Exact post-critical particle mass spectra in a family of gelling systems

The critical behaviour of the coagulating systems, where the coagulation efficiency grows with the masses of colliding particles g and l asK(g, l) = gl (1/2 < α 1) is studied. The instantaneous sink that removes the particles with masses exceeding G is introduced which allows one to define the gel as a deposit of particles with masses between G + 1 and 2G. This system displays critical behaviour (the sol–gel transition) as G −→ ∞. The exact post-critical particle mass spectrum is shown to be an algebraic function of g times a growing exponent. All critical parameters of the systems are determined as the functions of α including the critical times. PACS numbers: 79.60.Bm, 42.65.Lm, 03.65.−w, 52.90.+z Let us consider a system of N particles that move chaotically, collide, and on colliding coalesce producing a daughter particle with the mass equal to the sum of masses of parent particles. This process is commonly referred to as coagulation. Symbolically it can be presented as an irreversible binary chemical reaction (g) + (l) −→ (g + l). (1) Here g and l are the masses of colliding particles measured in units of a monomeric mass, i.e., the integers g and l are simply the numbers of monomers in the particles. The rate of the process (1) K(g, l) (the coagulation kernel) is supposed to be a known homogeneous function of its arguments, i.e., K(ag, al) = aK(g, l). From the first sight the coagulation process looks absolutely offenceless. It is difficult to imagine that such simple systems are able to display something unusual. And nevertheless they do. At λ > 1 the coagulating systems experience the sol–gel transition, i.e., they separate into sol and gel fractions after a finite interval of time tc (see e.g., recent articles Leyvraz (2003, 2006), Ben-Naim and Krapivsky (2005a, 2005b), Lushnikov (2006), and references therein). The sol part is a collection of g-mers whose concentrations cg(t) are the solutions to the kinetic equation of the process (1). Less clear is how to introduce the gel. It does not appear in the kinetic equation explicitly and can only be detected by the behaviour of the sol mass which begins to drop down with time after t = tc. The sol is considered to transfer a 1751-8113/07/040119+07$30.00 © 2007 IOP Publishing Ltd Printed in the UK F119 F120 Fast Track Communication part of its mass to the gel whose mass begins to grow with time after t = tc, although we see no gel in our equations. The sol part disappears either gradually or instantly. In the latter case the sol and the gel fractions cannot coexist. Below I consider only the first scenario. My starting point is the truncated Smoluchowski’s equation that describes the coagulating particles whose masses are limited with a maximal (cutoff) mass G. This truncated model describes the coagulation process in the system with a sink that instantly removes the particles with the masses exceeding G (Lushnikov and Piskunov 1982, 1983, Lushnikov 2006). The Smoluchowski equation for the number concentrations cg(t) of actively coagulating particles (g G) comprising exactly g monomers at time t looks as follows: dcg(t) dt = 1 2 g−1 ∑ l=1 K(g − l, l)cg−l (t)cl(t)− cg(t) G ∑ l=1 K(g, l)cl(t). (2) The first term on the right-hand side (RHS) of this equation describes the gain of g-mers due to the reaction (g − l) + (l) −→ (g). The second term is responsible for g-mer losses due to their sticking to all other particles. This second term contains the cutoff mass G, i.e., we assume that all particles with masses G+1,G+2, . . . , 2G do not participate in the coagulation process and form a passive deposit. We introduce its spectrum and denote the concentrations of deposited particles as c+ g(t). It is clear that only the gain term contributes to the rate of change to c+ g , dc+ g(t) dt = 1 2 G ∑ l=g−G K(g − l, l)cg−lcl . (3) In equations (2) and (3) the dimensionless units are used, i.e., the concentrations are measured in units of the initial particle concentration c1(0), and the unit of time is 1/K(1, 1)c1(0). The initial condition to equation (2) is chosen in the form cg(t = 0) = δg,1, (4) i.e., the coagulation process starts with a set of monomers whose total mass concentration M = 1. In equation (4) δg,l is the Kroneker delta. The coagulation kernel K(g, l) = gl (5) with λ = 2α > 1 is used below. So far only the case α = 1 is well studied, although many interesting results for more general coagulation kernels are reported in Hendriks et al (1983). In what follows I demonstrate how the sol–gel transition can be investigated for the kernels given by equation (5). To this end I apply the truncated model of coagulation and a rather artificial trick proposed in Lushnikov (1973). Although the problem cannot be resolved entirely exactly (like in the case α = 1), a full asymptotic analysis of the post-critical stage is possible. Let us introduce the new variable τ and the new unknown functions νg(τ ), τ = ∫ t 0 c1(t ′) dt ′, νg(τ ) = cg(τ )/c1(τ ). (6) On substituting this into equation (2) yields two equations for νg(τ ) and c1(τ ): dνg(τ ) dτ = 1 2 g−1 ∑ l=1 K(g − l, l)νg−l (τ )νl(τ )− νg(t) G ∑ l=1 [K(g, l)−K(1, l)]νl(τ ) (7) with ν1(τ ) = 1 and νg(0) = 0 (g > 1). This equation does not contain c1(τ ). The second Fast Track Communication F121 equation allows one to find c1(τ ), once νg(τ ) are known, dc1(τ ) dτ = −c1(τ ) G ∑ l=1 K(1, l)νl(τ ). (8) The initial condition to this equation is c1(0) = 1. For the kernel K(g, l) = gl equations (7) and (8) reduce to dνg(τ ) dτ = 1 2 g−1 ∑ l=1 (g − l)lνg−l (τ )νg(τ )− (g − 1)νg(τ ) G ∑ l=1 lνl(τ ), (9) dc1(τ ) dτ = −c1(τ ) G ∑ l=1 lνl(τ ). (10) Let us try to look for the solution to equation (9) in the form νg(τ ) = h(g−1)(τ )rg(τ ), (11) where the function h(τ) is introduced by the equation dh dτ + G ∑ l=1 lrlh (l) = κ, (12) with κ being yet an unknown constant and h(0) = 0. On substituting these νg(τ ) into equation (9) yields the set of differential equations for determining rg , κ(g − 1)rg + hg dτ = 1 2 g−1 ∑ l=1 (g − l)αlαrg−lrlh[(g−l)α+lα−gα]. (13) A very important identity c1(t)/h(t) = 1/κt (14) follows from equations (10) and (12). On combining these equations yields dτ ln(c1/h) = −κ/h. Next, applying the definition of τ (equation (6)) leads to the closed equation for c1/h: dt (c1/h) = −κ(c1/h) or c1/h = (κt)−1. Pay attention that the integration constant t0 = 0 in this equation, because c1(0) = 1 and h(0) = 0. Now let us do the decisive step: namely, we assume that after the critical time the functions rg(τ ) are strictly independent of τ , i.e., the term containing dτ rg can be crossed out from equation (13). I shall justify this statement later on. The above assumption allows us to find the coefficients rg from the recurrence, κ(g − 1)rghα = 1 2 g−1 ∑ l=1 (g − l)αlαrg−lrlh[(g−l)α+lα ] (15) with r1 = 1. For determining the dependence of rg on h let us introduce r̃g = rg(1). From equation (15) one has κ(g − 1)r̃g = 1 2 g−1 ∑ l=1 (g − l)lr̃g−l r̃l . (16) F122 Fast Track Communication Table 1. Parameter B of the post-critical particle mass spectrum, equation (29), critical time tc , equation (35), and the separation constant κ , equation (21). α 0.6 0.7 0.8 0.9 1.0 B 0.745 0.586 0.493 0.436 0.399 tc 3.486 2.157 1.530 1.194 1.000 κ 3.503 3.215 3.005 2.842 2.718 Noticing that the combination sg = rghα entering equation (15) also satisfies equation (16) with a different first term of the sequence (s1 = h) allows us to derive the explicit dependence of rg on h, rg(h) = r̃gh(g−gα). (17) Let us return to the separation constant κ and then determine the asymptotic behaviour of r̃g at large g. To this end we introduce two generating functions D0(z) and Dα(z), where Dσ(z) = ∞ ∑ g=1 g zr̃g (18) (pay attention that here the summation goes up to ∞). From equation (16) one finds 2κ[Dα(z)−D0(z)] = D α(z). (19) On solving this equation with respect to Dα yields Dα(z) = κ − √ κ2 − 2κD0(z). (20) The separation constant is chosen as κ = 2D0(1) = Dα(1). (21) This choice locates the singularity of both generating functions at z = 1 and removes thus the exponential factors from r̃g . Equation (20) was analysed in Hendriks et al (1983). The result is convenient to present in terms of Dσ(1): r̃g ≈ κ √ D1(1) 2πDα(1) g−(α+3/2). (22) This expression is found by expanding D0(z) with respect to z− 1, using the Stirling formula for the expansion coefficients of the function √ 1 − z, and the obvious formulaD′ 0(1) = D1(1). Here prime stands for the differentiation with respect to z. The values of Dα(1) and D1(1) can be determined from a numerical analysis of equation (16) (see table 1). Now we are ready to analyse the solution of equation (12). Since κ = Dα(1) = ∑∞ l=1 l r̃l (equation (21)), we can rewrite equation (12) in the form dh dτ + G ∑ l=1 lr̃l(h l − 1) = ∞ ∑ l=G+1 lr̃l . (23) First we put G = ∞. Then the solution to equation (23) can be expressed in terms of D0(h), ∫ h 0 dh′ Dα(1)−Dα(h′) = τ. (24) The integral on the left-hand side of this equation converges at h′ = 1 which means that h(τ) reaches unity during a finite interval of τ = τc. According to equations (6) and (14) Fast Track Communication F123 τ(t) −→ ∞ as t −→ ∞, so the function h(τ) = 1 and the particle mass spectrum remain algebraic after the sol–gel transition. Hence, the condition h(τc) = 1 (25) defines the critical value of τ . At finite G the situation is similar. The function h(τ), grows with τ , reaches the value h = 1 and then exceeds unity by a little becoming independent of τ . Let us try to find the limiting value of h in the form h(τ) = exp(ξ/G). We substitute this h into equation (23) and replace the sums with the integrals. After some simple algebra we obtain the condition for determining ξ ∫ 1 0 x−3/2 [exp(ξx)− 1] dx = 2. (26) Remarkably, that ξ = 0.854 is independent of α. In the post-critical period h = exp(ξ/G) which gives us the grounds to assume that rg are independent of τ in the post-critical period. Indeed, rg(τ ) = rg(h(τ )), because drg/dτ = (drg/dh)(dh/dτ) and dh/dτ is a function of h (see equation (12)). Hence, the spectrum has the form cg(t) = 1 t √ D1(1) 2πDα(1) g−(3+λ)/2 exp[ξ(g/G)]. (27) In order to understand what is going on in such systems in the post-critical period we calculate the rate of the sol mass transfer through the cutoff mass G, dMsol dt = − 2 ∑ l,m (l + m)K(l,m)clcm, (28) where Msol(t) = ∑G l=1 lcl(t) and the summation on the right-hand side of equation (28) goes over all integers l, m obeying the conditions: l, m G and l + m > G. On replacing the sum in this expression with the integral and keeping in mind the asymptotic structure of cg(t) in the post-critical period, cg = G−γ Bt−1c(x), (29) where B = D1(1)/(2πDα(1)), x = g/G and c(x) = x−γ e , we find dMsol dt = − 2 (B/t)2G3+λ−2γ ∫ 1 0 dx ∫ 1 1−x dy(x + y)K(x, y)c(x)c(y). (30) One immediately sees that the rate of mass transport through the cutoff mass is independent of G for any homogeneous kernels if γ = (3 +λ)/2 and the integrals on the RHS of this equation converge. The truncated model permits for calculating the spectrum of the deposit. This spectrum stretches from g = G to g = 2G (see equation (3)). At large masses the sum on the RHS of this equation can be converted to the integral. On doing this we get c g(t) = B2 2G2 ( 1 tc − 1 t ) F(s), (31)