Nucleation and growth in supersaturated monolayers

Advances in Colloid and Interface Scimce, 47 (1993) l-23 Elsevier Science F%blishers B.V., Amsterdam

00167 A

NUCLEATION AND GROWTH IN SUPERSATURATED MONOLAYERS D. Vollhardt Max-Planck-Institut fir Kolloid- undGrenzGchenforschung RudowerChaussee5, O-l 199 Berlin,Getmany

Abstract

A

nucleation-growth collision

transformation kinetics from

supersaturated

following the

monolayers theoretically.

key features:

nuclei,

(ii)

theory has been surveyed which can describe

for formation and growth of three-dimensional (i) single-step

are with

reviewed,

The models are based

the first

on the

nucleation and subsequent growth of

the total transformation rate as convolution of

rate and growth rate, and (iii)

the

centers

overlapping of the growing centres.

nucleation Two models

one for the limiting cases of nucleation and growth

assumed geometric shape and the second one for nucleation

according

to

the exponential law and the formation of lenticular

centers from a supersatur-

ated monolayer.

the effect

cial

In the more general second model,

tensions at the three-phase contact air/water/center

the contact angLes of the lenticular

center.

evidence

is

provided

for the adequacy of

is allowed for

Qplication

enables the determination of nucleation rate constants. the

of the interfaboth

on experimental data Manifold experimental

nucleation-growth

collision

theory.

Insoluble monolayers spread on the aqueous surface represent a physical system gaining

renewed attention.

monolayer for technique.

The main reason is that they form the

the manufacture of ultrathin films

Over the last decade,

by the

precursor

Langmuir-Blodgett

new inform&ion has been obtained

on

the

structure and morphology of insoluble mDnolayers at microscopic and molecular level

supported by the development of highly sensitive

ques, such as Brewster angle m.icroscopyS*‘, synchrotron X-ray diffraction’*“. ~01-%~~9~$~.00

01993 -

experminental

fluorescence

techni-

microscopyS,4,

Elsevier Science Pub&hem B.V. All rights reserved.

and

2 Fundamental macroscopic

information about molecular properties

is

based

on

surface pressure measurements. In general, information on the phase behaviour of amphiphilic monolayers can be obtained considering the surface pressure (I) -

molecular

area

(A) isotherms at different

temperature range around 20 V. three-dimensional

systems,

temperatures,

Neglecting details

however,

transitions

mostly

in

the

similar to the conventional from gaseous via liquid

to

solid phases has been observed with increasing molecular density.

Fig. 1 shows

generalized isotherms for different

According to

the

results

designated as gaseous (G), transitions

temperatures (TI < Ta x T.).

of recent morphological studies, coexisting

the single phase regions

condensed (C) and solid

phases are expected,

(S) state.

are

For first-order

as represented by the coexistence

region (C t G) of Fig. 1.

----

gaseous-condensed(GC) coexistence region ~~:equilibrium surface pressure IT, : irreversible

monolayer

collapse

A Figure

1.

Generalized

I

- A isotherm of an insoluble

monolayer based on

results of the morphological structure.

Recent

structure information on monolayers obtained by the methods mentioned

above has phases using various

led to a more detailed understanding of the nature of monolayer Based on morphological studies although there are further questions.

the Brewster angle microscopy7-s and the fluorescence

microscopylo,ll,

growth mechanismsof the condensed 2D phases which can be

controlled

by the experimental conditions during the phase transformations of monolayers, have been reported.

3 Another cases

important

feature of insoluble monolayers is the fact that

in

of equilibrium.

In general,

a large numberof the I - A isotherms has a wide

region of supersaturation above the equilibrium surface pressure (HP). this

region

irreversible

is confined by the ESP and the “collapse monolayer collapse starts (see Fig.

saturation,

such

1).

pressure”

Thus,

where the

In the region of super-

the insoluble monolayers held at constant area or constant surface

pressure frequently display definite For

many

only a part of the continuous s - A curves is in a thermodynamic state

relaxation phenomenaaa-15.

studying these phenomenamonolayer instabilities as

rearrangement of the amphiphilic

due to other

molecules,

processes,

dissolution

of

the

amphiphiles in the subphase or evaporation, should be experimentally excluded. Over the last decade, phenomenaof monolayers.

there have been attempts to explain In this context, different

mation into a 3D phase have been discussed. been assumed to

aspects of the transfor-

For example,

be a transformation of the

such relaxation

the relaxation

has

homogeneous monolayer into

a

“heterogeneous monolayer/ collapse phase system” which is defined as a highly complicated process including nucleation, leading

nuclear growth, and/or mass transfer

to concentration and surface tension gradients and to

the

Marangoni

effect’=. Some approaches have been proposed to describe the kinetics

of such transfor-

mation processes theoretically. The idea to model the monolayer collapse

as a mechanismof

nucleation

growth of the formed nuclei was suggested by Smith and DergX7. Their however, was limited to the initial assumption of the additivity concept

has

stability

of nucleation rate and growth rate. of

pH on

A similar

the

monolayer

in dependence on t&We. to describe the monolayer relaxation is the

the Pro&-Tompkins equationaX based on chain nucleationX6~ag~ao.

adaption

description

of nucleation and growth only if overlapping

chains can be excluded, i. e. it can be applied only for the initial the overall

of

The critical

of the Prout-Tompkins’ law has shown that it can be applied

theoretical

to

the

of

the

period of

nucleation and growth processa’.

De Keyser and Joosz5 studied the collapse kinetics modifying

theory,

stages of the process and was based on the

been pursued to describe the effect

Another possibility analysis

and

the classical

of spread monolayers by

theory for homogeneousnucleation and they

developed

4

an interesting of

theory describing the nucleation rate as a function of the

the surface tensions at the solid/air

collision

and solid/liquid

sum

and the

frequency.

In the present paper, we survey the potential collision

interfaces

of our nucleation-growth

model for the monolayer transformation

(3D) phasez4-26. Based on commonkey features,

into a three-dimensional

two nucleation-growth models

have been reviewed: (i) a model for the limiting cases of nucleation and overgrowth with assumed geometric shapeas-ns, and (ii) the formation of lenticular For

the

formation

a generalized model for

centersz9-Bo.

of 3D centers at monolayers the nucleation

rate

can be

descibed by the nexponential law”. The two limiting cases of special interest, namely instantaneous and progressive nucleation, the

case

of overgrown centers,

the growth has been assumed to

shape-preserving way in compact geometric forms, ders. The consideration lenticular

have been concerned with. In

of the air/water/center

e.g.

in

a

three-phase contact result in

growth.

The free energy change accompanying the formation of the centers in brief.

occur

hemispheres and cylin-

Finally,

in the theoretical

section the effect

is

reviewed

of overlapping in the

succeeding stages of the growth process is presented. The main concern relaxation. effect

of the experimental part is

Examples of

constant

pressure

typical nucleation-growth mechanisms

monolayer

and of

the

of system parameters have been included.

NUCLEATION -GROWTH-COLLISION THEORY The present nucleation-growth collision description

of

theory is focused on the

theoretical

the transformation kinetics of 2D monolayer material

to

an

overgrown 3D phase which is experimentally applicable to monolayer relaxation at

constant surface pressures.

formation

The problem to be solved is regarded

as

the

and the growth of three-dimensional centers from a supersaturated In the following subsections, the main

monolayer at the air/water interface.

aspects of the nucleation-growth collision

theory are highlighted.

by treating the nuclei as a seperate phase, the free energy for the First, formation of 3D clusters is derived. Then, models for the limiting cases of

5

nucleationand growthwith assumedgeometricshapeare presented. Further, a generaltheoryfor the formationof fluid lenticularcentersfrom a supersaturatedmonolayeris reviewed. Finally, the overlapping effectof the growing centersis demonstrated by comparisonwith unconfinedfree growth.

Free Energy for the Formation of a 3D Cluster

The free energy changeaccompanying the formationof a 3D clusterfrom the 21) monolayermaterial, AG(i) = i AG-

G(i),

can be expressed bySo

t AF

(I)

where AG- is the free energychange for the transformation of the monolayer parentphase to the nucleating3D phase,permonomer,i the numberof monolayer moleculesin a clusterand AF the free energyof the formationof the clusterto-surroundings surface. AP dependson the shape of the growingcenters. In all examples, the free energy of the formationof the cluster-to-surroundings surfacehas to consider the area which appearswhen the clusteris formedand thosewhich disappears in the course of its formation. Introduction of the normalizedfree energyof the formationof the cluster-tosurroundingssurfaceAFn AF

= A&

ja/s

(2)

leads to aG(i) = ilrG_ t i'/"AF,,

(3)

A maximumin AG' = G(i") occursat the criticalsize i" of the clusters.Hence follows d AG

(-)+-i* di

= 0

(4)

6 Inserting Eq. (3) into Eq. (4), it results 8 i”

AP,’

=

(5) 27 l&G-I=

4 AFn= bG” = -

(6)

27 aG,= As substantiated elsewhereso, AG- can be defined as the supersaturation,

S, of

the monolayer and thus be related to the stationary nucleation rate, k,.

(i) Overgrown clusters with assumed geometry (Fig. 2) : In this case only a new air-cluster

area is formed during the

growth.

the free energy of the formation of the cluster-to-surroundings

fore,

Theresurface

is A

F =

(la=

ha

(7)

- Oew &.w

For hemispheres: LaE = 2 I ra AF

and LW = rra

(8) (9)

= x rz (2 0-a - (I,)

For cylinders: Aa, = 2 x r (r/2 t h)

AF

(ii)

and A,.,, = x P

= 2 x r (r/2 t h) uao - I ra oaw

(11)

Lenticular clustersas,~o

The shape of the asymmetric lens, is

(10)

determined by

i.

e. of the two contact angles a1 and c#B=,

the three interfacial

tensions a_,,,, alo and cro

at

the

The expressions of the respective cluster/air/water three-phase contact. interfacial areas have been derived from the geometric conditions of an

7

asymmetric lens (Fig.

3).

R,

and 1&, are expressed by the contact angles &

and #* and by the radius, r, of the contact circle

of the lens.

Am? = x P,

2 I r2 (1 - cos &)

2 x rz (1 - co6 cp+) Lc

=

I

AWe= =

OF

= UPC L-z

+ UVXJLa

(12) siPg,

sin=*,

-

(131

oar IL,

Our general nucleation-growth collisiontheoryfor the formation of fluid lenticular centers from a supersaturated monolayer provides a theoretical possibilityto calculatenucleationrate constants, L, for differentsurface pressuresof a monolayer. Bence followsthat the free energyof formation of the criticalnucleus-to-surroundings surface,A F', the criticalsize of the nucleus, i”, AG',

and the free energyof the formationof the critical nucleus,

This work is in progress. can ba determinedSo.

General Aspects of the Nucteation-Growth-Collision Theory The nucleation-growth collisiontheOr’p~go

quantitatively describing the ZD-

30 transitionfor isothermsin the stateof supersaturation is based on three nucleationand subsequentgrowthof the nuclei, key features: i) single-step ii) the total relaxationrate as convolution of nucleationrate

and growth

rate, and iii) overlappingof the growingcentres. The formationof 3D centersat monolayershas been lookedupon as single-step nucleationso that the nucleationrate has been describedby the %xponential law". The two limitingcases of specialinterest, namalyinstantaneousand progressivenucleation, have been concernedwith. In a first approximation,the growthof the centershas been assumed to occur in a shapa-preserving way in compactgeometricforms, e.g. hemispheres and cylinders'". If the air/water/center three-phase contactis taken into consideration, lenticulargrowth

must result for fluid ceutersag~So.

the theoretical descriptionof the transient2D-3D transformation lowing route of calcalution

To obtain the

fol-

has been taken: (i) geometric study of the growing

8 center, total

(ii)

growth of a siugle muter,

centers,

law of aucleus formation, overlap of the

(iv)

growing

and (v) solution for the respective model.

Model fox the Limiting Cases of Nuhatioo In

(iii)

size of all centers for unconfined free growth,

aad Overgrowth with Assumed Shape

case of overgrown centers onto monolayers only a new air-cluster

area

is

formed during the growth (Fig. 2). There is

a uniform probability

nucleation law is of first N=Nm.x

of

forming nuclei

with

time.

Thus, the

orderf6sas (14)

(1 - expuw)

where N is the maher of nuclei at auy l&or, N-

the pot&Sal

uucleus-

forming sites aud L the nucleation rate constant. Two limiting

cases of special interest have been considered”~aD:

instantaneous nucleation for large k, N=Nmmx

progressive

(15)

nucleation for mall

k,

subphose oir ~

monolayer 2r

subphose Figure 2. Qvergrowth centers with assumed shape on au insoluble monolayer. The arrows indicate edge growth.

9

Based on these limiting cases of nucleation and the concept of centers of assumed geometric shape (Fig.

overgrown 3D

2), the following general expression

has been derivedas.a7 A,, - A

-

= 1 - exp(-KlJt - L)“)

(16)

Ao - R where A is the total monolayer area, A0 the initial

monolayer area, and k the

monolayer area for t + m. The exponent x of time, quantity

allowing us to determine the particular

t,

is a

nucleation

characteristic

mechanism which

describes the experimental data best. A specification

of

the

time exponent and of

the

overall

transformation

constant IL for the nucleation-growth model with assumed overgrown centers

is

given in Table 1”. Table I. Specification

of time exponent and overall transformation constant of

the nucleation-growth model with assumed geometric growth centers Exponent

Model details Geometry

Nucleation

K,

X

of centres 4rd”‘(2M/q)=‘=N,Ukl”= Hemisphere

Instantaneous

312 3n,,

81 Hemisphere

Progressive

512

-

d”= (2M/q) =“kz,N~k~“’

15n,, rhMN_kr = Cylinder

Instantaneous

2 n,- 9 rhMkzzN-k.’

cylinder

Progressive

3 3n,- 3

I

- molecular weight, g - molecular density,

nucleation rate constant,

kl,l.,

d - molecular diameter,

- growth rate constants, N-

of nuclei, n,- - total numberof material to 3D centers.

molecules

transformed

k,

-

- total number from monolayer

10

As can be seen in Table I, from each

other

the respective nucleation-growth mechanismsdiffer

by the time exponent x.

determined by the best fit

The prevailing

mechanism can be

of the experimental data evaluated by

the

least

squares approximation.

Generalized Model for the Formation of Lenticular Centers When a

fluid center is formed from an insoluble monolayer at

interface, conditions

the

shape of

of the nucleus/water/air

asymmetric lens. center

geometric

the new phase is

the

air/water

determined

by

the

three-phase contact and has the form of an

The contact angles and,

are determined by the interfacial

thus,

the shape of the

lenticular

tension at the three-phase

contact

(Fig. 3) applying the Neumann-Young eguationsS1 * =a.

Figure 3. henticular center formed by collision the air/water interface

from an insoluble monolayer at

for 0-v = 42.3 mNm-+, u,, = 22.1 mNIF,

and CL, =

33.6 mNm-l.

For

asymmetric lens growth,

a general solution of the exponential

nucleation includes the complete range from the totally (k,,t + 0) to the totally

law for

progressive nucleation

instantaneous nucleation (k,t + -) .

11

The final

result is

Ao - A Ao - k

= I - exp ( -c t=/= F(k&))

summarizing all constants 4

M -

C= -r( 3

(17)

by

N)X/I dS/a kW’ -

(Is)

II,-

?G

The geometry factor G defined as G = f(@l) the

lens from the spherical shape.

approximatively

+ f(&)

expresses the deviation

The function F&t)

by series expansion.

of

has been calculated

Its graphical representation

has been

shown elsewherezs. The two limiting

cases of nucleation considered in the models with growth in

assumed geometric shape follow

directly

from the properties

of the

function

F(k,t) : instantaneous nucleation lim F(U) k”k-8 -

= 1

(19)

Ao - A

-

=

1

-

exp

( -c

p/a)

(20)

Ao - k progressive

1i.m %f-‘0

A0 -A

A0 -R

nucleation F&l&) -=tt

2

(21) 5

2 = I - exp (- - c lC&=‘=) 5

(22)

12

Sguatious (17) and (la) allow the calculation

of the nucleation rate constant,

k,, from the data of moaofayer relaxation at constant surface pressure. For an application

to experimental data, it is noteworthy that besides the growth

rate constant, k, the constant C also comprises the geometry factor,

G, and

the mean numberof molecules contained in a center (Nnwc/&,_),

Overlapping Effect of the Growiag Centers In the succeeding stages of the growing process the centers cannot grow freely in

all directions

becomes limited.

since they ispiage on each other. The overlap of ~ee~nsional

growth centers

ly nucleated on the monolayer is symbolized in Fig. two centers are overlapping,

The size of the

centers

progressive-

da. In the hatched areas,

and in the dark regions, three centers are over-

lapping. According to Avrami~~~~.L,the calculation based on a splitting

in different

L,.~

as the fractional

is

defined

Statistical

of the true normalized volume, 2, is

classes of growth volumas (Fig.

approaches based on the

voluma by at least geometrical

4b).

Here,

n growth centers.

representation

of

these

extented volumes lead to (33)

exp ( -zx.uro)

z=lwhere z~,-~

is

the normalized volume of

freely

growing centers

without

nucleated

On the

overlap.

0

Figure 4. a) The overlap substrate:

of

three growth centers

progreSSiVely

uuhatched areas - Sx; hatched areas - L;

b) The component parts of &,,+, Sl,-C, and %,-t centers shown in Fig. 4a: unhatched areas - SI,~; dark area - S.,rXh.

dark area - SS. for the overlap of growth hatched areas - Sa,-t;

13

The unconfined

free

growth of the centers cau be expected

in

the initial

stages of the formation of 3D centers from a superaturated monolayer.

For the

nucleation-growth mechanismsdiscussed above we derive the respective

expres-

sions for free growth. The true volume in normalized form,

z,

cau ba expressed by the

experimental

data of the area relaxationas Ao - A

n, z=-_=-

(34) Ao - A..

n,-

The normalized volume of freely growing centers is defined asa6

nng Z1,wct

(35)

= -

n,and put in parentheses of the solutions for the models discussed above and

(16)

(17)).

Thus,

this expression is necessary to

derive

the

(hgs. models

considering the overlap of the growing centers. On the other hand, without the

the expression of a respective

nucleation-growth mechanism

overlap can be derived by series expansion of the final solutions

above models considering the overlapping effect.

For

small

of

exponential

terms the series can bs ruptured after the second term. The final result for unconfined free growth is (i)

for the model with the limiting cases of nucleation and growth aud assumed geometric shape Ao - A -

= Kx ( t - t,)x

(36)

Ao-R (ii)

for the generalized model for the formation of lenticular

centers

Ao - A

-

A0 -A..

=

C

t=‘= F&t)

(27)

14

The experimental profile data

data

of a typical

are shown in Fig.

5.

constant

surface

pressure

relaxation

The solid line represents the best fit

of the model with overlapping growth and the dotted lines for

to

the

the

same

model without overlap.

A/A,

o -

experimental

data

1 -

overlapping

centers

-

freely

growing

centers

0.75

0.5

0.25

0 0

Figure

200

Relaxation

5.

progressive

of

600

400

a

stearic acid monolayer at x

t/s

= 35 mN m-l

as

nucleation with hemispherical growth.

Applying an adequate nucleation-growth mechanismwith both freely growing and overlapping stearic with

centers

the fits

to the experimental data of the

acid monolayer demonstrate that, the

experimental

succeeding stages, however, an excellent mechanism to growing

as expected,

data in the initial

stages of

fitting

both models the

of

coincide

process.

In

the

of the true nucleation-growth

the experimental data is only obtained if the

centers is considered.

relaxation

On the other hand, the

overlap

of

corresponding

the model

derived for freely growing centers deviates increasingly.

EXPERIMENTAL RFWJLTS AND DISCUSSION In

this

section,

a

survey is given on the experimental

adequacy of the nucleation-growth collision

theory.

evidence

for

the

15

systematic information on the true nucleation-growth mechanismcan be obtained Constant surface pressure relaxations

by appropriate data analysis. satuated monolayers of different metastability For

the

of super-

amphiphiles have been studied over the

whole

rangea6~a7~S5.

model with assumed geometric growth centers

the

nucleation-growth

mechanism can be selected from those presented in Table I by the best fit the

experimental data.

As representative

constant surface pressure relaxations Pig.

6.

example,

of stearic

the results of

acid monolayers are shown in

The whole range with constant surface pressure relaxation

described

best

by progressive nucleation with

of

different

hemispherical

can be

edge

growth,

although the relaxation rate varies considerably with time.

1

A IA0

0

Figure

6.

1000

Progressive

change vs. time for stearic To obtain relaxation

2000

tls

nucleation with hemispherical edge growth for

area

acid monolayers on pH 3 water.

a convenient form for the presentation of data

3000

the

constant

pressure

Eg.(16) may be transformed as follows 1 ))%I% = &=‘“(t

(ln (

- t,)

(26)

1 - (A0 - A)/(Ao - A-) since straight t.

- A-))])‘/”

vs.

The constants K. and tl of the linear Eq.(28) can be determined by

lines can be obtained by plotting

(ln[l/(l-(A,

linear

16 regression. = 5/2

In Fig. 7 it has been demonstrated that, with the time exponent X

the experimental relaxation data of stearic acid

excellent

straight

lines due to progressive

monolayers

nucleation

with

lead

to

hemispherical

growth.

3-

3OmNm-’

4-34mNm-1

6 -4OmNm-’

0

Figure 7. Sq.

1000

2000

3000

t/S

Progressive nucleation with hemispherical edge growth according

(28) for constant surface pressure relaxation of stearic

acid

to

monolayers

on pH 3 water.

Sased on the constant

experimental relaxation data of

surface

pressures,

models discussed above.

stearic

acid

mcnolayers

a comparison can be made for applying

Somedata are compiled in Table II.

the

at two

17 Table II Overall rate constants, L rate constant, t,

(Eg.(16))

and 2/5 C k, (Kg. (17)),

and nucleation

obtained for progressive nucleation with assumed hemispher-

ical growth or lenticular

growth from supersaturated stearic 215 C k,

acid monolayers

x/mNm-l

L/s

IGC(x = 5/Z)

33

424.5

1.022*10-7

1.553’10-7

1.396*10-=

35

249.3

3.020*10-’

5.496*10-’

2.502*10-’

40

164.8

8.773’10~’

1.892*10-=

7.070*10-’

‘x - surface pressure, IL,,- inflection K - overall

k,

point of the A - t relaxation,

rate constant, x - time exponent, C - asymmetric lens

growth constant, k,, - nucleation rate constant As already shown in Figs.

6 and 7,

the model with the

assumed geometrical

growth centers leads to an overall constant IL for progressive nucleation hemispherical

The application

growth.

formation of lenticular k,

(limiting

calculated

case:

of the

generalized

centers provides values for an overall progressive nucleation)

and k,.

and

model for

the

constant 2/5

The overall

constants

by the two models agree reasonably in the order of magnitude. Hence

it is evident that the approach based on the limiting cases of nucleation assumed growth centers describes the monolayer relaxation able

C

way.

The model with a general solution of

the

and

in a rather

reason-

exponential

law of

nucleation and lenticular

growth centers corroborates the limiting case of the

progressive

The respective

calculated energy

nucleation. providing

the

basis for further calculations

of formation of the critical

critical

size of a nucleus,

critical

nucleus,

Most of

AG"

nucleation rate constants

i’,

such as

nucleus-to-surroundings

and the free energy

have been the

free

surface, r’F*, the

for the formation of

a

so.

the relaxating monolayers of amphiphiles can be described

best

by

these two mechanims. There are also monolayers the relaxation of which follows the other case of nucleation,

i.e.

limiting

very high nucleation rate constants imply instantan-

eous nucleationae*aS. An example for instantaneous nucleation is presented by the constant pressure

relaxation

Fig.

Information

8.

of 2-docosylamino-5-nitropyridine on orientation

and structure

surface

monolayers as shown in of

2-alkylamino-5-nitro-

18

pyridines

multilayers

promising

canditates

has been of special interest for integrated optics.

constant surface pressures in Fig. metastable

8,

as

they

the monolayer relaxation

monolayer range can be described by ins~n~~us

be

selected

of the

whole

nucleation

with

hemispherical growth although the transient relaxation profiles tive surface pressures differ

seem to

As demunstrated with

of the respec-

considerably from each other.

~

--_ ~

2

f

/

3 +----~_-_--+----_+-_-

I I

I I

I

I’ ! I

/ I ,,

1 I I I

--_-

t

I

II

i

1 i i I

-___-I___

_I.

---_-_-_I--_-_-_-_

T1

I

I

I

1 - 5 mNlm 2 -10 mNfm

I 1

i

4 -20 3 -25

2000

1

I I

4000

mN/m

6000

8000 tlS

Figure

8.

~ns~ntan~us

nucleation

constant surface pressure relaxation layers on water (20 V).

It

is

interesting

with he~spherical

Age of the spreading solution:

a

best

surface

mono-

1 hour.

the relaxation rate of the

(lo-”

m

ItIOnOlayer

Nevertheless, the experimental data of 2-dOCO~lamino-5-nitrO-

pyridine monolayers attained by spreading solutions of different fitted

growth for

to note that the age of the spreading solution

CHclS) although stored at -15 Oc effects substantially.

edge

of 2-docosylaeino-5-nitropyridine

age have been

by the instantaneous nucleation mcdels demonstrated in Pig. 9 for pressure of 10 mNBfl.

After a storage time of about one

days, however, the relaxation profiles

reach the final

fore.

to

two

19

0

1000

3000

2000

4000

5000 t/s

Figure

9.

Area relaxation at 10 mWm-l

of

2-docosylamino-5-nitropyridine

monolayers on water using spreading solution of different

Without going into detail,

age.

it should be mentioned that the monolayer relaxa-

tion is highly influenced by different

process parameters,

e.g.

the temperat-

ure and the components of the subsolution (pH, counterions)l’. Finally,

an experimental example is shown how to effect

the relaxation

arachidic acid monolayers by kind and valency of the counterions of the As presented in Fig. < Wg”, nucleation,

10, the order of increasing stability

rate of bulk.

is II+ x Li+ < Cal+

but yet in all cases the prevailing models are based on progressive in the presented example calculated with hemispherical growth.

20 7.0 AlAo 0.9

0.8

0.7

0.6

0.5

OX

0.3

0.2

Figure

Progressive

10.

relaxation different

I

I

I

3000

2000

1000

tis

nucleation with hemispherical edge growth for

of arachidic acid monolayers at x = 40 mNRI-+ on subsolutions

the of

10 ml4counterions.

CONCLUSIONS The relaxation above kinetics

of mouolayer isotherm

occuring in the su~r~t~ation

the sguilibriunt surface pressure can be regarded as the of

two-dimensional monolayer material to

The nucleation-growth collision quantitatively

transformation

three-dimensional

theory developed by us has the

describe the 2p - 3D transition

region phase.

potential

to

in nmnolayers.

The theoretical

concept is based on the following main features (i) single(ii) the total transforstep nucleation and subseguent growth of the nuclei,

mation rate

as convolution of nucleation rate and growth rate,

overlapping of the growing centers.

and

(iii)

21

The nucleation rate (kn) is

described by the Vxponential

law”.

cases

are instantaneous nucleation

for

of

special

progressive

interest

nucleation

for

The effect

small t.

growing

centers has to be considered theoretically

of

transformation kinetics

the

limited

in

the

excellent centers

of

overlap

k,, and

between the

in the succeeding

as the free growth of

succeeding stages.

Two limiting large

the

stages becomes

centers

Experimental evidence can be

given

agreement with the relaxation data when the overlap of the has been considered,

deviations

whereas in the

succeeding

have been encountered for theoretical

stages

models with

by

growing

increasing

freely

growing

centers. We have developed

two theoretical

models

(i) for

the

limiting

cases

nucleation and formation of overgrown 3D centers with assumed geometric and of

(ii)

for nucleation according to the exponential law and the

shape

formation

lenticular

formation

centers. For both models, the free energy accompanying the of a 3D cluster from the 2D monolayer material and the respective

formation of the cluster-to-surroundings

In the second, the

of

surface has been derived.

more general model, the effect

three-phase

contact

air/water/center

of the interfacial on both

asymmetric lens has been taken into consideration. range from totally

contact

tensions

angles

Furthermore,

progressive nucleation to totally

of

at the

the complete

instantaneous

nucleation

has been described for asymmetric lens growth.

The general nucleation-growth collision lity

to calculate

surface

pressure

calculations,

theory provides a theoretical

the nucleation rate constants from experimentally relaxation

data.

Thus the basis

surroundings surface,

the critical

the formation of the critical

size of the nuclei,

given

theory.

for

constant further

nucleus-to-

and the free energy for

nuclei.

Experimental evidence is provided for the adequacy of collision

is

such as the free energy of formation of the critical

possibi-

the

nucleation-growth

The approach based on the limiting cases of nucleation

assumed growth centers

describes the monolayer relaxation in

and

a reasonable

way.

Experimental monolayer relaxations

are found for both progressive and instan-

taneous nucleation,

but most of the experiPranta1 examples follow the limiting

case of progressive

nucleation.

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unpublished data