Screened Coulomb interaction and melting in two dimensions

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Screened Coulomb interaction and melting in two dimensions

E. Chang, D. Hone

To cite this version:

E. Chang, D. Hone. Screened Coulomb interaction and melting in two dimensions. Journal de

Physique, 1988, 49 (1), pp.25-34. �10.1051/jphys:0198800490102500�. �jpa-00210671�

25

Screened Coulomb interaction and melting ^{in two dimensions}

E. Chang and D. Hone

Department ^of Physics, University ^of California, Santa Barbara, ^CA 93106, ^U.S.A.

(Requ ^{le 29} juillet 1987, accept6 ^{le 22} septembre 1987)

Résumé.

Nous obtenons le potentiel électrostatique ^{d’un réseau} bidimensionnel formé d’une suspension

colloïdale de sphères ^{dans un} film mince d’eau comme une solution de l’équation ^de Debye-Hückel ^linéaire.

Nous discutons de l’applicabilité ^{de cette} solution à l’équation de Boltzmann-Poisson non-linéaire complète (avec charge renormalisée). Nous calculons les constantes élastiques d’un réseau triangulaire ^dans l’approxima-

tion harmonique ^{et nous} utilisons leurs valeurs pour prédire ^la densité de fusion par dislocation suivant la théorie de Kosterlitz-Thouless (KT). ^{L’accord avec une} expérience récente n’est pas ^très bon. Nous suggérons qu’une renormalisation de charge dépendant ^{de la} géométrie, des effets anharmoniques, l’écrange par paires

de dislocations ainsi que la valeur finie du ^{« contraste »} dielectrique 03B5r ^et la taille finie des particules peuvent

tous affecter de manière substantielle nos prédictions théoriques. ^{Il est} possible qu’après ^{toutes ces} corrections, la théorie KT décrive le comportement ^d’un système colloidal. Notons toutefois qu’en général, ^le potentiel ^d’un système bidimensionnel gouverné par des interactions coulombiennes écrantées ^{est très} peu sensible à la maille du réseau (densité ^de particules) ^{dans certaines} régions ^de l’espace ^des paramètres. ^{Pour un}

tel système ^à température fixée, ^{il est} possible que la théorie KT, ^ou toute autre théorie basée sur un

mécanisme purement thermodynamique, ^ne soient pas appropriées pour décrire la fusion ^en fonction de la densité de particules.

Abstract.

The electrostatic potential ^{of a two} dimensional lattice formed by ^{a colloidal} suspension ^of spheres

within a thin film of water is derived as a solution to the linear Debye-Hückel equation. ^The applicability ^of

this solution to the full nonlinear Boltzmann-Poisson equation (with ^a renormalized charge) is then discussed.

The elastic constants of a triangular ^{lattice are} calculated within the harmonic approximation ^{and used to} predict ^the density ^at which the lattice melts via dislocation unbinding according ^to the Kosterlitz-Thouless

(KT) theory. The results do not agree very well with those of ^{a recent} experiment. ^We suggest ^that geometry- dependent charge renormalization, anharmonic effects, screening by dislocation pairs, ^{as well as finite}

dielectric contrast, 03B5r, and finite particle ^{size effects, all can affect our} theoretical predictions substantially.

Although after these corrections the KT theory may ^account for the behaviour of the colloidal system, ^{we note} that in general ^the potential ^{of a two} dimensional system governed by screened Coulomb interactions can, in

some regions ^of parameter space, be very insensitive ^to lattice spacing (particle density). ^{For such a} system ^at fixed temperature, ^{the KT} theory ^or any other purely thermodynamic mechanism may ^not be appropriate ^for describing melting ^{as a} function of particle density.

J. Phys. ^{France 49} (1988) ^{25-34 JANVIER 1988,}

Classification

Physics ^Abstracts

64.70D

68.35R

82.70D

1. Introduction.

Colloidal suspensions ^of polystyrene spheres (« polyballs ») ^{in a} polar solvent such as water have received much attention ^as model many body sys- tems. Though ^the polyballs ^{are of} macroscopic size, their behaviour is governed by ^the fundamental interactions between them, ^{so that} they ^{form the}

classical analog of ’atomic scale solids and liquids.

Moreover, ^these interactions are electrostatic in

origin ^{and are in} principle well understood. Perhaps

most importantly, ^various means can be used to

control the range and strength ^of interaction between the polyballs. ^This presents ^{the rare} opportunity ^for

a detailed comparison ^between experiment ^and

theoretical prediction ^{of the} dependence ^{of the} properties ^{of a many} body system ^on its fundamental interaction parameters.

A solution of polyballs ^{in a} thin film of water (or

other polar liquid) ^{makes an} extremely ^attractive

class of system ^{in which to} study, ^both theoretically

and experimentally, ^the remarkably interesting phenomena associated with phase transitions in two dimensions

notably, melting ^moderated by ^the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198800490102500

unbinding of dislocations and the possible ^formation

of a hexatic phase. Interesting ^{work in the} past [1-5]

has focused on the case of electrons on the surface of

liquid He. There the long range force surely ^intro-

duces some special features. There is effectively only

the single parameter ^rto adjust (where ris the ratio of Coulomb _{energy of} a pair ^{of electrons at} average

separation ^to kB 1). ^The polyballs provide ^the oppor-

tunity ^not only ^to study the behaviour of a system with short range interactions, ^but again ^{to take} advantage ^{of a} possible ^detailed comparison ^between theory ^and experiment ^{as a} function of relative range and form of the interaction. For the experimental arrangement ⁱⁿ which the colloidal spheres ^are trapped ^at the air-water interface there is, in addition to a screened Coulomb potential, ^a dipolar ^interac-

tion [6]. ^{We will} study ^{here the more} symmetric arrangement ^of polyballs ^confined entirely ^within

the film, which has also been realized experimentally [7]. ^{In both} geometries there have been observed

triangular lattices and disordered two-dimensional arrays ^as parameters are varied, ^{and a hexatic} phase

may have been observed [7]. ^{It is our} purpose ^to

study melting ^{in this} system theoretically.

To carry ^out such a study ^{one needs first to}

understand the interaction between charged polyballs ^screened by surrounding counterions.

Even within a mean field approximation ^the problem

is described by ^the highly non-linear Boltzmann- Poisson equation. ^However, when all electrostatic

potential differences are much smaller than the thermal energy, ^{one can} approximate ^{the BP} equation by its linearized form, ^the Debye-Huckel equation. ^{This not} only provides great mathematical

simplification ; it introduces the essential feature of

superposition inherent in linear equations. ^{Then the}

static (thermodynamic) properties ^{of the} complex system ^of polyballs, counterions, and solvent ions

are reduced to those of a collection of classical

particles interacting through ^{an effective two} body potential. Although ^the potential gradients ^{near the} polyball ^{surface are, in} fact, generally ^too large ^to permit straightforward linearization of the equations there, it has been shown [8] ^{in the} three-dimensional geometry ^{that the} Debye-Huckel linearization can

be used if the electric charge ^{of the} polyball ^is suitably renormalized. We suggest ^{here that a similar} approach ^is possible in the two-dimensional geomet- ry ^we consider.

In the spherically symmetric ^case appropriate ^to

the three-dimensional system (within the standard

approximation ^which replaces ^the crystalline ^unit

cell by ^a sphere) the solution is both straightforward

and algebraically simple (Yukawa potential). ^{In the}

reduced symmetry of the film the form of the

potential ^{is more} complex. ^{This is true for the} polyballs trapped ^at the air-water interface, ^as considered in reference [6], ^{and to a lesser extent it}

is true for the case considered here, with the

polyballs restricted to the midplane ^{of the water}

film. However, ^the expressions ^are sufficiently ^tract-

able for numerical calculations, and they ^{do take on} expected simple limiting ^{forms, as we} shall indicate

explicitly ^below (Sect. 2).

With the effective two body interaction in hand we turn to a prediction ^{of the} melting ^{curve. In the} theory [9] introduced by Kosterlitz and Thouless

(KT) and extended by Halperin, ^{Nelson, and} Young [1, 2] melting is mediated by ^the dissociation of dislocation pairs. ^{In its} simplest ^{form the} theory gives ^a prediction ^{for the} melting temperature ⁱⁿ

terms of the bare (« zero temperature ») ^elastic

constants of the solid, ^{which can in turn} be calculated from the interparticle interaction. Application ^{to the}

two-dimensional electron solid [10] gave fair agree- ment with the experimental value of r. It was

subsequently recognized that renormalization of the elastic constants, associated both with anharmonicity [5] and with the dislocation unbinding process ^itself

[3, 4], ^were quantitatively important. Theory ^for

that case is now in excellent agreement ^{with both a} molecular dynamics simulation [3] ^and with exper- iment [5].

In section 3 we calculate the compressibility ^and

shear modulus for the polyballs ^{confined to the} midplane of the film. We then apply the results to the simple ^form of the KT theory ^to give ^an approximate melting ^{curve. We find some} qualitative disagreement ^with experiment, including ^the initially surprising prediction ^that the lattice does not melt for a sufficiently ^{thin water} film within this simplest approximation. ^We suggest ^{that the} screening ^effect

present ^{in this} system ^is responsible ^{for the} insensitiv-

ity ^{of its} potential ^{to lattice} expansion. Furthermore,

the effects of charge renormalization (and ^its depen-

dence on polyball density), and of lattice softening

due to the nonlinear phenomena mentioned in the

previous paragraph, ^{as well as} certain finite dielectric constant and finite size effects, ^can be substantial.

We discuss the expected ^effect of these corrections

on the melting ^curve in section 4.

2. Electrostatic interaction.

Consider a two dimensional array of polyballs ^sus- pended ^{at the} midplane ^{of a} thin film of water confined by parallel planes ^of glass, ^{as shown in} figure 1 ; this is close to the arrangement ^{of recent} related experiments [7]. ^We consider the ordered state, where the polyballs ^{form a} triangular ^lattice.

The thickness of the water film is 2 d. Upon ^immer-

sion in water the surface ionizable groups of the

polyball dissociate into counterions and leave behind

a uniformly charged polyball surface with total

charge Ze, where e ) I ^{= - e. On} the average (i.e.,

within a mean field approximation) the counterions

27

Fig. ^1.

The geomery studied here : ^a polyball ^{of radius}

R confined within a water film of thickness 2 d.

will distribute themselves according ^{to a Boltzmann}

distribution. Therefore, the electrostatic potential obeys ^the Boltzmann-Poisson equation

where p p represents ^the charge density ^{on the}

surface of the polyball, f3 =1 /kB ^{T is the inverse}

temperature, F is the local dielectric constant, ^and

Po e- el3tP is the counterion charge density. ^The

normalization constant po is determined from requir- ing charge neutrality within each unit cell of the full lattice :

where the integral ^{is over a} unit cell centred at the

polyball.

To have any reasonable hope ^of dealing ^{with this}

many particle system ^{we must at least be able to} describe the energy ^{as a} superposition ^of pairwise

interactions between the particles ; ^{we must seek an} appropriate ^linear approximation ^to equation (2.1).

From the arguments and calculations in three dimen- sions [8] ^{we can} expect ^{to be able to do} this, ^at the expense of ^a suitable renormalization of the polyball

effective charge. ^{The basic} argument ^{is that we want}

to known the potential ^{due to a} given polyball ^at relatively large distances from the ^source

namely,

at the positions ^{of other} polyballs

^where gradients

of that potential ^are expected ^{to be small and}

linearization possible. ^{Since the solution of} equation (2.1 ) has the full periodicity ^{of a two}

dimensional Bravais lattice, ^{it is} only necessary ^to consider a single Wigner-Seitz cell. That is, ^we define a unit cell surrounding ^a polyball by planes

which are perpendicular bisectors of the vectors to all neighbouring polyballs. By symmetry ^{the normal} derivative of the potential ^{vanishes on the surface of}

this cell. If we choose the arbitrary ^{zero of the} potential ^so that 0 vanishes somewhere ^on the cell surface then, ^at least in the neighbourhood ^{of the}

cell boundary, eQW « 1, ^{and we can linearize} equation (2.1 ) ^{to find the} Debye-Huckel equation :

where

is the square of the inverse screening length, ^{and the} prime ^on cP’ == cP - 1/ (3e ^{is in} recognition ^{of the} required ^shift by ^{an additive constant in the defini-}

tion of the potential upon linearization to give ^the

form (2.3). ^{We will} drop ^the prime henceforth.

Given a certain charge density parameter p o, ^we

can numerically integrate ^either equation (2.1) ^or equation (2.3) ^{inward towards the centre until the} polyball surface is reached. This is particularly straightforward ^{in the} three-dimensional problem,

once the Wigner-Seitz ^{cell has been} approximated by ^a sphere, ^{so that} by symmetry ^{all functions} depend only ^on the radial coordinate, ^{r. The electric}

field at the polyball ^surface gives the surface charge density ; that calculated from integration ^{of the full}

nonlinear BP equation (2.1) ^{is the true bare} charge density. On the other hand, the number obtained from integration ^of equation (2.3) ^{is an} effective charge density, which would give ^{the correct} poten- tial, electric field (and, ⁱⁿ fact, ^{the next} derivative,

the compressibility ^{or bulk} modulus) ^{at the cell} boundary, if the linearized Debye-Hilckel equation

were, in fact, ^valid everywhere ^{and the source had}

this effective (« renormalized ») charge.

We assume that charge renormalization is pos- sible, and that the system ^is satisfactorily ^described by ^the corresponding linearized DH equation. ^To

determine the effective interparticle interaction we

consider a single macroion in the film as shown in

figure ^{1. A} simple and useful particular solution of the DH equation (2.3) is that for the single polyball

in an unbounded solvent :

where Ao ^{is to be} determined by ^the boundary

condition at the polyball ^{surface, r}

^{R :}

(note that Ze is the appropriately renormalized

polyball charge). ^This expression differs from the

point ^{source result of} Ao

Ze/ F by ^{a term of order} ( K R )1/6. Typically rc R , 0.2. Therefore the finite size correction due to this term, although included in

our calculations, is usually negligible. ^{There are two}

other finite size corrections. One concerns the definition of _po, the other the potential energy between two polyballs. These effects will be discus- sed subsequently and in the appendix.

Within the film geometry ^{let us} again ^{consider a}

single ^isolated polyball. The solution to the homo- geneous part ^of equation (2.3) ^{separates in} cylindri-

cal coordinates :

inside the film, ^where

Outside the film K

0, ^{and 0 must vanish at}

Matching boundary conditions is simplified by ^the

standard representation :

The requirements that the normal component ^{of D} and the tangential component ^{of E be} continuous at the surface, ^z

d, ^then give

where e, ^{is the} ratio of internal to external dielectric constant (the ^« dielectric contrast »). Depending ^on

the material bounding ^{the film} E, ^can range ^from about 15 (for ^a water-glass interface) ^{to 80} (for water-air). ^{For the} larger values, ^{to a} good approxi-

mation we can simply ^{use the} large E, ^{limit in} equation (2.11) :

where we have assumed _E, tanh qd > ^{1 in} taking ^the

limit. Since q , K, this implies K d > 1 / ^{En which is}

well satisfied in all cases of interest (smaller ^{values of}

K d correspond ^to screening lengths ^{orders of} mag- nitude larger ^{than the} interparticle spacing ^{at de-}

nsities large enough ^to permit solidification at all). ^If

on the other hand e, is ^{as small as} 15, ^one might expect corrections of order 1/ Erin ^{the expansion of}

equation (2.11) ^{to become} important. ^A calculation

using ^some typical parameters, however, ^indicates

that the contribution of this term to the elastic constants is in fact only of the order of 1 or 2 percent

(the oscillatory behaviour of Jo reduces the size of the term). Therefore, ^{from now on we} will confine

our attention to the large limit. When the

polyballs ^{are confined to the z}

⁰ plane, ^{and if}

KR 1, then for the interparticle interaction we

need only ^{the behaviour of} equation (2.12) ^at

z

0 (in ^{fact, as} shown in the Appendix, the result is

equally tractable for general KR, ^{but this} unnecessar-

ily introduces yet ^another parameter ^{into the} prob- lem, ^{when K R} is, ⁱⁿ fact, ^small compared ^to unity ⁱⁿ

all cases of interest here). ^{Then for z}

^{0 we} expand

the integrand ^of equation (2.12) ^{in a} power series in

e- 2 qd to give

a result we could have invoked immediately ^{in this}

limit of large E, (so that 0 vanishes outside the film)

and symmetric placement ^{of the source at the film}

centre, so that the solution is given by ^{an infinite set}

of image ^{sources at z = ± 2 nd} (n = 1, 2, ..., oo)

within an unbounded medium of screening ^wave

vector K and dielectric constant E (the ^{second term in} Eq. (2.13)).

The series in equation (2.13) converges rapidly

when r is of the order of d or less. In general ^values

of n up ^to order J r / K d2 ^{are clearly required.}

However, ^{the slow rate} of convergence ^at large r / d ^{causes no} practical difficulty, because the slow variation itself permits ^the good approximation ^of

the sum over n in equation (2.13) by ^an integral :

This is, ^of course, nothing ^{more nor less than the}

screened potential ^{from a} thin wire (cylindrical)

source ; the discrete images ^{have been} merged ^{into a}

continuous wire by ^the replacement ^{of the sum with}

the integral ^over y.

Equation (2.13) is the basic linearized interaction that will be used throughout ^{the rest of} this paper.

Its relatively elementary ^{form makes it} simple ^{to use}

in numerical computations. ^{However, to have a}

better intuitive feel for its analytic ^{behaviour we will}

examine some of its approximate ^{forms in} physically interesting limits. First let us consider K d > 1 : the

screening length ^{is small} compared ^{to the film}

thickness. If r is not too large : r * ^{d, then the}

second term in equation (2.13) (or ⁱⁿ Eq. (2.12)) ^is negligible, ^and

That is, the potential in this limit is insensitive to the

presence of the water-glass interfaces and becomes

29

just ^{that of a} polyball ^{in an unbounded} medium, ^as

one would expect. ^At larger distances, r/d > ^Kd,

even though ^the screening length ^{is small} compared

to the film thickness d, ^the image charge ^contribu-

tions are substantial, ^{and we must} include them

(e.g., within the approximation (2.14)).

We find another interesting limit when r is large compared ^{to both} Kd2 and the screening length :

Kr >> (Kd )2 ; K r > 1. ^{The first} inequality ^{allows us to}

use equation (2.14), and the second to take the

asymptotic form of the Bessel function Ko :

Thus, ^the potential ^{is now} strongly ^modified by

the two water-glass interfaces. In this limit of infinite dielectric contrast (no ^field lines escape from the

film) ^{the distant field is} approximately ^{that of a} cylindrical ^{source. The} confining space has prevented

the potential ^from falling ^{off as} rapidly ^{as the}

Yukawa potential appropriate ^{to a} polyball ^{in an}

unbounded medium (Eq. (2.15)).

3. Melting ^criterion.

Assuming the dislocation pair dissociation mechan- ism of melting [9], Kosterlitz and Thouless gave ^a universal relationship between the elastic constants and the melting temperature,

where Tm ^{is the} melting temperature, kB

Boltzmann’s constant, a the lattice constant, and >

and A are the Lame coefficients, ^{which are related to}

the shear modulus and compressibility of the solid.

In terms of the usual stress and strain tensor

components, o-ij ^and Eij, respectively, ^{the Lame}

coefficients are defined by [11]

In the simplest form of this theory, ^{to which we will}

confine ourselves here, ^one ignores ^the softening ^of

the lattice and corresponding renormalization of the Lame coefficients from the effects of anharmonicity

and the unbinding of dislocation pairs.

The coefficients > ^{and A are} defined in terms of the properties ^{of an} elastic continuum. For ^a lattice of discrete particles, ^{such as we have here, we must}

look at the elastic behaviour on length ^scales large compared ^{with the} interparticle spacing. ^{It is often}

convenient to do this in terms of the long wavelength longitudinal ^and transverse sound velocities of the system. We will follow ^a similar path, though ^{in the} physical system ^the sound modes are damped by

viscous drag from the fluid. The melting criterion of

equation (3.1) ^{was derived} purely ^on thermodynamic grounds ^{based on} the static elastic properties ^{of the} system, ^as determined by ^the interparticle ^interac-

tions. We therefore introduce an artificial lattice with the same interparticle interactions, ^{but with no}

viscous damping, ^{and use standard} techniques ^to

calculate the normal modes - and, ⁱⁿ particular, ^the long wavelength ^sound velocities, ^and thereby ^the

desired Lame coefficients, ^{for this} system. ^The connection is :

where C; ^and CB ^{are the} longitudinal ^{and transverse}

sound velocities respectively, ^{and p} is the areal mass

density of the lattice (which ^can be chosen arbit-

rarily ; ^{there must} and will be no dependence ^{of the}

final results on p, which has been introduced for calculational convenience).

We calculate the sound velocities from the dynami-

cal matrix, Dij (k), the Fourier transform of the second spatial derivative matrix of the potential

energy function, in the usual way. It is convenient ^to introduce the set of dimensionless variables

where as represents the average separation ^between

the polyballs

i, e. , as

1/3 a 2/2 ^{is the area per}

polyball. ^The potential depends ^{further on the}

dimensionless parameter Kas, which ^measures the

screening length ^{relative to} the average interparticle spacing. ^{If we further extract suitable} coupling

constant parameters ^{from the} potential :

then in the long wavelength ^limit (kas 1 ) ^the dynamical ^{matrix can be written}

where the fundamental dimensionless lattice sums

xij ^now depend only ^on the above dimensionless parameters ^{and on} the lattice symmetry. ^{It is these}

sums which we will need to calculate numerically ^to

establish the predicted melting ^{curves. Note that a} triangular lattice is elastically isotropic [11], ^{so that}

the sound velocities do not depend ^on k. Pick

k to point ^{in the} positive x direction, ^chosen parallel

to a lattice vector. Then ly

0 due to symmetry,

and the sound velocities are given by

where the subscripts ^{refer to the} longitudinal ^and

transverse polarizations. Finally, ^{we have for the} melting temperature :

where Ao is defined by equation (2.6).

For the screened Coulomb interaction with fixed

polyball ^effective charge ^Z and radius R, ^{and for}

K R small enough ^{to be} neglected (which ^{is, as we}

have already indicated, ^{the situation} ordinarily ^ex- pected), the criterion (3.8) ^is remarkably insensitive

to the density 1 /as ^{which might be expected to}

determine stability ^{of the} liquid ^{vs the solid} phase.

The reasons are not hard to discover. First consider the dimensionless screening parameter,

obtained from equation (2.4) by approximating

p o by the average counterion charge density ^{within a}

unit cell and neglecting the small volume occupied by ^a polyball. (Corrections ^{due to the} spatial ^varia-

tions of charge density p (r ) ^are, by definition, ^not included in the linear approximation itself.) Then, ⁱⁿ

contrast to the 3-dimensional situation (where d ^is effectively replaced by as ^on the right hand side of

equation (3.9)), ^the quantity rcas is density indepen- dent ; ^the screening length ^is proportional ^{to the} particle separation. Moreover, if the interaction at

lengths large compared with film thickness (r > d)

dominate the sums X determining the elastic con-

stants (i. e. , ^{if the} screening ^is sufficiently weak),

when v (r)

^{Ze 0} (r) depends ^{on r} only through ^the

combination K r = ( K as ) (r/as ) ^{(see, e. g. ,}

Eq. (2.14)). ^{We have} just ^{seen that this} combination is scale, ^or density, independent. ^{Then to this extent}

the two sides of the melting ^condition (3.8) ^are independent of as, or of polyball density, ^{and the} solid, ^{if stable at} all, ^is predicted ^{not to} melt via the dislocation unbinding ^mechanism regardless ^{of how}

much that density is reduced ! 4. Results.

Murray and Van Winkle reported ^an experiment [7]

using polyballs 0. 15 > in radius and a pH ^determined

surface charge ^of approximately 4 x 104 e. ^Their

results gave the critical polyball separation (i.e., ^the separation ^at melting) ^{as a} function of the film thickness. Since the quoted value of surface charge

at best corresponds ^{to the bare} charge ^{of our} analysis, ^{we used a} typical ^data point from reference

[7] ^to estimate the effective charge ^{of this} system. ^A film thickness of 2 d = 1.9 R ^{and the} corresponding

critical separation of as = 1.38 J.L ^{were used as} inputs

to equation (3.8). The effective charge, ^determined by plotting both sides of this equation ^{as functions of} Z, ^was determined to be about 310. This charge ^was

then used to produce ^{the rest of the} melting curve, shown in figure ² along ^{with the} experimental ^curve

from reference [7]. ^One striking ^result immediately

apparent ^{is that no solution to} equation (3.8) ^exists

when d rx 0.6 _J.L, corresponding ^{to the} density ^inde- pendent regime mentioned in the previous ^section.

Fig. ^2.

^Critical separation as ^vs. film thickness 2 d. The small dashed curve is the result of the present calculation

using ^{the full} potential ^of equation (2.13). ^The large

dashed curve represents data from reference [7]. ^{The full}

curve is the result for a pure Yukawa potential.

Furthermore, ^{even in the} region where the lattice is

predicted theoretically ^to melt, ^the dependence ^of

the particle density ^{there on} film thickness d is very different from what is observed ; ^{it is not even} monotonic, ^as experiment ^{is. The} discrepancy ^{is so}

severe that one is certainly tempted simply ^to reject

the K-T mechanism and theory ^{as a suitable} descrip-

tion of melting ^{in this} system. ^{But we} pursue it

further, both because of the reported experimental

observation [7] ^{of a hexatic} phase ^{in these} systems, and because we have recognized ⁱⁿ doing ^these

calculations that corrections to the simple theory, including the anharmonic effects found important

for the 2-d electron solid, ^are essential for even a

qualitative understanding ^here. They play ^{such an} important role because this calculation is plagued by

an extraordinary sensitivity ^{to the} parameters ^which

enter. A relatively ^small change in the value of the left hand side of equation (3.8) can cause a large qualitative change ^{in the} melting ^{curve. To demon-}

strate this sensitivity ^{we have} plotted ⁱⁿ figure ^{3 the}

ratio of the left to right hand sides of equation (3.8)

for the experimentally observed values of as ^at

31

Fig. ^3.

Ratio of the left to right hand sides of

equation (3.8), ^{for the} experimentally observed values of as ^at melting, ^{as a} function of film thickness.

melting ^{as a} function of film thickness. It is seen

from figure ^{3 that over the full} experimental range of d only ^{a small monotonic} (with d) ^{shift is} required ^to bring theory ^into agreement ^with experiment.

To understand why equation (3.8) ^{is so sensitive}

to its parameters consider the right hand side of this

equation,

where the approximation results because .!XX >> lyy in all cases of interest. (Indeed ^{for an} infinite range

potential, ^as in the electron solid, ^A

^{and therefore} .!XX oc ..t ⁺ A

is infinite, ^{while ZYY} oc tk remains

finite.) ^The behaviour of .!YY as a function of as is shown in figure ^{4 for d}

^{0.62 tJL} (the ^{features to}

be described in the following ^are found for all values

Fig. ^{4. - -Vyy as a function} of as ^{for d}

^0.62 jjL. The solid circles represent the results of ^a calculation using ^{the full} potential ^of equation (2.13). The solid line uses only ^the

Yukawa portion ^{of the} potential. The dashed line

(Iyy

Cst. ) represents ^the limiting form associated with the approximate potential (2.16).

of d of interest here). Sums obtained from using ^the

two limiting ^forms (2.15) ^and (2.16) ^{are also shown}

in the figure. Except ^{for a} surprisingly ^narrow

transition region, ^{ZYY follows} closely ^{the two} limiting

forms. At small as ^the potential ^is nearly Yukawa, ^so

that lyy - lias ^(the lias dependence ^{is not exact}

because Ao itself has a slight as dependence). ^At large a5 ^the potential ^{behaves as} Ko ( K r ), ^{which is} independent ^of as, and lyy approaches ^{a constant. A} study of the numerical results verifies that, ^{as shown} explicitly for the case d

0.62 J.L ⁱⁿ figure 4, ^{for all d}

of interest .!YY as a function of as is very well

represented by ^these limiting ^{forms connected} by ^a

very ^narrow transitional region. If the left hand side of equation (3.8), ^which determines the melting point, is less than the constant asymptotic ^{value of}

.!YY at large as, then there is ^no solution to

equation (3.8) ; ^no melting ^is predicted. Such is the

case for our results at small film thickness : d -- 0.6 R. The upturn in the theoretical curve with

decreasing ^d below d = 1 _)JL comes as the left hand side of equation (3.8) ^rises slightly above this critical

asymptotic value and intersects the .!YY curve in the

narrow transition region, where the results are

clearly remarkably ^{sensitive to small} changes ⁱⁿ parameters. ^{We show in} figure 2, ^for example, ^how

the result changes dramatically if the Yukawa form of the potential is assumed a good approximation ^for

all d.

We therefore turn now to an estimate of the effects of the obvious corrections, ^{which we will}

show modify the values of both sides of

equation (3.8) ^{so as to} bring theory much closer to

experiment. These include the anharmonic effects, screening by dislocation pairs, ^finite e,, ^a third finite size correction, ^and charge renormalization.

In calculating the elastic constants above ^we have assumed that the harmonic approximation ^{to the} potential ^{near its} minimum is adequate. This ap-

proximation, however, ^is particularly poor for ^two dimensional systems ^near melting, ^{where the} parti-

cles have very large ^rms fluctuations about their

equilibrium positions. ^In fact, ^the fluctuations _grow without bound with the size of the system, ^which feature is reflected in the lack of true long range translational order. Certainly, anharmonic effects

can be expected ^{to be} important. ^{In Morf s molecular} dynamics simulation [3] for electrons on the surface of He the anharmonic correction accounted for

approximately 15 % of the reduction in r, ^the

dimensionless coupling constant at melting, ^relative

to its harmonic, ^{or low} temperature value. As is clear from the discussion above, effects of this

magnitude ^are adequate ^to bring theory ^{close to} experiment.

The result of the dynamical calculation, including

the anharmonic corrections, provides ^a starting

point ^{for the} renormalization _group analysis ^of

KTHNY [1, ^2, 9]. This takes into account the

screening ^of the interaction between members of a

dislocation pair by intervening pairs of smaller

separations, ^{in a} way completely analogous ^{to the} polarization screening ^{of a} pair of electric charges ⁱⁿ

a dielectric medium. The effect of this screening ^{is to}

reduce the interaction between dislocations, ^and therefore to give ^a softer lattice. Near melting ^the density of dislocations increases rapidly, decreasing

the screening length ^and reducing ^{the elastic cons-}

tants, ^{in a cascade} effect which precipitates melting.

The simulation [3] of the 2-d electron solid shows lattice softening ^near melting ^to be dominated by

this effect ; ^{the same is} likely ^{true for the} polyballs.

We discussed the finite c, correction briefly ⁱⁿ

section 2. Although it appears ^to be a rather small effect (on the order of a few _per cent), ^the depen-

dence of its magnitude ^{on film} thickness d works in the direction of bringing theory ^{closer to} experiment.

That is, ^this effect softens the lattice more at small d,

because a greater proportion of the field lines escape from the film (modification ^{of the} boundary ^condi-

tions is simply ^more significant for closer bound-

aries).

There is a third finite size effect that has been

neglected up ^{to now.} The definition of Kas in

equation (3.9) neglects the finite volume occupied by ^the polyball. This reduces the volume available to the screening charge, modifying equation (3.9) ^to

which now gives Kas ^a dependence on as, ^or density.

The correction is negligible ^{for the} parameters ^we have used above in the regime where the calculation shows melting. However, the effect can be substan- tial for sufficiently ^{small d, where its} tendency ^{is to}

weaken the elastic constants by enhancing ^the screening ^and thereby ^to promote melting. Its range of importance ^is clearly ^{limited to values of d, R, and}

as such that the polyball occupies ^a significant

fraction of the unit cell volume.

We have assumed that the polyball charge ^{used in}

the calculations has been suitably renormalized.

What we have not yet taken into account is that the renormalization may itself depend significantly ^on

the physical parameters, including ^lattice spacing,

film thickness, ^{and the} polyball radius, ^{as well as the} bare charge. ^{Since the} origin of the renormalization is the nonlinearity of the Poisson-Boltzmann

equation, ^{the size of the effect} depends ^{on the} region (near ^the polyball surface) ^where large potential gradients ^exist. Geometry implies ^a relatively greater fraction of the unit cell to be involved in 2 than in 3 dimensions, ^{and we} expect the effect to be more

pronounced in the film than in the bulk. Calculation

of the appropriate renormalization, ^as described in reference [8], requires the solution of the BP and DH equations within the relevant unit cell, ^{here a} cylinder ^{with the} spherical polyball ^{at its centre. This}

geometry ^{of mixed} symmetry implies ^nontrivial dependence ^{of the} potential 0 ^{on at least two} coordinates ; solution of the equations by simple

direct integration, e.g., is ^not feasible. Iterative numerical solution, ^on the other hand, ^{is rather} expensive ⁱⁿ required computer ^{time. There are two}

physically interesting limits of higher symmetry, however, corresponding ^{to the} approximate poten- tials of equations (2.15-16), which do allow for

simple ^direct calculation of the renormalization. In the first instance, if the film is thick enough ^{that the}

linearized potential ^is approximately of the Yukawa

form, equation (2.15), ^this simple ^radial dependence

suggests replacing ^{the actual} Wigner-Seitz ^cell by ^a spherical ^{one. Then the} problem ^becomes complete- ly one-dimensional (spatial dependence ^is only ^on

the single ^radial coordinate r), ^{and the} differential

equation (2.1) ^{can be} directly integrated ^{to solution} (precisely ^{what was} done in Ref. [8] ^{and outlined in} Sect. 2). The numerical results show only slight charge renormalization for a charge ^{as small as}

Z -- 310. This result suggests ^{that the} portion ^{of the} melting ^{curve at} large d ^is probably ^not appreciably

modified by variations in charge renormalization.

The second limiting situation, corresponding ^to equation (2.16), may be of ^more importance ^here.

As we argued ^at the end of section 3, ^{at small} enough d ^the potential has reached this limiting ^form

and further increases in polyball separation ^{are of no}

effect in promoting melting ^via dislocation unbin-

ding. Geometrically, ^the polyballs and their as-

sociated image charges ^can accurately by replaced by

a set of parallel charged ^{wires. We} again simplify ^the problem by approximating ^the hexagonal ^{unit cell} by

a circular cylinder, ^{so that the} dependence ^{of 0 is} again ^{on a} single (radial) coordinate and direct solution by integration ^becomes possible. ^{We can}

then study ^the dependence ^{of the} charge ^renormali-

zation on the relevant parameters, notably ^film

thickness d and particle separation as. Preliminary

results are encouraging, showing ^a reduction in Z*, the renormalized charge, ^with increasing as.

Thus the lattice is softened when the particle density

is reduced. Secondly, when the film thickness d is reduced, Z* is also reduced. Above we chose ^an intermediate point (at ^{2 d}

^1.9 R) ^as the reference to determine the constant Z* to be used in all calculations. This trend in Z* due to charge ^renor-

malization will therefore lead to a softer lattice at small d as well as a stiffer lattice at large ^{d, as is}

needed to account for the experimental data. More extensive analysis, ^{however, will be} required ^to

make detailed numerical comparisons.

We note that (as in the bulk [8J) ^the renormalized

33

charge saturates at a value Z * ,. ^No larger ^{Z* is} produced by ^{a bare} charge, ^however big it may be.

For unit cells with radii comparable ^{to those}

measured in reference [7] ^we find saturation values in the neighbourhood ^{of 300 to} 500, ^{at least when d}

is small enough ^{that the} cylindrical (wire) approxi-

mation discussed above can be used (the correspond- ing ^{number in the} opposite ^{extreme of a full three}

dimensional system [8] ^{is about} 1 500). ^This suggests that the effective charge ^we require ^{for the} theory ^is

not inconsistent with the much larger measured bare

charges.

The worst disagreement ^between theory ^{and ex-} periment ^occurs in the small d region ^of figure ^{2. For} theory ^{to match} experiment ^at, say d

0.6 _f.L, the combination of all the corrections listed above needs to reduce the right ^{hand side of} equation (3.8) by approximately ³⁵ %. The anharmonic corrections,

as we have noted before, can cause a 15 % or more

reduction in the elastic constants. The dislocation

screening ^caused softening ^{can be as} large ^{as 50 %.}

Although ^the magnitude ^{of these effects on tt and A}

do not have a clear dependence ^on d, ^{the softened}

elastic constants will cause the critical .!YY to occur in

a region where the Yukawa approximation ^is valid, leading ^{to a} qualitatively ^correct melting ^behaviour

at small d as we have discussed at the end of the

previous ^section.

Furthermore, ^{it is} important ^{to note} that these two corrections will enhance the effect of charge ^renor- malization, which weakens the lattice at small d as

well as strengthening ^{it at} large ^{d. The reason is that}

we have used the simple uncorrected theory ^to

extract Z* from an experimental ^data point. Using

this effective charge of 310 and the infinite wire

model, ^we estimated that charge renormalization amounts to about a 2 % reduction in Z at d

0.6 tJL.

If, however, the anharmonic and dislocation screen-

ing corrections were to be applied ^{to the} simple

result prior ^to charge renormalization, ^{then in re-} peating ^the procedure described above to extract the effective charge ^{for the same data} point, ^{we would}

have found a much higher Z*, resulting ^{in a} larger

bare charge. ^This large ^bare charge ^{will accentuate}

the effect of charge renormalization, giving ^{it a much}

stronger dependence ^{on d.}

The remaining ^two corrections are smaller in

magnitude. Since E, ^{can be as small as} 15, ^{it can}

contribute up ^to 5 or 6 % to the correction. The finite size effect is only ^at the 1 % level at d

0.6 _R, although it becomes increasingly ^more

important ^{at smaller} d and as. Again, ^{in view of the} sensitivity ^{of the} melting criterion, ^{even these correc-}

tions should be included in the analysis, especially

when the effect of these corrections is to weaken the lattice more at smaller d. Furthermore, they ^{too will}

enhance the effect of charge renormalization.

In conclusion then, although ^the simple ^model presented ^{in this} paper ^fails to account for the

experimental ^{data of reference} [7], ^we believe the combined effect of the corrections listed above is

large ^{and in} principle ^{can reconcile} the differences between theory ^and experiment. Furthermore, ^the simple ^model presented ^{in this} paper points ^{out an} interesting feature of a screened Coulomb system ⁱⁿ

two dimensions, namely ^the insensitivity ^{of its in-}

teraction to density changes. ^The validity ^{of the KT} theory ^for describing melting ^{in the} general ^{class of}

screened Coulomb systems ^{remains an} open ques- tion.

Appendix. Potential energy ; finite size effect.

In this section we evaluate the potential energy of

two interacting polyballs. Assume the potential given ⁱⁿ equation (2.13). ^The potential energy of ^a

polyball ^{situated at a} distance r from the origin ^is

obtained by integrating the contribution over the surface of this polyball.

where

and

This is an elementary integral, giving

In the limit of K R 1 this expression ^{reduces to the} expected ^{form of}

with corrections of order ( K R )2. ^{We used equation}

(A.2) in all the actual calculations.

References

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[2] YOUNG, ^A. P., Phys. ^{Rev. B 19} (1979) ^1855.

[3] MORF, ^R. H., Phys. ^{Rev. Lett. 43} (1979) ^931.

[4] FISHER, D., Phys. ^{Rev. B 26} (1982) ^5009.

[5] GRIMES, C. C. and ADAMS, G., Phys. ^{Rev. Lett. 42} (1979) ^795.

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[7] ^{VAN WINKLE, D. H. and MURRAY, C. A.,} Phys.

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