Supporting Information
Silicone dielectric elastomers based on radical crosslinked high molecular weight
polydimethylsiloxane co-filled with silica and barium titanate
Adrian Bele, George Stiubianu, Cristian-Dragos Varganici, Mircea Ignat, Maria Cazacu
Figure S1. Film formation procedure.
Theoretical estimation of dielectric permittivity
The following theoretical approaches were used for predicting the values of effective dielectric
permittivity of polymer composite systems.
1. Volume-fraction average:
εff = φpεp + φsεs + φbεb
(1)
where p, s and b are the siloxane polymer, the silica and the barium titanate phase, respectively, and
φ is the volume fraction of the constituents, with φp = 1 – (φs + φb). This model (Equation 1)
predicts a sharp increase of the effective dielectric constant starting at a low volume fraction of the
ceramic filler. In fact, contradictory results were found in both theoretical [38] and experimental
studies [39].
2. The Maxwell-Garnett equation; it is an approximation of a single spherical inclusion surrounded
by a continuous matrix of the polymer when the filler fraction goes to zero (infinite dilution) [40].
Since none of these conditions are fulfilled for the prepared elastomers – there is more than one
filler and the fractions of the fillers are significant – therefore this equation can be used for
calculating estimative values of εff only for samples containing just silica (S10 B0, S15 B0, S30
B0):
εff = εp· [εs + 2εp – 2(1 – φp)(εp – εs)]/[εs + 2εp + (1 – φp)(εp – εs)]
(2)
3. In the Bruggeman model, the binary mixture is made of repeated unit cells, each consisting of the
polymer matrix which has spherical inclusions in the center [40]. The effective dielectric constant
calculated with the Bruggeman equation increases sharply for filler volume fractions above 20%.
The effective dielectric constant (εff) of the composite mixture [41] is given by:
(εeff – εp)/(εeff + 2εp) = φs (εs – εp)/(εs + 2εp) + φb (εb – εp)/(εb + 2εp)
(3)
The Maxwell-Garnett equation (2) is just a particular case of the Bruggeman model.
4. The two-component Lichtenecker-Rother model [42] for complex permittivity is used for such
calculations in different systems, such as air-particulate composites [43], ceramic-ceramic
composites [44] and polymer-ceramic composites [45]. In this model the effective permittivity of
two-component systems is determined by introducing the volume fraction of each component
according to equation:
εeffβ = φp εpβ + φs εsβ + φb εbβ
(4)
β is a dimensionless parameter and its value is determined by the shape and orientation of the filler
particles within the bulk composite [46] and can take values between 1 and -1. Since the
microparticles of silica and barium titanate can be described as spherical inclusions and randomly
oriented ellipsoids, the value of β = 1/3, as determined by Landau, Lifshitz [47,48].
Figure S2 shows the values for the dielectric constant of the samples as resulted from the above
formulas and can be easily compared with the experimental values. The values for εp=2.9 and
εs=3.9 and εb=1700, density is 1 g/cm3 for the siloxane polymer, 2.2 g/cm3 for silica and 6.03 g/cm3
for barium titanate and the volume fractions are as follows:
-φp=1, φs=0 and φb=0 for S0B0,
-φp=0.9524, φs=0.0476 and φb=0 for S10 B0,
-φp=0.949, φs=0.043 and φb=0.0078 for S10 B5,
-φp=0.934, φs=0.042 and φb=0.0232 for S10 B15,
-φp=0.936, φs=0.0637 and φb=0 for S15 B0,
-φp=0.929, φs=0.0632 and φb=0.00771 for S15 B5,
-φp=0.915, φs=0.0623 and φb=0.0227 for S15 B15,
-φp=0.880, φs=0.120 and φb=0 for S30 B0,
-φp=0.873, φs=0.119 and φb=0.00725 for S30 B5,
-φp=0.861, φs=0.117 and φb=0.0214 for S30 B15.
εp=2.9 ; εs=3.9 ; εb=1700
2
S0B0: φp=1, φs=0 and φb=0,
S10 B0: φp=0.9524, φs=0.0476 and φb=0,
S10 B5: φp=0.949, φs=0.043 and φb=0.0078,
S10 B15: φp=0.934, φs=0.043 and φb=0.023,
S15 B0: φp=0.936, φs=0.064 and φb=0,
S15 B5: φp=0.929, φs=0.0632 and φb=0.00771,
S15 B15: φp=0.915, φs=0.0623 and φb=0.0227,
S30 B0: φp=0.880, φs=0.120 and φb=0,
S30 B5: φp=0.873, φs=0.119 and φb=0.00725,
S30 B15: φp=0.861, φs=0.117 and φb=0.0214.
0) Measured values
1) Volume fraction: εff = φpεp + φsεs + φbεb
2) Maxwell: εff = εp· [εs + 2εp – 2(1 – φp)(εp – εs)]/[εs + 2εp + (1 – φp)(εp – εs)]
3) Bruggeman: (εeff – εp)/(εeff + 2εp) = φs (εs – εp)/(εs + 2εp) + φb (εb – εp)/(εb + 2εp)
4) Lichtenecker-Rother: εeffβ = φp εpβ + φs εsβ + φb εbβ
Table S1. Dielectric permittivity values predicted by using different models
Sample
Silica,
Barium
Measured
Expected dielectric permittivity value
wt%
Titanate,
dielectric
Volume
Maxwell
permittivity fraction
Garnett
wt%
Bruggeman
LichteneckerRother
value
S0B0
0
0
3.02
2.90
2.90
2.90
2.90
S10 B0
10
0
3.41
2.95
2.94
2.94
2.95
S10 B5
10
5
3.67
16.18
-
3.01
3.46
S10 B15
10
15
3.95
41.89
-
3.14
3.69
S15 B0
15
0
3.48
2.96
2.96
3.15
2.98
S15 B5
15
5
3.66
16.04
-
3.01
3.48
S15 B15
15
15
4.09
41.48
-
3.16
4.68
S30 B0
30
0
3.45
3.02
3.01
3.01
3.01
S30 B5
30
5
3.89
15.32
-
3.07
3.50
S30 B15
30
15
4.26
39.33
-
3.2
4.63
3
Figure S2. Plotting experimental dielectric permittivity values as compared with those theoretical
estimated by using different models.
4
a
b
5
c
d
Figure S3. DSC curves for: a- PDMS and series S0; b – series S10; c – series S15; d – series S30
(H1 – first heating; H2 – second heating; C – cooling).
6
Figure S4. Water vapor sorption isotherms recorded at room temperature.
7
S10B0
S10B5
S10B15
8
S15B0
S15B5
S15B15
9
S30B0
S30B5
S30B15
Figure S5. The electric responses at an applied mechanical impulse.
10
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