We consider a wormlike polymer confined between two flat surfaces separated by a distance W. Using a wormlike chain formalism that couples the orientational and positional degrees of freedom, for a wormlike chain much longer than the persistence length 𝓁p, we calculate the free energy per segment as a function of W and compare it with the two scaling behaviors valid in the large and small W/𝓁p limits.

How do we search for the entire family tree without unwanted random guesses, starting from a high-index and high-energy stationary state on the energy landscape? Here we introduce a general numerical method that constructs the pathway map clearly connecting all stationary points branched from a single parent state. The map guides our understanding of how a physical system moves on the energy landscape. In particular, the method allows us to identify the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landaude Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

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The partial differential equation that yields the free energy and conformational properties of a long wormlike polymer in confinement is discussed and analyzed. In the strong confinement limit, the confinement free energy and polymer extension display the Odijk power laws; numerical solutions of the differential equations are obtained and analyzed to produce these power-law coefficients for tubes with square and circular cross sections; the result verifies recent determination of the coefficients, one of them by the same method and the others by the Monte Carlo simulation method. In the weak confinement limit, the free energy displays the typical De Gennes power law, which was obtained by examining the Gaussian polymer model; numerical solutions are obtained here for the free energy and segmental orientational properties in the crossover region from strong-to weak-confinement limits, for a long wormlike polymer confined in a tube with a circular cross section. Article pubs.acs.org/Macromolecules

We study the system of cylindrical colloids adhering to an originally flat fluid membrane, on the basis of a full treatment of the Helfrich model. Our approach allows for numerical calculation of the free energy in both shallow- and deep-wrapping conformations. We show that the free energy of two cylinders adhering to the same side of a membrane has two branches corresponding to shallow and deep wrapping and that the system of two cylinders adhering to opposite sides of a membrane can undergo a first-order phase transition between two membrane-mediated attractive states.