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Volumn 129, Issue 4, 2008, Pages

Osmotic pressure and polymer size in semidilute polymer solutions under good-solvent conditions

Author keywords

[No Author keywords available]

Indexed keywords

CONCENTRATION (PROCESS); OSMOSIS; POLYMER SOLUTIONS; SOLUTIONS; SOLVENTS;

EID: 49149090721     PISSN: 00219606     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.2955732     Document Type: Article
Times cited : (22)

References (34)
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    • At present the most accurate estimates of are =0.587 58±0.000 07 (Ref.), =0.5874±0.0002 [, 0305-4470 10.1088/0305-4470/34/43/102, ()]; =0.58765±0.00020 [, Macromolecules 0024-9297 10.1021/ma0355958 37, 4658 (2004)]; =0.5876±0.0002 [, J. Phys. A 0305-4470 10.1088/1751-8113/40/26/ F05 40, F539 (2007)]; =0.5876±0.0002 [, J. Phys. A 0305-4470 10.1088/1751-8113/40/36/003 40, 10973 (2007)] (assuming 0.50θ0.53); For an extensive list of older results, see, Phys. Re 368, 549 (2002).
    • At present the most accurate estimates of are =0.587 58±0.000 07 (Ref.), =0.5874±0.0002 [T. Prellberg, J. Phys. A 0305-4470 10.1088/0305-4470/34/43/102 34, L599 (2001)]; =0.58765±0.00020 [H. -P. Hsu, W. Nadler, and P. Grassberger, Macromolecules 0024-9297 10.1021/ma0355958 37, 4658 (2004)]; =0.5876±0.0002 [A. Pelissetto and E. Vicari, J. Phys. A 0305-4470 10.1088/1751-8113/40/26/F05 40, F539 (2007)]; =0.5876±0.0002 [N. Clisby, R. Liang, and G. Slade, J. Phys. A 0305-4470 10.1088/1751-8113/40/ 36/003 40, 10973 (2007)] (assuming 0.50θ0.53); For an extensive list of older results, see A. Pelissetto and E. Vicari, Phys. Rep. 368, 549 (2002).
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    • The residual error on K can be estimated by assuming Kest (M) =K+a/ M4. It follows Δ Kest (M) - Kest (2M) =15a/ (2M) 4, so that Kest (2M) -K=Δ/15. In the two cases we show, the expected systematic error on the value corresponding to M=128 is therefore much smaller than the statistical error.
    • The residual error on K can be estimated by assuming Kest (M) =K+a/ M4. It follows Δ Kest (M) - Kest (2M) =15a/ (2M) 4, so that Kest (2M) -K=Δ/15. In the two cases we show, the expected systematic error on the value corresponding to M=128 is therefore much smaller than the statistical error.
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    • The results of Ref. are expressed in terms of a variable X such that fZ (X) =1+X+O (X2) for X→0. The variable X is related to by X=1.313 [see Eq.].
    • The results of Ref. are expressed in terms of a variable X such that fZ (X) =1+X+O (X2) for X→0. The variable X is related to by X=1.313 [see Eq.].
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    • Equation (17.52) of Ref. gives the following expression for Z: Z=1+1.314 (H1 / H2) 0.309, H1 =1+2.15+1.00 2, and H2 =1+0.51. The variable s used in Ref. is related to by =1.169 s (see Sec. 13.3.2). For →∞ it predicts Z≈1.618 1.309.
    • Equation (17.52) of Ref. gives the following expression for Z: Z=1+1.314 (H1 / H2) 0.309, H1 =1+2.15+1.00 2, and H2 =1+0.51. The variable s used in Ref. is related to by =1.169 s (see Sec. 13.3.2). For →∞ it predicts Z≈1.618 1.309.
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    • This is true if the determination of the number of walks going through a given lattice point takes a time of order 1. This can be obtained by using, e.g., a hash table. A practical implementation is discussed in, (Addison-Wesley, Reading, MA), Vol., see Algorithm C, and Exercise 23 in Sec. 6.4.
    • This is true if the determination of the number of walks going through a given lattice point takes a time of order 1. This can be obtained by using, e.g., a hash table. A practical implementation is discussed in D. E. Knuth, The Art of Computer Programming (Addison-Wesley, Reading, MA, 1973), Vol. 3, see Algorithm C, p. 514, and Exercise 23 in Sec. 6.4.
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