After acquiring a fundamental understanding of the equilibrium and dynamic behavior of semi-flexible polymers in Chaps. The thermodynamic contribution of the conformational entropy dramatically changes the equilibrium behavior of semi-flexible polymer solutions with respect to their perfectly stiff analogue.
Introduction
Perhaps less obvious, finding the distribution function of the end-to-end distance of a polymer involves solving the partition function of a chain in an imaginary dipole field. 2.4, we use our result for the distribution function of the end-to-end distance to study the structure factor of a wormlike chain.
Two-Dimensional Solutions
Of particular interest is the first moment, which gives the force-elongation behavior of the chain. The solutions can be checked by finding the diagrams representing these moments, a process that further extends the usefulness of the diagrammatic representation.
Three-Dimensional Solutions
2.4, with a slight modification of the subsequent steps to account for the statistical behavior in three dimensions. With the expression R2n0 = (2n+ 1)!R2nz 0/(2n)!, our solution provides a convenient method to analytically generate arbitrary moments of the end-to-end distance.
Structure Factor of a Wormlike Chain
Since a wormlike chain is rigid at small enough length scales, the structure factor for the wormlike chain model approaches the rigid-rod limit kLS(k) → π as k → ∞ regardless of chain stiffness. On the rigid side of the spectrum (shown in Fig. 2.2), the structure factor of a worm-shaped chain becomes essentially indistinguishable from that of the perfectly rigid rod when N < 0.02 or when the length.
Polymer Nematics
The large scaling Λ can be understood by considering the conformational behavior of the chain in the quadrupole field. The Helmholtz free energy of the nematic phase relative to the isotropic phase is given by [9].
Conclusions
Analytical calculation of the scattering function for polymers of arbitrary flexibility using the Dirac propagator. 27] For more rigid chains, an accurate calculation of the structure factor requires higher levels of trimming in the continuous fraction.
Introduction
We develop a formalism to describe the kinematics of a worm-like chain confined to the surface of a sphere which simultaneously satisfies the spherical confinement and the inextensibility of the chain contour. We use this formalism to study the statistical behavior of the worm-like chain on a spherical surface.
Spherical Kinematics
Our mathematical description of chain kinematics is more clearly understood by considering the chain curvature vector. A physically intuitive description of the conformation is to consider the triad system as a spinning top.
Mean Square End-to-End Distance
A qualitatively different behavior occurs when the radius of the sphere is smaller than the persistence length. The behavior of the mean square end-to-end distance suggests the nature of the surface coverage of the polymer as the length of the polymer increases. Physically, strict confinement of the chain to a sphere surface corresponds to infinitely strong adsorption.
Introduction
Warner and co-workers studied both the phase behavior and chain conformational properties in solvent-free systems based on the exact solution of the mean-field equations using spheroidal functions [ 20 , 14 ]. 4.4, we present the bulk (zero wavenumber) fluctuation free energy in quadratic order around the mean-field solution in connection with the calculation of the limit of stability (spinodal) of the different phases. We then investigate a number of single-chain properties in the nematic state, with an emphasis on the anisotropy of the chain conformation and various energetic contributions.
Self-Consistent-Field Theory
The position and conformation of the ith polymer is described by the spatial curve ~ri(τ), where the path coordinate τ runs from zero to L. The first term in the Hamiltonian (equation 4.2) is the bending energy np of the polymer chains, which is assumed to be quadratic with respect to the curvature of polymer conformations. At the mean-field level, the above set of equations fully describes the thermodynamics and, via a single-chain distribution function (Eq. 4.9), the conformational properties of a liquid-crystalline polymer solution modeled by a Hamiltonian (Eq. 4.2). ).
Chain Statistics
Spheroidal functions asymptotically approach spherical harmonics in the limit γ → 0, i.e. in this limit Elm approaches l(l+ 1). Since the rigid rod limit represents the limit where all eigenfunctions are needed in the Green's function to correctly capture the chain statistics, the opposite limit occurs when only a few eigenfunctions need to be included in the Green's function, which we define as the ground state dominance limit [20, 44 ]. In the limit of flexible chains (/L → 0), the spheroid functions become spherical harmonics; therefore, the use of en.
Gaussian Fluctuations
The quadratic fluctuations in the auxiliary fields ∆Ws, ∆Wp and ∆Λ can be easily integrated to obtain the resulting energy without oscillation in terms of oscillation in the physical fields: the density ∆φp and the order parameter ∆S . For the nematic case, we see that the two modes are coupled in energy without oscillation; therefore, instability occurs due to a mixture of concentration and order parameter fluctuations characterized by the annihilation of the determinant Cij (i,j =φp,S). However, the free energy cost of oscillations around the saddle can only be properly calculated with free oscillations.
Results and Discussion
At total volume fractions above φp= 0.6404, the isotropic branch with the polymer-rich phase crosses the nematic branch at the point indicated by the circle. IV, the nematic state instability results from a combination of concentration and order parameter fluctuations. This scaling reflects the importance of small amplitude fluctuations in transverse directions for single-chain thermodynamics in this limit.
Summary and Conclusions
Analysis of the different spinodal curves in the phase diagram suggests rich and complex phase transition kinetics. We chose to calculate the spheroidal functions by expanding them in terms of the better known spherical harmonics [53]. In addition to the evaluation of the equilibrium potentials, knowledge of the chain statistics allows us to evaluate averages within the nematic phase [20].
Introduction
The spiral domain boundaries are eliminated from the relaxing structure by unwinding through the ends of the rod. The caliber of the nanometer-scale tube can be experimentally controlled in the range of 20 to 200 nanometers. An interesting consequence of the growth of the spiral domains is the phenomenon of self-propulsion [4] of the expanding body during the expansion process.
Elastic Rod Model
We will ignore the change in Young's modulus due to the expansion of the rod. The bending energy is due to curvature deformation of the rod with respect to its equilibrium value, which in our case is considered zero. IV, we further discuss the conditions under which thermal fluctuations contribute significantly to the expansion dynamics and justify the neglect of the thermal fluctuations in our study.
Expansion Dynamics
5.2 we show the evolution of the two energy contributions during the extension of the chain. 3], we define the slope-slope correlation function Kss to calculate the characteristic wavelengthλ of the breaking waves. 5.4 we show the time evolution of the breaking wavelength λ and the helical correlation length λH.
Linear Stability Analysis and Dynamic Scaling
This is clearly shown as the end of the plateau in the compression energy in figure. Therefore, we plot the bending energy as a function of ([∆ry −∆rz]/λ0); the result is shown in figure. The contribution of the compression energy is relatively unaffected by the presence of transverse buckles;
Conclusions
A recent theoretical analysis of DNA condensation [22] suggests that the chain collapse occurs via an Euler buckling instability, whereby the total energy is partitioned into bending and electrostatic energy. We are interested in analyzing the effect of helicity on the chain collapse dynamics and determining its role in the morphology of the condensed DNA strand. Numerical solution of the equations of motion for a twisted chain shows loop formation and subsequent plectone growth, localized near the chain ends.
Introduction
The study of the dynamics of twisted elastic yarns is rooted in the theory of elasticity [1, 2], which focuses on strain energy and mechanical stability conditions. 6.2, we find the conditions governing the stability of the twisted chain by performing a linear stability analysis. 6.3, we analyze the post-bending behavior of the twisted chain, focusing on the shape and energetics of a localized ring structure that predicts the formation of a plectonemic supercoil.
Linear Stability Analysis
The equations of motion for the dynamic variables ~r and ω3 are found, after some manipulations of the force and moment balance equations. As the spin increases, the wavelength of the initial instability is shortened, and the peaks become less significant as B becomes large. In the next section, we look at the post-buckling behavior of the twisted string, focusing on the formation of loops that lead to plectonomic supercoils.
Post-Buckling Behavior
The initiation of the transition occurs through spiral formation, similar to the structures analyzed in the previous section. This calculation uses the relationship between the pull-in distance ∆ and the tension T found in Ref. The total energy of the looped chain is given by the integral of the energy density found from Eqs.
Nonlinear Dynamics
Further growth of the instability shows the localization of the helical undulations in loops near the ends of the chain. We notice the plectonemic trace from the ends due to the viscous drag as the chain ends are pulled towards each other. If the ends are clamped between two bars, the chain will loop around the bars.
Introduction
Details of the packing motor of φ29 indicate the potential for rotation to play a role in the packing process. If the DNA rotates as it enters the capsid, then either the DNA relaxes the twisting deformation by axial rotation of the strand, or the conformation of the strand responds in some way to the twisting deformation. Provided that the packaging motor of a bacteriophage rotates the DNA as it enters the capsid, the conformation of the DNA within the capsid interior will be affected by the torsional deformation.
Results
The first simulation involves simultaneous rotation of the chain segment at the entry gantry as the chain is fed into the spherical confinement; thus the linking number increases linearly with the number of beads fed. The second simulation is performed without chain rotation at entry; so the connection number remains zero. It is therefore quite surprising that the conformational energy resulting from spin-packing is lower than the spin-free energy; this is due to the reduced bending energy of the coiled conformation in the former compared to the folded conformation in the latter.
Discussion
This is due to the uncertainty in the drag coefficient for DNA in a crowded, highly charged environment. Since we are interested in the behavior of the overall DNA conformation in the capsid, The topology of the chain is characterized by the bond number Lk, the twist T w and the twist W r of the curve [25].
