J. Fhys. Chem. 1882, 86, 2089-2094
2088
Potential Experienced by a Floating Cesium Ion in Zeolite A 1.Takalshl' and H. Hosol School of Materlals Science, Toyohashi University of Technology, Toyohashi, 440 Japan (Received: July 20, 1981; In Final Form: December 29, 198 1)
Calculations were made of the potential experienced by the cesium ion located at the center of the eight-membered oxygen ring, the window to the cavity, in cesium-containing zeolite A. The so-called floating of cesium ion was well described by adding the Lennard-Jones 6-12 potential to the electrostatic one. The 6-12 potential between cesium ion and the lattice oxygen was derived from a new empirical relation. The potential surface has a very gentle curvature at its valley, especially in the direction perpendicular to the window plane. The vibrational frequency of cesium ion was estimated to be very small in the perpendicular direction. This is in good harmony with results of the X-ray structural analysis and hydrogen encapsulation with cesium-containing zeolite A. The conditions necessary to the cation floating were investigated for various kinds of zeolite A. It is concluded that there is no possibility of the floating in Na-A and K-A zeolites but that there is in Rb-A zeolite. This conclusion is in agreement with X-ray structural results, supporting the correctness of the present method.
Introduction Dehydrated zeolite A has a large empty cavity, at the center of its unit cube, which is nearly spherical with a diameter of 11.4 A and is named the (Y cage. This cage is surrounded by six eight-membered oxygen rings which lie on the faces of the cube and constitute windows to the cage. In commercial molecular sieves 3A and 4A, the windows are blocked by potassium and sodium ions, respective1y.lv2 These ions are located so as to directly contact oxygen ions of the window framework. In cesium-containing zeolite A, on the other hand, Cs+ occupies the center of the Sring window, without directly contacting the oxygen ions. In (Cs,Nq)-A zeolite, by way of example, the distance between Cs+ and the nearest oxygen ion, d(Cs-0)) is larger by 0.2 8, than the sum of the ionic radii of Cs+ and oxygen Seff called such a situation "ion floating". However, the ion is not floating as a planet in space is, and the word "floating" is used in a less strict sense. Let V,, be the electrostatic potential experienced by Cs+. This cannot produce a potential valley by itself, because the conditions for a minimum (d2V,/dx2 > 0, d2Va/dy2 > 0, and a2Va/dz2 > 0) and the Laplace equation (VV,, = 0) cannot be satisfied simultaneously. The potential valley, in which Cs+ is floating, is produced through a cooperation between V , and another type of repulsive potential, V,,,. In the hydrogen encapsulation with Cs-A zeolites,4-8 window-blocking Cs+ must be displaced from its lowest state to make a way for a visiting hydrogen molecule. If the potential valley, in which Cs+ is placed, has such a gentle curvature that a larger displacement is easily allowed, then the encapsulation may be realized at a reasonable temperature. In the following we describe how the so-called floating of Cs+ is realized and in which direction (1) Peter C. W. Lung, Kervin B. Kunz, Karl Seff, and I. E. Maxwell, J. Phys. Chem., 79,2157 (1975). (2)R.Y. Yanagida, A. A. Amaro, and K. Seff, J. Phys. Chem., 77,805 (1973). (3)L. Pauling, "The Nature of the Chemical Bond", 3rd ed., Cornel1 University Press, Ithaca, NY, 1960. (4)D. Fraenkel and J. Shabtai, J. Am. Chem. SOC.,99, 7074 (1977). (5)P. L. Turi and J. F. Lakner, Report UCRL-52865(1979).Available from NTIS. From Energy Res. Abstr., 5(8), Abstr. No. 11406 (1980). (6)T. Takaishi and K. Itabashi, 'Recent Progress Reports and Discuasion, 5th International Conference on Zeolites", R. Sereale, c. Colella, and R. Aiello, Ed., Giannini, Napoli, 1980,p 95. (7)D. Fraenkel, J. Chem. Soc., Faraday Trans. 1 , 77, 2029 (1981). (8)D. Fraenkel, J. Chem. Soc., Faraday Trans. 1 , 77, 2041 (1981). 0022-365418212086-2089$0 1.2510
TABLE I : Positional Parameters and Occupancy Factors of Ions in (Cs,Na,)-A Zeolite Used in Calculations"
Wyckoff
occupancy
ions
position
x
Y
z
factor
T (A1 and Si)
24 (k)
0.00
0.18
0.37
1
01
12 (h) 12 (i) 24 ( m ) 3 (c) 8 (g) 1 (a)
0.00 0.00 0.11
0.23 0.29 0.11
0.50 0.29 0.34
1 1 1
0.00
0.50
0.50
1
0.20
0.20
0.20
1
0.00
0.00
0.00
1
011 0111
cs
Na1 NaII
* Lattice constant:
Q
= 12.2 A .
the potential valley has a gentle curvature.
Electrostatic Potential Structural Model. Let us assume an appropriate structural model to avoid unnecessary complexities. The real space group of zeolite A is R3?l0 but in the present study we adopt a pseudogroup Pm3m by neglecting the difference between A1 and Si ions and replacing them by T ions having their averaged properties. This is a crude first approximation, but it may give a general tendency of the potential map. A further refinement is a future problem, since the detailed structure of zeolite A is not known, which is compatible with its newly assigned space group R3.9J0 For reasons of symmetry, dehydrated Cs3Na9(A102Si02)12, which is abbreviated as (Cs3Na9)-A, is studied. Structural parameters adopted are given in Table I. As can be seen in the table, the ninth Na is located at the center of the sodalite unit. This is a rather artificial model giving good symmetry, but it may not introduce a large error into the potential calculation, since its contribution is small. The parameter values are chosen so as to be compatible with those of (Cs,Na,)- and (Cs,Ca)-A zeolites,"J2 inasmuch as no structural data on (Cs3Na,&-A are available. The electrostatic potential, V,,,depends both on the structural parameters of the constituent ions and on the (9)J. M. Thomas, L. A. Bursill, E. A. Lodge, A. K. Cheetham, and C. A. Fyfe, J. Chem. SOC., Chem. Commun. 276 (1981). (10)L. A. Bursill, E. A. Lodge, J. M. Thomas, and A. K. Cheetham, J . Phys. Chem., 85, 2409 (1981). (11) T. Blake Vance, Jr., and Karl Seff, J. Phys. Chem., 79, 2163 (1975). (12)V. Subramanian and Karl Seff, J. Phys. Chem., 84,2928(1980).
0 1982 American Chemical Society
2090
The Journal of Physical Chemistry, Vol. 86, No. 11, 1982
Takaishi and Hosoi
charge distribution of these ions. Our knowledge of the latter is very limited and only qualitative. Here, net charges on them are chosen as adjustable parameters but are confined in some ranges by their chemical properties. Let the net charges be -q(O1), -q(OII), and -q(OIII) for negative ions, and q(NaI), q(NaII), q(T), and q(Cs) for positive ions. Then the neutrality condition is 12d01) 12d011) + 244(0111)= 8dNa1) + d N a d + 2 4 0 3
+3qW
a
= 124, (1)
where qo is a measure of the ionicity of the zeolite lattice and is restricted to the range of 8ZqoZ2
(2)
if the charge is expressed in units of the elementary charge e . The upper limit, 8, means the ideal ionic crystal, and the lower limit, 2, corresponds to an ionicity of 25%. This lower limit is chosen in consideration of calculated results on charge distribution in an aluminosilicateframework.13J4 As cesium ion is not directly in contact with oxide ions of the framework, the back-donation of an electron from the latter to the former may be negligible. Its charge may be unity. Furthermore, we assume that q(Na1) = d N a d = d N a )
(3)
Then, eq 1 reduces to
dOI)
d0d
- _ _
40
+
40
2dOIII)
+-=-+40
3 d W 440
2dT) 40
1 +-=1 440 (4)
Equation 4 enables us to represent a distribution of positive charges and that of negative ones by respective triangular diagrams. Three kinds of oxygen ions have different respective charges, but such differences may not be so large. 0111 is in contact with Na+, and OI is nearest to Cs+,while On has no direct neighboring cations. The positive and negative charges are stabilized by neighbors. Thus, it is concluded that dOId
< do,) and d O I d
(54
Furthermore, the rejection of an extremely uneven charge distribution is tentatively expressed by s(O11) 2 0.24,
Flgure 1. Triangular diagram of the charge distribution on ions constikRing zeolite A: (a) charges on negative ions; (b) charges on positive ions. Hatched parts denote allowable regions; V-,- or V+,- has a constant value on the respective solid lines.
Then, the electrostatic potential experienced by the concerned Cs+ at point P is given by
(5b)
A point in the hatched region in Figure l a satisfies the conditions given by eq 5. As for positive charges, we have from their chemical properties q(T) I(3 + 4)/2 = 3.5 q(Na) I1 If zeolite is ideally ionic, i.e., qo = 8, then we have q(Na) = 1, and q(T) = 3.5. Even when the ionicity is extremely weak, i.e., qo = 2, the ratio q(T)/q(Na) may range from 3 to 7. Thus, the chemically plausible charge distribution is represented by a point in the hatched region in Figure lb. Points A’, B’, C’, D’, and E’, the corners of the hatched region, correspond to charge distribution (8, 1, 7/2), (61/8, 1/2, 7/2), (2, 7/59,49/59), (2, 7/27, 21/27), and (7, 1,3), respectively, in the (so, q(Na), q(T)) representation. Curvature in the Electrostatic Potential Surface. Let the coordinate of a given point P be r(P), and that of the ith ion of the uth lattice species be ri(v),and Ri(u) = r(P) - ri(v) (13) S. Beran and J. Dusky,J. Phys. Chem., 93, 2538 (1979). (14)W. J. Mortier, P. Geerlings, C. Van Alsenoy, and H. P. Figeys, J. Phys. Chem., 83, 855 (1979).
ve,
v-)
= 40(V+-
(6)
with qoV+ = q(Na)[u(NaI) + u(Na11)1+ u(Cs) + q(T) u(T) (7)
qov- = q(01) v(01) + q(O11)
4011)
+ q(0113 40111)
(8)
u ( v ) = C1/Ri(v) I
where the summation is carried out over the crystal. In the following we use a coordinate system whose origin is located at the center of the 8-ring and whose x and y axes are parallel to two edges of the square face of the cube, respectively. Then we have at the origin (d2u(v)/axz), =
( d Z U ( U ) /dy2)0
= -‘/(a~u(v)/dz2)o
from the Laplace equation and the site symmetry of D4h. Thus, only ( d z V + / d x 2 ) oand ( d 2 V - / d x 2 ) 0are required to inspect a general tendency of the potential surface at the center of the 8-ring. We have from eq 8 QOV-,xx = d0I)
~ ( 0 I ) X X
+ d 0 I I ) 4OII)xr + d 0 I I I )
U(OIII)rx
(9)
The Journal of Physical Chemistry, Vol. 86, No. 11, 1982 2091
Floating Cesium Ion in Zeolite A
TABLE 11: Numerical Values of uxx for Each Kind of Ionn no. of unit cells summed
100
294
648
1210
L J ( O I ) ~ ~115.811 113.828 112.788 112.148 89.878 89.238 90.919 u ( O I ~ ) ~92.903 ~ 113.044 110.962 109.682 u ( O I I I ) ~117.015 ~ 24.613 25.734 25.040 u(NaI),, 27.058 4.697 4.610 4.557 u(NaII),, 4.864 4CS)XX -2.749 -3.244 -3.504 -3.664 u(T),, 148.753 144.784 142.703 141.423 Values are given in units of a3, with a = 12.2 A .
where the subscript x x denotes the second partial derivative with respect to x at the origin. Values for ~(OI),,, u(O1~),,,and u(OIII),, are obtained by carrying out the summation over 1210 cells (see Table 11). (A convergence test of the lattice sum shows that even a summation over 100 cells gives satisfactory results.) Equation 9 represents a set of parallel straight lines in the triangular diagram in Figure la, and the position of each line is determined by the value of the parameter V-+,. The maximum value acceptable to V-&,is obtained from a line passing through the corner of the hatched region point C, while the minimum is obtained from that passing through another corner B. Their numerical values are tabulated in Table 111. Similarly, we have from eq 7
v+,,
=
which also represents a set of parallel straight lines in the triangular diagram of Figure lb. The maximum and minimum acceptable values for V+,x,are obtained from lines passing through the corners B' and D', respectively, and given in Table III. The value of V+$,strongly depends
upon u(T),, and qo, but weakly upon u(NaI),,, u(NaII),,, and ~(CS),,. In other words, the coordinate of T (A1 and Si) and the ionicity of the zeolite seriously affect the value for V+&,,while V-,,, is independent of the ionicity. The dependence of V++, on the coordinate of the T ion is quantitatively investigated by changing the structural parameters for T, results being given in Table IV. This table shows that the relation V-,x,(min) > V+,,,(max) holds except for the cases of no. 5-7, which are unacceptable from the structural point of view. Thus, we can safely conclude that (a2Ve,/axZ)0 = 40(v+,xx - V-,,,) < 0 To have a quantitative result, we must assign a plausible charge to each ion. Recently, several workers calculated a charge distribution in an aluminosilicate framework. According to their results, q(T) ranges between 1.30 and 1.55, q(Na) between 0.23 and 0.38, and q(0)between 0.55 and O.75.l3J4 These works concern systems consisting of only several six-membered oxygen rings. It is considered that their results describe, at least qualitatively, general features of zeolites. In view of this, we investigate the following case of the charge distribution as a representative one: qo = 3, q(T) = 5/4, q(Cs) = 1,q(Na) = 1/3, ~(01) = 241/300, q(OII) = 2/3, and q(OIII)= 153/200. The point representing the above negative charge distribution lies on the line passing through point A in Figure la.
Repulsive and van der Waals Potential Generally speaking, it is very difficult to estimate the repulsive and van der Waals forces between an ion pair at a medium distance, as these forces are masked by a much stronger electrostatic force. The pair potential between neutral atoms or molecules, U , is well described empirically by the Lennard-Jones 6-12 potential u = Xr-12 - pr-6 (11)
TABLE 111: Numerical Values of (a zVe,/ax')/qO for Various Charge Distributions" and Their Convergence Testb no. of unit cells summed V-JX ( A 1 V-,XX(B) v - ,xx ( c ) V+.XX(A'1 V+,XX(B') V+,XX(D'1 V+.XX(A')V+,XX(A')V+,XXP')V+,XXP)-
V-,XX(A) V-,XX(B) V-,XX(A) V-,XX(C)
100
294
648
1210
81.432 76.847 88.308 68.726 70.013 60.612 -12.706 -8.121 -20.820 -27.696
79.448 74.863 86.324 66.741 68.028 58.628 - 12.707 -8.122 - 20.820 -27.696
78.407 73.822 85.283 65.701 66.988 57.587 - 12.706 -8.121 -20.820 - 27.696
77.767 73.182 84.643 65.061 66.348 56.947 -12.706 -8.121 -20.820 -27.696
The charge distributions are as follows: at point A, q ( O I ) / q o= 1 / 4 , ~ ( O I I ) / =~ ,1 / 4 , 2q(oIII)/qo= 112; a t point B, q ( O I ) / q ,= 1/5, q ( 0 1 1 ) / q=, 1 / 5 , 24(0111)/4~= 315; at point C, q ( O I ) / q o= 2/5, q(OII)/q,= 115, 29(0111)/4~ = 2/5; at point A', 3q(Na)/4qo = 3/32, 2 q ( T ) / q 0= 7/8, 1/4q0= 1 / 3 2 ; at point B', 3q(Na)/4q0= 3/61, 2q(T)/q, = 56/61, 1/4q0= 2/61; and a t Values are given in units of a3, with a = 12.2 A . point D', 3q(Na)/4qO= 7/72, 2 q ( T ) / q ,= 7/9, 1/4q0= 1 / 8 . TABLE IV: Maximum and Minimum Values for V+,,, and Parameters of the T Iona
V+. and t h e Dependence of
V++ upon t h e Structural
v+,xx/a3 no. 1 2 3 4 5 6 7
structural parameters of T ion (0.00, 0.18, 0.37) (0.00, 0.185, 0.375) (0.00, 0.19, 0.38) (0.00, 0.195, 0.385) (0.00, 0.20, 0.39) (0.00, 0.205, 0.395) (0.00, 0.21, 0.40)
a Values are given in units of a3, with a = 12.2 A . Figure 1,a and b.
@'Ib
V-,,,/a3 (Qb
Wb
70.013 60.612 71.698 62.039 73.574 63.629 75.662 65.398 88.308 76.847 77.983 67.365 80.561 69.549 82.528 71.215 The letters in parentheses, B', D', C, and B, represent the points in
2092
Takaishi and Hosoi
The Journal of Physical Chemistry, Vol. 86, No. 11, 1982
or
U=
Uo[(r0/r)l2-
2(ro/r)6]
(12)
It is presumed that the same kind of potential operates between a pair of ions in addition to the classical electrostatic one. Here an unsolved problem is how to estimate values for the parameters in eq 11. Takaishi and Kobayashi have studied adsorption potentials of rare gas molecules in mordenite to determine parameter values for its lattice oxygen.15 They used familiar f ~ r m u l a s ' ~ J ~
TABLE V: Values for the Parameters Contained in Lennard-Jones 6-12 Potential between Cst and the Framework Oxygen r,/A
U,leV oIo-/A3
2.3 2.4 2.5 2.6
0--0'
0.0136 0.0140 0.0144 0.0148
cs+-cs+ cs+-0-
0--0-
cs+cs+ cs+-0-
0.0141 0.0141 0.0141 0.0141
4.07 4.10 4.14 4.17
4.11 4.11 4.11 4.11
0.0138 0.0140 0.0142 0.0144
4.09 4.11 4.13 4.14
+
XAB1/13= y2(XAA1/13 XBB1/13)
PAB
= (PAAPBB)'"
and newly developed empirical relations XAA
= (2.13 X 104)a~3.'0 eV -&12
(134
(2.58 X 10)aA'.88 eV A6
(13b)
pAA =
where A and B specify the kind of atom, AA, AB,and BB are the respective pairs, and a denotes the atomic polarizability. Equations 13 quite well describe values for rare gas atoms. The best-fit values determined for the lattice oxygen of mordenite are
0.1 0
0.05
>
a)
a. = 2.45 f 0.15 A3
Pi t
Am = (3.465 f 0.655) X lo5 eV poo
= (1.40 f 0.16)
X
lo2 eV A6
-0.05
Pading'* calculated quantum mechanically the polarizability of 02-to be 3.88 A3, and Barrer and his co-w~rkers'~ estimated that of neutral oxygen from the refractivity of potash feldspar to be 1.65 A3. The above value, 2.45 A3, is just intermediate between the two, and much nearer to the value of neutral oxygen. This is in good harmony with the before-mentioned CNDO result on aluminosilicate rings for which the charge on the framework oxygen is about 0.6e or so. Some questions might arise concerning the applicability of eq 13 to the lattice oxygen. Being encouraged by the success in the mordenite case, however, we use eq 13 in the following. It is considered that the nature of the framework oxygen in zeolite A may not seriously differ from that in mordenite. A possible small difference between the two may be checked by calculating for two extreme cases, e.g., for a0 = 2.6 and 2.3 A3. By aid of eq 13, we get
d-011
Flgure 2. Lennard-Jones 6-12 potential for a pair of Cs+-lattice oxygen. Solid curve referred to the polarizability of oxygen = 2.3 A3; dotted curve, referred to the polarizability of oxygen = 2.6 A3.
is used, as eq 13 has been derived on the basis of the a-value system due to Pauling. Four sets of parameter values are given in Table V, which range between eq 14 and 15. The pair potential curves are shown in Figure 2, in which arrows indicate the nearest, second-nearest, and third-nearest distances between the center of the 8-ring and the framework oxygen. One can see that Cs+, located at the center of the ring, may experience a repulsive potential. The potential VG12 is given by v6-12
for
a.
= 2.6
loo= 4.12 X lo5 eV
A12
(144
poo = 1.56 X lo2 eV
A6
(14b)
lo5 eV A'2 = 1.24 X lo2 eV A6
(15a)
A3, and Xoo = 2.81 poo
X
(15b)
for a. = 2.3 A3. As for ac t, two kinds of values have been proposed, that is, 2.42 by Pauling, and 3.34 A3 by Tessman, Kahn, and Shockley.20 Here, the former value
280, 466 (1964).
=
i,u
Vi,,
where Ui,uis a 6-12 potential between a pair of Cs+ and the uth kind of ion located at a position &(u) in the crystal. These forces are of rather short range, and one can safely neglect contributions from ions other than the nearest, second-nearest, and third-nearest oxygen ions. In view of eq 5, let us provisionally assign to 01,On,and OnI a values of 2.6, 2.3, and 2.5 A3, respectively. Then, through simple calculations, we get
A3
(15)T. Takaishi and H. Kobayashi, to be submitted for publication. (16) F. T. Smith, Proc. R. SOC.London, Ser. A , 5, 1708 (1972). (17)Chang Lyoul Kong, J . Chem. Phys., 59,2464 (1973). (18)L. Pauling, Proc. R. SOC.London, Ser. A , 114,181 (1927). (19)R. M.Barrer and D. L. Peterson, R o c . R . SOC.London, Ser. A ,
I I
(azv6-,,/axz), = 6.245 eV A-2
(16)
-0.857 eV A-2
(17)
(a2v,-,,/a~2)~=
at the origin. (20)J. R.Tessman, A. H. Kahn, and W. Shockley, Phys. Reo. 92,890 (1953).
Floating Cesium Ion in Zeolite A
The Journal of Physical Chemistry, Vol. 86, No. 11, 1982
TABLE VI: Depth and Curvature of Potential Experienced by Cs Ion and Its Vibrational Frequencies quantities
ideally ionica
extremeli less ionic
representative caseC
3.88 -17.681 1.093 -16.588 11.903
2.0 -6.740 0.165 -6.575 3.828
2.6, 2.3, 2.5 -8.593 0.360 -8.233 5.837
-0.203
0.126
-0.040
4.675 X 10’’ 2.651 X 10” 127.2 224.3
3.274 X 10” 157.1
a
Y
Y
t ............. ............. ............. .............
t ............. ............. .............
.............
.............
............. .............
::::::p::::: ...... :: --*x
......... ............. ............. .............
-2
0.::
............. 0
0.5
1.0
A
............. ............. .............
$ = -8.23eV 0 -7.6eV - 7 . 6 e V k @ > -7.7eV -7.7eV>#> -7.8eV 0 -7.8eV 2 @ > -7.9eV origin
0
q o = 8, and the charge distributions of points A and A’. q o = 2, and the charge distributions of points A and D’. a
“Representative case” refers to the plausible charge distribution q o = 3, q(T)= 5/4, q(Na) = 1/3, q ( C s ) = 1, q ( 0 I ) = 241/300, q ( 0 I I ) = 2/3, and q ( 0 I I I ) = 153/200.
2003
o>
- 7 . 9 e V k @ > -8.0eV -8.0eV>@> -8.leV - 8 . l e V & Q > -8.2eV -8.2eVLQ
. >
TABLE VII: Curvature of Potential Experienced by Cs Ion in (Cs3Na,)-, (Cs,,Na)-, and (Cs3Na K,)-A
zeolites (Cs,Na,)-A (Cs,,Na)-A (Cs,Na K,)-A
positional parameter of ions o n 8(g) position (0.20, 0.20, 0.20) (0.275, 0.275, 0.275) (0.231, 0.231, 0.231)
b
a,
(a’@/axz)o/ (a’@/az’)),/ (eV A - * ) (eV A-’) 5.837
-0.040
5.721
0.164
5.800
0.023
Potential Map The total potential experienced by the concerned Cs+, 9, is given by = Ves
+ Vel2 - f/2[((~Cs+)(VVes)~I
in which the last term refers to the polarization energy of Cs+. Maps of @ in the xy and yz planes are shown in Figure 3. The potential depth and the curvatures at the valley of the potential surface are tabulated in Table VI, for several cases. The vibrational frequency of Cs+ in the valley, v,, can be calculated by the approximate relation v, = (1/ 2*) { (a29 / dx2),/ mcs}1l2
where mcedenotes the mass of Cs. Calculated frequencies are rather small in the x and y directions. In the z direction, on the other hand, the frequencies cannot be calculated, inasmuch as the 9 vs. z curve has double minima as shown in Figure 3b, and a simple harmonic oscillator model cannot be applied. In this case, it is clear that Cs+ thermally oscillates with a large amplitude. Such soft vibrations are in good agreement with X-ray structural results. In conclusion, all values in the table are reasonable ones to ensure the floating of cesium ion and its large vibrational amplitude.
Other Zeolites. (CsllNa)-A and (Cs3NaK8)-A Now, NaI’s in (Cs,Na&A are replaced by Cs, designated as Csn, and a change introduced into @ is investigated. It is assumed, for simplicity, that CSII has the same charge as NaI but is slightly displaced into the a cage along the body diagonal. In this system, contributions to V, from Cs-Cs pairs must be taken into account, along with a change in Vee. Values for (d2@/dx2), and (d2@/dz2),are straightforwardly calculated and given in Table VII. A slight increase in ( ~ 3 ~ @ / d zis ~ )observed. , This means that the thermal factor of CsI in the z direction is larger in
Flgure 3. Maps of the potential surface experienced by Cs’ located
near the center of eight-membered oxygen ring: (a) two-dimensional potential maps in the planes of L = 0 and x = 0; (b) profile of the potential. Soli curves, a: dotted curve, V , iV,,,; chained curves, V,; dashed curves, V8-,*.
(Cs3Na9)-A than in (CsllNa)-A. If the Nais in (Cs3Na9)-A are replaced by potassium, a similar situation may occur with a much smaller displacement of the cation position. For a representative case, (d2@/dz2)ois calculated with aK = 0.83 A3 and a structural parameter (0.231, 0.231, 0.231) and given in Table VII. Even in this case, the increase in (d29/dz2), is appreciable.
Discussion Generally speaking, the calculation of the potential surface is a formidable task, especially in crystal phases. The potential seriously depends upon the charge distribution on constituent ions of the crystal. The charges are difficult to determine with confidence, either theoretically or experimentally. In the present work, studies were made for conceivable extreme cases of the charge distribution along with the palusible one. All of them give the qualitatively same result that Cs+ is floating at the center of the 8-ring and thermally vibrating with a large amplitude,
J. Phys. Chem. 1982, 86, 2094-2097
2094
TABLE VIII: Equilibrium Distances between Alkali Ions Framework Oxygen Ion in Lennard-Jones 6-12 Potential, r o , and the Distance between Oxygen Ion and the Center of the 8-Ring distance between and
oxygen ion and the center of the
small term of the polarization energy and consider the potential CP'. 9' =
CC[q(v)/ri(v) + UO(v)((ro(v)/ri(v))'* - 2(~,(v)/ri(v))~lI I
Y
We have by aid of the Laplace equation (V2CPrh=
132CC[(Uo(v)/ri(~)l~((rg(~)/ri(v))~~ - (5/11) X I
CS-A Rb-A K-A Na-A a
2.42 1.40 0.83 0.179
4.09 3.89 3.13 3.41
4.66 4.44 4.25 3.96
3.64 3.41 3.32 3.09
3.29 3.18 3.13 3.31
3.62 3.69 3.11 3.64
5.32 5.21 5.21 5.29
r* = r0/2'I6,at which V,-,, = 0.
especially in the direction perpendicular to the 8-ring plane. It may be said that the result is semiquantitatively reliable, as the covalent force may not operate so much in the present geometrical situation. Now let us discuss the existence of the double minima in the CP-z curve. The barrier height between the two minima is very small, e.g., 0.0086 eV, which is smaller than the limit of the reliability of the model used and the thermal energy at room temperature. In the real situation, the ninth sodium ion, NaII, is located not at the center of the sodalite unit but at another site with a lower symmetry. Then the CP-z curve is deformed to be asymmetric, and one of two minima becomes deeper than the other and the true potential bottom. Even in such a case, the CP-z curve may have a very gentle curvature, and Cs+ thermally vibrates with a large amplitude. Hence, at higher temperature, Cs+ is easily displaced to make a path for a visiting hydrogen molecule to pass through the Cs+-blocked window. Principally, the potential surface might be calculated for the window-passing process. In such a calculation, however, a contribution of covalent force must be taken into account, which is beyond the present scheme, and future work is desired. We now discuss a criterion for the occurrence of the potential minimum. For simplicity let us neglect a very
Y
( ~ o ( Y /) r i ( ~ ) ) ~(18) Il
The conditions for the occurrence of a minimum, CPxxr > 0, a;, > 0, and az; > 0, require that at least (VZCPr),> 0. The term in the square bracket in eq 18 becomes positive, if r i b ) < 2.21/6ro(v)
(19)
The distance between the center of the 8-ring and the framework oxygen, ri(v),is tabulated in Table VIII. The values for ro(v),in the table, are calculated by aid of eq 13 and the polarizabilities of respective cations. As can be seen from the table, the inequality 19 is always satisfied for pairs of 01-cation and 011-cation but not for those of OIrcation. Thus, this necessary condition is too loose to check the occurrence of the floating cation. A more severe measure is whether a cation at the origin experiences a repulsive V,,, potential or not, since (d2Ve,/dx2)0< 0. If the distance between the pair of cation-oxygen, d(C-0), is shorter than r* (=r0/2lI6),the repulsive potential operates. Values for r* are also tabulated in Table VIII. The larger the difference r* - ~(C-OI), the higher the possibility of the cation floating. Thus, the possibility sharply decreases from Rb-A to K-A and Na-A. This is in good agreement with X-ray structural results. Acknowledgment. The present work was financially supported by a Grant-in-Aid from the Ministry of Education of the Japanese Government, Contract No. 443001. H.H. sincerely thanks the Toso-Shogakukai for the scholarship which enabled him to carry out the present work.
Conductivity Studies in Search of Liquid-Liquid Phase Separation by Solutions of Lithium In Methylamine R. Hagedorn and M. J. Slenko" Baker Laboratory of Chemistry, Cornell University, Ithaca, New Ywk 14853 (Received: September 9, 1981; In Final Form: December 1 1, 198 1)
The ac electric conductivity as a function of temperature has been measured on a series of lithium-methylamine solutions spanning the metal-nonmetal transition. Ten solutions in the range 3.5-17.4 MPM (mole percent metal) were investigated between 180 and 230 K. No breaks in the resistivity vs. temperature curves signaling liquid-liquid phase separation were observed, even for the most concentrated solutions. All the data could be fitted to functions of the type u = uo exp(-hE/kT). AE showed a maximum value of 0.16 eV at 11 MPM. The analogous maximum in lithium-ammonia solutions has a value of 0.15 eV for AE and occurs at 4 MPM. Introduction A characteristic feature of metal-ammonia solutions (except for cesium) is the separation on cooling into two coexisting liquid phases.' Pitzer, in 1958, suggested that (1) For an excellent review, see J. C. Thompson, 'Electrons in Liquid Ammonia", Clarendon Press, Oxford, 1976, Chapter 5.
0022-3654/82/ 2086-2094$0 1.2510
this separation into two liquid phases is a "liquid-vapor phase separation of the sodium within the ammonia solvent",2thus connecting it to the metal-nonmetal transition that is also characteristic of these systems. In the case of lithium, the miscibility gap occurs at 210 K and (2) K. S. Pitzer, J. Am. Chem. SOC.,80, 5046 (1958).
0 1982 American Chemical Society
