On the determination of the average molecular weight,radius of gyration,and mass/length ratio of polydisperse solutions of polymerizing rod‐like macromolecular monomers by multi‐angle static light scattering

The angular dependence of the light scattered from polydisperse solutions of rod‐like or worm‐like linear polymers formed by the polymerization of rod‐like macromolecular monomers was studied with the aid of computer simulations. Except at very low conversion degrees, these ensembles are characterized by curved Zimm‐like plots. An appraisal of the use of polynomial fittings for the derivation of the weight‐average molecular weight (M_w) and of the z‐average square radius of gyration (<R_g²>_z) from such plots is presented and discussed. Depending on the average size and shape distribution, the use of polynomial fittings allows the applicability of the Zimm method well beyond the standard condition q²<R_g²>_z<<1, q being the scattering wavevector. In addition, the derivation of the w/z‐average mass/length ratio M_L of the polymers from only partially linear Casassa plots, from which large errors in the derived M_L values can be made, is analyzed. By combining the Casassa method with the complementary Holtzer plots, it is usually possible to assess the reliability of the results and give an estimate of their accuracy. However, it was also found that apparently linear Casassa plots, allowing a good estimate of the M_L value, may arise in particular situations even when the Holtzer plot would indicate otherwise.     

Kinetic model for tensile deformation of polymers. Part IV: Effect of polydispersity

A previously introduced molecular model for tensile deformation of solid, flexible chain polymers is used to study the effect of the molecular weight distribution on the strain at break, or maximum draw ratio. The parameters in the model are chosen to represent melt-crystallized linear polyethylene. We focus, in particular, on the relation between the maximum draw ratio and two distribution characteristics: theM _t molecular weight average, first introduced by Graessley, and the polydispersity ratioM _z /M _w . For a log-normal molecular weight distribution, an increase in polydispersity at constantM _t leads to a broadening of the optimum rate (or temperature) window for achieving maximum elongation, but is accompanied, however, by a substantial decrease in the maximum attainable draw ratio. Studies on the deformation of systems having a bimodal molecular weight distribution indicate that blends made ofequal weight fractions of long and short chains exhibit an unexpectedly high elongation at break. These results are explained in terms of the model and possible technological implications are discussed.Dedicated to Professor Hans-Henning Kausch on the occasion of his 60th birthday.     

Creep recovery of acrylate urethane oligomer/acrylate networks. 2. Investigation of different modes of molecular motion

The creep recovery and dynamic mechanical properties of acrylate urethane oligomer/acrylate networks were investigated. The retardation spectra L_CR obtained from the creep recovery experiments were significantly different from the corresponding retardation spectra L_DMA obtained from the dynamic mechanical measurements. The reduced frequency ω^* dependence of L_DMA and the relaxation spectra H_DMA in the higher ω^* region were approximately represented as L_DMA ∝_ω^*−p and H_DMA ∝_ω^*q, although L_CR decreased faster than L_DMA with an increase in ω^*. The exponents q were close to ½ characterizing the Rouse modes in the systems containing an acrylate urethane oligomer of M_w = 5000 but less than ½ in the system containing an acrylate urethane oligomer of M_w = 12,000. For the latter systems, significant thermorheological complexity was observed. © 1997 John Wiley & Sons, Inc. J Polym Sci B: Polym Phys 35 : 2543–2550, 1997     

The solution phase characterization of poly(ferrocenyldimethylsilane)s by small‐angle neutron scattering

A series of monodisperse (M_w/M_n < 1.1) poly(ferrocenyldimethylsilane)s was prepared with number‐averaged degrees of polymerization, 〈z〉_n, of 9, 33, 206, and 506 ( 2 – 5 , respectively), as determined by gel permeation chromatography (GPC). The polymers were studied by small‐angle neutron scattering (SANS) in solution with the aim of obtaining the radius of gyration, R_g, the weight‐averaged molecular weight, M_w, and the polydispersity index, M_w/M_n. Data were collected over the range 0.008 < Q/Å^?1 < 0.5 and for a series of concentrations (weight fraction, w = 0.0063, 0.0125, 0.025, and 0.05). The scattered intensity, I(Q), was fitted to a model based on a Schultz–Zimm distribution of isolated chains with excluded volume. A comparison of the molecular weight and size data determined by GPC and SANS indicated an acceptable agreement between the values for R_g, M_w and M_w/M_n. The results of this study demonstrate the potential utility of SANS to fully characterize metallopolymers, and other polymer systems where traditional methods cannot be applied. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013 , 51, 4011–4020     

Synthesis of arborescent styrene homopolymers and copolymers from epoxidized substrates

The synthesis of arborescent styrenic homopolymers and copolymers was achieved by anionic polymerization and grafting. Styrene and p‐(3‐butenyl)styrene were first copolymerized using sec‐butyllithium in toluene, to generate a linear copolymer with a weight‐average molecular weight M_w = 4000 and M_w/M_n = 1.05. The pendant double bonds of the copolymer were then epoxidized with m‐chloroperbenzoic acid. A comb‐branched (or arborescent generation G0) copolymer was obtained by coupling the epoxidized substrate with living styrene‐p‐(3‐butenyl)styrene copolymer chains with M_w ≈ 5000 in a toluene/tetrahydrofuran mixture. Further cycles of epoxidation and coupling reactions while maintaining M_w ≈ 5000 for the side chains yielded arborescent copolymers of generations G1–G3. A series of arborescent styrene homopolymers was also obtained by grafting M_w ≈ 5000 polystyrene side chains onto the linear and G0–G2 copolymer substrates. Size exclusion chromatography measurements showed that the graft polymers have low polydispersity indices (M_w/M_n = 1.02–1.15) and molecular weights increasing geometrically over successive generations. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012     

Characterization of polyurethane network chains by gel permeation chromatography

Starting materials, prepolymers, chain-extended oligomers, and polyurethane network chains were characterized by gel permeation chromatography in order to make clear the change of molecular distribution in the formation of polyurethane networks. The polyurethane networks were prepared from poly(oxypropylene)glycol (PPG 1000, M _n = 997, M _w/M _n = 1.04), 2,4-tolylene diisocyanate, and 1,4-butanediol by the prepolymer method. Polyurethane networks were degraded by the amine degradation method, by which allophanate groups as crosslinking sites were decomposed selectively. The prepolymer had four species. The polydispersity index of the prepolymer (M _w/M _n) was about 2, that is, the most probable distribution. The product of the chain-extending reaction of prepolymer with BD had five species. The molecular-weight distribution of this product was narrower than that of the prepolymer. The polydispersity of the interstitial chains between crosslinking sites was also narrower than that of the chain-extended product. The polyaddition mechanism in the formation of PPG–TDI–BD polyurethane networks was discussed.     

Behavior of cellulose in NaOH/Urea aqueous solution characterized by light scattering and viscometry

Cellulose was dissolved in 6 wt % NaOH/4 wt % urea aqueous solution, which was proven by a ¹³C NMR spectrum to be a direct solvent of cellulose rather than a derivative aqueous solution system. Dilute solution behavior of cellulose in a NaOH/urea aqueous solution system was examined by laser light scattering and viscometry. The Mark–Houwink equation for cellulose in 6 wt % NaOH/4 wt % urea aqueous solution at 25 °C was [η] = 2.45 × 10^?2 weight‐average molecular weight (M_w)^0.815 (mL g^?1) in the M_w region from 3.2 × 10⁴ to 12.9 × 10⁴. The persistence length (q), molar mass per unit contour length (M_L), and characteristic ratio (C_∞) of cellulose in the dilute solution were 6.0 nm, 350 nm^?1, and 20.9, respectively, which agreed with the Yamakawa–Fujii theory of the wormlike chain. The results indicated that the cellulose molecules exist as semiflexible chains in the aqueous solution and were more extended than in cadoxen. This work provided a novel, simple, and nonpollution solvent system that can be used to investigate the dilute solution properties and molecular weight of cellulose. © 2003 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 347–353, 2004     

Efficient synthesis of well‐defined amphiphilic cylindrical brushes polymer with high grafting density: Interfacial “Click” chemistry approach

Reported here is a novel approach toward efficient preparation of well‐defined cylindrical brushes polymer (CBPs) with both hydrophobic and hydrophilic side chains connected to the linear backbone by interfacial “click” chemistry in two immiscible solvents. The CBPs with high grafting density of more than 95% and molecular polydispersity (M_w/M_n) less than 1.12 can be readily synthesized using present approach. On contrary, the CBPs synthesized from the “click” reaction in a single solvent in homogenous state have low grafting density of less than 55% and molecular polydispersity (M_w/M_n) more than 1.78. © 2011 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2011     

Dilute solution properties of cellulose in LiOH/urea aqueous system

Cellulose was dissolved rapidly in 4.6 wt % LiOH/15 wt % urea aqueous solution and precooled to –10 °C to create a colorless transparent solution. ¹³C‐NMR spectrum proved that it is a direct solvent for cellulose rather than a derivative aqueous solution system. The result from transmission electron microscope showed a good dispersion of the cellulose molecules in the dilute solution at molecular level. Weight‐average molecular weight (M_w), root mean square radius of gyration (〈s²〉_z^1/2), and intrinsic viscosity ([η]) of cellulose in LiOH/urea aqueous solution were examined with laser light scattering and viscometry. The Mark–Houwink equation for cellulose in 4.6 wt % LiOH/15 wt % urea aqueous solution was established to be [η] = 3.72 × 10^?2 M in the M_w region from 2.7 × 10⁴ to 4.12 × 10⁵. The persistence length (q), molar mass per unit contour length (M_L), and characteristic ratio (C_∞) of cellulose in the dilute solution were given as 6.1 nm, 358 nm^?1, and 20.8, respectively. The experimental data of the molecular parameters of cellulose agreed with the Yamakawa–Fujii theory of the worm‐like chain, indicating that the LiOH/urea aqueous solution was a desirable solvent system of cellulose. The results revealed that the cellulose exists as semistiff‐chains in the LiOH/urea aqueous solution. The cellulose solution was stable during measurement and storage stage. This work provided a new colorless, easy‐to‐prepare, and nontoxic solvent system that can be used with facilities to investigate the chain conformation and molecular weight of cellulose. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3093–3101, 2006     

Microwave‐improved polymerization of ϵ‐caprolactone initiated by carboxylic acids

The ring‐opening polymerization of ε‐caprolactone (ε‐CL), initiated by carboxylic acids such as benzoic acid and chlorinated acetic acids under microwave irradiation, was investigated; with this method, no metal catalyst was necessary. The product was characterized as poly(ε‐caprolactone) (PCL) by ¹H NMR spectroscopy, Fourier transform infrared spectroscopy, ultraviolet spectroscopy, and gel permeation chromatography. The polymerization was significantly improved under microwave irradiation. The weight‐average molecular weight (M_w) of PCL reached 44,800 g/mol, with a polydispersity index [weight‐average molecular weight/number‐average molecular weight (M_w/M_n)] of 1.6, when a mixture of ε‐CL and benzoic acid (25/1 molar ratio) was irradiated at 680 W for 240 min, whereas PCL with M_w = 12,100 and M_w/M_n = 4.2 was obtained from the same mixture by a conventional heating method at 210 °C for 240 min. A degradation of the resultant PCL was observed during microwave polymerization with chlorinated acetic acids as initiators, and this induced a decrease in M_w of PCL. However, the degradation was hindered by benzoic acid at low concentrations. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 13–21, 2003     

Synthesis of poly(p‐phenylene) macromonomers and multiblock copolymers

Rigid‐rod poly(4′‐methyl‐2,5‐benzophenone) macromonomers were synthesized by Ni(0) catalytic coupling of 2,5‐dichloro‐4′‐methylbenzophenone and end‐capping agent 4‐chloro‐4′‐fluorobenzophenone. The macromonomers produced were labile to nucleophilic aromatic substitution. The molecular weight of poly(4′‐methyl‐2,5‐benzophenone) was controlled by varying the amount of the end‐capping agent in the reaction mixture. Glass‐transition temperatures of the macromonomers increased with increasing molecular weight and ranged from 117 to 213 °C. Substitution of the macromonomer end groups was determined to be nearly quantitative by ¹H NMR and gel permeation chromatography. The polymerization of a poly(4′‐methyl‐2,5‐benzophenone) macromonomer [number‐average molecular weight (M_n) = 1.90 × 10³ g/mol; polydispersity (M_w)/M_n = 2.04] with hydroxy end‐capped bisphenol A polyaryletherketone (M_n = 4.50 × 10³ g/mol; M_w/M_n = 1.92) afforded an alternating multiblock copolymer (M_n = 1.95 × 10⁴ g/mol; M_w/M_n = 6.02) that formed flexible, transparent films that could be creased without cracking. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 3505–3512, 2001     

Synthesis of arborescent polystyrene by “click” grafting

A method was developed for the synthesis of arborescent polystyrene by “click” coupling. Acetylene functionalities were introduced on linear polystyrene (M_n = 5300 g/mol, M_w/M_n = 1.05) by acetylation and reaction with potassium hydroxide, 18‐crown‐6 and propargyl bromide in toluene. Polymerization of styrene with 6‐tert‐butyldimethylsiloxyhexyllithium yielded polystyrene (M_n = 5200 g/mol, M_w/M_n = 1.09) with a protected hydroxyl chain end. Deprotection, followed by conversions to tosyl and azide functionalities, provided the side chain material. Coupling with CuBr and N,N,N′,N″,N″‐pentamethyldiethylenetriamine proceeded in up to 94% yield. Repetition of the grafting cycles led to well‐defined (M_w/M_n ≤ 1.1) polymers of generations G1 and G2 in 84% and 60% yield, respectively, with M_n and branching functionalities reaching 2.8 × 10⁶ g/mol and 460, respectively, for the G2 polymer. Coupling longer (M_n = 45,000 g/mol) side chains with acetylene‐functionalized substrates was also examined. For a linear substrate, a G0 polymer with M_n = 4.6 × 10⁵ g/mol and M_w/M_n = 1.10 was obtained in 87% yield; coupling with the G0 (M_n = 52,000 g/mol) substrate produced a G1 polymer (M_n = 1.4×10⁶ g/mol, M_w/M_n = 1.38) in 28% yield. The complementary approach using azide‐functionalized substrates and acetylene‐terminated side chains was also investigated, but proceeded in lower yield. © 2019 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2019, 57, 1730–1740     

SIZE EXCLUSION CHROMATOGRAPHIC COLUMN PACKED WITH REGENERATED CELLULOSE GELS*

Microporous regenerated cellulose gel particles were prepared by mixing cellulose cuoxam with silk fibroin aspore former, and the mean pore size and pore volume of the pallicles were 525 nm and 7.27 mL g~(-1), respectively. Apreparative size-exclusion chromatography (SEC) column (550 mm×20 mm) packed with the cellulose gel particles wasused for the fractionation of two polysaccharides Dextran 07 (M_w = 7.14×10~4, d= 1.7) and Dextran 50(M_w = 50.5×10~4,d = 3.8) in water phase. The fractionation range of the stationary phase covered M_w from 3×10~3 to 1.1×10~6. The dailythroughput was 2.9 g for Dextran 07 (D07) and 4.3 g for Dextran 50 (D50) with a flow-rate of 1.5 mL min~(-1). The fractionsobtained by using the SEC were analyzed by an analytical SEC combined with laser light scattering (LLS), and thepolydispersity indices of fractions for Dextran 07 and Dextran 50 were determined to be 1.34-1.57 and 1.53-3.36,respectively. The preparative SEC is a simple, rapid, and suitable means not only for the fractionation of polysaccharides inwater but also for other polymers in organic solvents.     

Poly(vinyl fluoride) solution characteristics

Nine unfractionated poly(vinyl fluoride) samples were characterized for molecular weight and polydispersity by means of sedimentation velocity, osmometry, and viscosity measurements. Molecular weights were in the range of 143,000–654,000 and M _w/M _n = 2.5–5.6. The Mark-Houwink (M-H) relation was established as [η] = 6.52 × 10^?5 M^0.80. The M-H exponent is at the Flory-Fox upper limit (0.80), as is characteristic of extended, polar polymers, in good solvents. The unperturbed chain dimensions, characteristic ratio and steric factor were derived by the methods of Stockmayer and Fixman and Kurata and Stockmayer. The steric factor is 1.7, which agrees with data reported for other poly(vinyl halides).     

Polydispersity effects in polymer self-diffusion measured by small-angle neutron scattering

The small-angle neutron scattering (SANS) method for measuring the self-diffusion coefficient D has been analyzed for effects of polydispersity in degree of polymerization for the case of linear polymers diffusing by reptation. Polydispersities corresponding to M_w/M_n = 1.0?10 were considered. It is shown that in all cases a meaningful effective diffusion coefficient D_e can be obtained from the short time recovery of the SANS intensity. This quantity D_e ≤ 1.3 D(M_w), where D(M_w) is the diffusion coefficient of a monodisperse polymer having molecular weight M = M_w. The method relies on SANS intensities extrapolated to zero scattering angle; realistic extrapolation is shown to give rise to quite acceptable errors on the order of 0.05 D_e.     

Arborescent polystyrene‐graft‐poly(tert‐butyl methacrylate) copolymers: Synthesis and enhanced polyelectrolyte effect in solution

A technique is described for the preparation of arborescent graft copolymers containing poly(tert‐butyl methacrylate) (PtBMA) segments. For this purpose, tert‐butyl methacrylate is first polymerized with 1,1‐diphenyl‐2‐methylpentyllithium in tetrahydrofuran. The graft copolymers are obtained by addition of a solution of a bromomethylated polystyrene substrate to the living PtBMA macroanion solution. Copolymers incorporating either short (M_w ≈ 5000) or long (M_w ≈ 30,000) PtBMA side chains were prepared by grafting onto linear, comb‐branched (G0), G1, and G2 bromomethylated arborescent polystyrenes. Branching functionalities ranging from 9 to 4500 and molecular weights ranging from 8.8 × 10⁴ to 6.3 × 10⁷ were obtained for the copolymers, while maintaining a low apparent polydispersity index (M_w/M_n ≈ 1.14–1.25). Arborescent polystyrene‐graft‐poly(methacrylic acid) (PMAA) copolymers were obtained by hydrolysis of the tert‐butyl methacrylate units. Dynamic light scattering measurements showed that the arborescent PMAA copolymers are more expanded than their linear PMAA analogues when neutralized with NaOH. This effect is attributed to the higher charge density in the branched arborescent copolymer structures. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 2335–2346, 2008     

Emulsion polymerization of vinyl acetate using iodine‐transfer and RAFT radical polymerizations

This study deals with control of the molecular weight and molecular weight distribution of poly(vinyl acetate) by iodine‐transfer radical polymerization and reversible addition‐fragmentation transfer (RAFT) emulsion polymerizations as the first example. Emulsion polymerization using ethyl iodoacetate as the chain transfer agent more closely approximated the theoretical molecular weights than did the free radical polymerization. Although ¹H NMR spectra indicated that the peaks of α‐ and ω‐terminal groups were observed, the molecular weight distributions show a relatively broad range (M_w/M_n = 2.2–4.0). On the other hand, RAFT polymerizations revealed that the dithiocarbamate 7 is an excellent candidate to control the polymer molecular weight (M_n = 9.1 × 10³, M_w/M_n = 1.48), more so than xanthate 1 (M_n = 10.0 × 10³, M_w/M_n = 1.89) under same condition, with accompanied stable emulsions produced. In the M_n versus conversion plot, M_n increased linearly as a function of conversion. We also performed seed‐emulsion polymerization using poly(nonamethylene L ‐tartrate) as the chiral polyester seed to fabricate emulsions with core‐shell structures. The control of polymer molecular weight and emulsion stability, as well as stereoregularity, is also discussed. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2013     

Additivity Factors in the Binding of the Diethyltin(IV) Cation by Ligands Containing Amino and Carboxylic Groups at Different Ionic Strengths

Complex formation constants were determined potentiometrically (by a ISE-H⁺, glass electrode) in the systems, M²⁺ – L^z – H⁺ [M²⁺ = (C₂H₅)₂Sn²⁺, L^z = malonate, glycinate and ethylenediamine] at t = 25 ^∘C and 0.1 mol-L⁻¹≤ I/ ≤ 1 mol-L⁻¹ in NaCl_aq (0.1 mol-L⁻¹ ≤ I ≤ 0.75 mol-L⁻¹ for the ethylenediamine system). Thermodynamic values of formation constants, at infinite dilution, are [± 95% confidence interval, ^Tβ_pqr refer to the equilibrium, pM²⁺ + qL^z + rH⁺ = M_pL_qH_r^(2+z+r)]: for malonate, log₁₀ ^Tβ₁₁₀ = (5.47 ± 0.10); for glycinate, log₁₀ ^Tβ₁₁₀ = (9.54 ± 0.08), log₁₀ ^Tβ₁₁₁ = (12.97 ± 0.10); and for ethylenediamine, log₁₀ ^Tβ₁₁₀ = (10.47 ± 0.10), log₁₀ ^Tβ₁₂₀ = (16.17 ± 0.12) and log₁₀ ^Tβ₁₁₁ = (15.46 ± 0.10). The dependence on ionic strength of the formation constants was modeled by a simple Debye–Hückel type equation and by the SIT approach. By analyzing the stability of the species in the three different systems we found a simple additivity rule that can be expressed by the relationship: log₁₀ K = 6.46 n_N + 3.96 n_O − 0.60 (n_N²+ n_O²), with a mean deviation, ε(log₁₀ K) = 0.15 (K = equilibrium constant for the interaction of the organometal cation with the unprotonated or protonated ligand, n_N = number of amino groups and n_O = number of carboxylic groups of the ligand(s) involved in the formation reaction of complex species).     

Dendritic growth of poly(ε-caprolactone) crystals from compatible blends with poly(t-butyl acrylate) at the air/water interface

Thermodynamic analyses of surface pressure-area (Π-A) isotherms and Brewster angle microscopy (BAM) reveal that poly(ε-caprolactone) (PCL) with a weight average molar mass of M_w = 10 kg mol⁻¹ and polydispersity index of M_w/M_n = 1.25 and poly(t-butyl acrylate) (PtBA, M_w = 25.7 kg mol⁻¹; M_w/M_n = 1.07) form compatible blends as Langmuir films below the dynamic collapse transition for PCL at Π = 11 mN m⁻¹. For PCL-rich blends, in situ BAM studies reveal growth of PCL crystals for compression past the PCL collapse transition. PCL crystals grown in the plateau regime of the Π-A isotherm exhibit a dendritic morphology presumably resulting from the rejection of PtBA from the growing PCL crystals and hindered diffusion of PCL from the surrounding monolayer to the crystal growth fronts. The ability to transfer the PCL dendrites as Langmuir–Schaefer films onto silicon substrates spincoated with a polystyrene layer facilitates detailed morphological characterization by optical and atomic force microscopy (AFM). AFM reveals that the dendritic branching occurs along the {100} and {110} sector boundaries and is essentially independent of composition. AFM also reveals that the average thickness of PCL dendrites formed at room temperature (22.5 °C), ∼7–8 nm, is comparable with that of PCL crystals grown from single-component PCL Langmuir films and spincoated thin films. In contrast, for PtBA-rich blend films PCL crystallization is suppressed. These findings establish PCL blends as an ideal system for exploring the interplay between chain diffusion and crystal growth in a two-dimensional confined geometry. © 2007 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 45: 3300–3318, 2007     

Characterization of a single sample of an unknown polymer by size exclusion chromatography

The relationship between viscosity constants, k', a and K_η from the equations of Huggins and Mark-Kuhn-Houwink has been considered. It is shown, theoretically, that the sum of k' and a must be constant for all flexible-chain macromolecules irrespective of the solvent used. On this basis, a combination of chromatography and viscometry measurements can be used to characterize a new species. The method has been applied to the new polymer, poly[methyl(pyridin-3-yl) siloxane] ( 1 ) where no suitable calibration standards are available. The value of a, k', and K_η for 1 has been calculated. The calculated constants enabled an estimation of different average molecular weights (M_nM_wM_z) and polydispersity (M_w/M_n) from a minimum of experimental data. The new method is general and can be applied to any homogeneous linear flexible-chain nondraining macromolecule.