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

Adsorption and collapse transitions of a linear polymer chain interacting with a surface adsorbed polymer chain

We study the problem of adsorption and collapse transition of a linear polymer chain situated in a fractal container represented by a 4-simplex lattice and interacting with a surface adsorbed linear polymer chain. The adsorbed chain monomers act as pinning sites for the polymer chain. This problem has been solved exactly using real space renormalization group transformation. The resulting phase diagram and critical exponents are given.Comment: 19 pages including 6 figures; LaTeX; to appear in Physica

Conformal Field Theories Near a Boundary in General Dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions $d$. Calculations of the universal function of a conformal invariant $\xi$ which appears in the two point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the \vep=4-d expansion for the the operator $\phi^2$ in $\phi^4$ theory. The form for the associated functions of $\xi$ for the two point functions for the basic field $\phi^\alpha$ and the auxiliary field $\lambda$ in the the $N\to \infty$ limit of the $O(N)$ non linear sigma model for any $d$ in the range $2<d<4$ are also rederived. These results are obtained by integrating the two point functions over planes parallel to the boundary, defining a restricted two point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two point function. Consistency of the results is checked by considering the limit $d\to 4$ and also by analysis of the operator product expansions for $\phi^\alpha\phi^\beta$ and $\lambda\lambda$. Using this method the form of the two point function for the energy momentum tensor in the conformal $O(N)$ model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two point functions are also derived making essential use of conformal invariance.Comment: Plain TeX file, 52 pages, with 1 postscript figur

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

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

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

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

Energy Momentum Tensor in Conformal Field Theories Near a Boundary

The requirements of conformal invariance for the two point function of the energy momentum tensor in the neighbourhood of a plane boundary are investigated, restricting the conformal group to those transformations leaving the boundary invariant. It is shown that the general solution may contain an arbitrary function of a single conformally invariant variable $v$, except in dimension 2. The functional dependence on $v$ is determined for free scalar and fermion fields in arbitrary dimension $d$ and also to leading order in the \vep expansion about $d=4$ for the non Gaussian fixed point in $\phi^4$ theory. The two point correlation function of the energy momentum tensor and a scalar field is also shown to have a unique expression in terms of $v$ and the overall coefficient is determined by the operator product expansion. The energy momentum tensor on a general curved manifold is further discussed by considering variations of the metric. In the presence of a boundary this procedure naturally defines extra boundary operators. By considering diffeomorphisms these are related to components of the energy momentum tensor on the boundary. The implications of Weyl invariance in this framework are also derived.Comment: 22 pages, TeX with epsf.tex, DAMTP/93-1. (original uuencoded file was corrupted enroute - resubmitted version has uuencoded figures pasted to the ended of the Plain TeX file