- Email: [email protected]
The structural characteristics of amorphous D2O ice by neutron diffraction
Journal of Non-Crystalline Solids 53 (1982) 247-265 North-Holland Publishing Company
247
T H E S T R U C T U R A L C H A R A C T E R I S T I C S OF A M O R P H O U S D20 ICE BY N E U T R O N D I F F R A C T I O N M.R. C H O W D H U R Y
* and J.C. D O R E
Physics Laboratory, University of Kent, Canterbury CT2 7NR, UK
J.T. W E N Z E L National Bureau of Standards, Washington DC, USA **
Received 13 May 1982 Neutron diffraction measurements have been made at several wavelengths for vapor-deposited D20 ice at 10 K. The observed structure factor exhibits more oscillatory features than corresponding measurements on liquid D20 water, indicating an amorphous phase which has strong intermolecular spatial correlations. A detailed examination of the data shows that this is strongly influenced by hydrogen-bonding effects and that the local environment for each molecule is very similar to that in crystalline ice. The predictions of Boutron and Alben, based on a continuous random network model, are found to be in excellent agreement with the data.
1. Introduction It has been k n o w n for some time [1] that water vapor c o n d e n s e d o n t o a substrate plate held at a sufficiently low temperature forms a n a m o r p h o u s ice deposit. T h e properties of this material have recently been the subject of an extended investigation b y Rice a n d collaborators. Structural studies involving b o t h n e u t r o n a n d X-ray diffraction have been reported in a d d i t i o n to a series of spectroscopic experiments such as R a m a n a n d infrared a b s o r p t i o n [2]. The m a i n purpose of this work has been to establish a detailed picture of the h y d r o g e n - b o n d i n g effects in a m o r p h o u s ice so that these can be used as a starting p o i n t for the description of the more complex interactions in liquid water. This presupposes a c o n t i n u i t y of state between a m o r p h o u s ice a n d supercooled water b u t this has been challenged [3] on the basis of thermodyn a m i c properties. A l t h o u g h a n u m b e r of early diffraction studies [4] have provided i m p o r t a n t X-ray data, the most thorough investigation is that of N a r t e n et al. [5]. The observed intensity d i s t r i b u t i o n was f o u n d to be d e p e n d e n t o n both the substrate temperature a n d the vapor deposition rate so that two forms (high * Attached to Materials Physics Division, Building 436, Atomic Energy Research Establishment, Harwell, Didcot, Oxon, OXI 10RA, UK. ** Now at Saint Gobain Recherche, 93304 Aubervilliers, France 0 0 2 2 - 3 0 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
M.R. Chowdhury et al. / Structure of amorphous D20 ice
248
and low density) of amorphous ice were produced. Subsequent investigations [6] were unable to reproduce the high-density form and it is now assumed that this requires special deposition conditions which have not been fully defined. Neutron diffraction measurements by Wenzel et al. [7] have been obtained for the low-density form of amorphous D20 ice. The data are sensitive to the positions of the deuterium atoms so that information on orientation of adjacent molecules can be studied. The two sets of diffraction data have been discussed in [5] and also compared with the continuous random network model of Boutron and Alben [8]. Due to the interest in the structural characteristics of amorphous ice and its possible relation to liquid water, we have made further neutron measurements on a D20 and extended the range to higher Q-values. The increased range of Q enables the short-range order to be studied in closer detail due to the better real-space resolution obtained from Fourier transformation of the diffraction data. Furthermore, it is possible to use the form-factor description conventionally adopted for the analysis of diffraction studies for molecular liquids and to extend the method to the cluster structure within the amorphous network. The new results and different analysis procedure therefore provide a critical test of the continuous random network (CRN) model proposed by Boutron and Alben [8]. Amorphous ice is also of more general interest as it represents a convenient link between the chemically-bonded network glasses such as amorphous germanium [9] or silica [10] and the non-bonded molecular glasses such as amorphous CC14 [11].
2. Theory The theory of neutron scattering by molecular liquids [12] and amorphous solids [13] has been reviewed previously and only a brief summary will be presented here. We adopt the formalism normally used for the analysis of diffraction data for molecular liquids and express the structure factor S,,(Q) as the sum of two terms,
Sm(Q)= Dm(Q) + f~(Q ) (1) where f~(Q) is the molecular form-factor corresponding to the scattering from isolated molecules (i.e. intramolecular contributions) and Dm(Q) represents the structural relationship between the molecular units (i.e. intermolecular contributions). The neutron form-factor for D20 is: 1
f,(Q) - (bo + 2bD)z. b• + 2b~ + 4bobDJo(QroD ) ×exp ~
Y°DQ2)'~2bZojo(QroD) exp~--YDD&~],
(2)
where bo and b D are the coherent scattering lengths for oxygen and deuterium respectively; jo(x) = (sin x ) / x is the spherical Bessel function of order zero and
M.R. Chowdhury et al. / Structure of amorphous DzO ice
249
rOD and rDD are the internal O D and D D distances within the molecule. The exponential terms in "~ODand ~DD correspond to Debye-Waller factors which account for thermal vibrations with ), = ½(u 2) where (u 2) is the mean square amplitude of vibration. The Dm(Q) function is mainly confined to low and intermediate Q-values so that the observed diffraction pattern at sufficiently high Q-values is characterised by the oscillatory nature of the ft(Q) form-factor. In the case of liquids this enables the structure of the basic unit to be very accurately established but the strong spatial correlations between adjacent units in the molecular glasses does not permit such an easy separation of the two contributions. In this case it is convenient to extend the form-factor to that of a cluster arrangement and this will be more fully developed in the analysis section. Information on the intermolecular structure may be obtained from the transform relation
d(r)=47roMr[ g(r)-- l]
2 o¢ = ~ f 0 QDM(Q)
sin QrOQ'
(3)
where PM is the molecular number density and g(r) is the composite atom pair-correlation function. The function g(r) is a weighted sum of partial correlation functions, and for neutron scattering by D20 may be written as: g ( r ) = 0.092 goo ( r ) + 0.486god ( r ) + 0.422 god ( r ) .
(4)
It is clear that the O - O contribution is relatively small and that the weighted function contains O - D and D - D contributions in nearly equal proportions. A complete determination of the partial functions requires three independent experiments with different weighting factors. This data can be obtained either by neutron diffraction using isotopic substitution [14], or by combining results from measurements using X-ray, neutrons an electrons [15]. Both the techniques have been used in an attempt to derive the partial g(r) functions for the liquid phase [14,15] and a paper on some recent neutron measurements is in preparation [16].
g(r)
3. Experimental procedure and data reduction
3.1. Sample preparation The cryostat used in the present experiment was the same as that used in the neutron studies of vapor-deposited a m m o n i a - N D 3 [17] and amorphous carbon tetrachloride I11]. The procedure was similar to that of Wenzel et al. [7], who used a l0 m m wide cadmium strip of 2 m m thickness as the substrate plate which was attached to the bottom of a helium cryostat and maintained at l0 K or lower. T h e liquid D 2 0 was held in an outer container and degassed by successive f r e e z e - p u m p - t h a w cycles. The ice sample was grown by vapor deposition over several days with a deposition rate restricted to several mg per
250
M.R. Chowdhury et al. / Structure of amorphous D20 ice
hour. The high absorption cross-section of cadmium offers the advantage of a very low substrate background but also restricts measurements to a reflection geometry. For measurements at small scattering angles the plate must be set at an oblique angle to the incident neutron beam and consequently the data are subject to large-shielding and absorption corrections, expecially when the uniformity of the sample cannot be accurately established. To overcome these difficulties in the present investigation two samples were studied using different substrate materials so that complementary measurements could be made on two separate diffractometers. 3.2. Measurements at 1.37 ,~
Neutron measurements up to Q-values of 7.3 A~ were made on the C U R R A N diffractometer at the D I D O reactor, AERE, Harwell. For these measurements a vanadium substrate plate of 40 m m width and 2 m m thickness was used. The scattering from vanadium is almost entirely incoherent and gives a constant contribution to the observed intensity. The incident wavelength was calibrated by nickel powder measurements and found to be 1.368 + 0.003 ,~. The scattered neutron distribution was recorded using transmission geometry for the range 2 < O < 85 ° and reflection geometry for the range 80 < 0 < 100 °. Three sets of diffraction patterns corresponding to three different angular settings of the substrate plate were collected. After the measurements on the sample were completed, the cryostat was allowed to warm up slowly until the deposit became detached from the plate. Measurements were then made on the scattering from the bare substrate plate. 3.2.1. Measurements at 0.7 and 0.5 fl. The high Q-value measurements were made on the D4 diffractometer [18] at the Institut Laue-Langevin, Grenoble, using two incident wavelengths. For these measurements the sample was deposited on a cadmium substrate plate of 40 m m width and 2 mm thickness. Data collection was made only in reflection geometry over the angular range 50-130 ° using three different settings of the plate angle. Several repeat scans of the detector were made and showed no variation in the intensity distribution. The data from the multidetector were combined in the usual manner to give an intensity distribution function for both wavelengths. The incident wavelengths were determined by nickel powder calibration and were found to be 0.696 + 0.002 A and 0.504 + 0.002 A. 3.3. Data reduction
The three sets of observed intensity distributions were corrected for scattering from the substrate plate and the cryostat head shields. Corrections for self-shielding, absorption and multiple scattering were applied in the usual manner and the resulting intensity distributions for different angular regions were combined to give a composite pattern for each of the three datasets. The
M.R. Chowdhury et al. / Structure of amorphous D20 ice
final results are shown in fig. 1 as a function of the elastic scattering vector, given by
251
Q,
Q = (4~r/),) s i n ( 0 / 2 ) . The intensity distributions cannot be placed on an absolute cross-section scale because the thicknesses of the sample deposits are not accurately known. Experimental corrections were applied assuming a uniformly thick sample deposit. An approximate normalization may be obtained by extrapolation of high-angle data to lower angles where the effects due to Placzek corrections
3 2
>,
0
>.,,
3
I
I
I
I
0.7A o
2 _c 1 I
I
I
I
5
10
15
20
÷+
1.37 A°
2 __>" u~
*
0
I
I
I
I
2
z.
6
8
O (Ao-1)
Fig. 1. The individual measurements of the diffraction pattern for neutron scattered from amorphous D20 ice; see the text for details.
M.R. Chowdhury et al. / Structure of amorphous D20 ice
252
[ 12] are much lower. This provides a convenient relative normalization of the data sets and confirms that each sample corresponded to approximately 10% scattering of the incident beam.
4. Analysis
4.1. The structure factor and single molecule form-factor The conversion of the corrected differential scattering cross-section do(0)/d~2 to the required structure factor Sm(Q) is not simple due to the effect of inelastic scattering. The most convenient approach based on an extension of the Placzek correction procedure has already been presented for liquid D20 [ 19] and a more sophisticated treatment has recently been made by Powles [20]. These results show that the fall-off in the region of high scattering angles is primarily due to the small mass of the deuterium atom and a shift in the oscillatory structure of the interference pattern arises from molecular recoil in the scattering process. It is inappropriate to present a detailed discussion of the various methods which may be used to correct for inelasticity effects particularly as there are certain simplifying features which apply in the case of amorphous ice. A full treatment of corrections applying to network systems has not yet been developed although the framework for such a theory has been presented [ 13,21 ]. We may write the observed cross-section per molecule in two terms corresponding to self and interference (distinct) scattering contributions
ao,0 =( do
do d ~ (0))se,f +
The Placzek expansion for D20 shows that the cross-section for self-scattering may be expressed as l
where a a is a detector constant and (Me,/ran) is the effective mass of scattering centers relative to the neutron mass. This produces the Q2-dependent fall-off in the overall level of the scattering intensity but overestimates the reduction for backward angle scattering. The factors which influence the M~fr parameter are strongly dependent on the motion of the deuterium atoms. In amorphous ice the temperature of the sample is low so that thermal excitation is negligible and the dynamics is entirely governed by zero-point motion. Under these conditions M~fr is expected to be about 4 a.m.u, as discussed by Wenzel et al. [7]. An accurate estimate of Meff cannot be made as the observed fall-off is dependent on the precise form of self-shielding corrections for the various geometries used in obtaining the full angular distribution. In certain respects, this difficulty does not pose serious problems for evaluating the
M.R. Chowdhury et al. / Structure of amorphouz" D20 ice
253
interference contributions since it is oscillatory shape which relates to the structural information and errors in the smooth monotonically-decreasing self-scattering function which do not introduce any spurious high-frequency components into the pattern. The interference contributions arising from distinct scattering are much simpler that for the liquid state since the molecules are presumably bound within the network. This means that recoil corrections are expected to be negligible and there is no need to separate the intra- and intermolecular components for different treatment. It follows that the intermolecular contribution can be treated within the static approximation so that
(b ° + 2bD)2D~(Q) and therefore the Dm(Q) function can be readily evaluated from the observed cross-section. The Q-ranges for the 0.7 and 0.5 A datasets were 7.8-15 A and 10.6-22.5 A. The largest Q-value of the 1.37 A set being 7.3 A l, there was a short gap between the largest Q-value of this set and the smallest Q-value of the 0.7 A set, and therefore in order to construct a composite dataset from the present measurements it was necessary to include data from the previous measurement [7]. Instead of using sections from four different datasets - three from the present measurements and one from Wenzel et al. [7], it was decided to combine the 1.37 ,~ and the 0.5 ,~ measurements with the missing section taken from the measurements of Wenzel et al. The final composite structure factor is shown in fig. 2. To investigate the structure of the D20 molecule in the amorphous phase the form factor f l ( Q ) (eq. (3)) was fitted to the large-Q region (8-16 ,~-1) of 1"2
-~
1"0
x,
~
J
I
[
~
r
F
I
I
[
I
I
I
I
S~(O)
1"8
0"6
0
0"2
0
4
I
8
I
Q ( A o4 )
12
16
20
Fig. 2. The composite structure factor, SIn(Q) for a-D20 and the fitted form-factor fl(Q) for a single molecule with parameters given in table 1.
254
M.R. Chowdhury et a L / Structure of amorphous D,O ice
Table 1 Parameter values obtained from the fl(Q) single molecule fit to the observed structure factor, SIn(Q) of amorphous D20 ice Parameter
Value
rod
0.998 + 0.003 1.578+ 0.005 (0.28 _O.03)xlO 2 (1.8 _+1.2)×10-1
rDD " YoD YDt~
,~2 ~2
Compiled for a DOD angle of 104.5°.
the Sm(Q) data using a nonlinear least-squares routine. Although the measured data has much more structure than the molecular form-factor the additional oscillations must be superimposed on the basic shape of the single molecule form-factor so the fitting routine effectively averages over these higherfrequency components to give approximate values of the roD and rDD parameters. The fit was found to be sensitive to the value of rOD, but relatively insensitive to the rDo value. This is because the intramolecular D - D distance ( - 1.6 ~,) is close to the intermolecular hydrogen-bond distance ( - 1.7 A). In the final analysis therefore the rDD parameter was constrained to the value expected from a D O D of 104.5 ° which is the value currently given for the vapor [22]. The form factor calculated from the listed parameter values given in table 1 is shown in fig. 2; the difference function Dm(Q) is presented in fig. 3 with the corresponding function for liquid D 2 0 at 21°C [19]. 0.6 0.~ 0-2 Amorphous D20
:,%
•
.~..
0.2 02' aE
•:
,.-,.
Liquid
D20
0
-0.2 -0.4 _0.60
1 4
i 8 O (A0-1)
J 12
I 16
20
Fig. 3. The inter-molecular function, DM(Q), for a-D20 obtained from fig. 2, compared with that for liquid D20 [27].
M.R. Chowdhury et aL / Structure of amorphous DeO ice
255
4.2. Molecular orientation and the cluster form-factor It is clear from fig. 2 that the diffraction data for an amorphous ice sample exhibit more structural features at intermediate Q-values than for liquid water and that a single molecule form-factor is a poor representation of the measurements in this Q-value region. This is due to an increased intermolecular contribution which results from a greater correlation between adjacent molecular units in the amorphous phase. If it is assumed that the orientational correlation is primarily influenced by hydrogen-bond effects, then the local order does not differ greatly from that of crystalline ice I. In either the hexagonal (Ih) or cubic (Ic) lattice arrangement there are four oxygen atoms with a tetrahedral configuration around another single oxygen atom. The conventional view is that the two deuterium atoms in the same molecule as the central oxygen atom lie on the connecting lines to two of the vertices which constrains the D O D angle to the tetrahedral value of 109.5 A. It has also been suggested that there is no fundamental reason to presuppose that the hydrogen bond is linear [23] so it is possible that the tetrahedral symmetry may be slightly distorted in the amorphous phase and the average D O D angle slightly reduced. The basic structure will still approximate to a tetrahedral unit of five molecules as shown in fig. 4. Two further deuterium atoms are situated close to the line joining the central oxygen atoms to the other two vertices of the tetrahedron. It follows that all atoms situated on or within the tetrahedron have relatively well-defined positions. The external deuterium atoms are not constrained to any specific direction and may be rotated about an axis drawn from the central oxygen atom to the vertex of the tetrahedron. The five-molecule cluster may therefore be considered as the basic building unit in crystalline ice. It is expected to play a significant role in the structure of
Dsb
05
D2a
....
..~
D2b
%
o4 i/
l'D3a Fig. 4. The tetrahedral arrangement of four hydrogen-bonded molecules around a reference molecule; this configuration is a basic unit in the ice Ih lattice.
256
M.R. Chowdhury et aL / Structure of amorphous D20 ice
amorphous ice where the tetrahedra are not arranged in a regular order but depend on the closure conditions required by a continuous random network. Under these circumstances it is convenient to model the structural properties on the basis of a fivefold molecular cluster rather than a single molecule form-factor: If tetrahedral symmetry is assumed there are only two parameters required to define the positions of the internal deuterium atoms, i.e. the bond-length (rOD) and the hydrogen (or deuterium) bond-length (rD). Since the external deuterium atoms are not specifically localized relative to the other atoms (intermolecular) in the unit, their contribution is small and can be neglected except for atoms on the rotation axis. The cluster form-factor can then be evaluated by pair-wise summation in the usual manner; the interference terms may be represented as 1
F~( Q ) - . E bi. 2
J*,E . bib/Jo( Qr, j
where the summation indices run over all atoms in the cluster. Using the ice values of 1.00 ,~ for rOD and 1.75 A for r D gives the oscillatory curve shown in fig. 5. The overall shape is in excellent agreement with the experimental curve which is shown below and provides a convincing demonstration that the local molecular environment in crystalline and amorphous ice is very similar. This confirms the earlier suggestions that the structure is based on strong hydrogen-bonding effects and implies that the disorder may be introduced by rotation of the external deuterium atoms which will influence the relative
0.4 0
O.B O-L 0 -0-~
I
5
O {A °-1 )
I
I
10
15
20
Fig. 5. A comparison of the experimental results with the calculated form-factor for a five-molecule cluster; see the text for details.
M.R. Chowdhuryet al. / Structureof amorphous D20 ice
257
orientation of neighbouring units. Further aspects of the network structure are discussed in §6.
5. Real-space distribution The DM(Q) function may be transformed to obtain the real-space distribution function d(r) according to eq. (3). The data are limited to a finite Q-range with a maximum value, Q . . . . and in order to minimise the truncation effects, the inversion was carried out incorporating a modification function, M(Q), [24] i.e.
d ( r ) = fQ~"'QDM(Q)M(Q ) sin(Qr) dQ a0
(5)
with
M ( Q ) =jo(~rQ/Qma,). The composite dataset was transformed with Qmax = 19 ,~ ~. The resulting d(r) curve is shown in fig. 6 together with the corresponding curve for liquid D 2 0 [24] for which the value of Qm,x is 12 ,~- 1. In the case of the amorphous ice, well-defined oscillations can be seen to extend up to 15 ,h, but for the liquid, the oscillations extend to only - 9 ,h,. This indicates a longer-range correlation in the amorphous solid than that in the liquid as expected. Numerical data for d(r) are given in table 2.
I
I
I
i
]
O.l, 0.2 d(r) 0 -0.2
-0-~ 0.2 =
oi -0.2 -0./, J
I
I
I
5
10
15
20
r IA °)
Fig. 6. The pair-correlation function, d(r), obtained for a-D20 and liquid D20 [27].
M.R. Chowdhury et al. / Structure of amorphous D20 ice
258
Table 2 The real-space distribution function d ( r ) for neutron diffraction by vapor-deposited D 2 0 ice
r(A)
d(r)
r(A)
d(r)
rtA)
d(r)
r(A)
d(r)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
-0.0391 -0.0778 -0.1168 -0.1560 -0.1947 - 0.2335 -0.2731 -0.3i15 -0.3499 - 0.3912 -0.4255 - 0.4307 - 0.4071 -0.3835 - 0.3601 - 0.2921 -0.1850 -0.1455 - 0.2437 -0.3714 -0.3590 -0.2142 -0.1210 -0.1927 -0.3442 -0.4440 - 0.4882 -0.5460 -0.5974 - 0.5352 -0.3336 -0.1199 -0.0242 -0.0247 0.0074 0.0708 0.1540 O. 1871 0.1719 0.1404 0.1265 0.1568 0.2301 0.3096 0.3555 0.3576 0.3296 0.2860 0.2377 0.1993
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0
0.1825 0.1763 0.1489 0.0844 0.0051 - 0.0554 -0.0918 -0.1239 -0.1637 - 0.2059 -0.2444 - 0.2800 - 0.3094 -0.3210 - 0.3050 - 0.2626 -0.2041 -0.1414 - 0.0840 -0.0383 -0.0049 0.0217 0.0478 0.0752 0.1013 0.1212 O. 1296 0.1231 0.1055 0.0886 0.0834 0.0912 0.1033 0.1091 0.1040 0.0910 0.0775 0.0700 0.0695 0.0708 0.0667 0.0531 0.0306 0.0026 -0.0260 -0.0510 -0.0702 -0.0851 -0.0993 -0.1145
10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14.0 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 15.00
-0.1285 -0.1366 -0.1352 -0.1236 -0.1035 - 0.0785 -0.0534 -0.0318 -0.0136 0.0039 0.0227 0.0412 0.0555 0.0632 0.0644 0.0619 0.0586 0.0562 0.0542 0.0516 0.0471 0.0411 0.0349 0.0293 0.0247 0.0214 0.0189 0.0157 0.0110 0.0050 -0.0009 -0.0059 -0.0100 -0.0148 -0.0211 - 0.0286 -0.0357 - 0.0408 -0.0432 -0.0531 -0.0418 -0.0403 -0.0383 -0.0345 -0.0276 -0.0182 -0.0079 0.0015 0.0085 0.0127
15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16.0 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17.0 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18.0 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.0 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 20.0
0.0151 0.0178 0.0218 0.0263 0.0293 0.0287 0.0240 0.0171 0.0114 0.0097 0.0117 0.0143 0.0140 0.0098 0.0039 0.0001 0.0 0.0031 0.0062 0.0060 0.0015 -0.0048 -0.0097 -0.0113 -0.0102 -0.0089 - 0.0095 -0.0123 -0.0158 - 0.0179 -0.0172 -0.0142 -0.0107 -0.0082 -0.0071 - 0.0065 -0.0051 - 0.0024 0.0012 0.0048 0.0073 0.0085 0.0089 0.0092 0.0096 0.0100 0.0100 0.0095 0.0089 0.0084
1.3
1.4 1.5 1.6
1.7 1.8 1.9
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
M.R. Chowdhury et al. / Structure of amorphous D20 ice
259
The real-space resolution is dependent on the value of Qmax and for the modification function used in the present transform procedure, the resolution width A r ( F W H M ) is given by [13] Ar = 5 . 4 4 / Q . . . .
which in the present case is 0.29 ,~. Since the DIn(Q) curve exhibits significant oscillations up to, and probably beyond, 19 ,,~- ~, termination of the integral at smaller Q-value is expected to have an effect on the details of the resulting d ( r ) function, especially at small r-values where it is possible to assign distances to particular intermolecular structure. The effect on the resolution is shown in fig. 7 where the d ( r ) curve is compared with other curves using lower truncation points. The peaks in the d ( r ) function at small r-values (1.5 < r < 3 ,~) can be related to the geometric structure of the network. The first physically significant peak occurs at 1.78 A and corresponds to the intermolecular hydrogen bond. The following peak at 2.31 ,~ arises from the shortest D - D distance between hydrogen-bonded molecules. These two peaks are well-resolved from each other and also from neighboring structural contributions. Further peaks are strongly overlapping and superimposed on a general distribution. For the region extending up to 5 ,~ it is still possible to identify specific distances within the network although a more detailed evaluation may be made be direct comparison of the full d ( r ) curve with the model predictions. These features will be discussed more fully in the following section but it is interesting to note that there is a good correspondence in the mean atomic positions obtained I
I
I
l
0-4 02 0
0.4 0.2 d(r)
0
04 0.2 0
- 0 "2 - 0 '4 0
I
I
4
8
I
I
12
16
20
r (A°)
Fig. 7. The effect of the truncation point, Qmax, on the evaluation of the d(r) function for a-D20.
260
M.R. Chowdhury et aL / Structure of amorphous D20 ice
Table 3 A comparison of interatomic distances obtained from the experimental data with predictions from the CRN model [6] and ice Ih. Network relation
a-D20 (experiment) Hydrogen bond Nearest neighbor (close) Nearest neighbor (tetrahedral) Second neighbor (symmetric) Nearest neighbor (remote) Second neighbor (close) Second neighbor (tetrahedral)
Corresponding distance (fig. 4)
r (,~) Ice Ih (or network)
1.78
1.76
O ~D2~, 03 D ~b, etc.
2.32
2.30
DlaD4, D2aD 1, D~aDI, etc.
( - 2.8)
2.76
OlO 2, OiO 3, etc.
( - 2.8)
2.87
D2aD 5
- 3.34
3.24
OiD2b, OlD 3, O2D I, O3Dla, etc.
- 3.80
3.74
O2Dsa, O3D2A, etc.
4.55
4.50
OiO 2, O103, etc.
f r o m the e x p e r i m e n t a l m e a s u r e m e n t s a n d those p r e d i c t e d b y either the C R N m o d e l of the a m o r p h o u s solid or the local a r r a n g e m e n t in the crystalline ice Ih phase. The values are listed in table 3 which also gives a suitable label to each of the c o n t r i b u t i o n s according to the schematic d i a g r a m of the h y d r o g e n b o n d e d local environment. The structural features in d ( r ) for r_< 1.5 ,~ are spurious a n d arise from errors in the overall level of the DIn(Q) function caused m a i n l y by small systematic errors in the correction p r o c e d u r e s a d o p t e d to derive the structure factor from t h e observed diffraction pattern. It seems likely that the small h u m p at 1.5 ,~ on the side of the h y d r o g e n - b o n d p e a k at 1.78 ,g, is an artefact resulting from the presence o f this d a m p e d oscillatory function. A l t h o u g h iterative techniques are sometimes used to remove the unphysical oscillations at low r-values it is i n a p p r o p r i a t e in this case as it can be seen that the effects will be negligible in the regions of interest.
6. The continuous random network model T h e d e s c r i p t i o n of a m o r p h o u s solids is c o m p l i c a t e d b y the strong local c o o r d i n a t i o n between a t o m s or molecules c o u p l e d to an absence of long-range order. One a p p r o a c h to this p r o b l e m has been t h r o u g h the c o n s t r u c t i o n of h a n d - b u i l t m o d e l s in which the local o r d e r is relaxed so that a c o n t i n u o u s space-filling n e t w o r k m a y be constructed. The c o o r d i n a t e s describing the a t o m p o s i t i o n s are refined b y c o m p u t e r p r o c e d u r e s which reduce the internal strains
M.R. Chowdhury et al. / Structure of amorphous D20 ice
261
within the network and enable the distribution of bondlengths and angle variations to be investigated by averaging over all atoms in the assembly. This method was first adopted by Polk [25] for amorphous semiconductors with tetrahedral coordination such as silicon and germanium. The success of this approach has led to further work on networks with different structural characteristics. The tetrahedral nature of the hydrogen bond lends itself naturally to a similar treatment where the oxygen atoms have a similar distribution to that of silicon or germanium. The proton positions may then be placed asymmetrically on the line connecting adjacent oxygen atoms provided the ice rules are satisfied, i.e. there are two protons to any one oxygen atom corresponding to the same molecule and also two protons at larger distances corresponding to hydrogen-bonded neighbors. Boutron and Alben [8] have adapted a relaxed version of the Polk-Boudreau model for germanium [2] to describe amorphous ice. The distribution function was scaled so that the oxygen-oxygen distances gave a mean value of 2.75 and the protons were placed at a distance of 1.0 A from one of the oxygen atoms and a mean value of 1.75 ,~ from the other oxygen atom. The final assembly consisted of 519 oxygen atoms and produced a spherical sample approximately 32 A in diameter. The pair correlation functions derived from the model were found to be in good agreement with the X-ray [5] and earlier neutron [6] data. The improved accuracy of the present data with a better resolution in real-space is capabl e of providing a more stringent test of the model. The model distribution function was found to be little influenced by the choice of molecular parameters corresponding ° to the vapor (rod = 0.96 A, 0 = 104 °) or the crystalline phase (roo = 1.0 A, 0 = 109.5°). The computed distribution function zSJ(r) may be converted to an equivalent d th(r) function by the relation
dth(r)=
b2 + 2b 2 (b o + 2 b D ) 2
zS./(r) r '
which may then be compared with the neutron results. The experimental distribution function is subject to a finite resolution, due to the limiting value of Qmax SO that the deXp(r) curve is effectively a convolution of the true distribution function with the transform of M(Q). Furthermore, the network model is based on a classical description in which the atom positions are defined by unique coordinates and no account is taken of the spread due to zero-point motion. The most convenient way to allow for these effects and to compare the two sets of data is to convolute the model function d th(r) with a suitable function which broadens the predicted pattern. We have chosen to use a Gaussian convolution and to adjust the width parameter so that the first two intermolecular peaks are in good agreement with the observed data. The required width ( F W H M ) is found to be 0.38 A, which is a little larger than the calculated resolution width of 0.3 ,~ resulting from the finite Qmax" Since the
M.R. Chowdhury et al. / Structure of amorphous DeO ice
262
observed widths result from the combination of the two terms in quadrature the natural width is approximately 0.2 ,~. The quantum corrections or zero-point motion for the deuterium atoms of an isolated D20 molecule at O K contributes an effective spread of 0.14 A for the O - D distance and 0.22 ,~ for D - D distances [26]. The values are in agreement with the apparent width deduced from the experimental results and suggest that the model predictions are consistent with the spread in the positions of the atoms within the network. The two d(r) curves are shown in fig. 8 where the peaks corresponding to the intermolecular distances (roD and rDD) have been omitted for clarity. There is excellent agreement in the general shape of the two distributions for the full range extending to 16 ,~. This provides a clear confirmation that the basic features of the model give a good representation for the structure of amorphous ice. The first two peaks correspond to the hydrogen-bonded O - D distance and the shortest intermolecular D - D distance respectively. Several other peaks in the d(r) curves can also be assigned to specific geometrical configurations and are shown schematically in table 3 for distances up to 5 ,~. Above this value there are many contributions and individual assignment is no longer feasible although there is good agreement between the model predictions and the experimental results. The results demonstrate the importance of hydrogen-bonding in the disordered condensed state of water. It is responsible for the formation of a completely hydrogen-bonded network. There is no evidence for the presence of non-bonded or partially-bonded interstitial molecules in the network. The formation of the bulk material from the vapor deposition process can therefore be considered as the positioning of individual molecules on a surface in such a
0.6
O.h 0.2
L
Amorphous
IExr I
DzO - Ice
nt
-0.2 -0.1. I 0
I t,-O
I
I 8.0
1
I 12-0
I
I 16.0
r{A ° ) Fig. 8. A comparison of the experimental results for a-D20 with the predictions of Boutron and
Alben [8] using a continuous random network model.
M.R. Chowdhury et al. / Structure of amorphous D20 ice
263
way that the local environment forms a hydrogen-bonded structure of minim u m energy. As the layer thickness is built up the interior structure cannot be modified. This process corresponds closely to the operations made in the initial hand-construction of the model framework. The low-density form of amorphous ice can therefore be characterized as a complete hydrogen-bonded continuous random network. The relation of this structure to that of liquid water remains somewhat controversial. In the liquid phase, the molecules have both translational and rotational freedom and the spread in the hydrogen-bond distance is known to be much greater [19,27]. Rice and Sceats [28] introduce a system in which the bond angles and positions of the oxygen atoms are allowed to vary in a continuous way as a function of temperature. This implies a continuity of state which is not realized in practice since the nucleation temperature for amorphous ice is reported to be in the range 120-130 K [29]. Supercooled water exhibits many anomalous properties [30] which can be explained as the basis of a type of critical-point behaviour with a temperature of approximately 230 K. Diffraction studies on supercooled water show a large shift in the main diffraction peak towards the value found in amorphous ice. This suggests an increasing proportion of hydrogen bonding as the temperature is lowered and is consistent with the recent percolation theories of water [31]. Recent small-angle X-ray scattering results [32] emphasize these particular features but are unable to distinguish between different models. It seems natural to view the amorphous ice structure as an end-point in the cooling of liquid water through the supercooled region to the glass formation point without crystallization. The experimental evidence cited above suggests that there is no temperature at which amorphous ice and supercooled water can coexist so that the structural relationship between amorphous ice and liquid water remains paradoxical. It is possible that supercooled water forms a different hydrogen-bonded polytype but this must also be of low density so that it is difficult to conceive an alternative structure. Work is already in progress [33] to try and elucidate this problem but the consensus view at the moment is that vapor-deposited amorphous ice is not equivalent to a glassy state of water. The only way to resolve these uncertainties would be by making diffraction measurements on super-cooled water near the transition point. It is hoped to extend the study by making these measurements but there are various technical difficulties such that an emulsion system [30,34] must be used to prevent crystallization at low temperatures.
7. Conclusion
The present results confirm that amorphous D20 may be represented as a continuous random hydrogen-bonded network. The hydrogen bond distance rod is well defined and has a narrow spatial distribution suggesting that it resembles a normal chemical bond at these low temperatures. The behaviour is
264
M.R. Chowdhury et al. / Structure of amorphous De0 ice
of particular interest in connection with the percolation models being developed [31] to explain anomalous properties of supercooled water. The predictions from the CRN model of Boutron and Alben are in excellent agreement with the data over a range extending above 10 ,~,. We wish to thank Norman Clarke at AERE, Harwell for assistance in the construction and operation of the cryostat. We also wish to thank Dr. Pierre Chieux for assistance during the course of the ILL experiments and Dr. Stewart Cummings for help with the initial computations. Two of us (MRC and JTW) wish to acknowledge support from the Science Research Council during the course of the work. The experimental study is part of a general programme which is supported by the Neutron Beam Reseach Committee of the SERC.
References [I] E.F. Burton and W.T. Oliver, Proc. R. Soc. A153 (1935) 166. [2] T.C. Sivakumar, D. Schuh, M.G. Sceats and S.A. Rice, Chem. Phys. Lett. 48 (1977) 212. T.C. Sivakumar, S.A. Rice and M.G. Sceats, J. Chem. Phys. 69 (1978) 3468. W.G. Madden, M.S. Bergren, R. McGraw and S.A. Rice, J. Chem. Phys. 69 (1978) 3497. [3] G.P. Johari, Phil. Mag. 35 (1977) 1077. S.A. Rice, M.S. Bergren and L. Swingle, Chem. Phys. Len. 59 (1978) 14. [4] P. Bondt, C.R. Acad. Sci., Ser. B 265 (1976) 316. C.G. Venkatesh, S.A. Rice and A.H. Narten, Science 186 (1974) 927. [5] A.H. Narten, C.G. Venkatesh and S.A. Rice, J. Chem. Phys. 64 (1976) 1106. [6] S.A. Rice, private communication (see also second of ref. 2). [7] J.T. Wenzel, C. Linderstrom-Lang and S.A. Rice, Science 187 (1975) 428. [8] P. Boutron and R. Alben, J. Chem. Phys. 62 (1975) 4848. [9] R.J. Temkin, W. Paul and G.A.N. Connell, Adv. Phys. 22 (1973) 581. G. Etherington, A.C. Wright, J.T. Wenzel, J.C. Dore, J.H. Clarke and R.N. Sinclair, J. Non-Cryst. Solids 48 (1982) 265. [10] A.C. Wright and R.N. Sinclair, in: The Physics of Silica and its Interfaces, Ed. J. Pantelides (Pergammon, Oxford, 1978) p. 133, and references cited therein. R.N. Sinclair, J.A.E. Desa, G. Etherington, P.A.V. Johnson and A.C. Wright, J. Non-Cryst. Solids 42 (1980) 107. [11] M.R. Chowdhury and J.C. Dore, J. Non-Cryst. Solids, in press. [12] J.G. Powles, Adv. Phys. 22 (1973) 1. L. Blum and A.H. Narten, Adv. Chem. Phys. 34 (1976) 203. [131 A.C. Wright, Adv. Str. Res. Diff. Math. 5 (1974) 1. C.N.J. Wagner, J. Non-Cryst. Solids 31 (1978) 1; 42 (1980) 3. [14] J.G. Powles, J.C. Dore and D.I. Page, Mol. Phys. 24 (1972) 1025. [15] G. Palinkas, E. Kalman and P. Kovacs, Mol Phys. 34 (1977) 525. W.E. Thiessen and A.H. Narten, in press. [16] J. Reed, J.C. Dore and M.R. Chowdhury, to be submitted to Mol. Phys. [17] M.R. Chowdhury, J.C. Dore and P. Chieux, J. Non-Cryst. Solids 43 (1981) 267. [18] Neutron Beam Facilities at the Institut Laue-Langevin, Ed. B. Maier (1974). [19] G. Walford, J.H. Clarke and J.C. Dore, Mol. Phys. 33 (1977) 25. I.P. Gibson, Thesis (University of Kent, 1978). I.P. Gibson and J.C. Dore, Mol. Phys. in press.
M.R. Chowdhury et al. / Structure of amorphous D20 ice
265
[20] J.G. Powles, Mol. Phys. 42 (1981) 757. [21] J.M. Carpenter and C.A. Pelizzari, Phys. Rev. BI2 (1975) 2391. R.N. Sinclair and A.C. Wright, Nuc. Instr. and Meth. 114 (1974) 451. [22] K. Kuchitsu and kS. Bartell, J. Chem. Phys. 36 (1962) 2460. [23] The structure aspects of crystalline ice phases have been reviewed in various papers such as: N.H. Fletcher, Rep. Prog. Phys. 34 (1971) 913. P.V. Hobbs, Ice Physics (Clarendon, Oxford, 1974) Chap. 1. The current views concerning molecular conformation in ice has been questioned by W. Whalley, Mol. Phys. 28 (1974) 1105 and remain unresolved. Recent neutron diffraction studies of ice by M. Lehmann (private communication) should help to clarify the situation. [24] J. Waser and V. Schomacher, Rev. Mod. Phys. 25 (1953) 671. [25] D.E. Polk, J. Non-Cryst. Solids 5 (1971) 365. D.E. Polk and D.S. Boudreaux, Phys. Rev. Lett. 31 (1973) 92. [26] S.J. Cyvin, Molecular Vibrations and Mean Square Amplitudes (Elsevier, Amsterdam, 1968). [27] LP. Gibson and J.C. Dore, submitted to Mol. Phys. [28] M.G. Sceats, M. Stavola and S.A. Rice, J. Chem. Phys. 70 (1979) 3927. M.G. Sceats and S.A. Rice, J. Chem. Phys. 72 (1980) 3236. A.C. Belch, S.A. Rice and M.G. Sceats, Chem. Phys. Lett. 77 (1981) 455. [29] A review of experiments on the stability of amorphous H20 ice with temperature is presented by P.V. Hobbs, in: Ice Physics (Clarendon, Oxford, 1974) pp. 44-57. [30] A. Angell, in: Water: A Comprehensive Treatise, Vol. 7, Ed. F. Franks, Chap. 5. [31] H.E. Stanley, J. Phys. AI2 (1979) L329. A. Geiger, F.H. Stillinger and A. Rahmann, J. Chem. Phys. 70 (1979) 4185. H.E. Stanley and J. Texeira, J. Chem. Phys. 73 (1980) 3404. H.E. Stanley, J. Texeira, A. Geiger and R.L. Blumberg, Physica 106A (1981) 260. [32] L. Bosio, J. Teceira and H.E. Stanley, Phys. Rev. Lett. 46 (1981) 597. [33] k Bosio, J. Texeira, J.C. Dore and H.E. Stanley, unpublished data. [34] J. Texeira and O. Conde, private communication.
247
T H E S T R U C T U R A L C H A R A C T E R I S T I C S OF A M O R P H O U S D20 ICE BY N E U T R O N D I F F R A C T I O N M.R. C H O W D H U R Y
* and J.C. D O R E
Physics Laboratory, University of Kent, Canterbury CT2 7NR, UK
J.T. W E N Z E L National Bureau of Standards, Washington DC, USA **
Received 13 May 1982 Neutron diffraction measurements have been made at several wavelengths for vapor-deposited D20 ice at 10 K. The observed structure factor exhibits more oscillatory features than corresponding measurements on liquid D20 water, indicating an amorphous phase which has strong intermolecular spatial correlations. A detailed examination of the data shows that this is strongly influenced by hydrogen-bonding effects and that the local environment for each molecule is very similar to that in crystalline ice. The predictions of Boutron and Alben, based on a continuous random network model, are found to be in excellent agreement with the data.
1. Introduction It has been k n o w n for some time [1] that water vapor c o n d e n s e d o n t o a substrate plate held at a sufficiently low temperature forms a n a m o r p h o u s ice deposit. T h e properties of this material have recently been the subject of an extended investigation b y Rice a n d collaborators. Structural studies involving b o t h n e u t r o n a n d X-ray diffraction have been reported in a d d i t i o n to a series of spectroscopic experiments such as R a m a n a n d infrared a b s o r p t i o n [2]. The m a i n purpose of this work has been to establish a detailed picture of the h y d r o g e n - b o n d i n g effects in a m o r p h o u s ice so that these can be used as a starting p o i n t for the description of the more complex interactions in liquid water. This presupposes a c o n t i n u i t y of state between a m o r p h o u s ice a n d supercooled water b u t this has been challenged [3] on the basis of thermodyn a m i c properties. A l t h o u g h a n u m b e r of early diffraction studies [4] have provided i m p o r t a n t X-ray data, the most thorough investigation is that of N a r t e n et al. [5]. The observed intensity d i s t r i b u t i o n was f o u n d to be d e p e n d e n t o n both the substrate temperature a n d the vapor deposition rate so that two forms (high * Attached to Materials Physics Division, Building 436, Atomic Energy Research Establishment, Harwell, Didcot, Oxon, OXI 10RA, UK. ** Now at Saint Gobain Recherche, 93304 Aubervilliers, France 0 0 2 2 - 3 0 9 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
M.R. Chowdhury et al. / Structure of amorphous D20 ice
248
and low density) of amorphous ice were produced. Subsequent investigations [6] were unable to reproduce the high-density form and it is now assumed that this requires special deposition conditions which have not been fully defined. Neutron diffraction measurements by Wenzel et al. [7] have been obtained for the low-density form of amorphous D20 ice. The data are sensitive to the positions of the deuterium atoms so that information on orientation of adjacent molecules can be studied. The two sets of diffraction data have been discussed in [5] and also compared with the continuous random network model of Boutron and Alben [8]. Due to the interest in the structural characteristics of amorphous ice and its possible relation to liquid water, we have made further neutron measurements on a D20 and extended the range to higher Q-values. The increased range of Q enables the short-range order to be studied in closer detail due to the better real-space resolution obtained from Fourier transformation of the diffraction data. Furthermore, it is possible to use the form-factor description conventionally adopted for the analysis of diffraction studies for molecular liquids and to extend the method to the cluster structure within the amorphous network. The new results and different analysis procedure therefore provide a critical test of the continuous random network (CRN) model proposed by Boutron and Alben [8]. Amorphous ice is also of more general interest as it represents a convenient link between the chemically-bonded network glasses such as amorphous germanium [9] or silica [10] and the non-bonded molecular glasses such as amorphous CC14 [11].
2. Theory The theory of neutron scattering by molecular liquids [12] and amorphous solids [13] has been reviewed previously and only a brief summary will be presented here. We adopt the formalism normally used for the analysis of diffraction data for molecular liquids and express the structure factor S,,(Q) as the sum of two terms,
Sm(Q)= Dm(Q) + f~(Q ) (1) where f~(Q) is the molecular form-factor corresponding to the scattering from isolated molecules (i.e. intramolecular contributions) and Dm(Q) represents the structural relationship between the molecular units (i.e. intermolecular contributions). The neutron form-factor for D20 is: 1
f,(Q) - (bo + 2bD)z. b• + 2b~ + 4bobDJo(QroD ) ×exp ~
Y°DQ2)'~2bZojo(QroD) exp~--YDD&~],
(2)
where bo and b D are the coherent scattering lengths for oxygen and deuterium respectively; jo(x) = (sin x ) / x is the spherical Bessel function of order zero and
M.R. Chowdhury et al. / Structure of amorphous DzO ice
249
rOD and rDD are the internal O D and D D distances within the molecule. The exponential terms in "~ODand ~DD correspond to Debye-Waller factors which account for thermal vibrations with ), = ½(u 2) where (u 2) is the mean square amplitude of vibration. The Dm(Q) function is mainly confined to low and intermediate Q-values so that the observed diffraction pattern at sufficiently high Q-values is characterised by the oscillatory nature of the ft(Q) form-factor. In the case of liquids this enables the structure of the basic unit to be very accurately established but the strong spatial correlations between adjacent units in the molecular glasses does not permit such an easy separation of the two contributions. In this case it is convenient to extend the form-factor to that of a cluster arrangement and this will be more fully developed in the analysis section. Information on the intermolecular structure may be obtained from the transform relation
d(r)=47roMr[ g(r)-- l]
2 o¢ = ~ f 0 QDM(Q)
sin QrOQ'
(3)
where PM is the molecular number density and g(r) is the composite atom pair-correlation function. The function g(r) is a weighted sum of partial correlation functions, and for neutron scattering by D20 may be written as: g ( r ) = 0.092 goo ( r ) + 0.486god ( r ) + 0.422 god ( r ) .
(4)
It is clear that the O - O contribution is relatively small and that the weighted function contains O - D and D - D contributions in nearly equal proportions. A complete determination of the partial functions requires three independent experiments with different weighting factors. This data can be obtained either by neutron diffraction using isotopic substitution [14], or by combining results from measurements using X-ray, neutrons an electrons [15]. Both the techniques have been used in an attempt to derive the partial g(r) functions for the liquid phase [14,15] and a paper on some recent neutron measurements is in preparation [16].
g(r)
3. Experimental procedure and data reduction
3.1. Sample preparation The cryostat used in the present experiment was the same as that used in the neutron studies of vapor-deposited a m m o n i a - N D 3 [17] and amorphous carbon tetrachloride I11]. The procedure was similar to that of Wenzel et al. [7], who used a l0 m m wide cadmium strip of 2 m m thickness as the substrate plate which was attached to the bottom of a helium cryostat and maintained at l0 K or lower. T h e liquid D 2 0 was held in an outer container and degassed by successive f r e e z e - p u m p - t h a w cycles. The ice sample was grown by vapor deposition over several days with a deposition rate restricted to several mg per
250
M.R. Chowdhury et al. / Structure of amorphous D20 ice
hour. The high absorption cross-section of cadmium offers the advantage of a very low substrate background but also restricts measurements to a reflection geometry. For measurements at small scattering angles the plate must be set at an oblique angle to the incident neutron beam and consequently the data are subject to large-shielding and absorption corrections, expecially when the uniformity of the sample cannot be accurately established. To overcome these difficulties in the present investigation two samples were studied using different substrate materials so that complementary measurements could be made on two separate diffractometers. 3.2. Measurements at 1.37 ,~
Neutron measurements up to Q-values of 7.3 A~ were made on the C U R R A N diffractometer at the D I D O reactor, AERE, Harwell. For these measurements a vanadium substrate plate of 40 m m width and 2 m m thickness was used. The scattering from vanadium is almost entirely incoherent and gives a constant contribution to the observed intensity. The incident wavelength was calibrated by nickel powder measurements and found to be 1.368 + 0.003 ,~. The scattered neutron distribution was recorded using transmission geometry for the range 2 < O < 85 ° and reflection geometry for the range 80 < 0 < 100 °. Three sets of diffraction patterns corresponding to three different angular settings of the substrate plate were collected. After the measurements on the sample were completed, the cryostat was allowed to warm up slowly until the deposit became detached from the plate. Measurements were then made on the scattering from the bare substrate plate. 3.2.1. Measurements at 0.7 and 0.5 fl. The high Q-value measurements were made on the D4 diffractometer [18] at the Institut Laue-Langevin, Grenoble, using two incident wavelengths. For these measurements the sample was deposited on a cadmium substrate plate of 40 m m width and 2 mm thickness. Data collection was made only in reflection geometry over the angular range 50-130 ° using three different settings of the plate angle. Several repeat scans of the detector were made and showed no variation in the intensity distribution. The data from the multidetector were combined in the usual manner to give an intensity distribution function for both wavelengths. The incident wavelengths were determined by nickel powder calibration and were found to be 0.696 + 0.002 A and 0.504 + 0.002 A. 3.3. Data reduction
The three sets of observed intensity distributions were corrected for scattering from the substrate plate and the cryostat head shields. Corrections for self-shielding, absorption and multiple scattering were applied in the usual manner and the resulting intensity distributions for different angular regions were combined to give a composite pattern for each of the three datasets. The
M.R. Chowdhury et al. / Structure of amorphous D20 ice
final results are shown in fig. 1 as a function of the elastic scattering vector, given by
251
Q,
Q = (4~r/),) s i n ( 0 / 2 ) . The intensity distributions cannot be placed on an absolute cross-section scale because the thicknesses of the sample deposits are not accurately known. Experimental corrections were applied assuming a uniformly thick sample deposit. An approximate normalization may be obtained by extrapolation of high-angle data to lower angles where the effects due to Placzek corrections
3 2
>,
0
>.,,
3
I
I
I
I
0.7A o
2 _c 1 I
I
I
I
5
10
15
20
÷+
1.37 A°
2 __>" u~
*
0
I
I
I
I
2
z.
6
8
O (Ao-1)
Fig. 1. The individual measurements of the diffraction pattern for neutron scattered from amorphous D20 ice; see the text for details.
M.R. Chowdhury et al. / Structure of amorphous D20 ice
252
[ 12] are much lower. This provides a convenient relative normalization of the data sets and confirms that each sample corresponded to approximately 10% scattering of the incident beam.
4. Analysis
4.1. The structure factor and single molecule form-factor The conversion of the corrected differential scattering cross-section do(0)/d~2 to the required structure factor Sm(Q) is not simple due to the effect of inelastic scattering. The most convenient approach based on an extension of the Placzek correction procedure has already been presented for liquid D20 [ 19] and a more sophisticated treatment has recently been made by Powles [20]. These results show that the fall-off in the region of high scattering angles is primarily due to the small mass of the deuterium atom and a shift in the oscillatory structure of the interference pattern arises from molecular recoil in the scattering process. It is inappropriate to present a detailed discussion of the various methods which may be used to correct for inelasticity effects particularly as there are certain simplifying features which apply in the case of amorphous ice. A full treatment of corrections applying to network systems has not yet been developed although the framework for such a theory has been presented [ 13,21 ]. We may write the observed cross-section per molecule in two terms corresponding to self and interference (distinct) scattering contributions
ao,0 =( do
do d ~ (0))se,f +
The Placzek expansion for D20 shows that the cross-section for self-scattering may be expressed as l
where a a is a detector constant and (Me,/ran) is the effective mass of scattering centers relative to the neutron mass. This produces the Q2-dependent fall-off in the overall level of the scattering intensity but overestimates the reduction for backward angle scattering. The factors which influence the M~fr parameter are strongly dependent on the motion of the deuterium atoms. In amorphous ice the temperature of the sample is low so that thermal excitation is negligible and the dynamics is entirely governed by zero-point motion. Under these conditions M~fr is expected to be about 4 a.m.u, as discussed by Wenzel et al. [7]. An accurate estimate of Meff cannot be made as the observed fall-off is dependent on the precise form of self-shielding corrections for the various geometries used in obtaining the full angular distribution. In certain respects, this difficulty does not pose serious problems for evaluating the
M.R. Chowdhury et al. / Structure of amorphouz" D20 ice
253
interference contributions since it is oscillatory shape which relates to the structural information and errors in the smooth monotonically-decreasing self-scattering function which do not introduce any spurious high-frequency components into the pattern. The interference contributions arising from distinct scattering are much simpler that for the liquid state since the molecules are presumably bound within the network. This means that recoil corrections are expected to be negligible and there is no need to separate the intra- and intermolecular components for different treatment. It follows that the intermolecular contribution can be treated within the static approximation so that
(b ° + 2bD)2D~(Q) and therefore the Dm(Q) function can be readily evaluated from the observed cross-section. The Q-ranges for the 0.7 and 0.5 A datasets were 7.8-15 A and 10.6-22.5 A. The largest Q-value of the 1.37 A set being 7.3 A l, there was a short gap between the largest Q-value of this set and the smallest Q-value of the 0.7 A set, and therefore in order to construct a composite dataset from the present measurements it was necessary to include data from the previous measurement [7]. Instead of using sections from four different datasets - three from the present measurements and one from Wenzel et al. [7], it was decided to combine the 1.37 ,~ and the 0.5 ,~ measurements with the missing section taken from the measurements of Wenzel et al. The final composite structure factor is shown in fig. 2. To investigate the structure of the D20 molecule in the amorphous phase the form factor f l ( Q ) (eq. (3)) was fitted to the large-Q region (8-16 ,~-1) of 1"2
-~
1"0
x,
~
J
I
[
~
r
F
I
I
[
I
I
I
I
S~(O)
1"8
0"6
0
0"2
0
4
I
8
I
Q ( A o4 )
12
16
20
Fig. 2. The composite structure factor, SIn(Q) for a-D20 and the fitted form-factor fl(Q) for a single molecule with parameters given in table 1.
254
M.R. Chowdhury et a L / Structure of amorphous D,O ice
Table 1 Parameter values obtained from the fl(Q) single molecule fit to the observed structure factor, SIn(Q) of amorphous D20 ice Parameter
Value
rod
0.998 + 0.003 1.578+ 0.005 (0.28 _O.03)xlO 2 (1.8 _+1.2)×10-1
rDD " YoD YDt~
,~2 ~2
Compiled for a DOD angle of 104.5°.
the Sm(Q) data using a nonlinear least-squares routine. Although the measured data has much more structure than the molecular form-factor the additional oscillations must be superimposed on the basic shape of the single molecule form-factor so the fitting routine effectively averages over these higherfrequency components to give approximate values of the roD and rDD parameters. The fit was found to be sensitive to the value of rOD, but relatively insensitive to the rDo value. This is because the intramolecular D - D distance ( - 1.6 ~,) is close to the intermolecular hydrogen-bond distance ( - 1.7 A). In the final analysis therefore the rDD parameter was constrained to the value expected from a D O D of 104.5 ° which is the value currently given for the vapor [22]. The form factor calculated from the listed parameter values given in table 1 is shown in fig. 2; the difference function Dm(Q) is presented in fig. 3 with the corresponding function for liquid D 2 0 at 21°C [19]. 0.6 0.~ 0-2 Amorphous D20
:,%
•
.~..
0.2 02' aE
•:
,.-,.
Liquid
D20
0
-0.2 -0.4 _0.60
1 4
i 8 O (A0-1)
J 12
I 16
20
Fig. 3. The inter-molecular function, DM(Q), for a-D20 obtained from fig. 2, compared with that for liquid D20 [27].
M.R. Chowdhury et aL / Structure of amorphous DeO ice
255
4.2. Molecular orientation and the cluster form-factor It is clear from fig. 2 that the diffraction data for an amorphous ice sample exhibit more structural features at intermediate Q-values than for liquid water and that a single molecule form-factor is a poor representation of the measurements in this Q-value region. This is due to an increased intermolecular contribution which results from a greater correlation between adjacent molecular units in the amorphous phase. If it is assumed that the orientational correlation is primarily influenced by hydrogen-bond effects, then the local order does not differ greatly from that of crystalline ice I. In either the hexagonal (Ih) or cubic (Ic) lattice arrangement there are four oxygen atoms with a tetrahedral configuration around another single oxygen atom. The conventional view is that the two deuterium atoms in the same molecule as the central oxygen atom lie on the connecting lines to two of the vertices which constrains the D O D angle to the tetrahedral value of 109.5 A. It has also been suggested that there is no fundamental reason to presuppose that the hydrogen bond is linear [23] so it is possible that the tetrahedral symmetry may be slightly distorted in the amorphous phase and the average D O D angle slightly reduced. The basic structure will still approximate to a tetrahedral unit of five molecules as shown in fig. 4. Two further deuterium atoms are situated close to the line joining the central oxygen atoms to the other two vertices of the tetrahedron. It follows that all atoms situated on or within the tetrahedron have relatively well-defined positions. The external deuterium atoms are not constrained to any specific direction and may be rotated about an axis drawn from the central oxygen atom to the vertex of the tetrahedron. The five-molecule cluster may therefore be considered as the basic building unit in crystalline ice. It is expected to play a significant role in the structure of
Dsb
05
D2a
....
..~
D2b
%
o4 i/
l'D3a Fig. 4. The tetrahedral arrangement of four hydrogen-bonded molecules around a reference molecule; this configuration is a basic unit in the ice Ih lattice.
256
M.R. Chowdhury et aL / Structure of amorphous D20 ice
amorphous ice where the tetrahedra are not arranged in a regular order but depend on the closure conditions required by a continuous random network. Under these circumstances it is convenient to model the structural properties on the basis of a fivefold molecular cluster rather than a single molecule form-factor: If tetrahedral symmetry is assumed there are only two parameters required to define the positions of the internal deuterium atoms, i.e. the bond-length (rOD) and the hydrogen (or deuterium) bond-length (rD). Since the external deuterium atoms are not specifically localized relative to the other atoms (intermolecular) in the unit, their contribution is small and can be neglected except for atoms on the rotation axis. The cluster form-factor can then be evaluated by pair-wise summation in the usual manner; the interference terms may be represented as 1
F~( Q ) - . E bi. 2
J*,E . bib/Jo( Qr, j
where the summation indices run over all atoms in the cluster. Using the ice values of 1.00 ,~ for rOD and 1.75 A for r D gives the oscillatory curve shown in fig. 5. The overall shape is in excellent agreement with the experimental curve which is shown below and provides a convincing demonstration that the local molecular environment in crystalline and amorphous ice is very similar. This confirms the earlier suggestions that the structure is based on strong hydrogen-bonding effects and implies that the disorder may be introduced by rotation of the external deuterium atoms which will influence the relative
0.4 0
O.B O-L 0 -0-~
I
5
O {A °-1 )
I
I
10
15
20
Fig. 5. A comparison of the experimental results with the calculated form-factor for a five-molecule cluster; see the text for details.
M.R. Chowdhuryet al. / Structureof amorphous D20 ice
257
orientation of neighbouring units. Further aspects of the network structure are discussed in §6.
5. Real-space distribution The DM(Q) function may be transformed to obtain the real-space distribution function d(r) according to eq. (3). The data are limited to a finite Q-range with a maximum value, Q . . . . and in order to minimise the truncation effects, the inversion was carried out incorporating a modification function, M(Q), [24] i.e.
d ( r ) = fQ~"'QDM(Q)M(Q ) sin(Qr) dQ a0
(5)
with
M ( Q ) =jo(~rQ/Qma,). The composite dataset was transformed with Qmax = 19 ,~ ~. The resulting d(r) curve is shown in fig. 6 together with the corresponding curve for liquid D 2 0 [24] for which the value of Qm,x is 12 ,~- 1. In the case of the amorphous ice, well-defined oscillations can be seen to extend up to 15 ,h, but for the liquid, the oscillations extend to only - 9 ,h,. This indicates a longer-range correlation in the amorphous solid than that in the liquid as expected. Numerical data for d(r) are given in table 2.
I
I
I
i
]
O.l, 0.2 d(r) 0 -0.2
-0-~ 0.2 =
oi -0.2 -0./, J
I
I
I
5
10
15
20
r IA °)
Fig. 6. The pair-correlation function, d(r), obtained for a-D20 and liquid D20 [27].
M.R. Chowdhury et al. / Structure of amorphous D20 ice
258
Table 2 The real-space distribution function d ( r ) for neutron diffraction by vapor-deposited D 2 0 ice
r(A)
d(r)
r(A)
d(r)
rtA)
d(r)
r(A)
d(r)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2
-0.0391 -0.0778 -0.1168 -0.1560 -0.1947 - 0.2335 -0.2731 -0.3i15 -0.3499 - 0.3912 -0.4255 - 0.4307 - 0.4071 -0.3835 - 0.3601 - 0.2921 -0.1850 -0.1455 - 0.2437 -0.3714 -0.3590 -0.2142 -0.1210 -0.1927 -0.3442 -0.4440 - 0.4882 -0.5460 -0.5974 - 0.5352 -0.3336 -0.1199 -0.0242 -0.0247 0.0074 0.0708 0.1540 O. 1871 0.1719 0.1404 0.1265 0.1568 0.2301 0.3096 0.3555 0.3576 0.3296 0.2860 0.2377 0.1993
5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0
0.1825 0.1763 0.1489 0.0844 0.0051 - 0.0554 -0.0918 -0.1239 -0.1637 - 0.2059 -0.2444 - 0.2800 - 0.3094 -0.3210 - 0.3050 - 0.2626 -0.2041 -0.1414 - 0.0840 -0.0383 -0.0049 0.0217 0.0478 0.0752 0.1013 0.1212 O. 1296 0.1231 0.1055 0.0886 0.0834 0.0912 0.1033 0.1091 0.1040 0.0910 0.0775 0.0700 0.0695 0.0708 0.0667 0.0531 0.0306 0.0026 -0.0260 -0.0510 -0.0702 -0.0851 -0.0993 -0.1145
10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 11.0 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 12.0 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13.0 13.1 13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14.0 14.1 14.2 14.3 14.4 14.5 14.6 14.7 14.8 14.9 15.00
-0.1285 -0.1366 -0.1352 -0.1236 -0.1035 - 0.0785 -0.0534 -0.0318 -0.0136 0.0039 0.0227 0.0412 0.0555 0.0632 0.0644 0.0619 0.0586 0.0562 0.0542 0.0516 0.0471 0.0411 0.0349 0.0293 0.0247 0.0214 0.0189 0.0157 0.0110 0.0050 -0.0009 -0.0059 -0.0100 -0.0148 -0.0211 - 0.0286 -0.0357 - 0.0408 -0.0432 -0.0531 -0.0418 -0.0403 -0.0383 -0.0345 -0.0276 -0.0182 -0.0079 0.0015 0.0085 0.0127
15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 16.0 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17.0 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 17.9 18.0 18.1 18.2 18.3 18.4 18.5 18.6 18.7 18.8 18.9 19.0 19.1 19.2 19.3 19.4 19.5 19.6 19.7 19.8 19.9 20.0
0.0151 0.0178 0.0218 0.0263 0.0293 0.0287 0.0240 0.0171 0.0114 0.0097 0.0117 0.0143 0.0140 0.0098 0.0039 0.0001 0.0 0.0031 0.0062 0.0060 0.0015 -0.0048 -0.0097 -0.0113 -0.0102 -0.0089 - 0.0095 -0.0123 -0.0158 - 0.0179 -0.0172 -0.0142 -0.0107 -0.0082 -0.0071 - 0.0065 -0.0051 - 0.0024 0.0012 0.0048 0.0073 0.0085 0.0089 0.0092 0.0096 0.0100 0.0100 0.0095 0.0089 0.0084
1.3
1.4 1.5 1.6
1.7 1.8 1.9
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0
M.R. Chowdhury et al. / Structure of amorphous D20 ice
259
The real-space resolution is dependent on the value of Qmax and for the modification function used in the present transform procedure, the resolution width A r ( F W H M ) is given by [13] Ar = 5 . 4 4 / Q . . . .
which in the present case is 0.29 ,~. Since the DIn(Q) curve exhibits significant oscillations up to, and probably beyond, 19 ,,~- ~, termination of the integral at smaller Q-value is expected to have an effect on the details of the resulting d ( r ) function, especially at small r-values where it is possible to assign distances to particular intermolecular structure. The effect on the resolution is shown in fig. 7 where the d ( r ) curve is compared with other curves using lower truncation points. The peaks in the d ( r ) function at small r-values (1.5 < r < 3 ,~) can be related to the geometric structure of the network. The first physically significant peak occurs at 1.78 A and corresponds to the intermolecular hydrogen bond. The following peak at 2.31 ,~ arises from the shortest D - D distance between hydrogen-bonded molecules. These two peaks are well-resolved from each other and also from neighboring structural contributions. Further peaks are strongly overlapping and superimposed on a general distribution. For the region extending up to 5 ,~ it is still possible to identify specific distances within the network although a more detailed evaluation may be made be direct comparison of the full d ( r ) curve with the model predictions. These features will be discussed more fully in the following section but it is interesting to note that there is a good correspondence in the mean atomic positions obtained I
I
I
l
0-4 02 0
0.4 0.2 d(r)
0
04 0.2 0
- 0 "2 - 0 '4 0
I
I
4
8
I
I
12
16
20
r (A°)
Fig. 7. The effect of the truncation point, Qmax, on the evaluation of the d(r) function for a-D20.
260
M.R. Chowdhury et aL / Structure of amorphous D20 ice
Table 3 A comparison of interatomic distances obtained from the experimental data with predictions from the CRN model [6] and ice Ih. Network relation
a-D20 (experiment) Hydrogen bond Nearest neighbor (close) Nearest neighbor (tetrahedral) Second neighbor (symmetric) Nearest neighbor (remote) Second neighbor (close) Second neighbor (tetrahedral)
Corresponding distance (fig. 4)
r (,~) Ice Ih (or network)
1.78
1.76
O ~D2~, 03 D ~b, etc.
2.32
2.30
DlaD4, D2aD 1, D~aDI, etc.
( - 2.8)
2.76
OlO 2, OiO 3, etc.
( - 2.8)
2.87
D2aD 5
- 3.34
3.24
OiD2b, OlD 3, O2D I, O3Dla, etc.
- 3.80
3.74
O2Dsa, O3D2A, etc.
4.55
4.50
OiO 2, O103, etc.
f r o m the e x p e r i m e n t a l m e a s u r e m e n t s a n d those p r e d i c t e d b y either the C R N m o d e l of the a m o r p h o u s solid or the local a r r a n g e m e n t in the crystalline ice Ih phase. The values are listed in table 3 which also gives a suitable label to each of the c o n t r i b u t i o n s according to the schematic d i a g r a m of the h y d r o g e n b o n d e d local environment. The structural features in d ( r ) for r_< 1.5 ,~ are spurious a n d arise from errors in the overall level of the DIn(Q) function caused m a i n l y by small systematic errors in the correction p r o c e d u r e s a d o p t e d to derive the structure factor from t h e observed diffraction pattern. It seems likely that the small h u m p at 1.5 ,~ on the side of the h y d r o g e n - b o n d p e a k at 1.78 ,g, is an artefact resulting from the presence o f this d a m p e d oscillatory function. A l t h o u g h iterative techniques are sometimes used to remove the unphysical oscillations at low r-values it is i n a p p r o p r i a t e in this case as it can be seen that the effects will be negligible in the regions of interest.
6. The continuous random network model T h e d e s c r i p t i o n of a m o r p h o u s solids is c o m p l i c a t e d b y the strong local c o o r d i n a t i o n between a t o m s or molecules c o u p l e d to an absence of long-range order. One a p p r o a c h to this p r o b l e m has been t h r o u g h the c o n s t r u c t i o n of h a n d - b u i l t m o d e l s in which the local o r d e r is relaxed so that a c o n t i n u o u s space-filling n e t w o r k m a y be constructed. The c o o r d i n a t e s describing the a t o m p o s i t i o n s are refined b y c o m p u t e r p r o c e d u r e s which reduce the internal strains
M.R. Chowdhury et al. / Structure of amorphous D20 ice
261
within the network and enable the distribution of bondlengths and angle variations to be investigated by averaging over all atoms in the assembly. This method was first adopted by Polk [25] for amorphous semiconductors with tetrahedral coordination such as silicon and germanium. The success of this approach has led to further work on networks with different structural characteristics. The tetrahedral nature of the hydrogen bond lends itself naturally to a similar treatment where the oxygen atoms have a similar distribution to that of silicon or germanium. The proton positions may then be placed asymmetrically on the line connecting adjacent oxygen atoms provided the ice rules are satisfied, i.e. there are two protons to any one oxygen atom corresponding to the same molecule and also two protons at larger distances corresponding to hydrogen-bonded neighbors. Boutron and Alben [8] have adapted a relaxed version of the Polk-Boudreau model for germanium [2] to describe amorphous ice. The distribution function was scaled so that the oxygen-oxygen distances gave a mean value of 2.75 and the protons were placed at a distance of 1.0 A from one of the oxygen atoms and a mean value of 1.75 ,~ from the other oxygen atom. The final assembly consisted of 519 oxygen atoms and produced a spherical sample approximately 32 A in diameter. The pair correlation functions derived from the model were found to be in good agreement with the X-ray [5] and earlier neutron [6] data. The improved accuracy of the present data with a better resolution in real-space is capabl e of providing a more stringent test of the model. The model distribution function was found to be little influenced by the choice of molecular parameters corresponding ° to the vapor (rod = 0.96 A, 0 = 104 °) or the crystalline phase (roo = 1.0 A, 0 = 109.5°). The computed distribution function zSJ(r) may be converted to an equivalent d th(r) function by the relation
dth(r)=
b2 + 2b 2 (b o + 2 b D ) 2
zS./(r) r '
which may then be compared with the neutron results. The experimental distribution function is subject to a finite resolution, due to the limiting value of Qmax SO that the deXp(r) curve is effectively a convolution of the true distribution function with the transform of M(Q). Furthermore, the network model is based on a classical description in which the atom positions are defined by unique coordinates and no account is taken of the spread due to zero-point motion. The most convenient way to allow for these effects and to compare the two sets of data is to convolute the model function d th(r) with a suitable function which broadens the predicted pattern. We have chosen to use a Gaussian convolution and to adjust the width parameter so that the first two intermolecular peaks are in good agreement with the observed data. The required width ( F W H M ) is found to be 0.38 A, which is a little larger than the calculated resolution width of 0.3 ,~ resulting from the finite Qmax" Since the
M.R. Chowdhury et al. / Structure of amorphous DeO ice
262
observed widths result from the combination of the two terms in quadrature the natural width is approximately 0.2 ,~. The quantum corrections or zero-point motion for the deuterium atoms of an isolated D20 molecule at O K contributes an effective spread of 0.14 A for the O - D distance and 0.22 ,~ for D - D distances [26]. The values are in agreement with the apparent width deduced from the experimental results and suggest that the model predictions are consistent with the spread in the positions of the atoms within the network. The two d(r) curves are shown in fig. 8 where the peaks corresponding to the intermolecular distances (roD and rDD) have been omitted for clarity. There is excellent agreement in the general shape of the two distributions for the full range extending to 16 ,~. This provides a clear confirmation that the basic features of the model give a good representation for the structure of amorphous ice. The first two peaks correspond to the hydrogen-bonded O - D distance and the shortest intermolecular D - D distance respectively. Several other peaks in the d(r) curves can also be assigned to specific geometrical configurations and are shown schematically in table 3 for distances up to 5 ,~. Above this value there are many contributions and individual assignment is no longer feasible although there is good agreement between the model predictions and the experimental results. The results demonstrate the importance of hydrogen-bonding in the disordered condensed state of water. It is responsible for the formation of a completely hydrogen-bonded network. There is no evidence for the presence of non-bonded or partially-bonded interstitial molecules in the network. The formation of the bulk material from the vapor deposition process can therefore be considered as the positioning of individual molecules on a surface in such a
0.6
O.h 0.2
L
Amorphous
IExr I
DzO - Ice
nt
-0.2 -0.1. I 0
I t,-O
I
I 8.0
1
I 12-0
I
I 16.0
r{A ° ) Fig. 8. A comparison of the experimental results for a-D20 with the predictions of Boutron and
Alben [8] using a continuous random network model.
M.R. Chowdhury et al. / Structure of amorphous D20 ice
263
way that the local environment forms a hydrogen-bonded structure of minim u m energy. As the layer thickness is built up the interior structure cannot be modified. This process corresponds closely to the operations made in the initial hand-construction of the model framework. The low-density form of amorphous ice can therefore be characterized as a complete hydrogen-bonded continuous random network. The relation of this structure to that of liquid water remains somewhat controversial. In the liquid phase, the molecules have both translational and rotational freedom and the spread in the hydrogen-bond distance is known to be much greater [19,27]. Rice and Sceats [28] introduce a system in which the bond angles and positions of the oxygen atoms are allowed to vary in a continuous way as a function of temperature. This implies a continuity of state which is not realized in practice since the nucleation temperature for amorphous ice is reported to be in the range 120-130 K [29]. Supercooled water exhibits many anomalous properties [30] which can be explained as the basis of a type of critical-point behaviour with a temperature of approximately 230 K. Diffraction studies on supercooled water show a large shift in the main diffraction peak towards the value found in amorphous ice. This suggests an increasing proportion of hydrogen bonding as the temperature is lowered and is consistent with the recent percolation theories of water [31]. Recent small-angle X-ray scattering results [32] emphasize these particular features but are unable to distinguish between different models. It seems natural to view the amorphous ice structure as an end-point in the cooling of liquid water through the supercooled region to the glass formation point without crystallization. The experimental evidence cited above suggests that there is no temperature at which amorphous ice and supercooled water can coexist so that the structural relationship between amorphous ice and liquid water remains paradoxical. It is possible that supercooled water forms a different hydrogen-bonded polytype but this must also be of low density so that it is difficult to conceive an alternative structure. Work is already in progress [33] to try and elucidate this problem but the consensus view at the moment is that vapor-deposited amorphous ice is not equivalent to a glassy state of water. The only way to resolve these uncertainties would be by making diffraction measurements on super-cooled water near the transition point. It is hoped to extend the study by making these measurements but there are various technical difficulties such that an emulsion system [30,34] must be used to prevent crystallization at low temperatures.
7. Conclusion
The present results confirm that amorphous D20 may be represented as a continuous random hydrogen-bonded network. The hydrogen bond distance rod is well defined and has a narrow spatial distribution suggesting that it resembles a normal chemical bond at these low temperatures. The behaviour is
264
M.R. Chowdhury et al. / Structure of amorphous De0 ice
of particular interest in connection with the percolation models being developed [31] to explain anomalous properties of supercooled water. The predictions from the CRN model of Boutron and Alben are in excellent agreement with the data over a range extending above 10 ,~,. We wish to thank Norman Clarke at AERE, Harwell for assistance in the construction and operation of the cryostat. We also wish to thank Dr. Pierre Chieux for assistance during the course of the ILL experiments and Dr. Stewart Cummings for help with the initial computations. Two of us (MRC and JTW) wish to acknowledge support from the Science Research Council during the course of the work. The experimental study is part of a general programme which is supported by the Neutron Beam Reseach Committee of the SERC.
References [I] E.F. Burton and W.T. Oliver, Proc. R. Soc. A153 (1935) 166. [2] T.C. Sivakumar, D. Schuh, M.G. Sceats and S.A. Rice, Chem. Phys. Lett. 48 (1977) 212. T.C. Sivakumar, S.A. Rice and M.G. Sceats, J. Chem. Phys. 69 (1978) 3468. W.G. Madden, M.S. Bergren, R. McGraw and S.A. Rice, J. Chem. Phys. 69 (1978) 3497. [3] G.P. Johari, Phil. Mag. 35 (1977) 1077. S.A. Rice, M.S. Bergren and L. Swingle, Chem. Phys. Len. 59 (1978) 14. [4] P. Bondt, C.R. Acad. Sci., Ser. B 265 (1976) 316. C.G. Venkatesh, S.A. Rice and A.H. Narten, Science 186 (1974) 927. [5] A.H. Narten, C.G. Venkatesh and S.A. Rice, J. Chem. Phys. 64 (1976) 1106. [6] S.A. Rice, private communication (see also second of ref. 2). [7] J.T. Wenzel, C. Linderstrom-Lang and S.A. Rice, Science 187 (1975) 428. [8] P. Boutron and R. Alben, J. Chem. Phys. 62 (1975) 4848. [9] R.J. Temkin, W. Paul and G.A.N. Connell, Adv. Phys. 22 (1973) 581. G. Etherington, A.C. Wright, J.T. Wenzel, J.C. Dore, J.H. Clarke and R.N. Sinclair, J. Non-Cryst. Solids 48 (1982) 265. [10] A.C. Wright and R.N. Sinclair, in: The Physics of Silica and its Interfaces, Ed. J. Pantelides (Pergammon, Oxford, 1978) p. 133, and references cited therein. R.N. Sinclair, J.A.E. Desa, G. Etherington, P.A.V. Johnson and A.C. Wright, J. Non-Cryst. Solids 42 (1980) 107. [11] M.R. Chowdhury and J.C. Dore, J. Non-Cryst. Solids, in press. [12] J.G. Powles, Adv. Phys. 22 (1973) 1. L. Blum and A.H. Narten, Adv. Chem. Phys. 34 (1976) 203. [131 A.C. Wright, Adv. Str. Res. Diff. Math. 5 (1974) 1. C.N.J. Wagner, J. Non-Cryst. Solids 31 (1978) 1; 42 (1980) 3. [14] J.G. Powles, J.C. Dore and D.I. Page, Mol. Phys. 24 (1972) 1025. [15] G. Palinkas, E. Kalman and P. Kovacs, Mol Phys. 34 (1977) 525. W.E. Thiessen and A.H. Narten, in press. [16] J. Reed, J.C. Dore and M.R. Chowdhury, to be submitted to Mol. Phys. [17] M.R. Chowdhury, J.C. Dore and P. Chieux, J. Non-Cryst. Solids 43 (1981) 267. [18] Neutron Beam Facilities at the Institut Laue-Langevin, Ed. B. Maier (1974). [19] G. Walford, J.H. Clarke and J.C. Dore, Mol. Phys. 33 (1977) 25. I.P. Gibson, Thesis (University of Kent, 1978). I.P. Gibson and J.C. Dore, Mol. Phys. in press.
M.R. Chowdhury et al. / Structure of amorphous D20 ice
265
[20] J.G. Powles, Mol. Phys. 42 (1981) 757. [21] J.M. Carpenter and C.A. Pelizzari, Phys. Rev. BI2 (1975) 2391. R.N. Sinclair and A.C. Wright, Nuc. Instr. and Meth. 114 (1974) 451. [22] K. Kuchitsu and kS. Bartell, J. Chem. Phys. 36 (1962) 2460. [23] The structure aspects of crystalline ice phases have been reviewed in various papers such as: N.H. Fletcher, Rep. Prog. Phys. 34 (1971) 913. P.V. Hobbs, Ice Physics (Clarendon, Oxford, 1974) Chap. 1. The current views concerning molecular conformation in ice has been questioned by W. Whalley, Mol. Phys. 28 (1974) 1105 and remain unresolved. Recent neutron diffraction studies of ice by M. Lehmann (private communication) should help to clarify the situation. [24] J. Waser and V. Schomacher, Rev. Mod. Phys. 25 (1953) 671. [25] D.E. Polk, J. Non-Cryst. Solids 5 (1971) 365. D.E. Polk and D.S. Boudreaux, Phys. Rev. Lett. 31 (1973) 92. [26] S.J. Cyvin, Molecular Vibrations and Mean Square Amplitudes (Elsevier, Amsterdam, 1968). [27] LP. Gibson and J.C. Dore, submitted to Mol. Phys. [28] M.G. Sceats, M. Stavola and S.A. Rice, J. Chem. Phys. 70 (1979) 3927. M.G. Sceats and S.A. Rice, J. Chem. Phys. 72 (1980) 3236. A.C. Belch, S.A. Rice and M.G. Sceats, Chem. Phys. Lett. 77 (1981) 455. [29] A review of experiments on the stability of amorphous H20 ice with temperature is presented by P.V. Hobbs, in: Ice Physics (Clarendon, Oxford, 1974) pp. 44-57. [30] A. Angell, in: Water: A Comprehensive Treatise, Vol. 7, Ed. F. Franks, Chap. 5. [31] H.E. Stanley, J. Phys. AI2 (1979) L329. A. Geiger, F.H. Stillinger and A. Rahmann, J. Chem. Phys. 70 (1979) 4185. H.E. Stanley and J. Texeira, J. Chem. Phys. 73 (1980) 3404. H.E. Stanley, J. Texeira, A. Geiger and R.L. Blumberg, Physica 106A (1981) 260. [32] L. Bosio, J. Teceira and H.E. Stanley, Phys. Rev. Lett. 46 (1981) 597. [33] k Bosio, J. Texeira, J.C. Dore and H.E. Stanley, unpublished data. [34] J. Texeira and O. Conde, private communication.