Mingyang Chen
email: minchen@ethz.ch

Benoit Coasne

Dominique Derome

Jan Carmeliet

Coupling of Sorption and Deformation in Soft Nanoporous Polymers: Molecular Simulation and Poromechanics

Keywords: nanoporous polymers, poromechanics, soft matter, molecular simulations

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Introduction

Sorption-induced deformation is a frequent occurrence in nanoporous polymers regardless of chemical components and micro-structures under study [START_REF] Lee | A Humidity-Sensing Composite Microfiber Based on Moisture-Induced Swelling of an Agarose Polymer Matrix[END_REF][START_REF] Loh | Modelling anomalous moisture uptake, swelling and thermal characteristics of a rubber toughened epoxy adhesive[END_REF][START_REF] Safari | Effect of Water Vapor Adsorption on Electrical Properties of Carbon Nanotube/Nanocrystalline Cellulose Composites[END_REF][START_REF] Vandamme | Coupling between adsorption and mechanics (and vice versa)[END_REF][START_REF] Yang | Moisture-Responsive Natural Fiber Coil-Structured Artificial Muscles[END_REF][START_REF] Zhao | Super Moisture-Absorbent Gels for All-Weather Atmospheric Water Harvesting[END_REF].

In nature, some plants use sorption-induced deformation for survival [START_REF] Dawson | How pine cones open[END_REF].

A plant material, wood, experiences significant swelling when exposed to moisture. In engineering, humidity-sensitive actuators, using nanoporous polymers as material, have been designed based on this feature [START_REF] Yang | Moisture-Responsive Natural Fiber Coil-Structured Artificial Muscles[END_REF]. For polymer-based composites, performance can be affected when exposed to moisture as the swollen polymers after moisture uptake generally undergo deterioration in both stiffness and strength [START_REF] Hahn | Residual Stresses in Polymer Matrix Composite Laminates[END_REF][START_REF] Loh | Modelling anomalous moisture uptake, swelling and thermal characteristics of a rubber toughened epoxy adhesive[END_REF]. Deformations can in turn impact the sorption process, which is generally termed as 'mechanosorptive effect' [START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF]Kulasinski et al., 2015a). For instance, the sorption capacity can be greatly enhanced by the swelling of the polymers, which enables some polymer-based gels to harvest atmospheric water [START_REF] Zhao | Super Moisture-Absorbent Gels for All-Weather Atmospheric Water Harvesting[END_REF]. This sorption-induced deformation of polymers arises at the molecular level, where water molecules enter the inter-chain space, break the hydrogen bonds and deform the polymers.

Upon deformation, the pores of the deformed polymers display a new energy landscape, which in turn impacts the sorption process (Kulasinski et al., 2015b(Kulasinski et al., , 2014b(Kulasinski et al., , 2014a)). This two-way coupling effect has been observed not only in polymers but also in many other porous media, such as metal-organic frameworks (MOF) and aerogels [START_REF] Gor | Adsorption-induced deformation of nanoporous materials-A review[END_REF][START_REF] Günther | Novel Insights into Nanopore Deformation Caused by Capillary Condensation[END_REF][START_REF] Neimark | Stress-Based Model for the Breathing of Metal-Organic Frameworks[END_REF]. However, as many nanoporous polymers undergo relative large sorption-induced deformation because of their low stiffness, the coupling effect is more noticeable in polymers [START_REF] Anand | A large deformation poroplasticity theory for microporous polymeric materials[END_REF][START_REF] Derome | Hygromorphic behaviour of cellular material: hysteretic swelling and shrinkage of wood probed by phase contrast X-ray tomography[END_REF][START_REF] Lee | A Humidity-Sensing Composite Microfiber Based on Moisture-Induced Swelling of an Agarose Polymer Matrix[END_REF][START_REF] Weber | Microporous Networks of High-Performance Polymers: Elastic Deformations and Gas Sorption Properties[END_REF][START_REF] Zhao | Super Moisture-Absorbent Gels for All-Weather Atmospheric Water Harvesting[END_REF].

Interest in modeling the coupled behavior between sorption and deformation of nanoporous materials is growing and different models have been proposed. Some of them are built on thermodynamic approaches considering well-defined pore structures such as slit pore or cylinder pore [START_REF] Balzer | Adsorption-Induced Deformation of Hierarchically Structured Mesoporous Silica-Effect of Pore-Level Anisotropy[END_REF][START_REF] Gor | Elastocapillarity in nanopores: Sorption strain from the actions of surface tension and surface stress[END_REF][START_REF] Gor | Adsorption-induced deformation of nanoporous materials-A review[END_REF][START_REF] Gor | Adsorption of n-Pentane on Mesoporous Silica and Adsorbent Deformation[END_REF][START_REF] Gor | Revisiting Bangham's law of adsorption-induced deformation: changes of surface energy and surface stress[END_REF][START_REF] Gor | Adsorption-Induced Deformation of Mesoporous Solids: Macroscopic Approach and Density Functional Theory[END_REF]. However, these approaches may not apply to polymers because they fail to capture the complex pore geometry in these materials. Indeed, pores in polymers refer to the morphologically distorted inter-chain space formed by disordered polymer chains. Another approach is to use the poromechanical framework, a macroscopic and continuum approach [START_REF] Brochard | Poromechanics of microporous media[END_REF]Kulasinski et al., 2015a;[START_REF] Vermorel | Enhanced continuum poromechanics to account for adsorption induced swelling of saturated isotropic microporous materials[END_REF][START_REF] Zhang | Mechanics of adsorption-deformation coupling in porous media[END_REF]. The mechanical effects of sorption are integrated in poromechanics by considering an extra stress term, often referred to as sorption-induced stress. Some researchers relate the sorption-induced stress to a surface excess amount of the adsorbate and successfully model the sorption-induced deformation of materials such as coal [START_REF] Zhang | Mechanics of adsorption-deformation coupling in porous media[END_REF]. For polymers, however, sorption cannot be described by surface covering, but rather by pore filling. Theories, such as Flory-Rehner theory, were built based on the concept of mixing polymer and liquid molecules rather than distinguishing the sorption surface [START_REF] Flory | Statistical Mechanics of Cross-Linked Polymer Networks II[END_REF][START_REF] Lai | A Triphasic Theory for the Swelling and Deformation Behaviors of Articular Cartilage[END_REF][START_REF] Yu | A three-dimensional transient mixed hybrid finite element model for superabsorbent polymers with strain-dependent permeability[END_REF]. The Flory-Rehner theory assumes that the elastic and mixing contributions to the free energy of swelling of dry polymer networks are separable and additive. The elastic contributions can be characterized knowing the mechanical constitutive relationship, while the mixing contribution is evaluated by considering the entropy and enthalpy of mixing. A solventpolymer interaction term is introduced in the theory, which is difficult to characterize directly by experiments or molecular simulations. Another way is to address the change in free energy of the system due to sorption by examining the sorption isotherm, as the state of the adsorbate can be fully characterized by the number of molecules adsorbed and its chemical potential. Following this line, Brochard et al. proposed a reformulation of poromechanics to account for sorption-induced stress in microporous solids as a function of the number of molecules adsorbed rather than by using the surface excess amount [START_REF] Brochard | Poromechanics of microporous media[END_REF]. A similar strategy was adopted by Kulasinski et al. (Kulasinski et al., 2015a). Constitutive equations derived based on this strategy are valid for a disordered porous medium of nanometer size and can thus be applied to polymers.

Poromechanics, as a macroscopic approach, can be enriched by the information acquired from studies at the microscopic scale. As the coupling between deformation and sorption occurs at the molecular level, molecular simulations such as molecular dynamics (MD) and Grand Canonical Monte Carlo (GCMC) simulations are appropriate tools for its investigation [START_REF] Chen | Molecular Simulation of Sorption-Induced Deformation in Atomistic Nanoporous Materials[END_REF][START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF]Kulasinski et al., 2014bKulasinski et al., , 2014a)). Researches based on these numerical approaches can provide a clear microscopic picture of the coupling and capture the governing mechanisms. These numerical studies rely on molecular configurations and the simulations are computationally expensive and time-consuming.

However, these microscopic mechanisms can be upscaled to the engineering scale using poromechanics, based on which a more general and efficient description of the coupled behavior can be expected.

Though the coupling between sorption and deformation is ubiquitous for polymers, it exhibits different features under different conditions. For instance, one can get different sorption isotherms under different mechanical conditions [START_REF] Bakhshian | Theoretical Model and Numerical Simulation of Adsorption and Deformation in Flexible Metal-Organic Frameworks[END_REF][START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF]. To better understand the interplay between sorption and mechanics in adsorbent-adsorbate systems with different sorption characteristics and mechanical properties, it is essential to develop a rational poromechanical model based on the microscopic coupling mechanism. In order to do this, in this paper, we first look at the coupling through molecular simulations, based on which a poromechanical model is proposed. The developed model allows us to study the sorption-induced deformation at different conditions and provides insights into the coupling phenomena. Finally, we demonstrate, using the developed model, the role of the different mechanical properties, sorption and porosity characteristics in the coupling and give corresponding physical interpretations

Molecular Simulation

Model and Method

In this study we use amorphous cellulose (AC) as the system representative of nanoporous polymers. To construct the molecular model of AC, cellobiose is firstly built as the repeat unit. The initial configuration of the cellulose chain is then generated using a Monte Carlo method within a periodic orthogonal box by employing the Amorphous Cell Module in Material Studio 8.0. The initial configuration of the AC has 4 chains and each chain has a degree of polymerization of 16. For the water molecules, we use SPC/E water model with the SHAKE algorithm [START_REF] Ryckaert | Numerical integration of the cartesian equations of motion of a system with constraints: molecular dynamics of nalkanes[END_REF]. The system is relaxed using MD simulation at constant temperature/stress with a Nose-Hoover thermostat/barostat with the help of LAMMPS (Large-Scale Atomic/Molecular Massively Parallel Simulator) [START_REF] Plimpton | Fast Parallel Algorithms for Short-Range Molecular Dynamics[END_REF]. The temperature and stress are set to 300 K and 0 Pa, respectively. The PCFF forcefield [START_REF] Sun | An ab Initio CFF93 All-Atom Force Field for Polycarbonates[END_REF] is employed to describe both intramolecular and intermolecular interactions. The Ewald summation method is employed to account for long-range electrostatic interactions with periodic boundary conditions. The integration time step and thermostat relaxation time of MD are set to be 1 fs and 500 fs respectively. The relaxation is carried out for 2 ns, after which the density converged well to a value of 1.40 g cm -3 , comparable to the reported experimental value 1.48 g cm -3 [START_REF] Chen | Molecular modeling of cellulose in amorphous state. Part I: model building and plastic deformation study[END_REF]. The prepared molecular configuration based on the approaches stated above is shown in dry and saturated states in FIG. 1 (a), with the box sizes of 2.73×2.75×3.07 nm 3 and 3.08×3.05×3.35 nm 3 for the dry and saturated state respectively. To model the coupling between sorption and deformation, a hybrid scheme of molecular simulation that consists of GCMC and MD is employed to perform simulations in the socalled osmotic statistical ensemble μσT where μ is the chemical potential of water, σ the stress applied to the cellulose material and T the temperature. Compared to regular MD or GCMC, such a hybrid strategy enables one to model the sorption and deformation quasisimultaneously as it allows the system to deform during sorption. The hybrid molecular simulations are performed at T=300 K using a Nose-Hoover thermostat/barostat and an anisotropic external stress σ = 0 Pa. Each simulation block consists of 2,000 GCMC insertion/deletion attempts and 200 MD timesteps. In total, 10 5 blocks were first performed to equilibrate the system, followed by 2×10 4 additional blocks to accumulate statistics. One can refer to [START_REF] Chen | Molecular Simulation of Sorption-Induced Deformation in Atomistic Nanoporous Materials[END_REF][START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF] for details and discussions on this hybrid technique. FIG. 1(b) shows the molecular configuration with swelling strain acquired from the simulation at saturated state.

Results

FIG. 2 (a) compares the simulated water sorption isotherm with its experimental counterpart [START_REF] Mihranyan | Moisture sorption by cellulose powders of varying crystallinity[END_REF]. The sorption amount n is defined as the number of water molecules per unit of initial (dry) volume, i.e. n=NW/V0, where NW is the number of adsorbed water molecule, V0 is the initial volume of AC. The relative humidity RH is defined as RH=p/p0, where p is the vapor pressure and p0 is the saturation vapor pressure (1017 Pa for the SPC/E water model in our simulation [START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF]). The experimental data are shifted up by n = +1.67 nm -3 , an equivalent of 0.05 moisture content, to account for the presence of residual water in the 'dried' sample. One can refer to [START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF] for further information about the motivation for shifting up the experimental results. FIG. 2 (a) shows that the simulated sorption isotherm agrees well with experimental data. Both the simulation and experiment show that the sorption amount n increases rapidly with RH in the low RH region and then less rapidly at medium RH. The data also show an inflection point after which significant subsequent sorption occurs, indicating that our molecular simulation grasps well the coupling between sorption and deformation. FIG. 2 (b) gives the sorption-induced strain of AC as a function of sorption amount. Here the strain is defined as volumetric strain, i.e. ε = (V-V0)/V0, where V is the current sample volume.

These data show that the strain is limited at very low sorption amount, after which the strain grows almost linearly with the sorption amount. These features indicate that the adsorbed water molecules at low RH mainly fill the initial porosity and thus do not lead to significant swelling. After the initial porosity is filled up, AC starts to swell and create more interchain space to accommodate water molecules. [START_REF] Mihranyan | Moisture sorption by cellulose powders of varying crystallinity[END_REF]. The solid line represents the solutions from the poromechanics model.

Adsorption isotherms with such features are classified as Type II isotherm in the nomenclature of International Union of Pure and Applied Chemistry (IUPAC) [START_REF] Thommes | Physisorption of gases, with special reference to the evaluation of surface area and pore size distribution (IUPAC Technical Report)[END_REF]. This adsorption isotherm shape indicates unrestricted monolayer-multilayer adsorption [START_REF] Sing | Reporting physisorption data for gas/solid systems with special reference to the determination of surface area and porosity (Recommendations 1984)[END_REF]. This explanation, however, does not apply to AC. According to [START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF], the pore size in AC is below 2 nm, even for the swollen state, which makes it a microporous material (in fact, it can be shown that in such small cavities water is above its capillary critical temperature Tcc [START_REF] Coasne | Molecular simulation of the adsorption and structure of benzene confined in mesoporous silicas[END_REF][START_REF] Schlaich | Dispersion truncation affects the phase behavior of bulk and confined fluids: Coexistence, adsorption, and criticality[END_REF]). In such small pores, water molecules are strongly confined, and unrestricted multilayer adsorption cannot happen. Microporous materials such as activated carbons, molecular sieve zeolites, and certain porous oxides, display Type I isotherms, which are concave to the RH axis and approach a limiting value as RH approaches 1. For such materials, the sorption amount grows fast at low RH because of the strong adsorbentadsorbate interaction. As the RH increases, the pore is getting filled, leading to a smaller increase in sorption amount upon further increasing RH. However, one distinct feature of AC compared to the other microporous materials mentioned above is its low mechanical modulus leading to a significant sorption-induced swelling. Swelling actually increases the inter-chain space (porosity) enhancing the sorption capacity. As a result, the sorption amount keeps increasing at high RH instead of being limited to a maximum value like in other microporous materials. 4 (a); we get adsorption isotherms for both cases that follow the Type I adsorption isotherm. The difference between the two adsorption isotherms comes from the different initial porosities. In the swollen state, the larger inter-chain space leads to larger porosity, which can accommodate more water molecules and thus has a higher sorption capacity. The Type I isotherms obtained here are consistent with other stiff microporous materials with limited sorption-induced deformations [START_REF] Fomkin | Adsorption of Gases, Vapors and Liquids by Microporous Adsorbents[END_REF][START_REF] Jin | The adsorption behaviour of CH4 on microporous carbons: effects of surface heterogeneity[END_REF][START_REF] Yakovlev | Adsorption and deformation phenomena at interaction of N2 and microporous carbon adsorbent[END_REF]. In the absence of significant deformation, the sorption process of AC would have followed the behavior of regular microporous materials. The comparison between FIG. 2 (a) and FIG. 4 (a) clearly displays the pronounced influence of deformation on the sorption process of AC. In practice, the sorption isotherms are often used to derive pore size distribution based on the assumption of constant pore sizes upon sorption [START_REF] Dollimore | An improved method for the calculation of pore size distribution from adsorption data[END_REF][START_REF] Groen | Pore size determination in modified micro-and mesoporous materials. Pitfalls and limitations in gas adsorption data analysis[END_REF]. Our results here indicate that one has to take the coupling between sorption and deformation into account for soft materials showing significant sorption-induced deformation. In these simulations with constrained deformation, the variation of sorption amount leads to change of stress applied to the solid. As shown in FIG. 4 (b), we measure the stress change (volumetric) as a function of the sorption amount. This stress is termed as sorptioninduced stress in some literature [START_REF] Gor | Elastocapillarity in nanopores: Sorption strain from the actions of surface tension and surface stress[END_REF]. Considering the fact that the maximum vapor pressure p0 in our simulation is only 1017 Pa, the sorption-induced stress is much larger than the vapor pressure in the reservoir. One can refer to [START_REF] Long | Under pressure: Quasi-high pressure effects in nanopores[END_REF] for a more detailed discussion on this pressure enhancement induced by nano-confinement. For both cases, the sorption-induced stress is convex to the sorption amount axis. This indicates that the adsorbed water molecules at low RH, which mainly fill the initial porosity, do not lead to a built up of significant stresses. After the free volume is exhausted, water molecules confined in the pores lead to severe crowding so that steric repulsion prevents further adsorption. As a result, further increase of the number of water molecules leads to a larger increase in sorption-induced stress. FIG. 4 (b) also shows that the swollen AC adsorbs more water molecules to achieve a similar sorption-induced stress as in unswollen AC. The swollen AC with larger porosity needs to host more water molecules to achieve the same degree of crowding.

Though molecular simulation results above agree with the experiments and show the coupling mechanism at molecular level, they are subjected to certain limitations. First, the results obtained here are specific to AC. As a matter of fact, not all polymers exhibit Type II sorption isotherms upon free swelling condition. Type I sorption isotherms are also reported [START_REF] Meng | Lignin-based Microporous Materials as Selective Adsorbents for Carbon Dioxide Separation[END_REF][START_REF] Park | Polymers with Cavities Tuned for Fast Selective Transport of Small Molecules and Ions[END_REF]. A model that is able to handle different conditions is needed. Second, the molecular simulation results provide a qualitative explanation of the coupling mechanism but a quantitative analysis based on energy balance is still needed. Last but not least, molecular simulations are generally unable to characterize the individual (i.e. independent, without coupling) contributions of different physical properties such as modulus and sorption capacity. We thus need to turn to poromechanics to overcome these limitations and gain better knowledge of the coupled behaviors.

Poromechanics

Derivation

Consider a porous material in contact with a vapor reservoir. The fluid phase has to be treated in the grand canonical ensemble as it can exchange molecules with the reservoir.

For the fluid at equilibrium, the state can be fully determined by its temperature T, volume Vf and chemical potential µk of the k-th component. An infinitely small change in the grand potential Ωf writes:

d d d d f f f k k k S T P V N        (1)
where Sf and P are the fluid entropy and pressure, Nk is the number of molecules of the kth component. In contrast, the solid phase, having a volume of V0, does not exchange molecules with the environment and must therefore be treated in the canonical ensemble, where the Helmholtz free energy Fs is minimized at equilibrium. An infinitely small change in the Helmholtz free energy balance is given by:

0 d d d d s s f F S T V P V     σ ε (2)
where Ss is the solid entropy while σ and ε are the stress and strain tensors respectively.

The term PdVf involves the work done by the fluid on the solid. The total grand potential of the solid-fluid system Ω can be considered as the summation of the grand free energy of the fluid and the free energy of the solid, i.e. Ω = Ωf + Fs. The grand potential density ω, which is defined as ω= Ω/V0, can be written as:

d d d d kk k s T n       σ ε (3)
where s is the entropy density of the system, nk the number of fluid molecules per volume of the k-th component. The use of Maxwell relation in Eq. ( 3) gives:

' ' , ,, k kk k T k T n         ε σ ε (4)
Integration with respect to the chemical potential gives:

1 ,..., 0 ( ,..., ) ,

d k kk k T n               σ σ ε ( 5
)
where σ0 is the stress that prevails in absence of fluid. Here we consider the sorptioninduced deformation as isotropic. Coupling between sorption and deformation is also considered isotropic and the stress deviator is assumed to be unaffected. Under isothermal conditions, by considering volumetric stress and strain and a single fluid, Eq. ( 5) is reduced to:

0 ( ) d n            (6)
where σ and ε are the volumetric stress and strain respectively. In the following part, the stress and strain deviators are not considered. Without further notification, the terms stress and strain refer to volumetric stress and strain respectively from now on.

To apply this thermodynamics-based derivation to a specific material, the sorption isotherm and the mechanical constitutive relationships have to be specified. First we consider the sorption isotherm, which is a function of both chemical potential and strain, i.e., n=n (µ,ε). Strain can influence the sorption process in many ways. Straining the material leads to a change in pore volume, as shown in the molecular simulations of AC; thus, the strained material will show a different sorption capacity compared to the undeformed material. Also, strain can change the density of adsorption sites and modify the energy landscape within the pore, leading to a change in sorption amount. Considering the fact that the deformation is large in AC and that sorption is greatly enhanced by the enlarged pore capacity, we assume that the influence of strain on the sorption isotherm solely comes on the enlarged sorption capacity. Then the sorption amount can be written

as 0 0 ( , ) ( ) ( , ) n n        (7)
where n0(µ) is the 'baseline' sorption isotherm, which corresponds to the sorption isotherm at zero strain. ϕ(ε,µ) is the current Lagrangian porosity depending on strain and chemical potential. ϕ0 is the initial (undeformed) porosity corresponding to the dry state of the solid.

The dependence of the current porosity on strain can be written based on the first-order expansion as:

0 ( , ) [1 ( ) ] C        (8)
where C(µ) is a parameter characterizing the dependence of porosity on strain. As shown in [START_REF] Coasne | Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica[END_REF], the exact dependence for (ε,µ) can be considered and the poromechanics model as described in Eq. ( 6) can be solved in an iterative fashion.

According to [START_REF] Brochard | Poromechanics of microporous media[END_REF], the dependence of C(µ) on chemical potential is not significant for amorphous materials. This also applies to nanoporous polymers as the polymer chains are in the amorphous state. Following this line, we neglect the dependence of C(µ) on chemical potential and simplify it into a constant C. Eqs. ( 7) and ( 8)

leads to 0 ( , ) [1 ] ( ) n C n      (9)
In Eq. ( 9), C links sorption and strain and is termed the sorption-strain coupling constant in some literature [START_REF] Zhang | Mechanics of adsorption-deformation coupling in porous media[END_REF]. For different adsorbent-adsorbate systems, C can be different depending on the coupling between sorption and deformation. To determine C for AC, we consider the fact that polymeric chains are not cross-linked so that the bulk material shows a much lower stiffness compared to the stiffness of cellulose chains. As a result, the cellulose chains will be barely strained and the deformation mainly comes from the change of the inter-chain space or change in porosity. Assuming the deformation of the solid is negligible, strain of the system can be expressed as: 0 ()

     (10)
Combining Eqs. ( 7), ( 9) and ( 10) gives C=1/ϕ0. These simplifications on C can be verified by comparing with the molecular simulations. The porosity of the molecular model is probed using the method given in (Kulasinski et al., 2015b). As shown in FIG. 5 (a), the porosity-strain relationship with C=1/ϕ0 agrees with that measured in the molecular model.

This indicates that C=1/ϕ0 is a reasonable simplification when considering the change of porosity with strain for AC. One should note that relating C to the porosity is not compulsory. The simplification C=1/ϕ0 only applies when all the deformation of the system is caused by the change of pore (inter-chain) space. For a more general consideration, C can be directly determined based on the molecular simulation results with the help of Eq.

(9). The other term to be determined is the 'baseline' sorption isotherm n0(µ) of the undeformed solid. As shown in the last section, the molecular simulation predicts a Type I sorption isotherm in the unswollen case that can be described using the Langmuir adsorption model:

0 0 0 1 n Bp n Bp   ( 11
)
where n0 0 and B are sorption parameters. n0 0 defines the sorption capacity while B defines the slope of the adsorption isotherm at very low vapor pressures (p/p0 = RH). The two parameters are calibrated by fitting the sorption isotherm of the unswollen model. The determined n0 0 and B values are given in Table 1. Combining Eq.( 6), Eq. ( 9) and ( 11), and taking into account dµ=(RT/p)dp, we have: s0 ()

     ( 12
)
where σs is the sorption-induced stress expressed as:

0 s0 ln(1 ) CRTn Bp   (13)
Apart from the sorption isotherm, we should also specify the mechanical constitutive relationship, i.e. how stress depends on strain. As shown in (Kulasinski et al., 2015b), a positive (swelling) strain can increase inter-chain distance of the AC, resulting in a decrease in the tangential bulk modulus. The tangential bulk modulus is thus assumed to decrease with increasing porosity, which can be written as:

00 d exp( )d Kb     ( 14
)
where K0 is the bulk modulus at zero porosity, whereas b is a parameter controlling the decaying behavior of the tangential modulus versus porosity. Specially, the material is linear elastic if b = 0. Combining Eqs. ( 10) and ( 14) and integrating over ε, we have:

0 0 0 [1 exp( )]exp( ) / K b b b        (15)
Finally, substituting Eqs. ( 13) and ( 15) into Eq. ( 12), the sorption-induced strain can be determined. Upon this point, a poromechanical framework covering both sorption-induced deformation and mechanosorptive effect has been established to describe the two-way coupling between sorption and deformation. To calibrate K0 and b, we extract the stress-strain curve in the solid part from molecular simulations with the following method. At each sorption amount (with different swelling strains) of the free swelling case, the system is fixed and all water molecules are deleted, then the stress prevailing in the solid is measured. Combining the stress-strain pairs for all the different sorption amounts, we get the stress-strain relationship of the solid part as seen in FIG. 5 (b). K0 and b are determined by fitting Eq. ( 15) against the data with the least square method; their values are given in Table 1. Both results show a smaller slope at very low sorption amount and a larger slope for the remaining sorption amounts. These comparisons show that, despite the assumptions made when conducting the derivation, the poromechanical model captures the governing mechanism of the coupled behavior between sorption and deformation.

Results

FIG

The poromechanical model is versatile as it is easy to use for the study of the coupled behavior under different conditions. We further compare the poromechanical results with molecular simulations for the fixed unswollen and swollen states respectively. The sorption isotherms are plotted in FIG. 4 (a). The Langmuir isotherm is seen to adequately describe the sorption isotherm at fixed strain. According to Eq. ( 7), the sorption amount is proportional to the current porosity. Thus if we scale the sorption amount with the porosity, the data from unswollen, swollen and free swelling cases should all collapse. That is what we see in FIG. 6 (a), therefore showing that the sorption isotherm for different cases can be described by a single Langmuir isotherm given by Eq. ( 11). The data in FIG. 6 (a)

suggest that the poromechanical solution agrees best with the unswollen case. This is because the parameters in the Langmuir adsorption isotherm are calibrated using data from the unswollen case in this work. The sorption-induced stress can be easily calculated from Eq. ( 13). Also, the sorptioninduced stress can be extracted from molecular simulations. In this case, the stress tensor in molecular simulation is calculated based on the viral expression. For both the unswollen and swollen cases, the sorption-induced stress σs is taken as the stress change of the system upon sorption. However, this strategy cannot be applied to the free-swelling case as the stress of the system is always 0 due to the free deformation. To extract the sorption-induced stress σs, we delete water molecules and measure the stress left in solid skeleton, which is taken as σs for the free swelling case. FIG. 4 (b) shows that the poromechanical solutions follow the same trend as the molecular simulation results for both the unswollen and swollen cases. It is worth noting that, according to Eq. ( 13), the sorption-induced stress is strain independent, which means that the stress obtained for different strains should collapse when plotted against RH. As shown in FIG. 6 (b), though the stresses calculated by molecular simulations do not strictly follow coincide, they are scattered around the poromechanical solutions. Deviations may arise from assumptions made on the coupling constant C. The strain-independent feature of sorption-induced stress comes from the fact that we simplify C to be a constant independent of strain. In reality, C is a function of both chemical potential and strain and the sorption-induced stress will show some strain dependence in molecular simulations. However, considering the fact that the difference between the sorption-induced stress with the minimum (unswollen case) and the maximum (swollen case) strain is at maximum 25%, we conclude that independence of C on strain is a reasonable assumption (again, see [START_REF] Coasne | Poroelastic Theory Applied to the Adsorption-Induced Deformation of Vitreous Silica[END_REF] for a treatment where such approximations are avoided). To further verify the validity of the poromechanical model, we compare results with the molecular simulations at different loading conditions. Two additional molecular simulations at µσT ensemble are conducted for stresses σ = 10MPa and σ = -10MPa

(corresponding to the sorption process under tension and compression respectively). For the poromechanical solution, the effect of external loading can be easily taken into account by setting σ to the corresponding values in Eq. ( 13). Comparisons between poromechanical solutions and molecular simulations at three different loading conditions (σ=-10MPa, 0, 10MPa) are shown in FIG. 7. In FIG. 7 (a), the sorption amount increases as the loading turns from compression to tension. The tension loading increases the pore space so that sorption is promoted (more free volume available). This feature is successfully captured by the poromechanical model, which shows good agreement with the molecular simulations in all three loading conditions. Moreover,FIG. 7 (b) shows that the data from the three different loading conditions collapse when plotting the strain versus sorption amount for both molecular simulations and poromechanical solutions. This is because the slope in strain-sorption amount curve is mainly determined by the dependence of the current porosity on the strain, i.e. C in Eq. ( 8). The influence of mechanical loading on C for AC is not significant.

One feature of the poromechanical model is that the sorption isotherm cannot reach RH=1 at high tensile loading. This feature can be seen in FIG. 7 (a) for the case σ = 10MPa: the sorption amount diverges before RH reaches 1. This corresponds to the fact that chains will just unfold and lose their stiffness and interactions under high tensile loading and high moisture content. This feature comes from the mechanical constitutive relationship employed in this work, i.e. Eq. ( 15). According to Eq. ( 15), there exists a maximum stress that the solid can sustain which is σmax=K0exp(-bϕ0)/b. If the total stress combining the sorption-induced stress and external stress is larger than σmax, the solution yields an infinite strain, equivalent to an evaporation of the material. It is worth noting that this feature is not just specific to the tension loading. As a matter of fact, even at zero mechanical loading, i.e. σ=0, this feature can emerge by adjusting model parameters listed in Table 1 to make the solid very soft or the sorption very strong. This point will be mentioned in the Discussion section below. In reality, this phenomenon is observed in some materials [START_REF] Barreiro | Water Sorption Isotherms of NPK 10-20-20/4 Muriate of Potash Fertilizer[END_REF] but it does not apply to all nanoporous polymers. In molecular simulations, RH=1 can be reached at this mechanical loading. This deviation can be attributed to two major reasons. First, polymers stop weakening at large strains because of the entanglement and straightening of polymer chains [START_REF] Bharadwaj | A strain stiffening theory for transient polymer networks under asymptotically nonlinear oscillatory shear[END_REF][START_REF] Jaspers | Ultra-responsive soft matter from strain-stiffening hydrogels[END_REF]. In other words, the mechanism governing the mechanical constitutive relationship changes at large strain. However, the constitutive relationship in Eq. ( 15) is proposed and calibrated based on the stress-strain curve of molecular simulations with strain below 0.4 (as shown in FIG. 5 (b))the weakening because of the enlarged interchain space is still the governing mechanism. Thus, features of AC at large strain cannot be captured by the Eq. ( 15). Second, pore size can become very large at large strains, leading to a change of sorption behavior. Thus, the Langmuir isotherm may become inappropriate in describing the sorption behavior for large pores with large strains. Thus, if one wants to address the coupling phenomenon at high tensile loading under the framework of poromechanics, a more comprehensive mechanical constitutive relationship capable of addressing the stiffening effect at high strain needs to be included [START_REF] Xiang | A physically based visco-hyperelastic constitutive model for soft materials[END_REF]. Moreover, the sorption isotherm needs to be recalibrated considering the possible change of sorption physics within large pores due to the large strain. However, although the current poromechanical model cannot recover the sorption isotherm over the entire RH range for high tension loading, it successfully captures the strong convexity at high RH and agrees with the molecular simulations at low and medium RH.

Discussion

Influence of Material Properties

By comparing with molecular simulations, we find that the poromechanical solutions capture the governing mechanism of the coupled behavior of sorption and deformation. We proceed in what follows to a parametric analysis studying how the material properties influence the coupled behavior. There are three groups of material-related parameters in the poromechanical model: the mechanical properties K0 and b, the sorption characteristics n0 0 and B and the structural information ϕ0 and C. In this section we discuss in detail the influence of these material parameters on the coupled behavior with the help of the poromechanical model.

First, we look at the mechanical properties. In the poromechanical model, the mechanical behavior of the solid is governed by two parameters: the modulus at zero porosity K0 and the weakening parameter b. A solid with lower K0 and larger b means a softer material.

Sorption isotherms at different K0 values are plotted in FIG. 8 (a). All the other parameters are kept constant. As K0 increases, the sorption amount is lower for the same RH. Indeed, a larger K0 corresponds to a stiffer material with less sorption-induced deformation. As a result, there is less additional pore space created by swelling to accommodate water molecules which finally leads to smaller sorption amount. The sorption isotherms change from Type II to Type I as K0 increases. For a better description of the transition, we define Type I and Type II as follows: a sorption isotherm is Type I if the sorption isotherm is concave for all the RH (vapor pressure) from 0 to 1 and Type II if the sorption isotherm turns from concave to convex below RH=1:

2 0 2 2 2 2 0 2 2 2 d Type-I : 0 [0, ] d d d d Type-II : | 0 , | 0, | 0 [0, ] d d d C C C p p p p p p C n pp p n n n pp p p p            (16)
By combing the classification for Type-I and Type-II in Eq. ( 16), we find that there is one critical state characterizing the transition between the two types of isotherms when pC=p0, i.e., d 2 n/dp 2 =0 at p=p0. This means that the sorption isotherm turns from concave to convex just at RH=1. Correspondingly we can define a critical modulus Kc at this critical state. The physical meaning of Kc is that a Type I sorption isotherm results from materials with a modulus above Kc while a Type II isotherm for materials with a modulus below Kc. In other words, one can use this critical value to characterize what type of sorption isotherms is to be expected for a given adsorbent-adsorbate system.

As mentioned in the Results section, one feature of the poromechanical model is that it gives infinite sorption amounts before RH=1 at certain mechanical conditions. Here for K0 specifically, a very low K0 can lead to this condition. This means that, for very soft materials, the solid cannot sustain the sorption-induced stress yielding very large strains. Thus, we can define another critical value of K0, Kcc, below which the strain goes to infinity when approaching RH=1. The poromechanical model is valid through the entire RH range, The other parameter influencing the mechanical property is b which can be seen as the weakening parameter. With a larger b, the tangential stiffness decreases more as strain increases. The impact of b on sorption isotherm is plotted in FIG. 8 (b). It shows that increasing b has a similar effect as decreasing K0, i.e., the sorption isotherm turns from Type I to Type II and the sorption amount increases. Indeed, larger b and smaller K0 have similar effects as both make the solid softer.

The effects of sorption related parameters, n0 0 and B, on sorption isotherm are plotted in FIG. 8 (c) and (d). n0 0 specifies how much water is adsorbed at saturated state, i.e. the sorption capacity for the undeformed solid, while B specifies how 'fast' sorption proceeds at low pressure/RH. Thus, a system with higher n0 0 and B corresponds to a stronger adsorbent-adsorbate interaction. FIG. 8 shows that, as n0 0 and B increases, sorption amount increases at same RH. Larger n0 0 and B lead to larger sorption-induced stress according to Eq. ( 13). As a result, the solid undergoes larger deformation and more additional pore space is generated, which finally leads to larger sorption amount. It also shows that the sorption isotherm changes from Type I to Type II as n0 0 and B increases.

There are two parameters in the poromechanical model characterizing the structure of the system, ϕ0 and C. ϕ0 characterizes the initial porosity and C characterizes the dependence of current porosity on the strain. In the previous part, C was taken as 1/ϕ0 assuming that the solid does not deform and all the strain comes from the change in porosity. However, this does not essentially hold to be true for other nanoporous polymers. Here we give up this constraint and discuss the contributions of the two parameters independently. Sorption isotherms at different ϕ0 values are plotted in FIG. 8 (e). The ratio n0 0 /ϕ0 is kept constant so that the sorption amount per unit pore volume is constant. It shows that increasing ϕ0 leads to an increase of the sorption and the change of sorption isotherm from Type I to Type II. This is because a system with larger initial porosity ϕ0 has larger sorption capacity n0 0 and thus has larger sorption amount, which leads to larger sorption-induced stress according to Eq. ( 13). Similar mechanism applies to the influence of C, which is plotted in FIG. 8 (f).

As sorption-induced stress is proportional to C according to Eq. ( 13), a larger C also leads to larger sorption amount and change of sorption isotherm from type I to type II as seen when studying ϕ0.

As we see from the discussion above, material properties significantly impact the coupling behavior. However, each material property is characterized by more than one parameter.

For instance, the stiffness of the material is characterized by both K0 and b. Though discussion based on each parameter sheds light on the specific effect, it is inconvenient to have a whole picture of the influence of different material properties and draw general conclusions based on these discussions. Here we consider the combined effect of all the parameters characterizing the same class of material properties. For instance, both K0 and b characterize the mechanical properties of the solid, thus we consider the combined effect of K0 and b. Instead of explicitly looking at the sorption isotherm, we concentrate more on the influence of the two parameters over the entire RH region from 0 to 1. A good choice for an indicator is the free energy associated with sorption, i.e., the area under the curve when plotting sorption amount against chemical potential:

Z 1 (1 ) b CRTn Bp K    .
A larger f means a larger free energy associated with sorption and corresponds to the larger total sorption amount. Moreover, it also corresponds to a larger deformation magnitude because of the coupling between sorption and deformation. For concision, f is normalized by f0 in the following discussion, where f0 is defined as the free energy corresponding to the case of zero strain. According to Eq. ( 9), the sorption amount with some swelling strain will always be larger than that of zero strain.

Thus, f /f0≥1 holds for all the swelling cases. The influence of the stiffness is characterized by studying the combined effects of K0 and b as plotted in FIG. 9 (a). Note that the compliance, i.e., 1/K0, is taken so that the stiffest solid sits on the origin. By going along the diagonal from origin to top right, the solid becomes softer and we see an increase in f. Indeed, the magnitude of deformation is larger for softer solid, which leads to a larger sorption amount. As a result, the free energy related to sorption is larger. We can also define three regions, I, II and III in a similar way as done in FIG. space created by the strain thus leads to a larger sorption amount. Thus, for a system with larger ϕ0 and C, stronger coupling between sorption and deformation can be expected.

Along this line, we can say that coupling turns from weak to strong by going along the diagonal of FIG. 9 (c) from origin to top right. The free energy f gets larger for strong coupling. This is because deformation facilitates sorption as the swelling leads to a higher sorption amount. A stronger coupling leads to more sorption-related free energy. The line of C=1/ϕ0 is also plotted as the red dashed line, which is the simplification we employ when addressing AC.

Absence of Hysteresis

We note that it is well-known that sorption in microporous polymers is accompanied with significant hysteresis. In this paper, however, only one sorption branch (desorption) is considered and hysteresis is not included into the poromechanical model. According to our research based on molecular simulations, the hysteresis is related to microscopic picture of hydrogen bonds [START_REF] Chen | Role of hydrogen bonding in hysteresis observed in sorption-induced swelling of soft nanoporous polymers[END_REF]. From the perspective of porous media, hysteresis relies on the mutual influence of the neighboring pores. Thus, we believe that a microscopic structure is needed for a physically-based model to account for hysteresis.

However, poromechanics is a macroscopic approach, where the structural information of the whole system is represented by a scalar porosity and the microscopic structure is inherently absent. As a result, the poromechanical model proposed in this paper is incapable of modeling the hysteresis physically. We are working on this issue with an attempt to take the microscopic structure into account with the help of Dependent Domain model [START_REF] Guyer | Hysteresis in modeling of poroelastic systems: Quasistatic equilibrium[END_REF].

Conclusion

In this work, the coupling of sorption and deformation in a soft nanoporous polymer was first investigated by means of molecular simulations. The atom-scale simulations, which are found to agree with experiments, reveal the coupling mechanism at the molecular level.

We find that the Type II sorption isotherm of amorphous cellulose upon free swelling is induced by the sorption-induced deformation, i.e. the mechanosorptive effect. The sorption isotherm changes to Type I if no deformation is allowed by applying fixed displacement boundary conditions. Moreover, we find by means of molecular simulations that the stress state of the system has a significant impact on the sorption process. A tensile/compressive stress will facilitate/restrict the sorption.

Based on the molecular simulations reported in the present work, a poromechanical model was proposed. We demonstrate that the poromechanical model agrees with molecular simulations and experiments, and captures the main feature of the coupling mechanism.

Three categories of parameters come into the poromechanical model: the mechanical properties, the sorption characteristics and the structural information. We find that different combinations of these parameters can lead to totally different sorption and deformation behaviors: the sorption isotherm can even change its type. Basically, systems with lower modulus, stronger adsorbent-adsorbate interaction and stronger coupling between sorption and deformation tend to have a Type II sorption isotherm with larger sorption amount and sorption-induced deformation, while systems with higher modulus, weaker adsorbentadsorbate interaction and weaker coupling between sorption and deformation tend to have a Type I sorption isotherm with limited sorption amount and sorption-induced deformation.

The poromechanical model shows its versatility in addressing the coupling between sorption and deformation of nanoporous polymers. Compared to molecular simulations, it can easily characterize the individual contributions of each physical property, which deepens our understanding of the coupled behavior. Besides, it is efficient and simple compared to the molecular simulations without loss of major coupling mechanism. Thus, this poromechanical model can be viewed as an upscaled model for nanoporous polymers at nanoscale, which can help to design and tailor materials for target coupled behaviors.

Moreover, it can also be easily applied to studies of polymer-based materials with more complex microstructures [START_REF] Safari | Effect of Water Vapor Adsorption on Electrical Properties of Carbon Nanotube/Nanocrystalline Cellulose Composites[END_REF]).

FIG. 1

 1 FIG. 1 Configurations in molecular simulations: (a) dry (unswollen) state at RH =0; (b) saturated (swollen) state at RH = 1. Carbon, oxygen and hydrogen atoms are colored by cyan, red and white. Atoms of cellulose chains and water molecules are plotted in segments and spheres respectively. The temperature T is 300K and external stress σ is 0 (stress free) for both cases.

FIG. 2

 2 FIG. 2 (a) Sorption isotherms n of water in AC at temperature T = 300K. (b) Relationship between strain ε and sorption amount n. The closed circles and open diamonds correspond to molecular simulations and experiments[START_REF] Mihranyan | Moisture sorption by cellulose powders of varying crystallinity[END_REF]. The solid line represents the solutions from the poromechanics model.

FIG. 3

 3 FIG. 3 Three boundary conditions considered when subjecting the polymer to a water vapor reservoir in molecular simulations and poromechanical model: (a) free swelling with system volume changing from V0 at RH = 0 to Vsat at RH = 1; (b) fixed displacement boundary conditions in unswollen state at volume V0; (c) fixed displacement boundary conditions in swollen state at volume Vsat. The blue dots and solid purple lines schematically represent water molecules and polymer chains respectively. The black solid line is the boundary of the system and the black dashed line represents the vapor reservoir. The black slash line represents the fixed boundary.

FIG. 4

 4 FIG. 4 (a) Sorption isotherms n of water in AC at fixed unswollen (red) and swollen (blue) states with temperature T=300K (b) Relationship between sorption-induced stress σs and sorption amount n at fixed unswollen (red) and swollen (blue) states. The closed circles and solid lines represent molecular simulations and poromechanics solutions respectively.

FIG. 5

 5 FIG. 5 (a) Dependence of porosity ϕ on strain ε based on molecular simulations (closed circle) and Eq. (10)

  FIG. 2 (a) gives the sorption isotherm under free swelling stemming from the poromechanical model. The predicted data show good agreement with both molecular simulations and experiments. The steeper sorption changes at low and high RHs framing

21FIG. 6

 6 FIG. 6 (a) Sorption isotherms n scaled by porosity  i.e. n/ (b) Dependence of sorption-induced stress σs on RH. The black, red and blue closed circles represent the free swelling, unswollen, and swollen cases respectively. The solid line represents the poromechanics solutions

FIG. 7

 7 FIG. 7Comparison of (a) sorption isotherm and (b) strain-sorption amount relationship between poromechanical solutions (solid line) and molecular simulations (closed circle) at three different loading conditions: σ=-10MPa (magenta), 0 (black) and 10MPa (green).

  FIG. 8 Sorption isotherms at (a) different K0 of 10 GPa, 11GPa, 14 GPa and 20 GPa; (b) different b of 2.8, 7.4, 10.2 and 10.8; (c) different n0 0 of 0.8nm -3 , 1.28nm -3 , 1.76nm -3 and 1.92nm -3 ; (d) different B of 0.0043 Pa -1 , 0.0130 Pa -1 , 0.0303 Pa -1 and 0.0389 Pa -1 ; (e) different ϕ0 of 0.03, 0.06, 0.075 and 0.079; (f) different C of 9, 10, 11 and 12. (g) Schematics of three different types of sorption isotherm. The black arrow indicates the direction of increasing corresponding parameters. Regions I (gray), II (pink) and III (white) correspond to sorption isotherms indicated by I, II and III in (g) respectively.

FIG. 9

 9 FIG. 9 Combined effects of (a) K0 and b, (b) n0 0 and B and (c) ϕ0 and C on the sorption-related free energy f. The regions I, II and III divided by red and black solid lines correspond to the three scenarios given in FIG. 8(g). The red dashed line in (c) represents C=1/ϕ0.

  8. Region I (down left of the red solid line) corresponds to stiff solids, the deformation of which is limited and thus corresponds to Type I isotherm. Region II (between the black and red solid lines) corresponds to softer solids and larger deformation, which gives Type II isotherm. Region III corresponds to the case where the solid is too soft to sustain the sorption-induced stress, thus RH could not reach 1.Following the same line, the combined effect of n0 0 and B is plotted in FIG.9(b) to characterize the influence of the adsorbent-adsorbate interaction strength. The strength increases by going along the diagonal from origin to top right, upon which we see more sorption amount as indicated by larger f and sorption isotherm turns from type I to type II. Similarly, the combined effects of ϕ0 and C are plotted in FIG. 9 (c). Though ϕ0 and C are parameters related to the structure (porosity), they can also be viewed as parameters characterizing the coupling strength as the two parameters define the mutual influence of deformation and sorption in the poromechanical model. ϕ0 characterizes the influence of the sorption on deformation. A larger ϕ0 corresponds to a larger domain of fluid within the system, thus leads to a stronger influence of sorption on deformation. C characterizes the influence of deformation on sorption. A larger C corresponds to a larger additional pore

Table 1

 1 Parameters used for the poromechanics model

	parameter	n	0 0	B	K0	b	ϕ0	C
	units	(nm -3 )	(Pa -1 )	(GPa)	-	-	-
	value	1.94	0.0433	9.98	10.8347	0.081	12.36

Acknowledgements

The authors acknowledge the support of the Swiss National Science Foundation (SNSF) (No.143601).


R Karcher

F Pereira

Dos Santos

S Merlet

Impact of direct digital synthesizer finite resolution on atom gravimeters

We report on the study of the impact of the finite resolution of the chirp rate applied on the frequency difference between the Raman lasers beamsplitters onto the phase of a free fall atom gravimeter. This chirp induces a phase shift that compensates the one due to gravity acceleration, allowing for its precise determination in terms of frequencies. In practice, it is most often generated by a direct digital synthesizer (DDS). Besides the effect of eventual rounding errors, we evaluate here the bias on the g measurement due to the finite time and frequency resolution of the chirp generated by the DDS, and show that it can compromise the measurement accuracy. However, this effect can be mitigated by an adequate choice of the DDS chirp parameters resulting from a trade-off between interferometer phase resolution and induced bias.

I. INTRODUCTION

Light pulse atom interferometry is now a mature technique, specially fit to design inertial sensors [START_REF] Riehle | Optical Ramsey spectroscopy in a rotating frame: Sagnac effect in a matter-wave interferometer[END_REF][START_REF] Kasevich | Atomic interferometry using stimulated Raman transitions[END_REF] which compete with state of the art classical sensors [START_REF] Gillot | Stability comparison of two absolute gravimeters: optical versus atomic interferometers[END_REF][START_REF] Freier | Mobile quantum gravity sensor with unprecedented stability[END_REF][START_REF] Asenbaum | Phase shift in an atom interferometer due to spacetime curvature across its wave function[END_REF][START_REF] Savoie | Interleaved Atom Interferometry for High Sensitivity Inertial Measurements[END_REF]. Among these, Cold Atom Gravimeters (CAG) based on three pulse Mach-Zehnder sequence [START_REF] Kasevich | Atomic interferometry using stimulated Raman transitions[END_REF] are absolute gravimeters which have demonstrated the ability to perform continuous measurements with better short term sensitivities [START_REF] Gillot | Stability comparison of two absolute gravimeters: optical versus atomic interferometers[END_REF][START_REF] Peters | Measurement of gravitational acceleration by dropping atoms[END_REF][START_REF] Hu | Demonstration of an ultrahigh-sensitivity atom-interferometry absolute gravimeter[END_REF] and long term stabilities than their classical counterparts [START_REF] Freier | Mobile quantum gravity sensor with unprecedented stability[END_REF]. They have also reached comparable or better accuracies by the careful evaluation of their systematic effects. The control of these systematic effects crucially depends on the control of the phase difference between the light beamsplitters, which get imprinted onto the atomic wavepackets at each pulse during the interferometer. Both the spatial and temporal laser phase fluctuations, which have been extensively studied in the past, such as in [START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF][START_REF] Louchet-Chauvet | The influence of transverse motion within an atomic gravimeter[END_REF][START_REF] Schkolnik | The effect of wavefront aberrations in atom interferometry[END_REF][START_REF] Tao | Note: Directly measuring the direct digital synthesizer frequency chirp-rate for an atom interferometer[END_REF][START_REF] Zhou | Observing the effect of wave-front aberrations in an atom interferometer by modulating the diameter of Raman beams[END_REF][START_REF] Karcher | Improving the accuracy of atom interferometers with ultracold sources[END_REF], need to be perfectly controlled to ensure the accuracy of the measurement of the local gravity acceleration g.

In gravimeters based on atom interferometry, the measurement of g is derived from the determination of the Doppler frequency chirp induced by the free fall of the atoms onto the lasers. In practice, the laser frequency difference can be adjusted at each pulse by switching between three fixed frequencies, a different one for each pulse, using an agile and stable synthesizer, such as in [START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF]. However, the finite frequency resolution of the synthesizer of 0.23 Hz allowed there for a coarse step size of 0.23 rad when scanning the interferometer phase.

Thus more than three frequencies were finally used to better control the interferometer phase.

In addition, a frequency sweep (over a limited frequency range) was repeatedly applied during the pulses, as a chirp was required to ensure the efficiency of the velocity preselection, which relied on a 380 µs long Raman pulse. This resulted in a complicated frequency control system, with four different synthesizers. An alternative allowed by current technology consists in applying a continuous frequency chirp over the whole interferometer duration, using a single synthesizer. Nevertheless, any parasitic frequency or phase shifts with respect to the ideal chirp will thus induce errors on the g measurement, such as for instance the bias due to frequency dependent radio-frequency delays in the electronics studied in [START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF], which can be rejected by a proper symmetrization of the chirp with respect to the mid pulse between both directions of the effective wave vector.

In the following, we investigate the impact of the finite frequency resolution of the os-cillator used to chirp the lasers frequency difference during the interferometer. This chirp being synthesized with discrete steps out of a clock reference signal which we consider here as perfectly stable, produces time-periodic frequency errors, leading to time-periodic errors on the interferometer phase. After a brief description of our measurement method of g, we first calculate the biases in the g measurements induced by rounding and frequency errors arising from the finite resolution of the direct digital synthesizer (DDS) we use. Then, we performed measurements of these biases, which we find in agreement with numerical calculations. This analysis allows us to accurately evaluate the impact of these effects on our absolute cold atom gravimeter.

II. NUMERICAL SIMULATIONS: FREQUENCY ERROR OF THE CHIRP

A. g measurement protocol

In the most mature atom gravimeters, g measurements are performed via Raman interferometry on free falling atoms. There, the interferometer is realized thanks to a sequence of three two-photon stimulated Raman transitions of duration τ -2τ -τ separated by a free evolution time T , which respectively split, redirect and finally recombine the atomic wave packets. As mentioned above, during the free fall, the Doppler shift modifies the resonance condition which has to be compensated for by applying a frequency chirp to the frequency difference between the Raman lasers. This chirp rate α adds a phase shift (αT 2 ) to the interferometer that, when properly tuned (α = α 0 = k.g), exactly compensates the phase shift induced by the gravity acceleration (-k.gT 2 ), k being the effective wave vector of the Raman transition. Remarkably, this leads to a dark fringe in the fringe pattern obtained when scanning α, whose position does not depend on the interferometer duration [START_REF] Farah | Underground operation at best sensitivity of the mobile LNE-SYRTE cold atom gravimeter[END_REF]. The measurement of the gravity acceleration, and its fluctuations, can thus finally be obtained from the precise tracking of this dark fringe and from the determination of the corresponding value of the chirp rate α 0 . Many of the systematic effects, such as related to one photon light shifts or the quadratic Zeeman effect, are rejected by averaging g measurements over opposite directions of the effective wave vector k [START_REF] Louchet-Chauvet | The influence of transverse motion within an atomic gravimeter[END_REF][START_REF] Weiss | Precision measurement of /M Cs based on photon recoil using laser-cooled atoms and atomic interferometry[END_REF]. Indeed, unlike the gravity phase shift, these systematics do not depend on the direction of k [START_REF] Peters | High-precision gravity measurements using atom interferometry[END_REF][START_REF] Louchet-Chauvet | The influence of transverse motion within an atomic gravimeter[END_REF]. However, the efficiency of this rejection depends on the superposition of the two trajectories for the two interferometer configurations. In practice, the difference in the momenta imparted to the atoms leads to small differences between the trajectories, of about a few mm. These are much smaller than the few tens of centimetres of the free fall distance, which guarantees the efficiency of this rejection technique [START_REF] Mehlstäubler | Systematic phase shifts in cold atom gravimeters[END_REF][START_REF] Cheng | Coherent population trapping in a Raman atom interferometer[END_REF].

B. Finite resolution of the DDS: rounding error

In our experiment, we generate the chirp thanks to a direct digital synthesizer (AD9852, from Analog Devices). It is a 48-bits DDS, clocked at 300 MHz, which corresponds to a frequency resolution of about 300 MHz/2 48 1 µHz. Within its chirped mode of operation, we can control both the temporal step δt and the frequency step δν, which can be tuned to design the desired chirp with a rate α (inserted picture in figure 1). The temporal step δt is defined by a 20-bit word, and can thus be varied in between 1 and 2 20 cycles of the master clock, ie between 3.3 ns and 3.5 ms. In our case, the frequency chirp required to compensate for gravity, of about 25 MHz/s, is realized with frequency steps of 125 Hz every 10 µs. Indeed, the DDS is compared in a phase frequency detector to the beat note between the Raman lasers (after a down conversion, realized by mixing the beat note with a 7 GHz oscillator, and a subsequent division by a factor 2). At last, this results in a frequency chirp resolution of about 2 × 1 µHz/10 µs 0.2 Hz/s. The corresponding resolution in terms of gravity acceleration is 8.3 µGal (1 µGal= 10 nm/s 2 ), which is smaller than the shot to shot noise on the g measurement (peak to peak of about ∼ 100 µGal). This resolution is thus not a limit in our measurement protocol, which consists in a digital integrator that steers the chirp rate onto the central fringe [START_REF] Merlet | Operating an atom interferometer beyond its linear range[END_REF]. However, the correction applied by the lock system onto the chirp rate is impacted by this resolution, as there is a difference between the corrections we calculate at each measurement cycle, and the ones that are actually applied, due to rounding errors. These errors, which amount on average to ±1/2 bit (or ± ∼ 4.15 µGal), could in principle be eliminated by recording the applied changes instead of the requested ones. On the other hand, and quite remarkably, these errors cancel when averaging over the two opposite k directions, as they lead to underestimating the chirp rate when the frequency is ramped up and overestimating it when ramped down.

Finally, more important than the effect of these roundings, the DDS cannot exactly produce the required change of the frequency/phase of the Raman phase difference because of its step wise character. This leads to systematic errors, which we discuss and evaluate in the following section.

C. Finite resolution of the DDS: frequency error

As shown in figure 1, the chirp is realized by the DDS by incrementing the laser frequency by δν every δt, leading to a chirp rate α. The deviation from a perfect linear frequency chirp induces a periodic frequency error represented by the sawtooth function displayed on figure 1. This function is determined by the two parameters δν and δt. Also, we name ∆t the time difference between the beginning of the chirp and the beginning of the interferometer sequence. Note that for the chirp displayed in figure 1, ∆t = 0. To calculate the bias on the interferometer phase, we use the sensitivity function g s [START_REF] Cheinet | Measurement of the sensitivity function in a time-domain atomic interferometer[END_REF],

which describes the impact of frequency fluctuations ∆f (t) onto the interferometer phase ∆Φ:

∆Φ = +∞ -∞ g s (t)2π∆f (t)dt (1)
It is an odd function: g s (-t) = -g s (t). It is given, for t > 0, by :

g s (t) =              sin(Ω R t) for 0 < t < τ 1 for τ < t < T + τ -sin(Ω R (T -t)) for T + τ < t < T + 2τ 0 for t > T + 2τ (2)
From now on and throughout the rest of the paper, we take, without loss of generality, the time origin t = 0 at the centre of the middle π pulse. Ω R is the Rabi frequency, given

by Ω R = π/2τ .

To evaluate the bias on the g measurement, we thus simply integrate equation ( 1) modulated by the sawtooth function over the duration of the interferometer. This condition also holds if adjusting, only very slightly, T for each δt. The corresponding biases are displayed as the solid red line in figure 2 b).

For δt = 10 µs the expected bias (0.06 µGal) is found to be negligible, but it tends to increase as a function of δt, leading to a bias as large as ∼ 100 µGal for δt = 100 µs. For a given δt, two parameters can be easily modified to modulate the impact of frequency errors :

• First, one can vary the delay ∆t between the beginning of the interferometer with respect to the beginning of the chirp. This parameter is varied between 0 and 100 µs on figure 3, which displays the results of the calculated bias as a function of ∆t for δt = 10 µs , 20 µs , 50 µs and 100 µs. As expected, the results are periodic with a period δt. The amplitude of the bias is lower than 2 µGal for δt ≤ 20 µs but reaches over -200 µGal for δt = 100 µs. Note that one can null the bias for specific values of the delay ∆t, which depend on δt.

• Second, one can change the Raman pulse durations. Figure 4 is the result of the calculated bias as a function of τ for δt = 50 µs and 100 µs. To illustrate the effect, we choose δt for which the effect is large. In this calculation, we use realistic durations of τ of the order of few tenth of µs and there is no delay (∆t = 0). The result is a periodic function which dampens for increasing values of τ . This illustrates the fact that when τ δt, the bias averages down to zero.

In a different manner than the rounding error which is rejected by the k-reversal algorithm, this bias can be cancelled with a proper choice of the Raman pulse duration.

A simple choice would be to use τ = δt/2, for which this effect is calculated to be null. In this section, we report on the experimental study of the impact of the DDS finite resolution on the measurements of our cold atom gravimeter, which we compare to the calculations described in the previous section.

A. Experimental setup

We present here briefly the experimental setup we have used to perform the measurements. A more detailed description can be found in [START_REF] Louchet-Chauvet | The influence of transverse motion within an atomic gravimeter[END_REF]. We start by trapping a sample of 87 Rb atoms in a 3D MOT, and further cool it down to 2 µK with far detuned molasses. About 10 8 atoms are released in free fall for 200 ms and prepared in the |m F = 0 magnetic state in a narrow vertical velocity distribution. We then apply the interferometer sequence which lasts a total time of 2T = 160 ms. As discussed earlier, the Raman lasers are kept resonant with the atoms during the free fall, by actually chirping one of the two Raman lasers [START_REF] Cheng | Influence of chirping the Raman lasers in an atom gravimeter: Phase shifts due to the Raman light shift and to the finite speed of light[END_REF].

The phase difference in this two-wave interferometer modulates the population N 1 and N 2 at the two output ports, which are measured at the bottom of the chamber by a internalstate selective fluorescence detection, using the state labelling property of Raman transitions [START_REF] Ch | Atomic interferometry with internal state labeling[END_REF]. From these measurements we derive the transition probability P , which is given by

P = N 1 /(N 1 + N 2 ) = 1 2 [1 + C cos(∆Φ)],
where C is the contrast of the interferometer and ∆Φ = -kgT 2 + αT 2 the total interferometer phase shift. Gravity measurements are performed by steering α towards kg, so as to nullify ∆Φ. The measurement repetition rate is about 3 Hz.

In the following, the measurements are performed in a differential way by alternating gravity measurements with different sets of parameters S = {δt, ∆t, τ } and directions of the effective wave vector. Our conventional parameters are S ref = {10 µs, 0 µs, 16 µs}.

In practice, we compare a first pair of configurations, g(k ↑ , S ref ) and g(k ↓ , S ref ), with opposite effective wave vectors and reference sets of parameters, with other pairs of configurations g(k ↑ , S) and g(k ↓ , S). In most of the measurements presented below, the quantity of interest is the difference between the average measurements of the pairs:

∆g = g(k ↑ , S) + g(k ↓ , S) 2 - g(k ↑ , S ref ) + g(k ↓ , S ref ) 2 (3)

B. Evaluation of the rounding error

At first, we tried to evaluate the rounding error. As it is rejected by the average over opposite directions of k, we consider here the differences between single-k directions such as:

∆g RE = [g(k ↑ , S) -g(k ↑ , S ref )] -[g(k ↓ , S)) -g(k ↓ , S ref )] 2 (4)
Results are represented on figure 5 (taking as a reference ∆g δt=10µs RE for which the bias is expected to amount to ≈ -4.15 µGal), together with the calculated errors in red. They show a good agreement with the results of the calculation, which are represented in red. The residues, also plotted on figure 6, are lower than 2 µGal even for the largest biases. Finally, we tried to cancel this bias for the particular case of δt = 50 µs, by using a Raman sequence of τ -2τ -τ = 25 µs-50 µs-25 µs. We use here the classical sequence τ -2τ -τ = 16 µs-32 µs-16 µs and δt = 50 µs as a reference. We obtain a difference of 37.3(1) µGal close to the expected bias of 37.1 µGal. The measured bias on the difference being comparable to the expected bias obtained with the simulation, we are confident that the bias is indeed null for the (25-50-25) configuration.

IV. CONCLUSION

We have evaluated the impact of the finite temporal resolution of the direct digital synthesizer we use to chirp the Raman laser frequency difference during the interferometer in our cold atom gravimeter. This chirp is essential to maintain the resonance condition of the Raman lasers during the free fall of the atoms. We have discussed and measured the rounding error, of 1/2 bit on average, arising from the finite frequency resolution of the DDS. Increasing the duration of the time step δt decreases the amplitude of the rounding, and improves the resolution on the interferometer phase, which could be useful in a very low noise environment.

However, this results in larger frequency deviations with respect to the ideal linear chirp, inducing potentially large biases on the interferometer phase. This effect can in principle be compensated for by a proper choice of either the duration of the Raman pulses or the time delay between the chirp sequence and the interferometer sequence. Our typical measurement conditions δt = 10 µs results from a compromise between the resolution and the bias on the interferometer phase. The rounding error, of 4.15 µGal, is rejected by the k-reversal algorithm, whereas the bias arising from the frequency error is as small as 0.06 µGal, well below the current accuracy of the instrument, of order of 2 µGal [START_REF] Karcher | Improving the accuracy of atom interferometers with ultracold sources[END_REF]. But, other choices for the time step can lead to much larger biases, from which g measurements would need to be corrected.

We finally stress that the calculations have been performed with an instrument transfer function which corresponds to square shaped Raman laser pulses. As a follow up study, one could investigate the impact of either more realistic or deliberately shaped pulses [START_REF] Fang | Improving the phase response of an atom interferometer by means of temporal pulse shaping[END_REF] onto the errors related to the finite resolution of the frequency chirp.

FIG. 1 :

 1 FIG. 1: Frequency chirp applied to the Raman lasers frequency difference. The perfect linear case is the straight line represented in black. The actual chirp is composed of frequency steps of δν = 125 Hz with temporal steps δt = 10 µs and a null delay ∆t. The insert shows the parameters of the DDS (δt, δν, ∆t). Bottom: residual or frequency difference ∆f between the ideal and actual chirps, represented as a sawtooth function.

Figure 2 a

 2 Figure2a) shows the calculated bias for δt ranging from 1 µs to 100 µs for our typical interferometer parameters (τ = 16 µs, T = 80 ms), and for a delay ∆t = 0 µs. The bias is found to exhibit rapid variations, inside an envelop that is growing with δt. A smoother behaviour is actually found when considering the particular case where the free evolution time T is a multiple of δt. There, the contributions in the above integral from the two free evolution phases compensate. For T = 80 ms exactly, this corresponds to specific values of δt only, for which the corresponding biases are displayed as red circles in figure2 a) and b).

FIG. 2 :

 2 FIG. 2: Calculated bias g δt as a function of the temporal step of the chirp δt for T = 80 ms, τ = 16 µs and ∆t = 0 µs. a) T fixed. b) T adjusted to the nearest integer multiple of δt.

FIG. 3 :

 3 FIG. 3: Calculated bias g ∆t as a function of the delay ∆t between the chirp sequence and the interferometer sequence for a) δt = 10 µs (thin black line) and 20 µs (thick red line). b) δt = 50 µs (thin black line) and 100 µs (thick red line).

FIG. 4 :

 4 FIG. 4: Calculated bias g τ as a function of the Raman pulse duration τ for a) δt = 50 µs and b) δt = 100 µs. The case where the duration τ equals half δt, for which the bias is null, is represented in blue.

  These differential measurements are found to converge towards 4.15 µGal, corresponding in fact to a null bias for asymptotically large values of δt. The measurements agree with the calculated values, within their uncertainties, except for δt = 1 µs.C. Measurement of the impact of the finite resolution of the DDSMore interesting, we performed differential g measurements (as in equation (3)) as a function of δt, using δt ref = 10 µs as a reference parameter, in order to probe the influence of the finite duration of the time step of the chirp. The results are displayed on figure6.

FIG. 5 :

 5 FIG. 5: Measurements of the rounding error ∆g RE (blue circles) as a function of the temporal step of the DDS δt. The results of the calculation are displayed as a continuous red line. The asymptotic line, which corresponds to a null bias and to a difference of 4.15 µGal with respect to the reference configuration of δt = 10 µs, is represented in black. The residues between theory and measurements are displayed at the bottom as blue open circles.

FIG. 6 :

 6 FIG.6: Measurement of ∆g as a function of δt the time step of the chirp (blue circles). The continuous line in red is the calculated bias, which is described in the previous section. The residues between theory and measurements are displayed at the bottom as blue open circles.

FIG. 7 :

 7 FIG. 7: Measurement of ∆g as a function the offset ∆t between the starts of the chirp and the interferometer (in blue circles). a) the reference is δt = 50 µs and b) δt = 100 µs.

This work has been supported by Region Ile-de-France in the framework of DIM SIRTEQ (CAUCAG project).

V. ACKNOWLEGMENTS