Karl Joulain Ensma

PHYSICAL PROCESSES DESCRIBING HEAT TRANSFER AT SHORT LENGTHSCALE

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Physical processes describing heat transfer at short lengthscale

Introduction

Heat transfer can be defined as thermal energy in transit due to a spatial temperature difference [START_REF] Incropera | Fundamentals of heat and mass transfer[END_REF]. Usually, three transfer modes are considered. Conduction, that refers to heat transfer across a material, convection occuring between a surface or a moving fluid and radiation describing the transfer in vacuum between different temperature bodies by means of electomagnetic waves. Classically, heat conduction is described by a diffusion equation. Radiative heat transfer is governed by the so called Radiative Transfer Equation (RTE). In convection, calculations are based on the existence of a boundary layer and the heat equation resolution in the layer.

At short scale, all these theories are invalidated when distances involved become smaller than characteristic lengths governing the phenomena physics. For example, in a fluid, when typical distances involved become smaller than particle mean free path, the concept of fluid cell is not pertinent anymore. In these conditions, hydrodynamics equations are not valid : particles in the fluid are flying from one side of the system to the other side. This phenomenon cannot be named convection anymore but molecular ballistic transfer. In the second section of this article, we will show through the example of transfer between two interfaces or between one tip and an interface how heat can be transmitted ballistically through air and reach a very high equivalent heat transfer coefficient.

The third section of this article will focus on heat conduction. In a solid, it is the principal heat transfer mode. Heat carriers can be free electrons of a metal or phonons in an insulator or a semiconductor. Like in gas, heat carriers in classical conditions are submitted to a lot of collisions. This leads to Fourier's law and to heat equation [START_REF] Kim | Nanostructuring expands thermal limits[END_REF]. When carriers mean free path is of the system size order, some carriers carry heat ballistically : heat transport is not a diffusion process. It resembles more to a ballistic process similar to that is occuring in thermal radiation. Heat flow obeys to the Boltzmann equation describing evolutions of a distribution function equivalent to a phonon or electron specific intensity. We will examine different collision processes occuring for phonons in solids. We will see through Boltzmann's equation solution in different geometries such as film, wires or tube that it is possible to describe and predict thermal properties in nanostructures. We will note that concepts such as thermal conductivity have to be used very cautiously. When distances approach the electron or the phonon wavelength, quantum effects such as interference and quantification of the modes has to be taken into account [START_REF] Chen | Phonon wave heat conduction in thin films and superlattices[END_REF]). We will see that in such nanoscale systems thermal conductance is quantified. [START_REF] Schwab | Measurement of the quantum of thermal conductance[END_REF]. Moreover, alternative stack of different material (superlattices) can form phononic crystals with forbidden frequency band gaps where phonons cannot propagate [START_REF] Narayanamurti | Selective transmission of high-frequency phonons by a superlattice: The "dielectric" phonon filter[END_REF].

Radiative transfer will be treated in the fourth section. In participating medium, radiation is classically described by Radiative Transfer Equation and radiometry laws. When the system size is of the order of the thermal wavelength, wave effects appear. Thermal radiation has to be treated with fluctuational electrodynamics theory [START_REF] Rytov | Principles of Statistical Radiophysics[END_REF]). The principles of the calculation are the following : thermal motions initiates currents in materials that behave like an antenna and radiate an electromagnetic field. Interference and tunelling effects can then be dominant. Nevertheless, it can be shown that radiative heat transfer can be retrieved from the Maxwell equations and that specific intensity can be defined under certain conditions from the electromagnetic field.

At subwavelength distances, thermal emission is very different that in far field [START_REF] Polder | Theory of radiative heat transfer between closely spaced bodies[END_REF][START_REF] Joulain | Surface electromagnetic waves thermally excited : Radiative heat transfer, coherence properties and casimir forces revisited in the near field[END_REF]. For example, contrary to classical thermal radiation, thermal emission in near field i.e at subwavelength distances can be both spatially and temporally coherent, when additional modes such as surface polaritons or guided modes exist. Experiments have shown that these thermal emission near-field properties can be exploited to built thermal coherent sources by microstructuring surfaces supporting surface waves. [START_REF] Greffet | Coherent emission of light by thermal sources[END_REF]. Thermal emission in near field can also be a tool for new type of imagery or spectroscopy. Near field optical microscopy experiments [START_REF] Dewilde | Thermal radiation scanning tunnelling microscopy[END_REF] have shown that thermal emission can map the electromagnetic density of states near a surface or image subwavelength objects.

Heat transfer can also be reformulated : two bodies at different temperatures separated by a vacuum distance d still exchange heat. But, in fluctuational electrodynamics, heat flux given by the Poynting vector contains different contributions, electric, magnetic, propagative or evanescent. In accordance with the type of material involved, these different contributions can be dominant or negligible so that the transfer can occur for some specific electromagnetic polarization or frequency.

Heat transfer in a fluid

Classically, heat transfer in a fluid is based on Navier-Stokes and heat equation resolution. The assumption that the Knudsen number is small, that is the fluid is in the collision regime, is required to derive these equations. When the fluid is rarefied, the system obeys a more general equation called the Boltzmann equation [START_REF] Reif | Statistical and thermal physics[END_REF]:

∂f ∂t + v.∇ r f + F.∇ v f = ∂f ∂t coll (1)
where f (r, v, t)d 3 rd 3 v is the average number of particles at a time t in an elementary volume

d 3 rd 3 v centered around (r, v). ∂f ∂t coll
is the collision contribution on the evolution of f . It is an integral over all the possible binary collisions that modifies f . Resolution of (1) is in general a difficult task. The Chapman Enskog development is the classical procedure to solve it [START_REF] Balian | De la mécanique statistique hors équilibre aux équations de transport[END_REF]. At the first two orders of approximation, this development recovers Navier-Stokes and heat equation. At higher orders, other equations over macroscopic quantities can be found : the so-called Burnett and super-Burnett equations [START_REF] Burnett | Proc. Math. Soc[END_REF]. Other resolution procedures have been recently proposed (Struchtrup & Torrilhon 2003). Non trivial rarefied fluid behaviour can be predicted from these procedures. For example, a rarefied viscous fluid in a channel has its velocity field discontinuous at the wall.

Ballistic Transfer

An easy limit is the ballistic regime. It occurs in a fluid when the fluid particles undergo no collision along a path of the system size. Heat flux calculation in such a system is easy. Let us consider two planar interfaces at two different temperatures T 1 and T 2 and separated by a gap of width d filled with a gas. When the gas particle mean free path is much larger than the system size, the particles fly ballistically from one side of the system to the other one. We consider that the particles touching a surface at temperature T adopt instantaneously a Maxwellian velocity distribution function. Using the fact that the heat flux

q = 1 2 mv 2 vf (v)d 3 v (2)
and that no particle flux exist between the interfaces ( vf (v)d 3 v = 0) the heat flux reads [START_REF] Carminati | Microscale and nanoscale heat transfer[END_REF])

qbal = n √ T 1 T 2 (2k B ) 3/2 √ πm( √ T 1 + √ T 2 ) (T 1 -T 2 ) (3)
Let us note that the heat flux does not depend on the distance between interfaces. A heat transfer coefficient can also written when T 1 ∼ T 2 ∼ T :

h bal = nT (2k B ) 3/2 2 √ πmT (4)
At ambient temperature and for gaseous nitrogen, h bal = 1.310 5 W m -2 K -1 . Note that the mean free path for nitrogen molecules in air is around 40 nm at ambient temperature. The diffusive regime is therefore valid until the distance is around 10 times the mean free path say 400nm. In this regime, air has roughly a thermal conductivity k of 0.024 W m -1 K -1 . For two interfaces separated by a 400 nm gap, a heat transfer coefficient h = k/d = 6 10 4 W m -2 K -1 . We therefore see that ballistic heat transfer do exist when particles mean free path is smaller than the distance involved. Such a situation occur in near field microscopy between a tip and a substrate. Consider for example the typical case of a heated tip situated at a distance d of a substrate (Fig. 1 left). This technology could be used for example in thermally assisted data storage. The principle is to heat a sample at a sufficient high temperature and at sufficiently high speed in order to write a bit on the sample (by vitreous transition or magnetic recording). The question to be solved is what is the heat flow deposited below the tip and at what scale? This problem can be addressed in the same way as the two interface situation in the ballistic regime. Nevertheless, the distance between the tip and the sample is often of the order of the molecules mean free path. In this mesoscopic regime, the effect of collision has to be taken into account. Chapuis et al. (Chapuis et al. 2006) have calculated the heat flux between a tip at 800 K and a sample at ambient temperature by means of a Monte Carlo method. Particles are emitted at the tip following a Maxwellian distribution. This particle may experienced a collision along a path of length L according to an exponential collision probability law exp(-l/L) where l is the mean free path. The results are presented in Fig. 1 right. A very high heat flux density is reached. It has been shown also that powers of the order of 0.1 nW can be deposited for a few tens of ps allowing the possibility for thermally assisted storage.

Conduction in solids

Thermal conduction in solids is also a phenomenon classically described by a diffusion process of heat carriers. An interesting feature to note about heat conduction in solids is that its magnitude does not change a lot with the materials concerned. For example, solids thermal conductivity spans only on four orders of magnitude contrary to electrical conductivity. Indeed, heat carriers can not only be electrons but also phonons in crystals. In metals, heat conduction due to electrons dominates over phonon conduction. In insulators, only phonon heat conduction is present, but it is impossible to be completely rid out of all heat carriers movement as it is the case for electrical conduction. Therefore, conduction due to the presence of acoustic waves in matter will always be present in the same way heat conduction is always present in gas due to the presence of molecules.

In a crystal, an analogy can be done with thermal radiation : instead of photons, heat is carried in solids by quasiparticles called phonons. These particles can be considered as boson in the same way photons are. At thermal equilibrium phonons are populated according to a distribution function that resembles to Planck's law. If the particles collides, the regime is diffusive and the heat flux follows the Fourier law (analogy with the Rosseland regime [START_REF] Modest | Radiative Heat Transfer[END_REF]). When the phonons are not submitted to collisions, heat conduction behaves like radiation in a transparent medium.

Diffusive regime : Fourier's law and limits

Fourier's law is the fundamental macroscopic equation from which general equations such as the heat equation are derived. q(r, t) = -k∇T (r, t)

The kinetic theory, which considers that heat carriers behave like a gas, is a way to establish the Fourier law. Let us come back to the Boltzmann equation reformulated for phonons or electrons.

In the collision regime, the collision integral is often written in the so-called relaxation time approximation. In this model, it is stated that the main collison integral contribution is to relax the distribution function to a local equilibrium distribution function. In steady state, Boltzmann's equation reduces to

v.∇ r f = f 0 -f τ (k) ( 6
)
where k is the particle wavenumber, f (r, k)d 3 rd 3 k is the number of phonons in the elementary volume d 3 rd 3 k. f 0 is thus a local equilibrium distribution function. For example, phonons in a solid are in local thermal equilibrium when they follow the following function (Bose-Einstein distribution)

f 0 (r, k) = 1 exp[hω(k)/k b T (r)] -1 (7)
τ (k) is the phonon relaxation time or collision time. Different physical phenomena are at the origin of phonon collisions. The presence of impurities, stack defaults, dislocations or isotopes scatters phonons in the same way molecules or small particles of dust scatter phonon in the atmosphere [START_REF] Klemens | [END_REF][START_REF] Holland | Phonon scattering in semiconductors from thermal conductivity studies[END_REF]. Phonons can also be scattered at the system boundaries due to their roughness. Finally, the fact that crystal potential is not purely harmonic makes the phonons not to be system eigenmodes. They can be scattered through a non-linear process where one phonon can split into two phonons or, on the reverse process two phonons interact to give a single one. In this last process, when the resulting phonon lies in the so called Brillouin zone, this is a normal process (N). When, the resulting phonon is outside the zone, this is an Umklapp process (U). Due to crystal periodicity, the final phonon is in the opposite direction of the initial phonons. This last process is at the origin of thermal resistance [START_REF] Landau | Physical kinetics : volume[END_REF].

Let us look for a solution of the form f = f 0 + ηf 0 where η ≪ 1. The Boltzmann equation ( 6) reads in presence of a temperature gradient ηf 0 = -τ (k)v.∇ r T ∂f 0 /∂T . Heat flux obtained from this Boltzmann equation solution is

q(r) = - k,p hω(k)τ k ∂f 0 ∂T (v.∇T )v = - 1 6π 2 p K M ax 0 Cv k,p v 2 g,p (k)τ p (k)dk (8)
for an isotropic medium. In this last expression, the summation is done on the polarization p and wavevector k. v g,p is the group velocity of the polarization branch p, K M ax is the maximum phonon wave vector (Brillouin zone limit), C v is the contribution to the thermal conductivity of one single mode C v = k B x 2 e x /(e x -1) 2 with x = hω/k B T . Therefore, the thermal conductvity k depends on the temperature but also of the modes group velocities and relaxation times. The higher is velocity and the lower are the collisions, the larger is the thermal conductivity.

Low thermal conductivities

Low thermal conductivities would be very useful for thermal barrier or for thermoelectric materials. In the diffusive regime, the simplest way to decrease crystal thermal conductivity is to introduce impurities that scatters phonons. This is the so called alloy limit. It can be shown that phonon scattering cross section is very analogous to Rayleigh light scattering when the phonon wavelength is much larger than the scatterer : σ ∼ a 6 /λ 4 where a is the impurity typical size. The smaller is phonon wavelength, the more it is scattered. In alloys, the impurity size is of the order of 10 -10 m. Impurities with larger sizes could scatter more efficiently phonon wavelength of the nanometer range that contribute the most to the thermal transport. This technique has been used [START_REF] Kim | Nanostructuring expands thermal limits[END_REF][START_REF] Chitirescu | Ultra-low thermal conductivity in disordered, layered crystals[END_REF] where nanometer size particles are introduced in a bulk medium in order to reach thermal conductivity approaching those of strong insulators. In the same idea, materials with a large disorder like amorphous materials exhibit a low thermal conductivity, because the heat carriers mean free path is of a few atomic spacing. For examples, glasses like fused silica have a typical thermal conductivity of 1 W m -1 K -1 .

It is possible to decrease phonon mean free path increasing phonon collisions with borders. A way is to confine phonons in nanostructure such as nanofilms or nanowires. Thermal conductivity is then mostly governed by nanometric boundary conditions. In Fig. 2, we show nanowire thermal conductvity calculation [START_REF] Terris | Numerical simulation of transient phonon heat transfer in silicon nanowires and nanofilms[END_REF]) compared with nanowire thermal conductivity measurements [START_REF] Li | [END_REF]. We see that Si nanowire thermal conductivity is highly reduced compared to bulk silicon. This is due to phonon scattering at wire boundaries. The same behaviour explains the in-plane conductivity decreasing in thin films, where the phonon mean-free path is of the order of the system width.

The other effect of phonon confinement is to modify phonon dispersion relation. The result is a reduction of phonon group velocity which has also a reduction effect on thermal conduction.

High thermal conductivities

High thermal conductivity materials are useful to design new heat sinks in electronics. Indeed, the increasing progress in nanotechnology has led to very high power dissipated in components. More efficient heat sinks are now required in order to ensure the electronic components thermal management. Following the reasoning done concerning low thermal conductivities, one can imagine two reasons for high conductivity. High phonon velocity and very long collision time. The phonon velocity, which can also be seen as a velocity of sound, is higher when molecular bound is stiff. For example, diamond is a well known thermal conductor with a thermal conductivity larger than 2300 W m -1 K -1 . Carbon nanotubes have been pointed to be very good thermal conductors from both theoretical calculations (Mingo & Broido 2005) and experiments (Kim et al. 2001[START_REF] Fujii | Measuring the thermal conductivity of a single carbon nanotube[END_REF][START_REF] Chang | Isotope effect on the thermal conductivity of boron nitride nanotubes[END_REF]. Indeed, the stiffness of carbon atoms bounds is even higher than in diamond. Morevoer, carbon nanotube cylindrical structure makes that phonons follow their way along the surface of the nanotube without seeing any border. Thus, their collision time is very long. Single walled carbon nanotube seems to have particular high thermal conductivity whereas the results is more controversial concerning Multiwalled carbon nanotubes.

Ballistic regime

Fourier's law is valid when heat transport is in the diffusive regime. When it is not the case, i.e when heat carriers have a mean free path of the order of the typical system size, the transport is mesoscopic or even ballistic when the mean free path is much lower than the system size. This last situation is particularly simple to analyse.

Let us consider as a simple illustration the following system : a solid which acoustic modes follow a single phonon branch in the Debye approximation [START_REF] Kittel | Introduction to solid state physics[END_REF]) and with plane parallel borders (orthogonal to z direction) separated by a distance d are at temperatures T 1 and T 2 small compared to the Debye temperature. In this approximation, the specific phonon intensity emitted by a wall at temperature T reads :

I 0 (ω) = hω 3 2π 3 v 2 g [exp(hω/k b T ) -1] (9)
Here the group velocity v g is constant. The heat flux emitted by a wall at temperature T 1 is therefore :

q = ∞ 0 dω Ω=2π I 0 (ω) cos θdΩ = k 4 b 2π 2 v 2 g h3 ∞ 0 x 3 dx e x -1 T 4 1 = σ ph T 4 1 ( 10
)
where θ is the angle between the direction z and the direction of the intensity considered, σ ph = k 4 b π 2 /(30h 3 v 2 g ). The total heat flux exchanged is obtained by subtracting the heat flux emitted by the wall at temperature T 2 so that ϕ = σ ph (T 4 1 -T 4 2 ). We see that the heat flux does not depend on the distance between the two interfaces. Phonons are flying from one interface to the other. Conductive heat flux behaves like thermal radiation. If T 1 ∼ T 2 ∼ T , the total heat flux can be approximated by ϕ ∼ 4σ ph T 3 (T 1 -T 2 ). Following the definition of the thermal conductivity by the Fourier law, a ballistic conductivity along the z direction can be written :

k bal z = 4σ ph dT 3 (11)
This conductivity depends on the system size. This is the opposite situation from the bulk where thermal conductance depends on the system size and thermal conductivity is an intrisic material property. At small distance, thermal conductance is the relevant quantity independent on the system size, contrary to thermal conductivity. Transition between diffusive and ballistic regime is exhibited in Fig. 3 where surface thermal conductance in silicon films is represented versus film thickness in log-scale. At distances larger than 10 µm, conductance decreases as 1/d where d is the film thickness. The film behaves like the bulk. When the thickness is of the mean free path order, the film is in the mesoscopic regime. Film conductance still increases with decreasing thickness. At extremely small distance, the conductance is ballistic. Temperature inside the film confirms this behaviour (Fig. 4). In the Fourier regime, the temperature profile is linear between extreme temperatures.

In the mesoscopic regime, there exists a temperature jump at boundaries similar to temperature jumps occuring in thermal radiation for optically thin media. In the ballistic regime, the profile is flat. One therefore sees that thermal conductivity, which is undoubtetly linked to the Fourier law can hardly be used in the last two regimes.

Wave effects

Due to Bose-Einstein statistics, phonons in a material at temperature T are distributed according to a function that resembles the Planck law. Their wavelength are peaked around a value given by an analog of Wien's law. For example at ambient temperature, phonons wavelength peak around a few nanometer. When the temperature is in a few Kelvins order, the wavelength is a fraction of a micrometer. When the system characteristic size is of the order of phonon wavelength, wave effects appear. For example, phonons can tunnel across a junction such as a quantum well. In the case of superlattices, that are periodic stacks of alternating different materials, forbidden band gaps may open in phonon dispersion relation : propagation is not possible at band gap wavelengths. This phenomenon is analog to Bragg reflection in optics. At low temperature, structures like nanowires can have their lateral size of the order of the most populated wire acoustic modes wavelength. At these wavelength, modes are quantified in the wire as optical modes are quantified in an optical fiber. Heat can only be carried at these special authorized modes. In a monodimensional structure, the heat flux between the system extremities separated by d at temperature T 1 and T 2 is :

φ = 1 L p,k hω(k)v g (k) 1 exp(hω/kT 1 ) - 1 exp(hω/kT 2 ) = p ωmax 0 hωdω π 1 exp(hω/kT 1 ) - 1 exp(hω/kT 2 ) (12) If T 1 ∼ T 2 ≪ ω max , the heat flux reads φ = πk 2 6h T 2 1 -T 2 2 ≈ G quant (T 1 -T 2 ) ( 13
)
where G quant = πk 2 T /3h is the quantum of conductance. Such a quantum of conductance has been observed by Roukes et al. [START_REF] Schwab | Measurement of the quantum of thermal conductance[END_REF] at low temperature for suspended wire.

Radiative heat transfer

Classical heat transfer is based on the concepts of radiometry i.e. of geometrical optics. Ray lights propagate in straight line and wave effects such as interferences and tunelling are not taken into account. This approximation fails when distances involved go below the typical wavelength. As in the case of light propagation, phenomena such as diffraction occur when light is confined at a subwavelength size. Fluctuational electrodynamics is the theory which has to be used to treat such problems.

Fluctuational electrodynamics

The framework of fluctuational electrodynamics has been introduced by Rytov [START_REF] Rytov | Principles of Statistical Radiophysics[END_REF]). The idea is that in any material at thermal equilibrium, random thermal fluctuations make charges such as electrons or ions moving. These charge motions initiate thermal currents that radiate an electromagnetic field. From an electromagnetic point of view, the problem will be solved if the currents characteristics are known as well as the radiation of an elementary dipole in the geometry considered. The currents are actually given through their cross-spectral correlation function by the fluctuation-dissipation theorem (FDT) whereas the system Green function gives the answer to the radiation problem.

In a medium at thermal equilibrium T characterized by its dielectric constant ǫ, thermal electric currents cross spectral correlation function is given by the FDT

j k (r, ω)j l (r ′ , ω ′ ) = 4πǫ 0 Im(ǫ)Θ(ω, T )δ kl δ(r -r ′ )δ(ω -ω ′ ) ( 14
)
where Θ is the mean energy of an oscillator Θ(ω, T ) = hω/[exp(hω/k b T ) -1]. The electromagnetic field is related to the currents by the system Green function. For example, the electric field reads

E(r, ω) = iµ 0 ω ↔ G (r, r ′ , ω).j(r ′ , ω)d 3 r ′ (15)
In thermal radiation, we are interested in quantities such as the emitted radiative flux given by the Poynting vector 1/2Re [ E(r, ω) × H * (r, ω)) ] and the density of energy. It can be shown that the specific intensity itself is related to a Wigner transform of the electric field cross spectral correlation function [START_REF] Apresyan | Radiation Transfer. Statisitical and wave aspects[END_REF]. Therefore, we have to calculate quantities such as E(r, ω)E * (r ′ , ω) . In all these equations, the brakets denote ensemble average over all the system realization.

Thermal radiation near an interface

The Green function of a system consisting of an interface separating a dielectric (medium 2) from a vacuum (medium 1) is well known. It is then straightforward to apply the FDT to currents in the heat flux or energy density expressions. The result is an integration over the wavevector parallel to the interface K and represents the summation of the individual plane wave contributions. The heat flux reads whereas the energy density expression reads

S z (r, ω) = hω 4π 2 1 e hω/(k b T ) -1 ω/c 0 KdK 1 -|r s 12 | 2 + 1 -|r p 12 | 2 (16)
u(ω, T ) = Θ(ω, T )ω 2π 2 c 2 ω/c 0 KdK γ 1 1 -|r s 12 | 2 2 + 1 -|r p 12 | 2 2 + Θ(ω, T )ω 2π 2 c 2 ∞ ω/c K 3 dK k 2 0 |γ 1 | [Im(r s 12 ) + Im(r p 12 )] e -2|γ 1 |z (17)
In the preceding expressions, r λ 12 is the Fresnel reflection coefficient in λ = s, p polarization. γ 1 is the wavevector component perpendicular to the interface.

The heat flux expression is similar to the classical one. Only propagative waves are emitted and carry energy from a heated body. Emissivity can be identified from the heat flux expression. It is equal to 1/2(1 -|r s 12 | 2 + 1 -|r p 12 | 2 ). Energy density has not only a propagative contribution, but also a contribution from the evanescent waves. Due to the exponentially decaying term, this contribution only exists if the distance to the interface is smaller than the wavelength considered. This supplementary term depends on the imaginary part of the surface reflection coefficient. A blackbody, for example, has no envanescent wave contribution to the energy density near its surface. If the reflection coefficient imaginary part is large, the difference in the energy density near or far from the surface is very important. This occurs in particular close to materials that exhibit surface waves such as polar materials (SiC or Silica). Surface waves are evanescent waves that propagate along the interface but decay exponentially in both directions perpendicular to the interface. They are associated to polarization waves in the material such as plasmon of phonon-polaritons [START_REF] Ashcroft | Solid state physics[END_REF]. In Fig. 5 from (Shchegrov et al. 2000), energy density spectrum above a semi infinite medium of SiC separated from vacuum by an interface is represented. We note that at distances large compared to the mean wavelength, the energy spectrum is very similar to a blackbody spectrum. When the distance to the interface is reduced, the energy density increases at a particular frequency. At a 100nm distance, the energy density spectrum is almost monochromatic. In the case of SiC, this peak is due to the presence of surface waves (phonon polaritons). At the peak frequency, the density of electromagnetic modes increases due to the presence of phonon polaritons.

At thermal equilibrium, energy density reads as the product of the electromagnetic density of states by the mean energy of an oscillator. Therefore, an instrument able to measure the electromagnetic energy density close to a surface also measures the electromagnetic density of states [START_REF] Joulain | Definition and measurement of the local density of electromagnetic states close to an interface[END_REF]. This instrument exists. This is an apertureless Scanning Near-Field Optical Microscope. An AFM tip approached at a nanometric distance of an interface can scatter the evanescent thermal field and convert evanescent waves into propagating waves that can be detected by a classical optical device. This experiment has been performed by De Wilde et al. [START_REF] Dewilde | Thermal radiation scanning tunnelling microscopy[END_REF]. Note that this kind of instrument is nothing else than an infrared near-field microscope where illuminated light is thermal radiation emitted by the sample.

Near-field thermal exchange between two semi-infinte bodies

Let us consider two flat interfaces delimiting two semi-infinite bodies separated by a distance d in vacuum. The two bodies are maintained at their temperature T 1 and T 2 and a heat flux is exchanged between the two bodies. This heat flux is obtained through the calculation of the Poynting vector ensemble average by means of the fluctuation dissipation theorem. As in the single plane interface case, the heat flux expresses as a summation of the individual plane waves. Two contributions can be identified:

q prop (ω) = i=s,p dωdΩ cos θ 2 (1 -|r i 31 | 2 )(1 -|r i 31 | 2 ) |1 -r i 31 r i 32 e 2iγ 3 d | 2 L 0 ω (T 1 ) -L 0 ω (T 2 ) q evan (ω) = i=s,p dω ∞ ω/c 2KdKe -2Im(γ 3 )d Im(r i 31 )Im(r i 32 ) |1 -r i 31 r i 32 e 2iγ 3 d | 2 [L 0 ω (T 1 ) -L 0 ω (T 2 )] k 2 0 (18)
where L 0 ω is the blackbody specific intensity. q prop denotes the propagative wave contribution. It is very close to the classical expression provided that the emissivity is identified to 1 -|r 12 | 2 . Nevertheless, the denominator, a Fabry-Perot like expression with a rapidly fluctuating term is slightly different from the classical expression. After summation on the angular frequency, it can be averaged so that the propagative contribution can be indentified to

q class (ω) = i=s,p dωdΩ cos θ 2 ǫ ′ 1 ǫ ′ 2 1 -ρ 1 ρ 2 L 0 ω (T 1 ) -L 0 ω (T 2 ) (19)
which is the classical radiative heat transfer flux between two opaque blackbodies. Here, ǫ ′ and ρ are the material emissivity and reflectivity. The evanescent contribution describes the heat flux due to tunnelling. This term is important when the reflection coefficient imaginary part is important. This can happen for material supporting surface waves when the surface waves supported by each of the interfaces interact each other. In Fig. 6(a), the heat transfer coefficient (h R (ω) = lim T 1 -T 2 q(ω)/(T 1 -T 2 )) is represented for SiC and glass versus sample separation distances. We note that h R (ω) increases as 1/d 2 at small distances. We also note in Fig. 6(b) that the heat transfer coefficient is almost monochromatic important at the resonant surface wave frequency, where the collision between the polaritons occurs. This collision process between polaritons also exists in bulk materials and is responsible of thermal conduction due to optical phonon modes [START_REF] Joulain | [END_REF] The heat transfer also increases when two metals are approched to each other but for completely different reasons : eddy currents appear at the surface metal creating an important magnetic near-field (Chapuis et al. 2008) Radiation spatial coherence properties Spatial coherence of a field is related to its cross spectral correlation function. The field is spatially coherent on a distance r if its component can interfere when taken at two different points separated by r. Electromagnetic field cross spectral correlation function can be obtained by the FDT. Thermal radiation field emitted by a blackbody is correlated on a maximum distance of the order λ/2. In the near-field, the electromagnetic field correlation length is radically changed depending on the nature of the thermal evanescent electromagnetic field (Fig. 7). At a surface plasmon wavelength, the electromagnetic field correlation function is determined by the surface wave propagation distance along the interface. In the extreme nearfield, the field is static. Retardation effects disappear and the field is correlated on the order of the distance to the interface. Therefore, thermal electromagnetic field coherence greatly changes depending on the regime considered.

These near-field correlation properties have been exploited to design and construct thermal sources that are spatially coherent (i.e are directional) in the far-field. The idea is to rule a grating on a material that support surface waves in a wavelength domain where thermal emission is important. The physical process is the following. Thermally excited surface waves propagate along the interface and are scattered by the grating lines that are regularly spaced. In the far field, and for a particular direction, scattered waves interfere constructively. This kind of grating has been succesfully ruled on SiC to give a spatially coherent thermal source [START_REF] Greffet | Coherent emission of light by thermal sources[END_REF]. The source emission diagram is represented in Fig. 8. Coherent thermal sources based on a waveguide rather than a material supporting surface waves have been designed. Guided waves propagate along the guide on which a grating has been ruled. Scattered guided waves interfere in the far-field to produce emission in a particular direction (Joulain & Loizeau 2007).

Conclusion

We have investigated in this paper physical phenomena that underlie heat transfer at microscopic scale. We have seen that depending on the distances involved, the phyics explaining heat transfer is not the same. Two typical lengths are determinant. Mean free path and wavelength heat carriers. When the carriers mean free path is small compared to the system size, the system is diffusive and obeys a diffusion equation like the heat equation. In the opposite situation, the transfer is ballistic and does not depend on the system size. At distances large compared to the carriers wavelength, wave effects are not present. Carriers behave like particles going in straight line and obey Boltzmann equation. At subwavelength distances on the contrary, band gaps open and transfer is forbidden at some frequency. Phonon filtering is then possible in order to reduce thermal transfer. The increasing improvement in nanotechnology has open the way to reach and go below these two characteristic lengths. It is now possible to engineer nanocomponent thermal properties in order to create new materials with exciting properties such as thermal barriers, heat sink or very high efficiency thermoelectric converters.

Figure 1 :

 1 Figure 1: Left: Pyramidal AFM tip above a substrate. Typical length used are l = 40 nm and h = 20 nm. The distance d is in the nanometric range. Right: Heat flux density along the section of the sample for different tip-sample distances.

Figure 2 :

 2 Figure 2: Silicon nanowire thermal conductivity for different wire diameter versus temperature. Calculations compared with experiments.

Figure 3 :

 3 Figure 3: Surface thermal conductance of silicon films at different temperatures versus film thickness.

Figure 4 :

 4 Figure 4: Temperature profile in a 2-micron-thick film for different temperatures. Note the transition between the Fourier regime (linear profile relating extreme temperatures) and the ballistic regime (flat profile).

Figure 5 :

 5 Figure 5: Energy density above a semi-infinite medium of SiC at T = 300 K.

Figure 6 :Figure 7 :

 67 Figure 6: (a) Heat transfer coefficient for two semi-infinite samples of SiC or glass at T = 300 K versus separation distance d. (b) Monochromatic heat transfer coefficient for two semi-infinite samples of SiC or glass at T = 300 K for d = 10 nm.

Figure 8 :

 8 Figure 8: SiC grating emission diragram at two different wavelengths. Grating period : Λ = 6.25 µm, filling factor F = 0.5 and height h = 0.285 µm.


Bruno Frka-Petesic

Kaspars Erglis

Jean-Francois F Berret

Andrejs Cebers

Vincent Dupuis

Jérôme Fresnais

Regine Perzynski
email: regine.perzynski@upmc.fr

Olivier Sandre

Dynamics of paramagnetic nanostructured rods under rotating field

Keywords: 82.70.Dd, 81.07-b, 83.80.Uv, 81.16.Dn Colloids, Nanocomposites, Block copolymers, Self-assembly

Introduction

Probing the local rheology of complex media, such as the internal compartments of a living cell [1], is a current stake of research. Magnetic rods gives the opportunity of such a local probing under a time-dependent field. Direct optical measurements are possible with sub-millimetric rods [2,3] and even at smaller scales, thanks to the under-field chaining of sub-micronic magnetic particles [4]. Such particles joined together in a chain through a temporary or a permanent link, can be used for mixing chemical moities in microfluidic devices [5] and allows a local rheological probing in the linear hydrodynamic regime [1,6,7]. However for large aspect ratios, these softly-linked chains present shape instabilities [4,8] analogous to the one observed with magnetic liquid needles [9]. Rigid magnetic rods would not present such limitations.

We present here a study by optical microscopy, of the rotational behavior in a simple liquid under an applied rotating field, of rigid magnetic nanostructured rods, obtained by the co-association of magnetic nanoparticles and di-block copolymers [10,11]. We first probe the system under a field abruptly rocked by 90 • and study the rotation of the rods as a function of field amplitude H. We then analyze the rotational behaviour of these rods under an applied field rotating at angular frequency ω as a function of H and ω, both in the linear regime and in the non-linear one, following the descriptions presented in [12].

Synthesis and shape characteristics of the microrods

The magnetic nanostructured rods result from the under-field auto-association of maghemite nanoparticles synthesized following Massart's general procedure [13] and block co-polymers denoted as PTEA 11kb-PAM 30k provided by Rhodia [14]. PTEA stands for poly(dimethyl ammonium ethylacrylate methyl sulfate) and is positively charged, while PAM stands for Author manuscript version of article available in its final form on the publisher's website: DOI 10.1016DOI 10. /j.jmmm.2010.11.036 .11.036 poly(acrylamide) and is neutral. PTEA blocks glue together the negatively charged nanoparticles and PAM blocks provide the rods with a neutral corona that takes the rods apart from each other. The process followed for the synthesis of the rods is detailed in [10]. The maghemite nanoparticles (NP's) used here are coated with 2000 g/mol PAA poly(acrylic acid). The distribution of magnetic NP diameters is well described by a log-normal distribution with a mean diameter 7.1 nm and a polydispersity 0.26. These superparamagnetic NP's have a dipolar magnetic interaction parameter 3χ 0 /24Φ = 0.48 where χ 0 is the initial magnetic susceptibility of a liquid dispersion of these nanoparticles at low volume fraction Φ (see [15]). Transmission Electron Microscopy (TEM) reveals the elongated shape of the obtained clusters, as illustrated in Figure 1. The observation by optical microscopy of dilute aqueous dispersions of these rods allows to determine their distribution of lengths L, as given by the inset of Fig. 1.

Experimental set-up

For the dynamical study under rotating field, dilute dispersions of magnetic rods are placed between two glass slides separated by 0.03 mm double-sided sticky tape with an aperture cut in the center. The rotating magnetic field is created by four water-cooled coils giving a field 0 -20 kA/m in the frequency range 0.1 -5 Hz. The rotation of the filaments is captured by a Zeiss Universal microscope and a JAI CV-S3200 video camera. With time, the rods slowly sediment under gravity and sometimes stick to the cell wall by one end. In the experiment which duration is short with respect to the sedimentation time, we retain only the rods which are freely rotating in the liquid carrier. The registered images are analyzed with the ImageJ code [17] .

Theoretical background

The paramagnetic rod submitted to a magnetic field H = H 0 h has a magnetic moment µ proportional to its magnetic susceptibility tensor χ : µ = µ 0 χV M H where V M is the magnetic volume of the rod. Projecting µ along its main axes e and e ⊥ and with β = ( h, e ), it writes :

µ = χ V M H 0 cos β e + χ ⊥ V M H 0 sin β e ⊥ (1)
the anisotropy of susceptibility of the rod being ∆χ = χχ ⊥ . If the magnetic field H rotates (abruptly or at the pulsation ω) in the plane ( e x , e y ), the rod experiences a magnetic torque Γ magn :

Γ magn = µ × H = µ 0 ∆χV M H 2 0 2 sin 2β e z (2)
Figure 3: Relaxation as a function of time t of the angle β between the field and the rod directions -same experimental conditions as in Figure 2. The full line is a best fit by expression (5).

and a viscous torque Γ visc [16] :

Γ visc = -f η Ω = - πηL 3 3ln(L/D) -2 Ω ( 3
)
where f η is the coefficient of hydrodynamic drag experienced by the rod, Ω the instantaneous angular velocity of the rod, η the viscosity of the fluid carrier and D the diameter of the rod section.

Field abruptly rotated by 90 •

If the re-orientation of the rod is forced under a quick rocking at right angle of the field direction, the relaxation is then driven by the following differential equation :

dβ dt = - 1 2τ sin 2β with 1 τ = µ 0 ∆χV M H 2 0 f η (4)
The equation has three solutions (mod π) : β = 0 (stable equilibrium), β = π/2 (unstable equilibrium) and :

β(t) = arctan exp - t -t π/4 τ (5)
where t π/4 is defined as β(t = t π/4 ) = π/4.

Field rotating at angular frequency ω

In the experiment under a field rotating at angular frequency ω, the angle θ = ( e , e x ) between the rod and the axis e x is equal to :

θ = ωt -β (6)
It leads to the differential equation in β :

dβ dt = ω -ω c sin 2β with ω c = µ 0 ∆χV M H 2 0 2 f η ( 7
)
where the critical pulsation ω c is introduced. . This equation has two solutions [18,12,19,20] : -for ω < ω c , there is a stationary solution in β : sin 2β = ω/ω c and thus

β = 1 2 arcsin ω ω c (8)
The expression of θ(t) is then affine :

θ(t) = ωt - 1 2 arcsin ω ω c (9)
-for ω > ω c , there is a periodic solution for β, and pseudo-periodic for θ.

β(t) = arctan ω c ω + 1 -ω c ω 2 tan 1 -ω c ω 2 ω(t -t 0 ) (10)
The mean angular velocity Ω = θ of the rod, taken over a period of time long with respect to 1/ω c , then writes as Ω = ωω 2 -ω 2 c [19].

Experimental results

Field abruptly rotated by 90 •

The β(t) relaxation under forced re-orientation after a quick rocking at right angle of the field direction is nicely reproduced by expression ( 5). An illustration of the good fit quality is given in Figure 3. Relaxation times τ measured for several rods are collected in Fig- ure 4. At the first order 1/τ scales as H 2 as predicted by expression (5). We should remark that if the field is not exactly rocked at θ init = 90 • , but at a smaller angle θ init , it is however possible to nicely determine the relaxation time τ.

Field rotating at angular frequency ω

Indeed with a field rotating at constant ω, the two rotational regimes are observed experimentally. Below a critical pulsation ω c , scaling as H 2 0 , each micro-rod synchronously follows the applied field with a phase-lag β (see Inset of Figure 5) as also observed in [16]. As the pulsation ω of the magnetic field increases, the angular velocity Ω of the rod increases (see main part of Figure 5) up to ω = ω c , for which the phase-lag β reaches the value π/4. Then the rotational dynamics of the rod becomes jerky, with back-and-forth rotations more and more pronounced as ω increases. As shown in Figure 5, the mean angular velocity of the rod Ω then decreases towards zero.

Polar representations, such as that of Figure 6, plotting the time t as radial coordinate and the angle θ as polar angle, clearly distinguish the two regimes. At low frequencies, the synchronous regime leads to Archimedes spirals while for ω > ω c some back-andforth rotations mark the jerky regime, they are more and more numerous as the pulsation increases above ω c .

Discussion

Identifying hydrodynamic and magnetic characteristics of the rods, we assume that the rod magnetic volume can be writen as :

V M = πLD 2 4 ( 11
)
Following expressions ( 3) and ( 4), we can rewrite the inverse relaxation time of forced re-orientation as :

1 τ = µ 0 ∆χ H2 4η (12)
where

H = H 0 D L 3 ln L D -2 (13)
Figure 7 presents the results of Figure 4, re-plotted as a function of H which is calculated with D = 150 nm, a value compatible with TEM observations (optical images have not enough accuracy for a precise determination of D). A master curve of slope 2 is then obtained in Figure 7, leading to a value the anisotropy of susceptibility ∆χ = χ 2 /(2 + χ) = 0.26 thus a susceptibility of the internal material χ 0.87. It is also possible to deduce the rod anisotropy of susceptibility ∆χ from the experimental threshold ω c using expressions (3) and (7). Figure 8 demonstrates for a particular rod of length L = 14 µm the pretty good agreement between the field-dependence of 2ω c and of 1/τ deduced from the two kinds of measurements. From these determinations, we find ∆χ = 0.26 (resp. 0.16) for D = 150 nm (resp. 200 nm), thus a susceptibility of the internal material χ = 0.87 (resp. 0.65). Taking in account the low value of the dipolar magnetic interaction parameter of the present superparamagnetic nanoparticles, this internal susceptibility is here directly proportional to the mean internal concentration Φ int of nanoparticles inside the rods [15]. With D = 150 nm (resp. 200 nm) we find Φ int = 23% (resp. 17%) of the same order as in [10]. If now a small ellipticity of the rotating field is introduced, it sensibly modifies the shape of the curve Ω = f(ω) of Figure 5, as predicted in [19], slightly reducing the experimental value of ω c while keeping its H 2 dependence. This behaviour will be detailled elsewhere. A remarquable point is that it introduces an anisotropy in the back and forth motion of the rods as it is here illustrated by the polar representation θ(t) of Figure 9 for the same rod as in Figure 6 with L = 10 µm. At high frequency, this anisotropy allows to observe localized motions of the rod around the direction H max as predicted in [19] (see Fig. 10).

Conclusion

Probing the rotational dynamics of our rigid magnetic rods, dispersed in a standard liquid, either by reorienation along field rocked at 90 • or by a steady rotating field, we demonstrate the paramagnetic nature of the rods and characterize their anisotropic susceptibility. The behaviour under steady rotating field fairly well follows in the linear and in the non-linear regime the theoretical predictions initially developped -for pairs of magnetic holes in a ferrofluid in [19,20] and -for paramagnetic ellipsoids in [12]. In [12], paramagnetic ellipsoids of aspect ratio up to 5 were tested, our magnetic rods have an aspect ratio more than 10 times larger.

These magnetic rods will thus find applications as active elements in microfluidic devices viz. mechanical stirrers or pump elements. A rheological probing of complex medium has now to be undertaken with these magnetic nanostructured rods.

Figure 1 :

 1 Figure 1: TEM picture of a nanostructured rod. Inset : Distribution of length L of the rods in a dilute aqueous dispersion as determined by optical microscopy. Full line : fit by a log-normal distribution with median length 12.3 µm and polydispersity 0.5

Figure 2 :

 2 Figure 2: Re-orientation of a rod with L = 15 µm under a field of amplitude H = 10 kA/m abruptly rotated by 90 • and observed as a function of time by optical microscopy.

4 .

 4 Forced re-orientation under a rocking field Experiments of rod re-orientations under a static applied field, abruptly rotated by 90 • are first performed. The rods follow the field orientation, as attested by Figure 2 with a rod of length 15 µm. The angle β between the field and the rod directions is registered as a function of time t in Figure 3. The inflexion point of the relaxation β = f (t) at β = 45 • confirms the paramagnetic nature of the rods as observed in [10, 11].

Figure 4 :

 4 Figure 4: Inverse of the relaxation time τ of various rods, measured under a field abruptly rotated by 90 • and plotted as a function of field amplitude H 0 ; Various symbols correspond to various rod lengths L. Dashed lines are guides for the eye of slope 2.

Figure 5 :

 5 Figure 5: Reduced representation of the mean angular velocity Ω of a rod of length L = 10 µm as a function of the pulsation ω of the field normalized at the limit of the synchronous regime where ω = ω c (H 0 = 8 kA/m and ω c = 6.2 rad/s). Inset : temporal evolution of the rod orientation (cos θ -thick line) compared to the orientation of the field (thin line) for a rod of length L = 14 µm under a field amplitude H 0 = 3.2 kA/m at ω = 0.94 rad/s just below ω c = 0.95 rad/s.

Figure 6 :

 6 Figure 6: Polar representation θ(t) (θ polar angle; time t radial coordinate) of the rotational dynamics of the same rod as in Figure 5 with L = 10 µm under a magnetic field of amplitude H 0 = 8 kA/m rotating at various frequencies (from top to bottom: left ω = 3.1, 7.5 and 12.6 rad/s; right ω = 5.0, 9.4 and 18.8 rad/s); here ω c = 6.2 rad/s

Figure 7 :

 7 Figure 7: Inverse of the relaxation time τ of various rods, measured under a field abruptly rotated by 90 • and plotted as a function of the reduced variable H given by expression (13). The line is an adjustment of the data by expression (12) with D = 150 nm.

Figure 8 :

 8 Figure 8: Log-log representation of 2ω c as a function of the amplitude of the rotational field H 0 for a rod of length L = 14 µm, compared to 1/τ determined under a field abruptly rotated by 90 • and plotted as a function of field amplitude H 0 .

Figure 9 :

 9 Figure 9: Polar representation θ(t) of the rotational dynamics of a rod with L = 10 µm under a magnetic field slightly elliptic of mean amplitude H max = 4.8 kA/m rotating at ω = 2.5 rad/s (field ellipticity H min /H max of the order of 0.87). The induced back and forth motion is anisotropic.

Acknowledgments

This work was supported by ANR "ITC-Nanoprobe", grant of University of Latvia n • ESS2009/6, Féd. F21 UPMC-Paris6, programs ECONET n • 16274QL and OSMOSE n • 22497YE. We thank Aude Michel (PECSA) for the TEM pictures. Figure 10: Polar representation of θ(t) of a localized mode of a rod with L = 10 µm, observed in an experiment with a field ellipticity H min /H max of the order of 0.87 (H max = 8 kA/m and ω = 4 Hz).


W Blum

Henri Rulquin

S Rigout

S Lemosquet

A Bach

J W Blum

Duodenal Infusion of Glucose Decreases Milk Fat Production in Grass Silage-Fed Dairy Cows

Keywords: mammary metabolism, milk fat, dairy cow)

Duodenal infusion of glucose decreases milk fat

production in grass silage-fed dairy cows

INTRODUCTION

Diets for dairy cows in intensive housing are often rich in carbohydrates, which are not completely fermented in the rumen, leading to a potential amount of starch hydrolyzed to glucose in the intestine. Duodenal glucose infusions are known to reduce milk fat production (Hurtaud et al., 1998b). The composition of milk fatty acids is also modified. There are decreases in short-and long-chain fatty acids but a concomitant increase in medium-chain fatty acids (Hurtaud et al., 1998b). Understanding how glucose modifies milk fat production is an important aspect of dairy cow nutrition, but mechanisms by which glucose regulates milk fat synthesis are poorly understood. Several hypotheses have been suggested [START_REF] Bauman | Regulation and nutritional manipulation of milk fat: Low-fat milk syndrome[END_REF]. The glucogenic-insulin theory, one of the first, suggested that increased release of insulin linked to an increased availability of glucose may slow down lipolysis and stimulate both NEFA reesterification and lipogenesis in the adipose tissue [START_REF] Vernon | Lipid metabolism in the adipose tissue of ruminant animals[END_REF]. However, in duodenal glucose infusion studies [START_REF] Lemosquet | Effects of a duodenal glucose infusion on the relationship between plasma concentrations of glucose and insulin in dairy cows[END_REF]Hurtaud et al., 1998b;[START_REF] Hurtaud | Effect of graded duodenal infusions of glucose on yield and composition of milk from dairy cows. 2. Diets based on grass silage[END_REF]Rigout et al., 2002a), it is difficult to relate the decreased availability of milk fat precursors to an increased insulin secretion because of the small varia-tion in insulin concentrations. Moreover, the insulin mechanism has recently been intensely evaluated using long-term euglycemic-hyperinsulinemic clamp studies [START_REF] Mcguire | Role of insulin in the regulation of mammary synthesis of fat and protein[END_REF][START_REF] Griinari | Role of insulin in the regulation of milk fat synthesis in dairy cows[END_REF][START_REF] Mackle | Effects of insulin and amino acids on milk protein concentration and yield from dairy cows[END_REF][START_REF] Bequette | Insulin regulates milk production and mammary gland and hind-leg amino acid fluxes and blood flow in lactating goats[END_REF], and a role for insulin in milk fat depression was not supported. On the other hand, it has also been postulated that trans 10-C 18:1 or related metabolites could be the cause of milk fat depression [START_REF] Bauman | Regulation and nutritional manipulation of milk fat: Low-fat milk syndrome[END_REF]; however, their implication in milk fat decrease during glucose infusions seems to be challenged, as transfatty acid percentage and production in milk decreased (Rigout et al., accepted). Nevertheless, specifically trans 10, cis 12 conjugated linoleic acid plays a bigger role in regulating milk fat synthesis than trans 10-C 18:1 [START_REF] Loor | Exogenous conjugated linoleic acid isomers reduce bovine milk fat concentration and yield by inhibiting de novo fatty acid synthesis[END_REF][START_REF] Chouinard | Milk yield and composition during abomasal infusion of conjugated linoleic acids in dairy cows[END_REF][START_REF] Baumgard | Identification of the conjugated linoleic acid isomer that inhibits milk fat synthesis[END_REF].

To our knowledge, no direct measurement of mammary metabolism has been reported to understand how milk fat precursors are utilized by the mammary gland in response to graded amounts of duodenal glucose. However, it is known that mammary uptake of milk fat precursors depends on their arterial availability [START_REF] Rulquin | Re ´gulation de la synthe `se et de la se ´cre ´tion des constituants du lait chez les ruminants[END_REF] and that jugular blood concentrations of milk fat precursors decreased in response to duodenal glucose infusions (Hurtaud et al., 1998b[START_REF] Hurtaud | Effect of graded duodenal infusions of glucose on yield and composition of milk from dairy cows. 2. Diets based on grass silage[END_REF]. Thus, in the present study, arteriovenous differences of milk fat precursors and blood flow were measured to describe the mechanisms involved in milk fat depression occurring during duodenal glucose infusions. Complementary data on glucose metabolism and milk production for this study have been presented previously (Rigout et al., 2002a).

MATERIALS AND METHODS

Cows, Treatments, and Feeding

Experimental details for this study have been reported (Rigout et al., 2002a). Briefly, four multiparous Holstein cows (635 ± 52 kg of BW; 81 ± 21 DIM; 32 ± 2 kg/d of milk) were used in the experiment. Treatments were continuous duodenal infusions of graded amounts of glucose with equal supplies of NE L and protein truly digested in the small intestine (PDI; Institut National de la Recherche Agronomique, 1989).

Glucose was infused at rates of 0, 443, 963, and 2398 g/d (G0, G1, G2, and G3, respectively). To avoid the confounding effect of an increasing supply of energy from glucose and to keep energy and protein constant among treatments, 0.46 kg of grass silage and 1.62 kg of energy concentrate were replaced with 1.0 kg of glucose and 0.50 kg of soybean meal, assuming that 1 kg of glucose provided 2.75 Mcal of NE L and 0 g of PDI [START_REF] Armstrong | The utilization of energy of carbohydrate by ruminants[END_REF]. The combinations of Journal of Dairy Science Vol. 85, No. 10, 2002 diet plus infusion were formulated to provide 100 and 110% of energy and protein requirements (Institut National de la Recherche Agronomique, 1989). The experiment was conducted according to a 4 × 4 Latin square design over successive 14-d periods (9 d of adaptation and 5 d of measure).

The basal diet (66% DM, 17% CP, 46% NDF, 1.74 MCal/d) consisted of 63.9% large bale grass silage, 7.1% formaldehyde-treated soybean meal, 27.7% energy concentrate (DM basis) supplemented with minerals and vitamins (300 g/d) and with L-Lys HCl (11 g/d; Ajinomoto Co. Inc., Tokyo, Japan) and DL-Met (16 g/d; Rho ˆne-Poulenc, Commentry, France) according to recommendations of [START_REF] Rulquin | Milk production and composition as a function of postruminal lysine and methionine supply: a nutrient-response approach[END_REF]. The energy concentrate was mainly composed of barley, wheat, and dehydrated beet pulps. The composition of feedstuffs and details of diet formulation has been reported in Rigout et al. (2002a).

Surgical Preparation

Six months before calving, cows were fitted with a proximal T-shaped duodenal cannula placed 10 to 15 cm from the pylorus. Two or three months before the beginning of the experiment, cows were surgically prepared to estimate the mammary uptake of nutrients according to the methods described by [START_REF] Guinard | Effect of graded levels of duodenal infusions of casein on mammary uptake in lactating cows. 1. Major nutrients[END_REF]. Two catheters were placed in the left carotid artery and the left subcutaneous abdominal vein to measure udder arteriovenous (AV) differences (Rigout et al., 2002a). A transit time ultrasonic flow probe (Transonic Systems Inc., Ithaca, NY) was placed around the left external pudic artery before the Sshaped bend in the artery to measure mammary blood flow (MBF). The four cows were housed in individual tie stalls. Grass silage was offered three times per day (25% at 0700 h, 25% at 1300 h, 50% at 1900 h) and concentrate eight times per day in equal portions. Access to feed was limited to 1 h every 3 h starting at 0700.

Measurements and Sampling

The MBF was measured throughout the experiment, but the other measurements were performed during the second week of each period. On d 13 at 0.5, 2.5, 4.5, 6.5, 8.5, and 10.5 h after the morning milking, blood was collected simultaneously from the artery and vein with syringes containing 21 ± 9 IU of heparin (S-Monovette, 7.5 ml; Sarstedt, Nu ¨mbrecht, Germany) for the determination of acetate [START_REF] Guynn | Enzymatic determination of acetate[END_REF], BHBA (McMurray et al., 1984), NEFA [START_REF] Duncombe | The colorimetric micro-determination of non-esterified fatty acids in plasma[END_REF], and total glycerides [START_REF] Bucolo | Quantitative determination of serum triglycerides by the use of enzymes[END_REF]. Blood was collected only from artery with syrin-ges containing 1.6 ± 0.4 mg/ml of potassium-EDTA (Sarstedt) for hormone determination. Cows were standing during blood sampling. Plasma concentrations of metabolites other than hormones were measured on a multiparameter analyzer (KONE Instruments Corporation, Espoo, Finland).

Heparinized plasma was analyzed for NEFA using an enzymatic kit (acyl-coA synthetase, acyl-coA carboxylase method; Wako kit, Oxoid S.A., Dardilly, France) and total glyceride (lipase, glycerol kinase, glycerol-3-P oxidase, peroxidase method; Biotrol Diagnostic, Chennevie `re les Louvres, France). Intra-assay coefficient of variation was 0.9% for NEFA and 0.1% for total glyceride.

Deproteinized plasma (with 50%, vol/vol, HCLO 4 by filtration) was analyzed for BHBA (BHBA dehydrogenase method; Sigma, Saint-Quentin Fallavier, France) and for acetate (acetyl-coA synthetase, citrate synthetase, malate dehydrogenase method; Sigma, Saint-Quentin Fallavier, France). Intraassay coefficient of variation was 0.4% for BHBA and 0.8% for acetate.

Cortisol, triiodothyronine, and thyroxine were measured by radioimmunoassay, as described by [START_REF] Hammon | Metabolic and endocrine traits of neonatal calves are influenced by feeding colostrum for different durations or only milk replacer[END_REF]. Intraassay coefficient of variation was <10% for these hormones.

Cows were milked twice daily at 0630 and 1830 h. Since blood was sampled on the left artery and vein, each half udder was milked separately to obtain a true balance for each nutrient on a half udder basis. Milk yield for each half udder was assayed for fat content by infrared analysis (Milkoscan; Foss Electric, Hillerod, Denmark). On d 14, 100 ml of milk was taken from the left udder of each cow at the morning milking for the determination of milk fatty acids and metabolites concentrations.

To determine milk fatty acids, we extracted lipids from 1 ml of milk fat according to Bauchard and Duboisset (1983) using 0.5 ml of ethanol/HCl (4/1, vol/ vol) solution followed by 0.5 ml of hexane. Milk fatty acids were then transesterified with 1 ml of a butanol/ HCl (100/5, vol/vol) followed by 2 ml of hexane. Fatty acid butyl esters in hexane were then injected into a gas chromatograph (Varian 3400, Les Ulis, France) equipped with an electron ionization detector. Separation of fatty acid butyl esters was performed with a OV-1 fused silica capillary column (25 m × 0.32 mm i.d.). The carrier gas was helium. The oven temperature was programmed from 70 to 220°C at 100°C/min and held for 32°min. Injector and detector were at 220 and 250°C, respectively.

For the determination of milk metabolites (citrate, isocitrate, oxoglutarate, glycerol-3-P, 6-phosphogluconate, malate, lactate and acetate), milk was immediately deproteinized with 5 M HCLO 4 before storage at Journal of Dairy Science Vol. 85, No. 10, 2002 -20°C and later analyzed according to the methods of [START_REF] Faulkner | The presence of cellular metabolites in milk[END_REF].

Statistical Analysis

The data were analyzed using a model for a Latin square with cow, period, treatment as independent variables. The residual error term was used to test for significance of cow, period, and treatment. In addition, the linear, quadratic, and cubic effects of glucose infusions were determined, using the residual error term. Because the treatment doses were not equally spaced, specific coefficients for each infusion amount (0, 1, 2, and 6) were used to avoid a biased comparison as described by [START_REF] Gill | Design and Analysis of Experiments in the Animal and Medical Sciences[END_REF]. The calculations were done using PROC GLM of SAS (1990). Results were expressed as least squares means with root means squares errors because of missing values for one period of one cow and two missing values for the ratios associated with glucose-6-P.

RESULTS

As planned, feed intake decreased linearly; however, total NE L intake was unaffected by treatments (Table 1). Duodenal infusions of glucose quadratically increased milk yield, linearly decreased (P < 0.001) milk fat yield (-20%) and curvilinearly decreased milk fat content (-19%). However, the major part of the milk fat yield decrease (-17%) occurred between G2 and G3, and the same was observed for FCM. As a consequence, NE L balance increased linearly (P < 0.05).

Milk Fat Precursors

Duodenal glucose infusions reduced on the whole, arterial concentrations of milk fat precursors (Table 2). Arterial concentrations of acetate, BHBA, and NEFA decreased linearly by 40, 63, and 55%, respectively. Arterial concentrations of total glycerides decreased curvilinearly by 35%.

Although arterial concentrations of milk fat precursors were dramatically decreased, their arterial fluxes (arterial concentrations × mammary plasma flow) were less affected because mammary plasma flow increased curvilinearly (Table 2). Arterial fluxes of acetate and total glycerides remained unchanged. Arterial flux of BHBA was decreased although not significantly by 45%, and arterial flux of NEFA tended (P = 0.093) to decrease linearly.

Mammary Utilization of Milk Fat Precursors

The AV differences of acetate, BHBA, and total glycerides decreased linearly or curvilinearly with in- Arterial concentrations and arteriovenous differences are expressed in mg/100 ml for acetate and BHBA, µmol/L for NEFA, and µmol/L for triglycerides. creasing infusions of glucose (Table 2). Mammary AV differences of NEFA were negative and tended (P = 0.11) to decrease linearly.

Results on mammary extraction rates varied according to the fat nutrients. Those of acetate and total glycerides tended to decrease or decreased linearly. Mammary extraction rates of BHBA remained unchanged, whereas mammary release rates of NEFA increased linearly.

Mammary uptake of acetate remained unchanged, whereas mammary uptake of BHBA decreased linearly (P = 0.057) by 37% with duodenal glucose infusions (Table 2). However, mammary uptake of BHBA was relatively constant between G0 and G2, and the major part of its decrease (-29%) occurred between G2 and G3. Mammary uptake of total glycerides decreased curvilinearly, and mammary release of NEFA concomitantly increased linearly with treatments. The ratio of mammary uptake of milk fat precursors to milk fat output was above 1 and remained unchanged with treatments (Table 2).

Milk Fatty Acid Composition and Production

The proportion of even short-chain fatty acids remained unchanged, despite a slight numerical decrease (Tables 3 and4). Moreover, the percentage of C 8:0 increased quadratically with duodenal infusions of glucose. The production of C 4:0 and C 6:0 decreased linearly and that of C 8:0 increased curvilinearly.

The percentage and production of even mediumchain fatty acids increased curvilinearly and quadratically (i.e., it decreased only at G3), respectively. The percentage of C 16 fatty acid group increased linearly, and its production tended to decrease linearly. Nevertheless, the yield of de novo synthesized fatty acids increased quadratically by 13%, reaching a maximum at G2.

The percentage and production of even long-chain fatty acids decreased curvilinearly and the percentage and production of C 17 fatty acid group and trans 10 and trans 11-C 18:1 decreased linearly.

Milk Metabolites

Milk glucose-6-P concentrations tended to increase linearly with treatments (Table 5). Glucose infusions curvilinearly decreased concentrations of citrate and malate in milk. Concentrations of isocitrate decreased quadratically and oxoglutarate, linearly. In contrast, concentrations of glycerol-3-P, 6-phosphogluconate, lactate and acetate in milk remained unchanged. The ratios of citrate to glucose-6-P and oxoglutarate to isocitrate decreased or tended to decrease linearly, Journal of Dairy Science Vol. 85, No. 10, 2002 whereas the ratio of isocitrate to glucose-6-P was unaffected. At the same time, ratios of glycerol-3-P and 6-phosphogluconate to glucose-6-P remained unchanged.

DISCUSSION

Duodenal Glucose Decreased Milk Fat

With equal energy supplied, duodenal glucose infusion of about 1 kg/d decreased milk fat yield (-4.5%) and composition (-12.5%). When about 2.4 kg/d was infused, glucose substantially decreased both milk fat yield and composition (∼20%).

The decrease in milk fat content was greater than is found in the literature [START_REF] Lemosquet | Effects of a duodenal glucose infusion on the relationship between plasma concentrations of glucose and insulin in dairy cows[END_REF][START_REF] Oldick | Abomasal infusion of glucose and fat effect on digestion, production, and ovarian and uterine functions of cows[END_REF]Hurtaud et al., 1998ab, 2000[START_REF] Hurtaud | Effect of the nature of energy source (propionic acid, glucose or starch) on milk yield and com-position in dairy cows[END_REF]. The decrease in milk fat yield obtained for G2 was lower than reported in the literature (-8%), but the decrease obtained for G3 was similar to that reported by Hurtaud et al. (1998b) when equivalent doses of glucose infused were associated with corn silage-based diets. But response was about twofold greater when the diet was based on grass silage [START_REF] Hurtaud | Effect of graded duodenal infusions of glucose on yield and composition of milk from dairy cows. 2. Diets based on grass silage[END_REF].

Duodenal glucose infusions modified milk fatty acid composition by decreasing long-chain fatty acids and increasing medium-chain fatty acids, as the percentage of short-chain fatty acids was quite constant between G0 and G2, confirming the previous results (Hurtaud et al., 1998b[START_REF] Hurtaud | Effect of graded duodenal infusions of glucose on yield and composition of milk from dairy cows. 2. Diets based on grass silage[END_REF]. To understand the mechanisms involved in the decrease in milk fat, results of G3 will be analyzed separately because it was an exacerbated physiological response.

Duodenal Glucose Decreased Arterial Concentrations of Milk Fat Precursors

In the present study, the reduced arterial concentrations of the milk fat precursors acetate, BHBA, and total glycerides were probably partially related to the decrease in feed DMI (-0.7 kg/d between G0 and G1 or G2) (Rigout et al., 2002a) as well as a decrease in the rate of adipose tissue lipolysis. The decrease in feed DMI is a direct consequence of the experimental design, since a part of VFA resulting from the ruminal fermentation of the diet was substituted by duodenal infusion of glucose. On the other hand, glucose has already been observed to directly decrease plasma arterial concentrations of NEFA, BHBA and acetate, resulting from an increase in fatty acid reesterification in adipose tissue [START_REF] Vernon | Lipid metabolism in the adipose tissue of ruminant animals[END_REF] or a decrease in the balance lipolysis/lipogenesis [START_REF] Griinari | Role of insulin in the regulation of milk fat synthesis in dairy cows[END_REF]. In this study, glucose infusions may have acted on the concentrations of milk fat precursors directly or alter- Represents the sum of percentages of even and odd fatty acids from C 5:0 to C 15 .

4 Calculated as the ratio of (C 10 + C 12 ) to (C 4 + C 6 ).

natively changing hormonal pathways. Indeed, variations in arterial concentrations of hormones existed but were too small to explain the overall response. In response to increasing duodenal glucose infusions, the rate of appearance of glucose and glucose utilization by portal drained-viscera increased (Rigout et al., 2002a). During this time, arterial concentrations of insulin tended to increase and growth hormone tended to decrease (Rigout et al., 2002a). But concentrations of triiodothyronine, thyroxine, and cortisol remained unchanged (Table 6). Hence, the increasing rate of appearance of glucose associated with little change in hormonal secretion may have favored fatty acid reesterification.

Journal of Dairy Science Vol. 85, No. 10, 2002

Although arterial concentrations of milk fat precursors were decreased, the dramatic increase in mammary blood flow (Rigout et al., 2002a) and the consistency of arterial fluxes (arterial concentrations × mammary plasma flow; Table 2) show that the total delivery of milk fat precursors to the mammary gland was only slightly affected.

Duodenal Glucose Decreased Mammary Uptake of Plasma Total Glycerides and Consequently Long-Chain Fatty Acids in Milk

The decreased production of C 18 fatty acid group accounted for 92% of the decreased fatty acids in milk Represents the sum of yields of even and odd fatty acids from C 6:0 to C 16:2 .

4 Calculated from the mammary uptake of milk fat precursors (acetate and BHBA) estimated over 12 h on one half udder and from the corresponding yield of de novo synthesized milk fatty acids.

Significantly different from 1 at P < 0.10. between G0 and G2. Milk long-chain fatty acids are known to come from fatty acids provided by the diet and adipose tissue (total glycerides and NEFA) in blood supplied to the mammary gland. Thus, the decrease of C 18 fatty acid group in milk mirrored the linear decrease in mammary uptake of total glycerides. As previously reported [START_REF] Miller | Patterns of nutrient uptake by the mammary glands of lactating dairy cows[END_REF][START_REF] Rulquin | Re ´gulation de la synthe `se et de la se ´cre ´tion des constituants du lait chez les ruminants[END_REF], the AV differences of total glycerides are positively correlated with their arterial concentrations (Figure 1). However, the slope between AV and arterial concentrations obtained in this study is about twofold greater than that noted by [START_REF] Rulquin | Re ´gulation de la synthe `se et de la se ´cre ´tion des constituants du lait chez les ruminants[END_REF]. This result indicates that the capacity of mammary tissue to extract total glycerides from blood circulation was rapidly decreased by duodenal glucose infusions. The decrease in mammary uptake of total glycerides probably reflects reduced lipoprotein lipase activity caused by glucose infusions. However, if the transport of total glycerides is diffusion limited, the increased MBF could have caused a reduction in uptake [START_REF] Vernon | The regulation of nutrient utilization: basic principles and mechanisms[END_REF][START_REF] Prosser | Regulation of blood flow in the mammary microvasculature[END_REF].

The decrease in long-chain fatty acids could also be due to a decrease in intramammary esterification process. Indeed, the linear increase in NEFA mammary release associated to the linear decrease of total glycerides mammary uptake also demonstrate that the hydrolyzed fatty acids could not be esterified to glycerol-3-P to form triglycerides within the mammary gland. In this experiment, the reduction of esterification was not related to a shortage in glycerol because milk concentration of glycerol-3-P increased. The causes for this impaired esterification are unknown; however, a mechanism could be proposed. [START_REF] Emery | Biosynthesis of milk fat[END_REF] suggested that acylation on sn-3 position of glycerol-3-P may regulate synthesis of milk triglycerides. Since this requires fatty acids like C 4:0 or C 18:1 , which in this study were utilized for de novo synthesis (C 4:0 ) and were less taken up by the udder (C 18 ), a shortage of these two substrates may have blocked the formation of milk triglycerides.

Duodenal Glucose Infusions Increased De Novo Synthesis of Milk Fatty Acids

The composition of milk fatty acids was also modified by the increase in milk fatty acid de novo synthesis and an increase in chain elongation, as the elongation index [(C 10:0 + C 12:0 )/(C 4:0 + C 6:0 )] linearly increased (Table 3). The changes in milk fatty acid composition were totally different to changes that occur in normal dietary caused low fat milk syndrome, where shortand medium-chain fatty acids decrease and long-chain fatty acids increased [START_REF] Davis | Low-fat milk syndrome[END_REF][START_REF] Bauman | Regulation and nutritional manipulation of milk fat: Low-fat milk syndrome[END_REF]. Although arterial concentrations of acetate and BHBA decreased linearly, their mammary uptake was not affected between G0 and G2. Moreover, although AV differences of BHBA and acetate linearly decreased, the efficiency of the mammary gland to extract them was not altered. The values of mammary extraction rates of acetate and BHBA were similar to values of [START_REF] Rulquin | Re ´gulation de la synthe `se et de la se ´cre ´tion des constituants du lait chez les ruminants[END_REF]Miller (1983). Additionally, the efficiency of mammary utilization of acetate and BHBA was increased, as the ratio of mammary uptake to milk output for short-chain fatty acids was greater than 1 for G0 and G1 and then was not different from 1 for G2.

The decrease in long-chain fatty acids may have allowed for an increase in de novo synthesis as it has been shown that longer chain fatty acids inhibit acetyl CoA carboxylase [START_REF] Palmquist | Feed and animal factors influencing milk fat composition[END_REF]. The increase of de novo synthesis was also supported by the decrease of milk citrate concentrations. Moreover, an increase in energy could have also allowed an enhancement in mammary de novo lipogenesis because glucose availability for the udder was improved (Rigout et al., 2002a). The supplemental mammary glucose was not only used for lactose synthesis but may have also been used to generate energy via the Krebs cycle and NADPH via pentose phosphate pathway. However, the greater energy supply did not arise from the Krebs cycle because ratios of citrate and isocitrate to glucose-6-P and oxoglutarate to isocitrate decreased with treatments. In contrast, milk concentrations of 6-phosphogluconate were greater in glucose treatments compared with G0 treatments and the ratio of 6-phosphogluconate to glucose-6-P remained unchanged. Thus, the glucose not utilized for lactose synthesis could have been used for greater synthesis of NADPH through the pentose phosphate pathway.

High Glucose and Mammary Cell Function

As indicated by a more severe decrease in feed DMI (-4 kg/d between G0 and G3;Rigout et al., 2002a), the decrease in milk fat precursors in plasma was highest for G3 compared with the other treatments. However, the shortage of milk fat precursors was not the unique reason for decreased fat yield with the G3 treatment. Indeed, the reduction in milk fatty acids yield between and G3 (-150 g/d of fatty acids) occurred with similarly decreased production of short-and medium-chain fatty acids (44/150 g/d = 29%), C 16 fatty acid group (41/ 150 g/d = 27%) and C 18 fatty acid group (65/150 g/d = 44%). The increase in mammary blood flow cannot explain the homothetic decrease in milk fatty acids in G3, because neither the efficiency of the mammary gland to extract acetate and BHBA, nor their mammary uptake were highly altered; thus, an intramammary disorder could be proposed as an explanation. As milk concentrations of glycerol-3-P were not affected by treatments, milk fatty acid decrease was not due to a reduced availability of this substrate to form milk triglycerides, because milk concentrations of this substrate was not affected by treatments. [START_REF] Ghyczy | Electrophilic methyl groups present in the diet ameliorate pathological states induced by reductive and oxidative stress: a hypothesis[END_REF] observed that an increase in the ratio of NADPH to NADP + resulted in a reducing stress that inhibited many enzymes including those of Embden-Meyehof pathway. Therefore, as illustrated by the evolution of the ratio of 6-phosphogluconate to glucose-6-P, an increase in the ratio of NADPH to NADP + might be responsible in the inhibition of fat synthesis.

CONCLUSIONS

Results suggested that the decrease in milk fat yield and content by duodenal glucose infusions was due to a decrease of lipoprotein lipase activity and a decrease of intramammary esterification process. However, because of the great increase in MBF, the decrease of total glycerides mammary uptake may have been blood flow-dependent. Duodenal glucose infusions led also to an increase of de novo synthesis and chain elongation, by increasing energy source availability.

Journal of Dairy Science Vol. 85, No. 10, 2002

Figure 1 .

 1 Figure 1. Relationship between arteriovenous (AV) differences and arterial concentrations (A) of plasma total glycerides (TG) of the present experiment (᭹); regression equation (-): AV differences TG (mg/dl) = 0.91 × A TG (mg/dl) -2.96. Regression equation of the relationship published by Rulquin (1997) (--): AV differences TG (mg/ dl) = 0.54 × A TG (mg/dl). Journal of Dairy Science Vol. 85, No. 10, 2002

Table 1 .

 1 Effect of duodenal infusions of glucose on feed and energy intake, energy balance, milk, FCM, milk fat content, and yield.

			Glucose 1				Effect
		G0	G1	G2	G3	RMSE 2	Linear	Quadratic	Cubic
								P
	Feed intake, kg/d	18.7	18.0	18.0	14.7	0.95	0.002	0.381	0.559
	NE L intake 3 , Mcal/d	32.4	32.4	33.9	32.1	1.99	0.743	0.317	0.566
	NE L balance, Mcal/d	0.7	1.4	2.4	3.7	1.59	0.044	0.561	0.826
	Milk, kg/d	25.5	26.6	27.8	25.0	1.48	0.314	0.043	0.791
	FCM, kg/d	27.5	26.8	27.7	23.8	0.99	0.002	0.111	0.254
	Fat
	%	4.55	4.04	3.98	3.69	0.110	<0.001	0.005	0.072
	g/d	1155	1078	1103	921	35.6	<0.001	0.568	0.098
	1 Duodenal infusions of glucose: G0 = 0 g/d; G1 = 443 g/d; G2 = 963 g/d; G3 = 2398 g/d.

2

Root mean squares error. 3

Including energy provided by glucose infusion.

Table 2 .

 2 Effect of duodenal infusions of glucose on mammary blood (MBF) and plasma flows (MPF), arterial concentrations, and mammary utilization of acetate, BHBA, NEFA and total glycerides.

			Glucose 1				Effect
		G0	G1	G2	G3	RMSE 2 Linear Quadratic Cubic
								P
	MBF, L/min	5.22	6.33	6.89	7.56	0.552	0.003	0.042	0.711
	MPF, L/min	3.96	4.89	5.26	5.88	0.416	0.002	0.039	0.561
	Arterial concentrations 3
	Acetate	13.0	10.4	10.0	7.8	1.39	0.005	0.146	0.395
	BHBA	13.5	6.8	6.8	5.0	4.24	0.077	0.144	0.401
	NEFA	92.6	74.7	50.7	41.4	16.99	0.010	0.075	0.647
	Total glycerides Arterial fluxes 4	72.3	62.6	54.9	47.2	3.23	<0.001	0.006	0.943
	Acetate	30.8	30.7	31.4	27.4	3.52	0.184	0.495	0.815
	BHBA	31.7	20.9	21.6	17.4	8.82	0.127	0.280	0.477
	NEFA	221	210	160	146	52.4	0.093	0.391	0.576
	Total glycerides	17.0	18.0	17.4	16.7	2.01	0.578	0.634	0.593
	Arteriovenous differences 3
	Acetate	8.99	7.53	6.69	5.10	1.135	0.006	0.195	0.834
	BHBA	4.47	3.17	2.94	1.85	0.898	0.014	0.224	0.533
	NEFA	-0.33	-8.31 -15.57 -17.56 11.008	0.105	0.227	0.924
	Total glycerides	32.3	23.4	16.7	8.8	4.33	0.001	0.033	0.987
	Extraction rates, %
	Acetate	69.1	71.6	66.6	65.1	3.33	0.075	0.896	0.179
	BHBA	37.8	43.2	42.7	37.3	5.12	0.449	0.133	0.574
	NEFA	-7.5	-16.2	-31.2	-42.4	17.45	0.038	0.353	0.737
	Total glycerides	44.2	37.0	30.6	18.8	4.48	0.001	0.125	0.942
	Mammary uptake 5
	Acetate	21.0	22.0	20.9	17.9	2.93	0.114	0.570	0.693
	BHBA	10.3	9.4	9.2	6.5	2.23	0.057	0.973	0.840
	NEFA	-0.07	-2.93	-4.86	-6.11	3.096	0.056	0.200	0.959
	Total glycerides Uptake/milk output 6	7.35 1.90	6.60 2.04	5.24 1.79	3.11 1.74	0.282 0.333	<0.001 0.385	0.019 0.965	0.119 0.467
	1 Duodenal infusions of glucose: G0 = 0 g/d; G1 = 443 g/d; G2 = 963 g/d; G3 = 2398 g/d.

2

Root mean squares error. 3

Table 3 .

 3 Effect of duodenal infusions of glucose on milk fatty acid composition.

			Glucose 1				Effect
		G0	G1	G2	G3	RMSE 2 Linear Quadratic Cubic
								P
	Fatty acids, %
	C 4	3.01	2.67	2.61	2.54	0.229	0.070	0.121	0.552
	C 5	0.018	0.021	0.024	0.030 0.003	0.002	0.284	0.842
	C 6	2.33	2.35	2.30	2.25	0.083	0.162	0.914	0.624
	C 7	0.019	0.027	0.035	0.046 0.005	0.001	0.078	0.864
	C 8	1.43	1.58	1.56	1.56	0.047	0.063	0.016	0.068
	C 9	0.023	0.035	0.047	0.071 0.012	0.003	0.348	0.914
	C 10	3.17	3.95	3.85	4.04	0.272	0.020	0.035	0.114
	C 11	0.041	0.063	0.088	0.132 0.019	0.001	0.218	0.802
	C 12	3.50	4.65	4.66	5.28	0.367	0.002	0.029	0.129
	C 13	0.077	0.103	0.128	0.158 0.017	0.001	0.078	0.847
	C 14:1	0.87	1.22	1.36	1.49	0.103	0.001	0.005	0.389
	C 14	12.1	13.8	13.9	13.4	0.71	0.222	0.017	0.326
	iso-C 15	0.28	0.25	0.27	0.25	0.015	0.044	0.776	0.070
	C 15:1	0.51	0.48	0.48	0.42	0.018	0.001	0.586	0.305
	C 15	1.11	1.23	1.42	1.55	0.143	0.008	0.144	0.593
	C 16:1	1.74	1.87	2.31	2.90	0.232	0.001	0.363	0.351
	C 16	37.6	37.4	39.6	41.9	1.25	0.003	0.696	0.218
	iso-C 17	0.38	0.35	0.35	0.33	0.050	0.295	0.531	0.760
	C 17:1	0.82	0.77	0.71	0.75	0.073	0.325	0.154	0.781
	C 17	0.62	0.56	0.53	0.53	0.054	0.103	0.131	0.822
	C 18:2	1.53	1.52	1.46	1.38	0.084	0.044	0.878	0.746
	C 18:3	0.39	0.44	0.41	0.45	0.057	0.286	0.880	0.401
	C 18:1	16.4	14.6	13.5	12.0	1.30	0.007	0.127	0.843
	trans-10 C 18:1	0.71	0.67	0.57	0.51	0.064	0.007	0.139	0.410
	trans-11 C 18:1	1.38	1.37	1.27	1.02	0.093	0.002	0.763	0.492
	C 18:0	10.02	8.13	6.48	4.97	0.625	0.001	0.005	0.818
	Fatty acids, %
	C 4:0 to C 8:0	6.77	6.61	6.47	6.35	0.304	0.131	0.452	0.983
	C 5:0 to C 9:0	0.060	0.083	0.106	0.147 0.018	0.001	0.193	0.880
	C 10:0 to C 14	19.6	23.6	23.8	24.2	1.25	0.012	0.011	0.160
	C 11:0 to C 15:0	2.02	2.12	2.39	2.51	0.186	0.012	0.170	0.471
	C 16:0 + C 16:1	39.3	39.2	41.9	44.8	1.28	0.001	0.584	0.180
	iso-C 17:0 + C 17:1 + C 17:0	1.82	1.68	1.59	1.60	0.161	0.188	0.175	0.964
	C 18:2 + C 18:3 + C 18:1 + C 18:0	28.3	24.6	21.9	18.8	1.88	0.001	0.036	0.984
	trans-10 C 18:1 + trans-11 C 18:1 De novo synthesis, % 3	2.09 25.5	2.05 29.7	1.83 30.2	1.53 30.7	0.139 1.38	0.002 0.010	0.580 0.012	0.406 0.196
	Unsaturated, %	24.3	22.9	22.1	21.0	1.61	0.042	0.296	0.904
	Elongation index 4	1.27	1.72	1.73	1.95	0.180	0.007	0.051	0.223

1 Duodenal infusions of glucose: G0 = 0 g/d; G1 = 443 g/d; G2 = 963 g/d; G3 = 2398 g/d. 2 Root mean squares error.

3

Table 4 .

 4 Effect of duodenal infusions of glucose on milk fatty acid production for an entire udder.

			Glucose 1				Effect
		G0	G1	G2	G3	RMSE 2 Linear Quadratic Cubic
								P
	Fatty acids, g/d
	C 4:0 to C 8:0 C 5:0 to C 9:0 , × 10 -3	0.64 4.67	0.57 5.97	0.57 7.72	0.47 9.09	0.038 1.52	0.023 0.011	0.379 0.172	0.282 0.723
	C 10:0 to C 14	0.92	1.04	1.07	0.92	0.087	0.354	0.031	0.657
	C 11:0 to C 15:0	0.083 0.082 0.096 0.085 0.010	0.777	0.155	0.291
	C 16:0 + C 16:1	1.54	1.42	1.55	1.39	0.080	0.067	0.583	0.077
	iso-C 17:0 + C 17:1 + C 17:0	0.065 0.057 0.056 0.046 0.005	0.005	0.292	0.444
	C 18:2 + C 18:3 + C 18:1 + C 18:0	0.97	0.80	0.7l	0.52	0.049	<0.001	0.019	0.276
	trans-10 C 18:1 + trans-11 C 18:1 0.072 0.066 0.061 0.042 0.004	<0.001	0.733	0.992
	Unsaturated, g/d De novo synthesis 3	0.85	0.76	0.76	0.60	0.040	<0.001	0.550	0.188
	Yield, g/d	2.08	2.13	2.24	1.94	0.132	0.100	0.061	0.638
	Uptake/milk output 4	1.27 5	1.33 5	1.15	1.13	0.197	0.272	0.762	0.420
	1 Duodenal infusions of glucose: G0 = 0 g/d; G1 = 443 g/d; G2 = 963 g/d; G3 = 2398 g/d.

2

Root mean squares error.

3

Table 5 .

 5 Effect of duodenal infusions of glucose on concentrations of milk metabolites (mmol/L) and their ratios (mmol/mmol).

			Glucose 1				Effect
		G0	G1	G2	G3	RMSE 2 Linear Quadratic Cubic
								P
	Metabolites (mmol/L)
	Glucose-6-P	0.024	0.034	0.033	0.044	0.010 0.079	0.674	0.467
	Citrate	9.83	7.97	7.64	7.60	0.761 0.027	0.019	0.351
	Isocitrate	0.27	0.26	0.21	0.28	0.030 0.266	0.022	0.258
	2-Oxoglutarate	0.20	0.13	0.11	0.10	0.040 0.039	0.052	0.630
	Glycerol-3-P	0.066	0.092	0.079	0.083	0.022 0.601	0.429	0.298
	6-Phosphogluconate	0.085	0.103	0.093	0.119	0.038 0.293	0.994	0.632
	L-malate	0.53	0.30	0.25	0.22	0.080 0.007	0.010	0.301
	Lactate	0.12	0.15	0.05	0.05	0.100 0.300	0.635	0.416
	Acetate	0.16	0.12	0.17	0.13	0.038 0.582	0.760	0.229
	Ratios (mmol/mmol)
	Citrate/Glucose-6-P	463	250	267	185	116.0	0.074	0.184	0.303
	Isocitrate/Glucose-6-P	12.1	8.3	7.2	7.0	2.78	0.146	0.148	0.665
	2-Oxoglutarate/Isocitrate	0.75	0.51	0.50	0.35	0.113 0.008	0.103	0.281
	Glycerol-3-P/Glucose-6-P	2.71	2.83	2.68	1.98	0.734 0.208	0.659	0.862
	6-P-gluconate/Glucose-6-P	3.82	3.44	3.38	2.73	1.706 0.472	0.929	0.915
	1 Duodenal infusions of glucose: G0 = 0 g/d; G1 = 443 g/d; G2 = 963 g/d; G3 = 2398 g/d.

2

Root mean squares error.

Journal of Dairy Science Vol.

85, No. 10, 2002

Table 6 .

 6 Effect of duodenal infusions of glucose on arterial blood concentrations of cortisol, triiodothyronine and thyroxine.

			Glucose 1				Effect
		G0	G1	G2	G3	RMSE 2 Linear Quadratic Cubic
								P
	Cortisol (ng/ml)	7.25	5.61	5.94	6.53	2.725	0.958	0.483	0.693
	Thyroxine (nmol/L)	43.0	40.1	42.0	37.5	6.76	0.336	0.947	0.642
	Triiodothyronine (nmol/L)	0.598	0.641	0.769	0.703 0.180	0.516	0.319	0.677
	1 Duodenal infusions of glucose: G0 = 0 g/d; G1 = 443 g/d; G2 = 963 g/d; G3 = 2398 g/d.

2

Root mean squares error.

Arterial fluxes (arterial concentrations × mammary plasma flow) are expressed for one half udder in g/ h for acetate and BHBA, in mmol/h for total glycerides and in µmol/h for NEFA.

Expressed in g/h for acetate and BHBA, mmol/h for NEFA and total glycerides.

Calculated from the mammary uptake of milk fat precursors (acetate, BHBA, and total glycerides) estimated over 12 h on one half udder and from the corresponding milk fat yield.Journal of Dairy Science Vol.85, No. 10, 2002

ACKNOWLEDGMENTS

The authors gratefully thank Y. Lebreton for assistance with surgeries; P. Lamberton and his team for their helpful assistance, care and feeding of cows; and M. Texier, S. Rigault, M. Ve ´rite ´and S. Kerbrat for technical assistance; and C. Hurtaud and J. Guinard-Flament for fruitful discussions in the preparation of the manuscript.