2
a
a
a
a
a
a
Table of contents
1. Optical images for the preparation process.
2. Yield of Ni(OH)
2
nanoparticles.
3. The relationship among the size, the band gap, and the conductivity of Ni(OH)
2
nanoparticles.
4. Structure characterization and spectrum analysis of Ni(OH)
2
nanoparticles.
5. Electrochemical characterization of Ni(OH)
2
nanoparticles.
6. Calculation R ct and D of Ni(OH)
2
from EIS and CV curves.
7. Reference.

1. Optical images for the preparation process.
Figure S1. Optical images for the preparation process: firstly, Ni(NO
3
)
2
was reduced to Ni nanoparticles; secondly, the Ni nanoparticles reacted with Na
3
C
6
H
5
O
7
to form Ni-Citrate; finally, the Ni(OH)
2 sediment was formed after adding NaOH.
2. Yield of Ni(OH)
2
nanoparticles. In experimental process, we noted that the upper solution was clear after the reaction (Figure S1), which indicated that the Ni ions had been consumed and the content of free Ni ions was negligible. The lower green sediment was purified by distilled water for several times, and then dried at 80 o C for 24 h. The obtained green solid was weighted and compared with the theoretical mass ( assuming that all Ni 2+ ions in Ni(NO
3
)
2
precursor are converted to Ni(OH)
2
). By the calculation, the yield of Ni(OH)
2
is ~92% at the temperature of 20 o C. The weight loss mainly comes from the centrifugation process because the Ni(OH)
2
colloidal cannot be completely collected by centrifugation even at 10000 rpm, indicating the ultra-small characteristic of the obtained Ni(OH)
2
. The weight loss of Ni(OH)
2 is alleviated since the reaction temperature increases. For example, at relatively high reaction temperatures (exceeding 100 o C), the yield of Ni(OH)
2
are stable at ~98%.
3. The relationship among the size, the band gap, and the conductivity of Ni(OH)
2 nanoparticles.
The band gap ( E g
) can be calculated on the basis of UV-vis adsorption spectrum by the following equation: s1
( αhv ) n = B ( hv E g
) (1)
Where hv is the photo energy, α is the absorption coefficient, B is a constant relative to the material and n is either 2 for the direct transition or 1/2 for the indirect transition. Here n is 2 for

the direct transition.
The effective band gap ( E g off
) can be generally defined by s2
E g off = E bulk
+ h 2 ( 1 / m e
* + 1 / m h
* )/ 8a 2 –
1.8e
2 /(
4πεε o r ) (2)
Where E bulk
is the bulk band gap energy, a is the particle radius, m e
* is effective mass of electron and m h
* is effective mass of hole, h is Plank’s constant, e is the charge of the electron, ε is the relative permittivity,
ε o
is the permittivity of free space. The polarization term ( 1.8e
2
/(
4πεε o r )) included in this model (or equation) is usually negligible, s2 thus the above equation (2) can be expressed as
E g off = E bulk
+ h 2 ( 1 / m e
* + 1 / m h
* )/ 8a 2 (3)
Or ∆E = h 2 (1/m e
* + 1/m h
*)/8a 2 ( ∆E derived from quantum confinement)
E g off
= E bulk
+
∆E
(4)
Ni(OH)
2
is recognized as a semiconducting material as proved by Hermet’ work via DFT approach.
s3 Size-dependent conductivity of Ni(OH)
2
nanoparticles can be deduced from the following equations: s4 n e
= n h
= 2 (
2πk
B
T / h
2
)
3/2
( m e m h
)
3/4 exp ( -E g
/ 2k
B
T ) (5)
Conductivity (
σ
):
σ = ( n e eμ e
+ n h eμ h
) (6) where n e is carrier concentration of electron and n h is carrier concentration of hole, respectively, T is the temperature, k
B
is the Boltzmann constant, m e
is mass of electron and m h
is mass of hole, h is
Planck constant, E g
is the band gap of nanoparticles measured by UV spectrometer, e is electron charge, and μ e and μ h are electron or hole mobility, respectively.
Insertion of the equation (5) to (6), yields the following equation:
σ = ( eμ e
+ eμ h
) [ 2 ( 2πk
B
T / h 2 ) 3/2 ( m e m h
) 3/4 exp ( -E g
/ 2k
B
T )]
σ
= 2 ( eμ e
+ eμ h
) (
2πk
B
T / h
2
)
3/2
( m e m h
)
3/4 exp ( -E g
/ 2k
B
T )
If s represents 2 ( eμ e
+ eμ h
) ( 2πk
B
T / h 2 ) 3/2 ( m e m h
) 3/4 , the conductivity can be written as
σ = s exp ( -E g
/ 2k
B
T ) (7)
Here, T is given at 293 K.
Inserting the equation (4) to (7), gives
σ = s exp [ ( E bulk
+ ∆E )/ 2k
B
T ]

σ = s exp(-E bulk
/ 2k
B
T) exp [ ( ∆E )/ 2k
B
T ] (8)
If
σ
0
= s exp(-E bulk
/2k
B
T) , the conductivity can be expressed as
σ
=
σ
0 exp ( -∆E / 2k
B
T ) (9)
Here, σ o represents the conductivity of bulk-state semiconductor.
σ/σ o
= exp ( -∆E / 2k
B
T ) (10)
Here, k
B
=1.38× 10 -23 J K -1 , T = 298 K, 1eV = 1.602 × 10 -19 J, then
σ/σ o is related to ∆E by:
σ/σ o
= exp [ 19.5(
∆E
)] (11)
Inserting the equation (3) to (7), gives
σ = s exp { -[E bulk
+ h 2 (1/m e
* + 1/m h
*)/8a 2 ] / 2k
B
T }
σ = s exp[-E bulk
/2k
B
T - h 2 (1/m e
* + 1/m h
*)/(16a 2 k
B
T)]
σ
= s exp(-E bulk
/2k
B
T)exp[-h
2
(1/m e
* + 1/m h
*)/(16a
2 k
B
T)]
If q represents h 2 (1/m e
* + 1/m h
*)/16k
B
T, the conductivity can be expressed as
σ = σ o exp (q / a 2 ) (12)
Here, σ o represents the conductivity of bulk-state semiconductor. Also, if T is given at 293 K, σ o
, and q can be recognized as the constants. Thus, the conductivity is related to a and can be expressed as
σ/σ o
= exp [ -1/(q -1/2 a) 2 ] (13)
From the equations (10) and (11), the conductivity ( σ ) of Ni(OH)
2
is inversely proportional to E g
.
The relationship between the conductivity and E g
is plotted as shown in Figure 1a. A larger E g value results in a smaller
σ value. For example, ~0.2 eV of
∆E
means a hundred times decrease in conductivity, which is consistent with Kim ’ work about GeO
2
nanoparticles.
s4 According to equation (3), the E g
value is inversely proportional to a ( a represents the particle size). So a smaller nanoparticle has a lager E g value, which is commonly ascribed to the quantum confinement effect.
s3,s4 The quantum size effect is most pronounced for semiconductor nanoparticles, where the band gap increases with a decreasing size, which leads to the conductivity shifts to the low value as shown in equation (10) and (13). The relationship between the conductivity and the particle size is plotted (Figure 1b), even though some important physical parameters for Ni(OH)
2 values such as m e
* and m h
* (recognized as constants) are unknown. From the character of exponential function of equation (13), it should be noted that these unknown

physical parameters do not impede our step to further discussion. Figure S10 shows the conductivity has a great change between 0.5 and 10 (for the value of q
-1/2 a ). Extremely reduced
(below 0.5 for q -1/2 a ) or increased particles size (over 10 for q -1/2 a ), the conductivity changes is little. Quantum confinement effect on conductivity of semiconductor materials usually occurs in shadow region in Figure 1b. Thus, Ni(OH)
2
nanoparticles with smaller size should have lower electrical conductivity.

4. Structure characterization and spectrum analysis of Ni(OH)
2
nanoparticles.
Figure S2. TEM and HRTEM images of different sized Ni(OH)
2 nanoparticles: (a) and (b) for 3.7 nm, (c) and (d) for 4.4 nm, (e) and (g) for 6.0 nm. Inserts are the corresponding the FFT images.

Table S1. A summary of the preparation condition and the size variation for Ni(OH)
2 nanoparticles calculated from XRD and TEM.
Reaction
Temperature
( o
C)
20
40
60
80
100
120
140
160
180
Crystallite
Size (nm) from (001)
1.4
1.3
1.6
1.7
2.5
3.3
4.1
4.3
4.9
Crystallite
Size (nm) from (100)
5.6
6.9
8.9
7.4
12.4
13.6
15.9
18.9
23.3
Crystallite
Size (nm) from (101)
2.8
3.6
3.8
3.7
4.5
5.8
6.6
7.2
8.7
Crystallite
Size (nm) from (102)
3.6
2.9
3.3
3.6
3
4.1
5.3
5.3
6.4
Crystallite
Size (nm) from (110)
5.1
8.0
9.9
13.7
9.3
12.1
14.7
16.3
20.7
Average
Crystallite
Size (nm)
3.3
3.7
4.4
6.0
6.3
7.9
9.4
10.0
12.2
Particle
Size (nm) from TEM
2.9
3.9
5.2
5.7
7.9
10.6
13.7
17.3
21.7
The above stated crystallite size of Ni(OH)
2
nanoparticles was estimated from major diffraction peaks, such as the stacking direction of for (001) peaks and the width of lamellar structures for
(100) peaks. As shown in Table S1, the crystallite sizes give the relatively small sizes from 1.4 to
4.9 nm for (001) peaks and the large sizes from 5.6 to 23.3 nm for the (100) peaks. Moreover, the crystallite sizes calculated from (100) peaks, especially 17.3 nm for 160 o C and 23.3 nm for 180 o C samples, are closed to the statistical particle size values estimated from TEM images.

Figure S3. Structure characterization of the 12.2 nm sized Ni(OH)
2
nanoparticles: (a) TEM image of platelet-like morphology Ni(OH)
2
nanoparticles, (b) HRTEM image indicating the hexagonal structure of Ni(OH)
2
nanoparticles, (c) Ni(OH)
2
depicted as lamellar structure stacks along the c-axis ((001) plane), (d) the corresponding FFT image indicating the presence of crystalline phase,
(e) TEM image of the mixture of Ni(OH)
2
nanoparticles and XC-72 (inset is a side view of
Ni(OH)
2
lamellar structure), (f) HTEM images of the edge of Ni(OH)
2 nanoparticles.
As shown in Figure S3a and e, most of Ni(OH)
2
nanoparticles are laid flat as marked by red rectangular, and the planes can be assigned to (100) lattice plane. Besides, a few Ni(OH)
2 nanoparticles are side mount as marked by red cycle, and the sides can be assigned to (001) lattice plane. As shown in Figure S3e, many Ni(OH)
2
nanoparticles are side mount since XC-72 is added.

HRTEM image (Figure S3f) shows that the edge of Ni(OH)
2 nanocrystalline is about 5.3 nm width and 30 nm long. The above stated results confirm that as-prepared relatively large sized Ni(OH)
2 nanoparticles are hexagonal lamellar structure, which is consistent with XRD analysis.
Figure S4 . Selected Raman spectra of Ni(OH)
2
nanoparticles with different sizes.
Figure S5. Optimal images showing that the Ni(OH)
2
gel can be laid flat and inverted in a beaker, even a chopstick can insert into the gel.

Figure S6. Stable Ni(OH)
2
colloidal solutions proved by Tyndall scattering effect via a red laser beam.
Figure S7.
The zeta potentials of the size-controlled Ni(OH)
2
nanoparticles dispersed in water.

5. Electrochemical characterization of Ni(OH)
2
nanoparticles.
Table S2. The relationship between the particle size and the characteristic potential data obtained from CV curves at a scan rate of 20 mV s -1 for the Ni(OH)
2
nanoparticles with the different sizes.
Particle Size
(nm)
3.3
3.7
4.4
6.0
6.3
7.9
9.4
10.0
12.2
E a
* (V)
0.3045
0.2839
0.2858
0.3017
0.2998
0.2860
0.2930
0.3326
0.3500
E c
* (V)
0.1815
0.1969
0.1930
0.1956
0.1959
0.1770
0.1780
0.1900
0.1864
E ac
* (V)
0.1230
0.0970
0.0928
0.1061
0.1039
0.1090
0.1150
0.1426
0.1636
E o
* (V)
0.4117
0.4003
0.4030
0.4086
0.4114
0.4016
0.4052
0.3978
0.4010
E oa
* (V)
0.1072
0.1064
0.1172
0.1069
0.1116
0.1156
0.1122
0.0652
0.0510
*E o
–OER potential, E a
–anodic peak potential, E c
–cathodic potential, E ac
= E a
– E c
, E oa
= E o
– E a
.
Figure S8. (a) Galvanostatic discharge curves of these Ni(OH)
2
nanoparticles at the current densities of 20 A g -1 , respectively. To display more clearly, the curves of Ni(OH)
2
nanoparticles with the sizes of 9.4, 10 and 12.2 nm are separately shown in the insets. (b) A schematic depicting the slope and plateau region in the typical discharge curve of Ni(OH)
2
nanoparticles.
Bruce Dunn et al.
suggest that nanostructured Ni(OH)
2 is the pseudocapacitive electrode for extrinsic pseudocapacitors, even the bulk state exhibits battery-type behavior.
s5 The sloping

regions can be ascribed to pseudocapacitive contribution from surface or near-surface layer charge storage. The potential plateau in all discharge curves indicates that energy storage is majorly contributed from the redox action between the Ni(OH)
2
and NiOOH phase. The slope proportion decreases with the decrease of size for Ni(OH)
2
nanoparticles, and it is more obviously at a high discharge current density of 20 A g -1 . To a certain extent, capacity contributed from surface or near-surface layer charge storage increases as the particle size of Ni(OH)
2 is reduced, indicating the nanosize effect on the Ni(OH)
2
.
Figure S9.
Galvanostatic charge curves for the different sized Ni(OH)
2
nanoparticles at the current density of 1 A g -1 .
The potential within the initial charging region for the Ni(OH)
2
nanoparticles with the average sizes of 3.3 and 4.4 nm is lower than that for Ni(OH)
2
nanoparticles with the average sizes of 6.6 and 10.0 nm, indicating the oxidation process of small Ni(OH)
2 nanoparticles occurring at lower voltage. In addition, the charge curve becomes steeper since the size of Ni(OH)
2
nanoparticles decreases. The steeper charge curve is due to the increased contribution of surface ion storage site or surface redox reactions caused by nanosize effect.
s5,s6

6. Calculation R ct
and D of Ni(OH)
2
from EIS and CV curves.
Figure S10. (a) The equivalent circuit used in Z-view software. (b) A simple model depicting the
EIS curves of Ni(OH)
2 nanoparticles.
The equivalent circuit ( R s
(R ct
Q
1
)Q
2
) was used to simulate the EIS curve of the different sized
Ni(OH)
2 electrodes. R s
represents the electrolyte resistance, the contact resistance at the interface among the Ni(OH)
2
nanoparticles, the intrinsic resistance of substrate, and the contact resistance between Ni(OH)
2
nanoparticles and current collector. R ct is the interfacial charge resistance. Q
(CPE) is the constant phase element. A simple model is used to depict the EIS curves of Ni(OH)
2 nanoparticles. EIS curves can be divided two parts: high frequency region (a typical semicircle curve) where charge transfer is dominated and low frequency region (a diagonal line) where diffusion control is dominated. Also, the charge transfer resistance ( R ct
) is easily estimated from this model.
EIS experiments were carried out with an amplitude of 5 mV over a frequency range of 100 kHz to 0.1 Hz. Z-View software was used to simulate the EIS spectra of Ni(OH)
2
nanoparticles

electrodes. The simulated circuit is as stated above R(RQ)Q . The charge transfer resistance ( R ct
) was obtained from Z-view software simulation. The standard rate constant ( k
0
) of the redox reaction occurred on Ni(OH)
2
can be calculated from the following equations.
s7 i
0
= RT ( nFR ct
) -1 (14) k
0
= i
0
( nFC ) -1 (15)
Where i
0
is the exchange current, R is the ideal gas constant, T is the temperature, n is the electron number of the redox reaction (about 1 for Ni(OH)
2
), F is the Faraday’s constant, and C is the ionic concentration.
Table S3. A summary of characteristic EIS data and the kinetic parameters of different sized
Ni(OH)
2
nanoparticles obtained from equivalent circuit model.
Particle size (nm)
3.3
3.7
4.4
6.0
6.3
7.9
9.4
10.0
12.2
R s
(Ω)
3.0
3.0
1.0
3.3
4.2
4.4
3.1
3.8
4.5
R ct
(Ω)
25.2
9.07
7.0
5.3
4.3
2.2
3.5
7.0
7.5
CPE1-T
(S·sec n )
0.00016
0.000073
0.00000046
0.000039
0.000138
0.000208
0.000160
0.000080
0.000027
CPE1-P
(n)(0<n<1)
0.71
0.78
0.86
0.82
0.76
0.71
0.71
0.78
0.84
CPE2-T
(S·sec n )
0.0018
0.0024
0.0028
0.0034
0.0037
0.0032
0.0028
0.0021
0.0018
CPE2-P
(n)(0<n<1)
0.76
0.81
0.88
0.90
0.90
0.90
0.91
0.86
0.85
Chi-Squa red
0.0044
0.0015
0.0197
0.0098
0.0186
0.0059
0.0103
0.0021
0.0010 k
0
(×10 -6 s -1 )
0.32
0.96
1.24
1.65
2.00
3.94
2.50
1.23
1.16

Figure S11. (a) CV curves of 7.9 nm sized Ni(OH)
2
nanoparticles with different scan rates ranging from 2 to 200 mV s -1 . Insert showing the b value calculated from peak anodic/or cathodic currents and CV scan rates. (b) The b values of the size-controlled Ni(OH)
2
nanoparticles from CV curves.
In order to study the relationship between the proton diffusion and the capacitance, the Ni(OH)
2 electrodes were cycled at various sweep rates ranging from 2 to 200 mV s -1 ( an example showing in Figure S12a). The peak current ( i ) obeys a power-law relationship with the scan rate ( v ), i = av b
( a and b are adjustable values).
s8 A b -value of 0.5 would indicate a battery-type character where the current is controlled by semi-infinite linear diffusion. In addition, a b -value of 1 indicates a capacitor-type character where the current is surface-controlled. More details can be seen in ref. s8 and s9 . As shown in Figure S12b, the b values for the size-tunable Ni(OH)
2
nanoparticles are

0.5~0.6, indicating a typical diffusion controlled redox process. Here, b -value is the slope of fitting line where the logarithm values of cathodic or anodic peak current is linear with the logarithm values of scan rate. It should be noted that some large b values from anodic process are over 0.75 for 10 nm and 12.2 nm sized Ni(OH)
2
, however, b values from cathodic process of these samples are still less than 0.6. This indicates that the whole redox process of Ni(OH)
2
electrode is limited by the cathodic process, so it is still majorly controlled semi-infinite proton diffusion. In this case, the proton diffusion coefficient ( D ) can be calculated from Randles-Sevcik equation
(more details in Experimental Section).
7. Reference
S1. Kulkarni, S. B., Jamadade, V. S., Dhawale, D. S. & Lokhande, C. D. Synthesis and characterization of β-Ni(OH)
2
up grown nanoflakes by SILAR method. Appl. Surf. Sci.
255,
8390-8394 (2009).
S2. Lin, K. F., Cheng, H. M., Hsu, H. C., Lin, L. J. & Hsieh, W. F. Band gap variation of size-controlled ZnO quantum dots synthesized by sol–gel method. Chem. Phys. Lett.
409 , 208-211
(2005).
S3. Hermet, P., Gourrier, L., Bantignies, J. L., Ravot, D., Michel, T., Deabate, S., Boulet, P. &
Henn, F. Dielectric, magnetic, and phonon properties of nickel hydroxide. Phys. Rev. B 84 , 235211
(2011).
S4. Son, Y., Park, M., Son, Y., Lee, J. S., Jang, J. H., Kim, Y. & Cho, J. Quantum confinement and its related effects on the critical size of GeO
2
nanoparticles anodes for lithium batteries. Nano Lett.
14 , 1005-1010 (2014).

S5. Augustyn, V., Simon, P. & Dunn, B. Pseudocapacitive oxide materials for high-rate electrochemical energy storage. Energy Environ. Sci.
7 , 1597-1614 (2014).
S6. Okubo, M., Hosono, E., Kim, J., Enomoto, M., Kojima, N., Kudo, T., Zhou, H. & Honma, I.
Nanosize effect on high-rate li-Ion intercalation in LiCoO
2
electrode. J. Am. Chem. Soc.
129 ,
7444-7452 (2007).
S7. Bard, A. J. & Faulkner, L. R. Electrochemical Methods: Fundamentals and Applications, 2nd
Edition, 2000, John Wiley& Sons, INC.
S8. Wang, J., Polleux, J., Lim, J. & Dunn, B. Pseudocapacitive contributions to electrochemical energy storage in TiO
2
(anatase) nanoparticles. J. Phys. Chem. C 111 , 14925-14931 (2007).
S9. Brezesinski, T., Wang, J., Tolbert, S. H. & Dunn, B. Ordered mesoporous-MoO
3
with iso-oriented nanocrystalline walls for thin-film pseudocapacitors. Nat. Mater.
9 , 146-151 (2010).