Arij Léo Farhat

Sandrick Le Maguer

Patrick Queffelec

Michel Ney

Patrick Quéffélec

TLM Extension to Electromagnetic Field Analysis of Anisotropic and Dispersive Media: A Unified Field Equation

Keywords: Anisotropic and dispersive media, ferrites, nonreciprocal circuits, permeability, time-domain analysis, transmission-line matrix (TLM) method

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

I. INTRODUCTION

O VER THE last few years, the rapid development of communication applications has generated a growing interest for miniaturization and cost reduction of microwave devices. The increase of operating frequencies and tunability are additional constraints that require more complex and accurate models for computer-aided design (CAD) of communication system components.

Among those components, circulators and isolators use the anisotropic properties of ferrite materials to insure the required nonreciprocal character of the wave propagation. Their field-dependent permeability provides some tunability of circuits, such as filters and phase shifters. More recently, their high refractive index has found interest in the size reduction of patch antennas. It has been demonstrated that antennas with a magneto-dielectric substrate that exhibits a permeability greater than its permittivity has better performances than those deposited on pure dielectric substrates [START_REF] Mosallaei | Magneto-dielectrics in electromagnetics: Concept and applications[END_REF].

To assist the design of ferrite-based microwave devices, one needs to have a proper design tool enabling the prediction of A. L. Farhat is with Cobham Microwaves, 91978 Villebon-sur-Yvette, France (e-mail: arij.farhat@telecom-bretagne.eu).

S. Le Maguer is with "Sprezzatura," 29200 Brest, France (e-mail: Sandrickl@emaguer.com).

P. Quéffélec is with the Laboratoire des Sciences et Techniques de l'Information, Communication et de la Connaissance (Lab-STICC, UMR CNRS), Université de Bretagne Occidentale, UFR Sciences, 29200 Brest, France (e-mail: patrick.queffelec@univ-brest.fr).

M. Ney is with the Laboratoire des Sciences et Techniques de l'Information, Communication et de la Connaissance (Lab-STICC, UMR CNRS), Telecom Bretagne, 29200 Brest, France (e-mail: michel.ney@telecom-bretagne.eu).

the microwave behavior of ferrite samples whatever their magnetization state. For instance, it is considered to use hexaferrite materials that operate at the remanent state to work out self-biased circulators [START_REF] Oliver | Integrated self biased hexaferrite microstrip circulators for millimeter-wavelength applications[END_REF]. It would also be very interesting to be able to predict the variation of the performances of patch antennas d e p o s i t e do naferrite substrate as a function of its magnetization state.

Magnetized ferrites are anisotropic media. Their electromagnetic properties must be represented by a tensor quantity, called the permeability tensor. The usual permeability model based on the Polder formulations [START_REF] Polder | On the theory of ferromagnetic resonance[END_REF] is only valid for a saturated sample. Designing ferrite microwave devices requires the knowledge of the permeability tensor of the magnetic materials used as a substrate, which directly influence the guided wavelengths and the performances of the devices.

The objective of this paper is twofold: first, to develop a rigorous model for field computation in the presence of complex media and second to insert a new model of nonsaturated ferrites [START_REF] Gelin | Generalized permeability tensor model: Application to barium hexaferrite in a remanent state for self-biased circulators[END_REF] into the algorithm.

Presently there is no commercial simulator capable to account for the complex physical phenomena appearing in the electromagnetic structures using magnetized magnetic materials, such as the following:

• nonhomogenous internal polarization field implying a space variation of the permeability; • nonsaturated zones in plate ferrite samples due to demagnetizing fields even for substrates biased with a strong dc field; • dynamic interactions between magnetic domains in nonsaturated regions of the ferrite substrate and between grains whatever the magnetization state; • magnetostatic modes. The above phenomena must be accounted for in the model, as they strongly affect the performances of the device in terms of bandwidth, insertion losses, etc. They can also preclude miniaturization of circulators. We have observed that a cutoff band appeared in the experimental response of miniaturized Y-junction circulators [START_REF] Guennou | Influence of the magnetic field nonuniformity on an -band microstrip Y-junction circulator bandwidth: Theory/experiment comparison[END_REF]. This degradation was missing in the transmission signal predicted by commercial softwares.

Finally, the time-domain character of the TLM not only allows a wideband characterization, but also accounts for the presence of potential nonlinearities.

To predict the dynamic behavior of polycrystalline ferrites for an arbitrary magnetization state, one proposes a theoretical approach that provides all tensor components. They are functions ofthedcbiasfield strength and direction, the sample geometry, and static magnetic characteristics such as saturation magnetization and magnetocrystalline anisotropy field. Finally, sample structural properties (magnetic domains and grains shape) are accounted for by the model. Details of the proposed theoretical approach can be found in [START_REF] Gelin | Generalized permeability tensor model: Application to barium hexaferrite in a remanent state for self-biased circulators[END_REF].

In this paper, the TLM is first extended to general dispersive and anisotropic media. Some pioneer work, based on circuit analogy, was presented for the TLM model [START_REF] Paul | Generalized material models in TLM-Part 1: Materials with frequency-dependent properties[END_REF]- [START_REF] Paul | Generalized material models in TLM-Part 3: Materials with nonlinear properties[END_REF]. This paper focuses on a formulation based directly on Maxwell's equations [START_REF] Peña | A general formulation of a three-dimensional TLM condensed node with the modeling of electric and magnetic losses and current sources[END_REF]. It gives a clear and systematic derivation that constitutes a general approach, which can be applied to any type of media. Two different types of media are then studied: unmagnetized plasma considered as a dispersive medium and a nonsaturated ferrite (with an arbitrary magnetization state).

II. TLM THEORETICAL MODEL

A. Formulation: Unified Field Equation

Let us consider a general anisotropic and dispersive medium in which electromagnetic quantities are governed by the general Maxwell curl equations in time domain [START_REF] Mosallaei | Magneto-dielectrics in electromagnetics: Concept and applications[END_REF] where subscript and stand for electric and magnetic, respectively, * denotes the convolution operation, double underlined quantities are tensors, is the speed of light in vacuum, and the other electromagnetic quantities are definedinT ableI.

For the symmetrical condensed node (SCN) TLM, the presence of materials affects the field values at the node center, while arm impedances are all kept at free-space value. The application of the general procedure described by Peña and Ney in [START_REF] Peña | A general formulation of a three-dimensional TLM condensed node with the modeling of electric and magnetic losses and current sources[END_REF] TABLE I ELECTROMAGNETIC QUANTITIES for general media is new and not straightforward. However, it has the advantage to allow the development for all TLM nodes such as hybrid symmetrical condensed node (HSCN) or super condensed symmetrical condensed node (SSCN) nodes, by following the same general procedure.

Equation ( 1) can be written as ( 2), shown at the bottom of this page.

1) Maxwell-Ampere: Consider the -component of the Maxwell-Ampere's curl equation in ( 2),

(2)

The above relation can be sampled in the plane , ac-cordingtotheillustrationshowninFig.1,andshownin(4)at the bottom of this page.

By multiplying this equation by , one can write ( 5), shown at the bottom of this page.

Rearranging ( 5) yields ( 6), shown at the bottom of this page. Now, by setting and [START_REF] Paul | Generalized material models in TLM\Part 2: Materials with frequency-dependent properties[END_REF] with and ( 8) Equation ( 6) becomes ( 9), shown at the bottom of the following page. Now, one can introduce the incident arm voltage (4)

(5) [START_REF] Paul | Generalized material models in TLM-Part 1: Materials with frequency-dependent properties[END_REF] at time and link them to field components at the node center and at time , [START_REF] Van Blaricum | A technique for extracting the poles and residues of a system directly from its transient response[END_REF] by setting [START_REF] Ko | A combination of FDTD and Prony's methods for analyzing microwave integrated circuits[END_REF] and

The subscript indicates field components computed from incident voltages , is the excitation that can exist at the note location and is the resulting field at the node center.

Equation ( 13) yields ( 14), shown at the bottom of this page. Convolution products can be eliminated via Laplace transform and ( 14) can be written in the compact form [START_REF] Green | Microwave characterization of partially magnetized ferrites[END_REF] where is the Laplace parameter. 2) Maxwell-Faraday: One can follow the similar procedure by using Faraday-Maxwell's equation with the field sampling illustrated in Fig. 2.

Consider the -component of the Maxwell-Faraday's curl e q ua t i oni n( 2) , [START_REF] Igarashi | Tensor permeability of partially magnetized ferrites[END_REF] (9) [START_REF] Schlömann | Microwave behavior of partially magnetized ferrites[END_REF] According to the illustration shown in Fig. 2

Equation ( 22) becomes ( 26), shown at the bottom of the following page.

Applying the Laplace transform eliminates the convolution products, and (26) can be written in the compact form

(27) (17) (21) (22)
In compact matrix notation, the process can be written as follows:

(28) with being the identity matrix, and where

Hence, the right-hand-side vector of (28) gives the updated value of all field components at the node center and time by inverting the matrix such as (31), shown at the bottom of this page. One defines the tensor (32) Usually, elements of (32) are complex valued expressions and approaches such as Prony's decomposition should be used.

One now has all quantities to compute updated field values at the node center through the use of the -transform to obtain in time domain (33)

Once the discrete-time system is characterized, one can determine the updated fields in the node arms for the next time iteration. Note that, in this formulation, these quantities are defined as updated to distinguish them from the reflected voltages on the transmission lines. In previous formulations [START_REF] Paul | Generalized material models in TLM-Part 1: Materials with frequency-dependent properties[END_REF]- [START_REF] Paul | Generalized material models in TLM-Part 3: Materials with nonlinear properties[END_REF], these quantities are called the reflected fields.

B. Digital Filtering

Once elements of the tensor (32) are determined, it is necessary to transform it to the discrete time domain to make it compatible with the TLM algorithm. This is achieved by using the -transform. However, tensor's element expressions have to be first approximated by polynomial ratios expressed in frequency domain.

1)

Frequency-Domain Prony Method: Given a set of NF complex frequency-domain data samples, , Prony's method [START_REF] Van Blaricum | A technique for extracting the poles and residues of a system directly from its transient response[END_REF], [START_REF] Ko | A combination of FDTD and Prony's methods for analyzing microwave integrated circuits[END_REF] yields a least square approximation by using exponential basis functions. Let be the number of poles, be the residue,

(31) and be the complex frequency of the th pole, the starting point for this procedure is the approximation (34) Taking the Laplace transform yields (35) Equation ( 35) is modified by increasing the order of the numerator by one for physical reasons. The goal of the Prony method is to identify the Padé coefficients 's and 's.

After applying the procedure detailed in [START_REF] Brittingharam | Pole extraction from real-frequency information[END_REF] to solve (35), Padé coefficients can be determined. Solving for the roots of the numerator and denominator lead to Using compact notation, the general system of these equations is (47)

The flow graph shown in Fig. 3 illustrates the general state space system corresponding to (47).

III. ANISOTROPIC AND DISPERSIVE MEDIA

To demonstrate the novelty and relevance of the proposed technique, two different types of dispersive and anisotropic materials are studied: an unmagnetized plasma considered as a dis- persive medium and a nonsaturated ferrite (with an arbitrary magnetization state) characterized by an anisotropic and dispersive extra diagonal tensor.

A. Isotropic and Dispersive Media

One can consider an unmagnetized plasma with the following electric susceptibility: The inverse matrix is given by inverting each diagonal element . . .

. . . (50)

The element of the tensor is estimated using the frequency-domain Prony method. Starting with the function known analytically in the frequency domain, which is, in this case, each element of the tensor , (51)

Note that this is a first-order function.

By applying Prony's decomposition, the approximation of the function is then performed. Fig. 4(a) shows a comparison of the analytic and approximated amplitude function. The Prony approximation also gives a very good fit of the phase of the function, as shown in Fig. 4(b).

Finally, a digital filtering is performed by applying a -transformation to obtain (33) in time domain. as described in Section II.

The studied plasma sample has the following characteristics:

rad s GHz

The medium fills a rectangular metal cavity and resonance frequencies have to be determined. One of the dimensions (along the height) is small so that only modes are excited. The cavity is excited by a Gaussian pulse and a fast Fourier transform (FFT) is performed to determine mode resonance frequencies.

Fig. 5 shows the mode spectrum, and resonance frequencies are compared with theoretical results. As can be observed in Table II, comparison between both methods yields some excellent agreement.

B. Anisotropic and Dispersive Magnetic Media

Depending on the strength of the applied magnetic dc field, a ferrite medium can be set in various states of magnetization: completely demagnetized, fully saturated [START_REF] Polder | On the theory of ferromagnetic resonance[END_REF], remanent, or partially saturated [START_REF] Rado | Theory of the microwave permeability tensor and Faraday effect in non saturated ferromagnetic materials[END_REF]- [START_REF] Green | Microwave characterization of partially magnetized ferrites[END_REF]. In the latter case, the ferrite is subdivided into Weiss domains, and thus must be considered as heterogeneous medium. This makes the calculation of the tensor quite complicated. To circumvent this difficulty, empirical models of permeability tensor were proposed [START_REF] Green | Microwave characterization of partially magnetized ferrites[END_REF], [START_REF] Igarashi | Tensor permeability of partially magnetized ferrites[END_REF]. These models have the disadvantage of not presenting all the elements of the permeability tensor in a single course. That is why a new statistic model that provides all the elements of the tensor of ferrites in a single calculation process and that takes the hysteresis phenomenon into account was needed.

The permeability tensor model developed in our laboratory [START_REF] Gelin | Generalized permeability tensor model: Application to barium hexaferrite in a remanent state for self-biased circulators[END_REF], [START_REF] Gelin | Effect of domain and grain shapes on the dynamic behavior of polycristalline ferrites. Application to the initial permeability[END_REF] has the advantage, compared to theoretical approaches previously proposed in the literature, of not considering the partly magnetized medium as composed of independent domains, but rather made of interactive grains composed of coupling domains.

Assuming that a dc magnetic field is applied along the -direction, the permeability tensorintheCartesiancoordinate system takes the following well-known form:

(52) where all tensor components and in a real medium are complex quantities owing to the existence of magnetic losses. Equating (52) and (32) gives the resulting tensor shown in (53) at the bottom of the this page, where the electric susceptibility is related to its relative permittivity by and the magnetic susceptibility is .

(53) (54) 6 shows that Prony's approximation gives an adequate fitto functions for . One can follow the same procedure detailed in Section II-B to perform the digital filtering for ferrites.

The above model is tested for validation by considering the reflection/transmission of a rectangular waveguide partly filled with ferrite, as shown in Fig. 7. The simulated -parameters at -band frequencies (8-12 GHz) will be compared with those measured for ferrites with well-known properties.

The experimental propagation structure is a waveguide loaded with ferrite that is positioned between the poles of an electromagnet to magnetize the sample as shown in Fig. 8.

When a uniform static magnetic field is applied along the side of the waveguide ( -axis of the Cartesian coordinate system), the field displacement occurs along the large side of the guide ( -axis) (Fig. 9). The nonreciprocity of the field displacement along the -axis, which depends on the wave propagation direction, breaks the symmetry and the reciprocity of the cell. The condition of nonreciprocity of the cell has been proven by experimentation for different ferrites and values of the applied static field [START_REF] Quéffélec | Nonreciprocal cell for the broad band measurement of tensorial permeability of magnetized ferrites: Direct problem[END_REF].

The validation of the proposed method consists in verifying that the -parameters simulated with the TLM are in accordance with the ones measured for the ferrite biased by various strengths of the applied dc field.

The experimental results were obtained with an -band rectangular waveguide (dimensions and mm) made of brass. -parameters were measured for different values of the applied static magnetic field. The cell was loaded with a ferrite of a saturation magnetization T, an anisotropy field A/m, and a resonance linewidth A/m. The network analyzer (HP 8510B) calibration procedure, necessary to achieve accurate measurements at high frequencies, is a thru-reflect-line (TRL) [START_REF] Engen | Thru-reflect-line: An improved technique for calibrating the dual six-port automatic network analyzer[END_REF].

Measurements illustrated in Fig. 10 proved the nonreciprocity of the measurement device. One can also observe a good agreement between the measurement and simulations over a wide frequency band and for different magnetization states of the ferrite. In the -band, the theoretical and experimental -parameters magnitudes are very close. Moreover, measurements and simulations show the same shift for the resonance frequency appearinginthereflection parameter.

The study of the cell sensitivity has allowed us to observe significant variations in the values and frequency behavior of -parameters when the electromagnetic characteristics of the ferrite or the magnetic static field magnitude changes. This confirms the good accuracy of the theoretical results.

IV. CONCLUSION

The permeability tensor model of polycrystalline ferrites as a function of the dc bias field has been, for the first time, derived from a self-consistent theoretical approach and integrated in the TLM algorithm.

The TLM algorithm was implemented in the case of general anisotropic and dispersive media. The theoretical derivation was revisited, starting with Maxwell's equations, without invoking circuit analogy. The procedure is general and can be applied to derive the algorithm for extended TLM nodes.

Preliminary results computed in the case of a dispersive plasma medium show that the model is accurate when compared with theoretical results.

To illustrate the interest of this theoretical approach for practical applications, an example of a waveguide partly filled with a ferrite in different polarization states has been given. Comparison between experimental measurements and TLM simulations yielded a good agreement.

The ultimate objective is to insert a new pseudoanalytical model for ferrites in different magnetization states that are used for planar nonreciprocal devices implemented in low-temperature co-fired ceramic (LTCC).

Fig. 1 .

 1 Fig. 1. Field component sampling for discretization of Maxwell-Ampere's curl's equation -component.

Fig. 2 .

 2 Fig. 2. Field component sampling for discretization of Maxwell-Faraday's curl's equation -component.

3 )Fig. 3 .

 33 Fig. 3. Representation of the state space equations.

Fig. 4 .

 4 Fig. 4. Prony approximation of the tensor element (amplitude and phase).

Fig. 5 .

 5 Fig. 5. Plasma-filled metal cavity resonance spectrum. Vertical lines show theoretical solutions.

Fig. 6 .

 6 Fig. 6. Prony approximation of tensor's elements (absolute values of

Fig. 7 .

 7 Fig. 7. Rectangular waveguide measurement cell.

Fig. 8 .

 8 Fig. 8. Experimental test device.

Fig. 9 .

 9 Fig. 9. Confining of the energy along the -axis in the waveguide cross section: waves propagated in the positive and negative directions.

Fig. 10 .

 10 Fig. 10. -parameters magnitude versus frequency for different values of the magnetic static field applied . Ferrite under test properties: T, A/m, A/m. (top) Magnetic static field applied A/m. (middle) Magnetic static field applied A/m. (bottom) Magnetic static field applied. One can write the inverse matrix as (54), shown at the bottom of the previous page, and where

TABLE II COMPARISON

 II BETWEEN TLM AND THEORETICAL RESULTS


Samira Fernandes Nassar

Sandra Domenek

Alain Guinault

Gregory Stoclet

Nicolas Delpouve

Cyrille Sollogoub
email: cyrille.sollogoub@lecnam.net

Structural and dynamic heterogeneity in the amorphous phase of Poly(L,L-lactide) confined at the nanoscale by co-extrusion process

Keywords: PLLA, poly(lactic acid), confinement, RAF, rigid amorphous fraction, CRR

Multi-nanolayered Polystyrene/Poly(L,L-lactide) (PS/PLLA) films were obtained by the layermultiplying co-extrusion process, with an individual PLLA layer thickness as thin as 20 nm. The confinement of the amorphous PLLA induced a change in the molecular mobility, evidenced by a drop of the Cooperative Rearranging Region (CRR) size at the glass transition. The annealing of confined PLLA layers revealed slower crystallization kinetics and two-dimensional crystalline growth geometry. Furthermore, the annealing of PLLA in confined layers allowed a decoupling between the amorphous and crystalline phase, evidenced by the absence of a Rigid Amorphous Fraction (RAF). As a consequence, the dynamic heterogeneity at the glass transition remained unaffected by the annealing procedure. In bulk polymers, where the level of coupling between amorphous and crystals is high, the glass transition temperature increased significantly whereas the CRR size fell. It is deduced that the glass transition dynamics in semi-crystalline polymers is strongly related to the mobility landscape at the interface with crystals.

Introduction

Confining a polymer to a scale comparable to its different characteristic lengths (gyration radius or dimensions of the crystalline structure for example) leads to strong deviations of the structural and dynamical properties from the bulk. Since the pioneering work of Jackson and McKenna [START_REF] Jackson | The glass transition of organic liquids confined to small pores[END_REF] on Tg deflection of organic liquids in nanopores, many studies [START_REF] Alcoutlabi | Effects of confinement on material behaviour at the nanometre size scale[END_REF][START_REF] Forrest | The glass transition in thin polymer films[END_REF][START_REF] Kremer | Glassy dynamics and glass transition in nanometric layers and films: A silver lining on the horizon[END_REF] have discussed the effect of geometrical confinement on the molecular mobility and the glass transition temperature deviations of amorphous polymers. Similarly, crystallization under confinement has been shown to generate distinct features from bulk crystallization, possibly leading to a great variety of crystalline morphologies, inducing unique preferential orientations [START_REF] Michell | Confined crystallization of polymeric materials[END_REF][START_REF] Prud'homme | Crystallization and morphology of ultrathin films of homopolymers and polymer blends[END_REF]. In the aim of searching for finite size effects, a variety of experimental systems and geometries has been proposed to confine both amorphous and semi-crystalline polymers: droplet dispersions [START_REF] Massa | Homogeneous crystallization of poly (ethylene oxide) confined to droplets: the dependence of the crystal nucleation rate on length scale and temperature[END_REF], thin films [START_REF] Forrest | Interface and chain confinement effects on the glass transition temperature of thin polymer films[END_REF] or nano-layers [START_REF] Carr | Confined crystallization in polymer nanolayered films: A review[END_REF], polymers infiltrated in inorganic nanotemplates (nanocylinders, nanolayers, nanospheres and nanopores) [START_REF] Jackson | Vitrification and crystallization of organic liquids confined to nanoscale pores[END_REF][START_REF] Michell | Confined crystallization of polymers within anodic aluminum oxide templates[END_REF], and self-assembled block copolymers [START_REF] Wang | Properties of dimethylsiloxane microphases in phase-separated dimethylsiloxane block copolymers[END_REF]. Confinement can also be present, probably in a less regular and controllable way, in material systems like polymer blends [START_REF] Wang | Confined growth of poly (butylene succinate) in its miscible blends with poly (vinylidene fluoride): morphology and growth kinetics[END_REF], nanocomposites [START_REF] Huang | Poly (L-lactic acid) crystallization in a confined space containing graphene oxide nanosheets[END_REF] and semi-crystalline polymers [START_REF] Aharoni | Increased glass transition temperature in motionally constrained semicrystalline polymers[END_REF].

In semi-crystalline polymers, the amorphous phase appears to be constrained by the crystalline lamellae. This constraint is the consequence of both a geometrical confinement and a covalent coupling between the lamellae and the non-crystalline regions through tie molecules. When this coupling is strong enough, a separate phase of nanometric size can be produced at the interface between the two phases. A three phase model is therefore necessary to fully describe the semi-crystalline structure, in which the amorphous phase is split into two differently mobile fractions: the rigid amorphous fraction (RAF) as opposed to the mobile amorphous fraction (MAF) [START_REF] Schick | Formation and disappearance of the rigid amorphous fraction in semicrystalline polymers revealed from frequency dependent heat capacity[END_REF][START_REF] Wunderlich | Reversible crystallization and the rigid-amorphous phase in semicrystalline macromolecules[END_REF]. While the latter relaxes at the glass transition, the former devitrifies at higher temperatures than Tg. The exact temperature at which the RAF vitrifies and devitrifies has been extensively studied and seems to mainly depend on the type of polymer investigated [START_REF] Wunderlich | Reversible crystallization and the rigid-amorphous phase in semicrystalline macromolecules[END_REF]. Moreover, the presence of RAF, having its own specific properties, not only modifies the mobility of the amorphous phase but may impact also the macroscopic properties, especially mechanical [START_REF] Kolesov | The rigid amorphous fraction of cold-crystallized polyamide 6[END_REF][START_REF] Rastogi | The three-phase structure and mechanical properties of poly(ethylene terephthalate)[END_REF] and gas barrier properties [START_REF] Hu | Oxygen transport and free volume in cold-crystallized and melt-crystallized poly (ethylene naphthalate)[END_REF][START_REF] Lin | Oxygen solubility and specific volume of rigid amorphous fraction in semicrystalline poly(ethylene terephthalate)[END_REF]. Therefore, the RAF must be taken into account and thoroughly characterized in order to propose a complete understanding of the structure -property relationships in semi-crystalline polymers.

Polylactide (PLA) is a polyester with slow crystallization rate, different polymorphic forms [START_REF] Pan | Polymorphism and isomorphism in biodegradable polyesters[END_REF] and a tendency to form RAF [START_REF] Del Rio | A PALS contribution to the supramolecular structure of poly (L-lactide)[END_REF][START_REF] Delpouve | Cooperativity length evolution during crystallization of poly (lactic acid)[END_REF][START_REF] Fernandes-Nassar | Multi-scale analysis of the impact of polylactide morphology on gas barrier properties[END_REF][START_REF] Guinault | Impact of crystallinity of poly (lactide) on helium and oxygen barrier properties[END_REF][START_REF] Righetti | Time and temperature evolution of the rigid amorphous fraction and differently constrained amorphous fractions in PLLA[END_REF][START_REF] Zuza | Glass transition behavior and dynamic fragility in polylactides containing mobile and rigid amorphous fractions[END_REF]. The RAF formation in PLA has been extensively studied and clearly correlated with the crystal growth [START_REF] Fernandes-Nassar | Multi-scale analysis of the impact of polylactide morphology on gas barrier properties[END_REF][START_REF] Righetti | Time and temperature evolution of the rigid amorphous fraction and differently constrained amorphous fractions in PLLA[END_REF][START_REF] Righetti | Crystalline, mobile amorphous and rigid amorphous fractions in poly (L-lactic acid) by TMDSC[END_REF]: at low crystallization temperature (near the glass transition) polymer chains have low mobility, which hampers the organization of the polymeric segments in ordered crystal structures. This causes coupling of the crystalline and amorphous phase by tie chains which can be observed by the existence of the RAF [24-26, 28, 29]. The RAF at low temperatures develops in parallel with the crystals, whereas for higher crystallization temperatures (near melting), the RAF appears at long crystallization time simultaneously with the secondary crystallization, occurring after spherulite impingement. Because of the existence of extensive knowledge, PLA is a good candidate for creating semi-crystalline systems with tailored crystalline morphology and controlled RAF in order to study the impact of the induced geometrical confinement on the amorphous phase dynamics. In this aim, different theoretical approaches and experimental techniques (differential scanning calorimetry, dielectric relaxation spectroscopy, and dynamic mechanical analysis) have been used and have sometimes led to contradictory results. While Mijovic et al. [START_REF] Mijovic | Normal and segmental mode dynamics of end-functionalized poly (propylene oxide) by dielectric relaxation spectroscopy and dynamic mechanical spectroscopy[END_REF] did not observe any change in the glass transition dynamics upon crystallization, Fritz et al. [START_REF] Fitz | T g Depression in poly (l (-)-lactide) crystallized under partially constrained conditions[END_REF] measured a significant Tg decrease when PLA crystallized under constrained conditions. Some other authors [START_REF] Zuza | Glass transition behavior and dynamic fragility in polylactides containing mobile and rigid amorphous fractions[END_REF][START_REF] Brás | Crystallization of poly (L-lactic acid) probed with dielectric relaxation spectroscopy[END_REF] have evidenced a clear confinement effect of the amorphous phase appearing during primary crystallization, inducing RAF formation, Tg increase and enhanced dynamic fragility around Tg. For example, Delpouve et al. [START_REF] Delpouve | Cooperativity length evolution during crystallization of poly (lactic acid)[END_REF][START_REF] Delpouve | Cooperative rearranging region size in semi-crystalline poly (l-lactic acid)[END_REF][START_REF] Saiter | Cooperative rearranging region size determination by temperature modulated DSC in semi-crystalline poly (L-lactide acid)[END_REF], using the concept of Cooperative Rearranging Regions (CRR), originally introduced by Adam and Gibbs [START_REF] Adam | On the temperature dependence of cooperative relaxation properties in glass-forming liquids[END_REF], and, following Donth's approach [START_REF] Donth | The size of cooperatively rearranging regions at the glass transition[END_REF], evidenced a confining effect due to the crystallization inducing a drastic reduction in the characteristic length of this CRR. Still, in such semi-crystalline systems, it is difficult to decorrelate the pure geometrical confinement from the coupling between the amorphous and the crystalline phases and alternative ways of confinement have to be used. Some authors measured the Tg variation by ellipsometry of spincoated PLLA thin films on different substrates. While Narladkar et al. [START_REF] Narladkar | Difference in glass transition behavior between semi crystalline and amorphous poly (lactic acid) thin films[END_REF] observed a decrease of the Tg for film thicknesses below 50 nm on non-interacting surface, Spièce et al. [START_REF] Spiece | Are polymers glassier upon confinement?[END_REF] reported an increase of several degrees for two different substrates. Those contradictory results point out the importance of polymer-surface interactions and free surfaces that can be responsible for enhanced molecular mobility in thin film [START_REF] Ediger | Dynamics near free surfaces and the glass transition in thin polymer films: a view to the future[END_REF][START_REF] Fu | Evidence of Enhanced Mobility at the Free Surface of Supported Polymer Films by in Situ Variable-Temperature Time-of-Flight-Secondary Ion Mass Spectrometry[END_REF]. Such free interfaces are suppressed when the thin polymer film is sandwiched between confining walls. The original layer-multiplying co-extrusion process allows creating multi-nanolayered systems in which one polymer is sandwiched between layers of another rigid confining polymer, creating capped symmetric interfaces. A further advantage is that those unique 'nanostructured bulk materials' can be characterized with conventional techniques for bulk materials. In conclusion, the nanolayer co-extrusion process is an interesting means for mimicking symmetric polymer confinement between hard walls in controlling geometrical constraints.

In our study, nanolayer co-extrusion has been used for the first time to confine PLA. Polystyrene (PS) is chosen as confining polymer since it provides non-interacting surface towards PLA and its Tg (around 100°C) allows for a post-annealing treatment at 85°C inducing PLA cold crystallization under confinement. The study of amorphous and crystallized PLA/PS nanolayered films should shed additional light above the macromolecular mobility in the amorphous phase and the coupling between the amorphous and crystalline phases in PLA under geometrical confinement.

Experimental

Process and crystallization parameters

Polylactide (PLLA) pellets were purchased from Corbion Purac (PLA REVODE 190). The content of L,L-lactide was higher than 99 mol-%. The mass average molar mass, measured by GPC (Agilent 220 HT) using PS standards, was Mw = 177 kg.mol -1 and the dispersity Đ, defined as Mw/Mn, was 1.64. The gyration radius, estimated from the plot of Fang et al. [START_REF] Fang | Bimodal architecture and rheological and foaming properties for gammairradiated long-chain branched polylactides[END_REF], was around 20 nm. Polystyrene (PS) pellets were obtained from Total Petrochemicals (PS 1340).

The layer-multiplying co-extrusion process was used to fabricate multi-layered films of 3, 513 and 2049 alternating layers of PLLA and PS. The process is presented in the Figure 1. It consists in two single screw extruders of 20 mm with gear pumps, a three-layer feed block (A-B-A), a series of layermultiplying elements, a flat die and chill rolls. The weight percentage of PLLA in the multilayer film is 25 %. The initial three-layer (PS/PLLA/PS) polymer flow enters a mixing section, followed by a sequence of layer-multiplying elements. The principle of the layer-multiplying elements is to cut the flow in half vertically, and to superpose, compress and stretch it to its original width, hence doubling the number of layers with each element. A series of n elements leads thus to 2 n+1 +1 alternating layers. The extrusion conditions are given in Supporting Information (SI 1).

Figure 1. Principle of the layer-multiplying co-extrusion process

The set of samples, presented in Table 1, consisted of a blank sample, being the initial three-layer coextruded film with an individual PLLA layer thickness of 30000 nm, an intermediate sample (PLLA layer thickness of 300 nm), and one confined sample (20 nm). The latter PLLA layer thickness is in the order of the gyration radius of PLLA. After extrusion, some films were sandwiched between two sheets of Teflon and two stainless steel plates, and annealed in a heating press (Gribite) during 180 minutes at 85°C and a pressure set to 5 x 10 6 Pa.

Flat die

Melt rheology. The melt viscosities of PLLA and PS were measured using an Anton Paar MCR 502 rheometer in an oscillatory mode between 0.01 and 100 rad/s. The domain of linear strain was determined by performing preliminary deformation sweeps, and the working strain was set to 1% for both polymers. The shear viscosity as a function of shear rate was calculated using the Cox-Merz rule.

Atomic force microscopy (AFM). AFM was used to characterize the layered morphology of the coextruded films. AFM images were obtained in tapping mode using a multimode microscope controlled by a Veeco Nanoscope V controller. The tips (silicon, spring constant 40 N/m, oscillation frequency ca. 300 kHz) had a radius of curvature less than 10 nm. Specimens were taken from the center of the extruded films and were cut from the cross section with an ultramicrotome 2088 Ultrotome V (LKB) equipped with a diamond knife at a cutting speed of 1 mm/s. 15 images were recorded at full resolution (4096 × 4096 pixels), with a scan rate of 0.9 Hz, throughout the thickness of the film. Following the method developed by Bironeau et al. [START_REF] Bironeau | Evaluation of morphological representative sample sizes for nanolayered polymer blends[END_REF], the analysis of these 15 images containing each around 20 layers is representative of the whole sample with an uncertainty of 10%.

Wide Angle X-Ray Scattering (WAXS). WAXS experiments were performed using a Genix microsource (XENOCS) equipment operating at 50 kV and 1 mA. The Cu-Kα radiation used was selected with a curved mirror monochromator. The 2D patterns were recorded on a CCD camera from Photonic Science and the working distance was calibrated using a PLA sample. Integrated intensity profiles were computed from the 2D patterns using the FIT2D software. Before analysis standard corrections were applied to the patterns such as dark current substraction and background correction.Analyzes of the isothermal crystallization kinetics were made using the same X-ray generator and heating in situ the sample using a Linkam heating stage.

Modulated Temperature Differential Scanning Calorimetry (MT-DSC).

The MT-DSC analyses were performed on a Thermal Analysis® Instrument DSC Q100. Nitrogen was used as purge gas (50 mL/min). The samples weights were about 5-10 mg, encapsulated in Tzero hermetic aluminum pans. Calibration in temperature and enthalpy was carried out using an indium standard. The specific heat capacity of each sample was measured using sapphire as a reference. The glass transition region of PLLA was analyzed using a heat-cool temperature modulation (oscillation amplitude of 3 K, oscillation period of 120 s, and heating rate of 1 K min -1 ). The crystallization and melting of PLLA were analyzed using a heat-only temperature modulation (oscillation amplitude of 0.32 K, oscillation period of 60 s, and heating rate of 2 K min -1 ). By MT-DSC, in addition of the modulated heat flow, the apparent specific complex heat capacity C* is obtained from the complete deconvolution procedure proposed by Reading and co-authors [START_REF] Lacey | Theory and practice of modulated temperature differential scanning calorimetry[END_REF]. Its in-phase component (C') versus temperature appears as a step, when its out-ofphase component (C'') represents a peak in the glass transition region.

Results and discussion

Optimization of the layer multiplying co-extrusion process

Both uniformity and continuity of the layers depend strongly on the viscosity ratio ρ between the two polymers, a viscosity ratio far from one can indeed result in phenomena of encapsulation of the polymer of high viscosity by the polymer of lower viscosity [START_REF] Ponting | Polymer nanostructures by forced assembly: process, structure, and properties[END_REF]. It is defined as follows:

where and are respectively the viscosities of the confined polymer (in our case PLLA) and confining polymer (PS). The viscosity curves (available in the SI 2) show that in the shear rate range of the co-extrusion process, typically between 1 and 50 s -1 , the viscosity ratio was close to 1, which ensures homogeneity of the stratified flow and prevents from layer rupture when decreasing layer thickness [START_REF] Bironeau | Existence of a critical layer thickness in PS/PMMA nanolayered films[END_REF].

Figure 2 depicts the AFM phase images obtained on as-extruded and annealed films. A reasonable contrast between the two components (PS corresponds to the larger layers) enabled to reveal the layered structure and to perform image analysis in order to measure a thickness distribution of the layers. The AFM images showed continuous and relatively uniform layers of PS and PLLA. Furthermore it can be observed that the layer continuity was maintained after crystallization. The observed distributions of thickness of PLLA layers for each sample are given in Figures 3a (2049 layers) and 3b (513 layers). The distribution of thickness can be represented by a log-normal distribution. The average layer thicknesses of PLLA was 220 ± 50 nm for the films containing 513 layers and 20 ± 5 nm for the films with 2049 layers, which was in reasonable agreement with the expected nominal thickness given in Table 2.

Crystallization of PLLA under confinement

Figure 4a shows the WAXS patterns obtained on as-extruded PS/PLLA films. Only the amorphous halos of PLLA and PS were visible, indicating that PLLA was amorphous after the extrusion, which was expected because of the slow crystallization kinetics of PLLA [START_REF] Courgneau | Effect of crystallization on barrier properties of formulated polylactide[END_REF][START_REF] Saeidlou | Poly (lactic acid) crystallization[END_REF]. The temperature window between the Tgs of PLLA (60°C) and PS (100°C) allowed for isothermal cold-crystallization of PLLA under hard confinement, i.e. between two glassy polymer walls. The DSC curves of the films annealed at 85 °C for 3 h (given in SI 3) did not exhibit any cold crystallization peak, indicating that the crystallization after 3 hours was complete for all the samples. Figure 4b shows the WAXS diffractograms of annealed PLLA. The strong reflections of PLLA at 2q around 16.4° and 2q = 18.5°, attributed to the 200/110 and 203 planes respectively, could be observed for all samples [START_REF] Stoclet | Thermal and strain-induced chain ordering in lactic acid stereocopolymers: influence of the composition in stereomers[END_REF][START_REF] Zhang | Disorder-to-order phase transition and multiple melting behavior of poly (Llactide) investigated by simultaneous measurements of WAXD and DSC[END_REF]. The present crystallization temperature (85 °C) would theoretically favor the a'-polymorph of PLLA [START_REF] Guinault | Impact of crystallinity of poly (lactide) on helium and oxygen barrier properties[END_REF][START_REF] Lorenzo | Crystal polymorphism of poly (l-lactic acid) and its influence on thermal properties[END_REF]. However the minor reflections characteristic of the a'-polymorph could not be observed here, most probably because of the sensitivity limit of the DRX measurement as PLLA accounts only for 25 wt% of the sample. As a consequence, the occurrence of the a'-polymorph cannot be asserted. In the aim to study the impact of the confinement on the crystallization kinetics of PLLA, the relative crystallinity of PLLA was recorded by WAXS during isothermal annealing at 85 °C for different layer thicknesses. For this experiment, several films were stacked in order to increase the WAXS signal, introduced in a heating stage and heated from the bottom side. Therefore, the temperature control in the sample thickness was not as efficient as it can be expected in the heating press. The results, shown in the Figure 5, are thus used for the sake of comparison of crystallization kinetics under these conditions.

The isothermal crystallization kinetics of PLLA was analyzed with the help of the Avrami equation:

, ( 2
)
where a(t) is the relative crystallinity fraction at time t, K is the crystallization rate constant and n is the Avrami exponent depending on the nucleation and the growth geometry (sphere, disc, etc.) [START_REF] Avrami | Kinetics of phase change. I General theory[END_REF]. Equation ( 2) can be transformed into the following form:

, ( 3
)
The kinetic parameters n and K can therefore be obtained by plotting ln(-ln(1-a(t))) as a function of ln(t). The Avrami model in Figure 5a, for 30000 nm sample, fitted well with experiments for relative crystallinity below 0.8, with an Avrami exponent close to 3, associated according to the Avrami theory [START_REF] Avrami | Kinetics of phase change. I General theory[END_REF], with a three dimensional crystallization growth with an instantaneous nucleation. However, for higher relative crystallinity (above 0.8), the experimental curve showed a slower crystallization than the one predicted by the Avrami model, which is typical for the occurrence of a secondary crystallization. The non-linear behavior for the double logarithm of the relative crystallinity in Figure 5b is another signature of this secondary crystallization. For 20 and 300 nm samples, this secondary crystallization is suppressed and the Avrami model fits perfectly with experiments, which corresponds to linear curves for the double logarithm of the relative crystallinity in Figure 5b. The Avrami exponents, n=1.7 and 1.6 for individual PLLA layer thicknesses of 20 nm and 300 nm respectively (Table 2), correspond to a two dimensional crystallization growth with an instantaneous nucleation. The crystallization rate (K) and crystallization half-time (t1/2) are given in Table 2. We observed a decrease in K when PLLA thickness decreases correlated with the increase in t1/2. The delay of the overall crystallization rate as the layer thickness decreases was already observed for other polymers in multi-nanolayered systems, such as PEO [START_REF] Wang | Crystallization kinetics of poly (ethylene oxide) in confined nanolayers[END_REF] or PET [START_REF] Flores | Confined crystallization of nanolayered poly (ethylene terephthalate) using Xray diffraction methods[END_REF].

To summarize, in the thin layers the crystallization rate was reduced, the crystalline growth was restricted to two dimensions and the secondary crystallization was not observed.

) exp( 1 ) ( n t K t × - - = a K t n t ln ) ln( ))) ( 1 ln( ln( + = - - a Figure 5
. Relative crystallization and Avrami plots versus time for 30000 nm, 300 nm and 20 nm isothermal crystallization at 85°C (a); Double logarithm versus ln(time) and the Avrami plots (black lines) for annealed samples (30000 nm, 300 nm and 20 nm) at 85°C (b).

Three-phase model of confined PLLA

As previously mentioned, PLLA can be described by a three-phase model and the quantification of each fraction is possible from calorimetric analysis. The degree of crystallinity (Xc) of the PLA samples was calculated from

, ( 4
)
where ΔHm is the enthalpy of melting after subtraction of the small exothermal event at the beginning of the melting peak, ΔHcc is the cold crystallization enthalpy (both ΔHm and ΔHcc are normalized to PLLA content), and ΔHm 0 is the standard melting enthalpy, i.e. melting enthalpy of the perfect crystal of infinite size (93.1 J/g) [START_REF] Fischer | Investigation of the structure of solution grown crystals of lactide copolymers by means of chemical reactions[END_REF]. The use of MT-DSC is specifically required in the present study as the glass transition of PS occurs in the same temperature range as the cold crystallization of PLLA. An example of the analysis procedure is given in Supporting Information (SI 4). The difference between the cold crystallization enthalpy and the melting enthalpy for the determination of the degree of crystallinity was double checked using the average heat flow signal which equals the classical DSC signal, and the reversing and the non-reversing heat flow signals from MT-DSC. The content of the RAF, XRA, can be calculated from:

with , ( 5
)
where XMA is the content of MAF, ΔCp is the heat capacity change measured at the glass transition and ΔCp 0 the heat capacity change of the fully amorphous sample. The value of ΔCp 0 was measured using the rejuvenated initial films. The numerical results are presented in

ln (time) (h)

30000 nm 300 nm 20 nm

X c = ΔH m -ΔH cc ΔH m 0 % 100 = + + RA MA c X X X 0 p p MA C C X D D =
not observe any difference in crystallinity for confined poly(ethylene oxide) (PEO). Our result is comparable to the one of Boufarguine et al. [START_REF] Boufarguine | PLA/PHBV films with improved mechanical and gas barrier properties[END_REF] for the couple PLA/PHBV. They observed an increase of the crystallinity degree for PHBV when the polymer was confined in thin layers, possibly attributed to the strong molecular orientation induced by the polymer flow in the multiplying elements or the development of crystalline orientations. In conclusion, the confinement retarded the kinetics of crystallization but allowed reaching a higher degree of crystallinity without secondary crystallization.

The RAF of PLLA decreased from about 40 to 10% as the layer thickness was reduced from 30000 to 300 nm, and it was not observed after 3h crystallization in confined PLLA layers of 20 nm thickness, although those samples had the highest degree of crystallinity (Table 2). For those films, a 25% content of RAF was recorded after a long annealing time (after 9h). This means that the decoupling between crystalline and amorphous phases was possible in thin layers (for 3h thermal annealing). Several hypotheses can be suggested to explain this result. First, it might be associated to a pure geometric constraint effect. It has been proposed that the domain of molecular segments involved in the coupling should reach a critical length to produce an intermediate phase [START_REF] Lin | Oxygen solubility and specific volume of rigid amorphous fraction in semicrystalline poly(ethylene terephthalate)[END_REF]. In a layer of 20 nm, the amorphous phase is besides strongly confined by the few existing crystalline lamellae that contribute to the further decrease of the available space. The change in the crystalline growth from a three dimensional to a two dimensional order, as deduced from the Avrami equation, was thus considered as a possible cause favouring the decoupling.

Another hypothesis is that the modification of the crystallization kinetics in thin layers delayed the formation of the RAF. In a previous work, it was reported that prolonging the annealing time favoured the gradual conversion of MAF into RAF with time [START_REF] Fernandes-Nassar | Multi-scale analysis of the impact of polylactide morphology on gas barrier properties[END_REF]. The decoupling of the amorphous and the crystalline phase in the confined configuration and the absence of secondary crystallization had thus an effect of retard on the formation of the RAF, the quantity of which even after 9 h of annealing is still smaller than that of 3 h of annealing of bulk PLLA (Table 2).

Table 2. Structural and thermal parameters of PLLA confined at different thickness obtained from insitu WAXS and MT-DSC heat-only analyses.

PLLA nominal thickness t1/2 To summarize, contrary to what was observed for bulk samples, the crystalline phase development under confinement occurred without simultaneous creation of RAF and secondary crystallization. The RAF was created at longer times with sufficient delay to allow the tuning of the content of RAF at high crystallinity degrees. The confined samples obtained after 3h annealing at 85°C offer a unique combination with maximal degree of crystallinity and no RAF at all, presenting a complete decoupling between amorphous and crystalline phases.

(h) n K (h -n ) ΔCp 0 (Jg -1 K -1 ) ΔCp (Jg -1 K -1 )

Dynamic heterogeneity of confined PLLA

The mobility landscape of the amorphous phase can be pictured by the size of the cooperative rearranging regions (CRR) that is impacted by structural constraints. As mentioned in the introduction, the decrease of the CRR size has been commonly reported in semi-crystalline polymers [START_REF] Delpouve | Cooperative rearranging region size in semi-crystalline poly (l-lactic acid)[END_REF][START_REF] Schick | Characteristic length of glass transition: experimental evidence[END_REF], revealing a restriction in amorphous chain segment mobility due to the presence of crystalline phase. However, in these systems, the strong interfacial interactions between crystalline and amorphous phases make it difficult to determine the exact nature of these constraints. Initially pure geometrical confinement was invoked to explain the disturbance of the amorphous phase dynamics [START_REF] Schick | Characteristic length of glass transition: experimental evidence[END_REF], but more recently the possibility of an additional alteration of dynamics induced by the RAF has been suggested [START_REF] Delpouve | Cooperativity length evolution during crystallization of poly (lactic acid)[END_REF]. It is nevertheless difficult to discriminate between the effects of the geometrical confinement and the "covalent coupling" of the amorphous phase to crystalline domains, corresponding to RAF. In this context, the samples obtained via nanolayer coextrusion in which PLLA is confined in thin layers of 20 nm and crystallized at its maximum value, in presence of RAF (25% when annealed 9 hours) or not (0% when annealed 3 hours), offer unique opportunity to gain insight in the impact of geometrical confinement and RAF on the dynamic landscape of the amorphous phase.

The size of the CRR can be calculated directly from MT-DSC measurements using Donth's approach [START_REF] Donth | The size of cooperatively rearranging regions at the glass transition[END_REF][START_REF] Hempel | Characteristic length of dynamic glass transition near T g for a wide assortment of glass-forming substances[END_REF]. An example of how to retrieve the required physical quantities is shown in Supporting Information (SI 5). The cooperativity volume of a CRR at the dynamic glass transition temperature Ta can be estimated from the equation below:

, ( 6
)
where kB is the Boltzmann constant, dT the average temperature fluctuation related to the dynamic glass transition of a CRR, r the density equal to 1.25 g.cm -3 for the amorphous phase of PLLA, and Cp the specific heat capacity at constant pressure. When applying the fluctuation-dissipation theorem (FDT) and fitting the out-of-phase component C''(T) or the derivative of C'(T) with respect to temperature with a Gaussian (supposing temperature variable and frequency constant), δT corresponds to the standard deviation of this Gaussian [START_REF] Hempel | Characteristic length of dynamic glass transition near T g for a wide assortment of glass-forming substances[END_REF], and Tα corresponds to its maximum. The values of Cp glass Ta and Cp liquid Ta are obtained from the C' signal by prolonging the glass and liquid lines to the dynamic glass transition temperature after normalization to the quantity relaxing at the glass transition, i.e. the MAF content. The real part and imaginary part of the complex heat capacity during the glass transition of PLLA are presented in Figure 6. The curves are plotted before any normalization; however the C' curves have been shifted for the consistency of Cp glass with the computed values linked to the vibrational molecular motions, as recommended by Pyda et al. [START_REF] Pyda | Heat capacity of poly (lactic acid)[END_REF]. For the thick layers (30000 and 300 nm), the increase of the crystallinity degree (Xc) of annealed PLLA layers caused the decrease of the amplitude of the heat capacity change because less amorphous phase was involved in the relaxation (Figure 6a). Besides, a shift of the glass transition to higher temperatures was observed due to the constraints generated by the crystalline environment, which caused the amorphous phase of bulk polymers to require more thermal energy for relaxation in the presence of crystallites. The Ta shift was concomitant with the presence of crystals and RAF [START_REF] Zuza | Glass transition behavior and dynamic fragility in polylactides containing mobile and rigid amorphous fractions[END_REF]. Moreover the glass transition domain was significantly broadened, meaning that dT strongly increased and that the peak of C" almost disappeared (Figure 6b). The same effect was observed in thin layers (20 nm) after 9 h of annealing, when 25% RAF was generated. However, the behaviour of PLLA after 3h annealing in the 20 nm layers was quite different. Due to the absence of RAF, the heat capacity step at the glass transition was in between the other semi-crystalline PLLA and the amorphous ones (Figure 6a). The C'' peak was furthermore perfectly discernible (Figure 6b). Finally, in comparison to the 20 nm amorphous PLLA, Ta did not shift to higher temperatures but 3°C below, and dT was only slightly higher.

The quantification of the changes in the dynamic heterogeneity of PLLA layers under increasing confinement is given in Table 3, presenting the variation of the dynamic glass transition temperature (Ta), the mean temperature fluctuation, and the cooperativity length for both amorphous (as-extruded films) and crystallized (annealed films) PLLA. No impact of the layer thickness on the Ta of amorphous PLLA samples was observed, which is in contradiction with the results of Narladkar et al. [START_REF] Narladkar | Difference in glass transition behavior between semi crystalline and amorphous poly (lactic acid) thin films[END_REF] and Spièce et al. [START_REF] Spiece | Are polymers glassier upon confinement?[END_REF]. To explain these results, one might suggest that in the absence of free surfaces in multi-nanolayered films, there was no change of the local free volume at the interface. Only a slight shift of 3°C towards low temperatures was observed for 20 nm PLLA after 3h annealing. In this sample, the confinement of the PLLA amorphous phase was stronger, although not quantified, since it was caused by both the layers of glassy PS and the crystallites. Attributing or not this decrease of Ta to a confinement effect is questionable. By contrast, the shift of Ta to higher temperatures recorded for the other semi-crystalline PLLA was clearly correlated with the apparition of RAF. This is coherent with earlier works that report that the Ta of PLA increases significantly when annealing conditions favoring strong coupling [START_REF] Saiter | Probing the chain segment mobility at the interface of semi-crystalline polylactide/clay nanocomposites[END_REF]. For all amorphous PLLA samples, the cooperativity length was close to three nanometers. Contrary to what was observed on other amorphous polymers (poly(methyl methacrylate), polycarbonate, PS) [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] Tomczak | Segment dynamics in thin polystyrene films probed by single-molecule optics[END_REF][START_REF] Tran | Nanoscale characteristic length at the glass transition in confined syndiotactic poly (methyl methacrylate)[END_REF], the cooperativity length in amorphous PLLA did not decrease when the PLLA layer thickness was reduced. This result is consistent with the absence of variations in Ta. As mentioned above, the effects of confinement were less easily apparent in these systems, even for layers as thin as 20 nm.

After 3h annealing, the 20 nm PLLA exhibited a cooperativity length around 2.7 nm. In this material there was no RAF, thus no interfacial area of coupling between amorphous zones and crystals. Moreover, due to the low compatibility of PLLA and PS, PS surfaces behaved as "slippery surfaces" and did not impose any supplementary restrictions. As a consequence, it is deduced that the pure geometric restrictions imposed by the neighboring layers of PS and the crystallites poorly impacted the dynamic heterogeneity in PLLA.

On the contrary, in all semi-crystalline PLLA exhibiting substantial contents of RAF, the cooperativity length decreased significantly, down to values ranging from 1.6 to 2.0 nm. Esposito et al. [START_REF] Esposito | From a Three-Phase Model to a Continuous Description of Molecular Mobility in Semicrystalline Poly (hydroxybutyrate-co-hydroxyvalerate)[END_REF] reported from MT-DSC and dielectric spectroscopy measurements that the width of the glass transition in semicrystalline materials was strongly related to the level of coupling between amorphous and crystalline phases. If the level of coupling is high, the structural constraints generated by the crystalline phase progressively transfer to the non-crystalline part of the material. This leads to a mobility gradient in the amorphous phase, the signature of which is a broad distribution of relaxation temperatures and therefore a significantly reduced CRR size.

To give a summary, the crystallization of PLLA under geometrical confinement offered favorable conditions for the decoupling of the amorphous and crystalline phases, characterized by the decrease of the coupled phase (RAF quantity). Consequently, the impact of crystallization on the relaxation phenomena in the amorphous phase was unique, leading to an almost negligible decrease of the glass a x T transition and reduction of the cooperativity length. In that case, the CRR size became independent from the degree of crystallinity of the samples. In bulk materials, where the crystal growth adds a supplementary constraint to the geometric confinement effect because one macromolecule is part of both the confining and the confined part, an usual increase of both the dynamic glass transition and the dynamic heterogeneity was recorded. In the mobility landscape of semi-crystalline polymers, the role of the RAF is thus crucial since the volume of the MAF involved in the relaxation processes depends on its degree of decoupling from the crystal. Consequently, even in the confined space, the increase of the RAF quantity with time reduced significantly the CRR size (Table 3).

Conclusion

While RAF is usually known to form concomitantly with crystallization for bulk PLLA, a complete decorrelation between these two phenomena has been observed in PLLA confined in nanolayered PLLA/PS films, fabricated with the layer multiplying co-extrusion process. This allowed to rationally control the quantity of created RAF and to fabricate unique samples with nanoconfined PLLA, crystallized at its maximum value and without any RAF content. Moreover, comparing samples of geometrically confined PLLA presenting different contents of RAF, allowed to discriminate the effects of confinement and coupling on the dynamic landscape of the amorphous phase. We observed that geometrical confinement impacted barely the amorphous dynamics, while phase coupling had a major influence on the dynamic heterogeneity and increased the alpha transition temperature. This knowledge can offer a novel strategy to optimize semi-crystalline polymers' performance properties that are impacted by the presence of rigid amorphous phases.

Figure 2 .

 2 Figure 2. Tapping mode AFM phase images of nanolayered PS/PLLA containing different number of layers as indicated.

Figure 3 .

 3 Figure 3. Distributions of PLLA layer thickness for the films containing 2049 layers (a) and 513 layers (b). The red lines represent the log-normal distribution curves.

Figure 4 .

 4 Figure 4. WAXS intensity profiles of PS/PLLA films (a) before annealing (b) after annealing at 85°C during 180 minutes.

Figure 6 .

 6 Figure 6. Temperature dependence of (a) real part (C') (b) imaginary part (C'') of complex heat capacity (C*) for different multilayered PS/PLLA films when PLLA is amorphous or crystallized.

Table 1 . PS/PLLA multilayered films characteristics and process conditions.

 1

	Number of	Number	Total film thickness	Nominal PLLA layer thickness
	multiplying elements	of layers	(µm)	(nm)
	0	3	120	30000
	8	513	300	300
	10	2049	80	20
	2.2. Characterization techniques

Table 2 .

 2 The

crystallinity degree of PLLA after annealing increased with the decrease of the layer thickness. The underlying physics are not yet established, but different types of behaviour were already observed. Mackey et al.

[START_REF] Mackey | Confined crystallization of PVDF and a PVDF-TFE copolymer in nanolayered films[END_REF]

showed a slight decrease in crystallinity for confined polyvinylidene fluoride (PVDF), while Wang et al.

[START_REF] Wang | Confined crystallization of PEO in nanolayered films impacting structure and oxygen permeability[END_REF]

did

Table 3 .

 3 Quantification of the changes in the dynamic heterogeneity of PLLA layers for different layer thickness and annealing time.

	PLLA nominal	XRA (%) Ta (°C)	dT (°C)	(nm)
	thickness
	30000 nm	0	59.0±0.5	3.0±0.2	2.9±0.3
	30000 nm*	39	65.5±0.5	5.3±0.2	2.0±0.2
	300 nm	0	59.5±0.5	2.8±0.2	3.0±0.3
	300 nm*	9	67.5±0.5	5.6±0.2	1.9±0.2
	20 nm	0	59.5±0.5	2.8±0.2	3.0±0.3
	20 nm*	0	56.5±0.5	3.1±0.2	2.7±0.3
	20 nm**	30	67.5±0.5	7.5±0.2	1.6±0.2

*Crystallized PLLA with annealing conditions: 85 °C during 3h, **85°C during 9 h


Benjamin Le Delliou

Olivier Vitrac

Mickael Castro

Stephane Bruzaud

Sandra Domenek
email: sandra.domenek@agroparistech.fr

Characterization of a new bio-based and biodegradable blends of poly(3hydroxybutyrate-co-3-hydroxyvalerate) and poly(butylene-co-succinate-co-adipate)

Melt blending of poly(3-hydroxybutyrate-co-3-hydroxyvalerate) and poly(butylene-cosuccinate-co-adipate) was investigated by means of batch mixing at different weight ratios (100:0, 70:30, 50:50, 30:70 and 0:100). PHBV and PBSA were immiscible. PBSA formed small nodules in PHBV, while PHBV formed large inclusions in PBSA. In 50/50 wt% blends, a cocontinuous structure was obtained. The crystallization rate of PHBV and PBSA increased in the blends, most probably due to mutual nucleation, except at the later stages, where PHBV crystallization rates slowed down inside the nodules and in the co-continuous structure. The mechanical properties were successfully modeled with the EBM model, including parallel and serial resistances and show that the rigidity of the material can be modulated using PBSA. The elongation at break is however governed by PHBV. The blends featured brittle fracture even if PBSA was the continuous phase. The analysis of the stress at break showed that the fracture could be ascribed to debonding at the interfaces and the fracture behavior of PHBV.

Introduction

Polymers are largely used in short-lived products, such as food packaging or single use containers because they are low-cost, light-weight, easy to process, flexible over an extensive range of temperatures. Their service life is thus much shorter than their total life time, which leads to persistent environmental pollution if collection and end-of-life of the materials are not managed adequately. 1 One means to prevent persistent pollution is to substitute commonly used polymers by biodegradable and compostable materials with satisfying performance.

Polyhydroxyalkanoates (PHAs) are biobased thermoplastic polyesters which are biodegradable even in marine environment, but they have poor mechanical performance and are difficult to process. They are synthesized as carbon and energy storage compounds by numerous microorganisms when macronutrients such as oxygen, nitrogen, or phosphorus are limited but a carbon source is still available in the culture medium. 2 Today, the main commercial polymers are short-chain-length PHAs, represented by poly(3-hydroxybutyrate) (PHB), and the copolymer poly(3-hydroxybutyrate-co-3-hydroxyvalerate) (PHBV). 2,3 Their glass transition temperature (Tg) ranges between 0 and 5 °C. The isotacticity of PHBV results in stiff and brittle materials with an elongation at break below 5 % and a stress at break between 30 and 40 MPa. 4 PHBV has, moreover, a very narrow processing window and degrades rapidly upon melting.

Its low shear viscosity and elastic melt strength impede its processing by extrusion-blowing. 5 To increase the material performance of PHBV, blending with other polymers was already proposed as a promising strategy. In the perspective of preserving the biodegradability of the mixture, binary blends were already studied with polylactic acid (PLA), [6][7][8] , polybutylene adipate-co-terephthalate (PBAT), 5,9 polybutylene succinate (PBS). 10 An improved toughness and the possibility of tuning the mechanical properties were found for blends based with PLA, PHBV, and PBS. 11 Poly(butylene-co-succinate-co-adipate) (PBSA) is an alternative to PBS and could be used to develop polymer blends with increased ductility. While great attention has been focused on the characterization of PHBV in literature, reports on PBSA are rare. The elongation at break of PBSA can reach up to 400 %. 12 It is a biodegradable polymer that can be partially or fully biobased. 13 The higher elongation at break and its lower Tg (-45 °C) make PBSA superior to PBS. In addition, the lower crystallinity of PBSA compared to its counterpart PBS, results in accelerated biodegradation. 14 Polyester blends of PBSA and PLA were already studied at ambient temperature and relative humidity. 15,16 Authors reported on immiscibility between both constituents, formation of small nodules of PBSA in PLA, a reduction in tensile modulus from about 2.7 GPa (neat PLA) to 1 GPa (PLA/PBSA 50/50) and a moderate improvement of the elongation at break, up to e=6.5 % for PLA/PBSA 70/30. 16 The combination of the brittle material PHBV, and the highly ductile PBSA is potentially interesting for obtaining bio-based and biodegradable blends with improved processing and mechanical properties.

The present study presents a detailed rheological, morphological, thermal and mechanical characterization of PHBV/PBSA blends. The impact of the blend composition and structure on viscosity, crystallinity and tensile modulus was quantified and interpreted using known models, such as mixing rules, Avrami-Jeziorny analysis for non-isothermal crystallization and Equivalent Box Model (EBM) model for tensile strength of blends.

Experimental section

Materials

Commercial PHBV (reference PHI 002) containing 3 mol% of hydroxyvalerate (HV)according to technical datasheet -and PBSA (reference PBE 001) were purchased from NaturePlast (France). Polymers were dried at 70 °C under vacuum for at least six hours before use.

Processing of PHBV and PBSA blends and sheets

PHBV and PBSA were blended at molten state within an internal mixer (model Rheoscam, Scamex, France) at 190 °C at 90 rpm for 9 min. PHBV was first introduced in the mixing chamber for 6 min to allow its melting. PBSA pellets were subsequently introduced and mixed for additional 3 min. Blended materials were processed by thermo-compression molding at 180 °C and 150 bar (thermocompression press model 15 T, Scamex, France). Typical torque curves recorded during mixing are presented in the supporting information S1 and thermogravimetric analysis in S2. Pellets were molten for 3 min and then compressed in successive steps at 80 and 150 bar for 1 min each. Aluminum foils of 200 µm were used to control the sample thickness.

Five blend ratios PHBV/PBSA were produced with the weight percent ratios: 100/0, 70/30, 50/50, 30/70, 0/100 wt%.

Characterization methods

Thermal characterization

Thermal properties of blends were assessed by Differential Scanning Calorimetry, DSC, (DSC1, Mettler Toledo, Switzerland) according to two complementary temperature programs shown in Figure 1. All measurements were carried out under nitrogen atmosphere (flowrate 50 mL min -1 ) using 5 to 10 mg of material sealed in 40 µL aluminum pans. Calibration was carried out with Indium and Zinc standards. All measurements were duplicated.

Program I (Figure 1) was used to measure melting and crystallization peaks and the glass transition temperature. Samples were heated from -80 °C to 200 °C at 10 °C min -1 and held for 3 min at 200 °C, then cooled down to -80 °C at -10 °C min -1 and held for 5 min at -80 °C, and subsequently heated to 200 °C at 10° C min -1 . The glass transition temperature (Tg) was measured by quenching the sample from 200 °C to -10 °C for 60 min to physically age PHBV then cooling down to -40 °C for 10 min and heating to 60 °C at 10 °C min -1 . Then physical aging of PBSA was carried out at -60 °C for 60 min, followed by cooling to -80 °C for 10 min, and heating to 60 °C at 10 °C min -1 . The crystallinity of each polymer [ci]i=PHBV,PBSA was determined from the melting endotherm DHm,i as:

𝜒 ! = ∆# !,# $∆# $$,# % # ∆# !,# % , (1)
where is the weight content of the corresponding polymer, DHcc,i the cold crystallization enthalpy and DH 0 m,i the melting enthalpy of a 100 % crystalline polymer with pure PHBV (146 J/g), 17 and PBSA (113.4 J/g ). 18 Program II (Figure 1) was used to analyze the non-isothermal crystallization of PHBV and PBSA. Samples were heated up to 190 °C at 10 °C/min, annealed for 3 min to erase thermal of PBSA was carried out for ratios 70:30 and 50:50, because at higher PBSA content the sample integrity was not conserved. Samples were sputter-coated with a thin gold layer (model Sputter coater Emitech K550, Emitech, UK) and imaged from their cryo-fracture edges.

Rheological properties

Rheological properties were measured at molten state on 25 mm disk-shaped specimens placed in a stress-controlled rheometer (model MCR 302, Anton Paar, Graz, Austria) mounted with a parallel-plate geometry (gap 750-800 µm). Dynamic frequency sweep experiments were performed at 185 °C with frequency varying from 0.01 to 100 Hz in the linear viscoelastic region.

Mechanical properties

Tensile properties were measured using a texture analyzer (model TAHD, Stable Micro Systems, UK) equipped with pneumatic grips at a 5 mm min -1 crosshead speed. Dumbbellshaped samples of type V (200 µm) were cut from the compression-molded sheets. The sample thickness was averaged from five measurements with a caliper. At least five samples were tested for each blend composition.

Dynamic mechanical properties

Viscoelastic behavior of blends was characterized at 1 Hz and a stretch ratio of 0.1 % with a dynamic mechanical analyzer (model Tritec 2000 DMA, Triton Technology Ltd., UK). A temperature scan was performed from -60 °C to 190 °C at a heating rate of 2 °C min -1 on samples cut from thermocompression molded sheets (length = 10.25 mm, width = 5 mm).

Results and discussion

Rheological properties

The The insert in Figure 2a shows that PBSA featured typical terminal flow behavior with a terminal slope close to 2. PHBV showed a strong drop in modulus at 0.3 Hz caused by the thermal degradation of PHBV. For confirmation, an isothermal dynamic time sweep test of PHBV is available in the supplementary information S3. PHBV is known to degrade quickly above its melting temperature. 19 The storage modulus (G') of PHBV/PBSA blends was situated between the PHBV and PBSA values. It increased with rising PBSA content. The complex viscosity spectra (Figure 2b) evidenced that neat PHBV had Newtonian flow behavior in the observed frequency range while PBSA showed shear-thinning behavior. The change from PBSA dominated viscosity to PHBV dominated behavior was observed for blends with a concentration of PHBV equal to or higher than 50 wt%. This was indicative of an inversion of the continuous phase.

The flow of blends is characterized by the ratio of their shear viscosities (h), the capillary number, and the ratio of viscous to surface forces. 20 The ratio of viscosity of one polymer with respect to the continuous phase, l, predicts possible morphologies of two immiscible polymers.

The critical capillary number gives the moment when shear forces overcome surface tension.

Using the rheological data between 30 and 100 Hz, hPBSA/hPHBV (for PHBV as theoretical continuous phase) and hPHBV/hPBSA (for PBSA as theoretical continuous phase) were evaluated to 3.6 -2.8 and 0.2 -0.1, respectively. Large lhPBSA/PHBV values are indicative for the high difficulty of dispersion of PBSA in PHBV as small droplets, whereas the formation of small droplets of PHBV in PBSA was very probable because of low lhPHBV/PBSA. The complex viscosity of PHBV-PBSA mixtures (h * mix) was calculated via a semi-logarithmic mixing rule: 21

𝑙𝑜𝑔𝜂 &!' * = 𝑤 )*+, 𝑙𝑜𝑔𝜂 )*+, * + (1 -𝑤 )*+, )𝑙𝑜𝑔𝜂 )#*- * (2)
Predictions shown in Figure 3 at different frequencies were in good agreement with experiments at PBSA concentration above 50 wt%, where small PHBV droplets were predicted.

Phase separation was the likely cause of the deviation observed for 30 wt% PBSA, where large droplets were predicted. The morphology of blends after melt-mixing and thermo-compression was observed on cryofractured samples by SEM (Figure 4a-e). Neat PBSA showed a smooth surface (Figure 4b), whereas neat PHBV presented a rugous surface (Figure 4a) with fractured crystallites and white items identified as inorganic fillers. 7 Phase separation was detected in PHBV/PBSA as expected from the rheological analysis. Figure 4c shows large PBSA droplets dispersed in PHBV for PHBV/PBSA 70/30 blend. For the PHBV/PBSA 50/50 blend, a co-continuous morphology was observed. Further increase in PBSA content led to a phase inversion, with small droplets of PHBV in PBSA as expected from viscosity ratios. The developed morphologies of PHBV/PBSA were typical of immiscible blends.

Morphology of PHBV/PBSA blends

Thermal properties of PHBV/PBSA blends

Melting behavior

Melting temperatures of neat PHBV and PBSA and of their blends were analyzed by DSC via protocol I (Figure 1). Two distinct melting points (Tm) were identified at 174 °C and 86 °C for (a) For calculation of the PBSA crystallinity degree, cold crystallization enthalpy was subtracted from the melting enthalpy.

Glass transition in the amorphous phase

Conventional heating scans are often not sensitive enough to reveal the glass transition of the amorphous phase of PHBV, because it accounts for a small quantity of the sample. The sensitivity problem is even higher in blends where the PHBV concentration is small. Therefore, a physical ageing treatment of PHBV or PBSA was applied to reveal the Tg using the enthalpy of relaxation peak. DSC thermograms are presented in Figure 6a,b and the quantified data are given in Table 1. Two separated Tg were identified in PHBV/PBSA blends, attributed to each of the polymers. The presence of two distinct Tg is consistent with the observation of a nodular blend structure due to immiscibility. In Table 1 a shift of the Tg of PHBV from 2 to -3 °C can be observed. This might indicate some level of compatibility between the polymers. The Tg of PBSA in the blends shifted also to lower temperatures. We suspect that thermal degradation of PBSA was be the reason of that.

Non isothermal crystallization kinetics of PHBV/PBSA blends

Crystallization kinetics of PHBV/PBSA blends were studied using non isothermal crystallization (program II, Figure 1). As shown in Figure 7, the crystallization peak temperature (Tc) of PHBV in PHBV/PBSA blends shifted towards lower temperature, while an increase in the PBSA crystallization temperature with addition of PHBV was observed. All raw data are shown in the supplementary information S5. The quantification of the non-isothermal crystallization kinetics of the blends was carried out using the Avrami-Jeziorny method and Liu & Mo's model. [23][24][25][26] Both models are presented in the supplementary information S6. The relative degree of crystallinity versus time for PHBV/PBSA blends showed a sigmoidal profile, indicative of the presence of secondary crystallization at higher crystallinity degrees (supplementary information S6). The linear part between 0.2 and 0.6 % relative crystallinity was used for the Avrami-Jeziorny analysis (supplementary information S6 27 and in PHBV/poly(styrene-co-acrylonitrile) blends. 28 In PHBV/poly(dicyclohexylitaconate) blends the PHBV crystallization kinetics seemed almost independent of the blend composition. 29 The acceleration of the PHBV crystallization kinetics was observed in PLA/PHBV blends using isothermal crystallization experiments at 125 °C when PHBV was in minority. 30 A small acceleration was also reported for isothermal crystallization of PHBV/PBS blends when the PHBV content decreased, which might be caused be a nucleating effect. 31 Because of the structural similarity of PBS and PBSA, we think that the small acceleration observed in the present work could be also linked to nucleation.

The acceleration of the PBSA crystallization kinetics in the blends was much more evident. The PBSA blank samples did not crystallize at a cooling rate higher than -30 °C min -1 , but the PBSA phase crystallized in all blends. A tendency to smaller t1/2 (PBSA) was observed when the PHBV quantity of the blend increase. We conclude that PHBV acted as a nucleating agent for PBSA. To go further in the kinetic analysis of the present study, the data were investigated with the model developed by Liu et al. 1997. 23 This model is characterized by two parameters, F(T) and m (ratio of Avrami and Ozawa exponents). F(T) represents the difficulty of the crystallization process, because it denotes the required cooling rate to achieve a given degree of crystallinity.

The analysis of the experimental data is shown in the supplementary information S6. Both parameters, m and F(T) were obtained from the slope and the intercept of the as-defined linear portion of log a versus log t respectively. The results are gathered in Table 3 and show that for both polymer phases, F(T) increased with increase of relative degree of crystallinity. This behavior is generally observed because the crystallization slows down at high crystallinity degrees. The comparison of the F(T) data of the blends shows that at the same level of crystallinity, the crystallization kinetics of the PHBV phase was sensitive to the blend morphology. The PHBV crystallization in blends was facilitated at the lower degrees of crystallinity, which is indicative of nucleation of PHBV by PBSA. At 20 % of crystallinity all F(T) values were decreased. At higher crystallinity degrees, the F(T) values of the PHBV/PBSA 50/50 and PHBV/PBSA 30/70 blends increased with respect to the blank value. The presence of PBSA in the co-continuous structure and, even more, the constraining of PHBV in small nodules hindered PHBV crystallization in its later stages. This was not true, if PBSA was dispersed as large nodules in the PHBV matrix. Here the crystallization was facilitated over the whole range of crystallinity degrees. All F(T) values of PBSA in blends were lower than the blank. Introduction of PHBV in PBSA facilitated the crystallization process notwithstanding the morphology. Nucleation might be the reason of that. 32 authors that the brittle behavior of PHBV can be caused by i) the cold crystallization of the amorphous phase at ambient temperature, ii) glass transition temperature close to room temperature, and iii) radial or circumferential cracks potentially contained in spherulites of PHBV with low content of 3HV. 33,34 The of these cracks is supposed to be caused by the difference in the thermal expansion coefficients of the material along the radius and the circumference of crystallites, which generates drastic internal stresses. 2 PBSA featured ductile behavior with a high elongation at break. Stress-oscillation of pure PBSA was observed. The occurrence of stress-oscillation was reported for PBS as a consequence of alternating regions of highly oriented crystalline zones and micro-cavities due to crazing and voiding. 35,36 The phenomenon generated alternating transparent/opaque bands as shown in Figure 9b. A video of the phenomenon is available in supplementary information S8.

Table 4 reports the tensile properties of PHBV, PBSA and its blends. The apparent Young modulus was in accordance with the E' values at room temperature obtained by DMA. The increase of the PBSA content lowered the stress at break. The obtained elongation at break of the formulated blends was still small, though. Even when the ductile polymer PBSA was the continuous phase, the samples broke at only 3 % elongation. Apparently, although there might be some interfacial compatibility between PBSA and PHBV, an efficient stress transfer between both phases could not be achieved. The evolution of the elastic modulus with PBSA content is shown in Figure 10a. To understand the result, the data were modelled with using parallel (Reuss model) or serial resistances (Voigt model):

𝐸 . = 𝜙 / 𝐸 / . + 𝜙 0 𝐸 0 . , (2)
Where E1 and E2 are the elastic modulus of phase 1 and 2, F1 and F2 the volume fraction of phase 1 and phase

E 5 = E / v /6 + E 0 v 06 + v 7 0 /( 8 &' 9 & + 8 (' 9 (
),

where E1 and E2 are the Young moduli of phases 1 and 2 and vi,p and vi,s are the total volume fractions of the parallel branch and series branch, respectively. The prediction of strength can be modeled as follows: 39

𝑆 : = 𝑆 / 𝑣 /3 + 𝑆 0 𝑣 03 + 𝐴𝑆 0 𝑣 4 , (4)
where S1>S2 and S1 and S2 are the strength of phase 1 and 2, respectively. A corresponds to the level of interfacial bonding.

The comparison of the experimental results with the different models (Figure 10a) shows that the evolution of the Young modulus (E) with increasing PBSA content in the blends was fairly well captured by the EBM model using the universal parameters. This observation further confirmed the immiscibility between both components. A similar observation was made on PHBV/PBS blends. 10 Figure 10b shows the modeling of the stress at yield of PHBV/PBSA blends. EBM is a predictive model, in that it indicates if the interfacial adhesion is strong (A=1)

or weak (A=0). The results showed that a low adhesion factor (A) needs to be assumed to describe the behavior of the blends at high PBSA content. The strain at break was at any composition lower than the strain at break of the stronger polymer, PHBV, which shows that the materials fractured by debonding at the interface.

3.5.Conclusion

This study presents a detailed investigation of thermo-mechanical and rheological properties of in the blends, most probably due to mutual nucleation. However, the crystallization rates observed at high PHBV crystallinity degree were slowed down when PHBV was dispersed in small nodules in PBSA or blends had co-continuous structure. The mechanical properties were successfully modeled with the EBM model, including parallel and serial resistances and showed that the rigidity of the material could be modulated using PBSA. The elongation at break was however governed by PHBV. Brittle fracture happened even in case PBSA was the continuous phase, in accordance with a model of serial resistances. The analysis of the adhesion factor showed that there was most likely debonding at the interfaces. In conclusion, PHBV/PBSA blends have improved melt viscosity and accelerated crystallization kinetics, but for successful improving mechanical properties supplementary compatibilizers will be necessary.

Figure 1 .

 1 Figure 1. Schematic representation of the different DSC protocols

  viscoelastic properties of neat PHBV, PBSA, and PHBV/PBSA blends were analyzed with dynamic frequency sweep experiments from high to low frequencies, as shown in Figure 2(a,b).

Figure 2 .

 2 Figure 2. Mechanical spectrum of (a) storage modulus and (b) complex viscosity for neat PHBV, PBSA (insert in (a)), and PHBV/PBSA blends at 185°C after batch mixing. Data were recorded starting at high frequencies.

Figure 3 .

 3 Figure 3. Complex viscosity of PHBV/PBSA blends as a function of PBSA content. Solid line represents the values calculated with the help of a semi-logarithmic mixing law.

Figure 4 .

 4 Figure 4. SEM images of cryo-fractured PHBV, PBSA and PHBV/PBSA blends: (a) neat PHBV, (b) neat PBSA, (c) PHBV/PBSA 70/30 after etching with THF at room temperature, (d) PHBV/PBSA 50/50 after etching with THF at room temperature, (e) PHBV/PBSA 30/70 (no etching).

1 .

 1 neat PHBV and PBSA, respectively. No significant change within the precision of the experiment in melting temperature was observed in binary blends. The tendency towards smaller Tm of PHBV might however suggest some compatibility. Details are reported in Table Cold crystallization of PBSA was observed at Tcc ≈ -5 °C. Cold crystallization due to incomplete crystallization of the copolymer PBS59A41 has been already described by Debuissy et al.22

Figure 5 .

 5 Figure 5. DSC curves of PHBV, PBSA and PHBV/PBSA blends recorded during the second heating scan following protocol I (exo up).

Figure 6 .

 6 Figure 6. DSC curves of PHBV, PBSA and PHBV/PBSA blends. Figure 6a shows the Tg of PHBV ager aging at -10 °C and Figure 6b the Tg of PBSA after aging at -40 °C.

Figure 7 .

 7 Figure 7. DSC thermogram of non-isothermal crystallization of PHBV, PBSA and PHBV/PBSA blends recorded during cooling at -10 °C min -1 (protocol II).

3. 4 .Figure 8 .

 48 Figure 8. Evolution of (a) the elastic modulus and (b) tan δ as function of temperature for PHBV, PBSA and PHBV/PBSA blends.

Figure 8a shows

  Figure8ashows the evolution of the elastic modulus (E') as function of temperature ranging from -60 °C to 180 °C. PHBV featured as expected higher storage modulus values over the whole temperature range than PBSA. The addition of PHBV to PBSA increased the height of the rubbery plateau modulus. It had only a moderate impact on the increase of the flow temperature of PBSA. Even in the co-continuous blend, the material flowed below 100 °C. In case of PHBV being the continuous phase, the decrease of the rubbery moduli of the nodular blends with respect to PHBV was observed without change in flow behavior. Table4highlights E' values at relevant service temperatures, i.e. at room temperature(20 °C) and in freezer conditions (-20 °C). Figure8bshows the evolution of tan δ as a function of temperature ranging from -60 °C to 60 °C. The peak of tan δ can be attributed to the dynamic glass transition. The

  dynamic glass transition temperature Ta was determined at the maximum value of the peak (Ta values are available in in supplementary information S7). Two distinct Ta were identified which is consistent with DSC data and shows the immiscibility of PHBV and PBSA.

  (a) experiments carried out without extensometer.

Figure 9 .

 9 Figure 9. (a) Typical stress-strain curves of tensile test of PHBV, PBSA and PHBV/PBSA blends, (b) stress-oscillation of PBSA samples during tensile testing.

Figure10.

  Figure10. (a) Modeling of the elastic modulus from tensile testing of PHBV/PBSA blends depending on PBSA content using the Voigt, Reuss, and EBM model, (b) evolution of the tensile stress of PHBV/PBSA blends with representation of EBM model at different levels of interfacial adhesion (A).

  PHBV/PBSA blends prepared by batch mixing. PHBV and PBSA were immiscible. As predicted by the viscosity ratio between the polymers and confirmed by SEM observation, PBSA formed small nodules in PHBV, while PHBV formed large inclusions in PBSA. A cocontinuous blend morphology was observed in 50/50 PHBV/PBSA blends. The investigation of the crystallization kinetics showed that the crystallization rate of PHBV and PBSA increased (b) a)

Table 1 .

 1 Thermal properties of PHBV/PBSA blends measured by DSC during the cooling scan and the second heating scan.

	PHBV/PBSA		Tm (°C)		Tc (°C)		c (%)	Tg (°C)
	Samples	PHBV PBSA PHBV PBSA PHBV PBSA PHBV	PBSA
	100/0	171±3 n.d	122± 1 n.d	68±1	n.d	2.4±0.4 n.d
	70/30	169±2 86±2 117±1 50±1 67±2	33±5	-1.5±0.1 -48.6±2.3
	50/50	169±2 87±1 116±2 49±5 61±1	39±6	-3.1±0.8 -47.0 ±1.3
	30/70	169±1 87±1 114±5 47±5 67±6	43±2	-3.3±0.3 -46.4±2
	0/100	n.d	88±3 n.d	38±3 n.d	45±4 (a) n.d	-45.9±1.6

Table 2 .

 2 ). The Avrami constants na, ka, and kc are gathered in a synoptic table available in supplementary information S5. The normalized crystallization enthalpy Hc, crystallization peak temperature Tc and half time of crystallization t1/2 calculated with the help of the Avrami-Jeziorny constants are presented in To help the observation, the data of Table 2 are repeated in graphical form in the supplementary information S6. No clear distinction could be made between the t1/2 (PHBV) and t1/2 (PHBV/PBSA 70/30). The t1/2 of the blends PHBV/PBSA 50/50 and 30/70 were smaller than the blank, except in case of the cooling rate 20 °C min -1 . At this cooling rate,

all t1/2 were very close and the PHBV blank value was smaller than the others. It seems that the PHBV crystallization is facilitated in the presence of interfaces with PBSA and in the smaller nodules. The change of the crystallization kinetics of PHBV in blends is much dependent on the nature of the second polymer. A slow-down of the isothermal PHBV crystallization kinetics was observed in PHBV/PCL blends,

Table 2 .

 2 Crystallization enthalpy normalized by the polymer content (Hc), crystallization

	temperature (Tc) and half time of crystallization (t1/2) calculated from the Avrami-Jeziorny
	parameters during non-isothermal crystallization of PHBV, PBSA and PHBV/PBSA blends.

Table 3 .

 3 Liu parameters during non-isothermal crystallization of PHBV, PBSA and PHBV/PBSA blends.

				PHBV			PBSA
	Samples	c(t) (%)	m	F(T)	R 2	m	F(T)	R 2
		20	0.92	10.11	0.97
	PHBV	40 60	0.97 1.04	12.87 15.88	0.97 0.98
		80	1.01	22.61	0.97
		20				1.02	14.39	0.92
	PBSA	40 60				1.1 1.18	27.51 43.78	0.92 0.91
		80				1.21	63.79	0.91
		20	1.26	6.2	0.99	1.34	4.72	0.99
	PHBV/PBSA	40	1.31	8.56	1	1.45	9.67	0.99
	70/30	60	1.44	11.12	0.99	1.51	14.96	0.99
		80	1.51	20.21	1	1.48	23.65	0.99

Table 4 .

 4 Determination of storage modulus E' and mechanical properties of PHBV/PBSA blends obtained from DMA and tensile test measurements.

		E' (MPa) from DMA		Tensile properties
	PHBV/PBSA
	samples	-20 °C	20 °C	Young modulus (a) (MPa)	Stress at yield (MPa)	Elongation at break (%)
	100/0	3625	2153	3485±63	22±2.0	0.98±0.1
	70/30	2247	1297	2227±108	19±2.5	1.48±0.4
	50/50	1438	829	1360±125	12±1.9	1.48±0.2
	30/70	966	543	833±147	12±1.6	3.10±0.8
	0/100	553	303	332±8	15±1.4	134.8±48

  2, and exponent n characterizes the type of model with n=1 (parallel or Reuss model) or n=-1 (Sseries or Voigt model). To account for a mixed behavior including parallel and serial resistances, the Equivalent Box Model (EBM)37 was used. It is detailed in supplementary information S9. Based on the percolation theory, critical volume fractions were set to 𝑣 /12 = 𝑣 012 = 0.16.38 The resulting tensile modulus of two components blend (Eb) is given as the sum 𝐸 3 𝑣 3 + 𝐸 4 𝑣 4 : 37

Acknowledgements

The authors acknowledge financial support from the company McCain Alimentaire S.A.S and from ANRT (French National Association for Research and Technology via the CIFRE agreement n°2017/1574 attributed to Benjamin Le Delliou. The authors greatly thank Pierre Gondé for his technical support. Pack. Technol. Sci.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.