Average Contour Length Research Articles

Molecular Weight and Polydispersity Estimation of Adsorbing Polymer Brushes by Atomic Force Microscopy

We have estimated the molecular weight, Mn, and polydispersity, PDI, of densely grafted poly(N-isopropylacrylamide) (PNIPAM) brushes using a novel atomic force microscopy (AFM) approach. When compression of a polymer brush induced adsorption of multiple chains to an AFM tip, the resulting decompression force profile exhibited a maximum attractive force at a separation, Lm, that decayed to zero with increasing tip-sample separation. We have found that the separation Lm approximates the average contour length, Lc, determined by gel permeation chromatography (GPC). The detection of a decaying attractive force at separations larger than Lc suggests that chains of above average length sequentially break free from the tip as they are stretched away from the grafting surface. The shape of the decompression profile in this region approximately paralleled the cumulative weight fraction of the grafted chains determined by GPC. The fraction of chains of a given molecular weight determined from a single force curve fit a log-normal distribution, having a standard deviation that provided an estimate of the PDI. We have characterized two PNIPAM brushes by this AFM technique as well as by GPC coupled to a multiangle laser light-scattering detector (MALLS). The values obtained by AFM-(1) Mn,AFM = (3.8+/-0.5) x 10(4), PDI,(AFM) = 1.3+/-0.1 and (2) Mn,AFM = (9.4+/-1.4) x 10(4), PDI,(AFM) = 1.3+/-0.1-agreed quite well with the corresponding GPC/MALLS values of (1) Mn,GPC = 4.77 x 10(4), PDI,GPC = 1.33 and (2) Mn,GPC = 9.49 x 10(4), PDI = 1.35. This technique requires only a single force curve to obtain a statistical distribution of contour lengths and provides a novel method for estimating the Mn and PDI of appropriate uniformly grafted dense polymer layers.

Investigation of particle-based and monolithic columns for cation exchange protein displacement chromatography using poly(diallyl-dimethylammonium chloride) as displacer

The overall topic of the investigation was the separation of basic proteins by cation exchange displacement chromatography. For this purpose two principal column morphologies were compared for the separation of ribonuclease A and α-chymotrypsinogen, two proteins found in the bovine pancreas. These were a column packed with porous particles (Macro-Prep S, 10 μm, 1000 Å) and a monolithic column (UNO™ S1). Both columns are strong cation exchangers, carrying SO 3 −-groups linked to a hydrophilic polymer support. Poly(diallyl-dimethylammonium chloride) (PDADMAC), a linear cationic polyelectrolyte composed of 100–200 quaternary pyrrolidinium rings, was used as displacer. The steric mass action (SMA) model and, in particular, the operating regime and dynamic affinity plots were used to aid method development. To date the SMA model has been applied primarily to simulate non-linear displacement chromatography of proteins using low molar mass displacers. Here, the model is applied to polyelectrolytes with a molar mass below 20 000 g mol −1, which corresponds to a degree of polymerization below 125 and an average contour length of less than 60 nm. The columns were characterized in terms of the adsorption isotherms (affinity, capacity) of the investigated proteins and the displacer.

Monitoring Diffusion of Reptating Polymer Chains by a Direct Energy Transfer Method: A Monte Carlo Simulation

Full Paper: A kinetic Monte Carlo method was used to simulate the diffusion of reptating polymer chains across an interface. A time-resolved fluorescence technique in conjunction with a direct energy transfer method was used to measure the extent of diffusion of dye-labeled reptating polymer chains. The diffusion of donor- and acceptorlabeled polymer chains between adjacent compartments was randomly generated. The fluorescence decay profiles of donor molecules were simulated at several diffusion steps to produce mixing of the polymer chains. Mixing ratios of donor- and acceptor-labeled polymer chains in compartments were measured at various stages (snapshots) of diffusion. It was observed that for a given molecular weight, the average interpenetration contour length was found to be proportional to the mixing ratio. Monte Carlo analysis showed that the curvilinear diffusion coefficient is inversely proportional to the weight of polymer chains during diffusion.

High Shape Persistence in Single Polymer Chains Rigidified with Lateral Hydrogen Bonded Networks

Exploiting tapping mode−scanning force microscopy (TM−SFM), we characterized single polymeric chains of poly(isocyanodipeptides) (PICs) equilibrated in quasi two-dimensions on the basal plane of mica surfaces. While the average contour length 〈L〉 of an acid-catalyzed PIC bearing l-alanine-d-alanine methyl ester groups was as high as 5.3 μm, the corresponding Ni-catalyzed product exhibited an 〈L〉 of 70 nm. With a newly devised method based on the statistical analysis of the curvature of polymeric chains on a length scale up to about 100 nm from SFM images, we determined their persistence length 𝓁p. The measured value of 𝓁p = 76 ± 6 nm for both products, independent of the contour length, indicates that the single polymer molecules are very rigid, i.e., even more rigid than the double-stranded DNA. This rigidity is attributed to the helical structure of the polymer backbone and, in particular, to the hydrogen-bonded networks that are present between the alanine moieties in the side chains.

Nano-control of self-assembled biomolecular structures

Abstract We have controlled the structure of self-assembled systems by introducing charges (charge effect) and polymeric tails (steric effects) on a spherical–cylindrical shape of nonionic surfactant micelles. In detail, we studied the effects of a phospholipid (DL-α-phosphatidycholine dimyristol: DMPC) on the shape of nonionic surfactant micelles (penta-ethyleneglycol mono- n -dodecyl ether: C 12 E 5 ), which has been studied in terms of an aggregation number, critical micellization concentration (CMC), second virial coefficient ( A 2 ), and hydrodynamic diameter ( D H ) by laser light scattering. DMPC, DOPC (DL-α-phosphatidylcholine dioleoyl), and DMPE (DL-α-phosphoethanolamin dimyristol) molecules added in C 12 E 5 micelle solutions decrease the spontaneous curvatures, leading to an increase of the end-cap energy E c that favors micellar growth. Based on the CMC values, the total free energy per micelle of C 12 E 5 /DMPC mixtures is estimated. The free energy per micelle of C 12 E 5 /DMPC mixtures decreases as DMPC is added. This is consistent with the decrease of A 2 and the strong hydrophobicity of DMPC compared with C 12 E 5 . The average contour length, the diffusion coefficient, and the end-cap energy of mixed micelles are estimated based on the CMC and molecular specific volumes of the moiety. The end-cap energy of the mixed micelles and the average contour length increase as DMPC is added, which is also reasonable considering the molecular structure of DMPC. Furthermore, the diffusion coefficients obtained from dynamic laser light scattering are in excellent agreement with the estimated diffusion coefficients obtained from a one-dimensional growth model based on static light scattering measurements. Charged lipid (1,2-dioleoyl-3-trimethylammonium-propane) and polymer lipid (1,2-dimyristoyl- sn -glycero-3-phosphoethanolamine-[poly(ethylene glycol)]) increase the spontaneous curvatures, resulting in breaking micelles into small size. When the lipids are added, the hydrodynamic diameter of the micelles of C 12 E 5 /lipids is nearly temperature independent due to the strong charge or steric repulsion.

Non-Newtonian viscosity in linear and star polymers

We theoretically investigate non-Newtonian viscosity and coil deformation of linear and (regular) star polymers in dilute solution subject to large shear rates. A bead-and-spring model with preaveraged hydrodynamic interaction, accounting also approximately for good-solvent expansion, is employed within the Rouse-Zimm approach. We impose a constraint on the average spring lengths, so as to keep constant the average contour length of the arms under shear: this corresponds to assuming that the springs become increasingly stiffer. For any topology and a very large molecular mass, coil deformation modifies the hydrodynamic interaction, that goes to a maximum, and then decreases with a crossover from the Zimm to the Rouse regime with increasing shear rate. Correspondingly, the intrinsic viscosity decreases and then raises above its low-shear value. This behavior is however much less pronounced under good-solvent conditions. At very large shear rate, the constraint on the spring lengths becomes the dominant factor. This leads to a decrease of intrinsic viscosity with an asymptotic –2/3 power law for any draining condition. Simultaneously, the strongly elongated coil becomes fully aligned with flow.

Force Measurements on Myelin Basic Protein Adsorbed to Mica and Lipid Bilayer Surfaces Done with the Atomic Force Microscope

The mechanical and adhesion properties of myelin basic protein (MBP) are important for its function, namely the compaction of the myelin sheath. To get more information about these properties we used atomic force microscopy to study tip-sample interaction of mica and mixed dioleoylphosphatidylserine (DOPS) (20%)/egg phosphatidylcholine (EPC) (80%) lipid bilayer surfaces in the absence and presence of bovine MBP. On mica or DOPS/EPC bilayers a short-range repulsive force (decay length 1.0–1.3 nm) was observed during the approach. The presence of MBP always led to an attractive force between tip and sample. When retracting the tip again, force curves on mica and on lipid layers were different. While attached to the mica surface, the MBP molecules exhibited elastic stretching behavior that agreed with the worm-like chain model, yielding a persistence length of 0.5 ± 0.25 nm and an average contour length of 53 ± 19 nm. MBP attached to a lipid bilayer did not show elastic stretching behavior. This shows that the protein adopts a different conformation when in contact with lipids. The lipid bilayer is strongly modified by MBP attachment, indicating formation of MBP-lipid complexes and possibly disruption of the original bilayer structure.

Self-Diffusion of Constituent Cationic Surfactants in Threadlike Micelles

Self-diffusion mechanisms of a surfactant molecule in entangled threadlike micelles formed from a mixture of cetyltrimethylammonium bromide (CTAB) and sodium salicylate (NaSal) in aqueous solution (CTAB:NaSal/W) are discussed. The self-diffusion coefficient (Dself) for the cetyltrimethylammonium (CTA+) cation in the micelle was calculated on the basis of a phantom network model. Two formulations for the self-diffusion coefficient are derived for two separate regimes with respect to time and space. When the relaxation time (τm) for the micellar system is short, Dself is represented as 〈ξ2〉{6τmP(Le,τm)}, where 〈ξ2〉 is the mean square of the entanglement spacing which depends on the CTA+ concentration and P(Le,τm) is a probability for the CTA+ cation to remain in the initial part of the threadlike micelle with the average contour length of Le between two entanglements. On the other hand, when τm is long, Dself is expressed as (2Q/3)(Dlat/2τm)1/2, where Q and Dlat are the persistence length of the threadlike ...

Rigid‐rod polyelectrolytes based on poly(p‐phenylene sulfonic acid)

AbstractThe synthesis and characterization of polyelectrolytes, composed of a shape persistent poly(p‐phenylene) backbone with regularly attached sulfonic acid‐ and n‐alkyl residues, is described. Polymers were obtained by Pd‐catalyzed coupling of substituted p‐phenylene‐bis‐boronic acid esters with 2,5‐dibromobenzenesulfonates or 4,4′‐dibromodiphenyl‐2,2′‐disulfonates, followed by saponification of the sulfonates. The polymers were found to have number average contour lengths (Ln) betwen 23 and 155 nm. The polymer sulfonic acids behave as strong acids and are soluble in water as their tetramethylammonium salts. The concentration dependence of the viscosity of aqueous solutions of the salts does not follow the behavior seen for conventional polyelectrolytes. The polymers show lyotropic behavior at low to moderate concentrations. Polymers with long alkyl substituents, if spread as monolayers on a Langmuir trough and subject to the Langmuir‐Blodgett technique, give LB‐multi‐layers, which have been characterized by X‐ray reflectometry. As a model compound, sodium 4,4″‐dimethyl‐1,1′:4′,1″‐terphenyl‐2′‐sulfonate was obtained as single crystal and its X‐ray structure is described.

Physicochemical properties of (1 → 6)‐branched (1 → 3)‐β‐D‐glucans. 1. Physical dimensions estimated from hydrodynamic and electron microscopic data

The physical dimensions of several (1-->6) branched (1-->3)-beta-D-glucan samples obtained from different organisms and their derivatives have been studied by electron microscopy, light scattering measurements, viscometry, and gel permeation chromatography. The electron micrographs indicate that in most samples these biopolymers are adequately described as linear worm-like coils. A sample reconstituted from alkaline media appeared as a blend of the linear, circular, and aggregated polymer morphologies. The average mass per unit length, ML = Mw/Lw for the macroscopically linear samples, was estimated to be 2100 +/- 200 g mol-1 nm-1. The parameter ML was determined from the contour lengths obtained by electron microscopy and the molecular weight by light scattering measurements. The observed ML was consistent with the triple-helical structure reported from x-ray diffraction studies and observed degree of side-chain substitution. From the molecular snapshots shown in the electron micrographs, the persistence lengths of these beta-D-glucans were determined to be 140 +/- 30 nm. The experimentally determined intrinsic viscosities were consistent with these estimates of ML and persistence length. Comparison of the molecular weight distributions obtained from gel permeation chromatography and those deduced from the electron micrographs indicates that number and weight average contour lengths are more reliable than z and z + 1 averages.

Influence of molecular structure on processing conditions and mechanical properties of graft rigid-rod copolymers

Dynamic and tensile mechanical properties of a series of graft rigid-rod copolymers were studied. The copolymers were single-component rigid-rod molecular composites consisting of a rigid-rod poly( p-phenylenebenzobisthiazole) backbone grafted with flexible-coil poly(oxy-1,3-phenylenecarbonyl-1,4-phenylene) side-chains. Dynamic storage moduli of the graft copolymers showed three characteristic regions: an initial plateau region, a transition zone and a second plateau region. The peak transition temperatures of the dynamic loss modulus curves were independent of the rigid-rod backbone length but decreased with increasing side-chain lengths, suggesting that the transition was a secondary transition associated with localized motions of the flexible side-chains. The graft rigid-rod copolymers were compression moulded into tensile test specimens at temperatures above their peak transition temperatures. Although the specimens appeared well consolidated as evidenced by the scanning electron micrographs of tensile-fractured surfaces, their tensile properties were relatively poor compared with those reported for rigid-rod molecular composite blends. The significance of poor tensile properties is discussed based on the structural characteristics of the graft copolymers, namely the length of rigid-rod backbone, the frequency of graft sites, and the average contour length of flexible-coil side-chains.

Creation, migration, and termination of conformational defects in poly(1,4-t r a n s-butadiene) in form II and in the inclusion complex in perhydrotriphenylene

The histories of the conformational defects created by transitions between anticlinal± states, either singly or correlated as next-nearest neighbors, have been studied using molecular dynamics trajectories computed for a chain of poly(1,4-trans-butadiene) in form II and in the inclusion complex with perhydrotriphenylene. The most likely fate of a newly created conformational defect is immediate reversion to the preferred conformation. The remaining defects can undergo diffusion. The average contour lengths for the paths covered in the diffusion are on the order of ten CH–CH2 bonds for a long poly(1,4-trans-butadiene) in the inclusion complex in perhydrotriphenylene. The diffusion is terminated either by loss of the defect out the end of the chain, or by the annihilation of two defects that encounter one another. The average time between hops is much shorter for the chain in the inclusion complex than for the chain in form II. The time-averaged number of defects in methyl terminated decamers is virtually identical in the two systems. For much longer chains, the poly(1,4-trans-butadiene) in the inclusion complex should contain more defects than the chain in form II.

Isolation and localization of a spectrin-like protein from echinoderm sperm

Thyone sperm undergo an explosive acrosome reaction resulting in the extension of a 90 microns long acrosomal process. In unreacted sperm, profilamentous actin is sequestered within the profilactin cup (Tilney: Journal of Cell Biology 69:73-89 1976), which consists of four major polypeptides: actin, profilin, and a 250/235 kDa equimolar doublet (TS 250/235). Dialysis of profilactin preparations into an actin assembly buffer resulted in the formation of acrosomal-like macromolecular aggregates containing actin, TS 250/235, and several other polypeptides as detected by SDS-PAGE. TS 250/235 was purified by subjecting extracts of pH solubilized profilactin cups to DEAE and phosphocellulose ion exchange chromatography. TS 250/235 demonstrated immunocrossreactivity with affinity purified polyclonal antibodies raised against S. purpuratus egg spectrin. As determined by biotinylated-calmodulin overlays, both subunits of TS 250/235 bound calmodulin in a Ca(++)-sensitive manner. Electron microscopy of low angle, rotary shadowed replicas of TS 250/235 revealed an elongate rod-shaped molecule with an average contour length of 203 nm. By indirect immunofluorescence, TS 250/235 was found to be uniformly distributed throughout the profilactin cup of the unreacted sperm. This distribution of TS 250/235 correlated with the location of monomeric actin as determined by localization studies utilizing fluorescent-DNase-1. Upon sperm activation, the cellular distribution of TS 250/235 dramatically changed and was observed both along the length and at the base of the extended acrosomal process.

Phase transitions of concentrated DNA solutions in low concentrations of 1 : 1 supporting electrolyte

Transitions between isotropic and liquid crystalline phases of concentrated solutions of DNA with an average contour length (500 A) near the persistence length were examined in 0.01 M supporting 1:1 electrolyte (predominantly NaCl). A quantitative phase diagram describing the transitions occurring over a DNA concentration range from 100 to 290 mg/mL and temperatures from 20 to 60 degrees C was constructed from solid-state 31P-nmr data and examination of the morphologies of the mesophases by polarized light microscopy. Three anisotropic phases were observed in solutions with DNA concentrations of 160-290 mg/mL: an unidentified, weakly birefringent phase termed "precholesteric," a true cholesteric phase with pitch approximately 2 microns, and a third, presumably more highly ordered phase. Comparison with previous studies showed that the critical concentration for anisotropic phase formation and the nature of the phases formed by these DNA molecules are not strongly affected by decreasing the supporting electrolyte concentration from approximately 0.2 M to 10 mM. There are, however, profound effects of decreasing the supporting electrolyte concentration on the width of the transition from isotropic to totally anisotropic solutions, and the nature of the transitions between phases. Decreasing the supporting electrolyte concentration significantly increases the concentration range of persistence of the isotrophic phase, and results in the formation of triphasic solutions (isotropic and two liquid crystalline phases). Values of the critical DNA concentrations for anisotropic phase formation from the theory of A. Stroobants et al. [(1986) Macromolecules 19, 2232 to 2238] were found to be significantly lower than the observed values for any reasonable estimate of the effective radius, probably because of the relatively short lengths of DNA fragments examined in the present study. Comparison of the experimentally determined DNA concentrations required for anisotropic phase formation with the values predicted from Flory's lattice statistics theory, which explicitly considers the rod length, permitted estimation of the effective DNA radius. The estimated radius was inconsistent with effective radii calculated from Poisson-Boltzmann (P-B) theory based on a supporting electrolyte concentration of 10 mM, but was in fair agreement with P-B theory assuming that Na+ DNA contributes approximately 0.24 Na+ counterions/nucleotide to the effective free sodium ion concentration.

Isolation and characterization of DNA from the entomopathogen Beauveria bassiana

Beauveria bassiana chromosomal (chr) and mitochondrial (mt) DNA were isolated, purified, and characterized. Modification of a recent rapid method of total DNA extraction was followed by DNA purification on CsCl-bisbenzimide gradients. Cytosine methylation at 5′-CpG-3′ doublets was undetectable in the total DNA as demonstrated by MspI and HpaII restriction enzyme digests. The chrDNA had a bouyant density of 1.7130 g/ml (G + C ratio of 53.4%) and a T m of 92.6°C (G + C ratio of 56.9%). Mitochondrial DNA was isolated from total DNA or purified mitochondria and had a buoyant density of 1.7120 g/ml (G + C ratio of 52.3%) and a biphasic DNA melting curve. A minor melt at 76.2°C (16.8% G + C ratio) yielded a 10% increase in absorbance while the major melt at 93.1°C (58% G + C ratio) yielded a 30% increase in absorbance. The mtDNA was estimated by gel electrophoresis to be 17.6 ± 0.6 MDa and determined to be 28.5 kb by restriction enzyme analysis. Electron microscopy of the mtDNA revealed circular molecules having an average contour length of 8.84 ± 0.51 μm. A physical map of the mtDNA was generated by double restriction enzyme digestion using the restriction enzymes ClaI, EcoRI, SalI, BstEII, and XhoI.

Conformation of apolipoprotein B after lipid extraction of low density lipoproteins attached to an electron microscope grid.

We have examined the shape of apolipoprotein B (apoB) from low density lipoproteins (LDL) using a new method to prepare the electron microscope grids. After adsorption of the lipoproteins to a carbon-coated copper grid, lipids were extracted with ethanol-ether 4:1; an aqueous negative stain was then applied. When the LDL residue was examined after this treatment, apoB, together with residual lipid, appeared as an elongated flexible structure about 600-700 A in length consisting of multiple domains of variable width from 20-70 A. Occasionally, the elongated apoB formed an irregular ring-shaped structure, but most of the rings were open. When LDL were pretreated with glutaraldehyde, then adsorbed, extracted, and stained, most of the images were closed rings with an average contour length of 700 A, again consisting of multiple domains of variable sizes. These results are consistent with apoB being composed of multiple domains arranged in an elongated structure on the surface of the LDL, and with distant domains possessing a mutual affinity that favors their cross-linking.

Mitochondrial DNA of the human malarial parasite Plasmodium falciparum

Covalently closed circular DNA molecules were isolated from Plasmodium falciparum total DNA by isopycnic centrifugation in CsCl gradients containing either ethidium bromide or 2′-6-diamidino-2-phenylindole. The circular molecules had an average contour length of 11.1 ± 0.5 μm, similar to the analogous molecules previously isolated from the simian malaria parasite P. knowlesi. Both circular molecules shared considerable sequence homology and conserved restriction sites. The nucleotide sequence of one 936 bp fragment of the P. falciparum molecule was determined and identified, by a data base homology search, as part of a mitochondrial small rRNA subunit, thus confirming the mitochondrial origin of the circular DNAs of both malarial species.

The precursor complex of Pf3 bacteriophage

Filamentous bacteriophage infections give rise to intracellular filamentous precursor complexes composed of a circular single-stranded DNA molecule and on the order of 103 copies of a viral-encoded, single-stranded DNA binding protein. A protocol was developed for the purification of the precursor complex from Pf3 phage-infected Pseudomonas aeruginosa cells, and existing protocols for Ff and Pfl complexes were amended by the introduction of a molecular sieving column step. The Pf3 precursor complex has an average contour length of 500 nm, which is shorter than that of the mature Pf3 virus. M/L (mass/ length) values were obtained from turbidity measurements, from scanning-transmission electron microscopy, and from the total particle mass divided by its length. The average M/L obtained was 19,300 Da/nm which is close to that of the Pf3 virion. The complex has a nucleotide/subunit ratio of 6.0. The protein component of the precursor complex has less than 10% α-helicity, whereas the protein component of the virus is greater than 90% α-helical. The DNA structure in the precursors is very different from that in the virus, so that during virus assembly the DNA structure must change dramatically. The results for the Pf3 system, together with the available information for the Ff and Pfl filamentous virus systems, indicate that different DNA-protein interactions and packing arrangements are involved in performing equivalent functions.

Rheological properties of rodlike polymers in solution. 2. Linear and nonlinear transient behavior

AbstractRheological and rheo‐optical studies are reported for isotropic solutions of the mesogenic rodlike polymer poly(1,4‐phenylene‐2,6‐benzobisthiazole) (PBT). Several PBT samples were used with average contour lengths from 95 to 135 nm. Concentrations were varied over a range just below the concentration Cc for the formation of an ordered (nematic) state. The predictions of a single‐integral constitutive equation of the BKZ type utilizing experimental estimates of the distribution of discrete relaxation times is compared with experimental data on the steady‐state viscosity ηκ, the recoverable compliance function Rκ, and the steady‐state flow birefringence as functions of the shear rate κ, with satisfactory results. The relaxation of the shear stress and the flow birefringence on cessation of steady‐state flow at shear rate κ are also compared with the single‐integral constitutive equation, and it is found that in the nonlinear response range the data can be superposed over a wide range in κ. The overall behavior is qualitatively similar to that for flexible chains, which can also be fitted by the single‐integral constitutive equations over similar ranges of η0R0κ, with η0 and R0 the limiting values of ηk and Rκ for small κ. Of course, the dependence of η0 and R0 on concentration and molecular weight differs markedly for rodlike and flexible‐chain polymers.

Concentration dependence of the primitive path step length

Recent experiments by Donald and Kramer have measured the extension ratio, λ craze, for polymers in crazes under tensile strain. In this paper it is shown that λ craze ∝I 1 2 e i where I e is the average contour length between entanglements in glassy, and not permanently crosslinked polymers and is obtained from melt elasticity experiments. It is also shown that I, the Rouse chain step length for a molten polymer, is independent of monomer density and that the average distance between entanglements, d ∝ ϱ − α , where ϱ is the number of monomers per unit volume with 1 4 ≲ α ≲ 1 2 . Finally these results are shown to provide confirmation for the configurational assumptions of the primitive path for reptation in which α appears as the determining factor for the concentration dependence of the relaxation modulus, the steady state viscosity and terminal relaxation time.