Spherical electric double layers containing mixed electrolytes: A case study for multivalent counterions

Accepted Manuscript Spherical electric double layers containing mixed electrolytes: A case study for multivalent counterions Chandra N. Patra PII: DOI: Reference:

S0009-2614(17)30771-6 http://dx.doi.org/10.1016/j.cplett.2017.08.010 CPLETT 35020

To appear in:

Chemical Physics Letters

Received Date: Accepted Date:

3 June 2017 7 August 2017

Please cite this article as: C.N. Patra, Spherical electric double layers containing mixed electrolytes: A case study for multivalent counterions, Chemical Physics Letters (2017), doi: http://dx.doi.org/10.1016/j.cplett.2017.08.010

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Spherical electric double layers containing mixed electrolytes: A case study for multivalent counterions Chandra N. Patra∗ Theoretical Chemistry Section, Chemistry Group, Bhabha Atomic Research Centre, Mumbai 400 085, India

Abstract Spherical electric double layers containing mixed electrolytes with multivalent counterions, is studied using density functional theory and Monte Carlo simulation. The macroion and small ions are represented as uniformly charged hard spheres within a continuum solvent. The theory involves an weighted density approximation for the hard-sphere contribution, whereas the electrical part is evaluated through a functional expansion around the uniform fluid. The system includes a number of parameters, viz. ionic concentrations, macroion charge density, and the valence of the counterion. This study points towards the distinctive evidence of size and charge correlations manifested through layering and charge reversal phenomena.

1

I.

INTRODUCTION

The structure and dynamics of ionic atmosphere around the macroion has been the subject of long standing research due to its utmost importance in resembling realistic systems that is encountered in daily life.1 The distribution of particulates in aerosol in the atmosphere is necessary to have control parameters for the air pollution.2 . Similarly, the concentration and valence of the supporting electrolytes in a colloidal system has its influence in the prediction of aggregation behavior that is useful in quality assurance of food products, paints and varnish, soaps and detergents, etc.3 The structural evolution of these systems not only helps in understanding the interactions at an atomic levels, but also throws light on the design of futuristic systems through delicate tuning4 of these interactions. The symmetry and the spatial correlations also play an important role in this.5 That the understanding of distribution behavior of supporting electrolytes in a colloidal solution and similar systems has generated exceptional interest in the last decade is quite convincing due to the development of a large number of fabricated nano-electromechanical devices,6 new drug delivery systems,7 and printing and imaging machines.8 In fact, theoretical investigations9 started at a later stage to have an understanding about the experimental condition necessary for various fabrication methods. The interfacial region near a charged surface that is surrounded by the electrolyte components gives rise to electric double layer (EDL),10 consisting of both the Stern and the diffused layers, the symmetry of the same being determined by that of the charged interface. The different electrolyte ions, their charge and valence as also the solvent play a major role in determining the structure of EDL and the electrode potential generated around it.11 The properties and phenomena related to EDL formation has been studied extensively in the last decade in many different geometries, viz., planar,12–17 cylindrical,18–22 and spherical one.23–31 Studies32 beyond these conventional geometries started only recently because of complexity of the problem involved. Spherical double layers (SDL) are the point of much discussion in recent times because of its application to fabrication of polyelectrolytes, placing tethers in various positions of macromolecules and preferential assembling of polymer nanoparticles.33 In such situations, not only the concentrations of the electrolyte ions as well as the macromolecules are important, but also the total charge of the nanoparticle and the valence of the constituent ions contribute quite significantly. It is quite well known that

2

presence of multivalent ions have a great influence on several macromolecular34 and biological phenomena.35 Hence a more rigorous understanding of the effect of multivalent ions on the SDL is the need of the hour. Study of SDL in presence of multivalent ions requires several levels of modeling description as studied through simulation and theory, the simplest one being the restricted primitive model (RPM),10 where the supporting electrolyte ions are charged hard spheres having the same diameter and the colloidal macroparticle is the larger hard sphere with uniform surface charge density and the solvent as a dielectric continuum with dielectric constant as that of water. Recent works on these systems are mostly based on modified PB approach,27 integral equation theory,23 and density functional theory25,30 at different versions alongside the Monte Carlo study.23,30 In a number of studies, we carried out detailed investigations of EDL in different geometries through Monte Carlo and density functional theory (DFT).17,30,36 As can be seen, the two important effects, viz. overcharging (OC)26 and charge reversal (CR)23 makes prominent contributions in determining the static structure of EDL. Mixed electrolytes having multivalent counterions have been studied in great detail because of of its utmost importance in several biological phenomena involving DNA.37 These studies also observe37,38 the OC and CR effects, as explained by the presence of strongly correlated layer of multivalent counterions at the macroion surface. However, the modeling seems to be too simplistic and hence a detailed theoretical description was necessary. Such a work was attempted from a combined experiment and theoretical study on model colloids.39 However, a systematic study with the gradual addition of multivalent counterion in a SDL formed from monovalent salt is not attempted so far. Although few, the effect of multivalent coions on the planar EDL40 and that of cylindrical one has been attempted20,21 quite recently. The effect of multivalent coion on the structural behavior of SDL has been studied through DFT and MC simulations.30,41 The results seems to be quite interesting in terms of variation of parameters and also for the OC and CR effects. In order to complete the comparisons within the SDL, we carry out a detailed DFT and MC study for these systems containing multivalent counterions. Since the theoretical prescriptions and the simulation method are described in a number of our earlier papers, we intend to restrict ourselves only in presenting the detailed results. The manuscript is divided into different sections with theoretical part as summary of the earlier formulation and special emphasis is given on results and discussions. A few concluding remarks alongwith future directions are given at the end. 3

II.

THEORETICAL FORMULATION

The model system considered here is a SDL formed around a macroion by small ions containing mixed electrolytes having the multivalent counterions within the restricted primitive model. The macroion is considered as a uniformly charged (total charge ZM e) hard sphere of radius R with surface charge density Q. The mixed electrolyte solution is treated within RPM, where the small ions are of charged hard spheres of equal diameter σ and the solvent is treated as a continuum with dielectric constant  = 78.5 at temperature, T = 298 K. Unless otherwise stated, R is taken as 1.5 nm and σ as 0.425 nm. DFT starts with the expression of the grand potential, Ω, as an exact functional of the density distribution ρα (r), that attains minimum at equilibrium, i.e., δΩ[{ρα (r)}] =0. δρα (r)

(1)

The application of this condition to look at the density distribution at interfaces requires approximation relating to the expressions for the free energy contributions in the grand potential. Without providing details, we write here the relevant expression for the calculation of the density profiles for the small ions constituting mixed electrolytes around a macroion as  hs hs ρα (r) = ρ0α exp −β0 zα ψ(r) + c(1) (r; [{ρα }]) − c(1) ([{ρ0α }]) α α el el + c(1) (r; [{ρα }]) − c(1) ([{ρ0α }]) , α α

(2)

where ψ(r) represents the mean electrostatic potential (MEP) of the system and (1)

cα (r; [{ρα }]) denote the first-order correlation function and the other symbols have their usual meaning. Although, formally exact, Eq. (2), requires several closure approximations to evaluate (1)hs

cα

(1)el

and cα

for the SDL system. Here, we resort to the Denton-Ashcroft version42 of (1)hs

weighted density approximation (WDA) for the hard-sphere contribution, cα the electrical contribution,

(1)el cα

, whereas

is evaluated using a perturbative expansion around the

uniform fluid.16,17 The evaluation requires uniform second order direct correlation functions, (2)hs

(2)el

c˜αβ and c˜αβ , taken from the analytical expression within the mean spherical approximation (1)

(MSA).43 Once these quantities, ψ(r) and cα (r; [{ρα }]) are known, the density distributions are calculated from Eq. (2) using a self-consistent iterative procedure. 4

We also carried out the detailed canonical Monte Carlo (CMC) simulations (N, V, T )44 for the same system of SDL containing mixed electrolytes for comparing our theoretical results. Here, the macroion is fixed at the center of a cubic simulation box surrounded by the mixed electrolytes with periodic boundary conditions employed in all three directions. The long-range Coulomb interactions are taken through the Ewald sum method45 and the acceptance ratio is always kept below 0.5. The equilibration process requires a total 8 × 106 MC moves and final averaging is carried out over 5 blocks each having 8 × 106 moves. With the density distributions of ions are at hand, integrated charge (IC) distribution function P (r) is given as28 r

Z

dr0

P (r) = ZM + 0

X

zα ρα (r0 ) ,

(3)

α

which represents the overall charge of SDL spread within the spherical distance r. In essence, the cumulative reduced charge (CRC), defined as28 Q∗ (r) =

P (r) , |ZM |

(4)

serves as a useful quantity in interpreting a number of structural informations about the double layers. Whereas, OC effect26 amounts to the increase in P (r), CR effect23 refers to the inversion of sign of P (r). The properties of the diffuse layer in a SDL is better understood in terms of zeta potential, ζ, which is the MEP value at interfacial location of the colloidal shear plane.46 Although, there is no direct way of locating this plane the choice is entirely model dependent, we limit to the definition of the zeta potential (ζ) that should undergo a change of sign if the reversal of the diffuse layer potential is considerable.47 With this representation, we assume that the approximation ζ ≡ ψ(R + σ/2) is quite acceptable in the present study.

III.

RESULTS AND DISCUSSION

The structure of SDL formed from the mixed electrolytes around the spherical macroion is presented here. At first, we attempt to see the effect of addition of bivalent counterions (SO2− 4 ) into the monovalent (1:1) electrolytes (NaCl). The singlet density profiles (with respect to the bulk), ρα (r)/ρ0α , of a 1:1:2 (NaCl/Na2 SO4 ) electrolyte in a SDL of positively charged macroion is presented in Figure 1(a) - (d) with the gradual increase in the concentration of the bivalent counterions. The essential feature of all the graphs is the accumulation 5

of counterions (Cl− and SO2− 4 ) and the depletion of coions at the macroion surface. With the − addition of SO2− 4 , there is a continuous decrease in the Cl concentration and an appreciable

increase in the coion (Na+ ) concentration at the interface. Also, the layering of the counterions starts decreasing and that of the coions starts increasing indicating even a second layer [cf. Fig. 1(d)]. The stronger electrostatic attraction between the multivalent counterion and the macroion, leads to an increased accumulation of counterions at the interface, thereby leading to effectively screening the macroion charge. This is also reflected through the increase in coion and decrease in monovalent counterion concentrations. Also, continu− − ous increase in [SO2− 4 ]:[Cl ], keeping Cl concentration to be the same, leads to a decrease in

SO2− 4 concentration at the surface due to stronger repulsions between overcrowded divalent counterions at the surface as compared to the monovalent counterions. These observations corroborate earlier studies on planar40 and cylindrical double layers.36 With gradual increase in multivalent counterion concentration, the system tends towards charge inversion, where the coion concentration crosses that of the counterions. Interestingly, the width as well as the depth of the inversion layer increases with increasing the multivalent counterion concentration [cf. Fig. 1(c) - (d)]. The mean electrostatic potential (MEP) profiles for the above system is depicted in Fig. 2, that clearly indicates a gradual reduction of zeta potential. This is because of increase of electrostatic attraction of bivalent counterions with that of the macroion and subsequent reduction of potential at the interface. This attraction started − becoming stronger with increase in [SO2− 4 ]:[Cl ] concentration ratio with the signature of

charge reversal. This is also evident from the dampening of the MEP profiles at a shorter distance from the interface with increase in bivalent counterions. The interplay between the size and the charge correlation effects on the density distribution is studied through the variation of surface charge density (Q) on the macroion for a 1 M NaCl with added 0.5 M Na2 SO4 . As can be seen from Fig. 3(a) - (d), there is considerable presence of coions (Na+ ) at lower Q, which gets reduced at higher Q. The CR phenomena is increased as Q increases. This is due to that fact that coions get repelled and counterions are attracted towards the macroion with larger Q. Because of increased charge correlations, the depth of CR is increased in passing to higher Q, alongwith an increase in its width. This is also due to depletion of coions and increase of counterions in the second layer from that of the first layer. The MEP profiles corroborates the findings as is clear from Fig. S1, it passes from positive to negative with increase in Q as if the macroion changes its sign due to 6

the presence of large number counterions at its vicinity. That CR effect marks prominence at larger Q is also evident from the CRC profile plotted in Fig. S2, which clearly reveals stronger charge correlations at larger Q. In order to see the effect of concentrations of the mixed electrolyte on the behavior of SDL, the bulk concentration of NaCl is varied from 0.01 M to 2 M with concentration of − Na2 SO4 at such that [SO2− 4 ]:[Cl ] ratio remains as 1:2. As can be visualized from Fig. 4(a)

- (d), the coion concentration at the interface continuously increases in going to a higher overall electrolyte concentration. This is followed by the continuous decrease in the densities of counterions at the interface upto 1 M of bulk concentration of NaCl, just because of accumulation of larger number of coions due to increase in concentrations. However, in passing to 2 M, the concentration of Cl− is increased at the interface because of formation of multiple layers due to size correlation. In fact, this is clearly corroborated in the MEP profiles of Fig. 5 as can be seen that for 2 M concentration, it crosses the zero line twice. It is also amply clear that CR effect starts become relevant at a higher concentration of 1 M onwards and its depth increases at 2 M concentration. The charge as well as the size correlations in SDL can better be studied by varying the size of the macroion. This is depicted in the ionic density profiles in Fig. 6(a) - (d) for − −2 1 M NaCl/Na2 SO4 mixed electrolyte with [SO2− as 4 ]:[Cl ] ratio as 1:2 at Q=0.102 Cm

the radius (R) of the macroion is varied from 0.5 nm to 6 nm. The coion concentration at the interface keeps decreasing although substantial accumulation of coions still exists for larger macroion. Because of increase in overall charge (ZM ) due to increase in the size of macroion, there is an appreciable increase in the counterion concentrations at the interface. This couples with the depletion of coions results in the CR effect, clearly noticeable for larger macroions. Although the strength of CR effect increases quite considerably, the width remains more or less constant indicating the extent of size correlations that affects the effect of charge correlations. It is also to be mentioned here that these SDL effects at larger macroion sizes are expected to resemble the planar and the cylindrical case and indeed these are found to be true. This is also clearly visible in the MEP profiles as displayed in Fig. S3, where the CR tends to occur at a lower distance from the macroion, for larger macroion compared to that of the smaller case. That the CR effect increases considerably with increase in the size of the macroion is also evident from the CRC profiles plotted in Fig. S4, which indicates a large negative P (r)[= Q∗ (r) × |ZM |] for R = 6 nm as ZM is quite 7

large for larger macroion. The effect of valence of the small ions on the structure of SDL is studied by gradual addition of Na3 PO4 at different ratios to bulk 1 M NaCl. Fig. 7 depicts the density profiles − of 1:1:3 NaCl/Na3 PO4 at four different ratios of PO3− 4 /Cl . Although at a lower ratio

of trivalent counterion, the presence of coion is more or less like that of the situation as − in divalent counterion, with increase in [PO3− 4 ]:[Cl ] ratio, the coion concentration at the

interface becomes substantial because of charge correlations. (cf. Fig. 1). The concentration of PO3− 4 is considerably larger at the interface at lower ratio of trivalent counterion, and continuously decreases, although it is always higher as compared to the divalent ion system. The Cl− counterion density in the mixed electrolyte system containing trivalent counterion is always lower as compared to the divalent counterion case. The resultant effect is the stronger charge inversion in case of 1:1:3 electrolyte as compared to 1:1:2 electrolyte. This is depicted in the MEP profiles in Fig. S5 and can be explained on the basis of large number of coions as compared to the divalent case. The CRC profiles also indicated the double inversion as is evident from Fig. S6, that fully supports the MEP profiles. We also calculated the zeta potential ζ of the SDL formed from the mixed electrolyte systems to get an idea of the behavior of the diffuse layer at different surface charge densities − (Q). The zeta potentials of the SDL formed from 1 M (NaCl/Na2 SO4 ) with [SO2− 4 ]:[Cl ]

ratio as 1:2 for R = 1.5 nm at different small ionic sizes is shown in Fig. 8 as obtained from both DFT and MC simulations. At all ionic sizes, ζ increases with Q, attends saturation and then starts decreasing in going from negative to positive Q. Because of overscreening of macroion charge, ζ starts decreasing with increasing ionic size. The maximum of the zeta curve shifts to lower Q, having completely negative ζ potential value at positive Q for larger ionic diameters (0.6 nm, in the present case).

IV.

CONCLUDING REMARKS

The structure of a spherical double layer formed around a positive macroion by a mixed electrolyte consisting of mono and multivalent anions is studied here. The multivalent anion is added gradually to study the effect specifically arising out in the interactions leading to layering and charge reversals. This study also completes our earlier work on this system in which case the multivalent ions are coions. The present work concerns with variations of a 8

number of physical parameters of the system, viz. the bulk electrolyte concentrations, ratios of mono- and multivalent counterions, the surface charge densities on the macroion, the size of the macroion and the valence of multivalent anion. In most of the parametric variations studied here, the DFT results go hand in hand to that of MC simulations indicating the robustness of the present theoretical formulations. This also provides the reliability of using the uniform second order direct correlation functions in calculating the density distribution of the nonuniform fluids even if it involves ionic interactions. The present work is aimed at to see the relative role of charge and size correlations on the double layer properties formed from a mixed electrolyte, where the multivalent counterion (SO2− 4 ) is gradually added in the system of 1:1 (NaCl) electrolyte. Increasing concentration of added multivalent counterions lead to the decrease in layering of the counterions increase in that of the coions alongwith enhancement in charge reversal effects. These effects are more for higher valency of the multivalent ions. The phenomena of layering as well as charge inversion also increases with increasing the surface charge density on the macroion or by increasing the size of the macroion, thereby increasing the effective charge on the macroion. Increasing the overall concentration of electrolyte with the ratio of multivalent to the monovalent counterion as constant leads to substantial layering and charge reversals. These effects are also manifested in the MEP profiles. However, proper parametrization is necessary to compare these results with real systems, as studied experimentally.48 The present formalism predicts the density distributions in a spherical double layer formed from mixed electrolytes in the presence of multivalent counterions quite well. It also predicts the zeta potentials over a wide range of surface charge densities for different small ion sizes. Although the theoretical predictions seems to be quite good, a more rigorous derivation involves different diameters of all the components. Although such a sophisticated version involving size-asymmetric electrolytes is already tested for positive macroion involving multivalent coions, a more generalization for the other type of ions is also necessary. It is quite well known that presence of solvent even in neutral form substantially alter the densities at the interface as well as the potential drop. It will be of interest to know the same effects in multivalent counterion or coion dominated double layers containing mixed electrolytes of different sizes. Since the system of electrical double layer is originally the nonuniform and locally non-neutral and the current WDA treats them through uniform locally neutral electrolyte, it will have its own limitations. Also MSA being a linear approximation, will 9

hold good as long as the interactions potential uαβ (r) is quite low. Both these conditions takes precedence, when the coupling parameter, defined as Γ = [β0 zα zβ e2 /σ] turns quite large (> 10). This obviously indicates that the theory will not perform well for concentrated electrolytes. and also when the macroion charge is quite high. There are behavioral similarities between the bivalent/trivalent counterions and ionic liquids, as both exhibit strong overscreening and layering. However, ionic liquids are large molecules with charges at different sites. Modeling ionic liquids as freely-jointed or fused hard chains of ions and carrying out an effective analytical calculations based on integral equations or density functional descriptions may throw lights on these. However, bond length and bond angle distribution also will have role to play on these. The study on these systems from molecular dynamics simulations are plenty.49 However, analytical theoretical descriptions based on density functional theory has started quite recently50 and is still an important open area.

Acknowledgments

The author gratefully acknowledges Swapan K. Ghosh for useful discussions. It is a pleasure to thank T.K. Ghanty for his kind interest and constant encouragement.

APPENDIX A: SUPPLEMENTARY MATERIAL

10

∗

Also at Homi Bhabha National Institute, Mumbai, India; Electronic mail: [email protected]

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FIGURES

FIG. 1: Ionic density profiles around a spherical macroion of R = 1.5 nm and Q = 0.102 Cm−2 for − 1 M bulk concentration of NaCl with added Na2 SO4 at different [SO2− 4 ]:[Cl ] concentration ratio

as: (a) 1:16, (b) 1:8, (c) 1:4, and (d) 1:2. Symbols are simulation results and lines represent DFT predictions. Na+ : (Black,

2), Cl−: (Red, 4), and SO2− 4 : (Green, #).

FIG. 2: Mean electrostatic potential profiles around a spherical macroion of R = 1.5 nm and − Q = 0.102 Cm−2 for 1 M NaCl with added Na2 SO4 at different [SO2− 4 ]:[Cl ] concentration ratios.

Symbols are simulation results and lines represent DFT predictions. (a) 1:16: (Green, (Red, 4), (c) 1:4: (Blue, ∗), and (d) 1:2: (Black,

2).

#), (b) 1:8:

− FIG. 3: Ionic density profiles for 1 M NaCl with added Na2 SO4 with [SO2− 4 ]:[Cl ] = 1:2, around a

spherical macroion of R = 1.5 nm at varying surface charge density: (a) Q = 0.102 C/m2 , (b) Q = 0.204 C/m2 , (c) Q = 0.306 C/m2 , and (d) Q = 0.408 C/m2 . The key is the same as in Fig. 1.

FIG. 4: Ionic density profiles around a spherical macroion of R = 1.5 nm and Q = 0.102 Cm−2 for − NaCl/Na2 SO4 with [SO2− 4 ]:[Cl ] = 1:2, having bulk concentration as: (a) 0.01 M, (b) 0.1 M, (c) 1

M, and (d) 2 M. The key is the same as in Fig. 1.

FIG. 5: Mean electrostatic potential profiles around a spherical macroion of R = 1.5 nm and − Q = 0.102 Cm−2 for NaCl with added Na2 SO4 with [SO2− 4 ]:[Cl ] = 1:2, having bulk concentration

as: (a) 0.01 M (Green,

#), (b) 0.1 M (Red, 4), (c) 1 M (Blue, ∗), and (d) 2 M (Black, 2). 13

− FIG. 6: Ionic density profiles for 1 M NaCl with added Na2 SO4 with [SO2− 4 ]:[Cl ] = 1:2, around

a spherical macroion of Q = 0.102 Cm−2 at different macroion radii: (a) R = 0.5 nm, (b) R = 1 nm, (c) R = 1.5 nm, and (d) R = 6 nm. The key is the same as in Fig. 1.

FIG. 7: Ionic density profiles around a spherical macroion of R = 1.5 nm and Q = 0.102 Cm−2 − for 1 M bulk concentration of NaCl with added Na3 PO4 with different [PO3− 4 ]:[Cl ] concentration

ratio as: (a) 1:16, (b) 1:8, (c) 1:4, and (d) 1:2. Symbols are simulation results and lines represent DFT predictions. Na+ : (Black,

2), Cl−: (Red, 4), and PO3− 4 : (Green, #).

FIG. 8: Zeta potentials around a spherical macroion of R = 1.5 nm for 1 M NaCl with added − Na2 SO4 with [SO2− 4 ]:[Cl ] = 1:2, with small ion diameters as σ = 0.2 nm (Black,

(Red, 4), σ = 0.4 nm (Blue, ast), σ = 0.5 nm (Green,

#)), and σ = 0.6 nm (Brick, ∇).

are simulation results and lines represent theoretical predictions.

14

2), σ = 0.3 nm Symbols

Graphical Abstract

50

 (mV)

25 0 -25 -50 -75 -0.50

-0.25

0.00

0.25

0.50

Q (Cm-2) Fig. 14

 

Zeta potential profiles around a spherical macroion of R=1.5 nm for NaCl with added Na2SO4 with [SO42-: Cl- ] = 1:2, at different small ion diameters clearly shows that zeta potential turns negative at larger ionic diameters.     

Highlights

Highlights of the Work carried out in the Present Manuscript submitted to Chemical Physics Letters

The contents of the present manuscript cover a wide spectrum of soft matter areas that includes colloidal suspensions, micelles, microemulsions, biological systems, to particulate matters in environments. The aim of the present manuscript is to provide sufficient information towards the present day understanding of electric double layers involving multivalent ions. The publication of such data is urgent due to following reasons: (1) The theoretical representation of electric double layers involving multivalent counrerions in a monovalent electrolyte requires a systematic study as such systems are readily encountered in experiments, regularly. (2) The data on the present study are based on exact models and are completely new and the same has been verified through extensive Monte-Carlo simulation. (3) The data provides distinctive evidence on the size and charge correlations manifested through singlet density profiles. Thus the present manuscript not only involves the application of theoretical methods but also provides rational explanation to many new phenomena related to multicomponent electrolytes. This necessitates the present manuscript to be published on an urgent basis.

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