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Muon spin relaxation study of the proton conductor HZr2(PO4)3
Solid State Ionics 170 (2004) 51 – 55 www.elsevier.com/locate/ssi
Muon spin relaxation study of the proton conductor HZr2(PO4)3 Nigel J. Clayden a,*, Upali A. Jayasooriya a, Stephen P. Cottrell b a
School of Chemical Sciences and Pharmacy, University of East Anglia, Norwich NR4 7TJ, UK b ISIS Facility, Rutherford-Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK
Received 13 August 2002; received in revised form 20 December 2002; accepted 20 December 2002
Abstract Muon spin relaxation has been used to study the implantation and dynamics of muons in the proton conductor HZr2(PO4)3 as a function of temperature. Muons are implanted at two different sites. One can be identified as H2OA+ on the basis of its low temperature second moment, Dx20. The other site is less well defined, Dx20 suggests P-O-A with an associated hydrogen. Dynamic averaging of the nuclear field is observed for H2OA+ above ca. 160 K, whereas no dynamic processes are observed for the P-O-A below 400 K. Above 400 K, further averaging of the nuclear field of both sites takes place which is attributed to translational diffusion of the muon through the lattice. D 2003 Elsevier B.V. All rights reserved. Keywords: Muon; Relaxation; Proton; Conductor
1. Introduction Proton conductors have been widely studied because of their applications in electrical devices such as fuel cells [1,2] and in sensors [3]. A number of crystalline proton conductors such as the heterometallates (H3PMo12O4030H2O) and HUO2PO44H2O are known. However, these tend to be unstable and difficult to prepare as thin films. For practical uses; low-cost, ease of fabrication and robustness are important considerations and for this reason, inorganic materials which can be prepared by sol – gel methods have become increasingly studied. A common feature of many of these inorganic proton conductors is the use of the phosphate group to provide the acidic hydrogen for proton conductivity. Amorphous phases such as P 2O 5 – MO 2 (M = Si, Ti, Zr) can be readily prepared by sol – gel methods and have been investigated for their proton conductivity [4 – 6]. These phases show a marked decrease in the conductivity with the temperature at which the phase is heat treated prior to its use, for example, from 10 3 S m 1 at 100 jC to 10 5 S m 1 at 300 jC for P2O5 – ZrO2 [6]. This is attributed to the loss of water. High conductivity around 10 3 S m 1 is often interpreted in terms of a high proton mobility which in turn is usually associated with the
* Corresponding author. Tel.: +44-1603592396; fax: +44-1603592003. E-mail address: [email protected] (N.J. Clayden). 0167-2738/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2738(02)00920-7
Grotthus mechanism [7] involving the transfer of a proton between water molecules and acid sites while the simple diffusive behaviour of an isolated proton is generally slower. Muons are potentially an excellent probe for studying proton conductors because the muon acts as a light isotope of hydrogen [8,9]. Like 1H NMR studies, analysis of the muon relaxation can yield information about the local nuclear field where the muon implants, that is the strength of the dipolar coupling arising from nearby nuclear spins, and the subsequent dynamics of the muon. Detailed analysis of muon dynamics may also offer an opportunity for studying isotope effects associated with the conductivity in view of the very substantial reduction in mass for the muon, being just 1/9 the mass of a proton [10]. The potential value of ASR rests in the important distinction between 1H NMR and ASR studies: Namely, 1H NMR relies on intrinsic 1 H spins in the material, whereas in ASR, the muon is extrinsic, it is implanted into the material. Consequently, the muon in the first instance must lie on an unoccupied lattice site, though subsequently, it may displace an intrinsic hydrogen atom. Furthermore, if the hypothesis that muons implant on sites involved in the proton conduction mechanism is valid, then the ASR experiments can then yield information about the lattice sites to which 1H NMR is blind. The aim of the current work was to determine the expected behaviour for an implanted muon in crystalline proton conductors where the possible sites for a muon are well known in order that proton conduction in amorphous
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systems such as P2O5 – SiO2 and P2O5 – ZrO2 can be analysed. HZr2(PO4)3 was chosen for the preliminary study of the muon behaviour in such systems for a number of reasons. First, it represents the end-member phase composition for the P2O5 – ZrO2 system, with a P/Zr ratio of 3:2. Second, the adventitious presence of H2O allows the effect of water to be studied while keeping a phase, which is stable to high temperatures. Third, thick film humidity sensors along the lines of those proposed for the amorphous materials have already been made using the HZr2(PO4)3 [11] and fourth, the proton transport properties are wellestablished, at 400 K, the proton conductivity is f 3 10 5 S m 1 rising to f 1 10 3 S m 1 at 600 K [12 – 14]. Lastly, the outline structure of HZr2(PO4)3 is known, consisting of a skeleton of ZrO6 octahedra sharing corners with PO4 tetrahedra. Based on the notation for the hexagonal NASICON family, the structure has four available sites (one M1 and three M2 sites). Within this structure, H2O molecules partially occupy the M2 site while the H+ ions are found in the M1 site [15].
2. Experimental HZr2(PO4)3 was synthesised by the thermal decomposition of NH4Zr2(PO4)3 at 450 jC in air [16]. As-prepared HZr2(PO4)3 is known to contain a fraction of water, corresponding to the composition HZr2(PO4)30.16H2O. NH4Zr2(PO4)3 itself was prepared by hydrothermal synthesis from the NH4+ substituted a-Zr(HPO4)2H2O [16] Confirmation of the purity of the H phase was obtained by IR spectroscopy and solid state 31P NMR spectroscopy. Both showed the absence of any residual ammonium form. Muon spin relaxation (ASR) measurements were made on the EMU spectrometer at the ISIS muon facility, RutherfordAppleton Laboratory, UK. In longitudinal field (LF)-ASR, an external magnetic field is applied parallel to the direction of the muon spin at the time of implantation with the asymmetry of the muon decay events as defined by Eq. (1) being measured by detectors placed in the forward ( F) and backward (B) directions with respect to the initial muon spin direction [17]. aðtÞ ¼
FðtÞ aBðtÞ FðtÞ þ aBðtÞ
ð1Þ
a is a correction factor which takes into account the relative efficiencies of the detectors. In zero field measurements, no magnetic field is applied. Muon beams are highly spin polarised, close to 100% and have intensities of f 105 A+s 1. Only a small proportion of this polarisation is lost on implantation, so that the observable polarisation is still far greater than thermal equilibrium. The decay of the polarisation with time, over and above the lifetime decay, is analogous to spin – lattice (T1) relaxation in nuclear or electron magnetic resonance.
HZr2(PO4)3 was packed into a 2-mm-deep and 40-mmdiameter recess in an aluminium plate and covered with a thin Mylar film to form an air tight seal. Muon experiments were carried out over a wide temperature range from 13 to 573 K in the zero field mode. In addition, longitudinal field and transverse field experiments were carried out at 13 K. Starting at room temperature, the sample was cooled to 13 K in a closed cryogenic refrigerator probe and then warmed to room temperature in 20 K steps. It was then transferred to the furnace probe for further heating to 573 K. In zero field experiments, the muon relaxation takes place within the local nuclear field and as such are the most appropriate for studying the dynamic processes of interest. Transverse field experiments were carried out at a number of temperatures to calibrate the alpha factor. The derived relaxation rates are corrected for the muon lifetime. Typically 20– 30 million muon decay events were accumulated for a relaxation decay at each temperature. Data analysis was carried out using inhouse nonlinear least squares fitting (UDA version 13.0) according to standard models for the muon implantation and decay process. The muon asymmetry is given by the amplitude of the exponentials used to fit the muon spin relaxation decay. When more than one exponential is required to fit the decay, the total asymmetry will be the sum of the amplitudes of each of the contributing exponential functions.
3. Results and discussion Preliminary LF-ASR experiments established that the main product of muon implantation is the diamagnetic muon, (equivalent to H+) rather than a paramagnetic species formed by abstraction of an electron from the surroundings since the asymmetry in a small longitudinal field of 20 G was close to the limiting value. In all cases, the slow relaxing component of the zero field decays was best represented by a Lorentzian function, G(t) = G0exp( kt) where k is the muon relaxation rate. Attempts to fit the data to Gaussian or Kubo –Toyabe [18] functional forms were unsuccessful; in each case, the fits failed to reproduce the initial decay of the muon signal. Although a Gaussian function, G(t) = G0exp( r2t2) where r is the muon relaxation rate, gave acceptable fits for the fast relaxing component, no consistent variation with temperature was seen. In particular, when the relaxation rate decreased towards the value seen for the slower relaxing component, a Lorentzian fit was required. The Lorentzian fits at low temperature are surprising, since the theoretical lineshape for the zero field decay of a muon in the presence of a static local nuclear dipolar field is given by the Kubo – Toyabe function. In the Kubo –Toyabe theory, Lorentzian character in a zero field decay only occurs with a modulation of the random local field by motion. More recently, a density matrix approach [19] has been proposed to explain the zero field decay in systems characterised by strong coupling to a small number
N.J. Clayden et al. / Solid State Ionics 170 (2004) 51–55
of nuclear spins, such as would be the case for HZr2(PO4)3. However, since the initial decay of the zero field muon signal is essentially identical for the classical Kubo – Toyabe and density matrix calculation, this cannot account for the Lorentzian fits. One possible explanation for the Lorentzian fits is that they represent a fitting artifact caused by the poor signal-to-noise ratio and the need to use two functions to get an acceptable v2. In support of this, excellent Gaussian fits were obtained for the muon relaxation in g-Zr(H2PO4)(PO4)2H2O with a muon relaxation rate, r = 0.5 As 1. In practise, the main difficulty arising from the Lorentzian character of the decays is that their interpretation in terms of the second moment, Dx02, is problematic. The temperature dependence of the asymmetry for the muon signal is shown in Fig. 1 and the relaxation rates in Fig. 2. For the most part, the analysis can be straightforwardly understood without taking into account the dehydration of the adventitious water. Only when the temperatures exceed 400 K does this need to be taken into consideration. At low temperature, the muon relaxation shows two components. One corresponding to roughly 70% of the muon signal has a low relaxation rate while the other remaining 30% is characterised by a fast relaxation. A quantitative analysis of the numbers of the two types of sites is not possible because the efficiency of muon implantation at these sites needs to be taken into account. In the zero field experiment, the relaxation rate is determined by the dephasing caused by the local nuclear dipolar fields. A high relaxation rate indicates a strong local dipolar field arising from the proximity of the muon to nuclear spins. In a proton conductor, the most important contribution to the local dipolar field will come from nearby hydrogens. Thus, one of the muon signals corresponds to a muon implanting at a site with a large dipolar field, a possible candidate being a water molecule. Muon implantation onto a water molecule to give the ion (H2OA)+ is consistent with the known presence of water in the, as prepared, HZr2(PO4)3 phase. A low relaxation rate, on the other hand, can indicate either fast dynamic averaging of the local nuclear field or the absence of such a field because the muon is relatively
Fig. 1. Fractions of the muons showing fast and slow relaxation.
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Fig. 2. Muon relaxation rates (k,r) for the two components in the muon spin relaxation.
isolated from nuclear spins. In this case, where a low relaxation rate is seen even at 13 K, the most plausible explanation is that the muon has implanted at a site remote from nuclear spins. Upon warming the sample from 13 K, a marked decrease is seen in the relaxation rate for the muon implanted at the site with a large dipolar field and essentially no change for the muon at the other site. By the time the temperature reaches 150 K, the relaxation rates are indistinguishable and the muon signal can then be fitted to a single exponential. This accounts for the discontinuity in the plot of asymmetry around 150 K. The reduction in the relaxation rate seen for the muon species identified as being (H2OA)+ indicates a fast dynamic process. There are three possible explanations: rotational diffusion or hops of (H2OA)+ about the lattice, M2, or surface site, translational diffusion of (H2OA)+ as an entity and chemical exchange involving the translational diffusion of the muon alone to another site. Rapid motion about the lattice site is the most likely reason, given that almost pseudo-isotropic motion is seen for the H3O+ ion at 200 K in the case of (H3O)Zr2(PO4)3 [20] by 2H NMR of the deuterated analogue, though it must be pointed out that the H3O+ ion here lies on the M1 site. It is not possible to say whether the reduction in the local nuclear field observed by muons is consistent with these H3O+ dynamics because of the unknown residual dipolar couplings between the muon and the lattice and the poor signal to noise of the muon signal limiting the accuracy of the multi-exponential fitting. (The 2H NMR experiments imply an order of magnitude reduction of the muon relaxation rate to ca. 0.03 As 1.) Further evidence for the rapid motion about the lattice site is provided by the limiting local nuclear field. For translational motion, the expectation is that the local nuclear field in the high temperature limit will be negligible. On the other hand, motion about a lattice site will lead to a residual local nuclear field, except in specific cases arising from the nature of the lattice site. Thus, the presence of a residual dipolar field after a marked reduction caused by a dynamic
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process is strong evidence that the motion must be a local rotation. The absence of translational diffusion of either (H2OA)+ as a whole or A+ alone is in line with the poor proton conductivity at low temperature. By analogy with 1H second moment studies made using NMR, analysis of the temperature dependence of the fast relaxing component in the region up to 160 K can be made, in principle, using the simple formula (2) [21] Dx2A ¼ Dx2R þ Dx20
2 arctanðaDxA sÞ: p
ð2Þ
Where DxA is the narrowed muon linewidth, a a constant of the order of unity, Dx02 the low temperature limiting second moment of the linewidth, DxR2 the high temperature residual second moment of the linewidth and s the correlation time for the dynamic process. For a Gaussian decay, the relaxation rate r can be straightforwardly related to the second moment: r2 = DxA2/2. In the present case though, as noted above, the Lorentzian character of the fast decaying component at the lowest temperature limits such an analysis. However, it is still possible to use Eq. (2) to estimate that the correlation time for the rotational motion must be of the order f 1 10 6 s since aDxAs will be of the order of unity at some stage during the line narrowing and DxA lies in the range [0.327, 0.099] As 1. The important feature being that the dynamics is too fast to be consistent with the observed ionic conductivity, as discussed below. Between 200 and 380 K, the relaxation rates are effectively constant at around 0.06 As 1, the value seen for the site with the small local nuclear dipolar field at the lowest temperatures. The slight rise between 160 and 200 K, which in any event is well within the fitting errors, can be attributed to the averaging of the relaxation rate between these two sites in the single exponential fitting. More accurate data would allow these two components to be separated, however, this was not feasible because of time constraints. Over this temperature range, the muon diffusion is too slow to modulate the muon dipolar coupling regardless of whether the muon implants onto an H2O or P-O. Such a view is consistent with the observed ionic conductivity of HZr2(PO4)3. Thus, using the Nernst – Einstein equation, taking the ionic conductivity at 300 K as 1 10 6 S m 1 [13], the concentration of HZr2(PO4)3 as 2 mol kg 1 and the cation transport number t+ = 1.0, the diffusion coefficient, D, can be estimated to be f 1 10 13 m2 s 1. For translational diffusion, the correlation time, sc, can be related to the diffusion coefficient by the expression [21]: sc ir2 =12D where sc is the time taken for the spins to diffuse a distance r apart, and D is the diffusion coefficient of the muon. Taking an order of magnitude estimate of 10 nm for r, then
sc f 1 10 4 s. Following Abragam, the condition for motional narrowing is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDx20 Þsc b1 where sc is the correlation time of the spin and Dx02, the second moment, as defined below and hence scb1 10 4 s. Thus, the condition for motional narrowing is qualitatively not met. Although a mechanism for sodium ion diffusion through either M1 – M2 or M2 – M2 pathways has been proposed for the sodium form of the NASICON [22], it is not clear whether similar restrictions apply to the smaller muon, where other pathways with smaller minimum openings may also be possible. Leaving aside the question of the pathway, it has been proposed that the dominant conduction mechanism for HZr2(PO4)3 at room temperature is a Grotthus type, involving a succession of H2O and/or H3O+ reorientation and proton tunneling [23]. In this context, the muon results reported here clearly show that facile reorientation of an H2OA+ species can occur. Perhaps of greater interest given the proposed contribution of proton tunneling would be a comparison of the muon and proton diffusion. However, uncertainty in the correct value to use for the diffusion distance r in the above calculation and its finely balanced nature precludes such a comparison. When the temperature increases above 400 K, the motional narrowing condition is now met and a further reduction in the muon relaxation rate occurs until at the highest temperatures studied, the muon shows almost complete averaging of the local nuclear dipolar field. At temperatures between 400 and 450 K, the analysis is complicated by the loss of the adventitious water as well as the poor counting statistics, which prevent an accurate analysis of the decay in terms of more than one exponential. Consequently over this temperature range, it was not possible to differentiate between the two possible muon sites and therefore say which of the sites or indeed whether both are involved in the diffusion. In contrast, above 450 K when the loss of water is complete, the analysis is very much simplified since the only site where the muon can implant will be the P-O. The reduction in the muon relaxation rate means that the P-O-A muon must now be undergoing some fast isotropic type motion or indeed translational motion. As with NMR, accurate analysis of the spin –lattice and spin –spin relaxation times at multiple static magnetic fields and as a function of temperature would allow differentiation between these types of motion [21] and in so doing give indirectly the muon diffusion coefficient and the activation energy for the dynamics. The more direct observation of translational motion through a muon equivalent of the NMR-pulsed field gradient experiment is not feasible with existing field gradients because of the short muon lifetimes (As). The muon results demonstrate that the muons in P-O-A sites are mobile at the temperatures expected for them to participate in the conductivity of the material and thus support the proposed conduction mechanism of the fully dehydrated HZr2(PO4)3, which is thought to be by an
N.J. Clayden et al. / Solid State Ionics 170 (2004) 51–55
activated classical hopping of the H+, or in this case A+, though M1 and M2 sites [22]. A quantitative assessment of the local nuclear field can be made using the following expression for the muon second moment, Dx02, which assumes powder averaging of the dipolar interaction [21]: X 4 A0 2 2 2 2 Dx20 ¼ cS cA t SðS þ 1Þ rj6 15 4p where cS and cA are the magnetogyric ratios of the nuclear spin, S and muon, A, respectively. rj is the distance between the nuclear spin j and the muon. Calculation of Dx02 was carried out with using the fractional co-ordinates and unit cell parameters for (H3O)Zr2(PO4)3 [24] since the co-ordinates for the hydrogen phase were not available, with the muon placed at different sites within the lattice. The second moment was calculated over a number of unit cells, although in practise, this was usually unnecessary, Dx02 being dominated by the nearest neighbours within the unit cell. Only the phosphorus (31P, I = 1/2, 100% abundant) and proton (1H, I = 1/2, 100% abundant) nuclear spins were included in the calculation, (16O, I = 0, 99.96% abundant, 91Zr, I = 5/2, 11.23%). Although Dx02 can be calculated with ease, there remains the difficulty caused by the Lorentzian nature of the decays found for the experimental data. One solution is to treat the decays as Gaussian and thus obtain Dx02 from the muon relaxation rate r of the Gaussian function (r2 = Dx02/2). Using this approximation, values for r of 0.114 and 0.41 As 1 are obtained for the slow and fast relaxing components at 13 K. These imply Dx02 is equal to 3.3 108 and 8.5 109 Hz2 for the slow and fast relaxing components, respectively. The latter is in good agreement with the value of 9.83 109 Hz2 calculated for the H2OA+ ion within the lattice. The identification of the slower decay as P-O-A is not clear cut since the predicted value for a P-O-A within the lattice but remote from a hydrogen is too small at f 2 107 Hz2. To obtain a larger value for Dx02, the muon must in fact add to a ˚ away. Such a site rather close to a hydrogen, only 2.3 A situation would occur if the muon implants into an occupied M1 site. However, the alternative of a grouping such as HOA can be ruled out because here the calculated Dx02 is 4.9 109 Hz2. The minimum second moment, Dx02 = 7.7 106 Hz2, is found for a muon at the centre of the M2 site, where three ˚ and six hydrogens at 6.1 A ˚. phosphorus are at 4.2 A
4. Conclusions Muon dynamics in HZr2(PO4)30.16H2O have been studied as a function of temperature. At low temperature, two different sites were seen for the muon implantation. One site was identified as H2OA+ on the basis of its large Dx02. Fast dynamics of this site was observed at temperatures above 13 K, with significant averaging of the local nuclear dipolar field. In the light of previous 2H NMR experiments, the
55
motional averaging was attributed to fast rotational reorientation about the lattice site. The other site is less well defined, Dx02 is consistent with P-O-A but only if there is an associated hydrogen. However, muon – hydrogen entities of the form HOA can be clearly ruled out. No dynamic processes were observed for the P-O-A+ site below 400 K. Above 160 K, because of the averaging of the local nuclear field of the H2OA+, the muon relaxation characteristics of the two sites become essentially identical. Above 400 K, further averaging of the nuclear field takes place attributable to translational diffusion of the muon through the lattice. Owing to the poor counting statistics, it was not possible to distinguish between the two sites and thus we cannot say which muons contribute to this averaging. The present work highlights the problems, which can arise in the analysis of muon relaxation data when more than one site is occupied, and the overall signal-to-noise ratio is poor. Acknowledgements We thank the Rutherford-Appleton laboratory for the use of the ISIS muon facility. References [1] T. Schober, H.H. Bohn, T. Mono, W. Schilling, Solid State Ionics 118 (1999) 173. [2] L. Depre, M. Ingram, C. Poinsignon, M. Popall, Electrochim. Acta 45 (2000) 1377. [3] P. Shuk, M. Greenblatt, Solid State Ionics 115 (1998) 229. [4] M. D’Apuzzo, A. Aronne, S. Esposito, P. Pernice, J. Sol-Gel Sci. Technol. 17 (2000) 247. [5] K. Makita, M. Nogami, Y. Abe, J. Mater. Sci. Lett. 16 (1997) 550. [6] Y. Abe, G. Li, M. Nogami, T. Kasuga, L.L. Hench, J. Electrochem. Soc. 143 (1996) 144. [7] N. Agmon, Chem. Phys. Lett. 244 (1995) 456. [8] J.S. Lord, S.P. Cottrell, K.S. Knight, W.G. Williams, Solid State Ionics 113 – 115 (1998) 341. [9] S.F.J. Cox, J.A.S. Smith, M.C.R. Symons, Hyperfine Interact. 65 (1990) 993. [10] K.-D. Kreuer, A. Fuchs, J. Maier, Solid State Ionics 77 (1995) 157. [11] P. Shuk, M. Greenblatt, Solid State Ionics 115 (1998) 229. [12] M. Ohta, A. Ono, J. Mater. Sci. Lett. 15 (1996) 1487. [13] M.A. Subramanian, B.D. Roberts, A. Clearfield, Mater. Res. Bull. 19 (1984) 1471. [14] A. Clearfield, Solid State Ionics 46 (1991) 35. [15] A. Ono, J. Mater. Sci. 19 (1984) 2691. [16] A. Clearfield, B.D. Roberts, M.A. Subramanian, Mater. Res. Bull. 19 (1984) 219. [17] E. Roduner, Lecture Notes in Chemistry vol. 49, Springer, Heidelberg, 1988. [18] R. Kubo, T. Toyabe, in: R. Blinc (Ed.), Magnetic Resonance and Relaxation, North-Holland, Amsterdam, 1967, p. 810. [19] J.S. Lord, S.P. Cottrell, W.G. Williams, Physica, B 289 (2000) 495. [20] N.J. Clayden, Solid State Ionics 24 (1987) 117. [21] A. Abragam, The Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961, p. 123, Chap. IV. [22] M. Ohta, A. Ono, F.P. Okamura, J. Mater. Sci. Lett. 6 (1987) 585. [23] M. Ohta, A. Ono, J. Mater. Sci. Lett. 15 (1996) 1487. [24] P.R. Rudolf, M.A. Subramanian, A. Clearfield, Solid State Ionics 17 (1985) 337.
Muon spin relaxation study of the proton conductor HZr2(PO4)3 Nigel J. Clayden a,*, Upali A. Jayasooriya a, Stephen P. Cottrell b a
School of Chemical Sciences and Pharmacy, University of East Anglia, Norwich NR4 7TJ, UK b ISIS Facility, Rutherford-Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, UK
Received 13 August 2002; received in revised form 20 December 2002; accepted 20 December 2002
Abstract Muon spin relaxation has been used to study the implantation and dynamics of muons in the proton conductor HZr2(PO4)3 as a function of temperature. Muons are implanted at two different sites. One can be identified as H2OA+ on the basis of its low temperature second moment, Dx20. The other site is less well defined, Dx20 suggests P-O-A with an associated hydrogen. Dynamic averaging of the nuclear field is observed for H2OA+ above ca. 160 K, whereas no dynamic processes are observed for the P-O-A below 400 K. Above 400 K, further averaging of the nuclear field of both sites takes place which is attributed to translational diffusion of the muon through the lattice. D 2003 Elsevier B.V. All rights reserved. Keywords: Muon; Relaxation; Proton; Conductor
1. Introduction Proton conductors have been widely studied because of their applications in electrical devices such as fuel cells [1,2] and in sensors [3]. A number of crystalline proton conductors such as the heterometallates (H3PMo12O4030H2O) and HUO2PO44H2O are known. However, these tend to be unstable and difficult to prepare as thin films. For practical uses; low-cost, ease of fabrication and robustness are important considerations and for this reason, inorganic materials which can be prepared by sol – gel methods have become increasingly studied. A common feature of many of these inorganic proton conductors is the use of the phosphate group to provide the acidic hydrogen for proton conductivity. Amorphous phases such as P 2O 5 – MO 2 (M = Si, Ti, Zr) can be readily prepared by sol – gel methods and have been investigated for their proton conductivity [4 – 6]. These phases show a marked decrease in the conductivity with the temperature at which the phase is heat treated prior to its use, for example, from 10 3 S m 1 at 100 jC to 10 5 S m 1 at 300 jC for P2O5 – ZrO2 [6]. This is attributed to the loss of water. High conductivity around 10 3 S m 1 is often interpreted in terms of a high proton mobility which in turn is usually associated with the
* Corresponding author. Tel.: +44-1603592396; fax: +44-1603592003. E-mail address: [email protected] (N.J. Clayden). 0167-2738/$ - see front matter D 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0167-2738(02)00920-7
Grotthus mechanism [7] involving the transfer of a proton between water molecules and acid sites while the simple diffusive behaviour of an isolated proton is generally slower. Muons are potentially an excellent probe for studying proton conductors because the muon acts as a light isotope of hydrogen [8,9]. Like 1H NMR studies, analysis of the muon relaxation can yield information about the local nuclear field where the muon implants, that is the strength of the dipolar coupling arising from nearby nuclear spins, and the subsequent dynamics of the muon. Detailed analysis of muon dynamics may also offer an opportunity for studying isotope effects associated with the conductivity in view of the very substantial reduction in mass for the muon, being just 1/9 the mass of a proton [10]. The potential value of ASR rests in the important distinction between 1H NMR and ASR studies: Namely, 1H NMR relies on intrinsic 1 H spins in the material, whereas in ASR, the muon is extrinsic, it is implanted into the material. Consequently, the muon in the first instance must lie on an unoccupied lattice site, though subsequently, it may displace an intrinsic hydrogen atom. Furthermore, if the hypothesis that muons implant on sites involved in the proton conduction mechanism is valid, then the ASR experiments can then yield information about the lattice sites to which 1H NMR is blind. The aim of the current work was to determine the expected behaviour for an implanted muon in crystalline proton conductors where the possible sites for a muon are well known in order that proton conduction in amorphous
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N.J. Clayden et al. / Solid State Ionics 170 (2004) 51–55
systems such as P2O5 – SiO2 and P2O5 – ZrO2 can be analysed. HZr2(PO4)3 was chosen for the preliminary study of the muon behaviour in such systems for a number of reasons. First, it represents the end-member phase composition for the P2O5 – ZrO2 system, with a P/Zr ratio of 3:2. Second, the adventitious presence of H2O allows the effect of water to be studied while keeping a phase, which is stable to high temperatures. Third, thick film humidity sensors along the lines of those proposed for the amorphous materials have already been made using the HZr2(PO4)3 [11] and fourth, the proton transport properties are wellestablished, at 400 K, the proton conductivity is f 3 10 5 S m 1 rising to f 1 10 3 S m 1 at 600 K [12 – 14]. Lastly, the outline structure of HZr2(PO4)3 is known, consisting of a skeleton of ZrO6 octahedra sharing corners with PO4 tetrahedra. Based on the notation for the hexagonal NASICON family, the structure has four available sites (one M1 and three M2 sites). Within this structure, H2O molecules partially occupy the M2 site while the H+ ions are found in the M1 site [15].
2. Experimental HZr2(PO4)3 was synthesised by the thermal decomposition of NH4Zr2(PO4)3 at 450 jC in air [16]. As-prepared HZr2(PO4)3 is known to contain a fraction of water, corresponding to the composition HZr2(PO4)30.16H2O. NH4Zr2(PO4)3 itself was prepared by hydrothermal synthesis from the NH4+ substituted a-Zr(HPO4)2H2O [16] Confirmation of the purity of the H phase was obtained by IR spectroscopy and solid state 31P NMR spectroscopy. Both showed the absence of any residual ammonium form. Muon spin relaxation (ASR) measurements were made on the EMU spectrometer at the ISIS muon facility, RutherfordAppleton Laboratory, UK. In longitudinal field (LF)-ASR, an external magnetic field is applied parallel to the direction of the muon spin at the time of implantation with the asymmetry of the muon decay events as defined by Eq. (1) being measured by detectors placed in the forward ( F) and backward (B) directions with respect to the initial muon spin direction [17]. aðtÞ ¼
FðtÞ aBðtÞ FðtÞ þ aBðtÞ
ð1Þ
a is a correction factor which takes into account the relative efficiencies of the detectors. In zero field measurements, no magnetic field is applied. Muon beams are highly spin polarised, close to 100% and have intensities of f 105 A+s 1. Only a small proportion of this polarisation is lost on implantation, so that the observable polarisation is still far greater than thermal equilibrium. The decay of the polarisation with time, over and above the lifetime decay, is analogous to spin – lattice (T1) relaxation in nuclear or electron magnetic resonance.
HZr2(PO4)3 was packed into a 2-mm-deep and 40-mmdiameter recess in an aluminium plate and covered with a thin Mylar film to form an air tight seal. Muon experiments were carried out over a wide temperature range from 13 to 573 K in the zero field mode. In addition, longitudinal field and transverse field experiments were carried out at 13 K. Starting at room temperature, the sample was cooled to 13 K in a closed cryogenic refrigerator probe and then warmed to room temperature in 20 K steps. It was then transferred to the furnace probe for further heating to 573 K. In zero field experiments, the muon relaxation takes place within the local nuclear field and as such are the most appropriate for studying the dynamic processes of interest. Transverse field experiments were carried out at a number of temperatures to calibrate the alpha factor. The derived relaxation rates are corrected for the muon lifetime. Typically 20– 30 million muon decay events were accumulated for a relaxation decay at each temperature. Data analysis was carried out using inhouse nonlinear least squares fitting (UDA version 13.0) according to standard models for the muon implantation and decay process. The muon asymmetry is given by the amplitude of the exponentials used to fit the muon spin relaxation decay. When more than one exponential is required to fit the decay, the total asymmetry will be the sum of the amplitudes of each of the contributing exponential functions.
3. Results and discussion Preliminary LF-ASR experiments established that the main product of muon implantation is the diamagnetic muon, (equivalent to H+) rather than a paramagnetic species formed by abstraction of an electron from the surroundings since the asymmetry in a small longitudinal field of 20 G was close to the limiting value. In all cases, the slow relaxing component of the zero field decays was best represented by a Lorentzian function, G(t) = G0exp( kt) where k is the muon relaxation rate. Attempts to fit the data to Gaussian or Kubo –Toyabe [18] functional forms were unsuccessful; in each case, the fits failed to reproduce the initial decay of the muon signal. Although a Gaussian function, G(t) = G0exp( r2t2) where r is the muon relaxation rate, gave acceptable fits for the fast relaxing component, no consistent variation with temperature was seen. In particular, when the relaxation rate decreased towards the value seen for the slower relaxing component, a Lorentzian fit was required. The Lorentzian fits at low temperature are surprising, since the theoretical lineshape for the zero field decay of a muon in the presence of a static local nuclear dipolar field is given by the Kubo – Toyabe function. In the Kubo –Toyabe theory, Lorentzian character in a zero field decay only occurs with a modulation of the random local field by motion. More recently, a density matrix approach [19] has been proposed to explain the zero field decay in systems characterised by strong coupling to a small number
N.J. Clayden et al. / Solid State Ionics 170 (2004) 51–55
of nuclear spins, such as would be the case for HZr2(PO4)3. However, since the initial decay of the zero field muon signal is essentially identical for the classical Kubo – Toyabe and density matrix calculation, this cannot account for the Lorentzian fits. One possible explanation for the Lorentzian fits is that they represent a fitting artifact caused by the poor signal-to-noise ratio and the need to use two functions to get an acceptable v2. In support of this, excellent Gaussian fits were obtained for the muon relaxation in g-Zr(H2PO4)(PO4)2H2O with a muon relaxation rate, r = 0.5 As 1. In practise, the main difficulty arising from the Lorentzian character of the decays is that their interpretation in terms of the second moment, Dx02, is problematic. The temperature dependence of the asymmetry for the muon signal is shown in Fig. 1 and the relaxation rates in Fig. 2. For the most part, the analysis can be straightforwardly understood without taking into account the dehydration of the adventitious water. Only when the temperatures exceed 400 K does this need to be taken into consideration. At low temperature, the muon relaxation shows two components. One corresponding to roughly 70% of the muon signal has a low relaxation rate while the other remaining 30% is characterised by a fast relaxation. A quantitative analysis of the numbers of the two types of sites is not possible because the efficiency of muon implantation at these sites needs to be taken into account. In the zero field experiment, the relaxation rate is determined by the dephasing caused by the local nuclear dipolar fields. A high relaxation rate indicates a strong local dipolar field arising from the proximity of the muon to nuclear spins. In a proton conductor, the most important contribution to the local dipolar field will come from nearby hydrogens. Thus, one of the muon signals corresponds to a muon implanting at a site with a large dipolar field, a possible candidate being a water molecule. Muon implantation onto a water molecule to give the ion (H2OA)+ is consistent with the known presence of water in the, as prepared, HZr2(PO4)3 phase. A low relaxation rate, on the other hand, can indicate either fast dynamic averaging of the local nuclear field or the absence of such a field because the muon is relatively
Fig. 1. Fractions of the muons showing fast and slow relaxation.
53
Fig. 2. Muon relaxation rates (k,r) for the two components in the muon spin relaxation.
isolated from nuclear spins. In this case, where a low relaxation rate is seen even at 13 K, the most plausible explanation is that the muon has implanted at a site remote from nuclear spins. Upon warming the sample from 13 K, a marked decrease is seen in the relaxation rate for the muon implanted at the site with a large dipolar field and essentially no change for the muon at the other site. By the time the temperature reaches 150 K, the relaxation rates are indistinguishable and the muon signal can then be fitted to a single exponential. This accounts for the discontinuity in the plot of asymmetry around 150 K. The reduction in the relaxation rate seen for the muon species identified as being (H2OA)+ indicates a fast dynamic process. There are three possible explanations: rotational diffusion or hops of (H2OA)+ about the lattice, M2, or surface site, translational diffusion of (H2OA)+ as an entity and chemical exchange involving the translational diffusion of the muon alone to another site. Rapid motion about the lattice site is the most likely reason, given that almost pseudo-isotropic motion is seen for the H3O+ ion at 200 K in the case of (H3O)Zr2(PO4)3 [20] by 2H NMR of the deuterated analogue, though it must be pointed out that the H3O+ ion here lies on the M1 site. It is not possible to say whether the reduction in the local nuclear field observed by muons is consistent with these H3O+ dynamics because of the unknown residual dipolar couplings between the muon and the lattice and the poor signal to noise of the muon signal limiting the accuracy of the multi-exponential fitting. (The 2H NMR experiments imply an order of magnitude reduction of the muon relaxation rate to ca. 0.03 As 1.) Further evidence for the rapid motion about the lattice site is provided by the limiting local nuclear field. For translational motion, the expectation is that the local nuclear field in the high temperature limit will be negligible. On the other hand, motion about a lattice site will lead to a residual local nuclear field, except in specific cases arising from the nature of the lattice site. Thus, the presence of a residual dipolar field after a marked reduction caused by a dynamic
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process is strong evidence that the motion must be a local rotation. The absence of translational diffusion of either (H2OA)+ as a whole or A+ alone is in line with the poor proton conductivity at low temperature. By analogy with 1H second moment studies made using NMR, analysis of the temperature dependence of the fast relaxing component in the region up to 160 K can be made, in principle, using the simple formula (2) [21] Dx2A ¼ Dx2R þ Dx20
2 arctanðaDxA sÞ: p
ð2Þ
Where DxA is the narrowed muon linewidth, a a constant of the order of unity, Dx02 the low temperature limiting second moment of the linewidth, DxR2 the high temperature residual second moment of the linewidth and s the correlation time for the dynamic process. For a Gaussian decay, the relaxation rate r can be straightforwardly related to the second moment: r2 = DxA2/2. In the present case though, as noted above, the Lorentzian character of the fast decaying component at the lowest temperature limits such an analysis. However, it is still possible to use Eq. (2) to estimate that the correlation time for the rotational motion must be of the order f 1 10 6 s since aDxAs will be of the order of unity at some stage during the line narrowing and DxA lies in the range [0.327, 0.099] As 1. The important feature being that the dynamics is too fast to be consistent with the observed ionic conductivity, as discussed below. Between 200 and 380 K, the relaxation rates are effectively constant at around 0.06 As 1, the value seen for the site with the small local nuclear dipolar field at the lowest temperatures. The slight rise between 160 and 200 K, which in any event is well within the fitting errors, can be attributed to the averaging of the relaxation rate between these two sites in the single exponential fitting. More accurate data would allow these two components to be separated, however, this was not feasible because of time constraints. Over this temperature range, the muon diffusion is too slow to modulate the muon dipolar coupling regardless of whether the muon implants onto an H2O or P-O. Such a view is consistent with the observed ionic conductivity of HZr2(PO4)3. Thus, using the Nernst – Einstein equation, taking the ionic conductivity at 300 K as 1 10 6 S m 1 [13], the concentration of HZr2(PO4)3 as 2 mol kg 1 and the cation transport number t+ = 1.0, the diffusion coefficient, D, can be estimated to be f 1 10 13 m2 s 1. For translational diffusion, the correlation time, sc, can be related to the diffusion coefficient by the expression [21]: sc ir2 =12D where sc is the time taken for the spins to diffuse a distance r apart, and D is the diffusion coefficient of the muon. Taking an order of magnitude estimate of 10 nm for r, then
sc f 1 10 4 s. Following Abragam, the condition for motional narrowing is: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðDx20 Þsc b1 where sc is the correlation time of the spin and Dx02, the second moment, as defined below and hence scb1 10 4 s. Thus, the condition for motional narrowing is qualitatively not met. Although a mechanism for sodium ion diffusion through either M1 – M2 or M2 – M2 pathways has been proposed for the sodium form of the NASICON [22], it is not clear whether similar restrictions apply to the smaller muon, where other pathways with smaller minimum openings may also be possible. Leaving aside the question of the pathway, it has been proposed that the dominant conduction mechanism for HZr2(PO4)3 at room temperature is a Grotthus type, involving a succession of H2O and/or H3O+ reorientation and proton tunneling [23]. In this context, the muon results reported here clearly show that facile reorientation of an H2OA+ species can occur. Perhaps of greater interest given the proposed contribution of proton tunneling would be a comparison of the muon and proton diffusion. However, uncertainty in the correct value to use for the diffusion distance r in the above calculation and its finely balanced nature precludes such a comparison. When the temperature increases above 400 K, the motional narrowing condition is now met and a further reduction in the muon relaxation rate occurs until at the highest temperatures studied, the muon shows almost complete averaging of the local nuclear dipolar field. At temperatures between 400 and 450 K, the analysis is complicated by the loss of the adventitious water as well as the poor counting statistics, which prevent an accurate analysis of the decay in terms of more than one exponential. Consequently over this temperature range, it was not possible to differentiate between the two possible muon sites and therefore say which of the sites or indeed whether both are involved in the diffusion. In contrast, above 450 K when the loss of water is complete, the analysis is very much simplified since the only site where the muon can implant will be the P-O. The reduction in the muon relaxation rate means that the P-O-A muon must now be undergoing some fast isotropic type motion or indeed translational motion. As with NMR, accurate analysis of the spin –lattice and spin –spin relaxation times at multiple static magnetic fields and as a function of temperature would allow differentiation between these types of motion [21] and in so doing give indirectly the muon diffusion coefficient and the activation energy for the dynamics. The more direct observation of translational motion through a muon equivalent of the NMR-pulsed field gradient experiment is not feasible with existing field gradients because of the short muon lifetimes (As). The muon results demonstrate that the muons in P-O-A sites are mobile at the temperatures expected for them to participate in the conductivity of the material and thus support the proposed conduction mechanism of the fully dehydrated HZr2(PO4)3, which is thought to be by an
N.J. Clayden et al. / Solid State Ionics 170 (2004) 51–55
activated classical hopping of the H+, or in this case A+, though M1 and M2 sites [22]. A quantitative assessment of the local nuclear field can be made using the following expression for the muon second moment, Dx02, which assumes powder averaging of the dipolar interaction [21]: X 4 A0 2 2 2 2 Dx20 ¼ cS cA t SðS þ 1Þ rj6 15 4p where cS and cA are the magnetogyric ratios of the nuclear spin, S and muon, A, respectively. rj is the distance between the nuclear spin j and the muon. Calculation of Dx02 was carried out with using the fractional co-ordinates and unit cell parameters for (H3O)Zr2(PO4)3 [24] since the co-ordinates for the hydrogen phase were not available, with the muon placed at different sites within the lattice. The second moment was calculated over a number of unit cells, although in practise, this was usually unnecessary, Dx02 being dominated by the nearest neighbours within the unit cell. Only the phosphorus (31P, I = 1/2, 100% abundant) and proton (1H, I = 1/2, 100% abundant) nuclear spins were included in the calculation, (16O, I = 0, 99.96% abundant, 91Zr, I = 5/2, 11.23%). Although Dx02 can be calculated with ease, there remains the difficulty caused by the Lorentzian nature of the decays found for the experimental data. One solution is to treat the decays as Gaussian and thus obtain Dx02 from the muon relaxation rate r of the Gaussian function (r2 = Dx02/2). Using this approximation, values for r of 0.114 and 0.41 As 1 are obtained for the slow and fast relaxing components at 13 K. These imply Dx02 is equal to 3.3 108 and 8.5 109 Hz2 for the slow and fast relaxing components, respectively. The latter is in good agreement with the value of 9.83 109 Hz2 calculated for the H2OA+ ion within the lattice. The identification of the slower decay as P-O-A is not clear cut since the predicted value for a P-O-A within the lattice but remote from a hydrogen is too small at f 2 107 Hz2. To obtain a larger value for Dx02, the muon must in fact add to a ˚ away. Such a site rather close to a hydrogen, only 2.3 A situation would occur if the muon implants into an occupied M1 site. However, the alternative of a grouping such as HOA can be ruled out because here the calculated Dx02 is 4.9 109 Hz2. The minimum second moment, Dx02 = 7.7 106 Hz2, is found for a muon at the centre of the M2 site, where three ˚ and six hydrogens at 6.1 A ˚. phosphorus are at 4.2 A
4. Conclusions Muon dynamics in HZr2(PO4)30.16H2O have been studied as a function of temperature. At low temperature, two different sites were seen for the muon implantation. One site was identified as H2OA+ on the basis of its large Dx02. Fast dynamics of this site was observed at temperatures above 13 K, with significant averaging of the local nuclear dipolar field. In the light of previous 2H NMR experiments, the
55
motional averaging was attributed to fast rotational reorientation about the lattice site. The other site is less well defined, Dx02 is consistent with P-O-A but only if there is an associated hydrogen. However, muon – hydrogen entities of the form HOA can be clearly ruled out. No dynamic processes were observed for the P-O-A+ site below 400 K. Above 160 K, because of the averaging of the local nuclear field of the H2OA+, the muon relaxation characteristics of the two sites become essentially identical. Above 400 K, further averaging of the nuclear field takes place attributable to translational diffusion of the muon through the lattice. Owing to the poor counting statistics, it was not possible to distinguish between the two sites and thus we cannot say which muons contribute to this averaging. The present work highlights the problems, which can arise in the analysis of muon relaxation data when more than one site is occupied, and the overall signal-to-noise ratio is poor. Acknowledgements We thank the Rutherford-Appleton laboratory for the use of the ISIS muon facility. References [1] T. Schober, H.H. Bohn, T. Mono, W. Schilling, Solid State Ionics 118 (1999) 173. [2] L. Depre, M. Ingram, C. Poinsignon, M. Popall, Electrochim. Acta 45 (2000) 1377. [3] P. Shuk, M. Greenblatt, Solid State Ionics 115 (1998) 229. [4] M. D’Apuzzo, A. Aronne, S. Esposito, P. Pernice, J. Sol-Gel Sci. Technol. 17 (2000) 247. [5] K. Makita, M. Nogami, Y. Abe, J. Mater. Sci. Lett. 16 (1997) 550. [6] Y. Abe, G. Li, M. Nogami, T. Kasuga, L.L. Hench, J. Electrochem. Soc. 143 (1996) 144. [7] N. Agmon, Chem. Phys. Lett. 244 (1995) 456. [8] J.S. Lord, S.P. Cottrell, K.S. Knight, W.G. Williams, Solid State Ionics 113 – 115 (1998) 341. [9] S.F.J. Cox, J.A.S. Smith, M.C.R. Symons, Hyperfine Interact. 65 (1990) 993. [10] K.-D. Kreuer, A. Fuchs, J. Maier, Solid State Ionics 77 (1995) 157. [11] P. Shuk, M. Greenblatt, Solid State Ionics 115 (1998) 229. [12] M. Ohta, A. Ono, J. Mater. Sci. Lett. 15 (1996) 1487. [13] M.A. Subramanian, B.D. Roberts, A. Clearfield, Mater. Res. Bull. 19 (1984) 1471. [14] A. Clearfield, Solid State Ionics 46 (1991) 35. [15] A. Ono, J. Mater. Sci. 19 (1984) 2691. [16] A. Clearfield, B.D. Roberts, M.A. Subramanian, Mater. Res. Bull. 19 (1984) 219. [17] E. Roduner, Lecture Notes in Chemistry vol. 49, Springer, Heidelberg, 1988. [18] R. Kubo, T. Toyabe, in: R. Blinc (Ed.), Magnetic Resonance and Relaxation, North-Holland, Amsterdam, 1967, p. 810. [19] J.S. Lord, S.P. Cottrell, W.G. Williams, Physica, B 289 (2000) 495. [20] N.J. Clayden, Solid State Ionics 24 (1987) 117. [21] A. Abragam, The Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961, p. 123, Chap. IV. [22] M. Ohta, A. Ono, F.P. Okamura, J. Mater. Sci. Lett. 6 (1987) 585. [23] M. Ohta, A. Ono, J. Mater. Sci. Lett. 15 (1996) 1487. [24] P.R. Rudolf, M.A. Subramanian, A. Clearfield, Solid State Ionics 17 (1985) 337.