Hydrochemical stability of an interface between two immiscible liquids : the role of Langmuir-Hinshelwood saturation law

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Hydrochemical stability of an interface between two immiscible liquids : the role of Langmuir-Hinshelwood

saturation law

J.L. Ibáñez, M. G. Velarde

To cite this version:

J.L. Ibáñez, M. G. Velarde. Hydrochemical stability of an interface between two immiscible liquids : the role of Langmuir-Hinshelwood saturation law. Journal de Physique, 1977, 38 (12), pp.1479-1483.

�10.1051/jphys:0197700380120147900�. �jpa-00208722�

LE JOURNAL DE PHYSIQUE

HYDROCHEMICAL STABILITY OF AN INTERFACE

BETWEEN TWO IMMISCIBLE LIQUIDS :

THE ROLE OF LANGMUIR-HINSHELWOOD SATURATION LAW

J. L.

IBÁÑEZ

and M. G. VELARDE

(*)

Departamento

^{de Fisica 2014}

C-3,

Universidad Autônoma de

Madrid,

^Canto

Blanco, Madrid, Spain (Reçu

^{le 11}

juillet 1977, accepté

^{le 17 août}

1977)

Résumé. ²⁰¹⁴ Nous étudions la stabilité de l’interface séparant ^deux

liquides

^{non miscibles et} siège

d’une réaction chimique

superficielle gouvernée

par ^une loi de saturation du type Langmuir-Hins-

helwood (Michaelis-Menten).

Abstract. ²⁰¹⁴ The

coupling

of chemical to

hydrodynamic instability

mechanisms is discussed for the case of an interface between two immiscible

liquids,

^{where a} saturation law of the Langmuir-

Hinshelwood (Michaelis-Menten) type governs ^a surface reaction _process at the interface.

Classification

Physics ^Abstracts

47.20

1. Introduction. ^-

Recently

Sanfeld and colla- borators

[1, 2]

have studied the

coupling

of chemical to

hydrodynamic phenomena

^{at an} interface.

They

discussed the role of a trimolecular scheme _upon the

stability

of the interface between two immiscible

liquid phases

^{at which} interface the reaction pro- ceeds. In the present ^{note we}

provide

^the

stability analysis

^{of a similar}

problem

but with consideration of a more realistic chemical _process,

namely

^an

autocatalytic

bimolecular scheme

involving

^{a satu-}

ration law of the

Langmuir-Hinshelwood type.

^This model is of relevance in reactor kinetics

[3]

^with

adsorption

^{at walls or at}

heterogeneous particles

that

play

the role of

catalysts.

^In

addition,

^{the satu-}

ration law that we use here

corresponds

^to the Michae- lis-Menten law in

enzyme-controlled

^reactions.

It is known that transfer of a solute between two immiscible

liquid phases,

^or

temperature and/or

concentration

inhomogeneities

^{at the}

interface,

^can

induce surface tension tractions and thus motions that

develop

^{first at the} interface and next within the bulk

phases.

Concentration

inhomogeneities

may be

brought

^about

by

chemical reactions at the inter- face. For a chemical scheme _may

yield

^sustained

temporal

oscillations or a

spatially

non-uniform distribution of the concentrations of reactants and this

provokes

variations of the surface tension at the interface

leading

^to Bénard-like cellular convective patterns

[4, 5].

(*) ^{Also at} Laboratoire de Dynamique ^et Thermophysique ^des Fluides, Université de Provence, Centre de Saint-Jérôme, 13397 Marseille cedex 4, France.

2. Reaction

scheme, steady

state, and the evolution

équations.

^{- We consider two}

incompressible

^New-

tonian

liquids separated by

^a deformable interface of

equation

in which z _z., refers to the lower fluid and z > z., the fluid above the interface. The coordinates x

and y

^{refer to the}

plane orthogonal

^{to z. In what}

follows, superscript

^{o 1 »} denotes the lower fluid.

The two bulk fluid

phases obey

the standard evolu- tion

equations, namely

^the

continuity equation

^and

the Navier-Stokes

equations.

The interface is also taken as a Newtonian two-dimensional fluid

layer

of mass

density ^TT.

^{The interface}

supports

^tractions

arising

^{from the}

adjacent parts

^{of the two bulk}

phases.

We assume that the

following

^chemical process takes

place

^on the interface

[6]

in which A and P are reactants whose concentration

we

keep

constant at the interface. Thus the interface is _open to flow of A and P. « S » stands for the

Lang-

muir saturation law

given

below. Whereas A and P

are soluble in either one of the two bulk

phases,

^,

neither X nor Y is soluble.

They

^are

present only

at the interface where

they

^{are free to diffuse}

according

to Fick’s law. When convection takes

place,

^{both X}

and Y are also carried with the

barycentric

^motion

of a surface element.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197700380120147900

1480

Under isothermal conditions the intermediate ^reac- tants

obey

^the

following

^evolution

equations

in which

ki. k2, k

^and

^k4

^are the kinetic constants of

reaction ; d 2 --_ ax 2 + ay 2

^{dénotes a two-dimen-}

;

!( a2 a2) ^{ax ay}

sional

Laplacian

^{on the}

interface ; Dx

^and

Dy

^{are the}

surface Fickian diffusion constants of

species

^X

and Y

respectively.

^{The vector VS =}

(vx, v’)

^describes

the

velocity

^of

points belonging

^to the interface.

Thus,

variations in the concentration of the inter- mediate reactants and

consequently

surface tension

gradients

^{are due either}

(i)

^to the chemical _process

(2)

or

(ii)

^{to diffusion}

along

the surface or

(iii)

^{to surface}

convective motions.

At the

interface,

^the

following boundary

^conditions

are considered :

(i) Continuity

^of

velocity

^and

velocity gradient along

the normal. Thus cavitation

phenomena

^are

excluded

(ii)

The difference of

hydrostatic

pressure ^across the interface is balanced

by

^the surface tension

(Laplace

condition on the normal

stresses)

in which

r T

^{accounts for the mass}

density

^{at the}

interface, Fz

denotes the vertical

component

of external forces

(here

^{reduced to}

gravity gl,

^(f is the surface

tension,

^and

Ri

^and

R2

^{are the}

principal

^{radii of curvature of the}

interface. ,u‘ (i

⁼

1, 2B

denotes the

viscosity

of either bulk

phases.

(üil

Surfacte tractions on both sides of the interface are correlated to surface tension

gradients

^{and to}

viscous effects on the surface

(Levich-Aris-Scriven

^condition

^{[7, 8,} 9])

In

equation (7)

^two viscosities are

incorporated :

^e, the surface shear

viscosity,

^and K, the surface dilata- tional coefficient

[9]. Equation (7)

^{accounts for the}

tangential

^stresses.

We shall discuss below the

stability

of the interface under the

joint

influence of chemical

reaction,

^diffu-

sion,

surface tension

gradients,

^and

gravity.

^{This case is a}

generalization

^{of the}

Rayleigh-Taylor problem [10].

Before

proceeding

^{to the}

stability analysis

^{we note} that the basic state or state of reference whose

stability

^will

be discussed in the next section is a state of rest

corresponding

^{to a level}

interface,

^{with no} motions in the bulk

phases (vl - ^{V2 ==}

^{v’ =}

0),

^{and with a} uniform distribution

of reactants,

^{i.e. a} uniform distribution of mass

fy along

the interface. This uniform mass distribution

corresponds

^{to the}

steady

solution of the chemical scheme

(2)

in

which q

⁼

k3 k4lk2

^{and a =}

k, k2 Alk 2 3-

3.

Stability analysis : generalized Rayleigh-Taylor problem.

^{- Let}

^t5vi (i

⁼

1,2), t5vs, t5X, t5 Y, ^t5zs,

^{b6 define}

small

perturbations

^to the reference motionless uniform

steady

^state described in the

previous section.

Insta-

bility

^to infinitesimal disturbances

yields

^a sufficient condition for

instability

of the interface. It suffices to test

stability

of a Fourier normal

mode, k,

of those disturbances

(see ^{[1, 4, 5, 10]}

^for details of the

technique).

Denoting

^a normal mode

by subscript

^{« k », in the}

neighbourhood

^of

(8)

^we

have,

from the evolution _equa- tions

(3),

... __

in which H denotes the

step

between the values of the function between brackets taken at the two sides of the interface.

k,

^{the wave}

number,

^{has two}

components k2 = k 2

⁺

kÿ,

^{and the}

^quantity

^m’ is defined as follows

co is the time constant that determines

stability [1, 2].

^In

equations (9)

^the variations in surface tension are consi- dered to

depend solely

^on the variation of the concentrations of the two

intermediate,

and non-soluble reactants.

We have

in which _aux and _ay denote

positive

parameters.

From

equations (9)

it follows that

stability

is determined

by

^an

eigenvalue problem

^that

yields

^the

following

determinantal

equation (for

details of the

technique

^see

[2, 4, 5, 10])

where

which accounts for the

coupling

^{between the chemical} process and diffusion of reactants ;

which

couples

the extemal force

(gravity)

^{to the} surface tension

gradient ;

^and

which

yields

^a

straightforward generalization

^{of the}

Rayleigh-Taylor problem [10]

^{with a} viscous interface

(8

^{+ K # 0) of mass}

density TT.

1482

If all bulk viscous

phenomena (in

the fluids 1 and

2)

may be

neglected

the secular

equation (12)

^reduces

to the

following

The first factor in

equation (17) merely represents

the standard non-viscous

Rayleigh-Taylor

pro- blem

,

[10].

^Thus

instability

^{occurs when}

p2

>

p’.

The

non-vanishing

values of

TT

^{in our case here}

merely

shift the value of the critical mode at the onset of

instability.

The determinantal factor

incorporates

^all

pheno-

mena that take

place

^at the interface : chemical

reaction,

^mass

diffusion,

surface tension inhomo-

geneities,

^surface

viscosity,

^and

possible

^convective

motions

along

the interface. It appears that for

vanishing k

^this determinantal factor

yields

^the

quantity

that controls the

stability

of the chemical process alone

[6].

^{Thus a} necessary condition for the reference motionless uniform state of the inter- face to be stable is that the

steady

^state

(8)

^{be stable.}

The

condition, however,

^{is not} sufficient as we shall discuss in the

follôwing

^section.

4. Numerical results and

stability diagram.

^{- For}

illustration we take a

and q

^{as state}

variables,

^and the

following

numerical values in C.G.S. units

{

(Xx = aY ⁼

108, Dx

^{= 5 x}

10-6, Dy

^{= 1 x}

10-6,

8 + ^x

= 1, k2

^{= 2 x}

103, k3

^{= 2 x}

10-4, FT

⁼

10-’, p2

⁼

1.0, pl

⁼

1.1 } (1).

With these values we solve the determinantal

equation given by ⁽¹⁷⁾

^as

done _by

Sanfeld and collaborators

[2]. Figure

^{1 shows the}

results. It describes the

following

^{cases :}

region

¹

is of

asymptotic stability

of the reference motionless uniform state with level interface. The dashed

portion corresponds

^{to stable states} of the chemical _process alone that become unstable with the

coupling

^of

the other

phenomena

^discussed above. Thus chemical

stability,

^even with mechanical

stability (p2 pi)

does not

preclude instability

^{of the interface when}

there is

interplay

between various

physicochemical

processes. The

instability

^can be traced back to the surface tension

inhomogeneities originating

^{in the}

variation in concentration of the reactants. Inci-

(1) ^{It is to} be noted that the ratio DX > Dy ^favors stability [6]

for the reaction-diffusion process alone. Thus ^new unstable states in the example ^{discussed are} genuinely ^{due to the} coupling ^between

the chemistry ^and hydrodynamics ^at the interface.

FIG. 1. ^- Stability diagram ^of system (3) ^{in the} neighbourhood ^of

the motionless uniform state (8). The dashed area corresponds ^to

the coupling of chemical to hydrodynamic processes according ^to

the discussion in section 4. Regions ^{1 and II are} respectively ^those

of stability ^and instability ^of (8). ^{With the} hydrochemical coupling region ^{II is} enlarged ^{at the} expense of region ^I. Region ^{III corres-} ponds ^to unphysical ^{states of} negative ^or undefined concentrations.

dentally,

^{this is not}

possible

if there is

only

^{one inter-}

mediate

species participating

in the chemical _pro-

cess

[11].

^For

one-species

processes chemical

stability

suffices to ensure

stability

^{of a}

mechanically

^stable

interface in the presence of the other

physical pheno-

mena. This is not so for two or more

species

^{at has}

already

^{been shown}

by

Sorensen et al.

[12]

^{in a}

problem involving quite

^a

general

reaction scheme.

5. Second-order

Langmuir-Hinshelwood

^{law. - In}

the

previous

^{sections we have} considered a non-

linear law

(Eq. (3a))

with n ⁼ 1. This case

corresponds

^{to a saturation}

law. With n ⁼ 2

(or

^more

generally n > ²⁾

^{the rela-}

tion

(18)

^{defines a case} of inhibition

following

^some

maximum of the

velocity

of reaction

(see [3]

^or

[13]

for motivation and further

details).

For the chemical process

alone,

^{there are the fol-}

lowing

main differences between the cases n ⁼ 1 and n ⁼ 2

[13].

(i)

^{With n = 1} either there is one

homogeneous steady

^{state or there is none. With n =}

^2,

^{two such}

states are admissible ^{or none.} When two

exist,

^one

is

always

^a

saddle,

and unstable.

(ii)

^{With n =}

1,

^{if the} admissible

steady

^state

is

asymptotically

^{stable to} infinitesimal disturbances it may also be

expected

^to be stable in the

whole, i.e.,

^{stable to}

arbitrary perturbations.

^{With n =}

2, however,

^{if one of the two}

steady

^{states is stable}

it is

surely

^{not stable to finite} disturbances. It suffices to

give

^a

perturbation

of the order of the distance between its coordinates and those of the unstable saddle

point

^{so as to} destabilize the former.

Then,

the concentration of the intermediate reactant X grows unbounded.

When the chemical _process

(2)

with the law

(18)

- case n ⁼ 2 ^- takes

place

^{at the}

interface,

^for

similar conditions to those described above there is

a

primary

motionless solution

VI = V2 =

Vs =

0,

with z ⁼ Zg, and uniform concentration of reactants at the

interface,

The

stability analysis

of this solution

(19)

^{leads to}

results

analogous

^to those found for

(8).

^The

phase diagram

^is

qualitatively

^{identical even}

though

^the

law here contains an

inhibitory

^section.

Again,

mechanical

stability

^{or chemical}

stability,

^{or both}

stabilizing

mechanisms taken

uncoupled,

^{do not}

guarantee

^the

stability

of the interface. The

coupling

of chemical to

hydrodynamic phenomena

^{can induce}

instability

^{of the}

interface,

^{due to the}

large separation

in the relaxation time scales involved. Chemical reaction is instantaneous whereas diffusion and con-

vection demand

non-negligible

^time

delays

^{to relax}

the fluctuations in concentration

originated by

^the

chemical process.

Acknowledgments.

^{- The results}

presented

ⁱⁿ

this note were obtained in the course of research

sponsored by

the Instituto de Estudios Nucleares

(Spain).

Our interest in the

problem

^{was stimulated}

by

fruitful discussions with A.

Sanfeld,

^{T. Sorensen}

and M.

Hennenberg,

^to whom the authors wish to express their thanks.

References

[1] STEINCHEN-SANFELD, ^{A. and} SANFELD, A., ^Chem. Phys. ¹ (1973) 156.

[2] HENNENBERG, M., SORENSEN, ^T. S., STEINCHEN-SANFELD, ^A.

and SANFELD, A., ^{J. Chim.} Phys. ⁷² (1975) ^1202.

[3] ARIS, R., The mathematical theory of diffusion ^{and reaction}

in permeable catalysts (Clarendon Press, Oxford, 1976)

Vol. II, Sect. 8.8.

[4] ^{For a} general ^{reference see}

GLANSDORFF, ^{P. and} PRIGOGINE, I., Structure, ^{stabilité et} fluctuations (Masson, Paris, 1971). ^{See also}

NICOLIS, ^{G. and} PRIGOGINE, I., Selforganization ⁱⁿ non-equi- librium systems (Wiley, ^New York, 1977).

[5] VELARDE, M. G., ^{in Fluid} dynamics ^edit. by ^{R. Balian and}

J. L. Peube (Gordon ^and Breach, ^New York, 1976)

pp. 469-527. See also

NORMAND, C., POMEAU, ^Y. and VELARDE, ^M. G., ^{Rev. Mod.}

Phys.49 (1977) ^581-624.

[6] IBANEZ, J. L., FAIREN, V. and VELARDE, M. G., Phys. ^Lett.

58A (1976) ^364.

[7] LEVICH, V., Physico-Chemical Hydrodynamics (Prentice-Hall, Englewood Cliffs), ^1967.

[8] SCRIVEN, L. E., ^Chem. Eng. ^{Sci. 12} (1960) ^98.

[9] ARIS, R., Vectors, tensors, and the basic equations of fluid

mechanics (Prentice-Hall, Englewood Cliffs), ^1962.

[10] CHANDRASEKHAR, S., Hydrodynamic ^and hydromagnetic ^sta- bility (Clarendon Press, Oxford, 1961) Chap. ^X.

[11] SANFELD, ^{A. and} STEINCHEN, A., Biophys. ^{Chem. 3} (1975) ^99.

[12] SØRENSEN, ^T. S., HENNENBERG, M., STEINCHEN, A. and SAN-

FELD,

A., ^J. Colloid Interface ^{Sci. 56} (1976) ^191.

[13] IBÁÑEz, ^{J. L. and} VELARDE, ^M. G., ^J. Physique Lett., to appear.