Lithium ion diffusion in Li -alumina single crystals measured by pulsed field gradient NMR spectroscopy Mohammed Tareque Chowdhury, Reiji Takekawa, Yoshiki Iwai, Naoaki Kuwata, and Junichi Kawamura Citation: The Journal of Chemical Physics 140, 124509 (2014); doi: 10.1063/1.4869347 View online: http://dx.doi.org/10.1063/1.4869347 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Measurement of laser heating in spin exchange optical pumping by NMR diffusion sensitization gradients J. Appl. Phys. 107, 094904 (2010); 10.1063/1.3371249 Microscopic anisotropy revealed by NMR double pulsed field gradient experiments with arbitrary timing parameters J. Chem. Phys. 128, 154511 (2008); 10.1063/1.2905765 Pulsed field gradient NMR study of phenol binding and exchange in dispersions of hollow polyelectrolyte capsules J. Chem. Phys. 127, 234702 (2007); 10.1063/1.2807239 Mesoscopic simulations of the diffusivity of ethane in beds of NaX zeolite crystals: Comparison with pulsed field gradient NMR measurements J. Chem. Phys. 126, 094702 (2007); 10.1063/1.2567129 Correlating the NMR self-diffusion and relaxation measurements with ionic conductivity in polymer electrolytes composed of cross-linked poly(ethylene oxide-propylene oxide) doped with LiN(SO 2 CF 3 ) 2 J. Chem. Phys. 113, 4785 (2000); 10.1063/1.1288801
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THE JOURNAL OF CHEMICAL PHYSICS 140, 124509 (2014)
Lithium ion diffusion in Li β-alumina single crystals measured by pulsed field gradient NMR spectroscopy Mohammed Tareque Chowdhury,a) Reiji Takekawa, Yoshiki Iwai, Naoaki Kuwata, and Junichi Kawamura Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, 2-1-1, Katahira, Aoba-ku Sendai 980-8577, Japan
(Received 27 December 2013; accepted 12 March 2014; published online 31 March 2014) The lithium ion diffusion coefficient of a 93% Li β-alumina single crystal was measured for the first time using pulsed field gradient (PFG) NMR spectroscopy with two different crystal orientations. The diffusion coefficient was found to be 1.2 × 10−11 m2 /s in the direction perpendicular to the c axis at room temperature. The Li ion diffusion coefficient along the c axis direction was found to be very small (6.4 × 10−13 m2 /s at 333 K), which suggests that the macroscopic diffusion of the Li ion in the β-alumina crystal is mainly two-dimensional. The diffusion coefficient for the same sample was also estimated using NMR line narrowing data and impedance measurements. The impedance data show reasonable agreement with PFG-NMR data, while the line narrowing measurements provided a lower value for the diffusion coefficient. Line narrowing measurements also provided a relatively low value for the activation energy and pre-exponential factor. The temperature dependent diffusion coefficient was obtained in the temperature range 297–333 K by PFG-NMR, from which the activation energy for diffusion of the Li ion was estimated. The activation energy obtained by PFG-NMR was smaller than that obtained by impedance measurements, which suggests that thermally activated defect formation energy exists for 93% Li β-alumina single crystals. The diffusion time dependence of the diffusion coefficient was observed for the Li ion in the 93% Li β-alumina single crystal by means of PFG-NMR experiments. Motion of Li ion in fractal dimension might be a possible explanation for the observed diffusion time dependence of the diffusion coefficient in the 93% Li β–alumina system. © 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4869347] I. INTRODUCTION
Lithium ion conducting electrolyte materials have attracted considerable interest because of their potential use in electrochemical devices such as batteries and sensors.1–3 It is well known that solid electrolytes are fundamentally safer than their liquid counterparts. As such, a solid electrolyte with high lithium ionic conductivity is a very attractive material for safe, high performance lithium ion batteries. Therefore, for the improvement of battery performance, study of the transport mechanisms of Li ions in solid electrolytes is of fundamental interest. Since the discovery of β-alumina,4 Li ion containing β-alumina has been recognized as one of the highest performance Li ion conductors among the Li ion containing solid electrolytes.5, 6 The molecular formula of β-alumina is (M2 O)x · 11Al2 O3 , where M represents a monovalent cation. For stoichiometric β-alumina, x = 1; however, the material is generally nonstoichiometric and x ranges between 1.2 and 1.3. β-alumina is a layered structure material with monovalent ion containing mirror planes situated between alumina spinel blocks. The mirror plane is typically referred to as the conduction plane because the monovalent cation demona) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. Tel.: +81 22 217 5347. Fax: +81 22 217 5347.
0021-9606/2014/140(12)/124509/9
strates two-dimensional (2D) conductivity along this plane. The mirror plane contains three crystallographic sites for the monovalent cation known as the Beevers–Ross (BR), antiBeevers-Ross (aBR), and mid-oxygen (mO) sites, as shown in Fig. 1. Because NMR spectroscopy is sensitive to the local environment as well as to the motion of ions, NMR spectroscopy can be used to study the structure and motion of ions in a crystal. Several NMR studies regarding Li ion motion in β-alumina crystals have been conducted.7–9 Highe et al. investigated the local environment of 7 Li by measuring angle dependent NMR spectra.7 NMR line shape and relaxation measurements provide important information about the correlation time, the jump frequency, and the activation energy of Li ions.10 However, the line narrowing process provides a low estimate of the frequency factor and the activation energy for 7 Li in β-alumina as discussed by Villa et al.9 Walstedt et al. reported on the microscopic co-ionic conductivity of 70% and 90% Li β-alumina single crystals by T1 relaxation measurements.8 Using both NMR and Raman spectroscopy, Villa et al. provided various details concerning the microscopic motion of Li ions in the β-alumina crystal.9 A very low estimate of the pre-exponential factor was also determined for 23 Na in Na β-alumina from line narrowing measurements.11 Line narrowing and relaxation data were also used to investigate the diffusion of Li ions in other Li containing solid electrolytes.12–15
140, 124509-1
© Author(s) 2014
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measurements and NMR line narrowing. Because all previous experiments were performed for polycrystalline/powder samples, we directly measured the anisotropic diffusion coefficient of 7 Li in β-alumina single crystals by changing the orientation of the crystal in a static magnetic field. II. THEORY
FIG. 1. The Li β-alumina crystal structure, as drawn by Vesta threedimensional visualization software. Red, blue, and green ball represent oxygen, lithium, and aluminum ion, respectively. The layered structure of the material is apparent with Li ions existing only in the mirror planes (conduction planes) separated by spinel blocks.
Spin alignment echo (SAE) NMR spectroscopy can be used to study Li ion diffusion parameters such as jump rate and activation energy in glassy and crystalline materials.16–19 SAE-NMR is sensitive to the motion of Li ions over longer time periods and therefore is a suitable technique to measure diffusion parameters at low temperature and probe Li ion motion over a microscopic length scale where other techniques are not appropriate. Although it is possible to estimate selfdiffusion coefficients using T1 relaxation data, the relaxation approach relies on a number of assumptions. Therefore, reliable estimation of the diffusion coefficient by this approach is problematic. The line narrowing process in many cases provides a lower activation energy for Li ion motion because the distribution of correlation times that might appear from the nonexponential correlation function of the motional relaxation process is not included in the typical model.20 Pulsed field gradient (PFG) NMR spectroscopy is a technique by which the self-diffusion coefficient of Li can be directly measured. Maekawa et al. studied mesoporous Al2 O3 using PFGNMR.21 Recently, a number of experimental results have been reported regarding 7 Li ion diffusion coefficient measurements using PFG-NMR.22–24 The Li ion diffusion coefficient in βalumina was first directly measured by Yao and Kumar et al. using a tracer diffusion process.4 Since then there have been no reports concerning measurements of the Li ion diffusion coefficient of Li β-alumina single crystals. Moreover, previous diffusion coefficient measurements conducted by PFGNMR were performed in the high temperature regime. In this work, we report the direct measurement of the diffusion coefficient of Li ions for the first time in a β-alumina single crystal at room temperature using PFG-NMR, and a comparative discussion is presented in connection with results derived from other experimental techniques such as impedance
The self-diffusion process involves a random translational motion of a molecule or ion. It is the most fundamental transport process. In the PFG-NMR method, the attenuation of the NMR signal resulting from the dephasing of the nuclear spins due to the combined effect of translational motion and the imposition of spatially well-defined gradient pulses is used to measure the motion of ions or molecules. The diffusion coefficient D is extracted from the NMR echo intensity as a function of the magnetic gradient field, as fit to Eq. (1) given by Stejskal and Tanner:25 δ , (1) I = I (0) exp −D(2πgγ δ)2 − 3 where I is the echo intensity which is a function of gradient field, I(0) is the intensity without any gradient field, gi is the magnetic field gradient along z direction (along the static magnetic field with unit of G/m), γ is gyromagnetic ratio, for 7 Li it is 1.6553 × 103 Hz/G, is the diffusion time, and δ is gradient pulse duration. The relation between the Larmor frequency ω0 (rad/s) and the static magnetic field B0 with unit G can be expressed as ω0 = γ B0 .
(2)
Since B0 is spatially homogeneous, the Larmor frequency is the same over the entire sample. However, if a magnetic field gradient is applied along the z direction in addition to the static magnetic field, the Larmor frequency becomes position dependent along the z direction. III. EXPERIMENTAL A. Sample preparation and composition analysis
Na β-alumina single crystals were as cut to size from a larger single crystal. Li containing β-alumina was prepared by an ion exchange procedure by heating the original Na βalumina sample in molten LiCl at 700 ◦ C for a total of 48 h. The ion exchange process was completed in two steps. At first, the Na β- alumina samples were kept in to the molten LiCl for 24 h at 700 ◦ C. The samples were then washed and cleaned. Second, the washed and cleaned samples were again kept in to fresh molten LiCl at 700 ◦ C for 24 h. Inductively coupled plasma atomic emission spectroscopy (ICPAES) was used to estimate the Na and Li contents in the β-alumina crystal. B. Impedance measurement
Ionic conductivity of 93% Li β-alumina single crystal along ab plane was measured by ac impedance method using Solartron SI 1260 impedance analyzer within a frequency
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range from 1 Hz to 1 MHz. The impedance measurement was performed by decreasing the temperature from 373 K to 293 K. The ionic conductivity was measured using bulk resistance and the cell constant. The sample size was 5.46 mm × 1.96 mm × 0.48 mm. The ionic conductivity along the c direction was measured using Alpha-A high Performance Frequency Analyzer of novocontrol Technologies in frequency range from 0.01 Hz to 10 MHz. The impedance measurement was performed by decreasing the temperature from 473 K to 293 K. The ionic conductivity measured along the ab plane of 93% Li β-alumina single crystal is directly correspond to the perpendicular orientation of the single crystal during the PFG NMR diffusion experiment.
J. Chem. Phys. 140, 124509 (2014) 1.2
experimental( perpendicular) fitting experimental (parallel) fitting
1.0
Intensity (a.u)
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0.8 0.6 0.4 0.2 0.0 -0.2 0
200 400 600 800 1000 1200 1400 1600 1800 2000
Magnetic field gradient(G/cm) C. NMR measurement
The NMR experiment was performed in a Bruker 400 Avance. The 7 Li one-dimensional spectra were obtained using 90◦ pulses with a Larmor frequency of 155.5 MHz with the crystallographic c axis parallel to the static magnetic field. The diffusion coefficient was measured by a stimulated echo pulse sequence. The gradient pulse duration δ was set to 5 ms and the diffusion time was varied from 10 ms to 300 ms. The maximum field gradient was 1492 G/cm. We measured the diffusion coefficient in the 297–333 K temperature range, which is around room temperature. The sample size is 8.26 mm × 4.00 mm ×0.5 mm. Two different orientations of the single crystal with respect to static magnetic field were employed, wherein the Li β-alumina crystal c axis was perpendicular and parallel to the static magnetic field. It should be noted that the magnetic field gradient was applied along the static field direction. IV. RESULTS AND DISCUSSION A. Diffusion measurement
The ICP-AES study indicated that the sample contained 93% Li and 7% Na. The 7 Li echo signal intensity was obtained as a function of the magnetic field gradient. The diffusion coefficient was obtained by fitting the gradient dependent echo intensity to Eq. (1). Because in β-alumina single crystals, monovalent cations possess 2D conductivity along the ab plane, the anisotropic diffusion coefficient of the Li ion was measured with the crystallographic c axis of the single crystal alternately oriented both perpendicular and parallel to the gradient field. A room temperature diffusion coefficient DPFG of 1.22 × 10−11 m2 /s was obtained in the case of the perpendicular orientation. In the case of parallel orientation at room temperature, the diffusion signal intensity was very weak and the signal to noise ratio very low. Therefore, in Fig. 2, we summarize the echo signal intensity as a function the field gradient for perpendicular and parallel orientations at a temperature of 333 K. For the perpendicular orientation of the crystal, a diffusion coefficient DPFG of 2.26 × 10−11 m2 /s was obtained at 333 K. On the other hand, the echo intensity was nearly constant as function of the magnetic field gradient in the case of the parallel orientation, which suggests that the diffusion coefficient along the c direction is very small which is estimated
FIG. 2. The echo signal intensity of 7 Li as a function of magnetic field gradient for the 93% Li β-alumina single crystal. B and g are the static magnetic field and magnetic field gradient directed along the z axis, respectively. The green and black solid squares are the experimental data obtained from PFGNMR spectroscopy with the crystallographic c axis parallel and perpendicular to the static magnetic field, respectively, at 333 K. The blue and red solid lines were obtained by fitting the experimental data to Eq. (1).
as 6.4× 10−13 m2 /s or less at 333 K. This represents the first experimental data ever presented for the diffusion coefficient of the Li ion in β-alumina single crystal by PFG-NMR. Direct measurement of the diffusion coefficient in the parallel and perpendicular directions suggests that macroscopic motion of the cation in β-alumina single crystal is 2D in nature. Here it is also important to mentioned that the present authors measured the ionic conductivity along ab plane and parallel to c axis which are 1.19 × 10−3 −1 cm−126 and 1.85 × 10−15 −1 cm−1 , respectively, at 303 K. Furthermore the temperature dependent conductivity measurement shows that the ab plane activation energy is 0.27 eV and the parallel to the c axis the activation energy is 0.93 eV. Therefore the anisotropic conductivity data are very much consistent with the PFG NMR results. However, a small diffusion coefficient along the parallel direction does not rule out the possibility of local microscopic motion of the Li ion along the c axis since we previously obtained a small activation energy corresponding to the local motion of Li ions from the temperature dependent quadrupole interaction.26 Indication of Li ion motion along the c axis has also been discussed previously based on the heat capacity and dielectric susceptibility measurement of β-alumina single crystal.27, 28 The present authors measured the ionic conductivity of 93% Li β-alumina single crystal single crystal along the ab plane by impedance measurement in the temperature range from 297 K to 373 K which is published previously.26 The temperature dependent ionic conductivity data were used to estimate the diffusion coefficient Dσ of Li ion in β-alumina using the Nernst–Einstein equation given by Eq. (3), Dσ =
kB T σ (T ). N q2
(3)
Here, σ (T) is the ionic conductivity of 93% Li β-alumina at temperature T. At room temperature σ is 1.19 × 10−3 −1 cm−1 reported previously,26 kB is the Boltzmann
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Diffusion Coefficient (m2/s)
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Impedance 10-10
PFG NMR
10-11
10-12
10-13 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-1
1000/T(K ) FIG. 3. Normalized echo signal intensity as a function of (2π γ gδ)2 ( − δ/3) at various temperatures for = 10 ms for the 93% Li β-alumina single crystal oriented perpendicular to the magnetic field. The symbols represent the experimental data. The solid lines correspond to Eq. (1).
constant, N is the charge carrier density, q is the charge of a carrier, and T is the temperature. For direct correspondence between conductivity and diffusion, it is assumed that the Li concentration in the crystal is equal to the Li ion charge carrier density. The room temperature diffusion coefficient Dσ estimated using Eq. (3) is 5.36 × 10−12 m2 /s. The diffusion coefficient of 6 Li in pure Li β-alumina has been previously measured by the tracer diffusion technique in the temperature range 200–400 ◦ C.4 The present authors estimated the room temperature diffusion coefficient DTR to be 6.58 × 10−14 m2 /s by extrapolation from an Arrhenius plot using an activation energy of 8.71 kcal/mole (0.38 eV) and a pre-exponential factor of 14.5 × 10−8 m2 /s which are listed in previous report.4 However, such an estimation of room temperature diffusion coefficient by extrapolation may not be correct, since tracer diffusion data do not include activation energy in the lower temperature regime. Whittingham et al. estimated the Li+ activation energy by DC conductivity which is 0.187 eV in the temperature range from −100 ◦ C to 180 ◦ C and 0.372 eV in the temperature range from 180 ◦ C to 800 ◦ C.29 Whittingham high temperature region activation energy 0.372 eV is almost similar to the tracer diffusion activation energy in the temperature range 200–400 ◦ C. Thus it is possible that tracer diffusion experiment might give new activation energy in the low temperature regime. Therefore, a reasonable approximation of room temperature diffusion coefficient by tracer diffusion is not possible due to the absence of room temperature activation energy data. Fig. 3 shows the normalized 7 Li echo signal as a function of (2π γ gδ)2 ( − δ/3) for = 10 ms, measured in the temperature range 297–333 K, where the slope corresponds to the diffusion coefficient. Because the temperature dependent diffusion coefficient follows an Arrhenius relation, it is possible to estimate the activation energy from the temperature dependent PFG-NMR diffusion data and impedance measurements which are shown in Fig. 4. The activation energy from the impedance measurements was estimated using the Arrhenius relation and was found
FIG. 4. The diffusion coefficient of 93% Li β-alumina as a function of inverse temperature. The blue squares represent the diffusion coefficients obtained from PFG-NMR (measured in the perpendicular orientation). The red squares represent the diffusion coefficient obtained from impedance data using Eq. (3). The blue and red solid line represent the diffusion coefficient estimated from the Arrhenius relation.
to be 0.27 eV, which is higher than the activation energy obtained from the PFG-NMR data (0.14 eV). Therefore, because of the smaller activation energy DPFG is larger than that of Dσ . Table I summarized the diffusion coefficient and the activation energy data obtained from the impedance and the PFG NMR measurement. Recently, Zhongli Wang et al.23 measured the diffusion coefficient of crystalline Li3 N and estimated the activation energy along the XY plane which is 0.15 eV shows good agreement with Impedance data which is 0.19 eV.30 However, Alpen et al.31 shows the Li3 N single crystal activation energy data obtained from impedance measurement is 0.29 eV which is two times higher than that of the PFG NMR data.23 It is also important to mentioned that the activation energy was obtained as high as 0.43 eV and 0.61 eV for polycrystalline Li3 N.23, 32 Nishida et al. observed lower activation energy among different impedance data, which might be due to the presence of oxygen impurity in the sample, since such impurity lead to the formation of vacancies which increases the ionic conductivity and decreased activation energy.30 Therefore defect formation energy might involve in the nearest neighbor position of Li ion in the β-alumina crystal along with site to site migration energy. This finding is discussed in some length as follows. The number of mobile ions, ND depends on the temperature and follows an Arrhenius type relation that can be expressed by Eq. (4) using the formation energy Ef . Ef is actually a defect formation energy that
TABLE I. Diffusion coefficients for the Li ion in 93% Li β-alumina at room temperature (303 K) estimated from impedance measurement and PFG-NMR measurement. Impedance Dσ (m2 /s) 5.36 × 10−12
PFG-NMR Eσ (eV)
DPFG (m2 /s)
EPFG (eV)
0.27
1.22 × 10−11
0.145
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facilitates Li ion motion.
ND (T ) = N0 exp −
Ef kB T
B. Jump rate estimation
.
(4)
On the other hand, the Li ion can jump from one site to another site with migration energy Em . This is actually the elementary jump process for ionic diffusion which corresponds to NMR diffusion. Here, the diffusion coefficient Dm is related to the jump rate ν of an ion from one site to a neighboring site and can be expressed as Em (5) ν(T ) = ν0 exp − kB T and
Em Dm (T ) = D0 exp − . kB T
(6)
The Nernst–Einstein equation (Eq. (3)) relates the conductivity with the diffusion coefficient in accordance with Eq. (7), σ (T ) =
NDq 2 Dm . kB T
Substituting Eqs. (4) and (6) into Eq. (7) provides q2 Eσ σ (T ) = N0 D0 exp − , kB T kB T
(7)
(8)
since Eσ = Ef + Em
q2 N0 D0 . kB T
D=
a2 . 4τ
(11)
In Eq. (11), a is the jump distance between sites which corresponds to the mo–BR, mO–aBR, BR–aBR, or mO–mO distance of the β-alumina single crystal. Because the crystallographic positions for 7 Li 93% Li β-alumina are unknown, the Na β-alumina inter site distances35 are used to estimate the jump period. For mO–BR or mO–aBR jumps (a = 1.5 Å), the jump period τ can be varies from 4.62 × 10−10 s to 2.48 × 10−10 s depending on the temperature, and for BR-aBR or mO-mO jumps (a = 3.2 Å), the jump period varies from 2.10 × 10−9 s to 1.13 × 10−9 s depending on the temperature in the temperature range from 297 K to 333 K. Pre-exponential factors of 6.2 × 1011 s−1 (a = 1.5 Å) and 1.3 × 1011 s−1 (a = 3.2 Å) are also estimated from the PFG-NMR data since the jump period follows the Arrhenius relation given below by Eq. (12),
(9)
1 −Ea 1 exp = , τ τ0 kB T
(10)
using Eqs. (3) and (11), the jump period (or jump rate τ −1 ) can be estimated by Eq. (13) from the previously measured ionic conductivity data obtained by the impedance measurements,26
and σ0 =
A simple model of ionic diffusion in a solid is that of an ion jumping from one site to another site. Because the motion of the Li ion in the β-alumina single crystal is mainly along the ab plane, it is a reasonable assumption that ion jumping occurs between the regular crystallographic BR, aBR, and mO sites. Therefore, the jump period from one crystallographic site to another can be estimated using the simple jump model given by Eq. (11) because the diffusion coefficient is already known from the PFG-NMR data,
Here, Eσ and Em are the activation energies obtained from the impedance measurement and PFG-NMR, respectively. Ef is the defect formation energy because the defect is essential for ion motion. Apart from this simple defect formation energy, Kamishima et al. discussed a different approach using MD calculation where a complex cluster type defect might form due to Ag-Ag interaction in case of Ag β-alumina.33 Such interaction between the ions in different time scale can influence the diffusion process, which is also important to understand the difference of activation energy and the diffusion coefficient obtained from PFG NMR and impedance measurement. The present authors investigated the T1 relaxation time of 7 Li in the low temperature region, and a low activation energy of 0.062 eV was obtained, which might be related to local motion of the Li ion in the β-alumina crystal.26 Such a low activation energy was also observed by Walstedt et al. in the low temperature region of T1 relaxation data.8 Kaneda et al. previously discussed such local motion from the standpoint of Raman spectroscopy.34 Apart from that, the present authors also observed temperature dependent quadrupole interaction, which suggests that out of plane motion for the Li ion might occur in the β-alumina crystal with a small activation energy of 0.029 eV.26
τ −1 =
4kB T σ (T ). N q 2a2
(12)
(13)
Since the jump time τ follow Arrhenius relation as Eq. (12), a pre-exponential factor τ10 can be roughly estimated for impedance and PFG NMR data. This pre-exponential factor or the attempt frequency might correspond to the optical phonon frequency (vibrational frequency) which can be measured directly by Raman or infrared spectroscopy.36 Table II summarizes the pre-exponential factors estimated using Eq. (12) derived from different experimental techniques. The PFG-NMR, impedance, are provided by the present authors, while the previously reported Raman37 and T1 relaxation9 data are included in Table II for the purpose of comparison. Because Raman spectroscopy provides a direct measurement of the vibrational frequency, it is reasonable to compare our data with Raman data. It appears from the Table II that even though simplified jump model was used, Impedance and PFG-NMR data show reasonable estimation of vibrational frequency as far as the Raman data for vibrational frequency of Na in beta alumina is concern which is 2 × 1012 s−1 .37
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TABLE II. The pre-exponential factor corresponds to the attempt frequency or vibrational frequency of 7 Li in 93% β-alumina as estimated from different experimental techniques.
Raman spectroscopy37 (Na β-alumina) (s−1 )
PFG-NMR spectroscopy (s−1 )
NMR T1 relaxation measurement9 (s−1 )
a = 3.2 Å
a = 1.5 Å
a = 3.2 Å
a = 1.5 Å
1012
1.41 × 1011
6.41 × 1011
9.09 × 1012
4.13 × 1013
2 × 1012
C. 7 Li NMR line shape analysis
Fig. 5 shows the temperature dependent 7 Li NMR spectra obtained with the perpendicular orientation. NMR spectral widths were observed to decrease as a function of temperature, which is directly related to the motion of the Li ion and is known as motional narrowing. The temperature dependent spectral width ν(T) (Hz) is fit with the Bluembergen–Purcell and Pound (BPP) relation shown in Eq. (14),20 π ν(T ) 2 1 tan τc (T ) = . (14) 2π αν(T ) 2 vRL Here, τ c (T) is the jump period at temperature T, where α is a constant of order unity and ν R L is the rigid lattice line width, which is 5800 Hz. Therefore, the jump rate ν or the jump period τ = 1/ν can be obtained from the fitting process. Because a jump model can correlate the jump period with the diffusion coefficient, Eq. (11) is used to estimate the diffusion coefficient for a given site to site distance a. Such a site to site distance corresponds to mO to BR, mO to aBR, BR to aBR, or mO to mO distances in the Na β-alumina crystal. The diffusion coefficient DLN obtained from the NMR line narrowing data at room temperature is 1.13 × 10−14 m2 /s, which
Line width(Hz)
6000 5000 4000 3000 2000 1000 0 100 150 200 250 300 350 400 450
Temperature (K)
285k 264k 235k 205k 175k 145k 130k -50
-25
0
Impedance measurements (s−1 )
25
50
75
Chemical shift(ppm) FIG. 5. Temperature dependent 7 Li NMR spectra of the 93% Li β-alumina single crystal with the c axis oriented perpendicular to the static magnetic field with a Larmor frequency of 155 MHz. The upper left corner illustrates the orientation of the single crystal c axis in the static magnetic field B. The graph in the inset shows the line width as a function of temperature. The black squares are the experimental line width data. The red line illustrates the fit of the experimental data to Eq. (15).
is 3 orders of magnitude lower than that obtained from the PFG-NMR and impedance data. The temperature dependent jump period τ (or jump rate τ −1 ), which follows the Arrhenius type relation given in Eq. (12), provides an activation energy of 0.0631 eV and a very low pre-exponential factor (1/τ 0 = 5.0 × 106 s−1 ). The low estimation activation energy (0.063 eV) and pre-exponential factor (3.0 × 106 s−1 ) was also reported for Li contained β-alumina crystal.9 Bishop and Bray reported low estimation of prefactor and activation energy in case of glassy and crystalline Lithium borates.38 Li these diffusion parameters such as diffusion coefficient D, activation energy Ea , pre-exponential factor obtained from NMR motional narrowing process show large disagreement with PFG and impedance data. Therefore, it is important to carefully look at the temperature dependent 7 Li 1D spectra. From the graph shown in the inset of Fig. 5, motional narrowing is nearly complete at approximately 300 K. From 113 K to 130 K the spectra were better fitted with single Gaussian line shape and above 130 K the sharp central peak was fitted with a single Lorentzian line shape. The satellite peak becomes slowly appearing in the higher temperature. In the motional narrowing region from 113 K to 300 K the broaden satellite peak merge with motional narrow central peak. The angle dependent 1D spectra of single crystal β-alumina show the first order weak quadrupole interaction where the central transition (CT) peak does not shift its position as a function of angle; instead the CT peak shift shows the chemical shift anisotropy. However, in the first order quadrupole interaction the satellite peak position shifts as a function of orientation the satellite separation represents the amplitude of the quadrupole interaction called quadrupole coupling constant at the nucleus which is correspond to the EFG along z direction, which is the largest component of the electric field gradient. From the temperature dependent spectra it appears that, the rigid lattice line width 5.8 kHz. As the temperature is increased the line width becomes started to narrowing and around 300 K the motional narrowing process is completed. Since, in the low temperature zone the satellite peak become very broad, the low temperature spectral width contributed by both diope-dipole interaction and quadrupole interaction. At high temperature, where the ST peak becomes sharp enough, the CT spectral width is corresponding to the dipole-dipole interaction. Since in the motional narrowing region, the broad spectrum corresponds to different relaxation mechanism due to the dipole and quadrupole interaction. Therefore different relaxation mechanism affects the spectra differently. Thus diffusion parameters estimated from single component fitting of the NMR spectra can be far from the actual diffusion parameters. The Gaussian line shape of the 1D spectrum
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suggests in the inhomogeneous line broadening. Therefore, the broad single one Gaussian line shape might actually summation of much individual line shape with each having a characteristic relaxation time, which corresponds to the structural inhomogenieity and fluctuation of the local field. Thus a generation expression for diffusion coefficient in the jump model can be written as
D=
g(τ )
a2 dτ . 4τ
(15)
Here g(τ ) is jump time distribution function. If g(τ ) = δ(τ − τ c ), then Eq. (11) is an special case of Eq. (15). The most rough estimation in this case is the mean square displacement (MSD) is approximated as a2 . In real case MSD is correspond to several random jump process after certain time scale. However, since the BPP model produce a single relaxation time from the fitting process, for the rough estimation we used the δ(τ − τ c ) approximation and the a2 corresponds to MSD. The distribution of relaxation time is given as g(τ ) in Eq. (15). Since τ in Eq. (11) is approximated from single broad Gaussian line shape it become overestimated (at room temperature it is 1.97 × 10−6 s), and the diffusion coefficient become very small. Inhomogeneity also results the energy barrier distribution in the jump process. The presence of this energy barrier distribution in Na β-alumina was discussed by Walstedt et al.39 According to Faske et al. the BPP model presented in Eq. (12) does not consider such a distribution and underestimates the activation energy.15 The spectral width in the BPP model is correspond to the spectral density with a exponential correlation function with a single correlation time/relaxation time. This exponential correlation function is corresponding to a single relaxation process which might be oversimplification of the actual relaxation process. However it is suggested that, an inherent disordered system like β-alumina is heterogeneous and that give a distribution of the relaxation time and energy barrier. A non exponential correlation function can deal such a distribution of relaxation time and energy barrier. Therefore the extracting diffusion data from the line narrowing experiment are not flaw less. The last problem in the BPP motional narrowing analysis is the low estimation of the pre exponential factor (1/τ 0 ) in Eq. (14) which is known as the “attempt frequency” related to the jumping motion of the ion. NMR line narrowing the pre-exponential factor appears to be 8.0 × 106 s−1 , which is 5 to 6 orders of magnitude lower than that derived from the Raman data.37 This is known as a prefactor anomaly40 and has been previously reported for the β-alumina crystals.9, 11 The reason behind this low estimate is still not clear. According to Huberman and Boyce et al., it is due to the breakdown of the absolute rate theory where the particle can return to the original position from the top of the barrier height.41 Richards suggests that low dimensionality of the system effects the relaxation processes.42 It suggests that we have to reconsider the dimension and inhomogenieity presence in the material and revised the BPP model that gives the correct description of diffusion phenomena for Li β-alumina single crystal.
D. Diffusion time dependence
It was observed that the diffusion coefficient obtained depended on the diffusion time , and the diffusion coefficient decreased with increasing . It appears that the Li ion can travel faster in the local environment over a short observation time. However, a slower Li ion motion, as demonstrated by a decreasing diffusion coefficient, with an increasing measurement time scale, suggests that a change in the local environment occurs with increasing diffusion time. This suggests the existence of a structural inhomogeneity in the β-alumina single crystal. This structural inhomogeneity might correspond to Li ions residing outside of the conduction plane and/or to the trace amounts of Na √ ions that exist in our sample. The mean squared distance, 4D, traveled by the Li ion at room temperature for various diffusion times from 10 ms to 300 ms are 1.3 μm to 2.7 μm, respectively. Even though the diffusion coefficient decreases with increasing , the total distance traveled by the diffusing ion is still increasing. Thus, interactions between the Li ion and the local environment are not straight forward. Hayamizu et al. found a similar diffusion time dependence for the solid electrolyte (Li2 S)x (P2 S5 )1−x (x = 0.7).24 According to their data, fast and slow moving Li ions can exist simultaneously depending on the thermal activation process involved. Furthermore, these researchers made the surprising observation that, while the fast moving Li ions moved freely, the slow moving ions were localized within small regions corresponding to a grain size of 1.5 μm. Fig. 6(a) shows the diffusion coefficient as a function of the inverse of the diffusion time at room temperature for the 93% Li β-alumina single crystal. The log-log plot suggests that the diffusion coefficient follows a low exponent power law behavior, as given by Eq. (16), n 1 , (16) D∝ where n = 0.138. Equation (16) is directly corresponds to the frequency dependent of conductivity. Therefore the present authors also measured Li β-alumina single crystal frequency dependent conductivity which shows small power law behavior with a exponent of 0.16–0.10, which is consistent with the recently reported experimental data.43 The frequency dependence conductivity of Li β-alumina is shown in Figure 6(b). This time dependent diffusion coefficient and the frequency dependent conductivity suggest the presence of a distribution of relaxation time which is given as g(τ ) in Eq. (15). Recently, Kamishima et al. reported that a power law behavior was observed for that the frequency dependent conductivity of Ag βalumina single crystal where the exponent is vary from 0.11 to 0.15 depends on the temperature which is consistent with the result obtained by the present authors. They suggest a scale invariance property of superionic β-alumina with small power law dependent. In terms of the transport mechanism, the mobile ion might move into a random motion in self repeating structure or a fractal dimension. Therefore, the time dependents diffusion data obtained from the PFG NMR experiment strengthen the arguments that motion of Li ion in the SIC (super ionic conductor) like β-alumina is moving randomly in
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FIG. 6. (a) Diffusion coefficients of 7 Li 93% Li β-alumina as a function of 1/ at room temperature with the perpendicular orientation. The red line was fit to the experimental data in accordance to the power law given by Eq. (16). (b) Frequency dependent conductivity at different temperature of 93% Li β-alumina.
a fractal dimension. Kamishima et al. investigated the many body effect on the ion dynamics in case of Ag β-alumina.33 Because of the inbuilt defect present in the β-alumina structure the ion can take various configurations in the presence of the defect and fast ionic motion is possible. They showed that the Ag-Ag weak repulsive interaction increases the ionic motion with high diffusion coefficient and strong repulsive interaction decrease the ionic transport with due to requirement of the additional energy for structural relaxation with increasing activation energy. This suggests that a relaxation time distribution present in the β-alumina type materials which are consistent with Ngai coupling model.44 Therefore time dependent diffusion of ion in the β-alumina crystal can also be explained by many body interaction or coupling model. It has been previously reported that, for a three-dimensional glass structure system, the frequency dependence of the conductivity follows a power law behavior with an exponent of 0.67 ± 0.03.45 The frequency dependence in a 3D system is explained in terms of relaxation mechanisms,32 coulombic interaction between ions,46 and random hopping of ions in a fractal dimension.47 Therefore, the time dependence of the diffusion coefficient give important information regarding the motion the motion of Li ions in 2D super ionic conductors like Li β-alumina. V. CONCLUSION
The diffusion coefficient of 7 Li in a 93% Li β-alumina single crystal was measured using PFG NMR in two different crystal orientations. This is the first reported attempt to measure the diffusion coefficient of the Li ion in β-alumina single crystals by PFG NMR. The Li ion shows a large diffusion coefficient along the ab plane, while diffusion along the c direction is very weak because the echo signal intensity does not change as a function of the magnetic field gradient. Therefore, Li ion diffusion is anisotropic, and the activation energy of diffusion along the ab plane was found to be 0.14 eV, as estimated from temperature dependent diffusion data. The diffusion coefficient and activation energy were also estimated from impedance measurements. The discrepancy
regarding the activation energies obtained from the PFGNMR and impedance data are explained in terms of the defect formation energy. 7 Li NMR line shape was analyzed using simple BPP model with single line shape fitting process which produce very low estimation of diffusion coefficient, activation energy and frequency pre-exponential factor. Low estimation of diffusion coefficient and activation energy might due to the improper treatment of the experimental line shape. The single Gaussian line shape corresponds to inhomogeneous line broadening which might be due to the relaxation time distribution as well as the energy barrier distribution which is not taken in to account in the BPP model.
ACKNOWLEDGMENTS
The first author, M.T.C., acknowledges MEXT, Japan for the Monbukagakusho scholarship. Thanks are also due to Professor Kamishima for a fruitful discussion concerning the frequency dependence of conductivity in the β-alumina single crystal. 1 G.
Y. Adachi, N. Imanaka, and H. Aono, Adv. Mater. (Weinheim, Ger.) 8, 127 (1996). 2 M. Duclot and J. L. Souquet, J. Power Sources 97–98, 610 (2001). 3 T. Minami, A. Hayashi, and M. Tatsumisago, Solid State Ionics 177, 2715 (2006). 4 Y.-F. Y. Yao and J. T. Kummer, J. Inorg. Nucl. Chem. 29, 2453 (1967). 5 J. L. Briant and G. C. Farrington, J. Electrochem. Soc.: Electrochem. Sci. Technol. 128, 1830 (1981). 6 G. C. Farrington, B. S. Dunn, and J. L. Briant, Solid State Ionics 3/4, 405 (1981). 7 A. Highe and R. W. Vaughan, J. Chem. Phys. 69, 4206–4208 (1978). 8 R. E. Walstedt, R. S. Berg, and J. P. Remeika, Solid State Commun. 37, 209 (1981). 9 M. Villa, J. L. Bjorkstam, G. Mariotto, A. Fontana, and E. Cazzanelli, J. Chem. Phys. 76, 2804 (1982). 10 A. Kuhn, S. Narayanan, L. Spencer, G. Goward, V. Thangadurai, and M. Wilkening, Phys. Rev. B 83, 094302 (2011). 11 W. Bailey, S. Glowinkowski, H. Story, and W. L. Roth, J. Chem. Phys. 64, 4126 (1976). 12 M. Wilkening, S. Indris, and P. Heitjans, Phys. Chem. Chem. Phys. 5, 2225 (2003). 13 P. Heitjans and S. Indris, J. Phys. Condens. Matter 15, R1257 (2003).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 128.118.88.48 On: Tue, 12 Aug 2014 14:12:42
124509-9 14 M.
Chowdhury et al.
Wilkening, D. Brork, S. Indris, and P. Heitjans, Phys. Chem. Chem. Phys. 4, 3246 (2002). 15 S. Faske, H. Eckert, and M. Vogel, Phys. Rev. B 77, 104301 (2008). 16 M. Wilkening and P. Heitjans, Chem. Phys. Chem. 13, 53 (2012). 17 M. Wilkening, A. Kuhn, and P. Heitjans, Phys. Rev. B 78, 054303 (2008). 18 M. Wilkening, J. Heine, C. Lyness, A. R. Armstrong, and P. G. Bruce, Phys. Rev. B 80, 064302 (2009). 19 M. Wilkening and P. Heitjans, Phys. Rev. B 77, 024311 (2008). 20 N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948). 21 H. Maekawa, Y. Fujimaki, H. Shen, J. Kawamura, and T. Yamamura, Solid State Ionics 177, 2711 (2006). 22 T. Ohkubo, M. Gobet, V. Sarou-Kanian, C. Bessada, M. Nozawa, and Y. Iwadate, Chem. Phys. Lett. 530, 61 (2012). 23 Z. Wang, M. Gobet, V. Sarou-Kanian, D. Massiot, C. Bessada, and M. Deschamps, Phys. Chem. Chem. Phys 14, 13535 (2012). 24 K. Hayamizu and Y. Aihara, Solid State Ionics 238, 7 (2013). 25 E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 42, 288 (1965). 26 M. T. Chowdhury et al., “The study of the lithium ion motion in β-alumina single crystal by NMR spectroscopy,” Solid State Ionics (published online, 2013). 27 D. B. McWhan, C. M. Varma, F. L. S. Hsu, and J. P. Remeika, Phys. Rev. B 15, 553 (1977). 28 P. J. Anthony and A. C. Anderson, Phys. Rev. B 19, 5310 (1979). 29 M. S. Whittingham and R. A. Huggins, in Solid State Chemistry, edited by R. S. Roth and S. J. Schneider, Jr. (National Bureau of Standards, Washington, D.C., 1972), Special Publication No. 364, p. 148.
J. Chem. Phys. 140, 124509 (2014) 30 K.
Nishida, T. Asai, and S. Kawai, Solid State Commun. 48, 701 (1983).
31 U. V. Alpen, A. Rabenau, and G. H. Talat, Appl. Phys. Lett. 30, 621 (1977). 32 B.
A. Boukamp and R. A. Huggins, Phys. Lett. A 58, 231 (1976). Kamishima, K. Kawamura, T. Hattori, and J. Kawamura, J. Phys.: Condens. Matter 23, 225404 (2011). 34 T. Kaneda, J. B. Bates, and J. C. Wang, Solid State Commun. 28, 469 (1978). 35 R. Collongues, D. Gourier, A. Kahn, J. P. Boilot, P. H. Colomban, and A. Wicker, J. Phys. Chem. Solids 45, 981 (1984). 36 M. B. Salamon, Topics in Current Physics: Physics of Superionic Conductors (Springer–Verlag, Berlin, 1979), p. 162. 37 W. L. Roth, F. Reidinger, and S. La placa, in Super Ionic conductors, edited by G. D. Mahan and W. L. Roth (Plenum Press, New York, 1976). 38 S. G. Bishop and P. J. Bray, J. Chem. Phys 48, 1709–1717 (1968). 39 R. E. Walstedt, R. Dupree, J. P. Rerneika, and A. Rodriguez, Phys. Rev. B 15, 3442 (1977) 40 M. Villa and J. L. Bjorkstam, Solid State Ionics 9–10, 1421 (1983). 41 B. A. Huberman and J. B. Boyce, Solid State Commun. 25, 843 (1978). 42 P. Richards, Solid State Commun. 25, 1019 (1978). 43 O. Kamishima et al., “Frequency dependence of ionic conductivity in a two-dimensional system of Ag β-alumina,” Solid State Ionics (published online, 2014). 44 K. L. Ngai and O. Kannert, Solid State Ionics 53, 936 (1992). 45 H. Kahnt, Ber. Bunsenges. Phys. Chem. 95, 1021 (1991). 46 P. Maass, J. Petersen, A. Bunde, W. Dieterich, and H. E. Roman, Phys. Rev. Lett. 66, 52 (1991). 47 J. Habasaki, K. L. Ngai, and Y. Hiwatari, J. Chem. Phys. 120, 8195 (2004). 33 O.
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THE JOURNAL OF CHEMICAL PHYSICS 140, 124509 (2014)
Lithium ion diffusion in Li β-alumina single crystals measured by pulsed field gradient NMR spectroscopy Mohammed Tareque Chowdhury,a) Reiji Takekawa, Yoshiki Iwai, Naoaki Kuwata, and Junichi Kawamura Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, 2-1-1, Katahira, Aoba-ku Sendai 980-8577, Japan
(Received 27 December 2013; accepted 12 March 2014; published online 31 March 2014) The lithium ion diffusion coefficient of a 93% Li β-alumina single crystal was measured for the first time using pulsed field gradient (PFG) NMR spectroscopy with two different crystal orientations. The diffusion coefficient was found to be 1.2 × 10−11 m2 /s in the direction perpendicular to the c axis at room temperature. The Li ion diffusion coefficient along the c axis direction was found to be very small (6.4 × 10−13 m2 /s at 333 K), which suggests that the macroscopic diffusion of the Li ion in the β-alumina crystal is mainly two-dimensional. The diffusion coefficient for the same sample was also estimated using NMR line narrowing data and impedance measurements. The impedance data show reasonable agreement with PFG-NMR data, while the line narrowing measurements provided a lower value for the diffusion coefficient. Line narrowing measurements also provided a relatively low value for the activation energy and pre-exponential factor. The temperature dependent diffusion coefficient was obtained in the temperature range 297–333 K by PFG-NMR, from which the activation energy for diffusion of the Li ion was estimated. The activation energy obtained by PFG-NMR was smaller than that obtained by impedance measurements, which suggests that thermally activated defect formation energy exists for 93% Li β-alumina single crystals. The diffusion time dependence of the diffusion coefficient was observed for the Li ion in the 93% Li β-alumina single crystal by means of PFG-NMR experiments. Motion of Li ion in fractal dimension might be a possible explanation for the observed diffusion time dependence of the diffusion coefficient in the 93% Li β–alumina system. © 2014 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.doi.org/10.1063/1.4869347] I. INTRODUCTION
Lithium ion conducting electrolyte materials have attracted considerable interest because of their potential use in electrochemical devices such as batteries and sensors.1–3 It is well known that solid electrolytes are fundamentally safer than their liquid counterparts. As such, a solid electrolyte with high lithium ionic conductivity is a very attractive material for safe, high performance lithium ion batteries. Therefore, for the improvement of battery performance, study of the transport mechanisms of Li ions in solid electrolytes is of fundamental interest. Since the discovery of β-alumina,4 Li ion containing β-alumina has been recognized as one of the highest performance Li ion conductors among the Li ion containing solid electrolytes.5, 6 The molecular formula of β-alumina is (M2 O)x · 11Al2 O3 , where M represents a monovalent cation. For stoichiometric β-alumina, x = 1; however, the material is generally nonstoichiometric and x ranges between 1.2 and 1.3. β-alumina is a layered structure material with monovalent ion containing mirror planes situated between alumina spinel blocks. The mirror plane is typically referred to as the conduction plane because the monovalent cation demona) Author to whom correspondence should be addressed. Electronic mail:
[email protected]. Tel.: +81 22 217 5347. Fax: +81 22 217 5347.
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strates two-dimensional (2D) conductivity along this plane. The mirror plane contains three crystallographic sites for the monovalent cation known as the Beevers–Ross (BR), antiBeevers-Ross (aBR), and mid-oxygen (mO) sites, as shown in Fig. 1. Because NMR spectroscopy is sensitive to the local environment as well as to the motion of ions, NMR spectroscopy can be used to study the structure and motion of ions in a crystal. Several NMR studies regarding Li ion motion in β-alumina crystals have been conducted.7–9 Highe et al. investigated the local environment of 7 Li by measuring angle dependent NMR spectra.7 NMR line shape and relaxation measurements provide important information about the correlation time, the jump frequency, and the activation energy of Li ions.10 However, the line narrowing process provides a low estimate of the frequency factor and the activation energy for 7 Li in β-alumina as discussed by Villa et al.9 Walstedt et al. reported on the microscopic co-ionic conductivity of 70% and 90% Li β-alumina single crystals by T1 relaxation measurements.8 Using both NMR and Raman spectroscopy, Villa et al. provided various details concerning the microscopic motion of Li ions in the β-alumina crystal.9 A very low estimate of the pre-exponential factor was also determined for 23 Na in Na β-alumina from line narrowing measurements.11 Line narrowing and relaxation data were also used to investigate the diffusion of Li ions in other Li containing solid electrolytes.12–15
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measurements and NMR line narrowing. Because all previous experiments were performed for polycrystalline/powder samples, we directly measured the anisotropic diffusion coefficient of 7 Li in β-alumina single crystals by changing the orientation of the crystal in a static magnetic field. II. THEORY
FIG. 1. The Li β-alumina crystal structure, as drawn by Vesta threedimensional visualization software. Red, blue, and green ball represent oxygen, lithium, and aluminum ion, respectively. The layered structure of the material is apparent with Li ions existing only in the mirror planes (conduction planes) separated by spinel blocks.
Spin alignment echo (SAE) NMR spectroscopy can be used to study Li ion diffusion parameters such as jump rate and activation energy in glassy and crystalline materials.16–19 SAE-NMR is sensitive to the motion of Li ions over longer time periods and therefore is a suitable technique to measure diffusion parameters at low temperature and probe Li ion motion over a microscopic length scale where other techniques are not appropriate. Although it is possible to estimate selfdiffusion coefficients using T1 relaxation data, the relaxation approach relies on a number of assumptions. Therefore, reliable estimation of the diffusion coefficient by this approach is problematic. The line narrowing process in many cases provides a lower activation energy for Li ion motion because the distribution of correlation times that might appear from the nonexponential correlation function of the motional relaxation process is not included in the typical model.20 Pulsed field gradient (PFG) NMR spectroscopy is a technique by which the self-diffusion coefficient of Li can be directly measured. Maekawa et al. studied mesoporous Al2 O3 using PFGNMR.21 Recently, a number of experimental results have been reported regarding 7 Li ion diffusion coefficient measurements using PFG-NMR.22–24 The Li ion diffusion coefficient in βalumina was first directly measured by Yao and Kumar et al. using a tracer diffusion process.4 Since then there have been no reports concerning measurements of the Li ion diffusion coefficient of Li β-alumina single crystals. Moreover, previous diffusion coefficient measurements conducted by PFGNMR were performed in the high temperature regime. In this work, we report the direct measurement of the diffusion coefficient of Li ions for the first time in a β-alumina single crystal at room temperature using PFG-NMR, and a comparative discussion is presented in connection with results derived from other experimental techniques such as impedance
The self-diffusion process involves a random translational motion of a molecule or ion. It is the most fundamental transport process. In the PFG-NMR method, the attenuation of the NMR signal resulting from the dephasing of the nuclear spins due to the combined effect of translational motion and the imposition of spatially well-defined gradient pulses is used to measure the motion of ions or molecules. The diffusion coefficient D is extracted from the NMR echo intensity as a function of the magnetic gradient field, as fit to Eq. (1) given by Stejskal and Tanner:25 δ , (1) I = I (0) exp −D(2πgγ δ)2 − 3 where I is the echo intensity which is a function of gradient field, I(0) is the intensity without any gradient field, gi is the magnetic field gradient along z direction (along the static magnetic field with unit of G/m), γ is gyromagnetic ratio, for 7 Li it is 1.6553 × 103 Hz/G, is the diffusion time, and δ is gradient pulse duration. The relation between the Larmor frequency ω0 (rad/s) and the static magnetic field B0 with unit G can be expressed as ω0 = γ B0 .
(2)
Since B0 is spatially homogeneous, the Larmor frequency is the same over the entire sample. However, if a magnetic field gradient is applied along the z direction in addition to the static magnetic field, the Larmor frequency becomes position dependent along the z direction. III. EXPERIMENTAL A. Sample preparation and composition analysis
Na β-alumina single crystals were as cut to size from a larger single crystal. Li containing β-alumina was prepared by an ion exchange procedure by heating the original Na βalumina sample in molten LiCl at 700 ◦ C for a total of 48 h. The ion exchange process was completed in two steps. At first, the Na β- alumina samples were kept in to the molten LiCl for 24 h at 700 ◦ C. The samples were then washed and cleaned. Second, the washed and cleaned samples were again kept in to fresh molten LiCl at 700 ◦ C for 24 h. Inductively coupled plasma atomic emission spectroscopy (ICPAES) was used to estimate the Na and Li contents in the β-alumina crystal. B. Impedance measurement
Ionic conductivity of 93% Li β-alumina single crystal along ab plane was measured by ac impedance method using Solartron SI 1260 impedance analyzer within a frequency
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range from 1 Hz to 1 MHz. The impedance measurement was performed by decreasing the temperature from 373 K to 293 K. The ionic conductivity was measured using bulk resistance and the cell constant. The sample size was 5.46 mm × 1.96 mm × 0.48 mm. The ionic conductivity along the c direction was measured using Alpha-A high Performance Frequency Analyzer of novocontrol Technologies in frequency range from 0.01 Hz to 10 MHz. The impedance measurement was performed by decreasing the temperature from 473 K to 293 K. The ionic conductivity measured along the ab plane of 93% Li β-alumina single crystal is directly correspond to the perpendicular orientation of the single crystal during the PFG NMR diffusion experiment.
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experimental( perpendicular) fitting experimental (parallel) fitting
1.0
Intensity (a.u)
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0.8 0.6 0.4 0.2 0.0 -0.2 0
200 400 600 800 1000 1200 1400 1600 1800 2000
Magnetic field gradient(G/cm) C. NMR measurement
The NMR experiment was performed in a Bruker 400 Avance. The 7 Li one-dimensional spectra were obtained using 90◦ pulses with a Larmor frequency of 155.5 MHz with the crystallographic c axis parallel to the static magnetic field. The diffusion coefficient was measured by a stimulated echo pulse sequence. The gradient pulse duration δ was set to 5 ms and the diffusion time was varied from 10 ms to 300 ms. The maximum field gradient was 1492 G/cm. We measured the diffusion coefficient in the 297–333 K temperature range, which is around room temperature. The sample size is 8.26 mm × 4.00 mm ×0.5 mm. Two different orientations of the single crystal with respect to static magnetic field were employed, wherein the Li β-alumina crystal c axis was perpendicular and parallel to the static magnetic field. It should be noted that the magnetic field gradient was applied along the static field direction. IV. RESULTS AND DISCUSSION A. Diffusion measurement
The ICP-AES study indicated that the sample contained 93% Li and 7% Na. The 7 Li echo signal intensity was obtained as a function of the magnetic field gradient. The diffusion coefficient was obtained by fitting the gradient dependent echo intensity to Eq. (1). Because in β-alumina single crystals, monovalent cations possess 2D conductivity along the ab plane, the anisotropic diffusion coefficient of the Li ion was measured with the crystallographic c axis of the single crystal alternately oriented both perpendicular and parallel to the gradient field. A room temperature diffusion coefficient DPFG of 1.22 × 10−11 m2 /s was obtained in the case of the perpendicular orientation. In the case of parallel orientation at room temperature, the diffusion signal intensity was very weak and the signal to noise ratio very low. Therefore, in Fig. 2, we summarize the echo signal intensity as a function the field gradient for perpendicular and parallel orientations at a temperature of 333 K. For the perpendicular orientation of the crystal, a diffusion coefficient DPFG of 2.26 × 10−11 m2 /s was obtained at 333 K. On the other hand, the echo intensity was nearly constant as function of the magnetic field gradient in the case of the parallel orientation, which suggests that the diffusion coefficient along the c direction is very small which is estimated
FIG. 2. The echo signal intensity of 7 Li as a function of magnetic field gradient for the 93% Li β-alumina single crystal. B and g are the static magnetic field and magnetic field gradient directed along the z axis, respectively. The green and black solid squares are the experimental data obtained from PFGNMR spectroscopy with the crystallographic c axis parallel and perpendicular to the static magnetic field, respectively, at 333 K. The blue and red solid lines were obtained by fitting the experimental data to Eq. (1).
as 6.4× 10−13 m2 /s or less at 333 K. This represents the first experimental data ever presented for the diffusion coefficient of the Li ion in β-alumina single crystal by PFG-NMR. Direct measurement of the diffusion coefficient in the parallel and perpendicular directions suggests that macroscopic motion of the cation in β-alumina single crystal is 2D in nature. Here it is also important to mentioned that the present authors measured the ionic conductivity along ab plane and parallel to c axis which are 1.19 × 10−3 −1 cm−126 and 1.85 × 10−15 −1 cm−1 , respectively, at 303 K. Furthermore the temperature dependent conductivity measurement shows that the ab plane activation energy is 0.27 eV and the parallel to the c axis the activation energy is 0.93 eV. Therefore the anisotropic conductivity data are very much consistent with the PFG NMR results. However, a small diffusion coefficient along the parallel direction does not rule out the possibility of local microscopic motion of the Li ion along the c axis since we previously obtained a small activation energy corresponding to the local motion of Li ions from the temperature dependent quadrupole interaction.26 Indication of Li ion motion along the c axis has also been discussed previously based on the heat capacity and dielectric susceptibility measurement of β-alumina single crystal.27, 28 The present authors measured the ionic conductivity of 93% Li β-alumina single crystal single crystal along the ab plane by impedance measurement in the temperature range from 297 K to 373 K which is published previously.26 The temperature dependent ionic conductivity data were used to estimate the diffusion coefficient Dσ of Li ion in β-alumina using the Nernst–Einstein equation given by Eq. (3), Dσ =
kB T σ (T ). N q2
(3)
Here, σ (T) is the ionic conductivity of 93% Li β-alumina at temperature T. At room temperature σ is 1.19 × 10−3 −1 cm−1 reported previously,26 kB is the Boltzmann
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Diffusion Coefficient (m2/s)
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Impedance 10-10
PFG NMR
10-11
10-12
10-13 2.50
2.75
3.00
3.25
3.50
3.75
4.00
-1
1000/T(K ) FIG. 3. Normalized echo signal intensity as a function of (2π γ gδ)2 ( − δ/3) at various temperatures for = 10 ms for the 93% Li β-alumina single crystal oriented perpendicular to the magnetic field. The symbols represent the experimental data. The solid lines correspond to Eq. (1).
constant, N is the charge carrier density, q is the charge of a carrier, and T is the temperature. For direct correspondence between conductivity and diffusion, it is assumed that the Li concentration in the crystal is equal to the Li ion charge carrier density. The room temperature diffusion coefficient Dσ estimated using Eq. (3) is 5.36 × 10−12 m2 /s. The diffusion coefficient of 6 Li in pure Li β-alumina has been previously measured by the tracer diffusion technique in the temperature range 200–400 ◦ C.4 The present authors estimated the room temperature diffusion coefficient DTR to be 6.58 × 10−14 m2 /s by extrapolation from an Arrhenius plot using an activation energy of 8.71 kcal/mole (0.38 eV) and a pre-exponential factor of 14.5 × 10−8 m2 /s which are listed in previous report.4 However, such an estimation of room temperature diffusion coefficient by extrapolation may not be correct, since tracer diffusion data do not include activation energy in the lower temperature regime. Whittingham et al. estimated the Li+ activation energy by DC conductivity which is 0.187 eV in the temperature range from −100 ◦ C to 180 ◦ C and 0.372 eV in the temperature range from 180 ◦ C to 800 ◦ C.29 Whittingham high temperature region activation energy 0.372 eV is almost similar to the tracer diffusion activation energy in the temperature range 200–400 ◦ C. Thus it is possible that tracer diffusion experiment might give new activation energy in the low temperature regime. Therefore, a reasonable approximation of room temperature diffusion coefficient by tracer diffusion is not possible due to the absence of room temperature activation energy data. Fig. 3 shows the normalized 7 Li echo signal as a function of (2π γ gδ)2 ( − δ/3) for = 10 ms, measured in the temperature range 297–333 K, where the slope corresponds to the diffusion coefficient. Because the temperature dependent diffusion coefficient follows an Arrhenius relation, it is possible to estimate the activation energy from the temperature dependent PFG-NMR diffusion data and impedance measurements which are shown in Fig. 4. The activation energy from the impedance measurements was estimated using the Arrhenius relation and was found
FIG. 4. The diffusion coefficient of 93% Li β-alumina as a function of inverse temperature. The blue squares represent the diffusion coefficients obtained from PFG-NMR (measured in the perpendicular orientation). The red squares represent the diffusion coefficient obtained from impedance data using Eq. (3). The blue and red solid line represent the diffusion coefficient estimated from the Arrhenius relation.
to be 0.27 eV, which is higher than the activation energy obtained from the PFG-NMR data (0.14 eV). Therefore, because of the smaller activation energy DPFG is larger than that of Dσ . Table I summarized the diffusion coefficient and the activation energy data obtained from the impedance and the PFG NMR measurement. Recently, Zhongli Wang et al.23 measured the diffusion coefficient of crystalline Li3 N and estimated the activation energy along the XY plane which is 0.15 eV shows good agreement with Impedance data which is 0.19 eV.30 However, Alpen et al.31 shows the Li3 N single crystal activation energy data obtained from impedance measurement is 0.29 eV which is two times higher than that of the PFG NMR data.23 It is also important to mentioned that the activation energy was obtained as high as 0.43 eV and 0.61 eV for polycrystalline Li3 N.23, 32 Nishida et al. observed lower activation energy among different impedance data, which might be due to the presence of oxygen impurity in the sample, since such impurity lead to the formation of vacancies which increases the ionic conductivity and decreased activation energy.30 Therefore defect formation energy might involve in the nearest neighbor position of Li ion in the β-alumina crystal along with site to site migration energy. This finding is discussed in some length as follows. The number of mobile ions, ND depends on the temperature and follows an Arrhenius type relation that can be expressed by Eq. (4) using the formation energy Ef . Ef is actually a defect formation energy that
TABLE I. Diffusion coefficients for the Li ion in 93% Li β-alumina at room temperature (303 K) estimated from impedance measurement and PFG-NMR measurement. Impedance Dσ (m2 /s) 5.36 × 10−12
PFG-NMR Eσ (eV)
DPFG (m2 /s)
EPFG (eV)
0.27
1.22 × 10−11
0.145
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facilitates Li ion motion.
ND (T ) = N0 exp −
Ef kB T
B. Jump rate estimation
.
(4)
On the other hand, the Li ion can jump from one site to another site with migration energy Em . This is actually the elementary jump process for ionic diffusion which corresponds to NMR diffusion. Here, the diffusion coefficient Dm is related to the jump rate ν of an ion from one site to a neighboring site and can be expressed as Em (5) ν(T ) = ν0 exp − kB T and
Em Dm (T ) = D0 exp − . kB T
(6)
The Nernst–Einstein equation (Eq. (3)) relates the conductivity with the diffusion coefficient in accordance with Eq. (7), σ (T ) =
NDq 2 Dm . kB T
Substituting Eqs. (4) and (6) into Eq. (7) provides q2 Eσ σ (T ) = N0 D0 exp − , kB T kB T
(7)
(8)
since Eσ = Ef + Em
q2 N0 D0 . kB T
D=
a2 . 4τ
(11)
In Eq. (11), a is the jump distance between sites which corresponds to the mo–BR, mO–aBR, BR–aBR, or mO–mO distance of the β-alumina single crystal. Because the crystallographic positions for 7 Li 93% Li β-alumina are unknown, the Na β-alumina inter site distances35 are used to estimate the jump period. For mO–BR or mO–aBR jumps (a = 1.5 Å), the jump period τ can be varies from 4.62 × 10−10 s to 2.48 × 10−10 s depending on the temperature, and for BR-aBR or mO-mO jumps (a = 3.2 Å), the jump period varies from 2.10 × 10−9 s to 1.13 × 10−9 s depending on the temperature in the temperature range from 297 K to 333 K. Pre-exponential factors of 6.2 × 1011 s−1 (a = 1.5 Å) and 1.3 × 1011 s−1 (a = 3.2 Å) are also estimated from the PFG-NMR data since the jump period follows the Arrhenius relation given below by Eq. (12),
(9)
1 −Ea 1 exp = , τ τ0 kB T
(10)
using Eqs. (3) and (11), the jump period (or jump rate τ −1 ) can be estimated by Eq. (13) from the previously measured ionic conductivity data obtained by the impedance measurements,26
and σ0 =
A simple model of ionic diffusion in a solid is that of an ion jumping from one site to another site. Because the motion of the Li ion in the β-alumina single crystal is mainly along the ab plane, it is a reasonable assumption that ion jumping occurs between the regular crystallographic BR, aBR, and mO sites. Therefore, the jump period from one crystallographic site to another can be estimated using the simple jump model given by Eq. (11) because the diffusion coefficient is already known from the PFG-NMR data,
Here, Eσ and Em are the activation energies obtained from the impedance measurement and PFG-NMR, respectively. Ef is the defect formation energy because the defect is essential for ion motion. Apart from this simple defect formation energy, Kamishima et al. discussed a different approach using MD calculation where a complex cluster type defect might form due to Ag-Ag interaction in case of Ag β-alumina.33 Such interaction between the ions in different time scale can influence the diffusion process, which is also important to understand the difference of activation energy and the diffusion coefficient obtained from PFG NMR and impedance measurement. The present authors investigated the T1 relaxation time of 7 Li in the low temperature region, and a low activation energy of 0.062 eV was obtained, which might be related to local motion of the Li ion in the β-alumina crystal.26 Such a low activation energy was also observed by Walstedt et al. in the low temperature region of T1 relaxation data.8 Kaneda et al. previously discussed such local motion from the standpoint of Raman spectroscopy.34 Apart from that, the present authors also observed temperature dependent quadrupole interaction, which suggests that out of plane motion for the Li ion might occur in the β-alumina crystal with a small activation energy of 0.029 eV.26
τ −1 =
4kB T σ (T ). N q 2a2
(12)
(13)
Since the jump time τ follow Arrhenius relation as Eq. (12), a pre-exponential factor τ10 can be roughly estimated for impedance and PFG NMR data. This pre-exponential factor or the attempt frequency might correspond to the optical phonon frequency (vibrational frequency) which can be measured directly by Raman or infrared spectroscopy.36 Table II summarizes the pre-exponential factors estimated using Eq. (12) derived from different experimental techniques. The PFG-NMR, impedance, are provided by the present authors, while the previously reported Raman37 and T1 relaxation9 data are included in Table II for the purpose of comparison. Because Raman spectroscopy provides a direct measurement of the vibrational frequency, it is reasonable to compare our data with Raman data. It appears from the Table II that even though simplified jump model was used, Impedance and PFG-NMR data show reasonable estimation of vibrational frequency as far as the Raman data for vibrational frequency of Na in beta alumina is concern which is 2 × 1012 s−1 .37
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TABLE II. The pre-exponential factor corresponds to the attempt frequency or vibrational frequency of 7 Li in 93% β-alumina as estimated from different experimental techniques.
Raman spectroscopy37 (Na β-alumina) (s−1 )
PFG-NMR spectroscopy (s−1 )
NMR T1 relaxation measurement9 (s−1 )
a = 3.2 Å
a = 1.5 Å
a = 3.2 Å
a = 1.5 Å
1012
1.41 × 1011
6.41 × 1011
9.09 × 1012
4.13 × 1013
2 × 1012
C. 7 Li NMR line shape analysis
Fig. 5 shows the temperature dependent 7 Li NMR spectra obtained with the perpendicular orientation. NMR spectral widths were observed to decrease as a function of temperature, which is directly related to the motion of the Li ion and is known as motional narrowing. The temperature dependent spectral width ν(T) (Hz) is fit with the Bluembergen–Purcell and Pound (BPP) relation shown in Eq. (14),20 π ν(T ) 2 1 tan τc (T ) = . (14) 2π αν(T ) 2 vRL Here, τ c (T) is the jump period at temperature T, where α is a constant of order unity and ν R L is the rigid lattice line width, which is 5800 Hz. Therefore, the jump rate ν or the jump period τ = 1/ν can be obtained from the fitting process. Because a jump model can correlate the jump period with the diffusion coefficient, Eq. (11) is used to estimate the diffusion coefficient for a given site to site distance a. Such a site to site distance corresponds to mO to BR, mO to aBR, BR to aBR, or mO to mO distances in the Na β-alumina crystal. The diffusion coefficient DLN obtained from the NMR line narrowing data at room temperature is 1.13 × 10−14 m2 /s, which
Line width(Hz)
6000 5000 4000 3000 2000 1000 0 100 150 200 250 300 350 400 450
Temperature (K)
285k 264k 235k 205k 175k 145k 130k -50
-25
0
Impedance measurements (s−1 )
25
50
75
Chemical shift(ppm) FIG. 5. Temperature dependent 7 Li NMR spectra of the 93% Li β-alumina single crystal with the c axis oriented perpendicular to the static magnetic field with a Larmor frequency of 155 MHz. The upper left corner illustrates the orientation of the single crystal c axis in the static magnetic field B. The graph in the inset shows the line width as a function of temperature. The black squares are the experimental line width data. The red line illustrates the fit of the experimental data to Eq. (15).
is 3 orders of magnitude lower than that obtained from the PFG-NMR and impedance data. The temperature dependent jump period τ (or jump rate τ −1 ), which follows the Arrhenius type relation given in Eq. (12), provides an activation energy of 0.0631 eV and a very low pre-exponential factor (1/τ 0 = 5.0 × 106 s−1 ). The low estimation activation energy (0.063 eV) and pre-exponential factor (3.0 × 106 s−1 ) was also reported for Li contained β-alumina crystal.9 Bishop and Bray reported low estimation of prefactor and activation energy in case of glassy and crystalline Lithium borates.38 Li these diffusion parameters such as diffusion coefficient D, activation energy Ea , pre-exponential factor obtained from NMR motional narrowing process show large disagreement with PFG and impedance data. Therefore, it is important to carefully look at the temperature dependent 7 Li 1D spectra. From the graph shown in the inset of Fig. 5, motional narrowing is nearly complete at approximately 300 K. From 113 K to 130 K the spectra were better fitted with single Gaussian line shape and above 130 K the sharp central peak was fitted with a single Lorentzian line shape. The satellite peak becomes slowly appearing in the higher temperature. In the motional narrowing region from 113 K to 300 K the broaden satellite peak merge with motional narrow central peak. The angle dependent 1D spectra of single crystal β-alumina show the first order weak quadrupole interaction where the central transition (CT) peak does not shift its position as a function of angle; instead the CT peak shift shows the chemical shift anisotropy. However, in the first order quadrupole interaction the satellite peak position shifts as a function of orientation the satellite separation represents the amplitude of the quadrupole interaction called quadrupole coupling constant at the nucleus which is correspond to the EFG along z direction, which is the largest component of the electric field gradient. From the temperature dependent spectra it appears that, the rigid lattice line width 5.8 kHz. As the temperature is increased the line width becomes started to narrowing and around 300 K the motional narrowing process is completed. Since, in the low temperature zone the satellite peak become very broad, the low temperature spectral width contributed by both diope-dipole interaction and quadrupole interaction. At high temperature, where the ST peak becomes sharp enough, the CT spectral width is corresponding to the dipole-dipole interaction. Since in the motional narrowing region, the broad spectrum corresponds to different relaxation mechanism due to the dipole and quadrupole interaction. Therefore different relaxation mechanism affects the spectra differently. Thus diffusion parameters estimated from single component fitting of the NMR spectra can be far from the actual diffusion parameters. The Gaussian line shape of the 1D spectrum
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suggests in the inhomogeneous line broadening. Therefore, the broad single one Gaussian line shape might actually summation of much individual line shape with each having a characteristic relaxation time, which corresponds to the structural inhomogenieity and fluctuation of the local field. Thus a generation expression for diffusion coefficient in the jump model can be written as
D=
g(τ )
a2 dτ . 4τ
(15)
Here g(τ ) is jump time distribution function. If g(τ ) = δ(τ − τ c ), then Eq. (11) is an special case of Eq. (15). The most rough estimation in this case is the mean square displacement (MSD) is approximated as a2 . In real case MSD is correspond to several random jump process after certain time scale. However, since the BPP model produce a single relaxation time from the fitting process, for the rough estimation we used the δ(τ − τ c ) approximation and the a2 corresponds to MSD. The distribution of relaxation time is given as g(τ ) in Eq. (15). Since τ in Eq. (11) is approximated from single broad Gaussian line shape it become overestimated (at room temperature it is 1.97 × 10−6 s), and the diffusion coefficient become very small. Inhomogeneity also results the energy barrier distribution in the jump process. The presence of this energy barrier distribution in Na β-alumina was discussed by Walstedt et al.39 According to Faske et al. the BPP model presented in Eq. (12) does not consider such a distribution and underestimates the activation energy.15 The spectral width in the BPP model is correspond to the spectral density with a exponential correlation function with a single correlation time/relaxation time. This exponential correlation function is corresponding to a single relaxation process which might be oversimplification of the actual relaxation process. However it is suggested that, an inherent disordered system like β-alumina is heterogeneous and that give a distribution of the relaxation time and energy barrier. A non exponential correlation function can deal such a distribution of relaxation time and energy barrier. Therefore the extracting diffusion data from the line narrowing experiment are not flaw less. The last problem in the BPP motional narrowing analysis is the low estimation of the pre exponential factor (1/τ 0 ) in Eq. (14) which is known as the “attempt frequency” related to the jumping motion of the ion. NMR line narrowing the pre-exponential factor appears to be 8.0 × 106 s−1 , which is 5 to 6 orders of magnitude lower than that derived from the Raman data.37 This is known as a prefactor anomaly40 and has been previously reported for the β-alumina crystals.9, 11 The reason behind this low estimate is still not clear. According to Huberman and Boyce et al., it is due to the breakdown of the absolute rate theory where the particle can return to the original position from the top of the barrier height.41 Richards suggests that low dimensionality of the system effects the relaxation processes.42 It suggests that we have to reconsider the dimension and inhomogenieity presence in the material and revised the BPP model that gives the correct description of diffusion phenomena for Li β-alumina single crystal.
D. Diffusion time dependence
It was observed that the diffusion coefficient obtained depended on the diffusion time , and the diffusion coefficient decreased with increasing . It appears that the Li ion can travel faster in the local environment over a short observation time. However, a slower Li ion motion, as demonstrated by a decreasing diffusion coefficient, with an increasing measurement time scale, suggests that a change in the local environment occurs with increasing diffusion time. This suggests the existence of a structural inhomogeneity in the β-alumina single crystal. This structural inhomogeneity might correspond to Li ions residing outside of the conduction plane and/or to the trace amounts of Na √ ions that exist in our sample. The mean squared distance, 4D, traveled by the Li ion at room temperature for various diffusion times from 10 ms to 300 ms are 1.3 μm to 2.7 μm, respectively. Even though the diffusion coefficient decreases with increasing , the total distance traveled by the diffusing ion is still increasing. Thus, interactions between the Li ion and the local environment are not straight forward. Hayamizu et al. found a similar diffusion time dependence for the solid electrolyte (Li2 S)x (P2 S5 )1−x (x = 0.7).24 According to their data, fast and slow moving Li ions can exist simultaneously depending on the thermal activation process involved. Furthermore, these researchers made the surprising observation that, while the fast moving Li ions moved freely, the slow moving ions were localized within small regions corresponding to a grain size of 1.5 μm. Fig. 6(a) shows the diffusion coefficient as a function of the inverse of the diffusion time at room temperature for the 93% Li β-alumina single crystal. The log-log plot suggests that the diffusion coefficient follows a low exponent power law behavior, as given by Eq. (16), n 1 , (16) D∝ where n = 0.138. Equation (16) is directly corresponds to the frequency dependent of conductivity. Therefore the present authors also measured Li β-alumina single crystal frequency dependent conductivity which shows small power law behavior with a exponent of 0.16–0.10, which is consistent with the recently reported experimental data.43 The frequency dependence conductivity of Li β-alumina is shown in Figure 6(b). This time dependent diffusion coefficient and the frequency dependent conductivity suggest the presence of a distribution of relaxation time which is given as g(τ ) in Eq. (15). Recently, Kamishima et al. reported that a power law behavior was observed for that the frequency dependent conductivity of Ag βalumina single crystal where the exponent is vary from 0.11 to 0.15 depends on the temperature which is consistent with the result obtained by the present authors. They suggest a scale invariance property of superionic β-alumina with small power law dependent. In terms of the transport mechanism, the mobile ion might move into a random motion in self repeating structure or a fractal dimension. Therefore, the time dependents diffusion data obtained from the PFG NMR experiment strengthen the arguments that motion of Li ion in the SIC (super ionic conductor) like β-alumina is moving randomly in
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FIG. 6. (a) Diffusion coefficients of 7 Li 93% Li β-alumina as a function of 1/ at room temperature with the perpendicular orientation. The red line was fit to the experimental data in accordance to the power law given by Eq. (16). (b) Frequency dependent conductivity at different temperature of 93% Li β-alumina.
a fractal dimension. Kamishima et al. investigated the many body effect on the ion dynamics in case of Ag β-alumina.33 Because of the inbuilt defect present in the β-alumina structure the ion can take various configurations in the presence of the defect and fast ionic motion is possible. They showed that the Ag-Ag weak repulsive interaction increases the ionic motion with high diffusion coefficient and strong repulsive interaction decrease the ionic transport with due to requirement of the additional energy for structural relaxation with increasing activation energy. This suggests that a relaxation time distribution present in the β-alumina type materials which are consistent with Ngai coupling model.44 Therefore time dependent diffusion of ion in the β-alumina crystal can also be explained by many body interaction or coupling model. It has been previously reported that, for a three-dimensional glass structure system, the frequency dependence of the conductivity follows a power law behavior with an exponent of 0.67 ± 0.03.45 The frequency dependence in a 3D system is explained in terms of relaxation mechanisms,32 coulombic interaction between ions,46 and random hopping of ions in a fractal dimension.47 Therefore, the time dependence of the diffusion coefficient give important information regarding the motion the motion of Li ions in 2D super ionic conductors like Li β-alumina. V. CONCLUSION
The diffusion coefficient of 7 Li in a 93% Li β-alumina single crystal was measured using PFG NMR in two different crystal orientations. This is the first reported attempt to measure the diffusion coefficient of the Li ion in β-alumina single crystals by PFG NMR. The Li ion shows a large diffusion coefficient along the ab plane, while diffusion along the c direction is very weak because the echo signal intensity does not change as a function of the magnetic field gradient. Therefore, Li ion diffusion is anisotropic, and the activation energy of diffusion along the ab plane was found to be 0.14 eV, as estimated from temperature dependent diffusion data. The diffusion coefficient and activation energy were also estimated from impedance measurements. The discrepancy
regarding the activation energies obtained from the PFGNMR and impedance data are explained in terms of the defect formation energy. 7 Li NMR line shape was analyzed using simple BPP model with single line shape fitting process which produce very low estimation of diffusion coefficient, activation energy and frequency pre-exponential factor. Low estimation of diffusion coefficient and activation energy might due to the improper treatment of the experimental line shape. The single Gaussian line shape corresponds to inhomogeneous line broadening which might be due to the relaxation time distribution as well as the energy barrier distribution which is not taken in to account in the BPP model.
ACKNOWLEDGMENTS
The first author, M.T.C., acknowledges MEXT, Japan for the Monbukagakusho scholarship. Thanks are also due to Professor Kamishima for a fruitful discussion concerning the frequency dependence of conductivity in the β-alumina single crystal. 1 G.
Y. Adachi, N. Imanaka, and H. Aono, Adv. Mater. (Weinheim, Ger.) 8, 127 (1996). 2 M. Duclot and J. L. Souquet, J. Power Sources 97–98, 610 (2001). 3 T. Minami, A. Hayashi, and M. Tatsumisago, Solid State Ionics 177, 2715 (2006). 4 Y.-F. Y. Yao and J. T. Kummer, J. Inorg. Nucl. Chem. 29, 2453 (1967). 5 J. L. Briant and G. C. Farrington, J. Electrochem. Soc.: Electrochem. Sci. Technol. 128, 1830 (1981). 6 G. C. Farrington, B. S. Dunn, and J. L. Briant, Solid State Ionics 3/4, 405 (1981). 7 A. Highe and R. W. Vaughan, J. Chem. Phys. 69, 4206–4208 (1978). 8 R. E. Walstedt, R. S. Berg, and J. P. Remeika, Solid State Commun. 37, 209 (1981). 9 M. Villa, J. L. Bjorkstam, G. Mariotto, A. Fontana, and E. Cazzanelli, J. Chem. Phys. 76, 2804 (1982). 10 A. Kuhn, S. Narayanan, L. Spencer, G. Goward, V. Thangadurai, and M. Wilkening, Phys. Rev. B 83, 094302 (2011). 11 W. Bailey, S. Glowinkowski, H. Story, and W. L. Roth, J. Chem. Phys. 64, 4126 (1976). 12 M. Wilkening, S. Indris, and P. Heitjans, Phys. Chem. Chem. Phys. 5, 2225 (2003). 13 P. Heitjans and S. Indris, J. Phys. Condens. Matter 15, R1257 (2003).
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124509-9 14 M.
Chowdhury et al.
Wilkening, D. Brork, S. Indris, and P. Heitjans, Phys. Chem. Chem. Phys. 4, 3246 (2002). 15 S. Faske, H. Eckert, and M. Vogel, Phys. Rev. B 77, 104301 (2008). 16 M. Wilkening and P. Heitjans, Chem. Phys. Chem. 13, 53 (2012). 17 M. Wilkening, A. Kuhn, and P. Heitjans, Phys. Rev. B 78, 054303 (2008). 18 M. Wilkening, J. Heine, C. Lyness, A. R. Armstrong, and P. G. Bruce, Phys. Rev. B 80, 064302 (2009). 19 M. Wilkening and P. Heitjans, Phys. Rev. B 77, 024311 (2008). 20 N. Bloembergen, E. M. Purcell, and R. V. Pound, Phys. Rev. 73, 679 (1948). 21 H. Maekawa, Y. Fujimaki, H. Shen, J. Kawamura, and T. Yamamura, Solid State Ionics 177, 2711 (2006). 22 T. Ohkubo, M. Gobet, V. Sarou-Kanian, C. Bessada, M. Nozawa, and Y. Iwadate, Chem. Phys. Lett. 530, 61 (2012). 23 Z. Wang, M. Gobet, V. Sarou-Kanian, D. Massiot, C. Bessada, and M. Deschamps, Phys. Chem. Chem. Phys 14, 13535 (2012). 24 K. Hayamizu and Y. Aihara, Solid State Ionics 238, 7 (2013). 25 E. O. Stejskal and J. E. Tanner, J. Chem. Phys. 42, 288 (1965). 26 M. T. Chowdhury et al., “The study of the lithium ion motion in β-alumina single crystal by NMR spectroscopy,” Solid State Ionics (published online, 2013). 27 D. B. McWhan, C. M. Varma, F. L. S. Hsu, and J. P. Remeika, Phys. Rev. B 15, 553 (1977). 28 P. J. Anthony and A. C. Anderson, Phys. Rev. B 19, 5310 (1979). 29 M. S. Whittingham and R. A. Huggins, in Solid State Chemistry, edited by R. S. Roth and S. J. Schneider, Jr. (National Bureau of Standards, Washington, D.C., 1972), Special Publication No. 364, p. 148.
J. Chem. Phys. 140, 124509 (2014) 30 K.
Nishida, T. Asai, and S. Kawai, Solid State Commun. 48, 701 (1983).
31 U. V. Alpen, A. Rabenau, and G. H. Talat, Appl. Phys. Lett. 30, 621 (1977). 32 B.
A. Boukamp and R. A. Huggins, Phys. Lett. A 58, 231 (1976). Kamishima, K. Kawamura, T. Hattori, and J. Kawamura, J. Phys.: Condens. Matter 23, 225404 (2011). 34 T. Kaneda, J. B. Bates, and J. C. Wang, Solid State Commun. 28, 469 (1978). 35 R. Collongues, D. Gourier, A. Kahn, J. P. Boilot, P. H. Colomban, and A. Wicker, J. Phys. Chem. Solids 45, 981 (1984). 36 M. B. Salamon, Topics in Current Physics: Physics of Superionic Conductors (Springer–Verlag, Berlin, 1979), p. 162. 37 W. L. Roth, F. Reidinger, and S. La placa, in Super Ionic conductors, edited by G. D. Mahan and W. L. Roth (Plenum Press, New York, 1976). 38 S. G. Bishop and P. J. Bray, J. Chem. Phys 48, 1709–1717 (1968). 39 R. E. Walstedt, R. Dupree, J. P. Rerneika, and A. Rodriguez, Phys. Rev. B 15, 3442 (1977) 40 M. Villa and J. L. Bjorkstam, Solid State Ionics 9–10, 1421 (1983). 41 B. A. Huberman and J. B. Boyce, Solid State Commun. 25, 843 (1978). 42 P. Richards, Solid State Commun. 25, 1019 (1978). 43 O. Kamishima et al., “Frequency dependence of ionic conductivity in a two-dimensional system of Ag β-alumina,” Solid State Ionics (published online, 2014). 44 K. L. Ngai and O. Kannert, Solid State Ionics 53, 936 (1992). 45 H. Kahnt, Ber. Bunsenges. Phys. Chem. 95, 1021 (1991). 46 P. Maass, J. Petersen, A. Bunde, W. Dieterich, and H. E. Roman, Phys. Rev. Lett. 66, 52 (1991). 47 J. Habasaki, K. L. Ngai, and Y. Hiwatari, J. Chem. Phys. 120, 8195 (2004). 33 O.
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