A Derivation of the reactive boundary condition for a planar surface

Effective surface motion on a reactive cylinder of particles that perform intermittent bulk diffusion

Abstract

In many biological and small scale technological applications particles may transiently bind to a cylindrical surface. In between two binding events the particles diffuse in the bulk, thus producing an effective translation on the cylinder surface. We here derive the effective motion on the surface, allowing for additional diffusion on the cylinder surface itself. We find explicit solutions for the number of adsorbed particles at one given instant, the effective surface displacement, as well as the surface propagator. In particular sub- and superdiffusive regimes are found, as well as an effective stalling of diffusion visible as a plateau in the mean squared displacement. We also investigate the corresponding first passage and first return problems.

pacs:
05.40.Fb,02.50.Ey,82.20.-w,87.16.-b
1

I Introduction

Bulk mediated surface diffusion (BMSD) defines the effective surface motion of particles, that intermittently adsorb to a surface or diffuse in the contiguous bulk volume. As sketched in Fig. 1 for a cylindrical surface, the particle, say, starts on the surface and diffuses along this surface with diffusion constant . Eventually the particle unbinds, and performs a three-dimensional stochastic motion in the adjacent bulk, before returning to the surface. Typically, the values of are significantly larger than . The recurrent bulk excursions therefore lead to decorrelations in the effective surface motion of the particle, and thus to a more efficient exploration of the surface.

Figure 1: A particle diffuses in the bulk (full lines) and intermittently binds to a cylinder surface on which it may also diffuse (broken lines). This produces an effective surface motion. We here consider the motion along the cylinder. Bottom right: frontal of diffusion between inner and outer cylinder.

Theoretically BMSD was previously investigated for a planar surface in terms of scaling arguments (1); (2), master equation schemes (3), and simulations (4). More recently the first passage problem between particle unbinding and rebinding for a free cylindrical surface was derived (5). Following our short communication (6) we here present in detail an exact treatment of BMSD for a reactive cylindrical surface deriving explicit expressions for the surface occupation, the effective mean squared displacement (MSD) along the surface, and the returning time distribution from the bulk. In this approach different dynamic regimes arise naturally from the physical timescales entering our description. Thus at shorter times we derive the famed superdiffusive surface spreading with surface MSD of the form (7); (8) and the associated Cauchy form of the surface probability density function (PDF). At longer times we obtain an a priori unexpected leveling off of the surface MSD, representing a tradeoff between an increasing number of particles that escape into the bulk and the increasing distance on the surface covered in ever-longer bulk excursions for those particles that do return to the surface. Only when the system is confined by an outer cylinder eventually normal surface diffusion will emerge. Apart from the Lévy walk-like superdiffusive regime the rich dynamic behavior found here are characteristic of the cylindrical geometry.

Nuclear magnetic resonance (NMR) measurements of liquids in porous media are sensitive to the preferred orientation of adsorbate molecules on the local pore surface, such that surface diffusion on such a non-planar surface produces spin reorientations and remarkably long correlations times (9). Apart from pure surface diffusion the experiment by Stapf et al. clearly showed the influence of BMSD steps and the ensuing Lévy walk-like superdiffusion (10). More recently NMR techniques were used to unravel the effective surface diffusion on cylindrical mineralic rods (5), supporting, in particular, the first passage behavior with its typical logarithmic dependence. BMSD on a cylindrical surface is also relevant for the transient binding of chemicals to nanotubes (11) and for numerous other technological applications (2). In a biological context, BMSD along a cylinder is intimately related to the diffusive dynamics underlying gene regulation (12); (13): DNA binding proteins diffuse not only in the bulk but intermittently bind non-specifically to the DNA, approximately a cylinder, and perform a one-dimensional motion along the DNA chain, as proved experimentally (15); (14). The interplay between bulk and effective surface motion improves significantly the search process of the protein for its specific binding site on the DNA. Similarly the net motion of motor proteins along cytoskeletal filaments is also affected by bulk mediation. Namely, the motors can fall off the cellular tracks and then rebind to the filament after a bulk excursion (16). Outside of biological cells the exchange behavior between cell surface and surrounding bulk is influenced by bulk excursions, the cylindrical geometry being of relevance for a large class of rod-shaped bacteria (bacilli) and their linear arrangements (17).

The dynamics revealed by our approach may also be important for the quantitative understanding of colonialization processes on surfaces in aqueous environments when convection is negligible: suppose that bacteria stemming from a localized source, for instance, near to a submarine hot vent, start to grow on an offshore pipeline. From this mother colony new bacteria will be budding and enter the contiguous water. The Lévy dust-like distribution due to BMSD will then make sure that bacteria can start a new colony, that is disconnected from the former, and therefore give rise to a much more efficient spreading dynamics over the pipeline.

In all these examples it is irrelevant which specific trajectory the particles follow in the bulk, the interesting part is the effective motion on the cylinder surface. We here analyze in detail this bulk mediated surface diffusion on a long cylinder.

Ii Characteristic time scales and important results

In this Section we introduce the relevant time scales of the problem of bulk mediated surface diffusion of the cylindrical geometry presented in Fig. 1 and collect the most important results characteristic for the effective surface motion of particles. As we are interested only in the motion along the cylinder axis , we consider the rotationally symmetric problem with respect to the polar angle , that will therefore not appear explicitly in the following expressions (compare also Sec. III). Since the full analytical treatment of the problem involves tedious calculations we first give an overview of the most important results, leaving the derivations to the forthcoming Sections and Appendices.

ii.1 Characteristic time scales

Using the result (48) for the Fourier-Laplace transform of the density of particles on the cylinder surface, we compute the Laplace transform of the number of surface particles,

(1)

where is a surface-bulk coupling constant defined below, is the bulk diffusion constant,

(2)

and . The and denote modified Bessel functions. We define the Laplace and Fourier transforms of the surface density through

(3)

and

(4)

Here and in the following we express the transform of a function by explicit dependence on the Fourier or Laplace variable, thus, is the Fourier-Laplace transform of .

From expression (1) we recognize that in the limit the number of particles on the cylinder surface does not change, i.e., . The coupling parameter according to Eq. (49) is connected to the unbinding time scale , the bulk diffusivity , and the binding rate through . Vanishing therefore corresponds to an infinite time scale for unbinding. This observation allows us to introduce a characteristic coupling time

(5)

Note that the binding constant has dimension , see Section III. As in the governing equations the coupling constant and the bulk diffusivity are the relevant parameters, the characteristic time in a scaling sense is uniquely defined. When the coupling between bulk and cylinder surface is weak, , the corresponding coupling time diverges. It vanishes when the coupling is strong, .

While the time scale is characteristic of the bulk-surface exchange, the geometry of the problem imposes two additional characteristic times. Namely, the inner and outer cylinder radii involve the time scales

(6)

and

(7)

respectively. By definition, is always larger than . For times shorter than the scale a diffusing particle behaves as if it were facing a flat surface, while for times longer then it can sense the cylindrical shape of the surface. Similarly, defines the scale when a particle starts to engage with the outer cylinder and therefore senses the confinement. With the help of these time scales we can rewrite expression (1) for the number of surface particles in the form

(8)

where

(9)

and

(10)

From the characteristic time scales , , and we can construct the three limits:

(i) Strong coupling limit

(11)

here the shortest time scale is the coupling time. This regime is the most interesting as it leads to the transient Lévy walk-like superdiffusive behavior.

(ii) Intermediate coupling limit

(12)

here the superdiffusive regime is considerably shorter, however, an interesting transition regime is observed.

(iii) Weak coupling limit

(13)

To limit the scope of this paper we will not consider this latter case in the following. However, we note that for the behavior will be similar to the part of the intermediate coupling limit (ii).

ii.2 Important results

We now discuss the results for the most important quantities characteristic of the effective surface motion. The dynamic quantities we consider are the number of particles , that are adsorbed to the inner cylinder surface at given time ; as well as the one-particle mean squared displacement

(14)

This quantity is biased by the fact that an increasing amount of particles is leaving the surface. To balance for this loss and quantify the effective surface motion for those particles that actually move on the surface, we also consider the ‘normalized’ mean squared displacement

(15)

The detailed behavior of these quantities will be derived in what follows, and we will also calculate the effective surface concentration itself. Here we summarize the results for the surface particle number and the surface mean squared displacements.

Strong coupling limit

In Table 1 we summarize the behavior in the four relevant time regimes for the case of strong coupling. The evolution of the number of particles on the surface turns from an initially constant behavior to an inverse square root decay when the particles engage into surface-bulk exchange. At longer times, the escape of particles to the bulk becomes faster and follows a law. Eventually the confinement by the outer cylinder comes into play, and we reach a stationary limit.

       Time regime                                           
Table 1: Effective surface diffusion, strong coupling limit. For the different regimes we list the number of particles on the surface, the surface mean squared displacement , and the normalized mean squared displacement . where is Euler’s constant.

The mean squared displacement has a very interesting initial anomalously diffusive behavior (7); (8). This superdiffusion arises due to mediation by bulk excursions resulting in the effective Cauchy distribution

(16)

In this initial regime we can use a simple scaling argument to explain this superdiffusive behavior, compare the discussion in Ref. (1). Thus, once detached from the surface a particle returns to the surface with a probability distributed according to . Due to the diffusive coupling in the bulk the effective displacement along the cylinder is then distributed according to , giving rise to a probability density .

Later, the mean squared displacement turns over to a square root behavior corresponding to subdiffusion. As can be seen from the associated normalized mean squared displacement, this behavior is due to the escaping particles. At even longer times the mean squared displacement reaches a plateau value. This is a remarkable property of this cylindrical geometry, reflecting a delicate balance between decreasing particle number and increasing length of the bulk mediated surface translocations. This plateau is the terminal behavior when no outer cylinder is present. That is, even at infinite times, when fewer and fewer particles are on the surface, the surface mean squared displacement does not change. In presence of the outer cylinder the mean squared displacement eventually is dominated by the bulk motion and acquires the normal linear growth with time.

Combining the dynamics of the number of surface particles and the mean squared displacement we obtain the behavior of the normalized mean squared displacement listed in the last column.

Intermediate coupling limit

In the intermediate coupling limit the results are listed in Table 2. Also in this regime we observe the initial superdiffusion and associated Cauchy form of the surface particle concentration. The subsequent regime of intermediate times splits up into two subregimes. This subtle turnover will be discussed in detail below. The last two regimes exhibit the same behavior as the corresponding regimes in the strong coupling limit.

       Time regime                                           
transition to transition to transition to
Table 2: Effective surface diffusion, intermediate coupling limit.

ii.3 Numerical evaluation

In Figs. 2 and 3 we show results from numerical Laplace inversion of the exact expressions for the number of surface particles and the surface mean squared displacement. We consider both the strong and intermediate coupling cases. The parameters fixing the time scales were chosen far apart from each other to distinguish the different limiting behaviors computed in the following Sections. In all figures a vanishing surface diffusivity () is chosen for clarity.

For strong coupling the selected time scales are , , and in dimensionless units. Therefore the bulk diffusion constant becomes for our choice . The coupling constant is , and the outer cylinder radius becomes .

In the intermediate regime we chose , , and . This sets the bulk diffusivity to and the outer cylinder radius to . These values are chosen such that we can plot the results for the intermediate case alongside the strong coupling case.

Figure 2: Time evolution of the number of surface particles obtained by numerical Laplace inversion, for the following cases: strong binding with (green dashed-dotted line) and without (red full line) outer cylinder, and intermediate binding in presence of the outer cylinder (blue dashed line). The indicated characteristic time scales correspond to the cases of strong (bottom) and intermediate (top) coupling.

Fig. 2 shows the time evolution of the number of surface particles, normalized to . For the strong coupling case the value remains almost constant until , and then turns over to an inverse square root decay that lasts until . Subsequently a behavior emerges. In presence of an outer cylinder, due to the confinement this inversely time proportional evolution is finally terminated by a stationary plateau. In the intermediate coupling case similar behavior is observed, apart from the two subregimes in the range at intermediate times.

Figure 3: Surface mean squared displacement obtained by numerical Laplace inversion. Left: The graphs show the various effective diffusion regimes along the cylinder. Note the characteristic transient plateau, which is the terminal behavior in the absence of a confining outer cylinder. Right: Normalized function . The subregime in the intermediate case distinctly shows a superdiffusive behavior that is even steeper than the initial scaling. We show the strong binding case with (red line) and without (green line) and outer cylinder, as well as the case of intermediate binding with outer cylinder (blue line). The characteristic time scales connected with the associated curves denote the cases of strong (bottom) and intermediate (top) coupling.

Fig. 3 depicts the behavior of the surface mean squared displacement. In the left panel the function shows the various regimes found in the strong and intermediate coupling limits. Remarkably the intermediate coupling regime exhibits a superdiffusive behavior in the range that is even faster than the initial scaling. The right panel of Fig. 3 shows the behavior of the normalized surface mean squared displacement. See Sections IV and V for details.

Iii Coupled diffusion equations and general solution

In this Section we state the polar symmetry of the problem we want to consider, and then formulate the starting equations for our model. The general solution is presented in Fourier-Laplace space. In the two subsequent Sections we calculate explicit results in various limiting cases, for strong and intermediate coupling.

iii.1 Starting equation and particle number conservation

The full problem is spanned by the coordinates measured along the cylinder axis, the radius measured perpendicular to the axis, and the corresponding polar angle . We are only interested in the effective displacement of particles along the cylinder axis and therefore eliminate the dependence. This can be consistently done in the following way. (i) As initial condition we assume that initially the particles are concentrated as a sharp peak on the inner cylinder surface, homogeneous in the angle coordinate . (ii) Our boundary conditions are independent.

For the bulk concentration of particles in the volume between the inner and outer cylinders this symmetry requirement simply means that we can integrate out the dependence and consider this concentration as function of , radius , and time : . The physical dimension of the concentration is . On the surface of the inner cylinder we measure the concentration by the density , which is of dimension . Note that does not explicitly depend on . We average this cylinder surface density over the polar angle, and obtain the line density :

(17)

such that . Note that on the inner cylinder with radius the expression is the cylindrical surface increment. The factor is important when we formulate the reactive boundary condition on the inner cylinder connecting surface line density and the volume density .

Given the line density , the total number of particles on the inner cylinder surface at given time becomes

(18)

We assume that initially particles are concentrated in a -peak on the cylinder surface at :

(19)

Consequently the initial bulk concentration vanishes everywhere on the interval such that

(20)

Let us now specify the boundary conditions at the two cylinder surfaces. At the outer cylinder () we impose a reflecting boundary condition of the Neumann form

(21)

In the case when we do not consider an outer cylinder () this Neumann condition may be replaced by a natural boundary condition of the form

(22)

The reactive boundary condition on the inner cylinder () is derived from a discrete random walk process in App. A (compare also Refs. (18)). Accordingly we balance the flux away from the inner cylinder surface,

(23)

by the incoming flux from the bulk onto the cylinder surface,

(24)

Here, with dimension is the characteristic time scale for particle unbinding from the surface. It is proportional to the Arrhenius factor of the binding free energy of the particles, , where denotes the thermal energy at temperature . The binding rate , in contrast, has physical dimension , which is typical for surface-bulk coupling in cylindrical coordinates, compare the discussions in Refs. (12); (13); (18). For convenience, we collect the coefficients in the reactive boundary condition (24) into the coupling constant

(25)

such that our reactive boundary condition finally is recast into the form

(26)

The time evolution of the bulk density is governed by the cylindrical diffusion equation

(27)

valid on the domain and . In Eq. (27), is the bulk diffusion coefficient of dimension . From a random walk perspective we can write , where is the average variance of individual jumps, and is the typical time between consecutive jumps. As shown in App. A the dynamic equation for the line density directly includes the incoming flux term and is given by

(28)

where denotes the surface diffusion coefficient. In many realistic cases the magnitude of is considerably smaller than the bulk diffusivity . The coupling term connects the surface density to the bulk concentration . The fact that here the bulk diffusivity occurs as coupling term stems from the continuum limit, in which the binding rate diverges, and therefore the binding corresponds to the step from the exchange site to the surface.

The diffusion equations (27) and (28) together with the boundary conditions (21) and (26) as well as the initial conditions (19) and (20) completely specify our problem. Moreover the total number of particles is conserved. Namely, the number of surface particles varies with time as

(29)

as can be seen from integration of Eq. (28) over and noting that . For the number of bulk particles we obtain

(30)

From these two relations we see that indeed the total number of particles fulfills

(31)

and therefore .

iii.2 Solution of the bulk diffusion equation

To solve Eq. (27) and the corresponding boundary and initial value problem we use the Fourier-Laplace transform method. The dynamic equation for is the ordinary differential equation

(32)

where we use the abbreviation

(33)

The reactive boundary condition becomes

(34)

and for the reflective condition we find

(35)

The general solution of Eq. (32) is given in terms of the zeroth order modified Bessel functions and in the linear combination

(36)

The constants and follow from the boundary conditions, such that

(37)

and

(38)

Using and , we can rewrite the latter relation:

(39)

The two coefficients are therefore given by

(40)

where we introduce the abbreviation

(41)

Note that, due to the definition of the variable the function indeed explicitly depends on the Laplace variable . The solution for the bulk density in Fourier-Laplace domain is therefore given by the expression

(42)

iii.3 Solution of the surface diffusion equation

In a similar fashion we obtain the Fourier-Laplace transform of the dynamic equation for the surface density , namely

(43)

Defining the propagator of the homogeneous equation,

(44)

we find

(45)

From Eq. (42) we obtain for the reactive boundary condition that

(46)

where

(47)

Insertion of relation (46) into Eq. (45) produces the result

(48)

Here we also define the coupling constant

(49)

which allows us to distinguish the regimes of strong, intermediate, and weak bulk-surface coupling used in this work. If we remove the outer cylinder, that enforces a finite cross-section in the cylindrical symmetry, we obtain the following simplified expression,

(50)

as in the limit , we have and . From the Fourier-Laplace transform (48) the number of particles on the cylinder surface is given by

(51)

following the definition of the Fourier transform.

Plugging the result (48) into Eq. (42) we obtain the closed form for the Fourier-Laplace transform of the bulk concentration,

(52)

We note that the solutions for and indeed fulfill the particle conservation,

(53)

Using the results for the surface propagator , Eq. (48), we characterize the effective surface diffusion on the cylinder in terms of the single-particle mean squared displacement

(54)

In Fourier-Laplace domain, we re-express this integral as

(55)

This mean squared displacement includes the unbinding dynamics of particles as manifest in the quantity . We can exclude this effect by defining the normalized mean squared displacement

(56)

From above results for the effective surface propagator we obtain the exact result for the surface mean squared displacement in App. B. In what follows, however, for simplicity of the argument we proceed differently. Namely we first approximate the effective surface propagator , and from the various limiting forms determine the surface mean squared displacement. Comparison to the limits taken from the general results derived in App. B, it can be shown that both procedures yield identical results.

Iv Explicit calculations: strong coupling limit

In this Section we consider the strong coupling limit , representing the richest of the three regimes. Based on the result for the effective surface propagator, Eq. (48), in Fourier-Laplace space obtained in the previous Section we now calculate the quantities characteristic of the effective motion on the cylinder surface, as mediated by transient bulk excursions. We consider the number of particles on the surface, the axial mean squared displacement, as well as the surface propagator. We divide the discussion into the four different dynamic regimes defined by comparison of the involved time scales , , and .

iv.1 Short times,

The short time limit corresponds to the Laplace domain regime

(57)

Surface propagator in Fourier-Laplace space

We first obtain the short time limit of the effective surface propagator in Fourier-Laplace space. To this end we note that the following inequalities hold:

(58)

and thus we have

(59)

For this case we use the following expansion of the Bessel functions contained in the abbreviations and . Namely, for ,

(60)

From expressions (41) and (47) we find

(61)

Therefore, the surface propagator in Fourier-Laplace in the short time limit reduces to the simplified form

(62)

Number of particles on the surface

From the relation we obtain the number of surface particles by help of the above expression for the limiting form of :

(63)

Since the leading behavior follows

(64)

i.e., we recover that the number of particles on the surface remains approximately conserved in the short time regime,

(65)

Surface mean squared displacement

The surface mean squared displacement is readily obtained from the limiting form of the surface propagator (62) by help of relation (55). Namely, we obtain

(66)

Since the leading behavior corresponds to

(67)

from which the time-dependence

(68)

yields after Laplace inversion. As in this short time regime , the normalized surface mean squared displacement follows the same behavior.

Remarkably, result (68) contains a contribution growing like . This superdiffusive behavior becomes relevant when , which is typically observed in many systems. Thus for DNA binding proteins the bulk diffusivity may be a factor of or more larger than the diffusion constant along the DNA: for Lac repressor the bulk diffusivity is of the order of , while the one-dimensional diffusion constant along the DNA surface ranges in between (15); (19).

Surface propagator in real space

We now turn to the functional form of the surface propagator in real space at short times in the strong coupling limit. We investigate this quantity in the limit of vanishing surface diffusion.

In the current short time limit we distinguish two parts of the surface density . Let us start with the central part defined by . The corresponding limiting form of Eq. (62) is then given by

(69)

The inverse Fourier-Laplace transform leads to the Cauchy probability density function

(70)

This central part of the surface propagator obeys the governing dynamic equation (7); (20)

(71)

with initial condition . Here, we defined the space fractional derivative in the Riesz-Weyl sense whose Fourier transform takes on the simple form (21)

(72)

Eq. (70) and the corresponding dynamic equation (71) are remarkable results, which are analogous to the findings in Ref. (2) for a flat surface obtained from scaling arguments (22). It says that the bulk mediation causes an effective surface motion whose propagator is a Lévy stable law of index 1. This behavior can be guessed from the scaling of the returning probability to the surface, together with the diffusive scaling . However, the resulting Cauchy distribution cannot have an infinite range, as the particle in a finite time only diffuses a finite distance. The question therefore arises whether there exists a cutoff of the Cauchy law, and of what form this is.

The advantage of our exact treatment is that the Cauchy law can be derived explicitly, but especially the transition to other regimes studied. To this end we introduce the time-dependent length scale

(73)

which turns out to define the range of validity of the Cauchy region. Namely, while at distances we observe a cutoff of the Cauchy behavior, for the Cauchy approximation is valid. Note that in this short time regime the Cauchy range scales as such that can indeed become significantly larger than at sufficiently short times, and thus the power law asymptotics in Eq. (70) become relevant. From this Cauchy part we obtain the superdiffusive contribution

(74)

to the mean squared displacement, that is consistent with the exact forms (68) and (199) [with ]. Calculation of the mean squared displacement however requires the limit and thus involves the extreme wings of the distribution. As the system evolves in time the central Cauchy part spreads. Already in the regime we have , and the asymptotic behavior can no longer be observed.

To show how at very large the Cauchy form of the propagator is truncated we consider Eq. (62) for small wave number ,

(75)

where . In this limit the surface propagator interestingly fulfills the time fractional diffusion equation (23); (24)

(76)

with the initial conditions and , the second defining the initial velocity field. Here, the fractional Caputo derivative is defined via its Laplace transform through (26); (25)

(77)

An equation of the form (76) can be interpreted as a retarded wave (ballistic) motion (23); (27). We choose that the initial velocity field vanishes. It is easy to show that Eq. (76) leads to the scaling of the surface mean squared displacement.

The inverse Fourier transform of Eq. (75) leads to

(78)

Inverse Laplace transform then yields

(79)

where we use the abbreviation

(80)

and where is the Mainardi function, defined in terms of its Laplace transform as (28); (25)