A Peigney
email: peigney@iris.ups-tlse.fr

Ch Laurent

E Flahaut

R R Bacsa

A Rousset

Specific surface area of carbon nanotubes and bundles of carbon nanotubes *

Keywords: A. Carbon nanotubes, C. BET surface area, D. Surface areas 2

The theoretical external specific surface area of single-and multi-walled carbon nanotubes and of carbon nanotube bundles is calculated as a function of their characteristics (diameter, number of walls, number of nanotubes in a bundle). The results are reported in diagrams and tables useful to correlate the microscopic characteristics and the specific surface area of samples. The calculated values are in good agreement with the microscopic characteristics and the specific surface area measurements which have been previously reported in the literature. The specific surface area is a macroscopic parameter which can be helpful to adjust the synthesis conditions of carbon nanotubes.

Introduction

electron microscopy (SEM), transmission electron microscopy (TEM) and high-resolution electron microscopy Carbon nanotubes (hereafter denoted as CNTs) have (HREM) observations are used for the characterization unique properties [START_REF] Dresselhaus | Science of Fullerenes and Carbon Nanotubes, San Diego: Academic Fig. 6. Distribution of the number of walls (a) and of the inner Press[END_REF] owing to their particular structure and [START_REF] Dai | Single-wall nanotubes produced by metalcatalyzed disproportionation of carbon monoxide[END_REF][START_REF] Hafner | Catalytic growth of single-2 theoretical SSA. The measured value (178 m / g) is in the wall carbon nanotubes from metal particles[END_REF][START_REF] Kong | Chemical vapor deposition of CNTs reported by the authors [18] could be only partially methane for single-walled carbon nanotubes[END_REF][START_REF] Cassell | Large scale catalytic metal particles covered with carbon probably CVD synthesis of single-walled carbon nanotubes[END_REF][START_REF] Colomer | Synthesis of single-wall carbon nanotubes by catalytic decomposition of hydrocarbons. iron-nickel during interaction with carbon monoxide-hydro[END_REF], sometimes in combination with Raman scatterare thus among the probable constituents of future materi-ing which also brings some information on the diameter of als. However, great quantities of well-characterized and SWNTs [START_REF] Cheng | of bundles of SWNTs, each 1.3 nm in diameter. The Marucci A et al. Bulk morphology and diameter distribution ratio between the experimental and theoretical values is ca. of single-walled carbon nanotubes synthesized by catalytic 0.22. From Table 2, we deduce that this value corresponds decomposition of hydrocarbons[END_REF]. However, the use of a measurable macroreproducible CNTs are needed to prepare such novel scopical parameter such as the specific surface area, in materials. For this reason, many research groups are addition to more local techniques, has been found to be working towards the low-cost, large-scale synthesis of very helpful [START_REF] Peigney | Carbon account the uncertainties on the diameter and thickness of nanotubes grown in situ by a novel catalytic method[END_REF][START_REF] Peigney | Carbon[END_REF][11][12][13]]. CNTs. In particular, the catalytic decomposition of hydro-Owing to its nanometric diameter, a CNT has a very carbons or CO over metal nanoparticles, also called carbon large specific surface area (denoted as SSA hereafter). In chemical vapour deposition (CCVD), is widely explored materials science, the SSA of macroscopic samples of with the aim to maximize the proportion of single-walled powders or porous materials is routinely determined by the nanotubes (SWNTs) in the products. The method is well measurement of gas adsorption (generally N at 77 K) and 2 known for the synthesis of carbon fibres (see Ref. [2] for a calculations using the Brunauer-Emmett-Teller (BET) review) and at first, specimens containing mostly multi-isotherm. This method is not destructive. Many authors walled nanotubes (MWNTs) with large diameters have conducted SSA measurements on carbon filaments nm) and multiple walls were prepared. More recently, and have correlated the measured values with the microsome works have shown that SWNTs can also be obtained scopical characteristics and / or synthesis parameters [START_REF] Dai | Single-wall nanotubes produced by metalcatalyzed disproportionation of carbon monoxide[END_REF][START_REF] Peigney | Carbon account the uncertainties on the diameter and thickness of nanotubes grown in situ by a novel catalytic method[END_REF][START_REF] Hafner | Catalytic growth of single-2 theoretical SSA. The measured value (178 m / g) is in the wall carbon nanotubes from metal particles[END_REF][START_REF] Kong | Chemical vapor deposition of CNTs reported by the authors [18] could be only partially methane for single-walled carbon nanotubes[END_REF][START_REF] Cassell | Large scale catalytic metal particles covered with carbon probably CVD synthesis of single-walled carbon nanotubes[END_REF][START_REF] Cheng | of bundles of SWNTs, each 1.3 nm in diameter. The Marucci A et al. Bulk morphology and diameter distribution ratio between the experimental and theoretical values is ca. of single-walled carbon nanotubes synthesized by catalytic 0.22. From Table 2, we deduce that this value corresponds decomposition of hydrocarbons[END_REF][START_REF] Colomer | Synthesis of single-wall carbon nanotubes by catalytic decomposition of hydrocarbons. iron-nickel during interaction with carbon monoxide-hydro[END_REF][START_REF] Peigney | Carbon[END_REF][11][12][13]. The main difficulty is to optimise the synthesis [2,[START_REF] Walker | Carbon forma[END_REF][START_REF] Park | Carbon deposition on[END_REF]. Some researchers have reported such measureconditions and the catalyst materials in order to avoid the ments on CNTs specimens [START_REF] Hernadi | Part I: influence of Lucas AA et al. Catalytic synthesis of carbon nanotubes 2 3 the Fe content on powder synthesis[END_REF][START_REF] Ye | Hydrogen adsorption and[END_REF]

MW e

The calculations are based on the following hypotheses:

The weight of this MWNT (w ) can be easily calculated The surface of all the graphene sheets which compose the network (Fig. 3). In agreement with the measurements of MWNT is:

Ge and Satller [START_REF] Ge | Vapor-phase grown bundles of carbon[END_REF], the distance between two adjacent CNTs is assumed to be d 5 0.3400 nm. The surface area

p.L[d 2 (n 2 1).d ],
the part of the accessible surface of each CNT. An e s -s example is given in Fig. 3 for N 5 3. We assume that the the different terms successively representing the surface of surface of the individual CNTs corresponding to the inner the external shell (number 1), of the first internal shell surface of the bundle (the dashed lines in Fig. 3) is not (number 2) and so on up to the innermost shell (n). This accessible to the adsorbate and therefore will not be expression can be simplified: included in the calculations of SSB3.

n 21
The surface area of a bundle made up of only two CNTs

s 5 pL nd 2 2d O i . (7)
F G (Fig. 4a) is SSB252 SSNT. For three CNTs (Fig. 3 and GS e s -s i 51

Fig. 4b): The specific surface area of the MWNT is:

p / 3 ] s SSB3 5 3 SSNT 2 3 SSNT 5 2.5 SSNT. MW 2p ]] SSA(MWNT) 5 . w MW
Starting with a bundle composed of two CNTs, each added Combining with Eqs. ( 4)- [START_REF] Cassell | Large scale catalytic metal particles covered with carbon probably CVD synthesis of single-walled carbon nanotubes[END_REF] gives:

CNT produces an SSBN increase equal to 0.5 SSNT, up to a total of six CNTs (Fig. 4c): SSA(GS).

d e ]]]]]] SSA(MWNT) 5 n 21 SSBN 5 [2 1 0.5 (N 2 2)] SSNT where 3 # N # 6. nd 2 2d O i F G e s -s i 51
However, the surface area of the bundle remains equal to 4 SSNT when the 7th CNT is added (Fig. 4d). This and thus corresponds to the completion of the first layer (noted as 1315.d e N9 5 1) surrounding the original CNT. CNTs. This equivalent number N is easily calculated for eq 2 where n $ 2, d is in nm and SSA(MW) is in m / g. e each CNT addition (Fig. 5, Table 1). The SSA of CNTs is plotted versus the diameter for Moreover, when N identical CNTs gather to form a different numbers of shells (Fig. 2). The internal diameters perfect bundle (i.e. N9 full layers), the SSA is modified by were considered to be in the range 0.7-10 nm. The SSA a correction factor f 5 N /N (Table 2). f decreases when eq 2 values are scattered on a very broad scale, from 1315 m / g more layers are added to the bundle, as illustrated by the larger increase of its weight. The diameter influences also the individual CNTs (Table 2). Since counting the CNTs the SSA of MWNTs (Fig. 2b andc), but to a lesser extent within a bundle is possible only on a HREM image of the than the number of shells.

]]]]] SSA(MWNT) 5 (8)
cross section of a bundle, which is sometimes difficult to obtain, the ratio d /d is more easily obtained from the b e

Specific surface area of bundles of CNTs

images than the number of layers N9.

In the case of bundles of SWNTs, SSBN depends only SWNTs and small MWNTs are seldomly obtained on the number of the involved SWNTs according to the isolated but most often form bundles, either because they expression: have grown in this configuration from a single catalytic particle, or because Van der Waals forces made them gather SSBN 5 SSA(SWNT).f 5 1315 f after each has grown from an individual catalytic particle.

2 Thus, it is necessary to calculate the reduction of SSA due which gives 751 m / g for a small bundle of seven 2 to the formation of the bundles. We consider that each SWNTs, 484 m / g for a bundle made up of 19 SWNTs, 2 bundle is composed of N identical carbon nanotubes, of and only 151 m / g for a very large bundle composed of diameter d , which are perfectly arranged in a triangular 217 SWNTs. e Fig. 2. Specific surface area of carbon nanotubes versus their diameter and number of walls.

Discussion

diameter. In addition, the SSA of a CNTs bundle decreases when the number of CNTs which compose it increases. The previous calculations show that the theoretical SSA Thus, large MWNTs as well as nanofibers, which are not of CNTs are dispersed on a very broad scale, mainly always tubular, will have a small SSA so far as they do not depending on the number of walls and also on the become porous [2,[START_REF] Walker | Carbon forma[END_REF]. Thus, one can conclude that max- imizing the SSA is a good criterion to synthesize samples in which a majority of the carbon filaments are SWNTs or bundles of SWNTs and that the measure of the SSA can provide a good representation of the characteristics of the CNTs in the specimen under examination.

The present laboratory has made use of a method based on SSA measurements and carbon analysis [START_REF] Peigney | Carbon[END_REF][11][12][13], in thus immediately available for CNT nucleation and growth that in high SSA samples, the filaments are only bundles of at the appropriate temperature. A CNTs-metal-oxide CNTs, whereas in low SSA samples most filaments are composite powder is thus obtained. The study of the carbon nanofibers, with irregular surfaces, and much influence of the iron content in the starting shorter than the CNT bundles. A study of the influence of a-Al Fe O powders [START_REF] Peigney | Carbon[END_REF] showed that the higher SSA the CH content in the H -CH reducing gas mixture [11]

222x 2x 3 4 2 4 2 2
of carbon (250 m / g), that we called the best carbon revealed that the SSA of carbon is maximized (230 m / g) quality, is obtained with only 5 cat.% Fe (i.e. x 5 0.5). Iron in the 9-18 mol.% CH range. The observed filaments 4 contents higher than 10 cat.% lead to much smaller SSA were CNTs, most having less than four walls, and generally tions, it was shown [12,13] that the SSA of carbon is hence that their inner surface area contributes to the 2 maximal (about 350 m / g) when using only cobalt (i.e. experimental value of SSA. However, the increase of SSA y 5 0), and for x 5 0.1. All filaments in that case are due to the opening of some CNTs may be balanced by the bundles of CNTs most of which are SWNTs.

decrease of SSA due to the gathering of some of the CNTs We have separated the CNT bundles from the remaining in bundles. part of the material by dissolving the oxide substrate Hernadi et al. [START_REF] Hernadi | Part I: influence of Lucas AA et al. Catalytic synthesis of carbon nanotubes 2 3 the Fe content on powder synthesis[END_REF] prepared mixtures of SWNTs and (MgO) and part of the catalytic Co particles in HCl [START_REF] Flahaut | Synthesis of using binary (Fe, Co, Ni) alloy nanoparticles prepared in situ single-walled carbon nanotube-Co-MgO composite powby the reduction of oxide solid solutions[END_REF].

MWNTs by CCVD on Co-zeolite catalyst. The measured 2 SEM observations have shown that the bundles are less SSA values were 653 m / g for specimens where the 2 than 20 nm in diameter and TEM observations revealed a zeolite was removed by HF treatment, and only 312 m / g distribution of the diameters and numbers of walls (Fig. 6).

after a further purification by KMnO oxidation. The TEM 4

The measured SSA of carbon in the final sample was equal and HREM observations revealed ropes of SWNTs and 2 to 365 m / g. From the histograms (Fig. 6) and the MWNTs having between 5 and 27 walls, and external diagrams (Fig. 2), we calculate that the theoretical SSA diameters between 8 and 28 nm. The theoretical SSA of would be equal to 940 m / g if all CNTs were individual. such MWNTs is between 400 and 70 m / g and if we The ratio between the experimental and theoretical values consider that the bundles of SWNTs can have a SSA of the 2 2 is 0.39, which corresponds to bundles of about 18 CNTs order of 300 m / g, then the reported value of 312 m / g is (Table 1) and d /d 5 5 (Table 2). Using the average CNT in acceptable agreement with the electron microscopy b e 2 diameter (2.5 nm) deduced from the histograms (Fig. 6), observations. However, 653 m / g seems to be a very high the bundle's diameter should be about [START_REF] Park | Carbon deposition on[END_REF] values are in good agreement with the microscopy observations [START_REF] Ye | Hydrogen adsorption and[END_REF] showing bundles about 10 nm in diameter, made up of ca. 50 SWNTs.

Conclusions

The theoretical external specific surface area of CNTs has been calculated as a function of their number of walls and diameter. Also, the theoretical specific surface area of CNT bundles has been calculated as a function of the number and characteristics of the CNTs which compose them. The theoretical specific surface area range over a 2 very broad scale, from 50 to 1315 m / g. The value mostly depends on the number of walls. The measured specific surface areas of CNTs samples which have been previously reported in the literature are in good agreement with the theoretical specific surface area values determined from the microscopic characteristics described by the respective authors. Moreover, specific surface area measurements have been efficiently used by the present authors to optimize the synthesis of CNTs. Thus, the specific surface area is a macroscopic parameter, relatively easy to measure, which gives a good representation of the characteristics of CNTs, either isolated or in bundles.

5 Fig. 1 .

 51 Fig. 1. Schematic representation of the arrangement of carbon atoms within the graphene layer (a) and of a multi-walled carbon nanotube made up of concentric graphene shells (b).

n 21 We

 21 can define N as the number of individual CNTs eq nd 2 0.68 O i F G e with a total SSA equal to that of a bundle made up of N i 51

2 for

 2 SWNTs to about 50 m / g for a MWNT 35 nm in following examples: diameter with 40 walls. Fig. 2b and c are enlarged charts 4 of Fig. 2a. ] f 5 5 0.571 for N9 5 1 (7 CNTs), 7 It appears that the number of shells is the predominant 2 parameter influencing the SSA (1315 m / g for SWNTs, 368 for N9 5 2 (19 CNTs). 680-850 m / g for 2WNTs, 295-430 m / g for 5WNTs, 19 etc.) because each shell addition does not provoke a strong A particular value of N9 corresponds to a value of the ratio increase of the surface area of the CNT but causes a much d /d , were d is the diameter of the bundle and d that of b e b e

Fig. 3 .

 3 Fig. 3. Schematic representation of the triangular arrangement of carbon nanotubes within a bundle.

Fig. 5 .

 5 Fig. 5. Schematic representation of a bundle, showing the succes-

Fig. 4 .

 4 Fig. 4. Schematic representation of the formation of bundles by successive additions of carbon nanotubes.

  Calculations were also done for the SSA of CNT bundles, as a function of the number and characs corresponds to two carbon atoms whose weight is,

	] OE 3 ] C2C 2 2 number of walls). C establishing the relations between the theoretical external 220 2 SSA of CNTs and their characteristics (diameter and s 5 3 d 5 5.246 3 10 m . (1) h 23 reported in the form of diagrams and tables which may be 12.01 g / mol) and the Avogadro number (156.023310
	useful to other researchers. It is furthermore shown that the	21 mol ):
	previously reported measured SSA of CNT specimens are in good agreement with the theoretical SSA, which has	w 5 (2 M /1 ) 5 3.988 3 10 h C	223	g.	(2)
	been estimated or calculated on the basis of the micro-scopic characteristics reported by the respective authors. We also give some examples on how the SSA measure-	Combining (1) and (2) gives the specific surface area of one side of a graphene sheet (GS):
	ments have been very efficient to optimize the parameters of a CCVD method [4,10-13]. Thus, it is demonstrated that the SSA is a macroscopic parameter which cheaply	SSA(GS) 5	s h ] w h	2 5 1315 m / g.	(3)
	and rapidly provides useful information on the microscopic	It follows that the SSA of a SWNT is that of one side of a
	characteristics of CNT specimens and is thus very helpful	graphene sheet, whatever its diameter:
	to adjust synthesis conditions of CNTs.
		2 SSA(SWNT) 5 1315 m / g.
	2. Calculations and diagrams	We now consider a MWNT (Fig. 1b) with an external diameter d , a length L and a number of shells n. Its e
	2.1. Specific surface area of individual CNTs	external surface is:
		s	5 p L d .
						. Other workers have
	formation of fibres and MWNTs. Very often, only scanning	studied the adsorption of N at 77 K, from which the
			In the present paper, we report geometrical calculations

h teristics of individual CNTs in the bundles. The results are taking into account the atomic weight of carbon (M 5

Table 1

 1 Equivalent number (N ) of CNTs with a specific surface area equal to that of a bundle made up of N CNTs, arranged in N9 layers

				eq
	1st layer	N	2	3	4	5	6	7
	(N951)	N eq	2	2.5	3	3.5	4	4
	2nd layer	N	8	9	10	11	12	13	14	15	16	17	18	19
	(N952)	N eq	4.5	5	5	5.5	5.5	6	6	6.5	6.5	7	7	7
	3rd layer	N	20	21	22	23	24	25	26	27	28	29	30	31	32	33	34	35	36	37
	(N953)	N eq	7.5	7.5	8	8	8	8.5	8.5	8.5	9	9	9	9.5	9.5	9.5	10	10	10	10
	4th layer	N	38	39	40	41	42	43	44	45	46	47	48	49	50	51	52	53	54	55
	(N954)	N eq	10.5	10.5	10.5	11	11	11	11	11.5	11.5	11.5	11.5	12	12	12	12	12.5	12.5	12.5
	arranged in very long bundles. Starting from		most of CNTs are individual. However, it cannot be
	Mg	12x 2 y	9 M M Al O (M, M95Fe, Co, Ni) solid solu-x y 2 4		entirely ruled out that some CNTs are open-ended and

Table 2

 2 Correction factor ( f ) used to calculate the specific surface area of bundles made up of N identical CNTs, arranged in N9 layers, from the

	specific surface area of the individual CNT	a
	N9		1	2	3	4	5	6	7	8
	f 5 N /N eq	4 / 7	7 / 19	10 / 37	13 / 61	16 / 91	19 / 127	22 / 169	25 / 217
			0.571	0.368	0.270	0.213	0.176	0.150	0.130	0.115
	d /d b	e	3	5	7	9	11	13	15	17

a N , equivalent number of CNTs (see text and Table

1

). d and d are the diameters of the bundle and the individual CNT, respectively. eq b e


Florence Collet
email: florence.collet@univ-rennes1.fr

Sylvie Prétot

Sylvie Pretot
email: sylvie.pretot@univ-rennes1.fr

VARIATION WITH FORMULATION, DENSITY AND WATER CONTENT

Keywords: experimental, hot wire, self-consistent scheme, hemp concrete

come

Introduction

In a context of sustainable development, green buildings aim at reducing the environmental impacts of buildings while also ensuring high indoor environmental quality (comfortable and healthy). The main impacts of buildings on the environment are due to consumption of resources (energy, raw material, water…) and to emissions (greenhouse gas, pollution, wastes...). Thus, green buildings should be energy efficient while showing a light foot-print on the environment over the entire life-cycle.

The energy efficiency of buildings depends on the hygrothermal behaviour of the building envelope and on the performance of systems. This behaviour is related to hygric and thermal properties of constitutive materials. Among these properties, the thermal conductivity is dependent on several parameters such as density, water content and temperature. The bibliography provides studies on the effect of density, of moisture content and of temperature on the thermal conductivity of building materials. For example, Uysal and al. studied the effect of cement dosage and of prumice aggregate ratio on density and thermal conductivity of concrete [Uysal and al., 2004]. In all studied cases, the thermal conductivity increases with density. The prumice aggregate allows reducing the thermal conductivity by 46% while reducing the density by 40%. Del Coz Díaz and al. measured the thermal conductivity of lightweight concrete produced from expanded clay. The thermal conductivity increases from 176 to 256 mW/(m.K) when the density increases from 973 to 1362 kg/m 3 at 23°C, 50% RH. Bederina and al. have shown that the addition of wood shaving in dune sand concrete allows reducing the thermal conductivity from 1.20 to 0.55 W/(m.K) while reducing the density from 2100 to 1400 kg/m 3 [START_REF] Bederina | [END_REF]. In addition, even if literature mainly provides dry state values, the thermal conductivity of hygroscopic materials increases with moisture content. For example, from dry state to saturated state, (i) the thermal conductivity of autoclaved aerated concrete increases up to six times [Jerman and al., 2013], (ii) the thermal conductivity of wood-concrete composite increases up to 2.76 times [Taoukil and al., 2013]. For lightweight concrete, the thermal conductivity increases by 2 from dry state to 100% RH [del Coz Díaz and al., 2013]. These authors give mathematical expression which defines the relationship between density, moisture content and thermal conductivity. Finally, Jerman and al. show that the dependence of thermal conductivity on temperature is less important than its dependence on moisture content for temperatures between 2 and 30°C [Jerman and al., 2013].

Bio-based building materials are derived from renewable biological resources. So, they are an answer to the problem of resource depletion. They are drawn from various raw materials such as wood [Aigbomian and al., 2013], coconut [Alavez-Ramirez and al., 2012], diss [Sellami and al., 2013], sunflower [Binici and al., 2013], hemp... Hemp straw provides two products used in building materials. Fibbers are used to produce insulating panels [Arnaud, 2000] [Collet andal., 2011] and shiv are used as bio-aggregate in hemp composites (concrete, render…). Hemp concrete is used for several applications: wall, floor or roof.

The main difference between applications is the hemp to binder ratio in the mix. Furthermore, hemp concrete can be produced by spraying, moulding or precasting.

Hemp concrete is environmentally friendly as it is made of a renewable raw material (hemp) and as it allows carbon sequestration [Boutin and al., 2005[START_REF] Miller | Life cycle greenhouse gas emissions of hemp-lime wall constructions in the UK[END_REF]][Pretot and al., 2014]. Hemp concrete is highly porous, with open and interconnected porosity. It is a lightweight material with densities between 200 and 600 kg/m 3 , depending on application [START_REF] Amziane | Bio-aggregate-based Building Materials, Applications to Hemp Concrete[END_REF]. Therefore, it shows quite low thermal conductivity, about 100 mW/(m.K) for a walling application [Arnaud, 2000] similar to other building materials with comparable density (cellular concrete for example). The thermal conductivity of hemp concrete depends on the amount of shiv which impacts density. For hemp percentage in weight between 20 and 40%, the density decreases from 611 to 369 kg/m 3 and the thermal conductivity decreases from 140.8 to 94.7 W/(m.K) [Benfratello and al., 2013]. The open and interconnected porosity allows moisture transfer and storage. The hygric characteristics of the material are moisture permeability and sorption curves that are representative of steady state. Previous studies have shown that hemp concrete is strongly hygroscopic, with high moisture transfer and storage capacities (water vapour permeability about 3.2 E-11 kg/(m.s.Pa) and sigmoid sorption curves with high hysteresis loop between adsorption and desorption curves) [Collet and[START_REF] Collet | [END_REF]al., 2013].

These properties allow hemp concrete to moderate ambient relative humidity variations. This ability is quantified through the Moisture Buffer Value of the material. Studies performed following the Nordtest Project classify hemp concrete from good to excellent hygric regulator (MBV around 2 g/(m².%RH)) [Rode, 2005][Collet and Pretot, 2012][Collet and al., 2013]. For comparison, the MBV of cellular concrete is about 1 g/(m².%RH) [Rode, 2005]. Studies performed at wall, or building scale have highlighted sorption / desorption or condensation / evaporation phenomena in hemp concrete wall [Pretot andCollet, 2012][Shea andal., 2012]. These phenomena involve binding or latent energy that impact energy balance of the wall. Thus, hemp concrete shows high hygrothermal performances which allow energy saving and high indoor comfort [Tran Le andal., 2010] [Evrard andde Herde, 2010]. For example, Tran Le and al. performed a numerical study to compare hemp concrete behaviour to that of cellular concrete. They found that hemp concrete induces a reduction ranging from 15 to 45 % in energy consumption, depending on ventilation strategy.

Actually, the energy needs (and indoor comfort) can be simulated thanks to dynamic thermal modelling which requires, among input data, the hydric and thermal properties of building materials. Such numerical models should take into account the variation of hydric and thermal properties with hygrothermal state like the variation of hydric properties with temperature or the variation of thermal properties with humidity.

This study deals with the thermal conductivity of five hemp concretes that differ from formulation.

Firstly, the thermal conductivity of hemp concretes is studied versus formulation. The impact of raw materials and hemp to binder ratio is highlighted. Then, the increase of thermal conductivity with dry density and water content is considered. The investigations are based on experimental measurements and on self consistent scheme modelling.

Materials

Formulations and production methods

Hemp concrete is a bio-aggregate-based building material made of hemp shiv and binder. This study investigates five hemp concretes that differ from formulation and production method. This work was supported by two industrial partners. The studied materials are representative of the materials usually produced by the partners.

The spraying method consists in mixing hemp shiv and lime-based binder to constitute a dry mix that is blown along a pipe by means of a flow of compressed air. Water is added just before the hose outlet, the quantity of which may be controlled by the operator via one valve. This allows reducing the quantity of water compared to conventional mixing which requires more water due to the absorption capacity of hemp shiv. In this study, hemp concrete is sprayed into moulds (figure 1). Three hemp to binder ratios are considered regarding practices: wall, floor and roof (table 1). Moreover, the spraying method induces small density variation due to the angle and the distance of spraying [START_REF] Elfordy | Mechanical and thermal 415 properties of lime and hemp concrete (''hempcrete'') manufactured by a projection process[END_REF], so three densities of sprayed hemp concrete are considered (light, medium and heavy) for the wall mix. The moulding method consists in mixing hemp shiv and lime-based binder in a mixer. Water content is adjusted to obtain a consistent workability of fresh hemp concrete. Moulds are filled with the mixture and the hemp concrete is slightly compacted.

The precasting method results from an industrial process. Firstly, slaked lime is produced from CaO and water. Hydraulic lime and hemp shiv are then added and the mixture is poured into moulds. Blocks are finally formed by compaction under vibrations (figure 2).

Figure 2. Precasting method (left : mixer, right : block press)

For this study, for all formulations and production methods, hemp concrete is manufactured in moulds of 30 cm×30 cm×16 cm (figure 1 and2). This size is representative of a wall thickness or of industrial precast blocks. Once produced, blocks are stabilized at 23°C, 50% RH. Then, the blocks are cut to obtain 15×10×5 cm 3 specimen. For SHC roof and for PHC, it was not possible to cut the blocks because they were exfoliating. For these materials the measurement are performed on blocks at (23°C; 50%RH) only.

Table 1 summarises the mix proportioning and manufacturing method of studied hemp concretes: Sprayed Hemp Concrete (SHC), Moulded Hemp Concrete (MHC) and Precast Hemp Concrete (PHC).

Density and porosity

Density is calculated from mass and dimensions of specimens. Porosity is measured by pycnometry.

Table 2 gives the apparent density at (23°C, 50%RH) and the total porosity of studied materials.

For sprayed hemp concretes, the apparent density ranges from 260 kg/m 3 for roof to 460 kg/m 3 for floor.

For wall mix, it ranges from 390 kg/m 3 for light wall to 460 kg/m 3 for heavy wall. The moulded hemp concrete shows an apparent density of 381 kg/m 3 while the precast one shows an apparent density of 457 kg/m 3 . These values are consistent with the values commonly found in literature [START_REF] Amziane | Bio-aggregate-based Building Materials, Applications to Hemp Concrete[END_REF].

The total porosity of studied hemp concretes ranges from 72% for precast hemp concrete to 85% for sprayed hemp concrete roof. This high porosity includes a wide range of pores from micrometric pores in binder matrix and hemp shiv to millimetric pores due to the arrangement between the hemp shiv and to the hemp-binder adhesion [START_REF] Collet | [END_REF]. 3 Methods

Experimental investigation of thermal conductivity

Hot wire method

The thermal conductivity λ (W.m -1 .K -1 ) was measured using the commercial CT-meter device. In this study, this device was equipped with a five-centimeter-long hot wire (figure 3). The measurement is based on the analysis of the temperature rise versus heating time. For cylindrical geometry, Blackwell [START_REF] Blackwell | Blackwell[END_REF] and Carslaw and Jaeger [Carslaw and Jaeger, 1959] solve the equation of heat conduction for a two media system including (i) the probe, assumed as an ideal infinitely thin and long line heating source, and (ii) the studied material, that constitutes an infinite surrounding and is supposed to be homogeneous and isotropic. For a sufficiently long time, there is a proportional relationship between temperature rise ∆ and logarithmic heating time (ln ( )) (figure 3)(1):

∆ 4. . (ln( ) ) (1)
Where is the heat flow per meter (W.m -1 ) and K is a constant including the thermal diffusivity of the material. 1). In this study, the heat flow ranges from 2.9 to 4.6 W/m and the heating time is 120 seconds. According to the manufacturer, the hot wire is well adapted for the measurement of thermal conductivities ranging from 0.02 to 5 W.m -1 .K -1 and the expected accuracy is 5%.

The main advantage of this method, compared to steady-state methods like hot plate, is that it is a transient method that does not induce (or that does limit) water migration during test [Hladik, 1990]. This allows the study of the effect of moisture content on thermal conductivity. The main disadvantage is that it is a localized measurement. Thus, measurements should be taken several times to ensure the representativeness of the thermal conductivity value. In this study, the given thermal conductivity is the average of five values with a variation coefficient lower than 5%. In addition, the correlation coefficient R² between experimental data and equation ( 1) is higher than 0.999 for all the measurements.

An infrared photography (figure 4), taken immediately after the measurement, allows viewing the cross section of the probed volume. The thermal wave doesn't reach the edges of the sample, so the hypothesis of infinite surrounding assumed in the model of the hot wire is valid. The radius of the probed volume is about 2.5 cm (figure 4), thus the probe volume is higher than the representative volume of hemp concrete (representative length about 5cm) [Collet and al., 2013].

Conditioning of specimens

In order to measure the effect of moisture content on thermal conductivity, the specimens are conditioned at 23°C to several ambient relative humidity (dry point, 33%RH, 50% RH, 81%RH and 90%RH) using dry chamber or climatic chamber Vötsch VC4060. Prior to the test, the specimens are discontinuously weighed until a constant mass is reached (difference lower than 0.01% for two consecutives weighing with 48 hours time step). The water content by mass w (g/g) is then calculated from wet mass and dry mass of the specimen.

During the test, specimens are kept at constant relative humidity in dry or climatic chamber.

Modelling thermal conductivity of hemp concrete

Modes of heat transfer in hemp concrete

In porous media, heat transfer can take place by three modes such as conduction, convection and radiation.

As soon as there is a temperature gradient, conduction takes place by means of molecular agitation within the material. The transfer occurs from higher temperature area to lower temperature area and acts to equalize temperature differences. Conduction takes place in all forms of matter such as solid, liquid and gas.

Heat transfer by convection results from convective currents of fluid that transfer heat from hotter place to cooler place. According to [Saatdjian, 1998], free convection takes place if the Rayleigh number is higher than 4π². The Rayleigh number is the product of the Prandtl number and the Grashof number

. The Prandtl number is the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity;

it is a characteristic of the fluid itself. The Grashof number approximates the ratio of the buoyancy to viscous force acting on a fluid (2) :

× × Δ × (2) With (i) the gravitational acceleration [m/s 2 ], (ii) ! the coefficient of thermal expansion [K -1 ], (iii)
the density of air [k/.m 3 ], (iv) the dynamic viscosity [kg/(m.s)], (v) the maximum length of pores

[m] and (vi) Δ the difference between the temperature of the surface and the temperature of the fluid

[°C], estimated from the temperature rise during the measurement Δ "#$ (about 10°C), the radius of the probed volume (2.5 cm) and the maximum length of pores (3mm) :

∆ Δ "#$ × (3)
For a temperature of 300 K, the density of air is 1.177 kg/m 3 , the dynamic viscosity is 1.85E-05 kg/(m.s) and the Prandtl number is 0.708 [START_REF] Sacadura | Initiation aux transferts thermiques, Techniques et documentation[END_REF]. The Rayleigh number is thus equal to 3.04.

This value is lower than 4π² for the maximum length of pores, so the free convection doesn't occur in hemp concrete.

Lastly, all matter with a temperature greater than absolute zero emits thermal radiation that is a consequence of thermal agitation of its composing molecules. The radiative effect in hemp concrete during the measurements is estimated from the model of Siegel and Howel [Siegel and Howel, 1972].

This model gives the radiative heat flux assuming that the porous medium is a set of parallel screens (solid phase) separated by a fluid phase (with absorption coefficient % & ). Gliksman, according to [Kalboussi, 1990], gives an equivalent thermal conductivity to radiative transfers considering that the fluid phase is non absorbent (% & =0), that the difference of temperature between the two sides of the sample is quite low, and that the distance between the screens is equal to the mean diameter of pores (4).

'#( 4) × * "+#, × #-. 2 / -1 (4)
Where (i) ) is the Stefan-Boltzmann constant (5.67 E -08 W/(m².K 4 )] ; (ii) * "+#, is the mean diameter of pores estimated to 1.2 E-05 m from mercury porosimetry results and from millimetric porosity of hemp concrete [Collet, 2004][Collet andal., 2008] ; (iii) #-. is the average temperature between the fluid and the solid phase (about 301 K) ; (iv) / is the emissivity, estimated to 0.9.

The thermal conductivity equivalent to radiative transfer is thus about 0.06 mW.m -1 .K -1 . This value is much lower than the thermal conductivity of hemp concrete (higher than 100 mW.m -1 .K -1 ), so the radiative transfer is negligible in hemp concrete.

Finally, the heat transfer in hemp concrete can be considered as only conductive. So, a conductive model can be used for modelling the thermal conductivity of hemp concrete.

Self-consistent scheme

The self consistent scheme allows the determination of macroscopic physical properties of heterogeneous materials. It was developed for the mechanical characterization [Hill, 1965] [Wu, 1966] and was then extended to electrostatics, magnetostatics, electric conduction and thermal conduction that are mathematically analogous [START_REF] Landauer | The electrical resistance of binary metallic mixtures[END_REF][START_REF] Kerner | [END_REF]] [Hashin, 1968].

The heterogeneous medium, with spherical inclusions, is assumed to be an assembly of spherical composite spheres of various sizes (figure 5). For a two-phase medium, a sphere of radius # (phase a) is embedded in a concentric spherical shell of external radius 2 (phase b). This composite sphere is embedded in a homogeneous and isotropic equivalent medium (figure 6). The approximation of the effective conductivity is based on the assumption that the energy embedded in the heterogeneous medium is equivalent to that of the equivalent homogeneous medium under the same boundary conditions. The effective conductivity is thus linked to the volume ratio n of the inner to the outer radius according (5):

+3. 2 × 4 5 6 1 , 7 89: ; < 8 = > = ? @ 98 A B C D ; E F G > G ? H (5)
Where # ('+IJ.2) is the thermal conductivity of the phase a (resp. b).

For a three-phase medium, taking into account a supplementary shell of external radius K (phase c), this expression becomes [Boutin, 1996](6):

+3. K × 4 5 6 1 L 89M ; < ;NOP = > = ? @ 98Q ;F = > = R @ 98H9OP = > = ? @ 98QPS = ? = R @ N8Q B C D ; / F G ? G R H and T 1 -F G > G ? H (6)
In this study, the two phase model is used for dry material considering air as phase and solid matrix (including hemp shiv and binder) as phase U, E is thus the porosity of the material. The three phase model is used for wet material considering air as phase , water as phase U and solid phase as phase V. / is the porosity of the material and T is calculated from water content and porosity. The thermal conductivity of the solid phase is calculated by fitting the experimental data with the self consistent scheme using the least square method. This value is then used to compute the variation of thermal conductivity with density of dry material, and the variation of thermal conductivity with water content of wet material.

Results and discussion

Variation of thermal conductivity of Hemp Concretes versus formulation at (23°C; 50%RH).

Figure 7 gives the thermal conductivity of the five studied hemp concretes versus density at (23°C; 50%RH). For the wall formulations, the thermal conductivity ranges from 120 to 150 mW/(m.K). For sprayed hemp concrete walls (diamonds, triangles and squares on figure 7), the variation of thermal conductivity versus density fit a linear regression curve.

The SHC roof shows the lowest thermal conductivity, 93 mW/(m.K), mainly due to its lowest density.

The SHC floor shows the highest thermal conductivity: 157 mW/(m.K). These values are not in line with the fitting curve of SHC wall. Actually, the hemp to binder ratio of SHC roof is higher than the hemp to binder ratio of SHC wall; and inversely the hemp to binder ratio of SHC floor is higher than the hemp to binder ratio of SHC wall. So, as it was shown in [Benfratello and al., 2013], the thermal conductivity of hemp concrete depends on the amount of shiv and is not linearly proportional to the increase of shiv in mixture. The amount of shiv actually impacts the thermal conductivity of the solid phase of hemp concrete.

The thermal conductivity of moulded hemp concrete is higher than the value given by the fitting curve of SHC wall. As moulded hemp concrete has a higher porosity, its higher conductivity must be due to higher conductivity of its solid phase, related to higher real density. This may also be due to a higher conductivity of fibered hemp shiv, compared to defibered hemp shiv, as the fibers are less porous than the shiv and are thus more conductive.

The thermal conductivity of precast hemp concrete is lower than the value given by the fitting curve of sprayed hemp concrete wall. On one hand, the hemp to binder ratio of precast hemp concrete is higher than the hemp to binder ratio of sprayed hemp concrete, like previously this induces a lower thermal conductivity. On the other hand, the type of binder also impacts the thermal conductivity of materials. It is shown in [Gourlay and Arnaud, 2010] that the thermal conductivity of hydraulic lime binder is higher than the thermal conductivity of hydrated lime binder. Furthermore, [START_REF] Stefanidou | [END_REF] underline that adding white cement increases the thermal conductivity of mixtures and that adding pozzolanic materials reduces the thermal conductivity in comparison with pure lime.

Finally, the values measured on wet hemp concretes (at 23°C; 50%RH) are in agreement with the values found in literature. Actually, de Bruijn and Johansson [de Bruijn and Johansson, 2013] studied the thermal conductivity of two lime-hemp mixes at 15%RH and 65%RH. At 65%RH, they give thermal conductivity values of 116 and 100 mW/(m.K) when the densities are respectively 394.8 and 298.1 kg/m 3 .

Evrard [Evrard, 2008] gives a thermal conductivity ranging from 122 to 137 mW/(m.K) in moist state for density about 460 kg/m 3 . According to [START_REF] Elfordy | Mechanical and thermal 415 properties of lime and hemp concrete (''hempcrete'') manufactured by a projection process[END_REF], the thermal conductivity of sprayed hemp concrete is 179 mW/(m.K) for a density of 417 kg/m 3 . This is higher than the values found in this study.

This may be due to the moist state that is not well known in [START_REF] Elfordy | Mechanical and thermal 415 properties of lime and hemp concrete (''hempcrete'') manufactured by a projection process[END_REF]. kg/m 3 . The thermal conductivity of moulded hemp concrete is 11% higher: 129.6 mW/(m.K) for ρ = 377 kg/m 3 . Lastly, the thermal conductivity of SHC floor is equal to 145 mW/(m.K) for a density of 450 kg/m 3 . Compared to the values found in literature, on the one hand, these results are higher than the values obtained from the relationship established by Cerezo [Cerezo, 2005](eq. 7), that gives a thermal conductivity of 99.4 mW/(m.K) when the density is 400 kg/m 3 . On the other hand, these results are in the range of the values given by Nguyen [Nguyen, 2010] for similar density. Cerezo studied hemp concrete manufactured by casting while Nguyen studied hemp concrete manufactured by compacting. It seems that the spraying method induces thermal conductivity similar to the values obtained from compaction.

0.0002 × 0.0194

The thermal conductivity of the solid phase of hemp concretes is calculated following the method given in section 3.2.2. The thermal conductivity of the solid phase is equal to 61.2 mW/(m.K) for the SHC wall while it is equal to 95.3 mW/(m.K) for the MHC and to 79.1 mW/(m.K) for the SHC floor. As assumed previously, the solid phase of moulded hemp concrete is more conductive than the solid phase of sprayed hemp concrete wall, related to the use of fibered hemp shiv. The solid phase of the SHC floor is 30 % more conductive than the solid phase of the SHC wall, in line with lower hemp to binder ratio.

The self consistent scheme models the increase of thermal conductivity with density. For SHC wall, the theoretical result fits well the experimentation. This allows validating the numerical model, which is supposed to be valid for SHC floor and MHC to. When the density increases by 2/3, from 300 to 500 kg/m 3 , the thermal conductivity increases from 97.1 to 149.2 mW/(m.K) for the SHC wall, from 107.6 to 165.9 mW/(m.K) for the MHC and from 103.4 to 159.3 mW/(m.K) for the SHC floor. So, for all these hemp concretes, the thermal conductivity increases in the same way with density (by about 54 % when the density increases by 2/3). Figures 9 and 10 give respectively the variation of thermal conductivity with water content for several density of SHC wall and for MHC and SHC floor. The points are experimental data and the lines model the variation of thermal conductivity with water content according the three phase self consistent scheme (equation 6). In all cases, there is a very high correlation between experimental data and self consistent scheme.

For the light SHC wall (dry density about 364 kg/m 3 ), the thermal conductivity increases by 25 % from 107 mW/(m.K) at dry state to 134.3 mW/(m.K) at 0.16 g/g in water content (reached at 90 %RH). For the medium and the heavy SHC wall, the thermal conductivity shows similar increase with water content.

The MHC and the SHC floor also show a similar increase of thermal conductivity with water content.

From dry state to 0.05 g/g in water content (reached around 81%RH), the thermal conductivity rises respectively by 6.3% and 6.9%.

In the region of highest relative humidity, the thermal conductivity of MHC and SHC floor are less impacted than the thermal conductivity of SHC wall due to their sorption curves that show lower water contents [Collet and al., 2013].

Synthesis -Variation of thermal conductivity of hemp concretes versus dry density and water content

Figure 11 and table 3 summarise the variation of thermal conductivity of SHC wall with dry density and water content.

According to [Collet and al., 2013], the water content of SHC wall (with a dry density around 400 kg/m 3 ) is lower than 0.10 g/g for a wide range of relative humidity (up to 90%RH). From dry state to 0.10 g/g in water content, the thermal conductivity of SHC wall increases by 12.8% for low density (250 kg/m 3 ) and by 17% for high density (600 kg/m 3 ). For highest values of water content (0.20 g/g), the increase of thermal conductivity from dry state is 25.7% for low density and 34.8% for high density.

Moreover, from low density (250 kg/m 3 ) to high density (600 kg/m 3 ) the thermal conductivity increases by 109% at dry state and by 117% at 0.10 g/g in water content.

So, the impact of density on thermal conductivity is much more important than the impact of moisture content. Similar results are obtained with MHC and SHC floor. However, once the hemp concrete is produced, its density remains the same while its water content varies according surrounding relative humidity. So, the variation of thermal conductivity with water content has to be taken into account.

Conclusions

The results presented herein show that the thermal conductivity of hemp concrete depends both on its formulation, its density and its water content. The classical increase of thermal conductivity with density is observed but it is also underlined that the binder type, the hemp shiv type (fibered or defibered) and the hemp to binder ratio deviate the value of thermal conductivity from the variation law. The study of thermal conductivity versus water content and density shows that the water content has a lower effect than density on thermal conductivity (increase lower than 15 to 20% for a wide range of relative humidity). However, this effect should be taken into account in the modelling of hygrothermal behaviour of the building envelope. The thermal conductivity for high density is higher than twice the value for low density with the same formulation.

Figure 1 .

 1 Figure 1. Spraying method (left : mixer, right : spraying hose)

Figure 3 .

 3 Figure 3. Measurement of thermal conductivity at dry state -left : CT Meter, Hot wire and dry chamber, right : experimental thermogram

Figure 4 .

 4 Figure 4. Infrared photography of the probed area, immediately after measurement.

Figure 5 .

 5 Figure 5. Heterogeneous medium assumed to be an assembly of spherical composite spheres of various sizes.

Figure 6 .

 6 Figure 6. Two-phase composite sphere embedded in a homogeneous equivalent medium.

Figure 7 .

 7 Figure 7. Thermal conductivity of hemp concretes versus density at (23°C; 50%RH), for sprayed hemp concretes SHC (walls, floor, roof), moulded hemp concrete MHC and precast hemp concrete PHC -experimental data and linear fit of sprayed hemp concrete walls

Figure 8 .

 8 Figure 8. Thermal conductivity of hemp concretes versus density at dry state for sprayed hemp concretes (wall and floor) and moulded hemp concrete -experimental data (points) and two-phase self-consistent scheme (lines).

Figure 9 .

 9 Figure 9. Thermal conductivity of hemp concretes versus water content for sprayed hemp concrete walls (light, medium and heavy) -experimental data and three phase self consistent scheme.

Figure 10 .

 10 Figure 10. Thermal conductivity of hemp concretes versus water content for sprayed hemp concrete floor and for moulded hemp concrete -experimental data and three phase self consistent scheme.

Figure 11 .

 11 Figure 11. Thermal conductivity of SHC wall versus dry density and water content.

Table 1 : Mixes proportioning and manufacturing method of studied hemp concretes

 1

	Material	Method of Production	Binder	Hemp shiv	Hemp/Binder mass ratio
	SHC Wall : Sprayed Hemp Concrete -wall	Spraying	Commercial lime-based binder *	Commercial defibered hemp shiv	0.5
	SHC Floor : Sprayed Hemp Concrete -floor	Spraying	Commercial lime-based binder*	Commercial defibered hemp shiv	0.4
	SHC Roof : Sprayed Hemp Concrete -roof	Spraying	Commercial lime-based binder*	Commercial defibered hemp shiv	1
	MHC : Moulded Hemp Concrete (wall)	Moulding	Commercial lime-based binder*	Commercial Fibered Hemp Shiv	0.5
	PHC : Precast Hemp Concrete (wall)	Precasting	CaO : 72% Hydraulic lime : 28%	Defibered hemp shiv	0.65

* lime-based binder made of 75% of hydrated lime (98% CaO), 15 % of hydraulic binder and 10 % of pozzolanic binder

Table 2 : Density and porosity of studied hemp concretes

 2

	Material	Apparent density at (23°C; 50%RH) [kg.m -3 ]	Total porosity n [%]
	SHC lw : Sprayed Hemp Concrete -	390-401	79.0-79.5
	light wall
	SHC mw : Sprayed Hemp Concrete -medium wall	420-426	77.8-78.0
	SHC hw : Sprayed Hemp Concrete -heavy wall	446-463	75.7-76.6
	SHC Floor : Sprayed Hemp Concrete -floor	460	78.7
	SHC Roof : Sprayed Hemp Concrete -roof	258	84.9
	MHC : Moulded Hemp Concrete (wall)	381	84.3
	PHC : Precast Hemp Concrete (wall)	457	72

Table 3 : Thermal conductivity of SHC wall versus dry density and water content

 3

	ρ0 [kg/m 3 ]	250	300	350	400	450	500	550	600
	n		0.86	0.83	0.80	0.77	0.75	0.72	0.69	0.66
		0.000	85.8	98.5	111.3	124.5	137.9	151.5	165.5	179.7
		0.040	90.2	103.8	117.7	131.9	146.4	161.2	170.9	185.7
		0.080	94.6	109.2	124.1	139.4	155.0	170.9	176.3	191.7
	w
	[g/g]	0.120	99.0	114.6	130.6	146.9	163.7	180.9	181.8	197.8
		0.160	103.4	120.1	137.1	154.6	172.5	190.9	187.3	204.0
		0.200	107.9	125.6	143.7	162.3	181.5	201.1	192.8	210.2

Acknowledgments

This article presents results following the French national projet ANR-06-MAPR-0002 Beton chanvre -Programme MAPR 2006 and the Breton regional project "PRIR Ecomatx". These projects were conducted with the financial support of the French National Agency of Research (ANR) and the Brittany Regional Council.

The authors thank the SMEs Easy Chanvre and SI2C for their contribution to the manufacturing of hemp concrete.


G Remenyi

S Sahling

K Biljakovic

D Staresinic

J.-C Lasjaunias

J E Lorenzo

P Monceau

A Cano

Incommensurate systems as a model approach to low temperature behavior of glasses

Keywords: Incommensurate systems, thermodynamics, vibrational states, low-energy excitations, glasses, Boson peak

The Debye model predicts a T 3 dependence of the specific heat C p at sufficiently low temperatures in insulators and it reproduces a wealth of data in isotropic crystals. However, this simple and general law fails to account for the low temperature specific heat of amorphous compounds and this matter is currently the focus of many scientific activities, yet unsettled. At low temperatures there are two anomalies that show up as an excess to the constant C p (T)/T 3 in glasses and amorphous systems: the upturn below 1 K and a broad bump at T≈10 K named Boson peak (BP). Whereas insulating crystals with low-lying optical modes and/or acoustic modes with non-linear dispersions can exhibit a similar bump in the specific heat, the Debye law is expected to re-emerge in these systems at sufficiently low temperatures. (Andres) One question that we want to address is to what extent this is the case. We show that, contrary to the expected, the Debye law is not fully obeyed as soon as translational periodicity is slightly broken like in insulating crystals with incommensurate superstructures. (Our suggestion) We show that, however, as soon as translational periodicity is slightly broken, like in insulating crystals with incommensurate superstructures, the Debye law is not the most important contribution to the low temperature specific heat anymore. On general grounds we show the specific heat of incommensurate modulated systems bears many similarities with that of amorphous systems.

Moreover we account for the excess of heat capacity within a simple model in which the energy dispersion and the damping of the lowest energy, non-acoustic, phonon branch are the main parameters.

Incommensurate crystals lack the translational periodicity, while retaining the structural order. In their simplest presentation of a zero-dimensional (0D) modulation the diffraction pattern is given by a single modulation wavevector expressed as an irrational combination of at least one of the three basic reciprocal lattice vectors that represents the average lattice (see Figures 1a and1b). Non-linear forces, anharmonicity, are at the very basis of the IC phase transitions. The same anharmonicity is responsible for the occurrence of discommensurations (or phase jumps giving rise to some kind of phase disorder) in the ground state of the modulation.

IC structures emerging in a displacive phase transition that is, a phase transition associated to a soft phonon mode, see Figure 1c) display a specific dynamics with the occurrence of two distinct excitations in the close neighbourhood of the modulation wavevector: the low energy phason mode and gapped amplitudon mode (see Figure 1d). They arise from the even and odd combination of the original soft phonon mode coordinates, respectively.

Although this decoupling is mathematically exact in the pure displacive limit of the phase transition, extensions to phase transitions including both soft mode and relaxation dynamics can also be made. Important for low temperature heat transport is the possibility that the phase mode is gapped (Figures 1b and1c), and thus contributing to the specific heat as a deviation from the T 3 law [1]. A second feature that is acknowledged to be present in many (if not all of) compounds is that the phason mode is damped [2] (Figure 1e), or even over-damped as in K 2 SeO 4 [3]. At this point, and for the sake of completeness, it is important to recall that many IC compounds do not have a soft mode, the pretransitional dynamics are rather of the relaxational type. In this case the decoupling soft phonon → phason+amplitudon is not observed in the inelastic neutron scattering (INS) spectra. However, the lowest energy soft mode of the correct symmetry acts as both the phason and the amplitudon and consequently the low temperature specific heat still bears similar features as those here described [4,5].

One of the best known examples of phonon softening is the Kohn anomaly due to electron-phonon coupling and the Fermi surface instability of 1D electron subsystem at the wavevector q IC =2k F . This is the Peierls mechanism for metal insulator phase transition.

In this case, the 2k F instability gives rise to an incommensurate (IC) charge density wave (CDW) and the concomitant lattice distortion that stabilizes the electronic phenomena [START_REF] Gruner | Charge Density Waves in Solids[END_REF][START_REF] Gruner | Density Waves in Solids[END_REF][START_REF] Monceau | [END_REF]. The bump in C p /T 3 in CDW systems such as NbSe 3 , TaS 3 , (TaSe 4 ) 2 I, K 0.3 MoO 3 was described as the contribution from the modified Debye spectrum with two cut-offs: a lower frequency corresponding to the pinned CDW state and an upper frequency corresponding to the effective Debye temperature for the collective modes (modified Boriack-Overhauser model) [1,9]. As shown in Figure 2, the bump is accompanied at low temperatures by an extra contribution scaling roughly as T -2 in the C p /T 3 plot. As in glasses it has been explained by invoking the presence of TLS and described as the tail of the corresponding Schottky anomaly. These metastable states, found also sensitive to the magnetic field [10], are attributed to the bisolitons generated at strong pinning centers in the CDW ground state [11,12]. They are decoupled from phonon modes, so that the specific heat exhibits non exponential relaxation with aging effects [13]. Surprisingly these two features bare many similarities with identical plots in glasses. In the upper middle panel of Fig. 2 we show the specific heat of vitreous silica [14] for comparizon.

In addition, a wealth of experimental evidence has been collected indicating that the CDW also forms a kind of charge density glass (CDG) at low temperatures [15,16]. The similarity of the phenomenology of CDW systems and other electronic crystals suggests that the low-T glass state is a universal feature in modulated electronic superstructures.

Interestingly enough, what makes the CDW state dynamics unique among IC compounds is that the CDW-phase fluctuations (or phasons) induce changes in the electron density which are coupled by long-range Coulomb interactions. Early theories soon recognized that at T=0 K the effect of such interactions is to transform the acoustic-like longitudinal phason mode FIG. 2. C P /βT 3 plots of several IC compounds, with β the coefficient of the Debye law. Left and middle panels are charge density wave compounds (K 0.3 MoO 3 [9], (TaSe 4 ) 2 I [20], KCP [9] and TaS 3 [16]). C P /βT 3 plot of vitreous silica is displayed as a comparison [14]. Right panel, insulating compounds, BCPS ((ClC 6 D 4 ) 2 SO 2 ) [21] and biphenyl [22] . FIG. 3. Temperature dependence of the specific heat divided by T 3 in a log-log plot for ThBr 4 .

Experimental data are displayed with full black circles. The black line represents the Debye contribution calculated from sound velocities determined in neutron scattering experiment [23] and using the Debye model of the dispersion with θ D ≈62 K and corresponding β=8•10 -3 J/mole K 4 .

The green line is the amplitudon contribution estimated from the measured INS dispersion. Full red circles represent the contribution remaining after subtraction of previous contributions, and assigned to the phason contribution.

into an optical one [17]. At finite temperatures however, the quasiparticles excited across the CDW gap screen out the Coulomb interactions and bring back the acoustic character of the phason [18,19].

In order to elucidate the key ingredient that induces a glassy dynamics in IC systems, we have extended our investigation to the case of modulated IC dielectrics for which the incommensurability is of a purely structural origin (and therefore electronic contributions, as in CDWs, are absent). A gapless phason is expected for an ideal, defect-free structure remaining incommensurate down to the lowest temperature. The work on dielectrics was initiated some years ago on (ClC 6 D 4 ) 2 SO 2 (BCPS) [21] and byphenil [22] that show also a deviation from the Debye behavior at low temperatures, as it is shown in Figure 2, and associated to the phason and amplitudon modes of the IC structure. An excess in the phonon density of states was observed as well, but a proper quantification of the anomalies is still lacking. Figure 2 illustrates at its best the fact that, as in glasses and amorphous systems, the Debye law approximation does not hold in IC compounds at low temperatures.

As we will show below the presence of excitations related to the breaking of translational periodicity is a more fundamental feature that determines the specific heat at low temperatures.

A proper account of the specific heat at low temperature calls for a quantification of the low energy excitations, phonons and the modes originating from the IC structure. Here we present the investigation of the low temperature specific heat of ThBr 4 . This system is an insulator that undergoes a displacive-like transition to the 1q-IC modulated superstructure at 95 K, with T-independent (between 80 K and 1.5 K) wave vector q IC = 0.31c * . This phase transition has been very well documented in the literature [23] and it is considered as one of the clearest examples of displacive dynamics and of the presence of phason and amplitudon excitations below the transition temperature. Interestingly the phase-mode dispersion has been measured along different directions [23] and therefore its contribution to the specific heat can be computed and straightforwardly compared with experiments.

The results of the specific heat measurements in ThBr 4 are shown in Figure 3 which can be decomposed as follows: (i) The background constant contribution in the C p (T)/T 3 plot due to acoustic phonons. The Debye temperature θ D =62 K has been calculated from the sound velocities measured by INS [23], yielding β = 8 • 10 -3 J/moleK 4 (see Supplementary Material). (ii) As in previous works the amplitudon contribution, as revealed by the bump in the C p (T)/T 3 plot, lies at around 10K. This contribution was estimated from the amplitudon dispersion with a gap of 570 GHz determined by Raman scattering measurements at low temperature [24] (see Supplementary Material). Red points in Figure 3 represent the specific heat after subtraction of these two contributions. It can be described as a bump followed by a rise as the temperature decreases. These features have been regularly considered in the literature as separated and having uncorrelated origins: the shoulder as mostly due to the phase gap and dispersion while the low temperature rise corresponding to the tail of a Schottky anomaly, originating from the presence of TLS as in glasses.

Cano and Levanyuk have proposed a plausible explanation for these striking features in insulating crystals [25]. In this picture, both the shoulder (that appears between 1 and 0.5 K in Figures 3 and4) and the subsequent rise can be described within the same formalism revealing that they are two different manifestations of the very same phason mode: gap and damping. The shoulder indicates that the phason dispersion in ThBr 4 is not completely acoustic-like as in textbook descriptions, but contains a low-frequency gap (the phason dispersion is shown as an inset of Fig. 4). This is not very surprising since, physically, the incommensurate superstructure is expected to be pinned to the underlying lattice rather than detached from it and totally free to move at no energy cost. According to Ref. 23, the upper limit of this gap can be estimated as ≈ 50 GHz. In addition, it was pointed out that the finite life-time of the phason excitations can play a non-trivial role in the low temperature properties of incommensurate systems. Such a finite life-time has been experimentally observed by INS [START_REF] Currat | Incommensurate Phases in Dielectrics[END_REF] within the temperature range of our specific heat data. This lifetime again evidences that the incommensurate superstructure cannot propagate freely. It will also be subjected to dissipation, which can be due to the scattering with impurities and/or disorder. This dissipation amounts to a redistribution of density of states that, as a rule, is ignored when addressing thermodynamic properties. However, and as pointed in Ref. 25, this redistribution cannot be neglected in incommensurate systems given the low-frequency of their characteristic phason modes, which can be the reason of their systematic low-temperature anomalies.

Figure 4 displays the results of a fit of the low temperature specific heat of ThBr 4 below 10 K, following Eq. 1 in the Supplementary Material section. Parameters extracted from the fits are phason gap of 46 GHz and phason damping of 3.8 GHz. The fit nicely reproduces the experimental behavior. We emphasize that the key ingredients are just the phason dispersion and the corresponding lifetime, which corresponds to those obtained in previous inelastic scattering experiments (independent from our measurements). Black curves in Fig. 4 show how the agreement with the INS data is sensitive to the actual value of the phason gap when ∆=22GHz and ∆=90 GHz for the same damping.

We have established that the excess of specific heat at low temperature (or bump in the C p (T)/T 3 plot) in ThBr 4 directly originates from gapped phase and amplitude modes of the incommensurate structure. Despite previous work in IC compounds this is the first time that we can quantitatively establish this correspondence beyond reasonable doubt [START_REF] Lorenzo | [END_REF]. As previously suggested [25], the two anomalies observed in the low temperature specific heat of ThBr 4 and, in general, of all studied incommensurate compounds can be described under the same and unique formalism by taking into account both the phason (or in general a low energy phonon mode of the appropriate symmetry) gap value and its damping. Anharmonic interactions acting at the IC wavevector are thus at the origin of both features.

Summarizing, we have demonstrated the intrinsic limitations of the Debye model for describing the low temperature heat capacity of solids lacking of translational periodicity.

For ThBr 4 , the simplest case of an incommensurate insulator, we have shown that low energy modes associated to the incommensurate superstructure dominate its low temperature heat capacity. This type of incommensurate systems represent the simplest "zero order approximation" of amorphous solids and glasses lacking crystalline order. Moreover, the concurrence between IC systems and glasses goes beyond a similar specific heat dependence at low temperatures. Incommensurate modulated structures can be conceptually viewed as the simplest approach to the structure (and dynamics) of an amorphous (glass) system. Starting from a crystalline structure, the amorphous state can be pictured as a multiply incommensurate modulation with every q-position being a modulation q-vector.

The modulation wavevectors thus span the 3D Fourier space. Conversely an incommensurate system is, in its simplest form, a 0D, just a single q-position, modulated structure.

The overly anharmonic interactions present in glasses, and certainly at the origin of the low temperature features in the specific heat data, are present in IC systems as well, in the form of modes such as phason and amplitudon, or more generally under the form of low lying phonon modes. Thus the intrinsic IC compounds low energy dynamics constitute a plausible simplification of the landscape of interactions giving rise to the characteristic low temperature anomalies in glasses. Such a view is compatible with recent molecular dynamics simulations in glasses [28][29][30], which we expect to make the link between these two seemingly unrelated areas of research.

Materials ThBr 4 crystals used in our experiment were the same used in the INS studies [23]. They were synthesized by P. Delamoye and kindly lent by R. Currat. As known to be highly hygroscopic, the sample has been kept in the vacuum-tight Pb container for the last 15 years. In order to avoid any contamination, its further handling for specific heat measurements was done in a glove box under a controlled nitrogen atmosphere. There, the sample of total mass 243 mg, broken in two pieces while removed from the Pb container, was embedded in 38 mg of Apiezon-N grease usually used to improve thermal contact. This grease is a very good protection from air and water contamination during transfer from the glove-box to the cryostat. C p data were obtained by a transient heat pulse technique [9].

The heat capacity of the addenda was below 30% of the total in the whole T-range, except for T≥ 20 K where it was between 30% and 40%.

FIG. 1 .

 1 FIG. 1. Overall sketch of the displacive dynamics in incommensurate compounds. (a) 2D picture of the reciprocal space for T>T C with the Bragg peaks in red. (b) Idem for T<T C , and the blue dots are the new peaks (or satellite peaks) originating from the IC phase transition. (c) Phonon softening above the transition temperature at an incommensurate position, namely q=0.31. (d) Decoupling of the phonon coordinate into an amplitudon and phason modes. The amplitudon frequency renormalizes very strongly at low temperatures. (e) The phason branch in gapped, with value ∆, and damping, Γ.

FIG. 4 .

 4 FIG. 4. Low energy contribution to the specific heat of ThBr 4 divided by T 3 in a semi-log plot. The red points correspond to the experimental data shown in Fig. 3 once the Debye and amplitudon contributions have been subtracted. The continuous black line is the result of the calculation from Eq. (1) in Supplementary Materials with the phason gap of 46 GHz and damping of 3.8 GHz. The black dash lines correspond to simulations using the same damping and different phason gaps of 20 and 90 GHz. Dispersion curve for the phason is drawn in the inset, with points representing the experimental data.

We acknowledge the funding obtained under French ECONET project and Croatian MSES project 035-0352827-2842.