# Space-resolved dynamic light scattering within a millimetric drop: from
Brownian diffusion to the swelling of hydrogel beads

Matteo Milani1 Ty Phou1 Guillame Prevot1 Laurence Ramos1
<EMAIL_ADDRESS>Luca Cipelletti1,2
<EMAIL_ADDRESS>1Laboratoire Charles Coulomb (L2C), Université
Montpellier, CNRS, Montpellier, France
2Institut Universitaire de France, Paris, France

###### Abstract

We present a novel dynamic light scattering setup to probe, with time and
space resolution, the microscopic dynamics of soft matter systems confined
within millimeter-sized spherical drops. By using an ad-hoc optical layout, we
tackle the challenges raised by refraction effects due to the unconventional
shape of the samples. We first validate the setup by investigating the
dynamics of a suspension of Brownian particles. The dynamics measured at
different positions in the drop, and hence different scattering angles, are
found to be in excellent agreement with those obtained for the same sample in
a conventional light scattering setup. We then demonstrate the setup
capabilities by investigating a bead made of a polymer hydrogel undergoing
swelling. The gel microscopic dynamics exhibit a space dependence that
strongly varies with time elapsed since the beginning of swelling. Initially,
the dynamics in the periphery of the bead are much faster than in the core,
indicative of non-uniform swelling. As the swelling proceeds, the dynamics
slow down and become more spatially homogeneous. By comparing the experimental
results to numerical and analytical calculations for the dynamics of a
homogeneous, purely elastic sphere undergoing swelling, we establish that the
mean square displacement of the gel strands deviates from the affine motion
inferred from the macroscopic deformation, evolving from fast diffusive-like
dynamics at the onset of swelling to slower, yet supradiffusive,
rearrangements at later stages.

††preprint: APS/123-QED

##  I Introduction

Soft matter systems comprising colloidal particles, emulsion drops, foam
bubbles and polymer and surfactant molecules are ubiquitous in industrial
processes and products and are actively studied in fundamental research [1].
The interest in soft systems owes to their high susceptibility to even modest
external stimuli, and to their typical structural length scale comparable to
the wavelength of visible light, which imparts them interesting optical
properties and makes them suitable for powerful experimental techniques such
as optical microscopy and light scattering. Furthermore, the tunability of the
interactions between the building blocks of soft materials allows for a large
variety of structures, resulting in a wide spectrum of physical, mechanical,
and optical properties.

Research in soft matter initially focused mostly on systems at thermodynamic
equilibrium, while in the last decades the investigation of out-of-equilibrium
systems, including active matter, has tremendously gained momentum [2, 3].

Soft matter, both at and out of equilibrium, is typically investigated using
bulk samples: the behavior of soft systems confined or manufactured in
specific shapes, such as in thin films or spherical drops, has received
comparatively less attention. However, these systems typically have
distinctive features as compared to their bulk counterparts, due to the larger
surface-to-volume ratio and specific boundary conditions, leading to
fascinating behavior and remarkable structural and use properties.

One prototypical example is provided by drops of colloidal suspensions, whose
drying has been investigated in fundamental studies [4, 5, 6, 7, 8], in view
of its interest in biology, e.g. in virus survival [9] or blood desiccation
[10], as well as for applications [11], including the design of functional
supraparticles [12, 13]. Another example of popular spherical soft systems are
hydrogel beads. They are widespread in applications ranging from food science
[14, 15] to the adsorption of dyes [16], acids [17] or metals [18], to drug
delivery [19].

The behavior of drop- or bead-like samples is often intimately related to
their very shape and to the absence of interactions with a sample container,
in contrast to conventional measurements on bulk materials. For instance,
polymeric hydrogels exhibit a wide variety of behaviors such as shrinkage upon
gelation, [20], slow syneresis after gelation [21], swelling when immersed in
a good solvent [22, 23, 24, 25, 26], which are different in a bulk sample and
in beads. Similarly, the interactions with a substrate were shown to play a
key role in the evolution of drying drops of colloidal suspensions [27, 5].

While there is a clear interest in characterizing the structure and dynamics
of spherically confined soft systems, the available experimental tools are far
more limited than for bulk samples. Methods used in previous works include low
magnification video imaging and shadowgraphy [4, 28, 26], imaging of a portion
of the drop by confocal microscopy [13, 29], post-mortem electron microscopy
[4, 30, 28], synchrotron X-ray scattering [31, 32], and nuclear magnetic
resonance (NMR) [33, 25]. These methods provide valuable information, however
they are restricted to structural quantities, they sometimes lack spatial
resolution (e.g. for x-ray scattering) and may be restricted to some portions
of the drop (e.g. in confocal micorscopy) or give no access to the system
evolution (as in post-mortem electron microscopy).

Measurements of the dynamics appear as an appealing alternative or complement
to structural studies. Dynamic light scattering (DLS) [34] is a well-
established and popular method for measuring the microscopic dynamics of soft
systems. However, in conventional DLS intensity correlation functions, which
quantify the microscopic dynamics, are obtained by extensively averaging over
time a signal issued from the whole sample volume illuminated by the incident
laser beam. In other words, conventional DLS lacks both space and time
resolution, two key features for investigating spherically confined systems,
where boundary conditions (and hence distance to the surface) play a key role,
and where time-dependent phenomena, such as evaporation or swelling are
ubiquitous.

Photon Correlation Imaging (PCI) [35] is a variant of conventional DLS that
affords both spatial and temporal resolution. It has been used to characterize
the dynamics of bulk samples, both spontaneous [35, 36, 37, 38] and under a
mechanical drive [39, 40, 41]. Very recently, we have reported a study of the
drying of drops of colloidal suspensions based on PCI [8]. Here, we present a
detailed description, validation and demonstration of the PCI setup used
therein.

The rest of the paper is organized as follows: in Sec. II we introduce the PCI
setup, discuss how to account for refraction effects associated with the
spherical shape of the sample, define the quantities introduced to
characterize the microscopic dynamics, and present measurements of the size of
the scattering volume and the setup stability. In Sec. III we validate the
setup and the data analysis procedure by measuring diluted colloids undergoing
Brownian motion. Section IV deals with the microscopic dynamics of a polymer
gel upon swelling. We first discuss theoretically the behavior expected for
uniform, affine swelling, a benchmark against which experimental data are then
compared. Our findings for the polymer gel are compared to previous works in
Sec. IV.4. Finally, Sec. V provides some concluding remarks. Details on some
calculations and data analysis methods are discussed in Appendixes A-C.

##  II Photon Correlation Imaging setup

###  II.1 Optical layout

Figure 1:  (a) Scheme of the dynamic light scattering setup optimized for
spherical, millimiter-sized samples. The incoming laser beam propagates along
the $x$ axis and is mildly focused by the lens $L_{1}$ in the middle of the
sample. The lens $L_{2}$ forms an image of the scattering volume on the sensor
of a CMOS camera. By combining a vertical slit, parallel to the $z$ direction,
and a diaphragm, we ensure that the image is formed by scattered light exiting
the drop along the $y$ direction, to within acceptance angles of $0.55$ deg
and $1.65$ deg in the $(x,y)$ and $(y,z)$ planes, respectively. The sample is
placed in a cubic glass box of side $5$ cm. Insets: schematic top view of the
relevant optical rays for a water-based spherical sample surrounded by water,
air and oil, respectively. Due to refraction, the scattering angle
$\theta_{s}$ for light exiting the drop along the $y$ direction depends on the
$x$ coordinate of the scatterer. (b) Scattering angle $\theta_{s}$ as a
function of $x_{p}$, the $x$ coordinate of the point $P$ on the bead surface
shown in the insets of a). Data for a bead of radius $1$ mm surrounded by air,
oil or water, as shown by the labels. (c) Difference $\Delta x=x_{P}-x_{S}$ as
a function of $x_{P}$, where $x_{S}$ is the abscissa of the point $S$ from
which a scattering ray originates, and $x_{P}$ is the abscissa of the point
$P$ where the scattered ray intersects the surface of the sphere. The curves
refer to the case of a bead of $1$ mm radius surrounded by air, oil or water,
see legend in b).

Figure 1a shows schematically the light scattering setup, specifically
designed to probe with space and time-resolution the microscopic dynamics
within millimeter-sized spherical samples. Due to the small radius of
curvature of the sample, whenever the refractive index of the solvent in the
drop, $n$, is different from that of the surrounding medium, refraction
effects considerably complicate the definition of the scattering angle. We
minimize these complications by illuminating the sample with a thin beam
impinging perpendicularly to the drop surface, such that the incoming beam is
essentially not refracted, and by collecting light exiting the sample along a
single, well-defined direction.

The light source is a single-mode continuous wave laser (Verdi V-2 by
Coherent) operating at an in-vacuo wavelength $\lambda_{0}=532.5$ nm with a
maximum power of $2$ W. A first lens ($L_{1}$ in Fig. 1a, with diameter $65$
mm and focal length $64$ mm), collimates the beam down to a beam waist
$w_{0}=59.5\pm 0.1~{}\mu$m, much smaller than the typical drop radius (see
Sec. II.4 for the measurement of $w_{0}$). Thus, the scattering volume is a
thin cylinder centered around a drop diameter, along the $x$ axis direction.
To achieve spatial resolution, the scattered light is not collected in the far
field, as in conventional scattering, but rather by forming an image of the
scattering volume on the detector of a CMOS camera, using lens L2 (diameter
$40$ mm, focal length $82$ mm). To insure that only light propagating along
the direction of the $y$ axis is collected by the camera, we place a slit and
a diaphragm in the back focal plane of L2, which reject scattered light
propagating at angles larger than $\theta_{yz}=1.65$ deg and
$\theta_{yx}=0.55$ deg with respect to the $y$ axis, in the $(y,z)$ and
$(x,y)$ planes, respectively. Note that the drop and the surrounding medium
are contained in a cubic sample chamber with walls perpendicular to the $x$
and $y$ directions, so as to avoid refraction effects at the chamber-air
interfaces. Finally, by taking the image of a ruler placed at the sample
position, we determine the system magnification to be $M=2.11\pm 0.02$. The
combination of the illumination and collection optical layout described above
allows for associating a well defined scattering angle $\theta_{s}$, and thus
a well defined magnitude of the scattering vector $q=4\pi
n\lambda_{0}^{-1}\sin(\theta_{s}/2$), to each point within the scattering
volume, as we will detail in Sec II.2.

Depending on the sample dynamics, we use either a fast camera (Phantom Miro
M310 by AMETEK Vision Research, e.g. for the Brownian dynamics described in
Sec. III) or a conventional camera (acA2000 340km by Basler, e.g. for the
polymer gel of Sec. IV). The fast camera is run at $19000$ fps, for a typical
run duration of 60 ms. The size of the acquired images is $896$ pixels $\times
184$ pixels, and the pixel size is $20\times 20~{}\mu\mathrm{m}^{2}$. By
contrast, to measure slow dynamics we use a CMOS camera (acA2000 340km by
Basler). In this case the collected images have a full frame size of $2048$
pixels $\times 1088$ pixels and a pixel size of $5.5$ $\mu$m. To allow for
faster image acquisition, we typically acquire only a fraction of the full
frame, selecting a region of interest of $2048$ pixels $\times 512$ pixels.
Here the acquisition is taken using the variable delay scheme of Ref. [42],
with shortest delay 1 s.

###  II.2 Refraction effects

Due to its spherical shape, the sample itself acts as an optical component of
the setup. As shown in the insets of Fig. 1a, the scattered light propagating
towards the detector is refracted at the interface between the sample and the
external medium, whenever the index of refraction of the surrounding medium is
different from that of the sample. As a consequence, each ray emerging from
the sample and directed towards the detector is associated with a particular
scattering angle $\theta_{s}$, such that there is an intrinsic coupling
between position and scattering vector
$\textbf{q}=\textbf{k}_{o}-\textbf{k}_{i}$, where $\textbf{k}_{i}$ is the wave
vector of the incident beam and $\textbf{k}_{o}$ that of the scattered light.

We now determine the components and magnitude of q as a function of the
coordinates $(x_{P},y_{P},z_{P}$) of a point $P$ at the intersection between a
scattered ray collected by the camera and the drop surface, where we take the
drop center as the origin of a Cartesian reference frame. The incident wave
vector is $\textbf{k}_{i}=2\pi n\lambda_{0}\hat{e}_{x}$, with $\hat{e}_{x}$
the unit vector along the x axis. Thanks to the combination of the slit and
diaphragm, only scattered light propagating along $\hat{e}_{y}$ after leaving
the drop contributes to the image formed on the camera detector. The
propagation direction of the scattered light within the drop,
$\hat{k}_{o}=\textbf{k}_{o}/k_{o}$, is found by imposing Snell’s law at the
drop-surrounding medium interface. By introducing the unit vector pointing at
$P$,
$\hat{r}_{P}=(x_{P}\hat{e}_{x}+y_{P}\hat{e}_{y}+z_{P}\hat{e}_{z})/\sqrt{x_{P}^{2}+y_{P}^{2}+z_{P}^{2}}$,
after some tedious but straightforward calculations, one finds

$\hat{k}_{o}=a\hat{r}_{P}+b\hat{e}_{y}\,,$ (1)

with $a$ and $b$ two prefactors that depend on the position of $P$ and the
refractive index mismatch between the drop and the surrounding medium, and
whose expression is given in Eqs. 27 of Appendix A. Once $\textbf{k}_{o}$ is
determined, the scattering vector and the scattering angle are calculated
according to

$\textbf{q}=\textbf{k}_{o}-\textbf{k}_{i}\,,$ (2)
$\theta_{s}=\arccos(\hat{k}_{o}\cdot\hat{e}_{x})\,,$ (3)

where we have used $\hat{k}_{i}=\hat{e}_{x}$. Figure 1b shows the scattering
angle $\theta_{s}$ as a function of $x_{P}$, for a water-based sample
($n=1.33$) in a one-millimeter drop surrounded by air ($n_{air}=1$), or by the
oil used for the experiments of Sec. III, $n_{oil}=1.403$. Note that the trend
is opposite for air and oil as a surrounding medium, and that the deviations
with respect to $\theta_{s}=90$ deg, the scattering angle for a ray
originating from the center of the drop, can be as large as $30$ deg for a
water-in-air drop. The corresponding scattering vectors have magnitude
$14\mu$m${}^{-1}\leq q\leq 28\mu$m-1, with $q\approx 22\mu$m-1 for
$\theta_{s}=90$ deg. Since DLS techniques are sensitive to relative
displacements of the order of $1/q$, our setup is sensitive to motion on
length scales of the order of a few tens of nanometers.

Another consequence of refraction at the drop surface is the fact that the
coordinate $x_{P}$ where a scattered ray intersects the drop surface differs
from the coordinate $x_{S}$ along the scattering volume from which that
scattered light originates, see also Fig. 4b. Note that $x_{P}$ is directly
obtained from the position on the image collected by the CMOS camera, but the
desired, physically relevant quantity is the position in the sample, i.e.
$x_{S}$. Assuming for simplicity that the incident beam is infinitely thin,
i.e. that the scattering originates from a point $S$ on the $x$ axis, and
using $\textbf{OS}=\textbf{OP}-\textbf{SP}$, one finds

$x_{S}=x_{P}+\frac{\hat{k}_{ox}}{|1-\hat{k}_{ox}|}\
\sqrt{y_{P}^{2}+z_{P}^{2}}\,,$ (4)

with $\hat{k}_{ox}=\hat{k}_{o}\cdot\ \hat{e}_{x}$. Figure 1c shows $\Delta
x=|x_{S}-x_{P}|$, as a function of $x_{P}$: deviations as large as 15% may be
observed for a water drop in air. In the following, we systematically use the
coordinate $x_{S}$ when showing space-dependent data.

###  II.3 Quantifying the dynamics: multispeckle intensity correlation
functions

To quantify the sample dynamics we take a time series of images of the speckle
pattern imaged onto the CMOS detector. Each image is divided into several
Regions of Interest (ROIs), typically rectangles of dimensions $(30\times
100)$ pixels2, corresponding to about $(70\times 260)$ $\mu$m2 in the sample.
The size of the ROIs is chosen so that each ROI contains around $100$
speckles, which provides an acceptable statistical noise [43], while keeping
the ROI size small enough to obtain about 10 ROIs over the bead diameter. The
local dynamics within a given ROI is quantified by a two-time degree of
correlation [44]:

$c_{I}(t,\tau,x)=\alpha\frac{\langle I_{p}(t)I_{p}(t+\tau)\rangle_{x}}{\langle
I_{p}(t)\rangle_{x}\langle I_{p}(t+\tau)\rangle_{x}}-1$ (5)

with $x$ the position along the diameter of the sample, $\tau$ the delay
between images, $I_{p}(t)$ the intensity of the $p$-th pixel of the ROI at
time $t$, and $\langle\cdot\cdot\cdot\rangle_{x}$ an average over all pixels
within a ROI centred around $x$. The multiplicative factor $\alpha$ is chosen
such that the intensity correlation function $g_{2}(\tau,x)-1$, obtained by
averaging $c_{I}(t,\tau,x)$ over a suitable time interval, tends to $1$ for
$\tau\to 0$.

For a dilute suspension of identical spheres of radius $a$ undergoing Brownian
motion, the intensity correlation function is a simple exponential decay:

$g_{2}(\tau)-1=\exp(-2q^{2}D_{0}\tau)=\exp\left(-2\frac{\tau}{\tau_{E}}\right)$
(6)

where $\tau_{E}=(q^{2}D_{0})^{-1}$, with $D_{0}=\frac{k_{B}T}{6\pi\eta a}$ the
diffusion coefficient given by the Stokes-Einstein relation, with $k_{B}$
Boltzmann’s constant, $T$ the temperature, and $\eta$ the solvent viscosity.
More generally, $g_{2}-1$ may be conveniently expressed by a stretched or
compressed exponential:

$g_{2}(\tau)-1=\exp\left[-2\left(\frac{\tau}{\tau_{E}}\right)^{\beta}\right]\,,$
(7)

where $\tau_{E}$ is the relaxation time of the coherent intermediate
scattering function $f_{coh}\propto\sqrt{g_{2}-1}$ [34]. The exponent $\beta$
depends on the sample type and dynamics. For Brownian diffusion of
monodisperse particles, as in Eq. 6, $\beta=1$, while for a sample exhibiting
a broad distribution of relaxation times, e.g. a polydisperse suspension or a
supercooled colloidal fluid, $\beta\leq 1$ (stretched exponential) [45]. By
contrast, compressed exponential decays, $\beta\geq 1$, are typically measured
for dynamics associated to the stress relaxation of jammed materials [46]. The
average relaxation time of a stretched or compressed exponential decay may be
defined as

$\tau_{D}=\int_{0}^{\infty}\exp\left[-2\left(\frac{\tau}{\tau_{E}}\right)^{\beta}\right]\textrm{d}\tau=\frac{\tau_{E}}{\beta
2^{1/\beta}}\Gamma\left(\frac{1}{\beta}\right)\,,$ (8)

with $\Gamma(x)$ the Gamma function. In particular, for the simple exponential
decay case of Eq. 6, $\beta=1$ and hence $\tau_{D}=\tau_{E}/2$.

###  II.4 Setup characterization

As an essential step before any further measurements, we characterize our
setup concerning two points that are important for the interpretation of the
data presented in the following: the size of the scattering volume and the
mechanical stability of the apparatus.

Dynamic light scattering is sensitive to the relative motion of the scatterers
within the scattering volume, $V_{S}$. In most cases, detailed knowledge of
the geometry of the scattering volume is not necessary, e.g. for the Brownian
particles of Sec. III, whose relative displacement is independent of $V_{S}$.
However, there are situations where the dynamics do depend on the size and
shape of $V_{S}$, e.g. for the beads of polymer gel of Sec. IV undergoing a
swelling strain $\epsilon$, where the relative motion between portions of the
gel separated by a distance $L$ scales as $\epsilon L$. In our setup, the
scattering volume is a thin “cylinder” centered around the $x$ axis. More
precisely, the incoming beam cross section has a Gaussian profile,
characterized by the beam waist $w_{0}$, the distance from the $x$ axis over
which the incident intensity decays by a factor $e^{-2}$ [47]. In order to
measure $w$ precisely, we replace the sample by a frosted glass positioned
perpendicularly to the incoming beam, at $x=0$. We place a lens with focal
length $f=40$ mm on the $x$ axis, right after the ground glass, and use a CMOS
camera placed in the focal plane of the lens to take an image of the speckle
pattern generated by the ground glass. Under these conditions, the speckle
size is directly related to $w$ through the spatial auto-correlation function
of the speckle pattern, $C_{I}(\Delta x,\Delta y)=\langle
I(x_{c},y_{c})I(x_{c}+\Delta x,y_{c}+\Delta y)\rangle$, with $I(x_{c},y_{c})$
the intensity measured at position $(x_{c},y_{c})$ on the detector. Indeed,
for a sample illuminated by a Gaussian beam with beam waist $w_{0}$ and
wavelength $\lambda$, the space-varying part of the auto-correlation function
measured at small angles reads [47]:

$C_{I}(\Delta x,\Delta
y)\propto\exp\left[-\frac{\pi^{2}w_{0}^{2}}{2f^{2}\lambda^{2}}(\Delta
x^{2}+\Delta y^{2})\right]\,.$ (9)

By fitting Eq. 9 to the measured $C_{I}(\Delta x,\Delta y)$, we find
$w_{0}=59.5\pm 0.1$ $\mu$m.

Figure 2: Intensity correlation functions for a glass cylinder (purple
diamonds), a colloidal glass bead (green crosses), and a quiescent bead of
polymer gel (light gray triangles). Solid lines are a fit of the slow decay to
Eq. 7. For the final decay of $g_{2}-1$, for the glass cylinder,
$\tau_{E}=434803$ s and $\beta=1$, for the colloidal glass $\tau_{E}=28036$ s
and $\beta=1.24$, and for the polymer gel $\tau_{E}=2649$ s and $\beta=1.37$.

Mechanical stability is another important issue, since DLS is sensitive to
displacements down to a few nanometers and the peculiar geometry of our
samples makes it difficult to immobilize them as firmly as for a standard
cuvette. Furthermore, samples like the gels of Sec. IV may have relaxation
times as long as several tens of seconds, making the requirement on mechanical
stability quite stringent. We first probe the overall stability of the
apparatus for a tightly fixed sample. We use a cylindrical piece of glass ($1$
cm diameter and $7$ cm height) firmly held by a clamp. The glass contains
frozen impurities that scatter sufficiently for $g_{2}-1$ to be measurable. We
plot in Fig. 2 the correlation function for a ROI in the center of the
cylindrical sample, measured during an experiment lasting ten days. By fitting
the final relaxation of $g_{2}-1$ to Eq. 7 and using Eq. 8 we obtain a decay
time $\tau_{D}=217401$ s (2.3 days), which demonstrates the excellent overall
mechanical stability of the apparatus.

We then repeat stability tests for a sample configuration similar to that in
experiments on spherical, water-based samples. The green crosses in Fig. 2
show $g_{2}-1$ for a bead of colloidal glass, prepared by depositing $10\mu$L
of an aqueous suspension of commercial silica nanoparticles (Ludox TM-50 from
Sigma Aldrich) on a hydrophobic surface immersed in silicon oil (47 V 100, VWR
chemicals). The oil has a small miscibility with water; thus, water in the
suspension slowly diffuses into the oil, decreasing the drop radius from 3.94
mm to 3.31 mm. Accordingly, the particle volume fraction grows from an initial
value $\varphi_{i}=0.31$ to $\varphi_{f}=0.52$, well within the glassy regime
for this sample [48]. Since there may still be some ultra-slow relaxations in
the bead of colloidal glass, the decay time obtained from a fit to the data,
$\tau_{D}=14956$ s, provides a lower bound for the slowest dynamics that can
be reliably measured on a drop sample. The compressing exponent $\beta=1.24$
is indicative of dynamics stemming from the relaxation of internal stresses,
as reported for the same system studied in the bulk [48] and for other jammed
or glassy soft materials [49, 50, 51].

Finally, the grey triangles in Fig. 2 show the $g_{2}-1$ function for a
quiescent bead of the polymer gel investigated in Sec. IV. In this case, the
surrounding medium is water-saturated oil, to prevent any changes of the drop
size. The measured dynamics are thus due to the spontaneous relaxation of the
gel, and are expected to be slower than the driven dynamics during swelling.
We find $\beta=1.37$, larger than one, as also reported in the past for the
slow dynamics of colloidal and polymer gels [46, 52]. The relaxation time is
$\tau_{D}=1460$ s, faster than that of the colloidal glass. This proves that
the quiescent dynamics of the polymer gel reported here, as well as the faster
dynamics during swelling discussed in Sec. IV, are not affected by any
artifact due to mechanical instabilities.

##  III Brownian particles in a drop

We first test our setup on a drop of a diluted suspension of Brownian
particles, whose dynamics are well known and can be easily measured in a
conventional DLS setup, for comparison. By running experiments that last a
fraction of a second, we insure that the drop radius does not change during
the measurements, regardless of the surrounding medium. Thus, the dynamics are
due solely to Brownian diffusion and should be isotropic and spatially
homogeneous. However, as discussed in Sec. II.2, the relaxation time of
$g_{2}-1$ is still expected to depend on the position of the ROI for which the
intensity correlation function is calculated, due to refraction effects that
lead to a spatial dependence of the scattering angle, see Fig. 1b. In the
following, we shall demonstrate that $g_{2}-1$ is fully consistent with data
obtained for the same sample using a conventional setup. Moreover, we will
show that the space dependence of the intensity correlation function is fully
accounted for by the $x$-dependence of the scattering vector $\mathbf{q}$
stemming from refraction effects.

###  III.1 Materials

The Brownian sample is a suspension of polystyrene particles from
Microparticles GmbH (diameter $2a=105$ nm), at a concentration $4\times
10^{-3}$ $\%$ w/v. The solvent is a $80/20$ w/w water/glycerol mixture, with
refractive index $n=1.3585$ [53], viscosity $\eta=2.3\cdot 10^{-3}$ Pa s, and
density $\rho=1048~{}\mathrm{kg~{}m^{-3}}$ that nearly matches that of the
particles [54]. A suspension drop of volume $30$ $\mu$L (radius $R_{0}=1.9$
mm) is gently deposited on a hydrophobic surface (contact angle $120$ deg). To
prepare the hydrophobic surface, we deposit on a microscope glass slide a
50/50 v/v water/ethanol solution of trimethoxy-(octadecyl)-silane ($20$
mg/mL). The glass slide is then heated in an oven, raising linearly the
temperature $T$ from $30^{\circ}$C to $70^{\circ}$C in 1 h, followed by $12$ h
at constant $T=70^{\circ}$C. To test refractive effects with opposite trends,
we use either air ($n_{m}=1$) or silicon oil (47 V 100 from VWR chemicals,
$n_{m}=1.403$) as the surrounding medium. The setup has no temperature
control, the room temperature is controlled and measured to be $T=22.0\pm
0.1^{\circ}$C, where the quoted uncertainty is that of the temperature probe.
Sequences of speckle images lasting 60 ms are acquired using a fast camera,
see Sec. II.1, and processed to calculate $g_{2}-1$.

###  III.2 Experimental results

Figure 3:  Intensity correlation functions for a drop of diluted suspension of
Brownian particles surrounded by air (a) and oil (b), respectively. Symbols
are experimental data and solid lines are fits to the data with a single
exponential decay, Eq. 6. (c) Decay time of $g_{2}-1$ as a function of
position inside the drop, with $x_{S}=0$ the center of the drop. (d) Inverse
of the decay time of $g_{2}-1$ as a function of $q^{2}$. The dotted line shows
$1/\tau_{D}=2D_{0}q^{2}$, as predicted for dilute particles undergoing
Brownian diffusion, with the diffusion coefficient $D_{0}$ independently
measured by conventional dynamic light scattering at $\theta_{S}=90$ deg on
the same suspension, but contained in a standard DLS cuvette. In c) and d),
error bars are estimated from the covariance matrix of the fit.

We plot in Fig. 3a the intensity correlation functions for a drop in air. Data
are acquired for different ROIs, centered around different positions $x_{S}$
along a drop diameter. The correlation functions are very well fitted with a
simple exponential decay as theoretically expected for monodisperse Brownian
particles (Eq. 6). We find that the decay time of the correlation function,
$\tau_{D}$, increases from $3.9$ to $8.1$ ms as $x_{S}$ grows. This trend is
consistent with what expected from the variation of the scattering vector q
associated with the position of the ROIs for $n>n_{m}$. The opposite case
$n<n_{m}$ is illustrated by the data in Fig. 3b, acquired for a drop immersed
in oil. In this case, the relationship between $\theta_{s}$ and $x_{S}$ has an
opposite trend, see Fig. 3c, leading to a faster decay for smaller $x_{S}$.
Moreover, since the refractive index mismatch between oil and water is smaller
than the one between air and water, the variation of the scattering angle, and
hence of $q$, is smaller for the drop in oil than for the same drop in air.
This explains the smaller variation of $g_{2}-1$ with ROI position for the
drop in oil, compare Figs. 3a and b. These observations are summarized in Fig.
3c, which shows the relaxation time $\tau_{D}$ along the drop diameter for
both surrounding media. To quantitatively check the relationship between
position $x_{S}$ and $q$, we plot in Fig. 3d, $1/\tau_{D}$ as a function of
$q^{2}$ for the data acquired with the drop in air and in oil, where $q$ is
calculated from the position of each ROI as explained in Sec. II.2 and
Appendix A. We find that the two sets of data nicely collapse on the
theoretical curve $1/\tau_{D}=2D_{0}q^{2}$, with $D_{0}$ the diffusion
coefficient measured independently on the same suspension, but in the bulk,
using a commercial DLS apparatus with a standard cuvette. These results
confirm that the setup and data analysis presented here allow for reliably
measuring the microscopic dynamics in millimeter-size drops.

##  IV Swelling of a Polymer Gel

In this section we investigate a bead of a water-based polyacrylamide gel that
is initially partially dried and then is swollen by immersing it in water. As
for the Brownian suspension, we use sequences of speckles images to measure
the microscopic dynamics. However, some caution is required when analyzing
$g_{2}-1$ functions taken during swelling. Due to the imaging geometry, the
speckle pattern associated to a given portion of the gel moves away from the
center of the bead as the gel swells. This drift motion would result in a
spurious fast decay of $g_{2}-1$. To avoid this artifact, we use the drift
correction scheme of Ref. [55], where the intensity correlation function is
calculated in a moving reference frame that compensates for the drift. A
second, more subtle effect concerns the nature of the displacements within the
gel. The microscopic dynamics are due to the strain field generated during
swelling, which in turn may be decomposed in a component associated to the
expansion of a perfectly elastic and homogeneous spherical body, and
additional displacements stemming from spatial heterogeneity in the elastic
response of the gel or strain-induced rearrangements [56, 57]. The latter
component is often the one of greater interest. Unfortunately, the geometry of
our experiments does not allow one to disentangle the two components. In order
to gauge the relative importance of each component, we first derive in Sec.
IV.1 a theoretical expression for the intensity correlation function during
the expansion or contraction at a macroscopic strain rate $\dot{\varepsilon}$
of an ideal, perfectly elastic and homogeneous spherical body. We then
describe the preparation protocol for the gel beads in Sec. IV.2 and compare
the experimental $g_{2}-1$ functions to those for the ideal case in Sec. IV.3.
Our results will be briefly discussed and compared to other studies of
swelling polymer beads in Sec. IV.4.

###  IV.1 Theory: intensity correlation function for the expansion or
compression of a purely elastic homogeneous sphere

Figure 4: Top view of the scattering from a water-based sample surrounded by
air. The green cylinder represents the incoming laser beam, which has a
Gaussian radial profile. Scattered light emerging in $P$ is issued by all
points lying on the segment $PT$ (red dotted line), which is colinear with the
scattered light wave vector $\textbf{k}_{o}$. The blue line shows the
propagation direction of the scattered light travelling outside the sphere
along the $\widehat{e}_{y}$ direction, before being imaged on the camera
sensor by the collection optics (not shown). The black dashed line shows the
perpendicular to the drop surface through point $P$; $\theta_{i}$ and
$\theta_{r}$ are the angles of incidence and refraction, respectively.

We discuss here the case of a perfectly elastic and homogeneous sphere
surrounded by a medium with arbitrary refractive index and undergoing
expansion or contraction. We shall derive a general expression for the decay
of $g_{2}-1$ in response to a strain increment (or decrement)
$\Delta\varepsilon=\Delta R/R$, with $\Delta R$ the change of the sphere’s
radius. This general expression will be used in numerical calculations; a
simpler analytical expression valid under some approximations will also be
introduced.

Operationally, the intensity correlation function is calculated pixel by
pixel, see Eq. 5. Any given pixel of the CMOS detector corresponds to a small
portion of the sphere surface centered around the point $P$ from which the
scattered light emerges from the sphere, see Fig. 4. The normalized intensity
correlation function can be written as an integral over the contributions of
all scatterers in the scattering volume $V_{P}$ associated to point $P$ [45]:

$g_{2}(\textbf{P},\Delta\varepsilon)-1\propto\left|\int_{V_{P}}e^{-i\textbf{q}\cdot\Delta\textbf{u}(\textbf{x},\Delta\varepsilon)}I(\textbf{x},\textbf{q})\mathrm{d}^{3}\textbf{x}\right|^{2}\,,$
(10)

where q is the scattering vector determined as shown in Sec. II.2,
$I(\textbf{x},\textbf{q})$ is the scattered intensity originating from a point
at position $\mathbf{x}=x\hat{e}_{x}+y\hat{e}_{y}+z\hat{e}_{z}$ and
$\Delta\textbf{u}(\textbf{x},\Delta\varepsilon)$ is the displacement field
generated by the strain increment $\Delta\varepsilon$. Note that in writing
Eq. 10 we have dropped inessential prefactors, keeping in mind that the final
expression for the correlation function has to be normalized such that
$g_{2}-1\rightarrow 1$ for $\Delta\varepsilon\rightarrow 0$.

In our setup, the pixel size corresponds roughly to the speckle size, and both
are $<<R$. Given that the intensity value does not change significantly within
a speckle [47], the integral in Eq. 10 may be simplified by recognising that
$I$ and $\Delta\mathbf{u}$ depend essentially only on position along the
segment $PT$. Thus, the 3D integration over $V_{P}$ may by replaced by a one-
dimensional integration over the scalar $l\in[0,1]$ that parametrizes the
position along $PT$:

$\textbf{x}(l)=l(\textbf{OP}-\textbf{OT})+\textbf{OT}\,,$ (11)

with $\textbf{x}(0)=\textbf{OT}$ and $\textbf{x}(1)=\textbf{OP}$ (see Appendix
B for expressions of the coordinates of $P$ and $T$). Furthermore, we note
that the scattered intensity $I(\mathbf{x},\mathbf{q})$ is proportional to
$I_{0}(\mathbf{x})$, the $\mathbf{x}$-dependent intensity of the incident
beam. The proportionality factor accounts for the $q$ dependence of the
scattered light; in this context, this proportionality factor is unimportant,
since it may be included in the normalization of $g_{2}-1$. Given that the
incident beam has a Gaussian profile and dropping again inessential
prefactors, one has

$I_{0}[\textbf{x}(l)]\propto\exp\left\\{-\frac{2\left[y(l)^{2}+z(l)^{2}\right]}{w_{0}^{2}}\right\\}\,.$
(12)

In writing Eq. 12, we have used the fact that the incoming beam is mildly
focused, such that the Rayleigh range is comparable to the drop diameter and
the cross section $w$ of the beam is nearly constant and equal to the beam
waist $w_{0}$. With these assumptions, Eq. 10 simplifies to

$g_{2}(\textbf{P},\Delta\varepsilon)-1\propto\\\
\left|\int_{0}^{1}e^{-i\textbf{q}\cdot\Delta\textbf{u}[\textbf{x}(l),\Delta\varepsilon]}\exp\left\\{-\frac{2\left[y(l)^{2}+z(l)^{2}\right]}{w_{0}^{2}}\right\\}\mathrm{d}l\right|^{2}\,.$
(13)

The displacement field for the compression or expansion of a homogeneous,
purely elastic sphere reads [58]

$\begin{cases}\Delta\textbf{u}(\mathbf{x},\Delta\varepsilon)=\left[C_{1}(\Delta\varepsilon)r+\frac{C_{2}(\Delta\varepsilon)}{r}\right]\hat{r}\,,r\geq
r_{c}\\\
\Delta\textbf{u}(\mathbf{x},\Delta\varepsilon)=0\,,r<r_{c}\end{cases}$ (14)

with $r=|\mathbf{x}|$, $\hat{r}=\mathbf{x}/r$, $r_{c}$ the radius of an
incompressible core, and $C_{1}$, $C_{2}$ two constants such that

$\begin{cases}C_{1}=-\frac{C_{2}}{r_{c}^{3}}\\\
C_{2}=R_{i}\Delta\varepsilon\left(-\frac{R_{i}}{r_{c}^{3}}+\frac{1}{R_{i}^{2}}\right)\end{cases}\
$ (15)

where $R_{i}$ is the initial radius of the sphere. Mathematically, $r_{c}$ is
introduced to avoid the divergence of $\Delta\mathbf{u}$ at the sphere center;
physically, is accounts for the fact that the continuum medium description of
Eq. 14 must break down at the structural length scale of the gel. In practice,
we expect $r_{c}$ to be of the order of the monomer size or the mesh size at
most; we checked that varying $r_{c}$ in the range $10^{-9}$—$10^{-5}$ m,
$\Delta\mathbf{u}$ and hence $g_{2}-1$ do not vary appreciably.

Equation 13, together with the expressions for the displacement field, Eqs.
14-15, may be solved numerically to obtain the decay of $g_{2}-1$ as a
function of the strain increment $\Delta\varepsilon$ or, assuming a constant
expansion or contraction rate $\dot{\varepsilon}$, for increasing time lags
$\tau=\Delta\varepsilon/\dot{\varepsilon}$. Note that $g_{2}-1$ is insensitive
to the sign of $\Delta\varepsilon$, i.e. the same correlation function is
recovered for compression or expansion, provided that $|\Delta\varepsilon|$ is
the same.

Equation 13 holds quite generally and may be used to compute $g_{2}-1$ for
various kinds of dynamics, using the suitable expression for
$\Delta\mathbf{u}$. For the case of expansion or contraction discussed here,
we show in Appendix B that an approximated analytical solution can be found
for Eq. 13, provided that one uses a simplified expression for the
displacement field, dropping the $1/r$ term in Eq. 14 and assuming

$\Delta\textbf{u}(\textbf{x},\Delta\varepsilon)=\textbf{x}\Delta\varepsilon\,.$
(16)

Under this hypothesis, we show in Appendix B that the correlation function
decays as a Gaussian function:

$g_{2}(\tau)-1=\exp\left[-2\left(\frac{\tau}{\tau_{E}}\right)^{2}\right]\,,$
(17)

where the relaxation time $\tau_{E}$ depends on the position of point $P$, the
strain rate $\dot{\varepsilon}$, the scattering vector $\mathbf{q}$ and the
beam waist $w_{0}$, see Eq. 36.

Figure 5: a) Symbols: intensity correlation function calculated numerically
for a water-based spherical bead of gel in water undergoing contraction or
expansion at a fixed strain rate, see text for details. The red solid line is
a fit to the data using Eq. 7, yielding $\tau_{E}=6.15$ s and $\beta=2$. The
black dotted line is $g_{2}-1$ obtained via the analytical approximation of
Eq. 17. b) Decay time $\tau_{D}$ as a function of position obtained from the
numerical $g_{2}-1$, for different surrounding media, as indicated by the
lablel.

The triangles in Fig. 5a show an example of the intensity correlation function
$g_{2}(\textbf{P},\tau)-1$ obtained numerically using the laser beam
parameters of our setup, for a point
$\textbf{P}=(0.001,0.9999,0.001)~{}\mathrm{mm}$ on a sphere with $R=1$ mm,
refractive index $n=1.33$, and surrounded by water, $n_{m}=1.33$. The strain
increment varies in the range $10^{-4}\leq\Delta\varepsilon\leq 6.3\times
10^{-3}$ and $g_{2}-1$ is plotted as a function of time delay $\tau$, assuming
expansion or contraction at a constant rate $\dot{\varepsilon}=5\times
10^{-4}~{}\mathrm{s}^{-1}$. The continuous line shows a fit to the the
numerical data with Eq. 7, yielding $\tau_{E}=6.15$ s and $\beta=2$. The
dotted line shows the approximated analytical solution, Eq. 17, where the
relaxation time $\tau_{E}$ has been calculated based on the same parameters as
for the numerical calculations, using Eq. 36. We find that the approximated
analytical form is indistinguishable from both the numerical calculation and
its fit, indicating that for the typical experimental parameters of our
measurements the simplified expression of the displacement field, Eq. 16, is
in excellent agreement with the full expression.

Finally, to demonstrate the impact on $g_{2}-1$ of the refractive index
mismatch between the gel bead and the surrounding medium, we show in Fig. 5b
the decay time $\tau_{D}$ obtained from the numerical $g_{2}-1$ as a function
of the position in the bead, for different surrounding media: air, water and
oil. In the center of the bead ($x_{S}=0$), we obtain the same decay time
independently of $n_{m}$, as expected since in this position the scattering
angle, and hence $q$, does not depend on $n_{m}$, see Fig. 1b. For $x_{S}\neq
0$, $\tau_{D}$ depends in general on the refractive index mismatch. For
$n_{m}=n=1.33$, which is the case of hydrogel beads immersed in water (green
triangles in Fig. 5b), $\tau_{D}$ is independent of position. By contrast, for
a sample surrounded by air, $n_{m}=1$, the decay time varies by more than a
factor of two along the bead diameter, reflecting the change in $q$. For a
sphere surrounded by oil, $n_{m}=1.403$, a reverse trend of $\tau_{D}(x_{S})$
is seen (compare the blue diamonds and the orange circles), since $n_{m}-n$
has an opposite sign compared to the case of a bead in air. Note also that the
variation of $\tau_{D}$ is smaller, since the refractive index mismatch is
lesser.

###  IV.2 Experiments: Materials

Polyacrylamide gels are prepared by copolymerization of acrylamide (46.03 g/L)
as a monomer and methylenebisacrylamide (0.373 g/L) as a co-monomer in the
presence of tetramethylenediamine (0.6 g/L) and sodium persulfate (0.93 g/L)
as initiators in water. All chemicals are quickly mixed together and a 30
$\mu$L drop of the solution is injected in an Petri dish, filled with silicon
oil (47V 100 from VWR chemicals), where polymerization occurs. The oil has
been previously saturated with nitrogen, since polymerization is inhibited by
oxygen. The oil mass density is $97~{}\mathrm{kg}~{}\mathrm{m}^{-3}$ at 25 °C,
nearly equal to that of water, such that gravity effects are negligible and
the drop has a spherical shape, due to surface tension. Changing the relative
amounts of the monomer and co-monomer allows one to tune the gel mechanical
properties; the beads used here have an elastic modulus $G_{0}=700$ Pa [59].
The mesh size of the gel, defined as the average distance between two
crosslinks and estimated from mechanical measurements, is
$\zeta\approx(k_{B}T/G_{0})^{\frac{1}{3}}\approx 20$ nm.

The gel bead is first shrunk by letting it dry in air for about 1 hour. During
this phase, the bead rests on a super-hydrophobic substrate prepared as
described in Sec. III.1 and placed in the sample chamber left open, resulting
in a $\sim 14\%$ decrease of the bead radius. The sample chamber, an optical
glass box of size $5\times 5\times 5~{}\mathrm{cm}^{3}$, is then filled with
water, initiating swelling and defining the experiment time $t=0$. All
measurements are performed in a room where $T=22.0\pm 0.1^{\circ}$C. In order
to keep illuminating the bead in its equatorial plane all along the swelling
process, the bead is translated upwards using a micrometric vertical
translation stage in between the acquisition of series of speckle images.

###  IV.3 Experimental results

We show in Fig. 6 the microscopic relaxation time measured during the last 14
minutes of the drying phase that precedes the swelling experiment. Remarkably,
we find that $\tau_{D}$ is in very good agreement with both the spatial trend
and the magnitude predicted for the microscopic relaxation time for a purely
elastic sphere undergoing isotropic contraction. In the following, for the
sake of simplicity, we shall refer to the radial displacement field associated
with the contraction or expansion of such a homogeneous, purely elastic sphere
as to the “affine dynamics”, and shall term “non-affine dynamics” any
microscopic displacements that may add up to the affine component. Note that,
although strictly speaking the strain field for an expanding or contracting
sphere is not purely affine due to the $1/r$ term in the r.h.s. of Eq. 14, in
practice in our experiments this term is negligible, justifying this
terminology. The data in Fig. 6 thus demonstrate that the microscopic dynamics
during the late stages of drying are essentially purely affine, with a local
strain rate that is identical to the macroscopic one, for all probed $x_{S}$.
This suggests that any radial gradients in solvent content that may have
developed in the initial stages of drying have relaxed by the time the
swelling experiment starts.

Figure 6: Decay time $\tau_{D}$ as a function of position for a bead of
polymer gel drying at a rate $\dot{\epsilon}=4.3\times 10^{-5}$ s-1. Empty
circles are experimental data, blue diamonds are numerical results obtained
using Eqs. 13-15 for a homogeneous, purely elastic sphere undergoing
compression at the same rate as in the experiments.

Figure 7 describes the gel behavior during the swelling phase. The time
evolution of the radius of the gel bead as obtained from the speckle images is
shown in Fig. 7a. At $t=0$ the bead radius is $R_{0}=2.08$ mm; as the gel
absorbs water and swells, $R$ grows reaching $R_{\rm{f}}=2.63$ mm at the end
of the experiment, corresponding to a maximum engineering strain
$\varepsilon_{\rm{sw}}=(R_{\rm{f}}-R_{0})/R_{0}=28\%$. During swelling, the
radius growth can be very well fitted by $R(t)=R_{0}+Ct^{\gamma}$, with
$R_{0}$ fixed to $2.08~{}\rm{mm}$, and fitting parameters $C=0.076\pm
0.022~{}\rm{mm}~{}\rm{min}^{-\gamma}$ and $\gamma=0.35\pm 0.06$ fitting
parameters. The macroscopic swelling rate
$\dot{\varepsilon}=R(t)^{-1}\mathrm{d}R/\mathrm{d}t$ obtained from the fit to
$R(t)$ is shown as a red line in the inset of Fig. 7a: it is essentially
indistinguishable from a power law decay $\dot{\varepsilon}\propto t^{-0.69}$
and is in very good agreement with the swelling ratio obtained at selected $t$
by measuring the increment of $R$ over a time interval $\Delta t$ directly
from the speckle images, using $\Delta t\approx 3.5~{}\mathrm{min}$ for the
first two points and $\Delta t\approx 14~{}\mathrm{min}$ for $t\geq
15~{}\mathrm{min}$.

Figure 7b shows the spatial dependence of the scattered intensity $I$, for
various $t$. Data are obtained by averaging $I(x,z)$ along the $z$ (vertical)
direction and over a sliding window of three pixels in the $x$ direction. We
find that $I$ is not uniform, revealing time-dependent spatial heterogeneity
in the gel structure and density. For a polymer gel, the scattered intensity
at a given $q$ depends on material parameters such as the monomer size and
refractive index, as well as the monomers concentration and their spatial
structure. This results in a complex behavior, such that in general $I(q)$ is
not simply proportional to monomer concentration and may even decrease with
increasing concentration. This is indeed the case for our gel: by calculating
the total scattered intensity, $I_{tot}(t)=\int I(x_{S},t)\mathrm{d}x_{S}$, we
find that $I$ increases as the gel swells, i.e. as the average monomer
concentration decreases (see Fig. 9a in Appendix C). The high-intensity
regions seen at the bead edges in Fig.7b thus correspond to a shell of swollen
gel that initially forms at the gel-solvent interface and then progressively
propagates inwardly. We determine the position of the shell-core interface by
analyzing the spatial dependence of $I$ (see Appendix C and Fig. 9b for
details) and plot in the inset of Fig. 7b its time dependence, together with
that of the bead radius. We find that the shell-core structure tends to fade
as the swollen region invades the whole bead.

Figure 7: (a) Radius of a swelling polyacrylamide gel bead as function of
time. The line is a power law fit to the increment of $R$, see the text for
details. Inset: strain rate as a function of time, measured directly from the
speckle images (symbols) or by differentiating the power law fit of the main
panel (line). Error bars represent the uncertainty of the gel boundary as
measured from the speckle images. (b) Scattered intensity as a function of
position for various $t$, indicative of the formation of a core-shell
structure upon swelling. For clarity, curves have been offset vertically in
increments of 5 gray units. From bottom to top, $t$ = 3.6, 7.5, 15, 45, 90,
180, and 360 min, same colors as in a). The inset shows the time evolution of
the bead radius (circles) and of the core-shell interface, as inferred from
the $I(x_{S})$ data shown in the main graph (triangles), or from the
relaxation time of panel d) (squares), see text for details. (c) Intensity
correlation functions measured during the swelling of the bead, for a ROI
located at the center of the bead, $x_{S}=0$, same $t$ values and color code
as in b). The solid lines are fits with Eq. 7, yielding $\tau_{D}=$ 4.44,
6.52, 7.25, 12.04, 27.94, 39.86, 47.34 s and $\beta=$ 1.20, 1.18, 1.04, 1.23,
1.45, 1.53, 1.52 for increasing $t$. The grey right triangles show $g_{2}-1$
for a gel where swelling is prevented, see Sec. II.4 (same data as in Fig. 2).
Inset: time dependence of the stretching exponent $\beta$. (d) Decay time
$\tau_{D}$ as a function of position $x_{S}$ along the bead diameter, same $t$
values, symbols and colors as in a), b) and c).

To gain insight on the swelling phenomenon at the microscopic level, we
inspect correlation functions measured at various $t$ and positions in the
gel. Since the hydrogel is immersed in water, the solvent refractive index is
the same of that of the surrounding medium, $n=n_{m}$. Thus, there are no
refractive effects: regardless of $x_{S}$, the dynamics are probed at the same
scattering vector q, corresponding to a typical length scale $1/q\approx 45$
nm, comparable to a few mesh sizes.

In Fig. 7b we plot the intensity correlation functions for a region in the
center of the bead, $x_{S}=0$, and various $t$. During swelling, the
correlations functions decay much faster than for a quiescent gel, compare the
colored symbols to the grey right triangles. This indicates that, due to the
interconnected structure of the gel, stresses due to swelling generate strain
fields and microscopic rearrangements throughout the whole network, even at
the beginning of the experiment, where swelling occurs only in a thin shell at
the gel periphery, as shown in Fig. 7b. Over the duration of experiment ($6$
hours), the dynamics slow down by more than one order of magnitude, mirroring
the decrease of the macroscopic expansion rate $\dot{\varepsilon}$ shown in
the inset of Fig. 7a. We quantify the microscopic dynamics by fitting
$g_{2}-1$ to a stretched exponential function, Eq. 7, expressing the
microscopic decay time as $\tau_{D}$, calculated using Eq. 8. We find that the
decay time $\tau_{D}$ increases with time by an order of magnitude, ranging
from $4.44$ s at the beginning of the swelling to $47.34$ s at $t=360$ min.
Concomitantly, the stretching exponent evolves from $\beta\approx 1$ in the
early stages of swelling to $\beta\approx 1.5$ for $t\geq 90$ min (inset of
Fig. 7b).

We find that the microscopic dynamics have a non-trivial spatio-temporal
dependence. Figure 7c displays the characteristic relaxation time $\tau_{D}$
at different times during the swelling process, as a function of position
$x_{S}$ along the bead diameter. The slowing down of the microscopic dynamics
as the gel swells, observed in the center of the bead, is seen at all $x_{S}$.
However, at the beginning of the swelling process the microscopic dynamics
clearly depend on position: $\tau_{D}$ is roughly two times smaller near the
bead edges as compared to in the center, consistent with the notion of a shell
that swells much faster than the bead core. This space dependence becomes
weaker with time, and eventually vanishes: nearly uniform microscopic dynamics
are recovered $180$ min from the beginning of the experiment. Remarkably, we
find that there is a difference in the shell-core structure as inferred from
static and dynamic measurements, Figs 7b and c,d, respectively. To quantify
this difference, we apply a procedure similar to that used for the $I(x_{S})$
data to the space-dependent relaxation time of Fig. 7d, determining the
position of the core-shell interface as inferred from measurements of the
dynamics. The results are shown in the inset of Fig. 7b (square symbols). The
core inferred from the dynamics is consistently smaller than that obtained
from the static $I(x_{S})$, and vanishes at $t=360$ min, when no core-shell
structure can be distinguished in the spatial dependence of the dynamics.

Figure 8:  (a) Decay time $\tau_{D}$ averaged over position $x_{S}$, as a
function of strain rate $\dot{\varepsilon}$. Error bars on $<\tau_{D}>$ are
the standard deviation of the data shown, for each $t$, in Fig. 7d. The green
line is the decay time expected for the expansion of a purely elastic,
homogeneous sphere, as obtained by fitting correlation functions calculated
numerically by inserting the experimental $\dot{\varepsilon}$ in Eq. 13. The
empty circle is $<\tau_{D}>$ for the shrinking gel measured during drying,
obtained form the data of Fig. 6. (b) Intensity correlation functions for a
polymer gels swelling at rate $\dot{\varepsilon}=5.73\times
10^{-6}~{}\mathrm{s}^{-1}$. Gray solid line: theoretical intensity correlation
function for purely affine dynamics; large yellow symbols: experimental
$g_{2}-1$ for $x_{S}=0$; small red symbols: corresponding intensity
correlation function for the non-affine displacements only, calculated using
Eq. 18. (c) Non-affine mean square displacement $\Delta r^{2}$ for $x_{S}=0$,
obtained via Eq. 19 as a function of the strain increment $\Delta\varepsilon$
during swelling. The solid lines are power law fits, $\Delta
r^{2}\propto\Delta\varepsilon^{\alpha}$, the exponent $\alpha$ is shown in the
inset (solid symbols), together with $\beta^{n.a.}$ (open symbols), the
stretching exponent obtained by fitting $g_{2}^{n.a.}-1$ with a stretched
exponential. In both the main plot and the inset the symbols are as in panel
a).

As discussed at the beginning of Sec. IV, our setup does not allow one to
separately measure the dynamics along the radial direction and perpendicularly
to it, which makes it difficult to directly assess the relative importance of
affine and non-affine dynamics. The data shown in Figs. 7c,d, however, already
point to deviations from affinity: indeed, we recall that for affine dynamics
$\beta=2$ as opposed to $\beta\lesssim 1.5$ as measured here, and that for
$n=n_{m}$ $\tau_{D}$ should be independent of position (see the triangles in
Fig. 5b), which is not the case in our experiments, at least in the initial
stages of swelling, see Fig. 7d. To further investigate deviations from
affinity, we compare $\tau_{D}$ as obtained in the swelling experiment to the
theoretical relaxation time for purely affine dynamics, which we calculate
using Eqs. 13-15 with $\Delta\varepsilon=\tau\dot{\varepsilon}$, where
$\dot{\varepsilon}$ is the experimental macroscopic strain rate and $\tau$ the
experimental time delay of $g_{2}-1$. Figure 8a compares the theoretical
relaxation time of affine dynamics (line) to the experimental relaxation time
$<\tau_{D}>$, obtained by averaging over space $\tau_{D}$. The experimental
dynamics are up to a factor of ten faster than the theoretical prediction.
Furthermore, for affine motion $\tau_{D}$ is inversely proportional to
$\dot{\varepsilon}$, see the line in Fig. 8a and Eq. 36, while the
experimental $<\tau_{D}>$ clearly deviates from such a
$\dot{\varepsilon}^{-1}$ dependence.

These discrepancies indicate that additional non-affine relaxations occur
during the swelling process. In order to quantify non-affine dynamics, we
assume that the affine and non-affine contributions to $g_{2}-1$ are
uncorrelated. Accordingly, the intensity correlation function may be
factorized:

$g_{2}(\Delta\varepsilon,\textbf{q})-1=\left[g_{2}^{aff.}(\Delta\varepsilon,\textbf{q})-1\right]\left[g_{2}^{n.a.}(\Delta\varepsilon,\textbf{q})-1\right]\,,$
(18)

where the superscripts $aff.$ and $n.a.$ indicate affine and non-affine
contributions, respectively. By dividing the experimental $g_{2}-1$ by the
affine correlation function $g_{2}^{aff.}(\Delta\varepsilon,\textbf{q})-1$
obtained numerically using the macroscopic strain rate $\dot{\varepsilon}$, we
obtain the non-affine correlation function. Figure 8b shows an example for
$\dot{\varepsilon}=5.74\times 10^{-6}~{}\mathrm{s}^{-1}$, the strain rate for
which the experimental and theoretical decay times are closest and thus the
experimental $g_{2}-1$ closest to the affine contribution. Even in this case,
the experimental correlation function is in fact very close to its non-affine
contribution, reflecting the fact that non-affine dynamics always dominate the
relaxation of $g_{2}-1$, as already inferred from the comparison of the
relaxation times shown in Fig. 8a. Note that this is in sharp contrast to the
behavior during the drying stage (open circle in Fig. 8a).

A thorough interpretation of the non-affine dynamics would require collecting
data at different $q$ vectors, which would allow one, e.g., to unambiguously
distinguish between diffusive and ballistic dynamics. Unfortunately, this is
not possible with our setup, due to the limitations imposed by the spherical
shape of the gel beads. To make progress, we assume that, for any given
$\Delta\varepsilon$ (or, equivalently, any $\tau$), the probability
distribution of the non-affine displacements is Gaussian. This hypothesis is
motivated by previous work on polyacrylamide gels seeded with micron-sized
tracer particles and submitted to a shear strain [56]. The particle
displacements were found to have both an affine and a non-affine component,
the latter being distributed according to a Gaussian function. Under this
assumption, the intensity correlation function depends on the (non-affine)
particle mean square displacement $\Delta r^{2}(\Delta\varepsilon)$ as

$g_{2}^{n.a.}(\Delta\varepsilon,\textbf{q})-1=\exp\left[-2(\Delta
r^{2}q^{2})/3\right]\,,$ (19)

a result easily obtained from standard dynamic light scattering theory [34,
60] by replacing the usual time delay $\tau$ by
$\Delta\varepsilon=\tau\dot{\varepsilon}$.

We show in the double-logarithmic plot of Fig. 8c the non-affine mean square
displacement $\Delta r^{2}(\Delta\varepsilon)$ for $x_{S}=0$, obtained from
Eq. 19 using $g_{2}^{n.a.}-1$ calculated according to Eq. 18 (data for $t=3.6$
min are omitted because they are too noisy). The lines are power law fits to
the data, yielding an exponent $\alpha$, whose time dependence is shown by the
solid symbols in the inset. In the initial stages of swelling, $\alpha\approx
1$, indicative of diffusive dynamics, consistent with the shape of
$g_{2}^{n.a.}-1$, for which a fit with Eq. 7 yields an exponent $\beta^{n.a.}$
close to $1$, as expected for a diffusive process [34, 60] (see the open
symbols in the inset of Fig. 8c). However, for $t\geq 90$ min, both $\alpha$
and $\beta^{n.a.}$ significantly deviate from unity, approaching or even
exceeding a value of $1.5$. These values are not compatible with diffusive
dynamics. Rather, $\beta\approx 1.5$ has been reported for the ballistic
dynamics attributed to the strain field resulting from the relaxation of
internal stresses in a variety of amorphous systems [61, 62, 63, 64]. In this
case, the probability distribution of the particle displacements $\Delta
r(\tau)$ at fixed $\tau$ (corresponding to fixed $\Delta\varepsilon$ in our
case) is described by a Levy law, with a power-law decaying wing $\sim\Delta
r^{-(\beta+1)}$ at large $\Delta r$ [46]. Strictly speaking, this non-Gaussian
distribution is incompatible with the assumptions underlying Eq. 19, which was
used to calculate the mean squared displacements shown in Fig. 8c.
Accordingly, one should be cautious in commenting the value of the exponent
$\alpha$ for the latest stages of swelling. However, it is worth pointing out
that $\alpha>1$ indicates supradiffusive behavior, thus confirming that in the
late stages of swelling the microscopic dynamics are non-diffusive.

###  IV.4 Discussion

Although the literature on swelling gels is abundant [65, 66, 67, 68], studies
dealing with spherical samples are not common, especially in experiments [69,
70, 71, 25, 33, 26, 22]. In this section, we briefly discuss our main findings
in view of Refs. [25, 33, 26, 22], where gels of similar composition were
investigated. When comparing results from different studies, it is important
to keep in mind that the exact gel composition may vary across studies (or may
not even be known precisely for commercially available beads), and that the
experimental protocol may somehow differ (preconditioning of the gel before
the swelling experiment, nature of the solvent in which the bead is immersed,
etc.). Hence, any comparison is likely to be relevant at a semi-quantitative
level only.

We find that the swelling of our gel beads is, at the macroscopic level,
similar to that found in previous works. The gel surface remains nearly
spherical, as in Ref. [22], without developing the shape instability reported,
e.g. in Refs. [26, 25, 33]. This is consistent with the observation that shape
instabilities develop in relatively rigid gels, while the elastic modulus of
our beads, $G_{0}=700$ Pa, is significantly smaller than that in Refs. [25,
33] (27—84 kPa). The growth of bead radius follows a sub-diffusive behavior,
$R\sim t^{\gamma}$ with an exponent $\gamma=0.35$, very close to
$\gamma=0.38$, as predicted theoretically by Bertrand et al. [26]. This value
is also close to, but somewhat lower than, $\gamma=0.45$—$0.46$, as found
experimentally in Refs. [25, 33] for the swelling of commercial polyacrylamide
hydrogel beads, and in Ref. [22], which measured beads of polyacrylamide gel,
synthesized by using acrylamide and N-methylene-bis-acrylamide as crosslink
agent.

Another similarity between the present study and previous works is the
presence of a core-shell structure that rapidly forms when the gel bead is
exposed to the solvent and gradually fades out as the gel tends to an
asymptotic equilibrium state. In previous works, the boundary between a
swollen shell and a dense core was identified by probing the local structure
with NMR [33, 25] or by shadowgraphy [26]. Our results obtained by analyzing
the spatial dependence of the scattered intensity $I$ indicate that a similar
core-shell scenario also applies to the gels studied here. We find that as the
gel swells the core-shell interface propagates towards the center of the bead
with an average speed of $1.1~{}\mu\mathrm{m}~{}\mathrm{s}^{-1}$, comparable
to $1~{}\mu\mathrm{m}~{}\mathrm{s}^{-1}$ as measured by [33]. Similarly to
Ref. [25, 33], we observe a continuous shrinking of the core. By contrast,
Bertrand et al. [26] reported a non-monotonic evolution of the core size,
which initially increased, before shrinking and eventually disappeared. This
discrepancy was explained as stemming from thresholds effect for identifying
the boundary, which may be different according to the experimental technique
[26].

Interestingly, we find that tracking the position of the interface between the
swollen shell and the core is not sufficient to fully account for the time
evolution of the gel. Indeed, Fig. 7b clearly shows that, as the shell
propagates inwardly, the magnitude of the spatial gradient in scattering
intensity corresponding to the core-shell interface becomes significantly
lower. This suggest that restructuring occurs in both regions, such that the
concentration difference between shell and core progressively vanishes. This
scenario is in contrast with the simple picture of a static core that remains
essentially unchanged until it is completely invaded by the shell, as
suggested by the data shown in [33, 25]. Indeed, our space-resolved data on
the microscopic dynamics demonstrate unambiguously that rearrangements do
occur in the core as well, from the very beginning of the swelling process
(see Figs. 7c,d). To our knowledge, this is the first experimental evidence
that the core-shell scenario should not be regarded as the growth of a swollen
soft layer at the expenses of an otherwise unchanged, rigid substrate. Rather,
the gel evolves globally as a material in which compressive stresses propagate
in the whole volume, as hypothesized by Bertrand et al. [26], based on
numerical results.

Finally, our PCI measurements have shed light on the dynamics associated with
the microscopic rearrangements that allow for gel swelling, a quantity that
was not available in previous studies. We find a striking difference between
the dynamics during shrinkage (in the gel preconditioning stage) and upon
swelling. While the former are essentially consistent with the homogeneous
shrinkage of a perfectly elastic sphere, the latter unveil substantial non-
affine microscopic rearrangements. These non-affine dynamics are reminiscent
of those observed for polymer gels under shear [56] and for many other jammed
systems upon applying an external strain or stress, e.g. colloidal gels and
glasses [57, 72, 73, 74] and polymer glasses [75]. For the gels studied here,
we speculate that the difference between affine dynamics upon shrinkage and
non-affine dynamics during swelling may be due to the additional
configurations that become accessible to the polymer chains as the gel swells,
due to the increase of free volume. Interestingly, the nature of the non-
affine mean squared displacement in the center of the bead evolves with time,
from diffusive-like behavior at the onset of swelling, to supradiffusive, non-
Gaussian behavior at later stages. These differences highlight, once again,
the notion that the gel core should not be considered as a static, unchanged
medium. They furthermore show that the microscopic dynamics exhibit a rich
behavior, in spite of the smooth, subdiffusive-like evolution of the
macroscopic bead size.

##  V Conclusions

We have presented a new dynamic light scattering apparatus specifically
designed for probing soft matter systems within a millimetric drop. By
combining Photon Correlation Imaging [35], the use of a thin incident beam,
and a carefully designed collection optics, we have addressed the challenges
posed by the small radius of curvature of the sample-surrounding medium
interface, which acts as an additional refractive element. The setup allows
for measuring the dynamics coarse-grained over regions of interest that
typically correspond to sections of the incident beam of length 250-500
$\mu\mathrm{m}$. For the central ROI, the scattering angle is 90 deg; the
scattering angle associated to the other ROIs depend on the refractive index
contrast between the sample and the surrounding medium; typical deviations
from 90 deg are up to $\pm 30$ deg. Thus, the length scales probed by the
setup are of the order of a few tens of nm. Given the sample geometry, spatial
resolution requires the scattered light to be collected from the side: in
practice, this limits the range of accessible $q$ vectors, since it would be
impossible to significantly extend the range of $q$, e.g. to smaller angles,
while retaining PCI capabilities. Another limitation concerns the accessible
time scales, which are dictated by the use of a CMOS detector: the smallest
delay time is of the order of $10^{-4}$ s, larger than in conventional DLS,
for which delay times down to a few tens of ns are available. On the other
hand, the setup is sufficiently stable to probe relaxation process as slow as
several thousands of seconds, which are unaccessible to conventional DLS.
Finally, the multispeckle method used to calculate $g_{2}-1$ drastically
reduces the averaging time, such that time-varying dynamics, e.g. during
gelation, drying or swelling, can be thoroughly characterized.

Measurements on a drop of non-evaporating, diluted Brownian suspension, for
which spatially homogenous dynamics are expected, have shown instead a
variation of the DLS relaxation time according to position within the drop.
This effect, however, has been fully accounted for by considering the $x_{S}$
dependence of the scattering vector: we have shown that data for the
suspension drop are in excellent quantitative agreement with measurements on
the same sample but using a conventional cuvette and a commercial DLS
apparatus.

After validating the setup on the Brownian system, we have demonstrated some
of its capabilities by studying the swelling of a polymer gel upon re-
hydration. We have found an asymmetry between shrinking and swelling. Thanks
to the comparison with a theoretical and numerical analysis of DLS from a
homogeneous elastic sphere under strain, we have shown that during shrinking
the dynamics are affine. Quite remarkably, this implies that the same
deformation field holds on length scales ranging from the macroscopic drop
radius down to a few tens of nm. By contrast, during swelling the microscopic
dynamics are up to one order of magnitude faster than those inferred from the
macroscopic affine strain field, presumably reflecting the extra degree of
freedom available to the chains as their local configurations become less
constrained during swelling. As in previous works [25, 26, 33], we have found
that swelling is spatially heterogeneous, starting from the gel periphery and
propagating inwardly as time proceeds. Our space-resolved measurements of the
dynamics, unavailable in previous works, point however to a scenario more
complex than expected: at the microscopic level, rearrangements and chain
displacements are not limited to the swelling shell, but also occur in the
core, albeit at a slower rate.

Overall, the results presented here demonstrate the potentiality of the setup
for investigating microscopic dynamics in millimeter-sized spherical samples.
A key feature is the possibility of collecting data with both temporal and
spatial resolution, making the apparatus an ideal tool for understanding the
rich phenomenology occurring in a wide range of applications and phenomena,
from the drying of drops of colloidal or biological suspensions [4, 8, 76, 9],
aggregation phenomena e.g. in the production of supraparticles for industrial
applications [12, 13], to polymer drops degradation [77] or gel syneresis [78]
and swelling.

###### Acknowledgements.

We thank J. Barbat for help in designing and building the sample environment,
C. Kindelberger for help in measuring the setup stability, and C. Ligoure and
P. Olmsted for enlightening discussions. We acknowledge financial support from
the French Agence Nationale de la Recherche (ANR) (Grant No.
ANR-19-CE06-0030-02, BOGUS). LC gratefully acknowledges support from the
Institut Universitaire de France.

##  Appendix A Calculation of the scattering vector

With reference to Fig. 4b, we calculate the scattering vector $q$ for a ray
emerging from the drop in $P=(x_{P},y_{P},z_{P})$ and propagating along the
$y$ axis. With no loss of generality, we shall assume that the camera is
placed in the $y>0$ half-space and that the scattering angle $\theta_{s}$, the
angle of incidence $\theta_{i}$ and the angle of refraction $\theta_{r}$ are
taken to be non-negative, implying $0\leq\theta_{s}\leq\pi$,
$0\leq\theta_{i}\leq\pi/2$, $0\leq\theta_{r}\leq\pi/2$. The unit vector
associated with $P$ is

$\hat{r}_{P}=\frac{x_{P}\hat{e}_{x}+y_{P}\hat{e}_{y}+z_{P}\hat{e}_{z}}{\sqrt{x_{P}^{2}+y_{P}^{2}+z_{P}^{2}}}\,,$
(20)

which also defines the outgoing normal to the drop surface in $P$. The angle
of refraction satisfies

$\cos\theta_{r}=\hat{e}_{y}\cdot\hat{r}_{P}\equiv A\,.$ (21)

The angles of refraction and of incidence are thus given by

$\theta_{r}=\arccos A\,,$ (22)
$\theta_{i}=\arcsin\left(\frac{n_{m}}{n}\sin\theta_{r}\right)\,,$ (23)

where in writing Eq. 23 we have used Snell’s law, with $n$ and $n_{m}$ the
refractive indices of the drop and of the surrounding medium, respectively.

For the special case $n=n_{m}$, the scattering angle is $\theta_{s}=\pi/2$ for
all $x_{P}$. Similarly, $\theta_{s}=\pi/2$ for $x_{P}=0$ and arbitrary $n$,
$n_{m}$. For these cases, $\hat{k}_{o}=\hat{e}_{y}$, with $\hat{k}_{o}$ the
unit vector along the direction of the scattered light in the drop. It is then
straightforward to calculate $\mathbf{q}$ using Eq. 2. For all the other
cases, it is convenient to express $\hat{k}_{o}$ as a linear combination of
the two unit vectors $\hat{r}_{P}$ and $\hat{e}_{y}$:

$\hat{k}_{o}=a\hat{r}_{P}+b\hat{e}_{y}\,,$ (24)

with $a$ and $b$ two coefficients to be determined. Figure 4b shows that

$\displaystyle\hat{k}_{o}\cdot\hat{r}_{P}$ $\displaystyle=\cos\theta_{i}$ (25)
$\displaystyle\hat{k}_{o}\cdot\hat{e}_{y}$
$\displaystyle=\cos(\theta_{r}-\theta_{i})\,.$

By inserting Eqs. 21 and 24 into Eqs. 25, one has

$\displaystyle a+bA$ $\displaystyle=\cos\theta_{i}$ (26) $\displaystyle aA+b$
$\displaystyle=\cos(\theta_{r}-\theta_{i})\,.$

For the ease of writing, we define $B=\cos\theta_{i}$ and
$C=\cos(\theta_{r}-\theta_{i})$ and solve Eqs. 26 for $a$ and $b$, finding

$\displaystyle a$ $\displaystyle=\frac{B-AC}{1-A^{2}}$ (27) $\displaystyle b$
$\displaystyle=\frac{C-AB}{1-A^{2}}\,.$

Equations 27, together with $\textbf{k}_{o}=2\pi n\lambda^{-1}\hat{k}_{o}$ and
Eqs. 20 and 24 allow for calculating the scattering vector for scattered light
being imaged on the camera detector after emerging from the drop in $x_{P}$
and having propagated along the $y$ direction, for the general case
$\hat{k}_{o}\neq\hat{e}_{y}$ .

##  Appendix B Determination of analytical $g_{2}-1$

We aim at calculating an analytical expression of the intensity correlation
function $g_{2}-1$ for the light scattered by a contracting or expanding
sphere and emerging in point $P$, as shown in Fig. 4. We assume in the
following that the strain field within the sphere is that of an ideal solid,
i.e. we calculate only the contribution to $g_{2}-1$ due to affine
displacements. As shown in Fig. 4, light emerging in $P$ is scattered by all
points on the $PT$ segment. In our experiments, the incident beam is Gaussian,
with a thin cross section (beam waist $w_{0}\approx 60~{}\mu$m), much smaller
than the drop radius $R$. Thus, most of the scattered light originates from a
small portion of $PT$ centered around point S on the $x$ axis. However, to
ease the calculation, it is convenient to consider the whole segment $PT$.
Since $w_{0}<<R$, refraction effects on the incoming beam may be neglected,
since they are only relevant for the portions of the incoming beam far from
the $x$ axis, whose intensity is vanishingly small. We thus assume that the
incident beam within the drop propagates along the $\hat{e}_{x}$ direction,
regardless of the distance from the $x$ axis. This insures that light
scattered by any point along $PT$ has the same wave vector $\textbf{k}_{o}$
and thus the same scattering vector q.

We start by determining the coordinates of $P$ and $T$. Experimentally,
$g_{2}-1$ is calculated for all pixels belonging to a ROI, i.e. for a set of
pixels identified by $x_{P}$ and $z_{P}$, the coordinates of $P$ in the
$(x,z)$ plane, which are obtained directly from the CMOS image. Since $P$ lies
on the surface of the drop, one has

$\textbf{OP}=x_{P}\hat{e}_{x}+\sqrt{R^{2}-(x_{P}^{2}+z_{P})^{2}}\hat{e}_{y}+z_{P}\hat{e}_{z}\,.$
(28)

The coordinates of point $T$ are found by imposing that $PT$ is colinear with
$\hat{k}_{o}$ and that $T$ lies on the surface of the sphere, in the $y<0$
half-space (see Fig. 4). One finds

$\textbf{OT}=\textbf{OP}-\left(2\textbf{OP}\cdot\hat{k}_{o}\right)\hat{k}_{o}\,.$
(29)

Equation 29, together with Eq. 28 and the expression of $\hat{k}_{o}$ provided
in Appendix A fully determines the coordinates of $T$ for any experimentally
chosen $(x_{P},z_{P})$.

To find an analytical expression for $g_{2}-1$, we need to write the scattered
intensity $I$ originating from an arbitrary point $\mathbf{x}$ along $PT$ as a
function of the parameter $l$, see Eq. 12. Using Eqs. 11-12 and the
coordinates of $P$ and $T$ given above, one finds

$I(\mathbf{x})\propto\exp\biggl{\\{}-(Dl^{2}+El+F)/(w_{0}^{2}/2)\biggr{\\}}$
(30)

where

$\displaystyle D$ $\displaystyle=(y_{T}-y_{P})^{2}+(z_{T}-z_{P})^{2}$ (31)
$\displaystyle E$ $\displaystyle=2z_{T}(y_{T}-y_{P})+2y_{T}(z_{T}-z_{P})$
$\displaystyle F$ $\displaystyle=y_{T}^{2}+z_{T}^{2}\,,$

and where we have omitted all unnecessary prefactors. Finally, we need an
expression for the displacement field
$\Delta\textbf{u}(\textbf{x},\Delta\varepsilon)$ in $\mathbf{x}$ corresponding
to a change of macroscopic strain $\Delta\varepsilon$. We use a simplified
version of Eq. 14, where we drop the second term in the r.h.s., which is
typically negligible, finding

$\Delta\textbf{u}[\textbf{x}(l),\Delta\varepsilon]=l\Delta\varepsilon(\textbf{OT}-\textbf{OP})+\Delta\varepsilon\textbf{OT}\,.$
(32)

The projection of the displacement onto the scattering vector is then

$\Delta\textbf{u}\cdot\textbf{q}=l\Delta\varepsilon\left[\textbf{q}\cdot(\textbf{OT}-\textbf{OP})\right]+\Delta\varepsilon\big{(}\textbf{q}\cdot\textbf{OT}\big{)}=\tilde{q}l+K$
(33)

where we have introduced the strain-dependent generalized wave vector
$\tilde{q}=\Delta\varepsilon\left[\textbf{q}\cdot(\textbf{OT}-\textbf{OP)}\right]$
and defined $K=\Delta\varepsilon\big{(}\textbf{q}\cdot\textbf{OT}\big{)}$. By
inserting Eqs. 30 and 33 into Eq. 13 and omitting temporarily any
normalization factor, we obtain:

$g_{2}-1\propto\left|\int_{0}^{1}e^{-i\tilde{q}l}\exp\left\\{-(Dl^{2}+El+F)/(w_{0}^{2}/2)\right\\}\mathrm{d}l\right|^{2}$
(34)

which, by extending to $\pm\infty$ the integration limits with no significant
change since the integrand is vanishingly small out of $[0,1]$, is the Fourier
transform of a generalized Gaussian function. This Fourier transform has an
analytical solution [79]:

$g_{2}-1=e^{-2\frac{(w_{0}^{2}/2)\tilde{q}}{4D}}\,,$ (35)

where we have used the correct normalization. Note that in the above
expression $g_{2}-1$ depends on $\Delta\varepsilon$ through $\tilde{q}$. Over
a time laps short enough for the strain rate $\dot{\varepsilon}$ to be
considered constant, one has $\Delta\varepsilon=\tau\dot{\varepsilon}$. Under
these conditions, Eq. 35 is of the form of Eq. 7 of the main text with
$\beta=2$ and

$\tau_{E}=\frac{2\sqrt{2}\sqrt{(y_{T}-y_{P})^{2}+(z_{T}-z_{P})^{2}}}{\dot{\varepsilon}\left[\textbf{q}\cdot(\textbf{OT}-\textbf{OP})\right]w_{0}}\,,$
(36)
$\tau_{D}=\frac{\tau_{E}}{2\sqrt{2}}\Gamma\left({\frac{1}{2}}\right)\approx
0.6267\tau_{E}\,.$ (37)

##  Appendix C Determination of the core-shell boundary in a bead of polymer
gel

Figure 9:  (a): Relative change of the average monomer concentration upon
swelling of the gel of Sec. IV (symbols and left axis), together with the time
evolution of the scattered intensity integrated over the whole scattering
volume (solid symbols and right axis). (b) Scattered intensity profile (left
axis and solid line) and its integral up to position $x_{S}$ (right axis and
dotted line). (c) Relaxation time as a function of position (left axis and
symbols), together with its integral up to position $x_{S}$ (right axis and
line). In b) and c), $t=45$ min and the vertical lines and red dots indicate
the core-shell boundaries.

We first show that, for our system, an increase of the scattered intensity $I$
corresponds to a decrease in monomer concentration, e.g. to gel swelling. This
is demonstrated by Fig. 9a, where we plot on the left axis the relative change
of the average monomer concentration due to swelling,
$c(t)/c_{0}=R_{0}^{3}/R(t)^{3}$, where the subscript 0 indicates quantities
measured at time $t=0$, and on the right axis the time dependence of
$I_{tot}$, the scattered intensity integrated over the whole scattering
volume. The two quantities follow an opposite trend.

In order to determine the core-shell boundary, we apply a similar method to
analyze the $x_{S}$ dependence of the static scattered intensity $I$ and of
the relaxation time $\tau_{D}$, respectively. For each of the $I(x_{S},t)$
curves shown in Fig. 7b, we plot the $x_{S}$ dependence of
$I_{cum}(t)=\int_{x_{min}}^{x_{S}}[I(x^{\prime},t)-I_{C}]\mathrm{d}x^{\prime}$,
where $I_{C}$ is the intensity value in the plateau region around the bead
center where the scattered intensity is roughly independent of position.
Figure 9b shows a typical $I_{cum}$ curve, for $t=45$ min (dotted line and
right axis), together with the corresponding $I(x_{S})$ data (solid line and
left axis). $I_{cum}$ initially raises steeply, in the $x_{S}<0$ part of the
shell, it then levels off in the core and raises again in the $x_{S}>0$ part
of the shell. By inspecting the profile of $I_{cum}$, we determine the core-
shell boundaries, shown by the red dots and vertical lines in Fig. 9b. From
their values, we determine the average position of the interface and its
uncertainty, reported in Fig. 7b.

We use the same approach for analyzing the $\tau_{D}(x_{S})$ data of Fig. 7d:
Figure 9c shows
$\tau_{cum}(t)=\int_{x_{min}}^{x_{S}}[\tau_{D}(x^{\prime},t)-\tau_{C}]\mathrm{d}x^{\prime}$
(right axis), where $\tau_{C}$ is the relaxation time in the plateau region
around the bead center 111We use linear interpolation to estimate $\tau_{D}$
in between the available data points. The corresponding $\tau_{D}(x_{S})$ data
are shown as symbols (left axis) and the red dots and vertical lines mark the
core-shell interfaces.

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