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

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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].
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 onedimensional 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 threedimensional 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].
Here we adapt the soluto-inertial beacons approach of Banerjee et al. [34,[40][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 contribu- tion predicted by Ref. [35] and explore strategies to maximise its effect.
The Derjaguin-Prieve-Anderson (DPA) theory [5,43] of electrolyte DP predicts that a suspended particle placed into a gradient in salt concentration, , will move with velocity U DP =  DP ∇ ln  . (1) Assuming an ideal solution, a 1:1 electrolyte, zeta potential , and an infinitesimal double layer [5,43], the DPA mobility is Here  and  are the permittivity and viscosity of the solvent,  B is Boltzmann's constant,  is absolute temperature and  is the elementary charge.The parameter encodes the difference in diffusion coefficients between the anion,  − , and the cation,  + .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  [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  to be sufficient [10,22,24,30,40,45,46].
According to the above,  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 , 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  .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 I shows  − ,  + , and  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  they collectively span.The contributions to  DP according to Eq. ( 2) are given in Table II, for  = −50 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  and .Fig. 2 shows  DP as a function of  for different values of particle  potential.When  DP > 0 (< 0), the net DP velocity is directed up (down) the concentration gradient.
For a given , there is a value  * for which  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  The sign of  * depends on the sign of the charge on the particle, and the magnitude increases with its  potential.The value for  * as a function of  obtained from Eq. ( 2) is plotted in the inset to Fig. 2. For example, KOAc has  = 0.28 and is approximately DP neutral when particles have  = −50 mV.But, if particles have the opposite charge,  = 50 mV, NaCl with  = −0.21 is a better choice for DP neutrality.

B. 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 where the index  labels each ion species.Here  =        is a weighted sum of ion densities (  = ±1 is the valence) and  =   2      is the conductivity.The second and third terms in the above are obtained by considering the ion current I, which in terms of the dimensionless electrostatic potential  and corresponding electric field E = −∇, is given by This result is derived by summing the set of Nernst-Planck equations (one per species) which govern how the concentrations   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 , and the current, 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 (∇  ).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  the same way as the second term.

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], ∇ • I = 0, which implies ∇•(∇)+∇ 2  = 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 (I, ) are uniquely specified up to an additive constant in  by the ion current through the boundaries (typically, in a closed system, I • n = 0 where n is the surface normal).
Taking the 'curl' of Eq. ( 7) and using the fact that where  ≡  −1 is the resistivity.This implies that if the gradients in  and resistivity  (equivalently conductivity ) are everywhere parallel (or antiparallel), then a solution exists with I = 0 everywhere, and by virtue of the uniqueness theorem, this is the only solution.
Thus I ≠ 0 requires 'crossed' gradients ∇ × ∇ ≠ 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  and , the current is irrotational [35]; the current satisfies ∇ × I = 0.So in the region without crossed gradients, the conditions are ∇ × I = 0 and ∇ • I = 0.They can be met if I is the gradient of some 'current potential', I = ∇, which is harmonic (∇ 2  = 0) in the region in question.
That a non-vanishing, irrotational, solenoidal current is perfectly possible can also be seen by an unexpectedly close analogy (Table III) 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 currentcarrying 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 [50].Note that just as magnetostatics demands ∇ • J = 0, the source term in Eq. ( 8) is divergence-free, ∇ • (∇ × ∇) = 0 ; this follows from standard vector calculus identities [49].
In the present problem therefore, provided there are crossed gradients (in the sense that ∇ × ∇ ≠ 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 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 magnetostatic.There is no equivalent to Maxwell's displacement current, the D/ 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,  = ( 1 −  2 ) and  = ( 1 +  2 ), because electroneutrality forces  1 =  2 = .Hence ∇ × ∇ = 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.

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  ≈ 1 mm across, so given the ambipolar salt diffusion coefficient  a ∼ 10 −9 m 2 s −1 , the time scale is  2 / a ≈ 10 3 s ≈ 16 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 B 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  apart, the gradients cross and nonlocal DP starts up after a time  2 / 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 ().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.

T I M E
experimentally accessible time scale space charge formation EDL charging salt diffusion FIG. 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  D ∼ 10 −9 m (100 mM salt), ambipolar diffusion coefficient for ions  a ∼ 10 −9 m 2 s −1 , and system size  ∼ 10 −3 m.Details are given in Appendix F.

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 I and II, 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.

A. Diffusiophoresis in 1d Salt Gradients
One-dimensional experiments use T-shaped microfluidic devices with integrated hydrogel sources containing 130 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, , as a function of the normal distance from a line identified at the edge of the hydrogel, , 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 .Tetrabutylammonium bromide (TBAB) and sodium chloride (NaCl) have  < 0 and are expected to drive DP in the same direction, with TBAB driving faster DP due to its larger absolute value of .Potassium chloride (KCl) has  ≈ 0 and therefore its DP is anticipated to be dominated by chemiphoresis.Potassium acetate (KOAc) has  > 0, meaning that its local electrophoretic component of DP is predicted to be oppositely directed to that due to TBAB or NaCl.To predict  () 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 .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  is of the order  2 / a , which for a distance of  ≈ 1000 µm 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 ( = 90 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  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 upgradient.Differences between salts are due to differences in their local electrophoretic terms, which depend on .
For TBAB and NaCl, the local electrophoretic contribution is positive, and acts in the same, up-gradient direction as chemiphoresis. 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  = 0.
KCl has  ≈ 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  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.
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  = −50 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.

B. Diffusiophoresis in 2d Salt Gradients
The one-dimensional gradients of Section III A 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 ( > 0) superposed with NaCl ( < 0); TBAB ( < 0) superposed with NaCl ( < 0); and KCl ( ≈ 0) superposed with KOAc ( > 0).A fourth combination, KCl ( ≈ 0) superposed with NaCl ( < 0), behaves very similarly to TBAB superposed with NaCl, and is therefore omitted from the article, but included in the Supplementary Information [55].

Orthogonal Gradients of KOAc and NaCl
Figure 5 (a) shows a snapshot of the experimental velocity field measured at  = 115 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 µm from the top edge of the field of view and the NaCl source is located approximately 865 µm from the right edge of the field of view.A Cartesian co-ordinate system, ( NaCl ,  KOAc ), is defined as illustrated in Fig. 1 (e) and the top right of Fig. 5.The co-ordinate  NaCl is the normal distance from a line drawn at the edge of the NaCl source and, similarly,  KOAc is the normal distance from a line drawn at the edge of the KOAc source.Unit vectors in the  NaCl and  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 µm 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.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 ∼ 4 µm s −1 whereas the model predic-tions are ∼ 1 µm 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.
Comparison of two-dimensional vector fields is challenging, measured ŪTBAB ( TBAB ) profiles shown in Fig. 7 (d) do not cross the  = 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.

Orthogonal gradients of KCl and KOAc
The final combination of salts considered is KCl ( ≈ 0) crossed with KOAc ( > 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 ŪKCl ( KCl ) crossing the  = 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  values (−0.21 and +0.28), and KOAc has a one-dimensional DP mobility  DP that is an order of magnitude smaller that of NaCl.Two very different  values are essential to generate nonlocal DP; the effect disappears if the two  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  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 mm) 2 / a ∼ 20 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  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,  = 0.4 [56]) and NaCl also predicts a clearly measurable nonlocal effect.NaTPB has larger  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 [53,54].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 µm 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 [2,22,23,34,40].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.
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× 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 µm by 1300 µm.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 ≲ 0.1 ms −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 Δ.An image at time  0 is divided into interrogation regions, and each interrogation region is compared to the image at later time  0 + Δ.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 (≈ 90 µm) centred on a square grid of locations separated by 32 pixels (≈ 45 µm), and image pairs separated by Δ = 5 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
The modeling of our 1-dimensional DP experiments is illus- (D1) The salt is treated as a single species which diffuses at a rate given by the ambipolar diffusion coefficient  a = 2 +  − /( + +  − ), 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), with spatial derivative (, ) The concentration immediately outside the source-sample interface, which is the effective value of (  +  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  0 = 1 mM to fix the units and model one-dimensional gradients of the form in Eq. (D3) using an estimated source concentration   = 100 mM.The time evolution of Eq.D3 is shown for NaCl in Fig. 9   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  = 3000 µm is considered.The model is initiated with crossed error function concentration distributions of two salts in the  and  directions.The error function in the -direction (-direction) is centred 300 µm from the right (top) edge of the domain and is 300 µm 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 ms steps by numerically solving the Nernst-Planck equations on a 50 × 50 grid subject to the condition of solenoidal current, ∇ • 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  = 15 s.This offset time is the time required for a typical ion with  = 1.5×10 −9 m 2 s −1 to establish a 300 m wide error function concentration profile from a step function at  = 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  = −50 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  a ∼ 10 −9 m 2 s −1 (Table I), the system size  ≈ 10 −3 m and solvent viscosity  ≈ 10 −6 m 2 s −1 .We use  to denote a generic salt (ion) concentration and take the Debye length  D ∼ 10 −9 m to correspond to  ≈ 100 mM.
a. Time scale for electromagnetic waves to propagate across the system Light, travelling at a speed of order 10 8 m s −1 , takes of order 10 ps to cross the system.This is extremely short meaning that one can assume electrostatics holds in these problems on longer time scales.
b. 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  ion < 1 nm this time scale is order  2 ion / ≲ 1 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).
c. 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 ∇ • E = −  /, where   the space charge density.The magnitude of E is at most of order the characteristic thermal potential difference  B /, divided by the system size .So the left-hand side of this equation scales as  B /( 2 ).The space charge density associated with this varying electric field is then   ∼   B /( 2 ) ∼  2 D / 2 , since the Debye length  2 D ∼   B /( 2 ).In units of the elementary charge, this is a tiny fraction ( 2 D / 2 ≪ 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 ℎ 2 , where ℎ is the chamber height and  the lateral size, the time scale can be estimated from the ratio of the charge  ∼   ℎ 2 ∼ ℎ 2 D  needed to establish the electric field, to the current  = / (not to be confused with the current density in the main text), as  ∼ / (if we write  = , this is equivalent to an -circuit charging time [52]).The voltage  ∼  B / as before, and the resistance  ∼  /(ℎ) ∼ (ℎ) −1 where the conductance  ∼  2 in terms of the ionic mobility  =  a / B . Hence  ∼  B /(ℎ a  2 ) and  = / ∼ ℎ a .Finally, / ∼ ℎ 2 D /(ℎ a ) ∼  2 D / a ∼ 10 −9 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 / D larger than the space charge formation time [52].The reason is that the amount of charge that must be transferred is / D larger.To see this, consider that the total charge in the EDLs will be of order the charge per unit area  D  (the charge density in the EDL times the EDL thickness), multiplied by the surface area ℎ (i.e. the perimeter of the sample chamber).This gives the estimate  ∼ ℎ D , which is indeed a factor / 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 / D ×  2 D / a ∼  D / a ∼ 10 −3 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.
d. 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 ≃ 50 µm, so the timescale is ℎ 2 /, 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.
e. 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  2 / a ∼ 10 3 s ≈ 16 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 ∇ ln , see Eq. where  = (  +  0 )/(  −  0 ).A crucial point is that  > 1 as long as some background ions are present ( 0 > 0), and the denominator tends to  − 1 > 0 for  → ∞.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  ≫ √  a  from the source, Eq. (F1) simplifies to The Gaussian in here means that when  ≫ √  a  the derivative of ln  is essentially zero.Thus both local DP terms are negligible whenever  ≫ √  a  and they only become appreciable when  ∼ √  a .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  after a time  2 / a .This is true even for the local electric field term, which depends on the local value of ∇ ln  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 √  a  ≪ , 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  2 / 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.

FIG. 1 .
FIG. 1. Microfluidic device schematics and micrographs.(a) Side view schematic showing microchannel construction using doublesided 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× 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  = 0, and the arrow indicates the co-ordinate, , defined as the normal distance from the  = 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 • .Green shaded ovals indicate the open channel ends which are sealed with glue after the experiment is initiated.(e) Brightfield micrograph at 5× 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× magnification.Navy blue dashed lines indicate the perpendicular hydrogel edges which form the axes of a Cartesian co-ordinate system, ( 1 ,  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).

FIG. 2 .
FIG.2.DPA diffusiophoretic mobility,    , as a function of the reduced difference in cation and anion diffusion constants,  for particles with  potentials indicated in the legend.These are calculated according to Eq. (2) for salts in water at 300 K.The asterisks represent  and  DP for the combination of the particles with  = −50 mV and salts employed in our experiments, (from left to right): TBAB, NaCl, KCl and KOAc.The inset shows the value of  necessary for diffusiophoretic neutrality,  * , defined as  DP ( * ) = 0, as a function of particle  potential.

FIG. 4 .
FIG. 4. (a) Average DP velocity towards / away from a hydrogel source containing 130 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  = 90 s after loading.Sources contain 130 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  = 90 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.

FIG. 5 .
FIG. 5. (a) Experimentally measured velocity field in orthogonal concentration gradients of KOAc and NaCl at  ≈ 115 s after initiation.The KOAc source is located ≈ 840 µm from the top of the field of view and the NaCl source is located ≈ 865 µm from the right of the field of view, as indicated in the diagram on the right.The spacing between velocity vectors is 44 µm.(b,c) Simulated velocity fields at  = 115 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 5 (
Figure 5 (a) shows only the snapshot at  = 115 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

FIG. 9 .
FIG. 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  representing the normal distance from the edge of the hydrogel.The micrograph is rotated to align  horizontally.(b) One dimensional concentration profiles according to Eq. D3 computed for NaCl.The shaded region represents the inside of the hydrogel source.The edge of the source is located at  = 0.The solid black curve represents the initial condition and dashed curves illustrate the time evolution of concentration, according to the legend.
D2)with   the salt concentration in the hydrogel source,  0 the initial concentration outside of the source, and   ≫  0 .The boundary conditions are (−∞, ) =   and (∞, ) =  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, .The diffusion equation is then solved by (b).To translate concentration gradients to velocities the values of  DP in Table II are used.Thus, by choosing   and  0 ,  (, ) =  DP ∇ ln (, ) is estimated for each salt.

FIG. 10 .
FIG. 10.Schematic showing the initial and the boundary conditions of the two-dimensional model.Ion concentrations are defined on a 50 × 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 n • I = 0 where n is the unit normal to the edge.Cartesian co-ordinates ( 1 ,  2 ) are defined from the top and right edges.

TABLE III .
[49]etostatic analogy for the ion current in the Nernst-Planck equations.Note that B = H.The boundary condition corresponds to a perfect conductor.The notation is from Jackson's Classical Electrodynamics, 3rd ed.[49].