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Orientational transitions in an adsorption layer of inter-acting molecules
J. Electroanal. Chem., 115 ( 1 9 8 0 ) 7 5 - - 8 8
75
© Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in T h e N e t h e r l a n d s
ORIENTATIONAL TRANSITIONS IN AN ADSORPTION LAYER OF INTERACTING MOLECULES
Yu.I. K H A R K A T S
Institute of Electrochemlstry, U.S.S.R. Academy of Sciences, Moscow (U.S.S.R.) ( R e c e i v e d 1 9 t h May 1 9 8 0 )
ABSTRACT T h e b e h a v i o u r o f a n a d s o r p t i o n s y s t e m o f i n t e r a c t i n g organic m o l e c u l e s p r e s e n t o n t h e surface in t w o d i f f e r e n t o r i e n t a t i o n s is discussed. T h e i n f l u e n c e o f t h e r a t i o b e t w e e n attract i o n c o n s t a n t s for various o r i e n t a t i o n s o n t h e f u n c t i o n linking coverage t o t h e b u l k ads o r b a t e c o n c e n t r a t i o n is e x a m i n e d . It is s h o w n t h a t a c c o r d i n g t o t h e r a t i o o f t h e i s o t h e r m p a r a m e t e r s o n e or t w o o r i e n t a t i o n a l t r a n s i t i o n s m a y be p r e d i c t e d . T h e critical c o n d i t i o n s for a b r u p t r e o r i e n t a t i o n s are o b t a i n e d .
INTRODUCTION
It is known that molecules of some organic substances m a y be differently oriented on the surface, and as pointed o u t by Frumkin [1], variation of volume concentration of the adsorbate may result in change of the predominant orientation of the adsorbed molecules. Reorientation may also occur when changing the potential of the adsorbing surface. The effect of reorientation of adsorbed organic molecules has been extensively studied, both theoretically and experimentally [2--20]. The general conclusion that can be drawn from these works is that for sufficiently strong attractional interaction between molecules, reorientation of adsorbed molecules as well as two-dimensional condensation occur abruptly and are first-order phase transitions. The equations describing the adsorption isotherm of interacting molecules which m a y occupy two different orientations on the surface were obtained by Parry and Parsons [2] by phenomenological generalization of the Frumkin and Flory--Huggins isotherms: 01
BlC = (1 -- 01 -- 02)" e x p ( - - 2 a l n O 1 - - 2a12n02)
(1)
02 e x p ( - - 2 a 2 r n 0 2 - - 2a12rn01) B2c = (1 - - 0 1 - - 0 2 ) m
(2)
where 01 and 02 are the coverages for the t w o orientations, n and m the number of adsorption sites occupied by a molecule in the corresponding orientation, c the volume concentration of the adsorbate, BI and B2 the adsorption coefficients and al, a2 and a~2 the constants describing interaction of molecules in
76
o r i e n t a t i o n 1, o r i e n t a t i o n 2 and cross-wise i n t e r a c t i o n . E q u a t i o n s (1) and (2) m a y also be derived f r o m the c o n d i t i o n s o f t h e r m o d y n a m i c e q u i l i b r i u m ~F/~01 = 0 and ~F/~02 = 0, w h e r e F is the free e n e r g y o f t h e a d s o r p t i o n s y s t e m which m a y be calculated b y m e t h o d s o f statistical t h e r m o d y n a m i c s (see, for e x a m p l e t h e A p p e n d i x in ref. 21 and f o r o u r s y s t e m takes the f o r m : F = k T F ~ [--alO~ --a20~
--
2a120102
ln(O1/Btc) + (02/m) ln(O2/B2c) +
+ (01/n)
(1 --01 --02) ln(1 --01 --02) + 0 1 ( n
-- 1)/n
+ 02(m --
1)/ml
(3)
w h e r e k is t h e B o l t z m a n n c o n s t a n t ; T t h e t e m p e r a t u r e and F~ the limiting n u m ber o f a d s o r p t i o n sites per u n i t area. Taking into a c c o u n t eqns. (1) and (2) t h e expression f o r F m a y be r e w r i t t e n in a simpler f o r m : F = k T F = [alO~ + a20~ + 2a120102 + ln(1 + Odm
--
--
0
1 --
02) + 01(n -- 1)/n
1)/ml
(4)
In this f o r m it coincides with t h e f o r m u l a f o r surface t e n s i o n o f an a d s o r p t i o n layer [8]. T h e set o f eqns. (1) and (2) d e t e r m i n e s t h e c o n e e n t r a t i o n a l d e p e n d e n c e o f the coverages 01 and 02 and o f the t o t a l n u m b e r o f m o l e c u l e s a d s o r b e d per u n i t area 17' = ['~(01/n + 02/m). T h e f u n c t i o n s 01(c), 02(c) and p(c) are m a n y - v a l u e d in certain intervals o f c, and a m o n g the possible states o f t h e a d s o r p t i o n s y s t e m physically realized t h e r e m u s t be the o n e t h a t c o r r e s p o n d s t o t h e m i n i m u m o f free energy F . T r a n s i t i o n f r o m o n e state i n t o a n o t h e r occurs in an a b r u p t m a n n e r and is similar t o a first-order phase transition, p r o v i d e d t h a t t h e free energies o f these states are equal. This t r a n s i t i o n involves n o t o n l y an a b r u p t change o f the t o t a l n u m b e r o f molecules per u n i t area in b o t h o r i e n t a t i o n s b u t also a significant variation o f the ratio b e t w e e n coverages 01 and 02, i.e. a b r u p t r e o r i e n t a t i o n o f molecules in t h e a d s o r p t i o n layer. B E H A V I O U R O F THE A D S O R P T I O N SYSTEM F O R a2 > 0 , al = 0, a12
=
0
Since the set o f eqns. (1) and (2) includes m a n y p a r a m e t e r s and its analytical s t u d y presents considerable difficulties, let us begin with t h e simplest case w h e n o n l y one o f t h e i n t e r a c t i o n c o n s t a n t s , say a2, is n o t equal t o zero. (Since we shall consider b o t h cases m > n and n > m, t h e results o b t a i n e d a f t e r some t r a n s f o r m a tions will also describe the case al > 0, a2 = 0 and a12 = 0.) Assuming in eqns. (1) and (2) al = 0 and a12 = 0 and designating x = B l c , a = a2rn and ~ = B1/B2, we obtain: x =
01
(1 --01 --02) n
(5)
~02 x - (1 -- 01 -- 02) m exp(--2a02)
(6)
We shall distinguish t h r e e main cases d e p e n d i n g o n the ratio b e t w e e n p a r a m e t e r s m and n: (1) m = n; (2) m < n; (3) m > n. T h e case m < n c o r r e s p o n d s t o pre-
77 d o m i n a n t i n t e r a c t i o n b e t w e e n m o l e c u l e s in t h e m o r e c o m p a c t a r r a y ( w h i c h m a y be o r i e n t e d n o r m a l t o t h e surface). T h e case m > n c o r r e s p o n d s t o pred o m i n a n t i n t e r a c t i o n b e t w e e n m o l e c u l e s in t h e less c o m p a c t a r r a y (which m a y be o r i e n t e d parallel t o t h e surface). O n e o f t h e possible m e c h a m s m s r e s p o n s i b l e f o r such i n t e r a c t i o n was e x a m i n e d in ref. 22. We shall s t a r t w i t h t h e case m = n. E q u a t i o n s (5) a n d (6) m a y be r e w r i t t e n as 01 = ~02 e x p ( - - 2 a 0 J
(7)
02 = 1 - - 01 - - ( 0 1 I x ) w ~
(8)
E q u a t i o n (7) describes a f u n c t i o n 01(02) w h i c h is r e p r e s e n t e d b y a c u r v e w i t h a m a x i m u m (Fig. 1). D e p e n d i n g on t h e values o f p a r a m e t e r s a and/3 this m a x i m u m m a y b e s i t u a t e d in t h e p h y s i c a l region 0 < 01 < 1, 0 < 02 < 1, 0 < 01 + 02 < 1 or o u t s i d e it. E q u a t i o n (8) is graphically r e p r e s e n t e d b y a set o f curves (in t h e specific ease w h e n m = n = 1 t h e y are s t r a i g h t lines) o r i g i n a t i n g in t h e p o i n t 01 = 0, 02 = 1, t h e " s l o p e " o f w h i c h s m o o t h l y varies w i t h x. I n t e r s e c t i o n o f t h e p l o t s o f t h e r e l a t i o n s (7) a n d (8) gives t h e graphical s o l u t i o n o f set (5) a n d (6) a n d allows t h e d e t e r m i n a t i o n o f t h e f u n c t i o n s 0 l(x) a n d 0 2 ( x ) . As can be seen f r o m Fig. 1, t h e i n t e r s e c t i o n o f t h e curves is possible in o n e p o i n t or in t h r e e p o i n t s c o r r e s p o n d i n g r e s p e c t i v e l y t o a single s o l u t i o n o f t h e set (5) a n d (6) or t o t h r e e s o l u t i o n s w h i c h we shall d e n o t e b y s u p e r s c r i p t s 1, 2 a n d 3. Curves a, b a n d e in Fig. 1, c o r r e s p o n d i n g t o a given value o f p a r a m e t e r a a n d t o t h r e e d i f f e r e n t values o f p a r a m e t e r / 3 , r e p r e s e n t t h r e e d i f f e r e n t b e h a v i o u r s o f t h e a d s o r p t i o n sys-
0
............................
5 a
0 ~
10 × FmO L~IF ~2,
d('
e~~
b
e' ~'
0 4
3
1
5
C
10
",
5
"~
r L-j
0 1
92
'e~J 1
2
3
4
Fig. 1. Graphic solution of set of eqns. (7) and (8) for m = n = 1 and a = 3. Curves a, b and c represent the functions 01 (02) for ~ = 30, 15 and 10 respectively. (1) Straight line 01 + 02 = 1; straight hnes (2), (3) and (4) correspond to x2 < x3 < x4. Fig. 2. The functions 01(X), 02(X ) and F(x) for cases (a), (b) and (e) depicted in Fig. 1. Arrows show the position of abrupt tranmtions on the dependence F(x).
78
tem. For curve a (sufficiently high values of/3) there is always one solution to the set (5) and (6) and the isotherm is single-valued (Fig. 2a). With decrease of/3, curve 01(02) drops off and its right-hand branch comes partially below the straight line 01 + 02 = 1. Isotherms 01(x), 02(x) and F(x) corresponding to this case are many-valued from value Xmln to x = oo (Fig. 2b). Eventually, with further decrease of/3, the m a x i m u m of the curve 01(02) comes below the straight line 01 + 02 = 1. Isotherms 0 ~(x), 02(x) and F(x) corresponding to this case are manyvalued in the interval Xmln < x < Xmax, which is determined from the condition of tangency of curves described by eqns. (7) and (8) (Fig. 2c). The critical conditions of transition from case (a) to case (b) and then to case (c) are found from the conditions of tangency of the straight line 01 + 02 = 1 to the curve 01(02): /3> ~max = aa- -+x /~~ - - 2 a
exp(a + v r ~ -- 2a),
~ < ~mln a+v~--2a a_ ~ e x p ( a -
- - x / ~ -- 2a),
~max(a)
~mm(a) < ~ <
case (a)
(9)
case (b)
(10)
case (c)
(11)
Graphically, conditions (9)--(11) correspond to division of the parameter plane (a, In fi) into regions separated by critical curves ai(ln fi) (or fl = timex(a)) and aii(ln fi) (or/3 = ~min(a)) originating from point a = 2, In/3 = 2 (Fig. 3). Note that the region where the isotherm is many-valued, determined by condition (9), occurs at a > 2 and additionally depends on the value of parameter ft. The simple fact that the functions 01, 02 and F of x are many-valued does n o t itself guarantee realization of a phase transition. Thus, abrupt reorientation is possible only in the part of the "many-valued" region in Fig. 3 which is situated to the left of the critical curve a111(ln/3) determined from condition F(O ~1), 0(21)) = F(O(~3), 0(23)) at x -+ ~ . To the right of the critical curve aHi(ln fi) inequality F(0(~1), 0~1)) < F(O~ 3), O~a>) is true for all x > Xm~n and, despite the fact that the isotherm is many-valued, transitions are absent. At m = n = 1, the values Xm~ and Xm~x, determining the region of manyvalued isotherms, may be f o u n d analytically and are given by the following equations: /3
Xm, n = ~ m a x ( a ) - -
Xmax -- ~ - - ~ m , n ( a )
(12) (13)
where timex(a) and fim~(a) are given by eqns. (9) and (10). The critical curve aiIi(ln/3) at m = n = 1 is determined from the set of equations derived from (4), (7) and (8) f o r x - ~ ~ o : a(0(21)) 2 * ln(1 -- 0(2~) ) = a(0(23)) 2 + ln(1 -- 02TM) (14) 1 -- O:(1'3) = ~0(2~'3) exp(--2a02(1'3))
(15)
the solution of which is aiIi=ln/3;
ln~/>2
(16)
79
Taking into a c c o u n t t h a t , during a b r u p t r e o r i e n t a t i o n , 02 usually changes f r o m v e r y small values 0(21) < < 1 to values close t o u n i t y (1 -- 0(23) < < 1), we m a y d e d u c e f r o m eqns. (4), (7) and (8) at m = n = 1 an a p p r o x i m a t e analytical expression for t h e c o n c e n t r a t i o n x* at which the o r i e n t a t i o n a l phase t r a n s i t i o n occurs:
x* =
fi e x p ( - - a ) 1 --/3 e x p ( - - a ) '
/3 e -2a < < 1, fie -a <~ 1
(17)
Let us n o w e x a m i n e t h e c o n d i t i o n s which lead t o t h e a p p e a r a n c e o f m a n y valued a d s o r p t i o n i s o t h e r m s due t o S-shaped sections. These c o n d i t i o n s , which m a y be w r i t t e n as dx/d02 -- 0 and d2x/dO~ = 0 (where x(02) is t h e inverse funct i o n o f 0 : ( x ) given b y eqns. (7) and (8), i m p o s e a relationship b e t w e e n values a and In ft. A t m = n = 1 this relationship reduces t o the c o n d i t i o n a = 2, In/3 ~< 2 and, in the general case m = n <> 1, it is r e p r e s e n t e d b y curve aiv(ln/3) o n plane (a, In/3) (Fig. 3) originating f r o m p o i n t a = 2, In/3 = 2 and t e n d i n g at In/3 -~ - - ~ to t h e a s y m p t o t e 1 al : ~(1 + x/~) 2
(18)
A b o v e the curve a~v(ln ~), the i s o t h e r m presents an S-shaped section and t h e r e is an o r i e n t a t i o n a l phase transition; for p a r a m e t e r s b e l o w t h e curve alv(ln/3) t h e i s o t h e r m is single-valued and transitions are absent. Hence, in this case (m = n) r e o r i e n t a t i o n a l phase transitions are possible in r e g m n (c) and p a r t o f region (b) on the p a r a m e t e r plane (a, In/3) r e s t r i c t e d b y the critical curves a~ii and aiv • In t h e case n > m t h e a d s o r p t i o n i s o t h e r m s given b y eqns. (5) and (6) are e i t h e r single-valued or have an S-shaped section on which a phase t r a n s i t i o n occurs. T h e f u n c t i o n s 0~, 02 and F o f x in this case are similar t o the plots s h o w n in Figs. 2a and 2c f o r t h e case m = n (see also ref. 8). T h e critical curve a(ln/3) c o r r e s p o n d i n g t o the a p p e a r a n c e o f S-shaped sections on t h e i s o t h e r m is determ i n e d b y eqns. (5) and (6) and the c o n d i t i o n s dx/d02 = 0, dZx/dO~ = O. A t n = 1 and m < n this curve is r e d u c e d to a h o r i z o n t a l straight line c o r r e s p o n d i n g
a
a~,
3
3
a~
2
J
J
T
-1
1
1
I
2
3
4
s
In~
2
I
0
I
2
3
Infl
Fig. 3. P o s i t i o n o f critical c u r v e s a i , a i i , a i i I a n d a i v o n t h e p a r a m e t e r p l a n e (a, In fi) a t m = n, n <> 1; a ~ ,2,3) c o r r e s p o n d t o m = n = 2, 1 a n d 0.5. Fig. 4. S h a p e o f t h e critical c u r v e a ( l n / 3 ) f o r t h e case m < n, n <> 1. ( 1 ) m = 1, n = 2; (2) _1 _1 1 m-~,n=l;(3) m-~,n=~.
80 £1
/ 3.
2
cl
1
I
5! a
2
o',"/
~':,,'
/
a'~
3
~ Inl]
.-"~.;'
82 1
-
4
3 a:'
z b
J 2
1
1
2
3
4 Ln~
1
2
3
4
x
Fig. 5. P o s i t i o n o f critical c u r v e s at, a i r , a r r [ , a l V a n d a v a t rn > n, n ~ l . (a) m = 2, n ~ 1; ( b ) s u p e r s c r i p t (1) c o r r e s p o n d s t o m = 3, n = 2, s u p e r s c r i p t ( 2 ) c o r r e s p o n d s t o m = 1, n = ~. Fig. 6. T h e f u n c t i o n 0 2 ( x ) f o r m = 2, n = 1 a n d a = 3.5. ( 1 ) fi = 5; (2) fi = 4 . 3 ; (3) ~ = 3.5; ( 4 )
,G= 3n.
to a = al given b y eqn. (18) and, in the general case n > m, t h e critical curve a(ln/3) s m o o t h l y changes f r o m t h e a s y m p t o t e a = al, where al is d e t e r m i n e d b y eqn. (18) at In/3 -~ - - ~ t o the a s y m p t o t e c o r r e s p o n d i n g t o In/3 -~ oo, i.e. ar = ½(1 + v/-m]-n) 2
(19)
In t h e specific case for which m = 1 and n = 2, studied in refs. 8 and 9, the a s y m p t o t i c value ar = 1.46 t o which the critical curve t e n d s as/3 -~ oo, is in agreem e n t with eqn. (19). T h e shape o f t h e critical curve a(ln/3) at n > m and n > 1, n = 1 and n < 1 is s h o w n in Fig. 4. In the region above the critical curve a b r u p t r e o r i e n t a t i o n s o c c u r , while b e l o w this curve t h e y are absent. L e t us n o w c o n s i d e r the case rn > n. T h e a d s o r p t i o n s y s t e m displays interesting features [ 1 1 - - 1 3 ] which are illustrated b y t h e plots o f t h e f u n c t i o n 02(x) and F ( x ) and b y the position o f critical curves o n t h e p a r a m e t e r plane (a, In/3) (Figs. 5--7). When values o f p a r a m e t e r s a and In/3 lie in t h e region situated t o t h e right o f t h e critical curve a i (ln/3) (Fig. 5) the a d s o r p t i o n isotherms 01(x), 0 ~(x) and F ( x ) are single-valued. On crossing t h e critical curve aT (ln/3) we e n t e r the region o f p a r a m e t e r s s i t u a t e d b e t w e e n aT (ln/3) and aiT (ln/3). T h e f u n c t i o n s 0 l(x) and 02(x) in this region consist o f t w o branches, t h e main b r a n c h beginning at t h e origin o f c o o r d i n a t e s and the o t h e r in the f o r m o f a closed l o o p . T h e critical curve ai(ln/3) c o r r e s p o n d s t o t h e a p p e a r a n c e o f a l o o p o f infinitely small size, which increases w h e n t h e critical curve a n (ln/3) is a p p r o a c h e d . When the parameters acquire values c o r r e s p o n d i n g t o critical curve a n (ln/3) t h e l o o p is t a n g e n t t o t h e main b r a n c h . In the region t o the left o f an(In/3) t h e f u n c t i o n s OT(x) and
81 f-IF_ 1
0
1
2
3
L
x
Fig. 7. T h e f u n c t i o n F ( x ) for m = 2, n = 1 a n d a = 3.5,/3 = 3.5. A r r o w s s h o w t h e calculated position of abrupt transitions.
02 (x) have two S-shaped sections, which correspond to two orientational phase transitions. It is evident from topological considerations that when the functions 0 l(x) and 02(x) and therefore the function F(x) have a loop branch, phase transitions are either absent or occur in pairs [11--13]. The absence of intersection of the two separate branches F(x) or their intersection at two points (in general, at an even number of points) correspond to such a situation. The tangency of the two separate branches F(x) * corresponds to the condition of the appearance of a pair of such reorientational transitions (bi-transition). This condition allows the calculation of the critical curve for the appearance of bi-transitions at1 ~(ln/3) on the plane (a, In/3) (Fig. 5). In the region between a I and a n i where the loop is still sufficiently small, the corresponding states are metastable and transitions do n o t occur. In the region between a l n and a n where the loop is sufficiently large a bi-transition is realized. In the general case there are two other critical curves aiv(ln/3) and av(ln/3) on the parameter plane (a, In/3) corresponding to the appearance (or disappearance) of the left- and right-hand S-shaped sections on curves 0~(x) and 02(x). In the specific case of m > n and n ~ 1 these two curves merge into one horizontal straight line a = a i v determined from eqn. (18). This straight line originates from the point of convergence of critical curves a~, aI[ and an~. In the region above critical line aiv, two S-shaped sections appear simultaneously on
01(x) and 02(x). The critical curves aiv(ln/3) and avon/3) originate from the point of convergence of curves ai, all a n d ' a n i and tend, for In/3 -~ --~, to horizontal asymptotes. The asymptotic value for alv corresponding to the appearance (or disappearance) of the left-hand S-shaped section is determined by eqn. (18) and that for a v corresponding to the appearance (or disappearance) of the right-hand S-shaped section is determined by eqn. (19). At n > 1 curve a w lies above curve av and at n < 1 below. Let us now consider the dependence F(x) for m > n. Analysis shows that * A n o t h e r e x a m p l e o f t a n g e n c y o f b r a n c h e s of free e n e r g y is a f f o r d e d b y t e m p e r a t u r e p h a s e t r a n s i t i o n s in a d s o r p t i o n s y s t e m s [ 2 1 , 2 3 ].
82
when the p a r a m e t e r s a and In fl are in the region b e t w e e n the critical curves a1 and a n and if the f u n c t i o n s 01(x) and 02(x) have a l o o p b r a n c h in a d d i t i o n to t h e main b r a n c h , t h e c o r r e s p o n d i n g closed b r a n c h F ( x ) is r e p r e s e n t e d b y a curve in t h e f o r m o f a h o r i z o n t a l " 8 " (Fig. 7). It s h o u l d be n o t e d t h a t , a l t h o u g h each t r a n s i t i o n f o r m i n g a bi-transition involves c o n s e c u t i v e r e o r i e n t a t i o n o f molecules (the ratio 01/02 varies f r o m high values to low and t h e n again t o high values), the t o t a l n u m b e r o f molecules a d s o r b e d on a u n i t area (in b o t h o r i e n t a t i o n s ) increases with each transition, and t h e r e f o r e in this case t w o - d i m e n s i o n a l c o n d e n sation is a t w o - s t e p process. B E H A V I O U R O F T H E A D S O R P T I O N S Y S T E M F O R al > 0, a2 > 0 A N D a12 = 0
T h e results described above m a y also be c o n s i d e r e d f r o m a slightly d i f f e r e n t v i e w p o i n t . L e t us use such designations t h a t p a r a m e t e r n will always e x c e e d p a r a m e t e r rn. T h e n , as s h o w n above, t h e b e h a v i o u r o f the a d s o r p t i o n s y s t e m f o r al = 0, a2 > 0 and a12 = 0 differs significantly f r o m the b e h a v i o u r for a~ > 0, a2 = 0 and a,2 = 0. A t t h e same t i m e t h e p a r t i c u l a r situations, a~ = 0, a2 > 0, a~2 = 0, and al > 0, a2 = 0, a12 = 0, m a y be c o n s i d e r e d as limiting cases o f t h e m o r e general s i t u a t i o n ax > 0, a2 > 0 and a12 = 0. Here we shall s h o w t h a t m t h e general case ax > 0, a2 > 0 and a12 = 0 the behaviour o f a d s o r p t i o n systems displays interesting features (first r e p o r t e d in refs. 1 1 - - 1 3 ) , including t h o s e characteristic o f the t w o limiting cases m e n t i o n e d above. F o r simplicity let us consider t h a t p a r a m e t e r s n and rn are equal t o 2 and 1 respectively. T h e critical c o n d i t i o n s c o r r e s p o n d i n g to o r i e n t a t i o n a l transitions in the general case will be given b y t h e surfaces/3 = fi (a~, a2) in the p a r a m e t e r space (al, a2, fi) (Fig. 8). I n t e r s e c t i o n s o f these surfaces with planes a2 = 0 and al = 0 give the critical curves and regions on t h e p a r a m e t e r planes, which are obtained f r o m Fig. 5a b y s u b s t i t u t i o n a = 2a~ and fi = 1/fi and f r o m curve 1 in Fig. 4. T h e y are s h o w n in a p l o t o f a vs. fi in Figs. 9a and 9b. In o r d e r t o d e m o n s t r a t e the m u t u a l a r r a n g e m e n t o f t h e critical surfaces in t h e space (a~, a2, fi) the calculated cross-sections o f these surfaces with the auxilary planes al = a2 and a2 = 2a~ are s h o w n in Figs. 9c and 9d. T h e plane ax = a2, fl is divided b y seven critical curves ar(fi)--awi (fi) into nine regions (Fig. 9c). T h e b e h a v i o u r o f the f u n c t i o n s 0 ~(x) and 02(x) in these regions is illustrated in Figs. 10--12.
1%
9a~, ,a~=az,F,g9c\ J
s,=O,
FLg
a2= 2a, F, g9d
i
,
\\,l , i
a2=O, F,g9b
/J
,/ Cl I
Fig. 8. M u t u a l a r r a n g e m e n t o f p a r a m e t e r planes s h o w n in Figs. 9a, 9b, 9c a n d 9d.
83
(1 =G2 O,
"
~ 2
Q:QI
Q~
°-Oz:20t
3
d
2
os Fig. 9. Position
p
of critical curves on parameter
o
:
2
p
p l a n e ( a , / 3 } , ( a ) a = a2 > 0 , a l = 0 , a 1 2 = 0 ; ( b )
a = al > 0, a2 = 0, a12 = 0; (c) a = al = a2 > 0, a12 = 0; (d) a = as = 2al > 0, a12 = 0.
Let us consider the variation of the functions 0 ~(x) and 02(x) at fixed values of parameters al = a2 = 1.5 and increasing fi (Fig. 10), when passing from region 2 into regions 4, 6 and 9 in Fig. 9c. In region 2, curves O~(x) and 02(x) are singlevalued at all values of x and abrupt orientational transitions are absent. On increase of fi, we cross the critical curve ali (fl), corresponding to the appearance of a loop solution to the set (1) and (2), and enter region 4. The functions 01(x) and 02(x) in region 4 consist of two disjointed branches: the main branch beginning at the origin of the coordinates and the loop branch which is situated above the main branch for 01(x) and below for 02(x). However, in so far as the loop branch is relatively small, the corresponding states of the adsorption system possess higher energy than the states on the main branch and consequently are metastable. When parameter ~ exceeds the critical values corresponding to the curve of bi-transition appearance, a i n (fi), the loop solution has a section in region 6 where the free energy is lower than on the main branch. This corresponds to an orientational bi-transition in the adsorption system. The first transition is accompanied by an abrupt increase of 01 and an abrupt decrease of 02, and the second transition by an abrupt decrease of 0~ and an abrupt increase of 02. Note, that as in the case examined previously (al > 0, a2 = 0 and a12 = 0) the value of P increases with each orientational transition. With further increase of fi we cross the critical curve aiv(~) corresponding to tangency of the loop to the main solutions of set (1) and (2) and we enter region 9. The functions 0 ~(x) and 0 ~(x) are described here by joint curves, each of them has two S-shaped sections for which an orientational transition occurs.
84
Hence, the behaviour of the adsorption system in regions 2, 4, 6 and 9 is very similar to that related to the corresponding regions shown in Fig. 9b. Let us now consider slightly higher values of al = a2, e.g. al = a2 = 2.25 (Fig. 11). The critical curve ai (/3) corresponds to the appearance of an S-shaped section on the main branch of the solutions of 0 l(X) and 02(x). At relatively small values of x corresponding to region 1 in Fig. 9c, the function O~(x) and 02(x) have one branch with an S-shaped section for which an orientational transition occurs. At higher/3 we enter region 3 where solutions consist of two disjointed branches. The main branch has an S-shaped section for which there is a transition, while states on the loop branch are metastable. In a certain range of concentrations, set (1) and (2) has five solutions. Analysis shows that the maximal n u m b e r of solutions is seven and that t h e y may exist at relatively high values of attraction constants al and a2. With further increase of fl we cross the critical bi-transition curve aiii (fi) and enter region 5 in Fig. 9c. The functions Offx) and 02(x) in this region, as in region 3, have main and loop branches. However, a considerable difference arises in the absence of the orientational transition on the S-shaped section of the main branch. This " b l o c k i n g " of the transition is due to the fact that the j u m p from the state preceding the S-shaped transition to the loop branch and then back to
~2 q
,
,
/
'J1 1
1hi J
0
1
,3
01
Fig. 1 0 . T i m e - f u n c t i o n 0 1 ( x ) a n d 02(x ) f o r a = al = a2 = 1.5. ( 1 ) ~ = 0 . 5 ; ( 2 ) ~ = 0 . 7 5 , ( 3 ) = 0 . 9 , ( 4 ) ~ = 1. C u r v e s 1 - - 4 c o r r e s p o n d t o r e g i o n s 2, 4, 6 a n d 9 in Fig. 9c. A b r u p t t r a n s i tions on curve 3 correspond to x = 0.145 and x = 0.33, on curve 4 to x = 0.16 and x = 0.22. Fig. 11. T h e f u n c t i o n 01(X ) a n d ~ 2 ( x ) f o r a = a l = a2 = 2 . 2 5 . ( 1 ) / 3 = 0 . 0 8 ( 0 . 0 0 8 5 ) , ( 2 ) = 0.2 (0.0213); (3)/3 = 0.5 (0.032, 0.088); (4)/3 = 0.75 (0.0287, 0.20); (5)/3 = 0.85 (0.0285, 0 . 2 6 ) . C u r v e s 1 - - 5 c o r r e s p o n d t o r e g i o n s 1, 3, 5, 7 a n d 9 In Fig. 9c. T h e v a l u e s o f x a t w h i c h a b r u p t o r m n t a t i o n a l t r a n s i t i o n s o c c u r a r e g i v e n in p a r e n t h e s e s .
85 ~2
0
i
118'
0
1
x
Fig. 12. T h e f u n c t i o n 0 1 ( x ) a n d 0 2 ( x ) f o r a = a 1 = a 2 = 1.25. (1)/3 = 1 . 0 5 ; (2)/3 = 1 . 1 2 ; (3) /3 = 1.2. C u r v e 1 c o r r e s p o n d s t o r e g i o n 2 a n d c u r v e s 2 a n d 3 t o r e g i o n 8 in Fig. 9c. A b r u p t t r a n s i t i o n s o c c u r o n c u r v e 2 a t x = 0.35 a n d o n c u r v e 3 at x = 0 . 4 6 .
the main branch, is thermodynamically more favourable. The region with a transition on the S-shaped section is thus by-passed along the loop solution. In region 7 situated to the right of critical curve a~v(~), (corresponding to the tangency of the main and loop solutions) the functions 0 ~(x) and 02(x) present two S-shaped sections for which orientational transitions occur. The critical curve av(~) separating regions 7 and 9 corresponds to the disappearance of the small S-shaped section which appeared after convergence o f the main and loop solutions. Hence, one orientational transition occurs in regions I and 3 and two orientational transitions in regions 5, 7 and 9. Two other critical curves avi and avi I originate from the point of convergence of curves aii, aiiI and a~v and correspond to the disappearance (or appearance) of S-shaped sections on curves 0 ~(x) and 02(x). Figure 12 illustrates the behaviour of 01(x) and 02(x) at al = a2 = 1.25. In region 8 a single orientational transition occurs. Therefore the parameter plane (at = a2, ~) (Fig. 9c) is divided by critical curves into nine regions. In regions 2 and 4 transitions are absent, in regions 1, 3 and 8 a single transition occurs and in regions 5, 6, 7 and 9 two orientati~nal transitions occur. The arrangement of the critical curves for the more general case a~ = a2 > 0 is more complex than in the two limiting cases a~ > 0, a2 = 0 and a~ = 0, a2 > 0 and may be approximately represented by superposition of the " b e a k " for case a~ > 0, a2 = 0 (Fig. 9b) and of the curve corresponding to the appearance of an S-shaped section characteristic of the case a~ = 0, a2 > 0 (Fig. 9a). As a result of this superposition the curve for the appearance of S-shaped sections is split into two parts, ai (/3) and avii(fi) (Fig. 9c).
86
In order to obtain a more complete picture of the arrangement of the critical curves in the (al, a2,/3) space and to reveal some additional features in the behaviour of the adsorption system, let us examine another cross-section of these surfaces, namely by the plane (a2 = 2al) (Fig. 9d). The parameter plane in this case is divided by the critical curves a~--av into five regions. In region 1 the functions 0 l(x) and 02(x) are single valued. The critical curve al corresponds to the appearance of an S-shaped section on 01(x) and 02(x). Critical curves aii and a n i correspond to the appearance of a loop solution and its convergence with the main branch. A specific feature of the case considered is that states on the loop branch t h r o u g h o u t region 3 (situated between a n and aii I) are metastable and a bitransition is not possible. Double transitions are nevertheless produced at sufficiently high values of/3. They appear in region 5, which is to the right of the critical curve of the appearance of a " h o p " -- aiv(~). The " h o p " p h e n o m e n o n may be described as follows. In regions 4 and 5, after convergence of the loop and the main branch solutions, functions 01(x) and 02(x) have a portion where the set (1) and (2) has five solutions. In this portion 02(x) displays a small, and 8 l(x) a large, S-shaped section (curves 3 and 4 in Fig. 13). In the case of five solutions the transition from the main state which occupies a middle position with respect to the other solutions may proceed in the direction of higher 02 (and lower 01) or in the direction of lower 02 (and higher 01). The first case occurs in region 4 and the second in region 5. It is easy to see that in region 5 the first orientational transition must be followed by a second one. The " h o p " of the orientational transition occurs when the free energies of three of the five solutions of set (1) and (2) corresponding to minima of free Le2
[
0
1
X
1
x
8~
4
/ f /
F i g . 1 3 . T h e f u n c t i o n s 01(x) a n d 02(x) f o r a = a2 = 2 a l = 3. ( 1 ) fi = 0 . 8 ; ( 2 ) / 3 = 1 . 2 ; ( 3 ) ~ = 1 . 6 ; ( 4 ) fl = 2 . 2 ; ( 5 ) / 3 = 5. T r a n s i t i o n s o n c u r v e 5 o c c u r a t x = 0 . 1 1 a n d x = 0 . 5 3 .
87
energy are equalized simultaneously, a situation which is represented by the critical curve a~v in Fig. 9d. Curve av(~) in Fig. 9d is similar to curve avi (~) in Fig. 9c and gives the condition o f disappearance of the small S-shaped section on 02(x) (large S-shaped section on 0 l(X)). Therefore, in the case a= = 2al, the parameter plane is divided by critical curves into five regions. In region 1 transitions are absent. In regions 2--4 a single transition occurs and in region 5 two transitions occur. The location o f the points on the parameter plane, from which critical curves ai (/3) originate and o f the as ym pt ot es to which critical curves a w and a v n in Fig. 9c and a~ and a v in Fig. 9d tend, may be f o u n d analytically. The origins of curves al for ~ -+ 0 are determined from eqn. (18). A sym pt ot i c values for curve ai in Fig. 9d and curve avn in Fig. 9c for/3 -~ ~ are given by eqn. (19). The a s y m pto tic behaviour of curves av in Fig. 9d and avi in Fig. 9c for ~ -~ ~ which determine the co n d i t i on of the appearance (or disappearance) of S-shaped sections on 0 l(x) and 02(x) is given by the equation of the asymptotic plane in space
(a~, a~, ~): a2 + a l - - = n
1 +
2
(20)
For given ratios between a~ and a2 the appropriate values of av and aw for/3 -~ m a y be obtained f r om eqn. (20). DISCUSSION
Analysis o f the behaviour of adsorption systems satisfying equations of isot h e r m (1) and (2) shows that, depending on the ratio between parameters, one or two orientational transitions m ay be observed, according t o t he adsorbate concentration. This is due in the first place to attraction interaction and, in the case of double transitions, also to the difference in geometrical properties of molecules adsorbed in different orientations. The latter factor is responsible for the "forcing o u t " effect which determines p r e d o m i n a n t coverage of the surface with molecules normally oriented t o the surface at high adsorbate concentrations. A specific feature of the system studied is that, along with t he state of "rarefied two-dimensional gas" at low adsorbate concentrations and the state of "condensed two-dimensional liquid consisting of adsorbed molecules normally oriented to the surface" at high adsorbate concentrations, the state of "condensed two-dimensional liquid consisting of adsorbed molecules arranged parallel to the surface" is also possible at intermediate concentrations. The occurrence o f double transitions corresponds to successive transitions into t he latter state and then into the state with normally oriented molecules. Depending on the ratio between the attraction constants and par a m et er ~, both abrupt and gradual transitions between the three states of the system are possible, as well as a behaviour of the adsorption system such that one o f the condensed states is n o t realized (e.g. the absence of a bi-transition in the case of the disjointed loop solution). It must be n o t e d that the specific features of the adsorption system presented here should still be observable for the m o st general case of arbitrary val-
88
ues o f a l , a2 and a12. It is also evident that they are not connected with the specific form of the isotherm equations and should be retained qualitatively in any other model which takes into account the attraction interaction between adsorbed molecules and the difference in geometrical properties of adsorbed molecules in different orientations. It is noteworthy that double transition may be predicted, even in the case when only one of the attraction constants (corresponding to interaction of molecules in the orientation parallel to the surface) is not equal to zero. In the present work we considered only the behaviour of an adsorption system as it depends on adsorbate concentration. In order to examine the behaviour in the case of fixed concentration and varying surface potential (¢) it is necessary to determine the dependence of the adsorption coefficients B1(¢) and B:(¢), for example within the framework of the parallel capacitors model. Owing to the Gaussian dependence B 1(~) and B: (~) the orientational transition resulting from the variation of concentration is revealed as a pair of transitions as the potential is varied [8,9]. Accordingly, the double orientational transitions described in this work may be revealed in the form of four (or fewer) abrupt changes on differential capacity curves. Such a dependence with four jumps was observed experimentally [19] in the study of adsorption of dipyridyl isomers. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
A . N . F r u m k m , Tr. F l z . - K h l m . Inst. lm. L. Y a . K a r p o v a , 4 ( 1 9 2 5 ) 5 6 . J.M. Parry and R. Parsons, J. E l e c t r o c h e m . S o c . , 1 1 3 ( 1 9 6 6 ) 9 9 2 . B.B. D a m a s k i n , A . N . F r u m k m a n d S.L. D y a t k l n a , Izv. A k a d . N a u k , Set. K h l m . , 1 0 ( 1 9 6 7 ) 2 1 7 1 . B.B D a m a s k m , J. E l e c t r o a n a l . C h e m . , 21 ( 1 9 6 9 ) 1 4 9 . B.B. D a m a s k m , E l e k t r o k h l m l y a , 5 ( 1 9 6 9 ) 3 4 6 . A.B. Ershler, Elektrokhlmlya, 9 (1973) 228. E.M. P o d g a e t s k n , E l e k t r o k h l m l y a , 1 0 ( 1 9 7 4 ) 6 6 6 , 11 ( 1 9 7 5 ) 1 7 5 9 . B.B. D a m a s k l n , E l e k t r o k h l m l y a , 13 ( 1 9 7 7 ) 8 1 6 . B.B. D a m a s k m a n d N . K . A k h m e t o v , E l e k t r o k h l m l y a , 1 5 ( 1 9 7 9 ) 1 6 9 1 . A . B . E r s h l e r , G A. T e d o r a d z e , I.M. L e v m s o n a n d E.M. P o d g a e t s k l l , E l e k t r o k h l m l y a , 7 ( 1 9 7 1 ) 1 0 8 3 . Y u . I . K h a x k a t s , Z h . E k s p . T e o r Flz., P l s ' m a v R e d a k t s l y u , 3 0 ( 1 9 7 9 ) 2 4 0 . Yu.I. Kharkats, Dokl. Akad. Nauk S.S.S.R., 252 (1980) 410. Yu.I. Khaxkats, Electrokhlmlya, 16 (1980), N12. U. R e t t e r , H. Jehrlng and V. Vetterl, J. E l e c t r o a n a l . C h e m . , 57 ( 1 9 7 4 ) 3 9 1 . H . K m o s h l t a , S.D. C h r i s t i a n a n d G. D r y h u r s t , J. E l e e t r o a n a l . C h e m . , 83 ( 1 9 7 7 ) 1 5 1 . V. B r a b e e , S.D C h r i s t i a n a n d G. D r y h u r s t , J. E l e c t r o a n a l . C h e m . , 8 5 ( 1 9 7 7 ) 3 8 9 . D. K r i n a r l d , P. V a l e n t a , H.W. Nurnberg and M. B r a n i c a , J. E l e e t r o a n a l . C h e m . , 9 3 ( 1 9 7 8 ) 4 1 . Y. T e m e k a n d P. V a l e n t a , J. E l e c t r o a n a l . C h e m . , 9 3 ( 1 9 7 8 ) 57. N . K . A k h m e t o v , R.I. K a g a n o v m h , B.B. D a m a s k m a n d E . A . M a m b e t k a z l e v , E l e k t r o k h l m i y a , 14 ( 1 9 7 8 ) 1761. N . K . A k h m e t o v , R . I . K a g a n o v l c h a n d B.B. D a m a s k m , Vest. M o s k . U m v . , Set. K h l m . , 2 0 ( 1 9 7 9 ) N 5 . Y u . Y a G u r e v m h a n d Y u . I . K h a r k a t s , J. E l e c t r o a n a l . C h e m . , 6 6 ( 1 9 7 8 ) 2 4 5 . M.I. U r b a k h a n d A.M. B r o d s k n , E l e k t r o k h l m l y a , 16 ( 1 9 8 0 ) 1 1 5 0 . Y u . Y a . G u r e v l c h a n d Y u . I . K h a r k a t s , D o k l . A k a d , N a u k S . S . S . R . , 2 3 0 (19"76) 1 3 2 .
75
© Elsevier S e q u o i a S.A., L a u s a n n e - - P r i n t e d in T h e N e t h e r l a n d s
ORIENTATIONAL TRANSITIONS IN AN ADSORPTION LAYER OF INTERACTING MOLECULES
Yu.I. K H A R K A T S
Institute of Electrochemlstry, U.S.S.R. Academy of Sciences, Moscow (U.S.S.R.) ( R e c e i v e d 1 9 t h May 1 9 8 0 )
ABSTRACT T h e b e h a v i o u r o f a n a d s o r p t i o n s y s t e m o f i n t e r a c t i n g organic m o l e c u l e s p r e s e n t o n t h e surface in t w o d i f f e r e n t o r i e n t a t i o n s is discussed. T h e i n f l u e n c e o f t h e r a t i o b e t w e e n attract i o n c o n s t a n t s for various o r i e n t a t i o n s o n t h e f u n c t i o n linking coverage t o t h e b u l k ads o r b a t e c o n c e n t r a t i o n is e x a m i n e d . It is s h o w n t h a t a c c o r d i n g t o t h e r a t i o o f t h e i s o t h e r m p a r a m e t e r s o n e or t w o o r i e n t a t i o n a l t r a n s i t i o n s m a y be p r e d i c t e d . T h e critical c o n d i t i o n s for a b r u p t r e o r i e n t a t i o n s are o b t a i n e d .
INTRODUCTION
It is known that molecules of some organic substances m a y be differently oriented on the surface, and as pointed o u t by Frumkin [1], variation of volume concentration of the adsorbate may result in change of the predominant orientation of the adsorbed molecules. Reorientation may also occur when changing the potential of the adsorbing surface. The effect of reorientation of adsorbed organic molecules has been extensively studied, both theoretically and experimentally [2--20]. The general conclusion that can be drawn from these works is that for sufficiently strong attractional interaction between molecules, reorientation of adsorbed molecules as well as two-dimensional condensation occur abruptly and are first-order phase transitions. The equations describing the adsorption isotherm of interacting molecules which m a y occupy two different orientations on the surface were obtained by Parry and Parsons [2] by phenomenological generalization of the Frumkin and Flory--Huggins isotherms: 01
BlC = (1 -- 01 -- 02)" e x p ( - - 2 a l n O 1 - - 2a12n02)
(1)
02 e x p ( - - 2 a 2 r n 0 2 - - 2a12rn01) B2c = (1 - - 0 1 - - 0 2 ) m
(2)
where 01 and 02 are the coverages for the t w o orientations, n and m the number of adsorption sites occupied by a molecule in the corresponding orientation, c the volume concentration of the adsorbate, BI and B2 the adsorption coefficients and al, a2 and a~2 the constants describing interaction of molecules in
76
o r i e n t a t i o n 1, o r i e n t a t i o n 2 and cross-wise i n t e r a c t i o n . E q u a t i o n s (1) and (2) m a y also be derived f r o m the c o n d i t i o n s o f t h e r m o d y n a m i c e q u i l i b r i u m ~F/~01 = 0 and ~F/~02 = 0, w h e r e F is the free e n e r g y o f t h e a d s o r p t i o n s y s t e m which m a y be calculated b y m e t h o d s o f statistical t h e r m o d y n a m i c s (see, for e x a m p l e t h e A p p e n d i x in ref. 21 and f o r o u r s y s t e m takes the f o r m : F = k T F ~ [--alO~ --a20~
--
2a120102
ln(O1/Btc) + (02/m) ln(O2/B2c) +
+ (01/n)
(1 --01 --02) ln(1 --01 --02) + 0 1 ( n
-- 1)/n
+ 02(m --
1)/ml
(3)
w h e r e k is t h e B o l t z m a n n c o n s t a n t ; T t h e t e m p e r a t u r e and F~ the limiting n u m ber o f a d s o r p t i o n sites per u n i t area. Taking into a c c o u n t eqns. (1) and (2) t h e expression f o r F m a y be r e w r i t t e n in a simpler f o r m : F = k T F = [alO~ + a20~ + 2a120102 + ln(1 + Odm
--
--
0
1 --
02) + 01(n -- 1)/n
1)/ml
(4)
In this f o r m it coincides with t h e f o r m u l a f o r surface t e n s i o n o f an a d s o r p t i o n layer [8]. T h e set o f eqns. (1) and (2) d e t e r m i n e s t h e c o n e e n t r a t i o n a l d e p e n d e n c e o f the coverages 01 and 02 and o f the t o t a l n u m b e r o f m o l e c u l e s a d s o r b e d per u n i t area 17' = ['~(01/n + 02/m). T h e f u n c t i o n s 01(c), 02(c) and p(c) are m a n y - v a l u e d in certain intervals o f c, and a m o n g the possible states o f t h e a d s o r p t i o n s y s t e m physically realized t h e r e m u s t be the o n e t h a t c o r r e s p o n d s t o t h e m i n i m u m o f free energy F . T r a n s i t i o n f r o m o n e state i n t o a n o t h e r occurs in an a b r u p t m a n n e r and is similar t o a first-order phase transition, p r o v i d e d t h a t t h e free energies o f these states are equal. This t r a n s i t i o n involves n o t o n l y an a b r u p t change o f the t o t a l n u m b e r o f molecules per u n i t area in b o t h o r i e n t a t i o n s b u t also a significant variation o f the ratio b e t w e e n coverages 01 and 02, i.e. a b r u p t r e o r i e n t a t i o n o f molecules in t h e a d s o r p t i o n layer. B E H A V I O U R O F THE A D S O R P T I O N SYSTEM F O R a2 > 0 , al = 0, a12
=
0
Since the set o f eqns. (1) and (2) includes m a n y p a r a m e t e r s and its analytical s t u d y presents considerable difficulties, let us begin with t h e simplest case w h e n o n l y one o f t h e i n t e r a c t i o n c o n s t a n t s , say a2, is n o t equal t o zero. (Since we shall consider b o t h cases m > n and n > m, t h e results o b t a i n e d a f t e r some t r a n s f o r m a tions will also describe the case al > 0, a2 = 0 and a12 = 0.) Assuming in eqns. (1) and (2) al = 0 and a12 = 0 and designating x = B l c , a = a2rn and ~ = B1/B2, we obtain: x =
01
(1 --01 --02) n
(5)
~02 x - (1 -- 01 -- 02) m exp(--2a02)
(6)
We shall distinguish t h r e e main cases d e p e n d i n g o n the ratio b e t w e e n p a r a m e t e r s m and n: (1) m = n; (2) m < n; (3) m > n. T h e case m < n c o r r e s p o n d s t o pre-
77 d o m i n a n t i n t e r a c t i o n b e t w e e n m o l e c u l e s in t h e m o r e c o m p a c t a r r a y ( w h i c h m a y be o r i e n t e d n o r m a l t o t h e surface). T h e case m > n c o r r e s p o n d s t o pred o m i n a n t i n t e r a c t i o n b e t w e e n m o l e c u l e s in t h e less c o m p a c t a r r a y (which m a y be o r i e n t e d parallel t o t h e surface). O n e o f t h e possible m e c h a m s m s r e s p o n s i b l e f o r such i n t e r a c t i o n was e x a m i n e d in ref. 22. We shall s t a r t w i t h t h e case m = n. E q u a t i o n s (5) a n d (6) m a y be r e w r i t t e n as 01 = ~02 e x p ( - - 2 a 0 J
(7)
02 = 1 - - 01 - - ( 0 1 I x ) w ~
(8)
E q u a t i o n (7) describes a f u n c t i o n 01(02) w h i c h is r e p r e s e n t e d b y a c u r v e w i t h a m a x i m u m (Fig. 1). D e p e n d i n g on t h e values o f p a r a m e t e r s a and/3 this m a x i m u m m a y b e s i t u a t e d in t h e p h y s i c a l region 0 < 01 < 1, 0 < 02 < 1, 0 < 01 + 02 < 1 or o u t s i d e it. E q u a t i o n (8) is graphically r e p r e s e n t e d b y a set o f curves (in t h e specific ease w h e n m = n = 1 t h e y are s t r a i g h t lines) o r i g i n a t i n g in t h e p o i n t 01 = 0, 02 = 1, t h e " s l o p e " o f w h i c h s m o o t h l y varies w i t h x. I n t e r s e c t i o n o f t h e p l o t s o f t h e r e l a t i o n s (7) a n d (8) gives t h e graphical s o l u t i o n o f set (5) a n d (6) a n d allows t h e d e t e r m i n a t i o n o f t h e f u n c t i o n s 0 l(x) a n d 0 2 ( x ) . As can be seen f r o m Fig. 1, t h e i n t e r s e c t i o n o f t h e curves is possible in o n e p o i n t or in t h r e e p o i n t s c o r r e s p o n d i n g r e s p e c t i v e l y t o a single s o l u t i o n o f t h e set (5) a n d (6) or t o t h r e e s o l u t i o n s w h i c h we shall d e n o t e b y s u p e r s c r i p t s 1, 2 a n d 3. Curves a, b a n d e in Fig. 1, c o r r e s p o n d i n g t o a given value o f p a r a m e t e r a a n d t o t h r e e d i f f e r e n t values o f p a r a m e t e r / 3 , r e p r e s e n t t h r e e d i f f e r e n t b e h a v i o u r s o f t h e a d s o r p t i o n sys-
0
............................
5 a
0 ~
10 × FmO L~IF ~2,
d('
e~~
b
e' ~'
0 4
3
1
5
C
10
",
5
"~
r L-j
0 1
92
'e~J 1
2
3
4
Fig. 1. Graphic solution of set of eqns. (7) and (8) for m = n = 1 and a = 3. Curves a, b and c represent the functions 01 (02) for ~ = 30, 15 and 10 respectively. (1) Straight line 01 + 02 = 1; straight hnes (2), (3) and (4) correspond to x2 < x3 < x4. Fig. 2. The functions 01(X), 02(X ) and F(x) for cases (a), (b) and (e) depicted in Fig. 1. Arrows show the position of abrupt tranmtions on the dependence F(x).
78
tem. For curve a (sufficiently high values of/3) there is always one solution to the set (5) and (6) and the isotherm is single-valued (Fig. 2a). With decrease of/3, curve 01(02) drops off and its right-hand branch comes partially below the straight line 01 + 02 = 1. Isotherms 01(x), 02(x) and F(x) corresponding to this case are many-valued from value Xmln to x = oo (Fig. 2b). Eventually, with further decrease of/3, the m a x i m u m of the curve 01(02) comes below the straight line 01 + 02 = 1. Isotherms 0 ~(x), 02(x) and F(x) corresponding to this case are manyvalued in the interval Xmln < x < Xmax, which is determined from the condition of tangency of curves described by eqns. (7) and (8) (Fig. 2c). The critical conditions of transition from case (a) to case (b) and then to case (c) are found from the conditions of tangency of the straight line 01 + 02 = 1 to the curve 01(02): /3> ~max = aa- -+x /~~ - - 2 a
exp(a + v r ~ -- 2a),
~ < ~mln a+v~--2a a_ ~ e x p ( a -
- - x / ~ -- 2a),
~max(a)
~mm(a) < ~ <
case (a)
(9)
case (b)
(10)
case (c)
(11)
Graphically, conditions (9)--(11) correspond to division of the parameter plane (a, In fi) into regions separated by critical curves ai(ln fi) (or fl = timex(a)) and aii(ln fi) (or/3 = ~min(a)) originating from point a = 2, In/3 = 2 (Fig. 3). Note that the region where the isotherm is many-valued, determined by condition (9), occurs at a > 2 and additionally depends on the value of parameter ft. The simple fact that the functions 01, 02 and F of x are many-valued does n o t itself guarantee realization of a phase transition. Thus, abrupt reorientation is possible only in the part of the "many-valued" region in Fig. 3 which is situated to the left of the critical curve a111(ln/3) determined from condition F(O ~1), 0(21)) = F(O(~3), 0(23)) at x -+ ~ . To the right of the critical curve aHi(ln fi) inequality F(0(~1), 0~1)) < F(O~ 3), O~a>) is true for all x > Xm~n and, despite the fact that the isotherm is many-valued, transitions are absent. At m = n = 1, the values Xm~ and Xm~x, determining the region of manyvalued isotherms, may be f o u n d analytically and are given by the following equations: /3
Xm, n = ~ m a x ( a ) - -
Xmax -- ~ - - ~ m , n ( a )
(12) (13)
where timex(a) and fim~(a) are given by eqns. (9) and (10). The critical curve aiIi(ln/3) at m = n = 1 is determined from the set of equations derived from (4), (7) and (8) f o r x - ~ ~ o : a(0(21)) 2 * ln(1 -- 0(2~) ) = a(0(23)) 2 + ln(1 -- 02TM) (14) 1 -- O:(1'3) = ~0(2~'3) exp(--2a02(1'3))
(15)
the solution of which is aiIi=ln/3;
ln~/>2
(16)
79
Taking into a c c o u n t t h a t , during a b r u p t r e o r i e n t a t i o n , 02 usually changes f r o m v e r y small values 0(21) < < 1 to values close t o u n i t y (1 -- 0(23) < < 1), we m a y d e d u c e f r o m eqns. (4), (7) and (8) at m = n = 1 an a p p r o x i m a t e analytical expression for t h e c o n c e n t r a t i o n x* at which the o r i e n t a t i o n a l phase t r a n s i t i o n occurs:
x* =
fi e x p ( - - a ) 1 --/3 e x p ( - - a ) '
/3 e -2a < < 1, fie -a <~ 1
(17)
Let us n o w e x a m i n e t h e c o n d i t i o n s which lead t o t h e a p p e a r a n c e o f m a n y valued a d s o r p t i o n i s o t h e r m s due t o S-shaped sections. These c o n d i t i o n s , which m a y be w r i t t e n as dx/d02 -- 0 and d2x/dO~ = 0 (where x(02) is t h e inverse funct i o n o f 0 : ( x ) given b y eqns. (7) and (8), i m p o s e a relationship b e t w e e n values a and In ft. A t m = n = 1 this relationship reduces t o the c o n d i t i o n a = 2, In/3 ~< 2 and, in the general case m = n <> 1, it is r e p r e s e n t e d b y curve aiv(ln/3) o n plane (a, In/3) (Fig. 3) originating f r o m p o i n t a = 2, In/3 = 2 and t e n d i n g at In/3 -~ - - ~ to t h e a s y m p t o t e 1 al : ~(1 + x/~) 2
(18)
A b o v e the curve a~v(ln ~), the i s o t h e r m presents an S-shaped section and t h e r e is an o r i e n t a t i o n a l phase transition; for p a r a m e t e r s b e l o w t h e curve alv(ln/3) t h e i s o t h e r m is single-valued and transitions are absent. Hence, in this case (m = n) r e o r i e n t a t i o n a l phase transitions are possible in r e g m n (c) and p a r t o f region (b) on the p a r a m e t e r plane (a, In/3) r e s t r i c t e d b y the critical curves a~ii and aiv • In t h e case n > m t h e a d s o r p t i o n i s o t h e r m s given b y eqns. (5) and (6) are e i t h e r single-valued or have an S-shaped section on which a phase t r a n s i t i o n occurs. T h e f u n c t i o n s 0~, 02 and F o f x in this case are similar t o the plots s h o w n in Figs. 2a and 2c f o r t h e case m = n (see also ref. 8). T h e critical curve a(ln/3) c o r r e s p o n d i n g t o the a p p e a r a n c e o f S-shaped sections on t h e i s o t h e r m is determ i n e d b y eqns. (5) and (6) and the c o n d i t i o n s dx/d02 = 0, dZx/dO~ = O. A t n = 1 and m < n this curve is r e d u c e d to a h o r i z o n t a l straight line c o r r e s p o n d i n g
a
a~,
3
3
a~
2
J
J
T
-1
1
1
I
2
3
4
s
In~
2
I
0
I
2
3
Infl
Fig. 3. P o s i t i o n o f critical c u r v e s a i , a i i , a i i I a n d a i v o n t h e p a r a m e t e r p l a n e (a, In fi) a t m = n, n <> 1; a ~ ,2,3) c o r r e s p o n d t o m = n = 2, 1 a n d 0.5. Fig. 4. S h a p e o f t h e critical c u r v e a ( l n / 3 ) f o r t h e case m < n, n <> 1. ( 1 ) m = 1, n = 2; (2) _1 _1 1 m-~,n=l;(3) m-~,n=~.
80 £1
/ 3.
2
cl
1
I
5! a
2
o',"/
~':,,'
/
a'~
3
~ Inl]
.-"~.;'
82 1
-
4
3 a:'
z b
J 2
1
1
2
3
4 Ln~
1
2
3
4
x
Fig. 5. P o s i t i o n o f critical c u r v e s at, a i r , a r r [ , a l V a n d a v a t rn > n, n ~ l . (a) m = 2, n ~ 1; ( b ) s u p e r s c r i p t (1) c o r r e s p o n d s t o m = 3, n = 2, s u p e r s c r i p t ( 2 ) c o r r e s p o n d s t o m = 1, n = ~. Fig. 6. T h e f u n c t i o n 0 2 ( x ) f o r m = 2, n = 1 a n d a = 3.5. ( 1 ) fi = 5; (2) fi = 4 . 3 ; (3) ~ = 3.5; ( 4 )
,G= 3n.
to a = al given b y eqn. (18) and, in the general case n > m, t h e critical curve a(ln/3) s m o o t h l y changes f r o m t h e a s y m p t o t e a = al, where al is d e t e r m i n e d b y eqn. (18) at In/3 -~ - - ~ t o the a s y m p t o t e c o r r e s p o n d i n g t o In/3 -~ oo, i.e. ar = ½(1 + v/-m]-n) 2
(19)
In t h e specific case for which m = 1 and n = 2, studied in refs. 8 and 9, the a s y m p t o t i c value ar = 1.46 t o which the critical curve t e n d s as/3 -~ oo, is in agreem e n t with eqn. (19). T h e shape o f t h e critical curve a(ln/3) at n > m and n > 1, n = 1 and n < 1 is s h o w n in Fig. 4. In the region above the critical curve a b r u p t r e o r i e n t a t i o n s o c c u r , while b e l o w this curve t h e y are absent. L e t us n o w c o n s i d e r the case rn > n. T h e a d s o r p t i o n s y s t e m displays interesting features [ 1 1 - - 1 3 ] which are illustrated b y t h e plots o f t h e f u n c t i o n 02(x) and F ( x ) and b y the position o f critical curves o n t h e p a r a m e t e r plane (a, In/3) (Figs. 5--7). When values o f p a r a m e t e r s a and In/3 lie in t h e region situated t o t h e right o f t h e critical curve a i (ln/3) (Fig. 5) the a d s o r p t i o n isotherms 01(x), 0 ~(x) and F ( x ) are single-valued. On crossing t h e critical curve aT (ln/3) we e n t e r the region o f p a r a m e t e r s s i t u a t e d b e t w e e n aT (ln/3) and aiT (ln/3). T h e f u n c t i o n s 0 l(x) and 02(x) in this region consist o f t w o branches, t h e main b r a n c h beginning at t h e origin o f c o o r d i n a t e s and the o t h e r in the f o r m o f a closed l o o p . T h e critical curve ai(ln/3) c o r r e s p o n d s t o t h e a p p e a r a n c e o f a l o o p o f infinitely small size, which increases w h e n t h e critical curve a n (ln/3) is a p p r o a c h e d . When the parameters acquire values c o r r e s p o n d i n g t o critical curve a n (ln/3) t h e l o o p is t a n g e n t t o t h e main b r a n c h . In the region t o the left o f an(In/3) t h e f u n c t i o n s OT(x) and
81 f-IF_ 1
0
1
2
3
L
x
Fig. 7. T h e f u n c t i o n F ( x ) for m = 2, n = 1 a n d a = 3.5,/3 = 3.5. A r r o w s s h o w t h e calculated position of abrupt transitions.
02 (x) have two S-shaped sections, which correspond to two orientational phase transitions. It is evident from topological considerations that when the functions 0 l(x) and 02(x) and therefore the function F(x) have a loop branch, phase transitions are either absent or occur in pairs [11--13]. The absence of intersection of the two separate branches F(x) or their intersection at two points (in general, at an even number of points) correspond to such a situation. The tangency of the two separate branches F(x) * corresponds to the condition of the appearance of a pair of such reorientational transitions (bi-transition). This condition allows the calculation of the critical curve for the appearance of bi-transitions at1 ~(ln/3) on the plane (a, In/3) (Fig. 5). In the region between a I and a n i where the loop is still sufficiently small, the corresponding states are metastable and transitions do n o t occur. In the region between a l n and a n where the loop is sufficiently large a bi-transition is realized. In the general case there are two other critical curves aiv(ln/3) and av(ln/3) on the parameter plane (a, In/3) corresponding to the appearance (or disappearance) of the left- and right-hand S-shaped sections on curves 0~(x) and 02(x). In the specific case of m > n and n ~ 1 these two curves merge into one horizontal straight line a = a i v determined from eqn. (18). This straight line originates from the point of convergence of critical curves a~, aI[ and an~. In the region above critical line aiv, two S-shaped sections appear simultaneously on
01(x) and 02(x). The critical curves aiv(ln/3) and avon/3) originate from the point of convergence of curves ai, all a n d ' a n i and tend, for In/3 -~ --~, to horizontal asymptotes. The asymptotic value for alv corresponding to the appearance (or disappearance) of the left-hand S-shaped section is determined by eqn. (18) and that for a v corresponding to the appearance (or disappearance) of the right-hand S-shaped section is determined by eqn. (19). At n > 1 curve a w lies above curve av and at n < 1 below. Let us now consider the dependence F(x) for m > n. Analysis shows that * A n o t h e r e x a m p l e o f t a n g e n c y o f b r a n c h e s of free e n e r g y is a f f o r d e d b y t e m p e r a t u r e p h a s e t r a n s i t i o n s in a d s o r p t i o n s y s t e m s [ 2 1 , 2 3 ].
82
when the p a r a m e t e r s a and In fl are in the region b e t w e e n the critical curves a1 and a n and if the f u n c t i o n s 01(x) and 02(x) have a l o o p b r a n c h in a d d i t i o n to t h e main b r a n c h , t h e c o r r e s p o n d i n g closed b r a n c h F ( x ) is r e p r e s e n t e d b y a curve in t h e f o r m o f a h o r i z o n t a l " 8 " (Fig. 7). It s h o u l d be n o t e d t h a t , a l t h o u g h each t r a n s i t i o n f o r m i n g a bi-transition involves c o n s e c u t i v e r e o r i e n t a t i o n o f molecules (the ratio 01/02 varies f r o m high values to low and t h e n again t o high values), the t o t a l n u m b e r o f molecules a d s o r b e d on a u n i t area (in b o t h o r i e n t a t i o n s ) increases with each transition, and t h e r e f o r e in this case t w o - d i m e n s i o n a l c o n d e n sation is a t w o - s t e p process. B E H A V I O U R O F T H E A D S O R P T I O N S Y S T E M F O R al > 0, a2 > 0 A N D a12 = 0
T h e results described above m a y also be c o n s i d e r e d f r o m a slightly d i f f e r e n t v i e w p o i n t . L e t us use such designations t h a t p a r a m e t e r n will always e x c e e d p a r a m e t e r rn. T h e n , as s h o w n above, t h e b e h a v i o u r o f the a d s o r p t i o n s y s t e m f o r al = 0, a2 > 0 and a12 = 0 differs significantly f r o m the b e h a v i o u r for a~ > 0, a2 = 0 and a,2 = 0. A t t h e same t i m e t h e p a r t i c u l a r situations, a~ = 0, a2 > 0, a~2 = 0, and al > 0, a2 = 0, a12 = 0, m a y be c o n s i d e r e d as limiting cases o f t h e m o r e general s i t u a t i o n ax > 0, a2 > 0 and a12 = 0. Here we shall s h o w t h a t m t h e general case ax > 0, a2 > 0 and a12 = 0 the behaviour o f a d s o r p t i o n systems displays interesting features (first r e p o r t e d in refs. 1 1 - - 1 3 ) , including t h o s e characteristic o f the t w o limiting cases m e n t i o n e d above. F o r simplicity let us consider t h a t p a r a m e t e r s n and rn are equal t o 2 and 1 respectively. T h e critical c o n d i t i o n s c o r r e s p o n d i n g to o r i e n t a t i o n a l transitions in the general case will be given b y t h e surfaces/3 = fi (a~, a2) in the p a r a m e t e r space (al, a2, fi) (Fig. 8). I n t e r s e c t i o n s o f these surfaces with planes a2 = 0 and al = 0 give the critical curves and regions on t h e p a r a m e t e r planes, which are obtained f r o m Fig. 5a b y s u b s t i t u t i o n a = 2a~ and fi = 1/fi and f r o m curve 1 in Fig. 4. T h e y are s h o w n in a p l o t o f a vs. fi in Figs. 9a and 9b. In o r d e r t o d e m o n s t r a t e the m u t u a l a r r a n g e m e n t o f t h e critical surfaces in t h e space (a~, a2, fi) the calculated cross-sections o f these surfaces with the auxilary planes al = a2 and a2 = 2a~ are s h o w n in Figs. 9c and 9d. T h e plane ax = a2, fl is divided b y seven critical curves ar(fi)--awi (fi) into nine regions (Fig. 9c). T h e b e h a v i o u r o f the f u n c t i o n s 0 ~(x) and 02(x) in these regions is illustrated in Figs. 10--12.
1%
9a~, ,a~=az,F,g9c\ J
s,=O,
FLg
a2= 2a, F, g9d
i
,
\\,l , i
a2=O, F,g9b
/J
,/ Cl I
Fig. 8. M u t u a l a r r a n g e m e n t o f p a r a m e t e r planes s h o w n in Figs. 9a, 9b, 9c a n d 9d.
83
(1 =G2 O,
"
~ 2
Q:QI
Q~
°-Oz:20t
3
d
2
os Fig. 9. Position
p
of critical curves on parameter
o
:
2
p
p l a n e ( a , / 3 } , ( a ) a = a2 > 0 , a l = 0 , a 1 2 = 0 ; ( b )
a = al > 0, a2 = 0, a12 = 0; (c) a = al = a2 > 0, a12 = 0; (d) a = as = 2al > 0, a12 = 0.
Let us consider the variation of the functions 0 ~(x) and 02(x) at fixed values of parameters al = a2 = 1.5 and increasing fi (Fig. 10), when passing from region 2 into regions 4, 6 and 9 in Fig. 9c. In region 2, curves O~(x) and 02(x) are singlevalued at all values of x and abrupt orientational transitions are absent. On increase of fi, we cross the critical curve ali (fl), corresponding to the appearance of a loop solution to the set (1) and (2), and enter region 4. The functions 01(x) and 02(x) in region 4 consist of two disjointed branches: the main branch beginning at the origin of the coordinates and the loop branch which is situated above the main branch for 01(x) and below for 02(x). However, in so far as the loop branch is relatively small, the corresponding states of the adsorption system possess higher energy than the states on the main branch and consequently are metastable. When parameter ~ exceeds the critical values corresponding to the curve of bi-transition appearance, a i n (fi), the loop solution has a section in region 6 where the free energy is lower than on the main branch. This corresponds to an orientational bi-transition in the adsorption system. The first transition is accompanied by an abrupt increase of 01 and an abrupt decrease of 02, and the second transition by an abrupt decrease of 0~ and an abrupt increase of 02. Note, that as in the case examined previously (al > 0, a2 = 0 and a12 = 0) the value of P increases with each orientational transition. With further increase of fi we cross the critical curve aiv(~) corresponding to tangency of the loop to the main solutions of set (1) and (2) and we enter region 9. The functions 0 ~(x) and 0 ~(x) are described here by joint curves, each of them has two S-shaped sections for which an orientational transition occurs.
84
Hence, the behaviour of the adsorption system in regions 2, 4, 6 and 9 is very similar to that related to the corresponding regions shown in Fig. 9b. Let us now consider slightly higher values of al = a2, e.g. al = a2 = 2.25 (Fig. 11). The critical curve ai (/3) corresponds to the appearance of an S-shaped section on the main branch of the solutions of 0 l(X) and 02(x). At relatively small values of x corresponding to region 1 in Fig. 9c, the function O~(x) and 02(x) have one branch with an S-shaped section for which an orientational transition occurs. At higher/3 we enter region 3 where solutions consist of two disjointed branches. The main branch has an S-shaped section for which there is a transition, while states on the loop branch are metastable. In a certain range of concentrations, set (1) and (2) has five solutions. Analysis shows that the maximal n u m b e r of solutions is seven and that t h e y may exist at relatively high values of attraction constants al and a2. With further increase of fl we cross the critical bi-transition curve aiii (fi) and enter region 5 in Fig. 9c. The functions Offx) and 02(x) in this region, as in region 3, have main and loop branches. However, a considerable difference arises in the absence of the orientational transition on the S-shaped section of the main branch. This " b l o c k i n g " of the transition is due to the fact that the j u m p from the state preceding the S-shaped transition to the loop branch and then back to
~2 q
,
,
/
'J1 1
1hi J
0
1
,3
01
Fig. 1 0 . T i m e - f u n c t i o n 0 1 ( x ) a n d 02(x ) f o r a = al = a2 = 1.5. ( 1 ) ~ = 0 . 5 ; ( 2 ) ~ = 0 . 7 5 , ( 3 ) = 0 . 9 , ( 4 ) ~ = 1. C u r v e s 1 - - 4 c o r r e s p o n d t o r e g i o n s 2, 4, 6 a n d 9 in Fig. 9c. A b r u p t t r a n s i tions on curve 3 correspond to x = 0.145 and x = 0.33, on curve 4 to x = 0.16 and x = 0.22. Fig. 11. T h e f u n c t i o n 01(X ) a n d ~ 2 ( x ) f o r a = a l = a2 = 2 . 2 5 . ( 1 ) / 3 = 0 . 0 8 ( 0 . 0 0 8 5 ) , ( 2 ) = 0.2 (0.0213); (3)/3 = 0.5 (0.032, 0.088); (4)/3 = 0.75 (0.0287, 0.20); (5)/3 = 0.85 (0.0285, 0 . 2 6 ) . C u r v e s 1 - - 5 c o r r e s p o n d t o r e g i o n s 1, 3, 5, 7 a n d 9 In Fig. 9c. T h e v a l u e s o f x a t w h i c h a b r u p t o r m n t a t i o n a l t r a n s i t i o n s o c c u r a r e g i v e n in p a r e n t h e s e s .
85 ~2
0
i
118'
0
1
x
Fig. 12. T h e f u n c t i o n 0 1 ( x ) a n d 0 2 ( x ) f o r a = a 1 = a 2 = 1.25. (1)/3 = 1 . 0 5 ; (2)/3 = 1 . 1 2 ; (3) /3 = 1.2. C u r v e 1 c o r r e s p o n d s t o r e g i o n 2 a n d c u r v e s 2 a n d 3 t o r e g i o n 8 in Fig. 9c. A b r u p t t r a n s i t i o n s o c c u r o n c u r v e 2 a t x = 0.35 a n d o n c u r v e 3 at x = 0 . 4 6 .
the main branch, is thermodynamically more favourable. The region with a transition on the S-shaped section is thus by-passed along the loop solution. In region 7 situated to the right of critical curve a~v(~), (corresponding to the tangency of the main and loop solutions) the functions 0 ~(x) and 02(x) present two S-shaped sections for which orientational transitions occur. The critical curve av(~) separating regions 7 and 9 corresponds to the disappearance of the small S-shaped section which appeared after convergence o f the main and loop solutions. Hence, one orientational transition occurs in regions I and 3 and two orientational transitions in regions 5, 7 and 9. Two other critical curves avi and avi I originate from the point of convergence of curves aii, aiiI and a~v and correspond to the disappearance (or appearance) of S-shaped sections on curves 0 ~(x) and 02(x). Figure 12 illustrates the behaviour of 01(x) and 02(x) at al = a2 = 1.25. In region 8 a single orientational transition occurs. Therefore the parameter plane (at = a2, ~) (Fig. 9c) is divided by critical curves into nine regions. In regions 2 and 4 transitions are absent, in regions 1, 3 and 8 a single transition occurs and in regions 5, 6, 7 and 9 two orientati~nal transitions occur. The arrangement of the critical curves for the more general case a~ = a2 > 0 is more complex than in the two limiting cases a~ > 0, a2 = 0 and a~ = 0, a2 > 0 and may be approximately represented by superposition of the " b e a k " for case a~ > 0, a2 = 0 (Fig. 9b) and of the curve corresponding to the appearance of an S-shaped section characteristic of the case a~ = 0, a2 > 0 (Fig. 9a). As a result of this superposition the curve for the appearance of S-shaped sections is split into two parts, ai (/3) and avii(fi) (Fig. 9c).
86
In order to obtain a more complete picture of the arrangement of the critical curves in the (al, a2,/3) space and to reveal some additional features in the behaviour of the adsorption system, let us examine another cross-section of these surfaces, namely by the plane (a2 = 2al) (Fig. 9d). The parameter plane in this case is divided by the critical curves a~--av into five regions. In region 1 the functions 0 l(x) and 02(x) are single valued. The critical curve al corresponds to the appearance of an S-shaped section on 01(x) and 02(x). Critical curves aii and a n i correspond to the appearance of a loop solution and its convergence with the main branch. A specific feature of the case considered is that states on the loop branch t h r o u g h o u t region 3 (situated between a n and aii I) are metastable and a bitransition is not possible. Double transitions are nevertheless produced at sufficiently high values of/3. They appear in region 5, which is to the right of the critical curve of the appearance of a " h o p " -- aiv(~). The " h o p " p h e n o m e n o n may be described as follows. In regions 4 and 5, after convergence of the loop and the main branch solutions, functions 01(x) and 02(x) have a portion where the set (1) and (2) has five solutions. In this portion 02(x) displays a small, and 8 l(x) a large, S-shaped section (curves 3 and 4 in Fig. 13). In the case of five solutions the transition from the main state which occupies a middle position with respect to the other solutions may proceed in the direction of higher 02 (and lower 01) or in the direction of lower 02 (and higher 01). The first case occurs in region 4 and the second in region 5. It is easy to see that in region 5 the first orientational transition must be followed by a second one. The " h o p " of the orientational transition occurs when the free energies of three of the five solutions of set (1) and (2) corresponding to minima of free Le2
[
0
1
X
1
x
8~
4
/ f /
F i g . 1 3 . T h e f u n c t i o n s 01(x) a n d 02(x) f o r a = a2 = 2 a l = 3. ( 1 ) fi = 0 . 8 ; ( 2 ) / 3 = 1 . 2 ; ( 3 ) ~ = 1 . 6 ; ( 4 ) fl = 2 . 2 ; ( 5 ) / 3 = 5. T r a n s i t i o n s o n c u r v e 5 o c c u r a t x = 0 . 1 1 a n d x = 0 . 5 3 .
87
energy are equalized simultaneously, a situation which is represented by the critical curve a~v in Fig. 9d. Curve av(~) in Fig. 9d is similar to curve avi (~) in Fig. 9c and gives the condition o f disappearance of the small S-shaped section on 02(x) (large S-shaped section on 0 l(X)). Therefore, in the case a= = 2al, the parameter plane is divided by critical curves into five regions. In region 1 transitions are absent. In regions 2--4 a single transition occurs and in region 5 two transitions occur. The location o f the points on the parameter plane, from which critical curves ai (/3) originate and o f the as ym pt ot es to which critical curves a w and a v n in Fig. 9c and a~ and a v in Fig. 9d tend, may be f o u n d analytically. The origins of curves al for ~ -+ 0 are determined from eqn. (18). A sym pt ot i c values for curve ai in Fig. 9d and curve avn in Fig. 9c for/3 -~ ~ are given by eqn. (19). The a s y m pto tic behaviour of curves av in Fig. 9d and avi in Fig. 9c for ~ -~ ~ which determine the co n d i t i on of the appearance (or disappearance) of S-shaped sections on 0 l(x) and 02(x) is given by the equation of the asymptotic plane in space
(a~, a~, ~): a2 + a l - - = n
1 +
2
(20)
For given ratios between a~ and a2 the appropriate values of av and aw for/3 -~ m a y be obtained f r om eqn. (20). DISCUSSION
Analysis o f the behaviour of adsorption systems satisfying equations of isot h e r m (1) and (2) shows that, depending on the ratio between parameters, one or two orientational transitions m ay be observed, according t o t he adsorbate concentration. This is due in the first place to attraction interaction and, in the case of double transitions, also to the difference in geometrical properties of molecules adsorbed in different orientations. The latter factor is responsible for the "forcing o u t " effect which determines p r e d o m i n a n t coverage of the surface with molecules normally oriented t o the surface at high adsorbate concentrations. A specific feature of the system studied is that, along with t he state of "rarefied two-dimensional gas" at low adsorbate concentrations and the state of "condensed two-dimensional liquid consisting of adsorbed molecules normally oriented to the surface" at high adsorbate concentrations, the state of "condensed two-dimensional liquid consisting of adsorbed molecules arranged parallel to the surface" is also possible at intermediate concentrations. The occurrence o f double transitions corresponds to successive transitions into t he latter state and then into the state with normally oriented molecules. Depending on the ratio between the attraction constants and par a m et er ~, both abrupt and gradual transitions between the three states of the system are possible, as well as a behaviour of the adsorption system such that one o f the condensed states is n o t realized (e.g. the absence of a bi-transition in the case of the disjointed loop solution). It must be n o t e d that the specific features of the adsorption system presented here should still be observable for the m o st general case of arbitrary val-
88
ues o f a l , a2 and a12. It is also evident that they are not connected with the specific form of the isotherm equations and should be retained qualitatively in any other model which takes into account the attraction interaction between adsorbed molecules and the difference in geometrical properties of adsorbed molecules in different orientations. It is noteworthy that double transition may be predicted, even in the case when only one of the attraction constants (corresponding to interaction of molecules in the orientation parallel to the surface) is not equal to zero. In the present work we considered only the behaviour of an adsorption system as it depends on adsorbate concentration. In order to examine the behaviour in the case of fixed concentration and varying surface potential (¢) it is necessary to determine the dependence of the adsorption coefficients B1(¢) and B:(¢), for example within the framework of the parallel capacitors model. Owing to the Gaussian dependence B 1(~) and B: (~) the orientational transition resulting from the variation of concentration is revealed as a pair of transitions as the potential is varied [8,9]. Accordingly, the double orientational transitions described in this work may be revealed in the form of four (or fewer) abrupt changes on differential capacity curves. Such a dependence with four jumps was observed experimentally [19] in the study of adsorption of dipyridyl isomers. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
A . N . F r u m k m , Tr. F l z . - K h l m . Inst. lm. L. Y a . K a r p o v a , 4 ( 1 9 2 5 ) 5 6 . J.M. Parry and R. Parsons, J. E l e c t r o c h e m . S o c . , 1 1 3 ( 1 9 6 6 ) 9 9 2 . B.B. D a m a s k i n , A . N . F r u m k m a n d S.L. D y a t k l n a , Izv. A k a d . N a u k , Set. K h l m . , 1 0 ( 1 9 6 7 ) 2 1 7 1 . B.B D a m a s k m , J. E l e c t r o a n a l . C h e m . , 21 ( 1 9 6 9 ) 1 4 9 . B.B. D a m a s k m , E l e k t r o k h l m l y a , 5 ( 1 9 6 9 ) 3 4 6 . A.B. Ershler, Elektrokhlmlya, 9 (1973) 228. E.M. P o d g a e t s k n , E l e k t r o k h l m l y a , 1 0 ( 1 9 7 4 ) 6 6 6 , 11 ( 1 9 7 5 ) 1 7 5 9 . B.B. D a m a s k l n , E l e k t r o k h l m l y a , 13 ( 1 9 7 7 ) 8 1 6 . B.B. D a m a s k m a n d N . K . A k h m e t o v , E l e k t r o k h l m l y a , 1 5 ( 1 9 7 9 ) 1 6 9 1 . A . B . E r s h l e r , G A. T e d o r a d z e , I.M. L e v m s o n a n d E.M. P o d g a e t s k l l , E l e k t r o k h l m l y a , 7 ( 1 9 7 1 ) 1 0 8 3 . Y u . I . K h a x k a t s , Z h . E k s p . T e o r Flz., P l s ' m a v R e d a k t s l y u , 3 0 ( 1 9 7 9 ) 2 4 0 . Yu.I. Kharkats, Dokl. Akad. Nauk S.S.S.R., 252 (1980) 410. Yu.I. Khaxkats, Electrokhlmlya, 16 (1980), N12. U. R e t t e r , H. Jehrlng and V. Vetterl, J. E l e c t r o a n a l . C h e m . , 57 ( 1 9 7 4 ) 3 9 1 . H . K m o s h l t a , S.D. C h r i s t i a n a n d G. D r y h u r s t , J. E l e e t r o a n a l . C h e m . , 83 ( 1 9 7 7 ) 1 5 1 . V. B r a b e e , S.D C h r i s t i a n a n d G. D r y h u r s t , J. E l e c t r o a n a l . C h e m . , 8 5 ( 1 9 7 7 ) 3 8 9 . D. K r i n a r l d , P. V a l e n t a , H.W. Nurnberg and M. B r a n i c a , J. E l e e t r o a n a l . C h e m . , 9 3 ( 1 9 7 8 ) 4 1 . Y. T e m e k a n d P. V a l e n t a , J. E l e c t r o a n a l . C h e m . , 9 3 ( 1 9 7 8 ) 57. N . K . A k h m e t o v , R.I. K a g a n o v m h , B.B. D a m a s k m a n d E . A . M a m b e t k a z l e v , E l e k t r o k h l m i y a , 14 ( 1 9 7 8 ) 1761. N . K . A k h m e t o v , R . I . K a g a n o v l c h a n d B.B. D a m a s k m , Vest. M o s k . U m v . , Set. K h l m . , 2 0 ( 1 9 7 9 ) N 5 . Y u . Y a G u r e v m h a n d Y u . I . K h a r k a t s , J. E l e c t r o a n a l . C h e m . , 6 6 ( 1 9 7 8 ) 2 4 5 . M.I. U r b a k h a n d A.M. B r o d s k n , E l e k t r o k h l m l y a , 16 ( 1 9 8 0 ) 1 1 5 0 . Y u . Y a . G u r e v l c h a n d Y u . I . K h a r k a t s , D o k l . A k a d , N a u k S . S . S . R . , 2 3 0 (19"76) 1 3 2 .