Mean Square Radius Research Articles

Polymer Chain Collapse upon Rapid Solvent Exchange: Slip-Spring Dissipative Particle Dynamics Simulations with an Explicit-Solvent Model

We investigate polymer precipitation dynamics using explicit-solvent dissipative particle dynamics. We present a method to partially exchange solvent by antisolvent. In a first set of simulations, we analyze the collapse of a single chain of 25 beads as both liquids are mixing. The variation of the chain’s mean-squared radius of gyration during collapse is found to be strongly influenced by the distance that the antisolvent must diffuse to the chain. This behavior is validated using an analytical model. We further report inclusion of solvent in the collapsed chains in equilibrium as well as the formation of a solvation layer. The influence of slip-springs and thus entanglements on the collapse of a single chain of 100 beads is evaluated in a second set of simulations. We did not find any influence of entanglements on the chain’s collapse nor can we report a two-stage collapse process when enforcing these self-entanglements onto the chain.

Probing relationship between structure and viscosity of unentangled polymers in steady shear flow

Structural responses of unentangled polymers of various architectures, including linear, ring, star, asymmetric star, and H-shaped polymers, in steady shear flow are investigated via nonequilibrium molecular dynamics simulations. As observed in our previous simulations, these polymers, when having an identical equilibrium mean-square radius of gyration exhibit the same viscosity curve in both the linear and nonlinear regimes. In this article, the polymer orientation and deformation and the rate of deformation are calculated as a function of shear rate. The differences between different architectures in most of these properties are found to be related to the polymer extensibility, the number of free ends and the ability of folding. The unique architecture-independent quantity is the gradient component of the radius of gyration tensor. We find that a simple relation η=C η (3 / )3/2 gives a near-perfect fit to shear viscosity η data, where C η is a constant. The results suggest that this relation between chain thickness and viscosity is universal for unentangled polymers of various architectures.

Form factor of the longitudinal component of weak vector current in pion beta decay

The form factor f- of the longitudinal component of the weak vector current in pion β-decay is derived using the generalized Ward identity. The form factor is expressed through the pion mass difference and mean-square radii in the channels with angular momenta J = 1, 0 and isospins T = 1, 2.

Shape evolution for neutron-deficient bismuth isotopes studied by resonance laser ionization spectroscopy

In-source laser spectroscopy experiments for bismuth isotopes at the 306.77 nm atomic transition has been carried out at the IRIS (Investigation of Radioactive Isotopes at Synchrocyclotron) facility of Petersburg Nuclear Physics Institute. New data on isotope shifts for 189–198,211Bi isotopes and isomers have been obtained. The changes in the mean-square charge radii were deduced. The large isomer shift has been observed for the intruder isomer states of Bi with spin I = 1/2 (A = 193, 195, 197). This testifies to the shape coexistence in these nuclei with the intruder isomer states more deformed than the ground states. Marked deviation from the nearly spherical behavior for ground states of the even-neutron Bi isotopes at N < 109 is demonstrated, in contrast to the Pb and Tl isotopic chains.

Synthesis and Solution Properties of a Rigid Helical Star Polymer: Three-Arm Star Poly(quinoxaline-2,3-diyl)

Six semiflexible helical three-arm star poly[5,8-dimethyl-6,7-bis(propoxymethyl)quinoxaline-2,3-diyl] (A3PQ) samples which have different arm lengths ranging in the weight-average molar mass Mw from 1.5 × 104 to 2.5 × 105 g mol–1 were synthesized via the living polymerization of the corresponding 1,2-diisocyanobenzene using a trifunctional palladium initiator. The particle scattering function P(q), the z-average mean-square radius of gyration ⟨S2⟩z, and the intrinsic viscosity [η] for the A3PQ samples were determined from light and small-angle X-ray scattering and viscosity measurements in tetrahydrofuran at 25 °C. The obtained data were compared with modern theories for the wormlike star. The experimental P(q) data were represented by the wormlike star model connected with a universal joint. The other data were comparably explained by the same model.

In-gas-cell laser spectroscopy of the magnetic dipole moment of the N≈126 isotope Pt199

Magnetic dipole moment and mean-square charge radius of $^{199}$Pt ($I^{\pi}=$ 5/2$^-$) have been evaluated for the first time from the investigation of the hyperfine splitting of the $\lambda_1=$ 248.792 nm transition by in-gas-cell laser ionization spectroscopy. Neutron-rich nucleus $^{199}$Pt was produced by multi-nucleon transfer reaction at the KISS where the nuclear spectroscopy in the vicinity of $N=$ 126 is planed from the aspect of an astrophysical interest as well as the nuclear structure. Measured magnetic dipole moment $+$0.63(13)$\mu_{\rm N}$ is consistent with the systematics of those of nuclei with $I^{\pi}=$ 5/2$^-$. The deformation parameter $|<\beta_2^2>^{1/2}|$ evaluated from the isotope shift indicates the gradual shape change to spherical shape of platinum isotopes with increasing neutron number toward $N=$ 126.

Structural properties of star-shaped polyions: Entropic sampling

The method of entropic sampling within the Wang–Landau algorithm (a modern variant of the Monte Carlo method) is used to determine the densities of energy states of a strongly charged star-shaped polyelectrolyte within the framework of the lattice model. The equilibrium thermal and structural properties of the polyelectrolyte with the number of arms f ≤ 6 and the length of arms N arm ≤ 73 in a wide temperature range are determined from the density of states. Such characteristics as the free energy, the mean-squared radius of gyration, its components, and the parameters characterizing the shape of a polyion are calculated. It is found that the concentration, length, and number of arms affect the thermal and structural properties. The main attention is focused on the effect of temperature on these properties. The coil-to-globule transition is detected, while for the polyion with the highest length of arms (N arm = 24) the transition from the liquid globule to the crystal-like one is observed. Differences in the properties of the star-shaped and corresponding linear polyions are characterized.

Statistical and Dynamical Properties of Topological Polymers with Graphs and Ring Polymers with Knots.

We review recent theoretical studies on the statistical and dynamical properties of polymers with nontrivial structures in chemical connectivity and those of polymers with a nontrivial topology, such as knotted ring polymers in solution. We call polymers with nontrivial structures in chemical connectivity expressed by graphs “topological polymers”. Graphs with no loop have only trivial topology, while graphs with loops such as multiple-rings may have nontrivial topology of spatial graphs as embeddings in three dimensions, e.g., knots or links in some loops. We thus call also such polymers with nontrivial topology “topological polymers”, for simplicity. For various polymers with different structures in chemical connectivity, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of the renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer–Grest model) expressed with complex graphs. We then address topological properties of ring polymers in solution. We define the knotting probability of a knot K by the probability that a given random polygon or self-avoiding polygon of N vertices has the knot K. We show a formula for expressing it as a function of the number of segments N, which gives good fitted curves to the data of the knotting probability versus N. We show numerically that the average size of self-avoiding polygons with a fixed knot can be much larger than that of no topological constraint if the excluded volume is small. We call it “topological swelling”.

Chain stiffness of cellulose tris(phenylcarbamate) in tricresyl phosphate (TCP)

Small-angle X-ray scattering (SAXS) measurements were carried out for two cellulose tris(phenylcarbamate) (CTPC) samples in tricresyl phosphate (TCP) at 25 °C to determine the particle scattering function P(q) and the z-average mean-square radius of gyration $$\left\langle {S^{2} } \right\rangle_{z}$$ . The obtained data were analyzed in terms of the wormlike chain model to estimate the Kuhn segment length λ −1 (the stiffness parameter, equivalent to twice the persistence length) and the helix pitch (or helix rise) per residue h. The resultant λ −1 and h were 11.5 ± 0.5 and 0.51 nm, respectively. While the latter value (h) is consistent with the previously reported values both for cellulose and cellulose derivatives, appreciably higher chain flexibility was found for CTPC in TCP than that in tetrahydrofuran at 25 °C (19–24 nm). The value is fairly close to that in anisol, cyclohexanol, and benzophenone, assuming an appropriate temperature coefficient. We may thus conclude that CTPC behaves as a semiflexible chain in TCP.

Topology-Dependent Chain Stiffness and Local Helical Structure of Cyclic Amylose Tris(3,5-dimethylphenylcarbamate) in Solution

Conformational properties of rigid and semiflexible cyclic chains are still unclear owing to few experimental researches on dilute solution properties. Five cyclic amylose tris(3,5-dimethylphenylcarbamate) (cADMPC) samples ranging in the weight-average degree of polymerization from 23 to 150 were prepared from enzymatically synthesized cyclic amylose. Light scattering and small-angle X-ray scattering measurements were made on the samples to determine the weight-average molar mass Mw, the particle scattering function P(q), and the z-average mean-square radius of gyration ⟨S2⟩z in methyl acetate (MEA), 4-methyl-2-pentanone (MIBK), and tetrahydrofuran (THF) at 25 °C. The obtained P(q) and ⟨S2⟩z data were analyzed on the basis of the cyclic wormlike chain model to determine the wormlike chain parameters, that is, the helix pitch (or helix rise) per residue h and the Kuhn segment length λ–1 (the stiffness parameter or twice of the persistence length) as a function of Mw. Although the chain stiffness parameter ...

Chain Dimensions and Stiffness of Cellulosic and Amylosic Chains in an Ionic Liquid: Cellulose, Amylose, and an Amylose Carbamate in BmimCl

Small-angle X-ray scattering measurements were made for cellulose, amylose, and amylose tris(ethylcarbamate) (ATEC) in an ionic liquid [1-butyl-3-methylimidazolium chloride (BmimCl)] at 25 °C to determine z-average mean-square radius of gyration ⟨S2⟩z and the particle scattering function P(q). The obtained data were analyzed in terms of the wormlike chain model to estimate the Kuhn segment length λ–1 (the stiffness parameter, equivalent to twice of the persistence length) and the helix pitch (or helix rise) per residue h. The chain stiffness (λ–1) was determined to be 7 ± 1 nm for cellulose, 3.5 ± 0.5 nm for amylose, and 7.5 ± 0.5 nm for ATEC. These values are almost the same or somewhat smaller than those of previously investigated systems, indicating these polymers have relatively high flexibility in the ionic liquid. It is reasonable to suppose that disruption of intramolecular hydrogen bonds of polysaccharide makes the main chain rather flexible in the ionic liquid.

Low-energy E0 transition between the components of the ground-state doublet in the muonic atom Th229

The probability of the $E$0 transition between the ground state and the unusually low-lying 3/2${}^{+}$ state of the thorium nucleus is calculated, and the mean-square charge radius of the latter state is estimated. Possible applications include a high-precision nuclear clock that is highly insensitive to its environment, and a nuclear laser.

Dimensional Characterizations from Rod Stars to Brushes of Polymers with a Low Degree of Polymerization

Dimensional characterizations from rod stars to brushes (PSt-g-HIC-32) with a relatively low degree of polymerization of the main chain have been studied in tetrahydrofuran (THF) at 25 °C by employing synchrotron radiation small-angle X-ray scattering, static and dynamic light scatterings, and viscosity measurements. The rod stars and brushes consist of a flexible linear polystyrene (PSt) main chain and 32 n-hexyl isocyanate (HIC) units in each side chain. Eight PSt-g-HIC-32 samples with a weight-average degree of polymerization (NM) of 11–495 of the main chain and a narrow molecular weight distribution (Mw/Mn) of 1.08–1.17 were prepared using a recycling preparative size-exclusion chromatography (SEC) system. The mean-square radius of gyration (⟨S2⟩), hydrodynamic radius (RH), and intrinsic viscosity ([η]) of the stars and brushes were tabulated. The absolute calibration curve obtained in SEC analysis, scattering form factor, and ρ (=⟨S2⟩1/2/RH) parameter indicated a continuous change in the shape from t...

Changes in nuclear structure along the Mn isotopic chain studied via charge radii

The hyperfine spectra of $^{51,53-64}$Mn were measured in two experimental runs using collinear laser spectroscopy at ISOLDE, CERN. Laser spectroscopy was performed on the atomic $3d^5\ 4s^2\ ^{6}\text{S}_{5/2}\rightarrow 3d^5\ 4s4p\ ^{6}\text{P}_{3/2}$ and ionic $3d^5\ 4s\ ^{5}\text{S}_2 \rightarrow 3d^5\ 4p\ ^{5}\text{P}_3$ transitions, yielding two sets of isotope shifts. The mass and field shift factors for both transitions have been calculated in the multiconfiguration Dirac-Fock framework and were combined with a King plot analysis in order to obtain a consistent set of mean-square charge radii which, together with earlier work on neutron-deficient Mn, allow the study of nuclear structure changes from $N=25$ across $N=28$ up to $N=39$. A clear development of deformation is observed towards $N=40$, confirming the conclusions of the nuclear moments studies. From a Monte Carlo Shell Model study of the shape in the Mn isotopic chain, it is suggested that the observed development of deformation is not only due to an increase in static prolate deformation but also due to shape fluctuations and triaxiality. The changes in mean-square charge radii are well reproduced using the Duflo-Zuker formula except in the case of large deformation.

Statistical and hydrodynamic properties of topological polymers for various graphs showing enhanced short-range correlation.

For various polymers with different structures in chemical connectivity expressed by graphs, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We call polymers with nontrivial structures in chemical connectivity and those of nontrivial topology of spatial graphs as embeddings in three dimensions topological polymers. We evaluate the two quantities both for ideal and real chain models and show that the ratios of the quantities among different structures in chemical connectivity do not depend on the existence of excluded volume if the topological polymers have only up to trivalent vertices, as far as the polymers investigated. We also evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer-Grest model) expressed with complex graphs.

Mean-square radii of two-component three-body systems in two spatial dimensions

We calculate root-mean-square radii for a three-body system confined to two spatial dimensions and consisting of two identical bosons ($A$) and one distinguishable particle ($B$). We use zero-range two-body interactions between each of the pairs, and focus thereby directly on universal properties. We solve the Faddeev equations in momentum space and express the mean-square radii in terms of first-order derivatives of the Fourier transforms of densities. The strengths of the interactions are adjusted for each set of masses to produce equal two-body bound-state energies between different pairs. The mass ratio, ${\cal A}=m_B/m_A$, between particles $B$ and $A$ are varied from $0.01$ to $100$ providing a number of bound states decreasing from $8$ to $2$. Energies and mean-square radii of these states are analyzed for small ${\cal A}$ by use of the Born-Oppenheimer potential between the two heavy $A$-particles. For large ${\cal A}$ the radii of the two bound states are consistent with a slightly asymmetric three-body structure. When ${\cal A}$ approaches thresholds for binding of the three-body excited states, the corresponding mean-square radii diverge inversely proportional to the deviation of the three-body energy from the two-body thresholds. The structures at these three-body thresholds correspond to bound $AB$-dimers and one loosely bound $A$-particle.

Mean-Square Radius of Gyration and Scattering Function of Semiflexible Ring Polymers of the Trefoil Knot.

A Monte Carlo study of the mean-square radius of gyration and scattering function with k the magnitude of the scattering vector for semiflexible ring polymers of the trefoil knot was conducted by the use of the discrete version of the Kratky–Porod (KP) wormlike ring model. The behavior of and as functions of the reduced contour length , defined as the total contour length L divided by the stiffness parameter , is clarified. A comparison is made of the results for the KP ring of the trefoil knot with those for the KP ring of the trivial knot and for the phantom KP ring without the topological constraints.

The effect of baking and enzymatic treatment on the structural properties of wheat starch

In this study, bread was baked with and without the addition of α-amylase. Starch was extracted from the baked bread and its molecular properties were characterized using 1H NMR and asymmetric flow field-flow fractionation (AF4) connected to multi-angle light scattering (MALS) and other detectors. The approach allows determination of molar mass, root- mean-square radius and apparent density as well as the average degree of branching of amylopectin. The results show that starch size and structure is affected as a result of the baking process. The effect is larger when α-amylase is added. The changes include both a decrease molar mass and size as well as an increase in apparent density. Moreover, an increase in average degree of branching and the number of reducing ends H-1(β-r) and H-1(α-r) can be observed.

Dilute solution properties of poly(di-tert-butyl fumarate)

The mean-square radius of gyration 〈S 2〉 is determined for poly(di-tert-butyl fumarate) (PDtBF) for a range of weight-average number of repeat units n w from 382 to 1520 in tetrahydrofuran at 30.0 °C. The data are analyzed on the basis of the Kratky–Porod (KP) wormlike chain with excluded volume (EV). It is shown that the average chain dimension of PDtBF is significantly larger than that of poly(diisopropyl fumarate) (PDiPF) because of a remarkable difference in chain stiffness between the two polymers.

Ion recombination correction in carbon ion beams.

In this work, ion recombination is studied as a function of energy and depth in carbon ion beams. Measurements were performed in three different passively scattered carbon ion beams with energies of 62 MeV/n, 135 MeV/n, and 290 MeV/n using various types of plane-parallel ionization chambers. Experimental results were compared with two analytical models for initial recombination. One model is generally used for photon beams and the other model, developed by Jaffé, takes into account the ionization density along the ion track. An investigation was carried out to ascertain the effect on the ion recombination correction with varying ionization chamber orientation with respect to the direction of the ion tracks. The variation of the ion recombination correction factors as a function of depth was studied for a Markus ionization chamber in the 62 MeV/n nonmodulated carbon ion beam. This variation can be related to the depth distribution of linear energy transfer. Results show that the theory for photon beams is not applicable to carbon ion beams. On the other hand, by optimizing the value of the ionization density and the initial mean-square radius, good agreement is found between Jaffé's theory and the experimental results. As predicted by Jaffé's theory, the results confirm that ion recombination corrections strongly decrease with an increasing angle between the ion tracks and the electric field lines. For the Markus ionization chamber, the variation of the ion recombination correction factor with depth was modeled adequately by a sigmoid function, which is approximately constant in the plateau and strongly increasing in the Bragg peak region to values of up to 1.06. Except in the distal edge region, all experimental results are accurately described by Jaffé's theory. Experimental results confirm that ion recombination in the investigated carbon ion beams is dominated by initial recombination. Ion recombination corrections are found to be significant and cannot be neglected for reference dosimetry and for the determination of depth dose curves in carbon ion beams.