Yves Garrabos

Carole Lecoutre

Characteristic parameters of xenon near its liquid-gas critical point

Keywords: numbers: 64.60.A-, 05.10.Cc, 05.70.Jk, 65.20.-w 1

The mean crossover functions estimated from the bounded results of the Massive Renormalization scheme applied to the Φ 4 d (n) model in three dimensions (d = 3) and scalar order parameter (n = 1) are used to represent the singular behaviors of the isothermal compressibility of xenon along the critical isochore in the homogeneous domain and the vapor-liquid coexisting densities of xenon in the nonhomogenous domain. The validity range and the Ising nature of the crossover description are discussed in terms of a single scale factor whose value can be analytically estimated beyond the Ising-like preasymptotic domain.

INTRODUCTION

It appears now well-established that the universal features of the one-component uids close to their vaporliquid critical point are Ising-like in nature [START_REF] Anisimov | Equations of State for Fluids and Fluid Mixtures[END_REF][START_REF] Zinn | Euclidean Field Theory and Critical Phenomena[END_REF], i.e., conform with the estimated universal features close to the non-trivial xed point of the so-called φ 4 d=3 (n = 1) model [START_REF] Zinn | Euclidean Field Theory and Critical Phenomena[END_REF][START_REF] Bagnuls | [END_REF] (d and n are the dimensions of the space and the order parameter density, respectively). The leading singular behaviors [4] are then governed by the universal values [5] of two independent critical exponents, while the contribution of the conuent corrections to scaling is governed by the universal lowest value ∆ 0.502 [5] of the conuent exponent. These Ising-like universal features only related to the contributions of two relevant scaling elds and a single irrelevant scaling eld [6,[START_REF] Privman | Universal critical point amplitude relations[END_REF] are strictly valid within the Ising-like preasymptotic domain, i.e., a domain where each complete Wegner-like expansion [START_REF] Wegner | [END_REF] can be approximated by its restricted (two-terms)

form [9]. Such an exact Ising-like asymptotical behavior appears as the essential tool to provide accurate determination of only three (two leading and one conuent)

amplitudes that characterize each Ising-like critical uid [START_REF] Privman | Universal critical point amplitude relations[END_REF].

However, as shown by a large set of results reported in the literature, the observed crossover behavior in onecomponent uids needs to use higher-order terms in the Wegner-like expansions to t carefully the measurements performed at nite distance to the critical point. In such an extended asymptotic domain covered by the tting analysis, the above three-parameter characterization of each Ising-like uid is not easy to probe carefully. For example, for xenon case, a relatively complete data set of leading amplitudes exists that is precisely conform to the two-scale-factor asymptotic universality [11,12]. In contrast, the uniqueness of the temperature-like crossover parameter (noted ϑ in the following) that complement the determination of two independent leading amplitudes has never clearly been demonstrated to dene the physical temperature range of validity of the three-parameter characterization.

The main goal of the present paper addresses to the analytic calculation of the xenon value of a scale factor (noted ϑ L in the following), which characterizes the singular behavior along the critical isochore. The calculation refers to the crossover functions estimated in

Refs. [9] and [10]. Therefore, this ϑ L -determination, which is performed beyond the Ising-like preasymptotic domain of xenon, maintains the conformity with the Ising-like universal features accounted for by the theoretical crossover functions calculated in the massive renormalization scheme of the φ 4 d=3 (n = 1) model [START_REF] Bagnuls | [END_REF]4]. We strictly avoid adjusting any xenon-dependent parameter by a minimization of tting errors over the experimental temperature range, as initially performed in Ref. [13].

Alternatively, only using the critical coordinates of the xenon critical point, we correlate ϑ L to the eective values [14] of the exponent and amplitude of a pure power law dened at a well-known nite temperature distance (T -T c ) to the critical temperature (T c ), largely beyond the Ising-like preasymptotic domain. We can then found ϑ L from the tangent envelop of any continuous function f P ∆τ * = T -Tc Tc that t the data measurements of the singular property P over a limited temperature range ∆τ * min ≤ ∆τ * ≤ ∆τ * max . To calculate each corresponding local value of ϑ L (∆τ * ), we use as two illustrative examples, the tting results of Guttinger and Cannell [15] for the susceptibility data and the tting results of Narger and Balzarini [16] for the symmetrical order parameter density data. That allows us to verify the uniqueness of the ϑ L value and its exact matching with the Ising asymptotic value ϑ when ∆τ * → 0. We simultaneously estimate the upper temperature limit of the extended asymptotic domain, providing thus the complete comparison between experimental results and crossover theories in a relative temperature range covering more than four decades above or below T c . Moreover, as expected from the denition of the master crossover functions valid for the one-component uid subclass [10], this comparison is made de facto without any adjustable parameters when four critical coordinates of xenon are known.

The paper is organized as follows. Section 2 gives a brief recall of the useful notations and denitions with the objective to obtain the three-parameter characterization of xenon within the Ising-like preasymptotic domain.

The three-parameter characterization of xenon beyond the Ising-like preasymptotic domain is dened in Section 3 analyzing the isothermal compressibility case and the order parameter density case. In Appendix A, we re- are the length and energy units, respectively (subscript c refers to a critical parameter). p (p c ) is the pressure (critical pressure). The superscript star labels dimensionless quantities obtained only using α c and (β c )

-1

when the thermodynamic properties are normalized per particle [1719]. On the other hand, in Eq. ( 4),

∆ρ LV,expt = ρ L -ρ V 2ρ c (7)
is the practical (symmetrical) dimensionless form of the order parameter density. ρ L (ρ V ) is the density of the liquid (vapor) coexisting phase, while ρ (ρ c ) is the (critical) mass density. Equation [START_REF] Privman | Universal critical point amplitude relations[END_REF] introduces the dimensionless form ρ = ρ ρc of the (mass) density [20, 21]. Correspond- ingly, the dimensionless ordering eld is dened as

∆μ = μρ -μρ,c (8)
where ∆μ is also written using a practical dimensionless form μ = µρρc pc [20,[START_REF] Levelt Sengers | Progress in Liquid Physics[END_REF] of the chemical potential. µ ρ (µ ρ,c ) is the (critical) chemical potential per mass unit.

The subscript ρ distinguishes the thermodynamic quan- the corresponding two-term restricted forms of, either the mean crossover functions given in Ref. [9], or the master crossover functions, as described in Ref. [10]. Therefore, among the three leading amplitudes ξ + , Γ + and B, only two are uid dependent [22], while among the three conuent amplitudes a + ξ , a + χ , and a M , only one is uid dependent. Selecting for example the amplitude set [9] S A,f = a + χ ; ξ + ; Γ +

to characterize each one-component uid f , the remaining amplitudes can then be calculated using the following universal amplitude ratios and combinations

ξ + -d Γ + B 2 = R C R + ξ d (10)
a + ξ a + χ = 0.67919 (11)
a M a + χ = 0.9 (12) with R C = 0.0574 and R + ξ = 0.2696 [5]. Now, in order to determine the values of S A,Xe = a + χ ; ξ + ; Γ + for the xenon case, we apply the mean and master theoretical crossover descriptions given in Refs. [9] and [10],

respectively. A mean crossover function valid for any three-dimensional (uniaxial) Ising-like system refers to a complete crossover function estimated in Ref. [9] (noted I in the following). A master crossover function only valid for any one-component uid refers to the modication of this complete function as proposed in Ref. [10] (noted II in the following).

Asymptotic characteristic parameters of xenon

The minimal material needed to calculate all xenon amplitudes is provided by the set of four critical coordinates The data sources and references are given in Appendix A. From Eqs. ( 14), ( 5) and ( 6), we obtain the following critical values of the energy and length units:

Q min c,ap = T c ; p c ; v p,c ; γ c (13)
(β c ) -1 = 4.0003 × 10 -21 J (15) α c = 0.881498 nm (16)
The values of the scale factors Y c and Z c are :

Z c = pcmp ρck B Tc = 0.28602 Y c = γ c Tc pc -1 = 4.93846 (17)
Following the results of II and Eqs. (15) to (17), the xenon values of the dimensionless amplitudes of the characteristic set S A,Xe are:

S A,Xe =      a + χ = (Y c ) ∆ Z 1,+ χ = 1.23709 ξ + = (Y c ) -ν Z + ξ = 0.209338 Γ + = (Z c ) -1 (Y c ) -γ Z + χ = 0.0578238      (18)
where the master amplitude set

S {1f } A =    Z 1,+ χ = 0.555 Z + ξ = 0.5729 Z + χ = 0.11975    ( 19
)
characterizes the master asymptotic singular behavior of the {1f }-subclass. The notations are dened in II. The value of the dimensional amplitude ξ + 0 is

ξ + 0 = α c ξ + = 0.184531 nm (20)
In addition, Eqs. (10) to (12) lead to B = 1.46762 a + ξ = 0.840217 a M = 1.11338 [START_REF] Levelt Sengers | Progress in Liquid Physics[END_REF] which closes the amplitude denitions of Eqs. (2) to (4) for the xenon case. Now introducing the scale factor set S {1f } SC that characterizes the master {1f }-subclass:

S {1f } SC =    Θ {1f } = 4.288 × 10 -3 L {1f } = 25.585 Ψ {1f } = 1.75505 × 10 -4    (22)
and the equations:

ϑ (Y c ) -1 = Θ {1f } g 0 α c = L {1f } ψ ρ (Z c ) 1 2 = Ψ {1f } (23)
where α c , Y c and Z c are given by Eqs. ( 16) and ( 17

t = Θ {1f } T * = Θ {1f } Y c ∆τ * = ϑ∆τ * (25)
when t → 0, T * = Y c ∆τ * → 0 and ∆τ * → 0, along the critical isochore (zero value of the magnetic-like elds) [START_REF] Bagnuls | [END_REF]6]. Similarly, the scale factors Ψ {1f } , ψ ρ and Z c are dened through the following linear relations betweeen the theoretical (h), the master (H * ), and the physical

(∆ μ) magnetic-like elds h = Ψ {1f } H * = Ψ {1f } Z c ∆μ = ψ ρ ∆μ (26)
when h → 0, H * = Z c ∆μ → 0 and ∆μ → 0, along the critical isotherm (t = T * = ∆τ * = 0) [START_REF] Bagnuls | [END_REF]6]. The third (dimensional) scale factor g 0 is the adjustable critical coupling constant of the Φ 4 term having a correct wave number dimension [START_REF] Bagnuls | [END_REF] to be introduced in the t-

ting equation th (t) = g 0 ξ expt (∆τ * ) (27)
The inverse wave number (g 0 )

-1

acts as the physical adjustable length in the link between the theoretical dimensionless correlation length ( th ) and the physical correla- tion length (ξ expt ) of each one-component uid. Then, by exchanging Eq. ( 27) and the following dimensionless

equation th (t) = L {1f } ξ * expt (∆τ * ) , (28)
we retrieve [see Eq. ( 23)]

L {1f } = α c g 0 (29)
L {1f } links both critical length units dened for {t = 0; h = 0} and {∆τ * = 0; ∆μ = 0}, respectively.

Equations ( 5), ( 6) and ( 29) are guarantee for uniqueness [START_REF] Privman | Universal critical point amplitude relations[END_REF] of the length and energy units in the dimensionless singular behaviors of thermodynamic and correlations functions (whatever the one-component uid).

After all, m th → 0 and ∆ρ → 0 in the nonhomogeneous domain are related by the equation

m th = L {1f } -d (ψ ρ ) -1 ∆ρ LV,expt , (30)
where m th is the theoretical magnetization-like order parameter.

Accordingly, the theoretical susceptibility

χ th (t) = ∂m th
χ th (t) = L {1f } -d (ψ ρ ) -2 κ * T,expt (∆τ * ) , (31)
with χT,expt (∆τ * ) ≡ κ * T,expt (∆τ * ) when ∆ρ = 0 (see above). The (three) tting Eqs. ( 28), [START_REF] Le Guillou | [END_REF] and (31) close the (three) parameter characterization of the Ising-like preasymptotic domain since they provide the unambiguous determination of ϑ, L {1f } (or g 0 ), and ψ ρ , and consequently, a + χ , ξ + , and Γ + . Thus, we can also retrieve the results given in Eqs. (18), using the following set of equations

a + χ = ϑ ∆ Z 1,+ χ ξ + = L {1f } -1 ϑ -ν Z + ξ -1 Γ + = L {1f } d (ψ ρ ) 2 ϑ -γ Z + χ -1 (32)
where the theoretical amplitude set (see I)

S {M R} A =      Z 1,+ χ = 8.56347 Z + ξ -1 = 0.471474 Z + χ -1 = 0.269571      (33)
characterizes the universal asymptotic singular behavior of the Φ 3 (1)-class.

However, we note the top-down hierarchy of Eqs. (32) that link the amplitude set S A = a + χ ; ξ + ; Γ + and the scale factor set S {M R} SF = ϑ; L {1f } (or g 0 ) ; ψ ρ for each one-component uid. The top equation shows that ϑ characterizes the lowest order of the uid conuent corrections to scaling in an unequivocal manner. Then the medium equation denes the dimensionles number L {1f } , which relates the uid microscopic wavelength g 0 and the uid thermodynamic length scale α c through Eq. [START_REF]The entropy per particle sp is related to the temperature derivatives γ = ∂p ∂T v p and δ = ∂µ p ∂T v p , which, at the critical point, leads to[END_REF].

Finally, the bottom equation introduces the scale factor ψ ρ of the order parameter density. ψ ρ can be estimated using any leading amplitude chosen among the ones of the extensive singular properties, as here above Γ + in the isothermal compressibility case. When the dimensionles number L {1f } takes its master value of Eq. ( 22), the uid scale factors ϑ and ψ ρ are well unequivocally related to the uid scale factors Y c and Z c respectively, as shown by Eqs. [START_REF] Garrabos | [END_REF] and (22).

Extension of the Ising-like preasymptotic domain of xenon

The most important result is that the value of ϑ enables the estimation of the extension L f PAD of the Ising- like preasymptotic domain of the one-component uid f .

In the xenon case we obtain:

L Xe PAD L Ising PAD ϑ (Xe) 0.9 × 10 -4 (34)
with

L Ising PAD 0.033 S 2 2 ∼ = 1.9 × 10 -6 (35)
In Eq. ( 35), L Ising PAD corresponds to the extension of the Ising-like preasymptotic t ≤ L Ising PAD , with S 2 = 22.9007

(see I). Obviously, this important result can be retreived from the following master extension of the Ising-like preasymtotic domain of the {1f }-subclass

L {1f } PAD = L Ising PAD Θ {1f } ∼ = 4.7 × 10 -4 (36)
using then the relation

L Xe PAD L {1f } PAD Y c (Xe) (37)
Now, the crucial problem to dene the temperature range of validity of Eqs. ( 2) to ( 4) is solved and Eq. (34) [or (37)] provides an essential tool for analyzing experimental data.

In xenon case, the Ising-like preasymptotic domain correspond to the temperature range

T -T c T c L Xe PAD = 26 mK (38)
However, in this small temperature range, the experimental control of the above three parameters characterization of xenon cannot easily performed for two main reasons: i) accurate measurements of the singular properties are generally made in a temperature range which does not reach the Ising-like preasymptotic domain (anticipating the results of the next section);

ii) data tting beyond the Ising-like preasymptotic domain generally use a Wegner-like expansion whose validity is questionable (see I).

As a result, the Ising-like equivalence between the three independent amplitudes of Eqs. (18) and the three independent scale factors of Eqs. ( 24) is generally obtained without the suitable precision. Therefore, a more appropriate analysis of the experimental results beyond the Ising-like preasymptotic domain, needs to introduce additional scaling rules which implicitely account for the asymptotic (three-parameter) characterization only valid inside the Ising-like preasymptotic domain. In the next section, we pay special attention to an analytic method of determination of the local value of a crossover parameter that introduces the contribution of the theoretical leading power law in conformity with the two-scale-factor universality, when it is applied beyond the Ising-like preasymptotic domain. The description of a xenon singular property P * (∆τ * ) by using a mean crossover function F P (t) (dened in I) in the reduced temperature range ∆τ * > L Xe PAD must involve contributions of the correction-to-scaling terms higher than the rst one. As a consequence, the value of ϑ introduced by the asymptotical analytic relation t = ϑ∆τ * , is associated to an unknown relative temperature extension L Xe EAD , which denes the extended asymp- totic domain such as ∆τ * ≤ L Xe EAD . In such a nite in- termediate range L Xe PAD < ∆τ * ≤ L Xe EAD , we are not able to evaluate the eective inuence of the numerous corrections neglected in the massive renormalization scheme of the φ 4 d=3 (n = 1) model (see I for details). We must solve new correlative diculties concerning the eective number (which thus can be greater than three) and the nature (which can originate from the neglected analytical and conuent eects in the critical massive renormalization scheme) of the uid-dependent parameters.

Therefore, in the absence of information concerning the extended asymptotic domain for an actual uid labeled f , it was proposed [4,10] to use the following tting equation of P * (∆τ * ):

P * (∆τ * ) = P * 0,L Z P (∆τ * ) -e P Π P t D(t) (39)
where the two adjustable parameters are the prefactor P * 0,L and a single eective crossover parameter ϑ L introduced through the linear relation:

t = ϑ L ∆τ * (40)
In Eq. ( 39), each function Π P t D(t) is given in the form of a three-term product of the variable t D(t) . The exponent function

D(t) = ∆ MF S2 √ t+∆ S2 √ t+1
is independent of P * and express the crossover of the eective conuent exponent which varies between ∆ and ∆ MF = 1 2 in the com- plete range t = {0, ∞}. Among the prefactors P * 0,L , only two (noted below L * 0,L , X * 0,L ), are uid-dependent to be conform with the two-scale-factor asymptotic universality. ϑ L is thus only characteristic of the extension L f EAD of the critical crossover. For the three singular properties P * = ξ * expt ; κ * T,expt ; ∆ρ LV,expt of present interest, the new tting equations thus read:

ξ * expt (∆τ * ) = ξexpt(∆τ * ) αc = L * 0,L Z + ξ (∆τ * ) ν Π + ξ (t D(t) ) (41)
κ * T,expt (∆τ * ) = X * 0,L Z + χ (∆τ * ) γ Π + χ (t D(t) ) (42) ∆ρ LV,expt (|∆τ * |) = M * 0,L Z M (|∆τ * |) β Π M |t| D(|t|) (43)
where the Ising-like nature of the asymptotic universality is maintained through the following combination of the three prefactors :

L * 0,L -d X * 0,L M * 0,L 2 = 1, (44)
For each uid f , the new (three-parameter) characteristic set reads

S {M R} 1CP,L f = ϑ L ; L * 0,L ; X * 0,L , (45)
where the subscript 1CP, L f recalls for the use of the single crossover parameter ϑ L that denes the Ising-like

extended asymptotic domain ∆τ * L f EAD = L Ising EAD ϑ L
of the uid. The set of Eq. ( 45) replaces the set S {M R} SF,f = {ϑ; g 0 ; ψ ρ } of Eq. ( 24), previously dened in the Isinglike preasymptotic domain.

In comparison to the previous tting Eqs. (28), [START_REF] Le Guillou | [END_REF], and (31), the noticeable modication of Eqs. ( 41) to (43) appears in each leading term in which ϑ L is no longer involved in the asymptotic scaling part of the critical behavior expressed in terms of the physical eld ∆τ * . Moreover, X * 0,L and L * 0,L are two characteristic prefac- tors of f , provided that the same length unit was used to dene the dimensionless quantities of Eqs. ( 41) and ( 42) [START_REF] Privman | Universal critical point amplitude relations[END_REF]. Correlatively, ϑ L is a pure crossover parameter, with the same value above and below T c , which exclusively controls the magnitude of many correction terms to scaling. In addition, ϑ L can also integrate some effects of the neglected terms linked to the supplementary conuent exponents, such as ∆ 2 or ∆ 3 , accounting for practical numerical approximations such as ∆ 2 ≈ 2∆ or ∆ 3 ≈ 3∆. ϑ L can also include the analytic contributions when T replaces T c in the energy unit or in the dimensionless form of the temperature distance to the critical temperature, and when we consider Massieu forms of the singular thermodynamic potential. For each uid f , the determination of ϑ L is then eectively equivalent to the determination of L f EAD . Therefore, as suggested in Ref. [4], from tting of the correlation length, the susceptibility, and the specic heat in the homogeneous and non homogeneous domains, and tting of the coexisting density measurements in the non-homogeneous domain, one must verify the uniqueness of the ϑ L value (along the critical isochore). Considering all these properties allows a consistent test for the determination of the set S {M R} 1CP,L f , in coherence with the basic hypotheses of the renormalization group approach at the origin of the theoretical crossover functions. To avoid this large task, we can thus use an alternative facet of the Isinglike universality of each mean crossover function F P (t).

Indeed, beyond the preasymptotic domain, we can introduce the eective universal behavior of its local exponent e P,th (t) to asymptotically transform each thermodynamic property P * (∆τ * ) into its theoretical universal function F P (e P,th ). Our following approach of the crossover universality illustrates this transformation in a self-consistent manner for the susceptibility and the order parameter density cases, using the universal and experimental eective amplitudes attached to the local power laws with eective exponents, as initially introduced by Kouvel and Fisher in Ref. [14] and already used in II.

The susceptibility case

The local values of the eective exponent γ e,th (t) and eective amplitude Z + χ,e (t) can be estimated by the equa-

tions γ e,th (t) = - ∂Ln [χ th (t)] ∂Lnt (46) Z + χ,e (t) = χ th (t) t -γe (47)
where χ th (t) is given in I. Eliminating t [then simultaneously eliminating the scale factor ϑ L (or ϑ), since t = ϑ L ∆τ * ], the theoretical classical-to-critical crossover is characterized by a universal theoretical curve Z + χ,e (γ e,th ) over the complete range γ MF ≤ γ e,th (t) ≤ γ (see the mixed red curve labeled Φ 3 (1)-MR in Fig. 1).

Our present interest is restricted to the Ising-like range

γ e,th (t) ≥ γ 1 2 = γ+γ MF 2 (see 3.
= Z + χ -1    1 + γ-γ e,th ∆|Z 1,+ χ | - γ-γ e,th ∆ 1 -log γ-γ e,th ∆|Z 1,+ χ | γ-γ e,th ∆ (48)
The .

As a matter of fact, the curve (S) corresponds to the lin-

earized slope ζ + χ,0 = (Z + χ,max ) -1 -(Z + χ,min ) -1
γmin-γmax = 0.007171 0.0025875

between the respective bounded coordinates of points

A and B (see inserted table in Fig. 1 and Ref. [4] for data sources).

Since only two parameters (X * 0,L and ϑ L ) are free in tting Eq. ( 42), Fig. 1 illustrates how the adjustable (metric) prefactor X * 0,L contributes to localize the uid Ising point of coordinates γ;

Γ + = X * 0,L Z + χ -1
. This latter point is represented by the lower cross in Fig. 1 < γe<γ) in xenon case, see Eq. ( 42) and text. Vertical double arrays: two-parameter transformation fγ e X * 0,L , ϑL from upper (T) and lower (P) points (circles) at constant γe = γ and one-parameter transformation fγ X * 0,L for the Ising-like limiting points (crosses) at γe ≡ γ. Curve labeled aT (aP ) : eective Ising-like behavior of Eq. ( 48) beyond the Ising-like preasymptotic domain (experimental case). Insert : details of the singular (logarithmic) behavior (vertical double array labeled (1)) of Eq. (( 48)) within the Ising-like preasymptotic domain of thickness L Ising PAD (see text). For other symbols and labels see text. Se also Fig. 4 in Ref. [10].

in Eqs. (18) or (), xes the value of the xenon prefactor using the following equation (see II)

X * 0,L = L {1f } d (ψ ρ ) 2 ϑ -γ = L {1f } d Ψ {1f } 2 Θ {1f } -γ (Yc) -γ Zc ( 49
)
On the top Eq. ( 49), the value of X * 0,L depends on the three asymptotic scale factors ϑ, L {1f } (or g 0 ), and ψ ρ .

However, L {1f } was a master constant which has xed g 0 knowing α c . Therefore, ϑ and ψ ρ (the latter being implicitly contained in X * 0,L only, due to the extensive na- ture of the susceptibility) are the two uid-dependent pa-rameters characterizing the two-scale factor universality.

This result is conrmed by the bottom Eq. ( 49) where

L {1f } , Ψ {1f } ,
κ * T,expt (∆τ * ) = Γ + (∆τ * ) -γ 1 + a + 1χ (∆τ * ) ∆ + a + 2χ (∆τ * ) 2∆ + a + 3χ (∆τ * ) 3∆ (50)
The critical exponents γ = 1.241 and ∆ = 0.496 [START_REF] Le Guillou | [END_REF] were xed to the theoretical values calculated at the time by Le Guillou and Zinn-Justin from the renormalizationgroup approach. The values of the adjustable parameters were Γ + = 0.0577 (±0.0001), a + 1χ = 1.29 (±0.03), a + 2χ = -1.55 (±0.2), a + 3χ = 1.9 (±0.5) (the error bars quoted are one standard deviation allowing for the correlation between parameters, with an uncertainty of ±0.5 mK on the T c value, see Appendix B). We can then dene the following eective exponent by : γ e,expt (∆τ * ) = -∂Ln κ * T,expt (∆τ * ) ∂Ln (∆τ * ) (51) and its attached eective amplitude by :

Γ + e (∆τ * ) = κ * T,expt (∆τ * ) (∆τ * ) -γe,exp (52)
The resulting single curve Γ + e (γ e,expt ) is illustrated in the lower part of Fig. 1 (see the full blue curve labeled Xe -GC 4 ). The expected (two parameter) transformation f X * 0,L , ϑ L schematized by a double array between the two points T and P on curves of well-dened nite slope, insures that the xenon theoretical curve (see the dotted red line labeled Xe -MR) matches the Xe -GC 4 curve. This transformation must contain both constraints needed to satisfy the (point) position and the related (tangent) direction. Therefore, the scaling nature of the matching beyond the Ising-like preasymptotic domain is signicantly dierent in the tting procedure which either eliminates or accounts for the contribution of the leading term. In the latter situation, we can then replace the prefactor X * 0,L by the true leading amplitude Γ + , as seen below.

In the rst case without contribution of the leading term at large temperature distance, the t procedure based on Eq. ( 42) is mainly equivalent to a predominant constraint in position given by the following relation between the two eective exponents:

γ e,expt (∆τ * ) ≡ γ e,th [ϑ L (∆τ * )] (53)
We numerically solve Eq. ( 53), using Güttinger and Cannell's tting results given by Eq. ( 50), then providing the γ e (∆τ * ) and ϑ L (∆τ * ) values as a function of ∆τ * . Both results are shown by the curve labeled GC 4 in Fig. 2a [γ e as a function of ∆τ * ], and the curve labeled 1 in Fig. 2b [ϑ L as a function of ∆τ * ], respectively.

In the second case, to account for the contribution of the leading term, it is necessary to use the following scaling relation between the two eective amplitudes

Γ + e = (ϑ L ) γ-γe X * 0,L Z + χ,e (54)
Now, the transformation

f γe X * 0,L , ϑ L = X * 0,L (ϑ L ) γ-γe (55)
is explicit in Eq. ( 54). Its takes an eective power law form of the crossover parameter ϑ L , while the prefactor X * 0,L has (as expected above) the same value whatever the γ e (= γ e,expt = γ e,th ) value is. Equation ( 55) distinguishes the metric nature of X * 0,L and the scale factor nature of ϑ L . The both constraints in position and direction are correctly taken into account. Therefore, we can use the following equation That infers the pure ϑ L -dependence of the righ-hand-side of equation

Γ + = X * 0,L Z + χ -1 (
Γ + e Γ + = (ϑ L ) γ-γe Z + χ,e Z + χ -1 (57)
and leads to the unequivocal determination of ϑ L when Γ + , γ e and Γ + e are known, through the equation

ϑ L = 1 Z + χ Z + χ,e × Γ + e Γ +

3.3.

The order parameter density case

The above approach for the susceptibility case can be easily duplicated to the order parameter density case. In this latter one, we consider the vapor-liquid coexisting density data measured by Närger and Balzarini [16] in two dierent samples of xenon, which were tted by the following (three term) Wegner-like expansion:

∆ρ LV,expt (|∆τ * |) = B |∆τ * | β 1 + a 1M |∆τ * | ∆ + a 2M |∆τ * | 2∆ (59)
For the ts, the exponents β = 0.327 and ∆ = 0.

f βe M * 0,L , ϑ L = M * 0,L (ϑ L ) βe-β , (60)
which is similar to the one

f γe X * 0,L , ϑ L = X * 0,L (ϑ L ) γ-γe
for the susceptibility case [see Eq.

M * 0,L = L {1f } d ψ ρ ϑ β = L {1f } d Ψ {1f } Θ {1f } β (Yc) β (Zc) 1 2
(61) which satisfy the two-scale factor universality through its analytical combination [see Eq. ( 44)] with L * 0,L of Eq.

(C14) and X * 0,L of Eq. ( 49). The implicit asymptotic dependence of the scale factors ϑ and ψ ρ is thus properly accounted for, i.e., in conformity with the two-scale factor universality of pure scaling. In that pure power law asymptotical scheme, the introduction of the true leading value of B to replace the prefactor M * 0,L complements our previous introduction of Γ + to replace the prefactor X * 0,L [see Eq. ( 56)]. We can also retrieve these amplitudes using the following equations

Γ + = L {1f } d Z + χ -1 (ψ ρ ) 2 ϑ -γ (62) B = L {1f } d Z M ψ ρ ϑ β (63)
Adding the fact that the prefactor L * 0,L leads to ξ + =

L {1f } -1 Z + ξ -1
ϑ -ν [see Eq. ( 32)], we can easily verify the validity of the Ising-like universal value of the amplitude combinations of Eq. ( 10). Accordingly, we can formulate two matching equations for β e and ϑ L , similar to the ones for γ e [see Eq. ( 53)] and ϑ L [see Eq. ( 58)].

In the rst case using the single constraint in position, we numerically solve the following equation between the two eective exponents:

β e,expt (|∆τ * |) ≡ β e,th [ϑ L (|∆τ * |)] (64)
The The curves ϑ L (|∆τ * |) of Eq. (58) are labeled 2 in Fig. 3(b) (with a full line for the NB 3a case and a dashed line for the NB 3b case). Then the parts (a) and (b) of Fig. 3 for the order parameter density case are similar to the ones of Fig. 2 for the isothermal compressibility case. In Fig. 3, the available (experimental) temperature range is also illustrated by a segment (here labeled NB).

As the theoretical crossover function of the correlation length in the nonhomogeneous domain is not given in Ref. [4], we have used the realistic approximation

th (t < 0) = th (t>0)

-3, as analyzed in

Refs. [28,3234] T c = 289.765 ± 0.005 K from Baidakov et al's [39,40]; T c = 289.790 ± 0.001 K from Güttinger and Cannell [15]; T c = 289.752 ± 0.001 K and T c = 289.789 ± 0.002 K from Balzarini et al's [16]).

Our calculated critical pressure p c = 5.84007 ± 0.00050 MPa accounts for thermodynamic continuity on pressure measurements crossing the critical temperature along the critical isochore. In such a calculation (see Ref. The value γ c = 0.1197 ± 0.0006 MPa K -1 (∼ ±0.5%)

was recently estimated [41] from the joint analysis of the pV T measurements of Habgood and Schneider [35] and Michels et al [44], to account for the small dierences on the critical density values. As a matter of fact, in spite of numerous values reported in the seventies literature, the determination of this nite critical derivative was never accurately analysed in xenon case [17]. We recall that, at the late sixties, the knowledge of the derivative ∂p ∂T ρ in the vicinity of the critical point was mandatory needed when the objectives were to dene the scaled forms of the equation of state (see Refs. Two conclusive remarks can be formulated.

(i) The values of Eq. ( 14) are in remarkable agreement with the ones dened by Gillis et al [26] in their recent analysis of the sound attenuation (in the frequency range 100 < f (Hz) < 7500) by thermoacoustic layers between solid surfaces and xenon at critical density.

(ii) The values of Eq. ( 14) are of basic interest using the scaled forms of the equation of state of xenon [45 50, 53, 54]. Especially in the linear-model parametric equation of state [48,50] and the Ho and Lister's [48] restricted cubic model of the equation of state, the singular behavior of each uid is characterized by only two dimensionless numbers (k and a in standard notations). Now we are able [10] to estimate the xenon parameters k and a only from our above values of the critical point coordinates. However, we recall that these parametric models are not quantitatively exact in regards to the Ising-like universal combinations of the leading amplitudes [55]. T data from κ * T,t = 0.07551466 (∆τ * ) -1.205879 . Segment labeled FIT and underlined gray area: restricted ∆τ * range and 1% error-bar of the t (see text). Segment labeled GC: full experimental temperature range of Ref. [15]. Red (full) circles: experimental data points obtained from Ref. [15], using ρc = 1113 kg m -3 . Pink (full) curve labeled 1: Eq. ( 50), with ρc = 1113 kg m -3 and a Tc shift of 0.5 mK. Pink (dotted) curve labeled 2: Eq. ( 50), with ρc = 1113 kg m -3 . Black (dotted) curve labeled GC4: Eq. ( 50) from Ref. [15], with ρc,GC = 1110 kg m -3 . Horizontal and vertical (pink is not aected by the increasing level of realistic uncertainties approaching T c and shows excellent agreement with the Güttinger and Cannell's one (the amplitude difference accounts for about -4% correction noted above in this restricted temperature range). In Fig. 4, we have reported the residuals %R (κ * T ) = 100

κ * T , κ * T ,t -1 (ex-
pressed in %) for each experimental data point κ * T . In the tting temperature range, the data dispersion is lowered at the ±1% level. Then, our Fig. 4, may be seen similar to Fig. 2 of Ref. [15] with a -4% zero shift in the vertical axis and a magnication of the high relative precision (∼ 0.2%) of the Guttinger and Cannell's measurements in the tting temperature range. In addition the mean (geometrical) value ∆τ * t = ∆τ * min ∆τ * max = 4.215 × 10 -3 of our selected temperature range is very close to the temperature value of the extremum of the residuals (see below). For easy link with our previous gures, the lower and upper horizontal axes are labeled in a similar manner.

Since our tting temperature range includes the value ∆τ * or = 4.215×10 -3 , it is now essential to show that the uncertainty of the amplitude value Γ e,t = 0.07551466 remains only related to the error-bar (∼±1.5%) on the calibration of the susceptibility data and the critical density value.

We have then estimated the residuals for three dierent tting results, using Eq. ( 50) with γ = 1.241, ∆ = 0.496, and the following set of parameters (i) Γ + = 0.0577, a + 1χ = 1.29, a + 2χ = -1.59, and a + 3χ = 1.9, i.e., the tting parameters obtained by Güttinger and Cannell with ρ c,GC = 1110 kg m -3 , which leads to the dotted black curve labeled GC 4 in Fig. 4; 1 and GC 4 combine the eects due to a dierence on the critical density values and a shift of 0.5 mK in T c . Finally, the well-dened extremum of the deviation curves in Fig. 4 is the most important consequence of the high relative precision of the Güttinger and Cannell's measurements.

(ii) Γ + = 0.0577 ρc ρ c,GC
That demonstrates that a well-dened local value (here γ e,t = 1.205879) of the eective exponent can be measured at a well-dened local value of the temperature distance to T c (here T c + 1.25 K). Now, we can rene the above power law analysis at nite distance to T c , accounting thus for the theoretical eective behavior Z + χ,e (γ e,th ) illustrated in Fig. 1, with γ e,th (t) and Z + χ,e (t) dened by Eqs. ( 46) and ( 47), respectively. The condition γ e,th = γ e,t = 1.205879 is observed at t γ e,th =1.205879 = 9.159 × 10 -5 , while the corresponding value of the eective theoretical amplitude is Z + χ,γ e,th = 0.396926. Using then t = ϑ∆τ * [Eq. (25) ] and the asymptotic Ising-like value ϑ = 0.0211752 dened in 2, we expect that the local value of the eective exponent must be observed at ∆τ * γ e,th =1.205879 = 4.347 × 10 -3 (i.e., T -T c +1.26 K), in excellent agreement with the reduced temperature position of the extremum of the curve 1 in Fig. 4. From Γ + γ e,th = Γ + ϑ γ-γ e,th Z + χ,γ e,th (Z + χ )

-1 [see Eq. ( 57)] where we introduce the asymptotic Ising-like values Z + χ -1 = 0.269571, Γ + = 0.057824 previously dened in 2, the corresponding local value of the effective amplitude is Γ + γ e,th =1.205879 = 0.0747481, i.e., a value only ∼ 1% lower than Γ + e,t . However, when the eective power law of Eq. (B1) is directly use to obtain the corresponding local value, hereafter noted ϑ e,t , of the scale factor, we must underline the combined eects of the temperature distance, here accounted for by the exponent dierence γ -γ e,t = 0.0337145, and the experimental uncertainty, here attached to the amplitude value Γ e,t = 0.07551466. Rewriting then Eq. (58) in the following form

ϑ e,t = Z + χ -1 Z + χ,γ e,th × Γ + e,t Γ + 1 γ-γ e,t (B2)
we obtain ϑ e,t = 0.028466, i.e., a value ∼ 35% higher than our asymptotic value ϑ = 0.0211752. Such important discrepancy needs to complement our understanding of the role of the experimental uncertainty on Γ + e,t in the determination of ϑ e,t , thanks to the high relative precision of the light scattering experiment of Güttinger and Cannell.

Indeed, a tting procedure where the eective exponent and amplitude are free in minimizing the mean deviations over a nite temperature range is uncorrect. It must be replaced by a local envelope representation of the κ * T (∆τ * )-curve when the contribution (which continously increases with ∆τ * ) of the conuent corrections to scaling is only characterized by a single parameter. As a practical result, the tting value Γ + e,t of the eective am- plitude is such as Γ + e,t = Γ + γ e,th =γ e,t

. For example, looking now at the 0.2% deviation level reported in Fig. 4, we can observe that the true tangent (pink) line of slope γ e,t = 1.205879 has eectively an amplitude ∼ 0.7% lower than the amplitude Γ + ϑ e,cor = ϑ e,t Γ + e,cor

Γ + e,t 1 γ-γ e,t ϑ e,t 1 + δΓ e,t γ -γ e,t (B3)
From the Güttinger and Cannell's results of Fig. 4 where δΓ e,t -0.007 and ϑ e,t 0.028466, we obtain ϑ e,cor 0.022556 which is now in better agreement (+6.5%) with our asymptotic value ϑ 0.211752. As previously underlined, the precise description by a local exponent value dening the slope of the tangent line to the singular behavior of the isothermal compressibility of xenon at a well-dened temperature distance to T c , is one major point of interest of the Güttinger and Cannell's results to validate the one-parameter crossover modelling predicted by the massive renormalization scheme. More- Literature results show that the experimental precision of the correlation length measurements along the critical isochore of xenon is not sucient to estimate accuratly the conuent corrections to scaling [24]. Accordingly, its singular behavior is often given as a pure power law [25 27] applied in a large temperature range. However, in such a large temperature range, it is also clearly established that the conuent corrections have a signicant contribution to the singular behavior of thermodynamic properties of xenon, as shwon above in the isothermal compressibility and order parameter density cases (see also Refs. [15,16]). Therefore, for the correlation length case, the universal values of the amplitude ratios of Eqs. (C3), (C1), and (C18). For other symbols and labels see text. See also Figs. 1 in Refs. [28] and [10].

tory condition T -T c 0.026 K, we will shown below how ϑ L and L * 0,L are involved in the geometrical con- struction of the point A e in Fig. 5, which corresponds to ξ * expt ∆τ * = 3.45 × 10 -3 = ξexpt αc 7.8.

In the tting Eq. ( 41), the Z + ξ Π + ξ t D(t) term is provided by the theoretical crossover function (see I)

th (t) = t -ν Z + ξ Π + ξ t D(t) (C1)
In the th ; t diagram of Fig.

* MR (∆τ * ) = L * 0,L (∆τ * ) -ν Z + ξ Π + ξ (ϑ L ∆τ * ) D(ϑ L ∆τ * ) (C3)
where the subscript MR recall for the massive renormalzation scheme. Equation (C3) contains the implicit geometrical link between the Ising-like theoretical point A u and the experimental point A e that is only depending on ϑ L and L * 0,L .

In Fig. 5, the relation t = ϑ L ∆τ * corresponds to the horizontal red arrow A e N (with label ϑ L ) between the x-coordinates of the points A e and A u . The singleness of ϑ L implies the unequivocal determination of any rstamplitude a + P,L of conuent corrections to scaling, using the following independent equation:

a ± P,L = -(ϑ L ) ∆ Z 1,± P (C4)
where Z

ξ + L = L * 0,L Z + ξ -1 (C5)
The metric nature of L * 0,L is then shown in Fig. 5 t -ν (for clarity only a limited part of the lines of slope ν is represented).

However, we can also re-introduce the scale factor nature of ϑ L in the pure asymptotic description. The new asymptotic parameter L * L is thus dened by the equation

L * L = L * 0,L (ϑ L ) ν -1 (C6)
A better understanding of the role of L * L , needs to dene simultaneously the dimensional prefactor L 0,L , the leading amplitude ξ + 0,L , and the microscopic wave number g 0,L of xenon by the following equations

L 0,L = α c L * 0,L , (C7) ξ + 0,L = L 0,L Z + ξ -1 , (C8) g 0,L = (L 0,L ) -1 (ϑ L ) -ν = α c L * 0,L (ϑ L ) ν -1 . (C9)
where the subscript L recalls for the non-asymptotic origin of these new parameters, which are ϑ L -dependent and L * 0,L -dependent. Accordingly, the dimensional power law

ξ M R (∆τ * ) = ξ + 0,L (∆τ * ) -ν is known, while the tting equation (with t = ϑ L ∆τ * ) ξ MR (∆τ * ) = (g 0,L ) -1 th (t) (C10)
is valid on the complete temperature range ∆τ * L Xe EAD .

Then, Eq. (C6) can be complemented as follows

L * L = g 0,L α c = α c ξ + 0,L Z + ξ (ϑ L ) ν -1 = ξ + L Z + ξ (ϑ L ) ν -1 (C11)
When α c is known and the singleness of ϑ L hypothesized, the parameter L * L can be substituted to the parameter L * 0,L by using Eq. (C11). The two dimensionless pa- rameters {ϑ L ; L * L } take a similar asymptotic scale factor nature to ϑ; L {1f } dened in S SF of Eq. ( 24)], except that their subscript L recalls for the non-asymptotic hypothesized introduction of ϑ L and L * 0,L , which was made considering the experimental range beyond the Ising-like preasymptotic domain. On other words, the extrapo-

ϑ L Y c = Θ {1f } (C12) L * L = g 0,L α c ≡ L {1f } (C13)
As a correlative result, the prefactor L * 0,L must be also unequivocally related to the critical parameter Y c through the equation (see II): However, the fact that Θ {1f } of Eq. ( 23) is also a master constant for all pure uids is already contained in our previous hypothesis that L {1f } is a master constant for all pure uids. The single needed material is the unique- occurs with t = 0.0211752∆τ * for the xenon case. Therefore, the condition ξ exp (∆τ * ) ∼ α c , where the xenon correlation length is of the order of the molecular interaction range in uids [17], corresponds to the condition th (t) ∼ L {1f } 25.7. This latter value is of the same order as th (t = t ∆ )

L * 0,L = Z + ξ Z + ξ (Y c ) -ν = 1 [L {1f } ×(Θ {1f } ) ν ] (Y c ) -ν (
28.8 at the crossover temperature t ∆ . As shown in I, t ∆ is a convenient sensor to localize the t-range of the critical-to-classical crossing temperatures where the eective theoretical exponents e P,e,th (t) = -∂Ln[F P (t)] ∂Ln(t) [14] crosse their mean crossover value e P, , is similar to the Ising-like prominent nature of the conuent function D(t) in the temperature-like range t t ∆ . Accordingly, the experimental condition ξ expt (∆τ * ) α c , or the measured eective exponents such as 0.63 > ν e 0.57 and 1.24 > γ e 1.12, can be well-understood as related to the Ising-like prominent nature of the crossover behavior. Therefore, Eq. (C19), with t = 0.0211752∆τ * , provides a useful tool to analyze the singular behavior of a xenon properties expressed, either as a function of ∆τ * (using for example an horizontal (lower) axes), or as a function of th (t) (using for example an horizontal (upper) axes) in a same diagram. Anticipating for example the discussion of the isothermal compressibility case given in 3.4 and Appendix C, we can immediatly undestand that the eective exponent value γ e,expt = 1.206 measured by Güttinger and Cannell [15] implies that the equation γ e,th (t) = γ e,expt (∆τ * ) is satised for t ∼ = 9.08 × 10 ii) only for a common dimensionless critical length L {1f } of all uids which obeys to this single parameter crossover description.

∆ ( 3 )∆α c = k B T c p c 1 d= (β c p c ) - 1 d

 311 call the main literature sources to estimate the critical coordinates of the vapor liquid critical point of xenon. In Appendix B, we give an illustration of the analytic determination of the local value of the crossover parameter ϑ L , based on the useful (pure power law) function f κ T (∆τ * ) = Γ e (∆τ * ) -γe to represent the eective singular behavior of the isothermal compressibility behavior in a restricted temperature range. In Appendix C, we give a geometrical interpretation of the nature and the role of the dierents scale factors analyzing the crossover behavior of the xenon correlation length.2. XENON CHARACTERIZATION WITHIN THE ISING-LIKE PREASYMPTOTIC DOMAIN. 2.1. Description of the Ising-like preasymptotic domain In the following we consider three singular properties along the critical isochore: the correlation length ξ * expt (∆τ * ) and the isothermal compressibility κ * T,expt (∆τ * ) in the homogeneous domain (∆τ * > 0), and the order parameter density ∆ρ LV,expt (|∆τ * |) in the nonhomogeneous domain (∆τ * < 0). ∆τ * = T -T c T c (1) is the reduced temperature distance. Within the Isinglike preasymptotic domain, the singular behaviors of the uid properties can be approximated by the two-term restricted forms of the complete Wegner-like expansions, i.e., ξ * expt (∆τ * ) = ξ + (∆τ * ) -ν 1 + a + ξ (∆τ * ) (∆τ * ) = Γ + (∆τ * ) -γ 1 + a + χ (∆τ * ) ∆ρ LV,expt (|∆τ * |) = B |∆τ * | β 1 + a M |∆τ * |

, which is thus lled with 1

 1 tities which are expressed per mass unit. The related practical dimensionless variables using critical parameters such as ρ c and ρc pc are then decorated by a tilde. The denition of μρ introduces another unit pc ρc ∼ energy mass for (specic) energy, which diers from (m pβ c ) Z c . m p is the (molecular) mass of the uid, while the subscript p now recalls for thermodynamic quantities expressed per particle. Correlatively, the total mass unit (M = 1) of a one-component uid system introduces the critical specic volume v c,M =1 = 1 ρc , which diers from v c,I = k B Tc pc = (α c ) d , i.e., the volume of the critical interaction cell [17]. That provides alternative choice between two energy units and two length units which originate from thermodynamics normalized, either per particle, or per mass unit. However, the comparison between the two volumes reects the extensive nature of the uid: at critical density, the amount of matter mp Zc is contained within the microscopic interaction cell of volume v c,I = (α c ) d Zc particles. Only v c,I = (α c ) d takes physical meaning in terms of the short ranged molecular interactions between the 1 Zc uid particles, leading to the appropriate understanding of the correlation length value, measured in unit of α c . For ξ * expt (∆τ * ), κ * T,expt (∆τ * ), and ∆ρLV,expt (|∆τ * |), we expect to dene the amplitudes of Eqs. (2) to (4) from

= 1 n

 1 which localizes the xenon critical point on the phase surface of equation of state Φ p ap (p, v p, T ) = 0. We have xed the value Λ * qe (xenon) = 1 (see Ref. [23] for detail) to neglect the quantum eects at the microscopic length scale. In Q min c,ap or Φ p ap , the subscript a p recall for the thermodynamic description starting from the Helmholtz free energy of the xenon particle, while v p = mp ρ is the molecular volume and γ c is the common critical direction at the critical temperature of the critical isochoric line and the saturation pressure curve in the p; T diagram. The selected critical coordinates of xenon are: T c = 289.733 K, p c = 5.84007 MPa, ρ c = 1113 kg m -3 or v p,c = 0.19596 nm 3 , γ c = 0.1197 MPa K -1 , with m p = 2.1805 × 10 -25 kg.

  g 0 = 29.0245 nm -1 ψ ρ = 3.28165 10 -4The scale factors Θ {1f } , ϑ and Y c are dened through the following linear relations betweeen the theoretical (t), the master (T * ), and the physical (∆τ * ) temperature-like elds

  isothermal susceptibility χT,expt (∆τ * ) = ∂∆ ρ ∂∆μ ∆τ * for uids, are related by the equation

1 (

 1 3 and the corresponding Ising-like behavior dened in the upper part of Fig. 1). The theoretical Ising-like limiting point takes universal coordinates γ; Z + χ upper cross in Fig. 1). The small extension γ -γ e,th Z 1,+ χ ∆ L Ising PAD ∆ ≈ 0.006 of the Ising-like preasymptotic domain is magnied in the insert of Fig. 1. On the other hand, the curve a T corresponds to the asymptotic singular behavior of the derivative ∂Z +

  vertical double arrow with label (1) indicates the logarithmic divergence of ∂Z + χ,e ∂γe γ e,th →γ [see Eq. (48) and II]. We note the signicant dierence between the curve a T and the curve (S) which results from analytic errorbar correlation between the Ising values of γ and Z + χ -1

Figure 1 :

 1 Figure 1: (Color on line) Eective dimensionless amplitudes -exponent diagram for the susceptiblity case. Upper mixed red curve labeled Φ3 (1) -MR : theoretical crossover function, see I and Eqs. (46) and (47). Lower full blue curve labeled Xe -GC4 : experimental (xenon) tting function, see Eqs. (50) to (52) [15]. Lower dotted red curve labeled Xe -MR: theoreticalexperimental matching equation of the Ising-like behavior (γ 1 2

  and Θ {1f } are master constants, while Y c and Z c are the two uid-dependent parameters unequivocally linked to ϑ and ψ ρ , respectively [see Eqs. (23)]. In principle, the critical divergence in the initial slope at the limiting points provides the second Ising-like constraint to determine unambiguously the true asymptotic scale factor ϑ [through the relation ϑ = , as previously mentioned in 2.2]. Nevertheless, our previous description of the Ising-like preasymptotic do-main and its geometrical illustration of Fig. 1 underline the challenging (theoretical and experimental) diculties to provide the exact characterization of the asymptotic scaling when a property reaches the Ising-like limiting point along a curve of universal, but innite, slope for all the physical systems. On the other hand, the description of the γ e -variations in the range γ -γ e 0.015 (i.e., in a temperature range signicantly beyond the Ising-like preasymptotic domain), can be explicited in terms of universal butnite quantities. As already noted in our Introduction, it is then only needed to t the isothermal compressibility data by any continuous function κ * T = f (∆τ * ) over the restricted temperature range L Ising PAD ≤ ∆τ * ≤ ∆τ * max covered by the experiments. In Appendix B for example, we will consider the useful case of an eective power law f (∆τ * ) = Γ + e (∆τ * ) -γe which was commonly used in data analysis at large distance from T c . Hereafter we use the Güttinger and Cannell's susceptibility data tted by the (four terms) Wegner-like expansion (labeled GC 4 , see also Appendix B].

  56) to eliminate X * 0,L and to introduce the true asymptotic amplitude [which constraints the position of the Isinglike limiting point of the experimental curve in Fig. (1)].

  58), applied in the extended asymptotic domain ∆τ * L Xe EAD , has equivalent Ising-like meaning as the equation ϑ = . (32)] applied within the Ising-like preasymptotic domain ∆τ * L Xe PAD . We have numerically solve Eq. (58), with Γ + = 0.057824 [see Eq.

( 18 )

 18 ], by the appropriate combination between Güttinger

Figure 2 : 1 2 2 ,

 212 Figure 2: (Color on line) Ising-like crossover behaviors (xenon case) of γe (a), ϑL (b), and R% (ϑL) ϑ = 100 ϑ L ϑ -1 (c), with ϑ = 0.021175 of Eq. (24), as a function of ∆τ * (lower horizontal axis) and th (upper horizontal axis). Part (a): black curve labeled MR: crossover function of Eq. (46); red curve labeled GC4: Güttinger and Cannell [15] tting Eq. (50); right corner and square: γe at temperature distances dened in Fig. 5; right side colored arrows : local values of the classical-to-critical crossing conditions dened in text. Parts (a) to (c), blue points : corresponding results at ∆τ * γ e,th = 4.347×10 -3 (see Appendix B). Parts (b) and (c), curve 1: Eq. (53); curve 2: Eq. (65); Curve m: mean value of curves 1 and 2. Part (c) : residual isoclines of Eq. (B7) with δΓ + e = ±0.2% and δΓ + e = ±0.02% (see Appendix B). For other symbols and labels see text.

(

  55)]. Thus the only needed material to obtain ϑ L concerns the universal feature of the matching between the Ising-like limiting points at β e = β. Here, the limiting transformation reads f β M * 0,L ≡ M * 0,L . The corresponding value of the xenon prefactor M * 0,L can be obtained using the following relations (see II)

  resulting curves β e [(|∆τ * |)] and ϑ L (|∆τ * |) are reported in Figs. 3(a) (with the label NB 3 ) and (b) (with the label 1), respectively (the case NB 3a corresponds tothe full line, while the case NB 3b corresponds to the dashed line). In the second case using both constraints in direction and position, the pure ϑ L -dependence is obtained solving numerically the equation

3 . 2 , β 1 2 (

 322 For the values of the eective exponent β e indicated by the black, red and brown horizontal arrows, respectively, which correspond to ∆ 1 order parameter density case), and γ -

1 2 (

 12 Fig. 2, respectively. More generally, the eective non asymptotic form of any parametric equation of state with only two-uid dependent parameters can be used to provide the local approximation of the crossover functions for each value of the eective exponent. The only needed Ising-like condition is that the correlation length satises the condition th 70 2.7L {1f } , i.e., ξexpt αc

  .

  xenon as a standard critical uid, and the mean crossover functions for both the susceptibility in the homogeneous domain and the order parameter density in the nonhomogeneous domain as illustrative examples, we have estimated the values of the uid-dependent parameters, which are compatible with the universal features predicted by the massive renormalization scheme. A special mention for the three (dimensionless) parameter characterization within the Ising-like preasymptotic domain was given only using a four parameter localisation of the xenon critical point. We have shown that, when T c and α c are known, the mean crossover functions take a convenient explicit form to determine analytically a single crossover parameter ϑ L in a temperature range beyond the Ising-like preasymptotic domain. We have also clearly shown that the value of this crossover parameter is entirely governed by the data measurements at the largest distance to the critical point. The Ising-like nature of this crossover parame-ter is then revealed using the dimensionless master value of a single characteristic length. Finally, the magnitude of the resulting deviations and the range of temperature where these deviations become signicant to invalidate the uniqueness of the crossover parameter are exactly accounted for. From this Ising-like standard situation provided by critical xenon, the real extension and amplitude of the singular behavior of the uid properties can be estimated for any one-component uid for which the vaporliquid critical point is localized in the pV T phase surface, thanks to the use of the master crossover functions given in II. Appendix A: XENON CRITICAL COORDINATES The xenon critical temperature was xed to the value T c = 289.733 ± 0.002 K recently recommended by Gillis et al [26, 27] from their critical temperature determination of the stirred xenon lling its acoustic resonator cell submitted to a ramp of temperature downward (in this experiment the absolute temperature precision is ±15 mK from reference to the ITS-90 temperature scale). This value agrees with the two respective values T c = 289.731 ± 0.0053 K and T c = 289.734 ± 0.003 K measured (with an absolute precision of ±50 mK) by Berg and al [25] from observation of the vapor-liquid meniscus appearance and disappearance in the Critical Viscosity of Xenon (CVX) experiment. These central values, and their relative uncertainties essentially due to the thermostat temperature control, compare well with T c = 289.740 ± 0.003 K obtained by Schneider et al from pV T measurements[35] and density measurements of coexisting liquid and vapor phases[36]. This latter value was generally used as a xenon critical temperature in previous review analyses[17,20,[START_REF] Levelt Sengers | Progress in Liquid Physics[END_REF] using the ITS-68 temperature scale. Indeed, the agreement was noticeable with T c = 289.747 ± 0.010 K obtained by Cannell et al[37] from measurements of Brillouin spectrum, and T c = 289.736 ± 0.002 K obtained by Smith et al[38] from light scattering intensity measurements. However, the Gillis et al's central value disagrees with some other values of similar relative precision (as for example:

[ 41 ] 9 - 3 . 4

 41934 for details), the Habgood and Schneider's isotherm p (ρ) at (T c ) HS = (273.15 + 16.59) K was used as the critical isotherm of xenon to estimate our above value of the critical pressure. Our selected value of ρ c = 1113 kg m -3 has an uncertainty of ±5 kg m -3 (∼ ±0.5%), which accounts for the ρ c values of Schneider et al's (ρ c = 1105 ± n.a. kg m -3 and ρ c = 1099 ± n.a. kg m -3 ) [35, 36], Cornfeld and Carr's (ρ c = 1111.2 +1.kg m -3 for three dierent estima- tions) [42], Baidakov et al's (ρ c = 1112.8 ± n.a. kg m -3 ) [39], and Balzarini et al's (ρ c = 1099 ± n.a. kg m -3 [43], ρ c = 1116.0 ± 1.7 kg m -3 and ρ c = 1114.7 ± 1.7 kg m -3 [16]).

  [4550]) and to test their related computations of the thermophysical property singularities (since the dimensionless quantity T pc ∂p ∂T ρc appears in many thermodynamic relations). In xenon case, the dimensionless value Tc pc γ c = 6.02 was initialy obtained by Vicentini-Misssoni et al [47] from their tting of the pV T measurements of Habgood and Schneider (with p c = 5.83 MPa, T c = 289.75 K, and ρ c = 1110 kg m -3 , as xenon critical coordinates). The related uncertainty on the dimensional value γ c = 0.1211 MPa K -1 was not given, in spite of the fact that this value was higher (∼ +1.8%) than the one γ c = 0.1189 MPa K -1 initially found by Habgood and Schneider from a self-consistent analysis of the derivative ∂p ∂T ρ graphically deduced from their pV T measurements. Subsequently, several published values [such as γ c = 0.11916 MPa K -1 from Cannell and Benedek [37], γ c = 0.12027 MPa K -1 from Smith et al [38], γ c = 0.1196 MPa K -1 from Swinney and Henry [51], γ c = 0.1192±0.0012 MPa K -1 from Garrabos [17]] were obtained from these Habgood and Schneider's data source and same xenon critical parameters. On the other hand, Badaikov et al have determined two values, γ c = 0.1865 MPa K -1 [39], and γ c = 0.1977 MPa K -1 [40], from their vapor pressure data below T c . Berg and al [25, 52] have used the dimensionless value Tc pc γ c = 5.65 (with an uncertainty of ±2.9%), in their viscosity data analysis of the CVX experiment with ρ c = 1110 kg m -3 . Their related value γ c = 0.113686 MPa K -1 is signicantly lower ( -5.%) than a selected value in the range γ c = 0.1195 -0.1197 MPa K -1 . More recently, Gillis et al, selecting the highest value ρ c = 1116 kg m -3 of the critical density, have used the dimensionless value Tc pc γ c = 5.9253 calculated by Swinney and Henry for the presumable critical isochore at ρ = 1110 kg m -3 , attributing then an uncertainty of 0.2% on the corresponding value γ c = 0.1195 MPa K -1 . Therefore, still today, the largest uncertainty in the xenon critical coordinates of Eq. (14) comes from the determination of γ c , which is then dependent of the selected value for the critical density. In the future, a better estimation of γ c needs that the two derivatives ∂p the vicinity of the critical point in order to account correctly for the contribution of the relative uncertainty in the ρ c value.

  Appendix B: EFFECTIVE POWER LAW ANALYSIS An useful mathematical function to t the isothermal compressibility data measured at nite distance to the critical temperature along the critical isochore is the simple power law: κ * T,t = Γ + e,t (∆τ * ) -γ e,fit , (B1) with an adjustable non-Ising exponent γ e,t and an adjustable eective amplitude Γ + e,t . The values of γ e,t and Γ + e,t are then associated to the limited experi- mental temperature range ∆τ * min ≤ ∆τ * ≤ ∆τ * max of the t. For example, Güttinger and Cannell have claimed that the correction to scaling terms are important by demonstrating in their Fig. 2 of Ref. [15], that their susceptibility measurements of high relative precision (∼ ±0.2%) deviate systematically from a simple power law behavior with γ e,GC = 1.206 and Γ + e,GC = 0.6390χ T,ρ (∆τ * or ) p c,GC (Tc) -1.206 (ρ c,GC ) 2 = 0.07867. Looking in detail their gure, we note that γ e,GC 1.206 corresponds precisely to the slope of the tangent line to the experimental behavior close to the relative temperature distance ∆τ * γ e,GC =1.206 4.35 × 10 -3 , i.e., the local temperature value T T c + 1.25 K where we observe the extremum (around -4%) of the deviation curve. Hereafter, we have magnied this observation considering the isothermal compressibility data which account

Figure 4 :

 4 Figure 4: (Color on line) Residuals R% (κ * T ) (expressed in %) of κ *T data from κ * T,t = 0.07551466 (∆τ * ) -1.205879 . Segment labeled FIT and underlined gray area: restricted ∆τ * range and 1% error-bar of the t (see text). Segment labeled GC:

  ) lines: from reference to the local value Γ + e,cor = 0.993 × 0.07551466 at ∆τ * γ e,th = 4.347 × 10 -3 where γ e,th = γ e,t = 1.205879 (see text). Horizontal blue line labeled R% Γ + e,exact : residual for the exact local value Γ + e,exact = 0.0747481 at ∆τ * γ e,th = 4.347 × 10 -3 (see text); Other symbols, marks and labels: see text and previous gures. for our values of the critical density ρ c = 1113 kg m -3 and critical pressure p c = 5.84007 MPa in the equation κ * T,expt = χ T,ρ,expt pc (ρc) 2 χ T,ρ (∆τ * or ). Using Eq. (B1), we have then tted the twelve data covering the restricted temperature range 9.115 × 10 -4 ≤ ∆τ * ≤ 1.95 × 10 -2 (i.e., 0.26 K ≤ T -T c ≤ 5.65 K) dened by the segment labeled FIT and the gray area in Fig. 4 (see also the upper axis of Fig. 5). As expected, our tting result κ * T,t = 0.07551466 (∆τ * ) -1.205879

2

  and same values of a + 1χ = 1.29, a + 2χ = -1.59, and a + 3χ = 1.9, leading to the full red curve labeled 1 which account for the present critical density value; (iii) the latter parameter set and a shift of 0.5 mK in T c , illustrated by the dotted pink curve labeled 2, which accounts for the realistic increase of the experimental uncertainty near T c . The relative dierences between curves 1 and 2 increases approaching the critical temperature, a result well-observed outside our temperature tting range. The shape dierences between curves

- 1 =Z 10 - 3 .

 1103 over, as indicated by the horizontal blue line corresponding to the residual 100 -1.015, the expected value, at ∆τ * γ e,th =1.205879 = 4.347 × 10 -3 , of the extremum which corresponds to the asymptotic value ϑ 0.211752, is well compatible with the realistic experimental uncertainty. Since the Güttinger and Cannell's results of highly relative precision 0.2% have illustrated the signicant role of the exponent dierence γ -γ e when the restricted temperature range is selected outside the Isinglike preasymptotic domain, we can also estimate the related temperature eect of the uncertainty level on the value of ϑ (as calculated in Section 2). We consider thus an eective tting procedure which provides the values of γ e , Γ + e , and ∆τ * e , where it is assumed that the eective amplitude Γ + e is determined with a relative error-value δΓ e , such as Γ + e,exact = Γ + e (1 + δΓ e ). Here Γ + e,exact is de facto the exact value when it is dened such that , the value of Z + χ , Γ + , γ e,th ≡ γ e , and Z + χ,e (γ e,th ) are known with zero uncertainty from the mean crossover function for the susceptibility case. Using then Eq. (B1) and (B4), we obtain ϑ γ e = γ e,th estimated at ∆τ * γ e,th =γe = tγ e,th ϑ with ϑ = 0.0211752, we can easily calculate the isocline ϑ e,δΓe ∆τ * γ e,th =γe at constant (small) value of δΓ e , using the following approximation ϑ e,δΓe ∆τ * γ e,th =γe ϑ 1 -δΓ e γ -γ e (B6) The corresponding isocline of the residuals is thus r%(ϑ e,δΓe ) = 100 ϑ e,δΓe ϑ -1 = -δΓ e γ -γ e (B7) Two pairs of symmetrical isoclines of Eq. (B7) are illustrated in Fig. 3(c) for δΓ e = ±0.2% (the relative precision of Güttinger and Cannell's data) and δΓ e = ±0.02%, respectively, thus evidencing the experimental challenge to validate the equation ϑ = Y c The needs for a critical increase of the experimental precision when ∆τ * decreases is now well quantied by Eq. (B7) when the objective is to test the asymptotic validity of linearized Eqs. (C15) or (C16).

( 11 )

 11 and (12) are not accounted for correctly and some inconsistencies are then introduced in the Ising-like universal features of xenon properties. Nevertheless, it remains possible to analyse the Ising-like consequences of the hypothetic existence of a single scale factor ϑ L for the singular behavior of the xenon correlation length in the homogeneous domain.

5 [ 5 .Figure 5 :

 555 Figure 5: (Color on line) Correlation lengths as functions of their respective thermal-like elds (log-log scale). Full curves: crossover functions. Tireted lines: corresponding asymptotic power laws of slope ν. Blue, e, 1, Ae, E : experimental (xenon) case ξ * expt (∆τ * ) [see Eq. (C3)]. Red, u, 2, Au, U : universal case th (t) [see Eq. (C1)]. Green, m, 3, Am, M : master case * (T * ) [see Eq. (C18)]. Horizontal and vertical, red (ϑL, L * L ), green (Θ {1f } , L {1f } ), and blue (Yc) arrows: scale factors of Eqs. (C15), (C16) and (C17)]. Colored circles, crosses, right corners, and squares : in same geometrical correspondance. Horizontal blue (label ED), red (UD) and green (MD) segments : intermediate crossover ranges of validity (see text). Vertical red arrow at ∆τ * = t = T * = 1: metric parameter L * 0,L of Eq. (C14). E, U, M, y-coordinates : respective prefactors parameters of Eqs.

20 and

 20 our following purpose concerns especially the point A u on this theoretical curve. Obviously, from our above hypothesized description, the experimental blue curve labeled e in Fig.5should be represented by the following theoretical crossover function ξ

  by the vertical arrow EU at t = ∆τ * = 1. The point E of y-coordinate ξ + L is located on the tireted blue line 1 of experimental pure power law ξ * MR (∆τ * ) = ξ + L (∆τ * )

  lated pure power laws illustrated by the tireted blue and red lines within the Ising-like preasymptotic domain are obtained without any reference to experimental measurements performed in the temperature range T -T c ≤ T c L Xe PAD = 0.026 K. In Fig. 5, this (nonasymptotic) scale factor nature of ϑ L and L * L is illus- trated by the horizontal and vertical red arrows corresponding to the ratios of xand y-coordinates of the points A e and A u of the curves e and u, beyond the Ising-like presymptotic domain. Obviously, in these intermediate Ising-like crossover domains, the same matching occurs for all points of the full curves e and u due to the singleness of ϑ L and the scale factor nature of ϑ L and L * L , as illustrated for example by the corresponding crosses, righ corners, and squares. The crosses correspond to the borderline of the Ising-like preasymptotic domains, while the right corners and squares dene the conditions ξ * MR 2.7 and ξ * MR = 1, respectively, which will be discussed below. The missing geometrical link with our previous description of the Ising-like preasymptotic domain made in Section 2, can be provided by the master description of the correlation length of any one-component uid [28]. As a matter of fact, if the xenon scale factor L * L takes the mas- ter value L {1f } of Eq. (29), thus the tting Eq. (28) is true and the identity ϑ L ≡ ϑ is valid. In that additional hypothetic situation for xenon, our previous description of the correlation length by the two-term Eq. (2) is valid within the Ising-like preasymptotic domain. The comparison with the tting Eq. (C10) obtained from measurement data beyond the Ising-like preasymptotic domain shows immediately that both non-asymptotic parameters ϑ L and (g 0,L ) -1 must be unequivocally related to the critical parameters Y c and α c respectively, through the equations

1

 1 C14) Finally, by assuming the singleness of ϑ L and the identity L {1f } L ≡ L {1f } , we observe that the initial hypothetic non-asymptotic parameters ϑ L and L * 0,L are only de- pendent of Y c . But Y c is a parameter strictly dened at the critical point that implies a pure Ising-like nature of the conuent corrections to scaling due to a single irrelevant uid eld. Indeed, in the scale dilatation method [19], the scale factor Y c has been obtained from a physical quantity expressed in units of k B (αc) d = pc Tc , which intro- duces a characteristic value of the entropy per particle [29]. Y c is then only dependent of our selected critical length unit of Eq. (5). For the correlation length measured in unit of the thermodynamic length unit α c , the conuent corrections to scaling are governed by a single characteristic parameter of the critical interaction cell of volume v c,I = (α c ) d , lled by n c,I = ρc k B mp (α c ) d = Zc particles. In other words, the asymptotic master contribution of the conuent corrections for the {1f }-subclass is well due to the master properties of the critical interaction volume only characterized by the two dimensionless numbers Y c and Z c (see also II). Our present hypothetic introduction of ϑ L and L * 0,L to describe the xenon cor- relation length by Eq. (C3) is then Ising-like equivalent to our previous introduction (see II) of the two dimensionless parameters Θ {1f } and L {1f } to obtain the asymptotic matching between (two-terms) master and universal crossover functions for correlation length.

Θ 2 . 5 - 3 .EAD 2 ×

 2532 ness of the length unit to express the correlation functions and the thermodynamic functions in dimensionless forms. Accordingly, the identity ϑ L ≡ ϑ is strictly Ising-like equivalent to any other identity L {1f } L ≡ L {1f } , L * 0,L ≡ L * 0 or L 0,L ≡ L 0 . That gives fundamental interest to recall that the following interrelated variable rescalingsξ * expt → th = L {1f } ξ * expt ∆τ * → t = ϑ∆τ * (C15) ξ * expt → * ≡ ξ * expt ∆τ * → T * = Y c ∆τ * (C16) * → th = L {1f } * T * → t = Θ {1f } T * (C17)provide the asymptotic theoretical matching of the experimental, universal and master singular behavior of the correlation length within the Ising-like preasymptotic domain, as described in 2.Therefore, accounting for Eqs. (C15) to (C17), Fig.5can be complemented by the full green curve (labeled m)of master crossover equation [28] * (T * ) = Z + ξ (T * ) {1f } T * D(Θ {1f } T * )(C18) applied in the intermediate Ising like crossover domain of present interest (with Λ * qe = 1 in the xenon case [23]). Accordingly, we have located the point A m of coordinates T * = Y c ∆τ * ; ξexpt(∆τ * ) αc , while the tireted green line (labeled 3) corresponds to the pure power law * = Z + ξ (T * ) -ν . The respective master scale factor role of L {1f } and Θ {1f } is illustrated by the vertical and horizontal green arrows corresponding to the ratios of yand x-coordinates of the points A m and A u of the curves m and u. The phsyical scale factor role of Y c is given by the horizontal blue arrow corresponding to the ratios of x-coordinates of the points A m and A u . As previously mentioned, the same matching occurs for the corresponding crosses, righ corners, and squares of the three curves. In an equivalent manner, at t = T * = 1, the respective metric nature of the prefactors of Eqs. (C1) and (C18) is given by the y-coordinates of the points M and U on lines 3 and 2, respectively. The intermediate Ising-like master crossover domain corresponds to the correlation length range 70 * (T * ) 1 dened for the temperaturelike range such as t∆ Θ {1f }T * t∆ Θ {1f } (see the green segment labeled MD in Fig.5).Therefore, the Ising-like equivalence between the ex-pected identities ϑ L ≡ ϑ, L {1f } L ≡ L {1f } , or L * 0,L ≡ L * 0 isgiven in Fig.5as a geometrical form of length equality between horizontal and vertical segments such as: PA u = A e A m + NA e , NA u = PA m , or EM = EU + UM [see Eqs. (47) to (49)]. Obviously, the results of Ref. [28] have shown that the Ising-like nature of these scale factors and prefactors is valid in the master extended temperature-like range T * 0.15 -0.2 which satises the condition * (T * ) The right corners in the three curves representing the dierent forms of the singular correlation length are representative of these Ising-like extended asymptotic domains. For the xenon case, the corresponding Ising-like extended asymptotic domain is such that ∆τ * L Xe 10 -2 , i.e., T -T c T c L Xe EAD 5.5 -6 K, as illustrated in the upper horizontal axis of Fig. 5. In the next 3.3, we discuss the Ising-like equivalent nature of the experimental and theoretical crossovers until the critical-to-classical crossing temperatures (square symbols).

2 .

 2 Critical-to-classical crossing temperaturesAs a direct consequence of the unequivocal link between ϑ ≡ ϑ L and Y c , the identity th (t)L {1f } ≡ ξ expt (∆τ * ) α c (C19)

ξexpt αc = th 25 . 7 ∼ = 6 .

 2576 -5 where th (γ e,th = 1.206) ∼ = 174.7, leading to match the experimental correlation length value 77 at ∆τ * ∼ = 4.3 × 10 -3 , i.e., T -T c ∼ = 1.25 K. In Appendix B, we will conrm that the eective value γ e,expt = 1.206 is precisely observed in the Güttinger and Cannell experiment at this temperature distance.More generally, the asymptotic scale factor nature of L {1f } in Eq. (C19) implies that the uniqueness of the critical length unit is a necessary condition for a better understanding of the extension range of the Ising-like universality. Indeed, the Ising-like asymptotic matching of the dierent expressions of the correlation length implies equivalent Ising-like asymptotic matching of any other forms of each singular thermodynamic property, by virtue of hyperscaling. In other words, from the hypothesis of a single crossover parameter and the use of a single length unit, all pure uid crossover functions are Ising-like universal: i) only over the temperature range where the crossover parameter is unique;

  5, th (t) is represented by the full red curve u in the intermediate Ising-like universal domain L Ising PAD ≤ t ≤ t ∆ , which corresponds to the red segment labeled UD (see also the ending vertical arrows

						the exchange between the prominent Ising-like nature of
						the conuent exponent due to the value ∆ = 0.50189
						approaching the non-trivial xed point [t	t ∆ ], to
						its prominent mean eld-like nature due to the value
						∆ MF = 1 2 approaching the Gaussian xed point [t t ∆ ]. As shown in I by introducing the practical relation L Ising ∼ = 10 -3 , we separate, PAD = t ∆ in Eq. (35), with either the Ising-like preasymptotic domain t t ∆ [see
						arrow labelled PAD(MR) in Fig. 5], or the intermediate
						Ising-like crossover domain	t ∆ < t	t ∆ [see horizon-
						tal red segment labeled UD in Fig. 5]. We observe that
						th ( t ∆ ) = 1916, while th (t ∆ ) = 28.8. Therefore, the
						Ising-like preasymptotic domain of interest in 2 corre-
	L Ising PAD and t ∆ ). L Ising PAD is given by Eq. (35). t ∆ is the	sponds to th matively two-decades of the curve u, i.e., 2000 1900. Here we are concerned by approxi-th
	theoretical crossover temperature (see I) given by
	t ∆	1 S 2	2	∼ = 1.9 × 10 -3	(C2)
	The universal function D (t) takes the eective mean
	value D (t ∆ ) = ∆+∆ MF 2	at t ∆ . Then t ∆ characterizes

  L , in conformity with the universal features calculated for the lowest exponent ∆. On the other hand, for ∆τ * → 0, the leading amplitude of the pure asymptotic power law ξ

	1,± P	is given in I. A single irrelevant eld respon-
	sible of the conuent corrections to scaling is then only
	characterized by the non-asymptotic crossover parame-
	ter ϑ

* MR (∆τ * ) = ξ + L (∆τ * ) -ν

can be calculated without reference to ϑ L , using the equation:

  Therefore, the Ising-like prominent nature of F P (e P,th ), dened by the equivalent condi-

	1 2 correlation length and susceptibility cases, t ν 1 t e P, 1 = e P +e P,MF (see Figure 4 in I). For the 2 2 ∼ = 3×10 -3
			2
	and t γ 1 2	2 ∼ = 4 × 10 -3 , respectively, with th t ν 1	= 22.2
	2 and th t γ 1	= 18.8.

tions th (t) L {1f } or e P,e,th (t) ⊂ e P , e P,

1 2

e,t used as a reference. More generally, from Eq. (B2) and a careful analysis of the t deviation curve to estimate δΓ e,t at the extremum position, we can write Γ + e,cor = Γ + e,t (1 + δΓ e,t ) and dene the corrected value ϑ e,cor related to the local envelope by the equation


A Kimbonguila

M Guessasma

P Coorevits

J Fortin

E Bellenger

Discrete modelling of the fragmentation of granular media under compressive forces: Application to grinding process

Keywords: Granular media, DEM, fragmentation, grinding process

The aim of this work is a numerical simulation of physical phenomena during the grinding process by using a discrete approach. The Discrete Element Method (DEM) has been used to model the fragmentation and mechanical behaviour of granular packing under compressive forces. Through this study, we show the effect of contact force chains on the stress field distribution into the granular packing. We have also studied the fragmentation mode of grains induced by the stress field and its influence on the granulometry. Numerical simulations of granular packing subjected to compressive loading have been carried out with MULTICOR software.

Introduction

The grinding process is commonly used in many industrial fields such as: pharmaceutical, mineral or food. However, such a process doesn't come cheap, essentially due to the energy cost and to the low quality of finished products (heterogeneity of grain size and shape). Hence, the economic stakes are large and the optimization of such process seems crucial. Due to this energy cost, understanding the grinding process for granular media under compressive forces could be a good way to reach a better fragmentation efficiency. In order to be able to get answers to many questions about the grinding process, numerical simulations seem to be a good way to predict the mechanical behaviour of granular media. To carry out such simulations, we must define the fragmentation mode of one grain subjected to a random contact forces distribution induced by the adjacent grains. The investigations carried out on the fracture of grain [START_REF] Aström | Fragmentation of grains in a two-dimensional packing[END_REF][START_REF] Morikawa | Analytical studies on local damage to reinforced concrete structures under impact loading by discrete element method[END_REF][START_REF] Tsoungui | Use of contact area trace to study the force distributions inside 2D granular systems[END_REF][START_REF] Tsoungui | Numerical model of crushing of grains inside two-dimensional branular materials[END_REF][START_REF] Morris | Simulations of fracture and fragmentation of geologic materials using combined FEM/DEM analysis[END_REF][START_REF] Lobo-Guerrero | Influence of pile shape and pile interaction on the crushable behaviour of granular material around driven piles: DEM analyses[END_REF] have shown that the fragmentation mechanism depends on several parameters, which are: the size, the shape, the material properties and the loading configuration. It has been concluded that the crack into the grain follows a transversal propagation leading to the fragmentation. In addition, the crack propagation depends on the stress field inside the grain [START_REF] Aström | Fragmentation of grains in a two-dimensional packing[END_REF] and the stress intensity factor of material [START_REF] Aström | Fragmentation of grains in a two-dimensional packing[END_REF][START_REF] Morikawa | Analytical studies on local damage to reinforced concrete structures under impact loading by discrete element method[END_REF][START_REF] Tsoungui | Use of contact area trace to study the force distributions inside 2D granular systems[END_REF][START_REF] Tsoungui | Numerical model of crushing of grains inside two-dimensional branular materials[END_REF].

Throughout this study, we have developed a discrete modelling for granular packing subjected to compressive loading by using DEM. We have used a grain fragmentation model depending on contact forces randomly spread [START_REF] Tsoungui | Numerical model of crushing of grains inside two-dimensional branular materials[END_REF][START_REF] Lobo-Guerrero | Influence of pile shape and pile interaction on the crushable behaviour of granular material around driven piles: DEM analyses[END_REF]. The criterion of fracture depends on the tensile stress inside the grain and on the stress intensity factor of the material. The proposed model of fragmentation has been implemented in MULTICOR software [START_REF] Fortin | Numerical simulation of granular materials by an improved discrete element method[END_REF] and carried out for numerical calculations of granular packing under oedometric compression. Through this mechanical test, the objective is to be able to reproduce similar effects arising during grinding process.

DEM approach

The 2D modelling of granular material is considered as a stacking of several discs subjected to a given mechanical solicitation. However, the discs are commonly called particles or grains in the literature. The conventional DEM allows to model really deformable particles as well as complex shapes (from the ellipsoid to the polygon) [START_REF] Cundall | The Development of Constitutive Laws for Soil Using the Distinct Element Method[END_REF][START_REF] Cundall | Modeling of Microscopic Mechanisms in Granular Material[END_REF][START_REF] Cambou | Numerical modelling of the mechanical behavior of non spherical crushable particles[END_REF]. Here, we have studied the simple case of non-deformable and non penetrable particles in 2D with the computational software MULTICOR. The coordinates and the rotations of Euler are the configuration parameters q. The gyroscopic and centrifugal terms are equal to zero in 2D. The matrix of generalized mass M of the system doesn't depend on q that is diagonal block. The mechanical equation ( 1) can be written in the following form:

   R t , q , q F q M ext      (1)
where F ext represents the known external forces and  R the unknown interior forces related to contact reactions and  the contact number.

In a system composed of p heterogeneous particles, the critical parameter for the modelling time is the maximum number of interactions between particles. The more the interaction range is important the more we have to test the possible interactions between particles (Figure 1). MULTICOR uses the partitioning method coupled to a connectivity table. This technique allows to reduce considerably the computational time. In that case, the computational time no longer increases like O(p 2 ) but only like O(p), which is almost optimal [START_REF] Fortin | Selecting contact particles in dynamics granular mechanics systems[END_REF]. For each couple of particles  i and  j which may enter in contact, is associated with a local reference whose axes are oriented according to the two unit vectors n and t, respectively normal and tangential vectors in the contact plan. The normal n is directed from  i and  j . The variables put in duality are ij u  , the relative local velocity of  i compared to  j , and the contact reaction ij r of  i and  j . In the local base, they are written by equations (2): The introduction of Coulomb's friction  leads to a non linear problem which can't be solved by a linear programming method. Unlike the usual approach, the bipotential method leads to a single variational principle and an inequality [START_REF] Fortin | Modelisation numerique des milieux granulaires par l'approchedu bi-potentiel[END_REF]. By using Usawa's algorithm, we obtain a resolution algorithm of the constitutive law based on the predictive-corrective scheme expressed by equations (3):

n r r r , n u u u ij n ij t ij ij n ij t ij        (2)
           K , proj r : corrector , n . u - u u - r : predictor ij ij ij t ij n ij t ij ij        ( 3
)
where  is a numerical parameter and  K the Coulomb's cone. The projection of the prediction ij  on the Coulomb's cone  K leads to one of these states: non contact, contact with adherence or sliding contact. Conventionally, at each time step, the contact forces in the system are determined repeatedly by the method of successive balances based on a Gauss-Seidel algorithm for the 2D version. Each contact force is calculated by adopting temporary values over the other contacts. The convergence is obtained when the force confirms the unilateral contact law with dry friction.

Stress tensor of granular material

In order to understand the mechanisms governing the fragmentation of the grains subjected to compressive loading, we must be able to compute the stress tensor inside each grain of the discrete media. The knowledge of stress field leads to the prediction of the grain fragmentation according to the material strength and the fracture criterion.

Previous studies [START_REF] Fortin | Construction of an average stress tensor for a granular medium[END_REF][START_REF] Guessasma | Numerical modelling of mechanical tests using discrete element method[END_REF] have been carried out in order to determine the stress tensor of granular material. The proposed stress tensor is averaged with respect to a Representative Elementary Volume (REV) containing N p grains separated by voids (Figure 2). The grains are considered as rigid solids in punctual contact for each grains p occupying the volume V p , the expression of stress tensor is given by the following equation ( 4):

               p V c p p r q dV q g q V 1        (4)
where g denote the gravity, q the position vector, q  its acceleration and  r the contact reaction acting on the particle at the point of contact  c q . According to the additivity on internal moments in the sense of Moreau [START_REF] Moreau | Numerical investigation of shear zones in granular materials, In: Grassberger[END_REF], the mean stress tensor  on the REV can be written as follows (5):

   V p p p V V 1   (5)
Therefore, the stress tensor p  is composed in three parts: contact reaction stress, gravity stress and stress part due to the inertial forces. Accordingly, the contribution of inertial forces insures the symmetry of the stress tensor in the sense of Cauchy.

In the present study, the stress tensor will be computed with respect to a REV volume containing only one grain. The stress tensor will be then determined inside each grain by considering only the contribution of contact forces (inertial and gravity effects are neglected).

Numerical modelling of grain fragmentation

The numerical modelling carried out in this study describes the mechanical behaviour of granular packing under compressive loading and the fragmentation mechanisms which brought in play by the contact force chains.

If we consider a single grain submitted to uniaxial compression by considering Hertz's assumptions [START_REF] Johnson | Contact Mechanics[END_REF], the analytical resolution permits to verify the existence of tensile stresses localised at the grain centre and perpendicular to the loading direction [START_REF] Kienzler | [END_REF]. Due to the existence of these traction forces, the first cracks start at the centre of the grain leading to the propagation of a transversal fracture following the direction of compression force. In the case of one grain subjected to many contact forces randomly spread, the loading on the grain can be considered as a biaxial compression following the principal direction of stress field as shown in Figure 3. The tensile stress inside the grain is then computed by taking into account the contribution of compression forces max r and min r .

Mechanical formulation

In 2D dimensions, the stress tensor inside the grain (6), resulting from the contact forces (Figure 3(a)), is given by the following relationship: [START_REF] Lobo-Guerrero | Influence of pile shape and pile interaction on the crushable behaviour of granular material around driven piles: DEM analyses[END_REF] with  r the contact forces on the grain,  the number of contact forces, p a the grain radius, p V the unitary volume of grain and  c n the unit normal vector.

       r n V a c p p p
The stress tensor inside the grain can be decomposed in two parts [START_REF] Fortin | Numerical simulation of granular materials by an improved discrete element method[END_REF]. The first one is represented by the hydrostatique stress tensor, while the second one is expressed by the deviatoric stress tensor:

1 2                               (7)
where max  and min  are respectively the major and minor principal stresses and  the principal direction with respect to the horizontal axe. Regarding the state of stress at the centre of grain, the analytic solution [START_REF] Johnson | Contact Mechanics[END_REF] leads to the following expressions (10): For the case of biaxial compression (Figure 3), the state of stress at the grain centre is obtained by considering the contribution of the both compression forces ( max r and min r ). The following relationships [START_REF] Fortin | Selecting contact particles in dynamics granular mechanics systems[END_REF]

The principal stresses

                      
                (11)
The tensile stress c tensile  is responsible of the beginning of cracks at the grain centre and its propagation following the major principal direction (direction of max r force). Therefore, when c tensile  overcomes a given critical stress, the fracture of grain can be started. Many experimental studies have leaded to several expressions of critical stress depending on the stress intensity factor, the geometric and problem dimensions [START_REF] Tsoungui | Numerical model of crushing of grains inside two-dimensional branular materials[END_REF][START_REF] Larson | The effect of size and contact geometry in single particle crushing[END_REF].

The force leading to the fracture of one grain, fracture r

, is given by the following equation ( 12 ).

By using the relation [START_REF] Cambou | Numerical modelling of the mechanical behavior of non spherical crushable particles[END_REF] expressing the tensile stress, the critical stress from which the fracture of grain is reached can be written in the following form [START_REF] Fortin | Construction of an average stress tensor for a granular medium[END_REF]:

   I p p fracture fracture K a 2 a r    (13)
Hence, when the tensile stress c tensile  at the centre of grain overcomes the critical stress fracture  , we have fracture of grain induced by traction.

However, we haven't taken into account in the proposed approach other phenomena which can occur during the grinding process. Indeed, some effects like attrition or heat dissipation (by friction and shock) have certainly an influence on the granular material behaviour during the grinding process.

Numerical model of grain fragmentation

Some numerical models for the grain fragmentation have been carried out in many studies [START_REF] Aström | Fragmentation of grains in a two-dimensional packing[END_REF][START_REF] Tsoungui | Numerical model of crushing of grains inside two-dimensional branular materials[END_REF][START_REF] Lobo-Guerrero | Influence of pile shape and pile interaction on the crushable behaviour of granular material around driven piles: DEM analyses[END_REF]. However, the principal deficiency of these models is the nonconservation of the grain mass at each fragmentation. Some solutions can be performed in order to reduce the percentage of loss mass. We can insert into the granular packing grains with small radius after each fragmentation. But, this solution isn't easy to perform due to the difficulty to satisfy the contact conditions between grains. Another way which can be used to make sure of the mass conservation is to consider grain density according to the time.

In our study, we have considered the fragmentation model as shown in Figure 4b [START_REF] Tsoungui | Numerical model of crushing of grains inside two-dimensional branular materials[END_REF], with no mass conservation. The proposed model is performed with a loss of mass from approximately 13%. Our works in progress will lead to reduce this percentage. The numerical model from which the fragmentation is taken into account is composed of 12 grains resulting from the subdivision of the initial grain. The grain fragmentation, following the principal directions (Figure 4(a)), will be performed as follows: the initial grain is divided in 4 gains with radius equal to a p /2 and a p /3, and 8 gains with radius equal to a p /6 and a p /11 (Figure 4b). The fragmentation grain is stopped for a critical size of grain. The relationship [START_REF] Guessasma | Numerical modelling of mechanical tests using discrete element method[END_REF] gives the equation from which the critical grain radius crit a is computed:

2 elastic 2 I crit K a    ( 14
)
where I K is the mode I stress intensity factor and elastic  the elastic strength of material.

Numerical simulations

The numerical simulations have been carried out with MULTICOR software in order to perform calculations with the proposed model. We have considered a granular packing under compressive loading (Figure 5). The granular bed contains 371 grains with radius equal to 4mm. The granular material is perfectly rigid with: density equal to 1400kg/m 3 , friction coefficient equal to 0.2 and tensile strength about 150MPa. The boundary conditions are the following: top wall velocity is imposed; bottom wall is fixe; periodic conditions are imposed on the lateral sides (Figure 5). At each time step, we compute for each grain the traction stress at the centre. When the condition ( 13) is satisfied, the grain is substituted by the numerical model (Figure 4(b)). Figure 6 shows different configurations of the granular packing during fragmentation process and the corresponding force chain distributions.

After an elapsed time of 2.5 10 -3 s, two shear bands appear into the granular packing inducing the fragmentation of grains across these bands. This phenomenon is often observed in the granular material subject to a large plastic deformation. At the next step (time=1.7 10 -2 s), the granular packing becomes less rigid leading to a lowest stress field inside the grains. During the oedometric test, segregation phenomenon has been observed with respect to the grain size. The new configuration (time=2.5 10 -2 s) of the granular bed shows that the small grains are subjected to a quasi-uniaxial compression because of their few contact forces number (Figure 7). The graph given below provides information about the traction stress evolution into the granular packing during the grinding process. The stress values have been plotted with respect to the grain position (Figure 8). The maximum value reached by the traction stress at different position means that we have fragmentation of grain. From these observations, the fragmentation is easier for the small grains than for the large grains during the grinding process. Indeed, the large grains are subjected to several contact forces inducing a quasi-hydrostatic effect, which reduces the possibility to start fragmentation mechanism. In conclusion, the fragmentation process of grain depends of both parameters: the grain size and the stress state inside the grain.

Conclusion

Throughout the present paper, we have investigated the mechanisms which brought in play by contact force chains during the grinding process by compression of granular material. In fact, the loading induces by this process type leads to traction forces inside the grain, which are responsible of the grain fracture. A numerical model of fragmentation has been performed by using criterion based on the stress intensity factor. The numerical simulations of oedometric test carried out with the proposed model have shown the influence of the grain size and the stress state on the fragmentation mechanism. Our works in progress will lead to better models in order to reduce the percentage of mass loss and taking into account other mechanical effects.

Figure 1 :

 1 Figure 1: Detection of contact.

Figure 2 :

 2 Figure 2: REV of granular material.

Figure 3 :

 3 Figure 3: (a) Contact forces randomly spread -(b) Biaxial compression.

  direction  are given respectively by the following relations (8):



  are respectively the tensile stresses according to the x and y axis and xy  is the in-plane shear stress.By considering the case of one grain subjected to biaxial compression, the principal stresses the compression forces following the major and minor directions.

  tensile and compression stresses at the grain centre.

K

  is the mode I stress intensity factor (I: tensile mode),

Figure 4 :

 4 Figure 4: (a) Biaxial compression following the principal directions -(b) Numerical modelling of fragmentation.

Figure 5 :

 5 Figure 5: Oedometric test of granular packing.

Figure 6 :

 6 Figure 6: (a) Granular packing configurations -(b) Contact force chains.

Figure 7 :

 7 Figure 7: Influence of the granulometry during the grinding process.

Figure 8 :

 8 Figure 8: Traction stress with respect to the grain position.

Time=2. 5

 5 10 -2 s