A note on the sign (unit root) ambiguities of Gauss sums in index 2 and 4 cases

Recently, the explicit evaluation of Gauss sums in the index 2 and 4 cases have been given in several papers (see [2,3,7,8]). In the course of evaluation, the sigh (or unit root) ambiguities are unavoidably occurred. This paper presents another method, different from [7] and [8], to determine the sigh (unit root) ambiguities of Gauss sums in the index 2 case, as well as the ones with odd order in the non-cyclic index 4 case. And we note that the method in this paper are more succinct and effective than [8] and [7]

Stability of Colloidal Quasicrystals

Freezing of charge-stabilized colloidal suspensions and relative stabilities of crystals and quasicrystals are studied using thermodynamic perturbation theory. Macroion interactions are modelled by effective pair potentials combining electrostatic repulsion with polymer-depletion or van der Waals attraction. Comparing free energies -- counterion terms included -- for elementary crystals and rational approximants to icosahedral quasicrystals, parameters are identified for which one-component quasicrystals are stabilized by a compromise between packing entropy and cohesive energy.Comment: 6 pages, 4 figure

Can Polymer Coils be modeled as "Soft Colloids"?

We map dilute or semi-dilute solutions of non-intersecting polymer chains onto a fluid of ``soft'' particles interacting via a concentration dependent effective pair potential, by inverting the pair distribution function of the centers of mass of the initial polymer chains. A similar inversion is used to derive an effective wall-polymer potential; these potentials are combined to successfully reproduce the calculated exact depletion interaction induced by non-intersecting polymers between two walls. The mapping opens up the possibility of large-scale simulations of polymer solutions in complex geometries.Comment: 4 pages, 3 figures ReVTeX[epsfig,multicol,amssymb] references update

Insights into the Kinetics of Supramolecular Comonomer Incorporation in Water

Multicomponent supramolecular polymers are a versatile platform to prepare functional architectures, but a few studies have been devoted to investigate their noncovalent synthesis. Here, we study supramolecular copolymerizations by examining the mechanism and time scales associated with the incorporation of new monomers in benzene-1,3,5-tricarboxamide (BTA)-based supramolecular polymers. The BTA molecules in this study all contain three tetra(ethylene glycol) chains at the periphery for water solubility but differ in their alkyl chains that feature either 10, 12 or 13 methylene units. C(10)BTA does not form ordered supramolecular assemblies, whereas C(12)BTA and C(13)BTA both form high aspect ratio supramolecular polymers. First, we illustrate that C(10)BTA can mix into the supramolecular polymers based on either C(12)BTA or C(13)BTA by comparing the temperature response of the equilibrated mixtures to the temperature response of the individual components in water. Subsequently, we mix C(10)BTA with the polymers and follow the copolymerization over time with UV spectroscopy and hydrogen/deuterium exchange mass spectrometry experiments. Interestingly, the time scales obtained in both experiments reveal significant differences in the rates of copolymerization. Coarse-grained simulations are used to study the incorporation pathway and kinetics of the C(10)BTA monomers into the different polymers. The results demonstrate that the kinetic stability of the host supramolecular polymer controls the rate at which new monomers can enter the existing supramolecular polymers

Exciton bimolecular annihilation dynamics in supramolecular nanostructures of conjugated oligomers

We present femtosecond transient absorption measurements on $\pi$-conjugated supramolecular assemblies in a high pump fluence regime. Oligo(\emph{p}-phenylenevinylene) monofunctionalized with ureido-\emph{s}-triazine (MOPV) self-assembles into chiral stacks in dodecane solution below 75$^{\circ}$C at a concentration of $4\times 10^{-4}$ M. We observe exciton bimolecular annihilation in MOPV stacks at high excitation fluence, indicated by the fluence-dependent decay of $1^1$B$_{u}$-exciton spectral signatures, and by the sub-linear fluence dependence of time- and wavelength-integrated photoluminescence (PL) intensity. These two characteristics are much less pronounced in MOPV solution where the phase equilibrium is shifted significantly away from supramolecular assembly, slightly below the transition temperature. A mesoscopic rate-equation model is applied to extract the bimolecular annihilation rate constant from the excitation fluence dependence of transient absorption and PL signals. The results demonstrate that the bimolecular annihilation rate is very high with a square-root dependence in time. The exciton annihilation results from a combination of fast exciton diffusion and resonance energy transfer. The supramolecular nanostructures studied here have electronic properties that are intermediate between molecular aggregates and polymeric semiconductors

Collisional and thermal ionization of sodium Rydberg atoms I. Experiment for nS and nD atoms with n=8-20

Collisional and thermal ionization of sodium nS and nD Rydberg atoms with n=8-20 has been studied. The experiments were performed using a two-step pulsed laser excitation in an effusive atomic beam at atom density of about 2 10^{10} cm^{-3}. Molecular and atomic ions from associative, Penning, and thermal ionization processes were detected. It has been found that the atomic ions were created mainly due to photoionization of Rydberg atoms by photons of blackbody radiation at the ambient temperature of 300K. Blackbody ionization rates and effective lifetimes of Rydberg states of interest were determined. The molecular ions were found to be from associative ionization in Na(nL)+Na(3S) collisions. Rate constants of associative ionization have been measured using an original method based on relative measurements of Na_{2}^{+} and Na^{+} ion signals.Comment: 23 pages, 10 figure

Effect of H-Bonding on Order Amplification in the Growth of a Supramolecular Polymer in Water

While a great deal of knowledge on the roles of hydrogen bonding and hydrophobicity in proteins has resulted in the creation of rationally designed and functional peptidic structures, the roles of these forces on purely synthetic supramolecular architectures in water have proven difficult to ascertain. Focusing on a 1,3,5-benzenetricarboxamide (BTA)-based supramolecular polymer, we have designed a molecular modeling strategy to dissect the energetic contributions involved in the self-assembly (electrostatic, hydrophobic, etc.) upon growth of both ordered BTA stacks and random BTA aggregates. Utilizing this set of simulations, we have unraveled the cooperative mechanism for polymer growth, where a critical size must be reached in the aggregates before emergence and amplification of order into the experimentally observed fibers. Furthermore, we have found that the formation of ordered fibers is favored over disordered aggregates solely on the basis of electrostatic interactions. Detailed analysis of the simulation data suggests that H-bonding is a major source of this stabilization energy. Experimental and computational comparison with a newly synthesized 1,3,5-benzenetricarboxyester (BTE) derivative, lacking the ability to form the H-bonding network, demonstrated that this BTE variant is also capable of fiber formation, albeit at a reduced persistence length. This work provides unambiguous evidence for the key 1D driving force of hydrogen bonding in enhancing the persistency of monomer stacking and amplifying the level of order into the growing supramolecular polymer in water. Our computational approach provides an important relationship directly linking the structure of the monomer to the structure and properties of the supramolecular polymer

Complete Solving for Explicit Evaluation of Gauss Sums in the Index 2 Case

Let $p$ be a prime number, $q=p^f$ for some positive integer $f$, $N$ be a positive integer such that $\gcd(N,p)=1$, and let \k be a primitive multiplicative character of order $N$ over finite field \fq. This paper studies the problem of explicit evaluation of Gauss sums in "\textsl{index 2 case}" (i.e. f=\f{\p(N)}{2}=[\zn:\pp], where \p(\cd) is Euler function). Firstly, the classification of the Gauss sums in index 2 case is presented. Then, the explicit evaluation of Gauss sums G(\k^\la) (1\laN-1) in index 2 case with order $N$ being general even integer (i.e. N=2^{r}\cd N_0 where $r,N_0$ are positive integers and $N_03$ is odd.) is obtained. Thus, the problem of explicit evaluation of Gauss sums in index 2 case is completely solved