WO2012022885A1

WO2012022885A1 - Method for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, and device for implementing such a method

Info

Publication number: WO2012022885A1
Application number: PCT/FR2011/051788
Authority: WO
Inventors: Christophe Dolabdjian; Sébastien SAEZ; Matthieu Denoual; David Menard; Paul Boyer
Original assignee: Universite De Caen
Priority date: 2010-07-26
Filing date: 2011-07-25
Publication date: 2012-02-23
Also published as: FR2963110A1; FR2963110B1

Abstract

The invention relates to a method for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, including the following steps: providing a device (1) including an excitation means (2) for generating a magnetic excitation H in a test space (3) at a plurality of pulses ω₁, of index i, and a magnetic sensor (4) for measuring a magnetic field B in said test space (3); measuring a vacuum magnetic field B₀ in the absence of nanoparticles in the test space, and a test magnetic field B₁ in the presence of nanoparticles; deducing a measured magnetic magnetization M_m by M_m = (Β₁-Β₀) /μ₀; determining optimum physical characteristics suitable for minimizing, for the plurality of pulses ω_i, differences between the Fourier transform of the measured magnetic magnetization of formula (I) and a Fourier transform of the calculated magnetic magnetization of formula (II).

Description

A method for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, and a device for carrying out such a method. The present invention relates to a method for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, and to a device for implementing such a method.

Methods and devices, such as magnetoresponse, so-called fluxgate magnetometers or so-called "SQUID" magnetometers make it possible to measure magnetic characteristics of nanoparticles.

However, such devices have magnetic sensors of large size and low spatial resolution. In addition, "SQUID" magnetometers use cryogenic means. These devices are therefore complex and very expensive.

The object of the present invention is to determine the physical characteristics of nanoparticles or agglomerates of nanoparticles in a precise and less expensive manner than the known techniques.

More particularly, the invention relates to a method for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, comprising the following steps:

providing a device comprising an excitation means for generating a magnetic excitation H in a test space at a plurality of pulses ac, index i, and a magnetic sensor for measuring a magnetic field B in said test space,

measuring with said magnetic sensor for said plurality of AC pulses, a vacuum magnetic field B ₀ in the absence of nanoparticles in the test space, and a magnetic test field Bi in the presence of nanoparticles in the test space,

- deduce a measured magnetic magnetization M _m from nanoparticles by:

M _m = (Bi-Bo) / μ ₀

μο being the magnetic permeability of the void,

determining optimum physical characteristics adapted to minimize for said plurality of pulses des, differences between the Fourier transform of the measured magnetic magnetization M _m (û) i) and a Fourier transform of a calculated magnetic magnetization M (co) ), the calculated magnetic magnetization M being obtained for nanoparticles subjected to the same magnetic excitation H and depending on the physical characteristics of the nanoparticles.

Thanks to these provisions, the physical characteristics of the nanoparticles are determined by a large number of measurements, so that such a determination is very precise. In addition, the magnetic sensor used to make the magnetic field measurements is inexpensive. The process is complex to implement and inexpensive.

In various embodiments of the method according to the invention, one or more of the following provisions may also be used:

the calculated magnetic magnetization M is obtained by solving a differential equation, said differential equation being a function of parameters and of a magnetic equilibrium magnetization M _e defined by a magnetization function M _e = f (H) of the nanoparticles, said parameters and said magnetic equilibrium function M _e being a function of the physical characteristics of the nanoparticles;

the calculated magnetic magnetization M is obtained by solving a differential equation of the first order of the type:

3 (t) 1 _{,, x}

- ^ = - - (M (t) - M _e )

and τ

or :

- t is the time, - τ is a magnetic relaxation time,

- M _e is the magnetic equilibrium magnetization, defined by a magnetization function M _e = f (H) nanoparticles;

the calculated magnetic magnetization M is obtained by solving a differential equation of the second order of the type:

or :

- t is the time,

- Oo is a characteristic pulsation,

- λ is a damping coefficient,

- M _e is the equilibrium magnetic magnetization, defined by a magnetization function M _e = f (H) nanoparticles;

the magnetic equilibrium magnetization M _e is defined by:

or :

- M _s is a magnetic moment per unit volume,

V is a volume of the nanoparticles,

- k _B is Boltzmann's constant,

- T is a temperature of the nanoparticles,

- L is the Langevin function defined for all x by L (x) = coth (x) -l / x;

the magnetic equilibrium magnetization M _e is linear and defined by M _e = χ.Η, where χ is called the magnetic susceptibility of the nanoparticles;

the magnetic susceptibility χ is determined by χ = | M (0) I / H ₀ , where H ₀ is an amplitude of the magnetic excitation H, such that H (t) = H ₀ .sin (cot), and

the magnetic relaxation time τ is equal to l / co _c , co _c being a frequency of cut-off frequency for which | ( ^Ω _Α ) | ⁼ | (Ο) | / Λ / 2 with | . | module operator of a complex number;

the magnetic equilibrium magnetization M _e is defined by:

M _e = χ.Η = ^μ ° ^s .H

3 «k _g .T

or :

- χ is the magnetic susceptibility of the nanoparticles,

- M _s is the magnetic moment per unit volume of the nanoparticles,

- V is the volume of the nanoparticles,

- k _B is Boltzmann's constant,

- T is the temperature of the nanoparticles;

the calculated magnetic magnetization M is equal to:

or

Ho is an amplitude of the magnetic excitation H, such that H (t) = H ₀ .sin (cot);

the physical characteristics of the nanoparticles are chosen from magnetic susceptibility, magnetic permeability, saturation magnetic magnetization, magnetic hysteresis, magnetic relaxation time, volume, concentration, mass and size of the nanoparticles.

The invention also relates to a device for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, comprising:

an excitation means adapted to generate a magnetic excitation H in a test space of the device at a plurality of pulses ca, of index i,

a magnetic sensor adapted to measure a magnetic field B in the test space, and

in which the following steps are carried out: measuring with said magnetic sensor for said plurality of pulses ω _± , a vacuum magnetic field B ₀ in the absence of nanoparticles in the test space and a magnetic field of test Bi in the presence of nanoparticles in the test space,

- deduce a measured magnetic magnetization M _m of the nanoparticles by:

M _m = (Βι-Βο) / μο

μο being the magnetic permeability of the void,

determining optimum physical characteristics adapted to minimize for said plurality of pulses des, differences between the Fourier transform of the measured magnetic magnetization M _m (a> i) and a Fourier transform of a calculated magnetic magnetization M (co) ), the calculated magnetic magnetization M being obtained for nanoparticles subjected to the same magnetic excitation H and depending on the physical characteristics of the nanoparticles.

In various embodiments of the device according to the invention, one or more of the following provisions may also be used:

the device further comprises a processing means connected to the excitation means for driving the magnetic excitation H in the test space and connected to the magnetic sensor, said processing means implementing the steps for determining the optimum physical characteristics; nanoparticles;

the excitation means comprises Helmholtz coils comprising at least two circular coaxial coils distant from each other by a predetermined distance in a longitudinal direction, the test space consisting of a volume situated between said two coils;

the excitation means generates the magnetic excitation H in a first direction and the magnetic sensor measures the magnetic field B in a second direction substantially perpendicular to the first direction;

Other features and advantages of the invention will become apparent from the following description of an embodiment, given by way of non-limiting example, with reference to the accompanying drawings.

On the drawings:

FIG. 1 is a schematic view of a device according to the invention,

FIG. 2 is an example of a frequency curve obtained with the device of FIG. 1 and making it possible to determine the physical characteristics of nanoparticles,

FIGS. 3a and 3b show configurations of placement of a magnetic sensor with respect to a sample volume of the device of FIG. 1,

FIG. 4 is an example of determination of nanoparticle concentrations of a sample volume of FIG. 3a or FIG. 3b.

In Figure 1, a device 1 for determining physical characteristics of nanoparticles or agglomerates of magnetic nanoparticles is shown.

The nanoparticles are assemblies of atoms of which at least one of its dimensions is of nanometric size, that is to say less than 1000 nm, and preferably less than 100 nm. These nanoparticles can be agglomerated into larger assemblies. In the remainder of this description, reference would be made to only nanoparticles for simplicity of writing, but agglomerates of nanoparticles can also be used. These nanoparticles can also be functionalized, that is to say chemical compounds can be linked to these nanoparticles to associate a chemical function nanoparticles.

This device 1 comprises:

an excitation means 2 adapted to generate a magnetic excitation H in a test space 3 of the device at a plurality of pulses co, with ω = 2 f, f being a corresponding frequency of a pulse co,

a magnetic sensor 4 adapted to measure a magnetic field B in the test space 3,

a processing means 5 connected to the excitation means 2 for driving the magnetic excitation H in the test space 3 and connected to the magnetic sensor 4 to obtain magnetic field measurements from said magnetic sensor.

Advantageously, the excitation means 2 is adapted to generate a substantially homogeneous magnetic excitation H in the test space 3.

In the embodiment shown, the excitation means 2 comprises Helmholtz coils: At least two coaxial coaxial coils 2a, 2b which are spaced from each other by a predetermined distance along a longitudinal direction X are These coils are known to generate a substantially constant and homogeneous magnetic field in the longitudinal direction X between the two coils 2a and 2b. The test space 3 then consists substantially of the volume located between said two coils 2a, 2b. The excitation means 2 thus generates a magnetic excitation H substantially collinear with the longitudinal direction X.

In other embodiments, the excitation means 2 comprises a single coil around the test space 3, of cylindrical or non-cylindrical shape.

The excitation means 2 can generate an excitation magnetic type H H (t) = H ₀ · sin (cot), that is to say comprising a single co pulsation, or simultaneously comprising a plurality of Oi pulsations, of index i. Optionally, the magnetic excitation H is any or produced by a broadband signal.

The plurality of pulses Oi have frequencies fi (fi = Oi / 2π) less than 1 MHz, and preferably less than 10 kHz. Thus, the equipment used (sensors, wave generators, etc.) is inexpensive.

The magnetic sensor 4 or magnetometer can be of any type, in particular based on a Hall effect cell, a magnetoresistance GMR (for "Giant Magnetoresistance"), a magneto-impedance GMI (for "Giant Magneto- Impedance "), or other. This magnetic sensor 4 has a high sensitivity to the magnetic field.

The magnetic sensor 4 thus measures a magnetic field B resulting from the magnetic excitation H of the excitation means 2, but also magnetic excitation of the magnetic nanoparticles when they are present in the test space 3. The field Magnetic B measured is then the sum of a first contribution of the magnetic excitation H generated by the excitation means 2 and a second contribution of the magnetic excitation generated by the magnetic nanoparticles.

More precisely, the magnetic excitation generated by the magnetic nanoparticles is a resultant or sum of contributions of each elementary magnetic particle of the test space. The magnetic sensor 4 therefore only measures an overall value, a kind of spatial average on the volume of the magnetic particles.

Optionally, the magnetic sensor 4 is oriented so that it measures a magnetic field B substantially perpendicular to the longitudinal direction X, that is to say in a direction substantially transverse to the longitudinal direction X.

Thanks to this provision, the contribution of the magnetic excitation H of the excitation means 2 in the magnetic field B measured by the magnetic sensor 4 is smaller (the magnetic sensor seeing only slightly the direct magnetic field B ₀ ), and the sensitivity of the process is increased.

As shown in FIGS. 3a and 3b, the nanoparticles are contained in a sample volume 6 which may be a container, a capsule or any other container adapted to contain said nanoparticles, such as biological samples. The nanoparticles are, for example, dissolved in a liquid that fills said volume of sample 6.

Advantageously, this volume of sample 6 is placed substantially in the center of the excitation means 2, so that the magnetic excitation H in the sample volume 6 included in the test space 3 is as homogeneous as possible.

The sample volume 6 generates or induces a magnetic field represented by the field lines 7 in FIG.

In addition, the magnetic sensor 4 is advantageously placed or positioned in the test space 3 so as to have its sensitivity direction tangent to a field line 7 of the magnetic field generated by the sample volume 6.

In particular, the magnetic sensor 4 can be oriented with a sensitivity direction perpendicular to the longitudinal direction X and tangential to a field line 7 of the magnetic field generated by the sample volume 6.

With this arrangement, the contribution of the magnetic excitation H of the excitation means 2 in the magnetic field B measured by the magnetic sensor 4 is zero, and the magnetic sensor 4 measures only the magnetic field generated by the magnetic nanoparticles of the sample volume 6. The device and the method is thus more sensitive. This device makes it possible to implement a method for determining the physical characteristics of nanoparticles described by the successive steps below:

1) Using the magnetic sensor 4 for a plurality of pulsations ω, a magnetic field B ₀ is measured in the absence of nanoparticles in the test space 3;

2) Place in the test space 3, the nanoparticles whose physical characteristics are sought;

3) Using the magnetic sensor 4 for the same plurality of pulsations On a magnetic field Bi is measured in the presence of the nanoparticles in the test space 3;

4) A measured magnetic magnetization Mm is deduced from the nanoparticles by the following calculation:

M _m = (Βι-Β ₀ ) / μ ₀ (where D μο is the magnetic permeability of the void,

μ ₀ = 4π. 10 ^"7 kg, m, A ^{" 2} . s ^"2 and

5) The optimal physical characteristics of the nanoparticles are determined by minimizing, for the plurality of pulsations ω, differences between the measured magnetic magnetization M _m and a calculated magnetic magnetization M. In practice, the comparison is made in the frequency domain, and it is calculated the differences between a Fourier transform of the measured magnetic magnetization M _m (a> i) and a Fourier transform of a calculated magnetic magnetization M (coi).

The magnetic magnetization measured Mm corresponds to a mean spatial value induced by a large number of nanoparticles present within the sample. Considering that magnetizations perpendicular to the direction of the applied magnetic excitation H cancel out on average, the measured magnetic magnetization can be treated as a scalar component.

The calculated magnetic magnetization M is calculated for the same magnetic excitation H of the excitation means 2. In addition, it depends or is a function of the physical characteristics of the nanoparticles. Thus, it is possible to determine said optimum physical characteristics.

The magnetic sensor 4 is optionally a multiaxis sensor measuring a magnetic field B along several axes. Alternatively, it is composed of a plurality of single-axis sensors.

The magnetic field B ₀ in the absence of nanoparticles and the magnetic field Bi in the presence of nanoparticles are then vectors comprising the components along the different axes. Equation (1) is vector. The measured magnetic magnetization Mm and the calculated magnetic magnetization M are vectors, as well as their Fourier transforms. Thus, all the relationships and equations established in the present description can be written in a vectorial manner without this presenting any difficulty. The notation below is simplified to help reading.

According to a first embodiment of the invention, the calculated magnetic magnetization M can be obtained or calculated by solving a first-order equation of the type:

> ₌ - I _Mt) - B., ₍₂ , and τ

or :

- t is the time.

- τ is a magnetic relaxation time,

M (t) is the calculated magnetic magnetization,

- M _e is the equilibrium magnetic magnetization, defined by a magnetization function M _e = f (H) nanoparticles.

This magnetization function f (H) can be modeled using physical characteristic quantities, which represent the magnetic behavior of the nanoparticles. These physical characteristics are, for example, magnetic characteristics of these nanoparticles such as susceptibility, permeability, saturation magnetization, hysteresis, relaxation time, but also physical characteristics in general, such as the volume, concentration, mass, size of nanoparticles. Examples of determination of these physical characteristics of the nanoparticles will be given in the course of the description below.

According to a first variant, for magnetic excitation pulsations Oi H less than l / τ, the magnetic equilibrium magnetization M _e of the nanoparticles can be defined by:

M _e = χ.Η, (3)

where χ is a constant called the magnetic susceptibility of nanoparticles.

The magnetic equilibrium magnetization M _e of the nanoparticles is a linear function. The use of such a magnetic equilibrium magnetization M _e is sufficiently accurate in a field in which the magnetic excitation H is of low amplitude H ₀ , such that the magnetic nanoparticles are not magnetically saturated.

According to a second variant, for magnetic excitation pulsations Oi H less than 1 / τ, the magnetic equilibrium magnetization M _e of the nanoparticles can be defined by

or :

- M _s is a magnetic moment per unit volume,

V is a volume of the nanoparticles,

- k _B is Boltzmann's constant,

k _B = 1, 3806504.10 ^"23 JK ^{" 1} ,

- T is a temperature of the nanoparticles,

in Kelvin degrees, - L is the Langevin function defined for all x by L (x) = coth (x) -l / x.

According to a third variant, for magnetic excitation pulsations Oi H less than 1 / τ, if the magnetic excitation H is of low amplitude H ₀ , the preceding equation can be simplified (L (x) ~ x / 3):

M _e = χ.Η = ^μ ° ^s .H, and (5)

3 «k _g .T

the magnetic equilibrium magnetization M _e is again a linear function.

In all these variants, the magnetic equilibrium magnetization M _e depends on the physical characteristics of the nanoparticles. The calculated magnetic magnetization M (t) solution of equation (2) also depends on these same physical characteristics of the nanoparticles.

We now study the resolution of equation (2) to determine the calculated magnetic magnetization M (t).

In the case of a magnetic magnetization of equilibrium M _e linear, the equation of the first order (2) of the magnetic magnetization M is written:

= _ i _{M (t) + H} , t), 6) ot τ τ

Such an equation has a general solution:

t

M (t) = H (t) .e ^{t / T} .dt (7) τ to

For a simply sinusoidal magnetic excitation, of the type H (t) = H ₀ .sin (cot) of Fourier transform

H (co) = H ₀ , this equation (6) is written:

t.

M (t) = - 1.e ^{~ t / T.} [H ₀ . sin (cot) .e ^{t / T} .dt which has the approximate value:

sin (cot) - ωτ cos (cot)

M (t) * χ.Η ₀ · _Ί fy (8)

1 + (ωτ)

The Fourier transform M (co) of the magnetic magnetization M (t) makes it possible to obtain a frequency curve magnetic magnetization. This Fourier transform M (co) of the magnetic magnetization depends on the pulsation ω and also on the physical characteristics of the nanoparticles.

The Fourier transform M (co) of the magnetic magnetization approximately modulates the following value:

with I. I module of a complex number,

Figure 2 is a schematic logarithmic representation of the pulse co.

According to a first variant of resolution of equation (2), the method uses the zero-pulse limit (ω = co ₀ = 0) of the preceding module is equal to | M (O) | = χ.Η ₀ = χ.Η (θ), which makes it possible to determine the magnetic susceptibility χ of the nanoparticles:

In addition, for a cutoff pulse co _c the module is | (CO _c ) | "Χ.Η ₀ . , which allows to determine the time

V2

relaxation τ of the nanoparticles:

/ co _c with co _c pulsation or i | M (O) _c Ju

τ = 1 | = | M (, - °

2

Such a method uses the module of the Fourier transform of the magnetic magnetization M (co) for two pulsations; the zero pulse C0o and the cutoff pulse co _c . This process is simple to implement.

According to a second variant of resolution of equation (2), the method uses more than two pulsations. For a plurality of pulses i, i being an index, i = 1 to N, N number of pulses, an optimization algorithm can then compare the complex values of the Fourier transforms M (coi) of the calculated magnetic magnetization with the values Fourier transform complexes M _m (coi) of the measured magnetic magnetization. The algorithm minimizes the differences between these values to determine optimal physical characteristics C _op t nanoparticles:

or

- ||. || is a standard mathematical operator, whose operator modulates a complex number is an example, and

- C _op t is a vector including all the desired physical characteristics.

Many optimization algorithms are known and usable for the present application: least squares algorithm or LMS (for "Least Mean Square"), gradient algorithm, simplex algorithm, ... These algorithms make it possible to search for a minimum, and if possible, a global minimum of an objective function, such as that given above.

In other words, the use of such algorithms makes it possible to find the parameters (physical characteristics) of the Fourier transform of the calculated magnetization M (co) so that its frequency curve passes as close as possible to the measured values of the Fourier transform of the measured magnetization M _m (co).

The method is then advantageously implemented by a processing means 5 which has a computing and memory unit for performing such a digital resolution.

In the case of a magnetic magnetization of equilibrium M _e nonlinear, equation (2) can be solved numerically by a numerical method as explained above.

According to a second embodiment of the invention, the calculated magnetic magnetization M can be obtained or calculated by solving an equation (10) of the second order of the type: (10)

or

ωο is a characteristic pulsation,

λ is a damping coefficient, and

T is the time.

Such a damping coefficient λ is related to an energy dissipation by the magnetic nanoparticles during the course of cycles of the hysteresis of the magnetization function f (H) of the magnetic nanoparticles. The area of the hysteresis of this magnetization function is related to the amount of energy dissipated. This hysteresis of the material of the magnetic nanoparticles comes for example from displacements or modifications of magnetic domains of the material which heat the material, and cause losses.

The rest of the method and the device are identical to the arrangements described for the first embodiment of the invention. According to other embodiments of the invention, the calculated magnetic magnetization M can be obtained or calculated by solving any differential equation, having parameters that are functions of the physical characteristics of the nanoparticles.

Such a differential equation is solved analytically or numerically to find optimal physical characteristic values C _opt .

For all the embodiments of the invention, the model of calculated magnetic magnetization M of the nanoparticles depends on the physical characteristics of the nanoparticles, so that it is possible to determine the optimal physical characteristics.

As mentioned earlier, these physical characteristics are magnetic characteristics of nanoparticles or general physical characteristics. By way of illustration, FIG. 4 is a curve showing the magnetization M as a function of the concentration C in nanoparticles. The points 10 represent samples having different nanoparticle concentrations. These C concentrations are calculated by C = V / V _e , where

V is the volume of the nanoparticles, as can be determined by the process described above and equations (4) or (5),

- V _e is the volume of the sample volume 6, known a priori.

The line 11 is an approximation of the evolution of the magnetization as a function of the concentration of nanoparticles. This evolution is substantially linear and proportional.

Knowing the density of the nanoparticles, one can also determine the mass of nanoparticles in the sample 6. The method and device according to the invention can have many applications.

First, they can be directly used for the characterization of nanoparticles or agglomerates of magnetic nanoparticles.

Secondly, they can be used to determine the drug concentration of a functionalized nanoparticle solution: A drug or drug compound is chemically grafted to magnetic nanoparticles, so that determining the concentration of magnetic nanoparticles is equivalent to determining the concentration of nanoparticles. drug of the solution.

Third, they can be used to analyze the kinetics of a chemical reaction. For example, a polymer is chemically grafted to magnetic nanoparticles. During the polymerization of the polymer, the polymer chains rigidly connect to each other the Magnetic nanoparticles and measured magnetic relaxation times change. Thanks to this measurement of the evolution of the relaxation time of the nanoparticles, it is possible to follow the evolution of the polymerization of the polymer.

Claims

A method for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, comprising the steps of:

- providing a device (1) comprising an excitation means (2) for generating a magnetic excitation H in a test space (3) at a plurality of pulses ac, index i, and a magnetic sensor (4) for measuring a magnetic field B in said test space (3),

measuring with said magnetic sensor (4) for said plurality of AC pulses, a vacuum magnetic field B ₀ in the absence of nanoparticles in the test space, and a magnetic test field Bi in the presence of nanoparticles in the test space,

- deduce a measured magnetic magnetization M _m of the nanoparticles by:

M _m = (Βι-Βο) / μ ₀

μο being the magnetic permeability of the void,

2. Method according to claim 1, wherein the calculated magnetic magnetization M is obtained by solving a differential equation, said differential equation being a function of parameters and a magnetic equilibrium magnetization M _e defined by a magnetization function M _e = f (H) nanoparticles, said parameters and said magnetic equilibrium function M _e being a function of the physical characteristics of the nanoparticles.

The method of claim 1 or claim 2, wherein the calculated magnetic magnetization M is obtained by solving a first order differential equation of the type:

3 (t) 1 _{,, x}

- ^ = - - (M (t) - M _e )

dt τ

or :

- t is the time,

- τ is a magnetic relaxation time,

The method of claim 1 or claim 2, wherein the calculated magnetic magnetization M is obtained by solving a second order differential equation of the type:

- 3 ² (t) - = ₂ ,,,, "dMÎt)

-ω. M t - M _e ) - 2.λ .- ^

dt ² dt

or :

- t is the time,

- coo is a characteristic pulstation,

- λ is a damping coefficient,

5. Method according to one of claims 2 to 4, wherein the equilibrium magnetic magnetization M _e is defined by:

or :

- M _s is a magnetic moment per unit volume,

V is a volume of the nanoparticles,

- k _B is Boltzmann's constant,

- T is a temperature of the nanoparticles,

- L is the Langevin function defined for all x by L (x) = coth (x) -l / x.

6. Method according to one of claims 2 to 4, wherein the magnetic equilibrium magnetization M _e is linear and defined by M _e = χ.Η, where χ is called the magnetic susceptibility of the nanoparticles.

The method of claim 6 wherein:

the magnetic relaxation time τ is equal to l / co _c , co _c being a frequency of cut-off frequency for which | (<B _C ) | = | M (O) / I with I. I module operator of a complex number.

8. Method according to one of claims 2 to 4, wherein the equilibrium magnetic magnetization M _e is defined by:

or :

- χ is the magnetic susceptibility of the nanoparticles, - M _s is the magnetic moment per unit volume of the nanoparticles,

- V is the volume of the nanoparticles,

- k _B is Boltzmann's constant,

- T is the temperature of the nanoparticles.

9. Method according to one of the preceding claims, wherein the calculated magnetic magnetization M is equal to:

or

- H ₀ is an amplitude of the magnetic excitation H, such that H (t) = H ₀ .sin (cot).

10. Method according to one of the preceding claims, wherein the physical characteristics of the nanoparticles are selected from magnetic susceptibility, magnetic permeability, saturation magnetic magnetization, magnetic hysteresis, magnetic relaxation time, volume, the concentration, mass and size of the nanoparticles.

Apparatus (1) for determining physical characteristics of nanoparticles or agglomerates of nanoparticles, comprising:

an excitation means (2) adapted to generate a magnetic excitation H in a test space (3) of the device at a plurality of pulses ca, of index i,

a magnetic sensor (4) adapted to measure a magnetic field B in the test space (3), and

in which the following steps are carried out:

measuring with said magnetic sensor (4) for said plurality of AC pulses, a vacuum magnetic field B ₀ in the absence of nanoparticles in the test space and a test magnetic field Bi in the presence of nanoparticles in the test space,

- deduce a measured magnetic magnetization M _m of the nanoparticles by:

M _m = (Bi-Bo) / μ ₀

μο being the magnetic permeability of the void,

12. Device according to claim 11, further comprising a processing means (5) connected to the excitation means (2) for controlling the magnetic excitation H in the test space and connected to the magnetic sensor (4), said processing means (5) implementing the steps for determining the optimal physical characteristics of the nanoparticles.

13. Device according to claim 11 or claim 12, wherein the excitation means (2) comprises Helmholtz coils comprising at least two circular coils (2a, 2b) coaxial and distant from each other. a predetermined distance in a longitudinal direction (X), the test space (3) consisting of a volume located between said two coils (2a, 2b).

14. Device according to one of claims 11 to 13, wherein the excitation means (2) generates the magnetic excitation H in a first direction and the magnetic sensor (4) measures the magnetic field B in a second direction substantially perpendicular to the first direction.

15. Device according to one of claims 11 to 14, wherein the physical characteristics of the nanoparticles are selected from magnetic susceptibility, magnetic permeability, saturation magnetic magnetization, magnetic hysteresis, magnetic relaxation time, volume, concentration, mass and size of nanoparticles.