between symbols and solid lines are observed beyond the statistical and
systematic uncertainties, especially upon increasing the correlation length
$\xi$, i.e., upon approaching the critical point. These discrepancies are
observed independently of the fact whether the third particle has the same
adsorption preference as the first two (realising $(---)$ BCs for the three
colloids) or not (realising $(--+)$ BCs), as shown in panels $(a)$ and $(b)$
of Fig. 3. This demonstrated the occurrence of genuine many-body effects. This
is particularly the case within the range of distances $l_{12}$ at which CCFs
provide the dominant contribution to the effective interaction — compared
with, say, the electrostatic repulsion. This contribution causes, upon
decreasing $l_{12}$, the sharp drop into the dip displayed by $U_{3}(l_{12})$
in Fig. 3 at short distances.

As anticipated in Sec. 4.3, besides many-body effects, the influence of the
CCF on the physical properties of colloidal suspensions is significant. This
is confirmed by a number of theoretical and experimental studies briefly
mentioned in that section and reviewed in Ref. 13, which the reader is
referred to for further details. Here we only mention that upon changing the
temperature-dependent range of the critical Casimir interaction between two
colloidal particles in a near-critical solvent, it is possible to induce and
access experimentally a liquid-gas phase separation of the colloidal fluid and
possibly the associated critical point. This was observed for solvents
consisting of a molecular binary liquid mixture134, 129 or of an aqueous
solution of globular solid micelles142, 143. Moreover, as a convenient way to
engineer and control the total interaction energy between colloids in
suspension, the CCF can be used in order to study static and dynamic
phenomena. They depend on the features of this interaction, such as the
formation of structures resulting from aggregation127, 130, which may change
in zero gravity.

###  5.4 Interplay with other forces

In Sec. 4.4 we anticipated that CCFs in soft matter usually act together with
forces of different nature such as electrostatic, van der Waals, or depletion
forces. Accordingly, it is natural to explore the extent to which their
corresponding effects simply add or if these various forces might influence
each other. In most of the cases of experimental relevance and in view of the
currently available experimental accuracy it turns out that this possible (and
somehow expected) mutual influence is quantitatively negligible within the
range of parameters studied so far. A notable exception was provided by the
investigations of CCFs emerging in binary liquid mixtures with added salt126,
144. In fact, in a sufficiently dilute colloidal suspension, one expects that
the only relevant interactions are provided by the electrostatic repulsion and
the critical Casimir attraction, the spatial ranges of which are controlled by
the Debye screening length $\kappa^{-1}$ and the correlation length $\xi$,
respectively. Depending on the ratio of these two lengths and thus on the
relationship between temperature and ion concentration in solution, one
expects the resulting total potential to be capable to drive aggregation or
not. The investigation of this phenomenon in the $\xi$–$\kappa^{-1}$ plane126
(see also Refs. 132 and 172) motivated a better understanding of the
interaction between a (charged) colloid and a surface in the presence of an
electrolyte.

Fig. 4: Effective interaction between $(a)$ a colloidal particle at a surface-
to-surface distance $z$ from a homogeneous substrate or $(b)$ two parallel
plates separated by a distance $L$, immersed in a critical binary liquid
mixture in the presence of electrolytes provided by certain salts. The two
surfaces involved have opposite adsorption preferences, as to realise $(+,-)$
BCs, giving rise to a _repulsive_ CCF. The salt concentration controls the
Debye screening length $\kappa^{-1}$, while the deviation of the temperature
$T$ from its critical value $T_{c}$ determines the correlation length $\xi$.
Panel $(a)$ reports the experimental data for the potential $\Phi(z)$
obtained144 via TIRM with a (hydrophilic) polystyrene colloid (realising $(-)$
BCs) of radius $R\approx 0.75\mu{\rm m}$ in a mixture of water and
2,6-lutidine containing 10 mM KBr. Panel $(b)$ shows the effective interaction
potential $\widetilde{\omega}(L)$ as a function of $L$ and for various values
of $\kappa\xi$, as obtained theoretically within the model discussed briefly
in Sec. 4.4. Although the theoretical predictions displayed here depend
quantitatively on the choice of the values of various parameters involved in
such a model (see Ref. 145 for details). Their qualitative features and the
dependence on $\kappa\xi$ are rather robust and agree with the experimental
observation shown in panel $(a)$. (Panels $(a)$ and $(b)$ are taken from Refs.
144 and 145, respectively.)

The corresponding potential $\Phi(z)$ was experimentally determined via TIRM
(see Sec. 5.2 and Ref. 144). As anticipated in Sec. 4.4, the attractive force,
observed for _similar_ adsorption preferences at the colloid and the surface,
i.e., for $(\pm,\pm)$ BCs, was qualitatively not affected by the presence of
the salt. The phenomenon observed in the presence of _opposite_ adsorption
preferences, i.e., for $(\pm,\mp)$ BCs (leading, in the absence of salt, to a
repulsive CCF) was richer, as shown in Fig. 4$(a)$. In fact, starting at a
temperature $T$ far from $T_{c}$, first one observes the expected
electrostatic repulsion (as in Fig. 2$(b)$). Then, upon approaching $T_{c}$,
an _attractive_ contribution to the total potential develops before this
gradually fades and a predominantly _repulsive_ component eventually emerges
close to criticality, as expected in the absence of salt and in line with the
concept that asymptotically for $T\to T_{c}$ the _universal_ scaling function
should be recovered. This experimental observation motivated the theoretical
investigations discussed in Sec. 4.4, which eventually turned out to explain
qualitatively the phenomenon, as shown in Fig. 4$(b)$. A quantitative
analysis, however, is still lacking, as it would require a careful
characterization of the material properties of the system (such as the surface
charges, etc.) which crucially determine the non-universal interactions within
this system.

##  6 Dynamics

The previous sections focussed on the _stationary_ properties of CCFs
occurring _in thermal equilibrium_. However, these forces are expected to
manifest themselves also in _non-stationary_ and possibly in _out-of-
equilibrium_ conditions, such as those generated by sudden perturbations of
the system due to temperature changes,173, 174, 175 the action of external
forces176, or due to the non-equilibrium nature of the stationary state in
which the confined fluctuations occur177, 178, 179. Considering non-
equilibrium conditions or time-dependent properties in equilibrium (such as
temporal correlation functions of the forces) poses a conceptual challenge to
the approaches discussed in Sec. 3. In fact, while the equilibrium expectation
value of the force can be derived from the effective free energy $\Omega$ or,
alternatively, from the expectation value of the stress tensor $T_{ij}$, only
the latter option is viable out of equilibrium173, taking into account
dependence of the OP field $\phi(\bm{x},t)$ on time $t$. (Note, however, that
the necessity of this choice has been debated.180, 181, 182) Accordingly, the
dynamical behaviour of CCFs can be predicted on the basis of the dynamics of
the OP field $\phi(\bm{x},t)$. For the sake of brevity, here we shall not
present in detail the well-established theory of dynamical critical
behaviour183, 184. Instead, we provide the theoretical prediction and the
experimental evidence that universality emerges also in this case, in the
sense that, on the long time scale, the collective dynamical behaviour of the
system is determined by the gross features of its dynamics, such as symmetries
and possible conservation laws, and is largely independent of the microscopic
details of the dynamics. In analogy to what occurs in equilibrium, this is due
to the fact that also the relevant timescale $\tau$ of the dynamics displays
an algebraic singularity as the critical point is approached, i.e.,
$\tau\sim\xi^{z}$, where $z$ is the so-called _dynamic critical exponent_ ,
which characterises the dynamic universality class. In particular, from the
theoretical point of view, the dynamics of the OP field is generically
expressed in terms of stochastic evolution equations with thermal white noise,
possibly involving additional fields beyond the OP itself. If the system under
investigation is in thermal equilibrium, these stochastic differential
equations satisfy the fluctuation-dissipation theorem, and the equilibrium
universality class might split into various dynamic universality classes,
depending on possible conservation laws.

###  6.1 Analytical and numerical studies

Among the simplest possible instances of _equilibrium_ dynamics for a scalar
OP field $\phi(\bm{x},t)$ are the so-called models A and B of Ref.183,
according to which

$\partial_{t}\phi(\bm{x},t)=-D(-\nabla^{2})^{a}\frac{\delta\mathscr{H}}{\delta\phi(\bm{x},t)}+\zeta(\bm{x},t).$
(9)

Here $\mathscr{H}$ is the Landau-Ginzburg effective free energy given in Eq.
(4) (omitting, in general, the boundary terms), $\zeta(\bm{x},t)$ is a
Gaussian white noise with $\langle\zeta(\bm{x},t)\rangle=0$ and correlation
function
$\langle\zeta(\bm{x},t)\zeta(\bm{x}^{\prime},t^{\prime})\rangle=2Dk_{\rm
B}T(-\nabla^{2})^{a}\delta(\bm{x}-\bm{x}^{\prime})\delta(t-t^{\prime})$, where
$a=0$ corresponds to model A and $a=1$ to model B. In the latter case, the OP
$\phi$ is locally conserved in the sense that
$\partial_{t}\phi=-\bm{\nabla}\cdot\bm{J}$ for a suitable stochastic current
$\bm{J}$, as it occurs if $\phi$ is the OP of a binary liquid mixture.
However, we note that the actual dynamics of these mixtures is significantly
more complicate, because one has to account also for the coupling of $\phi$ to
other slow modes, such as the fluid velocity. This is encoded in the so-called
model H of Ref. 183, briefly discussed further below.

As in the case of the stationary properties, the dynamics is affected by the
presence of boundaries (see, e.g., Ref. 17). The stochastic evolution of
$\phi(\bm{x},t)$ (and possibly additional conserved quantities) has to be
supplemented by the BCs discussed in Sec. 2.1. This possibly leads to a
further splitting of the dynamic universality class depending on the presence
or absence of locally conserved dynamics at the boundaries.32 Similarly, the
dynamics in the presence of spatial confinement can be theoretically
investigated by considering the simultaneous presence of various boundaries.
However, up to now, analytical progress has been possible only for the
aforementioned simplest cases of dynamics and for the film geometry. In this
respect, we note that the dynamics of CCFs is much less explored than their
stationary properties in spite of the fact that these dynamical properties are
within experimental reach and might be investigated in the near future.

_Film geometry.—_ In particular, the dynamics of a scalar Gaussian field
$\phi(\bm{x},t)$ (corresponding to $u=0$ in Eq. (4)), confined within a film
with ordinary BCs) was considered in Ref.176. If $\phi(\bm{x},t)$ is locally
perturbed inside the film, the resulting critical Casimir pressures, acting on
the two confining surfaces, become time- and space-dependent. These two
pressures are no longer necessarily equal as they are in thermal equilibrium.
Within model A, the associated scaling function was determined after switching
off the perturbation. This analysis was extended173 to the case of different
BCs at the two confining surfaces (Dirichlet/Neumann), and for a global quench
of the temperature of the medium, i.e., for a sudden change from a very high
value to the critical point. In both cases, the strength of the force on the
two surfaces turns out to be different while, as expected, it approaches the
same value as the relaxation towards an equilibrium condition occurs. In an
analogous setting, the case of a Gaussian field with short-ranged spatial
correlations (modelled by neglecting the term $\propto(\nabla\phi)^{2}$ in Eq.
(4)), conserved dynamics (model B), and no-flux condition $\bm{J}=\bm{0}$ at
the boundaries was considered in Ref. 174. In particular, upon quenching the
strength of the noise (i.e., the temperature) from an initial state with
$\phi=0$ (and therefore a vanishing fluctuation-induced force), the emergence
of a transient, long-ranged dynamical effective Casimir-like force also away
from criticality was observed. This is the case because the emergence of these
long-ranged correlations is due to the conservation law of the dynamics.
However, this force disappears in the stationary state, which is attained long
after the quench, because the correlation length of the fluctuations within
the medium was assumed to be vanishingly small. The case of inclusions
embedded in the fluctuating field and the possible realization of the
aforementioned quench in active matter (see also below) were discussed. The
interplay between the long-ranged correlations generated out of equilibrium,
in accordance with model B, and those emerging when the system is poised at
its critical point, becomes particularly rich after a quench to this point. In
Ref. 175, the resulting non-equilibrium critical Casimir effect in the film
geometry was investigated for a Gaussian field with non-symmetry-breaking BCs.
In particular, it turns out that complex transient regimes arise, including
correlation functions and forces with oscillatory evolution, depending on the
conditions in which the medium is prepared ahead of the quench to the critical
point and depending on the specific choice of the BCs.

Alternatively, a non-equilibrium condition can be generated by modulating
externally, as a function of time $t$, the distance $L(t)$ between a mobile
plate and a fixed one. The two plates confine a scalar, fluctuating medium
$\phi$ to a film with Dirichlet BCs. As a consequence, the force acting on the
fixed plate is also modulated and its dynamics and universal form were
determined analytically for a critical, scalar Gaussian field with model A
dynamics and for a harmonic dependence of $L(t)$ on $t$, under the assumption
of a small modulation amplitude compared to the average distance between the
boundaries.185 In this limit, the resulting force also depends harmonically on
time, with the same angular frequency $\omega_{0}$ as $L(t)$ does, and with a
$\omega_{0}$-dependent amplitude and phase shift. In addition, this force
displays two distinct contributions, one generated by diffusion of stress
within the medium and another one is related to resonant dissipation in the
cavity (see Ref. 185 for details).

The theoretical investigations mentioned above considered dynamical evolutions
in a confined geometry with BCs which do not break the
$\phi\leftrightarrow-\phi$ symmetry of the Gaussian model. The choice of the
latter properties are primarily motivated by the search for analytical
solutions. However, as extensively discussed in the previous sections, in many
cases of experimental interest the boundaries are responsible for critical
adsorption, which breaks such a symmetry and can be effectively represented by
introducing surface fields, such as in Eq. (4). As a consequence of this
choice, $\langle\phi(\bm{x},t)\rangle$ exhibits a non-trivial evolution, which
can be conveniently studied within a mean-field approximation of the evolution
equations. In Ref.186, the case of a scalar OP was studied with the effective
free energy $\mathscr{H}$ given by Eq. (4), with model B dynamics, and with
no-flux BCs at the $(+,+)$ boundaries of a film. The dynamics of the film was
considered after an instantaneous quench from a homogeneous, high-temperature
state with $\langle\phi(\bm{x},t)\rangle=0$ to $T_{c}$ or to still
supercritical but lower temperatures. The evolution of the CCF acting on the
confining surfaces was determined together with the corresponding dynamic
scaling functions. It turns out that such a force contains a non-equilibrium
contribution which, depending on the parameters of the system, might have a
non-monotonic dependence on time, while being generically _repulsive_. It
actually turns out187 that even the attractive or repulsive character of the
CCF in _equilibrium_ depends on whether the fluctuations of the field occur
within a grand canonical or canonical ensemble. (For the exactly solvable case
of the one-dimensional Ising model, the behaviour of the force emerging within
the two ensembles is analysed in Refs.188 and 189.) The grand canonical
ensemble is actually the one discussed in all previous sections. In
particular, for equal BCs the force is attractive within that ensemble, while
it is repulsive in the canonical case187. This is naturally realised as a
consequence of the conserved dynamics.

_Colloidal particles.—_ Beyond the instances discussed above with fixed or
slightly displacing boundaries, one might be interested in cases in which
these surfaces move according to the very same effect of the CCF acting on
them, as it occurs, e.g., for colloidal particles. Imposing BC of the kind
discussed in Sec. 2.1 on moving, non-planar boundaries turns out to be
challenging. Thus, it is convenient to think of the particles as localised,
compact parts of the medium with properties (e.g., compressibility or
viscosity), which differ from the rest of the medium190, 174. Within these
parts, a symmetry-breaking field is applied instead of imposing BCs.191, 192,
193 This portion of the medium then moves, e.g., according to an over-damped
Langevin dynamics, which takes into account the total force acting on each
particle due to its interaction with the (remaining part) of the medium and
which, in thermal equilibrium, satisfies the fluctuation-dissipation theorem.
Rather generically, even in the simplest case, in which the fluctuating field
is Gaussian and the particle acts on the medium only as an external field, the
effective dynamics of such a single particle turns out to be described by a
Langevin equation with a _non-linear_ memory kernel. This determines the
effective, time-dependent friction and the correlations of its
fluctuations,191 and it spans an increasingly wide time interval as the
critical point of the medium is approached. It causes, for example, an
algebraic — instead of exponential — long-time relaxation of the position of a
harmonically trapped particle when initially displaced from the centre of the
trap, independently of the dynamics of the OP field corresponding to model A
or B.192

The simple representation of a colloidal particle as outlined above allows one
to investigate193 the evolution of the effective interaction between two
particles when they interact with the same Gaussian field. This is the case
for the experiments discussed further below, in which two optically trapped
colloids are in contact with a near-critical medium. In particular, one can
consider the case that the position of one of the two particles is
periodically modulated in time with, say, a harmonic function. This might
induce (depending on the type of dynamics and whether the medium is critical
or not) a non-harmonic motion of the other particle, which is kept away from
the first one by a quadratic confining potential. At late times the field-
mediated interaction synchronises the motion of these two particles, and the
nonlinear response of the second particle can be determined as a function of
the driving frequency of the first one193. The response can be compared with
the adiabatic approximation which assumes an instantaneous relaxation of the
OP to its equilibrium configuration appropriate for the actual arrangement of
the particles. This comparison highlights the effects of _retardation_ in the
propagation of the force due to the slow critical dynamics of the medium.

Beyond the simplified models discussed above, the investigation of the
evolution of CCFs, which is relevant for the critical dynamics of actual
physical systems, has to rely on numerical calculations, even within the mean-
field approximation. This is due to the increasing complexity of the evolution
equations for more realistic models. For example, in the case of binary liquid
mixtures, in addition to the OP $\phi$ one has also to account for the
velocity field $\bm{v}$ of the mixture. The dynamics, which captures the
critical properties of $\phi$ and $\bm{v}$, corresponds to the so-called model
H,183 which includes hydrodynamics. Within the mean-field approximation one
considers the evolution equation without noise, in which $\phi$ is locally
conserved, i.e., it obeys the continuity equation with the current
$\bm{J}=\phi\bm{v}-D\bm{\nabla}\delta\mathscr{H}/\delta\phi$, where
$\mathscr{H}$ is the Landau-Ginzburg effective free energy given in Eq. (4).
The boundary terms are modified as described further below. The local velocity
$\bm{v}$ of the medium (which will eventually encompass also the particles
according to the idea discussed above) obeys the Navier-Stokes equation
$\rho(\partial_{t}+\bm{v}\cdot\nabla)\bm{v}={\bm{f}}-\bm{\nabla}\cdot{\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{\boldmath$\sigma$}}}}$,
with
$\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{$\bm{\sigma}$}}}=p\\!\\!\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{$\bm{I}$}}}-\eta[(\nabla{\bm{v}})^{\dagger}+\nabla{\bm{v}}]$,
where $\stackrel{{\scriptstyle\leftrightarrow}}{{\mbox{$\bm{I}$}}}$ is the
unit tensor, $\rho$ the mass density of the liquid (which is assumed to be
spatially constant), and $\eta$ its viscosity. The hydrostatic pressure $p$
facilitates to satisfy the incompressibility condition
$\nabla\cdot{\bm{v}}=0$, and ${\bm{f}}({\bm{x}})$ is the force density
${\bm{f}}=-\phi\bm{\nabla}\delta\mathscr{H}/\delta\phi$. According to the
approach outlined above, the presence of $N$ particles labeled by the index
$i\in\\{1,\ldots,N\\}$, at positions $\\{\bm{x}_{i}\\}$ in the medium is
represented by a spatial modulation $\chi(\bm{x}-\bm{x}_{i})$ of its local
properties, where $\chi(\bm{x})$ is a smooth function which vanishes for
$|\bm{x}|$ larger than the particle radius while it rapidly attains 1 upon
decreasing $|\bm{x}|$ below that radius. This function determines the boundary
contribution to $\mathscr{H}$ in the form
$\sum_{i=1}^{N}\int\mathrm{d}^{d}x[c_{i}\chi(\bm{x}-\bm{x}_{i})\phi^{2}(\bm{x})-h_{i}|\bm{\nabla}\chi(\bm{x}-\bm{x}_{i})|\phi(\bm{x})]$
where $c_{i}>0$ plays the role of the surface enhancement and $h_{i}$ that of
the surface field. We note that this expression generalises the boundary term
appearing in Eq. (4). In addition, $\chi$ controls the spatially varying
viscosity
$\eta(\bm{x})=\eta_{l}+(\eta_{c}-\eta_{l})\sum_{i=1}^{N}\chi(\bm{x}-\bm{x}_{i})$
where $\eta_{l}$ is the viscosity of the $l$iquid and $\eta_{c}\gg\eta_{l}$
the (common) viscosity of the $c$olloidal particles. The particles are assumed
to be a region of space filled by a highly viscous fluid. (From this approach
the notion of “fluid particle dynamics” earns its name; see, e.g., Ref. 190).
Similarly, the diffusivity $D$ acquires a space dependence,
$D(\bm{x})=D[1-\sum_{i=1}^{N}\chi(\bm{x}-\bm{x}_{i})]$, which suppresses
diffusion within the particles, while ${\bm{f}}$ (see above) receives a
contribution
$-\mathscr{V}^{-1}\sum_{i=1}^{N}\chi(\bm{x}-\bm{x}_{i})\bm{\nabla}_{\bm{x}_{i}}\mathscr{H}$,
where $\mathscr{V}=\int\mathrm{d}^{d}x\,\chi(\bm{x})$ is the effective volume
of each particle. The resulting evolution equations for $\phi(\bm{x},t)$ and
$\bm{v}(\bm{x},t)$ are solved numerically190 together with the dynamics of the
centre $\bm{x}_{i}$ of the particle $i$, which is determined by its
instantaneous velocity
$\bm{V}_{i}(t)=\int\mathrm{d}^{d}x\,\bm{v}(\bm{x},t)\chi(\bm{x}-\bm{x}_{i})/\mathscr{V}$.
In addition to the terms discussed above, a mechanical potential
$U(\bm{x}_{1},\ldots,\bm{x}_{N})$ might be introduced in $\mathscr{H}$ in
order to account for possible direct interactions among the particles.

Based on the approach described above, the authors of Ref. 190 investigated
the temperature dependence of the drag force on a single particle steadily
moving in a near-critical fluid mixture. They studied also the relative motion
of two particles due to the non-equilibrium CCF acting on them. Due to the
slow dynamics of the OP and depending on the Péclet number of the particle
motion, significant deviations of the instantaneous force acting on the
colloids from its equilibrium value at the instantaneous separation between
the two particles are observed. These retardation effects reduce the strength
of the force during particle aggregation.

Before concluding this part, we mention that various simplifications of the
model H described above, which are related to the velocity field $\bm{v}$,
were considered in the literature. They allowed one to determine in semi-
analytic form, e.g., the drag coefficient of a solid spherical colloid in the
near-critical binary liquid mixture (see Ref.194 and references therein) or
the amplitude of the fluctuations of the position of such a colloid when it is
confined by an harmonic trap (see Ref.195 and references therein). In the
latter case, it would be rewarding to investigate how such an amplitude
emerges from the equilibrium distribution of the particle displacement, which
is expected to be determined solely by the total potential of the mechanical
forces acting on the particle.

###  6.2 Experiments

The dynamical aspects of CCFs are experimentally still largely unexplored.
This is primarily due to the fact that the relaxation times $\tau$ within
experimental reach are still rather short compared to the time scales at which
the experimental measurements are performed. Accordingly, as we shall discuss
below, the available experimental evidences can still be interpreted in terms
of the occurrence of equilibrium CCFs, at least within the currently available
experimental accuracy. No retardation effects have been observed so far.
However, the possibility of using these forces also for controlling the
dynamics of colloidal particles was demonstrated.196, 197

Fig. 5:  Synchronisation of the motion of two trapped, identical colloidal
particles (with radii $R\approx 2.5\mu{\rm m}$) upon approaching, from left to
right, the (lower) critical temperature $T_{c}\approx 30^{\circ}{\rm C}$ of
the solvent, constituted by an aqueous, critical micellar solution of the
nonionic surfactant $\mbox{C}_{12}\mbox{E}_{5}$. The solid black lines
indicate the respective positions of the centres of the optical traps which
confine the two silica colloids. The red and blue lines, instead, correspond
to the experimentally determined centres of these two particles, while
$\varepsilon=(T_{c}-T)/T_{c}$ quantifies the deviation from criticality. Upon
approaching criticality from left to right, the two particles increasingly
displace from the centres of the corresponding optical traps under the action
of the attractive CCFs, reducing the inter-particle distance. This implies
that the red line is systematically below the upper black curve while the blue
line above the lower black one. Correspondingly, the motion of the particle,
which is in that trap fixed at $x=0$, is increasingly influenced and
synchronised with the motion of the particle in the moving trap. (Figure
adapted from Ref. 196.)

Critical Casimir forces can be used to synchronise the motion of two identical
colloidal particles which are immersed in a binary liquid mixture and are
trapped by two optical tweezers separated by a certain time-dependent
distance. Upon moving periodically the centre of one of the two traps and
therefore by moving one of the two colloids, it was shown196 — see Fig. 5 —
that it is possible to induce motion of the other colloid, in a way which
depends on the thermodynamic distance $\varepsilon$ from criticality. If the
mixture is far from being critical and thus the correlation length $\xi$ is
smaller than the typical inter-particle distance $\ell$, the motion of one
colloid does not actually affect the other, as expected and shown in Fig.
5$(a)$. This also demonstrates that the hydrodynamic interactions are
negligible in this setting.196 The situation changes when, upon approaching
criticality, $\xi$ becomes comparable to $\ell$ and one colloid senses the
presence of the other in terms of the action of CCFs, as shown in panels $(b)$
and $(c)$ of Fig. 5. Eventually, their motion synchronises and the average
$\ell$ decreases because of the occurrence of these attractive forces.

This setting and the periodic protocol allows one to study experimentally also
the energetics of the system composed of the two trapped colloidal particles
and therefore to determine the distribution of work done on this system with
the heat dissipated by each particle, which depends on the deviation from
criticality.198 In particular, the temperature variations are obtained in Ref.
198 by inducing local heating via a suitable laser. These relatively small
variations produce large effects due to the extreme sensitivity of CCFs on
temperature.

Alternatively, CCFs can also be used to influence the dynamics of diffusing
colloidal particles in the absence of optical trapping.197 In particular, by
using blinking optical tweezers, which can be periodically switched on and
off, the dynamics of two colloidal particles in a near-critical binary liquid
mixture was experimentally observed during diffusion. This way one can
highlight the effect of their mutual interaction, possibly due to CCFs. By
studying the position of the particles when both traps are switched on, one
can determine the non-universal parameters (e.g., trap stiffnesses, Debye
screening length, critical temperature, etc.), which control, inter alia, the
colloidal interaction in thermal equilibrium and which therefore fix its form.
This interaction potential can then be used to predict, via numerical
(Langevin or MC) simulations, the statistical properties of the dynamics of
the particles in the absence of the trapping. Then these predictions can be
compared with the actual experimental data for the relative velocity of the
two particles in order to highlight the possible emergence of retardation
effects. However, as anticipated above, no significant discrepancy was found
upon this comparison. A similar conclusion was reached on the basis of the
data for the distribution of the first passage time, i.e., of the first time
at which the inter-particle distance reaches a certain threshold value when
they are released from an initial distance larger or smaller than this
threshold. Depending on the temperature, one clearly observes that the
emergence of the (attractive) critical Casimir interaction between the
particles influences the distribution of these first passage times.197

##  7 Perspectives

In the previous sections we focused on Casimir-like effects induced by the
confinement of the critical fluctuations, which characterise a continuous
phase transition occurring in a liquid medium at thermal equilibrium. Before
concluding, we mention analogous instances in which Casimir-like forces emerge
because of the fluctuations of other relevant collective fields. These
include, for example, fluctuations in liquid crystals (see, e.g., Refs.199
and 200 and references therein), in polymers201, of the heights of
interfaces202, 203 and membranes204, 205, 206, 207, in complex fluids,208 and
in granular materials177, 209, as well as fluctuations associated with non-
equilibrium phase transitions178 or with transitions to non-equilibrium steady
states210, 211, 212. More recently, motivated by the increasing interest in
active matter, also this subject has been studied in the present context213,
179. In particular, an early numerical study of an ensemble of run-and-tumble
particles in $d=2$, confined by parallel lines of _finite_ length, revealed
the emergence of an attractive force, the magnitude of which increases upon
increasing the run length of the particles, but decreases _exponentially_ upon
increasing the distance $L$ between the lines. This attraction may be turned
into a repulsion via varying $L$ and the line length. In order to highlight
the effects associated with confining an emergent collective behaviour within
these systems, the phenomenon of flocking in active matter has been
considered179 in $d=2$. In fact, a system of interacting self-propelled
particles is known to undergo a flocking transition, characterised by the
emergence of a well-defined flocking direction of collective motion (see
Ref.179 and references therein). In the presence of two reflecting or partly
repelling parallel walls, the flocking direction is parallel to them. An
attractive Casimir-like effect emerges on the walls, which decays
_algebraically_ upon increasing their separation $L$, while displaying a
certain degree of universality. However, a systematic understanding of the
various features of such forces, caused by the long-ranged correlations of the
fluctuations in active matter systems which thus are similar to those at
criticality, is still lacking. Therefore this stands as an interesting problem
for future investigations.

In this review we discussed the emergence of CCFs in soft matter, starting
from very basic theoretical ideas up to reporting the first experimental
evidences in wetting films, and, most importantly, in colloidal suspensions.
These studies brought to light the following peculiar and useful features of
CCFs: (i) they can be reversibly switched on and off by controlling the
temperature of the solvent; (ii) their strength and spatial direction can be
controlled by suitable treatments of the _surfaces_ of the involved bodies;
(iii) they exhibit a universal character, and therefore they are largely
independent of the actual material properties of the involved bodies. These
features are rather unique to the CCFs, especially when compared with other
forces acting in soft matter, such as electrostatic, depletion, van der Waals,
hydrodynamics forces etc. Most of them sensitively depend on the microscopic
properties of the involved materials, and they can hardly be tuned, as far as
their strength, range, and spatial orientation are concerned, by varying an
external parameter. In this respect, CCFs may find a number of possible
applications in controlling and manipulating soft matter. In fact, this is the
scientific direction into which most of the corresponding research has been
developing recently. For example, the various studies mentioned in Sec. 4.3
focussed on exploiting CCFs in order to control the assembly of colloidal
dispersions and the possible structures of aggregates emerging from the
assembly, with an eye on concrete, technical applications. In addition, a
recent experimental study214 demonstrated that these forces provide a generic
method for size-selective nanoparticle purification/separation. The
aggregation they cause in a heterogeneous colloidal dispersion depends on the
relative particle sizes. The larger colloids aggregate while the smaller ones
remaining in solution. This allowed the authors to propose a novel approach
which turned out to be very efficient in purifying commercial silica
nanoparticles in a water-lutidine mixture. These investigations open the door
to a number of possible applications of CCFs to nanotechnologies. In this
respect, it is also of high importance to understand how these forces can be
used to counteract those which are usually at play in electromechanical
devices at the micrometer or nanometer scale, i.e., MEMS or NEMS,
respectively. Among the latter forces, the most prominent is the attractive
Casimir-Lifshitz force, usually responsible for the undesired stiction of
metallic parts. While in the experiments with colloidal particles described in
Secs. 5 and 6.2, such a van-der-Waals force turned out to be negligible, this
might not always be the case. This possibility motivated the experimental
study presented in Ref.215. It involves a microscopic gold flake suspended
above a flat gold-coated substrate and immersed in a critical water-lutidine
binary liquid mixture. In turns out that the attractive Casimir-Lifshitz force
acting on the flake — which, on its own, cannot be easily made repulsive — can
be conveniently counterbalanced by the repulsive critical Casimir force. This
offers an active control of the total force exerted on the flake. Beyond the
cases discussed here and in Secs. 4.4 and 5.4, the investigation of the
interplay of CCFs with other interactions in devices and in soft matter has
still to reach a satisfactory level of understanding and quantitative
accuracy. This stands as another interesting open problem for future studies.

For example, it has been realised only recently216 that in previous
theoretical studies of the wetting transition in systems with short-ranged
interactions, a low-temperature Casimir contribution had been overlooked in
the derivation of the interface potential describing the effective interaction
between the free interface of the wetting film and a substrate. This
contribution leads to a number of qualitative and quantitative changes
compared to long-standing predictions. This resolves also a controversy
concerning the comparison between corresponding theoretical predictions and
available numerical simulations.

In addition to some of the aforementioned open problems and directions for
future research as discussed in Sec. 6, the dynamics of CCFs in and out of
thermal equilibrium pose a theoretical and experimental challenge and are a
largely unexplored subject beyond simplistic models. Accordingly, any progress
in this direction, e.g., by accounting for the dynamics of fluctuations beyond
mean-field theory or by considering more realistic geometries for their
confinement, would be highly valuable, also in view of promising experimental
access to these phenomena.

The scientific activity reviewed in this contribution was largely motivated by
the inspiring work of Michael E. Fisher, and it enjoyed his support and
encouragement in the course of time. The wealth and the variety of theoretical
and experimental results, which originated from these studies, are therefore
part of his scientific legacy and pay tribute to his outstanding scientific
oeuvre.

## Acknowledgements

AG acknowledges support from MIUR PRIN project “Coarse-grained description for
non-equilibrium systems and transport phenomena (CO-NEST)” n. 201798CZL.

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