Mark Voorneveld

Jörgen W Weibull

OUTER MEASURE AND UTILITY

Keywords: preferences, utility theory, measure theory, outer measure JEL Classification : C60, D01

In most economics textbooks there is a gap between the non-existence of utility functions and the existence of continuous utility functions, although upper semi-continuity is sufficient for many purposes. Starting from a simple constructive approach for countable domains and combining this with basic measure theory, we obtain necessary and sufficient conditions for the existence of upper semi-continuous utility functions on a wide class of domains. Although links between utility theory and measure theory have been pointed out before, to the best of our knowledge this is the first time that the present route has been taken.

Introduction

In most economics textbooks there is a gap between the potential non-existence of utility functions for complete and transitive preference relations on non-trivial connected Euclidean domains -usually illustrated by lexicographic preferences [START_REF] Debreu | Representation of a preference ordering by a numerical function[END_REF] -and the existence of continuous utility functions for complete, transitive and continuous preferences on connected Euclidean domains; see, e.g. [START_REF] Mas-Colell | Microeconomic Theory[END_REF]. Yet, for many purposes, in particular for the existence of a best alternative in a compact set of alternatives, a weaker property -upper semi-continuity -suffices. Hence, the reader of such a textbook treatment might wonder if there exist upper semi-continuous utility functions, and whether this is true even if the domain is not connected. We here fill this gap providing necessary and sufficient conditions for the existence of upper semi-continuous utility functions on arbitrary domains; see Theorem 3.1 and Remark 3.2. Our approach is intuitive and constructive: we start by constructing utilities on a countable domain and then generalize the approach to arbitrary domains by way of basic measure theory.

Measure theory is the branch of mathematics that deals with the question of how to define the "size" (area/volume) of sets. We here formalize a direct intuitive link with utility theory:

given a binary preference relation on a set of alternatives, the "better" an alternative is, the "larger" is its set of worse alternatives. So if one can measure the "size" of the set of worse elements, for each given alternative, one obtains a utility function.

To be a bit more precise, measure theory starts out by first defining the "size" -measureof a class of "simple" sets, such as bounded intervals on the real line or rectangles in the plane, and then extends this definition to other sets by way of approximation in terms of simple sets.

The outer measure is the best such approximation "from above". This is illustrated in Figure 1, where a set S in the plane is covered by rectangles. The outer measure S is the infimum, over

S S

Figure 1: A set S and an approximation of its size using a covering.

all coverings by a countable number of rectangles, of the sum of the rectangles' areas. In more general settings, the outer measure is defined likewise as the infimum over coverings whose sizes have been defined; see, for instance, Rudin (1976, p. 304), [START_REF] Royden | Real Analysis[END_REF], Sec. 3.2), Billingsley (1995, Sec. 3), Ash (2000, p. 14).

We follow this approach by way of defining the utility of an alternative as the outer measure of its set of worse alternatives. We start by doing this for a countable set of alternatives, where this is relatively simple and then proceed to arbitrary sets.

Our paper is not the first to use tools from measure theory to address the question of utility representation: pioneering papers are [START_REF] Neuefeind | On continuous utility[END_REF] and [START_REF] Sondermann | Utility representations for partial orders[END_REF]. See Bridges and Mehta (1995, sections 2.2 and 4.3) for a textbook treatment. However, our approach differs fundamentally from these precursors. Firstly, we only use the basic notion of outer measure, while the mentioned studies impose additional topological and/or measure-theoretic constraints. 2 To the best of our knowledge, the logical connection between outer measure and utility has never been made before. This link between utility theory and measure theory is more explicit, intuitive and mathematically elementary than the above-mentioned approaches.

Secondly, our approach applies to a wider range of domains, in fact to all preference relations that admit real-valued representations.

The rest of the paper is organized as follows. Section 2 recalls definitions and provides notation. Our general representation theorem is given in Section 3. Its proof is in the appendix.

Definitions and notation

Preferences. Let preferences on an arbitrary set X be defined in terms of a binary relation ("weakly preferred to") which is:

complete: for all x, y ∈ X : x y, y x, or both;

transitive: for all x, y, z ∈ X: if x y and y z, then x z.

As usual, x ≻ y means x y, but not y x, whereas x ∼ y means that both x y and y x.

The sets of elements strictly worse and strictly better than y ∈ X are denoted

W (y) = {x ∈ X : x ≺ y} and B(y) = {x ∈ X : x ≻ y}.
2 [START_REF] Neuefeind | On continuous utility[END_REF] restricts attention to finite-dimensional Euclidean spaces and assumes that indifference sets have Lebesgue measure zero. [START_REF] Sondermann | Utility representations for partial orders[END_REF] assumes that preferences are defined on a probability space or a second countable topological space; see also Corollary 3.4 below.

For x, y ∈ X with x ≺ y, the "open interval" of alternatives better than x but worse than y is denoted

(x, y) = {z ∈ X : x ≺ z ≺ y}.
Topology. Given a topology on X, preferences are: Similarly, a function u :

continuous if for each y ∈ X, W (
X → R is usc if for each r ∈ R, {x ∈ X : u(x) < r} is open.
Three important topologies are, firstly, the order topology, generated by (i.e., the smallest topology containing) the collections {W (y) : y ∈ X} and {B(y) : y ∈ X}; secondly, the lower order topology, generated by the collection {W (y) : y ∈ X}, and thirdly, for any subset D ⊆ X, the D-lower order topology, generated by the collection {W (y) : y ∈ D}. By definition, the order topology is the coarsest topology in which is continuous; the lower order topology is the coarsest topology in which is usc.

As mentioned in the introduction, although one often appeals to continuity to establish existence of most preferred alternatives, the weaker requirement of upper semi-continuity suffices.

A short proof: consider a complete, transitive, usc binary relation over a compact set X. If X has no most preferred element, then for each x ∈ X, there is a y ∈ X with y ≻ x, i.e., the collection {W (y) : y ∈ X} is a covering of X with (by usc) open sets. By compactness, there are finitely many y 1 , . . . , y k ∈ X such that W (y 1 ), . . . , W (y k ) cover X. Let y j be the most preferred element of {y 1 , . . . , y k }. Then W (y j ) covers the entire set X, a contradiction.

Utility. A preference relation is represented by a utility function u :

X → R if ∀x, y ∈ X :    x ∼ y ⇒ u(x) = u(y),
x ≻ y ⇒ u(x) > u(y).

(1)

Upper semi-continuous utility via outer measures

A complete, transitive binary relation on a set X can be represented by a utility function if and only if it is Jaffray order separable3 [START_REF] Jaffray | Existence of a continuous utility function: An elementary proof[END_REF]: there is a countable set D ⊆ X such that for all x, y ∈ X:

x ≻ y ⇒ ∃d, d ′ ∈ D : x d ≻ d ′ y. (2)
apart: x lies on one side of d and d ′ , whereas y lies on the other. To make our search for a (usc)

utility representation at all meaningful, we will henceforth focus on preference relations that are Jaffray order separable.

The set D in the definition of Jaffray order separability is countable, so let n : D → N be an (3)

Notice that W is countable and that it is a covering of X. Extend ρ to an outer measure µ * on X in the usual way (recall Figure 1): for each set A ⊆ X, define µ * (A) as the smallest total size of sets in W covering A:

µ * (A) = inf{ i∈N ρ(W i ) : {W i } i∈N ⊆ W, A ⊆ ∪ i∈N W i }.
Define u : X → R for each x ∈ X as the outer measure of the set of elements worse than x:

u(x) = µ * (W (x)). ( 4
)
This outer measure gives the desired utility representation:

Theorem 3.1 Consider a complete, transitive, Jaffray order separable binary relation on an arbitrary set X. The outer-measure utility function u in (4) represents and is usc in the D-lower order topology.

Remark 3.2 We chose to formulate the theorem in its present form, as it stresses that the outer-measure utility function represents preferences whenever a utility representation is at all feasible, even if a usc representation is impossible. Yet it gives necessary and sufficient conditions for representation in terms of a usc utility function: there is a usc utility function representing preferences if and only if they are complete, transitive, Jaffray order separable, and usc in any topology equal to or finer than the D-lower order topology. ⊳ Corollaries 3.3 and 3.4 below provide applications of our general theorem. Consider preferences over a commodity space X = R n + (n ∈ N) with its standard Euclidean topology. 5 If is usc in this topology, it is Jaffray order separable [START_REF] Rader | The existence of a utility function to represent preferences[END_REF]. By assumption, W (y) is open for each y ∈ R n + , so the Euclidean topology is finer than the D-lower order topology. Hence, Theorem 3.1 applies:

Corollary 3.3 If is a complete, transitive binary relation over R n + (n ∈ N)
and usc in the Euclidean topology, the outer-measure utility function in (4) represents and is usc in the Euclidean topology.

The closest result we could find to Theorem 3.1 is a result in [START_REF] Sondermann | Utility representations for partial orders[END_REF], which is a special case. Call a preference relation on a set X perfectly separable if there is a countable set C ⊆ X such that for all x, y ∈ X, with x ∼ c and y ∼ c for all c ∈ C, the following holds:

x ≻ y ⇒ ∃c ∈ C : x ≻ c ≻ y.
Perfect separability implies Jaffray order separability [START_REF] Jaffray | Existence of a continuous utility function: An elementary proof[END_REF], so:

Corollary 3.4 [Sondermann, 1980, Corollary 2] Consider a complete, transitive, perfectly separable binary relation on a set X. Then there is a utility function representing , usc in any topology equal to or finer than the lower order topology.

Appendix: Proof of Theorem 3.1

Preliminaries. By definition,

∀d ∈ D : u(d) = µ * (W (d)) = ρ(W (d)) = d ′ ∈D:d ′ d w(d ′ ), (5)
and the outer measure µ * is monotonic: if A ⊆ B ⊆ X, then µ * (A) ≤ µ * (B).

Representation. We prove (1). Let x, y ∈ X. Semi-continuity. Let r ∈ R. We show that {x ∈ X : u(x) < r} is open. To avoid trivialities, let {x ∈ X : u(x) < r} = ∅, X. Hence, there is a y * ∈ X with r ≤ u(y * ) ≤ 1. Let x ∈ X have u(x) < r. It suffices to show that there is an open neighborhood V of x with u(v) < r for each

v ∈ V .
Case 1: There is no d ∈ D with d ∼ x. As D may be assumed to contain a worst element of X, if such exists (see footnote 4): W (x) = ∅. By definition of µ * , there are

{W i } i∈N ⊆ W with W (x) ⊆ ∪ i∈N W i and µ * (W (x)) ≤ i∈N ρ(W i ) < r ≤ 1. As W (x) = ∅, the set J = {i ∈ N : W i = ∅} is nonempty. As ρ(X) = 1 and i∈N ρ(W i ) < 1, W i = X for each i ∈ J. So for each i ∈ J there is a d i ∈ D with W i = W (d i ).
We show that d i ≻ x for some i ∈ J. Suppose, to the contrary, that

d i ≺ x for each i ∈ J. For each j ∈ J, the set {d i ∈ D : i ∈ J, d i d j } is infinite: otherwise, it has a best element d * , but then ∪ i∈N W i = ∪ i∈J W (d i ) = W (d * ) is a proper subset of W (x) by
Jaffray order separability, contradicting W (x) ⊆ ∪ i∈N W i . Let j ∈ J with ρ(W (d j )) := ε > 0. By the above, there are infinitely many i ∈ J with ρ(

W i ) = ρ(W (d i )) ≥ ρ(W (d j )) = ε, contradicting that i∈N ρ(W i ) < 1. We conclude that d i ≻ x for some i ∈ J. So x ∈ W (d i )
, an open set in the D-lower order topology, and for each

v ∈ W (d i ): u(v) < u(d i ) = ρ(W (d i )) < r.

  y) and B(y) are open; upper semi-continuous (usc) if for each y ∈ X, W (y) is open.

  injection. Finding a utility function on D is easy. Give each element d of D a positive weight such that the weights have a finite sum and use the total weight of the elements weakly worse than d as the utility of d. For instance, give weight 1 2 to the alternative d with label n(d) = 1, weight 1 4 to the alternative d with label n(d) = 2, and inductively, weight w(d) = 2 -k to the alternative d with label n(d) = k. In general, let (ε k ) ∞ k=1 be a summable sequence of positive weights; w.l.o.g. its sum ∞ k=1 ε k is one. Assign to each d ∈ D weight w(d) = ε n(d) . 4 Define u 0 : D → R for each d ∈ D by u 0 (d) = d ′ d w(d ′ ). Clearly, (1) is satisfied. We can extend this procedure from D to X as follows. Let W = {W (d) : d ∈ D} ∪ {∅, X} and define ρ : W → [0, 1] as follows: ρ(∅) = 0, ρ(X) = 1 and for d ∈ D: ρ(W (d)) = d ′ ∈D:d ′ d w(d ′ ).

  If x ∼ y, then W (x) = W (y) by transitivity of , so u(x) = u(y). If x ≻ y, there are d, d ′ ∈ D with x d ≻ d ′ y by (2). By monotonicity of µ * and (5): u(x) = µ * (W (x)) ≥ µ * (W (d)) > µ * (W (d ′ )) ≥ µ * (W (y)) = u(y).

Case 2 :

 2 There is a d ∈ D with d ∼ x. By (2) and y * ≻ x: B(d) ∩ D = {d ′ ∈ D : d ′ ≻ d} = ∅. Case 2A: There is a d ′ ∈ B(d) ∩ D with (d, d ′ ) = ∅. Then {z ∈ X : z d} = {z ∈ X : z ≺ d ′ } = W (d ′ ) is open in the D-lower order topology, contains x, and for each z ∈ W (d ′ ) : u(z) ≤ u(d) = u(x) < r. Case 2B: For each d ′ ∈ B(d) ∩ D, (d, d ′ ) = ∅. Then by (2), there is, for each d ′ ∈ B(d) ∩ D,a d ′′ ∈ B(d) ∩ D that is strictly worse: d ′′ ≺ d ′ . So B(d) ∩ D is infinite. Since the sequence of weights (ε k ) ∞ k=1 is summable, there is a k ∈ N such that ∞ ℓ=k ε ℓ < ru(x).Since there are only finitely manyd ′ ∈ D with n(d ′ ) < k, there is a sufficiently bad d * ∈ B(d) ∩ D such that n(d ′ ) ≥ k for each d ′ ∈ B(d) ∩ D with d ′ d * . Since d * ∈ B(d) ∩ D, x ∈ W (d * ), which is open in the D-lower order topology. By x ∼ d and the construction of d * :u(x) = d ′ ∈D:d ′ d w(d ′ ) and d ′ ∈B(d)∩D:d ′ d * w(d ′ ) = d ′ ∈B(d)∩D:d ′ d * ε n(d ′ ) ≤ ∞ ℓ=k ε ℓ < ru(x).Hence, for each v ∈ W (d * ): u(v) < u(d * ) = ρ(W (d * )) = d ′ ∈D:d ′ d w(d ′ ) + d ′ ∈B(d)∩D:d ′ d * w(d ′ ) < u(x) + ru(x) = r.

See Fishburn (1970, Sec. 3.1) orBridges and Mehta (1995, Sec. 1.4) for alternative separability conditions.

If there is a worst element in X (an x 0 ∈ X with x 0 x for all x ∈ X), one may assume w.l.o.g. that D contains one such element, say d. Its weight can be normalized to zero: w(d) = 0. This will assure that ρ(W (d)) = ρ(∅) = 0 in (3).

Or any other topological space with a countable base.

Klaus Ritzberger, and Peter Wakker for comments and to the Knut and Alice Wallenberg Foundation for financial support.


Aymeric Roux

Alexis Duchesne

Michael Baudoin
email: michael.baudoin@univ-lille.fr

Bubbles and liquid films resisting drainage, evaporation and nuclei-induced bursting

Soap bubbles are by essence fragile and ephemeral. Depending on their composition and environment, bubble bursting can be triggered by gravity-induced drainage and/or the evaporation of the liquid and/or the presence of nuclei. They can also shrink due to the diffusion of the inner gas in the outside atmosphere induced by Laplace overpressure. In this paper, we design bubbles made of a composite liquid film able to neutralize all these effects and keep their integrity for more than one year in a standard atmosphere. The unique properties of this composite film are rationalized with a nonlinear model and used to design complex objects. FIG. 1. Comparison of the lifetimes of three types of bubbles: a) Soap bubble bursting after 1 min. b) Water gas marble covered by partially-wetting micro-particles, whose shell rupture occurs after 6 min and complete drying after 9 min. Dashed line is used as a guide to visualize bubble opening and blue dye to visualize liquid drying. c) Water/glycerol gas marble perfectly intact in a standard atmosphere after ∼ 9 months (285 days). This bubble kept its integrity for more than one year (465 days) 1 .

As can be seen on Fig. 2, the shell of a water gas marble once dried collapses like a sand pile (Movie M1), while the shell of the water/glycerol gas marble is still liquid and spherical and reacts as a liquid film when punctured (Movie M2).

Under a standard atmosphere, soap bubbles typically burst in minutes (Fig. 1a) due to the effect of gravityinduced drainage and/or liquid evaporation and/or nucleï induced inception depending on their composition. "Bare" viscous bubbles bursting follows gravity-induced drainage of the liquid, resulting in shell thinning down to a limit thickness of the order of tens of nanometers, wherein spontaneous breaking occurs 2 . The sliding condition of the liquid on air leads to plug flow in the shell and hence characteristic lifetime τ = µ/ρ l gR, which does not rely on the liquid shell thickness e but rather on the radius of curvature of the bubble R, with ρ l the liquid density, g the gravitational acceleration and µ the liquid dynamic viscosity. The addition of surfactants to produce soap bubbles can lead to completely different pictures depending on the concentration of surfactants 3,4 : At intermediate surfactant concentrations, bubble ageing results from a complex interplay between gravity and capillary induced drainage, Marangoni stresses due to gradients of surfactant concentration and bubble evaporation. At large surfactant concentration, the surfac-tant rigidifies the film surface leading to no-slip boundary condition, Poiseuille flow and hence considerably slowed down drainage. Nevertheless, while surfactant drastically increase the lifetime of soap bubbles, evaporation and/or the presence of nuclei will eventually lead to their rupture. Hence long bubbles lifetimes are only reported in carefully controlled atmosphere 5 , wherein dusts are suppressed, the atmosphere is saturated to prevent evaporation, and the level of mechanical vibration is controlled. And even in these conditions, bubbles shrinkage resulting from the diffusion of the inner air in the outside atmosphere due to Laplace overpressure still occurs 5 .

Adding partially-wetting particles to liquid/air interfaces 6 has been investigated as a mean to strengthen their resistance to mechanical stresses providing a material with properties at the crossroad between solids and liquids 7 . When drops and bubbles are covered with such a composite interface, some extraordinary properties have been reported such as: non-stick droplets [START_REF] Aussillous | Liquid marbles[END_REF][START_REF] Quéré | Non-sticking drops[END_REF][START_REF] Mchale | Liquid marbles: principle and applications[END_REF] able to roll on solid surfaces, "armoured bubbles" 11 in a liquid stable to dissolution [START_REF] Manouk Abkarian | Dissolution arrest and stability of particle-covered bubbles[END_REF] and which can sustain metastable [START_REF] Prabhudesai | Nonspherical armoured bubble vibration[END_REF] non-spherical shapes [START_REF] Subramaniam | Non-spherical bubbles[END_REF][START_REF] Zoueshtiagh | Capillary tube wetting induced by particles: Towards armoured bub-bles tailoring[END_REF] or "gas marbles" (air bubbles) [START_REF] Timounay | Gas marbles: Much stronger than liquid marbles[END_REF][START_REF] Timounay | Low gas permeability of particulate films slows down the aging of gas marbles[END_REF] supporting positive and negative relative pressure one order of magnitude larger than the Laplace pressure. The addition of particles to an interface can also lead to some counter-intuitive behaviours such as surface energy driven fingering instability in reversed Saffman-Taylor configuration [START_REF] Bihi | Inverse Saffman experiments with particles lead to capillary driven fingering instabilities[END_REF] or films growing ahead of a liquid meniscus moving in a capillary tube [START_REF] Zoueshtiagh | Capillary tube wetting induced by particles: Towards armoured bub-bles tailoring[END_REF] .

Nevertheless, (i) the effect of such a composite interface on the lifetime of air bubbles has not been investigated and (ii) the single addition of particles to a bubble shell cannot prevent its evaporation (Fig. 1b) and hence its bursting. In this paper [START_REF] Roux | Bubbles and liquid films resisting drainage, evaporation and nucleiinduced bursting[END_REF] , we show that covering a bubble water shell with micro-particles inhibit gravityinduced drainage and that further adding glycerol leads to a stable state, wherein the evaporation of water is counterbalanced by the hygroscopicity of glycerol, which absorbs water molecules contained in the ambient air. This results in bubbles (Fig. 1c) which can keep their integrity in a standard atmosphere for more than one year, with no significant evolution of their radius 1 . This behaviour is rationalized with a nonlinear model able to reproduce quantitatively the evolution of the mass of the bubble and predict their fate depending on the initial composition of the bubble and the surrounding atmosphere. Neutralizing drainage. "Gas marbles" are by definition gas bubbles whose composite shell is made of liquid and partially-wetting particles [START_REF] Timounay | Gas marbles: Much stronger than liquid marbles[END_REF][START_REF] Timounay | Low gas permeability of particulate films slows down the aging of gas marbles[END_REF] . Here, we first study the mass evolution and lifetime of water gas marbles for different ambient humidities (Fig. 3a and3b). Gas marbles of radius 5.1 ± 0.1mm are produced with the sim-ple following process (movie M3): First, some partiallywetting particles (polyamide-11 particles, average radius r p = 80 ± 20 µm, contact angle with water 71 ± 3°1 5 ), are spread at the surface of a water bath to form a jammed granular raft. Second, a controlled volume of air is injected with a syringe below the granular raft, leading to the formation of air bubbles whose upper interface is covered with particle. Third, the bubble is pushed with a spoon toward the surrounding particle raft and made roll over it to cover its whole surface. Finally, the bubble is extracted and placed over a hydrophobic (Teflon) or superhydrophobic (nanopillar) substrate. The evolution of the gas marble mass and shape are respectively monitored with a FAS224 Fisherbrand high precision balance and a Nikon D850 camera with a 25 mm ultra macro Laowa lens. The relative humidity is measured with a A+life TH1818 hygrometer. Fig. 3a shows the evolution of the mass of a single water gas marble. Bubble drying is visualized with the addition of a blue dye and associated color change, while bubble opening (marked with a yellow triangle on Fig. 3a) can be seen by the brutal change in the bubble shape when the bubble shell is no more airtight (Movie M4). These data show (i) that the bubble mass decreases linearly until its complete drying (plateau region after 54 minutes on Fig. 3a) and (ii) that the gas marble opening (yellow triangle) occurs a few minutes before its complete drying, thus underlying that the water gas marble lifetime is mainly determined by the evaporation process. This scenario is further confirmed by numerous experiments performed for different ambient humidities (Fig. 3b). Hence, the presence of the particle shell inhibits the drainage of the liquid, which, in absence of the particles, would provoke bubble bursting after a characteristic time of the order of τ = µ/ρ l gR ∼ 20 µs 2 , with R the bubble radius, ρ l the liquid density, g the gravitational acceleration and µ the liquid viscosity. Note that the bubble lifetime variability on Fig. 3b at same ambient humidity is due to the fact that the initial film thickness is not controlled during the gas marble formation process.

Neutralizing evaporation. Designing "everlasting" bubbles, i.e. bubbles with very long lifetimes, requires to further neutralize evaporation. This problem is solved by adding glycerol to water. Indeed, glycerol is a liquid with a high concentration of hydroxyl groups, which have a strong affinity with water molecules with whom they create hydrogen bonds [START_REF] Chen | Hydrogen bonding analysis of glycerol aqueous solutions: A molecular dynamics simulation study[END_REF] . This mechanism is at the origin of glycerol's hygroscopicity (ability to absorb water molecules contained in air), which can compensate for water evaporation [START_REF] Lin | Influence of water evaporation/absorption on the stability of glycerol-water marbles[END_REF] . To test this hypothesis, bubbles were synthesized with the same process as described in the previous section but with a water/glycerol mixture. The evolution of the mass of these gas marbles for different initial glycerol mass ratio ω go and different values of the ambient relative humidity h (given in SI) is shown on Fig. 3c. This figure shows that the shell of water/glycerol gas marbles with the lowest initial glycerol concentrations (ω go ∈ [0, 0.15]) rupture after lifetimes of the order of 50 min, while the bubbles with higher glycerol concentration tend toward a steady state either by initially losing water (ω go ∈ [0.30, 0.65]), keeping a mass relatively constant (ω go = 0.85) or absorbing water from the humidity contained in the air (ω go = 0.9). Since glycerol is a non-volatile liquid, the mass variations of these gas marbles are solely due to water evaporation or absorption. The evolution toward a steady state with an equilibrium between the water and glycerol content is even clearer when we study the evolution of the normalized glycerol mass ratio (Fig. 3d), which tends toward a constant value (slight fluctuations are due to recorded fluctuations in the air humidity). We further studied the fate of gas marbles as a function of the initial glycerol mass ratio and relative humidity (Fig. 4). When a water/glycerol mixture is in contact with humid air with a relative humidity h, it will tend toward an equilibrium glycerol mass ratio ω geq materialized by the blue dotted line on Fig. 4. The diagram shows that when the initial glycerol mass ratio ω go is close to this equilibrium value, the gas marble evolves toward a stable equilibrium state (everlasting bubbles) either by losing some liquid (when ω go < ω geq ) or absorbing some liquid (when ω go > ω geq ) 22 . When the initial glycerol mass ratio is too far from this equilibrium value, the gas marble systematically ruptures.

Model. To rationalise this behaviour, we introduce a model able to quantitatively reproduce the experimental trends described in Fig. 3 and 4 without any adjustable parameter. Since glycerol is a non-volatile liquid, the liquid mass m l evolution of a water/glycerol gas marble is set by an equilibrium between the flux of water evaporation and the flux of water absorption due to the hygroscopicity of glycerol: dm l /dt = φ abs -φ evap . Water evaporation flux is driven by the vapor concentration gradient and hence is proportional to (1 -h), with h the relative humidity of air far from the bubble [START_REF] Dollet | Natural convection above circular disks of evaporating liquids[END_REF][START_REF] Miguet | Stability of big surface bubbles: Impact of evaporation and bubble size[END_REF] . Furthermore, since the shell liquid is a mixture of glycerol and water, the water evaporation flux will be proportional to the quantity of water molecules present at the bubble surface. Since the diffusion time τ d = e 2 /D wg 3 min (with e ∼ 150 µm the thickness of the film estimated from the surface of the bubble and its mass and D wg 10 -10 m 2 s -125 the diffusion coefficient of wa- ter molecules into the water/glycerol mixture) is short compared to the characteristic time τ c 100 min required for a bubble to reach its equilibrium state, the water/glycerol mixture in the film can be supposed to be homogeneous at the bubble mass evolution timescale. Hence, the water evaporation rate will be proportional to the concentration of water in the water/glycerol mixture (1 -ω g ) with ω g the glycerol mass ratio. We thus obtain: φ evap = k e (1 -h)(1 -ω g ), with k e a constant to be determined. Conversely the absorption of water through the formation of hydrogen bonds with glycerol molecules will be proportional to the concentration of water in the air h and to the concentration of glycerol in the liquid film ω g , so that φ abs = k a hω g . The final equation: dm l /dt = k a hω g -k e (1 -h)(1 -ω g ) relies on two unknown coefficients k e and k a , which depend on the evaporation regime, absorption efficiency and bubble geometry. The first coefficient can be determined by examining the case of pure water gas marble. In this case φ evap = k e (1 -h). A first validation of this model is that the linear trend is well recovered on Fig. 3b. The value of the coefficient k e = 2.3 ± 0.2 × 10 -8 kg s -1 is obtained by taking the median value of the slopes in Fig. 3b, which enables to obtain good fits for all the curves. Note that this coefficient is not a universal parameter and depends on the complex physics behind the evaporation process and the bubble geometry. The second coefficient is determined by examining the equilibrium state dm l /dt = 0. In this case, the mass concentration of glycerol in the water/glycerol mixture reaches a well established equilibrium value [START_REF]Association, Physical Properties of Glycerine and Its Solutions[END_REF] , which depends on the relative humidity of the ambient air. Our model predicts a relation between the equilibrium glycerol mass ratio ω geq and the relative humidity given by: ω geq = [1 + k a h/k e (1 -h)] -1 . A second validation of this model is that this law fits well the abacus available in the literature [START_REF]Association, Physical Properties of Glycerine and Its Solutions[END_REF] (Fig. 2 in SI), with k a /k e = 0.248, giving good confidence in the dependence of the evaporation and absorption fluxes on the glycerol concentration ω g and relative humidity h. Now, that the two coefficients k a and k e , are determined we can compare the model predictions to the experimentally measured gas marble mass evolution. This model recovers quantitatively the trends observed on Fig. 3c and3d. From this model, we can also derive a criterion for the transition between ephemeral and everlasting bubbles (Fig. 4) depending on the initial glycerol concentration ω go and relative humidity h. Assuming that bubbles rupture occurs when the bubble shell liquid mass reaches a critical value m lc , a bubble will be everlasting if its equilibrium liquid mass m leq is larger than m lc and ephemeral if m leq is smaller than m lc , with a transition between these regimes for m leq = m lc . Since the mass of glycerol m g is constant, we have m g = m leq ω geq = m lo ω go . Hence for m leq = m lc , we obtain ω goc = m lc ω geq (h)/m lo with ω geq (h) determined previously. The critical liquid mass m lc = 14 ± 4 mg is determined by measuring the bubble total mass m c = 32 ± 3 when its rupture occurs (measured here for 4 different gas marbles) and subtracting the particle mass m p = 18 ± 2 mg (value extracted from Fig. 3b after total evaporation of water). Since we cannot control the liquid content during the gas marble formation process, an ephemeral gas marble initial content m lo varies typically between 22 mg and 57 mg. The plot of the equation ω goc = m lc ω geq (h)/m lo on Fig. 4, shows that this model gives a good criterion for the transition between ephemeral and everlasting bubble (the continuous purple line corresponds to the average value m lo = 34 mg, while the purple dashed lines correspond to m lo = 22mg and 57mg respectively and account for the dispersion of the initial state).

Perspectives: composite films. In this paper, we have shown that air bubbles keeping their integrity for more than one year can be produced in a simple way by replacing surfactants by partially-wetting particles and water by a water/glycerol mixture. Indeed, partiallywetting particles prevent drainage of the liquid due to gravity, while the hygroscopicity of glycerol-water mixtures counterbalances evaporation, which are the two main mechanisms at the origin of classical bubble bursting. In addition, (i) the neutralization of these two phenomena and the presence of the particle shell enables to make bubble insensitive to the nuclei contained in the air and (ii) no evolution of the bubble radius over the bubble life was observed, indicating that bubble shrinkage due to Laplace overpressure is also suppressed. Beyond bubbles, this work unveils a robust composite liquid film, which can be used to create a wealth of different objects. Indeed, when a metallic frame is dipped below a liquid surface covered with a layer of jammed particles and lifted slowly up to the surface, it captures the particle-covered film (see Fig. 5). While marginal pinching due to the presence of the frame and surfactants leads to soap films bursting near the edge after a few seconds (Fig. 5.a), the replacement of surfactants by partially-wetting particles leads to the apparition of evaporation induced cracks after several minutes only (Fig. 5.b), and further addition of glycerol leads to long lasting liquid film stable for at least several months (Fig. 5.c). Following this process, complex objects such as everlasting pyramidal films with lifetimes of more than one year (378 days when these lines are written) can easily be manufactured (Fig. 5.d) paving the way toward the design of new fluidic objects with unexplored physical and chemical properties. This work would be ideally completed with a study on a large set of parameters of the effect of the composition of the shell and of the ambient atmosphere on the bubble stability. Finally, a puzzling question remains: does the Laplace overpressure inside the bubble vanishes over time and if it does, how does the bubble maintains its shape and size? We are currently trying to provide a convincing answer to this question.

  FIG. 2. (a) Lateral view of a water gas marble naturally collapsing after a few tens of minutes. (b) Top view and (c) lateral view of two one-month old water/glycerol gas marbles (initial glycerol mass ratio ωg0 = 0.60) punctured with a needle.

FIG. 3 .

 3 FIG. 3. Evolution of gas marbles in a standard atmosphere (temperature T = 21.0 ± 0.5 °C ). a) Evolution of the mass of a single water gas marble as a function of time at ambient humidity h = 0.52 ± 0.01. Drying is reflected by the change in the bubble color and bubble opening (yellow triangle on the curve) by the brutal change in the bubble shape (Movie M4). The plateau region (after 54 min) occurs when the water is entirely evaporated. b) Evolution of the mass of pure water gas marbles as a function of time at different relative humidity. c) Evolution of the mass of water/glycerol gas marbles as a function of time for different initial glycerol mass ratio ωgo. d) Normalized glycerol mass ratio (ωg/ωgeq) content of gas marble as a function of time (same experimental data as in Fig.3c). Black curves: evaporation/absorption model introduced in this paper.

FIG. 4 .

 4 FIG.4. Diagram of gas marble fate after six hours as a function of the initial glycerol mass ratio and the relative humidity. Each point corresponds to statistics made on at least ten gas marbles. Red: Systematic rupture of the bubble. Purple: Part of the bubble sample rupture. Blue: All gas marble keeping their integrity after six hours. The vertical bar indicates the variation of the relative humidity for each set of experiment. The dashed blue curve entitled "Eq. mass ratio" represents the equilibrium glycerol mass ratio ωgeq(h) and the purple continuous line the critical glycerol mass ratio ωgco.

FIG. 5 .

 5 FIG. 5. Lifetime of liquid films attached to a metallic frame (a,b,c: horizontal frame. d. pyramidal frame) a) Soap film made of water and surfactant. b) Water film covered with partially wetting particles. c) Water/glycerol liquid film (initial glycerol mass concentration ωgo = 0.80) covered with particles. d) 3D object created with a pyramidal frame supporting a water/glycerol liquid film covered with particles and perfectly intact after 203 days (∼ 6 months). When these lines are written the pyramidal film is still perfectly intact in our laboratory 378 days after its formation.

(), Note that our oldest bubble ruptured after 465 days. We believe that this rupture can be attributed to the development of "life" into our bubble since the bubbles became slightly green during the last month, which would not be surprising since (i) the bubble is made of water and glycerol which is a favorable environment for the development of fungi and bacteria and (ii) we did not take any precaution to avoid the pollution of the bubble with living

organisms. 2 G. Debrégeas, P.-G. de Gennes, and F. Brochard-Wyart, "The life and death of "bare" viscous bubbles," Science 279, 1704-

1707 (1998). 3 K.J. Mysels, K. Shinoda, and S. Frankel, Soap films, Studies of their thinning and a bibliography (

Pergamon, 1950). 4 L.W. Schwartz and R.V. Roy, "Modeling draining flow in mobile and immobile soap films," J. Colloid Interf. Sci.

218, 309-323 (1999). 5 A.V. Grosse, "Soap bubbles: two years old and six centimeters in diameter,"

Science 164, 291-293 (1969). 6 B.P. Binks, "Particles as surfactants -similarities and dif-

ferences," Curr. Opin. Colloid In. 7, 21-41 (2002). 7 D. Vella, P. Aussillous, and L. Mahadevan, "Elasticity of

Acknowledgements: We would like to thank F. Zoueshtiagh for useful discussions and technical assistance. This work was supported by ISITE-ULNE (ERC Generator program), Institut Universitaire de France and Renatech network.


Hervé Dekkiche

Mary-Ambre Carvalho

Christophe Jeandon

Lydia Karmazin

Corinne Boudon

Laurent Ruhlmann

Romain Ruppert
email: rruppert@unistra.fr.tel

Synthesis and electrochemistry of nickel(II)porphyrins bearing external palladium(II) or platinum(II) complexes

Keywords: porphyrinoids, nickel(II)porphyrin, electrochemistry, porphyrin dimers

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

Porphyrins are aromatic compounds with exceptional electronic properties widely present in Nature. Synthetic modifications of their structure at their peripheries were conducted to obtain new electronic properties or catalytic activities. These macrocyclic compounds possess a cavity well-suited to host many different metal ions. These metalloporphyrins, also found in natural systems, have been widely developed during the last few decades [START_REF] Kadish | The Porphyrin Handbook[END_REF] and have presented either catalytic activities or photochemical properties. The addition of metal ions to the periphery of the macrocycle was explored more recently, leading to porphyrins bearing two (or more) coordination sites, which can be described as internal (for the classical macrocyclic N4 ligand) and external coordination sites [START_REF] Richeter | Handbook of Porphyrin Science[END_REF].

Serendipity led to the simultaneous discovery of a new isomer of porphyrin now known as "N-confused or carbaporphyrin" by the groups of Prs. Hiroyuki Furuta and Lechoslaw Latos-Grazynski (see Fig. 1) [3,4]. In addition to novel spectroscopic properties and a modified internal coordination site CN3 giving access to organometallic species, the presence of an external nitrogen atom led to the development of new externally metalated porphyrinoids [5][6][START_REF] Pawlicki | Handbook of Porphyrin Science[END_REF][START_REF] Toganoh | Handbook of Porphyrin Science[END_REF]. Numerous other porphyrin derivatives bearing heteroatoms at their periphery have been described since the publication of these two seminal articles and coordination chemistry is now no longer restricted to the inner core of the macrocycle [START_REF] Wang | [END_REF][10][11][12][13][14][15][16][17][18][19].

Nickel(II)porphyrins bearing enamino(thio)ketones are easily accessible by an intramolecular Friedel-Craft cyclization followed by amination (and thionation)(see Chart 1). These porphyrinoids were used as ligands to bind metal ions. Deprotonation of the NH 2 part led to an external ligand isoelectronic with acetylacetonate and was therefore able to bind many metal ions. Numerous symmetric dimers linked by metal ions have been prepared by our group [20]. Strong electronic interactions were present in these dimers, due to electron delocalization through the coordination linkages. Magnetic interactions between metalloporphyrins were also found in porphyrin dimers linked by paramagnetic copper(II) or diamagnetic palladium(II) ions [21]. This article describes the synthesis, characterisation and electrochemical studies of a few nickel(II)porphyrins bearing at their periphery an external palladium(II) or platinum(II) acetylacetonate complex.

RESULTS AND DISCUSSION

Synthesis and characterization

The preparation of the palladium(II) appended nickel(II)porphyrins was performed in one step in the case of the enaminoketone functionalized starting material. Reaction of porphyrin 1 with a base and the complex Pd(PPh 3 )Cl(acac) gave compound 3 after purification (see Scheme 1). In the case of the enaminothioketone functionalized starting material 2, it was necessary to prepare an intermediate palladium(II) complex 4 bearing a phosphine and a chloride as additional ligands, before adding potassium acetylacetonate to isolate cleanly the desired compound 5. The phosphine ligand in complex 4 was located trans to the nitrogen atom of the enaminoketone and this coordination geometry was determined by NOESY experiments.

Preparing the analogous platinum(II) complexes proved to be more tedious, due to the known inertness of this metal ion. After numerous other unsuccessful attempts, compound 7 could be obtained in low yields by reacting the known porphyrin dimer 6 with an excess of acetylacetone in refluxing dichlorobenzene (see Scheme 2, left). Starting from nickel(II)porphyrin 2, reaction with a base and the complex KPt(DMSO) Cl 3 gave the intermediate compound 8, which could be transformed into the desired compound 9 by reaction with potassium acetylacetonate (see Scheme 2, right). Again the DMSO ligand was found trans to the enaminoketone nitrogen atom in complex 8 (determined by NOESY experiments).

All four new compounds 3, 5, 7, and 9 were characterized by NMR and mass spectrometry. A comparison between the 1 H NMR spectra (aromatic region) of the four compounds is shown in Fig. 2. The four spectra are quite similar, except for the acac signals (see experimental part), the aromatic proton next to the ketone (or the thio ketone) and the broad signals of the external N-H protons. The chemical shifts of these last signals moved from 6.97 ppm (compound 3) to 9.10 ppm (compound 9).

We could not grow single crystals of these complexes, but crystals suitable for X-ray could be obtained for an analogous porphyrin with different meso-aryl groups. The X-ray structure of the fragment palladium(II)acac attached to an enaminothioketone nickel(II)porphyrin with 3,5-di-tertbutylphenyl meso groups is shown in Fig. 3. We did not prepare all compounds with these meso-aryl groups, because the electrochemical studies were often more difficult with these substituents due to passivation of the electrodes.

The structure of the nickel(II)porphyrin was ruffled as expected for these porphyrins, because the four Ni-N distances were all rather short, ranging from 1.90 to 1.91 Å. The coordination geometry around the external palladium(II) was square planar, thus bringing the acetylacetonate ligand roughly in the same plane as the extended part of the porphyrin. In the crystal, π-π interactions were found between two molecules with a very small distance of 3.40 Å (see Fig. 4). Such small distances between two planes are present in planar poly-aromatic compounds and often formulated as π-π stacking. In the example shown here, the interaction between these planar parts of the two molecules was due to interactions involving p and d orbitals.

Electrochemical studies

The electronic spectra of the four new compounds are shown in Fig. 5. Bathochromic shifts were observed for the four compounds when compared to the spectra of the starting porphyrins 1 and 2. The lowest energy bands of compound 1 and 2 are found respectively at 651 nm (15361 cm -1 ) and 692 nm (14451 cm -1 ). Compounds 3 and 7 presented maxima around 680 nm (14706 cm -1 ): a shift of approximatively 30 nm compared to the spectrum of 1. Similarly, compounds 5 and 9 shifted bathochromically by approximatively 20 nm (from 692 to 710 nm (14085 cm -1 )) by comparison with 2. Changing the metal ion (Pd(II) or Pt(II)) led surprisingly to small differences in the electronic spectra. This observation contrasts notably with the earlier findings concerning symmetrical porphyrin dimers linked by palladium(II) or platinum(II), where much larger differences were observed between these dimers [20]. The electronic delocalization between the two porphyrins in dimers linked by metal ions is probably much stronger in this case because the two parts around the linking metal ion are identical. In the compounds described here, the acetylacetonate chelate is different from the porphyrin chelate found on the other side, hence limiting the delocalization. The same phenomenon has been observed earlier (compare results from the groups of DP Arnold and A Osuka with organometallic monomeric or dimeric derivatives) [22,23].

The redox potentials of the new porphyrinoids were measured in dichloromethane solutions containing 0.1 M NBu 4 PF 6 as supporting electrolyte by cyclic voltammetry (CV) and by rotating disk voltammetry (RDV). The measured potentials are summarized in Table 1. All electron exchange reactions were monoelectronic (number of electron exchanged determined by RDV) and chemically reversible for all compounds. As is known for nickel(II)porphyrins [START_REF] Kadish | The Porphyrin Handbook[END_REF], two oxidation and two reduction steps were observed for these porphyrinoids and no redox processes were observed for the external palladium(II) or platinum(II) complexes within the electrochemical window used (from -2.2 V to 1.0 V vs. Fc /Fc). The potential gap between the first oxidation and the first reduction (E ox1 -E red1 ) was very close to the optical gap measured in the electronic spectra. These gaps were smaller than the gap measured for the starting porphyrins 1 and 2 [20d].

A typical cyclic voltammetry is shown in Fig. 6 (top: CV of compound 9) and compared to previous electrochemical results obtained with a porphyrin dimer (two porphyrins 2 linked by a platinum(II) ion). The first oxidation potential measured for the porphyrin monomer can be found between the first two monoelectronic oxidation steps measured for the porphyrin dimer [20e]. The same observation can be made on the reduction side for the first reduction steps. As previously noticed for other dimers, the splitting of the first oxidation waves is much larger than the splitting of the first reduction waves. This was explained by the involvement of the "d" orbitals of the linking metal ion in the HOMO level but not in the LUMO level [20e].

EXPERIMENTAL General information

The HRMS ESI and MALDI were obtained on a Bruker MicrOTOF. UV-vis spectra were recorded in dichloromethane with a Cary 5000 UV-vis-NIR doublebeam instrument. Cyclic voltamograms (CV) were recorded in a standard three electrode electro chemical cell using glassy carbon (GC) as working electrode in dichloromethane containing 0.1 M NBu 4 PF 6 as supporting electrolyte. The cell was connected to an autolab PGSTAT30 potentiostat (Eco Chemie, Holland) driven by GSE software running on a personal computer. All potentials are vs. Fc /Fc used as external electrode and are uncorrected from ohmic drop. Chromatographic separations were performed with silica gel (Merck, 40-63 µm). Dichloromethane was distilled over calcium hydride and THF from sodium/benzophenone ketyl. NMR spectra were recorded on Bruker Avance 400 and 500 spectrometers equipped with a cryoprobe for 13 C. Chemical shifts are given in parts per million (ppm) by taking the solvent signals as a reference (δ 7.26 ppm or δ 77.16 ppm). All other solvents and reagents were used without further purification. The starting porphyrins 1 and 2 were prepared as previously described [20d-20e]. The dinuclear palladium(II) complex was prepared following a known procedure [START_REF] Uson | [END_REF]. The palladium(II) intermediate complex Pd(PPh 3 )Cl(acac) was prepared by following a procedure described previously for similar complexes (26).

Porphyrin 3. To a solution of porphyrin 1 (30 mg, 31 µmol) in dry THF (15 mL) was added potassium tertbutoxide (3.5 mg, 31 µmol, 1 eq.) under argon. After 5-10 Porphyrin 4. To a solution of porphyrin 2 (30 mg, 31 µmol) and toluidin (0.20 mL of a 100 mg/mL solution in THF, 31 µmol, 1 eq.) in dichloromethane (20 mL) was added [Pd(PPh 3 ) 2 Cl] 2 (20 mg, 18 µmol, 0.6 eq.) portionwise. The mixture was stirred for 30 min at room temperature and then evaporated to dryness. The residue was diluted with dichloromethane (30 mL) and filtered through alumina. The product 4 (45 mg, 31 µmol, quant.) was obtained after evaporation of the solvent. Porphyrin 8. To a solution of porphyrin 2 (50 mg, 51 µmol) and toluidin (0.33 mL of a 100 mg/mL solution in THF, 51 µmol, 1 eq.) in THF (20 mL) was added KPt(DMSO)Cl 3 (19 mg, 51 µmol, 1 eq.). The mixture was stirred for 1 day at room temperature and then 2 additional hours at 50 °C. After consumption of the starting materials, the solvents were removed under reduced pressure. The residue was chromatographed (silica gel, dichloromethane/cyclohexane, 3/1 then dichloromethane) and the product 8 (31 mg, 24 µmol, 48%) was isolated. UV-vis (CH 2 Cl 2 ): λ max (ε) 367 (41600), 396 (51200), 431 (97200), 466 (64300), 497 (76400), 652 (18900), 685 (20000), 705 nm (30600 M -1 . cm -1 ).

The crystals were placed in oil, and a single crystal was selected, mounted on a glass fibre and placed in a low-temperature N 2 stream. X-Ray diffraction data collection was carried out on a Bruker APEX II DUO Kappa-CCD diffractometer equipped with an Oxford Cryosystem liquid N 2 device, using Mo-Kα radiation (λ 0.71073 Å). The structure was solved by direct methods using the program SHELXS-2013 [27]. The refinement and all further calculations were carried out using SHELXL-2014 [28]. Two tert-butyl groups were disordered on two positions with a ratio of 0.6/0.4. The SQUEEZE instruction in PLATON [29] was applied and the residual electron density was assigned to five chlorobenzene molecules.

Crystal data. C 74 H 81 N 5 NiO 2 PdS, M 1269.60 g . mol -1 , crystal size 0.50 × 0.30 × 0.20 mm, triclinic, space group P-1, a 13.8668

CONCLUSION

The synthesis of a series of porphyrins bearing palladium(II) or platinum(II) complexes at their periphery was completed. Whereas the yields were quite good for the two palladium(II) containing compounds, the syntheses were less efficient for the platinum(II) complexes, due to the known inertness of the platinum ions. Bathochromic shifts were observed in the electronic spectra of these compounds when compared to the starting porphyrin derivatives. The electrochemical studies demonstrated that these porphyrinoids can be reversibly oxidized and/ or reduced twice, with the redox steps localized on the porphyrin rings.

Chart 1 .Scheme 1 .

 11 Scheme 1. Preparation of nickel(II)porphyrins 3, 4, and 5 bearing palladium(II) complexes at their periphery.

Fig. 2 .

 2 Fig. 2. 1 H NMR spectra (aromatic region) of the four nickel(II)porphyrins 3, 5, 7, and 9. Highlighted in pink are the broad signals of the external N-H protons.

Fig. 3 .

 3 Fig. 3. X-ray structure of a nickel(II) porphyrin bearing a peripheral palladium(II) complex.

Fig. 4 .

 4 Fig. 4. X-ray structure of a nickel(II) porphyrin bearing a peripheral palladium(II) complex: two orthogonal views showing the π-π interactions between the external complexes of two molecules in the crystal (all hydrogen atoms and the tert-butyl groups omitted for clarity).

Fig. 5 .

 5 Fig. 5. Electronic spectra of the four compounds 3, 5, 7, and 9 (in dichloromethane).

Fig. 6 .

 6 Fig. 6. Comparison between the cyclic voltametries of monomeric porphyrin 9 and the corresponding porphyrin dimer linked by a platinum(II) ion.

  (6) Å, b 17.0190 (7) Å, c 18.6347 (8) Å, α 82.888 (1)°, β 81.704 (1)°, γ 67.296 (1)°, V 4003.3 (3) Å 3 , Z 2, ρ calc 1.053 Mg/m 3 , T 173 (2) K, Mo-Kα 0.71073 Å, 1.601 < θ < 30.056, 109013 reflections measured, 23382 unique reflections, R 1 0.0474, wR 2 0.1171, GoF 1.033. CCDC Nr 2096258.

Scheme 2. Preparation of nickel(II)porphyrins 7 and 9 bearing platinum(II)(acac) fragments at their periphery.

	D	H. DEKKICHE ET AL.
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Table 1 .

 1 Electrochemical data: reduction and oxidation potentials of the four new compounds (Volts vs. Fc /Fc). a

	Compound	E red 2 (Volts)	E red 1 (Volts)	E ox1 (Volts)	E ox 2 (Volts)	E ox 1 -E red 1 (Volts)	Optical gap (eV)
	3	-1.88	-1.48	0.34	0.78	1.82	1.81
	5	-1.76	-1.34	0.41	0.80	1.75	1.75
	7	-1.86	-1.41	0.37	0.79	1.78	1.79
	9	-1.52	-1.39	0.37	0.79	1.76	1.74
	a Experimental conditions: Cyclic voltammetry, glassy carbon working electrode, v = 0.1 V/s in dichloromethane containing 0.1 M
	NBu 4 PF 6 as supporting electrolyte.

  min, the deprotonated porphyrin was added via cannula to a solution of Pd(PPh 3 )Cl(acac) (16 mg, 31 µmol, 1 eq.) in dry THF (15 mL) at 0 °C and the mixture was stirred at room temperature overnight. A few spatulas of NH 4 Cl were added and the solvent was evaporated. The products were separated by column chromatography (silica gel, cyclohexane/dichloromethane, from 2/1 to 1/1). Porphyrin 3 was obtained after recrystallization from dichloromethane/methanol (11 mg, 9.4 µmol, 30%). 1 H NMR (500 MHz, CDCl 3 , 25 °C): δ 9.03 (d, J 5.0 Hz, 1H, H pyrr ), 8.79 (dd, J 8.0, 1.5 Hz, 1H, H cyclPh ), 8.38 (d, J 5.0 Hz, 1H, H pyrr ), 8.32 (dd, J 8.2, 1.2 Hz, 1H, H cyclPh ), 8.08 (d, J 4.7 Hz, 1H, H pyrr ), 8.07 (d, J 4.9 Hz, 1H, H pyrr ), 8.00 (d, J 4.7 Hz, 1H, H pyrr ), 7.77 (ddd, J 8.3, 7.1, 1.5 Hz, 1H, H cyclPh ), 7.74 (d, J 4.8 Hz, 1H, H pyrr ), 7.53 (ddd, J 8.0, 7.0, 1.1 Hz, 1H, H cyclPh ), 7.41 (s, 2H, H m-Ar ), 7.32 (s, 2H, H m-Ar ), 7.27 (s, 2H, H m-Ar ), 6.97 (s, 1H, NH), 5.45 (s, 1H, CH -acac ), 2.20 (s, 3H, CH 3-acac ), 2.00 (s, 3H, CH 3-acac ), 1.98 (s, 6H, H Me ), 1.95 (s, 6H, H Me ), 1.90 (s, 6H, H Me ), 1.50 (s, 9H, H tBu ), 1.48 (s, 9H, H tBu ), 1.47 ppm (s, 9H, H tBu ).

		SYNTHESIS AND ELECTROCHEMISTRY OF NICKEL(II)PORPHYRINS BEARING EXTERNAL PALLADIUM(II)	G
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	22	13 C NMR (125 MHz, CDCl 3 , 25 °C): δ 187.5, 185.8,
	23	165.9, 163.3, 152.1, 150.8, 144.6, 143.6, 142.8, 141.8,
	24	140.5, 139.3, 139.1, 138.2, 138.0, 137.9, 137.2, 136.1,
	25	136.0, 134.4 (CH), 133.8 (CH), 133.3, 131.6, 131.3
	26	(CH), 130.8 (CH), 130.4 (CH), 130.0 (CH), 129.0 (CH),
	27	128.6, 127.8 (CH), 127.3 (CH), 125.6 (CH), 125.0 (CH),
	28	124.0 (CH), 123.9 (CH), 123.0, 120.0, 115.2, 114.5,
	29	101.1, 99.7, 34.7, 34.59, 34.56, 31.7 (CH 3 ), 31.61 (CH 3 ),
	30	31.57 (CH 3 ), 26.3 (CH 3 ), 26.2 (CH 3 ), 21.6 (CH 3 ), 21.44
	31 32	(CH 3 ), 21.36 ppm (CH 3 ). UV-vis (CH 2 Cl 2 ): λ max (ε)	419 (46300), 458
	33 34	(54800), 483 (77500), 629 (12200), 645 (11500), 684 nm (27100 M -1 . cm -1 ).
	35	HRMS, ESI-TOF: m/z	1167.3783. Calcd for
	36	C 68 H 69 N 5 NiO 3 Pd ([M] • ): 1167.3800.
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	56
	57

  NiPPdS (M • ): 1346.4040. Porphyrin 5. A solution of Kacac in THF (2.25 mL at 40 mM, 88 µmol, 3 eq.) was added to a solution of porphyrin 4 (42 mg, 29 µmol) in dry THF (30 mL) and the mixture was stirred overnight at 35 °C, then two additional hours at 60 °C. After evaporation of the solvent, the product was separated by column chromatography (silica gel, dichloromethane/cyclohexane, 1/1) and porphyrin 5 was obtained after recrystallization from dichloromethane/methanol(23 mg, 19 µmol, 66%). CDCl 3 , 25 °C): δ 9.17 (dd, J 8.1, 1.4 Hz, 1H, H cyclPh ), 9.00 (d, J 5.0 Hz, 1H, H pyrr ), 8.43-8.36 (m, 2H, NH H pyrr ), 8.36-8.29 (br dd, 1H, H cyclPh ), 8.11-8.04 (m, 2H, H pyrr ), 8.01 (d, J 4.7 Hz, 1H, NMR (500 MHz, CDCl 3 , 25 °C): δ 9.05 (d, J 5.0 Hz, 1H, H pyrr ), 8.88 (dd, J 8.0, 1.4 Hz, 1H, H cyclPh ), 8.40 (d, J 4.9 Hz, 1H, H pyrr ), 8.34 (d, J 8.1 Hz, 1H, H cyclPh ), 8.09 (d, J 5.0 Hz, 1H, H pyrr ), 8.08 (d, J 5.0 Hz, 1H, H pyrr ), 8.03-8.97 (m, J 4.7 Hz, 2H, H pyrr NH), 7.84 (ddd, J 8.4, 7.1, 1.5 Hz, 1H, H cyclPh ), 7.75 (d, J 4.8 Hz, 1H, H pyrr ), 7.54 (ddd, J 8.0, 7.0, 1.1 Hz, 1H, H cyclPh ), 7.41 (s, 2H, H m-Ar ), 7.32 (s, 2H, H m-Ar ), 7.27 (s, 2H, H m-Ar ), 5.56 (s, 1H, CH -acac ), 2.12 (s, 3H, CH 3-acac ), 1.99 (s, 6H, H Me ), 1.96 (s, 6H, H Me ), 1.90 (s, 6H, H Me ), 1.89 (s, 3H, CH 3-acac ), 1.50 (s, 9H, H tBu ), 1.49 (s, 9H, H tBu ), 1.47 ppm (s, 9H, H tBu ). 13 C NMR (125 MHz, CDCl 3 , 25 °C): δ 185.5, 184.5, 163.0, 159.2, 152.1, 150.8, 145.0, 143.2, 142.7, 141.3, 140.7, 139.6, 138.9, 138.2, 138.1, 137.8, 136.2, 136.1, 136.0, 134.3 (CH), 134.2 (CH), 133.1, 132.5, 131.4 (CH), 130.9 (CH), 130.3 (CH), 129.4 (CH), 129.1 (CH), 129.0, 127.9 (CH), 126.7 (CH), 125.6 (CH), 125.3 (CH), 124.0 (CH), 123.9 (CH), 122.5, 120.2, 116.2, 113.9, 102.8 (CH), 100.4, 34.7, 34.60, 34.57, 31.7 (CH 3 ), 31.62 (CH 3 ), 31.58 (CH 3 ), 26.5 (CH 3 ), 26.4 (CH 3 ), 21.6 (CH 3 ), 21.5 (CH 3 ), 21.4 ppm (CH 3 ).

	1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 16 17 18 19 20 21 22 23 24 25 26 27 28 30 29	1 H NMR (500 MHz, CDCl 3 , 25 °C): δ 9.58 (d, J 2.4 Hz, 1H, NH), 8.92 (d, J 4.9 Hz, 1H, H pyrr ), 8.42 (dd, J 8.2, 1.2 Hz, 1H, H cyclPh ), 8.35 (d, J 4.9 Hz, 1H, H pyrr ), 8.31 (dd, J 8.2, 1.2 Hz, 1H, H cyclPh ), 8.11 (d, J 4.8 Hz, 1H, H pyrr ), 8.05 (d, J 4.7 Hz, 1H, H pyrr ), 8.01 (d, J 4.7 Hz, 1H, H pyrr ), 7.93-7.84 (m, 7H, 6H o-PPh 1H pyrr ), 7.70 (ddd, J 8.3, 7.0, 1.3 Hz, 1H, H cyclPh ), 7.55 (ddd, J 7.2, 5.2, 1.7 Hz, 3H, H p-PPh ), 7.52-7.45 (m, 8H, 6H m-PPh 2H m-Ar ), 7.41 (ddd, J 8.2, 7.0, 1.2 Hz, 1H, H cyclPh ), 7.32 (s, 2H, H m-Ar ), 7.27 (s, 2H, H m-Ar ), 1.99 (s, 6H, H Me ), 1.95 (s, 6H, H Me ), 1.91 (s, 6H, H Me ), 1.52 (s, 9H, H tBu ), 1.50 (s, recrystallization from dichloromethane/methanol the porphyrin 7 was isolated (3 mg, 2.4 µmol, 13%). 9H, H 13 C NMR (125 MHz, CDCl 3 , 25 °C): δ 164.2, 153.0, 151.0, 146.0, 143.3, 142.3, 141.1, 139.6, 139.4, 138.4, 138.2, 138.1, 137.36, 137.36, 136.1, 136.0, 135.7 (CH), 135.56 (CH), 134.5 (CH), 134.1 (CH), 134.0, 132.6, 132.08 (CH), 132.05, 132.0, 131.1, 131.0 (CH), 130.7 (CH), 130.13, 130.10, 129.5 (CH), 129.3 (CH), 129.1, 128.8 (CH), 128.60 (CH), 128.59 (CH), 128.3 (CH), 128.2 (CH), 126.7 (CH), 126.2, 125.2 (CH), 124.11 (CH), 124.06 (CH), 123.2, 121.6, 114.4, 101.8, 34.9, 34.74, 34.71, 31.9, (CH 3 ) 31.7 (CH 3 ), 21.8-21.3 ppm (CH 3 ). 31 P NMR (160 MHz, CDCl 3 , 25 °C): δ 30.8 ppm. UV-vis (CH 2 Cl 2 ): λ max (ε) 380 (42100), 428 (86900), 466 (61000), 497 (82300), 637 (16800), 671 (18300), 704 nm (24700 M -1 . cm -1 ). HRMS, ESI-TOF: m/z 1346.4025. Calcd for (ddd, J 8.2, 7.0, 1.2 Hz, 1H, H cyclPh ), 7.55 (d, J 4.8 Hz, 1H, H pyrr ), 7.42 (s, 2H, H m-Ar ), 7.32 (s, 2H, H m-Ar ), 7.26 (s, 2H, H m-Ar ), 5.39 (s, 1H, CH -acac ), 2.09 (s, 3H, CH 3-acac ), 2.04 (s, 3H, CH 3-acac ), 1.99 (s, 6H, H Me ), 1.93 (s, 6H, H Me ), 1.89 (s, 6H, H Me ), 1.50 (s, 9H, H tBu ), 1.49 (s, 9H, H tBu ), 1.46 ppm (s, 9H, H tBu ). 13 C NMR (125 MHz, CDCl 3 , 25 °C): δ 186.4, 186.0, 165.1, 153.6, 152.2, 150.94, 150.91, 145.6, 143.4, 142.4, 141.0, 139.4, 139.2, 138.51, 138.45, 138.1, 137.9, 135.87, 135.85, 134.5 (CH), 134.1 (CH), 133.7, 133.0, 132.9, 131.8 (CH), 131.4, 130.9 (CH), 130.2 (CH), 129.24 (CH), 129.16 (CH), 129.0 (CH), 128.0 (CH), 127.6, 125.6 (CH), 125.4 (CH), 124.0 (CH), 123.9 (CH), 123.1, 121.3, 113.4, 101.8, 100.1, 92.8, 34.7, 34.60, 34.55, 31.60 (CH 3 ), 31.56 (CH 3 ), 27.8 (CH 3 ), 26.8 (CH 3 ), 21.6 (CH 3 ), 21.4 (CH 3 ), 21.3 ppm (CH 3 ). UV-vis (CH 2 Cl 2 ): λ max (ε) 385 (36300), 428 (64500), 467 (42600), 496 (54000), 645 (13600), 668 (13300), 707 nm (20800 M -1 . cm -1 ). HRMS, ESI-TOF: m/z 1183.3534. Calcd for C 68 H 69 N 5 NiO 2 PdS ([M] • ): 1183.3571. Porphyrin 7. A solution of platinum(II) dimer 6 (20 mg, 9.4 µmol) in a acetylacetone/1,2-dichlorobenzene mixture (1/3, 20 mL) was refluxed under argon overnight. The solvents were removed under vacuum and the residue was purified by column chromatography (silica gel, dichloromethane/cyclohexane, 2/1). After UV-vis (CH 2 Cl 2 ): λ max (ε) 370 (36000), 426 (60000), 462 (58500), 488 (84500), 636 (14300), 652 (12800), 691 nm (31300 M -1 . cm -1 ). HRMS, ESI-TOF: m/z 1257.4475. Calcd for C 81 H 77 N 5 H pyrr ), 7.79 (ddd, J 8.3, 7.0, 1.4 Hz, 1H, H cyclPh ), 7.59 C 68 H 70 N 5 NiO 3 Pt ([M H] ): 1257.4386.
	2150106.indd 7 2150106.indd 7	29-08-2021 20:43:19 29-08-2021 20:43:19

tBu ), 1.47 ppm (s, 9H, H tBu ). 1 H NMR (500 MHz, 1 H

  NH), 9.22 (dd, J 8.2, 1.3 Hz, 1H, H cyclPh ), 9.05 (d, J 4.9 Hz, 1H, H pyrr ), 8.44 (d, J 4.9 Hz, 1H, H pyrr ), 8.37 (dd, J 8.1, 1.2 Hz, 1H, H cyclPh ), 8.15 (d, J 4.8 Hz, 1H, H pyrr ), 8.13 (d, J 4.7 Hz, 1H, H pyrr ), 8.07 (d, J 4.7 Hz, 1H, H pyrr ), 7.89 (d, J 4.8 Hz, 1H, H pyrr ), 7.85 (ddd, J 8.3, 7.0, 1.3 Hz, 1H, H cyclPh ), 7.59 (ddd, J 8.2, 7.0, 1.CH), 124.03 (CH), 123.96 (CH), 122.9, 121.3, 114.6, 102.7, 46.3 (CH 3 ), 34.8, 34.62, 34.59, 31.8 (CH 3 ), 31.61 (CH 3 ), 31.57 (CH 3 ), 21.6 (CH 3 ), 21.5 (CH 3 ), 21.4 ppm (CH 3 ). NMR (500 MHz, CDCl 3 , 25 °C): δ 9.26 (dd, J 8.2, 1.3 Hz, 1H, H cyclPh ), 9.10 (s, 1H, NH), 9.02 (d, J 4.9 Hz, 1H, H pyrr ), 8.41 (d, J 4.9 Hz, 1H, H pyrr ), 8.34 (dd, J 8.2, 1.3 Hz, 1H, H cyclPh ), 8.10 (d, J 4.7 Hz, 1H, H pyrr ), 8.07 (d, J 4.8 Hz, 1H, H pyrr ), 8.03 (d, J 4.7 Hz, 1H, H pyrr ), 7.87 (ddd, J 8.3, 6.9, 1.3 Hz, 1H, H cyclPh ), 7.58 (ddd, J 8.1, 7.0, 1.2 Hz, 1H, H cyclPh ), 7.55 (d, J 4.8 Hz, 1H, H pyrr ), 7.42 (s, 2H, H m-Ar ), 7.32 (s, 2H, H m-Ar ), 7.26 (s, 2H, H m-Ar ), 5.49 (s, 1H, C H-acac ), 2.04 (s, 3H, CH 3-acac ), 1.99 (s, 6H, H Me ), 1.98 (s, 3H, CH 3-acac ), 1.93 (s, 6H, H Me ), 1.89 (s, 6H, H Me ), 1.50 (s, 9H, H tBu ), 1.49 (s, 9H, H tBu ), 1.46 ppm (s, 9H, H tBu ). Hz, H cyclPh ), 8.26 (d, 1H, J 4.8 Hz, H pyrr ), 8.16 (d, 1H, J 4.8 Hz, H pyrr ), 8.03 (d, 1H, J 4.8 Hz, H pyrr ), 7.93 (s, 1H, NH), 7.40-7.90 (broad, 6H, H Ar-ortho ), 7.79 (ddd, 1H, J 7.7 Hz, 7.7 Hz and 1.2 Hz, H cyclPh ), 7.73 (t, 1H, J 1.7 Hz, H Ar-para ), 7.65 (t, 1H, J 1.7 Hz, H Ar-para ), 7.61 (t, 1H, J 1.7 Hz, H Ar-para ), 7.57 (ddd, 1H, J 7.7 Hz, 7.7 Hz and 1.2 Hz, H cyclPh ), 5.37 (s, 1H, CH acac), 2.07 (s, 6H, CH 3 acac), 1.44 (s, 18H, CH 3 t-Bu), 1.41 (s, 18H, CH 3 t-Bu), 1.39 (s, 18H, CH 3 t-Bu). 13 C NMR (CDCl 3 , 125 MHz, 25 °C): δ C (ppm) 186.2, 185.9, 164.9, 153.3, 151.9, 149.4, 149.3, 146.1, 143.5, 143.1, 141.4, 139.6, 138.8, 138.7, 138.6, 136.2, 135.1 (CH), 134.5 (CH), 133.5, 132.7, 132.4 (CH), 131.4, 131.0 (CH), 130.9 (CH), 130.1 (CH), 129.2 (CH), 129.1 (CH), 128.7 (CH), 128.0 (CH), 127.6 (CH), 126.1 (CH), 125.9, 125.5 (CH), 124.2, 122.2 (CH), 121.5 (CH), 121.4 (CH), 116.0, 102.2, 100.2 (CH), 35.2, 35.02, 34.98, 31.7 (CH), 31.63 (CH 3 ), 31.60 (CH 3 ), 27.9 (CH 3 ), 27.0, 26.9 (CH 3 ). MALDI-TOF: m/z 1267.299 Calcd for C 74 H 81 N 5 NiO 2 PdS 1267.451 (M • ).

		SYNTHESIS AND ELECTROCHEMISTRY OF NICKEL(II)PORPHYRINS BEARING EXTERNAL PALLADIUM(II)	I
	1 2 3 4 5	dried filtrate by column chromatography (silica gel, 300 mL, cyclohexane/CH 2 Cl 2 (2:8)) and cristallization from hexane/dichloromethane/MeOH afforded the title compound (177 mg, 0.14 mmol, 92%) as green crystals. Crystals suitable for X-Ray diffraction analysis were	(CH), 125.9 (UV-vis (CH 2 Cl 2 ): λ max (ε) 385 (33800), 426 (69000),
	6 7	grown from chlorobenzene-methanol. 1 H NMR (CDCl 3 , 500 MHz, 25 °C): δ H (ppm) 9.15	470 (39600), 498 (56000), 641 (12200), 672 (15500), 700 nm (17300 M -1 . cm -1 ).
	8	(dd, 1 H, J 8.0 and 1.0 Hz, H cyclPh ), 9.00 (d, 1H, J	HRMS, ESI-TOF: m/z	1325.3220. Calcd for
	9	4.8 Hz, H pyrr ), 8.62 (d, 1H, J 4.8 Hz, H pyrr ), 8.35 (d,	C 65 H 68 ClKN 5 NiOPtS 2 ([M K] ): 1325.3191.
	10 11	1H, J 4.8 Hz, H pyrr ), 8.32 (dd, 1 H, J 8.0 Hz and 1.0	Porphyrin 9. A solution of porphyrin 8 (31 mg, 24 µmol) in dry THF (15 mL) was refluxed with Kacac
	12		(0.96 mL of a 0.125 M solution in THF, 120 µmol,
	13		5 eq.) for one day. The mixture was then washed
	14		with water (15 mL), the organic layer was separated,
	15		dried over Na 2 SO 4 and the solvent evaporated under
	16		reduced pressure. The product 9 was purified by
	17		column chromatography (silica gel, dichloromethane/
	18		cyclohexane, 2/1) and recrystallized from dichloro-
	19		methane/methanol (19 mg, 15 µmol, 30%).
	20
	21
	22
	23
	24
	25
	26
	27
	28
	29
	30
	31
	32		13 C NMR (125 MHz, CDCl 3 , 25 °C): δ	184.6,
	33		158.4, 152.2, 150.9, 145.8, 142.7, 142.3, 141.2, 139.8,
	34		139.0, 138.6, 138.4, 138.1, 137.8, 136.1, 136.0, 135.2
	35		(CH), 134.2, 134.0 (CH), 133.1, 133.0, 132.5, 131.8
	36		(CH), 131.2 (CH), 130.1 (CH), 129.5 (CH), 128.2
	37		(CH), 128.1 (CH), 127.7 (CH), 126.0 (CH), 125.6
	38		(CH), 124.0 (CH), 123.9 (CH), 122.1, 121.0, 112.7,
	39		102.8, 101.9, 34.8, 34.60, 34.56, 31.61 (CH 3 ), 31.57
	40		(CH 3 ), 27.9 (CH 3 ), 26.7 (CH 3 ), 21.6 (CH 3 ), 21.5 (CH 3 ),
	41 42		21.4 ppm (CH 3 ). UV-vis (CH 2 Cl 2 ): λ max (ε) 388 (45700), 436 (65700),
	43 44		467 (50000), 498 (63600), 649 (15100), 669 (14200), 712 nm (23400 M -1 . cm -1 ).
	45		MALDI-TOF: m/z	1272.432. Calcd for
	46		C 68 H 69 N 5 NiO 2 PtS (M • ): 1272.417.
	47	2 Hz,	X-Ray diffraction. Under argon, a solution of the
	48	1H, H cyclPh ), 7.51 (s, 2H, H m-Ar ), 7.34 (s, 2H, H m-Ar ), 7.29	nickel(II)porphyrin analogous to 2 but with 3,5-tBu 2 -
	49	(s, 2H, H m-Ar ), 3.66 (s, 6H, H Me-DMSO ), 1.99 (s, 6H, H Me ),	phenyl meso-aryl groups (160 mg, 0.15 mmol) in
	50	1.93 (s, 6H, H Me ), 1.91 (s, 6H, H Me ), 1.56 (s, 9H, H tBu ),	1,2-dichloroethane (130 mL) was added dropwise over
	51 52 53	1.51 (s, 9H, H tBu ), 1.48 ppm (s, 9H, H tBu ). 13 C NMR (125 MHz, CDCl 3 , 25 °C): δ 160.0, 153.2, 151.04, 151.02, 148.5, 145.3, 143.8, 142.9, 142.6, 141.3,	a period of 8.5 h to a solution of Pd(acac) 2 (930 mg, 3.05 mmol, 20 eq) in 1,2-dichloroethane (100 mL) at 60 °C. After cooling down to room temperature, the
	54	139.9, 138.74, 138.70, 138.1, 137.8, 137.4, 135.83,	solvent was removed under vacuum. Cyclohexane/
	55	135.81, 135.0 (CH), 134.2 (CH), 133.6, 133.4, 132.2,	CH 2 Cl 2 (8:2, 50 mL) were added to the mixture
	56	132.0 (CH), 131.9, 131.1 (CH), 130.6 (CH), 129.7 (CH),	and the obtained suspension was filtered. Pd(acac) 2
	57	129.2 (CH), 128.9 (CH), 128.6 (CH), 127.8, 126.9, 126.7	(783 mg) was recovered as a solid. Purification of the
	2150106.indd 8 2150106.indd 8				29-08-2021 20:43:19 29-08-2021 20:43:19

1 H NMR (500 MHz, CDCl 3 , 25 °C): δ 9.53 (s, 1H, 1 H

Acknowledgments

Continuous financial support from the University of Strasbourg and the CNRS are acknowledged. MAC thanks the FRC-Labex CSC for her PhD fellowship (ANR-10-LABX-0026 CSC). HD thanks the French Ministry of Research for a PhD fellowship.

Supporting information

Crystallographic data have been deposited at the Cambridge Crystallographic Data Centre (CCDC) under number CCDC-2096258. Copies can be obtained on request, free of charge, via https://www.ccdc.cam.ac.uk/ structures/ or from the Cambridge Crystallographic Data Centre, 12 Union Road, Cambridge CB2 1EZ, UK (fax: 44 1223-336-033 or email: deposit@ccdc.cam.ac.uk).


Mohammed Khalij

Boris Arcen

come

Improvement of the formulation of the dispersed phase boundary conditions for a rough wall needed in two-fluid models

INTRODUCTION

One of the approach which can be used for the numerical simulation of dispersed two-phase flows is the so-called Eulerian-Eulerian approach commonly named "the two-fluid model". It is easier to take the presence of the wall into account by Lagrangian simulation, just by using rebound laws and introducing the effect of particle shape or wall roughness by means of a virtual wall model (Tsuji et al. [START_REF] Tsuji | Numerical simulation of gas-solid two phase flow in a two dimensional horizontal channel[END_REF], Sommerfeld [START_REF] Sommerfeld | Modelling of particle/wall collisions in confined gas-particle flows[END_REF], Sommerfeld and Huber [START_REF] Sommerfeld | Experimental analysis and modelling of particle-wall collisions[END_REF]). In two-fluid models, the wall exists only through the boundary conditions needed for the solution of the differential system. In order to obtain accurate information about the influence of the wall on the behaviour of solid particles, it is important to develop a set of boundary conditions for the dispersed phase. Many works have been devoted to the study of the dispersed phase boundary conditions at the wall. Derevich and Zaichik [START_REF] Derevich | Particle deposition from turbulent flow[END_REF] have used the kinetic equation for probability density function (PDF) of particle coordinates and velocity distribution for the particle deposition. They have obtained a solution for their model at the wall by a perturbation method function of a reflection coefficient. Swailes and Reeks [START_REF] Swailes | Particle deposition from a turbulent flow. I. A steady-state model for High inertial particles[END_REF] have also used a PDF model (Kinetic equation) for steady-state transport and deposition of particle for partially or perfectly reflection at the wall . In the last case, elastic rebounds are considered and the boundary conditions have been obtained by a spectral method. Devenish et al. [START_REF] Devenish | A PDF model for dispersed particles with inelastic particle-wall collisions[END_REF] have extended this work for inelastic particle-wall collisions and for a restitution coefficients higher than 0.5. He and Simonin [START_REF] He | Non-equilibrum predictions of the particle phase stress tensor in vertical pneumatic conveying[END_REF] and Sakiz and Simonin [START_REF] Sakiz | Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow[END_REF] have established a set of boundary conditions for the dispersed phase at a smooth wall in terms of the Coulomb parameters (coefficients of friction and restitution), using a preassigned binormal PDF for the wall normal particle velocity. Their formalism is based on the determination of PDF of the particle velocities close to the wall. Using this PDF-two-fluid method, the mean value of any variable at the wall can be expressed as a function of the mean value before collision, and relationships between the various statistical moments of the particle velocity components can be derived, provided that a deterministic rebound law is prescribed. Alipchenckov et al. [START_REF] Alipchenkov | A comparison of two approaches to derivation of boundary conditiond for continuous equations of particle motion in turbulent flow[END_REF] have compared the results given by a perturbation method with the results obtained by a binormal PDF method for the phase dispersed. They showed that the approach using the resolution of the PDF equation by the perturbation method is only accurate for particle rebounds slightly inelastic whereas the binormal PDF method could be applied for a large range of the restitution coefficient. Nevertheless, all these studies are valid only with a smooth wall. In a rough case, no work exists about Eulerian dispersed phase boundary conditions taking the effect of the 3-D roughness into account. The work of Zhang and Zhou [START_REF] Zhang | A two-fluid particle-wall collision model accounting for the wall roughness[END_REF] is limited to a 2-D roughness case simulated with the help of a virtual wall model (Sommerfeld [START_REF] Sommerfeld | Modelling of particle/wall collisions in confined gas-particle flows[END_REF]) and without taking the shadow effect into account. This physical phenomenon has been revealed by Sommerfeld and Huber [START_REF] Sommerfeld | Experimental analysis and modelling of particle-wall collisions[END_REF]. According to us, to obtain a realistic simulation of the particle-wall interaction when a virtual wall model (Sommerfeld [START_REF] Sommerfeld | Modelling of particle/wall collisions in confined gas-particle flows[END_REF]) is used, some generated angles have to be eliminated since they can be unphysical. The establishment of boundary conditions incorporating the effect of wall roughness is identified as one of the objectives pursued in this paper. We propose here to introduce the effect of the 3-D roughness via the use of a virtual wall model based on the idea of Sommerfeld and Zivkovic [START_REF] Sommerfeld | Recent advances in the numerical simulation of pneumatic conveying trough pipe systems[END_REF]. No deposition phenomenon of particles was considered here. Before presenting and discussing the numerical simulation results issued from a statistical study on a large number of particle rebounds at the wall, the derivation of the dispersed phase boundary conditions for a rough wall is described. It is based on an analogy with the formalism established for a smooth

IMPROVEMENT OF THE FORMULATION OF THE DISPERSED PHASE BOUNDARY CONDITIONS FOR A ROUGH WALL NEEDED IN TWO-FLUID MODELS

M. Khalij, B. Arcen , A. Tanière* wall. Equivalent restitution and friction coefficients are introduced and their dependence on the roughness parameter is studied.

ROUGH WALL BOUNDARY CONDITIONS DERIVATION

Description of the proposal model

As Sakiz and Simonin [START_REF] Sakiz | Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow[END_REF], we consider a distribution function f for the velocity of the particles at the wall. The mean value of any function ψ of the velocity can be given by: ( ) ( ) ( )

1 fd N ψψ = ∫ uuuu (1)
where ( ) Nfd = ∫ uu is the particle number density at the wall. The fluctuation of ψ is defined by =-ψψψ ′ . In order to focus on the effect of the rebounds on the wall, as Sakiz and Simonin [START_REF] Sakiz | Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow[END_REF], we distinguish the mean values of ψ for incident .

-and reflected particles . + , linked by the following relation:

NNN ψψψ -+ -+ =+ (2)
where N -and N + are the number densities of incident and reflected particles at the wall, respectively, so that NNN -+ =+. Equation ( 2) can be written using the parameter ξ defined by NN ξ -= :

( )

1 ψξψξψ -+ =+- (3)
The relation (Eq. ( 3)), written for Y u ψ = , shows that the parameter ξ must therefore satisfy the following condition in order to have a zero particle mass flux at the wall ( i.e. 0

Y u = ): Y YY u N N uu ξ + - +- == - % % (4)
where

Y u + % is related to Y u - (keeping in mind that 0 Y u -<
) by the rebound law described below. The particle translational velocity after the collision is expressed by means of the impulsive equations and Coulomb's laws. Besides the coefficient of restitution e, the rebound law is expressed by means of the friction factor µ. In the case where sliding occurs in the longitudinal direction, and under the condition of negligible spanwise component of the impact velocity for all particles, the mechanical rebound laws for the translational velocity components are:

(1)

0 XXY YY Z uueu ueu u µ  =++   =-   =   % % % ⇒ = uCu % , ( ) 110
Such relations are always valid in the incident plane formed by the incident velocity and the wall normal vectors. The reflected velocity lies in this plane too. In this plane, the relation (Eq. ( 4)) becomes (knowing that

YY ueu +- =- % ):
1

e e ξ = + . ( 6
)
Note that this value of the parameter ξ is same as the one obtained in a smooth case since the formalism used by Sakiz and Simonin [START_REF] Sakiz | Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow[END_REF] is based on the following relation:

( ) ( ) ( ) 1 11 e ee ψψψ -- =+ ++ uuu % (7)
In opposite to us, they can express ( )

ψ + u % as a function of ( ) ψ - u %
since the rebounds are purely deterministic. By means of Eq. ( 5) and ( 7) and making an assumption about the distribution of the wall normal incident particle velocity, Sakiz and Simonin [START_REF] Sakiz | Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow[END_REF] obtained the following boundary conditions at the wall for all computational variables:

=  ′′′ =-    ′′′′′ =--   - ′′ =   (8)
where C depends on the distribution of Y u (in a Gaussian (similar to Eq. ( 6)). In the rough wall case, we then rewrite Eq. ( 3) in the following form, where the effects of the restitution coefficient, friction factor and roughness upon the particle phase boundary conditions at the wall are introduced via e * :

1 11 e ee ψψψ

Introduction of the 3-D roughness effect

The effect of wall roughness can be introduced into the Eq. ( 8) by means of the so-called "virtual wall" model of Sommerfeld [START_REF] Sommerfeld | Modelling of particle/wall collisions in confined gas-particle flows[END_REF]. In this model, the actual wall is replaced by a virtual wall, whose inclination angle β obeys a centered Gaussian distribution function with a given standard deviation σ β and a zero mean value. The generation of only one inclination angle traduces a 2-D roughness case. In order to extend it to a 3-D case, Sommerfeld & Zivkovic [START_REF] Sommerfeld | Recent advances in the numerical simulation of pneumatic conveying trough pipe systems[END_REF] have proposed a rotation of the virtual wall previously generated around the wall normal vector of the real wall. The second inclination angle α obeys a uniform distribution function (in the range [ ] , ππ -+ ). In order to facilitate the calculus, the rotation by α around Y-axis is first carried out and then the second rotation by β around z'-axis is applied as it can seen in Fig. 1. We have carried out successively three rotations following the scheme:

( ) ( ) ( ) ( ) ,,,,,,,, RXYZRxyzRxyzRxyz αβγ ′′′′′′′′′ →→→ (12)
The last rotation by γ around y"-axis permits to define an incident plane, this will be explained hereafter. The velocity components in the virtual wall co-ordinates system are:

In order to apply the Coulomb's laws, an incident plane containing the incident velocity vector has to be defined. A new inclination angle γ around y"-axis is then defined such as 0 z u = :

In this co-ordinates system, Coulomb's laws can be applied:

,,,, xyzxyz = uCu %

. According to Eq.( 5), Eq. ( 13) and

Eq. ( 15) the post-collisional velocity components with respect to the fixed co-ordinates X, Y and Z become:

11 ,,,, XYZXYZ -- = uPICIPu % (16)
Attention must be paid to the validity of such rebound laws. Sommerfeld and Zivkovic [START_REF] Sommerfeld | Recent advances in the numerical simulation of pneumatic conveying trough pipe systems[END_REF], Schade and Hadrich [START_REF] Schade | Investigation of influence of wall roughness on particle-wall collisions[END_REF] and Sommerfeld and Huber [START_REF] Sommerfeld | Experimental analysis and modelling of particle-wall collisions[END_REF] have shown that the distribution of the inclination angle depends on the impact angle of the particle with respect to the wall (the so-called shadow effect). In order to simulate a physical rebound, some values of the virtual wall inclination angle β must be avoided, because the incident and the reflected velocities must satisfy 0 y u ′′ < and 0 Y u > % , respectively. Thus, testing the signs of ′ ⋅ un and ⋅ un % (where n and ′ n are the unit vectors normal to the actual wall and virtual wall respectively) allows to take the shadow effect into account by disregarding the unphysical cases in the numerical simulation. Therefore, taking the shadow effect into account seems of primary importance in order to simulate realistic rebounds. However, up to now, the only work about the Eulerian modelling of the dispersed phase boundary conditions with wall roughness does not take this effect into consideration. This work is by Zhang and Zhou [START_REF] Zhang | A two-fluid particle-wall collision model accounting for the wall roughness[END_REF], who focused on the dispersed phase boundary conditions, but did not mention any test on unphysical impacts and reflected angle values, assuming that the inclination of the wall is so small that they can neglect the shadow effect.

NUMERICAL SIMULATION, RESULTS AND DISCUSSION

The main objective here is to numerically investigate the dependence of the equivalent coefficients e * and µ * on the primary collision parameters, mainly the roughness parameter, for a Gaussian distribution of the incident velocity, and to examine the suitability of the proposed model. For this reason, the numerical results about the dispersed phase boundary conditions at the wall are presented with comparison to the results issuing from the model (Eq. ( 9)). The collisions were assumed to be three-dimensional, i.e. the spanwise velocity of the particle was taken into account (

2 0;0.2 ZZ uu -- ′ ==
).

For each specification of the collision parameters, we simulated a large number of particle rebounds (10 6 ) by means of the rebound laws (Eq. ( 16)). The distribution of the virtual wall inclination angle was Gaussian with given standard deviation β σ and zero mean value. The values of β σ tested here were varied from 0 to 0.2 according to the results of Sommerfeld and Huber [START_REF] Sommerfeld | Experimental analysis and modelling of particle-wall collisions[END_REF], who showed that the optimum value of β σ decreases with increasing particle diameter for a given physical roughness. The choice of the value of β σ in practical problems can be made according to the recent work by Sommerfeld [START_REF] Sommerfeld | Analysis of collision effects for turbulent gas-particle flow in a horizontal channel: Part I. Particle transport[END_REF], who provided correlations of β σ as a function of the particle diameter (0 < d p < 500 µm) for spherical particles. At the present time however, it is not clear how to decide the value of β σ in order to apply such an irregular bouncing model for non spherical particles, although the results obtained by Sommerfeld [START_REF] Sommerfeld | Kinetic simulations for analysing the wall collision process of non-spherical particles[END_REF] by means of the exact impulse equations for various particle shapes are very promising in order to develop such a model. The Gaussian distribution of the incident velocity components was prescribed as follows: the tangential and the spanwise incident particle velocities were assumed to obey a Gaussian distribution. In order to close up practical problems, where usually the anisotropic character of the particle velocity variance is observed, therefore the effect of the variation of the ratio According to the model (Eq. ( 10)), ee * depends only on the distribution function f of the incident velocity of the particles at the wall. Whatever the case treated, the Gaussian distribution is used and the wall normal particle velocity variance is the same

Study of the equivalent restitution and friction coefficients

2 1 Y u - ′ =
, therefore such result is not surprising. In the plot, the ratio ee * is seen to increase with increasing β σ . The slope of the curves increases as the restitution coefficient e decreases. As can be observed, e * is always higher than e . Contrary to e , whose maximum value is equal to unity, e * can reach considerably higher values. This means t hat the reflected wall normal velocity can significantly exceed the particle impact velocity as soon as the roughness is present, since

YY ueu +- * =- % (with 0 Y u -< ).
Accordingly, the wall normal velocity after collision increases as β σ increases. Such a momentum redistribution due to collisions with a rough wall was noticed in earlier works where gas-solid flows were simulated by Lagrangian particle tracking, like the investigations by Fukagata et al. [START_REF] Fukagata | Effects of wall roughness in a gas-particle turbulent vertical channel flow[END_REF] o r Sommerfeld [START_REF] Sommerfeld | Analysis of collision effects for turbulent gas-particle flow in a horizontal channel: Part I. Particle transport[END_REF][START_REF] Sommerfeld | The importance of inter-particle collisions in horizontal gas-solid channel flows[END_REF]. As can be observed, the ratio ee * is only slightly influenced by the friction coefficient. The ratio µµ * first decreases with increasing β σ , and then it increases. In the isotropic case, the influence of e dominates in the first part of the curves, in opposite to the second part where the influence of µ is clearly prevailing (the slope change occurs around 0.05 β σ ≈ whatever the values of the friction and restitution coefficients).

Such observation is less emphasized in the anisotropic case where the influence of e and µ seems to be the same in all parts of the curve. In order to study the influence of the shadow effect on these results, β is plotted as a function of for the mean value of β , the obtained results are independent of the range chosen for α. We have also observed that the value of β is the same in the anisotropic and isotropic cases. A unique curve is obtained. Nevertheless, it is interesting to note that the shadow effect is more sensible to the value of the restitution coefficient than the value of the friction coefficient µ as it can be seen in Fig. 4. It does not depend on µ and decreases slightly with increasing e . Therefore, in that sense, the second part of the curves in Fig. 3(a) seems to be almost independent of the shadow effect since the influence of µ was seen to prevail, contrary to the first part of the curves, for which a small value of e corresponds to a maximum influence of the shadow effect. The shadow effect plays exactly the same role in Fig. 3 In this case, this tendency is more accentuated. This analysis shows that the equivalent parameters e * and µ * take into consideration the effects related to the roughness of the wall. .

Same caption as in Fig. 2. for which the transverse agitation is important. A rapid comparison between these results and those obtained in a two dimensional case permits to observe the effect of the 3-D character of the roughness. In the whole, all results are more attenuated which is physically correct.

Evaluation of the proposed model

In order to examine the pertinence of the proposed model, the results issuing from the model and from the numerical simulation are compared in Fig. 56, in what concerns the normalized third-order velocity correlations defined by To summarize, acceptable estimates of the velocity correlations at the wall are obtained by means of a formulation close to the smooth wall formulation of Sakiz and Simonin [START_REF] Sakiz | Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow[END_REF], using equivalent restitution and friction coefficients, whose values can be prescribed with the help of Fig. 2 and3.

CONCLUSIONS

In order to obtain the boundary conditions needed in twofluid models for the dispersed phase in case of irregular particle-wall bouncing, a numerical simulation was carried out by computing the particle statistics at the wall, handling the collisions by means of a 3-D virtual wall model and taking the so-called shadow effect into account. The computational results were analyzed by defining equivalent restitution and friction coefficients ( e * , µ * ) which may be used to replace the original coefficients in the available boundary conditions for smooth wall. The obtained results show that the wall roughness has a considerable influence on the equivalent coefficients. Thanks to these new coefficients, a model for the particulate phase boundary conditions is proposed ensuring zero mass flux at the wall. The proposed model leads to good agreement with the results of the numerical simulation. The next step of the study is to investigate the boundary conditions involving the particle rotational velocity components using the same technique, keeping in mind that the existing two-fluid models are still unable to take the rotational motion of the particle into consideration, however. 9)). (a), (b) as in Fig. 3.

NOMENCLATURE e

Normal restitution coefficient for wall-particle interaction (dimensionless) * e equivalent restitution coefficient (dimensionless) f

Particle velocity distribution function at the wall (m -6 .s 3 ) N

Particle number density at the wall (m -3 ) N -, N + incident and reflected particle number density at the wall (m -3 ) u instantaneous particle velocity vector (m.s -1 ) u % instantaneous reflected particle velocity vector (m.s -

  analogy with Eq. (8), a model is proposed for the particulate phase boundary conditions in the rough wall case, based on the equivalent Coulomb coefficients e * and µ * :

  rotation matrix which permits to obtain the velocity components in the incident plane, noted I , is:

Fig. 1 :

 1 Fig. 1: The 3-D virtual wall model. (a): rotation around Yaxis. (b): rotation around z'-axis. (c) rotation defining an incident plane.

5 .

 5 Whatever the case studied, the following values concerning the spanwise particle velocity and its variance have been imposed. The assumption, lying on the independence of the rotational and translation motion of the particles. Calculations were performed for one value of the mean tangential incident velocity, namely

Figure 2

 2 Figure 2 illustrates the ratio ee * versus the standard deviation of the virtual wall inclination angle β σ . Whatever the value of the ratio

Fig. 2 :=

 2 Fig. 2 : The ratio e * /e as a function of the roughness parameter β σ for 22 1

  Fig. 4 is related to the case , 22 ππ  -+   . Fortunately, except

Figure 3 (

 3 Figure 3(b) brings the anisotropic effect to light since in this case, the influence of µ is clearly noted whatever the value of β σ . Moreover, one can notice that the anisotropy to reduce the value of µµ * . Other tests have been carried out showing that a similar behavior is obtained for a value of

Fig. 3 :

 3 Fig. 3 : The ratio µ * /µ as a function of the roughness parameter β σ . (a): 22

Fig. 4 :

 4 Fig. 4: Influence of the Coulomb parameters on the shadow effect. Same caption as in Fig 2. Such parameters reproduce, on the one hand, the momentum re-distribution of the particle velocity characterized by e * , which is increasingly large as β σ is high, and, on the other hand, the antagonistic effects of µ * , i.e. a decrease in the equivalent friction for small values of β σ (since for such values the momentum re-distribution is not yet important) and an increase in µ * for larger values of β σ

  . 6 (a)-(b), which should be equal to 32 2232 2/ XYYY uuuu µµ * *  ′′′′ -- according Eq. (9). As it can be seen in Fig 5(a) and 5(b) the obtained values are qualitatively in good agreement with the simulations. The model allows to get a satisfactory estimation of correlation XXY Π is slightly underestimated by the model for high values of β σ , nevertheless the results are very satisfactory (Fig. 6(a) and 6(b)).

Fig. 5 :

 5 Fig. 5: Comparison between the present model and the numerical results for YYY Π . Same caption as in Fig. 2. Lines: model (Eq.(9)). (a), (b) as in Fig. 3.

Fig. 6 :

 6 Fig. 6: Comparison between the present model and the numerical results XXY Π . Same caption as in Fig. 2. Lines: model (Eq.(9)). (a), (b) as in Fig. 3.

  1 )

	X u ,	Y u ,	u	Z	streamwise, wall-normal and spanwise velocity components (m.s -1 )
	X u′ , Y u′ , Z u′ fluctuations of velocity components (m.s -1 )
	Greek letters
	α, β γ				inclination angles of the virtual wall (radian) inclination angle defining an incident plane
						(radian)
	σ	β				standard deviation of the inclination angle of the virtual wall (radian)
	µ					Friction coefficient for wall-particle interaction
						(dimensionless)
	µ *				equivalent friction coefficient (dimensionless)
	Subcripts
	,, xyz			direction of vector components in orthogonal
						system (,,) Rxyz
	''' ,, xyz			direction of vector components in orthogonal
						system	''' (,,) Rxyz
	""" ,, xyz		direction of vector components in orthogonal
						system	""" (,,) Rxyz
	Others
	.					global average
	.

-, .

+ average on respectively incident and reflected particles