Solution of a Characteristic Value Problem from the Theory of Chain Molecules

Zimm, Bruno H.; Roe, Glenn M.; Epstein, Leo F.

doi:10.1063/1.1742463

Cited by 150 publications

(50 citation statements)

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“…where N A is Avogadro's number, M is molecular weight of the polymer, and j Ј is an eigenvalue from diagonalizing the position-to-velocity transformation matrix in the analytic theory 4,20,93 which is given in the following:…”

Section: B the Viscositymentioning

confidence: 99%

Effect of cyclic chain architecture on properties of dilute solutions of polyethylene from molecular dynamics simulations

Jang

Çağın

Goddard

2003

The Journal of Chemical Physics

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We have used molecular dynamics methods to investigate the effects of cyclic chain architecture on the properties of dilute solutions. In order to include solvent effects in estimating these properties, we use a van der Waals scaling factor determined for each solvent by matching to the theta condition. We predict that the theta temperature ͑͒ of cyclic PE ͑c-PE͒ is ϳ10% lower than for the linear case ͑l-PE͒. This can be compared to the experimental results for polystyrene ͑PS͒, where for cyclic PS is 2% lower. For conditions corresponding to n-pentane solvent, we predict that ͗R g 2 ͘ cyclic /͗R g 2 ͘ linear is 0.59 for all temperatures above 350 K. The deviation from the ratio of 0.50-0.53 expected from analytic theory is due to the competition between chain stiffness and excluded volume effects. To calculate the intrinsic viscosity of c-PE and l-PE we extended the Bloomfield-Zimm type theory to include chain stiffness corrections. We find that for the theta temperature, the ratio of viscosities for c-PE and l-PE is 0.71, which is 7% higher than the value of 0.66 from the freely jointed chain model. This difference is caused by the larger value of ͗R g 2 ͘ cyclic /͗R g 2 ͘ linear from the simulations.

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Section: B the Viscositymentioning

confidence: 99%

Effect of cyclic chain architecture on properties of dilute solutions of polyethylene from molecular dynamics simulations

Jang

Çağın

Goddard

2003

The Journal of Chemical Physics

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“…Equation 3 completely differs from the ori gin a l e stim ate [3] that in first approximation the diagonal ele ments of the matrix M equal the eigenvalues ApR of th e Rouse model with va n ishing hydrodynamic interac tion , i. e. For h* = 0 (Rouse) the value s of both theories co inc id e.…”

Section: Thi Smentioning

confidence: 99%

“…Such a n estimate was based [3] on the s upposition that the tran sformation matrix Q c hanges so little by the introduc tion of hydrod ynamic interac tion that M remains practically th e same as in th e free draining case, i. e. M = A R ·…”

Section: Thi Smentioning

confidence: 99%

“…The main c ha nge introd uced by th e new soluti on as compared with th e old er incomplete soluti ons [3][4][5][6][7][8] is not in the eigenvalues Ap of H A Q -J H A Q = A (1) which were already calcul ated correctly in rece nt papers [6][7][8] and even tabulated for Z between 1 and 15 and h* = (3/7r)112 ah / bO = .01, .1, .2, .3, [6] and forZ = 250, h* = .3 and Z = 300, h* = 0.4 [1]. Here ah is the hydrodynamic radius of the bead, Z + 1 is the number of beads , Z is the number and bo the root mean square length of the links.…”

Section: Thi Smentioning

confidence: 99%

See 1 more Smart Citation

Frequency dependence of intrinsic stress and birefringence tensor of bead/spring model of polymer solutions

Peterlin¹,

Fong²

1977

J. RES. NATL. BUR. STAN. SECT. A.

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T he rece ntl y obtain ed c ompl e te so lution of Ih e s irn ult an eous diagonalizali on of matri ces H A a nd H in the hydrody na mi c diffu s ion equat ion has bas ic a ll y c han ged the diagona l va lues v" of the sy mme tri c malrix H of hydrod yna mic int e rac ti o n be lween a ll the beads of the e lasti c ra ndom c oil mod e l of th e iso la te d mac romo lecul e in soluti on. Si nce th ese va lu es ent er exp li citl y th e expression s for th e intrin sic stress and refracti ve in dex tensor in an a lt e rnating fl ow fi e ld if based on the conce pt o f inte rnal vi scos it y o f the model one had to reca lc ulat e a ll va lues obt a ined form erl y b y us in g the th e n ge ne ra ll y acce pt e d e rroneous se t of 11" data. Th e ne w vp e qua l uni ty independenl of p while th e old va lues were larger than 1 for small p and s ma ll e r for large p. He nce the ir too la rge contributi on s in th e fonn er range are partiall y compensated by th eir too small contributi on s in th e latt e r region. A s a c onsequence in the who le ran ge in vesti gat e d , be tween 3 and 3 00 c ha in link s, th e diffe re nces in rheo logica l and rhe oopti ca l e ffects are relat ive ly s mall, up to a fa c tor of 2, a lthough at hi ghe r link numbe r the diffe re nces le nd to grow wilh the logarithm of this numbe r.Key word s: Bead-sprin g mod el; eigenva lues; frequ ency res pon se ; intrin sic 0Pli cal tensor; intrinsic stress tensor; pol yme r soluti on.

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“…The function p defined in eq 1 satisfies the differential equation/-3 • 11 (4) with ( 5) where the operator L is split into three parts: the first an unperturbed self-ad joint operator A0 with a scalar product defined with weighting function P0 , the second (3A 1 which involves the excluded-volume potential and vanishes in the unperturbed state, and the third Lb which vanishes if the external force field (solvent flow) is absent. We choose as the basis set the eigenfunctions ¢k of the unperturbed free-draining time evolution operator A0°, i.e., the operator A0 without hydrodynamic interactions.…”

Section: Formulationmentioning

confidence: 99%