Colloidal dynamics in polymer solutions: optical two-point microrheology measurements

Advances in the microrheology of complex fluids

New developments in the microrheology of complex fluids are considered. Firstly the requirements for a simple modern particle tracking microrheology experiment are introduced, the error analysis methods associated with it and the mathematical techniques required to calculate the linear viscoelasticity. Progress in microrheology instrumentation is then described with respect to detectors, light sources, colloidal probes, magnetic tweezers, optical tweezers, diffusing wave spectroscopy, optical coherence tomography, fluorescence correlation spectroscopy, elastic- and quasi-elastic scattering techniques, 3D tracking, single molecule methods, modern microscopy methods and microfluidics. New theoretical techniques are also reviewed such as Bayesian analysis, oversampling, inversion techniques, alternative statistical tools for tracks (angular correlations, first passage probabilities, the kurtosis, motor protein step segmentation etc), issues in micro/macro rheological agreement and two particle methodologies. Applications where microrheology has begun to make some impact are also considered including semi-flexible polymers, gels, microorganism biofilms, intracellular methods, high frequency viscoelasticity, comb polymers, active motile fluids, blood clots, colloids, granular materials, polymers, liquid crystals and foods. Two large emergent areas of microrheology, non-linear microrheology and surface microrheology are also discussed.

Active Entanglement-Tracking Microrheology Directly Couples Macromolecular Deformations to Nonlinear Microscale Force Response of Entangled Actin

We track the deformation of discrete entangled actin segments while simultaneously measuring the resistive force the deformed filaments exert in response to an optically driven microsphere. We precisely map the network deformation field to show that local microscale stresses can induce filament deformations that propagate beyond mesoscopic length scales (60 μm, >3 persistence lengths l_p). We show that the filament persistence length controls the critical length scale at which distinct entanglement deformations become driven by collective network mechanics. Mesoscale propagation beyond l_p is coupled with nonlinear local stresses arising from steric entanglements mimicking cross-links.

Response of a polymer network to the motion of a rigid sphere

In view of recent microrheology experiments we re-examine the problem of a rigid sphere oscillating inside a dilute polymer network. The network and its solvent are treated using the two-fluid model. We show that the dynamics of the medium can be decomposed into two independent incompressible flows. The first, dominant at large distances and obeying the Stokes equation, corresponds to the collective flow of the two components as a whole. The other, governing the dynamics over an intermediate range of distances and following the Brinkman equation, describes the flow of the network and solvent relative to one another. The crossover between these two regions occurs at a dynamic length scale which is much larger than the network's mesh size. The analysis focuses on the spatial structure of the medium's response and the role played by the dynamic crossover length. We examine different boundary conditions at the sphere surface. The large-distance collective flow is shown to be independent of boundary conditions and network compressibility, establishing the robustness of two-point microrheology at large separations. The boundary conditions that fit the experimental results for inert spheres in entangled F-actin networks are those of a free network, which does not interact directly with the sphere. Closed-form expressions and scaling relations are derived, allowing for the extraction of material parameters from a combination of one- and two-point microrheology. We discuss a basic deficiency of the two-fluid model and a way to bypass it when analyzing microrheological data.

Extracting the dynamic correlation length of actin networks from microrheology experiments

The mechanical properties of polymer gels based on cytoskeleton proteins (e.g. actin) have been studied extensively due to their significant role in biological cell motility and in maintaining the cell's structural integrity. Microrheology is the natural method of choice for such studies due to its economy in sample volume, its wide frequency range, and its spatial sensitivity. In microrheology, the thermal motion of tracer particles embedded in a complex fluid is used to extract the fluid's viscoelastic properties. Comparing the motion of a single particle to the correlated motion of particle pairs, it is possible to extract viscoelastic properties at different length scales. In a recent study, a crossover between intermediate and bulk response of complex fluids was discovered in microrheology measurements of reconstituted actin networks. This crossover length was related to structural and mechanical properties of the networks, such as their mesh size and dynamic correlation length. Here we capitalize on this result giving a detailed description of our analysis scheme, and demonstrating how this relation can be used to extract the dynamic correlation length of a polymer network. We further study the relation between the dynamic correlation length and the structure of the network, by introducing a new length scale, the average filament length, without altering the network's mesh size. Contrary to the prevailing assumption, that the dynamic correlation length is equivalent to the mesh size of the network, we find that the dynamic correlation length increases once the filament length is reduced below the crossover distance.

Calibration of optical traps by dual trapping of one bead

We introduce a method for optical trap calibration that is suitable for viscoelastic material. The method is designed for use on experimental setups with two optical tweezers and is based on pulling a trapped particle with one trap while holding it with the other. No piezo stage is needed, and only one optical trap must be movable with galvo mirrors, piezo mirrors, or acousto-optical deflectors. The method combines advantages of commonly known PSD-fitting and fast-sweeping methods, allowing calibration of a completely fixed trap in a fluid of unknown viscosity/viscoelasticity. A detailed method description, a theoretical derivation, and an experimental comparison to other methods are reported.

Studying single red blood cells under a tunable external force by combining passive microrheology with Raman spectroscopy

The dynamic micromechanical and structural properties of single human red blood cells are studied using a combination of dual trap optical tweezers and confocal Raman spectroscopy. Such a combination permits us to show a direct relationship between the rheological properties and chemical structure conformation. The frequency dependence of the complex stiffness of the cells was measured using both one and two probe response functions under identical experimental conditions. Both the microrheology and Raman measurements were performed at different stretching forces applied to the cell. A detailed analysis of the auto- and cross-correlated probe motions allows exploring the local and overall viscoelastic properties of the cells over a controlled range of the deformations. The observed growth of the cell viscoelasticity with stretching was associated with structural changes in the cell membrane monitored via the Raman spectroscopy.

Auto- and cross-power spectral analysis of dual trap optical tweezer experiments using Bayesian inference

The thermal fluctuations of micron-sized beads in dual trap optical tweezer experiments contain complete dynamic information about the viscoelastic properties of the embedding medium and-if present-macromolecular constructs connecting the two beads. To quantitatively interpret the spectral properties of the measured signals, a detailed understanding of the instrumental characteristics is required. To this end, we present a theoretical description of the signal processing in a typical dual trap optical tweezer experiment accounting for polarization crosstalk and instrumental noise and discuss the effect of finite statistics. To infer the unknown parameters from experimental data, a maximum likelihood method based on the statistical properties of the stochastic signals is derived. In a first step, the method can be used for calibration purposes: We propose a scheme involving three consecutive measurements (both traps empty, first one occupied and second empty, and vice versa), by which all instrumental and physical parameters of the setup are determined. We test our approach for a simple model system, namely a pair of unconnected, but hydrodynamically interacting spheres. The comparison to theoretical predictions based on instantaneous as well as retarded hydrodynamics emphasizes the importance of hydrodynamic retardation effects due to vorticity diffusion in the fluid. For more complex experimental scenarios, where macromolecular constructs are tethered between the two beads, the same maximum likelihood method in conjunction with dynamic deconvolution theory will in a second step allow one to determine the viscoelastic properties of the tethered element connecting the two beads.

Conical diffraction of linearly polarised light controls the angular position of a microscopic object

Conical diffraction of linearly polarised light in a biaxial crystal produces a beam with a crescent-shaped intensity profile. Rotation of the plane of polarisation produces the unique effect of spatially moving the crescent-shaped beam around a ring. We use this effect to trap microspheres and white blood cells and to position them at any angular position on the ring. Continuous motion around the circle is also demonstrated. This crescent beam does not require an interferometeric arrangement to form it, nor does it carry optical angular momentum. The ability to spatially locate a beam and an associated trapped object simply by varying the polarisation of light suggests that this optical process should find application in the manipulation and actuation of micro- and nano-scale physical and biological objects.

Mechanical properties of cellularly responsive hydrogels and their experimental determination

Hydrogels are increasingly employed as multidimensional cell culture platforms often with a necessity that they respond to or control the cellular environment. Specifically, synthetic hydrogels, such as poly(ethylene glycol) (PEG)-based gels, are frequently utilized for probing the microenvironment's influence on cell function, as the gel properties can be precisely controlled in space and time. Synthetically tunable parameters, such as monomer structure and concentration, facilitate initial gel property control, while incorporation of responsive degradable units enables cell- and/or user-directed degradation. Such responsive gel systems are complex with dynamic changes occurring over multiple time-scales, and cells encapsulated in these synthetic hydrogels often experience and dictate local property changes profoundly different from those in the bulk material. Consequently, advances in bulk and local measurement techniques are needed to monitor property evolution quantatively and understand its effect on cell function. Here, recent progress in cell-responsive PEG hydrogel synthesis and mechanical property characterization is reviewed.

Light at work: the use of optical forces for particle manipulation, sorting, and analysis

We review the combinations of optical micro-manipulation with other techniques and their classical and emerging applications to non-contact optical separation and sorting of micro- and nanoparticle suspensions, compositional and structural analysis of specimens, and quantification of force interactions at the microscopic scale. The review aims at inspiring researchers, especially those working outside the optical micro-manipulation field, to find new and interesting applications of these methods.

Role of slip between a probe particle and a gel in microrheology

In the technique of microrheology, macroscopic rheological parameters as well as information about local structure are deduced from the behavior of microscopic probe particles under thermal or active forcing. Microrheology requires knowledge of the relation between macroscopic parameters and the force felt by a particle in response to displacements. We investigate this response function for a spherical particle using the two-fluid model, in which the gel is represented by a polymer network coupled to a surrounding solvent via a drag force. We obtain an analytic solution for the response function in the limit of small volume fraction of the polymer network, and neglecting inertial effects. We use no-slip boundary conditions for the solvent at the surface of the sphere. The boundary condition for the network at the surface of the sphere is a kinetic friction law, for which the tangential stress of the network is proportional to relative velocity of the network and the sphere. This boundary condition encompasses both no-slip and frictionless boundary conditions as limits. Far from the sphere there is no relative motion between the solvent and network due to the coupling between them. However, the different boundary conditions on the solvent and network tend to produce different far-field motions. We show that the far-field motion and the force on the sphere are controlled by the solvent boundary conditions at high frequency and by the network boundary conditions at low frequency. At low frequencies compression of the network can also affect the force on the sphere. We find the crossover frequencies at which the effects of sliding of the sphere past the polymer network and compression of the gel become important. The effects of sliding alone can lead to an underestimation of moduli by up to 33%, while the effects of compression alone can lead to an underestimation of moduli by up to 20%, and the effects of sliding and compression combined can lead to an underestimation of moduli by up to 43%.

Picoliter viscometry using optically rotated particles

Important aspects in the field of microrheology are studies of the viscosity of fluids within structures with micrometer dimensions and fluid samples where only microliter volumes are available. We have quantitatively investigated the performance and accuracy of a microviscometer based on rotating optical tweezers, which requires as little as one microliter of sample. We have characterized our microviscometer, including effects due to heating, and demonstrated its ability to perform measurements over a large dynamic range of viscosities (at least two orders of magnitude). We have also inserted a probe particle through the membrane of a cell and measured the viscosity of the intramembranous contents. Viscosity measurements of tears have also been made with our microviscometer, which demonstrate its potential use to study unstimulated eye fluid.

Motion of a sphere through a polymer solution

We present analytical solutions of the hydrodynamic resistance force a spherical particle experiences when it moves through a polymer solution containing nonadsorbing chains. Polymer depletion results in a reduced polymer concentration and fluid viscosity near the particle surface. The nonuniform physical properties in the fluid phase affect the transport behavior of a translating and rotating sphere as compared with the case of uniform properties. Based on Stokes' stream function theory, we develop a simplified two-layer approximation by using a step function to represent the viscosity profile. The presence of the polymer solution is formulated in terms of correction functions to the translational and rotational friction of a sphere in a pure solvent. The results are in fair agreement with systematic measurements of the friction of a colloidal sphere when it moves through a polymer solution [Koenderink, Phys. Rev. E 69, 021804 (2004)]. The analysis also predicts an apparent slip length in terms of the viscosity ratio and thickness of the depletion layer.

Investigation of short-time particle dynamics near an interface in the presence of nonadsorbed macro-ions

The optical technique of total internal reflection microscopy was used to study the normal Brownian motion of a single colloidal particle near an interface. The measurements were made using a recently developed technique in which the diffusion coefficient was determined by the variance of the short-time (Deltat --> 0) motion of the particle. Experiments were performed in solutions containing either silica nanospheres or clay platelets (Laponite RD) to investigate the effect of nonadsorbed material on the dynamics of near-contact particle motion. The change in the diffusion coefficient with separation distance between the particle and plate in solutions containing nonadsorbed macro-ions was well-described by the theory developed for simple fluids. These results suggest that, in dilute solutions of nonadsorbed material in which the bulk rheological properties remain similar to those of the pure fluid, the mobility and diffusion coefficient correction factors developed for simple fluids remain valid.

Colloid dynamics in semiflexible polymer solutions

We investigate the dynamics of monodisperse colloidal polystyrene particles suspended in solutions of the semiflexible polymer filamentous actin, over a range of filament lengths that either exceed or are substantially less than the particle radius. The filament length is controlled by the capping protein gelsolin, and particle surface chemistries that minimize the adsorption of filaments are used. The particle dynamics are measured on short time scales using diffusing wave spectroscopy. A sharp transition in the initial particle diffusivity marks the expected shift from a dilute to a tightly entangled polymer network as the filament average length increases. In both the dilute and entangled regimes, the measured particle dynamics are compared with the theories of rodlike and semiflexible polymer solution rheology using the generalized Stokes-Einstein relationship. In the dilute limit, the particle dynamics are in good agreement with theory. However, in the tightly entangled regime, the particle response is consistent with polymer depleted near the surfaces of the particles. The magnitude of the depletion layer thickness depends strongly on particle size and weakly on filament length. This behavior is in agreement with nonlocal entropic repulsions and the loss of conformational entropy associated with rodlike molecules near impenetrable particles. These results illustrate the use of microrheology as a method to investigate local structure and dynamics in colloid-polymer solutions.

Correlated fluctuations of microparticles in viscoelastic solutions: quantitative measurement of material properties by microrheology in the presence of optical traps

The Brownian motions of microscopic particles in viscous or viscoelastic fluids can be used to measure rheological properties. This is the basis of recently developed one- and two-particle microrheology techniques. For increased temporal and spatial resolution, some microrheology techniques employ optical traps, which introduce additional forces on the particles. We have systematically studied the effect that confinement of particles by optical traps has on their auto- and cross-correlated fluctuations. We show that trapping causes anticorrelations in the motion of two particles at low frequencies. We demonstrate how these anticorrelations depend on trap strength and the shear modulus of viscoelastic media. We present a method to account for the effects of optical traps, which permits the quantitative measurement of viscoelastic properties in one- and two-particle microrheology over an extended frequency range in a variety of viscous and viscoelastic media.

Colloid interaction and pair correlation function of one-dimensional colloid-polymer systems

The interaction and pair correlation function of weakly charged colloidal particles in quasi-one-dimensional colloid-polymer systems are determined by enhanced video microscopy and digital image analysis. The pair correlation function is found to depend not only on the polymer concentration, but also on the degree of confinement; in particular, it depends on whether the channel width is such that mutual passage of the colloid particles is possible or not. These findings are compared with exact results on short-range order in linear continuous systems.

Rotational diffusion of colloidal particles near confining walls

We study the rotational diffusion of a spherical colloid confined in a narrow channel between parallel plane hard walls. The walls damp translational diffusion much more than rotational diffusion so that there is expected to be little translation-rotation coupling. Using a recent calculation of the nonisotropic rotational mobilities arising from the hydrodynamic interactions with the walls, we set up the rotational Smoluchowski equation for either a particle with a permanent dipole moment or a polarizable particle with axisymmetric polarizabilities subject to an external electric field. Using the Smoluchowski equation dynamics we calculate the time-correlation functions of orientation that are measured in depolarized light scattering for the cases of no external field, external field normal to the walls, and external field parallel to the walls. The decay of correlations is shown to be given by a weighted sum of decaying exponentials and can be characterized by an initial and a mean characteristic decay time. The weights and decay rates of each component and the characteristic decay times are studied numerically for a range of field strengths. The nonisotropic rotational mobilities make these decay times highly sensitive to the distance of the particle from the confining walls. This position dependence can be used as a method of measuring the rotational mobilities or, conversely, the rate of decay of correlations can be used as a probe of particle position between the confining walls.

Characterization of optically driven fluid stress fields with optical tweezers

We present a controlled stress microviscometer with applications to complex fluids. It generates and measures microscopic fluid velocity fields, based on dual beam optical tweezers. This allows an investigation of bulk viscous properties and local inhomogeneities at the probe particle surface. The accuracy of the method is demonstrated in water. In a complex fluid model (hyaluronic acid), we observe a strong deviation of the flow field from classical behavior. Knowledge of the deviation together with an optical torque measurement is used to determine the bulk viscosity. Furthermore, we model the observed deviation and derive microscopic parameters.

Microrheology of polymeric solutions using x-ray photon correlation spectroscopy

We demonstrate the technique of XPCS microrheology on opaque polymeric solutions (1-20% w/w) using colloidal silica probes. The short time decay of the intensity correlation function provides the mean square displacement (MSD) of the colloidal probes. The MSDs of the probes are subsequently transformed using the generalized Stokes-Einstein equation, allowing the linear viscoelastic spectra of a biopolymer (gellan) and a synthetic polyelectrolyte (polystyrene sulfonate, PSS) to be calculated over two decades of frequency. MSDs can be measured that are two orders of magnitude smaller than those possible with video particle tracking microrheology, with a sensitivity of ∼10 nm s(-1) for displacements of ∼nms. The XPCS data for water, glycerol and PSS combs are in agreement with video particle tracking microrheology experiments performed at lower polymer concentrations.

Microrheology of entangled polymer solutions

Microrheology of semidilute polymer solutions is investigated. In this paper we calculate a response function of a probe particle embedded in a semidilute polymer solution by analyzing the two-fluid model. We find that when the size of the probe particle is comparable to the viscoelastic length, the response from the longitudinal compression modes becomes more important than that of the transverse shear modes. As a result, depending on the circumstances, the obtained complex shear modulus cannot be well approximated by that measured in macroscopic rheology experiments. The present results are due to the dynamical asymmetry coupling and the existence of the cooperative dynamics, which are intrinsic to entangled polymer solutions.

Optical microrheology using rotating laser-trapped particles

We demonstrate an optical system that can apply and accurately measure the torque exerted by the trapping beam on a rotating birefringent probe particle. This allows the viscosity and surface effects within liquid media to be measured quantitatively on a micron-size scale using a trapped rotating spherical probe particle. We use the system to measure the viscosity inside a prototype cellular structure.

Microrheology, stress fluctuations, and active behavior of living cells

We report the first measurements of the intrinsic strain fluctuations of living cells using a recently developed tracer correlation technique along with a theoretical framework for interpreting such data in heterogeneous media with nonthermal driving. The fluctuations' spatial and temporal correlations indicate that the cytoskeleton can be treated as a course-grained continuum with power-law rheology, driven by a spatially random stress tensor field. Combined with recent cell rheology results, our data imply that intracellular stress fluctuations have a nearly 1/omega2 power spectrum, as expected for a continuum with a slowly evolving internal prestress.

Rheological microscopy: local mechanical properties from microrheology

We demonstrate how tracer microrheology methods can be extended to study submicron scale variations in the viscoelastic response of soft materials; in particular, a semidilute solution of lambda-DNA. The polymer concentration is depleted near the surfaces of the tracer particles, within a distance comparable to the polymer correlation length. The rheology of this microscopic layer alters the tracers' motion and can be precisely quantified using one- and two-point microrheology. Interestingly, we found this mechanically distinct layer to be twice as thick as the layer of depleted concentration, likely due to solvent drainage through the locally perturbed polymer structure.