Open Access Article

This Open Access Article is licensed under a Creative Commons Attribution-Non Commercial 3.0 Unported Licence

Wei Kang
Lim
and
Alan R.
Denton
*

Department of Physics, North Dakota State University, Fargo, ND 58108-6050, USA. E-mail: alan.denton@ndsu.edu

Received
24th November 2015
, Accepted 14th December 2015

First published on 15th December 2015

Depletion-induced interactions between colloids in colloid–polymer mixtures depend in range and strength on size, shape, and concentration of depletants. Crowding by colloids in turn affects shapes of polymer coils, such as biopolymers in biological cells. By simulating hard-sphere colloids and random-walk polymers, modeled as fluctuating ellipsoids, we compute depletion-induced potentials and polymer shape distributions. Comparing results with exact density-functional theory calculations, molecular simulations, and experiments, we show that polymer shape fluctuations play an important role in depletion and crowding phenomena.

Complementary to depletion is the phenomenon of crowding upon mixing polymers or other flexible macromolecules with impenetrable obstacles. When colloids, nanoparticles, or other crowding agents are dispersed in a polymer solution or blend, flexible chains adjust their size and shape to conform to the accessible volume.^{45} The prevalence and importance of macromolecular crowding in biology was recognized over three decades ago.^{46} In the congested environment of a cell's cytoplasm or nucleoplasm, conformations of proteins, RNA, and DNA are constrained by the presence of other macromolecules, affecting biopolymer function.^{47–49} Crowding of polymers has been studied experimentally by neutron scattering,^{50,51} computationally via Langevin dynamics^{52–54} and Monte Carlo simulations of coarse-grained models,^{55–57} and by free-volume theories.^{45,56–58}

Attempts to interpret experimental or simulation data for depletion forces in colloid–polymer mixtures typically assume the spherical polymer model and treat the polymer size and concentration as free parameters.^{6–10} The fitted parameters invariably differ from measured values. Dependences on particle curvature and depletant concentration have been partially accounted for by introducing an effective polymer size or depletion layer thickness.^{19,36,38,43} The influence of depletant shape on interactions has been studied in mixtures of colloidal spheres and rods^{10,15} or ellipsoids,^{13,59} but only of fixed size and shape. Other workers have explored the impact of polymer conformations on relative stabilities of hard-sphere colloidal crystals^{60–62} and of crowding on polymer size^{50–58} (but not shape). Despite ample evidence that random-walk polymers exhibit significant asphericity,^{63–66} however, no studies have yet related shape fluctuations to depletion interactions and crowding. This paper presents the first consistent analysis of the role of depletant shape in mixtures of colloids and nonadsorbing polymers. By comparing results with exact theoretical calculations and with data from molecular simulations and experiments, we demonstrate the importance of polymer shape fluctuations for depletion and crowding.

A polymer coil of N segments has size and shape characterized by its gyration tensor, , where r_{i} is the position vector of segment i relative to the center of mass. A conformation with gyration tensor eigenvalues Λ_{1}, Λ_{2}, Λ_{3} has instantaneous radius of gyration . The experimentally measurable (root-mean-square) radius of gyration is an ensemble average over polymer conformations, R_{g} ≡ 〈R_{p}^{2}〉^{1/2}. If the average is defined relative to the polymer's principal-axis frame, the coordinate axes being labelled to preserve the eigenvalue order (Λ_{1} > Λ_{2} > Λ_{3}), then the average gyration tensor describes an aspherical object, whose average shape is an elongated (prolate), flattened ellipsoid.^{63–65} Each eigenvalue is proportional to the square of a principal radius of the general ellipsoid that best fits the shape of the polymer coil: x^{2}/Λ_{1} + y^{2}/Λ_{2} + z^{2}/Λ_{3} = 3.

Ideal, freely-jointed (random-walk) polymer coils can be modeled as soft Gaussian ellipsoids.^{67} For coils sufficiently long that extensions in orthogonal directions are essentially independent, the shape probability distribution is well approximated by the factorized form^{68}

P(λ_{1},λ_{2},λ_{3}) = P_{1}(λ_{1})P_{2}(λ_{2})P_{3}(λ_{3}), | (1) |

(2) |

The deviation of a polymer's average shape from a sphere is quantified by the asphericity^{65}

(3) |

(4) |

Depletion of polymers induces an effective interaction between colloids that reduces, in the dilute limit, to the potential of mean force (PMF), v_{mf}(r) = Ω(r) − Ω(∞), defined as the change in grand potential Ω(r) upon bringing two colloids from infinite to finite (center-to-center) separation r by working against the polymer osmotic pressure, Π_{p} = n_{p}k_{B}T (for ideal polymers). If we make the choice Ω(∞) = 0, then v_{mf}(r) = −Π_{p}V_{o}(r), where V_{o}(r) is the intersection of the excluded-volume regions surrounding the colloids. For spherical polymers (AOV model),

(5) |

Accurate calculation of the PMF requires calibrating the ellipsoidal polymer model to consistently match the polymer radius to the depletion layer thickness and to account for deformation of a polymer coil near a curved surface.^{29} A rational criterion for choosing the effective size ratio q_{eff} is based on equating the free energy to insert a hard sphere into a bath of ideal polymers, as predicted by polymer field theory,^{16} with the work required to inflate a sphere in the model polymer solution.^{36,38,43} For nonspherical polymers, we generalize this criterion to

(6) |

Fig. 1 Simulation snapshot (a) depicts colloids (blue spheres) and polymer (red ellipsoid), which induces potential of mean force v(r) at center–center distance r (units of colloid diameter σ_{c}) for polymer-to-colloid size ratios (b) q = 0.125, (c) q = 0.7776. Our MC simulation data for the fluctuating ellipsoidal polymer model (squares), and predictions of the spherical polymer (AOV) model (circles, dashed curves), are compared with (b) density-functional theory (DFT) predictions (solid curve) for continuum-chain polymers^{23} and (c) MC simulation data (solid curve) for lattice-chain polymers^{29} at corresponding effective size ratios q_{eff} [eqn (6)]. Also shown in (b) is the AOV model prediction for the bare size ratio q (dotted curve). Error bars are smaller than symbols. |

We turn next to the experiments of Verma et al.,^{9} who used an optical tweezer to measure interactions between silica microspheres of diameter σ_{c} = 1.25 ± 0.05 μm in aqueous solutions of λ-DNA (contour length 16 μm, radius of gyration R_{g} ≃ 500 nm), whose conformations are known to be random walks of ∼160 Kuhn segments.^{66} Taking the nominal size ratio of q = 0.8, we computed the PMF in both the ellipsoidal and spherical polymer models and here compare our results with data for a dilute DNA solution of concentration 25 μg ml^{−1} (n_{p} = 0.5 μm^{−3}), in which polymer interactions should be negligible (Fig. 2 and 3 of ref. 9). Since the experiments cannot accurately resolve the vertical offset of the potential, we varied the offset to most closely fit our simulation data. With this single fit parameter, the ellipsoidal polymer model, with effective size ratio q_{eff} = 0.8351 [from eqn (6)], is in close agreement with the measured interaction potential (Fig. 2), as is seen by comparing the least-squares fit to the experimental data with our simulation data (solid curve and squares in Fig. 2). In contrast, the AOV model, with q_{eff} = 0.7788, significantly overestimates the depth, and underestimates the range, of the potential. From visual inspection, it is clear that no vertical shift of the experimental data will yield close alignment with the AOV model (solid and dashed curves in Fig. 2).

Fig. 2 Potential of mean force between a pair of silica microspheres (diameter σ_{c} = 1.25 μm) induced by λ-DNA in water with R_{g} = 0.5 μm (q = 2R_{g}/σ_{c} = 0.8). Our MC simulation data for the fluctuating ellipsoidal polymer model (squares) at effective size ratio q_{eff} = 0.8351 [eqn (6)] are compared with both experimental data^{9} (circles) and predictions of the AOV model (dashed curve) [eqn (5)] for q_{eff} = 0.7788 [eqn (6)]. The solid curve is a least-squares fit to the experimental (not simulation) data of the function −exp(a_{0} + a_{1}r + a_{2}r^{2}) with a_{0} = 0.817, a_{1} = −0.167, a_{2} = −1.269. |

The close agreement of depletion potentials from our simulations of the ellipsoidal polymer model with, on the one hand, DFT calculations and simulations for explicit polymer models and, on the other hand, experimental data from optical tweezer measurements of colloid–DNA mixtures is strong evidence that aspherical polymer shapes play a significant role in depletion. Contrary to previous studies,^{1,19,25} we conclude that depletion interactions between hard-sphere colloids are not fully captured by modeling polymers simply as penetrable spheres of an effective size or, equivalently, with an effective depletion layer thickness. Moreover, our approach consistently accounts for fluctuations in polymer shape, in contrast to models of spheroidal depletants.^{13,59}

Our approach may be compared with the powerful and elegant method of Bolhuis and Louis et al.^{32–38} that models polymers as “soft colloids” by replacing a polymer coil with a single particle at the center of mass. These authors determined the effective pair potential between two polymers and between a polymer and a hard sphere by first computing the respective radial distribution function g(r) between the centers of mass, via MC simulation of explicit segmented polymers on a lattice, and then inverting g(r) via the Ornstein–Zernike integral equation. From subsequent simulations of a coarse-grained model of colloid–polymer mixtures governed by such effective pair potentials, they extracted polymer depletion-induced interactions between hard-sphere colloids. For polymers in a good solvent, whose excluded-volume interactions were modeled via self-avoiding walks, comparisons of effective pair potentials derived from simulations of the explicit and coarse-grained models were in close agreement for q ≃ 1 and in the dilute polymer concentration regime, with deviations emerging abruptly at higher concentrations. Moreover, a computationally practical superposition approximation that expresses two-body depletion interactions in terms of the radially symmetric density profile of a polymer around a single colloidal sphere, which for ideal depletants can be implemented as a simple convolution integral,^{75} proves nearly as accurate as simulations. Our analysis of polymer shape fluctuations, although here limited to ideal polymers, would suggest that the soft-colloid approach succeeds largely by capturing, in the effective colloid–polymer potential, an accurate representation of the distortion of a polymer coil near a hard, curved surface.

Our restriction thus far to the dilute limit, while intended to highlight the role of polymer shape fluctuations in depletion interactions, raises the important question of how such shape fluctuations may be modified in more crowded environments, as in concentrated suspensions and biological cells. As a first step toward assessing the impact of crowding on polymer shapes, we simulated polymers amidst many mobile colloids, now including trial displacements of both species. Previously, we computed polymer shape distributions, radii of gyration, and asphericities in the protein limit (q ≫ 1), using a coated-ellipsoid approximation for the excluded volume.^{57} By applying the exact overlap algorithm, we can now extend this analysis to the colloid limit (q < 1). Fig. 3 shows results from simulations of 216 colloids and one polymer at q = 0.8 in a cubic box with periodic boundary conditions, along with predictions of a free-volume theory based on a mean-field approximation for the average volume accessible to an ellipsoid in a hard-sphere fluid.^{57} Upon increasing the colloid volume fraction, ϕ_{c} ≡ (4π/3)n_{c}R_{c}^{3}, the polymer eigenvalue distributions shift toward contraction of the polymer along all three principal axes, while the radius of gyration and asphericity decrease, reflecting compactification of the polymer. These trends imply a decreasing range of pair attraction with increasing colloid concentration.

Fig. 3 (a) Simulation snapshot depicts colloids (blue spheres) and polymer (red ellipsoid) in a cubic simulation cell with periodic boundary conditions. (b) Probability distributions for eigenvalues (λ_{1}, λ_{2}, λ_{3}) of the gyration tensor of an ideal polymer coil with random-walk segment statistics. Simulation data (symbols) are compared with predictions of free-volume theory^{57} (solid curves) for an ellipsoidal polymer with uncrowded size ratio q = 0.8 amidst 216 colloids of volume fraction ϕ_{c} = 0.5. Dashed curves: uncrowded (ϕ_{c} = 0) distributions [eqn (1)]. Inset: Polymer asphericity A vs. ϕ_{c} [eqn (3)]. (c) Polymer radius of gyration R_{g}vs. ϕ_{c}. |

Our approach provides a new conceptual framework for interpreting experiments, is computationally more efficient than explicit polymer models, and may be adapted to model depletion and crowding in mixtures of colloids and excluded-volume polymers,^{35–40,43} represented as self-avoiding random walks^{64} in good solvents. It may be further extended to the protein limit of polymer–nanoparticle mixtures, by incorporating an appropriate penetration free energy.^{57,76} Models of shape-fluctuating particles also may be useful for exploring phase behavior in polymer nanocomposites and in dispersions of soft colloids, e.g., microgels, whose shapes deform at high concentrations.^{77}

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