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The application of neutron scattering methods to aqueous electrolyte solutions
Physica 120B (1983)325-334 North-Holland Publishing Company §4.1. NON-PERIODIC SYSTEMS
THE APPLICATION OF NEUTRON SCATTERING METHODS TO AQUEOUS ELECTROLYTE SOLUTIONS G.W. NEILSON H.H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 ITL, UK Invited paper
The following article is a review of recent developments in the application of neutron diffraction and quasi-elastic neutron scattering (QNS) to aqueous electrolyte solutions. The diffraction studies are based on the method isotopic differences which facilitates the extraction of detailed information regarding ionic hydration and ion-ion correlations respectively. The QNS results are derived from data obtained from the INI0 back scattering spectrometer of the ILL. IN10 provides a range of energy and momentum transfer ideal for studies of aqueous systems. The structural and dynamical information obtained from these investigations provide a series of crucial tests of liquid state theories and theoretical models currently used in solution science. Two models are discussed: the Frank-Wen dynamical model of ionic solution and the primitive model of ion-ion structure.
1. I n t r o d u c t i o n
T h e p r o p e r t i e s of electrolyte s o l u t i o n s have fascinated physicists a n d chemists for o v e r two c e n t u r i e s [l]. H o w e v e r , until the r e c e n t a d v a n c e s of diffraction a n d scattering t e c h n i q u e s [2] on the o n e h a n d a n d c o m p u t e r s i m u l a t i o n m e t h o d s on the other, o u r u n d e r s t a n d i n g of the s t r u c t u r e of these systems was e x t r e m e l y limited. This p a p e r will describe the r61e which n e u t r o n scattering (diffraction a n d quasi-elastic scattering, N Q S ) has played in p r o v i d i n g the i n f o r m a t i o n necessary for a d e e p e r u n d e r s t a n d i n g of ionic solutions. Basically, an a q u e o u s electrolyte s o l u t i o n consists of an ionic salt (solute) dissolved in w a t e r (solvent). Fig. 1 r e p r e s e n t s an atomistic view of such a system. This picture should be t h o u g h t of as a ' r e p r e s e n t a t i v e ' snap shot, so that from a series of such pictures o n e could d e d u c e a time a v e r a g e d spatial d i s t r i b u t i o n of ions a n d w a t e r molecules. T h e spatial d i s t r i b u t i o n for a liquid can be expressed m a t h e m a t i c a l l y in t e r m s of a h i e r a r c h y of c o r r e l a t i o n f u n c t i o n s [3], the most f u n d a m e n t a l of which are the pair dist r i b u t i o n f u n c t i o n s , g(r), w h e r e r r e p r e s e n t s an i n t e r p a r t i c l e s e p a r a t i o n . In the p r e s e n t case there
are ten such c o r r e l a t i o n f u n c t i o n s : those which relate to ionic h y d r a t i o n are gM-o, g~-H, gx~o, gX-H, those which describe i o n - i o n effects are gM-~, gM x, gM-M, a n d those which help to specify s o l v e n t - s o l v e n t i n t e r a c t i o n s are gH-H, gH-O, go-o. A k n o w l e d g e of these f u n c t i o n s is a p r e r e q u i s i t e to the c o m p l e t e u n d e r s t a n d i n g of a q u e o u s solutions. It was not until the m e t h o d of n e u t r o n diffraction m u l t i - p a t t e r n structural analysis (first applied to liquid alloys [4] a n d s u b s e q u e n t l y to m o l t e n salts [5]) was applied to solutions in the 1970s that it was realised that m a n y of the pair d i s t r i b u t i o n f u n c t i o n s listed a b o v e could be d e t e r m i n e d . T w o types of e x p e r i m e n t a l m e t h o d have b e e n d e v e l o p e d which are particularly suited to a q u e o u s solutions. T h e first o r d e r difference m e t h o d (section 2.1) can be used to identify l i n e a r c o m b i n a t i o n s of s t r u c t u r e factors so as to give i n f o r m a t i o n r e g a r d i n g ionic hydration. C o n s e q u e n t l y crucial tests can be m a d e of results from q u a n t u m m e c h a n i c a l cluster calc u l a t i o n s [61 a n d c o m p u t e r s i m u l a t i o n [7] used by theorists to predict t h e r m o d y n a m i c a n d structural p r o p e r t i e s of solutions. T h e s e c o n d o r d e r difference m e t h o d (section 2.2) can be used to d e t e r m i n e i n d i v i d u a l pair
0378-4363/83/0000~}(}00/$03.00 © 1983 N o r t h - H o l l a n d and Y a m a d a Science F o u n d a t i o n
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G. W. Neilson / The dynamical structure of ionic solutions
;LJ
interatomic potentials and distances and as such can be used to yield information regarding the structure and dynamics of atomic systems [11]. Neutrons are scattered characteristically by the nuclei of the system and it is this property which facilitates the extraction of detailed structural information for complex systems [12]. For the case of an aqueous solution, MXn in D2Ot it has been possible to focus attention on the structure around both the metal ion M and the anion X by means of the method of isotopic substitution described below.
2. I. First order differences Fig. 1. A microscopic picture of an a q u e o u s solution containing cations • (M+), anions © (X-) and water molecules.
correlation functions and has been successfully used to calculate ion-ion correlation functions. These experimental results provide a sensitive test of the various models and liquid state theories currently used to represent an ionic solution. A m o n g the most frequently used model is the primitive one which seeks to represent the system as hard spherical ions imbedded in a dielectric continuum. This model is particularly straightforward to use, both theoretically and in c o m p u t e r simulation, and has helped our understanding of ionic solutions [8]. Parallel to the structural (time average) investigations outlined above, Hewish, Enderby and Howells [9] have applied the methods of neutron quasi-elastic scattering to aqueous solutions. From their discoveries, a picture of solutions similar to that first proposed by Frank and Wen [10] is beginning to emerge. The idea of a variable second near neighbour zone which crucially governs the widely differing macroscopic properties of solutions has now been clearly demonstrated.
2. The method of isotopic differences
In a neutron diffraction experiment, the significant quantity which can be extracted is the function F(k), k being the amplitude of the scattering vector. F(k) is derived from raw intensity data which have been corrected for absorption, multiple scattering, and put on an absolute scale, barns sterad -j nucleus -1, by reference to a vanadium standard [12]. Although kinematic (Placzek) corrections pose a problem in the determination of absolute F(k)'s, for systems containing hydrogen or deuterium, it turns out that for the difference methods applied to heavier elements (i.e. atomic masses >5) difficulties with data normalisation are substantially overcome [12]. F(k) is a linear combination of partial structure factors, S~(k), and is given by
F(k) = ~'~ ~] c~cff~f~(S~(k)- 1),
(1)
B
where f~ is the coherent scattering amplitude of particle a and c~ is the atomic concentration of particle type a. There are ten partial structure factors for the system MX,, - D 2 0 and these are directly related to the pair distribution functions g~(r) by k sin kr(S~# (k) - 1) d k ,
g ~ (r) - 1 - 2~2r p
(2)
0
Thermal neutrons are a particularly appropriate p r o b e of condensed matter. They have energies and wavelengths c o m m e n s u r a t e with
t D e u t e r i u m is often used in place of hydrogen in neutron diffraction because of the large incoherent scattering of hydrogen.
G. W. Neilson / The dynamical structure of ionic solutions
where p = total n u m b e r density of the system, typically ~0.1 A - . The individual g~a(r)'s can be used to obtain certain useful p a r a m e t e r s for the system, e.g. near neighbour distances and coordination numbers. The n u m b e r of species a around fl in the element of volume 8r at r is given by o
327
AM(k) = 2CoCMfo(fM -- fh)(SMo(k ) - 1)
3
+ 2 c o c ~ f D ( & - fh)(S~:, - 1)
+ 2CxCMfx(fM -- fh)(SMx(k ) - 1) + ch(~-f~)(S~(k)-
Because there are ten g,o(r)'s it is necessary to carry out ten independent scattering experiments before a unique determination can be m a d e of the g~o(r)'s. For the case here this is not feasible at present, and one is limited to a few selective experiments. Eq. (1) shows that F ( k ) depends on the f~'s of the individual species. These f~'s are different for different isotopes of the same nucleus [12]. For example, replacement of the ion M (or X) by one of its isotopes 'M ('X) yields an appreciably different scattering function, F ' ( k ) , fig. 2. In this case, the difference between F ( k ) and F ' ( k ) , the so-called first order difference, gives pertinent information regarding the water conformation (hydration) around the substituted ion. It is straightforward to show [13] that substitution of the cation M by one of its isotopes 'M yields a difference equation of the form:
(4)
-----AI(SMo(k)- 1)+ BI(SMD(k ) - 1) +C~(SMx(k)- I)+ D~(S~(k)-
(3)
n ~ = c~p4rrr 2 8r g~a (r).
1)
1).
(4a)
A similar equation obtains when the anion X is replaced by one of its isotopes 'X. In order to study ion-water correlations in real space, it is necessary to construct the Fourier transform of A (k). The quantity
(~(r)=
1
2 rr2pr
f~(k)sin
(5)
krdk
can be obtained by standard numerical quadrature. It follows from eqs. (2) and (4) that (~rM(r) = A l g M O 4- B I g M D q- C l g M x q- DlgMM 4- E l ,
(6) where E~ -- - ( A ~ + B~ + C~ + D 0. Similarly t~x(r) := A2gxo + B2gxD + C2gxM + Dzgxx + E2 • (6a) In general A and B are much greater than C and
1'50
F (j) ÷Y, ci fi 2 i
(barns str "lnucteug 1)
×*. • x
x
0'90 & x
0.30
,J
I
i
8'.5
17
Fig. 2. Structure factor F(k)+ ~ cef~ for two 4.35 molal solutions of NiCI2 in heavy w a t e r with different isotopic contents of nickel. (N NiCI2 • D 2 0 - crosses; 62NICI2 - D 2 0 - dots).
328
G . W . N e i l s o n / T h e d y n a m i c a l structure o f ionic s o h a i o n s
/9, h e n c e G ( r ) is d o m i n a t e d by M - O ( X - O ) and M - D ( X - D ) correlations, thereby providing structural information regarding ionic hydration. Integration of 4rrrZ(~M(r) dr in the range rl ~ r2 can be used to extract c o o r d i n a t i o n n u m b e r s n~ or nN. F r o m eqs. (3) and (6)
j 4rrr2GM(r)dr ~ A l n ~ / p c o 4
MX,, 'MXn, M'X,,,, 'MX~' . If ,3 x is the difference b e t w e e n the F ( k ) ' s for the first two samples and ( k x ) ' is the difference b e t w e e n the F ( k ) ' s for the third and fourth samples, then
r2
rl
T h e cross correlation function SMX can be o b t a i n e d directly f r o m a 4 pattern analysis where the isotopic states are represented by
3
+ BlnM/pcD + Ei3rr(r>- r3).
(7)
SMx(k)
l-
Ax-(kx)'
(9)
2C~Cx(fM -- f6)(fx-- f ; 0 If peaks are clearly identified in GM(r) over a particular range r~ < r < r2 then n oM and n M can be calculated. 2.2. Second order differences
By extending the diffraction m e a s u r e m e n t s to include a second isotopic substitution of the species M (or X) in M X , , . D 2 0 it is possible to d e t e r m i n e the individual pair correlation functions gMM(r) and gxx(r). If this new system " M X n . D 2 0 (or M " X , . D 2 0 ) gives a structure factor F"(k), then it is straightforward to show that (SMM(k)--
1) =
AMffk)
It can be readily seen from the form of eqs. (4), (8) and (9) that Placzek corrections associated with the difference m e t h o d s will not affect the analysis of the results (for further details see A p p e n d i x to ref. [13]). Because of the n u m b e r of variables in the equations it is necessary that samples used in the difference m e t h o d s are well defined and the experiments are carried out with great care. T h e first o r d e r differences are usually on a scale of -4-0.1 b and the effective cross section of i o n - i o n terms is - 2 0.01 b.
3. Structural properties
AM2(k)
3.1. C a t i o n - w a t e r structure or
( S x x ( k ) - 1 ) - AXe(k) A~>
AX2(k) B(x~ '
w h e r e AMI, AM2 and the coefficients A <2) M ~" are given by AM1 = F -
F',
AM2 = F -
F",
,, , AM = c2(fM -- f'M)(f'M -- fM) B(~d) :
2 ,, , CM(fM-fM)(fM
AX1 = F - F ' , 2 *-X'dt (2) = C x ( f x --XR(2) =
2 CX(fX
--
-
-
t, fM),
AX2 = F - F " , t
t
tt
fx)(fx - fx)
tt t tt -- fx)(fx -- fx )
(8a)
]~(2)
" ''--X
T h e cation c o n f o r m a t i o n in a q u e o u s solution has been d e t e r m i n e d in a large n u m b e r of cases (table 1). It is perhaps not surprising that the first application of the first o r d e r difference m e t h o d was a 4.35 molal solution of NiC12 in heavy water [13]. Isotopes of 62Ni, a~Ni and SaNi were used in varying a m o u n t s so as to optimise the f o r m of ANi(k). Fig. 3 shows the first o r d e r difference function, (3(r), for a 4.35 m NiC12.D20 solution. T h e first two peaks represent N i - O and N i - D correlations respectively and being in the numerical ratio of 6 : 12 (eq. (7)) it is inferred that there are about 6 water molecules a r o u n d a Ni 2+ ion. F r o m the near n e i g h b o u r distances rhi-O and rhi D and assuming rOD = 1 A and / _ D O D = 104.5 ° it is calculated that the angle between the
G. W. Neilson / The dynamical structure of ionic solutions
329
Table I Scattering lengths and sample parameters for ionic hydration
Electrolyte solution
Substituted species
Isotopes
Scattering a) lengths (x 10 12cm)
LiCI
Li*
(6Li; 7Li)
CaCI2
Ca 2+
(NCa: 44Ca)
NiCl2
Ni2+
Molality
A (x 10 2 b)
B (x 11} 2 b)
C (x 10 3 b)
D (x 10 3 b)
((). 18; -0.233)
9.95 3.57
1.66 0.695
3.83 1.60
-
1.34 -0.20
4.38
(0.466; 0.18)
4.49 2.80 1.1}
0.30 0.20
0.91) 1t.38 0.05
0.16
0.07
0.68 0.47 0.18
4.41 3.05 1.46 0.85 0.42
1.74 1.26 0.64 0.385 0.194
4.00 2.90 1.46 0.885 1/.446
5.1)5 2.52 0.6 l 0.22
1!.32
(1.03; -0.79)
(NNi; 62Ni)
0.60
I).07 0.01
0.15 {1.04 0.013 I).1111338
CuCI2
Cu 2+
(63Cu; 6SCu)
(0.67; 1.11)
4.32
0.415
1/.954
0.054 ].2
ND4CI
N
(14N; 15N)
(0.936; 0.65)
5.1)
0.26
1}.716bl
0.43
LiCI
CI
(35C1; 37C1)
(1.17; 0.35)
9.95 3.57
1.66 0.695
3.83 1.60
- 1.34
4.38
-0.20
{1.60
0.55 0.25
NaCI
CI-
(35CI; 37C1)
(1.17; 0.35)
5.32
0.99
2.27
0.65
] .37
RbCI
CI
(350; 37C1)
(1.17; 0.35)
2.99
0.58
1.33
0.52
0.46
CaC12
C1-
(35C1; 37C1)
(1.17; 0.35)
4.49
1.60
3.68
1.16
3.77
NiCI2
CI-
(35C1; 37C1)
(1.17; 0.35)
4.35
1.57
3.63
2.4{}
3.62)
NaNO3
N
(14N; ISN)
(0.936; 0.65)
7.8
0.54b)
0.846
0.036
0.079
a Calculated from mass spectroscopic measurements on isotopic abundance. b Includes both intra- and inter-atomic oxygen.
angle
02o DN,(r }
000
I I' ....
-':,.%
-"'..._..... .-~.5_":J":.:
.
B.5 "'":"; r {A....)
-
,-. .................
17' S
--
"'IQ
~0.20
Fig. 3. Ni 2+ ion distribution function (3si(r) for a 4.35 molal NiCI: heavy water solution. The inset at top left represents the local Ni2÷ water molecule conformation inferred from the results.
N i - O axis a n d t h e p l a n e o f t h e w a t e r m o l e c u l e ( a n g l e o f tilt) is - 4 2 ° (fig. 3 inset)• T h i s a n g l e is intermediate between the lone pair configuration - 5 5 ° a n d t h e d i p o l e c o n f i g u r a t i o n 0 ° a n d is a s t r o n g f u n c t i o n o f c o n c e n t r a t i o n [14]. T h i s e f f e c t is, w e b e l i e v e , a c o n s e q u e n c e o f d i s t o r t i o n s o f t h e h y d r a t i o n s p h e r e s as t h e p a c k i n g f r a c t i o n is increased. There have been no theoretical studies a i m e d at u n d e r s t a n d i n g this o b s e r v a t i o n . In o r d e r t o d e t e r m i n e w h e t h e r it w a s a c o n s e q u e n c e o f C1- i o n p e r t u r b a t i o n o f t h e N i 2+ ions, an experiment was undertaken on an 3.8m N i C I O n . D 2 0 s o l u t i o n w h e r e it is k n o w n t h a t i o n p a i r i n g is u n l i k e l y . N e w s o m e et al. [15] s h o w e d t h a t (3Ni(r) is e s s e n t i a l l y i n d e p e n d e n t o f t h e a n i o n ( t a b l e II). S i m i l a r s t r u c t u r a l p a t t e r n s t o fig. 3 h a v e b e e n o b s e r v e d f o r t h e i o n s o f Li + a n d C a 2+ [2], in-
330
G. W. Neilson / The dynamical structure of ionic solutions
Table II Cation hydration determined by neutron diffraction
Ion
Solute
Molality
Ion-oxygen distance (A)
Ion~leuterium distance (,~)
0"1
Coordination number
Li +
LiC1
9.95 3.57
1.95 -+(I.02 1.95 -+0.0
2.5(/-+ 0.02 2.55 -+ 0.02
52° ± 5° 40 ° -+ 5°
3.3 + 0.5 5.5 ± 0.3
Ca 2+
CaCI:
4.49 2.80 1.0
2.41 -+0.03 2.39 ± 0.02 2.46 ± 0.03
3.04 _+(I.(/3 3.02 -+0.03 3.(/7 ± 0.03
34° ± 9° 34° ± 9° 38° -+9°
6.4 ± 0.3 7.2 ± 0.2 1(1.(t± (1.6
Ni2*
NiCI2
4.41 3.05 1.46 0.85 (I.46 0.086
2.07 -+0.02 2.07 ± 0.02 2.07 ± 0.(12 2.09 -+0.02 2.10 ± 0.02 2.07 _+0.03
2.67 ± 0.02 2.67 ± 0.02 2.67 + 0.02 2.76 -+0.02 2.80 ± 0.02 2.80 _+0.(/3
42 ° ± 8° 42° ± 8° 42 ° ± 8° 27° ± 10° 17° ± 10° 0° -+ 20°
5.8 ± 0.2 5.8 ± 0.2 5.8 ± 0.3 6.6 ± 0.5 6.8 ± 0.8 6.8 ± 0.8
Cu 2+
CuCI2
4.32
2.05 _+0.03
2.56 _+0.10
58° ± 10°
2.3 ± 0.3
a Computed on the basis of rot) = 1 ,~ and D 0 D = 104.5°.
d i c a t i n g an a l m o s t i d e n t i c a l i o n - w a t e r c o n f o r m a t i o n (inset fig. 3). H o w e v e r , in b o t h c a s e s t h e r e is an a p p r e c i a b l e c h a n g e in c o o r d i n a t i o n n u m b e r u p o n d i l u t i o n [16] ( t a b l e II). T h e r e h a v e b e e n s e v e r a l ab i n i t i o c l u s t e r c a l c u l a t i o n s [6] a n d c o m p u t e r s i m u l a t i o n s s t u d i e s [7] of t h e c o o r d i n a t i o n o f Li + t o w a t e r . G e n e r a l l y it is f o u n d that the ion-oxygen distances are better represented by ab i n i t i o q u a n t u m chemistry whereas the computer simulation tends to give a better agreement with coordination numbers. T w o o t h e r c a t i o n s h a v e b e e n s t u d i e d , N D ~ in 5 m o l a l N D 4 C I - D 2 0 w h e r e t h e i s o t o p e s 14N a n d ~SN w e r e u s e d [17] a n d C u 2* in 4.32 m o l a l C u C 1 2 . D 2 0 u s i n g t h e i s o t o p e s 63Cu a n d 6SCu [18]. In t h e f o r m e r c a s e a b r o a d u n r e s o l v e d f e a t u r e is o b s e r v e d in (~N(r). T h i s r e s u l t is in b r o a d agreement with a molecular dynamics simulation s t u d y [19], b o t h r e s u l t s g i v i n g a c o o r d i n a t i o n number of between 10-11 w a t e r m o l e c u l e s a r o u n d t h e N D I ion. T h e r e is, h o w e v e r , a l a r g e discrepancy arising from the overall shape of G s ( r ) . T h e s i m u l a t e d G y ( r ) is m u c h s h a r p e r a n d appears to over-estimate either the hydrating s t r e n g t h o f t h e N D ~ ion, o r t h e s p h e r i c a l n a t u r e of its e f f e c t i v e p o t e n t i a l . A m u c h m o r e c o m p l i c a t e d (3cu(r) is c a l c u l a t e d f o r t h e c o o r d i n a t i o n of C u 2+ in s o l u t i o n [18]. A
2 + 4 c o o r d i n a t i o n a r o u n d t h e C u 2+ ion is c l e a r l y e v i d e n t c o n f i r m i n g t h e v i e w t h a t local c h e m i c a l e f f e c t s d o m i n a t e t h e c o n f o r m a t i o n of t h e i o n water subsystem. Effective ion-water potentials u s e d in s i m u l a t i o n s t u d i e s m u s t b e sufficiently r e a l i s t i c to r e f l e c t this fact.
3.2. A n i o n - w a t e r
structure
T h e m o s t e x t e n s i v e s t u d i e s of a n i o n s h a v e b e e n c o n c e r n e d w i t h t h e C1- i o n w h e r e i s o t o p e s 35C1 a n d 37C1 w e r e u s e d . T h e r e a p p e a r s t o b e a u n i v e r s a l i t y of s t r u c t u r e of C1- in s o l u t i o n [20] as e v i d e n c e f r o m t y p i c a l Gc~(r)'s d e m o n s t r a t e s , fig. 4 a n d t a b l e III. T h e s k e t c h in t h e u p p e r r i g h t of fig. 4 s h o w s t h e c o n f o r m a t i o n of t h e w a t e r m o l e c u l e w i t h r e s p e c t to C1-. T h e d i s t a n c e s rc~o a n d rc~D are in e x c e l l e n t agreement with those obtained from quantum m e c h a n i c a l c a l c u l a t i o n s [21]. In g e n e r a l , r a o is greater than that obtained from X-ray diffraction, a c o n s e q u e n c e , w e b e l i e v e , of t h e i n h e r e n t l y l o w d i s c r i m i n a t i o n of t o t a l X - r a y p a t t e r n s of s o l u t i o n s [12]. Computer simulations based on the ST2 p o t e n t i a l p r e d i c t s i g n i f i c a n t c o u n t e r ion effects w h i c h a r e c l e a r l y at v a r i a n c e w i t h t h e o b s e r v a t i o n s h e r e . It will b e i n t e r e s t i n g to s e e if o t h e r
G. W. Neilson / The dynamical structure of ionic solutions
3.3. Ion-ion structure
0,10 I _D(1 } 0
c~--
D(I)
0
DI2i
i; -0,0
::
E
~4~._. ~ w ( 3
0,051
":
110
210
L r(~,)
6'°
;0
331
8°
Fig. 4. t~o(r) for two alkali halide heavy water solutions: Full curve-9.95 molal LiCI in D20; crossed curve-5.32 molal NaCI in DzO after scaling by concentration factor of 1.87.
types of potentials, e.g. the 'central force' model, d e m o n s t r a t e that (~o(r) is indeed independent of counter ion and concentration [7]. A recent study has been carried out on the anion NO~ in an 7 . 8 m solution of NaNO3 in heavy water [22]. Nitrogen isotopes 14N and 15N were used and the difference method was applied (eq. (4)). The results show that the NO5 has a unique local conformation. There are between one and two nearest neighbour D 2 0 molecules and about 3 D - b o n d e d next nearest neighbour DzO molecules in the immediate environment of the NO~. Results from X-ray diffraction [23] were unable to discern this structure because of the lower discrimination of the results together with the overlapping correlations from N and O atoms.
The first and best system on which to use the second order differences method (eqs. (8, 9)) is nickel chloride in heavy water. In a series of recent experiments on a 4.35 molal NiCI2 heavy water solution it has been possible to obtain the three ion-ion partial structure factors [24], SNiNi, SNio(k) and S a a ( k ) . H o w e v e r the experiments are at the limit of existing technology and there are large errors associated with the data (see, for example, fig. 5). The corresponding g(r)'s show that there is a well-defined structure for C1-CI and Ni-Cl coordination. However, the coordination for Ni-Ni is not well defined although the o cut-off distance at r = 4 . 1 A is remarkably insensitive to the large errors in the raw data. A distance comparable to this has been calculated theoretically in iron chloride solutions 1251. It will be interesting to see whether the various models and liquid state theories are capable of generating the above results. Preliminary calculations based on a primitive model within the mean spherical approximation M S A show large discrepancies between theory and experiment [261. S o a ( k ) has been obtained in two other solutions [2]. In a 5.32 molal NaCl solution the results were shown to be consistent with the M S A version of the primitive model [6]. A much more sensitive test of the theory has been carried out by Copestake and Neilson who determined S o o ( k ) for a 14.9 molal LiCl solution. A
T a b l e II1 A n i o n h y d r a t i o n d e t e r m i n e d by n e u t r o n diffraction
Ion
Solute
Molality
X-D(1) (A)
X-O (,~)
CI
LiCI LiCI NaCI RbCI CaCI2 NiC12
3.57 9.95 5.32 4.36 4.49 4.35
2.25 2.22 2.26 2.26 2.25 2.29
3.34 3.29 3.20 3.20 3.25 3.20
NaNO3
7.8
2.05 -+ 0.02 b)
NO~
± 0.02 ± 0.02 - 0.04 -+ 0.04 ± 0.02 ± 0.02
a C o m p u t e d on t h e basis of rOD = 1 A (see fig. 4). b D i s t a n c e s f r o m n i t r o g e n nucleus.
± 0.05 -+ 0.04 _+ 0.05 ± 0.05 ± 0.04 ± 0.04
2.65 ± 0. l0 b)
X-D(2) (,~)
~oa)
Coordination number
3.50-3.68 3.55-3.65 3.40-3.50
0° 0° 0°-20° 0°-20 ° 00-7 ° 5 ° - 11 °
5.9 5.3 5.5 5.8 5.8 5.7
-
20 ° -+ 10°
1.3 -+ 0.2
-+ 0.2 -+ 0.2 ± 0.4 ± 0.3 ± 0.2 ± 0.2
G.W. Neilson / The dynamical structure of ionic solutions
332
energy transfers accompany the neutron scattering p r o c e s s it is n e c e s s a r y to g e n e r a l i s e o u r c o n c e p t s of t h e s t r u c t u r e of a system. It was V a n H o v e [27] w h o s h o w e d t h a t t h e r e exists a link b e t w e e n the g e n e r a l i s e d c o r r e l a t i o n function G(r, t) for a liquid a n d its n e u t r o n s c a t t e r i n g cross section. (G(r, t) can b e t h o u g h t of as the p r o b a b i l i t y of finding a p a r t i c l e at t h e origin r = 0 at t = O a n d a p a r t i c l e at r at t i m e t.) It is convenient both physically and mathematically to d i v i d e G(r, t) into a self p a r t G S a n d a distinct part G d
(7)
G = Ga+ G s , 20
{ -30
/
t
$iol
[
1
I 2
I
Fig. 5. SNisi(k) for 4.35 molal nickel chloride in heavy water. SNiNi(0 ) is calculated from thermodynamics [12], and data between 0 ~ k ~ 0 . 5 , ~ -1 have been interpolated. The dots represent an SNiNi(k ) which is derived from a well behaved gNiNi(r).
t h e o r e t i c a l c a l c u l a t i o n b a s e d on t h e M S A with a p r i m i t i v e m o d e l in which t h e b u l k d i e l e c t r i c c o n s t a n t of the s o l u t i o n iSo25, t h e Li + a n d CI d i a m e t e r s a r e 5 A a n d 3 A r e s p e c t i v e l y shows t h a t S a c l ( k ) is in g o o d q u a l i t a t i v e a g r e e m e n t with t h e e x p e r i m e n t a l result [26]. It will b e n e c e s s a r y to c a r r y o u t m o r e e x p e r i m e n t s using t h e s e c o n d o r d e r d i f f e r e n c e techn i q u e so that o n e can establish t h e e x t e n t to which p r i m i t i v e m o d e l s a r e a p p l i c a b l e to ionic solutions.
4. Neutron quasielastic scattering T h e r m a l n e u t r o n s a r e a s u i t a b l e p r o b e of b o t h the s t r u c t u r a l a n d d y n a m i c a l p r o p e r t i e s of t h e p a r t i c l e s of a c o n d e n s e d m a t t e r system. W h e n
w h e r e G ~ r e p r e s e n t s t h e p r o b a b i l i t y of finding a p a r t i c l e at r = 0 a n d t = 0 a n d t h e s a m e p a r t i c l e at r at t i m e t. It t u r n s o u t that, [2], G s is directly r e l a t e d to t h e i n c o h e r e n t s c a t t e r i n g cross section. This is p a r t i c u l a r l y c o n v e n i e n t for a q u e o u s s o l u t i o n s w h e r e t h e d o m i n a n c e of the h y d r o g e n i n c o h e r e n t cross section is a distinct a d v a n t a g e . F o r e x a m ple, in the s y s t e m M X n ' H 2 0 t h o u g h t h e r e are f o u r t e e n t e r m s which can c o n t r i b u t e to the scatt e r i n g p a t t e r n (4 self t e r m s a n d 10 i n t e r f e r e n c e t e r m s ) the t e r m GS(r, t) d o m i n a t e s t h e s c a t t e r i n g patterns. F o r t r a n s l a t i o n a l diffusion, GS(r, t) t a k e s t h e f o r m of a G a u s s i a n [28]. T h e e x p e r i m e n t a l l y accessible s c a t t e r i n g law, SS(k, w), t h e r e f o r e has a Lorentzian form
SS(k, w ) -
1 Dk 2 rr ( D k )2 + w 2
w h e r e h w is t h e e n e r g y t r a n s f e r in the s c a t t e r i n g p r o c e s s , hk is the m o m e n t u m t r a n s f e r a n d D is t h e t r a n s l a t i o n a l diffusion coefficient. In o r d e r that this p a r t i c u l a r f o r m can b e obtained experimentally three conditions must be met. (i) T h e self t e r m ~> distinct t e r m s . This is easily m e t for s y s t e m s c o n t a i n i n g h y d r o g e n . (ii) Dk2>> rro~,, i.e. all r o t a t i o n a l m o t i o n has b e e n a v e r a g e d out. (iii) t h e r e s o l u t i o n of the s p e c t r o m e t e r is such that g~o ~ 2 D k 2.
O. W. Neilson I The dynamical structure of ionic solutions
Conditions (ii) and (iii) imply that experiments must be carried out with high resolution --1 ~ e V at k ~ 1.~-1. The backscattering technique exemplified by the I N I O spectrometer at I L L G r e n o b l e is ideally suited for such experiments. Hewish et al. [9] carried out a series of experiments on a variety of aqueous solutions. The results were derived from I N I O data and were arrived at after careful corrections for multiple scattering. F r o m their results they were able to identify the two limiting cases known to occur for proton rate processes in solution [29]. In the first case, fast exchange, the results can be analysed as a single Lorentzian which gives an average D, for all the protons in the solution. This type of behaviour is observed in weakly hydrating systems such as CsCI. In the second case, the slow exchange, it was possible to identify two types of proton behaviour. This observation was based on the necessity of a two Lorentzian fit to the experimental results (fig. 6). Two diffusion coefficients were identified, D~ = Dio., that characteristic of the strongly hydrating
333
It) ~a t~
N o o
Dion I
1
1
;
2 (molar)
C
3
to o
...... I
i
~
B2
- - DseC
Dion 1 1
I 3 C (motat)
Fig. 7. T h e c a t i o n i c d i f f u s i o n c o e f f i c i e n t Dio,, t o g e t h e r w i t h t h e fitted v a l u e s f o r £)2 a n d c a l c u l a t e d v a l u e s f o r D~c f o r t w o s o l u t i o n s s t u d i e d . (a) N i C I 2 - H 2 0 , (b) M g C I 2 . H 2 0 . D a t a measurements were used with the neutron results to produce c u r v e £)2-
~0
30
~
2O
/
.Z 10:1
-15
k. ,'%.Cbl
#
-10
-5
,
0
5
10
15
Fig. 6. C u r v e a is a n a t t e m p t t o fit t h e o b s e r v e d S(k, to) w i t h a single L o r e n t z i a n f o r 3 m o l a l NiCI2 in H 2 0 . C u r v e b is a t w o L o r e n t z i a n fit of t h e s a m e d a t a . D~ = DNI w a s t a k e n f r o m t r a c e r m e a s u r e m e n t s (0.29 × 10 -5 c m 2 s 1) a n d /32 w a s f o u n d to h a v e t h e v a l u e 1.24 x 10-5 c m 2 s -j.
ion, and 132 which is an average of protons in all other environments. Systems exhibiting this latter behaviour are solutions of NiCl2 and MgC12. The nickel chloride results were analysed using the value of Dio, known from tracer measuremeres and the diffraction results referred to in section 3.1 which were used to correctly weigh the two Lorentzians. It was thereby possible to extract the diffusion coefficient (Dsec) of a second zone of hydrated water (fig. 7). This result adds credence to a model proposed by Frank and Wen [8] which predicted that, in general, several zones of hydration, each with a characteristic mobility, will exist. The full implications of this work are still being assessed. 5. Conclusions and future prospects
T h e successful application of a variety of neutron scattering methods to ionic solutions is
334
G. W. NeiLgon / The dynamical structure of ionic" solutions
providing a clearer insight into the origins of their wide ranging properties. The close link between computer simulation methods and experiment at the microscopic level represents a m a j o r step forward in our understanding of these complex systems. With the advent of higher fluxes of neutrons from pulsed sources the above methods can be pushed to greater limits. Not only will it be possible to apply the isotope enrichment method to systems with less favourable nuclei but it will b e c o m e feasible to carry out investigations over wider ranges of pressure, t e m p e r a t u r e and concentration where interesting effects are known to occur [1].
Acknowledgements It is a pleasure to thank Professor J.E. Enderby w h o s e help and advice have been invaluable in writing this paper. I also ack n o w l e d g e the assistance of Alan Copestake w h o has carried out much of the theoretical studies on the ion-ion structure.
References [1] For a comprehensive review of the literature see Water, a Comprehensive Treatise, Vols. 1-7, ed. F. Franks (Plenum, New York, 1973--1981). [2] J.E. Enderby and G.W. Neilson, Rep. Prog. Phys. 44 (1981) 593. [31 J.P. Hansen and I.R. Macdonald, Theory of Simple Liquids (Academic Press, London and New York, 1976). [4] J.E. Enderby, D.M. North and P.A. Egelstaff, Phil. Mag. 14 (1966) 961. [5] D.I. Page and K. Mika, J. Phys. C4 (1971) 3034.
[6] H. Kistenmacher, H. Popkie and E. Clementi, J. Chem. Phys. 59 (1973) 5842. [7] M. Mezei and D.L. Beveridge, J. Chem. Phys. 74 (1981) 69{)2. [8] J.E. Enderby and O.W, Neilson, Adv. in Physics 29 (1980) 323. [91 N.A. Hewish, J.E. Enderby and W.S. Howells, Phys. Rev. Letts. 48 (1982) 756. {10] H.S. Frank and W.Y. Wen, Discuss. Faraday Soc. 24 (1957) 133. Ill] P.A. Egelstaff, An Introduction to the Liquid State (Academic Press, London and New York, 1967) p. 100. [12] J.E. Enderby and G.W. Neilson, in Water, a Comprehensive Treatise, Vol. 6, ed, F. Franks (Plenum, New York, 1979)p. 1. [13] A.K. Soper, G.W. Neilson, J.E. Enderby and R.A. Howe, J. Phys. C10 (1977) 1793. [14] G.W. Neilson and J.E. Enderby, J. Phys. Cll (1978) L625. [15] J.R. Newsome, M.R. Sandstr6m, G.W. Neilson and J.E. Enderby, Chem. Phys. Letts. 82 (1981) 399. [16] N.A. Hewish, G.W. Neilson and J.E. Enderby, Nature 297 (1982) 138. [17] N.A. Hewish and G.W. Neilson, Chem. Phys. Letts. 84 (1981) 425. [18] G.W. Neilson, J. Phys. C15 (1982) L233. [19] G. Szasz and K. Heinzinger, Z. Naturforsch. 34a (1979) 840. [201 S. Cummings, J.E. Enderby, G.W. Neilson, J.R. Newsome, R.A. Howe, W.S. Howells and A.K. Soper, Nature 287 (1980) 714. [21] H. Kistermasher, H. Popkie, E. Clementi and R.O. Watts, J. Chem. Phys. 6l (1974) 799. [22] G.W. Neilson and J.E. Enderby, J. Phys. C15 (1982) 2347. [23] R. Camaniti, G. Licheri, G. Paschina, G. Piccaluga and G. Pinna, J. Chem. Phys. 72 (1980) 4522. [24] G.W. Neilson and J.E. Enderby, submitted to Proc. Roy. Soc. [25] B.L. Tembe, H.L. Friedman and M.D. Newton, J. Chem. Phys. 76 (1982) 149{). [26] A. Copestake, private communication. [27] L. Van Hove, Phys. Rev. 95 (1954) 249. [28] J.R.D. Copley and S.W. Lovesey, Rep. Prog. Phys. 38 (1975) 461. [29] H.G. Hertz, in Water, a Comprehensive Treatise, Vol. 3, ed. F. Franks (Plenum, New York, 1973) p. 301.
THE APPLICATION OF NEUTRON SCATTERING METHODS TO AQUEOUS ELECTROLYTE SOLUTIONS G.W. NEILSON H.H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 ITL, UK Invited paper
The following article is a review of recent developments in the application of neutron diffraction and quasi-elastic neutron scattering (QNS) to aqueous electrolyte solutions. The diffraction studies are based on the method isotopic differences which facilitates the extraction of detailed information regarding ionic hydration and ion-ion correlations respectively. The QNS results are derived from data obtained from the INI0 back scattering spectrometer of the ILL. IN10 provides a range of energy and momentum transfer ideal for studies of aqueous systems. The structural and dynamical information obtained from these investigations provide a series of crucial tests of liquid state theories and theoretical models currently used in solution science. Two models are discussed: the Frank-Wen dynamical model of ionic solution and the primitive model of ion-ion structure.
1. I n t r o d u c t i o n
T h e p r o p e r t i e s of electrolyte s o l u t i o n s have fascinated physicists a n d chemists for o v e r two c e n t u r i e s [l]. H o w e v e r , until the r e c e n t a d v a n c e s of diffraction a n d scattering t e c h n i q u e s [2] on the o n e h a n d a n d c o m p u t e r s i m u l a t i o n m e t h o d s on the other, o u r u n d e r s t a n d i n g of the s t r u c t u r e of these systems was e x t r e m e l y limited. This p a p e r will describe the r61e which n e u t r o n scattering (diffraction a n d quasi-elastic scattering, N Q S ) has played in p r o v i d i n g the i n f o r m a t i o n necessary for a d e e p e r u n d e r s t a n d i n g of ionic solutions. Basically, an a q u e o u s electrolyte s o l u t i o n consists of an ionic salt (solute) dissolved in w a t e r (solvent). Fig. 1 r e p r e s e n t s an atomistic view of such a system. This picture should be t h o u g h t of as a ' r e p r e s e n t a t i v e ' snap shot, so that from a series of such pictures o n e could d e d u c e a time a v e r a g e d spatial d i s t r i b u t i o n of ions a n d w a t e r molecules. T h e spatial d i s t r i b u t i o n for a liquid can be expressed m a t h e m a t i c a l l y in t e r m s of a h i e r a r c h y of c o r r e l a t i o n f u n c t i o n s [3], the most f u n d a m e n t a l of which are the pair dist r i b u t i o n f u n c t i o n s , g(r), w h e r e r r e p r e s e n t s an i n t e r p a r t i c l e s e p a r a t i o n . In the p r e s e n t case there
are ten such c o r r e l a t i o n f u n c t i o n s : those which relate to ionic h y d r a t i o n are gM-o, g~-H, gx~o, gX-H, those which describe i o n - i o n effects are gM-~, gM x, gM-M, a n d those which help to specify s o l v e n t - s o l v e n t i n t e r a c t i o n s are gH-H, gH-O, go-o. A k n o w l e d g e of these f u n c t i o n s is a p r e r e q u i s i t e to the c o m p l e t e u n d e r s t a n d i n g of a q u e o u s solutions. It was not until the m e t h o d of n e u t r o n diffraction m u l t i - p a t t e r n structural analysis (first applied to liquid alloys [4] a n d s u b s e q u e n t l y to m o l t e n salts [5]) was applied to solutions in the 1970s that it was realised that m a n y of the pair d i s t r i b u t i o n f u n c t i o n s listed a b o v e could be d e t e r m i n e d . T w o types of e x p e r i m e n t a l m e t h o d have b e e n d e v e l o p e d which are particularly suited to a q u e o u s solutions. T h e first o r d e r difference m e t h o d (section 2.1) can be used to identify l i n e a r c o m b i n a t i o n s of s t r u c t u r e factors so as to give i n f o r m a t i o n r e g a r d i n g ionic hydration. C o n s e q u e n t l y crucial tests can be m a d e of results from q u a n t u m m e c h a n i c a l cluster calc u l a t i o n s [61 a n d c o m p u t e r s i m u l a t i o n [7] used by theorists to predict t h e r m o d y n a m i c a n d structural p r o p e r t i e s of solutions. T h e s e c o n d o r d e r difference m e t h o d (section 2.2) can be used to d e t e r m i n e i n d i v i d u a l pair
0378-4363/83/0000~}(}00/$03.00 © 1983 N o r t h - H o l l a n d and Y a m a d a Science F o u n d a t i o n
326
G. W. Neilson / The dynamical structure of ionic solutions
;LJ
interatomic potentials and distances and as such can be used to yield information regarding the structure and dynamics of atomic systems [11]. Neutrons are scattered characteristically by the nuclei of the system and it is this property which facilitates the extraction of detailed structural information for complex systems [12]. For the case of an aqueous solution, MXn in D2Ot it has been possible to focus attention on the structure around both the metal ion M and the anion X by means of the method of isotopic substitution described below.
2. I. First order differences Fig. 1. A microscopic picture of an a q u e o u s solution containing cations • (M+), anions © (X-) and water molecules.
correlation functions and has been successfully used to calculate ion-ion correlation functions. These experimental results provide a sensitive test of the various models and liquid state theories currently used to represent an ionic solution. A m o n g the most frequently used model is the primitive one which seeks to represent the system as hard spherical ions imbedded in a dielectric continuum. This model is particularly straightforward to use, both theoretically and in c o m p u t e r simulation, and has helped our understanding of ionic solutions [8]. Parallel to the structural (time average) investigations outlined above, Hewish, Enderby and Howells [9] have applied the methods of neutron quasi-elastic scattering to aqueous solutions. From their discoveries, a picture of solutions similar to that first proposed by Frank and Wen [10] is beginning to emerge. The idea of a variable second near neighbour zone which crucially governs the widely differing macroscopic properties of solutions has now been clearly demonstrated.
2. The method of isotopic differences
In a neutron diffraction experiment, the significant quantity which can be extracted is the function F(k), k being the amplitude of the scattering vector. F(k) is derived from raw intensity data which have been corrected for absorption, multiple scattering, and put on an absolute scale, barns sterad -j nucleus -1, by reference to a vanadium standard [12]. Although kinematic (Placzek) corrections pose a problem in the determination of absolute F(k)'s, for systems containing hydrogen or deuterium, it turns out that for the difference methods applied to heavier elements (i.e. atomic masses >5) difficulties with data normalisation are substantially overcome [12]. F(k) is a linear combination of partial structure factors, S~(k), and is given by
F(k) = ~'~ ~] c~cff~f~(S~(k)- 1),
(1)
B
where f~ is the coherent scattering amplitude of particle a and c~ is the atomic concentration of particle type a. There are ten partial structure factors for the system MX,, - D 2 0 and these are directly related to the pair distribution functions g~(r) by k sin kr(S~# (k) - 1) d k ,
g ~ (r) - 1 - 2~2r p
(2)
0
Thermal neutrons are a particularly appropriate p r o b e of condensed matter. They have energies and wavelengths c o m m e n s u r a t e with
t D e u t e r i u m is often used in place of hydrogen in neutron diffraction because of the large incoherent scattering of hydrogen.
G. W. Neilson / The dynamical structure of ionic solutions
where p = total n u m b e r density of the system, typically ~0.1 A - . The individual g~a(r)'s can be used to obtain certain useful p a r a m e t e r s for the system, e.g. near neighbour distances and coordination numbers. The n u m b e r of species a around fl in the element of volume 8r at r is given by o
327
AM(k) = 2CoCMfo(fM -- fh)(SMo(k ) - 1)
3
+ 2 c o c ~ f D ( & - fh)(S~:, - 1)
+ 2CxCMfx(fM -- fh)(SMx(k ) - 1) + ch(~-f~)(S~(k)-
Because there are ten g,o(r)'s it is necessary to carry out ten independent scattering experiments before a unique determination can be m a d e of the g~o(r)'s. For the case here this is not feasible at present, and one is limited to a few selective experiments. Eq. (1) shows that F ( k ) depends on the f~'s of the individual species. These f~'s are different for different isotopes of the same nucleus [12]. For example, replacement of the ion M (or X) by one of its isotopes 'M ('X) yields an appreciably different scattering function, F ' ( k ) , fig. 2. In this case, the difference between F ( k ) and F ' ( k ) , the so-called first order difference, gives pertinent information regarding the water conformation (hydration) around the substituted ion. It is straightforward to show [13] that substitution of the cation M by one of its isotopes 'M yields a difference equation of the form:
(4)
-----AI(SMo(k)- 1)+ BI(SMD(k ) - 1) +C~(SMx(k)- I)+ D~(S~(k)-
(3)
n ~ = c~p4rrr 2 8r g~a (r).
1)
1).
(4a)
A similar equation obtains when the anion X is replaced by one of its isotopes 'X. In order to study ion-water correlations in real space, it is necessary to construct the Fourier transform of A (k). The quantity
(~(r)=
1
2 rr2pr
f~(k)sin
(5)
krdk
can be obtained by standard numerical quadrature. It follows from eqs. (2) and (4) that (~rM(r) = A l g M O 4- B I g M D q- C l g M x q- DlgMM 4- E l ,
(6) where E~ -- - ( A ~ + B~ + C~ + D 0. Similarly t~x(r) := A2gxo + B2gxD + C2gxM + Dzgxx + E2 • (6a) In general A and B are much greater than C and
1'50
F (j) ÷Y, ci fi 2 i
(barns str "lnucteug 1)
×*. • x
x
0'90 & x
0.30
,J
I
i
8'.5
17
Fig. 2. Structure factor F(k)+ ~ cef~ for two 4.35 molal solutions of NiCI2 in heavy w a t e r with different isotopic contents of nickel. (N NiCI2 • D 2 0 - crosses; 62NICI2 - D 2 0 - dots).
328
G . W . N e i l s o n / T h e d y n a m i c a l structure o f ionic s o h a i o n s
/9, h e n c e G ( r ) is d o m i n a t e d by M - O ( X - O ) and M - D ( X - D ) correlations, thereby providing structural information regarding ionic hydration. Integration of 4rrrZ(~M(r) dr in the range rl ~ r2 can be used to extract c o o r d i n a t i o n n u m b e r s n~ or nN. F r o m eqs. (3) and (6)
j 4rrr2GM(r)dr ~ A l n ~ / p c o 4
MX,, 'MXn, M'X,,,, 'MX~' . If ,3 x is the difference b e t w e e n the F ( k ) ' s for the first two samples and ( k x ) ' is the difference b e t w e e n the F ( k ) ' s for the third and fourth samples, then
r2
rl
T h e cross correlation function SMX can be o b t a i n e d directly f r o m a 4 pattern analysis where the isotopic states are represented by
3
+ BlnM/pcD + Ei3rr(r>- r3).
(7)
SMx(k)
l-
Ax-(kx)'
(9)
2C~Cx(fM -- f6)(fx-- f ; 0 If peaks are clearly identified in GM(r) over a particular range r~ < r < r2 then n oM and n M can be calculated. 2.2. Second order differences
By extending the diffraction m e a s u r e m e n t s to include a second isotopic substitution of the species M (or X) in M X , , . D 2 0 it is possible to d e t e r m i n e the individual pair correlation functions gMM(r) and gxx(r). If this new system " M X n . D 2 0 (or M " X , . D 2 0 ) gives a structure factor F"(k), then it is straightforward to show that (SMM(k)--
1) =
AMffk)
It can be readily seen from the form of eqs. (4), (8) and (9) that Placzek corrections associated with the difference m e t h o d s will not affect the analysis of the results (for further details see A p p e n d i x to ref. [13]). Because of the n u m b e r of variables in the equations it is necessary that samples used in the difference m e t h o d s are well defined and the experiments are carried out with great care. T h e first o r d e r differences are usually on a scale of -4-0.1 b and the effective cross section of i o n - i o n terms is - 2 0.01 b.
3. Structural properties
AM2(k)
3.1. C a t i o n - w a t e r structure or
( S x x ( k ) - 1 ) - AXe(k) A~>
AX2(k) B(x~ '
w h e r e AMI, AM2 and the coefficients A <2) M ~" are given by AM1 = F -
F',
AM2 = F -
F",
,, , AM = c2(fM -- f'M)(f'M -- fM) B(~d) :
2 ,, , CM(fM-fM)(fM
AX1 = F - F ' , 2 *-X'dt (2) = C x ( f x --XR(2) =
2 CX(fX
--
-
-
t, fM),
AX2 = F - F " , t
t
tt
fx)(fx - fx)
tt t tt -- fx)(fx -- fx )
(8a)
]~(2)
" ''--X
T h e cation c o n f o r m a t i o n in a q u e o u s solution has been d e t e r m i n e d in a large n u m b e r of cases (table 1). It is perhaps not surprising that the first application of the first o r d e r difference m e t h o d was a 4.35 molal solution of NiC12 in heavy water [13]. Isotopes of 62Ni, a~Ni and SaNi were used in varying a m o u n t s so as to optimise the f o r m of ANi(k). Fig. 3 shows the first o r d e r difference function, (3(r), for a 4.35 m NiC12.D20 solution. T h e first two peaks represent N i - O and N i - D correlations respectively and being in the numerical ratio of 6 : 12 (eq. (7)) it is inferred that there are about 6 water molecules a r o u n d a Ni 2+ ion. F r o m the near n e i g h b o u r distances rhi-O and rhi D and assuming rOD = 1 A and / _ D O D = 104.5 ° it is calculated that the angle between the
G. W. Neilson / The dynamical structure of ionic solutions
329
Table I Scattering lengths and sample parameters for ionic hydration
Electrolyte solution
Substituted species
Isotopes
Scattering a) lengths (x 10 12cm)
LiCI
Li*
(6Li; 7Li)
CaCI2
Ca 2+
(NCa: 44Ca)
NiCl2
Ni2+
Molality
A (x 10 2 b)
B (x 11} 2 b)
C (x 10 3 b)
D (x 10 3 b)
((). 18; -0.233)
9.95 3.57
1.66 0.695
3.83 1.60
-
1.34 -0.20
4.38
(0.466; 0.18)
4.49 2.80 1.1}
0.30 0.20
0.91) 1t.38 0.05
0.16
0.07
0.68 0.47 0.18
4.41 3.05 1.46 0.85 0.42
1.74 1.26 0.64 0.385 0.194
4.00 2.90 1.46 0.885 1/.446
5.1)5 2.52 0.6 l 0.22
1!.32
(1.03; -0.79)
(NNi; 62Ni)
0.60
I).07 0.01
0.15 {1.04 0.013 I).1111338
CuCI2
Cu 2+
(63Cu; 6SCu)
(0.67; 1.11)
4.32
0.415
1/.954
0.054 ].2
ND4CI
N
(14N; 15N)
(0.936; 0.65)
5.1)
0.26
1}.716bl
0.43
LiCI
CI
(35C1; 37C1)
(1.17; 0.35)
9.95 3.57
1.66 0.695
3.83 1.60
- 1.34
4.38
-0.20
{1.60
0.55 0.25
NaCI
CI-
(35CI; 37C1)
(1.17; 0.35)
5.32
0.99
2.27
0.65
] .37
RbCI
CI
(350; 37C1)
(1.17; 0.35)
2.99
0.58
1.33
0.52
0.46
CaC12
C1-
(35C1; 37C1)
(1.17; 0.35)
4.49
1.60
3.68
1.16
3.77
NiCI2
CI-
(35C1; 37C1)
(1.17; 0.35)
4.35
1.57
3.63
2.4{}
3.62)
NaNO3
N
(14N; ISN)
(0.936; 0.65)
7.8
0.54b)
0.846
0.036
0.079
a Calculated from mass spectroscopic measurements on isotopic abundance. b Includes both intra- and inter-atomic oxygen.
angle
02o DN,(r }
000
I I' ....
-':,.%
-"'..._..... .-~.5_":J":.:
.
B.5 "'":"; r {A....)
-
,-. .................
17' S
--
"'IQ
~0.20
Fig. 3. Ni 2+ ion distribution function (3si(r) for a 4.35 molal NiCI: heavy water solution. The inset at top left represents the local Ni2÷ water molecule conformation inferred from the results.
N i - O axis a n d t h e p l a n e o f t h e w a t e r m o l e c u l e ( a n g l e o f tilt) is - 4 2 ° (fig. 3 inset)• T h i s a n g l e is intermediate between the lone pair configuration - 5 5 ° a n d t h e d i p o l e c o n f i g u r a t i o n 0 ° a n d is a s t r o n g f u n c t i o n o f c o n c e n t r a t i o n [14]. T h i s e f f e c t is, w e b e l i e v e , a c o n s e q u e n c e o f d i s t o r t i o n s o f t h e h y d r a t i o n s p h e r e s as t h e p a c k i n g f r a c t i o n is increased. There have been no theoretical studies a i m e d at u n d e r s t a n d i n g this o b s e r v a t i o n . In o r d e r t o d e t e r m i n e w h e t h e r it w a s a c o n s e q u e n c e o f C1- i o n p e r t u r b a t i o n o f t h e N i 2+ ions, an experiment was undertaken on an 3.8m N i C I O n . D 2 0 s o l u t i o n w h e r e it is k n o w n t h a t i o n p a i r i n g is u n l i k e l y . N e w s o m e et al. [15] s h o w e d t h a t (3Ni(r) is e s s e n t i a l l y i n d e p e n d e n t o f t h e a n i o n ( t a b l e II). S i m i l a r s t r u c t u r a l p a t t e r n s t o fig. 3 h a v e b e e n o b s e r v e d f o r t h e i o n s o f Li + a n d C a 2+ [2], in-
330
G. W. Neilson / The dynamical structure of ionic solutions
Table II Cation hydration determined by neutron diffraction
Ion
Solute
Molality
Ion-oxygen distance (A)
Ion~leuterium distance (,~)
0"1
Coordination number
Li +
LiC1
9.95 3.57
1.95 -+(I.02 1.95 -+0.0
2.5(/-+ 0.02 2.55 -+ 0.02
52° ± 5° 40 ° -+ 5°
3.3 + 0.5 5.5 ± 0.3
Ca 2+
CaCI:
4.49 2.80 1.0
2.41 -+0.03 2.39 ± 0.02 2.46 ± 0.03
3.04 _+(I.(/3 3.02 -+0.03 3.(/7 ± 0.03
34° ± 9° 34° ± 9° 38° -+9°
6.4 ± 0.3 7.2 ± 0.2 1(1.(t± (1.6
Ni2*
NiCI2
4.41 3.05 1.46 0.85 (I.46 0.086
2.07 -+0.02 2.07 ± 0.02 2.07 ± 0.(12 2.09 -+0.02 2.10 ± 0.02 2.07 _+0.03
2.67 ± 0.02 2.67 ± 0.02 2.67 + 0.02 2.76 -+0.02 2.80 ± 0.02 2.80 _+0.(/3
42 ° ± 8° 42° ± 8° 42 ° ± 8° 27° ± 10° 17° ± 10° 0° -+ 20°
5.8 ± 0.2 5.8 ± 0.2 5.8 ± 0.3 6.6 ± 0.5 6.8 ± 0.8 6.8 ± 0.8
Cu 2+
CuCI2
4.32
2.05 _+0.03
2.56 _+0.10
58° ± 10°
2.3 ± 0.3
a Computed on the basis of rot) = 1 ,~ and D 0 D = 104.5°.
d i c a t i n g an a l m o s t i d e n t i c a l i o n - w a t e r c o n f o r m a t i o n (inset fig. 3). H o w e v e r , in b o t h c a s e s t h e r e is an a p p r e c i a b l e c h a n g e in c o o r d i n a t i o n n u m b e r u p o n d i l u t i o n [16] ( t a b l e II). T h e r e h a v e b e e n s e v e r a l ab i n i t i o c l u s t e r c a l c u l a t i o n s [6] a n d c o m p u t e r s i m u l a t i o n s s t u d i e s [7] of t h e c o o r d i n a t i o n o f Li + t o w a t e r . G e n e r a l l y it is f o u n d that the ion-oxygen distances are better represented by ab i n i t i o q u a n t u m chemistry whereas the computer simulation tends to give a better agreement with coordination numbers. T w o o t h e r c a t i o n s h a v e b e e n s t u d i e d , N D ~ in 5 m o l a l N D 4 C I - D 2 0 w h e r e t h e i s o t o p e s 14N a n d ~SN w e r e u s e d [17] a n d C u 2* in 4.32 m o l a l C u C 1 2 . D 2 0 u s i n g t h e i s o t o p e s 63Cu a n d 6SCu [18]. In t h e f o r m e r c a s e a b r o a d u n r e s o l v e d f e a t u r e is o b s e r v e d in (~N(r). T h i s r e s u l t is in b r o a d agreement with a molecular dynamics simulation s t u d y [19], b o t h r e s u l t s g i v i n g a c o o r d i n a t i o n number of between 10-11 w a t e r m o l e c u l e s a r o u n d t h e N D I ion. T h e r e is, h o w e v e r , a l a r g e discrepancy arising from the overall shape of G s ( r ) . T h e s i m u l a t e d G y ( r ) is m u c h s h a r p e r a n d appears to over-estimate either the hydrating s t r e n g t h o f t h e N D ~ ion, o r t h e s p h e r i c a l n a t u r e of its e f f e c t i v e p o t e n t i a l . A m u c h m o r e c o m p l i c a t e d (3cu(r) is c a l c u l a t e d f o r t h e c o o r d i n a t i o n of C u 2+ in s o l u t i o n [18]. A
2 + 4 c o o r d i n a t i o n a r o u n d t h e C u 2+ ion is c l e a r l y e v i d e n t c o n f i r m i n g t h e v i e w t h a t local c h e m i c a l e f f e c t s d o m i n a t e t h e c o n f o r m a t i o n of t h e i o n water subsystem. Effective ion-water potentials u s e d in s i m u l a t i o n s t u d i e s m u s t b e sufficiently r e a l i s t i c to r e f l e c t this fact.
3.2. A n i o n - w a t e r
structure
T h e m o s t e x t e n s i v e s t u d i e s of a n i o n s h a v e b e e n c o n c e r n e d w i t h t h e C1- i o n w h e r e i s o t o p e s 35C1 a n d 37C1 w e r e u s e d . T h e r e a p p e a r s t o b e a u n i v e r s a l i t y of s t r u c t u r e of C1- in s o l u t i o n [20] as e v i d e n c e f r o m t y p i c a l Gc~(r)'s d e m o n s t r a t e s , fig. 4 a n d t a b l e III. T h e s k e t c h in t h e u p p e r r i g h t of fig. 4 s h o w s t h e c o n f o r m a t i o n of t h e w a t e r m o l e c u l e w i t h r e s p e c t to C1-. T h e d i s t a n c e s rc~o a n d rc~D are in e x c e l l e n t agreement with those obtained from quantum m e c h a n i c a l c a l c u l a t i o n s [21]. In g e n e r a l , r a o is greater than that obtained from X-ray diffraction, a c o n s e q u e n c e , w e b e l i e v e , of t h e i n h e r e n t l y l o w d i s c r i m i n a t i o n of t o t a l X - r a y p a t t e r n s of s o l u t i o n s [12]. Computer simulations based on the ST2 p o t e n t i a l p r e d i c t s i g n i f i c a n t c o u n t e r ion effects w h i c h a r e c l e a r l y at v a r i a n c e w i t h t h e o b s e r v a t i o n s h e r e . It will b e i n t e r e s t i n g to s e e if o t h e r
G. W. Neilson / The dynamical structure of ionic solutions
3.3. Ion-ion structure
0,10 I _D(1 } 0
c~--
D(I)
0
DI2i
i; -0,0
::
E
~4~._. ~ w ( 3
0,051
":
110
210
L r(~,)
6'°
;0
331
8°
Fig. 4. t~o(r) for two alkali halide heavy water solutions: Full curve-9.95 molal LiCI in D20; crossed curve-5.32 molal NaCI in DzO after scaling by concentration factor of 1.87.
types of potentials, e.g. the 'central force' model, d e m o n s t r a t e that (~o(r) is indeed independent of counter ion and concentration [7]. A recent study has been carried out on the anion NO~ in an 7 . 8 m solution of NaNO3 in heavy water [22]. Nitrogen isotopes 14N and 15N were used and the difference method was applied (eq. (4)). The results show that the NO5 has a unique local conformation. There are between one and two nearest neighbour D 2 0 molecules and about 3 D - b o n d e d next nearest neighbour DzO molecules in the immediate environment of the NO~. Results from X-ray diffraction [23] were unable to discern this structure because of the lower discrimination of the results together with the overlapping correlations from N and O atoms.
The first and best system on which to use the second order differences method (eqs. (8, 9)) is nickel chloride in heavy water. In a series of recent experiments on a 4.35 molal NiCI2 heavy water solution it has been possible to obtain the three ion-ion partial structure factors [24], SNiNi, SNio(k) and S a a ( k ) . H o w e v e r the experiments are at the limit of existing technology and there are large errors associated with the data (see, for example, fig. 5). The corresponding g(r)'s show that there is a well-defined structure for C1-CI and Ni-Cl coordination. However, the coordination for Ni-Ni is not well defined although the o cut-off distance at r = 4 . 1 A is remarkably insensitive to the large errors in the raw data. A distance comparable to this has been calculated theoretically in iron chloride solutions 1251. It will be interesting to see whether the various models and liquid state theories are capable of generating the above results. Preliminary calculations based on a primitive model within the mean spherical approximation M S A show large discrepancies between theory and experiment [261. S o a ( k ) has been obtained in two other solutions [2]. In a 5.32 molal NaCl solution the results were shown to be consistent with the M S A version of the primitive model [6]. A much more sensitive test of the theory has been carried out by Copestake and Neilson who determined S o o ( k ) for a 14.9 molal LiCl solution. A
T a b l e II1 A n i o n h y d r a t i o n d e t e r m i n e d by n e u t r o n diffraction
Ion
Solute
Molality
X-D(1) (A)
X-O (,~)
CI
LiCI LiCI NaCI RbCI CaCI2 NiC12
3.57 9.95 5.32 4.36 4.49 4.35
2.25 2.22 2.26 2.26 2.25 2.29
3.34 3.29 3.20 3.20 3.25 3.20
NaNO3
7.8
2.05 -+ 0.02 b)
NO~
± 0.02 ± 0.02 - 0.04 -+ 0.04 ± 0.02 ± 0.02
a C o m p u t e d on t h e basis of rOD = 1 A (see fig. 4). b D i s t a n c e s f r o m n i t r o g e n nucleus.
± 0.05 -+ 0.04 _+ 0.05 ± 0.05 ± 0.04 ± 0.04
2.65 ± 0. l0 b)
X-D(2) (,~)
~oa)
Coordination number
3.50-3.68 3.55-3.65 3.40-3.50
0° 0° 0°-20° 0°-20 ° 00-7 ° 5 ° - 11 °
5.9 5.3 5.5 5.8 5.8 5.7
-
20 ° -+ 10°
1.3 -+ 0.2
-+ 0.2 -+ 0.2 ± 0.4 ± 0.3 ± 0.2 ± 0.2
G.W. Neilson / The dynamical structure of ionic solutions
332
energy transfers accompany the neutron scattering p r o c e s s it is n e c e s s a r y to g e n e r a l i s e o u r c o n c e p t s of t h e s t r u c t u r e of a system. It was V a n H o v e [27] w h o s h o w e d t h a t t h e r e exists a link b e t w e e n the g e n e r a l i s e d c o r r e l a t i o n function G(r, t) for a liquid a n d its n e u t r o n s c a t t e r i n g cross section. (G(r, t) can b e t h o u g h t of as the p r o b a b i l i t y of finding a p a r t i c l e at t h e origin r = 0 at t = O a n d a p a r t i c l e at r at t i m e t.) It is convenient both physically and mathematically to d i v i d e G(r, t) into a self p a r t G S a n d a distinct part G d
(7)
G = Ga+ G s , 20
{ -30
/
t
$iol
[
1
I 2
I
Fig. 5. SNisi(k) for 4.35 molal nickel chloride in heavy water. SNiNi(0 ) is calculated from thermodynamics [12], and data between 0 ~ k ~ 0 . 5 , ~ -1 have been interpolated. The dots represent an SNiNi(k ) which is derived from a well behaved gNiNi(r).
t h e o r e t i c a l c a l c u l a t i o n b a s e d on t h e M S A with a p r i m i t i v e m o d e l in which t h e b u l k d i e l e c t r i c c o n s t a n t of the s o l u t i o n iSo25, t h e Li + a n d CI d i a m e t e r s a r e 5 A a n d 3 A r e s p e c t i v e l y shows t h a t S a c l ( k ) is in g o o d q u a l i t a t i v e a g r e e m e n t with t h e e x p e r i m e n t a l result [26]. It will b e n e c e s s a r y to c a r r y o u t m o r e e x p e r i m e n t s using t h e s e c o n d o r d e r d i f f e r e n c e techn i q u e so that o n e can establish t h e e x t e n t to which p r i m i t i v e m o d e l s a r e a p p l i c a b l e to ionic solutions.
4. Neutron quasielastic scattering T h e r m a l n e u t r o n s a r e a s u i t a b l e p r o b e of b o t h the s t r u c t u r a l a n d d y n a m i c a l p r o p e r t i e s of t h e p a r t i c l e s of a c o n d e n s e d m a t t e r system. W h e n
w h e r e G ~ r e p r e s e n t s t h e p r o b a b i l i t y of finding a p a r t i c l e at r = 0 a n d t = 0 a n d t h e s a m e p a r t i c l e at r at t i m e t. It t u r n s o u t that, [2], G s is directly r e l a t e d to t h e i n c o h e r e n t s c a t t e r i n g cross section. This is p a r t i c u l a r l y c o n v e n i e n t for a q u e o u s s o l u t i o n s w h e r e t h e d o m i n a n c e of the h y d r o g e n i n c o h e r e n t cross section is a distinct a d v a n t a g e . F o r e x a m ple, in the s y s t e m M X n ' H 2 0 t h o u g h t h e r e are f o u r t e e n t e r m s which can c o n t r i b u t e to the scatt e r i n g p a t t e r n (4 self t e r m s a n d 10 i n t e r f e r e n c e t e r m s ) the t e r m GS(r, t) d o m i n a t e s t h e s c a t t e r i n g patterns. F o r t r a n s l a t i o n a l diffusion, GS(r, t) t a k e s t h e f o r m of a G a u s s i a n [28]. T h e e x p e r i m e n t a l l y accessible s c a t t e r i n g law, SS(k, w), t h e r e f o r e has a Lorentzian form
SS(k, w ) -
1 Dk 2 rr ( D k )2 + w 2
w h e r e h w is t h e e n e r g y t r a n s f e r in the s c a t t e r i n g p r o c e s s , hk is the m o m e n t u m t r a n s f e r a n d D is t h e t r a n s l a t i o n a l diffusion coefficient. In o r d e r that this p a r t i c u l a r f o r m can b e obtained experimentally three conditions must be met. (i) T h e self t e r m ~> distinct t e r m s . This is easily m e t for s y s t e m s c o n t a i n i n g h y d r o g e n . (ii) Dk2>> rro~,, i.e. all r o t a t i o n a l m o t i o n has b e e n a v e r a g e d out. (iii) t h e r e s o l u t i o n of the s p e c t r o m e t e r is such that g~o ~ 2 D k 2.
O. W. Neilson I The dynamical structure of ionic solutions
Conditions (ii) and (iii) imply that experiments must be carried out with high resolution --1 ~ e V at k ~ 1.~-1. The backscattering technique exemplified by the I N I O spectrometer at I L L G r e n o b l e is ideally suited for such experiments. Hewish et al. [9] carried out a series of experiments on a variety of aqueous solutions. The results were derived from I N I O data and were arrived at after careful corrections for multiple scattering. F r o m their results they were able to identify the two limiting cases known to occur for proton rate processes in solution [29]. In the first case, fast exchange, the results can be analysed as a single Lorentzian which gives an average D, for all the protons in the solution. This type of behaviour is observed in weakly hydrating systems such as CsCI. In the second case, the slow exchange, it was possible to identify two types of proton behaviour. This observation was based on the necessity of a two Lorentzian fit to the experimental results (fig. 6). Two diffusion coefficients were identified, D~ = Dio., that characteristic of the strongly hydrating
333
It) ~a t~
N o o
Dion I
1
1
;
2 (molar)
C
3
to o
...... I
i
~
B2
- - DseC
Dion 1 1
I 3 C (motat)
Fig. 7. T h e c a t i o n i c d i f f u s i o n c o e f f i c i e n t Dio,, t o g e t h e r w i t h t h e fitted v a l u e s f o r £)2 a n d c a l c u l a t e d v a l u e s f o r D~c f o r t w o s o l u t i o n s s t u d i e d . (a) N i C I 2 - H 2 0 , (b) M g C I 2 . H 2 0 . D a t a measurements were used with the neutron results to produce c u r v e £)2-
~0
30
~
2O
/
.Z 10:1
-15
k. ,'%.Cbl
#
-10
-5
,
0
5
10
15
Fig. 6. C u r v e a is a n a t t e m p t t o fit t h e o b s e r v e d S(k, to) w i t h a single L o r e n t z i a n f o r 3 m o l a l NiCI2 in H 2 0 . C u r v e b is a t w o L o r e n t z i a n fit of t h e s a m e d a t a . D~ = DNI w a s t a k e n f r o m t r a c e r m e a s u r e m e n t s (0.29 × 10 -5 c m 2 s 1) a n d /32 w a s f o u n d to h a v e t h e v a l u e 1.24 x 10-5 c m 2 s -j.
ion, and 132 which is an average of protons in all other environments. Systems exhibiting this latter behaviour are solutions of NiCl2 and MgC12. The nickel chloride results were analysed using the value of Dio, known from tracer measuremeres and the diffraction results referred to in section 3.1 which were used to correctly weigh the two Lorentzians. It was thereby possible to extract the diffusion coefficient (Dsec) of a second zone of hydrated water (fig. 7). This result adds credence to a model proposed by Frank and Wen [8] which predicted that, in general, several zones of hydration, each with a characteristic mobility, will exist. The full implications of this work are still being assessed. 5. Conclusions and future prospects
T h e successful application of a variety of neutron scattering methods to ionic solutions is
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providing a clearer insight into the origins of their wide ranging properties. The close link between computer simulation methods and experiment at the microscopic level represents a m a j o r step forward in our understanding of these complex systems. With the advent of higher fluxes of neutrons from pulsed sources the above methods can be pushed to greater limits. Not only will it be possible to apply the isotope enrichment method to systems with less favourable nuclei but it will b e c o m e feasible to carry out investigations over wider ranges of pressure, t e m p e r a t u r e and concentration where interesting effects are known to occur [1].
Acknowledgements It is a pleasure to thank Professor J.E. Enderby w h o s e help and advice have been invaluable in writing this paper. I also ack n o w l e d g e the assistance of Alan Copestake w h o has carried out much of the theoretical studies on the ion-ion structure.
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