PHYSICAL REVIEW E 89, 032121 (2014)
Composition distributions of particles in a gelling mixture A. A. Lushnikov* Geophysical Center of Russian Academy of Science, 3, Molodezhnaya Street, 119296 Moscow, Russia and Karpov Institute of Physical Chemistry, 10 Vorontsovo Pole, 105064 Moscow, Russia (Received 14 January 2014; published 19 March 2014) Gelation in a two component disperse system wherein binary coagulation governs the temporal changes of particle composition spectra is studied for the crossproduct coagulation kernel proportional to m1 n2 + m2 n1 , with m,n being the numbers of monomers of the first and the second component in the coalescing pair of particles. This model reveals the sol-gel transition, i.e., after a finite interval of time the conservation of the total particle mass concentration violates because of the formation of giant particles (the gel). This paper reports on the exact solution of this model for arbitrary initial particle composition spectra. Exact expressions for the particle composition spectrum, the gel mass, and the second moments of the composition distribution are derived. Two scenarios of gelation, where the gel is either active or passive, are considered. DOI: 10.1103/PhysRevE.89.032121
PACS number(s): 02.50.−r, 05.90.+m, 68.03.Fg
I. INTRODUCTION
Binary aggregation processes of the type (m1 ,n1 ) + (m2 ,n2 ) −→ (m1 + m2 ,n1 + n2 ),
(1)
where the notation m,n stands for an (m + n)-mer comprising m and n monomers of the first and the second component respectively, are often met in nature. The subscripts 1 and 2 mark the two coalescing objects. The role of coagulation processes is commonly recognized and has been discussed in detail in the books and the review articles [1–10]. Still very little attention was given to coagulating mixtures. So far this problem remained clear of the mainstream of works on coagulation, although its importance remains beyond any doubt. The most evident example is coagulation of a binary mixture, where a gas of M + N monomers of two sorts begins to form clusters containing m and n monomers of each sort. This type of coagulation had been considered almost four decades ago in Ref. [11] and then in Refs. [12–23]. Another important example is the evolution of a random bipartite graph [24–32] and emergence of a giant component in it. I return to this problem in connection with my recent results in the theory of coagulation-gelation process in monocomponent coagulating systems [33]. It occurs that the extension of the results of my work [33] to a nontrivial and very instructive example of a coagulating mixture, while straightforward, is in no extent trivial. The approach applied in the present paper allows for obtaining all characteristics of the evolving coagulating system for arbitrary initial conditions. The analytical results reported here are sufficiently simple and transparent. The pregelation stage of coagulation in a binary mixture had been studied in Ref. [8] and later in Refs. [2,3] for initially monodisperse particles. In these papers coagulation of a binary mixture was studied within the traditional approach based on the Smoluchowski equation for the composition spectrum c(m,n; t), dt cm,n (t) = K(m,n|m1 ,n1 ; m2 ,n2 )cm1 ,n1 (t)cm2 ,n2 (t). (2)
*
[email protected]
1539-3755/2014/89(3)/032121(9)
Here c(m,n; t) is the concentration of coagulating particles comprising m and n monomeric units of the first and the second kind respectively. The summation in Eq. (2) goes over all nonnegative integers m1 , m2 and n1 , n2 and K(m,n|m1 ,n1 ; m2 ,n2 ) = 12 K(m1 ,n1 ; m2 ,n2 ) ×[δm,m1 +m2 δn,n1 +n2 − δm,m1 δn,n1 − δm,m2 δn,n2 ]
(3)
is the coagulation kernel, the transition rate for the process given by Eq. (1). The Kroneker δs (δk,l ) provide the mass conservation at each collision. The first term on the right-hand side (RHS) of Eq. (3) describes the gain in the (m + n)-mer concentration due to coalescence of (m1 + n1 )- and (m2 + n2 )-mers (m1 + m2 = m, n1 + n2 = n), while the second one is responsible for the losses of (m + n)-mers due to their sticking to all other particles. In this paper we investigate the coagulation process for the cross product coagulation kernel, K(m1 ,n1 ; m2 ,n2 ) = m1 n2 + m2 n1 .
(4)
This coagulation kernel describes the evolution of a random bipartite graph. Indeed, let us imagine a system comprising M0 and N0 functional units connected with K MN links allowing for an exchange of information between them (see Fig. 1). The initially empty graph comprising M0 hearts and N0 circles begins to evolve by randomly adding one edge at a time connecting a couple heart-circle. The final stage of this process is the complete graph, where it is impossible to add an extra edge. The intermediate stages of this process are displayed in Fig. 1. In this rather primitive picture one easily recognizes a schematic model of a computer, the Internet, a living organism, or the human brain. Other (more physical examples) of such systems are polymers, disordered materials, random electric nets. The structures of all these systems can be modeled by a random graph, where K edges randomly distributed among M0 N0 vertices form clusters (linked components). The final question to answer is to find the linked cluster distribution over their orders (the number of hearts and circles in a linked component) at time t [31]. The coagulation process with the kernel Eq. (4) is another example of the exactly solvable model of the sol-gel transition,
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©2014 American Physical Society
A. A. LUSHNIKOV
PHYSICAL REVIEW E 89, 032121 (2014)
Section VI contains the discussion of the main results and Sec. VII concludes the paper. II. BASIC EQUATION (a)
For the coagulation kernel Eq. (4) the Smoluchowski equation (2) looks as follows: dt cm,n (t) = (m − k)lcm−k,n−l (t)ck,l (t)
(b)
k,l
−(mN + nM)cm,n (t), where (c)
(d)
M(t) =
FIG. 1. Evolution of a random bipartite graph. Initially empty bipartite graph (a) begins to evolve. An edge linking two vertices of different sorts (hearts and circles) is added at a time. Heart-toheart and circle-to-circle edges are forbidden. An example of a graph formed after a time t is shown in (b). This graph comprises two linked components with 4 and 3 vertices. Next appearing edge can either link these two components and thus produce one component of order (4,3) as shown in (c) (coalescence of graphs) or to add the edge to one of these graphs not changing the distribution of the graphs over their order (d). The latter graph (d) thus contains a cycle.
where a giant particle (gel) with the mass comparable to the total mass of the whole system forms after a finite time interval 0 < t < tc . In normal (nongelling) systems tc = ∞. In gelling systems at t < tc the coagulation process develops as in normal ones. At t > tc the gel affects the coagulation kinetics either interacting with the sol particles (active gelation) or consuming the sol mass from larger sol particles whose mass flux to infinity remains finite after t > tc (passive gelation). The details of the gelation process are thoroughly discussed in [32] (see also earlier citations therein). The remainder of the paper is divided as follows. The next section introduces the reader to the statement of the problem. Here the basic equation and the initial condition to it are formulated. The bivariate generating function for the particle composition is introduced and found in Sec. III and Appendix A as the solution to a partial differential equation. The solution to this equation is expressed in terms of two subsidiary functions directly linked to the initial composition spectrum. In Sec. IV the exact expressions for the masses of the first and the second components in the gel are derived for two scenarios of gelation: passive gelation, where the gel does not interact with the sol, and active gelation, where the gel serves as a sink for the sol particles. The final result [Eqs. (25)–(27)] is extremely simple and expresses the gel mass through the generating functions for the initial particle composition. The expression for the particle mass spectrum [Eq. (42)] derived in Sec. V is then used here for the asymptotic analysis of the particle composition spectrum. The latter is shown to depend on the type of gelation. In the case of active gel all m,n-mer concentrations drop down exponentially with time after the transition point and are entirely absorbed by the gel. The passive gelation leaves a part of monomeric particles unreacted. For initially monodisperse particles the concentration of the residual monomer is found analytically.
N (t) =
mcm,n (t) and
m,n
(5)
ncm,n (t)
(6)
m,n
are the mass concentrations of two coagulating species. For the cross-product coagulation kernel Eq. (4) the total mass concentrations M and N are not always constants in time. Two scenarios of gelation can be realized: (i) After the critical time one giant particle (the gel) forms that interacts with the sol particles (active gelation). In this case the gel contributes to the mass balance of the coagulating system and M and N entering Eq. (5) remain constant at all t. (ii) There is a sink of particles with large masses. In this case the large particles do not contribute to M and N and after a critical time t = tc they drop down with time [32]. Equation (5) should be supplemented with the initial condition (0) cm,n (0) = cm,n ,
(7)
(0) is a known function of m and n. where cm,n Let us introduce t t M(t) = M(t )dt , N (t) = N (t )dt 0
(8)
0
and do the replacement in Eq. (5), cm,n (t) = bm,n (t)e−mN −nM . We come to the equation for b, dt bm,n (t) = (m − k)lbm−k,n−l (t)bk,l (t).
(9)
(10)
k,l
From this equation we find that bm,0 and b0,n do not depend on time, because neither coalescence act produces a particle comprising the monomeric units of one sort. Hence, the concentrations of pure m- and n-mers change because of their sticking to the particles containing the units of opposite sort, i.e., (0) −mN cm,0 (t) = cm,0 e ,
(0) −nM c0,n (t) = c0,n e .
(11)
III. BIVARIATE GENERATING FUNCTION
The generating function F (x,y; t) for bm,n (t) is introduced as
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F (x,y; t) =
m>0,n>0
bm,n (t)x m y n .
(12)
COMPOSITION DISTRIBUTIONS OF PARTICLES IN A . . .
PHYSICAL REVIEW E 89, 032121 (2014)
Equations (10), (12), and (5) serve for deriving the equation for the generating function, ∂F ∂F ∂F =x y . ∂t ∂x ∂y
(13)
The initial condition to this equation is F (x,y; 0) = F0 (x,y).
(14)
(15)
In order to solve Eq. (13) we introduce two subsidiary functions, A(x,y,t) = xFx ,
B(x,y,t) = yFy .
M0 − μ = A0 (Zν (t),Zμ (t)),
0
A˙ = xAx B + yAy A,
B˙ = yBy A + xBx B.
Hence,
B = B0 (ξ,η).
η = yetA0 (ξ,η) .
(20)
(21)
= A(e N(t) =
m,n
,e
−M
; t) = A0 (e−N +Nt ,e−M+Mt ),
ncm,n (t) =
(22)
(23)
ne−mN −nM bm,n (t)
m,n
= B(e−N ,e−M ; t) = B0 (e−N +Nt ,e−M+Mt ).
(31)
t dξ Zξ (t) = exp − t dt dt 0
(32)
0
Then
t
(30)
∂ν dt . ∂t
t
with ξ = μ, ν. Let us now differentiate Eq. (27). We find ∂A0 ∂A0 μ˙ 1 − tZμ − ν˙ tZν = 0, ∂y ∂x ∂B0 ∂B0 −μtZ ˙ μ + ν˙ 1 − tZν = 0. ∂y ∂x μ˙ = 0,
m,n −N
∂μ dt . ∂t
(33)
There are two solutions to this set of equations. The first one,
The current masses M(t) and N (t) are expressed through the generating functions A0 and B0 as follows [see Eqs. (20) and (21)]: M(t) = mcm,n (t) = me−mN −nM bm,n (t) m,n
t
N (t) − N (t)t =
(19)
The masses of m and n components are defined as the differences between the initial and current masses, ν(t) = N0 − N(t).
t
=
(18)
μ(t )dt − M0 t + μ(t)t
0
IV. GEL COMPOSITION
μ(t) = M0 − M(t),
t
Similarly,
The dependencies of ξ and η on x, y, and t are defined by the set of two equations ξ = xetB0 (ξ,η) ,
M(t) − M(t)t = M0 t −
0
The solution to these equations is reproduced in Appendix A. The result is A = A0 (ξ,η),
0
(29)
(17)
Differentiating Eq. (13) over x and y yields the set of the first-order partial differential equations for A and B,
(28)
Equation (28) can be simplified. The definition Eq. (22) of the gel masses gives t t μ(t )dt , N (t) = N0 t − ν(t )dt . M(t) = M0 t −
The evident identities will be of use, ∂B ∂ 2F ∂A =x = xy . y ∂y ∂x ∂x∂y
N0 − ν = B0 (Zν (t),Zμ (t)), (27)
where Zμ (t) and Zν (t) are defined as Zμ (t) = e−M+Mt , Zν (t) = e−N +Nt .
−M(t)
M(t) = A(e ,e ; t), N (t) = B(e−N (t) ,e−M(t) ; t).
(26)
(16)
The total mass concentrations are expressed through A and B as follows: −N (t)
N0 − ν(t) = B0 (e−νt ,e−μt ). If the gel is passive, then Eqs. (23), (24), and (28) yield
The generating function F(x,y; t) for cm,n (t) is linked to F (x,y; t) as follows: F(x,y; t) = cm,n (t)x m y n = F (xe−N (t) ,ye−M(t) ; t).
In the case of the active gel M(t) = M0 t and N (t) = N0 t. Hence, (25) M0 − μ(t) = A0 (e−νt ,e−μt ),
(24)
ν˙ = 0,
(34)
realizes if the determinant ∂A0 ∂B0 1 − tZν (Zμ ,Zν ) = 1 − tZμ ∂y ∂x − t 2 Zμ Zν
∂B0 ∂A0 ∂y ∂x
(35)
of the set Eq. (33) = 0. Otherwise the condition = 0 and one of Eqs. (33) determines μ and ν. As follows from Eq. (28), Zξ (tc ) = 1. Then the critical time is readily found from the conditions (1,1) = 0 and ∂x A0 (1,1) = Q2,0 ∂y A0 (1,1) = ∂x B0 (1,1) = Q1,1 and ∂y B0 (1,1) = Q0,2 , 1 . (36) tc = Q1,1 + Q2,0 Q0,2 Here Qr,s = m,n mr ns cm,n (0) are the moments of the initial composition distribution.
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In the above the integration goes counterclockwise along the circles surrounding the origin of coordinates in the complex planes ξ and η.
V. COMPOSITION SPECTRUM A. Exact results
According to Eqs. (12) and (9) the composition spectrum is expressed in terms of A and B as follows: (m + n)cm,n (t) = e
−nM(t)−mN (t)
× Coef x,y
A(x,y; t) + B(x,y; t) , x m+1 y n+1
(37)
where the symbol Coef stands for the operation (see Ref. [34]) am,n x m y n = a−1,−1 . Coef x,y (38) Here am,n are arbitrary objects and the series on the left of the above equation just enumerates them. The operation Coef displays many features of usual residues and is convenient to apply in the cases of many complex variables and divergent series. In particular, we can replace the variables x and y by ξ and η defined as ξ = xeB0 (ξ,η)t and η = yeA0 (ξ,η)t [see Eq. (21)]. Equation (20) allows us to express A and B through initial generating functions A0 and B0 . We thus have
B. Asymptotic analysis
At large m and n the integrals in Eq. (44) can be evaluated by the standard saddle-point method. To this end we rewrite the integrand in the form
[A0 (ξ,η) + B0 (ξ,η)]et[(m+1)B0 +(n+1)A0 ] . (39) ξ m+1 ηn+1
tξs A0 (ξs ) = α.
Now let us expand R(ξ ) in the exponent over ξ − ξs up to the terms of the second order,
R (ξ ) = tA0 (ξ ) + α/ξ 2 .
1 φA (n,m; t) = 2π i 1 φB (m,n; t) = 2π i
=
etmηg (η) dη. ηn+1
enRs
. 2π n tA0 (ξs )ξs2 + α
(50)
(42)
(44)
emPs
2π n tB0 (ηs )ηs2 + 1/α
(51)
with P (η) = tB0 (η) −
ln η α
(52)
and 1 . (53) α The asymptotic expression for the composition spectrum follows from Eq. (43), 1 cm,n (t) = ξ −n η−m 2π t(mn)3/2 s s tηs B0 (ηs ) =
e−n[M−tA0 (ξs )]−m[N −tB0 (ηs )] × . tA0 ξs2 + α tB0 ηs2 + 1/α
etnξf (ξ ) dξ, ξ m+1
(49)
According to the standard saddle-point procedure we deform the integration contour in such a way that it intersects the real axis at the saddle point at the right angle. Because only a small vicinity of the saddle point contributes to the integral, the latter is readily performed, ∞ enRs nR ξ 2 dξ φA (n,m; t) ≈ exp − s 2π ξs −∞ 2
(41)
1 −nM(t)−mN (t) e φA (n,m; t)φB (m,n; t), (43) mnt
where
(48)
Here Rs = R(ξs ) and
φB (m,n; t) ≈
then cm,n (t) =
(47)
Similarly we find
The concentrations of pure m or n-mers (m = 0, n = 0 or m = 0, n = 0) are given by Eq. (11). If the initial generating function does not contain the cross derivative, i.e., if F0 (x,y) = f (x) + g(y),
(46)
where the ratio α = m/n is assumed to be finite. The position of the saddle point ξs is determined from the condition R (ξs ) = 0 or
The calculation of cm,n (t) by applying Eq. (39) is rather tedious. The details are described at length in Appendix B. The final result is, however, surprisingly simple. At m,n > 0 1 −nM(t)−mN (t) e mnt ∂ 2 F0 et(nA0 +mB0 ) . × Coef ξ,η m+1 n+1 1 − tξ η ξ η ∂ξ ∂η
R(ξ ) = tA0 (ξ ) − α ln ξ,
R = Rs + 12 R (ξs )(ξ − ξs )2 .
The notation J in the above equation stands for the transition Jacobian, ∂B0 ∂(x,y) ∂A0 J = = e−(A0 +B0 )t 1 − tξ − tη ∂(ξ,η) ∂ξ ∂η ∂B0 ∂A0 ∂B0 ∂A0 2 − . (40) + t ξη ∂ξ ∂η ∂η ∂ξ
cm,n (t) =
(45)
with
(m + n)cm,n (t) = e−nM(t)−mN (t) J (ξ,η; t) × Coef ξ,η
etnA0 (ξ ) 1 = enR(ξ ) ξ m+1 ξ
(54)
In the case of the active gel M − tA0 (ξs ) = t[M0 − A0 (ξs )], N − tB0 (ηs ) = t[N0 − B0 (ηs )], and the asymptotic composi-
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PHYSICAL REVIEW E 89, 032121 (2014)
tion spectrum is cm,n (t) =
1 ξ −n η−m 2π t(mn)3/2 s s e−nt[M0 −A0 (ξs )]−mt[N0 −B0 (ηs )] × . tA0 ξs2 + α tB0 ηs2 + 1/α
(55)
For the passive gelation we find cm,n (t) =
1 ξ −n η−m Zμn Zνm 2π t(mn)3/2 s s 1 × . tA0 ξs2 + α tB0 ηs2 + 1/α
(56)
At the precritical period t < tc both these spectra coincide because μ(t) = ν(t) = 0. The condition ξs = ηs = 1 determines the critical line on the plane m,n, Q20 m = α= . (57) n Q02 Along this line the composition spectrum is thus algebraic, √ Q20 Q02 1 cm,n (tc ) = . (58) 2π (mn)3/2 (tc A0 + α)(tc B0 + 1/α) The combinations tc A0 (1) and tc B0 (1) are expressed in terms of the second and third moments of the initial composition distribution, Q30 − Q20 , tc A0 (1) = √ Q20 Q02
Q03 − Q02 tc B0 (1) = √ . Q20 Q02
(59)
For monodisperse initial conditions, B0 (x,y) = N0 y,
= 1 − t M0 N0 Zμ Zν . 2
(60)
Equations (27) and (35) allow us to find the composition of the gel. For passive gelation we have t (61) = 1 − t 2 M0 N0 exp − t (μ˙ + ν˙ )dt . tc
From this equation we find μ+ν =
2 2 − . tc t
and
ν = N0 (1 − Zμ )
1 xy = 2 . t
(72)
On the other hand, according to Eq. (25) one has c1,0 (∞) = M0 − μ(∞) = A0 [Zν (∞)].
(74)
On combining this equation with Eq. (73) finally yields c1,0 (0)Zν (∞) = A0 [Zν (∞)]. (75) It is easy to check that this equation converts to the identity c1,0 (0) = M0 for initially monodisperse mixtures. D. Active gel
(63) (64)
We substitute this to Eq. (64) and obtain the set of equations for x = M0 Zν and y = N0 Zμ . This set looks very simple, 2 2 x + y = M0 + N0 − + , tc t
If M0 = N0 then x+ = x− = 1/t and μ(t) = ν(t) = M0 (1 − 1/t).
(71)
If the gel interacts with the sol particles, then the masses μ(t) and ν(t) meet the following set of equations: (76) M0 − μ = M0 e−νt , N0 − ν = N0 e−μt .
or μ + ν = M0 + N0 − M0 Zν − N0 Zμ .
Here the notation p = N0 /M0 is introduced. The mass of N -gel goes to N0 as t −→ ∞. Indeed, 1 . ν ≈ N0 − x− = N0 1 − √ √ 2( M0 − N0 )t 2
(70)
(62)
On the other hand, μ = M0 (1 − Zν )
and the concentration of the residual sol is √ M0 − μ(∞) = M0 (1 − p)2 .
The part of residual monomer can also be found in the general case. Indeed, c1,0 (∞) = c1,0 (0)Zν (∞). (73)
C. Passive gel
A0 (x,y) = M0 x,
The solution to this set is readily found. Finally from Eq. (63) we have 1 2s μ(t) = M0 − s − + s 2 + , t t (66) 1 2s 2 ν(t) = N0 − s − − s + , t t √ 2 √ where s = ( M0 − N0 ) /2. At large t 2 (67) x+ ≈ 2s + , t 1 x− ≈ . (68) 4st 2 If M0 exceeds N0 then not all M-monomers will occur in the gel phase. It is possible to find the concentration √ √ of the residual monomer. At large t x+ ≈ 2s = ( M 0 − N 0 )2 and according to Eqs. (65) and (68) μ(∞) x+ √ = 2 p − p2 = 1− (69) M0 M0
Let us find the second moment, ∂ ∂F ∂A(xe−N ) Q2,0 (t) = x x = x ∂x ∂x x=y=1 ∂x x=y=1 =x
(65)
∂A0 (ζ ) = M0 ζx |x=y=1 , ∂x x=y=1
(77)
where ζ (x,y; t) and θ (x,y; t) are defined by the equalities x = ζ eN −N0 θt , y = θ eM−M0 ζ t . (78)
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We differentiate these two equations with respect to x, put x = y = 1, and find after some algebra M0 e−νt 1 − M0 N0 e−(μ+ν)t t 2 M0 − μ = . 1 − (M0 − μ)(N0 − ν)t 2
Q2,0 =
(79)
Here we used the fact that ζ = e−νt and θ = e−μt at x = y = 1. In the same way we find the other two moments, Q0,2 =
N0 − ν , 1 − (M0 − μ)(N0 − ν)t 2
Q1,1 =
t(M0 − μ)(N0 − ν) . 1 − (M0 − μ)(N0 − ν)t 2
(80)
The expressions for the second moments for general initial conditions are derived in Appendix C. E. Composition spectrum
We begin by considering the monodisperse initial conditions. In this case A0 (ξ ) = M0 ξ , B0 (η) = N0 η. The calculation of integrals is not difficult,
tnM0 ξ 1 (tnM0 )m e φA (n,m; t) = . (81) dξ = 2π i ξ m+1 m! Similarly, (tmN0 )n . (82) n! Hence according to Eq. (41) we find the composition spectrum, φB (m,n; t) =
the final expression for the composition spectrum. In the case of the active gel M(t) = M0 t and N (t) = N0 t. For the passive gelation we should subtract from this expression the contribution of the gel mass. From the first sight, this modification cannot lead to appreciable changes to the spectrum. However, it is not so. The postcritical spectrum is algebraic in the case of the passive gelation and it exponentially drops down otherwise. (iii) The time dependencies of the first and the second moments of the composition distribution found in Sec. V help one to understand the qualitative picture of the sol-gel transition in coagulating mixtures. Below the gelation time the total sol mass concentration is conserved together with the total mass of each component in the sol particles. On crossing the critical point the sol begins to lose particles spending them for the creation of a giant gel particle which either interacts with the sol particles (the active gelation) or the particles are lost because the total mass flux remains finite as the particle masses grow. The picture of the sol-gel transition in this case is not quite satisfactory, because it remains unclear what does it mean, the giant gel particle. This problem had been discussed in Refs. [22,32] in connection with coagulation kinetics in finite systems (the number of monomeric units is finite) and in the systems with sharp sink at large particle masses (the truncated models). (iv) Of special interest is the final stage of the gelation process. It occurs that a part of monomers does not have enough time to join to the gel. The component with lower concentration of monomers converts to gel earlier and leaves the rest of the sol without the partners allowing for further growth of the monomers. The analytical result [Eqs. (66) and
mn−1 nm−1 −nM−mN e . (83) m!n! This result reproduces the known composition spectrum, cm,n (t) = M0n N0m t m+n−1
mn−1 nm−1 −nM0 t−mN0 t e , (84) m!n! for active gelation [15]. In the case of passive gelation the postcritical composition spectrum looks different: we should t substitute M(t) = M0 (t) − tc μ(t )dt and N (t) = N0 (t) − t tc ν(t )dt into Eq. (83). The expressions for μ(t) and ν(t) are known [see Eq. (66)]. The respective integrals can be expressed in terms of elementary functions. The results are not reproduced here because they are rather cumbersome. cm,n (t) = M0n N0m t m+n−1
VI. RESULTS AND DISCUSSION
(i) The composition spectrum cm,n (t) is expressed in terms of the exponentiated bivariate generating function for the initial spectrum cm,n (0) [Eq. (41)]. Although the application of this result demands some efforts, they are not huge, especially when the initial sol does not contain mixed particles. In the latter case the problem reduces to the evaluation of a couple of ordinary contour integrals. The traditional saddle point method applies for the asymptotic estimation of these integrals in the limit of large m and n [Eq. (56)]. (ii) The difference between the postcritical behavior of the two scenarios (active and passive gelation) is only related to the different time behavior of the factor enM(t)+mN (t) entering
FIG. 2. The gel masses vs time for active and passive gelation. Calculations are done for M0 = 0.8 and N0 = 0.2. Solid lines correspond to the active gelation. In this case the gel grows faster than for passive gelation (dashed lines). The upper dashed curve never reaches the upper limit μ = 0.8 which means that a part of m monomers has not enough time for finding their partners for coagulation. Neither are they themselves able to agglomerate.
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PHYSICAL REVIEW E 89, 032121 (2014)
FIG. 3. Postcritical behavior of the second moments of the composition distribution vs time.
(76)] is found for the case of initially monodisperse particles and displayed in Fig. 2. (v) The second moments of the composition spectrum in the postcritical period are shown in Fig. 3 as the functions of time for active gel [Eqs. (79) and (80)]. In the case of passive gelation after the critical time the sums in the expressions for these moments diverge.
centration of the first component in the particle) instead of m and n. Respectively, the coagulation kernel depends on the particle masses g1 and g2 , and the particle compositions β1 and β2 , ˜ 1 ,β1 |g2 ,β2 ). (85) K(m1 ,n1 |m2 ,n2 ) → K(g If the function K(m1 ,n1 |m2 ,n2 ) is a homogeneous function of its variables, K(am1 ,bn1 |am2 ,bn2 ) = a μ bν K(m1 ,n1 |m2 ,n2 ), (86) ˜ 1 ,β1 |g2 ,β2 ) is the homogeneous function of g1 and then K(g g2 with the homogeneity exponent λ = μ + ν, i.e., λ ˜ ˜ K(ag (87) 1 ,β1 |ag2 ,β2 ) = a K(g1 ,β1 |g2 ,β2 ). The sol-gel transition should thus occur at λ > 1 or μ + ν > 1. The assumption that the particle compositions play a secondary role in the gelation process is adopted. In this paper the case μ = ν = 1 was considered. Starting with the approach based on the evolution equation for the composition spectrum c(m,n; t) (the concentration of m,n-mers) at time t (Sec. II) the sol-gel transition in the coagulating mixture has been considered. In my opinion, the most remarkable result of this paper is the expression of the composition spectrum through the exponentiated generating function for the initial composition spectrum [Eq. (41)]. Of principal interest are the results on the global characteristics of the coagulation system: composition of the gel and the second moments. The similarity with the gelation in one component systems is apparent, although the description of the process in mixtures is much more complex. ACKNOWLEDGMENT
The financial support from Project No. 14-03-00507 of Russian Foundation of Basic Research is acknowledged.
VII. CONCLUDING REMARKS
APPENDIX A: SOLUTION OF EQ. (19)
The shift that we experienced in the past decades in our understanding of the coagulation kinetics and especially of the coagulation-gelation processes was not swift, but still rather significant. First of all, the nature of the sol-gel transition became clear, especially the fact that there is nothing mysterious in the notion of “gel.” It turns out that the gel is either a giant particle whose concentration is zero in the thermodynamic limit (active gelation) or a passive deposit that forms due to the coagulation growth within a finite range of the particle sizes; I mean the truncated model of coagulation [32] where a sharp sink instantly removes the particles with masses exceeding a given very large value G. As was shown in Ref. [33] the dynamics of the sol-gel transition can be satisfactorily described by using the standard Smoluchowski approach that well reproduces the behavior of the sol even in the postcritical period except for a very small interval of time in the vicinity of the critical point. At present there exists the common opinion on the form of the coagulation kernels associated with the gelation processes in the single component systems. These kernels should grow faster than K(g,g) ∝ g [35]. Naturally, the question comes up: what type of coagulation kernels can lead to gelation in mixtures? Although nobody yet investigated this problem, a criterion can be formulated. Indeed, let us characterize each cluster by its total mass g = m + n and the particle composition β = m/g (the relative con-
Let us look for the solution to the set Eq. (19) in the form (A,B,x,y,t) = 0
and
(A,B,x,y,t) = 0, (A1)
where the functions A(x,y; t) and B(x,y; t) are the solutions to the set Eq. (A1) and and are yet unknown functions of x,y,t. On differentiating Eq. (A1) over t we come to the set of ˙ two linear algebraic equations for A˙ and B, ˙ = 0, A A˙ + B B˙ +
˙ = 0. (A2) A A˙ + B B˙ + ˙ The Let us solve the set Eq. (A2) with respect to A˙ and B. result is ˙ − ˙ B ) B˙ = D−1 (A ˙ − ˙ A ). (A3) A˙ = D−1 (B Here D = (A B − B A ).
(A4)
Similar equations come up in differentiating Eq. (A1) over x and y. We substitute thus found derivatives of A an B into Eq. (19) and come to the set of two homogeneous ˙ + xBx + yAy linear equation for the combinations − ˙ − xBx − yAy , and ˙ + xBx + yAy ]B + [ ˙ − xBx − yAy ]B = 0, [− ˙ − yAy − xBx ]A + [− ˙ + yAy + xBx ]A = 0. [
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(A5)
A. A. LUSHNIKOV
PHYSICAL REVIEW E 89, 032121 (2014)
dξ dη d Aη eaA 1 ebB n b η dξ ξ m m 1 Aη eZ dηdξ Aξ η eZ dηdξ = − b ξ m+1 ηn b ξ m ηn a Aξ Aη eZ dηdξ − . b ξ m ηn
Assuming that D = 0 we obtain two linear equations for the functions and ,
=−
∂ ∂ ∂ ∂ ∂ ∂ = Bx + Ay and = Ay + Bx . ∂t ∂x ∂y ∂t ∂y ∂x (A6) These equations are readily solved by the method of characteristics. The equations for the characteristics [for both of Eqs. (A6)] are dt = −
dy dA dB dx =− = = . xB yA 0 0
(A7)
Hence (x,y,t,A,B) = φ(xeAt ,yeBt ,A,B),
(A8)
(x,y,t,A,B) = ψ(xeAt ,yeBt ,A,B),
where φ and ψ are yet arbitrary functions. Now we can solve the set of Eqs. (A1) and find A(x,y; t) = A0 (xeBt ,yeAt ),
(A9)
B(x,y; t) = B0 (xeBt ,yeAt ).
Here and below the subscripts ξ and η denote the derivatives over these variables. Similarly, we have dξ dη aA+bB e Aξ Bη ξ m ηn aA 1 1 aA dξ bB bB dξ dη d Aξ e = de = − Aξ e e b ξ m ηn b ξ m dη ηn n 1 Aξ eZ dηdξ Aξ η eZ dηdξ = − m n+1 b ξ η b ξ m ηn a Aξ Aη eZ dηdξ − . (B3) b ξ m ηn Hence
It is convenient to introduce ξ (x,y; t) = xeBt ,
I3 = and
η(x,y; t) = yeAt .
(A10)
Then A = A0 (ξ,η),
B = B0 (ξ,η).
(A11)
APPENDIX B: EVALUATION OF THE INTEGRAL
In what follows, instead of double integrals,
1 1 dx dy, 2π i 2π i
in Eqs. (37) and (39) the single integration symbol will be used. Then Eq. (37) can be rewritten as follows: ∂ ∂ 1 + (I1 − tI2 + t 2 I3 ), cm,n (t) = m + n ∂a ∂b a=mt, b=nt
∂B0 ∂A0 eZ dξ dη eZ dξ dη ξ + η , , I = 2 ξ m+1 ηn+1 ξ m+1 ηn+1 ∂ξ ∂η Z ∂B0 ∂A0 e dξ dη ∂B0 ∂A0 I3 = − , ξ m ηn ∂ξ ∂η ∂η ∂ξ
and Z = aA0 + bB0 . Let us begin with I3 . Integrating by parts gives dξ dη Z e Aη Bξ ξ m ηn dη 1 = Aη eaA m n debB b ξ η
m b
n Aη eZ dηdξ − m+1 n ξ η b
Aξ eZ dηdξ . ξ m ηn+1
Let us convert Aξ to Bξ in the last integral, aA bB 1 1 Aξ eZ dηdξ bB de aA dη d e = = − e e ξ m ηn+1 a ξ m ηn+1 a ηn+1 dξ ξ m dξ dη dξ dη m b = eZ m+1 n+1 − Bξ eZ m n+1 . a ξ η a ξ η And finally we have dξ dη dξ dη m mn I3 = Aη eZ m+1 n − eZ m+1 n+1 b ξ η ab ξ η dξ dη nb + Bξ eZ m n+1 . ba ξ η Let us return to our integral, eaA+bB U (a,b) = J (ξ,η; t)dξ dη ξ m+1 ηn+1 2 t n t 2m eaA+bB 1+ + − 2t ξ ηFξ η = ξ m+1 ηn+1 a b t 2 mn dξ dη. − ab
(B1) where I1 =
(B2)
It is important to note that U (nt,mt) =
dxdy = 0, x m+1 y n+1
as should it be. Now we differentiate U (a,b) (m + n)c(m,n; t) = (∂a + ∂b )U (a,b)|a=nt, b=mt . Finally we have 1 cm,n (t) = mnt 032121-8
∂ 2 F0 et(nA0 +mB0 ) 1 − tξ η dξ dη. ξ m+1 ηn+1 ∂ξ ∂η
COMPOSITION DISTRIBUTIONS OF PARTICLES IN A . . .
PHYSICAL REVIEW E 89, 032121 (2014)
Solving these equation yields
APPENDIX C: SECOND MOMENTS IN GENERAL CASE
In the case of the general initial condition the gel masses should be found from the set of two equations, M0 − μ = A0 (e
−νt
,e
−μt
), N0 − ν = B0 (e
−νt
,e
−μt
).
ζx =
(C1) θx
The second moments are
∂ ∂F Q2,0 (t) = x x ∂x ∂x
(1 −
,
=
(1 − tθ A0θ )e−νt , − tθ A0θ ) − t 2 ζ θ A0ζ B0θ
tζ B0ζ )(1
tθ A0ζ e−νt (1 − tζ B0ζ )(1 − tθ A0θ ) − t 2 ζ θ A0ζ B0θ
(C4)
.
Similarly we have
x=y=1
∂ ∂F Q0,2 (t) = y y , ∂y ∂y x=y=1
θy =
∂ 2 F Q1,1 (t) = xy ∂x∂y
ζy
.
)e−μt (1 − tζ B0ζ (1 − tθ A0θ )(1 − tζ B0ζ ) − t 2 ζ θ A0θ B0ζ
,
−μt e tζ B0θ = . (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
(C5)
x=y=1
So we must find the derivatives of A, ∂A ∂A0 ∂A0 = ζ + θ . ∂x x=1,y=1 ∂ζ x ∂θ x
We thus find (C2)
In order to proceed we must find ζx and θx . On differentiating both sides of Eq. (A10) (both equations) we find the set of equations for ζx and θx , eνt = ζx (1 − tζ B0ζ ) − tθx ζ B0θ ,
0 = θx (1 − tθ A0θ ) − ζx tθ A0ζ .
A0ζ e−νt , (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
Q02 =
B0θ e−μt , (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
Q11 =
A0ζ B0θ tζ e−μt + A0,θ e−μt (1 − tζ B0,ζ ) . (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
and
(C3)
[1] N. A. Fuchs, Mechanics of Aerosols (Pergamon, New York, 1964). [2] J. M. Hidy and J. R. Brock, The Dynamics of Aerocollidal Systems (Pergamon, Oxford, 1970). [3] J. H. Seinfeld and S. Pandis, Physics and Chemistry of the Atmosphere (John Wiley & Sons, Hoboken, NJ, 2006). [4] M. M. R. Williams and S. K. Loyalka, Aerosol Science, Theory & Practice (Pergamon, Oxford, 1991). [5] S. K. Friedlander, Smoke, Dust and Haze, 2nd ed. (Oxford University Press, Oxford, 2000). [6] P. Meakin, Cambridge Nonlinear Science (Cambridge University Press, Cambridge, England, 1998), Vol. 5. [7] R. L. Drake, in Topics in Current Aerosol Research (Part II), edited by G. M. Hidy and J. R. Brock (Pergamon, Oxford, 1972), p. 201. [8] D. J. Aldous, Bernoully 5, 3 (1999). [9] H. R. Pruppacher and J. D. Klett, Microphysics of Clouds and Precipitation (Reidel, Dordrecht, 1978). [10] F. Leyvraz, Phys. Rep. 383, 95 (2003). [11] A. A. Lushnikov, J. Colloid Interface Sci. 54, 94 (1976). [12] F. M. Gelbard and J. H. Seinfeld, J. Colloid Interface Sci. 63, 472 (1978). [13] R. D. Vigil, and R. M. Ziff, Chem. Eng. Sci. 53, 1725 (1998). [14] T. Matsoukas, K. Lee, and T. Kim, AIChE J. 52, 3088 (2006). [15] E. R. Domilovskii, A. A. Lushnikov, and V. N. Piskunov, Dokl. Akad. Nauk SSSR 243, 407 (1978). [16] J. M. Fernandez-Diaz and G. J. Gomez-Garcia, Europhys. Lett. 78, 56002 (2007).
Q20 =
(C6)
[17] S. S. Chauhan, J. Chakraborty, and S. Kumar, Chem. Eng. Sci. 65, 3914 (2010). [18] I. J. Laurenzi, J. D. Bartels, and S. L. Diamond, J. Comput. Phys. 177, 418 (2002). [19] P. Alstrøm, Phys. Rev. A 38, 4905 (1988). [20] S. Qamar and G. Warnecke, Chem. Eng. Sci. 62, 679 (2007). [21] Y. P. Kim and J. H. Seinfeld, J. Colloid Interface Sci. 149, 425 (1992). [22] A. A. Lushnikov, Phys. Rev. E 73, 036111 (2006). [23] K. Lee, T. Kim, P. Rojniak, and T. Matsoukas, Chem. Eng. Sci. 63, 1293 (2008). [24] J. L. Spouge, J. Stat. Phys. 38, 573 (1985). [25] R. Albert and A.-L. Barab´asi, Rev. Mod. Phys. 74, 47 (2002). [26] B. Bollob´as, Random Graphs (Academic, London, 1985). [27] M. V. Simkin and V. P. Roychowdhary, Phys. Rep. 502, 1 (2011). [28] D. Han, J. Phys. A: Math. Gen. 36, 7485 (2003). [29] S. Janson, T. Luczak, and A. Rucinski, Random Graphs (Wiley, New York, 2003). [30] D. Knuth, Algorithmica 22, 561 (1998). [31] A. A. Lushnikov, J. Phys. A: Math. Gen. 38, L777 (2005). [32] A. A. Lushnikov, Physica D 222, 37 (2006). [33] A. A. Lushnikov, Phys. Rev. E 88, 052120 (2013). [34] G. P. Egorychev, Integral Representation and Counting of Combinatorial Sums (Nauka, Siberian branch, Novosibirsk, 1977) (in Russian). [35] I. S. Gradstein and I. M. Ryzhik, Tables of Integrals, Series, and Products (Elsevier, New York, 2007).
032121-9
Composition distributions of particles in a gelling mixture A. A. Lushnikov* Geophysical Center of Russian Academy of Science, 3, Molodezhnaya Street, 119296 Moscow, Russia and Karpov Institute of Physical Chemistry, 10 Vorontsovo Pole, 105064 Moscow, Russia (Received 14 January 2014; published 19 March 2014) Gelation in a two component disperse system wherein binary coagulation governs the temporal changes of particle composition spectra is studied for the crossproduct coagulation kernel proportional to m1 n2 + m2 n1 , with m,n being the numbers of monomers of the first and the second component in the coalescing pair of particles. This model reveals the sol-gel transition, i.e., after a finite interval of time the conservation of the total particle mass concentration violates because of the formation of giant particles (the gel). This paper reports on the exact solution of this model for arbitrary initial particle composition spectra. Exact expressions for the particle composition spectrum, the gel mass, and the second moments of the composition distribution are derived. Two scenarios of gelation, where the gel is either active or passive, are considered. DOI: 10.1103/PhysRevE.89.032121
PACS number(s): 02.50.−r, 05.90.+m, 68.03.Fg
I. INTRODUCTION
Binary aggregation processes of the type (m1 ,n1 ) + (m2 ,n2 ) −→ (m1 + m2 ,n1 + n2 ),
(1)
where the notation m,n stands for an (m + n)-mer comprising m and n monomers of the first and the second component respectively, are often met in nature. The subscripts 1 and 2 mark the two coalescing objects. The role of coagulation processes is commonly recognized and has been discussed in detail in the books and the review articles [1–10]. Still very little attention was given to coagulating mixtures. So far this problem remained clear of the mainstream of works on coagulation, although its importance remains beyond any doubt. The most evident example is coagulation of a binary mixture, where a gas of M + N monomers of two sorts begins to form clusters containing m and n monomers of each sort. This type of coagulation had been considered almost four decades ago in Ref. [11] and then in Refs. [12–23]. Another important example is the evolution of a random bipartite graph [24–32] and emergence of a giant component in it. I return to this problem in connection with my recent results in the theory of coagulation-gelation process in monocomponent coagulating systems [33]. It occurs that the extension of the results of my work [33] to a nontrivial and very instructive example of a coagulating mixture, while straightforward, is in no extent trivial. The approach applied in the present paper allows for obtaining all characteristics of the evolving coagulating system for arbitrary initial conditions. The analytical results reported here are sufficiently simple and transparent. The pregelation stage of coagulation in a binary mixture had been studied in Ref. [8] and later in Refs. [2,3] for initially monodisperse particles. In these papers coagulation of a binary mixture was studied within the traditional approach based on the Smoluchowski equation for the composition spectrum c(m,n; t), dt cm,n (t) = K(m,n|m1 ,n1 ; m2 ,n2 )cm1 ,n1 (t)cm2 ,n2 (t). (2)
*
[email protected]
1539-3755/2014/89(3)/032121(9)
Here c(m,n; t) is the concentration of coagulating particles comprising m and n monomeric units of the first and the second kind respectively. The summation in Eq. (2) goes over all nonnegative integers m1 , m2 and n1 , n2 and K(m,n|m1 ,n1 ; m2 ,n2 ) = 12 K(m1 ,n1 ; m2 ,n2 ) ×[δm,m1 +m2 δn,n1 +n2 − δm,m1 δn,n1 − δm,m2 δn,n2 ]
(3)
is the coagulation kernel, the transition rate for the process given by Eq. (1). The Kroneker δs (δk,l ) provide the mass conservation at each collision. The first term on the right-hand side (RHS) of Eq. (3) describes the gain in the (m + n)-mer concentration due to coalescence of (m1 + n1 )- and (m2 + n2 )-mers (m1 + m2 = m, n1 + n2 = n), while the second one is responsible for the losses of (m + n)-mers due to their sticking to all other particles. In this paper we investigate the coagulation process for the cross product coagulation kernel, K(m1 ,n1 ; m2 ,n2 ) = m1 n2 + m2 n1 .
(4)
This coagulation kernel describes the evolution of a random bipartite graph. Indeed, let us imagine a system comprising M0 and N0 functional units connected with K MN links allowing for an exchange of information between them (see Fig. 1). The initially empty graph comprising M0 hearts and N0 circles begins to evolve by randomly adding one edge at a time connecting a couple heart-circle. The final stage of this process is the complete graph, where it is impossible to add an extra edge. The intermediate stages of this process are displayed in Fig. 1. In this rather primitive picture one easily recognizes a schematic model of a computer, the Internet, a living organism, or the human brain. Other (more physical examples) of such systems are polymers, disordered materials, random electric nets. The structures of all these systems can be modeled by a random graph, where K edges randomly distributed among M0 N0 vertices form clusters (linked components). The final question to answer is to find the linked cluster distribution over their orders (the number of hearts and circles in a linked component) at time t [31]. The coagulation process with the kernel Eq. (4) is another example of the exactly solvable model of the sol-gel transition,
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©2014 American Physical Society
A. A. LUSHNIKOV
PHYSICAL REVIEW E 89, 032121 (2014)
Section VI contains the discussion of the main results and Sec. VII concludes the paper. II. BASIC EQUATION (a)
For the coagulation kernel Eq. (4) the Smoluchowski equation (2) looks as follows: dt cm,n (t) = (m − k)lcm−k,n−l (t)ck,l (t)
(b)
k,l
−(mN + nM)cm,n (t), where (c)
(d)
M(t) =
FIG. 1. Evolution of a random bipartite graph. Initially empty bipartite graph (a) begins to evolve. An edge linking two vertices of different sorts (hearts and circles) is added at a time. Heart-toheart and circle-to-circle edges are forbidden. An example of a graph formed after a time t is shown in (b). This graph comprises two linked components with 4 and 3 vertices. Next appearing edge can either link these two components and thus produce one component of order (4,3) as shown in (c) (coalescence of graphs) or to add the edge to one of these graphs not changing the distribution of the graphs over their order (d). The latter graph (d) thus contains a cycle.
where a giant particle (gel) with the mass comparable to the total mass of the whole system forms after a finite time interval 0 < t < tc . In normal (nongelling) systems tc = ∞. In gelling systems at t < tc the coagulation process develops as in normal ones. At t > tc the gel affects the coagulation kinetics either interacting with the sol particles (active gelation) or consuming the sol mass from larger sol particles whose mass flux to infinity remains finite after t > tc (passive gelation). The details of the gelation process are thoroughly discussed in [32] (see also earlier citations therein). The remainder of the paper is divided as follows. The next section introduces the reader to the statement of the problem. Here the basic equation and the initial condition to it are formulated. The bivariate generating function for the particle composition is introduced and found in Sec. III and Appendix A as the solution to a partial differential equation. The solution to this equation is expressed in terms of two subsidiary functions directly linked to the initial composition spectrum. In Sec. IV the exact expressions for the masses of the first and the second components in the gel are derived for two scenarios of gelation: passive gelation, where the gel does not interact with the sol, and active gelation, where the gel serves as a sink for the sol particles. The final result [Eqs. (25)–(27)] is extremely simple and expresses the gel mass through the generating functions for the initial particle composition. The expression for the particle mass spectrum [Eq. (42)] derived in Sec. V is then used here for the asymptotic analysis of the particle composition spectrum. The latter is shown to depend on the type of gelation. In the case of active gel all m,n-mer concentrations drop down exponentially with time after the transition point and are entirely absorbed by the gel. The passive gelation leaves a part of monomeric particles unreacted. For initially monodisperse particles the concentration of the residual monomer is found analytically.
N (t) =
mcm,n (t) and
m,n
(5)
ncm,n (t)
(6)
m,n
are the mass concentrations of two coagulating species. For the cross-product coagulation kernel Eq. (4) the total mass concentrations M and N are not always constants in time. Two scenarios of gelation can be realized: (i) After the critical time one giant particle (the gel) forms that interacts with the sol particles (active gelation). In this case the gel contributes to the mass balance of the coagulating system and M and N entering Eq. (5) remain constant at all t. (ii) There is a sink of particles with large masses. In this case the large particles do not contribute to M and N and after a critical time t = tc they drop down with time [32]. Equation (5) should be supplemented with the initial condition (0) cm,n (0) = cm,n ,
(7)
(0) is a known function of m and n. where cm,n Let us introduce t t M(t) = M(t )dt , N (t) = N (t )dt 0
(8)
0
and do the replacement in Eq. (5), cm,n (t) = bm,n (t)e−mN −nM . We come to the equation for b, dt bm,n (t) = (m − k)lbm−k,n−l (t)bk,l (t).
(9)
(10)
k,l
From this equation we find that bm,0 and b0,n do not depend on time, because neither coalescence act produces a particle comprising the monomeric units of one sort. Hence, the concentrations of pure m- and n-mers change because of their sticking to the particles containing the units of opposite sort, i.e., (0) −mN cm,0 (t) = cm,0 e ,
(0) −nM c0,n (t) = c0,n e .
(11)
III. BIVARIATE GENERATING FUNCTION
The generating function F (x,y; t) for bm,n (t) is introduced as
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F (x,y; t) =
m>0,n>0
bm,n (t)x m y n .
(12)
COMPOSITION DISTRIBUTIONS OF PARTICLES IN A . . .
PHYSICAL REVIEW E 89, 032121 (2014)
Equations (10), (12), and (5) serve for deriving the equation for the generating function, ∂F ∂F ∂F =x y . ∂t ∂x ∂y
(13)
The initial condition to this equation is F (x,y; 0) = F0 (x,y).
(14)
(15)
In order to solve Eq. (13) we introduce two subsidiary functions, A(x,y,t) = xFx ,
B(x,y,t) = yFy .
M0 − μ = A0 (Zν (t),Zμ (t)),
0
A˙ = xAx B + yAy A,
B˙ = yBy A + xBx B.
Hence,
B = B0 (ξ,η).
η = yetA0 (ξ,η) .
(20)
(21)
= A(e N(t) =
m,n
,e
−M
; t) = A0 (e−N +Nt ,e−M+Mt ),
ncm,n (t) =
(22)
(23)
ne−mN −nM bm,n (t)
m,n
= B(e−N ,e−M ; t) = B0 (e−N +Nt ,e−M+Mt ).
(31)
t dξ Zξ (t) = exp − t dt dt 0
(32)
0
Then
t
(30)
∂ν dt . ∂t
t
with ξ = μ, ν. Let us now differentiate Eq. (27). We find ∂A0 ∂A0 μ˙ 1 − tZμ − ν˙ tZν = 0, ∂y ∂x ∂B0 ∂B0 −μtZ ˙ μ + ν˙ 1 − tZν = 0. ∂y ∂x μ˙ = 0,
m,n −N
∂μ dt . ∂t
(33)
There are two solutions to this set of equations. The first one,
The current masses M(t) and N (t) are expressed through the generating functions A0 and B0 as follows [see Eqs. (20) and (21)]: M(t) = mcm,n (t) = me−mN −nM bm,n (t) m,n
t
N (t) − N (t)t =
(19)
The masses of m and n components are defined as the differences between the initial and current masses, ν(t) = N0 − N(t).
t
=
(18)
μ(t )dt − M0 t + μ(t)t
0
IV. GEL COMPOSITION
μ(t) = M0 − M(t),
t
Similarly,
The dependencies of ξ and η on x, y, and t are defined by the set of two equations ξ = xetB0 (ξ,η) ,
M(t) − M(t)t = M0 t −
0
The solution to these equations is reproduced in Appendix A. The result is A = A0 (ξ,η),
0
(29)
(17)
Differentiating Eq. (13) over x and y yields the set of the first-order partial differential equations for A and B,
(28)
Equation (28) can be simplified. The definition Eq. (22) of the gel masses gives t t μ(t )dt , N (t) = N0 t − ν(t )dt . M(t) = M0 t −
The evident identities will be of use, ∂B ∂ 2F ∂A =x = xy . y ∂y ∂x ∂x∂y
N0 − ν = B0 (Zν (t),Zμ (t)), (27)
where Zμ (t) and Zν (t) are defined as Zμ (t) = e−M+Mt , Zν (t) = e−N +Nt .
−M(t)
M(t) = A(e ,e ; t), N (t) = B(e−N (t) ,e−M(t) ; t).
(26)
(16)
The total mass concentrations are expressed through A and B as follows: −N (t)
N0 − ν(t) = B0 (e−νt ,e−μt ). If the gel is passive, then Eqs. (23), (24), and (28) yield
The generating function F(x,y; t) for cm,n (t) is linked to F (x,y; t) as follows: F(x,y; t) = cm,n (t)x m y n = F (xe−N (t) ,ye−M(t) ; t).
In the case of the active gel M(t) = M0 t and N (t) = N0 t. Hence, (25) M0 − μ(t) = A0 (e−νt ,e−μt ),
(24)
ν˙ = 0,
(34)
realizes if the determinant ∂A0 ∂B0 1 − tZν (Zμ ,Zν ) = 1 − tZμ ∂y ∂x − t 2 Zμ Zν
∂B0 ∂A0 ∂y ∂x
(35)
of the set Eq. (33) = 0. Otherwise the condition = 0 and one of Eqs. (33) determines μ and ν. As follows from Eq. (28), Zξ (tc ) = 1. Then the critical time is readily found from the conditions (1,1) = 0 and ∂x A0 (1,1) = Q2,0 ∂y A0 (1,1) = ∂x B0 (1,1) = Q1,1 and ∂y B0 (1,1) = Q0,2 , 1 . (36) tc = Q1,1 + Q2,0 Q0,2 Here Qr,s = m,n mr ns cm,n (0) are the moments of the initial composition distribution.
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In the above the integration goes counterclockwise along the circles surrounding the origin of coordinates in the complex planes ξ and η.
V. COMPOSITION SPECTRUM A. Exact results
According to Eqs. (12) and (9) the composition spectrum is expressed in terms of A and B as follows: (m + n)cm,n (t) = e
−nM(t)−mN (t)
× Coef x,y
A(x,y; t) + B(x,y; t) , x m+1 y n+1
(37)
where the symbol Coef stands for the operation (see Ref. [34]) am,n x m y n = a−1,−1 . Coef x,y (38) Here am,n are arbitrary objects and the series on the left of the above equation just enumerates them. The operation Coef displays many features of usual residues and is convenient to apply in the cases of many complex variables and divergent series. In particular, we can replace the variables x and y by ξ and η defined as ξ = xeB0 (ξ,η)t and η = yeA0 (ξ,η)t [see Eq. (21)]. Equation (20) allows us to express A and B through initial generating functions A0 and B0 . We thus have
B. Asymptotic analysis
At large m and n the integrals in Eq. (44) can be evaluated by the standard saddle-point method. To this end we rewrite the integrand in the form
[A0 (ξ,η) + B0 (ξ,η)]et[(m+1)B0 +(n+1)A0 ] . (39) ξ m+1 ηn+1
tξs A0 (ξs ) = α.
Now let us expand R(ξ ) in the exponent over ξ − ξs up to the terms of the second order,
R (ξ ) = tA0 (ξ ) + α/ξ 2 .
1 φA (n,m; t) = 2π i 1 φB (m,n; t) = 2π i
=
etmηg (η) dη. ηn+1
enRs
. 2π n tA0 (ξs )ξs2 + α
(50)
(42)
(44)
emPs
2π n tB0 (ηs )ηs2 + 1/α
(51)
with P (η) = tB0 (η) −
ln η α
(52)
and 1 . (53) α The asymptotic expression for the composition spectrum follows from Eq. (43), 1 cm,n (t) = ξ −n η−m 2π t(mn)3/2 s s tηs B0 (ηs ) =
e−n[M−tA0 (ξs )]−m[N −tB0 (ηs )] × . tA0 ξs2 + α tB0 ηs2 + 1/α
etnξf (ξ ) dξ, ξ m+1
(49)
According to the standard saddle-point procedure we deform the integration contour in such a way that it intersects the real axis at the saddle point at the right angle. Because only a small vicinity of the saddle point contributes to the integral, the latter is readily performed, ∞ enRs nR ξ 2 dξ φA (n,m; t) ≈ exp − s 2π ξs −∞ 2
(41)
1 −nM(t)−mN (t) e φA (n,m; t)φB (m,n; t), (43) mnt
where
(48)
Here Rs = R(ξs ) and
φB (m,n; t) ≈
then cm,n (t) =
(47)
Similarly we find
The concentrations of pure m or n-mers (m = 0, n = 0 or m = 0, n = 0) are given by Eq. (11). If the initial generating function does not contain the cross derivative, i.e., if F0 (x,y) = f (x) + g(y),
(46)
where the ratio α = m/n is assumed to be finite. The position of the saddle point ξs is determined from the condition R (ξs ) = 0 or
The calculation of cm,n (t) by applying Eq. (39) is rather tedious. The details are described at length in Appendix B. The final result is, however, surprisingly simple. At m,n > 0 1 −nM(t)−mN (t) e mnt ∂ 2 F0 et(nA0 +mB0 ) . × Coef ξ,η m+1 n+1 1 − tξ η ξ η ∂ξ ∂η
R(ξ ) = tA0 (ξ ) − α ln ξ,
R = Rs + 12 R (ξs )(ξ − ξs )2 .
The notation J in the above equation stands for the transition Jacobian, ∂B0 ∂(x,y) ∂A0 J = = e−(A0 +B0 )t 1 − tξ − tη ∂(ξ,η) ∂ξ ∂η ∂B0 ∂A0 ∂B0 ∂A0 2 − . (40) + t ξη ∂ξ ∂η ∂η ∂ξ
cm,n (t) =
(45)
with
(m + n)cm,n (t) = e−nM(t)−mN (t) J (ξ,η; t) × Coef ξ,η
etnA0 (ξ ) 1 = enR(ξ ) ξ m+1 ξ
(54)
In the case of the active gel M − tA0 (ξs ) = t[M0 − A0 (ξs )], N − tB0 (ηs ) = t[N0 − B0 (ηs )], and the asymptotic composi-
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tion spectrum is cm,n (t) =
1 ξ −n η−m 2π t(mn)3/2 s s e−nt[M0 −A0 (ξs )]−mt[N0 −B0 (ηs )] × . tA0 ξs2 + α tB0 ηs2 + 1/α
(55)
For the passive gelation we find cm,n (t) =
1 ξ −n η−m Zμn Zνm 2π t(mn)3/2 s s 1 × . tA0 ξs2 + α tB0 ηs2 + 1/α
(56)
At the precritical period t < tc both these spectra coincide because μ(t) = ν(t) = 0. The condition ξs = ηs = 1 determines the critical line on the plane m,n, Q20 m = α= . (57) n Q02 Along this line the composition spectrum is thus algebraic, √ Q20 Q02 1 cm,n (tc ) = . (58) 2π (mn)3/2 (tc A0 + α)(tc B0 + 1/α) The combinations tc A0 (1) and tc B0 (1) are expressed in terms of the second and third moments of the initial composition distribution, Q30 − Q20 , tc A0 (1) = √ Q20 Q02
Q03 − Q02 tc B0 (1) = √ . Q20 Q02
(59)
For monodisperse initial conditions, B0 (x,y) = N0 y,
= 1 − t M0 N0 Zμ Zν . 2
(60)
Equations (27) and (35) allow us to find the composition of the gel. For passive gelation we have t (61) = 1 − t 2 M0 N0 exp − t (μ˙ + ν˙ )dt . tc
From this equation we find μ+ν =
2 2 − . tc t
and
ν = N0 (1 − Zμ )
1 xy = 2 . t
(72)
On the other hand, according to Eq. (25) one has c1,0 (∞) = M0 − μ(∞) = A0 [Zν (∞)].
(74)
On combining this equation with Eq. (73) finally yields c1,0 (0)Zν (∞) = A0 [Zν (∞)]. (75) It is easy to check that this equation converts to the identity c1,0 (0) = M0 for initially monodisperse mixtures. D. Active gel
(63) (64)
We substitute this to Eq. (64) and obtain the set of equations for x = M0 Zν and y = N0 Zμ . This set looks very simple, 2 2 x + y = M0 + N0 − + , tc t
If M0 = N0 then x+ = x− = 1/t and μ(t) = ν(t) = M0 (1 − 1/t).
(71)
If the gel interacts with the sol particles, then the masses μ(t) and ν(t) meet the following set of equations: (76) M0 − μ = M0 e−νt , N0 − ν = N0 e−μt .
or μ + ν = M0 + N0 − M0 Zν − N0 Zμ .
Here the notation p = N0 /M0 is introduced. The mass of N -gel goes to N0 as t −→ ∞. Indeed, 1 . ν ≈ N0 − x− = N0 1 − √ √ 2( M0 − N0 )t 2
(70)
(62)
On the other hand, μ = M0 (1 − Zν )
and the concentration of the residual sol is √ M0 − μ(∞) = M0 (1 − p)2 .
The part of residual monomer can also be found in the general case. Indeed, c1,0 (∞) = c1,0 (0)Zν (∞). (73)
C. Passive gel
A0 (x,y) = M0 x,
The solution to this set is readily found. Finally from Eq. (63) we have 1 2s μ(t) = M0 − s − + s 2 + , t t (66) 1 2s 2 ν(t) = N0 − s − − s + , t t √ 2 √ where s = ( M0 − N0 ) /2. At large t 2 (67) x+ ≈ 2s + , t 1 x− ≈ . (68) 4st 2 If M0 exceeds N0 then not all M-monomers will occur in the gel phase. It is possible to find the concentration √ √ of the residual monomer. At large t x+ ≈ 2s = ( M 0 − N 0 )2 and according to Eqs. (65) and (68) μ(∞) x+ √ = 2 p − p2 = 1− (69) M0 M0
Let us find the second moment, ∂ ∂F ∂A(xe−N ) Q2,0 (t) = x x = x ∂x ∂x x=y=1 ∂x x=y=1 =x
(65)
∂A0 (ζ ) = M0 ζx |x=y=1 , ∂x x=y=1
(77)
where ζ (x,y; t) and θ (x,y; t) are defined by the equalities x = ζ eN −N0 θt , y = θ eM−M0 ζ t . (78)
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We differentiate these two equations with respect to x, put x = y = 1, and find after some algebra M0 e−νt 1 − M0 N0 e−(μ+ν)t t 2 M0 − μ = . 1 − (M0 − μ)(N0 − ν)t 2
Q2,0 =
(79)
Here we used the fact that ζ = e−νt and θ = e−μt at x = y = 1. In the same way we find the other two moments, Q0,2 =
N0 − ν , 1 − (M0 − μ)(N0 − ν)t 2
Q1,1 =
t(M0 − μ)(N0 − ν) . 1 − (M0 − μ)(N0 − ν)t 2
(80)
The expressions for the second moments for general initial conditions are derived in Appendix C. E. Composition spectrum
We begin by considering the monodisperse initial conditions. In this case A0 (ξ ) = M0 ξ , B0 (η) = N0 η. The calculation of integrals is not difficult,
tnM0 ξ 1 (tnM0 )m e φA (n,m; t) = . (81) dξ = 2π i ξ m+1 m! Similarly, (tmN0 )n . (82) n! Hence according to Eq. (41) we find the composition spectrum, φB (m,n; t) =
the final expression for the composition spectrum. In the case of the active gel M(t) = M0 t and N (t) = N0 t. For the passive gelation we should subtract from this expression the contribution of the gel mass. From the first sight, this modification cannot lead to appreciable changes to the spectrum. However, it is not so. The postcritical spectrum is algebraic in the case of the passive gelation and it exponentially drops down otherwise. (iii) The time dependencies of the first and the second moments of the composition distribution found in Sec. V help one to understand the qualitative picture of the sol-gel transition in coagulating mixtures. Below the gelation time the total sol mass concentration is conserved together with the total mass of each component in the sol particles. On crossing the critical point the sol begins to lose particles spending them for the creation of a giant gel particle which either interacts with the sol particles (the active gelation) or the particles are lost because the total mass flux remains finite as the particle masses grow. The picture of the sol-gel transition in this case is not quite satisfactory, because it remains unclear what does it mean, the giant gel particle. This problem had been discussed in Refs. [22,32] in connection with coagulation kinetics in finite systems (the number of monomeric units is finite) and in the systems with sharp sink at large particle masses (the truncated models). (iv) Of special interest is the final stage of the gelation process. It occurs that a part of monomers does not have enough time to join to the gel. The component with lower concentration of monomers converts to gel earlier and leaves the rest of the sol without the partners allowing for further growth of the monomers. The analytical result [Eqs. (66) and
mn−1 nm−1 −nM−mN e . (83) m!n! This result reproduces the known composition spectrum, cm,n (t) = M0n N0m t m+n−1
mn−1 nm−1 −nM0 t−mN0 t e , (84) m!n! for active gelation [15]. In the case of passive gelation the postcritical composition spectrum looks different: we should t substitute M(t) = M0 (t) − tc μ(t )dt and N (t) = N0 (t) − t tc ν(t )dt into Eq. (83). The expressions for μ(t) and ν(t) are known [see Eq. (66)]. The respective integrals can be expressed in terms of elementary functions. The results are not reproduced here because they are rather cumbersome. cm,n (t) = M0n N0m t m+n−1
VI. RESULTS AND DISCUSSION
(i) The composition spectrum cm,n (t) is expressed in terms of the exponentiated bivariate generating function for the initial spectrum cm,n (0) [Eq. (41)]. Although the application of this result demands some efforts, they are not huge, especially when the initial sol does not contain mixed particles. In the latter case the problem reduces to the evaluation of a couple of ordinary contour integrals. The traditional saddle point method applies for the asymptotic estimation of these integrals in the limit of large m and n [Eq. (56)]. (ii) The difference between the postcritical behavior of the two scenarios (active and passive gelation) is only related to the different time behavior of the factor enM(t)+mN (t) entering
FIG. 2. The gel masses vs time for active and passive gelation. Calculations are done for M0 = 0.8 and N0 = 0.2. Solid lines correspond to the active gelation. In this case the gel grows faster than for passive gelation (dashed lines). The upper dashed curve never reaches the upper limit μ = 0.8 which means that a part of m monomers has not enough time for finding their partners for coagulation. Neither are they themselves able to agglomerate.
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FIG. 3. Postcritical behavior of the second moments of the composition distribution vs time.
(76)] is found for the case of initially monodisperse particles and displayed in Fig. 2. (v) The second moments of the composition spectrum in the postcritical period are shown in Fig. 3 as the functions of time for active gel [Eqs. (79) and (80)]. In the case of passive gelation after the critical time the sums in the expressions for these moments diverge.
centration of the first component in the particle) instead of m and n. Respectively, the coagulation kernel depends on the particle masses g1 and g2 , and the particle compositions β1 and β2 , ˜ 1 ,β1 |g2 ,β2 ). (85) K(m1 ,n1 |m2 ,n2 ) → K(g If the function K(m1 ,n1 |m2 ,n2 ) is a homogeneous function of its variables, K(am1 ,bn1 |am2 ,bn2 ) = a μ bν K(m1 ,n1 |m2 ,n2 ), (86) ˜ 1 ,β1 |g2 ,β2 ) is the homogeneous function of g1 and then K(g g2 with the homogeneity exponent λ = μ + ν, i.e., λ ˜ ˜ K(ag (87) 1 ,β1 |ag2 ,β2 ) = a K(g1 ,β1 |g2 ,β2 ). The sol-gel transition should thus occur at λ > 1 or μ + ν > 1. The assumption that the particle compositions play a secondary role in the gelation process is adopted. In this paper the case μ = ν = 1 was considered. Starting with the approach based on the evolution equation for the composition spectrum c(m,n; t) (the concentration of m,n-mers) at time t (Sec. II) the sol-gel transition in the coagulating mixture has been considered. In my opinion, the most remarkable result of this paper is the expression of the composition spectrum through the exponentiated generating function for the initial composition spectrum [Eq. (41)]. Of principal interest are the results on the global characteristics of the coagulation system: composition of the gel and the second moments. The similarity with the gelation in one component systems is apparent, although the description of the process in mixtures is much more complex. ACKNOWLEDGMENT
The financial support from Project No. 14-03-00507 of Russian Foundation of Basic Research is acknowledged.
VII. CONCLUDING REMARKS
APPENDIX A: SOLUTION OF EQ. (19)
The shift that we experienced in the past decades in our understanding of the coagulation kinetics and especially of the coagulation-gelation processes was not swift, but still rather significant. First of all, the nature of the sol-gel transition became clear, especially the fact that there is nothing mysterious in the notion of “gel.” It turns out that the gel is either a giant particle whose concentration is zero in the thermodynamic limit (active gelation) or a passive deposit that forms due to the coagulation growth within a finite range of the particle sizes; I mean the truncated model of coagulation [32] where a sharp sink instantly removes the particles with masses exceeding a given very large value G. As was shown in Ref. [33] the dynamics of the sol-gel transition can be satisfactorily described by using the standard Smoluchowski approach that well reproduces the behavior of the sol even in the postcritical period except for a very small interval of time in the vicinity of the critical point. At present there exists the common opinion on the form of the coagulation kernels associated with the gelation processes in the single component systems. These kernels should grow faster than K(g,g) ∝ g [35]. Naturally, the question comes up: what type of coagulation kernels can lead to gelation in mixtures? Although nobody yet investigated this problem, a criterion can be formulated. Indeed, let us characterize each cluster by its total mass g = m + n and the particle composition β = m/g (the relative con-
Let us look for the solution to the set Eq. (19) in the form (A,B,x,y,t) = 0
and
(A,B,x,y,t) = 0, (A1)
where the functions A(x,y; t) and B(x,y; t) are the solutions to the set Eq. (A1) and and are yet unknown functions of x,y,t. On differentiating Eq. (A1) over t we come to the set of ˙ two linear algebraic equations for A˙ and B, ˙ = 0, A A˙ + B B˙ +
˙ = 0. (A2) A A˙ + B B˙ + ˙ The Let us solve the set Eq. (A2) with respect to A˙ and B. result is ˙ − ˙ B ) B˙ = D−1 (A ˙ − ˙ A ). (A3) A˙ = D−1 (B Here D = (A B − B A ).
(A4)
Similar equations come up in differentiating Eq. (A1) over x and y. We substitute thus found derivatives of A an B into Eq. (19) and come to the set of two homogeneous ˙ + xBx + yAy linear equation for the combinations − ˙ − xBx − yAy , and ˙ + xBx + yAy ]B + [ ˙ − xBx − yAy ]B = 0, [− ˙ − yAy − xBx ]A + [− ˙ + yAy + xBx ]A = 0. [
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dξ dη d Aη eaA 1 ebB n b η dξ ξ m m 1 Aη eZ dηdξ Aξ η eZ dηdξ = − b ξ m+1 ηn b ξ m ηn a Aξ Aη eZ dηdξ − . b ξ m ηn
Assuming that D = 0 we obtain two linear equations for the functions and ,
=−
∂ ∂ ∂ ∂ ∂ ∂ = Bx + Ay and = Ay + Bx . ∂t ∂x ∂y ∂t ∂y ∂x (A6) These equations are readily solved by the method of characteristics. The equations for the characteristics [for both of Eqs. (A6)] are dt = −
dy dA dB dx =− = = . xB yA 0 0
(A7)
Hence (x,y,t,A,B) = φ(xeAt ,yeBt ,A,B),
(A8)
(x,y,t,A,B) = ψ(xeAt ,yeBt ,A,B),
where φ and ψ are yet arbitrary functions. Now we can solve the set of Eqs. (A1) and find A(x,y; t) = A0 (xeBt ,yeAt ),
(A9)
B(x,y; t) = B0 (xeBt ,yeAt ).
Here and below the subscripts ξ and η denote the derivatives over these variables. Similarly, we have dξ dη aA+bB e Aξ Bη ξ m ηn aA 1 1 aA dξ bB bB dξ dη d Aξ e = de = − Aξ e e b ξ m ηn b ξ m dη ηn n 1 Aξ eZ dηdξ Aξ η eZ dηdξ = − m n+1 b ξ η b ξ m ηn a Aξ Aη eZ dηdξ − . (B3) b ξ m ηn Hence
It is convenient to introduce ξ (x,y; t) = xeBt ,
I3 = and
η(x,y; t) = yeAt .
(A10)
Then A = A0 (ξ,η),
B = B0 (ξ,η).
(A11)
APPENDIX B: EVALUATION OF THE INTEGRAL
In what follows, instead of double integrals,
1 1 dx dy, 2π i 2π i
in Eqs. (37) and (39) the single integration symbol will be used. Then Eq. (37) can be rewritten as follows: ∂ ∂ 1 + (I1 − tI2 + t 2 I3 ), cm,n (t) = m + n ∂a ∂b a=mt, b=nt
∂B0 ∂A0 eZ dξ dη eZ dξ dη ξ + η , , I = 2 ξ m+1 ηn+1 ξ m+1 ηn+1 ∂ξ ∂η Z ∂B0 ∂A0 e dξ dη ∂B0 ∂A0 I3 = − , ξ m ηn ∂ξ ∂η ∂η ∂ξ
and Z = aA0 + bB0 . Let us begin with I3 . Integrating by parts gives dξ dη Z e Aη Bξ ξ m ηn dη 1 = Aη eaA m n debB b ξ η
m b
n Aη eZ dηdξ − m+1 n ξ η b
Aξ eZ dηdξ . ξ m ηn+1
Let us convert Aξ to Bξ in the last integral, aA bB 1 1 Aξ eZ dηdξ bB de aA dη d e = = − e e ξ m ηn+1 a ξ m ηn+1 a ηn+1 dξ ξ m dξ dη dξ dη m b = eZ m+1 n+1 − Bξ eZ m n+1 . a ξ η a ξ η And finally we have dξ dη dξ dη m mn I3 = Aη eZ m+1 n − eZ m+1 n+1 b ξ η ab ξ η dξ dη nb + Bξ eZ m n+1 . ba ξ η Let us return to our integral, eaA+bB U (a,b) = J (ξ,η; t)dξ dη ξ m+1 ηn+1 2 t n t 2m eaA+bB 1+ + − 2t ξ ηFξ η = ξ m+1 ηn+1 a b t 2 mn dξ dη. − ab
(B1) where I1 =
(B2)
It is important to note that U (nt,mt) =
dxdy = 0, x m+1 y n+1
as should it be. Now we differentiate U (a,b) (m + n)c(m,n; t) = (∂a + ∂b )U (a,b)|a=nt, b=mt . Finally we have 1 cm,n (t) = mnt 032121-8
∂ 2 F0 et(nA0 +mB0 ) 1 − tξ η dξ dη. ξ m+1 ηn+1 ∂ξ ∂η
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Solving these equation yields
APPENDIX C: SECOND MOMENTS IN GENERAL CASE
In the case of the general initial condition the gel masses should be found from the set of two equations, M0 − μ = A0 (e
−νt
,e
−μt
), N0 − ν = B0 (e
−νt
,e
−μt
).
ζx =
(C1) θx
The second moments are
∂ ∂F Q2,0 (t) = x x ∂x ∂x
(1 −
,
=
(1 − tθ A0θ )e−νt , − tθ A0θ ) − t 2 ζ θ A0ζ B0θ
tζ B0ζ )(1
tθ A0ζ e−νt (1 − tζ B0ζ )(1 − tθ A0θ ) − t 2 ζ θ A0ζ B0θ
(C4)
.
Similarly we have
x=y=1
∂ ∂F Q0,2 (t) = y y , ∂y ∂y x=y=1
θy =
∂ 2 F Q1,1 (t) = xy ∂x∂y
ζy
.
)e−μt (1 − tζ B0ζ (1 − tθ A0θ )(1 − tζ B0ζ ) − t 2 ζ θ A0θ B0ζ
,
−μt e tζ B0θ = . (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
(C5)
x=y=1
So we must find the derivatives of A, ∂A ∂A0 ∂A0 = ζ + θ . ∂x x=1,y=1 ∂ζ x ∂θ x
We thus find (C2)
In order to proceed we must find ζx and θx . On differentiating both sides of Eq. (A10) (both equations) we find the set of equations for ζx and θx , eνt = ζx (1 − tζ B0ζ ) − tθx ζ B0θ ,
0 = θx (1 − tθ A0θ ) − ζx tθ A0ζ .
A0ζ e−νt , (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
Q02 =
B0θ e−μt , (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
Q11 =
A0ζ B0θ tζ e−μt + A0,θ e−μt (1 − tζ B0,ζ ) . (1 − tθ B0θ )(1 − tζ A0ζ ) − t 2 θ ζ A0θ B0ζ
and
(C3)
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Q20 =
(C6)
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