A Mohammadpour
email: a.mohammadpourshoorbakhlou@imperial.ac.uk

Net force in the absence of external forces in systems with retardation: Emphasis in electromagnetism

This paper demonstrates retardations can have significant implications for the kinetics and physical properties of electromagnetic systems, which should not be overlooked in general. It is shown that two electrical objects, besides attraction or repulsion, may exhibit an uncommon behavior resembling a leader-follower interaction, where one object follows the other. It is shown that the sum of forces within a system can be non-zero under certain conditions and the absence of external forces. Additionally, further analysis shows that accounting for retardation can reveal an uncommon oscillatory resistive reactance at high frequencies in circuits. Finally, an experimental setup is proposed to measure the one-way speed of light. In this proposal, clock synchronization is not needed, potentially allowing the identification of anisotropy of the speed of light.

Introduction

The origin of this work stemmed from a thought experiment. In this experiment, two magnetic bodies are imagined: one on Earth and the other on the Moon. They produce a magnetic field for a period of time and exert force on each other. If one of the objects suddenly loses its magnetism, there will be a delay before the other object detects this change. During this delay, the second object will continue to experience a force for the duration equivalent to the time it takes for an electromagnetic wave to travel from the Moon to Earth. In contrast, the other object ceases to experience any force. Claiming that the loss of force for both objects is immediate after the loss of magnet implies that information can travel instantaneously and is equivalent to having instant internet on the Moon with zero latency. On the other hand, accepting the delay in reaction leads to the conclusion that one object carries load while the other does not. A question immediately arises: What implications can arise from changing the state of electrical objects when there is a delay in the reaction? This leads to other thought experiments detailed in this paper.

In electromagnetism, the term "retardation" refers to the delay in the propagation of electromagnetic effects from a source to an observer. This delay is a crucial factor that should be incorporated into the laws of physics, as it can lead to significant consequences. One consequence of retardation in electromagnetism is illustrated in a thought experiment where two objects at a distance do not experience equal and opposite forces, seemingly defying Newton's Third Law. Several scientific literatures mention the violation of Newton's third law [START_REF] Cornille | Review of the application of newton's third law in physics[END_REF][START_REF] Pinheiro | On newton's third law and its symmetry-breaking effects[END_REF][START_REF] Ivlev | Statistical mechanics where newton's third law is broken[END_REF]. The question of whether this law universally holds ambiguously is answered. However, this alleged violation can be avoided by positing that the Third Law is locally and instantaneously applicable at the point of interaction between charged particles and the fields, where equal and opposite momentum is exchanged. Sebens, in his work [START_REF] Sebens | Forces on fields[END_REF], has taken a step further, proposing that Newton's Third Law can be preserved by assuming that fields themselves can carry force. It is stated that "The force from the electromagnetic field on matter is balanced by an equal and opposite force from matter on the field". Regarding distant interactions, it can be said that the reaction force depends on the state of the field and particles at their respective points. Consequently, no predetermined statements or laws should be made about them without considering the specific conditions at those locations. For instance, it is impossible to make any definitive statements about the magnitude and direction of the force acting on an object at a distance based on the force experienced at one's current time and projecting forward. This is because the force at any given moment depends on the specific conditions and interactions at that distance, which may not immediately depend on the other one's current situation, position, and time. Conclusively, while Newton's Third Law may hold in local interactions, it is not accurate to universally apply it to objects at a distance.

Despite the controversies, one may be less concerned about Newton's Third Law as long as the conservation of momentum within the field is maintained [START_REF] Feynman | The feynman lectures on physics[END_REF][START_REF] Griffiths | Introduction to electrodynamics[END_REF], and the exchange of force and momentum aligns with the fundamental laws of physics. In this work, in section 4, the Lorentz force is used for the applied force of electromagnetic fields, and the fields are derived from Maxwell equations, which inherently satisfies the conservation of momentum.

In section 5, the discussion centers on the oscillatory reactance of a circular loop, presenting a unique perspective on incorporating retardation effects. This illustration reveals a new aspect of the electromagnetic properties of circuits. To the author's best knowledge, this specific property has not been addressed in any existing literature until now.

In section 6, a new method for measuring the anisotropy of one-way speed of light is proposed. In the proposed method, clock synchronization is not needed, and the measurement is sensitive to the speed of light for both the outgoing and coming waves.

State change in retarded systems and actionreaction consequences

Let us consider a scenario in which two individuals, designated as A and B, engage in a shooting exchange. Both shooters, A and B, are situated at a distance d from each other within an inertial frame. They initiate gunfire simultaneously at time t = 0, with the bullets traveling at a velocity c. If either shooter is struck by the opposing party's bullet, they will experience a force of magnitude f in the direction opposite to their fired shot. Assume the bullet travel time is t 0 = d c . Shooter A, unaware of shooter B's actions, is instructed to lie down after firing, a maneuver that takes a duration t l where t l < t 0 . Under these circumstances, the outcome of the shooting incident is as follows: shooter B's reaction to shooter A's action results in receiving a force f . Conversely, shooter A does not experience a reactionary force due to shooter B's action. However, if the change in shooter A's stance occurs over a longer duration, t l > t 0 , then both A and B will experience forces of equal magnitude but in opposite directions. There is an equal and opposite momentum exchange between each shooter and the bullet they shoot, which happens nearly instantly. In the same way, there is an equal and opposite momentum exchange between the bullet and the victim. Now consider two electrical objects, A and B, positioned at a distance d. Following the principles of relativity, the fields generated by these objects cannot propagate faster than the speed of light c. Assume that both elements possess an equal electrical charge at time t = 0. It takes a duration t ret = d c for the fields from each side to reach and interact with the other's electrical charges. If the electric charge of object A dissipates within a time t l where t l < t ret , it will not experience any force when the electric field of B reaches A. In contrast, element B will experience an electrical force. There is an equal and opposite exchange of momentum between the electric charge and the field at the point of the electrical charge. This exchange of momentum results in a force being applied to the electrical charge.

These thought experiments clearly demonstrate that under certain circumstances, mainly due to the delay in the exchange of momentum in a distributed system, the sum of reactionary forces at specific time instances is not zero.

It can be stated that the reaction force experienced by any entity within a system at any given point is dependent on the state of the entity itself, as well as the state of other actors at a retarded time t ret (the time at which the state information is sent from the actor to the entity through the corresponding field and is available for the observer). In a distributed system, the force at point p is the function of the state of p and the state of the other entities in the system 3 and can be written as

F p (x, y, z, t) = n i=1,i̸ =p F p i (S p (x p , y p , z p , t); S i (x i , y i , z i , t ret = t -r i /c)) (1)
where r i is the distance between the point p and entity i, and r i is not necessarily an Euclidean distance (in specific conditions, the electromagnetic wave does not travel along the shortest path). In the classical Newtonian framework, F p n (S p ; S n ) = -F n p (S n ; S p ) and consequently, the sum of all internal forces in a system should be zero when there is no external force on the system. However, this is not true in general according to Equation (1), i.e. F p n ̸ = -F n p . If the speed of the wave c is an inherent property of the physics of the system and is related to the propagation of the field and, consequently, the propagation of momentum, then neglecting retardation means overlooking a critical aspect of the system's physics. Therefore, not accounting for the force in a retarding manner concerning the state of the system may violate the underlying physics.

Let's assume two electrical objects with their specific geometry and electrical state are positioned at a distance r from each other, and their state changes through time and also Let's assume the applied force is obtained thorough 1,2

F A B = D(r AB ) |S A ||S B |g A (t)g B (t -r/c)r AB ; F B A = D(r BA ) |S A ||S B |g B (t)g A (t -r/c)r BA ; (2)
where state of object A and B are defined by

S A = |S A |g A (t); S B = |S B |g B (t); (3)
where D(r) relates the force magnitude to the relative distance, g A and g B defines the dependency of the element state A and B to the time. S A and S B define the state of the electrical element and the unit distance vector are defined as rBA = -r AB . F A B and F B A will cancel out each other at time t 0 under the condition that there is almost no change in electromagnetic state of A and B during the time interval [t 0 -r/c, t 0 ] interval, i.e. g A (t 0 ) ≈ g A (t 0 -r/c) and g B (t 0 ) ≈ g B (t 0 -r/c). The sum of force acts on A and B at time t can be approximated by Taylor series expansion and neglecting second and higher order terms of r/c as

F A B + F B A ≈ D(r AB ) |S A ||S B | 0 -g A (t 0 ) dg B (t) dt t0 -g B (t 0 ) dg A (t) dt t0 r c r AB (4)
1 In general, the force between two electrical objects may not conform to this specific form. However, in this paper, the examples presented have this general form. The primary objective of this work is to explore the possibility of generating net force without external excitation, and the generality of the formulation is not intended to encompass all possible scenarios 2 An evident example can be two electromagnetically active celestial bodies separated by a distance r Equation ( 4) clearly shows when the rate of the change in the state of A and B is small with regard to r/c, the sum of the forces is nearly zero.

3 Time-state profiles and net force

Square wave

Without delving into the specifics, let's assume we have two electrical objects exerting forces on each other at a distance r. The goal is to define an electrical state-time profile such that net force is produced. The selection of objects A and B could be any charged plate or magnetic element, resulting in a force-time profile as described in Equation [START_REF] Pinheiro | On newton's third law and its symmetry-breaking effects[END_REF]. A simple choice would be to select g A and g B such that

g A (t)g B (t -r/c) = 1; g B (t)g A (t -r/c) = -1; (5)
If the time function is chosen such that g A (t) = g(t) and g B (t) = g(t + ϕ), it can be shown mathematically that g(t) is a periodic function with a constant magnitude |g(t)| = 1. This function changes its sign at the midpoint of its period, resembling a square wave (for the proof, see Appendix A). The period of the function is T = 4 × r c and the phase shift is ϕ = r c . The time profile of g A (t) = g(t) and g B (t) = g(t + ϕ) is depicted in Figure 1. In this case, the sum of all forces applied on two elements will be |F | = 2 D(r AB ) |S A ||S B |, which is twice the force exerted on each object when they are stationary at their maximum state value.

Sinusoidal wave

The square wave profile is not always suitable for real-world applications, as sharp load changes often result in very high impedance in circuits and inductors. Consequently, a very high voltage is required to overcome this impedance. In addition, the natural oscillation of most circuits is sinusoidal, and using square waves for these types of applications leads to imbalanced response to square wave excitation. Now lets consider a sinusoidal function for g A (t) = sin(ωt) and g B (t) = sin(ωt + ϕ). After some mathematical calculations, the sum of all internal forces will be

F = D(r AB ) |S A ||S B |sin(ϕ) sin(ω r/c)r AB (6)
If the angular velocity is given by ω = 2πf and f is the frequency, the Equation (6) will be maximized when

f = n c 2 r + c

Leader-follower interaction

In Equation (2), when g A = 1 and g B = -1, with these values remaining constant over time, objects A and B exhibit mutual attraction. Conversely, if they possess the same sign, they repel each other. In the case of square-wave and sinusoidal state-time profiles, different behaviors can be observed depending on the value of ϕ. For instance, object B follows object A if ϕ = π/2 in a sinusoidal wave. In contrast, if ϕ = -π/2, the situation reverses with object A following object B. Similarly, in a square-wave scenario, if ϕ = r/c, object B follows object A, and with ϕ = -r/c, object A follows object B. In these cases, both objects experience forces in the same direction, resulting in non-zero net force. This occurs without any external force being applied, illustrating a "leader-follower" behavior where one object dynamically responds to changes in the state of the other. Recent experiments, as reported in [START_REF] Mandal | A molecular origin of non-reciprocal interactions between interacting active catalysts[END_REF], illustrates such non-reciprocal force interactions between enzyme molecules where molecules experience nonreciprocal interactions akin to prey-predator(leader-follower) behavior in the absence of external forces. It's important to clarify that the intention here is not to explain this phenomenon with the present work. Rather, the goal is to highlight the existence of such non-reciprocal behavior as a phenomenon in the natural world.

Implication of assuming zero net force

On the contrary, it can be demonstrated that if the states of elements A and B change over a sufficiently long period of time3 , such that g A and g B are not constant. Consequently, there exists no time interval during which they are constant (including zero) 4 , the only possible scenario in which the sum of the forces on the two elements equals zero is by assuming that the speed of the traveling wave c is infinite.

To prove that, we also need to assume that the state of objects A and B are generally not dependent, i.e., there is no state entanglement for two objects at every time and position. To achieve a net force of zero on both elements, the following relation must hold in Equation ( 2)

g A (t)g B (t -r/c) = g B (t)g A (t -r/c) (8)
since the states of A and B are independent, Equation 8 can be written as

g A (t) g A (t -r/c) = g B (t) g B (t -r/c) = cte (9)
Equation ( 9) should hold for r = 0 and consequently cte = 1 and finally we have

g A (t) = g A (t -r/c); (10)
Since Equation [START_REF] Zhang | Special relativity and its experimental foundation[END_REF] should hold for all possible distances r between elements A and B, it follows that either g A is constant over time or r c → 0. The first conclusion contradicts our initial assumption; thus, the only possible scenario is when the speed of c is infinite.

Retarded field response to electromagnetic source

In this section, the electromagnetic field response to an infinitesimally small current source through retardation is investigated using Maxwell's electromagnetic equations. Some examples show that the force acting on currents can obey the general form presented in (2) or a combination of these forms in integral format. The general Maxwell wave equation for an element with charge density ρ and current density J can be expressed as

∇ 2 E - 1 c 2 ∂ 2 E ∂t 2 = 1 ϵ 0 ∇ρ + µ 0 ∂J ∂t (11)
and

∇ 2 B - 1 c 2 ∂ 2 B ∂t 2 = -µ 0 ∇ × J ( 12
)
where ϵ 0 is the permittivity and µ 0 is the permeability of the vacuum. The speed of light in vacuum is given by ϵ 0 µ 0 = 1/c 2 . The electric and magnetic field is presented by E and B, respectively. In [START_REF] Jefimenko | Causality, electromagnetic induction, and gravitation: a different approach to the theory of electromagnetic and gravitational fields[END_REF], it is demonstrated that the solution to the inhomogeneous wave equation ( 12) can be correlated with its cause. More specifically, it is related to the state of the cause at a retarded time, as expressed through the following:

E = - 1 4πϵ 0 ∇ s ρ(x s , y s , z s , t s ) + 1 c 2 ∂J(xs,ys,zs,ts) ∂ts ret r dV s (13)
and

B = µ 0 4π [∇ s × J] ret r dV s (14)
The integral over dV s represents the summation of interactions encompassing the entire volume of source points. When the retardation symbol, [] ret , is applied to a function, it indicates the state of the source element located at coordinates (x s , y s , z s ) as observed from the point (x, y, z). This observed state is a function of the distance r between these two points and the speed of the wave c. For example

[f (x, y, z, t)] ret = f (x s , y s , z s , t -r/c) (15)
In the subsequent examples presented in this paper, our focus will be solely on the electric objects that possess a current and no electric charge. Consequently, we consider the charge density ρ to be zero. Jefimenko [START_REF] Jefimenko | Causality, electromagnetic induction, and gravitation: a different approach to the theory of electromagnetic and gravitational fields[END_REF], has reformulated equations ( 13) and ( 14) in absence of charge density ρ as

E = - 1 4πϵ 0 1 r ∂J ∂t s ret dV s (16)
and

B = µ 0 4π [J] ret r 3 + 1 r 2 c ∂J ∂t s ret × rdV s (17)
The first term on the right-hand side of Equation (17) typically represents the classical Biot-Savart law. The second term on the right-hand side of this equation is typically negligible in everyday observations and can be difficult to measure at low-frequency AC currents. Furthermore, even in high-frequency circuits, the amplitude of currents is usually tiny, and this term remains insignificant. However, for the current work, the significance of this term is shown in the following examples and sections.

Two parallel wires

Consider a scenario involving two parallel wires spaced a distance d apart. Suppose there is a current I flowing through these wires. Additionally, we assume the presence of two sources for each wire, positioned sufficiently far from the cables, such that the phase difference in the alternating current (AC) along the wire is negligible due to the effect of retardation. The magnetic field around the wire is cylindrically symmetric. The magnetic field at a distance d around a wire carrying an alternating current of |I| sin(ωt + ϕ) can be derived from Equation (17). This derivation involves integrating over the wire by summing the effects of each wire segment. To simplify the calculation, the distance from the segment is chosen as the integration variable, and it is performed from the minimum distance to positive infinity for half of the wire.

B(wt, ϕ, d) = µ 0 |I|d 2π +∞ d sin(ω(t -r/c) + ϕ) r 2 √ r 2 -d 2 + ω c cos(ω(t -r/c) + ϕ) r √ r 2 -d 2 dr (18)
The first integral represents the short-range interactions, which decay rapidly with increasing distance from the wire. In contrast, the second integral corresponds to long-range interactions, with its magnitude being proportionally affected by the current's frequency. The right-hand rule determines the direction of the magnetic field at each point. For two wires A and B with current I A and I B that are out of phase with phase difference ϕ, the force on each wire would be 18) is the term commonly utilized in static electromagnetic equations, while the effect of the second term is typically overlooked. It's important to note that the net force approaches zero at low frequencies, aligning with daily observations and classical predictions. Figure 3 illustrates the effect of different distances on the net force of two infinite wires. The frequency of the wires is selected as C 2d0 and d 0 = 1m. It can be observed the long-range force decays very slowly over different distances due to the fact that the effect of long-range force accumulates over infinite wire, and the more extended portion of wire can contribute to magnifying the magnetic field.

F A B = B B (wt, ϕ, d)|I A |sin(ωt) F B A = -B A (wt, 0, d)|I B |sin(ωt + ϕ) (19)

Two concentric parallel loops

Let's consider two parallel loops, each with a radius R, positioned at a distance d from each other, with their connecting centers orthogonal to the loops. Let's also assume the alternating current I is passing through the loop. The magnetic

B(wt, ϕ, r, R) = µ 0 |I| 4π 2π 0 sin(ω(t -|r-R| c ) + ϕ) |r -R| 3 R ′ × (r -R) dθ + µ 0 ω|I| 4πc 2π 0 cos(ω(t -|r-R| c ) + ϕ) |r -R| 2 R ′ × (r -R) dθ (20)
assuming that the normal of the loop is in Z direction R and R ′ is given by

R = R [cos(θ), sin(θ), 0] R ′ = R[-sin(θ), cos(θ), 0] (21)
For two Loops of A and B with current I A and I B that are out of phase with phase shift ϕ, the force on each loop would be:

F A B = 2π 0 r ′ A B × B B (wt, ϕ, r A B , R)|I A |sin(ωt)dθ A F B A = 2π 0 r ′ B A × B A (wt, 0, r B A , R)|I B |sin(ωt + ϕ)dθ B ( 22
)
where

r A B = [R cos(θ A ), R sin(θ A ), d] r B A = [R cos(θ B ), R sin(θ B ), -d] (23)
Figure 4 illustrates the net force between two circular wires separated by various distances, operating at different frequencies, and carrying a current of 1000 amps with a phase shift of ϕ = -π/2. Figure 2(a) depicts the net force between two loops spaced 0.2m apart for various frequencies. The first term in Equation (20) results in a nearly constant force amplitude across different frequencies. However, the second term causes an amplification in the net force. Figure 2(b) demonstrates the net force between two loops at various distances at a frequency of ∼ 375 MHz. The force arising from the first term of the integral in Equation ( 20) is a short-range term whose amplitude decays rapidly over distance and is negligibly influenced by frequency changes. In contrast, the second term is responsible for the long-range force and has a milder decay over distance.

Oscillatory resistive reactance of circuits at high frequencies

The retardation in electromagnetic fields results in significant consequences, which are often overlooked in the daily application of electrical instruments.

One of these phenomena that can be magnified at high frequencies in circuits is oscillatory resistance and inductive reactance caused by the produced electromotive force (EMF). According to Maxwell-Faraday's law of induction, the EMF is given by:

ε = c E • dl = A ▽ × E • dA = - A ∂B ∂t • dA (24)
Assuming that the integration is done over a frame that is stationary, as it is extensively explained in [START_REF] Jefimenko | Causality, electromagnetic induction, and gravitation: a different approach to the theory of electromagnetic and gravitational fields[END_REF], the equation (24) can be written as:

∂B ∂t = µ 0 4π ▽ × 1 r ∂J ∂t s ret dV s (25)
considering the below identity for retarded fields(proof can be found in [START_REF] Li | Jefimenko made easy: Electromagnetic fields through retardation[END_REF])

∇ × [J] ret r = 1 r 3 [J] ret + 1 r 2 c ∂J ∂t s ret × r (26)
and using the equations ( 25) and ( 26), the emf produced in a closed circuit with current I is given

ε = - µ 0 4π A c 1 r 3 ∂I ∂t s ret + 1 r 2 c ∂ 2 I ∂t 2 s ret dl × r.dA (27)
Where r represents the radial distance from the closed loop at which the EMF is evaluated. In this context, the current i is expressed as |I|sin(ωt + ϕ) where |I| denotes the amplitude of the current, ω is the angular frequency. Hence, Equation ( 27) becomes

ε = -|I|ω [L (ω) cos(ωt) + R react (ω) sin (ωt)] (28)
The EMF has two parts: one acts as inductance and the other as resistance to the current. Inductance reactance opposes the change in the current, and resistive reactance opposes the current direction. It can be demonstrated that these elements exhibit behaviors contrary to their typical roles at certain frequencies. The L (ω) and R react (ω) in Equation ( 28) is given by

L (ω) = µ 0 4π A c 1 r 3 cos (ωr/c) + ω r 2 c sin (ωr/c) dl × r.dA (29)
and

R react (ω) = µ 0 4π A c 1 r 3 sin (ωr/c) - ω r 2 c cos (ωr/c) dl × r.dA (30)
It is evident that assuming an infinite speed of light (c) or no retardation in the system, the reactive resistance component (R react ) vanishes, and the inductance (L) takes its classical form. This conclusion is also valid at low frequencies with a finite speed of c. Figure 5 illustrates a circular loop's resistive and inductive reactance with a radius of R = 20cm for various frequencies. The wire's radius is chosen a = 2cm. The integrals in Equation ( 29) and (30) are calculated from the center to the inner radius of the wire R -a. Owing to the complexity involved in calculating the current distribution inside the wire, which requires detailed calculations beyond the scope of this work, only the approximation of the EMF at the inner radius is calculated here 5 .

The resistive and inductive reactance oscillate at various frequencies and can get negative values. Negative resistive reactance suggests that the energy stored in the field aids the current flow in its direction. This negative resistance arises from the delay in the magnetic field, which, at specific frequencies, generates an EMF opposing the current, while at other frequencies, it supports the current.

On one-way speed of light test

The results presented in previous sections are based on the assumption of the isotropic speed of light in both directions between two stationary electrical objects in their respective frames. According to Zhang [START_REF] Zhang | Special relativity and its experimental foundation[END_REF], in specific experiments, an anisotropic speed of light assumption yields results compatible with the isotropy of the one-way speed of light as long as the isotropy and constancy of the two-way speed of light are maintained in anisotropy assumption. However, it can easily be demonstrated that in retarded systems, the anisotropy can lead to distinct outcomes, such as different net forces. The measurement of the speed of light in one direction has been a subject of controversy and a significant challenge, primarily due to the conventions of clock synchronization and the philosophical debates surrounding these conventions [START_REF] Janis | Conventionality of Simultaneity[END_REF][START_REF] Reichenbach | The philosophy of space and time[END_REF]. Additionally, the methodologies employed to synchronize spatially separated clocks may rely on knowing the one-way speed of light or physics laws that assume the isotropic speed of light, which cannot be used for anisotropy tests [START_REF] Mansouri | A test theory of special relativity: I. simultaneity and clock synchronization[END_REF][START_REF] Selleri | Noninvariant one-way velocity of light[END_REF][START_REF] Anderson | Conventionality of synchronisation, gauge dependence and test theories of relativity[END_REF].

In this work, a test experiment is suggested to test the anisotropy of the speed of light without the need for clock synchronization. This test is based on the measurement of the net force between two electric objects, where the anisotropy of the speed of light affects the measured force.

It is assumed that two electrical objects, A and B, are positioned in an inertial frame, and their electromagnetic states change alternatively over time. There is no initial phase synchronization between the states of A and B, and their phases are independent. A and B can rotate( or reposition) within the inertial frame, and the net force on both can be measured (See Figure 6). However, force measurements are carried out when A and B are at rest after relocating. Although A and B can rotate, the positions of the sources feeding A and B are fixed in the inertial frame and not under accelerated movement. Consequently, no change in the clocking of sources A and B should occur. The fact that A and B move in the inertial frame will not affect the clocks at rest in the frame, and our signal generators are independent of the position of A and B. The sources A and B may have a phase difference of ϕ s . However, there is no consensus on the value of ϕ s as different observers may not concur on its relative value. This discrepancy arises because different observers experience delays due to their varying distances from the sources and potential anisotropy in the speed of light. The only way that all observers reach a unanimous agreement on the value of ϕ s is to know the speed of light in all directions. Despite this, the procedure will not depend on the measurement of the phase shift ϕ s (initial synchronization is not needed). The only agreement for all observers is that changing the phases of sources will change the observed net force. The phase difference between A and B, denoted as ϕ, differs from ϕ s due to the distances of the sources from their respective receivers. The distance from the source of A to receiver A is selected equal to the distance from the source of B to receiver B. This symmetry is deliberately chosen for simplification.

To further explain the procedure, let's assume c AB defines the speed of the wave traveling from A to B and c BA is the speed of the wave traveling from B to A (indicating anisotropy in the speed of light). The sources have an angular velocity of ω, and their states sinusoidally change, and there is a phase shift ϕ s between the two sources. The state of A is proportional to the time profile of + ϕ), and that of B is determined by sin(ωt). Where ϕ is the phase shift between A and B, the phase shift at the angle of θ = 0 • and θ = 180 • can be related to the phase shift of sources and the delay resulting from their distance from the sources through

ϕ 0 • = ϕ s + ω(d s /c BA -d s /c AB ) ϕ 180 • = -ϕ s -ω ((d s + d)/c BA -(d s + d)/c AB ) = -ϕ 0 • -ω(d/c BA -d/c AB ) (31)
Where d is the distance of objects from each other. Although time and phase synchronization may not be practically possible, observers in the inertial frame can agree on the angular velocity of objects and sources A and B at rest. The force exerted by object B on A is denoted as F A B , and F B A is the force exerted by A on B.

F A B = D(r AB ) |S A ||S B |sin(ωt + ϕ) sin(ω(t -r/c BA ))r AB ; F B A = D(r BA ) |S A ||S B |sin(ωt) sin(ω(t -r/c AB ) + ϕ)r BA ; (32)
With some mathematical calculation, the net force on two objects considering they have phase shift ϕ 0 • is given by

F net = 1 2 D(d) |S A ||S B | [cos(ϕ 0 • + ωd/c BA ) -cos(ϕ 0 • -ωd/c AB )] [cos(2ωt + ϕ 0 • -ωd/c AB ) -cos(2ωt + ϕ 0 • -ωd/c BA )] r AB (33)
When light speed is isotropic, there is no oscillatory component over time, which allows us to observe a fixed net force without oscillation. Conversely, when light speed is anisotropic, by filtering out the oscillatory component, the fixed part can be measured and is given by

|F | f ixed 0 • = 1 2 D(d) |S A ||S B | [cos(ϕ 0 • + ωd/c BA ) -cos(ϕ 0 • -ωd/c AB )] r AB (
34) By positioning A and B in opposite directions by rotating the system and keeping it running, now A and B are replaced, and they will be in position of each other but with phase shift ϕ 180 • . If the light speed is anisotropic in direction, different setups lead to different magnitudes of force at different angles. For the current setup and at ϕ 180 • the fixed component of the net force can be Using the equations (34), (35), and (36), a set of exact solutions can be obtained for t AB and t BA . However, to ensure the uniqueness of these solutions, it can be considered, from Equation (36), that t AB and t BA are bounded and the solutions can be limited to a specific range. The experimental setup described above ensures that the sources remain stationary to prevent any effect on their timing due to the theory of relativity. However, it is possible to rotate the sources within the rotating frame very slowly, allowing a more straightforward test setup while introducing a negligible change in clocks.

Conclusions

In this paper, it is demonstrated that changes in the electrical state can induce a net force on retarded systems, even in the absence of external forces. The magnitude of this net force depends on the rate of change in the state and, under specific conditions, can result in measurable forces. It is also shown that the impact of retardation extends beyond the kinetics of the systems. For instance, retardation effects can lead to oscillatory reactance. Additionally, we have presented arguments that anisotropy in the speed of light can lead to variations in the calculated force. This phenomenon could potentially be utilized to test whether light behaves anisotropically in different frames of reference.

Figure 1 :

 1 Figure 1: A schematic of the state-time profile of two electrical objects, resembling a square wave with the period of T = 4 × r c . The above and below functions have a phase difference of ϕ = r c .

Figure 2

 2 Figure2illustrates the net force per meter between two infinitely long wires separated by a distance of 1 meter and carrying a current of 1000 amps for various frequencies Figure2(a) and phase shifts ϕ Figure2 (b). As observed in Figure2(a), the force resulting from the first term of the integral in equation (18) is the term commonly utilized in static electromagnetic equations, while the effect of the second term is typically overlooked. It's important to note that the net force approaches zero at low frequencies, aligning with daily observations and classical predictions.Figure3illustrates the effect of different distances on the net force of two infinite wires. The frequency of the wires is selected as C 2d0 and d 0 = 1m. It can be observed the long-range force decays very slowly over different distances due to the fact that the effect of long-range force accumulates over infinite wire, and the more extended portion of wire can contribute to magnifying the magnetic field.

Figure 2 :

 2 Figure 2: The net force between two parallel wires at a distance of d 0 = 1m. (a) The variation in net force across different frequencies, with a phase difference set at ϕ = π/2. (b) The change in net force over different phase shifts, for a frequency of ω = πc/2d 0 . The value of ϕ can be the relative phase difference of the first or second wire with respect to the other.

Figure 3 :

 3 Figure 3: The net force between two parallel wires at different distances with a phase difference set at ϕ = π/2 and for a frequency of ω = πc/2d 0 and d 0 = 1m. The plot starts from distance d = 0.25m

Figure 4 :

 4 Figure 4: (a) The net force between two circular wires at a distance of d 0 = 1m. The variation in net force across different frequencies is plotted (b). The net force between two circular wires at different distances for a frequency of ω = πc/2d 0 and d 0 = 0.2m. The plot starts from a distance d = 0.1m. In both cases, the phase difference is set at ϕ = -π/2, and the radius of the circuit is R = 0.25m.

Figure 5 :

 5 Figure 5: The variation in reactance of a circular loop over different frequencies: (a) inductive reactance, (b) resistive reactance

Figure 6 : 2 D

 62 Figure 6: Diagram of the experimental setup for testing anisotropy of speed of light. Two electrical objects, A and B, are positioned within an inertial frame. These objects can rotate within the frame, and net forces are measured when they are at rest after relocation. The sources are fixed in the inertial frame and are connected to receivers. The sources are providing

r ϕ = nπ + π/2where n ∈ N 0 ;(7)The force tends toward zero if ω r/c ≈ 0, indicating that the net forces will be insignificant at low frequencies.

It should be longer than the time required for the information to travel from A to B and vice versa.

g A and g B can be zero a finite number of times on any time interval, and this does not affect the generality of the proof

To make this approximation valid, it can be assumed that the wire is composed of strands with a significantly smaller conductor radius. The voltage distribution across these strands is arranged to generate a concentric current, characterized by a greater current distribution at the center and less current near the circumference.

Acknowledgments

This work is conducted without any external financial support. During the preparation of this work, the author has used Chatgpt and Grammarly tools for grammar and readability enhancement, and no additional text has been generated by these tools. After using these tools, the author reviewed and edited the content as needed and takes full responsibility for the content of the publication.

Appendix A

Let's start from this point that the equation ( 5) should hold for an arbitrary time shift of r/c. By shifting the first row, we obtain

Similarly, by applying the same shift to the second row of (5), we can conclude that

This implies that the function changes its sign over a time shift of 2 × r/c, and it can be easily deduced that the period is T = 4 × r/c. The equation (A.2) by two times shifting can be expressed as

Given that g B (t) = g A (t+ϕ), and by choosing ϕ = r/c while taking into account (A.2), from (5) we can conclude that |g A | = 1 which represents a square wave with period T = 4 × r/c.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Call for collaboration

Ideas exist for developing and designing high-frequency electromagnetic systems capable of producing net forces. I want to extend an invitation for future collaboration and support to benefit the science community and humanity.


Sofia Arris-Roucan
email: sofia.roucan@univ-pau.fr

Fionn Mcgregor

Antonin Fabbri

Céline Perlot

Towards the determination of carbon dioxide retention in earthen materials

Keywords: Indoor air quality, Earthen materials, Carbon dioxide adsorption, Adsorption enthalpy, Thermogravimetry, Differential Scanning Calorimetry

Thanks to its hygroscopic properties, raw earth can passively buffer indoor humidity. The high sorption properties of clay minerals present in earth also attract other molecules, such as carbon dioxide (CO2), this can potentially lead to indoor CO2 buffering. It would improve air quality in buildings and their resilience towards mechanised ventilation systems. However, there is currently no protocol to evaluate this potential. This paper aims to present a innovative methodology based on thermogravimetric (TG) and calorimetric (DSC) analysis to characterise the interactions between clay and CO2 at the microscopic scale by focusing on the dry state. Results show a CO2 adsorption capacity of 134 mg/kg at 5000 ppm at 35°C associated with a reversible physisorption process and provides first evidence of a passive CO2 regulation capacity of the indoor air. Determination of the adsorption enthalpies as a function of the amount adsorbed revealed high values (from -74 kJ/mol to -40 kJ/mol) which are characteristic of a strong energetic heterogeneity of the material.

Introduction

Indoor Air Quality (IAQ) has become an issue of concern in the building and construction sector for several years [START_REF] Awada | Occupant health in buildings: Impact of the COVID-19 pandemic on the opinions of building professionals and implications on research[END_REF]. This has resulted in an increase in the number of scientific publications on the subject [START_REF] Abouleish | Indoor air quality and COVID-19[END_REF], but also in the recognition in the UN Agenda 2030 as one of the 17 Sustainable Development Goals.

It has been shown that poor indoor air quality can have negative health effects, ranging from simple discomfort to severe pathologies [START_REF] Jones | Indoor air quality and health[END_REF]. However, in the context of climate change and resource depletion, significant changes have taken place in the way buildings are designed [START_REF]Global Status Report for Buildings and Construction: Towards a Zero-emission, Efficient and Resilient Buildings and Construction Sector[END_REF], making them more airtight than older buildings in order to improve their energy efficiency [START_REF] Pérez-Lombard | A review on buildings energy consumption information[END_REF]. This air-tightness, coupled with the widespread use of synthetic building materials, contributes to higher pollutant concentrations indoors than those found outdoors [START_REF] Kirchner | État de la qualité de l'air dans les logements français. Environnement[END_REF]. This is of particular concern in high-income countries where people spend more than 90% of their time indoors, increasing exposure time to pollutants and associated health risks [START_REF] Klepeis | The National Human Activity Pattern Survey (NHAPS): a resource for assessing exposure to environmental pollutants[END_REF]. Thus, while buildings are primarily intended to enhance the well-being of occupants, they are associated with a range of hazards, which may be exacerbated by climate change [START_REF] Vardoulakis | Impact of climate change on the domestic indoor environment and associated health risks in the UK[END_REF]. It is therefore necessary to recognise the strong interdependence between indoor environmental quality, energy efficiency and health.

Among the many pollutants present in indoor air, carbon dioxide (CO 2 ) has been the subject of numerous recent studies. The presence of this gas in indoor air is mainly due to human metabolism and its concentration is therefore strongly related to occupancy levels. While it used to be considered harmless to health, several studies have shown that even at concentrations considered to be low, CO 2 can have a harmful impact on health, particularly on fragile populations (children, elderly, people with chronic illnesses) [START_REF] Jacobson | Direct human health risks of increased atmospheric carbon dioxide[END_REF] [START_REF] Azuma | Effects of low-level inhalation exposure to carbon dioxide in indoor environments: A short review on human health and psychomotor performance[END_REF]. These effects can range from cognitive impairment to chronic disease depending on the concentration and time of exposure [START_REF] Zhang | Physiological responses to elevated carbon dioxide concentration and mental workload during performing MATB tasks[END_REF] [START_REF] Zhang | The effects of elevated carbon dioxide concentration and mental workload on task performance in an enclosed environmental chamber[END_REF]. While the proposed limit for indoor air is 1000 ppm [START_REF]The standards For Ventilation And Indoor Air Quality. Technical report[END_REF], in inadequately ventilated areas its concentration often exceeds this threshold, which has been measured in schools in Europe [14] [START_REF] Szabados | Association of parent-reported health symptoms with indoor air quality in primary school buildings -The InAirQ study[END_REF].

One of the strategies being considered for the regulation of CO 2 in air is the buffering of these concentrations through the adsorption of CO 2 on adsorbent surfaces [START_REF] Shayegan | A review on an emerging solution to improve indoor air quality: Application of passive removal materials[END_REF] [START_REF] Mata | Indoor Air Quality: A Review of Cleaning Technologies[END_REF]. Adsorption is the accumulation of particles (adsorbate, here CO 2 ) at a surface (adsorbent, here earth-based construction material) due to the operation of surface forces. The reverse process is called desorption. Adsorption can be of two kinds: physisorption in which the forces involved are intermolecular forces (van der Waals forces) and chemisorption which results from chemical bond formation [START_REF] Gold | The IUPAC Compendium of Chemical Terminology: The Gold Book[END_REF].

By addressing indoor air quality through a passive strategy, buildings could provide healthier environments for their occupants, while also reducing their energy consumption and improving their overall resilience [START_REF] Gall | New directions: Potential climate and productivity benefits from CO 2 capture in commercial buildings[END_REF]. The difficulty in implementing this technique lies in the fact that indoor air concentrations are very low, that the material must be able to regenerate and thus release all or part of the CO 2 when the concentration drops and finally the presence of humidity in the air must not hinder the stability of the material. Several materials have been studied for this purpose, including: amine-impregnated silicas [START_REF] Rajan | Advanced buffer materials for indoor air CO 2 control in commercial buildings[END_REF] [START_REF] Krishnamurthy | Amine-Based Latex Coatings for Indoor Air CO 2 Control in Commercial Buildings[END_REF], metal organic frameworks [START_REF] Seok | Moisture-tolerant diamine-appended metal-organic framework composites for effective indoor CO2 capture through facile spray coating[END_REF], zeolites [START_REF] Shih | CO2 Adsorption by Y-Type Zeolite Impregnated with Amines in Indoor Air[END_REF] [START_REF] Kim | CO2 adsorption on zeolite 13X modified with hydrophobic octadecyltrimethoxysilane for indoor application[END_REF] and activated carbons [START_REF] Wang | Development of high-performance adsorbent using KOH-impregnated rice husk-based activated carbon for indoor CO2 adsorption[END_REF]. However, the difficulty of preparing these materials, their cost, as well as their stability at high humidity could compromise their widespread use.

On the other hand, raw earth, a material widely used in particular for interior plastering, could present ideal characteristics for this type of application. Earthen materials are known for their hygroscopic properties on the regulation of indoor humidity and their thermal inertia for heat storage [START_REF] Mcgregor | The moisture buffering capacity of unfired clay masonry[END_REF] [START_REF] Woloszyn | The effect of combining a relative-humidity-sensitive ventilation system with the moisturebuffering capacity of materials on indoor climate and energy efficiency of buildings[END_REF].

Through the presence of clay minerals and their microstructure they could regulate the concentration of CO 2 on the same principle as humidity. In addition, they are fulfilling a truly circular economy approach without affecting natural resources, the earth being sourced from waste (mainly excavation works) from the construction sector and can potentially be infinitely reused [START_REF] Morel | Earth as construction material in the circular economy context: practitioner perspectives on barriers to overcome[END_REF]. Some relevant publications on indoor air pollutants retention had already concluded that one of the most promising passive removal materials for indoor ozone control are clay-based plasters and clay paints [START_REF] Darling | Field-to-laboratory analysis of clay wall coatings as passive removal materials for ozone in buildings[END_REF] [START_REF] Darling | Impacts of a clay plaster on indoor air quality assessed using chemical and sensory measurements[END_REF]. However, there is currently no protocol for determining whether raw earth could passively regulate CO 2 even though some research is ongoing [START_REF] S Roucan | Indoor CO 2 buffering potential of clay-based building materials[END_REF][32] [START_REF] Santos | Earth, Gypsum and Cement-Based Plasters Contribution to Indoor Comfort and Health[END_REF].

In this context, the objective of this paper is to present a new experimental methodology, based on thermogravimetry (TG) and differential scanning calorimetry (DSC), to evaluate the potential of raw earth to buffer CO 2 . The experimental set-up, consisting of the TG-DSC apparatus, a gas mixer and a wet gas generator, allows the representation of a wide range of indoor environments. The measurement of the mass and heat flow variations enables the characterisation of the process taking place. As a first step in the understanding of this phenomenon are the interactions between the injected gas and the clay material, therefore only the dry state of the material will be presented and discussed in this paper.

Material and Methods

Earth characterisation

A single earth was selected for this study coming from the "Auvergne Rhône-Alpes" region in southeastern France. It was collected from a centenarian rammed earth construction located in the city of "Dagneux" [START_REF] Al | Efficient stabilization with hydraulic binders of local earth for building construction applications[END_REF] and referenced as DAG. This earth was selected for this study because it has been extensively studied in previous work ([35] [36]) and its properties are well known. Furthermore, coming from a real construction, this earth is suitable to be used as a building material. The selected earth was crushed and sieved at 5 mm. Its particle size distribution was determined according to the French Norms NF EN ISO 17892-4 [37]. The different properties determined are presented in table 1, in particular it revealed that the earth contains 18% of clay (particles lower than 2 µm). As clay minerals play a key role in adsorption mechanisms, to get more information about the activity of this clayey fraction the methylene blue value (M B) was used as an indicator since methylene blue is preferentially absorbed by clays and organic materials. The methylene blue tests were carried out on the fraction below 80 µm of this earth following the French Standard NF P 94-068 [START_REF] Nf P | Soils : Investigation and testing -Measuring of the methylene blue adsorption capacity of a rocky soil -Determination of the methylene blue of a soil by means of the stain test[END_REF]. The ratio between M B and the clay content in the 0-80 µm fraction of the earth enables the deduction of the clay activity index (A cb ). Results given in Table 1 indicate that the clay fraction of DAG has normal activity.

Then, powder X-ray diffraction (XRD) was used to identify the mineralogical composition of the earth. It was performed on oriented aggregates using two preparations: air dried or natural and after glycolation. Swelling clay minerals were detected in DAG. This seems in accordance with the measured clay activity index. The detected swelling clay minerals were characterized as interlayered illite/smectite minerals.

Finally nitrogen adsorption/desorption was performed to investigate the small pores. Approximately 3 g of earth (fraction below 2mm) was subjected to one nitrogen intrusion-extrusion cycle at a constant temperature of 77 K. This cycle consisted in the pressurisation of gaseous nitrogen up to the saturation value followed by depressurisation back to the initial value. Throughout the cycle, the amount of intruded nitrogen was continuously measured to determine the isothermal adsorption and desorption curves (see Fig. 1). The BET theory in the range of 0.05 to 0.35 p/p 0 was applied to determine the specific surface area of the 0-2mm fraction which is found to be 13 m 2 /g, typical values for pure clay minerals are in range of 5-20 m 2 /g for kaolinite, between 100 and 200 m 2 /g for illite and less than 1000 m 2 /g for semctite [START_REF] Fripiat | Surface Properties of Alumino-Silicates[END_REF]. The shape of the isotherm corresponds to a type II according to the IUPAC classification [START_REF] Thommes | Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF] which represent a non-porous or macroporous material. The presence of a type H3 hysteresis is usually given by non-rigid aggregates of plate-like particles such as clay minerals that expand during the cycle. It is therefore difficult to asses the pore size distribution since there is no rigid skeleton, the interlayer space of clay minerals can expand and adopt various size. Throughout the study, it is assumed that it is mainly the clay minerals in the earth that interact with the CO 2 , as is the case with water. To maximise the phenomena, the choice was made to focus on the fine fraction of the earth, below 80 µm which presents the highest surface area. The properties of the fraction below 80 µm are very similar to the total fraction: the 0-80 µm fraction contains 22% of sand and fine gravel, 59% of silt and 19% of clays. The nitrogen adsorption/desorption on that fraction and the application of the BET theory reveals a specific surface of 15.5 m 2 /g. Therefore, to work on this fine fraction, the earth was sieved at 80 µm with the help of water. It was then heated to a maximal temperature of 95 °C for drying until constant mass was reached. Until testing, dried samples were stored in a box with silica gel.

Experimental set-up for carbon dioxide retention measurement

The general principle of the device is based on the coupling of a simultaneous thermal analysis with a wet gas generator and a CO 2 /air mixer. It allows the study of materials in a wide range of indoor like conditions by precisely controlling the relative humidity and CO 2 concentration. The general objective of the experiment is to qualify and quantify CO 2 and water retention capacities of an earthen material. For that purpose mass variation and heat flow of samples are measured as a function of the CO 2 concentration, relative humidity and temperature. The set-up is composed of: gas cylinders, gas mixer to select the desired CO 2 concentration, pressure regulator, wet gas generator, TG-DSC apparatus, CO 2 sensors as shown in Fig. 2. The operation and role of each device is detailed below. MFC is 0 to 0.5 l/min. This system can produce a mixture of air and CO 2 between 0 and 50000 ppm.

It is particularly stable in producing low concentrations of CO 2 , around 1500 ppm, representative of those found in indoor air. The desired CO 2 concentration is calculated from the respective flow rates of the two mass flow controllers, which are themselves controlled via the associated Communicator Burkert software. At the outlet of the mixer, a controlled leakage flow generated by a manual flowmeter allows the flow rate to be reduced from 200 ml/min to 50 ml/min for the wet gas generator inlet while maintaining a sufficient inlet pressure. Part of the gas flow leaves the circuit and ends up in the beaker, allowing the flow to be reduced without changing the concentration of the mixture. A wet gas generator is then connected to carry out experiments in a humid environment. Inside the apparatus: the dry inlet gas is splitted in two lines, one remaining dry and the other one saturated with water. The two gas streams are mixed in a chamber. To reach the desired relative humidity, the respective flow rates are adjusted to obtain the required gas mixture. However, as we only consider the dry state in this study, the wetsys is programmed at 0%RH and only the dry line is opened. The gas is then transferred to the TG-DSC device with a flow rate of 50 ml/min.

The Setaram ® SENSYS Evo TG-DSC is a thermal analyser used for thermogravimetric (TG) and differential scanning calorimetry (DSC) measurements. The TG part is composed of a symmetrical beam balance connected to two columns, one for reference containing either nothing or an inert material, the other one containing the sample to be measured. The alumina crucibles containing the sample and the inert material are hung on each side and introduced in the columns. The symmetrical balance compensates the buoyancy effect, resulting in a very sensitive TG determination. The calorimetry part is a Tian-Calvet calorimeter which is a heat-flow type with a differential assembly. Heat-flow calorimeters use several thermocouples connected in serie (e.g. a thermopile) insuring thermal conduction between the calorimetric cell and its surroundings to measure the heat flow. Here each column (reference and measure) is surrounded by a thermopile. Heat is measured in all directions. The differential assembly eliminates the effect of residual thermal disturbances when they arrive in phase on the two thermopiles connected in opposition. This type of apparatus possesses high sensitivity, which is particularly suitable for microcalorimetric measurment (energy ≤ 110 mJ). In addition, to avoid steam condensation, a programmable thermostatic bath keeps the oven walls at a temperature above the dew point of the water vapor. For temperature conditions close to ambient, the experiment cells is regulated with this thermostatic bath.

Finally, a Vaisala ® CO 2 sensor is used to measure the CO 2 concentration of the gas mixture. Here it is placed at the outlet of the TG-DSC device.

Test procedure

For this study, it was chosen to work with the driest material possible in order to focus on the interactions between CO 2 and earth only, without water. The humidity of the experiment is set at 0.5%RH, the lowest possible value for the wet gas generator. The CO 2 concentration is chosen between 1500 ppm and 10000 ppm. The temperature of the experiment (inlet gas and measuring cell) is set at 35°C, the oven needs a sufficient temperature delta from the ambient to keep the heating power stable. This value may be considered high for the temperatures encountered in indoor air and knowing that adsorption is favoured at low temperatures, this choice will have an impact on the adsorbed CO 2 .

At the beginning of each experiment, the material is collected in the silica gel box in which it was stored. Then, the desired sample mass is introduced into the apparatus (usually 110 mg if not specified). The tests are divided into three stages. The first stage consists of a minimum of three days of stabilisation, the sample being swept by an air flow without CO 2 at 0.5% RH. Then, successive CO 2 concentration levels of 5 hours minimum are applied. Finally, to observe the reversibility of the process, the concentration is reduced to its initial state and applied for a minimum of 5 hours. The duration of 5 hours was chosen on the basis of preliminary studies. During this study, another protocol was also used, consisting of successive cycles of adsorption at CO 2 concentrations of 5000 ppm and desorption in between. The tests are carried out in steady state, the measuring cell is considered as an open system and is continuously swept by the gas mixture. The balance is itself swept by a protective gas (dry air).

Throughout the experiment, the acquisition time is set to 1 point every 2 seconds for the first and third stage and 1 point every 0.2 seconds for the second stage.

3 Results

Carbon dioxide adsorption and desorption results

Results are presented for a single sample, subjected to eight concentration steps of 12 hours: four steps of increasing CO 2 concentrations (1500, 3000, 5000, 10000 ppm) and the same steps with decreasing CO 2 concentrations. The temperature was maintained constant at 35°C. The results will be presented in two parts: the Thermogravimetry (TG) results and the Differential Scanning Calorimetry (DSC) results.

Thermogravimetry results

The Fig. 3 shows the recorded TG signal in µg and the corresponding CO 2 concentrations in function of time.

For each increase of concentration, a rapid gain in mass is observed, followed by a slower gain until stabilisation of the mass signal. This mass gain reflects the retention of CO 2 in the material. For each concentration level, a relatively stable mass is reached after about 10 hours. When the concentration is decreased, a similar phenomenon is observed: a quasi-instant loss of mass followed by a slower loss trending towards a stabilisation of the mass. It denotes the release of the retained CO 2 , which would indicate that all or part of the process is reversible. The reversibility of the process is characteristic of From these results, the maximum CO 2 retention capacity Q of the material can be calculated for each concentration, expressed in mg of adsorbed CO 2 per kg of material. It is calculated from the relative mass variation of the sample after and before the concentration step and divided by the initial reference sample mass. According to the curves, the results are: 60 mg/kg for 1500 ppm, 97 mg/kg for 3000 ppm 132 mg/kg for 5000 ppm and 214 mg/kg for 10000 ppm.

Differential scanning calorimetry results

The The exothermic peaks associated with a mass gain seem to increase in intensity with higher CO 2 concentration. Adsorption is an exothermic process [START_REF] Saha | Insight Into Adsorption Thermodynamics[END_REF], making it thermodynamically favorable which is consistent with these results. These heat flows correspond to the physisorption of CO 2 on the surface of the material. For each decrement of CO 2 concentration we observe an endothermic peak associated To go further on that point, the associated enthalpies will be calculated in section 4.2.

Reversibility between adsorption and desorption cycles

Previous experiments seem to indicate a reversible physisorption process. In order to confirm this hypothesis, a cyclic variation of CO 2 concentration was applied three times, each cycle corresponding to 9 hours at 5000 ppm and then 9 hours without CO 2 . The TG results are presented in Fig. 5. The maximum retention capacity, Q, is found to be 128 mg/kg for the first cycle, 110 mg/kg and 108 mg/kg for the second and the third cycle respectively. Despite a slight decrease, over the 3 cycles, the adsorption capacity remains high and we will assume here that this decrease is negligible. Thus, these results tend to support the hypothesis that the adsorption is mainly driven by reversible physisorption of CO 2 on the clay minerals. However, a clear distinction can not always be made between chemisorption and physisorption as intermeditate cases may exist such as adsorption involving strong hydrogen bonds or weak charge transfer.

Validation of the experimental procedure

Comparison with a modified Dynamic Vapor Sorption instrument

In order to confirm the robustness of the experimental set-up developed with respect to the measured mass gains, the thermal analysis was coupled to a DVS (dynamic vapour sorption). It will allow the comparison of the mass variations due to CO 2 adsorption/desorption data measured simultaneously on As the minimum in the DVS is 1%RH then it is a limiting factor for the Wetsys, which is then set to 1%RH and not 0.5%RH as before. The protocol applied was 5 hours of exposition at 5000 ppm and 5

hours of exposition at 0 ppm. The results are given in Fig. 6. The graph shows the relative mass gain in mg per kg for the TG-DSC (black) and the DVS (grey) as a function of time. The two samples in the two devices appear to behave exactly the same way. It should be noticed that this behaviour appears slightly different from the one observed in the previous section. Indeed, there is a rapid gain in mass followed by a small loss of mass before reaching equilibrium, which was not present in the previous results. This difference can be explained by the presence of water on the adsorption sites, which has an impact on the quantity and kinetics of CO 2 adsorption. The mass measurements are similar between the devices and confirm the validity of both the TG-DSC device and the use of the DVS device.

Repeatability of the experiments

In order to confront the repeatability of the measurements, the same protocol was repeated on 3 different samples of DAG earth with the same initial mass (110mg). The protocol consist of 5 days of stabilisation at 0 ppm, 5h of exposition to CO 2 level at 5000 ppm and a return to 0 ppm during 5h. The results are shown in Fig. 7, the graph a. shows the mass variation versus time of the three tested samples and the The tests were carried out on the 6th of May 2022 (test 1), the 4th of July 2022 (test 2) and the 26th of August 2022 (test 3) and will be compared with the experiment presented in section 3.1. Fig. 7 highlights a very similar behaviour for the three tests, especially test 1 and test 3 being almost identical.

The slight difference with test 2 can be explained by the intrinsic variability of the material, as earth is a natural material and the composition of this sample may have differed slightly from the other two tests.

However, the overall behaviour of the sample 2 remains similar in all respects, with a slight increase in mass variation, that is reflected in the heat flow. The values of the maximum retention capacity Q are: 132 mg/kg for test 1, 146 mg/kg for test 2, 128 mg/kg for test 3, which is similar to the value (132 mg/kg) determined for the previous test in the same conditions in section 3.1. From these results the mean value of Q can be calculated as: m = 1 n n i=1 m i with m i being the experimental data of the

s = 1 n-1 n i=1 (m i -m) 2 and results in: Q = m ± st √ n = 134 ± 5 mg/kg

Influence of the sample mass

The previous experiments were all carried out with 110 mg of material in the crucible of 170 µL, corresponding to a full crucible. Given the low density of the material in the crucible (ρ d = 0.64g/cm 3 ), the hypothesis was made that the CO 2 diffused perfectly inside the crucible and thus that the whole material reacts. In order to confirm this hypothesis experiments were repeated with different initial masses in the crucible: 40 mg associated with a nearly empty crucible, 60 mg associated with a slightly more than half filling and 110 mg for a full crucible. The results are given in Fig. 8. Figure 8 a. presents the TG signal and figure 8 b. the corresponding heat flows (with a 1h-zoom around the concentration change) for the three initial masses of material tested. As expected, Fig. 8 shows that the greater the initial mass of the sample is, the greater the mass gain is. This means that more CO 2 is adsorbed when more adsorption sites are available. This indicates that the CO 2 is diffusing within the porous material and that most of the volume of material is active. However, when analysing the results in mg of adsorbed CO 2 per kg of material it can be seen that the maximum adsorption capacity seems to increase while there is less sample in the crucible. Indeed, for 110 mg Q = 132 mg/kg, for 60 mg Q = 142 mg/kg and for 47 mg Q = 155 mg/kg. This can lead to three hypotheses:

• Either when the crucible is full, the lower part of the material at the bottom of the crucible does not react because the CO 2 does not diffuse to the bottom.

• Or when the crucible is full, drying is not complete in 5 days and water remains in the pores, blocking adsorption sites that are therefore not available. Whereas when the crucible is not full, the whole material is dried.

• Or we are in the uncertainty range of the measurements and this difference is only a measurement artefact.

Anyway, the difference remains quite small, and it will be assumed to be negligible. Therefore, thanks to the comparison with the DVS, the methodology for the determination of the mass of CO 2 adsorbed on DAG was validated. The repetition of the experiments made it possible to calculate the errors in the uptake capacity measurements and to prove the repeatability. Finally, the robustness to the initial mass of material was tested. The whole experimental protocol is validated.

Discussion

Carbon dioxide adsorption isotherm

Carbon dioxide uptake capacity

The results of section 3.1 concerning the CO 2 retention capacity Q in mg/kg and CO 2 concentration in ppm can be converted into mmol/g and kPa to allow an easier comparison with literature data. This results in: at 0.12 kPa (1500 ppm) the amount of CO 2 adsorbed is 0.001 mmol/g , at 0.3 kPa (3000 ppm) the amount adsorbed is 0.002 mmol/g, at 0.5 kPa (5000 ppm) it is 0.003 mmol/g and at 1 kPa (10000 ppm) it is 0.004 mmol/g. The uncertainty calculated before of 5 mg/kg becomes 0.0001 mmol/g.

To the best of our knowledge, no study has yet investigated the behaviour of raw earth and clay minerals with CO 2 at the usual CO 2 pressures of indoor air. The studies we found fall into two categories: studies on pure clay minerals at high pressure of CO 2 (usually for carbon capture and storage) and studies on transformed materials at low pressure representative of indoor air.

As expected, the results found here are much lower than those of the first group of papers, mainly because our material is less pure (only 19% of clay minerals in DAG) and the pressures investigated are lower. For example, Chen et al. [START_REF] Chen | CO2 capture by kaolinite and its adsorption mechanism[END_REF] [44] mentioned an adsorption capacity for raw kaolinite at 25°C and 1 bar between 0 and 0.07 mmol/g. Several authors also reported adsorption capacities for montmorillonite between 0.2 mmol/g at 45°C and 1 bar and 0.5 mmol/g at 10°C and 1 bar [START_REF] Gómez-Pozuelo | CO2 adsorption on amine-functionalized clays[END_REF] [START_REF] Stevens | Preparation and CO2 adsorption of diamine modified montmorillonite via exfoliation grafting route[END_REF].

Although the test temperatures are different and known to affect adsorption, the range of variation can be considered small and does not affect the order of magnitude of the adsorption results. The trend should remains the same. The impact of temperature should be assessed to confirm this statement.

Our results are also lower than those found in the second group of papers. These studies mainly focus on materials transformed in laboratories such as zeolites, metal-organic frameworks (MOF) or activated carbons. The table 2 gathers the main results obtained. The highest adsorption capacities (2.8 mmol/g for a MOF at 60%RH and 1000 ppm [START_REF] Seok | Moisture-tolerant diamine-appended metal-organic framework composites for effective indoor CO2 capture through facile spray coating[END_REF]) are found to be associated with materials with low regenerability (e.g the capacity to release the CO 2 retained at ambiant condition of temperature and humidity). Rajan et al. [START_REF] Rajan | Advanced buffer materials for indoor air CO 2 control in commercial buildings[END_REF] reported adsorption capacities ranging from 1.42 to 1.62 mmol/g at 0% RH and 3000 ppm for MOF, zeolite and silica-supported amine (PD-TEPA) powders, but their regeneration capacity was very low (30 % for the amines) or even nul for zeolite and MOF. According to the literature, it seems that some amines have a good regeneration capacity as analysed by Krishnamurthy et al. [START_REF] Krishnamurthy | Amine-Based Latex Coatings for Indoor Air CO 2 Control in Commercial Buildings[END_REF], but the cost as well as the stability of these materials to hygrothermal conditions of indoor air may be a limiting factor to their wide use as wall coating.

Thus, the adsorption capacity of DAG earth at different CO 2 concentrations is much lower than the capacities of other materials reported in the literature. It should be kept in mind that the material studied is a natural and unprocessed material and that it can be applied in large quantities on the wall and ceiling surfaces of a room, which would compensate for its relatively low adsorption capacity. For example, plasters made from the same earth in previous studies ( [35] [36]) had an apparent density between 1.5 and 1.9 g/cm 3 for a formulation containing 40% earth sieved at 2 mm. Reducing this percentage to the fraction studied here (< 80µm) and for a plaster thickness of 1 cm would give between 5 and 7 kg/m 2 of our material per square meter of wall.

Modelling the adsorption isotherm using Henry's law

As the experiments were carried out at low pressure (less than 1.2 kPa), the isotherm of CO 2 adsorption on DAG can be analysed using Henry's law. This empirical law translates the fact that at low pressure the adsorbed molecules are independent of each other. It can be expressed as: n = k h p. Where n represents the specific surface excess amount and k h the Henry's law constant. Thus, at sufficiently low pressures, n should vary linearly with the equilibrium gas pressure. It was originally defined to study the amount of gas dissolved in a liquid, but it reflects the general principle that can be applied to gas adsorption: when the adsorbate concentration tends to 0, the amount adsorbed also tends to 0.

In figure 9a there is the fitting of the experimental data with Henry's law, the results give a constant of 0.005 mmol/(g*kPa) with an R 2 of 0.90. This value is very low compared to those reported for other adsorbents such as: modified zeolites [START_REF] Arefi Pour | Adsorption separation of CO 2 /CH 4 on the synthesized NaA zeolite shaped with montmorillonite clay in natural gas purification process[END_REF] or pillared clays [START_REF] Garcés | Comparative Study of the Adsorption Equilibrium of CO 2 on Microporous Commercial Materials at Low Pressures[END_REF] which are in the range of 0.012 to 0.05 at 293 K and 310 K respectively. These results are in line with the values obtained for CO 2 uptake, i.e.

much lower compared to chemically modified materials. Even though the experiments are carried out at very low pressures where Henry's law should apply, from the figure it seems that the fitting is not very consistent. As stated in Rouquerol et al. [START_REF] Françoise Rouquerol | Adsorption by powders and porous solids: principles, methodology and applications[END_REF], the hypothesis needed for the application of Henry's law can be easily verified by plotting the isotherm on a semi-logarithmic scale. In figure 9b one can see the isotherm plotted on a semi-logarithmic scale. It appears clearly that the plot is not linear. At the beginning of the curve, a convex deviation could be attributed to the effect of either surface heterogeneity or microporosity in the interlayer space of clay minerals. Indeed, some specific adsorbent-adsorbate interactions can be associated with energetic heterogeneity. This case was identified by Rouquerol et al. [START_REF] Françoise Rouquerol | Adsorption by powders and porous solids: principles, methodology and applications[END_REF] and the adsorption energies should help to establish whether this bend is associated with adsorption on a small high-energy fraction of the surface or whether adsorbate-adsorbate interactions are detectable over the linear range. Anyway, this indicates the need for another form of approximation for the isotherm at low concentrations.

Thus, modelling the adsorption isotherm with Henry's law has shown the existence of non-linear phenomena at low pressures. These phenomena can be explained either by the entry of CO 2 into the micropores of clay minerals (interlayer space for example) or by CO 2 -CO 2 interactions or by a strong surface heterogeneity (presence of adsorption sites with more energy than others). of the background noise of the signal during integration, the choice was made to narrow the integration boundaries as much as possible while keeping the major part of the signal, this procedure is detailed in fig. 10. To do this, the maximum value of HF was calculated (10a: "HF max"). Then, 95% of this value was subtracted from the peak to reach the line "95% HF max" (10a: "95% HF max"). The intersection between this line and the HF curve determined the upper and lower boundaries for the integration.

Adsorption and desorption enthalpies

Finally, a time interval was subtracted to the lower boundary to include the beginning of the heat flow peak in the calculations. The signal obtained between the two boundaries corresponds to figure 10b.

In other words, integration was performed between the beginning of the peak and 95% of its maximum value. The integration result (10b: " HeatF low ") was then divided by the corresponding mass uptake (10b :"n a "). The same procedure was applied for the determination of the enthalpy of desorption.

The enthalpies have been calculated for all experiments conducted between 0 and 5000 ppm and are given in Table 3

Impact of the adsorbed quantity on the enthalpy

In order to follow the evolution of the enthalpy as a function of the amount of CO 2 adsorbed, the calculation of the adsorption and desorption enthalpies were repeated between the successive CO 2 concentration steps of the experiment in section 3.1 ( 1500, 3000, 5000 and 10000 ppm). The results are given in Figure 11, the differential enthalpy of adsorption (left) or desorption (right) between two states was calculated for the successive CO 2 steps.

The covering can be defined as the surface occupation by the CO 2 molecules. It increases with the amount adsorbed. From the graph it can be seen that ∆h ads is higher for the first range of concentrations and progressively decreases towards -40 kJ/mol as the coverage increases. Theoretically, if the concentration of CO 2 continues to increase, the values of adsorption enthalpy should tend towards the enthalpy of vaporisation of CO 2 (-17 kJ/mol at 288 K), but this comparison is difficult to make at 35°C because the triple point of CO 2 has passed and no change of state between the gas and liquid phase can take place. In all cases, the first values of -74 kJ/mol are very high and the last values of about -40 kJ/mol are in agreement with the values found in the literature for physisorption. The high initial values can be associated with the interaction of adsorptive molecules with highly active surface sites and/or their entry into narrow micro-pores. The reason that the enthalpy of adsorption decreased with increasing coverage is due to the progressive covering of the thermodynamically favorable sites, such that the most favorable sites are filled first, followed by the remaining sites. Going back to the Henry's law analysis, it can be All the analyses carried out in the discussion section show that the phenomenon is dominated by physical interactions (even if a sharp distinction is not possible and desired because intermediate cases exist). Secondly, at low pressures, the heterogeneity of the material surface leads to the adsorption of the first CO 2 molecules on very energetic sites (or micropore), followed by the adsorption on less energetic sites.

Conclusions and perspectives

The aim of this paper was to present a new experimental methodology based on thermogravimetry coupled with calorimetric analysis to evaluate the CO 2 retention capacities of raw earth. The coupling of the TG-DSC apparatus was done with a gas mixer capable of producing an air flow with low CO 2 concentration representative of indoor air and with a wet gas generator to represent several relative humidity. For the dry state: the set-up and experimental procedure were validated through repeatability testing and coupling to a DVS, while the robustness of the device was demonstrated with respect to the initial masses of materials tested. Considering the results obtained, at 5000 ppm the dry raw earth material has a maximum CO 2 retention capacity of about 134 mg/kg and an almost complete regeneration capacity at 35°C. The almost complete reversibility of the process shows that the CO 2 is physisorbed, most likely on the surface of the clay minerals present in the material. The determination of the enthalpies of adsorption allowed to characterize precisely the processes involved. At 5000 ppm the enthalpy of adsorption is -60 kJ/mol and the one of desorption is 55 kJ/mol on average, the value we found is higher than the enthalpies associated with physisorption which are between 20 and 40 kJ/mol.

However, when the enthalpies were examined as a function of the quantity adsorbed, it was found that ∆h ads varied from -70 kJ/mol to -40 kJ/mol when the quantity adsorbed was increased. This indicates the presence of a strong energetic heterogeneity of the material or the entry of CO 2 in micropores, which is confirmed by the non-linearity of the isotherm at low pressures. These results are in agreement with the fact that raw earth is a natural and highly heterogeneous material.

As indoor air also contains water vapour, it is crucial to determine the role of water in the CO 2 retention and release process. Research is underway to analyse the phenomena at at higher relative humidity (between 50%RH and 80%RH) in order to understand the possible competition between the water and CO 2 molecules adsorbed. This work provides the first values of CO 2 retention capacity for raw earth and thus provides first evidence of the potential passive regulation capacity of this material. A better understanding of the mechanisms will allow us to better understand the benefits of raw earth for using it in a passive strategy to improve IAQ and can therefore lead to its wider uses by the construction and building sector.

Figure 1 :

 1 Figure 1: Nitrogen adsorption desorption isotherm at 77 K for 3 g of DAG earth

Figure 2 :

 2 Figure 2: Schematic view of the experimental set-up for carbon dioxide retention measurement

Figure 3 :

 3 Figure 3: TG signal of the sample exposed to successive levels of CO 2 : 1500, 3000, 5000, 10000 ppm, 35°C and 0.5%RH

  DSC being a heat flow type the analysed results are heat flows. Results associated to the previous experiment are shown in Fig.4. The figure shows the recorded heat flow signal with time and the respective carbon dioxide concentration of the gas injected. Upward peaks corresponding to an increase of CO 2 concentration are exothermic while downward peaks corresponding to an decrease of CO 2 concentrations are endothermic. For the sake of clarity, only a close-up of 2 hours around the change of concentration level is represented in order to highlight the beginning of the process.

Figure 4 :

 4 Figure 4: Heat flow signal of the sample exposed to successive levels of CO 2 concentrations: 1500, 3000, 5000, 10000 ppm, at 35°C and 0.5% RH

Figure 5 :

 5 Figure 5: TG signal of the same sample exposed to three CO 2 cycles of (0 -5000 -0 ppm) at 35°C and 0.5%RH

Figure 6 :

 6 Figure 6: Mass uptake of the samples inside the TG-DSC apparatus (black) and DVS apparatus (grey) when exposed to 5000 ppm, 35°C and 1% RH.

Figure 7 :

 7 Figure 7: TG (a) and heat flow signals (b) with time of 3 samples exposed to the same conditions (5000 ppm, 35°C and 0.5% RH)

Figure 8 :

 8 Figure 8: TG (a) and heat flow (a) signals with time of 3 samples with different initial mass exposed to the same conditions (5000 ppm, 35°C and 0.5%RH)

  (a) CO2 adsorption isotherm and associate Henry's law fit (b) CO2 adsorption isotherm in semi-log scale

Figure 9 :

 9 Figure 9: Modelling of the adsorption isotherm using the Henry's law (a) and evidence of non-consistent fitting by plotting the adsorption isotherm in semi-log scale (b)

4. 2 . 1

 21 Methodology for the determination of CO 2 adsorption enthalpyThermodynamically, enthalpy of adsorption (or desorption) is the heat released (or absorbed) during the adsorption process. The enthalpy of adsorption, ∆h ads , was determined by calculating the area under the heat flow signal curve and dividing by the associated mass change during the adsorption phase. The same process was applied on the desorption phase to determine the desorption enthalpy. The baseline for the signal integration was chosen as the median of the points preceding the peak. In order to get rid

( a )

 a Full heat flow signal for a given CO2 concentration (b) Extracted heat flow signal between the limits of integration for a given CO2 concentration

Figure 10 :

 10 Figure 10: Limits of integration (vertical dotted lines) to determine CO 2 adsorption enthalpy from TG and DSC signals

Figure 11 :

 11 Figure 11: Absolute values of enthalpies of adsorption (black) and desorption (gray) for successive CO 2 concentration levels, at 0.5% RH and 35°C

Table 1 :

 1 Main properties of DAG earth

	Properties

Table 2 :

 2 Confrontation between DAG CO 2 uptake capacity and those reported in the literature for porous materials at room temperature

	Material		CO 2 uptake Desorption Exp. conditions	Ref.
			[mmol/g]	capacity [%]	[%RH, ppm]
	DAG	powder	0.003	100	0.5, 5000	This study
	Latex+TEPA70 a powder	1.33	92	15, 3000	[21]
		paint	1
	Latex+PEI50 b	powder	1.1
		paint	0.8
	epn-MOF@SBS c spray	2.8	0	60, 1000	[22]
	PD-TEPA-25% d powder	1.36	30	0, 3000	[20]
	MOF-74(Mg) e		1.62	0
	Zeolite-13X		1.42	0

a Silica including amine (TEPA), 40 wt% b Silica including amine (PEI) 40 wt% c Functionalized metal-organic-framework mixing with a hydrophobic polymer (SBS), 80 wt% MOF loading d Silica including amines (TEPA) 25%wt e Metal-organic framework

Table 4 :

 4 . The enthalpy of adsorption is negative because the flow is exothermic and the enthalpy of desorption is positive as associated with an endothermic flow. Results indicate that ∆h ads in found to be equal to approx. -60 kJ/kg and ∆h des to approx. 55 kJ/mol for all experiments. The experiments are therefore perfectly repeatable in terms of energy. It can be noticed that the initial mass of the sample has little influence on the final adsorption and desorption enthalpy result. For the adsorption and desorption cycles, ∆h ads decreases when the number of cycles increases which is in agreement with the results found for the CO 2 uptake capacity and has to be further investigated. In general, the results of adsorption enthalpies found are quite high for physisorption which is generally around 20 to 40 kJ/mol which would indicate a strong energetic interaction between the adsorption sites and the CO 2 . Enthalpies above 40 kJ/mol could indicate chemisorption, but the reversibility of the process, when the concentration decreases, seems to rule out this hypothesis. Moreover, chemisorption enthalpies found in the literature Comparison of the CO 2 adsorption enthalpy of DAG with those reported in the literature for porous material at low adsorption site occupancy

	Material	Exp. conditions	∆H ads	Ref.
		(method)	[kJ/mol]
	DAG 80µm	0.5%RH 35°C (calorimetric)	-60	This study
	SBA 15+ APS a	30°C (calorimetric)	[-40,-90]	[51]
	Al-, Zr-PILC b	[-10, 20]°C (isosteric)	-30	[48]
	Zeolite 13X	0°C (calorimetric)	-55	[52]
	CD-MOF-2 c	25°C (calorimetric)	-113.5 (irreverible)	[53]
			-65.4 (reverible)
			-40.1 (reversible)

a Mesoporous silica (SBA-15) support with amine (APS) loadings b Pillared clays c Metal-organic framework synthesized from γ cyclodextrin (γ-CD)

Acknowledgments

This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sk lodowska-Curie grant agreement No 945416, the Institut Universitaire de France, the IMPULSION PALSE program, the CARNOT institute and the Chair ConstrucTerr.

for clay minerals are generally above 80 kJ/mol, which is far from being the case here [START_REF] Schoonheydt | 1 -Clay minerals and their surfaces[END_REF]. To our knowledge, no study has yet published data on the enthalpy of CO 2 adsorption in raw earth or clay minerals. Hence, the results of this study are compared with those of processed materials (MOF, amines..) used for CO 2 regulation of indoor air, in the same way as in previous sections (see table 4). Two methods are mainly used to determine the enthalpy of adsorption of CO 2 on materials, the "indirect" isosteric method based on CO 2 adsorption and desorption isotherms at several temperatures and the calorimetric method (used here). The calorimetric method is the more accurate method as it provides a direct measurement. For silica-supported amines [START_REF] Alkhabbaz | Important Roles of Enthalpic and Entropic Contributions to CO 2 Capture from Simulated Flue Gas and Ambient Air Using Mesoporous Silica Grafted Amines[END_REF], pillard clays [START_REF] Garcés | Comparative Study of the Adsorption Equilibrium of CO 2 on Microporous Commercial Materials at Low Pressures[END_REF] and zeolites [START_REF] Miranda Da Silva | Adsorption microcalorimetry applied to the characterisation of adsorbents for CO 2 capture[END_REF] values of the enthalpy of CO 2 adsorption are ranging form -90 to -30 kJ/mol at low coverage (i.e. low surface occupancy, low amount adsorbed). Similarly, Wu et al. [START_REF] Wu | Direct Calorimetric Measurement of Enthalpy of Adsorption of Carbon Dioxide on CD-MOF-2, a Green Metal-Organic Framework[END_REF] has determined the enthalpy of adsorption at 25°C by calorimetric measurment of a MOF, at zero-coverage an irreverisble event of -113 kJ/mol occurs which is followed by a reversible event of -65 kJ/mol and finally a plateau at -40 kJ/mol is observed as coverage is increased. Our results, around -60 kJ/mol at adsorption, are thus in the range of values found in other studies. This confirms both the robustness of the experimental set-up with respect to the analysis of heat flow and the determination of CO 2 adsorption enthalpies. It also reinforces the hypothesis that the phenomena are mainly driven by physical interactions.