CENTRAL RESEARCH INSTITUTE FOR PHYSICSBUDAPEST

T U - л г - Г . 1 Ы .

K F K I - 1 9 8 0 - 1 2 0

SZ, V A S S

C O N N E C T I O N B E T W E E N C H E M I C A L R A T E C O E F F I C I E N T S

A N D T W O - P A R T I C L E C O R R E L A T I O N F U N C T I O N S IN A G G R E G A T E D S Y S T E M S

Hungarian Academy o f Sciences

CENTRAL RESEARCH

INSTITUTE FOR PHYSICS

BUDAPEST

2017

KFKI-1980-120

CONNECTION BETWEEN CHEMICAL RATE COEFFICIENTS

AND TWO-PARTICLE CORRELATION FUNCTIONS IN AGGREGATED SYSTEMS

Sz. Vass

Central Research Institute for Physics H-1525 Budapest 114, P.O.B. 49, Hungary

Presented at the meeting on Chemistry of Miaroemulsions"

Cambridge, Sept. 14-16, 1980

'Physioa l held in

HU ISSN 0368 5330 ISBN 963 371 766 3

A B S T R A C T

The forward direction rate coefficient X of irreversible bimolecular reaction A+B + C is expressed in terms of the average velocities and of the spatial correlation of reactants. The anisotropic conditions supposed in microaggregates result in a formula differing from that in isotropic liquids This difference ensures a conceptual possibility to explain the deviation of rate coefficients observed in microaggregates from those observed in conven­

tional solutions.

АННОТАЦИЯ

Постоянная скорости X необратимой бимолекулярной химической реакции типа А+В ■+ С выражена с помощью терминов средней скорости и пространственной корреляции реагентов. Анизотропные условия, предположенные в микроагрегате, приводят к формуле, отличающейся от выражения, полученного для изотропных жидкостей. Это отличие дает принципиальную возможность интерпретации разности постоянных скорости, наблюдаемых в микроагрегате и конвенциональных растворах.

K I V O N A T

Az А+В + С tipusu irreverzibilis bimolekuláris kémiai reakció X sebes­

ségi együtthatóját a reagensek átlagos sebességével és térbeli korrelációjá­

val hoztuk kapcsolatba. A mikroaggregátumban feltételezett anizotrop körül­

mények olyan formulát eredményeznek, amely különbözik az izotróp folyadékok­

ra kapott kifejezéstől. Ez az eltérés elvi lehetőséget ad arra, hogy segít­

ségével a mikroaggregátumokban, illetve a konvencionális oldatokban megfi­

gyelt sebességi együtthatók különbségét értelmezzük.

1. I N T R O D U C T I O N

The present paper deals with some aspects of chemical rate coefficients in the liquid phase in the case when reactants are incorporated into more or less stable and closed aggregates of surface active molecules. The term ag­

gregate covers here a wide range of theoretically different formations from micelles to microemulsions.

An ever increasing attention has been paid in the last decade to inves­

tigations of rate coefficients and reaction mechanisms in the presence of microaggregates [1-21]. The exact treatment of these reactions is very diffi­

cult for two reasons. First, the processes involved in aggregate formation [22-35] and in particle incorporation [36-39] are extremly complicated.

Second, the systems under consideration are so small [24,29-32] that the use of concepts of classical statistical physics elaborated for large systems

(e.g. diffusion [15,18,19]) can be questioned. Thus, it seems that in the case of microaggregates the classical concepts need a careful reformulation

[40-41].

Experimental evidence [13,14] suggests that random collision models [3, 13,20,21,42] are adequate to describe chemical kinetics in these systems.

Our aim is to express the coefficients of random collision models using the concepts of liquid physics and to discuss some features of the coefficients.

2. G E N E R A L C O N S I D E R A T I O N S

In order to describe observable phenomena due to chemical reactions in microaggregates, the first step is to determine the processes occuring in individual microaggregates. The chemical reaction is taken to be a result of reactive collisions between chaotically moving particles and the rate coef­

ficients correspond to collision densities [42,43] which are calculated from the average number of reactive collisions per unit time.

In order to simplify the very complicated calculations [44] the follow­

ing assumptions are made:

a) The processes are restricted to irreversible bimolecular reactions of type Л + В ^ C .

2

b) Reactant molecules of type A and В are taken to be hard-core spheres with radii гд and rß , respectively. The interaction between two reactant mol­

ecules depends only on their distance from each other, described e.g.

either by hard-core or by Lennard-Jones potentials and the spatial dis­

tribution of molecules is not affected by their internal degrees of free­

dom.

c) Colliding molecules combine a reaction product molecule with probability P r which is supposed to be independent of the collision probability.

d) The system is in equilibrium; this state is not perturbed by chemical reactions.

The Hamiltonian has the form:

H(q1 - • • Чад ' P ]_ • • • Рад»Q ]_ • • • Оад'P 1 ‘ ‘ ' PN ^

2 2

M p N P .

= У — i— + У — 3— + Í 2mi j 2mj + V(q^. . 'Чад/О^.’ "

(2.1)

where (q,p) are space coordinates and momenta of the reactants, (Q,P) are generalized space coordinates and momenta of all other molecules. The interaction potential is divided into three parts as

M N

v(g1 ...gM ,Q1 ...QN ) =

I

I I

I

l^iCjsJM 1 D l^iCj^N

) (2.2)

The first term of the potential function describes the interaction be­

tween reactants (see condition b.), the second term stands for the interac­

tion between the reactants and the other (among them the aggregate) molecules.

The last term takes into account the effect of the structure of the liquid.

Note that M and N are the numbers of reactant and of other molecules, respectively, the quantities q,p,Q,P are vectors and the integration with respect to dq,dp,dQ,dP denotes a threefold integration.

3. A V E R A G E N U M B E R O F C O L L I S I O N S P E R U N I T T I M E

In order to evaluate the average number of collisions per unit time, one has first to determine the geometrical condition of collision for a chosen pair of reactants. Let us introduce the concept of collision boundary

for particles with convex boundaries i,j as follows: let the particle j of type В move around the fixed particle i of type A in such a way that their relative orientation remains unchanged and that their surfaces remain in contact. The collision boundary is defined as the set of points repre-

3

senting all possible geometrical positions which can be taken by the centre of mass point of particle j .

In the simple case of hard-cQre spheres of radii гд and rß the collision boundary is a spherical surface of radius r = r + r (which is well known

A о

from the textbooks of statistical physics and physical chemistry; see Fig. 1/a).

Fig. l/а. Definition-of the collision boundary for hard-core spheres

Let = (Pj~P^)At be the relative displacement vector of particles i,j during At and let us define the volume element Дсск^ by the integral:

Дм

j " J

grad S

(pj~p i ) I grad S ±j dS (3.1)

°ij

where n is the unit normal vector of S

ij in the surface element dS and the integration has to be carried out over that region of the collision boundary where we have for the inner product the inequality (nAq^)<0.

If and only if the centre of mass point of particle j falls into the volume element Д м ^ , will particle j impinge on particle i during At, see Fig. 1/b. This event is represented by the characteristic function

•ij = fij (At,qi ,q.,pi ,pj )

1, if q . 6 Дм . i3

(3.2) 0, otherwise

4

Fig. 1/b. Determination of the volume element by the collision boundary and by the relative displacement

of moving hard-core spheres

Since a collision of two particles is a mutual event, the characteristic function of their collision ^ has to be a symmetric function in the indi­

ces; this condition is satisfied if it is expressed as the arithmetic mean of f. . and f .. in the form

V. . = •= (f. .+f ., ) iD 2 i] ji'

1, if а.еДсо,. and q.6Aw..

4 3 i} ]i

<

0, otherwise

(3.3)

The total number of collisions between reactants of type A and В observed in At is a random function of the reactant coordinates and momenta and it is given by the summation of the characteristic functions of two-particle col­

lisions as

W (At) = Ф (At,q

1* •qM'P l ‘ ■Pm5

n a n b

l l i j

^ij <A t *qi '4j 'P-i/Pj) + O(At)

(3.4)

5

N. and N are the numbers of the A and В type reactants involved,

A В

NA +NB = M ' describes the number of ternary, quarternary etc. colli­

sions. The average number of collisions u(At) is given as u(At) = T(At, q 1 ...qM ,p1 . ■ P ^ d W ^ .

,pM ,Qi- •QM ,P1* •PM> (3.5) where dW is the statistical weight of finding the reactants around coordinates and momenta q ^ ...qM>p ^ ...pM and the other particles around gen­

eralized coordinates and generalized momenta . .PN « For equilibrium systems the statistical weight dW is given by the Gibbs distribution function

dW = Z exp

2 m , + V (q

1* ,qM ,Qr

• V ,

In Eq. (3.6) Z is the system integral, к is the Boltzmann-constant, T is the absolute temperature and dr is the volume element of the phase space.

Substituting this formula into Eq. (3.5) and integrating with respect the variables Q^...QN , P^...PN which have no direct effect on the integrand, Eq. (3.5) transforms to

U (At) C V ■M

W i A t ^ . . . q ^ p ^ . .pM ) exp

. M p 2 -i- У 1 kT V 2nu

(3.7) x FM (qr

•qM )dpr •dPMdqr .dq M

where C is a constant, V is the volume of the system, FM (q^...qM ) is the M-particle correlation function derived from the interaction part of the Hamiltonian in Eq. (2.2) (for details see Refs. [45-47]) in the form

FM (q1*•*4MЧм> = -Vм.exp

" ^ <Di ( iq r q j i )

exp

■ /М N \

l$i<j<N

dQ-^. . .dQ,

N (3.8)

=VM®Q (q

1* •qM )"exp - 5 ^ 1 ф 1 (lq i'q j I) ' l^i<j^M

Since Ф (At) is a sum of characteristic functions depending on two- particle coordinates and FM (q^...qM ) is a symmetrical function of its argu­

ments, u(At) can be expressed by an integral of the two-particle correlation function F 2 (q^,q2), (see Refs. [45-47]) as

6

U (At) = CV-M

N N А В

х dp^. . .dpj^dq^. . . dq

n a n b

= CV м l l

J

1 ^Mг

2 kT l

Z 2m p

M

1 ^Mг P *

kT 1 l 2ra*

x (3.9) x dpx ..•dpM dq1 ...dqM

- cv"2 “a % ’W A t 'qA'q B'PA'PB )F2 (qA'qB ) eXp

1 ÍPA

2 \ h рв 2kT Га mB l\

x dpA dpB dqA dqB

In the last line of Eq. (3.9) the subscripts A and В denote coordinates of different type reactants. The two-particle correlation function F2 <q^,q2) can be reduced to the product of a one-particle function F^(q^) and a condi­

tional two-particle function F2 (q2 |q^) [45-47] of the form

F 2 (q2'q l ) = F 1 (q!)F2 (q2lq !) (3.10)

Substituting Eq. (3.10) into Eq. (3.9) and taking into consideration that the characteristic function Тдв is nonzero only in the region A«AB where it is equal to 1, the final form to be discussed in the following is obtained as

U(At) = C ^{n a n b} V

F l (qA ) V d<i,

- / 2 ²

qA )dqB eXp 1

~2kT b + к

PB m B

*a b (At' qA ' PA'PB >

dpAdpB

(3.11)

4. R A T E C O E F F I C I E N T S IN D I F F E R E N T M E D I A

Until this point, our considerations apply to arbitrary gas and con­

densed phase systems. In Eq. (3.8) function <t>Q contains the boundary informa­

tion for reactants: in gases and liquids, where no aggregates are present, 4>q

is non-zero for the entire reaction vessel and V is equal to its volume. In this case F^ is constant and in isotropic medium F 2 (q2 |q^^ ds a function of the coordinate differences [45-47], thus

F ^ q ^ = 1

F 2 (q2 I q i > = 1<32-д1 I 1 = g(q)

(4.1)

where g(q) is the so called pair-correlation function. The integration in Eq. (3.11) in this case reduces to a simple integration over the colli-

7

sión boundary, using spherical polar-coordinates on the collision boundary SAB the inner integral in Eq. (3.11) leads to the simple expression

2n n/2

Лео

F 2 (qBlqA )dqB = A t IPABI AB

2 2

v g(r)sind cosd dd dep = r ng(r)

p A B iAt

and g(r) is the value of the pair-correlation function at the collision boundary . Hence, integrating with respect to and over the momentum space, and finally, taking into account that the average number of reactive collisions in At corresponds to the increase N r = XN^NgAt in reaction product molecules, from comparision of this formula and of the result of the integra­

tion the rate coefficient X is given as X = r Tig.(r)

V *r lpAB'exp

/ 2 ^2\

1 r A + рв 2kT im. m J

\A ^B/J

dpA dpB

■ s -sP i Pr c 5a b<t>

(4.3)

Some features of bimolecular rate coefficients are apparent from E q . (4.3).

The rate coefficient X is inversely proportional to the volume of the reac­

tion vessel and it shows Maxwellian-type temperature-dependence in Cp.D (T).

2 AB

The factor r ng(r) shows the effect of reactant size and of solution struc­

ture, pr is defined in Section 2, in condition c. If the pair-correlation function is expressed in terms of particle density (i.e. as a virial series,

[45]), in the case of hard-core potentials for very dilute systems Eq. (4.3) reproduces the classical expression for gas phase rate coefficients [48].

In the case of aggregate formation, the function in Eq. (3.8) is sup­

posed to differ from zero in the finite volume of the aggregate VA ; the basic assumptions listed in Section 2 are completed with this condition. The evaluation of Eq. (3.11) is not a simple task and therefore we restrict ourselves to its qualitative discussion.

According to theoretical considerations and to experimental evidence [49, 50] an aggregate is characterized by rapidly varying forces in the neigh­

bourhood of its boundary; because of its small size the boundary effect can­

not be neglected and thus it cannot be considered isotropic [45-47,49,50]. As a consequence of this fact, F^ is not a constant function inside the microag­

gregate and the conditional two-particle correlation function cannot be re­

placed by the pair-correlation function g(r). Instead of g(r) the form

F 2 = F 2 (г,9,ф,0о,ф о Iq ^ ) has to be used which generally depends on the orienta­

tion of q^ and the relative momentum рдв (see Fig. 2). The inner integral in Eq. (3.11) can be written as

2n n/2

F 2 (qB lqA )dqB = 1рАв1Д Ь | F 2 (r,0,cp,9o ,cpo |qA )r 'cosö sinö dö dtp (4.4) AcoAB

8

and, in order to express the rate coefficient it has to be averaged over the possible orientations of дд and рдв as

A g V

Ag '>rC' W T >

г1<Ча>

A g

2n u/2 d q A X

(4.5)

f

J

Fig. 2. Spherical polar coordinate system connected to the relative momentum vector of the colliding hard-core

spheres for the calculation of rate coefficients

The explicit temperature dependence of the rate coefficient is of the same form, as in the case of isotropic solutions (cf. Eq. (4.3)). The rate coefficient is inversely proportional to the aggregate volume ; if the

9

size distribution of microaggregates is not so narrow as it is usually sup­

posed, its effect on the reaction kinetics cannot be neglected.

The integrals in Eq. (4.5) define an average spatial correlation, which can strongly depend on the type of aggregating molecules and of reactants.

If reactants are found in the boundary region of the microaggregate, the ef­

fect of the fields on the reaction probability p^ has to be taken into ac­

count. These effects - which I cannot discuss in detail at the present level of my knowledge - can result in a further deviation of from the rate coefficient determined in isotropic solutions.

A C K N O W L E D G E M E N T

The author is indebted to Dr. L.Bata, Dr. G.Jancsó and to Dr. R.Schiller from the Central Research Institute for Physics, and to Dr. G. Putir'skaya from the Central Research Institute for Chemistry of the Hungarian Academy of Sciences for valuable discussions.

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[14] G. Rothenberger, P. Infelta and M. Grätzel, J. Phys. Chem. (УЗ, 1871 (1979) .

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Plenum Press, New York, 1977.

[26] K.S. Birdi, i b i d ., v o l . 1, p. 151.

[27] H.F. Eicke, i b i d ., v o l . 1, p. 429.

[28] A .S . Kertes , ibid., v o l . If P . 445 [29] P. Mukerjee , ibid., v o l . If P . 171 [30] E. Ruckenstein, ibid. vol. 2, p. 755.

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Benjamin, New York, 1963.

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[41] L.R. Fisher and D.G. Oakenfull, Chem. S o c . Rev. 6, 25 (1977).

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[44] Sz. Vass, to be published.

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