Marianne Gaborieau

Robert G Gilbert

Angus Gray-Weale

Javier M Hernandez

Patrice Castignolles
email: pcastignolles@usyd.edu.au

Theory of multiple detection size exclusion chromatography of complex branched polymers

Keywords: multiple detection Size Exclusion Chromatography (SEC), viscometry, light scattering, branching, band broadening

Size exclusion chromatography (SEC) separates complex branched polymers by hydrodynamic volume, rather than by molecular weight or branching characteristics.

Equations relating the response of different types of detectors are derived including band broadening, by defining a distribution function N'(M,V h ), the number of chains with molecular weight M and hydrodynamic volume V h . While the true molecular weight distribution of complex polymers cannot be determined by SEC whatever detector is used, the formalism enables multiple detection SEC data to be processed to both analyze the polymer sample and reveal mechanistic information about the polymer synthesis. The formalism also

Introduction

Size Exclusion Chromatography (SEC) is a major technique for analyzing synthetic and natural polymers. [START_REF] Mori | Size Exclusion Chromatography[END_REF][START_REF] Balke | Characterization of complex polymers by SEC and HPLC[END_REF] Benoit [START_REF] Grubisic | [END_REF]4] and others [5][6][7] have provided evidence that SEC separation occurs by the product of [η] × M, where [η] is the intrinsic viscosity and M the molecular weight (MW), and not according to MW only. Zimm and others demonstrated that the presence of long-chain branches significantly changes the hydrodynamic volume at a given MW. [8,9] While for linear polymers there exists a one-to-one relation between MW and the elution time, neglecting band broadening (the conventional calibration curve), this is not the case for complex branched polymers: a sample wherein all chains have the same hydrodynamic volume may contain a distribution of molecular weights. [5,[10][11][12] Complex branched polymers not only have a distribution of MWs but also a distribution of chain branch lengths and topologies. They are different from regular branched polymers, like regular stars or combs. Complex branched polymers are present in several important families of polymers: (i) dendritic polymers, both biopolymers (e.g. amylopectin [START_REF] Daniel | Starch[END_REF][START_REF] Galinsky | [END_REF] ) and synthetic polymers [15] (ii) statistically (or randomly) branched polymers [9] such as polyethylene, poly(vinyl acetate) and polyacrylates (some biopolymers like amylose having a similar structure containing a few long-chain branches [16] ).

Many of the fundamentals needed for processing SEC data for complex branched polymers have been given by Hamielec and co-workers. [5,[10][11][12] The objectives of the present paper are to extend the work of Hamielec and others as follows:

(a) to define distribution functions which provide the basis for interpretation of SEC and similar data for complex branched polymers; these distribution functions are a generalization of the familiar number and mass distributions for the MW of linear polymers;

(b) to show how SEC data for both linear and branched polymers using a range of different types of online detectors (refractometer, viscometry, light scattering etc.) can be expressed in a single formalism which incorporates current and potential future modes of detection;

(c) to show how this formalism enables band broadening to be taken into account for distributions from any mode of detection, thus enabling new knowledge to be obtained from the detailed shape of these distributions [17][18][19][20][21] (rather than just averages such as w M and n M , which are usually not strongly affected by band broadening in contemporary SEC devices [22] ), and (d) to present details of how true average molecular weights, n M and w M , can be obtained. For complex branched polymers, the true molecular weight distribution (MWD) cannot be obtained; thus the averages obtained from these molecular weight distributions are not true values, while they are however calculated this way by many (including some commercial SEC software). True hydrodynamic volume distributions will be defined and can be obtained using the methods given here, leading in turn to the calculation of the true values of n M and w M .

While the details given here are for data from SEC, the expressions derived here are equally applicable to data for linear and branched polymers using other separation techniques, particularly other chromatographic techniques, for example ion chromatography, [START_REF] Haddad | Ion Chromatography -principles and applications[END_REF][START_REF] Koch | [END_REF] field flow fractionation [25,26] and capillary gel electrophoresis. [27] For complex branched polymers assuming no band broadening, universal calibration is valid using the product [η] w × n M , [5,[10][11][12]28] (as proved also below using the new formalism), where

[η] w is the weight-average intrinsic viscosity and n M the number-average molecular weight of the distribution of chains eluting at one elution time t el (or equivalently elution volume); this V h -dependent n M will be denoted n M (V h ) in the case of the separation of complex branched polymers assuming no band broadening. Similarly, light-scattering detectors yield w M (V h ) [5,11] . Other types of detectors will yield different types of average response (e.g.

NMR [START_REF] Hatada | Online SEC/NMR analysis of polymers[END_REF] and online osmometry [START_REF] Lehmann | [END_REF] which give n M (V h )). As well known but often unappreciated, the true molecular weight distribution cannot be obtained for complex branched polymers using SEC, even using multiple detection. [START_REF] Balke | Characterization of complex polymers by SEC and HPLC[END_REF]5] An important question to ask in the characterization of complex branched polymers is just what sort of information is desired and for what purpose. (a) Frequently, useful information comes from the molecular weight averages, such as n M and w M . [START_REF] Ferry | Molecular theory for undiluted amorphous polymers and concentrated solutions; networks and entanglements[END_REF][START_REF] Fukuda | [END_REF] However, there is also useful information in detailed distribution functions, as exemplified for linear polymers in the elucidation of the mechanisms of conventional [17,20] and controlled [21] free radical polymerization and starch biosynthesis [18] . More detailed distribution functions for complex branched polymers also have the potential to yield information on the synthetic processes involved in the formation of the target polymer. (b) Another use for appropriate SEC data is simply characterization to provide a means of differentiating polymer samples. For complex branched polymers, hydrodynamic volume distributions provide an interesting tool for comparative studies. (c) A third use is to utilize such data as a basis for structure-property relations.

For linear polymers, SEC data can be expressed entirely in terms of the number molecular weight distribution N(M) (or equivalently the mass distribution W(M) or the SEC distribution w(logM) [17,33] ). N(M) (sometimes also noted P(M) [17] ) is the number distribution of chains with molar mass M; the normalization of this and all other distributions is arbitrary. The other distributions are defined by:

W(M) = M N(M) (1) w(logM) = M 2 N(M) (2)
Two of the issues considered here are (1) how the true n M and w M can be obtained from appropriate SEC data for complex branched polymers, and ( 2) what is the appropriate form of the distribution function corresponding to N(M) for complex branched polymers. With regard to the former, we generalize the SEC expressions presented by Hamielec et al. [5] for obtaining n M and w M for complex branched polymers.

We now develop the new formalism, first for the case of linear polymers and then for branched ones. Account is taken of the variety of different detectors in current and potential use: those which are sensitive to the number of polymer chains eluting at a given elution time (such as UV or fluorescent detection where each chain bears a single fluorophore), to their mass (refractive index detector, UV when sensitive to the monomer unit), to a product of mass and a size-related parameter (viscometric detection, online osmometry [START_REF] Lehmann | [END_REF] , light scattering), to their molar mass (NMR for oligomers [START_REF] Hatada | Online SEC/NMR analysis of polymers[END_REF] ) and to their composition or tacticity (NMR [START_REF] Hatada | Online SEC/NMR analysis of polymers[END_REF]34] ).

The branching level could also be determined by SEC-NMR, especially in the case of hyperbranched polymers, which exhibit a high branching level. Using mass spectrometry as a detector, the equations would need to be refined, since mass spectrometry not only detects but also separates the sample coming through the detector. A complete separation may thus be used for some complex branched polymers coupling to MALDI TOF MS or Electrospray MS to SEC. However, MS techniques are biased toward high molecular weight, which thus may lead to biased results even when coupled with SEC. The present development of the formalism is confined to homopolymers; extension to copolymers may require to consider variations in the refractive index increment dn/dc [START_REF] Balke | Characterization of complex polymers by SEC and HPLC[END_REF] . Using multiple detection SEC, it is important to correct for inter-detector delay; this problem is not treated in the present work since it is already detailed in the literature [35][36][37] .

All the equations are derived taking band broadening into account, and then simplified for the case when band broadening is negligible or corrected. We thus treat four different cases depicted in Figure 1: a) linear chains (or regular branched chains) with band broadening, b) linear chains without band broadening, c) complex branched chains with band broadening, d) complex branched chains without band broadening.

The reason that it can be necessary to include the possibility of band broadening is the following. While band broadening is often considered as minimal in many modern SEC set-ups, [22] it has been found to significantly broaden molecular weight distribution in several cases. [38,39] It has been shown [40] that this broadening is sufficient to affect the fine details of the shape of the molecular weight distribution (or hydrodynamic volume distribution), and this detailed shape can be used to provide mechanistic information. [17][18][19][20][21] Moreover, band broadening is quite significant as soon as low flowrates have to be used (which is the case for high molecular weights or viscous eluents, see e.g. [41] ).

New Formalism for the Response of SEC Detectors with Linear Polymers

While all the results in this section are well known, the new formalism necessary for proper interpretation of multiple detection SEC of complex branched polymers can be better understood if the distribution/observable relations for linear polymers are incorporated into this new formalism which takes band broadening into account.

The SEC signal S det (t el ) obtained at elution time t el using a detector of type 'det' for a linear polymer is given by the Tung convolution equation: [42][43][44] S det (t el )

= A det ∫ ∞ 0 G(t el ,M) f det (M) N(M) dM (3)
where f det (M) is the 'detector function' defined below for chains with MW M, G(t el ,M) is the band-broadening function (which is independent of the nature of the polymer [45] ) and A det is a calibration constant for the particular instrument, eluent and polymer. Detailed derivations including all the relevant constants (refractive index increment, calibration constant for each detector etc.) are developed in the appendix. All these constants must be included in any treatment of SEC data but they are ignored in the main text for sake of clarity. For the special case of no band broadening, one has:

G(t el ,M) (no band broadening) = δ( el t (M) -t el ) (4)
where the Dirac function δ is defined by:

g(a) = ∫ +∞ ∞ - δ(x -a) g(x) dx ( 5
)
where g is any mathematically well-behaved function and the function el t (M) is the 'calibration curve' relating t el and M.

In a real system with band broadening, G(t el ,M) is the distribution of elution times of a perfectly monodisperse distribution in M.

The function f det (M) is different for each detector. For a refractometer (which is masssensitive) one has:

f DRI (M) = M (6)
For a number-sensitive detector (either UV or fluorescence with a single fluorophore per chain), one has:

f NS (M) = 1 (7)
For molecular-weight sensitive detectors (light scattering, viscometer etc.), one has to consider that, at one particular elution time t el , a mixture of different molecular weights elute through the detector and this detector can thus only yield a molecular weight average [case a)

of Figure 1]. The function G(t el ,M) is however likely to be the same as the one used for massand number-sensitive detectors, since the band broadening in the detector cells is commonly negligible compared to band broadening before the detectors (axial dispersion in the columns etc.) [46] Band broadening can also be important after the viscometer, which is therefore always located at the end of the detectors line to overcome this problem.

Expression for the signal of the viscometer

While the software provided with a commercial online viscometer often processes the data to give an intrinsic viscosity and a molecular weight, this data processing uses the signal from two detectors: the viscometer and a concentration detector (e.g., refractometer). We here consider these signals separately, which is necessary to obtain the general formulae which, for example, can be used to correct band broadening, including for the case of complex branched

polymers. An online viscometric detector measures a pressure drop between a line containing the polymer solution and one containing the pure eluent, [47] and this ratio is proportional to the specific viscosity η sp = (η/η 0 ) -1, where η is the viscosity of the solution and η 0 that of the pure eluent. In a sample in an increment of elution time from the SEC separation, one has a mixture of species with different MWs. From the Einstein impenetrable sphere result, [START_REF] Kulicke | Chapter 4: The intrinsic viscosity[END_REF] if a component i with mass concentration (mass per unit volume) c i in a solution has a specific viscosity i sp η (that is the specific viscosity of a solution containing only this component at this mass fraction), then:

η sp = ∑ i i sp η (8)
Thus at a particular elution time, one has:

S visc (t el ) = ( ) ∫ ∞ o M M d sp η (9)
where η sp (M) is the specific viscosity of a sample of chains with molar mass M.

The intrinsic viscosity [η] of a mixture is given by η sp /Σc i (in the limit of low concentration). Furthermore, the mass in an increment G(t el ,M) dM is G(t el ,M) M N(M) (assuming for the moment that N(M) is normalized). Thus the intrinsic viscosity can be expressed as: (10) where the volume as other constants is omitted.

η sp (Μ) = [η(Μ)] G(tel,M) M N(M)
The signal of the viscometer can then be expressed as a function of [η(M)], the intrinsic viscosity of a chain of molecular weight M (treated as a continuous rather than a discrete variable, so that an integral replaces the summation):

S visc (t el ) = ∫ ∞ 0 G(t el ,M) [η(M)] M N(M) dM ( 11
)
For the viscometer, the function f visc (M) is thus:

f visc (M) = [η(M)] M ( 12
)
f visc (M) is proportional to the hydrodynamic volume V h (Equation ( 14)) . G(t el ,M) can be determined using the concentration-sensitive detector and deconvolution methods.

Baumgarten et al. have given a comprehensive and critical review of the different methods

used to correct band broadening. [46] Castro et al. proposed recently a new method using capillary electrophoresis to obtain the true (i.e. unbroadened) molecular weight distribution; [41] in the latter method, G can be any sufficiently flexible function, e.g. an exponentially modified gaussian. Equation (11) gives the means to correct band broadening on the online viscometer signal separately.

On combining the viscometer signal with the refractometer signal without correcting band broadening, an average intrinsic viscosity is determined:

( ) ( ) el DRI el visc t S t S =[η(t el )] = ( ) [ ] ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 el 0 el d , d , M M N M M t G M M N M M t G M η (13)
This average intrinsic viscosity, [η(t el )], of the mixture of chains eluting at t el , is thus proved here to be the mass-weighted average of the intrinsic viscosities of the components eluting at t el , as also stated by Balke and Hamielec et al. [START_REF] Balke | Characterization of complex polymers by SEC and HPLC[END_REF]5] The viscometer can also yield a molecular weight, using a universal calibration curve. The universal calibration curve is established by injecting standards of known molecular weight and intrinsic viscosity (or known Mark-Houwink-Sakurada parameters in the eluent) to relate the product [η] × M to the elution time via a polynomial function. If band broadening is not corrected, the signal of the detector at t el corresponds to the response to a mixture of chains with different hydrodynamic volumes (Figure 1 a). It is thus necessary to introduce an average hydrodynamic volume of the chains eluting at t el. Flory gives the following expression for the hydrodynamic volume of a chain of intrinsic viscosity [η(Μ)] and molecular weight M: [START_REF] Flory | Configurational and frictional properties of the polymer molecule in dilute solution[END_REF] V

h = γ M [η(Μ)], γ = Φ 2 3 6 1 3 4 π ( 14
)
where Φ is the Flory constant. Φ is indeed known to vary with polymer architecture. [28] The number-average hydrodynamic volume of the chains eluting at t el is defined as:

( ) el n h, t V = ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 el 0 h el d , d , M M N M t G M M N V M t G ( 15
)
Equation ( 11), ( 14) and ( 15) can be used to express the viscometer signal as follows:

S visc (t el ) = ( ) ( ) ( ) ∫ ∞ 0 el el n h, d , M M N M t G t V ( 16
)
If one uses a refractometer in conjunction with a viscometer, then combining Equation (16) with Equation ( 3) and ( 6) leads to:

( ) ( ) el DRI el visc t S t S = ( ) el n h, t V ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 el 0 el d , d , M M N M M t G M M N M t G (17)
Combining the signal of a viscometer, a refractometer and using an universal calibration curve, one thus determines at each elution time a number-average molecular weight defined as:

n

M (t el ) = ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 el 0 el d , d , M M N M t G M M N M M t G ( 18
)
through the following relation:

( ) ( ) el DRI el visc t S t S = ( ) ( ) el n el n h, t M t V ( 19
)
If band broadening is corrected or negligible (Figure 1 b), then the ratio of the signals of the viscometer and the refractometer gives simply:

( ) ( ) el * DRI el * visc t S t S = M V h ( 20
)
where S * det (t el ) corresponds to the signal of a detector corrected for band broadening. The true molecular weight distributions of linear (or regular branched) homopolymers can thus be obtained using universal calibration with an online viscometer and a refractometer if the band broadening is corrected or negligible.

Note that n M (t el ) can also be obtained from the ratio of the signal of a mass-sensitive detector (Equation ( 3) and ( 6)) to that of a number-sensitive detector (Equation ( 3) and ( 7)):

( ) ( ) el NS el DRI t S t S = n M (t el ) (21)
or

( ) ( ) el * NS el * DRI t S t S = M (22)
The latter equation has already been used to determine molecular weight, e.g. ref. [START_REF] Takeda | [END_REF] .

Expression for the signal of light-scattering detectors

For a light-scattering (LS) based detector, using the Rayleigh relation [START_REF] Flory | Determination of molecular weights[END_REF] and assuming the injected concentration to be low enough, the function f LS (M) can be expressed as:

f LS (M) = M 2 P(θ) (23)
where P(θ) is the form factor. The signal of the light scattering is:

S LS (t el ) = ∫ ∞ 0 G(t el ,M) M 2 P(θ) N(M) dM (24)
Equation ( 24) can be used to correct band broadening on the light-scattering signal.

The light scattering can yield a molecular weight if one determines the form factor P(θ). If band broadening is significant but not corrected (Figure 1 a), then one takes a z-average form factor into account:

( ) z θ P = ( ) ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 2 el 0 2 el d , d , M M N M M t G M M N M P M t G θ (25)
If the band broadening is not corrected, the signal of the light scattering can be expressed by combining Equation ( 24) and ( 25):

S LS (t el ) = ( ) z θ P ∫ ∞ 0 G(t el ,M) M 2 N(M) dM (26)
Combining the signals of the light scattering and the refractometer, one obtains:

( ) ( ) el DRI el LS t S t S = ( ) z θ P ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 el 0 2 el d , d , M M N M M t G M M N M M t G ( 27
)
The weight-average molecular weight determined at each elution time is thus defined as:

w

M (t el ) = ( ) ( ) ( ) ( ) ∫ ∫ ∞ ∞ 0 el 0 2 el d , d , M M N M M t G M M N M M t G (28) ( ) ( ) el DRI el LS t S t S = ( ) w z M P θ (t el ) (29)
If band broadening is corrected or negligible, the ratio of the signal of the light scattering and the refractometer is then no longer a function of average form factors and molecular weights:

( ) ( ) el * DRI el * LS t S t S = P(θ) M ( 30
)
To deduce w M (t el )from the light-scattering signal which is the same as M in the case of negligible or corrected band broadening, one needs to know ( ) z θ P , which is P(θ) without band broadening. Three different methods are used. The use of multiple-angle laser light scattering (MALLS) makes it possible to extrapolate to θ = 0 ° where the form factor is unity. [START_REF] Wyatt | [END_REF] Low-angle laser light scattering (LALLS) uses an angle small enough to consider that the form factor is equal to 1 (there is however a critical molecular weight above which P(θ) decreases to values significantly below 1). [53] Triple detection uses the 90° angle, RALLS, and calculates the form factor using relations given by Flory [START_REF] Flory | Configurational and frictional properties of the polymer molecule in dilute solution[END_REF][START_REF] Flory | Determination of molecular weights[END_REF]54] :

P(θ) = ( ) 2 1 2
x x e x --- (31) with:

x(θ) = 3 8 R g 2 2 sin π ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ θ λ n ( 32
)
where n is the refractive index of the eluent, λ is the wavelength of the laser and the radius of gyration R g is obtained from:

R g = 2 1 6 1 [ ] 3 1 Φ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ η M ( 33
)
In the case of triple detection (θ = 90°):

x(90°) = 3 4 R g 2 π ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ λ n ( 34
)
In the case of triple detection, assuming the form factor P(θ) is equal to 1, an approximate molecular weight is calculated using RALLS (Equation ( 29) and ( 30)). This approximate MW value allows to estimate the radius of gyration R g (using as well the intrinsic viscosity obtained from an online viscometer, Equation ( 33)) and then a better approximation of P(θ)

(Equation ( 31) and ( 34)). More and more accurate approximations of the form factor and the molecular weight can be calculated through iterations until the values obtained are considered constant. If one does not correct for band broadening and if one proceeds through the iterations described above, then the numerical value for R g in Equation ( 33) is calculated from the weight-average molecular weight, w M (t el ) (Equation ( 28)) and the weight-average viscosity from the viscometer signal (Equation ( 13)). If band broadening is not corrected, the iteration process will be based on ill-defined R g values and it appears thus of particular importance to correct all detector signals for band broadening before calculating any molecular weight by multiple-detection SEC techniques.

LALLS, MALLS and triple detection are the light-scattering methods most commonly used to determine molecular weight, and combine the light-scattering detector signal together with the signal of a concentration-sensitive detector, generally a refractometer. One can theoretically also deduce a molecular weight from the MALLS data alone: instead of extrapolation at θ = 0 °, one then uses the slope of some relation between the Rayleigh ratio and the angle. The zaverage radius of gyration and z-average molecular weight are then obtained. [START_REF] Wyatt | [END_REF]55]

Molecular weight distribution

Taking now the specific case of refractometer detection without band broadening (Figure 1 b)) when Equation ( 4) is applicable, substituting this result and Equation ( 6) into Equation (3), and changing variable of integration from M to t el , yields (using Equation ( 5)) the result (derivation detailed in appendix):

S DRI * (t el ) = ( ) ( ) ( ) el el ẽl d d t M t M M t M N M = (35)
By definition of w(logM) (Equation ( 2)

w (logM) = ( ) ( ) ( ) el el ẽl el DRI d dlog t M t M t M t S = (36)
The simplest case is obtained for a linear calibration curve and no band broadening:

( ) M t el ~ (linear calibration) = a + b lnM (37)
The SEC distribution is then simply the familiar result: w(logM) =S DRI (t el ).

Response of SEC Detectors for Complex Branched Polymers

General equations for detector responses

For complex branched polymers, even without band broadening, at a given elution time one may detect polymer chains having the same V h , but different M as depicted in Figure 1 d). [5,10,12] To take this into account, we define N'(M,V h ) as the number distribution of chains with molar mass M and hydrodynamic volume V h ; this distribution corresponds to N(M) for linear polymers. There is a corresponding definition for the mass distribution W'(M,V h ) = M N'(M,V h ). To take band broadening into account, the function G(t el ,M) is replaced by

G'(t el ,V h ). G'(t el ,V h )
is the distribution of elution times of a perfectly monodisperse distribution in V h . We thus assume in this case that band broadening is not significantly affected by the structure of the polymer chain. The general equation for the signal of a detector, replacing Equation (3), is thus:

S det (t el ) = ∫ ∞ 0 G'(t el ,V h ) ( ∫ ∞ 0 f' det (M,V h ) N'(M,V h ) dM) dV h (38)
Equation ( 6) and ( 7) for number-sensitive detectors (UV, fluorescence detecting end-groups etc.), refractometer and light-scattering detectors remain unchanged, except that formally one replaces f det (M) by f' det (M,V h ), because none of the signals from these various detectors have a dependence on V h .

It is important to be aware that, except where specifically stated, the inclusion of G'(t el ,V h )

enables signals for all detectors to be corrected explicitly for band broadening. Again it is emphasized that while band-broadening corrections for modern SEC devices have little effect on averages such as n M , [22] they are essential to take into account if one wants detailed mechanistic information from the actual shape of the various distribution functions. If one knows the broadening function G'(t el ,V h ), then a number of procedures have been given [46] for numerically inverting the convolution equation, Equation (38). The form of G'(t el ,V h ) for a

given SEC set-up can be obtained using ultra-narrow standards [46] or a new procedure [41] using broad standards which is widely applicable.

One can readily extend the definition of the distribution function N'(M,V h ) to take account of detectors which can measure, for example, branching fraction φ (which could be determined by NMR as a function of hydrodynamic volume), in which case one would define a distribution N'(M,φ,V h ). Similar extensions for composition can also be made (possible dn/dc variations have then also to be considered).

While complex branching prevents from having a unique relation between M and t el , a unique relation still exists between V h and t el as long as universal calibration holds (Figure 1 d)).

However, band broadening also precludes obtaining a unique relation between V h and t el (Figure 1 c)).

Expression for the signal of the viscometer

The viscosity-sensitive detector is more complex. Recall that this detector measures a pressure drop ratio, which is proportional to the specific viscosity η sp . In a sample in an increment of elution time from the SEC separation, one has a mixture of species for a complex branched polymer, particularly chains with different MWs. From the Einstein impenetrable sphere result at a particular elution time, one has:

S visc (t el ) = ( ) ( ) h 0 h sp h el 0 d d , ' , ' V M V M V t G ∫ ∫ ∞ ∞ η (39)
where η' sp (M,V h ) is the specific viscosity of a sample of chains with molar mass M and hydrodynamic volume V h . The intrinsic viscosity [η] is given by η sp /Σc i (in the limit of low concentration). Furthermore, the mass in an increment G'(t el ,V h ) dM dV h is G'(t el ,V h ) M N'(M,V h ) (assuming for the moment that N'(M,V h ) is normalized). Thus the intrinsic viscosity can be expressed as:

η sp (Μ,V h ) = [η(Μ)] G(tel,V h ) M N'(M,V h ) (40)
where the volume as other constants is omitted.

S visc (t el ) = ( ) ( ) [ ] ( ) h 0 h h h el 0 d d , ' , ' , ' V M V M N M V M V t G ∫ ∫ ∞ ∞ η (41)
Equation ( 41) enables one to correct the viscometer signal for band broadening.

For the viscometer, the function f' visc (M,V h ) is:

f' visc (M,V h ) = [η'(M,V h )] M (42)
If one combines the viscometer signal with the refractometer one without correcting for band broadening, an average intrinsic viscosity is determined:

( ) ( ) el DRI el visc t S t S =[η'(t el )] = ( ) [ ] ( ) ( ) ( ) ( ) h 0 h h el 0 h 0 h h el h 0 d d , ' , ' d d , ' , ' , ' V M V M N M V t G V M V M N M V t G V M ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ η (43)
This average intrinsic viscosity, [η(t el )], of the mixture of chains eluting at t el , is the massweighted average of the intrinsic viscosities of the components. Note that the dependence from average viscosity on t el prevents from comparing [η'(t el )] measured on different SEC setups or even using different columns.

The viscometer combined with a refractometer can also yield a molecular weight, using a universal calibration curve. If band broadening is not corrected, one needs to introduce the number-average hydrodynamic volume of the chains eluting at t el :

n h, ' V (t el ) = ( ) ( ) ( ) ( ) h 0 h h el 0 h 0 h h h el 0 d d , ' , ' d d , ' , ' V M V M N V t G V M V M N V V t G ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ (44)
Equation( 38), ( 41) and ( 44) can be used to express the viscometer signal as follows:

S visc (t el ) = n h, ' V (t el ) ( ) ( ) h 0 h h el 0 d d , ' , ' V M V M N V t G ∫ ∫ ∞ ∞ (45)
If one uses a refractometer in conjunction with the viscometer, then combining Equation (45) with Equation ( 38) and ( 6) yields:

( ) ( ) el DRI el visc t S t S = n h, ' V (t el ) ( ) ( ) ( ) ( ) h 0 h h el 0 h 0 h h el 0 d d , ' , ' d d , ' , ' V M V M N M V t G V M V M N V t G ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ (46)
Combining the signal of a viscometer, a refractometer and using a universal calibration curve, one thus determines at each elution time a number-average molecular weight defined as:

n ' M (t el ) = ( ) ( ) ( ) ( ) h 0 h h el 0 h 0 h h el 0 d d , ' , ' d d , ' , ' V M V M N V t G V M V M N M V t G ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ (47)
through the following relation:

( ) ( ) el DRI el visc t S t S = ( ) ( ) el n el n h, ' ' t M t V ( 48
)
If band broadening is corrected or negligible, then all the chains detected at one elution time have the same hydrodynamic volume. This one-to-one relation between hydrodynamic volume and elution time makes it possible to express the ratio of the signal of the viscometer and the refractometer as:

( ) ( ) el * DRI el * visc t S t S = ( ) h n h V M V = [η(V h )] ( 49
)
The combination of the signals of the refractometer and the viscometer thus allows the determination of a weight-average intrinsic viscosity:

[ η(V h )] = ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) h 0 h el h el 0 h 0 h h el h el 0 d d , ' δ d d , ' , ' δ V M V M N M t V t V M V M N M V M t V t ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ - - η (50)
When band broadening is corrected or negligible, combining the signal of a viscometer, a refractometer and using an universal calibration curve, one thus determines at each elution time a number-average molecular weight defined as:

n M (V h ) = ( ) ( ) ∫ ∫ ∞ ∞ 0 h 0 h d , ' d , ' M V M N M V M N M ( 51
)
In contrast to n ' M (t el ) and [η'(t el )] (Equation ( 47) and ( 43)), the quantities n M (V h ) and [η(V h )] can be compared even if obtained from different SEC systems, as long as universal calibration is valid.

With complex branched polymers, it is possible to determine the hydrodynamic volume but not the true molecular weight distribution using SEC even after correction of band broadening. In the case where band broadening has been corrected for, to obtain meaningful results, universal calibration has to be applied using an online viscometer to obtain n M (V h ).

If some Mark-Houwink-Sakurada (MHS) parameters of the complex branched polymer are used to obtain the intrinsic viscosity, the situation is completely different. The use of MHS parameters instead of on-line viscometry will yield different values, since it refers to the intrinsic viscosity of an equivalent chain, while the online viscometer yields the average intrinsic viscosity [η(V h )]. The MHS relation is an empirical relation that relates a weightaverage intrinsic viscosity to the viscometric-average molecular weight:

[ η(Vh)]= K ( ) α h V M v ( 52
)
where K and α are the MHS parameters and v M is the viscometric-average molecular weight. It is important to note here that v M is a dimensionless quantity because of its empirical nature. By using Equation ( 52) and ( 14), the following relationship is obtained:

V h = 3 4 π ( ) ( ) Φ 6 2 3 n v h h V M V M K α (53)
Thus, it is not possible to obtain physically meaningful values of molecular weight using MHS parameters for complex branched polymers. Note that also MHS parameters vary with branching topology and distributions [28] and thus they are very likely to be quite inaccurate for complex branched polymers. [56] Furthermore, as in the case of linear polymers (Equation ( 21) and ( 22)), the ratio of the signals of a mass-sensitive and number-sensitive detectors yield the number-average MW:

( ) ( ) el NS el DRI t S t S = n ' M (t el ) (54) or ( ) ( ) el *
NS el * DRI t S t S = n M (V h ) (55)

Expression for the signal of light-scattering detectors

For a light-scattering based detector, the Rayleigh relation leads to the same expression of f' LS (M,V h ) as the function f LS (M) in Equation (23). The signal of a light scattering detector can thus be expressed as:

S LS (t el ) = ∫ ∫ ∞ ∞ 0 0 G'(t el ,M) M 2 P(θ) N'(M,V h ) dM dV h (56)
Equation ( 56) enable band broadening to be corrected on the light-scattering signal.

A light scattering detector can yield molecular weight if one determines the form factor. If band broadening is not corrected, then one has to take a z-average form factor into account:

( ) z θ P = ( ) ( ) ( ) ( ) ( )
h 0 h 2 el 0 h 0 h 2 el 0 d d , ' , ' d d , ' , ' V M V M N M M t G V M V M N M P M t G ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ θ (57)
If the band broadening is not corrected, the signal of the light scattering can be expressed as:

S LS (t el ) = ( ) z θ P ∫ ∫ ∞ ∞ 0 0 G'(t el ,M) M 2 N'(M,V h ) dM dV h (58)
Combining the signal of a light scattering detector with a refractometer, one obtains:

( ) ( ) el DRI el LS t S t S = ( ) z θ P ( ) ( ) ( ) ( ) h 0 h el 0 h 0 h 2 el 0 d d , ' , ' d d , ' , ' V M V M N M M t G V M V M N M M t G ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ (59)
One thus determines at each elution time a weight-average molecular weight defined as:

w ' M (t el ) = ( ) ( ) ( ) ( ) h 0 h el 0 h 0 h 2 el 0 d d , ' , ' d d , ' , ' V M V M N M M t G V M V M N M M t G ∫ ∫ ∫ ∫ ∞ ∞ ∞ ∞ (60)
via the equation:

( ) ( ) el DRI el LS t S t S = ( ) z θ P w ' M (t el ) (61)
If band broadening is corrected or negligible, the ratio of the signal of the light scattering and the refractometer is then a function of less complex average form factors and molecular weights:

( ) ( ) el * DRI el * LS t S t S = ( ) z θ P w M (V h ) (62)
where w

M (V h ) = ( ) ( ) ∫ ∫ ∞ ∞ 0 h 0 h 2 d , ' d , ' M V M N M M V M N M (63) and ( ) z θ P = ( ) ( ) ( ) ∫ ∫
∞ ∞ 0 h 2 0 h 2 d , ' d , ' M V M N M M V M N M P θ (64)
The expressions stated by Hamielec and co-workers [5,11] are thus proved to give the correct results for universal calibration and light scattering of complex branched polymers. The rigorous derivation given here explicitly expresses the hydrodynamic volume and form factor and makes it possible to correct for band broadening. Assuming no band broadening or correcting it, LALLS, MALLS or triple detection can then be applied to obtain w M (V h ).

There are no apparent reasons for LALLS not to be valid for complex branched polymers.

The situation is far more complex for MALLS and triple detection. First, the validity of the equations allowing one to calculate or extrapolate P(θ) can be questioned in the case of complex branched polymers. For example, the form factor of 6-arm stars or comb polymers has a significantly different dependence on R g compared to linear chains. [57] The scattering of light from polymers in solution is essentially the same physical process for complex branched polymers as for linear ones. The scattered intensity is related to the polymers' form factor, in effect its average density profile. If we have many different chains present in a sample, we will obtain the average form factor, or equivalently the average R g 2 if we expand the form factor about small angles. Equation ( 31), ( 32) and ( 34) are valid for linear and branched polymers. However, the validity of the expression of R g in Equation ( 33) is questionable for complex branched polymers.

Second, the expression used for P(θ) in triple detection (see Equation (31) to (33) in the case of linear chains) uses the molecular weight, M, and the intrinsic viscosity [η] to calculate hydrodynamic volume. The product [η] × M has dimensions of volume, and though strictly this volume is related to the hydrodynamic radius, it is reasonable for a linear polymer to replace it with the radius of gyration; both almost certainly vary in the same way with degree of polymerization. In the case of the separation of complex branched polymers, the input is w M (V h ) and [η(V h )] if band broadening is corrected or negligible. However, as seen previously when dealing with a viscometer signal, the hydrodynamic radius is only obtained

through the product n M (V h ) × [η(V h )]. Using w M (V h )
, the hydrodynamic radius is overestimated and thus P(θ) is underestimated (P(θ) is a monotonic decreasing function of x).

more work is needed to establish the relations between branching structure and different measures of the size of a hyperbranched polymer, it seems likely that the combination of light scattering and viscometry will prove an informative probe of the structure of hyperbranched polymers such as amylopectin.

Hydrodynamic volume distributions

In the following, it is considered that band broadening is negligible or has been corrected for.

True molecular weight distributions cannot be obtained for complex branched polymers, even using multiple detection SEC. Raw chromatograms prevent any comparison, since they depend on a type of column at a given time. In order to obtain comparable data, one can use some arbitrary distributions, e.g. distribution of linear equivalent molecular weight. [59] As discussed in the literature [START_REF] Huber | Molecular Characteristics of Glucans: High-amylose cornstarch[END_REF] but presently rarely implemented, from SEC one can also obtain true distributions expressed explicitly in terms of hydrodynamic volume. Using a refractometer, one can define the distribution w(logV h ) that corresponds to w(logM) in Equation ( 2) and (36) as:

w (logV h ) = V h W(V h ) = ( ) ( ) ( ) el h el h el h el * DRI d dlog t V t V t V t S = (65)
The derivation is given in the appendix. The number distribution N(V h ) can be readily shown to be obtained from

N (V h ) = ( ) ( ) h n h V M V W ( 66
)
Thus the number distribution N(V h ) can be obtained with a mass-sensitive detector (refractometer), universal calibration and an online viscometer.

The value of V h corresponding to a particular SEC elution time can be obtained through Equation ( 49) establishing the universal calibration curve from monodisperse standards and online viscometry. Instead of a viscometer, one can also use the Mark-Houwink-Sakurada parameters K and α for the standards (which then have to be linear or regularly branched), the hydrodynamic volume in one's particular SEC set-up at the elution time at which this standard elutes can then be found from:

V h = 3 4 π Φ 6 2 3 1 + α M K ( 67
)
It is preferable to determine V h directly using online viscometry without relying on the Mark-Houwink-Sakurada empirical relation.

Instead of using online viscometry, the number distribution N(V h ) can also be obtained from a number-sensitive detector and universal calibration. Following the same type of calculations as the one previously described, the signal of this detector can be expressed as:

S NS * (t el ) = ( ) ( ) ( ) el h el h h el h h dlog d t V t V V t V N V = (68) Note that while M N(M) = W(M), V h N(V h ) does not correspond to W(V h ); as shown in Equation (66), W(V h ) = n M (V h ) N(V h ). n

M and w M for the whole molecular weight distribution of complex branched polymers

The number hydrodynamic volume distribution for a complex branched polymer is given by:

N(V h ) = ∫ ∞ 0 N'(M,V h ) dM (69)
with a similar definition for the mass distribution:

which can be re-written as

w M = ( ) ( ) ( ) h 0 h h 0 h h w d d V V W V V W V M ∫ ∫ ∞ ∞ (74)
which also corresponds to the expression quoted by Hamielec. [5] The true Statistically branched polyacrylates have been analyzed by multiple detection SEC [START_REF] Castignolles | [END_REF][START_REF] Gaborieau | [END_REF] and all the data treatment presented here have been successfully tested and will be presented in a forthcoming publication. [START_REF] Gaborieau | [END_REF] The importance of band broadening and complex branching in this specific case will be discussed.

Conclusions

As is well known, but not universally appreciated, data from size separation detectors such as SEC yield distributions in terms of hydrodynamic volume, rather than in terms of molecular weight. A new formalism for expressing observables obtained by SEC or other size separation technique of complex branched polymers such as polyacrylates, polyethylene and starch has been developed. The new formalism is based on a multi-variable distribution function N'(M,V h ) giving the number distribution of polymers with molar mass M and hydrodynamic volume V h (the formalism using this distribution function could be extended to additional independent variables such as composition in a copolymer or degree of branching). We also make use of the corresponding mass distribution W'(M,V h ) = M N'(M,V h ). The generalized convolution equation is derived relating this distribution to the various observables using multiple detection SEC, such as online light scattering, viscometry, number-sensitive detection and mass-sensitive detection (refractometer etc.). These detectors yield signals as a function of elution time which are generically written as S det (t el ), encapsulated in Equation (38). This formalism shows how data from the various detectors can be corrected for band broadening, which is important for new techniques [17][18][19][20][21] which enable mechanistic information to be deduced from the detailed form of appropriate distributions. The formalism shows (e.g. in Equation ( 65)) how the distributions in terms of hydrodynamic volume can be obtained from appropriate data. The formalism also shows how data from appropriate detectors can be treated to yield quantities of interest such as the distribution for the dependence of number-and weight-average molecular weights on hydrodynamic volume, e.g.

in Equation ( 51) and (63). The new formalism is also used to rigorously derive expressions whereby the true overall n M and w M can be determined from such data (Equation ( 72) and (74)), and these expressions show that some current commercial software for obtaining w

M

and n M from SEC data will not yield correct results for complex branched polymers.

with a distribution of compositions.

The refractometer signal is expressed as:

S DRI (t el ) = A DRI c n d d ∫ ∞ 0 G(t el ,M) M N(M) dM (75)
In the case of no band broadening, a change in variable from M to t el yields:

S * DRI (t el ) = A DRI c n d d ∫ ∞ 0 δ( el t (M)-t el ) ( ) ( ) M M t M N M d d el dt el (76)
or

S * DRI (t el ) = A DRI c n d d ∫ ∞ 0 δ( el t (M)-t el ) ( ) ( ) M M t M N M dlog d el 2 dt el (77)
Thus by definition of the Dirac function

S * DRI (t el ) = A DRI c n d d ( ) ( ) ( ) el el ẽl 2 dlog d t M t M M t M N M = (78)
The molecular weight distribution obtained with a refractometer is thus:

w (logM) =M W(M)= M 2 N(M) = ( ) ( ) ( ) el
( ) el h el h h el h h dlog d t V t V V t V W V = (89)
which can also be expressed as

w(logV h ) =V h W(V h )= V h n M (V h ) N(V h )= ( ) ( ) ( ) el h el h el h DRI el * DRI d dlog d d t V t V t V c n A t S = (90)
Similarly, the signal of the number-sensitive detector can be expressed as:

S NS * (t el ) = A NS ( ) ( ) ( ) el h el h h el h h dlog d t V t V V t V N V = (91)
The ratio of the signals of the viscometer and the refractometer can thus be expressed as:

TABLE OF CONTENTS

wMM

  of the whole distribution can thus be obtained by multiple detection SEC using universal calibration (using online viscometry or the Mark-Houwink-Sakurada relation) with a refractometer yielding W(V h ), plus a light-scattering detector yielding w M (V h ). of the whole MWD of a complex branched polymer can thus be obtained by multiple detection SEC using the hydrodynamic volume distributions. It is very important to emphasize that the data treatment used in many current commercial SEC software packages considers that the separation is complete and that at each elution time only one MW is detected and measured. The n M and w M values are thus calculated by these software packages by using some distribution of n M (V h ) and w M (V h ) rather than the actual hydrodynamic volume distributions W(V h ) and N(V h ). Some current commercial SEC software packages thus yield incorrect n M and w M values for complex branched polymers.

  to the expression of the signal of the viscometer and light scattering: molecular weight can also be determined from the ratio of the signals of a mass-sensitive detector to a number-sensitive detector: of complex branched polymers without band broadeningIn the case of complex branched polymers, the dependence of the particular detector on mass and hydrodynamic volume f' det (M,V h ) is not changed for the weight-or numbersensitive detectors.-for the viscometer:f' visc (M,V h ) = A visc [η(V h )] inj V M ,-for a light-scattering based detector at angle θ:f' LS (M,V h ) = A LS M P(θ) inj V M ,In the case of no band broadening, the refractometer signal is expressed asS DRI * (t el ) = A DRI c t (V h )-t el ) M N'(M,V h ) dM dV h(85)Using the definition of the weight hydrodynamic volume distribution (Equation (70)): h )-t el ) W(V h ) dV h (86)A change in variable from the hydrodynamic volume V h to elution time t el results in: t (V h )-t el ) V h W(V h ) the definition of the Dirac function δ it becomes:

FIGUREFigure 1 :

 1 FIGURE

  SEC of complex branched polymers (polyethylene, polyacrylates, starch etc.) cannot yield the true molecular weight distributions but can yield hydrodynamic volume distributions. The equations relating the response of different detectors are derived, by defining a number distribution N'(M,V h ) of chains which have molecular weight M and hydrodynamic volume V h . The true w M and nM can be obtained from correct processing of the hydrodynamic volume distributions.

Acknowledgements

PC thanks Professor Bernadette Charleux for stimulating discussions and the Australian Research Council for an International Linkage Fellowship. We greatly appreciate the additional support of an ARC Discovery grant and of an ARC LIEF grant. The Key Centre for Polymer Colloids was established and supported by the ARC's Research Centres program.

The authors also appreciate fruitful discussions with Dominik Konkolewicz and constructive comments from referee 1.

The numerical value of w M (V h ) obtained by triple detection may thus be overestimated (when the molecular weight is high enough to have P(θ) different from one). As shown above, in the most general case of complex branched polymers and significant band broadening, [η] × M = V h is replaced by the number-average hydrodynamic volume n h, ' V (t el ) (Equation ( 44) and ( 45)). It is essential that this number-average is over the polymers physically present in the viscometer, whose molecular weight may vary both due to band broadening and to actual variations in the polymer microstructure.

We can obtain an estimate of the average size of polymers in a sample from either light scattering (we would obtain ( )

R ) or from a viscometer (which would yield n h, ' V 1/3 ). The broader the distribution of polymer properties present, the less likely these two estimates are to agree, and the less useful is Equation (33). In cases of weak branching (polyethylene, polyacrylates, amylose etc.), one might simply allow for a Flory constant (Φ in Equation ( 33)) specific to branched polymers, and this may even be all that is necessary for statistical branching, since the variation of the Flory constant with statistical branching have been measured to be relatively minor. [28] Another new effect is present when we consider the variation of n h,

V

with degree of polymerization for complex branched polymers. For hyperbranched polymers, both experimental and theoretical evidence point to different scaling behaviors of the number-average hydrodynamic volume. [19,58] The packing of shortchain branches into the polymer may make volume grow slowly with degree of polymerization, so that the intrinsic viscosity shows a maximum. Simulations and mean-field theory also predict qualitatively different variations in n h, ' V at different sizes, as for example, the repulsions between the chains become more important as more are added. [19] Although

It is useful to determine the true average MWs for the whole MW distribution (MWD). The expression for n M for a complex branched polymer is:

which corresponds to the expression quoted by Hamielec. [5] While the true MWD, N(M) or W(M), cannot be obtained for a complex branched polymer by any size separation technique, it is possible to obtain the true n M using multiple detection SEC and universal calibration: for example, using a refractometer yielding W(V h ) and a viscometer yielding n M (V h ). Any mass-sensitive detector could theoretically replace the refractometer, also yielding W(V h ); instead of a viscometer, one could use a number-sensitive detector, yielding N(V h ) directly ( n M (V h ) is then not used in the calculation).

The definition of w M in the case of complex branched polymers is:

Appendix A. Case of linear polymers

All the derivations and equations presented in the main text are detailed here including all the constants that need to be used and known to process SEC signals.

In the case of linear or regular branched polymers, the dependence of the particular detector where A det is the calibration constant for each detector, dn/dc is the refractive index increment, and V inj is the injection volume. Note that the dn/dc is not a constant in the case of copolymers


Pierre Borgnat

Patrice Abry

Patrick Flandrin

Jean-Baptiste Rouquier

Studying Lyon's Vélo'V: A Statistical Cyclic Model

Keywords: Complex System, Community bicycle program, Vélo'v, Cyclostationarity, Auto-Regressive Process

Lyon's community bicycle program called Vélo'v is a major initiative in shared public transportation, in activity since May 2005. It is studied here at a global level, to assess the evolution with time of the number of hired bikes. Based on the entire Vélo'v data set, up to December 2007, a statistical model is proposed to describe the daily and weekly patterns in a cyclostationary manner, jointly with the non-stationary evolutions over larger time-scales larger. Combining this model with linear statistical regression, a procedure is developed for the prediction of the number of bikes hired per hour. This prediction method involves several explanation factors such as the number of subscribed users, the time in the week, the occurrence of holidays or strikes, and weather parameters (temperature, volume of rain). The conclusion is that, for most days, the observation of the number of actually hired bicyles is satisfyingly explained and predicted by the model proposed here.

Vélo'v is a program of community shared bicycles that is deployed in the cities of Lyon and Villeurbanne [1]. It is one of the leading program of this kind in France, running actively by JCDecaux from May 2005 on. It consists now of 4000 available bicycles that can be hired at any of the 334 stations, spread all over the towns, and returned back later at any station. With the increasing need of green and versatile public transportation in cities, there is a current development of such community shared equipement programs in many european cities, e.g., the Vélib' program in Paris since July 2007, or the Bicing program launched in Barcelona in March 2008 [START_REF] Froehlich | Measuring the Pulse of the City through Shared Bicycle Programs[END_REF]. The specificity of those programs, as compared to old-fashioned rental systems, is that stations can be accessed freely at any time and are fully automatized and computerized. This makes possible a global management and potentially some real-time survey of the state of the system. Currently, automatized station reports are collected into a central data set and mostly used a posteriori. Yet, there is a strong incentive to evolve toward less empirical management of the system, for instance by being able to increase or redeploy in realtime the available bicycles if need be at a specific time one day. Therefore, a challenge consists of quantitatively studying the behaviour of the system, so as to be able, in the future, to regulate and optimize it (e.g., increase bicycle availability). The present work proposes a global and statistical study of the evolution of the number of bicycles hired along time. Objectives are not only to identify its temporal patterns but, going way further than [START_REF] Froehlich | Measuring the Pulse of the City through Shared Bicycle Programs[END_REF] in the modeling, to propose a statistical model for the temporal patterns that encompass their cyclostationarity and their non-stationarity. Finally, this model is used to predict the number of bicycle rentals on a daily or hourly basis.

Anonymized data were made available to us by JCDecaux and the Grand Lyon. The data set consists of the records of all bike rental operations (starting and ending points and times), over more than two years of exploitation. Companion studies are made from this data set about the specific spatial patterns that can be exhibited in journeys [START_REF] Borgnat | Spatial analysis of dynamic movements of Vélo'v, Lyon's shared bicycle program[END_REF]. Here, the focus will be on aggregated data over the whole network, and its evolution along time, as shown in Fig. 1 (a).

In Fig. 1 (b), the number of available bikes and stations and the number of subscribed users (note that bikes can also be used without subscription, with short-term registration cards) are shown. This sketches the progressive deployment and the increase in popularity of the program.

Empirical patterns of the number of rentals

A first question is to choose a time scale ∆ to aggregate the number of new rentals. The trade-off here is usual: the smaller ∆ is, the larger the fluctuations are, whereas a larger ∆ may smooth the signal with the risk of losing some relevant temporal feature. Fig. 1 (c) displays the distribution of rental duration. This distribution is large, yet there is a mode and the median of this duration, equal to 11 minutes, is representative of it. Choosing ∆ = 1h is enough to smooth out some of the differences of individual rentals, while keeping the global evolution of their collection.

On Fig. 1 (a) is shown the number of hired bikes, aggregated on ∆, from May 2005 to Dec. 2007, superimposed with the per hour hiring numbers averaged over the day and over the week. Two features are striking. First, the mean is non-stationary and evolves with time. The first interpretation is to be related to the increase of the size and of the popularity of the program, as already seen in Fig. 1 (b). A complementary explanation is that the use of Vélo'v also depends on the season (with less users during winter, or during holidays). The second feature is a strong modulation of the hirings with the moment in the week: this cyclic evolution (more properly called cyclostationarity) comes from the evident fact that from a social point of view, days and hours are not equivalent for people. Those two features, non-stationarity and cyclostationarity, are those the model proposed below aims at accounting for.

Model for the cyclic temporal patterns

Let us first study non-stationary patterns on time scales larger than the day. Empirically, we put them in evidence by computing, from the hourly hirings L(t), the mean A d (d) over each day (d is the variable of day). Then, inspired from cyclostationary methodologies [START_REF] Gardner | Cyclostationarity: Half a century of research[END_REF], we estimate with a cyclic mean over the week, the weekly temporal pattern for L(t) (displayed on Fig. 2 (a)):

< L(t) >c= 1 Nw Nw-1 X d=0 L(t + 168 d), for t = 0 .. (24 • 7 -1)h = 0 .. 167h. (1)
Here, Nw is the number of weeks of data used. The result shows patterns also observed in the Barcelona program [START_REF] Froehlich | Measuring the Pulse of the City through Shared Bicycle Programs[END_REF]. During week-days, three peaks are seen: in the morning (8am-9am), noon (12am-1pm) and end of afternoon (6pm-7pm, this one being the biggest). During week-ends, the pattern changes, with mostly a large peak spread during the afternoon, having a maximum around 5pm (with only a small increase on top of that at noon). These features match intuitive interpretations about the fact that people use bicycle transportations mostly during the day to go to and come back from work, or during lunch break, whereas during the week-end, the major trend is to take an afternoon pleasure ride.

From the cyclic means, let us write A mod (d 7 ) = P (d7) < L(t) >c the average number of rentals per day d 7 , that depends only on the position of the day in the week (and we write d 7 to represent the day in the week). For a quantitative approach about the temporal pattern, we propose the following model:

L(t) = L mod (t) + F (t) = A d (d) < L(t) >c A mod (d 7 ) + F (t), (2)
where F (t) is a fluctuation not accounted for by the cyclic model. In Fig. 2 (b), we illustrate the model for a specific range of days, to show that it usually holds well, even when specific occasions change the flow of days, such as holidays or festivities (here we illustrate that on December, the 8th, which is a specific festivity day in Lyon). It has not yet been discussed so far how to predict of the fluctuations F and the amplitude A d . This is the objective of the next Section. model in eq. ( 2), prediction is split into two subparts: First, the prediction of the nonstationary amplitude A d (d) for a given day; Second, the prediction of the fluctuations F (t) at a specific hour. The corresponding time scales being different, it is sound to try to predict them separately.

Prediction of A d (d). Our starting proposition is that the factors explaining A d (d) are to find among the following ones: (i) the weather conditions summarized by the average temperature T (d) over one day and the volume of rain R(d) (in mm); to remove the mean from the temperature, we will use δT (d) = T (d)-< T (d) > (in o C); (ii) the development and popularity of the program: the number of subscribed users Ns(d), the number of bicycles available Nv(d); here again, we take deviations δNs(d) and δNv(d) between the real value and the value at the end of the data (XII.2007) where the system is supposed to have reached its final state; (iii) specific conditions such as holidays, with a marker J h (d) taking value 0 usually and 1 for those specific days, or strikes with marker Js(d). A linear regression model is written as:

A d (d) = α 0 (d 7 ) + α 1 δNs(d) + α 2 δNv(d) + α 3 δT (d) + α 4 R(d) + α 5 J h (d) + α 6 Js(d), (3)
where features δNs to R have been normalized to variance 1, and where the term α 0 (d 7 ) describes the mean with an additional linear dependence on the position of day d 7 during the week (from Monday to Sunday):

α 0 (d 7 ) = A 0 + c 1 `Amod (d 7 )-< A mod (d 7 ) > d7 ´. ( 4
)
This dependence is needed because, as seen on Fig. 2 (a), the expected number of hirings each day varies from Monday to Sunday; it is, for instance, smaller during the week-ends. Solving this problem of linear regression using standard least square minimization, we obtain the results reported in Table 1. Confidence intervals are reported along with the estimated values of the coefficients because, even though computed under Gaussian hypothesis, which does not hold for many factors, it assists us in the interpretation of the relevance and importance of each factor. Results call for the following comments. First, the term depending on the day α 0 (d 7 ) is simple enough: it consists of a constant A 0 whose value is close to the average number of hired bikes per day during the last months in the data set (17 500 during the last 4 months of 2007), with a linear correction (with factor close to 1) that takes into account the dependence with the day in the week. Second, some factors play an important role in controlling A d (d): a larger number of subscribers increases it; weather factors act in an expected manner: the warmer, the larger the number of bikes used (and conversely), under heavy rain, A d (d) decreases. The factor pertaining to holidays J h also impacts A d (d): there is a decrease (whose relevance is assessed by the confidence interval) during holidays-a feature that appears qualitatively in Fig. 1 (a) and is explained by the fact that people are out of town during holidays. The two remaining factors show non conclusive effects; the number of available bikes does not impact much A d (d) and this can interpreted by looking again at Fig. 1 (b): the numbers of subscribers and the number of bikes follow roughly the same evolution at the beginning, then are constant. This lack of influence hence results from the fact that a part of the evolution is already accounted for by the evolution of Ns, and by the fact that there seems to be no major depletion of bikes as confronted to subscribers. Finally, strikes are a non conclusive factor, mostly because of the scarce number of such events in the current data set.

Using this linear regression model, it has been possible to predict the amplitude of the number of bikes rented per day, depending on all the external factors proposed here. If one would use only the average number of hirings A mod (d 7 ) adjusted only for the position of the day d 7 in the week and without the other non-stationary factors, the rms of the error value between observed data A d (d) and this number, divided by the mean of this amplitude, would conduct to 30% of mean relative error. Using the model A d (d), it decreases to 12%. Clearly there is still room for improvement, yet the quantitative gain is non negligible and, more importantly, the interpretation of the dependence with the various factor shows their relevance.

Prediction of fluctuations and its anomalies. Let us know turn to the fluctuation term F (t), whose standard deviation is 210 (in bikes hired per hour; it can be compared to the mean of L(t) equal to 655 hired bikes per hour). A standard empirical spectrum analysis shows that is is well modeled by a parametric AR(1) process:

F (t) = a 1 F (t -1) + I(t), (5)
where a 1 ≃ 0.81 and I(t) is a white innovation of standard deviation equals to 120. The coefficient a 1 and the order of the model were estimated using the classical Levinson-Durbin algorithm [START_REF] Priestley | Spectral analysis and times series[END_REF]. This leads to a general prediction scheme for the number of hourly rentals that follows eq. ( 2) with A d (d) obtained from Eq. ( 3) and d F (t) = a 1 (L(t -1) -L mod (t-1)). In Fig. 2 (b), the displayed model is built using these estimates A d (d) and d F (t) and eq. ( 2). It works satisfactorily in following the observed variations of L(t) along time. Using this improved scheme including prediction of the fluctuations, the standard deviation of the error of the global prediction decreases from 210 bikes to 120 bikes per hour, i.e., the std. of the innovation I, which, by nature of the approach, cannot be predicted. The improvement, which is clear, is not larger because of many anomalies existing in the data that are not explained by the factors used here. We propose that, for instance, taking into account the rain at the time scale of the hour (rain being often an event that does not last several hours), would improve the prediction of the fluctuation from one time to the next one. This is not done here.

Conclusion and on-going work

This study is a preliminary step toward a better and quantitative understanding of the usage of the Vélo'v program in Lyon. The combination of non-stationarity and cyclostationarity that drives the number of hourly hirings was modeled here using suitable statistical signal processing methods. A first result is to confirm that the temporal pattern over the week that was already exhibited in studies on other bicyle sharing programs seems to hold in different cities (when corrected for non-stationary evolutions and short-lived anomalies). A new result is the possibility to model the number of per hour rental along time and predict its evolution based both on characteristics of the deployed program (number of bikes and or subscribers), and on external conditions such as weather. Combined together, they explain most of the remaining deviations from the model. Spatial patterns are studied with complementary techniques in a joint work [START_REF] Borgnat | Spatial analysis of dynamic movements of Vélo'v, Lyon's shared bicycle program[END_REF]. The objective is that these quantitative studies of the data set of all trips with Vélo'v could help on the one hand sociological studies on community shared transportation systems and, on the other hand, managers that have to start or run such programs.

Fig. 1

 1 Fig. 1 Global features of Vélo'v. (a) Number of bikes hired per hour, L(t), and its average per day A d and per week Aw. (b) Evolutions along time of the numbers of subscribers Ns of the program (dashed line, in blue) and of available bikes Nv (solid line, in red). (c) Rental duration distribution (in %).

4Fig. 2

 2 Fig. 2 Cyclic models and comparison to data. (a) Model < L(t) >c giving the typical expected evolution over the week. (b) Examples of L(t) for some chosen days, compared to the model L mod (t) + d F (t). Here, we show the days around December, the 8th, in Lyon (which was a Saturday in 2007), showing that the model holds qualitatively well here.

Table 1

 1 Statistical model for A d (d) as per eq. (3). For the different linear coefficients associated to the factors in play, we report the estimated value (est.) and its Confidence Interval (under Gaussian assumption), given by [CI -, CI + ].

	Factor	α 0 (d 7 )	δNs(d)	δNv(d)	δT (d)	R(d)	J h (d)	Js(d)
	coeff.	A 0	c 1	α 1	α 2	α 3	α 4	α 5	α 6
	est.	17 370	1.05	1 860	-120	2270	-1280	-2900	20
	CI -	17 050	0.90	1 210	-720	1980	-1520	-3700	-2900
	CI +	17 680	1.18	2 560	+490	2560	-1030	-2100	+2900

Acknowledgments

This work was made by possible by the kind help of JCDecaux and the Grand Lyon. This work is part of an on-going project of scientific studies about the Vélo'v program and is supported by the IXXI (Institut des Systèmes Complexes -Complex Systems Institute) of Lyon. The authors would like to thank Antoine Scherrer, Céline Robardet, Eric Fleury and Pablo Jensen for interesting discussions within this project.