B C Toth
email: btoth@uni-wuppertal.de

Sz Borsanyi

, , , , J. N Z Fodor

C Guenther

S D Hoelbling

L Katz

T Lellouch

K Lippert

Miura

L Parato

K K Szabo

F Stokes

Cs Torok

L Varnhorst

Muon -2: BMW calculation of the hadronic vacuum polarization contribution

We compute the leading order hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. The calculations are performed using four flavors of stout smeared staggered quarks, with quark masses at their physical values. The continuum limit is taken using six different lattice spacings ranging from 0.132 fm down to 0.064 fm. All strong isospin breaking and electromagnetic effects are accounted for to leading order. A controlled infinite volume limit is taken thanks to dedicated simulations performed in box sizes up to 11 fm. Putting all these ingredients together, we find [( -2)/2] LO-HVP = 707.5[5.5] × 10 -10 , which has a total uncertainty of 0.8%. Compared to determinations based on the + -→ hadrons cross section, our result significantly reduces the tension between the standard model prediction for the muon -2 and its experimental value.

Introduction

The muon is an ephemeral sibling of the electron. It is 207 times more massive, but has the same electric charge and spin. Similarly to the electron, it behaves like a tiny magnet, characterized by a magnetic moment. This quantity is proportional to the spin and charge of the muon, and inversely proportional to twice its mass. Dirac's relativistic quantum mechanics predicts that the constant of proportionality, , should be 2. However, in a relativistic quantum field theory such as the standard model, this prediction receives small corrections due to quantum, vacuum fluctuations. These corrections are called the anomalous magnetic moment and are quantified by ( -2)/2. They were measured to an exquisite 0.54 ppm at the Brookhaven National Laboratory in the early 2000s [START_REF] Bennett | Final Report of the Muon E821 Anomalous Magnetic Moment Measurement at BNL[END_REF], confirmed recently by Fermilab [START_REF] Abi | Measurement of the Positive Muon Anomalous Magnetic Moment to 0.46 ppm[END_REF], and have been calculated with a comparable precision (see [START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF] for a recent review).

At this level of precision, all of the interactions of the standard model contribute. The leading contributions are electromagnetic and described by quantum electrodynamics (QED), but the one that dominates the theory error is induced by the strong interaction and requires solving the highly non-linear equations of quantum chromodynamics (QCD) at low energies. This contribution is determined by the leading-order, hadronic vacuum polarization (LO-HVP), which describes how the propagation of a virtual photon is modified by the presence of quark and gluon fluctuations in the vacuum. Here we compute this LO-HVP contribution to ( -2)/2, denoted by LO-HVP , using ab initio simulations in QCD and QED. In the present work, we include both QED and QCD, as well as four non-degenerate quark flavors (up, down, strange and charm), in a lattice formulation taking into account all dynamical effects. We also consider the tiny contributions of the bottom and top quarks.

We compute LO-HVP in the so-called time-momentum representation [START_REF] Bernecker | Vector Correlators in Lattice QCD: Methods and applications[END_REF], which relies on the following, zero three-momentum, two-point function in Euclidean time :

( ) = 1 3 2 =1,2,3 ∫ 3 ( ì , ) (0) , (1)
where is the quark electromagnetic current with / = 2 3 ¯ -

1 3 ¯ -1 3 ¯ + 2 3 ¯
. , , and are the up, down, strange and charm quark fields and the angle brackets stand for the QCD+QED expectation value to order 2 . It is convenient to decompose ( ) into light, strange, charm and disconnected components, which have very different statistical and systematic uncertainties. Integrating the one-photon-irreducible (1 I) part of the two-point function (1) yields the LO-HVP contribution to the magnetic moment of the muon [START_REF] Bernecker | Vector Correlators in Lattice QCD: Methods and applications[END_REF][START_REF] Lautrup | Recent developments in the comparison between theory and experiments in quantum electrodynamics[END_REF][START_REF] Rafael | Hadronic contributions to the muon g-2 and low-energy QCD[END_REF][START_REF] Blum | Lattice calculation of the lowest order hadronic contribution to the muon anomalous magnetic moment[END_REF]:

LO-HVP = 2 ∫ ∞ 0 ( ) 1 I ( ) , (2)
with the weight function,

( ) = ∫ ∞ 0 2 2 2 2 2 - 4 2 sin 2 2 , (3)
and where ( ) = [ + 2 -( + 4)] 2 / ( + 4), is the fine structure constant in the Thomson limit and is the muon mass. Since we consider only the LO-HVP contribution, for brevity we B. C. Toth drop the superscript and multiply the result by 10 10 , i.e. stands for LO-HVP × 10 10 throughout this work.

The subpercent precision, that we are aiming for, represents a huge challenge for lattice QCD. To reach that goal, we have to address the following critical issues: scale determination; noise reduction; QED and strong-isospin breaking; infinite-volume and continuum extrapolations. We briefly discuss these one by one. For a more detailed exposition we refer the reader to the Supplementary Information in [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF].

Scale determination

The quantity depends on the muon mass. When computing (2) on the lattice, has to be converted into lattice units,

, where is the lattice spacing. A relative error of the lattice spacing propagates into about a twice as large a relative error on , so that has to be determined with a few permil precision. We use the mass of the Ω -baryon, Ω -= 1672.45 [START_REF] Lehner | RBRC Workshop on Lattice Gauge Theories[END_REF] MeV [START_REF] Tanabashi | Review of Particle Physics[END_REF], to set the lattice spacing. Here we describe how we determine its QCD contribution; QED corrections are described below.

To extract the mass of the positive-parity, ground-state Ω baryon, a number of different operators are available in the staggered formalism. First, there are two operators from the pioneering work of Golterman and Smit [START_REF] Golterman | Lattice Baryons With Staggered Fermions[END_REF]. To label these operators we use the convention of [START_REF] Ishizuka | Operator dependence of hadron masses for Kogut-Susskind quarks on the lattice[END_REF]:

Ω VI ( ) = even [ 1 12 13
-2 21 23

+ 3 31 32 ] ( ), (4)
Ω XI ( ) = even [ 1 2 3 ] ( ). (5)
Here, ( ) is the strange-quark field with color index . The operator performs a symmetric, gauge-covariant shift in direction , while ≡ . Both Ω VI and Ω XI couple to two different tastes of the Ω baryon, which become degenerate in the continuum limit. At finite lattice spacing however, there is a splitting between the two tastes. In principle they could be disentangled by carrying out an analysis involving the correlators of both Ω VI and Ω XI and also their cross terms. Later, Bailey successfully constructed an operator which only couples to a single taste [START_REF] Bailey | Staggered baryon operators with flavor SU(3) quantum numbers[END_REF]. To achieve this, two additional (valence) strange quarks are introduced: the strange-quark field gets an additional "flavor" index: with = 1, 2, 3. The operator is then given as

Ω Ba ( ) = 2 1 2 3 -3 1 2 -2 3 1 + (. . . ↔ . . . ) • • even 1 12 13
-2 21 23

+ 3 31 32
( ). ( 6
)
The mass of this state becomes degenerate with the above two taste partners in the continuum limit. We investigated the difference between these three operators on an ensemble with large statistics. At = 3.7000, corresponding to our coarsest lattice spacing, we computed the corresponding Ω propagators on about 3000 configurations in addition to those used for the current propagator measurements. The effective masses for the above three operators are shown in Figure 1. In the asymptotic regime we see deviations below 0.1%, which gives an estimate of the taste violation. We expect that these will get smaller as we go to finer lattice spacings. In this work we chose the Ω VI operator for our scale setting measurements. This is justified, since typical statistical and systematic errors on our ensembles are around 0.1%, and thus cover the taste-violation effects estimated here.

On the Ω propagators we perform a four-state fit using the fit function ℎ, with two positive and two negative parity states:

ℎ( , , ) = 0 ℎ + ( 0 , ) + 1 ℎ -( 1 , ) + 2 ℎ + ( 2 , ) + 3 ℎ -( 3 , ) (7)
with ℎ + ( , ) = -+ (-1) -1 -( -) and ℎ -( , ) = -ℎ + ( , -) describing the time dependence of the positive and negative parity states. Here 0 and 0 are the mass and amplitude of the ground state. In order to stabilize the fit, a prior term was introduced, containing priors on the masses except for the ground state. The prior for the negative parity ground state, 1 = 2012 MeV, is motivated by the recent observation from the Belle collaboration [START_REF] Yelton | Observation of an Excited Ω -Baryon[END_REF]. The excited states, 2 = 2250 MeV and 3 = 2400 MeV, have not been discovered in experiments so far, so their priors follow from the quark model [START_REF] Capstick | Baryons in a Relativized Quark Model with Chromodynamics[END_REF]. The existence of these undiscovered states is also motivated by lattice thermodynamics below the chiral transition [START_REF] Bazavov | Additional Strange Hadrons from QCD Thermodynamics and Strangeness Freezeout in Heavy Ion Collisions[END_REF][START_REF] Alba | Constraining the hadronic spectrum through QCD thermodynamics on the lattice[END_REF].

In addition to the above four-state fit to the Ω propagator we also used a mass extraction procedure proposed in [START_REF] Aubin | A new approach for Delta form factors[END_REF], which is based on the Generalized Eigenvalue Problem (GEVP). The method has the advantage of not using priors.

For each time slice we construct a 4 × 4 matrix H , ( ) = +( -1)+( -1) ( , = 1, . . . , 4), from the hadron propagator . Then for a given and let ( , ) be an eigenvalue and ( , ) an eigenvector solution to this 4 × 4 generalized eigenvalue problem:

H ( ) ( , ) = ( , )H ( ) ( , ). (8)
Here we select the smallest eigenvalue and use the corresponding eigenvector to project out the ground state:

+ ( , )H ( ) ( , ), (9)
B. C. Toth which then can be fitted to a simple exp(-) type function. The mass extracted using the GEVP gives a third Ω value for each ensemble, beside the results with the four-state fit procedure with two fit ranges. We will use the deviation between these three values as a systematic error in the Ω mass determination. More details on the scale determination can be found in Ref. [START_REF] Varnhorst | High precision scale setting on the lattice[END_REF] and in the Supplementary Information of [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF].

Isospin-breaking effects

Our staggered path integral includes four flavors of quarks, = { , , , }, gluon fields and photon fields and is given by:

= ∫ [ ] exp(-[ ]) ∫ [ ] exp(-[ ]) det 1/4 [ exp( ), ]. (10)
The photon integral measure [ ] and action are defined in the QED L scheme [START_REF] Hayakawa | QED in finite volume and finite size scaling effect on electromagnetic properties of hadrons[END_REF]. The one-hop staggered matrix in a background field can be written as

[ , ] = [ ] + = [ ] + , (11)
where is the covariant differentiation in the direction involving and its adjoint † together with the obligatory staggered phases. In the path integral the fermions are coupled to a gauge field that is a product of the exponentiated photon field and of the smeared gluon gauge field . ∈ {+ 2 3 , -1 3 , -1 3 , + 2 3 } stand for the quark electric charges in units of the positron charge , for the quark masses and = 2 /(4 ). We use the notation ≡ for the difference in the up and down quark masses and ≡ 1 2 ( + ) for their average. To simplify later formulas we also introduce the notations

≡ [ , ] and dets[ , ; { }, { }, ] ≡ det 1/4 , (12)
where the latter is the product of all fermion determinants. In this work isospin-breaking is implemented by taking derivatives with respect to the isospinbreaking parameters and by measuring the so obtained derivative operators on isospin-symmetric configurations [START_REF] Rm | Leading isospin breaking effects on the lattice[END_REF]. We introduce a set of notations for isospin-symmetric observables and their isospin-breaking derivatives. Consider the observable ( , ), which is a function of and . Then we define 0 ≡ (0, 0), ′ ≡ (0, 0),

′ 1 ≡ (0, 0), ′′ 2 ≡ 1 2 2 2 (0, 0). ( 13
)
We take into account only leading-order isospin-breaking in this work, so no higher derivatives are needed.

In the case of the fermion determinant, the isospin-symmetric value is denoted by dets 0 . Since dets is symmetric under the exchange ↔ , the strong-isospin-breaking of dets is zero at leading B. C. Toth order: dets ′ = 0. The electromagnetic derivatives are

dets ′ 1 dets 0 = 4 Tr -1 [ ] , (14)
dets

′′ 2 dets 0 = 1 2       dets ′ 1 dets 0 2 - 2 4 Tr -1 [ 2 ] - 2 4 Tr -1 [ ] -1 [ ]       ,
where Tr is trace over color and spacetime indices and the argument of the operator is a 3 × 3 complex matrix valued field, e.g. 2 has components 2

, [ ] , . The implementation of these derivatives is detailed in Refs. [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF][START_REF] Parato | QED and strong isospin corrections in the hardonic vacuum polarization contribution to the anomalous magnetic moment of the muon[END_REF].

We also make a distinction between the electric charge in the fermion determinant and in the operator that we measure. We call the former sea electric charge and denote it by , the latter is the valence electric charge and is denoted by . For an observable that depends on both the valence and sea charges, ( , ), the second order electric charge derivatives are defined as follows:

′′ 20 ≡ 1 2 2 2 (0, 0), ′′ 11 ≡ 2 (0, 0), ′′ 02 ≡ 1 2 2 2 (0, 0). ( 15
)
For functions that depend on either or , but not on both, we use the single digit notations of Equation [START_REF] Yelton | Observation of an Excited Ω -Baryon[END_REF].

The expectation value of an operator is calculated by inserting [ , ] into the integrand of the path integral of Equation ( 10) and normalizing the integral by . Here we consider operators whose photon field dependence arises entirely from the photon-quark interaction, i.e.

= [ , ]. The expectation value of this operator depends on , and , and the expansion in isospin breaking corrections can be written as:

= [ ] 0 + 2 ′′ 20 + ′′ 11 + 2 ′′ 02 + ′ . (16)
Here, the individual terms can be expressed as expectation values obtained with the isospinsymmetric path integral, which we denote by . . . 0 . The concrete expressions are: isospin-symmetric: [ ] 0 = 0 0 qed valence-valence:

′′ 20 = ′′ 2 0
qed sea-valence:

′′ 11 = ′ 1 dets ′ 1 dets 0 0 qed sea-sea: ′′ 02 = 0 dets ′′ 2 dets 0 0 -0 0 dets ′′ 2 dets 0 0 strong-isospin-breaking: ′ = ′ 0 ( 17
)
In the derivation of these expressions we use dets ′ 1 dets 0 0 = 0. Note that Equation ( 16) is an expansion in bare parameters and not what we consider a decomposition into isospin-symmetric and isospin breaking parts. The latter involves derivatives with respect to renormalized observables and our prescription for that is given in Section 3.1. There is no need to introduce a renormalized electromagnetic coupling though: its running is an ( 4 ) effect, i.e. beyond the leading order isospin approximation that we consider here.

B. C. Toth

Isospin-breaking decomposition

For various purposes it is useful to decompose the observables into isospin-symmetric and isospin-breaking parts. This requires a matching of the isospin symmetric and full theories, in which we specify a set of observables that must be equal in both theories. Of course, different sets will lead to different decompositions, which is commonly referred to as scheme dependence. Only the sum of the components, i.e. the result in the full theory, is scheme independent.

A possible choice for the observables are the Wilson-flow-based 0 scale and the masses of mesons built from an up/down/strange and an anti-up/down/strange quark, / / . These mesons are defined by taking into account only the quark-connected contributions in their two-point functions [START_REF] Borsanyi | Isospin splittings in the light baryon octet from lattice QCD and QED[END_REF]. Their masses are practical substitutes for the quark masses. Also, they are neutral and have no magnetic moment, so they are a reasonable choice for an isospin decomposition. These masses cannot be measured in experiments, but have a well defined continuum limit and thus a physical value can be associated to them. In particular, we use the combinations 2 ≡ 1 2 ( 2 + 2 ) and Δ 2 ≡ 2 -2 . For the determination of the physical values of 0 , and Δ 2 , see the Supplementary Information of [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF].

For the decomposition we start with the QCD+QED theory and parameterize our observable with the quantities defined above:

( 0 , 0 , 0 , Δ 0 , ) . (18)
Here, the continuum limit is assumed. We can isolate the electromagnetic part by switching off the electromagnetic coupling, while keeping the other parameters fixed. The strong-isospin-breaking part is given by the response to the Δ parameter, and the isospin-symmetric part is just the remainder:

qed ≡ 2 • 2 0 , 0 , 0 ,Δ 0 , =0 (19)
sib ≡ (Δ 0 ) 2 • (Δ 0 ) 2 0 , 0 , 0 ,Δ 0 =0, =0 (20)
iso ≡ ( 0 , 0 , 0 , 0, 0).

One can also define the decomposition at a finite lattice spacing, for which 0 in lattice units can be additionally fixed. In doing so the isospin symmetric part iso has to be distinguished from the value of the observable at the bare isospin-symmetric point [ ] 0 .

Noise reduction techniques

In this section we consider quantities at the isospin-symmetric point; noise reduction techniques for the isospin-breaking part are discussed in Refs. [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF][START_REF] Parato | QED and strong isospin corrections in the hardonic vacuum polarization contribution to the anomalous magnetic moment of the muon[END_REF]. For the strange and charm connected contributions, strange 0 and charm 0 , and for the disconnected contribution disc 0 we use the same measurements that are presented in our previous work [START_REF] Borsanyi | Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles[END_REF]. A new measurement procedure is implemented for the light connected component light 0 . It is used to reanalyze the old configurations and make measurements on new ensembles. This plays a key role in reducing the final statistical error in .

Low Mode Averaging

The technique utilizes the lowest eigenmodes of the fermion matrix; for an early work with low eigenmodes, see [START_REF] Neff | On the low fermionic eigenmode dominance in QCD on the lattice[END_REF]. The way in which we use these modes here is essentially the same as in [START_REF] Li | Overlap Valence on 2+1 Flavor Domain Wall Fermion Configurations with Deflation and Low-mode Substitution[END_REF], where it is called Low Mode Substitution. In the space orthogonal to these modes, the computational effort is reduced considerably by applying imprecise (aka. sloppy) matrix inversions. This is called the Truncated Solver Method [START_REF] Bali | Effective noise reduction techniques for disconnected loops in Lattice QCD[END_REF] or All Mode Averaging [START_REF] Blum | New class of variance-reduction techniques using lattice symmetries[END_REF]. Here we describe the technique for the connected part of the current propagator. The same technique was applied recently for magnetic moment computations in [START_REF] Aubin | Light quark vacuum polarization at the physical point and contribution to the muon -2[END_REF] also.

We consider the connected current propagator for timelike separation, and perform an averaging over the source positions, together with a zero spatial-momentum projection at the sink:

( , ¯ ) ≡ 1 3 3 ì ¯ , ì , =1,2,3 conn , , , ¯ ( , 0) = - 1 12 3 =1,2,3 ReTr D , -1 D , ¯ -1 , (22)
where

D , = ì [
] is an operator that performs a symmetric, gauge-covariant shift on a vector :

[D , ] = , , + + † , - - • 4 , , (23)
where , are the staggered phases. We use the simplifying notation D = D , and D = D , ¯ in following. In Equation ( 22), we apply the real part to reduce noise, because the imaginary part vanishes anyway after averaging over gauge configurations.

Using the lowest eigenmodes of we split the quark propagator into an eigenvector part and into its orthogonal complement, denoted by "e" and "r", respectively:

-1 = -1 + -1 , -1 = 1 † and -1 = -1 1 - † , ( 24
)
where / is the -th eigenvector/eigenvalue of the operator . Correspondingly, splits into eigen-eigen, rest-eigen and rest-rest contributions:

= + + = - 1 4 3 = , = , ReTr D -1 D -1 , (25)
where an average over is assumed but not shown explicitely. The benefit of this decomposition is that the trace in the eigen-eigen part can be calculated exactly, and is thus equivalent to calculating the propagator with all possible sources in position space. This is the main ingredient for the noise reduction. Though no extra inversions are needed in this part, it has to be optimized carefully, since there is a double sum over the eigenmodes, where each term is a scalar product † D . In the resteigen part we have terms † D -1 D and also terms where D and D are exchanged. Therefore, this part is only a single sum over the eigenmodes, and each term involves one matrix inversion. Note that these inversions are preconditioned by the eigenvectors, so they need many fewer iterations than standard inversions. Additionally, we speed up the inversions by running them with a reduced precision, and for some randomly selected eigenvectors we correct for the small bias by adding the difference between a high precision solver and the reduced precision one [START_REF] Bali | Effective noise reduction techniques for disconnected loops in Lattice QCD[END_REF][START_REF] Blum | New class of variance-reduction techniques using lattice symmetries[END_REF]. Finally, the rest-rest part is evaluated using random source vectors : we calculate † D -1 D -1 , which requires two inversions per random source. The reduced precision inverter technique is used here too. The improvement achieved by using these noise reduction techniques is shown in Figure 2.

Upper and lower bounds on

In the case of the light and disconnected contributions to the current propagator, the signal deteriorates quickly as distance is increased. To calculate the HVP, a sum over time of the propagator has to be performed. As was suggested in [START_REF] Lehner | RBRC Workshop on Lattice Gauge Theories[END_REF][START_REF] Borsanyi | Slope and curvature of the hadronic vacuum polarization at vanishing virtuality from lattice QCD[END_REF], we introduce a cut in time , beyond which the propagator is replaced by upper and lower bounds, thereby reducing the statistical noise. Our estimate is given by the average of the bounds at a where the two bounds meet. The bounds are derived from the fact that the current propagator is a sum of exponentials with positive coefficients.

For the light connected propagator at the isospin-symmetric point the bounds express the positivity (lower bound) and that the propagator should decay faster than the exponential of two pions (upper bound). They are given as

0 ≤ light ( ) ≤ light ( ) ( ) ( ) , (26)
where ( ) = exp(-2 ). For 2 we use the energy of two non-interacting pions with the smallest non-zero lattice momentum 2 / . The larger the the better the upper bound, but it comes with more statistical noise. The exponential decay above assumes an infinite time extent, = ∞. We incorporate the effects of a finite-using next-to-leading-order chiral perturbation theory. There the exponential decay with the two-pion energy gets replaced by the following cosh-type form:

exp(-2 ) -→ cosh[ 2 ( -/2)] + 1 cosh( 2 /2) -1 . ( 27
)
In the case of the isospin-symmetric disconnected propagator the bounds are

0 ≤ -disc ( ) ≤ 1 10 light ( ) ( ) ( ) + strange ( ) + charm ( ). ( 28
)
Since the strange and charm terms fall off much faster than the light and disconnected one, their contribution does not change the value of obtained.

Finite-size effects

We compute finite-size effects on in a systematic way, which includes dedicated lattice simulations, chiral perturbation theory and phenomenological models. The concrete goal is to provide a single number that is to be added to the continuum-extrapolated lattice result obtained in a reference box, which is defined by a spatial extent of ref = 6.272 fm and a temporal extent of ref = 3 2 ref .

For a more extensive description see Ref. [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]. We perform dedicated lattice simulations with two different lattice geometries: one is a 56 × 84 lattice with the reference box size and the other is a large 96 × 96 lattice with box size = big = 10.752 fm and = big = big . Since taste violations distort the finite-size effects, we designed a new action with highly-suppressed taste breaking, which we call 4HEX (see [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]). Our strategy is then to compute the finite-size correction as the following sum:

(∞, ∞) -( ref , ref ) = = [ ( big , big ) -( ref , ref )] 4HEX + [ (∞, ∞) -( big , big )] XPT . ( 29
)
The first difference on the right hand side is taken from the dedicated 4HEX simulations. The second difference is expected to be much smaller than the first and is taken from a non-lattice approach: two-loop chiral perturbation theory. We consider four non-lattice approaches to compute both differences on the right hand side of Equation ( 29). In the case of the first difference, the results obtained are compared to our 4HEX simulations. The first approach is chiral perturbation theory (XPT) to next-to-leading and nextto-next-to-leading orders (NLO and NNLO), the second is the Meyer-Lellouch-Luscher-Gounaris-Sakurai model (MLLGS), the third approach is that of Hansen and Patella (HP) [START_REF] Hansen | Finite-volume effects in ( -2) HVP,LO[END_REF] and the fourth is the rho-pion-gamma model of [START_REF] Chakraborty | The hadronic vacuum polarization contribution to from full lattice QCD[END_REF], which we abbreviate as RHO here.

We compute the first difference in Equation ( 29) using dedicated simulations with the 4HEX action. We use the harmonic-mean-square (HMS) to set the physical point:

-2 ,HMS ≡ 1 16 -2
, , defined as an average over the masses of the 16 pion tastes, , . We set ,HMS to the physical value of the pion mass, which requires lowering the Goldstone-pion mass to 110 MeV. This way of fixing the physical point results in much smaller lattice artefacts than the usual setting with the Goldstone-pion, at least for an observable like the finite-size effect. To generate the 4HEX data set, we performed simulations with two different Goldstone pion masses:

= 104 MeV and 121 MeV. To set the physical point as described above, we perform an interpolation from these two pion masses to = 110 MeV. To compute light from the current propagator in our 4HEX simulations we use the upper and lower bound technique described in Section 4.2. Results for the = 121 MeV simulation point are plotted in Figure 3. The bounds meet at around 4.2 fm and 4.7 fm on the small and large volumes, respectively. At these distances we take the average of the two bounds as an estimate for light .

We only have one lattice spacing with the 4HEX action, so the finite-size effects cannot be extrapolated to the continuum limit. We estimate the cutoff effect of the result by comparing with the 4HEX action at this single lattice spacing to the continuum extrapolated 4stout lattice result, both in the ref volume. The 4HEX result is about 7% larger than the continuum value. Therefore we reduce the measured finite-size effect by 7%, and assign a 7% uncertainty to this correction step. For the difference we get

( big , big ) -( ref , ref ) = 18.1(2.0) stat (1.4) cont . (30)
The result includes a 9 10 charge factor, and the first error is statistical and the second is an estimate of the cutoff effect.

The finite-size effects computed in various non-lattice approaches are collected in Table 1. Except for the NLO result, the different models give finite-size effects of similar size, which agree well with the lattice determination of Equation [START_REF] Borsanyi | Slope and curvature of the hadronic vacuum polarization at vanishing virtuality from lattice QCD[END_REF]. We also see that, according to the models, the finite-effect is much smaller than the finite-effect.

The good agreement for the finite-size effect in the reference box, between the models and the lattice, gives us confidence that the models can be used to reliably compute the very small, residual, finite-size effect of the large box. The corresponding model estimates can be found in Table 1. For an infinite-time extent the NNLO XPT, the HP and RHO approaches agree nicely. As a final value for the large-box, finite-size effect we take the NNLO XPT result including finite-effects:

(∞, ∞) -( big , big ) = 0.6(0.3) big ,
where the uncertainty is an estimate of higher-order effects, given here by the difference of the NNLO and NLO values.

For our final result for the finite-size effect in the reference box, we also include the contribution of isoscalar channel and isospin-breaking effects giving: (1.4) The first error is the statistical uncertainty of our 4HEX computation, the second is an estimate of the 4HEX cutoff effects, the third is the uncertainty of the residual finite-size effect of the "big" lattice, the fourth is a XPT estimate of the = 0 finite size effect and the fifth is an estimate of the isospin-breaking effects. The last, total error in the square-brackets is the sum of the first five, added in quadrature.

(∞, ∞) -( ref , ref ) = 18.7(2.0) stat

Taste improvement

As is well known, some of the most important cutoff effects of staggered fermions are taste violations. At long distances, these violations distort the pion spectrum. Since is predominantly a long-distance observable, dominated by a two-pion contribution, including the resonance, we expect these effects to be largest in the light-quark terms.

We investigate various physically motivated models for reducing long-distance taste violations in our lattice results. We consider three techniques: next-to-next-to-leading order chiral perturbation theory (NNLO XPT), a Meyer-Lellouch-Lüscher-Gounaris-Sakurai model (MLLGS) and the rhopion-gamma model (RHO). A detailed exposition of these models can be found in the Supplementary Material of Ref. [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]. We investigate and discuss the suitability of their staggered versions for reducing the taste violations present in our lattice data. We call the resulting corrections taste improvements, because they improve the continuum extrapolation of our lattice data without, in principle, modifying the continuum-limit value. Indeed, these corrections vanish in that limit, as taste-breaking effects should. These improvements are applied on light-quark observables at the isospin-symmetric point, whose taste violations have the largest impact on our final uncertainties.

The models NNLO XPT, MLLGS and RHO describe the long-distance physics associated with finite-volume effects, as measured in our simulations. One can also define corresponding models describing the taste violations, they are denoted NNLO SXPT, SMLLGS and SRHO. We find that they describe the physics associated with taste violations, at least at larger distances. This is illustrated in Figure 4, where cutoff effects in the integrand of light are plotted as a function of Euclidean time. More specifically, we define the physical observable, obtained by convoluting the integrand of light with a smooth window function

( ; 1 , 2 ) ≡ Θ( ; 1 , Δ) -Θ( ; 2 , Δ) with Θ( ; ′ , Δ) ≡ 1 2 + 1 2 tanh[( -′ )/Δ] (31)
of a width of 2 -1 = 0.5 fm and starting at a time of 1 . Then we consider the difference in the value of this observable, obtained on a fine and a coarse lattice at a sequence of 1 separated by The plot shows the difference between a fine and a coarse lattice, whose sizes are = 6.14 fm and = 6.67 fm. The black squares with errors are obtained from the simulation. The colored curves are the predictions of NLO and NNLO SXPT, the SRHO and the SMLLGS models. They are computed at the parameters (pion mass, taste violation, volume) of the simulations.

2 . Because the determination of this quantity does not require overcoming many of the challenges described above, other lattice groups have obtained it with errors comparable to ours [START_REF] Aubin | Light quark vacuum polarization at the physical point and contribution to the muon -2[END_REF][START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF]. This allows for a sharper benchmarking of our calculation of this challenging, light-quark contribution that dominates . Our light ,win differs by 0.2 and 2.2 from the lattice results of [START_REF] Aubin | Light quark vacuum polarization at the physical point and contribution to the muon -2[END_REF] and [START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF], respectively. Moreover, ,win can be computed in the R-ratio approach [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF]. However, here we find a 3.7 tension with our lattice result.

Conclusions

Combining all of these ingredients we obtain, as a final result, = 707.5(2.3) (5.0) [5.5]. The first, statistical error comes mostly from the noisy, large-distance region of the current-current correlator. The second, systematic error is dominated by the continuum extrapolation and the finitesize effect computation. The third, total error is obtained by adding the first two in quadrature. In total we reach a relative accuracy of 0.8%.

Figure 6 compares our result with previous lattice computations and also with results from the R-ratio method, which have recently been reviewed in [START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF]. As one can see, there is a tension between our result and those obtained by the R-ratio method. For the total, LO-HVP contribution to , our result is 2.0 , 2.5 , 2.4 and 2.2 larger than the R-ratio results of = 694.0(4.0) [39], = 692.78(2.42) [START_REF] Keshavarzi | -2 of charged leptons, ( 2 ) , and the hyperfine splitting of muonium[END_REF],

= 692.3(3.3) [START_REF] Colangelo | Two-pion contribution to hadronic vacuum polarization[END_REF][START_REF] Hoferichter | Three-pion contribution to hadronic vacuum polarization[END_REF] and the combined result = 693.1(4.0) of [START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF], respectively. It is worth noting that the R-ratio determinations are based on the same experimental data sets and are therefore strongly correlated, though these data sets were obtained in several different and independent experiments that we have no reason to believe are collectively biased. Clearly, these comparisons need further investigation.

To conclude, when combined with the other standard model contributions (see eg. [START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF]), our result for the leading-order hadronic contribution to the anomalous magnetic moment of the muon, ] iso . The data points are extrapolated to the infinite-volume limit. Two different ways to perform the continuum extrapolations are shown: one without improvement, and another with corrections from a model involving the -meson (SRHO). In both cases the lines show linear, quadratic and cubic fits in 2 with varying number of lattice spacings in the fit. The continuum extrapolated result is shown with the results from other lattice groups, RBC'18 [START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] and Aubin'19 [START_REF] Aubin | Light quark vacuum polarization at the physical point and contribution to the muon -2[END_REF]. Also plotted is our R-ratio-based determination, obtained using the experimental data compiled by the authors of [START_REF] Keshavarzi | -2 of charged leptons, ( 2 ) , and the hyperfine splitting of muonium[END_REF] and our lattice results for the non light connected contributions. This plot is convenient for comparing different lattice results with each other. Regarding the total ,win , for which we also have to include the contributions of other-than-light flavors and isospin-breaking effects, we obtain 236.7 [1.4] on the lattice and 229.7 [1.3] from the R-ratio, the latter is 3.7 or 3.1% smaller than the lattice result. for the leading-order, hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. See [START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF] for a recent review. Green squares are lattice results: this work's result, denoted by BMWc'20 and represented by a filled symbol at the top of the figure, is followed by LM'20 [START_REF] Lehner | Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio[END_REF], Mainz'19 [START_REF] Gerardin | The leading hadronic contribution to ( -2) from lattice QCD with f = 2 + 1 flavours of O( ) improved Wilson quarks[END_REF], FHM'19 [START_REF] Davies | Hadronic-Vacuum-Polarization Contribution to the Muon's Anomalous Magnetic Moment from Four-Flavor Lattice QCD[END_REF], ETM'19 [START_REF] Giusti | Electromagnetic and strong isospin-breaking corrections to the muon -2 from Lattice QCD+QED[END_REF], RBC'18 [START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] and our earlier work BMWc'17 [START_REF] Borsanyi | Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles[END_REF]. Red circles were obtained using the R-ratio method: the combined result WP'20 [START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF] is followed by DHMZ'19 [START_REF] Davier | A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to (m 2 Z )[END_REF], KNT'19 [START_REF] Keshavarzi | -2 of charged leptons, ( 2 ) , and the hyperfine splitting of muonium[END_REF] and CHHKS'19 [START_REF] Colangelo | Two-pion contribution to hadronic vacuum polarization[END_REF][START_REF] Hoferichter | Three-pion contribution to hadronic vacuum polarization[END_REF]; these results use the same experimental data as input. The blue shaded region is the value that LO-HVP would have to have to explain the experimental measurement of ( -2), assuming no new physics.

B

Figure 2 :

 2 Figure 2: Comparison of a conventional random source based technique, as we applied it in our earlier work [23], and the low-mode improvement technique of this work on a = 3.9200 4stout ensemble for the case of [ light ] 0 's upper and lower bounds (see Section 4.2).

Figure 3 :

 3 Figure 3: Upper and lower bounds on the light isospin-symmetric component of . The results shown here are obtained with the 4HEX action on two different volumes at = 0.112 fm lattice spacing and = 121 MeV Goldstone-pion mass. We also have another simulation with = 104 MeV mass. From these two we interpolate to = 110 MeV.

Figure 4 :

 4 Figure 4: Isospin-symmetric component of light computed with a sliding window. The window starts at 1 and ends 0.5 fm later. The plot shows the difference between a fine and a coarse lattice, whose sizes are = 6.14 fm and = 6.67 fm. The black squares with errors are obtained from the simulation. The colored curves are the predictions of NLO and NNLO SXPT, the SRHO and the SMLLGS models. They are computed at the parameters (pion mass, taste violation, volume) of the simulations.

BFigure 5 :

 5 Figure 5: Continuum extrapolation of the isospin-symmetric, light, connected component of the window observable ,win , denoted by [ light ,win] iso . The data points are extrapolated to the infinite-volume limit. Two different ways to perform the continuum extrapolations are shown: one without improvement, and another with corrections from a model involving the -meson (SRHO). In both cases the lines show linear, quadratic and cubic fits in 2 with varying number of lattice spacings in the fit. The continuum extrapolated result is shown with the results from other lattice groups, RBC'18[START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] and Aubin'19[START_REF] Aubin | Light quark vacuum polarization at the physical point and contribution to the muon -2[END_REF]. Also plotted is our R-ratio-based determination, obtained using the experimental data compiled by the authors of[START_REF] Keshavarzi | -2 of charged leptons, ( 2 ) , and the hyperfine splitting of muonium[END_REF] and our lattice results for the non light connected contributions. This plot is convenient for comparing different lattice results with each other. Regarding the total ,win , for which we also have to include the contributions of other-than-light flavors and isospin-breaking effects, we obtain 236.7[1.4] on the lattice and 229.7[1.3] from the R-ratio, the latter is 3.7 or 3.1% smaller than the lattice result.

Figure 6 :

 6 Figure 6:Comparison of recent results for the leading-order, hadronic vacuum polarization contribution to the anomalous magnetic moment of the muon. See[START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF] for a recent review. Green squares are lattice results: this work's result, denoted by BMWc'20 and represented by a filled symbol at the top of the figure, is followed by LM'20[START_REF] Lehner | Consistency of hadronic vacuum polarization between lattice QCD and the R-ratio[END_REF], Mainz'19[START_REF] Gerardin | The leading hadronic contribution to ( -2) from lattice QCD with f = 2 + 1 flavours of O( ) improved Wilson quarks[END_REF], FHM'19[START_REF] Davies | Hadronic-Vacuum-Polarization Contribution to the Muon's Anomalous Magnetic Moment from Four-Flavor Lattice QCD[END_REF], ETM'19[START_REF] Giusti | Electromagnetic and strong isospin-breaking corrections to the muon -2 from Lattice QCD+QED[END_REF], RBC'18[START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] and our earlier work BMWc'17[START_REF] Borsanyi | Hadronic vacuum polarization contribution to the anomalous magnetic moments of leptons from first principles[END_REF]. Red circles were obtained using the R-ratio method: the combined result WP'20[START_REF] Aoyama | The anomalous magnetic moment of the muon in the Standard Model[END_REF] is followed by DHMZ'19[START_REF] Davier | A new evaluation of the hadronic vacuum polarisation contributions to the muon anomalous magnetic moment and to (m 2 Z )[END_REF], KNT'19[START_REF] Keshavarzi | -2 of charged leptons, ( 2 ) , and the hyperfine splitting of muonium[END_REF] and CHHKS'19[START_REF] Colangelo | Two-pion contribution to hadronic vacuum polarization[END_REF][START_REF] Hoferichter | Three-pion contribution to hadronic vacuum polarization[END_REF]; these results use the same experimental data as input. The blue shaded region is the value that LO-HVP would have to have to explain the experimental measurement of ( -2), assuming no new physics.

  Effective mass of the ground state of the Ω baryon in lattice units on our coarsest ensemble with = 3.7000. Results with three different staggered operators, Ω VI , Ω XI and Ω Ba are shown. The horizontal lines and the shaded regions represent the fit values and the errors obtained with a four-state fit, Equation[START_REF] Blum | Lattice calculation of the lowest order hadronic contribution to the muon anomalous magnetic moment[END_REF]. The dashed lines are the effective masses computed from the fitted functions.

		1.122
		1.120 1.121	Ω VI	Ω XI	Ω Ba
	M Ω effective mass	1.116 1.117 1.118 1.119
		1.115
		1.114
		1.113
			8 10 12 14 16 18	8 10 12 14 16 18	8 10 12 14 16 18
			t/a	t/a	t/a
	Figure 1:

Table 1 :

 1 cont (0.3) big (0.6) =0 (0.1) qed[2.5] . Finite-size effect in the reference box of the isospin-symmetric component of . The figures are obtained in various model approaches.

	B. C. Toth

B. C. Toth

weakens the longstanding discrepancy between experiment and theory. However, as discussed above and can be seen in Figure 6, our lattice result shows some tension with the R-ratio determinations. Obviously, our findings should be confirmed -or refuted-by other collaborations using other discretizations of QCD.

B. C. Toth 0.1 fm. These are compared to the NLO SXPT, NNLO SXPT, SRHO and SMLLGS predictions for this quantity, evaluated at the exact parameters of the ensembles.

The SMLLGS, the SRHO and the NNLO SXPT taste improvements describe the numerical data very nicely for 1 > ∼ 2.0 fm, fairly well for 1 > ∼ 1.0 fm and all the way down to 1 ≃ 0.4 fm in the case of SRHO. All three slightly overestimate the observed cutoff effects, the rho-meson based approach performing best, whereas NNLO displays a large deviation from the lattice results in the 1 ≤ 0.8 fm region. The lattice results have a maximum at 1 = 1.4 fm, as does the SRHO improvement, reinforcing our confidence that this model captures the relevant physics.

These findings lead us to apply the following taste corrections to our simulations results for light ( , , ), obtained on an 3 × lattice with lattice spacing , before performing continuum extrapolations:

with sep = 0.4, 0.7, 1.0, 1.3 fm, and where the factor (10/9) is related to the quark charges. Note that by using ref and ref in the above Equation, we are applying a very small volume correction to interpolate all of our simulation results to the same reference, four-volume so that they can be extrapolated to the continuum limit together. The taste-improved data is then extrapolated to the continuum using our standard fit procedure, in the course of which isospin-breaking effects are also included. For estimating the systematic error we use a histogram technique [START_REF] Borsanyi | Leading hadronic contribution to the muon magnetic moment from lattice QCD[END_REF].

The procedure described above does not yet take into account the systematic uncertainty associated with our choice of SRHO for taste improvement for > 1.3 fm. Since applying no taste improvement in that region is not an option, because of the nonlinearities introduced by two-pion, taste violations, we turn to NNLO SXPT, only as a means to estimate the uncertainty associated with this choice. Thus, we define this systematic uncertainty as ERR = (SRHO -NNLO SXPT) for > 1.3 fm. Then, we perform the same histogram analysis but with SRHO, SRHO-ERR and SRHO+ERR improvements. From this histogram we extract the contribution which comes from the variation in the improvement model from SRHO-ERR to SRHO+ERR. We assign this full spread to the systematic uncertainty associated with the taste-improvement procedure. We add this error in quadrature to the error given by the histogram technique discussed in the previous paragraph.

Intermediate window

The work [START_REF] Rbc | Calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment[END_REF] defined a particularly useful observable ,win , in which the current propagator is restricted to a time window [ 1 , 2 ], using the smooth weight function ( ; 1 , 2 ) defined in Equation [START_REF] Hansen | Finite-volume effects in ( -2) HVP,LO[END_REF]. The advantage of ,win over is that, by choosing an appropriate window, the calculation can be made much less challenging on the lattice than for the full . Here we will be interested in the window between 1 = 0.4 fm and 2 = 1.0 fm, i.e. in an intermediate time range. By this choice we eliminate both the short-distance region, where large cutoff effects are present, and the long-distance region, where the statistical uncertainties and finite-size effects are large. Moreover, in the case of staggered fermions, it has reduced taste-breaking artefacts. This is shown in Figure 5, where the light, connected component of ,win is plotted as a function of


Nouh Zeggai

Zohra Bouberka

Frédéric Dubois

Lamia Bedjaoui

Tewfik Bouchaour

Chems Eddine Gherdaoui

Jonathan Potier

Philippe Supiot

Ulrich Maschke
email: ulrich.maschke@univ-lille.fr

Tuning of thermally-induced shape memory properties of low-cost biocompatible linear and chemically crosslinked isobornyl/isobutylacrylate copolymers

Keywords: Isobornyl acrylate, isobutyl acrylate, entanglements, shape memory polymers, thermomechanical characterization

A series of binary low-cost biobased copolymers was elaborated from isobornylacrylate (IBOA) and isobornyl (IsoBA) monomers applying UV-visible induced free radical polymerization, yielding linear and crosslinked architectures. Thermo-mechanical measurements revealed that the elaborated random copolymers exhibit single glass transition temperatures, which can be adjusted on a broad temperature range by varying the amount of IBOA-and HDDA-units. Governed by the glass transition, linear high molecular weight IBOA-isoBA copolymers exhibit good thermally induced shape memory properties related to the occurrence of strong physical entanglements, reinforced by the presence of the voluminous isobornylene groups. Fixity and recovery ratios were found to be in the same order of magnitude when compared to crosslinked thermoplastics and thermosetting polymers. Linear IBOA-isoBA copolymers maintain their form memory properties even after several application cycles, confirming the efficiency of the presence of physical crosslinking.

These findings can be explained by the high entanglement molecular weight of 44 kg/mol providing an important elasticity as well as by an elevated glassy/rubbery moduli ratio of around 3000, facilitating deformation and fixing modes.

Introduction

Shape memory polymers (SMPs) are materials that possess the ability to regain their permanent shapes upon exposure to external stimuli such as temperature [START_REF] Hu | A review of stimuli-responsive polymers for smart textile applications[END_REF], electrical field [START_REF] Asaka | Bending of polyelectrolyte membrane platinum composites by electric stimuli Part II . Response kinetics[END_REF], light [START_REF] Jiang | Polymers Move in Response to Light[END_REF], magnetic field [START_REF] Xulu | Preparation and Responsive Properties of Magnetically Soft Poly ( Nisopropylacrylamide ) Gels[END_REF], or humidity [START_REF] Yang | Effects of moisture on the thermomechanical properties of a polyurethane shape memory polymer[END_REF]. Thermoset SMPs deform mechanically at a temperature above a certain transition temperature (T trans ), and can be fixed to a temporary shape before being cooled to a temperature below T trans [START_REF] Kim | Polyurethane having shape memory effect[END_REF]. Heating a SMP above T trans allows to regain its permanent form.

The thermomechanical behavior of SMPs requires generally the existence of two phases which present different thermomechanical properties. The first phase is rigid allowing the fixing of the shape of the polymer at low temperature and the second phase is flexible and represents some mobility, allowing the deformation of the material above a transition temperature, from the first to the second phase. SMPs can be used as an alternative to shape memory alloys (SMAs) in many applications and most importantly, they can be used in applications where SMAs cannot be used. SMPs are less expensive to produce and they can easily be worked into a variety of different shapes and sizes. These performances make it possible to consider industrial applications in several fields such as aerospace [START_REF] Liu | Shape memory polymers their composites in aerospace applications : a review[END_REF] namely deformable airplane wings [START_REF] Yu | Application of SMP composite in designing a morphing wing[END_REF], and intelligent textiles [START_REF] Hu | A review of stimuli-responsive polymers for smart textile applications[END_REF]. SMPs are classified into two main categories according to the physical and chemical nature of the crosslinking points. For SMPs based on chemical crosslinks, shape memory properties strongly depend on the degree of covalent crosslinking [START_REF] Hu | A review of stimuli-responsive polymers for smart textile applications[END_REF][START_REF] Asaka | Bending of polyelectrolyte membrane platinum composites by electric stimuli Part II . Response kinetics[END_REF][START_REF] Jiang | Polymers Move in Response to Light[END_REF]. The key aspect of SMPs based on physical crosslinking is crystalline or amorphous rigid segment, and the high molecular weight of their main chains, forming strong entanglements. Among the physically crosslinked SMPs, thermoplastic segmented polyurethanes have been extensively investigated for their shape memory effects [START_REF] Lin | Study on shape-memory behavior of polyether-based polyurethanes. I. Influence of the hard-segment content[END_REF][START_REF] Li | Crystallinity and morphology of segmented polyurethanes with different soft-segment length[END_REF][START_REF] Kim | Polyurethane ionomers having shape memory effects[END_REF][START_REF] Jeong | Shape memory polyurethane containing mesogenic moiety[END_REF]. Amorphous viscoelastic polymers with high molecular weight (> 10 6 g/mol) also show shape memory effects, due to the absence of flow above the glass transition, and their high elasticity ratio E g (glassy state modulus)/E r (rubbery state modulus). Polynorbornene (Norsorex ® ) [START_REF] Jeon | Shape memory and nanostructure in poly (norbornyl-POSS) copolymers[END_REF][START_REF] Sakurai | Strain-induced crystallization in polynorbornene[END_REF] and poly (methyl methacrylate) (PMMA) with high molecular weight, for example, possess a significant number of molecular entanglements in their chains, serving as (non-covalent) physical crosslinks [START_REF] Yang | Impression recovery of amorphous polymers[END_REF]. These entanglements form a 3D network leading to excellent elasticity in the glassy domain. Polynorbornene and PMMA exhibit efficient shape recovery due to their high entanglement density, highlighted by the lack of flow and large rubbery plateaus [START_REF] Liu | Review of progress in shape-memory polymers[END_REF]. It has already been reported from reptation theory as well as from tube model, that molecular chains tend to form entanglements when the chain length exceeds a critical value M c [START_REF] De Gennes | Scaling concepts in polymer physics[END_REF][START_REF] Doi | The theory of polymer dynamics[END_REF]. Although the role of hard/soft segments, crystallinity, and molecular entanglements in SMPs has already been discussed in many studies, there is a lack of reports dealing with the influence of entanglement density and elasticity ratio on shape memory performance of amorphous linear acrylic copolymers.

Regardless of the SMP type, the glass transition temperature (T g ) corresponding to T trans represents a crucial parameter for their operation. It can be tailored around body temperature making SMPs attractive for their use in biomedical applications [START_REF] Fatiha | Shape Memory Materials for Biomedical Applications[END_REF] such as catheters [START_REF] Ortega | A shape memory polymer dialysis needle adapter for the reduction of hemodynamic stress within arteriovenous grafts[END_REF], stents [START_REF] Wache | Development of a polymer stent with shape memory effect as a drug delivery system[END_REF], neural electrodes [START_REF] Sharp | Toward a self-deploying shape memory polymer neuronal electrode[END_REF], aneurysms [START_REF] Salazkin | Cold hibernated elastic memory foams for endovascular interventions[END_REF], microactuators to remove clots from blood vessels [START_REF] Maitland | Photothermal Properties of Shape Memory Polymer Micro-Actuators for Treating Stroke[END_REF], artificial muscles [START_REF] Takashima | Sensors and Actuators A : Physical McKibben artificial muscle using shape-memory polymer[END_REF], and sutures [START_REF] Lendlein | Biodegradable , Elastic Shape-Memory Polymers for Potential Biomedical Applications[END_REF]. SMPs are currently under study for use as fixation of a fractured bone [START_REF] Yang | Hydrolytic and enzymatic degradation of poly ( trimethylene carbonate-co-D , L -lactide ) random copolymers with shape memory behavior[END_REF]. Each application requires a combination of different properties and functions, therefore, SMPs with well-defined properties are required. However, it is difficult in practice to develop new monomers or crosslinking agents with adjusted properties, for example to obtain appropriate T g 's. In this context, the use of copolymers allow tuning the thermomechanical properties by playing with their composition and morphology [START_REF] Suchao-In | Grafting to' as a novel and simple approach for tripleshape memory polymers[END_REF]. Acrylate-based copolymers are particularly advantageous because of their ease of use.

Their mechanical and rheological properties can be adjusted by varying composition, chemical structure, and molecular weight of the monomers [START_REF] Genix | Enhancing the mechanical properties of glassy nanocomposites by tuning polymer molecular weight[END_REF]. Many reports in literature focus on chemically crosslinked polymer networks for synthesizing SMPs [START_REF] Zhuohong | The study of crosslinked shape memory polyurethanes[END_REF], which can be triggered near the temperature of the human body. However, there are still several limitations in the synthesis of suitable SMPs, such as the choice of segments and the adjustment of mechanical properties. The most relevant drawbacks of SMPs are high recovery temperatures [START_REF] Zhenjie Ding | High-Temperature Triple-Shape Memory Polymer with Full Recovery through Cross-Linking All-Aromatic Liquid Crystalline Poly(ester imide) under Reduced Molding Temperature[END_REF] and long recovery periods [START_REF] Rousseau | Challenges of shape memory polymers: A review of the progress toward overcoming SMP's limitations[END_REF].

In this work, a simple strategy for modulating T g of new versatile acrylate-based thermally controlled SMPs functioning near the human body temperature will be proposed, via elaboration of linear and chemically crosslinked random copolymers based on isobornylacrylate (IBOA) and isobutylacrylate (IsoBA) units. Variation of the copolymer composition allows tuning of the glass transition and thus the operational temperature domain of these SMPs. Thermo-mechanical studies by DMA will be carried out to characterize the behavior of the elaborated linear and crosslinked copolymers under the effect of temperature.

The effect of shape memory and recovery as well as the efficiency of the SMP performance after several cycles will be investigated.

Materials and methods

Materials

Isobutylacrylate (IsoBA), Isobornylacrylate (IBOA) and 1,6 hexanedioldiacrylate (HDDA) were considered as acrylic compounds. 2-hydroxy-2-methyl-1-phenyl-propan-1-one (commercial designation: Darocur 1173) was used as photoinitiator. All these products were purchased from Sigma-Aldrich, France, and used as received.

Preparation of copolymers

Four IsoBA/IBOA based copolymers were elaborated by varying the weight percentage of IsoBA/IBOA/HDDA as follows: 40/60/0, 50/50/0, 60/40/0, 59.75/39.75/0.5, leading to linear copolymers except for the last composition, which generates a chemically crosslinked network using experimental method described in our previous work [33a, 33b]. Linear acrylic homopolymers were prepared as well, based on n-butylacrylate (nBA), 2-ethylhexylacrylate (2-EHA), and tert-butyl acrylate (tBA). The amount of photoinitiator was set at 0.5% by weight related to the total weight of monomers. After a stirring time of 3h at room temperature, the monomeric solutions were poured into Teflon molds, and then exposed to UV-visible light at a wavelength of =365nm and an intensity of I 0 =1,5mW/cm 2 (Philips black light) (Fig. 1). The monomeric mixtures were exposed to UV-visible light during 40 min under nitrogen atmosphere to ensure maximum conversion of acrylic monomeric double bonds.

Characterization techniques 2.3.1. Molecular weight determination

Gel permeation chromatography (GPC) analysis was carried out at room temperature on a Waters Alliance e2695 system using differential refractive index (Wyatt RI) and Multi Angle Light Scattering (Wyatt MALS, λ (laser) = 670 nm) detectors. Three columns were coupled in series (Styragel HR1, Styragel HR3, Styragel HR4). THF was used as solvent (flow rate 1mL/min), and calibration was established with polystyrene (PS) standards from Polymer Laboratories.

Differential scanning analysis

Thermal properties of the elaborated copolymers were investigated by means of a DSC (Perkin-Elmer model 8000). The experiments were carried out under nitrogen flow (20 mL/min) applying a heating rate of 10°C/min. Calibration of the instrument was performed with indium and zinc as standards. 8-10 mg of copolymers were introduced from arbitrary positions of the disk-shaped samples into aluminum crucibles.

The program consisted first in cooling the sample, followed by three heating and cooling cycles at a rate of 10°C/min in the temperature range from -72°C to 120°C, separated by oneminute isotherms at maximum/minimum temperatures. Only thermograms from the second heating cycles were considered for the analysis. T g was determined from the midpoint of the transition range of the thermograms by using a software program from Perkin-Elmer (Pyris).

Thermomechanical studies

Dynamic Mechanical Analysis (DMA) was performed using a Perkin Elmer DMA 8000 device. The thermomechanical studies of the copolymer samples were carried out under nitrogen atmosphere in traction mode at a frequency of 1Hz applying a heating rate of 5°C/min in the temperature range from -60°C to 200°C. Test samples were cutted into rectangular shapes with dimensions of 29 mm x 7mm x 2 mm.

Shape memory polymer evaluation

The dual shape memory experiments were conducted in a controlled force mode measuring strain and stress, using a Perkin Elmer DMA 8000 device. The sample exhibiting identical dimensions as described in section 2.3.3., was heated to a temperature of 80°C (well above T trans equal to T g ), by application of a constant heating rate of 5°C/min, followed by an isothermal step of 10min, to ensure a good homogeneous temperature distribution throughout the sample. Then, a static force of 0.8N was applied during a period of 10 min to deform the sample uniaxially. The next step consists in cooling down the sample (cooling rate 5°C/min) to 0°C using liquid nitrogen, followed by an isothermal step of 10 min. Then, the applied force was removed and the fixing of the temporary shape was completed. The sample was subsequently heated up again to 80°C, using a constant heating rate of 5°C/min, allowing to recover the initial (permanent) shape. Shape fixity (R f ) and shape recovery (R r ) ratios were calculated using Eqs. ( 1) and ( 2):

100%   (1) 100%    (2)
where ε stands for the fixed strain after sample cooling when the external load is removed, ε Load represents the maximum strain under external load, while ε rec corresponds to the strain measured after recovery.

Nota bene : Good fixation of the samples might result in a sudden increase in stress shown in the figures presenting shape memory effects. In fact, polymer chains in this unstable step apply a force in the opposite direction to that applied during fixation, except that this force is not large enough for the chains to relax completely, allowing the sample to return to its stable permanent state. This concurrency between applied force and reverse force causes an increase in the stress of the sample.

Stress-strain behavior of shape memory polymers

Tensile tests to evaluate the yield strain were carried out on a VERSA® machine, equipped with a force transducer of ±100N. The reproducibility of the experiments was evaluated by comparing the response curves of a few specimens analyzed under the same loading condition. The stretching speed used was 200 mm/s.

Stress-relaxation of shape memory polymers

Stress-relaxation experiments were performed on SMPs using the Perkin Elmer DMA 8000 device. A constant strain was applied on SMPs to induce a fixed deformation (10%) for a defined time, and the change in stress to maintain that strain was measured. weights for all linear copolymers were high enough for physical entanglements to be created [START_REF] Sukumaran | Modeling Entangled Dynamics: Comparison between Stochastic Single-Chain and Multichain Models[END_REF]. A slight upward trend of the molecular weights was observed when increasing the amount of IBOA in the copolymers.

Results and discussions

Molecular weights

Glass transition

The elaborated acrylic copolymers considered here were previously characterized by DSC to investigate their phase behavior and to determine their T g 's [33b,33c]. In order to compare results from DSC with those obtained from DMA, Fig. 2 reproduces selected thermograms from reference [33b,33c], and the corresponding T g -data were collected in Table 2. Addition of 0.5wt-% of the bifunctional acrylic HDDA to the monomeric mixture composed of

[IsoBA] 40 and [IBOA] 60 yields a higher T g of the crosslinked copolymer than that of the linear one. This result was expected, since crosslinking restricts free chain motion, and the polymer requires more energy i.e. higher temperature to transform from glassy to rubbery state.

Dynamic mechanical properties

[IsoBA] X -[IBOA] Y copolymers were examined by DMA applying a temperature sweep mode to examine the thermomechanical properties of the samples. At temperatures above T g , polymer chains follow micro-Brownian motion, leading to rubbery behavior, whereas below T g , this type of motion is frozen and the copolymers become glassy. The evolution of the storage (E') modulus of [IsoBA] x -[IBOA] y copolymers as function of temperature is shown in Incorporation of HDDA as crosslinking agent into the copolymer increases the storage modulus compared to the linear copolymer, due to the establishment of covalent permanent crosslinking points (Table 2). Storage moduli values were found to be comparable to those of photochemically synthesized acrylic polymers [START_REF] Zhu | Preparation of environmentally friendly acrylic pressure-sensitive adhesives by bulk photopolymerization and their performance[END_REF].

Dissipation factors (tan ) were obtained by calculating the ratio of loss to storage moduli, and T g 's were determined from the temperatures corresponding to the peak maxima of the tan curves. Fig. 3b shows the evolution of tan δ as function of temperature, revealing that an increase of the IBOA concentration in the copolymers shifts the glass transition to higher temperatures, similar to the results obtained by DSC (Fig. 2). The presence of chemical crosslinking points raised T g to 57°C, 7°C higher than the corresponding value of the linear copolymer. : storage moduli and tan determined at (T=-25°C). b: entanglements/m 3 calculated using Bersted entanglement model (Eq. 4). T g data obtained by DMA were found to exceed the corresponding values from DSC by several degrees. Indeed, the peak maximum of tan corresponds roughly to the transition midpoint or inflection point of the decreasing log E' curve, while T g values determined by DSC can be related to the initial drop of E' at the transition of glassy to rubbery states. Fig. 4 illustrates an important difference of the temperature dependant evolution of E' for linear and crosslinked [IsoBA] x -[IBOA] y copolymers in the rubbery region. All linear [IsoBA] x -[IBOA] y copolymers show pseudo-plateau rubbery moduli on a broad temperature range (from 90°C to 170°C). Existence of such plateau moduli indicates the presence of molecular entanglements [START_REF] Sukumaran | Modeling Entangled Dynamics: Comparison between Stochastic Single-Chain and Multichain Models[END_REF]. Indeed, these entanglements play the role of efficient physical crosslinking points, representing one of the salient features leading to desirable shape memory behavior. Above 170°C, the storage moduli of the linear copolymers drop significantly, since the increased free volume leads to chain sliding and the material starts to flow. The ability to flow generally depends on the polymer molecular weight, however, no such dependence was observed from Fig. 4, keeping in mind that the molecular weights were roughly situated in the same range of values (Table 1). On the other hand, the starting flow temperature seems to be composition-dependent: increasing concentration of IBOA units in the copolymers shifts this temperature to higher values, probably related to the increased sterical hindrance effect of the bulky IBOA units, reducing thus chain sliding. In the case of the [IsoBA] 40 -[IBOA] 60 -R copolymer, the rubbery modulus remains constant over the entire temperature range from 90°C to 200°C and the material behave like a rubber. This copolymer does not flow under the effect of temperature contrary to the linear copolymers.

Ratios of E g /E r of more than 100 facilitate the deformation of thermally responsive SMP materials at high temperatures and their fixing at low temperatures [START_REF] Ratna | Recent advances in shape memory polymers and composites: A review[END_REF]. E g and E r values were extracted from Fig. 3a at T=-25°C and T=100°C, respectively, and Table 3 represents the corresponding data showing that all copolymers exhibit ratios around 3000. This means that the elaborated linear as well as the crosslinked [IsoBA] x -[IBOA] y copolymers are potentially interesting SMP materials and should present good shape memory properties.

The magnitude of the molecular weight between entanglements (M e ) represents one of the key parameters influencing the mechanical properties of SMP. For linear and chemically crosslinked polymer chains M e is defined as molecular weight between adjacent temporary and permanent entanglement points, respectively, and is mainly controlled by the molecular architecture. Zosel et al. [START_REF] Zosel | Adhesion and tack of polymers: Influence of mechanical properties and surface tensions[END_REF] as well as Jullian et al. [START_REF] Jullian | Rheological characterization and molecular modeling of poly(n-butyl acrylate)[END_REF] demonstrated that high M e would provide an important elasticity, which is a crucial parameter to obtain optimal SMP properties. Moreover, high M e values should provide an easy shape recovery. As the physical meaning of rubbery plateau refers to the stress maintain during deformation [START_REF] Zhuohong | The study of crosslinked shape memory polyurethanes[END_REF] that would be necessary for an optimal recovery, the lowest shape recovery would be expected for the crosslinked [IsoBA] 40 -[IBOA] 60 -R copolymer. M e can be estimated by writing ( [START_REF] Jullian | Rheological characterization and molecular modeling of poly(n-butyl acrylate)[END_REF], and references cited therein) :

(

) 3
where E r corresponds to the storage modulus of the rubber plateau taken at T=T g + 40°C, stands for the polymer density [START_REF] Van Krevelen | Properties of polymers: Properties of polymers: their correlation with chemical structure; their numerical estimation and prediction from additive group contributions[END_REF], R is the gas constant, and T represents the absolute temperature. Table 2 and

where M c =2M e , and stands for the Avogadro number. The results for  e are given in Table 2 presenting the same tendency as for M e , i.e. high values for linear copolymers and a low value for the crosslinked one. It should be pointed out that physical entanglements formed by linear chains are considered efficient only if the polymer molecular weight exceeds at least 8-10 times the value of M e , which is roughly the case here.

Shape memory behavior

High molecular weight [IsoBA] x -[IBOA] y copolymers

Investigation of the shape memory effect of [IsoBA] x -[IBOA] y copolymers was carried out as a function of time by performing fixity and recovery tests in pseudo linear deformation regime (see Fig. S2). Fig. 5 shows the result obtained for the [IsoBA] 60 -[IBOA] 40 copolymer, and the graphical presentations belonging to the two other copolymers can be found in the supplementary material (Figs. S3 andS4). These figures show that the three [IsoBA] x -

[IBOA] y copolymers possess the ability to maintain programmed temporary shapes, due to the presence of a significant number of physical entanglements per chain caused by high molecular weight (M n > 200kg/mol). Increasing the amount of IBOA or adding a crosslinking agent (HDDA) show similar effects on shape fixity which was determined as 89%, 88% and 91% for linear [IsoBA] x -[IBOA] y copolymers with 40%, 50% and 60% of respectively (see Table 3).

Similar data were obtained for SMP materials based on polyurethane, which are known for their high efficiency [START_REF] Pretsch | Triple-shape properties of a thermoresponsive poly(ester urethane)[END_REF]. In contrast, [IsoBA] 40 -[IBOA] 60 -R presented a lower fixity ratio (81%) compared to linear SMP. This decrease can be attributed to the presence of chemical when polymer chains are stretched between two adjacent chemical crosslinking points, entropy will be largely reduced so that the chains will be forced to return to the energetic favorable equilibrium unstretched state [START_REF] Treloar | Physics of Rubber Elasticity[END_REF].

Fig. 6 and Table 3 show that all ([IsoBA] x -[IBOA] y and [IsoBA] 40 -[IBOA] 60 -R) samples recovered nearly their original shape when the applied stress was removed at a temperature above T trans , and the recovery rates were found greater than 88%. Unlike the traditional SMP structure, which requires rigid and flexible segments, the existing entanglements as physical crosslinks in the linear copolymers, thus promoting recovery shape while preventing flow at 80°C. Like shape fixity, the shape recovery values of the investigated [IsoBA] x -

[IBOA] y copolymers were found to be close to those mentioned in literature for thermoplastics [START_REF] Jeong | Miscibility and shape memory property of poly(vinyl chloride)/thermoplastic polyurethane blends[END_REF], thermosets [START_REF] Alteheld | Biodegradable, amorphous copolyesterurethane networks having shape-memory properties[END_REF], and reinforced composites [START_REF] Nishikawa | Effect of fiber arrangement on shape fixity and shape recovery in thermally activated shape memory polymerbased composites[END_REF], which means that

[IsoBA] x -[IBOA] y copolymers can be considered as potential candidates for SMP applications. the evolution of ε from the end of the shape fixing period at T=0°C up to the shape recovery cycle at a temperature of 80°C. It can be noticed that shape memory recovery can be tuned as function of temperature, in particular around body temperature, depending on the copolymer composition. It was found that the shape recovery of a given SMP rises gradually with increasing temperature. A sharp increase of shape recovery was observed when the temperature approaches T g , until reaching plateau values corresponding to maximum recovery around 80°C. With the increase of IBOA content the copolymers, the shape recovery curves shift to higher temperatures, in relationship with the corresponding increase of T g of the examined [IsoBA] x -[IBOA] y copolymers, and the storage modulus in the vitreous state (the larger the modulus, the better the recovery) [START_REF] Xiao | Shape memory polymers with high and low temperature resistant properties[END_REF]. A difference in the recovery rate is clearly apparent while comparing linear with chemically crosslinked [IsoBA] x -[IBOA] y copolymers, indicating that shape recovery was governed by the crosslinking type: physical entanglements in the case of linear SMP and covalent bonds and entanglement for crosslinked SMP. As the temperature increases above 70°C, the shape recovery increases to more than 90% and stabilizes at this percentage for linear [IsoBA] x -[IBOA] y copolymers, suggesting that the irreversible deformation of these copolymers presents less than 10%. On the other hand, shape recovery of [IsoBA] 40 -[IBOA] 60 -R was found at a lower value (82%) compared to the linear one. Fig. 6 also shows that the crosslinked SMP tends to exhibit a relatively wide shape recovery temperature range, which may originate from its broader glass transition zone compared to the linear copolymers.

Stress relaxation measurements

The stress relaxation of high molecular weight copolymers was investigated under application of constant strain (10% of strain in flexion configuration). Fig. S5 shows the evolution of stress as function of time for different SMPs at room temperature. The corresponding curves do not present a disentanglement effect, which could be caused by the deformation applied to the films. These findings thus reinforce the results obtained by DMA. The stress-relaxation behavior clearly depends on the amount of IsoBA present in the copolymer. In particular it was found that the copolymer with 60% IsoBA presents the best set properties since it takes less time than the other copolymers to relax. With the increase of the amount of IBOA in the copolymer, the relaxation becomes more complex and takes longer time compared to the 60%

IsoBA film. This is due to the steric hindrance caused by the isobornyl group, which makes it difficult for the chains to rotate by this covalent bond, especially for the crosslinked copolymer.

To describe the relaxation under constant strain, a two-dimensional Maxwell model (Maxwell-Weichert) was applied, which was already employed to study stress-relaxation phenomena [START_REF] Heuchel | Relaxation based modeling of tunable shape recovery kinetics observed under isothermal conditions for amorphous shape-memory polymers[END_REF]. The time-dependent relaxation will thus be described by two characteristic relaxation times instead of one (classical Maxwell model), because of the complexity of the structure, high molar weight and presence of entanglements. The Maxwell-Weichert model can be presented in the following form :

  (5)
where represents the applied stress, and , are the characteristic relaxation times. Fig. S5 shows a good agreement between theory and experimental data, together with high R 2 values around 0.99 for all compositions.

Additional stress recovery experiments were performed on linear and crosslinked [IsoBA] x -

[IBOA] y copolymers. The corresponding experimental results as well as data fitting by the Maxwell-Weichert model are displayed in Fig. S6 showing the same general tendencies for the relaxation behavior as presented in Fig. S5.

Linear polymers and low molecular weight copolymers

For a better understanding of the effect of molar mass and formulation (using only soft segments) on shape memory effects and elasticity ratios E g /E r , an additional series of polyacrylates were synthesized and characterized, including linear homopolymers (poly nBA, [IBOA] 60 copolymers. The results obtained for the E g /E r ratios are presented in Table 3. The two main properties of shape memory (fixity and recovery) have been investigated (Table 3, 3).

These two values are higher than 100, but rather low compared to those from copolymers with high molecular weight. The low molar mass of the polymer is responsible for the low values obtained of the elasticity Ratio E g /E r (Fig. S9).

Interestingly, linear poly nBA (Fig S8b ), poly 2-EHA (Fig S8c ) and poly tBA, exhibit shape memory effects, with a fixity of (~90%) and recovery of (~94%), despite the absence of rigid segments and chemical crosslinking. These homopolymers are characterized by high molecular weights (M n > 260 kg/mol), presenting thus viscoelastic properties (a large rubbery plateau and a high storage modulus). The elasticity ratios E g /E r were calculated as 3097, 2723, 2944, for poly nBA, poly 2-EHA and poly tBA, respectively (Table 3), and are in the same order of magnitude as those of the high molecular weight copolymers. The important difference between glassy and rubbery moduli allows an easy shaping (good fixing properties)

of the polymers at T>T g (rubbery state) and great resistance to deformation at T<T g (glassy state).

An evaluation of cyclic stability is often required to probe the robustness of shape memory performance over several consecutive cycles. This type of experiment is particularly necessary before the SMP could be intended for practical applications. Fig. 7b presents the shape recovery percentage derived from the thermomechanical cyclic tests (Fig. 7a). As a result, shape recovery remains almost stable during the four cycles and was found equal to more than 90%, demonstrating high stability of cycling performance which is essential for device applications involving multiple cycles.

Conclusions

In this work, a simple strategy for modulating the transition temperature of linear and

Number of recovery cycles

crosslinking points, slowing the flow of the material at high temperature to allow elastic deformation for fixation of the shape. Shape fixity and shape recovery ratios reveal that the SMPs maintain their form memory properties even after four successive recovery cycles, confirming thus the efficiency of the presence of strong physical entanglements in the linear copolymers. The SMP properties of the investigated [IsoBA] x -[IBOA] y copolymers were found to be in the same order of magnitude when compared to crosslinked thermoplastics and thermosetting polymers, resulting from high entanglement molecular weight of 44 kg/mol and elevated glassy/rubbery moduli ratio of around 3000.

Fig. 1 .

 1 Fig. 1. Schematic representation of the elaboration of linear [IsoBA] x -[IBOA] y copolymers using UV-visible light induced polymerization of the corresponding monomers in the presence of a photoinitiator. A more detailed presentation of the reaction mechanism is given in Fig S1.

Fig. 2 .

 2 Fig. 2. Thermograms of linear and crosslinked [IsoBA] x -[IBOA] y copolymers [33b,33c].

Fig. 3a .

 3a Fig.3a. The linear copolymers show apparent rubbery plateaus confirming the formation of physical entanglements of the polymer chains[START_REF] Petisco-Ferrero | The relevance of molecular weight in the design of amorphous biodegradable polymers with optimized shape memory effect[END_REF]. The storage modulus E' in the glassy state increased by raising the [IBOA]/[IsoBA] ratio, which is mainly due to the specific chemical structure of IBOA, containing the voluminous isobornylene group. The latter reduces the free volume of the copolymer and consequently increases the storage modulus.

Fig. 3 .

 3 Fig. 3. (a) Storage modulus and (b) tan as functions of temperature for linear and crosslinked (R) [IsoBA] x -[IBOA] y copolymers.

c:

  The value of M n taken to calculate  e is that of the linear copolymer.

Fig. 4 .

 4 Fig. 4. Storage modulus of [IsoBA] x -[IBOA] y copolymers in the rubbery region. The insert shows the dependence of the entanglement molecular weight M e on the sample composition for linear copolymers.

  the insert of Fig. 4 present the calculated M e data : The values obtained for the linear copolymers were found to be almost identical; M e = 44 kg/mol for [IsoBA] 60 -[IBOA] 40 and [IsoBA] 50 -[IBOA] 50 , and M e = 45 kg/mol for [IsoBA] 40 -[IBOA] 60 However, M e for [IsoBA] 40 -[IBOA] 60 -R was found to be much lower at M e = 16 kg/ mol, close to the value of 12.5 kg/mol obtained by Voit et al. [41] from DMA of a low crosslinked methacrylate/IBOA containing copolymer.  e values (number of entanglements/m 3 ) of SMPs were calculated using Bersted's entanglement model [42]  1

Fig. 5 .

 5 Fig. 5. Shape fixing and recovery for [IsoBA] 60 -[IBOA] 40 copolymer. ε was determined at the end of the shape fixing period whereas ε rec was obtained at t=80min.

Fig. 6 .

 6 Fig. 6. Tuning of thermally-induced SMPs operating around body temperature: Shape recovery of several linear and crosslinked (R) [IsoBA] x -[IBOA] y copolymers.

Fig. 6

 6 Fig. 6 represents an extraction of data particularly from Figs. 5, S3 and S4, corresponding to

poly 2 -

 2 EHA, poly tBA), and low molecular weight linear [IsoBA] 50 -[IBOA] 50 and [IsoBA] 40 -

Figs.

  Figs.S7 and S8a-c). Both low molecular weight copolymers failed to keep their shape in the fixing step, according to the strong decrease of the deformation while decreasing the temperature (Figs.S7 and S8b). This result was expected, since the low-molecular weight chains cannot form sufficiently strong entanglements to fix the copolymer to a desired deformation in the rubbery state (T>T g ). The elasticity E g /E r ratios for the low-molecular weight [IsoBA] 50 -[IBOA] 50 and[IsoBA] 40 -[IBOA] 60 copolymers revealed values of 858 and 349, respectively (Table3).

  Fig.7ashows the result for fixity and shape recovery determined by four consecutive thermomechanical cyclic tests of [IsoBA] 40 -[IBOA] 60 copolymer. It can be concluded from Fig.7athat the shape memory effect can be repeated several times without risk of any negative effect on thermomechanical properties, indicating that physical crosslinking introduced by strong entanglements maintains its efficiency, in terms of reproducing several times the SMP effect.

Fig. 7 .

 7 Fig. 7. (a) Deformation-recovery cyclic test and b) shape recovery of linear [IsoBa] 40 -[IBOA] 60 copolymer as function of the number of recovery cycles.

  photochemically crosslinked[IsoBA] x -[IBOA] y copolymeric SMPs operating near the temperature of human body was proposed. It should be highlighted that no report is known up to now investigating SMPs based on linear poly(acrylate). Characterization of molecular weights by GPC revealed sufficiently high molecular weights of the linear[IsoBA] x -[IBOA] y copolymers to form strong physical molecular entanglements. Unlike conventional SMP structures, requiring rigid and flexible segments, these entanglements served as physical

Table 1 .

 1 Number and weight average molecular weights, M n and M w , as well as molecular weight distributions (polydispersities) M w /M n of linear[IsoBA] x -[IBOA] y copolymers and other linear acrylic homopolymers were gathered in Table1. M n of the copolymers varied from Designation, molecular weights and polydispersities of[IsoBA]

	Materials	IsoBA/IBOA/HDDA ratio (wt%)	M n (kg/mol) M w (kg/mol)	M w /M n
	[IsoBA] 60 -[IBOA] 40	60/40/0	322	533	1.65
	[IsoBA] 50 -[IBOA] 50	50/50/0	224	528	1.6
	[IsoBA] 50 -[IBOA] 50 (low M n )	50/50/0	61	122	2
	[IsoBA] 40 -[IBOA] 60	40/60/0	374	750	2
	[IsoBA] 40 -[IBOA] 60 -R	39.75/59.75/0.5	ND	ND	ND
	[IsoBA] 40 -[IBOA] 60 (low M n )	40/60/0	58	119	2.05
	nBA	-	282	402	1.4
	2-EHA	-	263	417	1.6
	tBA	-	260	423	1.62

x -[IBOA] y copolymers and nBA-, 2-EHA-, and tBA-homopolymers. ND = Not Determined (crosslinked).

224 kg/mol to 374 kg/mol, and their polydispersities from 1.6 to 2. The obtained molecular

Table 2 .

 2 Some thermo-mechanical properties of[IsoBA] x -[IBOA] y copolymers.

	Materials	E' (Pa) a	T g DSC (°C)	T g tan delta (°C) a	M e (kg/mol)	 (E/m 3 ) b
	[IsoBA] 60 -[IBOA] 40	3.7* 10 8	7.7	27.1	44.2	2.57×10 25
	[IsoBA] 50 -[IBOA] 50	4.2*10 8	15.6	39.1	44.3	2.15×10 25
	[IsoBA] 40 -[IBOA] 60	1.2*10 9	24.2	50	45.6	2.76× 10 25
	[IsoBA] 40 -[IBOA] 60 -R	2*10 9	29.1	57	16.2	1.18×10 25c
	a

Table 3 .

 3 Some properties of SMPs based on[IsoBA] x -[IBOA] y copolymers and nBA-, 2-EHA-, and tBA-homopolymers.

	Materials	E g /E r	Shape fixity (%)	Shape recovery (%)
	[IsoBA] 60 -[IBOA] 40	3111	89	93.7
	[IsoBA] 50 -[IBOA] 50	2877	88	93
	[IsoBA] 50 -[IBOA] 50 -low M n	858	No fixity	No recovery
	[IsoBA] 40 -[IBOA] 60	3274	97	90.5
	[IsoBA] 40 -[IBOA] 60 -R	3130	82	88.4
	[IsoBA] 40 -[IBOA] 60 -low M n	349	No fixity	No recovery
	nBA	3097	91	94
	2-EHA	2723	89	91
	tBA	2948	91	95

Acknowledgements

This work has accomplished in the of an international research program. The authors gratefully acknowledge the support of the Algerian Ministry of Higher Education and Scientific Research (MESRS), the Universities of Tlemcen and Oran (USTOMB) in Algeria, the French Ministry of Higher Education and Research (MENESR), the CNRS, the University and the CROUS of Lille/France.