THE LATTICE GAS INTERFACE
A contribution to the theory of crystal growth
/
C. van Leeuwen
P1129
7057
C10025
73986
THE LATTICE GAS INTERFACE
A contribution to the theory of crystal growth
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOC-T O R IN DE DOC-TECHNISCHE WEDOC-TENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS PROF. IR. L. HUISMAN, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DE-KANEN, TE VERDEDIGEN OP WOENSDAG 20
APRIL 1977 TE 16.00 UUR
DOOR
CORNELIS VAN LEEUWEN
scheikundig ingenieurgeboren te R o t t e r d a m
/^^9 y^^7
KRIPS REPRO MEPPEI
\ % DOELENSiR.101 -'^'
BIBLIOTHEEK TU Delft P 1129 7057
DIT PROEFSCHRIFT I S GOEDGEKEURD DOOR DE PROMOTOR DR. P . BENNEMA
AAN COSY, TEN HALVE SLACHTOFFER, TEN HALVE MEDESCHULDIG. VOOR MARC
Met dank aan allen, die aan de totstandkoming van dit proefschrift hebben bijgedragen.
VOORWOORD EN SAMENVATTING
De recente toepassing van de Monte Carlotechniek in het vakge-bied kristalgroei heeft het inzicht in de processen verdiept en de uitbreiding van theorieën gestimuleerd. Dit proefschrift geeft een overzicht van deze ontwikkelingen, waarbij het accent ligt op veelal in samenwerking met collega's door de auteur verkregen resultaten.
De dissertatie bestaat uit twee gedeelten, elk van drie hoofd-stukken. In het eerste gedeelte wordt het grensvlak tussen twee fasen in thermostatisch evenwicht behandeld; in het tweede gedeelte wordt de niet-evenwichtstoestand bekeken. Beide gedeelten beginnen met een algemeen inleidend hoofdstuk. Specifieke onderwerpen worden apart ingeleid in het desbetreffende hoofdstuk.
Als model voor een unair systeem kiezen we het roostergas. Dit eenvoudige statistische model wordt ingeleid in hoofdstuk 2. Hierin worden ook enige eigenschappen van het roostergas berekend in een Bragg-Williamsbenadering. Eveneens wordt aangegeven op welke wijze de thermodynamische funkties van het grensvlak te berekenen zijn. Vervolgens wordt de vast-boven-vast-benadering in het roostergas in-gevoerd, waardoor het op een Kosselkristal (blokkendoos) gaat lijken. In hoofdstuk 3 worden enige resultaten van Monte Carlosimulaties ge-presenteerd en besproken.
De laatste drie hoofdstukken zijn gewijd aan kristalgroei, met de nadruk op de groei van Kosselkristallen. In hoofdstuk 5 wordt de groei beschreven van perfekte oppervlakken. Speciale aandacht wordt besteed aan het kristalgroeimechanisme, waarbij groei plaats vindt door de vorming van twee-dimensionale kiemen. In hoofdstuk 6 wordt de groei van kristallen met treden beschouwd. Hierbij worden het kon-tinuüm model en het Monte Carlomodel vergeleken.
PREFACE AND SUMMARY
The recent application of the Monte Carlo technique in the field of crystal growth has led to a deeper imderstanding of the processes and has stimulated the development of theories. This thesis reviews these developments, emphasizing the results which were obtained by the author mostly in cooperation with his colleagues.
The dissertation consists of two parts, each containing three chapters. In the first part the interface between two phases in ther-mostatic equilibrium is dealt with. In the second part non-equili-brium states are considered. Both parts start with a general intro-ductory chapter. Specific subjects art introduced separately in the relevant chapters.
The lattice gas is used to model an imary system. This simple statistical model is introduced in chapter 2. Some properties of the lattice gas are calculated in a Bragg-Williams approximation. The method of calculating interface excess thermodynamic functions is indicated. The lattice gas becomes similar to the Kossel crystal when particles are only allowed on top of other particles. In chap-ter 3 some results of Monte Carlo simulations of this restricted model are presented and discussed.
The chapters 4, 5 and 6 are devoted to crystal growth with the emphasis on the growth of Kossel crystals. Chapter 5 describes the growth of perfect interfaces. Special attention is paid to the growth mechanism of two-dimensional nucleation. In chapter 6 the continuum model and the Monte Carlo model for the growth of stepped interfaces are compared.
CONTENTS
VOORWOORD EN SAMENVATTING V PREFACE AND SUMMARY VII
CONTENTS VIII INTERFACES: AN INTRODUCTION 1
1 Phase transitions 1 2 Plane interfaces 2 3 Geometrical thermodynamics 6
LATTICE GAS INTERFACES: PART I 9 1 Introduction / 9 2 The homogeneous lattice gaé 11 3 The two-phase lattice gas 16 4 The restricted lattice gas 18
5 Discussion 20 LATTICE GAS INTERFACES: PART II 24
1 Introduction 24 2 The restricted lattice gas: isotropic interactions 26
3 The restricted lattice gas: anisotropic interactions 36
4 Stepped interfaces 40
5 Discussion 47 CRYSTAL GROWTH: AN INTRODUCTION 51
1 The growth affinity 51 2 Metastability 53 3 The growth process 56 THE GROWTH OF PERFECT INTERFACES 60
1 Introduction 60 2 The nucleus 67 3 Nucleation growth 75 4 Discussion 80
6 THE GROWTH OF STEPPED INTERFACES 87
1 I n t r o d u c t i o n 87 2 The p a r a l l e l sink approximation 89
3 Stepped l a t t i c e gas i n t e r f a c e s 91
4 D i s c u s s i o n 9 5
1 INTERFACES: AN INTRODUCTION
1.1 PHASE TRANSITIONS
Let us consider a unary system of N particles of a simple sub-stance A with a volume V. The state of this system is determined by the state variables pressure (p), temperature (T) and density (c). In principle the state variables are related by the equation of state, i.e. only two of the state variables can be chosen indepen-dently, or -according to the phase rule- there are generally two de-grees of freedom. We can also say that p, T and c are the thermody-namic coordinates, representing a point in p-T-c space.
According to thermostatics two phases (denoted by ' and ") can coexist in stable (thermostatic) equilibrium (in absence of external forces), when there is thermal (T' = T " ) , hydrostatic (p' = p") and chemical (y' = y") equilibrium. Here y' is the chemical potential of component A. In two-phase equilibrium, a unary system has one degree of freedom, i.e. the points representing the coexistence lie on lines -the coexistence lines- in the p-T projection of the p-T-c space. When the solid phase occurs in only one modification, there are three two-phase curves: the sublimation curve, the fusion curve and the vaporization curve. The three two-phase curves meet in one point, the triple point, where all three phases coexist.
The vaporization curve terminates in a critical point (c' = c " ) . Below the critical temperature the gas phase is usually called vapor phase. The existence of a critical point implies that a vapor can be converted into a liquid (via the gas phase) without crossing the phase transition line. It is widely believed that the fusion curve does not also terminate in a (second) critical point . In this way the vapor-liquid equilibrium distinguishes itself; it is called a fluid system. It seems not unreasonable to assume that the third party -the solid phase- is responsible for this difference.
2)
According to Ehrenfest first and second order phase transi-tions are defined by the occurrence of discontinuities in all first and second order derivatives of the Gibbs potential G, respectively,
discontinuities in G being unphysical. Now let us consider the ex-trapolated curves G'(p,T) on both sides of the transition point. In a first order phase transition the Gibbs potential must and can fulfil G < G', and thus a positive latent heat is warranted. In a second order transition always G > G' at the low heat capacity side of the transition point, i.e. there must be a natural reason why the phase of high heat capacity can have no extrapolated existence
3) beyond the transition point
Since G(p,T) is continuous, a phase transition may be accom-plished by (i) the formation of (infinitesimal) small areas of new phase -called clusters- with different properties (eg. specific volume) or (ii) formation of new phase throughout the system with
4)
the same properties (at the transition point) . It seems reasonable that in a second order transition the new phase is not formed at the transition temperature but before it. This temperature only marks the complete conversion. An "abnormal" rise in heat capacity before the transition point accounts for the formation of new phase in the old one. In a dynamic description of a first order phase transition we may distinguish the formation and the subsequent growth of clus-ters. In the field of crystal growth these processes are usually called nucleation and growth.
Above the boundary between the two phases of a two-phase system, the interface, is disregarded. This is generally justified when we are interested in the overall extensive thermodynamic quantities, because these are not notably influenced by the contribution of the interface. Nevertheless, some properties of a two-phase system are solely due to the existence of an interface, eg. the isothermal com-pressibility of both separate phases composing the system is finite, whereas for the two-phase system it is not. When we want to describe an interface in thermodynamics, it is not allowed to treat it as a separate system because it possesses no identity of its own: the in-terface is merely the consequence of two phases in the system.
1.2 PLANE INTERFACES
fluid interfaces , Perhaps the best method is due to Goodrich Again we consider a two phase unary system. Combining the first and second law of thermodynamics we may write for the total system:
dE = TdS - pdV + ydN + TdA, (1.2.1)
where A is the interfacial area and i is an intensive variable, which is thermostatically conjungated with A. The other symbols have their conventional meanings. Next we assume that the system is built up by imaginary slices perpendicular to the interface with entropies dS, volumes dV, etc. Or in other words, we suppose that the system can grow in extent from a suitable external source while its nature and state remains constant. Eq.(1.2.1) can now be integrated to give
E = TS - pV + yN + FA (1.2.2)
From eqs.(1.2.1 and 2) Gibbs-Duhem's equation can be obtained:
SdT - Vdp + AdF + Ndy = 0 (1.2.3)
The above equation for one mole of homogeneous ' and " phase at the same temperature and pressure reads:
s'dT - v'dp + dy = 0 (1.2.4a)
s"dT - v"dp + dy = 0, (1.2.4b)
where s and v are molar quantities. Goodrich multiplied eq.(1.2.4a) by x, eq.(1.2.4b) by y and subtracted both from eq.(1.2.3). The re-sulting equation is valid for any value of x and y. The multipliers can be chosen in such a way that the terms in dp and dy vanish. This gives:
dr
- A — = (S - xs' - ys") E S (1.2.5a) dT
with
X + y = N (1.2.5b)
and
XV' + yv" = V. (1.2.5c)
The general idea is that we have obtained an interface excess ther-modynamic function; here the interface excess entropy S. Specific interface excess thermodynamic functions (i.e. per unit interface area) are independent of the extent of the system. In a similar way an expression for the interface excess energy can be derived. Now the interface excess free energy F, defined in the usual way
F = È - TS (1.2.6)
turns out to be •
F = FA (1.2.7)
Thus for the multipliers defined by eqs.(1.2.5b and c) F equals the specific interface excess free energy. In the next chapter we will need the relation between the interface excess of the grand potential Ü, defined by
fi = F - yN (1.2.8)
and F. It is easy to see that for the multipliers chosen above
n = F A (1.2.9)
Nowhere in the above derivation the actual position of the interface enters the argument. And indeed interface properties should not de-pend on it. Now we will briefly discuss the relation between
Goodrich's method and two other methods of defining interface excess thermodynamic functions.
7)
According to Gibbs , interface excess thermodynamic functions can be obtained from a geometrical construction. Somewhere in the system we draw a plane parallel to the interface. This dividing sur-face divides the volume V into two parts V' and V". We assume that any extensive thermodynamic function equals the sum of the contri-butions of the interface and parts ' and ". Hence, the interface excess volume V = 0. Moreover, we require that the interface excess of moles N = 0, thus fixing the position of the dividing surface. Above nothing has been said about the position of the interface,
8 ^ which does not necessarily coincide with the dividing surface . It follows from Goodrich's description that extensive thermodynamic functions labeled ' should now be interpreted as to relate to a number of moles N' in a volume V' of a phase which remains homo-geneous up to the dividing surface. Clearly, the dividing surface does geometrically what Goodrich's multipliers do algebraically. Therefore, there are more dividing surfaces, each one corresponding to a particular choice of the multipliers. The one discussed above is commonly referred to as Gibbs' equimolar dividing surface.
9)
A method due to Guggenheim implies the creation of a new mental phase: the interface phase. It extends from somewhat below till somewhat above the real interface to places where the density gradients vanish. The interface phase has its own volume. In Good-rich's terminology we would indeed need two dividing surfaces to create the geometrical analogy of the case in which the multipliers were chosen to eliminate all but the dp terms.
The above treatment was given for fluid interfaces. When one of the phases is crystalline, the interface thermodynamic functions become dependent on the orientation of the interface. Another con-sequence is that the area of the interface can be changed now in two ways: (i) by making new interface of the same nature and state, and
(ii) by deforming the interface lattice. It is clear that the work term introduced in eq.(1.2.1) corresponds to the first manner. Generally, it is applied to fluid interfaces. When the interface
lattice is also deformed the interface work is rather given by dFA instead of FdA. Putting gdA = dFA gives
g = F + A ^ • ^ (1.2.10)
Following Herring we have assumed here that g is an interface excess thermodynamic function. A complete derivation reveals that eq.(1.2.10) is only valid in the case of high symmetry crystal sur-faces . g is then a diagonal element of the symmetric, diagonal surface stress tensor. It is commonly called the surface stress. It is interesting to read Linford's comparison of the naming of in-terface thermodynamic functions by various authors, to get an idea of the wide-spread confusion in this field. In the field of crystal growth one considers local surface stress fields in connection with
12) 13) dislocations and with crystal surface steps
1.3 GEOMETRICAL THERMODYNAMICS
When one of the phases has a lattice structure the interface excess thermodynamic functions depend upon the orientation of the interface, i.e. some interface orientations may be more favourable than others. A crystallite will be in stable equilibrium if the surface integral of the excess interface free energy is minimal at constant crystal volume. The shape of the crystal which fulfils this condition is called the equilibrium form. When the specific inter-face excess free energy f is known for all orientations of the in-terface, the equilibrium form can be found from a geometrical con-struction. A fundamental concept in geometrical thermodynamics is the Wulff ' plot: the set of all tips of end-points of radius vectors with length f(9,$) (<}) and 9 are polar coordinates). Planes drawn through the end-point of a radius vector and perpendicular to the radius vector are called Wulff planes. The inner envelope of all Wulff planes compose the equilibrium form. The proof of this theorem is rather complicated and the reader is referred to Landau and Lifshitz . The construction can also be reversed, i.e. the Wulff plot can be obtained if the equilibrium form is known. However, this
can only be done if the equilibrium form possesses continuously turning tangent planes. In this context it is important to note that only if the Wulff plot shows cusps (singularities), flat faces are present on the equilibrium form. And conversily, when the equilibrium form has flat faces, cusps must be present in the Wulff plot. Faces corresponding to singular and near singular orientations in the Wulff plot are called singular and vicinal, respectively. All other
laces are called non-singular.
Now suppose a crystal contains a plane not belonging to the equilibrium form. Obviously, the system can reduce its free energy by changing the shape of the crystal to the equilibrium form. Such a change may require the transport of a "large" amount of material as compared to the process of faceting. To answer the question whether a face of any orientation is stable against faceting, we con-sider the corresponding Wulff plane. Next we shift this plane paral-lel to itself until it becomes a contact plane of the equilibrium form. Whenever the distance f'(6,(ti) from the origin to the contact plane is smaller than the distance f(9,(t)) to the Wulff plane, the face is unstable against faceting. The specific interface excess free energy of the faceted face is f'(9,(j)). The orientation of the facelets of the facets correspond to the orientations of faces on the equilibrium form. When f'(9,(J)) = f(9,(j)) the face is said to be stable against faceting. So all singular faces are stable against faceting.
REFERENCES
1 H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Clarendon, Oxford, 1971).
2 p. Ehrenfest, Commun. Kamerlingh Onnes Lab. Univ. Leiden, Suppl. 75b (1933) 7.
3 J.E. Mayer and S.F. Streeter, J. Chem. Phys. 7 (1939) 1019. 4 R. Brout, in: Fazovie Perekhodi (Mir, Moskow, 1967) preface. 5 R.G. Linford, in: Solid State Surface Science, vol. 2, Ed.
M. Green (Marcel Dekker, New York, 1973) p.1.
E. Matijevi6 (Wiley, New York, 1969) p.1.
7 J.W. Gibbs, The Scientific Papers of J. Willard Gibbs, vol. 1, (Longmans-Green, London, 1906) p.219.
8 R.C. Tolman, J. Chem. Phys. 17 (1949) 118. 9 E. Guggenheim, Trans. Faraday Soc. 36 (1940) 397.
10 C. Herring, in: Structure and Properties of Solid Surfaces, Eds. R. Gomer and C.W. Smith (Univ. Chicago Press, Chicago, 1952) p.5.
11 R. Shuttleworth, Proc. Roy. Soc. A63 (1950) 444.
12 I. Sunagawa, P. Bennema, B. van der Hoek and J.P. van der Eerden, poster presented at the ECCG-1 in Zurich (1976).
13 J.M. Biakely, Introduction to the Properties of Crystal Surfaces, vol. 12, Ed. W.S. Owen (Pergamon, Oxford, 1973) p.104.
14 G. Wulff, Z. Krist. 34 (1901) 449. 15 C. Herring, Phys. Rev. 82 (1951) 87.
16 L.D. Landau and E.M. Lifshitz, Statistical Physics, vol. 5 (Pergamon, Oxford, 1969) p.458.
2. LATTICE GAS INTERFACES: PART I
2.1. INTRODUCTION
Following Goodrich's arguments concerning interfaces in unary systems, we cannot simply model an interface. Instead we have to model the two phase system and both corresponding one phase systems. From a statistical point of view all three systems are similar. For a system with identical particles the configuration integral in the canonical partition function can be approximated by a configuration sum over all configurations if we assume that the particles fill the cells of an imaginary lattice. Here we will use the lattice gas
2) model :
( i ) The system volume i s d i v i d e d i n t o a l a t t i c e of c e l l s . ( i i ) Each c e l l i s e i t h e r f i l l e d or empty.
( i i i ) Nearest n e i g h b o u r i n t e r a c t i o n s e x i s t between f i l l e d c e l l s .
We d e f i n e the f o l l o w i n g q u a n t i t i e s :
y chemical p o t e n t i a l of a f i l l e d c e l l e n e a r e s t neighbour p a i r p o t e n t i a l N number of broken bonds
N- number of f i l l e d c e l l s N number of empty c e l l s
N number of p a i r s of f i l l e d c e l l s M number of c e l l s
V c e l l volume
q cell coordination number
It is frequently easier to work in an open system. The appro-priate partition function for such a system is the grand partition function H, given by
where the summation is performed over all configurations and the con-3)
t r i b u t i o n of k i n e t i c energy i s o m i t t e d . Following H i l l we use M as t h e e x t e r n a l parameter i n s t e a d of V = Mv. The grand p o t e n t i a l fi = -pM (now t h e p r e s s u r e p has the dimensions of energy) i s r e l a t e d to H by
f2(y,M,T) = - kT In - ( 2 . 1 . 2 )
The potential energy E of a particular configuration is simply N e, or using the broken bond convention
E = (4qN^ - JN^^) £ C2.1.3)
Hence the grand partition function can be expressed as
5 = E exp {(y - Jqe) N^/kT + èN^^E/kT} (2.1.4)
As yet it has not been possible to evaluate 5 analytically for the three dimensional lattice gas without introducing approximations. Examples of such approximations will be discussed below: the mean field approximation in section 2.2 and the low density series expan-sion method in section 3.2. For small systems an exact result can be
4)
obtained by generating all configurations . However, we can also estimate an expectation value by averaging over a limited number of configurations. As unbiased selecting of configurations gives very poor results, one is forced to use a biased selection scheme. The bias is designed to favor those configurations which make the impor-tant contributions in the evaluation of the expectation value. In importance sampling techniques, the configurations may be chosen to give a minimum variance estimate, but usually one directly uses the
5, 6)
distribution of the statistical mechanical ensemble . Then the averages over the sample ensemble are simply arithmetic means. As there does not appear to exist an equivalent of the "urn" containing
7)
the required distribution, a Markov chain of configurations is generated, with the property that in the limit as the length of the sequence becomes infinite, the probability of occurrence of
configu-rations converges to the correct statistical mechanical one. When P(liJ.) is the probability of occurrence of a conf iguration ijj. and P.
j J J"' is an element of the stochastic matrix of the process, i.e. the con-ditional probability that i|i . is transferred to ip , then the conver-gence of the stationary process is ensured as
P(ijj ) P , = P(.4i, ) P, . for all j, k (2.1.5)
3 Jk "^k kj
E P., = 1 for all j (2.1.6) k J''
provided the configurations are all in the same ergodic class. We will apply the above method of importance sampling, using the distri-bution of the statistical ensemble and refer to it as the Monte Carlo
(MC) simulation technique. In the appendix somewhat more detailed information is given.
Above we have tried to describe the generation of a Markov chain of configurations as a mathematical trick to obtain a proper distri-bution of configurations. In principle it should not be mixed up with a dynamic approach of the problem, although it can be conceived as such. The master equation describing the relevant time dependent stochastic process of Markov cannot be solved analytically and these processes can be simulated equally well with the Monte Carlo tech-nique (see section 5.1). Under equilibrium conditions the solution of the master equation is expected to converge to the solution of the equilibrium state. We have used this method to calculate some proper-ties of the interface in the restricted lattice gas in a first order mean field approximation.
2.2. THE HOMOGENEOUS LATTICE GAS
We will evaluate H for a homogeneous lattice gas in a mean field approximation, i.e. we assume that N depends on N only. Hence, we sum over states rather than configurations. This results in a combi-natorial factor P in H
P-J
c
I i _
0 0.5 1
Fig. 2. 2.1. This figure shows an isotherm (T - -qe/Sk) of the lattice gas, calculated in a 0th order mean field approximation (eq. (2.2.4)). The solid branches correspond to stable states, the dashed part of the curve corresponds to unstable states.
-0,5
-1.0
-1.5
2 4 6
Fig. 2. 2. 2. The y-T projection of the spinodal and the coexistence line of the lattice gas divide the y-T space in four regions: (A) stable dense phase, (B) stable dense phase and metastable dilute phase, (C) stable dilute phase and metastable dense phase, and (D) stable dilute phase. The curves are the result of 0th order mean field calculations (eq. (2.2.5)).
il)
(2.2.2) In a zeroth order or Bragg-Williams approximation N is given byN^P = qN^(M - N^)/H ' (2.2.3)
In this approximation any cell forms pairs in accordance with the average composition, eg. a filled cell has qN /M empty neighbours, thus the number of broken bonds in the system is N qN /M.
In the thermodynamic limit (M ->•<») H is sharply peaked for a particular value of N . Taking only this term H (N ) into account
in eq.(2.2.1), we can find E by maximizing 5 with respect to N . However, minimizing Q, = - kT In E is more convenient since Stir-ling's formula for logarithms of faculties can be applied. Stationary points in Ü fulfil the equality
qEc - y + kT In -^- = 0 (2.2.4)
where c = N./M.
Eq.(2.2.4) is the equation of state of this lattice gas system. In Fig.2.2.1 an isotherm is presented for T = -q£/8k. The solid lines
2 X 2
correspond to stable states (d Q /dN > 0 ) , the dotted line to un-stable states. At the stationary points, fi (N ) has a saddle point. The set of all transition points form the spinodal. In Fig.2.2.2 the T-y projection of the spinodal, given by
;, = ^ {1 ^ •l^4kT/q£} ^kT in ^ ^ *"' ^ ^'^^^J^ (2.2.5) 1 + /l + 4kT/qE
9) is presented
The phases described by the left and right branches in Fig.2.2.1 show a difference in density; we will distinguish their coordinates by the superscripts ' and ", respectively. Both types of phases are in thermal equilibrium (T' = T") and can be in chemical equilibrium (y' = y") because of the overlap in y. The low and high density
08
OM
O
-0 4
.1.6 -1 -0.4
Fig.2.2.3. The chemical potential dependence of the lattice gas pressure at T = -qe/Bk in a 0th order mean field approximation
(eq. (2.2.6)).
0.50'
0.25-0 0.25-0.5 1
Fig.2.2.4. Temperature-density projection of the p-T-c space figure of the lattice gas, calculated in a 0th order mean field approxima-tion (eq. (2.2.4)). The dashed line is the projecapproxima-tion of the spinodal
phases coexist when they are also in hydrostatic equilibrium, i.e. when p' = p" or Q' = fi" where
n = M [ (èqe-y)c- iqEc(l-c) + kT{(l-c)ln(l-c) + c In c} ] (2.2.6)
This condition is fulfilled when
y = iqe E y'^ (2.2.7)
This is the coexistence condition in the lattice gas model. It is not so easy to see that this is the only solution. Therefore we plotted p versus y, also for T = -qe/Sk (Fig.2.2.3). Clearly, parts of the "stable" branches in Fig.2.2.1 are metastable.
The coexistence condition can be obtained also by considering the process of condensation starting at a point on the left branch in Fig.2.2.1. Physically the condensation terminates at a point on the right branch and the process follows a straight horizontal line in Fig.2.2.1, since dy = 0 during the process. The final state can be reached also by following the path given by the equation of (here un-stable) state of the lattice gas. Since both states are stable states we equate
c" c"
I ydc = y'^ j dc (2.2.8) c' c'
where y in the left hand side is given by eq.(2.2.4) and the right hand side is the physical result. Now we have obtained Maxwell's
geo-10)
metrical construction . Because of the symmetry properties of eq. (2.2.4) it can easily be seen that again the coexistence condition is given by eq.(2.2.7).
The coexistence condition can be derived also in a much simpler and more general way, although it is more difficult to understand its physical meaning. Following Leamy, Gilmer and Jackson we state that in an open lattice gas system two phases coexist if the probabi-lity of finding any configuration is equal to the probabiprobabi-lity of finding its filled/empty inverted configuration. Since the number of
broken bonds is essentially the same in both configurations, the factor of N , in eq.(2.1.4) must be zero.
Upon defining T = -q£/4k, for coexisting phases eq.(2.2.4) has
(i) one solution for T > T c
(ii) two coinciding solutions for T = T (iii) three solutions for T < T .
c
The solutions for T <^ T , corresponding to minima in Ü , are shown in Fig.2.2.4. This figure is the T-c projection of the p-T-c space figure. Clearly the lattice gas model has a critical temperature T .
2.3 THE TWO-PHASE LATTICE GAS
The object of treating a lattice gas with uni-directional densi-ty differences is to obtain interface thermodynamic functions of
9,12)
planar interfaces . So we must be able to model homogeneous phases at the same y, p and T, too (cf. section 2.2). This
necessa-c
rily implies that y = y for all three systems.
We divide the system of M cells into layers of N cells and as-sume that the layers themselves are homogeneous, N . being the number of filled cells in the ith layer. Distinguishing pairs of cells and pair potentials within and between the layers by the superscripts xy and z, in the zeroth order mean field approximation E can be express-ed as
= = En P^ exp f4N^Q ^ E^VkT + i N^ E^/kT} (2.3.1) i=_co
P, =( J 1 (2.3.2)
% i = * ^ ^ ^ i ^ ' ^ - ^ i . l > ^ ^ i . l < ^ - ^ i ) ^
+ N., . (N-N, . , ) + N., . ., (N-N., . )}/N (2.3.4) I l 1 1-1 1 1-1 1 1
Again we seek { N .} which fulfils the requirement that dfi /dN . = 0 for all i. This gives a set of coupled difference equations
, xy xy ^ z z c
1
Two types of solutions f u l f i l these equations
( i ) ^\, • ~ ^ ~ ^u ^ even solution (2.3.6)
( i i ) c, = è; c. . = 1-c, . odd solution (2.3.7) k k + i k - i
In the even and odd solution k i s an a r b i t r a r y constant. The boundary conditions for the difference equations are
lim c. = c' and lim c. = c" (2.3.3)
1 1
where c' and c" are the densities of the homogeneous phases at the
same y, p and T.
We will now consider even solutions because they correspond to stable states, {c.} can be found numerically by iteration on c till c , i > 1 converges to c'. To obtain the interface excess of the grand potential Ü, given by
n = Q - n- - fi" (2.3.9)
we have to find the grand potentials H' and Ü" of homogeneous phases at the same y, p and T with volumes M' and M", and N' and N" filled cells such that the interface excess volume M and number of filled cells N are zero (Gibbs' convention)
E N , . = M'c' + M"c" (2.3.11) i 1 1
These equations can be solved for M' and M" and thus Q can be calcu-lated using the results of the previous section.
Note that like Goodrich we succeeded in calculating interface excess thermodynamic functions without using the concept of an inter-face. Below we will speak of interface thermodynamic functions, i.e. for briefness we omit the word "excess".
2.4 THE RESTRICTED LATTICE GAS
Now we will describe a model which is essentially similar to the model described above except for one restriction: filled cells are only allowed on top of other filled cells. When we compare this model with the unrestricted one we arrive at the following conclusions:
(i) Homogeneous phases are either completely filled or empty, i.e. they are badly modelled at higher temperatures and the model does not show critical behaviour.
(ii) When uni-directional density differences are present, over-hanging interface configurations are not accounted for. Never-theless we may hope that interface excess thermodynamic func-tions calculated from this model are not too bad even at rather high temperatures, because the incorrect homogeneous phase con-tributions cancel in the process of the calculation.
Physically the restriction implies an anisotropy in the pair poten-z XV poten-z XV
tials and clearly £ << £ or £ , £ "^ << - kT.
The grand partition functions for the restricted and unrestrict-13) ed Bragg-Williaras models are similar (cf. eq.(2.3.1)), except for
/N
?=.c» % i = <! ~/2 (2.4.2)
The constant number of broken bonds in the z direction and the ex-pression for P. mathematically reflect the imposed restriction. Following the same procedure as above, we minimize Ü by putting dn^'/dN . = 0. This yields
l n | ^ ~ - : : ^ j - — ^ ^ ^ ( l - 2 c . ) = 0 (2.4.3)
Again we have (stable) even and (unstable) odd solutions (cf. eqs. (2.3.6-7)) and again the problem is one of finding {c.} satisfying the proper boundary conditions, i.e.
lim c. = 0 and lim c. = 1 (2.4.4)
i-H» i-*-_cQ
Temkin showed that the boundary conditions can be introduced in the difference equations: the transformed difference equations generate a solution from which afterwards the values of {c.} and -Jq £ /kT
13) ^ can be calculated . However, this method does not reduce the
com-putational effort to obtain solutions with respect to the numerical method described above. Temkin also found that the even and odd solu-tions were the only types of solusolu-tions he could obtain from numerical computations. The evaluations of Ü is easy because Ü' and Ü" are simply zero
fi = N {-iq E + E kT(c._ -c.) In (c._ -c.) +
- iq^^^'E^'^c. (1-c.)} (2.4.5)
Note that when y deviates from y , the pressure of the homoge-neous filled phase deviates from the pressure of the homogehomoge-neous empty phase, which remains essentially zero. This implies that in the restricted model two phases cannot be in metastable thermal and hy-drostatic equilibrium.
2.5 DISCUSSION
For the development of the theory of crystal growth knowledge of the interface properties is important. It is obvious that we are in-terested especially in interfaces separating phases of which one can be conceived as a crystal. The lattice structure of the lattice gas invites us to choose the cell volume equal to the volume per particle in the crystal, although the minimum hard core of interaction is overestimated this way. Also, there seems to be only a good corres-pondence between a real crystal and the lattice gas when the dense phase is almost completely filled. Therefore, the solid-vapour system may be modelled well at low values of -JqE/kT. But then the restrict-ed lattice gas also gives a reasonable description of the interface. Disregarding other physical considerations ' , the lattice gas does not model the solid-liquid system because it does not des-cribe two high density phases of almost equal density. In 1966 Temkin modelled the solid-liquid interface with what we will call the
lat-13)
tice melt model . Instead of using filled and empty cells Temkin introduced cells occupied by solid and liquid particles. In this way he was able to model two equally dense but different phases. Although this is allowed within the general framework of the Ising model -cells can be occupied in two alternative ways- one may feel that the artificial distinction between solid and liquid particles gives rise to inconsistencies in the description. We will try to circumvent these inconsistencies by regarding the lattice melt model as an ex-tended interpretation of the restricted lattice gas model. In the argument of the exponent of the restricted lattice gas model we re-place 1 xy xy xy . z z z 4 q ^ E ^ N ^ ^ _ . . i q £ N^^ . by xy xy xy z z z q ' ' w ^ Ng[ . . q w N ^ ^ . (2.5.1)
where
*' = ^SL - * ^ss - * ^LL ""^^^ ^ = ^y. ^
It is easy to see that the number of solid-liquid pairs N replaces SL
the number of broken bonds N . w , the formation energy of a solid-liquid pair according to the "reaction"
SS + LL ^- 2 SL
replaces -ie . Hence interface thermodynamic functions in the res-tricted lattice gas and in the lattice melt model are the same func-tions of - J E and w , respectively. Note that within this interpre-tation no metastable states are described in the lattice melt model (cf. section 2.4).
We have already seen that the introduction of dimensionless thermodynamic functions facilitates the presentation of the results. In Table 2.5.1 the interaction parameter u is defined for the various
Table 2.5.1. Definition of the interaction parameter o) (i = x, y, z) and its relation to the latent entropy parameter a - L/kT (L is the latent heat) for the various models.
^Assuming complete wetting, i.e. e = £„,. This does not always 15) ^^ ^^
hold, eg. for Ga
u a = L/kT
unrestricted lattice gas - J E /kT (2c'-1) E q W
i r e s t r i c t e d l a t t i c e gas - J E /kT E q w
i i i * l a t t i c e melt w /kT E q CO
models. The relation between o) and the latent entropy parameter a is also given in this table. In the next chapter interface thermodynamic functions are frequently plotted against to or its reciprocal value. The interpretation of these graphs depends on which model we have in mind. Upon modelling the solid-liquid interface (with the lattice melt model), to every substance corresponds only one point in the graph. For a solid-vapour interface, however, the plot may show the effect of changing the temperature. For convenience, occasionally we will refer to 0) as the temperature. For the same reason we de-fine a quantity y hy
Y = F/NkT (2.5.2)
and refer to it as the specific interface free energy.
REFERENCES
1 F.C. Goodrich, in: Surface and Colloid Science, vol, 1, Ed. E. Matijevi6 (Wiley, New York, 1969) p.1.
2 T.D. Lee and C.N. Yang, Phys. Rev. 87 (1952) 404.
3 T.L. Hill, Statistical Mechanics (McGraw-Hiii, New York, 1956). 4 A hardware model with 36 lattice positions has been constructed by the Laboratory of Applied Physics of the Delft University of Technology.
5 L.D. Fosdick, Phys. Rev. 116 (1959) 565.
6 L.D. Fosdick, in: Methods of Computational Physics, vol. 1 (Academic, New York, 1963) p.245.
7 J.L. Doob, Stochastic Processes (Wiley, New York, 1953). 8 W. Bragg and E. Williams, Proc. Roy. Soc. (London) A145 (1934)
609.
9 C. van Leeuwen, P. Bennema and D.J. van Dijk, Acta Met. 22 (1974) 687.
10 R. Brout, Phase Transitions (Benjamin, New York, 1965).
11 H.J. Leamy, G.H. Gilmer and K.A. Jackson, in: Surface Physics of Materials, vol. 1, Ed. J.M. Biakely (Academic, New York, 1975) p.121.
12 J.L. Heijering, Acta Met. 14 (1966) 251.
13 D.E. Temkin, in: Crystallization Processes, Eds. N.N. Sirota,
F.K. Gorski and V.M. Varicash (Consultants Bureau, New York, 1966) p.15.
14 R.M. Cotterill, E.J. Jensen and W.D. Kristensen, in: Anharmonic Lattices, Structural Transitions and Melting, Ed. T. Riste
(Noordhoff, Leiden, 1974) p.405.
15 D.P. Woodruff, The Solid-Liquid Interface (Cambridge Univer-sity Press, London, 1973) p.171.
3. THE LATTICE GAS INTERFACE: PART II
3.1 INTRODUCTION
In this chapter we will use the tools described in the proceed-ing chapter to get an idea about the structure and thermodynamic properties of the interface in the restricted simple cubic lattice gas with lateral isotropic (section 3.2) and anisotropic (section 3.3) interactions. In section 3.4 we will consider the properties of the stepped interface. In these sections we will use the restricted model because it has some computational advantages and because it resembles the old and famous Kossel crystal, a familiar model in the field of crystal growth theories. In this introduction some
properties of the interface in the restricted and unrestricted model will be compared on the basis of the zeroth order mean field approxi-mation. The tendencies, however, are believed to be general.
At low temperatures the structural properties of the interfaces in the restricted model and in the unrestricted model are equal. As an example we consider the density profile across the interface. At low temperatures these density profiles coincide (cf. eqs.(2.3.5) and (2.4.3)) and are very steep. At higher temperatures the density differences across the interface become smaller. Also a difference between both models appears due to the differences in the densities of the homogeneous phases. This can be seen in Fig. 3.1.1. In the un-restricted model the density profile is a horizontal straight line at the critical temperature.
The thermodynamic properties of both models cannot be compared, unless the restricted lattice gas is slightly modified. The imposed solid-on-solid restriction implies that E = - co and this makes the interface free energy infinite for all temperatures. Therefore we require that the restricted model has the same low temperature
be-z xy
haviour as the unrestricted model, i.e. we put oj = lo = to and re-gard the restricted model as an approximation for the unrestricted model.
1.0
0.5
0.0
Cj-X
+
1x
+
1. X+
1a
*
*
i X + +
1 -^ X X 1 1 1 1 11.0-Q5
0.0
* * *
i * * * *
I 1 _Fig. S. 1.1. Comparison of the density profiles across the interface calculated for the unrestricted model + and for the restricted model X. The 0th order mean field calculations show a clear difference for t o ^ ^
(b).
0) -0.5 (a), but already a minor difference for to ^ = o) =0.1
1.0 0.8 0.6 0.4 0.2 0
Y
y
/ 1 / 1 / 1 / 1 / 1 / 1 1 1 • // // n /f /1 / 1 f 1 f 1f
a
Fig.3.1.2. The latent entropy (a) dependence of the interface free energy (y) in the lattice gas model with isotropic
inter-actions: unrestricted model, restricted model and -•-.-.- asymptote (0th order mean field calculations).
In Fig.3.1.2 the latent entropy (a) dependence of the interface free energy (y) is shown. The curves in both models converge to the
asymptote Y = ct/6 for a > 6. For low values of the latent entropy the curves are dissimilar. In the unrestricted model y = 0 for a = 0 at the critical temperature. In the restricted model Y = 0 for a = 2.70 . For smaller values of a, Y becomes negative and the model does not describe physical interfaces anymore. Although the in-terface free energy becomes zero at high temperatures in both cases, there is a difference in the way this is accomplished. In the unres-tricted model both E and S go to zero; in the resunres-tricted model E = TS ?* 0 when F = 0. This difference in behaviour is illustrated in Fig.3.1.3. Curves of similar shape were obtained by Leamy et
2 3)
al. ' using the Monte Carlo method. Their Monte Carlo estimate for the critical temperature seems in fair agreement with the value obtained by Essam and Fischer 4 ) (1) = 0 . 4 3 3 , w h i l s t t h e zeroth o r d e r
c
mean f i e l d approximation y i e l d s o) = 0 . 3 ^ .
Fig. 3,1. 3. The differences between the temperature depen-dences of the interface energy in the restricted model (dashed curve) and in the unrestricted model (solid curve) becomes significant at high temperatures (0th order mean field calcula-tions) ,
3.2 THE RESTRICTED LATTICE GAS: ISOTROPIC INTERACTIONS
In this section several structural interface properties will be introduced. The results of Monte Carlo simulations of the (001) in-terface of the restricted isotropic simple cubic lattice gas will be compared with analytical approximations. Of course, the interface properties are related. Their relations will be discussed in the discussion section of this chapter.
Prior to the treatment of the results of Monte Carlo simulations, we will comment on the simulations themselves. The results mentioned
in this section and in section 3.3 are generally based on the latter three of four runs, each of which contains 5000 Monte Carlo steps. In one Monte Carlo step the number of events (trials) equals the number of matrix positions. Every 10 Monte Carlo steps the matrix is scanned and thus the run average is an average over 500 configura-tions. The matrix size dependence of interface properties was inves-tigated by using matrices of the following dimensions: 18x18, 18x38, 30x30, 38x38 and 58x58. Results for to''^ = 1.3, 1.0 and 0.7 are pre-sented in Table 3.2.1.
Table 3.2.1. MC results for the restricted lattice gas. The system size dependence of several properties is shown. From top to bottom: to^2^ = I.3 N 18x18 18x38 30x30 38x38 58x58 N 18x18 18x38 30x30 38x38 N 18x18 18x38 30x30 38x38 58x58 , ^"^ = 1 s 0.056 0.057 0.056 0.057 0.057
s
0.253 0.251 0,250 0.251 s 1.0 1.01 1.01 1.02 1.01 0 and to^ -1 0.12 0.124 0.124 0.126 0.126 1 0.29 0.29 0.30 0.30 1 0.8 0.9 0.88 0.9 0.94 ^ 0.7. XxlO^ 7 3.5 2.7 1.7 0.76 XxlO^ 2 0.5 0.4 0.3 0 XxlO 0.3 0.2 0.1 X'xlO^ 7 3.5 2.7 1.7 0.76 X'xlO^ 2 0.5 0.4 0.3 X'xlO° 0.1 0.1 0.1 0.1 p'^d) 0.08 0.10 0.10 0.10 0.10 p'^d) 0.24 0.24 0.24 0.24 P^'d) 0.5 0.55 0.56 0.58 0.66Comparison of Fig.3.1.1.a and b shows that the density profile across the interface becomes steeper as the temperature decreases. Consequently the interface width increases. Cahn and Hilliard defined the interface width as the width given by the straight line extrapo-lation of the density profile at mid-density to the homogeneous phase densities . A better definition may be obtained if we consider the second moment about the mean y^ of the distribution of tops of columns of filled cells. The theoretical frequency distribution being t(i),
^- 6) y IS given by :
oo o
y, = E (i - y') t(i) (3.2.1)
^ i = _co •••
in which y' denotes the theoretical mean:
yj = E l t d ) (3.2.2)
1 i=—oo
In the restricted Bragg-Williams model t(i) can be chosen to be an even function for even interfaces (cf. eq.(2,3.6)) and then a simple formula results:
y„ = E i^-(c. , - c.) (3.2.3) 2 i=_oo 1-1 1
Now the interface width 1 is defined as the theoretical standard deviation :
1 = v ^ (3.2.4)
In Monte Carlo simulation experiments the interface width can be ob-tained as the average of the interface widths 1. of a series of gene-rated interfaces (J) for each of which
°° - 2 i
1. = {E (i - i) t.} (3.2.5)
1 i=_oo 1
jth interface, which mean interface position i is given by
i = E it.
i=:—CO ^
(3.2.6)
7)
Weeks et al. expect a divergence of the interface width at some finite temperature. According to their higher order low temperature series expansion calculations this phenomenon takes place at
xy
to = 0 . 7 8 . Swendsen arrived at the same conclusion, although his extended zeroth order mean field model gives an artificially high
XV 8 ^
transition temperature: to = 0.25 . No singularities are present in the thermodynamic properties. Our open system Monte Carlo simula-tions indicate that there is no system size dependence of the inter-face width at low temperatures, whilst at higher temperatures a de-finite correlation exists (see Fig.3.2.1).
Fig.3.2.1. The temperature dependence of the interface width in the restricted lattice gas model. Solid line and dots: 0th and 1st order mean field approximation. MC results: xfl=38x38, +N=38xl8 andoN=18xl8. At low temperatures the system size effect is small and the MC re-sults are given by the dashed line.
Our results are in agreement with the more accurate results obtained
recently by Swendsen, who used ten times as many Monte Carlo steps 9)
per point . His results indicate a logarithmic divergence for xy
to < 0.87 + 0.04. The zeroth and first order mean field approxima-tions do not show a divergence of the interface width (see also Fig.3.2.1). The first order mean field results presented in this chapter are obtained as a numerical solution of the dynamic pair equations (see section 5.1) for a twenty layer system.
We recall that we measured the configuration average interface width and not the interface width of the average profile. At higher temperatures this profile has a more or less flat part due to delo-calization of the interface. The delodelo-calization is measured by the variance x of the distribution of average interface positions. When the distribution function is a well-behaved function, a finite value of X implies that there is a finite probability for the interface to hop one lattice spacing. Hence the interface is delocalized. When X = 0 clearly this hop probability is zero and the interface is lo-calized. The quantity X defined above equals the reduced interface
10)
susceptibility as defined by Hunt and Gale . The interface suscep-tibility determines the response of the system (by varying the inter-face position) to fluctuations of the chemical potential about its equilibrium value, x ~ 0 evidences the existence of metastable states and finite values of x indicate their absence. Some results of Monte Carlo experiments on the system size dependence of the in-terface susceptibility are presented in Fig.3.2.2. Our data suggest that at low temperatures the interface is localized, i.e. X = 0, but no conclusions can be drawn from the higher temperature data. Pro-bably no finite interface is localized and locked in. The large scatter and the matrix size effect imply that a profound investiga-tion can only be performed at the expense of a considerable amount of computational effort.
The nature of the distribution of interface positions was also determined by examining its second moment x' about a variable origin determined by the nearest integer position. As to be expected at low temperatures our measurements show that X' - X> thus indicating even
profiles. At high temperatures X' becomes 0.08 +^ 0.02 (see Fig. 3.2.3), indicating that the distribution is uniform. Clearly, a uniform dis-tribution implies that the interface is not constrained to any par-ticular position, i.e. delocalized.
Fig.3.2.2. The system size dependence of the variance of the interface position in the restricted lattice gas indicates that the interface is localized in an infinite system at (j = 1.3.
Fig.3.2.3. The interface be-comes delocalized for to' when the distribution of
in-terface positions becomes uniform. MC results: o ff-18x18, xN=18x38 and-l-N=38x38. 0.8 in'
v'
Iff ióf 10="• t
J *
- x'
-_
1 1t6
® * ^ 1P
9
$ x OIs
+ X 1 1 1 1 1 0,6The density profile and the interface width do not give much information about the lateral structure of the interface. This
in-X formation is contained in the correlation coefficient. Let t (i,j) be the frequency distribution of the interface positions z = i and z = j at a distance of n (elementary distances) in the x direction.
X 6) We define a correlation function p (n) by
p''(n) = {E E (i - y') (j - y M t'' (i,j)}/y„ (3.2.7) i j 1 1 " 2
Upon assuming that the column heights are independent when the dis-tances between the columns are large, we find p («>) = 0. Also, ob-viously, p (0) = 1. Due to the periodic boundary conditions, used to provide the cells at the edges of the matrix with neighbours, the theoretical correlation function becomes symmetric. Cluster forma-tion implies that the correlaforma-tion funcforma-tion is positive for nearby positions. Experimentally for square matrices, we distinguish corre-lation functions which converge to zero or become clearly negative. In the latter case the most distant surface positions are correlated and it is concluded that the matrix size is too small in the sense
xy
that this should not hold for an infinite surface. For to = 1. 3 an xy
18x18 matrix seems already sufficiently large. For to = 1 an 18x18 matrix is too small but a 38x38 matrix seems sufficient. In both cases the positive part of the correlation function probably does
xy not depend on the matrix size for larger matrices. But for to = 0.7 it is questionable whether even a 58x58 matrix is large enough and the positive part of the correlation function is clearly increasing with matrix size (see Fig.3.2.4).
We will now introduce roughening functions for the restricted simple cubic lattice models and start by writing the general expres-sion for the interface excess of the grand partition function (eq. (2.1.4)) under the coexistence condition (eq.(2.2.7))
S = E exp E è N^ (j) £^ / kT
{j}
i
^°
= E E exp E è (N^ (j) - N^^ (0)) E"- / kT (3.2.8) Ijl 1
075 05 025 0
\ p^p>'
i
k
- ^ ^ N \ .\^^;:;s==^^
1 1 1 1 1n
10 15 20 25Fig.3.2.4. The correlation function defined by eq.(3.2.7) is shown for several system sizes: N - 18x18 dotted curve, N - 38x38 dashed curve and N = 58x58 solid curve. Each pair of correlation functions in the x and y direction is from the same MC run for to ^ = 0.7.
in which i runs over all types of pairs and N (0) is the number of broken pairs of type i in the ground state configuration (in which the interface is perfectly flat). It follows from the above equation that E i {N 10 N ^ Q ( O ) } £ (E/E^) d(l/kT) (3.2.9) Upon defining
*;»
10 N J , ( 0 ) (3.2.10) eq.(3.2.9) becomes d In (È/S^) = {E è ^\Q E ^ d(èE/kT)} / \ l (3.2.11)where J E is a suitable function of E which will be chosen such that the roughness s, defined by
s = (E è N^^ E^) / i £ N (3.2.12)
can be expressed in terms of N /N and the ratios between the pair potentials 6 :
6-''' = £*' / E^ (3.2.13)
Integration of eq.(3.2.11) from T = 0 K to T gives
Ü = NkT / s dto' (3.2.14)
i'
where
sE / kT (3.2.15)
and n = Q - Q (3.2.16) o
Q, is the interface excess of the grand potential relative to the ground state, i.e. due to roughening of the perfectly flat interface. In general we will call such functions roughening functions. For (lateral) isotropic nearest neighbour interactions within the simple cubic lattice geometry we choose
s = N?|^^ / N (3.2.17)
ii = ie^^ (3.2.18)
Thus the roughness equals the number of lateral broken bonds per site. It appears that the roughness is almost independent of the sys-tem size in Monte Carlo simulations (see Table 3.2.1). In Fig.3.2.5 the temperature dependence of the roughness is plotted. The mean field approximations give a smaller roughness than the Monte Carlo
Fig. 3. 2. 6. The rough-ness-temperature depen-dence in the restricted lattice gas model. Solid curve and dots: 0th and 1st order mean field approximation. The dashed curve is drawn through MC data points (o).
XV simulations, the difference being most pronounced for 0.5 < to < 1. Low temperature series expansion yields
s = 8TI {1 + 3n^ + o(n^ )} xy xy 'xy
(3.2.19)
in which
n = exp -to xy (3.2.20)
This relation was derived for a flat interface on which only adunits and vacancies and pairs of adunits and pairs of vacancies are present and overhangs are not allowed . For higher order series expansion we refer to Refs.10 and 11. Another approximation for the roughness of the interface in the three-dimensional lattice gas is the two
12) 13)
layer version of Onsager's two-level two-dimensional Ising model. In this model overhanging positions are not excluded. The re-sults of all these approximations are compared in Table 3.2.2. for
several temperatures. It can be seen in Table 3.2.2 that the diffe-rence in roughness between a two-layer and an infinite-layer system
xy xy is small for to = 1 and to = 1 . 3 .
Table 3.2.2. Comparison of the roughness calculated according to several methods.
Method Number Roughness for to
of layers 1.3 1.0 xy 0.7 Monte Carlo two-layer Onsager
Ist order series expansion 0th order mean field 0th order mean field 1st order mean field
OO 2 2 2 00 0 0.0569 0.05692 0.05275 0.04606 0.04659 0 0 0 0 0 0 251 2544 1926 1623 1689 23 1 1 0 0 0 0 01 1202 6985 5884 6838 95
3.3 THE RESTRICTED LATTICE GAS: ANISOTROPIC INTERACTIONS
When we consider an anisotropy in the lateral pair potentials In the simple cubic geometry, i.e
it is useful to define \z as follows
X y within the simple cubic geometry, i.e. when we distinguish e and E
è£ = è (JE'' + i£^) (3.3.1)
This allows us to compare the properties of interfaces with the same z
latent heat (when E is the same in both cases) or kink site removal energy but different anisotropy factors 6 xy.
^^y y / X
£ / £ with 6"^ < 1 (3.3.2)
The roughness now becomes (cf. eq.(3.2.12) 2 (N
10
^^'
<o>
N (1 + 6''^)
(3.3.3)
Fig. 3. 3.1. The to dependence of the roughness in the res-tricted lattice gas. MC re-sults for 18x38 matrices: S^ =0.2 (top curve), 0.4, 0.6, and 1.0. The inset shows MC curves for 6 =0.2 (a) and 1.0 (b) and the high ani-sotropy limit ijf = 0 ((a), eo. (3.3.7)).
broken bonds, in that weak bonds contribute less to the roughness. In Fig.3.3.1 Monte Carlo results are presented for several ani-sotropy factors. The data are somewhat more accurate than those
pre-14)
sented before . At high temperatures the roughness does not de-pend on the anisotropy factor within our accuracy. For very low temperatures -when only adunits and vacancies are present- the roughness is independent of the degree of anisotropy, since then
8 exp -4to (3.3.4)
In between these two extreme cases the roughness increases with the degree of anisotropy. First order, low density series expansion gives the following expressions for the numbers of broken bonds in the X and y direction
X 2 2 2 2 4 4 2 2 N _ / N = 4n n (1 + n + 2n + o ( n ,n .n n ) ) 1 0 x y y X x y x y ( 3 . 3 . 5 ) N ^ ^ / N 2 2 4n n ( 1 X y 2 4 4 2 2 2n + o ( n ,n ,n n ) ) y x y x y
in which
n = e x p -to for i = x, y (3.3.6)
14) These expressions are almost identical to those obtained before Their region of validity is smaller the lower the anisotropy factor.
y
In the limit of very strong anisotropic interactions (to ->- 0) , the crystal consists of independent vertical slices and thus (cf. section 3.4)
4 exp -to (3.3.7)
It can be seen in Fig.3.3.1 that this behaviour is not approached in the simulation experiments.
0.6
X y 10 10 -0-0--0--0--°'0-c>o--0--0--0 °' t), •OQ Oa-o-o-o-O" O. A. .--O" •e _ ^ - ' "000®''-'° - - - - o — l/öj0.5
1.0
1,5
Fig. 3. 3.2. The to dependence of the ratio of numbers of broken bonds in the x and y direction. MC results for 18x38 matrices —o ; 1st order mean field data A and low density series expansion results (eq. (3.3.5)) •— . All curves from top to bottom: ST^ = 0.8, 0.6, 0.4, and 0.2.
The influence of the anisotropy shows out rather well in Fig. 3.3.2, in which the temperature dependence of the ratio of broken
bonds in the x and y direction is plotted. The low density model (eq.(3.3.5)) shows the convergence of this ratio to unity in the limit of low temperatures. At higher temperatures Monte Carlo data indicate the existence of a minimum. Some preliminary results of first order mean field calculations (cf. section 5.1) are rather well in agreement with the Monte Carlo results. The temperature dependence of the ratio of broken bonds tells a little story about the interface structure. At low temperatures the ratio is almost unity because only adunits and vacancies are present. With increasing temperature the ratio decreases as a result of the formation of oblong clusters.
XV —
Fig. 3.3.3. MC interfaces for 6 "^ =0.2. From top to bottom: M = 1.1, 0.95, and 0.85.
The subsequent increase of the ratio after passing its minimum value may be explained either by the breakdown of oblong clusters or by an
increase jn the adunit / vacancy concentration. The Monte Carlo mea-3)
surements of Leamy et al. for isotropic systems show that the ad-unit concentration decreases with temperature in the high temperature
region. Thus the latter possibility can be disregarded. Hence the minimum can be associated with the breakdown of structure. The
inter-faces shown in Fig.3.3.3 are ment to illustrate this story.
3.4 STEPPED INTERFACES
Crystals are generally not perfect. Steps may be present in the 15)
interface, eg. due to screw dislocations in the crystal . The step edges -ledges- are important for the crystal growth process in that they provide sinks for particles. In a low temperature (and low step density) approximation, a ledge can be conceived as the one-dimensional interface in a two-one-dimensional system. In 1951, Burton,
12)
Cabrera and Frank evaluated the properties of the (10) interface in the two-dimensional restricted lattice gas applying the laborious method of the detailed balance. Thereafter more elegant methods were presented by Temperley and Leamy et al. . Here we will follow the method used by Leamy, but we introduce a slight modification with respect to the boundary conditions. The grand partition function
for the restricted two-dimensional lattice gas can be written 1 X m X y m
= = = o / , - P t ^ ^ . I W l l ^ ^ ^ " ^ 1 ^ ^ ^i^ <^-^-^>
iy> i=-m+i i=-m Here E is the contribution of the ground state configuration y = 0
o i for all i and y. measures the position of the interface in the ith
column with respect to the ground state configuration (see Fig. 17)
3.4.1a). We will apply the boundary conditions
y = y = 0 (3.4.2) -m m
Physically, this corresponds to a ledge on a surface between two screw dislocations with Burgers vectors of opposite sign (+1 and -1) as can be seen in Fig.3.4.lb.
Fig.3.4.1. (a) The one-dimensional interface in the two-dimensional
restricted lattice gas. Here the end-points are fixed at the
posi-tions (-m,0) and (m,0). (b) Physically, this corresponds to a step
between two screw dislocations of opposite sign, terminating in the
surface at P and Q.
Using the definition of to (see Table 2.5.1) and upon introducing
A^ = y, 'i-l (3.4.3) and = (P £ - E-') / kT X y (3.4.4) we obtain exp {A} m
E n
{A} i=-{-to'' E i=-m+l {exp- to'AJ-
iA.} 1 \ \ -iÉ (3.4.5)Neglecting the weak correlation induced by the boundary conditions -only one of the jumps A. cannot be chosen independent of the others-we find
E exp (-0) | j | - i S j ) i=-m+l J=-"=
( 3 . 4 . 6 )
i=-m+l
1 (1-n )/{l+ri^-n {exp (-16) + exp d B ) } } ]
L X X X -'
We will use this general result in section 5.2. Here we will summarize some properties of the interface under the coexistence con-dition, i.e. when 3 = 0 . The interface length being denoted by N
(= 2 m ) , eq.(3.4.6) reduces to 1 + n \N
o I 1 - n
\ 3
So the interface excess of the grand potential becomes
o 1 - n " y , X Y = "TT^ = to-' + In ' NkT 1 + n (3.4.7) (3.4.8) 1.0 0.5 n
-_
Y- ^
yy y / y / y / y / y //
/
1 1Fig. 3.4.2. The to dependence of the interface free energy in the two-dimensional res-tricted lattice gas (solid curve) and the ledge free energy obtained from Leamy 's data 21) (dashed curve).
0.8 ID 1.2 1.4
In Fig. 3.4.2 the to dependence of y is shown for a system with iso-X y
tropic interactions, i.e. to = üJ = to. It is easy to see that y = 0 for to = - In (/2 - 1) = 0.88137. Because of the absence of jump cor-relation no interface profile can be associated with the one-dimen-sional interface. For interfaces of infinite length the interface
Fig. 3. 4. 3. Illustration of the influence of the interface length on the width of the interface in the restricted two-dimensional lattice gas with periodic boundary conditions. Interface lengths shown are N = 40, 80, 160 and 480 for uf^ = 1.4 (top line) and to =1.0. The long interface at the bottom has a length N = 960 for to = 1.0. The configurations are the result of closed system MC simulations.
width ( s e e s e c t i o n 3.2) i s i n f i n i t e except for T = 0 K. The i n t e r f a c e width has only some meaning for f i n i t e systems. F i g . 3 . 4 . 3 i l l u s t r a t e s the i n c r e a s e of t h e i n t e r f a c e width with temperature and system s i z e . The i n t e r f a c e c o n f i g u r a t i o n s a r e obtained from closed system Monte
18)
Carlo simulations w i t h p e r i o d i c boundary c o n d i t i o n s . These i n t e r -faces a r e d e l o c a l i z e d for non-zero t e m p e r a t u r e s . Other s t r u c t u r a l p r o p e r t i e s of i n t e r e s t are t h e jump d e n s i t y d i s t r i b u t i o n r of jumps of s i z e j
r . = {exp -00 | j | } / E exp -u) | i |
J i
( 3 . 4 . 9 )
and the average distance 1 between kinks (filled cells with different types of coupled neighbours in all directions)
1 = 2 cosh^ ih^^) (3.4.10) k
Fig. 3.4.4. (a) An edge self-avoiding walk on a square lattice, (b)
The interfacial region in the two-dimensional lattice gas. (c)
Stretched out version of the interface in (b) in which jumps can be
defined as in the interface of the restricted two-dimensional lattice
gas.
The one-dimensional interface treated above can be conceived as a special case of the edge self-avoiding random walk problem on a
19)
square lattice . In Fig.3.4.4a an example of such a path is shown. Obviously, the only difference lies in the absence of overhanging configurations in the restricted model. Apart from overhangs the un-restricted two-phase two-dimensional lattice gas also describes clusters of filled and empty cells. It seems worthwhile to note that the exact solution of the homogeneous two-dimensional lattice
12,13)
gas has a critical point for
(0^^ = - la (y/2 - 1) 0.88137 (3.4.11)
It is a remarkable coincidence that the temperature at which the edge free energy becomes zero in the restricted model equals the critical temperature in the unrestricted model (see also Ref.20).
Now we will consider the properties of ledges in their natural environment, i.e. (here) in the interface of the restricted three-dimensional lattice gas. Stepped interfaces can be simulated by using
18 21 ^ appropriate boundary conditions in a Monte Carlo model ' . Being interested in the structural ledge properties, we conceive a ledge as a random walk path. But in the interfacial layer containing the ledge also clusters are present. Before 18) we decided that clusters
