Downloaded by guest on September 30, 2021 otwsenUiest,Easo,I 00;and 60208; IL Evanston, University, Northwestern a Bagchi Debarshee solutions polyelectrolyte confined in effects polarization Surface lcrdshv ensuiduigcntn oeta bound- www.pnas.org/cgi/doi/10.1073/pnas.2007545117 potential constant metallic using by studied and confined corrugated been Electrolytes is have (17). it when electrodes multivalent or are (16) ions charged the weakly is when except surface negligible, the are on effects Studies it polarization discon- that charged. dielectric found that one tinuity are only assumption account surfaces into taking the the electrolytes simple when on negligible include based not be do would effects, polar- date, surface to polarization the studies, as surface in most manifests However, jump which (13–15). interface, a ization the in the at results of field that discontinuity electric from dielectric different very Such typically permit- solution. is relative electrodes the the batteries, interest of and tivity fundamental supercapacitors In and 12). practical 11, of (1, are surfaces charged with attractions electrophoretic or nonmonotonic and (7) (10). local (9), mobility repulsions mobility of enhanced enhanced breakdown (8), (6), (5), overcharging simple neutrality and such both charge inversion phenomena, systems, intriguing charge confined exhibit as such electrolytes In molecular (4). and reported investigated been performance Cur- being battery has impressive are batteries. and materials, electrolytes lithium–ion supercapacitor of as standard solutions aqueous to rently, environmentally compared more storage and as safer, energy friendly cheaper, are competitive they be since devices can like electrodes Supercapacitors carbon-based two graphene. solutions between electrolyte confined electri- or typically as liquids are to ionic capacitors, referred double-layer also cal supercapacitors, In (3). technologies P capacitance differential negative the polyelectrolytes and constant. species, dielectric lower charged with surfaces the the from between repulsion between the interplay polyelectrolytes, interaction confined the the Coulombic of to entropy attributed conformational the the are on behaviors impor- These depend quality. the interesting solvent strongly the of and ions flexibility capacitance, all backbone polyelectrolyte of differential that and absorption find energy amplification, also storage, charge the as We such is properties, surface. double-layer tant which like-charged amplification, a increase on charge an energy with the electrostatic enhanced is the in structure effects, double-layer Furthermore, the polarization capacitance. in of stored differential presence density, negative the charge in a surface sur- increasing of confining with decrease the indicative to at found layer is two across double face polyelectrolyte potential between adsorbed electrostatic confined the the solution to Surprisingly, this surfaces. polyelectrolyte due double-layer charged In a the effects influence of devices. polarization remarkably structure that storage mismatch dielectric show energy such we for of paper, and efficiency con- processes, contributed for biological the crucial and is improving environmental Qin) constants Jian many dielectric and between Messina different trolling Rene interface by with reviewed the media 2020; 21, at two April review interactions for nanoscale (sent 2020 Understanding 24, June Cruz, la de Olvera Monica by Contributed eateto aeil cec n niern,Nrhetr nvriy vntn L60208; IL Evanston, University, Northwestern Engineering, and Science Materials of Department h hsclpoete fpleetoyesltosi contact in solutions polyelectrolyte of properties physical The s opyia n iesine 1 ) swl st modern to as well as 2), (1, sciences life inter- and great physical of to are est confinement spatial under olyelectrolytes | confinement a rn a Nguyen Dac Trung , | ilcrcmismatch dielectric c eateto hsc n srnm,Nrhetr nvriy vntn L60208 IL Evanston, University, Northwestern Astronomy, and Physics of Department b | nrystorage energy n oiaOvr el Cruz la de Olvera Monica and , | doi:10.1073/pnas.2007545117/-/DCSupplemental at online information supporting contains article This 1 ulse ne the under Published interest.y competing no declare University. authors Stanford The J.Q., and and T.D.N., Lorraine; of D.B., University and R.M., T.D.N. Reviewers: data; and analyzed D.B. paper. M.O.d.l.C. y research; the and wrote designed T.D.N., M.O.d.l.C. M.O.d.l.C. D.B., and research; T.D.N., performed D.B., contributions: Author Equilibrium layer. double the across double voltage respect the a with in storage change of charge the the to capacitance in change differential the characterizes the layer definition, capac- By differential negative itance. of issue the is imposed capacitors double-layer the of function con- a of case as potential. the electrodes, surface in metallic and, increases, by density finement charge (21), surface inversion charge the and as (5) possibility amplification the charge of physical including degree the and solutions, on polyelectrolyte polarization of surface determine of properties We effect surfaces. debated contrast confining highly the dielectric the and and solution density, the charge between surface fraction, polyelectrolyte flexibility, charge of degree polyelectrolyte effects the polarization varying by surface analyze we solutions, polyelectrolyte polyelectrolyte a of solution. features col- and structural surfaces the the parallel on is two mismatch study by dielectric present confinement spatial the of of effects focus lective The charge (20). unex- surface increases to the density leads as mismatch conformations, symmetry-breaking dielectric a pected that by showed polyelectrolyte cavity of confinement spherical dielectric interest. on of studies are graphene, Recent electrolytes settings as such molecular biological materials, applications, dielectric many by supercapacitor confined in for as However, well 19). as (18, conditions ary owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To imthehne ufc hreapicto nflexi- in storage energy amplification their capabilities. 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APPLIED PHYSICAL SCIENCES thermodynamics and statistical mechanics (22, 23) suggest that The paper is organized as follows. We first study flexible the differential capacitance should always be strictly positive, but polyelectrolytes for various surface charge densities, and then this conclusion was challenged by later works (24, 25). Attard et we investigate the roles of chain flexibility, solvent dielectric al. (24) pointed out that there exists no rigorous proof that the constant, and boundary conditions at the confining surfaces, in differential capacitance has to be positive for an isolated double determining the capacitative behavior and energy storage of the layer. Simulations of 1 : 2 electrolyte solutions also demonstrate double-layer structure. that the differential capacitance can become negative (25). Sim- Atomistic simulations typically allow investigation of small sys- ilar results were reported for charge- and size-symmetric elec- tems. In order to draw conclusions based on large systems and trolytes, in the presence of a large spherical macroion, and the a broader range of system parameters, we use here an implicit emergence of the negative differential capacitance was explained solvent–explicit ions coarse-grained model of the confined poly- in terms of the compactness of the double layer (26). Moreover, electrolyte system. Typical configurations of the polyelectrolyte subsequent studies (27, 28) suggest that, in a uniform charge solution are depicted in Fig. 1. Each polyelectrolyte is rep- density controlled system, the differential capacitance of one resented by a linear bead-spring chain with randomly placed double layer can become negative, as long as the differential charged beads (29). The charge fraction is defined as fq = n/N , capacitance of both of the double layers taken together remains where n is the total number of charged monomer beads, each positive, whereas, when the electrodes are held at constant with one electron charge −Ze, and N is the total number of electrostatic potentials, the differential capacitance will always monomers. For overall electroneutrality, we have added n pos- be positive. itive counterion beads with +Ze charge. The polyelectrolytes Here, we show that dielectric mismatch between the polyelec- and counterions are confined between two planar surfaces, each trolyte solution and the confining walls exhibits nontrivial effects composed of spherical beads arranged into a square lattice. on the electrical double layers near the charged interfaces. We The positively charged surface is placed at z = 0, and the neg- demonstrate that such effects lead to the negative differential atively charged surface is placed at z = H . Unless otherwise capacitance of the double layer for a certain range of the surface stated, we employ the fixed-charge boundary conditions, where charge density. Furthermore, we observe that polarization effects each surface bead is assigned a partial charge such that the sur- result in the noticeably enhanced energy storage in the electrical face charge density is Σ. The confining surfaces are composed double layer, which can be useful for supercapacitor design. of a material of low dielectric constant, 1 = 2. The solvent is

Fig. 1. Typical configurations of the polyelectrolyte solution confined between two dielectric planar surfaces, for polyelectrolyte charge fraction fq = 0.9. The green, orange, and red beads represent charged monomers, neutral monomers, and counterions, respectively. The three snapshots correspond to (A) uncharged surfaces with polarization, (B) charged surface without polarization, and (C) charged surface with polarization. In B and C, the region near the positively charged surface is shown.

2 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2007545117 Bagchi et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 mlfiaini u oteitrlybtentegi nthe in gain the between interplay charge the that to suggests solution due This electrolyte amplification. is the charge amplification for show profile not density how- does charge system, net polyelectrolyte the the Unlike ever, of all removed. that are except bonds solution the polyelectrolyte the to identical is show which we 2B, Fig. disappears. amplification In charge and large, becomes charge counterions densities However, charge they higher surface surface. low for repulsion for like-charged the only observed than the is amplification larger to is due counterions higher experience the for of entropy pronounced in more 2A, becomes (Fig. and ues 31), referred 21, is (5, surface as like-charged a to near in counterions shown configuration of typical mulation the from 1B seen sur- Fig. be the can of This both near faces. accumulate counterions at surface charged charged positively positively the accu- near chains polyelectrolyte mulate charged negatively The fraction baseline. monomers, the charge the charged of plot negatively values first the We and respectively. counterions charged where acie al. et Bagchi is density (C charge fraction amplification. surface charge The charge the to of leads values that different for surface charged positively the near same. the being parameters all with 2. Fig. the by characterized are profiles layer density double charge net electrical time-averaged the in species Profiles. Density Charge Discussion and Results in given (30). are work method Methods simulation recent and and our Materials model in our of described details methods Specific (MD) the we dynamics using molecular solution, step every the at time charges and induced the surfaces for dielectric the solve between uniform mismatch with dielectric background continuum constant a as modeled hreamplification charge (A ρ o oylcrlt hrefraction charge polyelectrolyte for +  and 2 Σ and otk noacutteplrzto u to due polarization the account into take To 80. = e hredniyprofile density charge Net B) aus h euso ewe h ufc n the and surface the between repulsion the values, .Ti hnmnnaie eas h gain the because arises phenomenon This Inset). ρ − r h hredniiso h positively the of densities charge the are ρ(z and h tutrlfaue ftecharged the of features structural The ) AB CD = Σ (z) / (z) [reduced unit] c solution, electrolyte equivalent an for IAppendix. SI -1.4 -1.2 -0.8 -0.6 -0.4 -0.2 0.2 ρ(z -0.2 -0.1 0.1 0.2 as eerdt as to referred (also 4C·m 0.04 -1 0 0 f Insets q ) 0 ihu oaiainefcsas effects polarization without , 0 o he different three for 2A, Fig. in anf h est rfiena h w ufcs h e trin star red The surfaces. two the near profile density the magnify −2 -0.2 -0.1 f 0.1 0.2 q 0 0.2 . ρ(z o (C for 0 o ifrn auso h hrefraction charge the of values different for ) f f f ρ(z q q q 0.4 =0.90 =0.60 =0.30 0.05 lcrlt ouin optdfo h aain data the from computed solution, electrolyte (D) and solution polyelectrolyte ) 0.4 f q = ) z /H z -0.1 z /H 0.1 0.2 0 = 0 = 0 0.9 0.1 ρ 0.6 0.6 + hsaccu- This .9 hra the whereas , overcharging (z 0.95 f f f q q q ) =0.90 =0.60 =0.30 − 0.8 0.8 ρ f 1 q − (z val- Σ; 1 1 ), and ) uuaiecag density charge Cumulative D) h ali h rsneo oaiainefcs n h space polyelectrolyte the counterions. the more and and by effects, occupied surface is polarization charged layer from of positively away presence the repelled the between are in chains wall polyelectrolyte dif- the the for that 3B, seen 1 Fig. Fig. in be Comparing seen densities. be charge can surface as ferent heights surface, peak charged increased positively the the polar- the by of of indicated presence Second, effects, the top surfaces. in ization on stronger impenetrable becomes the counterions amplification to charge and due polyelectrolytes confinement spatial the to effect confine- dielectric (when Therefore, more present. mismatch experiences are 3A, effects solution polarization Fig. when in polyelectrolyte ment from two seen the away be in is, shifted can (as is that altered surface profile charged density is positively the the solution charge First, the 3A). polyelectrolyte account, (Fig. into aspects the taken of are solution distribution the are and effects surface the these 2D. Fig. of in all shown system while electrolyte if amplification, the even in charge absent surfaces no the polyelec- at is The observed inversion surfaces. there is charge but both exhibits 2C) surface, near solution charges Fig. observed trolyte positively in is the near curve inversion layer charge the double of the in part only positive (the cation sncsayt etaieiscag,wihi phenomenon a is what which than charge, sign its opposite as neutralize the known density to of charge necessary charges be surface is more to the found attracts than is surface 2C) magnitude (Fig. in surfaces larger both near solution electrolyte (z) [reduced unit] the between in layers difference double the the in into systems. difference two stored translates the energy later, profiles Coulombic shown density the be charge will the As in surface. the polyelectrolytes to the of adsorbed earlier) entropy conformational (mentioned in penalty counterions the and the of entropy configurational hnplrzto fet u odeeti imthbetween mismatch dielectric to due effects polarization When h uuaiecag density charge cumulative The -0.3 -0.2 -0.1 0.1 0.2 0.3 0 0 hreinversion charge f q -0.3 -0.2 -0.1 0.1 lcrlt solution, electrolyte (B) and solution polyelectrolyte (A) for 0 0.2 0  1 0.05 0.4  < -0.1 z /H 0.1 0.2 0.3 A 0 2 0.9 fetvl dsa xr confinement extra an adds effectively ) hw h oiino oneinaccumulation counterion of position the shows 0.1 ρ 0.6 c (z 2,3) oeta hreamplifi- charge that Note 32). (21, 0.95 cldb h nefc hredensity charge interface the by scaled ) f f f q q q =0.90 =0.60 =0.30 0.8 ρ 1 c (z 1 = ) NSLts Articles Latest PNAS R 0 z ρ (s A and )ds B and respectively. B, o h poly- the for Each Σ. tcan it C, h + | Inset); f8 of 3 near Σ

APPLIED PHYSICAL SCIENCES 2 α α 2 A Rgα = h(~ri −~rcm ) i, where α ≡ x, y, z; h·i denotes the aver- 0.2 0.2 age over all of the polyelectrolyte chains. As can be seen from 2 2 0.1 Fig. 3C, in the bulk, 3Rgz (z)/Rg ≈ 1, and the chains, on aver- age, have a spherically symmetric conformation. However, next 0 2 2 0.1 to the positively charged surface 3Rgz (z)/Rg < 1, the polyelec- -0.1 trolyte chains are compressed along the z− direction into an 2 2 0.9 0.95 1 oblate spheroid conformation. Thereafter, 3Rgz (z)/Rg > 1 over- 0 shoots the bulk value. In the overshoot region, the chains are 0.1 more elongated along the z direction and assume a prolate 0 -0.1 spheroid conformation. The flipping from a z-compressed oblate (z) [reduced unit] -0.1 to a z-elongated prolate conformation occurs in a region that -0.2 w/Polr mostly has a net positive charge, that is, with excess counteri- -0.2 ons (Fig. 3C, Inset). The oblate–prolate flipping happens more 0 0.05 0.1 wo/Polr than once, indicating that the polyelectrolyte chains arrange themselves in alternate layers of z-compressed and z-elongated 0 0.2 0.4 0.6 0.8 1 conformations under confinement. The conformational flipping z / H therefore can be attributed to the polyelectrolyte chains trying B to minimize the electrostatic repulsion of the neighboring lay- 0.12 ers of polyelectrolyte chains. The overshooting peak is more Polyelectrolyte w/Polr prominent in the presence of polarization, where the confine- 0.1 Polyelectrolyte wo/Polr ment effects are stronger. This phenomenon is reminiscent of the cubatic phase that was observed originally in simulation stud- 0.08 ies of hard cut spheres (33), although the scenario is much more complicated here. + 0.06 We note that, for confining surfaces that have a higher dielec- h tric constant than the solution (i.e., 1 > 2), the charges are 0.04 attracted toward the surfaces in the presence of polarization effects, even when the surface charge density is zero. The charge 0.02 density near the surfaces exhibits higher peaks near the surface compared to the case where there is no dielectric mismatch, 0 1 = 2 (SI Appendix, Fig. S1). 0 0.05 0.1 0.15 0.2 Differential Capacitance and Energy Storage in the Double -2 [Cm ] Layers. We first obtain the electrostatic potential profile Φ(z) C between the two surfaces from the net charge density pro- 1.4 file ρ(z) by numerically integrating the Poisson’s equation, 2 Polyeletrolyte w/Polr ∂z Φ(z) = −ρ(z)/(02), where 2 is the solvent dielectric con- 1.2 Polyeletrolyte wo/Polr stant and 0 is the vacuum permittivity. The boundary condi- tions are chosen as Φ(z = 0) = 0 and E(z = H /2) = −∂z Φ(z =

(z) 1 H /2) = 0. 2 Typical potential profiles Φ(z) obtained from the simula- 0.8 tions for the polyelectrolyte and an electrolyte solution (which 1 2 2 (z)/Rg 3 Rgz (z)/Rg (z) is the same as the polyelectrolyte solution but with all of the 2 0.6 z bonds between the monomers removed) are shown in Fig. 4A 0.4 0 (see also SI Appendix, Fig. S2). We denote the potential drop (z) between the positively charged surface and the bulk as Φ+ (and 3 Rg 0.2 similarly for Φ−) which is the voltage drop across the electric double layer (EDL) and is referred to as the EDL potential. 0 0 0.1 0.2 0.3 0.4 0.5 Here the EDL is identified from the charge density profile, 0 0.1 0.2 0.3 0.4 0.5 where ρ(z) first approaches the bulk value from the positively charged surface (marked by an arrow in Fig. 3A). For the z / H polyelectrolyte solution, Φ+ 6= Φ−, unlike for the electrolyte solution. Here, we will focus on the EDL potential Φ+, since Fig. 3. (A) Comparison of the charge density profile ρ(z) of the polyelec- this is the voltage drop across the double layer formed near the trolyte solution for fq = 0.90 with (w/Polr) and without (wo/Polr) polariza- positively charged surface where the polyelectrolyte chains get tion effects for surface charge density Σ = 0.04 C·m−2. The region between the positively charged surface and the point marked by the arrow is consid- absorbed. The differential capacitance of the double-layer structure, Cd , ered as the double layer. Insets magnify the charge density profile near the −1 charged surfaces. (B) The height of the charge amplification peak for differ- is defined as Cd = (dΦ+/dΣ) , where the double-layer poten- ent values of the surface charge density Σ, with and without polarization. tial Φ+ for different values of the surface charge density Σ is 2 2 (C) The normalized radius of gyration of the polyelectrolytes 3Rgz(z)/Rg with shown in Fig. 4B. When polarization effects are neglected, Φ+ 2 2 and without polarization for fq = 0.40. In Inset,3Rgz(z)/Rg is shown against increases monotonically with increasing Σ, similar to what is the net charge density ρ(z) for the polyelectrolyte solution with polarization observed with electrolyte solutions. When polarization effects effects. are included, however, the relationship between Φ+ and Σ −2 becomes nonmonotonic. Specifically, for Σ < 0.1 C·m , Φ+ Interesting features in the conformation of the polyelec- increases with increasing Σ. In this regime, the difference in Φ+ trolyte chains are observed in the double layer formed near with and without polarization effects is due to the increase in the positively charged surfaces. The average squared radius of the charge amplification peak h+ in the presence of polarization 2 2 gyration is defined as Rg = h(~ri −~rcm ) i and its components effects (Fig. 3B).

4 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2007545117 Bagchi et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 acie al. et Bagchi ssfcetysrn odpeealo h onein rmthe from counterions the of all deplete nega- to the strong surface between sufficiently charged positively is attraction the and Coulombic polyelectrolytes charged the tively that means This by indicated as vanishes, surface charged increasing capacitance, ihu oaiainefcs h ufc hredniyis density charge surface in The effects. polarization without f fet.(C effects. potential determined. EDL are the bulk density charge the of to variation surfaces The charged (B) the from drops fraction potential charge the for solutions electrolyte 4. Fig. C B A q A For o h lcrlt ouinadtepleetoyeslto,wt and with solution, polyelectrolyte the and solution electrolyte the for U+ [mJ] [mV] (z) [V] and + 55 25 30 35 40 45 50 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0.5 0.6 0.7 0.8 0.9 1.1 0.1 lcrsai oeta profiles potential Electrostatic (A) C. 1 h obelyreetottcenergy electrostatic double-layer The ) 0 0 0.2 ≤ nti eie hreapicto ttepositively the at amplification charge regime, this In Σ. Σ Σ C 0 ih(le qae)adwtot(le ice)polarization circles) (filled without and squares) (filled with d ≤ Polyelectrolyte: w/Polr < 0. as 0, C·m 2 0.05 Electrolyte: w/Polr 0.2 Polyelectrolyte wo/Polr + 0.4 Polyelectrolyte w/Polr Φ + −2 Polyelectrolyte erae rm5 Vt 5m with mV 45 to mV 55 from decreases eosreangtv differential negative a observe we , 0.4 [Cm Electrolyte 0.1 Φ 0.6 z /H + f q f Φ q = o ifrn auso surface of values different for -2 (z 0 h rosso how show arrows The 0.90. ftepleetoyeand polyelectrolyte the of ) 0.6 ] h + U en eo(i.3B). (Fig. zero being + 0.15 ihcag fraction charge with , wo/Polr , wo/Polr 0.8 0.8 = Σ - 4C·m 0.04 0.2 1 1 −2 h oylcrlts ecntanteage ewe the between angles the constrain we polyelectrolytes, the the Flexibility. discuss Chain will sections. We following the surfaces. in factors the these at constant, of effects dielectric conditions solvent boundary flexibility, and chain design as key of such functions parameters as structure double-layer storage the energy of and behaviors capacitative the understand to importance cal storage energy favorable for more electrolytes is over devices. it polyelectrolytes that in choose suggests those to This the than solution. for stronger electrolyte polarization the appreciably the of account are effects into solution the takes polyelectrolyte Also, one polarization. when of enhanced effects further polyelectrolyte the is for elec- stored solution the energy in the stored Moreover, energy EDL. the trolyte than higher is EDL polyelectrolyte density fraction storage charge energy polyelectrolyte the the plot profile We density EDL. charge the net inside the integrating numerically from area dielectric below. solvent of shown and as function flexibility constant, a chain be as such would parameters important various are effects polarization which this of of vicinity side the in prominent effect most drop is the EDL potential result, electrostatic a the on As polarizability investigation. of under system that particular find the We respectively. polarization, ∆Φ without and Let where with potential. effects, values electrostatic polarization the of for importance true 4B not Fig. define is therein), us this references and that 34 strates as ref. unimportant (see increasingly increases become sur- effects on polarization dependence that its density and charge effects face polarization of importance and size explored. spherical) the versus regimes to (planar parameter due surfaces the confining is the electrolytes of confined shape in and between findings difference our The and (26). those macroion solution of spherical electrolyte with- for a behavior density reported surrounding charge previously nonmonotonic surface was solu- The polarization the electrolyte out with S7). 1:3 potential Fig. al. electrostatic and differential the Appendix , that 1:1 et find for (SI Partenskii We positive tions by (28). always Jordan suggestion is This and capacitance positive. the Partenskii remains with and of capacitance (27) consistent total value indeed the is, is all that S6B); for Fig. monotonically increases D n ihtecnnn surfaces. confining the the with in and charges EDL conformational the between interactions polyelectrolyte electrostatic and certain entropy, correlations, a volume within excluded capacitance differential of negative range emergence EDL The S6A). the Fig. Appendix, of (SI included are effects tion Φ and dominates, attraction Coulombic polyelectrolyte–surface the S5). Fig. where Appendix, case (SI the excluded are to the effects compared For polarization value mean S4). the and around distribution S3 of Figs. of range Appendix, density given (SI charge distributed net the geneously the in of EDL distribution the spatial The surfaces. + rmtespraaio einpito iw ti fpracti- of is it view, of point design supercapacitor the From surface unit (per EDL the in stored energy electrostatic The rca on htw att aehr srgrigthe regarding is here make to want we that point crucial A oeta h oa oeta drop potential total the that Note density, charge surface of values greater For + lasicesswith increases always L ece h aiu au around value maximum the reaches 2 sgvnby given is ) 4C·m 0.04 = Σ Σ ∆Φ Σ em orsl rmacmlxitrlybetween interplay complex a from result to seems ∗ + au.W xetta both that expect We value. h npaecag este aeabroader a have densities charge in-plane the Σ, = x–y orpeetteflxblt o tfns)of stiffness) (or flexibility the represent To |Φ Σ + p U ln eel httecagsaehetero- are charges the that reveals plane −2 + − .Atog h eea eifis belief general the Although 4B). (Fig. 1 = Φ Σ h nrysoaei the in storage energy The 4C. Fig. in + 0 eadeso hte rntpolariza- not or whether of regardless /2Φ |/(Φ + + p Q Φ + + f q /L o xdsraecharge surface fixed a for + 0 2 Σ ) NSLts Articles Latest PNAS where , ∗ Σ sa niao fthe of indicator an as Φ = Φ Σ n ssalo either on small is and Φ ∗ U ∗ + p n h ag over range the and 0 = Σ + and Σ + safnto of function a as Q 7C·m .07 IAppendix, (SI Φ > Φ + + + Φ C·m 0.2 sobtained is + 0 costhe across − demon- r the are | always −2 f8 of 5 ρ(z −2 for Σ ) ,

APPLIED PHYSICAL SCIENCES adjacent bonds along the polyelectrolyte backbone with a har- pact EDL that is formed in the lower dielectric constant solvent, 2 monic potential U (θ) = Kθ(θ − θ0) , where the stiffness con- because of the stronger electrostatic interactions between the ◦ stant Kθ = 100, and equilibrium angle θ0 = 180 . Here, θ is charged monomers and the confinement surface. We also find the angle between two adjacent bond vectors ri,i−1 and ri,i+1, that, as Σ increases, the solvent with a higher dielectric constant where ri,j is the vector pointing from monomer i to monomer stores more electrostatic energy in the polyelectrolyte EDL, as j . The effects of the backbone rigidity in the emergence of the shown in Fig. 6B. negative differential capacitance can be realized by comparing the EDL potential for the stiffer chains (Fig. 5A, red squares) Constant Potential Boundary Conditions. We study a limiting case with that for the fully flexible chains (shown in Fig. 4B, red of surface polarization where 1 approaches infinity; that is, squares). For any given value of Σ, the EDL potential for the we consider the surfaces (electrodes) to be metallic, as in bat- stiffer chains is smaller than for the fully flexible chains. Also, teries. In this limit, the electrical field inside each electrode the negative differential capacitance is observed over a larger vanishes, and the electrostatic potential at the electrode surface −2 range 0.04 < Σ < 0.16 C·m , where Φ+ decreases from 40 mV is constant. Under the applied electrostatic potential difference to 15 mV. Notably, for stiffer polyelectrolyte chains, we find, between the electrodes, ∆Φ, the electrode charge fluctuations from Fig. 5A, that negative differential capacitance emerges even can be sampled from Boltzmann statistics (23). Although there in the absence of polarization effects for 0.04 < Σ < 0.12 C·m−2. are numerous approaches for handling the constant potential Recall that, for the fully flexible chains (i.e., Kθ = 0), Cd is always boundary conditions in MD simulations (18, 19, 35–37), here positive without polarization effects, suggesting that the reduced we follow the method proposed by Petersen et al. (19) for conformational entropy of the stiffer chains is another factor that computational efficiency. For our simulations, we approximate leads to the negative slope and nonmonotonic behavior of the the metallic materials by setting the dielectric constant of the Φ+ versus Σ curve. electrodes to be 1 = 5,000 and computing the contributions The net charge density profiles (SI Appendix, Fig. S8) further from the primary and higher-order image charges to the fluc- reveal that the EDL formed by the rigid polyelectrolytes is more tuating electrode charges using the energy functional approach compact than the fully flexible polyelectrolyte EDL. This is pre- (30, 38). sumably because the rigid chains have a lower conformational For small potential differences ∆Φ, charge amplification is entropy, and therefore can form a more compact double layer present at the positive electrode similar to what is observed with near the surface. Moreover, charge amplification is also found carbon-based electrodes (Fig. 2A). However, a negative differ- to be more pronounced in the case of rigid polyelectrolytes. ential capacitance was not observed; that is, the charge stored in These results suggest that the higher double-layer compactness the polyelectrolyte EDL always increases monotonically with Φ+ and charge amplification are strongly related to the more pro- (SI Appendix, Fig. S9) for all of the different constant potential nounced negative differential capacitance that is observed for differences studied. This validates the claim, mentioned earlier, rigid polyelectrolytes. that negative differential capacitance emerges only in the uni- Regarding the Coulombic energy stored in the EDL, we find form charge density ensemble, but not in the constant potential that, while U+ for the fully flexible chains increases roughly ensemble, where differential capacitance plays a role similar to linearly with the surface charge density Σ, it appears to be non- susceptibility (23, 27). Thus, the result obtained from the uni- monotonic for the stiffer chains (Fig. 5B), which is correlated form charge density ensemble is inequivalent to the constant with the variation of Φ+. Moreover, U+ for the stiffer chains potential ensemble. This is reminiscent of the specific heat in is considerably smaller than for the fully flexible chains. This systems with long-ranged interactions: The specific heat can is because the number of the (positively charged) counterions become negative in the microcanonical ensemble, but is always in the EDL for the stiffer chains is smaller than that for the positive in the canonical ensemble, thus giving rise to ensemble fully flexible counterparts, or, in other words, the number of inequivalence in long-ranged interacting systems (39). Although counterions released to the bulk is greater for the stiffer chains. the two boundary conditions, fixed charge density and constant potential, may yield different differential capacitance behav- Solvent Dielectric Constant. For a solvent with a lower dielectric iors, they are not completely unrelated: A negative differential constant 2 = 40 (e.g., ethylene glycol), the electrostatic interac- capacitance in the uniform charge density boundary conditions tions are twice as strong as in water. As can be seen in Fig. 6A, implies interfacial instabilities and phase transitions in a constant the nonmonotonic relationship between Φ+ and Σ persists in potential controlled system (27). Note that negative differen- the presence of polarization effects. In fact, the region of neg- tial capacitance was also predicted to occur in a third kind of ative differential capacitance, for 2 = 40, begins at a lower value thermodynamic constraint where the total charge on the con- of the surface charge density Σ ≈ 0.05 C·m−2 as compared to fining surfaces is held constant and the local charge density is −2 Σ ≈ 0.1 C·m for 2 = 80. This is attributed to a more com- fluctuating (27).

AB 5 40 w/Polr Rigid 35 4 wo/Polr Flexible 30 3 [mV] [mJ] + 25 + 2 20 U 15 1 10 0 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2 [Cm-2] [Cm-2]

Fig. 5. (A) Comparison of the EDL potential Φ+ for a solution of stiff polyelectrolytes, in the presence and absence of polarization effects. (B) Comparison of energy storage U+ in the double layer near the positively charged surface for the rigid and the fully flexible polyelectrolyte solutions, in the presence of polarization effects.

6 of 8 | www.pnas.org/cgi/doi/10.1073/pnas.2007545117 Bagchi et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 acie al. et Bagchi constant. dielectric solvent the and solvent, the of flexibil- quality chain on the of depends ity, turn, entropy capaci- in conformational that, chains, differential the polyelectrolyte on as the and dependent such strongly storage, properties, are double-layer tance, energy crucial amplification, the con- of charge simulations the all a our Finally, inside that have neglected. indicate distribution be surfaces cannot charge This thus charged density. and the finement, charge for on surface effect effects of considerable range polarization certain that a implies for storage EDL energy the increased of in and enhancement capacitance, to or differential lead emergence negative mismatch energy. amplification, dielectric electrostatic charge to more enhanced due store effects tunable to polarization more ability Second, com- with the behavior under First, and richer solutions remarks. parameters, much polyelectrolyte a few exhibit solutions, a confinement electrolyte with to discussion pared our conclude We double-layer polyelectrolyte Conclusion enhances S10D). Fig. It Appendix, same (SI solvents: the storage is energy poor solvent good poor in a to in as compared polarization of when effect conditions The solvent solvents. charge good layer weaker under double this polyelectrolyte lower the to is in Due storage counterions. energy the amplification, that by good surface confining occupied the a the near be space in approach can less leaving can chains thus and closely, polyelectrolyte more out surface the stretched because, more is solvent, This are poor S10). solvent Fig. a Appendix, in (SI solvents unlike compared poor when weaker in is that solvent to good that using a find in to We amplification for monomers. switch charge the we potential between interaction solvent, Weeks–Chandler–Andersen nonbonded good the repulsive a purely of effect the the interaction. investigate (LJ) the Lennard-Jones To the conditions between by solvent represented interaction is poor non-Coulombic monomers under nonbonded performed the are where far so Solvent. presented Good a in Polyelectrolytes storage energy of Comparison (B) effects. 6. Fig. .J .Zaikn .Ovr el rz ual otsrcuei hre fluids charged in structure soft Tunable Cruz, la de Olvera M. Zwanikken, screening the of W. analysis J. Scaling in Perkin, S. 8. breakdown Smith, M. A. neutrality Perez-Martinez, S. Charge C. Lee, A. Levin, A. Y. 7. Santos, dos P. A. Girotto, M. Colla, T. 6. Jim F. 5. Suo L. 4. polyelectrolyte on perspective Gonz A A. perspective: 3. anniversary 50th Muthukumar, surfaces. at M. and solutions 2. in polyelectrolytes of Theory Rubinstein, M. Dobrynin, V. A. 1. ofie ydeeti interfaces. dielectric (2013). by confined electrolytes. concentrated in length simulation. and (2016). Theory electrolytes: aqueous macroions. confined by inversion charge and B reversal, Chem. charge Overcharging, wall: chemistries. materials. and Technologies solutions. Sci. Polym. Prog. enez- ´ oprsno h D potential EDL the of Comparison (A) Wtri-at lcrlt nbe ihvlaeauoslithium-ion aqueous high-voltage enables electrolyte “Water-in-salt” al., et lz .Gioe,J .Brea .Msk eiwo supercapacitors: on Review Mysyk, R. Barrena, A. J. Goikolea, E. alez, 2679 (2004). 7286–7296 108, ´ Macromolecules nee,M oaaCso,Amdlmcoo ouinnx oacharged a to next solution macroion model A Lozada-Cassou, M. Angeles, ´ Science 0911 (2005). 1049–1118 30, 3–4 (2015). 938–943 350, 5896 (2017). 9528–9560 50, ee.Ssan nryRev. Energy Sustain. Renew. AB + [mV] 15 20 25 30 35 40 45 50 55 hs e.Lett. Rev. Phys. rc al cd c.U.S.A. Sci. Acad. Natl. Proc. 0 l ftesmlto results simulation the of All U 0.04 + Φ ntedul ae ertepstvl hre ufc o h w cases. two the for surface charged positively the near layer double the in + 202(2017). 026002 119, o oylcrlt ouinwt ovn ilcrcconstants dielectric solvent with solution polyelectrolyte a for 0.08 .Ce.Phys. Chem. J. 1910 (2016). 1189–1206 58, [Cm 2 2 =40,w/Polr =80,w/Polr 0.12 -2 ] 5301–5308 110, 0.16 094704 145, .Phys. J. 0.2 7 .W,H i .J oi,M lead aCu,E uje,Aymti lcrltsnear electrolytes Asymmetric Luijten, E. Cruz, la de Olvera M. Solis, J. F. Li, H. surface. Wu, charged H. a at 17. multiple-scattering adsorption polyelectrolyte the on forces and image of Effect interfaces Messina, R. dielectric near 16. polarization Simula- Charge surfaces: charged Qin, dielectric liquids J. between in 15. Electrolytes structure Levin, Ionic Y. Cruz, Santos, la dos de P. Olvera A. M. 14. Zwanikken, W. a J. to Jadhao, adsorption V. polyelectrolyte Jing, of Y. Simulations 13. Levin, Y. polyelectrolyte Girotto, M. on Santos, effects dos P. surface A. of 12. study Atomistic Holm, C. Cerda, J. induced J. polyampholytes Qiao, of B. reversal mobility 11. Electrophoretic Holm, C. Hickey, A. O. 10. e uhta h jru eghi h solution the in length Bjerrum the that such sen yteSemnFicidFudto.DB rtflyakolde fruitful manuscript. acknowledges Jim gratefully F. the D.B. with Foundation. and discussions of Fairchild 70NANB19H005, part Award Sherman as under the Technology Design by Materials and Hierarchical Standards for of Center Institute National Commerce, ACKNOWLEDGMENTS. debarshee Appendix is unit time dimensionless The simulations. as as our defined throughout defined 1.0 is be temperature to dimensionless set and The depth. of well constant LJ the spring elas- the nonlinear extensible with finitely 10 springs the tic of by modeled accuracy is particle–particle monomers an the the between with using computed method is mesh interaction particle Coulombic of The distance nm. 0.7 cutoff the with modeled potential is LJ particles 12–6 where charge the standard overall between the ensure interaction to by system nonbonded the The to neutrality. added are Counterions simulation. of chains, polyelectrolyte composed the represent to each model bead-spring linear the use We Methods and Materials general. in systems study charged at heterogeneous this accurately in included in and be interfaces reported should effects mismatch polarization dielectric that suggest to non- due the flexible Thus, effects effects. for polarization trivial surface higher constant of dielectric presence be the larger in a to and with solvent found poor is a in storage polyelectrolytes energy particular, In aaAvailability. Data simulation and model were our simulations on in details the given More are of 2018. method July All Simulator 16 30. Parallel version Massively ref. time (LAMMPS) Atomic/Molecular every in Large-Scale at described with particle methods performed substrate the each for using charges step induced the compute we substrates at two the fixed between distance is The counterions. and monomers the of .H i .Ebs .W wnke,M lead aCu,Inccnutvt in conductivity Ionic Cruz, la de Olvera M. Zwanikken, W. J. Erbas, A. Li, H. 9. U+ [mJ] tutrddeeti interfaces. dielectric structured E Rev. Phys. formalism. theory. and tions interfaces. dielectric by confined surface. like-charged dielectric monolayer. (styrenesulfonate) poly (2011). a 1707–1718 of study Case adsorption: confinement. or fields electric strong by hydrogels. polyelectrolyte 0 1 2 3 4 5 0 σ stelnt cl fteL oeta.Tepril charge particle The potential. LJ the of scale length the is 2 2 aebe eoie oBtukt(https://bitbucket.org/NUaztec/ Bitbucket to deposited been have =40,w/Polr =80,w/Polr τ trung H otMatter Soft 0.04 = 582(2004). 051802 70, = σ 100σ p monica .Ce.Phys. Chem. J. l ftedt hw ntemnsrp and manuscript the in shown data the of All m/ 0.08 N IAppendix SI hnplrzto fet r ae noaccount, into taken are effects polarization When . enez- p ´ 1523 (2019). 2125–2134 15, where , = hswr a upre yteU eatetof Department US the by supported was work This [Cm pnas Macromolecules 0bas n hr r 0can natypical a in chains 60 are there and beads, 40 nee,adA he o aeu edn fthe of reading careful for Ehlen A. and Angeles, 0.12 ´ -2 .Py.Ce.B Chem. Phys. J.  .Ce.Phys. Chem. J. ] 914(2015). 194104 142, 2 2020/src/master/). .Ce.Phys. Chem. J. m = . stepril asbigietclfrall for identical being mass particle the is 0ad4,i h rsneo polarization of presence the in 40, and 80 0.16 .Ce.Phys. Chem. J. k 2994 (2016). 9239–9246 49, = 671(2018). 164701 149, 0.2 30/σ 08–09 (2016). 10387–10393 120, 958(2015). 194508 143, −4 NSLts Articles Latest PNAS h oddinteraction bonded The . and 995(2013). 194905 138, l B = R (Ze) 0 = Macromolecules 2 1.5σ /(4π where , T * r 0 c  Z = | 2 = k scho- is k B f8 of 7 2.5σ B T T  = ) 44, / SI is ,

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