# Diffusion-controlled reactions: an overview

Denis S. Grebenkov<EMAIL_ADDRESS>Laboratoire de Physique
de la Matière Condensée,
CNRS – Ecole Polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau,
France

###### Abstract

We review the milestones in the century-long development of the theory of
diffusion-controlled reactions. Starting from the seminal work by von
Smoluchowski who recognized the importance of diffusion in chemical reactions,
we discuss perfect and imperfect surface reactions, their microscopic origins,
and the underlying mathematical framework. Single-molecule reaction schemes,
anomalous bulk diffusions, reversible binding/unbinding kinetics and many
other extensions are presented. An alternative encounter-based approach to
diffusion-controlled reactions is introduced, with emphasis on its advantages
and potential applications. Some open problems and future perspectives are
outlined.

diffusion; surface reaction; heterogeneous catalysis; confinement; geometric
complexity; biochemistry; reversible reactions; encounter-based approach;
Brownian motion; encounter-dependent reactivity

##  I Introduction

The nineteenth century was marked by impressive advances in the theory of
chemical reactions, even though the existence of atoms and molecules, the
(quantum) origins of chemical bonds and many other fundamental aspects
remained to be clarified. Understanding of stoichiometric relations between
reactants and the development of a mathematical theory of ordinary
differential equations (ODE) provided a powerful tool to describe the kinetics
of very sophisticated reactions. On a basic level, stoichiometric relations
allow one to calculate the right proportions of ingredients and the masses of
produced reactants at the end. Moreover, they determine the form of the ODEs
that govern the time evolution of concentrations of the reactants. For
instance, upon disintegration of a substance $A$, its concentration $[A]$
obeys the simplest ODE,

$\frac{d[A]}{dt}=-k_{A}[A],$ (1)

where $k_{A}$ is the disintegration rate; here, change in time of the
concentration on the left-hand side is proportional to the remaining
concentration on the right-hand side. The solution of this equation,
$[A](t)=[A]_{0}\exp(-k_{A}t)$, shows an exponential decay of the concentration
from the initial level $[A]_{0}$. The simplicity of this solution is caused by
linearity of the equation. For instance, the dynamics of a bimolecular
synthesis reaction,

$A+B\rightarrow AB,$ (2)

is described by nonlinear differential equations such as

$\frac{d[A]}{dt}=-k_{AB}[A][B],$ (3)

in which the rate of decrease of the concentration $[A]$ is proportional to
the product of concentrations of both substances, i.e., to the likelihood of
meeting between reactants $A$ and $B$. More generally, stoichiometric
relations, which determine how many copies of each reactant molecule are
involved in chemical reaction, set the powers of the involved concentrations.
The nonlinearity of ODEs describing chemical kinetics presents one of the
major mathematical challenges for their analysis but also the origin of many
peculiar features (e.g., non-existence or non-uniqueness of the solution, a
finite time to the extinction of some reactants, etc.). These features and
their implications in chemistry and biology have been thoroughly investigated
in the twentieth century [1, 2].

The above description totally ignores spatial aspects of chemical reactions,
as if the concentrations of reactants were homogeneous in space at any time.
This is known as the well-mixing assumption when the reactants are assumed to
be well mixed so that reaction occurs in different points of space in the same
way. However, there are numerous situations, in which the spatial aspects are
critically important. For instance, many biochemical reactions in living cells
involve proteins and macromolecules that are produced in one spatial location
but have to diffuse to another location to find their reaction partners (e.g.,
receptors, enzymes, or specific sites on DNA chains). Even for small particles
such as oxygen molecules, ions and metabolites, there is generally a gradient
of concentration between their “source” and “sink” that drives their
directional transport in space. Moreover, even if the concentrations $[A]$ and
$[B]$ are macroscopically homogeneous but low, single molecules $A$ and $B$
have to meet each other to form an aggregate $AB$ according to the reaction
(2), and this transport step takes time and can be the limiting factor in the
overall reaction rate. The crucial role of diffusion was put forward by M. von
Smoluchowski, who formulated in 1917 the first mathematical description of the
coagulation dynamics [3], which later became the cornerstone of the theory of
diffusion-controlled reactions in a much broader context [4, 5, 6, 7, 8, 9].
Note that such reactions bear other names as diffusion-limited, diffusion-
mediated or diffusion-influenced reactions.

In this concise review, we focus on the spatial aspect of chemical reactions.
In Sec. II, we describe a chemical transformation on a catalytic surface and
emphasize the role of diffusion and the consequent spatial dependence of the
concentration (e.g., the formation of a depletion zone). Section III presents
a more realistic setting of imperfect surface reactions, which combine
diffusion in the bulk and chemical kinetics on the surface. In Sec. IV, we
briefly overview various extensions such as anomalous diffusions, reversible
binding/unbinding reactions, reactions in dynamically heterogeneous media,
etc. Section V describes an alternative approach to diffusion-controlled
reactions based on the statistics of encounters between the reactant and the
catalytic surface, while Sec. VI concludes the review.

##  II The role of diffusive transport

For the sake of clarity, we focus on heterogeneous catalysis when a reactant
$A$ can be transformed into a product $B$ in the presence of an immobile
catalyst $C$:

$A+C\rightarrow B+C.$ (4)

If the catalytic germs were uniformly dispersed in a chemical reactor, one
could still rely on Eq. (1). However, in many practical situations, catalytic
germs have specific locations, most often on a surface of a porous medium, so
that the reactant $A$ should first reach this spatial location. As the
reactants near the catalytic germs have higher chances to reach on them and
thus to be transformed to $B$, the concentration $[A]$ becomes space-
dependent. In particular, a depletion zone with low concentration of $A$ is
formed near the catalytic surface (Fig. 1, top row). This is a direct
consequence of the transport step, which can be described, as in the case of
coagulation dynamics, by the diffusion equation (also called Smoluchowski
equation or heat equation),

$\frac{\partial[A]}{\partial t}=D\Delta[A],$ (5)

where $\Delta=\partial^{2}/\partial x^{2}+\partial^{2}/\partial
y^{2}+\partial^{2}/\partial z^{2}$ is the Laplace operator, and $D$ is the
diffusion coefficient of reactant $A$ in a liquid. In analogy to Eq. (1), this
equation describes the time evolution of the concentration $[A](\bm{x},t)$ in
each spatial point $\bm{x}$ due to diffusive displacements of the reactants
$A$ in the bulk. In turn, the reaction itself that occurs on the catalytic
surface $C$ is implemented via a boundary condition on that surface. If $A$ is
transformed into $B$ immediately upon the first encounter with the catalyst
$C$ (so-called perfect reactions), the concentration $[A]$ is set to zero on
$C$. This so-called Dirichlet boundary condition was first imposed by von
Smoluchowski and still remains the most well-studied and frequently used
boundary condition. Its effect is illustrated on Fig. 1 (top row) by dark
color near the surface of a spherical catalyst. Note that the overall reaction
rate is determined by the diffusive flux of reactants $A$ onto the catalytic
surface $C$:

$J(t)=\int\limits_{C}d\bm{x}\biggl{(}-D\frac{\partial[A](\bm{x},t)}{\partial
n}\biggr{)},$ (6)

where $\partial/\partial_{n}=(\vec{n}\cdot\nabla)$ is the normal derivative
along the normal direction $\vec{n}$ to the surface.

Figure 1:  (Top row) Rescaled concentration
$[A](\bm{x},t)/[A]_{0}=1-\frac{R}{|\bm{x}|}\mathrm{erfc}\bigl{(}(|\bm{x}|-R)/\sqrt{4Dt}\bigr{)}$
of reactants $A$ near a perfectly reactive catalytic sphere of radius $R$ (in
gray) at different time instances (here $\mathrm{erfc}(z)$ is the
complementary error function) [3]. (a) Homogeneous concentration at $t=0$; (b)
Formation of a thin depletion zone at short time $Dt/R^{2}=0.1$; (c,d)
Progressive growth of the depletion zone at larger times $Dt/R^{2}=1$ and
$Dt/R^{2}=10$; (e) Approach to a steady-state concentration
$[A](\bm{x},\infty)/[A]_{0}=1-R/|\bm{x}|$ as $t\to\infty$. (Bottom row)
Rescaled concentration
$[A](\bm{x},t)/[A]_{0}=1-\frac{R-R_{\kappa}}{|\bm{x}|}\left\\{\mathrm{erfc}\left(\frac{|\bm{x}|-R}{\sqrt{4Dt}}\right)+e^{Dt/R_{\kappa}^{2}+(|\bm{x}|-R)/R_{\kappa}}\mathrm{erfc}\left(\frac{|\bm{x}|-R}{\sqrt{4Dt}}+\frac{\sqrt{Dt}}{R_{\kappa}}\right)\right\\}$
of reactants $A$ near a partially reactive catalytic sphere of radius $R$,
with reactivity $\kappa R/D=1$, at the same time instances [23] (with
$R_{\kappa}=R/(1+\kappa R/D)$).

The inclusion of space dependence into the theory of chemical kinetics led to
many fundamental changes. As reaction does not occur homogeneously in space
anymore, there are two consecutive steps: the diffusion step (transport
towards the catalytic surface described by the diffusion equation) and the
reaction step (chemical transformation from $A$ to $B$ on it described by the
boundary condition). The dependence of these two steps on the shape of the
catalytic surface introduces a new geometric dimension to the theory [10, 11,
12]. How efficient are catalytic surfaces of different shapes? Since reaction
occurs on the catalytic surface, can irregularly shaped catalysts speed up the
overall production due to their higher surface area? Can one optimize the
shape to increase the production? Have the large reactive surfaces of exchange
organs such lungs and placentas been optimized by evolution for more efficient
oxygen capture? These and many other questions have been intensively studied
since the 80’s of the twentieth century (see [13, 14, 15, 17, 16, 18, 19, 20,
21] and references therein).

##  III Imperfect surface reactions

Despite “popularity” among theoreticians, perfect surface reactions ignore
intrinsic chemical kinetics during the reaction step and therefore may lead to
paradoxical predictions. For instance, the overall reaction rate on a
perfectly reactive sphere of radius $R$,

$J(t)=4\pi RD[A]_{0}\biggl{(}1+\frac{R}{\sqrt{\pi Dt}}\biggr{)},$ (7)

found by von Smoluchowski [3], is infinitely large at the very first time
instance (as $t\to 0$). This divergence is caused by the molecules in an
immediate vicinity of the catalyst that react instantly. As a consequence, if
one searches to maximize the overall production by distributing a given amount
of a catalytic material, the optimal solution consists in dispersing this
material into a “dust”, i.e., a uniform arrangement of tiny catalytic germs.
Moreover, if the subdivision of this material into smaller and smaller germs
could be repeated up to infinity, such a fractal dust would transform all the
reactants $A$ in the bulk instantly [22]. From a mathematical point of view,
this is not surprising because any reactant $A$ would have in its immediate
vicinity a tiny catalytic germ, thus eliminating the diffusion step. However,
such a behavior does not make sense from a practical point of view.
Limitations of perfect reactions have been recognized in 1949 by Collins and
Kimball [23] who proposed to replace Dirichlet boundary condition by so-called
Robin or radiative boundary condition on the catalytic surface:

$-D\frac{\partial[A](\bm{x},t)}{\partial n}=\kappa\,[A](\bm{x},t).$ (8)

This condition postulates that the (net) diffusive flux of reactants $A$
coming onto the catalytic surface from the bulk (the left-hand side) is
proportional to their concentration $[A]$ on that surface at each surface
point. The proportionality coefficient $\kappa$, called a “reactivity” of the
catalytic surface, can range from $0$ for an inert surface to infinity for a
perfectly reactive surface. In the former case, the diffusive flux of
reactants is zero, meaning that no reaction occurs on that surface. In the
latter case, the division by $\kappa$ and the limit $\kappa\to\infty$ reduce
Eq. (8) back to the Dirichlet boundary condition $[A](\bm{x},t)=0$ on the
surface of $C$. Note that the reactivity $\kappa$ (in units m/s) can also be
expressed in terms of a forward reaction constant $k_{\rm on}$ (in units
m3/s/mol or 1/M/s) as $k_{\rm on}=\kappa N_{A}S_{C}$, where $N_{A}$ is the
Avogadro number, and $S_{C}$ is the surface area of the catalytic surface.
Figure 1(bottom row) illustrates the effect of partial reactivity onto the
concentration of reactants near the catalytic sphere of radius $R$. The
depletion zone is thinner and grows slower than in the case of perfect
reactions. Moreover, the overall reaction rate $J(t)$ is finite in the short-
time limit: $J(0)=4\pi R^{2}\kappa[A]_{0}$. Indeed, only the molecules near
the catalyst (of surface area $4\pi R^{2}$) can react at short times, and
their contribution is now limited by chemical kinetics, i.e., by the time
needed for chemical transformation (4), which is controlled by the reactivity
$\kappa$. As time increases, molecules from further locations arrive onto the
catalyst and can thus contribute. At long times, the region near the catalyst
is depleted, and reactants $A$ from very distant locations need to diffuse
towards the catalyst. In this limit, one gets $J(\infty)=4\pi
RD[A]_{0}/(1+D/(\kappa R))$, i.e., the overall production is therefore
diffusion-limited. In other words, the overall production exhibits a
transition from the reaction-limited regime at short times to the diffusion-
limited regime at long times.

The partial reactivity of the surface, described by Robin boundary condition
(8), can model various microscopic mechanisms of imperfect reactions [12, 24],
as illustrated by Fig. 2. In physical chemistry, once the reactant $A$ arrives
onto the catalytic surface, it has to overcome an activation energy barrier in
order to react [25, 26]. This activation energy determines the probability $p$
of the reaction attempt to be successful. However, the reactant may fail its
reaction attempt (with probability $1-p$) by leaving the proximity of the
catalytic surface and thus resuming its diffusion until the next encounter,
and so on. In this setting, the microscopic interaction determines the
probability $p$, which, in turn, fixes the effective macroscopic reactivity
$\kappa=\frac{D}{a}\,\frac{p}{1-p}$, where $a$ is the width of the reactive
layer near the catalytic surface (i.e., the interaction range, which is
typically of the order of a nanometer) [27]. Varying $p$ from $0$ to $1$
covers the whole range of reactivities from $0$ to $+\infty$. In the
biochemical context, conformational changes of a macromolecule between nearly
isoenergetic folded states can alter its function; this mechanism is primarily
important for protein-ligand and protein-protein recognition [29, 30, 28].
When such a protein arrives onto the catalytic surface (its reaction partner),
it has to be in an appropriate conformational state (with probability $p$) to
be able to initiate the reaction (4); otherwise, the protein leaves the
catalytic surface and restarts its bulk diffusion [31]. Even small particles
such as calcium ions can spontaneously loose their reactivity via reversible
binding to buffer molecules. This is the basis of one of the regulatory
mechanisms in neuron signaling when the tunable concentration of buffer
molecules inside a presynaptic bouton allows to control the ability of calcium
ions to reach calcium-sensing proteins that trigger the vesicular release of
neurotransmitters (see [32] and references therein). In the microcellular
context, the catalytic surface may represent a plasma membrane of a cell or of
a nucleus, while the reaction event may consist in the passage through a
channel on that membrane; such “reaction” occurs if the channel is open (with
probability $p$), while the reactant is reflected back from a closed channel
[33, 34, 35]. Even if the channel is always open (e.g., just a hole in a
container or in a filter), there is an entropic barrier that may prohibit the
escape from the confining domain and lead to reflection and resumed diffusion
[36, 37, 38]. In heterogeneous catalysis, the macroscopic reactivity $\kappa$
may account for micro-heterogeneity of the catalytic surface, which is not
fully covered by catalytic germs; in this case, $p$ is the probability to hit
the catalytic surface at the catalytic germ (and thus to react), while $1-p$
is the probability of arriving at the inert part of the surface and thus being
reflected. Homogenization of spatially heterogeneous catalytic surfaces leads
to Robin boundary condition (8), in which the reactivity $\kappa$ effectively
accounts for distributed reactive spots [39, 40, 41, 42, 43, 44]. For
instance, in the seminar work by Berg and Purcell [39], the probability $p$
was found for a spherical cell of radius $R$ covered by $N$ disk-shaped
receptors of radius $a$: $p=Na/(Na+\pi R)$.

Figure 2:  Various microscopic origins of imperfect surface reactions. (a)
When the reactant $A$ arrives onto the catalytic surface $C$, an activation
energy barrier $E_{a}$ has to be overcome for a chemical transformation of $A$
into $B$; if failed, the reactant leaves the vicinity of $C$ and thus resumes
its bulk diffusion; (b) A macromolecule can spontaneously switch its
conformational state from “active” (in red) to “passive” (in blue) with the
rate $k_{a}$, and back (with the rate $k_{d}$), while its reaction on the
catalytic surface (in gray) or with another macromolecule (a receptor, an
enzyme, a DNA strand, etc.) is only possible in the “active” conformational
state; (c) The reactant can be temporarily trapped by a buffer molecule (in
green) that makes it inactive for the considered surface reaction; their
association/dissociation kinetics is usually described by forward and backward
rates $k_{a}$ and $k_{d}$; (d) An ion can pass through an open channel, while
it is reflected back from the close channel; (e) An escape of a semi-flexible
polymer through a small hole can be described by an entropic barrier that
leads to partial reactivity when the first arrival to the hole does not
guarantee the passage; (f) An inert (gray) surface is covered by reactive
catalytic germs (black spots) so that the reactant may fail to react upon the
first arrival, and thus resumes its bulk diffusion until the next encounter,
and so. Similarly, a protein can search for a specific (target) site on a DNA
chain for successful binding.

The partial reactivity adds an important intermediate step to diffusion-
controlled reactions: after the first arrival onto the catalytic surface, the
reactant executes a sequence of diffusive explorations of the bulk near the
catalytic surface after each failed reaction attempt. This step may
considerably slow up the overall production, while the shape and reactivity of
the catalytic surface are entangled through diffusion in a sophisticated way.
Note that the same problem emerges in the context of semi-permeable membranes
in biology and blocking electrodes in electrochemistry [45, 46, 47]. The role
of reactivity (or, equivalently, permeability or resistivity) onto the overall
production was thoroughly investigated [48, 49, 50, 51, 52, 53]. For instance,
B. Sapoval and co-workers discussed the role of the “reaction length”
$D/\kappa$ as a physical scale for oxygen capture efficiency in human lungs
[54].

##  IV Various extensions

The basic description of diffusion-controlled reactions via Eqs. (5, 8) has
been generalized in different ways. Most efforts were dedicated to extensions
of the diffusion equation (5) that describes the simplest diffusive motion of
reactants, the so-called Brownian motion. For instance, the Fokker-Planck
equation allows one to incorporate the effects of external potentials (e.g.,
an electric field acting on a charged particle), anisotropy, and space- and/or
time-dependence of the diffusion coefficient [55, 56, 57, 58]. Fractional
space and time derivatives can further include nonlocal displacements and
memory effects in continuous-time random walks [59, 60, 61, 62]. Diffusing
diffusivity and switching diffusivity models were proposed to describe the
diffusive transport in dynamically heterogeneous media or in the presence of
buffer molecules that may reversibly bind the reactant and thus randomly
change its diffusion coefficient [63, 64, 65, 66]. The addition of a linear
term proportional to $[A]$ to the right-hand side of the diffusion equation
(5) can account for first-order disintegration mechanisms such as photo-
bleaching, bulk relaxation, radioactive decay, or a finite lifetime of the
reactant [67, 68, 69], as well as the effect of diffusion-sensitizing magnetic
field gradient encoding in diffusion magnetic resonance imaging [70].
Moreover, the diffusion equation with nonlinear terms in $[A]$ can describe
reaction waves and many out-of-equilibrium chemical reactions involving
“activators” and “inhibitors” (e.g., Belousov-Zhabotinsky reaction), paving a
way to the theory of pattern formations initiated by A. Turing [1, 71].

The above extensions generally employ the canonical Dirichlet or Robin
boundary conditions. Such a “persistence” can partly be explained by two
mathematical reasons: (i) the Laplace operator with either of these boundary
conditions is known to be self-adjoint (Hermitian) that allows one to rely on
powerful methods of spectral theory and to borrow numerous tools from quantum
mechanics; (ii) the diffusion equation with these boundary conditions has a
straightforward probabilistic interpretation that provides strong intuition
onto the behavior of the studied diffusion-reaction processes, offers
efficient Monte Carlo simulations, and helps to extend the macroscopic
description in terms of concentrations to single-molecule experiments. In
fact, many biochemical reactions involve proteins that are not abundant inside
living cells. When the number of proteins is relatively small (e.g., few tens
or few hundred of transcription factors in a bacterium [72]), the macroscopic
notion of concentration may be inapplicable, the overall reaction rate may be
uninformative or even misleading, while fluctuations become critically
important. Such reactions require therefore a probabilistic description in
terms of the survival probability of a single reactant molecule and the
probability density of the first-reaction time [73, 74]. In many settings, the
survival probability of a single molecule obeys the same equations (5, 8) and
hence is equal to the rescaled concentration $[A](\bm{x},t)/[A]_{0}$. This
equivalence bridges the macroscopic and probabilistic descriptions, providing
complementary insights and opening efficient ways to analyze and interpret
single-molecule experiments [75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85].

At the same time, the Robin boundary condition (8) remains limited to modeling
rather simple surface reactions with a constant reactivity. Consideration of
time- and/or space-dependent reactivity is one natural extension (see [86] and
references therein). Another important extension concerns reversible reactions
such as binding/unbinding, association/dissociation, and adsorption/desorption
kinetics, in which case the reactant can be temporarily bound to the surface
(or to another molecule). The exchange between free particles and those bound
on the surface can be incorporated through the “back-reaction” boundary
condition, also known as “generalized radiation” or “generalized Collins-
Kimball” boundary condition [87, 88, 89, 90, 91, 92, 93, 94, 95]. Application
of the Laplace transform with respect to time,
$\tilde{[A]}(\bm{x},s)=\int\nolimits_{0}^{\infty}dt\,e^{-st}\,[A](\bm{x},t)$,
reduces this boundary condition to Robin boundary condition (8) with
$s$-dependent reactivity $\kappa(s)$ (see details in [95]). In this way,
reversible and irreversible diffusion-controlled reactions admit essentially
the same mathematical description in Laplace domain (in terms of $s$); in
turn, the $s$-dependent reactivity results in fundamentally different
behaviors in time domain (in terms of $t$). In addition, one can further relax
the assumption of an immobile bound state and allow for diffusion on the
surface. The efficiency of such intermittent search dynamics with alternating
phases of bulk and surface diffusion was thoroughly investigated [96, 97, 98,
99, 100, 101, 102, 103, 104, 105] (see also a review [106]).

##  V Beyond the conventional framework

To handle more general surface reaction mechanisms such as, e.g., deactivation
or passivation of catalysts [107, 108], or progressive activation of enzymes,
an alternative theoretical description of diffusion-controlled reactions was
proposed [109]. This so-called encounter-based approach originates from the
theory of reflected stochastic processes in confined domains and relies on the
concept of the boundary local time $\ell$ – a rescaled number of encounters
between the reactant and the catalytic surface. In this approach, one can
disentangle the respective roles of the shape and reactivity of the catalytic
surface. In fact, the concentration of reactants $A$ can be represented as

$[A](\bm{x},t)=\int\limits_{0}^{\infty}d\ell\,e^{-\ell\kappa/D}\,\rho(\ell,\bm{x},t),$
(9)

where $\rho(\ell,\bm{x},t)$ describes the statistics of encounters with an
inert surface. In other words, the function $\rho(\ell,\bm{x},t)$ encodes how
the shape of the catalytic surface affects the diffusive dynamics, whereas the
exponential factor $e^{-\ell\kappa/D}$ incorporates the reactivity $\kappa$
that was implicitly imposed via Robin boundary condition (8) in the
conventional approach. As the successful surface reaction is preceded by a
sequence of failed reaction attempts at each encounter, the exponential factor
in Eq. (9) can be interpreted as the exponential probability law,
$\mathbb{P}\\{a\hat{n}>\ell\\}=e^{-\ell\kappa/D}$, for the random number
$\hat{n}$ of encounters in that sequence. Due to the self-similar nature of
Brownian motion, the number of encounters has to be rescaled by the width $a$
of a thin surface layer, in which the molecule can interact with the catalytic
surface (see details in [109]). While the statistics of encounters was
investigated for simple confinements [110, 111, 112, 113, 114, 115, 116], its
shape dependence for porous media representing industrial catalysts or
biological environments remains still unknown.

Most importantly, one can replace the exponential factor in Eq. (9), which
incorporated the effect of a constant reactivity $\kappa$, by another
probability law $\mathbb{P}\\{a\hat{n}>\ell\\}=\Psi(\ell)$, to model more
sophisticated surface reaction mechanisms with an encounter-dependent
reactivity

$\kappa(\ell)=D\frac{-\frac{d}{d\ell}\Psi(\ell)}{\Psi(\ell)}\,.$ (10)

If $\Psi(\ell)=e^{-\ell\kappa/D}$, this formula yields the constant reactivity
considered above: $\kappa(\ell)=\kappa$. However, another choice of the
function $\Psi(\ell)$ allows one to implement the reactivity of the catalytic
surface that depends on how many times the reactant has encountered it. To
illustrate this idea, let us consider the gamma model, by choosing
$\Psi(\ell)=\Gamma(\nu,q\ell)$, where $q>0$ and $\nu>0$ are two parameters,
and $\Gamma(\nu,z)=\int\nolimits_{z}^{\infty}dx\,x^{\nu-1}e^{-x}$ is the upper
incomplete gamma function. For $\nu=1$, one has $\Gamma(1,z)=e^{-z}$ and thus
retrieves the above setting of constant reactivity $\kappa=qD$. Figure 3
illustrates the corresponding encounter-dependent reactivity $\kappa(\ell)$,
given by Eq. (10) (panel a), and the overall reaction rate $J(t)$ on a
spherical catalyst of radius $R$ (panel b) that can be found in the framework
of the encounter-based approach [109]. When $0<\nu<1$, the catalytic surface
is highly reactive at the beginning and then reaches a constant reactivity
$qD$. This situation can model a progressive passivation of the catalytic
surface by repeated encounters with the reactant, up to a constant level.
Expectedly, the diffusive flux is high at short times and then decreases to a
constant steady-state level. Note that $\nu=0$ formally corresponds to a
perfect reaction, with the Smoluchowski’s rate (7). The particular value
$\nu=1$ yields the constant reactivity, independent of the number of
encounters, for which the diffusive flux is constant at short times, $4\pi
R^{2}qD[A]_{0}$, and slowly decreases to another constant at long times, as
predicted by Collins and Kimball [23]. In turn, if $\nu>1$, the catalytic
surface is passive at the beginning and then reaches a constant reactivity.
This situation can model progressive activation of that catalytic surface.
Accordingly, the overall reaction rate is zero at short times and then
increases to a constant steady-state level. Choosing an appropriate function
$\Psi(\ell)$, one can produce the desired shape of the encounter-dependent
reactivity $\kappa(\ell)$ that opens a way to model various surface reaction
mechanisms.

Figure 3:  (a) Encounter-dependent reactivity $\kappa(\ell)$ from the gamma
model, with $q=1$ and three values of $\nu$. (b) The overall reaction rate
$J(t)$ on a spherical catalyst of radius $R$, rescaled by the Smoluchowski’s
rate $J_{S}=4\pi DR[A]_{0}$, with $q=1$ and three values of $\nu$. Dotted
curve represents Eq. (7) for a perfectly reactive sphere (it formally
corresponds to $\nu=0$).

The encounter-based approach goes far beyond the conventional theory of
diffusion-controlled reactions described by Dirichlet or Robin boundary
conditions. From the mathematical point of view, the description of a general
surface reaction with an encounter-dependent reactivity $\kappa(\ell)$ is not
reducible to the Robin boundary condition. As a consequence, Laplacian
eigenfunctions that are conventionally used in spectral expansions, need to be
replaced by so-called Steklov eigenfunctions [109, 117]. Though being less
known in the context of chemical reactions, these eigenfunctions turn out to
be particularly well suited for describing diffusive explorations near a
catalytic surface between successive reaction attempts. Several extensions of
the encounter-based approach have already been explored such as (i) inclusion
of an external potential that leads to a biased or drifted motion [118]; (ii)
the effects of stochastic resetting [119, 120] of the position and of the
boundary local time onto diffusion-controlled reactions [121, 122]; (iii) the
cooperative search by multiple independent particles and the related extreme
first-passage statistics [123]; (iv) the escape problem [124]; (v) non-
Markovian binding/unbinding kinetics [95]. Moreover, the same concepts can be
applied to describe diffusive permeation across membranes [125, 126, 127].
Despite these recent advances, there are many open questions and promising
perspectives for future developments, such as merging anomalous bulk
diffusions with generalized surface reactions, the effect of sophisticated
geometric confinements onto the encounter statistics, competition of multiple
reactive centers for capturing a limited amount of diffusing reactants,
indirect coupling of different reactants through encounter-dependent catalytic
surfaces, inference of appropriate surface reaction models from experimental
data, to name but a few.

##  VI Conclusion

In summary, we reviewed the major steps in the long history of developments in
the theory of diffusion-controlled reactions. M. von Smoluchowski first
recognized the importance of the diffusion step, during which the reactants
have to meet each other. He also put forward the diffusion equation to
describe of the dynamics of reactants in the bulk and boundary conditions to
account for the reaction on the surface. His mechanism of perfect reactions
upon the first encounter was then improved by Collins and Kimball. While most
later theoretical efforts were dedicated to improvements of the bulk dynamics,
an encounter-based approach was recently developed to enable more general
surface reaction mechanisms. This approach has already shown many advantages
such as probabilistic insights onto surface reactions, disentanglement of the
impacts of shape and reactivity of the catalytic surface, flexibility in
characterization of diffusive explorations near the reactive surface, etc. In
particular, the concept of encounter-dependent reactivity allows one to
describe an action of reactants onto the catalytic surface, and such a
feedback may potentially be relevant in various biochemical and
electrochemical settings. There are still many open questions and current
developments, aiming at understanding the mathematical formalism of the
encounter-based approach, relating the shape of the catalytic surface to the
spectral properties of the underlying operators, elaborating various
extensions, and uncovering potential applications in chemistry and
biochemistry.

###### Acknowledgements.

The author acknowledges the Alexander von Humboldt Foundation for support
within a Bessel Prize award.

## References

  * Murrey [2003] Murrey, J. D. Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd Ed.; Springer: Berlin, Germany, 2003.
  * Volpert [2014] Volpert, V. Elliptic Partial Differential Equations. Vol. 2. Reaction-Diffusion Equations; Monographs in Mathematics, 104; Springer: Basel, Switzerland, 2014.
  * Smoluchowski [1918] von Smoluchowski, M. Versuch einer Mathematischen Theorie der Koagulations Kinetic Kolloider Lousungen. Z. Phys. Chem. 1918, 92U, 129–168.
  * North [1966] North, A. M. Diffusion-controlled reactions. Q. Rev. Chem. Soc. 1966, 20, 421–440.
  * Wilemski [1973] Wilemski, G.; Fixman, M. General theory of diffusion-controlled reactions. J. Chem. Phys. 1973, 58, 4009–4019.
  * Berg [1985] Berg, O. G.; von Hippel, P. H. Diffusion-Controlled Macromolecular Interactions. Ann. Rev. Biophys. Biophys. Chem. 1985, 14, 131–160.
  * Rice [1985] Rice, S. Diffusion-Limited Reactions; Elsevier: Amsterdam, The Netherlands, 1985.
  * Bressloff [2013] Bressloff, P. C.; Newby, J. M. Stochastic models of intracellular transport. Rev. Mod. Phys. 2013, 85, 135–196.
  * Lindenberg [2019] Lindenberg, K.; Metzler, R.; Oshanin, G. (Eds.) Chemical Kinetics: Beyond the Textbook; World Scientific: New Jersey, 2019.
  * Benichou [2010] Bénichou, O.; Chevalier, C.; Klafter, J.; Meyer, B.; Voituriez, R. Geometry-controlled kinetics. Nat. Chem. 2010, 2, 472–477.
  * Benichou [2014] Bénichou, O.; Voituriez, R. From first-passage times of random walks in confinement to geometry-controlled kinetics. Phys. Rep. 2014, 539, 225–284.
  * Grebenkov [2019] Grebenkov, D. S. Imperfect Diffusion-Controlled Reactions. in Chemical Kinetics: Beyond the Textbook; Lindenberg, K.; Metzler, R.; Oshanin, G., Eds.; World Scientific: New Jersey, 2019, pp. 191–219.
  * Blender [1990] Blender, R.; Dieterich, W.; Kirchhoff, T.; Sapoval, B. Impedance of Fractal Interfaces. J. Phys. A: Math. Gen. 1990, 23, 1225–1232.
  * Coppens [1999] Coppens, M.-O. The effect of fractal surface roughness on diffusion and reaction in porous catalysts: from fundamentals to practical applications. Cat. Today 1999, 53, 225–243.
  * Filoche [2000] Filoche, M.; Sapoval, B. Transfer Across Random versus Deterministic Fractal Interfaces. Phys. Rev. Lett. 2000, 84, 5776–5779.
  * Andrade [2007] Andrade, J. S. Jr.; Araujo, S. D.; Filoche, M.; Sapoval, B. Screening Effects in Flow through Rough Channels. Phys. Rev. Lett. 2007, 98, 194101.
  * Grebenkov [2005] Grebenkov, D. S.; Filoche, M.; Sapoval, B.; Felici, M. Diffusion-reaction in Branched Structures: Theory and Application to the Lung Acinus. Phys. Rev. Lett. 2005, 94, 050602.
  * Serov [2016] Serov, A. S.; Salafia, C.; Grebenkov, D. S.; Filoche, M. The Role of Morphology in Mathematical Models of Placental Gas Exchange. J. Appl. Physiol. 2016, 120, 17–28.
  * Galanti [2016] Galanti, M.; Fanelli, D.; Traytak, S. D.; Piazza, F. Theory of diffusion-influenced reactions in complex geometries. Phys. Chem. Chem. Phys. 2016, 18, 15950–15954.
  * Grebenkov [2019f] Grebenkov, D. S.; Traytak, S. D. Semi-analytical computation of Laplacian Green functions in three-dimensional domains with disconnected spherical boundaries. J. Comput. Phys. 2019, 379, 91–117.
  * Grebenkov [2020f] Grebenkov, D. S. Diffusion toward non-overlapping partially reactive spherical traps: fresh insights onto classic problems. J. Chem. Phys. 2020, 152, 244108.
  * Nguyen [2010] Nguyen, B.-T.; Grebenkov, D. S. A Spectral Approach to Survival Probability in Porous Media. J. Stat. Phys. 2010, 141, 532–554.
  * Collins [1949] Collins, F. C.; Kimball, G. E. Diffusion-controlled reaction rates. J. Coll. Sci. 1949, 4, 425–437.
  * Piazza [2022] Piazza, F. The physics of boundary conditions in reaction-diffusion problems. J. Chem. Phys. 2022, 157, 234110.
  * Weiss [1986] Weiss, G. H. Overview of theoretical models for reaction rates. J. Stat. Phys. 1986, 42, 3–36.
  * Hanggi [1990] Hänggi, P.; Talkner, P.; Borkovec, M. Reaction-rate theory: fifty years after Kramers. Rev. Mod. Phys. 1990, 62, 251–341.
  * Grebenkov [2003] Grebenkov, D. S.; Filoche, M.; Sapoval, B. Spectral Properties of the Brownian Self-Transport Operator. Eur. Phys. J. B, 2003, 36, 221–231.
  * Bressloff [2017] Bressloff, P. C. Stochastic switching in biology: from genotype to phenotype. J. Phys. A.: Math. Theor. 2017, 50, 133001.
  * Cortes [2010] Cortés, J.; Le, D. T.; Iehl, R.; Siméon, T. Simulating ligand-induced conformational changes in proteins using a mechanical disassembly method. Phys. Chem. Chem. Phys. 2010, 12, 8268–8276.
  * Luking [2022] Lüking, M.; Elf, J.; Levy, Y. Conformational Change of Transcription Factors from Search to Specific Binding: A lac Repressor Case Study. J. Phys. Chem. B 2022, 126, 9971-9984.
  * Galanti [2016b] Galanti, M.; Fanelli, D.; Piazza, F. Conformation-controlled binding kinetics of antibodies. Scient. Rep. (2016), 18976.
  * Reva [2021] Reva, M.; DiGregorio, D. A.; Grebenkov, D. S. A first-passage approach to diffusion-influenced reversible binding: insights into nanoscale signaling at the presynapse. Sci. Rep. 2021, 11, 5377.
  * Benichou [2000] Bénichou, O.; Moreau, M.; Oshanin, G. Kinetics of stochastically gated diffusion-limited reactions and geometry of random walk trajectories. Phys. Rev. E 2000, 61, 3388–3406.
  * Reingruber [2009] Reingruber, J.; Holcman, D. Gated Narrow Escape Time for Molecular Signaling. Phys. Rev. Lett. 2009, 103, 148102.
  * Lawley [2015] Lawley, S. D.; Keener, J. P. A new derivation of Robin boundary conditions through homogenization of a stochastically switching boundary. SIAM J. Appl. Dyn. Syst. 2015, 14, 1845–1867.
  * Zhou [1991] Zhou, H.-X.; Zwanzig, R. A rate process with an entropy barrier. J. Chem. Phys. 1991, 94, 6147–6152.
  * Reguera [2006] Reguera, D.; Schmid, G. Burada, P. S.; Rubí, J.-M.; Reimann, P; Hänggi, P. Entropic Transport: Kinetics, Scaling, and Control Mechanisms. Phys. Rev. Lett. 2006, 96, 130603.
  * Chapman [2016] Chapman, S. J.; Erban, R.; Isaacson, S. Reactive boundary conditions as limits of interaction potentials for Brownian and Langevin dynamics. SIAM J. Appl. Math. 2016, 76, 368–390.
  * Berg [1977] Berg, H. C.; Purcell, E. M. Physics of chemoreception. Biophys. J. 1977, 20, 193–219.
  * Berezhkovskii [2004] Berezhkovskii, A. M.; Makhnovskii, Y. A.; Monine, M. I.; Zitserman, V. Y.; Shvartsman, S. Y. Boundary homogenization for trapping by patchy surfaces. J. Chem. Phys. 2004, 121, 11390–11394.
  * Berezhkovskii [2006] Berezhkovskii, A. M.; Monine, M. I.; Muratov, C. B.; Shvartsman, S. Y. Homogenization of boundary conditions for surfaces with regular arrays of traps. J. Chem. Phys. 2006, 124, 036103.
  * Muratov [2008] Muratov, C. B.; Shvartsman, S. Y. Boundary homogenization for periodic arrays of absorbers. Multiscale Model. Simul. 2008, 7, 44–61.
  * Bernoff [2018] Bernoff, A.; Lindsay, A.; Schmidt, D. Boundary homogenization and capture time distributions of semipermeable membranes with periodic patterns of reactive sites. Multiscale Model. Simul. 2018, 16, 1411–1447.
  * Punia [2021] Punia, B.; Chaudhury, S.; Kolomeisky, A. B. Understanding the Reaction Dynamics on Heterogeneous Catalysts Using a Simple Stochastic Approach. J. Phys. Chem. Lett. 2021, 12, 11802–11810.
  * Sapoval [1994] Sapoval, B. General Formulation of Laplacian Transfer Across Irregular Surfaces. Phys. Rev. Lett. 1994, 73, 3314-3316.
  * Sapoval [1996] Sapoval, B. Transport Across Irregular Interfaces: Fractal Electrodes, Membranes and Catalysts, in Fractals and Disordered Systems, Bunde, A.; Havlin, S., Eds.; Springer-Verlag: Berlin, 1996, p. 233-261.
  * Grebenkov [2006] Grebenkov, D. S.; Filoche, M.; Sapoval, B. Mathematical Basis for a General Theory of Laplacian Transport towards Irregular Interfaces. Phys. Rev. E 2006, 73, 021103.
  * Sano [1979] Sano, H.; Tachiya, M. Partially diffusion-controlled recombination. J. Chem. Phys. 1979, 71, 1276–1282.
  * Brownstein [1979] Brownstein, K. R.; Tarr, C. E. Importance of Classical Diffusion in NMR Studies of Water in Biological Cells. Phys. Rev. A 1979, 19, 2446–2453.
  * Powles [1992] Powles, J. G.; Mallett, M. J. D.; Rickayzen, G.; Evans, W. A. B. Exact analytic solutions for diffusion impeded by an infinite array of partially permeable barriers. Proc. R. Soc. London A 1992, 436, 391–403.
  * Bressloff [2008] Bressloff, P. C.; Earnshaw, B. A.; Ward, M. J. Diffusion of protein receptors on a cylindrical dendritic membrane with partially absorbing traps. SIAM J. Appl. Math. 2008, 68, 1223–1246.
  * Grebenkov [2017a] Grebenkov, D. S.; Oshanin, G. Diffusive escape through a narrow opening: new insights into a classic problem. Phys. Chem. Chem. Phys. 2017, 19, 2723–2739.
  * Guerin [2021] Guérin, T.; Dolgushev, M.; Bénichou, O.; Voituriez, R. Universal kinetics of imperfect reactions in confinement. Commun. Chem. 2021, 4, 157.
  * Sapoval [2002] Sapoval, B.; Filoche, M.; Weibel, E. Smaller is better - but not too small: A physical scale for the design of the mammalian pulmonary acinus. Proc. Nat. Acad. Sci. USA 2002, 99, 10411–10416.
  * Gardiner [1985] Gardiner, C. W. Handbook of stochastic methods for physics, chemistry and the natural sciences; Springer: Berlin, Germany, 1985.
  * Risken [1996] Risken, H. The Fokker-Planck equation: methods of solution and applications, 3rd Ed.; Springer: Berlin, Germany, 1996.
  * VanKampen [1992] Van Kampen, N. G. Stochastic Processes in Physics and Chemistry; Elsevier: Amsterdam, the Netherlands, 1992.
  * Schuss [2013] Schuss, Z. Brownian Dynamics at Boundaries and Interfaces in Physics, Chemistry and Biology; Springer: New York, USA, 2013.
  * Metzler [2000] Metzler, R.; Klafter, J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 2000, 339, 1–77.
  * Metzler [2004] Metzler, R.; Klafter, J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 2004, 37, R161–R208.
  * Sokolov [2012] Sokolov, I. M. Models of anomalous diffusion in crowded environments. Soft Matter 2012, 8, 9043–9052.
  * Krapf [2019] Krapf, D.; Metzler, R. Strange interfacial molecular dynamics. Phys. Today 2019, 72, 48–54.
  * Chubynsky [2014] Chubynsky, M. V.; Slater, G. W. Diffusing Diffusivity: A Model for Anomalous, yet Brownian, Diffusion. Phys. Rev. Lett. 2014, 113, 098302.
  * Chechkin [2017] Chechkin, A. V.; Seno, F.; Metzler, R.; Sokolov, I.M. Brownian yet Non-Gaussian Diffusion: From Superstatistics to Subordination of Diffusing Diffusivities. Phys. Rev. X 2017, 7, 021002.
  * Lanoiselee [2018] Lanoiselée, Y.; Moutal, N.; Grebenkov, D. S. Diffusion-limited reactions in dynamic heterogeneous media. Nature Commun. 2018, 9, 4398.
  * Grebenkov [2019d] Grebenkov, D. S.; A unifying approach to first-passage time distributions in diffusing diffusivity and switching diffusion models. J. Phys. A: Math. Theor. 2019, 52, 174001.
  * Yuste [2013] Yuste, S. B.; Abad, E.; Lindenberg, K. Exploration and Trapping of Mortal Random Walkers. Phys. Rev. Lett. 2013, 110, 220603.
  * Meerson [2015] Meerson, B.; Redner, S. Mortality, Redundancy, and Diversity in Stochastic Search. Phys. Rev. Lett. 2015, 114, 198101.
  * Grebenkov [2017] Grebenkov, D. S.; Rupprecht, J.-F. The escape problem for mortal walkers. J. Chem. Phys. 2017, 146, 084106.
  * Grebenkov [2007] Grebenkov, D. S. NMR Survey of Reflected Brownian Motion. Rev. Mod. Phys. 2007, 79, 1077–1137.
  * Turing [1952] Turing, A. The Chemical Basis of Morphogenesis. Phil. Trans. R. Soc. Lond B 1952, 237, 37–72.
  * Milo [2015] Milo, R.; Rhillips, R. Cell Biology by the Numbers; Garland Science: New York, USA, 2015.
  * Redner [2001] Redner, S. A Guide to First Passage Processes; Cambridge University Press: Cambridge, UK, 2001.
  * Metzler [2014] Metzler, R.; Oshanin, G.; Redner, S. (Eds), First-Passage Phenomena and Their Applications; World Scientific: Singapore, 2014.
  * Yu [2006] Yu, J.; Xiao, J.; Ren, X.; Lao, K.; Xie, X. S. Probing Gene Expression in Live Cells, One Protein Molecule at a Time. Science 2006, 311, 1600–1603.
  * Raj [2008] Raj, A.; van Oudenaarden, A. Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences. Cell 2008, 135, 216–226.
  * Xie [2008] Xie, X. S.; Choi, P. J.; Li, G.-W.; Lee, N. K.; Lia, G. Single-Molecule Approach to Molecular Biology in Living Bacterial Cells. Annu. Rev. Biophys. 2008, 37, 417–444.
  * Li [2011] Li, G.-W.; Xie, X. S. Central dogma at the single-molecule level in living cells. Nature 2011, 475, 308–315.
  * Kastantin [2012] Kastantin, M.; Walder, R.; Schwartz, D. K. Identifying Mechanisms of Interfacial Dynamics Using Single-Molecule Tracking. Langmuir 2012, 28, 12443–12456.
  * Wang [2017] Wang, D.; Wu, H.; Schwartz, D. K. Three-Dimensional Tracking of Interfacial Hopping Diffusion. Phys. Rev. Lett. 2017, 119, 268001.
  * Norregaard [2017] Norregaard, K.; Metzler, R.; Ritter, C. M.; Berg-Sørensen, K.; Oddershede, L. B. Manipulation and Motion of Organelles and Single Molecules in Living Cells. Chem. Rev. 2017, 117, 4342–4374.
  * Sungkaworn [2017] Sungkaworn, T.; Jobin, M.-L.; Burnecki, K.; Weron, A.; Lohse, M.-J.; Calebiro, D. Single-molecule imaging reveals receptor-G protein interactions at cell surface hot spots. Nature 2017, 550, 543–547.
  * Grebenkov [2018c] Grebenkov, D. S.; Metzler, R.; Oshanin, G. Strong defocusing of molecular reaction times results from an interplay of geometry and reaction control. Commun. Chem. 2018, 1, 96.
  * Grebenkov [2018d] Grebenkov, D. S.; Metzler, R.; Oshanin, G. Towards a full quantitative description of single-molecule reaction kinetics in biological cells. Phys. Chem. Chem. Phys. 2018, 20, 16393–16401.
  * Elf [2019] Elf, J.; Barkefors, I. Single-molecule kinetics in living cells. Ann. Rev. Biochem. 2019, 88, 635–659.
  * Grebenkov [2019c] Grebenkov, D. S. Spectral theory of imperfect diffusion-controlled reactions on heterogeneous catalytic surfaces. J. Chem. Phys. 2019, 151, 104108.
  * Goodrich [1954] F. C. Goodrich, F. C. Random walk with semi-adsorbing barrier. J. Chem. Phys. 1954, 22, 588–594.
  * Mysels [1982] Mysels, K. J. Diffusion-controlled adsorption kinetics. General solution and some applications. J. Phys. Chem. 1982, 86, 4648–4651.
  * Agmon [1984] Agmon, N. Diffusion with back reaction. J. Chem. Phys. 1984, 81, 2811–2817.
  * Agmon [1989] Agmon, N.; Weiss, G. H. Theory of non-Markovian reversible dissociation reactions. J. Chem. Phys. 1989, 91, 6937–6942.
  * Agmon [1990] Agmon, N.; Szabo, A. Theory of reversible diffusion-influenced reactions. J. Chem. Phys. 1990, 92, 5270–5284.
  * Kim [1999] Kim, H.; Shin, K. J. Exact solution of the reversible diffusion-influenced reaction for an isolated pair in three dimensions. Phys. Rev. Lett. 1999, 82, 1578–1581.
  * Prüstel [2013] Prüstel, T.; Tachiya, M. Reversible diffusion-influenced reactions of an isolated pair on some two dimensional surfaces. J. Chem. Phys. 2013, 139, 194103.
  * Scher [2023] Scher, Y.; Lauber Bonomo, O.; Pal, A.; Reuveni, S. Microscopic theory of adsorption kinetics. J. Chem. Phys. 2023, 158, 094107.
  * Grebenkov [2023b] Grebenkov, D. S. Diffusion-controlled reactions with non-Markovian binding/unbinding kinetics. J. Chem. Phys. 2023, 158, 214111.
  * Chechkin [2009] Chechkin, A. V.; Zaid, I. M.; Lomholt, M.; Sokolov, I. M.; Metzler, R. Bulk-mediated surface diffusion along a cylinder: Propagators and crossovers. Phys. Rev. E 2009, 79, 040105(R).
  * Benichou [2010b] Bénichou, O.; Grebenkov, D. S.; Levitz, P. E.; Loverdo, C.; Voituriez, R. Optimal Reaction Time for Surface-Mediated Diffusion. Phys. Rev. Lett. 2010, 105, 150606.
  * Benichou [2011b] Bénichou, O.; Grebenkov, D. S.; Levitz, P. E.; Loverdo, C.; Voituriez, R. Mean first-passage time of surface-mediated diffusion in spherical domains. J. Stat. Phys. 2011, 142, 657–685.
  * Rojo [2011] Rojo, F.; Budde, C. E. Enhanced diffusion through surface excursion: A master-equation approach to the narrow-escape-time problem. Phys. Rev. E 2011, 84, 021117.
  * Chechkin [2011] Chechkin, A. V.; Zaid, I. M.; Lomholt, M. A.; Sokolov, I. M.; Metzler, R. Effective surface motion on a reactive cylinder of particles that perform intermittent bulk diffusion. J. Chem. Phys. 2011, 134, 204116.
  * Chechkin [2012] Chechkin, A. V.; Zaid, I. M.; Lomholt, M.; Sokolov, I. M.; Metzler, R. Bulk-mediated diffusion on a planar surface: Full solution. Phys. Rev. E 2012, 86, 041101.
  * Rupprecht [2012a] Rupprecht, J.-F.; Bénichou, O.; Grebenkov, D. S.; Voituriez, R. Exact mean exit time for surface-mediated diffusion. Phys. Rev. E 2012, 86, 041135.
  * Rupprecht [2012b] Rupprecht, J.-F.; Bénichou, O.; Grebenkov, D. S.; Voituriez, R. Kinetics of active surface-mediated diffusion in spherically symmetric domains. J. Stat. Phys. 2012, 147, 891–918.
  * Berezhkovskii [2015] Berezhkovskii, A. M.; Dagdug, L.; Bezrukov, S. M. A new approach to the problem of bulk-mediated surface diffusion. J. Chem. Phys. 2015, 143, 084103.
  * Berezhkovskii [2017] Berezhkovskii, A. M.; Dagdug, L.; Bezrukov, S. M. Bulk-mediated surface transport in the presence of bias. J. Chem. Phys. 2017, 147, 014103.
  * Benichou [2011] Bénichou, O.; Loverdo, C.; Moreau, M.; Voituriez, R. Intermittent search strategies. Rev. Mod. Phys. 2011, 83, 81–129.
  * Filoche [2005] Filoche, M.; Sapoval, B.; Andrade J. S. Jr, Deactivation Dynamics of Rough Catalytic Surfaces. AIChE 2005, 51, 998–1008.
  * Filoche [2008] Filoche, M.; Grebenkov, D. S.; Andrade A. S. Jr.; Sapoval, B. Passivation of irregular surfaces accessed by diffusion. Proc. Nat. Acad. Sci. USA 2008, 105, 7636–7640.
  * Grebenkov [2020] Grebenkov, D. S. Paradigm Shift in Diffusion-Mediated Surface Phenomena. Phys. Rev. Lett. 2020, 125, 078102.
  * Grebenkov [2007b] Grebenkov, D. S. Residence times and other functionals of reflected Brownian motion. Phys. Rev. E 2007, 76, 041139.
  * Grebenkov [2019b] Grebenkov, D. S. Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains. Phys. Rev. E 2019, 100, 062110.
  * Grebenkov [2020b] Grebenkov, D. S. Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets. J. Stat. Mech. 2020, 103205.
  * Grebenkov [2020c] Grebenkov, D. S. Surface Hopping Propagator: An Alternative Approach to Diffusion-Influenced Reactions. Phys. Rev. E 2020, 102, 032125.
  * Grebenkov [2021] Grebenkov, D. S. Statistics of boundary encounters by a particle diffusing outside a compact planar domain. J. Phys. A.: Math. Theor. 2021, 54, 015003.
  * Bressloff [2022] Bressloff, P. C. Narrow capture problem: An encounter-based approach to partially reactive targets. Phys. Rev. E 2022, 105, 034141.
  * Grebenkov [2022] Grebenkov, D. S., Statistics of diffusive encounters with a small target: Three complementary approaches. J. Stat. Mech. 2022, 083205.
  * [117] Levitin, M.; Mangoubi, D.; Polterovich, I. Topics in Spectral Geometry (Preliminary version, May 29, 2023; https://www.michaellevitin.net/Book/TSG230529.pdf)
  * Grebenkov [2022a] Grebenkov, D. S. An encounter-based approach for restricted diffusion with a gradient drift. J. Phys. A: Math. Theor. 2022, 55, 045203.
  * Evans [2011] Evans, M. R.; Majumdar, S. N. Diffusion with Stochastic Resetting. Phys. Rev. Lett. 2011, 106, 160601.
  * Evans [2020] Evans, M. R.; Majumdar, S. N.; Schehr, G. Stochastic resetting and applications. J. Phys. A: Math. Theor. 2020, 53, 193001.
  * Bressloff [2022d] Bressloff, P. C. Diffusion-mediated surface reactions and stochastic resetting. J. Phys. A: Math. Theor. 2022, 55, 275002.
  * Benkhadaj [2022] Benkhadaj, Z.; Grebenkov, D. S. Encounter-based approach to diffusion with resetting. Phys. Rev. E 2022, 106, 044121.
  * Grebenkov [2022b] Grebenkov, D. S. Depletion of Resources by a Population of Diffusing Species. Phys. Rev. E 2022, 105, 054402.
  * Grebenkov [2023a] Grebenkov, D. S. Encounter-based approach to the escape problem. Phys. Rev. E 2023, 107, 044105.
  * Bressloff [2022c] Bressloff, P. C. A probabilistic model of diffusion through a semipermeable barrier. Proc. Roy. Soc. A 2022, 478, 20220615.
  * Bressloff [2023a] Bressloff, P. C. Renewal equation for single-particle diffusion through a semipermeable interface. Phys. Rev. E. 2023, 107, 014110.
  * Bressloff [2023b] Bressloff, P. C. Renewal equations for single-particle diffusion in multilayered media. SIAM J. Appl. Math. 2023, 83, 1518–1545.