J. Stat. M

ech. (2014) P08019

Fractional diffusion equation, boundary conditions and surface effects

E K Lenzi1, A A Tateishi1,2, H V Ribeiro1,3, M K Lenzi4, G Gonçalves5 and L R da Silva6

1 Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Estadual de Maringá. Avenida Colombo 5790, 87020-900, Maringá, PR, Brazil

2 Departamento de Física, Universidade Técnologica Federal do Paraná, Pato Branco, PR-85503-390, Brazil

3 Departamento de Física, Universidade Tecnológica Federal do Paraná, Apucarana, PR 86812-460, Brazil

4 Departamento de Engenharia Química, Universidade Federal do Paraná, Setor de Tecnologia—Jardim das Américas, Caixa Postal 19011, 81531-990, Curitiba, PR, Brazil

5 Departamento de Engenharia Química, Universidade Tecnológica Federal do Paraná, Ponta Grossa, PR 84016-210, Brazil

6 Departamento de Física and National Institute of Science and Technology for Complex Systems, Universidade Federal do Rio Grande do Norte, 59072-970 Natal, RN, Brazil

E-mail: eklenzi@dfi.uem.br

Received 12 May 2014Accepted for publication 26 June 2014 Published 22 August 2014

Online at stacks.iop.org/JSTAT/2014/P08019doi:10.1088/1742-5468/2014/08/P08019

Abstract. We investigate a system governed by a fractional diffusion equation with an integro-differential boundary condition on the surface. This condition can be connected with several processes such as adsorption and/or desorption or chemical reactions due to the presence of active sites on the surface. The solutions are obtained by using the Green function approach and show a rich class of behaviors, which can be related to anomalous diffusion.

Keywords: exact results, Brownian motion, diffusion

E K Lenzi et al

Fractional diffusion equation, boundary conditions and surface effects

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10.1088/1742-5468/2014/08/P08019

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1. Introduction

The dynamic behavior of diffusing particles subjected to sorption and/or reaction pro-cesses on a surface is an important issue that has been investigated in several contexts. For instance, gas–solid systems [1], liquid crystals media [2], biological systems [3, 4], kinetics of polymers [5], bulk-surface exchange in the diffusion of adsorbed molecules at liquid-solid and liquid-fluid interfaces [6], hydrogen interactions with nanomaterials [7, 8], and even as in biotechnological application in wastewater treatment [9–11]. In par-ticular, diffusion and adsorption-desorption are processes essential to molecular transport and interactions in living cells. Recent investigations regarding molecular diffusion in vivo, based on single-particle tracking approaches, have shown that the occurrence of anomalous diffusive behavior is common in these systems (e.g. [12–14]), i.e. when the mean-square displacement is characterized by !(x " !x#)2# $ t!, being superdiffusion (! > 1) commonly related to active transport [15–18], while a subdiffusive behavior (! < 1) may be related, for example, to the molecular crowding [19] and fractal struc-ture [20]. These anomalous dynamics have been investigated through continuous time random walk, fractional Brownian motion and fractional diffusion equations (see [12, 21–26]). In particular, the last approach has been successfully applied to describe memory effects in membrane cells [27–29], intracellular transport [30], ergodicity breaking [31, 32], subdiffusion in thin membranes [33] and chemotaxis diffusion [34]. This is mainly due to the possibility of modeling the boundary conditions, leading to exact solutions, i.e. it is possible to specify the geometry [35–37] of the system or even to consider nonusual boundary conditions such as moving [38] or reactive boundary conditions [39]. Moreover, as reported in [40], a fractional reaction-diffusion equation with a reactive term (related with adsorption-desorption processes) exhibiting memory effects can be derived from a continuous time random walk approach. Situations characterized by an effective motion along a cylinder of a particle that freely diffuses in the bulk and intermittently binds to the cylinder have been considered in [41, 42]. In [43, 44], generalizations of the reactive boundary conditions, as well as fractional diffusion equations, are also derived based on a continuous time random walk model [45]. In this context, we investigate the solutions

Contents

1. Introduction 2

2. Diffusion and surface effects 4

3. Conclusions 11

Acknowledgments 11

Appendix A. H Function and generalized Mittag–Leffler function 11

References 13

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of one-dimensional fractional diffusion equation by taking nonusual boundary conditions into account, which can be related to an adsorption and/or desorption process. We also investigate a situation characterized by active sites, where reaction processes can occur.

The fractional diffusion equation considered here is

K D! !!! = "

#$$$!!

%&''''" "(

tx t

xx t( , ) ( , ) ,t0

12

2 (1)

where K! is the diffusion coefficient, 0 < " ! 1 (for " = 1 usual diffusion, 0 < " < 1 subdiffusion), and the fractional time derivative is the Riemann-Liouville one [46], i.e.

D !! " #!= "

" # #$ #

##

$$x t

tt

x t

t t( ( , ))

1

( )d

( , )

( ).t

t

t

01

10

(2)

The presence of the fractional time derivative in equation (1) has been used to inves-tigate situations where properties of the region (or media) can be connected to, for example, fractal structure [47, 48], porous media [49], and disordered systems [50, 51]. The boundary conditions considered are

K D D !! " !"#$$$%%

&'((( = "

#$$$ ) * ) ) &'(((# # $*=

*x

x t t t t t( , ) d ( ) (0, ) andtx

t

t

01

0

01

0 (3)

!!! =="x

x t( , ) 0x

(4)

with 0 ! # ! 1. The initial condition is ! != !x x( , 0) ( ), where !! x( ) is an arbitrary func-tion normalized. Note that the boundary conditions represented by equations (3) and (4), which define the system in a half-space, may be connected to situations character-ized by adsorption and/or desorption processes depending on the choice of ", # and $(t). Thus, these quantities originate an unified framework for investigating several sit-uations and others scenarios such as the one presented in [52], where a mixing between physisorption and chemisorption processes are considered. In addition, the presence of a fractional time derivative in the diffusion equation enables us to consider anomalous processes in the bulk, which are not suitably described in terms of the usual diffusion equation. It may be obtained in the context of the continuous time random walk [53] if a reactive boundary condition is considered, similarly to the developments performed in [39, 44]. In this sense, a particular choice for the kernel is related to the physical process underlying the system. From a phenomenological point of view, the form of the boundary condition could be related, for example, to the surface irregularity [54], which is an important factor in adsorption–desorption [55, 56], diffusion, and catalysis [57, 58]. Similar boundary conditions have also been used to investigate the electrical response, e.g. water [59] and liquid crystals [60]. In particular, the experimental data for these cases have shown that in the low frequency limit the surface effects (boundary condition) have a relevant contribution for the electrical response [61]. For instance, a typical adsorption–desorption process is characterized by a Debye relaxation where $(t) % e"t/%. Another choice is ! !! +" "!t t t( ) 1/ /1 2, which can be connected with a non-Debye relaxation; in particular, this choice with # = 0 enables us to reproduce the

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behavior of the electrical response presented in [59 and 60], suggesting how are the underlying processes of the ions at the layers on (or near) the surface of the electrodes.

Equation (3) or (4) may also incorporate, for instance, the presence of a substance flux by performing suitable changes. In addition, it can be also shown, by using the previous equations, that

! ! !! " # $ !+ " # " # # =#$

"x x tt

t tt t t td ( , )

1

( )

d

( )d ( ) (0, ) const . ,

t t

0 0 1 0 (5)

where the first term represents the quantity of substance present in the bulk. The sec-ond term gives the quantity of substance adsorbed which, in contact with the surface, may be converted in another substance by a chemical reaction (here considered of first order). Note that &(x, t) is not normalized for t > 0 when $(t) & 0.

This work is organized as follows. Next section, section 2, is devoted to investigat-ing the solutions and the processes related to equation (1), accomplishing the bound-ary conditions given by equations (3) and (4). In this context, we also investigate the scenario characterized by a surface with active sites, where a first order reaction may occur with the formation of a substance which can move from the surface to the bulk. In section 3, we present a summary of our results and conclusions.

2. Diffusion and surface effects

Let us start the discussion on the time dependent solutions of equation (1) subjected to an adsorption/desorption process described in terms of the boundary conditions given by equations (3) and (4) on the surface present in x = 0. To face this prob-

lem, we use the Laplace transform L{ } !! ! !"#$$ = =

% &x t t x t x s( , ) d e ( , ) ( , )0

st and

L !! " ! !"#$$%$$&'$$($$= = )

*+++,

, -+

-+x s

is x s x t( , )

1

2d e ( , ) ( , )

i c

i c1 st and the Green function approach.

Applying the Laplace transform, the previous equations can be rewritten as

K ! ! !!! = "" ""sx

x s s x s x( , ) ( , ) ( , 0)12

2 (6)

and

K ! " !!! " =# # $"

=xx s s s x s( , ) ( ) ( , ) 0 ,

x 0 (7)

!!! =="x

x s( , ) 0 .x

(8)

In terms of the Green function approach, the solution of equation (6), subjected to equations (7) and (8), is formally given by

G!! !" = # "$

x s x x x s x( , ) d ( , ; ) ( , 0) ,0

(9)

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with the Green function obtained from the equation

K G G !!! " # " = # "" "#sx

x x s s x x s x x( , ; ) ( , ; ) ( ) ,12

2 (10)

with the boundary conditions

K G G!!! " # " =" " ##

=xx x s s s x x s( , ; ) ( ) ( , ; ) 0 ,

x 0 (11)

G!! " ==#x

x x s( , ; ) 0 .x

(12)

The effect produced by the surface is characterized by the choice of # and the function $(s) present in equation (11). They determine the rates of adsorption and/or desorption on the surface. By solving equation (10), we obtain that the Green function is given by

GK

K

K K

K

x x ss s

s s s s

( , ; )1

2 /e e

1

/ ( )e

s x x s x x

s x x

1 !

! ="#

$%%%%%%

"&

'((((((

"+

" "

" " #

" " ! " + !

"" + !

""

""

""

(13)

and performing the inverse Laplace transform it is possible to show that

G G G G!" = # # " # + " # " $ # " + " "! ! !( )x x t x x t x x t t t t x x t( , ; ) ( ; ) ( ; ) 2 d ( ) ( ; )f f

t

f( ) ( )

0

( ) (14)

with

GK K

=!

"

#####

$

%

&&&&&

!! ! ! !

! !'( )x t

tH

x

t( ; )

1

4f( )

1,11,0

(0, 1)

12,2

(15)

where !"##

$%&&H xp q

m n

b B

a A,,

( , )

( , ) is the Fox H function [62] (see appendix A for details) and

K

!

!! !"! " # $ %

% %

= +#

$%%%%%&

'

())))) &

* & &!

$

=

+

& & &

t tn

t t t

t t t t t t t

( ) ( )1 1

( )d ( )

d ( ) d ( ) ,

n

n t

n n

t

n n n

tn

1 0

0 1 1 0 1 11

n 1

(16)

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with ' = # " "/2 for # > "/2. In particular for $(s) = $'/% = constant, equation (14) can be simplified to

G G G

K KG

( )x x t x x t x x t

t

t tE t t x x t

( , ; ) ( ; ) ( ; )

2d

( )( ) ( ; ) ,

f f

t

f

( ) ( )

0 1 /2 ,( )!!

"!"

" =# # " + + "

# " "# "

$%&&&&#

" # "'())))) + " "

# #

# # $ % %#

% #+ #

(17)

where E!,((x) is the generalized Mittag–Leffler function [46]. Note that the presence of the Fox H function and the generalized Mittag–Leffler function in previous equa-tions can be connected to the anomalous spreading of the initial condition due to the bulk effects which are governed by the fractional time derivative present in the dif-fusion equation. The effects produced by the processes governed by the function $(t) are present in the last term of equation (14) and represent how the surface interacts with particles present in the bulk. In figure 1, we illustrate the behavior of the solution when different boundary conditions are considered so as to illustrate how the processes present on the surface may modify the spreading of the system. For this, we consider

&(x, 0) = )(x " x') as the initial condition with ! ! ="

x xd ( , 0) 10

, which implies that

the substance was initially concentrated in the bulk. Other situations, such as the sub-stance initially concentrated on the surface, can also be investigated. This case can find applications in drug dissolution [63] or delivery [64].

From the previous results, we can obtain the survival probability S t( ) and, consequently, the quantity of substance which was absorbed by the surface

Figure 1. The behavior of the solution for two different choices, " and #, for $(t) = ($'/%))(t/%). For simplicity, we consider K =! 1, $' = 1, % = 1, x' = 1, and t = 1.8 in arbitrary unities.

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M S= !t t( ) 1 ( ). The survival probability is defined as S ! !="

t x x t( ) d ( , )0

and, for

our case, is given by

S K K! !! ! "= " # # + # # +$ " # $ " ##

###s

s sx x x x

s s( )

1 1d ( , 0) e d ( , 0)

1

( )e ,

s x s x

0 0 (18)

with K! != " # #!s s s s( ) ( ) /1 . By performing the inverse Laplace transform and the

Fox H function, equation (18) can be written as

SK

K

!

! !

!

" !

= " # #$

%

&&&&&

#'

(

)))))

+ # " # # #$

%

&&&&&

##

'

(

)))))

" "

"

" "

"

*

*

( )

( )

t x x Hx

t

t t t x x Hx

t

( ) 1 d ( , 0)

d ( ) d ( , 0) .t

0 1,11,0

(0, 1)

1,2

0 0 1,11,0

(0, 1)

1,2

(19)

The behavior of survival probability S t( ) is illustrated in figure 2. In figure 3, we consider the case # = 0, which implies that, after some time, the dynamics is governed by a desorption process, in contrast to the case # = 1, illustrated in the same figure. Thus, depending on the choice of parameters or the time dependent function $(t), the

Figure 2. (a) The behavior of the survival probability and (b) the behavior of the particles adsorbed by the surface. The parameters are the same used in figure 1 as well as the time dependent function $(t).

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adsorption or desorption process may present a pronounced effect on the dynamics of the particles interacting with the surface.

By analyzing the spreading of the system via the mean square displacement, ! = !x x( )x

2 2 , we can investigate this point. Figure 4 illustrates the behavior of !x2, for three situations; it shows that the spreading is initially usual, ~! tx

2 , for small times, and when the adsorption process by the surface has a pronounced effect the behavior of !x2 is not usual, i.e. the diffusion is anomalous. The usual diffusion for long times is obtained for the case illustrated by the green dotted line, where # = 1/2, in contrast to the cases characterized by # = 1 with different $(t), i.e. straight and dashed black lines.

Now, let us incorporate a first order reaction process on the surface which can be related to the presence of active sites on the surface. The presence of these active regions may be obtained by a surface treatment such as the sol–gel method. This is a typical situation present in heterogeneous catalysis and in other processes taking place in gas–solid interfaces [65–69] where the surface processes play an

Figure 3. Behavior for the distribution and the survival probability for " = 1, # = 0 and # = 1. Note that when # = 0 the process on the surface, for long time, is governed by a desorption process, in contrast with the case # = 1, which is essentially characterized by an adsorption process. For simplicity, the parameters used in figure 3 are the same used in figure 1 as well as the time dependent function $(t).

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important role. In this context, we consider that the substance represented by *r, the result of a reaction process derived from the contact of the substance present in the bulk with active sites on the surface, can leave the surface. In the bulk, we assume for this substance that diffusion is governed by the following fractional diffusion equation

K D! !!! = "

#$$$!!

%&''''" "(

tx t

xx t( , ) ( , ) ,r r t r, 0

12

2 (20)

where K! r, is the diffusion coefficient and !< !0 1. For the kinetic process, we assume, for simplicity, that the reaction process is of first order, i.e.

M! != !t

t k t k td

d( ) ( ) ( ) ,r I II r (21)

(kI and kII are connected with the rates of formation by chemical reaction and desorp-tion, respectively) which has the solution given by

M!! = " "# # "t k t t( ) ( )e d .r I

tk t t

0

( )II (22)

Figure 4. The behavior of the mean square displacement for # = 1 and # = 1/2 by considering, for simplicity, D = 1, $' = 1, " = 1, and % = 1, with the initial condition given by &(x, 0) = )(x " 1).

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For the boundary conditions, we consider that

K D ! "!"###$$

%&''' =(# #(=x

x tt

t( , )d

d( )r t r

x

r, 01

0

(23)

!!! =="x

x t( , ) 0 .rx

(24)

At this point, it is important to note that we are considering there is no reaction between &r(x, t) and &(x, t) in the bulk, i.e. the diffusive process governs the dynamic of the substance in the bulk and the reaction process is restricted to the surface. For the substance obtained from a reaction process on the surface, we obtain the solution for equation (20)

G M G! !! "= " # " " # " # " "# #x t k t x t t t k t x t t t( , ) d ( ; ) ( ) d ( ; ) ( ) .r I

t

f II

t

f r0

( )

0

( ) (25)

Figure 5 illustrates the behavior of equation (25) for the case characterized by an adsorption process and, consequently, a reaction resulting on the formation of the substance characterized by &r(x, t) which leaves the surface according to the boundary

Figure 5. The behavior of equation (9) and equation (25) when a kinetic reaction is present on the surface for two different times in order to show the time evolution of the solutions. We consider, for simplicity, K =! 1, $(t) = ($'/%) )(t/%), $' = 1, ! = 1 # = 1, % = 1, kI = 1, and kII = 1, in arbitrary unities.

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condition given by equation (23). This figure also indicates that the amount of sub-stance in the bulk produced by a reaction process depends on the quantity of substance absorbed by the surface for the reaction process take place.

3. Conclusions

We have investigated solutions for a fractional diffusion equation subjected to an integro-differential boundary condition which can be connected to several scenarios, in particular, to adsorption-desorption processes. Analytical solutions were found by using the Laplace transform and Green function approach for an arbitrary initial condition. This solution was used to show the influence of the surface effects on the evolution of the initial condition, which may be characterized by an adsorption and, in particular, followed by a desorption process (see, for example, the blue line in figure 3). These effects observed in the solution change the behavior of the mean square displacement (see figure 4) which has an usual behavior for small times and, for long times, may exhibit different behavior depending on the characteristics imposed by the surface through the boundary condition. We have also incorporated a reaction process on the surface and the solution was illustrated in figure 5. It is also possible to consider other situations for the kinetic process on the surface such as the one considered in [2] where memory effects are incorporated.

Acknowledgments

We would like to thank CNPq (Brazilian agency) and Fundaçao Araucária.

Appendix A. H Function and generalized Mittag–Leffler function

The Fox H function (or H-function) may be defined in terms of the Mellin–Branes type integral [22, 62]

!!

H x H xi

x

b B a A

b B a A

1

2( ) d

( )( ) (1 )

(1 ) ( )

p qm n

b B

a Ap qm n

b B b B

a A a A

L

jm

j j jn

j j

j mq

j j j np

j j

,,

( , )

( , ),,

( , ), ,( , )

( , ), ,( , )

1 1

1 1

q q

p p

q q

p p

1 1

1 1 !! " # #

" # $ % # $ % #$ % # $ % #

"#$$

%&'' =

"#$$

%&'' =

= ( ( +( + (

#(

= =

= + = +

(A.1)

where m, n, p and q are integers satisfying 0 ! n ! p and 1 ! m ! q. It may also be defined by its Mellin transform

! ! ""#$$

%&'' =" "( ) )H ax x x ad ( ) .p q

m n

b B

a A

0,,

( , )

( , )1

q q

p p

(A.2)

Here, the parameters have to be defined such that Aj > 0 and Bj > 0 and aj(bh + +) & Bh(aj " , " 1) where +, , = 0, 1, 2, ..., h = 1, 2, ..., m and j = 1, 2, ..., m. The contour

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L separates the poles of *(bj " Bj') for j = 1, 2, ..., m from those of *(1 " aj + Aj') for j = 1, 2, ..., n [62]. The H-function is analytic in x if either (i) x & 0 and M > 0 or (ii)

0 < |x| < 1/B and M = 0, where ! != "= =M B Aj

qj j

pj1 1 and = ! != =

"B A Bjp

jA

jq

jB

1 1j j.

Some useful properties of the Fox H function found in [22, 62] are listed below.

(a) The H-function is symmetric in the pairs (a1, A1), ···, (ap, Ap), likewise (an + 1, An + 1), ···, (ap, Ap); in (b1, B1), ···, (bq, Bq) and in (bn + 1, Bn + 1), ···, (bq, Bq).

(b) For k > 0

!"##

$%&& =

!"##

$%&&H x kH xp q

m n

b B

a Ap qm n k

b kB

a kA,,

( , )

( , )

( , )

( , )

q q

p p

q q

p p

(A.3)

(c) The multiplication rule is

!"##

$%&& =

!"##

$%&&+

+x H x H xk

p qm n

b B

a Ap qm n

b kB B

a kA A,;,

( , )

( , ),,

( , )

( , )

q q

p p

q q q

p p p

(A.4)

(d) For n " 1 and q > m,

!"##

$%&& =

!"##

$%&&' '

'' ' ' '!!

!!

H x H xp qm n

b B b B a A

a A a A a A

p qm n

b B b B

a A a A,,

( , ) ( , )( , )

( , )( , ) ( , )

1, 1, 1

( , ) ( , )

( , ) ( , )

q q

p p

q q

p p

1 1 1 1 1 1

1 1 2 2

1 1 1 1

2 2

(A.5)

(e) The relation between the generalized Mittag–Leffler function and the Fox H func-tion is given by

! ! " #= + = "#$$ %

&'((# " " #=

)

%E x

x

nH x( )

( )n

n

,0

1,21,1

(0,1)(1 , )

(0,1)

(A.6)

(f) If the poles of ! "! "= b B( )jm

j j1 are simple, the following series expansion is valid:

#

( )( )

( )( )

H xx

B

b b

b b

a b

a b

( 1)

!

( )

1 ( )

1 ( )

( ).

p qm n

b B

a A

h

m b B

h

j j hm

jB

B h

j mq

jB

B h

jn

jA

B h

j np

jA

B h

,,

( , )

( , )

1 0

( )/ 1,

1

1

1

q q

p ph h

j

h

j

h

j

h

j

h

! ! !" # !" # !

" # !" # !

"#$$

%&'' =

( ( +( + +

)( + +

( +

!

! !

= =

* + = +

= +

=

= +

(A.7)

(g) The generalized Mittag–Leffler can also be represented in terms of a Mellin–Branes type integral as follows

!!" "" # $= "" "$ # %

%

" #

+ # "E xi

s s

sx s( )

1

2

( ) (1 )

( )( ) d .

i

is

, (A.8)

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(h) The asymptotic expansion for the generalized Mittag–Leffler function is [46]

~ ! ! " #" "# "=

#E x

n x( )

1

( )nn,

1 (A.9)

for 0 < ! < 2, ( arbitrary, |x| ( ), and µ being a real number such that -!/2 < µ < min [-, -!] with µ ! |arg x| ! -. For small arguments (|x| ! 1) the generalized Mittag–Leffler function can be approximated by E!, ((x) $ 1 + x/*(( + !) $ exp(x/*(( + !)). Thus, the generalized Mittag–Leffler function may interpolate between the initial expo-nential form and the long-time inverse power-law behavior.

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