Determination of Ionic Conductivity and Its Impact on Proton Diffusion

The ionic conductivity is an important but previously ignored aspect for the nickel hydroxide used in alkaline batteries. With a specially designed de...
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J. Phys. Chem. B 2006, 110, 2057-2063

2057

Determination of Ionic Conductivity and Its Impact on Proton Diffusion Model for Nickel Hydroxide Liang Xiao, Juntao Lu, Peifang Liu, and Lin Zhuang* Department of Chemistry, Wuhan UniVersity, Wuhan 430072, China ReceiVed: August 26, 2005; In Final Form: NoVember 15, 2005

The ionic conductivity is an important but previously ignored aspect for the nickel hydroxide used in alkaline batteries. With a specially designed device, the ionic conductivity is determined for single beads of spherical nickel hydroxide in KOH solutions. The apparent ionic conductivity is found on the order of 10-3-10-2 S cm-1 in 6 M KOH and to change with the conductivity of the solution in which the bead is immersed. The ionic conductivity of the bead can be mainly attributed to the electrolyte absorbed in the bead. On the basis of these findings, the dual structure model for proton diffusion in spherical nickel hydroxide is refined by specifying nanoparticles to be the component showing a large apparent proton diffusion coefficient (on the order of 10-7 cm2 s-1). This refined model is able to interpret the main features of the diffusion coefficients reported in the literature, including the unusually large scattering (up to 6 orders of magnitude) and inconsistency in the dependence of proton diffusion coefficient on the state of the charge. Besides, this refined model is supported by the influence of bulk KOH concentration on chronoamperometry and transmission electon microscopy observations.

1. Introduction Nickel hydroxide is the active material for the positive electrode of important alkaline rechargeable battery systems, including nickel-cadmium, nickel-hydrogen, nickel-metal hydrides and nickel-zinc.1 The charge and discharge of nickel hydroxide involves proton transport in the material, and the proton diffusion in solid phase is commonly considered as the rate-determining step.2-4 (As will be shown below, however, the commonly considered homogeneous solid phase is actually porous and heterogeneous.) Therefore, the proton diffusion coefficient in this material has attracted much research interest, but the reported values for the proton diffusion coefficient in nickel hydroxide were unusually scattered, spanning about 6 orders of magnitude.2-15 In most previous papers concerning the proton diffusion in nickel hydroxide, a homogeneous model was adopted, but this model cannot explain the huge differences in the reported values for the diffusion coefficient. In our recent paper,16 we reported the determination of a proton diffusion coefficient using potential step measurement on single beads of spherical nickel hydroxide, and a dual structure model was proposed. In this model, the bead was thought to consist of densely packed grains showing a small apparent proton diffusion coefficient (on the order of 10-10 cm2 s-1) and an intergrain component showing a large apparent diffusion coefficient (on the order of 10-7 cm2 s-1). This model is able to interpret the main features of existing data, including the unusually large difference in reported diffusion coefficient values. However, no experimental basis was available at that time to discuss why the diffusion coefficient in a “solid phase” could be so large. The work reported in this paper concerns a crucial but previously ignored aspect of nickel hydroxide, i.e., the ionic conductivity due to the absorbed electrolyte. With a specially designed device, the ionic conductivity was determined for single beads of spherical nickel hydroxide in KOH solutions. The ionic conductivity thus obtained provides a new clue to * Corresponding author. E-mail: [email protected].

Figure 1. Device used for determining ionic conductance in a single bead of spherical nickel hydroxide: A, the cell; B, the bead in the capillary; C, simplified current distribution in the capillary in the presence of a bead.

the understanding of proton diffusion within the previously assumed “solid” phase of nickel hydroxide. 2. Experimental Section The device used for determining the ionic conductance inside a single bead of spherical nickel hydroxide is shown in Figure 1. The cell was divided by a piece of silicon rubber membrane which had a pinhole (a capillary) through its thickness, and a nickel hydroxide bead (ranging 60-150 µm in diameter) was held in the hole. Voltage steps with different amplitudes were imposed across the two nickel foam electrodes using an

10.1021/jp0548467 CCC: $33.50 © 2006 American Chemical Society Published on Web 01/13/2006

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TABLE 1: Representative Data of the Single Bead Measurements (6 M KOH) Fs/Ω cm

Rt/Ω

1.82

5.19 × 10

Rs/Ω 4

5.00 × 10

3

Rsb/Ω

Rb/Ω

82.8

4.70 × 10

electrochemical analyzer (CHI 634a, Shanghai Chenhua Instruments). Because the interfacial impedance at the nickel foam electrode was negligible compared with the ionic resistance in the capillary at high frequencies, the current extrapolated to zero time was determined by the capillary resistance. The latter was calculated from the slope of the straight line of applied voltage versus the extrapolated current, and the ionic resistance of the bead was deduced from the resistance change of the capillary brought about by inserting the bead. The true density of the pristine nickel hydroxide was measured using a pycnometer filled with water and found to be 3.7 g cm-3. According to the weight difference between wet (in saturated water vapor for 2 days) and dry samples (treated in oven at 120 °C for 7-8 h), a porosity 2-3% was estimated for pristine beads, assuming all the pores being filled with absorbed water in the saturated water vapor atmosphere. All electrochemical measurements were carried out at room temperature.

Fb/Ω cm 4

1.03 × 10

κb/S cm-1 3

9.67 × 10-4

pristine and charged beads are given in Figure 2. In both cases, the bead conductance increases with the solution conductance linearly and the straight line goes through the origin within experimental error. This implies that the measured ionic conductivity of the bead is mainly due to the electrolyte solution contained in the bead.

3. Results and Discussion 3.1. Deduction of Ionic Conductance for Single Beads. The experimental data of the extrapolated zero time current versus applied voltage showed satisfactory linearity. From the slope of the linear plot, the ionic resistance of the capillary can be straightforwardly calculated. For the sake of simplicity, the current distribution in the capillary was assumed undistorted by the inserted bead, as shown in Figure 1C. In the capillary, the solution resistance and the bead resistance are connected in series and, therefore, additive to each other. The bead resistance Rb can be found from the change of capillary resistance caused by inserting the bead

Rb ) Rt - (Rs - Rsb)

(1)

where Rt and Rs are the capillary resistances with and without the bead inserted, respectively, and Rsb is the resistance of the solution replaced by the bead. The bead resistance is related to the ionic conductivity of the bead (κb) by the following equation (The derivation of eq 2 is given in the Appendix.)

Rb ) 1/(π κb rb)

(2)

where rb is the radius of the bead. By analogy, Rsb can be calculated from the solution conductivity κs available in the literature19 using the following equation

Rsb ) 1/(π κs rb)

(3)

A set of representative data is shown in Table 1 for a bead of 140 µm diameter in 6 M KOH. The ionic conductivity values for pristine beads changed in the range of approximately 1-5 mS cm-1, depending on individual beads, while the samples from different suppliers did not show variations larger than those found between different beads from the same source. However, the ionic conductivities of fully charged beads were found to be about an order of magnitude larger than those of pristine beads. This increase in apparent ionic conductivity may be attributed to the increase in porosity. 3.2. Dependence of Bead Ionic Conductivity on Solution Conductivity. The ionic conductance of single beads was measured in different KOH concentrations. Typical data for

Figure 2. Ionic conductivity of single beads versus ionic conductivity of KOH solutions in equilibrium with beads: (A) pristine bead; (B) charged bead.

From the measured conductivity, the porosity accessible to the electrolyte solution can be estimated and compared with that found by water absorption experiments. The apparent ionic conductivity of a porous body (κb) filled with electrolyte solution can be related to the conductivity of the solution (κs) through a simple equation

ν ) β2κb/κs

(4)

where ν is the porosity and β2 is the tortuosity of the electrolyte channels in the porous body. For simple cases β2 is about 3 and the ratio κb/κs can be found from the slope in Figure 2. The ν values were found in the range from about 0.5 to 2.5% for pristine beads, in reasonable agreement with the values found in the water absorption experiments. The charged beads all showed much increased ionic conductivities, corresponding to porosities of about 7-10%. The increase in porosity is in line with the commonly reported expansion during charge for nickel hydroxide.17,18 3.3. Comparison of Ionic Conductivity and Diffusion Coefficient Data. Both D and κ reflect the movement of a species in the gradient of a potential. In diffusion, the force driving the species is due to the gradient of chemical potential which is related to the concentration of the diffusing species; in ionic conduction, the driving force is due to the gradient of electric potential. If the driving forces in these two cases are equivalent, the same mass transport fluxes will result. An equation can thus be derived to correlate D and κ

κ ) DcF2/RT

(5)

Ionic Conductivity

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Figure 4. Schematic illustration of dual structure for spherical nickel hydroxide featuring grains and nanoparticles.

Figure 3. The fields for proton diffusion and ionic conduction: A, diffusion in the whole bead; B, diffusion within the particles composing the bead; C, proton conduction across the whole bead as measured in this work.

where c is the concentration of the mobile ions and other symbols have their usual meanings. Now that measurements have been done for both ionic conductivity and proton diffusion coefficient of nickel hydroxide, it will be interesting to compare the results by using eq 5. Because the bead contains both solid and liquid (electrolyte solution) phases, the measured ionic conductivity is, in principle, a sum of the contributions of the two phases. The contribution from the solution phase is proportional to the conductivity of the bulk solution in equilibrium with bead while the contribution from the solid phase is independent of the bulk solution conductivity. Therefore, the intercepts of the lines in Figure 2 represent the ionic conductivities in the solid phase. Figure 2 shows that the ionic conductivity of the solid phase is immeasurably small within the experimental error which was a few tenths of mS cm-1. In our previous paper,16 it was found that the nickel hydroxide bead is composed of a slow component and a fast component with corresponding D and c values being 10-10 cm2 s-1 and 0.017 mol cm-3 and 10-7 cm2 s-1 and 0.006 mol cm-3, respectively. When the D and c values of the slow component were substituted into eq 5, κ came out to be 7 × 10-6 S cm-1, which is negligibly small. However, when the relevant value for the fast component was used, κ turned out to be 2.3 mS cm-1, which is an order of magnitude larger than the experimental error and would cause the straight line in Figure 2B to move upward substantially, as indicated by the dashed line, in conflict with experiment. This obvious conflict has significant impact on understanding the proton diffusion mechanism in nickel hydroxide. As mentioned in our previous paper, the fast component might be either a continuous solid phase with a large proton diffusion coefficient or a large number of nanoparticles with a rather small proton diffusion coefficient. The two possible versions are schematically shown in Figure 3. For the continuous phase version, the proton diffusion in the potential step experiments is assumed to occur in the radial direction of the bead (Figure 3A). For the nanoparticles version, the fast component consists of a large number of nanoparticles and the proton diffusion occurs within individual nanoparticles (Figure 3B) instead of in the radial of the bead. Though the true proton diffusion coefficient is rather small, the apparent coefficient as deduced in the potential step measurement is large because of the very

large total surface area and very small size of the nanoparticles. These two possible models are not distinguishable on the basis of the diffusion coefficient measurement alone. Now a combination of diffusion coefficient and ionic conductivity data allows one to make judgments. Because the ionic conductivity calculated from the continuous phase model (dashed line in Figure 2B) is in obvious contradiction with experimental results (solid line in Figure 2B), the large apparent diffusion coefficient (10-7 cm2 s-1) does not seem to correspond to the proton diffusion in the radial direction of the bead. In contrast, according to the nanoparticle model for the fast component, the true diffusion coefficient is about 8 orders of magnitude smaller that the apparent diffusion coefficient (see section 3.6). Therefore, according to eq 5, the ionic conductivity due to the proton mobility in solid phase will be at most on the order of 10-11 S cm-1, which is in agreement with the essentially zero intercept in Figure 2. 3.4. Dual Structure Model Featuring Solid Grains and Nanoparticles for Nickel Hydroxide. According to the discussion in the last paragraph, it seems that both the fast and slow components are subparticles in the bead. The main difference between the two components is the particle size. The refined model for the bead of spherical nickel hydroxide is schematically depicted in Figure 4. The bead is composed of relatively large particles (called grains) for the slow component and nanoparticles for the fast component. The size of the grains ranges from about 50 to 100 nm, and in the same single bead all grains have about the same size. The nanoparticles are also assumed to have approximately uniform size in a single bead. The grains and nanoparticles are both roughly spherical. The apparent volume of a bead is mainly determined by the number of rather densely packed grains, and the intergrain space is filled with nanoparticles. The grains and nanoparticles are electronically well connected to each other in a bead (except for the pristine bead which is poorly conducting). The bead is slightly porous, and the absorbed electrolyte forms an ionic network to connect all the grains and nanoparticles in the bead to the solution outside the bead. In discharge, the grains and nanoparticles act as many tiny individual electrodes, and the recorded current is a sum of all the currents generated by the grains and nanoparticles. Because of distinctly different particle sizes, the two components have different time scales in discharge. The nanoparticles have small sizes but a large total surface area so that they produce a large total current in the short interval of time at the beginning of potential-step measurement, but this current decays quickly because of fast exhaustion of the nanoparticles. This process is reflected by the large apparent diffusion coefficient Df corresponding to the so-called fast component. In contrast, because of relatively large size, the discharge of the grains can last much longer than that of the nanoparticles and dominates the recorded current in the long time response of potential-step measurement. This process is reflected by an apparent diffusion coefficient

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Xiao et al. There are two good approximate equations for deducing D from potential step measurements, one for the initial time and the other for the long time. The two equations correspond to two typical diffusion modes, i.e., semi-infinite diffusion and finite diffusion, respectively. The actual applicable time regions for the two equations depend on the value of D and the dimension of the sample in particular cases. When r2/πD < 1, the diffusion is in the semi-infinite mode, and this time region is referred to as the initial short time in this paper. According to our model, in the initial short time of a potential step measurement, the recorded current is overwhelmingly governed by the currents generated from the nanoparticles and can be described by the following equation

ip ) 4nFπrp2Dpc[(πDpt)-1/2 - rp-1][fp(rb/rp)3]

Figure 5. TEM image of ground nickel hydroxide.

Ds, which is much smaller than Df, in the potential step measurement. 3.5. Transmission Electron Microscopy Observation of Nanoparticles. To check the presence of the proposed nanoparticles, transmission electron microscopy (TEM) examination was attempted. The spherical nickel hydroxide was manually ground with an agate mortar and pestle. The sample was wetted with water to prevent aggregation, and the grinding lasted about half an hour. The suspension thus formed was loaded on the screen for TEM. Figure 5 shows a large number of nanoparticles with an approximately uniform size, about 3 nm in diameter. It should be pointed out that it is usually impossible to disperse a material into uniform nanoparticles by simply grinding with mortar and pestle. Therefore the observed nanoparticles should originally exist in the bead. The function of manual grinding here was only to break the spherical nickel hydroxide beads into smaller pieces suitable for TEM observation. In our previous work,16 atomic force microscopy and scanning electron microscopy revealed spherical features of dimensions in the range of about 50-100 nm. These observations and the TEM image shown in Figure 5 proved the existence of grains and nanoparticles in a bead. However, the internal structure of the grains is not yet clear. Up to now, we failed to see both grains and nanoparticles at the same time under TEM. It may imply that a grain is also composed of nanoparticles. The nanoparticles in grains and the nanoparticles in intergrain spaces may differ mainly only in their packing densities. The nanoparticles may be more densely packed in grains than in the intergrain spaces, but this density difference may be hard to recognize under TEM. Efforts are needed in the future to see more microscopic details. 3.6. Apparent versus Intrinsic Diffusion Coefficients. According to the dual structure model, the experimental apparent diffusion coefficient can be related to the intrinsic diffusion coefficient in the grains and nanoparticles if the sizes for the grains and nanoparticles are known. For the sake of simplicity, the effective proton (or proton vacancy in charged samples) concentrations inside the grains and nanoparticles are assumed both equal to the average concentration over the whole bead. Because the correlation between the apparent and intrinsic coefficients for the grains is analogous to that for the nanoparticles, these correlations can be considered in a generalized way.

(6a)

where Dp is the intrinsic diffusion coefficient in the nanoparticles and rp and fp are the radius and volume fraction for the nanoparticles. On the right-hand side of eq 6a, the quantity in the second pair of brackets is the number of nanoparticles in the bead, and the rest represents the current produced by a single nanoparticle. In previously published work (except for ref 16), the presence of the grains and nanoparticles substructures was ignored and all the experimentally deduced diffusion coefficients are apparent diffusion coefficients, Dapp. The equation used to calculate Dapp from the initial current in those works was

i ) 4nFπrb2Dappcapp[(πDappt)-1/2 - rb-1]

(6)

Equation 6a can be rearranged to fit the format of eq 6 for easy comparison

ip ) 4nFπrb2Dp(rb/rp)2c fp{[πDp(rb /rp)2t]-1/2 - rb-1} (6b) Comparing eq 6b with eq 6 reveals

Dapp ) Dp(rb /rp)2

(7a)

capp ) fpc

(7b)

and

A similar derivation for the long time response current also leads to the same conclusion. In this case, the subscript p in eqs 7a and 7b refers to the grains. Equation 7a indicates that the apparent diffusion coefficient is proportional to the square ratio of the bead radius to the particle radius and is not influenced by the volume fraction fp. It should be pointed out that for film samples an equation similar eq 7a is available with rb being replaced by the film thickness. In our previous potential step measurements on over 30 samples,16 the average bead diameter was 26 µm, and average apparent diffusion coefficients for the fast and slow components were Df ) 2 × 10-7 and Ds ) 6 × 10-10 cm2 s-1, respectively. Taking a diameter of 3 nm for the nanoparticles and 75 nm for the grains and recalling that the grains and nanoparticles are responsible for the fast and slow diffusion coefficients, respectively, the intrinsic diffusion coefficients for the nanoparticles and grains can be calculated using eq 7a

Dnano ) Df(3 nm/26 µm)2 ) 3 × 10-15 cm2 s-1

(8a)

Dgrain ) Ds(75 nm/26 µm)2 ) 5 × 10-15 cm2 s-1 (8b)

Ionic Conductivity In view of the approximate nature of the above estimations, it can be concluded that the intrinsic proton diffusion coefficients in the grains and nanoparticles are essentially the same. The proton diffusion coefficients reported in the literature for nickel hydroxide are all apparent values, Dapp. According to the refined dual structure model and eq 7a, the main features of reported Dapp values can be explained straightforwardly. First of all, because rb /rp is always larger than unity, the apparent diffusion coefficient must be larger than the intrinsic one which was estimated on the order of 10-15 cm2 s-1. The smallest D value found in the literature is 3.4 × 10-13 cm2 s-1 for a cathodically precipitated film of 20-40 nm thickness.11 It is noted that the film thickness is an order of magnitude larger than the nanoparticles (3 nm) and the apparent diffusion coefficient is 2 orders of magnitude larger than the intrinsic value, in good agreement with eq 7a. Second, depending on the time scale of measurements and other experimental conditions, the electrochemical response of the sample may be dominated by either the grains or the nanoparticles. Because the grains and the nanoparticles differ by about 1.5 orders of magnitude in size, according to eq 7a, two apparent diffusion coefficients differing by 3 orders of magnitude may be obtained for the same sample. This was just what we found for the fast and slow components in the same sample. Third, the unusually large difference in reported D values can mainly be attributed to the difference in the ratio of the sample dimension (bead radius or film thickness) to the dimension of the particles whose electrochemical response dominates the measurements. For example, this ratio for the smallest reported D (3.4 × 10-13 cm2 s-1)11 may be estimated to be 101 while the ratio for the fast component (Df ) 2 × 10-7 cm2 s-1) of our spherical nickel hydroxide was about 104. It can be seen that the apparent diffusion coefficients in the two cases differ by a factor of 106 while the dimensional ratios differ by 103, again in agreement with eq 7a. Finally, the refined model can easily explain the inconsistency in the SOC (the state of charge) dependence of the apparent diffusion coefficient. The intrinsic diffusion coefficient for either the grains or the nanoparticles is SOC independent. Therefore, when the electrochemical response is determined by either the grains or the nanoparticles, the deduced apparent diffusion coefficient will be SOC independent. The initial response of a fully charged sample is determined by the nanoparticles and the long time response in any SOC is determined by the grains. In these two cases the deduced D should be SOC independent. However, for partially discharged samples, the initial response is governed by both grains and nanoparticles in a way depending on SOC, and therefore the diffusion coefficient deduced from the initial response should be SOC. These predictions are in agreement with experimental findings.16 3.7. Influence of Solution Concentration on the PotentialStep Measurements. The refined dual model (“grains plus nanoparticles”) is identical with the previous version (“grains plus continuous fast component”) in all aspects except for specifying the nanoparticles to be the fast component. These two versions both can rationalize the main discrepancies in the literature, including the unusually large scattering of the reported D values, and the inconsistency in D dependence on the state of charge. However, the new data given in this section can only be interpreted by the “grains plus nanoparticles” version. Figure 6 shows the chronoamperometric curves for the same bead recorded in 1 M and 6 M KOH solutions, respectively. It can be seen that the plot of current versus reciprocal square root of time is linear in the time interval corresponding to t-1/2

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Figure 6. Potential-step chronoamperometric curves of a single bead (diameter 28.6 µm) in 1 and 6 M KOH. Potential stepped from 1.45 to 1.15 V (vs RHE).

from about 6 to 1 s-1/2 for 6 M KOH (curve a) while the curve for 1 M KOH (curve b) largely deviates to lower values for t-1/2 > 4 s-1/2. This phenomenon can be well interpreted in the framework of the refined dual structure model as follows. According to the refined dual structure model, the bead is porous and composed of grains and nanoparticles. All grains and nanoparticles are connected to the bulk electrolyte solution through the electrolyte absorbed in the bead. In potential step measurements, the particles (grains and nanoparticles) were discharged independently. The discharge process of each particle depended on the electrode potential experienced by that particle. Because of the internal resistance in the bead, the electrode potential of a particular particle depended on the location of the particle. The distribution of local electrode potential inside a bead can be estimated using Ohm’s law. For the sake of simplicity, the bead is assumed to be well conductive electronically; i.e., the electric potential in the solid phase is uniform throughout the bead. (Note that the word “electric potential” and “electrode potential” are conceptually different here.) In contrast, the electric potential of the solution phase in the bead changes in the radial direction because of ohmic drop. Assuming, for simplicity, a uniform volumetric current density I throughout the bead, the current generated by the particles inside the sphere of radius r (the upper in Figure 7) is 4πr3I/3. According to Ohm’s law, the potential gradient in the solution phase at r is

dφl/dr ) FbI(4πr3/3)/4πr2

(9)

where φl is the internal electric potential of the solution phase in the bead and Fb is the apparent ionic resistivity of the bead. The sign of eq 9 implies that the ionic current and the radial axis r are opposite in direction. It is well-known that the change in electrode potential is equal to the change in the internal electric potential difference between the solid (electrode) and solution phases, i.e., dE ) d(φs - φl), where φs is the internal electric potential of the solid phase. Because the electronic conductivity of the bead is assumed to be sufficiently high, φs is independent of r. Therefore, dφs ) 0 and dE ) -dφl. The distribution of φl and, in turn, E can be found by solving eq 9 with appropriate boundary conditions, and the results are shown in the lower part of Figure 7. In discharge, a lower electrode potential means a larger polarization (a larger driving force for the electrode reaction). The curve indicates that the particles near the bead center are less polarized than those near the bead surface. The difference of local electrode potential between the

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Figure 8.

Figure 7. Schematic illustration of ohmic drop and potential distribution inside a bead: top, illustration of eq 11; bottom, local electrode potential distribution.

center and the surface of the bead may serve as an indicator for the unevenness of local electrode potential in the bead and can be obtained by solving eq 9

∆E ) IFbrb2/6

(10)

The value of ∆E can be estimated by substituting representative values for the variables in eq 10. For example, the bead shown in Figure 6 had a radius rb ) 14.3 µm and at t-1/2 ) 6 s-1/2 the current was i ) 15 µA. The volumetric current density came out to be I ) 1229 A cm-3, which is very large. The Fb value was about 80 Ω cm (Figure 2B). Thus, ∆E ) 0.03 V. The linearity of curve a in Figure 6 indicates that a 0.03 V difference in local electrode potential did not cause appreciable unevenness of current distribution within the bead. However, curve a also indicates that 0.03 V is about the upper limit for a linear i vs t -1/2 plot. Curve a in Figure 6 reveals a trend that the data points for t-1/2 > 6 (i.e., t < 28 ms) would fall below that predicated by the linear relationship. At shorter times the particles were able to generate larger currents because of thinner effective diffusion thickness. In general, the shorter the time, the larger the bead current. However, according to eq 10, this would cause a larger ∆E. In this case, the particles in the central part of the bead would not be sufficiently polarized and could not produce currents as much as the particles near the bead surface did. As a result, the total current of the bead was smaller than that predicated by the linear relationship. Apparently, whether the linearity holds depends on the ohmic drop in the bead and the latter depends on both the current and ionic resistivity of the bead. The higher the ionic resistivity, the lower the upper current limit for the linearity. Because the ionic resistivity of 1 M is about 3 times that of 6 M KOH, it may be expected that the upper limit current for linearity in 1 M KOH would be about one-third of that for 6 M KOH. This prediction is in reasonable agreement with experimental findings (Figure 6). Thus the data in Figure 6 serve as a favorable support of the refined dual structure model. 4. Summary Using a specially designed device, ionic conductivity was successfully determined for single beads of nickel hydroxide. The data indicate that the ionic conductivity in a bead is mainly

due to the electrolyte solution absorbed in the bead and the proton mobility in the solid phase has little contribution, in conflict with the rather high ionic conductivity converted from the large apparent proton diffusion coefficient. On the basis of this new finding, the dual structure model for nickel hydroxide was refined by specifying the component of fast proton diffusion to be the nanoparticles filling in the spaces between grains. In the refined model, a bead is considered to be a porous assembly consisting of grains (diameter 50-100 nm) and nanoparticles (diameter round 3 nm). All the gains and nanoparticles are connected to the bulk electrolyte solution through the ionic network formed by the absorbed electrolyte in the bead. The grains and nanoparticles each behave like individual tiny electrodes and contribute to the total current of the bead during change and discharge. The ohmic drop in the solution phase causes unevenness of electrode potential and local current density along the radial of the bead. This refined model can interpret the main features of the proton diffusion coefficient data reported in the literature as the previous version did. Besides, the “grains plus nanoparticles” model is also supported by the chronoamperometry in different concentrations of KOH and TEM observations. Acknowledgment. This work was supported by the National Natural Science Foundation of China (NSFC No. 20073223). Appendix: Derivation of Equation 2 When the bead is placed in a capillary, for simplicity, the current distribution is assumed not to be distorted by the presence of the bead (Figure 1C). Accordingly, the bead can be imagined to be composed of a series of coaxial differential tubes, and the effective conductance of the bead in the capillary is the sum of the conductances of all the coaxial tubes. Denoting the radius and thickness of the differential tube as x and dx, respectively, the length L and cross section area A of the tube are given by eqs A and B (Figure 8).

L ) 2(r2 - x2)1/2

(A)

A ) 2πx dx

(B)

Denoting the apparent ionic conductivity of bead as κ, the differential conductance of the tube is

dσ ) Aκ/L

(C)

The total resistance of the bead is the reciprocal of the integration of eq C from x ) 0 to x ) r

R ) 1/

x)r dσ ) 1/πκr ∫x)0

(D)

References and Notes (1) Linden, D. Handbook of Batteries; McGraw-Hill: New York, 1995.

Ionic Conductivity (2) Weidner, J. W.; Timmerman, P. J. Electrochem. Soc. 1994, 141, 346. (3) Motupally, S.; Streinz, C. C.; Weidner, J. W. J. Electrochem. Soc. 1998, 145, 29. (4) Paxton, B.; Newman, J. J. Electrochem. Soc. 1996, 143, 1287. (5) Srinivasan, V.; Weidner, J. W.; White, R. E. J. Solid State Electrochem. 2000, 4, 367. (6) MacArthur, D. M. J. Electrochem. Soc. 1970, 117, 422. (7) Zhang, C.; Park, S. J. Electrochem. Soc. 1987, 134, 2966. (8) Yoon, Y.; Pyun, S. Electrochim. Acta 1997, 42, 2465. (9) Briggs, G. W. D.; Snodin, P. R. Electrochim. Acta 1982, 27, 565. (10) MacArthur, D. M. J. Electrochem. Soc. 1970, 117, 729. (11) Ta, K. P.; Newman, J. J. Electrochem. Soc. 1998, 145, 3860.

J. Phys. Chem. B, Vol. 110, No. 5, 2006 2063 (12) Kim, H.-S.; Itoh, T.; Nishizawa, M.; Mohamedi, M.; Umeda, M.; Uchida, I. Int. J. Hydrogen Energy 2002, 27, 295. (13) Motupally, S.; Weidner, J. W. J. Electrochem. Soc. 1995, 142, 1401. (14) Watanabe, K.; Kikuoka, T. J. Appl. Electrochem. 1995, 25, 219. (15) Gille, G.; Albrecht, S.; Meese-Marktscheffel, J.; Olbrich, A.; Schrumpf, F. Solid State Ionics 2002, 148, 269. (16) Xiao, L.; Lu, J.; Liu, P. Zhuang, L.; Yan, J.; Hu, Y.; Mao, B.; Lin, C. J. Phys. Chem. B 2005, 109, 3860. (17) Chen, J.; Bradhurst, D. H.; Dou, S. X.; Liu, H. K. J. Electrochem. Soc. 1999, 146, 3606. (18) Singh, D. J. Electrochem. Soc. 1998, 145, 116. (19) Fu, X.; Shen, W.; Yao, T. Physical Chemistry, 4th ed.; Higher Education Press: p 328.