Anomalous diffusion of surface-active species at liquid-fluid and liquid-solid interfaces

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Anomalous diffusion of surface-active species at liquid-fluid and liquid-solid interfaces

Oleg Bychuk, Ben O’Shaughnessy

To cite this version:

Oleg Bychuk, Ben O’Shaughnessy. Anomalous diffusion of surface-active species at liquid-fluid and liquid-solid interfaces. Journal de Physique II, EDP Sciences, 1994, 4 (7), pp.1135-1156.

�10.1051/jp2:1994192�. �jpa-00248034�

Classification Physic-s Abstracts

68.10J 68.45D 05.40

Anomalous diffusion of surface-active species at liquid-fluid

and liquid-solid interfaces

Oleg

V.

Bychuk (I)

and Ben

O'shaughnessy (2)

(') Department

of

Physics,

Columbia

University,

New York, NY10027. U-S-A-

(2)

Department

of Chemical

Engineering,

Materials Science and

Mining Engineering,

Columbia

University,

New York, NY10027, U-S-A-

(Received 23 Febrl~aiy 1994, accepted I

April1994)

Abstract. We study the role of bulk-surface exchange in the density relaxation kinetics and self-

diffusion of surface-active molecules _at liquid surfaces. In «strongly adsorbing» systems, relaxation _occurs

through

bulk-mediated effective surface diffusion characterized by one-step distributions with

long

tails _; molecules execute

Ldvy

walks _on the surface.

Correspondingly,

at times before

particles

_are finally lost _to the bulk, surface

displacement

_r is non-Fickian and

exhibits anomalous

scaling

moments grow as

(r~)

^~t'~~~, where ((q)=q for q~l,

((q (q + 1)/2 for _q

~ l and

jr )

t In t. The width of _an

initially

localized density disturbance

increases

linearly

in time with _{a «}

speed

_» c which is

universally

related to other observables.

Numerical simulations confirm the

family

of exponents ((q), and

reproduce

the observable

c. We consider _a

simple example

where end-functionalised macromolecules adsorb _{at a} solid surface,

finding

_c l/s where _s is the surface _« stickiness _» parameter. ^At

liquid-fluid

interfaces

viscoelastic effects compete. ^{For sub-micron} scales, _{we argue} that self-diffusion will

typically

remain dominated at high _coverages by the anomalous bulk-mediated mechanism, while surface

viscoelasticity

will dominate the relaxation of density

perturbations.

1. Introduction.

Adsorption-desorption

kinetics _at interfaces _are involved in many

phenomena

of fundamental and

practical importance. Adsorption

at the

solid-liquid

interface

arises,

for

instance,

in many

biological

contexts

involving protein deposition [1-3],

in solutions _or melts of

synthetic

macromolecules

[4-7],

in colloidal

dispersions [8],

and in the manufacture of

self-assembly

mono and

multi-layers [9-12].

Meanwhile, soluble surfactant molecules

adsorbing

and

desorbing

at

liquid-liquid

_or

liquid-gas

interfaces

[9, 13-16]

_are

important

in the relaxation and

collision

dynamics

of

microemulsions,

emulsions and foams. In fact, when any

liquid

containing

soluble molecules is

brought

into contact with _a surface for which the

panicles

have

a certain

affinity,

such processes arise.

Amongst

the

experimental

methods suited _to the measurement of these

kinetics,

surface

elasticity

measurements

[17, 18]

and

pendant drop tensiometry

studies

[19]

have

provided

valuable information. In

general, however, quantitative

results _are in short

supply, though

many other methods have the

potential

to

yield

further information _on kinetics.

Examples

include surface

light

_or neutron

scattering [20],

second harmonic

generation [21, 22], ellipsometry [23, 24]

and various luminescent

techiliques

such _as fluorescence recovery after

photobleaching (FRAP) [20, 25]

and others

[26-30].

This paper is _a

study

of the influence of bulk-surface

exchange

in the surface

density

relaxation kinetics and surface self-diffusion in such systems. ^{The role of}

adsorption-

desorption

^{kinetics remains}

relatively unexplored

both

experimentally

and

theoretically.

In the various

examples

mentioned

above,

surface

density inhomogeneities

arise

(either produced by

external agents ^such as flow _or

by

intrinsic thermal

fluctuations)

and _one would like to

establish the relaxation time

of,

for

example,

a localized

perturbation.

A related issue is the

time-dependence

of the surface

displacement

of individual adsorbed molecules, I-e- their self- diffusion. There _{are a} number of relevant mechanisms _: in-surface

diffusion,

_{as at}

liquid-solid

interfaces _; surface viscoelastic effects

[13, 20, 31-35]

in the

liquid-fluid

_{cases ;} and bulk-

surface

exchange

whenever surfactants _are soluble. Bulk-surface

exchange

_may lead _to

ej$ectiiie

bulk-mediated diffusion. It is this process, in which

particles

move from _one surface location to another

through desorption

followed

by

diffusion in the bulk and

readsorption,

which is the

principal subject

of the present

study

(see

Fig. I).

At

liquid-solid

interfaces, in

particular,

it is

expected

in many cases to

provide

the

primary

^mechanism for both self- diffusion _as well _as the relaxation of

density

fluctuations

(cooperative diffusion)

on timescales

before the adsorbed molecules _are

permanently

released _to the bulk.

Meanwhile,

at many

liquid-liquid

and

liquid-gas

interfaces we expect self-diffusion to remain dominated

by

this mechanism _at

high

_coverages where in-surface diffusion is

heavily suppressed. Experimental

FRAP studies of self-diffusion

[25]

are of

particular potential

in this respect. ^In

principle, they

can

provide

detailed and _accurate surface diffusion information with which _to compare the present theoretical work.

viscoelasticity (these

effects will be addressed in Sect.

7),

and _we seek the relaxation kinetics

of the surface coverage r

(number

of molecules per unit

area)

for small deviations from

equilibrium.

The relaxation of surface

density inhomogeneities

is

closely

related to that of

homogeneous perturbations

ⁱⁿ r, which involve _{a net} transfer of _mass between surface and bulk these bulk-surface

exchange

kinetics _are of _two

possible

_types,

depending

_on the relative

magnitudes

of the

microscopic parameters goveming adsorption, desorption

and bulk diffusion. We will define _«

strongly adsorbing

_» bulk-interface systems as those which retain adsorbate molecules for _a sufficient time that _« diffusion-control _»

(DC)

onsets in the bulk. In

« weak _» systems, on the other hand, the

desorption

time is smaller than that

required

for bulk

DC effects to

develop. Strong systems

are characterized

by

slow bulk-surface

exchange

kinetics,

in the _sense that the time for the interface to renew its adsorbate

population

is much greater ^{than the}

desorption

time for _a

single

molecule. Given that molecules _are

desorbing,

and yet are not

being permanently

lost

by

^the

interface,

it follows that

they

must be

readsorbing

elsewhere _on the surface that

is,

the interface must be

shuffling

its occupants ^{around the}

surface

by

bulk-mediated effective surface diffusion. Each excursion into the bulk is

characterized

by

distributions of

possible

times and

displacements

which tum out to have

long algebraic

tails each molecule

performs

_a

Ldvy

walk

[36-40]

_on the surface. The central limit

theorem is

inapplicable

in such _cases, and _we will find that the _net surface

displacement

^for strong systems ^is

correspondingly

non-Fickian for times less than the bulk-surface

exchange

time. In

fact,

the

particle displacement statistics,

which at low coverages determine the

relaxation of

r,

exhibit _« anomalous

scaling

_»

[41]

in that the different moments are

govemed by

a non-trivial

family

of exponents.

The essential feature of _a strong system ^{is that the}

desorption

time is much greater ^{than the}

«

typical

_»

readsorption

time

during

the _« anomalous

»

regime.

Thus, _a

typical

molecule

spends virtually

all of its time _on the surface and its bulk-mediated surface

displacement

is much less than the free Fickian diffusion result: the _«

sticky

_» interface

strongly

retards

particle

motion. We will show, in fact, that the time spent ^{in the bulk}

(for

most

molecules)

grows like t~ and the surface

displacement

evolves _{as r} t ; the surface

density

distribution

(starting

from _a localised

patch)

is _a 2-dimensional

Cauchy

^{form whose width} grows

linearly

in time. However, an

important

consequence of the

algebraic

tails of the step distributions is that

a small fraction of molecules make very

big jumps, lasting

of order t, and suffer Fickian

displacement

which scales _as t'~~ It _turns out that this

minority

^{determines the}

higher

moments

of _r.

The

emphasis

of this paper is _on theoretical results. In

addition,

_we have

performed

extensive numerical simulations which have substantiated _our

predictions.

Some of these

results, which

provide

_{strong support} for the anomalous

scaling,

are

presented

in the

following.

Details will appear elsewhere

[42].

The paper is structured _as follows. In section 2 the definitions of _« weak

» and _« strong _» are

made

precise,

and in section 3 the relaxation of surface

density inhomogeneities

is introduced.

Weak systems ^{relax such}

perturbations

in _a

single desorption time, simply replacing

the old

population

with _{a new one,} while strong systems are

subject

to the anomalous diffusion

described

previously

and for short times exhibit _a non-Fickian relaxation spectrum,

r~ l/k for surface wave-vector k. The

remaining

sections

specialize

to strong systems and

follow the motion of _a

single

molecule, _or

equivalently

the

spreading

out of _an

initially

localized

patch

of surface

density

_excess or deficit. In sections 4 and 5 we

study

the

long-tailed

distributions of total time

elapsed

and total

displacement, respectively,

^after n excursions into the bulk, I-e- after _n steps of the molecule's

Ldvy flight.

Numerical results are

presented

^which corroborate the theoretical

picture.

Section 6 deals with those

properties corresponding

to

experimental

observables,

namely

the statistics of the

displacement

r after _a

given

time

t, which _are

closely

related _to statistics at fixed _n. At short times we find anomalous

scaling

of moments,

(r~) ^t~"J

with _« non-trivial _»

((q),

and _we present computer ^{simulation results}

which _are in accord with this behaviour. In section 7 _we consider the influence of other

relaxation mechanisms. We argue that surface viscoelastic

effects, coupled

_to the

bulk,

overwhelm the _« anomalous

» effects at

typical liquid-fluid

interfaces _as far _as surface

density

relaxations _are concerned.

However,

at

high

_coverages self-diffusion is

likely

_to remain

dominated

by

^the bulk-mediated mechanism. We conclude with _a summary in section 8.

2. _« Weak _» and _« strong » adsorbers.

The

simplest description

of _an

adsorbing-desorbing

interface

interacting

with _a dilute bulk must invoke the bulk

diffusivity

D, a characteristic

desorption

time

Q~ ',

a surface _« capture range » b and surface

adsorption

time

Qid(.

These last two define _an

adsorption

_« sink _» of

width b and

strength uwo~~~b.

To

appreciate

the

meaning

of _« weak» and _« strong»,

consider

firstly

the pure

adsorption

_case

(Q

" 0 the interaction of the bulk with the interface

is then _a I-d

reaction-diffusion problem (ignoring

transverse variations for

simplicity).

Reaction kinetics in I-d exhibit _« diffusion-controlled

»

(DC)

behaviour in consequence of the compactness ^{of the}

spatical exploration

of _a

particle [43, 44].

This is _a true statement, at

large enough

times t, even for _a very weak reaction sink _{one can} say that kinetics _are driven to DC form

by increasing

t. The _cross-over time, t

*,

is estimated

by equating

to

unity

the

adsorption probability

of _a

particle

close to the interface

(I.e.

within _a range

m,I)

~

~

~ _j~~)1/2

~~d~ _~h

'

~~~

in which

b/(Dt)~~~

is the fraction of

« steps »

(each lasting _t~)

which lie

adjacent

to the interface,

during

each of which the

adsorption probability

is

roughly Qads

t~. ^This

implies

t ^* ₌

(r* )~/D

,

r*

=

2 DIM

,

(2)

where

r*,

the diffusion

length

associated with

t*,

is the

« renormalized _» sink size

(the

factor

of 2 is introduced for

convenience).

That is,

coarse-grained

over the scales r* and

t ^* in space and time, kinetics _{are «}

perfectly adsorbing

_{» any}

particle

which strays to within r* of the interface almost

certainly

« reacts _».

Clearly

then, an

initially homogeneous

bulk

must

develop

a hole of size

,$

_for

t _~ t*, and

delivery

of material _to the interface becomes controlled

by

diffusion.

Switching desorption

back _on, if it

happens

that

Q~

^' » t ^* the DC behaviour described above has had time to

develop

before the surface releases any adsorbed

particles,

and _one

anticipates

serious bulk disturbance in response to a surface

perturbation.

This is _our definition of _a

« strong »

adsorber, Qt*

« I. In _{a «}

weakly adsorbing

_{» system,}

Qt*

» I,

desorption

onsets before _a bulk hole has

developed

the bulk remains close _to

equilibrium

and this govems the

delivery

of material to, for

example,

a bare surface. As _we will demonstrate, this

phenomenology

^is indeed exhibited

by

the

following

bulk-interface

dynamics,

where

r m (.<, y,

z)

and _x is the coordinate

orthogonal

to the

planar

surface, _x

=

0

j-D v)p

=

-u&(xjp+Q&(,<jr f

=

Qr

+

ups (3j

Here

p~wp(.r=0)

is the bulk concentration _at the surface,

boundary

conditions

p(x-± oJ)

= p~~

pertain,

and _{we use} the domain _{oJ ~x~ oJ} with initial conditions

symmetric

about x=0 _to mimic

reflecting boundary

conditions. The sink

strength,

u =

Qads b,

can then be written _as

req

u =

2

Qh,

h

= -,

(4)

P~~

where r~~ is the

equilibrium

surface coverage

corresponding

to p~~

Physically,

the

«

adsorption depth

_» h

corresponds

to the amount of bulk

required

to fill _an empty ^surface.

Generally,

_p and r should be

interpreted

as deiiiations from the

equilibrium

values,

equation (3)

is then valid

provided

these deviations _are small, and h is _more

generally

defined to be the

slope

of the

equilibrium adsorption

isotherm.

Consider an

initially empty

^bulk and

homogeneous

surface

(r

=

r°,

_p

=

0). Equation (3)

is then

easily

^solved

by Laplace transforming,

t _- E

~ iS~

(x)

~~~~~

~

l ₊

E/Q

+

/

^'

ro E/Q

₊

/

~~

E _I

+

E/Q

+

/~

^{' ~~~}

where

iS~(x)= (4ED)~'°exp(- x(E/D)'~~)

is the bulk diffusion Green's function and

t~ m h

~/D

_{the diffusion time}

corresponding

to the

adsorption depth.

This time-scale will feature

prominently

in what follows _; in fact

« strong » and _« weak

» can

equivalently

be defined _as

Qt~

» I and

Qt~

w I

respectively

since

equation (2) implies

~~~ ~t~

~~~

It is _now

straightforward

to demonstrate that

equation (5) reproduces

^the «

no-desorption

_»

reaction kinetics discussed above in the domain

E/Q

» I. Our main

interest, however,

is in

longer

times when the

E/Q

term can be

dropped.

Then for time-scales much less than tj~, ^or

Et~

W I, the inverse

Laplace

transform of the

expression

in

equation (5) yields

diffusion-

controlled behaviour

ro

~

~~~'

~~

h ~~~~

2,$

^'

~ ^)/2

r(t

m

r°

_I

(Q~

^' « t « tj,

(7)

th

Clearly,

the above

regime only

exists for _a strong ^adsorber

(Qt~

» I

).

The response in the _case of _a weak system ^is very different then the sequence of time scales reads t~ «

Q~

« t ^* and

for times less than _t *

(Et

^* » one can discard the

/

_{terms in}

_{equation (5) (being}

_much

less than

E/Q)

and _one finds

e,rponential

recovery, ^r r~~~ e~Q~ with the bulk almost

unaffected, _pi

p~~.

Thus the relaxation time for w'eak systems ^is

Q~~

the surface has

completely

recovered before any DC effects _{can occur.}

Returning

to the _« strong » result,

equation (7),

one can say ^that for times much less than t~ the relative number of solute

particles

on the surface is

virtually unchanged

and _a front of

desorbed

particles

extends

@

_{into the bulk. Unlike}

a weak system, ^{the surface holds} onto its

initial occupants ^well

beyond

the

desorption

time

Q~ exchange

^{kinetics with the bulk} are

very slow,

requiring

a time t~ ^for

completion

_{as can} be deduced from

equation (5)

for

Etj~ w I when _one finds that r and _p have recovered their

equilibrium

levels. The relaxation time is t~.

Why

does the surface

(for

_strong

cases)

retain its occupants

during

the

regime

Q~

_~ t _~ t~ ^?

Roughly, equation (I)

defines _t *

as the

readsorption

time since t ^* «

Q~ ', particles spend

almost all _on the time of the surface

during

_{a sequence} of

desorptions

and

readsorptions.

At any

instant, therefore, nearly

all of the initial

population

must be _on the surface _even for t »

Q~

^' However, this statement is rather loose in fact (see Sect.

4)

the distribution of

readsorption times,

_r, has _a

long

_r ^~~~ tail and _t * is

merely

the median time

(the

mean does _not

exist).

This tail

eventually dominates,

and

particles

leave the surface _as follows.

After _n

hops

on the surface

(I.e.

_n

desorption-readsorption events)

the number of

hops

for

which the time spent ^{in the bulk lies between} r and r+dr is

approximately

n(t*/r

f~~

(dr/t*).

The total time spent ^{in the} bulk, t~~'~, therefore

obeys '~~~

^d~ t ^{* 3/2}

n r = t~~~~

,

(8)

t* t ^* _T

whence

t~~~~

m

n~

_t *

(n

»

) (9)

The _sum is dominated

by

the

small_number

^of excursions

(of

order

unity)

into the bulk whose duration is of the _same order _as the total time, t~~'~.

Comparing

t~~'~ with the total time spent on

the

surface, nQ~

_{~, one} deduces that for small

enough

times

(and n)

the fraction of time spent

on the surface is

virtually unity,

but the surface loses its

particles

when the number of steps reaches _a value _n

w

Qt~

_or t

m t~.

In summary, there _{are two} natural candidates for the interface relaxation time the

desorption

time

Q~ ',

and the diffusion time t~

required

to transport

enough

_mass from the bulk to restore surface

equilibrium.

For weak systems the relaxation time is the former, for strong

systems the latter.

3. Relaxation of surface

density inhomogeneities.

The

previous

section suggests ^that

spatial

variations in r should relax very

differently

in the weak and strong cases. Let _us follow the evolution of _an initial harmonic surface _wave, r

=

r)

_e~~ ~

(r

and k

lying

in the

surface)

confronted

by

an empty ^bulk.

Equation (3)

_can be solved _to

give [45]

E/Q

+

EhS~~ r)

~~~

l ₊

E/Q

+

EhS~~

E ^' ~~~~

in which the transform of the _{« return}

probability

_»

[45]

satisfies

~ j

_{ET W I}

(I II

Er~

^"~ _~h

~' ~ '

EhSEk

~

~

l ₊

ETk /

'

~~~

~

'

where r~ m I

/Dk~

_{is the Fickian relaxation time. Motivated}

by

our

previous observations,

let _us examine wavevectors in the range t*

~ r~ ~ t~

(which only

exists for _a strong

system).

In the

bulk,

of _course, such _{a wave} would have relaxed

by

t _m r~; in contrast,

by choosing Er~

_~ l in

equation (10),

_one can show that the surface _wave is little

changed

from its initial

amplitude

_on these time scales. We will _see later that this is due to the _«

clinging

_» effect of the

surface which reduces the surface

displacement

after time r~ ^to a value much less than k~ _~. Relaxation in fact

proceeds

_on a much

longer

scale _; from

equations (10)

and

(11)

_one

finds

r~(t j

_m

r) e~'/~k

_{(t »}

r~j, ^(12j

where the _« anomalous

» relaxation time is

f~

₌

rim _{(ck)~ (13)}

and is characterized _not

by

_a diffusion _constant but rather

by

a _«

speed

_{» c,}

cm

~. ₍₁₄₎

h

The harmonic surface

imprint

is remembered for much

longer

than the bulk memory

time,

r~, This is _a very different

decay

to that for the weak _case which _{one can} demonstrate _to be

independent

of wavevector,

r~

e~Q~ The surface loses its harmonic _wave after _a

single desorption

time.

Corresponding

to this anomalous surface relaxation is _an unusual

development

of the bulk field p~

(xi. Specialising

to the _same k values and times greater than the

desorption time,

_one

can show

[45]

that the bulk is

locally equilibrated

with the surface

(hp~(x

₌ ⁰

=

r~)

and the surface influences the bulk up ^to a

penetration depth

k~

~m0 p~ 1

~ e~ ^~~ e~ ^~~~

(t

» T~

j (15)

11

The total _mass in the bulk for _a

given

value of _r is much less than the amount on the surface at that

point, by

a factor I/kh.

Let _us translate into real space. In

particular,

we ask how _a localized

patch

of _excess surface

density

_«

spreads

_» on the surface in _a strong ^{adsorber. Provided the initial}

patch

dimension is much less than h, _«

spreading

_» is _a reasonable _term since _we know the surface retains

particles

up to t~. ^{Now the}

expression

in

equation (12)

is

precisely

the Fourier transform of the

Cauchy

distribution

[46, 47].

Since its

validity

in

k-space

^{is restricted to k~ '} w

@

_for

a

given

^time t, one can

justifiably

transform to _r

provided

the result is

only interpreted

on

corresponding

scales

ct ~r w (Dt)'~~ t w

th1

~~

~~~'

2gr

j(ct

_{)~ +} r~]~~~

Equation (16)

describes _a

patch

which remains attached to the surface and grows

linearly

in time with

«

speed

_{» c.} The

patch density

follows _a 2-d

Cauchy distribution, falling

off

slowly

at

large distances, r~l/r~,

with

amplitude

at the

origin decaying

_as

r~l/t~.

_The

normalization up ^to scales _r

m

(Dt

_{)~'~ is} close to

unity

when _{t w} t~ ^{and it is}

straightforward

to

show that the contribution from wavevectors

k~' ~,$

_is

correspondingly

small when

t w t~.

However,

the behaviour of r _on the

largest

scales, _r »

,$,

_{is in fact of}

_importance

despite representing only

_a small fraction of the surface

particles.

As will be shown later, it determines the

higher

moments of _r.

In _a similar _manner, the real space bulk

profile

is obtained _as

~

2~h

j(~~

/ )

,,2j3/2 ^~~

~ ~" ~~~~

This result is valid

only

^for,x, r _~

@.

_In

_fact, integrating

_over transverse coordinates _r

gives

I/h for any value of.<, ^{and the} x

integration

would then

diverge

_were ^it not cut off _at

x w

,$. _(The

_{behaviour in}

x after the _r

integration

is of _course

just

the result

(7), viz.,

the evolution for the _case of

homogeneous

^surface coverage, and is _cut off

by

the

complementary

error

function).

The effect _on the bulk of the

initially

localized surface

perturbation, depicted

in

figure 2,

is thus _a

right

_cone

extending

_a distance

~$

_both

orthogonally (x

and

transversally (r).

The

angle

between the interface and the _cone

edge,

_r

= x + ct, is

45°,

and the top ^section of the _cone, where it _meets the

patch

of width _c-t, is

« cut off _»

by

the surface. For each value of

x the

profile

in _r is _a

Cauchy

distribution of width.< ₊ ct, and each such slice contains the _same

« mass »,

~

x +

cl

<

r=x+ct

Fig.

2. Effect of

spreading

surface

patch

_on bulk for _a strong ^adsorber at times _t « t~. ^{A slice} y 0 is depicted, where _r

m (j~, z lies in the surface. Most _mass is contained in the surface patch whose size grows with _« speed » c, whilst the _mass in the bulk occupies a cone of _extent

,$

» c-t and edge

r .i + c-t. A disk

~ Cte, is shown particles contained in this disk follow _a Cauchy distribution in _r of width.r ₊ c-t, Each such disk contains equal _mass.

In the

following

sections _we will

study

this bulk-mediated motion of

particles

_on the surface

more

thoroughly,

The

Cauchy distribution, equation (16),

describes _an anomalous non-Fickian effective surface

diffusion,

r~t. It will be shown that the distribution _on

larger

scales

corresponds

to a different but

equally

anomalous

dynamics.

A clear

picture

_emerges when _one follows the motion of _a

single initially

adsorbed solute

particle

_; such _a

particle performs

_a

Ldvy

walk _on the surface.

4. Time

elapsed

after _n surface hops :

Lkvy flight.

We

begin by asking

how much time has

elapsed

after _a

particle, specified

to be adsorbed at t ₌

0,

has

performed

_n steps on the surface. This and the next two sections will deal

exclusively

^with strong systems. ^A new « step»

begins

when _a

particle

is readsorbed, continues for the duration of its stay on the

surface,

continues further _as the

particle

desorbs

and diffuses in the

bulk,

and terminates _at

readsorption.

The distribution of times for _one step,

~ (t ),

is characterized

by long tails,

that

is,

the

particle performs

_a directed

Ldvy flight [48]

in time,

4.I ONE-STEP DISTRIBUTION OF TIMES SPENT IN BULK. To obtain

~(t),

_one must account

both for time _on the surface and in the bulk. The distribution of times in the bulk,

~~~~~(r ),

is related _to

H(r),

the

probability

a

particle starting

in the bulk _{at x}

= 0 has been

readsorbed

by

the time _r, which is the solution

(for r)

to the one-dimensional version of

equation (3)

with

Q

set to zero and initial conditions r

=

0,

_p

=

(x)

The

probability density

that a

particle's

excursion into the bulk lasts _r is

just

the

derivative,

~~~'~

₌ H, which from

equation (18)

has the

long

time tail

(r

» t* ) ^mentioned

previously,

~~~'~

_{r ~~~,}

It is worth

noting

that at

long

times ~~~~~ exhibits the _same behaviour _as for _a

peifiectly adsorbing

interface with the

particle initially

at _{x m} r*, rather than _x

=

0, _{as may} be shown _as follows. This latter

problem

is

simply

solved

by introducing

_an

image (of

normalization

I)

at.<

=

r* _; the normalization in the

bulk,

HP~'~, is then

just

the difference of the

areas

(in

_x~

0)

of _two Gaussians with relative

displacement

2 r* which for TN t*

gives

I

HP"~wi~*/,$= (t*/T)"~.

_{This has the}

same form _as the

long

time behaviour of

equation (18).

That the _r

~ t* behaviour in _our

problem,

where the sink

strength

^is finite, should

reproduce

the infinite

strength

result is

expected

in view of the discussion of section 2 for times

r ~ t* and

spatial

scales _r

~ r* the sink is

effectively perfectly adsorbing.

4.2 _n-STEP DISTRIBUTION OF TIMES SPENT IN BULK. Consider

firstly

the distribution of total

amounts of time t~~'~ spent ^{in the bulk after} _n steps, ~,$~~~(t~~'~). The

long

tail,

~~~'~ r~~/~,

immediately

tells _us that the central limit theorem breaks down and the one-step distribution for bulk excursions

belongs

to the domain of attraction of the

(directed)

stable distribution with characteristic exponent 1/2

(Smirnov distribution) [46, 48],

To _see

this,

let _us define the distribution of scaled times _as

I[~'~(tbU'L/n2)

=

n2 ~,)U'k(tbU'k), (191

which

obeys I)~'~(E)= ~,$~'~(E/n~).

In

Laplace

_space the n-step distribution is

just

[~~~'~(E)]~,

whence

where

equation (18),

_was used. This

gives

the Smirnov distribution

[49]

with characteristic scale n~

t*,

~ j- ^{1/2 2 * 3/2} ~ ~j~

~,)~~(t~~'~

₌

)

~

~~(~

^e- ~''*"'~

(2 ii

n t t

in accord with the argument ^{of section} 2, where it _was found that the characteristic value of

t~~'

was n~ _t*. The above

simplified

derivation of

~)~'

is made

rigorous

in the

theory

^of stable distributions

[46].

In conclusion, the distribution of total times spent ^{in the bulk has} a

long tail,

just

_as for the one-step distribution.

4.3 _n-STEP DISTRIBUTION OF TIMES SPENT ON SURFACE. Since _a

particle

desorbs with

constant

probability Q

_Per unit time, the distribution of residence times is

Q

e~ Q'.

Having

no

long

tail, the central limit theorem is _now

applicable. Indeed,

if _one repeats ^the

procedure

used to obtain

~(~'~,

^but

replaces

n~

- n, it is

straightforward

to show that the one-step distribution is driven _{to a}

simple

delta function after _n » I steps, &(t

nQ~ ').

4.4 n-STEP DISTRIBUTION OF TOTAL TIMES ELAPSED.

Having

derived

~~~'~,

the one-step

distribution of total times

elapsed

is

immediately

obtained _as

~

(t

=

o

dt'

Q

e~ Q"

~~~'~(t

_t'

(22)

It follows that the n-step distribution,

~~(t),

of total times

elapsed

is driven _to the

product

of

the above Smimov and delta functions in

Laplace

_space. In real space, therefore,

~~(t)

is the Smirnov function

shifted by nQ~

_~, and

vanishing

for t _~

nQ~

^'

~n (t i

₌

~j~'~(t _nQ~ i

°

(t nQ~ i (231

Provided _{one can}

ignore

the

long

tail

(which

is

permissible only

for moments q ~

l/2)

then the above n-step distribution of times is

essentially

_a delta

function,

I-e- _one has _t

w

nQ~

^{' for such}

lower _moments. The distribution is illustrated in

figure

3. Thus

(in

some

sense)

a

particle spends

most of its time _on the surface

provided

_n w

1/Qt*

or

equivalently

t w t~. ^For

larger

n and _t, the width of the _« cut-off _»

region

of the Smimov distribution

(the

median total time spent ^{in the}

bulk)

becomes

larger

than the time spent on the

surface,

n~ _t *

~

nQ~

^' However,

such _{statements are} loose in fact any ^moment

(t~)

for

qm1/2

does _not exist, I-e- is

dominated

by long

bulk excursions. The

preceding

remarks _are thus

meaningful only

in

reference _to

sufficiently

small _moments.

~-3/2

nQ-~ n~t*

Fig. 3. Distribution of times elapsed after _n steps ^for a strong system. The _case t « t~ ^{is shown for} which the time spent on the surface is much greater than the time spent ^{in the bulk} :

nQ~

^' » n~ t*.

To _test the above

predictions

and those which

follow,

_we have measured

[42]

the bulk- mediated

spreading

of

particles

in _a number of lattice computer simulations for _a range of strong systems. ^Particles

simply

random walk _on the cubic lattice and

adsorb/desorb

with

probabilities

_per unit time Qads ^and

Q respectively.

These latter _are varied _to

give

a range of

«

strengths

_{» as} measured

by Qt~

(the reader is referred to Ref.

[42]

for

details).

In

figure 4a,

total times

elapsed

_versus number of surface

hops

_{n are} shown for _a

strength Qt~

=

6 _x

10~,

as

measured

by

the moment q = I/4. The

anticipated

linear

relationship,

t n, is almost

exactly reproduced.

300

j~~

^I-O

jb)

~t

<

200

I

O.5

~ L

~A J

j I

O-O +

100 £

-o.5

~

0 100 200 300 ~°~ ~

n

1-o (C)

"

<

+ f

«

I«~

) ff

0.5 ~

log tit

Fig.

4.- Numerical results for Qt~ 6x10~. (al

Q(t~)'~~

_{as a} function of number of steps

n for q =

I/4 (.), The linear dependence

expected

for times much less than th (see text) is shown for

comparison

(solid line). (b)

log (r~)

/r*~) _versus log _n ^for _{q =} ^0.2 ( + ) and _q

=

0.5 (Ill. Solid lines _are theoretical

predictions (having slopes

q), All results _are shifted

by

the q-dependent constant

A(q) to force theoretical

intercept

_to

origin

(see Ref, [42] for details). (cl

log ((r~)/r*~)

_versus

log

Qt for the _{same q} values _as in (b). Results _are again shifted by A (q), The coincidence of numerical

and theoretical intercepts is _a precise test of the speed _c.

To summarize this

section,

the total time

elapsed

after _n steps ^scales at t n for times smaller than t~, ^and as t

n~

_{for times greater than t~,}

provided

one considers

sufficiently

low moments. In section 6 we will _see that this _{t n} relation

plays

a crucial role for lower _moments

of the anomalous surface

displacement,

while

higher

_{moments are}

governed by

the

long

t~~~~ tail of the distribution of total times spent ^{in the bulk.}

5.

Displacement

on surface after _n

hops

_:

Lkvy flight.

In this section _we

forget

about

time, seeking

the

particle's displacement

after _n steps. ^{We will} show that the

displacement

_grows

linearly

with the number of steps, r n,

provided

one

considers _moments lower than the first.

This,

too, is _a consequence of _a

long

tail one-step distribution _{; a}

symmetric Ldvy flight [47]

is executed. The one-step distribution of surface

displacements

is

simply

related to the bulk time distribution

through

the two-dimensional diffusion Green's function

oJ

~ (k

j

=

dT ~~~~~(T

j

e~ ^{~~ ~}

(24j

0

Observing

that the

Laplace

transform of the

integrand

is

~~~'(E+Dk~),

the _use of

equation (18)

leads _to

~

(k j

=

~

~

(25j

Clearly,

the variance of the above distribution

(- V(

~ _[~

~)

does not

exist,

_so when iterated

the central limit theorem does not

apply.

The small k behaviour

(~

l kr* indicates that

~(r) belongs

to the domain of attraction of the

(symmetric)

stable distribution with

characteristic exponent

(Cauchy distribution).

The small and

large

k behaviours

imply

~(r)~

l/r for

rwr*, ~(r)~ l/r~

_{for r»r*. (In fact}

explicit

inverse transform of

equation (25)

is

possible

in _terms of the zeroth order Struve and Neumann functions

[50].

As _we would expect, ^for r » r* the one-step distribution ~

(r)

is the _{same as} for a surface of infinite

adsorbing strength given

an initial

particle

location at x - r*.

(In

this _case,

using

_an

image

sink at,<

=

r*,

the

I/r~

_{behaviour emerges}

as the field of _an electric

dipole.)

For _our

finitely adsorbing

_case, the

particle

_« bounces

» many times _on the surface,

moving

_a distance of order r* before it gets ^adsorbed.

Before

formally deriving

the n-step distribution ~,~

(r),

let _us estimate the

displacement

after

n steps

using

some

simple

but

physically revealing

arguments. For this purpose, consider

firstly

_a

general

distribution with characteristic exponent a in d-dimensional space,

~

(r

r~ ^{'~ + ~} Then the

probability

distribution for

jr

behaves like p

(r

r~ ^{' + "} and after

a total of _n steps ^{the number of} steps

having

size of order _r is

roughly

n

(r

*/r)". An estimate of their contribution _to the total

displacement

is

j.* _« 1/2

d(r

)

= n i~ r

' "~~

(26)

r

since the

n(r*/r)~

steps are

randomly

oriented. For

a ~ 2, the smallest scales, _r

m r*, are

dominant and the total

displacement

is

approximately d(r*

m

n'~~ _{r*. This is the central limit} theorem. However, for

long

tails, a ~ 2, the

largest

scales dominate,

namely

_r

-

r~°',

where

1"°~ is the total

displacement.

That is, the size of the

biggest

steps

(of

which there _are order

unity

in

number),

and the total

displacement

are one and the _{same :} d(r~°~

ii

r~°~ Thus

tot (la _~

~) (~~)

r in 1' £Y~

Applying

this to the present case, for which

~

l

/r~

_{in d}

=

2, that is _a

=

I,

_one expects ^linear

growth

r~°~ _n.

To demonstrate this linear behaviour in _{a more concrete} way, let _us calculate

~~(r)

explicitly. Analogously

to

equation (19),

_we define the distribution of scaled total

displace-

ments as

i~(r/n ₎

m

n~ ~~(r), implying i~(k)

=

~~(k/n ).

Once

again,

the n-step distribution in

reciprocal

_space is

~~(k),

and

using equation (25)

_one has

i~(k

= e~ ^{~ '~ l' + ~'*~~J} _m e~ ^~~*

(n

»

(28)

This leads _{to a}

Cauchy

distribution of width nr* _on the

surface,

~~~~~

_"

l

((nr~)1) _r2j3/2

^{' ~~~~}

exhibiting

the

expected

_{r n} and,

asymptotically, ^I/r~

behaviour, The numerical results

[42]

of

figure

4b, obtained for the _same parameter ^values as

figure

4a, _are in close agreement ^with

the _moments for _q

=

0.2 and _q

=

0.5 _as deduced from

equation (29),

after _a transient of about 10 steps.

Niively,

the t _-n and _r~n behaviours which have been established in this and the

preceding section, respectively,

_suggest the r~t

dynamics

of section 3.

However,

the

relationship

between statistics _at fixed

n and _at fixed _t is rather subtle and will be clarified in

the next section.

Clearly,

these _are

fundamentally

different. For instance,

according

to

equation (29)

moments of _r for q m I do _not

exist,

for _a fixed _{n ;} the

origin

of this is the

long

tail in the bulk time distribution,

equation (21).

In contrast, for _a

given

time _t the time in the

bulk cannot, of _course, exceed _t itself

correspondingly,

one can show that all _moments of

displacement

exist _at fixed _t.

6.

Displacement

^after time t

Lkvy

walk.

Experimental

observables

correspond,

^of _course, to

specifying

time rather than number of steps. ^{Our aim in this section is the}

particle displacement

after time _t which results from its

Ldvy

walk _on the surface. That

is,

_we wish _to determine the surface

density

distribution

r(r, t)

^for a delta-function initial condition.

Before

deriving

the walk statistics, _a

simple

_argument

helps

connect the fixed _n and fixed

t

relationships,

The argument ^{which follows} suggests r t and _r

t'~~,

for _t

respectively

less than and greater ^than t~. We remind the reader that t~ ^{is the timescale after which the interface} in _a

strongly adsorbing

system ^{loses its} occupants, ^Let us

imagine

a « transparent »

surface,

such that after _a time which _we

(suggestively)

_name t~~'~, the

particle (initially

located _at

x = 0) would have visited a

region

^{of size} r m

r*(t~~'~/t*)'~~

and its

trajectory

would have

intersected the «renormalized» surface

(of

width

r*)

of order _nm

(t~~'~/t*)'~~

times.

Switching

back on

desorption

and

adsorption,

this

corresponds

to an n-step ^{walk with} a time t~~'~ spent ^{in the bulk} and _a net surface

displacement

_r. Thus _r

m nr

*,

t~~'~

m

n~

_t * and the total surface residence time is t~~~

=

nQ~

^' For t _~ tj~, therefore, the total time t is

approximated by

t~~~ and _{we recover r}

m c-t with _c

w

Qr

^* ₌ ^D/h, In contrast, for _t

~ t~ one has _t

m t~~'~,

leading

to the familiar Fickian result _r

=

r*

(t/t*

_)~~~ We remark that the essential component ^{of this} argument ^{is the}

coarse-graining

over scales r* and t* after which the surface is,

effectively, perfectly adsorbing.

Now in section 3 the distribution of surface

displacements r(r, t) during

the anomalous

regime

t w t~ was derived. This _was the distribution of

equation (16) particles spread

out on

the- surface with _a

Cauchy

distribution of

growing

width _ct.

Accordingly,

the

q-th

moment