On the convergence to critical scaling profiles in submonolayer deposition models

In this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $n\geq 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $x=\tau$ in the cluster size $x$ vs. time $\tau$ plane. In this paper we consider a different similarity variable, $\xi := (x-\tau)/\sqrt{\tau}$, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $\Phi_{2,n}(\xi)$ when $x, \tau\to +\infty$ with $\xi$ fixed, as well as the rate at which the limit is approached.


Introduction
The deposition of a monolayer of atoms on top of a single crystal facet is a process of clear technological importance, and its theoretical understanding is a theme of current scientific interest (see, e.g., [5]).
In the early stages of the deposition process the new monolayer is still far from being completed and only islands, or j-clusters, made up of a number j of adatoms exist, which, being far apart from each other on the crystal facet, do not interact among them.
One of the mathematical descriptions of this stage of the deposition process consists in a coagulation type differential equation [3,4] obtained as follows: denote by c j = c j (t) the concentration of j-clusters at time t on the crystal facet. The surface is hit at a constant rate α > 0 by 1-clusters (also called monomers). Consider the non interacting assumption above. Then, the only reactions of the clusters on the crystal surface are those in which a monomer takes part, namely (1) + (j) → (j + 1), with rate coefficients a 1,j ≥ 0, for j ∈ {1, 2, . . .}. Thus, the differential equations governing the dynamics of the population of clusters {c j (t)} are a Smoluchowski coagulation system with Becker-Döring like coagulation kernel (see, e.g., [1,8]).
An additional assumption relevant to some applications is the existence of a critical cluster size n below which the clusters on the crystal facet are not stable and do not exist in any significant amount in the time scale of the coagulation reactions. Mathematically, this behaviour can be described as follows, [4]: clusters of size larger than 1 and smaller than or equal to n − 1 cannot exist, and thus the smaller cluster that is not a monomer has size n and is formed when n monomers come together and react into an n-cluster. These "multiple collisions" can be though of as follows [2]: if, as in Monte Carlo simulations, we consider the monomers sitting in the vertices of a lattice, and if we have n − 1 monomers surrounding an empty site -as in the four nearest neighbours in a square lattice -which is suddenly hit by a monomer, we have one of these "multiple" collisions and the creation of an n-cluster in a single n body 1-cluster reaction.
Considering all coagulation rates time and cluster size independent (a 1,j ≡ 1, say), the mean-field model for the submonolayer deposition with a critical cluster size n is given by the following coagulation type ordinary differential equation system: The convergence of solutions of (1) to a scaling profile was studied in [4]. Using the new time scale (2) τ (t) := t t0 c 1 (s)ds it was proved that solutions of (1) in this time scale, satisfy the following scaling behaviour: This extended the result in [3], valid for the absence of critical cluster sizes (i.e., when n = 2). Exploiting techniques based on center manifold theory used in [3], we recently proved in [2] results about the rate of convergence to this scaling behaviour.
corresponds to look at them with η → 1 as τ → ∞ and so, in a certain sense, the new scaling variable ξ blows up to the whole real line the singular behaviour identified at the critical value η = 1. In [3] the convergence to a similarity profile in this new similarity variable was proved for the case with no critical size clusters (n = 2). In this paper we extend that result to general n ≥ 2 by proving that The rate of convergence to similarity profiles in several types of coagulating systems has recently attracted some attention (see, e.g., [6,7]). As stated above, for the deposition model (1) we recently studied in [2] the rate of convergence to the profile Φ 1,n in the similarity variable η. In the present paper we complement that work by studying the rate of convergence to Φ 2,n in the variable ξ, thus completing the study in [2,3,4]. Our main result is the following: For every solution (c j ) of (1) with initial condition (c j0 ) consider the change of variables (t, c j ) → (τ, c j ) introduced in (2) and (3). If there exists ρ 1 , ρ 2 > 0, µ > 1 − 1 n such that for all j ≥ n, j µ c j (0) ∈ [ρ 1 , ρ 2 ] then, as j, τ → +∞, with ξ = j−τ √ τ fixed, the following holds: As a consequence of the proof of this theorem, we obtain also the following result valid for an important case in applications: the deposition of monomers on top of a crystal facet having no j-cluster at the initial time t = 0, i.e., with initial conditions c j (0) = 0 for all j ≥ n: Theorem 1.2. With the notation used in Theorem 1.1 consider initial data with c j0 = 0 for all j ≥ n. Then, as j, τ → +∞, with ξ = j−τ √ τ fixed, it holds:

Preliminaries
We briefly present our approach to the study of (1), which follows the one used in [2,3,4] and consists of the exploration of the following two observations: (1) First note that the equation for c 1 depends only on c 1 and on the "bulk" quantity y(t) := ∞ j=n c j (t), which (formally) satisfies the differential equationẏ = c n 1 . Thus, the definition of this bulk variable allow us to decouple the resulting infinite dimensional system into a closed two-dimensional system for the monomer-bulk variables (c 1 , y), from which we get all the needed information about the behaviour of c 1 . (2) Secondly, the remaining equations for c j , with j ≥ n, depend only on those same variables c j , and on c 1 . However, the way they depend on c 1 is such that, by the change of time variable (2), the system is transformed into a linear lower triangular infinite system of ordinary differential equations for c j (τ ) = c j (t(τ )) , which can be recursively solved, in terms of c 1 , using the variation of constants formula.
In fact, writing the second and third equations of (1) in the form and introducing the new time scale (2) along with scaled variables (3) system (6) reads where (·) ′ = d/dτ . The equation for c n can be readily solved in terms of c 1 resulting in We can then use this solution to solve recursively for j ≥ n + 1. The following expression for c j (τ ) is thus obtained : and (10) The information about c 1 needed to study (10) is extracted from the two-dimensional system for (c 1 , y) referred to in (1) above, which is . This was done in [2,4] using asymptotic and center manifold methods. The result that we need below is the following: Theorem 6] With τ and c 1 (τ ) as before, the following holds: In the following two sections we prove Theorem 1.1 by studying separately the contributions of the two terms (9) and (10) to the limit (8). As a consequence of the estimates in section 4 a proof of Theorem 1.2 will also be obtained.

3.
Rates of convergence to Φ 2,n : contribution of I 1 In this section we study the limit Part of the analysis was presented in [3] for the case n = 2 but the study of S 4 below is new and proves the conjecture left open in section 6.2 of [3]. Using the definition of ξ we can write j = τ + ξ √ τ , which, solved for τ gives τ = j∆ j where the function ∆ j : R −→ R is defined by A number of properties of this function can be easily established, in particular, Assuming c k (0) ≤ ρ/k µ in (9), changing the summation variable to ℓ := j − k, writing τ in terms of j as indicated, and not taking multiplication constants into account, the limit (13) can be written as In order to evaluate (17) we decompose the sum into a "small ℓ" and a "large ℓ" contribution. It is natural to consider a cut-off size separating the two sums that scales like τ as a function of j, which means that the cut-off value should behave as and write the expression in (17) as The "small ℓ" sum, S 3 , that corresponds to the contribution of large cluster in the initial data (remember the change of variable k → ℓ), can be estimated in the same way as the sum S 1 in the proof of Theorem 5.1 of [3], and was already done in section 6.2 of [3] with n = 2. The general n case is no different: The existence of a positive constant C is due to the fact that We now consider the "large ℓ" sum, S 4 , corresponding to the contribution of small clusters. Recalling Stirling's expansion Now, observing that the function x → xe 1−x , for positive x, is positive and has an absolute maximum at x = 1 with value equal to 1; recalling (16); and using the expression of j ⋆ in (18) to conclude that, when j ⋆ + 1 ≤ ℓ ≤ j − n, we have To estimate the sum in the right-hand side observe that and the last sum is the lower Darboux sum of the integral of x → x −µ between n − 1 and j − j ⋆ − 1. Thus, denoting by C constants independent of j that can vary from case to case, we have: when µ ∈ (0, 1), and when µ > 1, Hence (21) can be estimated as Together with the estimate (19) we conclude that, for µ > 1 − 1 n , the limit (13) is equal to zero and the rate of convergence to the limit is the same as for S 4 above, namely, 4. Rate of convergence to Φ 2,n : contribution of I 2 We now consider the limit with I 2 (j, τ ) defined by (10). Analogously to what was done in [2,3], instead of using (10) with j ∈ N we consider the function I 2 (x, τ ) defined for (x, τ ) ∈ [n, ∞) × [0, ∞) by the same expression as (10) and thus will consider instead the limit of (24) ϕ(x, τ ) =  (24), the problem is reduced to the study of the limit of ϕ(τ + ξ √ τ , τ ) as τ → +∞.
It is not difficult to see that a change of the integration variable and a few algebraic manipulations allow us to write (24) in the following form where, and the J j are the contributions corresponding to taking the following in each of the bracketed terms in the integrand of (25): Term Contribution Contribution from (1 + f n ) from (1 + g) J 1 1 1 We shall now estimate the behaviour as τ → +∞ of each of these terms P and J j , paying attention not only to the limit but actually to the rate at which that limit is approached.
Now, suppose that q > 0. We have to find lower estimates for each polynomial p k (·, ξ). If ξ ≥ 0, then the minimum of for all 0 ≤ x ≤ ηθ, and for some positive constant C 2 only depending on n and η. If ξ < 0, then the minimum of for all 0 ≤ x ≤ ηθ, for some positive constant C 3 , only depending on n, ξ and η.
For both cases (i) and (ii) we thus obtain, with C = C j , j = 2, 3, accordingly, for all 0 ≤ x ≤ ηθ, for some positive constant C 4 , only depending on n, ξ and η.
Taking in account all the previous estimates, we conclude that, regardless the sign of ξ, there is a positive constant C 5 only depending on n, ξ and η such that, for all 0 ≤ x ≤ ηθ, Therefore, where M > 0 only depends on ξ, n and η. By taking together (36) and (37), we obtain |J 3 | = O(θ −1 ), as θ → +∞, and considering the definition of θ, the lemma is proved.
Proof. By the continuity of f n in [0, +∞[ and Theorem 2.1, we know that f n is bounded so that there is M > 0, which is independent of w and τ such that, for all w ≥ 0 and τ ≥ 0, |f n (w n √ τ )g(w, τ, ξ)| ≤ M |g(w, τ, ξ)|, and therefore we can take the estimates from the proof of lemma 4.3 to obtain the result.
For the next result let ν 0 := ∞ k=n c k (0). Lemma 4.5. The following holds true for τ sufficiently large: Proof. We estimate By performing the change of variable w = x 1/n τ 1/2n , we obtain, where, By (26), On the other hand, from (11) we have the equation, from which we obtain, for each τ > 0, Now, we estimate the asymptotics of the sum in the second member of this equality. First, we observe that, from Theorem 2.1, as τ → +∞. But, by the center manifold estimate (3.6) in Proposition 3 of [2], and considering the definitions of the variables therein, we obtain, as τ → +∞, From the two previous expressions we deduce that, as τ → ∞,

Now, it remains to estimate
Let us start by observing that 1 − e −s ≤ min{1, s}. To apply this to the exponential in I 2 observe that s + := −ξ + ξ 2 + 2 is the value of s = x √ τ > 0 for which ξs + 1 2 s 2 = 1. Thus, we can start by estimating I 2 as follows: Now, since in the second integral in (41) we have x > √ τ s + , if τ is sufficiently large we can substitute the asymptotic form of f n (x) obtained from using (12) in (26), where C is a constant depending on n and ξ (through s + ) and from now on can vary from line to line. Using the monotonicity of the logarithm to write log u < log √ τ /s + we can continue the estimate above as Now let us consider the first integral in (41). It is convenient to write it as an integral between fixed limits: In order to apply the asymptotic form of f n , as in (42) we need to decompose the integral in (44) into a sum of two integrals such that in one we can apply the result about the asymptotic behaviour of f n and the other can be adequately controlled.
Proof. The result follows directly from the lemmata 4.2-4.4.
If there are no clusters initially present on the crystal facet, c j (0) = 0 for j ≥ n, and thus ν 0 = 0. In this case, since we know that τ = j∆ j with ∆ j = 1 + o(1) as j → ∞ (see section 3), we immediately conclude that Theorem 4.6 reduces to Theorem 1.2.

Conclusion
To conclude the proof of Theorem 1.1 it is enough to put together the result given in Theorem 4.6 with the estimate (22) proved in section 3 for the contribution of the initial condition. Under the assumption of Theorem 1.1 that the initial condition c j (0) decays like j −µ , we conclude that the slowest decaying term arising from ϕ decays like j − 1 2n , while by (22) we see that the contribution coming from the initial data is in all cases never faster than this decay. Thus, the contribution coming from the initial condition is the dominant one for the decay rate, which proves Theorem 1.1.
We end by observing that Theorem 1.1 implies that when the tail of the initial condition decays sufficiently slow at infinity, with a rate µ ∈ 1 − 1 n , 1 , the rate of convergence to the similarity profile of the solution of the deposition model (1) will depend on µ. This lack of forgetfulness of the rate of convergence to the scaling profiles about some features of the initial condition was also observed recently in [2] for this deposition model along noncritical directions η = 1. So, although the similarity profile itself lacks any information about the initial condition, the rate at which it is approached by the solutions of (1) can still convey some information.