the 2-scale CB state does not diminish as compared to the one of the 1-scale model is as follows: suppose now the state CW 1 CB 2 loses over CB 12 on the whole CB range of cos θ 0 . For the 2-scale model, this occurs if -tan θ 2 < r1-1

1-φ 1
, where θ 2 is given by -cos θ 2 = r 2 1-φ 2 r2-φ 2

. This condition together with 1-φ 1 r1-1 ≤ 1-φ 2 φ 2 (r2-1) will also meet the desired requirements. One example fulfilling these constraints is φ 1 = φ 2 = 1/2, r 1 = r 2 = 2.

The graph is displayed in Fig. 14 showing that the CB 12 state in the 2-scale model is not polluted by mixed states; here the states CW 1 CB 2 , CW 1 W 2 do not contribute at all.

In the two latter cases, the mixed states will not screen the CB 12 state.

Conclusions

We have shown that quite generally a surface roughness with some aspect ratios will present a mixed state, besides the pure Cassie-Baxter and Wenzel cases. Combining now two different length scales to generate additively the surface on top of their flat elements, we describe rigorously the variety of states, pure or mixed, which will appear versus the aspect ratios. We show how the macroscopic contact angle will vary versus the given affinity between the liquid and the solid. Finally, we consider the scaling limit of the latter two-scale model and compute explicitly the final Young-Dupré curve. We prove that there are only 6 states present. We determine which of these states is the winner and in which range of θ 0 : We show that in any case, the contact angle θ of the two-scale model is larger than the one occurring in any of its one-scale counterpart, meaning an increase of hydrophobicity induced by the presence of two scales. However we observe that in a 2-scale model, the Cassie-Baxter state can be screened by the parasite mixed states which is a drawback for slipperiness and thus superhydrophobicity. Conditions under which the range of cos θ 0 of the 2-scale Cassie-Baxter state does not shrink as compared to the one of the 1-scale model are given at the end of the paper. These are the most favorable instances for an increase of both slipperiness and an increase of the macroscopic contact angle and they are attained only in a specific range of the aspect ratios that we exhibit.

Figure 1 .

 1 Figure 1. Macroscopic contact angle cos θ on rough surface versus contact angle cos θ 0 on flat surface, in Cassie-Baxter and Wenzel states regimes. The CB line has slope φ, the W line has slope r.

Figure 2 .

 2 Figure 2. A microscope view of a two-scale pattern.

Figure 3 .

 3 Figure 3. a = b = 1, c = 7, A case of coexistence of the 3 states CB (red), W (blue) and CW (green).

Figure 4 . 2 .

 42 Figure 4. Unit cell of a periodic substrate. c = 5b and cos θ 0 ∈ -21/29, -1/ √ 2 . The CW state minimizes the energy.

3. 1 .

 1 Mixed Cassie-Baxter and Wenzel states. Let x =integral part of x and k = (k 1 , k 2 ). Let CW k (l), l = 1, ..., (k 2 + 1) /2 be the mixed Cassie-Wenzel states obtained while appending linearly the liquid interface on the top-left corner of the l-th rectangular protrusion of height b 2 , starting from the top-right corner of the first rectangular protrusion of height b 1 and symmetrically, while appending linearly the liquid interface on the top-right corner of the (k 2 -l + 1) -th rectangular protrusion of height b 2 , ending at the top-left corner of the second rectangular protrusion of height b 1 of the pattern.

Figure 6 .

 6 Figure 6. The state hitting the well inside a valley.

Figure 7 .

 7 Figure 7. The roof state CW roof k (2).

Figure 8 .

 8 Figure 8. The state W C 22

3. 7 .

 7 Summary of the results. Consider a general 2-scale periodic pattern model (the grooves model in 1+1 dimensions) defined as a large-scale micropattern (a 1 , b 1 , c 1 ) within which small-scale nanopatterns (a 2 , b 2 , c 2 ) were nested additively on their flat horizontal parts, both plateau (on top of the large scale structure) and well inside the valley.

4. 2 .

 2 Summary of the results and examples. Taking the limit min (k2 , k 1 ) → ∞, k 1 /k 2 → ρ and b 1 = (k 2 + k 1 ) b, b > 0, with

Figure 9 .

 9 Figure 9. φ 2 = 1/2, r 2 = 2 and φ 1 = 1/6 and r 1 = 1 + 1/3 : the state CW 1 CB 2 does not contribute.

Figure 10 .

 10 Figure 10. φ 2 = 1/2, r 2 = 2 and φ 1 = 1/9 and r 1 = 1 + 2/9: the state CW 1 CB 2 contributes.

Figure 11 .Figure 12 .

 1112 Figure 11. φ 2 = 1/2, r 2 = 2 and φ 1 = 1/6 and r 1 = 1 + 1/3 : the state W 12 in blue dominates the two other ones.

Figure 13 .

 13 Figure 13. Two scales with φ 2 = 1/2, r 2 = 2 and φ 1 = 1/9 and r 1 = 1 + 2/9 in blue. The one scale model with φ = 1/2, r = 2 is in red.

2 = θ W 2 2 r2-φ 2 . 2 = cos θ W 2 =: cos θ 2 = -r 2 1-φ 2 r2-φ 2 .

 2222222222 defined by cos θ 0 = -1-φ For this value of θ 0 , cos θ CB

2 -Figure 14 .sin 2 θ CB 2 ≥ ( 1 -φ 1 ) 2 1 + cos θ CB 2 2 or(r 1 - 1 ) 2 1 -cos θ CB 2 ≥ ( 1 -φ 1 ) 2 1 + cos θ CB 2 .-cos θ CB 2 ≤ ( 1 -φ 1 ) 2 -(r 1 - 1 ) 2 ( 1 -φ 1 ) 2 + (r 1 - 1 ) 2 ,with -cos θ CB 2 = -cos θ 2 = r 2 1-φ 2 r2-φ 2 at the value of θ 0 for which θ CB 2 = θ W 2 .

 21421112112111221121211212222222 Figure 14. Two scales with φ 2 = 1/2, r 2 = 2 and φ 1 = 1/2 and r 1 = 2 in blue. The one scale model with φ = 1/2, r = 2 is in red.

Acknowledgments: The present work was initiated when F. Dunlop and T. Huillet were visiting professors at the University of Mons. This work is partially supported by the Ministère de la Région Wallonne and the Belgian Funds for Scientific Research (FNRS). The authors would also like to thank the BELGIAN FEDERAL SCIENCE POLICY OFFICE for the financial support under the DOLFIN ISO PRODEX (2014-2015) project.


Evans Gouno
email: evans.gouno@univ-ubs.fr

Lise Guérineau
email: lise.guerineau@edf.fr

Lise Gu Érineau

FAILURE RATE ESTIMATION IN A DYNAMIC ENVIRONMENT

Keywords: Reliability, failure rate, self-exciting point process, piecewise exponential distribution, Cox's model, maximum likelihood

We present a method to assess the reliability of a material operating in a dynamic environment. The dynamic environment is represented as a sequence of shocks governed by a self-exciting point process. Different distribution models for the time-to-failure of the material are suggested. The case of a piecewise exponential model where a Cox's model is integrated to take into account the effect of the stress is investigated. Maximum likelihood estimates of the model parameters are studied.

Introduction

Many models have been suggested to describe the life distribution of a device subjected to a sequence of shocks occurring randomly. Esary and al. [START_REF] Esary | Shock models and wear processes[END_REF] consider a Poisson process. Singpurwalla [START_REF] Singpurwalla | Survival in dynamic environments[END_REF] reviews the stochastic-process-based approach to failure modeling. Gaudoin et al. [START_REF] Gaudoin | Failure rate behavior of components subjected to random stresses[END_REF] present a theoretical study of different classes of stochastic models to depict the influence of stress on lifetime. In the situation considered here, we assume that shocks undergone by the device result in damage (not necessarily a failure) so that its failure rate increases. We also assume that the device keeps memory of the previous shocks. We consider that the shocks process is a stochastic self-exciting point process (SEPP) also called Hawkes' process [START_REF] Hawkes | Spectra of some self-exciting and mutually exciting point process[END_REF][START_REF] Snyder | Miller Random Point processes in Time and Space[END_REF]. The jumps of this SEPP define the steps of a piecewise exponential model [START_REF] Colvert | Estimation in the piece-wise constant hazard rate model[END_REF] which is assumed to be the distribution of the time-to-failure. Thus the failure rate is supposed to be constant in each step and related to the environment according to a Cox's model [START_REF] Cox | Regression models and life-table[END_REF][START_REF] Friedman | Piecewise exponential models for survival data with covariates[END_REF][START_REF] Gamerman | Dynamic bayesian models for survival data[END_REF]. We address the problem of fitting a SEPP and estimating the Cox's model parameters. Some tests of hypotheses procedures are suggested to assess the shocks effect.

The shocks process

We consider that the shocks process is a self-exciting point process (SEPP). Self-exciting point processes are counting processes with an intensity depending on all or part of the history of the process itself. The intensity of SEPP is not only a function of time but it is also a function of the number of jumps previously occurred on the process. In other words, a given number of more recent events will influence the evolution of the process; the process is said to be self-excited processes. This type of processes can be especially suitable to model some dynamic environments; earthquakes, thunderstorm for example.

Let {N (t), t ≥ 0} be the shocks process, its intensity is considered to be of the form :

(1)

µ(t) = µ 0 + t 0 g(t -s)dN (s) with g(t) = ηe -γt , 0 < η < γ.
If the sequence t 1 , . . . , t n is available, one can easily check the goodness-of-fit to a SEPP using the residual method ([10], p.258). Maximum likelihood estimate of η and γ can be obtained with the method suggested by Ozaki [START_REF] Ozaki | Maximum likelihood estimation of Hawkes self-exciting point processes[END_REF]. Some other response functions g(t) can be considered. In particular, one can consider a link between the covariate characterising the environment and the shocks process. The SEPP can also be used to define the time-to-failure distribution.

The time-to-failure distribution

One can consider that after a shock, the instantaneous probability of failure is high and then decreases. A Weibull distribution can be assumed with a shape parameter depending on the history of the shocks process. The shape parameter can be assumed to be µ(t), the intensity of the SEPP and the level of the shock can be integrated in the function g. In the present work, we consider a sample of time-to-failure (X 1 , . . . , X n ). We assume that the X i 's are independent and follow non-identical piecewise exponential distributions with change-points equal to the dates of the shock. Thus, the failure rate for X i , the time-to-failure of the device i is of the form:

(2)

λ i (x) = +∞ j=0 λ i,j 1 [ti,j ; ti,j+1[ (x).
where t i,j is the date of the j-th shock undergone by the device i, t i,0 = 0 and λ i,j is the failure rate after the j-th shock and before the (j + 1)-th shock. Successive shocks increase the instantaneous probability of failure by a multiplicative factor depending on the covariate characteristics associated with the shocks. We assume a Cox's model such that the failure rate after the j-th shock is constant equal to:

(3)

λ i,j = exp{β ′ z i,j }λ i,j-1
where z i,j ∈ R q is a covariate associated with the j-th shock sustained by device i and λ i,0 = λ 0 , the baseline failure rate.

By induction, we have:

(4) λ i,j = exp{β ′ Z i,j }λ 0 where Z i,j = j ℓ=1 z i,ℓ .
β and λ 0 are the parameters to be estimated on the basis of the observation of a sequence (t i,j , z i,j ) representing the dates and the levels of shocks experienced by the device i, associated with X i , the time-to-failure of the device.

Likelihood

Let us denote θ i,ℓ (β) = exp{β ′ Z i,ℓ } and ∆ i,ℓ+1 = t i,ℓ+1t i,ℓ , ℓ = 0, . . . , k i with k i the total number of shocks occurring on device i. From (2), for x i ∈ [t i,j ; t i,j+1 [, the reliability for the device i is:

(5) R j (x i ) = exp {-λ 0 A i,j (x i , β)} , where A i,j (x i , β) = j-1 ℓ=0 θ i,ℓ (β)∆ i,ℓ+1 + θ i,j (β)(x i -t i,j ).
The probability density function of X i is thus:

(6) f j (x i ) = λ 0 θ i,j (β) exp {-λ 0 A i,j (x i , β)} , for x i ∈ [t i,j ; t i,j+1 [.
Let δ i,j = 1 if the event (failure or censoring) for device i occurs in [t i,j ; t i,j+1 [, 0 else. Let d i = 1 if the device i fails, 0 else. Then the expression of the likelihood is:

L(λ 0 , β) = n i=1 ki j=1 f j (x i ) di R j (x i ) 1-di δi,j = n i=1 ki j=1 λ 0 θ i,j (β) δi,j di exp {-δ i,j λ 0 A i,j (x i , β)}
The log-likelihood is:

(7) log L(λ 0 , β) = k log λ 0 + i,j δ i,j d i β ′ Z i,j -λ 0 i,j δ i,j A i,j (x i , β)
where i,j stands for

n i=1 ki j=1 and k = i,j δ i,j d i .
The likelihood equations are:

(8) ∂ ∂λ 0 log L(λ 0 , β) = k λ 0 - i,j δ i,j A i,j (x i , β) = 0. For u = 1, . . . , q, (9) ∂ ∂β u log L(λ 0 , β) = i,j δ i,j d i Z i,j,u -λ 0 i,j δ i,j ∂ ∂β u A i,j (x i , β) = 0.
Existence and unicity of the maximum likelihood estimate (m.l.e.) for λ 0 and β can be shown applying the theorem 2.2 from Mäkeläinen et al. [START_REF] Mäkeläinen | On the existence and uniqueness of the maximum likelihood estimate of vector-valued parameter in fixed-size samples[END_REF]. One has to prove that the Hessian matrix of the log-likelihood is negative definite. The proof requires then the computation of the second partial derivatives of [START_REF] Cox | Regression models and life-table[END_REF]. The Hessian matrix is negative definite if and only if odd upper-left minors are negative and even upper-left minors are positive. In our case, the first upper-left minor is obviously negative. Conditions on vectors Z can be elicited to satisfy the conditions on the other minors. An expression of λ 0 depending on β can be obtained from [START_REF] Friedman | Piecewise exponential models for survival data with covariates[END_REF]. Replacing λ 0 with this expression in (9), the m.l.e. for β is computed using a numerical method.

Tests of hypotheses

It is possible to test whether the shocks have an effect on the reliability device by considering the null hypothesis H 0 : β = 0. A first possibility is to use the score statistic. Let I * -1 (λ 0 , β) denotes the inverse of minus the matrix of the second partial derivatives of [START_REF] Cox | Regression models and life-table[END_REF]. The score statistic to test β = 0 is: U( λ0 , 0) ′ I * q ( λ0 , 0) -1 U( λ0 , 0) where I * -1 q (., .) is the suitable submatrix of I * -1 (., .), λ0 is the m.l.e. of λ 0 under H 0 , and U(0

) = ∂ ∂β log L( λ0 , β) ( λ0,0)
is the vector of the first partial derivatives of [START_REF] Cox | Regression models and life-table[END_REF]. The distribution of this statistic converges to a χ 2 (q) distribution. The test can also be based on the asymptotic normal approximation:

( λ0 , β) ′ ∼ N (λ 0 , β) ′ , I * ( λ0 , β) -1 . Thus under H 0 : β = 0, ( λ0 , β) ′ I * (0)( λ0 , β)
is approximately χ 2 (q). A third approach is to consider the likelihood ratio statistic that is

Λ = -2 log L( λ0 , 0) L( λ0 , β) .
Again this statistics is asymptotically χ 2 (q) under H 0 . Remark that other tests can be considered to evaluate which shocks characteristics have an influence; the test is conducted on a portion of β.

Concluding remarks

We have described a methodology to assess the failure rate of equipment under a dynamic environment. We have considered a piecewise exponential model with random steps combined with a Cox's model. An application of this approach has been conducted to study the reliability of equipments exposed to thunderstorm. The shocks are lightning strikes. It is possible to model the occurrence of lightning strikes in a given area with a self-exciting point process. Thus the technique described here can be used as a simulation tool to obtain chronicles of strikes and to assess the parameters of the time-to-failure distribution leading to a better understanding of the electrical surge effect on materials and allowing to elaborate strategy to manage a fleet of equipments subjected to such constraint.