Controlled Chemical Kinetics in Porous Membranes

Article Cite This: J. Phys. Chem. B 2018, 122, 8269−8273

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

J. Phys. Chem. B 2018.122:8269-8273. Downloaded from pubs.acs.org by UNIV OF NEW ENGLAND on 01/17/19. For personal use only.

Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel ABSTRACT: We investigate theoretically the kinetics of chemical reactions in polar solvents in the vicinity of charged porous membranes. When the pore charge (or potential) exceeds a critical value, the pores undergo a filling transition that can be first or second order depending on the ambient temperature, mixture composition, and other parameters. This filling transition leads to a dramatic acceleration or slowing down of the reaction. Such control of reaction kinetics by an external potential may be useful in applications where catalysts are absent or when fast spatiotemporal response is required.

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their presence does not affect the mixture’s phase diagram due to their small number. To stress the new mechanism, the rate constant k is assumed to be independent of the local field and temperature; the field manipulates the solvent composition, and thereby the local densities of A and B molecules, leading to an increase or decrease of the reaction by an indirect physical mechanism. The reaction is highly temperature-sensitive even if k is temperature-independent (see below). A chemical species i (i = A, B, or C) has different solubility potentials in the water and in the cosolvent, uiw and uics, respectively. The Gibbs transfer energy in units of kBT (where kB is the Boltzmann constant), Δui = uiw − uics, is the free energy required to move molecule i from the cosolvent to the polar solvent. In many liquids, Δui is of the order 1−10.9 A molecule i thus “feels” an effective potential of the form −ui(ϕ) = uiwϕ + uicsϕcs, and this potential is an important driving force in the following discussion. The dimensionless mass balance equations for the chemical species are

he quest to increase the rate, selectivity, and conversion of chemical reactions has motivated many studies in basic and applied research. Traditionally, efforts are made to control parameters such as pressure, temperature, surface area, concentration, and different reaction pathways using catalysts. Control of reactions using such parameters is bounded by known thermodynamic limits and new methods are sought after. Electric fields can be easily switched on/off and their amplitude and frequency can be controlled. They can modify reactions by bringing two or more charged molecules close to each other, or by changing the electron energy levels in molecules.1−5 It has been shown recently that the direction and strength of the field can catalyze Diels−Alder reactions.6 Electric fields are especially suited to modern advanced chemistry methods in microfluidic systems,7 since their amplitude 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 control of the reaction rate.8 For carbonaceous surfaces or membranes, the surface potential can be easily controlled by external means, and this allows us to achieve slowing down or acceleration of reactions by means of manipulation of the solvent composition. Such accelerated kinetics is particularly advantageous in cases where addition of catalysts is undesirable due to its adverse effects or difficulties in its removal from the reaction products. Importantly, this method does not rely on a specific chemistry.

∂C̃i(r ̃, t )̃ 2 = ∇̃ C̃i(r ̃, t )̃ + ∇̃ ·[C̃i(r ̃, t )̃ ∇̃ u i(ϕ)] ∂t ̃ ̃ ã (r ̃, t ) C̃b(r ̃, t ) − νikC

Here, C̃ i = Ci/C0 is the concentration scaled by C0, the initial average bulk concentration of the reagents A and B. To elucidate the effect of field gradients, we focus on reactions taking place near porous membranes. The pores are cylinders with diameter L which is much smaller than their length so that edge effects can be neglected. L is larger than both the debye length λD and the correlation length modified by the presence of salt ξ(T, n0, Δu) (curvature is smaller than the

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MODEL Consider a binary mixture of a polar solvent (e.g., water) and a cosolvent. Their volume fractions are ϕ and ϕcs, respectively. The mixture contains a small amount of dissociated salt and reagents. The chemical reaction takes place between two species, A and B, to irreversibly form a molecule C:

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

k

A + B → 2C. The reagents are assumed to be nonionic, and © 2018 American Chemical Society

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DOI: 10.1021/acs.jpcb.8b05175 J. Phys. Chem. B 2018, 122, 8269−8273

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

corresponding inverse lengths).10,11 This means that the composition of the mixture, the ion density, and electrostatic potential all decay from their high value at the wall to their bulk values if the distance from the wall is longer than ξ and λD. That is, the interior area of the cylinder (its center included) is so far from the walls that it has the same composition, ion density, and (vanishing) potential as the bulk reservoir. Consequently, we model the pore as two flat and parallel surfaces. The first term on the right-hand side is Fickian diffusion, where for simplicity we assume the diffusivities of the reagents are equal, Da = Db = Dc = D, and independent of the local solvent composition. All lengths are scaled by L and time is scaled as t ̃ = Dt/L2. The second term originates from selective solvation (Gibbs transfer energy). The rate constant in the third term of the equation is scaled as k̃ = kC0L2/Da, νA = νB = 1 and νC = −2. This scaling allows us to focus on the influence of temperature, composition, and external potential on the rate of reaction, putting aside the obvious dependence on concentration of reactants squared, as in any second-order chemical reaction. We assume the chemical reaction takes place on a slow time scale as compared to the phase transition kinetics. This means that the reaction in eq 1 occurs in a polar binary mixture whose composition is the equilibrium one. One therefore needs to describe the equilibrium behavior of the pore. When the pore potential is increased from zero, the initially homogeneous mixture demixes such that the polar solvent is enriched near the pore walls and depleted far from it. The thermodynamics of such mixtures is given by the grand potential integrated over the pore volume Ω=

f ± = kBT[n±(ln(n±v0) − 1) − (u w±ϕ + ucs±ϕcs)n±]

with a molecular volume v0 assumed common to all molecules. Similar to the definition above for molecules, the parameters u±w and u±cs are the solvation energies of cations/anions in water and in the cosolvent, respectively. The difference Δu± = u±w − u±cs is the corresponding Gibbs-like transfer energy for the cations/anions. In our simple formalism, the boundary condition for ϕ at the pore wall is ∂ϕ/∂x|x=±L/2 = 0. The boundary condition for ψ is ψ(x = ±L/2) = ±ψs/2 (fixed potential) or ψ′(x = ±L/2) = ±σ/ε (fixed surface charge σ).16 A direct and specific short-range interaction of ions with the surfaces is therefore not accounted for.17−19 In the grandcanonical ensemble, the system is coupled to a reservoir at composition ϕ0 where the potential vanishes and n±0 are the bulk ion densities, which define the chemical potentials.20

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RESULTS AND DISCUSSION The system has three distinct time scales. The first one characterizes the fast ionic response time. In this “adiabatic” (quasi-static) approximation, the ions adjust rapidly to their surrounding, and therefore, they obey a Boltzmann distribution including the solvation energy terms:20

∫ [fm + fe + f + + f − − λ+n+ − λ−n− − μϕ] d3r (2)

n± =

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

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where the amplitudes P are given by P ± = exp( ∓eψ /kBT + (Δu± + χ )ϕ + λ±/kBT )

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The second time scale characterizes the mixture’s diffusive dynamics and is much longer than the ionic response time. The third time scale characterizes the reaction kinetics and is assumed to be the longest. This means that the concentrations C̃ i 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 both distribute in space such that the total energy is minimized. The chemical species migrate to their “preferred” location due to their preferential solvation. In this view, the chemical species simply experience an energy landscape that depends on the mixture. Thus, the spatial and temporal behavior of the reaction follows trivially from the thermodynamic state of the mixture. We give here a concise description of this state based on our previous work.20 When an initially homogeneous polar mixture is found near charged walls, the polar solvent (water) is enriched near the surfaces (cosolvent is depleted) . If the surface potential ψs is too small, the composition ϕ varies slowly 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. Depending on the temperature and average composition, this demixing transition can be first or second order. At yet larger values of

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where χ is the Flory−Huggins interaction parameter.12 The phase with lower grand potential is more preferable. In the bulk, mixing free energy has a critical composition ϕc = 1/2 and the binodal curve is χb(ϕ) = ln(ϕ/(1 − ϕ))/(2ϕ − 1). For large surface potentials, the ion density near the surfaces can be high and their volumes cannot be neglected; therefore, we use the constraint ϕ + ϕcs + v0n+ + v0n− = 1. The reagents do not appear in this constraint since their number is assumed to be small, even in the bulk, and they are uncharged. The (∇ϕ)2 term is required to account for the interface between coexisting phases and the resultant surface tension. In our work, the interface is not found at the walls of the pores, and hence, the volume conservation ϕ + ϕcs ≈ 0 holds without inclusion of the ion densities sufficiently far from the walls. That is, when it is far from the walls, the mixture essentially has only two components, and only one gradient term is required. The electrostatic free energy density is given by fe = −(1/2)ε0ε(ϕ)|∇ψ |2 + q(n+ − n−)ψ

P ±(1 − ϕ) v0(1 + P+ + P−) ±

kT fm = B (ϕ log(ϕ) + ϕcs log(ϕcs) + χϕϕcs v0 + (1/2)χv0 2/3(∇ϕ)2 )

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DOI: 10.1021/acs.jpcb.8b05175 J. Phys. Chem. B 2018, 122, 8269−8273

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The Journal of Physical Chemistry B ψs, ion affinities, temperature, or composition, the pore fills with water and the sharp interface is lost (inside the pore; it exists between the pore and the outer bulk) . The physical driving forces for filling of the pore with a water-rich phase are 2-fold: the first is the destruction of an interface between water-rich and water-poor domains, and the second is the large energy gained by the ions, attracted to the pore walls, when they are in contact with much less cosolvent. The shift of the filling temperature T(ϕ0) away from the binodal curve is stronger when Δu± or n0 are large.20 Figure 1a shows the change in composition profiles as the pore potential increases from zero. Profiles were obtained by a

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 average concentration of reaction product at time t ̃ = 10 scaled by the value in the absence of potential. We used pore diameter L = 5 nm, hydrophilic ions Δu± = 4, T = 0.97Tc, and both reactants prefer the polar solvent, ΔuA = ΔuB = 10.

filling curve, bulk mixing−demixing occurs. The curves displace to higher temperatures when more salt is added at given values of n0 and ϕ0. The temperature corresponding to these parameters is referred to as bulk mixing−demixing temperature Tbmd. In b, we show the corresponding mixture composition profiles for fixed pore potential ψs. The filling transition can be achieved for higher salt concentration or lower applied electric potential. The average pore composition ⟨ϕ⟩ depends sensitively on the pore potential (part c)at small values of ψs the pore’s composition equals the bulk value; as ψs increases past the critical value, ⟨ϕ⟩ has a discontinuous jump in its value. The threshold value of ψs decreases with increasing value of n0. In d, we examine the influence of ψs on the product by showing ⟨*c⟩ = ⟨Cc̃ ⟩/⟨Cc̃ ⟩ψs = 0 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 critical voltage. As in (c), the critical voltage decreases with increasing salt content n0. It is clear from Figure 2 that the average mixture composition ϕ0 has a great influence on the state of the pore. 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, the difference between the “empty” and “filled” pores is quite small. However, for smaller values of ϕ0, away from the critical point, the difference can be very large. This is due to the basic shape of the zero-field binodal curve, whose coexisting compositions, as obtained by the classical common-tangent construction, are increasingly different as temperature is reduced. As is clear from Figure 3b, at a constant surface potential ψs , the chemical reaction product increases significantly 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 Figure 4, we examine how the filling transition and the subsequent acceleration or deceleration of a chemical reaction are influenced by the pore size. Two pore sizes are shown at three different temperatures, which are all above the bulk demixing temperature in the absence of external potential. In general, filling occurs at smaller potentials when the pore

Figure 1. Control of the effective reaction rate in pores using external potentials. 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 the composition of the polar solvent inside the pore for different electric potentials ψs. A filling transition occurs as the pore potential increases. (b) Change in time of the product concentration averaged over the pore volume, ⟨C̃ c⟩. Both reactants, A and B, are more soluble in the polar phase: ΔuA = ΔuB = 10 (accelerated reaction). Colors and line styles correspond to the same values of ψs in (a). The concentrations i of the molecular species i = A, B at time t ̃ = 0 was Ci = C0e−u (ϕ). (c) A The same as in (b), but reactant A is polar (Δu = 10) and reactant B is 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, we took L = 10 nm, Δu± = 3, ϕ0 = 0.36, ΔuC = 0, and T = 0.99Tc. In all the figures, we took molecular length a = v01/3 = 3.4 Å.

simultaneous numerical solution to the Euler−Lagrange and Poisson’s nonlinear equations. At small ψs, ϕ is large only close to the pore walls, but above the filling transition, ϕ is large also in the pore’s center. Part b shows the spatially averaged product in the pore, ⟨C̃ c⟩, at different times for the case where both reactants prefer the polar phase. Before the filling transition, the changes to ⟨C̃ c⟩ induced by the external potential are small. However, the filling transition leads to large concentrations of molecules A and B inside the pore and, consequently, to a dramatic 10−30-fold acceleration in the reaction kinetics (Figure 1b). Reduction of ⟨C̃ c⟩ (slowing down) of a similar scale occurs when molecules A and B are antagonistic (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 n0 are shown in Figure 2a, which affect the stability of mixtures in the pore. For points in the phase diagram that are below the 8271

DOI: 10.1021/acs.jpcb.8b05175 J. Phys. Chem. B 2018, 122, 8269−8273

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The Journal of Physical Chemistry B

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) the average concentration of reaction product when both reactants prefer the polar solvent. The pore diameter is L = 5 nm, n0 = 0.1 M, and the temperature is taken above the mixing−demixing curve, T(ϕ0) = Tbmd(ϕ0) + 0.01Tc.

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

increased, the average volume fraction of the polar solvent increases inside the pore. At a certain potential, an abrupt firstorder filling transition is observed, and for the system with a higher affinity to the polar phase, the required potential for transition is lower. When the electric potential is increased beyond the transition potential, the average composition remains constant. However, we find a small increase in ⟨ϕ⟩ with electric potential, and then a further increase can result in a slight decrease in ⟨ϕ⟩ due to the nonlinearity of the Poisson− Boltzmann equation. The higher ⟨ϕ⟩ at an abrupt transition results in a many-fold increase in reaction rate and average composition of reaction products as shown in Figure 5b . We can see that for a system with Δu = 1, the ⟨*c⟩ is about 40 times 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 or slowed down depending on the affinity of the reacting species toward the solvents. This new mechanism depends on the pore size, temperature, and solvent composition, and thus, reaction rates may vary considerably across the surface of heterogeneous membranes. The effect may occur unintentionally near rough or porous surfaces. An external potential may be a useful new addition to the existing arsenal to control the temporal and spatial extent of reactions. This kind of control is especially suited for devices that guide reagents flowing in microfluidic channels adjacent to micrometer-scale electrodes.

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

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

diameter decreases or the temperature is reduced close to the filling temperature (a) . The pore diameter at which filling occurs is determined by a nonlinear combination of two length scales: the debye length and the correlation length ξ (in this work, depending also on n0 and Δui).10,11,20,21 At higher temperatures, the filling occurs gradually and second-order filling transition is observed, while at lower temperatures, it is a first-order phase transition. The spatially averaged product of the reaction grows correspondingly for A and B molecules preferred at the polar solvent (b). A dramatic increase of the reaction, by 40−60 times as compared to the no-field case, is shown. An opposite trend, of chemical slowing down, is shown in panels c and d for antagonistic molecules and when both A and 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 polar solvent Δu. One can see that as the applied potential is

ORCID

Yoav Tsori: 0000-0003-3664-6498 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by the Israel Science Foundation Grant No. 56/14 and the Frankel Family Chair in Energy and Chemical Engineering.

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