Optical porosimetry and investigations of the porosity experienced by light interacting with porous media.

Abstract

We investigate how light samples disordered porous materials such as ceramics and pharmaceutical materials. By combining photon time-of-flight spectroscopy and sensitive laser-based gas sensing, we obtain information on the extent to which light interacts with solid and pore volumes, respectively. Comparison with mercury intrusion porosimetry shows that light predominantly interacts with the solid. Analysis based on a two-state model does not fully explain observations, revealing a need for refined modeling. Nonetheless, excellent correlation between actual porosity and the porosity experienced by photons demonstrates the potential of nondestructive optical porosimetry based on gas absorption.

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Optical porosimetry and investigations of the porosity
experienced by light interacting with porous media
Tomas Svensson,1,* Erik Alerstam,1Jonas Johansson,2and Stefan Andersson-Engels1
1Department of Physics, Lund University, 221 00 Lund, Sweden
2Astra Zeneca R&D Mölndal, 431 83 Mölndal, Sweden
*Corresponding author: tomas.svensson@fysik.lth.se
Received February 9, 2010; revised April 19, 2010; accepted April 19, 2010;
posted April 23, 2010 (Doc. ID 123966); published May 17, 2010
We investigate how light samples disordered porous materials such as ceramics and pharmaceutical materials. By
combining photon time-of-flight spectroscopy and sensitive laser-based gas sensing, we obtain information on the
extent to which light interacts with solid and pore volumes, respectively. Comparison with mercury intrusion por-
osimetry shows that light predominantly interacts with the solid. Analysis based on a two-state model does not fully
explain observations, revealing a need for refined modeling. Nonetheless, excellent correlation between actual por-
osity and the porosity experienced by photons demonstrates the potential of nondestructive optical porosimetry
based on gas absorption. © 2010 Optical Society of America
OCIS codes:
120.4290, 300.6500, 300.6320, 290.4210, 160.2710.
Understanding of the interaction of light and turbid por-
ous materials is of fundamental importance in many
areas—from fundamental studies of photon localization
in porous semiconductor materials [1,2] to analysis of
chemistry, pore structure and optics of powders and
pharmaceutical materials [3–5], porous ceramics [6–8],
and porous silicon [9,10]. Throughout the years, massive
efforts have been put into developing models for the
macroscopic aspects of photon migration in turbid mate-
rials. Radiative transport theory and derivatives (e.g.,
Monte Carlo simulation and diffusion theory) are the
most commonly used models. These models are based on
photon conservation, ignore wave properties, and
introduce average optical properties to account for
microscopic heterogeneity that causes light scattering.
Understanding of how photons actually sample turbid
heterogenous structures is, however, limited, and the
interpretation of average optical properties remains
debated.
InthisLetter,wereportonexperimentalwork aimedat
improving fundamental understanding of light propaga-
tion in porous materials. We show that photons propagat-
ing in porous structures predominantly interact with the
solid,andthatthisphenomenoncanbestudiedindetailby
carefully combining photon time-of-flight spectroscopy
(PTOFS) and high-resolution gas in scattering media ab-
sorption spectroscopy (GASMAS). Our experiments, in
combination with mercury intrusion porosimetry (a por-
osimetry gold standard), provide new insight on the rela-
tion between photon travel habits, average optical
properties, and material pore structure. In addition, we
present a quantitative study of direct optical assessment
of porosity. Being based on gas absorption, the method
stands in great contrast to work on optical porosimetry
basedonmeasurementsoflightscatteringincombination
with Mie theory and presupposing negligible material
absorption and low porosity [6,7].
Ceramics and stacked crystals are examples of mate-
rials with only interparticle porosity, while both interpar-
ticle and intraparticle porosity is exhibited by, e.g.,
zeolite powders and granulated pharmaceutical filler ma-
terials. Structure, pore sizes, and porosity strongly influ-
ence optical pathlengths and attenuation. In this work,
we assume that light propagation can be described using
a two-state model where light at a given time propagates
in either solid or pore. The average pathlength through
solid, Ls, is of paramount importance, since it determines
spectroscopic signatures. Unfortunately, Lsis not easily
assessed. Another fundamental question concerns the re-
lation between the material porosity ϕ and the division of
the total photon pathlength L in pathlength through the
solid, Ls, and pathlength through pore volumes, Lp. In
previous work based on PTOFS and GASMAS, it had
been assumed that L is divided according to the volume
fractions, i.e., Ls=L ¼ Vs=V and Lp=L ¼ Vp=V (where V
is the total sample volume and Vsand Vpare the material
and pore volume, respectively) [11,12]. Note, however,
that the validity of this assumption was questioned al-
ready in the first paper, where PTOFS and GASMAS
was combined, and further scrutiny was suggested
[11]. The present work investigates this important matter
and includes studies of photon travel habits in both por-
ous ceramics and complex granulated materials.
PTOFS is employed to determine the photon time-of-
flight (TOF) distribution and thus also the average total
TOF t ¼ tsþ tp, where tsand tpare the time spent in
the solid and the pore volume, respectively. Assuming a
unitary refractive index in pore volumes, the relation be-
tween TOFs and physical pathlengths is
L ¼ Lsþ Lp¼ c0n−1
stsþ c0tp;
ð1Þ
where c0is the speed of light in vacuum and nsis the re-
fractive index of the solid. To reach deconvoluted
information about Lsand Lp, we utilize the absorption ex-
hibited by free gas located in the pores. The spectrally
sharp (gigahertz) gas absorption is distinguished from
the dull imprints of the solid and is measured by means
of GASMAS [13,14]. GASMAS allows measurement of
Lp¼ c0tp. Combining GASMAS and PTOFS, Lscan be
determined using Ls¼ c0n−1
ments are conducted by tuning a 0:3 mW vertical-cavity
surface-emitting diode laser over the R9Q10 absorption
line of molecular oxygen (760:654 nm). A large-area
photodiode detects diffuse light transmitted through the
s × ðt − tpÞ. GASMAS experi-
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samples.Highsensitivityisreachedbyemployingsecond-
harmonic wavelength modulation spectroscopy (2f
WMS). The GASMAS setup is described in detail in
[12,14]. Note that since we study gas confined in micro-
meter-sized pores at atmospheric pressure, wall collision
broadening (which would appear in nanopores) is negli-
gible[8].Theabsorptionlineshapeoftheconfinedoxygen
is therefore the same as free oxygen [see, e.g., Fig. 1(b)].
OurPTOFSexperimentsarebasedonpicosecondpulsed-
light sources, a fast microchannel plate–photomultiplier
tube, and time-correlated single-photon counting, allow-
ing measurement of TOF distributions and average
absorption and scattering of the material [15,16]. The dif-
ference in detector size between GASMAS and PTOFS is
takenintoaccountbycalculatingtfortheGASMASdetec-
tor using the optical properties derived from PTOFS.
The ceramic material investigated is a macroporous
alumina ceramic with a porosity of ϕHg¼ 34:0% and pore
diameters mainly in the 1 to 5 μm range (as given by
mercury intrusion porosimetry). The sample was made
to 92.5% from a 10 μm α-alumina powder (Al2O3), the re-
maining part being a 40 nm silica powder (SiO2) used as
binder. The bulk material was reached by sintering and
was polished to a thickness of 2:85 mm (14 mm in diam-
eter). Figure 1 presents experimental data. PTOFS at
760 nm [16] showed that the average total TOF t is
2163 ps, while GASMAS revealed that Lpis 58:8 mm
(tp¼ 196 ps). Since the refractive indexes of α-alumina
andsilica is1.765and1.5, respectively,nscanbe approxi-
mated by 1.75. This means that Ls¼ 337:2 mm and L ¼
Lsþ Lp¼ 396 mm.Theporosityexperiencedbyphotons,
ϕopt¼ Lp=L (from now on termed optical porosity), is
thusabout14.9%.Thisdatarevealamajordiscrepancybe-
tween optical and actual porosity (ϕopt¼ 0:44 × ϕHg).
Turning to complex porous materials with both
interparticle and intraparticle porosity, we conducted a
detailed study of 12 granulated pharmaceutical tablets
with microcrystalline cellulose (MCC) as the main consti-
tuent. All had a total weight of 300 mg and were manufac-
tured from two sieve fractions (granule sizes less than
150 μm in group A, and 150–400 μm in group B) using dif-
ferent compression forces (i.e., a subset of the samples
described in [12]). The refractive index of the solid mate-
rial was about 1.5. Photon TOF was measured at 786 nm
[12], but the difference in TOF between 786 nm and
760 nm is negligible for these samples (investigated using
a tunable instrument [16]). The pore structure was inves-
tigated by means of mercury intrusion porosimetry and is
presented in Fig. 2. The porosity, ϕHg, was between 10%
and 30%. Evaluation of PTOFS data [12] yields reduced
scattering of ∼500 cm−1and absorption of ∼0:03 cm−1
(varying with compression and granular size). Interest-
ingly, our data again show that ϕoptsignificantly deviates
from actual porosity. A comparison of optical and actual
porosity is given in Fig. 3. Note that simple gravimetric
considerationsagreewithϕHg.Thedatathusclearlyshow
that photons are predominantlyconfined tothe solid, ϕopt
being 0:48 × ϕHg.
These results show that the use of ϕopt¼ Lp=L as a
measure of porosity, as suggested in previous work com-
bining GASMAS and PTOFS [11,12], can be highly inac-
curate. Nonetheless, as shown in Fig. 4, our data give first
evidence of an excellent correlation between ϕoptand
ϕHg. Since the precision in Lpis better than 2%, and better
than 1% for t, the precision in ϕoptshould be ∼5% for the
samples considered in this work. This shows that our op-
tical approach has a great potential as a tool for rapid and
nondestructive porosimetry.
The physical origin of the discrepancy between
optical and actual porosity needs further attention. Total
internal reflection can potentially explain why photons
are predominantly confined to the solid, a phenomenon
recently discussed in relation to photon channeling in
aqueous foams [17,18]. When geometrical optics is valid,
and assuming random angles of incidence and polariza-
tion, one can calculate probabilities for photon transfers
Fig. 1.
the macroporous alumina. Diffusion modeling [12] yields a re-
duced scattering of ∼1300 cm−1and negligible absorption. The
gas absorption signal is a 2f WMS signal. Evaluation using an air
spectrum [12] yields a pathlength of 66:0 mm (including a
7:2 mm offset).
(Color online) (a) PTOFS and (b) GASMAS data from
Fig. 2.
blets. 1–6 μm pores dominate the pore volume. Compression
removes large pores, shifting large-pore cutoff from 5 to 2 μm.
(Color online) Cumulative mercury intrusion for ta-
Fig. 3.
and porosity given by mercury intrusion,ϕHg, for 12 pharmaceu-
tical samples. The solid line shows porosity expected from as-
suming an MCC density of 1:46 g=cm3. To avoid artifacts due to
deformation at high mercury pressures, ϕHgis based on the cu-
mulative intrusion at 0:1 μm and may thus be a slight underes-
timation of the true porosity.
(Color online) Comparison of optical porosity, ϕopt,
June 1, 2010 / Vol. 35, No. 11 / OPTICS LETTERS 1741

Page 3

between solid and pore volumes. A simple two-state Mar-
kov model can then provide the stationary distribution
(fraction of steps in solid, πs, versus steps in pores,
πp). For the ceramic, ns¼ 1:75, the probabilities for in-
ternal reflection in the solid and pore, respectively, are
pss¼ 0:4219 and ppp¼ 0:1166 (subscripts referring to
solid-to-solid and pore-to-pore, respectively). The sta-
tionary distribution is then πs¼ 0:6044 and πp¼ 0:3956.
Since ϕ and 1 − ϕ are measures of the relative length of
steps in pores and solid compartments, respectively, the
optical porosity predicted by this model is
ϕopt¼
πpϕ
πpϕ þ πsð1 − ϕÞ:
ð2Þ
For our ceramic (ϕ ¼ 34%) this model suggests that ϕopt
should be about 25%, i.e., 0.74 times the actual porosity.
Since our heterogeneities are not much larger than the
wavelength, the breakdown of geometrical optics may
explain why the observed ratio is as low as 0.44.
For the case of the pharmaceutical samples, where the
solid has a lower refractive index, one might expect a
smaller difference between ϕoptand ϕHg. Although this
agrees with experimental data, the discrepancy between
model and experiment remains large. Calculations analo-
goustothosepresentedabove,forns¼ 1:5andϕbetween
10%and30%,yieldsopticalporositiesbetween7%and22%
and ratios between 0.70 and 0.78 (values increasing with
increasing porosity). Experimental data gives a ratio be-
tween 0.44 and 0.54. Besides a breakdown of geometrical
optics, the discrepancy may here also be influenced by a
difference between intraparticle and interparticle poros-
ity in combination with multiple intraparticle scattering.
To conclude, we have presented an experimental ap-
proach for investigation of how light samples pores
and solid parts of porous media. It allows nondestructive
assessment of porosity and improves fundamental under-
standing of the interaction of light and porous media. Our
observations cannot be fully explained by the utilized
two-state model, revealing a need for refined modeling.
In addition, our results show that the average refractive
index cannot be based on volume fractions only, a com-
plication previously discussed in connection with extre-
mely scattering porous semiconductors [2,19]. Our
approach may be used to test refined models. Ultimately,
knowledge on how light samples porous media allow im-
proved analysis of signals obtained in e.g., absorption,
fluorescence, and Raman spectroscopy of porous media.
This work was funded by the Swedish Research
Council.
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Fig. 4.
porosity (correlation coefficient 0.98, ϕopt≃ 0:48ϕ×Hg).
(Color online) Correlation between optical and actual
1742OPTICS LETTERS / Vol. 35, No. 11 / June 1, 2010