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879 — Oscillations generated by pulsating membrane crossing
505
Bioelectrochemistry and Bioenergetics, 15 (1986) 505-511 A section of J. Electroanal. Chem., and constituting Voi. 211 (1986) Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
879 - - O S C I L L A T I O N S MEMBRANE
CROSSING
GENERATED *
BY PULSATING
FRANCESCA D'ALBA lstituto di lngegneria Chimica, Viale delle Scienze, 90128 Palermo (Italy) SERGIO DI LORENZO lstituto Tecnico Industriale di Stato per la Chimica, Viale Europa, 24100 Bergamo (Italy) (Manuscript received July 20th 1985)
SUMMARY In this paper we develop the theoretical background of oscillations due to pulsating membrane crossing, and give evidence for the role of some resisiances in their generation. We examine different membrane types: (1) porous septum; (2) membrane without pores; (3) membrane without pores and with carrier. The solvent and the channel walls in a porous septum are chemically bonded. These bonds must be broken in order to allow a chemical component to pass from one side of the septum to the other. Their energy is the resistance that allows oscillations. If a membrane has no pores, the chemical component must dissolve in it in order to cross. This involves rupturing the surface and work against the surface tension, generating a resistance able to allow oscillations. The mechanisms are analogous in a membrane with a carrier.
INTRODUCTION Oscillatory phenomena are common in biological systems. As they are very c o m p l i c a t e d , t h e i r d i r e c t s t u d y is a l m o s t i m p o s s i b l e a n d i t is n e c e s s a r y t o e l a b o r a t e a simple chemical and mathematical model, able to explain them. We'have demonstrated the necessity of an irreversible step, related to mass transfer, to generate o s c i l l a t i o n s [1]. T h e a i m o f t h i s p a p e r is t o a p p l y o u r m o d e l t o o s c i l l a t o r y p h e n o m e n a membranes or porous septa.
through
* Contribution presented at the VIIIth International Symposium on Bioelectroehemistry and Bioenergetics, Bologna, June 24th-29th 1985. 0302-4598/86/$03.50
© 1986 Elsevier Sequoia S.A.
506 CHEMICAL
AND
We consider
ENERGETIC
a perfectly
FEATURES
mixed solution.
An irreversible
reaction
(1)
A sol -+ Bs,, with first-order
(I) kinetics
(2)
(2) takes place inside the solution. At time t = 0 we have [A] = [A,,] and [B] = 0. The solution is separated from some pure solvent by a membrane, impermeable for component A and permeable for component B. As the (Gibbs) free enthalpy difference associated with this transformation is negative, component B should pass through the membrane. Yet, depending on the type of membrane, either an intermediate state exists in the crossing of the membrane, with a free enthalpy larger than the one associated with an increase in concentration of component B in solution, or work is required in order to pass through the membrane. Neither phenomenon allows component B to pass through the membrane, until the free enthalpy difference between solution and pure solvent overcomes the barrier created by the phenomenon. In order to calculate the free enthalpy of the system, we assume, as standard states, pure solvent, pure component B and pure component A at a temperature equal to that of the solution. We indicate as AGsolTA the free enthalpy difference, associated with the dissolution of one mole of A, and as of one mole of AGS01.a the free enthalpy difference, associated with the dissolution B, and suppose that their values are not influenced by concentration, as the solution is very diluted. We indicate as AG,,,,,,, the free enthalpy difference per mole of A, associated with reaction (3) (3)
A,+BCl at standard state. Hence the free enthalpy solution, is AG,,,,, = -AG,,,%,
+ AG,,,,,,,
difference,
associated
with reaction
+ AG,,,,,
(1) in
(4)
Only the free enthalpy difference associated with the presence of B in solution an effect on the crossing of the membrane, and under our conditions it is
has
A%,,= A%,,Eml
(5)
where V is the volume MECHANISMS
of the solution.
OF OSCILLATION
AND TYPES OF MEMBRANE
The mechanisms of oscillation depend on the membrane models, of which we examine several types, but they take place only if the rate of pulsating crossing is larger than that of the chemical reaction.
507
Porous septum We consider a membrane with the characteristics of a septum, crossed by channels, having the same radius r,, as component B; the latter has to push the solvent out of the channels in order to pass through them. If e is the degree of vacuum of the septum, V, is the volume of the septum and p is the molar density of the solvent, there are n = V1ep
(6)
moles of solvent inside the septum and we can consider every molecule to be in touch with the walls of the channels, since their radii are the same. Chemical bonds take place between the solvent and the channel walls with which a free enthalpy of formation AG P,M, per mole of solvent, is associated. Hence, AG, AG, = V,cp AG,,,
(7)
is the amount of free enthalpy associated with the formation of these chemical bonds. In order for them to be broken, the amount of free enthalpy, associated with an increase in concentration of component B in solution, must be negative and larger (in absolute value) than the amount associated with the breaking. When crossing of a membrane occurs, the mass transfer balance of B in solution can be written as follows:
(%I.%,,,= (%)rhem+ (T?)d,... Substituting the appropriate equation (8) becomes =&[A,,]
exp(-k
overall
values,
1
t)-k
if the pure
2
SIBI
solvent
(8) is renewed
continuously,
(9)
v
where S is the pore surface reduced in order to account for back diffusion of some solvent from some pores. Referring it to 8, i.e. the time which has passed since crossing started, equation (9) can be written as follows:
overall
= k,
[Aoqj] exp( -k,8)
with boundary
condition
[Bl = LB,]
at 8=0
- k,Su
V
(11) (II)
in which [B,] is the threshold value of [B] at which crossing of B starts; and [A,,j] is the concentration of component A at time 0 = 0 and identifies the cycle. If the last term on the right hand side of equation (10) is larger than the first one, [B] decreases and equation (10) holds true until [B] reaches a value [Bsrop] which is so low that the motion becomes very slow and chemical bonds occur again between solvent and channels walls. In the mass-transfer balance of B, the last term on the right hand
508
side of equation (10) vanishes is given by equation (12):
[B]=&_]e~p(-y)+
and [B] increases
Fiky!k
[exp(-k,8)-exp(-y)]
k 2
if crossing
of the membrane
again. The solution
of equation
(10)
(12)
1
takes place, and by equation
(13)
ex~(-‘,‘)I
(13)
[Bl = [~srop] + [Ao,j][exP(-‘15) if crossing of the membrane does not take place. Here, 5 is the time which, when substituted in equation (12) gives the value [B] = [Bsrop].
Membrane without pores If the membrane is not a porous septum, component B must dissolve in it or must open some channels in order to pass through. If the membrane is permeable to the solvent, component B dissolves together with its solvation shell. Every surface has some weak points in its structure, at which dissolution or opening of a channel may occur. The solution pushes against the walls of the membrane to open it, as the free enthalpy difference would impose crossing. Curving of the surface in the failing points allows the membrane to oppose a pressure pour
where u is the surface tension and r is the radius of the curve. As r decreases until the curve becomes a hemisphere, the membrane can oppose crossing by the increasingly concentrated B. If the radius of the hemisphere is larger than the thickness of the membrane, a channel is formed. The solution passes through as in a pore and the value of the concentration of B is governed by equations (12) and (13) where [B,] and [Bsrop] are obtained by the condition of dynamic equilibrium of the membrane. If the membrane has a thickness larger than the radius of the hemisphere, channels do not form. When curving goes beyond a hemisphere, however, the radius increases again, the pressure on the membrane-side becomes lower than that on the solution-side and dynamic equilibrium cannot exist. Rapid dissolution, involving some molecules of B and some solvent, occurs and leads to a decrease in the concentration of B, which is more marked at larger values of the surface/volume ratios, as it is in a sphere with a very small radius. An analogous phenomenon takes place at the membrane/solvent interface, though with another value of [B] i.e. [Blc2] above which crossing of B starts. If the solvent is renewed continuously, the system will be governed by the mathematical equations
= khl
exp(-ht) - k2%([Bso,l - [Bi.J~~
05) (16) (17)
509
with the boundary conditions [B,,,] = [B,,,,]
= [B,,,,] = 0
W
at t = 0
in membrane = k,([B].,,in membrane = k, [Bi,d] R,
[Bi,,])R,
(19) (20)
Equation (15) represents the mass-transfer balance of B in the solution, R, is a coefficient with two values, 0 when crossing does not take place and 1 when crossing does take place, S, is a surface reduced in order to account for the fact that mass transfer does not involve the entire geometrical surface of the membrane and [Bi,,] is the concentration of B (at the interface solution/membrane, solution side) equilibrating the concentration of B at the same interface, membrane side, [Bi.,]. Equation (16) represents the mass-transfer balance of B inside the membrane. Equation (17) is a relationship, analogous to Henry’s law, of equilibrium between the two sides of the interface where H symbolizes a simple proportionality constant. In equation (18), [B,,,] symbolizes the concentration of B in the solution contacting the membrane, [Bmemb] the concentration of B within the membrane and [Bdiss] the concentration of B in the initially B-free solvent contacting the other side of the membrane. Equation (19) expresses the equality of the fluxes on the two sides of the membrane/solution interface. Equation (20) expresses the equality of the fluxes on the two sides of the membrane/solvent interface. R, is a coefficient with two values, 0 when crossing does not take place and 1 when crossing does take place. [Bi d] is the concentration of B in the initially B-free solvent at the surface that equilibrates the concentration of B in the membrane at the membrane surface. As [Bi .I and PmemJ increase at every cycle, before B crosses into the pure solvent, we have a progressive flattening’of the oscillations, due to an increase in the minimum. Yet pulsating crossing of B into the solvent restores the oscillations. Membrane
with carrier
If component B is insoluble in the membrane, a carrier is necessary for crossing to occur. The carrier cannot go out of the membrane and interaction between component B and the carrier takes place only at the interface. As the carrier cannot go out of the membrane, no free enthalpy difference drives it towards the interface and we have to admit that statistical fluctuations around an average value give some carrier molecules a free enthalpy difference larger than the work necessary to reach the interface. Statistical fluctuations cannot generate oscillations, which depend only on component B. Component B must reach the interface in order to react and this implies work against the surface tension of the solution. As the maximum area in the interface occurs at the breakage of the surface, when curving is hemispherical, the work per mole of B is expressed by equation (21) w = 21rr&N,
(21)
510
where r, is the radius of B and NA is Avogadro’s number. For crossing to start, the free enthalpy difference, associated with an increase in the concentration of B in solution, must become larger than the work necessary for B to reach the interface. The physico-chemical system is governed by the following mathematical equations:
d[Bl dt
i
)
= kl[Aol exd-kd - k&4PlR3
ouerall,sol
(22)
in which M symbolizes the carrier, and for which [B]=Oand[M]=[M,] D -=a*[MB] ’ D
(23)
a[MB]
(24)
at
ax*
a*[Ml a[Mi -=* ax* at
(25)
at t=O
[MB]=• _D
att=O
ww
1 ax
=D
(26)
a[Mi= kMM~3
at x=0
(27)
m-u= k_,[MB]R,
at x=1
(28)
2- ax
while _D
m-1
1 ax
=D
-
* ax
Equation (22) represents the mass-transfer balance in solution, where k, is the rate constant of reaction between component B and carrier M, R, is a coefficient with two values, 0 and 1, which gives significance to the last term on the right hand side, when crossing occurs. Equation (24) represents the mass-transfer balance of the complex BM (component B + carrier) in the membrane. Equation (25) represents the mass-transfer balance of carrier in the membrane. Equations (26), (27) and (28) are boundary conditions, where R, is a coefficient with two values, 0 and 1, to account for pulsating crossing from membrane to pure solvent. As generation of B must be slow to generate oscillations, we can consider the concentration of carrier to be in excess and simplify the system as given by the following equation (29). =k,[A,]
oueroll.sol
exp(-k,t)
- k,a[B]R,
where (Y stands for a dimensional coefficient required to make equation (29) dimensionally correct. Oscillations generated by these mechanisms, are very small in amplitude. Yet, if cxis large, as it is for example in a sphere with a very small radius, the amplitude relative to the average value can also be large. As mass transfer involves only the external layer, the oscillations have a sawtooth shape.
511 CONCLUSIONS
The above models are able to explain some types of oscillation generation and evidence the role of some resistances in mass transfer in order to account for oscillations. It is clear that higher resistances lead to a higher amplitude of oscillations. REFERENCES 1 F. D’Alba and S. Di Lorenm, J. Chem. Sot., Faraday Trans. 1, 79 (1983) 38.
Bioelectrochemistry and Bioenergetics, 15 (1986) 505-511 A section of J. Electroanal. Chem., and constituting Voi. 211 (1986) Elsevier Sequoia S.A., Lausanne - Printed in The Netherlands
879 - - O S C I L L A T I O N S MEMBRANE
CROSSING
GENERATED *
BY PULSATING
FRANCESCA D'ALBA lstituto di lngegneria Chimica, Viale delle Scienze, 90128 Palermo (Italy) SERGIO DI LORENZO lstituto Tecnico Industriale di Stato per la Chimica, Viale Europa, 24100 Bergamo (Italy) (Manuscript received July 20th 1985)
SUMMARY In this paper we develop the theoretical background of oscillations due to pulsating membrane crossing, and give evidence for the role of some resisiances in their generation. We examine different membrane types: (1) porous septum; (2) membrane without pores; (3) membrane without pores and with carrier. The solvent and the channel walls in a porous septum are chemically bonded. These bonds must be broken in order to allow a chemical component to pass from one side of the septum to the other. Their energy is the resistance that allows oscillations. If a membrane has no pores, the chemical component must dissolve in it in order to cross. This involves rupturing the surface and work against the surface tension, generating a resistance able to allow oscillations. The mechanisms are analogous in a membrane with a carrier.
INTRODUCTION Oscillatory phenomena are common in biological systems. As they are very c o m p l i c a t e d , t h e i r d i r e c t s t u d y is a l m o s t i m p o s s i b l e a n d i t is n e c e s s a r y t o e l a b o r a t e a simple chemical and mathematical model, able to explain them. We'have demonstrated the necessity of an irreversible step, related to mass transfer, to generate o s c i l l a t i o n s [1]. T h e a i m o f t h i s p a p e r is t o a p p l y o u r m o d e l t o o s c i l l a t o r y p h e n o m e n a membranes or porous septa.
through
* Contribution presented at the VIIIth International Symposium on Bioelectroehemistry and Bioenergetics, Bologna, June 24th-29th 1985. 0302-4598/86/$03.50
© 1986 Elsevier Sequoia S.A.
506 CHEMICAL
AND
We consider
ENERGETIC
a perfectly
FEATURES
mixed solution.
An irreversible
reaction
(1)
A sol -+ Bs,, with first-order
(I) kinetics
(2)
(2) takes place inside the solution. At time t = 0 we have [A] = [A,,] and [B] = 0. The solution is separated from some pure solvent by a membrane, impermeable for component A and permeable for component B. As the (Gibbs) free enthalpy difference associated with this transformation is negative, component B should pass through the membrane. Yet, depending on the type of membrane, either an intermediate state exists in the crossing of the membrane, with a free enthalpy larger than the one associated with an increase in concentration of component B in solution, or work is required in order to pass through the membrane. Neither phenomenon allows component B to pass through the membrane, until the free enthalpy difference between solution and pure solvent overcomes the barrier created by the phenomenon. In order to calculate the free enthalpy of the system, we assume, as standard states, pure solvent, pure component B and pure component A at a temperature equal to that of the solution. We indicate as AGsolTA the free enthalpy difference, associated with the dissolution of one mole of A, and as of one mole of AGS01.a the free enthalpy difference, associated with the dissolution B, and suppose that their values are not influenced by concentration, as the solution is very diluted. We indicate as AG,,,,,,, the free enthalpy difference per mole of A, associated with reaction (3) (3)
A,+BCl at standard state. Hence the free enthalpy solution, is AG,,,,, = -AG,,,%,
+ AG,,,,,,,
difference,
associated
with reaction
+ AG,,,,,
(1) in
(4)
Only the free enthalpy difference associated with the presence of B in solution an effect on the crossing of the membrane, and under our conditions it is
has
A%,,= A%,,Eml
(5)
where V is the volume MECHANISMS
of the solution.
OF OSCILLATION
AND TYPES OF MEMBRANE
The mechanisms of oscillation depend on the membrane models, of which we examine several types, but they take place only if the rate of pulsating crossing is larger than that of the chemical reaction.
507
Porous septum We consider a membrane with the characteristics of a septum, crossed by channels, having the same radius r,, as component B; the latter has to push the solvent out of the channels in order to pass through them. If e is the degree of vacuum of the septum, V, is the volume of the septum and p is the molar density of the solvent, there are n = V1ep
(6)
moles of solvent inside the septum and we can consider every molecule to be in touch with the walls of the channels, since their radii are the same. Chemical bonds take place between the solvent and the channel walls with which a free enthalpy of formation AG P,M, per mole of solvent, is associated. Hence, AG, AG, = V,cp AG,,,
(7)
is the amount of free enthalpy associated with the formation of these chemical bonds. In order for them to be broken, the amount of free enthalpy, associated with an increase in concentration of component B in solution, must be negative and larger (in absolute value) than the amount associated with the breaking. When crossing of a membrane occurs, the mass transfer balance of B in solution can be written as follows:
(%I.%,,,= (%)rhem+ (T?)d,... Substituting the appropriate equation (8) becomes =&[A,,]
exp(-k
overall
values,
1
t)-k
if the pure
2
SIBI
solvent
(8) is renewed
continuously,
(9)
v
where S is the pore surface reduced in order to account for back diffusion of some solvent from some pores. Referring it to 8, i.e. the time which has passed since crossing started, equation (9) can be written as follows:
overall
= k,
[Aoqj] exp( -k,8)
with boundary
condition
[Bl = LB,]
at 8=0
- k,Su
V
(11) (II)
in which [B,] is the threshold value of [B] at which crossing of B starts; and [A,,j] is the concentration of component A at time 0 = 0 and identifies the cycle. If the last term on the right hand side of equation (10) is larger than the first one, [B] decreases and equation (10) holds true until [B] reaches a value [Bsrop] which is so low that the motion becomes very slow and chemical bonds occur again between solvent and channels walls. In the mass-transfer balance of B, the last term on the right hand
508
side of equation (10) vanishes is given by equation (12):
[B]=&_]e~p(-y)+
and [B] increases
Fiky!k
[exp(-k,8)-exp(-y)]
k 2
if crossing
of the membrane
again. The solution
of equation
(10)
(12)
1
takes place, and by equation
(13)
ex~(-‘,‘)I
(13)
[Bl = [~srop] + [Ao,j][exP(-‘15) if crossing of the membrane does not take place. Here, 5 is the time which, when substituted in equation (12) gives the value [B] = [Bsrop].
Membrane without pores If the membrane is not a porous septum, component B must dissolve in it or must open some channels in order to pass through. If the membrane is permeable to the solvent, component B dissolves together with its solvation shell. Every surface has some weak points in its structure, at which dissolution or opening of a channel may occur. The solution pushes against the walls of the membrane to open it, as the free enthalpy difference would impose crossing. Curving of the surface in the failing points allows the membrane to oppose a pressure pour
where u is the surface tension and r is the radius of the curve. As r decreases until the curve becomes a hemisphere, the membrane can oppose crossing by the increasingly concentrated B. If the radius of the hemisphere is larger than the thickness of the membrane, a channel is formed. The solution passes through as in a pore and the value of the concentration of B is governed by equations (12) and (13) where [B,] and [Bsrop] are obtained by the condition of dynamic equilibrium of the membrane. If the membrane has a thickness larger than the radius of the hemisphere, channels do not form. When curving goes beyond a hemisphere, however, the radius increases again, the pressure on the membrane-side becomes lower than that on the solution-side and dynamic equilibrium cannot exist. Rapid dissolution, involving some molecules of B and some solvent, occurs and leads to a decrease in the concentration of B, which is more marked at larger values of the surface/volume ratios, as it is in a sphere with a very small radius. An analogous phenomenon takes place at the membrane/solvent interface, though with another value of [B] i.e. [Blc2] above which crossing of B starts. If the solvent is renewed continuously, the system will be governed by the mathematical equations
= khl
exp(-ht) - k2%([Bso,l - [Bi.J~~
05) (16) (17)
509
with the boundary conditions [B,,,] = [B,,,,]
= [B,,,,] = 0
W
at t = 0
in membrane = k,([B].,,in membrane = k, [Bi,d] R,
[Bi,,])R,
(19) (20)
Equation (15) represents the mass-transfer balance of B in the solution, R, is a coefficient with two values, 0 when crossing does not take place and 1 when crossing does take place, S, is a surface reduced in order to account for the fact that mass transfer does not involve the entire geometrical surface of the membrane and [Bi,,] is the concentration of B (at the interface solution/membrane, solution side) equilibrating the concentration of B at the same interface, membrane side, [Bi.,]. Equation (16) represents the mass-transfer balance of B inside the membrane. Equation (17) is a relationship, analogous to Henry’s law, of equilibrium between the two sides of the interface where H symbolizes a simple proportionality constant. In equation (18), [B,,,] symbolizes the concentration of B in the solution contacting the membrane, [Bmemb] the concentration of B within the membrane and [Bdiss] the concentration of B in the initially B-free solvent contacting the other side of the membrane. Equation (19) expresses the equality of the fluxes on the two sides of the membrane/solution interface. Equation (20) expresses the equality of the fluxes on the two sides of the membrane/solvent interface. R, is a coefficient with two values, 0 when crossing does not take place and 1 when crossing does take place. [Bi d] is the concentration of B in the initially B-free solvent at the surface that equilibrates the concentration of B in the membrane at the membrane surface. As [Bi .I and PmemJ increase at every cycle, before B crosses into the pure solvent, we have a progressive flattening’of the oscillations, due to an increase in the minimum. Yet pulsating crossing of B into the solvent restores the oscillations. Membrane
with carrier
If component B is insoluble in the membrane, a carrier is necessary for crossing to occur. The carrier cannot go out of the membrane and interaction between component B and the carrier takes place only at the interface. As the carrier cannot go out of the membrane, no free enthalpy difference drives it towards the interface and we have to admit that statistical fluctuations around an average value give some carrier molecules a free enthalpy difference larger than the work necessary to reach the interface. Statistical fluctuations cannot generate oscillations, which depend only on component B. Component B must reach the interface in order to react and this implies work against the surface tension of the solution. As the maximum area in the interface occurs at the breakage of the surface, when curving is hemispherical, the work per mole of B is expressed by equation (21) w = 21rr&N,
(21)
510
where r, is the radius of B and NA is Avogadro’s number. For crossing to start, the free enthalpy difference, associated with an increase in the concentration of B in solution, must become larger than the work necessary for B to reach the interface. The physico-chemical system is governed by the following mathematical equations:
d[Bl dt
i
)
= kl[Aol exd-kd - k&4PlR3
ouerall,sol
(22)
in which M symbolizes the carrier, and for which [B]=Oand[M]=[M,] D -=a*[MB] ’ D
(23)
a[MB]
(24)
at
ax*
a*[Ml a[Mi -=* ax* at
(25)
at t=O
[MB]=• _D
att=O
ww
1 ax
=D
(26)
a[Mi= kMM~3
at x=0
(27)
m-u= k_,[MB]R,
at x=1
(28)
2- ax
while _D
m-1
1 ax
=D
-
* ax
Equation (22) represents the mass-transfer balance in solution, where k, is the rate constant of reaction between component B and carrier M, R, is a coefficient with two values, 0 and 1, which gives significance to the last term on the right hand side, when crossing occurs. Equation (24) represents the mass-transfer balance of the complex BM (component B + carrier) in the membrane. Equation (25) represents the mass-transfer balance of carrier in the membrane. Equations (26), (27) and (28) are boundary conditions, where R, is a coefficient with two values, 0 and 1, to account for pulsating crossing from membrane to pure solvent. As generation of B must be slow to generate oscillations, we can consider the concentration of carrier to be in excess and simplify the system as given by the following equation (29). =k,[A,]
oueroll.sol
exp(-k,t)
- k,a[B]R,
where (Y stands for a dimensional coefficient required to make equation (29) dimensionally correct. Oscillations generated by these mechanisms, are very small in amplitude. Yet, if cxis large, as it is for example in a sphere with a very small radius, the amplitude relative to the average value can also be large. As mass transfer involves only the external layer, the oscillations have a sawtooth shape.
511 CONCLUSIONS
The above models are able to explain some types of oscillation generation and evidence the role of some resistances in mass transfer in order to account for oscillations. It is clear that higher resistances lead to a higher amplitude of oscillations. REFERENCES 1 F. D’Alba and S. Di Lorenm, J. Chem. Sot., Faraday Trans. 1, 79 (1983) 38.