Percolation models of free and bound water under the influence of nearest neighbour interactions

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Percolation models of free and bound water under the influence of nearest neighbour interactions

W. Weiss

To cite this version:

W. Weiss. Percolation models of free and bound water under the influence of nearest neighbour interactions. Journal de Physique, 1987, 48 (5), pp.877-883. �10.1051/jphys:01987004805087700�.

�jpa-00210507�

Percolation models of free and bound water

under the influence of nearest neighbour interactions

W. Weiss

II. Physikalisches Institut, Universität

Köln, D-5000 Köln 41, ^F.R.G.

(Requ le 8 décembre 1986, révisé le 30 janvier 1987, accept6 ^{le 30} janvier 1987)

Résumé.

On modélise l’eau

généralisant le modèle de percolation ^de Stanley ^et Teixeira. Les calculs montrent que l’eau liée ^entre des bicouches de lécithine et soumise à des interactions entre proches ^voisins

comporte ^{comme dans les} systèmes

couplage intermoleculaire. Les potentiels chimiques calculés pour l’eau libre et pour l’eau liée diffèrent d’une quantité qui ^croît lorsque le nombre de molécules liées entre les bicouches de lécithine décroît. Cette prédiction ^est

^accord

les résultats expérimentaux.

Abstract.

It is shown on

computer ^water model » based

correlated site percolation calculations

generalizing ^those performed by Stanley and Teixeira that water bound between lecithin bilayers has similar

properties under the influence of nearest neighbour interactions

in systems without intermolecular coupling.

In agreement ^with experimental ^data calculations of the chemical potential of free and bound water showed

increasing difference in chemical potential ^with decreasing ^{number of water} molecules bound between lecithin

bilayers.

Classification

Physics ^Abstracts

05.50

68.15

87.20

1. Introduction.

Water can be interpreted ^{as a} space filling ^network

of hydrogen ^{bonded water} molecules. This network in liquid ^{water can} be modelled on the computer by

simulations such as correlated site percolation ^cal-

culations. The connectivity properties ^{of four-}

bonded water molecules have been studied by ^ex- amining ^the weight distribution functions W, ^of

water molecules belonging ^to clusters of s molecules.

With these distributions one is able to calculate

connectivity properties ^{of the} hydrogen ^{bond net-}

work in liquid ^water [1-4]. ^{These results were}

obtained with percolation calculations and with molecular dynamic calculations. In their percolation

model Stanley and Teixeira occupied randomly ^the

bonds connecting the sites of a lattice with four

neighbours per site and examined the connectivity properties of the sites with four incident bonds. In this paper ^we concentrate on correlated site percola-

tion calculations, ^where interactions in thermal

equilibrium ^{are included.}

We have examined the properties ^of four-bonded

water molecules in bulk water and in water bound between thin lecithin bilayers, the latter being rep-

JOURNAL DE PHYSIQUE. - T. 48, ^? 5, MAI 1987

resentative of biological ^model membranes. Only

interactions in thermal equilibrium ^were considered.

From experiments ^we know that the properties ^of

lecithin bilayers ^are influenced by ^{the amount of the}

so-called bound water between adjacent bilayers [5, 6]. ^{In this} way ^an enhanced chemical potential ^{of the}

water molecules between phospholipid bilayers ^was

found with decreasing water contents in the bilayer.

This is interpreted ^{as a decreased} entropy ^{due to a} higher degree ^{of order of} hydrogen ^bonded H20

molecules at the lecithin bilayer ^surface [5-7]. ^{In the}

present work, correlated site percolation calculations

are presented ^{on the} ice-like diamond lattice. The results are compared ^{with those obtained} by ^earlier

calculations on a simple cubic lattice [8, 9] ^without

interactions. Furthermore, ^{we examined} the influ-

ence of Ising-like next-neighbour interactions on the distribution functions.

2. Model.

There is a maximum of four hydrogen bonds per water molecule. For our calculations it is therefore

more realistic to use the diamond lattice with four

neighbours in space per site instead of the simple

57

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004805087700

878

cubic lattice [1]. ^The diamond lattice is similar to the ice lattice, ^the only difference being ^the angles

between the bonds connecting the sites. For percola-

tion calculations where relations between neighbour- ing ^{sites are} examined, ^these differences are negli-

gible.

In the diamond-like lattice one interpretes ^the occupied ^{sites as} oxygen ^atoms and the bonds between two sites as hydrogen ^bonds (Fig. 1) [1].

Therefore we can look at two neighbouring occupied

sites as two H20 ^{molecules connected} by ^{an intact} hydrogen ^{bond, and we denote as}

^{water » those}

molecules with four intact bonds per oxygen ^atom.

In our model these water molecules are represented by occupied ^{sites where all four} neighbour ^{sites are} occupied, ^{too. The lattice sites are} occupied ^random- ly ⁱⁿ difference to the Stanley-Teixeira model where the bonds are occupied by ^{random. This model is an} example ^for correlated site percolation ^because

clusters of occupied ^{sites with four} occupied neighbours ^{are examined} only. ^{Later we} will include effects from an Ising model interaction.

We investigated the behaviour of the clusters,

built of nearest neighbour ^water molecules with four

Fig. ^1.

a) Clusters of four-coordinated water molecules in bulk water (solid circles). b) The introduction of two boundaries characterized by forming

permanently

intact hydrogen bond per occupied site increases markedly

number and size of clusters.

intact bonds, ^{as a} function of the fraction p ^{of intact}

H-bonds. We obtained p ^{from :}

where fj is the fraction of water molecules with j

intact hydrogen bonds. We have calculated the

weight ^functions of the cluster size, Ws(P), ^{and the}

second moment of the cluster size distribution with- out the biggest cluster, C (p ) _ 2: sWs (p). ^{Ws (p) is}

s

calculated from

where s is the number of water molecules, ^{these are} those sites with four occupied neighbours, ^{in a}

cluster and ns the number per occupied ^{site of}

clusters of size s. Thus E ^{W, is the} fraction of water

. s

molecules and differs from p. In order ^to avoid

analysing ^too many data, ^{and to} reduce statistical fluctuations, ^{one can} combine classes of neighbour- ing cluster sizes in one bin, e.g. all clusters with 8 ^to 15 sites or with 16 to 31 and so on. The results are

plotted ^{at the} geometric ^{mean of the two border}

sizes. For further information see [10]. ^The percola-

tion threshold is determined from the maximum of

C (p ).

The results without interaction for bulk water were found from randomly occupied lattices with up to 216 000 sites.

So far

computer ^{water »} has been treated as an

example of random and correlated percolation neglecting forces between the water molecules, ^but

real water molecules interact. The formations built of hydrogen ^{bonded water molecules} depend ^{on the}

influences of neighbouring molecules. It is therefore useful to examine the influence of nearest neighbour

interactions on the correlated site percolation ^calcu-

lations described above.

To implement ^{the nearest} neighbour interactions

we proceeded ^as is usual in the Ising ^Model [11]

where just two states exist, up spins Si = ^{+ 1 and}

down spins Si = - 1. ^{Here we} interpret ^an up spin ^as

an occupied ^{site and a down} spin ^{as an} empty ^{site of}

the lattice. Transferring ^the properties ^{of the} Ising

Model to our water model, ^two occupied ^{or two}

empty ^{sites i} and k have the energy - like ^If just ^one

site is occupied they ^have the energy ⁺ like ^Further-

more, ^{we can} add an external energy ^± H represent-

ing ^{an external} magnetic ^{field in the} Ising Model,

which is connected to the chemical potential. ^Thus

each occupied site has the additional energy - H ^and

for empty ^{sites we add +} H ; ^{2 H is} apart ^{from an} additive constant, ^the chemical potential. ^In this way

we find the total energy

where Si

^{+ 1} represents occupied ^{sites and} Si = - 1 empty ^{sites. While in the random} percola-

tion model each site is occupied randomly, ^{in this}

model the occupation depends ^{on the state of the} neighbouring molecules. We start with an initial

configuration, ^{which can, but need not, be one with}

all sites occupied, and calculate for each site the energy change ^AE conncected with a flip ^from occupied ^to empty, ^{or vice versa} (« Glauber kinetic

Ising ^model »). ^Then the transition probability

T : Temperature kB : ^{Boltzmann constant}

AE : Energy change

is calculated for that flip. Only ^{if W is} greater ^{than a}

randomly chosen number the state is changed. ^In

this _way we can consider the influence of the

neighbours ^{on the state of a} lattice site. If all

neighbour ^{sites are} occupied, ^the probability ^{that a}

certain site is occupied ^{is rather} high. ^With increasing temperature ^the influence of the neighbours ^decrea-

ses, ^see (4). ^At very high temperatures ^{we should} get the same results as with random site percolation.

From Ising calculations on the diamond lattice we

know that there is a phase separation ^{in the} system

for values of coupling strength K

J/kB ^{T which}

are greater ^{than 0.37} [12]. ^As liquid ^{water is}

examined we have to take values for this coupling ^K

which are below the phase separation. ^{We varied K}

from 0.18 to 0.34. To compare the results of these calculations with those from correlated site percola-

tion and to examine the influence of the interaction,

we varied the energy H ^so that the probability p ^for

intact hydrogen ^{bonds for each K} is identical to that in the random case (K

0).

To apply this model to the bound water between lecithin bilayers, the first and the last water layer perpendicular ^{to the} layer ^{thickness were assumed to}

be fixed to the lecithin molecules by occupying ^each

site of these two layers.

3. Results.

3.1 INFLUENCE OF NEAREST NEIGHBOUR INTERAC- TIONS ON BULK WATER. - The results for the bulk water and non interacting ^site percolation ^were

obtained for a system ^{of size 60 x 60 x 60. For}

«

Ising » ^{results we used} systems with up ^to 34 x

34 x 34 sites and periodic boundary conditions.

To investigate the influence of nearest neighbour

interaction on this model of liquid ^{water we com-} pared ^the weight ^{functions and the} percolation

threshold of the four-coordinated water molecules of both calculations. For very high temperatures

(K

0) ^{we obtained, as} expected, ^{the same results}

for both models. For lower temperatures (K :::. 0.15 )

we found small effects on the weight functions. With

increasing influence of the neighbours, ^{the maxima}

of all weight functions decreased. For high ^{values of} coupling ^{the maxima in the curves were} slightly

shifted towards lower values of the probability p.

Under the influence of interactions the weight

functions increase at lower values of p than for the

non interacting system, ^{but for} larger p ^{we obtained}

the same values. The weight ^functions Wi, W3 ^and W5 ^are the clusters of size 1, ^{4 to 7 and 16 to 31 and}

are plotted ⁱⁿ figure ^{2 for different} values of K. To

Fig. ^2.

Weight functions of cluster sizes 1 (WI), ^{4 to 7} (W3) ^{and 16 to 31} (W5) plotted

^the fraction p ^{of intact}

H-bonds for strenghts ^K

⁰ ( * ), ^K

^0.34 (0) ^and

K

0.18 (+).

880

measure the influence of nearest neighbour ^interac-

tions we compared p,, which represents ^the probabi- lity ^{that a} lattice site is occupied, ^{with the fraction of}

intact hydrogen ^bonds p. An intact hydrogen ^bond

occurs if two neighbouring lattice sites are occupied.

In systems ^without interactions, ^each neighbour ^{of a} randomly chosen site is occupied ^{with the} probability

p,. Accordingly ^{we found the same values} of p ^and

p, for correlated site percolation ^{without nearest} neighbour interactions. Under the influence of nea-

rest neighbour interactions, ^{the state of each site} depends ^{on the} neighbouring ^{sites. The} probability

that a site is occupied ^{is the} higher, ^{the more} neighbouring ^{sites are} occupied. ^This probability ^is

influenced by ^the strength ^of the forces between nearest neighbours. ^{Therefore we can take the}

deviation of p from ps ^{as a measure of the} strength ^of coupling ^{in the} system. ^In interacting systems ^the values of p ^are greater ^than P, if _p 0.85 (Fig. 3).

The differences are greater ^{for smaller} p. For the

occupied lattice sites we can see the tendency ^to clump together under the influence of nearest

neighbour interactions. For the four-coordinated

H20 ^{molecules we found the same} properties.

Fig. 3. - ps

function of p for three different strengths

of coupling (K = 0 * , ^{K =} 0.34 Q, K = 0.18 +).

The percolation ^{threshold of} the four-coordinated sites, calculated from the position ^of the maximum in C (p ) ^{for bulk water without nearest} neighbour

interactions is found at W, ^{= Pc}

^{0.795 ± 0.005}

s

in good agreement with the results of Blumberg

et al. [1].

3.2-WATER ^IN THIN LAYERS.

To examine the water bound between lecithin bilayers with correla- ted site percolation ^on the diamond lattice we varied the thickness of the layer ^{from 2 to 10 water}

molecules per lecithin molecule corresponding ^{to a} layer thickness of four ^to twenty molecules, respecti-

vely. ^The extension in the two other directions

(60 ^{x 60} molecules) ^was still the same as for bulk water. Decreasing ^the layer ^thickness yielded ^an

increase of the weight ^functions (Fig. 4). ^Further-

more, the curves for greater ^{clusters were} slightly

shifted towards greater values of _p. These are

qualitatively ^{the same results as} found with calcula- tions on the simple cubic lattice [8, 9]. ^In percolation

Fig. ^4.

Weight ^functions WI, W3 ^and W5 ^{for bulk water} (*) compared with 5 bound water molecules per lecithin molecule (0). ^The Wi

calculated by correlated site

percolation.

theory ^the percolation ^threshold corresponds ^{to a} macroscopic phase transition. At and above the threshold _pc there exists one infinite cluster among other smaller clusters of hydrogen bonded molecu- les. If p decreases from values slightly ^above

p, ^to values slightly below Pc, just ^a small number of

hydrogen ^{bonds breaks} and there is no longer ^an

infinite cluster. When ice melts, just ^a small fraction of the intact hydrogen bonds in the system ^breaks [1, 17]. ^{So we} tentatively interpret ^the percolation

threshold of the four-coordinated sites as the melting temperature of the ice phase. ^{Fortuin and} Kasteleyn [13] derived the following relation between probabi- lity p ^and temperature :

We assume the same relation for the percolation

threshold p, and the phase transition temperature Tc ^{which we} identify tentatively ^{with the} melting point. ^After equating ^{terms we} get:

As p increases, ^the temperature ^{is seen to decrease} in agreement ^{with the} growth of intact H-bonds with

decreasing temperature ^{in water.}

We found _{that Pc} increases with decreasing layer

thickness D. In figure 5 pc is plotted ^{versus D. The}

calculations are in qualitative agreement ^{with the} experimental finding, ^{that with} decreasing ^number

of water molecules _per lecithin molecule the subzero

,

phase transition temperature decreases, ^too [7, ^14, 16].

The weight functions of the systems ^{with interac-} tions show the same properties ^as the functions of the non interacting ^{model. We found} increasing weight ^{functions with} decreasing layer ^thickness

Fig. ^5.

Percolation threshold p, and temperature ^T

function of layer thickness D for K

0.18. Percolation calculations (0). Experimental results for DPPC (cooling runs) (*) [16].

Fig. ^6.

Weight ^functions Wl, W3, ^and W5 ^{for bulk water} (*) and for bound water with 5 water molecules per lecithin molecule (0) under the influence of nearest

neighbour interactions of strengh ^K

^0.18.

(Fig. 6). ^{These curves} show that with decreasing

content of water bound between lecithin bilayers,

there are more clusters of four-coordinated water molecules related to the total number of molecules than in free water. We interpret ^{this as a} higher degree ^{of order. The bound water is more viscous or}

ice-like. This interpretation is confirmed by exper-

imental results obtained for the lecithins dimyristoyl-

882

phospatidylcholine (DMPC) ^and dipalmitoyl-phos- phatidylcholine (DPPC) ^deduced from dielectric and pressure experiments [8] which showed an increasing

difference in chemical potential ^with decreasing

water contents. This fact was interpreted ^{as a}

decrease in entropy ^{due to a} higher degree ^{of order}

of hydrogen ^{bonded water molecules.}

To compare ^the results of our calculations with these experimental data it is useful to calculate the differences in chemical potential A ^{between bulk and}

bound water for different concentrations of water molecules _per lecithin molecule.

The magnetic ^{field H in the} Ising ^{Model corre-} sponds ^{to the chemical} potential ^{of the} lattice gas model. Each state of the system ^was generated by

well known values of AE/kB ^T (see Eq. (4)). ^{So we}

Fig. ^7.

Difference in chemical potential tL between bulk and bound water

function of the number of water molecules per lecithin molecule. The results of percolation

calculations with nearest neighbour interactions of

strength ^K

^0.34 (+) ^{and for} experiments

given ⁱⁿ

dimensionless form IL / kB T and in meV. (+) Percolation calculations (21 °C) ; (0) ^DPPC (25 °C).

get ² H/kB ^T respectively u /kB ^{T and are able to}

calculate from H the differences in A /kB T ^{for bulk}

water and water bound between lecithin bilayers.

As a result, JL decreases with the layer thickness, furthermore we found an increasing difference in chemical potential ^{of free} and bound water with

decreasing ^{number of water} molecules per lecithin molecule. The results are in fair agreement with

experimental ^data. Figure ^{7 shows the} difference in chemical potential ^{vs. the} number n of water molecu- les _per lecithin molecule from percolation ^calcula-

tions with nearest neighbour interactions of strength

K

0.18 and from experimental ^data [5, 15]. ^The experimental ^{values were} deduced from the thermal activation of the dipole relaxation time of the water

molecules. The differences between different mate- rials are of the same order as those between theory

and experiment.

4. Conclusion.

Percolation calculations on the diamond lattice which is similar to the ice lattice gave qualitatively

the same results of the properties ^{of water bound}

between lecithin bilayers ^as previous calculations on

the simple cubic lattice [8]. The influence of nearest

neighbour interactions on the shape ^of the examined cluster numbers is negligible, ^{but for} higher degrees

of coupling ^{the curves were shifted to} lower values of the probability p. The calculated differences in chemical potential ^{of free and bound water con-}

firmed the model [5, 6] ^{that water bound at the}

surface of a lipid bilayer ^{is more ordered or behaves}

ice-like compared ^{with free water.}

Acknowledgments.

I would like to acknowledge ^the valuable discussions and support ^{of this work} given by ^{G. Nimtz and D.}

Stauffer, ^{and the financial} _support by the Deutsche

Forschungsgemeinschaft.

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