Nano Res.
Electronic Supplementary Material
Theoretical study on rotary-sliding disk triboelectric
nanogenerators in contact and non-contact modes
Tao Jiang1,§, Xiangyu Chen1,§, Keda Yang2, Changbao Han1, Wei Tang1, and Zhong Lin Wang1,3 ()
1
Beijing Institute of Nanoenergy and Nanosystems, Chinese Academy of Sciences, Beijing 100083, China
Computer Network Information Center, Chinese Academy of Sciences, Beijing 100190, China
3
School of Materials Science and Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0245, USA
§
These authors contributed equally to this work.
2
Supporting information to DOI 10.1007/s12274-016-0997-x
S1
Effect of tribo-surface spacing on the basic characteristics of TENG
Figure S1 shows the real short-circuit charge and open-circuit voltage without choosing a charge reference state.
Different from the case of reference state, the potential difference between two electrodes increases with
increasing h. That is because the potential difference arising from the vertical separation between two tribosurfaces is also included. The real VOC still has a maximum peak at the half-cycle point, but the peak becomes
wider when h increases. The real QSC has the same trend as the QSC under the reference state, which is just a
vertical translation of that under the reference state. Specifically, the real QSC has the equal maximum at α = θ0
for any h.
Figure S1 (a) Short-circuit charge, (b) open-circuit voltage as functions of rotation angle for the dielectric RD-TENG with N = 4 at
various tribo-surface spacings h in the real cases.
The basic properties of metal RD-TENG at various tribo-surface spacings were also calculated under the charge
reference state, as shown in Fig. S2. The curve shapes of QSC, C and VOC are the same as the case of dielectric
RD-TENG, and the variable trends of QSC, C and VOC with the change of h are similar. There are only slight
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Nano Res.
differences between the dielectric and metal types: The metal type has a higher capacitance, a lower OC voltage
and a lower SC charge.
Figure S2 (a) Short-circuit charge, (b) capacitance, and (c) open-circuit voltage as functions of rotation angle for the metal RD-TENG
at various tribo-surface spacings h under the charge reference state.
To better demonstrate the dependence of basic properties on h, we derived the analytic equations for QSC, C,
and VOC for the ideal case (neglecting the edge effect). The SC transferred charges (QSC) was obtained by
deducting Qinitial from total transferred charges under SC (Qnet,SC) at a certain angle α. For the non-contact mode,
the Qinitial was assigned to the total amount of charges transferred at α = 0 under SC condition. At the initial state,
the lower surface of dielectric 1 has the charge density of +σ, and the upper surface of dielectric 2 has the
density of –σ. The Qinitial equals the short-circuit charge of a contact-separation TENG with a separation distance
of h, which is given by
Qinitial 
N 0 h(r2 2  r12 )
d

d
2 1  2  h
  r1  r2

(S1)
When the dielectric 1 layer rotates for an angle α, the QSC was derived from the charge distribution under SC
condition for different α ranges.
(1) When 0 ≤ α≤ θ0
Under SC condition, we assume the charge densities on the upper surface of metal 2 for regions I and II (Fig. S3)
are respectively σ2,I and σ2,II. Since the electric field inside metal is always 0, σ2,I satisfies
 2,I  
(S2)
From the charge conservation and zero electric field inside metal 2, we obtain
 2,II 
h
d1
 r1

d2
 r2
(S3)
h
Therefore, the total charges Qnet,SC on metal 2 can be given by
 d

d 
N (r2 2  r12 )  1  2     0 h 
N (r2  r1 )
  r1  r2 

  2,I   2,II ( 0   )  

2
d

d
2 1  2  h


 r1

r2
2
Qnet,SC
2
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(S4)
Nano Res.
QSC is the difference between Qnet,SC and Qinitial, given by
QSC  Qnet,SC  Qinitial
d
d 
N  1  2  (r2 2  r12 )


r2 
 r1

 d1 d2

2

 h
  r1  r2

(S5)
Figure S3 Schematic charge distribution model for the dielectric RD-TENG at 0 ≤ α ≤ θ0.
(2) When θ0 ≤ α ≤ 2θ0
The center angles of regions I and II for metal 2 are respectively α − θ0 and 2θ0 – α (Fig. S4). Similar to above, the
charge densities σ2,I and σ2,II satisfy
 1,I 
d1
 r1

h
d2
 r2
,
h
 2,II  
(S6)
Therefore, the total charges Qnet,SC on metal A is given by
Qnet,SC
 d1 d2 

2
2




N
(
r
r
)

(2


)

h




2
1
0
0
N (r2 2  r12 )
  r1  r2 

  1,I (   0 )   2,II (2 0   )  

2
 d1 d2

2

 h
  r1  r2

(S7)
Then
QSC  Qnet,SC  Qinitial
d
d 
N (2 0   )  1  2  (r2 2  r12 )


r2 
 r1

 d1 d2

2

 h
  r1  r2

(S8)
Figure S4 Schematic charge distribution model for the dielectric RD-TENG at θ0 ≤ α≤ 2θ0.
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As long as two dielectrics are not close to fully separated, the capacitor at the matched region (region II in
Fig. S3 and region I in Fig. S4) is the dominate part of the capacitance. The total C can be represented by the
local capacitance at region II or I. Of course, this is the ideal case neglecting the edge effect. The C can be given by
C
N 0  0    (r2 2  r12 )
d

d
2 1  2  h
  r1  r2

(S9)
Then the open-circuit voltage can be obtained by
VOC
S2
 
 d1 d2 



,
  0 ( 0   )   r1  r2 

 (2 0   )  d1  d2  ,

  (   )  
 r2 
0  r1
 0
0 ≤   0
(S10)
 0   ≤ 2 0
Effect of grating number for the metal RD-TENG
For the metal RD-TENG in contact-mode, the QSC, C, and VOC as functions of α at various N are shown in Fig. S5.
It can be found that under the ideal condition, the influences of N on the QSC, C, and VOC are similar to the
dielectric RD-TENG. The difference is that the metal RD-TENG has a lower VOC, a higher C, and a lower QSC.
Figure S5 (a) Short-circuit charge, (b) capacitance, and (c) open-circuit voltage as functions of rotation angle for the metal RD-TENG
with various grating numbers N under the ideal condition.
S3
Continuous fraction interpolation results based on FEM data
Table S1 shows the values of 1/C and VOC at different α that are extracted through the linear fitting of FEM
results for the dielectric RD-TENG in contact-mode with N = 4, r1 = 5 mm, and r2 = 50 mm. Table S2 shows the
values of coefficients in the interpolation equations captured by the interpolation calculations based on the
FEM results. Figure S6 shows the comparison of 1/C and VOC obtained through the FEM calculation and semianalytical interpolation. It can be seen the interpolation results are accurate and have little error.
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Table S1 Values of 1/C and VOC that are extracted from the linear V–Q relationship at different 
–1
 (rad)
1/C (F )
VOC (V)
0
2.14890E+09
0.00000
/60
2.28451E+09
5.26981
/30
2.45363E+09
11.92007
/20
2.65105E+09
19.68467
/15
2.88358E+09
28.83053
/12
3.16115E+09
39.74826
/10
3.49806E+09
53.00059
7/60
3.91539E+09
69.41878
2/15
4.44573E+09
90.28591
3/20
5.14195E+09
117.68379
/6
6.09589E+09
155.23180
11/60
7.48233E+09
209.81727
/5
9.67868E+09
296.32217
13/60
1.36757E+10
453.83120
7/30
2.31146E+10
826.13105
29/120
3.47638E+10
1,286.24579
/4
5.79608E+10
2,206.01748
Table S2 Interpolation results of parameters mj and nj in the interpolation equations
j
j (rad)
mj
nj
0
0
2.148900E+09
0.000000E+00
1
/60
3.861063E–10
9.935819E–03
2
/30
–1.233197E+09
–4.550437E+01
3
/20
–2.676859E–10
–6.518350E–03
4
/15
–9.034102E+08
–3.859934E+01
5
/12
–2.387911E–08
–6.441766E–01
6
/10
–1.543327E+07
–5.724412E–01
7
7/60
–6.032284E–09
–2.973582E–01
8
2/15
–1.662672E+07
–5.583324E–01
9
3/20
–1.862521E–08
–1.387255E–01
10
/6
7.609109E+06
1.216946E+00
11
11/60
2.607198E–08
–3.745003E–01
12
/5
5.682030E+06
–1.199226E+00
13
13/60
5.206273E–08
1.358660E–01
14
7/30
1.528261E+06
4.430243E–01
15
29/120
–4.190050E–08
–2.803265E–01
16
/4
3.164847E+06
2.228046E–01
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Figure S6 Comparison of (a) 1/C and (b) VOC obtained through the FEM calculation and semi-analytical interpolation.
S4
Formula derivation of periodic boundary condition and output characteristics
We calculated the resistive load characteristics of RD-TENG through the basic differential equation of Eq. (9),
which is shown as below
R
dQ
1

 Q  VOC ( )
dt
C( )
(S11)
Considering the first half cycle between t = 0 and t = θ0/ (0 ≤ α≤ θ0), the solution of above differential equation
with the boundary condition Q(t = 0) = Q0 at constant angle velocity  can be given by
tV
(t )
 t dt  
 t dz  
Q  exp   
Q0   OC
exp  


 dt 
0
R
 0 C(t )R  
 0 C( z)R  
t V
( )
 1  t d  
 1  dz  
OC
 exp  
Q0  
exp 


 d 


0
R
 R 0 C( )  
 R 0 C( z)  
(S12)
Therefore, at the end of this half cycle, Q is given by


Q1  Q  t  0




 1 0 d  
 1  dz 
1 0
Q0 
VOC ( ) exp 
d 
  exp  






R 0
 R 0 C( )  
 R 0 C( z) 


(S13)
During the second half cycle between t = θ0/ and t = 2θ0/ (θ0 ≤ α≤ 2θ0), the solution of governing equation
with the boundary condition Q(t = θ0/) = Q1 at constant  is given by
t V
(t )
 t
 t
dt  
dz  
OC
Q  exp    0
exp   0
 Q1   0
 dt 
R

  C(t )R  
  C( z)R  
t V

( )
 1  t d  
 1  dz 
OC
 exp  
exp 
 Q1   
 d 




0
R
 R 0 C( )  
 R 0 C( z) 

Therefore, at the end of the second half cycle, Q is given by
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2 0 V

2 
( )

 1 20 d  
 1  dz 
OC
exp 
Q2  Q  t  0   exp  
 Q1   
 d 




0
 
R
 R 0 C( )  
 R 0 C( z) 


0 V
 0 dz 


(
)
 1 0 d  

1
OC
 exp  
exp 
 Q1   0
 d 


0 C ( )

R
R


 R  C( z) 

0 V
 0 dz
 dz 


(
)
 1 0 d  

1
1
OC
 exp  

exp 
 d 
 Q1   0



0
0
0
C( z) R
C( z)  
C( )  
R
 R
 R
 1 0 d 
 1  dz 
1 0
 exp  
VOC ( ) exp  
 Q1 
 d



0
0
C( ) 
R
 R
 R 0 C( z) 
(S15)
By applying the periodic boundary condition Q0 = Q2, we can obtain the equation for Q0, which is the same as
Eq. (13), given by
 2 0 d 
exp  


 R 0 C( )   1 0 V ( ) exp  1  dz  d
Q0 


OC

 2 0 d  R  0
 R 0 C( z ) 
1  exp  


 R 0 C( ) 

(S16)
 1  dz 
1 0
VOC ( ) exp  
 d


0
 2 0 d  R
 R 0 C( z) 
1  exp  


 R 0 C( ) 
1

Then the current I was derived from the differentiation of Q by t
 t dt 
exp   

V (t )
 0 C(t )R   Q  t VOC (t ) exp  t dz  dt  , 0 ≤ t ≤  0

I  OC
 0 0
0
 
R
C(t )R
R

 C( z)R  

I
VOC (t )
R
 t
dt 
exp    0

t V
C
(
t )R  
(t )

 t
dz  
 

 Q1   0 OC
exp   0
 dt  ,
C(t )R
R

  C( z)R  

2
0
≤t ≤ 0


(S17a)
(S17b)
And the voltage output can be calculated by Ohm’s law and is given by
 t dt 
exp   

 0 C(t )R   Q  t VOC (t ) exp  t dz  dt  , 0 ≤ t ≤  0
V  VOC (t ) 
 0 0
0
 
C(t )
R

 C( z)R  

 t
dt 
exp    0

t V

C
(
t )R  
(t )
 t
dz  
 
 Q1   0 OC
V  VOC (t ) 
exp   0
 dt  ,
C(t )
R

  C( z)R  

0
2
≤t ≤ 0


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(S18a)
(S18b)
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