ORDER FORMATION IN BINARY MIXTURES OF MONODISPERSE LATEXES

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ORDER FORMATION IN BINARY MIXTURES OF MONODISPERSE LATEXES

S. Yoshimura, S. Hachisu

To cite this version:

S. Yoshimura, S. Hachisu. ORDER FORMATION IN BINARY MIXTURES OF MONODIS- PERSE LATEXES. Journal de Physique Colloques, 1985, 46 (C3), pp.C3-115-C3-126.

�10.1051/jphyscol:1985310�. �jpa-00224627�

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JOURNAL DE PHYSIQUE

Colloque Ci, supplément au n°3, Tome k6, mars 1985 page C3-115

ORDER FORMATION IN BINARY MIXTURES OF MONODISPERSE LATEXES

S. Yoshimura and S. Hachisu**

Institute of Applied Physics, University of Tsukuba, Sakura, Ibavaki 305, Japan Résumé - On a étudié la formation de structures ordonnées dans des mélanges binaires de latex à particules de tailles diffé- rentes. Les latex binaires, aux concentrations moyennes, pour lesquelles les particules sont distribuées au hasard, sont de couleur blanche comme le lait. Quand la concentration en particules des latex binaires excède une certaine limite, on obser- ve une séparation de phase dans les latex, conduisant la sédimen- tation de la phase (ou des phases) à la structure (ou aux structures) d'alliages. Ces structures ont été examinées avec un microscope optique. Elles sont du type A1B2, NaZnj3, CaCUs, et MgCu2 ; une autre de type 'AB^ a la symétrie d'espace P63/mmc (le cristal de ce type n'est pas encore trouvé). L'approximation de sphère dure est employée pour analyser ces résultats. On a con- clu que (1) le type de structure est déterminé par la propor- tion des diamètres effectifs des sphères dures pour les particules composantes, (2) chacune des structures réalise un état de bon empilement (si ce n'est celui d'empilement le plus compact) et a une densité plus grande que celle de la phase désordonnée dont elle découle. Cette densité plus grande est une des causes de la transition de phase.

Abstract - Order formation in binary mixtures of monodisperse latexes of different particle sizes is studied. A binary latex of a moderate particle concentration is milky white with random distribution of the particles. When the concentration exceeds certain limit, phase separation takes place to deposit an ordered phase or phases with alloy structures. The structures were examined by a light microscope. Structures thus far observed are AIB2, NaZn1 3, CaCu5, MgCu2 and another type which has a composition of ABU with P6 3/mmc symmetry. The structures are analysed basing on hard sphere model. It is concluded that (1) structure type is determined by the ratio of effective diameters of the constituent latexes, (2) an alloy structure is a realization of a good state of packing if not of the state of the closest packing

and has a higher density than the density of the disordered phase from which it has deposited. This higher density is a cause of the phase transition from disordered to ordered state.

I. INTRODUCTION

Formation of ordered structure in monodisperse latexes is well known and has been studied extensively.1,2'-^ it has been established that the order formation is a result of a phase transition and that the trans- tion behavior is understood by the concept of Alder transition^ which was found in computer calculation of hard sphere systems. Recently, it has been found that the order formation is not restricted within monodisperse colloids, but be able to occur in mixtures of two different

Present addresses : *Izumi High School, Takane-cho 875-1, Chiba-shi, 280-01, Japan

**15-3, Minamisawa 5, Higashikume, Tokyo 203, Japan Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985310

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JOURNAL DE PHYSIQUE

monodisperse colloids5,6~he first work was done by Sanders T o n the structures in opals consisting of two or more kinds of silica spheres.

The authors studied binary mixtures of monodisperse latexes and found several alloy structures formed in them596~he behavior of binary, systems is supposed to be far more complicated than that of uni-component systems. However, the result thus far obtained is rather simple and evid- ent suggestingthat the fundamental properties of binary systems is rather simple. In view of the facts that the interactions between stably dispersed latex particles is effectively repulsive, and that in monodisperse case, the behavior was, in the main, understood by hard sphere approximation (Alder transition), the property of binary systems would also be understood basing on hard sphere model.

In the present paper, several alloy structures formed in latex mixtures are briefly described (for the details readers are refered to the ref- erences5 9 6, , and then an attempt is made to understand the mechanism of the structure-formation by the packing model of hard spheres, in which the latex particles are approximated by hard spheres with effective diameters. The underlying idea is that the structure type is determined by the principle of sphere packing and further that the fundamental transition behavior of binary latexes is essentially the same as that of binary hard sphere systems.

I I. EXPERIMENT

Method : Colloids used were polystyrene latexes, whose diameters were ranging from 2000A to 8000A. They were purified by dialysis and sub- squent treatment by ion-exchange resin. As the particle concentrations were from 0.1 to 0.3 in volume fraction, they were all iridescent. Any electrolyte was not added for adjustment of their electrolyte concentrations. Therefore, the electrolyte concentrations of these stock latexes would be at the level of conductivity water. Merit of this low electrolyte concentration was that the interparticle repulsive potential was long ranging to produce large interparticle distance in the alloy structures to facilitate the observation by a light microscope.

A demerit was that it caused difficulties in analysing the structures.

The alloy structures were produced as follows: Two stock latexes were mixed; the mixture was in disordered state, or to say, in glassy

state, not to show any iridescence. This original mixture was carefully (not to make overdilution) diluted to bring about the state of phase separation or a state very-near to it. Under an optimum condition, an ordered structure(or structures) appeared in 10 to 50 hours. Depending upon the condition, sometimes several weeks were needed. These structures were observed by metallugical microscope of inverted type. Sever- al mixtures of different combinations of latexes were examined at var- ious particle number ratios.

In the description of the observations given below, each latex is specified by its particle diameter. Therefore, a mixture is expressed by a combination of the diameters of the constituent latexes. From theore- tical point of view, this way of specification of a mixture is not ade- quate. As mentioned earlier, the repulsive interaction in the latexes are so long ranging that the core diameter of the particles does not reflect their property. For analysis of the structure the "effective diameter" which is introduced in hard sphere approximation of latex systems becomes important and takes the place of the core diameter. The proceedure of the measurement of effective diameter will be mentioned after the description of alloy structures.

Obseration : In most cases, ordered structures were seen as islands in the sea of disordered phase. Sometimes, the mixture was entirely ordered and consisted of one or more different structures. Alloy structures found were A1B2 ,NaZnl , ^{C a C y}, MgCu;! and another hexagonal one which is not yet identified to any of existing real crystals. They are expla- ned by photographs and drawings of the lattice strucbures shortly, the details of which are given elsewhere 5 3 6 .

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A l B , - s t r u c t u r e : T h i s was found i n f o u r m i x t u r e s , t h e combinations o f 8000A-2800A, 5600A-2100A, 5500A-2500A, and 4700A-20OOA l a t e x e s . I n t h e l a t t e r two m i x t u r e s u n d e r l i n e d , a n o t h e r s t r u c t u r e Naznlswas found t o c o e x i s t . The t y p i c a l p a t t e r n i s shown i n F i g . 1 ( a ) . Large p a r t i c l e s a r e packed c l o s e l y i n s q u a r e l a t t i c e w i t h s m a l l p a r t i c l e s i n t h e c e n t e r . B e s i d e t h i s , s e v e r a l p a t t e r n s o f o t h e r n e t p l a n e s were observed. From t h e s e p a t t e r n s , t h e s t r u c t u r e was determined a s shown i n F i g . 1 ( b ) . I t c o n s i s t s o f a l t e r n a t i v e s t a c k o f two t y p e s o f hexagonal n e t p l a n e s ; a s m a l l p a r t i c l e s i t u a t e s i n t h e c e n t e r of a t r i g o n a l prism formed by t h e l a r g e o n e s . The photograph shows (1010) n e t p l a n e .

Fig.1. ( a ) shows (10T0) p l a n e o f A l B z - s t r u c t u r e in8000A-2800A m i x t u r e . I n ( b ) , t h e l a t t i c e i s i l l u s t r a t e d . The l a r g e b l a c k c i r c l e s and s m a l l d o t t e d c i r c l e s a r e s e e n i n t h e photograph. Window s u r f a c e of t h e o b s e r - v a t i o n c e l l p a s s e s t h e c e n t e r of c i r c l e s drawn by broken l i n e s .

N a Z n l r s t r u c t u r e : T h i s was found i n 5500A-2500A, 4700A-2000A, and 6000A- 3100A m i x t u r e s . A s mentioned above, i n t h e f i r s t two u n d e r l i n e d mix- t u r e s , A l B 2 - s t r u c t u r e was found t o c o e x i s t . Observed p a t t e r n s a r e shown i n Fig.2 ( a ) and ( b ) . They r e p r e s e n t r e s p e c t i v e l y (110) and (100)' p l a n e s o f a s i m p l e c u b i c l a t t i c e made of l a r g e p a r t i c l e s . I n ( c ) , t h e l a t t i c e o f e n t i r e s t r u c t u r e i s shown. I n each o f t h e s i m p l e c u b i c c e l l o f t h e l a r g e p a r t i c l e s , an i c o s a h e d r o n of s m a l l p a r t i c l e s a r e c o n t a i n e d . Neighboring i c o s a h e d r o n s a r e s t a g g e r e d by 90' d e g r e e s . A s w i l l be men- t i o n e d l a t e r , t h i s i c o s a h e d r o n may be a b i t d i s t o r t e d t o produce a b e t t e r packing.

F i g . 2 . P a t t e r n s observed i n 5500A-2500A m i x t u r e . ( a ) shows (110) and (b) d o e s (100) p l a n e . I n ( b ) , a rhombus of s m a l l p a r t i c l e s i s s e e n i n e v e r y s q u a r e c e l l . They s t a g g e r w i t h each o t h e r by 90°degrees.

C a C u , - s t r u c t u r e : T h i s was formed i n 4700A-3100A, 5500A-3100A m i x t u r e s . T y p i c a l p a t t e r n s a r e shown i n Fig.3. ( a ) and ( b ) . I n ( a ) , s m a l l p a r t - i c l e s a r e i n t h e c e n t e r s o f t r i a n g l e s made o f l a r g e p a r t i c l e s , w h i l e i n

( b ) , s m a l l p a r t i c l e s a r e between t h e two l a r g e p a r t i c l e s . I n microscope

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C3-118 JOURNAL DE PHYSIQUE

o b s e r v a t i o n , p a t t e r n ( a ) a p p e a r e s i n t h e f i r s t p l a c e b e c a u s e it s i t u - a t e s j u s t i n s i d e o f t h e window s u r f a c e . By moving f o c u s toward i n s i d e of t h e s t r u c t u r e , t h e ( b ) p a t t e r n a p p e a r e d . T o t a l s t r u c t u r e i s t h e s t a c k of t h e s e two n e t p l a n e s . Each n e t p l a n e and t h e e n t i r e s t r u c t u r e i s shown i n t h e drawing g i v e n on t h e r i g h t s i d e o f t h e F i g . 3 .

F i g . 3. Photographs a r e t h e p a t t e r n o b s e r v e d i n 5000A-3100A m i x t u r e ; ( a ) shows t h e f i r s t p l a n e ( s e e t h e t e x t ) , and ( b ) t h e second p l a n e . I t i s t o b e n o t e d t h a t t h e s m a l l p a r t i c l e s i n ( b ) a r e between two l a r g e p a r t i c l e s and t h a t t h e l a r g e p a r t i c l e s a r e s e e n a s b l u r s p o t s , i n d i c a t - i n g t h a t t h e y a r e n o t i n t h e second p l a n e .

Fig.4 ( a ) i s t h e photograph o f one o f t h r e e p a t t e r n s o b s e r v e d i n 4500A-3800A m i x t u r e . A u n i t c e l l i s marked by b l a c k i n k . I t i s i l l u s t - r a t e d by t h e drawing i n (b). I t c o n t a i n t h r e e k i n d s o f n e t p l a n e s ( s e e t h e t e x t ) i n d i c a t e d by numerals 4 , 5 and 6 . E n t i r e u n i t c e l l i s shown i n ( C ) , where 12 n e t p l a n e s a r e s t a c k e d one on a n o t h e r . P o s i t i o n s o f t h e n e t p l a n e s i n t h e photograph c a n Le known by t h e numerals.

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MgCu2-structure : This was found in 4500A-2800A, 8000A-5600A,4500A- 2800A and in 4500A-3100A mixtures. One of the patterns is shown in Fig.4 (a). A peculiar feature is that large particles form hexagonal net work and in every hexagon a small particle situates in the center.

Close observation revealed that this pattern consisted of three net planes of different depth. A primitive cell is marked on the photograph; in which three different kinds of particles are distinguished, Indicating the presence of three different net planes. Beside this pattern three patterns of different kinds were observed. The lattice type was finally determined as shown in Fig.4 (c). It is a stack of twelve net planes of hexagonal symmetry. The large particles form a diamond lattice, and small particles forming tetragons are packed in the interstitial vacancies of the diamond lattice.

AB,-structure : This was found in 6500A-3100A mixture. Photograph in Fig.5 (a) shows the pattern, in which the region of typical structure is marked by black ink. By moving microscope focus, some of the key features of the particle arrangement were known; from which firstly obtained was the framework made of large particles as shown in Fig.5

(b). It is an alternative stack of hexagonal planes staggering each other by 60 degrees. It has two kinds (wide and narrow) of vacancies, denoted in the figure as W and N. Small particles are packed in these vacancies. In a W-vacancy, a linear linkage of centered icosahedrons, and in N-vacancy a linear array of small particles are accommodated as shown in Fig.6. It is the entire structure looked from [0001]

direction, it is constructed as a package of effective hard spheres.

A unit cell is indicated in the figure by bold lines; it has a composition of ABs and a symmetry type of P6s/mmc. Fig.6 (b) is a photograph of the model made by two sets of plastic spheres with diameters of 5cm and 3cm each. We have not yet found its counterpart in the realm of

Fig.5. (a) is a typical pattern of m 4 . (b) is the frame work made of large particles, black circles is the first plane, and white ones are

Fig.G.(a) ~ n t i r e structure of AB,-structure. (b) is a model made of plastic spheres.

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JOURNAL DE PHYSIQUE

Effective Diameter : In order to approximate latex particles by hard spheres, the repulsive potential between the particles is to be repla- ced by a hard core potential. The diameter of this core is the effective diameter of the particles. For a monodisperse latex, the effective diameter is obtained experimentally from its ordered state by assuming surface to surface contact between the spheres. Here, a care must be taken that an ordered latex is a system of high compressibility. There- fore, to determine the effective diameter, it is to be measured at a specified state, say, the state of phase separation.

The method is as follows: A monodisperse latex (a stock latex) was brought to the state of phase separation by diluting with ion exchanged water, and the ordered phase was photographed. The center to center distance between two adjacent particles was taken as the effective diameter. However, this method sometimes gave unreasonable results. It was probably because the ion-concentrations of stock latexes and of their mixture were not equal. Drift of electrolyte concentration was quite possible to occur in these latex systems. Thus, we adopted another method; it was to make use of a fact that an alloy structure can coexist with a pure phase under a suitable condition. From this coexisting pure phase, we obtained the effective diameter, and this vdlue was applied to this alloy structure. This method has a drawback that a pure phase of the large particles of the alloy was difficult to form, since the small particles were far more abandant in number (ten times or more usually) than the large particles. In order to get the effective diameter of the-large particles, we introduced rather an arbitrary assumption that the difference (AD) between effective diameter and act- ual core diameter is the same for small and large particles in the concerning mixture. Basing on this assumption, effective diameter of large particles was obtained. The result is given in Table I.

Incidentally, the value of effective diameter depends on the way of approximation. In the above, surface to surface contact is assumed, but in more refined approximation, finite separation between the

spheres in the ordered phase is to be taken into account (in Alder transition, there is separation about 10% of the diameter of the sphere diameter). However, accuracy of the present study does not need such a refinement.

Structure and Effective Diameter Ratio : The structure type is guessed to be determined by the ratio (y-value) of the effective'diameters of the constituent latex particles. In Fig.7, Y-valbes of the observed structures are plotted on a linear scale. Because of limited number of observations, the values are sparcely distributsd, but it is dout- less that there is a zone of Y-value for a particular structure to appear. A l B 2 Y-zone and NaZnls-zone overlap one on the other. In this overlapped region (from Y =o. 56 to 0.61) , coexistence of the two structures were actually observed. For ABs-structure, which was obsrved only at one point Y=0.62, we can not estimate the zone width. Around this Y -value region, three zones (for A1B2 , ^NaZnl,, ândÂB5⁾ ôverlap^;

the behavior of the system would be very delicate and complex here.

Remarkable fact is that in a region from 0.63to 0.73, no structure was found. It is not certain at present whether this is a real fact or an artifact produced by some errors in Y-value estimation. Another possibility is that this region might have been jiust overlooked in our experiment which was not carefully designed to cover the whole region of Y-value. Another noticeable fact is that MgCuz-structure found at

=0.77 situates very near CaCus-structures being isolated from the rest of its class. This MgCuz-structure might be a mis-identifiaation of CaCus-structure.

Anyway, the fact that there is a relatively narrow Y-zone for a particular structure to appear is a strong support to the proposition that the structure is determined by Y-value and that the structure formation is understood by sphere packing priciple.

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Structure composition composition

TYPe in core dia. (A) in ef f .-dia. (A) AD y=% PO

DA DB DA DB DB

[8000-28001 [10400-52001 2400 0.50 0.76 A1B 2 [5600-21001 [9500-51001 3000 0.54 0.78 f.4700-20001 t6900-42001 2200 0.61 0.75 [5500-25001 16809-38001 1300 0.56 0.78 [47OO-20001 [6900-42001 2200 0.61 0.72 NaZnl, [5500-25001 [6800-38001 1300 0.56 0.73 [6000-31001 [7900-50001 1900 0.63 0.71 CaCu [5500-31001 [8500-61003 3000 0.72 0.69 [4700=3100] 16300-47001 1600 0.75 0.67

Table I. In the 2nd column, the mixture are expressed by the combination of core diameters as done in the text. In the 3rd column,they are given by effective diameters. It is noticeable that 3100A latex appears in four different mixtures(inthe 2nd column),but its effective diameter varies from one mixture to another.

P- blank 4

Fig.7 : Y-value diagram of the observed structures. Each Y-zone is shown by a horizontal arrow bound by two vertical bars. Circles represent the structures, respective lattice type of which are indicated by short broken arrows.

Miscellaneous Features : There were several observations other than the search for alloy structures. Noticeable features were as follows:

1) Ordered phases appeared at the bottom of the obervation cells, indicating that they had higher densities than those of coexisting disordered phases. 2) Number of coexisting phases did not exceed three after the system attained equilibrium, being in conformity with the phase rule. 3) Particle number ratio in the observation cell changed in a long run by sedimentation of large particles. Some alloy structure for example CaCu5 was unstable and tended to decompose into pure phase or phases. This may be due to the sedimentation effect. 4) As seen in the table, a latex (3100A latex for example) has five different effective diameters in five mixtures, indicating that electrolyte concentration varied from one mixture to another over a wide range. This fact was employed to vary the y-value of a particular mixture by deionizing heavily by ion exchange resin. In the table,mixture 14500-28001 in MgCu2-forming class has two y-values, 0.77 and 0.83.

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JOURNAL D E .PHYSIQUE

111. THEORY

Phase Transition : The problem how these structures are formed and are able to persist stably is dealt with here basing on hard sphere approximation. The most simple and resonable interpretation is that the structures are the realization of the state of maximum packing under high (osmotic) pressure. In fact, alloy structures in opals are explained by the principle of maximization of the packing density, where the gravitational force works to condense a silica sol, the math- er phase of opals, up to the maximum state of packing. Sanders argues that when a binary mixture of hard spheres separate into two phases, each with volume fraction of 0.74, the whole mixture will also has the volume fraction of 0.74. Therefore, at the state of maximum density, alloy structure with volume fraction less than 0.74 can not exist; it will decompose into two pure phases to attain the state with volume fracion of 0.74. Then, only alloy structure with volume fraction higher than 0.74 can survive in this enviroment. Basing on this criterion, Sanders explained the stability of alloy structures in 0~als.8 In latex case, however, this principle does not work, because in latexes , graviational effect is negligible, no attraction force works between the particles (electrolyte concentration was very low), and furthermore, ordered structures are formed in the state of phase separation or in a state very near to it, where the system still has a sig- nificant room for further compression. Thus, the principle of maximization of packing density does not work here. In fact, as shown later the stability of NaZnis , CaCu5, MgCuz and AB4 are not explained by the maximization principle. Some other criterion than Sanders' is necess-

ary. In view of the fact that the ordered structures are formed as the result of phase separation, the answer to this problem is to be sought in the mechanism of the phase transition.

The phase transition in concentrated binary systems can not be delt with in analytical way. However, by use of hard sphere approximation, and by adopting semiempirical way, we are able to have a perspective about the general property of the phase transition. ~ i c e 9 argued that when glass spheres are randomly thrown together, the density did not exceed about 85% of the density of closely packed ordered state, therefore, the disordered state at high enough pressure must be thermodyna- mically unstable with respect to ordered state. As argument lacks the consideration on entropy, it does not give the point of transition.

Computer calculation4 revealed that transition occurs actually and takes.place at 0.5 in volume fraction toward the ordered state with volume fraction 0.55. These values are naturally less than 0.64(=0.74 X0.85) and 0.74, the corresponding values of glass sphere case.

The same logic will apply to binary systems. If a binary mixture of hardspheres in a state of orderly packing has a larger density than that of the random packing, we could say that transition from disordered to ordered state will occur at high enough pressure, and the state of phase separation will appear. Although, we do not have at present a computer calculation corresponding to that in monodisperse case, we could postulate this phase transition as the fundamental property of binary hard sphere systems.

Then, our tasks is to know the packing density of binary (macroscopic) hard sphere systems in random state and in ordered states. .The random packing of binary sphere systems was extensively studied by

Yerazunis, Cornel and Winterlo. They summarized their their result in a semi-empirical formla, which gives the maximum value Pro£ random packing as a function of diameter ratio y and particle number ratior)

= n ~ / n ~ (A indicates large particle, B does small particle) as shown below.

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From t h i s formula, we can getp7-Y r e l a t i o n f o r d e s i r e d v a l u e o f I l l

t h e p a r t i c l e number r a t i o . Example a r e shown i n Fig.8. f o r q = 1 0 and f o r n = 2 0 , t h e e x p e r i m e n t a l c o n d i t i o n most f r e q u e n t l y employed. Curve I means t h a t i n a m i x t u r e w i t h q = 1 0 , t h e d i s o r d e r e d s t a t e i s s t a b l e i n t h e r e g i o n below i t s e l f b u t n o t i n t h e upper r e g i o n . Namely it r e p r e - s e n t s t h e boundary l i n e o f t h e d i s o r d e r e d r e g i o n t o o r d e r e d r e g i o n , and works a s a c r i t e r i o n f o r t h e s t a b i l i t y o f an o r d e r e d s t r u c t u r e . A t b o t h e n d s , y=0 and y = l , o f t h e c u r v e s , t h e b i n a r y systems d e g e n e r a t e i n t o monodisperse s t a t e and p-value becomes t o b e 0.64, t h e v a l u e f o r random p a c k i n g o f h a r d s p h e r e s . I n c i d e n t a l l y , S a n d e r s ' c r i t e r i o n i s r e p r e s e n t e d by p=0.74 l i n e .

Next t a s k i s t o g e t t h e volume f r a c t i o n p o f observed a l l o y s t r u c t u r e s r e g a r d e d a s t h e packing o f h a r d s p h e r e s ( w i t h o u t t h e r m a l m o t i o n ) , a n d t o p l o t them t o g e t h e r w i t h a c u r v e i n Fig.8. I f t h e p o i n t S f a l l i n t h e up- e r r e g i o n o f t h e boundary c u r v e , t h e s t a b i l i t y o f t h e s t r u c t u r e i s pro- ved. I t i s done i n t h e n e x t s e c t i o n .

Fig.8. Y.C.W.-curve f o r 17 = l 0 ( I ) and q=20. A b i n a r y m i x t u r e o f h a r d s p h e r e s c a n n o t t a k e o r d e r e d s t r u c t u r e above t h i s c u r v e . The o r d i n a t e i s volume f r a c t i o n and a b s c i s s a i s d i a m e t e r r a t i o .

Volume F r a c t i o n o f A l l o y S t r u c t u r e s : By t h e word volume f r a c t i o n , we mean t h e volume f r a c t i o n o f h a r d s p h e r e s w i t h e f f e c t i v e d i a m e t e r , and we assume t h a t t h e y a r e packed w i t h no t h e r m a l motion i n t h e s t r u c t u r e . The volume f r a c t i o n s o f o b s e r v e d a l l o y s t r u c t u r e s can b e o b a t a i n e d by s p a c e f i l l i - n g c u r v e method by ~ a n d e r s . 8 H e c a l c u l a t e d volume f r a c t i o n

p f o r a few a l l o y s t r u c t u r e s made o f two s e t s o f h a r d s p h e r e s a s func- t i o n s o f Y , t h e d i a m e t e r r a t i o o f t h e c o n s t i t u e n t s p h e r e s , and p l o t t e d a s s p a c e f i l l i n g c u r v e P-Y. On t h e o t h e r hand, he o b t a i n e d a c t u a l d i a - m e t e r r a t i o ?ex i n t h e same t y p e s t r u c t u r e s i n o p a l s . From t h e s e two k i n d s o f q u a n t i t i e s , he found t h e volume f r a c t i o n o f t h e a l l o y s t r u c t - u r e s i n o p a l s . (He d i d n o t u s e e f f e c t i v e d i a m e t e r , and g o t s u c c e s s f u l r e s u l t ; t h i s i s p r o b a b l y b e c a u s e t h e e l e c t r o l y t e c o n c e n t r a t i o n i n math- e r phase o f o p a l s , s i l i c a s o l s , was v e r y h i g h , ho make t h e t h i c k n e s s o f e l e c t r i c a l d o u b l e l a y e r n e g l i g i b l y t h i n ) . I n t h e same way, we g o t t h e e f f e c t i v e volume f r a c t i o n s of t h e o b s e r v e d s t r u c t u r e s .

I n F i g . s 9,10 and 11, s p a c e f i l l i n g c u r v e s a r e shown. S h o r t v e r t i c a l l i n e s e x p r e s s e x p e r i m e n t a l Y - v a l u r e s from T a b l e I . They a r e drawn SO

a s t o i n t e r s e c t t h e r e l e v a n t c u r v e s . I n t h e f o l l o w i n g , i n d i v i d u a l c a s e s a r e e x p l a i n e d .

A I B z - s t r u c t u r e : Space f i l l i n g c u r v e c a l c u l a t e d by s a n d e r s 8 i s g i v e n i n Fig.9 t o g e t h e r w i t h Y.C.ru.-curve f o r r ) = l O . From t h e i n t e r s e c t s w i t h v e r t i c a l l i n e s f o r Y=0.50, 0.54, 0.56 and 0.61, e f f e c t i v e volume f r a c - t i o n s a r e known t o b e p=0.76, 0.78, 0.78 and 0.76 r e s p e c t i v e l y ; t h e y a r e g i v e n i n t h e 6 t h column o f t h e t a b l e . I n t h e f i g u r e , it is remark- a b l e t h a t a l l t h e i n t e r s e c t i o n s l i e on t h e t o p p a r t o f t h e c u r v e . They

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0.2 0.4 0.6 0.8 1.0 Y

Fig.9. Space filling curve for A1B2-structure and YCW-curve.

Ordinate is volume fraction and abscissa is y.

situate not only above Y.S.1T.-curve, but above Y=0.74 line. This result is quite natural, since a state of good packing is expected more stable, and very encouraging in view of the crudeness of our way of determing the effective diameters, especially those of larger particles.

NaZnls-structure : Two space filling curves i) and ii) are shown in Fig.10. Curve i) represents the result of our calculation taking into the distortion of icosahedrons of small particles (detail will be published elsewhere), while curve ii) is for regular icosahedron calcula- ted by Sanders 8 It is obvious that by this distortion the packing eff- iciency is significantly improved. The short vertical lines represent

?-values 0.56,0.61 and 0.63. p-values obtained from the intersections with curve i) are 0.73, 0.72 and 0.71 respectively and qiven in table, and are used for the stability consideration. The p-values from curve i) are all below those from curve i), especially, that atT=o.63 is very small and very near Y.C.W.-curve. If the icosahedrons are not distorted, this structure may not appear: the distortion of icosahedrons are very probable. $$ ^isof interest that in real crystalls this distortion is recognized. Important result is that all the Y-values lie below the 0.74-line, indicating that the principle of maximization of density does not work here.

Fig.10. Space filling curve for NaZnls-structure. Curve i) is for distorted case and curve ii) is for undistorted case. All intersections are below 0.74-line showing that Sanders criterion is not useful here.

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CaCu,-structure : The space filling curve is shown in Fig.11, together with that of MgCu2and Y.C.W.-curve. It is noteworthy that these two curves lie well below ~=0.74 line, indicating that Sanders criterion does not work here as was already seen with NaZn13. The curve for CaCu, has two peaks and one shoulder. Y-vaues are 0.72 and 0.75; and corresponding p are 0.69 and 0.67 respectively. Different from AlB, and NaZnla cases, Y-values are not found around the highest peak at(y=0.65) but situate near the shoulder which is significantly lower than the highest peak. The absence of Y-values in the Y-zone from 0.63 to 0.72 was questioned earlire from another point of view. However, this may be an artifact coming from errors involved in the process of effective diameter determination.

I r

0.20 0.40 0 60 0.80

-y- 0.60 0.80

Fig.11. (a) and (b) show space filling curves for CaCus- and MgCuz-structures. In (b), relevant part is given in magnification;

MgCuz-structure : Space filling curve in Fig.11 (a), (b) has only one distinct peak at Y=o.82 (p=0.71). Y-values from Table I are 0.77,0.80 0.83 and 0.84; all values situate near arround the peak. This result fs reasonable like the situation in AlBz-structure, and seems to give support to the hard sphere approximation. p-values are respectively 0.65, 0.68,~. 0.69 and 0.67. For Y=0.77, intersection point situates almost on Y.C.Y.-curve; its stability is somewhat doubtful. The presence of MgCuz at Y=o.77 was already suspected in earlier consideration and guessed to be a misidentification of CaCus. It 5s probable from the present point of view too, because 7=0.77 line intersects with CaCus-space filling curve atP=o.66, which is fairly above the Y.C.W.- curve as seen in the figure.

AB4-structure : The calculation of space filling curve is on the way.

Although, com?lete curve is not yet obbained, they-value would be very near 0.62 with P-value of 0.70.

Stability of the Structure : In the above described proceedure of obta- ining P-values, we already dealt with the 'stability problem of the observed structures. Here in Fig.12, the result is plotted on the phase diagram; it is seen that all the points situates above the Y.C.W.-

curve, satisfying the stability requirement. Thus, the fundamental as- pect of the mechanism of order formation is solved.

Conclusion is that: I) The order formation in a binary latex system is generally governed by the density principle. Because of the fact that the ordered-phase has a higer density than the disordered phase, trans- tion f ~ o m disordered .state bo ordered state' occures under high pressure

(or at hiqh particle concentration), and the transition codition is

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C3-126 JOURNAL DE PHYSIQUE

Fig.12 : p-)phase diagram of binary hard sphere systems (no thermal motion). Lowest point fo MgCu, can be raized to position of the dotted

circle, if it be a point of CaCu5.

given by Y.C.W.-function O(Y Q 1. 11)Struature type is determined by Sanderes principle of maximization of density at the extreme of high pressure (or high density), but iii) in the intermediate pressure, in other words, in the region between Y=0.74 line and Y.C.W.-curve, the situation is complex; a structure appears if its space filling curve has a peak or a shoulder near around the T-value of the concerning sys- kern.

REFERENCES

[l] Luck, W.,Klier, I4. and Wesslau, H., Ber. Bunsenges. Phys. Chem. 67 (1963) 75, 84.

[2] Hiltner, P. A. and Krieger, I.M., J. Chem. 73 (1969) 2386.

Hiltner, P. A., Papier,Y. S. and Krieger, I. M., J. Phys. Chem. 75 (1971) 1831; Krieger, i. M. and Hiltner P. A., in "Polymer Colloid" ed. by Fitch, R., Plenum Press. N.Y. 1971.

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[L] Alder, B. J. and Weinright,T.,E.,Phys. Rev., 127 (1962) 352;

Alder, B. J., Hoover,H. G. and Young,D. A., J. Chem. Phys. Q (1968) 3688.

[ 5 ] Yoshimura, S. and Hachisu, S., Nature 283 (1980) 188;

[6] Yoshimura, S. and Hachisu, S., Progr. Colloid Interface Sci. 62 (1983) 59.

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[8] Sanders, J.,B. and Murray,M. J., Phil. Mag. 2 (1980) 721 [g] Rice, 0. K., J.Chem. Phys. 12 _-(1944)l.

1101 Yerazunisl S - . Cornel, S. W. and Winter, B., Nature 207 (1965)835.

[l11 shoemaker,^. P., Marsh, R. E., Ewing, F. J. and Pauling, L., Acta Cryst. 5 (1952) 637.