Diffusion-controlled death of A-particle and B-particle ...

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PHYSICAL REVIEW E 77, 030101共R兲 共2008兲

Diffusion-controlled death of A-particle and B-particle islands at propagation of the sharp annihilation front A + B \ 0 Boris M. Shipilevsky Institute of Solid State Physics, Chernogolovka, Moscow district, 142432, Russia 共Received 28 September 2007; revised manuscript received 21 December 2007; published 6 March 2008兲 We consider the problem of diffusion-controlled evolution of the A-particle-island–B-particle-island system at propagation of the sharp annihilation front A + B → 0. We show that this general problem, which includes as particular cases the sea-sea and island-sea problems, demonstrates rich dynamical behavior from selfaccelerating collapse of one of the islands to synchronous exponential relaxation of both islands. We find a universal asymptotic regime of the sharp-front propagation and reveal the limits of its applicability for the cases of mean-field and fluctuation fronts. DOI: 10.1103/PhysRevE.77.030101

PACS number共s兲: 05.40.⫺a, 82.20.⫺w

For the last decades the reaction-diffusion system A + B → 0, where unlike species A and B diffuse and annihilate in a d-dimensional medium, has acquired the status of one of the most popular objects of research. This attractively simple system, depending on the initial conditions and on the interpretation of A and B 共chemical reagents, quasiparticles, topological defects, etc.兲, provides a model for a broad spectrum of problems 关1,2兴. A crucial feature of many such problems is the dynamical reaction front—a localized reaction zone which propagates between domains of unlike species. The simplest model of a reaction front, introduced almost two decades ago by Galfi and Racz 共GR兲 关3兴, is a quasi-onedimensional 共quasi-1D兲 model for two initially separated reactants which are uniformly distributed on the left side 共x ⬍ 0兲 and on the right side 共x ⬎ 0兲 of the initial boundary. Taking the reaction rate in the mean-field form R共x , t兲 = ka共x , t兲b共x , t兲, GR discovered that in the long-time limit kt → ⬁ the reaction profile R共x , t兲 acquires the universal scaling form R = RfQ

冉 冊

x − xf , w

共1兲

where x f ⬀ t1/2 denotes the position of the reaction front center, R f ⬀ t−␤ is the height, and w ⬀ t␣ is the width of the reaction zone. Subsequently, it has been shown 关4–8兴 that the mean-field approximation can be adopted at d ⬎ dc = 2, whereas in 1D systems fluctuations play the dominant role. Nevertheless, the scaling law 共1兲 takes place at all dimensions with ␣ = 1 / 6 at d ⬎ dc = 2 and ␣ = 1 / 4 at d = 1, so that at any d the system demonstrates a remarkable property of the effective dynamical repulsion of A and B: on the diffusion length scale LD ⬀ t1/2 the width of the reaction front asymptotically contracts unlimitedly: w / LD → 0 as t → ⬁. Based on this property a general concept of the front dynamics, the quasistatic approximation 共QSA兲, has been developed 关4,5,8,9兴 which consists in the assumption that for sufficiently long times the kinetics of the front is governed by two characteristic time scales. One time scale tJ = −共d ln J / dt兲−1 controls the rate of change in the diffusive current, J = JA = 兩JB兩, of particles arriving at the reaction zone. The second time scale t f ⬀ w2 / D is the equilibration time of the reaction front. Assuming that t f / tJ Ⰶ 1 from the QSA in 1539-3755/2008/77共3兲/030101共4兲

the mean-field case with DA,B = D it follows that 关4,5,9兴 R f ⬃ J/w,

w ⬃ 共D2/Jk兲1/3 ,

共2兲

whereas in the 1D case w acquires the k-independent form w ⬃ 共D / J兲1/2 关4,5兴. On the basis of the QSA a general description of spatiotemporal behavior of the system A + B → 0 has been obtained for arbitrary nonzero diffusivities 关10兴 which was then generalized to anomalous diffusion 关11兴, diffusion in disordered systems 关12兴, diffusion in systems with inhomogeneous initial conditions 关13兴 and to several more complex reactions. Following the simplest GR model 关3兴 the main attention has been traditionally focused on the systems with A and B domains having an unlimited extension—i.e., with an unlimited number of A and B particles, where asymptotically the stage of monotonous quasistatic front propagation is always reached: t f / tJ → 0 as t → ⬁. Recently, in 关14兴 a new line in the study of the A + B → 0 dynamics has been developed under the assumption that the particle number of one of the species is finite; i.e., an A-particle island is surrounded by a uniform sea of B particles. It has been established that at sufficiently large initial number of A particles, N0, and a sufficiently high reaction rate constant k the death of the majority of island particles N共t兲 proceeds in the universal scaling regime N = N0G共t / tc兲, where tc ⬀ N20 is the lifetime of the island in the limit k , N0 → ⬁. It has been shown that while dying, the island first expands to a certain maximal amplitude xM f ⬀ N0 and then begins to contract by the law x f = xM f ␨ f 共t / tc兲 so that on reaching xM f 共the turning point of the front兲 t M /tc = 1/e,

N M /N0 = 0.19886 . . . ,

共3兲

and, therefore, irrespective of the initial particle number and dimensionality of the system ⬇4 / 5 of the particles die at the stage of the island expansion and the remaining ⬇1 / 5 at the stage of its subsequent contraction. In this Rapid Communication we consider a much more general problem of the A + B → 0 annihilation dynamics with the initially separated reactants under the assumption that the particle number of both species is finite. More precisely, we consider the problem of the diffusion-controlled death of A-particle and B-particle islands at propagation of the sharp

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©2008 The American Physical Society

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BORIS M. SHIPILEVSKY

annihilation front A + B → 0. We show that this island-island 共II兲 problem, of which particular cases are the GR sea-sea 共SS兲 problem and the island-sea 共IS兲 problem 关14兴, exhibits rich dynamical behavior and we reveal its most essential features. Let in the interval x 苸 关0 , L兴 particles A with concentration a0 and particles B with concentration b0 be initially uniformly distributed in the islands x 苸 关0 , ᐉ兲 and x 苸 共ᐉ , L兴, respectively. Particles A and B diffuse with diffusion constants DA and DB, and when meeting they annihilate A + B → 0 with a reaction constant k. We will assume, as usual, that concentrations a共x , t兲 and b共x , t兲 change only in one direction 共flat front兲, and we will consider that the boundaries x = 0 , L are impenetrable. Thus, our effectively one-dimensional problem is reduced to the solution of the problem

⳵a/⳵t = DAⵜ2a − R,

⳵b/⳵t = DBⵜ2b − R,

共4兲

in the interval x 苸 关0 , L兴 at the initial conditions a共x , 0兲 = a0␪共ᐉ − x兲 and b共x , 0兲 = b0␪共x − ᐉ兲 and the boundary conditions 兩 ⵜ 共a , b兲兩x=0,L = 0 where ␪共x兲 is the Heaviside step function. To simplify the problem essentially we will assume DA = DB = D. Then, by measuring the length, time, and concentration in units of L, L2 / D, and b0, respectively—i.e., assuming L = D = b0 = 1—and defining the ratio of initial concentrations a0 / b0 = r and the ratio ᐉ / L = q, we come from Eqs. 共4兲 to the simple diffusion equation for the difference concentration s = a − b,

⳵s/⳵t = ⵜ2s,

共5兲

in the interval x 苸 关0 , 1兴 at the initial conditions s0„x 苸 关0,q兲… = r,

s0„x 苸 共q,1兴… = − 1,

共7兲

According to the QSA for large k → ⬁ at times t ⬀ k−1 → 0 there forms a sharp reaction front w / x f → 0 so that the solution s共x , t兲 defines the law of its propagation, s共x f , t兲 = 0, and the evolution of particle distributions, a = s共x ⬍ x f 兲 and b = 兩s兩共x ⬎ x f 兲. In the limits sea-sea 关3兴 共ᐉ → ⬁ , L → ⬁兲 or island-sea 关14兴 problem 共ᐉ finite, L → ⬁兲 the corresponding solutions sSS共x , t兲 and sIS共x , t兲 describe the initial stages of the system’s evolution at times 冑t Ⰶ q, 1 − q, and q Ⰶ 冑t Ⰶ 1, respectively. The general solution to Eqs. 共5兲–共7兲 for arbitrary r, q, and t has the form ⬁

s共x,t兲 = ⌬ + 兺 An共r,q兲cos共n␲x兲e−n

2␲2t

,

共8兲

n=1

where coefficients An共r , q兲 = 2共r + 1兲sin共n␲q兲 / n␲ and ⌬共r , q兲 = NA − NB = rq − 共1 − q兲 is the difference of the reduced number of A and B particles which remains constant. At t ⬎ 1 / ␲2 the main mode in Eq. 共8兲 becomes dominant, so neglecting the contribution of small-scale modes we find 2

s = ⌬ + A1共r,q兲cos共␲x兲e−␲ t + ¯ .

共9兲

Taking s共x f , t兲 = 0 we obtain from Eq. 共9兲 the law of the front motion,

共10兲

where coefficient C can be represented in the form C = − ⌬/A1 = q共r쐓 − r兲/A1 = 共q쐓 − q兲/q쐓A1 ,

共11兲

where q쐓 = 1 / 共r + 1兲 and r쐓 = 共1 − q兲 / q are the critical values of q and r at which C reverses its sign. From Eq. 共10兲 it follows that at 兩C兩 ⬍ 1 / e and r ⫽ r쐓 , q ⫽ q쐓, when the ratio of the initial particle numbers,

␳=

NA0 r 共1 − q쐓兲q = = ⫽ 1, NB0 r쐓 共1 − q兲q쐓

共12兲

the front x f 共t兲 moves either towards the boundary x = 0 共␳ ⬍ 1兲 or towards the boundary x = 1 共␳ ⬎ 1兲 so that in the limit k → ⬁ the island of a smaller particle number 共A or B, respectively兲 dies within a finite time tc = 共1/␲2兲兩ln兩C兩兩.

共13兲

From Eqs. 共10兲 and 共13兲 in the time interval 1 / ␲2 ⬍ t 艋 tc we obtain x f = 共1/␲兲arccos共⫾e␲

2共t−t 兲 c

兲

共14兲

共here and in what follows the upper sign corresponds to ␳ ⬍ 1 and the lower sign corresponds to ␳ ⬎ 1兲, from which for the front velocity v ˙f = x˙ f we find v f = − ␲ cot共␲x f 兲 = ⫿ ␲/共冑e2␲

2共t −t兲 c

− 1兲.

共15兲

Making use then of Eq. 共13兲, for the distribution of particles 关a = s共x ⬍ x f 兲 , b = 兩s兩共x ⬎ x f 兲 关14兴兴 at ␳ ⫽ 1 we obtain s = ⌬共1 ⫿ cos共␲x兲e␲

共6兲

with the boundary conditions 兩 ⵜ s兩x=0,1 = 0.

2

cos共␲x f 兲 = Ce␲ t + ¯ ,

2共t −t兲 c

兲+ ¯ .

共16兲

Thus from the condition NA = 兰x0f sdx = NB + ⌬ we find the laws of decay of the A and B particle numbers, NA = 共兩⌬兩/␲兲共冑e2␲

2共t −t兲 c

− 1 ⫿ ␲x f 兲,

共17兲

and then we derive finally the diffusive boundary current in the vicinity of the front, J = 兩 − ⳵s/⳵x兩x=x f = ␲兩⌬兩冑e2␲

2共t −t兲 c

− 1,

共18兲

which according to 共2兲 defines the evolution of the amplitude R f 共t兲 and of the width of the front w共t兲. From Eqs. 共13兲–共18兲 we immediately come to the following important conclusions: for arbitrary r and q which satisfy the condition 兩C共r , q兲兩 ⬍ 1 / e, at ␳ ⬍ 1 or ␳ ⬎ 1, 共i兲 the motion of the front is the universal function of the “distance” to the collapse time tc − t with the remarkable property ⬎ x⬍ f 共tc − t兲 = 1 − x f 共tc − t兲; moreover, the front velocity v f is the unique function of x f with the remarkable symmetry x f ↔ 1 − x f , v f ↔ −v f ; 共ii兲 the reduced particle number NA / 兩⌬兩 and the reduced boundary current J / 兩⌬兩 are universal functions of tc − t with the remarkable properties NA⬍共tc − t兲 = NA⬎共tc − t兲 − 兩⌬兩 and J⬍共tc − t兲 = J⬎共tc − t兲. Introducing the relative time T = tc − t, from Eqs. 共13兲–共18兲 in the vicinity T Ⰶ 1 / ␲2 of the critical point tc we come to the universal power laws of self-accelerating collapse 共兩v f 兩 ⬀ T−1/2兲:

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⬎ 冑 x⬍ f ,1 − x f = 2T + ¯ ,

共19兲

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DIFFUSION-CONTROLLED DEATH OF A-PARTICLE…

NA⬍,NB⬎ = 共冑8/3兲␲2兩⌬兩T3/2 + ¯ ,

共20兲

J = 冑2␲2兩⌬兩冑T + ¯ .

共21兲

At large tc Ⰷ 1 / ␲2 far from the critical point T ⬎ 1 / ␲2 according to Eqs. 共13兲–共18兲 there is realized the intermediate 2 exponential relaxation regime 共兩v f 兩 ⬀ e−␲ T兲 = 1/2 ⫿ e−␲ T/␲ + ¯ , x⬍,⬎ f

共22兲

NA⬍,⬎ = 共兩⌬兩/␲兲e␲ T共1 ⫿ ␲e−␲ T/2 + ¯ 兲,

共23兲

2

2

2

2

2

J = ␲兩⌬兩e␲ T共1 − e−2␲ T/2 + ¯ 兲,

共24兲

which in the limit tc → ⬁共兩C兩 , 兩
− 1兩 → 0兲 becomes dominant. Thus, at large tc Ⰷ 1 / ␲2 the point x f ⬇ 1 / 2 共stationary front兲 is an “attractor” of trajectories. Exactly at the critical point ␳쐓 = 1 from Eqs. 共9兲 and 共10兲 we find x쐓f = 1 / 2 and obtain N쐓/N0 =

冉 冊

2 sin共␲q兲 −␲2t e + ¯, ␲2 q共1 − q兲 2

J쐓 = 2关sin共␲q兲/q兴e−␲ t + ¯ .

共25兲 共26兲

In order to answer the question of when and how the attractor x쐓f = 1 / 2 is reached it is necessary to retain the next term 共n = 2兲 in the sum 共8兲. With allowance for the first two terms one can easily obtain x쐓f = 1/2 − D共q兲e−3␲ t + ¯ , 2

共27兲

where D共q兲 = 共A2 / ␲A1兲 = sin共2␲q兲 / 2␲ sin共␲q兲. According to Eq. 共27兲 at q = 1 / 2 the coefficient D reverses its sign; therefore, as is to be expected, at q ⬍ 1 / 2 and q ⬎ 1 / 2 the front reaches the attractor x쐓f = 1 / 2 from the left and right, respectively. By combining Eqs. 共22兲 and 共27兲, at small but finite 2 2 兩C兩 we have x⬍,⬎ = 1 / 2 − Ce␲ t / ␲ − De−3␲ t + ¯. We thus conf clude that under the condition DC ⬎ 0 there arises the turning point of the front 共vM f = 0兲 with the coordinates

FIG. 1. 共Color online兲 Evolution of the front trajectories x f 共t兲 with growing ␳, calculated from Eqs. 共30兲 共blue lines兲 and 共31兲 共red circles兲 at ␳ = 0.5, 0.7, 0.9, 0.98, 1, 1.02, 1.1, and 2 共from left to right兲. The region of the scaling IS regime is shaded.

of ␳. Therefore, the system’s evolution at t Ⰷ q2 is determined by the sole parameter ␳. At q2 Ⰶ t Ⰶ 1 we have the scaling IS regime 关14兴 x f = 冑2t ln1/2共␳2/␲t兲,

t c共 ␳ 兲 = ␳ 2/ ␲ ,

共30兲

2 2 冑 with xM f = ␳ 2 / ␲e and t M = ␳ / ␲e. For t ⬎ 1 / 2␲ with allowance for two principal modes 共n = 1 , 2兲 we obtain from Eq. 共8兲 2

x f = 共1/␲兲arccos关G共␳,t兲e3␲ t/4兴,

冑

2

共31兲

2

and where G共␳ , t兲 = 1 + 8C共␳兲e−2␲ t + 8e−6␲ t − 1 C共␳兲 = 共1 − ␳兲 / 2␳. For the time of collapse tc共␳兲 we derive from Eq. 共8兲 the general equation for arbitrary ␳ ⫽ 1, ⬁

t M = 共1/4␲2兲ln共␭M 兩D/C兩兲 + ¯ ,

共28兲

共⫾1兲ne−n ␲ t 共␳兲 = ⫾ 兩C共␳兲兩, 兺 n=1

3/4 + ¯, xM f = 1/2 − m M D兩C/D兩

共29兲

from which in accordance with Eq. 共31兲 for the leading 共at small 兩C兩兲 correction to Eq. 共13兲 we find

where ␭ M = 3␲ , m M = 4 / 共3␲兲 , whereas at DC ⬍ 0 there arises the inflection point of the front trajectory 共兩vsf 兩 = min兩v f 兩兲 with the coordinates ts and xsf which are determined by Eqs. 共28兲 and 共29兲 with the coefficients ␭s = 3␭ M and ms = 2m M / 共3兲3/4. The analysis presented demonstrates the key points of the evolution of the island-island system at arbitrary r and q which satisfy the condition 兩C共r , q兲兩 ⬍ 1 / e 关according to Eqs. 共11兲 and 共12兲 this condition restricts the interval ␳l ⬍ ␳ ⬍ ␳u to the values of ␳l,u which are not too different from unity: at q Ⰶ 1 we find ␳l ⬇ 0.6 and ␳u ⬇ 4兴. Below we will focus on a detailed illustration of this evolution from the initial island-sea configuration 共q Ⰶ 1兲. A remarkable property of the island-sea configuration q Ⰶ 1 is that at r Ⰷ 1 the ⌬共␳兲 = ␳ − 1 value and all the coefficients An共␳兲 = 2␳ up to n ⬀ 1 / q Ⰷ 1 become unique functions 3/4

2 2

c

tc共␳兲 = 共兩ln兩C兩兩 ⫾ 兩C兩3 + ¯ 兲/␲2 .

共32兲

共33兲

Using small-t representations of the series 共32兲, one can easily show that, with growing ␳, tc initially grows by the law 2 tc共␳兲 = ␳2共1 + 4e−␲/␳ + ¯ 兲 / ␲; then, it passes through the critical point tc共␳쐓兲 → ⬁ according to Eq. 共33兲 and finally at large ␳ decays by the law tc共␳兲 ⬀ 1 / ln ␳. From Eqs. 共31兲 and 共17兲 for the starting points t M,s of front self-acceleration at small 兩C兩 we find t M,s/tc = 1/4 + ␤ M,s/兩ln兩C兩兩 + ¯ ,

共34兲

where ␤ M with the number of A particles, = ␤s / 2 = ln 3 / 4. Remarkably, the same as for the scaling IS regime 共3兲 and 共30兲 in the vicinity 兩␳ − ␳쐓兩 Ⰶ 1 the ratio t M / tc

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NAM,s / NA0 ⬀ 兩C兩1/4,

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BORIS M. SHIPILEVSKY

reaches the universal limit tM / tc = 1 / 4. In Fig. 1 are shown the calculated from Eqs. 共30兲 and 共31兲 trajectories of the front x f 共t兲, which illustrate the evolution of the front motion with the growing ␳. It is seen that to ␳ ⬇ 0.7 the death of the island A proceeds in the scaling IS regime 共30兲 共t M / tc = 1 / e兲; then, the x f 共t兲 trajectory begins to deform, and at small 兩␳ − ␳쐓兩 Ⰶ 1 the regime of the dominant exponential relaxation 共22兲–共24兲 and 共34兲 共t M / tc ⬇ 1 / 4兲 is reached. After the critical point ␳쐓 = 1 has been crossed, the death of the island A is superseded by the death of the island B, so the front trajectory becomes monotonous and the stopping point M of the front xM f , t M 共v f = 0兲 “transforms” to the point of maximal deceleration of the front xsf , ts 共vsf = min v f ⬀ 兩C兩3/4兲 which at large ␳ shifts by the law 1 − xsf ⬀ 1 / 冑ln ␳ with ts ⬀ 1 / ln ␳. One of the key features of the island-island problem is a rapid growth of the front width w while the islands are dying. Therefore, to complete the analysis we have to reveal the applicability limits for the sharp front approximation ␩ = w / min共x f , 1 − x f 兲 Ⰶ 1. By substituting Eq. 共21兲 into 共2兲 we obtain for the self-accelerating collapse ␩ ⬃ 共TQ / T兲␮ where MF = 1 / 冑兩⌬ 兩 k. For a for the mean-field front ␮MF = 2 / 3 and TQ perfect diffusion-controlled 3D reaction k ⬃ Dra where ra is the annihilation radius. Thus, as our k is measured in units of D / L2b0 关14兴 for the dimensionless k we have k = raL2b0. Substituting here ra ⬃ 10−8 cm, L ⬃ 10 cm, and b0 ⬃ 1022 cm−3 MF we find k ⬃ 1016 and derive TQ ⬃ 10−8 / 冑兩⌬兩 so that for not −8 too small 兩⌬兩 共兩␳ − ␳쐓兩 Ⰷ 10 兲 the sharp front is not destroyed almost down to the point of collapse. Clearly at small 兩⌬兩 → 0 the “destruction” of the front has to occur already at the stage of exponential relaxation 共22兲–共26兲. Substituting

Eq. 共26兲 into 共2兲 for the exponential relaxation we find 2 MF 共t−tQ兲 where ␯MF = 1 / 3 and tQ = 共ln k兲 / ␲2. SubstitutMF ing here k ⬃ 1016 we obtain tQ ⬃ 3.7 and then from Eq. 共25兲 we find N쐓MF共␩ = 0.1兲 / N0 ⬃ 10−13. An analogous calculation for the fluctuation 1D front gives ␮F = 3 / 4, F F ⬃ 1 / 共兩⌬兩n0兲2/3 and ␯F = 1 / 2, tQ = 共ln n0兲 / ␲2 where TQ F 6 n0 = Lb0. Substituting here n0 ⬃ 10 we find TQ ⬃ 10−4 / 兩⌬兩2/3, F F −4 tQ ⬃ 1.4, and n쐓共␩ = 0.1兲 / n0 ⬃ 10 . We conclude that both for the mean-field and the fluctuation fronts the vast majority of the particles die in the sharp-front regime; therefore, the presented theory has a wide applicability scope. In summary, the evolution of the island-A–island-B system at the sharp annihilation front A + B → 0 propagation has been considered and a rich dynamical picture of its behavior has been revealed. The theory presented may have a broad spectrum of applications—e.g., in the description of electron-hole luminescence in quantum wells 关15兴, the formation of nontrivial Liesegang patterns 关16兴, and so on. Of special interest is the analogy of the island-island problem with the problem of annihilation on the catalytic surface of a restricted medium where for unequal species diffusivities in a recent series of papers 关17兴 the phenomenon of annihilation catastrophe has been discovered. Study of the much more complicated case of unequal diffusivities and comparison with the annihilation dynamics on the catalytic surface is a generic and challenging problem for the future.

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This research was financially supported by the RFBR through Grant No. 05-03-33143.

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