Control of chemical reactions using electric field gradients

0.2 MB·

by  Shivaraj D. Deshmukh

Control of chemical reactions using electric field gradients
Shivaraj D. Deshmukh1 and Yoav Tsori1,a)
Department of Chemical Engineering, Ben-Gurion University of the Negev, Beer-Sheva 84105,
Israel
We examine theoretically a new idea for spatial and temporal control of chemical reactions. When chemical
reactionstakeplaceinamixtureofsolvents,anexternalelectricfieldcanalterthe localmixture composition
therebyacceleratingordeceleratingthe rateofreaction. Thespatialdistributionofelectricfieldstrengthcan
be non-trivialanddepends onthe arrangementofthe electrodesproducingit. Inthe absenceofelectricfield,
the mixture is homogeneous and the reaction takes place uniformly in the reactor volume. When an electric
fieldisappliedthesolventsseparateandreactantsareconcentratedinthesamephaseorseparatetodifferent
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1 phases, depending ontheir relative miscibility in the solvents,andthis canhave a large effect on the kinetics
0 of the reaction. This method could provide an alternative way to control runaway reactions and to increase
2 the reaction rate without using catalysts.

n
a
J Chemical reactions are conventionally controlled by case the reactionwill take place at the interface separat-
pressure,temperature,surfacearea,concentration,orby ing liquid domains. This interfacial reaction is expected
8
1 usingacatalyst1,2. Attentionhasturnedtoactivemodifi- to be slowed down compared to the no-field (homoge-
cationofmolecularcollisionprocesses3andtothemanip- neous) state. The product will initially be created at

] ulationofactivationenergybarriers4,5. Laserswereused the interface between coexisting domains. In both cases,
t
f forselectivelymakingandbreakingchemicalbonds6 and switchingoffthefieldwillallowre-mixingofthe solvents
o
current research looks at ways to exploit ultrafast lasers and return to the homogeneous state and to “normal”
s
. for mode-selective chemistry, stereodynamic, and quan- kinetics.
t
a tum control of molecular processes3,7,8 in cost effective Thermodynamics of liquid mixtures is underlying the
m ways. Electric9–11 and magnetic fields and ultrasound reactionsinelectricfields. Considerforsimplicityamix-
- are also used to control molecular collisions and thereby tureoftwosolvents. Thetotalfreeenergyofthemixture
d the chemistry12,13. Electric and magnetic fields modify is given on the mean-field level as
n
the orientationofmolecules andmaychangethe ionand
o
c moleculartransportrates,they canmodify the quantum F = [fm(φ,T)+fe(φ,E)+fi(φ,T)]dΩ , (1)
[ states of molecules12,13 and they may lead to Stark and ZΩ

Zeeman shifts in the energy levels. In addition, large where Ω is the volume occupied by the mixture.
1
electric fields at chargedmetal surfaces are important in
v The volume fraction of solvent 1 is given by φ
the chemisorption of atoms and molecules14.
0 and the volume fraction of solvent 2 is 1 − φ,
8 Here we propose a new direction for spatial and tem- T is temperature and fm, fe and fi are mixing,
9 poral control of chemical reactions. Our idea relies on electrostatic and interfacial energies. The mixing
4 the use of a mixture of two or more solvents. Reactions free energy per unit volume is given by17,18 fm =
0 taking place in such mixture are influenced by the com-
. (kBT/v0)[φlogφ+(1−φ)log(1−φ)+χφ(1−φ)],
1 position of the solvents. An external field can lead to where kB is the Boltzmann constant and v0 is a
0 “electroprewetting”transitionofthesolventsevenwhen molecular volume assumed here to be equal for the two
7 their initialstateis homogeneous15,16. Inthis transition,
solvents. χistheFloryparametervaryinginverselywith
1 the solvents separate from each other and migrate to lo-
T: χ∼1/T. For positive values of χ the phase diagram
:
v cationsthatminimizethetotalfreeenergyofthemixture hasuppercriticalsolutiontemperature19 andthecritical
i and depend on the electrode design. An interface thus
X point is givenby (φc,χc)=(1/2,2). For such symmetric
appears separating the formed domains.
free energy expressions, the binodal curve Tb(φ) in the
r There are now two scenarios and for clarity we focus
a φ-T phase diagramis given by ∂f(φ,Tb)/∂φ=0. Above
on a binary mixture of solvents and two reactants. In
this curve, the mixture is stable in the homogeneous
the first, due to their Gibbs transfer energy, the reac-
phase. A quench to temperatures T below Tb leads
tants are more miscible in the same solvent. In that
to phase separation of two coexisting phases with
case field-induced demixing will lead to concentration of
compositions given by the values of the binodal at T.
the reactants in a small volume and to accelerated re- The interfacial free energy density is given by18 fi =
action kinetics compared to the no-field, homogeneous (kBT/2v0)χλ2|∇φ|2, where λ is a constantrelated to in-
state. Theproductwillbeinitiallyproducedinthesame
terface width. The electrostatic free energy density is
small volume. In the second scenario, the two reactants given by fe = −(1/2)ǫ0ǫ(φ)|∇ψ|2, where ǫ0 is permit-
are preferentially miscible in different solvents. In that
tivity of free space, ǫ is the relative permittivity of the
mixture depending on its relative composition, and ψ(r)
is the local electric potential.
Inorder for the electric fieldto be effective in separat-
a)[email protected] ing the liquids, it must have large spatial gradients. We
2
chosethe simple “wedge”geometryto illustrate the con- pounds B and C:
ceptofchemicalreactions. Thewedgeiscomprisedoftwo
flβabteatnwdeecnontdhuecmtin(gβp=lat0e◦scoorrierenstpedonwdisthtoanaoppaernailnlegl-apnlagtlee ∂∂C˜t˜b = DDab h∇˜2C˜b+∇˜ ·(cid:16)C˜bUb′∇˜φ(cid:17)i−k˜C˜aC˜b (4)
cthaepaecleitcotrr)ic. fiIenldtdheispeenffdesctoinvelylyontwtoh-eddimisetnansicoenarlfrsoymstetmhe, ∂∂C˜t˜c = DDac h∇˜2C˜c+∇˜ ·(cid:16)C˜cUc′∇˜φ(cid:17)i+2k˜C˜aC˜b (5)
imaginary meeting point of the plates and is oriented in
the azimuthal θˆdirection20. with C˜b = Cb/Ca0, C˜c = Cc/Ca0, Ub′ = (ub2 −ub1)/kBT
In the wedge geometry chosen, all quantities depend and Uc′ =(uc2−uc1)/kBT.
on r alone and the hydrodynamic flow velocity v van- Inthispaper,weassumethatthereactionisslowcom-
ishes. The equations governing the demixing dynamics pared to the kinetics of phase separation. If this con-
are then20,21 dition holds, the initially homogeneous mixture phase-
separates on a fast time-scale to two coexisting domains
∂φ δf with uniform densities of the reactants. The composi-
=D∇2 , ∇·[ǫ(φ)∇ψ]=0 , (2a)
∂t δφ tion φ(r) attains its equilibrium profile, minimizing the
sum of mixture, interfacial, and electrostatic energies.
On a much longer scale, the chemical reaction then pro-
where f = fm+fi+fe is the total mixture free energy
ceeds, with the compounds A, B, and C experiencing a
density and D is an Onsager diffusivity constant taken
spatially-dependentforce derivable from the u’s as is ex-
here to be independent of φ.
plained above. We further distinguish between the two
k
WeconsiderasimpleirreversiblereactionA+B −→2C,
cases whether the molecules A and B prefer the same
where molecules of compound A and B react to give
phase or not.
productC andk istherateconstant. Whenthereaction Reactants A and B prefer the same phase. Fig-
takesplaceinamixtureoftwosolvents,eachmoleculeA,
ure1(a)-(c)showsconcentrationprofilesatvarioustimes
B, and C feels a spatially-dependent potential that de- when the reactants and product prefer the more polar
pends onits relativesolubility denotedby ua, ub, anduc phase (high value of φ). At t˜= 0 the mixture has two
respectively. Forthe Amoleculeua(r)=ua1φ+ua2(1−φ) coexisting domains with an interface at r/R1 ≃ 2.5. As
and similar expressions for ub(r) and uc(r). The param- time progresses the A and B molecules diffuse towards
a a
etersu1 adu2 indicate the solubilitiesofcompoundAin the more polar region (r/R1 < 2.5, large φ) and sharp
liquids 1 and 2 respectively. The difference in the solu-
gradient occurs at the interface between the coexisting
a a a
bility parameters ∆u =u2−u1 is related to the Gibbs phases. The concentrations of A and B molecules is the
transferenergy∆Gt fortransferringoneAmoleculefrom samebecausewetookUa′ =Ub′. Astimeincreases,Aand
asolventwithcompositionφ1 toasolventwithcomposi- B continuetodiffusetothepolarsolventbutonthesame
tionφ2 via∆Gt =∆u±(φ2−φ1). Experimentsshowthat time are consumed due to the reaction and the creation
Gt is on the order of 1−10kBT in aqueous mixtures22 of C. At long times C˜a and C˜b become very small; the
and hence ∆u± ∼1−10.
concentration of C obeys the Boltzmann’s distribution
The mass balance of compound A gives the modified and is found preferentially in the polar phase where it is
reaction-diffusion equation: ∂Ca/∂t=Da∇2Ca+Da∇· more soluble.
[Ca∇(ua/kBT)]−kCaCb. Here Da is the diffusion co- In Figure 1(d) we calculate the spatially-averaged
efficient of compound A in the mixture, assumed to be product amount C¯c ≡Ω−1 C˜cdΩ vs time. We compare
independent of T and φ. This equation can be recast in the case with electric fieldRand two coexisting domains
dimensionless form (lines) with the zero-field case and a homogeneous mix-
ture (lines with symbols), for constant electric field and
∂∂C˜t˜a =∇˜2C˜a+∇˜ ·(cid:16)C˜aUa′∇˜φ(cid:17)−k˜C˜aC˜b (3) vfiaerldyiningcrreeaascetsiotnheraetffeesctk˜iv.eArastecaonf tbheesreeeanc,titohne. eItlecdtoreics
somore effectivelyfor the slowreactions(small valuesof
where C˜a = Ca/Ca0 is the concentration scaled by k˜) and less effectively for the fast reactions (large values
Ca0, the initial (uniform) concentration of A molecules, of k˜). Irrespective of k˜, material conservation dictates
k˜ = kCa0R12/Da is a scaled reaction rate, and Ua′ = that C¯c →2 when t˜→∞.
a a
(u2−u1)/kBT. The length is scaled using R1 (the min- Fig. 2examineshowtheeffectivereactionratedepends
imal distance r from the imaginary meeting point of the on the magnitude of the external potential. A phase-
plates): r˜ = r/R1, and time is scaled via t˜= Da/R12t. separation transition occurs when the applied voltage in
When a potential difference of magnitude V is applied thewedgeislargerthanthecriticalvalue,whichdepends
across wedge electrodes, the ratio of the electrostatic onthe temperature andmixture composition. When the
energy stored in a molecular volume v to the thermal voltage increases past this threshold, the “contrast” be-
0
energy is given by the dimensionless number23 Mw ≡ tween the phases increases (φ increases at small values
V2v0ǫ0/(4β2kBTcR12), where Tc is critical temperature. ofr anddecreasesfor largevalues ofr) andthe interface
Similar mass balance equation can be written for com- displacestolargervaluesofr. Threeprofilesofφfordif-
3
2
(a) (b)
ntration12 t˜=0.01 CCC˜˜˜abc ntration12 t˜=0.10 CCC˜˜˜abc 1.5 1
e e
onc onc Cc 1 φ0.5
C0 C0
1 2 3 4 5 1 2 3 4 5 0
r/R1 r/R1 1 2 3 4 5
r/R1
(c) (d) 0.5 MMww==00..0036
n8 t˜=1.00 C˜a 2 MFuwlly=S0e.p1a4r4atedsolventsand
entratio46 CC˜˜bc Cc1 0 10initiallyreact2an0tsinsamesolv3e0nt 40 50
nc2 t˜
o
C
0 0 FIG. 2: Dependence of the average product
1 2 3 4 5 10 20 30 40 50
r/R1 t˜ concentration C¯c on the applied electric potential Mw.
Inset shows the mixture profiles for the same Mw’s.
FIG. 1: Concentration profiles C˜a, C˜b, and C˜c vs time U′ =U′ =10, φ =0.33, and k˜ =0.1
a b 0
when both A and B molecules are soluble in the polar
phase. (a)-(c): temporal progressionobtained from a
numerical solution of Eqs. 3, 4, and 5. (d):
volume-averageproduct C¯c vs time for varying values of 2
k˜. Solid, dashed, dash-dot and dotted lines correspond
to k˜ =0.01,0.1, 1, 10, respectively. Circles are the same
1.5
but without field. The solubility potentials are
U′ =U′ =U′ =10, the diffusion constants are
a b c
Da =Db =100×Dc, the averagemixture composition Cc 1
Miswφ0==0.01.4343,,athneddthimeednk˜ism=ioenn1ls.eisosnelelessctrreicacptoiotnenrtaiatel iiss 0.5 UUUUaaaa′′′′====UUUUbbbb′′′′====041400
0
ferent values of the potential are shown in the inset. As 0 10 20 30 40 50
t˜
canbe seen, C¯c is largerwith increasingpotential. How- FIG. 3: Dependence of the average product
ever, an increase of Mw from 0.036 to 0.144 has only a concentration C¯c on the solubility parameters.
modesteffect onC¯c. Indeed, evena hypotheticalinfinite φ0 =0.33, Mw =0.144, and k˜ =0.1
potential would lead to a finite effective reaction rate.
The preferential solubility of the reactants in the sol-
vents is an important factor determining the rate of the
reaction. InFig. 3 we plotC¯c for differentsolubility val- mainly due to diffusion (first term in the right-handside
ofequations3and4). Nearaninterfacethehighgradient
uesatafixedvalueofMw. While theasymptoticbehav-
ior at long times dictates C¯c → 2 clearly, the dynamics in φ leads to a large force from the solubility potential
and transport is dominated by the solubility difference
are faster as the solubility difference increases.
(second term in the right-hand side of equations 3 and
Reactants A and B prefer different phases. We
4).
turn to the interesting case where the reactants are “an-
tagonistic”inthesensethattheypreferdifferentsolvents. In Fig. 4(d) we plot the average product C¯c vs time
for different values of the potential. In contrast to Fig.
Figure 4 shows the concentration profiles vs time and
should be compared to Fig. 1. As before, C˜a and C˜b 1(d)heretheeffectivereactionisslower withfield(lines)
startfromauniformdistributionatt˜=0. Butherethey ascomparedtotheno-fieldcase(lineswithsymbols). As
diffuse in opposite directions – A to the polar solvent at in Fig. 1(d), fast reactions are less affected by the field
small r’s and B to the less polar solvent at large r’s. At (large values of k˜). The total product C˜c tends to 2 due
earlytimesthegradientsinC˜a andC˜b occurattheinter- to mass conservation.
facebetweenthe solvents. Theprofilesevolveintime; as The effective slowing-down of the reaction saturates
the reactants migrate according to their solubility they with the magnitude of the applied potential, see Fig. 5.
are consumed and C is created. In this calculation we As Mw increases (Mw ∝ V2) the more polar solvent is
assumed the product is equally soluble in both solvents pulled to the region with higher electric field, thereby
and thus at long times C˜c is uniform. Far from the in- raising the value of φ at small r’s and reducing it at
terface, in the bulk liquid, the transport of A and B is large r’s (low electric field region). Curves show C˜c vs
4
responding to diffusion over the interface, accompanied
(a) (b)
ntration12 t˜=0.01 CCC˜˜˜abc ntration1.512 t˜=0.10 CCC˜˜˜abc bsthcyoepariecsallceotnwagntrthesslat.hxTeahtliaeornsgterdrointchtgaeetrdetidffheebryienncdcoiemffiunpsaicotoinbnicoleivtnyetrrbamettiawocnereoon-f
e e A and B molecules in the two domains and the sloweris
c c
n n0.5
o o the overallrelaxation.
C C
0 0
In Fig. 7 we show the non-trivial dependence of the
1 2 3 4 5 1 2 3 4 5
r/R1 r/R1 rate of reaction on the average mixture composition φ0.
Atafixedtemperatureandelectricpotentialandforval-
(c) (d) uesofφ smallerthanthecriticalvalue,anincreaseinφ
0 0
ntration46 t˜=1.00 CCC˜˜˜abc Cc1.512 tilneossvtahpleoulelaosrccaplothisaoesnrestoof(itnthhseeetb)ii.nntHoedrofawalcecevuerbrv,eetawtleetaehdneststhaomeaenpotiilnmacrreeatanhsdee
nce2 0.5 “contrast”between the phases diminishes and the thick-
o
C nessofthe interfaceincreases. As aresult, forthe values
0 0
1 2 3 4 5 10 20 30 40 50 presented in this calculation, at long times the reaction
r/R1 t˜ whichisslowestisonewithintermediatevalue, φ =0.2,
0
FIG. 4: Concentration profiles C˜a, C˜b, and C˜c and and not one of the extreme values φ0 =0.1 or φ0 =0.4.
average product C¯c for the same conditions as in Fig. 1
except that molecule A is soluble in the polar phase and
B in the less polar phase: U′ =10 and U′ =−10. 2
a b
1.5
2
Cc 1
1.5
Cc 1 0.5 UUUUaaaa′′′′====−−−−UUUUbbbb′′′′====041400
1
0
10 20 30 40 50
φ0.5 t˜
0.5 Mw=0.0 FIG. 6: The influence of the preferential solubility of A
0 MMMwww===000...013342644 01 2 r/3R1 4 5 arnedacBtiomnoilseccuolnessidoenratbhleyasvloewraegdedtoowtanl parso|dUua′c|taCn¯dc.|UTbh′|e
10 20 t˜ 30 40 50 increase. φ0 =0.33, Mw =0.144, and k˜ =1.
FIG. 5: The influence of the electric potential Mw on
the averageproduct concentration C¯c. Larger values of
Mw decrease the effective reaction rate. Inset shows the
equilibrium profiles φ(r) for the same Mw’s. Ua′ =10, 1.6
U′ =−10, φ =0.33 and k˜ =1. 1.4
b 0
1.2
1
Ct˜imc e=fo0radtifft˜er=en0t vtaoluC˜ecs o=f M2 wat. Ainlfilnciutyrv.esCiunrcvreesaseexhfriobmit Cc0.8 1
an early increase on a small timescale dictated by diffu- 0.6
φ0.5
sion over the mesoscopic width of the interface between
coexisting phases. After the rapid increase, the curves 0.4 φ0=0.10
increase slowly on a timescale dictated by diffusion over 0.2 φφ00==00..2303 01 2 3 4 5
the macroscopic size of the wedge. φ0=0.40 r/R1
0
The overall reaction rate is slowed down when the re- 0 500 1000 1500 2000
t˜
actants are preferentially soluble in different solvents, as
FIG. 7: Influence of the averagemixture compositionφ
0
can be seen in Fig. 6. Clearly the electric potential is on the average total product C¯c at a fixed electric
crucial here because in its absence the mixture is homo-
potential given by Mw =0.144, solubilities
geneous andwith it two domains exist. All curves in the Ua′ =−Ub′ =10 and k˜=0.1. Inset is the equilibrium
Figure start from zero and increase to 2 at long times.
profiles φ(r) for the same values of φ .
0
Hereagaintherearetwotimescales–afasttransientcor-
5
In summary, electric field gradients cancontrolthe ef- 56/14.
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