Abdelouahab Sedeki

Pascale Auban-Senzier

Shingo Yonezawa

Claude Bourbonnais

Denis Jerome

Influence of carrier lifetime on quantum criticality and superconducting T c of (TMTSF) 2 ClO 4

This work presents and analyzes electrical resistivity data on the organic superconductor (TMTSF) 2 ClO 4 and their anion-substituted alloys (TMTSF) 2 (ClO 4 ) 1-x (ReO 4 ) x along the least conducting c axis. Nonmagnetic disorder introduced by finite size domains of anion ordering on non-Fermi-liquid character of resistivity is investigated near the conditions of quantum criticality taking place at x = 0. The evolution of the T -linear resistivity term with anion disorder shows a limited decrease in contrast with the complete suppression of the critical temperature T c as expected for unconventional superconductivity beyond a threshold value of x. The resulting breakdown of scaling between both quantities is compared to the theoretical predictions of a linearized Boltzmann equation combined to the scaling theory of umklapp scattering in the presence of disorder-induced pair breaking for the carriers. The theory shows that quantum antiferromagnetic fluctuations, which are at the core of unconventional Cooper pairing and inelastic scattering of T -linear resistivity in these systems, are weakly affected by disorder while the phase coherence responsible for a finite T c can be completely suppressed.

I. INTRODUCTION

Quantum criticality that becomes unstable against the emergence of superconductivity is a common feature of most recent unconventional superconductors. It happens when the ordering temperature of an antiferromagnetic (AFM) phase, which can be metal-like, semimetallic, or insulating, ends towards zero temperature, as a control parameter doping or pressure is varied [START_REF] Norman | The challenge of unconventional superconductivity[END_REF]. Such is the situation encountered in the vast family of pnictide superconductors [START_REF] Paglione | High-temperature superconductivity in iron-based materials[END_REF][START_REF] Johnston | The puzzle of high-temperature superconductivity in layered iron pnictides and chalcogenides[END_REF]. That may also be the situation met in hole-doped cuprates, although there the quantum critical point (QCP) corresponding to the termination at zero temperature of the pseudogap phase is beneath the superconducting dome of their phase diagram [START_REF] Taillefer | Scattering and pairing in cuprate superconductors[END_REF]. This is also found in electron-doped cuprates and heavyfermion superconductors, which both exhibit an AFM phase terminating toward zero temperature when the electron doping level or pressure is varied respectively in the former [START_REF] Dagan | Evidence for a Quantum Phase Transition in Pr 2-x Ce x CuO 4-δ from Transport Measurements[END_REF] or latter compounds [START_REF] Jaccard | Pressure induced heavy fermion superconductivity of CeCu 2 Ge 2[END_REF][START_REF] Mathur | Magnetically mediated superconductivity in heavy fermion compounds[END_REF]. The proximity of an AFM Mottinsulating phase to superconductivity is also present in layered organic superconductors bearing much resemblance with the phase diagram of cuprates [START_REF] Kanoda | Recent progress in NMR studies on organic conductors[END_REF][START_REF] Lefebvre | Mott Transition, Antiferromagnetism, and Unconventional Superconductivity in Layered Organic Superconductors[END_REF][START_REF] Mackenzie | Similarities between organic and cuprate superconductors[END_REF].

Most experimental investigations performed near QCP conditions have revealed that several physical properties deviate significantly from the canonical Fermi liquid behavior, in particular the resistivity which instead of the expected T 2 dependence exhibits a linear-T dependence [START_REF] Jin | Link between spin fluctuations and electron pairing in copper oxide superconductors[END_REF], a puzzling feature shared by essentially all the aforementioned unconventional superconductors close to a magnetic QCP.

Moreover, there exists another class of materials displaying both antiferromagnetism and superconductivity that has not been as much highlighted, namely, the quasi-one-dimensional (Q1D) Bechgaard organic superconductors [START_REF] Bechgaard | The properties of five highly conducting salts[END_REF][START_REF] Bourbonnais | Interacting electrons in quasione-dimensional organic superconductors[END_REF]. Although superconductivity in heavy fermions [START_REF] Steglich | Superconductivity in the Presence of Strong Pauli Paramagnetism: CeCu 2 Si 2[END_REF] was discovered before that in Q1D organics, it is in these latter materials that superconductivity was first revealed following the suppression of an insulating phase under pressure which was subsequently identified as a spin-density-wave (SDW) state [START_REF] Jérome | Superconductivity in a synthetic organic conductor (TMTSF) 2 PF 6[END_REF][START_REF] Scott | Magnetic Properties of the Organic Conductor Bis-Tetramethyltetraselenafulvalene Hexafluorophosphate, (TMTSF) 2 PF 6 : A New Phase Transition[END_REF][START_REF] Andrieux | Spin-density wave ground state in the one-dimensional conductor (TMTSF) 2 PF 6 : Microscopic evidence from 77 Se and 1 H NMR experiments[END_REF]. A strong argument in favor of Q1D superconductors is the relative simplicity of their electronic structure, comprising a single nearly flat Fermi surface as well as a temperature-pressure phase diagram which can be considered as a textbook example of magnetic quantum criticality that becomes unstable to the formation of superconductivity [START_REF] Jerome | The physics of organic superconductors[END_REF][START_REF] Sedeki | Extended quantum criticality of low-dimensional superconductors near a spindensity-wave instability[END_REF].

An unusual behavior of metallic transport in (TMTSF) 2 PF 6 under pressure at low temperature has been observed in the first studies [START_REF] Jérome | Superconductivity in a synthetic organic conductor (TMTSF) 2 PF 6[END_REF] related to a pronounced T -linear tendency between 0.1 and 10 K and even a downward curvature of the resistivity in the pressure range corresponding to the vicinity of the QCP [START_REF] Schulz | Possibility of superconducting precursor effects in the quasi-one-dimensional organic conductors: Theory and experiments[END_REF]. While superconducting fluctuations were first considered as a possible interpretation in the three-dimensional (3D) temperature regime very close to T c where the transverse coherence length is larger than the interstack distance, paraconductivity from superconducting origin failed to account for the correct temperature dependence observed up to 10 K or so [START_REF] Schulz | Possibility of superconducting precursor effects in the quasi-one-dimensional organic conductors: Theory and experiments[END_REF]. Thirty years have been required to clarify the problem when similar behaviors of transport have been detected in other superconductors close to a QCP, in particular in cuprates and pnictides [START_REF] Taillefer | Scattering and pairing in cuprate superconductors[END_REF][START_REF] Jin | Link between spin fluctuations and electron pairing in copper oxide superconductors[END_REF].

As far as organic superconductors are concerned, detailed investigations of transport at low temperature were conducted first on (TMTSF) 2 PF 6 under pressure [START_REF] Doiron-Leyraud | Correlation between linear resistivity and T c in the Bechgaard salts and the pnictide superconductor Ba(Fe 1-x Co x ) 2 As 2[END_REF]. In this case the temperature dependence of the inelastic scattering in conditions very close to the QCP is a clear-cut behavior. ρ(T ) follows a T -linear law over a decade in temperatures above T c , already visible on a log-log plot of the resistivity. The behavior is becoming quadratic as expected in a Fermi liquid above 10 K [START_REF] Doiron-Leyraud | Correlation between linear resistivity and T c in the Bechgaard salts and the pnictide superconductor Ba(Fe 1-x Co x ) 2 As 2[END_REF]. However, as the location of the compound in the T -P phase diagram is moved away from the QCP conditions the resistivity acquires a quadratic component even at low temperature and becomes fully quadratic when T c is suppressed. The log-log plot of the resistivity reveals a power law ρ ∝ T β with an exponent β evolving from 1 to 2 between 11.8 and 20.8 kbars [START_REF] Doiron-Leyraud | Correlation between linear resistivity and T c in the Bechgaard salts and the pnictide superconductor Ba(Fe 1-x Co x ) 2 As 2[END_REF]. Subsequently, another procedure has been followed to analyze the inelastic scattering of (TMTSF) 2 PF 6 , namely, a sliding fitting procedure [START_REF] Doiron-Leyraud | Linear-T scattering and pairing from antiferromagnetic fluctuations in the (TMTSF) 2 X organic superconductors[END_REF] with a second-order polynomial form such as ρ(T ) = ρ 0 + A(T )T + B(T )T 2 . Here, the fitting procedure enables the detection of a possible temperature dependence of the A and B prefactors, whereas the value of ρ 0 depends on pressure only [START_REF] Doiron-Leyraud | Linear-T scattering and pairing from antiferromagnetic fluctuations in the (TMTSF) 2 X organic superconductors[END_REF]. This procedure has shown that the T -linear contribution is dominant below 10 K compared to the Fermi liquid contribution close to the QCP, but otherwise becomes of the order of the quadratic contribution.

The same log-log analysis failed to provide a clear-cut picture in (TMTSF) 2 ClO 4 , but the polynomial fitting procedure conducted on the resistivity along c led to conclusions that A and B are only weakly temperature dependent in this compound between T c and 15 K (though still pressure dependent) [START_REF] Auban-Senzier | The metallic transport of (TMTSF) 2 X organic conductors close to the superconducting phase[END_REF]. The detailed analysis of the temperature-dependent resistivity over an extended range of pressures has shown that ρ c (T ) can thus be fitted to the polynomial expression ρ c 0 + AT + BT 2 where all parameters are evolving under pressure but the experimental feature, A → 0 as T c → 0, is preserved [START_REF] Auban-Senzier | The metallic transport of (TMTSF) 2 X organic conductors close to the superconducting phase[END_REF], as found in other materials like the cuprates for instance [START_REF] Cooper | Anomalous criticality in the electrical resistivity of La 2-x Sr x CuO 4[END_REF].

Although a precise functional relation between A and T c is uncertain, the salient result of these pressure studies is the existence of a T -linear contribution to the resistivity being finite only when T c is finite, suggesting in turn the existence of a common origin for pairing and the T -linear inelastic scattering [START_REF] Doiron-Leyraud | Linear-T scattering and pairing from antiferromagnetic fluctuations in the (TMTSF) 2 X organic superconductors[END_REF].

While there is no general consensus about the microscopic origin of T -linear resistivity appearing near a QCP [START_REF] Coleman | Quantum criticality[END_REF], theoretical efforts have been displayed to shed some light on its origin in the context of the Q1D Bechgaard salts [START_REF] Sedeki | Extended quantum criticality of low-dimensional superconductors near a spindensity-wave instability[END_REF][START_REF] Meier | Resistor model for the electrical transport in quasi-one-dimensional organic (TMTSF) 2 PF 6 superconductors under pressure[END_REF][START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF]. The electron-electron scattering rate, as extracted from the imaginary part of the one-particle self-energy, has been derived by a renormalization group procedure. The calculation that takes into account the quantum interference between electron-electron and electron-hole scattering channels was able to show in the first place the emergence of unconventional (d-wave) superconductivity from quantum SDW fluctuations [START_REF] Bourbonnais | Link between antiferromagnetism and superconductivity probed by nuclear spin relaxation in organic conductors[END_REF][START_REF] Duprat | Interplay between spin-densitywave and superconducting states in quasi-one-dimensional conductors[END_REF][START_REF] Nickel | Triplet Superconducting Pairing and Density-Wave Instabilities in Organic Conductors[END_REF], in a way that goes beyond the standard spin fluctuation scenarios in which both scattering channels are not dynamically coupled [START_REF] Béal-Monod | Possible superconductivity in nearly antiferromagnetic itinerant fermion systems[END_REF][START_REF] Scalapino | d-wave pairing near a spin-density-wave instability[END_REF][START_REF] Bourbonnais | New mechanisms for phase transitions in quasi-one-dimensional conductors[END_REF]. In the second place, it allowed the possibility to link the strength of the linear resistivity term to the one of pairing as a function of pressure [START_REF] Sedeki | Extended quantum criticality of low-dimensional superconductors near a spindensity-wave instability[END_REF]. Another strategy has been to connect the solution of the transport Boltzmann equation to the same renormalization group approach to nonconserving momentum scattering amplitudes. This procedure has confirmed the existence of a T -linear term in resistivity and the correlation between its strength and the size of superconductivity pairing mediated by SDW fluctuations [START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF].

These findings have motivated the present work devoted both experimentally and theoretically to the influence of a finite elastic lifetime on the T -linear resistivity in the Bechgaard salts. On the theoretical side impurity scattering, assumed to be absent so far in the renormalization group calculations of the scattering amplitudes, is a source of pair breaking for both unconventional superconductivity and density-wave correlations. We show that this affects to a certain extent the constructive interference between both types of correlations at low energy. Although this can be sufficient for the complete suppression of phase coherence responsible for a finite T c , it turns out that SDW correlations that build up at higher energy are less affected by disorder-induced pair breaking and remain relatively strong in amplitude. As the source of umklapp scattering for carriers in the Boltzmann equation, these correlations sustain to a great extent the non-Fermi-liquid character of resistivity in spite of the scaling breakdown between the T -linear term and T c .

On experimental side, previous investigations have shown that T c in the (TMTSF) 2 X series is strongly affected by disorder of nonmagnetic origin [START_REF] Choi | Effect of radiation damage on the metalinsulator transition and low-temperature transport in the tetramethyltetraselenofulvalinium PF -6 salt (TMTSF) 2 PF 6[END_REF][START_REF] Tomić | Influence of the disorder potential of the anions on the ground state of the organic alloy (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF][START_REF] Joo | Suppression of superconductivity by non-magnetic disorder in the organic superconductor (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF][START_REF] Joo | Impurity-controlled superconductivity/spin density wave interplay in the organic superconductor: (TMTSF) 2 ClO 4[END_REF], providing in turn some hint for the existence of a nonconventional superconducting gap displaying both signs over the Fermi surface in this spin singlet superconductor [START_REF] Shinagawa | Superconducting State of the Organic Conductor (TMTSF) 2 ClO 4[END_REF], pointing for instance to a d-wave (or g-wave) type of superconductivity. As the linear term of transport at low temperature was found to be related to T c , we feel it is legitimate to test the robustness of the established connection between A and T c when T c can be modified by other means but pressure, namely, by the influence of nonmagnetic disorder [START_REF] Suzumura | Thermodynamic properties of impure anisotropic quasi-one-dimensional superconductors[END_REF]. It is worth mentioning that in Sr 2 RuO 4 , an equal sensitivity to nonmagnetic disorder has been observed [START_REF] Mackenzie | Extremely Strong Dependence of Superconductivity on Disorder in Sr 2 RuO 4[END_REF], suggesting that spin triplet pairing is a possible candidate for the superconductivity (SC) ground state in this material [START_REF] Maeno | Superconductivity in a layered perovskite without copper[END_REF].

Fortunately, (TMTSF) 2 ClO 4 is the unique system among other superconductors in the (TMTSF) 2 X series in which structural disorder can control the elastic scattering time. This property is based on the peculiarity of the tetrahedral symmetry of the anion ClO - 4 . The structure of (TM) 2 X salts belongs to the triclinic space group P 1 with every anion site located on an inversion center. Hence, as long as the local symmetry of anions like PF - 6 is centrosymmetric, their actual orientation always fits the overwhole crystal symmetry. Thus, no additional disorder is introduced. However, noncentrosymmetric anions such as tetrahedral ClO - 4 or ReO - 4 may have their oxygen atoms pointing towards methyl groups or Se atoms of one or the other neighboring donor molecules [START_REF] Pouget | Structural aspects of the Bechgaard salts and related compounds[END_REF][START_REF] Lepevelen | Temperature and pressure dependencies of the crystal structure of the organic superconductor (TMTSF) 2 ClO 4[END_REF]. These (at least two) possible directions lead in turn to a potential source of disorder. While there is no disorder on average at elevated temperature since these tetrahedral anions are rotating as shown by NMR data [START_REF] Takahashi | An NMR study of the organic superconductor: (TMTSF) 2 ClO 4[END_REF], the minimization of entropy due to the reduction of degrees of freedom in thermodynamical equilibrium triggers an anion ordering at low temperature with the concomitant occurrence of a superstructure. In addition, the relative orientation at low temperature of neighboring anions depends on the nature of the anions. For (TMTSF) 2 ReO 4 , the orientation of ReO - 4 alternates along the three directions providing a periodic lattice distortion of wave vector (1/2, 1/2, 1/2) (in units of reciprocal lattice vector in each direction) with a concomitant metal insulator transition below T AO = 176 K [START_REF] Moret | X-Ray Scattering Evidence for Anion Ordering and Structural Distortions on the Low Temperature Phase of Di(Tetramethyltetraselenafulvalenium) Perrhenate (TMTSF) 2 ReO 4[END_REF], whereas the alternation of ClO - 4 occurs solely along the b axis in (TMTSF) 2 ClO 4 below T AO = 24 K, leading in turn to the (0, 1/2, 0) order [START_REF] Pouget | X-ray evidence of a structural phase transition in ditetramethyltetraselenafulvalenium perchlorate (TMTSF) 2 ClO 4 , pristine and slightly ReO 4 substituted samples[END_REF][START_REF]According to structural studies, the anion ordering inside the methyl group cavity is also accompanied by a shift of the anion itself either towards methyl groups for (TMTSF) 2 ClO 4 [43,83] or towards[END_REF]. Such a doubling of periodicity along b is thus responsible for the doubling of the Fermi surface folded along b in slowly cooled samples. These samples retain in turn metallic properties at low temperature.

A possible means of introducing controllable disorder in an otherwise well-ordered (TMTSF) 2 ClO 4 crystal is the substitution of ClO - 4 by ReO - 4 anions. The consequences of the isostructural anion alloying on the electronic properties such as superconductivity were revealed long ago [START_REF] Tomić | Influence of the disorder potential of the anions on the ground state of the organic alloy (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF], even though the actual structural reasons were not elucidated. For instance, ReO - 4 replacing ClO - 4 in (TMTSF) 2 ClO 4 affects T c so severely that a 1% ReO - 4 concentration is large enough to shift T c from 1.3 to 0.9 K and 10% suppresses T c totally [START_REF] Tomić | Influence of the disorder potential of the anions on the ground state of the organic alloy (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF]. Additional investigations of structural and electronic properties of (TMTSF) 2 (ClO 4 ) (1-x) (ReO 4 ) x have been conducted subsequently [START_REF] Joo | Suppression of superconductivity by non-magnetic disorder in the organic superconductor (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF][START_REF] Ravy | Competition between structural instabilities in organic conductors and superconductors[END_REF]. What structural studies have revealed in the ClO - 4 alloys at low concentration of ReO - 4 is the existence of domains in which the (0, 1/2, 0) ClO - 4 order persists, the size of them controlling the carrier elastic mean free path [START_REF] Ilakovac | Substitutional disorder and anion ordering transition in the (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x solid solution[END_REF].

Another way to create defects in (TMTSF) 2 ClO 4 is to beat the slow anion orientation process by rapid cooling of the sample, as first detected by NMR, specific heat, EPR, and transport measurements [START_REF] Takahashi | Observation of a magnetic state in the organic superconductor (TMTSF) 2 ClO 4 : Influence of the cooling rate[END_REF][START_REF] Garoche | Influence of the Cooling Rate on the Superconducting Properties of the Organic Solid Di-Tetramethyltetraselenafulvalenium-Perchlorate, (TMTSF) 2 ClO 4[END_REF][START_REF] Tomić | EPR and electrical conductivity of the organic superconductor di-tetramethyltetraselenafulvalenium-perchlorate, (TMTSF) 2 ClO 4 and a metastable magnetic state obtained by fast cooling[END_REF]. A recent reinvestigation of magnetic and transport properties of (TMTSF) 2 ClO 4 under a well-controlled cooling procedure in the vicinity of the anion ordering temperature has revealed at increasing cooling speed above 1 K/min a crossover from a rather homogenous localized disorder to a granular situation in which anion-ordered puddles are embedded in an anion-disordered background [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. This picture is reminiscent of the present situation in alloys where the carrier mean free path of the well-ordered grains is also limited by their own size.

The fact that the elastic mean free path of (TMTSF) 2 ClO 4 can be controlled by disorder is a remarkable property of the material and makes (TMTSF) 2 ClO 4 a unique system in which one can check simultaneously the effect of a change of the mean free path on T c and on quantum criticality behavior possibly related to the onset of superconductivity [START_REF] Doiron-Leyraud | Correlation between linear resistivity and T c in the Bechgaard salts and the pnictide superconductor Ba(Fe 1-x Co x ) 2 As 2[END_REF].

The present report attends to fulfill this goal with a quantitative study of both superconducting and metallic state properties in two situations where disorder can be introduced with a concomitant control of the mean free path: first, via studying lightly ReO - 4 substituted (TMTSF) 2 ClO 4 samples, and second, studying the fast cooling of pure (TMTSF) 2 ClO 4 . Theoretically, we broaden the renormalization group (RG) approach of the Q1D electron gas model to include impurity pair-breaking effects on the renormalization of umklapp scattering that enters the linearized Boltzmann equation of electrical transport.

Our study will show that the correlation between the non-Fermi-liquid resistivity and T c , which is well established under pressure [START_REF] Doiron-Leyraud | Correlation between linear resistivity and T c in the Bechgaard salts and the pnictide superconductor Ba(Fe 1-x Co x ) 2 As 2[END_REF], breaks down when T c in this d-wave superconductor is suppressed by nonmagnetic disorder. This result emphasizes the strong pair-breaking role of a limited elastic mean free path on T c as opposed to the much weaker influence on antiferromagnetic fluctuations, which are believed to be a major ingredient for pairing in this superconductor. The positive connection between theory and experiment widens the consistency of the model of magnetically driven mechanisms for quantum criticality and superconductivity in the Bechgaard salts.

In Sec. II, the low-temperature transverse resistivity measurements of (TMTSF) 2 (ClO 4 ) 1-x (ReO 4 ) x alloys are presented and their superconducting and non-Fermi-liquid properties described. Section III is devoted to the results of the numerical solution of the linearized Boltzmann equation of resistivity for the Q1D electron gas model with umklapp scattering and pair breaking, as yielded by the renormalization group method detailed in the Appendix. In Sec. IV, we discuss the results and their connection with theory. In Sec. V, we summarize and conclude this work.

II. EXPERIMENT

The present study is based on transport measurements of (TMTSF) 2 (ClO 4 ) (1-x) (ReO 4 ) x single crystals grown electrochemically by Bechgaard in Copenhagen. All data have been taken at Orsay except for one (TMTSF) 2 ClO 4 sample already measured in a previous study at Kyoto [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF] for its superconducting properties.

The measurement of the transverse resistivity ρ c along the least conducting c direction has been privileged because of the requested quantitative comparison between samples with different ReO - 4 concentration. The frequent cracks occurring on cooling for the resistivity along a or b axes prevent any comparison between samples whereas ρ c is known to be less influenced by such cracks even after multiple cooling processes [START_REF] Cooper | Magnetoresistance of the organic conduct-ing tetramethyltetraselenafulvalene salts (TMTSF) 2 ClO 4 and (TMTSF) 2 PF 6 : Search for the coherent-diffusive transition or localization effects with increasing temperature[END_REF]. In addition, magnetoresistance measurements have shown that the band theory should apply below 10 K as also supported by the existence of a Drude edge along the c axis in the same temperature range [START_REF] Henderson | Anisotropic electrodynamics of low dimensional metals: Optical studies of (TMTSF) 2 ClO 4[END_REF]. A previous comparative study of transport along a and c has shown that in spite of a very large anisotropy, electron scattering times measured along a or c exhibit a similar temperature dependence at least up to 30 K [START_REF] Auban-Senzier | The metallic transport of (TMTSF) 2 X organic conductors close to the superconducting phase[END_REF].

For (TMTSF) 2 ClO 4 samples studied in Kyoto, the resistivity was measured with the resistivity option of a PPMS with a dc current of 10 μA reversed to cancel thermoelectric voltages. For (TMTSF) 2 (ClO 4 ) (1-x) (ReO 4 ) x studied in Orsay, a lock-in amplifier technique was used with an ac current of 10 μA.

One of the major difficulties in interpreting lowtemperature transport in most (TMTSF) 2 X superconductors is the existence of a negative curvature in the R(T ) curve approaching T c below 4 K [START_REF] Jérome | Superconductivity in a synthetic organic conductor (TMTSF) 2 PF 6[END_REF]. This negative curvature seems to be fairly robust as it can be observed even when SC is suppressed either by a magnetic field or in alloyed samples [START_REF] Auban-Senzier | Fluctuating spin density wave conduction in (TMTSF) 2 X organic superconductors[END_REF]. While precursors due to SC could contribute to a possible collective paraconductive contribution up to about 1.2T c [START_REF] Auban-Senzier | Fluctuating spin density wave conduction in (TMTSF) 2 X organic superconductors[END_REF], there is still much uncertainty as to the origin of paraconductivity up to 3.5-4 K remaining a pending problem. Since precursor effects are not discussed in the present article the analysis of the ρ c (T ) curves has been limited to the range FIG. 1. Interlayer resistivity data for the (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x solid solutions all measured in the relaxed state and normalized to 28 cm at ambient temperature. Samples have been labeled according to their nominal ReO - 4 concentration which may deviate in some cases from the actual defects concentration given by an electron microprobe analysis which in turn agrees with the value of the residual resistivity. The clean sample measured at Kyoto displayed a slightly smaller residual resistivity (30 m cm) than the Orsay one (50 m cm). This is in line with T c = 1.35 K at Kyoto while T c is only 1.18 K in the Orsay sample. We also show the data for the quenched 6% sample. The data above 8 K or so may be affected by some annealing occurring on warming up [START_REF]The SDW (anion-disordered) state being metastable[END_REF]. The clean but quenched (TMTSF) 2 ClO 4 data have been taken from Ref. [START_REF] Ya | Role of anion ordering in the coexistence of spin-density-wave and superconductivity in (TMTSF) 2 ClO 4[END_REF]. [START_REF] Taillefer | Scattering and pairing in cuprate superconductors[END_REF]-10 K since our purpose is a study of the single-particle non-Fermi-liquid properties only.

The study of transport has been conducted in the alloy series (TMTSF) 2 (ClO 4 ) (1-x) (ReO 4 ) x where the size of the ordered regions is controlled by the amount of the ReO - 4 substituent. The resistivity data are displayed in Fig. 1. We also emphasize that the samples in Fig. 1 have been labeled according to the nominal concentration of ReO - 4 which may differ significantly from the results obtained in an electron microprobe analysis [START_REF]A determination of the anion concentration by electron microprobe analysis in (TMTSF) 2 X solid solutions has shown that the final anion concentration may deviate up to 30% from the initial concentration in the electrochemical cell[END_REF].

The resistivity of (TMTSF) 2 ClO 4 is at variance with that of (TMTSF) 2 PF 6 since no clear T -linear resistivity behavior is emerging from the raw data of (TMTSF) 2 ClO 4 [60].

Note that every sample in Fig. 1 differs by the behavior of its resistivity at low temperature. Both 0% samples exhibit a complete superconducting transition although their T c and their residual resistivity ρ c 0 are slightly different. The SC transition is still complete in the 7% sample. We also notice a complete transition for the 6% sample (green curve in Fig. 1) but the broadening at low temperature is a signature of a proximity effect between superconducting puddles [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. A further depression of T c is observed in the 10% sample, but there the finite zero temperature value of the resistivity suggests the existence of superconducting islands far from each other precluding global SC coherence through the proximity effect. The 8% indicates the absence of any transition at finite temperature. The value of its ρ c 0 appears to be reliable, although a tiny break in the ρ c (T ) data (not visible in Fig. 1) occurring around 5 K made the polynomial analysis of the temperature dependence less reliable. In the 15% sample a metal-insulator transition is observed at 2.5 K which can be related to a SDW state occurring in the anion-disordered sample after fast cooling [START_REF] Tomić | Influence of the disorder potential of the anions on the ground state of the organic alloy (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF]. Similarly, the sample with a nominal concentration of 17% (not shown in Fig. 1) displays the onset of a SDW at 3.9 K.

The residual resistivity ρ c 0 can be extracted relatively easily at different ReO - 4 concentrations from the data in Fig. 1, as it is only weakly dependent on the determination of the other prefactors in the polynomial fit (vide infra), but the correct analysis of the inelastic temperature dependence is a very delicate operation even in pure samples as already discussed in several previous publications [START_REF] Doiron-Leyraud | Correlation between linear resistivity and T c in the Bechgaard salts and the pnictide superconductor Ba(Fe 1-x Co x ) 2 As 2[END_REF][START_REF] Auban-Senzier | The metallic transport of (TMTSF) 2 X organic conductors close to the superconducting phase[END_REF].

The present study utilizes a polynomial fitting procedure keeping A and B fixed within the fitting window for each sample in Fig. 1. As it is difficult to put error bars on such fits we have privileged the data coming from three different fitting ranges between 4.5 and 10 K leading to values for ρ c 0 , A, and B from the temperature dependence in Fig. 1 which are displayed in Fig. 2. The fitting window has been adequately chosen in order to avoid the onset of the as yet uncharacterized paraconductive contribution at low temperature and the proximity of the anion ordering at high temperature [61]. Transition temperatures defined by the onset temperatures have been normalized to our highest observed transition for superconductivity, namely, 1.51 K, and to 6 K for the SDW transition as observed in the case of strong anion disorder [START_REF] Ya | Coexistence of superconductivity and spin density wave in (TMTSF) 2 ClO 4 : Spatial structure of the two-phase state[END_REF]. All samples in Fig. 2 refer to the very slowly cooled regime except for the 6% which has been measured both under relaxed and quenched conditions (see Sec. IV for a further discussion). The data of A for the 15% sample are departing from the slow decrease with x observed at lower concentrations. This may actually be the consequence of the resistivity upturn due to the SDW phase leading in turn to an underestimation of A using the present fitting window.

The x axis of Fig. 2 refers to the residual resistivity as shown in the fits of Fig. 1 and not to the actual scattering rate. We do not expect this resistivity to be directly related to the elastic scattering rate whenever the conducting medium departs from homogeneity. This is likely true for the present data at high ReO - 4 concentrations when the superconducting transition becomes incomplete at, say, 10%.

In the solid solution (TMTSF) 2 (ClO 4 ) (1-x) (ReO 4 ) x two different anion orders are competing according to x-ray diffuse scattering experiments [START_REF] Ravy | Competition between structural instabilities in organic conductors and superconductors[END_REF][START_REF] Ilakovac | Substitutional disorder and anion ordering transition in the (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x solid solution[END_REF]. The (1/2, 1/2, 1/2) ReO - 4 ordering is responsible for a metal-insulator transition at 176 K in (TMTSF) 2 ReO 4 [START_REF] Moret | X-Ray Scattering Evidence for Anion Ordering and Structural Distortions on the Low Temperature Phase of Di(Tetramethyltetraselenafulvalenium) Perrhenate (TMTSF) 2 ReO 4[END_REF] while the (0, 1/2, 0) ClO - 4 ordering at 24 K folding the Q1D Fermi surface leads to a metallic phase at low temperature with a reduced value of the residual resistivity [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. At concentrations for x between zero and 1, both anion orders are competing, but on the ClO - 4 rich side of the solid solution (0, 1/2, 0) long-range order of ClO - 4 persists up to about x = 3% [START_REF] Ravy | Competition between structural instabilities in organic conductors and superconductors[END_REF][START_REF] Ilakovac | Substitutional disorder and anion ordering transition in the (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x solid solution[END_REF]. Increasing the ReO - 4 content further, the ClO - 4 ordering becomes short ranged above 7% or so and a SDW ground state is stabilized instead of superconductivity [START_REF] Tomić | Influence of the disorder potential of the anions on the ground state of the organic alloy (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x[END_REF]. The picture which can be inferred from the x-ray analysis is thus a homogenous medium with local impurities at very low ReO - 4 concentration only becoming heterogenous above 3% where each impurity is surrounded by a disordered volume. The mean free path has been shown to be very large, of the order of 1600 nm in a pristine sample [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF][START_REF] Yonezawa | Magnetic-field variations of the pair-breaking effects of superconductivity in (TMTSF) 2 ClO 4[END_REF]. As the size of the ordered domains in alloys is likely to be much smaller than 1600 nm, we may assume that the mean free path in alloys is determined by the typical domain size. The predominant role of the domain size limiting the mean free path is also corroborated by a similar study performed on the solid solution (TMTSF) 2 (AsF 6 ) (1-x) (SbF 6 ) x when the symmetry of the anions is central, which has shown that even a 15% concentration of SbF 6 has no detectable effect on T c [START_REF] Traetteberg | Le mode collectif de l'onde de densité de spin dans les composés (TMTSF) 2 (AsF 6 ) 1-x (SbF 6 ) x[END_REF], being at variance with the present alloying effect in (TMTSF) 2 ClO 4 .

The Abrikosov-Gor'kov theory [START_REF] Abrikosov | Contribution to the theory of superconducting alloys with paramagnetic impurities[END_REF] of a conventional superconducting T c in the presence of pair breaking has also been reported in Fig. 2 together with the results of the renormalization group theory (Sec. III and the Appendix). We observe a satisfactory agreement between the onset temperatures and theories when the residual resistivity is considered as a representative of the scattering rate [START_REF] Suzumura | Thermodynamic properties of impure anisotropic quasi-one-dimensional superconductors[END_REF]. Such an agreement, however, may be surprising as the residual resistivity is the macroscopic resistivity of the entire sample which we found to become heterogenous at high ReO - 4 concentration, while T c is the onset transition temperature of well-ordered domains. Consequently, this agreement between experiment and theories infers in turn the existence of some connection, ρ c 0 ∝ 1/λ, between the mean free path λ determined by domain size entering the theory and the macroscopic resistivity.

III. THEORY: RENORMALIZED LINEARIZED BOLTZMANN EQUATION FOR RESISTIVITY

In this section we sketch out the Boltzmann-RG approach to the calculation of electric resistivity in the presence of pair breaking. The calculation combines the linearized Boltzmann equation with the renormalization group method in the framework of the Q1D electron gas model for the Bechgaard salts [START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF]. Prior to doing so, it is important to draw attention on the fact that on experimental grounds, both longitudinal ( ρ a ) and transverse ( ρ c * ) inelastic contribution to resistivity, though differing by a factor of ∼10 4 in amplitude for a compound like (TMTSF) 2 ClO 4 , exhibit essentially the same temperature dependence below 30 K or so [START_REF] Auban-Senzier | The metallic transport of (TMTSF) 2 X organic conductors close to the superconducting phase[END_REF]. Such a concurrence indicates that in this temperature range, they are apparently governed by the same scattering mechanism. In the following we then focus on the calculation of longitudinal resistivity for an array of weakly coupled chains in the F ig4 plane perpendicular to the c direction.

We first consider the Boltzmann equation of the Fermi distribution function f k ,

∂f k ∂t coll = eE • ∇ hk f k , (1)
which stands for coherent carriers of charge e coupled to an external static electric field E = E â. The collision term on the left depends on the interparticle (umklapp) scattering and collisions to impurities or defects. The linearization of the equation standardly proceeds by looking at small dimensionless deviations φ k to the equilibrium Fermi distribution function f 0 k = 1/(e β p k + 1), which in leading order yields the small variation

δf k f 0 k (1 -f 0 k )φ k . Here ε p k = hv F (pk -k F ) -2t ⊥ cos k ⊥ d ⊥ -2t ⊥ cos 2k ⊥ d ⊥ (2)
is the electron spectrum of the Q1D electron gas model, where v F is the Fermi velocity for right-(p = +) and leftmoving (p = -) carriers along the chain, ±k F are the onedimensional (1D) Fermi points in the absence of interchain hopping t ⊥ , the second nearest-neighbor interchain hopping term of amplitude t ⊥ acts as an antinesting term that simulates the effect of pressure in the model, and d ⊥ is the interchain distance along the b direction.

The linearization of (1) then yields

- ∂φ k ∂t coll = k L kk φ k = eβE • v k (3)
where v k is the carrier velocity in the k direction. The collision operator is given by

L kk = L u kk + L imp kk = (π hv F ) 2 (LN P ) 2 k 2 ,k 3 ,k 4 1 2 g 3 k p F , k p 2 F,2 ; k -p 3 F,3 , k -p 4 F,4 -g 3 k p F , k p 2 F,2 ; k -p 4 F,4 , k -p 3 F,3 2 2π h δ k+k 2 ,k 3 +k 4 +pG × δ ε p k + ε p 2 k 2 -ε p 3 k 3 -ε p 4 k 4 × f 0 k 2 1 -f 0 k 3 1 -f 0 k 4 1 -f 0 k × (δ k,k + δ k 2 ,k -δ k 3 ,k -δ k 4 ,k ) + π hv F LN P 2π h g 2 imp δ ε p k -ε p k (1 -δ k,k ), (4)
where g 2 imp is the square of the impurity scattering matrix element (normalized by π hv F ) times the impurity concentration and N P is the number of transverse momentum wave vectors. In the framework of the Q1D electron gas model, the electron-electron umklapp scattering amplitude is given by g

3 (k p F , k p 2 F,2 ; k -p 3 F,3 , k -p 4
F,4 ) (normalized by π hv F ). For an array of quarter-filled but weakly dimerized chains, this momentum dissipative process has its origin in the scattering of two carriers from one side of the Fermi surface to the other, which is made possible if momentum conservation in (4) involves the longitudinal reciprocal lattice vector G = (4k F , 0) at half filling (k F = π/2a) [START_REF] Emery | Electron-Electron Umklapp Scattering in Organic Superconductors[END_REF]. Umklapp scattering amplitude is evaluated on the Fermi surface sheets

k p=± F = (k p F (k ⊥ ), k ⊥ )
, which is determined by the equation ε p k F = 0 and parametrized by the transverse wave vector k ⊥ . As we will see below, its momentum-dependent renormalization at temperature T will be obtained by the RG technique at the one-loop level (see also the Appendix).

The longitudinal electric current is given in leading order by the expression

j a 2e LN P d ⊥ k v F f 0 k 1 -f 0 k φ k ⊥ , ( 5
)
where φ k F ≡ φ k ⊥ can been taken on the Fermi surface at low temperature. The conductivity σ a or the inverse of the resistivity along the chains can then be written in the form

σ a = ρ -1 a = e 2 c h φk ⊥ FS , (6)
where we have rescaled the deviation φk ⊥ = φ k ⊥ /(βeEd ⊥ ) and inserted the lattice constant c in the c * direction. Here • • • FS corresponds to an average over the Fermi surface. Therefore the numerical solution of (3) for φk ⊥ using the momentum and temperature dependence of g 3 provided by the RG method leads to the resistivity as a function of the temperature [START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF]. We can complete the standard description of interactions in the Q1D electron gas model entering in the RG calculations of the Appendix. Thus besides the spectrum (2) and the aforementioned umklapp term, we have the backward and forward scattering amplitudes

g 1 (k - F,1 , k + F,2 ; k - F,3 , k + F,4 ) and g 2 (k + F,1 , k - F,2 ; k - F,3 , k + F,4
), also defined on the Fermi surface. For a superconducting compound like (TMTSF) 2 ClO 4 , which at ambient pressure is close to a magnetic QCP, typical values of the bare parameters of the model have been assessed in previous works [START_REF] Sedeki | Extended quantum criticality of low-dimensional superconductors near a spindensity-wave instability[END_REF][START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF][START_REF] Shahbazi | Seebeck coefficient in correlated low dimensional organic metals[END_REF]. For example the Fermi temperature T F = hv F k F /k B 3000 K and interchain hopping t ⊥ /k B 200 K are representative band parameters for the Bechgaard salts [START_REF] Grant | Electronic structure of the 2:1 charge transfer salts TMTCF[END_REF][START_REF] Ducasse | Temperature dependence of transfer integrals in the (TMTSF) 2 X and (TMTTF) 2 X families[END_REF]. The bare couplings (normalized by hπ v F ) we use are g 1 = g 2 /2 0.32 and g 3 0.033, consistent with uniform susceptibility measurements and weak dimerization of the organic stacks. In these conditions the RG calculations for relatively weak antinesting parameter t ⊥ lead to SDW ordering temperatures that are compatible with those found in the antiferromagnetic Bechgaard salts at ambient pressure.

To suppress the SDW state and bring the system close to the QCP, t ⊥ /k B is raised to 42 K or so, yielding a critical temperature T 0 c ∼ 1 K for (d-wave) superconductivity [START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF], which is congruent to the situation that prevails in (TMTSF) 2 ClO 4 in slow cooling. This set of figures will fix the initial conditions with zero pair-breaking effects (τ -1 0 = 0) in the RG calculations [Eqs. (A1)] and the numerical solution of the Boltzmann equation ( 3). Finally a normalized impurity scattering matrix element g 2 imp = 0.001 has been used in order to simulate a small residual resistivity term from (4); its value smoothly increases with τ -1 0 . It is worth noticing that the interplay between the impurity and inelastic parts in (4) has been found to be negligible. In Fig. 3 we show the resistivity calculated from Eqs. ( 6) and (A1) in the low-temperature range down to the onset of critical superconducting domain occurring close to T c where the renormalization flow goes to strong coupling.

Thus for zero pair breaking (τ -1 0 = 0), the system is very close to the QCP and the resistivity becomes essentially T linear up to 10 K or so where a crossover to a sublinear temperature dependence takes place (see Ref. [START_REF]evolve towards a T 2 behavior outside the quantum critical domain at high temperature[END_REF]). The origin of T linearity stems from the anomalous temperature dependence of umklapp scattering along the Fermi surface. Actually, despite the absence of SDW long-range order, spin correlations remain important and keep growing as temperature is lowered [START_REF] Bourbonnais | Link between antiferromagnetism and superconductivity probed by nuclear spin relaxation in organic conductors[END_REF]. This derives from Cooper pairing terms in the RG flows [Eq. (A1)] which interfere constructively with the electron-hole (Peierls) pairing ones and then SDW fluctuations. This quantum fluctuation effect results in an anomalous temperature dependence of umklapp scattering that transforms the Fermi liquid quadratic T 2 dependence of resistivity into a T -linear behavior. Roughly speaking, the resistivity goes like ρ a ∼ T 2 g 2 3 FS at low temperature where the mean-square value of umklapp scattering on the Fermi surface, which is representative of spin fluctuations, goes as 1/T at the critical t ⊥ [START_REF] Shahbazi | Electrical transport near quantum criticality in low-dimensional organic superconductors[END_REF].

When pair-breaking τ -1 0 departs from zero, T c gradually decreases and follows relatively closely the Abrikosov-Gor'kov mean-field result down to the critical hτ -1 0 /k B 1.25 K where T c → 0 (see Fig. 2). Concomitant with the fall of T c , the temperature dependence of resistivity shows comparatively slow, albeit noticeable, alterations. We have proceeded to the fit of calculated resistivity curves of Fig. 3 to the polynomial form, ρ a = ρ 0 + AT + BT 2 , in the interval from 10 K down to the lowest temperature. According to Fig. 4, the linear coefficient A of resistivity weakens with FIG. 4. A (left scale) and B (right scale) coefficients of the polynomial fit ρ a (T ) = ρ a0 + AT + BT 2 of the calculated resistivity (Fig. 3) as a function of T c . the size of τ -1 0 or the reduction of T c . When T c → 0 as we approach the threshold τ * -1 0 , the reduction in the ratio A/A 0 reaches about 25%; it continues a steady decrease beyond that point. As for the Fermi liquid BT 2 term of resistivity, it becomes visible and weakly grows, but regularly for T c /T 0 c < 1. It follows that when pair breaking is present the scaling of A with T c no longer holds. It must be stressed, however, that the threshold energy scale for pair-breaking hτ * -1 0 , which is of the order of k B T 0 c , only affects electronic states within a narrow energy shell around the Fermi surface. Although the corresponding decline of the Cooper loop (L C ) entering the RG flow [Eq. (A1)] is sufficient to suppress the instability against d-wave superconductivity, scattering processes taking place at higher energy distance from the Fermi surface remain virtually unaffected. These still contribute to the mixing of Cooper pairing and SDW correlations. The latter alongside umklapp scattering continue to be strong and are responsible for sustaining the amplitude of the T -linear coefficient of resistivity despite the absence of a finite T c .

Before closing this section, it is worth noticing that the suppression of T c as x approaches the critical concentration x c in Fig. 2 gives in principle rise to a quantum critical point. According to Figs. 3 and4, however, this point does not give rise to any irregular change in the variation of linear coefficient for resistivity. This comes from the fact that the electronic degrees of freedom involved in the suppression of T c are confined to a small energy shell, that is of the order of T c itself, near the Fermi surface. These have essentially no additional influence whatsoever on the strength of spin fluctuations responsible for anomalous resistivity.

IV. DISCUSSION

Before we proceed with a comparison between theory and experiments in these alloys, it is important to notice that the present study is not dealing with a random distribution of strongly localized defects. Instead, we are facing a situation where the carrier lifetime in ordered regions is governed by the size of extended disordered domains. The transport data FIG. 5. The anion-ordered volume fraction p derived from the residual resistivity and the effective medium model. The inset shows the x dependence of the inverse correlation length for the (0, 1/2, 0) ordering [START_REF] Ravy | Competition between structural instabilities in organic conductors and superconductors[END_REF][START_REF] Ilakovac | Substitutional disorder and anion ordering transition in the (TMTSF) 2 (ClO 4 ) (1-x ) (ReO 4 ) x solid solution[END_REF] and the inverse length between ReO - 4 sites located at random. We note that around x = 10% the (0, 1/2, 0) anion-ordered coherence length becomes shorter than the average distance between centers. It is above such a concentration that a granular behavior prevails. The superconducting coherence spreads over the whole volume in the 6% sample with p = 0.18 whereas the percolation threshold for ordered regions should arise around p = 0.3, i.e., close to the 7% sample. This is a consequence of the proximity effect for superconductivity.

showing the remanence of nonsuperconducting domains at high ReO - 4 concentration suggest we are facing a problem of granular materials. Hence, the value of the residual resistivity cannot be simply extracted from the knowledge of the ReO - 4 concentration. The situation in the solid solution has similarities with that encountered in pure (TMTSF) 2 ClO 4 when the disorder is caused by rapid cooling of the sample [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. Therefore, we use a similar procedure to extract the fraction of the respective ordered and disordered regions from the resistivity data according to the effective-medium theory for a two-component conductor [START_REF] Landau | Electrodynamical of Continous Media[END_REF][START_REF] Creyssels | Pressure dependence of the electrical transport in granular materials[END_REF],

1/ρ eff = (p(1/ρ Min ) 1/3 + (1 -p)(1/ρ Max ) 1/3 ) 3 , ( 7
)
where p is the volume fraction of the anion-ordered domains. We take ρ Min = 50 m cm according to the data in Fig. 1 and ρ Max is given by ρ Max = ρ Min + ρ c = 280 m cm, where ρ c is the drop in residual resistivity coming from the anion ordering at 24 K which is known from the pure (TMTSF) 2 ClO 4 data ( ρ c = 230 m cm) [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. The derived values for p are displayed in Fig. 5. Using the value for the residual resistivity of sample 0% in Eq. ( 7) would provide p = 1 for that sample in which the residual defects are not under control. As x-ray data have shown that full anion ordering does not exist even in pure and the best relaxed samples [START_REF] Pouget | Structural aspects of the Bechgaard and Fabre salts: An update[END_REF][START_REF]Another sign supporting the existence of residual disorder in (TMTSF) 2 ClO 4 due to anions can be found in the NMR relaxation data in the superconducting state of (TMTSF) 2 ClO 4 recovering a Korringa-like relaxation, namely, the existence of a finite density of states at low temperature ascribed to residual disorder[END_REF], we feel entitled to proceed differently for the determination of p in sample 0%. The derivation of p from Eq. ( 7) is expected to become meaningful only in samples where the additional disorder provided by alloying is large compared to the residual disorder in a pristine sample.

Therefore we have proceeded for p of the 0% sample with a linear interpolation between ρ c 0 = 0 and 100 m cm (x = 3%) in Fig. 5 leading to the determination of p ( 0.78) in the sample 0% of residual resistivity ρ c 0 = 50 m cm. From Fig. 1, the 10% sample exhibits a superconducting transition ending at a finite value for the resistivity at zero temperature, which means that ordered regions are not percolant. Furthermore, in the 6% corresponding to p = 0.18 according to Fig. 5, the transition becomes complete, meaning that pairing coherence becomes infinite. The volume fraction for ordered domains in 6% is smaller than the actual value expected for the percolation threshold, namely, p = 0.3 at which the first cluster of 3D ordered anions with infinite length develops [START_REF] Stauffer | Introduction to Percolation Theory[END_REF]. Such a situation can be understood taking into consideration the proximity effect which enables superconducting coherence over the entire volume even though ordered regions are still separated by small gaps of anion-disordered matter.

Regarding the SDW transition in Fig. 2, the onset of such a ground state in fast cooled samples has first been reported by Tomic et al. [START_REF] Tomić | EPR and electrical conductivity of the organic superconductor di-tetramethyltetraselenafulvalenium-perchlorate, (TMTSF) 2 ClO 4 and a metastable magnetic state obtained by fast cooling[END_REF]. This phenomenon has been ascribed to the presence of Fermi surface folding whenever ClO - 4 anions do not order. The behavior of fast cooled samples has been revisited recently leading in turn to the highest SDW transition temperature at 6.6 K in a clean although fully anion-disordered sample [START_REF] Ya | Role of anion ordering in the coexistence of spin-density-wave and superconductivity in (TMTSF) 2 ClO 4[END_REF]. Such a SDW transition temperature is actually lower than the transition occurring around 12 K in (TMTSF) 2 PF 6 or even 15 K in salts with tetrahedral anions such as (TMTSF) 2 ReO 4 when these latter anions remain ordered at low temperature [START_REF] Tomić | A hidden low-temperature phase in the organic conductor, (TMTSF) 2 ReO 4[END_REF]. As far as the clean but fully disordered (TMTSF) 2 ClO 4 is concerned, the decrease of the mean free path is likely responsible for the lowering of the transition temperature. In the quenched-6% sample more scattering coming from the disordered ClO - 4 anions adds to the original scattering in the relaxed 6% ReO - 4 substituted sample leading in turn to a further lowering of the transition temperature. As emphasized in Sec. III, nonmagnetic defects provide alongside antinesting some additional pair-breaking reducing density-wave correlations and then T SDW , in the same way as nonmagnetic defects act in d-wave superconductors [START_REF] Chang | Anion ordering and superconducting and spin-density wave transition temperatures in Bechgaard salts[END_REF][START_REF] Dora | Impurity effects in unconventional density waves in the unitary limit[END_REF][START_REF]anion potential has been studied on the SDW state of (TMTSF) 2 (AsF 6 ) (1-x ) (SbF 6 ) x where the question of anion symmetry is irrelevant showing a 7% decrease of the transition temperature in a 50% solid solution[END_REF].

We can now plot the dependence of A on T c , as shown in Fig. 6 comparing experimental data provided by the analysis of the resistivity of the metallic phase above T c to the Boltzmann-RG calculation. We must be cautious because A is related to a homogenous medium in the two extreme cases of high and low T c only where the sample is anion ordered or disordered, respectively, but in between (say, 7%) we face a more complex situation where A is related to the mixture of two metallic phases whereas T c is given by the onset of superconductivity in the ordered phase only.

Experiments and theory of Fig. 6 agree on the fact that the suppression of A is no more than 25% once superconductivity in alloys is fully suppressed. In these conditions a sustained strength of a T -linear component for resistivity indicates that in the presence of nonmagnetic pair breaking the system moves away only slowly from the magnetic QCP. The amplitude of SDW correlations and with it the enhancement of umklapp scattering persist to a great extent in the anomalous temperature dependence of resistivity. The comparisons between theory and experiments for the growth of the Fermi FIG. 6. Dependence of A/A 0 on T c , from experiment and theory (solid line). The colored square, triangle, and circle symbols correspond to experimental data according to the different fitting ranges described in Fig. 2. The green diamonds are estimated A/A 0 derived from the two-conductor model in Eq. ( 7) for the inelastic contribution. A/A 0 = 0.75 at T c = 0 K and the volume fractions p according to Fig. 5 have been used. The dashed green line is a guide for the eye. The inset displays the dependence of the unnormalized B on T c . The green diamonds are provided by the two-conductor model and the dashed green line is a guide for the eye. liquid B coefficient of resistivity are displayed in Fig. 4 for the theory and in the inset of Fig. 6 for the experiment. The rise of B as T c decreases is steady and compatible with calculations; its amplitude remains small.

Similarly, in the case of pure (TMTSF) 2 ClO 4 , the evolution of the T -linear term A as a function of the cooling rate can be extracted from a polynomial fit of the raw data of previous works in Ref. [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF], which we reproduce in Fig. 7 for representative cooling rates. The fit is in the window between 4.5 and 10 K, namely, outside the paraconductive downturn of resistivity whose amplitude increases with the cooling rate. The suppression of A is at most 25% when the volume fraction of anion-ordered regions is monitored by the cooling rate leading in turn to a complete suppression of T c , as displayed in Fig. 7.

Finally, we comment on the behavior of ρ c (T ) in pristine (TMTSF) 2 ClO 4 at various cooling rates in Fig. 7. These data come from the extension to finite temperature of a previous work investigating residual resistivity and superconductivity in (TMTSF) 2 ClO 4 as the cooling rate is varied following a procedure already presented in Ref. [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. The resistivity has been normalized to the value of 28 cm at ambient temperature following reports of previous ρ c (T ) measurements [START_REF] Joo | Impurity-controlled superconductivity/spin density wave interplay in the organic superconductor: (TMTSF) 2 ClO 4[END_REF][START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. What is clear at first sight in Fig. 7 is the marked contrast between the strong effect of cooling rate on the residual resistivity and the much weaker effect on the inelastic part, a feature captured by the calculations of Sec. III.

The pure (TMTSF) 2 ClO 4 study [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF] has shown that the material acquires a granular texture above a critical cooling rate of 1 K/min where disordered regions separate anionordered domains in which bulk superconductivity arises and possibly spread over the whole sample via proximity effect. The volume fraction p related to ordered domains has been FIG. 7. ρ c (T ) measured on (TMTSF) 2 ClO 4 and displayed in the temperature range between 12 K and the superconducting region for 0.020, 0.52, 2.5, 7.6, and 18 K/min cooling rates across the anion ordering transition. The same polynomial fitting procedure as for the data in Fig. 1 has been used to extract the A, B, and ρ c 0 parameters at each cooling rate within the fitting window of 5-10 K. The results are plotted as broken lines. estimated at each cooling rate using the model of a twocomponent conductor [START_REF] Yonezawa | Superconductivity in a quasi-one-dimensional metal with impurities[END_REF]. The data for the fitted values of A are displayed on Fig. 8 versus the anion-disordered volume fraction 1p. A has been normalized to its value at the smallest cooling rate. We also show in the same figure the prediction of the effective medium model using Mathiessen's law applied to the inelastic contribution, taking A/A 0 = 1 and 0.75 at 1p = 0 and 1, respectively, following the results from the alloys series. The solid squares with the dotted line are theoretical Boltzmann-RG points corrected for the evolution of the mean free path within each puddle at concentration p. These values provide an improvement for the agreement with the experimental data. The variation of the T -linear resistivity of the metallic phase above T c at various cooling rates appears to be controlled nearly equally by lifetime and phase mixture effects.

V. CONCLUSION

In this work, we have examined the influence of nonmagnetic disorder generated by finite domains of anion ordering on the temperature dependence of resistivity in both (TMTSF) 2 (ClO 4 ) 1-x (ReO 4 ) x alloys and pristine (TMTSF) 2 ClO 4 at different cooling rates. These domains, presumably of different electronic structure, were found to have a strong impact on the onset of superconducting order. T c is suppressed pointing to the existence of pair-breaking effects congruent with unconventional Cooper pairing. At variance with T c , the pronounced T -linear resistivity in (TMTSF) 2 ClO 4 observed in slow cooling conditions does not reveal as a strong decline in alloying. A polynomial decomposition of resistivity temperature dependence has revealed that the linear term is not strongly weakened with x, indicating that quantum critical features remain strong despite the suppression of superconducting long-range order. The scaling between the T -linear term strength and T c , which is seen under pressure in pure samples, was found to no longer hold as a function of disorder tuned by x or to a certain extent the cooling rate.

These results have been compared to a theoretical calculation based on the combination of the linearized Boltzmann equation of transport and the renormalization group approach to the umklapp vertex function. In the framework of the Q1D electron gas model, we have shown that a range of relatively small energy scales for pair breaking is sufficient to instill a characteristic fall in the metallic instability against d-wave superconductivity. The calculated drop in T c agrees with the one observed with x. Although small pair-breaking energy is enough to suppress a scale like T c , its impact on the strength of SDW fluctuations that feeds the amplitude of interparticle umklapp scattering remains relatively weak. The anomalous growth of umklapp at low temperature was found to be weakly altered along with the linear resistivity term whose amplitude remains sizable, in agreement with experiment. Although the correlation between T c and linear resistivity apparently breaks down, the correlation of the latter with spin fluctuations, which are the core of d-wave pairing, prevails.

Institut Quantique (IQ) of the Université de Sherbrooke for its financial support.

APPENDIX: RENORMALIZATION GROUP EQUATIONS FOR THE ELECTRON GAS WITH PAIR-BREAKING EFFECT

In this Appendix we give the one-loop RG flow equations for the momentum scattering amplitudes g i=1,2,3 in the presence of pair-breaking effects due to a finite lifetime τ 0 of the carriers. The RG approach to the quasi-1D electron gas is detailed in Refs. [START_REF] Sedeki | Extended quantum criticality of low-dimensional superconductors near a spindensity-wave instability[END_REF][START_REF] Bourbonnais | Link between antiferromagnetism and superconductivity probed by nuclear spin relaxation in organic conductors[END_REF][START_REF] Nickel | Superconducting and density-wave instabilities in quasi-onedimensional conductors[END_REF]. Each energy shell of thickness 1 2 E 0 ( )d , located at ± 1 2 E 0 ( ) from either sides of the Fermi sheets, is segmented into N p patches, each centered at a particular value of the transverse momentum k ⊥ , where E 0 ( ) = E 0 e -is the scaled bandwidth at step and E 0 ≡ 2E F is the initial bandwidth. The successive partial trace integration of electron degrees of freedom in the partition function leads to the renormalization or flow of the scattering amplitudes g i as a function of . At the one-loop level, the flow combines corrections from the electron-hole (Peierls) and electron-electron (Cooper) interfering channels of scattering. Impurity (back) scattering introduces pair breaking in the form of a carrier lifetime τ 0 in both 2k F electron-hole (Peierls) [START_REF] Patton | Fluctuation Conductivity in the Incommensurate Peierls System[END_REF] and (d-wave) Cooper [START_REF] Sun | Impurity effects in d-wave superconductors[END_REF] channels. This leads to the flow equations for the g i (k

p 1 F 1 , k p 2 F 2 ; k p 3 F 3 , k p 4 F 4 ) → g i (k ⊥1 , k ⊥2 , k ⊥3 , k ⊥4
) on the Fermi surface, which can be written in the compact form

∂ g 1 (k ⊥1 , k ⊥2 , k ⊥3 , k ⊥4 ) = (-2g 1 • g 1 + g 1 • g 2 + g 2 • g 1 )∂ L P -(g 1 • g 2 + g 2 • g 1 )∂ L C , ∂ g 2 (k ⊥1 , k ⊥2 , k ⊥3 , k ⊥4 ) = -(g 1 • g 1 + g 2 • g 2 )∂ L C + (g 2 • g 2 + g 3 • g 3 )∂ L P , ∂ g 3 (k ⊥1 , k ⊥2 , k ⊥3 , k ⊥4 ) = (-g 1 • g 3 -g 3 • g 1 + g 2 • g 3 + g 3 • g 2 )∂ L P + 2g 2 • g 3 ∂ L P , ( A1
)
where ∂ = ∂/∂ . L ν=P ,C are the Peierls and Cooper loops whose derivative at finite temperature comprises an integration over the patch. These take the form where A ν (k ⊥ , q ( ) ⊥ν ) = -⊥ (k ⊥ )η ν ⊥ (η ν k ⊥ + q ( ) ⊥ν ) + η ν ⊥ (η ν k ⊥2(4) + q ( ) ⊥ν ) + ⊥ (k ⊥2( 4) ) (A3) with

∂ L ν (k ⊥ , q ( ) ⊥ν ) = E 0 ( ) 4
⊥ (k ⊥ ) = -2t ⊥ cos k ⊥ d ⊥ -2t ⊥ cos 2k ⊥ d ⊥ and q ( ) ⊥P = k ⊥3 -k ⊥2 = k ⊥1 -k ⊥4 (k ⊥3 -k ⊥1 = k ⊥2 -k ⊥4
), q ⊥C = k ⊥1 + k ⊥2 = k ⊥3 + k ⊥4 ; η P = 1 and η C = -1. θ (x) is the Heaviside function [θ (0) ≡ 1 2 ]. The momentum dependence of couplings in the discrete convolution products "•" over the internal k ⊥ loop variable on the right-hand side of Eqs. (A1) are in order g(k ⊥ , k ⊥4 , k ⊥1 , k ⊥q ⊥P )g(k ⊥ , k ⊥2 , k ⊥3 , k ⊥q ⊥P ) for the Peierls channel, g(k ⊥1 , k ⊥2 , k ⊥ , q ⊥Ck ⊥ )g(k ⊥3 , k ⊥4 , k ⊥ , q ⊥Ck ⊥ ) for the Cooper channel, and g(k ⊥ , k ⊥4 , k ⊥2 , k ⊥q ⊥P )g(k ⊥1 , k ⊥ , k ⊥3 , k ⊥1q ⊥P ) for the "•" product of the off-diagonal Peierls channel.

The integration of Eqs. (A1) up to → ∞ gives the values of the moment-dependent scattering amplitudes g i at temperature T . A singularity in the scattering amplitudes signals an instability of the electron gas against the formation of an ordering state at a critical temperature. For the band parameters and repulsive interactions fixed in Sec. III, the most probable instabilities are against either SDW or d-wave SC ordering. At low antinesting t ⊥ and zero pair breaking τ -1 0 = 0, an SDW instability occurs with a singularity in both g 2 and g 3 amplitudes at the modulation wave vector q 0 = (2k F , q ⊥P = π ) and T SDW ∼ 10-20 K. By tuning the antinesting at the QCP near t * ⊥ 42 K, the SDW is suppressed and the instability occurs in the d-wave superconducting channel at T 0 c ∼ 1 K, which is associated to a singularity for both g 2 and g 1 couplings at zero Cooper pair momentum q 0 = (0, q ⊥C = 0), but with a cosine modulation in k ⊥ space. T 0 c stands as the maximum T 0 c at τ -1 0 = 0 which we liken to the situation of the pure and slowly cooled (TMTSF) 2 ClO 4 compound.

By increasing τ -1 0 it is the weakening of the Cooper loop (L C ) singularity in Eq. (A2) that affects the instability against d-wave superconductivity. T c /T 0 c goes down and follows relatively closely the mean-field Abrikosov-Gor'kov result [START_REF] Sun | Impurity effects in d-wave superconductors[END_REF] as shown in Fig. 2. The concomitant impact of τ -1 0 on the momentum and temperature profile of umklapp scattering and in turn on resistivity as obtained from the Boltzmann equation ( 4) is discussed in Secs. III and IV.

FIG. 2 .

 2 FIG. 2. Normalized value of A from the two conductor model (open circles) versus the residual resistivity and from the fits with different fitting ranges (red triangles, 4.5-10 K; blue squares, 5-9 K; magenta circles, 5-10 K). A has been normalized to the value linearly extrapolated to ρ c 0 = 0. T c for the onset of superconductivity from experiment and theories and T SDW versus ρ c 0 . Data for the 0%Q are provided by Ref. [58].

FIG. 3 .

 3 FIG. 3. Calculated temperature-dependent resistivity as a function of pair-breaking parameter hτ -1 0 /k B . The continuous lines are fits to the polynomial expression ρ a (T ) = ρ a0 + AT + BT 2 .

FIG. 8 .

 8 FIG. 8. Dependence of A/A 0 on the disordered volume fraction 1p in pure (TMTSF) 2 ClO 4 at various cooling rates (blue dots). The model calculations for an effective conductor with A/A 0 = 0.75 at p = 0 are shown as red open squares. The model calculations taking into account the evolution of the mean free path in anionordered regions are the red solid squares.

2 + 1 0×

 21 μA ν | -E 0 ( )/2) E 0 ( ) + μA ν + i hτ -[tanh[βE 0 ( )/4] + tanh[β(E 0 ( )/4 + μA ν /2)]] , (A2)

ACKNOWLEDGMENTS

The samples studied for this work had been synthesized and grown by the late Prof. Klaus Bechgaard. This work in France has been supported by CNRS and in Japan by Japan Society for the Promotion of Science (JSPS) Grant-in-Aids KAKENHI Grants No. JP26287078 and No. JP17H04848. C.B. thanks the National Science and Engineering Research Council of Canada (NSERC) under Grant No. RGPIN-2016-06017 and the Réseau Québécois des Matériaux de Pointe (RQMP) for financial support. Computational resources were provided by the Réseau québécois de calcul de haute performance (RQCHP) and Compute Canada. A.S. thanks the


A Iordan

A Duperray

C Verdier
email: verdier@ujf-grenoble.fr

A fractal approach to the rheology of concentrated cell suspensions

Keywords: numbers: 83.60.La 47.57.Qk 47.57.E-83.85.Cg rheology, CHO cells, Yield stress, Viscoelasticity, Fractal, aggregates

New results on the rheological behavior of CHO cell suspensions in a large range of concentrations are reported. The concentration-dependent yield stress and elastic plateau modulus are formalized in the context of fractal aggregates under shear, and quite different exponents are found as compared to the case of red blood cell suspensions. This is explained in terms of intrinsic microscopic parameters such as the cell-cell adhesion energy and cell elasticity but also the cell's individual dynamic properties, found to correlate well with viscoelastic data at large concentrations (φ ≥ 0.5).

I. INTRODUCTION

The rheology of complex fluids has been studied extensively over the past decades [START_REF] Larson | The and rheology of complex fluids[END_REF] and has revealed very intriguing behaviors, in particular properties of suspensions, either micronic or colloidal, are still a subject of interest [START_REF] Coussot | Rheometry of pastes, suspensions and granular materials[END_REF][START_REF] Flatt | [END_REF]4,5]. Classical behaviors of suspensions usually reveal shear-thinning effects, but other unusual ones like shear-thickening [6] (i.e. viscosity increase with shear rate) or yield stress have been observed [START_REF] Coussot | Rheometry of pastes, suspensions and granular materials[END_REF][START_REF] Flatt | [END_REF]. The yield stress is the critical value of the shear stress needed to induce flow for a given fluid.

It is closely related to the internal structure of the fluid therefore its ability to form (or break) particle clusters under flow. In this respect most studies have focused on solid sphere suspensions.

On the other hand, there are much less works dedicated to suspensions of deformable particles, such as biological cell suspensions. The main works can be found in the field of blood rheology. Suspensions of Red Blood Cells (RBC) within plasma were first investigated by Chien [7,8] and revealed a shear-thinning behavior, but a more detailed inspection of the viscosity-shear rate diagrams showed that at low shear rates, the stress level is close to a constant σ s (Pa), called the yield stress. The well-known Casson's model [START_REF] Casson | A flow equation for pigment-oil suspensions of the printing ink type[END_REF] relating the shear stress σ to the shear rate γ (µ being a constant viscosity) can be used to determine the yield stress:

√ σ = √ σ s + µ γ (1)
Chien and co-authors obtained σ s for a large range of hematocrit (H), i.e. the RBC volume concentration [START_REF] Chien | [END_REF]. They showed a relationship of the type σ s ∼ (Hb) 3 (b being a constant hematocrit).

It is still not known yet whether this type of behavior is universal, or if it could depend on cell type, cell shape or other biological effects such as cell adhesion or cell elasticity.

In particular, one proposed explanation of the yield stress in RBCs suspensions is based on the existence of 'rouleaux' which build due to cell interactions and exhibit large shape aspect ratios [8] and a fractal dimension D. Therefore it is necessary to apply strong enough stresses in order to break such aggregates, in close relation with the yield stress.

In this work we propose to investigate the rheology of a new cell suspension, consisting of CHO cells (Chinese Hamster Ovary cells) in a large range of concentrations. Such cells are commonly used in biology, easy to culture, and can be genetically modified to induce different adhesive properties. These cells are spherical when suspended in a culture medium, and organized in a specific manner leading to particular aggregation patterns of fractal type.

This leads to the determination of scaling laws based on fractal exponents (for the yield stress σ s and elastic modulus G 0 ) which are seen to be non universal but dependent on cell type.

The flow curves constitute a basis to test classical empirical models (Bingham, Casson, Herschel-Bulkley models) and other ones [11,12] based on kinetic theories describing the rupture and formation of particle clusters. The latter ones successfully relate macroscopic effects to microscopic parameters, such as the cell-cell adhesion energy and the cell elasticity.

The microscopic parameters that we find match well the ones found in the literature using other techniques. This is important in the context of recent studies related to tumour growth [13,14,15] which consider cell assemblies with interactions as well as cell elastic deformations. Furthermore, this study emphasizes the relationship between the dynamic rheological properties of suspensions [16] and the single cell properties.

The paper is organized as follows. In §II, we describe the materials and methods of investigation (i.e. mainly rheometry and microscopy). Then steady shear and dynamic oscillatory shear results are presented in §III. In the light of the typical scaling laws obtained, we suggest the use of the model of Snabre and Mills [11] presented in §IV, to analyze our data, and find the corresponding microscopic parameters. Finally, we present an aternative approach based on structural similarity [17] in §V.

II. MATERIALS AND METHODS

In our model system, adherent CHO cells are grown in culture medium (DMEM containing 10% Fetal Calf Serum) using standard T 75 boxes under proper conditions (37 • C, 5% CO 2 ), until they are detached using trypsin, when they reach a confluency of ∼ 70%.

Suspended cells are centrifuged at 1200 rpm, a high enough velocity to get a concentrated suspension, but slow enough in order to maintain the cells alive. Cell volume concentration φ (i.e. similar to the hematocrit H) is determined accurately after centrifugation in hematocrit tubes containing the CHO cells. Then the right amount of remaining supernatant is removed until the desired concentration is obtained (between 0 and 60 %). Different experiments were carried out on a conventional Rheometer (Bohlin Gemini 150). Both steady shear and oscillatory measurements were made at T = 20 o C. Due to the large amount of cells needed (we usually require twelve T 75 flasks in order to obtain a volume of roughly 0.3 mL of cells), we chose to use a plate-plate geometry (20 mm diameter) with a small gap (between 400 µm and 1 mm) for the concentrated suspensions whereas the smaller concentrations (below 10 %) were tested using the 60 mm cone-plane geometry (2 o angle). Typically in our fluid, the suspended cells are spherical and monodisperse with a radius a ∼ 10µm.

III. RESULTS

Experimental results for constant steady state shear rate γ are presented in Fig. 1. The viscosity η is shown to vary over several decades, within shear rates typically between 10 -3 s -1 and 10 3 s -1 . In some cases, we limited ourselves to the higher shear rates because of experimental reasons (i.e. steady state not reached). By a first inspection of the curves, we recognize the signature of a yield stress fluid as depicted by the slope close to -1 in the viscosity-shear rate diagram (or equivalently a constant shear stress at low shear rates), especially at the largest concentrations φ, which will be particularly of interest here. The existence of this yield stress is attributed to weak interactions which can exist after preparation of the system. Already existing proteins are available on cell membranes and can be recruited to form bonds, leading to particular structure arrangments. This explains the presence of a yield stress related to the formation of such structures. The yield stress is found to depend on volume concentration φ in a manner to be discussed later.

A second series of experiments was carried out in order to study the systems under oscillatory strains at frequency f . Small deformations (1% or less) within the linear regime were performed in order to characterize the elastic modulus G ′ (f ) and the loss modulus G"(f ). The frequency values were limited to the narrow range corresponding to fast modes, in order to limit a possible time-dependence of the results, due to sedimentation, protein expression, or cluster formation-destruction. We find an interesting behavior as shown in Fig. 2. Moduli G"(f ) usually prevails over G ′ (f ) at small concentrations (e.g. φ = 0.2), but as φ increases, the system becomes elastic with a much larger G ′ (f ). This behavior is the signature of a viscoelastic medium, due to the fact that interactions between elastic cells become effective at large concentrations (φ ≥ 0.4). The slow increase of the elastic modulus G ′ against frequency reveals the presence of a so-called 'elastic plateau' modulus The presence of elasticity has been observed previously for RBC suspensions [18], above a critical volume fraction around φ = 0.2, and is believed to come from the elasticity of the cells as they are packed more closely at large concentrations such as the ones also encountered in tumour spheroids [14]. Finally, we observe that the trends in the G ′ -G" plots for large concentrations (φ ≥ 0.5) are remarkedly similar to previous microrheological results obtained on single cells [19,20,[START_REF] Hoffman | Proc. Natl Acad. Sci. USA[END_REF]. Indeed, they show a slowly increasing G ′ and G" (increasing slightly faster at frequencies above 1Hz), where the elasticity dominates (G ′ ≥ G").

As in the case of suspensions, we define a maximum packing fraction φ 0 (which is usually 0.64 or even 0.74 for solid spheres in a face-centered-cubic crystal), depending on cell elasticity, i.e. their compactness [START_REF] Quemada | [END_REF]. Due to the presence of soft spherical cells, it is expected that the value of φ 0 will be in this range. φ 0 is determined using the reduced viscosity plot η η 0 as a function of φ (η at a shear rate of 10 2 s -1 , η 0 = 0.0014P a.s the solvent viscosity). In our case, this data (not shown) is found to match the well-known equation

η η 0 = (1 - φ φ 0 ) -2.5φ 0 (2)
proposed by Krieger and Dougherty [23], this providing the value φ 0 ∼ 0.65. Note that this relationship is interesting because it matches Einstein's viscosity for hard spheres [24,25] η/η 0 = 1 + 2.5 φ as well as Batchelor's correction for non colloidal spherical particles [26] η/η 0 = 1 + 2.5 φ + 5.2φ 2 (here an expansion of (2) for small φ gives a second coefficient in the expansion of ∼ 5 instead of 5.2, when using φ 0 = 0.65). Finally, (2) diverges as expected when φ → φ 0 , the limiting packing fraction.

In order to investigate the effect of the volume concentration φ, we first need to obtain the flow curve σ( γ) of the suspensions, as well as the relevant parameters, such as the yield stress σ s . From the viscosity curve in Fig. 1, we plot the stress σ = η( γ) γ vs. shear rate γ in Fig. 3, and fit the data with the Herschel-Bulkley law [2] σ = σ s + M γn , where M is a constant, and n is a shear-thinning exponent ranging between 0 and 1 (1 is for a Bingham frequency f = 1 Hz) are plotted in Fig. 4 as a function of volume concentration. This plot shows power law dependences of the form σ s ∼ φ m 1 and G 0 ∼ φ m 2 and reminds previous results [11] obtained in the case of the rheology of RBCs suspensions, at least for the yield stress σ s . From Fig. 4 we find that m 1 ∼ 8.4 and m 2 ∼ 11.6. The m 1 exponent is quite different from the one obtained in the case of RBCs suspensions (m 1 ∼ 3) as this will be discussed below.

IV. MODELLING

As seen above, rheological modelling of such suspensions should therefore predict shearthinning behavior, as well as yield stress properties at low shear rates γ → 0 and a concentration dependence of σ s and G 0 . In addition, cell suspensions correspond to aggregated systems (see Fig. 5). Under flow, their structure is based on the persistent remodelling of the cells with respect to each other as they exhibit deformations, rotations, possible rolling and/or separation. During such events, cells may form clusters of size R f to be compared with the cell size a (radius). The formation and destruction of cell clusters is the major ingredient to understand the rheological properties of the cell system, in order to explain our data.

For example, when sheared under stress σ, clusters break into smaller ones, leading to shear-thinning effects. On the other hand, the possible encounter of clusters leads to the formation of larger structures, increasing the viscosity. Clusters are organized in a fractal way. First, one needs to consider a cluster at rest. Its size is R f and it contains N cells, linked by the following relationship [START_REF] De Gennes | Scaling concepts in polymer physics[END_REF]:

R f a ∼ N 1/D ( 3
)
where D is the fractal dimension. To determine D, we follow a previous approach [START_REF] Allain | [END_REF] and consider circles (instead of rectangles) of radius R f containing a cluster. Then cells (radius a) are counted for two rather small concentrations (10% and 20%). Clichés like the left one in Fig. 5 are used to draw circles and count the number of cells N. For larger clusters containing more cells, we obtain a linear relationship between log 10 (R f /a) and log 10 (N), as shown in Fig. 6. Note that the two cases studied (10% and 20%) give the same slope for large values of N, this justifying the fractal hypothesis indicated by equation (3). For our system, we determine D ∼ 1.47 from the 2D images. Thus, in three dimensions, we expect a fractal dimension of the order D ∼ 2 [30]. This number is similar to the ones found for RBCs suspensions, although the scaling exponents for yield stresses are quite different.

In the semi-empirical model proposed by Snabre and Mills [11,12], the formation and dissociation of clusters under flow is taken into account. A change in R f as a function of the applied shear stress is assumed :

R f a = 1 ( σ * σ ) m (4)
where m is a dimensionless parameter. σ * is a critical stress related to the interfacial adhesion between cells : σ * = Γ/a, and Γ is the cell adhesion free energy. Using the concept of effective medium with volume fraction

φ A = φ ( R f a ) 3-D (5)
one assumes an effective viscosity :

η(σ) = η 0 1 -φ A (1 -φ A φ 0 ) 2 (6)
and obtains the constitutive equation [START_REF]The 1D-constitutive equation is given by η(σ) = σ γ = η 0[END_REF] which contains the yield stress given by :

σ s ∼ σ * ( φ φ 0 ) 1 m(3-D) (7)
The last parameter to be used in the formula, φ 0 , is the maximum packing concentration found previously.

We use the previous model to explain our experimental data. The exponent m 1 = 8.4 found for the yield stress σ s is plugged into the previous scaling law (7) for determination of the parameter m = 0.078. This is smaller than the values of m found for RBC suspensions (typically m ∼ 0.3). This means that the size of clusters is not so sensitive to the applied stress, indeed one can consider that the cell aggregates are easy to form (or hard to break) because of the round shape of the cells, in contrast with RBCs which need to bind in a very special way to form 'rouleaux'. Thus, once broken by stress, 'rouleaux' are difficult to reform. We have obtained the value of the critical stress σ * = 1.4 N/m 2 , and a corresponding value of Γ = 1.4 10 -5 N/m. This value of σ * is higher than the ones obtained for RBCs [11] but the interfacial energy Γ is in the range of the small values indicated for vesicles [START_REF] Guttenberg | [END_REF]. This is in favor of the initial assumption that few adhesion molecules are involved in the region of contact between the cells. Finally, we postulate a similar relationship [START_REF] De Gennes | Scaling concepts in polymer physics[END_REF] for the shear elastic modulus

G 0 ∼ G * ( φ φ 0 ) 1 p(3-D) (8)
where G * is an effective elastic modulus, but we include an additional exponent n to be determined. We come up with p = 0.056 and G * = 234P a. This value of the reference modulus G * , as explained in the concept of fractal exponents [START_REF] De Gennes | Scaling concepts in polymer physics[END_REF], is to be related to typical values for single cells. In particular, it corresponds to a Young's elastic modulus E * = 702P a (assuming that the cell is incompressible) which is typical for adherent Wild Type CHO cells, of the order 0.5 -1 kP a as measured by AFM [33,34].

V. AN APPROACH BASED ON STRUCTURAL SIMILARITY

Another method for having access to parameters like the yield stress σ s and the viscosity η of such suspensions has been proposed earlier [17]. It can be of interest to mention such an approach since it is relevant to our case, in the context of concentration-dependent laws.

The idea consists in assuming a dependence of the reduced shear stress T = σ/σ s as a function of the reduced shear rate S = η γ/σ s . This similarity is interesting because it can allow to superpose the different curves onto a single master curve. Such an approach has been used previously with success in the case of clay-water suspensions. The master curve is shown in our case for CHO cell suspensions (Fig. 7).

We note again a good superposition of the data, although the few available data points for low reduced shear rates do not allow very accurate results for the parameters under investigation. In an attempt to model the first part of the curve (low shear rates), a relationship of the following kind was found :

T = 1 + 6.13 S 0.47 (9)
The value of the exponent close to 1/2 recalls the well-known Casson's equation but in a slightly different form. In fact, this form is a limiting case of Casson's equation, corresponding to an asymptotic expansion of equation ( 1) for small enough shear rates. The use of equation ( 1) instead of (9) does not fit the whole data. Proceeding further, we can obtain values of the yield stress σ s and viscosity η from the 'ad-hoc' data reduction. This has led us to similar relationships for the yield stress dependence vs. concentration (as in section IV). Similarly, the analysis of the viscosity (η) dependence against concentration φ also shows that equation ( 2) and the following equation η/η 0 = [1 + 0.75/(φ 0 /φ -1)] 2 from Chong and coauthors [35] both predict a correct evolution of the viscosity η leading to a packing fraction of the order φ 0 ∼ 0.65. Therefore we can conclude that this approach is complementary to the previous one in the sense that it can lead to an increased accuracy, when sufficient data is available, although it does not provide physical correlations between microscopic and macroscopic parameters, like the model that we chose to use in the above analysis [11].

VI. CONCLUSIONS AND PERSPECTIVES

To sum up, the system studied here provides unique features important for the rheology of biological suspensions and tissues. These concentrated cell suspensions behave as yield stress fluids (also called visco-plastic materials), for which a fractal approach has been used.

Under shear, the fractal structure changes and can be modelled using a yield stress σ s and elasticity modulus G 0 related to the fractal dimension D. Two other microscopic parameters of interest have been introduced in the model: the cell adhesion energy Γ, and the cell's effective elastic modulus E * found to be : Γ ∼ 10 -5 N/m, E * ∼ 700 P a [START_REF] Chien | [END_REF] The first is in the range of typical values of cell adhesion energies, and the second in agreement with previous microrheology experiments. We also found a similar behavior between the dynamic shear moduli G ′ (f ) and G"(f ) in this study (at φ ≥ 0.5), and the ones obtained from microrheological studies on single cells [19,20,[START_REF] Hoffman | Proc. Natl Acad. Sci. USA[END_REF] using various techniques.

Both show slowly increasing dynamic moduli in terms of frequency, with the same relative positions. This idea probably deserves more attention and should be tested in the future, in particular further works may focus on the characterization of other cellular suspensions including cells with different elastic properties.

Finally, such a study can naturally lead to the understanding of biological tissues, by including stronger adhesion properties between the cells, or by taking into account the addition of Extra-Cellular Matrix components.

FIG. 1 :

 1 FIG. 1: Color online. Viscosity η (Pa.s) vs. shear rate γ (s -1 ) at different volume concentrations φ from 0 to 60%

FIG. 2 :

 2 FIG. 2: Color online. Shear moduli G ′ and G" (Pa) vs. frequency f at different volume concentrations φ ranging from 20% to 60%

FIG. 3 :FIG. 4 :

 34 FIG. 3: Color online. Determination the yield stress σ s using Herschel-Bulkley's model.

FIG. 5 :

 5 FIG. 5: Color online. Phase microscopy of CHO cell suspension : 10% and 52%. Same scale for both images.

FIG. 6 :

 6 FIG. 6: Color online. Plot of R f a as a function of N to determine the fractal dimension D. Two concentrations are used: 10% and 20%. There are noticeable differences at low values of N but there is an increased accuracy for large N , where the two concentrations give rise to the same slope. D = 1.47.

FIG. 7 :

 7 FIG. 7: Color online. Master of the reduced shear stress T vs. reduced shear-rate S.

Acknowledgments -The authors wish to thank the EC Marie Curie Research Training Network MRTN-CT-2004-503661 on "Modelling, mathematical methods and computer simulation of tumour growth and therapy" (http://calvino.polito.it/˜mcrtn/) for financial support, and C. Misbah for fruitful discussions about the manuscript.