Depletion interactions of non-spherical colloidal particles in polymer solutions

We consider anisotropic colloidal particles immersed in a solution of long, flexible, and nonadsorbing polymers. For the dumbbell shapes of recently synthesized particles consisting of two intersecting spheres and for lens-shaped particles with spherical surfaces we calculate the isotropic and anisotropic interaction parameters that determine the immersion free energy and the orientation-dependent depletion interaction between particles that are induced by the polymers. Exact results are obtained for random-walk like (ideal) polymer chains

Casimir interaction of rod-like particles in a two-dimensional critical system

We consider the fluctuation-induced interaction of two thin, rod-like particles or "needles" immersed in a two-dimensional critical fluid of Ising symmetry right at the critical point. Conformally mapping the plane containing the needles onto a simpler geometry in which the stress tensor is known, we analyze the force and torque between needles of arbitrary length, separation, and orientation. For infinite and semi-infinite needles we utilize the mapping of the plane bounded by the needles onto the half plane, and for two needles of finite length the mapping onto an annulus. For semi-infinite and infinite needles the force is expressed in terms of elementary functions, and we also obtain analytical results for the force and torque between needles of finite length with separation much greater than their length. Evaluating formulas in our approach numerically for several needle geometries and surface universality classes, we study the full crossover from small to large values of the separation to length ratio. In these two limits the numerical results agree with results for infinitely long needles and with predictions of the small-particle operator expansion, respectively.Comment: 68 pages, 9 figure

Polymer depletion effects near mesoscopic particles

The behavior of mesoscopic particles dissolved in a dilute solution of long, flexible, and nonadsorbing polymer chains is studied by field-theoretic methods. For spherical and cylindrical particles the solvation free energy for immersing a single particle in the solution is calculated explicitly. Important features are qualitatively different for self-avoiding polymer chains as compared with ideal chains. The results corroborate the validity of the Helfrich-type curvature expansion for general particle shapes and allow for quantitative experimental tests. For the effective interactions between a small sphere and a wall, between a thin rod and a wall, and between two small spheres quantitative results are presented. A systematic approach for studying effective many-body interactions is provided. The common Asakura-Oosawa approximation modelling the polymer coils as hard spheres turns out to fail completely for small particles and still fails by about 10% for large particles.Comment: 68 pages, 6 figure

Two-dimensional critical systems with mixed boundary conditions: Exact Ising results from conformal invariance and boundary-operator expansions

With conformal-invariance methods, Burkhardt, Guim, and Xue studied the critical Ising model, defined on the upper half plane $y>0$ with different boundary conditions $a$ and $b$ on the negative and positive $x$ axes. For $ab=-+$ and $f+$, they determined the one and two-point averages of the spin $\sigma$ and energy $\epsilon$. Here $+$, $-$, and $f$ stand for spin-up, spin-down, and free-spin boundaries, respectively. The case $+-+-+\dots$, where the boundary conditions switch between $+$ and $-$ at arbitrary points, $\zeta_1$, $\zeta_2$, $\dots$ on the $x$ axis was also analyzed. In this paper the alternating boundary conditions $+f+f+\dots$ and the case $-f+$ of three different boundary conditions are considered. Exact results for the one and two-point averages of $\sigma$, $\epsilon$, and the stress tensor $T$ are derived. Using the results for $\langle T\rangle$, the critical Casimir interaction with the boundary of a wedge-shaped inclusion is analyzed for mixed boundary conditions. The paper also includes a comprehensive discussion of boundary-operator expansions in two-dimensional critical systems with mixed boundary conditions. Two types of expansions - away from switching points of the boundary condition and at switching points - are considered. The asymptotic behavior of two-point averages is expressed in terms of one-point averages with the help of the expansions. We also consider the strip geometry with mixed boundary conditions and derive the distant-wall corrections to one-point averages near one edge due to the other edge using the boundary-operator expansions. The predictions of the boundary-operator expansions are consistent with exact results for Ising systems.Comment: 50 pages, 1 figur

Polymers interacting with spherical and rodlike particles

The interaction of a long flexible polymer chain with mesoscopic particles of spherical or elongated cylindrical shape is investigated by field-theoretic methods using the polymer-magnet analogy. In the case that these particles are immersed in a dilute polymer solution and exhibit purely repulsive surfaces we study density profiles for monomers and chain ends near such a particle, the change of configurational entropy by immersing a particle into the solution, and the depletion interaction between a particle and a distant planar wall. Both ideal chains and chains with an excluded-volume interaction are considered. We also analyze particle surfaces with a short-ranged attraction and the adsorption-desorption transition for an ideal polymer chain. Properties such as the number of surface contacts are evaluated both in the adsorbed limit, in which the thickness of the adsorbed layer is much smaller than the unperturbed polymer size so that ground-state dominance applies, and at the adsorption threshold

Critical Casimir torques and forces acting on needles in two spatial dimensions

We investigate the universal orientation-dependent interactions between non-spherical colloidal particles immersed in a critical solvent by studying the instructive paradigm of a needle embedded in bounded two-dimensional Ising models at bulk criticality. For a needle in an Ising strip the interaction on mesoscopic scales depends on the width of the strip and the length, position, and orientation of the needle. By lattice Monte Carlo simulations we evaluate the free energy difference between needle configurations being parallel and perpendicular to the strip. We concentrate on small but nonetheless mesoscopic needle lengths for which analytic predictions are available for comparison. All combinations of boundary conditions for the needles and boundaries are considered which belong to either the "normal" or the "ordinary" surface universality class, i.e., which induce local order or disorder, respectively. We also derive exact results for needles of arbitrary mesoscopic length, in particular for needles embedded in a half plane and oriented perpendicular to the corresponding boundary as well as for needles embedded at the center line of a symmetric strip with parallel orientation.Comment: 33 pages, 15 figure

Influence of long-range correlated quenched disorder on the adsorption of long flexible polymer chains on a wall

The process of adsorption on a planar wall of long-flexible polymer chains in the medium with quenched long-range correlated disorder is investigated. We focus on the case of correlations between defects or impurities that decay according to the power-low $x^{-a}$ for large distances $x$, where ${\bf x}=({\bf r},z)$. Field theoretical approach in $d=4-\epsilon$ and directly in $d=3$ dimensions up to one-loop order for the semi-infinite $|\phi|^4$ m-vector model (in the limit $m\to 0$) with a planar boundary is used. The whole set of surface critical exponents at the adsorption threshold $T=T_a$, which separates the nonadsorbed region from the adsorbed one is obtained. Moreover, we calculate the crossover critical exponent $\Phi$ and the set of exponents associated with them. We perform calculations in a double $\epsilon=4-d$ and $\delta=4-a$ expansion and also for a fixed dimension $d=3$, up to one-loop order for different values of the correlation parameter $2<a\le 3$. The obtained results indicate that for the systems with long-range correlated quenched disorder the new set of surface critical exponents arises. All the surface critical exponents depend on $a$. Hence, the presence of long-range correlated disorder influences the process of adsorption of long-flexible polymer chains on a wall in a significant way.Comment: 4 figures, 2 table

Random Walk with a Boundary Line as a Free Massive Boson with a Defect Line

We show that the problem of Random Walk with boundary attractive potential may be mapped onto the free massive bosonic Quantum Field Theory with a line of defect. This mapping permits to recover the statistical properties of the Random Walks by using boundary $S$--matrix and Form Factor techniques.Comment: 17 pages, Latex, 3 figures include

Critical Casimir Forces between Spherical Particles in Fluids

Long-ranged correlations in a fluid close to its critical point Tc cause distinct forces between immersed colloidal particles and the container walls. We calculate such a force and its temperature dependence for the generic case of a spherical particle located at a distance D from a planar wall and find that the force attains a maximum at a temperature Tmax(D) above Tc, which facilitates quantitative experimental tests. The corresponding effective pair interaction between the colloidal particles themselves, potentially leading to aggregation, is also discussed

Exact results for the adsorption of a semiflexible copolymer chain in three dimensions

Lattice model of directed self avoiding walk has been solved analytically to investigate adsorption desorption phase transition behaviour of a semiflexible sequential copolymer chain on a two dimensional impenetrable surface perpendicular to the preferred direction of the walk of the copolymer chain in three dimensions. The stiffness of the chain has been accounted by introducing an energy barrier for each bend in the walk of the copolymer chain. Exact value of adsorption desorption transition points have been determined using generating function method for the cases in which one type of monomer is having interaction with the surface viz., (i) no interaction (ii) attractive interaction and (iii) repulsive interaction. Results obtained in each of the case show that for stiffer copolymer chain adsorption transition occurs at a smaller value of monomer surface attraction than a flexible copolymer chain. These features are similar to that of a semi-flexible homopolymer chain adsorption.Comment: 8 pages with one figur