Electrostatic screening length in “soft” electrolyte solutions

Electrostatic screening length in “soft” electrolyte

solutions

Vahid Adibnia a,b_{, Budha Ratna Shrestha} a,‡_{, Marziye Mirbagheri} a,b_{, Frederic Murschel} a_, Gregory De Crescenzo b_{, Xavier Banquy} a,_*

a _{Faculty of Pharmacy, Université de Montréal, Université de Montréal, 2900 Édouard-Montpetit,}

Montreal, Quebec H3C 3J7, Canada

b _{Department of Chemical Engineering, Ecole Polytechnique de Montreal, P.O. Box 6079, succ.}

Centre-Ville, Montreal, Quebec H3C 3A7, Canada

AUTHOR INFORMATION

‡_{Current address: 4700 King Abdullah University of Science and Technology, Thuwal}

23955-6900, Kingdom of Saudi Arabia

Corresponding Author

*xavier.banquy@umontreal.ca, Tel.: (514) 343-2470

ABSTRACT. Using the Surface Forces Apparatus on solutions of polymeric ions, the effect of specific ion-ion correlations is evaluated on the characteristic decay length, λ , of screened electrostatic interactions between charged surfaces. Electrolyte solutions composed of point charges surrounded by repulsive polymeric shells were used to elucidate the role of ions size and size asymmetry between co- and counter-ions on the screening of electrostatic forces. In

electrolytes composed of large polymeric cations and small point-charge anions, because of the steric and excluded volume effects, the screening length follows the simple scaling relation λ d , where d is the characteristic size of the large cation. It is also reported that both co- and counter-ion size affect the thickness of the electrical double layer, and influence the screened electrostatic interactions. In solutions of polymeric cation/anion pairs, the screening length is shown to depend on an asymmetry factor. These results provide insights into correlation effects in electrolytes, which were so far unreachable experimentally, and help elucidate such effects in electronics, energy storage devices and biomedical systems.

TOC GRAPHICS

KEYWORDS. Colloids, Debye length, Electrical double layer, DLVO.

Studying screened electrostatic potential in electrolytes near charged surfaces is crucial to understand nanoscale interactions in soft matter. Such interactions are of interest in renewable

energy1,2_{, electronics}3 _{and biomedical applications}4,5_{. In the dilute electrolyte regime and when}

the ion-ion correlations are negligible (including the effect of ion size), the well-known Debye-Hückel theory predicts that these interactions decay exponentially away from the charged surface with a decay length

λD=

(

∑

)

known as the Debye length. Here ε is the dielectric constant of the solvent, kB is the Boltzmann constant, T is the absolute temperature, ni and zi are the number density and the ion valency of ion type i , respectively, and e is the elementary charge. For 1:1 aqueous electrolytes at room temperature Equation (1) becomes λD=0.304/

√

D in nm. The width of the region with variable electrostatic potential, from the charged surface to the bulk electrolyte, is of the order of 3 λD to 4 λD 6_{. The Debye-Hückel framework has been successfully tested on a myriad of systems}

over the past few decades. Nevertheless, there are many natural or man-made systems that deviate from this limiting regime because of ion-ion correlations7,8_{, such as concentrated}

electrolytes including ocean’s water, blood plasma, and many electrolytes used in energy storage applications and electronics. In such systems, although the electrical potential is still theoretically expected to decay exponentially, the characteristic decay length is often different from the Debye length9–13_{, and the electrostatic interaction between charged surfaces is governed according to a}

more general decay length. In fact, it has been shown experimentally that in several pure ionic liquids and concentrated electrolyte solutions, the decay length is significantly larger than the Debye length14,15_{. Moreover, the decay length has been shown to increase strongly with ion}

concentration when the ratio d / λD>1 16,17, where d is the ion diameter. This phenomenon was reported for different ionic liquids and concentrated chloride salts with d <¿ 0.5 nm.

Interestingly, the transition point d / λD≈ 1 , where the classical framework of uncorrelated electrolyte solutions was experimentally shown to fail, is close to the theoretical Kirkwood line d / λD≈

√

chloride salts increases similarly when d / λD>1 , Smith et al.16 _{hypothesized that bulk} nanostructuring in ionic liquids or ion size and shape anisotropy do not affect the observed qualitative behavior. Since these observations have only been reported for small ions at high electrolyte concentrations (typically c >¿ 0.5 M), it remains unclear if this behavior is still

true in dilute electrolyte solutions of sufficiently large ions ensuring d / λD>1 .

Numerous theoretical studies have reported that finite ion size could significantly change the electrostatic interactions in the double layer due to steric interaction effects or by screening the ion charge10,11,20,21_{. In addition, the ion size has been found to influence ion distribution around}

nucleic acids, critically affecting biological and physical processes such as RNA folding and molecular recognition22,23_{. Other experimental studies have also reported ion size-dependent}

electrostatic interactions close to charged surfaces20,24,25_.

In this study, the electrostatic interactions are measured using the surface force apparatus (SFA) on mica surfaces, which are negatively charged in aqueous media. In addition to evaluating the concentration dependence of the electrostatic interactions in an organic-ion electrolyte with a small ion size, the electrostatic screening in electrolytes with anomalously large ion sizes are investigated. Protonated ethanolamine cations were chosen as small organic ions ( d=0.46

nm26_{). The large cations are constituted of a single ammonium group at the center of a neutral}

polyethylene oxide (PEO) chain with molecular weights Mw = 6, 10 and 40 kDa (Flory diameters

of 14, 19, and 44 nm). In water, the PEO chain adopts a random coil conformation around the ammonium group, resulting in a structure that can be interpreted as a cation surrounded by a “soft” steric shell with a characteristic size determined by the Flory radius of the PEO chain. In aqueous media, the cation charge is activated in the presence of an equimolar concentration of a strong acid (HNO3 in this study). Surface interaction forces are measured across these electrolyte

solutions with various concentrations in the range 0.1-100 mM. The use of “soft” ions also allows to experimentally explore the role of cation/anion size asymmetry on the electrostatic interactions, which is previously unexplored. Here, electrolyte solutions with equimolar concentrations of soft cations (protonated polymers with molecular weight of 6 kDa or 10 kDa) and soft anions (PEO chains with molecular weights of 10 kDa or 40 kDa bearing a single deprotonated carboxylic acid group instead of ammonium group) were used. The chemical structure of these ions and their schematic ionic form are shown in Figure 1a. The SFA setup and an example of interaction force profiles for 10 kDa cation and NO3- anion are also shown in

Figure 1b and c, respectively. In the force profile shown in Figure 1c, the long range exponentially decaying interaction forces are associated to pure electrostatic forces with a characteristic decay length . The short range steric forces are associated to weakly adsorbed polymer layers on the mica surfaces, which can be squeezed out from the contact by a large enough force (see inset of Figure 1c).

Figure 1. (a) Chemical structure of the anion and cation used as soft ions with their

corresponding schematic representation (b) Schematic representation of the SFA setup used to assess electrostatic forces across soft ions solutions. The top and bottom surfaces are negatively charged mica pieces, and R=1 cm. (c) Typical interaction force profile across a solution of 10 kDa polymeric cation and NO3- anion at a concentration of 10 mM. The red solid line shows

the long-range electrostatic interaction exponential decay as F /R exp (−λ−1D) . The inset

highlights the short-range steric interactions.

As the polymer chains are mainly composed of ethylene oxide units, it is reasonable to use available empirical equations and parameters for polyethylene oxide to estimate the ion radius and polymeric interactions. Water is expected to be a good solvent for the polymers27_{. Therefore,}

Flory radius, RF=l . n

0.6

l=0.365 nm is the length of a segment28_{, and number of segments n=M}

w/Mw 0, where M_{w 0}=44 Da is the molecular weight of the segment. The variation of λ for the electrolytes composed of organic ions with respect to the ion concentration, c, is shown in Figure 2a. For the small ethanolamine cations, the decay length is equal to the Debye length up to c = 0.5 M, whereas above this value, the experimental decay length is significantly larger than the Debye length ( λ /λD≈ 12 , at c=2 M). When the reduced decay length λ /λD is plotted against the ion correlation parameter d / λD (see Figure 2b), the limiting concentration of 0.5 M corresponds to d / λD≈ 1 , as was also observed by Smith et al.16 and Lee et al. for different point-charge salts.17 _{In addition, for ethanolamine, the deviation of  from the Debye length at}

d / λ_D>1 increases sharply as was reported previously16,17_{, and was consistent with the scaling}

relation λ /λ_D

(

)

3

proposed by Lee et al.17 _{For the large soft ions, a different behavior}

was observed leading to a different scaling relationship. First, the decay length across the soft electrolytes deviates from the Debye length only when these polymerc ions enter a correlated regime at a concentration where small electrolytes are still well within the semi-dilute polymer regime, i.e. when c is greater than the overlapping concentration c¿ _{(see Figure 2a). For}

random coil polymer chains, the critical overlapping concentration is defined as 29

c¿

= Mw

4/3 π R_g3N_A, where NA is the Avogadro number, and Rg=0.0215 Mw

0.583 _{, is the}

gyration radius27_{. Secondly, when plotting the scaled decay length λ /λD versus d / λD , the}

data from Figure 2a collapses onto a single master curve which is different from the master curve observed for small ions. For the large polymeric ions, at d / λD>1 , the measured scaling law is λ /λ_D d / λ_D , and the deviation begins at d / λD≈ 2 , when d = 2RF is used (see Figure 2b),

and at d / λD≈ 1 , when d = 2RG is used to estimate the ion diameters (see Figure S1). When

calculating the radius in the semi-dilute regime, a correction factor of c¿ c /¿ ¿ 8 1/¿ ¿ −¿ ¿ ¿ was applied to

account for the reduced blob size due to shielding of the polymer excluded volume30_.

Figure 2. (a) The experimental decay length across the soft electrolytes versus the concentration.

The red line indicates the Debye length values. (b) The variation of scaled decay length versus the ion-correlation parameter. The diameters here are calculated according to the Flory radius. The Kirkwood limit d / λD≈

√

Above c¿

, steric forces between ions are expected to affect the interaction forces between the mica surfaces. To evaluate if the steric forces can influence the force profile in the electrostatic

range, the experiments with d / λD>1 were repeated without the protonation of amine groups (effectively uncharged polymer solutions). In this case, the steric interactions were found to be short range compared to the electrostatic interactions. By increasing the polymer concentration,

the steric interaction range increased as d ¿ λ_D ¿ ¿ ¿ ¿ λ_D ¿ λ_s/¿

, where λ_s is the decay length of the steric

forces (see Figure S2 and S3). Therefore, the linear increase in the decay length with the ion diameter ( λ d at d / λD>1 ) can be attributed to the redistribution of ions in the electrical double layer due to electrostatic effects (see Figure S4 for the EDL expansion force profiles with increased polymer size). To explain the linear expansion of the double layer with respect to the ion diameter, it should be noted that the charged surface and the ion cloud around it form an electrically neutral system, requiring a fixed number of ions to neutralize the surface charge. By increasing the PEO chain length (increasing the polymeric ion blob diameter), the volume around the point charge ion that is excluded from other ions increases. Above c¿

, where polymer chains partially overlap, this causes an increase in the volume required to contain the ions in the double layer, and distributes counterions farther from the charged surface. The observed increase in the electrostatic decay length is attributed to such expansion of the double layer.

It is important to note that coions (here coions are the anions for the negatively charged mica) are also present in the double layer6_{. Therefore, their size could also affect the ion distribution in the}

double layer. In this study, anion-cation size asymmetry is investigated by varying the molecular weights of both soft anions and cations. The use of soft ion allows to explore the effect of the size asymmetry on the double layer thickness and electrostatic forces by providing highly asymmetrical cation/anion pairs. Following Frosberg et al.11_{, the ion asymmetry factor is defined} as d ¿ −¿ +¿+d−¿, d¿ 2¿ ¿ β=¿ (2)

where d is the ion diameter, with + and - superscripts referring to the cation and anion species, respectively. At the limiting condition that the anion is a point charge, β=2 .

The scaled electrostatic decay length across solutions of asymmetric electrolytes at a fixed concentration of 5 mM is plotted versus the asymmetry factor in Figure 3, keeping the cation size constant and increasing the anion size from a point charge (NO3-) to a significantly larger soft

anion ( −¿_d=43.5 Flory

¿ nm). With decreasing the anion size, for a cation of size

+¿=¿ dFlory

¿ 18.9 nm,

the relative decay length appears to rapidly decrease for -1 <  < 0 and then increase for  > 0,

with a minimum value of λD=¿

λ /¿ 1.38 ± 0.02 at  ~ 0. For smaller soft cation ( +¿

d_Flory¿ = 13.9

nm), the results were similar except that the increase of λ /λ_D at  > 0 did not occur. This could be explained by noting that among the six samples evaluated in Figure 3, only the 6 kDa cation - NO3- anion pair was in the dilute polymer regime (see Figure 2a for c¿ values).

Therefore, the decay length for this sample was very close to the Debye length. To explain the variation of the scaled decay length with the asymmetry factor, we refer to Frosberg et al.11 _to describe the effective ion charge with respect to its counterion size. The presence of counterions around an ion screens the ion effective charge density. By increasing the counterion size, the number of the counterions that can sufficiently approach the ion to screen its charge decreases, causing an increase in the effective charge of the ion. This effect is manifested in Figure 3, where the decay length first decreases with increasing the anion size at 0<β <2 (from right to left in Figure 3). However, above c¿

, increasing the anion size, similarly to the cation size, expands the double layer due to the increase in the polymer excluded volume. Although this effect is less pronounced for anions than cations because of their lower number density in the double layer, it eventually causes the decay length to increase with further increasing the anion size when

β≲0 .

Figure 3. The scaled decay length versus the asymmetry factor defined in Equation (2). The

Flory diameter of cations are 13.9 nm (green data points) and 18.9 nm (blue data points), for 6 kDa and 10 kDa polymers, respectively. The anions from right to left are NO3- (assumed a point

charge), 10 kDa polymer ( −_d¿=¿

Flory

¿ 18.9 nm) and 40 kDa polymer ( −_d¿=¿

Flory

In summary, the present study showed that large soft polymeric ions can be used to investigate the effect of ion-ion correlation on electrostatic interaction forces between charged surfaces. Similarly to the experimental works of Smith et al.16 _{and Lee et al.}17_{, which demonstrated that}

the screened electrostatic interactions in concentrated electrolyte solutions increases with concentration, we demonstrated that, at significantly lower ion concentrations, the decay length of electrostatic interactions across soft ion solutions shows a similar increase due to steric interactions between ions. However, we also demonstrated that the relation between the scaled decay length and the ion correlation parameter is not universal and depends on the type of the correlation. For example, in concentrated electrolytes, the decay length has been proposed to

increase as d ¿ λ_D ¿ ¿ ¿ ¿ λ_D ¿ λ /¿

, due to the ion-ion correlations arising from the extreme proximity of the

ions to each other17_{. However, in solutions of point charges with anomalously large soft shells,}

the steric and excluded volume effects lead to an increase in the decay length of the form λ /λD d / λD . Interestingly, this is not limited to the counterion size, since the coion size can also influence these ion-ion correlations in the double layer, and affect the interactions close to a charged surface.

Supporting Information available: including a description of the materials and methods, and

supplementary figures and force profiles.

Notes

The authors declare no competing financial interests.

ACKNOWLEDGMENT

This research was undertaken thanks, in part, to funding from the Canada First Research Excellence Fund through the TransMedTech Institute. V. A. and F. M. appreciates the financial support from the Fonds de Recherche du Québec (FRQ-NT). B.R.S. is grateful for the financial support from GRUM. XB acknowledges support from NSERC (Discovery) and the Canada Research Chair program.

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