Natalia Kushik
email: ngkushik@gmail.com

Husnu Yenigun
email: yenigun@sabanciuniv.edu

Heuristics for Deriving Adaptive Homing and Distinguishing Sequences for Nondeterministic Finite State Machines

Keywords: Nondeterministic Finite State Machines, Adaptive Homing Sequence, Adaptive Distinguishing Sequence, Novel heuristics

Distinguishing Sequence (DS) and Homing Sequences (HS) are used for state identification purposes in Finite State Machine (FSM) based testing. For deterministic FSMs, DS and HS related problems are well studied, for both preset and adaptive cases. There are also recent algorithms for checking the existence and constructing Adaptive DS and Adaptive HS for nondeterministic FSMs. However, most of the related problems are proven to be PSPACEcomplete, while the worst case height of Adaptive DS and HS is known to be exponential. Therefore, novel heuristics and FSM classes where they can be applied need to be provided for effective derivation of such sequences. In this paper, we present a work in progress on the minimization of Adaptive DS and Adaptive HS for nondeterministic FSMs.

Introduction

Distinguishing Sequences (DS) and Homing Sequences (HS) are used for state identification purposes in Finite State Machine (FSM) based testing [START_REF] Dorofeeva | FSM-based conformance testing methods: a survey annotated with experimental evaluation[END_REF][START_REF] Gill | State-identification experiments in finite automata[END_REF][START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF]. A DS identifies the initial state of the FSM under investigation, while an HS is used to identify the final state after the sequence has been applied. A sequence is adaptive if the next input to be applied to an FSM under investigation is chosen based on the previously observed outputs, and the sequence is preset if the outputs need to be observed only after the entire sequence applied. The methods to derive preset/adaptive HS/DS are well elaborated for complete and deterministic FSMs [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF][START_REF] Kohavi | Switching and Finite Automata Theory[END_REF][START_REF] Moore | Gedanken-experiments on sequential machines[END_REF]. Even though the length of most of these sequences is polynomial w.r.t. the number of FSM states, the current complexity of digital systems and software makes it almost impossible to derive a complete deterministic behavior of the system integrated into the overall software and/or hardware environment. Moreover, current specifications of telecommunication protocols and other digital systems include an optionality of output responses under the same queries. That is the reason why researchers turn their attention towards special FSM types, and in particular, nondeterministic FSMs.

A method for deriving a DS for two states of an observable nondeterministic FSM has been proposed in [START_REF] Spitsyna | Studying the separability relation between finite state machines[END_REF], where the length of this sequence is shown to be exponential for nondeterministic FSMs. The upper bound on the length of an HS for an observable nondeterministic FSM is shown to be exponential as well, and this upper bound is reachable [START_REF] Kushik | On the Length of Homing Sequences for Nondeterministic Finite State Machines[END_REF]. As for adaptive experiments for nondeterministic FSMs, it has been shown that the length of a shortest adaptive DS for two states of an observable nondeterministic FSM with n states is at most n(n -1)/2 [START_REF] Alur | Distinguishing tests for nondeterministic and probabilistic machines[END_REF]. Whenever such an adaptive sequence is derived to distinguish m > 2 states of an observable nondeterministic FSM, the length of this sequence grows exponentially [START_REF] Kushik | On Adaptive Experiments for Nondeterministic Finite State Machines[END_REF]. The problem of checking the existence of a preset DS is known to be PSPACE-complete even for complete deterministic FSMs [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF]. The latter means one cannot directly apply these techniques to effectively derive test sequences against FSM specifications. That is the reason why a number of heuristics to decrease such complexity have been proposed. In particular, various heuristic methods to construct reduced adaptive DS for complete deterministic FSMs have been proposed in [START_REF] Türker | Lookahead-Based Approaches for Minimizing Adaptive Distinguishing Sequences[END_REF]. However, it has been also proven that constructing a minimal adaptive DS for complete deterministic FSMs is an NP-hard problem [START_REF] Türker | Hardness and inapproximability of minimizing adaptive distinguishing sequences[END_REF]. Therefore, novel heuristics need to be provided for effective derivation of adaptive HS and adaptive DS for nondeterministic FSMs.

In this paper, we focus on two techniques for deriving adaptive DS and adaptive HS for nondeterministic FSMs. In particular, we present a class of nondeterministic FSMs for which it is possible to construct an adaptive DS without using any nondeterministic transitions. We also argue how existing adaptive DS minimization approaches can be used for such nondeterministic FSMs. As for adaptive HS, we address a method proposed in [START_REF] Kushik | On Adaptive Experiments for Nondeterministic Finite State Machines[END_REF] that constructs an adaptive HS for an observable nondeterministic FSM when each state pair is adaptively homing. We discuss how this method can be improved so that the resulting adaptive HS can be shorter. Therefore, the main contributions of this work in progress are two novel heuristics for effective derivation of adaptive DSs for nondeterministic FSMs and adaptive HSs for observable nondeterministic FSMs.

The paper is organized as follows. Section 2 contains preliminaries. A heuristic method for effective adaptive DS derivation for nondeterministic FSMs is given in Section 3 while Section 4 presents an approach for minimizing adaptive HS for observable nondeterministic FSMs. Section 5 concludes the paper.

Preliminaries

In this paper, we focus on minimizing adaptive HS and DS for nondeterministic FSMs. As usual, an FSM S is a 4-tuple (S, I, O, hS), where S is a finite set of states; I and O are finite non-empty disjoint sets of inputs and outputs, respectively

; hS ⊆ S × I × O × S is a transition relation, where a 4-tuple (s, i, o, sʹ′ ) ∈ hS is a transition. An FSM is complete if for each pair (s, i) ∈ S × I there exists (o, sʹ′ ) ∈ O × S such that (s, i, o, sʹ′ ) ∈ hS. Otherwise it is called partially specified. If for some pair (s, i) ∈ S × I , there exist two transitions (s, i, o 1 , s 1 ), (s, i, o 2 , s 2 ) ∈ hS, such that o 1 ≠ o 2 or s 1 ≠ s 2 then S is called nondeterministic.
For a nondeterministic FSM the nondetermin-ism is observable if each input/output pair i/o uniquely identifies the successor of each FSM state (if it exists). Given a state s of S and an input/output sequence α/β, the α/βsuccessor of state s is the set of all states that are reached from s via an application of α when an output reaction β is produced. Note that for an observable FSM S, the cardinality of the α/β-successor of state s is at most one for any input/output sequence α/β.

A sequence α is a distinguishing (homing) sequence (DS/HS) for the FSM S if after applying α and observing output reaction β one can uniquely conclude about the initial (final/current) state of S. The length of preset DS and HS is exponential for nondeterministic FSMs, however sometimes this length can be shorter when adaptivity is used. An adaptive DS/HS can be represented as a tree or as a specific acyclic singleinput output-complete FSM that are called test cases [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF][START_REF] Petrenko | Conformance Tests as Checking Experiments for Partial Nondeterministic FSM[END_REF].

A number of methods for deriving adaptive DS and HS for nondeterministic FSMs have been proposed (see, for example [START_REF] Kushik | On Adaptive Experiments for Nondeterministic Finite State Machines[END_REF]). However, the length of the corresponding sequence in general case remains exponential and thus, novel heuristics need to be provided for minimizing the length of such sequences.

Nondeterministic FSMs with a Deterministic Adaptive DS

For a complete deterministic FSM with n states, it is known that n(n -1)/2 is the tight upper bound for the height of an adaptive DS [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF]. However, for nondeterministic FSMs, the height of an adaptive DS is exponential in general. In this section, we present a class of nondeterministic FSMs for which the height of an adaptive DS is at most 2n 2 -n -1 and existing algorithms of adaptive DS minimization can be readily applied.

A transition (s, i, oʹ′ , sʹ′ ) ∈ hS is a deterministic transition if for any transition (s, i, oʹ′ʹ′, sʹ′ʹ′) ∈ hS we have oʹ′ʹ′= oʹ′ and sʹ′ʹ′= sʹ′ . Intuitively, S d is the same FSM as S where the deterministic transitions are preserved but all other transitions are removed. Hence, S d is a deterministic FSM by definition. It is easy to see that an adaptive DS for S d can be directly used as an adaptive DS for S as well, and it will be a deterministic adaptive DS in the sense that only the deterministic transitions are used throughout the application of the adaptive DS in S. In the lucky case that S d is a complete deterministic FSM, the existence check and the adaptive DS construction algorithms given in [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF] can be directly applied. However, in general, S d is a partially specified deterministic FSM. Moreover, in the worst case the transition relation of S d can be empty, and in this case, the question of existence of an ADS for S remains open.

Although there usually is a complexity jump for the algorithms when one considers partially specified FSMs, this is not the case for the problems related to adaptive DS. In [START_REF] Hierons | Distinguishing Sequences for Partially Specified FSMs[END_REF] a polynomial time algorithm is provided to check if a partially specified FSM has an adaptive DS or not. In this paper, we adapt this algorithm for efficient derivation of an ADS for a nondeterministic FSM that requires a partial deterministic projection. We also improve the upper bound on the length of such ADS compared to the one given in [START_REF] Hierons | Distinguishing Sequences for Partially Specified FSMs[END_REF]. In fact, the approach given in [START_REF] Hierons | Distinguishing Sequences for Partially Specified FSMs[END_REF] is based on constructing a complete deterministic FSM C(S d ) from a given partially specified deterministic FSM S d such that there exists an adaptive DS with the height H for C(S d ) iff there exists an adaptive DS with height H -1 for S d . The number of states in C(S d ) is 2n, where n is the number of states in S d . Therefore, it is possible to check if there exists an adaptive DS for S d (and hence a deterministic ADS for S) in O(pnlgn) time [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF], where p is the cardinality of I d .

If there exists an adaptive DS for C(S d ), one can then use the LY algorithm (the adaptive DS construction algorithm given in [START_REF] Lee | Testing finite-state machines: state identification and verification[END_REF]) to construct an adaptive DS. Although [START_REF] Hierons | Distinguishing Sequences for Partially Specified FSMs[END_REF] provides an upper bound of π 2 n 2 /3 for the height H of the adaptive DS constructed for C(S d ), using the result in [START_REF] Sokolovskii | Diagnostic experiments with automata[END_REF], the LY algorithm actually constructs an adaptive DS with the height at most H = 2n 2 -n for C(S d ). Therefore, the height of the adaptive DS for S d is at most H -1 = 2n 2 -n -1.

Note that the LY algorithm does not aim for the minimization of the adaptive DS it constructs. However, there exist heuristics for the minimization of adaptive DSs with respect to different metrics (e.g. height, total external path length, etc.) for complete deterministic FSMs [START_REF] Türker | Lookahead-Based Approaches for Minimizing Adaptive Distinguishing Sequences[END_REF]. Since C(S d ) is a complete deterministic FSM, one can directly use these heuristics in order to construct minimized adaptive DSs for C(S d ), and hence for the nondeterministic FSM S. Note that S can be nonobservable as well.

Minimizing adaptive HS for Nondeterministic FSMs

In this section, we discuss how to optimize the procedure for deriving an adaptive HS for an observable FSM S. The procedure is taken from [START_REF] Kushik | On Adaptive Experiments for Nondeterministic Finite State Machines[END_REF] while the complexity of the related problem is given in [START_REF] Kushik | Adaptive Homing is in P[END_REF]. The homing test case derivation strategy is based on the condition that each state pair of S is adaptively homing. In other words, there exists a homing test case for S if there exists an adaptive homing sequence for each subset {s i , s j } ⊆ S of states of the observable nondeterministic FSM S. A test case P i,j is a homing test case for the subset {s i , s j } ⊆ S of states if for every input/output sequence α/β defined in P i,j , the α/β-successor of the subset {s i , s j } ⊆ S has at most one state.

If there exists a homing test case for the FSM S then the set S is a homing set and the test case P is a homing test case for the set S or the test case P homes states of the set S. Otherwise, the set S is not homing. The homing test case for the set S = {s 1 , .., s n } is derived iteratively. As mentioned above, P i,j is a homing test case for the subset (pair) {s i , s j } ⊆ S of FSM states. The procedure starts with a homing test case P 1,2 for the set {s 1 , s 2 }, then state s 3 is added to the subset {s 1 , s 2 } ⊆ S. Each input/output sequence α/β that is defined in the test case P 1,2 is applied at state s 3 . If the sequence α/β is defined at state s 3 , then the deadlock state q in the test case P 1,2 is replaced with a test case P q,z , where q is the α/β-successor of state pair {s 1 , s 2 }, while z is the α/βsuccessor of state s 3 . All the input/output sequences α/β′ that are defined at state s 3 but not defined in the test case P 1,2 are also included into the test case P 1,2,3 . Proceeding in this way by iteratively adding the remaining states s 4 , s 5 , ..., s n , the test case P 1,2,…,n is derived.

We note that there exist various homing test cases P i,j for the same state pair {s i , s j } ⊆ S. Therefore, the first optimization step can be to consider a somehow opti-mal test case P i,j that is chosen for the first state pair and for those test cases P q,z that are appended iteratively, when P 1,2,…,n is being derived. On the other hand, the length of the resulting test case can significantly depend on the choice of state s j that is added to the test case P 1,2,…,j -1 . Thus, the second optimization criterion can be on an optimal choice of the next state s j when the test case P 1,2,…,j -1 is already constructed. We further discuss how these two ideas can be taken into account to effectively derive a test case P 1,2,…,n . We mention that when optimizing the length of the corresponding adaptive HS, represented as a test case, we focus on optimizing not only the height of the corresponding tree but also the number of transitions in the acyclic FSM.

The first type of heuristics is related to making an optimal choice between possible initial state pairs {s i , s j } ⊆ S, i < j, as well as defining the best homing test case P i,j for this pair. As it is shown in [START_REF] Kushik | On Adaptive Experiments for Nondeterministic Finite State Machines[END_REF], the length of the overall homing test case P 1,2,…,n significantly depends even on the first input to be included into the test case P 1,2 . As the test case P 1,2,…,j is defined based on the test case P 1,2,…,j -1 , the height / the number of transitions of it significantly depends on the number of sequences that are defined in the test case P 1,2,…,j -1 and in state s j that will be added to the root of the corresponding successor tree. That is the reason why we suggest to choose a pair {s i , s j } ⊆ S to start with in such a way that its homing test case P i,j is a homing test case for some other state pairs. It is naturally to assume that the more state pairs are 'covered' by a homing test case P i,j , the better is the resulting test case P 1,2,…,n . Therefore, we suggest to choose the first state pair {s i , s j } ⊆ S, i < j, such that there exists a homing test case P i,j for this pair that homes as much state pairs in S as possible.

The idea behind the second optimization step is the same as in the previous case. In particular, we suggest choosing the next state s j to be added to a test case P 1,2,…,j -1 in such a way that the number of input/output sequences that are defined in P 1,2,…,j -1 and are not defined at state s j would be minimal. This fact can help to reduce the number of homing test cases P q,z that have to be appended at each state z, where z is the α/β-successor of state s j , and q is the α/β-successor of the set {s 1 , …, s j -1 }. Moreover, as for each pair of states {q, z} there can exist various adaptive homing test cases, the best choice of the test case P q,z can affect the overall height / transition number of the test case P 1,2,…,n . Therefore, it is also necessary to consider which test case P q,z should be chosen for an intermediate state pair {q, z} ⊆ S.

We mention that both optimization steps for deriving an adaptive homing test case P 1,2,…,n need to be thoroughly estimated. On one hand, theoretical investigation on FSM classes that have shorter adaptive HS need to be elaborated, and on the other, experimental evaluation on the efficiency of proposed heuristics needs to be performed.

Conclusion

In this paper, we proposed two heuristic methods for optimizing the size of adaptive DS and adaptive HS for nondeterministic FSMs. The topic of such optimization is always motivated by the fact that the length of such sequences is exponential for nondeterministic FSMs. Moreover, related decision problems on the existence of such sequences even for the preset case are known to be PSPACE-complete. We note that this current work in progress only presents the ideas behind optimization criteria and optimization techniques to achieve shorter resulting trees for adaptive DS or HS. As a future work, we plan to perform experimental evaluation over machines of various types (random, protocol specifications, etc.) to estimate the efficiency of the proposed techniques.

  For a given FSM S = (S, I, O, hS), we define the deterministic projection S d = (S, I d , O d , h d S) of S as follows. S d and S have the same set S of states. For a transition (s, i, oʹ′ , sʹ′ ) ∈ hS, (s, i, oʹ′ , sʹ′ ) ∈ h d S, if and only if (s, i, oʹ′ , sʹ′ ) is a deterministic transition in S. I d and O d consist of the inputs and outputs used in the transitions in h d S.

Acknowledgments

The authors gratefully acknowledge Prof. N. Yevtushenko for interesting discussions.

This work is partially supported by RFBR Grant# 15-58-46013 CT_a and TÜBİTAK Grant# 114E921.


Sylvain Rat

Venonika Nagy

Iurii Suleimanov

Gábor Molnár

Lionel Salmon

Philippe Demont
email: demont@cict.fr

Levente Csóka

Azzedine Bousseksou
email: azzedine.bousseksou@lcc-toulouse.fr

G Molna ´r

L Cso

Elastic coupling between spin-crossover particles and cellulose fibers

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

Composite materials made of cellulose fibers and spin crossover micro-particles were investigated by magnetic measurements and dynamic mechanical analysis (DMA). The storage modulus of the cellulose handsheet (0.6 GPa at room T) is significantly enhanced in the composite (1.7 GPa). The latter also displays a reversible increase of ca. 10% when switching the magnetic spin state of the particles from the low spin (LS) to the high spin (HS) form. Around the spin transition temperature a loss modulus peak is also observed, highlighting the strong viscoelastic coupling between the particles and the cellulose matrix. These results pave the way for the development of a novel family of actuator materials based on spin crossoverpolymer composites.

Spin crossover (SCO) complexes of transition metal ions 1 have lately received increasing attention for their mechanical properties and in particular for the possibility of harvesting a very significant volume change, which accompanies the SCO phenomenon. Indeed, this property provides remarkable prospects for the use of SCO materials in mechanical actuator technology. 2 The spontaneous strain during the spin transition was recently exploited through the integration of SCO materials into bimorph cantilevers, which were actuated either thermally or by light irradiation. 3 Strain-mediated coupling of SCO with electrical, 4,5 optical 6 and magnetic properties 7-10 has also been reported in hybrid SCO materials. In this context, we realize that the development of ''artificial muscles'' using polymer-SCO composites appears to be particularly promising. An artificial muscle is a kind of actuator generally defined by analogy with the skeletal muscle, and as stated by Tondu, 11 any material or device whose shape can change in response to a stimulus can potentially be a candidate for the label ''artificial muscle''. This was clearly demonstrated through the development of an electromechanical actuator made of particles of the SCO complex [Fe(Htrz) 2 (trz)](BF 4 ) (Htrz = 1,2,4-4H-triazole and trz = 1,2,4-triazole) dispersed in a poly(methylmethacrylate) matrix. 12 For further progress with this novel class of actuator materials, it will be, however, crucial to understand how the elastic strain associated with the spin transition is transmitted to the macroscopic scale in composite materials.

Here we report on the mechanical analysis of cellulose-SCO composites, revealing remarkable elastic strain coupling phenomena. The linter cellulose fibre composite 1 with B50 w% loading of [Fe(Htrz) 2 (trz)](BF 4 ) microparticles was prepared using a procedure described in a previous article. 13 The fairly homogenous distribution of SCO particles over the linter cellulose surface is clearly observed on the SEM images in Fig. 1. The magnetic properties of the composite are shown in Fig. 2. At room temperature, the composite is diamagnetic, but upon heating it becomes paramagnetic around 110-115 1C. The return to the diamagnetic phase occurs around 80 1C with a large thermal hysteresis. This behaviour corresponds to the well-known spin transition of [Fe(Htrz) 2 (trz)](BF 4 ). 14 The spin transition of the composite is also accompanied by a pronounced thermochromism between the LS (pink) and HS (white) phases. 13 Fig. 1 SEM images of composite 1 (the inset shows a zoomed image). Scale bars are 10 mm.

The mechanical properties of the composite and pure cellulose handsheets were characterized by DMA, wherein an oscillatory force (stress) is applied to a material and the resulting displacement (strain) is measured. An ARES G2 rheometer from TA Instruments was used in tensile mode in the temperature range between 55 and 130 1C with heating/cooling rates of 2 1C min À1 , a preload force of 0.7 N, a static force/dynamic force amplitude ratio (''force tracking'') of 120%, and an oscillation frequency of 1 Hz. Samples were cut in a rectangular shape (25 Â 7 mm), with an approximate thickness of 170 mm, and were loaded at a starting gap height of 10 mm between the clamps. The composite was cycled several times before reaching stability. Preliminary tests were performed at 23 1C in a dry nitrogen atmosphere to determine optimal mechanical conditions as well as stress-strain curves (Fig. 3). The latter indicate a reinforcement, i.e. higher stiffness of the handsheets loaded with SCO particles. The slope of the stress-strain curves below B0.1% strain gives the Young's modulus, which increases from 0.6 GPa for the pure cellulose to 1.7 GPa for composite 1. Besides the intrinsically higher stiffness of the SCO material, one may suggest that the SCO particles also affect the free OH groups of the cellulose and thus participate in inter-fibre hydrogen bonding, which in turn increases the stiffness of the network. 15,16 Hence the reinforcement imparted by the [Fe(Htrz) 2 (trz)](BF 4 ) particles allows greater stress transfer from the cellulose to the SCO particles at the interface.

Application of a dynamic, vibrational load to paper sheets will result in a combination of elastic and viscous responses. Various theoretical approaches were suggested to model this behaviour. Researchers usually employ linear and non-linear continuum models in order to explain stress relaxation and creep, since these are the standard methods for investigation of paper viscoelastic properties. [17][18][19][20][21] In the visco-elastic model the dynamic modulus E* is the ratio of stress to strain under vibratory conditions defined as: [START_REF] Mark | Handbook of Physical Testing of Paper[END_REF] E* = E 0 + iE 00

The real part E 0 , the storage modulus (elastic response), is proportional to the energy stored by the material upon reversible elastic deformation. The imaginary part E 00 , the loss modulus (viscous response), is proportional to the energy dissipated as heat (mainly) by the friction of macromolecular chains. The latter is an irreversible phenomenon. Fig. 4 shows the temperature dependence of the storage and loss moduli of the pure cellulose and composite sheets for a complete heating-cooling cycle. As can be expected, the storage modulus of the pure cellulose sheet exhibits a gradual decrease while increasing the temperature from 55 to 130 1C. This phenomenon is mostly related to the increase in polymer chain mobility at higher temperatures. While the magnitude of the storage modulus obviously depends on the type of fibre and its refining state, the decreasing storage modulus is a general tendency. The composite sheets display a similar monotonous decrease of the storage modulus up to ca. 110 1C, wherein E 0 abruptly increases by ca. 10%. Upon cooling, these phenomena are reversible with a decrease of E 0 around 80 1C. This hysteresis of the elastic modulus in Fig. 4(ii) is obviously correlated with the magnetic properties (Fig. 2). We can thus conclude that the spin transition leads to a reversible change in elastic properties of the whole cellulose composite. At first glance, the increase of the storage modulus in the HS state of the composite might seem to be surprising since the stiffness of the SCO material itself decreases at the same time. [START_REF] Fe ´lix | [END_REF] However, one should also consider that the volume fraction of the spin crossover particles increases by ca. 11% while going from the LS to the HS state. 24 The strain induced by this volume change on the cellulose fibres seems to be the main contribution to the storage modulus increase upon the SCO in this particular case.

In addition to the elastic response to time dependent stress, irreversible plastic deformation and associated energy dissipation will also take place in the composite -mainly due to the mechanical relaxation processes of the cellulose fibres as they are stretched. These phenomena are reflected by the loss modulus whose temperature dependence is also shown in Fig. 4 for the two samples. The E 00 curve of the sheet made from the as received cellulose fibres shows a decrease of the loss modulus with increasing temperature. According to the results obtained by Roylance et al., this tendency can be associated with the high temperature part of a broad mechanical relaxation centred around À50 1C which is related to the relaxation of restricted amorphous chains in the crystalline phase of cellulose. 25 The mechanical losses are weak between 80 and 130 1C, indicating that no significant relaxation occurred in this range reflected also by the weak temperature dependence of E 00 . The E 00 curves of the composite material reveal two well defined loss peaks around the spin transition temperatures. These two peaks can be assigned to heat dissipation due primarily to the friction of cellulose fibres during the spin transition.

In summary, DMA analysis revealed an abrupt and reversible stiffening of the cellulose-[Fe(Htrz) 2 (trz)](BF 4 ) composite when going to the high temperature HS phase and an associated peak in the mechanical losses. These results demonstrate that it is possible to combine the functionality of compliant, free-standing polymer materials with the actuating force arising upon the spin transition in SCO particles. This approach thus provides access to functionalities (e.g. manufacturability), which are not easily attainable with monolithic spin crossover materials and, vice versa, it also allows endowing the matrix with advanced mechanical and optical properties. As a perspective, it is important to notice that by simply changing the filler microstructure (particle shape, arrangement, etc.) one may obtain very different mechanical couplings with the matrix. In the case of highly oriented high aspect ratio particles, one may even anticipate opposite responses (i.e. DE HS-LS 0 4 0 or DE HS-LS 0 o 0) depending on the direction in which the force is applied to the composite. The empirical Halpin-Tsai equations 26 may provide a very useful frame for the design of these systems with tailored mechanical responses.

Notes and references

15

 15

Fig. 2

 2 Fig. 2 Magnetization of composite 1 as a function of the temperature recorded at 4 1C min À1 . Arrows indicate heating and cooling.

Fig. 3

 3 Fig. 3 Stress-strain curves of the pure and composite cellulose sheets at 23 1C. The error bars are smaller than the symbol size.

Fig. 4

 4 Fig. 4 Storage and loss moduli of the pure (i and iii) and composite (ii and iv) cellulose sheets at different temperatures in the heating (red open symbols) and cooling (blue closed symbols) modes. Arrows indicate heating and cooling.