C De Souza

V J S Leite

S Tarbouriech

E B Castelan

Emulation-based Dynamic Output-Feedback Control of Saturating Discrete-time LPV Systems

Keywords: Emulation-based control, Linear parameter-varying systems, Actuator saturation, Stability analysis

This paper addresses the problem of emulation-based dynamic output-feedback control for discrete-time linear parameter varying systems under actuator saturation. Sufficient conditions based on linear matrix inequalities (LMI) are derived to design the parameters of two dependent event-triggering rules handling the sensor-to-controller and controller-to-actuator communication and to guarantee the regional asymptotic stability of the closed-loop system. A convex optimization scheme incorporating these conditions is also formulated to reduce the output and control signals' update rate through communication channels. A numerical example illustrates the usefulness of the proposal.

I. INTRODUCTION

Recently, networked control systems (NCSs) have attracted significant attention due to their potential advantages comprising high reliability and low-cost installation and maintainability [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF]. Although the network infrastructure brings us numerous benefits, some problems arise with its introduction, such as network congestion and elevated energy consumption on wireless systems. In this sense, it is meaningful to introduce a mechanism to reduce the unnecessary waste of communication resources and computation as in a periodic framework. As stated in [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF], the so-called "event-triggering" scheme can reduce the number of data transmissions while guaranteeing a certain performance of the closed-loop system. Event-triggered control (ETC) is a strategy in which the data are transmitted only when a certain event (generated by a specified eventtriggering scheme) indicates an update is required. The existing approaches can be divided into two general categories, namely, emulation based [START_REF] Heemels | Periodic event-triggered control for linear systems[END_REF]- [START_REF] Heemels | Model-based periodic event-triggered control for linear systems[END_REF] and co-design based [START_REF] Peng | Event-triggered communication and H∞ control co-design for networked control systems[END_REF], [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for linear systems[END_REF]. The co-design performs the design of both the controller and C. de Souza (carla.souza93@hotmail.com) is with PPGEAS of Universidade Federal de Santa Catarina (UFSC), DAS/CTC/UFSC, 88040-900, Florian ópolis, SC, Brazil.

S. Tarbouriech (sophie.tarbouriech@laas.fr) is with LAAS-CNRS, Universit é de Toulouse, CNRS, Toulouse, France.

V. J. S. Leite (valter@ieee.org) is with the Department of Mechatronics Engineering, Campus Divin ópolis -CEFET-MG, R. Álvares Azevedo, 400, 35503-822, Divin ópolis, MG, Brazil. V. J. S. Leite is also with the the Graduate Program on Electrical Engineering (PPGEL) of CEFET-MG / UFSJ, Av. Amazonas 7675, Belo Horizonte, MG, Brazil. E. B. Castelan (eugenio.castelan@ufsc.br) is of the Department of Automation and Systems, Universidade Federal de Santa Catarina (UFSC), PPGEAS -DAS/CTC/UFSC, 88040-900, Florian ópolis, SC, Brazil. the event-triggering rule, simultaneously. On the other hand, the emulation considers that the controller is given a priori, and the goal is to design only the event-triggering rule.

In the current paper, we focus on the emulation design of event-triggered control for linear-parameter varying (LPV) systems with actuator saturation, as shown in Figure 1. In this case, a parameter-dependent dynamic output-feedback controller that stabilizes the system is previously computed without taking the event-triggering nature of the control system into account. Two dependent event-triggering mechanisms (ETM) are introduced in the loop to reduce the number of data transmissions between both sensor-to-controller and controller-to-actuator channels. Unlike the works in the literature, we consider that each ETM can use information from the other transmitted signal, which provides an enlarged set of information about the behavior of the system. As a consequence, the usage of the communication resources can be significantly saved.

Despite significant advancements in LPV systems analysis and control synthesis, only a few works on LPV eventtriggered control design are available in the literature. The event-based reference tracking control problem for discretetime LPV systems is addressed in [START_REF] Golabi | Event-triggered constant reference tracking control for discrete-time LPV systems with application to a laboratory tank system[END_REF], where a state-feedback controller and three event-triggering conditions are simultaneously designed. [START_REF] Li | Co-design of event-triggered H∞ control for discrete-time linear parameter-varying systems with network-induced delays[END_REF] investigates the event-triggered H ∞ control for discrete-time LPV systems by jointly designing a mixed ETM and a set of state feedback controllers. [START_REF] Braga | Discretization and event triggered digital output feedback control of LPV systems[END_REF] addresses the problem of discretization and digital output feedback control design for continuous-time LPV systems subject to a time-varying networked-induced delay. The codesign problem of a state-feedback controller and an event generator is studied for discrete-time LPV systems in [START_REF] Li | Event-triggered control for discrete-time uncertain linear parameter-varying systems[END_REF], where the scheduling variables are assumed to be not precisely known, but their estimates satisfy known uncertainty levels. However, the saturation is not considered in these works.

Since the actuator saturation exists in almost every practical application, its effect on event-triggered control has been studied in recent publications. In [START_REF] Seuret | Event-triggered control with LQ optimality guarantees for saturated linear systems[END_REF], a local exponentially stability is guaranteed by a designed event-triggering condition concerning an LQ cost function. The dynamic event-triggered and self-triggered output feedback strategy is investigated in [START_REF] Zuo | Dynamic event-triggered and selftriggered control for saturated systems with anti-windup compensation[END_REF] for linear plants subject to input saturation and antiwindup compensation. [START_REF] Quan | Output-based event-triggered control for linear systems with actuator saturation[END_REF] designs an output-based event generator and an anti-windup compensator for a linear system with actuator saturation. [START_REF] Moreira | PI event-triggered control under saturating actuators[END_REF] proposes a methodology to design stabilizing event-triggering strategies for PI-controlled linear continuous-time plants subject to input saturation. The co-design problem of an event-triggered controller based on a static/dynamic feedback is investigated in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] for stabilization of discrete-time systems subject to actuator saturation.

Given the previous discussions, the main contributions of this paper can be summarized as: i) An event-triggering strategy is proposed for discrete-time LPV systems with input saturation controlled by LPV dynamic output-feedback controllers; ii) Two ETMs are designed for transmitting the sensor measurements and the controller outputs through a communication network; iii) The proposed ETMs are dependent, i.e. each ETM can use information from the other transmitted signal; iv) The event-triggering rules are designed to ensure the asymptotic stability of the closed-loop system. An estimate of the origin's basin of attraction is characterized, and the data transmission activity is then indirectly reduced thanks to adequate optimization procedures. The convex conditions are based on a parameter-dependent Lyapunov function, which yields a linear matrix inequality (LMI) based formulation. A numerical example borrowed from the literature allows for depicting our approach's effectiveness compared with other literature methods.

Notation: R is the set of real numbers. 0 the null matrix of appropriate dimension I n corresponds to the identity matrix with dimensions n × n. R m×n stands for the set of matrices with real entries of dimensions m × n. A block-diagonal matrix A with blocks A 1 and A 2 is denoted as A = diag{A 1 , A 2 }. The th row of the vector or matrix A is indicated by A ( ) , and for a square matrix A its diagonal element at ( , ) is noted A ( , ) . The set of integer numbers belonging to the interval from a

∈ N to b ∈ N, b ≥ a, is denoted by I[a, b]. The symbol
represents the transpose blocks in relation to the diagonal of real square and symmetric matrices. For x ∈ R n , x is the Euclidean norm, and

x 2 Q is defined by x Qx with 0 < Q = Q ∈ R n×n .

II. PROBLEM FORMULATION

Consider the saturating discrete-time LPV system represented by:

x p,k+1 = A(θ k )x p,k + B(θ k )sat(û k ), y k = Cx p,k , (1)
where x p,k ∈ R n is the state space vector, y k ∈ R p is the measured output, ûk is the updated control signal, and sat(û k ) is the standard decentralized saturation function, which is given by

sat(û k( ) ) = sign(û k( ) ) min(|û k( ) |, ū( ) ), (2)
with ū( ) > 0, ∈ I[1, m], the symmetric magnitude bound relative to the th update control input. The parameter-varying matrices

A(θ k ) ∈ R n×n and B(θ k ) ∈ R n×m belong to a polytope A, B (θ k ) = N i=1 θ k(i) A i , B i , ∀θ k ∈ Θ (3)
i.e., the vector of parameter-varying θ k ∈ R N lies in the unit simplex given by

Θ N i=1 θ k(i) = 1, θ k(i) ≥ 0, i ∈ I[1, N ] . (4)
As usual in emulation-based approach [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], we assume that the following parameter-dependent dynamic output-feedback (DOF) controller is given and stabilizes the system (1), as represented in Figure 1,

x c,k+1 = A c (θ k )x c,k + B c (θ k )ŷ k -E c (θ k )Ψ(û k ), u k = C c (θ k )x c,k + D c (θ k )ŷ k , (5)
where

x c,k ∈ R nc is the control state, Ψ(û k ) : R m → R m is the dead-zone non-linearity defined by Ψ(û k ) = ûk -sat(û k ),
and ŷk is the updated output signal. It is worth to say that the matrix E c (θ k ) only mitigates the windup effects of the saturation and, therefore, can be set to zero in this approach if it has not been considered a priori. The controller matrices verify the following assumption:

Assumption 1: The parameter-varying matrices A c (θ k ) ∈ R nc×nc , B c (θ k ) ∈ R nc×p , C c (θ k ) ∈ R m×nc , D c (θ k ) ∈ R m×p
, and E c (θ k ) ∈ R n×m of the DOF controller (5) have the following structure:

A c , B c (θ k ) = N i=1 N j=i (1 + ν ij )θ k(i) θ k(j) Acij 2 , Bcij 2 , C c , D c (θ k ) = N i=1 θ k(i) C ci , D ci , E c (θ k ) = N i=1 θ k(i) E ci ,
with θ k ∈ Θ and ν ij = 1 if i = j and ν ij = 0 otherwise. Note that, whenever the matrices of a dynamic controllers have a polytopic structure, they can always be cast in the structure given in Assumption 1.

To reduce the number of data exchange on sensor-tocontroller and controller-to-actuator channels, we are employing two dependent ETMs, as shown in Figure 1. In this case, the decision for output updates is made by the following rule:

ŷk := y k , f (y, ŷ, û) > 0, ŷk-1 , otherwise, (6)
where f (y, ŷ, û) is the triggering condition defined by

ŷk-1 -y k 2 Q∆y -y k 2 Qy -ûk 2 Q û > 0 , (7)
and the decision for the control updates is made by the following rule:

ûk := u k , g(u, û, ŷ) > 0, ûk-1 , otherwise, (8)
where g(u, û, ŷ) is the triggering condition defined by

ûk-1 -u k 2 Q∆u -u k 2 Qu -ŷk 2 Q ŷ > 0 (9)
The positive definite matrices Q ∆y , Q ŷ , Q y ∈ R p×p , and Q ∆u , Q û, Q u ∈ R m×m are variables to be designed, and their choice directly affects the ETM performance. The interest of the structure of ETM ( 6)-( 9) resides in the use of more information from the system behavior to decide to transmit or not the signals, expanding the event-triggering rules proposed in [START_REF] Souza | Event-triggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF]. Therefore, the problem we intend to solve can be summarized as follows.

Problem 1: Consider the DOF controller (5) that regionally stabilizes the saturating LPV system (1) in the absence of a communication network. Design the two ETMs ( 6) and ( 8), such that the closed-loop asymptotic stability is preserved and the number of data transmissions on the sensor-to-controller and controller-to-actuator channels is as low as possible.

An implicit objective in solving Problem 1 is to characterize an estimate of the basin of attraction of the origin of the closedloop system [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF].

III. PRELIMINARY RESULTS

Let us define the augmented state vector

x k = x p,k x c,k
∈ R n+nc , the output error e y,k = ŷk -y k , and the control error e u,k = ûk -u k . Thus, the closed-loop system can be written as

x k+1 = A(θ k )x k -B(θ k )Ψ(û k ) + E y (θ k )e y,k + E u (θ k )e u,k , u k = K(θ k )x k + D c (θ k )e y,k , y k = Cx k . ( 10
)
where, as in Assumption 1, the closed-loop matrices verify

A, E y (θ k ) = N i=1 N j=i (1 + ν ij )θ k(i) θ k(j) Aij 2 , Eyij 2 , B, E u , K (θ k ) = N i=1 θ k(i) B i , E ui , K i , with θ k ∈ Θ, ν ij = 1 if i = j and ν ij = 0 otherwise,

and are given by

A ij = A i + A j + (B i D cj + B j D ci )C B i C cj + B j C ci B cij C A cij , B i = B i E ci , E ui = B i 0 , E yij = B i D cj + B j D ci B cij , K i = D ci C C ci , and C = C 0 .
Note that if [START_REF] Peng | Event-triggered communication and H∞ control co-design for networked control systems[END_REF] activates an update at instant k, then from [START_REF] Abdelrahim | Co-design of output feedback laws and event-triggering conditions for linear systems[END_REF] we have that e y,k = ŷk -y k = y k -y k = 0. Otherwise, if y k is not updated at instant k, then from [START_REF] Peng | Event-triggered communication and H∞ control co-design for networked control systems[END_REF] we have that e y,k = ŷk -y k = ŷk-1 -y k . Thus, the following inequality is always satisfied.

e y,k 2
Q∆y ≤ y k 2 Qy + ûk 2 Q û . (11)
Similarly, if [START_REF] Golabi | Event-triggered constant reference tracking control for discrete-time LPV systems with application to a laboratory tank system[END_REF] activates an update at instant k, then from (9) we have that e u,k = ûk -u k = ûk-1 -u k = 0. Otherwise, and if u k is not updated at instant k, then from [START_REF] Golabi | Event-triggered constant reference tracking control for discrete-time LPV systems with application to a laboratory tank system[END_REF] we have that e u,k = ûk -u k = ûk-1 -u k . Consequently, we always have e u,k

2 Q∆u ≤ u k 2 Qu + ŷk 2 Q ŷ . (12)
To investigate the emulation-based design, we use the Lyapunov theory with the following candidate Lyapunov function

V (x k ) = x k P (θ k )x k , (13)
where

P (θ k ) = N i=1 θ k(i) P i , with 0 < P i = P i ∈ R (n+nc)×(n+nc) , i ∈ I[1, N ], and θ k ∈ Θ.
From the function V (x k ) defined in [START_REF] Zuo | Dynamic event-triggered and selftriggered control for saturated systems with anti-windup compensation[END_REF], one can define the unitary level set

L V = {x k ∈ R n+nc : V (x k ) ≤ 1, ∀θ k ∈ Θ}.
In Theorem 1 below, sufficient conditions are provided to ensure that the level set L V is an invariant and contractive set with respect to the trajectories of the closed-loop system [START_REF] Braga | Discretization and event triggered digital output feedback control of LPV systems[END_REF], and, therefore, constitutes an estimate of the basin of attraction for the system [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF]. Furthermore, by taking inspiration from [START_REF] Jungers | Gain-scheduled output control design for a class of discrete-time nonlinear systems with saturating actuators[END_REF], we can state the following lemma showing how to compute the set L V from matrices P i .

Lemma 1: The level set L V , associated with V (x k ) defined in [START_REF] Zuo | Dynamic event-triggered and selftriggered control for saturated systems with anti-windup compensation[END_REF], is computed as

L V = ∀θ k ∈Θ E(P (θ k )) = ∀i∈I[1,N ] E(P i ), (14)
where E(P i ) denotes the ellipsoidal sets

E(P i ) = x k ∈ R n+nc : x k P i x k ≤ 1, ∀i ∈ I[1, N ] . Proof: x ∈ L V is equivalent to ∀θ k ∈ Θ, V (x k ) < 1, which is equivalent to x ∈ ∀θ k ∈Θ E(P (θ k )). In addition ∀θ k ∈Θ E(P (θ k )) ⊂ ∀i∈I[1,N ] E(P i ). On the other hand, to prove that ∀i∈I[1,N ] E(P i ) ⊂ ∀θ k ∈Θ E(P (θ k )), consider x ∈ ∀i∈I[1,N ] E(P i ), then x P i x < 1 ∀i ∈ I[1, N ]. This im- plies that x ∈ E(P (θ k )), ∀θ k ∈ Θ or x ∈ ∀θ k ∈Θ E(P (θ k )).
The dead-zone function Ψ(û k ) enjoys a generalized sector condition as in Lemma 1.6 page 43 of [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF] or its equivalent formulation proposed in Equation (7.20) in Remark 7.4 in [START_REF] Tarbouriech | Stability And Stabilization Of Linear Systems With Saturating Actuators[END_REF], as summarized in the lemma below.

Lemma 2:

Consider a matrix G(θ k ) = N i=1 θ k(i) G i with G i ∈ R m×n+nc for I[1, N ], θ k ∈ Θ and define the set S(ū) {x k ∈ R n+nc : |G(θ k )x k | ≤ ū}. (15)
If x k ∈ S(ū), then for any diagonal positive definite matrix S ∈ R m×m , then the following inequality holds

Ψ(û k ) S G(θ k )x k -sat(û k ) ≤ 0. (16)

IV. MAIN RESULTS

The following theorem states a first step to solve Problem 1.

Theorem 1: Suppose that there exist symmetric positive definite matrices P i ∈ R (n+nc)×(n+nc) , Q ∆u , Q û, Qu ∈ R m×m and Q ∆y , Q ŷ , Qy ∈ R p×p , a positive definite diagonal matrix S ∈ R m×m , and matrices F ∈ R (n+nc)×(n+nc) and Z i ∈ R m×(n+nc) , with i ∈ I[1, N ], such that (17) (given at the top of the next page) and

P i Z i( ) 2S ( , ) -ū2 ( ) > 0, i ∈ I[1, N ] ∈ I[1, m], (18)
          -P r -F -F 1 2 A ij F 1 2 (P i + P j ) -C (Q y + Q ŷ )C -1 2 (K i + K j ) (Q u + Q û)(K i + K j ) 1 2 (E ui + E uj ) F -1 2 Q û (K i + K j ) Q ∆u -Q û 1 2 E yij F -1 2 (D ci + D cj ) (Q u + Q û)(K i + K j ) -1 2 (D ci + D cj ) Q û -Q ŷ C -1 2 (B i + B j ) F -1 2 (S(K i + K j ) -Z i -Z j ) -S Q ∆y -Q ŷ -1 2 (D ci + D cj ) (Q u + Q û)(D ci + D cj ) -1 2 S(D ci + D cj ) 2S           > 0, r, i ∈ I[1, N ], j ∈ I[i, N ]; (17)
are feasible. Then, the closed-loop system (10) subject to the ETMs ( 6) and ( 8) with matrices Q ∆u , Q u , Q û, Q ∆y , Q y , and Q ŷ is regionally asymptotically stable. Furthermore, the region L V , defined in [START_REF] Quan | Output-based event-triggered control for linear systems with actuator saturation[END_REF], is an estimate of the basin of attraction for the closed-loop system. Proof: Suppose first the feasibility of ( 18) and use the fact that (ū -2

( ) -S ( , ) )ū 2 ( ) (ū -2 ( ) -S ( , ) ) ≥ 0 or equivalently ū2 S ( , ) S ( , ) ≥ 2S ( , ) -ū-2 ( )
. By considering the change of variables Z(θ k ) = SG(θ k ), the satisfaction of ( 18) means that

P i S ( , ) G i( ) S 2 ( , ) ū2
( ) > 0, i ∈ I[1, N ], ∈ I[1, m]. Now,
multiply the left-hand side of the above inequality by θ k(i) and sum it up for i ∈ I[1, N ]. Then, pre-and post-multiply it by diag{I n , S -1 ( , ) }. Next, by applying the Schur complement and pre-and post-multiplying the resulting inequality by x k and x k , respectively, one obtains

x k P (θ k )x k -x k G(θ k ) ( ) (ū 2 ( ) ) -1 G(θ k ) ( ) x k ≥ 0, (19
) which, from ( 13), [START_REF] Quan | Output-based event-triggered control for linear systems with actuator saturation[END_REF], and [START_REF] Moreira | PI event-triggered control under saturating actuators[END_REF], ensures the positivity of V (x k ) and the inclusion L V ⊆ S(ū). Now consider that ( 17) is also verified, multiply its lefthand side by θ k+1(r) , θ k(i) , and θ k(j) , and sum it up for r, i ∈ I[1, N ] and j ∈ I[1, N ]. Next, replace Z(θ k ) by SG(θ k ) and rearrange the terms to obtain (20) (given at the top of the next page). Then, pre-and post-multiply the first term on left hand-side of (20) by the augmented vector

ζ k = x k+1 x k e u,k e y,k Ψ(û k )
, and replace

A(θ k )x k - B(θ k )Ψ(û k ) + E y (θ k )e y,k + E u (θ k
)e u,k by x k+1 according to [START_REF] Braga | Discretization and event triggered digital output feedback control of LPV systems[END_REF]. Finally, observe that x k+1 P (θ k+1 )x k+1 -

x k P (θ k )x k is equivalent to ∆V (x k ) = V (x k+1 ) -V (x k ).
Then, by using condition [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] in Lemma 2 and conditions ( 11) and [START_REF] Seuret | Event-triggered control with LQ optimality guarantees for saturated linear systems[END_REF], it follows:

∆V (x k ) ≤ ∆V (x k ) -2Ψ(û k ) S Ψ(û k ) - (K(θ k ) -G(θ k ))x k -D c (θ k )e y,k -e u,k < e u,k Q ∆u e u,k - u k Q u u k -ŷ k Q ŷ ŷk + e T y,k Q ∆y e y,k -y T k Q y y k -ûk Q û ûk ≤ 0.
Therefore, we conclude that the feasibility of ( 17) and [START_REF] Souza | Input-to-state stabilization of discrete-time LPV systems with bounded time-varying state delay and saturating actuators through a dynamic controller[END_REF] ensures the positivity of V (x k ) and the negativity of ∆V (x k ) for any x k ∈ L V . In other words, L V is an invariant and contractive set with respect to the closed-loop system (10), and, therefore, constitutes an estimate of the basin of attraction for the system. That concludes the proof.

V. OPTIMIZATION PROCEDURE

The objective here is to present an optimization procedure to find the parameters of the two triggering functions f (y, ŷ, û) and g(u, û, ŷ) that lead to a reduced number of events on both the sensor-to-controller and controller-to-actuator channels. An intuitive method to reduce the transmission activity is to minimize tr(Q ∆y , Q ∆u ), and to maximize tr(Q ŷ , Q y ) and tr(Q û, Q u ) [START_REF] Groff | Event-triggered control codesign for discrete-time systems subject to actuator saturation[END_REF], [START_REF] Souza | Event-triggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF]. However, based on experience, this may not be the most effective due to the objective function's generality and it appears better to impose restrictions on some matrices. In this sense, we propose the following optimization procedure [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], [START_REF] Souza | Input-to-state stabilization of discrete-time LPV systems with bounded time-varying state delay and saturating actuators through a dynamic controller[END_REF], tr(diag{Q ∆y , Q ∆u }) ≤ ηI p+m (21) with η ∈ R + , η > 1 small enough for well conditioning purposes.

min -tr(Q ∆u + Q u ) -tr(Q ∆y + Q y ) subject to
Another objective optimization consists of reducing sensor and control update rates considering a given region of admissible initial condition E(R), defined similarly as in [START_REF] Quan | Output-based event-triggered control for linear systems with actuator saturation[END_REF]. In this case, we have to ensure that E(R) ⊆ L V , which can be done by imposing P i < R, for i ∈ I[1, N ]. If we consider only the part referring to the system states x k , the inequality becomes I n 0 P i I n 0 < R n , where R n consists of the first n rows and n columns of R. Note that the greater the transmission saving, the smaller the attraction domain size, thus establishing a trade-off between attraction domain size and transmission saving.

VI. SIMULATIONS RESULTS

Consider the inverted pendulum system whose model is described by equation ( 1

          -P (θ k+1 ) 0 P (θ k ) -C (Q y + Q ŷ )C -K(θ k ) (Q u + Q û)K(θ k ) 0 -Q û K(θ k ) Q ∆u -Q û 0 -D c (θ k ) (Q u + Q û)K(θ k ) -D c (θ k ) Q û Q ∆y -Q ŷ -Q ŷ C -D c (θ k ) (Q u + Q û)D c (θ k ) 0 -S(K(θ k ) -G(θ k )) -S -SD c (θ k ) 2S           + He(φψ) > 0, (20)
where φ = F 0 0 0 0 and ψ = -

I n+nc A(θ k ) E u (θ k ) E y (θ k ) -B(θ k ) .
in the literature [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], [START_REF] Groff | Event-triggered control codesign for discrete-time systems subject to actuator saturation[END_REF], [START_REF] Wu | Event-triggered control for linear systems subject to actuator saturation[END_REF], but without taking into account possible parameter variations, i.e., with ϕ k = 0. This model can be cast as in [START_REF] Hespanha | A survey of recent results in networked control systems[END_REF] by computing the vertex matrices with the extreme values of ϕ k . We assume that ϕ k is available and that the following dynamic output feedback controller has been designed with the conditions proposed in [START_REF] Souza | Input-to-state stabilization of discrete-time LPV systems with bounded time-varying state delay and saturating actuators through a dynamic controller[END_REF] (by disregarding delay and perturbation): -0.0067 -0.0003 , C c2 = -0.0067 -0.0010 , D c1 = 7.7120 1.3814 , and D c2 = 7.7053 1.3193 , which ensures the regional asymptotic stability of the LPV system [START_REF] Souza | Event-triggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF] in the absence of a communication network.

A c11 = -0.
Problem 1 is addressed by considering both the LPV and linear time-invariant (LTI) cases. Then, by using the optimization procedure [START_REF] Groff | Event-triggered control codesign for discrete-time systems subject to actuator saturation[END_REF] and the parameter-varying ϕ k = | sin(0.01kπ)|. The average rate of sensor updates and of control updates were 12% and 18.67%, respectively. Also, by considering the initial conditions x p,0 = 0.0023 0.6639 and x c,0 = 0 0 , which is in the L V border, we simulated the closed-loop system response, and plotted the states, the control input, the inter-events on the sensor and the inter-events on the controller as shown in Figure 2. We can see the regional asymptotic stability of the system with an output update rate of 13.67% and a control update rate of 21.33%. Therefore, using an ETC strategy, we were able to significantly reduce the number of communications between process and controller, and controller and actuator.

Assuming time-invariant and know parameters (ϕ k = 0), the inverted pendulum system is also investigated in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], where both the static and dynamic feedback controllers under Fig. 2: The closed-loop response of system [START_REF] Souza | Event-triggered dynamic output-feedback controller for discrete-time LPV systems with constraints[END_REF] for x p,0 = 0.0023 0.6639 and x c,0 = 0 0 . an event-triggering mechanism are designed. For the dynamic controller, [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] obtained a sensor update rate of 6.7% in the simulation of the closed-loop response considering the initial condition x p,0 = 0.2 0.8 and x c,0 = 0 0 . The control update rate in this case is the worst possible, 100%, since [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] does not consider the presence of a communication network on the controller-to-actuator channel. The controller proposed by [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF]

Q û ≈ Q u ≈ 2 × 10 -10 .
Figure 3 depicts the projection (-), the cut (-) of L V , and E(R) (-). Note that E(R) is a subset of L V as required. Also, the projection of some divergent (-) and some convergent (-) trajectories stating from the points marked with (+) and ( * ), respectively, are illustrated. By considering the initial condition x p,0 = 0.2 0.8 , used in [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF], one obtains a closed-loop convergent trajectory (dashed black line) with a sensor update rate of 6.6% and a control update rate of 18.20%. Although [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF] has achieved a sensor update rate 1.52% smaller than ours, its control update rate is significantly higher than ours, in the order of 449.45%.

Fig. 3: The estimate of the basin of attraction L V , projection (-) and cut (-), and the region of admissible initial conditions E(R), (-), found by [START_REF] Ding | Event-triggered static/dynamic feedback control for discrete-time linear systems[END_REF].

VII. CONCLUSION

paper handled the problem of emulation-based dynamic output-feedback control for discrete-time linear parameter varying systems under actuator saturation. Two eventtriggering mechanisms were proposed to deal with the data transmission on the sensor-to-controller and controller-toactuator channels, while preserving the closed-loop stability. Sufficient conditions based on linear matrix inequalities (LMI) allowed to design the parameters of the two dependent eventtriggering rules and to characterize an estimate of the basin of attraction of the origin. A convex optimization scheme incorporating these conditions as constraints was proposed to reduce the number of updates on the sensor-to-controller and the controller-to-sensor channels. The effectiveness of the method has been tested in an illustrative example borrowed from the literature. Future work could be dedicated to expanding the results to the co-design problem, for example, by taking inspiration from preliminary results in [START_REF] Souza | Eventtriggered policy for dynamic output stabilization of discrete-time LPV systems under input constraints[END_REF].

Fig. 1 :

 1 Fig. 1: Emulation-based control loop.

  ) with matrices A = 1.0018 0.01 0.04ϕ k + 0.36 1.0018 , B = -0.001 0.025ϕ k -0.184 , (22) and C = I n , where |ϕ k | ≤ 1, and the symmetric saturation level ū = 1. This system has been extensively addressed

  with η = 19, we got the following ETM matri-Q ∆u = 19, and Q û ≈ Q u ≈ 5 × 10 -11 . The efficiency of the event-triggering strategy was assessed by simulating 1000 initial conditions generated with uniformly distributed random x 1 and x 2 in the intervals contained in L V : x 1 ∈ I[-0.04, 0.04] and x 2 ∈ I[-0.3, 0.3],

	ces Q ∆y =	18.3651 3.1401 3.1401 0.6349	, Q ŷ =	0.1255 -0.0239 -0.0239 0.0045	,
	Q y =	8.9332 -0.9033 -0.9033 0.0913	,

ACKNOWLEDGMENT

This work has been supported by the Brazilian Agencies CAPES under the project Print CAPES-UFSC "Automation 4.0" and CNPq (Grants 311208/2019-3 and 306927/2017-9); and by ANR, project HANDY 18-CE40-0010.


Henri Gouin
email: henri.gouin@univ-cezanne.fr

Variational Theory of Mixtures in Continuum Mechanics

Keywords: Variational methods in continuum mechanics, Mixing, Multiphase flows, Phase equilibria of fluid mixtures PACS: 46.15.Cc, 47.51.+a, 47.55.-t, 64.75Cd

. Only immiscible mixtures were investigated. We have chosen a different approach.

In this paper, we first write the equations of motion for each constituent of an inviscid miscible mixture of fluids without chemical reactions or diffusion. The theory is based on Hamilton's extended principle and regards the mixture as a collection of distinct continua.

The internal energy is assumed to be a function of densities, entropies and successive spatial gradients of each constituent. Our work leads to the equations of motion in an universal thermodynamic form in which interaction terms subject to constitutive laws, difficult to interpret physically, do not occur.

For an internal energy function of densities, entropies and spatial gradients, an equation describing the barycentric motion of the constituents is obtained.

The result is extended for dissipative mixtures and an equation of energy is obtained. A form of Clausius-Duhem's inequality which represents the second law of thermodynamics is deduced. In the particular case of compressible mixtures, the equations reproduce the classical results.

Far from critical conditions, the interfaces between different phases in a mixture of fluids are layers with strong gradients of density and entropy. The surface tension of such interfaces is interpreted.

Introduction

The knowledge of equations describing processes occuring in fluid mixtures in one or several phases is scientifically and industrially very important [START_REF] Barrere | Equations fondamentales de l'aérothermochimie[END_REF]; [START_REF] Ishii | Thermo-fluid dynamic theory of two-phase flow[END_REF]). Many papers concerning theoretical and experimental works for mixtures have been produced in continuum mechanics and physical chemistry. References in [START_REF] Bedford | Recent advances. Theories of immiscible and structured mixtures[END_REF]] cover most of the works in continuum theory. The book by [START_REF] Rowlinson | On the viscosities of liquid mixtures[END_REF] gives the statistical methods. [START_REF] Truesdell | Sulle basi della termomeccanica[END_REF][START_REF] Truesdell | The rational mechanics of material, Continuum Mechanics II, Gordon and Breach[END_REF] first derived equations of balance and motion through the use of a continuum theory of mixtures. The mixture is then considered as a distribution of different continuous media in the same physical space, at time t. Each constituent of the mixture is identified with its reference space. The thermal and mechanical equations of balance are introduced for each constituent. They keep the usual form for a fluid description but they must include numerous terms of interaction between the different constituents. The study of the average motion, the motion of the barycentre of constituents with coefficients equal to their respective densities, are deduced from the sum of momenta, energy and entropy of each constituent. The terms including interaction between different constituents of the mixtures are difficult to interpret physically. They require constitutive postulates that are difficult to interpret experimentally.

Several assumptions are open to doubt. How must we choose an entropy? (Mixture entropy or entropy for each constituent?) What is the form of the second law of thermodynamics? Is it symbolized with one or several Clausius-Duhem's inequalities? Must we consider a temperature of the mixture or a temperature for each constituent? [START_REF] Ishii | Thermo-fluid dynamic theory of two-phase flow[END_REF]; [START_REF] Williams | On the theory of mixtures[END_REF]; [START_REF] Atkin | Continuum theories of mixtures: basic theory and historical development[END_REF]; Sampio [START_REF] Rowlinson | On the viscosities of liquid mixtures[END_REF]; [START_REF] Bowen | A theory of constrained mixtures with multiple temperatures[END_REF]; [START_REF] Nunziato | On ideal multiphase mixtures with chemical reactions and diffusion[END_REF]; [START_REF] Drew | Mathematical modelling of two-phase flow[END_REF] and many others argue about the various points of view. In several papers [START_REF] Bedford | A variational theory of immiscible mixtures[END_REF]; [B & D, 1979]; [B & D, 1980] presented a variational theory for immiscible mixtures (a continuum theory for mixtures whose constituents remain physically separated, such as a mixture of immiscible liquids or a fluid containing a distribution of particles, droplets, or bubbles). That the constituents remain physically separated has several implications for the theory. For instance, the authors use a variational principle in which a variation is added to the motion of only one constituent. However it is necessary to add to the variations of kinetic and potential energies a virtual work due to the forces not represented by a potential. Such forces are difficult to take into account in a variational principle deduced from the one by Hamilton. It is not possible to use such a method in a more general model. In this paper a new mathematical systematic method that leads to the equations of motion and energy for miscible fluid mixtures without chemical reac-tion and diffusion is propounded. The presented work is essentially different from the one by Bedford and Drumheller. The mixture is represented by several distinct continuous media that occupy the same physical space at time t. The novelty consists in the fact that the used variational principle is applied to a lagrangian representation associated with a reference space for each component. The proposed method uses the knowledge of the internal energy of the mixture at every point of the physical space. This is not the case in [B & D]. For classical fluids, the method corresponds to Hamilton's principle in which one makes a variation of the reference position of the particle. It was proposed by [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF]. The inferred equations are written in a different form that facilitates the study of first integrals for conservative motions ( [START_REF] Casal | Principes variationnels en fluide compressible et en magnétodynamique des fluides[END_REF]; [Gouin, 1981]; [Casal & Gouin, 1985b, 1988a[START_REF] Casal | Invariance properties of inviscid fluids of grade n[END_REF]). For mixtures, the principle appears far more convenient than the method of only taking the variation of the average motion into consideration: it separately tests each component and leads to a thermodynamic form of the equation of motion for each of them. The internal energy is assumed to be a function of different densities or concentrations of the mixture. Because of the form of the principle we use, one must consider an entropy for each component in the mixture. The total entropy of the system is the sum of the partial entropies. To consider areas where strong gradients of density occur -for example shocks or capillary layers -the internal energy is chosen as a function of the successive derivatives of densities and entropies. Two cases of conservative motions for mixtures are considered: isentropic motion and isothermal motion. These are only limiting mathematical cases for testing the method. The physical motion is intermediate, between these extremes. An important medium is the one of thermocapillary mixtures. It corresponds to an energy taking into account the gradients of densities and entropies. Then, one obtains an equation of the barycentric motion of constituents. For a compressible mixture, the equations of motion involve a hydrostatic stress. In the same way as for thermocapillary fluids, an additional term that has the physical dimension of a heat flux may be added to the energy equation ( [START_REF] Eglit | A generalization of the model of an ideal compressible fluid[END_REF]; [START_REF] Berdichevskii | Construction of models of continuous media by means of the variational principle[END_REF]; [Casal & Gouin, 1985a]; [START_REF] Dunn | On the thermodynamics of interstitial working[END_REF]; [START_REF] Emschwiller | Modelisation of fluid-fluid interfaces with material properties[END_REF]; [START_REF] Seppecher | Etude d'une modélisation des zones capillaires fluides[END_REF]). A generalization of the Clausius-Duhem inequality is obtained for non-conservative mixtures. The inequality is deduced from the heat conduction inequality by introducing an irreversible stress tensor associated with a general dissipation function. The theory fits with the second law of thermodynamics. From these results one generalizes equilibrium conditions obtained through interfaces in compressible fluid mixtures [START_REF] Rocard | Thermodynamique[END_REF]. An interpretation of capillarity stresses is deduced (rule of Antonov) [START_REF] Emschwiller | Modelisation of fluid-fluid interfaces with material properties[END_REF], and generalization of results by [START_REF] Vignes-Adler | A micromechanical derivation of the differential equations of interfacial statics[END_REF] may be investigated.

2 The motion of a continuous medium

Motion of a fluid with only one constituent

Recall that the motion of a continuous medium can be represented by a surjective differentiable mapping:

z → X = M(z),
where z = (t, x) belongs to W , an open set in the time-space occupied by the fluid between time t 1 and time t 2 . The position of a particle in a reference space D o is denoted by X; its position at time t in D t is denoted by x [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF]. Hamilton's principle -variational form of the principle of virtual powers -allows us to investigate the equation of motion.

The variations of motion of particles are deduced from:

X = Ψ(x, t; β)
for which β is a parameter defined in a neighborhood of zero; it is associated with a family of virtual motions of the fluid. The real motion corresponds to β = 0. Virtual material displacements associated with any variation of real motion can be written [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF]:

δX = ∂Ψ ∂β β=0 . ( 1
)
Such a variation is dual with Serrin's [S, 1959]. This has been studied in the case of compressible perfect fluids [START_REF] Gouin | Etude géométrique et variationnelle des milieux continus[END_REF] and corresponds to the natural variation of the motion in a lagrangian representation.

The lagrangian of the fluid is:

L = ρ( 1 2 V * V -α -Ω),
where V denotes the velocity vector of the particles , Ω the extraneous force potential defined in D t , α the specific internal energy, ρ the density and * the transposition in D t .

Between time t 1 and time t 2 , the hamiltonian action is written:

a = t 2 t 1 Dt L dvdt. (2)
The use of virtual displacements (1) provides the equations of motion of compressible perfect fluids in a thermodynamic form. For fluids of grade n, the calculus is swift. It generalizes the equation (29,8) of fluids in [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF] that may be written in a form that is independent of their complexity [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF]:

Γ = θ grad s -grad(h + Ω),
where Γ denotes the acceleration, s the entropy, θ a temperature and h a specific enthalpy.

Motion of a fluid mixture

For the sake of clarity, we study a mixture of two fluids. The method can be immediately extended to any number of constituents. No assumption has to be made about their composition or their miscibility.

The motion of a two-fluid continuum can be represented using two surjective differentiable mappings:

z → X 1 = M 1 (z) and z → X 2 = M 2 (z), (3)
(Subscripts 1 and 2 are associated with each constituent). X 1 and X 2 denote the positions of each constituent in reference spaces D o1 and D o2 . The variations of motion of particles are deduced from:

X 1 = Ψ 1 (x, t; β 1 ) and X 2 = Ψ 2 (x, t; β 2 ). ( 4
)
This is a generalization of the one in paragraph 2.1 where β 1 and β 2 are defined in a neighborhood of zero; they are associated with a two-parameter family of virtual motions of the mixture. The real motion corresponds to β 1 = 0 and β 2 = 0; the associated virtual displacements can be written:

δ 1 X 1 = ∂Ψ 1 ∂β 1 β 1 =0 and δ 2 X 2 = ∂Ψ 2 ∂β 2 β 2 =0 . (5)
They generalize relation (1) given for a fluid with one constituent. The principle tests separately each constituent of the mixture and so is very efficient.

Remark. -It follows from the classical representation of a two-constituent mixture, that formula (4) can be written

x = ϕ 1 (X 1 , t; β 1 ) and x = ϕ 2 (X 2 , t; β 2 ). ( 6
)
A variation of the motion like Serrin's (p. 145, [S, 1959]) cannot be extended to any two-parameter family of virtual displacements: as a matter of fact, any field of virtual displacements associated with ϕ 1 must be such that δx = ∂ϕ 1 ∂β 1 (X 1 , t; 0) and there is no reason for being identifiable with a vector field written in the form δx = ∂ϕ 2 ∂β 2 (X 2 , t; 0).

In the case of compressible fluids, writing (1) for the virtual displacement associated with a one-parameter family of virtual motion of the fluid is deduced from the variation defined by Serrin by the way of vector spaces isomorphism [START_REF] Gouin | Noether theorem in fluid mechanics[END_REF]. For mixtures, by using (5), a virtual displacement of miscible fluid mixtures can be considered.

The lagrangian of the mixture is:

L = 1 2 ρ 1 V * 1 V 1 + 1 2 ρ 2 V * 2 V 2 -ε -ρ 1 Ω 1 -ρ 2 Ω 2 .
Here, V 1 et V 2 denote the velocity vectors of each constituent, ρ 1 and ρ 2 are the densities, Ω 1 and Ω 2 are the extraneous force potentials depending only on z = (t, x) and ε is the internal energy per unit volume.

The expansion of the lagrangian is general. In fact dissipative phenomena imply that V 1 is almost equal to V 2 . Because of the interaction of the constituents, ε does not always divide into energies related to each constituent of the mixture, as it does in the case of simple mixtures of fluids [START_REF] Müller | Theory of mixtures of fluids[END_REF].

More generally, the lagrangian may be written

L = 1 2 ρ 1 V * 1 V 1 + 1 2 ρ 2 V * 2 V 2 -η
where η denotes the volumetric potential energy of the mixture. In our particular case, η is the sum of internal energy and extraneous force potentials.

In fact, the following calculus would be similar (but even simpler). We use the first form of the lagrangian in order to obtain classical expressions used for fluids.

Hamilton's action between time t 1 and time t 2 can be written as in (2). The mixture is assumed not to be chemically reacting. The conservation of masses requires that:

ρ i det F i = ρ oi (X i ) (7 1 )
for the densities of each constituent, with subscript i belonging to {1, 2}. At fixed t, the jacobian mapping associated with each M i is denoted by F i ; ρ oi is the reference specific mass in D oi .

In differentiable cases, equation (7 1 ) is equivalent to:

∂ρ i ∂t + div ρ i V i = 0. (7 2 )
The volumetric internal energy ε is given by the thermodynamical behavior of the mixture and will appear in the equations of motion. Each constituent has a specific mass; similarly, two specific entropies s 1 and s 2 are supposed to be associated with constituents 1 and 2. For a specific internal energy depending on derivatives of gradients of densities and entropies up to the n-th order:

ε = ε(s 1 , grad s 1 , ..., (grad) n-1 s 1 ; s 2 , grad s 2 , ..., (grad) n-1 s 2 ; ρ 1 , grad ρ 1 , ..., (grad) n-1 ρ 1 ; ρ 2 , grad ρ 2 , ..., (grad) n-1 ρ 2 ), (8)
(we say, the mixture is of grade n), the calculus is similar to than in ( [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF]; [Casal & Gouin, 1988a]). The orders of gradients of ρ 1 , ρ 2 , s 1 , s 2 can be chosen differently, but the exposition would be less clear. Let us note that the volumetric potential energy can be written:

η = η(s 1 , grad s 1 , ..., (grad) n-1 s 1 ; s 2 , grad s 2 , ..., (grad) n-1 s 2 ; ρ 1 , grad ρ 1 , ..., (grad) n-1 ρ 1 ; ρ 2 , grad ρ 2 , ..., (grad) n-1 ρ 2 , z)
with z = (t, x).

Examples of motions of a fluid mixture

Let us consider two examples of Hamilton's principle. Obviously, they are two limit cases deduced from a variational principle and incompatible with any dissipative process.

First case: Isentropic motions.

We assume that the specific entropy of each particle is constant:

ds i dt = 0, or ∂s i ∂t + ∂s i ∂x V i = 0.
(Lie's derivative with respect to the velocity field of constituent i is equal to zero. The average motion of the mixture does not have Lie's derivative, so henceforth, we can expect each constituent of the mixture to possess entropy).

The equation

s i = s oi (X i ) (9)
defines an isentropic motion. Motion equations are deduced from Hamilton's principle. The two variations of hamiltonian action are:

δ i a = a ′ (β i ) β i =0 .
For i ∈ {1, 2}, we deduce:

δ i a = t 2 t 1 Dt 1 2 V * i V i -ε ,ρ i -Ω i δ i ρ i + ρ i V * i δ i V i -ε ,s i δ i s i - ε ,s i,γ δ i s i,γ + ... + ε ,s i,γ 1 ,...,γ n-1 δ i s i,γ 1 ,...,γ n-1 - ε ,ρ i,γ δ i ρ i,γ + ... + ε ,ρ i,γ 1 ,...,γ n-1 δ i ρ i,γ 1 ,...,γ n-1 dvdt.
Subscript γ corresponds to spatial derivatives associated with gradient terms; as usually, summation is made on repeated subscripts γ.

The quantity

h i = ε ,ρ i + n-1 p=1 (-1) p (ε ,ρ i,γ 1 ,...,γ p ) ,γ 1 ,...,γ p (10 1 )
defines the specific enthalpy of the constituent i of the mixture, and

θ i = 1 ρ i   ε ,s i + n-1 p=1 (-1) p (ε ,s i,γ 1 ,...,γ p ) ,γ 1 ,...,γ p   (10 2 )
defines the temperature of the constituent i of the mixture.

The motion of constituent i of the mixture is expressed by the equation

Γ i = θ i grad s i -grad(h i + Ω i ). ( 11
)
Appendix 1 gives details of the calculation.

The fact that an entropy is introduced for each component implies the existence of a temperature for each fluid constituent. This is in accordance with [START_REF] Green | A dynamic theory of interacting continua[END_REF]. In this first limit case, the variational principle does not lead to the equality of temperatures. The behavior of the two components corresponds to their thermodynamical independence (no exchange of heat between particles). The two components are only bounded by effects of compressibility. Such a model can be used in practice only if one includes the fact that some heat conduction exists and adjusts the values of the two temperatures. So, one verifies that equations obtained with the variational principle must be completed.

Second case: Only the whole entropy of the mixture is conserved.

We have:

Dt (ρ 1 s 1 + ρ 2 s 2 ) dv = K, ( 12 1 )
where K is constant (independent of the time).

Let us use the method applied in [Casal & Gouin, 1988b] for compressible fluids. The variation of the entropy of each component is assumed to be the sum of a variation associated with the virtual motion and another one, δ oi s i related to the particle:

δ i s i = ∂s oi ∂X i δ i X i + δ oi s i ,
where δ oi s i and δ i X i are independent and δ oi must verify, with respect to component i of the mixture:

δ oi t 2 t 1 Dt (ρ 1 s 1 + ρ 2 s 2 )dv = 0. ( 12
2 )
Classical methods of variational calculus provide the variation of the hamiltonian action: the variations δ i X i yield a motion equation of the form of (11). Relation (12 2 ) implies: there exists a constant Lagrange multiplier θ o , constant in space and in time, such that:

δ oi a = t 2 t 1 Dt -ε ,s i δ oi s i -ε ,s i,γ δ oi s i,γ -... -ε ,s i,γ 1 ,...,γ n-1 δ oi s i,γ 1 ,...,γ n-1 + ρ i θ o δ oi s i dvdt.
Let us point out that adiabatic condition (12 1 ) implies that even if the mixture is moving, θ o remains constant in space and time since θ o is a Lagrange multiplier associated with an integral constraint.

Considering variations δ oi s i vanishing on the boundary of D t ; an integration by parts yields:

δ oi a = t 2 t 1 Dt ρ i (θ o -θ i ) δ oi s i dvdt.
The principle: "for any δ oi s i vanishing on the boundary of D t , δ oi a = 0 " leads to the equation for the temperature:

θ i = θ o , (13)
i.e., all the constituents have the same temperature θ o , constant in all the flow: the flow is isothermal.

This variational principle can be compared with the static one expressing that the equilibrium of a compressible fluid kept in a fixed adiabatic reservoir without exchange of mass and entropy with outside is obtained by writing the uniformity of temperature and pressure. It can be interpreted as a limit case of a mixture with an infinite heat conductivity. Equation ( 11) is replaced by

Γ i + grad(g i + Ω i ) = 0, ( 14
)
where g i = h iθ i s i is the free specific enthalpy (or chemical potential) of constituent i.

When the fluid is compressible, Eqs ( 11) or ( 14) yield first integrals and theorems of circulation [START_REF] Serrin | Mathematical principles of classical fluid mechanics[END_REF]; [START_REF] Casal | Principes variationnels en fluide compressible et en magnétodynamique des fluides[END_REF]; [Gouin, 1981]; [Casal & Gouin, 1985b[START_REF] Casal | Invariance properties of inviscid fluids of grade n[END_REF]]. These properties can easily be extended to each constituent of an inviscid mixture. Each constituent of the mixture in an isothermal equilibrium state satisfies

G i = C i ( 15
)
where G i = g i + Ω i and C i is a constant.

An interface in a two-phase mixture is generally modelled by a surface without thickness. Far from critical conditions, this layer is of molecular size; density and entropy gradients are very large. A continuous model deals with such areas by using energy in the form (8) extending forms given in ( [START_REF] Rocard | Thermodynamique[END_REF]; [START_REF] Cahn | Free energy of a non-uniform system[END_REF]) for compressible fluids.

Equation ( 15) is satisfied for each constituent of a mixture in an isothermal equilibrium state. The fact that C i values are the same in each bulk is also a consequence of Eq.( 15). It follows that with extraneous forces being neglected in a two-phase mixture, if the two different phases are in isothermal equilibrium, then specific free enthalpies (or chemical potentials) of each bulk are equal. This result generalizes the classical law to mixtures of grade n: the different constituents keep their own chemical potential through a change of phases [START_REF] Rocard | Thermodynamique[END_REF].

Thermocapillary mixtures

The previous equations yield the equations of motion for each of the two fluids constituting the mixture. Usually, the chemical potentials h i and g i are rarely considered except for equilibrium; Eqs. ( 11) and ( 14) make use of them in the study of the motion.

In order to compare results given by the variational technics and the usual approach of mixture motions, we can try to study the mean motion and diffusive motions for two components. But how do we define these motions? In this paper, we consider the mean motion using an approach which is independent of the number of constituents.

For the mean motion, ρ = i ρ i and ρ Γ = i ρ i Γ i (using Γ = i Γ i has no physical meaning).

On the contrary, ρ Γ has the physical dimension of a volumetric force and is an additive quantity in the same way as body forces or the stress tensor divergences. So, in this first article, we have not attempted to investigate the difficult question of the relative motion associated with " Γ 1 -Γ 2 " and hence to see how diffusion behaves in miscible mixtures.

Conservative motions for mixtures of thermocapillary fluids

In the case n = 2, we write the equations of motion by using the stress tensor. The mixture is said to be a thermocapillary mixture if the internal energy ε is written as:

ε = ε(s 1 , grad s 1 , s 2 , grad s 2 , ρ 1 , grad ρ 1 , ρ 2 , grad ρ 2 ). ( 16
)
The expressions

ρ = 2 i=1 ρ i and ρ Γ = 2 i=1 ρ i Γ i
introduce the total density and acceleration of the mixture. With denoting by α the specific energy of the mixture, we have ρα = ε, and,

dα = 2 i=1 P i ρρ i dρ i + ρ i ρ Θ i ds i + 1 ρ Φ iγ dρ i,γ + 1 ρ Ψ iγ ds i,γ , with P i = ρρ i α ,ρ i , Θ i = 1 ρ i ε ,s i , Φ iγ = ε ,ρ i,γ , Ψ iγ = ε ,s i,γ .
Four new vectors Φ i and Ψ i (i ∈ {1, 2}) are introduced by the theory; it is an extension of the calculus written in [Casal & Gouin, 1988a].

For a compressible fluid (with only one constituent), P and Θ are respectively the pressure and the Kelvin temperature. The fluid is isotropic, so the internal energy is of the form ε = ε(ρ i , s i , β ij , χ ij , γ ij ) with β ij = grad ρ i . grad ρ j , χ ij = grad ρ i . grad s j , γ ij = grad s i . grad s j (i ∈ {1, 2}). It follows that:

Φ i = 2 j=1 C ij grad ρ j + E ij grad s j , Ψ i = 2 j=1 D ij grad s j + E ij grad ρ j , with C ij = (1 + δ ij )ε ,β ij , D ij = (1 + δ ij )ε ,γ ij , E ij = ε ,χ ij ,
where δ ij is the Kronecker symbol. The simplest model is for C ij , D ij and E ij constant.

Assuming that Ω 1 = Ω 2 = Ω (this is the case for gravity forces), we obtain a general formulation of the equation of motion for thermocapillary mixtures:

ρ Γ = div σ -ρ grad Ω (17) or ρ Γ γ = σ νγ ,ν -ρ Ω ,γ
. The sum of σ 1 and σ 2 obtained by successive integrations by parts is conventionally called the generalization of the stress tensor. This is simply for the convenience of writing (in fact, the sum of stresses associated with different media is a non-sense). Let

σ = 2 i=1 σ i , where σ iνγ = -p i δ νγ -Φ iν ρ i,γ -Ψ iν s i,γ , (18)
with

p i = P i -ρ i div Φ i .
The tensorial natures of entropy and density are different; that is the reason why only div Φ appears in the pressure term. The calculus for this case appears in appendix 2. In appendix 4, we deduce Antonov's rule related to the surface tensions of an interface between two immiscible fluids from the expression of the stress tensor.

For mixtures with internal capillarity [START_REF] Emschwiller | Modelisation of fluid-fluid interfaces with material properties[END_REF]; [START_REF] Seppecher | Etude d'une modélisation des zones capillaires fluides[END_REF]), only Φ i , i ∈ {1, 2}, are not equal to zero. In the case of classical compressible mixtures, terms of gradients are null and Φ i and Ψ i are zero; σ νγ is written:

-Pδ νγ with P = 2 i=1 P i defining the pressure of the mixture. Let us remark that P i with i ∈ {1, 2} does not correspond to the classic notion of partial pressures in ideal mixtures. This result fits with I. Müller's paper ([1968], p. 37): according to Müller terminology, if ε = ρα = ρ 1 α 1 + ρ 2 α 2 with α i = α i (ρ i , s i ), such an internal energy corresponds to a simple mixture of fluids. If Π i = ρ 2 i α i,ρ i (ρ i , s i ), then

P 1 = Π 1 + ρ 1 ρ 2 ρ (α 1 -α 2 )
and

P 2 = Π 2 + ρ 1 ρ 2 ρ (α 2 -α 1 );
with assuming that for every point in the mixture all the constituents have the same temperature θ i = θ 0 , then, P = Π 1 + Π 2 and Π i represents the partial pressure of constituent i [START_REF] Bruhat | Thermodynamique[END_REF]].

Example of binary system with complete miscibility: liquid-vapor equilibrium

At a given temperature, the numerous constitutive equations represent P as a function of densities of the constituents ( [START_REF] Soave | Equilibrium constants from a modified Redlich-Kwong equation of state[END_REF]; [START_REF] Simonet | A modified Redlich-Kwong equation of state for accurately representing pure components data[END_REF]; [START_REF] Peng | A new two-constant equation of state[END_REF]). These equations, very useful in petroleum industry, derive from subtle calculus on mixtures of Van der Waals fluids. Taking into account Eq. ( 15), they allow the study of a liquid bulk and a vapour bulk for two fluids that can be completely mixed.

A plane interface lies between two phases of a two-fluid mixture in isothermal equilibrium. This is an unidimensional problem with reference to the direction perpendicular to the interface. Body forces being neglected, Eq. ( 17) can be written: dσ dz = 0.

Vectors Φ i and Ψ i are zero in the bulks in which P l = P v = P 0 (l and v refer to the bulks with the same pressure P 0 ). For equilibrium, eight variables ρ il , ρ iv , s il , s iv with i ∈ {1, 2} determine the properties of the two bulks. They are subject to six relations:

g 1l = g 1v , g 2l = g 2v , θ 1l = θ 1v = θ 2l = θ 2v ,
and

P l = P v .
This fits with Gibbs law of phases that gives the system as "divariant" [START_REF] Rocard | Thermodynamique[END_REF]. Two additional conditions, θ il = θ iv = θ 0 (i ∈ {1, 2}) and P l = P v = P 0 , set an equilibrium of the mixture.

3.2 Equation of motion and equation of energy for a thermocapillary mixture

Conservative motions

Some properties of mixtures are directly deduced from the equation ( 17) of motion.

Let us define

M i = Γ i -θ i grad s i + grad(h i + Ω i ), B i = dρ i dt + ρ i div V i , S i = ρ i θ i ds i dt , E i = ∂e i ∂t + div [(e i -σ i )V i ] -div U i -ρ i ∂Ω i ∂t , with e i = 1 2 ρ i V 2 i + ε i + ρ i Ω i , ε i = ρ i ε ρ and U i = dρ i dt Φ i + ds i dt Ψ i dρ i dt = ∂ρ i ∂t + dρ i dx V i and ds i dt = ∂s i ∂t + ∂s i ∂x V i .
Theorem 1 For a thermocapillary internal energy and for any motion of the mixture, the relation

2 i=1 E i -ρ i M * i Vi - 1 2 V 2 i + h i + Ω i B i -S i = 0 (19)
is identically satisfied.

(Proof in appendix 3). Limit cases (the isentropic and isothermal motion of Sec. 2) yield the two corollaries:

Corollary 2 -Any isentropic motion of an thermocapillary mixture satisfies the equation of balance of energy:

2 i=1 ∂e i ∂t + div[(e i -σ i )V i ] -div U i -ρ i ∂Ω i ∂t = 0. ( 20
)
This results from the simultaneity of equations S i = 0, B i = 0 and M i = 0.

Corollary 3 -For any conservative motion of a thermocapillary perfectly heat conducting mixture, Eq. ( 13) is equivalent to:

2 i=1 ∂f i ∂t + div[(f i -σ i )V i ] -div U i -ρ i ∂Ω i ∂t = 0 ( 21
)
with f i = 1 2 ρ i V 2 i + ϕ i + ρ i Ω i and ϕ i = ε iρ i θ i s i corresponding to the specific free energy of constituent i at temperature θ 0 .

Equation ( 21) is called the equation of balance of free energy.

Dissipative motions

If we try to introduce dissipative phenomena, while keeping the form of the equations, we may only add an irreversible stress tensor σ Ii for each constituent. So, for each fluid, (i ∈ {1, 2}), supplementary energy flux vectors div(σ Ii V i ) and dissipative forces tr(σ Ii ∆ i ) have additive properties (∆ i is the deformation tensor of constituent i) and are introduced and summed. The only assumption for functions of dissipation, 2 i=1 tr(σ Ii ∆ i ) ≧ 0, leads to a Clausius-Duhem inequality. So, the model fits with the second law of thermodynamics independently of any constitutive behavior for σ I1 and σ I2 [START_REF] Chapman | The Mathematical Theory of Non-Uniform Gases[END_REF]).

Equation of motion (17) can be written: ρ Γ = div(σ + σ I )ρ grad Ω (as in Eq.( 17), we are led to formally writing σ I = σ I1 + σ I2 ).

Afterwards, the heat flux vector will be denoted by q and heat supply by r. It follows, Corollary 4 -For any motion of a thermocapillary mixture, the equation of energy

2 i=1 ∂e i ∂t + div[(e i -σ i -σ Ii )V i ] -div U i -ρ i ∂Ω i ∂t + div q -r = 0 (22)
is equivalent to the equation of entropy:

2 i=1 ρ i θ i ds i dt -tr(σ Ii ∆ i ) + div q -r = 0.
Assuming that the work of dissipative forces satisfies 2 i=1 tr(σ Ii ∆ i ) ≧ 0, we obtain a generalized form of Planck's inequality:

2 i=1 ρ i θ i ds i dt + div q -r ≧ 0.
At any point of the fluid, each constituent is supposed to have the same temperature (not necessarily uniform). This requires that the time for relaxation of molecules is short in comparison with the time characterizing the flow. Then, θ = θ i .

By use of Fourier's law in the form:

q. grad θ ≦ 0, we deduce a generalization of Clausius-Duhem's inequality:

2 i=1 ρ i ds i dt + div q θ - r θ ≧ 0.
This Clausius-Duhem's inequality is a form of the second law of thermodynamics. Let us note that some authors separate q and r in q i and r i ; this does not seem necessary.

Appendix 1

Equations of motion for mixtures of perfect fluids

Isentropic motion

Let us denote by ξ i the variation δ i X i , we deduce from the two relations in lagrangian variables [START_REF] Gouin | Thermodynamic form of the equation of motion for perfect fluids of grade n[END_REF]:

δ i V i = -F i ∂ξ i ∂t , δ i ρ i = ρ i div oi ξ i + ρ i ρ oi ∂ρ oi ∂X i ξ i (23)
and

δ i s i = ∂s i ∂X i δ i X i ,
where ∂ ∂X i is the linear form associated with the gradient and div oi is the divergence operator on D oi . Variations of entropies satisfy relation (9). With assuming that terms on the edge of W are zero, we obtain:

δ i a = t 2 t 1 Dt 1 2 V * i V i -ε ,ρ i -Ω i δ i ρ i + ρ i V * i δ i V i -ε ,s i δ i s i -   n-1 p=1 (-1) p (ε ,ρ i, γ 1 ,...,γ p ) ) ,γ 1 ,...,γ p   δ i ρ i -   n-1 p=1 (-1) p (ε ,s i, γ 1 ,...,γ p ) ) ,γ 1 ,...,γ p   δ i s i    dvdt, or δ i a = t 2 t 1 D oi ρ oi R i div oi ξ i + R i ρ oi ∂ρ oi ∂X i ξ i -V * i F i ∂ξ i ∂t -θ i ∂s i ∂X i ξ i dv oi dt with R i denotes 1 2 V * i V i -h i -Ω i . But, div oi (ρ oi R i ξ i ) = ρ oi R i div oi ξ i + R i ∂ρ oi ∂X i ξ i + ρ oi ∂R i ∂X i ξ i .
So,

δ i a = t 2 t 1 D oi ρ oi - ∂R i ∂X i + ∂ ∂t (V * i F i ) -θ i ∂s i ∂X i ξ i dv oi dt,
(terms given by integration on the edge of D oi are zero).

Finally, for each constituent,

∂ ∂t (V * i F i ) = ∂R i ∂X i + θ i ∂s i ∂X i ,
and by using of the relation

∂ ∂t (V * i F i ) = Γ * i F i + V * i ∂V i ∂X i ,
we obtain

Γ * i F i + ∂ ∂X i ( 1 2 V * i V i ) = ∂R i ∂X i + θ i ∂s i ∂X i or Γ * i = θ i ∂s i ∂x - ∂ ∂x (h i + Ω i ).
This leads to the vectorial form:

Γ i = θ i grad s i -grad(hi + Ω i ) (11) Appendix 2
Equation of motion of thermocapillary mixtures

Equation (11) implies 2 i=1 ρ i Γ iγ + ρ i Ω i,γ = 2 i=1 ε ,s i s i,γ -(ε ,s i,ν ) ,ν s i,γ -ρ i (ε ,ρ i ) ,γ + ρ i [(ε ,ρ i,ν ) ,ν ] ,γ .
By noting that

ε ,γ = 2 i=1 ε ,s i s i,γ + (ε ,s i,ν ) ,ν s i,νγ + ε ,ρ i ρ i,γ + ε ,ρ i,ν ρ i,νγ , we obtain 2 i=1 ρ i Γ iγ + ρ i Ω i,γ = ε ,γ - 2 i=1 ε ,s i,ν s i,νγ + ε ,ρ i ρ i,γ + ε ,ρ i,ν ρ i,νγ + (ε ,s i,ν ) ,ν s i,γ + ρ i (ε ,ρ i ) ,γ -ρ i [(ε ,ρ i,ν ) ,ν ] ,γ , or 2 i=1 ρ i Γ iγ + ρ i Ω i,γ = 2 i=1 ε -ρ i ε ,ρ i + ρ i (ε ,ρ i,ν ) ,ν ,γ -Φ iν ρ i, γ + Ψ iν s i, γ ,ν .
In the case Ω 1 = Ω 2 = Ω, this agrees with Eq. ( 17) when ρ α = ε.

Appendix 3

Proof of relation (19)

Relation (10 1 ) yields

E i = ∂e i ∂t + div ρ i 1 2 V 2 i + h i + Ω i V i -div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i -ρ i ∂Ω i ∂t .
On the left hand side of Eq.( 19), the extraneous force potential and inertia terms cancel out. With e i , h i , Φ i and Ψ i having been replaced by their respective values, it remains to be proved that the terms from the internal energy also cancel out. By using the convention that terms subscripted by i are summed on {1, 2}, the internal energy involves: a) in E i :

∂ε ∂t + div(ρ i h i V i ) -div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i , b) in ρ i M * i V i : -ρ i θ i grad * s i V i + ρ i grad * h i V i , c) in 1 2 V 2 i + h i + Ω i B i : h i dρ i dt + ρ i div V i , d) in S : ρ i θ i ds i dt .
This leaves the following in the first term of (19):

∂ε ∂t + div(ρ i h i V i ) -div ∂ρ i ∂t Φ i + ∂s i ∂t Ψ i -ρ i θ i ∂s i ∂t -ρ i grad * h i V i -h i dρ i dt + ρ i div V i ,
or:

∂ε ∂t -(h i + div Φ i ) ∂ρ i ∂t -(ρ i θ i + div Ψ i ) ∂s i ∂t -Φ * i ∂ grad ρ i ∂t -Ψ * i ∂ grad s i ∂t .
Definitions ( 16) of ε and (10) of h i and θ i show immediately that this expression is identically zero.

with C i = 2ρα , β i and D = ρα ,γ . Let us denote Π = i =j ρρ i α ,ρ i -ρ i div(C i grad ρ i +D grad ρ i )+C i (grad ρ i ) 2 +Dgrad * ρ i gradρ j .

In each bulk, gradient terms are zero and Π takes the values:

Π 1 = ρ 2 1 α ,ρ 1 (ρ 1 , 0, 0, 0, 0) and Π 2 = ρ 2 2 α ,ρ 2 (0, ρ 2 , 0, 0, 0) that correspond to the pressure in each phase of fluid. For a flat interface normal to grad ρ 1 and grad ρ 2 , the eigenvalues of stress tensor σ are

λ 1 = -Π + C 1 (grad ρ 1 ) 2 + C 2 (grad ρ 2 ) 2 + 2Dgrad * ρ 1 grad ρ 2 ,
(associated with the plane of interface), and

λ 2 = -Π
(associated with the direction normal to the plane of interface).

In a system of coordinates suitable for the interface (i.e. an orthonormal sys-tem whose third direction is z, the stress tensor is written:

σ =        λ 1 0 0 0 λ 1 0 0 0 λ 2       
.

The equation of balance momentum in the planar interface implies:

λ 2 = -Π 0 ,
where Π 0 denotes the pressure that must be common to the two bulks.

The line force per unit of length on the edge of the interface is (see Fig. 3):

F = z 2 z 1 λ 1 dz = -Π 0 (z 2 -z 1 ) + z 2 z 1
{C 1 (grad ρ 1 ) 2 + C 2 (grad ρ 2 ) 2 + 2Dgrad * ρ 1 grad ρ 2 }dz, where z 2z 1 corresponds to the physical size of the interface. Let us note that, due to the thickness of the interface, -Π 0 (z 2 -z 1 ) is negligible.

Conditions on the edge are not the ones of first gradient fluid. Nevertheless, the problem concerns a plane interface and the mean radius of curvature R m is such that R m = ∞. Therefore, the constraint condition on the edge is in the form n * σ where n * is normal to the edge of the interface [START_REF] Casal | Equations à potentiels en magnétodynamique des fluides[END_REF]. Let

Fig

  Fig. 1.

Fig. 2 .

 2 Fig. 2. Profiles of densities through the interface

Fig. 3 .

 3 Fig. 3. Interpretation of the mixture surface tension

Appendix 4 Interpretation of capillary tensions associated with two immiscible fluids

We consider an interface between two immiscible fluids. The assumption that two fluids remain absolutely separated is not reflected in physical experiments. The two fluids interpenetrate each other in a thin layer (thickness about few molecular diameters). Energetic reasons justify the assumption [START_REF] Brin | Contribution à l'étude de la couche capillaire et de la pression osmotique[END_REF]. The density of each constituent regularly changes between the value in bulk 1 and the one in bulk 2, with a profile in accordance with the one of a liquidvapour interface; each density vanishes asymptotically in the complementary phase (see Fig. 2 ). In the interfacial layer, density gradients are important. In the isothermal case, the zone of mixture of two isotropic fluids can be represented with an internal energy of the form:

The stress tensor reduced from Eq. ( 18) is:

us note:

(The integrals converge in the two bulks). The line force per unit of length on the edge is:

Here, H represents the surface tension of the flat interface in its equilibrium state expressed as a function of the surface tensions H 1 and H 2 of each bulk and a supplementary term H 1.2 associated with the effect of the mixture. This is the expression of the surface tension of an interface between two immiscible fluids known as Antonov's rule [START_REF] Emschwiller | Modelisation of fluid-fluid interfaces with material properties[END_REF].


Hisashi Oishi

Sayaka Mashima

Yutaka Kuwahara

Makoto Takafuji

Kyohei Yoshida

Reiko Oda

Hongdeng Qiu

Hirotaka Ihara
email: ihara@kumamoto-u.ac.jp

Polymer encapsulation and stabilization of molecular gel-based chiroptical information for strong, tunable circularly polarized luminescence film †

We demonstrate a strategic approach for a completely organic, chiroptical polymer film, exhibiting strong (g lum = 0.01-0.05) and broad-band circularly polarized luminescence that can be realized by the stabilization of self-assembly-driven secondary chirality through polymer encapsulation.

Circularly polarized luminescence (CPL) has attracted attention as a next-generation light signal because it carries more information due to its chirality and no angular effect compared to normal and linearly polarized lights. A simple example can be seen in its use as a communication tool by some animals. 1 One technical approach for producing CPL is by converting polarized light from normal to linear using a quarter-wave plate. However, this approach can suffer from serious reductions in light energy with removal of light. The other solution for CPL generation is the use of chiral luminescent compounds that are totally organic. [2][3][4][5][6][7][START_REF] Bosson | [END_REF] Studies using this approach have realized stronger CPL by precise molecular design, skilled organic synthesis, and careful purification. Therefore, gram-scale synthesis has not yet been achieved without few exceptions. [START_REF] Bosson | [END_REF]9 In this work, we describe the fabrication of chiroptical polymer films that exhibit strong CPL and anticipate their possible applications in bio-sensing, chiroptical memory, detector system for security devices, and storage units. 4,10,11 For such purposes, gram-scale synthesis becomes increasingly more important.

We selected a binary system for CPL generation, 6,7 which has a distinct advantage on larger scale. Instead of using specially designed chiral luminescent materials, commercially available, non-chiral luminescent dyes can be used and chirality can be induced by chiral templates. Therefore, the essential problem becomes, rather than the scalability of synthesis and fabrication, the retention of chirality induction in the polymer system. In this study, we demonstrate that strong CPL-exhibiting polymer films can be fabricated by incorporating into the polymer a chirally self-assembled CPL-inducing system consisting of a nonchiral luminescent dye and associated chiral template.

We applied several cyanine dyes including anionic NK-2012 and cationic NK-77 as non-chiral luminescent species for CPL generation. Because these cyanine dyes are very sensitive to the chiral microenvironment, they function as good indicators for chirality evaluation. However, cyanine dyes are often unstable, especially in oxygen atmosphere under sunlight, and therefore, these systems are not suitable for our present purpose. Fig. 1 presents several commercially available fluorescent dyes as new chirality-induction candidates. To induce chirality in these non-chiral dyes, G-ca, having a carboxy head group, was chosen as a self-assembling organogelator (Fig. 1). The advantages of G-ca can be summarized in terms of (1) its good solubility in various organic solvents, which is very important because most of a bulk polymer system is hydrophobic and used in organic media; (2) the enhanced chirality due to the formation of When an organic amine such as pyridine or triethylamine (TEA) is added to the G-ca/dye system, a distinct increase in CD intensity is observed. Fig. 3b shows the effect of TEA on the CD spectrum. The peak intensity is amplified ca.20 times with the addition of ten equimolar of G-ca. This is probably due to the promotion of electrostatic interactions between the chiral G-ca and non-chiral NK-77 by neutralization of the iodide counter ion of NK-77. Therefore, in this study, TEA was applied in the promotion of induced CD for the other non-chiral dyes.

Polystyrene (PS) was selected as the bulk polymer for film fabrication. The typical fabrication procedure is as follows: a 5 wt% solution of PS was combined with G-ca in hot toluene, and then the methanolic dye solution was added to the mixture. Part of the mixture (400 mL) was drop-cast onto a glass plate (12 mm  25 mm) and dried in air. The thickness of the resultant polymer film was adjusted to be approximately 60 mm. The main composition rate was adjusted to be 2.0 wt% in the G-ca-in-polymer, and to be 1 : 20 in the molar ratio of dye and G-ca. Fig. 2d shows the confocal microscopy images of the PS film containing NK-77 with G-ca. The fibrillar fluorescence image indicates that not only does G-ca self-assemble in the polymer, but also that NK-77 is associated with the G-ca fibrils. Similar fluorescence images (Fig. S1,ESI †) were observed with the other dyes, including acriflavine (AC, l max = 530 nm) and Basic Red-5 (BR-5, l max = 591 nm). Notably, all the PS films are transparent, which means that the diameters of the aggregates are sufficiently small to escape the problem of undesirable light scattering. This is an essential and important requirement for optical applications, and therefore, we conclude that our strategic approach is unquestionably suitable for our research purposes.

Fig. 4 shows the CD spectra of the PS films containing the G-ca/dye systems. Large CD intensities are observed near the absorption bands of the dyes in all films. Since no CDs are observed in the absence of G-ca and the CD intensity is secondary chirality through chiral arrangement; (3) its ionizability to promote electrostatic interactions with the cationic dyes shown in Fig. 1; and ( 4) its possible gram-scale synthesis via aone-step coupling reaction between L-glutamide 12 and glutalic anhydride.

Mixed systems of G-ca and dyes were prepared by the following procedure: a 0.5 mM solution of G-ca in toluene was prepared by heating at 85 1C; cooling to 25 1C produced a clear solution. If the concentration was higher than a few millimolar in toluene, the solution became a clear gel. This phenomenon has been recognized as molecular gelation. 7,13,14 Fig. 2a shows the TEM image of the 0.5 mM G-ca solution after staining with uranyl acetate. G-ca forms nano-fibrillar aggregates whose smallest diameter is within 50 nm in the photo. This indicates that the gelation behavior is the result of entanglement among the well-developed nano-fibrillar aggregates of G-ca. Next, a methanolic solution of the dye was mixed in a given molar ratio with the G-ca solution. No significant change is observed by TEM, but confocal microscopy provides fibrillar fluorescence images, as shown in Fig. 2b. Since G-ca does not itself fluoresce, the obtained confocal image indicates that the dye is combined with G-ca and the fibrillar aggregation of G-ca is maintained after mixing with the dye.

The chirality of the G-ca/dye mixed systems was investigated by CD spectroscopy. As shown in Fig. 3a, the G-ca/NK-77 system exhibits a new exciton coupling ([y] 573 =5Â 10 4 deg cm dmol À1 ) near the absorption band (l max =570nm)ofNK-77.Since NK-77is non-chiral, the observed CD can be attributed to induced chirality from the G-ca aggregates. Although this induced CD intensity is not so small, whilewehaveconfirmed much larger values (41 Â 10 6 deg cm dmol À1 ) 15 in the case of an anionic cyanine dye, NK-2012, on poly(L-lysine). Therefore, we examined the effect of additives on the G-ca/dye system to promote their interaction. distinctly decreased without TEA, it is clear that the dyes are chirally perturbed by the G-ca chiral aggregates. Fig. 4 also includes the CPL spectra. The emission bands agree with the fluorescence bands of the dyes (Fig. S2, ESI †), and the values of g lum reach À0.012, À0.048 and À0.040 with NK-77, AC and BR-5, respectively. A |g lum | value over 0.04 corresponds to the highest value in the polymer thin film system.

Our CPL generation system has multiple advantages. First, mixed-dye systems can be employed to provide broad-band CPL over a wide range of emission bands. For example, when AC and BR-5 are mixed, the resultant PS film provides two CD signals (Fig. S3, ESI †). Second, broad-band CPL can also be generated by adjusting the chiral microenvironment. For example, when the mixing ratio of AC to G-ca is adjusted, the luminescent peak width increases as shown in Fig. 5: that is, the Stokes shift can be made tunable by promoting the heterogeneity of chiral perturbation. Finally, by encapsulating G-ca in the polymer as well as selecting the proper dye, the CPL phenomenon can be remarkably stabilized. For example, we confirmed that the G-ca/AC polymer system maintained strong CPL over at least a month.

Conclusions

In conclusion, we have established a strong CPL (g lum E 0.05) generation system in a polymer thin film. Our system is advantageous because no chirality is needed in the fluorescent component, meaning that it is free from precise molecular design and complicated synthetic limitations, and therefore possible for industrial-scale application. In addition, the emission bands can be readily tuned by adjusting the molecular orientation of the chiral template system as well as the dye selection. These advantages could expand possible system applications, such as in optical uses (e.g., chiroptical detector for bio-signal sensing), in agriculture, or in solar cells as a reduced-angular wavelength converter.

Fig. 1

 1 Fig. 1 Chemical structures of non-chiral fluorescent compounds and organogelator G-ca used in this study.

Fig. 2

 2 Fig. 2 TEM and confocal micrographs of G-ca aggregates. (a) Initial concentration of G-ca: 0.1 mM in toluene. The dried sample was obtained by drop-casting on Cu mesh, stained with uranyl acetate and dried at room temperature. The confocal microscopy for the solution state (b) and the film (d) were observed with excitation wavelength at 488 nm. (b) G-ca: 0.5 mM and NK-77: 0.01 mM in toluene. (c) and (d) G-ca: 2 wt% in polystyrene. G-ca : NK-77 = 20 : 1.

Fig. 3

 3 Fig. 3 (a) CD spectra of the G-ca with NK-77 in toluene. [G-ca] = 0.5 mM; [NK-77] = 0.01 mM; [TEA] or [pyridine] = 5 mM. (b) Effect of TEA additive in the G-ca/dye system on the CD intensity.

Acknowledgements

This work was supported by the French-Japanese International Associated Laboratory, Chiral Nanostructures for Photonic Applications (LIA-CNPA), and a Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research and the Bilateral Joint Research project.

Conflicts of interest

There are no conflicts to declare.

Notes and references