# Polymeric Liquids in Nanoporous Photonic Structures: From Precursor Film
Spreading to Imbibition Dynamics at the Nanoscale

Guido Dittrich  Institute for Materials and X-Ray Physics, Hamburg University
of Technology, 21073 Hamburg-Harburg, Germany  Luisa G. Cencha  IFIS-Litoral
(Universidad Nacional del Litoral-CONICET), Guemes 3450, 3000 Santa Fe,
Argentina  Martin Steinhart  Institute of Chemistry of New Materials,
Osnabrück University,49076 Osnabrück, Germany  Ralf B. Wehrspohn  Institute of
Physics, Martin Luther University of Halle-Wittenberg, 06120 Halle (Saale),
Germany  Korea Institute of Energy Technology (KENTECH), 21 Kentech-gil, Naju
58330, South Korea  Claudio L. A. Berli  INTEC (Universidad Nacional del
Litoral-CONICET), Predio CCT CONICET Santa Fe, RN 168, 3000 Santa Fe,
Argentina  Raul Urteaga  IFIS-Litoral (Universidad Nacional del Litoral-
CONICET), Guemes 3450, 3000 Santa Fe, Argentina  Patrick Huber  Institute for
Materials and X-Ray Physics, Hamburg University of Technology, 21073 Hamburg-
Harburg, Germany  Center for X-Ray and Nano Science CXNS, Deutsches
Elektronen-Synchrotron DESY, 22603 Hamburg, Germany

(August 28, 2024)

###### Abstract

Polymers are known to wet nanopores with high surface energy through an
atomically thin precursor film followed by slower capillary filling. We
present here light interference spectroscopy using a nanoporous membrane-based
chip that allows us to observe the dynamics of these phenomena in situ with
sub-nanometer spatial and milli- to microsecond temporal resolution. The
device consists of a mesoporous silicon film (average pore size 6 nm) with an
integrated photonic crystal, which permits to simultaneously measure the phase
shift of the thin-film interference and the resonance of the photonic crystal
upon imbibition. For a styrene dimer, we find a flat fluid front without a
precursor film, while the pentamer forms an expanding molecular thin film
moving in front of the menisci of the capillary filling. These different
behaviors are attributed to a significantly faster pore-surface diffusion
compared to the imbibition dynamics for the pentamer and vice versa for the
dimer. In addition, both oligomers exhibit anomalously slow imbibition
dynamics, which could be explained by apparent viscosities of six and eleven
times the bulk value, respectively. However, a more consistent description of
the dynamics is achieved by a constriction model that emphasizes the
increasing importance of local undulations in the pore radius with the
molecular size and includes a sub-nanometer hydrodynamic dead, immobile zone
at the pore wall, but otherwise uses bulk fluid parameters. Overall, our study
illustrates that interferometric, opto-fluidic experiments with nanoporous
media allow for a remarkably detailed exploration of the nano-rheology of
polymeric liquids.

††preprint: AIP/123-QED

##  I Introduction

Polymeric liquids exhibit a variety of transport mechanisms during capillary
filling of nanoporous materials. Understanding them is essential for many
applications. Immobile layers at the pore walls or confinement effects are
important for separation processesKrutyeva  _et al._ (2013); Kusmin  _et al._
(2010), the formation of precursor films can be used for templated
tubesSteinhart  _et al._ (2002) and temperature-dependent imbibition for
passive temperature sensors Cencha  _et al._ (2022), to just name a few.

Discussions describing capillary rise in nanopores often start with the Lucas-
Washburn equation (LWE).Lucas (1918); Washburn (1921) It relates the driving
capillary pressure to the counteracting viscous drag in a ${\rm\sqrt{t}}$
versus imbibition length ${\rm L}$ dependency. Although it is not suitable to
solve every capillary filling problem, in particular if there are systematic
variations in the hydraulic permeability, Reyssat  _et al._ (2008) it is
fairly close to many solutions of more complex porous systems and
liquids.Gruener  _et al._ (2009, 2015); Huber (2015) The prefactor of the LWE
can be divided into the fluid-centered properties; surface tension
${\rm\sigma}$, viscosity ${\rm\mu}$, and the liquid-solid contact angle
${\rm\theta}$, and into the geometrical ones; an effective pore radius ${\rm
r_{\rm eff}}$ whose definition captures the nature of the porous structure.

$\displaystyle{\rm L(t)=\Gamma\sqrt{r_{\rm eff}t}}$ (1)

where $\Gamma=\sqrt{\frac{\sigma\cos{\theta}}{2\mu}}$.

Many experimental studies have been performed on melt infiltration of
polymers, with a molecular weight above the critical value (${\rm M_{c}}$) for
entanglements, into nanopores.Vazquez Luna  _et al._ (2021); Luna  _et al._
(2022); Cencha  _et al._ (2019); Yao  _et al._ (2017) In such studiesVazquez
Luna  _et al._ (2021); Luna  _et al._ (2022) on melt infiltration of
polystyrene into anodized aluminum oxide with straight pores and into the
sponge-like structure of controlled porous glass, a square-root-of-time law
was used to model the dynamics. Pearson correlation coefficients for different
exponents of a power-law fit suggested though an even better fit for exponents
deviating from the famous LWE like ${\rm\sqrt{t}}$ proportionality.

Yao et al. Yao  _et al._ (2018) proposed a unified theory of capillary rise
dynamics of polymers ${\rm>M_{c}}$ involving an immobile layer and reptation
under pressure model. The latter explains the relative increase of imbibition
dynamics with molecular mass, in contrary to the increasing bulk viscosity, by
disentanglement under confinement. In the immobile layer model the pore radius
behind the imbibition front is reduced by adsorbed molecules. This phenomenon
is also observed for polymeric liquids with molecular weight below ${\rm
M_{c}}$ and other liquids.

Polymeric liquids below ${\rm M_{c}}$ under confinement exhibit imbibition
phenomena that are not well understood yet. A molecular dynamics simulation of
Lennard-Jones (LJ) fluids and a decane melt in a cylindrical nanopore is in
close agreement with the LWE.Dimitrov, Milchev, and Binder (2007) In case of
slip flow they suggest to modify the LWE by virtually increasing the radius by
a slip length. Also a much faster precursor film spreading obeying a square-
root-of-time law has been inferred in atomistic simulations Chibbaro  _et al._
(2008).

Engel and StühnEngel and Stuehn (2010) studied polyisobutylene and
poly-${\rm\epsilon}$-caprolactone in nanoporous polycarbonate (both <${\rm
M_{c}}$) by in-situ small angle X-ray scattering. They found a fast first
wetting of the pore walls by a precursor film, followed by a much slower than
expected complete filling. The fast initial wetting is attributed to a
stronger adhesive than cohesive force. Although the aspect ratio of pore size
to the radius of gyration was not small at all, they discovered a strong
confinement effect on the complete filling.
In-situ nanodielectric spectroscopy is simultaneously sensitive to molecular
dynamics and capillary rise. An imbibition studyTu  _et al._ (2020) on cis-
polyisoprene (<${\rm M_{c}}$) in anodized aluminum oxide (AAO) has measured
two time regimes decelerated compared to the LWE. An increasing molecule-wall
interaction with time for native AAO was concluded, as less freely fluctuating
chains were contributing to the dielectric signal. Modifying the surface by
silanization successfully reduced the interaction.

Recent single-pore molecular dynamics simulations of LJ fluids predict slower
capillary filling for short polymer chains, below ${\rm M_{c}}$ for
entanglements, compared to the LWE prediction, while longer chains show the
opposite trend.Zhang  _et al._ (2023) The slowed down filling dynamics of
short polymer chains is explained by a nearly immobile layer with finite slip
and lower free energy due to confinement. It is quantified by an effective
viscosity, which is twice the bulk viscosity for a radius-to-chain length
ratio of one and, interestingly, increases for higher ratios.

We focus here on the imbibition dynamics of styrene oligomers in nanoporous
silicon and will see that our observation fit such a slowdown seen in the
simulations. Moreover, we are going to scrutinize the relevance of precursor
film spreading for the imbibition phenomenology.

Nanoporous silicon (pSi) is an established functional material Sailor (2012);
Canham (2015); Brinker  _et al._ (2020); Brinker and Huber (2022); Brinker
_et al._ (2022), most prominently in optical sensing applications for
biochemistryPacholski (2013), gasesLin  _et al._ (2004) and fluid
dynamicsCencha  _et al._ (2020). It can also be used to explore the effect of
nanoconfinement on condensed matter. Henschel  _et al._ (2007); Hofmann  _et
al._ (2012); Calus  _et al._ (2012); Huber (2015); Kondrashova  _et al._
(2017) A common optical sensing mechanism relies on the change of the
effective refractive index of porous layers during the displacement of air by
an analyte. Porous silicon is electrochemically etched in a self-organizing
top-down process with a strong anisotropy of the pores in the etching
direction. When the pore size and interpore distances are much smaller than
the probing wavelength of an electromagnetic wave, the effective optical
properties of pSi are adjustable by the porosity and thickness. The latter
parameters depend on the synthesis conditions. Continuous or abrupt changes in
the in-depth porosity can be accomplished by the course of applied current
density, whereas the time applied determines the thickness. This enables the
synthesis of layered structures, Bragg- and Rugate-filters, as well as 1-D
photonic crystals.Pacholski (2013) These structures can be applied for the in-
situ investigation of polymer infiltration into nanoporous scaffold
materials,Cencha  _et al._ (2020) providing an improved resolution in the
plane perpendicular to the pore orientation.

##  II problem statement

It has been demonstrated that in-situ white light reflection spectroscopy of
pSi photonic crystals (PC) provides the precision to resolve precursor film
spreading. A thin film pSi layer in front of the PC (1a,b) acts as a column to
investigate different imbibition lengths and thereby times. The shape of fluid
fronts can be inferred from re-scaling the filling dynamics of the PC with a
capillary-filling model for the fluid transport of the in-front homogeneous
porous layer.Cencha  _et al._ (2020)

As outlined above, polymers exhibit a variety of different dynamics for the
infiltration of nanoporous scaffolds Yao  _et al._ (2018), depending on the
molecular weight, temperature, entanglements, shear thinning/thickeningCao,
Yang, and Hu (2016) behavior or confinement. Therefore, the capillary filling
model that best fits the experimental data needs to be adapted for each
individual fluid. The geometrical descriptor of the scaffold imbibed should on
the contrary be unanimous.

In the following a method is introduced, where the infiltration of the stacked
pSi-layer (thin film + PC) is measured simultaneously with two spectrometers
comprising two different wavelength ranges; visible and NIR. In this way, the
infiltration curve corresponding to the PC can be decoupled from the
infiltration curve of the stacked layer. Both curves, combined, allow us to
infer not only the meniscus dynamics but also the precursor film profile and
dynamics. In this study we have used two oligostyrenes as imbitition fluids.
Overall, we introduce an exceptionally precise technique to characterize fluid
dynamics, particularly relevant for polymer infiltration into nanopores.

##  III materials

Probing liquids are mono disperse oligostyrenes (OS) (from Polymer Standards
Service) with two (OS-2) and five (OS-5) monomer repetition units, having a
sec-butyl and proton end group. Because of their well defined properties, OS
molecules are commonly used as a standard for gel-chromatography. The
viscosity at room temperature is analysed by a rheometer (Kinexus Prime
Series) in plate to plate geometry, resulting in a viscosity of
34,000${\rm{\pm}570\leavevmode\nobreak\ mPas}$ in a shear rate range of ${\rm
0.01-100\leavevmode\nobreak\ Hz}$ and 9.4${\rm{\pm}2.1\leavevmode\nobreak\
mPas}$ at ${\rm 100-51600\leavevmode\nobreak\ Hz}$ for OS-5 and OS-2
respectively. Both liquids provide Newtonian fluid behavior in the shear-rate
range of the capillary filling. Other parameters used in the analysis are
listed in Tab. 1.

Porous silicon is fabricated by electrochemical anodization. Monochristalline
100-orientation, boron doped (p-type) wafers with resistivity of ${\rm
1-4\leavevmode\nobreak\ m{\Omega}}$cm are treated with a HF(${\rm
50\leavevmode\nobreak\ w\%}$):EtOH 1:2 volumetric solution and different
current densities. By variation of the latter, layers with different
thicknesses and porosity can be obtained. For the current method, a thin film
layer of largely homogeneous porosity is etched first, subsequently followed
by a PC. The PC is designed with its band gap and width in the visible range,
whereas the overall stack of the layers has a detectable interference pattern
in the NIR range of the used setup. The PSi thin films were fabricated using a
current density of 12.7 mA $\rm cm^{-2}$, while the PC was fabricated by
alternating current densities of 12.7 and 50.9 mA $\rm cm^{-2}$. More
information about the synthesis can be found in a previous publication Cencha
_et al._ (2020).

Table 1:  Literature values used in the calculations. The refractive indices are the mean between ${\rm 1150-1650\leavevmode\nobreak\ nm}$ for the RIFTS assessment of porosity and thickness. ${\rm\sigma}$Wu (1970) [${\rm mJ/m^{2}}$] |  ${\rm cos(\theta)}$Xu  _et al._ (2022) |  ${\rm r_{h}}$ [nm] | ${\rm n_{\rm Si}}\cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{green2008self}{\@@citephrase{(}}{\@@citephrase{)}}}$ |  ${\rm n_{\rm air}}$Peck and Reeder (1972) |  ${\rm n_{\rm OS}}$Zhang  _et al._ (2020)
---|---|---|---|---|---
40.7 | 1 | 5.8 | 3.51 | 1 | 1.57

##  IV Setup

Measurements are conducted with a white light reflection spectroscopy setup
(Fig. 1) consisting of a glass fiber bundle reflection probe with 17
illumination and two read fibers. Illumination is provided by a balanced
deuterium-halogen light source (AVALIGHT-DH-S-BAL) with a spectral range of
${\rm 215-2500\leavevmode\nobreak\ nm}$. One read fiber is connected to an UV-
VIS-spectrometer (AvaSpec-2048CL) equipped with a ${\rm 10\leavevmode\nobreak\
\mu m}$ slit, ${\rm 200-1100\leavevmode\nobreak\ nm}$ spectral range and a
spectral resolution of 1 nm. The other to an, via a coax cable synchronized,
NIR-spectromter (AVASPEC-NIR512) with a ${\rm 25\leavevmode\nobreak\ \mu m}$
slit, ${\rm 850-1650\leavevmode\nobreak\ nm}$ spectral range and a spectral
resolution of 4 nm.

Figure 1:  a. Illustration of the measurement setup, starting from the top
consisting of a reflection probe, glass carrier, oligomer film, and sample. b.
Scanning electron micrograph of the sample regions indicated in the
illustration. c. PSi structure with a strong anisotropic pore orientation in
the direction of the etching although the displayed pore walls on the given
scale appear dendritic. d. Simplified representation by a constriction model.
e. Single representative, constricted tube with a dead zone for the fluid
velocity profile during spontaneous imbibition.

##  V Method

The imbibition of oligostyrenes is monitored by in-situ white light
spectroscopy at normal incidence and conducted at room temperature. Before the
experiment PSi samples are cleaned with toluene and allowed to dry. The
oligomers are drop-cast on a glass substrate and fixed in front of the
reflection probe (see Figure 1a). Spontaneous capillary rise is initiated once
the polymer film is brought in direct contact with the sample. The sampling
rates of the spectrometers where chosen to 500 ms for OS-5 and ${\rm
1-5\leavevmode\nobreak\ ms}$ for the lower viscosity OS-2, respectively.
At first, the PSi thin film is infiltrated. The substitution of air by OS with
its higher refractive index increases the effective refractive index ${\rm
n_{\rm eff}}$ of the material, which can be measured by a proportional shift
of the optical thickness (OT) ${\rm e_{\rm ot}}$ according to ${\rm e_{\rm
ot}=2d{\cdot}n_{\rm eff}}$, where d is the physical thickness. Assuming an
endless supply of liquid during the capillary filling, a normalized OT
${\rm{\Delta}e_{\rm ot}(t)}$ can be determined by the values at the starting
and completed filling time ${\rm t_{0}}$ and ${\rm t_{\rm end}}$,
respectively. This normalized OT reflects the filling fraction of the
imbibition. In the case of homogeneous porosity, which implies a homogeneous
$\rm n_{\rm eff}$, through the entire thickness, ${\rm{\Delta}e_{\rm ot}(t)}$
is directly related to the imbibition fluid front position. For other pore
morphologies, ${\rm{\Delta}e_{\rm ot}(t)}$ accounts for the relative pore
volume infiltrated. With the range of layer thicknesses between ${\rm
3-20\leavevmode\nobreak\ \mu m}$, it was found that the OT is determined with
the best signal to noise ratio in the wavelength range of ${\rm
1150-1650\leavevmode\nobreak\ nm}$.

For a single thin film a simple sinusoidal interference pattern would be
measured that undergoes a phase shift during the capillary filling. This can
be tracked by several methods. Because of the stacked structure of PSi thin
film and PC, a superposition of sinusoids is measured. A mathematical
transformation method facilitates differentiating the sinusoids and
attributing them to the respective physical layer. Therefore, a Hann-windowed
fast Fourier transformation (FFT) is used for the analysis of the OT. This
method is known as reflective interferometric Fourier transform spectroscopy
(RIFTS)Sailor (2012). Fig. 2 displays the FFT of an empty and subsequently OS-
filled sample. The three most prominent peaks, indicated by vertical dotted
lines, can be attributed to the PC, the thin film, and the stack of both from
left to right. This is conducted by effective medium approximation with the
orthogonal Maxwell-Garnett model Eq. 3Jalas  _et al._ (2014), where
${\rm\Phi_{i}}$ is the average porosity of the layer i, ${\rm n_{\rm\rm 1/2}}$
is the refractive index of the filling medium and ${\rm n_{\rm\rm Si}}$ the
one of silicon, in conjunction with ${\rm e_{\rm\rm ot}=2d{\cdot}n_{\rm\rm
eff}}$.
The results of the thicknesses are in good agreement with scanning electron
microscopy and the porosities are consistent (Tab. 2). An approach to
measuring the capillary filling would be to follow the particular OT peak in
the FFT over time. This has been found challenging for the PC and thin film
peak, as they are not always apparent, since the thin film peak is too close
to the peak of the overall stack. Here, a major problem arises from windowing,
which increases the width of the peaks, but is also a necessity to reduce
spectral leakage and thereby side lobes of the main peaks. The solution chosen
is to follow the OT of the stack (see Figure S2). An equivalent alternative
technique is to follow the wavelength temporal evolution of a certain peak of
the interference pattern in the spectrum, $\rm\lambda_{\rm peak}(t)$. In this
method, the resulting data series is normalized by initial and final
wavelength. The latter method requires less processing time and because of
that is the chosen one in this work. A comparison of the curves obtained by
both methods can be found in Figure S2.

$\displaystyle{\rm{\Delta}e_{\rm ot}(t)=\frac{e_{\rm ot}(t)-e_{\rm
ot}(t_{0})}{e_{\rm ot}(t_{\rm end})-e_{\rm ot}(t_{0})}}$ (2)

Figure 2:  Fast Fourier transform (FFT) of the initial (blue) and final (red) spectrum in the NIR region from ${\rm 1150-1650\leavevmode\nobreak\ nm}$. For illustration purposes the magnitude of the final curve is shifted by a magnitude of one. The dashed lines indicate the center of the main peaks by full width half maximum (FWHM) peak determination. Start (left) and end (right) are used to solve the orthogonal Maxwell-Garnett effective medium approximation (Eq. 3) for the layers empty and filled state as described in the method section. Thickness and porosity of the layers are listed in Tab. 2.  Table 2:  Thickness ${\rm d_{\rm oMG}}$ and porosity ${\rm\Phi_{\rm oMG}}$ by orthogonal MG using the RIFTS methodSailor (2012) for the measurement displayed in Fig. 2. The thickness is in reasonable agreement with SEM measurements and the equation ${\rm d_{\rm tot}\Phi_{\rm tot}=d_{\rm film}\Phi_{\rm film}+d_{\rm PC}\Phi_{\rm PC}}$ is fulfilled. layer |  ${\rm d_{\rm SEM}}$ [${\rm\mu m}$] |  ${\rm d_{\rm oMG}}$ [${\rm\mu m}$] |  ${\rm\Phi_{\rm oMG}}$ [${\rm\%}$]
---|---|---|---
PC | 3.4 | 3.5 | 71
thin film | 9.3 | 8.3 | 55
stack | 12.7 | 11.9 | 61

$\displaystyle{\rm n_{\rm 1/2}^{2}=\frac{\Phi_{i}\left(n_{\rm
1/2}^{2}\frac{2n_{\rm Si}^{2}}{n_{\rm 1/2}^{2}+n_{\rm Si}^{2}}-n_{\rm
Si}^{2}\right)+n_{\rm Si}^{2}}{\Phi_{i}\left(\frac{2n_{\rm Si}^{2}}{n_{\rm
1/2}^{2}+n_{\rm Si}^{2}}-1\right)+1}}$ (3)

Subsequently the PC is filled by the invading OS. Here the filling of the
central cavity can be measured by the increasing wavelength of its resonance
valley.Cencha  _et al._ (2020) It can be normalized to start and end
analogously to the OT in Eq. 2. Abstractly viewed, the PC acts as a sensor of
the local filling, since its resonance position ${\rm\lambda_{0}}$ is
extremely sensitive to the optical thickness of its central defect layer. This
means that when even a small fraction of air is replaced by an invading
liquid, the microcavity will change the original syntonization wavelength
showing a shift in ${\rm\lambda_{0}}$.

##  VI analysis

In the following, capillary rise in pSi is treated as its structure consists
of an unconnected array of pores, which can be described by a representative
effective pore radius $\rm r_{\rm eff}$. The dynamics of fluids for this kind
of problem are often described or at least compared to the LWE. Since many
studies of more complex liquids and geometries have discovered deviations from
this law, several modified LWE like square-root-of-time laws have been
proposed to adapt.

Cencha et al. Cencha, Urteaga, and Berli (2018) have demonstrated that the
melt infiltration of an ethyl vinyl acetate copolymer into pSi can be
described by taking into account a mean hydraulic radius ${\rm\langle r_{\rm
h}\rangle=2A/P}$, defined by the ratio of pore perimeter P and area of the
cross-section A, and a tortuosity parameter ${\rm\tau_{\rm L}}$ (Eq. 4). The
latter is the elongation factor relating the imbibition length, i.e. the
normal of the materials surface plane and the center of the imbibition front,
to the actual path of a fluid through a meandering pore. In previous
experiments with short chained alkanes a value of 2.6 was determined for this
tortuosity in pSi.Acquaroli  _et al._ (2011)

$\displaystyle{\rm r_{\rm eff}=\frac{\langle r_{h}\rangle}{\tau_{L}^{2}}}$ (4)

In that approach, the concept of tortuosity comes to mathematically explain
the abnormally low effective radius $\rm r_{\rm eff}$ that is required to
describe the experimental results. Actually, $\rm r_{\rm eff}$ represents a
straight cylindrical pore with the same imbibition dynamics. In the present
work, we adopt a different strategy to describe the polymer infiltration in
the mesoporous silicon: we start by defining a pore network that better
represents the observed physical structure (Figure 1d), and then incorporate
corrections that compensate for some nanoscale effects, notably the presence
of a stagnant polymer layer on the pore walls. Calculations are made in the
framework of the sub-continuum approach Kavokine, Netz, and Bocquet (2021),
where continuum equations are still valid, although emerging confinement
effects are to be considered.

More precisely, we consider an array of nanotubes with periodic modulation of
the cross-section, where the pore radius varies between $\rm r_{\rm min}$ and
$\rm r_{\rm max}$ (Figure 1e). Then, momentum and mass conservation equations
are applied to calculate the spontaneous imbibition dynamics. Assuming that
modulations randomly repeat along the tube, the effective radius can be
written as

$\displaystyle{\rm r_{\rm eff}=\frac{1}{\langle r^{-4}\rangle\langle
r^{3}\rangle}}$ (5)

where $\rm{\langle r^{n}\rangle=\frac{1}{h}\int_{\rm 0}^{h}r(z)^{n}\,dz}$ is
the n-th probability moment of the radius distribution, being $h$ the distance
across the porous material Franck  _et al._ (2022). It is worth noting that
Eq. 5 had been previously reported, though with a less general formalism
Sharma and Ross (1991). In particular, for an array of tubes with periodic
step changes of radius, the averages in Eq. 5 are $\langle\rm
r^{n}\rangle=(\rm r^{n}_{\rm min}+r^{n}_{\rm max})/2$ and the effective radius
results,

$\displaystyle\rm{\frac{r_{\rm eff}}{r_{\rm min}}=\frac{4}{{\left(\frac{r_{\rm
min}}{r_{\rm max}}\right)^{4}+\left(\frac{r_{\rm max}}{r_{\rm
min}}\right)^{3}+\frac{r_{\rm min}}{r_{\rm max}}+1}}}$ (6)

This expression clearly shows that the periodically constricted tube yield an
effective radius that can be even lower than that of a uniform cylinder of
radius ${\rm r_{min}}$. Further, for ${\rm r_{max}\textfractionsolidus\rm
r_{min}\gg 1}$, Eq. 6 can be written as follows,

$\displaystyle{\rm\frac{r_{\rm eff}}{r_{\rm min}}\approx 4\left(\frac{r_{\rm
min}}{r_{\rm max}}\right)^{3}}$ (7)

The functionality of Eq. 7 suitably captures the characteristic length scales
governing the capillary filling of mesoporous materials Berli, Mercuri, and
Bellino (2017). In addition, the trend predicted by Eq. 7 has been
experimentally observed in different porous media with bimodal pore size
distributions Dullien, El-Sayed, and Batra (1977); Patro, Bhattacharyya, and
Jayaram (2007).

In the case of the pSi thin film used in this work (Fig. 1), $\rm\langle
r_{h}\rangle=5.8$ nm is obtained from the analysis of SEM images Cencha  _et
al._ (2020). Moreover, imbibition measurements with glycerol indicate that
this structure has $\rm r_{\rm eff}\sim 0.9$ nm, which is consistent with a
tortuosity of 2.6 (Eq. 4).Cencha  _et al._ (2020) Then, by combining Eqs. 4, 5
and 7, it is found that the morphology of this layer can be represented
considering the model of periodically constricted tubes with $\rm r_{\rm
min}=3.5$ nm and $\rm r_{\rm max}=8.5$ nm.

If one further considers the presence of an adsorbed layer of polymer, which
forms a dead zone of thickness $b$ in the flow field (Fig. 1e), then Eq. 6
results

$\displaystyle{\rm\frac{r_{\rm eff}}{r_{\rm
min}}=\frac{4(1-\beta)}{\alpha^{-4}+\alpha^{3}+\alpha^{-1}+1}}$ (8)

where ${\rm\alpha=[(r_{\rm max}\textfractionsolidus r_{\rm
min})-\beta]\textfractionsolidus(1-\beta)}$ and
${\rm\beta=b\textfractionsolidus r_{\rm min}}$ . Figure 3 illustrates the
prediction of Eq. 8, where one readily observes the strong diminution of ${\rm
r_{\rm eff}}$ induced by a uniform layer of stagnant polymer on the pore
walls.

Figure 3:  Theoretical prediction of the effective pore radius ($\rm r_{\rm
eff}$) from Lucas-Washburn dynamics in periodically constricted tubes. The
continuous line shows the dependence of $\rm r_{\rm eff}$ with the pore size
ratio ( $\rm r_{\rm max}\textfractionsolidus r_{\rm min}$, Eq. 6). The dashed
lines show the model results when an immobile layer of size $\rm b$ is
incorporated (Eq. 8). Results are shown for two values of $\rm b$ that fit the
measured behavior of the oligomers OS-2 and OS-5. The vertical full line
stands for the pore ratio (2.43) that characterizes the pSi sample used in the
experiments.

In what follows we use Eq. 1 with this effective radius (Eq. 8) to interpret
the capillary filling curves obtained from styrene oligomers.

Figure 4:  a,c. Contour plots of the capillary filling of pSi samples for OS-2
and OS-5, respectively. The color scale represents the reflectance intensity.
Solid lines show the evolution of one reflectance peak $\rm\lambda_{\rm peak}$
during imbibition while dashed lines indicate the evolution of the microcavity
position $\rm\lambda_{0}$. b. Normalized evolution of $\rm\lambda_{\rm peak}$
and $\rm\lambda_{0}$ as a function of time normalized with $\rm t_{\rm fill}$.
The solid line shows a fitting with a $\rm\sqrt{t}$ dependence.

Figure 4 shows the results obtained for the capillary imbibition of the
oligomers. Figure 4c show a contour map of the evolution of the spectral
reflectance for OS-5. A superimposed dashed line indicates the evolution of
the resonance wavelength of the cavity formed by the PC. The first increase in
wavelength begins almost immediately and is associated with the arrival of the
fluid precursor film to the PC. The second abrupt increase that occurs around
3100s is associated with the arrival of the fluid front to the PC. Figure 4b
shows in dashed lines the evolution of the microcavity normalized position for
measurements carried out with OS-2 and OS-5 in a pSi sample comprised of a
thin film of 9.3 $\mu$m thick. The experimental points obtained for the
evolution of the stack filling fraction are also included in the same figure.
The procedure to determine the time at which the thin film imbibition is
completed and the imbibition of the PC starts ($\rm t_{\rm fill}$), was the
following: first, $\rm t_{\rm fill}$ was determined for OS-5 by identifying
the maximum of the second derivative of $\rm\lambda_{0}(t)$. This filling time
was used to normalize the time axis for OS-5 experimental data
($\rm\lambda_{0}(t)$ $\rm\&$ $\rm\lambda_{\rm peak}(t)$). Then, the time axis
of OS-2 experimental data was normalized with the $\rm t_{\rm fill}$ that best
collapses $\rm\lambda_{\rm peak}(t)$ OS-2 and OS-5 curves. This method is
accurate since the compared imbibition experiments were measured in different
pieces of the same pSi sample, eliminating any possible morphological
variation of the porous structures. It is relevant to note that the filling
fraction corresponding to the thin film (indicated with a horizontal dotted
line in Figure 4b) effectively occurs in coincidence with the arrival of the
fluid front at the PC, which is evident by a slope change in
$\rm\lambda_{0}(t)$. Moreover, the filling fraction of 0.8 matches the
thickness measured through SEM images.

In Figure 4b, a fitting with the LWE (Eq. 1) has been included (solid line) to
show that the capillary filling of both oligomers follows the classical
$\rm\sqrt{t}$ dynamics in the uniform porosity thin film. The analysis
continues by extracting $\rm r_{\rm eff}^{OS-2}$ and $\rm r_{\rm eff}^{OS-5}$
from Eq. 1 with the corresponding $\rm t=t_{\rm fill}$ and $\rm L$ equal to
the thin film thickness. The obtained results are summarized in Tab. 3. They
indicate that $\rm r_{\rm eff}^{OS-2}$ and $\rm r_{\rm eff}^{OS-5}$ are 5.8
and 11 times smaller than the $\rm r_{\rm eff}$ estimated for the glycerol
imbibition, respectively. Using Eq. 8, the immobile layer thickness that is
consistent with that change in $\rm r_{\rm eff}$ in each case can be
estimated. The values obtained in this way are $\rm b_{\rm OS-2}=1.6$ nm and
$\rm b_{\rm OS-5}=1.95$ nm. Quantum mechanical molecular size estimations
resulted in ${\rm 0.9-1.2\leavevmode\nobreak\ nm}$ for OS-2 and ${\rm
1.4-1.8\leavevmode\nobreak\ nm}$ for OS-5.Jensen and Kromann (2013) The ratio
of these sizes of 1.5 is comparable to the ratio of the immobile layer of 1.2.
A way to interpret the immobile layer is to consider ${\rm 1-3}$ densely
packed molecular layersZhang  _et al._ (2021), with molecules stretched in the
direction of drag and a slow diffusion at the wall. Note that the existence of
such an hydrodynamically dead layer agrees with findings on other (simpler)
molecular liquids Gruener  _et al._ (2009); Vincent, Szenicer, and Stroock
(2016); Gruener and Huber (2019).

It is interesting to compare these results with the tortuous pore model (Eq.
4), which requires $\tau$=6.3 for OS-2 and $\tau=$8.6 for OS-5 to predict the
above effective radii. Beyond the extremely high values, $\tau$ results are
different for each measurement, which is not compatible with the fact that the
morphology of the thin film is the same in both cases.

Table 3:  Results: The decelerated dynamics in capillary filling can be illustrated by apparent viscosities ${\rm\mu_{\rm app}}$, six and eleven fold higher than the bulk for OS-2 and OS-5, respectively. In the constriction model the different capillary rise dynamics of the oligomers is explained by the immobile layer thickness $b$. The squared fit parameter "a" of the main meniscus can be compared to the diffusion-like coefficient ${\rm D_{\rm PF}}$ of the precursor film spreading. Additionally, a thickness ${\rm d_{\rm PF}}$ for the precursor film is derived. |  ${\rm\mu_{\rm app}}$ [${\rm mPa\leavevmode\nobreak\ s}$] |  ${\rm b}$ [${\rm nm}$] |  ${\rm a^{2}}$ [${\rm m^{2}/s}$] |  ${\rm D_{\rm PF}}$ [${\rm m^{2}/s}$] |  ${\rm d_{\rm PF}}$ [${\rm nm}$]
---|---|---|---|---|---
OS-2 | 54 | 1.60 | ${\rm 3.3{\cdot}10^{-10}}$ | - | -
OS-5 | ${\rm 3.7{\cdot}10^{5}}$ | 1.95 | ${\rm 5.1{\cdot}10^{-14}}$ | ${\rm 9.4{\cdot}10^{-13}}$ | 0.9

Fig. 5 displays the fluid front shapes of OS-2 and OS-5 for two samples each,
with different porous layer thicknesses in front of their PC. Thinking in the
chronological sequence in the experiment the figure must be read from right to
left, since it has been re-scaled in space and time by ${\rm
x_{0}/(a\sqrt{t})}$. Here ${\rm x_{0}}$ stands for the middle of the PC and
"a" is the measured coefficient for capillary rise in the preceding thin film.
We calculated ${\rm x_{0}}$ for the abscissa position of one, by the point in
time where the filling fraction is converging to a value of one. It can be
noted, that the fluid front shape for the individual oligomers maintains their
shape independent of the thin film layer thickness. OS-2 shows a sharp front
whereas OS-5 has a broad slope, i.e. a precursor film, starting at a value
${\rm x_{\rm PF}}$ around 4.3 of the abscissa. The onset of the precursor film
is determined by the intercept of a linear fit of the constant initial region
and the tangent where the filling fraction starts to increase.
A diffusion like coefficient for the precursor film spreading can be derived
by ${\rm D_{\rm PF}=(\tau\leavevmode\nobreak\ a\leavevmode\nobreak\ x_{\rm
PF})^{2}}$. For the experiments displayed in Fig. 5 a mean of ${\rm 9.4\cdot
10^{-13}\leavevmode\nobreak\ m^{2}/s}$ is obtained. Similar to MD simulations
on nanoporesChibbaro  _et al._ (2008) the spreading dynamics of the precursor
film follows a faster ${\rm\sqrt{t}}$-law than the main meniscus, as can be
seen by comparing the squared prefactor of the capillary rise in Tab. 3. The
value for the main meniscus of OS-5 is about twenty times smaller than its
precursor film, which in turn is more than two orders of magnitude smaller
than the capillary rise of OS-2. The latter explains why there is no precursor
film observed for OS-2, even if one assumes a faster precursor film formation
for this shorter oligomer. If one assumes that ${\rm D_{\rm PF}}$ scales
similarly as a function of the degree of polymerization $N$ as in polymer
surface diffusion, where Sukhishvili et al. found a proportionality of ${\rm
D\sim N^{-3/2}}$, there is barely any increase in ${\rm D_{\rm PF}}$ expected
for OS-2 compared to OS-5 Sukhishvili  _et al._ (2002).
The end of the precursor film region is determined by deviation from a linear
fit of the meniscus region by more than the standard deviation in 10
consecutive points. Taking the difference of the filling fraction limiting the
precursor region, its maximal filling fraction ${\rm\Phi_{\rm PF}}$ can be
assessed. A simple tube filling model starting from the pore wall leads to an
approximation of the precursor thickness by ${\rm\Phi_{\rm PF}=\frac{2\langle
r_{h}\rangle d_{\rm PF}-d_{\rm PF}^{2}}{\langle r_{h}\rangle^{2}}}$, resulting
to ${\rm d_{\rm PF}\approx 0.9\leavevmode\nobreak\ nm}$.

Figure 5:  Filling fraction of the photonic crystal with OS-2 and OS-5 versus
a spatio-temporal rescaled variable. "a" is the prefactor of a square-root-of-
time fit of the filling dynamics of the thin film in front of the PC. The
filling time $t$ of the PC is rescaled with respect to its central position
${\rm x_{0}}$, which is adjusted for same starting positions of menisci of the
respective oligomer. The thickness $d$ in the legend belongs to the pSi layer
in front of the PC. It is measured by SEM.

##  VII conclusion

The combined thin-film interference and photonic crystal analysis of pSi
during liquid imbibition allows the detection of capillary rise and precursor
film dynamics in the same experiment. Especially for polymeric liquids, the
necessity to measure capillary rise directly becomes clear, as they exhibit
different fluid dynamics when the pore size of the scaffold becomes comparable
to the molecule size. In the experiments performed with OS-2 and OS-5, a
strong slow-down was observed. This can be explained by an apparent viscosity
of about six and eleven times the bulk value, respectively. We, however,
explain this by the increasing importance of local undulations in the pore
radius with the molecular size of the infiltrating liquid, but otherwise
unchanged bulk fluid parameters. Indeed, a simplified constriction model with
a dead zone consistently describes imbibition in the pSi thin films for both
oligomers. This model is superior to the concept of a modified LWE with a
tortuosity by having a consistent geometrical description. Moreover, the
tortuosities were found to be unreasonably high, suggesting much more
meandering pores than can be deduced from SEM images.

In the present study, the constriction model gives a very good description of
the capillary rise dynamics. Even in the framework of the semi-continuum
approach, the model relates the measured (macroscopic) variables to parameters
that characterize the most important nanoscale features, namely the thickness
of the hydrodynamic dead zone. Furthermore, the predicted values are in
reasonable agreement with estimates of the molecular size of the oligomers.

Following the resonance wavelength of the PC, filling phenomena can be
resolved with sub-nanometer precision. The coefficient of capillary rise
dynamics determined in the same experiment is used to visualize the shape of
the fluid front. For OS-2 the fluid front is flat, while for OS-5 a precursor
film is detected. We attribute this to the fact that the pore surface
diffusion dynamics in OS-5 are fast enough to allow a thin film to move beyond
the main capillary front. In contrast, the fast imbibition rate of OS-2
precludes precursor film formation. We also show that our opto-fluidic
technique allows to measure the maximum thickness and a surface diffusion-like
coefficient of the OS-5 precursor film with remarkably high spatio-temporal
resolution.

From an application point of view, the parameters obtained by the presented
method could be used to rationally design polymer nanotubes by melt
infiltration. Moreover, we envision that our experimental platform can be
employed to determine the surface diffusion coefficients of polymers, which is
generally a challenging task.

###### Acknowledgements.

The authors acknowledge the funding support received from the following
institutions: Deutsche Forschungsgemeinschaft (DFG, German Research
Foundation) Project number 519853330, CRC 986 "Tailor-Made Multi-Scale
Materials Systems" Project number 192346071, ”Dynamic Electrowetting at
Nanoporous Surfaces: Switchable Spreading, Imbibition, and Elastocapillarity”,
Project number 422879465 (SPP 2171), Consejo Nacional de Investigaciones
Cientificas y Tecnicas, grant PIP-2020-1049, Universidad Nacional del Litoral,
grant CAID 2020-50620190100114L.

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