Supplemental materials
Manuscript title:
Candle Soot Nanoparticles-PDMS Composites for Novel Laser Ultrasound Transducers
To better justify the improved photoacoustic conversion efficiency of CSNPs-PDMS composite,
the maximum achievable acoustic pressure by PDMS based transducer was estimated theoretically.
The attenuation and loss effects are neglected in this discussion.
The achieved acoustic pressure strongly relies on the material properties of the absorption layer
and adjacent layers. This can be understood better using an analytical solution. By assuming the
100 % light absorption and 100 % thermal energy dissipation into the surrounding PDMS, the
transducer can be modeled by a surface-heat-source model. This is based on the assumption that
the thermal diffusion occurs within an infinitesimal amount of time, requiring a large thermal
conductivity and small light penetration depth of the absorption materials.1,2 Based on this model
and neglecting the attenuation, the amplitude of the elastic wave in the PDMS under a harmonic
heat source with density he jt can be expressed as
A
 r1
1 j


2
c
v
a
(1  r ) 2 (1a1   2 a2 )  1 1 1
2
(1  j )h
 1
  
 2 

 a1c1 a2 c2  
where r  1v1 / ( 2 v2 ) is the ratio of the characteristic acoustic impedance in the PDMS to that of
the glass substrate, and ρ, ν, t, ω stands for the density, acoustic velocity, time and frequency,
respectively. a  (1 /  )1/ 2 where α is the thermal diffusion coefficient. κ is the thermal conductivity
and c is the elastic modulus.   T B where  T is the thermal expansion coefficient and B is the
bulk modulus. The subscripts
1, 2
correspond to the PDMS and glass backing layer, respectively.
The materials properties of glass and PDMS are listed in Table S1Table S. Based on these materials
properties, the acoustic pressure in PDMS under a 6 ns laser pulse with a Gaussian profile and an
energy density of 3.57 mJ/cm2 can be calculated to be 8.35 MPa. The experimentally measured
pressure is around half of this estimation under ideal assumptions. This is thus practically reliable
by considering the neglected attenuation and loss during both thermal and acoustic propagations in
the model.
Table S1. The material properties of transducer layers.3
κ
α × 10-7
βT × 10-6
ν
ρ
B
(Wm-1K-1)
(m2s-1)
(K-1)
(ms-1)
(kgm-3)
(GPa)
Glass
1.38
8.48
0.55
5900
2200
50
PDMS
0.15
1.06
920
1076
970
7.7
The explanation of the improved photoacoustic conversion efficiency in CSNPs-PDMS composite
is described below. Based on the one dimension model the most effective elastic-wave generation
occurs when materials in or adjacent to the heated region have large values of βT(D)1/2 where D is
the thermal diffusion distance which equals D   2 /   .1 It should be noted that these parameters
1/ 2
are the effective values of the composite thus determined by not only the materials composition but
also the spatial structure configuration. At 10 MHz, the thermal diffusion distance in PDMS is 60
nm while as high as 11 µm in CSNPs. The hierarchical and arborized structure of the CSNPs allow
the thermal energy to transfer to a large distance within a short amount of time and dissipate into
the surrounding PDMS more uniformly and faster. In contrast, the aggregated CBs are isolated by
the surrounding PDMS matrix which blocks the efficient thermal transfer. Therefore, higher
effective thermal diffusion distance can be expected in CSNPs-PDMS composite compared that of
the CB-PDMS composite, resulting in a higher photoacoustic conversion efficiency.
The observed narrower frequency bandwidth of CB-PDMS composite is believed to arise from the
non-uniform mixing and dispersion of CB particles in the PDMS matrix. The generated pulse shape
is most dependent on the optical attenuation coefficient β and the acoustic attenuation property. As
β-1 increases, the pulse width becomes narrow and sharper profile is expected until saturation occurs.
The pulse width is around 5β-1 before saturation, or five penetration depths.1 The penetration depth
of CB alone resides between 10 ~ 15 nm where pulse width already saturates.4 However, with low
CB concentration in the PDMS polymer, the penetration depth could reach hundreds of µm. 5,6 A
100 µm penetration depth yield a 300 ns pulse width thus a narrow frequency bandwidth. This can
be induced by the aggregation of CB, leaving numerous porous PDMS areas where light could
travel deep. The acoustic attenuation is unlikely to be the reason for the narrower bandwidth of CBPDMS as the total thickness of each composite is similar thus the attenuation levels are expected
to be equivalent.
1
2
3
4
5
6
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