Static relaxations of puckering distortions in intercalated layered solids

Journal of Non-Crystalline Solids 131-133 (1991) 1213-1217 North-Holland

1213

Static relaxations of puckering distortions in intercalated layered solids S.A. Solin NEC Research Institute, 4 Independence Way, Princeton, NJ 08540, USA

H. Miyazaki 1, S. Lee 2 a n d S.D. M a h a n t i Center for Fundamental Materials Research and Department of Physics and Astronomy, Michigan State University, East Lansing, M1 48824-1116, USA

Layered solids have been qualitatively characterized according to their transverse layer rigidity with respect to point-like puckering distortions. In this scheme graphite is a 'floppy' layer system, layered aluminosilicates contain 'rigid' layers and layer dichalcogenides fall between these extremes. The primary signature of layer rigidity is the dependence of the basal spacing, d(x), on composition, x, in a ternary system A 1_xBxL where L represents the host layers while A and B are distinct guest species with radii rA < r n. Previous attempts to quantify the layer rigidity (i.e. account for d(x)) using models which employed infinitely rigid layers have been singularly unsuccessful. In the present paper a one-parameter finite layer rigidity model, which yields an extremely good fit to the composition dependent basal spacing of a wide variety of layered solids which includes the floppy and rigid extremes, is described. In this model, a healing length is introduced to quantify the static spatial relaxation of a point-like puckering distortion. Both a discrete and continuous version of the model are presented and shown to yield equivalent results. It is found that, as expected, the more rigid the host layer, the larger the healing length, the values of which are deduced for several important layered solids.

1. Introduction It has recently been shown [1,2] that layered solids can be classified qualitatively according to their transverse layer rigidity with respect to distortions that are perpendicular to the layer planes. Thus, graphite [3] and boron nitride constitute class I compounds whose mono-atomically thin layers are relatively 'floppy' with respect to transverse distortions. The class II group consists of layer dichalcogenides, iron oxychloride and other similar materials whose layer units are composed of three layers of interconnected atoms. This intralayer interconnectivity endows the class II materials with a higher degree of transverse rigidPermanent address: Department of Applied Physics, Tohoku

University, Sendai, 980, Japan 2 Current address: Department of Physics, Harvard University, Cambridge, MA, USA

ity than those of class I. Finally, compounds such as layered ahiminosilicate clays whose relatively 'rigid' host units consist of highly interconnected networks of seven or more atomic layers constitute the class III compounds. It is, of course, desirable to quantify layer rigidity in terms of quantities that cannot only be readily measured but also employed as a test of microscopic models. To this end Solin and coworkers have shown [4,5] that valuable rigidity information can be extracted from a measurement of the normalized basal spacing, dn(x ), of stage-1 ternary intercalated layered compounds [6] A~_xBxL, where A and B represent distinct guest species, L represents the layered solid whose rigidity is of interest and x is the gallery composition, i.e. the fraction of B ions which occupy non-defect sites that contribute to the expansion of the gallery between host layers. (A stage-n intercalation compound is one in which n host layers separate

0022-3093/91/$03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland)

1214

/ Puckering distortions in intercalated layered solids

S.A. Solin et al.

. . . . ".-7-.•

/,,,,,,,-""" •• ~ . " ~ •° • •

/" /

/

t

/

J

.~ I ~ ; ; ; ;

/

!I

/ !

t tt t~ ! t ,

t

• .• • ° •

; t" ;

.

I

°

."

~

••" . . . . . .

d.(x) =x/[(1

; ; " "

x Fig. 1. A schematic representation of the possible functional forms of the composition dependence of the basal spacing of an intercalated layered solid. These forms are labeled as follows: . . . . , superlinear; . . . . . , sigmoidal; - - , Vegard's law; . . . . . , sub-linear.

nearest pairs of guest layers. Thus every gallery in a stage-1 compound contains a layer of guest species.) The normalized basal spacing is given by dn(x) - ~

Z d--~'

- x)a + x]

(2)

l°

•" 00

formed from vermiculite ( = V), a prototypical layered aluminosilicate clay [9]. Several authors have attempted to account for the superlinear behavior illustrated in fig. 2 [10-12] by employing rigid layer models in which the host layers are treated as non-deformable sheets coupied by pairs of springs of sp ,ring constant k and K representing, respectively, the guest-host and the vacancy or host-host interactions. The composition dependence of the normalized basal spacing calculated using the rigid layer model is [11]

where a is the ratio of the spring constants. A1though this form satisfies the boundary conditions d(0) 0, and d(1) 1 and yields Vegard's law when a = 1, it clearly fails to fit the data shown in fig. 2 a s c a n be seen from the dotted lines which correspond to various values of et. This failure is especially significant in the c a s e o f t h e c l a s s I I I compound f o r w h i c h the rigid layer m o d e l s h o u l d be most appropriate [5] Solin and co-workers [4,5] reasoned that layer deformations played an essential role in the de-

:

(1) 0.8

where d ( x ) is the c-axis repeat distance of the alloy and d(1) and d(O) are the repeat distances of the pure end-member compounds. The possible forms of d ~ ( x ) are illustrated in fig. 1. These include the linear form which is associated with Vegard's law, a superlinear form, a sub-linear form and a sigmoidal form. To our knowledge there is to date no layered solid which exhibits the sub-linear form while there is only one reported example of Vegard's law behavior and that result is controversial [7,8]. By contrast, the majority of ternary layered intercalation compounds exhibit the superlinear form which encom' passes the behavior of key compounds from the three classes of host structure discussed above, This is illustrated in fig. 2 which shows the n o r m a l i z e d b a s a l spacing o f v 1 _~Li~C6 (v = vacancy), a class I compound, vI _xLi~TiS2, a class II compound and Csa_xRbxV, a class III compound

~ 06 ~ 0.4

"

~

--

o.2 00

I ...... 0

0.2

0.4

i ...... ×

0.6

0.8

]

1

Fig. 2. The normalized basal spacing versus expanding site gallery composition for Rb I _xCsxVm (12]) [41, vI _xLixTiS2 ( o ) [4] and v1_xLixC 6 ((>) [12]. The solid lines are fits to the data using the finite layer rigidity model (see text) and yield the rigidity parameters, P, given in table 1. The dotted lines are fits to the data using the rigid layer model (see text) with a = 0.5, 0.2, 0.1, and 0.05, top to bottom, respectively. The straight dash-dotted line represents Vegard's law and corre-

spondstoP=lora=l.

S.A. Solin et al. / Puckering distortions in intercalated layered solids

1215

Table 1 Important parameters for several ternary layered intercalation compounds Stage and structure

Class

Sample

Rigidity parameter, V

Site ratio, fl

Intersite distance, a0 (A)

Stage-1 Triangular

1 II 111

Val -~ LixC6 Va1_ ~LixTiS2 Rb I _ xCs~Vm

2 3.5 7

3 1 1

2,46 3.41 5.34

Healing lengths (,~) ~d 3.16 3.35 7.42

~c 2.45 2.60 5.77

termination of the normalized basal spacing. They introduced a finite layer rigidity model in which the relaxations of the static distortions of the bounding layer, which are caused by the introduction of the larger guest (the B ion in A l_xBx L) into a gallery region occupied by smaller guest ions, are treated as discrete entities or as a continuum. In the discrete version of the finite layer rigidity model, the above-described distortion takes the form of a pillbox of height, h a, at the site of the B ion. The A ions in the catchment area around B will have height, h B. It can be shown from a simple statistical analysis [13,4] that the basal spacing in this case is given by

Although the parameter P is useful to quantify rigidity, it is more appropriate to define a healing length, A, which is a measure of the lateral range over which a point-like puckering distortion heals. Lee et al. [5] showed that the healing length could be computed from both a discrete and a continuum approach. In what follows, we briefly review their initial results and then introduce additional support for the finite layer rigidity model by showing explicitly how the continuum healing length can be computed from elastic stiffness constants or from phonon dispersion curves. We also show that healing lengths computed in this way are in reasonable agreement with those deduced

h=hA(1-x)

using the rigidity parameter.

e +ha[1-(1-x)e],

(3)

where P is a rigidity parameter that depends on the number of A sites surrounding the B ion whose gallery heights are raised to h a as a result of the finite (discrete) relaxation length of the point-like puckering distortion caused by B. Then, using eq. (3) to compute the normalized basal spacing from the discrete version of the finite layer rigidity model, we find

2. Results 2.1. The discrete finite layer rigidity model The healing length, Xd, can be simply estimated from the ratio of the puckered area near a large impurity to the effective area per site. Thus P = ~X~/~A o,

do(x)

h (ha x ) _ - hhA, ,

--

1 - (1 - x ) e.

(4)

The solid lines shown in fig. 2 are linear leastsquares fits to the normalized basal spacing data using eq. (4). As can be seen, the fits are very good and indicate that a model which includes distortions can successfully account for the composition dependence of the basal spacing of all three classes of layered solids. Also note from table 1 that the rigidity parameters derived from the finite layer rigidity model increase with class number as expected,

(5)

where A 0 is the area per host lattice site and r , the site ratio, is the number of host lattice sites per guest superlattice site. (Note that the guest species in the compounds from all three classes addressed in fig. 2 and table 1 form triangular superlattices so that the inter-class comparison discussed here is valid.) If a 0 is the intersite distance in the host lattice, then A 0 = (v~-/2)a02 and [Pfl/3.6311/2ao . (6) The discrete healing lengths computed from the measured rigidity parameters are listed in table 1. Xd =

S.A. Solin et al. / Puckering distortions in intercalated layered solids

1216

2.2. The continuum f i n i t e layer rigidity model

1.5

We now treat the host layers as an elastic continuum in the form of a deformable plate of finite thickness. For a single point-like B ion at r0 in a gallery of vacancies or A ions, the gallery height, W ( r - to), is obtained from [14]

{DWn-KW2

+G}W(r-ro)=for(r-ro),

(7)

= 2[(C?,

-

C72)H31/3C,,,

r- . . . . .

~

T~7

, , , ,

t 1.4

1.3 N

where D is the flexural rigidity, K is the transverse rigidity, G is the c-axis compressibility, and f0 represents the g-function-like force from the B ion. Each of the coefficients in the brackets in eq. (7) can be expressed in terms of the effective thickness of the host layers, 2H, the basal spacing, d, and the host elastic constants, cij [14]. (Note, the thickness 2 H is equal to the basal spacing in the pristine or unintercalated form of the host system.) Thus

, , - t ~

1.2

1.1

ro Fig.

.....

[. . . . . . . ] . . . . 0.1 02

3. A n u m e r i c a l

8=

solution

d

[ .... 0.3

for the function

J

I ..... 0.4 0.5 Z(8)

where

K / J 2 D~/ffG] (see text).

(8)

r = 2HC•,

(9)

and

(10)

G = C33/d. Equation (7) can be solved to give

fo ~oo qjo(qp/lo) dq, W ( p ) - 2~r D~/-D-G-o q4 + 28q2 + 1

(11)

For a single defect, one can define the continuum healing length, ~,¢, as the distance at which the gallery height relaxes to half its maximum value, i.e. W()~¢)= ½[W(0)]. F r o m eq. (11) we obtain [14] A~(8) = Z ( 8 ) l o where Z ( 8 ) is a slowly decreasing function of 8 as can be seen from the numerical solution shown in fig. 3. One can obtain he from 8 and 10 through the dependence of the latter two on D, K, and G. For graphite K - 0 and D and G can be obtained from the known stiffness constants. Thus with

)~ = Z ( 8 ) l o = l . 3 0 2 ( D / 2 G ) 1/4 [ t

C2 C2 ]1/4 _~__~12 "'11"'33

J

in-plane T A dispersion of layered solids given by ~02(q) = Dq 4 + g q 2 + G.

where p = I r - r0 I, lo = ( D / G ) 1/4 gives the length scale and 8 = K/[2(GD) 1/2] < 1 gives the relative strength of the transverse rigidity,

= 1.302](1/3)H3d

and d = 3.35 A, 2 H = 1.0 .~, while Cll, C12 and C33 = 106, 18, and 3.65 × 1011 dyon/cm2, respectively [15], we find that ~,¢ = 1.83 A. The values of D, K, and G can also be determined from the

(12)

(13)

For graphite K, G << D so the T A dispersion [16] yields D directly and it is found to be - 9 . 9 × 10 -13 dyn cm which together with the value of G computed from the C33 stiffness constant yields Ac - 1.9 ,~. This value compares very favorably to the value computed from the stiffness constants alone. Moreover, both of these values are also in reasonable agreement with the value 2.46 .~ obtained from the experimental value of P. Note that h c for a defect of finite size should be somewhat larger than that for a point object. For most layered materials, neither the elastic constants nor the phonon dispersions are available. Therefore, it is useful to derive a relationship between ~,~ and P by extending the continuum theory to the case of a dilute distribution of B ions. Using the superposition principle, we can obtain the average gallery height, ( W ) , as a function of the concentration x. The result for ( W )

S.A. Solin et al. / Puckering distortions in intercalated layered solids

normalized by the maximum gallery height, = 0), due to a single B ion yields

W(p

d,(x) = flg(8)[hc(6)/ao] x,

(14)

where

g(6)

is given by 16 ( 1 - 82) 1/2 g ( 6 ) = /~_ 1 (2/qr) tan_ 1[ 8 / ( 1 -

62)1/2] . (15)

By comparing eq. (14) with eq. (4), we have ~c(6) =

[ ~P a l '/z a 0

(16)

3. Discussion Not surprisingly, eq. (16) is identical to within a constant factor to eq. (6), the only difference being the replacement of the factor 3.63 in the bracketed denominator with the factor g(6). These factors are geometric. The former results from the step-function-like decay of the puckering distortion at the pillbox boundary in the discrete finite layer rigidity model, whereas the latter results from the gradual decay over the range of several healing lengths in the continuum model. The values of ~c(8) obtained for stage-1 ternary layered intercalation compounds compare reasonably well with those ~'d (see table 1). Here the value of 6 is assumed to be the same as for graphite in the cases of TiS 2 and vermiculite. The short healing length found for graphite is consistent with the results of X-ray studies of disordered alkali intercalation compounds [17] which indicate that the host potential is strongly modulated in the vicinity of a guest ion. 4. Conclusions We have shown that the composition dependence of the basal spacing of layered intercalation

1217

compounds can be quantitatively accounted for with the finite layer rigidity model. Values of the rigidity parameter and of the continuum and discrete healing lengths have been deduced. The latter have been obtained from the elastic stiffness constants which are macroscopic parameters. These

however depend on microscopic quantities such as the pair potentials that determine the guest-host and guest-guest interactions. Thus it is these microscopic quantities that ultimately govern the basal spacing response. The authors gratefully acknowledge useful discussions with M.F. Thorpe and T.J. Pinnavaia. This work was supported by the N S F and in part by the MSU CFMR.

References [1] S.A. Solin, in: Intercalation in Layered Materials, ed. M.S. Dresselhaus (Plenum, New York, 1986) p. 291. [2] S.A. Solin, J. Molec. Catal. 27 (1984) 293. [31 S.A. Solin, Adv. Chem. Phys. 49 (1982) 455. [4] H. Kim, W. Jin, S. Lee, P. Zhou, T.J. Pinnavaia, S.D. Mahanti and S.A. Solin, Phys. Rev. Lett. 60 (1988) 2168. [5] S. Lee, H. Miyazaki, S.D. Mahanti and S.A. Solin, Phys.

Rev. Lett. 62 (1989) 3066. [61 J.E. Fischer and H.J. Kim, Phys. Rev. B35 (1987) 3295. [7] D. Medjahed, R. Merlin and R. Clarke, Phys. Rev. B36

(1987) 9345. [81 P. Chow and H. Zabel, Phys. Rev. B35 (1988) 3363.

[9] R.E. Grim, Clay Mineralogy, 2nd Ed. (McGraw-Hill, New

York, 1968). [101 J.R. Dahn, D.C. Dahn and R.R. Haering, Solid State Commun. 42 (1982) 179. [11] S.A. Safran, in: Solid State Physics, eds. D. Turnbull and H. Ehrenreich (Academic Press, New York, 1987). [12] S.A. Solin and H. Zabel, Adv. Phys. 37 (1988) 87. [13] M.F. Thorpe, Phys. Rex,., in press. [14] H. Miyazaki, S. Lee, S.D. Mahanti and S.A. Solin, to be published. [15] A.W. Moore, private communication. [16] R.N. Nicklow, N. Wakabayashi and H.G. Smith, Phys. Rev. B5 (1972) 4951. [17] S.C. Moss, G. Reiter, J.L. Robertson, C. Thompson, J.D. Fan and K. Ohshima, Phys. Rev. Lett 57 (1986) 3191.