Jean-Yves Fortin
email: jean-yves.fortin@univ-lorraine.fr

Pierrick Lample

Itinerant electrons on dilute frustrated Ising lattices

Keywords: numbers: 05.50.+q, 71.10.-w, 71.27.+a, 75.10.Hk

We consider itinerant spinless fermions as moving defects in a dilute two-dimensional frustrated Ising system where they occupy site vacancies. Fermions interact via local spin fluctuations and we analyze coupled selfconsistent mean-field equations of the Green functions after expressing the spin and fermion operators in terms of Grassmann variables. The specific heat and effective mass are analyzed with the solutions satisfying the symmetry imposed by the coupling layout. At low temperature, we find that these solutions induce stripes along the lines of couplings with the same sign, and that a low fermion density yields a small effective mass.

Introduction

Cooperative phenomena in correlated electronic systems yield to the emergence of spatial structures of the charge densities due to environmental constraints. For example, hole segregation was evidenced in the superconductor La 1.6-x Nd 0.4 Sr x CuO 4 and interpreted as the presence of a stripe modulation [1,2] which is assumed to be responsible for the suppression of superconductivity. One important question is how to probe for the possible existence of coherent states leading to superconductivity without the help of phonons [3], possibly using the indirect coupling induced by magnetic fluctuations. In this paper we focus on the physical properties that emerge between itinerant fermions in a dilute frustrated spin environment. The indirect coupling between fermions is mediated by spin fluctuations via site dilution where fermions replace the missing spins, mimicking hole doping in twodimensional cuprate materials, but with classical spin fluctuations. The itinerancy leads to spin disorder however it is favorable for the fermions to move along the boundary of spin domains in order to lower the energy. In a different context, similar models such as the migration of electrons on ionic bonds have been studied theoretically [4,5], also with classical moving defects [6], and itinerant electrons coupled to a spin environment are known to generate various domain structures [7,8]. Models of coupled lattices between spins and fermions were introduced to investigate metal-insulator interfaces [9] with the help of quantum Monte Carlo methods. In a previous paper [10], we analyzed a similar system of itinerant fermions occupying the empty sites of a two-dimensional dilute Ising model. The dilution is known to suppress the magnetic order, but spin fluctuations induce an indirect correlation between electrons. For this kind of model, we observed the divergence of the effective mass in the critical region, and a diminution of its value at low temperature, whereas the fermions behave like slightly heavy quasiparticles in the high temperature region. In the present paper, we would like to investigate the properties of these itinerant fermions in the presence of frustration which does not display criticality but instead a macroscopic ground state degeneracy.

The paper is organized as follow. In section 2, we introduce the quantum Hamiltonian describing the interaction between fermions that occupy the vacancies of a classical spin model displaying frustration at low temperature with a highly degenerate ground state. The method used in the present paper is based on Grassmann techniques [11] for the spin and fermion sectors [START_REF] Lichtenstein | Path integrals and dual fermions The Physics of Correlated Insulators[END_REF]. The resulting action can be partially reduced by integrating over the spin components and self-consistent equations for the Green functions and self-energies are analyzed under the meanfield approximation on an effective action, see section 3. A homogeneous solution is given in section 3.1, and an investigation of a stripe solution satisfying the symmetry of the spin coupling configurations is developed in subsection 3.2. Section 4 is dedicated to the Monte-Carlo study of the classical equivalent of the quantum model, with fermions replaced by itinerant classical particles obeying Markov dynamics, and finally a discussion in given in the conclusions, section 5.

Model Hamiltonian for the frustrated dilute spin system

In this section we introduce the main Hamiltonian of a system composed of Ising spins σ r = ±1 and spinless fermions on a frustrated lattice of size N = L × L. Fermions are described by operators (c r , c † r ) and fermion numbers n r = c † r c r . When a fermion occupies a site r = (m, n), the corresponding spin becomes inactive. Also, every fermion moves to a neighboring site with a kinetic or transfer energy t, which will be set to unity in the numerical applications. The Hamiltonian is then defined by

H = - J 2 r,δ=±x,±y ǫ r σ r σ r+δ (1 -n r )(1 -n r+δ ) -t r,δ=±x,±y c † r+δ c r , (1)
where ǫ r = ±1 is the sign function of the local couplings Jǫ r between spins, and δ the discrete unit vectors of the lattice. Couplings are antiferromagnetic if ǫ r = -1, and ferromagnetic otherwise. We consider two simple models of Ising lattices in which the geometry of the couplings leads to spin frustration. By definition, a system is frustrated if the product of the four couplings around each plaquette is negative. This can be achieved by considering two simple configurations of couplings. The first one corresponds to the piled up dominoes model (PUD), and the second one to the zig-zag dominoes model (ZZD), as represented in figure 1. These models were previously extensively studied and exact solutions were given in the non dilute case [START_REF] Andre | [END_REF] using transfer matrix methods. Among the principal results is the macroscopically degeneracy of the ground state and a non-zero entropy per site at zero temperature with a non-trivial value. There is also no phase transition at non zero temperature for the case where all the couplings have the same absolute value, and the system remains paramagnetic. The local couplings are defined for these two models by the function ǫ r = ǫ m,n such that ǫ m,n = (-1) n for the PUD model, ǫ m,n = (-1) m+n for the ZZD model.

(

) 2
The non-frustrated Ising case where ǫ m,n = 1 was studied in a previous reference [10] within a mean-field approximation on the Green functions of the fermionic sector.

J -J

(a) Piled up dominoes model

J -J (b) Zig-zag model

Figure 1: Two frustrated lattice configurations with periodic and antiferromagnetic couplings -J (red) and ferromagnetic ones J (black). Case (a) corresponds to the piled up dominoes model (PUD) , and the second to the zig-zag dominoes model (ZZD), see [START_REF] Andre | [END_REF]. Each plaquette is frustrated. The double arrow indicates a ground state spin with two allowed orientations.

We first consider only the spin Hamiltonian for the dilute case without the kinetic part of equation ( 1), with a coupling configuration displayed in figure 1(a) or (b)

H F = -J m,n ǫ m,n y m,n σ m,n y m+1,n σ m+1,n + y m,n σ m,n y m,n+1 σ m,n+1 , (3)
where y m,n = 1 -n m,n = 0 when a defect (hole or fermion) is present and y m,n = 1 otherwise when the site is occupied by a spin. The total Hamiltonian including kinetic part of the fermions, see equation (1), will be studied in section 3 below and an effective action will be derived. We should notice that the spin part of the Hamiltonian (3) which contains the number operators n r does not commute with the kinetic part of the fermions in general.

Introducing the partition function Z F = σm,n=±1 e -βHF , we can rewrite the Boltzmann weights as

Z F = σm,n=±1 cosh(K) 2N m,n (1 + uǫ m,n y m,n σ m,n y m+1,n σ m+1,n )(1 + uy m,n σ m,n y m,n+1 σ m,n+1 ),
with parameters K = βJ and u = tanh(K). The general case without vacancies can be solved using Grassmann variables and was derived extensively by Plechko [14,15,16,17] in the case of various lattice geometries, and the partition function can be expressed as the Pfaffian of a quadratic form. In presence of vacancies, the Hamiltonian can be mapped onto an action with a quadratic and quartic parts in Grassmann variables representing the interaction contribution [18,[START_REF] Plechko | Fermionic path integrals and two-dimensional Ising model with quenched site disorder Path Integrals from PeV to TeV: 50 Years after Feynman[END_REF][START_REF] Plechko | [END_REF], and therefore the partition function can not be expressed by a simple Pfaffian. However we can show using perturbative methods that the quartic part leads only to a small contribution ‡, see also [11].

In order to derive an effective action for the dilute frustrated models (2), we follow in detail the method given by Plechko in a previous reference [18] and introduce two pairs of Grassmann variables (a m,n , ām,n ) and (b m,n , bm,n ) such that

Z F = σm,n=±1 cosh(K) 2N m,n dā m,n da m,n e am,nām,n (1 + a m,n y m,n σ m,n )(1 + uǫ m,n ām,n y m+1,n σ m+1,n ) × d bm,n db m,n e bm,n bm,n (1 + b m,n y m,n σ m,n )(1 + u bm,n y m,n+1 σ m,n+1 ) = σm,n=±1 cosh(K) 2N m,n
d bm,n db m,n dā m,n da m,n e am,nām,n+bm,n bm,n (A m,n Ām+1,n )(B m,n Bm,n+1 ), (4) where the link operators in both directions are given by

A m,n = 1 + a m,n y m,n σ m,n , Ām,n = 1 + uǫ m-1,n ām-1,n y m,n σ m,n B m,n = 1 + b m,n y m,n σ m,n , Bm,n = 1 + u bm,n-1 y m,n σ m,n . (5)
Moving the horizontal and vertical Grassmann factors inside the general product using mirror ordering principle, for which the computational details are not presented here, we obtain the partition function as a series of ordered products

Z F = cosh(K) 2N m,n d bm,n db m,n dā m,n da m,n e am,nām,n+bm,n bm,n × σm,n=±1 n -→ n m -→ m Ām,n Bm,n A m,n m ←- m B m,n (6)
where the arrows indicate the order of the product, right or left. This allows for the summation over each individual spin variable using for example the fact that only even powers of σ m,n do not vanish: σm,n=±1 σ 2 m,n = 2. We can then perform the exponentiation of the quadratic Grassmann factors

Z F = 2 N cosh(K) 2N m,n d bm,n db m,n dā m,n da m,n exp a m,n ām,n + b m,n bm,n (7)
× exp y m,n a m,n b m,n + u(ǫ m-1,n ām-1,n + bm,n-1 )(a m,n + b m,n ) + u 2 ǫ m-1,n ām-1,n bm,n-1 . ‡ unpublished results, work to be submitted

We then integrate over the pairs (a m,n , b m,n ) in order to reduce the number of variables, using the general identity for any linear Grassmann factors L m,n and Lm,n db m,n da m,n e ym,nam,nbm,n+am,nLm,n+bm,n Lm,n = y m,n + Lm,n L m,n .

This yields the following integral over the remaining 2N Grassmann variables (ā m,n , bm,n ) which can be relabeled using the substitutions ām,n → -a m,n and bm,n → ām,n

Z F = 2 N cosh(K) 2N m,n dā m,n da m,n exp -u 2 y m,n ǫ m-1,n a m-1,n ām,n-1 × m,n y m,n + a m,n ām,n + uy m,n (a m,n + ām,n )(ǫ m-1,n a m-1,n -ām,n-1 ) .
When a vacancy is present y m,n = 0, we have the simple result dā m,n da m,n a m,n ām,n = 1, and when y m,n = 1 we obtain

dā m,n da m,n exp -u 2 ǫ m-1,n a m-1,n ām,n-1 + a m,n ām,n + u(a m,n + ām,n )(ǫ m-1,n a m-1,n -ām,n-1 ) = dā m,n da m,n e SF m,n . (9)
The total action S F = m,n S F m,n corresponding to the undiluted spin system can be written as

S F = m,n a m,n ām,n + u(a m,n + ām,n )(ǫ m-1,n a m-1,n -ām,n-1 ) -u 2 ǫ m-1,n a m-1,n ām,n-1 . (10)
Therefore we can write the partition function as

Z F = 2 N cosh(K) 2N m,n dā m,n da m,n e SF m,n (1 -y m,n )a m,n ām,n e -SF m,n + y m,n , (11)
where

a m,n ām,n e -SF m,n = a m,n ām,n + u 2 ǫ m-1,n a m,n ām,n a m-1,n ām,n-1 . ( 12
)
Because the variables y m,n can take only values 0 or 1, the expression (11) can not in principle be exponentiated. Indeed, we first need to factorize the terms y m,n in the brackets in order to exponentiate the quadratic and quartic terms exactly, following ( 12)

y m,n 1 + 1 -y m,n y m,n a m,n ām,n e -SF m,n = y m,n exp 1 -y m,n y m,n a m,n ām,n e -SF m,n .
This presents in principle a singularity in the exponential term at y m,n = 0. But this expression is formally true since, after integrating over the Grassmann variables in the overall integral (11), the result is a polynom in y m,n and does not contain any singularity when y m,n = 0. However, we can assume that the y m,n variables can be extended to real variables that take any positive value less than unity, as we will show in the density approximation presented in the section 3, and we obtain

Z F = 2 N cosh(K) 2N m,n y m,n dā m,n da m,n × exp S F m,n + v m,n a m,n ām,n + v m,n u 2 ǫ m-1,n a m,n ām,n a m-1,n ām,n-1 , (13)
where v m,n = (1 -y m,n )/y m,n . The variables v m,n can be viewed as local coupling parameters between the nondilute model and the vacancies whose contribution is represented by a quadratic and quartic forms. In general, we assume that the contribution of the quartic form in ( 13) is small, since it contains in the continuous limit a double space derivative whose amplitude should be small for low momenta.

Partition function for the non-dilute case

We consider for simplification the PUD case where ǫ m,n = (-1) n , see figure 1(a), and use the Fourier decomposition a r = L -1 k a k e ik.r , ār = L -1 k āk e -ik.r for r = (m, n) and k = (k x , k y ) = (2πp/L, 2πq/L). Using the identity L-1 n=0 e 2iπqn/L (-1) n = Lδ q,±L/2 , we obtain after some algebra

S F = kx,ky
a kx,ky ākx,ky + u a kx,ky a -kx,-ky±π e ikx + ākx,ky a kx,ky±π e -ikx -u a kx,ky ākx,ky e iky + ākx,ky ā-kx,-ky e -iky + u 2 a kx,ky ākx,ky±π e i(ky-kx) . (

) 14
We can decompose the momentum zone by considering the reduced domain -π ≤ k x ≤ π and 0 ≤ k y ≤ π/2, see figure 2(a), of independent momenta. This implies that only the following set of 4 pairs of independent Grassmann variables is necessary to decompose the partition function into independent blocks

a k = a kx,ky , a -k = a -kx,-ky , b k = a kx,ky-π , b -k = a -kx,-ky+π , āk = ākx,ky , ā-k = ā-kx,-ky , bk = ākx,ky-π , b-k = ā-kx,-ky+π . ( 15
)
In the reduced grey zone of figure 2(a), and represented by a prime symbol on the sum over momenta, we obtain the following decomposition of the action into independent blocks of 8 variables, after regrouping the different terms and further simplifications

S F = kx,ky ′ a k āk (1 -ue iky ) + a -k ā-k (1 -ue -iky ) + b k bk (1 + ue iky ) + b -k b-k (1 + ue -iky ) + 2iu sin k y āk ā-k -2iu sin k y bk b-k + a k 2iu sin k x b -k -bk ue -ikx (1 -ue iky ) + a -k -2iu sin k x b k -b-k ue ikx (1 -ue -iky ) + āk b k ue -ikx (1 + ue iky ) + ā-k b -k ue ikx (1 + ue -iky ). ( 16
)
The total partition function is therefore equal to the following set of integrals, where the L × L/2 vectors k are located inside the reduced zone only

Z F = 2 N cosh(K) 2N k ′ dā k da k dā -k da -k d bk db k d b-k db -k e SF . ( 17
)
We obtain after integration and simplifications

Z F = 2 N cosh(K) 2N (1 -u 2 ) N/2 kx,ky ′ (1 + u 2 ) 2 -2u 2 (cos(2k x ) + cos(2k y )) . (18)
This result can be further simplified by noting that 1 -u 2 = cosh -2 (K), and we finally obtain a simple product

Z F = 2 N kx,ky ′ 1 + sinh 2 (2K) sin 2 k x + sin 2 k y . (19)
In the thermodynamical limit, the free energy per site is then equal to

-βF/N = ln 2 + 1 π 2 π/2 0 dk x π/2 0 dk y ln 1 + sinh 2 (2K) sin 2 k x + sin 2 k y , (20)
which corresponds to the result (6) of reference [START_REF] Andre | [END_REF]. The entropy is given by S = -∂F/∂T and the specific heat by C v = T ∂S/∂T . In the limits T → 0 and L → ∞, a series expansion gives the entropy per site lim

T →0 lim N →∞ S N ≃ 1 π 2 π/2 0 dk x π/2 0 dk y ln 4 sin 2 k x + sin 2 k y = G π ≃ 0.291 561,
where G is the Catalan constant. The entropy is non zero as the ground state is macroscopically degenerate. In figure 3 we have plotted the specific heat and entropy per site. The specific heat presents no phase transition but a Schottky peak around T ≃ J as expected for a gaped system. We can notice that for the ZZD model, equivalent results are found. We obtain the same expression for the free energy as equation [START_REF] Plechko | [END_REF] except that the terms sin 2 k x are replaced by cos 2 k x . This does not change the overall expression of the free energy since we can always perform the change of variables k x → π/2 -k x within the domain of allowed momenta. In addition, since the function ǫ m,n of the ZZD model depends on both m and n, the subsequent calculations imply that the momenta are restricted in the grey area of the Brillouin zone represented in figure 2(b), with the introduction of 16 variables for each block. However each block can be divided into two independent blocks of 8 variables and therefore the calculations are similar to the PUD case and can be simplified accordingly.

Coupling with fermions

We would like to construct the total partition function Z = Tr e -βH ≃ e S eff , where the effective action S eff includes the Ising part and the electron kinetic term in (1). This can be reformulated as a Grassmannian functional integral by substituting the Fermi operators with dimensionless Grassmann variables c r → c r (τ ) and c † r → cr (τ ), see for example the technical details in references [START_REF] Altland | Condensed Matter Field Theory 2[END_REF][START_REF] Lichtenstein | Path integrals and dual fermions The Physics of Correlated Insulators[END_REF], where τ is the imaginary time variable taking values between 0 and β

Z = 2 N cosh(K) 2N r dā r da r db r r,τ dc r (τ )dc r (τ ) exp S F + r b r 1 2 -β -1 β 0 dτ cr (τ )c r (τ ) + r b r 1 2 + β -1 β 0 dτ cr (τ )c r (τ ) a r ār + r β 0 dτ -c r (τ )∂ τ c r (τ ) + t δ cr+δ (τ )c r (τ ) . ( 21
)
The additional nilpotent variables b r in the integral are introduced in order to impose the hole constraints inside equation (11). Instead of a nilpotent variable, a pair of Grassmann variables could also have been introduced as well. The following prescription was considered whenever the product of two Fermi operators on the same site are encountered:

c † r c r = (c † r c r -c r c † r + 1)/2 → cr (τ )c r (τ ) + 1/2.
Moreover, we assume that only the quadratic terms a r ār in equation [START_REF] Lichtenstein | Path integrals and dual fermions The Physics of Correlated Insulators[END_REF] contribute to the interaction between the spins and fermions as discussed previously. This will simplify the calculations and the integrals are reduced to Pfaffians or determinants. In the following section we study the extrema of this action using Lagrange multipliers in the mean-field approximation. From the action S eff defined in equation ( 21), we impose antisymmetric boundary conditions c r (0) = -c r (β) and cr (0) = -c r (β). These operators can be expressed as a Fourier series using Matsubara frequencies

ω n = (2n + 1)π/β, n = 0, ±1, ±2, • • •, such that c r (τ ) = β -1/2 ωn e -iωnτ c r (ω n ), cr (τ ) = β -1/2 ωn e iωnτ cr (ω n ). (22)
To simplify the problem, we observe that the functional (21) depends on two types of fluctuating operators, coming from the representation of the spins and fermions, with a coupling between them that depends on the local hole constraints. The spin fluctuations are described by the quadratic part with the pairs (a r , ār ), and the dynamics of the fermions also by a quadratic part with the time dependent variables (c r (τ ), cr (τ )). Integrating over (a r , ār ) and the coupling operators b r introduces nonlocal interactions between the fermions and is not treatable exactly because of the strong vacancy constraints since the density operators in factor of b r have only zero or one as possible values. This strong condition can be lifted if we consider local mean-field density for the fermion sector. In that case, the singularities coming from the b r couplings are smoothed out and self-consistent local mean-field equations can be expressed in function of the local densities and self-energies of the fermions. Under these assumptions, we replace the operators cr (ω n )c r (ω n ) by a Green function G r (ω n ) with conjugate variable Σ r (ω n ), or self-energy, which serves as a Lagrange multiplier. After integration over the b r variables, the total effective action, including the spin and fermion parts with their interaction is finally given by a quadratic Grassmannian depending on (a r , ār ) and (c r (ω n ), cr (ω n ))

S eff = N ln 2 + 2N ln cosh(K) + S F + r ln 1 2 -β -1 ωn G r (ω n ) + r 1 + 2β -1 ωn G r (ω n ) 1 -2β -1 ωn G r (ω n ) a r ār + r,ωn Σ r (ω n ) G r (ω n ) -cr (ω n )c r (ω n ) + r ωn (iω n + µ)c r (ω n )c r (ω n ) + t δ cr+δ (ω n )c r (ω n ) . (23)
In this expression, we have introduced a chemical potential µ. The second term in the second line corresponds to the local constraints imposed by the Lagrange multipliers. The effective thermodynamical potential -βΩ = ln( e S eff ) after integration over the Grassmann variables is given by

-βΩ = N ln 2 + 2N ln cosh(K) + r ln 1 2 -β -1 ωn G r (ω n ) + r,ωn Σ r (ω n )G r (ω n ) + 1 2 ln det A F + V r,r ′ + ωn ln det [iω n + µ -Σ r (ω n )]δ r,r ′ + t δ δ r ′ ,r+δ r,r ′ , (24)
where A F is the connectivity matrix corresponding to the spin sector in the real space, see equation (10):

S F = 1 2 r,r ′ Ψ ⊤ r (A F ) r,r ′ Ψ r ′ ,
in the basis of vectors Ψ r = (a r , ār ) ⊤ , and V the operator corresponding to the hole coupling at the quadratic approximation

(V ) r,r ′ = 1 + 2β -1 ωn G r (ω n ) 1 -2β -1 ωn G r (ω n ) iσ 2 δ r,r ′ = v r iσ 2 δ r,r ′ , v r = n r 1 -n r , n r = 1 2 + β -1 ωn G r (ω n ). (25)

Homogeneous solution

We consider the mean field approach of the effective action [START_REF] Asgari | [END_REF] with an homogeneous solution for the Green function and self energy:

G r (ω n ) = G(ω n ), Σ r (ω n ) = Σ. This implies that v r = v = n e /(1 -n e )
where n e = v/(1 + v) is the fermion average density. By adding the action S F (16) and the quadratic interaction part of equation ( 13), after discarding the quartic terms, we find that all the contributions depending on (a r , ār ) are included in the following action

S F +int = S F + v m,n a m,n ām,n -N ln(1 + v). (26)
We can express the interaction part using Fourier modes in the reduced Brillouin zone of figure 2

(a) v m,n a m,n ām,n = v kx,ky ′ a k āk + a -k ā-k + b k bk + b -k b-k , (27)
and we obtain after integration Z F (v) = 2 N cosh(K) 2N e SF+int , where

Z F (v) = 2 N cosh(K) 2N (1 + v) N kx,ky ′ (1 -u 4 ) 2 + (1 + v) 4 -4vu 4 -1 (28) + 2u 4 v 2 cos 2(k y -k x ) -2u 2 (1 + v -u 2 ) 2 cos 2k x + cos 2k y .
The Grand Potential given by ( 24) can be written in the mean-field approximation as

- β N Ω = ln 2 + 2 ln cosh K + ln(1 -n e ) + n e - 1 2 βΣ + 1 N k ln cosh β 2 (µ -Σ -ǫ k ) + 1 N k ′ ln (1 -u 4 ) 2 + (1 + v) 4 -4vu 4 -1 (29) +2u 4 v 2 cos 2(k y -k x ) -2u 2 (1 + v -u 2 ) 2 cos 2k x + cos 2k y .
The self-consistent equations are obtained by extremization of ( 24) with respect to the Green functions G(ω n ) and self energy Σ. This yields the two equations

βΣ(µ) = 1 1 -n e - F (v) (1 -n e ) 2 , (30)
n e = 1 2 + β -1 ωn G(ω n ), G(ω n ) = 1 N k 1 iω n + µ -Σ -ǫ k . (31)
In the last expression, we have redefined the kinetic energy of the fermions by ǫ k = -t(cos k x + cos k y -2) by shifting µ. Function F (v) is given by Function F (v) is plotted in figure 4 for different temperatures. Physically, F (v) is defined for v ≥ 0, whereas for v < 0 (negative densities) the function presents different singularities which are not real. However we have represented these singularities for an illustration purpose only, as the mathematical domain of F (v) is not restricted to positive values. F (v) can be viewed as a local chemical potential due to the spin environment and proportional to the fermion self-energy. This quantity is also related to the energy to insert a hole inside the spin bath. In the limit of low density of fermions v ≃ 0, we have indeed Σ(µ) ≃ k B T (1 -F (0)) which is not zero and is temperature dependent. When T = 0, a singularity appears at v = 0 and has the shape of a cusp, see the green curve for T close to zero (u = 0.9). This corresponds to the phase transition of the non-dilute frustrated model at T = 0. We estimate the effective mass from the derivative of Σ with respect to the chemical potential using the definition m * = 1 -∂Σ/∂µ [START_REF] Mahan | Many-particle physics 3rd ed Physics of solids and liquids[END_REF][START_REF] Asgari | [END_REF]. Σ ′ (µ) can be computed and expressed with F (v) and its derivative

F (v) = 4 N k ′ (1 + v) 3 -u 4 + u 4 v cos 2(k y -k x ) -u 2 (1 + v -u 2 ) cos 2k x + cos 2k y (1 -u 4 ) 2 + (1 + v) 4 -4vu 4 -1 + 2u 4 v 2 cos 2(k y -k x ) -2u 2 (1 + v -u 2 ) 2 cos 2k x + cos 2k y . (32)
F ′ (v) = ∂ v F (v) β ∂Σ ∂µ = n ′ e (1 -n e ) 4 (1 -n e ) 2 -2(1 -n e )F (v) -F ′ (v) = n ′ e Λ, (33)
with the density of states n ′ e = ∂ µ n e defined by

n ′ e = 1 - ∂Σ ∂µ β 4N k 1 -tanh 2 β 2 (µ -Σ -ǫ k ) = 1 - ∂Σ ∂µ β ñ′ e 4 , (34)
where 0 < ñ′ e < 1 for any temperature. It is known that the density of states is proportional to the effective mass. We can finally express the effective mass as m * = (1 + ñ′ e Λ/4) -1 . In the high temperature regime, we can perform a series expansion for u ≪ 1 of function F (v)

F (v) ≃ 1 1 + v + 3u 4 (1 + v) 4 , (35)
and obtain an estimate of the self-energy, Σ ≃ -3(1 -n e ) 2 J 3 /(k B T ) 3 < 0, from which

m * ≃ 1 - 3 2 u 4 (1 -n e ) < 1. ( 36
)
For the non-frustrated Ising model, ǫ r = 1, it is worth noting that we obtained instead the opposite sign for this approximation in the same limit m * ≃ 1 + 3 2 u 4 (1 -n e ) > 1 [10].

Solution with stripes

The previous solution does not incorporate the symmetry of the spin couplings. We expect that the charge density will follow a modular structure similar to the local couplings Jǫ r . We consider therefore the set of trial mean-field solutions with alternating values, n m,n = n 1 if n is even and n m,n = n 2 otherwise when n is odd

n m,n = 1 2 n 1 (1 + (-1) n ) + 1 2 n 2 (1 -(-1) n ). ( 37
)
We also define the quantities v 1 = n 1 /(1 -n 1 ) and v 2 = n 2 /(1 -n 2 ). The quadratic interaction part of the action ( 26) can be decomposed as

m,n v m,n a m,n ām,n = v 1 + v 2 2 kx,ky ′ a k āk + a -k ā-k + b k bk + b -k b-k + v 1 -v 2 2 kx,ky ′ a k bk + a -k b-k + b k āk + b -k ā-k . ( 38
)
After including these terms to S F , we obtain the following partition function for the frustrated Ising part that includes the fermion coupling

Z F (v 1 , v 2 ) = 2 N cosh(K) 2N (1 + v 1 ) N/2 (1 + v 2 ) N/2 k ′ (1 -u 4 ) 2 + (1 + v 1 ) 2 (1 + v 2 ) 2 -2(v 1 + v 2 )u 4 -1 + u 2 (v 1 -v 2 ) 2 + 2u 4 v 1 v 2 cos 2 (k y -k x ) -2u 2 (1 + v 1 -u 2 )(1 + v 2 -u 2 ) cos 2k x + cos 2k y -2u(v 1 -v 2 )(u 2 -(1 + v 1 )(1 + v 2 )) cos (k x ) + 2u 3 (u 2 -1)(v 1 -v 2 ) cos (k x -2k y ) = 2 N cosh(K) 2N (1 + v 1 ) N/2 (1 + v 2 ) N/2 Q F (v 1 , v 2 ). ( 39
)
When v 1 = v 2 = v, the last two terms in the product vanish and we recover equation (28

), Z F (v, v) = Z F (v).
The space asymmetry is revealed by the terms proportional to (v 1 -v 2 ). The local self-energies and Green functions are expressed similarly

Σ r (ω n ) = 1 2 Σ 1 (ω n )(1 + (-1) n ) + 1 2 Σ 2 (ω n )(1 -(-1) n ) = S 1 (ω n ) + (-1) n S 2 (ω n ), G r (ω n ) = 1 2 G 1 (ω n )(1 + (-1) n ) + 1 2 G 2 (ω n )(1 -(-1) n ), (40)
and the determinant ∆ ωn (Σ 1 , Σ 2 ) of the fermionic kinetic part defined by

∆ ωn (Σ 1 , Σ 2 ) = det [iω n + µ -Σ r (ω n )]δ r,r ′ + t δ δ r ′ ,r+δ r,r ′ , (41)
can be evaluated by considering the eigenvalues λ of the corresponding matrix in the Fourier space. If we note ϕ k the eigenvectors of λ, we have the set of secular equations

iω n + µ -S 1 -ǫ k -λ ϕ k -S 2 φk = 0, S 2 ϕ k -iω n + µ -S 1 -ǭk -λ φk = 0, (42)
where φk = ϕ k=(kx,ky±π) , and similarly for ǭk = ǫ k=(kx,ky±π) . This leads to the solutions

λ = λ ± k (ω n ) = iω n + µ -S 1 - 1 2 (ǫ k + ǭk ) ± 1 2 (ǫ k -ǭk ) 2 + 4S 2 2 , (43)
and the determinant ∆ ωn is directly equal to

∆ ωn (Σ 1 , Σ 2 ) = k ′ λ + k (ω n )λ - k (ω n ) 2 , (44)
where the prime symbol corresponds as before to the sum reduced to the Fourier modes of the Brillouin zone of figure 2(a), and the square takes into account the remaining modes that correspond to negative values of k y . The Grand potential can then be expressed from equation ( 24) by replacing the different parameters with the trial solution. Thus

- βΩ N = ln 2 + 2 ln cosh(K) + 1 2 ln(1 -n 1 ) + 1 2 ln(1 -n 2 ) + 1 2 ωn Σ 1 (ω n )G 1 (ω n ) + 1 2 ωn Σ 2 (ω n )G 2 (ω n ) + 1 N ln Q F (v 1 , v 2 ) + ωn 2 N k ′ ln λ + k (ω n )λ - k (ω n ) . (45)
From this expression, we obtain the mean-field solutions by considering the different derivatives with respect to the quantities

{Σ 1 (ω n ), G 1 (ω n ), Σ 2 (ω n ), G 2 (ω n )}
, which leads to the following expressions for the pair of Green functions

G 1 (ω n ) = 2 N k ′ σ=±1 1 λ σ k (ω n )   1 - 2σS 2 (ǫ k -ǭk ) 2 + 4S 2 2   , G 2 (ω n ) = 2 N k ′ σ=±1 1 λ σ k (ω n )   1 + 2σS 2 (ǫ k -ǭk ) 2 + 4S 2 2   . (46)
The self-energies satisfy moreover the equations

0 0.5 1 1.5 2 T 0 0.2 0.4 0.6 0.8 1 effective mass m * 1 , n e =0.3 m * 2
, n e =0.

, n e =0.

βΣ 1 (ω n ) = 1 1 -n 1 - 2 (1 -n 1 ) 2 ∂ ∂v 1 ln Q F (v 1 , v 2 ), βΣ 2 (ω n ) = 1 1 -n 2 - 2 (1 -n 2 ) 2 ∂ ∂v 2 ln Q F (v 1 , v 2 ), (47)
with the density average relation n 1 + n 2 = 2n e . They depend on the Green functions by the identities

n 1 = 1 2 + β -1 ωn G 1 (ω n ), n 2 = 1 2 + β -1 ωn G 2 (ω n ). (48)
The previous homogeneous case n 1 = n 2 = n e is a particular solution of this set of equations. The associated effective masses m i * = 1 -∂ µ Σ i are evaluated analytically although the expressions are cumbersome. They can ). The dotted line is the specific heat for the spins of the non-dilute PUD system, and the dashed line is the sum of the previous specific heat with the specific heat of free fermions at µ = 0, which corresponds to low density of particles. At high temperature it coincides with the correlated system. In insets are represented the relative difference of the densities, and the ratio C v /T . This ratio is diverging at low temperature for low density or low chemical potential. be derived directly from the two following equalities involving the densities of states n i ′ = ∂ µ n i , the derivatives

F i = ∂ vi log Q F (v 1 , v 2 ), and F i,j = ∂ vi ∂ vj log Q F (v 1 , v 2 ) β ∂Σ 1 ∂µ = n 1 ′ (1 -n 1 ) 4 (1 -n 1 ) 2 -4(1 -n 1 )F 1 -2F 1,1 - 2n 2 ′ (1 -n 1 ) 2 (1 -n 2 ) 2 F 2,1 , β ∂Σ 2 ∂µ = n 2 ′ (1 -n 2 ) 4 (1 -n 2 ) 2 -4(1 -n 2 )F 2 -2F 2,2 - 2n 1 ′ (1 -n 1 ) 2 (1 -n 2 ) 2 F 1,2 . (49)
The homogeneous solution (30) is recovered when

Σ 1 = Σ 2 = Σ by noting that F ′ (v) = F 1,1 + F 1,2 + F 2,1 + F 2,2 .
We have plotted in figure 5 the effective masses m 1 * and m 2 * as function of the average density n e . Their value is always less that unity, and decreases to a minimum at low temperature. The minimum is more accentuated at low density. In the same figure we have also plotted in dashed lines the homogeneous solution m * , see equation (33). We observe in all the temperature regime that m * 1 > m * 2 > m * and that Λ > 0. For a given density n e , we define the free energy using the relation F = Ω+N µ(n e -1/2). In figure 6 we have plotted the specific heat C v = -T ∂ 2 Ω/∂T 2 for different chemical potentials, and in figure 7 the specific heat in the canonical ensemble

C v = -T ∂ 2 F/∂T 2 .
The relative density (n 2 -n 1 )/(n 2 + n 1 ) shows that the fermions predominantly align along the frustrated coupling lines, depending on the initial conditions. If we consider the ratio C v /T at low temperature, which, in the theory of Fermi liquids, tends to a constant proportional to the effective mass [24], it appears that this coefficient decreases with density in the canonical ensemble, see inset of figure 7. This is consistent with an effective mass decreasing with the density of fermions. However, in inset of figure 6, the ratio C v /T at low densities or for low chemical potential seems to diverge in the grand canonical ensemble, which is inconsistent with a finite entropy and probably due to the mean-field approximation. This behavior can be seen experimentally in the ground state of some spin-ice systems such as DyTi 2 O 7 [25], where a non-vanishing specific heat at zero temperature was observed, suggesting a spin-glass transition due to the increase of the spin relaxation time and macroscopic degeneracy of the ground state. This problem is not solved in our mean-field approximation, and requires a more precise analysis. However at higher temperature and above the Schottky peak, the decoupling between spins and fermions is evidenced in figure 6 by the dashed line which corresponds to the sum of the two independent specific heats of the non-dilute spin system and free fermions. The magnitude of the Schottky peak seems to suggest a strong coupling effect between the spins and fermions in low temperature regime near T ≃ J.

Numerical results with a classical model

In this section we present a Monte-Carlo algorithm based on the diffusion of classical particles on the frustrated PUD lattice. The particles in the model (1) are considered as classical in order to simplify the numerical difficulties inherent to Quantum Monte Carlo algorithms. We expect however qualitative results that can be applied to the quantum case. Monte-Carlo simulations on classical particle-spin models were performed [6] to mimic the physics of "telephone-number" or ladder compounds such as cuprates (La,Sr,Ca) 14 Cu 24 O 41+δ [26]. Their idea is to construct a spin-1 Hamiltonian with, in particular, ferromagnetic couplings between spins σ r = ±1 on the horizontal lines and antiferromagnetic couplings on the vertical lines. The holes or defects are represented by σ r = 0, and the two horizontal spins surrounding the hole are connected by one antiferromagnetic coupling. The phase diagram displays vertical lines of holes as they tends to form stripes at low temperature. Although the model does not incorporate frustration, the appearance of stripes seems to depend strongly on the experimental coupling values and the choice of their configurations on the lattice. The corresponding classical Hamiltonian includes quadratic spin-spin couplings as well as a quartic term that represents the presence of the itinerant holes. The algorithm presented in this section is performed at fixed particle density. Each move is described by the following procedure: Each site is occupied by a spin or hole, and at every update a site is randomly chosen. If the site is occupied by a spin, its direction is reversed and the new configuration is updated depending on the acceptation rate. If the site is occupied by a hole, it is exchanged randomly with one of the four neighboring spins (or possibly holes), mimicking the kinetic transfer but without energy scale t. The simulations consists in 10000 × N updates or 10000 MC steps at different temperatures and concentrations of particles. In figure 8 are plotted two final configurations at n e = 0.4 and for two different temperatures. At low temperature, the holes tend to align along the ferromagnetic couplings and they form an alternating structure of stripes. For higher temperatures, this alignment breaks into smaller linear hole structures. The system seems to minimize the frustration by forming linear arrangements along the ferromagnetic couplings. We should also notice that positioning the holes along the antiferromagnetic couplings should also be a solution as this should lower the energy as well, and this solution appears indeed to depend on the initial conditions for the set of self-consistent equations. In order to probe the stripe formation and the coherence of these structures, we have plotted in figure 9 horizontal and vertical correlators C right and C up which are defined as follow: We count the number of sets of two fermions that are aligned horizontally or vertically respectively, which provides an insight of the stripe structure. We observe in this figure that indeed the horizontal correlations are larger than the vertical ones for temperatures approximately below the coupling value J, which is the threshold for the formation of stripes. We also note that, for low densities, these observables are very fluctuating as the probability to find two fermions close to each other is smaller than for denser configurations.

Conclusion

Properties of itinerant fermions in a frustrated dilute spin medium with indirect interaction induced by the doping were analyzed in order to probe the effect of spin fluctuations and frustration. The fermions are found to organize themselves into lines or stripes along the antiferromagnetic or ferromagnetic lines in order to reduce the frustration. These lines arise because of the geometrical constraint imposed by the PUD couplings which are the lines along which frustration can be lowered by distribute the holes. In the ZZD model, these holes would self-organize differently. The effective mass in the mean-field approximation decreases in the regime of low temperature and low density of fermions. This is consistent with the limiting value of C v /T in the canonical ensemble at low temperature. In the high temperature regime, the effective mass is less than unity, contrary to the non-frustrated Ising case. This suggests that spin fluctuations decrease the effective mass when the system possesses some degree of frustration. The presence of low temperature anomalies at low density in the grand canonical specific heat could be explained by a possible spin-glass solution of the ground state which is not reported by our approximation, or an artifact of this approximation. Such anomalies were however previously observed in the specific heat of spin-ice materials [25], which could point to a complex ground state. Our model can be generalized with negative couplings different from -J, which induces either a ferromagnetic or antiferromagnetic phase transition. It is worth considering in this case the influence of the proximity of a phase transition on the effective mass at low temperature. A more precise theoretical analysis would effectively lead to more insight, using either diagrammatic methods or dynamical mean field theory. The latter method may be implemented starting from the original Hamiltonian which appears to be similar to a model of spinless fermions with repulsive or attractive next-nearest interactions.

  (a) Piled up dominoes model (b) Zig-zag model

Figure 2 :

 2 Figure 2: Brillouin zone restricted to the domain (k x , k y ) (grey area) such that -π ≤ k x ≤ π and 0 ≤ k y ≤ π/2 for the PUD model, and 0 ≤ k x ≤ π and 0 ≤ k y ≤ π/2 for the ZZD model. When the vector k sweeps over the grey area, the other vectors fill the complementary Brillouin zone.

Figure 3 :

 3 Figure 3: Entropy per site and specific heat as function of temperature for the PUD model (L = 512). The thermodynamical functions are also valid for the ZZD model since the expression of the free energy is equivalent, see text.

Figure 4 :

 4 Figure 4: Function F (v) for different temperatures (L = 64). The physical domain corresponds to v > 0. All the singularities are located in the domain v ≤ 0.

Figure 5 :masses m 1 * and m 2 *

 512 Figure 5: Effective masses m 1 * and m 2 * as function of the temperature for different densities n e (L = 150). The dotted lines represent the solution m * in the homogeneous case.

1 )Figure 6 :

 16 Figure6: Specific heat as function of the temperature for different chemical potentials (L = 200). The dotted line is the specific heat for the spins of the non-dilute PUD system, and the dashed line is the sum of the previous specific heat with the specific heat of free fermions at µ = 0, which corresponds to low density of particles. At high temperature it coincides with the correlated system. In insets are represented the relative difference of the densities, and the ratio C v /T . This ratio is diverging at low temperature for low density or low chemical potential.

1 )Figure 7 :

 17 Figure 7: Specific heat in the canonical ensemble as function of the temperature for different fermion densities (L = 200). In insets are represented the relative difference of the densities, and the ratio C v /T . This ratio tends to a constant at low temperature which decreases with density.

  (a) Low temperature configuration (b) High temperature configuration

Figure 8 :

 8 Figure 8: Two MC final configurations: T = 0.1 (a) and T = 0.36 (b), for N = 100 sites and hole density n e = 0.4. The spins up or down are represented by blue and green arrows respectively. The red disks represent the vacancies.

Figure 9 :

 9 Figure 9: Horizontal and vertical correlation funtions at densities n e = 0.1 and n e = 0.4. See text for the definition of the correlators.

Acknowledgments

The authors would like to thank Christophe Chatelain for useful discussions.


Dr Mathieu Ciancone

Dr Nathalie Bellec

Dr Franck Camerel
email: franck.camerel@univ-rennes1.fr

I PT : An Index to Rank Molecular Photothermal Agents

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In the last two decades, the photothermal effect has aroused strong interest to develop light-responsive materials such as actuators, [1] drug delivery system for chemo-photothermal combination cancer therapy [2] as well as smart materials for water purification [3] or contrast agents for bioimaging. [4,5] Many photothermal agents have been used spanning from inorganic nanoparticles to conducting polymers and molecular entities. [6,7,8,9] The effectiveness of the materials strongly depends on their ability to convert light into heat. However, until now, no real figure of merit has been properly established to account for the photothermal performance of a given compound, especially for well-defined molecular entities. Most of the time, the temperature increase (T being the difference between the steady-state final temperature reached after laser irradiation and the starting temperature before irradiation) is given to attest of the photothermal performance of a compound. [9] Of course, a good photothermal agent should lead to higher temperature increase. However, the temperature increase doesn't represent a reliable data since it is heavily impacted by parameters intrinsic of the experimental setup such as the concentration of the photothermal agents, the laser power density, the irradiation time, the nature of the solvent, the irradiation wavelength… Consequently, comparison of the photothermal agents according to ΔT would require the measurements to be made using the exact same experimental conditions. In 2007, Roper et al. proposed a system of equations allowing to calculate the photothermal efficiency (), which reflects the ability of a compound to convert photons into heat, in a similar way that luminescence quantum yield are calculated. [10] Despite being a big step toward a better characterization of photothermal performances, it appears that the photothermal efficiency is not sufficient for comparing the relative photothermal performances of a list of agents. In fact, some compounds (including some in the present paper) have shown that a high photothermal efficiency when coupled with a low absorption at the irradiation wavelength results in a weak photothermal effect. On the contrary, a compound showing a weak photothermal efficiency but a strong absorption at the laser irradiation wavelength can lead to a temperature increase far more important. Thus, definition of an intrinsic data that could properly account for the photothermal strength of a compound is still needed. Herein, we demonstrate through the study of seven molecular photothermal agents that a new photothermal index IPTmax, referring to the brilliance used to evaluate luminescence, can be used in order to compare molecular photothermal compounds reliably while limiting experimental constraints. Scheme 1. Chemical structure, molecular weight and abbreviation of the investigated compounds (the same colour coding will be used all along the manuscript).

The seven molecules selected for this study are presented in Scheme 1. Two dithiolene complexes formed by complexation of two diphenylethylenedithiolate ligands bearing eight C12 carbon chains with a nickel (Ni8C12) or a platinum (Pt8C12) metal center as well as two nickel-trithione derivatives carrying C12 (Ni-timdtC12) or C16 (Ni-timdtC16) carbon chains have been selected for their well-known photothermal properties in the NIR region. [11] A copper derivative (Cu-ISQ) [12] and a sulfurated perylene derivative (PerDIS) [13] absorbing in the NIR region have also been investigated as potential photothermal agent. These compounds have been synthesized and purified as previously described (see SI). Finally, the commercially available prototypical photothermal agent, IndoCyanine Green (ICG), was introduced in this study. [14] Equimolar solution of each compound (10 M) have been prepared in toluene with 1 %v/v of CHCl3 (or EtOH for ICG). The absorption spectra of each solution measured between 310 and 1300 nm are presented in Figure 1. The absorption maxima and the molar absorption coefficients are gathered in table S1. All the studied compounds show a single broad absorption band with a maximum in the NIR-I therapeutic window, commonly considered as ranging from 650 to 1000 nm, except for the two Ni-timdt complexes for which the absorption maxima are located at the start in the NIR-II window, around 1010 nm. [15,16] The Cu-ISQ complex absorbs over a large wavelength range from 600 to 1000 nm but its molar absorptivity remains low (5300 M -1 .cm -1 at 792 nm). The perylene derivative (PerDIS) also has a wide absorption band from 700 nm to 1000 nm, with a maximum molar absorption coefficient of around 18000 M -1 .cm -1 . Among the absorbers studied, the dithiolene complexes (Pt8C12, Ni8C12, Ni-timdtC12 and Ni-timdtC16) as well as the ICG have the highest absorbance, with molar extinction coefficients between 40000 and 65000 M -1 .cm -1 . The temperature increase of the solutions under laser irradiation and the photothermal efficiency of the compounds were measured using two different lasers emitting at 808 (±3) and 940 (± 10) nm (3 W.cm -2 , 10 min ON, 10 min OFF, spot size 0.79 cm 2 ). The solutions were irradiated during 10 min which is a good compromise to reach a quasi-steady state and avoid evaporation of the toluene. All the temperature profiles are presented in Figures S1 andS2 and the temperature increases observed at 808 and 940 nm are compiled in Figure S3. All the parameters associated to the photothermal activity of the compounds are summarized in Table 1 and2. The dithiolene complexes and the PerDIS compound were found stable under laser irradiation, while ICG and Cu-ISQ showed some pronounced photodegradation (Figure S4).

All the solutions showed temperature increases higher than the one measured when irradiating pure toluene, testifying to the photothermal activity of each compounds. As it can be seen in Figure 1, ICG, Cu-ISQ and PerDIS solutions mainly absorb at 808 nm while Ni8C12, Ni-timdtC12 and Ni-timdtC16 solutions mainly absorb at 940 nm. Only the Pt8C12 solution shows good absorption at both wavelengths. As it may be expected, the solutions showing the greatest temperature increase ΔT are those for which the absorbance at the irradiation wavelength is the highest. Thus, no significant temperature increase was observed for the Ni-timdtC12 or the Ni-timdtC16 complexes irradiated at 808 nm, their absorbance at 808 nm being almost zero. No marked temperature increase has also been measured for ICG, PerDIS and Cu-ISQ solutions irradiated at 940 nm. As expected, in regard to the molar absorption coefficient at each wavelength, the temperature increase for Pt8C12 solution is higher at 940 nm (ΔT = 39 °C) than at 808 nm (ΔT = 29 °C). However, it should also be noticed that the temperature increase obtained is not directly proportional to the absorbance of the compound. Indeed, at 940 nm, the ΔT obtained for Ni8C12 is 45 °C while it is only 39 ° C for the Pt8C12 complex, although the absorbance of the Ni8C12 solution at 940 nm (ε940 = 32500 M -1 .cm -1 ) is lower than that of the Pt8C12 solution (ε940 = 41450 M -1 .cm -1 ). Likewise, Ni-timdtC12 displays a ΔT of 36 °C against 27.5 °C for Ni-timdtC16 while their absorbance at 940 nm are very close (16600 M -1 .cm -1 and 13600 M -1 .cm -1 , respectively). Figure 2a clearly shows, that despite a slight linear tendency, there is no real correlation between the value of the molar absorption coefficient measured at the irradiation wavelength and the temperature increase recorded. Table 1. Absorbance, temperature increase, photothermal efficiency and heat-transfer coefficient (h) measured for each solution irradiated at 808 nm.

Table 2. Absorbance, temperature increase, photothermal efficiency and heat-transfer coefficient (h) measured for each solution irradiated at 940 nm. Some values do not appear in the table because the measured temperature increase and / or absorbance were too low to obtain a reliable result.

(*) photodegradation of the compound leads to a slight underestimation of photothermal efficiency.

The photothermal efficiency of the compounds was also determined by recording the complete temperature profile during the irradiation and after turning off the laser power. The photothermal efficiency was calculated using the equation developed by Roper et al. (see SI). [10] The photothermal efficiency values obtained for each compound, irradiated at 808 and 940 nm, are listed in Tables 1 and2, respectively. Since the heat dissipation coefficient h is an intrinsic characteristic of the experimental setup, its value varies little from one measurement to another (Tables 1 and2). The Ni-timdtC12 and Ni-timdtC16 complexes display a photothermal efficiency of around 60 % at 940 nm. This value is higher than the best energy conversion yields found in the literature for organic structures, such as the recently developed terrylenediimide (41 %), [17] or the frequently used polypyrrole nanoparticles (35 %). [18] To our knowledge, these are the highest values reported for molecular photothermal agents, placing them at the level of the most efficient inorganic agents in NIR, such as the tungsten oxides W18O49 (59 % at 808 nm), [19] gold nanorods (21-50 % at 808 nm), [20] germanium nanocrystals (20-47 % at 770 nm), [21] or Ti3C2 nanosheets (MXene) (30 % at 808 nm). [22] The substitution of the nickel metal center in Ni8C12 by platinum metal center leads to a decrease of about 15 % in the photothermal efficiency of the complex. The PerDIS compound displays conversion yield comparable to the one of Ni8C12 complex, i.e. around 40 %. This value is consistent with what was found by the team of Zhang et al. in their work on the formulation of nanoparticles by self-assembly of an amphiphilic perylene diimide analog (42 %). [23] However, the extinction coefficient of PerDIS being, at the wavelengths used, lower than that of the other molecules studied, the temperature increase generated is lower than the one observed for Ni8C12. Finally, the ICG and the Cu-ISQ complex display the lowest photothermal efficiencies measured (25 % and 20 % respectively) among the 7 compounds. It is also important to note that the photothermal efficiency values calculated at 808 nm and 940 nm are fairly close for compounds absorbing at both wavelengths, confirming that the photothermal efficiency is indeed intrinsic to a molecule and does not depend on the irradiation wavelength. The plot of the temperature increase as a function of the photothermal efficiency demonstrates that there is no direct correlation between the temperature increase measured and the photothermal efficiency of the compound (Figure 2b). As an example, the temperature increase measured for Ni8C12 under 940 nm laser irradiation is 45.5 °C while it is only 9.8 °C for PerDIS, despite these two compounds have similar photothermal efficiency (48 and 47 %, respectively). As already stated, when taken independently, neither the photothermal efficiency nor the absorptivity can account for the real photothermal activity of a compound which in fine is only revealed by the temperature increase measured under identical experimental conditions. However, interestingly, by plotting the temperature increase as a function of the product of the molar absorption coefficient at a given irradiation wavelength by the photothermal efficiency, a clear linear relation emerges (Figure 3). Thus, based on this observation, a photothermal index (IPT) is proposed to reflect the photothermal activity of a molecular compound and is defined as follow:

I PT = ε  x η x 10 -3
IPT was multiplied by a factor 10 -3 to obtain more convenient values. This photothermal index allows to properly evaluate the photothermal strength of a molecular compounds. Irradiation of compounds showing high photothermal indexes will lead to higher temperature increases, i.e. higher energy conversion. Thus, IPT allows to compare photohermal agent irradiated at the same wavelength.

The IPTmax (IPT calculated with εmax) could represent reliable quantification of the strength of a photothermal agent, independently of the experimental setup used for the measurement, thus allowing for the direct comparison of the molecular photothermal agents that can be found in the literature. Figure 4 and table S2 show the IPTmax for the molecular agents presently studied. IPTmax allows to compare, in a more general way, the strength of photothermal agents if they are irradiated at their absorption maxima.

I PTmax = ε max x η x 10 -3 It should also be noticed that despite the fact that the photothermal efficiency is an intrinsic characteristic of the photothermal agent, the molar absorption coefficient will depend on the nature of the solvent. Thus, it is required to compare IPTmax values determined in the same solvent or mixture of solvents.

In conclusion, through the study of seven molecular photothermal agents, a clear relation could have evidenced between the temperature increase measured under laser irradiation and the product of the photothermal efficiency by the molar absorption coefficient ( x ). This product can be compared with the brilliance used in luminescence studies and allows to quantify properly the photothermal activity of a molecular photothermal agent. From this observation, a new photothermal index has been defined IPTmax = max x  x 10 -3 which provides a reliable tool for quantifying the photothermal strength of molecules and which allows to compare and rank the photothermal agents. The only experimental constraints remaining, in order to be properly compared, is that the IPTmax should be determined in the same solvent. This work and the concept of photothermal brilliance can certainly be extended to inorganic nanoparticles or conducting polymers probably by using the mass extinction coefficient instead of molar coefficient but additional parameters such as the particle size and the surface functionalization will have to be taken into account to properly evaluate the photothermal efficiency. This photothermal index will undoubtedly be of strong interest in the future development and screening of photothermal agents.

Figure 1 .

 1 Figure 1. Absorption spectra of the various compounds in toluene at 10 μM (the wavelength of the two lasers used for the photothermal studies are indicated by vertical lines).

Figure 2 .

 2 Figure 2. Temperature increase measured for each solution under laser irradiation at 808 or 940 nm (a) as a function of the molar absorption coefficient (measured at 808 or 940 nm) or (b) as a function of the photothermal efficiency (808 nm: ICG, Cu-ISQ, Pt8C12, PerDIS; 940 nm: Ni8C12, Pt8C12, Ni-timdtC12, Ni-timdtC16).

Figure 3 .

 3 Figure 3. Temperature increase as a function of the product of the molar absorption coefficient by the photothermal efficiency (correlation coefficient of the linear regression R = 0.97506).

Figure 4 .

 4 Figure 4. Photothermal efficiency index IPTmax of the various agents studied (based on εmax measured in toluene / CHCl3 or EtOH (99: 1, v/v). From the IPTmax values, dihiolene complexes appear to be the most efficient photothermal agents among the agents studied, with N,N'-dialkoxyimidazolidine-2,4,5-trithione derivatives showing better performances than the 1,2-diphenylethylene-1,2dithiolate derivatives, whereas Cu-ISQ has the worse photothermal activity. The photothermal performance of ICG is between the one of 1,2-diphenylethylene-1,2-dithiolate complexes and the one of the perylene derivative. The IPTmax of IGC is 10, the ones 1,2-diphenylethylene-1,2-dithiolate complexes are around 20 and the ones N,N'dialkoxyimidazolidine-2,4,5-trithione derivatives are above 30.It should also be noticed that despite the fact that the photothermal efficiency is an intrinsic characteristic of the photothermal agent, the molar absorption coefficient will depend on the nature of the solvent. Thus, it is required to compare IPTmax values determined in the same solvent or mixture of solvents.

[23] S. Zhang, J. Li, J. Wei, M. Yin, Sci. Bull. 2018, 63, 101-107.

Acknowledgements

The authors thank the CNRS, the University of Rennes 1 and the "Ligue Contre le Cancer Grand Ouest" for financial support. Drs. Loïc Charbonnière and Fabien Gauffre are also acknowledged for their helpful discussions.

Entry for the Table of Contents

The study of several molecular photothermal agents has allowed to define a new photothermal index IPT, comparable to the brilliance in luminescence, which allow to properly rank the photothermal agents as a function of their ability to convert light into heat, with minimal experimental constraints. The calculation of this photothermal index only implies to know the intrinsic photothermal efficiency of the compound and the molar absorption coefficient at the chosen irradiation wavelength in a given solvent.


Deniz Vardar

Hale Ocak
email: hocak@yildiz.edu.tr

Akdas Huriye

Olivier Kilic

Franck Jeannin

Belkiz Camerel

Eran Bilgin
email: bbilgin@yildiz.edu.tr

Huriye Akdaş-Kılıç

Olivier Jeannin

Franck Camerel

Belkız Bilgin

Synthesis and characterization of new pyridine-based chiral calamitic liquid crystals

Keywords: Liquid crystal, pyridine, calamitic compounds, nicotinate, picolinate, chiral chain

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Introduction

A huge interest in the field of liquid crystals has been grown rapidly due to their unique properties which have led to useful and technologically promising materials. [START_REF] Tschierske | Development of Structural Complexity by Liquid Crystal Self-assembly[END_REF][START_REF] Kohout | Effect of alkyl chain length in the terminal ester group on mesomorphic properties of new chiral lactic acid derivatives[END_REF][START_REF] Lehmann | High Charge-Carrier Mobility in π-Deficient Discotic Mesogens: Design and Structure-Property Relationship[END_REF][START_REF] O'neill | Liquid Crystals for Charge Transport, Luminescence, and Photonics[END_REF] This special class of materials, which keep a combination of order and mobility at the molecular level, are used in many fields such as electro-optical displays and temperature sensors. [START_REF] Girotto | New liquid crystals derived from thiophene connected to the 1,2,4-oxadiazole heterocycle[END_REF][START_REF] Ghosh | Recent Advances in Heterocycle-Based Metal-Free Calamitics[END_REF] The use of six-membered N-heterocyclic systems in mesogenic molecules impacts the mesomorphic properties due to the fact that the introduction of nitrogen atom in the molecular structures mainly effect the molecular geometry, polarity and polarisability. [START_REF] Ong | Heterocyclic pyridine-based liquid crystals: synthesis and mesomorphic properties[END_REF] Pyridine is one of the simplest heterocyclic aromatics used as a core system in the design of liquid crystalline molecules. [START_REF] Ong | Heterocyclic pyridine-based liquid crystals: synthesis and mesomorphic properties[END_REF][START_REF] Campbell | Nematic 2,5-disubstituted thiophenes[END_REF][START_REF] Burrow | The synthesis and liquid crystal properties of some 2,5-disubstituted pyridines[END_REF][START_REF] Chia | Synthesis and mesomorphic properties of a series of phenyl 6-(4alkoxyphenyl) nicotinates[END_REF][START_REF] Chia | Synthesis thermotropic studies of a novel series of nematogenic liquid crystals 2-(6-alkoxynaphthalen-2-yl)-5-cyanopyridines[END_REF] The position of the nitrogen atom in the molecular core considerably influences the liquid crystalline properties of pyridine derivatives [START_REF] Petrov | Nitrogen-containing fused heterocycles as the structural fragments in calamitic liquid crystals[END_REF] and has a significant impact on the dispersion forces, the dipole moment, the polarizability of the molecule and inevitably, on the dielectric nature of the mesogen. [START_REF] Nash | Studies of Some Heterocyclic Mesogens[END_REF] Additionally, the geometry and the value of the dihedral angle between the two aromatic rings of the phenyl-pyridine moiety, [START_REF] Rogojerov | Determination of the average orientation of 4-phenylpyridine in nematic solvent by means of infrared linear dichroism: Study of its conformational dependence on the dihedral angle between aromatic rings[END_REF] depending of the position of the nitrogen, give rise to an effect on the mesophase molecular stacking. For this purpose, interesting mesomorphic behaviors are revealed by using pyridine ring in the central unit in calamitic molecules.

Recently, a variety of pyridine containing liquid crystalline compounds were synthesized, [START_REF] Ong | Heterocyclic pyridine-based liquid crystals: synthesis and mesomorphic properties[END_REF][START_REF] Yu | Computational Chemistry-Guided Design of Selective Chemoresponsive Liquid Crystals Using Pyridine and Pyrimidine Functional Groups[END_REF][START_REF] Chia | Effect of Pyridine on the Mesophase of Teraryl Liquid Crystals: A New Series of Nematic Liquid Crystals Named 2-(4-Alkoxybiphen-4'-yl)-5-methylpyridines[END_REF][START_REF] Byron | Properties of the Liquid Crystals Formed by Certain 4-(2'-Pyridyl)phenyl and 4-(4'-Pyridyl)phenyI 4''-n -Aikoxybenzoates[END_REF] most of them were rod-like calamitic liquid crystals, but only few examples presenting nicotinate and picolinate as building blocks were reported, in order to determine the effect of 2-, 3-, and 4-pyridyl rings relative to phenyl in promoting mesophase thermal stability. [START_REF] Chia | Synthesis and mesomorphic properties of a series of phenyl 6-(4alkoxyphenyl) nicotinates[END_REF][START_REF] Kamogawa | Synthesis of mesomorphic aryl esters bearing a pyridine ring[END_REF] However, several Japanese patents were published concerning optically active liquid crystal based on pyridine carboxylate, demonstrating the importance of this type of patterns for technological applications. [START_REF] Nobuhiro | Preparation of optically active 4alkoxycarbonylphenyl 5-phenyl-3-pyridinecarboxylate derivatives as liquid crystals and antiferroelectric liquid crystals having tristable state[END_REF][START_REF] Yuzo | Optically active pyridine carboxylate derivatives as liquid crystals[END_REF][START_REF] Yoshitake | Optically active 3-phenylpyridine-2-carboxylate esters as ferroelectric smectic liquid crystals[END_REF] The mentioned compounds in that patents mainly exhibit two liquid crystal mesophases such as SmC and SmA. They especially promise a potential to be used in display devices.

Previously, various biphenyl esters with a rod-like core, with interesting properties, were prepared and studied by optical and thermal methods. [START_REF] Canli | Conductivity Mechanism of Ethyl-4-(7,7,8,8,9,9,10,10,10-nonafluorodecyloxy)Biphenyl-4'-Carboxylate[END_REF][START_REF] Yildiz | The determination of the phase transition temperatures of a semifluorinated liquid crystalline biphenyl ester by impedance spectroscopy as an alternative method[END_REF] Recently, we reported the compounds that the internal phenyl ring of the biphenyl system replaced with a 3-pyridyl ring to generate functional materials with an interesting mesogenic behavior. [START_REF] Karanlik | New pyridine based liquid crystalline esters with different terminal chains[END_REF] Herein, we report the synthesis and the mesomorphic properties of novel chiral calamitic mesogens, consisting of n-dodecyloxyphenyl substituted pyridine carboxylate core linked to one or two benzene rings through ester linkers. Into the other terminal, a flexible (S)-2-methylbutyloxy or (S)-3,7-dimethyloctyloxy chiral chain was introduced to focus on the effect of molecular chirality on mesophases of pyridine-based calamitic molecules.

Experimental

Materials and instrumentation

New pyridine-based chiral calamitic molecules (4, 5, 7a,b, 10a,b and 11a,b) were characterized by [START_REF] Tschierske | Development of Structural Complexity by Liquid Crystal Self-assembly[END_REF] H NMR and [START_REF] Nash | Studies of Some Heterocyclic Mesogens[END_REF] C NMR (Bruker Avance III 500 spectrometer, in CDCl 3 solution, with tetramethylsilane as internal standard). Microanalysis was performed using a Thermo Fischer Scientific FlashEA 1112 Series elemental analyzer. Fourier Transform Infrared (FT-IR) spectra of the newly synthesized molecules were recorded on a "Perkin Elmer Spectrum One".

The mesomorphic properties of compounds were investigated by using a Mettler FP-82 HT hot stage with its control unit coupled with a Leica DM2700P polarizing microscope. DSCthermograms of chiral pyridine derivatives were recorded on a Perkin-Elmer DSC-6, heating and cooling rate: 10 °C min -1 in a nitrogen atmosphere. X-ray scattering experiments (SAXS) were performed using a Microstar Bruker rotating anode X-ray generator operated at 40 kV and 40 mA with monochromatic Cu Kα radiation (λ = 1.541 Å) and point collimation. The patterns were collected with a Mar345 Image-Plate detector (Marresearch, Norderstedt, Germany). The exposure time at each temperature was 3600 s and the heating or cooling speed between two temperatures was 10 °C/min. The samples were held in Lindeman glass capillaries (1.5 mm diameter). The capillaries were placed inside a Linkam HFX350-Capillary X-Ray stage which allow measurements from -196 °C up to 350 °C with an accuracy of 0.1 °C.

Methyl 5-bromopyridine-2-carboxylate (ABCR), methyl 6-chloropyridine-3-carboxylate (Aldrich), 1-bromododecane (Merck), 4-bromophenol (Alfa Aesar), K 2 CO 3 (Merck), n-Butyllithium solution 2.5 M in hexanes (Alfa Aesar), trimethyl borate ≥98% (Alfa Aesar), hydrochloric acid technical grade, tetrakis(triphenylphosphine)-palladium (0) (Aldrich), 1,2-Dimethoxyethane anhydrous 99.5% (Merck), sodium hydrogen carbonate (Merck), were purchased from commercial sources. 2-Butanone (Merck) and THF for HPLC, ≥99.9% (Riedel de Häen) were purchased commercially as dry solvents and were used without further purification. Solvents used in the purification step such as crystallization and column chromatography (hexane, ethyl acetate, chloroform, dichloromethane and ethanol) were distilled prior to use. Analytical thin-layer chromatography (TLC) was carried out on aluminium plates coated with silica gel 60 F254 (Merck). Column chromatography was performed using silica gel 60 (Merck, pore size 60 Å, 230-400 mesh particle size).

Synthesis and Characterization

The synthesis of new pyridine based chiral calamitic compounds ( The synthesis of the precursor molecules 6a,b, 8a,b and 9a,b via using multi-step procedures, their characterizations and the mesomorphic properties were reported in our previous study. [START_REF] Vardar | Pyridine Based Chiral Smectogens: Effects of Polar End Groups on liquid crystal Properties[END_REF] 4-Dodecyloxybenzen boronic acid [START_REF] Huo | Synthesis and characterization of 4-dodecyloxyphenyl and (4'-dodecyloxy-4-biphenyl)methylene-substituted bispyrrolylvinylthiophene-based polysquaraines having good solubility and very low bandgap for light absorption[END_REF] (1) was synthesized starting from 1-bromo-4dodecyloxybenzen [START_REF] Huo | Synthesis and characterization of 4-dodecyloxyphenyl and (4'-dodecyloxy-4-biphenyl)methylene-substituted bispyrrolylvinylthiophene-based polysquaraines having good solubility and very low bandgap for light absorption[END_REF] by using previously reported procedure. [START_REF] Karanlik | New pyridine based liquid crystalline esters with different terminal chains[END_REF][START_REF] Huo | Synthesis and characterization of 4-dodecyloxyphenyl and (4'-dodecyloxy-4-biphenyl)methylene-substituted bispyrrolylvinylthiophene-based polysquaraines having good solubility and very low bandgap for light absorption[END_REF] Methyl 5-bromopyridine-2carboxylate (2) and methyl 6-chloropyridine-3-carboxylate (3) were commercially purchased. 0.5 mmol 4-Dodecyloxybenzene boronic acid (1), 0.5 mmol the initial halogenated derivatives (2, 3, 6a,b, 8a,b and 9a,b) and 0.022 mmol Pd(PPh 3 ) 4 were dissolved in 10 mL of 1,2-dimethoxyethane under argon atmosphere. After the addition of 0.75 mL of saturated NaHCO 3 solution, the reaction mixture was boiled under reflux for 2.5 hours and the evolution was monitored by TLC (H: EA/3:1). The solvent was removed under reduced pressure and the residual mixture was extracted three times with CHCl 3 . The combined organic phases were washed with saturated NaCl solution and dried over Na 2 SO 4 . The solvent was evaporated in a rotary evaporator. The residue was dissolved in CHCl 3 and filtered over silica gel in order to remove the catalyst from the medium. The product was purified by column chromatography (silica gel 60, H: EA/3:1).

[C 12 H 7 NO 2 ].

Results and Discussions

Liquid crystalline properties

The mesomorphic properties of the pyridine-based chiral calamitic compounds 4 24 , 5, 7a,b, 10a,b and 11a,b were investigated by using optical polarizing microscope (POM) and differential scanning calorimeter (DSC). The phase transitions temperatures observed on heating and cooling cycles and corresponding transition enthalpies of the compounds are given in Table 1. The peak temperatures are given in degree Celsius and the numbers in parentheses indicate the transition enthalpy (ΔH).

DSC and POM analyses have revealed that the nicotinate ester 5 compound displays only a monotropic SmA phase upon cooling from the isotropic state (see Table 1, Fig. 1). The DSC traces displays two endotherms upon heating associated to a Cr 1 -Cr 2 -Iso phase sequence and two exotherms upon cooling associated to the phase sequence Iso-(SmA)-Cr 1 . On the contrary, the methyl picolinate ester 4, which was previously reported, [START_REF] Karanlik | New pyridine based liquid crystalline esters with different terminal chains[END_REF] exhibits enantiotropic smectic A mesophase. The mesophase of both simple pyridine-based methyl esters 4 and 5 carrying a ndodecyloxy chain is characterized by the typical fan-shaped texture (see Fig. 2a,b). The comparison of the nicotinate-based compound 5 with the picolinate-based compound 4 [START_REF] Karanlik | New pyridine based liquid crystalline esters with different terminal chains[END_REF] ,

shows that the position of the nitrogen does not affect the type of the mesophase but increases the mesophase stability. A stable molecular stacking is possible with the intra-molecular interactions of nitrogen atom placed at the ortho position of the ester group instead of inner position.

Table 1 Mesophase, phase transition temperatures (°C) and the corresponding transition enthalpies (ΔH kJ/mol) a of the compounds 4 [START_REF] Karanlik | New pyridine based liquid crystalline esters with different terminal chains[END_REF] The DSC thermograms of compound 7a carrying a (S)-2-methylbutyloxy end chain shows three endotherms for a phase transition sequence of crystal (Cr)-chiral smectic C* (SmC*)smectic A (SmA) -isotropic phase (Iso) and three exotherms for the reverse phase transition sequence (Table 1, Fig. 1). An enantiotropic chiral tilted mesophase (SmC*) with a Schlieren texture with some striped regions was clearly (Fig. 3a). Additionally, SmA phase appeared with a homeotropic texture between the SmC* phase and the isotropic phase (Fig. 3b).

Compound 7b terminated with a (S)-3,7-dimethyloctyloxy chiral chain, which is a longer and more branched chain, shows SmC* and SmA mesophases with at lower transition temperatures compared to compound 7a (Table 1). Compound 7b showed only two endotherms and two exotherms corresponding to Cr -SmC* and SmA -Iso transitions on the DSC heating and cooling traces. The SmA -SmC* phase transition is not detectable by on the DSC thermogram but is clearly evidenced by POM observations (see Fig. 3c,d). It can be noticed that the increase in chain length and branching give rise to a decrease of the phase transition temperatures as well as the temperature range of chiral tilted mesophase.

The four-ring calamitic nicotinate-based compound 10a and 10b exhibits a mesophase on heating. However, both compounds 10a and 10b decompose around 215 °C and 228 °C, respectively. For compound 10a, due to decomposition, the mesophase was assigned as uncharacteristic mesophase M 1 (see Fig. 24, ESI). The DSC heating curve of compound 10b b) a)

displays to clear endotherms centered at 65 and 121 °C attributed to a crystal to crystal phase transition and to a crystal to an uncharacteristic M 1 mesophase . A third endotherm, associated to the decomposition of the compound, is observed around 218 °C. No exotherm could be detected after decomposition during the cooling process (Fig. 1).

Interestingly, the four-ring calamitic picolinate-based compounds 11a and 11b exhibit a more stable mesomorphism as compared to compounds 10a and 10b. The DSC thermograms of 11a

show three endotherms which corresponds to Cr 1 → Cr 2 , Cr 2 → SmC* and SmA → Iso phase transition. The exotherm associated to the SmC*-SmA phase transition was not observed on the heating curve. However, POM observations clearly revealed that when the temperature reaches 122° C, an characteristic texture associated to a SmC* mesophase readily develops and when the temperature reaches 248 °C, the texture transforms into a characteristic SmA fan-shaped texture (Fig. 3e,f). On cooling, two exotherms associated to Iso-SmA and SmC*- In comparison, the corresponding biphenyl analogues present a richer polymorphism [START_REF] Heppke | New chiral side chains for ferro-and antiferroelectric liquid crystals derived from the preen-gland wax of the domestic goose[END_REF] with at least three different mesophases (SmC, SmA and higher ordered smectic phase). In reported series here, we can notice that the clearing point is drastically increasing with the number of rings within the core. Indeed, mesophase-to-isotropic transition temperature increases up to 273 °C for compound 11a. Additionally, an increase in the mesomorphic range was observed when the number of aromatic cores is increased.

The position of the nitrogen in the aromatic ring gives rise to drastic effects on the mesomorphic properties. Comparison of the pyridine-based methyl esters 4 and 5 compounds, which differ only by the position of the nitrogen atom, shows that compound 5 present a monotropic smectic A phase over a very narrow mesomorphic range while 4 shows a welldefined enantiotropic SmA phase over a larger temperature range. Similarly, the picolinate based compound 10a and nicotinate based compound 11a, which both of them carrying a short chiral chain but differing only by the position of the nitrogen, an uncharacteristic mesophase, which decomposed at high temperatures, appeared for 10a whereas the nicotinate aromatic ring in compound 11a allows the formation of lamellar mesophases (SmC* and SmA) at high temperatures (see Fig. 3g,h) without decomposition. A similar behavior was observed with 10b and 11b, presenting a long chiral chain and differing only by the position of the nitrogen, the nicotinate core leads to the occurrence of more stable mesophases such as SmC* and SmA without a decomposition of sample at high temperatures. These results demonstrate that the position of the nitrogen in four-rings calamitic mesogens prevails on To further confirmed, the mesophase formations and the phase assignments deduced for compounds 5, 7a,b and 11b by POM observations, temperature-dependent XRD analyses were also conducted. More information, about the distinct phases, derived from XRD are presented in the following section.

SAXS Measurements

The temperature-dependent SAXS patterns recorded on compound 5 confirm that two thermal transitions occur between 90-110 °C and between 125-130 °C upon heating (Fig. 4 The temperature-dependent SAXS patterns confirm that compound 11b undergoes several reversible phase transitions (Fig. 7a). On heating, below 100 °C, this compound is in a crystalline state as confirmed by the observations of numerous sharp diffraction peaks over the whole 2 range explored. Above 120 °C, the compound is clearly in a smectic liquid crystalline phase. Two sharp reflections, indexed as the (001) and (002) reflections of a lamellar phase, are observed in the small angles region as well as a broad peak around 19 ° in 2 in the wide angles region. By POM observation, this mesophase was identified as a smectic C* phase. The length of the molecule in a fully extended conformation is around 45 Å. So, the mesophase is constituted of monolayers of tilted molecules. The molecular tilt estimated at 140 °C from an interlayer distance of 39.5 Å is around 28.5 °and this tilt decrease with the temperature increase (Fig. 7b). The molecules are likely arranged in head-to-tail fashion such as the space symmetry is broken into a phase of C2 symmetry. [START_REF] Goodby | Twisted and frustrated states of matter[END_REF] No information about the helical pitch could have been extracted from the SAXS patterns. Entering in the smectic A phase around 230 °C, the intensity of the first order reflection strongly decrease and shift to higher distance, at a distance close to the molecular length dl, and the second order reflection completely vanished. Finally, at 245 °C, the compound is in an isotropic state.

The reverse sequence was observed upon cooling from the isotropic state and the phase transitions were detected in the same temperature range as determined by DSC. The Cr 1 -Cr 2 and the Cr 2 -SmC* transitions were too close to properly isolate the SAXS pattern of the Cr 2 phase.

Conclusion

In this article, chiral multi-ring calamitic molecules with a specific architecture have been designed in order to examine structure-mesogenicity relationship and obtain new liquid crystal materials that may be useful for potential applications. The effects of the introduction of a heteroatom via nicotinate and picolinate based cores on the mesomorphic properties of calamitic compounds with a chiral end chain such as (S)-2-methylbutyloxy or (S)-3,7dimethyloctyloxy have been investigated by POM observation, DSC measurement and XRD analysis.

All the calamitic compounds exhibited liquid crystalline behaviour by forming ordered monolayer smectic phases depending on type of hetero aromatic rings, number of aromatic cores and the type of chiral branched chain. Firstly, the extension of the core via an increase of the number of aromatic rings leads to an increase of the mesophases stability and an expansion of the mesophase range. The length and branching of the chiral terminal chains play an effective role on transition temperatures and on the mesomorphic ranges.

Additionally, it can be concluded that the inner position of the nitrogen in three or four-rings calamitic mesogens gives rise to more stable smectic phases due to enhanced lateral interactions.

4 , 5 , 2 :Scheme 1 .

 4521 Scheme 1. Synthesis of the pyridine-based chiral calamitic compounds (4 24 , 5, 7a,b, 10a,b and 11a,b).

  Suzuki Cross-coupling reaction between the initial halogenated derivatives (2, 3, 6a,b, 8a,b and 9a,b) and the boronic acid 1 gave the target pyridine-based chiral calamitic compounds 4, 5, 7a,b, 10a,b and 11a,b by using tetrakis(triphenylphosphine)-palladium (0) as catalyst in 1,2-dimethoxyethane. Spectroscopic data and mesomorphic properties of methyl 5-(4-ndodecyloxyphenyl)pyridine-2-carboxylate (4) are given in ref. 27 . The general procedure for the synthesis of the target compounds 4, 5, 7a,b, 10a,b and 11a,b:

(

  CH 3 ). FT-IR: γ (cm -1 ) = 1721 (C=O stretching). C 25 H 35 O 3 N (397.55); Anal. Calc. (%): C, 75.52; H, 8.89; N, 3.52. Found (%): C, 75.35; H, 9.04; N, 3.13. (S)-4-(2-Methylbutoxy)phenyl 6-(4-dodecyloxyphenyl)pyridine-3-carboxylate (7a) Yield: 57 %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 9.41 (d, J  2.0 Hz, 1H, Ar-H), 8.48 (dd, J 1  8.3 Hz, J 2 = 2.0 Hz, 1H, Ar-CH), 8.10 (d, J  8.5 Hz, 2H, Ar-CH), 7.84 (d, J  8.3 Hz, 1H, Ar-CH), 7.16 (d, J  9.0 Hz, 2H, Ar-CH), 7.05 (d, J  8.5 Hz, 2H, Ar-CH), 6.97 (d, J  9.0 Hz, 2H, Ar-CH), 4.06 (t, J  6.6 Hz, 2H, OCH 2 ), 3.85-3.77 (m, 2H, OCH 2 ), 1.93-1.88 (m, 1H, CH), 1.87-1.82 (m, 2H, CH 2 ), 1.65-1.29 (m, 20H, CH 2 ), 1.05 (d, J  6.8 Hz, 3H, CH 3 ), 0.98 (t, J  6.5 Hz, 3H, CH 3 ), 0.91 (t, J  6.5 Hz, 3H, CH 3 ). 13 C NMR (125 MHz, CDCl 3 ): δ (ppm) = 164.51 (COO), 161.

, 5 ,

 5 7a,b, 10a,b and 11a,b.

11bFig. 2 .

 2 Fig. 2. Optical textures of SmA mesophase as observed between crossed polarizers (indicated by arrows) between ordinary glass plates on cooling; (a) fan-shaped texture of compound 4 at T = 111 °C; (b) fan-shaped texture of compound 5 obtained at T = 113 °C.

  Cr 1 phase transition were observed and similarly the SmA → SmC* transition was only detected at 234 °C by POM. For compound 11b carrying a (S)-3,7-dimethyloctyloxy chiral chain, four endotherms and four exotherms are observed on the DSC traces and correspond to a reversible Cr 1 -Cr 2 -SmC*-SmA-Iso phase transition sequence. Typical finger-print texture of SmC* and fan-shaped texture of SmA phase observed for compounds 11b are shown in Fig. 3g,h.

Fig. 3 .

 3 Fig. 3. Optical textures of the mesophases as observed between crossed polarizers (indicated by arrows) between ordinary glass plates; (a) Schlieren texture of the SmC* mesophase at T = 165 °C and (b) homeotropic texture of the SmA mesophase at T = 170 °C of compound 7a on cooling; (c) finger-print texture of the SmC* mesophase at T = 140 °C on cooling and (d) focal-conic texture of the SmA mesophase at T = 147 °C of compound 7b on heating; (e) texture characteristic of a SmC* mesophase at T = 223 °C and (f) fan-shaped SmA texture at T = 257 °C of compound 11a on cooling; (g) finger-print texture of the SmC* mesophase at T = 221 °C and (h) fan-shaped SmA texture at T = 232 °C of compound 11b on cooling.

  ), agreeing with the transition temperatures detected by DSC. The diffractograms recorded below 125 °C display several sharp peaks over the whole 2 range explored, typical of a crystalline compound. Above 125 °C, the diffractograms present only two broad halos at 26.8 and 4.6 Å associated to the mean distances between molecules in a disordered state. The low temperature transition is thus attributed to a crystal to crystal transformation whereas the high temperature transition is attributed to a crystal to isotropic phase transition. Upon cooling from the isotropic state, a first thermal transition is detected between 130 °C and 120 °C and is associated to a crystallization of the compound. In fact, the SAXS patterns recorded below 120 °C, display several sharp peaks in the small and wide-angle regions (Fig.4b). Another crystal to crystal phase transition has also been detected between 110 °C and 90 °C.Transition temperature discrepancies can be observed between SAXS and DSC measurements due to experimental kinetic aspects. During the cooling, the formation of a monotropic Smectic A phase could not have been confirmed by SAXS measurements and only crystalline states have been detected below the isotropic phase. The observation by POM of the monotropic SmA phase is likely attributed to kinetic effects.

Fig. 4 .

 4 Fig. 4. SAXS diffractograms recorded of compound 5 upon heating (a) and cooling (b).

Fig. 5 .

 5 Fig. 5. a) SAXS patterns obtained as a function of the temperature with compound 7a upon cooling; b) SAXS pattern obtained in the smectic C* phase of compound 7a at 110 °C upon cooling.

Fig. 6 .

 6 Fig. 6. SAXS patterns recorded on compound 7b upon cooling.

Fig. 7 .

 7 Fig. 7. a) Temperature-dependent SAXS patterns measured on compound 11b compound on heating; b) SAXS pattern measured at 140 °C on heating.

Methyl 6-(4-dodecyloxyphenyl)pyridine-3-carboxylate (5)

  Ar-C), 150.86, 137.80, 128.76, 118.91, 114.83 (Ar-CH), 68.18 (OCH 2 ), 52.26 (OCH 3 ), 31.92, 29.66, 29.64, 29.60, 29.58, 29.40, 29.35, 29.21, 26.02, 22.69 (CH 2 ), 14.12

	Yield: 42 %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 9.24 (d, J ≈ 2.1 Hz,
	1H, Ar-CH), 8.31 (dd, J 1 ≈ 8.4 Hz, J 2 ≈ 2.2 Hz 1H, Ar-CH), 8.03 (d, J 1 ≈ 8.9 Hz, 2H, Ar-CH),
	7.76 (d, J ≈ 8.4 Hz, 1H, Ar-CH), 7.02 (d, J ≈ 8.9 Hz, 2H, Ar-CH), 4.04 (t, J ≈ 6.5 Hz, 2H,
	OCH 2 ), 3.98 (s, 3H, OCH 3 ), 1.86-1.77 (m, 2H, CH 2 ), 1.61-1.25 (m, 18H, CH 2 ), 0.89 (t, J ≈
	6.5 Hz, 3H, CH 3 ). 13 C NMR (125 MHz, CDCl 3 ): δ (ppm) = 166.00 (COO), 160.95, 160.56,
	130.44, 123.36 (

-4-(3,7-Dimethyloctyloxy)phenyl 6-(4-dodecyloxyphenyl)pyridine-3-carboxylate (7b) Yield: 36

  10, 161.01, 157.25, 143.90, 130.36, 122.96 (Ar-C), 151.36, 138.27, 128.88, 122.31, 118.98, 115.17, 114.89 (Ar-CH), 73.32, 68.21 (OCH 2 ), 34.73 (CH), 31.93, 29.67, 29.65, 29.61, 29.59, 29.41, 29.36, 29.22, 26.14, 26.03, 22.70 (CH 2 ), 16.52, 14.13, 11.33 (CH 3 ). FT-IR: γ (cm -1 ) = 1731 (C=O stretching). C 35 H 47 O 4 N (545.75); Anal. Calc. (%): C, 77.02; H, 8.70; N, 2.56. Found (%): C, 76.71; H, 8.37; N, 2.24. %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 9.32 (d, J  2.2 Hz, 1H, Ar-CH), 8.37 (dd, J 1  8.4, J 2  2.2 Hz, 1H, Ar-CH), 7.99 (d, J  8.9 Hz, 2H Ar-CH), 7.74 (d, J  8.4 Hz, 1H, Ar-CH), 7.07 (d, J  8.9 Hz, 2H, Ar-CH), 6.95 (d, J  8.9 Hz, 2H, Ar-CH), 6.87 (d, J  8.9 Hz, 2H, Ar-CH), 3.96-3.93 (m, 4H, OCH 2 ), 1.78-1.73 (m, 1H, CH), 1.55-1.46 (m, 3H, CH, CH 2 ), 1.40-1.07 (m, 26H, 13 CH 2 ), 0.88 (d, J  6.6 Hz, 3H, CH 3 ), 0.81 (m, 9H, 3CH 3 ). 13 C NMR (125 MHz, CDCl 3 ): δ (ppm) = 164.51 (COO), 161.08, 161.04, 157.05, 143.93, 130.41, 122.94 (Ar-C), 151.39, 138.24, 128.87, 122.33, 118.96, 115.16, 114.88 (Ar-CH), 68.20, 66.77 (OCH 2 ), 39.25, 37.29, 36.21, 31.93, 29.67, 29.65, 29.60, 29.58, 29.40, 29.36, 29.21, 26.03, 24.67, 22.72 (CH 2 ), 29.85, 27.98 (CH), 22.70, 22.62, 19.66, 14.13

	(S)82, 163.41 (COO), 160.32,
	157.16, 155.07, 144.87, 144.13, 140.44, 133.35, 127.56 (Ar-C), 134.49, 131.87, 127.66,
	126.22, 122.36, 122.13, 115.44, 115.16, 114.74 (Ar-CH), 73.31, 68.28 (OCH 2 ) 34.73, 31.94,
	29.69, 29.66, 29.63, 29.60, 29.41, 29.38, 29.21, 26.15, 26.05 (CH 2 ), 22.72, 16.55, 14.15,
	11.35 (CH

(CH 3 ). FT-IR: γ (cm -1 ) = 1731 (C=O stretching). C 40 H 57 O 4 N (615.88); Anal. Calc. (%): C, 78.00; H, 9.35; N, 2.27. Found (%): C, 77.67; H, 9.02; N, 1.89. (S)-[4-(2-Methylbutoxy)phenoxycarbonyl]phenyl 5-(4-dodecyloxyphenyl)pyridine-2carboxylate (10a) Yield: 20 %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 8.98 (d, J  2.1 Hz, 1H, Ar-CH), 8.29-8.20 (m, 3H, Ar-CH), 7.99 (dd, J 1  7.8 Hz, J 2  2.1 Hz, 1H, Ar-CH), 7.55 (d, J  7.8 Hz, 2H, Ar-CH), 7.36 (d, J  8.4 Hz, 2H, Ar-CH), 7.05 (d, J  8.4 Hz, 2H, Ar-CH), 6.98 (d, J  7.8 Hz, 2H, Ar-CH), 6.87 (d, J  8.4 Hz, 2H, Ar-CH), 3.96 (t, J  6.6 Hz, 2H, OCH 2 ), 3.76-3.67 (2m, 2H, OCH 2 ), 1.85-1.72 (m, 1H, CH), 1.56-1.43 (m, 2H, CH 2 ), 1.40-1.22 (m, 20H, CH 2 ), 0.95 (d, J  6.8 Hz, 3H, CH 3 ), 0.89 (t, J  7.5 Hz, 3H, CH 3 ), 0.81 (t, J  6.7 Hz, 3H, CH 3 ). 13 C NMR (125 MHz, CDCl 3 ): δ (ppm) = 164.3 ). FT-IR: γ (cm -1 ) = 1751 and 1735 (2 C=O stretching). C 42 H 51 O 6 N (665.86); Anal. Calc. (%): C, 75.75; H, 7.73; N, 2.10. Found (%): C, 75.43; H, 7.42; N, 1.76. (S)-[4-(

3,7-Dimethyloctyloxy)phenoxycarbonyl]phenyl 5-(4-dodecyloxyphenyl)pyridine- 2-carboxylate (10b) Yield: 29

  %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 9.41 (d, J  2.1 Hz, 1H, Ar-CH), 8.46 (d, J 1  8.4 Hz, J 2  2.1 Hz, 1H, Ar-CH), 8.31 (d, J  8.6 Hz, 2H, Ar-CH),

	(S)-[4-
	8.08 (dd, J  8.4 Hz, 1H, Ar-CH), 7.84 (d, J  8.4 Hz, 2H, Ar-CH), 7.41 (d, J  8.6 Hz, 2H,
	Ar-CH), 7.13 (d, J  8.9 Hz, 2H, Ar-CH), 7.03 (d, J  8.4 Hz, 2H, Ar-CH), 6.95 (d, J  8.9
	Hz, 2H, Ar-CH), 4.12-3.94 (m, 4H, OCH 2 ), 1.88-1.83 (m, 1H, CH), 1.54-1.47 (m, 3H, CH,
	CH 2 ), 1.44-1.15 (m, 26H, CH 2 ), 0.95 (d, J  6.6 Hz, 3H, CH 3 ), 0.92-0.84 (m, 9H, CH 3 ). 13 C
	NMR (125 MHz, CDCl 3 ): δ (ppm) = 164.75, 163.35 (COO), 160.32, 156.97, 155.08, 144.65,
	144.20, 140.23, 128.36, 127.56 (Ar-C), 148.19, 134.43, 131.83, 128.57, 126.14, 122.34,
	122.06, 115.44, 115.16 (Ar-CH), 68.27, 66.78 (OCH 2 ), 39.25, 37.29, 36.22, 31.91, 29.64,
	29.62, 29.58, 29.57, 29.37, 29.33, 29.19, 26.02, 24.65, 22.70 (CH 2 ), 29.86, 27.97 (CH), 22.67,
	22.60, 19.65, 14.10 (CH = 164.79, 163.56 (COO), 161.36, 161.29, 157.18, 154.71, 144.11, 130.13, 129.00, 127.54
	(Ar-C), 151.43, 138.45, 131.89, 130.05, 122.35, 121.95, 119.10, 115.16, 114.95 (Ar-CH),
	73.32, 68.24 (OCH 2 ), 34.74 (CH), 31.94, 29.69, 29.66, 29.63, 29.60, 29.42, 29.38, 29.22,
	26.15, 26.05, 22.72 (CH 2 ), 16.56, 14.16, 11.35 (CH 3 ). FT-IR: γ (cm -1 ) = 1732 (2 C=O
	stretching). C 42 H 51 O 6 N (665.86); Anal. Calc. (%): C, 75.75; H, 7.73; N, 2.10. Found (%): C,
	75.47; H, 7.38; N, 1.81. MS (ESI) (+): m/z (%) = 666 (100) [M + ], 498 (81) [M + -C 12 H 25 ], 366
	(70) [M + -C 18 H 19 O 4 ].

3 ). FT-IR: γ (cm -1 ) = 1734

(

2 C=O stretching). C 47 H 61 O 6 N (735.99); Anal. Calc. (%): C, 76.70; H, 8.37; N, 1.90. Found (%): C, 76.41; H, 8.09; N, 2.19. (S)-[4-(2-Methylbutoxy)phenoxycarbonyl]phenyl 6-(4-dodecyloxyphenyl)pyridine-3carboxylate (11a) Yield: 33 %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 9.43 (d, J  2.1 Hz, 1H, Ar-CH), 8.48 (dd, J 1  8.4 Hz, J 2  2.1 Hz, 1H, Ar-CH), 8.32 (d, J  8.5 Hz, 2H, Ar-CH), 8.10 (d, J = 8.5 Hz, 2H, Ar-CH), 7.86 (d, J  8.4 Hz, 1H, Ar-CH), 7.43 (d, J  8.5 Hz, 2H, Ar-CH), 7.15 (d, J = 8.5 Hz, 2H, Ar-CH), 7.05 (d, J  8.5 Hz, 2H, Ar-CH), 6.97 (d, J  8.5 Hz, 2H, Ar-CH), 4.07 (t, J  6.6 Hz, 2H, OCH 2 ), 3.85-3.77 (m, 2H, OCH 2 ), 1.95-1.88 (m, 1H, CH), 1.89-1.78 (m, 2H, CH 2 ), 1.63-1.31 (m, 20H, CH 2 ), 1.05 (d, J  6.5 Hz, 3H, CH 3 ), 0.98 (t, J  6.8 Hz, 3H, CH 3 ), 0.91 (t, J  6.8 Hz, 3H, CH 3 ). 13 C NMR (125 MHz, CDCl 3 ): δ (ppm)

(3,7-Dimethyloctyloxy)phenoxycarbonyl]phenyl 6-(4-dodecyloxyphenyl)pyridine- 3-carboxylate (11b) Yield: 19

  %; colorless crystals. 1 H NMR (500 MHz, CDCl 3 ): δ (ppm) = 9.40 (d, J  2.1 Hz, 1H, Ar-CH, 8.46 (dd, J 1  8.4, Hz, J 2  2.1 Hz, 1H, Ar-CH), 8.31 (d, J  8.6 Hz, 2H, Ar-CH 2 ), 1.48-1.16 (m, 26H, CH 2 ), 0.95 (d, J  6.5 Hz, 3H, CH 3 ), 0.89-0.78 (m, 9H, CH 3 ). 13 C NMR (125 MHz, CDCl 3 ): δ (ppm) = 164.77, 163.60 (COO), 161.43, 161.21, 156.96, 154.70, 144.11, 130.21. 127.49, 122.34 (Ar-C), 151.52, 138.34, 131.87, 128.94, 122.29, 121.94, 119.02, 115.12, 114.89 (Ar-CH), 68.20, 66.73 (OCH 2 ), 39.24, 37.28, 36.20, 31.92, 29.67, 29.64, 26.60, 29.58, 29.40, 29.36, 29.20, 26.02, 24.66, 22.73 (CH 2 ), 29.83, 27.98 (CH), 22.70, 22.61, 19.65, 14.14 (CH

	CH), 8.07 (d, J  8.7 Hz, 2H, Ar-CH), 7.83 (d, J  8.4 Hz, 1H, Ar-CH), 7.41 (d, J  8.6 Hz,
	2H, Ar-CH), 7.13 (d, J  8.9 Hz, 2H, Ar-CH), 7.03 (d, J  8.7 Hz, 2H, Ar-CH), 6.95 (d, J 
	8.9 Hz, 2H, Ar-CH), 4.05-3.97 (m, 4H, 2H, OCH 2 ), 1.93-1.86 (m, 1H, CH), 1.84-1.62 (m,
	3H, CH,

3 ). FT-IR: γ (cm -1 ) = 1734 (2 C=O stretching). C 47 H 61 O 6 N (735.99); Anal. Calc. (%): C, 76.70; H, 8.37; N, 1.90. Found (%): C, 76.40; H, 8.54; N, 1.89. MS (ESI) (+): m/z (%) = 736 (100) [M + ], 366 (35) [M + -C 23 H 29 O 4 ], 198

  These transitions were determined by POM. c 1 st heating scan at a rate of 10 °C min -1 was only carried out due to decomposition.

	Heating Cr 105.90 [48.22] SmA 118.98 [5.14] Iso Iso 114.90 [6.22] SmA 91.36 [52.65] Cr Cooling Cr 1 119.55 [3.89] Cr 2 126.01 [32.63] Iso Iso 117.26 [2.36] (SmA) 113.66 [49.88] Cr 1 Cr 102.80 [38.63] SmC*161[XX] SmA 0 50 100 150 200 250 Endo ® T / °C 173.27 °C 102.80 °C 83.35 °C 170.23 °C DH= 38.63 kJ/mol DH= 3.64 kJ/mol DH= 3.43 kJ/mol DH= 39.55 kJ/mol 7a 40 60 80 100 120 140 160 180 T / °C A D 119.55 °C 126.01 °C DH= 3.89 kJ/mol DH= 32.63 kJ/mol 117.26 °C DH= 2.36 kJ/mol 113.66 °C DH= 49.88 kJ/mol 50 100 150 200 250 DH= 26.38 kJ/mol 20 5 94.56 °C T / °C B D 154.14 °C 150.78 °C 65.44 °C DH= 2.74 kJ/mol DH= 24.74 kJ/mol DH= 2.77 kJ/mol 7b 0 50 100 150 200 250 300 Endo ® T / °C 65.33 °C 121.10 °C 228.77 °C DH= 8.22 kJ/mol DH= 37.35 kJ/mol DH= 9.32 kJ/mol 10b 0 50 100 150 200 250 300 350 T / °C B D 112.65 °C 122.89 °C 273.85 °C 270.09 °C 91.72 °C DH= 2.71 kJ/mol DH= 24.75 kJ/mol DH= 29.51 kJ/mol DH= 10.17 kJ/mol DH= 3.30 kJ/mol 11a 50 100 150 200 250 300 Endo ® T / °C 104.46 °C 236.97 °C 234.52 °C 82.99 °C DH= 30.88 kJ/mol DH= 3.92 kJ/mol DH= 3.61 kJ/mol DH= 32.92 kJ/mol 224.59 °C liquid phase. Fig. 1. DSC thermograms of compounds 5, 7a,b and 11a,b on 2 nd heating and cooling and 4 24 5 7a Endo ® 0 Endo ® Endo ® 222.21 °C	B D A A C
	DSC thermogram of 10b on 1 st heating (10 °C.min -1 ).

b

Acknowledgement

This research has been supported by The Scientific and Technological Research Council of Turkey (TÜBİTAK), Turkey with the Project Number 116Z465. The authors are grateful to the TÜBİTAK-2232 programme for a support with Project Number 118C273. F.C. and O.J. also thank the Institut des Sciences Chimiques de Rennes, the University of Rennes 1 and Rennes Metropole for their financial support. Appendix A. Supplementary data Electronic supplementary information (ESI) is available.