Céline Hubert

Cyril Chomette

Anthony Désert

Ming Sun

Mona Tréguer-Delapierre

Stéphane Mornet

Adeline Perro

Etienne Duguet

Serge Ravaine

Synthesis of multivalent silica nanoparticles combining both enthalpic and entropic patchiness

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

ARTICLE

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Introduction.

Colloidal particles anisotropic in shape and/or in surface chemical functionality have attracted recent interest because of their potential use as building units for directed assembly into new materials. [1][2][3] Among others, Glotzer et al. numerically studied the self-assembly of non-spherical and patchy colloids, i.e. particles with at least one portion of its surface area which differs in chemical functionality. [4][5][6] In particular, they have predicted the formation of the diamond lattice, which could exhibit a full photonic band gap, 7 according to the shape or the surface functionality of building blocks involved. They also distinguished the complementary concepts of entropic and enthalpic patches particles: the former describes topological surface irregularities capable to mechanically block directed interactions with other building blocks, e.g. corners, edges, dimples, etc. while the latter concerns "conventional" patches made of sticky chemical functions. 6 Therefore, the selforganization of colloidal rods into nematic or smectic phases 8,9 or of micron-sized cubic particles into colloidal crystals with a simple cubic symmetry 10 may be seen as entropically controlled.

Other recent examples concern the use of microparticles bearing several dimples capable of assembly with conventional spheres under depletion interactions. [11][12][13] Several examples of enthalpically patchy particles have also been reported. Most of their synthesis routes are templating methods. 14 Velegol et al. patterned particles with regions of predetermined size by the particle lithography technique. 15 Kretzschmar et al. used the glancing angle deposition technique to create various patches in shape and/or in composition on polystyrene (PS) beads. [16][17][18] This method was also used by Granick et al. to synthesize particles with two hydrophobic patches in order to self-assemble them in a kagome lattice. 19 Starting from multipod-like silica/PS clusters [20][21][22] and using the PS pods as temporary masks, 23 we reported the synthesis of silica nanoparticles with 4, 6 and 12 dimples ideally arranged according to the repulsion figures of tetrahedron, octahedron and icosahedron, respectively. 24 We also shown that the silica surface between these entropic patches may be regioselectively functionalized.

Here, we extend this work to silica particles with 1, 2 and 3 dimples, because they are just as complementary building blocks for assembly into functional materials. We also report that we are able to selectively functionalize the organic residues located at the bottom of the dimples that remain after the dissolution of the PS nodules. Therefore, the as-obtained silica nanoparticles combine both enthalpic and entropic patchiness that can be tuned in order to either enhance particular desired particle alignment or produce competing forces.

Materials and methods

Materials. Tetraethoxysilane (TEOS, ≥ 99 %, Fluka), N- [3-(trimethoxysilyl)propyl]ethylenediamine (EDPS, 97%, Aldrich), tetrakis(hydroxymethyl)phosphonium chloride (THPC, 80%, Aldrich), n-propyltriethoxysilane (PTES, 97%, Aldrich), paraformaldehyde (reagent grade, crystalline, Sigma-Aldrich), tin(IV) chloride (98%, Aldrich), ethylene diamine (99.5%, Fluka), hydrochloric acid (37%, Sigma-Aldrich) and ammonia (28-30% in water, SDS) were used as received. Deionized water with a resistivity of 18.2 Ohm.cm at 25°C was obtained with a MilliQ system (Millipore). Tetrahydrofuran (THF), dimethylformamide (DMF), chloroform and absolute ethanol were purchased from Sigma-Aldrich. Synthesis of dimpled silica particles. Multipods made of a central silica core and one, two, three, four, six or twelve PS satellite nodules, respectively, were prepared by seeded-growth emulsion polymerization of styrene, according to an already published procedure. 20,21 The regrowth of the silica cores (step A) was performed in the following conditions: a certain volume of TEOS, VTEOS, was diluted ten times in absolute ethanol. The resulting solution was introduced continuously in a mixture made of 1 mL of an aqueous suspension of multipods, 45.5 mL of absolute ethanol and 3.2 mL of ammonia thanks to a singlesyringe pump (0.5 mL.h -1 ). The mixture was stirred at room temperature during 12 h. The dissolution of the PS nodules (step B) was performed in 100 mL of THF at room temperature under stirring for 1 h. After washing by three cycles of centrifugation/redispersion in THF, the multivalent silica particles were finally redispersed in absolute ethanol.

Functionalization of the surface area between the dimples.

To graft amino groups onto the unprotected silica surface (step C), a given volume of EDPS corresponding to a nominal surface density of 20 functions.nm -2 was added directly into the particle suspension and the mixture was let to age at 40°C for 2 hours. The particle dispersion was purified by 3 cycles of centrifugation/redispersion in absolute ethanol. For the step C1, 5-nm gold nanoparticles, synthesized as previously described, 25 were added into 20 mL of the functionalized multipod-like particles dispersion. The mixture was stirred at room temperature overnight, and then the particle assemblies were collected by centrifugation and washed three times with absolute ethanol. The dissolution of the PS nodules (step C2) was performed in 100 mL of THF at room temperature under stirring for 1 hour. After washing by three cycles of centrifugation/redispersion in THF, the multivalent silica particles were finally redispersed in absolute ethanol. Functionalization of the PS residues at the bottom of the dimples. To graft propyl groups onto the unprotected silica surface (step D), a given volume of PTES corresponding to a nominal surface density of 20 functions.nm -2 was added dropwise into the particle suspension and the mixture was let to age at room temperature for 3 h. The solvents were removed by using a rotavapor. An excess of THF was added to the solid residue to dissolve the PS nodules (step D1). After washing by three cycles of centrifugation/redispersion in THF, the silica particles were redispersed in chloroform. Finally, the particles were washed by three cycles of centrifugation/redispersion in 40 mL of chloroform. Step D2 consisted first in the chloromethylation of the PS chains by adding 5 mL of a 3M butyl chloromethylether solution in chloroform, prepared according to Warshawsky et al., 26 and 0.3 mL of SnCl4 into 20 mL to the silica particles and the mixture was aged overnight at 45°C. After washing by three cycles of centrifugation/redispersion in HCl solution (4 wt.% in water) and three cycles of centrifugation/redispersion in a water/EtOH (50/50 wt %), the silica particles were redispersed in 20 mL of DMF. Then, 1 mL of ethylenediamine was added and the mixture was aged overnight at 90°C. After three cycles of centrifugation/redispersion in water, the pH was adjusted to 4 with HCl. Finally, 5-nm gold nanoparticles were added into the suspension (step D3). The mixture was stirred at room temperature overnight, and then the particle assemblies were collected by centrifugation and washed three times with absolute ethanol. Characterizations. TEM experiments were performed with a Hitachi H600 microscope operating at 75 kV. The samples were prepared as follows: colloidal dispersions were diluted in ethanol and one drop was deposited on a copper grid coated with a carbon membrane and left to evaporate.

Results and Discussion

Figure 1 exemplifies the multi-step approach developed to synthesize multivalent dimpled silica particles from PS/silica tetrapods. The number and position of the dimples depend directly on the number and position of the PS satellite nodules. The precursor PS/silica multipods are made of a silica core decorated by 1, 2, 3, 4, 6 or 12 PS nodules (Figure 3a-l, Table 1) and were synthesized by a seeded-growth emulsion polymerization of styrene. [20][21][22][23] Their conversion into dimpled silica particles consists first in growing the silica core of the multipods used as endotemplates (step A). The PS nodules of the multipods were then dissolved in order to reveal the dimples (step B). The non-protected surface of the grown silica core can be selectively functionalized with amine groups (step C), as revealed by the regioselective absorption of gold nanoparticles (step C1), before the PS nodule dissolution (step C2). Because some PS are chemically grafted at the silica surface, they are present at the bottom of the dimples and can also be derivatized to improve their functionality. To do so, small alkyl chains have first to be grafted onto the non-protected silica surface (step D) before the PS nodules dissolution (step D1) to facilitate the dispersion of the silica particles in chloroform. The PS residues that remain in each dimple are then functionalized with chlorine and amine groups (step D2) through the chloromethylation of the phenyl groups followed by an amination reaction (Figure 2). Once again, the success of this regioselective functionalization may be revealed by the adsorption of gold nanoparticles (step D3). The slow addition of a dilute solution of tetraethoxysilane in ethanol into a hydroalcoholic suspension of multipods, in presence of ammonia, led to the growth of the silica core conforming to the shape of the polystyrene nodules. As shown in Figure 3, the size of the silica cores can be tuned by adjusting the quantity of injected TEOS solution. Nevertheless, when this quantity is too high (250 µL), the formation of silica protuberances at the surface of the polystyrene nodules is observed Figure 3 d3 and e3). The dependence of the size of the silica core on VTEOS is reported in Table 2. The subsequent dissolution of the PS nodules (Figure 1, step (B)) gives rise to silica particles with a fixed number of dimples (Figure 4). As these concave imprints are precisely localized around the particles, they can be used as entropic patches to endow the particles with directional interactions and predetermined "instructions" for assembly, making them similar to atoms which have valences. Morphology yields were equal to 94%, 67%, 66%, 82%, 84% and 69% for mono-, bi-, tri-, tetra-, hexa-and dodecavalent particles, respectively, as determined by statistical analysis of TEM images over at least 100 dimpled silica particles per batch.

Precursors

We have already demonstrated that polystyrene nodules can be used as protecting masks to functionalize specifically the unprotected silica surface. 23,24 According to this strategy, we functionalized the silica surface of the multipods after its growth with amino groups by adding EDPS at the end of the TEOS condensation (Figure 1, step (C)). The efficiency of this functionalization and of its regioselective character was evidenced by mixing the functionalized multipods with an aqueous suspension of THPC-stabilized 5-nm gold nanoparticles, 25 which are known to adsorb specifically on amine groups (Figure 1, step (C1) and Figure 5 top). The last step of the synthesis was the dissolution of the polystyrene nodules revealing the presence of gold nanoparticles only on the unprotected silica surface (Figure 1, step (C2) and Figure 5 bottom). A closer look at Figure 4 shows that a bump remains in the bottom of each dimple after the PS nodules dissolution. The organic nature of these bumps was confirmed by using energy-dispersive X-ray spectroscopy (EDX) analyses coupled with scanning transmission electron microscopy (STEM) (Figure 6 a). We selectively modified these bumps through a chloromethylation reaction of their phenyl groups (Figure 1, step D2). The success of this functionalization step was first confirmed by EDX mapping (Figure 6 b) which shows the presence of chlorine atoms essentially at the bottom of the concave notches. We also grafted a diamine onto the chloromethylated macromolecules and revealed this regioselective functionalization by the specific adsorption of gold nanoparticles within the dimples (Figure 1, step (D3) and Figure 7 a-f).

Conclusions

In summary, we generalized a method for preparing silica particles with a controlled number of dimples from 1 to 12. We also showed that the size of these new dimpled silica particles can be easily controlled by adjusting the volume of the injected TEOS solution. The successful functionalization of both the nonprotected surface of the grown silica core and of the organic residues remaining at the bottom of each dimple was also demonstrated, thereby providing the possibility of getting particles with patches combining both enthalpic and entropic characters. These particles are promising building blocks for the fabrication of new lattices by self-assembly or as "lock" particles able to be combined with a precise number of "key" particles in each dimple in order to obtain new supracolloids.

Notes and references

a CNRS, Univ. Bordeaux, CRPP, UPR 8641, F-33600 Pessac, France. b CNRS, Univ. Bordeaux, ICMCB, UPR 9048, F-33600 Pessac, France. c Univ. Bordeaux, CNRS, CBMN, UMR 5248, F-33600 Pessac, France. d School of Chemical Engineering and Light Industry, Guangdong University of Technology, Guangzhou 510006, PR China. e Univ. Bordeaux, CNRS, ISM, UMR 5255, F-33405 Talence, France.

ARTICLE | 3 Fig. 1

 31 Fig.1Schematic representation of the multi-step approach for the synthesis and the regioselective functionalization of tetravalent silica particles.

Fig 2

 2 Fig 2 Reactions of chloro-methylation and amination of the PS residues.

Fig 3

 3 Fig 3 TEM images of the multipods before (a-l) and after the addition of different quantities of TEOS: (a1-f1) VTEOS = 50 µL, (a2-f2) VTEOS = 100 µL and (a3-f3) VTEOS = 250 µL. Scale bars: 100 nm.

Fig 4 5 Fig 5

 455 Fig 4 TEM images at different magnifications of silica particles with a) 1, b) 2, c) 3, d) 4, e) 6, f) 12 dimples. Scale bars: 100 nm.

Fig 6

 6 Fig 6 STEM EDX mapping a) before and b) after chloromethylation of the organic bumps inside the dimples.

Fig 7

 7 Fig 7 TEM images of silica particles with a) 1, b) 2, c) 3, d) 4, e) 6 and f) 12 gold-decorated bumps at the bottom of each dimple. Scale bars: 100 nm.

Table 1

 1 Characteristics of the multipod-like precursors

		Diameter of silica core (nm)	Diameter of PS nodules (nm)	Morphology yield
	Monopods	85	220		96 % monopods, 4 %, bipods
	Bipods	41	182		68 % bipods, 16 % tripods, 16 % monopods
	Tripods	43	157		66 % tripods, 21 % bipods, 13 % monopods
	Tetrapods	54	129		84 % tetrapods, 13 % tripods , 3 % bipods
	Hexapods	85	136		83 % hexapods, 13 % pentapods, 4 % tetrapods
	Dodecapods	85	67		70 % dodecapods, 17 % decapods, 13 % nonapods
		Precursors	VTEOS = 50 µL VTEOS = 100 µL VTEOS = 250 µL Average edge-length values of the silica core (nm)
	Monopods (7.3 10 12 part.)	105	175	210
		Bipods (3.7 10 13 part.)	100	150	200
		Tripods (3.7 10 13 part.)	90	140	190
	Tetrapods (1.8 10 13 part.)	110	150	180
	Hexapods (7.3 10 12 part.)	150	200	240
	Dodecapods (7.3 10 12 part.)	100	150	250

Table 2

 2 Average edge-length values of the silica core after the injection of various volumes of TEOS into the reactive medium containing the indicated number of multipods.

Acknowledgements

TEM experiments were performed at the Plateforme de Caractérisation des Matériaux and Bordeaux Imaging Center of the University of Bordeaux. This work was supported by the Conseil Régional d'Aquitaine, the Agence Nationale de la Recherche (grant ANR-07-BLAN-0271), and the LabEx AMADEus (ANR-10-LABX-42) in the framework of IdEx Bordeaux (ANR-10-IDEX-03-02), i.e. the Investissements d'Avenir programme of the French government managed by the Agence Nationale de la Recherche.


Tomás Aquino

Marco Dentz

Kinetics of Contact Processes under Segregation

come

I. INTRODUCTION

Mass-action reactions find applicability as population dynamics models of contact processes between agents, spanning biological processes [1], epidemiology [2], ecology [3], quantum molecular dynamics [START_REF] Nightingale | Quantum Monte Carlo Methods in Physics and Chemistry[END_REF], and chemical reactions in geological media [START_REF] Steefel | [END_REF]. The large-scale dynamics of such processes are determined by the interplay between local mechanisms, spatial heterogeneity, and segregation. Transport limitations under segregation and spatial heterogeneity lead to inter-reaction times which are related to the first encounter time between reactants [START_REF] Ben Avraham | Diffusion and Reactions in Fractals and Disordered Systems[END_REF][START_REF] Meroz | [END_REF][8][9][10]. Broad reaction times result in mesoscopic dynamical coupling between transport and reaction due to memory effects [11,12]. In turn, the large-scale behavior may involve non-classical, fluctuation-driven kinetics [13][14][15], broad reaction rate distributions [16], time-dependent reaction rates [17], and time-nonlocal kinetics [12]. The concept of residence time and its role in reactive transport have received much attention, in particular in hydrogeological applications [18,19]. This has led to the formulation of large-scale dynamics in terms of so-called stochastic-convective streamtube models, where reactive transport in the presence of physical and chemical heterogeneity is represented in terms of an ensemble of streamtubes [20][21][22][23].

The classical picture for stochastic reactions, corresponding to the Gillespie algorithm [24], assumes complete reactant mixing. It leads to exponential inter-reaction times and predicts the classical mass-action rate laws for large reactant numbers [25][26][START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF]. The recently proposed chemical continuous time random walk (chCTRW) framework allows for more general interreaction times, leading to a broader class of large-scale rate laws including time-nonlocal kinetics [START_REF] Aquino | [END_REF]. However, linking disorder properties, mass transfer, and inter-reaction times, a fundamental step towards the understanding and quantification of the emergence of largescale kinetics in the presence of spatial heterogeneity and reactant segregation, remains in general an open problem. This work develops this link for advective transport under spatial chemical disorder. We consider segregated immobile reactants, which react with mobile components. Inter-reaction times relate directly to first-passage times (FPTs) of mobile reactants to and across reactive regions. Upon encountering a reactive region, mobile components come into contact with the locally-available reactants. As we will see, this corresponds to a restart of the immobile reactants according to the original resident copy numbers. Processes under restart have received considerable attention in the context of chemical reactions and in particular as a framework for optimizing search strategies [29][START_REF] Kautz | Proc. 18th Nat. Conf. on Artificial Intelligence[END_REF][START_REF] Reuveni | [END_REF][32].

We derive a generalized master equation [33] for the chemical kinetics using the chCTRW framework, which we generalize to account for restart and parameterize in terms of the heterogeneity. The corresponding large-scale kinetics exhibit memory effects, and fluctuations about the average concentration play a role at large scales.

II. MODEL

We consider mobile reactants advected along separate trajectories. Advective velocity V is constant in each trajectory, and independent and identically distributed (i.i.d.) according to a probability density function (PDF) ξ(•) across trajectories. Additional reactants are confined to certain regions, see Fig. 1. We term these reactants immobile and these regions reactive. The latter are treated as well-mixed batch reactors, such that the chemical dynamics within proceed according to the classical Gillespie algorithm [24].

The length L r of reactive regions is an i.i.d. random variable distributed according to the PDF ρ r (•). The well-mixed assumption is difficult to justify if ρ r has infinite moments.

Instead, we assume L r is characterized by a finite mean r , and for simplicity we set ρ r ( ) = e -/ r / r . Segregation of immobile reactants is characterized by the PDF ρ c (•) of nonreactive region lengths L c between reactive regions, which are also i.i.d. We distinguish the cases of mild segregation, corresponding to finite inter-reactive-region mean distance c , and strong segregation, corresponding to infinite mean distance, ρ c ( )

≈ ( / c ) -1-β /[ c |Γ(-β)|]
for c , where Γ(•) is the gamma function, c is a characteristic length, and 0 < β < 1 [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF].

We assume in what follows that mobile reactants begin within a reactive region to simplify the exposition. The general case of start in an arbitrary region, as illustrated in Fig. 1, can readily be treated in the framework developed below by including an additional waiting time until the first reaction, corresponding to the time to reach the first reactive region.

The idealized model set forth here retains the key features of segregation and will allow us to develop a quantitative model of its impact on chemical reactions. We consider in general a set of m s species S j , j = 1, . . . , m s , which are subject to m r reactions, ms j=1 r ij S j → κ i ms j=1 p ij S j , i = 1, . . . , m r . Each reaction is characterized by its stoichiometry, with r ij (p ij ) denoting the number of reactants (products) of species j consumed (produced) by reaction i. Furthermore, each reaction is associated with an intrinsic (microscopic) reaction rate κ i which fully determines its behavior under well-mixed conditions. All reactions involve immobile reactants, so that no reaction occurs outside reactive regions. The chemical state N (t) describes the number of mobile and immobile reactants of each species at time t. The net change in chemical state due to reaction i is described by the stoichiometry vectors s i = p i -r i . Sub-or superscripts M, I mark quantities relating to mobile and immobile components, respectively. For example, the chemical state is decomposed into mobile and immobile components as N = (N M , N I ) and the stoichiometry vectors as

s i = (s M i , s I i ).
Throughout, we denote the Laplace transform by a tilde, the Laplace variable by λ, and ensemble averages (across trajectories) by angled brackets. A vertical bar is used to denote conditioning.

We introduce also some key quantities governing the dynamics. The mean time spent in a reactive region at velocity v is µ(v) = r /v, so that 1/µ(v) is the probability per unit time for mobile reactants to exit a given reactive region and experience the delay induced by the subsequent non-reactive region. Reactive patches are associated with a Damköhler number Da(v) = µ(v)a(n)/n, where n are characteristic copy numbers of each species (e.g., the initial state) and n is an overall characteristic copy number (e.g., the average of n across components or a component of interest). This dimensionless number is the ratio of the characteristic transport and reaction times in a single reactive region. Finally, the ratio of characteristic non-reactive and reactive region lengths is denoted by α = c / r .

III. QUALITATIVE DYNAMICS OF REACTION UNDER SEGREGATION

In order to motivate the general theory developed below, we first illustrate the impact of segregation on the large-scale kinetics of the mobile-immobile degradation reaction

S M + S I → κ ∅
, where S M is mobile and S I is immobile. We consider both mild and strong segregation, specifically with exponential and Lévy-stable [START_REF] Feller | An Introduction to Probability Theory and Its Applications[END_REF] inter-reactive region lengths. For simplicity, we set a fixed velocity V = v and a fixed number of initial immobile components n 0,I in each reactive region.

We consider an instantaneous, point initial injection of mobile components, and we simulate their average degradation under segregation for large particle numbers. When characteristic reactant numbers n → ∞, we may define continuous (number) concentrations C(t) = N (t)/n. We take n as the average initial number of reactants. The reaction in reactive patches then proceeds according to the well-mixed rate laws. For n M,I mobile/immobile particles under well-mixed conditions, there are n M n I mobile-immobile particle pairs available for reaction. The reaction between each pair proceeds at the microscopic reaction rate κ. The reaction between some pair thus proceeds at a rate a(n) = κn M n I . In terms of concentrations, the well-mixed rate laws are thus given by

dc wm M,I (t) dt = -κ C c wm M (t)c wm I (t), (1)
where κ C = nκ is a macroscopic reaction rate associated with well-mixed concentration decay. The analytical solution of this equation, along with details on the numerical simulations under segregation, is given in Appendix A.

The solid lines in Fig. 2 in this example is lower than the initial mobile concentration, reaction then slows down and the mobile concentration approaches equilibrium.

The impact of segregation on reaction dynamics, illustrated here for this simple degradation reaction, is in general due to two factors. First, reaction is punctuated by times spent in non-reactive regions, which leads to an effective reaction slowdown. Second, mobile components react only with the locally-available immobile reactants in a given region.

When a reactive region is left and a subsequent one is entered, there is a restart of available immobile reactants, according to their initially-available concentration. In order to highlight the role of immobile reactant depletion and restart, the markers in Fig. 2 show the evolution of mobile concentrations for the catalytic degradation reaction S M + S I → κ S I , using the same parameter values as before. In this case, the immobile component acts as a catalyst, and it is thus neither consumed nor subject to the effect of restart. For a single well-mixed patch, in the Da → ∞ limit, the mobile species' decay is then purely exponential and always faster than under segregation. However, as Da → 0, corresponding to negligible reaction within each single region, the decay of mobile concentration becomes identical under the two reactions.

In what follows, we will discuss and quantify the mechanisms discussed in the previous section. To this end, we develop a general framework for arbitrary reaction dynamics under spatial segregation.

A. Catalytic reactions

We start by examining the case where the immobile components serve as catalysts, i.e., they participate in the reactions but are not consumed or produced (s I i = 0). We further assume for the moment that the immobile reactants are present in identical copy numbers in each reactive region. In this case, the dynamics proceed identically with the classical wellmixed scenario, except the periods of availability of reactants are punctuated by waiting times due to segregation. We proceed to find the chCTRW description of this system in terms of a generalized master equation, starting from the determination of the inter-reaction times.

Inter-reaction times

The lack of memory of the exponential distribution implies that the distribution of distances to the end of reactive regions, starting from any point within a reactive region, has the same length distribution as the full regions [START_REF] Van Kampen | Stochastic Processes in Physics and Chemistry[END_REF]. Thus, it is sufficient to study inter-reaction times starting at the beginning of reactive regions.

Consider a given chemical state N (t) = n and velocity V = v. For advective transport, the FPT across reactive/conservative regions is given by the crossing time L r,c /v. Thus, the corresponding PDFs are given by ψ r,c (t|v) = vρ r,c (vt). Following chCTRW theory, we write the inter-reaction times as

τ (n, v) = τ r (n) + τ c (τ r , v
= φr i [λ/ K(λ|v)|n, v], (2)
where

K(λ|v) = 1 + 1 -ψc (λ|v) µ(v)λ -1 (3)
is the Laplace transform of a memory kernel, as we will see shortly.

Generalized master equation

We first define the probabilities γ i (n|n ) of reaction i yielding the chemical state n, given the starting state n . The change in state is given by the stoichiometry vector s i . Thus,

γ i (n|n ) = δ n,n +s i , (4)
where δ •,• is the Kronecker delta. According to the general theory developed in [START_REF] Aquino | [END_REF], the generalized master equation for the ensemble-averaged propagator of the chemical state, given initial state n 0 and velocity v, is then given by

∂p(n, t|n 0 , v) ∂t = n 0 mr i=1 t 0 dt [γ i (n|n ) -δ n,n ] × M i (t -t |n , v)p(n , t |n 0 , v), (5)
where the Laplace transforms of the memory functions are given in terms of the inter-reaction time densities as

Mi (λ|n, v) = λ φi (λ|n, v) 1 -mr i=1 φi (λ|n, v) . (6)
Using Eq. ( 13) together with the well-mixed description, we obtain

Mi (λ|n, v) = K(λ|v)a i (n), (7)
with a i (n) = κ i h i (n) and a(n) = mr i=1 a i (n), where the κ i are microscopic rate constants and the h i (n) encode the dependency of the rates on the state. In the absence of segregation, ψ c (t) = δ(t). Equation (3) with ψc (λ) = 1 leads to K(λ|v) = 1, and therefore K(t|v) = δ(t).

This recovers the classical chemical master equation [25]. We thus see that K plays the role of a memory kernel describing reaction slowdown due to segregation. Reactions under compound-Poisson delay, and corresponding simulation techniques to solve the generalized master equation ( 5), are discussed in general terms in [START_REF] Aquino | [END_REF]. Note that for catalytic reactions, s I i = 0 implies immobile copy numbers do not change, and it is sufficient to consider the evolution of the mobile components.

As an example, consider the catalytic degradation reaction introduced in Section III.

We have h 1 (n) = n M n I , the number of pairs of mobile-immobile particles, and a(n) = a 1 (n) = κn M n I . The intrinsic waiting time density is given by φ r

1 (t|n) = a(n) exp[a(n)t],
and the single memory function is M1 (λ|n, v) = K(λ|v)a(n). The stoichiometry vector is

s 1 = (-1, 0
). Thus, the master equation for the mobile components becomes

∂p(n M , t) ∂t = - t 0 dt K(t -t ) a(n M , n 0,I )p(n M , t ) -a(n M + 1, n 0,I )p(n M + 1, t ) , (8)
where we have omitted the dependency on the initial condition and v for notational brevity.

N (t) = n at time t is p(n, t) = p[n, t|N 0 (V ), V ] ,
where for each V = v the initial condition N 0 (v) is distributed according to γ(•|v). We note also that the propagator contains all necessary information to compute spatial quantities. Spatial distributions may be obtained by multiplying the propagator by the probability that mobile reactants are at position x at time t before averaging, which, assuming mobile species start at x = 0 at t = 0, is given here by the Dirac delta δ(x -V t). Similarly, concentrations fluxes restart, we write

φ i (t|n, v, L r , L c ) = b i (n, v, L r )φ |i (t|n, v, L r , L c ), (9)
where b i (n, v, L r ) is the propensity of reaction i, i.e., the probability that it fires next, and

φ |i (•|n, v, L r , L c
) is the inter-reaction time PDF of reaction i given that it will fire next.

Restart occurs if the minimum reaction time is larger than the restart time L r /v. This happens with probability

∞ Lr/v dt φ r (t|n) = exp[-a(n)L r /v]. Otherwise, with probability 1 -exp[-a(n)L r /v],
the normal reaction with the minimum inter-reaction time fires. Thus,

b i (n, v, L r ) =      e -a(n)Lr/v 1 -e -a(n)Lr/v a i (n) a(n) . ( 10
)
Here and in what follows, the first case refers to i = 0 (restart), and the second to 1 i m r (regular reactions). The inter-reaction time PDFs must also be conditioned on t < L r /v for the regular reactions, and the waiting time associated with restart is (L r + L c )/v, so that

φ |i (t|n, v, L r , L c ) =      δ t -Lr+Lc v a(n) exp[-a(n)t] 1-exp[-a(n)Lr/v] H Lr v -t , (11)
where H is the Heaviside step function. Thus, according to Eq. ( 9),

φ i (t|n, v, L r , L c ) =      e -a(n)Lr/v δ t -Lr+Lc v a i (n)e -a(n)t H Lr v -t . (12)
Defining φ i (t|n, v) = φ i (t|n, v, L r , L c ) , we find the Laplace transforms

φi (λ|n, v) =      ψr [λ + a(n)|v] ψc (λ|v) a i (n) 1-ψr[λ+a(n)|v] λ+a(n) . ( 13
)
The inter-reaction times are thus fully determined by first-passage properties together with the rates a i (n).

Generalized chemical master equation under restart

We turn to the generalized master equation incorporating the effect of restart. The transition probabilities γ i (n|n ) corresponding to the regular reactions remain given by Eq. ( 4), as the effect of these reactions does not change. The effect of the restart reaction, conditioned on a given velocity v, is characterized by:

γ 0 (n|n , v) = γ I (n I |v)δ n M ,n M , (14)
meaning that mobile reactants remain unaffected, and immobile copy numbers are redrawn from the initial distribution as discussed above. As shown in detail in Appendix D, the generalized master equation corresponding to the dynamics under restart retains the same form as Eq. ( 5), with the memory functions again given in terms of the inter-reaction time densities according to Eq. ( 6). However, sums over reactions in both these equations now extend to i = 0, the reaction describing restart, and the inter-reaction time densities are given by Eq. ( 13). Direct computation shows that the modified inter-reaction times lead to the same memory functions for the regular reactions, as given by Eq. ( 7). Restart is associated with the memory function

M 0 (n|n , v) = K(λ|v) ψc (λ)/µ(v). ( 15
)
These results follow from the fact that normal reactions proceed at rate a i (n) in reactive regions, whereas restart occurs at rate 1/µ(v) and is associated with an additional delay corresponding to traversing a non-reactive region. Realizations of these dynamics may be simulated with recourse to a generalized Gillespie algorithm under restart, which we outline in Appendix E.

Consider as an example the full degradation of Section III, with equal initial copy numbers of immobile components n 0,I in each reactive region. The stoichiometry vector is s 1 = (-1, -1), and γ 0 = δ n I ,n 0,I δ n M ,n M . Similarly to Eq. ( 8) for the catalytic degradation example, we find the master equation

∂p(n M , n I , t) ∂t = - t 0 dt K(t -t ) a(n M , n I )p(n M , n I , t ) -a(n M + 1, n I + 1)p(n M + 1, n I + 1, t ) (16)
- t 0 dt K c (t -t ) a(n M , n I )p(n M , n I , t ) -a(n M , n 0,I )p(n M , n 0,I , t ) ,
where the memory kernel associated with restart is given by the convolution K c (t) = t 0 dt K(t -t )ρ c (t ), and we have again omitted dependencies on the initial condition and v.

C. Restart and catalytic reactions

Consider equal initial immobile copy numbers in all reactive regions as in Section IV A.

Since for catalytic reactions s I i = 0, immobile copy numbers do not change due to either restart or regular reactions, the i = 0 term in the master equation is null, and we recover catalytic dynamics, Eq. ( 5).

The catalytic description also plays a role as the limiting behavior for slow reaction. For small Damköhler number, Da 1, the dynamics are transport-dominated at the scale of a single region, meaning that many reactive regions must be visited before appreciable change due to reaction can occur. For fixed initial immobile copy numbers, the reset mechanism ensures there is no appreciable change in immobile copy numbers. The catalytic description is then valid for arbitrary reactions, and the subordination formulation of [START_REF] Aquino | [END_REF] holds. Note also that under these conditions, for arbitrary initial copy numbers, the dynamics are independent of the specific reactive region length distribution as long as it has a finite mean, see

Dynamical equations

In Appendix F, we show that the ensemble-averaged concentrations obey the integrodifferential dynamical equations

dc M (t|c 0 , v) dt = t 0 dt K(t -t |v) mr i=1 s M i a C i [C(t )]|c 0 , v , dc I (t|c 0 , v) dt = mr i=1 s I i a C i [C(t)]|c 0 , v -µ(v) -1 [c I (t|c 0 , v) -c 0,I (v)] . ( 17
)
Segregation induces memory effects in the form of a convolution with a memory kernel for the mobile components. For the immobile components, restart leads to a mean-reverting forcing term. For mild segregation, the memory is short-term, and the late-time rate equations are time-local, whereas memory is long-range under strong segregation and the late-time rate equations involve non-local fractional derivatives [START_REF] Meerschaert | Stochastic Models for Fractional Calculus[END_REF].

In the classical rate laws, fluctuations vanish in the large-particle-number limit and

a C i [C(t)]|c 0 , v = a C i [c(t|c 0 , v)] [27].
Here, this is not the case. Under strong segregation, realizations of the dynamics break ergodicity weakly due to large fluctuations in the inter-reaction times, leading to persistent fluctuations about the average concentrations [START_REF] Aquino | [END_REF].

Under mild segregation, short-term memory effects coupled with the discontinuous changes in immobile concentrations caused by restart also prevent the fluctuations from vanishing.

While equations for higher moments of the concentration can be found by appropriate averaging of the generalized master equation ( 5), these depend on still higher moments, and closed rate equations for the average components do not exist in general. In other words, fluctuations play an important role in the mean behavior, analogous to Ovchinnikov-Zeldovich segregation in a bimolecular annihilation reaction among diffusing components [13,[START_REF] De Anna | [END_REF].

A common approach is to employ moment closure approximations [37,38], a technique that must be adapted to specific reactions and which we do not explore here. We employ stochastic algorithms, outlined in Appendix E, to numerically solve for the exact average concentration. We note here that the mean-reverting term in the large-scale description of immobile concentration depends only on average concentrations. Thus, the catalytic description (for catalytic reactions or the low-Damköhler limit of general reactions, see Sections IV A and IV C) holds at the large scale even if initial immobile copy numbers vary across reactive regions.

Asymptotics

We now examine the asymptotic behavior of the large-scale kinetics. For mild segregation, under which the inter-reactive-region lengths have a mean, the Laplace transform of the corresponding FPT is approximated by ψc (λ|v) ≈ 1 -αµ(v)λ for λ 1/[αµ(v)], corresponding to large times compared to the mean time to traverse a non-reactive region. To leading order in λ 1/[(1 + α)µ(v)] (corresponding to large times compared to the mean time to traverse a reactive and a non-reactive region),

Mi (λ|n, v) = 1 1 + α      1/µ(v) a i (n) . ( 18
)
This leads to time-local equations for the mobile components at late times,

dc M (t|c 0 , v) dt = (1 + α)µ(v) 2 mr i=1 s M i a C i [C(t)]|c 0 , v . (19)
For strong segregation, under which the inter-reactive region lengths do not have a mean, we have instead the small-λ expansion ψc (λ|v

) ≈ 1 -[αµ(v)λ] β , 0 < β < 1. To leading order in λ 1/[(1 + α)µ(v)], Mi (λ|n, v) = µ(v)λ [αµ(v)λ] -β      1/µ(v) a i (n) , (20)
yielding, at late times, the time-nonlocal equations

d β c M (t|c 0 , v) dt β = [αµ(v)] β µ(v) mr i=1 s M i a C i [C(t)]|c 0 , v , (21)
where d β /dt β denotes the Riemann-Liouville fractional derivative of order β [START_REF] Meerschaert | Stochastic Models for Fractional Calculus[END_REF].

V. LARGE-SCALE DYNAMICS OF REACTION UNDER SEGREGATION

In order to illustrate the main features of the theoretical developments in the context of a particular reaction, let us return to the large-scale dynamics of mobile concentration for the degradation reaction S M + S I → κ ∅ introduced in Section III. As the Damköhler number Da(v) = κ C µ(v)c 0,I (v) → 0, the changes in immobile concentration due to reaction in each region become arbitrarily small, so that C I C M ≈ c 0,I (v) C M at all times. Thus, for small Da, the late-time rate equation for the mobile component under mild segregation is, according to Eq. ( 19),

dc M (t|c 0 , v) dt = - Da(v) (1 + α)µ(v) c M (t|c 0 , v). (22)
There is no appreciable reaction before the late-time equation is valid, so that the initial condition c 0,M may be employed, and

c M (t|c 0 , v) = c 0,M exp - Da(v)t (1 + α)µ(v) . ( 23
)
Under strong segregation, the Laplace transform of the late-time equation ( 21) is, for

Da(v) 1, [αµ(v)λ] β Da(v) cM (t|c 0 , v) -λ -1 c 0,M = -cM (λ|c 0 , v). ( 24
)
10 -2 10 -1 10 0 10 1 10 2 10 3

Nondim. time t $ = Da 1=-t=(,7) Noticing that, for small λ, the initial-condition term on the left-hand side dominates, and inverting the Laplace transform,

c M (t|c 0 , v) ≈ c 0,M Da(v)Γ(1 -β) t αµ(v) -β . (25)
Simulation results for mild segregation are shown in Fig. 3 and for strong segregation in Fig. 4. We consider for concreteness exponential ρ c for mild and Lévy-stable ρ c for strong segregation, as before. Note that, unlike in Fig. 2, time is nondimensionalized so as to highlight the collapse of the low-Damköhler behavior onto the Da-independent curve valid for both the full degradation reaction and the catalytic degradation reaction S M + S I → κ S I . Under this nondimensionalization, higher Da leads to slower decay due to the effect of depletion of the immobile component. We compare a full stochastic algorithm employing the Gillespie method in reactive regions to a more efficient algorithm, valid for large particle numbers, which makes use of the well-mixed rate equations as in Section III (see Appendix E). The results are in very good agreement.

For mild segregation, simulations suggest that C I C M ≈ c 0,I (v) C M holds at late times, as expected under finite-mean inter-reaction times (see the right panel of Fig. 3). Thus, the late-time concentration for each velocity v decays in general exponentially as argued above, but the leading coefficient differs because the appropriate initial condition for the late-time equation depends on the dynamics before it becomes valid. However, for large Da, the exponential asymptotic regime is not observed, as the concentration reaches very small values before it occurs. We show also the solution obtained by numerically integrating the closed rate equations obtained by setting C M C I = c M c I in Eq. ( 17) (see Appendix F).

Its breakdown for Da 1 is due to the role of concentration fluctuations. Even though the fluctuations vanish at late times, they have an irreversible impact on the total reaction.

Under strong segregation, the leading coefficient varies with Da for two reasons. First, the initial condition to be used with the asymptotic equation differs as above. Second, the weak ergodicity breaking displayed by the inter-reaction times impedes C I (t)C M (t) → c 0,I C M (t) for late times, because there is a sufficiently high probability that C M (t) remains large due to long non-reactive regions. Nonetheless, simulations suggest that C I (t)C M (t) ∝ c I (t)c M (t) ≈ c 0,I c M (t) at late times for all values of Da, see the right panel of Fig. 4, so that the power-law behavior ∝ t -β remains unaffected.

VI. CONCLUSIONS

The link between first-passage and inter-reaction times connects the kinetics of contact processes to spatial heterogeneity and segregation. We have quantified this link for advective transport under spatial segregation of immobile components and formulated the reaction dynamics in terms of a generalized master equation. The evolution of total mass may be obtained from this description, and mass fluxes and spatial reactant distributions may also be easily computed.

In contrast to the classical picture for well-mixed reactions, the resulting large-scale kinetics cannot be fully quantified in terms of the dynamical equations obtained by averaging over the chemical master equation. This is due to the presence of concentration fluctuations on the order of the average values, which result from a combination of the restart mechanism and memory effects, both caused by segregation. For this reason, closed-form rate laws valid for all times do not exist in general. In the case of strong segregation, characterized by infinite-mean inter-reactive-region distances, memory is long-range and induces weak ergodicity breaking across trajectories, a typical feature of anomalous transport [START_REF] Klages | Anomalous Transport: Foundations and Applications[END_REF][START_REF] Bel | [END_REF][41][42][43][44]. (F4)

We solve these equations numerically using a combination of the forward Euler method for the derivative with the trapezoidal rule for the convolution integral. We use a discretization time step ∆t = 10 -2 min{µ, 1/(κ C c 0,M ), 1/(κ C c 0,I )}.

Figure 1 .

 1 Figure 1. Illustration of the model segregation structure. The velocity V of mobile species (proportional to black bar length) is constant along each separate trajectory and distributed across trajectories. Mass evolves according to reaction in reactive regions (blue), interspersed with nonreactive regions (yellow), both of distributed length.

  show the ensemble-averaged mobile concentrations c M (t), normalized by the initial concentration c 0,M , as a function of time t for different values of Damköhler number Da = κ C µc 0,I , where c 0,I is the initial immobile concentration in each reactive region. The evolution of concentration presents qualitative differences in the functional form of the decay with varying Damköhler number. For mild segregation (left panel), the decay at low Da is always exponential, but slower-than-exponential decay is present for longer times as Da increases. Under strong segregation (right panel), we observe power-law decay for all finite Da, characterized by the exponent β associated with the inter-reactiveregion lengths. The solutions for a single well-mixed patch, corresponding to the Da → ∞ limit, are also shown. In this case, the concentration initially decays faster than in the segregated systems. However, due to depletion of the initial immobile concentration, which

Figure 2 .

 2 Figure 2. Temporal evolution of concentration under the full degradation reaction S M + S I → κ ∅ (solid lines) and the catalytic degradation reaction S M + S I → κ S I (markers), for different values of Damköhler number Da. The initial conditions for mobile and immobile concentrations are c 0,M = 4/3 and c 0,I = 2/3. Fixed parameters are α = 2, v = 4, and κ C = 2, and Da is set by varying r . Concentrations are averaged over 10 5 realizations. Left: Mild segregation, with exponentiallydistributed non-reactive region lengths. Right: Strong segregation, with Lévy-stable-distributed non-reactive region lengths with exponent β = 0.7.

  The ensemble-averaged probability of a given state at a given time is in general obtained by averaging over velocities and initial conditions. Denote by γ M (•|v) the initial distribution of mobile components at injection and by γ I (•|v) the initial distribution of immobile reactant numbers across reactive regions, given velocity v. The initial copy number distribution at the first reactive region is thus γ(n|v) = γ M (n M |v)γ I (n I |v). For equal initial immobile component copy numbers n 0,I in each reactive patch, γ I (n|v) = δ n,n 0,I . The probability of finding the state

Flucts 1 Da = 10 ! 1 Da = 10 ! 2 Figure 3 .

 1123 Figure 3. Temporal evolution of concentration for S M +S I → κ ∅ under mild segregation for different values of Damköhler number Da. Symbols represent stochastic simulations based on the Gillespie algorithm (10 7 initial mobile particles and 5 • 10 6 immobile particles per reactive region, averaged over 10 5 realizations) and solid lines employ the well-mixed rate equations in reactive regions (10 6 realizations). Non-reactive region lengths are exponentially-distributed. Fixed parameters are α = 2, v = 4, and κ C = 2, and Da is set by varying r . Dash-dotted lines (overlapping the solid lines for low Da) are numerical solutions of Eq. (17) using C M C I = c M c I . Left: Average mobile concentration. The dashed line is the analytical solution in the limit of small Damköhler. Right: Concentration fluctuations.

Flucts

  Nondim. time t$ = Da 1=-t=(,7) . hcM (t$)cI (t$)i!cM (t$)cI (t$) hcM (t$)cI (t$)i Da = 1 Da = 10 !1 Da = 10 !2

Figure 4 .

 4 Figure 4. Temporal evolution of concentration for S M + S I → κ ∅ under strong segregation for different values of Damköhler number Da. Symbols represent stochastic simulations based on the Gillespie algorithm (10 7 initial mobile particles and 5 • 10 6 immobile particles per reactive region) and solid lines employ the well-mixed rate equations in reactive regions. All results are averaged over 10 5 realizations. Non-reactive region lengths are Lévy-stable-distributed with exponent β = 0.7. Fixed parameters are α = 2, v = 4, and κ C = 2, and Da is set by varying r . Left: Average mobile concentration. The dashed line is the late-time analytical solution in the limit of small Damköhler. Right: Concentration fluctuations.

  Appendix F: Large-scale kinetics Following[START_REF] Aquino | [END_REF], we define p C (c, t|c 0 , v) = np(nc, t|nc 0 , v), M C i (t|c, v) = M i (t|nc, v)/n, i = 1, . . . , m r , and M C 0 (t|c, v) = M 0 (t|nc, v). We multiply Eq. (D5) by, and sum over, n, and write n = n dc in the limit n → ∞. Integrating by parts, we obtainλ c(λ|c 0 , v) -c 0 = mr i=1 s i L t→λ M C i [λ|C(t)]|c 0 , v -L t→λ [C(t) -(C M (t), c 0,I (v))] M C 0 [λ|C(t)]|c 0 , v ,(F1)where L t→λ f (t) = f (λ). Substituting Eq. (6) for the memory functions, we have,λ cM (λ|c 0 , v) -c 0,M = mr i=1 s M i K(λ|v)L t→λ a C i [C(t)]|c 0 , v , λ cI (λ|c 0 , v) -c 0,I (v) 1 + K(λ|v) ψc (λ|v) λ|v)L t→λ a C i [C(t)]|c 0 , v .(F2)Using Eq. (3) for the memory kernel, we find that 1 -K(λ|v)[1 -ψc (λ|v)]/[µ(v)λ] = K(λ|v), so that the second equation may be rearranged and divided by K(λ|v) to giveλ cI (λ|c 0 , v) -c 0,I (v) = -λ cI (λ|c 0 , v) -c 0,I (v) a C i [C(t)]|c 0 , v . (F3)Inverse-Laplace-transforming yields Eq.(17).In order to integrate Eqs.(17) for the degradation reaction S M + S I → ∅, under the assumption C I C M = c I c M and mild segregation, note that the memory kernel (3) readsK(λ|v) = 1 -α/[1 + α + αµ(v)λ], so that, inverting the Laplace transform, K(t|v) = δ(t) -exp{-(1 + α)t/[αµ(v)]}/µ(v). Equations (17) then read dc M (t|c 0 , v) dt = -κ C c M (t|c 0 , v)c I (t|c 0 , v) 1+α αµ(v) (t-t ) c M (t |c 0 , v)c I (t |c 0 , v), dc I (t|c 0 , v) dt = -κ C c M (t|c 0 , v)c I (t|c 0 , v) -c I (t|c 0 , v) -c 0,I µ(v) .

  non-reactive regions, which is fully determined by the current τ r and does not depend further on the chemical state. Specifically, a given time τ r = t r corresponds to fully traversing a certain number η(t r , v) of reactive regions, and τ c (t r , v) =

	η(tr,v) k=1	L c,k /v,
	where the L c,k are i.i.d. according to ρ c . For exponential reactive lengths, η(t r , v) is Poisson-
	distributed, as shown in Appendix C. Thus, segregation leads to a compound-Poisson inter-
	reaction delay. According to [28], the full inter-reaction time density is then related to the
	intrinsic inter-reaction time density by
	φi (λ|n, v)

). The intrinsic reaction time τ r represents reaction in the absence of segregation. It depends on the current chemical state and corresponds to time spent in reactive regions, where reaction proceeds according to the classical well-mixed theory. Let φ r i (t|n) dt represent the joint probability that, in the absence of segregation, reaction 1 i m r fires next and after a waiting time in [t, t + dt]. Then, τ r (n) has PDF φ r (t|n) = i φ r i (t|n) (see Appendix B for a brief overview of the well-mixed description). The additional global delay τ c is caused by segregation. It corresponds to time spent in

Future work will focus on extending this approach to more complex transport processes, including effects such as variable velocity within each trajectory [45,46] and local mixing (e.g., diffusion). Moment closure approximations for the rate equations will also be the subject of further study.

ACKNOWLEDGMENTS

The authors acknowledge the support of the European Research Council (ERC) through the project MHetScale (617511).

at a control plane at distance x as a function of time t are obtained by multiplying by the FPT to distance x, given here by δ(t -x/V ).

B. General reactions

We now study the general case where reactions may involve net production or consumption of immobile components, and where different reactive regions may initially comprise different copy numbers of immobile reactants. In this case, as the mobile reactants reach each reactive region, they encounter the initial resident copy numbers. These are then depleted or produced according to reaction. Once the mobile components exit a given reactive region and arrive at the subsequent one, they again encounter resident copy numbers according to their initial distribution. This leads naturally to the concept of restart of immobile components. Developing a generalized master equation for these dynamics thus requires generalizing the inter-reaction times to account for restart.

Inter-reaction times under restart

In order to make use of the techniques developed in [START_REF] Aquino | [END_REF], we require that the dynamics be a Markov process in reaction step. For exponential reactive region lengths, whenever a reaction fires, the leftover reactive region length is identically distributed with the full region length, as discussed above. However, if the initial number of immobile reactants depends on the reactive region length, the number of immobile reactants then gives information about the region length, and the inter-reaction times are no longer independent of past history.

We assume here that this is not the case.

Consider a given chemical state N (t) = n, velocity V = v, and lengths of a consecutive reactive/non-reactive region pair L r,c . The effect of restart may be treated as a special reaction, which we number i = 0. It fires after a time L r /v and leads to restart of the immobile components after a time (L r + L c )/v. The next reaction to fire is the one with the minimum inter-reaction time, including restart. The inter-reaction time densities, given the chemical state, velocity, and region lengths, are defined such that φ i (t|n, v, L r , L c ) dt is the joint probability of reaction i firing and the inter-reaction time being in [t, t + dt]. Under D. Large-scale kinetics Next, we obtain the large-scale description corresponding to the mesoscopic master equations developed in the previous sections. In the large particle number limit, we work in terms of concentrations C(t) = N (t)/n as introduced in Section III. Correspondingly, we define

Note that these quantities are fully defined in terms of their microscopic equivalents. For example, considering again the degradation reaction of Section III, we have

the usual rate in the well-mixed rate laws for concentration. The average initial numbers of immobile components for velocity v are denoted by c 0,I (v) and the average concentrations for initial condition c 0 and velocity v by c(t|c

Appendix A: Well-mixed degradation kinetics

Consider the degradation reaction S M + S I → κ ∅ introduced in the main text. In a well-mixed reactor, the kinetic rate laws are given by Eq. ( 1). For equal initial conditions,

For c 0,M = c 0,I , setting c max = max{c wm M , c wm I } and c min = min{c wm M , c wm I },

where

The simulations of the evolution of concentration due to this reaction in the presence of segregation proceed as follows. A reactive region length is generated according to the PDF ρ r . The solution of the well-mixed rate laws is then applied to obtain the evolution of concentrations for a time interval /v. Then, a non-reactive region length is generated according to ρ c , and no reaction occurs for a time interval /v. This procedure is iterated up to a desired time, with the initial condition for the well-mixed rate laws in each reactive patch being set according to the current mobile concentration and the initial resident immobile concentration.

If one considers instead the catalytic degradation reaction S M + S I → κ S I in a well-mixed reactor, the immobile species concentration does not change and remains equal to c 0,I . The rate law for the mobile component is the same as before, and we obtain

The simulations under segregation proceed in the same manner as above, using this solution for the mobile concentration.

Appendix B: Intrinsic inter-reaction times

This Appendix provides a brief review of the intrinsic inter-reaction times, which characterize reactions in the absence of segregation. These correspond to the classical stochastic theory of well-mixed reaction [24]. Consider a given chemical state n. Each reaction i = 1, . . . , m r considered in isolation has an exponential inter-reaction time with rate

, where κ i is a (microscopic) rate constant and h i (n) encodes the dependency on the chemical state. For mass-action reactions,

It follows from the exponential character of reaction times, and the fact that the next reaction to fire is the one with the minimum waiting time, that the inter-reaction time density is

where a(n) = mr i=1 a i (n), and

with φ r (t|n) dt the probability that the inter-reaction time is in [t, t + dt[. The Laplace transform of the inter-reaction time density is thus

Appendix C: Number of traversed reactive regions

Here, we determine the distribution ν r (•|t r , v) of the number η r (t r , v) of fully traversed reactive regions between reactions, given time t r spent in reactive regions and velocity v.

The length traversed in time t r is vt r . The number η r (t r , v) of reactive regions traversed in this time is such that their total length is smaller than vt r , but the total length of η r (t r , v)+1

regions is larger than vt r . Thus,

where H is the Heaviside step function and the L r,k are i.i.d. according to ρ r . Conditioning on the total length of the first k regions, we obtain

Taking the Laplace transform with respect to t r ,

Thus, using ρ r ( ) = e -/ r / r and ψ r (t|v) = vρ r (vt), we have νr

Note that, as long as reactive region lengths have a finite mean r , we have, for small

the Laplace transform of the exponential density. Thus, the distribution of the number of traversed reactive patches is always approximately Poisson for large t r . Since typical reaction times are large when the Damköhler number is low, as explained in the main text, the specific distribution of reactive lengths does not play a role in that case.

Appendix D: Generalized master equation under restart

Consider the process K(t), which describes the number of reactions as a function of time, and write N K(t) = N (t). We have

with the inter-reaction times τ k = T k+1 -T k independent of T k . Thus,

where

with R 0 (n, t|n 0 , v) = δ n,n 0 δ(t), γ i (n|n , v) = γ i (n|n ) given by Eq. ( 4), and γ 0 (n|n , v) given by Eq. ( 14). Laplace transforming Eqs. (D2) and (D3) summed over k,

where R(n, t|n 0 , v) = k 0 R k (n, t|n 0 , v). Eliminating R, we find

where Mi (λ|n, v) is given according to Eq. ( 6), for all 0 i m r and with the sum extending to i = 0. Laplace inversion leads to the same form as the generalized master equation ( 5), with the sum extending to i = 0.

Appendix E: Stochastic simulation algorithms

We describe a generalized Gillespie algorithm that takes into account restart as described in the main text. Velocity v is to be sampled from the PDF ξ(•). In order to simulate dynamics up to time t m (or distance m ), the following algorithm should be repeated for a prescribed number of realizations:

1. Set time t = 0 (or distance = 0). Generate n according to the initial state distribution.

2. Generate 1 according to ρ r and ∆t r according to φ r (•|n), see Eq. (B3).

3. Find the next reaction i: If ∆t r > 1 /v set i = 0. Else generate 0 < r a(n) from the uniform distribution and set i such that a i (n) < r a i+1 (n), see Eq. (B3).

4. If i = 0 generate 2 according to ρ c and set ∆t = ( 1 + 2 )/v. Else set ∆t = ∆t r .

Increment t by ∆t ( by v∆t).

5. If t < t m ( < m ) update n according to γ i (•|n, v), see Eq. ( 4), and go to 2. Else set t = t m ( = m ) and end.

For large particle numbers, this procedure may be replaced by a more efficient method to find the concentrations:

1. Set time t = 0 (or distance = 0). Generate c according to the initial concentration distribution.

2. Generate 1 according to ρ r and ∆t r according to φ r (•|n), see Eq. (B3). Set ∆t = min{∆t r , t m -t}. Note that this algorithm reduces to the one outlined in Appendix A for the reactions considered therein.