J. Phys. Chem. 1992, 96, 1437-1444
4 D, D* 8
g' G Hi
H
H k I
a2/[#(0*
Ji, Ji
L
n
P P
P R t
diffusion coefficients in the volume-fixed reference frame subscript, the quantity is due only to the diffusion (all velocities are zero) matrices of diffusion coefficients in mass-fixed and volume-fixed reference frame, respectively (the number of elements of matrices D and D* is n X n) gravity aderation component along they axis; g is negative; its component along the x axis is zero = @'/PO assumed as a constant in all equations dealing with perturbations vector of constants of eq 15 concentration cocffcients of density, Hi = ap/aCi (see eq 1) vector of H / s matrix whose elements are H,, = ti& (ti,,, Kroneker symbol)
+ 1131
identity matrix (iij = tiij, Kronecker symbol) flow of component i in the mass-fixed and volume-fixed reference frames, respectively reference length;corresponds to the height of vortices given by the convective solution (eqs 14 and 15); can assume any positive value within the dimensions of the diffusion layer number of independent components (2 in a ternary system) pressure constant appearing in eqs 14 and 15 pL2/[xf(a2 + l)], constant, eq 16, must be negative for gravltationally stable diffusion boundaries ?
- [g*kL4(PD)zz/2P'~l~eq 31
time
1437
horizontal axis vector whose components are defined by eq D22 vertical axis, positive in the upward direction subscripts, indicate derivatives (a/&, a/ax, a/ay) components of velocity, Y , along axes x and y partial specific volume of component i convective velocity vector of mass center velocity of component i reduced length z = y/2v't Po Hi &I AP
Kroneker symbol; 6,, = 1, 6, = 0 for i # j differencebetween top and bottom properties through the diffusion boundary coefficient of cinematic viscosity (cm2s-l) coefficient of dynamic viscosity, cc = qp (g cm-I s-l) eigenvalues of matrix D erf ot/v'U density density at the average concentration, (pboctom+ p,,)/2 = (p)/(l + x H i ( C i ) ) ,constant of eq 2 p'
HjC,
vector whose elements are pi superscript, refers to a perturbed quantity (convective term) mass fraction of component i, wi = Ci/p flow function, aq/ay = U, a3/ax = -0, v29 = rot Y trace of D = C,D, scalar product divergence operator Laplace operator
Hydrogen Intercalation within Transition Metal Oxides: Entropy, Enthalpy, and Charge Tr ansfet J. J. Fripiat* and X. Lin Department of Chemistry and Laboratory for Surface Studies, University of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201 (Received: April 2, 1991; In Final Form: August 21, 1991)
Intercalation isotherms of hydrogen within transition metal oxides, through H spillover, are quantitatively accounted for by a statistical model where the average enthalpy is the weighted sum of the energy of occupied, prevented, and vacant sites of an idealized network of interstitial sites ("interstitial lattice") and where the entropy contains two contributions. The first corresponds to the number of configurationsof occupied, prevented, and vacant sites within the interstitial lattice, whereas the second contribution contains the nonconfigurational contributions, including a charge-transfer term. This term can be approximated from the volume of the intercalated hydrogen. The charge transfer consists in the donation of H 1s electron to the oxide lattice. The enthalpy of formation, obtained from calorimetric measurements,correlates with the charge transfer as well. Filling the interstitial lattice predicts nonstoichiometric H contents, as observed experimentally. The observed stepwise intercalation isotherms can be fitted rather nicely using the calculated energy and entropy terms. It is also shown that the calculated intercalation free energy predicts nicely the observed order of stability. In the appendix it is suggested that this treatment can be extended to transition metal hydrides.
Introduction Transition metal oxides intercalate a large number of hydrogen atoms at pressures lower than 1 atm and in the room temperature range, provided atomic hydrogen is available. An adequate source of atomic hydrogen is small particles of metals able to chemisorb dissociatively molecular hydrogen, such as Pt,Pd, Rh,etc.' The overall intercalation reaction can be sketched as a twwtep process, as follows: H2 s 2H XH
+ ox
~,4+0x-x4
(1)
where OX stands for the transition metal oxide lattice. The resulting H-intercalated lattice, H,f+OX-x', is commonly called To whom correspondence should be addressed.
H-bronze. H-bronzes with final compositions such as &.4sW03? H1.6M00~,3HWb6V205$and H,3.3VMo05.? have been described. At constant temperature, x is a function of the equilibrium pressure P(HJ and this function is the intercalation isotherm. In the steady state the chemical potential p, of H within the solid is equal to the chemical potential in the gas phase, p = In [P(H,)/P*(H,)] where P*(H,) is the absolute activity of H2 and P(HJ the pressure needed to achieve the intercalation of xH (1) Sermon, P. A,; Bond, G. C. Catal. Rev. 1973, 8, 211. (2) Berzins, A. R.;Sermon, P. A. Nature 1983, 303, 506. (3) Tinet, D.; Ancion, C.; Poncelet, G.; Fripiat, J. J. J. Chim. Phys. 1987, 809. (4) Tinet. D.; Fripiat, J. J. Reu. Chim. Miner. 1982, 19, 612. (5) Lin, X.;Lambert, J. F.; Fripiat, J. J.; Ancion, C. J. Carol. 1989, 119, 215.
0022-3654/92/2096-1437$03.00/00 1992 American Chemical Society
1438 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
atoms. The activity of the intercalated H guest is In A, = p / R T . In the temperature range 300-400 K the intercalation isotherms are most often step isotherms, which means that In Ag = A x ) is constant in a considerable domain of x. This charactenstic shape is also observed in crystalline metal hydrides.6 The main difference between intercalation isotherms in hydrogen bronzes and crystalline metal hydrides is in the extent of the hysteresis loop. Lowering P(H2) 0 in metal hydrides lowers x (here x = [H]/[M]) to a large extent (say by more than 50%), whereas x decreases much less in hydrogen bronzes. For instance, deintercalation from H-3.4VM005.5is less than (1/3)xM5and less than ( I / 1o)xM from Hl,6M003,3xMbeing the maximum hydrogen uptake. This poor reversibility is most probably of kinetic origin, since a larger chemical driving force such as reacting H bronzes with O2 or C2H4(forming C2H6) depletes the reservoir to a much greater The noble metal particles (Pt, for instance) are the gates by which intercalation and deintercalation proceed^?^^ whereas in metal hydrides the metal atoms at the solid-gas interface play that role. In metal hydrides, Griessen and Driessen'o.'' have found a relationship between the intercalation enthalpy and the difference between the energy at the Fermi level and the energy for which the integrated density of states of the host metal is equal to (1/2)&, n, being the number of electrons per atom in the lowest s-like conduction band. In hydrogen bronzes, depending upon the nature of the host lattice, the negative charge given by the intercalated species to the oxide lattice can be either localized and transferred to the transition metal cation Mrt or it can be delocalized in a conduction band.4s5s12-13 For instance, H1.6M003is a meta1I3J4with a Fermi temperature ca. 1.5 X lo4 K whereas H,3.3VMo05,55is a semiconductor with a rather small gap, ca. 0.15 eV. Interestingly, it seems that there is a correlation between the H mobility in these intercalation compounds and the metallic c h a r a ~ t e r . ' J ~ 'The ~ band structures of the starting oxide and of the bronze being unknown, there is no way presently to predict intercalation enthalpy as suggested for metal hydrides. In fact, the band structure of the perovskite like alkali-metal bronzes of W 0 3 has only been sketched.I The main goal of this paper is to calculate a general equation accounting for the shapes of H intercalation isotherms within transition metal oxide. By fitting experimental and theoretical isotherms it will be shown that the differences in energy between an interstitial site occupied by hydrogen and a vacant site as well as the interaction energy between intercalated species can be calculated, and that the charge-transfer contribution to the enthalpy term can be estimated. The extent of charge transfer is obviously related to the insulatopmetal or insulatorwmb"iuctor transition induced by the H intercalation. Pre~iously,'~ we had suggested a statistical model which allowed us to calculate the configurational entropy and to simulate very satisfactorily the shape of the isotherm, but not the actual position
-
(6) Griessen,R. Phys. Reo. B 1983,27,7575. (7) Marcq, J. P.; Wispenninckx, X.; Poncelet, G.;Keravis, D.; Fripiat, J. J. j . Catal. i982,73, 309. (8) Marq, J. P.; Poncelet, G.; Fripiat, J. J. J . Caral. 1984, 84, 339. (9) Sermon, P. A.; Bond, G.C. Trans. Faraday Sm. 1980,76, 889. (10) Griessen, R.; Driessen, A. Phys. Rev. B 1984,30, 4372. (1 1) Griessen, R.; Driessen, A. J. Less-Common Met. 1984,103, 245. (12) Tinet, D.; Legay, M. H.; Gatineau, L.; Fripiat, J. J. J . Phys. Chem. 1986,90, 948. (13) Tinet, D.; Canesson, P.; Estrade, H.; Fripiat, J. J. J . Phys. Chem. Solids 1979,41, 583. (14) Erre, R.; Legay, M. H.; Fripiat, J. J. Surf. Sci. 1983,127, 68. (15) Bullet, D. W. Surface Properties and Catalysis by Non-Metals; Bonnelle, J. P., Delmon, B., Derouane, E., Eds.;NATO AS1 Series C; Reidel: Dordrecht, 1982; No. 105, p 47. (16) Vannice, M. A.; Boudart, M.; Fripiat, J. J. J. Catal. 1970,17, 359. (17) Taylor, R. E.; Ryan, L. M.; Tindall, P.; Gerstein, B. G.J . Phys. Chem. 1980,73. 5500. (18) Cirillo, A. C.; Ryan, L.; Gerstein, B. C.; Fripiat, J. J. J . Chem. Phys. 1980,73, 3060. (19) Fripiat, J. J.; Lambert, J. F. J . Phys. Chem. 1989,93, 2083.
Fripiat and Lin of In A, vs x. It will be shown that,by adding a noncodigurational entropy term, both the shape and the position of the isotherm along the In A, axis can be reconciled with experimental data. Theory
The basic assumption is that the intercalated species are located on a lattice of interstitial sites. This lattice may or may not be commensurate with the real lattice of the starting oxide. It is used to calculate the configurational entropy during the hydrogen uptake. SlmulatiOO of the hte~~tithl Lattice Codlgumtion. The theory has been published previo~sly.'~It will be briefly summarized here for the sake of clarity. Whatever the particular model of "interstitial lattice" we choose, it is completely defined by specifying its "neighboring graphnmin which (i) each vertex represents a site, and (ii) two vertices are joined by an edge, if and only if the corresponding sites are considered as "neighbors". We restrict ourselves to graphs in which all vertices are equivalent and have Z neighbors, i.e., to 2-regular graphs. Furthermore, among the possible graphs, we have considered only those representing real simple crystallographic lattices and we have named them according to their resemblance; Table I in ref 19 summarizes their characteristics. Each graph is, thus, characterized by the dimension n of the associated lattice and the number of neighbors of each site Z. In order to fill these "lattices", simulations were run on an IBM PC. To each lattice site was associated an element of a n-dimensional array (n = 1,2, or 3) containing N elements overall. Each of these N elements could have one of three values corresponding to vacant, (noted v), prevented (p), or occupied sites (o), respectively. At the beginning of a simulation all the array elements were set vacant. Then the following process was repeated until no more vacant sites were present. Each of the array subscripts was assigned a random value and the state of the element with the corresponding address was checked. If it was already occupied or prevented, another random selection was made. If it was vacant, it switched to being occupied. At any time the numbers of occupied, prevented, or vacant sites (No,N,, and N,, respectively) could be monitored as well as the corresponding mole fractions X, = N,/N, X, = N,/N, and X, = N,/N with, of course,
x,+ x,+ X" = 1 Border effects were eliminated by "sticking" opposite edges of the lattice. For instance, the first row of a bidimensional array was adjacent to the last one, etc. Two salient features were observed for all simulations. (a) Even if the final codiguration displayed for repeated filling of the same lattice appeared very different, the maximum number of occupied sites Nm, with X,, = Nm/N, is typical of the lattice under consideration. When compared to Xom(th)which is the theoretical maximum fractional occupation in a perfectly ordered lattice, Xm/Xm(th) ranges between 0.6 and 0.872, according to the lattice (Table I, ref 19). (b) Even if the simulation technique is modified in order to incorporate diffusion from a privileged region of the lattice (one face or one side for instance), the final value of X, remains unchanged. This is an answer to the anticipated objection that the random aCCeSS model, in which all vacant sites have the same probability of being occupied at any given time, is not physically realistic. Obtaining for X, a noninteger or a rational fraction of integer when compared with X,(th) which is rational, is not trivial. For the development which follows, it was very desirable to obtain an analytical function X, = F(Xo). We found a good agreement between the X, values coming from the simulations and those (20) Bollobis, B. Graph Theory, An Introductory CourstGraduate Texts in Mathematics; Springer-Verlag: New York, 1979; p 63. (21) Lin, X.; Davis, J.; Fripiat, J. J. Coral. Lett. 1990,6 , 1. (22) Lin, X.; Fripiat, J. J., unpublished work. (23) Tinet, D.; Partyka, S.; Rouquerol, J.; Fripiat, J. J. Mater. Res. Bull. 1982,17, 561.
The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1439
Hydrogen Intercalation in Transition Metal Oxides
where
obtained by using the following empirical relationship
x, = (1 -X,,)[l - (1 - a ) I
(3)
AI =
with (4)
LaIn [
+(
sXom ) ( l -a){-']d a
So = 2 1 . 4 ~ ~
So = 15.6 eu
a = Xo/Xom
OX
One is tempted to believe that the nonstoichiometriccharacter of the H bronzes is related to the irrational values computed for X,. The values of X , and tare repeated in Table I of the present paper. Calculation of the Cbemical Potential of tbe Intercalated H Specks. The chemical potential of the guest species p, is (aF,/dNo)N,pwhere F, is its free enthalpy or Gibbs free energy and No the number of occupied sites among the N interstitial sites of a specified interstitial lattice. Each possible "structure" is characterized by the t parameter of this lattice. If W , is the average energy of an averaged configuration, Q the total of possible configurations, and AP' the transfer entropy accounting for the nonconfigurational terms, then
L(%)N,c-(E)N,f-(-) N ax, a"/R RT NaX,
N,f
+ '/2H,(g)
+
(5) = ( E , - E , ) + (-
- a)f-I(EP- E,)
NJ
(7) as long as (E, - E,) and ( E - E,) are a independent, that is, in the framework of the mean ield assumptionunderlying the present theory. Since the size N of the interstitial lattice is arbitrary, it can be made equal to the size of the interstitial lattice in one stoichiometric unit H,OX, in which case NX, = x , at the completion of the intercalation reaction. The differential enthalpy AI? is equal to (BWa,/NaXo)N,for to (aW,,/x, aa)N,pIf the integral enthalpy of the intercalation reaction as a goes from 0 to 1 is AH or if Ai?? = AH/x, is the intercalation enthalpy per mole of H, it follows
M / x , (E,
- E,) +
1 -Xom (EP- E,)XO,
-- AH
aR = M t ( 1 - a)[-'+ (E, - E,)[1 - ((1 - a){--'](9) As shown in the Experimental Section, because the calorimetric measurement of AH vs x or a = x / x , shows a quasi-linear
variation, (8) is a good approximation. (E, - E,) is a function of the interaction energy between the intercalated guests. This term plays a central role in determining the shape of the isotherm, as it will appear later. Its value is easily obtained from (8).
[Ai?? - (E,
.,
- E,)]- 1 =om -Xom
Note that if Ai?? C (E, - E,), E, C E, and vice versa. In eq 6 (In Q / N ) is the configurationalentropy hsoo*/NR the calculation of this term has been carried out previo~sly'~ with the following result W
U
'
NRx,m
= -a In a - (1 - a) In (1 - a) + hl
OX
+ H(g)
*I 7&OXlR
vg'vs
In the first step, the known values of S0(1/2H2) and of So(H) are used. The second step represents a compression term bringing the volume of H in the gas phase (V,) to its volume as un-ionized intercalated H or V,. The last step consists in transferring a partial negative charge e from H to the guest lattice, or to the transition metal cation Mz+. Thus, per mole of H
= 11.8 eu
+ R In (V,/V,) + A$?
(13)
where is the charge-transfer contribution. It will be shown later how the second term and third term can be estimated. In addition, we assume that the integral and differential "transfer" entropies are equal, or that hs' as"' = a(R)
(14) NRX,, Then, the total entropy change (both configurational and nonconfigurational) AS, is
--As, - -a In a - (1 - a ) In (1 - a ) + AZ+
a
XomNR and
.
.
- In a +In 1-a
[
('
1 + l(): ;
-
+ $ (16)
Then, by combining eq 16, 9, and 6 we obtain the isotherm equation
-h- - In A, = D'F
(8)
Therefore, from (7) and (8) the link between the differential and integral enthalpy is
E , - E,
-c
@oxS
(6)
The average energy per site, W,,/N, is the sum of three terms, corresponding to the fractionsX,, X , and X, to which the energies E,, E,, and E, per occupied, prevented, and vacant sites are assigned respectively: Wav/N= X J , + X&, X,E,, and . .
(12)
The calculation of the nonconfigurational entropy APr can be operated in three steps as suggested by the following thermodynamic cycle:
and where
lnA,=
1
(11)
The variation of the integral Gibbs free energy per intercalated guest is obtained in a similar way
AF, -A F-B --Xm
NXom
+ R q a In a + (1 - a) In (1 - a) - lu) - aTas"'
aAi??
(18)
Let us illustrate the use of the isotherm equation (17) by a few examples. Assume that the integral intercalation enthalpy Ai?? is known. After choosing the interstitial lattice, characterized by the cou le of parameters (and Xm,possible values for (E, - E,) and A '!, are introduced into eq 17 and the numerical results displayed as In A, vs x, that is, the isotherm at temperature T . Figure 1 shows two sets of simulated intercalation isotherms obtained in making hR = -12.5 kcal per mole of H and = -15.5 cal/(mol of H)/K for the three values of E, - E, shown in the caption (Chart I). Two interstitial lattices, hexagonal and fcc, were selected for this calculation and T was 333 K in this example.
1440 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992
Fripiat and Lin
CHART I
E, - E,,
interst. lattice
isotherm
kcal/(mol of -12.19 -14.19 -16.19
fcc fcc fcc
1
2 3
E , - E,, kcal/(mol of H) -0.057 +0.31 +0.68
H)
E -4,
interst. lattice hexagonal hexagonal hexagonal
isotherm 4 5 6
kcal/hnol of H) -0.19 ' +1.03 +2.26
TABLE I: Maximum O c ~ u p a t i o Probability ~l of an Interstitial Site (X,) and 1Parameter (Q 4) of the Studied Interstitial Lattices (See Table I, Ref 19)'
lattice linear (Z= 2) hexagonal (Z= 3) square (Z= 4) triangular (Z= 6) cubic (Z= 6)
bcc (Z= 8) fcc (Z= 12)
wo,, MOO^
Xo,
j-
0.436 0.380 0.363 0.232 0.308 0.300 0.156
1.546 1.839 2.279
1.808 2.658 3.424 2.218
wo3 MOO, v205 VMOQ.5 (x, = 0.45 0.5, ( x , = 1.65 0.05, (x, = 3.7 0.1, (x, = 2.8 0.05, V205 VMoO5.5 X M = 0.45) XM 1.70) XM -3.8) X M = -3.4) 4.8 4.4 4.2 3.8 4.0 3.6 2.6 2.3 3.4 3.0 3.4 3.0 1.7 1.6
Nx,,
*
+ + + + +
+ + + + + + +
2.6 2.3 2.2 1.4 1.8
1.8 0.9
+ + +
+ + + + +
a From the theoretical maximum hydrogen uptake x, = NX,, experimental x, (a = I), and maximum uptake xM,the possible interstitial lattices obeying the rule x, I X,,, are indicated by the symbol
+.
limit. Between a = 0 and some value of a close to 1 it will be shown in the next section that the simulation is quite satisfactory. Within these limits the following considerations apply. In the absence of interaction and, thus, of prevented sites, t would be zero and (17) would reduce to
4
-10
Eo - E, a AP = -+In--RT RT 1-CY R from which the following expression for a is obtained a = 1/(1 + exp(-A/RT))
-M,- - In A,
-12
-14
where
-16
.I8
(19)
I' 0
,
I
I
1
I
0.2
0.4
0.6
0.8
A 1
alpha Figure 1. Parameters used in the calculation of these hypothetical intercalation isotherms AI? = -12.5 kcal/(mol of H); 0= -15.5 cal deg-'/(mol of H). T = 333 K. (See Chart I.)
1, A step isotherm is obtained only within a narrow range of (E, - E,) and within this range, E, - E, is positive. If E , - E, is negative the intercalation isotherm is not stepwise. It looks more like a Halsey-Hill isotherm in the region 0.2 I a I 0.8. 2. For a given set of (E, - E,) and values, the number of lattices yielding step isotherms is always limited. By anticipating the application of these calculations to experimental isotherms, it is useful to keep in mind that the experimental error in determining In ABin the plateau region is about f0.1 logarithmic unit. This error is larger at small pressure (a 0) because of the instrumental limitation and at large pressure (a 1) because of the difficulty of determining accurately the maximum hydrogen uptake as illustrated later on. The presented theory must be considered as a first approximation, since all interstitial sites are assumed to be equivalent. Each of them is surrounded by the same 2 neighbors for a specified interstitial lattice. The segregation between the sites which remain vacant till X, X,, and those called prevented is the result of H-H interaction and this interaction is repulsive, since E, - E, is positive. It is clear that when H content becomes high the prevented sites must overlap in some more complex manner, not taken into account here. In addition, there is a limit to the applicability of the mean field concept underlying the previous development. At about that limit, (aAH/ax)must go to zero while the variations @(E,, - E,)/ax),, (d(E, - E , ) / ~ Xand ) ~(@/ax) assumed to be zero can no longer be neglected. Thus, it may be anticipated that the isotherm eq 17 will not simulate the experimental intercalation isotherm beyond some
p,
- (E, - E,)
+ TAP
Thus,,a is not an explicit function of the concentration in intercalated guests and the intercalation isotherm is not stepwise. When H-H interaction is present, t > 0 and
where
or
--
-
and where A(a) = pg - (E, - E,)
+ c[l - f ( a ) ] ( E , - E,) + T O '
a is now an explicit function of the concentration. In that case the intercalation isotherm may show a plateau or be stepwise, if E, - E, is positive and not too large. In addition, Eo, E,, and E, scale in the order E? > E, > E,. Locally, the intershtial lattice regions containing prevented and vacant sites are metastable, since they have a larger energy than those surrounding an occupied site.
Simulation of Experimental Intercalation Isotherms The experimental intercalation isotherm gives the variation of In A, = In P(H2)/P(H& with respect to x in HxdOX-xf. The simulated isotherm is obtained in using eq 17 and by choosing an interstitial lattice. Since AI? is obtained through calorimetry, the simulation yields E, - E, and The first step in fitting the experimental isotherm is in the calculation of u,which requires knowing x,. Consider the experimental variation of In A and of AH with respect to x obtained for H,VMOO~.~ shown in hgure 2. For the reasons explained before, it is clear that, in this case, the simulation of the experimental isotherm cannot be performed for x > 2.8, since the
or.
The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1441
Hydrogen Intercalation in Transition Metal Oxides TABLE 11: Experimental Calorimetric Enthalpy Shown in Figure 3 O
-M,b
T,K
wo3
kcal mol-' H 8.2523
MOO,
12.523
333
v2°5
14.623
343
VMQ.5
15.65
342
oxide
(AR), Adjustable Parameters ( E , - E"),and AS" Obtained for the Two Best-Fitting Isotherms -(Eo- E,),
kcal mol-' H 8.94 8.75 13.6 14.1 15.8 16.3 17.1 16.6
373
-0,
cal
E,
K-'mol-' H
- E,,
kcal mol-' H 0.09 0.30 0.46 0.86 0.68 1.04 0.85 0.45
8.5 1.2 16.0 16.8 24.2 24.2 21.4 26.6
diff 0.10 0.11 0.22 0.24 0.13 0.19 0.31 0.33
lattice fcc bCC
cubic square square hexagonal square cubic
' E . - E. is calculated using ea 15. "diff" is the average difference, in logarithmic units, between the experimental points and simulated isotherms. *Su&rscr{pts are reference numbers. I
I
I
I
I
I
4
,
4
4
8'
-
.10
.12
I .14
0
0.5
1
1.5
2
2.5
3
0.2
0.4
0.0
I 0.2
I
I
I
I
0.4
0.6
0.8
1
0.8
1
-12 3.5
X
Figure 2. Variation of enthalpy (AH, kcal) (open symbols) and of chemical potential (filled symbols) p,/RT = In A vs x in H,VMoO5,, for different sourca of atomic hydrogen. (0 and mj Pt particles; (A and A) Pd particles; (0)Pt particles deposited on q-A1203and mixed with VMOO~,~.*~ In the latter case the H spillover is indirect. In the former cases, it is direct. The vertical solid line at x = 2.8 corresponds to a = 1 or x / x , = 1. The linear regression AH 0: xAZ7 is shown by a solid line.
variation of In A, vs x has a saddle point at x = 2.8. Thus, in this case x, in a = x/x, is 2.8, although the maximum uptake xM is about 3.45. Thus, for 2.8 < x'l 3.45 the mean field approximation breaks down. By contrast, no saddle points were observable in the experimental intercalation isotherm obtained for H,W0,,2 H , M o O ~or ~ H,V205? Table I shows the value of x, observed (within f0.05 unit) for four H bronzes. The second step consists in the selection of the interstitial lattice. Irrespective of any crystallographic consideration, it is most likely that in transition metal oxides the intercalated H moieties must occupy potential wells neighboring 0" anions and that, at the best, the total number of interstitial sites per stoichiometricunit, N,cannot be larger than twice the number of oxygens. For example, the experimental x, should be smaller than 6X0,in HmMo03. The calculated NX, for the seven interstitial lattices are shown in Table I as well as the possible interstitial lattices which can be retained. It can be seen that, on the basis of the criterion x, < NX,,, all the seven lattices are possible for WO,, whereas at the other extreme, only three can be explored for HxV205a
AR and a being known experimentally, a simple computer program is used to fit the calculated isotherm, eq 17, to the experimental isotherm. The convergence is very rapid and the margin of uncertainty on E, - E, and A P is within the second decimal. Figure 3 shows the two best fitting isotherms for the four bronzes, whereas Table I1 contains the corresponding E, - E, and fitting parameters. It also gives the calorimetric AI? and the values of E - E, calculated from eq 10. No conclusion on the dimensionalty of the interstitial lattice can be reached. For instance, the square or the cubic lattices with dimensionality 2 and 3, respectively, are best fitting for H,Mo03, whereas crys-
ar
8'
-
-10
-12
0
alpha
Figure 3. Solid and interrupted lines: best fitting isotherms yielding the E,, - E, and shown in Table 11. Dots: experimental values. (Bottom) VMOO~,~: -, square lattice; - -,cubic lattice. MOO,: -, square lattice; - -,cubic lattice. (Top) W03: -, BCC lattice; -,fcc lattice. V,05: -, hexagonal lattice; -,square lattice. Dotted line: theoretical isotherm calculated using eqs 13, 17, 21, and 22 and the values of charge-transfer function @e shown in Table 111.
-
-
--
--
tallographic studies suggest that the interstitial sites should form a bidimensional lattice within the van der Waals gap in Moo3. It should be reemphasized that the choice of any interstitial lattice is no more than choosing neighboring relationships between occupied, prevented, or vacant sites. There are probably lattices, or graphs, with lower symmetry whose and X,, characteristic parameters could fit experimental isotherms. Figure 4A-D shows the variations of AW,, = am,of T W (Il), of TASt (15), and of AF, (18) with respect to a,calculated from the best fitting values in Table 11. The compensation effect of the entropy term on the enthalpy term is obvious. With respect to the intercation free enthalpy per mole of H (not shown), the order of stability scales as MOO, > VzO5 H VMOO5.5 > WO, The extent of H deintercalation under vacuum observed experimentally follows the opposite trend.2-3*5
Fripiat and Lin
1442 The Journal of Physical Chemistry, Vol. 96, No. 3, 1992 1
I
............................................................................. ...................... .
A
,
--.-.....
B
--
'
4
0
1
'
oa
1
0.1
01
1
.
OJ
11
O
0
0
.10
.IO
do
B
Ba do
do 40
a
do 0
0.8
0.
01
0
OA
01
01
04
a
I
O
Figure 4. Variation of AFB (---). TAS, AW,, (-), T W " ' ( - - - ) vs CY calculated in using the best fitting parameters E, - E, and the corresponding interstitial lattice in Table 11. (A) WO, (fcc); (B) MOO, (cubic); (C) V20s (square); (D) V M O O ~(square). ,~ (-e),
TABLE III: Lattice Parameters, Number of Stoichiometric UNts per Unit Cell (n/uc), Volume of the Stoichiometric Units (v), and Volume Occupied per Intercalated Mole of H (Vi)" sym* ref 0, A b, A c, A nluc v,A' V,, cm3 @e 0 28 7.501 3.824 7.274 4 52.16 0.1021 wo3 H0.45W03 T 2 3.82 3.75 3.75 1 53.72 2.09 Moo3 0 28 13.855 (9) 3.6964 (9) 3.9628 (8) 4 50.735 0.5524 D1.6SM003
M
29
13.986
3.78
VMOOSS H3.4VMoOs.s
0 0 0
30 30 28 4
19.549 20.497 11.519 (6)
3.635 3.74 3.564 (7)
v205 H3.6V20S
am
4.065 ( p = 93.99) 4.097 3.811 4.371 (2)
4
53.595
1.04
4 4 2
72.784 73.037 89.723
0.045
0.9807 0.9254
"The charge-transfer function @, is from eq 22. *Symmetry: 0, orthorhombic; T, tetragonal; M, monoclinic; am, amorphous.
The configurationalentropy does not contribute much to AF, except for imposing an upward curvature as a approaches 1. Actually, AF, should be minimum for a = 1 and at that point In A, and p, should be zero. In conclusion, this paragraph shows that the theory leads to a very acceptable fit between experimental and calculated isotherms in using the calorimetric enthalpy. The choice of the adjustable parameters E, - E, and is very restricted: departures by more than 0.2 kcal for E, - E, or 0.2 cal mol-' K-' for hsv lead to deviations which are larger than the experimental error in the measurement of In A, vs a. With respect to the first paper on the subject,19 the major improvement brought about by the present contribution is in the use of the experimental intercalation enthalpy in the calculation of the isotherm. This leads to an evaluation of the nonconfigurational entropy,
e'.
Physical Meaning of the Adjustable Parameters ASb and E , -E, In order to understand the physical meaning of hsv, we rewrite eq 13 obtained from the cycle in the Theory section. A&? =
- 11.8 eu - R In (V,/V,)
(13')
The problem is to find the relation between A&* and the extent of charge transfer. As a guideline we will examine some of the ideas put forth for the interpretation of the thermodynamic properties of metal hydrides. We have already stressed the analogy between hydrogen bronzes and metal hydrides in the Introduction. The comparison does not seem too far-fetched, since, for example, the experimental enthalpies for hydrogen bronzes are in the same
range of values of those observed for stable metal hydrides.24 Miedema2s*27 and Miedema et have studied in a systematic way the heats of formation of solid alloys and the results of this theory have been extended to the interpretation of the heats of formation of binary transition metal hydrides, in which hydrogen is considered as a metal forming an alloy with the transition Thus, the basic model is that suggested for alloys of transition metals for which the heat of formation can be accounted for by means of a cellular model. The Wigner-Seitz concept of atomic cells, extensively used in the description of pure metals, was shown to be still significant for the two types of atomic cells in a binary alloy, if (i) the discontinuity in the density of electrons at the boundary between dissimilar cells and if (ii) the charge transfer between cells, which equalizes the difference in electron chemical potentials, are accounted for. In the extension of the theory to transition metal hydrides, the heat of formation contains two terms proportional to the interface area between the cells that is to VH2f3. (24) Bouten, P. C. P.; Miedema, A. R. J. Less-Common Mer. 1980, 71, 147. (25) Miedema, A. R.J. Less-Common Met. 1973, 32, 117. (26) Miedema, A. R.; Boom, R.; DeBoer, F. R. J. Less-Common Met. 1975, 41, 283. (27) Miedema, A. R. J. Less-Common Met. 1976, 46, 67. (28) Donnay, J. D. H.; Donnay, G.; Cox, E.-G.; Kennard, 0.;King, M. V. Crystal Daza, Determinative Tables; American Crystallographic Association: 1963), ACA Monograph No. 5. File no.: WO3:O9697; MoO3:0286O VZO,: 037967-30940. (29) Anne, M.; Fruchart, D.; Derdour, S.;Tinet, D. J. Phys. Fr. 1988,49, 505. (30) Ancion, C.; Poncelet, G.; Fripiat, J. J. C. R. Acad. Sc., Paris Ser. 2 1983, 296, 1509.
The Journal of Physical Chemistry, Vol. 96, NO.3, 1992 1443
Hydrogen Intercalation in Transition Metal Oxides
4
Let us assume that A&? is a function of ( 6 / V s ) 2 / 3that , is, of the ratio of the areas of the intercalated hydrogen after electron In the first transfer, ( VJ2l3,and before electron transfer ( Vg)Z/3. approximation, we suggest that
AS,? a
[(VJV,)2/3- I ]
(21)
For crystalline bronzes, namely, H0.45W03, D1,65M~03, and H3,4VMO05,5, the lattice parameters are known, as shown in Table I11 and, therefore, can be calculated from the unit cell volume expansion and x,. Since is known from the simulation of the experimental isotherm, eq 13’ contains one unknown, the volume V,of the intercalated H, atom before ionization. V, should not vary appreciably with the nature of the bronze, whereas the extent of the charge transfer may be very different. Thus, if eq 13’ is equalized to the empirical eq 21
or
0‘-11.8 e u - R In
v,
= k[
(
;)’3-
.10
E
I -12
3
y .I4
11
one should obtain one value for V, in using the and values obtained for the three crystalline bronzes. This is, indeed, the case and V, = 2.33 cm3/(mol of H), the corresponding atomic radius being 0.973 A or 1.84 (ao = Bohr radius) for un-ionized H and the proportionality constant k being 22.314 cal/(mol of H)/K. With these values the agreement between the two members of the above equation is excellent. The extent of charge transfer to the lattice can, therefore, be represented by a function 9,described as
and AS,? could be as well equalized to 22.3 14 [( 1 - 9,)2/3 - 11 eu. From eq 21 we can determine by interpolation 9,for an amorphous bronze, such as H,V205. Values of are shown in Table 111. Thus, the chargetransfer function is relatively low for bronzes with metallic character (as H,WO, or H,MO03) and large for semiconducting bronzes such as H,V205 and H,VMOO~,~.A rough check of the meaning of 9,can be obtained for H1.65M003 in which the number offree electrons obtained from the variation of the static susceptibility upon H intercalation had been found to correspond with 0.27 e transferred from each intercalated hydrogen to the conduction band.I3 This charge transfer must be smaller than a, since a fraction of the charge is localized on MO(+~)+ centers. Thus, about ~ ‘ / ~electrons 9 , are, indeed, transferred to those centers from each intercalated guest in Hl,65M003.
Since 9,seems to account reasonably for the charge transfer entropy, we can now use it for determining the contribution of the charge transfer to the intercalation enthalpy aR. As we did for the nonconfigurational entropy, we consider again the following cycle
-J
P=O OX
+ 1/2H2(g)
R = 52.09 kcal
OX
+ H(g)
-14 .
0.2‘
0.4
alpha 2
0.0
0 0.8
1
~
Figure 6. (Open circles) Experimental data: intercalation isotherm of H2 within ZrNiH?I T = 373 K. a = 1 corresponds to ZrNiH2.9. The steep rise observed for 0 < CY < 0.31 is not shown. (Solid line) Simulated isotherm: triangular interstitial lattice; AI? = -9.7 kcal mol-l H, E, E, = -12.3 kcal mol-’ H, = -7.7 cal/(mol of H)/K.
kcal mol-’, whereas the proportionality constant in eq 24 is -9.3 kcal mol-’ H. The constancy of the Micontribution has to be tied up with the constancy of V, in the expression of the nonconfigurational entropy. Finally, it was observed that E, - E, can also be expressed as a function of
E, - E, = (-6.69
- 10.279.,2/3)kcal mol-’
(26)
with a standard error of 0.22 kcal mol-’. The recalculated hR and E, - E, are shown to agree very well with the corresponding “experimental” values in Figure 5. From (25) and (26), it results that E, E, increases with 9,C2/3, e.g., becoming more positive as the charge of the intercalated H increases. In conclusion, the two adjustable parameters E, - E , and used for fitting the simulated isotherm (17) to the experimental isotherms can be calculated in using eqs 21 and 13, and eq 26. In Figure 3 theoretical isotherms, recalculated in that way, are compared to the simulated isotherms. The discrepancies are within the experimental errors.
-
d i
ur
with
AI? = (52.09 + Mi+
I
I
kcal mol-’
(23)
and
aRela (24) Using the experimental values of AI? (Table 11) the following empirical equation is obtained aR = (-6.21 - 9.39.,2/3)kcal mol-’
(25) the standard error being 0.23 kcal mol-’. Thus, A& must be practically constant, irrespective of the bronze and equal to -58.3
General Conclusion The interesting aspect of the theoretical treatment presented here is that, in feeding the isotherm equation with the experimental enthalpy, the nonconfigurational entropy can be obtained from the fitting of the calculated to the experimental intercalation isotherm of hydrogen within transition metal oxides. The nonconfigurational contribution, which is much larger than the configurational entropy, seems to be related to a function (ae) describing the charge transfer from the hydrogen to the host lattice,
l
J . Phys. Chem. 1992,96, 1444-1448
1444
by a simple empirical equation (21). Using @e it has also been shown that the experimental intercalation enthalpy and the repulsion energy between intercalated hydrogen contain contributions of the charge transfer. The metallic character of the hydrogen bronzes seems to decrease as the charge transfer increases. Finally, the calculated intercalation free energies account well for the observed order of stability of these intercalation compounds.
Acknowledgment. J.J.F.thanks Professors R. Vanselow and W. Tysoe for stimulating discussions. Appendix
content. The very steep slope near composition Z T N ~ His ~re-, ~ sponsible for the relatively large value of E,, - E, = 0.78 kcal mol-’, the best fit being obtained for a “triangular” interstitial lattice. The simulations yield E, - E, as -12.3 kcal mol-’ H and AP = -7.7 eu. Again the interesting feature is @‘, since it may be related to the charge-transfer function according to the theory developed for the hydrogen bronzes. If a metal hydride may be considered as an alloy of metallic hydrogen (HM) and the other metal(^),^^ it seems logical in the calculation of the nonconfigurationalentropv 13 and 21) to -_( .a *s replace V, by VM, VMbeing the volume of metallic hydrogen; or VM = 1.85 cm3.32 Then
Since the similarity between transition metal hydrides and hydrogen bronzes has been outlined several times, it was interesting [(q/VM)2/3- 11 (27) to check if the theory developed here could be generalized to where is the molar volume of intercalated H, or 1.7 cm3 actransition metal hydrides. The system drconium-nickel-hydrogen cording to Libowitz et aL3’ Thus, must be proportional to has been chosen for that purpose, because it had been well -0.055, since @e = [ 1 - (Vi/ VM)] = 0.08 1. This is the partial characterized by Libowitz et al.31 In particular, intercalation electron transfer from intercalated H to the ZrNi lattice, metallic isotherms have been measured at several temperatures, in the range hydrogen being taken as reference. Assuming that the proporof stability of the hydrogen bronzes. These isotherms exhibit an extended plateau in the region going from Z r N 5 b 9 to Z T N ~ H ~ . ~ ; tionality coefficient in eq 21 is still valid in eq 27, it turns out that the intercalation enthalpy is about -9.2 kcal mol-’ H and the AP = -8.2 eu volume of the intercalated hydrogen in ZrNiH2.9is ca. 1.7 cm3. Compared with the value obtained from fitting the isotherm, We simulated the isotherm using hR = -9.7 kcal mol-’ H with namely, -7.7 eu, the agreement is good. the result shown by the best fitting curve in Figure 6. The first Thus, on the limited basis of this single example, it seems that point on the left (a = 0.31) corresponding to composition the framework of the theory presented here could be extended ZrNiH,,9 is at the end of the steep slope observed at lower H to transition metal hydrides.
-
(31) Libowitz, G. L.; Hayes, H. F.; Gibbs, Jr., T. R. P. J . Phys. Chem. 1958,62,16.
(32) Ross, M.; McMahar, A. K. Phys. Rev. B 1976, 13, 5154.
Observation of Mixed AI(O,N),, Structural Units by 27AI Magic Angle Spinnlng NMR Mark E.Smith Division of Materials Science and Technology, CSIRO, Locked Bag 33, Clayton, Victoria 3168, Australia (Received: August 12, 1991)
It is shown for the first time that the application of high-speed magic angle spinning (MAS) (up to 16 kHz) and high applied magnetic fields (up to 11.7 T) allow unequivocal identification of specific mixed aluminum local coordinations AlO,”, (y = 1,2, or 3) in a bulk solid using 27A1NMR spectroscopy. In combination with 29SiMAS NMR the structures of three silicon aluminum oxynitride ceramics (sialon), 15R and 21R AIN-polytypoid and 8’-sialon, are refined, confirming strict bonding preferences of Si-N and AI-0 in these materials.
Introduction
A number of silicon aluminum oxynitride ceramics (sialons) exist that have considerable practical application as components used in high-temperature engineering.’ Sialons are three-dimensional structures of mainly (Si,AI)O,N,, (0 Iy I4) tetrahedra with a few phases also having some octahedrally coordinated aluminum. Many of these compounds show significant compositional variations due to the interchangeability of silicon for aluminum and nitrogen for oxygen. The two examples studied here are the series of polytypoids based on AlN and the extensive solid solution of A1 and 0 in BSi3N4 to form j3’-sialon, Si3-~1,O,N4_, (0 Ix I2). Ordering of (Si,Al)O,N, (0 Iy I 4) tetrahedra is difficult to probe by conventional diffraction techniques, particularly X-ray diffraction since the pairs of elements silicon, aluminum and oxygen, nitrogen have very similar X-ray scattering factors. Magic angle spinning (MAS)2 NMR
is a powerful technique that has provided unique insights into the atomic level structures and chemistry of ceramic phases, particularly when complex phase mixtures of crystalline and amorphous components are present. The technique can distinguish local structural units in solids by their isotropic chemical shift and has been used to identify SiO,,N, (0 I yI 4) and AN4, NOs,NO4, and A106 coordinations in ceramic^.^-'^ The present work for (3) Butler, N. D.; Dupree, R.; Lewis, M. H. J . Muter. Scl. Lett. 1984,3, 469. (4) Dupree, R.; Lewis, M. H.; Leng-Ward, G.; Williams, D. S.J . Mater. Sci. Lett. 1985, 4, 393. (5) Klinowski, J.; Thomas, J. M.; Thompson, D. P.; Korgul, P.; Jack, K. H.; Fyfe, C. A.; Gobbi, G. C. Polyhedron 1984.3, 1767. (6) Marshall, G. L.; Harris, R. K.; A p p l y , D.; Yeung, R. Proc. Symp. Sci. Ceram. 1987, 14, 341. (7) Faman, I. E.; Dupree, R.; Forty, A. J.; Jeong, Y. S.;Thompson,G. E.; Wood, G. C. Philos. Mag. Lett. 1989, 4, 189. (8) Dupree, R.; Lewis, M. H.; Smith, M. E. J . Appl. Crystallogr. 1988, 7 1_ in9 _ _, - _ .
(1) Jack, K. H. J . Mater. Sci. 1976, 11, 1135. (2) Andrew, E. R. Int. Rev. Phys. Chem. 1981, I , 195.
0022-3654/92/2096-1444%03.00/0
(9) Dupree, R.; Lewis, M. H.; Smith, M. E. J . Am. Chem. Soc. 1989, I l l , 5125.
0 1992 American Chemical Society
