Diffusion tensor and molecular jump frequencies in naphthalene single crystals

HAL Id: jpa-00246765

https://hal.archives-ouvertes.fr/jpa-00246765

Submitted on 1 Jan 1993

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Diffusion tensor and molecular jump frequencies in naphthalene single crystals

A. Bendani, A. Dautant, L. Bonpunt

To cite this version:

A. Bendani, A. Dautant, L. Bonpunt. Diffusion tensor and molecular jump frequencies in naphthalene single crystals. Journal de Physique I, EDP Sciences, 1993, 3 (3), pp.887-901. �10.1051/jp1:1993170�.

�jpa-00246765�

Classification

Physics

Abstracts

61.708 66.30H

Diflusion tensor and molecular jump frequencies in

naphthalene single crystals

A. Bendani, A. Dautant and L.

Bonpunt

Laboratoire de

Cristallographie

_et

Physique

Cristalline, Universit£ Bordeaux 1, 351 cours de la Libdration, 33405 Talence Cedex, France

(Received 2 June 1992, accepted ⁱⁿ

final form

29

September1992)

R4sum4.-Cet article propose une mdthode

compl~te

d'obtention, h

partir

d'un _tenseur

macroscopique

de diffusion, d'un modble microscopique permettant ^de

quantifier

la mobilit6 moldculaire _ou

atomique.

Il _a dt£ choisi, _pour cette Etude, ^le cas de la diffusion du

naphtaldne

ou du

naphtol-2

dans le naphtal~ne monocristallin. Un mdcanisme lacunaire h 3 _sauts moldculaires

possibles

_permet

d'expliquer

le _tenseur observd ainsi que son

anisotropie.

Les

fl6quences

r6elles de saut

(comprises

entre 20 et 80s-' _pour 338K<T<348K) _sont d6termin6es. L'effet de corr61ation est pris en compte et ddtermind. Les

dnergies expdrimentales

de saut sont

compardes

h celles calcul£es

th60riquement.

Abstract. A

complete

method is

presented

in order to obtain from the

macroscopic

diffusion tensor a

quantitative

microscopic ^{model of atomic} or molecular

displacements.

The _case of self- diffusion and hetero-diffusion of

naphthol-2

in single crystals ^of

naphthalene

_was chosen. A vacancy mechanism with three

possible

molecular jumps ^is found to explain ^{the observed} tensor, and its anisotropy. ^The

realjump frequencies

(in the range 20 to 80 s~ ^' for 338 K _< T

< 348 K) _are

determined. Experimental jump

energies

_are compared with the theoretical _ones.

1. Introduction.

In low

symmetry

^molecular

crystals,

rates of molecular

jumps

which allow matter

transport depend

_on the

crystallographic

direction. This fact involves the

macroscopic anisotropy

of

matter diffusion for these

crystals. Then,

diffusion must be

expressed by

_a tensorial form. In

retum, in cubic

crystals,

the diffusion tensor is reduced _{to a}

single

constant, the diffusion coefficient. In this _case, it is easy to deduce, from this diffusion

coefficient,

the

unique jump frequency

characteristic of the cubic environment of the molecule. This is _no

longer

true where the symmetry ^{of the} studied

crystal

^{is lowered. The}

analysis

^{of diffusion} tensor in order _to

obtain the

jump frequencies

is _not classical. We _{want to} present here, for the first time _{to our}

knowledge,

_a method of determination of characteristic data of the

microscopic

level of diffusion

jump frequencies

^and correlation

factors,

from diffusion tensor measurements.

This

analysis

is

presented

in the _case of the

naphthalene crystal,

used _{as an}

example.

^The

choice of

naphthalene

is related to its denomination of «molecular

crystal

model _»

by

Kitaigorodski [I]

the space group

(P~~la)

^{and the molecular}

packing

_are those of _a great number of

organic crystals. Very

_numerous

physical properties

of

naphthalene crystals

are

known.

Finally

the

hypothesis

^of _vacancy mechanism

allowing

bulk molecular

mobility

_seems

for this structural type reasonable. Moreover, three self-diffusion tensors

corresponding

to three

temperatures [2]

and

naphthol

heterodiffusion tensor at 343 K _were measured

[3].

The

conditions of the establishment of _a realistic

microscopic

model of molecular

mobility

are

filled up.

In this article first _we want to present ^the characteristics of the vacancy mechanism

(vacancy

formation energy, notions of

possible

and

likely jumps),

then

temperature dependence

and

anisotropy

of the

jump frequencies. Finally

the

experimental

activation

energies

_are

determined and

compared

with their theoretical values.

2. The vacancy

migration

and formation

energies

ⁱⁿ the

naphthalene single crystal.

Following Kitaigorodskii,

^the interaction

potential

between two molecules I and 2 in _a molecular

crystal

can be written _{as a} summation of the interactions between the

Ni

atoms of molecule I and the

N~

atoms of molecule 2

[1]

~~ ~~

~~~ i I ~jj

~~~

'"~ f"~

The

exponential-6

function

V~~ ⁼

Arj

^~ + B exp

(- Cr;~ ^{) (2)}

has been chosen for the

potential

V~~ between two non-bonded _atoms I and

j,

situated _at the distance

r,~,

^{with the number IV set of Williams}

[4]

for parameters A, B and C.

Then,

the lattice energy is

w

Ej~~ =

£ _Uj~ (3)

j 2

This atom-atom

potential

method has been used to calculate the vacancy formation energy.

A molecule _{on a}

given

site is removed and the lattice energy of _a

crystallite containing

this defect is determined. The vacancy formation energy

AE~

^{is the difference between the lattice} energy and the energy modification

AE~i

due to the relaxation of molecule around the defect.

The

migration

_energy

required

for _a

jump

is also calculated

by

this

technique

_: the

study

consists in

moving, step by

_{step, a} molecule from its initial site to _an

initially

vacant one. At each step ^{the lattice} energy of _a

crystallite containing

this mobile molecule is minimized. A first calculation is

performed

in which the mobile molecule alone is allowed _{to move a} second

calculation allows the relaxation of nearest

neighbour

molecules. In this way, the minimum energy

path

_can be determined. The

migration

_energy

corresponds

to the difference between

the maximum and the minimum value of the energy

along

^this

trajectory ^[5].

The

naphthalene crystal

is monoclinic

(space

_{group P~~,~,} a =

8.262h,

_b

=

5.984h,

c = 8.678

h, _{p =122.8°)}

_{with two molecules}

_(Z

=

2)

in the unit cell

[6].

There is _no

orientationally

disordered

phase

known for

naphthalene

_up to its

melting point.

These unit cell

parameters (measured

at 293

K)

_were chosen to carry out the calculations. The

dependence

^of

results on this choice is very weak. Calculations

performed

on a lattice of I lo molecules

[3]

gave the

following

values _: Ej~~ = 71.6 kJ. mol~

AE~I

3 kJ. mol~ Thus

AE~

= 68.6 kJ. mol~ ^'

3.

Analysis

of molecular

jumps

ⁱⁿ

naphthalene.

3.I GEOMETRICALLY _{POSSIBLE JUMPS.} Let _us consider the bulk

composed

of

eight

neighbouring cells,

centred _{on a} vacancy ^at the 000 site of the lattice. A molecular

jump

from _a site _{uvw to an}

initially

vacant site 000 is written

[uvw].

The number of

possible jumps

is 34.

If _we take the _two

possible

molecular orientations A and B into account, we can

distinguish

two types of

jumps

_:

(A ~A)

_or

(B ~B)

with the _same orientation before and after the

jump (A

_~

B)

_or

(B

~ A

)

where the orientation is different before and after the

jump.

The first type

corresponds

to a pure

translation,

and the second _{one to a} translation

accompanied by

a rotation.

So,

for symmetry reasons

(presence

of _a screw-axis and _an inversion

center),

some

jumps

are

equivalent.

For

example,

the

jumps [0 0]

and

[0 0]

_are

equivalent.

A similar relation

occurs between the

jumps

122~l'122~~'~22~~

^~~

(22~~'

This fact induces _us to

assign

a

multiplicity

a~ _to each

jump

numbered _q.

For

example

:

a '

= 2 for the

jump [0

0

(q

=

I

)

a

~

=

4 for the

jump

0

(q

=

2

a

~

= 2 for the

jump [0

0

(q

₌ ³

) Finally,

the number of distinct

jumps

is reduced to eleven

(Fig. I).

5 ₄

5 ₄

j

' _I I

' I

~ i

I

I /

~ i

11 <-

,1'

' ~ ⁱ

i j j

I j j

I j j

I j j

I

~

~,l'

_l'

5

'~'~

,

~

l' 4

, 5

Fig. I.

Geometrically

possible

jumps

and

energetically-likely jumps

in

naphthalene.

3.2 ENERGETICALLY _{LIKELY JUMPS.}

By

the _atom-atom

potential method,

_a

rigid

lattice

calculation of the

migration

_energy showed that among the eleven

jumps,

seven are

characterised

by energies

much

larger

than 000 kJ.mol~ _~, and

therefore, they

_are

unlikely.

The energy of

jump

number 4 is

approximately

I loo kJ.mol~ and is much

larger

than the

jump energies

for q =

1,

2 and 3.

Therefore,

_we considered that

only

3

jumps

_are

probable (Tab. I).

This number of

probable jumps

is denoted

f.

In

considering

the relaxation of the

neighbouring

molecules

during

the

jump,

_more reliable

migration energies, ES,

were calculated for

jumps 1,

2 and 3. The results

[2]

are

reported

in table1.

Table I. The

energetically-likely jumps,

their

multiplicity

and theoretical

migration

energies

in the

naphthalene crystal.

Type

of

jump lo 0]

o

lo

0 1]

22

Jump

_q number 2 3

Multiplicity

aQ 2 4 2

AE$ ~kJ.mol~ )

²² 40 30

4. Diffusion tensor and molecular

mobflity.

We _{want to} determine the real

jump frequencies

and the correlation factors _or matrix

(microscopic parameters).

From _a

comprehensive

view of the

results,

_we expect to obtain _a better

knowledge

of the molecular

mobility

and therefore of

crystalline

defects in

naphthalene crystals.

The relation between the

macroscopic

measurable

quantity,

the diffusion coefficient (or

tensor),

and the

microscopic mobility,

the _mean free

path,

is due to Einstein

[7]

_:

D

= lim

~~ (4)

rsmall 2

where

(X~)

is the _mean square

displacement

in time

r for _{an atom}

(or molecule),

and D the diffusion coefficient.

When the medium is _a

crystal

with

trajectories

defined

by

the lattice

periodicities,

the Einstein relation _can be written _{as :}

j ^I

~"j

~

§ i

_" ^~

_{~l ~l}

~~

(~)

q =1

where D;~ ^{is the diffusion tensor} component, ^{aQ the}

multiplicity

of the

jump

of type

q,

if

and

iJ

_are the

projections

of the

jump

vector on axes I and

j

and rQ is the

jump frequency.

There _are f

probable jumps.

This number

depends

on the

crystalline

structure and _on the

atom or molecule arrangement ^{in the unit cell. The number of data}

depends only

_on the

crystalline

system. ^{It is the number of}

independent

coefficients of the diffusion tensor

(equal

to one for the cubic system, up to six for the triclinic

system).

So,

depending

on the _{case, one can} have _an

N-equation

_system with

f

unknowns where N is

lower, equal

_or

higher

than

f.

Therefore the best way ^to determine rQ

depends

on the _nature of the

equation

system. ^{We do} not want to examine all the

possible

cases, but

only

the _case of the monoclinic

crystal

^of

naphthalene.

4. I DETERMINATION _OF JUMP FREQUENCIES. As _an

application

of the

three-jump

model

(the

most

probable jumps) presented

in section

3,

the data of the diffusion tensor at

temperature

^T

j2],

and relation

(5) yield

_a

system

^{of 4}

equations (due

to the

4-independent

coefficients of the

tensor)

and the 3 unknowns

r', r~, r~ (relative

to

jumps

_q

=

1, 2, 3).

If _{we name} a~,

p~

^and

_y~

^the

^angles

^{between the direction of the}

^jump,

^of

^length

iQ,

and the monoclinic reference _{axes a,} b and _c *

respectively,

the

system

^{described above is} written

Djj

=

(4 I( cos~

a~

r~

_{+ 2}

I( cos~

a ~

r~) ₍₆₎

2

D~~

=

(2 I( cos~ pi ^r'

+ 4

I( _{cos~ p~} r~) ₍₇₎

~~'~

D~~ ₌

I( cos~

y~

r3 (8)

The values of a~,

p~, _y~

^{and i~} are

given

in table II. The _tracer self-diffusion coefficients

D~~ [2] give

the

D,~

^values,

corresponding

to a

given

temperature.

(Sj )

is _an over-determined system ^{and it} must be solved

by

a least squares method

by introducing

a parameter r,, the _« residual

» of

equation

I

(I

= 3,

4).

The system

(Sj)

becomes _:

(6), (7)

and _:

i( cos~

_y3

r~ D33

₌ r3

(8')

~~~~

ij

CDS a 3 CDS y3

l~~ D13

" ~4

~~'~

Table II.

Angles

benveen the

jump

direction and the

crystallographic

_axes and

lengths of jUmps.

Jump

_{q a,} in

degrees p,

in

degrees

_y, in

degrees I, (A)

90 0 90 5.98

2 36 54 90 5.10

3 122 90 32 8.68

The solution

r~

_{must minimize}

r~

where

r~

=

r(

₊

r(

By replacing

the calculated value of

r~

obtained in the other

equations

of

(S~),

we deduce

r~

_and

r~.

In order _to get ^{the best coherence between the} set of values of

r~, r~

and

r~

_{and the}

experimental

data, an iteration process is

performed.

The _tensor

(DA

~)~~i~ is calculated from _an

initial _set of three

frequencies.

The calculated _tensor (DA~)~~j~ must be included within the confidence range of the observed tensor (DA~)~~~. ^{A refinement} process of the values of the

model revealed that several sets of r~

satisfy

this

condition;

in other words the

rQ _are not

independent.

From these sets, one of them is chosen _so that the variation of each r~ with temperature

(as

_{we can} calculate it _at the three studied

temperatures)

follows _an Arrhenius law.

As,

in this _case, the AT

interval,

338-348

K,

is

small,

this

assumption

is

quite

reasonable. Therefore _a coherent _set of rQ which does _not constitute _a

unique

solution but _a

plausible

_one, is

proposed.

4.2 RANDOM-WALK _AND CORRELATION.

If,

for _a molecule within the

lattice,

_no

jump

direction is

preferred,

_{we can say} that the

trajectory

of this molecule is _a random-walk.

However,

this cannot be the _case for diffusion

by

a vacancy mechanism where the directions of two successive

jumps

_are not

entirely independent.

This is the

phenomenon

of correlation

between

jumps.

As _a

general

rule there is _a correlation between successive

jumps

if _a least 3 different

particles (in

_{our case ; vacancy,} tracer

A*,

and molecule

A)

are to be considered in the

elementary

process of diffusion.

4.2.I Correlation

factor. Considering

the diffusion _tensors for _a correlated-walk,

D~~,

and a

random-walk, D~,

the correlation factor is

given by

:

D~*

#

D~ ~f~j). (10)

This factor

f,

which is _a

(3

_x 3 _non

symmetrical

matrix

[5],

is deduced

by

_a numerical simulation based _{on a} Monte Carlo method.

In _a

parallelepipedic

box which has the structure of the

naphthalene lattice,

_{a vacancy} is

placed initially

at the _center. Successive

jumps

of this vacancy are decided from

drawing

successive random numbers, so that each

jump frequency

is

equal

to its

probability.

A

given trajectory

consists of

10~ jumps.

The

geometrical

characteristic X of each

trajectory

is calculated. A _set of 10~

trajectories

allows _to calculate the _mean free

path (X)

of the vacancy, and then to determine the vacancy self-diffusion tensor. The

trajectories

of _a

given

molecule

can be determined in the _{same manner.} Therefore the different types ^{of diffusion} tensor can be attained

by

this simulation program.

Equations (10)

and

(11) (see later)

are used to deduce

correlation factor and matrices.

4.2.2 Determination

of

real

jump frequencies ri.

The simulation program allows _{us to}

calculate,

from _a set of

frequencies~

not

only

the

quantity describing

the correlation

(matrix,

correlation factor of

jumps fi),

but also the

corresponding

diffusion tensor. In

practice,

the initial _set is made up of effective

frequencies ri~ (tracer jump frequencies

calculated

by

^the

method indicated in

4).

The first simulation

yields

an initial value of the data relative to the correlation. The values of

ri

deduced _are :

ri~

ri

=

(I I)

fi

These values of

ri

_are used for _{a new} simulation and _{so on.} The process converges very

quickly

to two sets of

ri

and of

fi.

From the

ri

values the real _tensor

D(

is calculated. It describes the

mobility

of _a non-labelled molecule A into the

crystal

A. From

ri~

and

fi

the calculated tracer diffusion _tensor

(D(~

_)~~j~ is calculated. It has _to be

compared

with the observed one

(D(~)~~~,

^{in order} to

verify

the coherence of the model with the

experimental

data.

4.3 EXPERIMENTAL _{DIFFUSION TENSORS.} The diffusion _tensors measured

by

_us

[I] (self-

diffusion in

naphthalene

at

temperatures 338,

343 and

348K)

and

by

Faure _et al.

[2]

(heterodiffusion

at infinite dilution of

naphthol-2

in

naphthalene

at 343

K)

_are

given

in table III.

Table III. Tensor

coejfiicients of

bulk

dijfiusion

in

naphthalene,

^relative to

Ro (crystallogra- phic referential)

and R

~principal referential),

_w is the rotation

angle

benveen

Xi

and _a.

D in 10-17 m2 s-1

~~~ ~~ ~~~ ~~~ ~~ ~~

in

d~rees

348K

D~~~~~

^{3.3 2.9 3.4 -0.5 3.8 2.9 58}

D~~~~j~ ^{3.0 3.2 2.9 1.8 4.8 1.2 44}

343 K

D~~

~~~

l.9 2.6 2.4 0.5 2.7 1.6 60

D~~~~j~ ^{1.9 2.I 1.6 1.0 2.8 0.7 41}

DB~~~~

^{2.5 2.9 2.2 -0.7 3,I 1.6 39}

DB~~~~

^{2.3 2.9 1.9 1.2 3.3 0.9 40}

338 K

D~~

~~~

l.5 1.7 0.7 0.6 1.8 0.4 28

D~.

~~j~ 1.2 1.4 0.9 0.5 1.6 0.5 37

The self-diffusion coefficients _are

designed by DA*.

The hetero-diffusion coefficients _are

designed by DB~.

The _{tensors are}

presented

^{in the form} :

Djj

⁰

^Dj3

0

D~~

^{0 referred} to the

crystallographic

_axes

(a, b, c*)

1~13 ° 1~33

and

Di

^{0 0}

0

D~

0 referred _to its

principal

_axes

(Xi _X~ X~)

0 0

D~

w is the

angle

between the

principal

direction

Xi

of the _tensor and the

crystallographic

axis

a of the _structure. Due to the choice of

crystallographic

_axes for the measurements,

1~22

~1~2.

4.4 CALCULATED _DIFFUSION COEFFICIENTS. RELIABILITY OF THE MODEL. The values of

D~~~~j~

^{for the three} temperatures are within the confidence range of the

experimentally

determinated tensor

D~~~~~.

^{Therefore the}

proposed

set of

frequencies

constitutes _a coherent set, and _a

possible

solution to the

problem.

As _an

example,

in

figure 2,

_{one can see} the sections of the flux

ellipsoids by

the ab and ac*

planes, corresponding

to the measured and calculated _tensors at 338 K. There is _a small gap between the _two

quadrics. However,

the calculated tensor

is,

within

experimental

_errors, the

same as the measured _one.

I

a a

- corresponds _{to a} flux of

lx10'l?

_molecules

m

2f~

Fig.

2. Sections

by

the ab and ac*

planes

of the flux

ellipsoids

of the tracer self-diffusion tensors at 338 K,

measured-(D~.~~~)

and calculated (D~~~~j~).

5. Effective

jump frequencies.

The effective _{or tracer}

jump frequencies r(~, r(~

and

r(~

_{relative to} the

jumps

[0 0], 0j

^and

^[0

⁰

^1]

are

given

in table IV

together

with the effective

jump

2 2

frequencies

^of

^naphthol-2

ⁱⁿ

naphthalene

at 343 K.

It must be noted that the

frequencies rB~

have been obtained

directly

from the data for

heterodiffusion at 343K. It is

possible

that the cohesion

imposed

_on the

frequencies

r~~

introduced _a bias which is _not found with the

frequencies rB~.

We therefore note that the

r$~'s

are

systematically higher

than the

ri~'s.

A _common feature of the _two series of

frequencies

^{in the whole} temperature range is that

r'

_and

r~

_have

a similar

magnitude

which is

always higher

than

r~.

We will _come back _to this result when _we examine the real

jump frequencies (ri).

The

representation

of

Log (ri~)

variations _versus T~ allows _{us to} determine the apparent activation

enthalpies

AHQ.

AH~

=

72 kJ,mol~

AH~

=

84 kJ.mol~

AH~

=

116 kJ.mol~

The AH~ do not have _an immediate

physical

_sense, because

they

_are derived from

ri~ frequencies

which contain _two

physical quantities

that vary

independently

with temperature: ^real

jump frequencies

and correlation effects. Nevertheless these apparent activation

enthalpies

can be used for

interpolations

or limited

extrapolations.

Table IV.

Ejfiective jump frequencies of

tracer in

naphthalene.

Temperature (K)

Effective

jumps

338 343 348

(S-I)

r(~

26 40 59

r(~

55

r(~

_{26 39 60}

r2

~j

r(~

16 30 54

r3

3~

6. Correlation effect.

6.I PARTIAL CORRELATION FACTORS OF THE JUMP. In order to

clearly

determine the

influence of correlation _on the observed diffusion and _to be able _to obtain the real

jump frequencies,

the

partial

correlation factor

~f~)

relative _to each

jump (Tab. V)

is calculated

by

the method indicated in section 4.2.I. As _seen in table

V,

all the correlation factors

display

values in the range

[0.70

_;

0.80].

Table V. Partial correlation

factors ~f~) of jumps.

Temperatures

in

(K)

Correlation factors 338 343 348

(

f(

0.70 0.72 0.73

f(

^0.70

fj

^{0.77 0.76 0.77}

fj

0.77

f(

0.79 0.76 0.74

fj

0.79

6.2 VARIATION _{OF THE JUMP} CORRELATION FACTORS WITH TEMPERATURE. Irl CUbiC symmetry, ^{the factor}

f

is _a

purely geometrical parameter corresponding

to a

given

mechanism.

In _a low symmetry structure, it is

dependent

on the

mechanism,

on the

geometrical

environment of the vacancy and _on the various

jump frequencies. Therefore,

it

depends

on

temperature.

The variations with

temperature

^{of the three}

partial

correlation factors _are of different

types

:

f(

increases,

f(

is constant and

f(

decreases with

temperature.

^{There is} a clear correlation with the real

jump frequencies (see

Sect.

7).

The relative contribution of _a

given jump

at a

given

temperature can be

easily

calculated _as :

ri

£ ri (12)

q

When this contribution increases with

temperature

^{in the}

(case

of the

r~ jump),

the

partial

correlation factor of the

jump

decreases.

Similarly

when

r~ decreases, f~

increases with

temperature.

^The near constancy ^{of the} contribution of

r~

_is

accompanied by

that of

f2

hypothesis

:

a more frequent

jump

is _{more orrelated.}

But

here, the comparison

on

hole set

of jumps

at _a given emperature, due to

the

fact that the

correlated

with each other ;

I.e.

it is

their

relative ontribution

which must be taken into

account.

Another

f~ values decrease when temperature _increases. t is one

of

the _causes

of

the ncreasing

sotropy of the observed self-diffusion ^{tensor rom} 338 to 348 K. _An

interesting

fact

is

constancy

of the mean value _of the f~'s _: 0.75 at the

three

temperatures. This may

with ompaanand _Haven [8]

where the orrelation in

cubiccrystals for the vacancy

mechanism

was

0.78 and _0.73 for FCC and BCC, The crystalline _structure ^of

naphthalene _can be

described by a dense _packing

of molecules

pproaching

a

« one

centered

_cubic » arrangement and our mean

value

of the

correlation

_{actor is}

consistent with these geometrical

In hetero-diffusion ^at

olecule types are present (A olecules, B _olecules and _B

jumps

cease

random. _{Moreover we}

can no longer

nsider than the

two

ypes

of

_{molecules are}

quasi-

identical (as _{A and A*}

in

^the first pproximation) ; in other words _B*

(or

different _ehaviour

with point

defects. For

example, a

vacancy

be more easily

attached

to

molecule B* than

to

molecule A.

But

_{B* cannot migrate unless the vacancy} itself _igrates,

is to say hanges place _with a olecule A. Otherwise ^{there will} be an

incessant

_exchange

etween the

vacancy

and B*,

in _ther words no diffusion at all. herefore, the diffusion

of

depends not only on

the

umps of B* _but also _{on those}

of

A.

Although

we cannot give a

simple

ormula for the

hetero-diffusion

orrelation factors in

relation

with the

jump the

simulation does _provide a numerical value for

correlation factors f$.

The factors

f( or f(

seem

and

temperature independent.

We

note without

any

xplanation, that

f(

₍₃₄₃

K

₎ = f( (338

K

)

this is not

a

simpleunction

of

_the

r( and

r( r r(. ^hesepartial_jump

factorsepend on _all

the

frequencies.

6.3 ORRELATION

MATRIX.

- _Using,

in

(DA *< j

~

l~ AI k

~fkj

) (

the

following

correlation matrices _at the three studied temperatures are obtained _:

0.70

^{0 0.00}

0 0.74 0 _at T

=

348 K

0.02 0 0.74

0.71

^{0 0.01}

0 0.73 0 _at T

= 343 K

0.04 0 0.76

0.69

^{0 0.00}

0 0.72 0 _at T

= 338 K

0.07 0 0.80

These matrices _{are not}

symmetrical. They

_are, to _our

knowkedge,

the first of this kind obtained for self-diffusion in _a low symmetry structure.

7. Real

jump frequencies.

The real

frequencies corresponding

to random walk

jumps

_are calculated

by equation (I

I

).

The

ri

_are

given

in

figure

3.

All the real

jump frequencies display

values between 20 and 80 s~

However,

these

jump frequencies

do _not

correspond

to the duration of the

jump

itself but rather reflect the low

probability

of _a

jump occurring.

The characteristic time for the

jump

itself is thus _a very small fraction of the time between

jumps (12

to 50

ms).

r

s-i

.

p(

2.95 K~'

'°~

Fig. 3. Variation of the real jump

frequencies

with temperature.

So, the

difficulty

of

observing

such

jumps by

methods

giving

access to the characteristic

jump

times is due _more to the _rare

phenomenon

_occurrence than _to their

long

« lifetime _»

compared

with

« the time window of the method

»

(molecular dynamics, Quasi

elastic _neutron

scattering).

Recently,

Gullion and Conradi

[9] published

an NMR

study

of diffusion in benzene.

They

obtain

jump frequencies

of

1.25,

6.50 and 8.25

s~~.

_With

spin-echo

and stimulated echo

techniques,

these

frequencies

^become measurable.

Up

to now, few results in this

frequency

range have been

published.

Having

obtained the real

jump frequencies,

_we have

completed

_our

analysis

of the measured diffusion _tensor. The three

physical quantities

used _to describe the

microscopic

_vacancy model

quantitatively

have been determined _{: a}

spatial

_component of structural

origin (length

and direction of

jumps),

_a

temporal

_component, the real

jump frequencies

and _a mixed component, the correlation

proceeding

from

spatial

and

temporal origins.

Their

respective

contributions to the

macroscopic

diffusion

anisotropy

have been determined. The

jump frequencies

have _a

large effect,

and _{we can} say that the

tendency

towards

isotropy

observed with temperature increase is

due,

to a great extent, to

them,

since the

geometrical

characteristics _are almost invariable. However the correlation factors also follow this

tendency,

since the

fi's approach

one another _as the temperature ^{is raised. For}

example,

as

already

noted for the

ri~,

the

jumps [001]

remain less active than the other contained in the compact ^sheets ab. This difference

between the

jump frequencies

has to be

explained

from the structure. A step ^{in this direction}

can be made

by studying

the influence of temperature on the

frequencies.

i =

^b

R~

_~

j

ASf

+

ASS

here )o

R

~

AS~ ^and

ASS

_are the vacancy formation and

migration entropies respectively. wJ

is the

presentation frequency.

AH~

and

AH$

_are the vacancy formation and

migration enthalpies.

From the data of

figure 4,

one obtains _:

ioioi r(

=

1.4 _x

io12

_exp

(-

^{~~.~ ~~. ~°'~} _s- ⁱ

RT

jl/2

^1/2

0] r2

~ ~ ~ _~ ~~13 _~ ^{76.3 kJ.mol~ _1}

~ ~~

RT ^~

loon r(

= 4.5 _x

io19

_exp

(-

^{~~~.~ ~~.~°'~} s- ⁱ

RT

The uncertainties in the estimates of

(ri

_)o and AHf~~ are very

large.

The

experimental

_errors,

the

complex procedure

to obtain the

ri

^and

finally

the limited temperature range lead to results with limited

reliability.

Nevertheless, it _seems

interesting

to present ^them here,

given

the

scarcity

of such determinations.

j

_°j

.

a _. a

/

-

corresponds to a flux of

lx10'~7

_molecules

m 2s'~

Fig.

4. Sections by ^{the ab and} ac* planes ^{of the} flux

ellipsoids

of _tracer... (D~~~~~) and real- (D~~~i~) self-diffusion _tensors calculated in naphthalene at 338 K.

7.2 COMPARISON _BETWEEN THEORETICAL AND EXPERIMENTAL JUMP ENERGIES. The

activation

enthalpies

have been obtained from

experiments

carried _{out at} constant pressure.

The vacancy formation and

migration energies

have been calculated with conditions of

constant volume. Nevertheless _a

comparison

between theoretical and

experimental

ther-

modynamical

characteristics of

jumps

_can be achieved difference between energy ^and

enthalpy

must be small in this _case, and less than the

experimental

errors.

Table VI

gives

the _sum of

E~

+

ES

for _a

given jump

_q which _can be

compared

with the

experimental

_AH$~~.

Table VI.

Comparison

benveen

jump energies.

Jump

E~ +

E$

in kJ, mol~ ^'

AHj

j~ kj ~~j-

[0 0]

69 ₊ 22

= 91 68

[1/2

1/2

0]

69 ₊ 40

= 109 76

[0

0

1]

60 ₊ 30

=

99 119

A difference of 20 _to 30 fb is observed, between the calculated and

experimental jump

energies. Although

the

experimental

values have limited

precision,

_part of the difference may also be due to the

approximations employed

in the model.

First,

_{an energy} barrier with _a

single

saddle

point

is

only

_a first

approximation

because the

jump

_energy

profiles

of the three studied

jumps

_possess two maximum values. Thus the

probability

of _a

given jump

cannot be

rigorously expressed

_as the

product

of _a

unique presentation frequency

times _a

simple

function of the energy barrier

height.

On the second

hand,

it is also

possible

that the

migration

of

molecules is assisted

by phonons,

collective

displacement

of the molecules which constitute the environment of the saddle

point positions.

So the real energy barrier is lower than the

calculated

E~.

7.3 COMPARISON _WITH NMR DATA RELATIVE TO MOBILITY OF NAPHTHALENE MOLECULE.

Molecular motions in

non-plastic crystals

can be

essentially

detected

by

radio-tracer

measurements and

by

NMR. Our results of the three

jump frequencies along

with their

temperature ^variations

(between

338 and 348

K)

cannot be

directly compared

with similar data issued from the literature. The work of Sherwood and White

[10]

_was relative _{to tracer} self diffusion measurements in a

unique

direction c* of the

naphthalene crystal.

An activation

enthalpy

of

179kJ.mol~~

was found. The

correspondence

with _a

jump

activation energy

cannot be achieved without the

complete knowledge

of the self-diffusion tensor. Two studies

of the

naphthalene

molecule motion

by

NMR have been

published (Von

Schutz and Wolf

[I I]

and Mac

Guigan

_et al.

[12]). Naphthalene samples

_were studied _as

powder

and the

anisotropy

could _not be detected. Molecular motion characterised

by

activation

enthalpies

of

105kJ.mol~~ _[11]

_and

91kJ.mol~' _[12]

was shown. A

pre-exponential

factor of

10~~~*~

_{second for the} characteristic correlation time _was

given by MacGuigan

et al. The order of

magnitude

of _our result of

r(

_{is the same.} Nevertheless the author's

interpretation

is that the observed motion is

probably

an

in-plane

molecule reorientation. Measurements _on

single crystals

similar _to those carried out on benzene

by

Gullion and Conradi should be useful for _a better

analysis

of molecular motion in

naphthalene.

8. Real diffusion coefficientso

Given the

previous

data it is

possible

to calculate the real self-diffusion coefficients

D~~~j~.

^{The values of the}

D~~~i~

are shown with those of

D~~~~j~

^{and the} hetero-diffusion

coefficients calculated _at 343 K (DB~~j~ ^and

D~~~~j~)

^{in table VII.}

Table VII. -Bulk real

self-dijfiusion

and

hetero-dijfiusion of 2-naphthol

in

naphthalene,

relative _to

Ro (crystallographic referential)

and R

~principal referential).

_w is the rotation

angle

benveen

Xi

and _a.

Din10-17 m2 s-I

Dll ~2

_{1~33 1~13}

~l ~3

in

d~rees

348 K

D~

~~~ 4.2 4.3 4.0 2.5 6.6 1.6 43

D~~~~j~ ^{3.0 3.2 2.9 1.8 4.8 1.2 44}

D~~~~

2.6 2.9 2.I 1.3 3.7 1.0 39

343 K

D~~~~ic

^{1.9 2.I 1.6 1-o 2.8} 0.7 41

D~~~~

^{3.2 4.0 2.4} 1.5 4.4 1.3 37

D~~~~~

^{2.3 2.9 1.9} 1.2 3.3 0.9 40

338 K

D~~~ic

1.6 2.0 1-1 0.7 2.1 0.6 34

D~~

_~~~ ^{l.2 1.4 0.9 0.5 1.6 0.5 37}

As an

example, figure

4 shows the sections of the flux

ellipsoids

for the tracer self-diffusion tensor and the true self-diffusion tensor at 338 K

by

the ab and ac*

planes.

The _true self-diffusion

(and hetero-diffusion)

tensors, inaccessible

by experiment, always correspond

to a greater

mobility

than those measured. This is due to correlation factors which

display

values

ranging

between 0.7 and 0.8.

Is this type ^of

quantity

accessible to

experimentation

? When _{we can} follow _a molecule

moving

within its

lattice,

without

marking it,

we will have to measure such tensors

taking

_an

average of _a great ^{number of} observations. The calculated tensor is

equal

to that

representing

diffusion

by

vacancies with _a

multiplication coefficient,

the vacancy concentration, which is unknown.

9~ Conclusion~

In this paper, we present a method of

analysis

of

anisotropic

diffusion. The diffusion _tensor is

decomposed

into three

microscopic quantities,

_one of _a

geometrical

_nature,

lengths

and direction of

jumps,

one of _a

temporal

nature,

jump frequency

_; and the third _{one, a} correlation effect. Each component was determined and _a coherent vacancy model of self-diffusion was

obtained. For the first time in _a low

symmetry crystal, jump frequencies

and

partial

correlation factors _were calculated.

Along

with these results, _we

present

^{the method itself of}

obtaining microscopic

parameters ^{from the}

macroscopic

measured diffusion _tensor.

References

ill KITAIGORODSKI A. I., Molecular

Crystals

and Molecules (Academic Press, New York, 1973) p. 357.

[2] BENDANI A. and BONPUNT L., to be

published.

[3] FAURE F., DAUTANT A., BENDANI A., BONPUNT L., J.

Phys.

Chem. Solids 51(1990) 1005-1010.

[4] WILLIAMS D. E., J. Chem.

Phys.

45 (1966) 3770.

[5] DAUTANT A., Thdse d'dtat-Bordeaux

(1988).

[6] CRUICKSHANK D. W. J., Acta Crysi. 10 (1957) 504-508.

[7] EINSTEIN A., Ann.

Phys.

17 (1905) ^549-560.

[8] COMPAAN K., HAVEN Y., Trans. Faraday Sac. 52 (1956) 768.

[9] GULLION T. and CONRADI M. S., Phys. Rev. B 32 (1985) 7076-7082.

[10] SHERWOOD J. N., WHITE D. J., Philos.

Mag.

is (1967) 745-753.

[I I] VbN SCHUTz J. U., WOLF H. C., Z. Naturf. (a) 27 (1972) 42.

[12] MCGUIGAN S., STRANGE J. H, and CHEzEAU J. M., Molec.

Phys.

49 (1983) 275-282.