Description of monolayers of discotic molecules at air-water interface with spin one models including vacancies and nesting of pairs

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Description of monolayers of discotic molecules at air-water interface with spin one models including

vacancies and nesting of pairs

M. Banville, A. Caille, G. Albinet

To cite this version:

M. Banville, A. Caille, G. Albinet. Description of monolayers of discotic molecules at air-water

interface with spin one models including vacancies and nesting of pairs. Journal de Physique, 1985,

46 (1), pp.101-107. �10.1051/jphys:01985004601010100�. �jpa-00209938�

Description ^of monolayers of discotic molecules ^at air-water interface with spin ^{one models} including vacancies and nesting ^of pairs (*)

M. Banville, ^{A. Caille}

Département ^de Physique, Université de Sherbrooke, Sherbrooke, Québec ^J1K 2R1, ^Canada and G. Albinet

Département ^de Physique ^des Liquides, Université de Provence, 13331 Marseille Cedex 3, France

(Reçu ^le 9 juillet ^1984, accepté ^{le 12} septembre 1984)

Résumé.

Nous proposons des modèles _{de gaz} sur réseaux pour essayer de décrire les isothermes de compression

de monocouches constituées de molécules de benzène-hexa-n-pentanoate (BH-5) disposées ^sur support aqueux.

Ces modèles sont résolus par la méthode de Bragg-William ^{et ils} permettent ^{de tenir} compte ^des lacunes, ^d’une réduction d’aire du coeur dur moléculaire suite à un emboîtement de paires. ^Nous considérons plusieurs ^méca-

nismes de formation des paires donnant des transitions tantôt du premier, tantôt du second ordre.

Abstract.

Spin ^{one models} based upon ^a lattice Bragg-William’s ^{method are} applied ^{to the} description ^{of the} surface-pressure isotherms of disc-like molecules of benzene-hexa-n-pentanoate (BH-5) forming monolayers ^at

air-water interface. The models include vacancies and consider a reduction of hard core area due to nesting ^when pairing ^{occurs. Various} assumptions about the way pairs ^{form lead to different models} exhibiting ^either first order

or second order phase transitions.

Classification Physics ^Abstracts

87.20C

1. Introduction.

Disc-like molecules of benzene-hexa-alcanoates have been shown [1] ^to form stable Langmuir monolayers behaving ⁱⁿ many respects ^as monolayers ^of fatty ^acids.

The benzene ring plays ^the role of the polar ^head group while the alcanoate chains that of the long aliphatic

tails. These results indicate that the benzene rings ^lie

flat at the interface. The projection of the molecular diameter onto the interface is practically ^{identical to}

the lattice spacing ^{measured in the} liquid-crystalline

columnar mesophases ^{formed in bulk whenever} they

exist [2]. ^The area-pressure isotherms exhibit the well known three states characteristic of liquid-condensed, liquid-expanded and gaseous films according ^to

surface concentrations and temperatures.

Comparison between the lengths of the alcanoate chains deduced at equilibrium spreading pressure 1te and from molecular models show that the chains

are not fully ^{extended at least} beyond the fifth or

sixth methylene group and that there is a certain

chain rigidity ^which prevents ^{the first} methylene

groups ^to stand directly upright ^{For BH-5, there is}

some indications that their fully ^extended length

is about 1.5 times their length ^at 03C0e.

A lattice model proposed by Bell et al. [3] ^with

two types ^of particles differing by their orientations

including ^vacancies gave ^a second order phase

transition but did not explain ^the very large ^com- pressibilities after the transition as we compress the system.

The purpose of the present ^{work is to offer an} explanation ^{for the} very large compressibility ^after

the main transition as we compress monolayers

of BH-5 molecules. We propose spin ^{one models}

similar to Bell’s [3] ^{and we assume} further that an area reduction occurs when pairs ^{are formed in some}

relative orientations [4]. We consider here two diffe- rent models which differ by ^the way pairing ^occurs

between molecules. In part ^{2 we} will discuss the fea- tures common to the two models followed by ^a description of their differences. The first model,

called model 1, ^is based upon ^an assumption ^for pair

formation that is not symmetric ^with respect ^{to the} change ^of sign ^{of the so-called} «magnetization»

order parameter, (to ^{be defined} later) hence leads (*) This research was partially supported by the Natural

Sciences and Engineering ^Research Council of Canada, ^and

le Fonds FCAC of Quebec.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01985004601010100

102

to a first order transition. The second model assumes a symmetric ^{mechanism as} in Bell’s model allowing

for a second order phase transition. In part 3, ^the parameters ^{for the two models are} determined to

give ^{the best} agreement ^{with the} experimental data;

the results are analysed ^and compared ⁱⁿ part ^4.

The experimental area-pressure isotherms for

benzene-hexa-n-pentanoate (BH-5) used for this study

were measured [5] by the movable barrier method for several temperatures between 0 °C and 28 °C ; they

are shown on figure ^1. Important relaxation times

were observed near the main transition especially

for the lowest temperatures. ^{A real} plateau ^{is never}

observed after the transition although ^{it is} nearly

so for the lowest temperatures.

Fig. ^1.

^Surface pressure isotherms of benzene-hexa-n- pentanoate ^for temperature 1) 0.6, 2) 5.4, 3) 10.4, 4) 15.0, 5) 20.9, 6) 24.9, 7) 27.2, 8) 28.6, ^and 9) ^{31.5 °C.}

The experimental procedure ^{used to} observe the relaxations in the transitions is the same as the one

described for fatty acids in reference [6].

2. The models.

We consider models in which rigid ^{molecules are}

assumed to be in one of two orientations with respect

to the triangular ^lattice depicted ⁱⁿ figures ^{2. The}

lattice is introduced for the ’purpose ^of facilitating

the counting ^of configurations ^{and to take into}

account some of the local order; ^{it is} important

to keep in mind that there exists no substrate lattice.

The lattice sites also approximately ^{take into account}

the hard core repulsive interaction between molecules.

Molecules in the first orientation, ^{called a} have chains

pointing ^towards the middle of ^{the lines} joining ^the

lattice points, those in the second, ^called P, ^have

chains pointing ^to the lattice points.

Vacancies or unoccupied triangles ^{are called} y.

An index i

1, 2, ³ will be used to designate ^the

states a, 03B2 ^and y respectively. ^A given configuration

for the system ^{will be} specified by the numbers N1 ¹

and N2 of molecules in the two orientations and the number N3 ^{of vacant sites.}

Provided we assume that the extra entropy ^due

to the _presence of molecules in the orientation P

does not stabilize a mixed configuration, ^the fully

condensed state will be formed when all the mole- cules are in the orientation a. At other concentra-

tions, vacancies, a-a, a-fl, ^and p-p pairs ^{will be} present The area for the vacancies will be A3, and A1 = A2

will be the area of isolated molecules in either orien-

tation ; differences in hard core area are assumed to appear when pairs ^are formed due to nesting ^of

the radially extending ^chains leading ^{to an area}

reduction 03B4Aij ^{when a} molecule of type i pairs ^with

a molecule of type j.

The total area for a given configuration ^{of N}

molecules is

where J is the _average area per molecule and Nij

are the number of i-j pairs averaged ^{over a} statistical ensemble of configurations characterized by N19 N2, N3. We work with a statistical ensemble where the number of molecules is fixed but the area varies;

this is the isothermal-isobaric ensemble [7]. ^The probability Pi ^of occupation ^{of a} given site in the ith state is given by

not taking ^{into account} correlations. Here, ^z

^{6 is} the number of sites occupied by ^a molecule and

N.,

^{zN +} N3 is the total number of sites. Simi-

larly, the numbers of pairs ^are given by

where ^z

6 is the coordination of a molecule in that case.

The models being analog ^{to an} Ising spin ^one system, it is desirable to introduce order parameters ^M and Q ^{that are the} analogs ^{of the} magnetization ^and

the proportion ^of occupied ^sites respectively ^as

The probabilities expressed ^{in terms of M and} Q

are

and the area per molecule is

The isotherm-isobar potential [7] describing ^the equilibrium ^{of this} system ^{will be}

where p ^{is the chemical} potential per molecule whose minimum with respect ^{to M and} Q at constant and T determines the equilibrium configuration, H;n, is the energy of interaction between pairs

and S is the entropy ^{of the} system

It is convenient to introduce factors Uij ^{that will}

be ^{used to} define different models involving ^a single adjustable parameter ^{K such that}

When the area reduction terms in [8] ^{are included}

with the interaction terms, the interaction is seen to be renormalized; ^{we define a} renormalized inter- action

where To ^{is room} temperature.

We rewrite (7) ^{in terms of the} probabilities Pi

The _pressure isotherms _may be obtained by ^eva- luating ^from (6) after the values of M and Q ^mini- mizing (12) ^{for fixed x and T} have been found

The first model, called model 1 is based _upon the type ^of pairing depicted ^on figure 2-a, ^a-a pairs

share 4 alcanoate chains, Lx-P pairs share 3 chains and fl-fl pairs ^share only ^{two hence the a-a Van der}

Waals bonds are expected ^{to be} stronger ^than a-03B2

bonds and a-fl ^bonds stronger ^than P-P ^{bonds. The}

model is rather insensitive to the exact relative values of the interaction parameters Kij ^{as will be explained}

in the discussion, ^{hence we select a} priori ^the following

relative values U11 ^{= 3,} U12 = ^2, U22

^{1. We}

assume no area reduction (03B4A22

0) ^when P-P pairs form and the same area reduction 03B4A 11

03B4A 12

for a-a and cx-P pairs; ^{this is} suggested by ^{a crude} geometrical ^{model of} nesting ^{and this} assumption

will be later justified in the discussion.

For this model, ^the potential (12) ^{is not an even}

function of the ^« magnetization ^{» M since} K 11 # lz 22

and we can immediately predict that it will lead to a

first order transition.

Model 2 is based _upon the pairing depicted ^on figure 2-b; ^a-a pairs ^and P-P pairs ^{share 4 alcanoate}

chains and cx-p pairs ^{share 3} chains; pairs ^{of similar}

molecules will have a higher ^Van der Waals inter- action and will exhibit the same area reduction hence

we set U11

U22 ^and A11

6A22- ^{We further}

assume no interaction ( U 12

^{0) and no area reduc-}

tion (ðA12

0) ^for Lx-# pairs; ^this assumption ^is

seen in part ^{3 to} be consistent with the adjustment

of the parameters ^that give the best fit to the experi-

mental data. This model gives ^a potential (12) ^which

is symmetric under the inversion of the sign ^{of M}

Fig. ^2.

Configurations of BH-5 molecules a and {1, ^and vacancies y showing ^various types ^of pairing ^{on a} triangular

lattice : a) ^for model 1, b) ^{for model 2.}

hence second order phase transitions are possible

with this model.

104

3. Effect of the adjustable parameters.

3.1 MODEL 1. - For model 1, ^the isotherm-isobar potential (12) is ^{seen to} ccniain four adjustable para- meters, namely A1, A3, Ö II ^{t and} KlkB To. ^{We will}

select KlkB To ^{in such a v ay as to} obtain the critical temperature ^{around 28 °C. Now there are a number} of features in the experimental surface-pressure ^iso-

therms that can be used to determine the three remain-

ing parameters A 1, A 3 ^and 6A 11. ^We depict ^on figure ³

Fig. ^3.

Characterization of the surface isotherms at critical temperature by ^{the critical area} (Jc and critical pressure 1tc, and the extremities Qp ^{and am} of the coexistence

region ^{at some} temperature ^T.

the parameters nc, ac and Tc that characterize the critical point, the limits ap ^and am of the coexistence

AT A7T

region, ^the variations - # ^and An ^{near a given}

temperature lower than Tc; ^Aa

am - ap, ^and An is the rise in _pressure in the coexistence region

for a variation AT of the temperature. ^{We also exa-} mine Anm/Aam, ^{the slope} of the coexistence line.

The experimental ^{data on} figure ¹ suggest ^the

following ^values for those parameters : nc

10 ± ¹ dynejcm, Ue

^{150 ± 5} A2, Tc

^{28 ± 5 °C,}

ap =130±10A2, am =250±5A2 ^{A a} 4.5+

up = 130 :f: 10

^m

2iT

A2 ^An A7rm

0.5 JOC, ^" N-T

^{0.12 +}

^0.02 dyne/cm. OC, ^{Y /}

^and Aorm ^{OQ m}

-

0.05 ± ^0.002 dyne/cm. ^A2.

The area cho per molecule in the fully ^condensed

state can be obtained by setting P,

1, P2

P3

⁰ yielding

When the _pressure isotherms are calculated

requiring the critical temperature ^{to be} Tc

^{28 OC,}

the critical _pressure is numerically ^{seen to} depend

almost exclusively upon the ^area A 3 of the vacancies :

We take 1tc

¹⁰ dyne/cm ^{from the} data, ^hencf A3

^{9.7 A2 from} (14).

We also notice that the critical area (J c depends

almost entirely upon the values of A1 ^and ao. When

we require ^{the fit to be best near} condensation, ^we

find the approximate ^relation

to hold, ^where Qo is defined through (13) and the value of _Jo

80 A2 is nearly independent of the values of

A1 ^and 6A 11 . We take the critical area ac

150 A2

from the data, ^hence A1

^{115 A2 from (15), and} 8A 11

^{11.6 A 2 from} (13). There is another relation that is seen to hold between the adjustable para- meters from the constraint T,

^{28 °C; it is}

that determines uniquely the interaction parameter

kB K To = 1.116, ^{using A3 =9.7A 2} and 03B4A 1 1 = 1 1 .6 A2.

A slight readjustment ^{of those} parameters pro- vides a better fit with the width of the coexistence

region ; ^{we show on} figure ^{4 the} pressure isotherms

Fig. ^4.

^The model 1 area-pressure isotherms for T =0 °C to 28 °C by steps of 4 °C for the following ^{values of the} adjustable parameters : A ¹

¹²⁵ A’, A 3 ^{= 9.7} A2 ,

for T

0 °C to 28 °C by steps ^{of 4 °C} calculated with the parameters A1

¹²⁵ A2, A3

^9.7 A2, ball

15A2, kB K To

^1.092. The limits of the coexistence

B 0

region ^{are i}

0.132 dyne/cm.

in good quantitative agreement ^{with the} experimental

d Au .

in d I.. a reement but the

data, ET- ^{IS in} good qualitative agreement ^{but the}

A7r.

d ^{2 ,}

slope e Q - - !.lam ^0.0208 dyne/cm. A ^{is too small} by ^a

factor 2 and the area of the vacancies compared ^to A1/z, ^{the area per} site, ^{is too small} by ^{a factor 2.}

The examination of the _way the populations ^of

molecules and pairs vary ^{as we} compress the system clarifies the nature of the transition. ^{We have shown}

on figures ^{5 and} 6, the variations of N 11 ^and N 12

Fig. ^5.

Variation of the number of pairs ^{of the same} type N11/Ng

with surface pressure for model 1.

Fig ^6.

^Variation of the number of pairs of different type N,21N. with surface pressure for model 1.

with _pressure. At the critical point, Z 2 1 N2 - ^{= 2.25 and} N. 2013

^{5.62; the degree of ordering is important as}

well as the proportion of vacancies. At a temperature

N N3

near 0 oC, ^at N ’ = ^{1.50, and} N3 ^{19.1 ; at up’}

N1 NN2 ^N

-

_3.07, and L3 - _{4 hence the} main transition

2013

^{3.07, and 2013}

^{1 54 ;} hence the main transition

N2 ^N

is dominated by ^the expulsion of vacancies. The

number of pairs ^at am ^are N 11 =0.011 and N 12 _

NS NS

0.014; ^at _03C3p N-11 0. 135 and N12

0.087; ^hence 0.014;

^at _up’ NNs ^{s 0.135 and} N12 - . N ^{s 0.087;}

^{he n ce}

the expulsion of vacancies is accompanied by ordering (i.e. : formation of a-a pairs); the number of a-fl pairs stays relatively ^{small as can be seen on} figure ⁶

and this explains why the model is not sensitive to the exact values of K12 ^and K22 ^as long ^as they ^are

smaller than K 11; ^{the same remark} holds for the difference between bAl2 ^and 03B4A11 ; the model is not sensitive to a variation of 6A 12

Similarly, ^{the nature of the} changes ^{in the com-} pressibility anomaly region ^between _up ^{and the area}

where the monolayer collapses may be clarified;

N1/N2 changes ^{from 3.07 to 4.8 from} up ^to collapse,

and N3/N changes ^{from 1.54 to 0.60, hence the} expul-

sion of vacancies is $till the dominant mechanism

compared ^{to the} ordering. ^{Near T}

⁰ °C, ^{the number}

of pairs N11/Ng increases from 0.135 to 0.26 and

N12/Ns increases from 0.087 to 0.12 as we go from up ^{to the} collapse ^{of the} monolayer; ^the ordering ^is accompanied by ^{the area} reduction. If we extra-

polate ^{the model to} pressures above collapse, 1V12/1Vs

starts decreasing towards ^zero past 20 dynes/cm

while N11I1Vs keeps increasing towards its limiting

value of 0.5 indicating ^{that the} fully ^{condensed state}

if is existed would be completely ^{ordered and not}

a mixed phase.

3.2 MODEL 2.

Model 2 gives isotherms with a very small slope after the transition as we compress

provided ^{we are close or} below the tricritical point indicating that the tricritical temperature ^is pro-

bably somewhere between 00 and 28 °C. There is a

possibility ^to go directly ^{from an} isotropic phase

to an ordered phase ^{with a} plateau having ^finite slope ^{or from the} isotropic phase ^{to the ordered} phase through ^a coexistence region ^{with a true} plateau.

In model 2 we also have four adjustable para- meters namely A 1, A 3, bA l l, ^and K/kB To ^{and we}

use the same criteria as for model 1 to find the values

giving the best fit to the experimental ^{data. It is}

seen that the slope of the critical line is quite ^insensi-

tive to the value of the parameters therefore there remains some freedom to choose bAll ^and K/kB To.

We have shown on figure ^{7 the} pressure isotherms for the following ^{values of the} parameters : A 1

150 A2, A3 = 25 A2, 03B4A11 ^{= 25 A2 and} K/kB To

2.0 (here U11

U22 = ^1, U12

^{0, and} bAl2

^0).

Equation (13) gives ^{a condensed area} ao

75 A2

in that case. The nature of the transition is clarified

by ^the figures ⁸ and 9 where we have shown the variation of N11 ^and N12 with pressure respectively;

in the expanded phase, pairs ^{of all} types ^{are in} equal

number for a given pressure and ordering suddenly

takes place ^{as a-a} pairs rapidly form while a-fl pairs

106

Fig. ^7.

^Model 2 area-pressure isotherms for ^T

0 OC to 28°C by steps of 4 OC for the following ^{values of the} adjustable parameters : A1 ¹

¹⁵⁰ A2, A3 ^{= 25} A2, bAi ^l

²⁵ A2, ^and K/kB To

^2.0.

Fig. ^8.

^Variation of the number of pairs ^{of the same} type N11/Ns

with surface pressure for model 2.

Fig. ^9.

Variation of the number of pairs of different type N12/Ns

^{with surface} pressure for model 2.

decrease in number leading ^{to almost} complete ordering ^near collapse ^{of the} monolayer.

The isotherms are closer together ^{near transition}

than observed, ^{we obtain} An/AT

^{0.075 in} quali-

tative agreement ^with experiment; e Qm/e T

0.76,

very much smaller than observed, ^and An m

2i (T-M

-

0.10 dyne/cm. A’, ^{half the} experimental ^{value. The}

area of the vacancies is equal ^to A 1/6, ^{the area} per site.

For a symmetric model like our model 2, ^{it is} possible ^{in some cases to have a} first order critical

point c 1 determined by the conditions a

p y

aQ

~203C0

(~203C0)

^{0 and M}

^{0 provided that such a point}

~2 T

does not lie within the coexistence region of the second order transition. The conditions for the critical point correspond ^to QC1 ¹

^{1 /2 and}

The parameters chosen for model 2 give ^{for that} point ^the coordinates T C = ^103.3 K, _7rcl

1.12 dyne/cm, ^and 03C3C1

^281.2 Ä2; ^{it is located well}

within the second order phase transition coexistence

region therefore it is not a stable state for this model.

Adjustable parameters necessary ^to locate this first order critical _{10.5 dyne/cm, and} point ^at TCt ¹

^{300 K, nCt 1}

O’Ct

¹⁴⁰ A2 require ^{such a} high

interaction that the system is condensed even at

03C0

1 dyne/cm. ^{Such an} interpretation ^{must be}

dismissed

It is seen that the two models give predictions ^for

Ay Ayr Aye-

that d.

^t: h

I

Au AT ^{and A7r m that} deviate from the experimental

AT’AT Aum

values in opposite directions; ^{the two models are}

in some sense extreme opposites ^{and we} might expect ^that intermediate models would give ^better

agreement ^{with the} experimental ^data.

We have examined other models not symmetric

with respect ^{to the} « magnetization » by allowing A2 ^to be different from A1 but the results deviate

even more from the observation. Other models exhi-

biting ^{second order} phase transitions with either

03B4A12 "# ^{0 or} U,2 :0 ^{0 or both # 0 have been exa-} mined ; ^the slope of the critical line is not significantly

affected and we do not improve ^{the fit} significantly,

the main effect being the reduction of am ; ^{one can}

in fact fit _am either at 0 °C at 28 °C but not at both

temperatures.

4. Conclusion.

The nature of the transitions given by ^{the two models}

considered is seen to depend upon the inclusion of vacancies as is made evident by (14) ^where A3

and K almost exclusively determine the critical and tricritical temperatures; reducing A 3 would make the critical _pressure to grow indefinitely. ^{On the con-} trary, lattice models with no vacancies exhibiting

transitions usually predict pressures much lower than observed

The introduction of an area reduction due to nesting

when pairs ^{form seems to} provide ^an explanation

for the large ^anomalous compressibility ^{with or}

without a true pressure plateau after the transition when a second order phase transition takes place.

Model 2 in fact is compatible with the observed relaxations near the transition; ^{moreover the area} A3

that must be assumed is compatible with the size of triangles in the lattice used for the model.

Neither model is completely satisfactory ^{as is}

shown by ^the large deviations in the slopes ^{of the}

coexistence regions, from the observed value. Also both models predict ^{a condensed area} Qo around 80 A2 which is about one-half the value obtained

by extrapolating ^{to n}

^{5 the area measured} by X-ray (2) ^{for BH-n} existing ^{in the} mesophase (discs irregularly spaced forming liquid-like ^{columns) that}

is n

7 and 8. Since the monolayers ^{can be com-} pressed ^to nearly ^{100 A2 before} collapse, it is believed

that the chains in the monolayer ^{are not extended}

as they ^{are in the} mesophase ^{but are} instead in a

tighter configuration stabilized by ^the strong anchoring

of the C02 groups ^to the water substrate thus allowing

a smaller closed packing ^area.

The anomalous features observed on BH-5 are

also present for the other members of the BH-n

family and also for phospholipids, ^see references [8-10]

for the literature on that subject; phospholipids ^are

sometimes described [10] ^{as a} system ^where rigid

molecules undergo melting of their chains as the concentration is lowered and a model similar to ours could be used provided the differences in internal energy and entropy ^are included in the isotherm- isobar potential. Their transition could be partly explained by ^a nesting concommittent with the

ordering of the molecules through rigidization upon condensation thus showing ^an anomalous compres-

sibility ^{after a} transition where mostly ^{vacancies are} expelled ^and rigid ^{molecules are} pairing ^{at a} high

rate.

Of course, ^our models are not versatile enough

to explain the richness of features observed for the BH-n’s or for the phospholipids, ^{hence we cannot}

exclude a priori explanations ^based upon cooperative phenomena involving formation of clusters [11] leading

to continuous transitions that seem to have been

recently ^confirmed experimentally by electron micro- scopy diffraction studies _upon phospholipid ^mono- layers transferred from water to solid substrates [12].

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