Control of chemical reactions using electric field gradients

TL;DR

This paper proposes a theoretical method to control chemical reaction rates by applying electric field gradients in solvent mixtures, affecting local composition and phase separation to accelerate or decelerate reactions.

Contribution

It introduces a novel approach for reaction control using electric fields to manipulate solvent distribution and phase behavior in chemical processes.

Findings

Electric fields can alter local solvent composition and phase separation.

Reaction rates can be controlled without catalysts using electric field gradients.

Potential to prevent runaway reactions and enhance reaction efficiency.

Abstract

We examine theoretically a new idea for spatial and temporal control of chemical reactions. When chemical reactions take place in a mixture of solvents, an external electric field can alter the local mixture composition thereby accelerating or decelerating the rate of reaction. The spatial distribution of electric field strength can be non-trivial and depends on the arrangement of the electrodes producing it. In the absence of electric field, the mixture is homogeneous and the reaction takes place uniformly in the reactor volume. When an electric field is applied the solvents separate and reactants are concentrated in the same phase or separate to different phases, depending on their relative miscibility in the solvents, and this can have a large effect on the kinetics of the reaction. This method could provide an alternative way to control runaway reactions and to increase the reaction rate without using catalysts.

Equations

$$\mathcal{F}={\int }_{\Omega }\left[f_m(\varphi ,T)+f_e(\varphi ,\mathbf{E})+f_i(\varphi ,T)\right]d\Omega ,$$

$$\frac{\partial \varphi }{\partial t}=D{\nabla }^2\frac{\delta f}{\delta \varphi },\nabla \cdot [ϵ(\varphi )\nabla \psi ]=0,$$

$$\frac{\partial {\tilde{C}}_a}{\partial \tilde{t}}={\tilde{\nabla }}^2{\tilde{C}}_a+\tilde{\nabla }\cdot \left({\tilde{C}}_a{U}_a^{\prime }\tilde{\nabla }\varphi \right)-\tilde{k}{\tilde{C}}_a{\tilde{C}}_b$$

$$\frac{\partial {\tilde{C}}_b}{\partial \tilde{t}}$$

$$\frac{\partial {\tilde{C}}_c}{\partial \tilde{t}}$$

Full text

Chemical reactions are conventionally controlled by pressure, temperature, surface area, concentration, or by using a catalystSanderson (2011); Atkins and de Paula (2006). Attention has turned to active modification of molecular collision processesBrumer and Shapiro (1992) and to the manipulation of activation energy barriersRatschbacher et al. (2012); Tseng, Wang, and Zocchi (2010). Lasers were used for selectively making and breaking chemical bondsAssion et al. (1998) and current research looks at ways to exploit ultrafast lasers for mode-selective chemistry, stereodynamic, and quantum control of molecular processesBrumer and Shapiro (1992); Zare (1998); Sussman et al. (2006) in cost effective ways. ElectricTachiya (1987); Hilczer, Traytak, and Tachiya (2001); Reigh, Shin, and Kim (2010) and magnetic fields and ultrasound are also used to control molecular collisions and thereby the chemistryLemeshko et al. (2013); Brouard, Parker, and van de Meerakker (2014). Electric and magnetic fields modify the orientation of molecules and may change the ion and molecular transport rates, they can modify the quantum states of moleculesLemeshko et al. (2013); Brouard, Parker, and van de Meerakker (2014) and they may lead to Stark and Zeeman shifts in the energy levels. In addition, large electric fields at charged metal surfaces are important in the chemisorption of atoms and moleculesKreuzer (1991).

Here we propose a new direction for spatial and temporal control of chemical reactions. Our idea relies on the use of a mixture of two or more solvents. Reactions taking place in such mixture are influenced by the composition of the solvents. An external field can lead to “electro prewetting” transition of the solvents even when their initial state is homogeneous Tsori, Tournilhac, and Leibler (2004); Tsori (2009). In this transition, the solvents separate from each other and migrate to locations that minimize the total free energy of the mixture and depend on the electrode design. An interface thus appears separating the formed domains.

There are now two scenarios and for clarity we focus on a binary mixture of solvents and two reactants. In the first, due to their Gibbs transfer energy, the reactants are more miscible in the same solvent. In that case field-induced demixing will lead to concentration of the reactants in a small volume and to accelerated reaction kinetics compared to the no-field, homogeneous state. The product will be initially produced in the same small volume. In the second scenario, the two reactants are preferentially miscible in different solvents. In that case the reaction will take place at the interface separating liquid domains. This interfacial reaction is expected to be slowed down compared to the no-field (homogeneous) state. The product will initially be created at the interface between coexisting domains. In both cases, switching off the field will allow re-mixing of the solvents and return to the homogeneous state and to “normal” kinetics.

Thermodynamics of liquid mixtures is underlying the reactions in electric fields. Consider for simplicity a mixture of two solvents. The total free energy of the mixture is given on the mean-field level as

[TABLE]

where $\Omega$ is the volume occupied by the mixture. The volume fraction of solvent $1$ is given by $\phi$ and the volume fraction of solvent $2$ is $1-\phi$, $T$ is temperature and $f_{m}$, $f_{e}$ and $f_{i}$ are mixing, electrostatic and interfacial energies. The mixing free energy per unit volume is given byDoi (1996); Safran (1994) $f_{m}=(k_{B}T/v_{0})\left[\phi\log\phi+(1-\phi)\log(1-\phi)+\chi\phi(1-\phi)\right]$, where $k_{B}$ is the Boltzmann constant and $v_{0}$ is a molecular volume assumed here to be equal for the two solvents. $\chi$ is the Flory parameter varying inversely with $T$: $\chi\sim 1/T$. For positive values of $\chi$ the phase diagram has upper critical solution temperatureDebye and Kleboth (1965) and the critical point is given by $(\phi_{c},\chi_{c})=(1/2,2)$. For such symmetric free energy expressions, the binodal curve $T_{b}(\phi)$ in the $\phi$-$T$ phase diagram is given by $\partial f(\phi,T_{b})/\partial\phi=0$. Above this curve, the mixture is stable in the homogeneous phase. A quench to temperatures $T$ below $T_{b}$ leads to phase separation of two coexisting phases with compositions given by the values of the binodal at $T$.

The interfacial free energy density is given bySafran (1994) $f_{i}=(k_{B}T/2v_{0})\chi\lambda^{2}|\nabla\phi|^{2}$, where $\lambda$ is a constant related to interface width. The electrostatic free energy density is given by $f_{e}=-(1/2)\epsilon_{0}\epsilon(\phi)|\nabla\psi|^{2}$, where $\epsilon_{0}$ is permittivity of free space, $\epsilon$ is the relative permittivity of the mixture depending on its relative composition, and $\psi({\bf r})$ is the local electric potential.

In order for the electric field to be effective in separating the liquids, it must have large spatial gradients. We chose the simple “wedge” geometry to illustrate the concept of chemical reactions. The wedge is comprised of two flat and conducting plates oriented with an opening angle $\beta$ between them ($\beta=0^{\circ}$ corresponds to a parallel-plate capacitor). In this effectively two-dimensional system, the electric field depends only on the distance $r$ from the imaginary meeting point of the plates and is oriented in the azimuthal $\hat{\theta}$ directionGalanis and Tsori (2014).

In the wedge geometry chosen, all quantities depend on $r$ alone and the hydrodynamic flow velocity ${\bf v}$ vanishes. The equations governing the demixing dynamics are thenImaeda, Furukawa, and Onuki (2004); Galanis and Tsori (2014)

[TABLE]

where $f=f_{m}+f_{i}+f_{e}$ is the total mixture free energy density and $D$ is an Onsager diffusivity constant taken here to be independent of $\phi$.

We consider a simple irreversible reaction $A+B\xrightarrow{k}2C$, where molecules of compound $A$ and $B$ react to give product $C$ and $k$ is the rate constant. When the reaction takes place in a mixture of two solvents, each molecule $A$, $B$, and $C$ feels a spatially-dependent potential that depends on its relative solubility denoted by $u_{a}$, $u_{b}$, and $u_{c}$ respectively. For the $A$ molecule $u_{a}({\bf r})=u_{1}^{a}\phi+u_{2}^{a}(1-\phi)$ and similar expressions for $u_{b}({\bf r})$ and $u_{c}({\bf r})$. The parameters $u_{1}^{a}$ ad $u_{2}^{a}$ indicate the solubilities of compound $A$ in liquids $1$ and $2$ respectively. The difference in the solubility parameters $\Delta u^{a}=u_{2}^{a}-u_{1}^{a}$ is related to the Gibbs transfer energy $\Delta G_{t}$ for transferring one $A$ molecule from a solvent with composition $\phi_{1}$ to a solvent with composition $\phi_{2}$ via $\Delta G_{t}=\Delta u^{\pm}(\phi_{2}-\phi_{1})$. Experiments show that $G_{t}$ is on the order of $1-10k_{B}T$ in aqueous mixtures Marcus (1985) and hence $\Delta u^{\pm}\sim 1-10$.

The mass balance of compound $A$ gives the modified reaction-diffusion equation: $\partial C_{a}/\partial t=D_{a}\nabla^{2}C_{a}+D_{a}\nabla\cdot\left[C_{a}\nabla(u_{a}/k_{B}T)\right]-kC_{a}C_{b}$. Here $D_{a}$ is the diffusion coefficient of compound $A$ in the mixture, assumed to be independent of $T$ and $\phi$. This equation can be recast in dimensionless form

[TABLE]

where $\tilde{C}_{a}=C_{a}/C_{a0}$ is the concentration scaled by $C_{a0}$, the initial (uniform) concentration of $A$ molecules, $\tilde{k}=kC_{a0}R_{1}^{2}/D_{a}$ is a scaled reaction rate, and $U^{\prime}_{a}=(u_{2}^{a}-u_{1}^{a})/k_{B}T$. The length is scaled using $R_{1}$ (the minimal distance $r$ from the imaginary meeting point of the plates): $\tilde{r}=r/R_{1}$, and time is scaled via $\tilde{t}=D_{a}/R_{1}^{2}t$. When a potential difference of magnitude $V$ is applied across wedge electrodes, the ratio of the electrostatic energy stored in a molecular volume $v_{0}$ to the thermal energy is given by the dimensionless numberSamin and Tsori (2009) $M_{w}\equiv V^{2}v_{0}\epsilon_{0}/(4\beta^{2}k_{B}T_{c}R_{1}^{2})$, where $T_{c}$ is critical temperature. Similar mass balance equation can be written for compounds $B$ and $C$:

[TABLE]

with $\tilde{C}_{b}=C_{b}/C_{a0}$, $\tilde{C}_{c}=C_{c}/C_{a0}$, $U^{\prime}_{b}=(u_{2}^{b}-u_{1}^{b})/k_{B}T$ and $U^{\prime}_{c}=(u_{2}^{c}-u_{1}^{c})/k_{B}T$.

In this paper, we assume that the reaction is slow compared to the kinetics of phase separation. If this condition holds, the initially homogeneous mixture phase-separates on a fast time-scale to two coexisting domains with uniform densities of the reactants. The composition $\phi({\bf r})$ attains its equilibrium profile, minimizing the sum of mixture, interfacial, and electrostatic energies. On a much longer scale, the chemical reaction then proceeds, with the compounds $A$, $B$, and $C$ experiencing a spatially-dependent force derivable from the $u$’s as is explained above. We further distinguish between the two cases whether the molecules $A$ and $B$ prefer the same phase or not.

Reactants $A$ and $B$ prefer the same phase. Figure 1(a)-(c) shows concentration profiles at various times when the reactants and product prefer the more polar phase (high value of $\phi$). At $\tilde{t}=0$ the mixture has two coexisting domains with an interface at $r/R_{1}\simeq 2.5$. As time progresses the $A$ and $B$ molecules diffuse towards the more polar region ($r/R_{1}<2.5$, large $\phi$) and sharp gradient occurs at the interface between the coexisting phases. The concentrations of $A$ and $B$ molecules is the same because we took $U^{\prime}_{a}=U^{\prime}_{b}$. As time increases, $A$ and $B$ continue to diffuse to the polar solvent but on the same time are consumed due to the reaction and the creation of $C$. At long times $\tilde{C}_{a}$ and $\tilde{C}_{b}$ become very small; the concentration of $C$ obeys the Boltzmann’s distribution and is found preferentially in the polar phase where it is more soluble.

In Figure 1(d) we calculate the spatially-averaged product amount $\bar{C}_{c}\equiv\Omega^{-1}\int\tilde{C}_{c}d\Omega$ vs time. We compare the case with electric field and two coexisting domains (lines) with the zero-field case and a homogeneous mixture (lines with symbols), for constant electric field and varying reaction rates $\tilde{k}$. As can be seen, the electric field increases the effective rate of the reaction. It does so more effectively for the slow reactions (small values of $\tilde{k}$) and less effectively for the fast reactions (large values of $\tilde{k}$). Irrespective of $\tilde{k}$, material conservation dictates that $\bar{C}_{c}\to 2$ when $\tilde{t}\to\infty$.

Fig. 2 examines how the effective reaction rate depends on the magnitude of the external potential. A phase-separation transition occurs when the applied voltage in the wedge is larger than the critical value, which depends on the temperature and mixture composition. When the voltage increases past this threshold, the “contrast” between the phases increases ($\phi$ increases at small values of $r$ and decreases for large values of $r$) and the interface displaces to larger values of $r$. Three profiles of $\phi$ for different values of the potential are shown in the inset. As can be seen, $\bar{C}_{c}$ is larger with increasing potential. However, an increase of $M_{w}$ from $0.036$ to $0.144$ has only a modest effect on $\bar{C}_{c}$. Indeed, even a hypothetical infinite potential would lead to a finite effective reaction rate.

The preferential solubility of the reactants in the solvents is an important factor determining the rate of the reaction. In Fig. 3 we plot $\bar{C}_{c}$ for different solubility values at a fixed value of $M_{w}$. While the asymptotic behavior at long times dictates $\bar{C}_{c}\to 2$ clearly, the dynamics are faster as the solubility difference increases.

Reactants $A$ and $B$ prefer different phases. We turn to the interesting case where the reactants are “antagonistic” in the sense that they prefer different solvents. Figure 4 shows the concentration profiles vs time and should be compared to Fig. 1. As before, $\tilde{C}_{a}$ and $\tilde{C}_{b}$ start from a uniform distribution at $\tilde{t}=0$. But here they diffuse in opposite directions – $A$ to the polar solvent at small $r$’s and $B$ to the less polar solvent at large $r$’s. At early times the gradients in $\tilde{C}_{a}$ and $\tilde{C}_{b}$ occur at the interface between the solvents. The profiles evolve in time; as the reactants migrate according to their solubility they are consumed and $C$ is created. In this calculation we assumed the product is equally soluble in both solvents and thus at long times $\tilde{C}_{c}$ is uniform. Far from the interface, in the bulk liquid, the transport of $A$ and $B$ is mainly due to diffusion (first term in the right-hand side of equations 3 and 4). Near an interface the high gradient in $\phi$ leads to a large force from the solubility potential and transport is dominated by the solubility difference (second term in the right-hand side of equations 3 and 4).

In Fig. 4(d) we plot the average product $\bar{C}_{c}$ vs time for different values of the potential. In contrast to Fig. 1(d) here the effective reaction is slower with field (lines) as compared to the no-field case (lines with symbols). As in Fig. 1(d), fast reactions are less affected by the field (large values of $\tilde{k}$). The total product $\tilde{C}_{c}$ tends to $2$ due to mass conservation.

The effective slowing-down of the reaction saturates with the magnitude of the applied potential, see Fig. 5. As $M_{w}$ increases ($M_{w}\propto V^{2}$) the more polar solvent is pulled to the region with higher electric field, thereby raising the value of $\phi$ at small $r$’s and reducing it at large $r$’s (low electric field region). Curves show $\tilde{C}_{c}$ vs time for different values of $M_{w}$. All curves increase from $\tilde{C}_{c}=0$ at $\tilde{t}=0$ to $\tilde{C}_{c}=2$ at infinity. Curves exhibit an early increase on a small timescale dictated by diffusion over the mesoscopic width of the interface between coexisting phases. After the rapid increase, the curves increase slowly on a timescale dictated by diffusion over the macroscopic size of the wedge.

The overall reaction rate is slowed down when the reactants are preferentially soluble in different solvents, as can be seen in Fig. 6. Clearly the electric potential is crucial here because in its absence the mixture is homogeneous and with it two domains exist. All curves in the Figure start from zero and increase to $2$ at long times. Here again there are two timescales – a fast transient corresponding to diffusion over the interface, accompanied by a slow relaxation dictated by diffusion over macroscopic lengths. The stronger the incompatibility between the reactants the larger the difference in concentration of $A$ and $B$ molecules in the two domains and the slower is the overall relaxation.

In Fig. 7 we show the non-trivial dependence of the rate of reaction on the average mixture composition $\phi_{0}$. At a fixed temperature and electric potential and for values of $\phi_{0}$ smaller than the critical value, an increase in $\phi_{0}$ to values closer to the binodal curve leads to an increase in the location of the interface between the polar and less polar phases (inset). However, at the same time the “contrast” between the phases diminishes and the thickness of the interface increases. As a result, for the values presented in this calculation, at long times the reaction which is slowest is one with intermediate value, $\phi_{0}=0.2$, and not one of the extreme values $\phi_{0}=0.1$ or $\phi_{0}=0.4$.

In summary, electric field gradients can control the effective reaction rate in reactions taking place in mixtures of solvents. Whether the reaction in the ’on’ state is faster or slower than in the ’off’ state depends on whether the reactants are miscible in the same solvent or in different ones. The effective reaction rate depends sensitively on the electric potential, mixture composition, and distance from the binodal curve of the mixture, and is independent of the preferential solvation of the product $C$. We believe this method could provide an alternative way to control runaway reactions and to increase the reaction rate without using catalysts.

In the current study, we assumed that the liquid flow due to phase separation is very fast compared to the reaction kinetics but it may be interesting to relax this assumption. Electric fields are especially suited for use in microfluidic devices where they have been used to transport liquids in small channels. Such systems pose great promise for field-induced chemical reactions. Indeed, the challenge is to better understand how reactions are coupled with flow, either from pressure gradients in the channel or from the phase-separation itself. Future work will also consider ionic reagents, mixtures that have a Lower Critical Solution Temperature (e.g. Water and 2,6 Lutidine), Samin et al. (2014) and reversible reactions. In such reactions the preferential solubility of $C$ is important and the effective reaction rate can increase or decrease depending on whether $C$ is soluble in the same solvent as $A$ and $B$ or not.

This work was supported by the European Research Council “Starting Grant” No. 259205, COST Action MP1106, and Israel Science Foundation Grant No. 56/14.