Ideal Polymer Chain Research Articles

Conformation of a tethered polymer in a leaky nanocavity

The conformation of an ideal polymer chain confined in a box with a D x D hole is studied by N-step random walk simulations. For a polymer chain tethered inside the box, three conformational states are observed when the chain length is increased. For short chains the polymer assumes "mushroom" configurations. Increasing the chain length leads to a state in which the polymer fills the box. When the chain is long enough, the polymer escapes from the box, assuring an overall random walk state. Probability distributions of the free end and end-to-end distance are used to characterize polymer conformations. The escape transition is determined by the disappearance of the bimodal feature in the end-to-end probability density function. A crossover of the free energy difference between the confined and escaped states signals an escape transition analogous to the coil-globule transition of a free ideal chain.

Analytical theory of finite-size effects in mechanical desorption of a polymer chain

We discuss a unique system that allows exact analytical investigation of first- and second-order transitions with finite-size effects: mechanical desorption of an ideal lattice polymer chain grafted with one end to a solid substrate with a pulling force applied to the other end. We exploit the analogy with a continuum model and use accurate mapping between the parameters in continuum and lattice descriptions, which leads to a fully analytical partition function as a function of chain length, temperature (or adsorption strength), and pulling force. The adsorption-desorption phase diagram, which gives the critical force as a function of temperature, is nonmonotonic and gives rise to re-entrance. We analyze the chain length dependence of several chain properties (bound fraction, chain extension, and heat capacity) for different cross sections of the phase diagram. Close to the transition a single parameter (the product of the chain length N and the deviation from the transition point) describes all thermodynamic properties. We discuss finite-size effects at the second-order transition (adsorption without force) and at the first-order transition (mechanical desorption). The first-order transition has some unusual features: The heat capacity in the transition region increases anomalously with temperature as a power law, metastable states are completely absent, and instead of a bimodal distribution there is a flat region that becomes more pronounced with increasing chain length. The reason for this anomaly is the absence of an excess surface energy for the boundary between adsorbed and stretched coexisting phases (this boundary is one segment only): The two states strongly fluctuate in the transition point. The relation between mechanical desorption and mechanical unzipping of DNA is discussed.

Depletion of ideal polymer chains near a spherical colloid particle beyond the Dirichlet boundary conditions

We reconsider the depletion interaction of an ideal polymer chain, characterized by the gyration radius R(G) and bond length a , and an impenetrable spherical colloid particle of radius R . Forbidding the polymer-colloid penetration explicitly (by the use of Mayer functions) without any other requirement we derive and solve analytically an integral equation for the chain partition function of a long ideal polymer chain for the spherical geometry. We find that the correction to the solution of the Dirichlet problem depends on the ratios R/R (G) and R/a . The correction vanishes for the continuous chain model (i.e. in the limit R/R (G) --> 0 and R/a --> infinity but stays finite (even for an infinite chain) for the discrete chain model. The correction can become substantial in the case of nano-colloids (the so-called protein limit).

Finite- N effects for ideal polymer chains near a flat impenetrable wall

This paper addresses the statistical mechanics of ideal polymer chains next to a hard wall. The principal quantity of interest, from which all monomer densities can be calculated, is the partition function, G(N)(z) , for a chain of N discrete monomers with one end fixed a distance z from the wall. It is well accepted that in the limit of infinite N , G(N)(z) satisfies the diffusion equation with the Dirichlet boundary condition, G(N)(0) = 0, unless the wall possesses a sufficient attraction, in which case the Robin boundary condition, G(N)(0) = - xi G(N)(')(0), applies with a positive coefficient, xi. Here we investigate the leading N(-1/2) correction, Delta G (N)(z). Prior to the adsorption threshold, Delta G(N)(z) is found to involve two distinct parts: a Gaussian correction (for z approximately <or= aN(1/2) with a model-dependent amplitude, A, and a proximal-layer correction (for z approximately <or= a described by a model-dependent function, B(z).

Excitonic versus electronic couplings in molecular assemblies: The importance of non-nearest neighbor interactions

We demonstrate that for a range of phenylene- and thiophene-based conjugated polymers of practical relevance for optoelectronic applications, exciton couplings in one-dimensional stacks deviate significantly from the nearest neighbor approximation. Instead, long-range interactions with non-nearest neighbors have to be included, which become increasingly important with growing oligomer size. While the exciton coupling vanishes for infinitely long ideal polymer chains and provides a sensitive measure of the actual conjugation length, the electronic coupling mediating charge transport shows rapid convergence with molecular size. Similar results have been obtained for very different molecular backbones, thus highlighting the general character of these findings.

The ideal polymer chain near planar hard wall beyond the Dirichlet boundary conditions

We present a new ab initio approach to describe the statistical behavior of long ideal polymer chains near a plane hard wall. Forbidding the solid half-space to the polymer explicitly (by the use of Mayer functions) without any other requirement, we derive and solve an exact integral equation for the partition function G (D)(r,r', N) of the ideal chain consisting of N bonds with the ends fixed at the points r and r'. The expression for G(r,r', s) is found to be the sum of the commonly accepted Dirichlet result G (D)(r,r', N) = G (0)(r,r', N) - G (0)(r,r'', N) , where r'' is the mirror image of r', and a correction. Even though the correction is small for long chains, it provides a non-zero value of the monomer density at the very wall for finite chains, which is consistent with the pressure balance through the depletion layer (so-called wall or contact theorem). A significant correction to the density profile (of magnitude 1/[Formula: see text]is obtained away from the wall within one coil radius. Implications of the presented approach for other polymer-colloid problems are discussed.

Conformational Entropy of an Ideal Cross-Linking Polymer Chain

We present a novel analytical method to calculate conformational entropy of ideal cross-linking polymers from the configuration integral by employing a Mayer series expansion. Mayer-functions describing chemical bonds within the chain and for cross-links are sharply peaked over the temperature range of interest, and, are well approximated as statistically weighted Dirac delta-functions that enforce distance constraints. All geometrical deformations consistent with a set of distance constraints are integrated over. Exact results for a contiguous series of connected loops are employed to substantiate the validity of a previous phenomenological distance constraint model that describes protein thermodynamics successfully based on network rigidity.

Kinetics of Loop Formation in Polymer Chains

We investigate the kinetics of loop formation in ideal flexible polymer chains (the Rouse model), and polymers in good and poor solvents. We show for the Rouse model, using a modification of the theory of Szabo, Schulten, and Schulten, that the time scale for cyclization is tau(c) approximately tau(0)N(2) (where tau(0) is a microscopic time scale and N is the number of monomers), provided the coupling between the relaxation dynamics of the end-to-end vector and the looping dynamics is taken into account. The resulting analytic expression fits the simulation results accurately when a, the capture radius for contact formation, exceeds b, the average distance between two connected beads. Simulations also show that when a < b, tau(c) approximately N(alpha)(tau), where 1.5 < alpha(tau) < or = 2 in the range 7 < N < 200 used in the simulations. By using a diffusion coefficient that is dependent on the length scales a and b (with a < b), which captures the two-stage mechanism by which looping occurs when a < b, we obtain an analytic expression for tauc that fits the simulation results well. The kinetics of contact formation between the ends of the chain are profoundly effected when interactions between monomers are taken into account. Remarkably, for N < 100, the values of tau(c) decrease by more than 2 orders of magnitude when the solvent quality changes from good to poor. Fits of the simulation data for tau(c) to a power law in N (tau(c) approximately N(alpha)(tau)) show that alpha(tau) varies from about 2.4 in a good solvent to about 1.0 in poor solvents. The effective exponent alpha(tau) decreases as the strength of the attractive monomer-monomer interactions increases. Loop formation in poor solvents, in which the polymer adopts dense, compact globular conformations, occurs by a reptation-like mechanism of the ends of the chain. The time for contact formation between beads that are interior to the chain in good solvents changes nonmonotonically as the loop length varies. In contrast, the variation in interior loop closure time is monotonic in poor solvents. The implications of our results for contact formation in polypeptide chains, RNA, and single-stranded DNA are briefly outlined.

Phase behavior of a suspension of hard spherocylinders plus ideal polymer chains

We study isotropic-isotropic and isotropic-nematic phase transitions of fluid mixtures containing hard spherocylinders (HSC) and added non-adsorbing ideal polymer chains using scaled particle theory (SPT). First, we investigate isotropic-nematic (I -N phase coexistence using SPT in the absence of polymer. We compare the results obtained using a Gaussian form of the orientational distribution function (ODF) to minimize the free energy versus minimizing numerically. We find that formal numerical minimization gives results that are much closer to computer simulation results. In order to describe mixtures of HSC plus ideal chains we studied the depletion of ideal chains around a HSC. We analyze the density profiles of ideal chains near a hard cylinder and find the depletion thickness delta is a function of the ratio of the polymer's radius of gyration R(g) and the cylinder radius R(c). Our results are compared with a common approximation in which the depletion thickness is taken equal to the radius of gyration of the polymer chain. We incorporate the correct depletion thickness into SPT and find that for R (g)/R (c) < 1.56 using ideal chains gives phase transitions at smaller polymer concentrations, whereas for R (g)/R (c) > 1.56 , which is a common experimental situation, the phase transitions are found at larger polymer concentrations with respect to delta = R (g) . The differences are significant, especially for R (g) >> R (c) , so we can conclude it is essential to take into account the properties of ideal polymer chains and the resulting depletion near a cylinder. Finally, we present phase diagrams for rod-polymer mixtures which could be realized under experimental conditions.

Depletion Interaction Mediated by a Polydisperse Polymer Studied with Total Internal Reflection Microscopy

Total internal reflection microscopy (TIRM) was applied to measure depletion forces between a charged colloidal sphere and a charged solid wall induced by dextran, a nonionic nonadsorbing polydisperse polysaccharide. The polymer size polydispersity is shown to greatly influence the depletion potential. Using the theory for the depletion interaction due to ideal polydisperse polymer chains, we could accurately describe the experimental data with a single adjustable parameter.

Phase behaviour of charged colloidal sphere dispersions with added polymer chains

We study the stability of mixtures of highly screened repulsive charged spheres andnon-adsorbing ideal polymer chains in a common solvent using free volume theory. Theeffective interaction between charged colloids in an aqueous salt solution is described by ascreened Coulomb pair potential, which supplements the pure hard-sphere interaction. Theideal polymer chains are treated as spheres that are excluded from the colloids by ahard-core interaction, whereas the interaction between two ideal chains is set tozero. In addition, we investigate the phase behaviour of charged colloid–polymermixtures in computer simulations, using the two-body (Asakura–Oosawa pairpotential) approximation to the effective one-component Hamiltonian of the chargedcolloids. Both our results obtained from simulations and from free volume theoryshow similar trends. We find that the screened Coulomb repulsion counteractsthe effect of the effective polymer-mediated attraction. For mixtures of smallpolymers and relatively large charged colloidal spheres, the fluid–crystal transitionshifts to significantly larger polymer concentrations with increasing range of thescreened Coulomb repulsion. For relatively large polymers, the effect of the screenedCoulomb repulsion is weaker. The resulting fluid–fluid binodal is only slightly shiftedtowards larger polymer concentrations upon increasing the range of the screenedCoulomb repulsion. In conclusion, our results show that the miscibility of dispersionscontaining charged colloids and neutral non-adsorbing polymers increases uponincreasing the range of the screened Coulomb repulsion, or upon lowering the saltconcentration, especially when the polymers are small compared to the colloids.

Mechanical stability of end-linked polymer gel

The mechanical stability of end-linked polymer network (gel) at large deformation has been studied with the aid of a simplified model. The model consists of square network of end-linked Hookean spring, each representing an ideal polymer chain. To mimic the structure prevailing at the gel point, the square network of springs was self-similarly repeated to generate a deterministic fractal structure having fractal dimensions, d f = 1.56 – 1.8 . For any given d f, the distribution of pores (measured by lacunarity of the network) in the structure also serves as an important parameter that characterizes the morphology of the network. Our preliminary results show that the present minimalist model qualitatively reproduces the non-monotonous structure–stability relationship, as observed in radiation-induced cross-linked polymer networks. Structures with highest lacunarity for a given fractal network have also been found to exhibit highest mechanical stability against an external deforming force before it is finally crumpled and found to form a wrinkled network.

Depletion interactions of non-spherical colloidal particles in polymer solutions

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

Shrunk loop theorem for the topology probabilities of closed Brownian (or Feynman) paths on the twice punctured plane

The shrunk loop theorem presented here is an integral identity which facilitates the calculation of the relative probability (or probability amplitude) of any given topology that a free, closed Brownian or Feynman path of a given 'duration' might have on the twice punctured plane (the plane with two marked points). The result is expressed as a scattering series of integrals of increasing dimensionality based on the maximally shrunk version of the path. Physically, this applies in different contexts: (i) the topology probability of a closed ideal polymer chain on a plane with two impassable points, (ii) the trace of the Schroedinger Green function, and thence spectral information, in the presence of two Aharonov-Bohm fluxes, (iii) the same with two branch points of a Riemann surface instead of fluxes. Our theorem starts with the Stovicek expansion for the Green function in the presence of two Aharonov-Bohm flux lines, which itself is based on the famous Sommerfeld one puncture point solution of 1896 (the one puncture case has much easier topology, just one winding number). Stovicek's expansion itself can supply the results at the expense of choosing a base point on the loop and then integrating it away. The shrunk loop theorem eliminates this extra two dimensional integration, distilling the topology from the geometry.

Polymer networks between two parallel planar surfaces

The formation and mechanical properties of a polymer network on and between two flat parallel surfaces are investigated. Most treatments of surface-attached polymers have been limited to scaling theory. In the present investigation we probe the physics of the system by means of a mathematical description of the random crosslinking of ideal (or phantom) chains. We modify an existing bulk model of network formation by Deam and Edwards, with polymer–polymer crosslinks, to include surfaces and polymer–surface crosslinks. We investigate two variations of this model: in the first place, the polymer–surface links are formed anywhere along the contours of the long, ideal polymer chains. In the second brush network model, the surface links are restricted to one endpoint of each macromolecule. Within the framework of replica theory, we compute statistical averages and the elastic properties of the systems such as the stress–strain relationship. In both cases the elastic modulus of the bulk network is altered, and has a characteristic form due to the confinement. Furthermore, we find that the stress–strain relationship depends on the manner of crosslinking.

Exactly solvable model with stable and metastable states for a polymer chain near an adsorbing surface.

We report on the conformational properties and transitions of an ideal polymer chain near a solid surface. The chain is tethered with one of its ends at distance z(0) from an adsorbing surface. The surface is characterized by an adsorption parameter c. The exact expression for the partition function is available. We obtained the distribution of complex zeros of this function. The comparison with the Yang-Lee theory allows the characterization of the phase transitions. A first-order conformational transition from a coil to a (adsorbed) flower conformation occurs at c(*)=6z(0)/N. The flower is composed of a strongly stretched stem and a pancake that collects the remaining adsorbed segments. The degree of stretching of the coil or of the stem serves as an order parameter which parametrizes the analytical expressions of the Landau free energy. The phase diagram with one binodal and two spinodal lines is presented. The height of the barriers between metastable and stable states is obtained and the lifetime of metastable states is estimated. A two-state ansatz is used to develop scaling arguments to account for the effects of excluded volume.

Theory of liquid chromatography of mono- and difunctional macromolecules: I. Studies in the critical interaction mode

The theory of liquid chromatography of mono- and difunctional polymers based on the model of ideal polymer chain in wide slit-like pores is presented. Analytical equations describing chromatographic behavior of functional macromolecules in both adsorption, exclusion and critical modes are derived and compared with experiments. The focus of this experimental study was on the verification of the theory in chromatography at critical conditions. Chromatographic behavior of low molar mass end-functionalized polyethylene glycols was found to be in a very good qualitative and in a reasonable quantitative agreement with the theory.

Center-of-mass distribution of a polymer near a repulsive wall

An exact analytic expression is obtained for the center-of-mass density profile of free ideal polymer chains in a half-space bounded by a hard planar wall. The result differs qualitatively from the monomer density profile and the density profiles of chain ends or midpoints.

Luminescence kinetics of linear polymer molecules with chromophores randomly distributed along the chain

Luminescence kinetics of chromophores randomly attached to an isolated flexible polymer chain is studied. In the static case, when the polymer chain does not change its conformation during the donor excited-state lifetime, analytical equations are derived for the luminescence kinetics of both ideal and nonideal polymer chains. The conditions are formulated when the inhomogeneous broadening should be taken into account. For the dynamic case, the diffusion and hopping mechanisms of the polymer motion are studied.

Mapping of continuum and lattice models for describing the adsorption of an ideal chain anchored to a planar surface

An ideal polymer chain anchored to a planar surface is considered by using both lattice and continuum model approaches. A general equation relating the lattice and continuum model adsorption interaction parameters is derived in a consistent way by substituting the exact continuum solution for the free chain end distribution function into the lattice model boundary condition. This equation is not mathematically exact but provides excellent results. With the use of this relation the quantitative equivalence between lattice and continuum results was demonstrated for chains of both infinite and finite length and for all three regimes corresponding to attractive, repulsive and adsorption-threshold energy of polymer-surface interaction. The obtained equations are used to discuss the distribution functions describing the tail of an anchored macromolecule and its adsorbed parts. For the tail-related properties the results are independent of the microscopic details of the polymer chain and the adsorbing surface. One interesting result obtained in the vicinity of adsorption threshold point is a bimodal tail length distribution function, which manifests chain populations with either tail or loop dominance. The properties related to the number of surface contacts contain, apart from universal scaling terms, also a nonuniversal factor depending on microscopic details of polymer-surface interaction. We derived an equation for calculating this nonuniversal factor for different lattice models and demonstrated excellent agreement between the lattice results and the continuum model.