Dynamic Double Layer Force Between Charged Surfaces

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PHYSICAL REVIEW RESEARCH 2, 013138 (2020)

Dynamic double layer force between charged surfaces

Bhavya Balu * and Aditya S. Khair Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA

(Received 12 December 2019; accepted 9 January 2020; published 7 February 2020)

We develop a theory for the “dynamic double layer force” between charged surfaces in an electrolyte under a time-dependent voltage. Speciﬁcally, the force between two planar surfaces is calculated within the Poisson- Nernst-Planck framework for dilute electrolytes, accounting for unequal ionic diffusivities. Due to the inherent nonlinear dependence of the force on the electric potential, a sinusoidal voltage oscillating with frequency ω gives rise to a nonzero time-averaged force, along with a component oscillating with frequency 2ω. The time- averaged force is always attractive, as expected for surfaces of opposite polarity. However, the instantaneous force switches between attractive and repulsive over an oscillation cycle at certain frequencies. Next, the force is quantiﬁed for suddenly applied and pulsed voltages. In the former case, the force approaches its long-time limit exponentially: on a resistor-capacitor or bulk diffusion time scale for an electrolyte with equal or unequal ionic diffusivities, respectively. The long-time decay of the force for a pulsed voltage also decays exponentially on the same timescales. The theory developed here is a dynamic generalization of the equilibrium double layer force, with potential applications to electrochemical devices, force-based spectroscopy, and colloidal directed assembly.

DOI: 10.1103/PhysRevResearch.2.013138

I. INTRODUCTION modification to account for dynamic interactions that may arise due to charge regulation [12–14], electrokinetic particle Charged surfaces in contact with electrolytes are abundant motion [15–17], and external charging of electrochemical in electrochemical [1], colloidal [2], and biological [3]sys- cells [7,18–20]. Research on charge regulation models tems. The ions in the electrolyte form a diffuse charge layer focuses on systems in quasiequilibrium; i.e., the ions in the that screens the surface charge. The thickness of this layer electrolyte reach their equilibrium distribution much faster for sufficiently dilute solutions is characterized by the Debye √ than a charge regulating reaction on the surface. Studies on κ−1 = ε / 2 ε length, kBT (2I0e ), where is the permittivity the effect of electrokinetic motion focus on the modified of the electrolyte, kB is the Boltzmann constant, T is the hydrodynamic force due to the relative motion of the ions in temperature, I0 is the ionic strength, and e is the charge on the diffuse layer and the surface. a proton. The net (counterion) charge density results in an Recent surface force apparatus experiments by Perez- electrical body force on a fluid element within the layer, which Martinez and Perkin [7] externally apply an oscillating voltage must be balanced by an osmotic pressure gradient to maintain and simultaneously measure the time-dependent surface force mechanical equilibrium. The presence of such diffuse layers between two electrodes across an electrolyte solution. Tivony implies that the interaction between charged surfaces in an et al. [20] measure the time-dependent surface force due electrolyte is fundamentally different to that in a dielectric to a suddenly applied, or step, voltage in a similar appa- medium. Quantifying this interaction is crucial in predicting ratus. Motivated by these recent studies, here we develop the macroscopic stability of dispersions [4,5], designing elec- a theoretical model for the “dynamic double layer force” trochemically active devices [6–8], and engineering self- and between two surfaces whose potential difference oscillates directed assembly of colloids and macromolecules [9,10]. in time. We also consider the response of the system to a The seminal Derjaguin-Landau-Verwey-Overbeek suddenly applied voltage and a single voltage pulse. Our (DLVO) theory [2] describes the long range electrostatic and focus is on quantifying how the ion transport across the short range van der Waals forces between charged surfaces electrolyte, via electromigration and diffusion, determines the at thermodynamic equilibrium. State of the art experimental time-dependent force on the surface. A recent study [21] techniques [4,11] use it to analyze measurements of the calculated the double layer force between two planar sur- force between charged surfaces. The DLVO theory needs faces in a symmetric, binary electrolyte under an ac voltage with a dc bias. Here, we consider an asymmetric electrolyte with unequal ionic diffusivities and calculate the force for *[email protected] a suddenly applied dc voltage and a voltage pulse in addition to an ac voltage. We do not consider the effect of Published by the American Physical Society under the terms of the surface reactions; thus the time dependence solely arises Creative Commons Attribution 4.0 International license. Further from the dynamic formation of double layers in response to distribution of this work must maintain attribution to the author(s) the external voltage. Our theory contributes a dynamic, out- and the published article’s title, journal citation, and DOI. of-thermodynamic-equilibrium generalization of the classical

2643-1564/2020/2(1)/013138(12) 013138-1 Published by the American Physical Society BHAVYA BALU AND ADITYA S. KHAIR PHYSICAL REVIEW RESEARCH 2, 013138 (2020)

them; hence the ion transport is solely in the direction (x) between the electrodes. We study the response of this system to an applied alternating voltage 2V (t ) = 2V0 cos(ωt ), where V0 is the amplitude, ω is the angular frequency of the signal, and t represents time. From the solution to this problem, we also derive the response of the cell to a suddenly applied voltage and a pulsed voltage. This effectively allows us to determine the dynamic double layer force between such surfaces. We assume that V0 is small compared to the thermal voltage scale kBT/e, such that the dimensionless voltage V˜0 = V0e/kBT 1. Henceforth, the tilde decoration denotes a dimensionless variable. We define the charge density ρ and ionic strength I in terms of the ionic concentrations n± as ρ = e(n+ − n−), and 2I = n+ + n−. The normalized ρ ˜ φ˜ FIG. 1. Schematic of a model electrochemical cell under an charge density ˜ , ionic strength I, and electric potential in the electrolyte are governed by the Poisson-Nernst-Planck oscillating voltage 2V0 cos(ωt ). A time dependent force of magnitude (PNP) equations. For our model system, the dimensionless, FT is exerted on each electrode due to the ion transport under this potential difference. Regions of net ionic charge are confined linearized PNP equations are − to a distance of the order of the Debye length, κ 1, from each 2 ∂ρ˜ DA ∂I˜ ∂ ρ˜ κ−1 κL + = − κ2L2ρ,˜ electrode. Typically, is much smaller than the distance between 2 ∂t˜ DF ∂t˜ ∂x˜ electrodes, L. D ∂ρ˜ ∂I˜ ∂2I˜ κL A + = , (1) ∂ ∂ ∂ 2 electrostatic double layer force on which DLVO theory is DF t˜ t˜ x˜ founded. ∂2φ˜ =−κ2L2ρ.˜ We show that the force per unit area on a surface scales as ∂x˜2 εV 2/L2, where ε is the permittivity of the electrolyte, V is the The first two equations arise from linear combinations of the applied voltage, and L is the distance between the electrodes. cationic and anionic species balances [24,25], and the third is Thus taking V = k T/e ≈ 25 mV at 298 K across L = 1 μm B Poisson’s equation describing the variation in the local electric in an aqueous electrolyte generates a stress of about 1 Pa. This field due to the ionic space charge density. A derivation of (1) stress exerts a force of O(10 pN) on a spherical particle of is presented in Appendix A. Here, D = 2D+D−/(D+ + D−) one micron radius, which is a typical force scale in colloidal A is the ambipolar diffusivity, D = 2D+D−/(D− − D+), and and biological soft matter. Further, the stress is comparable F D+ and D− are the (distinct) diffusivities of the cations to the elastic modulus of a nervous membrane [22]. We and anions, respectively. Note, we do not assume ambipolar suggest, therefore, that the dynamic double layer force is rel- diffusion. The ambipolar diffusion coefficient D naturally evant in soft materials, for example, in biocompatible flexible A arises when expressing the PNP equations (1) in terms of the electronics that aim to mimic the mechanical properties of charge density and ionic strength, rather than the individual biological membranes [23]. Further, our analysis could be ion concentrations n±. The charge density is normalized by used to develop new experimental techniques, such as a force 2I e, and the ionic strength by I , where I is the equilibrium based spectroscopy, in analogy to electrochemical impedance 0 0 0 ionic strength when there is no applied voltage, the potential spectroscopy, to probe physicochemical electrolyte properties. by k T/e, the distance by L, the length of the half cell, Finally, we show that the dynamic double layer force can vary B and the time by the resistor-capacitor (RC) charging time by orders of magnitude with frequency. Hence estimation of τ = κ−1L/D [18,26]. The system is subject to boundary the mechanical force on electrochemically active surfaces via RC A conditions at the blocking electrodes,x ˜ =±1; namely, DLVO theory can lead to wildly erroneous results. The system considered here represents a minimal model ∂ρ˜ ∂φ˜ ∂I˜ + = , = , φ˜ =±˜ , for dynamic forces between charged surfaces exposed to time 0 0 ±1 V (t˜) (2) ∂x˜ ∂x˜ ± ∂x˜ ± varying voltages. Our work serves as a starting point to in- 1 1 vestigate more complex systems that have temporally varying corresponding to zero flux of charge and neutral salt, and an ± ˜ surface charges, where nonideal ion interactions and irregular applied voltage of V (t˜) at opposite electrodes, and hence ˜ particle shapes could play a significant role in determining the a net potential difference of 2V (t˜). We consider three forms ˜ ˜ = ˜ ω double layer force. of V (t˜): an oscillating voltage, V V0 cos ( ˜ t˜), a suddenly applied voltage, V˜ = V˜0H(t˜) where H(t˜) is the Heaviside step function, and a voltage pulse, V˜ = V˜0δ(t˜), where δ(t˜)isthe II. PROBLEM FORMULATION Dirac delta function. A. Ion dynamics We consider a model electrochemical cell with a di- B. Double layer force lute, monovalent, binary electrolyte between planar, parallel, The force on an electrode is calculated as the integral of the blocking electrodes (Fig. 1). It is assumed that the electrode traction on its surface originating from osmotic pressure and dimensions are large compared to the distance (2L) between electrical (Maxwell) stress. For the one-dimensional system

013138-2 DYNAMIC DOUBLE LAYER FORCE BETWEEN CHARGED … PHYSICAL REVIEW RESEARCH 2, 013138 (2020) considered here, the force per unit area, normalized as shown Here, below, is given by 2 2 2 2 κ L λ1 v2 κ L Q = λ1 − − λ2 − 2 λ λ λ κ2L2 1 ∂φ˜ 1 2 v1 2 F ≡ F˜ = [˜p(˜x) − p˜ ] − . (3) T T 0 ∂ κ2L2 λ κ2L2 I0kBT 2 x˜ + λ − 1 v2 λ , 2 tanh 1 2 tanh 2 (6) λ λ2 v1 λ In the right side of (3), the first term represents the difference 1 2 in the local osmotic pressurep ˜ due to an external voltage and a and λ1 and λ2 are the eigenvalues of the system of ODEs , bulk reservoir valuep ˜0 under no voltage. This pressure is due governing charge density and ionic strength (1); [v1 1] and to an increase in ionic strength between the electrodes relative [v2, 1] are the corresponding eigenvectors. These are to the value I0 that defines the (dimensional) reservoir pressure = κ2L2 κ2L2 4˜ω2 D2 p0 I0kBT . The second term in the right side of (3) is due λ2 = + ιωκ˜ L ± 1 − A (7) i κ2 2 2 to electric stresses. Note that although the pressurep ˜ and the 2 2 L DF potential φ˜ vary with locationx ˜, their combination in (3)is and a spatial constant. The scale of the force can be understood as a product of the reservoir osmotic pressure I kBT and the ω2 D2 0 − − 4˜ A 2 2 1 1 κ2 2 2 square of the normalized Debye length 1/(κ L ). Since the L DF v2 = . Debye length is inversely proportional to the square root of (8) v1 ω2 D2 + − 4˜ A the ionic strength I0, the force scale is independent of I0. 1 1 κ2L2 D2 The permittivity of the electrolyte ε enters the force scale F through the Debye length. We assume that the permittivity of Our Eqs. (5)–(8) are in agreement with Barbero and Lelidis the electrolyte is the same as that of the solvent, neglecting [30] who derive equivalent expressions in their study on the influence of the ions, which can alter the permittivity ambipolar diffusion in impedance spectroscopy. To proceed, [27–29]. It would be an interesting future direction to consider we first obtain the pressure distributionp ˜(˜x) by invoking the influence of a frequency or ion-concentration dependent mechanical equilibrium. Then, the force per unit area F˜T on permittivity on the dynamic double layer force. To calculate the electrode atx ˜ = 1 is calculated using (3). As detailed p˜ we assert that the system is in mechanical equilibrium; in Appendix B, this yields a force with a steady, time- thus the pressure gradient is balanced by the divergence of independent component and a component that oscillates with the electric stress. As shown in Appendix B, this leads to an twice the applied frequency, which we partition as ∂ p˜ ∂ ∂φ˜ 2 osmotic pressure gradient, = 1 ( ) . Importantly, we do 2ιω˜ t˜ ∂x˜ 2 ∂x˜ ∂x˜ F˜T = F˜S + Re FÕe . (9) not assume that the ions follow a Boltzmann distribution (as is standard in DLVO theory) since the force we calculate is not Here, the steady, time-averaged force is thermodynamic at equilibrium. ˜ κ2 2 λ κ2 2 2 V0 L 1 v2 L F˜S =− λ1 − − λ2 − , (10) Q λ1 λ2 v1 λ2 III. RESULTS AND DISCUSSION where |z| represents the modulus of the complex number z. A. Force spectroscopy The oscillating component of the force has amplitude For an applied oscillating voltage, 2V˜ (t˜) = 2Re(V˜ eιωt˜), 0 V˜ 2 sech2 λ λ v sech2 λ with frequencyω ˜ normalized by 1/τ , we expect the solu- F˜ =− 0 κ4L4 1 + 1 2 κ4L4 2 RC O 2 λ2 λ λ2 tions of the linearized PNP equations to be of the formg ˜ = Q 2 1 2 v1 2 2 ιω˜ t˜ 2Re(˜g1√e ), whereg ˜1 is a complex valued function ofx ˜ and κ2L2 λ v κ2L2 2 ω ι = − ρ ˜ + λ − − 1 2 λ − . ˜ , 1, Re is for the real part, andg ˜ represents ˜ , I,or 1 λ λ 2 λ (11) φ˜. This is equivalent to performing a Fourier transform of the 1 2 v1 2 PNP equations and yields the long-time oscillatory response Thus there is a nonzero time averaged force over an oscillation of the electrolyte, i.e., after the initial transients have died out. cycle even though the applied voltage averages out to zero. Details of the solution process are provided in Appendix A. This is due to the inherently nonlinear (quadratic) dependence The force on the electrode depends on the electric potential of force on the applied potential. Note that because we con- gradient, sider a purely oscillating field with no dc bias, our expression for force has no component that oscillates with a frequency ∂φ˜ ∂φ˜ 1 ιω˜ ω˜ , which is different from a recent calculation [21], where = 2Re e ˜ t , (4) ∂x˜ ∂x˜ a dc bias is considered. The total force F˜T as a function of time is plotted in Fig. 2(a) for small, intermediate, and large where the potential gradient in Fourier space is frequenciesω ˜ . The dependence of the maximum value of the force during an oscillation cycle on the normalized inverse ∂φ˜ V˜ cosh (λ x˜) λ v cosh (λ x˜) κ ω 1 = 0 −κ2L2 1 − 1 2 2 Debye length L and on the applied frequency ˜ is plotted ∂x˜ Q λ1 cosh (λ1) λ2 v1 λ2 cosh (λ2 ) in Figs. 2(b) and 2(c). The dependence of the time averaged κ ω κ2 2 λ κ2 2 force on L and ˜ is shown in Fig. 3. L 1 v2 L ω → κ + λ1 − − λ2 − . (5) At small frequencies, ˜ 0, with L fixed, the amplitude λ1 λ2 v1 λ2 | ˜ |→ ˜ 2κ2 2 2 κ of oscillations FO V0 L cosech ( L), which precisely

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FIG. 3. Steady, time-averaged component of force (a) as a function of κL for low, intermediate, and high driving frequencies. Force is independent of κL at high frequencies, indicating the dielectric limit, and (b) as a function of frequency for three values of κL. In (a) the open circles represent the quasisteady limit of the time averaged force. In both plots, D+ = D− and V˜0 = 1.

recovers the equilibrium double layer force between surfaces held at constant potentials of equal magnitude and opposite polarity [2]. This is demonstrated in Fig. 2(b) by the open circles (representing magnitude of equilibrium force) over- lapping with the blue curve for sufﬁciently large κL (our dynamic force forω ˜ → 0). Also in this limit, the time-averaged 2 2 2 component F˜S →−ω˜ /[tanh (κL) + ω˜ ]. This is shown in Fig. 3(a) by the open circles (asymptotic expression) over- FIG. 2. (a) Total force on the electrode as a function of time lapping with the blue curve (full expression of steady force). over one oscillation cycle, for small, intermediate, and large voltage Physically, here the frequency is small compared to the inverse frequenciesω ˜ . The dashed gray curve represents the applied voltage. of the double layer charging time. As such, double layers Here, κL = 1. (b) The maximum value of force (including steady and adjacent to each electrode form and relax quasisteadily over oscillating components) over an oscillation cycle as a function of κL an oscillation cycle. Further, for the practically important limit for small, intermediate, and large driving frequencies. The symbols of thin double layers, κL 1, the double layers near either represent the quasisteady limit of equilibrium double layer force, and surface do not overlap, and the surface potential is almost the κL independent force at high frequencies indicates the dielectric completely screened. The amplitude of the oscillating force | ˜ |→ ˜ 2κ2 2 − κ limit. (c) The maximum value of force over an oscillation cycle as a FO V0 L exp ( 2 L) is exponentially small, and the κ function of the driving frequency for three values of L. In all three time-averaged force F˜S → 0, as κL →∞, ω˜ → 0. Thus, in = = plots, D+ D− and V˜0 1. the quasisteady double layer limit, for the practically relevant

013138-4 DYNAMIC DOUBLE LAYER FORCE BETWEEN CHARGED … PHYSICAL REVIEW RESEARCH 2, 013138 (2020) values of κL, the force switches sign within a cycle, indicat- exact expression for the Laplace space potential gradient is ing a switch between an attraction (F˜T < 0) and repulsion given in Appendix B. The inverse Laplace transform is not (F˜T > 0). This is shown in Fig. 2(a) by the blue curve for possible to perform analytically for all times. However, we ω˜ = 0.01. obtain the asymptotic behavior of the force at long times from In the opposite limit of high frequencies,ω ˜ →∞,the the Laplace inverse of the potential gradient in the limit of ˜ →−˜ 2 → steady force FS V0 and the amplitude of oscillations s˜ 0. The long time asymptotic expression for the force is | ˜ |→ ˜ 2 FO V0 , which is the force expected from a dielectric given by (13). To obtain this, we first calculate the pressure capacitor. This is observed in Figs. 2(c) and 3(b) for all distribution from the electric potential by invoking mechanical values of κL asω ˜ →∞. The total force therefore ranges equilibrium. Similar to the frequency response, the force is − ˜ 2 from 2V0 to 0, and is always attractive [red curve for independent of positionx ˜ for the model system considered ω˜ = 100 in Fig. 2(a)]. Physically, the large frequency does here. Thus a numerical inverse Laplace transform (using the not allow for the formation of double layers; thus the force algorithm from De Hoog et al. [33]) of the x-independent part is independent of κL, as is suggested by the red curves for of the potential gradient is used to calculate the dynamics of ω˜ = 100 in Figs. 2(b) and 3(a). Additionally, as κL → 0, the force at all times, and is plotted in Fig. 4. the force approaches the dielectric limit irrespective of the The total (normalized) force on an electrode initially jumps − ˜ 2/ frequency. to the dielectric value, V0 2 [insets in Figs. 4(a) and 4(c)]. For intermediate frequencies and thin Debye layers This is expected as the ions are yet to form double layers. This (κL 1) the bounds of the force are in between the two is analogous to the high frequency limit in the spectroscopy limiting cases, as shown by the curves forω ˜ = 0.5, 1, and 2 analysis. The force decays exponentially at long times, and − 1 ˜ 2κ2 2 2 κ in Figs. 2(b) and 3(a). However, for thicker Debye layers finally reaches the equilibrium value, 4V0 L cosech ( L) [κL ∼ O(1)], the maximum and average forces could be [inset in Fig. 4(a)], corresponding to fully formed double greater than their dielectric limits. The instantaneous force layers, which is simply the quasisteady limit of our dynamic switches signs between attractive and repulsive for a small force spectroscopy. Note that, while all the curves approach range of frequencies, following which it is always attractive the equilibrium value, it is only large enough to distinguish [Fig. 2(a)]. For some frequencies and κL, the amplitude of numerically for the smallest κL. For this κL = 10, the total oscillations of the force vanishes; thus the maximum force is force drops by six orders of magnitude from the initial appli- just the time-averaged value. This approach to zero appears cation of voltage. When unequal diffusivities are considered, nonsmooth in Figs. 2(b) and 2(c) as we plot the absolute we have a dynamic force composed of two distinct decays. value of the force. Finally, for intermediate frequencies, the Thus, after the initial jump to the dielectric value, we see a total force undergoes a phase shift [black curve in Fig. 2(a)], second plateau in the force before it approaches its equilib- relative to either extreme limit of frequency. Indeed it is rium value [inset in Fig. 4(c)]. interesting that the extremes of dielectric and quasisteady An asymptotic analysis of the Laplace space potential gra- response are in phase with one another, despite the entirely dient fors ˜ → 0, corresponding to t˜ →∞, reveals that the po- different physics. tential, and hence the force, approaches its equilibrium value exponentially. Specifically, the evolution of force is composed of two exponential decays: an initial one with a shorter B. Force chronometry resistor-capacitor (RC) charging time,τ ˜RC, corresponding to We now consider the model cell subject to a suddenly double layer charging, and a second exponential decay with a applied voltage. This scenario is relevant in the charging longer bulk diffusion time,τ ˜D, due to the difference in ionic of energy storage devices like supercapacitors. Further, a diffusivities. The equilibrium force is chronoamperometry experiment [32] measures the current as ˜ 2 V0 2 2 2 a function of time, which is analyzed using models relating F˜eq =− κ L cosech κL, (12) the current decay to the physicochemical properties of the 4 electrolyte. Below, we suggest that, in addition to the current, and the asymptotic approach to equilibrium, one could monitor the force on an electrode to extract the − /τ − /τ 2 F˜ − F˜ ≡ F˜ ∼−1 f˜ e t˜ ˜RC − f˜ e t˜ ˜D . charging time scales and hence the ionic diffusivities and tot eq dyn 2 RC D (13) charge carrier concentration. Here the time scales, normalized by the RC charging time, are The dynamics of the force between two charged surfaces D under a suddenly applied voltage is governed by the same τ = A τ ≈ κ ˜RC − RC coth L (14) PNP formulation (1), albeit with a different potential bound- κ 1L ary condition, namely, φ˜x˜=±1 =±V˜0H(t˜). We calculate the and electric potential in Laplace space which is a function of the DA κL /τ τ˜ = τ ≈ + coth κL, (15) normalized Laplace frequencys ˜ (scaled by 1 RC). This is D κ−1L D 3 obtained from the potential in Fourier space by replacing ιω˜ ˜ ˜ withs ˜. The Laplace space potential boundary condition at and the coefficients fRC and fD are =± ˜ =±˜ −1 ˜ x˜ 1is V0s˜ , where is the normalized potential 3 D 2 f˜ = V˜ , f˜ = V˜ A κL. in Laplace space. RC 0 D 0 κ coth (16) Since the force is quadratic in the potential gradient, the L DF solution for the potential in Laplace space is transformed The RC time is smaller than the bulk diffusion time by a back to the time domain before calculating the force. The factor of κL. However, the coefficient of the diffusive decay

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is smaller than that of the RC decay by a factor of κL. Thus, at initial times, one only observes the RC decay, but at longer times, the RC decay becomes subdominant and thus one observes only the diffusive decay. A detailed calculation of the long time behavior of the force from the potential gradient in Laplace space is provided in Appendix B. In a log-linear plot of F˜dyn [Figs. 4(b) and 4(c)]thetwo exponential decays are represented by straight lines with negative slopes. The initial slope is proportional to the reciprocal −1 of the RC time scale (τRC ∼ κ L/DA for κL 1), as shown in Fig. 4(b). Hence a measurement of the time dependent force at short times allows one to infer the RC time scale of the system. At long times it depends on the reciprocal of the 2 diffusive time scale (τD ∼ L /DA for κL 1), as shown by Fig. 4(c), where the time is scaled by the bulk diffusion time scale. The measurements of the force-time curve at long times can thus be used to infer the ambipolar diffusivity of the salt. The ratio of the RC time and diffusive time scales depends on the Debye length [τRC/τD ∼ 1/(κL)]. Thus, from a single force-time curve, one could infer both the ambipolar diffusion coefﬁcient and the Debye length, and therefore the number of charge carriers in the electrolyte. In the special case of equal ionic diffusivities the long time diffusive time scale disappears and the force reaches its equilibrium value in the order of the RC time [Fig. 4(a)]. This is in agreement with the asymptotic expression where fD = 0 when DA/DF = 0forD+ = D−. Thus the force evolves solely on the RC time. Considering that all electrolytes have unequal ionic diffusivities on some degree, the bulk diffusion time scale should be expected in general.

C. Force pulse An impulse response measurement under a voltage V˜ = V˜0δ(t˜) could also allow one to infer the ambipolar diffusion coefficient and the number of charge carriers. Such a measurement has the advantage of not requiring a sustained applied voltage. We calculate the electric potential gradient in Laplace space and its long time limit from the solution in Fourier space by replacing ιω˜ withs ˜. The corresponding (Laplace space) potential boundary condition at x =±1 becomes ˜ = V˜0.The long time asymptotic behavior of an impulse voltage is given − /τ − /τ 2 ˜ ∼−1 ˜ t˜ ˜RC − ˜ t˜ ˜D by Fimp 2 ( fRCe fDe ) . This is the same as the dynamic force obtained for a suddenly applied voltage, with the values ofτ ˜RC,˜τD, f˜RC, and f˜D given by (14), (15), and (16), FIG. 4. Deviation from the long-time, equilibrium value of force respectively. Thus the impulse response is also composed of on the electrode as a function of time for three values of κL for two exponential decays, an initial RC decay followed by a (a) D+ = D−, where the force exponentially approaches its long time bulk diffusive decay. It differs from the response of the force −1 limit on an RC time, κ L/DA.Theinset(t˜ scaled logarithmically to a step voltage in two ways: (i) the initial force is infinite as for clarity) shows the total force going from the dielectric value opposed to a finite jump discontinuity and (ii) the force decays (dashed lines) to the equilibrium double layer value. Here the force to zero at long times, instead of the equilibrium double layer is rescaled by 1/κ2L2 to separate the three lines which otherwise value obtained for a step voltage. overlap. (b) D+ = D−, where the initial decay (inset) is on the RC time, followed by a second decay on a bulk diffusion time. (c) D+ = 2 D−, with time rescaled by the diffusive time scale, L /D .The A D. Comparison to experiments parallel slopes at longer times indicate a diffusive decay. The inset (t˜ scaled logarithmically) shows the total force undergoing a two-step We now make qualitative comparisons to two recent ex- decay from the dielectric to the equilibrium double layer values. In perimental measurements of the dynamic double layer force. (b) and (c), DA/DF = 0.18, corresponding to measurements of ionic Note that neither of these experiments admit all the simpli- diffusivities of sodium and chloride ions in sea water [31]. fying assumptions, namely dilute electrolyte, weak applied

013138-6 DYNAMIC DOUBLE LAYER FORCE BETWEEN CHARGED … PHYSICAL REVIEW RESEARCH 2, 013138 (2020) voltage, and a large separation between electrodes, made by one also has to account for the fluid flow and a hydrodynamic our theory. Hence we cannot make quantitative comparisons. pressure along with the osmotic pressure considered here. Surface force measurements performed by Perez-Martinez Recent work regarding rectified electric fields in asymmetric and Perkin [7] measure the instantaneous time-dependent electrolytes under an oscillatory voltage [42] suggests higher force between two charged surfaces where the applied voltage order contributions to the force beyond the limit of weak oscillates in time. The applied potentials are large (≈7V) voltages considered here. That is, beyond the weak-voltage compared to the thermal voltage (≈25 mV), and the surfaces limit the frequency response will contain a host of higher are separated by an ionic liquid. They report a nonzero steady harmonics in addition to the zeroth and second mode we state force at long times. While the existence of a steady time consider. averaged force is captured by our theory, the magnitude of the predicted force is significantly different from what they ACKNOWLEDGMENTS measure. We attribute this to the assumptions of a dilute electrolyte and weak applied voltage, making their experiments We acknowledge support from the Camille Dreyfus very much outside the realm of our calculations. Teacher-Scholar Award program (A.S.K.) and the Mahmood Measurement of the surface force in response to a suddenly I. Bhutta Fellowship in Chemical Engineering, Carnegie Mel- applied voltage has been performed by Tivony et al. [20]ina lon University (B.B.) similar surface force balance. They investigate the charging dynamics of a dilute electrolyte in the case of very narrow APPENDIX A: DERIVATION AND SOLUTION OF PNP electrode separation, mimicking charging in nanopores. The EQUATIONS UNDER OSCILLATING VOLTAGE dynamics of force follows a long time exponential decay, Here, we derive (1) in the main text. Consider a dilute which is in agreement with our theory for force chronometry. binary monovalent electrolyte with unequal ionic diffusivities. However, the time scale measured by their experiments is The ionic fluxes are thus given by significantly larger than the ones predicted by our theory for the same electrode dimensions and spacing. They attribute n±e j± =−D± ∇n± ± ∇φ , (A1) the increased time scale to ion migration from the reservoir kBT outside the thin gap between the two charged surfaces in their where n± is the number concentration of the positive or apparatus. This transport process, therefore, involves ionic negative ions, φ is the electric potential, k is the Boltzmann fluxes that are not solely normal to the electrode surfaces B constant, T is the temperature, and e is the charge on a proton. and is represented as a transmission line model in [20]. We The species balances are consider a closed, one-dimensional system, where there is no transport of ions beyond the gap between the electrodes. ∂n± =−∇·j±. (A2) Hence we do not capture this larger time scale. ∂t For the planar parallel electrodes considered here, the gra- IV. CONCLUSION dients are present only along the direction perpendicular to the electrodes (say x). The ionic species balance equations are We developed a theory for the dynamic force between thus charged surfaces in an electrolyte under a time-dependent 2 ∂n+ ∂ n+ ∂ n+e ∂φ voltage. Our work represents a nonequilibrium generaliza- = D+ + (A3) ∂ ∂ 2 ∂ ∂ tion of the double layer force used in DLVO theory. We t x x kBT x envision applications to electrochemical, colloidal, and bi- and ological problems in which interactions between bodies or 2 ∂n− ∂ n− ∂ n−e ∂φ particles with time-dependent surface charges or potentials = D− − . (A4) ∂ ∂ 2 ∂ ∂ are encountered. For example, in the directed assembly of t x x kBT x colloidal particles under an ac field [34–37], a time-dependent Note that this is the classical PNP model used in previous double layer polarization alters the force and, hence, the works [24,25]. These equations are linearized in the limit of interaction between the particles. Further, our work can serve small applied voltages, V0/(kBT/e) = 1, where kBT/e as a theoretical foundation for new force-based spectroscopy is the thermal voltage, V0 is the applied voltage, and is a techniques to analyze electrochemically active surfaces, e.g., perturbation parameter. Thus the ion concentrations are ap- during charging or discharging of batteries or supercapacitors. proximated as a small deviation from their equilibrium value, 2 The model problem we have solved could be furthered in a n± = neq + η± + O( ). The linearized O( ) equations are, number of interesting directions. Specifically, one could go from (A3) and (A4), beyond our analysis using ideal, dilute solution theory in pla- 2 2 ∂η+ ∂ η+ neqe ∂ φ nar geometries under a weak voltage. For instance, steric and = D+ + (A5) ∂ ∂ 2 ∂ 2 electrostatic correlations between ions in concentrated solu- t x kBT x tions have been extensively studied in the context of charging and and discharging dynamics [38–40]; it would be interesting to 2 2 ∂η− ∂ η− neqe ∂ φ quantify their impact on dynamic double layer forces. Another = D− − . (A6) ∂ ∂ 2 ∂ 2 useful direction would be to explore the effect of curvature of t x kBT x the electrodes [41], since the surface force apparatus in [7] We define the O( ) charge density, ρ = e(η+ − η−), and the uses a crossed cylinder geometry. For nonplanar electrodes, O( ) ionic strength, 2I = η+ + η−. To reformulate (A5) and

013138-7 BHAVYA BALU AND ADITYA S. KHAIR PHYSICAL REVIEW RESEARCH 2, 013138 (2020) √ (A6) in terms of these newly deﬁned quantities, we consider a complex valued function ofx ˜ andω ˜ , ι = −1, Re is for the the linear combination D− (A5) ± D+ (A6) to obtain real part, andg ˜ representsρ ˜ , I˜,orφ˜, yields a set of ordinary 2 differential equations independent of time given by ∂η+ ∂η− ∂ I D− + D+ = 2D+D− (A7) ∂t ∂t ∂x2 ∂2ρ˜ D 1 = ιωκ˜ Lρ˜ + κ2L2ρ˜ + ιωκ˜ L A I˜ , ∂ 2 1 1 1 and x˜ DF ∂η ∂η ∂2ρ ∂2φ ∂2I˜ D + − 1 neqe 1 = ιωκ + ιωκ A ρ . D− − D+ = D+D− + 2 . (A8) ˜ LI˜1 ˜ L ˜1 (A18) 2 2 ∂ 2 ∂t ∂t e ∂x kBT ∂x x˜ DF

Using η+ = I + ρ/(2e) and η− = I − ρ/(2e) in the above The corresponding potential and flux boundary conditions in equations, and dividing through by 2D+D− we get Fourier space are 2 (D− + D+) ∂I (D− − D+) 1 ∂ρ ∂ I φ˜1|±1 =±V˜0 (A19) + = (A9) ∂ ∂ ∂ 2 2D+D− t 2D+D− 2e t x and and ∂ρ˜ ∂φ˜ ∂I˜ 2 2 + = 0, = 0. (A20) (D− − D+) ∂I (D− + D+) 1 ∂ρ 1 ∂ ρ neqe ∂ φ ∂ ∂ ∂ + = + . x˜ x˜ ±1 x˜ ±1 2D+D− ∂t 2D+D− 2e ∂t 2e ∂x2 k T ∂x2 B The linearized ordinary differential equations can be writ- (A10) ten as = / + = / D Defining DA 2D+D− (D+ D−) and DF 2D+D− d2 ρ˜ κ2L2 + ιωκ˜ L ιωκ˜ L A ρ˜ 1 = DF 1 . (D− − D+), we have DA (A21) dx˜2 I˜1 ιωκ˜ L ιωκ˜ L I˜1 DF 1 ∂I 1 1 ∂ρ ∂2I + = (A11) This is of the form u = Au. To decouple the two equations, D ∂t D 2e ∂t ∂x2 A F we perform an eigenvalue decomposition of the matrix A. and That is, A = PDP−1, where the eigenvalues of A form the 2 2 1 ∂I 1 1 ∂ρ 1 ∂ ρ neqe ∂ φ elements of the diagonal matrix D, and the corresponding + = + . (A12) −1 D ∂t D 2e ∂t 2e ∂x2 k T ∂x2 eigenvectors form the columns of P and P is the inverse F A B of P. Thus we have u = PDP−1u. Defining a transformed ρ φ / − We normalize I with neq, with 2eneq, with kBT e, x with variable vector, y = P 1u, we get y = Dy. Since D is a τ = κ−1 / L, and t with RC L DA. The normalized equations are diagonal matrix, this represents a decoupled set of equations 2 that are readily solved. The solution in terms of the original ∂I˜ DA ∂ρ˜ ∂ I˜ κL + κL = (A13) variables is then obtained using the inverse transformation, ∂t˜ D ∂t˜ ∂x˜2 F Py = u. The eigenvalues and eigenvectors of A are obtained and using D ∂I˜ ∂ρ˜ ∂2ρ˜ ∂2φ˜ κL A + κL = + . (A14) Tr(A) ± [Tr(A)]2 − 4[Det(A)] D ∂t˜ ∂t˜ ∂x˜2 ∂x˜2 λ2 = (A22) F i 2 Finally, φ˜ is governed by the Poisson equation for electrostat- and the corresponding eigenvectors, vi,are ics, whose dimensionless form is given by √ 2 A11−A22± [Tr(A)] −4[Det(A)] ∂2φ˜ v = . =−κ2L2ρ.˜ (A15) i 2A12 (A23) ∂x˜2 1 Substituting (A15) into the species balances (A13) and (A14), The eigenvalues and eigenvectors are [as given in (7) and (8) we have of the main text] 2 ∂I˜ DA ∂ρ˜ ∂ I˜ κL + κL = (A16) κ2 2 κ2 2 ω2 2 2 2 L L 4˜ DA ∂t˜ DF ∂t˜ ∂x˜ λ = + ιωκ˜ L ± 1 − (A24) i 2 2 κ2L2 D2 and F D ∂I˜ ∂ρ˜ ∂2ρ˜ and κL A + κL = − κ2L2ρ,˜ (A17) 2 D ∂t˜ ∂t˜ ∂x˜ ω2 D2 F − − 4˜ A 1 1 κ2L2 D2 v2 F which correspond to (1) in the main text. These are solved = . (A25) with the boundary conditions given by (2). v1 ω2 D2 + − 4˜ A 1 1 κ2 2 2 We first consider an oscillating applied voltage V˜ = L DF V˜0 cos(ω ˜ t˜), whereω ˜ is the angular frequency of oscillations. To solve (A16) and (A17), we recognize that these are coupled The solution in terms of the transformed variables y is linear, second order partial differential equations. The Fourier y1 = a1 sinh(λ1x˜), transform of (A16) and (A17), defined for this case of a ιω˜ t˜ y = a sinh(λ x˜). (A26) single oscillating frequency asg ˜ = 2Re(˜g1e ), whereg ˜1 is 2 2 2

013138-8 DYNAMIC DOUBLE LAYER FORCE BETWEEN CHARGED … PHYSICAL REVIEW RESEARCH 2, 013138 (2020)

Here, we have invoked antisymmetry due to the applied From the solution of the PNP equations, we have the voltage being an odd function about the center of the cell. potential gradient in Fourier space [as (5)inmaintext] Transforming back to the original variables gives ∂φ˜ V˜ cosh (λ x˜) λ v cosh (λ x˜) 1 = 0 −κ2L2 1 − 1 2 2 u1 = ρ˜1 = v1a1 sinh(λ1x˜) + v2a2 sinh(λ2x˜), ∂x˜ Q λ1 cosh (λ1) λ2 v1 λ2 cosh (λ2 ) u2 = I˜1 = a1 sinh(λ1x˜) + a2 sinh(λ2x˜). (A27) 2 2 2 2 κ L λ1 v2 κ L + λ1 − − λ2 − . (B3) The potential and potential gradient are calculated from the λ1 λ2 v1 λ2 above solution using (A15)as Here, κ2 2 λ κ2 2 ∂φ˜ v a v a L 1 v2 L 1 =−κ2 2 1 1 λ + 2 2 λ + Q = λ − − λ − L cosh( 1x˜) cosh( 2x˜) c1 1 λ λ 2 λ ∂x˜ λ λ 1 2 v1 2 1 2 2 2 2 2 (A28) κ L λ1 v2 κ L + tanh (λ ) − tanh (λ ) . (B4) λ2 1 λ v λ2 2 and 1 2 1 2 The potential gradient can be expressed as 2 2 v1a1 v2a2 φ˜ =−κ L sinh(λ x˜) + sinh(λ x˜) + c x˜ + c . ∂φ˜1 1 λ2 1 λ2 2 1 2 = α cosh (λ x˜) + α cosh (λ x˜) + β, (B5) 1 2 ∂x˜ 1 1 2 2 (A29) where −κ2 2 ˜ The values of a , a , c , and c are found using the boundary L V0 1 2 1 2 α1 = , (B6) conditions (A19) and (A20). λ1 cosh (λ1)Q λ κ2 2 ˜ α = 1v2 L V0 , APPENDIX B: CALCULATION OF THE DYNAMIC 2 (B7) λ2v1 λ2 cosh (λ2 )Q DOUBLE LAYER PRESSURE and The pressurep ˜ is calculated from the solution for the 2 2 2 2 V˜0 κ L λ1 v2 κ L electric ﬁeld by invoking mechanical equilibrium β = λ1 − − λ2 − . (B8) Q λ1 λ2 v1 λ2 ∇ p˜ +∇·σ˜ E = 0, (B1) Since the pressure calculation involves the square of the σ σ = potential gradient, we have to convert from Fourier space to where E is the electric stress tensor, deﬁned by E the time domain. For a single frequency response we have ε(∇φ∇φ − 1 (∇φ) · (∇φ)I), where I is the identity matrix. 2 ∗ For the one dimensional system considered here, the mechan- ∂φ˜ ∂φ˜ ιω ∂φ˜ ιω ∂φ˜ −ιω = 2Re 1 e t˜ = 1 e ˜ t˜ + 1 e ˜ t˜ , (B9) ical equilibrium reduces to ∂x˜ ∂x˜ ∂x˜ ∂x˜ ∗ 2 φ˜ φ˜ ∂ p˜ ∂ 1 ∂φ˜ where 1 is the complex conjugate of 1. The square of the = . (B2) potential gradient is then ∂x˜ ∂x˜ 2 ∂x˜

∂φ˜ 2 ∂φ˜ ∂φ˜∗ 2 = 1 eιω˜ t˜ + 1 e−ιω˜ t˜ ∂x˜ ∂x˜ ∂x˜ = α λ + α λ + β ιω˜ t˜ + α∗ λ∗ + α∗ λ∗ + β∗ −ιω˜ t˜ 2 [ 1 cosh ( 1x˜) 2 cosh ( 2x˜) ]e [ 1 cosh ( 1x˜) 2 cosh ( 2x˜) ]e = α λ + α λ + β 2 2ιω˜ t˜ + α∗ λ∗ + α∗ λ∗ + β∗ 2 −2ιω˜ t˜ [ 1 cosh ( 1x˜) 2 cosh ( 2x˜) ] e [ 1 cosh ( 1x˜) 2 cosh ( 2x˜) ] e ∗ ∗ ∗ ∗ ∗ + 2{[α1 cosh (λ1x˜) + α2 cosh (λ2x˜) + β][α cosh (λ x˜) + α cosh (λ x˜) + β ]} 1 1 2 2 α2 α2 α2 cosh (2λ x˜) α2 cosh (2λ x˜) = 2Re 1 + 2 + β2 + 1 1 + 2 2 + 2α α cosh (λ x˜) cosh (λ x˜) 2 2 2 2 1 2 1 2 + α β λ + α β λ 2ιω˜ t˜ + α α∗ λ λ∗ + α α∗ λ λ∗ 2 1 cosh ( 1x˜) 2 2 cosh ( 2x˜) e 2[ 1 1 cosh ( 1x˜) cosh ( 1x˜) 2 2 cosh ( 2x˜) cosh ( 2x˜) + α α∗ λ λ∗ + α α∗ λ λ∗ + α∗ λ∗ β + α λ β∗ 1 2 cosh ( 1x˜) cosh ( 2x˜) 2 1 cosh ( 2x˜) cosh ( 1x˜) 1 cosh ( 1x˜) 1 cosh ( 1x˜) + α λ β + α∗ λ∗ β + ββ∗ . 2 cosh ( 2x˜) 2 cosh ( 2x˜) ] (B10)

This can be written, for brevity, as ∂φ˜ 2 α2 α2 = 2Re 1 + 2 + β2 + γ (˜x) e2ιω˜ t˜ + 2[ββ∗ + ν(˜x)], (B11) ∂x˜ 2 2

013138-9 BHAVYA BALU AND ADITYA S. KHAIR PHYSICAL REVIEW RESEARCH 2, 013138 (2020) where with the Laplace variables ˜. Thus 2 2 s s α cosh (2λ1x˜) α cosh (2λ2x˜) ∂ ˜ V˜ cosh λ x˜ λs vs cosh λ x˜ γ (˜x) = 1 + 2 = 0 −κ2L2 1 − 1 2 2 2 2 ∂ s λs λs λs s λs λs x˜ Q 1 cosh 1 2 v1 2 cosh 2 + 2α α cosh (λ x˜) cosh (λ x˜) + 2α β cosh (λ x˜) 1 2 1 2 1 1 κ2L2 λs vs κ2L2 + α β λ + λs − − 1 2 λs − . (B19) 2 2 cosh ( 2x˜) (B12) 1 λs λs s 2 λs 1 2 v1 2 and Here, ∗ ∗ κ2 2 λs s κ2 2 ν = α α λ λ s L 1 v2 s L (˜x) [ 1 1 cosh ( 1x˜) cosh ( 1x˜) Q = λ − − λ − 1 λs λs vs 2 λs + α α∗ λ λ∗ 1 2 1 2 2 2 cosh ( 2x˜) cosh ( 2x˜) κ2L2 λs vs κ2L2 ∗ ∗ + λs − 1 2 λs , + α α cosh (λ x˜) cosh (λ x˜) s tanh 1 tanh 2 (B20) 1 2 1 2 λ2 λs vs λs 2 1 2 1 2 + α α∗ λ λ∗ 2 1 cosh ( 2x˜) cosh ( 1x˜) λs λs 1 and 2 are the eigenvalues of the system of ODEs govern- + α∗ λ∗ β + α λ β∗ 1 cosh ( 1x˜) 1 cosh ( 1x˜) ing charge density and ionic strength in Laplace space, and [vs, 1] and [vs , 1] are the corresponding eigenvectors. Thus + α λ β + α∗ λ∗ β . 1 2 2 cosh ( 2x˜) 2 cosh ( 2x˜) ] (B13) 2 2 2 2 2 2 2 κ L κ L 4˜s D We choose to express the square of the potential gradient in λs = + s˜κL ± 1 + A (B21) i 2 2 κ2L2 D2 this way, separating the x-dependent and x-independent parts F to clarify the integration and differentiation in the next steps. and The mechanical equilibrium (B2)gives 2 D2 − + 4˜s A s 1 1 κ2L2 D2 ∂ α2 α2 v2 F 1 2 2 2ιω˜ t˜ = , (B22) p˜ = Re + + β + γ (˜x) e s v 2 D2 x˜ ∂x˜ 2 2 1 + + 4˜s A 1 1 κ2 2 2 L DF + ββ∗ + ν [ (˜x)] dx˜ wheres ˜ = sDAκ/L is the normalized Laplace variable. The electric potential gradient in Laplace space can be expressed dγ ιω dν ιω as = Re e2 ˜ t˜ + dx˜ = Re[γ (˜x)e2 ˜ t˜] + ν(˜x) + c. x˜ dx˜ ∂ ˜ x˜ = αs cosh λs x˜ + αs cosh λs x˜ + βs, (B23) (B14) ∂x˜ 1 1 2 2 Here c is an integration constant, which we identify as a where 2 2 “reservoir” pressure,p ˜0. This is equivalently the osmotic −κ L V˜ αs = 0 , (B24) pressure in the electrolyte when the surface is uncharged. In 1 λs cosh λs Qs this limit, γ (˜x) = 0 and ν(˜x) = 0. Thus 1 1 s s 2 2 ιω λ v κ L V˜ p˜ − p˜ = Re[γ (˜x)e2 ˜ t˜] + ν(˜x). (B15) αs = 1 2 0 , (B25) 0 2 λs s λs λs s 2v1 2 cosh 2 Q The total force per unit area on the electrode is given by and 2 2 2 s s 2 2 1 ∂φ˜ V˜0 κ L λ v κ L ˜ = − − . β = λs − − 1 2 λs − . (B26) FT p˜ p˜0 (B16) s 1 λs λs s 2 λs 2 ∂x˜ Q 1 2 v1 2 Using (B11) and (B15) yields To obtain the force, we ﬁrst invert this back to the time domain. This is not possible to do in closed form; however, α2 α2 we obtain an asymptotic behavior at long times by taking the F˜ =−Re 1 + 2 + β2 e2ιω˜ t˜ − [ββ∗], (B17) T 2 2 limit of the potential gradient in Laplace space as the Laplace variables ˜ → 0, and then inverting it to the time domain. This where the ﬁrst term represents the oscillating part of the force gives and the second term is the steady, time-averaged value. In ∂ ˜ f cosh (κLx˜) f cosh (κLx˜) terms of the notations used in the main text, these are lim = x − x → s˜ 0 ∂x˜ s˜ s˜ + 1/τ˜x 2 2 α α ∗ F˜ =− 1 − 2 − β2, F˜ =−ββ =−|β|2. (B18) fRC fD O 2 2 S + + , (B27) s˜ + 1/τ˜RC s˜ + 1/τ˜D α α β Substituting for 1, 2, and , we have the expressions for the where force as given in the main text; namely (10) and (11). To obtain the step response of the force, the potential fx = V˜0κL cosech (κL), (B28) gradient in Laplace space (∂ /∂˜ x˜) is obtained from that in fRC = V˜0, (B29) the Fourier space (∂φ˜1/∂x˜) by substituting the frequency ιω˜

013138-10 DYNAMIC DOUBLE LAYER FORCE BETWEEN CHARGED … PHYSICAL REVIEW RESEARCH 2, 013138 (2020) and 2 3 DA fD =−V˜0 coth (κL). (B30) κL DF

DA Here, fx, fRC, and fD are independent of x. The time scales (normalized by the RC time κ−1 )are L tanh(κL) tanh(κLx˜) 1 τ˜ = coth (κL) + − + O , (B31) x 2 2 κL 1 τ˜ = coth (κL) + O , (B32) RC κL and κL 1 τ˜ = + coth (κL) + O . (B33) D 3 κL The inverse Laplace transform of (B27) thus gives the long-time behavior

∂φ˜ − /τ − /τ − /τ ∼ cosh (κLx) f − f e t˜ ˜x + f e t˜ ˜RC + f e t˜ ˜D . (B34) ∂x˜ x x RC D The long-time pressure gradient is again calculated using mechanical equilibrium (B2) ∂ p˜ 1 ∂ −t˜/τ˜x −t˜/τ˜RC −t˜/τ˜D 2 ∼ cosh (κLx) fx − fxe + fRCe + fDe dx˜. (B35) ∂x˜ x˜ 2 ∂x˜ Thus

∂ p˜ 1 ∂ 1 + cosh 2κL − /τ 2 − /τ − /τ 2 = f − f e t˜ ˜x + f e t˜ ˜RC + f e t˜ ˜D ∂x˜ 2 ∂x˜ 2 x x RC D x˜ −t˜/τ˜x −t˜/τ˜RC −t˜/τ˜D + 2 cosh (κLx) fx − fxe fRCe + fDe dx˜. (B36)

We integrate to obtain the long-time pressure. The integration constant is determined similar to the frequency dependent case: cosh(2κL) − /τ 2 1 − /τ − /τ − /τ − /τ − /τ p˜ − p˜ = f − f e t˜ ˜x + f 2e 2t˜ ˜x − 2 f 2e t˜ ˜x + 2 cosh (κLx) f − f e t˜ ˜x f e t˜ ˜RC + f e t˜ ˜D . 0 2 x x 2 x x x x RC D (B37) Thus we have the total force, 2 2 1 ∂φ˜ f 1 − /τ − /τ 2 F˜ = p˜ − p˜ − =− x − f e t˜ ˜RC + f e t˜ ˜D . (B38) tot 0 2 ∂x˜ 4 2 RC D Here the steady equilibrium value of the force is given by f 2 V˜ 2 F˜ =− x =− 0 κ2L2 cosech2(κL), (B39) eq 4 4 and the dynamic part of the force is

− /τ − /τ 2 ˜ ≡ ˜ − ˜ ∼−1 t˜ ˜RC + t˜ ˜D . Fdyn Ftot Feq 2 fRCe fDe (B40) This corresponds to (13) in the main text.

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