# Colloidal diffusiophoresis in crossed electrolyte gradients: experimental
demonstration of an ‘action at a distance’ effect predicted by the Nernst-
Planck equations

Ian Williams  School of Mathematics and Physics, University of Surrey,
Guildford GU2 7XH, United Kingdom  Patrick B. Warren
<EMAIL_ADDRESS>The Hartree Centre, STFC Daresbury Laboratory,
Warrington WA4 4AD, United Kingdom  Richard P. Sear<EMAIL_ADDRESS>School of Mathematics and Physics, University of Surrey, Guildford GU2 7XH,
United Kingdom  Joseph L. Keddie  School of Mathematics and Physics,
University of Surrey, Guildford GU2 7XH, United Kingdom

###### Abstract

In an externally imposed electrolyte (salt) concentration gradient, charged
colloids drift at speeds of order one micrometre per second. This phenomenon
is known as diffusiophoresis. In systems with multiple salts and ‘crossed’
salt gradients, a nonlocal component of the electric field associated with a
circulating (solenoidal) ion current can arise. This is in addition to the
conventional local component that depends only on the local salt gradients.
Here we report experimental observations verifying the existence of this
nonlocal contribution. To our knowledge this is the first observation of
nonlocal diffusiophoresis. The current develops quasi-instantaneously on the
time scale of salt diffusion. Therefore, in systems with multiple salts and
crossed salt gradients, one can expect a nonlocal contribution to
diffusiophoresis which is dependent on the geometry of the system as a whole
and appears as a kind of instantaneous ‘action-at-a-distance’ effect. The
interpretation is aided by a magnetostatic analogy. Our experiments are
facilitated by a judicious particle-dependent choice of salt (potassium
acetate) for which the two local contributions to diffusiophoresis almost
cancel, effectively eliminating conventional diffusiophoresis. This enables us
to clearly identify the novel, nonlocal effect and may be useful in other
contexts, for example in sorting particle mixtures.

##  I Introduction

Diffusiophoresis (DP) is the directed transport of suspended particles driven
by a gradient in solution concentration [1, 2]. Originally described by
Derjaguin [3], and developed by Prieve and Anderson [4, 5, 6], DP is a
consequence of the interaction between the particle surface and molecules in
solution. In an externally-imposed concentration gradient, the particle-
surface interfacial free energy depends on position and this induces motion
towards where this free energy is lower [4]. This is known as chemiphoresis.
If the solute in question is ionic, an additional electrophoretic contribution
to particle motion arises in response to the electric field spontaneously
established when anions and cations diffuse at different rates [5].

Recent research has revealed the ubiquity of DP in processes ranging from
biological transport [7, 8] and pattern formation [9], to dialysis [10], water
purification [11] and laundry detergency [12]. Furthermore, researchers are
harnessing DP to develop technologies in areas including self-stratifying
coatings [13, 14], enhanced oil recovery [15], and colloidal sorting and
separations [16, 17, 18, 19, 20]. Traditionally, researchers have
experimentally and theoretically explored DP in linear, one-dimensional
concentration gradients of electrolytes, charged nanoparticles, or neutral
molecules [21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32]. Squires and
coworkers have further demonstrated how colinear concentration gradients of
multiple molecules can effect complex manipulation of suspended particles [33,
34]. However, gradients in any biological or technological context are
unlikely to be so simple.

In a recent Letter [35], one of us (PBW) theoretically described DP in
orthogonal concentration gradients of two different salts in two dimensions.
For closed systems with one-dimensional gradients, there are electric fields
but no electric currents. This is also true in two or three dimensions if
there is only one salt. However, this work showed that two- or three-
dimensional gradients of two or more salts will necessarily generate a
nonlocal solenoidal current throughout the solution, with an associated
electric field. This field is in addition to the local electric field present
in conventional one-dimensional DP.

Local here means determined by local salt concentrations. Nonlocal means that
as soon as orthogonal salt gradients are developed anywhere in the solution,
the solenoidal current and hence the electric field appear everywhere in the
solution. This electric field will move charged particles, and so this
nonlocal DP behaves essentially as an almost instantaneous action at a
distance. This is qualitatively different behaviour from local DP which only
acts at a point in solution once the salt has diffused to that point. Here we
present the first experimental evidence of this action-at-a-distance effect on
colloidal particles.

The phenomenon may have applications in charge-sensitive colloidal sorting and
separation, including electrode-free nanomaterial recovery and recycling [36,
37, 38]. In addition to being much faster than conventional DP to start up, it
acts to separate particles of different surface potentials as particles with
different surface potentials follow different, diverging trajectories. Two-
dimensional DP has further potential as an inexpensive and portable
characterisation tool for concentrated suspensions, capable of extracting
particle charge and isoelectric point from DP trajectories [18]. Precise
control over colloidal trajectories by two-dimensional DP may also lead to
advances in self-assembly by electrophoretic deposition [39].

Figure 1:  Microfluidic device schematics and micrographs. (a) Side view
schematic showing microchannel construction using double-sided adhesive tape.
(b) Top view T-shaped channel schematic for one-dimensional DP experiments. A
poly(ethylene glycol) diacrylate (PEG-DA) hydrogel containing salt is formed
in the source channel.  Green shaded ovals indicate the open channel ends
which are sealed with glue after the experiment is initiated. (c) Brightfield
micrograph at $5\times$ magnification showing the region close to the
T-junction. Hydrogel and tape are labeled.  The navy blue dashed line
indicates the hydrogel edge, defined as $r=0$, and the arrow indicates the co-
ordinate, $r$, defined as the normal distance from the $r=0$ line. Green
shaded region shows the region of interest considered for velocity profile
calculations. (d) Top view schematic of the branched channel for two-
dimensional DP experiments. PEG-DA hydrogels containing salts are formed in
perpendicularly oriented source channels. Sample inlet channel is located
between the hydrogels and oriented at $45^{\circ}$.  Green shaded ovals
indicate the open channel ends which are sealed with glue after the experiment
is initiated. (e) Brightfield micrograph at $5\times$ magnification showing
the branch junction. Hydrogel sources are labeled 1 and 2. Tape walls are
labeled T. Green shaded region indicates the region of interest for
experiments performed at $10\times$ magnification.  Navy blue dashed lines
indicate the perpendicular hydrogel edges which form the axes of a Cartesian
co-ordinate system, $(r_{1},r_{2})$, indicated by the arrows. The origin of
this co-ordinate system is in the top right of the image and labeled O. Scale
bar in (c) also applies to (e).

Here we adapt the soluto-inertial beacons approach of Banerjee et al. [40, 34,
41, 42] to realise electrolyte concentration gradients in microfluidic
devices, as illustrated in Fig. 1. We first characterise the DP of negatively
charged polystyrene colloids in one-dimensional gradients of four salts using
T-shaped microfluidic channels, Fig. 1 (b, c). We subsequently superpose
orthogonal gradients of pairs of salts in branched devices, Fig. 1 (d, e), and
provide the first experimental measurements of two-dimensional DP. We identify
the current-induced contribution predicted by Ref. [35] and explore strategies
to maximise its effect.

##  II The Theory of Electrolyte Diffusiophoresis

###  II.1 Electrolyte Diffusiophoresis in 1d

The Derjaguin-Prieve-Anderson (DPA) theory [5, 43] of electrolyte DP predicts
that a suspended particle placed into a gradient in salt concentration, $c$,
will move with velocity

${\mathbf{U}}_{\text{DP}}=D_{\text{DP}}\,\nabla\ln c\,.$ (1)

Assuming an ideal solution, a 1:1 electrolyte, zeta potential $\zeta$, and an
infinitesimal double layer [5, 43], the DPA mobility is

$D_{\text{DP}}=\frac{\epsilon k_{\mathrm{B}}T}{\eta
e}\left(\frac{4k_{\mathrm{B}}T}{e}\ln\cosh\frac{e\zeta}{4k_{\mathrm{B}}T}+\zeta\beta\right)\,.$
(2)

Here $\epsilon$ and $\eta$ are the permittivity and viscosity of the solvent,
$k_{\mathrm{B}}$ is Boltzmann’s constant, $T$ is absolute temperature and $e$
is the elementary charge. The parameter

$\beta=\frac{D_{+}-D_{-}}{D_{+}+D_{-}}$ (3)

encodes the difference in diffusion coefficients between the anion, $D_{-}$,
and the cation, $D_{+}$.

More general forms of Eq. (2) can be formulated [5, 23, 43]. When
concentration gradients lead to gradients in solution pH, it is necessary to
account for the concentration dependence of $\zeta$ [43, 44]. Researchers have
also further developed DP theory to account for multivalent electrolytes [27]
or high ion concentrations [26, 28]. However, the majority of researchers have
found the expression of Eq. (2) and the assumption of constant $\zeta$ to be
sufficient [10, 22, 24, 40, 45, 46, 30].

According to the above, $D_{\text{DP}}$ is the sum of two contributions. The
first term in Eq. (2) is a chemiphoretic contribution arising as a consequence
of the free energy of the particle surface depending on the local salt
concentration. This term is always positive and independent of $\beta$,
meaning that chemiphoresis is always directed up-gradient. The second term in
Eq. (2) is a local electrophoretic contribution, which is a consequence of the
electric field established when the anion and cation diffuse at different
rates. This depends on the product $\zeta\beta$. Thus, depending on the sign
of the particle surface potential and which ion diffuses more rapidly, the
local electrophoretic contribution can be directed in either direction, up or
down the gradient.

Table 1 shows $D_{-}$, $D_{+}$, and $\beta$ for the salts used in this
research. These are tetrabutylammonium bromide (TBAB), sodium chloride (NaCl),
potassium chloride (KCl), and potassium acetate (KOAc), chosen for the range
of $\beta$ they collectively span. The contributions to $D_{\text{DP}}$
according to Eq. (2) are given in Table 2, for $\zeta=-50\,\mathrm{mV}$,
corresponding to the particles used in our experiments.

Because the local electrophoretic term can have either sign, it can add to or
partially cancel the chemiphoretic term, depending on the values of $\zeta$
and $\beta$. Fig. 2 shows $D_{\text{DP}}$ as a function of $\beta$ for
different values of particle $\zeta$ potential. When $D_{\text{DP}}>0$ ($<0$),
the net DP velocity is directed up (down) the concentration gradient.

Table 1: Anion and cation diffusion coefficients at infinite dilution, $D_{-}$ and $D_{+}$, and the diffusivity difference parameter $\beta=(D_{+}-D_{-})/(D_{+}+D_{-})$. Data from Refs. [2] and [47]. | $D_{-}$ | $D_{+}$ |
---|---|---|---
Electrolyte | $({}\times 10^{-10}\,\mathrm{m}^{2}\,\mathrm{s}^{-1})$ | $({}\times 10^{-10}\,\mathrm{m}^{2}\,\mathrm{s}^{-1})$ | $\beta$
TBAB | 20.8 |  05.2 |  $-$0.60
NaCl | 20.3 | 13.3 |  $-$0.21
KCl | 20.3 | 19.6 |  $-$0.02
KOAc | 10.9 | 19.6 |  $-$0.28
Table 2: Theoretical $D_{\text{DP}}$ according to the DPA model, Eq. (1), calculated with $\zeta=-50\,\mathrm{mV}$ and $\beta$ given in Table 1, assuming solvent viscosity $\eta=$1\text{\,}\mathrm{mPa}\text{\,}\mathrm{s}$$ and permittivity $\epsilon=80\,\epsilon_{0}$.  | Contribution to $D_{\text{DP}}$ $({}\times 10^{-10}\,\mathrm{m}^{2}\,\mathrm{s}^{-1})$ |
---|---|---
Electrolyte | Local Electrophoresis | Chemiphoresis | Total
TBAB |  $-$5.50 | 2.13 |  $-$7.63
NaCl |  $-$1.90 | 2.13 |  $-$4.03
KCl |  $-$0.17 | 2.13 |  $-$2.30
KOAc |  $-$2.60 | 2.13 |  $-$0.48
Figure 2:  DPA diffusiophoretic mobility, $D_{DP}$, as a function of the
reduced difference in cation and anion diffusion constants, $\beta$ for
particles with $\zeta$ potentials indicated in the legend. These are
calculated according to Eq. (2) for salts in water at 300 K. The asterisks
represent $\beta$ and $D_{\text{DP}}$ for the combination of the particles
with $\zeta=-50\,\mathrm{mV}$ and salts employed in our experiments, (from
left to right): TBAB, NaCl, KCl and KOAc. The inset shows the value of $\beta$
necessary for diffusiophoretic neutrality, $\beta^{*}$, defined as
$D_{\text{DP}}(\beta^{*})=0$, as a function of particle $\zeta$ potential.

For a given $\zeta$, there is a value $\beta^{*}$ for which $D_{\text{DP}}=0$
(Fig. 2). This corresponds to the two terms inside the parentheses in Eq. (2)
perfectly cancelling, a condition we refer to as _diffusiophoretic
neutrality_. If we linearise the $\ln\cosh$ term we obtain the approximate
expression

$\beta^{*}\approx-\frac{e\zeta}{8k_{\mathrm{B}}T}\,.$ (4)

The sign of $\beta^{*}$ depends on the sign of the charge on the particle, and
the magnitude increases with its $\zeta$ potential. The value for $\beta^{*}$
as a function of $\zeta$ obtained from Eq. (2) is plotted in the inset to Fig.
2. For example, KOAc has $\beta=0.28$ and is approximately DP neutral when
particles have $\zeta=-50\,\mathrm{mV}$. But, if particles have the opposite
charge, $\zeta=50\,\mathrm{mV}$, NaCl with $\beta=-0.21$ is a better choice
for DP neutrality.

###  II.2 Electrolyte Diffusiophoresis in 2d

To the best of our knowledge, DP in two dimensions when perpendicular
gradients of two salts are superposed has been described only in Ref. [35].
The theory predicts a DP velocity given by

$\begin{split}{\mathbf{U}}_{\text{DP}}=&\frac{\epsilon}{\eta}\Bigl{(}\frac{k_{\mathrm{B}}T}{e}\Bigr{)}^{2}\,\Bigl{[}4\ln\cosh\left(\frac{e\zeta}{4k_{\mathrm{B}}T}\right)\nabla\ln\sum_{i}c_{i}\\\
&\hskip 80.00012pt{}+\frac{e\zeta}{k_{\mathrm{B}}T}\frac{\nabla
g}{\sigma}+\frac{e\zeta}{k_{\mathrm{B}}T}\frac{{\mathbf{I}}}{\sigma}\Bigr{]}\,,\end{split}$
(5)

where the index $i$ labels each ion species. Here
${g=\sum_{i}z_{i}D_{i}c_{i}}$ is a weighted sum of ion densities ($z_{i}=\pm
1$ is the valence) and ${\sigma=\sum_{i}z_{i}^{2}D_{i}c_{i}}$ is the
conductivity.

The second and third terms in the above are obtained by considering the ion
current ${\mathbf{I}}$, which in terms of the dimensionless electrostatic
potential $\varphi$ and corresponding electric field
${\mathbf{E}}=-\nabla\varphi$, is given by

${\mathbf{I}}=-\nabla g+\sigma{\mathbf{E}}\,.$ (6)

This result is derived by summing the set of Nernst-Planck equations (one per
species) which govern how the concentrations $c_{i}$ of the salt ions change
with time [35, 48]. Inverting Eq. (6), one can therefore associate the
electric field with a local contribution from a gradient in $g$, and the
current,

${\mathbf{E}}=\frac{\nabla g}{\sigma}+\frac{{\mathbf{I}}}{\sigma}\,.$ (7)

Injecting this into the standard electrophoresis model gives rise to the
second and third terms in Eq. (5) (the first term is just chemiphoresis, as in
the one-dimensional case). With this decomposition, the first and second terms
of Eq. (5) are generalisations of the chemiphoretic and local electrophoretic
terms of the standard DPA theory to the case of multiple salts. Both are local
in the sense that they are functions only of the local gradients in the ion
concentrations ($\nabla c_{i}$). The new (third) term cannot be written as a
function only of local gradients and is therefore nonlocal. Because this term
is also fundamentally electrophoretic in origin, it scales with $\zeta$ the
same way as the second term.

####  II.2.1 Conditions for currents and hence action-at-a-distance
diffusiophoresis, within the Nernst-Planck equations

What are the conditions for a non-vanishing ion current? First, imposing
electroneutrality in the Nernst-Planck equations shows that the current must
be _solenoidal_ [35], $\nabla\cdot{\mathbf{I}}=0$, which implies
$\nabla\cdot(\sigma\nabla\varphi)+\nabla^{2}g=0$ [35, 48]. This is an
inhomogeneous Poisson equation for the electrostatic potential, which replaces
the true electrostatic Poisson equation in the problem (obviated by
electroneutrality). One can show [35] that solutions $({\mathbf{I}},\varphi)$
are uniquely specified up to an additive constant in $\varphi$ by the ion
current through the boundaries (typically, in a closed system,
${\mathbf{I}}\cdot{\mathbf{n}}=0$ where ${\mathbf{n}}$ is the surface normal).

Taking the ‘curl’ of Eq. (7) and using the fact that
$\nabla\times{\mathbf{E}}=0$ (because ${\mathbf{E}}=-\nabla\varphi$) obtains

$\nabla\times\left(\varrho{\mathbf{I}}\right)=\nabla g\times\nabla\varrho\,,$
(8)

where $\varrho\equiv\sigma^{-1}$ is the resistivity. This implies that if the
gradients in $g$ and resistivity $\varrho$ (equivalently conductivity
$\sigma$) are everywhere parallel (or antiparallel), then a solution exists
with ${\mathbf{I}}=0$ everywhere, and by virtue of the uniqueness theorem,
this is the only solution.

Thus ${\mathbf{I}}\neq 0$ requires ‘crossed’ gradients
$\nabla\sigma\times\nabla g\neq 0$ _somewhere_ in the system. However this
does not imply that a current is restricted to such regions. It follows from
Eq. (8) that in regions where there are no gradients in $\varrho$ and $g$, the
current is _irrotational_ [35]; the current satisfies
$\nabla\times{\mathbf{I}}=0$. So in the region without crossed gradients, the
conditions are $\nabla\times{\mathbf{I}}=0$ and $\nabla\cdot{\mathbf{I}}=0$.
They can be met if ${\mathbf{I}}$ is the gradient of some ‘current potential’,
${\mathbf{I}}=\nabla\omega$, which is harmonic ($\nabla^{2}\omega=0$) in the
region in question.

Table 3: Magnetostatic analogy for the ion current in the Nernst-Planck equations. Note that ${\mathbf{B}}=\mu{\mathbf{H}}$. The boundary condition corresponds to a perfect conductor. The notation is from Jackson’s Classical Electrodynamics, 3rd ed. [49]. Nernst-Planck | magnetostatic analogy
---|---
$\nabla\cdot{\mathbf{I}}=0$ | $\nabla\cdot{\mathbf{B}}=0$ | Gauss’ law
$\nabla\times(\varrho{\mathbf{I}})=\nabla g\times\nabla\varrho$ | $\nabla\times{\mathbf{H}}={\mathbf{J}}$ | Ampère’s law
${\mathbf{I}}\cdot{\mathbf{n}}=0$ | ${\mathbf{B}}\cdot{\mathbf{n}}=0$ | boundary condition

That a non-vanishing, irrotational, solenoidal current is perfectly possible
can also be seen by an unexpectedly close analogy (Table 3) to magnetostatics
[49]. This means that the ion currents around a region where there are crossed
salt gradients look like the magnetic flux lines around a current-carrying
conductor in the same place, and in two dimensions the ion current circulates
around the crossed gradients like magnetic field lines encircling a current-
carrying wire 111There are some restrictions: Eq. (7) implies
$\oint\\!\varrho({\mathbf{I}}+\nabla g)\cdot\mathrm{d}{\mathbf{l}}=0$. Hence
$\oint\\!{\mathbf{I}}\cdot\mathrm{d}{\mathbf{l}}=0$ along a path where
$\varrho$ is constant and
$\oint\\!\varrho{\mathbf{I}}\cdot\mathrm{d}{\mathbf{l}}=0$ along a path where
$g$ is constant. Since $\varrho>0$ this implies a circulating ion current must
pass, possibly separately, through regions where $\varrho$ and $g$ are
spatially varying [35]. This result can also be derived by observing that
$\nabla
g\times\nabla\varrho=\nabla\times(g\nabla\varrho)=-\nabla\times(\varrho\nabla
g)$ which implies via Stokes’ theorem $\int(\nabla
g\times\nabla\varrho)\,\cdot{\mathbf{n}}\mathrm{d}{S}=\oint\\!g\nabla\varrho\cdot\mathrm{d}{\mathbf{l}}=-\oint\\!\varrho\nabla
g\cdot\mathrm{d}{\mathbf{l}}$. Hence, if there are net crossed gradients
within an area $S$, there must be places on the perimeter where
$\nabla\varrho\neq 0$ and where $\nabla g\neq 0$.. Note that just as
magnetostatics demands $\nabla\cdot{\mathbf{J}}=0$, the source term in Eq. (8)
is divergence-free, $\nabla\cdot(\nabla g\times\nabla\varrho)=0$ ; this
follows from standard vector calculus identities [49].

In the present problem therefore, provided there are crossed gradients (in the
sense that $\nabla\sigma\times\nabla g\neq 0$) somewhere in the system, one
expects generically that an ion current will appear (quasi-instantaneously,
see below) throughout the whole system, even in places where there are no
crossed gradients. According to Eq. (5) this ion current drives nonlocal DP,
which should therefore in principle be observed quasi-instantaneously
throughout the system. A point to note, which will be important in the sequel,
is that the effect should be largest where the conductivity is least, because
this is where ${\mathbf{E}}$ is largest; this follows immediately from Eq.
(7).

The Nernst-Planck equations describe the evolution of the salt gradients, so
we are always looking at the current distribution as a snapshot in time, as it
were. The magnetostatic analogy evolves as the salt gradients evolve, but it
remains magneto-_static_. There is no equivalent to Maxwell’s displacement
current, the $\partial{\mathbf{D}}/\partial t$ term in Maxwell’s equations.

In any system with one-dimensional gradients all gradients are parallel and
there are no currents, although the gradients may be coupled [51]. It is also
true however, that the current vanishes when there is only one salt present.
This follows from electroneutrality. For example, for a 1:1 electrolyte,
$\sigma=(D_{1}-D_{2})c$ and $g=(D_{1}+D_{2})c$, because electroneutrality
forces $c_{1}=c_{2}=c$. Hence $\nabla\sigma\times\nabla g=0$ everywhere. To
summarise: in strictly one-dimensional systems, or when there is only one salt
present, there is no current. However, if there are two or more salts present
in a geometry that is not one-dimensional, ‘crossed’ gradients are likely to
arise, if not initially then almost certainly as the concentrations relax,
making the accompanying ion currents and action-at-a-distance diffusiophoresis
inevitable.

####  II.2.2 Timescales in experiments in two-dimensional diffusiophoresis

Before we discuss our experimental results, it is useful to consider the time
scales relevant to our experimental system and elucidate precisely what is
meant by ‘quasi-instantaneously’ in the above discussion. The details are in
Appendix F; here we summarise key conclusions. The longest time scales, and
the only ones accessible in experiment, are that of diffusion of salt across
the system. Our system is of order $L\approx$1\text{\,}\mathrm{mm}$$ across,
so given the ambipolar salt diffusion coefficient $D_{\mathrm{a}}\sim
10^{-9}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$, the time scale is
$L^{2}/D_{\mathrm{a}}\approx 10^{3}\,$\text{\,}\mathrm{s}$\approx
16\,\mathrm{minutes}$. As we show in Appendix F, salt diffusion sets the time
scale of both local diffusiophoresis terms, i. e. including the local-
diffusiophoresis contribution to the electric field. So all local DP
propagates across our system at the speed of salt diffusion.

In II.2.1 we showed that the Nernst-Planck equations predict that nonlocal DP
starts as soon as there are crossed salt gradients somewhere in the system.
These salt gradients are established by diffusion, and so for salt sources $d$
apart, the gradients cross and nonlocal DP starts up after a time
$d^{2}/D_{\mathrm{a}}$. Our experimental setup is in Fig. 1(e). The distance
between the sources is about 2.5 times smaller than the distances from the
sources to the farthest point from the sources imaged (bottom left of green
shaded region). So nonlocal DP will reach the bottom-left point about five
times faster than local DP.

Nonlocal DP starts, in effect, as soon as salt gradients cross because the
nonlocal currents and electric fields are established and propagate very
rapidly. It is this that makes the nonlocal DP look like instantaneous action-
at-a-distance. The two relevant timescales are shown in Fig. 3, and are
discussed by Bazant et al. [52], and in Appendix F. The timescales are for
establishment of a space charge and charging of the electric double layers
(EDLs). Both involve moving relatively tiny amounts of charge density, much
less than that in the salt solution ($ce$). So they are much faster than salt
diffusion and much faster than we can observe with microscopy. The space
charge is needed to establish electric fields, and so the current. And the
EDLs control the boundary conditions for these electric fields.

Figure 3: Schematic of diffusion-limited time scales relevant to the system,
from smallest (left) to largest (right). The shaded region is approximate
range accessible in our experiments. We assume the Debye length
$\lambda_{\text{D}}\sim 10^{-9}\,\mathrm{m}$ ($100\,\mathrm{mM}$ salt),
ambipolar diffusion coefficient for ions $D_{\mathrm{a}}\sim
10^{-9}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$, and system size $L\sim
10^{-3}\,\mathrm{m}$. Details are given in Appendix F. Figure 4:  (a) Average
DP velocity towards / away from a hydrogel source containing
$130\,\mathrm{mM}$ NaCl as a function of distance from the source, measured in
experiments. Colour indicates time elapsed between loading the the
microfluidic device and data acquisition. Each data set is the average of
three repeated experiments. Error bars indicate the standard error in the
mean. (b) Theoretical DP velocity towards / away from an NaCl source as a
function of distance from the source according to the DPA model, Eq. (1).
Colour indicates time elapsed since gradient initiation. (c) Average DP
velocity towards / away from a hydrogel source as a function of distance from
the source measured in experiment at $t=90\,\mathrm{s}$ after loading. Sources
contain $130\,\mathrm{mM}$ of different electrolytes, as indicated in the
legend. Each data set is the average of three repeated experiments. Error bars
indicate the standard error in the mean. (d) Theoretical DP velocity towards /
away from electrolyte sources at $t=90\,\mathrm{s}$ after gradient initiation
as a function of distance from the source according to the DPA model, Eq. (1).
Colour represents electrolyte, as indicated in the legend.

##  III Results and Discussion

We seek to directly observe the nonlocal, current-induced electrophoretic
contribution to DP predicted in Ref. [35]. To this end, soluto-inertial
beacons [40] are used to create electrolyte gradients. Fluorescence microscopy
and particle imaging velocimetry are employed to observe and quantify the
motion of negatively charged colloids in these gradients.  Particle velocities
are interpreted as arising exclusively as a result of diffusiophoresis,
although in general we expect an additional contribution due to particle
advection in the fluid flows generated by diffusioosmosis at the channel walls
[53, 54]. This is discussed in greater detail in Section IV.

Working in the microfluidic devices illustrated in Fig. 1, we first measure DP
in one-dimensional gradients of the four salts summarised in Tables 1 and 2,
and compare these experiments to the predictions of the DPA model.
Subsequently, we superpose orthogonal gradients of pairs of these salts and
measure DP in the two-dimensional concentration field. These experiments are
compared to simulations implementing the model described in Ref. [35]. Full
methods are provided in Appendices A, B and C.

###  III.1 Diffusiophoresis in 1d Salt Gradients

One-dimensional experiments use T-shaped microfluidic devices with integrated
hydrogel sources containing $130\,\mathrm{mM}$ of salt, as described in
Appendix B and shown in Fig. 1 (b, c). Particle velocity fields are estimated
by PIV, as described in Appendix C, and the components of velocity towards /
away from the salt source are spatially averaged to obtain velocity, $U$, as a
function of  the normal distance from  a line identified at the edge of the
hydrogel, $r$, as indicated in Fig. 1 (c). The region of interest is
restricted to be directly in front of the hydrogel (green shaded region).
Negative velocities indicate motion up-gradient. The experiment is repeated
three times with each salt and velocity profiles are averaged over the three
repeats.

The salts considered span a range of $\beta$. Tetrabutylammonium bromide
(TBAB) and sodium chloride (NaCl) have $\beta<0$ and are expected to drive DP
in the same direction, with TBAB driving faster DP due to its larger absolute
value of $\beta$. Potassium chloride (KCl) has $\beta\approx 0$ and therefore
its DP is anticipated to be dominated by chemiphoresis. Potassium acetate
(KOAc) has $\beta>0$, meaning that its local electrophoretic component of DP
is predicted to be oppositely directed to that due to TBAB or NaCl. To predict
$U(r)$ for each salt using Eq. (1), the time evolution of the salt
concentration is estimated as described in Appendix D.

Results are shown in Fig. 4. The time evolution of the velocity profile
measured in a NaCl gradient is shown in Fig. 4 (a), and model predictions are
in Fig. 4 (b). Since our particles are negatively charged, DP speeds are
always negative, i. e. towards the source of the salt. Analogous plots for
TBAB, KCl, and KOAc are in the Supplementary Information [55].

The DP velocity profile changes with time because the salt concentration
gradient is transient. Fast DP is measured at early times when the
concentration gradient is steep but short-ranged. Initially, a peak in DP
speed is measured at finite $r$. Especially at early times, far from the
source, DP speed drops to zero as it takes time for salt to diffuse outwards
and establish the local concentration gradient necessary to drive DP. The time
for the salt to diffuse a distance $r$ is of the order $r^{2}/D_{\mathrm{a}}$,
which for a distance of $r\approx$1000\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
is of the order ten minutes. We show in Appendix F that this local DP
propagates at the speed of salt diffusion. At longer times the gradients
weaken and so does DP. The trends in the experimental data are also seen in
the model calculations in Fig. 4 (b).

Experimentally measured DP velocity profiles for different salts at a single
time ($t=90\,\mathrm{s}$) are compared in Fig. 4 (c), with the corresponding
theoretical predictions in Fig. 4 (d). There is good agreement between the
model and experiment. Salts with larger $D_{\text{DP}}$ drive faster DP, as
expected. The chemiphoretic term is positive and identical for all monovalent
electrolytes, meaning that chemiphoresis is predicted to always act up-
gradient. Differences between salts are due to differences in their local
electrophoretic terms, which depend on $\beta$.

For TBAB and NaCl, the local electrophoretic contribution is positive, and
acts in the same, up-gradient direction as chemiphoresis. $D_{\text{DP}}$ for
TBAB is approximately twice as large for NaCl, and therefore, in identical
gradients, TBAB is predicted to drive DP at approximately twice the speed of
NaCl. The difference measured in experiment is not so large, but the
measurements do show faster DP with TBAB. We speculate that  the main source
of this discrepancy is due to the unknown association and diffusion constants
for the ions in the hydrogel. If TBAB associates with the hydrogel more
strongly than NaCl, it will be emitted more slowly, meaning that the effective
source concentration will be smaller for TBAB, and the concentration gradient
will be more shallow.

An additional source of quantitative discrepancy between the experiments and
modeling may be that the model is strictly 1-dimensional, while the
experiments are not. The finite width and often slightly curved shape of the
hydrogel surfaces, both mean that the iso-concentration contours in
experiments will not be perfectly parallel to the line chosen as $r=0$.

KCl has $\beta\approx 0$ as its anion and cation have very similar diffusion
coefficients. Consequently, the local electrophoretic term for KCl is very
small and KCl is predicted to drive DP primarily by chemiphoresis. The total
$D_{\text{DP}}$ for KCl is still positive but smaller than for TBAB and NaCl.
This is consistent with the experiments, which show up-gradient motion at
slower speeds than both NaCl and TBAB.

Figure 5:  (a) Experimentally measured velocity field in orthogonal
concentration gradients of KOAc and NaCl at $t\approx 115\,\mathrm{s}$ after
initiation. The KOAc source is located
$\approx$840\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from the top of the field
of view and the NaCl source is located
$\approx$865\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ from the right of the
field of view, as indicated in the diagram on the right. The spacing between
velocity vectors is $44\text{\,}\mathrm{\SIUnitSymbolMicro m}$. (b,c)
Simulated velocity fields at $t=115\,\mathrm{s}$ computed using (b) the full
model, including nonlocal electrophoresis, and (c) the local model including
only chemiphoresis and local electrophoresis. The red dashed rectangle
corresponds to the location of the experimental field of view.  Top right
illustrates the Cartesian co-ordinate system in which position is defined by
the normal distances from lines drawn at the edges of the two sources and the
origin is located between the sources. Figure 6:  KOAc superposed with NaCl.
Projected velocity profiles in the region for which all three experiments
overlap. Colour represents time after initiation, as indicated in legends.
Horizontal dashed lines indicate $U=0$. (a–d) Experimental velocity profiles
computed as an average over three repeated experiments. (e–h) Velocity
profiles predicted by the full model, including the nonlocal contribution to
DP. (i–l) Velocity profiles predicted by the local model: (a), (e), and (i)
show $\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{NaCl}})$ ; (b), (f), and (j) show
$\bar{U}_{\mathrm{KOAc}}(r_{\mathrm{NaCl}})$ ; (c), (g), and (k) show
$\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{KOAc}})$ ; (d), (h), and (l) show
$\bar{U}_{\mathrm{KOAc}}(r_{\mathrm{KOAc}})$.

The local electrophoretic contribution for KOAc is negative, oppositely
directed to chemiphoresis. The two terms have comparable magnitude and
virtually cancel one another out, making KOAc an almost DP-neutral salt for
our particles. DP is predicted to be an order of magnitude slower with KOAc
than for the other salts, and indeed we observe negligible motion in a one-
dimensional KOAc gradient.

In summary, we have measured the DP motion of negatively charged particles
under gradients of TBAB, NaCl, KCl, and KOAc. Experimental measurements are in
good agreement with the predictions of Eq. (1). We therefore consider the one-
dimensional DP driven by these salts to be established and understood. If the
objective is to drive fast DP in one dimension, then (for particles with
$\zeta=-50\,\mathrm{mV}$) KOAc is almost the worst possible salt. However,
this feature makes KOAc almost perfect for studying DP effects that only occur
with superposed orthogonal salt gradients, as we demonstrate in the following
Section.

###  III.2 Diffusiophoresis in 2d Salt Gradients

The one-dimensional gradients of Section III.1 are combined by superposing two
different gradients in orthogonal directions. This is achieved using branched
microfluidic devices containing two hydrogel sources, as described in Appendix
B and shown in Fig. 1 (d, e). The only experimental observable is the particle
velocity field, which Ref. [35] predicts to arise as a vector sum of
chemiphoresis, local electrophoresis, and nonlocal electrophoresis. We measure
this velocity field, and compare it to the fields predicted by modelling with
and without the current-induced contribution to DP predicted to exist in two
dimensions. Three salt combinations are chosen based on their qualitatively
different one-dimensional DP behaviour: KOAc ($\beta>0$) superposed with NaCl
($\beta<0$); TBAB ($\beta<0$) superposed with NaCl ($\beta<0$); and KCl
($\beta\approx 0$) superposed with KOAc ($\beta>0$). A fourth combination, KCl
($\beta\approx 0$) superposed with NaCl ($\beta<0$), behaves very similarly to
TBAB superposed with NaCl, and is therefore omitted from the article, but
included in the Supplementary Information [55].

####  III.2.1 Orthogonal Gradients of KOAc and NaCl

Figure 5 (a) shows a snapshot of the experimental velocity field measured at
$t=115\,\mathrm{s}$ after initiation in an experiment superposing a vertical
gradient of KOAc with a horizontal gradient of NaCl. The KOAc source is
located approximately $840\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the top
edge of the field of view and the NaCl source is located approximately
$865\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the right edge of the field
of view.  A Cartesian co-ordinate system,
$(r_{\mathrm{NaCl}},r_{\mathrm{KOAc}})$, is defined as illustrated in Fig. 1
(e) and the top right of Fig. 5. The co-ordinate $r_{\mathrm{NaCl}}$ is the
normal distance from a line drawn at the edge of the NaCl source and,
similarly, $r_{\mathrm{KOAc}}$ is the normal distance from a line drawn at the
edge of the KOAc source. Unit vectors in the $r_{\mathrm{NaCl}}$ and
$r_{\mathrm{KOAc}}$ directions are perpendicular to one another, and the
origin of the co-ordinate system is located between the two sources in
experiment, or in the top-right corner in the model. The location of the
microscope field of view relative to the sources varies between experiments,
and so  the origin of the co-ordinate system is measured for each experiment
using composite images as described in the Supplementary Information [55].

Figure 5 (a) reveals that the direction and magnitude of particle velocity are
dependent on position relative to the salt sources. At this instant, particles
on the right of the field of view are moving towards the NaCl source with
speeds between 1 and $2\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$. This motion is reminiscent of that expected in
a one-dimensional gradient of NaCl, which drives up-gradient DP of these
particles. Ahead of this NaCl-attractive front, there is a region in which
velocities are smaller and directed upwards, towards the KOAc source. Further
still from the NaCl source, velocities are directed towards the upper left
corner.

Figure 5 (a) shows only the snapshot at $t=115\,\mathrm{s}$, but the velocity
field evolves with time, see Movies 1 and 2. As time proceeds, the NaCl-
attractive front moves outwards and the velocity vectors rotate past the
vertical, until the attraction towards the NaCl source dominates the field of
view and all vectors point towards the right. Along with this change in
direction, the overall DP speed decreases with time, consistent with the
measurements of one-dimensional DP shown in Fig. 4.

The experimental results, Fig. 5 (a), are now compared with the predictions of
the full model, Fig. 5 (b), and the model without the nonlocal term, Fig. 5
(c). Model vector fields are shown over the full simulated region, with the
red boxes indicating the the approximate location of the experimental field of
view.

Within the red boxes and close to the NaCl source, DP in both full and local-
only models is dominated by an up-gradient NaCl-attraction, consistent with
the experimental velocity field. However,  in the bottom left of the red box
in Fig. 5 (c), the local-only model predicts that particles are almost
stationary. This contrasts with the full model, Fig. 5 (b) which qualitatively
agrees with the experiment and predicts motion up and to the left. This
particle motion can only be reproduced by a model incorporating the nonlocal,
current-induced, contribution to DP. To the best of our knowledge, these data
are the first experimental observation of this effect.

The full dynamics may be compared by comparing the vector fields for
experimental particle motion in Movie 2 with those predicted by the full
(Movie 3) and local-only (Movie 4) models. Only the full model shows the same
transient motion up and to the left as observed in experiment.

However agreement between this experiment and the full model is not
quantitative. At its fastest, the motion in this particular experiment is
$\sim$4\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$$
whereas the model predictions are $\sim$1\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{\,}{\mathrm{s}}^{-1}$$. The quantitative discrepancy between
experiments and model results likely results from the combined effect of many
differences. The geometry of the model does not attempt to accurately
replicate the geometry of the branched microfluidic devices and salt sources.
The initial conditions in modeling and experiment are necessarily different.
In the experiments an initial liquid flow is needed for the colloidal
suspension to flow into the device. This flow will transport some salt away
from the sources. In the model, the chemical potential is fixed at the
boundaries and so the models tend towards steady-state crossed gradients,
while the experiments approach an equilibrium of uniform concentrations.
Modeling neglects any advective particle transport in fluid flows. In
experiment such flows could occur, driven by diffusioosmosis at the device
walls, convection, or any other imperfections in the construction of the
microfluidic devices. Despite all these sources of quantitative discrepancy,
the qualitative agreement between the experimental observations and the DP
model incorporating the nonlocal, current-driven contribution is striking.

Figure 7:  TBAB superposed with NaCl. Projected velocity profiles in the
region for which all three experiments overlap. Colour represents time after
initiation, as indicated in legends. Horizontal dashed lines indicate $U=0$.
(a-d) Experimental velocity profiles computed as an average over three
repeated experiments. (e-h) Velocity profiles predicted by the full model,
including the nonlocal contribution to DP. (i-l) Velocity profiles predicted
by the local model: (a), (e), and (i) show
$\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{NaCl}})$ ; (b), (f), and (j) show
$\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{NaCl}})$ ; (c), (g), and (k) show
$\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{TBAB}})$ ; (d), (h), and (l) show
$\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{TBAB}})$. Figure 8:  KCl superposed with
KOAc. Projected velocity profiles in the region for which all three
experiments overlap. Colour represents time after initiation, as indicated in
legends. Horizontal dashed lines indicate $U=0$. (a-d) Experimental velocity
profiles computed as an average over three repeated experiments. (e-h)
Velocity profiles predicted by the full model, including the nonlocal
contribution to DP. (i-l) Velocity profiles predicted by the local model: (a),
(e), and (i) show $\bar{U}_{\mathrm{KOAc}}(r_{\mathrm{KOAc}})$ ; (b), (f), and
(j) show $\bar{U}_{\mathrm{KCl}}(r_{\mathrm{KOAc}})$ ; (c), (g), and (k) show
$\bar{U}_{\mathrm{KOAc}}(r_{\mathrm{KCl}})$ ; (d), (h), and (l) show
$\bar{U}_{\mathrm{KCl}}(r_{\mathrm{KCl}})$.

Comparison of two-dimensional vector fields is challenging, so in Fig. 6 each
velocity field is projected onto four one-dimensional velocity profiles  in
the $(r_{\mathrm{NaCl}},r_{\mathrm{KOAc}})$ co-ordinate system, which are more
easily compared. The three repeated implementations of each experiment are
averaged. However, since the location of the field of view differs between
experiments, we  compute average velocities only the subregion where all
experiments overlap  as illustrated in Section X of the Supplementary
Information. Over this subregion, four velocity profiles are computed: (i) the
average velocity towards / away from the NaCl source as a function of distance
from the NaCl source, $\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{NaCl}})$; (ii) the
average velocity towards / away from the KOAc source as a function of distance
from the NaCl source, $\bar{U}_{\mathrm{KOAc}}(r_{\mathrm{NaCl}})$; (iii) the
average velocity towards / away from the NaCl source as a function of distance
from the KOAc source, $\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{KOAc}})$; and (iv)
the average velocity towards / away from the KOAc source as a function of
distance from the KOAc source, $\bar{U}_{\mathrm{KOAc}}(r_{\mathrm{KOAc}})$.
The same quantities are computed from the full and local model velocity
fields, in the subregion corresponding to the overlapping region of the three
experiments.

The projected profiles are shown in Fig. 6 at six times between
$t=60\,\mathrm{s}$ and $t=360\,\mathrm{s}$. The first row (a–d) shows the
average projected velocity profiles computed over three independent
experiments. The second (e–h) and third (i–l) rows show the projected profiles
predicted by the full and local models, respectively. Negative (positive)
velocities represent motion towards (away from) the corresponding electrolyte
source.

From Fig. 6 (a), we see that at the earliest times, profiles are positive,
indicating an effective NaCl-repulsion. At the latest times, profiles are
negative, indicating an NaCl-attraction. The same qualitative behaviour is
evident in the model predictions incorporating the current-induced
contribution to DP, see Fig. 6 (e). By contrast, the local model predicts
profiles that are always negative, i. e. motion is always towards the NaCl
source. Our findings for the other three sets of projected velocity profiles
are similar, the full model agrees qualitatively with experiment while the
local model often predicts velocities with the wrong sign, especially at early
times. In the case of motion towards / away from the KOAc source, the local
model predicts almost zero velocity due to KOAc’s DP-neutrality with our
particles, while experiments show clear negative velocities. These negative
velocities emerge in the modelling when the current-induced term is included.

In all projections, the differences between the full model (middle row) and
local model (bottom row) predictions are most pronounced at earlier times. The
current-driven component of DP scales with the reciprocal of conductivity,
which is largest when ion concentration is small. In these experiments, for
which the concentration gradient is transient, ion concentration is lowest at
initiation and increases as ions are released from the hydrogel sources.
Therefore, nonlocal DP effects are increasingly suppressed as time progresses,
gradients become more shallow and conductivity increases.

####  III.2.2 Orthogonal gradients of TBAB and NaCl

The second combination considered is TBAB superposed with NaCl. Both of these
salts have $\beta<0$, and so both drive up-gradient DP of negatively charged
colloids in one-dimensional gradients (Fig. 4). Movie 5 shows the colloidal
motion observed in orthogonal TBAB and NaCl gradients. The projected average
velocity profiles measured in experiment and generated by modelling are shown
in Fig. 7. Compared to the previous case of KOAc superposed with NaCl, the
differences between the full (middle row) and local (bottom row) model
predictions are much less apparent. The model predictions for
$\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{NaCl}})$ (first column),
$\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{NaCl}})$ (second column), and
$\bar{U}_{\mathrm{NaCl}}(r_{\mathrm{TBAB}})$ (third column) show the same
qualitative behaviour as one another, and either model could be said to be in
agreement with the experimental data (top row).

The only qualitative difference between the full and local models is in the
$\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{TBAB}})$ profiles. The local model
predicts that $\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{TBAB}})<0$ for all times,
representing a TBAB-attraction. The effect of adding the current-induced
contribution to DP is that $\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{TBAB}})$
crosses the $U=0$ line and shows a weakly TBAB-repulsive region at early
times. The magnitude of this predicted down-gradient DP motion is
significantly smaller than the up-gradient, TBAB-attraction predicted closer
to the TBAB source. The experimentally measured
$\bar{U}_{\mathrm{TBAB}}(r_{\mathrm{TBAB}})$ profiles shown in Fig. 7 (d) do
not cross the $U=0$ line and do not show a region of down-gradient, TBAB-
repulsive motion. For this pair of salts, the predicted signature of nonlocal
DP is very weak. It is therefore not surprising that the experiments do not
reveal it.

####  III.2.3 Orthogonal gradients of KCl and KOAc

The final combination of salts considered is KCl ($\beta\approx 0$) crossed
with KOAc ($\beta>0$). Movie 6 shows the colloidal motion observed in
orthogonal KCl and KOAc gradients. The projected average velocity profiles
measured in experiment and generated by modelling are shown in Fig. 8. These
are similar to those shown in Fig. 6 for KOAc superposed with NaCl. This is
unsurprising, given that, like NaCl, a one-dimensional KCl gradient drives up-
gradient DP of negatively charged colloids (Fig. 4).

Figure 8 (i, k) show that, without the current-induced contribution to DP,
only very slow motion towards or away from the KOAc source is predicted. When
the nonlocal term is included, (e) and (g), KOAc-attractive motion is
predicted, and this is precisely what is observed in the experimental data,
(a) and (c). Furthermore, the full model predicts a transient KCl-repulsive
region characterised by $\bar{U}_{\mathrm{KCl}}(r_{\mathrm{KCl}})$ crossing
the $U=0$ line at early times, Fig. 8 (h). This signature is observed in the
experimental data shown in (d), albeit the measured KCl-repulsive motion is
very slow. As was the case for KOAc superposed with NaCl, the experimental
observation of these signatures which are unique to the full model is taken as
confirmation of the existence of the nonlocal, current-induced contribution to
DP predicted by Ref. [35].

##  IV Conclusion

We have directly observed the motion of charged colloidal particles in
orthogonal concentration gradients of a range of salt pairs. We have compared
the results with the predictions of the model described in Ref. [35] and found
that our observations can only be explained with a nonlocal contribution to DP
that is absent in one-dimensional gradients or with only one salt. These are
the first experimental observations of this phenomenon and strongly support
the theory in Ref. [35]. We also experimentally measured DP in one-dimensional
concentration gradients of our four salts and verified its consistency with
the DPA model.

The nonlocal effect was strongest with NaCl and KOAc. This pair of salts have
very different $\beta$ values ($-0.21$ and $+0.28$), and KOAc has a one-
dimensional DP mobility $D_{\text{DP}}$ that is an order of magnitude smaller
that of NaCl. Two very different $\beta$ values are essential to generate
nonlocal DP; the effect disappears if the two $\beta$ values are equal.

The nonlocal effect is most easily identified at early times, and far from the
sources. See Supplementary Movie 2 where the motion of the particles due to
the nonlocal term is easiest to see after around 10% (i. e. two minutes) of
the twenty-minute experimental run shown in the movie, and in the bottom-left
corner, farthest away from the two sources of salt. This is also shown in Fig.
5. There are two effects at play here that aided us in clearly identifying the
nonlocal contribution to DP: first the time scale for the propagation of local
DP is much larger than that for nonlocal DP, and second the small value of
$D_{\text{DP}}$ of one salt (KOAc).

Starting with the first, the local electrophoresis and chemiphoresis terms are
both proportional to _local_ gradients in the logarithm of salt concentration.
For our millimetre field of view it takes of order
$($1\text{\,}\mathrm{mm}$)^{2}/D_{\mathrm{a}}\sim$20\text{\,}\mathrm{min}$$
for salt, and hence local DP, to move across the system, and this can be seen
as a front moving from right to left in Supplementary Movie 2, which follows
the system for 20 minutes. By contrast, the time scales for the appearance of
a current and its associated electric field are all much less than a second
(Fig. 3). This is much too fast for us to observe so we detect the nonlocal
effect as instantaneous action at a distance on the colloids. For a further
discussion, see Appendix F.

For the second, unambiguously attributing particle motion to the nonlocal
effect is easiest when one of the salts (KOAc) has $\beta$ such that the two
terms in conventional salt diffusiophoresis are oppositely directed and almost
cancel. For example, KOAc provides the vertical gradient in Fig. 5 and while
there is strong motion to the right at the right-hand edge (towards the NaCl
source) there is no vertical motion along the top edge, towards or away from
the KOAc source. This cancellation of chemiphoresis and local electrophoresis
allows the nonlocal term to dominate, making its effect easier to identify.

However, this cancellation does not need to be perfect. Additional modelling
(see Supplementary Information [55]) of the salt pair sodium tetraphenylborate
(NaTPB, $\beta=0.4$ [56]) and NaCl also predicts a clearly measurable nonlocal
effect. NaTPB has larger $\beta$ than KOAc, meaning that cancellation of the
two local terms is much less good than for KOAc. We cannot verify this
prediction in experiment as NaTPB inhibits the ability of the PEG-DA precursor
solution to form a hydrogel.

Another advantage of nonlocal DP is due to the fact that it can occur in
locations where there is no salt gradient. The local driving force for
convection is the mass density gradient which will be absent at points where
there is no concentration gradient. So nonlocal DP can occur in locations in a
solution where the local driving force for convection is zero. In our system
we suppressed convection — to make it easier to see DP — by making the
vertical dimension small. However, it may be possible to see nonlocal DP in
systems where salt gradients do drive significant convection but this
convection is far from where there is nonlocal DP.

There are several differences between the experiments and the model. The model
geometry and boundary conditions do not perfectly recreate those of the
experiments. The electrolyte sources in the model span the entire top and
right edges of the square domain, while in experiment, the sources have a
finite width. Furthermore, the model boundaries are maintained at constant
chemical potentials, meaning that the models are approaching a steady state
concentration gradient. By contrast, the experiments are evolving towards an
equilibrium characterised by uniform concentration. Nevertheless the
concordance between experiments, and predictions combining the electroneutral
Nernst-Planck equations and the DPA theory of DP, is naturally very pleasing.

Additional phenomena may contribute to the motion of the colloids in
experiments which are excluded by design in modelling, where colloidal motion
is determined entirely by diffusiophoresis. Both natural convection due to the
dependence of mass density on solution concentration and diffusioosmosis at
the top and bottom walls of the device can generate fluid flows that move
particles. In one-dimensional concentration gradients in sealed devices, both
of these generate circulating flows of known form [54, 53]. Such flows would
add a height-dependent advective component to colloidal motion, in addition to
the height-independent diffusiophoresis. Advection of particles in these
circulating flows due to diffusioosmosis or natural convection is expected to
drive particles in different (opposite) directions at different heights, which
we do not see significant evidence of. Hence it is likely that diffusioosmosis
is weaker than the motion due to diffusiophoresis. But the flows should be
present, and, we assume, account for some of the quantitative discrepancy
between modeling and experiment. Diffusiophoresis and diffusioosmosis have the
same source so it is typically not possible to eliminate one entirely while
keeping the other. We estimate the maximum convection flow speeds in Section
VII of the Supplementary Information and find that they should be less than
$0.1\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{\,}{\mathrm{s}}^{-1}$ which is
at the limit of what we can detect in experiment.

Because of its effectively instantaneous propagation speed, nonlocal DP opens
new applications where local DP would take too long to start up. The DP
velocity also does not have to be towards, or away from, salt sources. This in
contrast to a concentration gradient of a single salt, where DP is necessarily
directed either up or down the local gradient [2, 22, 23, 40, 34]. We hope
that this work inspires future research into using DP to move particles with
faster start ups, and in more complex, more controlled ways.

## Data Availability

Experimental and modelling data supporting this article are freely available
at Figshare [57] https://doi.org/10.6084/m9.figshare.23579262.v1.

###### Acknowledgements.

The authors thank Anirudha Banerjee for advice regarding soluto-inertial
beacons and Bertie Woodward-Rowe for advice regarding hydrogel formation and
measurements of the UV lamp spectrum and power density. Funding for IW was
provided by the EPSRC through a New Horizons grant (Grant No. EP/V048473/1).

##  Appendix A Materials

Sodium chloride (NaCl, 99.9% pure), potassium acetate (KOAc, 99.0% pure) and
tetrabutylammonium bromide (TBAB, 98.0% pure) were purchased from Sigma
Aldrich and used as received. Potassium chloride (KCl, 99% pure) was purchased
from Fisher Scientific and used as received. Fluorescently labeled polystyrene
particles of diameter $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and
$\zeta$-potential $\approx-50\,\mathrm{mV}$ [58] (Invitrogen FluoSpheres™,
carboxylate-modified, yellow-green fluorescent) were purchased from
ThermoFisher as a 2% solids suspension and diluted to 1 drop
($\approx$30\text{\,}\mathrm{\SIUnitSymbolMicro L}$$) per 5 mL. Poly(ethylene
glycol) diacrylate (PEG-DA), number-average molecular weight
$M_{n}=$700\text{\,}\mathrm{g}\text{\,}{\mathrm{mol}}^{-1}$$ was obtained from
Merck. Photoinitiator 2-hydroxy-2-methylpropiophenone was obtained from Sigma
Aldrich.

##  Appendix B Microfluidic Device Fabrication

Single-use microfluidic devices are made by sandwiching double-sided adhesive
tape (VK3220, Viking Industrial) between a microscope slide and a coverslip
[59, 60] as shown in Fig. 1 (a).  Good adhesion between the tape and the glass
is ensured by clamping the device with a pair of bulldog clips and placing it
in an oven at $80^{\circ}\,\mathrm{C}$ for 20 minutes. Channel depth is set by
the tape thickness and is approximately $50\text{\,}\mathrm{\SIUnitSymbolMicro
m}$, which is sufficiently shallow to suppress circulating convective flows
due to the mass density gradient associated with a concentration gradient
[54].  This is justified in more detail in Section VII of the Supplementary
Information. Channels are either T-shaped, Fig. 1 (b, c), for studies of one-
dimensional DP with a single salt source, or have a branched shape, Fig. 1 (d,
e), for studies of two-dimensional DP in crossed gradients with sources of two
salts.

Ionic gradients are created by adapting the soluto-inertial beacons of
Banerjee et al. [40, 42]. This approach uses fixed hydrogel structures
containing a high concentration of solute. When a low concentration solution
is brought into contact with the hydrogel, solute effluxes at a rate set by
the diffusive and associative properties of the hydrogel and solute [41].

A hydrogel precursor stock solution is created by mixing deionised (DI) water
with 40% by volume PEG-DA and 4% by volume photoinitiator. This stock is mixed
at a 1:1 volume ratio with a $260\,\mathrm{mM}$ salt solution to create a
precursor solution containing 20% by volume PEG-DA, 2% by volume
photoinitiator, and $130\,\mathrm{mM}$ salt. The source channel(s) of the
microfluidic devices are filled with this precursor  via capillary action,
carefully adding one drop at a time to avoid overfilling the source channel
and leaking out past the channel intersection. The device is then illuminated
with UV light for 60 seconds to crosslink the PEG-DA and form a hydrogel. The
UV source spectrum has power density of approximately
$5\,\mathrm{mW}\,\mathrm{cm}^{-2}$ at $365\,\mathrm{nm}$.

##  Appendix C Microscopy and Image Analysis

The suspension of fluorescent polystyrene particles is loaded by capillary
action into the device through one end of the sample channel (one-dimensional
devices) or through the inlet channel between the two sources (two-dimensional
devices).  As the suspension is drawn into the device from the inlet, air is
forced out via the outlet(s). It is important to expel all the air and not
trap any bubbles in the device. The open  external ends of  inlet, outlet, and
source channels are sealed using Araldite Rapid 5 Minute epoxy adhesive  at
the locations indicated by the green, dashed ovals in Fig. 1 (b) and (d). The
device is therefore isolated during data acquisition rather than connected to
an external reservoir of the particle suspension.

The sample is placed on an upright microscope (Olympus BX3M) operating in
epifluorescence mode with a $10\times$ magnification objective. In one-
dimensional experiments, the hydrogel source is included in the microscope
field of view, while in two-dimensional experiments, the hydrogel sources are
located outside of the field of view, as indicated by the green region in Fig.
1 (e). The field of view spans a region of approximately
$1700\text{\,}\mathrm{\SIUnitSymbolMicro m}$ by
$1300\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Images of resolution 1224 by 960
pixels are acquired at 1 frame per second for 20 minutes.  The time between
loading the device and beginning acquisition is approximately 60 seconds.
Within this time, residual flows resulting from the loading procedure have
subsided to speeds of $\lesssim 0.1\,\mu\mathrm{m}\mathrm{s}^{-1}$, as
demonstrated by an experiment performed with two hydrogels formed without any
salts, described in the Supplementary Information.

During an experiment, a subpopulation of the particles adhere to the lower
glass surface and appear stationary. Including stationary particles in
subsequent data analysis would lead to an underestimate of DP speed and they
are therefore removed by subtracting the time-averaged image over the whole
acquisition period from each frame of the video. However, in two-dimensional
experiments, the stationary particles are useful for aligning composite images
used to locate the microscope field of view relative to the hydrogel sources,
as described in the Supplementary Information [55].

The particle velocity field is quantified using particle imaging velocimetry
(PIV) [61, 62] implemented in ImageJ [63]. This approach estimates a velocity
field through two-dimensional correlations in image intensity in two images
separated by a time interval $\Delta t$. An image at time $t_{0}$ is divided
into interrogation regions, and each interrogation region is compared to the
image at later time $t_{0}+\Delta t$. The displacement of the interrogation
region which maximises the correlation between the two images gives an
estimate of the local velocity. Repeating this for each interrogation region
and each pair of images gives a velocity field that evolves in time. Here,
overlapping square interrogation regions of side length 64 pixels
($\approx$90\text{\,}\mathrm{\SIUnitSymbolMicro m}$$) centred on a square grid
of locations separated by 32 pixels
($\approx$45\text{\,}\mathrm{\SIUnitSymbolMicro m}$$), and image pairs
separated by $\Delta t=5\,\mathrm{s}$ are employed.

In the absence of DP, particles exhibit Brownian motion, and under this
scenario it is incorrect to assume that the PIV velocity in one interrogation
region is correlated to the velocity in an adjacent region. Consequently, to
ensure the procedure is agnostic to the existence of DP, no normalised median
or dynamic mean test is performed in post-processing of the PIV velocity
fields. Instead, noise is suppressed by applying a moving average over a time
interval of 10 seconds.

##  Appendix D One-dimensional Diffusion Model

Figure 9:  One-dimensional diffusion model. The experimental geometry shown in
the micrograph in (a) is reduced to a 1-dimensional diffusion problem in the
co-ordinate $r$ representing the normal distance from the edge of the
hydrogel. The micrograph is rotated to align $r$ horizontally. (b) One
dimensional concentration profiles according to Eq. 11 computed for NaCl. The
shaded region represents the inside of the hydrogel source. The edge of the
source is located at $r=0$. The solid black curve represents the initial
condition and dashed curves illustrate the time evolution of concentration,
according to the legend.

The modeling of our 1-dimensional DP experiments is illustrated in Fig. 9. The
diffusion equation in one dimension for salt concentration is

$\frac{\partial c(r,t)}{\partial
t}=D_{\mathrm{a}}\,\frac{\partial^{2}c(r,t)}{\partial r^{2}}\,.$ (9)

The salt is treated as a single species which diffuses at a rate given by the
ambipolar diffusion coefficient $D_{\mathrm{a}}=2D_{+}D_{-}/(D_{+}+D_{-})$,
which accounts for the coupling between anion and cation diffusion [28]. The
initial condition is modeled as a step function,  shown as the solid black
line in Fig. 9 (b),

$c(r,0)=\begin{cases}c_{s}\,,&(r<0)\\\ (c_{s}+c_{0})/2\,,&(r=0)\\\
c_{0}\,,&(r>0)\end{cases}$ (10)

with $c_{s}$ the salt concentration in the hydrogel source, $c_{0}$ the
initial concentration outside of the source, and $c_{s}\gg c_{0}$. The
boundary conditions are $c(-\infty,t)=c_{s}$ and $c(\infty,t)=c_{0}$.  This
models an idealised, strictly 1-dimensional version of experiments such as are
shown in Fig. 1 (c) and Fig. 9 (a), in which salt concentration depends only
on time and the normal distance from the edge of the hydrogel, $r$.

The diffusion equation is then solved by

$c(r,t)=\frac{c_{0}+c_{s}}{2}+\frac{c_{0}-c_{s}}{2}\,\mathrm{erf}\left(\frac{r}{\sqrt{4D_{\mathrm{a}}t}}\right),$
(11)

with spatial derivative

$\frac{\partial c(r,t)}{\partial r}=\frac{c_{0}-c_{s}}{\sqrt{4\pi
D_{\mathrm{a}}t}}\,\exp\left(-\frac{r^{2}}{4D_{\mathrm{a}}t}\right).$ (12)

The concentration immediately outside the source–sample interface, which is
the effective value of $(c_{s}+c_{0})/2$, depends on the association constant
characterising the affinity between the ions and the hydrogel, and the ion
diffusion coefficients within the hydrogel [41]. These quantities are unknown.
Diffusiophoresis depends on the gradient of the logarithm of the concentration
so is determined by ratios. For present purposes we assume the ratio between
the source and background concentrations is 100:1. Therefore, we can set the
concentration far from source at $c_{0}=1\,\mathrm{mM}$ to fix the units and
model one-dimensional gradients of the form in Eq. (11) using an estimated
source concentration $c_{s}=100\,\mathrm{mM}$.  The time evolution of Eq. 11
is shown for NaCl in Fig. 9 (b). To translate concentration gradients to
velocities the values of $D_{\text{DP}}$ in Table 2 are used. Thus, by
choosing $c_{s}$ and $c_{0}$, $U(r,t)=D_{\text{DP}}\nabla\ln c(r,t)$ is
estimated for each salt.

##  Appendix E Computational Methods

Figure 10:  Schematic showing the initial and the boundary conditions of the
two-dimensional model. Ion concentrations are defined on a $50\times 50$ grid
represented by the grey dots. Blue (orange) contours show the initial
concentration gradient of salt 1 (salt 2). Constant chemical potential
boundary conditions fix the product of salt 1 ion concentrations at the top
and bottom edges and the product of salt 2 ion concentrations at the left and
right edges. The boundary condition on the current at all edges is
$\mathbf{n}\cdot\mathbf{I}=0$ where $\mathbf{n}$ is the unit normal to the
edge. Cartesian co-ordinates $(r_{1},r_{2})$ are defined from the top and
right edges.

Two-dimensional DP is modelled computationally using methods developed for
Ref. [35]  and code available at Ref. [64].  The simulated geometry, and
initial and boundary conditions are illustrated in Fig. 10. A square domain of
side length $L=$3000\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ is considered.
The model is initiated with crossed error function concentration distributions
of two salts in the $x$ and $y$ directions. The error function in the
$x$-direction ($y$-direction) is centred
$300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the right (top) edge of the
domain and is $300\text{\,}\mathrm{\SIUnitSymbolMicro m}$ wide.  These are
represented by the blue and orange contours in Fig. 10. The initial
concentration at the right (top) edge is 100 times that at the left (bottom)
edge, and these edge concentrations are maintained by fixed chemical potential
boundary conditions.  As described in Supplementary Material to Ref. [35],
fixing the chemical potential at the boundary amounts to fixing the product of
the anion and cation concentrations for a neutral ion pair. The boundary
condition on the current is that the current normal to each edge vanishes.

The model is iterated through time in $90\,\mathrm{ms}$ steps by numerically
solving the Nernst-Planck equations on a $50\times 50$ grid subject to the
condition of solenoidal current, $\nabla\cdot{\mathbf{I}}=0$. The model is
iterated through $4000$ steps, corresponding to $6$ minutes.

To facilitate comparison with experiment and to account for the finite width
of the initial concentration conditions, the first output of the model is
considered to represent $t=15\,\mathrm{s}$. This offset time is the time
required for a typical ion with $D=1.5\times
10^{-9}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$ to establish a $300\,\mu\mathrm{m}$
wide error function concentration profile from a step function at $t=0$ within
the 1d problem described in Appendix D. Initial conditions of finite width are
chosen to ensure that the initial concentration gradients are well-represented
over multiple grid points. The impact of the choice of initial condition width
is discussed in the Supplementary Information.

Using the two-dimensional concentration fields of each ion, the three
contributions to the DP velocity of a particle with $\zeta=-50\,\mathrm{mV}$
are computed according to Eq. (5). Depending on the choice of crossed
electrolytes, there are either 3 or 4 unique ion species. A complete
description of the computational methods is provided in Ref. [35]; see also
Ref. [64]

##  Appendix F Time scales

A number of time scales, spanning a large dynamic range, are relevant to DP.
These are outlined here, in approximate order from fastest to slowest. In the
calculations we assume an ambipolar diffusion coefficient for ions
$D_{\mathrm{a}}\sim 10^{-9}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$ (Table 1), the
system size $L\approx 10^{-3}\,\mathrm{m}$ and solvent viscosity $\nu\approx
10^{-6}\,\mathrm{m}^{2}\,\mathrm{s}^{-1}$. We use $c$ to denote a generic salt
(ion) concentration and take the Debye length $\lambda_{\text{D}}\sim
10^{-9}\,\mathrm{m}$ to correspond to $c\approx 100\,\mathrm{mM}$.

##### Time scale for electromagnetic waves to propagate across the system

Light, travelling at a speed of order $10^{8}\,\mathrm{m}\,\mathrm{s}^{-1}$,
takes of order $10\text{\,}\mathrm{ps}$ to cross the system. This is extremely
short meaning that one can assume electrostatics holds in these problems on
longer time scales.

##### Time scales for ions and colloids to respond to an electric field

When the local electric field changes, it takes time for the ions to respond
and reach a new steady velocity. For an ion of radius
$r_{\mathrm{ion}}<1\,\mathrm{nm}$ this time scale is order
$r_{\mathrm{ion}}^{2}/\nu\lesssim$1\text{\,}\mathrm{ps}$$. This is also very
fast, and on longer time scales means that the steady-state assumption in the
Nernst-Planck equations can be considered to be valid. On the other hand the
corresponding time scale for micron-sized colloids (the momentum relaxation
time) is of the order microseconds. This is the time it takes for the colloid
to reach its steady-state drift velocity in phoresis. Since this is also fast,
it validates the use of the DPA theory on the salt diffusion time scale
(below).

##### Time scale for formation of electric fields

We next turn to the question of the time scales for the formation of electric
fields in the system, due to space charges or EDL charging. The physics here
is discussed in detail by Bazant et al. [52]; see also Refs. [35] and [65]. We
outline the essential scaling arguments.

Our problem is essentially an electrostatic one (see a. above), so the
electric field obeys $\nabla\cdot{\mathbf{E}}=-\rho_{Q}/\epsilon$, where
$\rho_{Q}$ the space charge density. The magnitude of ${\mathbf{E}}$ is at
most of order the characteristic thermal potential difference
$k_{\mathrm{B}}T/e$, divided by the system size $L$. So the left-hand side of
this equation scales as $k_{\mathrm{B}}T/(eL^{2})$. The space charge density
associated with this varying electric field is then $\rho_{Q}\sim\epsilon
k_{\mathrm{B}}T/(eL^{2})\sim\lambda_{\text{D}}^{2}ce/L^{2}$, since the Debye
length $\lambda_{\text{D}}^{2}\sim\epsilon k_{\mathrm{B}}T/(ce^{2})$. In units
of the elementary charge, this is a tiny fraction
($\lambda_{\text{D}}^{2}/L^{2}\ll 1$) of the salt ion concentration. Thus
establishing the space charge density requires only a few ions to move a small
distance and so is many orders of magnitude faster than the salt diffusion
time itself. Assuming sample volume $hL^{2}$, where $h$ is the chamber height
and $L$ the lateral size, the time scale can be estimated from the ratio of
the charge $Q\sim\rho_{Q}hL^{2}\sim h\lambda_{\text{D}}^{2}ce$ needed to
establish the electric field, to the current $I=V/R$ (not to be confused with
the current _density_ in the main text), as $\tau\sim Q/I$ (if we write
$Q=CV$, this is equivalent to an $RC$-circuit charging time [52]). The voltage
$V\sim k_{\mathrm{B}}T/e$ as before, and the resistance $R\sim
L\varrho/(Lh)\sim(h\sigma)^{-1}$ where the conductance $\sigma\sim\mu ce^{2}$
in terms of the ionic mobility $\mu=D_{\mathrm{a}}/k_{\mathrm{B}}T$. Hence
$R\sim k_{\mathrm{B}}T/(hD_{\mathrm{a}}ce^{2})$ and $I=V/R\sim
hD_{\mathrm{a}}ce$. Finally, $Q/I\sim
h\lambda_{\text{D}}^{2}ce/(hD_{\mathrm{a}}ce)\sim\lambda_{\text{D}}^{2}/D_{\mathrm{a}}\sim
10^{-9}\,\mathrm{s}$. This is the origin of the  space-charge formation time
scale in Fig. 3.

The other relevant time scale is the time to charge the electric double layers
(EDLs). This is a factor of $L/\lambda_{\text{D}}$ larger than the space
charge formation time [52]. The reason is that the amount of charge that must
be transferred is $L/\lambda_{\text{D}}$ larger. To see this, consider that
the total charge in the EDLs will be of order the charge per unit area
$\lambda_{\text{D}}ce$ (the charge density in the EDL times the EDL
thickness), multiplied by the surface area $hL$ (i. e. the perimeter of the
sample chamber). This gives the estimate $Q\sim hL\lambda_{\text{D}}ce$, which
is indeed a factor $L/\lambda_{\text{D}}$ larger than above. Then, the same
line of argument can be followed as for the space charge: the potential,
resistance, and current estimates remain the same, making the EDL charging
time of the order
$L/\lambda_{\text{D}}\times\lambda_{\text{D}}^{2}/D_{\mathrm{a}}\sim
L\lambda_{\text{D}}/D_{\mathrm{a}}\sim 10^{-3}\,\mathrm{s}$. This is the next
time scale shown in Fig. 3.

These time scales are much faster than the salt diffusion time (below). Thus
we can consider that the space charge giving rise to the electric fields, and
the EDLs which control the boundary conditions, are fully formed on  salt
diffusion time scale.

##### Time scale for viscous flow relaxation

The suspension of colloidal tracers is loaded into the experimental
microfluidic device as described in Appendix C. The relaxation time for
residual flows is set by the time scale for momentum to diffuse across the
shortest distance to a wall; walls function as momentum sinks. The shortest
distance is the height $h\simeq$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, so
the timescale is $h^{2}/\nu$, which is of order milliseconds. This is
comparable to the EDL charging time, see above and as shown in Fig. 3. Again
this timescale is fast compared to the experimental time window. Therefore,
one can consider that any residual flows from loading the device will have
decayed before data acquisition starts.  Flowing the colloidal suspension into
the microfluidic device to fill it takes a few seconds.

##### Time scale for salt diffusion and local DP

We now consider the longest remaining time scale in the problem, which is the
time it takes for the salt gradients themselves to relax. This time scale is
order $L^{2}/D_{\mathrm{a}}\sim 10^{3}\mathrm{s}\approx 16\,\mathrm{minutes}$,
as shown in Fig. 3. In a nutshell, salt takes tens of minutes to diffuse
across the field of view of our microscope.

An important consideration though is that this means that local DP also
propagates at the same rate. To see this, note that both local DP terms scale
as $\nabla\ln c$, see Eq. (1). Using Eqs. (11) and (12) for the concentration
profile of a diffusing salt we have that

$-\frac{\partial\ln c(r,t)}{\partial r}=\frac{1}{\sqrt{\pi
D_{\mathrm{a}}t}}\times\frac{\exp(-{r^{2}}/{4D_{\mathrm{a}}t})}{\gamma-\mathrm{erf}({r}/{\sqrt{4D_{\mathrm{a}}t}})}\,,$
(13)

where $\gamma=(c_{s}+c_{0})/(c_{s}-c_{0})$. A crucial point is that $\gamma>1$
as long as some background ions are present ($c_{0}>0$), and the denominator
tends to $\gamma-1>0$ for $r\to\infty$. In water, a background ion
concentration will always be present due to the dissociation of water
molecules to form hydrogen and hydroxyl ions. Even pure water at pH 7 has an
ionic concentration of $10^{-7}$moles/litre.

Under these conditions, for distances $r\gg\sqrt{\pi D_{\mathrm{a}}t}$ from
the source, Eq. (13) simplifies to

$-\frac{\partial\ln c(r,t)}{\partial
r}\approx\frac{\exp(-{r^{2}}/{4D_{\mathrm{a}}t})}{(\gamma-1)\sqrt{\pi
D_{\mathrm{a}}t}}\,.$ (14)

The Gaussian in here means that when $r\gg\sqrt{\pi D_{\mathrm{a}}t}$ the
derivative of $\ln c$ is essentially zero. Thus both local DP terms are
negligible whenever $r\gg\sqrt{\pi D_{\mathrm{a}}t}$ and they only become
appreciable when $r\sim\sqrt{\pi D_{\mathrm{a}}t}$.

The implication is that when a source of salt is introduced into a system,
local DP propagates away from the source at a speed set by salt diffusion, and
so reaches a distance $r$ after a time $r^{2}/D_{\mathrm{a}}$. This is true
even for the local electric field term, which depends on the local value of
$\nabla\ln c$ which propagates via diffusion. This contrasts completely with
the nonlocal electric field which comes into existence when the space charge
forms, on a much faster time scale. The significance is that at least at early
times when $\sqrt{D_{\mathrm{a}}t}\ll L$, local DP will be confined to regions
around the ion sources, and outside these regions DP will be due to the
nonlocal action-at-a-distance effect. The caveat is that this effect requires
crossed salt gradients somewhere in the system, so that unless the ion sources
are close together, there is only a time window of order
$L^{2}/D_{\mathrm{a}}$ where nonlocal DP can be cleanly observed (i. e. the
ions have to diffuse far enough for the gradients to meet, but not so far that
they swamp the whole system). Fortuitously, this is exactly the time window
probed by our experiments.

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