Diffusion in narrow channels on curved manifolds Guillermo Chacón-Acosta, Inti Pineda, and Leonardo Dagdug Citation: The Journal of Chemical Physics 139, 214115 (2013); doi: 10.1063/1.4836617 View online: http://dx.doi.org/10.1063/1.4836617 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A Brownian dynamics algorithm for colloids in curved manifolds J. Chem. Phys. 140, 214115 (2014); 10.1063/1.4881060 Unsteady peristaltic transport in curved channels Phys. Fluids 25, 091903 (2013); 10.1063/1.4821355 The effect of magnetisation and Lorentz forces in a two-dimensional biomagnetic channel flow AIP Conf. Proc. 1522, 496 (2013); 10.1063/1.4801167 Communication: Propagator for diffusive dynamics of an interacting molecular pair J. Chem. Phys. 134, 121102 (2011); 10.1063/1.3565476 Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels Appl. Phys. Lett. 85, 1855 (2004); 10.1063/1.1785854
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THE JOURNAL OF CHEMICAL PHYSICS 139, 214115 (2013)
Diffusion in narrow channels on curved manifolds Guillermo Chacón-Acosta,1,a) Inti Pineda,2,b) and Leonardo Dagdug2,c) 1
Department of Applied Mathematics and Systems, Universidad Autónoma Metropolitana-Cuajimalpa, Artificios 40, México D. F. 01120, Mexico 2 Department of Physics, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México D. F. 09340, Mexico
(Received 30 September 2013; accepted 15 November 2013; published online 5 December 2013) In this work, we derive a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, embedded on a curved surface, in the simple diffusion of non-interacting, point-like particles under no external field. To this end, we extend the generalization of the Kalinay–Percus’ projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic twodimensional diffusion equation on a curved manifold, into an effective one-dimensional generalized Fick-Jacobs equation that is modified according to the curvature of the surface. For such purpose we construct the whole expansion, writing the marginal concentration as a perturbation series. The lowest order in the perturbation parameter, which corresponds to the Fick-Jacobs equation, contains an additional term that accounts for the curvature of the surface. We explicitly obtain the first-order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley’s coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of the metric elements of the surface. In a straightforward manner, we study the perturbation series up to the nth order, and derive the full effective diffusion coefficient for two-dimensional diffusion in a narrow asymmetricchannel, with according tothe metric terms. This expression is given modifications g1 g2 w (ξ ) D0 as D(ξ ) = w (ξ ) g2 {arctan[ g1 (y0 (ξ ) + 2 )] − arctan[ gg21 (y0 (ξ ) − w 2(ξ ) )]}, which is the main result of our work. Finally, we present two examples of symmetric surfaces, namely, the sphere and the cylinder, and we study certain specific channel configurations on these surfaces. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4836617] I. INTRODUCTION
The transport of molecules and small particles spatially constrained within pores, channels, or other quasi-onedimensional systems has gained increasing attention over the last decade, as such systems are ubiquitous in both nature and technology.1 Examples in biology, chemistry, and nanotechnology include diffusion of ions and macromolecular solutes through biological membrane channels,2, 3 transport in zeolites,4 nanostructures of complex geometry,5 carbon nanotubes,6 serpentine channels in microfluidic devices,7 artificially produced pores in thin solid films,8 and protein and solid-state nanopores as single-molecule biosensors for the detection and structural analysis of individual molecules such as DNA and RNA.9, 10 However, many of these problems have been addressed as diffusive processes on flat Euclidean spaces. Inspired by an array of biological situations, intense activity has emerged in the study of diffusion on curved manifolds occurring on cell surfaces. The plasma membrane is composed of hundreds of different lipid species and a huge a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
0021-9606/2013/139(21)/214115/7/$30.00
diversity of proteins, spatially and temporally organized, as a requirement for its biological function.3, 11 Lateral diffusion of curved biological membrane components is vital for numerous cellular processes and has been studied both theoretically and experimentally. Cellular events via membranes, such as signaling transduction, exo- and endocytosis, and reorganization of membrane cytoskeletons, are influenced by lateral diffusion because many chemical reactions in biological membranes are caused by collision among membrane molecules therein.12 These diffusion processes of molecules in cell membranes can also be hindered by the presence of impermeable lateral heterogeneities, patches, rafts, microdomains, holes, and tubular networks, with the effective diffusion constant thus being reduced.13 In this case, lateral motion of membrane components will resemble the problem of particles in a confined system in the presence of obstacles, embedded on a curved surface. In the past few years, the boom of new techniques to investigate the kinetic properties of membrane molecules, especially their diffusion rates, such as fluorescence recovery after photobleaching (FRAP), single-particle tracking (SPT), and fluorescence correlation spectroscopy (FCS), has allowed for unprecedented ways to study the motion of proteins, molecular receptors, and lipids on cell surfaces. These techniques have greatly furthered our
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understanding of its crucial role in the dynamic structure of biological membranes and cellular function.14 Analysis of experimental data has revealed, in some cases, that diffusion proceeds at a different rate than conventional diffusion on the plane.15 Along with progress in experimental techniques, the problem of particle transport on Euclidian spaces through confined structures containing narrow openings, bottlenecks, and obstacles has led to recent theoretical efforts to study diffusion dynamics appearing in those geometries.16 Previous studies by Jacobs18 and Zwanzig17 ignited a revival of research on this topic. The so-called Fick-Jacobs (FJ) approach dramatically simplifies the problem if one assumes that solute distribution in any transverse direction of the channel is uniform as at equilibrium.17, 18 This fast equilibrium assumption allows one to reduce a two-dimensional problem with a complex boundary to the one-dimensional problem of diffusion along the longitudinal direction of a channel in the presence of an entropy barrier. Significant progress in understanding such reduction has been made in recent years.17–26 For wide channels, one can map particle motion onto an effective one-dimensional description in terms of diffusion along the centerline of the channel y0 (x), in the presence of the entropy potential Uent (x) given by βUent (x) = ln(1/w(x)), where x is the particle coordinate measured along the x axis, w(x) is the channel width as a function of x, β = 1/(kB T), kB is the Boltzmann constant, and T is the absolute temperature. It is well known that confinement in higher dimensions gives rise to an effective entropic potential in reduced dimensions.17, 18 The associated approximate description relies on the generalized FJ equation for the probability density in the channel p(x, t), ∂ ∂ p(x, t) ∂ p(x, t) = D(x)w(x) , (1) ∂t ∂x ∂x w(x)
sults for symmetric channels are recovered,19
which is formally equivalent to a Smoluchowski equation,17 where the potential is replaced by the entropic barrier and the diffusion coefficient becomes space dependent D(x): ∂ ∂ ∂ D(x)e−βU (x) eβU (x) p(x, t) . (2) p(x, t) = ∂t ∂x ∂x
is the expression that was improved by Reguera and Rubí27 (RR) based on heuristic arguments, by suggesting that D(x) entering into Eq. (1) is given by
This equation with a position-independent diffusion coefficient, D(x) = D0 , is known as the FJ equation.18 Recently, using the projection method proposed by Kalinay and Percus (KP),19 a robust effective diffusion coefficient in two dimensions that can be used to study wide channels with a nonstraight midline and varying width (i.e., an asymmetric channel) was obtained by Dagdug and Pineda (DP),20 D0 w (x) arctan y0 (x) + D(x) = w (x) 2 w (x) − arctan y0 (x) − , (3) 2 where y0 (x) is the midline of the channel. Equation (3) generalizes all known effective diffusion coefficients theoretically derived so far, and was validated both in channels with straight walls21 and in channels with a non-straight midline and varying width.22 Setting y0 (x) = 0 in Eq. (3), KP’s re-
D(x) ≈ DKP (x) =
1
w (x) 2 1 w (x) 2
arctan
(4)
D0
is the expression that was rederived using a different method by Martens et al.23 On the other hand, if w (x) = 0, the following case of a serpentine channel previously studied by Yariv and co-workers24 is obtained, D(x) ≈ DYBK (x) =
D0 . 1 + y0 (x)2
(5)
Furthermore, when the Taylor expansion of Eq. (3) is kept up to the first order in w (x) and y0 (x), the diffusion coefficient computed by Bradley25 (Br) is recovered,
1 2 2 D(x) ≈ DBr (x) = D0 1 − y0 (x) − w (x) , 12
(6)
which is essentially the same as that obtained two years later by Berezhkovskii and Szabo,26 D(x) ≈ DBS (x) =
D0 1 + y0 (x)2 +
1 w (x)2 12
.
(7)
Note that if y0 (x) = 0 in Eq. (7), the first original proposal of an effective diffusion coefficient given by Zwanzig17 is obtained, D(x) ≈ DZw (x) =
1+
D0 1 w (x)2 12
D0 D(x) ≈ DRR (x) = 1/3 . 1 1 + 4 w (x)2
(8)
(9)
Reduction of 2D diffusion in a channel to the effective 1D description in terms of the generalized FJ equation is reasonable when |w (x)| 1, and the difference between the expressions for D(x) given in Eqs. (4), (8), and (9) can be ignored, 1 DZw (x) ≈ DRR (x) ≈ DKP (x) ≈ 1 − w (x)2 D0 . (10) 12 In Sec. II of this paper, we derive a Fick-Jacobs-like equation for diffusion in a narrow channel embedded on a curved surface. In Sec. III, we apply KP’s19 method to obtain a general diffusion coefficient for channels with variable midline and width on symmetric manifolds. As far as we know, this is the most general expression for an effective diffusion coefficient in 2D obtained to date. In Sec. IV, we study certain
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specific channel configurations embedded on two symmetric surfaces, namely, spheres and cylinders. Finally, Sec. V summarizes our results and conclusions. II. FICK-JACOBS EQUATION ON A SYMMETRIC CURVED SURFACE
The diffusion equation on a curved surface is usually written as a manifestly covariant Fokker-Planck equation in an arbitrary reference frame.28, 29 An important point is that, as the concentration is related with probability density, we need to construct a scalar function constrained to the definition of the total number of particles as the integral on the surfaces of such a function. On a curved surface, the integration measure is weighted by the determinant of the metric tensor gαβ of the surface. Then, it is appropriate to consider the product of the concentration and the determinant of the met√ ric C˜ = gC as the corresponding scalar function. This is required to ensure the scalar nature of the diffusion equation being used, and all quantities involved are changed accordingly. For isotropic diffusion, the diffusion equation on a curved surface is obtained by replacing the Laplacian by the LaplaceBeltrami operator.28, 29 In such a case, the mean square path is modified by a series in the curvature scalar.30–34 There are plenty of works showing similar results obtained by the Monge or height parametrization.35, 36 In order to study diffusion in narrow channels, Kalinay and Percus proposed a mapping procedure that allows one to obtain higher order corrections in terms of an expansion parameter λ = Dx1 /Dx2 , which is the ratio of the diffusion constants in the longitudinal and transverse local directions.19 Using this scaling, the fast transverse modes (transients) separate from the slow longitudinal ones and can be projected out by integration over the transverse directions. In the anisotropic case, diffusion coefficients can be arranged in the following diffusion tensor: Dx1 0 α . (11) [D β ] = 0 Dx2 Then, in this case, the corresponding diffusion equation can be written as
∂ C˜ 1 ∂ √ α βγ ∂ C˜ , (12) gDβ g =√ ∂t g ∂x α ∂x γ where xα are the local coordinates on the surface and g is the determinant of the metric gαβ of the surface. For isotropic diffusion D αβ = D δβα , the diffusion equation reduces to an equation with the Laplace-Beltrami operator.30–32 Given that this is a scalar equation, the quantities appearing in Eq. (12) transform properly under the transformations in the corresponding manifold. Let us proceed with the derivation of a Fick-Jacobs-like equation on a curved surface, whose metric only depends on one of the two generalized coordinates (ξ , η). Although at first glance this might seem as a very strong constraint, we can see that most of the surfaces of interest have this property. Let us use Eq. (12) for a two-dimensional curved surface, whose
J. Chem. Phys. 139, 214115 (2013)
metric can be written as g1 (ξ ) 0 , [gαβ ] = 0 g2 (ξ )
[g αβ ] =
g1−1 (ξ )
0
. g2−1 (ξ ) (13) For anisotropic diffusion on the curved surface, the diffusion tensor (11) in local coordinates x1 = ξ and x2 = η is such that Dξ = Dη . The corresponding diffusion equation on the twodimensional curved surface (12) for the invariant concentration, in local coordinates is ˜ η, t) ∂ C(ξ, g2 ∂ ˜ Dξ ∂ C(ξ, η, t) = √ ∂t g1 g2 ∂ξ g1 ∂ξ
g1 ∂ 2 ˜ Dη C(ξ, η, t). (14) +√ g1 g2 g2 ∂η2 0
Let us assume that diffusion takes place in a narrow channel that, in principle, is oriented in the direction of one of the local coordinates of the surface. In this situation, we will use the projection method of Kalinay and Percus19 and extend the previous result for a general asymmetrical channel obtained by Dagdug and Pineda20 for the flat surface case, when g1 = g2 = 1. Let us choose η as the variable to be integrated, and assume that the metric does not depend on this coordinate. Then, the projected one-dimensional marginal concentration is defined as follows: f2 (ξ ) ˜ η, t)dη, C(ξ, (15) c(ξ, t) ≡ f1 (ξ )
where f1 (ξ ) and f2 (ξ ) are functions on the surface that define the shape of the channel. Let us now integrate Eq. (14) on η over the whole width of the channel, by means of the fundamental theorem of Calculus and the Leibnitz rule, which takes us to
∂ Dξ g2 ∂ ∂c(ξ, t) =√ c(ξ, t) ∂t g1 g2 ∂ξ g1 ∂ξ g2 ∂ f2 (ξ )C˜ f2 (ξ ) − f1 (ξ )C˜ f1 (ξ ) − ∂ξ g1
g2 ∂ C˜ g2 ∂ C˜ − f (ξ ) − f2 (ξ ) 1 g1 ∂ξ f2 (ξ ) g1 ∂ξ f1 (ξ ) f (ξ ) Dη ∂ 2 ˜ 2 C + . (16) g2 ∂η2 f1 (ξ ) For a manifold, the flux components are given by J α = −D αξ
1 ∂ C˜ 1 ∂ C˜ − D αη . g1 ∂ξ g2 ∂η
(17)
On the boundaries, this flux is proportional to the tangent vector, J α |fi (ξ ) ∝ t α , where tiα =
1 1+
fi (ξ )2
(ξˆ α + fi (ξ )ηˆ α ),
(18)
ξˆ α , ηˆ α are the local basis, and i = 1, 2 corresponds to the upper or lower boundary. Then, the parallel boundary condition can
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be written as follows: Dξ fi (ξ ) ∂ C˜ D η ∂ C˜ = . g2 ∂η fi (ξ ) g1 ∂ξ fi (ξ )
J. Chem. Phys. 139, 214115 (2013)
Using Eqs. (23) and (24), Eq. (20) becomes (19)
Dξ ∂ ∂c(ξ, t) = √ ∂t g1 g2 ∂ξ
Applying these boundary conditions to (16), we arrive at the following projected effective 1D diffusion equation:
(20) In order to proceed, we can use the FJ approach as a first approximation to provide an infinite transverse diffusion rate and infer from (15) that c(ξ, t) . f2 (ξ ) − f1 (ξ )
(21)
Inserting this last expression into (20) after defining the difference f2 (ξ ) − f1 (ξ ) = w(ξ ) as the function of the width of the channel, we obtain
Dξ ∂ g2 ∂ c(ξ, t) ∂c(ξ, t) =√ , (22) w(ξ ) ∂t g1 g2 ∂ξ g1 ∂ξ w(ξ ) which is the corresponding Fick-Jacobs-like equation on a symmetric curved surface.
III. GENERALIZATION OF THE KALINAY AND PERCUS METHOD ON A CURVED SURFACE
In this section, KP’s19 method is applied to obtain corrections to the FJ equation on curved surfaces (Eq. (22)). The expansion parameter is λ = Dξ /Dη . Kalinay and Percus suggested an operator procedure to map the solutions of the corrected FJ equation back into the space of solutions of the original n-dimensional problem.19 This recurrence scheme provides systematical corrections to the FJ equation. Readers interested in further details of the projection method are referred to Ref.19, and references therein. To construct the ˜ η, t) as a perturbawhole expansion, we have to write C(ξ, tion series, ˜ η, t) = C(ξ,
∞
λj ρj (ξ, η, t),
∞
g2 w(ξ ) g1
λj (f2 (ξ )ρˆj (ξ, f2 (ξ ), ∂ξ )
j =1
∂ ∂c(ξ, t) g2 ∂ Dξ = √ c(ξ, t) ∂t g1 g2 ∂ξ g1 ∂ξ g2 ∂ ˜ ˜ (f (ξ )C|f2 (ξ ) − f1 (ξ )C|f1 (ξ ) ) . − ∂ξ g1 2
˜ η, t) ∼ C(ξ, =
−
−f1 (ξ )ρˆj (ξ, f1 (ξ ), ∂ξ ))
∂ c(ξ, t) . ∂ξ w(ξ )
(25)
The diffusion equation on a curved surface, Eq. (14), can be rewritten using (23) and (24) and replacing the expression of ∂c(ξ , t)/∂t from (25). To get the recurrence relation for the operators ρˆj (ξ, η, ∂ξ ), the coefficients at the same order of λ are compared, and we obtain 1 ∂2 ρˆj +1 (ξ, η, ∂ξ ) g2 ∂η2 j
1 ∂ ∂ ρˆj −k (ξ, η, ∂ξ ) √ =− ∂ξ g1 g2 w(ξ ) ∂ξ k=1
g2 g1
×[f2 (ξ )ρˆk (ξ, f2 (ξ ), ∂ξ ) − f1 (ξ )ρˆk (ξ, f2 (ξ ), ∂ξ )]
1 ∂ g2 ∂ w(ξ ) + ρˆj (ξ, η, ∂ξ ) √ ∂ξ g1 g2 w(ξ ) ∂ξ g1
∂ g2 ∂ 1 ρˆj (ξ, η, ∂ξ ). (26) −√ g1 g2 ∂ξ g1 ∂ξ Double integration over η follows: the first integration constant is set to fulfill the boundary condition (19) and the second one must provide a normalization condition according to definition (21),
f2 (ξ )
ρˆj (ξ, η, t) f1 (ξ )
∂ c(ξ, t) dη = 0, ∂ξ w(ξ )
j > 0.
(27)
As an example, we show the first operator of this recurrence scheme, ρˆ1 (ξ, η, ∂ξ )
g2 1 (y0 (ξ ) − η)(f1 (ξ )f2 (ξ ) − f2 (ξ )f1 (ξ )) = g1 w(ξ ) w (x) 2 1 2 η − (f1 (ξ ) + f1 (ξ )f2 (ξ ) + f22 (ξ )) , + 2 3
(23)
(28)
where ρ j (ξ , η, t) will have the form of some operators acting on c(ξ , t), to make Eq. (20) self-consistent,
where y0 (ξ ) = [f2 (ξ ) + f1 (ξ )]/2. Using KP’s mapping procedure,19 we can gain systematically higher order corrections in the parameter λ. Let us write Eq. (25) in the form
j =0
ρj (ξ, η, t) = ρˆj (ξ, η, ∂ξ )
∂ c(ξ, t) . ∂ξ w(ξ )
(24)
˜ η, t) does not depend on η, so ρ0 (ξ, η, t) For λ = 0, C(ξ, = c(ξ, t)/w(ξ ) according to the FJ approximation, therefore ρˆ0 (ξ, η, ∂ξ )∂/∂ξ = 1.
∂ ∂ ˆ ∂ξ )] ∂ c(ξ, t) , c(ξ, t) = D0 w(ξ )[1 − λZ(ξ, ∂t ∂ξ ∂ξ w(ξ )
(29)
ˆ ∂ξ ) is an operator.19 There is no difference in uswhere Z(ξ, ing either this operator or the effective diffusion coefficient
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D(ξ ) in the steady state, where ∂c(ξ , t)/∂t = 0. Both Eqs. (22) and (29) represent the 1D mass conservation law, so we can immediately write formulas for net flux J = −w(ξ )D(ξ )
∂ c(ξ ) ∂ξ w(ξ )
(30)
and ˆ ∂ξ )] ∂ c(ξ ) . J = −w(ξ )[1 − λZ(ξ, ∂ξ w(ξ )
(31)
To find D(ξ ) such that the solution of Eq. (30) also satisfies Eq. (31), we substitute ∂/∂t[c(ξ, t)/w(ξ )] from the former equation in the latter, finding that ˆ ∂ξ )] 1 = w(ξ )[1 − λZ(ξ,
1 . w(ξ )D(ξ )
(32)
This relation enables us to express the series of the function D(ξ ) uniquely within a recurrence scheme coming from the mapping procedure. If we relax the rule stating that operators act on everything to the right in products and assume that ˆ ∂ξ ) acts only on the following w −1 (ξ ), we can perform Z(ξ, the final inversion and write ˆ ∂ξ ) D(ξ ) 1 − λw(ξ )Z(ξ,
1 . w(ξ )
D(ξ ) = D0 1 −
1+
g2 y (ξ ) g1 0
2
1 − 3
g2 w (ξ ) g1 2
D0 2 , 2 g2 g2 w (ξ ) 1 y (ξ ) + g1 0 3 g1 2
2
(34)
2 g2 g2 w (ξ ) 2 1 y0 (ξ ) − D(ξ ) = D0 1 − g1 3 g1 2
4
2
g2 g2 g2 w (ξ ) 2 + y (ξ ) + 2 y (ξ ) g1 0 g1 0 g1 2
4 g2 w (ξ ) 1 + − ... . (35) 5 g1 2 Neglecting the second and higher derivatives of w(x) and y0 (ξ ), terms depending only on w (ξ ) and y0 (ξ ) can be summed up to infinity,
i 2n ∞ (−1)n g2 w (ξ ) y (ξ ) + D(ξ ) = D0 2n + 1 i=0 g1 0 2 n=0
2n−i g2 w (ξ ) y0 (ξ ) − . g1 2
IV. ILLUSTRATIVE EXAMPLES
In this section, we study the behavior of the effective diffusion coefficient given by Eq. (37) on two curved surfaces, sphere and cylinder, when diffusion takes place in different channel configurations. To this end, we first have to determine the width function w(ξ ) and the midline y0 (ξ ) for the chosen channel, with ξ being the longitudinal coordinate across each surface.
First, we study effective diffusion on the surface of the sphere, which can be parameterized by the polar and azimuthal angles θ and φ, respectively. The metric components are g2 (θ ) = r2 sin 2 θ and g1 (θ ) = r2 , where r is the radius. Each value of r is considered fixed and only θ is varied. Accordingly, we take the arc length s = rθ as the longitudinal local coordinate. We consider the symmetrical conical channel made up by a straight line given by f2 (θ ) = r(mθ + φ 0 ) and f2 (θ ) = −f1 (θ ). In this case, Eq. (37) reduces to Dsph (θ ) = D0
which is nothing but an analogous to Bradley’s result when g1 = g2 . Up to second order, this yields
×
Equation (37) is the main result of this work and gives a general expression for the effective diffusion coefficient along the slow ξ -coordinate for an asymmetric 2D channel embedded on a curved surface.
A. Diffusion in narrow channels on the sphere
(33)
For the first order correction, setting Dξ = D0 and λ = 1 gives
This last expression can be reduced to the following formula:
g1 g2 w (ξ ) D0 arctan y0 (ξ ) + D(ξ ) = w (ξ ) g2 g1 2
g2 w (ξ ) y0 (ξ ) − . (37) − arctan g1 2
(36)
arctan [rm sin θ ] . rm sin θ
(38)
Figure 1 shows a diagram of this system and plots the effective diffusion coefficient as a function of the angular variable for different values of the radius. Surprisingly, unlike flat surfaces, where dependence is only on slope m, here we observe dependence on θ , which increases as effective diffusion decreases, with the same behavior being observed for large radii. As is well known, symmetric channels with zero slope boundaries have an exactly constant diffusion coefficient. However, in this case, we have additional dependence on both the radius and θ . At the limit of very large radii, the flat surface case20 can be recovered. Furthermore, for small enough angles, the boundaries have no influence on diffusion, and therefore, diffusion is constant as well. Next, we study the asymmetrical simple conical channel, formed by the two lines f1 (θ ) = r(m1 θ − φ 0 ) and f2 (θ ) = r(m2 θ + φ 0 ). In this case, the effective diffusion coefficient is Dsph (θ ) = D0
arctan [rm1 sin θ ] − arctan [rm2 sin θ ] . (39) r(m1 − m2 ) sin θ
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FIG. 1. A symmetric channel formed by straight walls. Dependence on the angular coordinate is shown. The yellow hue is for small radii, changing to red as the radius increases.
Figure 2 shows a schematic representation of the conical tubes studied. We keep constant the upper slope m2 , while varying the lower slope m1 . We also show dependence of Dsph on the lower slope m1 ; for large r we note that the behavior of the flat surface case20 is recovered. In the first plot in Figure 2, we are plotting the value of the effective diffusion coefficient near the beginning of the channel at θ = π /20, while in the second plot of Figure 2, we show its value in θ = π /2. We note that as we move through the channel, diffusion decreases such that for positive slopes, it is close to zero and almost constant. Finally, in this subsection, we study diffusion in a tube made up of a serpentine tube connected to a symmetrical one formed with the following boundaries: f1 (θ ) = −0.07 + 0.3(e−2(2θ−1) − e−2(2θ−3) ), 2
2
f2 (θ ) = 0.07 + 0.3(e−2(2θ−1) + e−2(2θ−3) ). 2
2
This configuration is shown in Figure 3. An important feature of this channel is the transition from asymmetric to symmetric regions. This figure shows dependence on θ and the behavior for different values of the radius. One can note that as r grows, effective diffusion decreases. Also worth noting is that the values of both maxima are maintained because diffusion at those points depends solely on the shape of the channel. However, the peak of region II that represents the transition between asymmetric region I and symmetric region III, changes depending on the radius of the sphere. Again, Figure 3 clearly shows that the known result for the flat surface case20 is reproduced.
FIG. 2. An asymmetric channel formed by straight walls. The upper boundary is fixed at m2 = 1 and the lower one varies from −1 ≤ m1 ≤ 1. The first plot shows effective diffusion near the beginning of the channel at θ = π /20, while the second plot shows its behavior at θ = π /2. For both figures, the increase in radius is shown with the change from yellow to red hues.
B. Diffusion on the cylinder
The next surface considered is a cylinder of radius r and height z. In this case, both components of the metric are constants gθ = r2 and gz = 1. In principle, we can orient our channel along the z axis, as well as around the cylinder. However, by substituting the metric components in the effective diffusion coefficient Eq. (37), the factor gi /gj is set, where i and j are equal to 1 and 2, respectively, depending on the orientation chosen for the channel. This occurs because Eq. (37) is obtained by integrating over the transversal variable, producing a different result for each separate choice. If we focus either on a channel of varying width formed by straight walls defined by z1 (θ ) = m1 rθ − z0 and z2 (θ ) = m2 rθ + z0 , oriented along the angular variable, or on a channel of varying width formed by straight walls defined by the functions θ1 (z) = m1 rz − θ0 and θ2 (θ ) = m2 rz + θ0 , oriented along the z axis, we find in both cases that Dcyl (θ ) = D0
arctan [m2 ] − arctan [m1 ] . m2 − m1
(40)
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fusion coefficient is the same equation as for the flat surface case.
ACKNOWLEDGMENTS
This work was partially funded by Programa de Mejoramiento del Profesorado (PROMEP) under Grant No. 47510283 (GCA), and by Consejo Nacional de Ciencia y Tecnología (CONACyT) under Grant No. 176452. We extend our gratitude to C. Reynaud for her numerous illuminating discussions and comments on the paper. 1 J.
FIG. 3. A serpentine channel connected to a symmetrical channel embedded on a sphere. Effective diffusion coefficient as a function of θ ; small radii are shown in yellow hues, which change to red as the radius increases.
As seen, this is the same equation as for the flat surface case.20 This is not surprising, given that the metric components are constant with zero curvature.
V. SUMMARY AND CONCLUSIONS
Using the KP projection method,19 we found the corresponding FJ operator for a one-dimensional generally asymmetric channel with varying width w(ξ ) and a non-straight midline y0 (ξ ), embedded on a symmetric curved manifold. The coordinate-dependent diffusion coefficient up to the first order in the metric determinant is given by Eq. (37), and is the main result of this work. For a flat surface, this expression reduces to DP’s20 result for a general asymmetric channel given in Eq. (3), which also contains all known results for specific cases. We studied two specific channel configurations on the surface of a sphere. For the symmetric conical channel, the diffusion coefficient decreases with angle θ for a fixed slope. Considering different radii, the effective diffusion coefficient decreases as r increases. For the asymmetrical conical channel, we recovered the well-known results obtained by DP20 with additional dependence on the angle. Finally, for the serpentine channel, we found the transition between symmetric and asymmetric regions of a tube along the angular coordinate. In both cases, the diffusion tends to a constant as r decreases. Finally, we showed that when a channel is embedded on a cylinder and oriented along both axes, the effective dif-
Kärger, D. M. Ruthven, and D. N. Theodorou, Diffusion in Nanoporous Materials (Wiley-VHC, Weinheim, 2012), Vols. 1 and 2. 2 B. Hille, Ion Channels of Excitable Membranes (Sinauer, Sunderland, 2001). 3 B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Molecular Biology of the Cell (Garland Science, New York, 2007). 4 J. Kärger and D. M. Ruthven, Diffusion in Zeolites and Other Microporous Solids (Wiley, New York, 1992). 5 A. Berezhkovskii and G. Hummer, Phys. Rev. Lett. 89, 064503 (2002). 6 I. D. Kosinska, I. Goychuk, M. Kostur, G. Schmid, and P. Hänggi, Phys. Rev. E 77, 031131 (2008). 7 J. C. T. Eijkel and A. van den Berg, Microfluid. Nanofluid. 1, 249 (2005). 8 M. Gershow and J. A. Golovchenco, Nat. Nanotechnol. 2, 775 (2007). 9 C. Dekker, Nat. Nanotechnol. 2, 209 (2007). 10 U. Keyser, B. Koeleman, S. V. Dorp, D. Krapf, R. Smeets, S. Lemay, N. Dekker, and C. Dekker, Nat. Phys. 2, 473 (2006). 11 T. Fujiwara, K. Ritchie, H. Murakoshi, K. Jacobson, and A. Kusumi, J. Cell Biol. 157, 1071 (2002); R. G. Parton and J. F. Hancock, Trends Cell Biol. 14, 141 (2004); S. Semrau and T. Schmidt, Soft Matter 5, 3174 (2009). 12 T. Yoshigaki, Phys. Rev. E. 75, 041901 (2007). 13 M. J. Saxton, Biophys. J. 39, 165 (1982); A. Kusumi, Y. Sako, and M. Yamamoto, ibid. 65, 2021 (1993); A. D. Douglass and R. D. Vale, Cell 121, 937 (2005); C. A. Day and A. K. Kenworthy, Biochim. Biophys. Acta 1788, 245 (2009). 14 J. Hwang, L. A. Gheber, L. Margolis, and M. Edidin, Biophys. J. 74, 2184 (1998); J. Capoulade, M. Wachsmuth, L. Hufnagel, and M. Knop, Nat. Biotechnol. 29, 835 (2011). 15 J. Lippincott-Schwartz, E. Snapp, and A. Kenworthy, Nat. Rev. Mol. Cell Biol. 2, 444 (2001). 16 P. S. Burada, P. Hänggi, F. Marchesoni, G. Schmid, and P. Talkner, ChemPhysChem 10, 45 (2009). 17 R. Zwanzig, J. Chem. Phys. 96, 3926 (1992). 18 M. H. Jacobs, Diffusion Processes (Springer, New York, 1967). 19 P. Kalinay and J. K. Percus, J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006). 20 L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012). 21 I. Pineda, J. Álvarez-Ramírez, and L. Dagdug, J. Chem. Phys. 137, 174103 (2012). 22 F. Li and B. Ai, Phys. Rev. E 87, 062128 (2013). 23 S. Martens, G. Schmid, L. Schimansky-Geier, and P. Hänggi, Phys. Rev. E 83, 051135 (2011). 24 E. Yariv, H. Brenner, and S. Kim, SIAM J. Appl. Math. 64, 1099 (2004). 25 R. M. Bradley, Phys. Rev. E 80, 061142 (2009). 26 A. M. Berezhkovskii and A. Szabo, J. Chem. Phys. 135, 074108 (2011). 27 D. Reguera and J. M. Rubí, Phys. Rev. E 64, 061106 (2001). 28 H. Risken, The Fokker-Planck Equation. Methods of Solution and Applications (Springer, Berlin, 1989). 29 N. G. van Kampen, J. Stat. Phys. 44, 1 (1986). 30 P. Castro-Villarreal, J. Stat. Mech.: Theory Exp. (2010) P08006. 31 J. Balakrishnan, Phys. Rev. E 61, 4648 (2000). 32 J. Faraudo, J. Chem. Phys. 116, 5831 (2002). 33 C. Chevalier and F. Debbasch, J. Stat. Phys. 131, 717 (2008); F. Debbasch and C. Chevalier, IMS Collect. 4, 85 (2008). 34 P. H. Roberts and H. D. Ursell, Philos. Trans. R. Soc. London, Ser. A 252, 317 (1960). 35 B. M. Aizenbud and N. D. Gershon, Biophys. J. 38, 287 (1982); S. Gustafsson and B. Halle, J. Chem. Phys. 106, 1880 (1997). 36 A. Naji and F. L. H. Brown, J. Chem. Phys. 126, 235103 (2007).
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THE JOURNAL OF CHEMICAL PHYSICS 139, 214115 (2013)
Diffusion in narrow channels on curved manifolds Guillermo Chacón-Acosta,1,a) Inti Pineda,2,b) and Leonardo Dagdug2,c) 1
Department of Applied Mathematics and Systems, Universidad Autónoma Metropolitana-Cuajimalpa, Artificios 40, México D. F. 01120, Mexico 2 Department of Physics, Universidad Autónoma Metropolitana-Iztapalapa, San Rafael Atlixco 186, México D. F. 09340, Mexico
(Received 30 September 2013; accepted 15 November 2013; published online 5 December 2013) In this work, we derive a general effective diffusion coefficient to describe the two-dimensional (2D) diffusion in a narrow and smoothly asymmetric channel of varying width, embedded on a curved surface, in the simple diffusion of non-interacting, point-like particles under no external field. To this end, we extend the generalization of the Kalinay–Percus’ projection method [J. Chem. Phys. 122, 204701 (2005); Phys. Rev. E 74, 041203 (2006)] for the asymmetric channels introduced in [L. Dagdug and I. Pineda, J. Chem. Phys. 137, 024107 (2012)], to project the anisotropic twodimensional diffusion equation on a curved manifold, into an effective one-dimensional generalized Fick-Jacobs equation that is modified according to the curvature of the surface. For such purpose we construct the whole expansion, writing the marginal concentration as a perturbation series. The lowest order in the perturbation parameter, which corresponds to the Fick-Jacobs equation, contains an additional term that accounts for the curvature of the surface. We explicitly obtain the first-order correction for the invariant effective concentration, which is defined as the correct marginal concentration in one variable, and we obtain the first approximation to the effective diffusion coefficient analogous to Bradley’s coefficient [Phys. Rev. E 80, 061142 (2009)] as a function of the metric elements of the surface. In a straightforward manner, we study the perturbation series up to the nth order, and derive the full effective diffusion coefficient for two-dimensional diffusion in a narrow asymmetricchannel, with according tothe metric terms. This expression is given modifications g1 g2 w (ξ ) D0 as D(ξ ) = w (ξ ) g2 {arctan[ g1 (y0 (ξ ) + 2 )] − arctan[ gg21 (y0 (ξ ) − w 2(ξ ) )]}, which is the main result of our work. Finally, we present two examples of symmetric surfaces, namely, the sphere and the cylinder, and we study certain specific channel configurations on these surfaces. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4836617] I. INTRODUCTION
The transport of molecules and small particles spatially constrained within pores, channels, or other quasi-onedimensional systems has gained increasing attention over the last decade, as such systems are ubiquitous in both nature and technology.1 Examples in biology, chemistry, and nanotechnology include diffusion of ions and macromolecular solutes through biological membrane channels,2, 3 transport in zeolites,4 nanostructures of complex geometry,5 carbon nanotubes,6 serpentine channels in microfluidic devices,7 artificially produced pores in thin solid films,8 and protein and solid-state nanopores as single-molecule biosensors for the detection and structural analysis of individual molecules such as DNA and RNA.9, 10 However, many of these problems have been addressed as diffusive processes on flat Euclidean spaces. Inspired by an array of biological situations, intense activity has emerged in the study of diffusion on curved manifolds occurring on cell surfaces. The plasma membrane is composed of hundreds of different lipid species and a huge a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
0021-9606/2013/139(21)/214115/7/$30.00
diversity of proteins, spatially and temporally organized, as a requirement for its biological function.3, 11 Lateral diffusion of curved biological membrane components is vital for numerous cellular processes and has been studied both theoretically and experimentally. Cellular events via membranes, such as signaling transduction, exo- and endocytosis, and reorganization of membrane cytoskeletons, are influenced by lateral diffusion because many chemical reactions in biological membranes are caused by collision among membrane molecules therein.12 These diffusion processes of molecules in cell membranes can also be hindered by the presence of impermeable lateral heterogeneities, patches, rafts, microdomains, holes, and tubular networks, with the effective diffusion constant thus being reduced.13 In this case, lateral motion of membrane components will resemble the problem of particles in a confined system in the presence of obstacles, embedded on a curved surface. In the past few years, the boom of new techniques to investigate the kinetic properties of membrane molecules, especially their diffusion rates, such as fluorescence recovery after photobleaching (FRAP), single-particle tracking (SPT), and fluorescence correlation spectroscopy (FCS), has allowed for unprecedented ways to study the motion of proteins, molecular receptors, and lipids on cell surfaces. These techniques have greatly furthered our
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© 2013 AIP Publishing LLC
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understanding of its crucial role in the dynamic structure of biological membranes and cellular function.14 Analysis of experimental data has revealed, in some cases, that diffusion proceeds at a different rate than conventional diffusion on the plane.15 Along with progress in experimental techniques, the problem of particle transport on Euclidian spaces through confined structures containing narrow openings, bottlenecks, and obstacles has led to recent theoretical efforts to study diffusion dynamics appearing in those geometries.16 Previous studies by Jacobs18 and Zwanzig17 ignited a revival of research on this topic. The so-called Fick-Jacobs (FJ) approach dramatically simplifies the problem if one assumes that solute distribution in any transverse direction of the channel is uniform as at equilibrium.17, 18 This fast equilibrium assumption allows one to reduce a two-dimensional problem with a complex boundary to the one-dimensional problem of diffusion along the longitudinal direction of a channel in the presence of an entropy barrier. Significant progress in understanding such reduction has been made in recent years.17–26 For wide channels, one can map particle motion onto an effective one-dimensional description in terms of diffusion along the centerline of the channel y0 (x), in the presence of the entropy potential Uent (x) given by βUent (x) = ln(1/w(x)), where x is the particle coordinate measured along the x axis, w(x) is the channel width as a function of x, β = 1/(kB T), kB is the Boltzmann constant, and T is the absolute temperature. It is well known that confinement in higher dimensions gives rise to an effective entropic potential in reduced dimensions.17, 18 The associated approximate description relies on the generalized FJ equation for the probability density in the channel p(x, t), ∂ ∂ p(x, t) ∂ p(x, t) = D(x)w(x) , (1) ∂t ∂x ∂x w(x)
sults for symmetric channels are recovered,19
which is formally equivalent to a Smoluchowski equation,17 where the potential is replaced by the entropic barrier and the diffusion coefficient becomes space dependent D(x): ∂ ∂ ∂ D(x)e−βU (x) eβU (x) p(x, t) . (2) p(x, t) = ∂t ∂x ∂x
is the expression that was improved by Reguera and Rubí27 (RR) based on heuristic arguments, by suggesting that D(x) entering into Eq. (1) is given by
This equation with a position-independent diffusion coefficient, D(x) = D0 , is known as the FJ equation.18 Recently, using the projection method proposed by Kalinay and Percus (KP),19 a robust effective diffusion coefficient in two dimensions that can be used to study wide channels with a nonstraight midline and varying width (i.e., an asymmetric channel) was obtained by Dagdug and Pineda (DP),20 D0 w (x) arctan y0 (x) + D(x) = w (x) 2 w (x) − arctan y0 (x) − , (3) 2 where y0 (x) is the midline of the channel. Equation (3) generalizes all known effective diffusion coefficients theoretically derived so far, and was validated both in channels with straight walls21 and in channels with a non-straight midline and varying width.22 Setting y0 (x) = 0 in Eq. (3), KP’s re-
D(x) ≈ DKP (x) =
1
w (x) 2 1 w (x) 2
arctan
(4)
D0
is the expression that was rederived using a different method by Martens et al.23 On the other hand, if w (x) = 0, the following case of a serpentine channel previously studied by Yariv and co-workers24 is obtained, D(x) ≈ DYBK (x) =
D0 . 1 + y0 (x)2
(5)
Furthermore, when the Taylor expansion of Eq. (3) is kept up to the first order in w (x) and y0 (x), the diffusion coefficient computed by Bradley25 (Br) is recovered,
1 2 2 D(x) ≈ DBr (x) = D0 1 − y0 (x) − w (x) , 12
(6)
which is essentially the same as that obtained two years later by Berezhkovskii and Szabo,26 D(x) ≈ DBS (x) =
D0 1 + y0 (x)2 +
1 w (x)2 12
.
(7)
Note that if y0 (x) = 0 in Eq. (7), the first original proposal of an effective diffusion coefficient given by Zwanzig17 is obtained, D(x) ≈ DZw (x) =
1+
D0 1 w (x)2 12
D0 D(x) ≈ DRR (x) = 1/3 . 1 1 + 4 w (x)2
(8)
(9)
Reduction of 2D diffusion in a channel to the effective 1D description in terms of the generalized FJ equation is reasonable when |w (x)| 1, and the difference between the expressions for D(x) given in Eqs. (4), (8), and (9) can be ignored, 1 DZw (x) ≈ DRR (x) ≈ DKP (x) ≈ 1 − w (x)2 D0 . (10) 12 In Sec. II of this paper, we derive a Fick-Jacobs-like equation for diffusion in a narrow channel embedded on a curved surface. In Sec. III, we apply KP’s19 method to obtain a general diffusion coefficient for channels with variable midline and width on symmetric manifolds. As far as we know, this is the most general expression for an effective diffusion coefficient in 2D obtained to date. In Sec. IV, we study certain
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specific channel configurations embedded on two symmetric surfaces, namely, spheres and cylinders. Finally, Sec. V summarizes our results and conclusions. II. FICK-JACOBS EQUATION ON A SYMMETRIC CURVED SURFACE
The diffusion equation on a curved surface is usually written as a manifestly covariant Fokker-Planck equation in an arbitrary reference frame.28, 29 An important point is that, as the concentration is related with probability density, we need to construct a scalar function constrained to the definition of the total number of particles as the integral on the surfaces of such a function. On a curved surface, the integration measure is weighted by the determinant of the metric tensor gαβ of the surface. Then, it is appropriate to consider the product of the concentration and the determinant of the met√ ric C˜ = gC as the corresponding scalar function. This is required to ensure the scalar nature of the diffusion equation being used, and all quantities involved are changed accordingly. For isotropic diffusion, the diffusion equation on a curved surface is obtained by replacing the Laplacian by the LaplaceBeltrami operator.28, 29 In such a case, the mean square path is modified by a series in the curvature scalar.30–34 There are plenty of works showing similar results obtained by the Monge or height parametrization.35, 36 In order to study diffusion in narrow channels, Kalinay and Percus proposed a mapping procedure that allows one to obtain higher order corrections in terms of an expansion parameter λ = Dx1 /Dx2 , which is the ratio of the diffusion constants in the longitudinal and transverse local directions.19 Using this scaling, the fast transverse modes (transients) separate from the slow longitudinal ones and can be projected out by integration over the transverse directions. In the anisotropic case, diffusion coefficients can be arranged in the following diffusion tensor: Dx1 0 α . (11) [D β ] = 0 Dx2 Then, in this case, the corresponding diffusion equation can be written as
∂ C˜ 1 ∂ √ α βγ ∂ C˜ , (12) gDβ g =√ ∂t g ∂x α ∂x γ where xα are the local coordinates on the surface and g is the determinant of the metric gαβ of the surface. For isotropic diffusion D αβ = D δβα , the diffusion equation reduces to an equation with the Laplace-Beltrami operator.30–32 Given that this is a scalar equation, the quantities appearing in Eq. (12) transform properly under the transformations in the corresponding manifold. Let us proceed with the derivation of a Fick-Jacobs-like equation on a curved surface, whose metric only depends on one of the two generalized coordinates (ξ , η). Although at first glance this might seem as a very strong constraint, we can see that most of the surfaces of interest have this property. Let us use Eq. (12) for a two-dimensional curved surface, whose
J. Chem. Phys. 139, 214115 (2013)
metric can be written as g1 (ξ ) 0 , [gαβ ] = 0 g2 (ξ )
[g αβ ] =
g1−1 (ξ )
0
. g2−1 (ξ ) (13) For anisotropic diffusion on the curved surface, the diffusion tensor (11) in local coordinates x1 = ξ and x2 = η is such that Dξ = Dη . The corresponding diffusion equation on the twodimensional curved surface (12) for the invariant concentration, in local coordinates is ˜ η, t) ∂ C(ξ, g2 ∂ ˜ Dξ ∂ C(ξ, η, t) = √ ∂t g1 g2 ∂ξ g1 ∂ξ
g1 ∂ 2 ˜ Dη C(ξ, η, t). (14) +√ g1 g2 g2 ∂η2 0
Let us assume that diffusion takes place in a narrow channel that, in principle, is oriented in the direction of one of the local coordinates of the surface. In this situation, we will use the projection method of Kalinay and Percus19 and extend the previous result for a general asymmetrical channel obtained by Dagdug and Pineda20 for the flat surface case, when g1 = g2 = 1. Let us choose η as the variable to be integrated, and assume that the metric does not depend on this coordinate. Then, the projected one-dimensional marginal concentration is defined as follows: f2 (ξ ) ˜ η, t)dη, C(ξ, (15) c(ξ, t) ≡ f1 (ξ )
where f1 (ξ ) and f2 (ξ ) are functions on the surface that define the shape of the channel. Let us now integrate Eq. (14) on η over the whole width of the channel, by means of the fundamental theorem of Calculus and the Leibnitz rule, which takes us to
∂ Dξ g2 ∂ ∂c(ξ, t) =√ c(ξ, t) ∂t g1 g2 ∂ξ g1 ∂ξ g2 ∂ f2 (ξ )C˜ f2 (ξ ) − f1 (ξ )C˜ f1 (ξ ) − ∂ξ g1
g2 ∂ C˜ g2 ∂ C˜ − f (ξ ) − f2 (ξ ) 1 g1 ∂ξ f2 (ξ ) g1 ∂ξ f1 (ξ ) f (ξ ) Dη ∂ 2 ˜ 2 C + . (16) g2 ∂η2 f1 (ξ ) For a manifold, the flux components are given by J α = −D αξ
1 ∂ C˜ 1 ∂ C˜ − D αη . g1 ∂ξ g2 ∂η
(17)
On the boundaries, this flux is proportional to the tangent vector, J α |fi (ξ ) ∝ t α , where tiα =
1 1+
fi (ξ )2
(ξˆ α + fi (ξ )ηˆ α ),
(18)
ξˆ α , ηˆ α are the local basis, and i = 1, 2 corresponds to the upper or lower boundary. Then, the parallel boundary condition can
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be written as follows: Dξ fi (ξ ) ∂ C˜ D η ∂ C˜ = . g2 ∂η fi (ξ ) g1 ∂ξ fi (ξ )
J. Chem. Phys. 139, 214115 (2013)
Using Eqs. (23) and (24), Eq. (20) becomes (19)
Dξ ∂ ∂c(ξ, t) = √ ∂t g1 g2 ∂ξ
Applying these boundary conditions to (16), we arrive at the following projected effective 1D diffusion equation:
(20) In order to proceed, we can use the FJ approach as a first approximation to provide an infinite transverse diffusion rate and infer from (15) that c(ξ, t) . f2 (ξ ) − f1 (ξ )
(21)
Inserting this last expression into (20) after defining the difference f2 (ξ ) − f1 (ξ ) = w(ξ ) as the function of the width of the channel, we obtain
Dξ ∂ g2 ∂ c(ξ, t) ∂c(ξ, t) =√ , (22) w(ξ ) ∂t g1 g2 ∂ξ g1 ∂ξ w(ξ ) which is the corresponding Fick-Jacobs-like equation on a symmetric curved surface.
III. GENERALIZATION OF THE KALINAY AND PERCUS METHOD ON A CURVED SURFACE
In this section, KP’s19 method is applied to obtain corrections to the FJ equation on curved surfaces (Eq. (22)). The expansion parameter is λ = Dξ /Dη . Kalinay and Percus suggested an operator procedure to map the solutions of the corrected FJ equation back into the space of solutions of the original n-dimensional problem.19 This recurrence scheme provides systematical corrections to the FJ equation. Readers interested in further details of the projection method are referred to Ref.19, and references therein. To construct the ˜ η, t) as a perturbawhole expansion, we have to write C(ξ, tion series, ˜ η, t) = C(ξ,
∞
λj ρj (ξ, η, t),
∞
g2 w(ξ ) g1
λj (f2 (ξ )ρˆj (ξ, f2 (ξ ), ∂ξ )
j =1
∂ ∂c(ξ, t) g2 ∂ Dξ = √ c(ξ, t) ∂t g1 g2 ∂ξ g1 ∂ξ g2 ∂ ˜ ˜ (f (ξ )C|f2 (ξ ) − f1 (ξ )C|f1 (ξ ) ) . − ∂ξ g1 2
˜ η, t) ∼ C(ξ, =
−
−f1 (ξ )ρˆj (ξ, f1 (ξ ), ∂ξ ))
∂ c(ξ, t) . ∂ξ w(ξ )
(25)
The diffusion equation on a curved surface, Eq. (14), can be rewritten using (23) and (24) and replacing the expression of ∂c(ξ , t)/∂t from (25). To get the recurrence relation for the operators ρˆj (ξ, η, ∂ξ ), the coefficients at the same order of λ are compared, and we obtain 1 ∂2 ρˆj +1 (ξ, η, ∂ξ ) g2 ∂η2 j
1 ∂ ∂ ρˆj −k (ξ, η, ∂ξ ) √ =− ∂ξ g1 g2 w(ξ ) ∂ξ k=1
g2 g1
×[f2 (ξ )ρˆk (ξ, f2 (ξ ), ∂ξ ) − f1 (ξ )ρˆk (ξ, f2 (ξ ), ∂ξ )]
1 ∂ g2 ∂ w(ξ ) + ρˆj (ξ, η, ∂ξ ) √ ∂ξ g1 g2 w(ξ ) ∂ξ g1
∂ g2 ∂ 1 ρˆj (ξ, η, ∂ξ ). (26) −√ g1 g2 ∂ξ g1 ∂ξ Double integration over η follows: the first integration constant is set to fulfill the boundary condition (19) and the second one must provide a normalization condition according to definition (21),
f2 (ξ )
ρˆj (ξ, η, t) f1 (ξ )
∂ c(ξ, t) dη = 0, ∂ξ w(ξ )
j > 0.
(27)
As an example, we show the first operator of this recurrence scheme, ρˆ1 (ξ, η, ∂ξ )
g2 1 (y0 (ξ ) − η)(f1 (ξ )f2 (ξ ) − f2 (ξ )f1 (ξ )) = g1 w(ξ ) w (x) 2 1 2 η − (f1 (ξ ) + f1 (ξ )f2 (ξ ) + f22 (ξ )) , + 2 3
(23)
(28)
where ρ j (ξ , η, t) will have the form of some operators acting on c(ξ , t), to make Eq. (20) self-consistent,
where y0 (ξ ) = [f2 (ξ ) + f1 (ξ )]/2. Using KP’s mapping procedure,19 we can gain systematically higher order corrections in the parameter λ. Let us write Eq. (25) in the form
j =0
ρj (ξ, η, t) = ρˆj (ξ, η, ∂ξ )
∂ c(ξ, t) . ∂ξ w(ξ )
(24)
˜ η, t) does not depend on η, so ρ0 (ξ, η, t) For λ = 0, C(ξ, = c(ξ, t)/w(ξ ) according to the FJ approximation, therefore ρˆ0 (ξ, η, ∂ξ )∂/∂ξ = 1.
∂ ∂ ˆ ∂ξ )] ∂ c(ξ, t) , c(ξ, t) = D0 w(ξ )[1 − λZ(ξ, ∂t ∂ξ ∂ξ w(ξ )
(29)
ˆ ∂ξ ) is an operator.19 There is no difference in uswhere Z(ξ, ing either this operator or the effective diffusion coefficient
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Chacón-Acosta, Pineda, and Dagdug
J. Chem. Phys. 139, 214115 (2013)
D(ξ ) in the steady state, where ∂c(ξ , t)/∂t = 0. Both Eqs. (22) and (29) represent the 1D mass conservation law, so we can immediately write formulas for net flux J = −w(ξ )D(ξ )
∂ c(ξ ) ∂ξ w(ξ )
(30)
and ˆ ∂ξ )] ∂ c(ξ ) . J = −w(ξ )[1 − λZ(ξ, ∂ξ w(ξ )
(31)
To find D(ξ ) such that the solution of Eq. (30) also satisfies Eq. (31), we substitute ∂/∂t[c(ξ, t)/w(ξ )] from the former equation in the latter, finding that ˆ ∂ξ )] 1 = w(ξ )[1 − λZ(ξ,
1 . w(ξ )D(ξ )
(32)
This relation enables us to express the series of the function D(ξ ) uniquely within a recurrence scheme coming from the mapping procedure. If we relax the rule stating that operators act on everything to the right in products and assume that ˆ ∂ξ ) acts only on the following w −1 (ξ ), we can perform Z(ξ, the final inversion and write ˆ ∂ξ ) D(ξ ) 1 − λw(ξ )Z(ξ,
1 . w(ξ )
D(ξ ) = D0 1 −
1+
g2 y (ξ ) g1 0
2
1 − 3
g2 w (ξ ) g1 2
D0 2 , 2 g2 g2 w (ξ ) 1 y (ξ ) + g1 0 3 g1 2
2
(34)
2 g2 g2 w (ξ ) 2 1 y0 (ξ ) − D(ξ ) = D0 1 − g1 3 g1 2
4
2
g2 g2 g2 w (ξ ) 2 + y (ξ ) + 2 y (ξ ) g1 0 g1 0 g1 2
4 g2 w (ξ ) 1 + − ... . (35) 5 g1 2 Neglecting the second and higher derivatives of w(x) and y0 (ξ ), terms depending only on w (ξ ) and y0 (ξ ) can be summed up to infinity,
i 2n ∞ (−1)n g2 w (ξ ) y (ξ ) + D(ξ ) = D0 2n + 1 i=0 g1 0 2 n=0
2n−i g2 w (ξ ) y0 (ξ ) − . g1 2
IV. ILLUSTRATIVE EXAMPLES
In this section, we study the behavior of the effective diffusion coefficient given by Eq. (37) on two curved surfaces, sphere and cylinder, when diffusion takes place in different channel configurations. To this end, we first have to determine the width function w(ξ ) and the midline y0 (ξ ) for the chosen channel, with ξ being the longitudinal coordinate across each surface.
First, we study effective diffusion on the surface of the sphere, which can be parameterized by the polar and azimuthal angles θ and φ, respectively. The metric components are g2 (θ ) = r2 sin 2 θ and g1 (θ ) = r2 , where r is the radius. Each value of r is considered fixed and only θ is varied. Accordingly, we take the arc length s = rθ as the longitudinal local coordinate. We consider the symmetrical conical channel made up by a straight line given by f2 (θ ) = r(mθ + φ 0 ) and f2 (θ ) = −f1 (θ ). In this case, Eq. (37) reduces to Dsph (θ ) = D0
which is nothing but an analogous to Bradley’s result when g1 = g2 . Up to second order, this yields
×
Equation (37) is the main result of this work and gives a general expression for the effective diffusion coefficient along the slow ξ -coordinate for an asymmetric 2D channel embedded on a curved surface.
A. Diffusion in narrow channels on the sphere
(33)
For the first order correction, setting Dξ = D0 and λ = 1 gives
This last expression can be reduced to the following formula:
g1 g2 w (ξ ) D0 arctan y0 (ξ ) + D(ξ ) = w (ξ ) g2 g1 2
g2 w (ξ ) y0 (ξ ) − . (37) − arctan g1 2
(36)
arctan [rm sin θ ] . rm sin θ
(38)
Figure 1 shows a diagram of this system and plots the effective diffusion coefficient as a function of the angular variable for different values of the radius. Surprisingly, unlike flat surfaces, where dependence is only on slope m, here we observe dependence on θ , which increases as effective diffusion decreases, with the same behavior being observed for large radii. As is well known, symmetric channels with zero slope boundaries have an exactly constant diffusion coefficient. However, in this case, we have additional dependence on both the radius and θ . At the limit of very large radii, the flat surface case20 can be recovered. Furthermore, for small enough angles, the boundaries have no influence on diffusion, and therefore, diffusion is constant as well. Next, we study the asymmetrical simple conical channel, formed by the two lines f1 (θ ) = r(m1 θ − φ 0 ) and f2 (θ ) = r(m2 θ + φ 0 ). In this case, the effective diffusion coefficient is Dsph (θ ) = D0
arctan [rm1 sin θ ] − arctan [rm2 sin θ ] . (39) r(m1 − m2 ) sin θ
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214115-6
Chacón-Acosta, Pineda, and Dagdug
J. Chem. Phys. 139, 214115 (2013)
FIG. 1. A symmetric channel formed by straight walls. Dependence on the angular coordinate is shown. The yellow hue is for small radii, changing to red as the radius increases.
Figure 2 shows a schematic representation of the conical tubes studied. We keep constant the upper slope m2 , while varying the lower slope m1 . We also show dependence of Dsph on the lower slope m1 ; for large r we note that the behavior of the flat surface case20 is recovered. In the first plot in Figure 2, we are plotting the value of the effective diffusion coefficient near the beginning of the channel at θ = π /20, while in the second plot of Figure 2, we show its value in θ = π /2. We note that as we move through the channel, diffusion decreases such that for positive slopes, it is close to zero and almost constant. Finally, in this subsection, we study diffusion in a tube made up of a serpentine tube connected to a symmetrical one formed with the following boundaries: f1 (θ ) = −0.07 + 0.3(e−2(2θ−1) − e−2(2θ−3) ), 2
2
f2 (θ ) = 0.07 + 0.3(e−2(2θ−1) + e−2(2θ−3) ). 2
2
This configuration is shown in Figure 3. An important feature of this channel is the transition from asymmetric to symmetric regions. This figure shows dependence on θ and the behavior for different values of the radius. One can note that as r grows, effective diffusion decreases. Also worth noting is that the values of both maxima are maintained because diffusion at those points depends solely on the shape of the channel. However, the peak of region II that represents the transition between asymmetric region I and symmetric region III, changes depending on the radius of the sphere. Again, Figure 3 clearly shows that the known result for the flat surface case20 is reproduced.
FIG. 2. An asymmetric channel formed by straight walls. The upper boundary is fixed at m2 = 1 and the lower one varies from −1 ≤ m1 ≤ 1. The first plot shows effective diffusion near the beginning of the channel at θ = π /20, while the second plot shows its behavior at θ = π /2. For both figures, the increase in radius is shown with the change from yellow to red hues.
B. Diffusion on the cylinder
The next surface considered is a cylinder of radius r and height z. In this case, both components of the metric are constants gθ = r2 and gz = 1. In principle, we can orient our channel along the z axis, as well as around the cylinder. However, by substituting the metric components in the effective diffusion coefficient Eq. (37), the factor gi /gj is set, where i and j are equal to 1 and 2, respectively, depending on the orientation chosen for the channel. This occurs because Eq. (37) is obtained by integrating over the transversal variable, producing a different result for each separate choice. If we focus either on a channel of varying width formed by straight walls defined by z1 (θ ) = m1 rθ − z0 and z2 (θ ) = m2 rθ + z0 , oriented along the angular variable, or on a channel of varying width formed by straight walls defined by the functions θ1 (z) = m1 rz − θ0 and θ2 (θ ) = m2 rz + θ0 , oriented along the z axis, we find in both cases that Dcyl (θ ) = D0
arctan [m2 ] − arctan [m1 ] . m2 − m1
(40)
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214115-7
Chacón-Acosta, Pineda, and Dagdug
J. Chem. Phys. 139, 214115 (2013)
fusion coefficient is the same equation as for the flat surface case.
ACKNOWLEDGMENTS
This work was partially funded by Programa de Mejoramiento del Profesorado (PROMEP) under Grant No. 47510283 (GCA), and by Consejo Nacional de Ciencia y Tecnología (CONACyT) under Grant No. 176452. We extend our gratitude to C. Reynaud for her numerous illuminating discussions and comments on the paper. 1 J.
FIG. 3. A serpentine channel connected to a symmetrical channel embedded on a sphere. Effective diffusion coefficient as a function of θ ; small radii are shown in yellow hues, which change to red as the radius increases.
As seen, this is the same equation as for the flat surface case.20 This is not surprising, given that the metric components are constant with zero curvature.
V. SUMMARY AND CONCLUSIONS
Using the KP projection method,19 we found the corresponding FJ operator for a one-dimensional generally asymmetric channel with varying width w(ξ ) and a non-straight midline y0 (ξ ), embedded on a symmetric curved manifold. The coordinate-dependent diffusion coefficient up to the first order in the metric determinant is given by Eq. (37), and is the main result of this work. For a flat surface, this expression reduces to DP’s20 result for a general asymmetric channel given in Eq. (3), which also contains all known results for specific cases. We studied two specific channel configurations on the surface of a sphere. For the symmetric conical channel, the diffusion coefficient decreases with angle θ for a fixed slope. Considering different radii, the effective diffusion coefficient decreases as r increases. For the asymmetrical conical channel, we recovered the well-known results obtained by DP20 with additional dependence on the angle. Finally, for the serpentine channel, we found the transition between symmetric and asymmetric regions of a tube along the angular coordinate. In both cases, the diffusion tends to a constant as r decreases. Finally, we showed that when a channel is embedded on a cylinder and oriented along both axes, the effective dif-
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