Rodolfo Orjuela

Benoît Marx

José Ragot

Didier Maquin

Proportional-Integral observer design for nonlinear uncertain systems modelled by a multiple model approach

In this paper, a decoupled multiple model approach is used in order to cope with the state estimation of uncertain nonlinear systems. The proposed decoupled multiple model provides flexibility in the modelling stage because the dimension of the submodels can be different and this constitutes the main difference with respect to the classically used multiple model scheme. The state estimation is performed using a Proportional Integral Observer (PIO) which is well known for its robustness properties with respect to uncertainties and perturbations. The Lyapunov second method is employed in order to provide sufficient existence conditions of the observer, in LMI terms, and to compute the optimal gains of the PIO. The effectiveness of the proposed methodology is illustrated by a simulation example.

I. INTRODUCTION

In many real world engineering applications, the knowledge of the system state is often required not only for control purpose but also for monitoring and fault diagnosis. In practice however, the measurements of the system state can be very difficult or even impossible, for example when an appropriate sensor is not available or economically viable. Model-based state estimation is a largely adopted strategy used in order to cope with this important problem. Typically a state estimation is provided by means of an observer whose inputs are the inputs and the outputs of the system and the outputs are the estimated states. Note that the structure of an observer is based on the mathematical model of the considered system. Therefore, the accuracy of the state estimation depends on the accuracy of the used mathematical model and the quality of the employed measurements. However, a mathematical model is an abstract representation of the real world and it only provides an approximation to dynamic behaviours of the actual system. Consequently, modelling errors between the system and its model are unavoidable. Besides, the employed measurements are also affected by external disturbances due to the interactions between the system and its environment.

Hence, many efforts have been made in the past two decades to improve robustness of the state estimation of linear systems affected by disturbances and parametric uncertainties (e.g. in [START_REF] Weinmann | Uncertain Models and Robust Control[END_REF]- [START_REF] Xie | Robust Kalman filtering for uncertain discrete-time systems[END_REF] norm-bounded uncertainties are considered). However, dynamic behaviour of most of real systems is nonlinear and consequently a linear model is not This work is partially supported by the Conseil Régional de Lorraine (France).

The authors are with Centre de Recherche en Automatique de Nancy (CRAN), Nancy-Université, CNRS, 2 avenue de la Forêt de Haye F-54516, Vandoeuvre-lès-Nancy {rodolfo.orjuela, benoit.marx, jose.ragot, didier.maquin}@ensem.inpl-nancy.fr able to provide a good characterisation of the system in the whole operating range. On the other hand, the observer design problem for generic nonlinear models is delicate and so far this problem remains unsolved in a general way.

Multiple model approach is an appropriate tool for modelling complex systems using a mathematical model which can be used for analysis, controller and observer design. The basis of the multiple model approach is the decomposition of the operating space of the system into a finite number of operating zones. Hence, the dynamic behaviour of the system inside each operating zone can be modelled using a simple submodel, for example a linear model. The relative contribution of each submodel is quantified with the help of a weighting function. Finally, the approximation of the system behaviour is performed by associating the submodels and by taking into consideration their respective contributions. Note that a large class of nonlinear systems can accurately be modelled using multiple models.

The choice of the structure used to associate the submodels constitutes a key point in the multiple modelling framework. Indeed, the submodels can be aggregated using various structures [START_REF] Filev | Fuzzy modeling of complex systems[END_REF]. Classically, the association of submodels is performed in the dynamic equation of the multiple model using a common state vector. This model, known as Takagi-Sugeno multiple model, has been initially proposed, in a fuzzy modelling framework, by Takagi and Sugeno [START_REF] Takagi | Fuzzy identification of systems and its application to modelling and control[END_REF] and in a multiple model modelling framework by Johansen and Foss [START_REF] Johansen | Constructing NARMAX models using ARMAX model[END_REF]. This model has been largely considered for analysis, modelling, control and state estimation of nonlinear systems (see among others [START_REF] Tanaka | Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, Hinf control theory, and Linear Matrix Inequalities[END_REF]- [START_REF] Assawinchaichote | Fuzzy control and filter design for uncertain fuzzy systems[END_REF] and references therein).

In this paper, an other possible way for building a multiple model is employed. The used model, known as decoupled multiple model, has been suggested in [START_REF] Filev | Fuzzy modeling of complex systems[END_REF] and results of the association of submodels only in the output equation of the multiple model. Note also that this multiple model has been successfully employed in modelling [START_REF] Venkat | Identification of complex nonlinear processes based on fuzzy decomposition of the steady state space[END_REF], [START_REF] Vinsonneau | Extended global total least square approch to multiple-model identification[END_REF], control [START_REF] Gawthrop | Continuous-time local state local model networks[END_REF]- [START_REF] Gregorcic | Control of highly nonlinear processes using self-tuning control and multiple/local model approaches[END_REF] and state estimation [START_REF] Uppal | A hybrid neuro-fuzzy and decoupling approach applied to the DAMADICS benchmark problem[END_REF], [START_REF] Orjuela | State estimation for nonlinear systems using a decoupled multiple model[END_REF] of nonlinear systems. The main feature of the decoupled multiple model is that submodels of different dimensions (e.g. number of states) can be used. This fact introduces some flexibility degrees in the modelling stage in particular when the model is obtained using a black box modelling strategy. Indeed, the dimensions of the submodels can be well adapted to each operating zone and consequently the total number of parameters necessary for describing the system can be reduced.

This paper deals with the design of a Proportional-Integral Observer (PIO) for a class of nonlinear systems modelled by a decoupled multiple model with parameter uncertainties.

The parameter uncertainties are assumed to be unknown, time-varying and norm bounded. With respect to the classic proportional observer, the PIO offers more additional degrees of freedom which can be used for improving its robustness properties with respect to perturbations and imperfections in the model (details are given in section III). The PIO design problem consists in finding the gains of the observer such that the state estimation error converges toward zero or at least remains globally bounded for all admissible uncertainties and perturbations. Furthermore, the PIO design based on the multiple model representation does not seem to be reported previously to the best authors' knowledge.

The outline of this paper is as follows. Discussion about decoupled multiple model is proposed in section II. In section III, the PIO design problem is investigated and the gains of the observer are obtained by LMI optimization. Finally, in section IV, a simulation example illustrates the state estimation of a decoupled multiple model.

II. ON THE DECOUPLED MULTIPLE MODEL

REPRESENTATION

In this paper, an uncertain nonlinear system described by a decoupled multiple model is considered. The state space representation of this multiple model is given by:

ẋi (t) = (A i + ∆A i )x i (t) + (B i + ∆B i )u(t) + D i w(t) , (1a) y i (t) = C i x i (t) , (1b)
y(t) = L ∑ i=1 µ i (ξ (t))y i (t) + W w(t) , (1c)
where x i ∈ R n i and y i ∈ R p are respectively the state vector and the output of the i th submodel; u ∈ R m is the input, y ∈ R p the output and w ∈ R r the perturbation. The matrices

A i ∈ R n i ×n i , B i ∈ R n i ×m , D i ∈ R n i ×r , C i ∈ R p×n i and
W ∈ R p×r are known and appropriately dimensioned. The parametric uncertainties in the system are represented by matrices ∆A i and ∆B i (details are given in section II-A).

The complete partition of the operating space of the system is performed using a characteristic variable of the system called decision variable ξ (t) that is assumed to be known and real-time available (e.g. the inputs and/or exogenous signals). Note that the contribution of the submodels are quantified by the weighting functions µ i (ξ (t)) which are associated with each operating zone. They satisfy the following convex sum constraints:

L ∑ i=1 µ i (ξ (t)) = 1 and 0 ≤ µ i (ξ (t)) ≤ 1 , ∀i = 1...L, ∀t. (2)
Thanks to the above properties, the contributions of several submodels can be taken into account simultaneously and therefore the dynamic behaviour of the multiple model can be truly nonlinear instead of a piecewise linear behaviour.

Note that the contributions of the submodels are taken into account via a weighted sum in the output equation of the multiple model. Consequently, dimensions of the submodels can be different and therefore this multiple model form is suitable for black box modelling of complex systems with variable structure and/or variable complexity in each operating zone. The model parameters can be obtained from a set of measured input and output data using appropriate black box identification tools proposed for instance in [START_REF] Venkat | Identification of complex nonlinear processes based on fuzzy decomposition of the steady state space[END_REF], [START_REF] Vinsonneau | Extended global total least square approch to multiple-model identification[END_REF], [START_REF] Orjuela | Nonlinear system identification using uncoupled state multiple-model approach[END_REF].

Remark 1: It should be mentioned that the outputs y i (t) of the submodels are intermediary modelling signals only used in order to provide a representation of the real system behaviour. The submodel outputs y i (t) are internal signals of the multiple model. They are not physically available and consequently no measurement is possible. Hence, they cannot be employed for driving an observer. Only the global output y(t) of the multiple model can be used for this purpose.

A. Model uncertainties

The parametric uncertainties in the system are represented by the following norm-bounded matrices:

∆A i = µ i (ξ (t))M i F i (t)N i , (3)
∆B i = µ i (ξ (t))H i S i (t)E i , (4)
where M i , N i , H i and E i are known constant matrices of appropriate dimensions and F i (t) and S i (t) are unknown, real and possibly time-varying matrices with Lebesguemeasurable elements satisfying:

F T i (t)F i (t) ≤ I and S T i (t)S i (t) ≤ I ∀t . (5)
Note that the uncertainties of each submodel are taken into consideration according to the validity degree of each submodel via its associated weighting function µ i (ξ (t)). Indeed, the uncertainties of a submodel can be neglected when its respective contribution is not taken into consideration for providing the overall multiple model output.

Notations: the following notations will be used all along this paper. P > 0 (P < 0) denotes a positive (negative) definite matrix P; X T denotes the transpose of matrix X, I is the identity matrix of appropriate dimension and diag{A 1 , ..., A n } stands for a block-diagonal matrix with the matrices A i on the main diagonal. The L 2 -norm of a signal, quantifying its energy is denoted and defined by e(t) 2 2 =

∞ 0 e T (t)e(t)dt.

Finally, we shall simply write µ i (ξ (t)) = µ i (t).

III. ON THE PROPORTIONAL-INTEGRAL OBSERVER

The conventional Luenberger or proportional observer only uses a proportional correction injection term given by the output estimation error. In the PIO an additional injection term z(t), given by the integral of the output estimation error, is included in the dynamic equation of the observer. Thanks to this additional degree of freedom some robustness degrees of the state estimation with respect to the system uncertainties and perturbation are introduced [START_REF] Weinmann | Uncertain Models and Robust Control[END_REF], [START_REF] Beale | Robust control system design with a proportional integral observer[END_REF], [START_REF] Linder | Rejecting disturbances to flexible structures using PI Kalman filters[END_REF]. The PIO has also been successfully employed in the synchronization of a chaotic system by [START_REF] Hua | Synchronization of chaotic systems based on PI observer design[END_REF]. The extension of the PIO design, based on dissipativity framework, to a particular nonlinear system whose non linearity is assumed to satisfy a sector bounded constraint, has been recently proposed in [START_REF] Jung | Dissipative Proportional-Integral Observer for a clas of uncertain nonlinear systems[END_REF].

In this section, sufficient conditions for ensuring convergence and optimal disturbance attenuation of the estimation error are established in LMI terms [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF] using the Lyapunov method. Note that the classic observer design cannot be employed directly in the multiple model framework because the interaction between submodels must be taken into consideration in the observer design procedure for ensuring the observer stability for any blend between the submodels.

Firstly new notation of the decoupled multiple model needed to design a PIO is introduced. The suggested PIO is then presented and its design is proposed by introducing some H ∞ performances.

A. Augmented form of the decoupled multiple model

Consider the following augmented state vector:

x(t) = x T 1 (t) • • • x T i (t) • • • x T L (t) T ∈ R n , n = L ∑ i=1 n i ( 6
)
and the supplementary variable z(t) = t 0 y(ξ )dξ needed for the PIO design. Thus, the decoupled multiple model ( 1) may be rewritten in the following compact form:

ẋ(t) = ( Ã + ∆ Ã)x(t) + ( B + ∆ B)u(t) + Dw , (7a) ż(t) = C(t)x(t) + W w(t) , (7b)
y(t) = C(t)x(t) + W w(t) , (7c)
where

à = diag {A 1 • • • A i • • • A L } , (8)
B = B 1 T • • • B i T • • • B L T T , (9)
D = D 1 T • • • D i T • • • D L T T , (10)
C(t) = L ∑ i=1 µ i (t) Ci , (11)
Ci = 0 • • • C i • • • 0 (12)
with the parametric uncertainties given by:

∆ Ã = L ∑ i=1 µ i (t) Mi F i (t) Ñi , (13)
∆ B = L ∑ i=1 µ i (t) Hi S i (t)E i , (14)
Mi = 0 • • • M i T • • • 0 T , ( 15
) Ñi = 0 • • • N i • • • 0 , (16)
Hi = 0 • • • H i T • • • 0 T . ( 17
)
Finally, the equations ( 7) can be rewritten in the following augmented form:

ẋa (t) = ( Ãa (t) + C 1 ∆ ÃC T 1 )x a (t) + C 1 ( B + ∆ B)u(t) + Da w(t) , (18a)
y(t) = C(t)C T 1 x a (t) + W w(t), ( 18b
)
z(t) = C T 2 x a (t) , (18c)
where

x a (t) = x(t) z(t) , Ãa (t) = Ã 0 C(t) 0 , Da = D W , C 1 = I 0 T , C 2 = 0 I T .
Let us notice that, by using the convex properties of the weighting functions, the matrix Ãa (t) can be rewritten as:

Ãa (t) = L ∑ i=1 µ i (t)A i , (19)
where

A i = Ã 0 Ci 0 . ( 20
)

B. PIO structure

The state estimation of the decoupled multiple model ( 18) is achieved by using the following PIO:

ẋa (t) = Ãa (t) xa (t) + C 1 Bu(t) + K P (y(t) -ŷ(t)) +K I (z(t) -ẑ(t)) , (21a) ŷ(t) = C(t)C T 1 xa (t) , (21b)
ẑ(t) = C T 2 xa (t) (21c)
which has a similar structure to the PIO used in [START_REF] Hua | Synchronization of chaotic systems based on PI observer design[END_REF]. Notice that the use of the auxiliary integral signal z(t) in the dynamic equation is at the origin of the designation Proportional-Integral Observer. The matrix K I introduces a freedom degree in the observer design.

C. Design of the PIO

Consider the state estimation error defined by:

e a (t) = x a (t) -xa (t) (22)
and its dynamics by:

ėa (t) = ( Ãa (t) -K P C(t)C T 1 -K I C T 2 )e a (t) + C 1 ∆ Ãx(t) + C 1 ∆ Bu(t) + ( Da -K P W )w(t) . (23)
Finally, (7a) and ( 23) can be gathered as follows:

ε(t) = A obs (t)ε(t) + Φ w(t) , (24)
where

ε(t) = e T a (t) x T (t) T , (25)
w(t) = w T (t) u T (t) T , (26)
A obs (t) = Ãa (t) -K P C(t)C T 1 -K I C T 2 C 1 ∆ Ã 0 Ã + ∆ Ã ,( 27
) Φ = Da -K P W C 1 ∆ B D B + ∆ B . (28)
Notice that the proportional gain K P can be used to reduce the impact of the perturbation on the estimation error e a (t).

On the other hand, the observer dynamics can be improved with the help of the integral gain K I . Note also that, from equation (24), ε(t) is stable if and only if the decoupled multiple model [START_REF] Tanaka | Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, Hinf control theory, and Linear Matrix Inequalities[END_REF] with admissible uncertainties ∆ Ã is stable and the observer gains K P and K I are chosen so that Ãa (t)

-K P C(t)C T 1 -K I C
T 2 is also stable. In the sequel, the two following assumptions will be considered:

Assumption 1: The decoupled multiple model [START_REF] Tanaka | Robust stabilization of a class of uncertain nonlinear systems via fuzzy control: quadratic stabilizability, Hinf control theory, and Linear Matrix Inequalities[END_REF] with admissible uncertainties ∆ Ã is stable.

Assumption 2: The input and the perturbation are bounded energy signals, i.e. u(t) 2 2 < ∞ and w(t) 2 2 < ∞. The robust PIO design problem can thus be formulated as finding the matrices K P and K I such that the influence of w(t) on the estimation error e a (t) is attenuated and the state estimation error remains globally bounded for any blend between the submodels. To this end, the following objective signal which only depends on the estimation error e a (t) is introduced:

ν(t) = Y 0 ε(t) , ( 29
)
where Y is a matrix of appropriate dimension chosen by the designer. Finally, the expected performances of the PIO can be formulated by the following H ∞ performances:

lim t→∞ e a (t) = 0 for w(t) = 0, F i (t) = 0, S i (t) = 0 , (30a) ν(t) 2 2 ≤ γ 2 w(t) 2 2
for w(t) = 0 and ν(0) = 0 , (30b) where γ is the L 2 gain from w(t) to ν(t) to be minimized.

Theorem 1: Consider the uncertain model ( 18) and assumptions 1 and 2. There exists a PIO [START_REF] Jung | Dissipative Proportional-Integral Observer for a clas of uncertain nonlinear systems[END_REF] ensuring the objectives (30) if there exists symmetric positive definite matrices P 1 ∈ R (n+p)×(n+p) and P 2 ∈ R n×n , matrices L P ∈ R (n+p)×p and L I ∈ R (n+p)×p and positive scalars γ, τ i 1 and τ i 2 such that the following condition holds for i

= 1...L min γ subject to        Γ i + Γ T i +Y T Y 0 Ψ 0 P 1 C 1 Mi P 1 C 1 Hi 0 Λ i P 2 D P 2 B P 2 Mi P 2 Hi ( * ) ( * ) -γ I 0 0 0 0 ( * ) 0 φ i 0 0 ( * ) ( * ) 0 0 -τ i 1 I 0 ( * ) ( * ) 0 0 0 -τ i 2 I        < 0 ,
where

Γ i = P 1 A i -L P Ci C T 1 -L I C T 2 , Ψ = P 1 Da -L P W , Λ i = P 2 Ã + ÃT P 2 + τ i 1 ÑT i Ñi , φ i = -γ I + τ i 2 E
T i E i for a prescribed matrix Y . The observer gains are given by K P = P -1 1 L P and K I = P -1 1 L I ; the L 2 gain from w(t) to ν(t) is given by γ = γ.

Proof: The proof is deferred to the appendix.

IV. A SIMULATION EXEMPLE Consider the decoupled multiple model with L = 2 submodels with different dimensions (n 1 = 3 and n 2 = 2), given by:

A 1 =   -0.1 -0.3 0.6 -0.5 -0.4 0.1 -0.3 -0.2 -0.6   , A 2 = -0.3 -0.1 0.4 -0.2 , B 1 = 0.3 0.5 0.6 T , B 2 = 0.4 0.3 T , D 1 = 0.1 -0.1 0.1 T , D 2 = -0.1 -0.1 T , C 1 = -0.4 0.3 0.5 0.5 0.3 0.4 , C 2 = 0.4 -0.2 0.3 0.2 , M 1 = -0.1 0.2 -0.1 T , M 2 = -0.2 0.1 T , N 1 = 0.1 -0.2 0.3 , N 2 = 0.1 0.2 , H 1 = 0.3 -0.1 0.2 T , H 2 = -0.1 -0.2 T , E 1 = -0.2 , E 2 = -0.3 , W = 0.1 -0.1 , Y = I (7×7) .
Here, the decision variable ξ (t) is the input signal u(t) ∈ [-1, 1]. The weighting functions are obtained from normalised Gaussian functions:

µ i (ξ (t)) = η i (ξ (t))/ L ∑ j=1 η j (ξ (t)), (31)
η i (ξ (t)) = exp -(ξ (t) -c i ) 2 /σ 2 , (32)
with the standard deviation σ = 0.6 and the centres c 1 = -0.3 and c 2 = 0.3. The perturbation w(t) is a normally distributed random signal with zero mean and standard deviation equal to one. The input, the weighting functions and the outputs are shown in figure 1. The time-varying signals F i (t), S i (t) and the perturbation w(t) are plotted in figure 2. Notice that for 0 < t < 120 no uncertainties in the multiple model are considered.

Fig. 2. F i (t), S i (t) and w(t)

A solution satisfying conditions of theorem 1 is obtained using YALMIP interface and SEDUMI solver. The gains of the PIO are: T the minimal attenuation level is γ = 0.8654 and τ 1 1 = 0.74, τ 1 2 = 1.37, τ 2 1 = 7.67, τ 2 2 = 2.08. In the simulation the initial conditions of the multiple model are x(0) = 0.1 -0.1 0.1 -0.1 0.1 and the initial condi-tions of the observer are equal to zero. Figures 3 and4 display the comparison between the states of the submodels and their estimates. Note that the interaction between submodels is at the origin of some compensation phenomenons in the state estimation. For example, if the output of submodel 1 is only taken into consideration (i.e. µ 1 (t) ≈ 1) then naturally a bad state estimation of the submodel 2 is provided by the observer. However, the overall output estimation of the multiple model is not truly affected by this bad estimation. Finally, a comparison between the outputs of the multiple model and their estimates is shown in figure 5. Note that the output estimation errors remain globally bounded despite that model uncertainties and perturbations appear in the model. V. CONCLUSIONS In this paper a PIO design is presented for a class of uncertain nonlinear system which can be modelled with the help of a decoupled multiple model. This model is suitable for modelling variable structure systems because the dimension of the submodels can be different in each operating zone. Sufficient conditions, in LMI terms, for ensuring H ∞ performances of the estimation error are established using Lyapunov method. The effectiveness of the proposed approach is illustrated via a simulation example. Further research, in a fault diagnosis perspective, will be to investigate the sensitivity of the state estimation with respect to perturbations, model uncertainties and faults in order to establish the sensitivity of the fault symptoms of the system. APPENDIX PROOF OF THE THEOREM Lemma 1: For any constant real matrices X and Y with appropriate dimensions, a matrix function F(t) boundednorm, i.e. F T (t)F(t) ≤ I, then the following property holds for any positive matrix Q

K P = -3.
XF(t)Y + Y T F T (t)X T ≤ XQ -1 X T + Y T QY .
Consider the following quadratic Lyapunov function:

V (t) = e T a (t)P 1 e a (t) + x T (t)P 2 x(t) , (33)
where P 1 = P T 1 > 0 et P 2 = P T 2 > 0. The objectives (30) are guaranteed if there exists a Lyapunov function (33) such that [START_REF] Boyd | Linear Matrix Inequalities in System and Control Theory[END_REF]:

V (t) < -ν T (t)ν(t) + γ 2 w T (t)w(t) . (34)
The time-derivative of (33) along the trajectories of (24) and (7a) is given by: V (t) = Ω T (t) PA obs (t) + A T obs P(t) PΦ ( * ) 0 Ω(t) , (35)

P = diag {P 1 , P 2 } , (36)
Ω(t) = ε T (t) w T (t) T . (37)
Now, by taking into consideration (35), the condition (34) becomes:

Ω T (t) PA obs (t) + A T obs (t)P + Y T Y 0 0 0 PΦ ( * ) -γ 2 I Ω(t) < 0 , (38)
which is a quadratic form in Ω(t). By using the definitions of A obs and Φ given respectively by ( 27) and (28), the inequality (38) is also guaranteed if:

    Γ + Γ T + Y T Y P 1 C 1 ∆ Ã Ψ P 1 C 1 ∆ B ( * ) X 1 + X 2 P 2 D P 2 ( B + ∆ B) ( * ) ( * ) -γ 2 I 0 ( * ) ( * ) 0 -γ 2 I     < 0, (39)
where

Γ = P 1 ( Ãa (t) -K P C(t)C T 1 -K I C T 2 ) , (40)
Ψ = P 1 ( Da -K P W ) , (41) X 1 = P 2 Ã + ÃT P 2 , (42)
X 2 = P 2 ∆ Ã + ∆ ÃT P 2 , (43)
Notice that by using the definition of Ãa (t) and C(t) given respectively by ( 19) and ( 11), Γ can be rewritten as :

Γ = L ∑ i=1 µ i (t)Γ i , (44)
Γ i = P 1 (A i -K P Ci C T 1 -K I C T 2 ) . (45)
At this point, by considering (44) and ( 43), the nominal and the uncertain terms in (39) may be dissociated as follows:

L ∑ i=1 µ i (t)     Γ i + Γ T i + Y T Y 0 Ψ 0 0 X 1 P 2 D P 2 B ( * ) ( * ) -γ 2 I 0 0 ( * ) 0 -γ 2 I     + Z + Z T < 0, ( 46
)
where

Z =     0 P 1 C 1 ∆ Ã 0 P 1 C 1 ∆ B 0 P 2 ∆ Ã 0 P 2 ∆ B 0 0 0 0 0 0 0 0     . (47)
Now, by introducing the definitions of ∆ Ã and ∆ B given by ( 13) and ( 14) then Z + Z T becomes:

Z + Z T = L ∑ i=1 µ i (t) Xi Ỹi + Ỹ T i XT i , (48)
where

Xi =     P 1 C 1 Mi P 1 C 1 Hi P 2 Mi P 2 Hi 0 0 0 0     , (49)
Ỹi = F i (t) 0 0 S i (t)

0 Ñi 0 0 0 0 0 E i . ( 50
)
Notice that the dependence of the unknown functions F i (t) and S i (t) upon (48) can be removed, by using the lemma 1 with Q i = diag τ i 1 , τ i 2 , as follows:

Z + Z T ≤ L ∑ i=1 µ i (t) Xi τ i 1 0 0 τ i 2 -1 XT i + Ỹi τ i 1 0 0 τ i 2 Ỹ T i . (51)
Finally, using the definition (42) of X 1 , the inequality (46) is guaranteed if for i = 1...L the following inequality holds: where

       Γ i + Γ T i +Y T Y 0 Ψ 0 P 1 C 1 Mi P 1 C 1 Hi 0 Λ i P
Λ i = P 2 Ã + ÃT P 2 + τ i 1 ÑT i Ñi , ( 53
) φ i = -γ 2 I + τ i 2 E T i E i . (54)
This condition follows from the use of (51) in (46), the use of the well known Schur complement and the convex sum properties of µ i (t). Note that asymptotic convergence towards zero of the estimation error, when no uncertainties and no perturbations affect the system, is guaranteed by the negativity of the block (1, 1) in (52).

Finally, let us notice that (52) is not a LMI in P 1 , K P , K I and γ. However, it becomes a LMI by setting L P = P 1 K P , L I = P 1 K I and γ = γ 2 . Now, standard convex optimization algorithms can be used to find matrices P 1 , P 2 L P and L I minimising γ. This completes the proof of theorem 1.

Fig. 1 .

 1 Fig. 1. Input, weighting functions and outputs

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. States of submodel 1 and its estimates


Steven Araujo

Nicolas Delpouve
email: nicolas.delpouve1@univ-rouen.fr

Laurent Delbreilh

Dimitry Papkov

Yuris Dzenis

Eric Dargent

Dielectric and calorimetric signatures of chain orientation in strong and tough ultrafine electrospun polyacrylonitrile

Ultrafine diameter fibers of polyacrylonitrile (PAN), obtained from electrospinning, have huge potential for structural applications since they exhibit an unusual combination of strength and toughness. However, the difficulty to characterize their supramolecular architecture limits their production at the industrial scale. In this work, the glass transition of electrospun nanofiber mats of PAN was investigated by means of thermal analysis techniques.

Modulated temperature differential scanning calorimetry (MT-DSC) and dielectric relaxation spectroscopy (DRS) were used, and relaxation parameters characteristic of the glass transition were obtained. Reduction in average fiber diameter resulted in broadening of the glass transition and a shift of its midpoint to higher temperatures as observed by MT-DSC, revealing additional level of constraints in the amorphous phase. The DRS curves, obtained above the calorimetric signature of the glass transition, superimpose independently on the fiber diameter. This result, which contrasts with MT-DSC observations, shows that the

INTRODUCTION

Electrospinning is a process which allows creating continuous micro-or nanofibers by jetting polymer solution in high electric field [START_REF] Bhardwaj | Electrospinning: A fascinating fiber fabrication technique[END_REF][START_REF] Ewaldz | Molecular Interactions in Electrospinning: From Polymer Mixtures to Supramolecular Assemblies[END_REF][START_REF] Agarwal | Functional materials by electrospinning of polymers[END_REF][START_REF] Jiang | Electrospinning of polymer nanofibers for tissue regeneration[END_REF][START_REF] Ding | Electrospun polymer biomaterials[END_REF][START_REF] Liao | Progress in electrospun polymeric nanofibrous membranes for water treatment: Fabrication, modification and applications[END_REF][START_REF] Séon-Lutz | Electrospinning in water and in situ crosslinking of hyaluronic acid / cyclodextrin nanofibers: Towards wound dressing with controlled drug release[END_REF][START_REF] Xu | Melt electrospinning: Electrodynamics and spinnability[END_REF][START_REF] Inagaki | Carbon Nanofibers Prepared via Electrospinning[END_REF][START_REF] Govinna | Thermal properties and structure of electrospun blends of PVDF with a fluorinated copolymer[END_REF]. Due to the growing interest in the fabrication of continuous nanofibers for structural and functional applications, electrospinning was used in different fields such as tissue engineering [START_REF] Jiang | Electrospinning of polymer nanofibers for tissue regeneration[END_REF][START_REF] Liao | Electrospun aligned PLLA/PCL/HA composite fibrous membranes and their in vitro degradation behaviors[END_REF], medical [START_REF] Kwon | Electrospun nano-to microfiber fabrics made of biodegradable copolyesters: structural characteristics, mechanical properties and cell adhesion potential[END_REF], filtration [START_REF] Liao | Progress in electrospun polymeric nanofibrous membranes for water treatment: Fabrication, modification and applications[END_REF][START_REF] Yoon | High flux ultrafiltration membranes based on electrospun nanofibrous PAN scaffolds and chitosan coating[END_REF], food industry [START_REF] Aydogdu | Nanostructured poly(lactic acid)/soy protein/HPMC films by electrospinning for potential applications in food industry[END_REF] and opto-electronics [START_REF] Serrano | Electrospun composite poly(lactic acid)/polyaniline nanofibers from low concentrations in CHCl3: Making a biocompatible polyester electro-active[END_REF]. Fibers obtained by electrospinning can exhibit outstanding mechanical properties, such as combined strength and toughness, when their diameter is less than 250 nm [START_REF] Papkov | Simultaneously Strong and Tough Ultrafine Continuous Nanofibers[END_REF]. To explain this result, it was hypothesized that electrospinning can generate high level of orientation while preserving low degree of crystallinity [START_REF] Papkov | Simultaneously Strong and Tough Ultrafine Continuous Nanofibers[END_REF]. Indeed, the macromolecular chains undergo both electrical and mechanical stretching during electrospinning [START_REF] Bhardwaj | Electrospinning: A fascinating fiber fabrication technique[END_REF]. On the other hand, the crystalline growth occurs in a confined environment which can slow down the crystallization kinetics [START_REF] Zong | Structure and process relationship of electrospun bioabsorbable nanofiber membranes[END_REF]. In addition, fast solvent evaporation may hinder the crystallization process [START_REF] Wu | Modeling of Solvent Evaporation from Polymer Jets in Electrospinning[END_REF]. As a result, the microstructure generated by electrospinning is complex, with the possible coexistence of crystalline and amorphous phases, as well as mesophase [START_REF] Liu | Effects of hard and soft components on the structure formation, crystallization behavior and mechanical properties of electrospun poly(l-lactic acid) nanofibers[END_REF]. Mesophase is defined as an intermediate order structure which is a precursor to crystallization [START_REF] Stoclet | New Insights on the Strain-Induced Mesophase of Poly(d,l-lactide): In Situ WAXS and DSC Study of the Thermo-Mechanical Stability[END_REF][START_REF] Wasanasuk | Structural Regularization in the Crystallization Process from the Glass or Melt of Poly(l-lactic Acid) Viewed from the Temperature-Dependent and Time-Resolved Measurements of FTIR and Wide-Angle/Small-Angle X-ray Scatterings[END_REF][START_REF] Wang | Orientation and structural development of semicrystalline poly(lactic acid) under uniaxial drawing assessed by infrared spectroscopy and X-ray diffraction[END_REF]. Ma et al. [START_REF] Ma | Relationship between the rigid amorphous phase and mesophase in electrospun fibers[END_REF] investigated the formation of mesophase in electrospun polylactide. They show [START_REF] Ma | Relationship between the rigid amorphous phase and mesophase in electrospun fibers[END_REF] that mesophase behaves similarly to rigid amorphous fraction (RAF), i.e., the part of the amorphous phase, resulting from the coupling between crystal and amorphous phases, which does not relax at the glass transition [START_REF] Wunderlich | Thermal Analysis of Polymeric Materials[END_REF][START_REF] Chen | Vitrification and devitrification of rigid amorphous fraction of PET during quasi-isothermal cooling and heating[END_REF][START_REF] Wunderlich | Reversible crystallization and the rigid-amorphous phase in semicrystalline macromolecules[END_REF].

The possibility to process nanofibers that are both strong and tough is very appealing. To reach this objective, the microstructure in the nanofibers should be perfectly controlled. The evaluation of orientation is particularly critical since the polymer chain alignment is one probable needed condition for the nanofiber to exhibit high values of modulus, true strength, and toughness, as previously reported [START_REF] Papkov | Simultaneously Strong and Tough Ultrafine Continuous Nanofibers[END_REF]. Despite the interest to characterize polymer chain orientation in electrospun fibers, access to such information is highly challenging [START_REF] Richard-Lacroix | Molecular Orientation in Electrospun Fibers: From Mats to Single Fibers[END_REF].

Relevant spectroscopic techniques exhibit significant artefacts due to light interaction with sub-wavelength diameter fibers, exhibiting high surface curvature [START_REF] Papkov | Polarized Raman analysis of polymer chain orientation in ultrafine individual nanofibers with variable low crystallinity[END_REF]. In addition analysis of individual fibers requires designing new experimental protocols. Consequently, most of the studies are currently carried out on nanofiber mats. The question of the quantification of polymer chain orientation in nanofibers has been reviewed recently [START_REF] Papkov | Quantifying Polymer Chain Orientation in Strong and Tough Nanofibers with Low Crystallinity: Toward Next Generation Nanostructured Superfibers[END_REF]. Various techniques, including X-Ray diffraction, Polarized Fourier Transform, infrared spectroscopy, thermal analyses, NMR, selected area electron diffraction, polarized optical light spectroscopy, polarized Raman spectroscopy, and also emerging techniques for evaluation of orientation, were investigated. Although some techniques are promising for quantification of orientation, most of them are restricted to the analysis of the orientation in the crystalline phase [START_REF] Papkov | Quantifying Polymer Chain Orientation in Strong and Tough Nanofibers with Low Crystallinity: Toward Next Generation Nanostructured Superfibers[END_REF].

However superfine nanofibers are characterized by their low crystallinity. Therefore the interrogation of the amorphous phase is crucial and it is of interest to propose new approaches to characterize polymer chain alignment in electrospun systems. The goal of this study is to investigate whether the orientation of the amorphous phase in the electrospun nanofibers could be tracked from thermal analysis. To do so, characterization of polyacrylonitrile (PAN) nanofiber mats has been performed by both modulated temperature differential scanning calorimetry (MT-DSC) and dielectric relaxation spectroscopy (DRS). In PAN, degradation occurs before melting. Thus the analysis of the calorimetric data is jeopardized by the degradation which modifies or suppresses the response of the sample during the melting. In this context it becomes difficult to assess properties regarding the crystalline phase of PAN from DSC curves. Only the amorphous state can be safely interrogated from usual thermal analysis protocols. Our idea is to investigate whether the glass transition signature in nanofiber mats changes as a function of the average fiber diameter. The cooperative rearranging region (CRR) concept introduced by Adam and Gibbs [START_REF] Adam | On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids[END_REF] has been applied at the glass transition of PAN. A CRR is defined [START_REF] Adam | On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids[END_REF] as a subsystem which can rearrange its configuration into another, independently of its environment upon a sufficient thermal fluctuation. Each subvolume can be then considered as a thermodynamic system in metastable equilibrium with fluctuating variables [START_REF] Adam | On the Temperature Dependence of Cooperative Relaxation Properties in Glass-Forming Liquids[END_REF]. Calculating the CRR size when the glass transition occurs in restricted volume, i.e., confined environment at the nanoscale, has been the object of previous studies [START_REF] Tran | Nanoscale characteristic length at the glass transition in confined syndiotactic poly(methyl methacrylate)[END_REF][START_REF] Arabeche | Study of the cooperativity at the glass transition temperature in PC/PMMA multilayered films: influence of thickness reduction from macro-to nanoscale[END_REF][START_REF] Schick | Characteristic length of glass transition: experimental evidence[END_REF]. A decrease of the CRR size has been reported when the amorphous phase is confined between the layers of organophilic clay [START_REF] Tran | Nanoscale characteristic length at the glass transition in confined syndiotactic poly(methyl methacrylate)[END_REF], in multi nanolayer polymer films [START_REF] Arabeche | Study of the cooperativity at the glass transition temperature in PC/PMMA multilayered films: influence of thickness reduction from macro-to nanoscale[END_REF], and by crystals [START_REF] Schick | Characteristic length of glass transition: experimental evidence[END_REF]. Furthermore, many investigations on drawn films also evidenced a decrease in the cooperativity length [START_REF] Hamonic | Contribution of chain alignment and crystallization in the evolution of cooperativity in drawn polymers[END_REF][START_REF] Furushima | The characteristic length of cooperative rearranging region for uniaxial drawn poly(ethylene terephthalate) films[END_REF][START_REF] Delpouve | Structural Dependence of the Molecular Mobility in the Amorphous Fractions of Polylactide[END_REF][START_REF] Delpouve | Amorphous Phase Dynamics at the Glass Transition in Drawn Semi-Crystalline Polyester[END_REF][START_REF] Lixon | Evidence of Cooperative Rearranging Region Size Anisotropy for Drawn PET[END_REF] as well as anisotropy in the CRR size [START_REF] Lixon | Evidence of Cooperative Rearranging Region Size Anisotropy for Drawn PET[END_REF]. Monnier et al. [START_REF] Monnier | Molecular Dynamics in Electrospun Amorphous Plasticized Polylactide Fibers[END_REF] investigated the CRR size in electrospun fibers of polylactide. They showed that the very early stage of orientation of electrospinning is characterized by the cohesive role of mesophase which leads to a slight increase in the CRR size [START_REF] Monnier | Molecular Dynamics in Electrospun Amorphous Plasticized Polylactide Fibers[END_REF]. Cho et al. [START_REF] Cho | The glass transition and interfacial dynamics of single strand fibers of polymers[END_REF] showed by employing Adam-Gibbs theory that the fast interface dynamics of polymer fibers should influence the cooperative motion of monomers.

Furthermore, the calculation of CRR size has recently be applied to new biobased polymers [START_REF] Codou | Glass transition dynamics and cooperativity length of poly(ethylene 2,5-furandicarboxylate) compared to poly(ethylene terephthalate)[END_REF].

In this work the relaxation dynamics of PAN nanofibers mats are investigated at the calorimetric glass transition from MT-DSC, and in the liquid state from DRS. The temperature dependence of the relaxation time is obtained from DRS, and the degree of deviation from Arrhenius behaviour, i.e., the fragility index [START_REF] Angell | Spectroscopy simulation and scattering, and the medium range order problem in glass[END_REF] is calculated. Since the improvement of mechanical properties, essentially attributed to the high orientation in the amorphous phase linked to low degree of crystallinity, is observed for diameter less than 250 nm, the average diameter of the PAN fiber in the mat goes down to 232 nm.

EXPERIMENTAL Materials

PAN fibers were electrospun at ambient conditions from PAN (Pfaltz and Bauer, Inc.®; cat# P21470, MW = 150 000 g/mol in average) solutions in N,N-dimethylformamide (DMF) (Sigma-Aldrich®; cat# 271012) in a 1 mL syringe using a 20 gauge needle and a feed rate equal to 0.6 mL/h. Fibers were collected on a stationary target covered with aluminium foil.

The applied voltage was 12 kV; the distance between the spinneret and collector was 20 cm.

The samples were electrospun from 8, 10, and 12 wt% of PAN in DMF.

Scanning electron microscopy (SEM)

Nanofiber diameter was examined by scanning electron microscopy (SEM) in an environmental apparatus (Quanta 200 FEG from FEI®) with no coating and no additional preparations. Several images of each mat were taken at different locations. Nanofiber diameters were measured for at least 200 fibers in each sample using ImageJ® software, average diameter was calculated, and a histogram was constructed.

Modulated temperature differential scanning calorimetry (MT-DSC)

Experiments were performed on a DSC Q100 Thermal Analysis® instrument. The samples, cut from fiber mats, then folded, were about 2 mg, encapsulated in Tzero® DSC aluminum alloy pans, and disposed to recover the surface of the pan in contact with the thermal sensor.

Calibration in temperature was carried out using standards of indium and benzophenone.

Calibration in energy was carried out using standard values of indium. The specific heat capacity for each sample was measured using sapphire as a reference. The modulation parameters were chosen to obtain a clear view of the glass transition (oscillation amplitude of 2.5 K, oscillation period of 100 s, and heating rate of 1 K.min -1 ). The furnace was permanently swept by nitrogen flow (50 mL/min).

Broadband Dielectric Spectroscopy (DRS)

Dielectric relaxation spectra were measured with an Alpha Analyzer from Novocontrol® (frequency interval: 10 -1 -10 6 Hz). A portion of the mat was cut and placed between parallel electrodes, and the temperature was regulated through a heated flow of nitrogen gas, by means of a Quatro Cryosystem®, from 343 K to 473 K. The sample geometry was ranged from 80 to 100 μm thick and 20 mm diameter (upper electrode) for each sample. The thickness was measured when the nanofiber mat was placed between the electrodes. The complex permittivity ε* is calculated from:

* = - = * (1)
Where Z* = U* / I* is the impedance calculated from the Novocontrol system, U* is the applied voltage and I* the measured current, ε' and ε'' are the real and imaginary parts of ε*, i is the unit imaginary number, ω (=2πf) is the angular pulsation and f the frequency, C0 is the ideal empty cell capacity:

= ( 2
)
Where ε0 is the vacuum permittivity, D is the diameter of the capacitor (upper electrode) and d is the thickness of the sample.

The conductivity σ * of the material is calculated from the permittivity:

* = - = 2 * -1 (3)
To analyze dielectric relaxation curves, the imaginary part of the permittivity has been fitted thanks to Havriliak-Negami (HN) complex function [START_REF] Havriliak | A complex plane analysis of α-dispersions in some polymer systems[END_REF]. Grafitylab software was used to fit the experimental data using the least squares method. The correlation coefficient of the Havriliak-Negami fits was higher than 0.99 for the whole sample series. The relaxation times were extracted isothermally using a HN contribution and a conductivity term.

* = - + ∑ ∆ $ %&' ( $ ) $* + $ + , - (4)
Where k is the number of contributions for the fitting procedure (equal to 1 in the present study), n is a fitting parameter related to the conductivity slope, Δε is the dielectric strength, τ is the relaxation time, and α and β are the broadening and asymmetric shape factors. ε∞ is the permittivity at high frequency of the relaxation process, with Δε = εS -ε∞, where εS is the permittivity at low frequency (static). The errors associated to the fitting procedure are estimated about 5% for α and β and 3% for σ, n, τ and Δε.

RESULTS AND DISCUSSION

SEM pictures of the electrospun fibers are given in Figure 1. For all compositions, the fiber shape is globally homogeneous and the majority of the fibers do not show any beading effect.

For 8, 10, and 12 wt% of PAN in DMF, the average diameter of fibers in the mats are 232 nm (Fig. 1a), 451 nm (Fig. 1b), and 785 nm (Fig. 1c), respectively. In the following sections, the samples are named as PAN_232, PAN_451, and PAN_785 accordingly. The diameter is less dispersed when decreasing the concentration of polymer in solvent.

PAN_785 exhibits the widest dispersion in the fiber diameter, whereas the diameter profile is the narrowest for PAN_232. Besides, the fiber diameter distribution of PAN_232 does not extend beyond 375 nm, whereas 375 nm is close to the minimum fiber diameter in PAN_785.

Therefore the role of fiber size on both thermal characteristics and relaxation properties can be reliably investigated through the analysis of the three nanofiber mats. From the average heat flow curves recorded from MT-DSC (shown in Supporting Information Figure SI.1), the calorimetric signature of the glass transition is indiscernible since it is covered by parasitic thermal events. A clear view of the glass transition can however be obtained by applying the complete deconvolution procedure, suggested by Reading [START_REF] Lacey | Theory and practice of modulated temperature differential scanning calorimetry[END_REF], to obtain the in-phase component C' of the complex heat capacity. MT-DSC recorded variations of C' in the glass transition domain are presented in Figure 2a. Characteristic parameters obtained from the glass transition investigation are given in Table 1. The curves have been shifted to coincide at 333 K with the solid heat capacity values given by Gaur et al. [START_REF] Gaur | Heat capacity and other thermodynamic properties of linear macromolecules VI. Acrylic polymers[END_REF]. The corresponding equation of solid heat capacity that was reported is Cp = exp[0.222649 (ln T) 3 -3.36593 (ln T) 2 + 17.6347 (ln T) -28.1638] J mol -1 K -1 . This procedure can be applied since the solid heat capacity is expected to be largely independent on the microstructure [START_REF] Wunderlich | Heat capacities of solid polymers. The Advanced THermal Analysis System[END_REF]. This means that, although the degree of crystallinity and the degree of orientation should be different between samples, we assume the solid heat capacity to remain invariant. It is helpful when the heat capacity variation with temperature has to be accurately known, as for the CRR calculation. The glass transition temperature measured at midpoint Tg mid is shifted towards higher temperature (from 351 to 360K) with the decrease of the fiber diameter. Furthermore, the glass transition significantly broadens. According to Hempel et al. [START_REF] Hempel | Characteristic Length of Dynamic Glass Transition near Tg for a Wide Assortment of Glass-Forming Substances[END_REF], the temperature fluctuation δT associated with the main relaxation process (α-relaxation) reflects the heterogeneity in the amorphous phase dynamics. δT was estimated by fitting dC'/dT (shown in Figure 2b) with a Gaussian peak shape and determining its standard deviation. δT increases from 4.0 K for PAN_785 to 6.2 K for PAN_232. Many studies report an enlargement of the glass transition, i.e., an increase of δT, when the amorphous phase mobility is hampered by geometrical confinement [START_REF] Tran | Nanoscale characteristic length at the glass transition in confined syndiotactic poly(methyl methacrylate)[END_REF][START_REF] Arabeche | Study of the cooperativity at the glass transition temperature in PC/PMMA multilayered films: influence of thickness reduction from macro-to nanoscale[END_REF], crystalline growth [START_REF] Schick | Characteristic length of glass transition: experimental evidence[END_REF][START_REF] Delpouve | Structural Dependence of the Molecular Mobility in the Amorphous Fractions of Polylactide[END_REF], or orientation [START_REF] Hamonic | Contribution of chain alignment and crystallization in the evolution of cooperativity in drawn polymers[END_REF][START_REF] Furushima | The characteristic length of cooperative rearranging region for uniaxial drawn poly(ethylene terephthalate) films[END_REF][START_REF] Delpouve | Structural Dependence of the Molecular Mobility in the Amorphous Fractions of Polylactide[END_REF][START_REF] Delpouve | Amorphous Phase Dynamics at the Glass Transition in Drawn Semi-Crystalline Polyester[END_REF][START_REF] Lixon | Evidence of Cooperative Rearranging Region Size Anisotropy for Drawn PET[END_REF]. From Figure 2a, one can observe that the heat capacity step at the glass transition is significantly higher for PAN_232 (∆Cp = 0.40 J g -1 K -1 versus approximately 0.30 J g -1 K -1 for the two other materials). This indicates that the relative content of amorphous phase relaxing at the glass transition increases with the decrease of the average fiber diameter. This means that the broadening and the shift of the glass transition for PAN_232 are not related to higher crystallinity content. This result is however consistent with previous work reporting that the reduced diameter fibers have less crystallinity but improved chain orientation [START_REF] Papkov | Simultaneously Strong and Tough Ultrafine Continuous Nanofibers[END_REF]. Similar results were obtained by characterizing electrospun PAN fibers from several techniques including X-Ray diffraction, Infrared and Raman spectroscopy [START_REF] Papkov | Quantifying Polymer Chain Orientation in Strong and Tough Nanofibers with Low Crystallinity: Toward Next Generation Nanostructured Superfibers[END_REF]. One concern for the analysis of the permittivity results is that PAN mats between parallel electrodes are not bulky materials; instead they contain a certain volume of non-controlled air (nanofibers + air between nanofibers). The real part of the permittivity of air, equal to 1, is different from the one of PAN bulk material or fiber alone. Thus, even knowing the perfect thickness of the NF mats, the measurement will not provide the accurate permittivity value of the PAN fiber alone. Electromagnetic mixing models may help to calculate the average permittivity of the two-phase material. However, the exact amount of air in the system is not known. Consequently, a normalization of the permittivity was performed to compare the shape and position of the peak maximum of the α-relaxation, i.e., real and imaginary part of the permittivity were divided by the value of ε' and ε'' at the related α-relaxation time (at 419K). This leads to ε''/ε''max = 1 for the relaxation time corresponding to the maximum response associated with the dielectric manifestation of the glass transition. Therefore, both real and imaginary parts of the complex dielectric permittivity recorded at 419 K (random choice in the investigated temperature range) are presented after normalization in Figure 3a and Figure 3b At low frequencies, conductivity phenomena are superimposed to the relaxation. One can notice that the observation of relaxation is extended to 10 3 Hz for the fiber mats with largest diameter nanofibers, whereas it is restricted by conductivity to high frequencies (10 4 to 10 6

Hz) for PAN_232, which also exhibits the highest conductivity values. Figure 4 depicts the real (a) and imaginary (b) parts of the conductivity. It can be seen that the DC plateau corresponding to the nearly linear behavior of the real part of the conductivity, increases when the fiber diameter decreases which reveals the higher conductive character when decreasing the fiber diameter. These results are consistent with the study of Zhang and Rutledge [START_REF] Zhang | Electrical Conductivity of Electrospun Polyaniline and Polyaniline-Blend Fibers and Mats[END_REF],

which reports that the conductivity of polyaniline fibers increase by solid state drawing which both increases orientation, and decreases the average diameter. The increase in the real part of the permittivity is most likely related to the undesirable effect of electrode polarization which is observable in Figure 4b by the change of slope in the imaginary part of the conductivity In Figure 7, ∆ε is normalized to its maximum value, and plotted as a function of temperature for all the samples. This allows comparing directly the trend of the dielectric strength for each sample as a function of temperature. However, we do not recommend comparing the absolute value of the dielectric strength as the amount of air is influencing this value. The variation of the normalized dielectric strength with temperature follows a similar trend for all the samples and converges at high temperature. The expression of ∆ε is given by the Kirkwood-Fröhlich equation [START_REF] Böttcher | Theory of electric polarization[END_REF][START_REF] Onsager | Electric Moments of Molecules in Liquids[END_REF]:

∆ = & . / 0 -1 2 3 4 ( 5
)
Where ε0 is the vacuum permittivity, g is the Kirkwood correlation factor, µ 2 is the timecorrelation function of the total dipole moment, kB is Boltzmans's constant, T is the temperature, and N/V is the volume density of dipoles. While Eq. 5 shows that ∆ε is inversely proportional to the temperature, Figure 7 reveals in contrast that the dielectric strength monotonically increases with temperature. An overall increase in ∆ε might be related to an increase of the correlation factor (which is however known to be inversely dependent on the temperature), an increase in the dipole moment (which seems unlikely since all the fiber mats are made of PAN), or an increase in the density of dipoles N/V. The increase of N/V with temperature has previously been reported for semi-crystalline polymers [START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly(Hydroxybutyrate-Co -Hydroxyvalerate)[END_REF] and is associated with the progressive mobilization of constrained amorphous phase which devitrifies in a temperature domain that is not limited to the calorimetric glass transition. For example this could indicate the presence of RAF in the material. Indeed RAF is the part of the amorphous phase that does not mobilize during the glass transition but devitrifies in a temperature domain, often broad, which stands above the glass transition. Electrospinning induces an orientation of the polymer chains. Thus, we hypothesize that this behavior is the signature of initial orientation, which severely constrains the mobility in the amorphous phase close to the calorimetric glass transition, but then progressively vanishes with the temperature increase due to polymer chain relaxation. Until merging of the curves, the dielectric strength is the lowest for PAN_232. This is consistent with the idea that polymer chains are more oriented in this sample. In Figure 8a the relaxation time is plotted as a function of the inverse of temperature. As expected, the samples exhibit a non-linear dependence of the relaxation time with temperature, which is typical of the α-relaxation process associated with the glass transition.

The Arrhenius plots were fitted with the Vogel-Tamman-Fulcher (VTF) law:

5 = 5 678 2 2 2 (6)
Where τ0 is a pre-exponential factor, D is a dimensionless parameter related to the slope variation (steepness strength), and T0 is the Vogel temperature. VTF fitting parameters are given in Table 2. We observe a good concordance of the fits with, and without, adding the calorimetric glass transition data at the relaxation time corresponding to a period of 100s of MT-DSC analysis.

At high temperature the relaxation time is the same for all materials, while the curves start deviating from each other with the decrease in temperature. PAN_232 exhibits the highest relaxation times close to the glass transition. From a Tg normalized representation of the relaxation time variations (Figure 8b), it is possible, as proposed by Angell [START_REF] Angell | Spectroscopy simulation and scattering, and the medium range order problem in glass[END_REF], to compare the materials by the slope of the deviation from the Arrhenius behavior, using the fragility index m.

=

:;< = ( >2/2 @ A B 2C2 @ [START_REF] Séon-Lutz | Electrospinning in water and in situ crosslinking of hyaluronic acid / cyclodextrin nanofibers: Towards wound dressing with controlled drug release[END_REF] Despite the shift of the glass transition, PAN_451, and PAN_785 exhibit the same fragility index (about 100). On the other hand the slope is more steep for PAN_232 leading to a fragility index of 133. The uncertainties on kinetic fragility are about 10% of the calculated index (see Table 1). According to the approach of Donth [START_REF] Donth | The Size of Cooperatively Rearranging Regions at the Glass Transition[END_REF], the cooperativity length ξ can be calculated from the temperature fluctuation associated with the glass transition, which can be estimated from calorimetric measurements such as MT-DSC using Equation 8:

D 2 ) . = E = FG @HIJJ L) = FG HMNOMP L) Q-1 2 ) RS2 (8)
Where Cp is the specific heat capacity at constant pressure, kB is the Boltzmann's constant, Tα is the dynamic glass transition temperature, ρ is the density equal to 1.2 [START_REF] Hurley | Density of polyacrylonitrile[END_REF], and δT is the mean square temperature fluctuation associated with the glass transition. The Cp glass and liquid values as well as the δT were extracted first by MT-DSC measurements from the calorimetric glass transition temperature for every sample. Values of ξ = 1.7 nm, 1.9 nm, and 2.1 nm were obtained for PAN_232, PAN_451, and PAN_785 respectively. Saiter et al. [START_REF] Saiter | Temperature dependence of the characteristic length scale for glassy dynamics: Combination of dielectric and specific heat spectroscopy[END_REF] proposed to extend the calculation of the cooperativity length from the calorimetric glass transition to the rise of cooperativity at the temperature of crossover, by combining DRS and MT-DSC measurements. To do so, δT was estimated as a function of temperature from DRS results. Cp glass was estimated as a function of temperature from the equation proposed by Gaur et al. [START_REF] Gaur | Heat capacity and other thermodynamic properties of linear macromolecules VI. Acrylic polymers[END_REF]. Cp liquid was recorded as a function of temperature from MT-DSC measurements. Data are reported in the supporting information as Table SI.2, SI.3 and SI.4 for PAN_232, PAN_451 and PAN_ 785 respectively. As can be seen in Figure 9a, for temperatures significantly above the calorimetric glass transition, the cooperativity length is identical between samples. Moreover, the cooperativity decreases with the increase of temperature, consistently with previous studies [START_REF] Saiter | Temperature dependence of the characteristic length scale for glassy dynamics: Combination of dielectric and specific heat spectroscopy[END_REF][START_REF] Rijal | Dynamic heterogeneity and cooperative length scale at dynamic glass transition in glass forming liquids[END_REF]. The cooperativity is plotted in Figure 9b as a function of relaxation time. For τ = 16 seconds, which corresponds to the calorimetric glass transition, cooperativity differs between the samples. ξ decreases with the fiber average diameter. A decrease of cooperativity was previously reported when increasing the polymer chain orientation [START_REF] Hamonic | Contribution of chain alignment and crystallization in the evolution of cooperativity in drawn polymers[END_REF][START_REF] Furushima | The characteristic length of cooperative rearranging region for uniaxial drawn poly(ethylene terephthalate) films[END_REF][START_REF] Delpouve | Structural Dependence of the Molecular Mobility in the Amorphous Fractions of Polylactide[END_REF][START_REF] Delpouve | Amorphous Phase Dynamics at the Glass Transition in Drawn Semi-Crystalline Polyester[END_REF][START_REF] Lixon | Evidence of Cooperative Rearranging Region Size Anisotropy for Drawn PET[END_REF].

On the other hand, for lower relaxation times, the cooperativity is the same between PAN_232, PAN_451, and PAN_785. This again suggests that the relaxation dynamics are only influenced by orientation close to the calorimetric glass transition.

Hong et al. [START_REF] Hong | Is there a connection between fragility of glass forming systems and dynamic heterogeneity/cooperativity?[END_REF] separated the fragility index m in two contributions (see Eq. 9). The index mV depicts the pure thermal variations of the relaxation time in isochoric conditions. Its counterpart (m -mV) represents the fragility term that is sensitive to volume variations.

9 = 9 4 + T4 # V & -1 W L X ( 9
)
Where κ is the compressibility and αT is the coefficient of thermal expansion of the supercooled liquid at Tg. The ratio αT/κ goes from 0.5 to 3.0 MPa / K for a wide range of glass formers, and ∆V # is equal to 5% of the cooperativity volume. Therefore (m -mV) depends on the cooperativity length. For an average value αT/κ = 1.5 MPa / K, (m -mV) = 9, 13, and 19 for PAN_232, PAN_451, and PAN_785 respectively were obtained. Thus PAN_451 and PAN_785 exhibit close values for mV, 93 and 81 respectively, while mV equal to 124, for PAN_232, is significantly higher. As shown in a previous study on plasticized polylactide [START_REF] Araujo | Reducing the gap between the activation energy measured in the liquid and the glassy states by adding a plasticizer to polylactide[END_REF], the variations of cooperativity and fragility are correlated under the condition that mV almost remains invariant. On the other hand, relaxation parameters m and ξ can also exhibit opposite variations [START_REF] Delpouve | Structural Dependence of the Molecular Mobility in the Amorphous Fractions of Polylactide[END_REF][START_REF] Sharma | Unusual fragility and cooperativity in glass-forming and crystalline PVDF/PMMA blends in the presence of multiwall carbon nanotubes[END_REF] like in the present study. By investigating the relaxation dynamics in interpenetrated networks [START_REF] Araujo | A structural interpretation of the two components governing the kinetic fragility from the example of interpenetrated polymer networks[END_REF], we recently gave a structural interpretation of each fragility contribution in polymers. (m -mV), which follows cooperativity, depicts the level of interchain interactions, whereas mV is a parameter characteristic of the backbone rigidity, increasing with intrachain interactions, that could develop during orientation. It is assumed that mV is the highest for PAN_232 because the amorphous phase is more oriented in this sample. One can consider that in the present study mV gives qualitative information regarding the degree of orientation in fiber mats. More generally, we observe that the average fiber diameter has a strong impact on several relaxation parameters obtained from thermal analysis. The combined use of MT-DSC and DRS is potentially interesting to supplement data obtained from spectroscopy techniques such as X-Ray diffraction.

CONCLUSION

Characterizing polymer chain orientation in electrospun fibers is of high interest since their outstanding mechanical properties result from a combination of low crystallinity and high polymer chain alignment. This study shows the possibility to obtain relative information, i.e., to compare electrospun fiber mats, by investigating the relaxation dynamics, from the combination of modulated temperature differential scanning calorimetry (MT-DSC) and dielectric relaxation spectroscopy (DRS). The mat with smaller average nanofiber diameter exhibits both a decrease of the cooperativity length at the calorimetric glass transition, and an increase of the thermal contribution of fragility. Our interpretation is that these results are caused by a constraining of amorphous phase dynamics and by an increase in the polymer chain rigidity respectively. The potential explanation that these results could be related to an increase of the degree of crystallinity is inconsistent with the experimental results since the material exhibiting the lowest cooperativity also has the highest content of amorphous phase relaxing at the glass transition. More probably the change in the glass transition dynamics is associated with a stronger polymer chain alignment. Moreover, the relaxation dynamics, investigated from DRS, merge far above the calorimetric glass transition but significantly below the melting domain, suggesting a constraint slackening in the amorphous phase when crossing the glass transition. These results are also consistent with the decrease in crystallinity for smaller average nanofiber diameters previously observed by X-Ray diffraction studies [START_REF] Papkov | Quantifying Polymer Chain Orientation in Strong and Tough Nanofibers with Low Crystallinity: Toward Next Generation Nanostructured Superfibers[END_REF]. It is worth mentioning that the influence of the orientation on the relaxation dynamics is only detected for the mat with the smallest average fiber diameter. This result is in agreement with previous investigations [START_REF] Papkov | Simultaneously Strong and Tough Ultrafine Continuous Nanofibers[END_REF] showing major improvements in mechanical properties for nanofibers with diameter below 250 nm.

Figure 1 .

 1 Figure 1. SEM pictures along with the fiber diameter distribution of as-spun solutions containing 8 (a), 10 (b), and 12 (c) wt% of PAN in DMF respectively. Samples are named PAN_x, where x is the fiber average diameter in the PAN.

Figure 2 .

 2 Figure 2. (a) MT-DSC in-phase (C') component of the complex heat capacity and (b) its derivative dC'/dT versus temperature for PAN_232, PAN_451, and PAN_785.

Figure 3 .

 3 Figure 3. Real (a) and imaginary (b) parts of the complex dielectric permittivity recorded at 419 K from DRS for PAN_232, PAN_451, and PAN_785. Data have been normalized to the relaxation maximum response.

  respectively (raw data are provided in Supporting Information Figure SI.3). Nevertheless, according to Figure 3b there is no significant change in the position of the maximum (refer to Supporting Information Figure SI.3 for cross comparison between curves).

[ 50 ]

 50 . It can be seen in Fig 3 and Fig 4 that the deviation (in frequency) from AC regime in both real and imaginary part of the conductivity coincide with the increase in the imaginary and real part of the permittivity respectively, related respectively to the conductivity of the material and the parasite electrode polarization effect.

Figure 4 .

 4 Figure 4. Real (a) and imaginary (b) parts of the conductivity as a function of frequency at 146°C for PAN_231, PAN_451 and PAN_785.

Figure 5 .

 5 Figure 5. Example of the fitting procedure with a conductivity term and a HN contribution performed for PAN_785 at 130°C. Empty squares are the experimental points, red dash line is the HN function, blue dash line is the conductivity term and dark cyan dash line is the sum of the two contributions. The obtained correlation coefficient is equal to 0.998 after 2 iterations.

Figure 6 .

 6 Figure 6. Fitting parameters α and β, obtained from Havriliak-Negami equation, versus inverse temperature for PAN_232, PAN_451, PAN_785.

Figure 7 .

 7 Figure 7. Evolution of the normalized dielectric strength with temperature obtained from DRS measurements for PAN_232, PAN_451, and PAN_785. The error bars were estimated as 10% of the recorded value.

Figure 8 .

 8 Figure 8. (a) Evolution of the relaxation time with the inverse of temperature from DRS measurements and (b) Tg normalized variations of the relaxation time for PAN_232, PAN_451, and PAN_785. Dashed curves correspond to VTF fit from DRS data and solid curves correspond to VTF fit from the combination of DRS and MT-DSC data.

Figure 9 .

 9 Figure 9. (a) Cooperativity length as a function of temperature, and (b) as a function of the relaxation time, calculated for PAN_232, PAN_451, and PAN_785.

Table 1 .

 1 Relaxation parameters obtained from MT-DSC and DRS for PAN_232, PAN_451, and PAN_785. Fragility values are given with an uncertainty of 10 %. mV and (m -mV) are calculated for

	αT/κ = 1.5 MPa / K.
		MT-DSC			DRS
	∆Cp	Tg mid	δT	ξΤα	m	mV	m -mV
	(J g -1 K -1 )	(K)	(K)	(nm)
	PAN_232 0.40 ± 0.04	360 ± 1	6.2 ± 0.2	1.7 ± 0.2	133 ± 14	124 ± 13	9 ± 1
	PAN_451 0.27 ± 0.04	354 ± 1	4.5 ± 0.2	1.9 ± 0.2	106 ± 11	93 ± 10	13 ± 2
	PAN_785 0.30 ± 0.04	351 ± 1	4.0 ± 0.2	2.2 ± 0.2	100 ± 10	81 ± 9	19 ± 2

Table 2 .

 2 Fitting parameters of the VTF equation for PAN_232, PAN_451 and PAN_785 obtained from the combination of MT-DSC and DRS results (solid line).

		log τ0	D	T0
	PAN_232	-10.5 ± 0.1	3.1 ± 0.1	321.2 ± 0.8
	PAN_451	-10.9 ± 0.1	3.8 ± 0.1	314.0 ± 0.5
	PAN_785	-11.0 ± 0.1	4.2 ± 0.2	307.5 ± 1.2

ACKNOWLEDGEMENTS

Dimitry Papkov and Yuris Dzenis gratefully acknowledge partial support of this work by grants from ONR (N000141410663), NSF (DMR-1310534, CMMI-1463636), NIH (1R01HL125736-01), and NCESR.


H Kanso

R Patte

D Ledue

Antiferromagnetic thickness and temperature dependence of the exchange bias properties of Co/IrMn nanodots and continuous films: A Monte Carlo study

Motivated by the challenge of understanding the complex influence of the antiferromagnetic (AF) thickness and the temperature on exchange bias (EB) properties, and by the necessity of miniaturization of devices, we investigate EB properties of Co/IrMn nanodots and of continuous films by using kinetic Monte Carlo simulations. To that purpose, we use a granular model, which takes into account disordered interfacial phases in the AF layer and, in the case of nanodots, disordered phases at the edges in the AF layer. Our results show that the AF thickness dependence of the exchange field HE (measured at room temperature) in both nanodots and continuous films exhibits a maximum in agreement with experimental results. We explain these results in terms of superparamagnetic and blocked grains in the AF layer at room temperature and also not polarized AF grains during the initial field-cooling. The simulated values of HE in nanodots are smaller than that in continuous films for small AF thicknesses and larger for larger ones due to the contribution of the disordered phases at the edges in the AF layer. Also, we investigate the temperature and AF thickness effects on HE and on the coercive field HC. We found that HE slightly decreases at low temperatures due to the disordered interfacial phases. Importantly, at the maximum blocking temperature of the AF grains, HE vanishes and HC exhibits a maximum. Our numerical results are successfully compared to experimental data on Co/IrMn bilayers for various IrMn thicknesses and all temperatures. In addition, our results indicate that HE is smaller in nanodots at low measurement temperature due to the presence of disordered phases at the edges. Concerning HC, our data show that it can be either larger or smaller in nanodots depending on the measurement temperature.

I. INTRODUCTION

The phenomenon of EB [1][2][3][4] has been extensively investigated in the last few decades, mainly in continuous films, from both experimental and fundamental points of view. EB occurs due to the coupling at the interface between ferromagnetic (F) and AF materials and depends on various parameters such as bulk and interfacial exchange couplings, bulk anisotropies, layer thicknesses, measurement and cooling temperatures ... Among these parameters, the measurement temperature and the AF layer thickness (tAF) dependence of EB properties are the most frequently investigated from an experimental point of view. It should be noted that most of the theoretical models for EB properties do not account for thermal effects [5][6][7][8][9]. Recently, many experimental studies were conducted to well understand the thermal effects on the EB properties and they have shown some common behaviors of EB properties and some contradictory behaviors [10][11][12][13][14][15][16][17]. The main common behavior is that the exchange field HE and the coercive field HC decrease as the temperature increases at low temperatures. As the temperature increases, HC increases and reaches a peak, while HE still decreases in some studies [10][11][12][13][14] or increases and reaches a peak in other studies [15][16][17]. At higher temperatures, both HE and HC decrease again with increasing temperature. Now, considering the effect of tAF the results indicate either a decrease [18,19] or a maximum [20,21] of HE as tAF increases. The decrease of HE is usually ascribed, in a random field model, to the presence of AF domains, whose structure strongly depends on tAF [7,22,23]. The presence of a maximum in HE is attributed to a thermal effect on the stability of the grains.

More recently, since EB is used in spintronic devices such as spin valves and magnetic tunnel junctions [24,25], and because of the necessity of increasing the magnetic storage density and reducing devices sizes [26], investigations have been carried out on nanodots with lateral sizes of few hundred nanometers [27]. Then, in addition to the various parameters which affect EB properties in continuous films, size and boundary effects play an important role in nanodots which complicates the understanding of EB properties. Indeed, contradictory results on HE in nanodots compared to that of continuous films have been reported. For example, in NiFe/IrMn bilayers, it was observed that HE at room temperature is larger in nanodots for thicknesses tIrMn > 11 nm and smaller below this thickness compared to continuous films [18,28]. On the other hand, in Ref. [29] opposite results have been found. In another recent study on Co/IrMn nanodots at room temperature [20], it was shown that the dot lateral size has no significant effect on HE (3 nm ≤ tIrMn ≤ 15 nm). For the same system Co/IrMn (tIrMn = 7 nm), it was observed that HE at 4 K is smaller in nanodots than in 24-06-2019 3 continuous films [30]. An important difference between nanodots and continuous films is the dot edges which consist of additional locations for the formation of disordered magnetic phases [30]. Nevertheless, few numerical studies on EB properties in F/AF nanodots have been carried out. A granular approach has shown a significant difference in HE between nanodots and continuous films [31]. However, we wish to emphasize that the model used is rather simple since it does not include disordered phases neither at the F/AF interface, nor at the edges in the AF layer although these disordered phases have significant effect on EB properties [7,32,33]. Finally, let us mention an atomistic model which has shown an increase of HE for systems patterned with small FM dots compared to continuous films [34].

In this study, we investigate EB properties of square F/AF nanodots and continuous films based on a granular model which accounts for the disordered interfacial phases by considering less stable magnetic grains at the interface in the AF layer [35,36]. In addition, we model the effect of the nanofabrication process by the presence of less stable grains at the edges of the nanodots in the AF layer as experimentally demonstrated [30]. We first study the effect of the AF thickness on the EB properties of nanodots in comparison with continuous films at room temperature. Then we investigate the effect of temperature on the EB properties of nanodots and continuous films. In these two cases, our results are compared to experimental data on Co/IrMn bilayers. Our work is carried out by kinetic Monte Carlo simulations.

The remaining of the paper is organized as follows: The model and simulation technique are described in Sec. II. Numerical results of the effect of the AF thickness and temperature in Co/IrMn nanodots and continuous films in comparison with experimental data are given in Sec. III, and a conclusion is given in Sec. IV.

II. MODEL AND SIMULATION

The model is essentially that described in details in Refs [35,36]. The F and AF layers are modeled by sets of grains coupled by exchange interactions. Since we assume columnar growth, they have the same granular microstructure which is created using a Voronoi tessellation [37] in two-dimensions. In nanodots, we model the effect of the nanofabrication process by the presence of grains with reduced areas at the edges due to grain cutting in the two layers. In Fig. 1 we present the top view of the granular microstructure for the continuous film and a nanodot, and the corresponding grain volume distribution for each case.

Importantly, the presence of smaller grains at the edges is noticeable in the nanodot compared to the continuous film. In addition to F and AF grains, small grains (SG) of thickness tSG < tAF are spread randomly over the F/AF interface within the AF layer in both nanodots and continuous films (Fig. 2) [35,36]. These SG take into account disordered interfacial phases 33] due to magnetic frustration produced by defects at the F/AF interface (e. g. interlayer diffusion and stacking faults). So, we consider that these SG exhibit altered magnetic properties (reduced anisotropy and coupling compared to the bulk AF grains). Assuming that the disordered interfacial magnetic phases extended over 3-4 atomic planes [38,39], we set tSG= 2 nm. Since, the fraction of SG at the F/AF interface xSG may be varied between about 20% and 80%, as it depends on the fabrication process [40], we set xSG to an average value of 50% in both nanodots and continuous films. In addition to that, we assume that the grains which are at the edges in the AF layer in nanodots (which will be referred as SGE) exhibit the same magnetic 24-06-2019 5 properties as the SG (Fig. 2) [30]. Accordingly, the fraction of grains with altered magnetic properties (SG and SGE) is larger in nanodots since there is no SGE in continuous films. Note that since tSGE = tAF is larger than tSG, the blocking temperature (TB) distribution of a nanodot differs from that a continuous film.

The exchange energy between two grains is

j i ij ex J E σ σ . - =
where Jij is the exchange constant (JF-AF, JF-SG or JF-F) and i σ , j σ are unit vectors representing the magnetization orientation of a F grain, a SG, a SGE or the interfacial uncompensated magnetization orientation of an AF grain. Note that the AF grains are not linked to each other [41,43], and nor are the SG and the SGE (Fig. 2). Indeed the SG (and SGE) and the AF grains are not linked to each other because such couplings do not affect directly the exchange bias properties of the bilayer. The couplings which contribute directly to HE are the interfacial couplings: between the F grains and the AF grains (JF-AF) and between the F grains and the smaller grains (JF-SG) (the smaller grains are the SG in continuous films and the SG and SGE in case of nanodots). Actually, a coupling between SG or SGE or between AF grains would slightly affect their blocking temperature and thus could affect very slightly the exchange bias properties, so we have neglected them. In addition, concerning SG (or SGE), couplings between them are expected to be small because of the non-collinearity of the spins in the disordered interfacial and edges phases, we expect that they do not affect the exchange bias properties. Assuming an uniaxial anisotropy energy along a common easy axis (y-axis) in the plane of the layer, the anisotropy energy is

Ea = -Ki Vi 2 ) . ( y i e σ
where Ki is the anisotropy constant per unit volume (KF, KAF or KSG) and Vi is the volume of the grain. We make this assumption for simplicity and also because we think that the results would not change qualitatively. Indeed, in case of random easy axes, the energy barriers to overcome during the reversal of the grains would be slightly modified which result in a small change in the blocking temperature of the grains. So, only a small change in our results could be observed. Finally, let us mention that such a magnetic texture could be obtained by applying an external field along the y-axis during the fabrication of the samples. A linear temperature dependence of the anisotropy constants per unit volume is implemented [44,14] with TN = 690 K [3]. Also, a Zeeman term Some parameters of our model can be estimated from experimental results. For example, from experimental values of HE [19], we found that jF-AF = 4.5×10 -4 J m -2 is a reasonable value in Co/IrMn bilayers. Likewise, the 0 K anisotropy constant for the AF grains was estimated to be = 0 AF K 4×10 5 J m -3 [START_REF] Vallejo-Fernandez | [END_REF]. By contrast, the coupling per unit area, jF-SG, and the effective anisotropy of SG and SGE are unknown. However, jF-SG is assumed to be smaller than jF-AF, so we set jF-SG = 3×10 -4 J m -2 . The effective anisotropy of the SG was also assumed to be weaker than that of the AF grains [30], we arbitrarily set 0 SG K to 0.75 0 AF K . In order to be in the weak coupling regime where the exchange field is independent on the F coupling (jF-F) and there is 24-06-2019 7 no variability between one nanodot to another, we choose jF-F = 10 -4 J m -2 [47]. The fixed parameters of all our simulations are summarized in Table 1.

jF-F = 10 -4 J m -2 jF-AF = 4.5×10 -4 J/m 2 jF-SG= 3×10 -4 J/m 2 TN = 690 K KAF 0 = 4×10 5 J/m 3 KSG 0 = 3×10 5 J/m 3 xSG = 50% tSG = 2 nm tSGE = tAF Table 1.
Fixed parameters of all our simulations.

Our Monte Carlo simulations [14,35,48,49] are capable of simulating cooling or heating under an external magnetic field. Thus, we can calculate the temperature dependence of the magnetic properties such as hysteresis loops from which the exchange and the coercive fields can be extracted.

III. RESULTS AND DISCUSSION

For all results presented below, we have simulated an initial field cooling (FC) from T0 down to Tf. The hysteresis loops are simulated at a fixed TM = 298 K in Sec. III.A or at increasing TM (Tf ≤TM < T0) in Sec. III.B. To ease the understanding, we want to emphasize that the grains (in the AF layer) in contact with the F layer can be separated into three groups depending on their TB and only one group contributes to HE: (i) the grains with TB > T0, are not polarized by the external field and thus have a zero net magnetization and do not contribute to HE; (ii) the grains with TM < TB < T0, are polarized by the external field and are blocked at the measuring temperature TM, so they contribute to HE; (iii) the grains with TB < TM, are polarized by the external field but they are superparamagnetic at TM so they do not contribute to HE. Since, the experimental data that we compare with them correspond to an average over numerous nanodots, we therefore average our simulation results over several nanodots. We found that 60 nanodots of averaging is enough for convergence.

A. Effect of the AF thickness (TM = 298 K)

In this part, we study the influence of the AF thickness on the EB properties, and then we compare our results with experimental data presented in Ref. [20]. To do so, we model Co/IrMn bilayers with tCo = 5 nm and various tIrMn (3 nm < tIrMn < 14 nm). For each value of tIrMn, the hysteresis loops are simulated at TM = 298 K after FC under HFC = 2 kOe from 24-06-2019 8 T0 = 473 K to Tf = 298 K. Figure 3 presents the hysteresis loops for tIrMn = 4, 8 and 14 nm for the continuous film and nanodots. The tIrMn-dependence of HE in comparison with experimental data is shown in Fig. 4 [20]. Similar to the experimental data, our results show that the HE curve exhibits a maximum in both cases of nanodots and continuous film. Note that these maxima are found at a slightly larger thickness than that reported experimentally (6.5 nm). HE since their TB are smaller than 298 K (Fig. 5), while only a fraction of AF grains does contribute where this fraction depends on tIrMn. For small thicknesses (Fig. 5.a), the fraction of AF grains which contributes to HE is small and thus the value of HE is small. When tIrMn increases, this fraction increases and reaches a maximum at tIrMn ≈ 9 nm (Fig. 5.b) leading to a maximum in HE. For tIrMn > 9 nm, the fraction of the contributing AF grains decreases and thus HE (Fig. 5.c). The small shift of our curves toward higher thicknesses compared to the experimental data is due to the small shift of the blocking temperature distribution of the AF grains in our model compared to the experimental one. The tIrMn-dependence of HC is plotted in Fig. 6 in comparison with experimental data [20].

Our results reproduce qualitatively the experimental observations in a continuous film with a maximum of HC at tIrMn = 3-4 nm. In regards to HC, the grains which contribute are those which are in contact with the F layer and having TB ≈ TM. Since all SG have TB < TM for all IrMn thicknesses (see Fig. 5), only a fraction of the AF grains in contact with the F layer contribute to Hc. Then according to Fig. 5, it can be seen that the fraction of these grains is Comparing the case of the continuous film with nanodots, our results indicate that HC is smaller in nanodots for tIrMn < 7 nm while it is larger when tIrMn > 7 nm. This is because the fraction of grains which contribute to HC is smaller in nanodots for tIrMn < 7 nm due to the presence of the SGE which are superparamagnetic and thus do no contribute to HC. However, for 7 nm < tIrMn < 12 nm, HC is larger in nanodots because, the blocking temperatures of SGE becomes closer to TM, so they contribute to HC.

B. Effect of the temperature and AF thickness

In this section, we study the effects of the AF thickness on the temperature dependence of the exchange and coercive fields in comparison with experimental data [19]. For that we consider a Co/IrMn bilayer with tCo = 4 nm and various IrMn layer thicknesses (tIrMn = 3 nm, 9 nm and 15 nm). The simulated procedure consists of a FC under HFC = 4 kOe from T0 = 550 K down to Tf = 2 K. Then successive hysteresis loops are simulated at increasing temperatures TM ≥ 2 K. The simulated temperature dependence of HE and Hc for various tIrMn in the continuous film is shown in Fig. 7 in comparison with those of Ref. [19].

AF AF N B N AF AF B V K n T k T V K T + =
where n is the number of Monte Carlo steps [36] (Fig. 8). The vanishing temperature of HE exhibits, as expected, the same tIrMn-dependence as Regarding HC, a good qualitative agreement with experimental data is also obtained. In particular, a maximum in HC is observed simultaneously at the vanishing temperature of HE.

This maximum shifts toward higher temperatures and its amplitude decreases as tIrMn increases. Actually, this maximum results from both the shift of the blocking temperature to TM contributes to HC (see Fig. 9).

In order to see the effects of reducing the lateral size, we perform the same simulations for nanodots with tIrMn = 9 nm. The temperature dependence of HE and HC is plotted in Fig. 10 in comparison with that of a continuous film. The same behavior is observed in nanodots as in the continuous film. Note that at TM = 2 K, all grains in contact with the F layer contribute to HE, where HE is smaller in nanodots due to the presence of SGE at the edges (which are less coupled with the F layer than the AF grains). Note also that the location of the minimum and the maximum of Hc is shifted towards lower temperatures in nanodots. This is because the fraction of grains contributing to Hc, at low temperatures, increases in the nanodots due to presence of SGE (Fig. 11). shows a decreases at low temperatures due to the disordered phases and vanishes at the maximum blocking temperature of the AF grains for which HC is maximum. Our results show that the simulated values of HE in nanodots can be smaller or larger than that in continuous films depending on the AF thickness and the measurement temperature due to contribution of the disordered phases at the edges in the AF layer. Note that these results provide an explanation of the various experimental results found in literature about the comparison between EB properties of nanodots and continuous films. To achieve a deeper understanding of the disordered interfacial phases, it would be of great interest to use an atomic approach which takes into account grain boundaries and atomic interdiffusion in the AF layer. Note that to avoid huge computational time, such a study would be restricted to nanodots.

Fig. 1 .

 1 Fig. 1. (Color online) Top view of the granular microstructure created by Voronoi tessellation for the continuous film (10 4 grains) and a nanodot (289 grains), and the corresponding grains volume distribution.

  account where mi is the magnetic moment of a F grain or a SG or a SGE (in the nanodots) and H is the field applied along the y-axis. The magnetic moments mi of F grains, SG and SGE are calculated according to mi = M Vi where M is the magnetization of the grain and Vi is its volume. For the F grains, we have taken the Co magnetization M = 1,44×106 A m -1[START_REF] Coey | Magnetism and Magnetic Materials[END_REF].

Fig. 2 .

 2 Fig. 2. (Color online) Sketch of the F/AF nanodot with SG randomly spread over the F/AF interface (in green) and SGE at the edges in the AF layer (in green too). The small sketch above represents the collinear atomic moments of one F grain and one AF grain, and the noncollinear atomic moments of the disordered interfacial phase which are on-average along the y-axis as shown in the large sketch.

Fig. 3 .

 3 Fig. 3. (Color online) Simulated hysteresis loops at 298 K for various tIrMn, for (a) the continuous film and (b) nanodots (averaged over 60 nanodots).

Fig. 4 .

 4 Fig. 4. (Color online) (a) Simulated tIrMn-dependence of HE measured at TM = 298 K for Co/IrMn bilayers in comparison with (b) experimental data [20].

Fig. 5 .

 5 Fig. 5. (Color online) Intrinsic blocking temperature distributions for the grains in contact with the F layer at the interface in a Co/IrMn continuous film for (a) tIrMn = 4 nm, (b) tIrMn = 9 nm and (c) tIrMn = 14 nm (the grey rectangles indicate the grains which contributes to HE)

  tIrMn = 4 nm and decreases for tIrMn > 4 nm which explains the behavior of HC.

Fig. 6 .

 6 Fig. 6. (Color online) (a) Simulated tIrMn-dependence of HC measured at TM = 298 K for Co/IrMn bilayers in comparison with (b) experimental data [20].

Fig. 7 .

 7 Fig. 7. (Color online) (a) Simulated temperature dependence of HE and Hc in Co/IrMn bilayers (for various tIrMn =3 nm, 9 nm and 15 nm) in the continuous film in comparison with (b) experimental data [19].

Fig. 8 .

 8 Fig. 8. Comparison between the vanishing temperature of the simulated HE and the maximum of the intrinsic blocking temperature versus tIrMn.

Fig. 9 .

 9 Fig. 9. (Color online) Intrinsic blocking temperature distributions for the grains in contact with the F layer at the interface in a Co/IrMn continuous film for (a) tIrMn = 3 nm, (b) tIrMn = 9 nm and (c) tIrMn = 15 nm (the grey rectangles indicate the grains which contribute to HE at TM = 2 K).

  distribution of AF grains as tIrMn increases and the fact that only the AF grains having TB close

Fig. 10 .

 10 Fig. 10. (Color online) Simulated temperature dependence of (a) HE and (b) HC in Co/IrMn bilayers (tIrMn =9 nm) in both nanodots and continuous film.

Fig. 11 .

 11 Fig. 11. (Color online) Intrinsic blocking temperature distributions of the grains in contact

  Note that no difference between nanodots and the continuous films can be detected in experimental data due to error bars whereas our results indicate a smaller HE in nanodots for tIrMn < 12 nm, and a larger one if

tIrMn > 12 nm. This can be explained by the contribution of SGE in nanodots which becomes more pronounced as tIrMn increases. For tIrMn < 12 nm, all SGE are superparamagnetic and do not contribute to HE, therefore the simulated values of HE are slightly smaller than that in continuous films. However, for tIrMn > 12 nm, most of SGE contribute to HE, and thus the simulated values of HE are slightly larger than that in continuous films.

Acknowledgments

This project is funded by the Région Normandie and the European Union. Europe invests in Normandy with the European Regional Developement Fund (ERDF) -MAGMA project.

The authors acknowledge the Centre Régional Informatique et d'Applications Numériques de Normandie (CRIANN) where simulations were performed as Project No. 2010006. We also thank Ahmed Albaalbaky for critical reading of the manuscript. 24-06-2019