[Paper Review] Pair Interaction Potentials of Colloids by Extrapolation of Confocal Microscopy Measurements of Collective Structure

Christopher R. Iacovella, Reginald R. Rogers+2 more|arXiv (Cornell University)|May 14, 2010

Material Dynamics and Properties39 references5 citations

TL;DR

This paper presents a method to extract pair interaction potentials $U(r)$ of colloidal particles by extrapolating the potential of mean force $W(r,\phi)$, derived from 3D confocal microscopy measurements of radial distribution functions $g(r)$, to infinite dilution ($\phi \to 0$). The approach enables accurate reconstruction of $U(r)$ for refractive index-matched, fluorescent colloids, with validation via Monte Carlo simulations and experiments on PMMA/PHSA particles in DOP, showing good agreement with screened Coulomb potentials, especially for longer-ranged interactions.

ABSTRACT

A method for measuring the pair interaction potential between colloidal particles by extrapolation measurement of collective structure to infinite dilution is presented and explored using simulation and experiment. The method is particularly well suited to systems in which the colloid is fluorescent and refractive index matched with the solvent. The method involves characterizing the potential of mean force between colloidal particles in suspension by measurement of the radial distribution function using 3D direct visualization. The potentials of mean force are extrapolated to infinite dilution to yield an estimate of the pair interaction potential, $U(r)$. We use Monte Carlo (MC) simulation to test and establish our methodology as well as to explore the effects of polydispersity on the accuracy. We use poly-12-hydroxystearic acid-stabilized poly(methyl methacrylate) (PHSA-PMMA) particles dispersed in the solvent dioctyl phthalate (DOP) to test the method and assess its accuracy for three different repulsive systems for which the range has been manipulated by addition of electrolyte.

Motivation & Objective

• To develop a method for measuring the true pair interaction potential $U(r)$ between colloidal particles using collective structural data from 3D confocal microscopy.
• To overcome the limitation of conventional methods that only yield the potential of mean force $W(r,\phi)$ at finite volume fractions.
• To assess the accuracy of the extrapolation method under varying conditions, particularly polydispersity and screening length.
• To validate the method experimentally using refractive index-matched, fluorescent PMMA/PHSA colloids in dioctyl phthalate (DOP) with tunable electrostatic repulsion via added electrolyte.
• To establish guidelines for the method’s applicability, especially in systems where traditional techniques like TIRM or AFM are inapplicable due to index matching.

Proposed method

• Measure 3D particle positions in colloidal suspensions using confocal laser scanning microscopy to determine the radial distribution function $g(r)$ at multiple volume fractions $\phi$.
• Calculate the potential of mean force $W(r,\phi)$ from $g(r)$ using the relation $W(r,\phi)/k_B T = -\ln[g(r)]$.
• Extrapolate $W(r,\phi)$ to $\phi \to 0$ to estimate the true pair interaction potential $U(r)$, assuming the limit recovers the pairwise potential.
• Use Monte Carlo simulations to test the method’s accuracy and explore the effects of polydispersity, screening length, and surface charge on the extrapolation.
• Compare the extrapolated $U(r)$ with theoretical screened Coulomb potentials derived from electrokinetic measurements to validate the method.
• Apply the method to experimental systems with varying electrolyte concentrations (e.g., 10 $\mu$M and 2 mM TBAC) to tune the range of repulsion and assess accuracy.

Experimental results

Research questions

• RQ1 Can the pair interaction potential $U(r)$ be reliably extracted from collective structural data (via $g(r)$) by extrapolating to infinite dilution?
• RQ2 How does polydispersity affect the accuracy of the extrapolated $U(r)$, especially for short-ranged potentials?
• RQ3 To what extent does the method agree with independently measured screened Coulomb potentials in refractive index-matched colloidal systems?
• RQ4 How does the screening length influence the validity and precision of the extrapolation method?
• RQ5 Can the method be applied to systems where conventional direct-pair-potential techniques (e.g., TIRM, AFM) are inapplicable due to index matching?

Key findings

• The extrapolation method accurately recovers $U(r)$ for monodisperse colloids in Monte Carlo simulations, demonstrating near-perfect agreement with the true potential.
• For the PMMA/PHSA system in DOP with 10 $\mu$M TBAC, the extrapolated $U(r)$ shows excellent agreement with the theoretical screened Coulomb potential ($M = 0.35$).
• At 2 mM TBAC, the extrapolated $U(r)$ deviates from the theoretical potential due to short-ranged repulsion (Debye length = 49 nm), but truncating the potential at $r \sim 1.25\mu$m improves agreement.
• Polydispersity has a stronger effect on accuracy when the potential range is comparable to the particle size distribution (4%), particularly for short-ranged interactions.
• The method maintains accuracy for $U(r)/k_B T < 10$, even when the absolute magnitude is not fully resolved, and successfully captures the shape of the potential.
• The method is well-suited for refractive index-matched, fluorescent colloids where traditional techniques fail, and can be used to validate or replace electrokinetic measurements when key parameters are inaccessible.