Direct measurement of the attractive electrosolvation force between a pair of colloidal particles

TL;DR

This work directly measures the pair potential between isolated like-charged colloidal particles and discovers a strong, long-range attractive component exceeding the nominal Debye length. Through high-resolution optical imaging and analysis, the authors extract two decay lengths, κ_1^{-1} and κ_2^{-1}, with κ_2^{-1} ≈ 0.7–1.5 μm and κ_1^{-1} ≈ 60–100 nm, and show that κ_2^{-1} depends on particle size and surface chemistry. The attraction arises for diverse anionic coatings, including DNA and lipid bilayers, and even in alcohols for positively charged particles, challenging DLVO theory and pointing to nonlocal solvent effects and interfacial water polarization as key factors. Brownian Dynamics simulations using the measured potentials validate the extracted decay lengths and support a scenario where solvent-mediated interfacial structuring governs long-range attraction, with potential implications for biological organization such as chromatin condensation and biomolecular phase separation.

Abstract

In solution, electrically like-charged particles can experience a strong and long-ranged attraction that leads to the formation of stable, slowly reorganizing clusters. The attractive force underpinning this spontaneous organization process has been shown to depend on both the sign of charge of the particle and the nature of the solvent medium. The origin of the attraction has been ascribed to the preferential orientation of solvent molecules at the object-electrolyte interface. Here, we use optical imaging to directly measure the spatial profile of the potential of mean force between isolated pairs of charged microspheres. Working with particles carrying a variety of surface chemistries we find that the range of the electrosolvation attraction is substantially longer than previously held. In particular we show that particles carrying strongly anionic surface coatings composed of DNA or phospholipid bilayers display long-range attraction. We further find that the length scale governing the decay of the attractive force can depend on the properties of the interacting particles. This contrasts with the canonical expectation that the screening length governing the interaction of charged particles in solution depends exclusively on the properties of the intervening electrolyte medium. The observations point to significant departures from current thinking, and the likely need for a model of interactions that accounts for the molecular nature of the solvent, its interfacial behaviour, and spatial correlations. Finally, a strong and long-ranged attraction mediated by anionic matter constituting lipid membranes and chromatin could carry far-reaching implications for biological organization and structure formation.

Figures (5)

• Figure 1: Experimental setup and procedure for pair-potential measurements. (A) Schematic representation of two interacting microspheres of radius $R$, at an interparticle distance $r$, and intersurface separation $x=r-2R$ observed using bright-field microscopy. (B) Rescaled radial probability density function, $P(r)$, measured for a typical pair of interacting particles. (C) Left: Snapshot of well separated pairs of negatively charged SiO$_2$ particles of radius $R=2.4\ \mu$m in DI water ($c\approx 10^{-5}$ M) engaged in "bound states" (white dashed boxes) Scalebar: 10 $\mathrm{\mu m}$; right: Time-series of images for pair #1 displaying bound (blue outline) and "free" states. Scalebars: 5 $\mathrm{\mu m}$. (D) Measured time-dependent separation $r$ for pair #1 engaged in a bound-state. The start and end of a bound state are characterized by $r<r_{1}$ and $r>r_{2}$, which occur at $t\approx 1$ min and $t\approx 50$ min in the displayed trace, respectively. (E) Measured interaction potential for pair #1, given by $U(x) = -k_\mathrm{B} T \ln P(x)+w$, where $w$ and $x_{\mathrm{min}}$ denote the depth and location of the minimum (symbols). $U(x)$ data are fit with piecewise screened Coulombic functions, $U_1$ and $U_2$, indicating the repulsive (red) and attractive (blue) regions of the interaction, respectively. The van der Waals ($U_\mathrm{vdW}$) contribution to the total pair interaction (dashed grey line) is calculated as described in Refs. RN101RN14. (F) Log-linear plot of measured data and fit-functions $U_1$ and $U_2$ as shown in (D) with fitted parameter values and errors listed (inset).
• Figure 2: Salt-concentration and particle-size dependence of the pair interaction potential. (A) Measured pair potentials $U(x)$ for ca. 15 pairs of $R$ = 2.4 $\mu$m silica particles (thin red and blue lines) in aqueous solution with salt concentration $c$ ranging from $10^{-5}$ M to $1.1 \times 10^{-4}$ M, displaying representative measurements of individual pairs (symbols) and the corresponding piecewise fits (thick lines). Displayed values of $\kappa_1^{-1}$ and $\kappa_2^{-1}$ represent average and standard error on mean fitted values across all particle pairs in a given dataset. (B) A bi-exponential function representing the average of all individual fitted curves for particle pairs in (A) displays systematic shifts of both $x_{\mathrm{min}}$ and well depth $w$ to lower values with increasing salt concentration. (C) Interaction potentials measured on ca. 12 pairs of $R$ = 2.0 $\mu$m (left) and $1.6$$\mu$m (right) silica particles in water ($c \approx 10^{-5}$ M). Smaller particles ($R$ = 1.6 $\mu$m) display a significant decrease in $\lvert w\rvert$ and $\kappa_2^{-1}$ under the same experimental conditions. (D) Bi-exponential fits to averaged pair potential data from measurements in (C) display systematic shifts of $\kappa_2^{-1}$ to larger values with increasing particle radius $\it{R}$. (E) Average bound-state lifetimes $\tau$ for interactions in (A) display the expected factor $\approx 2$ reduction based on a reduction in magnitude of the well depth $w$ with increasing salt concentration. Solid lines denote fits of the data to the form ${P(\Delta t)} = (1/{\tau})$exp$(\Delta t/\tau)$. Note that a measurement of $\tau$ at $c \approx 10^{-5}$ M was infeasible owing to bound-state durations $\Delta t\approx 1$ hour (see Fig. 1D). Zeta potentials of $\approx-50$mV were measured for all particle sizes under the stated solution conditions.
• Figure 3: (Caption next page.)
• Figure 3: (Previous page.) The attractive electrosolvation interaction measured for microspheres surface-functionalised with DNA or supported lipid bilayers (LB). (A) Left: schematic illustration of layer-by-layer assembly of positive polyelectrolyte (poly(diallyldimethyl ammonium chloride), or PDADMAC, shown in pink), and negative polyelectrolyte (100 bp double-stranded DNA, shown in blue) on silica spheres of radius $R=2.4\ \mu$m. Illustration of the expected arrangement of surface-adsorbed polyelectrolyte. Bright-field images show absence of clustering (indicative of monotonic repulsion) when the outermost coating is PDADMAC (image II), to be contrasted with the formation of ordered clusters when the outermost layer is dsDNA (image III) (bottom). Scalebars: 10 $\mathrm{\mu m}$. Right: Piecewise fits of the attractive and repulsive arms of the potential for 21 pairs of particles (thin blue and red lines, respectively). Also presented are data for a representative particle-pair (symbols) and corresponding fits (thick lines) with fitted parameter values noted, similar to Fig. 2. An exponential fit of the residence time histogram $P(\Delta t)$ for all bound pairs reveals an average lifetime of $\tau = 679\pm 54\ \mathrm{s}$. (B) Schematic illustration of LB formation on aminated silica particles (radius $R=2.0\ \mu$m). Bright-field images show the absence of attraction between the positively charged aminated silica particles (image I), as opposed to the ordered clusters that form when the particles are coated with negatively charged LBs (image II). Scale bars: 10 $\mathrm{\mu m}$. Measured pair-potentials and histograms of bound-state duration measured for 17 pairs of LB-coated particles (right). The average bound-state lifetime for LB-coated particles ($\tau = 178 \pm 17\ \mathrm{s}$) is about a factor 3 smaller than that of the DNA coated capturing the reduction of $\approx 1.1 k_{\mathrm{B}}T$ in the average magnitude of the well depth. (C) Table listing the nominal Debye length, $\kappa^{-1}$, fitted parameters ($\kappa_1^{-1}$, $\kappa_2^{-1}$ and $w$), and $\zeta$-potentials for each anionic surface coating: dsDNA, LBs, polyglutamic acid - poly-E, polystyrene sulfonate - PSS.
• Figure 4: Dependence of the the long-range attraction on salt concentration, particle size and surface properties. (A) Plot of averaged measured decay lengths $\kappa_1^{-1}$ (squares) and $\kappa_2^{-1}$ (circles) compared with the Debye screening length $\kappa^{-1}$ (solid grey line) reveals that whilst both decay lengths respond to the salt concentration, $\kappa_2^{-1}$ displays a weaker dependence than $\kappa_1^{-1}$. (B) Varying particle radius at a fixed salt concentration $c=10^{-5}$ M shows that $\kappa_1^{-1}$ and $\kappa_2^{-1}$ respond differently to changes in particle size: $\kappa_1^{-1}\approx 100$ nm remains relatively constant, similar to the nominal Debye length $\kappa^{-1} \approx$ 90 nm (grey band), while $\kappa_2^{-1}$ decreases with decreasing particle radius (dashed line is a guide to the eye). (C) At a similar salt concentration $c\approx 10^{-5}$M, particles of radius $R=2.0\ \mu$m coated with LBs with increasing negative charge density do not display a significant difference in $\kappa_2^{-1}$ compared to uncoated silica particles of similar radius. DNA-coated particles of radius $R=2.4 \mu$m however show significantly higher $\kappa_2^{-1}$ values compared to the corresponding uncoated silica particles, possibly pointing to a role for surface properties in the long range attraction. (D) Possible heuristic relationship between decay lengths $\kappa_2^{-1}$ and $\kappa_1^{-1}$, and particle radius $R$, highlighting the weak observed dependence of the decay length $\kappa_2^{-1}$ on particle radius under the experimental conditions of this study. Identity line provided for reference (solid grey).