Three-Body Intrachain Collisions in a Single Polymer Chain

Oyama, Tomo o.; Oono, Yoshitsugu

doi:10.1143/jpsj.42.1348

Cited by 22 publications

(12 citation statements)

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“…5 From eq 3, 8, and 11, it is obvious in the @-region that although <R 2 ) N, the distribution function W(R) is not gaussian, since the {33 term is relevant as already pointed out in a previous paper. 3 Equation 14 derived from eq 8 predicts the collapse of a chain as first shown by Ptitsyn. 4 However, this prediction is incomplete, since all terms in eq 3 become relevant in the same order ( /3;N) at v = 1/3.…”

Section: Theory On T -C Diagram Of Polymer Solutionsmentioning

confidence: 88%

“…In eq 3, (N/R 3 )' is a measure of the probability of isegments contact and z, = (N/ R 3 )'v( C) gives the average number of effective i-segments contacts. 3 W 0 (N, R) represents the gaussian character of the unperturbed state and P(N, R, {3,, C) is the perturbation term. Since -kT In W(R) corresponds to the Landau free energy in phase transitions, equilibrium properties are determined by the minimum of -kT In W(R) with respect to the order parameter R.…”

Section: Assumptionsmentioning

confidence: 99%

“…Similarly to well known facts in imperfect gases, 3 it is assumed that near the EJ temperature, {3 2 is proportional to r=(T-EJ)/EJ (EJ is the temperature of /32 = 0) and /33 is positive and nearly constant. Therefore, the r dependence is equivalent to the {3 2 dependence.…”

Section: Assumptionsmentioning

confidence: 99%

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Mean-Field Theory on Temperature–Concentration Diagram of Polymer Solutions

Oyama

Shiokawa

1980

Polym J

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ABSTRACT:The temperature__,;oncentration diagram of polymer solutions was studied by application of the mean-field theory on the basis of certain simple assumptions. In concentrated solutions, it was assumed that the excluded-volume effect decreases proportionally with CRIC, where C R is the concentration of segments belonging to the assigned chain in its sphere and Cis the macroscopic segment concentration. The mean-square radius, cross-over line, osmotic pressure, and correlation length were derived for each of Regions I, I', II, III, and IV defined by Daoud and Jannink. These results fully agreed with those obtained by the scaling laws.KEY WORDS Mean-Field Theory I Excluded-Volume Effect Concentrated Solution I Mean-Square Radius I Cross-Over Line I Osmotic Pressure I Correlation Length I Daoud and Jannink 1 (DJ) derived the temperature--<:oncentration diagram of polymer solutions. A magnetic analogue and the scaling laws were used for the derivation of the diagram. In another paper, 2 DJ et a!. gave more intuitive derivations and pointed out the deviation from the mean-field theory in semidilute solutions.Recent experimental results, especially by neutron scattering, 2 seem to support more the relationships derived by the scaling laws than those derived by the mean-field theory. This means that assumptions used in the mean-field theory up to the present are not applicable to semidilute solutions. The purpose of this paper is an attempt to reconcile these two different methods on the basis of certain simple assumptions. ASSUMPTIONSA polymer chain is assumed to be a sphere of radius R in which N segments (N, the number of segments per chain) are uniformly distributed. The distribution function W(R) is assumed to be as follows.were {3, is the ith virial coefficient of the segments, C(=.NNP/V, NP; the number of polymer chains in the solution) is the segment concentration, b is the segment length and v( C) is the effective volume which contributes to the excluded-volume effect in a chain sphere at concentration C (see the following). We also define the polymer concentration as CP=NP/V and the segment concentration of the appointed chain in its chain sphere as C R = N/ R 3 . In eq 3, (N/R 3 )' is a measure of the probability of isegments contact and z, = (N/ R 3 ) 'v( C) gives the average number of effective i-segments contacts. 3 W 0 (N, R) represents the gaussian character of the unperturbed state and P (N, R, {3,, C) is the perturbation term. Since -kT In W(R) corresponds to the Landau free energy in phase transitions, equilibrium properties are determined by the minimum of -kT In W(R) with respect to the order parameter R.Similarly to well known facts in imperfect gases, 3 it is assumed that near the EJ temperature, {3 2 is proportional to r=(T-EJ)/EJ (EJ is the temperature of /32 = 0) and /33 is positive and nearly constant. Therefore, the r dependence is equivalent to the {3 2 dependence. 507

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Section: Theory On T -C Diagram Of Polymer Solutionsmentioning

confidence: 88%

Section: Assumptionsmentioning

confidence: 99%

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Mean-Field Theory on Temperature–Concentration Diagram of Polymer Solutions

Oyama

Shiokawa

1980

Polym J

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“…Miyaki and Fujita's values 13 of {3 0 (7.2 x 10-23 cm 3 ) and (S 2 ) 0 /M (8.3 x 10-18 cm 2 ) for computing z and f3 from eq 12 and 13. The solid lines in Figure 4 show the relations between -A 2 and T-1 calculated in this way from eq 10 and 11 for the molecular weights of the samples studied.…”

Section: The Second Virial Coefficientmentioning

confidence: 99%

The Second Virial Coefficient of Monodisperse Polystyrene in Cyclohexane below the Theta Point

Tong

Ohashi

Einaga

et al. 1983

Polym J

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ABSTRACT:A light scattering study was made on monodisperse polystyrene + cyclohexane solutions in the range of molecular weight from 10000 to 500000 and that of temperature from 34.5°C (8) down to a value about I oc above the cloud point. The second virial coefficient A2 was found to depend essentially on temperature only, at least in the range of molecular weight examined. This result is not in conformity with any current two-parameter theories for A2 of polymer solutions.KEY As will be noted later, Wolf and Adam's finding does not conform to the prediction of the two-parameter regime in regard to molecular weight dependence. However, it seems too early to accept this as definitive, and more experiments must be undertaken before a general conclusion, if any, may be established on A 2 of polymer solutions below e.In the course of our recent experimental evaluation of the Flory-Huggins interaction parameter, we had an opportunity of carrying out light scattering measurements to estimate A 2 on six monodisperse polystyrene samples in cyclohexane below 8. 8 These samples ranged in molecular weight from 10 x 10 3 to 500 x I 0\ and the measurement on each was effected down to a temperature as close to the cloud point as possible. The finding is presented in this paper. EXPERIMENTAL Polystyrene SamplesWe started with five standard polystyrene samples Fl, F2, F4, F20, F40 supplied by Toyo Soda Co., and one laboratory-made sample F7 (anionically polymerized by Mr. H. Watanabe of our department). These samples were subjected to precipitation and/or column fractionation to sharpen further their already sufficiently narrow molecular weight distributions. The main fractions of the respective series of fractions were chosen as the test samples. By GPC measurements, their ratios of weight-average to number-average molecular weights were found to be smaller than 1.03. Hence, in the present data analysis, any of the samples was treated as completely monodisperse in molecular weight.The molecular weight M determined by light scattering in benzene at 25°C was 10000, 21500, 43600, 181000, 498000, for the samples F1, F2, F4, combined with small angle laser light scattering gave an M of 74000 for the sample F7. Preparation of SolutionsEach polystyrene sample was dissolved in benzene and repeatedly filtered through a Millipore membrane until no light scattered by dust particles could be observed in the laser beam passing through the solution. The filtered solution was then freezedried in a vessel covered with a Millipore film. A proper amount of the dry polymer was weighed into a cylindrical light scattering cell and again thoroughly dried. The solvent cyclohexane was distilled and immediately poured into the cell in a dry bag filled with nitrogen. The cell was then sealed with a ground glass stopper. The solution was left standing overnight at a temperature between 30 and 40°C with occasional shaking. Complete dissolution of the polymer was confirmed by observing transparency of the solution, and also by the absence of unusual inten...

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“…-(e/T)dx*2 + In (1 -d,*) +$,* = o(37) Now it is easy to show that As &* approaches zero, q"(&) approaches zero, and condition 36 no longer holds. The behavior of dx* as a function of TI8 as determined from eq 37 is shown inFigure 1.For T < 8, dX* is finite, and the approximation, eq 31, holds for sufficiently large r. Under these conditions, eq 30 yields (39) (S2) N dr2i3 (40) where c and d are functions of T I 8 only.…”

mentioning

confidence: 99%

Phase Transition Behavior of the Isolated Polymer Chain

1979

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I , , " I 271 (1976).R. Guillard and A. Englert, Biopolymers, 15, 1301 (1976). V. V. Zelinski, V. P. Kolobkov, and L. G. Pikulik, Opt. Spektrosk., 1, 560 (1956). (38) As to the active sphere radius r, ' for intermolecular electron transfer, there are some arguments. If one assumes a uniform distribution of energy acceptors around an energy donor, the condition that the total transfer efficiency calculated by the active sphere model is the same as that for the exact case, yields, ro6 Jrd4*r2 dr = 4*r2 drwhere r, denotes the Forster distance. Integration gives r, ' = 1.16rp The above equation means the radius of the active sphere should be a little larger than the Forster distance. Jabionski (A. Jabronski, Bull. Acad. Pol. Sci., Ser. Sci. Math., Astron. Phys., 6,663 (1958)) reported a similar calculation and suggested r, ' = 1.33r0. On the other hand, in the intramolecular case, the distribution of acceptors cannot be uniform and,. strictly speaking, there can be no simple way to define r,'. In fact, a comparison of static transfer efficiency for the active sphere model (broken line in Figure 3) with that for the exact case (solid line, ro = 24 A) indicates that r, ' should be smaller than ro for n < 12 but larger than ro for n < 12. However, since the difference between the two lines is not significant, it is sufficient to use the Forster distance as an active sphere radius in our approximate analysis. A small change in the active sphere radius did not change any conclusions in the text. The authors wish to thank the referee for informine us of this Doint.ABSTRACT: A mean field theory of chain dimensions is formulated which is very similar to the van der Waals theory of a simple fluid. In the limit of infinite chain length, the chain undergoes a Landau-type second-order phase transition. For finite chains, the transition is pseudo-second-order. At low temperatures, the chain is in a condensed or globular state, and the mean square gyration radius (S2) varies as r2/3 where r is proportional to chain length. At high temperatures, the chain is in a gaslike or coil state where (p) varies as r6/5. In the globular state, fluctuations in (S2) are very small, whereas they are very large in the coiled state. A characteristic feature of the theory is that ternary and higher order intramolecular interactions are approximated. At high temperatures, only binary interactions are important, but a t low temperatures, many of the higher order terms contribute. An important conclusion of this study is that a polymer chain does not obey ideal chain statistics a t the 8 temperature. Although the second virial coefficient vanishes at 8, the third virial coefficient does not; its presence is responsible for the perturbation of the chain statistics.For an infinite chain, 8 and the second-order phase-transition temperature are identical. For finite chains, the pseudo-second-order transition temperature is less than 8. When generalized to d dimensions, the theory yields at low temperatures ( S 2 ) d / 2r for all d and at high temperatur...

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Three-Body Intrachain Collisions in a Single Polymer Chain

Cited by 22 publications

References 26 publications

Mean-Field Theory on Temperature–Concentration Diagram of Polymer Solutions

Mean-Field Theory on Temperature–Concentration Diagram of Polymer Solutions

The Second Virial Coefficient of Monodisperse Polystyrene in Cyclohexane below the Theta Point

Phase Transition Behavior of the Isolated Polymer Chain

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