The formation of cluster vibrations in imperfectly structured materials

chemical Ph_ysics91 (1984) 183-199 Worth-Holland_ Amswrdam

Is3

THE FORMATION

OF UUSTER

VIBRATiONS

IN IMPERFECTLY

SI-RUCI-URED

MATERIALS

L_A. DISSADO

Rcceiwd Z Nowmbcr

I9S3. in tinsi form 11 July 1984

The dynamics of ctusrer formation in imperfecfl) stvcrured materials is dkusxd in Iof their cwlution from a prepared state fluctuation. It is shoan thar this is accompanied by the generation of group oscdlations of dqlaaxxnu in the ccntres/axes of motion of Lheduster componenk A fracticxk index defimin~ the pazzitioningof the viirational,/2ibrational cnbctuxax SIX group &lIation+ and the 0mu-c of motim vibrxtiou is shoum to be rckxcd co the swrural rqqkri+- of the duster-_The gcncn lion of &roup ~lhtions hy also beal &led x0 she ctunge in ConiiguRlio~ CSropy a1tcndantupon the irreversible formation of imperfectly structured dusters. expressed as rhc same fraction of the change arising from compke dissociation of Lhc ideal site

L Introduction

A general description of the atomic or molecular dynamics of materials in a condensed state presents a major theoretical problem because of the large number of degrees of freedom Possessed by a many-body system. Exact solutions to the problem can only be obtained in the case of an ideal crystal for which translational symmetry assembles the degrees of freedom into a smaller number of bands, with each member of a band being an irreducible representation of the translational symmetry group and each band an irreducible representation of the unit cell symmetry group [1.2]_ In the case of an ideal liquid a steady state distribution of particle velocities exists and the problem can again be reduced to that of a small number of ma croscopic modes [3j_ When atoms are tightly bound forming molecules the same ideal states can be defined, with a subset of atomic degrees of freedom abstracted as intra-mokcular vibrations_ Real crystals, however. contain lattice imperfections [4] and reaI liquids possess a local structure [5] so that such ideal states define unrealisable extremes_ In general a wide range of intermediate material morphologies may exist inbetween these extremes which are generally described

0301-0104/84/$03.00 (North-Holland

by a prominent physical feature or mechanical property_ As a result such macroscopic terms as paracrystalline, amorphous [6], glassy [7], plastic. rubbery [S], etc. have come into currency without any precise microscopic defiition. Some general features of non-ideal morphologies may be elucidated. Firstly, the loss of translational >m.metty means that the local structure will cease to exist over a ftite distance_ Secondly, unless the material is a super-lattice, the array formed by translation of the local region as an entity, does not itself possess translational symmetry. In this way very complex morphologies can be constructed with each imperfectly structured group on one level of organisation forming an element in the structure of the succeeding, grosser. array. Such a situation occurs for semi-crystalline [S] polymers where polymer side-group structure influences inter-segmental otggsation, whose array forms crystalline lamella which themselves may be organised into spherulites to complete the macroscopic picture_ The spatial extent of each element of structural orggtion is often used to describe the system through an exponentially decaying spatial correlation function [lo]_ This approach, however. treats the system as a continuum and therefore omits the local structure and its effect upon the dynamics of

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Physics PublZshing Division)

the system_ IF. honever_ some means of defining and determining the regularity of the local structure could be established then a complete morphological description could be made_ It is the intention here to outline a may of achieving this aim whilst retaining both dynamic and thermodynamic validity_ As a starting point it is proposed that the thermal equilibrium state of imperfectly structured materials can be described in tetms of clusters within which localised modes of vibration can be established Ill]_ Such a concept has been suggested by molecular d_vnamics simulations of both liquids [12] and ferroelectrics [13]_ and some properties have been considered theor&cally [11.14]_ The structure of this imperfect state will be defined with respect to the ideal state from which it can be considered as originating. For example through distortion-forming stresses in solids or aggregating forces in liquids. Although the material itself need not have been made in such a manner. this construction has the advantage of relating the imperfect state to an ideal state whose properties are at least formally understood_ Nor is this a purely formal construct since structural fluctuations of the imperfect state will exist which possess locally the structure of the ideal state. Such fluctuations describe non-cquilibrium stattz of the imperfect material since local stress is generated by their formation leading to an increase in local stored energy H hich must be supplied by the heat bath. They are thus thermally generated material fluctuations_ In the following secticns the r+gression of the ideal state fluctuations to the equilibrium imperfect state is described in terms of the evolution of cluster-iocalisecl motions. A microscopic approach is adopted. in vvhich the stress present in the non-equihbrium local structure modifies the ideal state lattice or molecular motions into cluster group vibrations_ An index is defined which is a measure of the correlation of the group motions within the average cluster and therefore describes its degree of structural ordering It is shown that the microscopic description of the time evolution can be recast into a form appropriate to the thermodynamics of the formation/dissociation of an imperfectly structured cluster or aggnzgate. Furthermore the dynamically defined correlation index is shown

to be related to the configurational entropy change on cluster formation and the cluster binding energy expressed as fractions of these properties for the ideal system. Since material fluctuations of the kind described will couple to a spatially uniform conjugate field [15]. the observed response [16]. will be governed by their regression to the cluster structure of the imperfect state [li]. Therefore analysis of the temporal or frequency dependence of the response observed in such common techniques as acoustic. mechanical or dielectric relaxation provides the experimental verification [18] of the time evoluticn derived here and the means of determining the value of the correlation index_ The relnxation function derived here can also be expected to apply to the time dependence of lattice distortion. u hen introduced into a perfect crystal by an external influence such as photoexcitation 1191. and u hich plays au important role in the production of excimtrs [2@] and aitematix-e photochemistq.

2_ Theory of cluster oscillations Z_I. The srrucrw-ai nmure of chsren Cluster structures can be produced in a variety of ways. For example an ideal lattice may be distorted by the prcsene of interstitials [21] or substitutional impurities, or solute molecules may form salvation compiexes with a solvent [22]. Alternatively topographical imperfections [4] can arise as a result of steric. kinetic [7] and mechanical restraints [S] of formation_ All such structures are charactez-ised by a spatial con-elation function 1% hich is non-monotonic over short ranges j23.241. reflecting the inter-element spacing. before falling off exponentially at distances of the order of the correlation length 5,. The physical picture of a cluster is therefore that of a group of discretely spaced structural elements. with the element at any site being reproduced from a chosen origin element only by site-dependent displacement and/or rotation in addition to an integral number of unit translations_ Because of the loss of translational symmetry there will be a number of equivalent constructions with the same local internal

energy and thus the position and orientation of a site eltment with respect to the origin must be described through a site probability density consequently reducing the probability of finding the origin elements position specifications at the site to below unity. as described in the spatial correlation function. A cluster array can be constructed in like manner through the translations of the cluster itself as a single entity. an imperfect structural organisation resulting when cluster modilications in size. density. orientation. etc.. are combined with the translations_ Dynamically such a cluster is characterised by its internal vibrations. which are localised modes with Iocalisation length .$< &_ These cluster cigenfrequencies will be intermediate between those appropriate to the structural extremes of a cluster. namely the ideal originating lattice and the dissociated cluster. In this hypothetical latter state the cluster elements are unbound from one another along the coordinates of structural imperfection. which will only be isotropic in liquids. A definition of the equilibrium state in which absolute values of local frequency are eliminated can be obtained by taking the relativmefrequency change \\ith respect to one of the ideal structural extrema [Xl_ In solids it is convenient to choose the ideal lattice for this purpose_ This state can be thought of as being generated by applying a constant stress to the equilibrium cluster in such a way as to remove the local lattice distortion (strain). with the magnitude and form of the stress required being characteristic of the type of cluster. In this way a centre is created whose struc:ure and lattice frequencies can be defined. but uhich on removal of the applied constraint will distort (strain) under the action of the local conjugate stress. The result of this cluster forming stress can be illustrated with respect to the simple example of fig_ 1_ Here an interstitial ion in the ideal lattice. fig. la, causes neighbouring ions to be displaced from their ideal lattice sites producing a reggon of focal distortion. fig_ 1b. In so doing the vibration frequencies of the affected ions about their displaced sites will be altered from those of the ideal lattice. because the local potential will be changed as a result of the lattice distortion. The particular local distortion shown in fig. lb is not however

0

ca-

(al

1. Diim tic repr-ntion of 3 dipoiar impunr) cakium fhonde cr\s~~I contaming tn~aknt impurn> Er”. and charge compensating F-. (a) Diibctment actors of ideal h~ticc ~wu~t~re due to the ccntre mdtcated b> the -o-s (b) A r~~plal diston~on of this cltsra fonmng an:rc Fig_

centrein

unique. but one of a continuous range. with each possible strained configuration resulting from the sanze magnitude of local stress and therefore being equally probable. The observed distortion must therefore be regarded as an average of all these possibilities, each of which will contribute to its formation from the ideal state_ As a result the displacement of a representative ion (site structural clement) during thr formation of the average cluster can he organised into a hierarchy of contributions in each of which it joins uith a differenr number of elements in displacing and/or rotating as a stngle body_ For example the local stress can be used to cau_sc the chosen site to displace independently of the other sites ‘;+hich act as a fixed background. The minimum number of elements involved in a group displacement notion is thus unity. and :he motion of the element is by definition free or unbounded from its environment. Similarl_v, groups of more than one element can be formed in which the elements move so as to disiplace the centr-um or rotate the axes of the group #while retaining their relative internal configuration_ The maximum number of elements that can take part in such a group oscilr’arion will be fixed by the correlation length J, and although it may be

possible to generate group oscillations with larger numbers these will be unstable and become overdamped because of the ioss of correlation of their elements. During the formation of the equilibrium cluster structure from the ideal state fluctuation each representative site wi;illtake part in the same combination of ail possible group oscillations and as a result it will describe a complicated tmjec:ory which at no point repeats itself_ At the same time the centre as a whoIe will progressively distort passing through a series of instantaneous distortions. of which that of fig_ lb is one. such that it can only be described by an avarge distortion with a lower local density than the ideal state. In the general case the coordinates in v.hich the group oscillations take place wil! be constructed from a subset only of the tota! degrees of freedom of the individual elements_ The particular subset involved u-ill depend upon the type of cluster considered and a wide range may be expected. for example. ion clusters in the channels of Hollandites will be confined to displace.ments along a single translational coordinate only. u htreas vve may expect liquid complexes to displace in all coordinates2-2 The Jtamdtonian for chster-locolised

cibrarions

It is equally possible to regard a cluster as an arbitrarily strained rcggon of idcal lattice or part& bound region of ideal liquid, with the choice of viewpomt being largely determined by the material state. The former viewpoint u-ii1 be adopted here for convenience and any pertinent differences with the latter wil! be pointed out The natural starting point for a theoretical formulation of the problem is to consider all the cluster elements as vibrating about the ideal lattice positions appropriate to the local structure. and to introduce the cluster forming stress as a perturbation_ In this state the centres of motion and axes of rotation of the elements together with their vibration/libration frequencies can be formally speckfied. it should however be noted that this state can be generated from the equilibrium cluster structure by means of an applied local stress which cancds the cluster forming stress. The ideal state in imper-

fect structured materials requires work for its formation and is thus a local thermal fluctuation, in which energy is stored in a local structural change. The cluster forming stress which generates site displacements of centres/axes of motion together with a concomitant change in vibration frequency as it converts the ideal state to the equilibrium cluster state, requires the use of creation. a,+(l). a/‘(l), and annihilation. o,(l), a,(l) operators for its description in second quant&ation form. Here the subscripts refer to the ideal state fluctuation_ f_ and a specified cluster eigenstate. c. and the operators act on a state vector so that for example a,( I) annihilates the element I at its chuster centre of motion together with its simultaneously defined vibrations about that centre for the eigenstate c. whereas a/+(l) creates the same element in its ideal crystal position with ideal vibrations_ The ideal state. which is an excitation of the cluster eigenstates can be described by operators AZ(f), A,(I), anaicgous to those of electronic excitation [25] in the sate representation. with A;(I)=n;(I)n,(l). 4(i)

= of(Oa,(O-

(la)

Although identical excitations may be generated for each element, the displacement of the centre of motion makes it impossible for more than one excitation of a given type to exist at a particular site. The excitations are therefore similar to site excitons [25] and the operators obey the commutation relationship [/ff(l).A,.(I’)J

=6,;6,,._

(lb)

In this formalism the harmonic limit can be recovered by eliminating centre of motion displacements and the stress giving rise to it. The eigenstates f and c now become successive members of the Iocal vibration stack and the excitation operators as defined in (la) arc now site phonon operators_ The site vibrations about the cluster centres of motion are desctiw by the hamiltonian, HO, with a system of units adopted in which ii is taken to be unity_

Here the site index I ranges over all elements that are potentially able to take part in a given cluster. and E,(f) is a selected cluster eigenmode with index c_ The value of E,(I) will lie between the site vibration frequency of the ideal lattice, 3, and zero, according to the number of cluster elements taking part in the eigenmode c. Since a continuum of possible eigenmodes exists each of which may be realised either simultaneously in spatially separated clusters of the array, or consecutively in the same cluster, the ideal lattice fluctuation must be referred to the average cluster frequency_ (EC)_ as ground state_ The total hamiltonian of the system must also allow for local excitations to the ideal state structure which is created from the average cluster by the operators A ‘(I). with an excitation energy ,i? per site I. and is given by

the ideal state fluctuation exists_ Although different Iypes of cluster forming centres may exist in the same material such as different interstitials or aggregates in solids or solutes in liquids, each centre of the same type will generate the same stress in the same ideal lattice- i\s a result the total perturbation available to produce any particular cluster eigenmode c at a given cenue will be a constant typical of the type of eentre_ It is useful to express this in the form ~~‘K,*12=

,

U’,

(6)

where I and I’ run over allthe sites involved in any particular distortion of a given type of centre and U is an intra-cluster intensive property. The form of H’ can be expressed more compactly by defining operators B,'(! ) and B,(I)given

by I A’(I)

B,'(I)=A,(I)=afjl)a,(Z). =
(3)

with z = tr - )-

(4)

The perturbation H’ originates nith the structural stress that causes the ideal state fluctuation to form clus:ers. It therefore acts so as to annihilate elements in their ideal state and create them with displaced eentres of motion and shifted eigenfrequencies. Each possible distortion will involve a specified set of elements, I’, in a particular cluster eigenmode c_ Since the stress that causes each element to displace is provided by the u hole set of elements involved in a particular distortion, the perturbation N’ must be written in terms of a double sum over a two-site operator V,,.. Hence H’=CCCV,,*A’(I)A(f) c I I’ x Ia:(r~)n,(r~)~a,(r~)n~(r~)]_

B,(I)=A~(I)=~~(l)a,(l),

(7)

which generate the specific cluster eigenmode t from the ideal state at site I and vice versa_ The significance of these operators ear-rbest be appreciated by referring to the example shown in fig. I, in which an impurity-interstitial pair induces site displacements in the ideal lattice 1171.In this case the operators i?:(l) create the distorted local structure illustrated in fig lb, whereas A,( I) annihilate the displacements shown by arrows in fig_ la to convert this particular distortion (c) to the ideal lattice. The operators A,(I) therefore create a structural displacement in the equilibrium cluster structure. c. Retaining the notation A-(1)/A(I) only for the creation/annihilation of the ideal state from the average cluster eigenstate. the total hamiltonian takes the general form H=~ti’(I)A(l)th‘,

(5)

where the sul~ls over I and I’ run over all the sites involved in the distortion that produces the particular cluster eigenmode c_ The excitation number operator A '(I)A(I) appears as a factor because the cluster forming stress is only non-zero when

:xXx

V,,-A'(I)A(I)[R~(I')+B,(f')].

c 1 r (8) which represents a system of excitations caused by centre-of-motion displacements from the equi-

librium cluster ground state whose hamiltonian is The furrn of the potential in which these excitations take place can be determined by defining a genera&xl coordinate, R,(I), for the centre of motion displacement that transfers element I in cluster eigenstate c into the ideal state. R,(f) is given by HO.

R,(Oa

[K(1)

f4(01.

(9)

which in the harmonic case reduces to the phonon displacement operator_ With the use of expressions (9) and (7) the first term of the hamiltonian of expression (8) is found to be the cluster average harmonic contribution to motion in the coordinate R,(I). whereas the third term originates in an anharmonic portion of the potential_ Denoting the potential for the element I as V( R(f)) its form is found to be V(R(I))=g:R’(l)i-CCh,.R’(I)R,(I’), I’ =

(10)

in which K(I) is averaged over all cluster eigenstates_ Here V( R(I)) has a minimum at the equilibrium cluster site for which R(I) is zero, and in

the absence of the displacement of other elements maximum occurs at a negative value of R(I). Hence the element can be dissociated from the cluster if suitably activated. Non-zero displacements of other elements shift the maximum and may either reduce or increase its magnitude. but do not affect the position of the minimum The motion of any single element in this potential is therefore influenced by the displacement motions of all other elements and their coupling will lead to the formation of the chrster group oscillations. The form of V(R(I)) also allows for cooperative effects in the facilitation of dissociation of an element from the cluster through the reduction or even elimination of the potential barrier_ Such processes lead to cluster fragmentation and coalescence through transfer of an eiement or displacement between clusters and may also contribute to transport properties_ This possibility will be excluded here both in the subsequent development and by the omission of the hermitean conjugate of the third term in the harniltonian. expression (S). with attention being confined soiely to the formation of group oscillations. 2

23. The ecolurion of rhe ideal stare rnro a ciuver The perturbation H’ in eq_ (5) causes the ideal state fluctaation of a particular cluster ccntre to decay temporahy through the generation of lattice distorting site displacements. described by the operators B,‘(f). The time development of the ideal state can be dete_brined from that of the fluctuation operators A *(i). A(I) given in the interaction representation by [X] id/dr(oP)=[oP,(~-~O)].

(II)

as A-(f)=i~~-(I)-_TA*(f)+CCiV;,.[B,’(I’)+B,(I’)]A’(i), c I’

(122) (I2b)

Here a phenomenoIogical damping constant r has been introduced since the ideal state is a fluctuation with an energy E per site in excess of the true ground state to which it must therefore relax by returning eneqq to the heat bath through random processes_ The magnitude of r wilI be dependent upon the detail of any specific centre [27] among which will be the maximum number of elements involved in the cluster group motions. For the example of fig 1 ionic rearrangements within the cluster will facilitate activated transfer of the interstitial between alternative sites. In general no restriction to single-phonon processes can be made and a number of alternative relaxation routes [27l will be available Consequently T may range from = 1 to many orders of magnitude smaller than 5, -&is latter being particularly the case when the

average cluster contains a large number of elements, all of which must be involved in thermal pm. In the interaction representation the time dependence of an operator is given by exp(iHerjO,

- 03)

exp(--Wcrj,

which leads directly to the unperturbed time-dependent perturbation becomes X,(r)

= exp(iH,I)(

time development of A ‘(I)

C C I$*[ &?(I’) c I’

+ B,(f’j])

,

and A(l)

in eqs. (12.a) and (12b)_ The

exp( --i&r).

04;

and the perturbed solutions to eqs. (12a) and (12b) take the Form A’(f.r)=exp[(iE-_)r]

(Isa)

exp[iX,(r)]A-(I).

A(f.r)=A(f)exp[-(iEiT)r]

(15b)

exp[-i&(r)].

which approach the exact (Heisenberg) behaviour when the cluster eigenstates in f& are the selfconsistent solutions for the strained state_ The evolution of an ideal state fluctuation at a given centre into a cluster can be described by determining the two-time correlation function
rjA’(I.

0)).

(16)

where ( ) indicates a matrix element. and f is any site in the cluster-formin, Q centre. 1 ‘sing eqs. (14) and (15) it is found [28] that

(A(f,r)A*(f,O))=exp(-iEt)

exp(-G)

exp[F(t)j(A(f.O)A’(f.O))_

07)

with

where ( ), denotes a cumulant 1281or connected diagram in a graphical representation_ The form of the perturbation X(t) is such that only terms of even order in the sum over p contribute- Since

B,‘(f’,t)=A,(f’,r)=exp(-i[~-_E,(f’)]r}A,(f’).

(19a)

B,(f’,f)=R,‘(f’,f)=exp{i[~-Ec(f’)]t3A~(f’).

Wb)

the first non-zero contribution FJ 1) is given by

*
[l -

I’

ztf,) -

-r

it -+ ( 5-

drl.

(20)

the integrals becomes

W)=CClt;l-12 c

exp(-i[c-E,(f’)](r,-rIzj})df,

&(I’)

1 -exP~ik--ml~l

1 --up(--i[r-~mb~

-

[GE,(f’#


[s-Ec(fq2

*

1 ,

=

(f’)A’(f’))

‘@

1

(f’j)

c

=

_

1 _

(21)

The correlation function eq_ (17)_ describes a situation in which the cluster elements are prepared in the ideal state at zero time. Under the influence of the cluster forming stress. the centres/axes of motion of the cluster elements are displaced and their vibration quanta reduced to EJZ’). The frequency shift, 5 - E,(I’) defines a site quanta of vibration in the displacement coordinate rather than about the centre/axes of motion, and is created by the operator _4,C(I’). Such displacement oscillations cannot exist in the centre when it is in its ideal state fluctuation and thus on!y the stress induced generation term A,(I’)Af(f’) is non-zero in F,(t). As a result the ideal state fluctuation becomes clothed [29] by localised displacement oscillations whose average frequency is E, and F?(f) is given by

In order to compIete the derivation it is only necesmry to determine the form and density of the cluster eigenstates Here use is made only of general properties of cluster formation which can be expected to apply in ai1 such systems As previously described each cluster eigenstate c describes a state of the stress center in which ail the stress has been used to distort the lattice over a number of sites whose centre-of-motion vibrations remain in resonance and hence form a localised mode of vibration_ Therefore for a given state c, E,( I’) is independent of the site I’ involved in the cluster eigenmode. and for this reason the site frequencies of the centre-of-motion dispiacemems are also in resonance and form a group oscillation of the cluster with a frequency Ed given by E&=3-E,_

(23)

cluster eigenstate is thus formed from the ideal state by the generation of a grcup displacement oscillation. and hence the same total stress is used in the formation of each group oscillation which therefore possess the same total energy_ For this reason each duster eigenstate-group oscillation represents one possible description of the distorted centre all of which have equal a priori probability. and differ only in the number of sites taking part_ Denoting the number of sites taking part in the group oscillation of frequency Ej by N( Ed). the total energy restraint gives

Each

Ed N( Ed) = constant = {_

(24)

The identification of the constant with i follows when a cluster is restricted to a single site (element)_ in which situation the site vibration about its centre oi motion must be completely decoupkd from the rest of the Iattice and its energy 1 totally converted to a site oscillation in the displacement coordinate_ The cluster eigenstates should therefore be reo,arded as possible states of the centre in the presence of an-v magnitude of stress, with each state corresponding to a specific form and range of distortion. The role played by the strength of the stress is in determining the proportion of such states that is created at each cluster site. This is given by the perturbation strength

in eq_ (22) where I’ runs over those sites involved in the group oscillation of frequency E,, and therefore implies an undisclosed dependence upon Ed_ This causes no problems however when the number of elements involved in a group oscillation of frequency E,,, eq. (24), is taken into account, together with the constant total stress, eq. (6), generating each cluster eigenstate. The contribution of a singie participating site to the generation of this group mode is then found to be

U.DUQ&/lXC/

onnalionofclxzsfff abmuw

19i

Since there must be a continuum of possible distortions each described by a sir&e group oscillation. the sum over eigenstates c in eq. (22) can be converted to an integral over the oscillation frequency E( = E,) giving &(~)=~t(Ul/s)[ir-(1

-e-‘E’)/Ejp(E)dE,

(26)

with P(E)

( 27)

= l/3-

The constant value of p(E). eq. (27). results because each group oscillation-cluster probable in the irequency range zero to C_ ‘The integral in F2(r) can be rephrased in a conceptually dearer form as Fz(r)=irfZLV(E)dE-f_N(E)(l

eigenstate is equally

-ee-‘&‘)dE.

(28)

H here

( 29)

K(E)=n/E.

is the number [30) of group oscillations of frequency E generated per site by the ideal state fluctuation_ and the numerical index is n = uz/cz_

0-J)

The first term in eq. (28) is thus the average frequency of the group oscillation which must equal E by definition. and thus fEN(E)

dE=nc=E,

(31)

giving O
(32)

It should be noted that if any residual structure exists at all in the cluster subsequent to distortion the value of n will approach unity bttt not achieve it. The second term in eq. (28) is the average number of group oscillations generated per site by the ideal state fluctuation. and is equivalent to the overlap of the site oscillator in the ideal state with a packt t of displaced oscillators where the displacement (24) itself forms an oscillator. i-e_. the group oscillation, of frequency E. in the time-dependent form the overlap (311 is given by

exp -

C [A( E)/p]‘fi

-

ebtE’ 11 =QP[

-k’[A(E)/B]‘(l

(33)

-em’“)dE/l].

E

where j3 and A(E) are the root-mean-square displacement oscilhtion respectively, with /I = (tir_ZS)?

A(E)

= (h/2I*E)?

amplitudes

of

the ideal state site oscillator

and

the

34

where Z and I’ are the momenta of inertia or mass equivalents for the site element in the two types of motion. Substituting for j3 and A(E) in eq. (33) leads to the form of the second term of eq. (28) with n a numerical constant givenby Z/Z *.

LA.

192 The

integg

Dtzmdo

/ 77te fomtat~m

of cIustcr nbratiom

in eq_ (28) can be determined anaIyticaIly 1321to give

F2(t)=iEtLFi(t).

(35a)

F;(r)=

(35b)

with

where F:(r)

-n[E,(ijf!tlnCiSr)ty;.

results from the infrared divergent number integraI [30_33]. with y Euler’s constant and E,(iSr) inted [32]_ At short times_ cz < 1, E,(z) has the asymptotic expansion

the exponential

E,(z)= -y-In(z)-

e

(-1)‘s.

(36)

p--i

and F,(t)=

- rlpr=./4,


(37a)

whereas at Iong times. SK>, 5 6(r)

= iEr - n-l-n

In(i~r),

(37b)

{f 2 5.

Substitution of F,(r)in the correlation function. cq. (17). gives (A([,

r)A’(I_

0)) = (d(f,

O)A‘(f.

0)) exp(--Tr)

exp(-nc’r’/4)

= (A([,

0)A’il.

0)) exp(--Tr)

exp( -ny)(ilf)-“_

exp( -iEf). 5115;


1;

(%a) (3gb)

v.here A ‘(I) creates the ideaI state from the average cluster state and describes a fluctuation in which the chrster elements have been displaced from their cluster ground state centres of motion aIong the distortion coordinate. with an excess energy z per site utilised to raise the frequency of the centre-of-motion vibration to its ideaI lattice vaIue of I_ The corresponding classicaI correlation function for such a displacement [34] is given by C(r)E~[(~(!_f)A-(l.0)>f(A*(I_f)A(I.0)>] =f[
~~p(-_r)Re(~~p[E(r)]).

(39)

and the time dependence of C(f) [17], for the exact function F;(r) is shown in fig. 2 with r set equal to zero for simplicity_ The function C(r) describes the time development of an ideal state fluctuation following its creation at zero time_ InitialIy &c cluster fo rming stress causes a cluster element to convert the zero-point energy of its centre of motion into the kinetic energy of oscillation along a site distortion coordinate_ This gives rise to oscillatory behaviour in C( r )_ As time progresses the stress is shared by larger numbers of sites with a smaller osciliation frequency in the disturbed coordinate_ As a result the initial state is not reproduced perfectIy over a period of the average displacement oscillation frequency E_ in C(r) this appears as imperfect recurrence cycles on the time scale 1 < gr 5 lo_ At longer times the progressive involvement of Iarger numbers of sites in group oscihations along the distortion coordinate with srnakr frequencies causes the ideal state fluctuation to decay monotonicahy following a time power law t-“. eq. (38b). In this phase of development, memory of the initiai ideaI state fluctuation is progressively lost and its excess enem, E per site, converted into kinetic energy of a range of IucaIised group oscillations involving increasing numbers of sites. At Ionger times still, t = I’-‘. the ce n tre is invoIved in an exchange of energy with the heat bath which relaxes the ideal state fluctuation, and allows the kinetic energy of the group-oscillations to be dissipated in the lattice_ &r a result alI grouposcillations wiIl be damped with the damping rate r of the fluctuation and

293 b

03 04 06 --

10’

102

10

0

IO6

Time

tgi’

Time in units of

6

16

In units of

24

Csii

Fig_ 2. The ~mc dcpcndcna of tic cvmtutionof group oscilhrions in rhc oxri3p ficror C(r). (a) P1otra.iloggrithm~crfl~sti-xns cwlution into the time poser law bAGour_ (b) Shon time bchkour plotred ImurIy to cmptiisc the ozdihroc behaviwr.

rhc

those with a frequency smaller than r will have insufficient time to establish themselves as an oscillation and u-ill therefore be overdamped_ For this reason the lower limit of *he integrals in eq. (28) should be r rather than zero. and the remaining portion will supplement the dampin, 0 term uith the conLribution [35] -(~z/23’)[rf-(1

-ehn)]

= -[U’/2lN(r)][r-(1

-eWrz)/T]_

(40)

The physical origin of this change in the damping constant lies in the structural disordering of the duster and the shift in ground state energy_ A time-independent multiplicative factor is also generated for the same reason_ The self-consistent solution for the damping constant denoted by T, is equal to the magnitude for the minimum group oscillation frequency that can be established- Substituting this value into eq- (24) thtxfore @KS the maximum number of cluster elements, &, that can establish a group osciliation as N, = N(T,)

= s/r,,

(41)

which identify the relationship between r, and cluster size_ Although r, cuts off the integmls in r’,(z) at long times. the definition of n will remain effectively unchanged because E will normally be or&s of magnitude greater than r,_ The selfxonsistent exponential damping factor of C(r) also fulfills the same role in terminating the time power law as does truncation of F,(r) in eq_ (28) and thus no exor is introduced by representing C(r) as C(r)

=+[@(I.

O)A’(I.

0)) + (A’([.

O)A(L

O))]

exp( -T,r)(st)-“_

(42)

The two time scales in eq. (42), 3-l and r; ‘. refer respectively to the onset of group oscillations and the lifetime of the group oscillation packet describing the fluctuation_ By referring all times to the longer time xale. r=-‘. C(r) can be written as C(r)=;(c/r,)-“[(A@

O)A+(Z. 0)) + (A*(l,

O)A(I,

O))]ff(Q),

(43)

u here

ff(T,t)=

(r=r)-”

exp(-rT,r),

is the decay function [17] describing the relaxation of the ideal state fluctuation_ As previously mentioned this function is an important co_mponent in the theory of relaxation in condensed phase systems [15.17_18.36] and a physical origin for it in the motions of structurally distorted centres has been suggested [15,17,37]_ Its contribution to the observed response has been verified experimentally [KS]. and provides a means of determining n for a given material_ ?he time-independent amplitude factor of C(r)

f(g/r,)-“[(A(I,

O)A’(I.

0)) f
O)A(i.

O)>l.

(45)

is the renormalised contribution of a sin& site (element) to the ideal state fluctuation of the cluster_ Using eq. (41) for the number of sites involved in clustering. the renormalised operators x(I) and A’( I) can be expressed as

IA’(U= The

(K) -““A+(f),

iA(l)i=

(N__)-“‘zA(l)_

index n therefore mcasurcs the degree

to which

(W the group

oscillations

in the centre-of-motion

displacements required to construct the ideal state from the average cluster form a normal mode of the stress

centre_ It should be remembered that when the ideal state is an ideal crystal the group oscillations are modes of vibration of the d=tortiun structure [37]_ Therefore in the limit of n approaching zero the elements displace almost independently of each other whereas when n approaches unity the elements displace almost in tinison as a normal mode of a structural entity nearly divorced from its ideal crystalline environment_ ‘The former situation may be expected of interstitials and weakly bound species such as protons in hydrogen bonds. v+hercas the latter case will be found in general for slip-planes and topological imperfections_ This feature of the cluster forming dynamics is reinforced when reference is made to fig. 2 Here the values of n approaching unity show pronounced oscillations as a normal mode in the displacement coordinate is almost established_ The qualitative relationship between this function and the spatial correlation function [24] of the dispiacenrerzr sn-ucrure follows by noting that the time development of C(r) arises from the progressive increase in the number of sites (elements) that are involved in the group oscillations_ Thus longer times are equivalent to a sampling of greater spatial separations leading to both the discrete local strncture and the progressive weakening of correlation observed. The results in this section have been obtained in a completely self-consistent framework in which: (i) all the stress of the centre has bun utilised to form each possible distortion. (ii) the necessary changes in centre-of-motion vibration frequencies have been allowed for without violating energy conservation. (iii) the total stress has been related to the average energy required for lluctuation formation through energy conservation. (iv) the lifetime of the group oscillations has been self-consistently determined as the lifetime of the fluctuation, (v) the size of the cluster has been self-consistently related to the lifetime of the fluctuation_ It is these features of seif-consistency that has made it possible to achieve a precise and comprehensive description of a complex structural entity_ Z-4. CIusren in liqaidr IJnlike the structural distortion of solids. clusters in liquids are the result of nett attractive

forces which produce the local structure of pure liquids 1241 or solute complexes. The ideal state fluctuation is formed by dissociating the cluster into independent constituent elements which col-

lide at random_ In this state the local forces are the origin of the cluster forming perturbation V,). of the previous section and generate group oscillations during the conversion of the kinetic energy of individual elements into the kinetic energy of the cluster as a rigid body. Both translational and rotational motion [35] can be involved in group oscillations but the conceptually clearer physical picture of cluster formation is given by the latter_ Here the molecules in the ideal state fluctuation can be considered as independently rotating about their molecular axes. allowing 1 to be identified with the thermally average molecular rotation frequency [38] (kT/I)‘/‘. where I is the moment of inertia. The group oscillations should be understood to be librations of the molecular axis itself rather than rotations about the axis. created by the operators A,‘(f’) with a frequency E inversely proportional to the number of molecules participating- When only one molecule. the minimum number. is involved its rotational energy is converted to molecular axis libration in a rigid structural cage of surrounding molecules consttucted as a result of the cluster forming perturbation V,,._ As the group libration extends over a larger number of molecules its localisation and frequency E decrease because of the imperfect translational symmetry of the cluster. The cluster can be rea,arded as a rigid body with an overdamped rotation about a cluster axis. when the group osciliation range covers the whole cluster. The thermally average rotation frequency of an average cluster can be identified as (E,). and will be given by (kT/P)‘r-. with I l the cluster moment of inertia. The definition of the index n given by eq. (32) shows it to be dependent only upon I and r’* and to approach zero when the degree of structural regularity in the cluster becomes small and the liquid approaches ideality25

Ihe influence of neglecred remts

These include non-zero Az(I’)A,(I’) terms in eq. (21) which allow displacements in one clus:er to be transferred to another and also the cumulams Fa( I) and higher in the series of eq. (18) which describe the effect of the residual stress/ attractive perturbations after allowing for the for-

mation of cluster group oscillations. These latter represent an effect on a weaker level than &(t) and in many problems arising from a finite range of correlation can be shown to approach zero [39,40] as the cluster eigenv-alues approach selfconsistency_ The physical features of the present model require a more careful assessment of their joint infIuenc.e to be made_ A quahtatir~e picture can be achieved by recognising that the residual perturbations rear-range the displacement coordinate of the intra-cluster group oscillations into a new composite displacement i41]. Group oscillations in this rean-anged coordinate with range .$ less than the cluster size .& are unstable with respect to the more strongly correlated group oscillations previously described. For a range greater than 5, the correlation of oscillations in the intracluster displacement coordinate is weakened [42] and becomes unstable with respect to oscillations in the rearranged coordinate. As a result a nea+ independent manifold of modes is generated in which the clusters oscillate in the rearranged coordinate with respect to one another- Such inter-chrster oscillations define group oscillaticns of the cluster QJTUJ-as defined in section 21, and their development can be described in a qualitative manner similar to those of the intra-cluster oscillations [17]_ The appearance of such a feature can be illustrated by considering a molecule to be the extreme limit of a cluster of atoms, in which three of the atomic translational degg of freedom are reconstructed into molecular rotations which generate the librational modes of a condensed system. In the case of clusters with diffuse boundaries such inter-cluster motions are equivalent to the transport of strain and in the case of fluids. molecules [lS], betueen clusters_ indeed inter-molecular librations sometimes couple to intra-molecular motions even in van der Waals crystals ]43]_ As a result the inter-cluster motions cause the structure and density of the clusters within the average array to fluctuate in time so as to adopt all possibilities with a given steady state probability density [15,17]. When applied to liquids the molecular transport involved causes temporary fragmentation and coalescence of clusters_ The results of the preceding sections and particularly the index n should be understood to relate to the average cluster in the

steady state distribution [15&l]_ A more detailed consideration of these points will be found in ref_ 1151.

3_ The thermodynamics of duster formation In the prccedmg section a microscopic description of cluster formation has been given in terms of the dynamic development of a stressed ideal structure into a region of strain. In this process the vibrations of cluster elements about ideaI structure sites are converted to vibrations with altered quanta about the displaced centres of motion in the strained lattice together with group oscillations of the cluster along the displacement coordinate_ The vibrational frequency of the system is thereby partitioned between vibrations about centres of motion in a subset of ideal state coordinates. and group oscillations which displace the centres of motion along coordinates constructed from the same ideal state subset, with index n defining the fraction in the group oscillation_ Thus n describes the degree to u hich motions in a particular coordinate become unbound from the idcal lattice in crystals or bound together in liquids. It was also demonstrated that n defined the extent to which group oscillations of the cluster could be regarded as a fully correlated normal mode of the cluster. and its relationship to the structural reggularity of the cluster outlined_ It should be remembered here that in distorted solids the cluster has been defined as the dirtorrion structure of the ideal lattice. It is this interpretation of n that is the basis of the definition derived in eq. (32). in which n is given by the average fractional change of ideal state frequency on cluster formation_ This frequency change is equal or proportional to the amount of energy required per cluster eIement to generate the ideal state from the average cluster_ As a result the cluster binding energy per element can be identified as proportional to .??(=n{) and will be the energy required to activate an element from its average cluster ground state whose energy is proportional to (E,). into the ideal state fluctuation which can be thought of as the dissociated cluster_ During formation of the cluster from the ideal state fluctuation the binding energy per element is

converted from the zero-point motions or free rotations into group oscillations. which therefore contain the energy required to dissociate the cluster in the form of inrra-cluster excitations of the cluster ground state_ As a result the process of cluster evohrtion described in section 2 follows the increase in cluster heat content, AH,(,), as pro gressively wider-ranged group oscillations are generated. At a time I = rC-’ all the energy required to dissociate the cluster will have been converted to a cluster heat content aH(r;‘) surplus to the cluster ground state energy_ with M-f( r:

’ ) a ng_

147)

Over the same period of time the system will have suffered a change in configurarionai entropy. UC,< I), as the idcal state evolves into a package of cluster states. with the overlap factor of eq. (39) given by the probability of finding each possible cluster state in the package- Just as the first integg of cq. (25) gives the av-erage energy shift per cluster element_ the second integral which gives the total number of ways of sharing the ideal state energy (5) of a single element among groups of elements selected from the cluster. defines the change in cluster configurational entropy per element as a function of time. After allowing for the truncation of the integral at the value of E for which the cluster attains maximum extent and the group oscillations become overdamped. namely r,. the configurational entropy change is given by as,(

r;‘)/@

=/;X( c

E)dE

= n In({/r,)

= n ln( iv,).

(4Sa)

leading to a&(

r=-‘)/k

= nZVc ln( IV,)_

(4Sb)

If the number of cluster elements N, is large. UT,_(r;‘> approximates to the well-known form a&_( r=-‘)/k

- n ln( N,!).

(49)

giving a statistical weighting PC of PC= (N,!)“.

(504

LA. Dusado / 7%~/ ormonon

I

which becomes Z’, = (NC)“*‘*.

(50b)

when N is smaJL Eqs. i47) and (4Sb) show that the magnitudes of the changes in enthalpy and configurational entropy that accompany cluster formation are a fraction R of the amounts required to dissociate a perfectIy ordered structure into its randomly ordered component elements_ Under conditions of constant temperature rhe change in Gibbs free energy. AG, occurring when the ideal state has fully evolved into the cluster together with attendant group oscillations, at I = rC-‘. is given by ZtG(r;‘)=M(r,-‘)-ms(I--‘)=O.

(511

The latter equality follows as a result of the constant energy constraint applied to this phase of cluster evolution in section 2. and the identification of MT(T;‘) and JS(rC-‘) as identical fractions. n_ of the dissociation enthalpies and configurationai entropies_ The zero vaJue of AG applies for all times I < rC- ’ and implies Lhat this phase of evolution is thermodynamically reversible. The damping of the group oscillation at t = rg t however arises from an irreversible transfer of cluster enera to the heat bath The surpius heat content of the ideal state fluctuation. now con\-erted to cluster group oscillations is therefore removed from the cluster giving M(r>r,-‘)=o.

(52)

and the Gibbs free energg change at times t > r:’ becomes 1G(r>T,-‘)=

-7iiS(-r;‘).

(53)

when the effect of the higher cumulants in forming a cluster distribution (441 are neglected_ The microscopic description of cluster formation presented in section 2 has therefore been demonstrated to obey the second Jau of thermodynamics as is necesmry if the stressed ideal state is to spontaneously evolve into clusters. The duction in the Gibbs free energy accompanying this process has also been expressed as a change of configurational entropy as is appropriate to the imperfect structure of the clusters as defined.

of clzuter

idrottons

197

Summary and condusions

A dynamicaJ definition of clustering in structurally imperfect condensed phase materials has been given in terms of a ftite sized group of elements whose Jibration/vibration Frequencies about a site ccntrc/axis of motion remain in resonance although shifted from those of the perfect structure. The formation of such cJusters from a stressed ideaJ state has been followed using infinite-order time perturbation theory and shown to be equivalent in form to the overlap of the ideal state oscillator with a continuum of displaced oscillators representing ali JxxsibJe ciuster structures. In this approach, motion along the displacement coordinate itself forms into oscillations of groups of cIuster demerits, which become overdamped when all the cluster is involved_ in this. way the vibrational energy of the ideal state in coordinates that combine to form the distortion is partitioned between the group oscillations and the residuai centre/axes of motion vibrations. with the fraction n in the group osdlations determining the approach of the displacement motion to a normaI mode of the distortion structure_ The change in configurational entropy resulting from cluster formation has been derived from the same theoretical formalism and is the same fraction n of the change that would occur if the perfect state were fully dissociated/associated. A similar result applies to the increase of enthalpy required to form the ideal state by dissociating the cluster of distortion_ The fractional index n is independent of cluster size and is thus an intensive property of the cluster. Its definition in eq. (30) shows it to be determined solely by the type of cluster forming centre, and its influence upon the displacement group oscillations, eq. (46) and fig. 2. indicates its relationship to me perfection of structuraJ reguJarity in the site displacements It can be concluded therefore that reggons of imperfect structure in a condensed phase material recjuire both a size. denoted here by the number of elements N,, and an intensive property, the fraction n, for their compiete deftition. The physical origin for the index n lies in the form of the structuraJ correlation function, which it describes

in a

coarse grailled manner [15], and allows structural regularity to be linked to cluster dynamics, group oscillation sperm and formation thermodynamics_ It is experimentally accessible from frequency-dependent response measurements, and wiil provide an exact measure of the rype of structural imperfection that exists. since it will have different ranges of values for the various materials that can occur such as -gIassW, “amorphonC. “plastiC. etc_ The formalism that has been outlined here is primarily intended to facilitate the interpretation of experiments in which structural relaxation occurs by describing the manner in which correlated displacements are formed in regions of structural irreo,ularity. In the process it has aIso been demonstrated that a microscopic approach to the dyzunics of imperfect structures can act as a bridge to the thermodynamics of cluster formation, when deviations from the perfect state properties are incorporated in a realistic and self-consistent manner under the constraint of a constant total energy_ Therefore if the model were to be applied to a particular system in detail so that parameters such as the ideal lattice coordinates involved in clustering, the number density of clusters. r. etc_ couid be determined, the possibility arises of interrelating relaxation properties with the residual entropy ami specific heat of the system_ The seIfcorsisten~ constraints that have been introduced here as a feature of the model should however be regarded as a guide to the necessary and sufficient conditions for the formulation of a more sophisticated self-consistent 139,401description of cluster formation in imperfectly structured materials of a general character_

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