Keywords: mechano-bactericidal activity, nanoparticles, nanotoxicity

Antibacterial Action of Nanoparticles by Lethal Stretching of Bacterial Cell Membranes

Denver P. Linklater, Vladimir A. Baulin, Xavier Le Guével, Jean-Baptiste Fleury, Eric Hanssen, The Hong Phong Nguyen, Saulius Juodkazis, Gary Bryant, Russell J. Crawford, Paul Stoodley, and Elena P. Ivanova* Nanoparticles (NPs) are heavily used in biomedical, industrial, and commercial applications due to the benefits associated with the specific physical and chemical properties of both the bulk and the nanoscale material. The antimicrobial activity of NPs is widely recognized, but the mechanisms of their underlying toxicity remain unclear despite repeated attempts to establish a structure-function relationship between their physicochemical properties and their interactions with biological systems. [1,2] NP uptake in mammalian cells is generally considered to be an active process, mediated by endocytosis. Indeed, transport across the cell membrane and intracellular accumulation dictates the nanoparticle fate and cytotoxicity. [3] The critical size for NPs non-disruptively (passively) crossing cellular membranes is below 10 nm, irrespective of surface functionalization, [4][5][6][7] Figure 1. For this reason, there is a slowly forming consensus that smaller NPs bear greater toxicity than larger ones. [8,9] Similarly, the antibacterial properties of small NPs have It is commonly accepted that nanoparticles (NPs) can kill bacteria; however, the mechanism of antimicrobial action remains obscure for large NPs that cannot translocate the bacterial cell wall. It is demonstrated that the increase in membrane tension caused by the adsorption of NPs is responsible for mechanical deformation, leading to cell rupture and death. A biophysical model of the NP-membrane interactions is presented which suggests that adsorbed NPs cause membrane stretching and squeezing. This general phenomenon is demonstrated experimentally using both model membranes and Pseu-domonas aeruginosa and Staphylococcus aureus, representing Gram-positive and Gram-negative bacteria. Hydrophilic and hydrophobic quasi-spherical and star-shaped gold (Au)NPs are synthesized to explore the antibacterial mecha-nism of nontranslocating AuNPs. Direct observation of nanoparticle-induced membrane tension and squeezing is demonstrated using a customdesigned microfluidic device, which relieves contraction of the model membrane surface area and eventual lipid bilayer collapse. Quasi-spherical nanoparticles exhibit a greater bactericidal action due to a higher interactive affinity, resulting in greater membrane stretching and rupturing, corroborating the theoretical model. Electron microscopy techniques are used to characterize the NP-bacte-rial-membrane interactions. This combination of experimental and theoretical results confirm the proposed mechanism of membrane-tensioninduced (mechanical) killing of bacterial cells by nontranslocating NPs.

Figure 1. Two separate mechanisms of NP-membrane interactions. A) Small NPs or clusters of NPs, with overall diameter less than the thickness of the phospholipid bilayer (10 nm), are either trapped inside the bilayer or form a pore, enabling translocation. [34] B) Large NPs, with diameter greater than the thickness of the cell membrane (10 nm) are wrapped by the membrane, regardless of surface chemistry. [6] The adhesion of large NPs to the lipid membrane leads to cell deformation and/or crumpling as the elasticity of the membrane comes into play. The degree of membrane deformation depends on the balance between the energetic gain attributed to maximization of the contact area between the lipid membrane and the NP and the free energy cost of the membrane deformation. [77] been linked to the formation of large irreversible pores during their translocation across the bacterial cell membrane. [10][11][12] At the same time, there has been an increasing number of communications that have reported the killing of bacteria with NPs in the range of 80-100 nm that are unable to freely translocate across the bacterial cell membrane. [13][14][15][16][17] The mechanism of mechanical forces to bacterial cells. The mechano-bactericidal antimicrobial effect was first exhibited in nature, as a topographical characteristic of insect wings. [21,24,25] The pioneering research in this area has mostly focused on the synthesis of biomimetic nanopillars and nanospikes on surfaces to create mechano-bactericidal topographies. [26][27][28] Nanostructures at the surfaces of bactericidal patterns can rupture bacterial cell walls by stretching the membrane that is adsorbed on the pattern. [1] This mechanism also works for gold-coated natural bactericidal surfaces, [2] showing that it is geometrical parameters of nanostructures that are key factors for this mechanism, not specific interactions or chemical components that are usually attributed to be responsible for the nanotoxicity of nanomaterials. The rupture of cell membranes under mechanical stress is an integral component of the toxicity of nanomaterials. However, the fundamental mechanisms by which NPs might induce mechanical rupture of biological membranes have not yet been determined.

In this study, we provide evidence of the (mechano-)bactericidal mechanism of non-translocating NPs. The development of a biophysical model shows that the adhesion of large, chemically inert NPs leads to increased tension and consequent rupture of the bacterial cell membrane, Figure 1B. This proposed mechanism was studied in two independent systems: a model membrane in a custom-designed microfluidic platform, and in vitro bacterial cell experiments. (Cryo)-transmission and scanning electron microscopy were employed to visualize the impact of spherical and star-shaped NPs of approximately 100 nm in diameter on the viability of Pseudomonas aeruginosa ATCC 9721 and Staphylococcus aureus CIP 65.8 T bacterial cells. The contraction (squeezing) of the lipid bilayer by adsorbed AuNPs of varying surface hydrophobicity and shape and eventual rupture was demonstrated in both systems.

We present a biophysical model of the adhesion of large nanoparticles to a lipid bilayer to evaluate the effects of clusters of different size versus individual NPs on the stretching of the elastic layer, and determine the effects of NP aggregation (clustering) on global membrane stretching. Within this model, the membrane is described as an infinite elastic layer characterized by elastic modulus k. The adsorption of NPs to the elastic layer is described by the interaction parameter, ε per adsorption site, while the density of adsorption sites n0 is assumed to be constant on the surface of NPs. It turns out that these parameters can be absorbed in a dimensionless control parameter antibacterial activity of this size NP has yet to be clearly identified, since the variability in parameters such as concentration,  = n 0 k that describes the interaction with a surface of a NP.

dimensions, physical and chemical properties, as used in different studies, have not produced consistent results. [2,18] The mechano-bactericidal activity of nanomaterials is a new area of research, which promises to combat bacterial adhesion, biofilm formation, and infections with comparable effectiveness to traditional antibacterial methods. [19][20][21] The emergence of antibiotic resistance in pathogenic bacteria over the past few decades has posed a severe risk to public health. [22,23] Likewise, the decrease in the development of new antibiotics, for which bacteria have not yet developed a resistance, calls for the development of innovative techniques to combat multi-drug resistance. This new route of designing antibacterial nanomaterials exploits nanostructure geometry to deliver deadly Within this model ζ describes the interplay between attractive forces and elastic deformation of the membrane bilayer that leads to an equilibrium stretching of the elastic layer and partial wrapping of the spherical NPs by the elastic layer, that is, membrane stretching-compression.

The interaction of non-interacting individual NPs with an elastic layer is considered in the approach similar to the construction of Wigner-Seitz cells as a result of Voronoi decomposition of the surface with individual NPs in the center of the cell. In average, an area that corresponds to an adsorbed NP is determined by the average density of adsorbed NPs at the membrane (Figure 2). Then the area that corresponds to each NP is divided in two regions for the sake of simplicity: A

+ + = S     k  
is the region of cell membrane interacting with the surface of the NP and B is the region of the cell membrane between each adsorbed NP (Figure 2A). Local stretching-compression of the elastic layer is determined by the balance between two energies: the attraction energy to the surface of the NPs, characterized by the energy gain of NPs diameters is shown in Figure 2 for different dimensionless parameters ζ describing the attraction to the bilayer and initial stretching αi.

To evaluate the effect of clustering on the application of forces to the membrane, we assume that the AuNPs are grouped in clusters of spherical shape and C(N) is the distribution function of clusters containing N NPs.

F gain = n (r )d A (1)
For equilibrium distribution of clusters, the free energy of clusters is given by the mean field expression and energy loss due to stretching of the area between NPs and compression on the surface of the NP,

F = kT N = 1 C (N )lnC (N ) 1 + NN  (6)
F loss = + k  2 (r ) n (r ) d ( 2
)
where εN is the energy in terms of kBT of the cluster containing N NPs, kB is Boltzmann constant, and T is the temperature. For A+B 2 n0 spherical clusters it is given by:  =  0 +  . 5 Here  0 is the Here n(r) is the density of attraction points at the surface in the position r and dσ is the element of the surface of the NP, α(r) is the local stretching at the point r. The resulting free energy after minimization with the constraint that the projec-

N 1
N 3 bulk energy and δ is the cohesive energy per NP. Minimization of the free energy with respect to C(N) leads to the equilibrium cluster distribution tion of unperturbed and stretched areas on the surface plane remains constant, yields

C (N )   C (1)e    N (7) n S k   2 S  2 S   S S S 
with the average size of the cluster N   e  , which depends

F = 0 A + A A + 1 +  A 2  1 +  A B B +  A + 1 + B   1 +  A B i 1 + B 1 + i 
(3) on the surface density of NPs at the membrane σ. For spherical aggregates, the distribution C(N) is sharply peaked around the where index i indicates the initial values of area S and stretching α. Here λ is the Lagrange multiplier, which ensures the conservation of the area to the initial area S0 average size. Thus, in the following, we will consider a monodisperse distribution of clusters around the average size where, N is the average size of the cluster. Thus, assuming dense packing of NPs in the clusters, the radius of a spherical cluster

1 d A+B 1 +  (r ) 0 (4)
is given by R = rN 3 , where r is the radius of an individual nanoparticle. With this, the surface density of the clusters φ of equal size is related to the surface density of the nanoparticles σ, The above equations allow for calculation of the stretching and compressions of two regions, A and B, considering that the stretching is uniform throughout the whole region, which is a reasonable approximation for an elastic layer.

The tension of the membrane © is related to the relative stretching of the non-adsorbed part of the membrane αB  (S S0)/S0 as adsorbed at the surface as

 =   R  3 (8) r
In the case of individual NPs, φ = σ, but if the NPs are aggregated in large clusters, the surface density of clusters is significantly lower than surface density of individual NPs.

©= k S S0 S0

(5) However, the stretching of the membrane by large clusters is larger than that imposed by the individual NPs. The model of interaction of clusters with the elastic layer has where S is the actual area of the membrane under tension and S0 is the area of a unstressed membrane. Since the lipid bilayer area compressibility modulus is known, k = 100 mN m 1 , the tension of the membrane and the relative area change are directly related.

Non-aggregating individual NPs interact with the bilayer independently, thus providing an additive effect on stretching and scaling linearly with the number of adsorbed nanoparticles. Therefore, to compare the stretching of NPs of different sizes it is practical to fix the surface fraction of NPs at the membrane, thus comparing equal area fractions of NPs at the surface. As a reference, we choose the density close to saturation of 100 nm NPs, when the distance between each NP is equal to the NP diameters and use this corresponding fraction for NPs of other diameters. Stretching of the membrane as a function the following parameters: the elastic layer is described by the elastic modulus k, and the clusters are described by the diameter, D = 2R, the attractive interaction to the surface, ε, and the surface density of clusters, ϕ. For a fixed interaction parameter ε, layer stretching as a function of the surface density of clusters and cluster diameter is shown in Figure 2D.

We assume that individual NPs and clusters do not interact at the surface and are distributed equally in repetitive arrangement (pure translation) in unit cells. This approximation allows us to study a single cluster of variable size in a square box with periodic boundary conditions (Figure 2). The area of the box L 2 is then translated in the density of clusters,   1 . Although L 2 this approximation neglects cooperative effects and interactions between nanoparticles and clusters, this is a good first approximation for comparison of effects of clustering on the local and global stretching of the lipid bilayer.

As expected, membrane stretching is increased with increased cluster diameter and the density of clusters, which corresponds to a higher concentration of NP. In turn, higher interaction parameters lead to greater membrane stretching. However, it is more interesting to represent these results as a function of a surface density of the nanoparticles composing the cluster, σ as seen in Figure 2C. These plots indicate that if the number of NPs attached to the surface is fixed, the cumulative stretching is higher for dispersed individual NPs (maximum at smaller cluster diameters) than for NPs aggregated in clusters (Figure S2, Supporting Information). Nevertheless, the stretching exerted by one large cluster increases with the size of the cluster (Figure 2D, φ = 0) and one large cluster alone can rupture the membrane. [29] Nonspecific attractive forces that promote nanomaterialmembrane interactions are mediated by inherent nanomaterial characteristics such as surface charge, hydrophobicity, and roughness. Hydrophobic NPs tend to agglomerate and are therefore less likely to undergo membrane interactions. However, for more hydrophilic nanomaterials, surface membrane interactions may be determined by their relative hydrophilicities. Membrane interactions can also be affected by nanoscale surface roughness. NP surface protrusions or depressions have been shown to influence affinity for the cell membrane. Simulations of NP-membrane interactions show that NP surface roughness greatly minimizes repulsive interactions, thereby promoting adhesion. [30,31] In the present work, hydrophilic and hydrophobic 100 nm quasi-spherical (AuNSPs) and star (AuNSTs) shape AuNPs were synthesized to explore the antibacterial mechanism of non-translocating AuNPs as a function of size, shape, and surface charge. [5,32] The surface charge and chemical properties of the NPs were determined by capping agents which play a crucial role in controlling the interactions between bacteria and the NPs (Figure 3). Hydrophilic AuNPs were coated with mercaptohexanoic acid (MHA) while hydrophobic AuNPs were stabilized with a modified sulfobetaine ligand (ZwBuEt). [33] To visualize the adsorption of quasi-spherical and nanostar AuNPs onto a model lipid membrane, a novel 3D chip was developed (Figure S3, Supporting Information), in which a lipid bilayer was formed through the sandwiching of lipid in oil phase (DOPC:DOPS+squalene) between two water phases. The bilayer is stable at least for 2 h. The lipid bilayer formed (final diameter 200  10 m) upon drainage of the oil phase at the intersection of the upper well and the bottom channel and can be viewed optically. AuNPs were introduced into the chip upper well and attachment to the bilayer was visually recorded over a period of 30 min by dark field microscopy.

The AuNSP and AuNST functionalized with MHA or ZwBuET were studied independently. In Figure 4, the lipid bilayer annulus is clearly visible, and its shape could be approximated to a free-standing quasi-2D circle with radius, r, and area A = πr². As seen in Figure 4A,B, the bilayer area diminished with time due to the presence of the AuNSP that were adhering permanently to the entire surface of the lipid bilayer (for MHA and ZwBuET ligand). By contrast, AuNST were found to adhere permanently only at the lipid bilayer annulus (just at the lipid-oil/water interface), and they did not strongly adhere to the lipid membrane (see Figure S4, Supporting Information). AuNST were unable to place the same amount of tension on the lipid membrane with time (for AuNST with either MHA or ZwBuET ligands) to cause a contraction of the bilayer area. The spontaneous bilayer contraction by AuNSP adsorption was also investigated for smaller AuNSPs with effective diameters of 20 and 50 nm (Figures S5, Supporting Information). Thus, the phenomena of the NP-induced contraction of the free-standing lipid bilayer exists for a large of NP size range.

As seen in Figure 4C,D, by plotting the bilayer radius as a function of time, we calculated that the bilayer radius diminished at a rate of 1.4 m s 1 for both AuNSP-ZwBuEt and AuNSP-MHA (at the concentration of 100 g Au mL 1 ), which corresponded to a reduction of approximately 0.7% (of the bilayer total radius) per second. Under our experimental setup, on average, both NPs appeared to consistently shrink the bilayer at a similar rate with the bursting time between 100 and 200 s when the bilayer radius reached 75 m. The estimated tension (T) force was in the order of approximately 0.1-1 mN m 1 (see section Results, Supporting Information).

To quantify the stretching and disruption of the membrane because of quasi-spherical NP attachment, the model membrane was supported in a custom-designed microfluidic set-up (Figure 4A); the formation of the DOPC:DOPS lipid bilayer was confirmed by electrophysiological measurements, which give the capacitance of the bilayer. Capacitance values could be normalized according to the bilayer area. To express the results in specific capacitance Cs values, with Cs = 5.9 mF m 2 for the DOPC: DOPS (9:1) bilayer. Using the electrical replacement circuit of a plate capacitor, the DOPC layer thickness could be calculated as d  ε0εL/Cs  3.3 nm, with vacuum permittivity ε0  8.85  10 12 F m 1 and εL  εoil  2.2. [33,34] Thus, the specific capacitance of the pure bilayer should decrease if AuNPs were either fully or partially inserted into the membrane. [33] From the values of the surface tension and the bilayer contact angle θ, which were obtained from pendant drop measurements or optical microscopy images (Figure 5), respectively, the bilayer tension could be calculated using Young's equation (see Experimental Section).

As can be seen in Figure 5B, there is a significant increase in the as-measured bilayer tension between experiments with no NPs and with AuNSPs. AuNSP-MHA achieved the largest increase in membrane tension. When the experiment was conducted over a range of concentrations (50-200 g Au mL 1 ) there was a linear increase in the measured bilayer tension (Figure S6, Supporting Information). Similarly, when analyzing the capacitance measurements, there is a general decline in membrane bilayer capacitance when AuNSP functionalized with ZwBuEt and then MHA is introduced to the system. This corresponds to attachment, or partial insertion of the NP into the model bilayer. [35] Furthermore, to confirm physical contact with the membrane by AuNSP-MHA/ZwBuEt, a fluorescent lipid bilayer composed of 3% DOPC-Rho and 5 mg mL 1 DOPS extract, was monitored using a fluorescence microscope. The bilayer appeared bent for mechanical stability (Figure 5A). The AuNSP functionalized with MHA appeared to be fluorescent for 30 min following interaction with the lipid bilayer (these interactions were recorded, Movie S1, Supporting Information), thus indicating that the AuNSPs were adsorbed to the cell membrane.

Synthetic lipid bilayers are a convenient model system for NP-membrane interactions that enable detailed physical insight to be obtained thanks to their relative structural simplicity and well-characterized properties; however, real biological membranes are far more complex in structure. In this section we describe the interaction of large NPs with bacterial cells and demonstrate the stretching effect on bacterial cell membranes.

To assess the antimicrobial activity of the two types of AuNPs studied in the model system, the following parameters/factors were considered and investigated: i) the influence of surface hydrophobicity, ii) the impact of the concentration of the NPs (25, 50, 100, and 200 g Au mL 1 ), iii) kinetics of bactericidal activity over time, and iv) the effect of shape (quasi-spherical or star).

Analysis of the bactericidal activity (Figure 6) indicated that quasi-spherical AuNPs with increased hydrophobicity showed less antimicrobial efficiency toward both gramnegative P. aeruginosa and gram-positive S. aureus. Indeed, no significant reductions in viable bacteria were noted for P. aeruginosa when (Figure 6A). By contrast, hydrophilic AuNSP-MHA achieved complete inactivation of both bacterial species after a 30 min incubation period at concentrations 100 g Au mL 1 (Figure 6A,B); significant reductions (p  0.02) in CFU mL 1 were observed at concentrations 10 g Au mL 1 (Figure S7, Supporting Information). The differences in antibacterial activity may be attributed to the variable affinities (due to altered hydrophobicity) of the AuNPs toward bacterial cells. Bidentate zwitterionic (hydrophobic) ligands exhibit excellent colloidal stability under conditions of high concentration, however, when the functionalized NPs are serially diluted, an increase in non-specific adsorption and aggregation has previously been noted. [36] The zwitterionic coating may also impart novel antibiofouling properties to the nanomaterials and functionalized NPs would not exhibit non-specific adhesion to living cells. [37] In this work, under physiological conditions, MHA-NPs (hydrophilic) showed higher stability and less aggregation than ZwBuEt-NPs (hydrophobic) (Figure S1, Supporting Information). The antibacterial efficacy of the AuNPs used in this study was also confirmed via fluorescence microscopy (Figure 6E).

In accordance with the model membrane experiments above, Au nanostars exhibited considerably less antibacterial efficacy toward both P. aeruginosa and S. aureus compared to that of the quasi-spherical NPs and the antimicrobial activity was not greatly affected by surface chemical functionalization. Matching data for the antibacterial effects of AuNSTs are shown in Figure S8, Supporting Information.

To visualize the AuNP-induced disruption of bacterial cells by AuNSPs and AuNSTs on both P. aeruginosa and S. aureus, electron microscopy was used to examine the morphology of the bacterial cells (Figure 7). Bacterial cells that were treated for 2 h with AuNPs showed severe morphological changes. Analysis of the SEM and transmission electron microscopy (TEM) images revealed the attachment of NPs onto P. aeruginosa and S. aureus cell membranes, and in some cases, complete encapsulation of bacterial cells leading to the destruction of the original cell morphology (Figure 7B,C,D). NPs were observed to attach directly onto the membrane causing stretching (Figure 7A) or indentations (Figure 7B) in the membrane or complete cell lysis (Figure 7C).

AuNSPs (and to a lesser extent AuNSTs) adversely affect bacterial cells by inducing high local stress on the bacterial cell membrane at the time of the interactions, eventually causing lysis of the cell (as seen in Figure 7C). It was presumed that the cell damage was not incurred through "internalization" of the nanoparticles due to their large size (approximately 100 nm) and no evidence of internalization was observed, in agreement with previous studies of AuNP anchorage onto gram-positive and gram-negative bacterial membranes. [38] TEM experiments (Figure S8, Supporting Information) show that the AuNST nanospikes are embedded in the bacterial capsule but do not have sufficient length to interact with the bacterial membrane.

Nanoparticle affinity for the bacterial membrane was determined using Cryo-TEM. The brief incubation period allowed for the initial interactions between NPs and bacteria to be visualized, as they would appear in liquid/native conditions (Figure 7G, Figure S9, Supporting Information). The functionalization of the NPs greatly affected the behavior of the NPs toward bacteria and toward each other in bacterial suspension. For quasi-spherical NPs, the hydrophobic zwitterionic ligand (ZwBuEt) did not enhance the particle affinity for bacterial membranes, indeed there seemed to be almost no interaction between bacteria and AuNSP-ZwBuEt. By contrast, AuNSP-MHA appeared greatly attracted to the bacteria in suspension and could be seen encapsulating the cell, as shown in Figure 7G. Weak interactions between the phospholipid bilayer and AuNSTs were additionally confirmed using cryo-TEM. This result is supported by our experimental model demonstrating that nanostars exhibit weaker antibacterial activity because of reduced contact area with the cell membrane. For AuNSTs, those capped with ZwBuEt were observed in large clusters while those capped with MHA were distributed across the grid with no interaction with bacteria. This directly corresponds to the increased rates of antibacterial efficiency observed for AuNSP-MHA over AuNSP-ZwBuEt.

It is known that the exposure of NPs to bacterial cells can lead to membrane damage caused by NP adsorption. In the case of TiO2, CeO2, MgO, and Au-NPs, the interaction with the cell wall is the main source of toxicity to bacteria even though no cell penetration occurs (Table S3, Supporting Information). In such cases, the NPs electrostatically adsorb onto the bacterial cell wall and destroy the cell wall due to the NP aggregation observed only at the cell surface. [39][40][41][42] These studies suggest that NP adsorption on the cell wall followed by its disintegration is the primary reason for toxicity, although no definitive mechanisms have been put forward.

The adsorption of large NPs (where the NP diameter is greater than the thickness of the membrane) on model lipid bilayers has been studied both theoretically and experimentally. This includes studies of wrapping of NPs by lipid membranes, [43][44][45][46] effective interactions at the surfaces and collective behavior of adsorbed NPs. [47][48][49] Electrostatic interactions between the NPs and the membrane are found to be the most important factor for inducing membrane disruption. [29,[50][51][52] In particular, it was shown that large NPs adsorbed on lipid bilayers will always at least partially wrap, [6,53] while membrane wrapping may in turn lead to eventual collapse of the lipid bilayer. [54] Wrapping and membrane tension in the adhesion of NPs are coupled such that low tension membranes allow for higher wrapping and vice versa. [55] Nevertheless, most membrane wrapping studies are focused on NP uptake by mammalian cells whereby membrane wrapping is triggered by many NP-ligand-surface receptor interactions for NP engulfment. Less attention has been focused on NP-membrane interactions of bacterial cells.

We recently demonstrated a mechanism of tension induced permeation of cell membranes by the adhesion of large nontranslocating apatite nanoparticles that provoked nondisruptive permeation of red blood cells (that do not undertake endocytosis). [56] In another study the internalization of large silica nanoparticles into giant unilamellar vesicles (GUVs) was confirmed to be mediated by the spreading forces of the lipid bilayer on the silica surface. The lipid membrane was observed to strongly adhere to the NP surface and "creep" along the silica surface, causing deformation of the liposome. [57] Similar behavior was observed for the adherence of silica particles on red blood cells. [58] The haemolytic activity of silica NPs has been found to increase with increasing particle size due to preferential adsorption of the NPs onto the red blood cell membrane, causing deformation until rupture. [58,59] In the work presented here, the antibacterial performance of the hydrophilic quasi-spherical AuNPs appears to be mediated by NP-membrane interaction which is driven by the attractive force between the NPs and the phospholipid bilayer.

As the particle size increases, the adhesion energy increases, leading to a greater area of contact and overall stretching of the membrane. Individually adhered NPs provoke an additive stretching effect where the total membrane stretching is the sum of individual NP contributions. In the case of NP clusters, the stretching effect is cumulative at the attachment points but does not exceed the tensions delivered to the membrane by single, spatially dispersed NPs. This agrees with a previous model of the hemolytic potential of silica NPs. [60] In the case of clustering between individual NPs, the elastic response of the membrane depends on the size of the clusters and their number. In addition, if the clusters are placed close to each other, the membrane acts as a medium, affecting the interactions between NPs and even inducing their clustering. [48] The developed theoretical model provides insight to explain the results for both experiments: NPs induced tension in the model lipid bilayer and the in vitro bacterial cell killing mechanism showing that low dispersity of NPs achieved greater antibacterial efficiency. Individual NPs were visualized attached to the exterior of the cell, encapsulating it and deforming the shape, by independent experimental techniques, and in some cases, complete cell lysis was observed.

The in vitro model used to mimic the interactions of a phospholipid membrane and NPs demonstrated the delivery of tension forces that resulted in cell perturbation by the attachment of NPs onto the membrane. Previous theoretical models have predicted that NPs can translocate a tethered DOPC lipid bilayer membrane and/or become wrapped in by the membrane, depending on the balance of contact-adhesion energy and curvature energies. [61] The minimum nanoparticle size for such wrapping effects is estimated to be 30 nm, which represents a state of transition between free NP and partially lipid coated NP. [62] In the work presented here, direct physical interactions between the quasi-spherical NPs (both hydrophilic and hydrophobic) and the phospholipid membrane resulted in an increase in membrane tension and decrease in capacitance of the synthetic lipid bilayer that correlated to partial insertion of the NPs into the lipid bilayer. Using fluorescently tagged DOPC, AuNPs were confirmed to contact the lipid bilayer (verified by consequent fluorescence of the NPs through removal of phospholipids from the bilayer); however, no transport across the bilayer was observed, yet the bilayer appeared "bent" for mechanical stability. Hydrophilic NPs delivered the greatest forces to the lipid bilayer upon membrane adsorption, achieving the greatest resultant membrane tension measurements. Hydrophilic quasi-spherical NPs also elicited the largest antibacterial response amongst both shaped particles achieving 100% bacterial cell death at a concentration of 100 g Au mL 1 . Nevertheless, at high concentrations (100 g Au mL 1 ) of both hydrophilic and hydrophobic quasi-spherical NPs large numbers of attached NPs to the lipid bilayer produced enough force to contract the membrane until it collapsed. This is analogous to the squeezing of a balloon until it bursts.

It was assumed that a greater hydrophobicity of the particle would results in greater affinity for bacterial membranes. [38] Chemical modification by bidentate sulfobetaine zwitterionic molecules led to an increase in hydrophobicity; [63] however, NP-membrane interactions were offset by the NP tendency to aggregate into clusters. [64] Herein, the NPs that exhibited low dispersity, such as hydrophilic quasi-spherical NPs, also demonstrated the greatest antimicrobial effects whereas hydrophobic quasi-spherical NPs did not achieve statistically significant reductions in viable cells. Thus, the affinity of the NP toward bacterial cells played an important role in facilitating bacterial cell inactivation. It was further hypothesized that the introduction of spikes to the nanomaterial surface would result in greater cell membrane perturbation. We have demonstrated previously that nanostructuring of surfaces to exhibit an array of high aspect ratio nanoprotrusions, similar to the surface topography of insect wings, can mechanically lyse contacting bacterial cells. [65] Bacterial cell death is mediated by the membrane adsorption to the nanopillars, which leads to wide-spread stretching of the adsorbed membrane [24] (similar to the findings for global stretching of bacterial cell membranes by colloidal nanomaterials presented in this study) by which the mechanism is independent of surface chemical functionality. For example, silicon nanopillar surfaces (black silicon) were the first synthetic surfaces demonstrated to mechanically kill attaching bacterial cells. [19] More recently, nanostructured surfaces of vertically aligned carbon nanotubes, [20] graphene nanoflakes, [66] adsorbed NPs, [67,68] TiO2 nanowires, [28] and more have been demonstrated to kill bacteria by nanomaterialinduced rupture. However, in this work the surface spikes did not enhance the killing effects of the AuNPs, in fact star-shaped AuNPs displayed only very modest antibacterial performance. It is assumed that the introduction of surface spikes actually led to a reduction in the available contact area of the NP surface, as confirmed by model membrane experiments where AuNSTs exhibited a reduced ability to adsorb to the lipid bilayer and were not observed to exert the same forces delivered by quasispherical NPs.

The mechanism of mechanical killing of bacterial cells by non-translocated NPs was elucidated using a combination of complementary experimental and theoretical studies. A biophysical model was developed to explain the induced stretching of a membrane by large with diameters much larger than the thickness of the membrane. Interactions between individual NPs and the elastic bacterial membrane induces an increase in membrane tension due to adsorption of the NPs to the membrane. [44,45,69] The importance of membrane tension for biological membranes is not always considered for fluid membranes, while tension can be even more important for elastic membranes such as bacterial cell walls due to their lack of cytoskeleton to compensate for external mechanical forces. Attractive forces acting between each NP and the membrane tend to increase the area of contact, [55] thus provoking the overall stretching of the membrane. Spatially distributed individual NPs provoke an additive effect, when the overall stretching is the sum of individual NP contributions, while clustered NPs created cumulative stretching at the attach-model lipid bilayer and kill bacteria via adsorption onto the cell membrane and increase of membrane tension. Such an elastic model is not limited by the type of NPs and the origin of the elastic layer. The universality of the model suggests that the described phenomenon of the stretching of the membranes by adsorbing NPs can go beyond the studied NPs and bacteria and it can be applied as a universal mechanism of mechanical stretching and rupture of cells, vesicles, droplets, and other soft objects with NPs. The demonstration of this stretching mechanism in both model and real bacterial system suggests its generality and applicability to a wide range of biological membranes. The results of this work may provide new guidelines for the synthesis of universal antibacterial colloids.

Experimental Section

Design and Fabrication of Microfluidic Device: Microchannels with a rectangular cross section were fabricated using typical soft lithography protocols. [START_REF] Lipomi | Polymer Science: A Comprehensive Reference[END_REF] Channel dimensions were 300 m in width and 140 m in height. The device was molded with an SU-8 photoresist on a silicon wafer using Sylgard 184 (Dow Corning, USA). The surface of the Sylgard 184 devices was exposed to nitrogen plasma (Diener electronic GmbH, Germany) and sealed with a plasma treated glass cover slide. The sealed device was rendered hydrophobic by heating it to 135 C overnight. The liquids were dispensed from syringes (Hamilton Bonaduz AG, Switzerland), which were connected to the microfluidic device by Teflon tubing. Computer-controlled syringe pumps were used to control the injection of the water and oil phases, respectively (Figure S9, Supporting Information). DOPC (1,2-dimyristoyl-sn-glycero-3-phosphocholine) and DOPS (1,2-dioleoyl-sn-glycero-3-phospho-l-serine) were purchased from Avanti Polar Lipids. Experiments were performed at room temperature to have the lipids in a fluid phase. Using a volume-controlled system with syringe pumps, two water droplets were injected face to face into microchannels to produce a free-standing lipid bilayer. After a few seconds, the water oil interface of each finger was covered with a monolayer of lipids molecules. Once the two liquid fingers were brought into contact, the two lipid monolayers interacted to form a lipid bilayer within a short time. [34,[START_REF] Bayley | [END_REF]72] The system was then stable and could be analyzed, simultaneously, using optical microscopy (axiovert Z1, Zeiss) and by electrophysiological inspection. The 3D-microfluidic device was produced based on the principle detailed in ref. [73]. Dark-field measurements were recorded on a Leica DM2700 microscope.

Measurement of Bilayer Tension: The surface tensions of the various lipid monolayers at oil/water interfaces were measured with the pendant drop method using a commercial measurement device (OCA 20, DataPhysics Instruments GmbH, Filderstadt, Germany). An oil solution with a concentration of 5 mg mL 1 lipids concentration (90% DOPC, 10% DOPS) was produced by introducing a droplet from a steel needle into the surrounding oil phase. The shapes of all droplets were fitted with the Young Laplace equation to obtain their interfacial tension γ. The experiments were then re-done in the presence of the two types of NPs (ZwBuEt and MHA). From the values of the surface tension and the bilayer contact angle θ, which were obtained from pendant drop measurements or optical microscopy images (see Figure 6), respectively, the bilayer tension can be calculated using Young's equation. [74] ment points. The cumulative effect of clustered particles may be locally stronger than the tensions delivered by a single NP;

 = 2 © cos  (9)
however, the combined forces administered by separated individual NPs encapsulating a cell would still be greater. Although the nature of the membranes used in both experiments are very different: the DOPC:DOPS lipid bilayer is a fluid membrane, whereas the bacterial cell wall is an elastic layer, both experiments confirmed that chemically inert AuNPs can rupture Capacitance Measurements: Ag/AgCl electrodes were prepared by inserting an electrode in a borosilicate glass pipet (outer diameter 1.5 mm, inner diameter 0.86 mm, Sutter) containing an electrolyte agarose solution (Sigma). The electrodes were carefully introduced into the buffer compartment of the Sylgard 184 device using a micromanipulator. The lipid membrane conductance was measured using the standard function provided by the patch clamp amplifier EPC 10 USB (HekaElectronics). A 10 mV sinusoidal wave with a frequency of 20 kHz was used as an excitation signal.

Characterization of AuNSP and AuNST: Spectrophotometry: UV-vis absorption spectra of the AuNSP and AuNSTs sols were collected using a UV-1800 spectrophotometer (Shimadzu). The spectra were collected in transmission mode.

Dynamic Light Scattering: Hydrodynamic size measurements of the different AuNPs were performed in water using a zetasizer (Malvern Instruments).

Bacterial Cell Growth Conditions: P. aeruginosa ATCC 9721 and S. aureus CIP 65.8 T bacterial strains were obtained from American Type Culture Collection (Manassas, VA, USA) and Culture Institute Pasteur (Paris, France), respectively. Prior to each experiment, bacterial cultures were refreshed on nutrient agar (Oxoid) from stocks (B.D., USA) at 37 C for 24 h. From the stock plate, fresh bacterial suspensions were grown overnight at 37 C in mL of nutrient broth (B.D., USA). Bacterial cells were collected at the logarithmic stage of growth (data not shown). Bacterial cells were then pelleted using centrifugation at 5000 rpm for 3 min and resuspended in 10 mm phosphate buffered saline (pH 7.4). The density of bacterial suspensions was adjusted to OD600 = 0.1 using UV vis spectrophotometry (Dynamica HALO RB-10) at a wavelength of 600 nm.

Antibacterial Assay: Standard plate count methods were used to estimate the number of viable following treatment with AuNPs, P. aeruginosa and S. aureus stock solutions were serially diluted in phosphate buffered saline (10 mm PBS, pH 7.4) until infectious doses were achieved (1000 CFU mL 1 P. aeruginosa [75] and 100 000 CFU mL 1 S. aureus [76] ). The AuNSP and NSTs sols were sonicated for 2 min and added to suspensions of P. aeruginosa and S. aureus (500 L) at concentrations of 5, 10, 25, 50, 100, and 200 g mL 1 . Nanoparticle/ bacterial suspensions were then incubated in a rotary shaker at 37 C for up to 2 h at 220 rpm. Bacterial suspensions in PBS, with no added nanoparticles, but an equal volume of MilliQ H2O, were used as controls for each given concentration. The NP/bacterial suspensions and controls were then spread plated on five separate nutrient agar (Oxoid) plates and incubated at 37 C overnight to estimate the number of viable cells. Statistical Analysis: All plate count experiments were performed in replicates of five and repeated a minimum of three times. Data are expressed as a mean  standard deviation of at least five independent samples. Statistical comparisons were made between results using a two-tailed Student's t-test where p  0.05 was considered significant.

Scanning Electron Microscopy: The morphology of synthesized AuNPs of various shapes and sizes were analyzed using SEM (Raith, GmBH). AuNSPs and NSTs were drop-cast onto plain silicon wafers and allowed to air dry under ambient conditions overnight. SEM images were obtained using a Raith150 Two direct write EBL instrument (Raith, GmBH) with SEM capabilities at 5 kV.

The morphology of bacterial cells incubated with nanoparticles was also examined using SEM (Raith, GmBH). The AuNPs/bacterial suspensions were incubated for 2 h in a rotary shaker at 37 C at 220 rpm and then prepared for SEM by placing a 100 L aliquot on clean silicon wafer, leaving for 1 h for cells to adhere to the substrate, then washing once briefly with MilliQ H2O to remove salt and other contaminants. Cells were then fixed on the substrate with 2.5% glutaraldehyde and the surfaces were then dehydrated with ethanol in 30%, 50%, 70%, 90%, and 100% concentrations. The substrata were kept in 100% ethanol and transferred to a critical point dryer (Polaron E3100, Quorum Technologies Ltd.). The samples were immersed in liquid CO2 via a series of ethanol/liquid CO2 exchanges, and the liquid CO2 was then heated to supercritical temperature and pressure (31.1 C and 1000 psi). The dried samples were then sputtered with gold for 2 min using a NeoCoater MP-19020NCTR (JEOL, Frenchs Forest, Australia). SEM images were obtained using the Raith150 Two direct write EBL instrument (Raith, GmBH) with SEM capabilities at 5 kV at magnifications of 5 k, 10 k,  25 k, and 50 k.

Transmission Electron Microscopy: Suspensions of AuNSP and AuNST sols were drop-cast onto TEM grids and viewed at an accelerating voltage of 200 kV using a Jeol JEM-1010 machine. TEM (JEM-1010, JEOL) was used to view cross sections of bacterial cells incubated with nanoparticles. Samples were incubated with bacteria at concentrations of 200 g mL 1 in a rotary shaker at 37 C at 220 rpm for 2 h. After incubation, cell suspensions were washed twice with PBS (10 mm, pH 7.4) using centrifugation at 1300 rpm for 5 min at 25 C. The pellets were then suspended in 2 mL of 1% glutaraldehyde in PBS for 1 h, and then washed twice in PBS for 5 min. After the final washing step, the cell suspensions were mixed thoroughly with 0.5 mL of molten 4% agarose gel by stirring. The agar was then immediately cooled to 4 C by refrigeration for 30 min, then cut into 1 mm 3 cubes and fixed with 1 mL of 1% osmium tetroxide (OsO4) for 1 h. Samples containing the cell suspension were washed twice in nanopure H2O (with a resistivity of 18.2 MW cm 1 ) for 20 min each. Samples were then dehydrated by passing them through a graded ethanol series (20%, 40%, and 60%) (2 mL) for 20 min on ice and then stained for 8 h with 2% uranyl acetate in 70% ethanol (2 mL). After staining, the cell blocks were further dehydrated by passing the samples through another graded ethanol series (80%, 90%, and 100%) for 15 min each (2 mL) on ice.

The embedding medium was prepared using London Resin Gold (ProSciTech). To embed the samples, each agar sample containing the cell suspension was incubated in 2 mL of 100% ethanol and LR gold monomer (1:1 ratio) for 8 h on ice, followed by a transfer to 100% ethanol and LR gold monomer (1:3 ratio) for 8 h, then finally a transfer into the pure LR gold monomer for 8 h. Each sample was then transferred into a gelatin capsule containing fresh LR gold monomer mixed with 1% dry benzoyl peroxide, which was then polymerized for 24 h at 4 C. The final blocks were trimmed, then cut into ultrathin sections (50 nm thickness) using a Leica EM UC7 Ultramicrotome (Leica Microsystems, Wetzlar, Germany) with a diamond knife (Diatome, Pennsylvania, USA). Sections were placed onto 300 mesh copper grids and examined using a JEM 1010 instrument (JEOL). Approximately 40 TEM images were taken at 10 000 and 20 000 magnifications for each sample analyzed.

Figure 2 .

 2 Figure 2. Global stretching of a lipid bilayer by individual and clustered NPs. A) Schematic representation of a layer adsorbed to NPs of diameter, D, and two membrane regions: adsorbed, A, and suspended, B. Individual NPs are arranged in repetitive cell units. The average surface density of adsorbed NPs corresponds to a distance between NPs, which determines the area of the layer per NP. The same number of NPs grouped in a cluster of nine NPs for the same surface density of NPs, σ. NPs may group in clusters to reduce tension. B) Membrane stretching as a function of NPs diameters for a fixed surface density of NPs at the membrane, σ = 10 4 nm 2 (i.e., the distance between centers that corresponds to full packing for 100 nm NPs) for tensionless membrane and for a membrane under tension. C) Membrane stretching as a function of cluster diameter for fixed surface density of NPs. D) Cumulative membrane stretching αB as a function of the cluster diameter and surface density of clusters φ. E) Cumulative membrane stretching as a function of the cluster diameter and surface density of NPs.

Figure 3 .

 3 Figure 3. Characterization of AuNSTs and AuNSPs. A) Schematic depiction of gold surfaces coated with zwitterionic ligands MHA and ZwBuEt. B) DLS measurements of the different AuNPs in water (pH 7). C) TEM images of AuNSPs and AuNSTs. AuNSTs show stars of uniform conformation with average diameter of 103.9  11.9 nm, an average of 12  2 nm spikes and spike length of 18.2  3.7 nm. AuNSPs are uniformly 100  20 nm.

Figure 4 .

 4 Figure 4. Optical microscopy visualization of the forces placed on the phospholipid membrane. A) Time series of the bilayer contraction due to the presence of 100 g Au mL 1 of 100 nm AuNSP-MHA in contact with the bilayer (bilayer diameter is 1 mm). An AuNSP agglomerate is visible in the last image (t = 183 s), indicated by the black arrow. The scale bar is 500 m and the magnification is 5. At this magnification single NPs are not visible under dark-field microscopy. B)Time-lapse series of microscopy images of the lipid bilayer interacting with the AuNSPs over 6 min observed from top view with dark-field microscopy, magnification 50 in the presence of 100 g Au mL 1 of AuNSP-MHA. C) The measured radius as function of time for five separate lipid bilayers in contact with 100 g Au mL1 AuNSP-MHA and D) the measured radius as function of time for a lipid bilayer in contact with 100 g Au mL 1 . AuNSP-ZwBuEt. Measurements were recorded when the bilayer radius was approximately 200 m. At this high concentration, the bilayer bursts due to interaction with the AuNPs when the radius of the bilayer has been reduced to approximately 75 m (sharp drop of bilayer radii to 0 correlated to bursting, indicated by the cyan arrow). The bursting effect of the phospholipid membrane is consistently observed between 100 and 200 s as confirmed using the two different types of the NPs. The movie showing the squeezing and bursting of the lipid bilayer can be found as Movie S2, Supporting Information.

Figure 5 .

 5 Figure 5. A) The lipid membrane in a microfluidic device observed by optical microscopy, indicated by the black arrow and the lipid bilayer in contact with AuNSP-ZwBuEt and AuNSP-MHA. A distinct difference in the contact angle is visible in the presence of the two AuNP. B) Membrane tension as function of the two types of quasi-spherical AuNSP adsorption on the lipid membrane. C) The measured specific capacitance of the bilayer in contact with AuNSP-MHA and AuNSP-ZwBuEt. Capacitance measurements, Cs = C/A. A concentration of 100 g Au mL 1 of AuNSP-MHA and ZwBuEt was used to measure the changes in membrane tension of the model membrane.From the electrostatic point of view, this is equivalent to a plane dielectric (i.e., the bilayer) in series with a spherical dielectric (AuNP), which is leading to a decrease in the overall capacitance.

Figure 6 .

 6 Figure 6. A,B) Concentration-dependent antibacterial activity of AuNSP toward P. aeruginosa (A) and S. aureus (B) expressed as the number of colonyforming units per mL (CFU mL 1 ). NPs were incubated with bacteria at concentrations of 25, 50, 100, and 200 g Au mL 1 for 2 h. AuNSP showed strong concentration-dependent antimicrobial activity against both bacterial strains. C,D) The kinetics of bactericidal efficiency of the AuNPs over time investigated at specific time points of 0.5, 1, 1.5, and 2 h. AuNSPs showed a steady increase in their antibacterial efficiency over the course of 2 h. AuNSP at concentrations of 50, 100, and 200 g Au mL 1 achieved complete inactivation of S. aureus at the maximum incubation time. AuNSP-MHA achieved 100% inactivation of P. aeruginosa cells at concentrations 50 g Au mL 1 . Data are displayed as a % reduction as calculated from the log10CFU. E) Representative confocal laser scanning microscopy (CLSM) images of P. aeruginosa and S. aureus incubated with AuNSP-MHA and AuNSP-ZwBuEt. CLSM scale bar is 5 m. Bacterial suspensions were incubated for 2 h with 100 g Au mL 1 AuNPs and subsequently stained with propidium iodide (PI) and Syto 9. PI (red) indicates non-viable bacterial cells caused by membrane disruption and Syto 9 (green) indicates viable bacteria.

Figure 7 .

 7 Figure 7. Morphology of bacterial cells post-incubation with AuNPs. A-F) SEM and TEM images of the changes in cell morphology of P. aeruginosa and S. aureus bacteria incubated with 200 g mL 1 AuNSPs for a period of 2 h in comparison to healthy cell morphology. The white arrows (SEM images) and blue arrows (TEM images) indicate areas of membrane deformation due to AuNP adsorption. Interaction between P. aeruginosa and AuNSPs lead to the formation of membrane blebbing (indicated by blue arrow, (A)) and the complete rupture of the bacterial cell (C). AuNSPs were observed to adsorb onto the S. aureus cell membrane (D,D) and cause membrane deformation (B) and eventual loss of cellular integrity (B, inset). E) Cryo-TEM analysis of the AuNP affinity for and attachment to bacterial membranes. P. aeruginosa was incubated with each NP AuNSP-ZwBuEt and AuNSP-MHA in PBS buffer for a brief period (15 min) and then 10 L aliquots were vitrified and G) visualized using Cryo-TEM at an accelerating voltage of 200 kV. The red arrows indicate bacterial cells and blue arrows point out single or clustered NPs. The bacteria/NP suspensions are vitrified on a lacy carbon grid, which provides the lattice structure in the background. SEM scale bars are 1 m.

Acknowledgements X.L.G. would like to thank Cancéropôle Lyon Auvergne Rhône-Alpes (CLARA), Plan Cancer (C18038CS), and ARC (R17157CC) for their financial support. Funding from the Australian Research Council (ARC) Industrial Transformation Research Hubs Scheme (Project Number IH130100017) and ARC Industrial Transformation Training Centre (ITTC) scheme (Project number IC180100005) are gratefully acknowledged. D.P.L would like to acknowledge the Australian Institute of Nuclear Science and Engineering (AINSE) for travel support. J.-B.F. acknowledges funding from the SFB1027 (DFG). P.S. acknowledges funding by NIH R01GM124436 National Institutes of Health. The authors acknowledge at the RMIT University Microscopy & Microanalysis Facility.

Supporting Information

Supporting Information is available from the Wiley Online Library or from the author.

Conflict of Interest

The authors declare no conflict of interest.


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Calibration of the liquid argon ionization response to low energy electronic and nuclear recoils with DarkSide-50

Calibration of the liquid argon ionization response to low energy electronic and nuclear recoils with DarkSide-50

I. INTRODUCTION

Dual-phase noble liquid time projection chambers (TPCs) have yielded, for more than a decade, ever increasing world leading sensitivity for the search for Weakly Interacting Massive Particles (WIMPs) with mass greater than 10 GeV/c 2 [1][START_REF] Cui | PandaX-II)[END_REF][3][4][5]. In recent years, the dual-phase technology has been extended to search for "light" dark matter candidates using the ionization component only, which allows to explore the sub-keV energy range. Experiments like XENON1T with a liquid xenon target [6,7] and DarkSide-50 with liquid argon (LAr) [8,9] have set the best limits on WIMP-nucleus interactions for M χ >1.8 GeV/c 2 (M χ >0.1 GeV/c 2 exploiting the Migdal effect), WIMP-electron scattering for M χ >30 MeV/c 2 , and absorption of dark photons and axion-like particles for M χ > ∼ 0.2 keV/c 2 . This was possible thanks to the intrinsic radiopurity, and high ionization yield and resolution, allowing for detection of single electrons, of noble liquids.

Unlike xenon, for which there exists a rich set of measurements that characterize its ionization response in the keV range (see, for instance. Refs [START_REF] Sorensen | XENON10[END_REF][11][12][13][14]), the argon response is almost unexplored. In spite of the smaller cross section of WIMP scattering on argon compared to xenon, due to the lower atomic number, interactions in argon produce more energetic recoils, with a higher probability of being detected above the threshold. The potential of dual-phase LAr TPC technology in direct dark matter search can therefore be significantly enhanced through a better understanding of the ionization response, especially in the sub-keV region. This represents the focus of this paper, through the measurement of the ionization * Deceased.

response to electronic (ER) and nuclear (NR) recoils with DarkSide-50, using β-decay sources intrinsic to LAr, i.e. 37 Ar and 39 Ar, and neutron sources located just outside the TPC, specifically 241 Am- 13 C and 241 Am- 11 Be. In addition, external datasets from beam test experiments are included in the analysis to better constrain the NR ionization response. This approach, already adopted in the so-called DarkSide-50 low-mass analyses published in 2018 [8,9], is presented in this paper in detail after being highly refined and improved.

Such calibrations not only have the potential to improve current limits with DarkSide-50, but also provide a basis for future detectors specifically designed to search for light dark matter candidates with LAr.

II. THE DARKSIDE-50 EXPERIMENT

The DarkSide-50 experiment is located in Hall C of the Gran Sasso National Laboratory (LNGS) in Italy. The TPC has a cylindrical active LAr target of ∼46 kg The lateral walls are made of polytetrafluoroethylene (PTFE), surrounded by field shaping copper rings. Two arrays of 19 3-in diameter photomultiplier tubes (PMTs) are located on both ends behind the transparent anode and cathode respectively. They observe light signals from both primary scintillation (S1) and electroluminescence (S2) from ionization electrons, extracted in the gas phase, after being drifted with a 200 V/cm field in liquid. The extraction efficiency is estimated >99.9% at the extraction field of 2.8 kV/cm. The PTFE and the fused silica windows at the top and bottom of the cylinder are coated with tetraphenyl butadiene (TPB), a wavelength shifter, that absorbs the 128 nm photons emission and re-emits softer photons with a peak wavelength at 420 nm.

The TPC was first operated, between 2013 and 2014, with atmospheric argon, and then until 2019, with low-radioactivity argon extracted from deep underground, naturally shielded against cosmogenic isotope production. For a more detailed description of the TPC, see ref. [15,16].

The TPC is surrounded by a hermetic neutron veto, a 4 m diameter stainless steel sphere filled with 30 t of liquid scintillator, loaded with trimethyl borate molecule (TMB) at 5% mass fraction. 110 8-inch PMTs are mounted on the inner surface of the sphere to detect the scintillation light. Neutrons are mostly captured by 10 B present in TMB via the 10 B(n, α) 7 Li * and 10 B(n, α) 7 Li reactions, with a mean capture time of ∼22 µs [17], thanks to the large 10 B cross section for thermal neutron. The first reaction, with 93.6% branching ratio, is also responsible for the emission of a characteristic 478 keV γ from the 7 Li * de-excitation, efficiently detected thanks to the relatively high light yield of the scintillator (∼530 pe/MeV).

The neutron veto is in turn immersed in 1000 t of ultrapure water, instrumented as a Cherenkov veto against cosmic muons, and passive shield against external background. See ref. [17] for more details on the veto system.

III. DETECTOR RESPONSE MODEL

The DarkSide-50 ionization signals are affected by instrumental effects, like the smearing induced by gasphase electroluminescence and by the PMT charge response. The measured S2 yield (g 2 ), defined as the mean number of photoelectrons per ionization electron extracted in the gas pocket, and the associated relative resolution are 23±1 pe/e -and ∼27%, respectively, for events localized beneath the central PMT [8]. An additional instrumental effect is related to the electron lifetime, i.e. the survival time of an electron to the capture from impurities in LAr along the drift. This was measured to be about 10 ms, almost 30 times longer than the maximum drift time in the TPC (376 µs [5]). The distortion of the ionization signal induced by the electron lifetime is therefore limited to a few percent.

The dominant instrumental effect is the dependence of the S2 response on the event radial position on the xyplane, orthogonal to the electric field. The S2 response yields a factor of ∼4 difference between the centre and the edges of the TPC. This has been assessed using the 41.5 keV line from the 83m Kr gas source injected into the active mass, and applying the position reconstruction algorithm [18], which provides a sub-cm-level resolution. Such a distortion may be due either to a nonuniformity of the TPB thickness, on the top fused-silica window, which would produce non-uniform conversion of VUV scintillation, or to the sagging of the window itself, resulting in a varying thickness of the gas pocket, and, in turn, of the number of electron-luminescence photons produced.

S2 pulses can be corrected using the radial-dependent efficiency measured with 83m Kr data. However, the en-ergy range of interest for the low-mass analysis extends down to ∼100 eV, where the reconstruction algorithm is inefficient because of the low number of S2 photoelectrons. The analysis published in 2018 [8] overcame this issue by exploiting a channel-based correction that defines the event position as that of the top-plane PMT (Ch max ), which observes the largest fraction of photoelectrons. The position correction based on this definition is rather coarse, because it does not take the finite size (3-inches) of the DarkSide-50 PMTs into account.

To improve the accuracy of the measurement of the intrinsic LAr response to ionization signals, all effects mentioned in this section have been incorporated into a Monte Carlo simulation. This approach has been validated on a sample of 37 Ar decays, naturally present in LAr. This cosmogenic background decays via a single electron capture transition, ground-state to ground-state, with a half-life of 35.01 (2) d [START_REF] Bé | Table of Radionuclides, Monographie BIPM-5[END_REF].

The dominant 37 Ar decay branches are from the electron capture on K (2.83 keV) and L1 (0.277 keV) atomic shells, with branching ratios of ∼90.4% and ∼8.4%, respectively, determined with the BetaShape code [START_REF] Mougeot | [END_REF]21] using the latest recommended Q-value of 813.87 [START_REF] Mougeot | [END_REF] keV [22]. The emitted cascades of electrons, X-rays, and UV photons are evaluated with the RELAX software [START_REF] Cullen | Program RELAX: A code designed to calculate atomic relaxation spectra of x-rays and electrons[END_REF], based on the latest EADL2017 library of atomic transition data [START_REF] Cullen | A Survey of Atomic Binding Energies for Use in EPICS[END_REF]. RELAX calculates atomic relaxation spectra of UV photons, X-rays and Auger electrons, hereafter referred to as "primaries", due to bound-state-to-boundstate transitions for a single initial vacancy in the different sub-shells. From the atomic transition probabilities, a deterministic step-by-step propagation of the vacancies was made up to the valence shell and to the neutralization. Atomic configuration has been accounted for, rejecting transitions that would require more electrons in a sub-shell than those actually present. It should be underlined that the only information available in the EADL2017 library is for a single atomic vacancy. In principle, each additional vacancy due to an Auger transition would require a complete recalculation of the atomic energies, wave functions and transition probabilities, which is not considered in this work.

The relaxation paths estimated by RELAX are 5213 and 72 for the K and L1 shells, respectively. Each contribution is determined per initial vacancy and weighted by the corresponding capture probability. The average energies of Auger electrons, UV photons and X-rays grouped by shell, are quoted in Table I. We can assume that the primary particle is almost always an electron, either because directly emitted or because extracted by X-rays or UV photons via photoelectric effect. The K-and L1shells emit, on average 3.9 and 2.8 primaries, respectively, neglecting UV photons with energy not sufficient to photoionize an 40 Ar atom. To evaluate the ionization yield, the mean number of primaries will be subtracted from the number of detected electrons, as primaries are also drifted and extracted in the gas pocket and contribute to the S2 signal. 37 Ar K and L1 shells as determined using RELAX [START_REF] Cullen | Program RELAX: A code designed to calculate atomic relaxation spectra of x-rays and electrons[END_REF] and the EADL2017 library [START_REF] Cullen | A Survey of Atomic Binding Energies for Use in EPICS[END_REF]. The capture probabilities were evaluated with BetaShape [START_REF] Mougeot | [END_REF]21].

All energies are expressed in eV.

The two lines from K and L1 electron capture are observed in the DarkSide-50 data by subtracting the latest ∼500 days of the underground argon campaign dataset (from 2015 to 2018), where 37 Ar is almost entirely decayed, from the first ∼100 days, normalizing the two samples by their livetimes. The subtracted spectrum is fitted with simulated spectra, generated by independently varying the average number of detected electrons (N e ) for each of the two 37 Ar lines. Events are simulated with a uniform spatial distribution in the TPC. The intrinsic fluctuations induced by the number of emitted particles and by the LAr ionization response to ERs are modelled with an empirical fudge factor implemented as a Fano factor [START_REF] Fano | [END_REF]. In the Monte Carlo, a fraction of electrons is suppressed according to the electron lifetime. The surviving free electrons are extracted in the gas pocket with 100% efficiency, and converted into S2 photoelectrons with g 2 that varies depending on the xy-position of the event, according to relative efficiency derived with the 83m Kr calibration source.

The fit is performed with a χ 2 analysis, where the free parameters are the numbers of extracted electrons induced by the two 37 Ar lines and the Fano factor. It is worth mentioning that no additional correction is applied to the resolution model. The shoulder-like structure observed at ∼35 e -in Figure 1 originates from the non-uniform radial response of the detector and is well reproduced by the Monte Carlo starting from input parameters only.

The resulting numbers of electrons are 12.0±0.1(stat.)±0.5(syst.) and 48.2±0.2(stat.)±2.1(syst.) for the 37 Ar L1-and K-shell, respectively. The systematic errors are calculated by propagating the uncertainty on the g 2 value. The best-fitted simulated spectrum is shown in Figure 1 together with data, as a function of the reconstructed number of electrons. The fit returns a reduced χ 2 of 82.4 / 64. The ratio between the measured amplitudes of the two lines, equal to 0.10±0.01, is in excellent agreement with ∼0.093, the expected value determined with BetaShape, and with measured values in literature (0.103±0.003 [26], 0.102±0.004 [27], 0.098±0.003 [28]).

The fitted Fano factor, 0.10±0.03, is compatible with predictions on the LAr ionization fluctuation from the Shockley (0.107) and the Alkhazov (0.116) models [29]. This result suggests that intrinsic fluctuations of the number of particles emitted in atomic cascades are negligible compared to the ionization fluctuation.

As already mentioned, in order to extract the number of ionization electrons from the two lines, the number of emitted primaries in the atomic cascades must be subtracted from the number of detected electrons. In the L1-shell cascade, the mean number of primaries is 2.8. The 25 eV UV has high probability of extracting an electron by photoionization, but with insufficient energy to induce additional ionization electrons.

The 51 eV M-shell electron is sufficiently energetic to ionize up to two atoms. The Thomas-Imel model, discussed in the following section, predicts a suppressed ionelectron recombination probability at such low ionization densities [30], a finding confirmed by numerical simulations [31]. Yet the energy lost to excitations is unknown at such low energies. Therefore because of uncertainties on the ionization mechanism, we conservatively assume that the 51 eV line contributes with 1±1 ionization electrons to the L1-shell cascade. In addition, we assume negligible interactions between the 51 eV and the L-shell 179 eV electron-ion clouds because of their low ionization density. The resulting number of ionization electron at 179 eV is 8.2±1.3, where the uncertainty takes into account both statistical and systematic errors, corresponding to a ionization yield of 45.7±7.0 e -/keV er .

Unlike the L1-shell, the superposition of ∼3 ionelectron clouds from K-shell electrons of 2414 eV (or 2634 eV X-rays) and 179 eV (1.8 multiplicity) cannot be ne-glected. However, the lack of a model able to describe the complex event topology does not allow to estimate the overall recombination effect. For this reason, we do not include data from 37 Ar K-shell in the analysis, discussed in the following section, for the determination of the ER ionization yield.

IV. ELECTRONIC RECOIL IONIZATION YIELD

The calibration of the ER energy scale relies on 37 Ar data, discussed in the previous section, and on the cosmogenic 39 Ar β --decay sample from the 2013-2014 atmospheric argon (AAr) campaign [15], acquired with the same drift field of 200 V/cm. In AAr, 39 Ar has a specific activity of ∼1 Bq/kg [15], and dominates the event rate. To suppress the "external" background from radioactivity in detector materials surrounding the active mass, events are selected within a central cylinder with 2 cm radius and 21.6 cm height, 16.8 cm far from the lateral walls and 7 cm from the top and bottom of the TPC. The very narrow cut in radius selects events corresponding to the innermost area of the central PMT only, minimizing the non-uniformity of the detector response. To further remove residual external contamination, events with more than one S2 pulse are rejected. These are multiple scatter events, and not compatible with the topology of the 39 Ar β-decay signature.

The kinetic energy of each event is reconstructed exploiting the full anti-correlation between the S1 and S2 signals, through the so-called "rotated" energy variable

E er = w S1 g 1 + S2 g 2 , (1)
where w=19.5±1.0 eV [32] is the average energy required to produce a quantum (excitation or ionization) and g 1 the S1 collection efficiency (0.16±0.01 [18]). The number of ionization electrons escaping the ion-electron recombination process, N i.e. , is calculated for each event of the 39 Ar sample as

N i.e. = S 2 g 2 -1, (2)
which accounts for the subtraction of the primary electron from the β-decay. The mean value of the ER ionization yield, Q ER y , is estimated using eqs. 1 and 2 for each 0.2-keV er bin. A lower threshold of E er >1.7 keV is applied to guarantee 100% efficiency in the identification of the S1 pulse. The uncertainty is dominated by the systematics from the g 2 parameter.

The ionization yield per unit of ER energy from 37 Ar and 39 Ar data is defined as where N i is the number of produced electron-ion pairs. The electron recombination probability (r) is predicted at low energies, O(keV er ), by the Thomas-Imel box model [30],

Q ER y = N i.e. E er = (1 -r)N i E er , (3)
1 -r = 1 γN i ln(1 + γN i ), (4)
where γ is a free parameter describing the recombination of the initial electron-ion pairs contained in a box and immersed in an electric field. This model has proven to work well in noble liquids for spatially short tracks [33], in the O(keV er ) range for ERs. In this range, Eq. 3 can be parametrized as

Q ER y = 1 γ ln(1 + γ ρ E er ) E er , (5)
with ρ=N i /E er and γ the free parameters of the model. γ can be expressed as C box /F , where F is the drift field (200 V/cm) and C box depends on the mean ionization electron velocity and on the size of the ideal box containing the electron-ion cloud. In this parameterization, we assume the approximation of a constant excitationto-ionization ratio, which implies that N i is proportional to the deposited energy [18]. The impact of this approximation is discussed at the end of this section. 39 Ar data, together with the 37 Ar calibration line, are fitted with Eq. 5 up to 3 keV er . As shown in Figure 2, the Thomas-Imel box model is in good agreement with 37 Ar and 39 Ar data points with E er <3 keV er .

The extension of the model to higher energies through the empirical Doke-Birks parameterization [34], in good agreement with data from the ARIS experiment above ∼40 keV er [35], is not compatible with 39 Ar data in the 3-20 keV er range. Instead, the agreement with 39 Ar data is recovered in the whole data range, as shown in Figure 2, by adding a custom term to Eq. 5, as follows

Q ER y = 1 γ + p 0 (E er /keV er ) p1 ln(1 + γ ρ E er ) E er , (6)
with two free parameters, p 0 and p 1 .

The fit yields C box =9.2±0.9 V/cm, ρ=54.4± 7.3 keV -1 er , p 0 =0.11±0.03, and p 1 =1.71±0.08. The C box and ρ parameters are compatible within 1 σ with 8.6±1.5 V/cm and 52.7±10.9 keV -1 er , respectively, obtained from the fit with the Thomas-Imel box model up to 3 keV er . It is worth highlighting that the extrapolation of the ER ionization yield below the lowest measured energy (179 eV er from the 37 Ar L1-shell electron) is weakly dependent on the custom term introduced in Eq. 6 as it is mainly driven by the Thomas-Imel box model.

To test the impact of the constant excitationto-ionization ratio assumption, an energy-dependent parametrization was introduced in Eq. 6, so that

ρ → ρ 1 + α(E er ) , (7)
where

α(E er ) = 0.21 1 + e (Eer-b1)/b2 , (8)
is a sigmoid function tending to the excitation-toionization ratio (N ex /Ni) of 0.21, as measured at high energies [34]. The fit of 37 Ar and 39 Ar data, assuming the change of variable as in eq. 7, requires two additional free parameters, and does not lead to significant variations of Q ER y with respect to the constant ρ approximation. Therefore, we conclude that data are not sensitive to either assumption and opt to retain the formalism with constant N ex /Ni that represents the simplest model.

V. NUCLEAR RECOIL IONIZATION YIELD

The NR ionization yield (Q N R y ) is formalized in analogy with the ER one in eq. 3,

Q N R y = N i.e. E nr = (1 -r)N i E nr (9)
where the electron recombination probability, r, is described by the Thomas-Imel box model in eq. 4.

Under the assumption that the excitation-to-ionization ratio is constant, N i can be expressed [36] as:

N i = β κ( ) = β s e ( ) s n ( ) + s e ( ) ( 10
)
where β is a normalization constant taken as the second free parameter of the model together with C box . The dimensionless parameter κ represents the energy lost in electronic excitations giving rise to ionization and scintillation signals. s e is the rate at which electrons are excited by inelastic collisions while s n is the rate at which energy is transferred to recoiling nuclei by elastic collisions. They all depend on the dimensionless parameter ( ) defined as:

= a 2e 2 Z 2 E nr /keV 0.0135 E nr /keV (11)
where the Thomas-Fermi screening length a = 0.626 • a 0 • Z -1/3 [36] is used with a 0 = h/(αm e c) 0.529 • 10 5 fm.

The stopping power, s e , can be expressed as [36]:

s e ( ) = 0.133 Z 2/3 A 1/2 F (v/v 0 ) √ 0.145 F (v/v 0 ) √ ( 12
)
where F (v/v 0 ) is a correction factor dependent on the nuclear (v) and Bohr (v 0 = e 2 /h) velocities. With no available theoretical calculation backing either a suppression or an enhancement of the electronic stopping power, we assume F (v/v 0 )=1. However, at the end of this section, we define an ad-hoc function as in Ref. [36] to test the sensitivity of the calibration data to a potential suppression of s e ( ) at low energies. The nuclear stopping power, s n , is modelled by Ziegler et al. using an universal screening function [START_REF] Biersack | Ion Implantation Techniques[END_REF]:

s n ( ) = ln(1 + 1.1383f Z ) 2[f Z + 0.01321(f Z ) 0.21226 + 0.19593(f Z ) 0.5 ]
(13) where f Z 0.953 is a conversion factor for argon that accounts for the slightly different dimensionless energy definition used in Ref. [START_REF] Biersack | Ion Implantation Techniques[END_REF] compared to Eq (11).

The NR response model is applied to simulated events analogously to what done for ERs in Section III. The only difference is in the intrinsic resolution of the NR ionization process. Following the procedure used in Ref. [8], we considered two extreme models: one allowing for fluctuations in energy quenching, ionization yield, and recombination processes obtained with binomial distributions, and an other where the fluctuations in energy quenching are set to zero. The analysis described in the below cannot distinguish between the two models, as the difference in the results is negligible. For this reason, in the following, we only consider the model without quenching fluctuations.

The model from Eq. 9 is constrained by fitting DarkSide-50 calibration data, using 241 Am- 13 C and 241 Am-9 Be neutron sources [START_REF] Agnes | [END_REF], acquired during the underground argon (UAr) campaign, and external datasets from the SCENE [39], ARIS [35], and Joshi et al. [40] experiments, as described in the following. FIG. 3. Schematic top view of the detector. The source holder is located in the liquid scintillator veto. Source events have to cross the cryostat, the LAr buffer and the PTFE walls before reaching the active mass. Events whose position is associated to channels in the outer ring (white) are not included in this analysis. Only events with maximum fraction of light observed by PMTs highlighted in green in the figure are included in the AmC analysis. The analysis of the AmBe source is extended also to events selected by PMTs highlighted in red.

A. 241 Am-13 C data selection

The 241 Am-13 C (from now on AmC) neutron source [41] is located outside the DarkSide-50 cryostat, in the liquid scintillator veto, as shown in Figure 3. The source emits neutrons via (α, n) on 13 C, producing 16 O in the ground level in the final state. First or second excited states, accompanied by γ emission from 16 O * deexcitation, are suppressed by a thin degrader, which reduces the α energy below that needed to reach the lowest excited state of 16 O.

A 2 mm thick lead shielding absorbs 241 Am X-rays, resulting in a neutron source with very low correlation with γ and X emission. However, to compensate for the low efficiency in neutron production, the 241 Am activity is rather high (∼3.6 MBq), producing pile-up X-rays and γs with a non-negligible probability of escaping the shielding and reaching the active volume of DarkSide-50, crossing the cryostat, the LAr buffer surrounding the TPC, the field-shaping copper rings, and the PTFE walls housing the active volume.

The 241 Am γ-ray with the highest branching ratio (35.9%) has an energy of 59.5 keV that falls in the regime dominated by the photoelectric effect and is fully absorbed in the LAr buffer and preceding materials. 241 Am γ-rays with an energy >99 keV and branching ratio Comparison between AmC spectrum (black) and contamination from TPC intrinsic events and γs from 241 Am, as described in the text. The low-energy excess is due nuclear recoils induced by neutrons.

>10 -9 , as quoted in [START_REF] Firestone | Table of isotopes: 1996 CD ROM Edition[END_REF], are simulated with G4DS [18], the DarkSide-50 Monte Carlo, to derive the spectrum of AmC uncorrelated events. The detector response is applied as described in Section III.

The AmC NR analysis is performed on events with Ch max corresponding to channels 24, 25, 30, and 31, the 4 central PMTs less exposed to the source, as shown in Figure 3, with 241 Am γ contamination minimized to 5.2%, as estimated from a Monte Carlo simulation. Events selected by channels 26, 29, and 35 (see Figure 3) are excluded from the analysis because of the high rate of 241 Am γs. Events not correlated with the source are subtracted using data from the UAr campaign, selected with the same Ch max cuts and normalized by the livetime.

Figure 4 shows AmC and UAr data spectra, the latter normalized by the livetime, and the simulated spectrum of γ-events from 241 Am reaching the TPC. The uncertainty on the source position affects the amplitude of the simulated spectrum, which is then normalized to the UAr-subtracted AmC spectrum in the [250, 900] N e range, where no NR event is expected. The excess of events from the source, shown in Figure 4, with respect to the TPC intrinsic contamination and to simulated 241 Am γ-rays, is attributed to neutron scatterings.

B. 241 Am-9 Be data selection

The 241 Am- 9 Be source (from now on AmBe) emits neutrons in association with the emission of, among others, 4.4 MeV γs. The prompt γ signal is detected in the liquid scintillator veto [17], where the source is deployed. Most neutrons, once scattered in the TPC LAr target, escape the TPC given the low neutron capture cross section in 40 Ar. Scattered neutrons that reach the neutron veto are mostly captured by 10 B with 22 µs mean time.

NRs in the TPC are selected with a three-fold coincidence by looking at the prompt 4.4 MeV γ, followed by a single-scatter signal in the TPC within a few tens of nanoseconds, in turn followed by the delayed neutron capture in the veto. In order to apply this selection, events with both S1 and S2 pulses are required. The events with S2 signal only, in fact, are delayed by the drift time up to a maximum of 376 µs, much longer than the coincidence time with the prompt signal in the veto, which, in most cases, falls outside the acquisition window. On the contrary, the S1 signal is very fast, and allows not to miss the coincidence with the prompt signal in the veto. The requirement of an S1 signal, however, implies an energy threshold, dictated by the S1 trigger and pulsefinder efficiencies. The overall efficiency for NR events is defined as the ratio between 2-pulses events (S1+S2) and all events with either S2-only or S1+S2. The efficiency is extracted from the AmC dataset, by subtracting the ER contamination from each of the two S2-only and S1+S2 samples. The resulting efficiency, which from now on is named S1 detection efficiency, is fitted with an error function, which takes into account Gaussian fluctuations expected in the S1 efficiency, as shown in Figure 5. Prompt signals in the veto are selected by requiring an electron equivalent energy in the [4,5] MeV range, and a time difference with respect to the TPC between -600 and -500 ns, taking into account the time offset of -550 ns between veto and TPC. The delayed signal is selected in the [START_REF] Sorensen | XENON10[END_REF]45] µs time window, and with event energy in the [370, 510] keV range, a range properly tuned to account for quenching effects and γ energy deposition in passive materials. The 10 µs low time threshold is applied to avoid veto PMT afterpulses. The selection cuts are shown in Figure 6.

The AmBe events selected in triple coincidence still suffer from a residual contamination due to accidental ERs. Such a contamination is account for by looking in control region where no NR is expected. The control region is defined selecting events with N e in the [200,500] range and with f 90 <0.4, where f 90 is the pulse shape discrimination estimator of DarkSide-50 [15], less than 0.4, a region where no NR is expected. The UAr spectrum is normalized with respect to the same selection criteria, and subtracted from the AmBe one. The contributions from AmBe and non-source events are shown in Figure 7.

C. External datasets

A further constraint to the LAr response to the ionization signal is provided by "external" datasets, i.e. measurements perfomed with small scale LAr detectors exposed to neutron beams. The SCENE collaboration [39] measured the ionization yield for 4 NR energies, between 16.9 and 57.3 keV, with g 2 =3.1±0.3 pe/e -. The drift field at 193 V/cm is very close to one used in DarkSide-50 (200 V/cm), and the difference is assumed negligible in this analysis. SCENE results are normalized to the DarkSide-50 response by the ratio between the corresponding g 2 's.

The ARIS collaboration [35] characterized the LAr scintillation response at 200 V/cm for 8 NR energies, between 7.1 and 117.8 keV. ARIS S1 data at 200 V/cm are rescaled to the DarkSide-50 response by the ratio between field-off S1 yields of DarkSide-50 (8.0±0.1 pe/keV) and ARIS (6.35±0.05 pe/keV). This allows to associate the NR nominal energies from ARIS to the correspondent S1 at 200 V/cm in DarkSide-50. The final step is the conversion of the so-obtained S1 values to S2, by looking at the S2/S1 ratio from NRs selected with the triple coincidence in the AmBe dataset. The correlation between S1 and S2 signals for the AmBe source was accounted for by means of Monte Carlo simulations, which embeds the DarkSide-50 energy and optical response models [18]. The resulting ionization yield is slightly lower at low energies than that published in 2018 [8]. This difference is due to the improved modelling of the detector response.

Joshi et al. [40] have measured the ionization yield of NRs at 6.7 keV using the endpoint of a spectrum induced by monochromatic 70 keV neutrons from a beam. The measurement at 240 V/cm, the closest field to the DarkSide-50 one, results in Q y =3.6 +0. 5 -1.1 e -/keV. However, after consulting with the authors, the data point is corrected for their single electron yield using the 2.82 keV K-shell capture 37 Ar line from their experiment and DarkSide-50 as a cross-calibration point. The corrected value is then Q y =6.0 +0.8 -1.8 e -/keV. However, because of this correction, we preferred not to include the measurement by Joshi et al. in this analysis, but to quote it in the comparison with the final result.

D. Global fit to data

The energy Monte Carlo spectra for fitting both AmC and AmBe data samples are generated with G4DS, which accurately describes detector and source geometries, with a statistics of about 10 5 neutron events, between 1 and 2 orders of magnitude higher than the data statistics. The detector response is applied with a toy Monte Carlo approach, as described earlier, by varying β and C box parameters from the model with a fine scan. In addition, in order to take into account the inefficiency on S1 pulses, the efficiency curve shown in Figure 5 is applied to AmBe simulated events as a function of N e .

The AmC and AmBe datasets are simultaneously fitted with a χ 2 analysis in the [3,250] N e range. The fit includes g 2 as an additional fit parameter constrained to the measured value within its uncertainty. To minimize the statistical fluctuations, the χ 2 value is averaged over 800 toy simulations for each (β, C box ) parameter pair. The resulting χ 2 map in the (β, C box ) parameter space is summed to the one from the simultaneous fit of the external datasets from SCENE and ARIS presented in the previous section.

The resulting best parameters of such a global fit are C box =8.1 +0.1 -0.2 V/cm and β=(6.8 function of the nuclear recoil energy in Figure 8. The errors on the model are from statistical uncertainties and systematic error on g 2 . The Q N R y extracted from external datasets only is in excellent agreement with the one from AmC and AmBe calibrations, as shown in Figure 8, and the reduced χ 2 from combined fit of all the datasets is equal to 1.34 (χ 2 /NDF = 676/506). Figure 9 shows the comparison between data and model, for both AmC and AmBe, assuming best fitted parameters. The model is constrained at low energy by the AmC calibration data, with a minimum energy value of 435 +47 -34 eV nr , corresponding to 3 electrons. This is the lowest NR calibration threshold ever achieved in LAr.

Regarding potential systematics, we have investigated the impact of S1 detection efficiency on the result for 5 low thresholds, ranging from 0 to 40 N e , applied to the AmBe dataset. The results obtained by repeating the fit for each threshold are in agreement within 1 σ with the one without threshold, confirming that the S1 detection efficiency does not induce any appreciable systematics.

E. Sensitivity to other theoretical models

We have investigated the sensitivity of our data to different models of the nuclear stopping power as a function of energy stemming from different assumption on screening effects from atomic electrons. This is encapsulated in screening functions arising from different models that can be written as [36] f

(η) = λ η 1-2m 1 + 2 λ η 2(1-m) q 1/q , (14)
with each model characterised by a different set of parameters. The additional tested models, as suggested by Bezrukov et al. [36], are Molière (m = 0.216, q = 0.570, λ = 2.37) [START_REF] Moliere | [END_REF], and Lenz-Jensen (m = 0.191, q = 0.512, λ = 2.92) [44,45]. Each of them leads to a different nuclear stopping power through

s n ( ) = 1 0 f (η) dη, (15)
with from Eq. 11. The Thomas-Fermi screening function, also investigated by Bezrukov et al. [36], is not included in this study as it is relevant when the projectile is a naked nucleus or an elementary particle, but not for partially ionized atoms. The above-described analysis was performed for the Molière and Lenz-Jensen models and was found that both can fit successfully the data, with no statistically significant difference among them. The Q N R y 's obtained from the fit using the different screening functions are shown in Figure 10. The comparison between model and data for two cases, Ziegler et al. and Molière, is shown in Figure 9.

The Ziegler et al. model is the one yielding the lowest Q N R y in the region of interest for WIMP analysis. Therefore the adoption of this model will result in the more conservative choice for the sensitivity to WIMPs in future DarkSide searches. FIG. 10. Fit of DarkSide-50 AmBe and AmC data, and ARIS and SCENE datasets using Ziegler et al. (eq. 13) [START_REF] Biersack | Ion Implantation Techniques[END_REF], Molière [START_REF] Moliere | [END_REF], and Lenz-Jensen [44,45] screening functions. The model bands correspond to 1σ uncertainty.

Following the same conservative approach, we also explored the impact of a low-energy s e suppression by introducing the functional form of F (v/v 0 ) in eq. 12 as suggested in [36]:

F (v/v 0 ) = 1/2 (1 + tanh(50 -z)), (16)
where F (v/v 0 )→1 for z → -∞. Setting z = 0.25 enables to reproduce the attempt in [46] to include Coulomb effects in the calculation of the electronic stopping power. Such a suppression, demonstrated to be not compatible with existing LXe datasets [36], affects the energy range in LAr below ∼3 keV nr , as shown in Figure 11. Through a null hypothesis test, where the null hypothesis corresponds to F (v/v 0 )=1, we verified that low energy AmC data have the power to constrain suppression effect to be within ∼18% (2 σ) at 1 keV nr below the nominal charge yield derived from the universal function. This test allows to exclude z > 0.04 or Q N R y (1 keV nr ) < 5.6 e -/ keV nr at 95% C.L..

VI. CONCLUSIONS

In this work, we presented the calibration of the LAr ionization response to electronic and nuclear recoils in the keV region at 200 V/cm with DarkSide-50. The electronic recoil one is measured down to 179 eV er , the energy of the L1-shell Auger electron from 37 Ar, and extrapolated down to few tens of eV by fitting data with the Thomas-Imel box model. The nuclear recoil ionization response is measured with a low threshold of ∼500 eV nr , corresponding to 3 ionization electrons, the lowest ever performed in LAr. The measured ER and NR ionization yields will impact direct dark matter searches with LAr, extending the observation window to low-mass candidates, like Weakly Interacting Massive Particles of few GeV/c 2 mass, axion-like particles, dark photons and sterile neutrinos, and to neutrino bursts from core-collapse supernovae [START_REF] Agnes | DarkSide 20k)[END_REF]. Dedicated campaigns of measurement

FIG. 1 .

 1 FIG.1. Comparison between the best fitted simulated spectrum and37 Ar data as a function of the reconstructed number of electrons. The free parameters in the model are the mean number of ionization electrons induced by the 37 Ar L1-and K-shell electrons, and the Fano factor.

FIG. 2 .

 2 FIG.2. Fit of the ER ionization yield, measured from AAr (black) and37 Ar (teal) data with a drift field of 200 V/cm, with the Thomas-Imel box model up to 3 keVer (eq. 4), and with the custom model from eq. 6, which extends the Thomas-Imel model up to 20 keVer. The model bands correspond to 1σ, also accounting for the correlation from the g2 systematics, which dominates the experimental uncertainties.

  FIG. 4. Comparison between AmC spectrum (black) and contamination from TPC intrinsic events and γs from 241 Am, as described in the text. The low-energy excess is due nuclear recoils induced by neutrons.

FIG. 5 . 10 FIG. 6 .

 5106 FIG.5. Efficiency in detecting S1+S2 events (N2p) with respect to S1+S2 and S2-only (N1p), as a function of the reconstructed number of electrons. Data are fitted with an error function (red line). The error band is propagated from the uncertainties of the fitted parameters.

FIG. 7 .

 7 FIG.7. Spectrum of events selected in triple coincidence (black), compared with accidentals with f90<0.4 from the UAr sample, normalized to events selected in the AmBe sample with Ne in the [200, 500] range.

FIG. 9 .

 9 FIG. 9. Best fitted simulated Ne spectra using Ziegler et al. (blue) and Molière (orange) screening functions, compared to AmC (top) and AmBe (bottom) data. The model bands correspond to 1σ uncertainty.

TABLE I

 I

	. Average numbers ( N ) and energies ( E ) of pri-
	maries (electrons, X-rays and UV photons) emitted in electron
	capture-induced cascades on

  +0.1 -0.3 )×103 . The corresponding model of the ionization yield is shown as a

		12					Fit of DS50 AmC+AmBe, ARIS, SCENE
							Fit of DS50 AmC+AmBe
		10					Fit of ARIS and SCENE
	Q NR y [e /keVnr]	4 6 8
			ARIS
		2	SCENE
			Joshi et al.
		0.4	1	2	3	5	10	20	50	100	250
						Energy [keVnr]
	FIG. 8. Fit of the NR ionization yield at 200 V/cm from
	the combined fit of DarkSide-50 AmBe and AmC calibration
	data, together with datasets from SCENE [39] and ARIS
	measurements [35], the latter combined with the DarkSide-50
	ionization-to-scintillation ratio. The measured Q N R y	by Joshi
	et al. [40] at 6.7 keVnr is reported for comparison. The model
	bands correspond to 1 σ uncertainty.

ACKNOWLEDGMENTS

We are grateful to Francesc Salvat and Pascal Lablanquie for their comments and suggestions. The Dark-Side Collaboration offers its profound gratitude to the LNGS and its staff for their invaluable technical and logistical support. We also thank the Fermilab Particle Physics, Scientific, and Core Computing Divisions. Construction and operation of the DarkSide-50 detector was supported by the U.S. National Science Foundation (NSF) (Grants No. PHY-0919363, No. PHY-1004072, No. PHY-1004054, No. PHY-1242585, No. PHY-1314483, No. PHY-1314501, No. PHY-1314507, No. PHY-1352795, No. PHY-1622415, and associated collaborative grants No. PHY-1211308 and No. PHY-1455351), the Italian Istituto Nazionale di Fisica Nucleare, the U.S. Department of Energy (Contracts No. DE-FG02-91ER40671, No. DEAC02-07CH11359, and No. DE-AC05-76RL01830), the Polish NCN (Grant No. UMO-2014/15/B/ST2/02561) and the Foundation for Polish Science (Grant No. Team2016-2/17). We also acknowledge financial support from the French Institut National de Physique Nucléaire et de Physique des Particules (IN2P3), the IN2P3-COPIN consortium (Grant No. 20-152), and the UnivEarthS LabEx program (Grants No. ANR-10-LABX-0023 and No. ANR-18-IDEX-0001), from the São Paulo Research Foundation (FAPESP) (Grant No. 2016/09084-0), from the Interdisciplinary Scientific and Educational School of Moscow University "Fundamental and Applied Space Research", from the Program of the Ministry of Education and Science of the Russian Federation for higher education establishments, project No. FZWG-2020-0032 (2019-1569), and from IRAP AstroCeNT funded by FNP from ERDF. Isotopes used in this research were supplied by the United States Department of Energy Office of Science by the Isotope Program in the Office of Nuclear Physics.

with setups exposed to neutron beams are highly desirable in the future to improve and better constrain response models at the keV scale.