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Influence of an external field on the stochastic theory of polymerization
Physica
116A (1982) 543-559 North-Holland
Publishing Co.
INFLUENCE OF AN EXTERNAL FIELD ON THE STOCHASTIC THEORY OF POLYMERIZATION M. MOREAU and L. VICENTE Laboraroire
de Physique
ThCorique des Liquides*, Uniuersiti Pierre et Marie Curie, 4, Place Jussieu, 75230 Paris Cedex 0.5, France
Received 4 May 1982
A stochastic model is proposed to take into account the effect of a weak external field associated to diffusion, a discrete Master Equation is written down for the spatial distribution of particles. Approximate solutions are found for the first two cumulants, and applied to an example of polymerization reactions.
1. Introduction Fluctuations in both equilibrium and non-equilibrium chemical systems have been recognized to be. of considerable interestlm3). These fluctuations may be treated within the frame of a stochastic description. In the multivariate Master Equation formalism&‘) the system is divided into cells between which the molecules may diffuse; the random variables are then the number of particles of each species in each cell. Such a discrete description has been discussed elsewherelV8). Our purpose is to extend the description to take into account the effect of an external field especially in the case of reactions of polymerisation. In a previous paper? we have developed a general treatment for a model of kinetics of polymerization, by means of the WKB approximation and the method of factorial cumulants: the equations for the first two cumulants were derived and the asymptotic behaviour of the correlations between the number of molecules was deduced. It was shown that the relaxation time of the second factorial cumulant is half the macroscopic relaxation time, which is a general result of this kind of approximation; furthermore the system exhibits a globally non-poissonian and locally poissonian behaviour. We turn our attention now to consider the effect of a field on the kinetics of relaxation. We will restrict to considering an external field, for instance the gravitational field, which causes the phenomenon of sedimentation, or an external electrical field acting on a polyelectrolyte solution, when the internal fluctuating field may be neglected. In section 2 we briefly recall the * Equipe Associte au C.N.R.S. 0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
544
M. MOREAU AND L. VICENTE
mechanism of reaction and diffusion and the equations for the first two cumulants. The effect of an external field is introduced in section 3 as a perturbation of the mechanism of diffusion; after giving the corresponding Master Equation we find the asymptotic solutions in section 4. The discrete representation of the external field used here is discussed in Appendix A. 2. Reaction-diffusion
model
In ref. 9 we consider a system of chemical reactions
of the type
xp+ xqpP4xp+q,p-q=1,2,...m or generally
any system with r elementary
Tv;x,*Tfi;x,,
a=l,2
)...
reactions
r,
(Y: (1’)
where X, denotes both the p-mer and the number of p-mer molecules; the v are the stoichiometric coefficients. Following Nicolis and Prigogine’) we suppose that in one dimension diffusion is modeled by the discrete mechanism:
xp,i %xp,i,
(2)
DP
where the system is divided into cells of length I ; Xp,i is the number of molecules p in cell i, and i’ denotes the cells adjacent to i (if= i _+1). Dp is a diffusion constant related to Fick’s constant d, by’) Dt, = d,112.
(3)
In this model, considering each chemical reaction and each diffusion exchange process between two adjacent cells as a birth and death process, the equations for the first two cumulants. near macroscopic equilibrium, may be written4S9SI’): $(G-CP)=@+u,*(Cr-G3,
(44
&E;=@+Q.G,
(4.b)
where
(5)
9 = (Gp,i), p,q=l,... G=(Gp,i;q,i),
m;i,j=l,
n.
(5’)
STOCHASTIC
G,,i is the concentration
THEORY
OF POLYMERIZATION
54s
of species p in cell i, and GO,.i its equilibrium value;
G,. i; q,j = j (~X~,iS&,j) - G,i$&j
(6)
is the nonpoissonian correlation between (p, i) and (q, j)“); 8 = (RlfGj) is the linearized chemical operator; linearizing around any value Goof G yields, with the general system (1’):
ma and Z” being the forward and backward rates for G = Go; here, naturally, we take for Go the uniform diffusion operator:
equilibrium
distribution
b = (D,S;A(), A{ = 6j+, - 2Sj + Sj-,,
Gi,i = 6,.
fi is the
(8) except for: At = Ai = 1.
(8’)
In these notations we denote a tensor (T$)blyJ;:. n,,,by ‘: the number of signs up corresponds to the number of pairs of indices (p, i) in these positions. Furthermore if I = (SZS!;) is the Kronecker tensor. k=Ri+iE I --
_-
(9)
Q=fii+iD. ---
(9’)
and similarly
Here Ri is the tensorial product of the tensors R and i. A dot (v) denotes the interior-product, consisting in a sum on repeated alternated indices; two dots (:) are used for a double interior product, etc . . . . If all species have the same diffusion coefficient D, the diffusion operator is @ = DA = D(G;Aj),
(10)
where A{ is given by (9’). We shall adopt this condition in what follows. If (V&=,.. m is a right eigenvector of eigenvalue 77of the reaction operator (R$p,4=1.. ,,,, it is shown in ref. 9 that (VP,),,,. ,n is a left eigenvector of this operator, with the same eigenvalue, if VP, ‘x V,“lG,,
(11)
GP being the equilibrium concentration of species p. We assume that the eigenvectors are normalized in the sense that X, VP,V;’ = 8:‘. Clearly (V,P a p) and (V”, = ~6,) are the left and right (unit) eigenvectors of (R,4). On the other hand (W?)i,l, ,.n denotes a right eigenvector of eigenvalue A of
M. MOREAU
546
AND L. VICENTE
the diffusion operator (Di) = D(Ai), and (Wk) the corresponding left eigenvector. It is known that the possible eigenvalues and normalized eigenvector are A=&=-20
(
krr l-COST)
)
Wi = W? = (2n))U2[eij(kn/n)+ .h
k = 0, 1, . . . n - 1, e-iW)(kdn)
I
-
I>
(12)
(12’)
where i = d-1. Then4) U”,” = (Vg Wf) is a right eigenvector of eigenvalue q + A of E- + D and the tensorial product UU’ of such two eigenvectors is a right eigenvector of g + p, the eigenvalue of which is the sum of the eigenvalues of U and LJ’. Then spectral theory permits to write the solutions of eq. (4a) and(4b) G-Go=
2 C,,,v” V+h#o
e’V+A”,
(13) (13’)
The constants C have been studied, and these results discussed in ref. 9. It should be noticed that G and G are not independent in a closed system because of the chemical conservition laws. For the polymerization reactions (1) the total number N = X,, i pXp, i of “monomer units” is conserved; we may suppose that it is exactly determined, so that its fluctuation 6N is 0, and eq. (6) implies: 2 PGP.i;q,j = - qG,j. P.1
(14)
3. Action of an ext.ernal field We consider that the action of the external field may be modelled by a modification of the diff usional mechanism, the chemical kinetics being unchanged; of course this is only an approximation but it may be justified if the field is weak. Thus we consider that the external field causes a directional preference for the motion of particles, which may be accounted for by introducing forward and backward diffusion coefficients D’ and D-, generally depending on the species p and on the cell i according to the scheme:
STOCHASTIC
THEORY
OF POLYMERIZATION
547
This discrete representation of the motion of particles under the field influence is discussed in appendix A and it gives a reasonable stochastic representation; we shall see that it agrees with the usual continuous macroscopic description. Here I&, is the diffusion coefficient introduced in section 2 by eq. (3). F,,i is the force acting on a molecule p in cell i; in the case of gravitation, it is given by: Fp, i = Fp = mp(l-
pvp),
(16)
the mass of the molecules of species p, its specific volume, and the total density of mass. In the case of a uniform electrical field E, if the internal field is neglected, we have q,,
up and p being, respectively,
Fp,i = Fp = zplq(E,
(17)
z, being the (algebraic) number of charges of an ion p, and q the charge of the electron. Finally the coefficient I’, may be deduced from microscopic considerations as done in appendix A; it is given by r, = d,/(2lkT).
(18)
From now on we shall write P, = P, since we assume d, to be independent
of
P. Then the probability
distribution
satisfies a multivariate
Master Equation
which reads: -$P(XJ)=R+zD,
(
P +
cxp,,
+
(X,,,+l)P(x,,,+l,x,,,-
ww,,.-,
-
1) - X,, 1P
1, X,,” + 1) - X,,“P
n-l +
?! I=
CXp,i
+
lNptxp,i
1,
Xp,i+i
-
1)
1)]-2~~Xp,iP}
+p(xp,i-l-l~Xp,i+
+ 2 r,
+
Fp. dCXp,I+ Op(X,, 1
, + 1, x,,, - 1) - x,, ,P]
+~~,“rcx,..+l)P~xp,“-,-l,x~,rl+I)-xp,”P] n-1 +
1= 7
-p(xp,i-I-
Fp,i(Xp,i
+
l)[P(Xp,i
l,Xp,i+
+
I)]
, I
1, Xp,i+l- 1) (19)
where R represents the (unchanged) chemical contribution. In the right hand side of eq. (19) we have only written those arguments of P which differ from (X,).
548
M. MOREAU AND L. VICENTE
Then, following the methods of ref. 9 we obtain the (non-linearized) equations for the first two cumulants in the case of a uniform field:
$G=K(G)+D.G+FG -----
(20) (20’)
where K is the non-linearized r=(T,F,G;B&
y= Z r?+ --
chemical operator;
furthermore
?r -I
(21)
with B{ = S{_, - 6j+,, except for: B! = - 1, Bi = 1;
(21’)
6 and e are defined by eqs. (9’) and (10) of section 2, and 5 by eqs. (9) and (7) where Go - = G. - Furthermore: (22)
E = (EM) with, using the notations
of eq. (7):
In order to pass to the continuum limit, we notice that the coordinate of cell i along the axe of the one dimensional system is z = il. When the length I of each cell tends to zero, eq. (20) yields the usual continuous macroscopic equation for the concentration G,(z) of species p at point z:
& G&,
t) = K,({G,(z)H+
4($ Gp- -$(5 G,))
(23)
with the no-flux limit condition: $GP-/$GP)=O
forz=Oandz=l,
which confirms our discrete stochastic
(23’)
description.
4. The solutions
4.1. First cumulant
(macroscopic
concentrations)
We study the case of weak external fields: more precisely, we suppose that the effect of the external field is small compared to diffusion. Thus we may consider this effect as a perturbation’of the reaction and diffusion terms, and
STOCHASTIC
THEORY
549
OF POLYMERIZATION
solve the evolution equations by successive for G as
approximations.
We write eq. (20)
(24) E being a formal perturbation parameter which is equated results. Then we expand G in powers of E:
to 1 in the final
G=G’+eG’+..., I-
(25)
Go being the uniform equilibrium GOp,j=Gp
@cl,...
state in the absence of field:
m;j=l,...
n).
(26)
Inserting (25) into (24), we linearize the exact chemical operator K around Go and obtain the equations of the successive perturbations by equating the terms on the same power in (E). We find for the first order perturbation: $Gl=(E+Q).
cl+C.GO
(27)
with, according to the previous notations: ii = (R;;i)= (%(Ip)).
(28)
4.1
If Q’ is the stationary
state of G’, we have
-&(G’-F’)=(“+@).(G’-e). Comparing with eq. (4), we see when there is no external field, times. From eq. (27) we find the null-vector o. of R + 6 is also notations of section 2: e = -
that G’- c’ satisfies the same equation as G and thus tends to 0 with the same relaxation
2 (q + h)-‘~A(OI,, q+h+O
or explicitly, B,\
(29)
= -r
stationary state 8’; noticing that the left a left null-vector of F, we obtain; with the
* ‘.
$>,
(30)
using eq. (12), (12’) and (21’) and taking 3
v:(z
i = fl:
Va,FqGq) 4
(17 + hk)-1(2n)-l[eij(k”n)
X
+ e -iU-Mkmln)l[
1 + e-i(hr/n)l
k3 d It
is
seen from (30’) that only odd values of k, corresponding
.
(30’)
to polar modes,
550
M. MOREAU
AND L. VICENTE
contribute to c’, as it should be expected. We obtain a more physical result by taking the continuous limit I + 0, with jl = z:
This expression agrees with the corresponding results of the macroscopic theory. In particular, let us consider the total concentration of monomer units: n(z)=CpG,(z)=n’+~n’(z)+...,
(32)
P
where no = X pep is the uniform equilibrium concentration in the absence of field. Since the left null-vector of the local chemical operator (V,P m p) is orthogonal to the non-null right eigenvectors of the same operator, the first order stationary perturbation n’ of n is, by (31):
(33) which is easily obtained from the macroscopic case where Fq is proportional to q:
equation (23). In the particular
Fq = qF
(34)
(which is reasonable if, for instance, Fq is the gravitational sion (33) of Gi is greatly simplified since
force), the expres-
so that
w> =;gq nl(z)=pBp& (z-i). II
(35)
Indeed the macroscopic equations, with the limit conditions (23), admit this stationary solution where local chemical equilibrium is realized everywhere (which may only be realized in this particular case, as shown in appendix B). 4.2. Second cumulant In the perturbation
treatment
adopted above, eq. (20’) becomes (36)
STOCHASTIC
THEORY OF POLYMERIZATION
We suppose again that G stays near the stationary without external field and we write:
551
value q of the problem
G=$++‘+***.
(37)
By substituting (37) into (36) and by denoting the different as co, I 8’ I we find: &=
@+4:
stationary
GO+g(:(Gq
values
(38)
~I;‘=(~+8:c’+wiCrlGO+E’.G’~:GO,
(39)
where (40) The physical relation (14) between G and G must be true for the solutions of the evolution equations (38) and (39) that satisfy (14) at the initial time. The same property holds for the approximations of all orders. Thus eq. (14) is verified at order 0 and 1, provided that it is verified at time 0, which may be realized by taking the following initial conditions: GO,,i(O)= GO,,i(t)= dp; Gi,i(O)= Gp,i(O)-&,;
(41)
Gi, i;,,j(O) = G,,i; q.j(O)+ GA i(O)&&j; GL,i; q,i(O) = - G~,i(o)&q&~
(41’)
Because gg = 0, eq. (38) is identical to eq. (4’) for g in the absence of field. Thus Go tends to its stationary value p = v v with the relaxation constants oP + wILobtained by adding two eigenvaluei i of g- + 0. 4.2.1. Stationary
solution
The evolution of 9’ is given by the non-homogeneous (39), which may be written:
differential
equation
(42) with C=ii+Q
(43)
&f(t) = g i G’@@“(t) + @’. G’(t) + r. g’(t).
(43’)
The stationary
soltuion p satisfies:
%: Cj’+ V(m) = 0,
(4)
M. MOREAU AND L. VICENTE
552
we know that L admits an unique left null-vector - zT,= LJ,LJ, and the compatibility zr, : Ap)
condition (14) imposes:
= 0.
(45)
This relation is verified, because the conservation 60,: 4=
relations implies
&: E= r&r=0
and consequently &:?ll = 0. The stationary 6’ = -
(46)
solution F may be written
2 (Cl++ wJ’[ 0$” or+o,#o
: M(q
yy,
(47)
where o,(p = 0, 1, . . . ) is the eigenvalue of the eigenvector r of “+ @, with - Denoting both the species index (p) and cell index (i) o=wo>o~>w*>” by one symbol a = (p, i), the component of c’ on U” _ U” - is (with the Einsteinsummation on repeated indices): b~,UaUb=-(WP+W”)-lUaUb~ c lr y
Y
(48)
ab
with
(49) (R’)zd =
the explicit relatior?):
&
R:(G
= $,
expressions
(E’):b =
&- E,,(G
=
3;
(50)
c
being easily derived
from (7) and (22’). From the
Q= ~o(~> a IpLjO, it results that if Y and p differ from 0, then u;u;n;i,,
= u;u”,Ec,b@.
(5 1)
From relations (22’) and (50) we find the component the polymerization reactions (1):
- k,,(&~-j,
+ 8,cL>>
(cl),,
for of 0, V# 0.
of 6’ on yy
for
(52)
STOCHASTIC
THEORY
OF POLYMERIZATION
553
If I_L # 0 and v = 0, eq. (14) implies (C’)Q=-7
U~U~G~=-~
WlWbVP,ViGi,i.
(52’)
Here again only the perturbation of the macroscopic stationary state participates in the result. The last bracket in the right hand side of eq. (52) represents the deviation from local chemical equilibrium. It vanishes in the particular case considered previously, where F4 = qF; then we have: Gi,b = -(U,“G$“,+
u,“G:u”,>.
(53)
The continuum limit is easily obtained for eq. (52) and (52’). In the particular case of eq. (53) the perturbation g&,, ZJ to the correlation function between species p and q at coordinates zl and z2 is 1 pqM .&7(Zl~
z2)
=
-Lx
f_
p:&;
z,+z*
L
2
kT
2
(
(53’)
7 )
P
which adds to the uniform correlation -0
g P.4
--
function in the absence of field?:
1 pq&&
= L
l$
4.2.2. Asymptotic
P2dp’ time dependent
We can now examinate
the relaxation
G’(t) - Q’ = x VT
of g’(t),
given by:
e (°F+Uv)*[(G’)p, LO) - (@), .I
P.v
f
+~~U” P.Y
solution
-
I
dse (w,++s)[M& “(S) - li;r,, “1
(54)
0
with (G’),.(t)
- (@),”
= o&
: [~‘W
(55)
- 6’1,
(56)
M/L,.(t) - fi&L,Y= &C:[~(t)-@I, M(t)
and i@ = M(w) being given by (43’) and (49).
Making use of the asymptotic expressions obtain an expression of the type:
of G’(t) - c’ and c’(t) - p
we
(57)
554
M. MOREAU AND L. VICENTE
where the coefficients R, E, r are constants. Let us analyse the terms of eq. (54). (i) The first term on the right hand side gives the proper evolution of G’(t). Its relaxation time is [o’[~‘, (w’( being the smallest (in absolute value) non-null eigenvector of @+ u; thus it does not depend on the external field. The corresponding term in G’(t) is: (U’v”+ tiU’)[(G’)‘,,(O) - (d’),J e”” --_ = (U’@+ v”U’) c U;lU,“[Gi(O) - GL] e”l’; -I-
(58)
(ii) The second term ii the induced evolution of G’(t), and, after (57), we see that resonances can be produced since some relaxation frequencies of M&,“(S)- fii, Y are proper relaxation frequencies of c’(t) as given by the terms (i) above. However it may be shown that there is no resonance for the smallest relaxation frequency lo,\ (see appendix C). Thus the term which dominates the evolution of G’(s) - 6’ is 0~e”“, whereas Go(s) - p relaxes with the frequency 21wll. Thus it is reasonable, if t B (wll-‘, to assume that co(t) has reached its stationary value; at this degree of approximation, the influence of the external field on the correlations disappears. Thus the method used to incorporate an external field in the stochastic model of a reactive system, and a very simple perturbation technique have permitted to analyse the effect of the field on the macroscopic concentrations, in agreement with the usual determinist equations and on the correlations, with the result that the external field only changes the final stationary state, but not the relaxation constants in a first approximation. The extension of the method to internal fields, depending on the molecular concentrations, is in progress; it should permit to treat the case of reactions in electrolytes which have been studied recently by Turq and coworkers’3) from a macroscopic point of view.
Appendix A Cell model in the presence of a field Following the usual method’V3.‘4)a one-dimensional system is divided into cells of length 1, the cell i containing Ni particles. We want to calculate the flux +i of particles passing from cell i to cell i + 1 under the influence of diffusion and of a weak uniform field E. We suppose that a particle moving with a velocity v suffers elastic collisions at regular intervals of time. The interval between two collisions 7 is velocity dependent; in a first approximation we suppose that the mean free path A = VT
STOCHASTIC
is independent
of velocity,
THEORY
555
OF POLYMERIZATION
because:
no being the number of molecules of solvant per volume unit and u the collision cross section. We assume that after every collision the velocity distribution cp is nearly maxwellian; between two collisions the particle evolves according to classical mechanics; thus if r and 2, are the position and velocity after collision at time t, just before the next collision its position is r+u7+2m
-L FT2,
(A. 1)
where F’s the force exerted on the particles by the field E. The condition for a particle to pass from cell i to cell i + 1 between times t and t + 7 is zi -
(
V,T
+ G
1
Fzr 2 =
Zi -
a C
Z C
)
Zi,
(A.3
zi being the coordinate of the cells separation along the main dimension of the system, z the coordinate of the particle at time t, v, and F, the corresponding components of v and F. Thus, the number of particles with velocity v satisfying condition (A.2) is: N(u) = cp(v) I dP i dz n(z), (A.3) (Z) *i-a where (2) is the boundary surface between the cells and n(z) the concamition at coordinate z. If we take cells to be cubes of side 1, eq. (A.3) becomes: N(V) s q(u)12an
(
zi -;
(A.4)
>
At this point, we notice that if I is significantly larger than a, the term n(zi -(a/2)) may notably differ from the mean concentration ni of cell i. It is necessary to know the gradient of n in the neighbourhood of the wall; thus the model of uniform cells is not physically significative and does not yield the real flux in the continuous limit, unless we correct it as will be shown later. If 1 is itself of the order of the mean free path A, we can replace (A.4) by: N(n) = q(o)12an
Zi -i (
1: >
q(v)l*Uni;
(A.3
thus, the number of particles with velocity v passing from cell i to cell i + 1 per time unit is:
64.6)
556
The
M. MOREAU flux
4i
AND L. VICENTE
from i to i + 1 is then:
rp($(u, +$+).
d_v
64.7)
u,>-(F,/Zm)r
If we replace the condition v, > - F,7/2m by V, > 0, the error may be neglected and we obtain from (A.7):
L4.8) with c =
(2 1I’?,
(A.9)
k being Boltzmann’s constant and T the temperature, assumed to be uniform. Similarly, the flux of particles from cell i + 1 to cell i is: (A. 10) and thus the total flux between cells is wi =
oji - qi = c - Ni+lI- Ni [
Fzh Ni + Ni+l +kTI 2
and finally we find that the total flux between cells is: Ni_l-
2Ni + Ni+l
l2
_-L-
1
(A. 11)
cell i and the neighbourhood
1
F A Ni+, - Ni_1
kT1
21
’
(A. 12)
Clearly this equation is also satisfied if the numbers of particles Ni are replaced by the particle-densities ni. If A = I (which is the condition adopted by Nicolis and Prigogine in’), eq. (A.12) must reproduce the physical phenomenon; indeed, the right hand side of (A.12) can be approximated by second and first derivatives, and we obtain the classical equation I
(A. 13)
7
where (A.14) is Fick’s diffusion coefficient. Eq. (A.8) shows that the particle
transport
between
cells i and i + 1 is
STOCHASTIC
THEORY
OF POLYMERIZATION
557
similar to the chemical reaction of first order with reaction constant (A. 15)
ki = D + I’FZ with
J-=C=d=)I~ 2kT
2kTA
(A. 16)
2kT
in agreement with the formulae utilised in section 3. Now if 1> A, eq. (A.ll) does not yield correct results. In (A.ll) the term corresponding to the field is 1/2(ni + Iti+,), which is a good approximation to the density of particles on the separation wall between the cells, or of [n(zi - (a/2)) + n(zi + (a/2))]/2; this term is thus significative. But on the other hand the diffusion term .contains the difference ni+l - ni whereas it should contain the quantity
Clearly this error can be corrected
by replacing eq. (A.8) by (A. 17)
with CA
D’=7=p;
d
1 d P=j-@z.
(A. 18)
as it is done in section 3.
Appendix B Local chemical equilibrium
in presence
of a uniform field
Let us assume that the stationary concentrations (G,(z)) satisfy local chemical equilibrium. Then, with the notations of sections 3 and 4, the chemical operator KP vanishes: Kp(O = 0
(B.1)
and G(z) is completely determined, for all z, by the concentration of monomer units-n(z) = Z, pG,,(z). Since G is also a stationary solution of eq. (23), we
558
M. MOREAU AND L. VICENTE
have aG,__CG az kT
=o p
Differentiating
03.2)
’
eq. (B. 1) with respect to n yields
a&JaGL@Lo aG, an
(B.3)
p an
SO that (ag,/an) is proportional
to the unique right null vector of CR;). Since
we obtain: aGP
PG
-
an
(B.4)
7 4’6’
from which Gp 0: (WP. Now eq. (B.2) may be written
aG,?!!_F,G an
az
kT
=()
p
and implies, by (B.4): an
az-
--.
lXq2Gq 5 kT
p’
which may only be true for all p if F = F’,/p is independent G,(z) = G,(O) exp($)
CC[Gd~)l~.
of p. In this case (B.3
The approximate result given by eq. (35) agrees with this formula up to the first order in FlIkT.
Appendix C
On the evolution of G’(t) The evolution of g’(t) is given by eq. (54). In the second term of the right hand side, resonances may occur between the proper relaxation frequencies o, + CO,and the relaxation frequencies of M,,(s) - fiPy given by eq. (57),
STOCHASTIC
559
THEORY OF POLYMERIZATION
leading to terms of type t e (“~+WJf. But such resonances do not appear for the smallest (in absolute value) relaxation frequency 10~1,which is obtained for p = 1, v = 0 or the contrary. Indeed, from (42) we have: &T
U’“Gk = T (R + D)‘,(z
UO”G:,) + T U,&f,,.
(C.1)
Using eqs. (14) (27) and (42) we obtain: x U”“(M,,(t) - fi& a
= - x (R + D);;(U’” - Uob)(G:(t) - Gf), C
(C.2)
from which M,,(s) - fiP0 = - c U;(R + D):(U; ,C
- U,b)(Gl.(s) - G:).
But the term in ewVSof G’(s) - G’ is proportional M,,(s) - tifio is proportional to c.C
Ui(R + D);(u;
- u,b)u;
= (0, - w,)
c
to U”; its contribution
(C.3) to
u;u;;u,b,
which vanishes for p = Y. Thus there is no resonance
at the frequencies
jw,\.
References 1) G. Nicolis and I. Prigogine, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
Self-Organization in Nonequilibrium Systems (Wiley Interscience, N.Y., 1977). I. Oppenheim, K.E. Shuler and G.H. Weiss, Stochastic Processes in Chemical Physics: The Master Equation (MIT Press, Cambridge, MA, 1977). N.G. Van Kampen, Physics Letters 59A (1976) 33. H. Lemarchand, Doctorat es-Sciences, Universite Paris VI (1977). H. Lemarchand and G. Nicolis, Physica 82A (1976) 521. G. Nicolis, J. Stat. Phys. 6 (1972) 195. M. Malek-Mansour and G. Nicolis, J. Stat. Phys. 13 (1975) 197. G. Nicolis, M. Malek-Mansour, K. Kitahara et A. Van Nypelseer, Phys. Letters 48A (1974) 217. M. Moreau and L. Vicente, On the Stochastic Theory of Polymerization Reactions, Physica 1llA (1982) 139. L. Vicente, Doctorat de 3tme Cycle, Universite Paris VI (1981). G. Nicolis and I. Prigogine, Proc. Nat. Acad. Sci. (USA) 68 (1971) 2102. C.W. Gardiner, K.J. McNeil, O.F. Walls and I.S. Matheson, J. Stat. Phys. 14 (1976) 307. P. Turq, L. Orcil, J. Barthel, M. Chemla, Ber. Bunsenges. Phys. Chem. 85 (1981) 535. G. Nicolis, P. Allen and A. Van Nypelseer, Prog. Theo. Phys. 52 (1974) 481.
116A (1982) 543-559 North-Holland
Publishing Co.
INFLUENCE OF AN EXTERNAL FIELD ON THE STOCHASTIC THEORY OF POLYMERIZATION M. MOREAU and L. VICENTE Laboraroire
de Physique
ThCorique des Liquides*, Uniuersiti Pierre et Marie Curie, 4, Place Jussieu, 75230 Paris Cedex 0.5, France
Received 4 May 1982
A stochastic model is proposed to take into account the effect of a weak external field associated to diffusion, a discrete Master Equation is written down for the spatial distribution of particles. Approximate solutions are found for the first two cumulants, and applied to an example of polymerization reactions.
1. Introduction Fluctuations in both equilibrium and non-equilibrium chemical systems have been recognized to be. of considerable interestlm3). These fluctuations may be treated within the frame of a stochastic description. In the multivariate Master Equation formalism&‘) the system is divided into cells between which the molecules may diffuse; the random variables are then the number of particles of each species in each cell. Such a discrete description has been discussed elsewherelV8). Our purpose is to extend the description to take into account the effect of an external field especially in the case of reactions of polymerisation. In a previous paper? we have developed a general treatment for a model of kinetics of polymerization, by means of the WKB approximation and the method of factorial cumulants: the equations for the first two cumulants were derived and the asymptotic behaviour of the correlations between the number of molecules was deduced. It was shown that the relaxation time of the second factorial cumulant is half the macroscopic relaxation time, which is a general result of this kind of approximation; furthermore the system exhibits a globally non-poissonian and locally poissonian behaviour. We turn our attention now to consider the effect of a field on the kinetics of relaxation. We will restrict to considering an external field, for instance the gravitational field, which causes the phenomenon of sedimentation, or an external electrical field acting on a polyelectrolyte solution, when the internal fluctuating field may be neglected. In section 2 we briefly recall the * Equipe Associte au C.N.R.S. 0378-4371/82/0000-0000/$02.75
@ 1982 North-Holland
544
M. MOREAU AND L. VICENTE
mechanism of reaction and diffusion and the equations for the first two cumulants. The effect of an external field is introduced in section 3 as a perturbation of the mechanism of diffusion; after giving the corresponding Master Equation we find the asymptotic solutions in section 4. The discrete representation of the external field used here is discussed in Appendix A. 2. Reaction-diffusion
model
In ref. 9 we consider a system of chemical reactions
of the type
xp+ xqpP4xp+q,p-q=1,2,...m or generally
any system with r elementary
Tv;x,*Tfi;x,,
a=l,2
)...
reactions
r,
(Y: (1’)
where X, denotes both the p-mer and the number of p-mer molecules; the v are the stoichiometric coefficients. Following Nicolis and Prigogine’) we suppose that in one dimension diffusion is modeled by the discrete mechanism:
xp,i %xp,i,
(2)
DP
where the system is divided into cells of length I ; Xp,i is the number of molecules p in cell i, and i’ denotes the cells adjacent to i (if= i _+1). Dp is a diffusion constant related to Fick’s constant d, by’) Dt, = d,112.
(3)
In this model, considering each chemical reaction and each diffusion exchange process between two adjacent cells as a birth and death process, the equations for the first two cumulants. near macroscopic equilibrium, may be written4S9SI’): $(G-CP)=@+u,*(Cr-G3,
(44
&E;=@+Q.G,
(4.b)
where
(5)
9 = (Gp,i), p,q=l,... G=(Gp,i;q,i),
m;i,j=l,
n.
(5’)
STOCHASTIC
G,,i is the concentration
THEORY
OF POLYMERIZATION
54s
of species p in cell i, and GO,.i its equilibrium value;
G,. i; q,j = j (~X~,iS&,j) - G,i$&j
(6)
is the nonpoissonian correlation between (p, i) and (q, j)“); 8 = (RlfGj) is the linearized chemical operator; linearizing around any value Goof G yields, with the general system (1’):
ma and Z” being the forward and backward rates for G = Go; here, naturally, we take for Go the uniform diffusion operator:
equilibrium
distribution
b = (D,S;A(), A{ = 6j+, - 2Sj + Sj-,,
Gi,i = 6,.
fi is the
(8) except for: At = Ai = 1.
(8’)
In these notations we denote a tensor (T$)blyJ;:. n,,,by ‘: the number of signs up corresponds to the number of pairs of indices (p, i) in these positions. Furthermore if I = (SZS!;) is the Kronecker tensor. k=Ri+iE I --
_-
(9)
Q=fii+iD. ---
(9’)
and similarly
Here Ri is the tensorial product of the tensors R and i. A dot (v) denotes the interior-product, consisting in a sum on repeated alternated indices; two dots (:) are used for a double interior product, etc . . . . If all species have the same diffusion coefficient D, the diffusion operator is @ = DA = D(G;Aj),
(10)
where A{ is given by (9’). We shall adopt this condition in what follows. If (V&=,.. m is a right eigenvector of eigenvalue 77of the reaction operator (R$p,4=1.. ,,,, it is shown in ref. 9 that (VP,),,,. ,n is a left eigenvector of this operator, with the same eigenvalue, if VP, ‘x V,“lG,,
(11)
GP being the equilibrium concentration of species p. We assume that the eigenvectors are normalized in the sense that X, VP,V;’ = 8:‘. Clearly (V,P a p) and (V”, = ~6,) are the left and right (unit) eigenvectors of (R,4). On the other hand (W?)i,l, ,.n denotes a right eigenvector of eigenvalue A of
M. MOREAU
546
AND L. VICENTE
the diffusion operator (Di) = D(Ai), and (Wk) the corresponding left eigenvector. It is known that the possible eigenvalues and normalized eigenvector are A=&=-20
(
krr l-COST)
)
Wi = W? = (2n))U2[eij(kn/n)+ .h
k = 0, 1, . . . n - 1, e-iW)(kdn)
I
-
I>
(12)
(12’)
where i = d-1. Then4) U”,” = (Vg Wf) is a right eigenvector of eigenvalue q + A of E- + D and the tensorial product UU’ of such two eigenvectors is a right eigenvector of g + p, the eigenvalue of which is the sum of the eigenvalues of U and LJ’. Then spectral theory permits to write the solutions of eq. (4a) and(4b) G-Go=
2 C,,,v” V+h#o
e’V+A”,
(13) (13’)
The constants C have been studied, and these results discussed in ref. 9. It should be noticed that G and G are not independent in a closed system because of the chemical conservition laws. For the polymerization reactions (1) the total number N = X,, i pXp, i of “monomer units” is conserved; we may suppose that it is exactly determined, so that its fluctuation 6N is 0, and eq. (6) implies: 2 PGP.i;q,j = - qG,j. P.1
(14)
3. Action of an ext.ernal field We consider that the action of the external field may be modelled by a modification of the diff usional mechanism, the chemical kinetics being unchanged; of course this is only an approximation but it may be justified if the field is weak. Thus we consider that the external field causes a directional preference for the motion of particles, which may be accounted for by introducing forward and backward diffusion coefficients D’ and D-, generally depending on the species p and on the cell i according to the scheme:
STOCHASTIC
THEORY
OF POLYMERIZATION
547
This discrete representation of the motion of particles under the field influence is discussed in appendix A and it gives a reasonable stochastic representation; we shall see that it agrees with the usual continuous macroscopic description. Here I&, is the diffusion coefficient introduced in section 2 by eq. (3). F,,i is the force acting on a molecule p in cell i; in the case of gravitation, it is given by: Fp, i = Fp = mp(l-
pvp),
(16)
the mass of the molecules of species p, its specific volume, and the total density of mass. In the case of a uniform electrical field E, if the internal field is neglected, we have q,,
up and p being, respectively,
Fp,i = Fp = zplq(E,
(17)
z, being the (algebraic) number of charges of an ion p, and q the charge of the electron. Finally the coefficient I’, may be deduced from microscopic considerations as done in appendix A; it is given by r, = d,/(2lkT).
(18)
From now on we shall write P, = P, since we assume d, to be independent
of
P. Then the probability
distribution
satisfies a multivariate
Master Equation
which reads: -$P(XJ)=R+zD,
(
P +
cxp,,
+
(X,,,+l)P(x,,,+l,x,,,-
ww,,.-,
-
1) - X,, 1P
1, X,,” + 1) - X,,“P
n-l +
?! I=
CXp,i
+
lNptxp,i
1,
Xp,i+i
-
1)
1)]-2~~Xp,iP}
+p(xp,i-l-l~Xp,i+
+ 2 r,
+
Fp. dCXp,I+ Op(X,, 1
, + 1, x,,, - 1) - x,, ,P]
+~~,“rcx,..+l)P~xp,“-,-l,x~,rl+I)-xp,”P] n-1 +
1= 7
-p(xp,i-I-
Fp,i(Xp,i
+
l)[P(Xp,i
l,Xp,i+
+
I)]
, I
1, Xp,i+l- 1) (19)
where R represents the (unchanged) chemical contribution. In the right hand side of eq. (19) we have only written those arguments of P which differ from (X,).
548
M. MOREAU AND L. VICENTE
Then, following the methods of ref. 9 we obtain the (non-linearized) equations for the first two cumulants in the case of a uniform field:
$G=K(G)+D.G+FG -----
(20) (20’)
where K is the non-linearized r=(T,F,G;B&
y= Z r?+ --
chemical operator;
furthermore
?r -I
(21)
with B{ = S{_, - 6j+,, except for: B! = - 1, Bi = 1;
(21’)
6 and e are defined by eqs. (9’) and (10) of section 2, and 5 by eqs. (9) and (7) where Go - = G. - Furthermore: (22)
E = (EM) with, using the notations
of eq. (7):
In order to pass to the continuum limit, we notice that the coordinate of cell i along the axe of the one dimensional system is z = il. When the length I of each cell tends to zero, eq. (20) yields the usual continuous macroscopic equation for the concentration G,(z) of species p at point z:
& G&,
t) = K,({G,(z)H+
4($ Gp- -$(5 G,))
(23)
with the no-flux limit condition: $GP-/$GP)=O
forz=Oandz=l,
which confirms our discrete stochastic
(23’)
description.
4. The solutions
4.1. First cumulant
(macroscopic
concentrations)
We study the case of weak external fields: more precisely, we suppose that the effect of the external field is small compared to diffusion. Thus we may consider this effect as a perturbation’of the reaction and diffusion terms, and
STOCHASTIC
THEORY
549
OF POLYMERIZATION
solve the evolution equations by successive for G as
approximations.
We write eq. (20)
(24) E being a formal perturbation parameter which is equated results. Then we expand G in powers of E:
to 1 in the final
G=G’+eG’+..., I-
(25)
Go being the uniform equilibrium GOp,j=Gp
@cl,...
state in the absence of field:
m;j=l,...
n).
(26)
Inserting (25) into (24), we linearize the exact chemical operator K around Go and obtain the equations of the successive perturbations by equating the terms on the same power in (E). We find for the first order perturbation: $Gl=(E+Q).
cl+C.GO
(27)
with, according to the previous notations: ii = (R;;i)= (%(Ip)).
(28)
4.1
If Q’ is the stationary
state of G’, we have
-&(G’-F’)=(“+@).(G’-e). Comparing with eq. (4), we see when there is no external field, times. From eq. (27) we find the null-vector o. of R + 6 is also notations of section 2: e = -
that G’- c’ satisfies the same equation as G and thus tends to 0 with the same relaxation
2 (q + h)-‘~A(OI,, q+h+O
or explicitly, B,\
(29)
= -r
stationary state 8’; noticing that the left a left null-vector of F, we obtain; with the
* ‘.
$>,
(30)
using eq. (12), (12’) and (21’) and taking 3
v:(z
i = fl:
Va,FqGq) 4
(17 + hk)-1(2n)-l[eij(k”n)
X
+ e -iU-Mkmln)l[
1 + e-i(hr/n)l
k3 d It
is
seen from (30’) that only odd values of k, corresponding
.
(30’)
to polar modes,
550
M. MOREAU
AND L. VICENTE
contribute to c’, as it should be expected. We obtain a more physical result by taking the continuous limit I + 0, with jl = z:
This expression agrees with the corresponding results of the macroscopic theory. In particular, let us consider the total concentration of monomer units: n(z)=CpG,(z)=n’+~n’(z)+...,
(32)
P
where no = X pep is the uniform equilibrium concentration in the absence of field. Since the left null-vector of the local chemical operator (V,P m p) is orthogonal to the non-null right eigenvectors of the same operator, the first order stationary perturbation n’ of n is, by (31):
(33) which is easily obtained from the macroscopic case where Fq is proportional to q:
equation (23). In the particular
Fq = qF
(34)
(which is reasonable if, for instance, Fq is the gravitational sion (33) of Gi is greatly simplified since
force), the expres-
so that
w> =;gq nl(z)=pBp& (z-i). II
(35)
Indeed the macroscopic equations, with the limit conditions (23), admit this stationary solution where local chemical equilibrium is realized everywhere (which may only be realized in this particular case, as shown in appendix B). 4.2. Second cumulant In the perturbation
treatment
adopted above, eq. (20’) becomes (36)
STOCHASTIC
THEORY OF POLYMERIZATION
We suppose again that G stays near the stationary without external field and we write:
551
value q of the problem
G=$++‘+***.
(37)
By substituting (37) into (36) and by denoting the different as co, I 8’ I we find: &=
@+4:
stationary
GO+g(:(Gq
values
(38)
~I;‘=(~+8:c’+wiCrlGO+E’.G’~:GO,
(39)
where (40) The physical relation (14) between G and G must be true for the solutions of the evolution equations (38) and (39) that satisfy (14) at the initial time. The same property holds for the approximations of all orders. Thus eq. (14) is verified at order 0 and 1, provided that it is verified at time 0, which may be realized by taking the following initial conditions: GO,,i(O)= GO,,i(t)= dp; Gi,i(O)= Gp,i(O)-&,;
(41)
Gi, i;,,j(O) = G,,i; q.j(O)+ GA i(O)&&j; GL,i; q,i(O) = - G~,i(o)&q&~
(41’)
Because gg = 0, eq. (38) is identical to eq. (4’) for g in the absence of field. Thus Go tends to its stationary value p = v v with the relaxation constants oP + wILobtained by adding two eigenvaluei i of g- + 0. 4.2.1. Stationary
solution
The evolution of 9’ is given by the non-homogeneous (39), which may be written:
differential
equation
(42) with C=ii+Q
(43)
&f(t) = g i G’@@“(t) + @’. G’(t) + r. g’(t).
(43’)
The stationary
soltuion p satisfies:
%: Cj’+ V(m) = 0,
(4)
M. MOREAU AND L. VICENTE
552
we know that L admits an unique left null-vector - zT,= LJ,LJ, and the compatibility zr, : Ap)
condition (14) imposes:
= 0.
(45)
This relation is verified, because the conservation 60,: 4=
relations implies
&: E= r&r=0
and consequently &:?ll
(46)
solution F may be written
2 (Cl++ wJ’[ 0$” or+o,#o
: M(q
yy,
(47)
where o,(p = 0, 1, . . . ) is the eigenvalue of the eigenvector r of “+ @, with - Denoting both the species index (p) and cell index (i) o=wo>o~>w*>” by one symbol a = (p, i), the component of c’ on U” _ U” - is (with the Einsteinsummation on repeated indices): b~,UaUb=-(WP+W”)-lUaUb~ c lr y
Y
(48)
ab
with
(49) (R’)zd =
the explicit relatior?):
&
R:(G
= $,
expressions
(E’):b =
&- E,,(G
=
3;
(50)
c
being easily derived
from (7) and (22’). From the
Q= ~o(~> a IpLjO, it results that if Y and p differ from 0, then u;u;n;i,,
= u;u”,Ec,b@.
(5 1)
From relations (22’) and (50) we find the component the polymerization reactions (1):
- k,,(&~-j,
+ 8,cL>>
(cl),,
for of 0, V# 0.
of 6’ on yy
for
(52)
STOCHASTIC
THEORY
OF POLYMERIZATION
553
If I_L # 0 and v = 0, eq. (14) implies (C’)Q=-7
U~U~G~=-~
WlWbVP,ViGi,i.
(52’)
Here again only the perturbation of the macroscopic stationary state participates in the result. The last bracket in the right hand side of eq. (52) represents the deviation from local chemical equilibrium. It vanishes in the particular case considered previously, where F4 = qF; then we have: Gi,b = -(U,“G$“,+
u,“G:u”,>.
(53)
The continuum limit is easily obtained for eq. (52) and (52’). In the particular case of eq. (53) the perturbation g&,, ZJ to the correlation function between species p and q at coordinates zl and z2 is 1 pqM .&7(Zl~
z2)
=
-Lx
f_
p:&;
z,+z*
L
2
kT
2
(
(53’)
7 )
P
which adds to the uniform correlation -0
g P.4
--
function in the absence of field?:
1 pq&&
= L
l$
4.2.2. Asymptotic
P2dp’ time dependent
We can now examinate
the relaxation
G’(t) - Q’ = x VT
of g’(t),
given by:
e (°F+Uv)*[(G’)p, LO) - (@), .I
P.v
f
+~~U” P.Y
solution
-
I
dse (w,++s)[M& “(S) - li;r,, “1
(54)
0
with (G’),.(t)
- (@),”
= o&
: [~‘W
(55)
- 6’1,
(56)
M/L,.(t) - fi&L,Y= &C:[~(t)-@I, M(t)
and i@ = M(w) being given by (43’) and (49).
Making use of the asymptotic expressions obtain an expression of the type:
of G’(t) - c’ and c’(t) - p
we
(57)
554
M. MOREAU AND L. VICENTE
where the coefficients R, E, r are constants. Let us analyse the terms of eq. (54). (i) The first term on the right hand side gives the proper evolution of G’(t). Its relaxation time is [o’[~‘, (w’( being the smallest (in absolute value) non-null eigenvector of @+ u; thus it does not depend on the external field. The corresponding term in G’(t) is: (U’v”+ tiU’)[(G’)‘,,(O) - (d’),J e”” --_ = (U’@+ v”U’) c U;lU,“[Gi(O) - GL] e”l’; -I-
(58)
(ii) The second term ii the induced evolution of G’(t), and, after (57), we see that resonances can be produced since some relaxation frequencies of M&,“(S)- fii, Y are proper relaxation frequencies of c’(t) as given by the terms (i) above. However it may be shown that there is no resonance for the smallest relaxation frequency lo,\ (see appendix C). Thus the term which dominates the evolution of G’(s) - 6’ is 0~e”“, whereas Go(s) - p relaxes with the frequency 21wll. Thus it is reasonable, if t B (wll-‘, to assume that co(t) has reached its stationary value; at this degree of approximation, the influence of the external field on the correlations disappears. Thus the method used to incorporate an external field in the stochastic model of a reactive system, and a very simple perturbation technique have permitted to analyse the effect of the field on the macroscopic concentrations, in agreement with the usual determinist equations and on the correlations, with the result that the external field only changes the final stationary state, but not the relaxation constants in a first approximation. The extension of the method to internal fields, depending on the molecular concentrations, is in progress; it should permit to treat the case of reactions in electrolytes which have been studied recently by Turq and coworkers’3) from a macroscopic point of view.
Appendix A Cell model in the presence of a field Following the usual method’V3.‘4)a one-dimensional system is divided into cells of length 1, the cell i containing Ni particles. We want to calculate the flux +i of particles passing from cell i to cell i + 1 under the influence of diffusion and of a weak uniform field E. We suppose that a particle moving with a velocity v suffers elastic collisions at regular intervals of time. The interval between two collisions 7 is velocity dependent; in a first approximation we suppose that the mean free path A = VT
STOCHASTIC
is independent
of velocity,
THEORY
555
OF POLYMERIZATION
because:
no being the number of molecules of solvant per volume unit and u the collision cross section. We assume that after every collision the velocity distribution cp is nearly maxwellian; between two collisions the particle evolves according to classical mechanics; thus if r and 2, are the position and velocity after collision at time t, just before the next collision its position is r+u7+2m
-L FT2,
(A. 1)
where F’s the force exerted on the particles by the field E. The condition for a particle to pass from cell i to cell i + 1 between times t and t + 7 is zi -
(
V,T
+ G
1
Fzr 2 =
Zi -
a C
Z C
)
Zi,
(A.3
zi being the coordinate of the cells separation along the main dimension of the system, z the coordinate of the particle at time t, v, and F, the corresponding components of v and F. Thus, the number of particles with velocity v satisfying condition (A.2) is: N(u) = cp(v) I dP i dz n(z), (A.3) (Z) *i-a where (2) is the boundary surface between the cells and n(z) the concamition at coordinate z. If we take cells to be cubes of side 1, eq. (A.3) becomes: N(V) s q(u)12an
(
zi -;
(A.4)
>
At this point, we notice that if I is significantly larger than a, the term n(zi -(a/2)) may notably differ from the mean concentration ni of cell i. It is necessary to know the gradient of n in the neighbourhood of the wall; thus the model of uniform cells is not physically significative and does not yield the real flux in the continuous limit, unless we correct it as will be shown later. If 1 is itself of the order of the mean free path A, we can replace (A.4) by: N(n) = q(o)12an
Zi -i (
1: >
q(v)l*Uni;
(A.3
thus, the number of particles with velocity v passing from cell i to cell i + 1 per time unit is:
64.6)
556
The
M. MOREAU flux
4i
AND L. VICENTE
from i to i + 1 is then:
rp($(u, +$+).
d_v
64.7)
u,>-(F,/Zm)r
If we replace the condition v, > - F,7/2m by V, > 0, the error may be neglected and we obtain from (A.7):
L4.8) with c =
(2 1I’?,
(A.9)
k being Boltzmann’s constant and T the temperature, assumed to be uniform. Similarly, the flux of particles from cell i + 1 to cell i is: (A. 10) and thus the total flux between cells is wi =
oji - qi = c - Ni+lI- Ni [
Fzh Ni + Ni+l +kTI 2
and finally we find that the total flux between cells is: Ni_l-
2Ni + Ni+l
l2
_-L-
1
(A. 11)
cell i and the neighbourhood
1
F A Ni+, - Ni_1
kT1
21
’
(A. 12)
Clearly this equation is also satisfied if the numbers of particles Ni are replaced by the particle-densities ni. If A = I (which is the condition adopted by Nicolis and Prigogine in’), eq. (A.12) must reproduce the physical phenomenon; indeed, the right hand side of (A.12) can be approximated by second and first derivatives, and we obtain the classical equation I
(A. 13)
7
where (A.14) is Fick’s diffusion coefficient. Eq. (A.8) shows that the particle
transport
between
cells i and i + 1 is
STOCHASTIC
THEORY
OF POLYMERIZATION
557
similar to the chemical reaction of first order with reaction constant (A. 15)
ki = D + I’FZ with
J-=C=d=)I~ 2kT
2kTA
(A. 16)
2kT
in agreement with the formulae utilised in section 3. Now if 1> A, eq. (A.ll) does not yield correct results. In (A.ll) the term corresponding to the field is 1/2(ni + Iti+,), which is a good approximation to the density of particles on the separation wall between the cells, or of [n(zi - (a/2)) + n(zi + (a/2))]/2; this term is thus significative. But on the other hand the diffusion term .contains the difference ni+l - ni whereas it should contain the quantity
Clearly this error can be corrected
by replacing eq. (A.8) by (A. 17)
with CA
D’=7=p;
d
1 d P=j-@z.
(A. 18)
as it is done in section 3.
Appendix B Local chemical equilibrium
in presence
of a uniform field
Let us assume that the stationary concentrations (G,(z)) satisfy local chemical equilibrium. Then, with the notations of sections 3 and 4, the chemical operator KP vanishes: Kp(O = 0
(B.1)
and G(z) is completely determined, for all z, by the concentration of monomer units-n(z) = Z, pG,,(z). Since G is also a stationary solution of eq. (23), we
558
M. MOREAU AND L. VICENTE
have aG,__CG az kT
=o p
Differentiating
03.2)
’
eq. (B. 1) with respect to n yields
a&JaGL@Lo aG, an
(B.3)
p an
SO that (ag,/an) is proportional
to the unique right null vector of CR;). Since
we obtain: aGP
PG
-
an
(B.4)
7 4’6’
from which Gp 0: (WP. Now eq. (B.2) may be written
aG,?!!_F,G an
az
kT
=()
p
and implies, by (B.4): an
az-
--.
lXq2Gq 5 kT
p’
which may only be true for all p if F = F’,/p is independent G,(z) = G,(O) exp($)
CC[Gd~)l~.
of p. In this case (B.3
The approximate result given by eq. (35) agrees with this formula up to the first order in FlIkT.
Appendix C
On the evolution of G’(t) The evolution of g’(t) is given by eq. (54). In the second term of the right hand side, resonances may occur between the proper relaxation frequencies o, + CO,and the relaxation frequencies of M,,(s) - fiPy given by eq. (57),
STOCHASTIC
559
THEORY OF POLYMERIZATION
leading to terms of type t e (“~+WJf. But such resonances do not appear for the smallest (in absolute value) relaxation frequency 10~1,which is obtained for p = 1, v = 0 or the contrary. Indeed, from (42) we have: &T
U’“Gk = T (R + D)‘,(z
UO”G:,) + T U,&f,,.
(C.1)
Using eqs. (14) (27) and (42) we obtain: x U”“(M,,(t) - fi& a
= - x (R + D);;(U’” - Uob)(G:(t) - Gf), C
(C.2)
from which M,,(s) - fiP0 = - c U;(R + D):(U; ,C
- U,b)(Gl.(s) - G:).
But the term in ewVSof G’(s) - G’ is proportional M,,(s) - tifio is proportional to c.C
Ui(R + D);(u;
- u,b)u;
= (0, - w,)
c
to U”; its contribution
(C.3) to
u;u;;u,b,
which vanishes for p = Y. Thus there is no resonance
at the frequencies
jw,\.
References 1) G. Nicolis and I. Prigogine, 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14)
Self-Organization in Nonequilibrium Systems (Wiley Interscience, N.Y., 1977). I. Oppenheim, K.E. Shuler and G.H. Weiss, Stochastic Processes in Chemical Physics: The Master Equation (MIT Press, Cambridge, MA, 1977). N.G. Van Kampen, Physics Letters 59A (1976) 33. H. Lemarchand, Doctorat es-Sciences, Universite Paris VI (1977). H. Lemarchand and G. Nicolis, Physica 82A (1976) 521. G. Nicolis, J. Stat. Phys. 6 (1972) 195. M. Malek-Mansour and G. Nicolis, J. Stat. Phys. 13 (1975) 197. G. Nicolis, M. Malek-Mansour, K. Kitahara et A. Van Nypelseer, Phys. Letters 48A (1974) 217. M. Moreau and L. Vicente, On the Stochastic Theory of Polymerization Reactions, Physica 1llA (1982) 139. L. Vicente, Doctorat de 3tme Cycle, Universite Paris VI (1981). G. Nicolis and I. Prigogine, Proc. Nat. Acad. Sci. (USA) 68 (1971) 2102. C.W. Gardiner, K.J. McNeil, O.F. Walls and I.S. Matheson, J. Stat. Phys. 14 (1976) 307. P. Turq, L. Orcil, J. Barthel, M. Chemla, Ber. Bunsenges. Phys. Chem. 85 (1981) 535. G. Nicolis, P. Allen and A. Van Nypelseer, Prog. Theo. Phys. 52 (1974) 481.