**3. Results**

In brief, experiments were interpreted to yield information regarding the frequency dependence of these parameters: capacitance and dielectric values below 0.8 V, energy density, and power density over the range of 0.1–2.3 V. For all nine capacitors containing aqueous solutions with dissolved ions, but not the control using distilled water, the data permitted excellent power law fits to all parameters.

#### **3.1. Control**

A capacitor employing distilled water as the electrolyte had such low values of relevant parameters that it was difficult to determine any parameters with precision, given the small capacitor size and the parameter ranges chosen for this study. A reliable power law 'roll-off' function was not obtained for any parameter as the absolute values were so small that the signal/noise ratio was large; however, it can be stated with certainty that the highest measured energy density was less than 0.03 J/cm<sup>3</sup> clearly demonstrating that anodized titania-based T-SDM containing distilled water are not SDM.

### **3.2. KOH**

In order to illustrate general trends for all three solutes, the data on all three capacitors created with KOH-based SDM are presented in detail. The trends of energy and power density as well as dielectric and capacitance values determined below 0.8 V are shown on log-log plots, and in each case, it is clear that the data are well represented by simple power law relations over a wide range of discharge times.

Energy and power values are derived directly from data over the entire discharge voltage, thus may be considered as the most reliable. As shown in **Figure 2**, all the energy density data for KOH are well fit by simple power law relationships over four orders of magnitude of discharge time, that is from 20 to 0.002 s. Moreover, the curve fit is clearly of a quality that permits reasonable extrapolation to the energy density anticipated even for a 1000 s discharge. This value of energy density at this very slow discharge rate is suggested herein as a reasonable comparison point with battery energy densities (**Table 1**). It is notable that energy density is not a linear function of KOH concentration, but the 30 wt% sample was clearly superior.

for many pulsed power applications. The data for NaNO<sup>3</sup>

1.00E-01

1.00E+00

**Energy**

**Density**

**[J/cm3]**

1.00E+01

1.00E+02

noted with all other SDM-based capacitors [9–13].

representing a summary of all nine capacitors. Greater detail is available elsewhere [24].

interest as it shows remarkably high values for very small volume systems.

Capacitance as a function of discharge time for all three KOH-based systems is shown in **Figure 4**. These values were computed directly from the slope of the curves below ~0.8 V (**Figure 1**) and hence are only valid below this value. Despite this limitation, the data are of

**Figure 2.** Energy density vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the energy density 'rolls off' as a very specific function of discharge time. This allows determination of energy density with high precision over a broad discharge time range. Note the curve slope increases, and the energy density at all concentrations increases with increasing solute concentration. Employing the linear fitting equations with DT, in seconds, yields energy in J/cm<sup>3</sup>

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as…

**Discharge Time [sec]**

It is notable that none of the parameters, including capacitance, show a clear pattern with salt concentration. The fact that key parameters do not track with salt concentration has been

The final parameter of interest is the dielectric constant, generally an excellent engineering value as it permits the selection of capacitors, based on this single number, with a high degree

and NH<sup>4</sup>

Cl are only shown in a figure

.

KOH 10% Energy = 7.04(DT)^0.49 KOH 20% Energy = 6.33(DT)^0.52 KOH 30% Energy = 11.74(DT)^0.57

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75

Power density, following a trend observed previously for capacitors constructed with fabric-SDM (13), increases as the discharge time decreases. The data for the KOH SDM-based capacitors are shown in **Figure 3**. The trend shown was also found for SDM based on NaNO<sup>3</sup> and NH<sup>4</sup> Cl aqueous solutions. The absolute values are also very informative. For example, the capacitors can provide of the order 100 W/cm<sup>3</sup> for discharges of 0.01 s, a remarkably high value appropriate

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as… http://dx.doi.org/10.5772/intechopen.71003 75

could fit the data collected in this fashion (see Results) demonstrates the efficacy of this method for determination of frequency response. An error analysis of this approach available elsewhere [13] suggests that all data for energy and power density is accurate to within 10% absolute.

In brief, experiments were interpreted to yield information regarding the frequency dependence of these parameters: capacitance and dielectric values below 0.8 V, energy density, and power density over the range of 0.1–2.3 V. For all nine capacitors containing aqueous solutions with dissolved ions, but not the control using distilled water, the data permitted excel-

A capacitor employing distilled water as the electrolyte had such low values of relevant parameters that it was difficult to determine any parameters with precision, given the small capacitor size and the parameter ranges chosen for this study. A reliable power law 'roll-off' function was not obtained for any parameter as the absolute values were so small that the signal/noise ratio was large; however, it can be stated with certainty that the highest measured

In order to illustrate general trends for all three solutes, the data on all three capacitors created with KOH-based SDM are presented in detail. The trends of energy and power density as well as dielectric and capacitance values determined below 0.8 V are shown on log-log plots, and in each case, it is clear that the data are well represented by simple power law relations over

Energy and power values are derived directly from data over the entire discharge voltage, thus may be considered as the most reliable. As shown in **Figure 2**, all the energy density data for KOH are well fit by simple power law relationships over four orders of magnitude of discharge time, that is from 20 to 0.002 s. Moreover, the curve fit is clearly of a quality that permits reasonable extrapolation to the energy density anticipated even for a 1000 s discharge. This value of energy density at this very slow discharge rate is suggested herein as a reasonable comparison point with battery energy densities (**Table 1**). It is notable that energy density is not a linear function of KOH concentration, but the 30 wt% sample was clearly superior.

Power density, following a trend observed previously for capacitors constructed with fabric-SDM (13), increases as the discharge time decreases. The data for the KOH SDM-based capacitors

aqueous solutions. The absolute values are also very informative. For example, the capacitors

are shown in **Figure 3**. The trend shown was also found for SDM based on NaNO<sup>3</sup>

clearly demonstrating that anodized titania-based

for discharges of 0.01 s, a remarkably high value appropriate

and NH<sup>4</sup>

Cl

**3. Results**

**3.1. Control**

**3.2. KOH**

lent power law fits to all parameters.

74 Supercapacitors - Theoretical and Practical Solutions

energy density was less than 0.03 J/cm<sup>3</sup>

a wide range of discharge times.

can provide of the order 100 W/cm<sup>3</sup>

T-SDM containing distilled water are not SDM.

**Figure 2.** Energy density vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the energy density 'rolls off' as a very specific function of discharge time. This allows determination of energy density with high precision over a broad discharge time range. Note the curve slope increases, and the energy density at all concentrations increases with increasing solute concentration. Employing the linear fitting equations with DT, in seconds, yields energy in J/cm<sup>3</sup> .

for many pulsed power applications. The data for NaNO<sup>3</sup> and NH<sup>4</sup> Cl are only shown in a figure representing a summary of all nine capacitors. Greater detail is available elsewhere [24].

Capacitance as a function of discharge time for all three KOH-based systems is shown in **Figure 4**. These values were computed directly from the slope of the curves below ~0.8 V (**Figure 1**) and hence are only valid below this value. Despite this limitation, the data are of interest as it shows remarkably high values for very small volume systems.

It is notable that none of the parameters, including capacitance, show a clear pattern with salt concentration. The fact that key parameters do not track with salt concentration has been noted with all other SDM-based capacitors [9–13].

The final parameter of interest is the dielectric constant, generally an excellent engineering value as it permits the selection of capacitors, based on this single number, with a high degree


The values at 10 s (discharge time) are in the measured range. The values at 1000 s are extrapolated values based on using the power law fits.

**Table 1.** Select parameters.

of certainty they will perform as anticipated. However, for SDM-based capacitors employed for energy storage, for which dielectric constant is not a constant of voltage or frequency, it is not a quantitative predictor of performance. Notably, the dielectric constant also does not serve any role in rating EDLC for which dielectric constants in the traditional sense can-

**Figure 4.** Capacitance vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the capacitance clearly follows a simple power law in all cases, permitting determination of its value with high precision over a broad

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

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77

KOH 10% Capacitance = 9.40E-03(DT)^0.42 KOH 20% Capacitance = 9.20E-03(DT)^0.47 KOH 30% Capacitance = 15.50E-03(DT)^0.47

**Discharge Time [sec]**

'dielectric' value cited [31–34]. Still, there are two good reasons for measuring and reporting this value. First, it provides a qualitative predictor of energy and power density. Second, the values (**Figure 5**) permit a quantitative comparison with the historic database of dielectric materials, including other super dielectric materials. For example, the far greater values

is the only

greater than any solid

not really be measured. In fact, for EDLC a dielectric constant with units, F/cm<sup>2</sup>

of dielectric constants for SDM below ~1 V, generally more than 10<sup>5</sup>

dielectric, show them to be a distinct class of materials.

1.00E-04

discharge time range.

1.00E-03

**Capacitance [F]**

1.00E-02

1.00E-01

**Figure 3.** Power density vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the power density increases with decreasing discharge time. The data clearly follow a simple power law in all cases, permitting determination of power density with high precision over a broad discharge time range. Employing the linear fitting equations, with DT, in s, yields power in W/cm<sup>3</sup> .

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as… http://dx.doi.org/10.5772/intechopen.71003 77

**Figure 4.** Capacitance vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the capacitance clearly follows a simple power law in all cases, permitting determination of its value with high precision over a broad discharge time range.

of certainty they will perform as anticipated. However, for SDM-based capacitors employed for energy storage, for which dielectric constant is not a constant of voltage or frequency, it is not a quantitative predictor of performance. Notably, the dielectric constant also does not serve any role in rating EDLC for which dielectric constants in the traditional sense cannot really be measured. In fact, for EDLC a dielectric constant with units, F/cm<sup>2</sup> is the only 'dielectric' value cited [31–34]. Still, there are two good reasons for measuring and reporting this value. First, it provides a qualitative predictor of energy and power density. Second, the values (**Figure 5**) permit a quantitative comparison with the historic database of dielectric materials, including other super dielectric materials. For example, the far greater values of dielectric constants for SDM below ~1 V, generally more than 10<sup>5</sup> greater than any solid dielectric, show them to be a distinct class of materials.

1.00E+00

equations, with DT, in s, yields power in W/cm<sup>3</sup>

1.00E -03 1.00E -02 1.00E -01 1.00E+00 1.00E+01 1.00E+02

KOH 10% Power = 7.04(DT)^-0.51 KOH 20% Power = 6.33(DT)^-0.48 KOH 30% Power = 11.74(DT)^-0.43

**) Power density (Watt/cm<sup>3</sup>**

**)**

**Discharge Time [sec]**

**Figure 3.** Power density vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the power density increases with decreasing discharge time. The data clearly follow a simple power law in all cases, permitting determination of power density with high precision over a broad discharge time range. Employing the linear fitting

.

1.00E+01

**Power Density [W/cm3]**

1.00E+02

NH<sup>4</sup>

NH<sup>4</sup>

NH<sup>4</sup>

NaNO<sup>3</sup>

NaNO<sup>3</sup>

NaNO<sup>3</sup>

using the power law fits.

**Table 1.** Select parameters.

**Solute (wt%) Dielectric constant<0.8V Energy density (J/cm3**

76 Supercapacitors - Theoretical and Practical Solutions

KOH (10) 1.1 E+8 7.3 E+8 22 208 74 KOH (20) 1.2 E+8 1.0 E+9 21 230 58 KOH (30) 2.1 E+8 1.8 E+9 44 602 85

Cl (10) 2.7 E+8 4.1 E+9 35 556 56

Cl (20) 2.2 E+8 2.9 E+9 30 356 72

Cl (30) 2.1 E+8 1.9 E+9 34 363 98

(10) 1.3 E+7 9.8 E+7 4 23 26

(20) 3.1 E+7 3.0 E+8 12 125 37

(30) 3.0 E+7 2.8 E+8 12 142 36

**10 s 1000 s 10 s 1000 s 0.01 s**

The values at 10 s (discharge time) are in the measured range. The values at 1000 s are extrapolated values based on

**Figure 5.** Dielectric constant vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the dielectric constant follows, below ~0.8 V, a simple power law, permitting determination of its value with high precision over a broad discharge time range. The absolute values of the dielectric constant are greater than 106 even at a discharge time of order 10−3 s, indicating these materials were super dielectric materials over the full range tested.

observed for both the aqueous KOH- and NH<sup>4</sup>

**Figure 6.** Dielectric constant vs. discharge time for NH<sup>4</sup>

materials were super dielectrics over the full range tested.

to those of KOH at all discharge times, and greater than 106

to be true in the next section.

1.00E+06

1.00E+07

Dielectric [-]

1.00E+08

1.00E+09

sity of the capacitors employing NaNO<sup>3</sup>

provide a reliable integrated energy density.

shown in **Figure 9**. Once again, the KOH and NH<sup>4</sup>

**3.5. Energy and power**

indicator that the energy and power density of capacitors built T-SDM employing this solution will not perform as well for storing energy and providing power. This is shown

dielectric constant follows, below ~0.8 V, a simple power law. The absolute values of the dielectric constant are similar

1.00E-03 1.00E-02 1.00E+01 1.00E+02

1.00E-01 1.00E+00 Discharge Time [sec]

NH4Cl 10% Dielectric = 7E+07(DT)^0.59 NH4Cl 20% Dielectric = 6E+07(DT)^0.56 NH4Cl 30% Dielectric = 7e+07(DT)^0.48

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as…

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79

Cl-based capacitors. At all three NH<sup>4</sup>

The energy density for all three aqueous salt solutions with 30 wt% concentration is shown

across the entire range of discharge times collected. This is consistent with the observations that they have very similar dielectric values over the same tested time range. The energy den-

either of the other two solutions at any given frequency and also qualitatively consistent with the relatively low dielectric value of this solution. Also notable is the clear indication that the method does not provide reliable data for discharge times less than approximately 0.001 s. At this high rate of discharge, the method does not capture a sufficient number of data points to

A comparison of the power density for the three capacitors built with 30 wt% salt solutions is

in **Figure 8**. The energy density for two of these solutions, KOH and NH<sup>4</sup>

Cl-based dielectrics. This is a qualitative

even at a discharge time of order 10−3 s, indicating these

is less than a third the value of capacitors built using

Cl behavior is very similar, as anticipated

Cl, are very similar

Cl concentrations, the

#### **3.3. NH4 Cl**

The only complete data set shown for the aqueous NH<sup>4</sup> Cl-based dielectric is the dielectric constant as a function of discharge time (**Figure 6**). The values of this parameter are similar to those of the aqueous KOH-based dielectrics and the other values are as well. Only dielectric constant is displayed as this parameter is most easily compared to the historic data set of dielectric materials. It is also notable that the data derived from aqueous NH<sup>4</sup> Cl solutions show greater variability than data from capacitors made with either of the other solutions.

### **3.4. NaNO3**

For the aqueous NaNO<sup>3</sup> -based dielectric, the only complete data set provided is the dielectric constant as a function of discharge time (**Figure 7**). The values of this parameter are distinctly less, on the order of a factor of five at any given discharge time, than those

**Figure 6.** Dielectric constant vs. discharge time for NH<sup>4</sup> Cl-based capacitors. At all three NH<sup>4</sup> Cl concentrations, the dielectric constant follows, below ~0.8 V, a simple power law. The absolute values of the dielectric constant are similar to those of KOH at all discharge times, and greater than 106 even at a discharge time of order 10−3 s, indicating these materials were super dielectrics over the full range tested.

observed for both the aqueous KOH- and NH<sup>4</sup> Cl-based dielectrics. This is a qualitative indicator that the energy and power density of capacitors built T-SDM employing this solution will not perform as well for storing energy and providing power. This is shown to be true in the next section.

#### **3.5. Energy and power**

**3.3. NH4**

tested.

**3.4. NaNO3**

For the aqueous NaNO<sup>3</sup>

**Cl**

1.00E+06

1.00E+07

**Dielectric [-]** 1.00E+08

1.00E+09

78 Supercapacitors - Theoretical and Practical Solutions

The only complete data set shown for the aqueous NH<sup>4</sup>

materials. It is also notable that the data derived from aqueous NH<sup>4</sup>

variability than data from capacitors made with either of the other solutions.

stant as a function of discharge time (**Figure 6**). The values of this parameter are similar to those of the aqueous KOH-based dielectrics and the other values are as well. Only dielectric constant is displayed as this parameter is most easily compared to the historic data set of dielectric

**Figure 5.** Dielectric constant vs. discharge time for KOH-based capacitors. At all three KOH concentrations, the dielectric constant follows, below ~0.8 V, a simple power law, permitting determination of its value with high precision over a broad discharge time range. The absolute values of the dielectric constant are greater than 106

at a discharge time of order 10−3 s, indicating these materials were super dielectric materials over the full range

1.00E-03 1.00E-02 1.00E-01 1.00E+00 1.00E+01 1.00E+02

**Discharge Time [sec]**

tric constant as a function of discharge time (**Figure 7**). The values of this parameter are distinctly less, on the order of a factor of five at any given discharge time, than those


Cl-based dielectric is the dielectric con-

KOH 10% Dielectric = 4E+07(DT)^0.42 KOH 20% Dielectric = 4E+07(DT)^0.47 KOH 30% Dielectric = 7E+07(DT)^0.47

Cl solutions show greater

even

The energy density for all three aqueous salt solutions with 30 wt% concentration is shown in **Figure 8**. The energy density for two of these solutions, KOH and NH<sup>4</sup> Cl, are very similar across the entire range of discharge times collected. This is consistent with the observations that they have very similar dielectric values over the same tested time range. The energy density of the capacitors employing NaNO<sup>3</sup> is less than a third the value of capacitors built using either of the other two solutions at any given frequency and also qualitatively consistent with the relatively low dielectric value of this solution. Also notable is the clear indication that the method does not provide reliable data for discharge times less than approximately 0.001 s. At this high rate of discharge, the method does not capture a sufficient number of data points to provide a reliable integrated energy density.

A comparison of the power density for the three capacitors built with 30 wt% salt solutions is shown in **Figure 9**. Once again, the KOH and NH<sup>4</sup> Cl behavior is very similar, as anticipated

**Figure 7.** Dielectric constant vs. discharge time for NaNO<sup>3</sup> -based capacitors. At all three concentrations, the dielectric constant follows, below ~0.8 V, a simple power law. The absolute values of the dielectric constant are about an order of magnitude less than the KOH and NH<sup>4</sup> Cl-based capacitors at any given discharge time. Still, at all discharge times tested the dielectric constant was greater than 10<sup>5</sup> , indicating these materials were super dielectrics over the full range tested.

based on the similarity in the reported dielectric values over the full range of discharge times

**Figure 9.** Power density comparisons. The power density for capacitors built with three different 30 wt% salt solutions

0.0001 0.0010 0.0100 1.0000 10.0000 100.0000

0.1000 **Discharge Time [sec]**

a factor of approximately three at all discharge times, as expected given the lower dielectric values reported. Although not shown here, the relative energy and power densities observed for the 30% solutions are exemplary of the relative values of these parameters at all concentrations.

The following were observed for all nine ion capacitors containing dissolved ions: (i) All the capacitors displayed 'roll-off' of capacitance (<0.8 V), dielectric constant (<0.8 V), and energy density (0.1–2.3 V) as discharge time decreased. (ii) The roll-off, of all these parameters, is well described by simple power law expressions derived from data covering more than three orders of magnitude of discharge times. (iii) Power density is also well described by a simple power law, but in contrast to all other parameters of interest, increased in all

capacitor yields the lowest power densities, by

NH4Cl 30% Power = 10.43(DT)^-0.49 NaNO3 30% Power = 3.93(DT)^-0.48 KOH 30% Power = 11.74(DT)^-0.43

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81

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as…

studied. It is also clear that the aqueous NaNO<sup>3</sup>

is shown over more than three orders of magnitude of discharge time.

**4. Discussion**

1.0000

10.0000

**Power**

**[W/cm3]**

100.0000

1000.0000

**Figure 8.** Energy density comparisons. The energy density for capacitors built with three different 30 wt% salt solutions is shown over more than three orders of magnitude of discharge time. The data below 0.001 s discharge time are considered inaccurate due to insufficient data collection times.

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as… http://dx.doi.org/10.5772/intechopen.71003 81

**Figure 9.** Power density comparisons. The power density for capacitors built with three different 30 wt% salt solutions is shown over more than three orders of magnitude of discharge time.

based on the similarity in the reported dielectric values over the full range of discharge times studied. It is also clear that the aqueous NaNO<sup>3</sup> capacitor yields the lowest power densities, by a factor of approximately three at all discharge times, as expected given the lower dielectric values reported. Although not shown here, the relative energy and power densities observed for the 30% solutions are exemplary of the relative values of these parameters at all concentrations.

#### **4. Discussion**

1.00E+05

**Figure 7.** Dielectric constant vs. discharge time for NaNO<sup>3</sup>

1.00E-04 1.00E-03 1.00E+00 1.00E+01

NaNO3 10% Dielectric = 5E+06(DT)^0.43 NaNO3 20% Dielectric = 1E+07(DT)^0.49 NaNO3 30% Dielectric = 1E+07(DT)^0.48

Cl-based capacitors at any given discharge time. Still, at all discharge times tested

, indicating these materials were super dielectrics over the full range tested.


NH4Cl 30% Energy = 10.43(DT)^0.51 NaNO3 30% Energy = 3.93(DT)^0.52 KOH 30% Energy = 11.74(DT)^0.57

1.00E-02 1.00E-01 Discharge Time [sec]

constant follows, below ~0.8 V, a simple power law. The absolute values of the dielectric constant are about an order of

0.0001 0.0010 0.0100 0.1000 1.0000 10.0000

**Figure 8.** Energy density comparisons. The energy density for capacitors built with three different 30 wt% salt solutions is shown over more than three orders of magnitude of discharge time. The data below 0.001 s discharge time are

**Discharge Time [sec]**

1.00E+06

Dielectric [-]

magnitude less than the KOH and NH<sup>4</sup>

the dielectric constant was greater than 10<sup>5</sup>

0.1000

considered inaccurate due to insufficient data collection times.

1.0000

**Energy Density [J/cm3]** 10.0000

1.00E+07

80 Supercapacitors - Theoretical and Practical Solutions

The following were observed for all nine ion capacitors containing dissolved ions: (i) All the capacitors displayed 'roll-off' of capacitance (<0.8 V), dielectric constant (<0.8 V), and energy density (0.1–2.3 V) as discharge time decreased. (ii) The roll-off, of all these parameters, is well described by simple power law expressions derived from data covering more than three orders of magnitude of discharge times. (iii) Power density is also well described by a simple power law, but in contrast to all other parameters of interest, increased in all cases as the discharge time was reduced. (iv) The identity and concentrations of the solutes had a strong impact on the value of all capacitor performance parameters. (v) The value of all parameters was not a clear function of solute concentration, although the highest weight concentration, 30%, performed the best. (vi) In general, capacitors based on KOH and NH<sup>4</sup> Cl were similar in behavior, but the NaNO<sup>3</sup> -based capacitors consistently showed the lowest values.

measured energy densities reported herein, for both KOH- and NH<sup>4</sup>

1000 s, the energy density is a remarkable ~600 J/cm<sup>3</sup>

studies.

comparable to the values obtained in prior studies. Those were obtained using a different solution, 30% NaCl, and a different measurement method, the traditional RC time constant method. In fact, using the simple power law dependencies obtained here and extrapolating to

Performance of Aqueous Ion Solution/Tube-Super Dielectric Material-Based Capacitors as…

nearly identical titania matrix containing 30 wt% NaCl aqueous solution yielded nearly 400 J/ cm3 for similar discharge times. Given the different ionic solutions, the different measurement protocols and other minor differences, there is an excellent agreement between the two

The basic model of T-SDM presented elsewhere [7, 8, 10] predicts high capacitance that decrease as the discharge time decreases. To understand both, a brief review of the static model of SDM and a qualitative review of the dynamics of SDM is required. Regarding the former: As illustrated by the cross-section model of an anodized titania filled with aqueous solution, **Figure 11**, dipoles created by the movement of ions in solution toward oppositely polarized electrodes create 'giant dipoles'. These dipoles, opposite in polarization to the electrodes, reduce the field, everywhere, created by charges on the electrodes. As voltage is the line integral of field from ground to electrode, the lowering of field everywhere reduces the voltage. Thus, it takes more charge on the electrodes to reach the same voltage when these giant dipoles are fully (static conditions) aligned. More charge, at the same voltage, means a higher capacitance, by definition. In essence, dipole formation is the basis for capacitance enhancement for all types of dielectrics; however, for SDM the dipoles are orders of magnitude longer than in any solid so the field reduction, and consequently the increase in capacitance, is more dramatic. Next, it is necessary to reflect on the dynamics of dipole formation, that is the impact of frequency, or period, on dipole strength. Specifically, if the electrode polarization is switched too quickly for the ions in solution to 'swim' to the maximal (static) dipole positions, the net or effective, dipole length, and concomitantly the dielectric and capacitance values, are reduced. The

**Figure 11.** X-Section model of dipole formation in SDM. 1. Top electrode, Grafoil. 2. Tube filled with aqueous ion solution. 3. 90 nm × 8 μm titania tubes formed by anodization. 4. Titanium metal electrode. Upon the application of a field, the ions in solution migrate to form dipoles oppositely polarized to the electrodes. The effective dipole strength in the dielectric is a function of time. Given sufficient time (static) dipoles of maximum length and charge separation form. The effective dipole length/strength is a function of net length and thus is discharge time/frequency dependent.

Cl-based capacitors, are

83

. In the earlier RC time constant work a

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The data presented herein provide the first report on the behavior of T-SDM as a function of discharge time. This information is critical for assessing the value of any type of capacitor for application to 'pulsed power'. Indeed, the measured power densities, just greater than 100 W/cm<sup>3</sup> for both the aqueous KOH- and NH<sup>4</sup> Cl-based capacitors, for discharges of 0.01 s, are exceptional. As shown in **Figure 10**, the KOH-based capacitor parameters fall above the 'range' of operation anticipated for EDLC-based supercapacitors and are far better than the performance determined using the identical methodology employed in this work to assess real commercial 'supercapacitors' [24] in our laboratory. Three supercapacitors were tested and the best passed through the bottom range of values anticipated by the plot shown (**Figure 10**), and the other two were completely below the 'bubble' of performance anticipated for 'double layer capacitors'.

It is also important to compare the data obtained in this study with earlier work on T-SDM. That earlier work was undertaken with a different objective: Study very slow discharges (>1000 s) appropriate for determining their potential use of T-SDM as energy storage devices. The

**Figure 10.** KOH 30 wt% NPS energy/power performance. On a modified US Defense Logistics Agency Ragone chart, it is clear that the 30 wt% KOH-based capacitor (solid line) is superior to that anticipated for EDLC or double layer capacitors. Also, the data fall on a line, which can reasonably be extrapolated using the power law fits (dashed curve). It was assumed that the dielectric is half salt water and half titania with density 2.6 g/cm<sup>3</sup> .

measured energy densities reported herein, for both KOH- and NH<sup>4</sup> Cl-based capacitors, are comparable to the values obtained in prior studies. Those were obtained using a different solution, 30% NaCl, and a different measurement method, the traditional RC time constant method. In fact, using the simple power law dependencies obtained here and extrapolating to 1000 s, the energy density is a remarkable ~600 J/cm<sup>3</sup> . In the earlier RC time constant work a nearly identical titania matrix containing 30 wt% NaCl aqueous solution yielded nearly 400 J/ cm3 for similar discharge times. Given the different ionic solutions, the different measurement protocols and other minor differences, there is an excellent agreement between the two studies.

cases as the discharge time was reduced. (iv) The identity and concentrations of the solutes had a strong impact on the value of all capacitor performance parameters. (v) The value of all parameters was not a clear function of solute concentration, although the highest weight concentration, 30%, performed the best. (vi) In general, capacitors based on KOH

The data presented herein provide the first report on the behavior of T-SDM as a function of discharge time. This information is critical for assessing the value of any type of capacitor for application to 'pulsed power'. Indeed, the measured power densities, just greater

of 0.01 s, are exceptional. As shown in **Figure 10**, the KOH-based capacitor parameters fall above the 'range' of operation anticipated for EDLC-based supercapacitors and are far better than the performance determined using the identical methodology employed in this work to assess real commercial 'supercapacitors' [24] in our laboratory. Three supercapacitors were tested and the best passed through the bottom range of values anticipated by the plot shown (**Figure 10**), and the other two were completely below the 'bubble' of

It is also important to compare the data obtained in this study with earlier work on T-SDM. That earlier work was undertaken with a different objective: Study very slow discharges (>1000 s) appropriate for determining their potential use of T-SDM as energy storage devices. The

**Figure 10.** KOH 30 wt% NPS energy/power performance. On a modified US Defense Logistics Agency Ragone chart, it is clear that the 30 wt% KOH-based capacitor (solid line) is superior to that anticipated for EDLC or double layer capacitors. Also, the data fall on a line, which can reasonably be extrapolated using the power law fits (dashed curve). It

.

was assumed that the dielectric is half salt water and half titania with density 2.6 g/cm<sup>3</sup>


Cl-based capacitors, for discharges

Cl were similar in behavior, but the NaNO<sup>3</sup>

performance anticipated for 'double layer capacitors'.

for both the aqueous KOH- and NH<sup>4</sup>

and NH<sup>4</sup>

the lowest values.

82 Supercapacitors - Theoretical and Practical Solutions

than 100 W/cm<sup>3</sup>

The basic model of T-SDM presented elsewhere [7, 8, 10] predicts high capacitance that decrease as the discharge time decreases. To understand both, a brief review of the static model of SDM and a qualitative review of the dynamics of SDM is required. Regarding the former: As illustrated by the cross-section model of an anodized titania filled with aqueous solution, **Figure 11**, dipoles created by the movement of ions in solution toward oppositely polarized electrodes create 'giant dipoles'. These dipoles, opposite in polarization to the electrodes, reduce the field, everywhere, created by charges on the electrodes. As voltage is the line integral of field from ground to electrode, the lowering of field everywhere reduces the voltage. Thus, it takes more charge on the electrodes to reach the same voltage when these giant dipoles are fully (static conditions) aligned. More charge, at the same voltage, means a higher capacitance, by definition. In essence, dipole formation is the basis for capacitance enhancement for all types of dielectrics; however, for SDM the dipoles are orders of magnitude longer than in any solid so the field reduction, and consequently the increase in capacitance, is more dramatic. Next, it is necessary to reflect on the dynamics of dipole formation, that is the impact of frequency, or period, on dipole strength. Specifically, if the electrode polarization is switched too quickly for the ions in solution to 'swim' to the maximal (static) dipole positions, the net or effective, dipole length, and concomitantly the dielectric and capacitance values, are reduced. The

**Figure 11.** X-Section model of dipole formation in SDM. 1. Top electrode, Grafoil. 2. Tube filled with aqueous ion solution. 3. 90 nm × 8 μm titania tubes formed by anodization. 4. Titanium metal electrode. Upon the application of a field, the ions in solution migrate to form dipoles oppositely polarized to the electrodes. The effective dipole strength in the dielectric is a function of time. Given sufficient time (static) dipoles of maximum length and charge separation form. The effective dipole length/strength is a function of net length and thus is discharge time/frequency dependent.

data suggest that effective dipole length follows a very simple pattern as a function of discharge time.

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It is notable that this is only the second time [13] the constant current charge/discharge method has been employed to determine the power law relationship for 'supercapacitor' parameters, specifically capacitance, dielectric constant, and energy and power density, over orders of magnitude of discharge times. This method arguably provides higher fidelity, more reliable, insight into 'frequency' dependence of this type of capacitor than other measurement protocols.
