Article

# .THETA. State, Transition Curves, and Conformational Properties of Cyclic Chains

Department of Experimental Sciences Didactics (Physics, Chemistry, Biology and Geology), Complutense University of Madrid, Madrid, Madrid, Spain

Macromolecules (Impact Factor: 5.8). 03/1995; 28(7). DOI: 10.1021/ma00111a019 **ABSTRACT**

We have performed a Monte Carlo study of dimensions and intrinsic viscosities for isolated three-dimensional cyclic chains of different lengths in solution. These chains are modeled as a number of Gaussian units interacting through a 6-12 Lennard-Jones potential. The reduced energy in the potential well represents the thermodynamic quality of solvent (or the reduced inverse temperature). From this study, the Theta state for this model has been characterized. Then, transition curves and scaling plots have been obtained. Using extrapolations to the long chain limit, different conformational parameters have been estimated and compared with existing experimental data.

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**ABSTRACT:**Ring diblock copolymers with different lengths of the A and B blocks have been investigated using theoretical and numerical simulation methods. The theoretical calculations are based on a first-order expansion in the parameter epsilon = 4 - d and extend to chains in a common Theta solvent, in a common good solvent, and in selective solvents, though only the first case is discussed in detail. The Monte Carlo calculations are performed for a common Theta solvent case with an off-lattice model of Gaussian units which interact through a Lennard-Jones potential. Averaged dimensions for each of the blocks and for the whole chain are obtained, and the theoretical and simulation results are analyzed and discussed. From the results, we have obtained general trends for all these different properties, and the most important differences and similarities with linear or homopolymer chains have been identified. -
##### Article: The collapse transition of a single polymer chain in two and three dimensions: A Monte Carlo study

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**ABSTRACT:**The collapse transition of a single polymer chain in two and three dimensions was studied using the bond‐fluctuation model. The obtained exponents ν of the scaling law 〈S2N〉∼N2ν agree with values proposed in the literature as well as above, at and below the Θ‐temperature TΘ. Transition curves and scaling analysis plots are presented. The scaling function α3SτN1/2 vs τN1/2 has a pronounced maximum before leveling off in the fully collapsed regime in accordance with the theory [α2S=〈S2N〉/〈S2N〉Θ, τ=‖(T−TΘ)/TΘ‖]. An analyzing of the subchain distances leads to disagreements with the blob model. The subchains are locally swollen for T≳TΘ and shrunken for T<TΘ. The probability distribution function of internal distances for T≥TΘ can be described by scaling functions of the form fs(x)∼xκs exp(−Dsxδs) for large x, x being the scaled distance. In contrast for T<TΘ none of these functions describe the data. The dynamic properties above TΘ are in agreement with the Rouse model, but below TΘ differences occur; the center of mass diffusion becomes anomalous and the relaxation times rise with a power law in N of the form τi(N)∼N2+3/d (d being the dimension of space). © 1996 American Institute of Physics. -
##### Article: Collapse of a ring polymer: Comparison of Monte Carlo and Born-Green-Yvon integral equation results

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**ABSTRACT:**The equilibrium properties of an isolated ring polymer are studied using a Born–Green–Yvon (BGY) integral equation and Monte Carlo simulation. The model polymer is composed of n identical spherical interaction sites connected by universal joints of bond length σ. In particular, we study rings composed of up to n = 400 square-well spheres with hard-core diameter σ and well diameter λσ (1⩽λ⩽2). Intramolecular site–site distribution functions and the resulting configurational and energetic properties are computed over a wide range of temperatures for the case of λ=1.5. In the high temperature (good solvent) limit this model is identical to a tangent-hard-sphere ring. With decreasing temperature (worsening solvent) both the radius of gyration and the internal energy of the ring polymer decrease, and a collapse transition is signaled by a peak in the single ring specific heat. In comparison with the Monte Carlo calculations, the BGY theory yields quantitative to semiquantitative results for T≳Tθ and is qualitatively accurate for T≲Tθ, where Tθ is the theta temperature. The thermal behavior of an isolated square-well ring is found to be quite similar to the behavior of an isolated square-well chain. The BGY theory indicates that rings and chains have comparable theta and collapse transition temperatures. In the low temperature limit (collapsed state) the microscopic structure of rings and chains becomes nearly identical.

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