# Inelastic Neutron Scattering Analysis with Time-Dependent Gaussian-Field
Models

Cedric J. Gommes  Current affiliation: Department of Chemical Engineering,
University of Liège B6A, allée du Six Août 3, B-4000, Liège, Belgium
<EMAIL_ADDRESS>Forschungszentrum Jülich GmbH, Jülich Center for
Neutron Science, 52425 Jülich, Germany  Reiner Zorn  Forschungszentrum Jülich
GmbH, Jülich Center for Neutron Science, 52425 Jülich, Germany  Sebastian
Jaksch  Henrich Frielinghaus  Olaf Holderer  Forschungszentrum Jülich GmbH,
Jülich Center for Neutron Science at the Heinz Maier Leibnitz Zentrum,
Lichtenbergstrasse 1, 85747 Garching, Germany

###### Abstract

Converting neutron scattering data to real-space time-dependent structures can
only be achieved through suitable models, which is particularly challenging
for geometrically disordered structures. We address this problem by
introducing time-dependent clipped Gaussian field models. General expressions
are derived for all space- and time-correlation functions relevant to coherent
inelastic neutron scattering, for multiphase systems and arbitrary scattering
contrasts. Various dynamic models are introduced that enable one to add time-
dependence to any given spatial statistics, as captured e.g. by small-angle
scattering. In a first approach, the Gaussian field is decomposed into
localised waves that are allowed to fluctuate in time or to move, either
ballistically or diffusively. In a second approach, a dispersion relation is
used to make the spectral components of the field time-dependent. The various
models lead to qualitatively different dynamics, which can be discriminated by
neutron scattering. The methods of the paper are illustrated with oil/water
microemulsion studied by small-angle scattering and neutron spin-echo. All
available data - in both film and bulk contrasts, over the entire range of $q$
and $\tau$\- are analyzed jointly with a single model. The analysis points to
static large-scale structure of the oil and water domains, while the
interfaces are subject to thermal fluctuations. The fluctuations have an
amplitude around 60 Å and contribute to 30 % of the total interface area.

##  I Introduction

Neutron scattering is one of the few experimental techniques that allow one to
probe both the structure and the dynamics of physical systems at the Ångström
scaleSivia (2011); Squires (2012). Typically, structural information is
obtained through the elastic scattering of cold or thermal neutrons (SANS).
The dynamic information is obtained through inelastic and quasi-elastic
scattering, as the neutrons gain or lose energy when they interact with moving
phases in the system. As for most scattering techniques, however, converting
experimental data to real-space and time-dependent structures can be
challenging. This is particularly the case for complex and disordered
structures that cannot be described in simple geometrical terms.

When studying disordered systems, stochastic models often provide a practical
compromise between geometrical realism and mathematical simplicity. The former
is necessary to account for as many geometrical features as possible, and the
latter improves the robustness of the analysis by avoiding unnecessarily large
number of parameters Serra (1982); Torquato (2002); Lantuéjoul (2002). In that
spirit, stochastic models have often been used to analyze small-angle
scattering data from a variety of physical systems and reconstruct their
structure Sonntag, Stoyan, and Hermann (1981); Roberts (1997); Gille (2011);
Gommes (2018); Prehal  _et al._ (2020). In the present paper, we generalize
this type of approach to analyze and model time-dependent structures
investigated by inelastic neutron scattering.

The paper focuses specifically on a family of descriptive models based on
clipped Gaussian random fields. These models originate in the work of Cahn on
spinodal decomposition Cahn (1965), but they have since been used as general
geometrical models of disordered structures in a variety of contexts,
including porous materials Quiblier (1984); Roberts and Teubner (1995); Gommes
(2018), polymers Chen  _et al._ (1996); D’Hollander  _et al._ (2010),
emulsions Berk (1987); Teubner (1991), gels Gommes and Roberts (2008),
confined liquids Gommes (2013); Gommes and Roberts (2018), nanoparticles
Gommes, Chattot, and Drnec (2020), etc. Gaussian random fields are
comprehensively characterized by their correlation function, which makes them
particularly useful in the context of scattering studies.

The theoretical developments of the present paper are illustrated on
previously-published elastic and inelastic neutron scattering data measured on
water/oil microemulsion, which are presented shortly in Section II, together
with some general results pertaining to elastic and inelastic neutron
scattering. Section III covers some classical results of static Gaussian-field
models, which are generalized to time-dependent structures in Section IV.
Three families of dynamic models are proposed, which are applicable to any
static Gaussian field and endow it with qualitatively different time-
dependence. In Section V, some aspects of the models relevant to inelastic
scattering are discussed, and the models are used to analyze the microemulsion
data.

##  II Neutron small-angle scattering and spin-echo data

The methods and models developed in the paper are illustrated with published
neutron small-angle scattering (SANS) patterns and neutron spin-echo (NSE)
data measured on a water/oil microemulsion stabilised with a
surfactantHolderer  _et al._ (2005); Holderer, O.  _et al._ (2007). The
relevant data are available on the authors institutional repositoryHolderer
(2021) and they are displayed in Fig. 1.

The bicontinuous phases of the microemulsion consisted in water and decane
with decyl-polyglycol-ether (C10E4) as a surfactant. The volume fractions of
the three phases were $\phi_{o}\simeq 0.4075$, $\phi_{s}\simeq 0.185$ and
$\phi_{a}\simeq 0.4075$ for oil, surfactant and aqueous phases, respectively.
Small amounts (0.25 wt.%) of homopolymers were dispersed in the continuous
phases in order to slightly modify their viscosity and the efficiency of the
surfactant (see Refs. Holderer  _et al._ (2005); Holderer, O.  _et al._
(2007)), namely polyethylene oxide (PEO) in water and polyethylene propylene
(PEP) in decane. The molecular weights slightly differed in SANS (10 kg/mol)
and NSE (5 kg/mol) experiments, which has only minor effects for the purposes
of this study on the relaxation rate in the NSE experiments ($<10$ %), but
gave a complete set of SANS and NSE data. The microemulsion was prepared in
two different neutron scattering contrasts, by exchanging hydrogen with
deuterium. In the so-called bulk contrast, deuterated water (D2O) was used
with protonated surfactant and oil, which results in a contrast between the
water domains and the oil-surfactant-domains. In film contrast, the decane was
deuterated as well, leaving only the protonated surfactant film visible in the
deuterated water/oil surrounding. The SANS experiments were conducted on the
KWS-2 small angle scattering instrument at the DIDO reactor of
Forschungszentrum Jülich, the NSE experiments were conducted on the IN15
instrument at the Institut Laue-Langevin in Grenoble. The resolutions the SANS
and NSE data in Fig. 1 are $\sigma^{SANS}_{q}=0.0034$ Å-1 and
$\sigma^{NSE}_{q}=0.0085$ Å-1, respectively.

Figure 1: Neutron Small-Angle Scattering data (SANS, $a$) measured on a
microemulsion in bulk (grey) and film (red) contrasts, together with structure
reconstructed from it as a clipped Gaussian field model ($b_{1}$: field,
$b_{2}$ clipped structure with oil in grey and surfactant in red). The neutron
spin-echo (NSE) data measured in the same conditions are shown in $c_{1}$ and
$c_{2}$. In the SANS patterns (a) the dots are the experimental values, and
the solid lines are the fitted model. The values in the Gaussian field shown
in $b_{1}$ range from -2.5 (blue) to + 2.5 (yellow). The error bars are $\pm
2\sigma$ for both SANS and NSE.

Microemulsions are strong coherent scatterers, so that incoherent scattering
from individual atoms (mainly hydrogen) does not play a significant role at
the length scales discussed in this paper. Therefore, the central structural
characteristic of the microemulsion relevant to both the SANS and NSE data is
the scattering-length correlation functionVan Hove (1954); Squires (2012)

$C_{\rho}(r,\tau)=\langle\rho(\mathbf{x},t)\rho(\mathbf{x}+\mathbf{r},t+\tau)\rangle-\langle\rho\rangle^{2}$
(1)

which characterises the statistical correlation between the scattering length
density $\rho$ at two points at a distance $r$ apart, and time lag $\tau$.
Throughout the paper we assume statistical isotropy, so that correlation
functions depend only on the modulus of the distance $r=|\mathbf{r}|$. In Eq.
(1) the brackets $\langle\rangle$ stand for the average value, evaluated over
all accessible positions $\mathbf{x}$ and times $t$. For the type of ergodic
models considered later in the paper, they can also be thought of as ensemble
averages.Torquato (2002); Lantuéjoul (2002); Gommes (2018)

When working with stochastic models it is convenient to introduce the concept
of covariance Serra (1982); Lantuéjoul (2002), which is occasionally also
referred to as 2-point probability functionsTorquato (2002) or stick-
probability functionsCiccariello  _et al._ (1981). The covariance of, say the
oil phase $o$ of the microemulsion, is defined as the probability for two
points at distance $\mathbf{r}$ from one another to belong to that phase at
two moments separated with time lag $\tau$, namely

$C_{oo}(r,\tau)=\textrm{Prob}\left[\left(\mathbf{x}\in o\ \textrm{at time}\
t\right)\&\left(\mathbf{x}+\mathbf{r}\in o\ \textrm{at time}\
t+\tau\right)\right]$ (2)

As this generalizes to cross-covariances for two points belonging to two
distinct phases, the name self-covariance is occasionally used to insist that
the two points belong to the same phase. Because each of the three phases of
the microemulsion - oily, aqueous and surfactant - has a specific scattering-
length density, the correlation function $C_{\rho}(r)$ is a linear combination
of the covariances of the phases. Out of the six self- and cross-covariances
that are defined for a three-phase system, only three are linearly
independent.Torquato (2002) A convenient expression for $C_{\rho}$ is
therefore Gommes (2013)

$\displaystyle C_{\rho}(r,\tau)$
$\displaystyle=(\rho_{o}-\rho_{s})(\rho_{o}-\rho_{a})[C_{oo}(r,\tau)-\phi_{o}^{2}]+(\rho_{s}-\rho_{o})(\rho_{s}-\rho_{a})[C_{ss}(r,\tau)-\phi_{s}^{2}]$
(3)
$\displaystyle+(\rho_{a}-\rho_{o})(\rho_{a}-\rho_{s})[C_{aa}(r,\tau)-\phi_{a}^{2}]$
(4)

where $\rho_{o}$, $\rho_{s}$ and $\rho_{a}$ are the scattering-length
densities of the oil, surfactant, and aqueous phases, respectively; $C_{oo}$,
$C_{ss}$ and $C_{aa}$ are the corresponding self-covariances. A derivation of
Eq. (3) is provided in the Supplementary Material (Sec. SM-1). The bulk
contrast relevant to Fig. 1 correspond to $\rho_{o}=\rho_{s}\neq\rho_{a}$, in
which case $C_{\rho}$ is proportional to $C_{aa}$. The film contrast
corresponds to $\rho_{o}=\rho_{a}\neq\rho_{s}$, and in that case $C_{\rho}$ is
proportional to $C_{ss}$.

The coherent inelastic neutron scattering data is expressed in terms of the
intermediate scattering function $I(q,\tau)$. The latter is defined as the
Fourier transform of the correlation function $C_{\rho}(r,\tau)$, namely Sivia
(2011); Squires (2012)

$I(q,\tau)=\int_{0}^{\infty}\frac{\sin(qr)}{qr}C_{\rho}(r,\tau)4\pi
r^{2}\textrm{d}r$ (5)

and the instrument resolution is accounted by multiplying $C_{\rho}$ by a
spread function with width $\sigma_{q}$, prior to Fourier transform. The
situation relevant to SANS is elastic scattering corresponding to $I(q,0)$, to
which we refer simply as $I(q)$ when there is no ambiguity. The data measured
in NSE instruments is $I(q,\tau)/I(q)$, as given in Fig. 1c1 and 1c2 for the
microemulsion.

In Fig. 1 the SANS data in both film and bulk contrasts were fitted jointly
with a clipped Gaussian field model, adapting a procedure developed
elsewhereGommes (2018). For the sake of completeness, the detailed procedure
is described in the Supplementary Material (Sec. SM-3.3). A realisation of the
model is shown in Fig. 1b.

##  III Clipped Gaussian-field models

###  III.1 Static Gaussian random fields

We focus here on static, that is time-independent, Gaussian random fields
(GRF) and we introduce time-dependence in Sec. IV. A convenient and classical
way to think of GRFs is as a superposition of random sine waves Berk (1991);
Levitz (1998)

$W(\mathbf{x})=\sqrt{\frac{2}{N}}\sum_{n=1}^{N}\sin\left[\mathbf{q}_{n}\cdot\mathbf{x}-\varphi_{n}\right]$
(6)

where the phases are uniformly distributed over $[0,2\pi)$ and the wavevectors
$\mathbf{q}$ are drawn from a user-specified density distribution over
reciprocal space $f_{W}(\mathbf{q})\textrm{d}V_{q}$, referred to as the
spectral density of the field. For asymptotically large values of $N$, the
central limit theorem ensures that $W(\mathbf{x})$ is Gaussian-distributed at
any point $\mathbf{x}$ with average equal to zero, and the factor in Eq. (6)
ensures that the variance is equal to one.

A central characteristic of the GRF in the context of elastic scattering is
its correlation function $g_{W}(r)$, defined as the statistical correlation
between the values of $W(\mathbf{x})$ at two points at distance $r$ apart

$g_{W}(r)=\langle W(\mathbf{x})W(\mathbf{x}+\mathbf{r})\rangle$ (7)

where the brackets have the same meaning as in Eq. (1), and the dependence is
only on the modulus $r=|\mathbf{r}|$ for isotropic fields. The field
correlation function is obtained as the Fourier transform of the spectral
density, namelyBerk (1987, 1991)

$g_{W}(r)=\int_{0}^{\infty}\frac{\sin(qr)}{qr}f_{W}(q)4\pi q^{2}\textrm{d}q$
(8)

In principle any integrable and positive function can be used as a spectral
density. In practice, in order to ensure that the structures modelled by
clipping the field have finite surface areasBerk (1991); Teubner (1991), it is
necessary to impose that the second moment of $f_{W}(q)$ be finite. This
enables one to define $l_{W}$ as

$\frac{1}{l_{W}^{2}}=\frac{1}{6}\int_{0}^{\infty}q^{2}f_{W}(q)4\pi
q^{2}\textrm{d}q$ (9)

which we refer to as the field characteristic length. The finiteness of
$l_{W}$ corresponds to a quadratic behavior of the correlation function
$g_{W}\simeq 1-(r/l_{W})^{2}+\ldots$ for small distances, and is a condition
for the modelled structures to have finite surface areasTeubner (1991); Berk
(1991).

Table 1: Examples of static Gaussian random fields (GRFs) with their spectral
densities $f_{W}(q)$, field correlation functions $g_{W}(r)$, and
characteristic lengths $l_{W}$. The function $w(|\mathbf{x}|)$ is the
corresponding elementary wave relevant to a dilution approach (see text, $l$
and $\mu$ are model parameters). These functions are plotted in Figs. SM-1 to
SM-8 of the Supplementary Material.

GRF# | $f_{W}(q)$ | $g_{W}(r)$ | $l_{W}$ |  $w(\mathbf{x})$a | Ref.
---|---|---|---|---|---
1 | $\frac{l^{2}}{16\pi^{3}}\delta[q-\frac{2\pi}{l}]$ | $\frac{\sin[2\pi r/l]}{2\pi r/l}$ | $\sqrt{6}l/(2\pi)$ | -b | Berk (1987)
2 | $\left(\frac{l}{2\sqrt{\pi}}\right)^{3}e^{-[ql]^{2}/4}$ | $e^{-[r/l]^{2}}$ | $l$ | $e^{-2\left(\frac{|\mathbf{x}|}{l}\right)^{2}}$ | Lantuéjoul (2002); Gelfand  _et al._ (2010)
3 | $\frac{l^{2}}{4q}\frac{\sinh[\pi ql/2]}{1+\cosh[\pi ql]}$ | $\frac{1}{\cosh[r/l]}$ | $\sqrt{2}l$ | -b | Gommes and Roberts (2008)
4 | $\left(\frac{l}{\sqrt{\pi}}\right)^{3}\frac{1}{480}(ql)^{4}e^{-[ql]^{2}/4}$ | $\left[1-\frac{4}{3}\left(\frac{r}{l}\right)^{2}+\frac{4}{15}\left(\frac{r}{l}\right)^{4}\right]e^{-\left(\frac{r}{l}\right)^{2}}$ | $\sqrt{3/7}l$ | $\left[\left(\frac{|\mathbf{x}|}{l}\right)^{2}-\frac{3}{4}\right]e^{-2\left(\frac{|\mathbf{x}|}{l}\right)^{2}}$ |
5 | $\frac{l^{3}}{4\pi^{2}\mu}\frac{\sinh[\pi^{2}/\mu]\sinh[\pi ql/(2\mu)]/(ql)}{\cosh[2\pi^{2}/\mu]+\cosh[\pi ql/\mu]}$ | $\frac{\sin[2\pi r/l]}{(2\pi r/l)\cosh[\mu r/l]}$ | $l/\sqrt{\frac{2\pi^{2}}{3}+\frac{\mu^{2}}{2}}$ | -b | Gommes and Roberts (2008)
6 | $\left(\frac{l}{\sqrt{\pi}}\right)^{3}\frac{\Gamma(\mu+\frac{3}{2})}{\Gamma(\mu)[1+(ql)^{2}]^{\frac{3}{2}+\mu}}$ | $2^{1-\mu}\frac{(r/l)^{\mu}K_{\mu}(r/l)}{\Gamma(\mu+1)}$ | $2l\sqrt{\mu-1}$ | $\left(\frac{|\mathbf{x}|}{l}\right)^{\frac{\mu}{2}-\frac{3}{4}}K_{\frac{\mu}{2}-\frac{3}{4}}\left(\frac{|\mathbf{x}|}{l}\right)$ | Gelfand  _et al._ (2010); Lantuéjoul (2002)
7 | $\left(\frac{l}{\sqrt{\pi}}\right)^{3}\frac{\Gamma(\mu+1)}{\Gamma(\mu-\frac{1}{2})}[1-(ql)^{2}]^{\mu-\frac{3}{2}}$ | $2^{\mu}\Gamma(\mu+1)\frac{J_{\mu}(r/l)}{(r/l)^{\mu}}$ | $2l\sqrt{\mu+1}$ | $\left(\frac{|\mathbf{x}|}{l}\right)^{-\frac{\mu}{2}-\frac{3}{4}}J_{\frac{\mu}{2}+\frac{3}{4}}\left(\frac{|\mathbf{x}|}{l}\right)$ |  Lantuéjoul (2002) -c
8 | Piecewise Linear | Cf. Sup. Info. | Cf. Sup. Info. | -b | Gommes (2018)

a: within unspecified normalizing factor; b: not available; c: a typo in the
formula provided for $f_{W}(q)$ has been corrected in the present table.

The methods developed in the paper apply to any static Gaussian field. A few
examples are given in Tab. 1 with explicit spectral densities, correlation
functions, and characteristic lengths $l_{W}$, which are also plotted in Figs.
SM-1 to SM-8 of the Supplementary Material. These fields are referred to in
the rest of the paper by the number in the first column. GRF-1 contains a
single spectral component, and is arguably the simplest possible Gaussian
field. By contrast, GRF-2 is extremely polydispersed and is referred to in
geostatistics as the squared-exponential correlation function. GRF-3 is also
polydispersed: its correlation function is exponential for asymptotically
large distances, but the $1/\cosh$ function ensures quadratic shape at the
origin and hence finite $l_{W}$. GRF-4 is introduced in Sec. V.2, and leads to
structures with a scattering peak. GRF-5 is obtained by multiplying the
correlation functions of GRF-1 and GRF-3, and provides one with a parameter to
control the polydispersity of the structure, which makes it convenient for SAS
data fittingGommes and Roberts (2008); Prehal  _et al._ (2017). Other examples
discussed in the scattering literature can be found e.g. in Refs. Teubner and
Strey (1987); Chen  _et al._ (1996); Roberts (1997). The following two entries
in Tab. 1 are classical in geostatistics but are seldom used in scattering
studies. GRF-6 is the Matérn model where $K_{\mu}$ is a modified Bessel
function. The parameter $\mu$ controls the smoothness of the field, which is
$\mu-1$ times differentiableRasmussen and Williams (2006), and GRF-2 is
obtained as a particular case for $\mu\to\infty$. By contrast to GRF-6, field
GRF-7 introduces strong correlations through Bessel function $J_{\mu}$. It
leads to peaked scattering functions (see Fig. SM-7) and coincides with GRF-1
in the limit $\mu\to 1/2$.

When it comes to analyzing experimental scattering patterns, the simple
analytical expressions in Tab. 1 seldom provide sufficient flexibility for
data fitting. Therefore, a convenient approach consists in linearly combining
independent Gaussian fields $W_{i}(\mathbf{x})$, with spectral densities
$f_{W}^{(i)}(q)$, so as to create a composite field

$W(\mathbf{x})=\sum_{i}\sigma_{i}W_{i}(\mathbf{x})$ (10)

where $\sigma_{i}$ are constants. The spectral density of the resulting field
is

$f_{W}(q)=\sum_{i}\sigma^{2}_{i}f_{W}^{(i)}(q)$ (11)

and a similar relation holds for $g_{W}(r)$. Because the integral of
$f_{W}(q)$ over the entire reciprocal space is the variance of the field, the
parameter $\sigma_{i}^{2}$ can be thought of as the contribution of
$W_{i}(\mathbf{x})$ to the total variance of the composite field
$W(\mathbf{x})$. In that spirit, a possible approach to data fitting would
consist in combining a large number of monodispersed fields (e.g. GRF-1 in
Tab. 1), so as to approximate an experimental spectral density as a sum of
Dirac peaks. As an unpractically large number of peaks might be needed to
approximate a continuous function, a more practical approach consists in
replacing the Dirac peaks by broader functions. The piecewise-linear model
(GRF-8 in Tab. 1) corresponds to such an approach, which was developed in
earlier workGommes (2018). As this approach was used here to fit the SANS data
in Fig. 1a, it is described in detail in the Supplementary Material (Sec.
SM-3). In particular, the influence of the number of nodes for the SANS fit
shown in Fig. 1a is illustrated in Fig. SM-10.

When generalising the Gaussian-field modelling to time-dependent structures it
will prove useful to use another construction of Gaussian fields, which is
mathematically equivalent to Eq. (6). In so-called dilution random functions,
Serra (1982); Lantuejoul (1991); Lantuéjoul (2002) a field is created as a sum
of localised elementary waves $w(\mathbf{x})$, randomly positioned in space,
namely

$W(\mathbf{x})=\sum_{s}A_{s}w(\mathbf{x}-\mathbf{x}_{s})$ (12)

where the sum is on all the seeds $\mathbf{x}_{s}$ of a Poisson point process
with density $\theta$, and $A_{s}$ is any random amplitude satisfying $\langle
A\rangle=0$ and $\langle A_{s}A_{s^{\prime}}\rangle=\langle
A^{2}\rangle\delta_{ss^{\prime}}$. The latter condition corresponds to
uncorrelated wave amplitudes. In the limit of a large density of the Poisson
process, many elementary waves overlap at any given point of space so that the
values of the field defined in Eq. (12) become Gaussian distributed.

In the context of a dilution approach, the correlation function of the field
is calculated asSerra (1982); Lantuejoul (1991); Lantuéjoul (2002)

$g_{W}(\mathbf{r})=\theta\langle A^{2}\rangle K(\mathbf{r})$ (13)

where

$K(\mathbf{r})=\int\textrm{d}V_{x}\ w(\mathbf{x})w(\mathbf{x}-\mathbf{r})$
(14)

is the self-convolution of the elementary wave. In order to ensure that the
variance of the field is equal to one, one has to impose $g_{W}(0)=1$, which
requires adjusting the amplitudes so that $\theta\langle A^{2}\rangle K(0)=1$.
Although Eqs. (6) and (12) are conceptually different constructions, the two
approaches are mathematically equivalent. The spectral density of the dilution
model is indeed obtained as

$f_{W}(\mathbf{q})=\theta\langle A^{2}\rangle\left|\int\textrm{d}V_{x}\
w(\mathbf{x})\exp(i\mathbf{q}\cdot\mathbf{x})\right|^{2}$ (15)

which results from evaluating the Fourier transform of Eq. (13). Among the
static Gaussian fields presented in Tab. 1, the shape of the elementary wave
is know for GRF-2, GRF-4, GRF-6 and GRF-7. Conceptually, however, any field
such that $\sqrt{f_{W}(q)}$ is integrable can be thought of as resulting from
a superposition of a large number of randomly positioned elementary waves.

###  III.2 Clipping procedure

Figure 2: Microemulsion modelling as a clipped Gaussian field, with the
underlying field shown in (a) and the structure in (b). The clipping
thresholds are $\alpha=-0.234$ and $\beta=+0.234$, resulting in aqueous
(white) and oil (grey) phases with volume fractions
$\phi_{a}=\phi_{o}=0.4075$, and $\phi_{s}=0.185$ for the surfactant (red). The
figure is a 2D cut out of a 3D realization obtained from GRF-2 of Tab. 1, with
distances normalized to $l_{W}$.

According to a classical approach, the phases of disordered systems can be
modelled as excursion sets of a Gaussian field $W(\mathbf{x})$, which is also
referred to as clipped-Gaussian-field models.Quiblier (1984); Berk (1987) In
the particular case of emulsionsTeubner (1991) a convenient clipping procedure
is based on two thresholds $\alpha\leq\beta$, as sketched in Fig. 2. The oil
phase is modelled as the points of space where $\beta\leq W(\mathbf{x})$, the
surfactant film-like phase as $\alpha\leq W(\mathbf{x})<\beta$, and the
aqueous phase as $W(\mathbf{x})<\alpha$.

Because the values of $W(\mathbf{x})$ are Gaussian distributed, the values of
the thresholds control the volume fractions of the phases. The volume fraction
of the oil $\phi_{o}$, is obtained as

$\phi_{o}=\Lambda_{1}[\beta]$ (16)

where the function $\Lambda_{1}[x]$ is the probability for a univariate
Gaussian variable to take values larger than $x$, which can be calculated as

$\Lambda_{1}[x]=\frac{1}{2}\left(1-\textrm{erf}[x/\sqrt{2}]\right)$ (17)

where erf is the error function. With the same notation, the volume fraction
of the surfactant phase is

$\phi_{s}=\Lambda_{1}[\alpha]-\Lambda_{1}[\beta]$ (18)

and the volume fraction of the remaining aqueous phase is
$\phi_{a}=1-\phi_{o}-\phi_{s}$. Relevant values for the microemulsion of Sec.
II are $\alpha=-0.234$ and $\beta\simeq+0.234$, corresponding to
$\phi_{f}\simeq 0.185$ and $\phi_{a}=\phi_{o}=0.4075$, as used in Fig. 2.

The scattering functions are obtained from the covariances of the various
phases of the microemulsion, which are calculated from the field correlation
function $g_{W}(r)$ and the clipping thresholds $\alpha$ and $\beta$. In line
with Eq. (3), we consider here the covariances $C_{oo}(r)$, $C_{ss}(r)$ and
$C_{aa}(r)$, defined as the probabilities for two randomly chosen points are
distance $r$ from one another to belong both to the oil, surfactant, and
aqueous phases, respectively. The covariances are expressed in terms of the
bivariate error function $\Lambda_{2}[\alpha,\beta,g]$, defined as the
probability for two correlated Gaussian variables, with correlation $g$, to
take values larger than $\alpha$ and $\beta$, respectively Roberts and Teubner
(1995). Explicitly the expressions areTeubner (1991); Levitz (1998)

$C_{oo}(r)=\Lambda_{2}[\beta,\beta,g_{W}(r)]$ (19)

for the oil phase,

$C_{ss}(r)=\Lambda_{2}[\alpha,\alpha,g_{W}(r)]+\Lambda_{2}[\beta,\beta,g_{W}(r)]-2\Lambda_{2}[\alpha,\beta,g_{W}(r)]$
(20)

for the surfactant phase, and

$C_{aa}(r)=1-2\Lambda_{1}[\alpha]+\Lambda_{2}[\alpha,\alpha,g_{W}(r)]$ (21)

for the aqueous phase. The values used to calculate the scattering functions
through Eqs. (3) are the centred covariance $\bar{C}_{oo}(r)$,
$\bar{C}_{ss}(r)$ and $\bar{C}_{aa}(r)$, obtained by subtracting the
corresponding squared volume fraction, so as to enable their Fourier
transformation through Eq. (5).

In principle the function $\Lambda_{2}[\alpha,\beta,g]$ can be calculated as
two-dimensional integral of a bivariate Gaussian distribution. Based on
Dirichlet’s representation of Heaviside’s step function, it can be calculated
in the following simpler wayBerk (1991); Teubner (1991); Roberts and Teubner
(1995); Levitz (1998)

$\displaystyle\Lambda_{2}[\alpha,\beta,g]$ $\displaystyle=$
$\displaystyle\Lambda_{1}[\alpha]\Lambda_{1}[\beta]+\frac{1}{2\pi}\int_{0}^{\textrm{asin}[g]}\exp\left[-\frac{\alpha^{2}+\beta^{2}-2\alpha\beta\sin(\theta)}{2\cos^{2}(\theta)}\right]\
\textrm{d}\theta$ (22)

which requires numerically evaluating only a one-dimensional integral. For any
given volume fraction of the phases, that is for given $\alpha$ and $\beta$,
Eqs. (19), (20) and (21) define non-linear relations between the field
correlation $g_{W}$ and the corresponding covariances. These relations are
illustrated in Fig. 3 for the centred covariances $\bar{C}_{oo}$,
$\bar{C}_{aa}$ and $\bar{C}_{ss}$, for the values of $\alpha$ and $\beta$
relevant to the microemulsion data. Note that the values satisfy
$\alpha=-\beta$, corresponding to $\phi_{o}=\phi_{a}$ and
$\bar{C}_{oo}(r)=\bar{C}_{aa}(r)$.

Figure 3: Effect of clipping: non-linear relation between the field
correlation $g_{W}$ and the centered covariance $\bar{C}=C-\phi^{2}$ of the
aqueous, oil, and surfactant phases, calculated from Eqs. (19-21) with
clipping constants $\alpha\simeq-0.234$ and $\beta\simeq+0.234$. The relations
are highly non-linear for $g_{W}\simeq 1$ and linear for asymptotically small
$g_{W}$, which limit is relevant for asymptotically large values of $r$ or
$\tau$. The dashed lines are the asymptotic approximations for $g_{W}\to 1$
calculated through Eq. (24), relevant to vanishingly small $r$ and $\tau$.

As visible in Fig. 3, for small field correlations $g_{W}$ the centred
covariances $\bar{C}_{oo}$ and $\bar{C}_{aa}$ are proportional to $g_{W}$. In
that region - corresponding to large $r$ or $\tau$ \- the bivariate error
function $\Lambda_{2}$ is approximated by

$\displaystyle\Lambda_{2}[\alpha,\beta,g]$ $\displaystyle\simeq$
$\displaystyle\Lambda_{1}[\alpha]\Lambda_{1}[\beta]+\frac{g}{2\pi}\exp\left(-\frac{\alpha^{2}+\beta^{2}}{2}\right)+\ldots$
(23)

which is the first term of a general development in terms of Hermite
polynomialsLantuéjoul (2002); Gommes (2013). However, in general $\Lambda_{2}$
is a non-linear function. In particular the relation between $g_{W}$ and the
covariances of any clipped structure is vertical when $g_{W}$ approaches 1
(see Fig. 3). The following asymptotic relation is useful for further purposes

$\displaystyle\Lambda_{2}[\alpha,\beta,1-\epsilon^{2}]=\Lambda_{1}\left[\textrm{max}\\{\alpha,\beta\\}\right]$
$\displaystyle-$
$\displaystyle\frac{\epsilon}{\pi\sqrt{2}}e^{-\frac{\alpha\beta}{2}}e^{-\frac{(\alpha-\beta)^{2}}{4\epsilon^{2}}}$
(24)
$\displaystyle+\frac{|\alpha-\beta|}{2\sqrt{2\pi}}e^{-\frac{\alpha\beta}{2}}\Big{(}1$
$\displaystyle-$
$\displaystyle\textrm{erf}\left[\frac{|\alpha-\beta|}{2\epsilon}\right]\Big{)}$
(25)

It is obtained by setting $g=1-\epsilon^{2}$ in Eq. (22), through a first-
order expansion in $\epsilon$. This equation controls the shape of the
covariances for asymptotically small $r$ and $\tau$, and therefore the
asymptotic shape of the scattering functions for large $q$ and small $\tau$,
as we discuss in detail later. The centred covariances approximated through
Eq. (24), are shown as dashed lines in Fig. 3.

##  IV Time-dependent clipped Gaussian-field models

We now introduce three qualitatively different dynamic models to construct
time-dependent Gaussian fields, starting from any static Gaussian field. This
is achieved by adapting Eqs. (6) or (12), by which the fields are constructed.
Although the models are quite general, the discussion is centred on static
field GRF-2 of Tab. 1, which has the following spectral density

$f_{W}(q)=\left(\frac{l}{2\sqrt{\pi}}\right)^{3}\exp\left[-\frac{(ql)^{2}}{4}\right]$
(26)

and field correlation function

$g_{W}(r)=\exp\left[-\left(\frac{r}{l}\right)^{2}\right]$ (27)

where $l$ is model parameter that coincides with the characteristic length
$l_{W}$. With this specific field the main results can be expressed in
analytical form. Moreover the corresponding elementary wave $w(\mathbf{x})$ is
also known analytically (see Tab. 1), which enables one to construct
realizations and visually illustrate all considered dynamic models.

All results of Sec. III remain valid for time-dependent Gaussian fields. This
is notably the case for the clipping relations between the field correlation
and the covariances of the water, oil and surfactant phases. However, the
field correlation function describes here the statistical correlation between
the values of $W$ at two points at distance $r$ apart, with a time lag $\tau$,
namely

$g_{W}(r,\tau)=\langle W(\mathbf{x},t)W(\mathbf{x}+\mathbf{r},t+\tau)\rangle$
(28)

In this case, the covariances obtained through Eqs. (19-21) are Van-Hove
correlation functions,Sivia (2011); Squires (2012) and the intensity obtained
subsequently through Eqs. (3 \- 5) is the coherent intermediate scattering
function $I(q,\tau)$.

Throughout this section, the qualitative geometrical properties of the
Gaussian fields are illustrated by clipping them at the value $\alpha=0$. This
yields two-phase morphologies with volume fractions $\phi=1/2$, different from
the emulsion in Fig. 1. All the mathematical results, however, are quite
general and remain valid for any clipping procedure. The specific case of the
three-phase emulsion, with finite surfactant volume, is considered again in
the discussion section.

###  IV.1 Dynamic model 1: independent time and space fluctuations

In the first approach, a field is created with statistically-independent space
and time fluctuations. This is achieved by starting from a series of
independent static fields $W_{n}(\mathbf{x})$, with $n=1,\ldots,N$, and
combining them linearly with time-dependent coefficients. The fields $W_{n}$
can be thought of as independent realisations of Eq. (6) each with different
random numbers but the same spectral density $f_{W}(q)$. The statistical
independence is expressed as

$\langle
W_{n}(\mathbf{x})W_{m}(\mathbf{x}+\mathbf{r})\rangle=g_{W}(r)\delta_{mn}$ (29)

where $\delta_{mn}=1$ for $m=n$ and 0 otherwise. Based on the set of
$W_{n}(\mathbf{x})$ the time-dependent field is built as

$W(\mathbf{x},t)=\sqrt{\frac{2}{N}}\sum_{n=1}^{N}W_{n}(\mathbf{x})\cos(\omega_{n}t-\varphi_{n})$
(30)

where the phases $\varphi_{n}$ are random and uniform over $[0,2\pi)$, and the
frequencies $\omega_{n}$ are drawn from a temporal spectral density
$f^{\prime}(\omega)\textrm{d}\omega$. With this first dynamic model, the space
and time field correlation function in Eq. (28) is found to be

$g_{W}(r,\tau)=g_{W}(r)g^{\prime}(\tau)$ (31)

in the limit of large $N$, with

$g^{\prime}(\tau)=\int_{0}^{\infty}\cos[\omega\tau]f^{\prime}(\omega)\textrm{d}\omega$
(32)

In geostatistics, models satisfying Eq. (31) are referred to as being
separable.Gelfand  _et al._ (2010)

A physical interpretation of separable models is obtained by noting that they
can be constructed in mathematically-equivalent way through a dilution
approach, as in Eq. (12). Indeed, the field constructed as

$W(\mathbf{x},t)=\sqrt{2}\sum_{s}A_{s}w(\mathbf{x}-\mathbf{x}_{s})\cos(\omega_{s}t-\varphi_{s})$
(33)

with $\varphi_{s}$ uniformly distributed in $[0,2\pi)$, and $\omega_{s}$
distributed according to temporal spectral density $f^{\prime}(\omega)$, has
the same correlation function as in Eq. (31). Therefore, dynamic model 1 can
be interpreted as resulting from incoherently fluctuating elementary waves.

Figure 4: Two-dimensional $(x,t)$ cuts through four-dimensional $(x,y,z,t)$
realizations of dynamic model 1, with static field GRF-2 from Tab. 1 and
exponential (a) and hyperbolic secant (b) temporal correlation functions. The
threshold assumed in the figure is $\alpha=0$, corresponding to $\phi=0.5$ for
the phases shown in white and grey.

For the purpose of data modelling, a natural choice for the temporal
correlation function is the exponential
$g^{\prime}(\tau)=\exp[-\tau/\tau_{c}]$, where the correlation time $\tau_{c}$
is a model parameter. This choice corresponds to the following spectral
density

$f^{\prime}(\omega)=\frac{1}{\pi}\frac{2\tau_{c}}{1+(\omega\tau_{c})^{2}}$
(34)

A realization of the clipped Gaussian field obtained from this specific time-
correlation function and static field GRF-2 is shown in Fig. 4a. The
corresponding correlation function $g_{W}(r,\tau)$ is plotted in Fig. 5,
together with the covariance assuming a single clipping threshold $\alpha=0$,
and the corresponding intermediate scattering function in the form of
$I(q,\tau)/I(q)$.

Figure 5: Correlation function $g_{W}(r,\tau)$ for static field GRF-2 and
independent space and time fluctuations, with exponential (a1) and hyperbolic-
secant (a2) time-correlation functions. The corresponding covariance (clipping
threshold $\alpha=0$) and intermediate scattering function are shown in b1/b2
and c1/c2. The two solid lines in b1/b2 and c1/c2 highlight the values for
$\tau=0$ and for small yet finite $\tau$.

The type of dynamics obtained from the exponential time correlation function
in Fig. 4a is extremely rugged. Smoother dynamics is obtained by modelling the
temporal correlation function as a hyperbolic secant
$g^{\prime}(\tau)=1/\cosh[\tau/\tau_{c}]$, which behaves like an exponential
for asymptotically large times but differs for short times. Its temporal
spectral density is

$f^{\prime}(\omega)=\frac{2\tau_{c}\cosh[\pi\omega\tau_{c}/2]}{1+\cosh[\pi\omega\tau_{c}]}$
(35)

A realization of the clipped Gaussian field obtained with this expression is
shown in Fig. 4b. The corresponding correlation function $g_{W}(r,\tau)$,
covariance (for $\alpha=0$), and intermediate scattering function are plotted
in Fig. 5a2 to 5c2.

###  IV.2 Dynamic model 2: dispersion relation

The second dynamic model introduces correlations between space and time
fluctuations, and belongs to the class of non-separable models.Cressie and
Huang (1999); Gneiting (2002) This is achieved through a dispersion relation
that deterministically assigns a specific temporal frequency $\omega$ to any
spatial frequency $q$ of the Gaussian field, and leads to the following
generalization of Eq. (6)

$W(\mathbf{x},t)=\sqrt{\frac{2}{N}}\sum_{n=1}^{N}\sin\left[\mathbf{q}_{n}\cdot\mathbf{x}+\omega(|\mathbf{q}_{n}|)t-\varphi_{n}\right]$
(36)

where $\omega(|\mathbf{q}|)$ is the dispersion relation, which we assume to be
isotropic. Based on the statistical independence of the various components of
the field in Eq. (36), the field correlation function is calculated as

$g_{W}(r,\tau)=\int_{0}^{\infty}f_{W}(q)\cos[\omega(q)\tau]\frac{\sin(qr)}{qr}4\pi
q^{2}\textrm{d}q$ (37)

in the limit of asymptotically large $N$.

In principle any suitable function can be used to model a dispersion relation.
We consider here two simple analytical forms $\omega=cq$ and $\omega=Dq^{2}$,
where $c$ and $D$ are constants with dimensions of velocity and diffusion
coefficient, respectively. Realizations obtained with static field GRF-2 and
these two dispersion relations are given in Fig. 6a and 6b. In the case of the
linear dispersion relation the structures propagate at constant velocity $c$,
which appears as slanted features with slopes $\pm 1$ on the scales of the
figure. In the case of quadratic dispersion the structures propagate with
size-dependent velocity, which leads to more complicated temporal evolution.

Figure 6: Two-dimensional $(x,t)$ cuts through four-dimensional $(x,y,z,t)$
realizations of model 2 of time-dependent Gaussian field, with GRF-2 from Tab.
1, and linear (a) and quadratic (b) dispersion relations. The threshold
assumed in the figure is $\alpha=0$, corresponding to $\phi=0.5$ for the
phases shown in white and grey.

In the particular case of static field GRF-2, analytical expressions are
obtained for the field correlation function through Eq. (37). For the linear
dispersion relation, one finds

$\displaystyle
g_{W}(r,\tau)=\exp\left[-\frac{r^{2}+(c\tau)^{2}}{l_{W}^{2}}\right]\times\Big{\\{}\cosh\left[\frac{2rc\tau}{l_{W}^{2}}\right]$
$\displaystyle-$ $\displaystyle
2\left(\frac{c\tau}{l_{W}}\right)^{2}\sinh\left[\frac{2rc\tau}{l_{W}^{2}}\right]/\left[\frac{2rc\tau}{l_{W}^{2}}\right]\Big{\\}}$
(38)

and for the quadratic dispersion, the relation is

$g_{W}(r,\tau)=\frac{\exp\left[-\frac{(r/l_{W})^{2}}{1+\left[4D\tau/l_{W}^{2}\right]^{2}}\right]}{\left(1+\left[4D\tau/l_{W}^{2}\right]^{2}\right)^{3/4}}\cos\left[\frac{(r/l_{W})^{2}4D\tau/l_{W}^{2}}{1+\left[4D\tau/l_{W}^{2}\right]^{2}}-\frac{3}{2}\tan^{-1}\left[\frac{4D\tau}{l_{W}^{2}}\right]\right]$
(39)

Detailed derivations of these equations are given in the Supplementary
Material (Sec. SM-4). The correlation functions and corresponding intermediate
scattering functions are plotted in Fig. 7.

Figure 7: Correlation function $g_{W}(r,\tau)$ for GRF-2 from Tab. 1 with
linear (a1) and quadratic (a2) dispersion relations, together with
corresponding intermediate scattering functions (b1 and b2) for clipping
threshold $\alpha=0$.

An interesting characteristic of the time-dependent structure in Fig. 6a is
that it displays temporal order in spite of being spatially disordered. The
absence of any feature in $g_{W}(r,0)$ testifies to spatial disorder (see Fig.
7a1 and Fig. SM-3). By contrast, the correlation function $g_{W}(0,\tau)$
displays a sharp minimum at $\tau=\sqrt{3/2}\times l_{W}/c$. This corresponds
to situation where a fixed point of space is visited alternatively by one
phase and the other with quasi periodicity.

The temporal order is not obvious in the correlation function of the quadratic
dispersion (Fig. 7a2), but the intermediate scattering function $I(q,\tau)$
exhibits marked oscillations for both dispersion relations (Figs. 7b1 and b2).
This can be understood by noting that for asymptotically large values of
$\tau$ the covariance is proportional to $g_{W}(r,\tau)$ (see the discussion
of Fig. 3), so that $I(q,\tau)$ is approximately the Fourier transform of
$g_{W}(r,\tau)$. It therefore results from the Fourier inversion of Eq. (37)
that $I(q,\tau)$ is proportional to $f_{W}(q)\cos[\omega(q)\tau]$ for large
values of $\tau$. It is the cosine in this expression that is responsible for
the observed oscillations in Fig. 7b1 and 7b2. For the particular
representation as $I(q,\tau)/I(q)$ the oscillations are further amplified by
the small value of $I(q)$ for large $q$.

###  IV.3 Dynamic model 3: moving waves

Experimental scattering functions seldom display the type of marked
oscillations obtained with dynamic model 2 and displayed in Fig. 7. In dynamic
model 3, the temporal correlations and corresponding oscillations are
naturally damped through a dilution approach, i.e. by using Eq. (12) instead
of Eq. (6) to describe the Gaussian field. Explicitely, the Gaussian field is
made time-dependent by allowing the elementary waves to propagate

$W(\mathbf{x},t)=\sum_{s}A_{s}w(\mathbf{x}-\mathbf{x}_{s}-\mathbf{j}_{s}(t))$
(40)

where $\mathbf{x}_{s}$ is the initial position of wave $s$, and
$\mathbf{j}_{s}(t)$ is the vectorial distance it has travelled at time $t$. We
consider two qualitatively different cases: the ballistic or diffusive motions
of waves with velocity $c$ or diffusion coefficient $D$. For static field
GRF-2 the shape of the elementary wave $w(\mathbf{x})$ is known mathematically
(see Tab. 1), which enables one to construct realizations as shown in Fig. 8a
and b. The diffusive model displays the same type of rugged dynamics as in
Fig. 4a, which we analyze in detail in the discussion section.

Figure 8: Two-dimensional $(x,t)$ cuts through four-dimensional $(x,y,z,t)$
realizations of dynamic model 3 with static field GRF-2 from Tab. 1, and (a)
ballistic and (b) diffusive propagation of elementary waves. The threshold
assumed in the figure is $\alpha=0$, corresponding to $\phi=0.5$ for the
phases shown in white and grey.

The field correlation function corresponding to Eq. (40) is calculated from
the time-dependent distributions $f_{t}(\mathbf{j})\textrm{d}V_{j}$ of the
wave position $\mathbf{j}$ as follows

$g_{W}(\mathbf{r},\tau)=\theta\langle A^{2}\rangle\int
K(\mathbf{r}-\mathbf{j})f_{\tau}(\mathbf{j})\ \textrm{d}V_{j}$ (41)

which generalizes Eq. (13) to time-dependent dilution processes. If the
elementary waves are compact in real space, then $K(\mathbf{r})$ has a range
comparable to the characteristic length $l_{W}$ of the field. It therefore
results from Eq. (41) that all correlations disappear as soon as the
elementary waves have travelled a distance comparable to $l_{W}$, which
happens in a time $l_{W}/c$ or $l_{W}^{2}/D$ for the ballistic or diffusive
case, respectively.

The case of ballistic propagation corresponds to density distribution

$f_{\tau}(\mathbf{j})=\frac{\delta(j-c\tau)}{4\pi j^{2}}$ (42)

where $j=|\mathbf{j}|$, $\delta(.)$ is Dirac’s function, and the denominator
accounts for the normalization of the probabilities. With such distribution,
Eq. (41) reduces to an integration on the unit sphere, and leads to

$g_{W}(r,\tau)=\theta\langle
A^{2}\rangle\frac{1}{2}\int_{-1}^{+1}K\left(\sqrt{r^{2}+(c\tau)^{2}-2rc\tau\mu}\right)\
\textrm{d}\mu$ (43)

In the case of static field GRF-2, this can be calculated explicitly as

$g_{W}(r,\tau)=\exp\left[-\left(\frac{r-c\tau}{l_{W}}\right)^{2}\right]\frac{1-\exp\left[-4rc\tau/l_{W}^{2}\right]}{4rc\tau/l_{W}^{2}}$
(44)

which is plotted in Fig. 9a1. The corresponding intermediate scattering
function (assuming $\alpha=0$) is plotted in Fig. 9b1.

Figure 9: Correlation function $g_{W}(r,\tau)$ of model 3 for Gaussian fields
GRF-2 with (a1) ballistic and (a2) diffusive motion of elementary waves,
together with corresponding intermediate scattering functions (b1 and b2) for
clipping threshold $\alpha=0$.

In the diffusive case, the probability distribution of $\mathbf{j}$ is given
by the classical expression for the position of a random walkerBerg (1993);
Cussler (2009)

$f_{t}(\mathbf{j})=\left(4\pi
Dt\right)^{-3/2}\exp\left[-\frac{|\mathbf{j}|^{2}}{4Dt}\right]$ (45)

For the case of static field GRF-2, Eq. (41) then leads to the following field
correlation function

$g_{W}(r,\tau)=\left(1+\frac{4D\tau}{l_{W}^{2}}\right)^{-3/2}\exp\left[-\frac{r^{2}}{l_{W}^{2}+4D\tau}\right]$
(46)

This function is plotted in Fig. 9a2, with the corresponding intermediate
scattering function (assuming $\alpha=0$) in Fig. 9b2.

The asymptotic behaviour of the intermediate scattering function $I(q,\tau)$
for large $\tau$ can be understood by expressing the field correlation
function in Eq. (41) in terms of the spectral density as follows

$g_{W}(r,\tau)=\int_{0}^{\infty}f_{W}(q)F_{\tau}(q)\frac{\sin[qr]}{qr}4\pi
q^{2}\textrm{d}q$ (47)

where $F_{\tau}(q)$ is the Fourier transform of $f_{\tau}(\mathbf{j})$. This
expression results directly from Eq. (41) by equating $\theta\langle
A^{2}\rangle K(r)$ to the Fourier transform of $f_{W}(q)$. For the same reason
as for model 2, the intermediate scattering function is proportional to the
Fourier transform of $g_{W}(r,\tau)$ in the limit of asymptotically large
$\tau$. In the case of model 3, this implies $I(q,\tau)\simeq
f_{W}(q)F_{\tau}(q)$. In the case of the ballistic motion described by Eq.
(42), the relevant value of $F_{\tau}(q)$ is

$F_{\tau}(q)=\frac{\sin[qc\tau]}{qc\tau}$ (48)

which explains the mild oscillations in Fig. 9b1. In the case of the diffusive
motion described by Eq. (45), the relevant function is

$F_{\tau}(q)=\exp\left[-q^{2}D\tau\right]$ (49)

which explains why no oscillations are observed at all in Fig. 9b2. Finally,
note that the expression of the field correlation function in Eq. (47) relies
on the spectral density and makes no explicit reference to the elementary wave
used to build the model. Therefore the usability of dynamic model 3 is not
limited to static Gaussian fields for which the form of the elementary wave is
known explicitly.

##  V Discussion

###  V.1 Temporal crossing rate

An interesting characteristic of the Neutron Spin-Echo (NSE) data in Fig. 1c1
and 1c2 is the very steep $\tau$-dependence for small $\tau$ and large $q$.
Among the three dynamic models discussed in Sec. IV, this type of behavior was
observed for model 1 with exponential time correlation function (Fig. 5c1),
and for model 3 with diffusive motion of elementary waves (Fig. 9b2). In both
cases, the realisations testify to extremely rugged dynamics as illustrated in
Figs. 4a and 8b.

A useful mathematical concept to describe the two types of dynamics in both
Fig. 4 and Fig. 8 is the temporal crossing rate $n_{t}$, which characterizes
how often a fixed point in space is crossed by moving interfaces. As discussed
shortly, this concept is the temporal equivalent of the spatial notion of
specific surface area. The surface area $a_{V}$ is defined as the total area
of an interface per unit volume of the system. For an isotropic system it is
mathematically related to the notion of average chord length, which
characterizes how frequently one crosses the interface when traveling along
any straight line crossing the system.Gommes  _et al._ (2020) The significance
of the surface area for scattering was first acknowledged by Debye, who
related $a_{V}$ to the small-$r$ behaviour of the covariance asDebye, Anderson
Jr., and Brumberger (1957)

$C(r,0)\simeq\phi-\frac{a_{V}}{4}r+\ldots$ (50)

which converts in reciprocal space to the well-known Porod’s lawGuinier
(1963); Ciccariello, Goodisman, and Brumberger (1988); Ciccariello and Sobry
(1995)

$I(q,0)\simeq\frac{2\pi a_{V}}{q^{4}}$ (51)

In the particular case of clipped Gaussian-field structures, the surface area
of an isosurface, say at $W(\mathbf{x})=\alpha$, is calculated asTeubner
(1991); Berk (1991)

$a_{V}=\frac{2\sqrt{2}}{\pi l_{W}}e^{-\alpha^{2}/2}$ (52)

where $l_{W}$ is the characteristic length of the field defined in Eq. (9).

The crossing rate $n_{t}$ is related to the covariance via

$C(0,\tau)=\phi-\frac{n_{t}}{2}\tau+\ldots$ (53)

which illustrates further the similarity between $n_{t}$ and the surface area
$a_{V}$ in Eq. (50). The factor $1/2$ differs from the factor $1/4$ in Eq.
(50) because the temporal process considered here is one-dimensional.Torquato
(2002) From Eq. (53) the crossing rate can be calculated as the limit of
$-2(\partial C/\partial\tau)$ for vanishingly small $r$ and $\tau$. In the
case of a clipped Gaussian field model, this limit corresponds to values of
the field correlation function $g_{W}(r,\tau)$ asymptotically close to 1. The
derivative can then be calculated using the asymptotic result in Eq. (24). The
following expression is then obtained for the crossing rate

$n_{t}=\frac{\sqrt{2}}{\pi}e^{-\alpha^{2}/2}\lim_{\tau\to
0}\frac{\sqrt{1-g_{W}(0,\tau)}}{\tau}$ (54)

which provides a physical interpretation to the small-$\tau$ behavior of
$g_{W}(0,\tau)$. The condition for having finite $n_{t}$ is that the field
correlation function should be quadratic at the origin

$g_{W}(0,\tau)\simeq 1-(\tau/\tau_{W})^{2}+\ldots$ (55)

which defines a natural characteristic time $\tau_{W}$. Note that the two
dynamic models of Sec. IV displaying rugged dynamics both have correlation
functions that are linear at the origin. In the case of model 1 with
exponential correlation function, Eq. (31) leads to

$g_{W}(0,\tau)=1-\tau/\tau_{c}+\ldots$ (56)

and in the case of model 3 with diffusive wave motion, Eq. (46) leads to

$g_{W}(0,\tau)=1-\frac{6D\tau}{l_{W}^{2}}+\ldots$ (57)

In both cases $\tau_{W}$ is not defined and Eq. (54) predicts infinite
crossing rate. The effect of linear versus quadratic correlation at short
times is illustrated further in Fig. 10.

Figure 10: Field correlation functions at a fixed point in space
$g_{W}(0,\tau)$ for dynamic model 1 with exponential (blue) and hyperbolic-
secant (black) temporal correlation functions. The insets are realizations of
the time-dependent Gaussian fields at a fixed point in space, together with
clipped structure with $\alpha=0$, for the exponential (a) and hyperbolic-
secant (b) models. The dashed line highlights the quadratic shape of the
hyperbolic-secant model at the origin.

Whether the crossing rate $n_{t}$ is finite or infinite controls the shape of
the intermediate scattering function $I(q,\tau)$ for asymptotically large $q$
and small $\tau$ (see Figs. 5 and 9). The analysis builds on the following two
mathematical facts. (i) First, the non-linearity of the clipping function at
$g_{W}=1$ is of the type $C\simeq 1-\sqrt{1-g_{W}}$ (see Fig. 3 and Eq. 24).
In particular, this converts a quadratic field correlation
$g_{W}(r,\tau)\simeq 1-r^{2}$ into a linear covariance $C(r,\tau)\simeq 1-r$.
This also converts a linear field correlation $g_{W}(r,\tau)\simeq 1-\tau$
into a singular covariance $C(r,\tau)\simeq 1-\sqrt{\tau}$. (ii) Second, the
asymptotic behavior of $I(q,\tau)$ for large $q$ is controlled by the
small-$r$ behavior of $C(r,\tau)$. This results from a generalization of the
Riemann-Lebesgue lemma by LighthillLighthill (1958); Berk (1991), and is
shortly discussed in the Supplementary Material (Sec. SM-6). In particular, a
covariance that is linear at the origin $C(r,0)=1-r$ leads to a $1/q^{4}$
scattering, in line with Porod’s law in Eq. (51). By contrast, a covariance
whose derivatives all vanish at $r=0$ leads to a scattering that decreases
faster than any power law.

Consider now the case where $\tau_{W}$ exists, i.e. where $g_{W}(r,\tau)$ is
quadratic in $\tau$ at the origin, and $C(r,\tau)$ is linear in $\tau$ (e.g.
Figs. 5b2). In that case Porod’s law holds not only for $I(q)$ but also for
$I(q,\tau)$ for small $\tau$s, because $C(r,\tau)$ is a smooth function of
$\tau$. As a consequence $I(q,\tau)/I(q)$ approaches the value 1 horizontally
for $\tau\to 0$ (Fig. 5c2). By contrast, if $\tau_{W}$ is not defined the
covariance $C(r,\tau)$ varies like $\sqrt{\tau}$, which has infinite slope for
$\tau\to 0$. Accordingly $C(r,\tau)$ passes from being linear in $r$ to having
vanishing derivative over infinitesimally short interval of $\tau$ (Fig. 5b1).
The intermediate scattering function $I(q,\tau)$ passes discontinuously from
Porod’s law for $\tau=0$ to decreasing faster than $q^{-4}$ for arbitrarily
small $\tau>0$. This explains the very steep intermediate scattering function
$I(q,\tau)$ for large $q$ and small $\tau$ in Fig. 5c1. The same explanation
holds for Fig. 9b2.

In the case of model 1 the existence $\tau_{W}$ and the finiteness of $n_{t}$
can be ascertained by direct examination of $g^{\prime}_{W}(\tau)$. In the
case of model 2, one has to examine both the dispersion relation and the
spectral density. Assuming a dispersion relation of the type

$\omega(q)=a_{n}q^{n}$ (58)

where $a_{n}$ and $n$ are constants, a truncated expansion of the cosine
factor in Eq. (37) leads to

$\frac{1}{\tau_{W}^{2}}=\frac{a_{n}^{2}}{2}\int_{0}^{\infty}f_{W}(q)q^{2n}4\pi
q^{2}\textrm{d}q$ (59)

In the particular case of a linear dispersion relation, with $a_{1}=c$, the
characteristic time is proportional to the characteristic length
$\tau_{W}=l_{W}/(c\sqrt{3})$. However, for exponents $n$ larger than one the
conditions are more stringent for $\tau_{W}$ than for $l_{W}$. The condition
for non-vanishing $\tau_{W}$ is that the spectral density $f_{W}(q)$ should
decrease faster than $q^{-\nu}$ with $\nu=2n+3$ (see also Sec. SM-VII in the
supporting information) . Finally, in the case of model 3, it results from Eq.
(47) that the ballistic case always leads to finite $n_{t}$, provided $l_{W}$
is finite. Replacing Eq. (48) by a truncated expansion for small $\tau$, the
following expression is indeed obtained

$g_{W}(0,\tau)\simeq 1-\left(\frac{c\tau}{l_{W}}\right)^{2}+\ldots$ (60)

which shows that $\tau_{W}=l_{W}/c$. By contrast, in the diffusive case,
$g_{W}(0,\tau)$ is linear in $\tau$ and $n_{t}$ is always infinite, as already
shown in Eq. (57). The latter equation holds for all spectral densities, and
it is not limited to static field GRF-2.

###  V.2 NSE data analysis

Figure 11: (a) Fitting of the microemulsion SANS patterns of Fig. 1 as a sum
of two static GRF contributions, with bulk and film contrasts in red and grey,
respectively. The dots are the data, and solid lines are the fits. The two
contributions to the spectral density and the correlation functions are shown
in b and c, with background contribution (GRF-2, dotted), correlated
contribution (GRF-4,dashed) and their sum shown as a solid line.

The developed approach offers the possibility of decomposing a given static
structure into distinct contributions, and building composite time-dependent
models whereby each structural contribution is animated according to different
dynamic models. In the particular case of the microemulsion, the static SANS
data hint at two types of structures as illustrated in Fig. 11. A distinctive
feature of the SANS is the presence of a sharp scattering peak in the bulk
contrast data around 0.03 Å-1, corresponding to strongly correlated
structures. That peak alone, however, does not describe the entire SANS as it
is superimposed with a more diffuse and featureless background scattering that
extends over the entire $q$ range. We now endeavour to model these two
contributions with suitable Gaussian fields, and use these SANS-born models to
explore the NSE data.

A natural choice for modelling the background-like contribution in the SANS is
the static field GRF-2, which was used for illustrative purposes throughout
Sec. IV. As the spectral density of GRF-2 (see Eq. 26) displays no peak, it is
not suitable to model the correlated part of the structure. For the latter
contribution, we introduce a new static field, the elementary wave of which is
built as the Laplacian of GRF-2, namely

$w(\mathbf{x})=\left[\left(\frac{|\mathbf{x}|}{l}\right)^{2}-\frac{3}{4}\right]\exp\left[-2\left(\frac{|\mathbf{x}|}{l}\right)^{2}\right]$
(61)

The static properties of this field are given in Tab. 1 under the name GRF-4.
By construction the elementary wave in Eq. (61) satisfies

$\int w(\mathbf{x})\textrm{d}V_{x}=0$ (62)

This is equivalent to $f_{W}(0)=0$ by virtue of Eq. (15), which leads to the
desired scattering peak in the SAS patterns (See also Fig. SM-4).
Interestingly, in the context of moving-wave dynamic models (model 3) when a
given volume is crossed by any wave satisfying Eq. (62) the local average
value of the Gaussian field $W(\mathbf{x})$ remains unchanged, in a
neighbourhood of size larger than $l_{W}$. Therefore, the propagation of the
elementary wave in Eq. (61) preserves locally the volumes of the phases. We
therefore refer to GRF-4 as a deformation mode. The volume-preservation
property can also be understood by noting that the low-$q$ limit of the
scattered intensity is proportional to the compressibility of the phases
Glatter and Kratky (1982) so that a spectral density that vanishes for $q\to
0$ corresponds to incompressible phases. By contrast, GRF-2 leads to local
modifications of the the volumes, and we refer to it as a breathing mode. The
spectral densities of these two modes are illustrated in Fig. 11b.

To analyze the microemulsion SANS data the breathing and deformation modes
were combined into a single Gaussian field, following Eq. (10). This leads to
a three-parameter static-field model, with the characteristic lengths of each
mode and their relative contribution to the field variance. The clipping
thresholds $\alpha=-0.234$ and $\beta=+0.234$ are imposed by the volume
fractions. The least-square fit of both bulk- and film-contrast data is
illustrated in Fig. 11. The breathing mode contributes 70 % of the variance
($\sigma_{b}^{2}\simeq 0.7$) with lengths $l_{d}\simeq$ 98 Å and $l_{b}\simeq$
65 Å for the deformation and breathing modes. These numerical values of
$l_{d}$ and $l_{b}$ coincidentally correspond to the same characteristic
lengths $l_{W}\simeq 65$ Å(see Tab. 1).

Figure 12: Fitting of the microemulsion NSE data with the composite breathing-
and-deformation model (GRF-2 and GRF-4), with the dynamics of the modes
modelled either as (i) fluctuations, (ii) ballistic waves, or (iii) diffusive
waves. From top to bottom: the breathing mode is assigned fluctuation (a),
ballistic (b), or diffusive dynamics (c). From left to right: the deformation
mode is assigned fluctuation (1), ballistic (2), or diffusive dynamics (3). In
each case the dots are the data with bulk (grey) and film (red) contrasts, and
the surface is the model. The corresponding parameters are in Tab. 2.

In order to analyze the NSE data, a dynamic model has to be assumed for each
mode. As the notion of breathing and deformation is inspired by an elementary-
wave interpretation we restrict the analysis to model 1 (fluctuating waves)
and model 3 (ballistically or diffusively propagating waves). Moreover, as the
NSE data exhibit steep slope for large $q$ and small $\tau$ (Fig. 1c1-c2), we
consider only the exponential correlation function for model 1 with infinite
crossing rate $n_{t}$. In the following, we explore systematically all
combinations of the three types of dynamics for the two modes, which leads to
nine composite time-dependent Gaussian fields. The least-square fits of the
NSE data are illustrated in Fig. 12, and the values of the corresponding
parameters and $\chi^{2}$ are reported in Tab. 2. The fitting required the
correlation function $g_{W}(r,\tau)$ to be known for the deformation mode
(GRF-4), for both ballistic and diffusive wave propagation. All details are
provided in Sec. SM-V of the Supplementary Material.

| Deformation mode
---|---
| Fluct. ($\tau_{d}$) | Ball. ($c_{d}$) | Diff. ($D_{d}$)
Breathing mode |  Fluct. ($\tau_{b}$) |  $\tau_{b}=1000$ ns∗ $\tau_{d}=519$ ns ($\chi^{2}=10$) |  $\tau_{b}=130$ ns $c_{d}=0.088$ Å/ns ($\chi^{2}=11$) |  $\tau_{b}=1000$ ns∗ $D_{d}=1.3$ Å2/ns ($\chi^{2}=8.2$)
Ball. ($c_{b}$) |  $c_{b}=0.001$ Å/ns∗ $\tau_{d}=429$ ns ($\chi^{2}=9.6$) |  $c_{b}=0.82$ Å/ns $c_{d}=0.003$ Å/ns ($\chi^{2}=14$) |  $c_{b}=0.001$ Å/ns∗ $D_{d}=1.7$ Å2/ns ($\chi^{2}=7.4$)
Diff. ($D_{b}$) |  $D_{b}=4.0$ Å2/ns $\tau_{d}=1000$ ns∗ ($\chi^{2}=8.4$) |  $D_{b}=8.6$ Å2/ns $c_{d}=0.081$ Å/ns ($\chi^{2}=7.6$) |  $D_{b}=0.001$ Å2/ns∗ $D_{d}=1.7$ Å2/ns ($\chi^{2}=7.4$)
Table 2: Parameters of the composite model, whereby each mode (breathing or
deformation, both with $l_{W}\simeq 65$ Å) is assigned either a fluctuating,
ballistic or diffusive dynamics. The values were obtained from the least-
square fits in Fig. 12, and the $\chi^{2}$ values are also reported. The stars
highlight values that have converged to the lower bound allowed for the fit.

Globally, none of the composite models in Fig. 12 is able to quantitatively
account for both the bulk- and film-contrast NSE data. The model that fares
worst is the one that assumes ballistic wave propagation for both breathing
and deformation modes. This model has finite crossing rate $n_{t}$, and is
unable to capture the steep experimental intermediate scattering function.
Based on the $\chi^{2}$ values in Tab. 2, the data is best described when both
modes have diffusive dynamics. More generally, it is interesting to note that
the calculated intermediate scattering functions in Fig. 12 are generally
smaller than the data, particularly for the film-contrast scattering, so that
the breathing and deformation-mode approach overestimates the dynamics. This
is also manifest in the values of Tab. 2, which often converge towards the
lower limits allowed on the parameters. This means that one of the two modes
is practically static and does not contribute to the dynamics.

As an alternative to the composite model based on breathing and deformation
modes, the stochastic models offer the possibility of a more general approach
based on a dispersion relation (dynamic model 2). This enables one to better
match experimental data by tuning the dynamics in a scale-dependent way
through the value of $q$. Moreover, the dispersion-relation approach does not
require one to explicitly decompose the SANS into substructures, so that the
same piecewise-linear spectral density (GRF-8) as in Fig. 1a and SM-10 can be
used to describe the underlying static Gaussian field.

Figure 13: Least-square fit of the NSE data from Fig. 1 with a dispersion
relation (dynamic model 2) and a static field with piecewise-linear spectral
density (GRF-8, same as Fig. 1a), for both bulk contrast (a) and film contrast
(b). The dispersion relation is Eq. (63) with parameter $a\simeq 2053\
\AA^{3}$/ns and $q_{c}\simeq 0.047\ \AA^{-1}$, resulting in $\chi^{2}=6.3$.
The surfaces are the model and the dots are the data, with errorbar $\pm
2\sigma$.

Although any dispersion relation can in principle be chosen to model the NSE
data, it is desirable that it should lead to an infinite crossing rate $n_{t}$
so as to capture the steep intermediate scattering function. As discussed in
Sec. SM-III of the Supplementary Material, the spectral density of the
piecewise-linear model decreases asymptotically as $q^{-6}$ so that any
dispersion relation with order $n>3/2$ would lead to infinite $n_{t}$. In
practice, the following dispersion relation is found to describe fairly the
NSE data

$\omega=a(q-q_{c})^{3}H[q-q_{c}]$ (63)

where $H()$ is Heaviside’s step function, $q_{c}$ is a cutoff frequency, and
the parameter $a$ sets the value of $\omega$. The least-square fit is
illustrated in Fig.13 and in Fig. SM-12 as 2D plots. The values of the fitted
parameters are $a\simeq 2050\pm 60\ \AA^{3}$/ns and $q_{c}\simeq 0.047\pm
0.0002\ \AA^{-1}$, resulting in $\chi^{2}=6.3$, which is a significant
improvement compared to the values in Tab. 2. The reported uncertainties on
$a$ and $q_{c}$ were obtained from a Monte-Carlo estimation, with a $\pm
2\sigma$ normal-distributed error on each NSE data point. The small errors on
the parameters results from the fact that all the NSE data - over the entire
$q$ and $\tau$ ranges, and for the two contrasts - are fitted jointly with
only two adjustable parameters which makes it quite robust. This also offers
the prospect of reducing the number of experimental data points needed to
reliably adjust a model (see Fig. SM-15).

As an alternative to Eq. (63) the NSE data were also fitted with a dispersion
relation modelled as a sum of power laws (from $n=1$ to $n=4$) with adjustable
factors. Such fit converges to a situation where factors with alternating
signs contribute to keeping $\omega$ close to zero for small $q$, leading to
an overall shape similar to the cutoff frequency used in Eq. (63) (see Fig.
SM-13).

The $q^{3}$ dependence assumed in Eq. (63) appears in a variety of dynamic
structure factors involving hydrodynamic interactions, although the specifics
of the relaxation curve are system-dependent. The exponent $3$ notably appears
in the Zimm model of polymers in solvents to describe the thermally driven
fluctuations of a polymer chain Zimm (1956); Dubois-Violette and de Gennes
(1967). It also appears in the Zilman-Granek analysis of membrane fluctuations
with bending rigidity Zilman and Granek (1996). In the present context this
can be understood from the following general scaling argument. The very
observation of infinite $n_{t}$ hints at thermal random motion. By itself,
this would lead to a quadratic dispersion relation $\omega=Dq^{2}$, where the
diffusion coefficient $D$ is related to the characteristic size $L$ by Stokes-
Einstein relation $D\simeq k_{B}T/(\eta L)$, where $k_{B}T$ is the thermal
energy and $\eta$ is the viscosity of the medium. In the case of microemulsion
deforming over a variety of length scales, inversely related to the scattering
vector $L\sim q^{-1}$, this suggests the following cubic dispersion relation

$\omega\sim\frac{k_{B}T}{\eta}q^{3}$ (64)

Using the value $\eta\simeq 10^{-3}$ Pa.s for the effective viscosity of the
water/decane microemulsion as reported in ref.Holderer  _et al._ (2005) the
factor in this scaling law takes the value 4000 Å3/ns, which is of the same
order of magnitude as the value inferred from the NSE data fitting.

Figure 14: (a) Time-dependent realizations of the model of Fig. 13 over 50 ns,
with oil in grey, surfactant in red, and water in white; (b) realization from
the static components of the field only, corresponding to spectral density
limited to $q<q_{c}$; (c) average oil density calculated over the duration of
the simulation; (d) average oil density profile as a function of distance to
the interface of the static components ($d$ is positive into the oil). The
blue area highlight the 80 % confidence interval, with width $\simeq 60$ Å.
The four curves in d were obtained from independent realizations.

The time-dependent structure of the microemulsion is illustrated in Fig. 14
with a particular realization of the dispersion-relation model over a time
interval of 50 ns. One notes in particular the stability in time of large-
scale structures, larger than approximately $2\pi/q_{c}\simeq 130$ Å, where
$q_{c}$ is the cutoff frequency from Eq. (63). The average position of the oil
and water phases does not change over the timescale of the figure. However,
the interfaces deform in random and very rugged way at smaller scale, as
typically expected from a system with infinite crossing rate $n_{t}$. This
picture matches the physical intuition as the redistribution of oil and water
over long distances occurs through slow hydrodynamic flow, while no such
obstacle exists for the local fluctuations. This scale-dependence is captured
by the Stokes-Einstein relation, and is responsible for the observed $q^{3}$
dynamics. On the other hand, the physical origin of the cutoff size $q_{c}$
remains unclear, although its phenomenology is reminiscent of de-Gennes
narrowing, whereby relaxation times are often observed to scale with the
scattering structure factorsMyint  _et al._ (2021).

In order to better understand the realizations of the fitted model, the
structure was further decomposed into its static and time-dependent
components. The realisation in Fig. 14b was obtained by setting to zero all
components of the spectral density $f_{W}(q)$ with $q>q_{c}$, which results in
much smoother interface. This is manifest in the characteristic lengths of the
field (Eq. 9), which pass from $l_{W}\simeq 52$ Å to $l_{W}\simeq 73$ Å. Based
on Eq. (52) this corresponds to surface areas $a_{V}\simeq 170$ m2/cm3 and
$a_{V}\simeq 120$ m2/cm3, respectively. The thermal fluctuations of the
interface therefore contribute to as much as 30 % of the area of the
interfaces. Due to the symmetry of the model (with clipping constants
$\alpha=-\beta$) the surfactant/oil and surfactant/water interfaces have
identical areas. Another interesting aspect of the fluctuations is their
amplitude, which can be estimated by evaluating first the average density,
say, of oil over the entire duration of a simulation. This is illustrated in
Fig. 14c, where the smooth transition between the white and black areas
correspond to all the successive positions of the interface over time. The
extent of the transition in the direction locally orthogonal to the interface
is given in Fig. 14d. During 80 % of the time, the interface fluctuates within
a 60 Å-thick layer that extends on both sides of the average position. It is
interesting to compare that value to the size of the oil and water phases,
estimated as an average chord lengthGommes  _et al._ (2020) as
$4\phi/a_{V}\simeq 130\ \textrm{\AA}$ for the average structure in Fig. 14b.
In other words, the interface fluctuates over distances as large as half the
size of the phases.

The dispersion-relation analysis hints at reasons why the breathing and
deformation-mode analysis was unable to account for the NSE data of the
emulsion. The dispersion relation points indeed at two dynamic regimes but
they are separated by a cutoff frequency, which is much sharper a transition
than between GRF-2 and GRF-4. It has also to be noted that the very concept of
independent modes contributing additively to the dynamics, is strictly
justified only as a linear approximation. Given the observed large amplitude
of the interface fluctuations, non-linear effects could be expected which
would rule out any possibility of linear-mode decomposition.

##  VI Conclusion

Clipped Gaussian field models have been extensively used to analyze the
elastic small-angle scattering data of disordered systems. When applied to
dynamical systems, such classical approach provides static snapshots of a
structure. In the paper, the models are generalised to make them time-
dependent, which enable one to analyze consistently both the instantaneous
spatial structures and their dynamics within a single statistical description.
General expressions are derived for all the space- and time-correlation
functions relevant to coherent inelastic neutron scattering, for multiphase
systems and arbitrary scattering contrasts between the phases.

With the proposed approach, for any given static structure inferred e.g. from
small-angle scattering, a variety of distinctly different dynamics can be
modelled. In a first family of models, the Gaussian field underlying the
structure is decomposed into a large number of localised elementary waves.
Qualitatively different dynamics are obtained by letting the waves randomly
fluctuate, or propagate ballistically or diffusively through the system. In
another family of models, the spectral components of the Gaussian field are
assigned any desired dynamics through a suitable dispersion relation. The
various types of dynamics lead to qualitatively different intermediate
scattering functions, which enables one to discriminate them through neutron
scattering. Moreover, all these approaches can be combined to yield models
with composite and possibly realistic dynamics.

A central characteristic of the dynamic models, which controls the shape of
the intermediate scattering functions, is their temporal crossing rate. This
is defined by considering a fixed point in space, and evaluating how often it
is passed through by a moving interface of the time-dependent structure.
Systems undergoing Brownian-like thermal fluctuations have infinite crossing
rate, which converts to infinitely steep intermediate scattering functions for
asymptotically large $q$ and small $\tau$.

The methods of the paper were illustrated with the analysis of neutron small-
angle scattering and spin-echo data measured on oil/water microemulsions. The
methodology consisted in analyzing first the SANS data in order to determine
the spectral density of the Gaussian field underlying the static structure,
corresponding to snapshots of the time-dependent structure. As a second step,
the NSE data was analyzed by complementing the so-determined static spectral
density with few dynamic parameters. This enabled us to analyze jointly the
entire SANS and NSE data, in both film and bulk contrasts and over the entire
range of $q$ and $\tau$, with a single coherent model. The small number of
adjusted parameter contributes to the robustness of the NSE analysis, and
offers the prospect of reducing the number of experimental points required to
reliably adjust a model.

From a physical perspective, the SANS and NSE data of the emulsion point to a
static large-scale structure of the oil and water domains, with thermal
fluctuations of the interfaces. The interface fluctuations take place over
distances as large as 60 Å, corresponding to half the domain size, and
contribute to 30% of the total interface area. In future work the stochastic
approach will be explored further to analyze the wavelike dynamics observed by
neutron spin-echo in lipid membranesJaksch  _et al._ (2017).

## Supplementary Material

See supplementary material for the mathematical derivation of some equations,
for numerical data-analysis procedures, as well as for additional figures.

## Data Availability

All SANS and NSE data discussed in the paper can be downloaded from the
authors’ institutional repository at https://doi.org/10.26165/JUELICH-
DATA/DJ3LIN.

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