Controlled Chemical Kinetics in Porous MembranesShivraj D. Deshmukh and Yoav Tsori*

Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel

ABSTRACT: We investigate theoretically the kinetics of chemical reactionsin polar solvents in the vicinity of charged porous membranes. When the porecharge (or potential) exceeds a critical value, the pores undergo a fillingtransition that can be first or second order depending on the ambienttemperature, mixture composition, and other parameters. This fillingtransition leads to a dramatic acceleration or slowing down of the reaction.Such control of reaction kinetics by an external potential may be useful inapplications where catalysts are absent or when fast spatiotemporal responseis required.

The quest to increase the rate, selectivity, and conversionof chemical reactions has motivated many studies in basic

and applied research. Traditionally, efforts are made to controlparameters such as pressure, temperature, surface area,concentration, and different reaction pathways using catalysts.Control of reactions using such parameters is bounded byknown thermodynamic limits and new methods are soughtafter. Electric fields can be easily switched on/off and theiramplitude and frequency can be controlled. They can modifyreactions by bringing two or more charged molecules close toeach other, or by changing the electron energy levels inmolecules.1−5 It has been shown recently that the directionand strength of the field can catalyze Diels−Alder reactions.6Electric fields are especially suited to modern advancedchemistry methods in microfluidic systems,7 since theiramplitude increases proportionally to the inverse system size,if the potentials are fixed.The new approach we employ relies on electrostatic

manipulation of the solvents to enable spatiotemporal controlof the reaction rate.8 For carbonaceous surfaces or membranes,the surface potential can be easily controlled by externalmeans, and this allows us to achieve slowing down oracceleration of reactions by means of manipulation of thesolvent composition. Such accelerated kinetics is particularlyadvantageous in cases where addition of catalysts is undesirabledue to its adverse effects or difficulties in its removal from thereaction products. Importantly, this method does not rely on aspecific chemistry.

■ MODELConsider a binary mixture of a polar solvent (e.g., water) and acosolvent. Their volume fractions are ϕ and ϕcs, respectively.The mixture contains a small amount of dissociated salt andreagents. The chemical reaction takes place between twospecies, A and B, to irreversibly form a molecule C:

+ →A B 2Ck

. The reagents are assumed to be nonionic, and

their presence does not affect the mixture’s phase diagram dueto their small number. To stress the new mechanism, the rateconstant k is assumed to be independent of the local field andtemperature; the field manipulates the solvent composition,and thereby the local densities of A and B molecules, leading toan increase or decrease of the reaction by an indirect physicalmechanism. The reaction is highly temperature-sensitive evenif k is temperature-independent (see below). A chemicalspecies i (i = A, B, or C) has different solubility potentials inthe water and in the cosolvent, uw

i and ucsi , respectively. The

Gibbs transfer energy in units of kBT (where kB is theBoltzmann constant), Δui = uw

i − ucsi , is the free energy

required to move molecule i from the cosolvent to the polarsolvent. In many liquids, Δui is of the order 1−10.9 A moleculei thus “feels” an effective potential of the form −ui(ϕ) = uw

i ϕ +ucsi ϕcs, and this potential is an important driving force in thefollowing discussion.The dimensionless mass balance equations for the chemical

species are

∂∂

ϕ

ν

∇ ∇

= ∇ + ·[ ]

−

C r tt

C r t C r t u

kC r t C r t

( , )( , ) ( , ) ( )

( , ) ( , )a b

i 2i i

i

i (1)

Here, Ci = Ci/C0 is the concentration scaled by C0, the initialaverage bulk concentration of the reagents A and B. Toelucidate the effect of field gradients, we focus on reactionstaking place near porous membranes. The pores are cylinderswith diameter L which is much smaller than their length sothat edge effects can be neglected. L is larger than both thedebye length λD and the correlation length modified by thepresence of salt ξ(T, n0, Δu) (curvature is smaller than the

Received: May 30, 2018Revised: August 7, 2018Published: August 8, 2018

Article

pubs.acs.org/JPCBCite This: J. Phys. Chem. B 2018, 122, 8269−8273

© 2018 American Chemical Society 8269 DOI: 10.1021/acs.jpcb.8b05175J. Phys. Chem. B 2018, 122, 8269−8273

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corresponding inverse lengths).10,11 This means that thecomposition of the mixture, the ion density, and electrostaticpotential all decay from their high value at the wall to theirbulk values if the distance from the wall is longer than ξ andλD. That is, the interior area of the cylinder (its centerincluded) is so far from the walls that it has the samecomposition, ion density, and (vanishing) potential as the bulkreservoir. Consequently, we model the pore as two flat andparallel surfaces.The first term on the right-hand side is Fickian diffusion,

where for simplicity we assume the diffusivities of the reagentsare equal, Da = Db = Dc = D, and independent of the localsolvent composition. All lengths are scaled by L and time isscaled as t = Dt/L2. The second term originates from selectivesolvation (Gibbs transfer energy). The rate constant in thethird term of the equation is scaled as k = kC0L

2/Da, νA = νB =1 and νC = −2. This scaling allows us to focus on the influenceof temperature, composition, and external potential on the rateof reaction, putting aside the obvious dependence onconcentration of reactants squared, as in any second-orderchemical reaction.We assume the chemical reaction takes place on a slow time

scale as compared to the phase transition kinetics. This meansthat the reaction in eq 1 occurs in a polar binary mixture whosecomposition is the equilibrium one. One therefore needs todescribe the equilibrium behavior of the pore. When the porepotential is increased from zero, the initially homogeneousmixture demixes such that the polar solvent is enriched nearthe pore walls and depleted far from it. The thermodynamicsof such mixtures is given by the grand potential integrated overthe pore volume

∫ λ λ μϕΩ = [ + + + − − − ]+ − + + − −f f f f n n rdm e3

(2)

where λ± and μ are chemical potentials for ions and mixture,respectively. n± are the number densities of cations and anions.fm is the mixture free energy density given by

ϕ ϕ ϕ ϕ χϕϕ

χ ϕ

= + +

+ ∇

fk T

v

v

( log( ) log( )

(1/2) ( ) )

mB

0cs cs cs

02/3 2

(3)

where χ is the Flory−Huggins interaction parameter.12 Thephase with lower grand potential is more preferable. In thebulk, mixing free energy has a critical composition ϕc = 1/2and the binodal curve is χb(ϕ) = ln(ϕ/(1 − ϕ))/(2ϕ − 1). Forlarge surface potentials, the ion density near the surfaces can behigh and their volumes cannot be neglected; therefore, we usethe constraint ϕ + ϕcs + v0n

+ + v0n− = 1. The reagents do not

appear in this constraint since their number is assumed to besmall, even in the bulk, and they are uncharged. The (∇ϕ)2term is required to account for the interface between coexistingphases and the resultant surface tension. In our work, theinterface is not found at the walls of the pores, and hence, thevolume conservation ϕ + ϕcs ≈ 0 holds without inclusion ofthe ion densities sufficiently far from the walls. That is, when itis far from the walls, the mixture essentially has only twocomponents, and only one gradient term is required.The electrostatic free energy density is given by

ε ε ϕ ψ ψ= − |∇ | + −+ −f q n n(1/2) ( ) ( )e 02

(4)

ε(ϕ) is the permittivity of the mixture, assumed to dependlinearly on composition: ε(ϕ) = εcsϕcs + εwϕ, where εw and εcsare the permittivities of water and cosolvent, respectively,scaled by the vacuum permittivity ε0. Recently, the directdependence of ε on the ionic densities n± has been taken intoaccount.13−15 The effects considered there are outside of thescope of the current work. The electrostatic potential ψ obeysthe Poisson’s equation ∇·(ε0ε(ϕ)∇ψ) = −e(n+ − n−), where eis the elementary charge.The cation and anion free energy densities are given by

ϕ ϕ= [ − − + ]± ± ± ± ± ±f k T n n v u u n(ln( ) 1) ( )B 0 w cs cs (5)

with a molecular volume v0 assumed common to all molecules.Similar to the definition above for molecules, the parametersuw± and ucs

± are the solvation energies of cations/anions in waterand in the cosolvent, respectively. The difference Δu± = uw

± −ucs± is the corresponding Gibbs-like transfer energy for thecations/anions. In our simple formalism, the boundarycondition for ϕ at the pore wall is ∂ϕ/∂x|x=±L/2 = 0. Theboundary condition for ψ is ψ(x = ±L/2) = ±ψs/2 (fixedpotential) or ψ′(x = ±L/2) = ±σ/ε (fixed surface charge σ).16

A direct and specific short-range interaction of ions with thesurfaces is therefore not accounted for.17−19 In the grand-canonical ensemble, the system is coupled to a reservoir atcomposition ϕ0 where the potential vanishes and n0

± are thebulk ion densities, which define the chemical potentials.20

■ RESULTS AND DISCUSSIONThe system has three distinct time scales. The first onecharacterizes the fast ionic response time. In this “adiabatic”(quasi-static) approximation, the ions adjust rapidly to theirsurrounding, and therefore, they obey a Boltzmann distributionincluding the solvation energy terms:20

ϕ= −+ +

±±

+ −nP

v P P(1 )

(1 )0 (6)

where the amplitudes P± are given by

ψ χ ϕ λ= ∓ + Δ + +± ± ±P e k T u k Texp( / ( ) / )B B (7)

The second time scale characterizes the mixture’s diffusivedynamics and is much longer than the ionic response time.The third time scale characterizes the reaction kinetics and isassumed to be the longest. This means that the concentrationsCi obey eq 1 with profiles ϕ that are the minimizers of Ω.With this separation of time scales, the fast ions adjust

instantaneously to the mixture composition, and bothdistribute in space such that the total energy is minimized.The chemical species migrate to their “preferred” location dueto their preferential solvation. In this view, the chemical speciessimply experience an energy landscape that depends on themixture. Thus, the spatial and temporal behavior of thereaction follows trivially from the thermodynamic state of themixture. We give here a concise description of this state basedon our previous work.20 When an initially homogeneous polarmixture is found near charged walls, the polar solvent (water)is enriched near the surfaces (cosolvent is depleted) . If thesurface potential ψs is too small, the composition ϕ variesslowly in space and the system is close to mixing. However,there is a critical value of ψs above which true demixing occurs,with a sharp interface between coexisting domains. Dependingon the temperature and average composition, this demixingtransition can be first or second order. At yet larger values of

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ψs, ion affinities, temperature, or composition, the pore fillswith water and the sharp interface is lost (inside the pore; itexists between the pore and the outer bulk) . The physicaldriving forces for filling of the pore with a water-rich phase are2-fold: the first is the destruction of an interface betweenwater-rich and water-poor domains, and the second is the largeenergy gained by the ions, attracted to the pore walls, whenthey are in contact with much less cosolvent. The shift of thefilling temperature T(ϕ0) away from the binodal curve isstronger when Δu± or n0 are large.20

Figure 1a shows the change in composition profiles as thepore potential increases from zero. Profiles were obtained by a

simultaneous numerical solution to the Euler−Lagrange andPoisson’s nonlinear equations. At small ψs, ϕ is large only closeto the pore walls, but above the filling transition, ϕ is large alsoin the pore’s center. Part b shows the spatially averagedproduct in the pore, ⟨Cc⟩, at different times for the case whereboth reactants prefer the polar phase. Before the fillingtransition, the changes to ⟨Cc⟩ induced by the externalpotential are small. However, the filling transition leads tolarge concentrations of molecules A and B inside the pore and,consequently, to a dramatic 10−30-fold acceleration in thereaction kinetics (Figure 1b). Reduction of ⟨Cc⟩ (slowingdown) of a similar scale occurs when molecules A and B areantagonistic (c) or nonpolar (d).How much salt should be dissolved in the solution to render

the filling transition effective? This is examined in Figure 2.The bulk mixing−demixing curves at varying salt content n0are shown in Figure 2a, which affect the stability of mixtures inthe pore. For points in the phase diagram that are below the

filling curve, bulk mixing−demixing occurs. The curvesdisplace to higher temperatures when more salt is added atgiven values of n0 and ϕ0. The temperature corresponding tothese parameters is referred to as bulk mixing−demixingtemperature Tbmd. In b, we show the corresponding mixturecomposition profiles for fixed pore potential ψs. The fillingtransition can be achieved for higher salt concentration orlower applied electric potential. The average pore composition⟨ϕ⟩ depends sensitively on the pore potential (part c)atsmall values of ψs the pore’s composition equals the bulk value;as ψs increases past the critical value, ⟨ϕ⟩ has a discontinuousjump in its value. The threshold value of ψs decreases withincreasing value of n0. In d, we examine the influence of ψs onthe product by showing ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ψ =C C/c c c 0s

at t = 10. This is

the product normalized by the product in uncharged pores (ψs= 0). All curves start from the value ≈1 at small values of ψs.There is a very large increase in the product at the criticalvoltage. As in (c), the critical voltage decreases with increasingsalt content n0.It is clear from Figure 2 that the average mixture

composition ϕ0 has a great influence on the state of thepore. In Figure 3a we plot ⟨ϕ⟩ vs ψs at different values of ϕ0.For all values of ϕ0 shown there is a critical filling voltage,signaled by a sharp increase in ⟨ϕ⟩. When ϕ0 is close to ϕc, thedifference between the “empty” and “filled” pores is quitesmall. However, for smaller values of ϕ0, away from the criticalpoint, the difference can be very large. This is due to the basicshape of the zero-field binodal curve, whose coexistingcompositions, as obtained by the classical common-tangentconstruction, are increasingly different as temperature isreduced. As is clear from Figure 3b, at a constant surfacepotential ψs, the chemical reaction product increasessignificantly as ϕ0 decreases farther away from ϕc = 1/2.In most experimental settings, porous membranes are

heterogeneous and the pore-size distribution is wide. In Figure4, we examine how the filling transition and the subsequentacceleration or deceleration of a chemical reaction areinfluenced by the pore size. Two pore sizes are shown atthree different temperatures, which are all above the bulkdemixing temperature in the absence of external potential. Ingeneral, filling occurs at smaller potentials when the pore

Figure 1. Control of the effective reaction rate in pores using externalpotentials. The pore is modeled as two parallel flat plates located at x= ±L/2 across which potential ψs is applied. (a) Profiles ϕ(x) for thecomposition of the polar solvent inside the pore for different electricpotentials ψs. A filling transition occurs as the pore potential increases.(b) Change in time of the product concentration averaged over thepore volume, ⟨Cc⟩. Both reactants, A and B, are more soluble in thepolar phase: ΔuA = ΔuB = 10 (accelerated reaction). Colors and linestyles correspond to the same values of ψs in (a). The concentrations

of the molecular species i = A, B at time t = 0 was Ci = C0e−ui(ϕ). (c)

The same as in (b), but reactant A is polar (ΔuA = 10) and reactant Bis nonpolar (ΔuB = −10). Reaction is slowed down by the potential.(d) The same as in (b), but both reactants are nonpolar, ΔuA = ΔuB =−10, and the reaction is slowed down by the potential. In all parts, wetook L = 10 nm, Δu± = 3, ϕ0 = 0.36, ΔuC = 0, and T = 0.99Tc. In allthe figures, we took molecular length a = v0

1/3 = 3.4 Å.

Figure 2. Effect of bulk ion concentration n0 on (a) bulk mixing−demixing curve (ψs = 0), (b) volume fraction of the polar solvent, (c)average volume fraction ⟨ϕ⟩ in the pore, and (d) the averageconcentration of reaction product at time t = 10 scaled by the value inthe absence of potential. We used pore diameter L = 5 nm,hydrophilic ions Δu± = 4, T = 0.97Tc, and both reactants prefer thepolar solvent, ΔuA = ΔuB = 10.

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diameter decreases or the temperature is reduced close to thefilling temperature (a) . The pore diameter at which fillingoccurs is determined by a nonlinear combination of two lengthscales: the debye length and the correlation length ξ (in thiswork, depending also on n0 and Δui).10,11,20,21 At highertemperatures, the filling occurs gradually and second-orderfilling transition is observed, while at lower temperatures, it is afirst-order phase transition. The spatially averaged product ofthe reaction grows correspondingly for A and B moleculespreferred at the polar solvent (b). A dramatic increase of thereaction, by 40−60 times as compared to the no-field case, isshown. An opposite trend, of chemical slowing down, is shownin panels c and d for antagonistic molecules and when both Aand B are preferred in the cosolvent.In Figure 5a, we plot the average composition of the polar

liquid inside the pore for different affinities of ions to the polarsolvent Δu. One can see that as the applied potential is

increased, the average volume fraction of the polar solventincreases inside the pore. At a certain potential, an abrupt first-order filling transition is observed, and for the system with ahigher affinity to the polar phase, the required potential fortransition is lower. When the electric potential is increasedbeyond the transition potential, the average compositionremains constant. However, we find a small increase in ⟨ϕ⟩with electric potential, and then a further increase can result ina slight decrease in ⟨ϕ⟩ due to the nonlinearity of the Poisson−Boltzmann equation. The higher ⟨ϕ⟩ at an abrupt transitionresults in a many-fold increase in reaction rate and averagecomposition of reaction products as shown in Figure 5b . Wecan see that for a system with Δu = 1, the ⟨ ⟩c is about 40times higher as compared to the no potential case.In summary, chemical reactions in microreactors are affected

by a novel filling transition. The reaction is accelerated orslowed down depending on the affinity of the reacting speciestoward the solvents. This new mechanism depends on the poresize, temperature, and solvent composition, and thus, reactionrates may vary considerably across the surface of heteroge-neous membranes. The effect may occur unintentionally nearrough or porous surfaces. An external potential may be a usefulnew addition to the existing arsenal to control the temporaland spatial extent of reactions. This kind of control is especiallysuited for devices that guide reagents flowing in microfluidicchannels adjacent to micrometer-scale electrodes.

■ AUTHOR INFORMATIONCorresponding Author*E-mail: tsori@bgu.ac.il.ORCIDYoav Tsori: 0000-0003-3664-6498NotesThe authors declare no competing financial interest.

■ ACKNOWLEDGMENTSThis work is supported by the Israel Science Foundation GrantNo. 56/14 and the Frankel Family Chair in Energy andChemical Engineering.

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Figure 3. Influence of the bulk volume fraction of the polar phase ϕ0on (a) the average volume fraction of the polar solvent in the poreand (b) the average concentration of reaction product ⟨ ⟩c at time t =10. Both reactants prefer the polar solvent, ΔuA = ΔuB = 10,L = 10nm, n0 = 0.1 M, Δu± = 4, and T(ϕ0) = Tbmd(ϕ0) + 0.005Tc, whereTbmd(ϕ0) is the temperature corresponding to ϕ0 on the bulk mixing−demixing curve, in a confined nanopore, with ions but no externalpotential (ψs = 0).

Figure 4. Chemical reactions in external potentials in different porediameters L and temperatures. (a) Average composition of the polarsolvent in the pore. (b) Average concentration of reaction product attime t = 10 scaled by the zero-field product when both reactantsprefer the polar solvent (ΔuA = ΔuB = 10). (c) Average concentrationof reaction product at time t = 10 scaled by the zero-field productwhen the reactants are antagonistic (ΔuA = −ΔuB = 10). (d) Averageconcentration of reaction product at time t = 10 scaled by the zero-field product when both reactants prefer the nonpolar solvent (ΔuA =ΔuB = −10). ϕ = 0.3, Δu± = 4, and n0 = 0.1 M. All temperatures areabove the demixing temperature, Tbmd = 0.969Tc. † denotesdiscontinuous filling and ‡ denotes continuous filling.

Figure 5. Effect of variation of affinity of ions to the polar phase Δu±on (a) the average volume fraction of polar solvent and (b) theaverage concentration of reaction product when both reactants preferthe polar solvent. The pore diameter is L = 5 nm, n0 = 0.1 M, and thetemperature is taken above the mixing−demixing curve, T(ϕ0) =Tbmd(ϕ0) + 0.01Tc.

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