Related Experiment Video
Updated: May 18, 2026
10:56
Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures
Published on: May 20, 2014
Exact partition function zeros of a polymer on a simple cubic lattice
Jae Hwan Lee1, Seung-Yeon Kim, Julian Lee
1School of Systems Biomedical Science and Department of Bioinformatics and Life Science, Soongsil University, Seoul 156-743, Korea.
Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 26, 2012
Summary
This study reveals two polymer transitions on a lattice: coil-globule collapse and melting-freezing. Analysis of partition function zeros suggests distinct behaviors for each transition.
Area of Science:
- Polymer physics
- Statistical mechanics
- Computational chemistry
Background:
- Understanding polymer conformational transitions is crucial in materials science.
- Lattice models provide simplified yet insightful frameworks for studying polymer behavior.
- Partition function zeros offer a rigorous method to identify phase transitions.
Purpose of the Study:
- To investigate polymer conformational transitions on a simple-cubic lattice.
- To identify and characterize distinct phase transitions using the exact partition function.
- To analyze the scaling behavior and critical phenomena associated with these transitions.
Main Methods:
- Calculation of exact partition function zeros for polymer chains up to length 24.
- Analysis of the loci of zeros in the complex temperature plane.
- Application of finite-size scaling techniques to estimate transition temperatures.
- Supplementary analysis of specific heat to identify transition characteristics.
Main Results:
- Two distinct loci of partition function zeros were identified for longer polymer chains.
- One locus clearly indicates a coil-globule collapse transition, with estimated temperature via finite-size scaling.
- The second locus suggests a melting-freezing transition, exhibiting a first-order-like pseudotransition via specific heat analysis.
Conclusions:
- The study provides strong evidence for two distinct phase transitions in the studied polymer model.
- Finite-size scaling successfully characterizes the coil-globule transition, including logarithmic corrections.
- The melting-freezing transition appears as a pseudotransition, highlighting complex behavior in lattice polymer models.
Related Concept Videos
01:27Real Zeros of Polynomials
Polynomials are algebraic expressions of terms with variables raised to non-negative integer powers. A central aspect of analyzing polynomial functions is determining their real zeros—values of the variable for which the polynomial evaluates to zero. These values represent the x-intercepts of the polynomial’s graph.The Rational Zeros Theorem lists possible rational solutions for a polynomial equation with integer coefficients. If f(x)=anxn+....+a0, then every rational zero is of the form p/q,...
02:33Lattice Centering and Coordination Number
The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
Types of Unit Cells
Imagine taking a large number of identical...
Types of Unit Cells
Imagine taking a large number of identical...
01:29Complex Zeros
Complex zeros are the solutions to polynomial equations that include imaginary numbers, specifically, numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit defined by i2=-1. These zeros satisfy the equation P(x) = 0, where P(x) is a polynomial with real or complex coefficients. Since the complex number system includes all real numbers, it provides a complete framework for analyzing all possible roots of a polynomial.Every polynomial of degree n≥1 can be...
02:46Crystal Field Theory - Tetrahedral and Square Planar Complexes
Tetrahedral Complexes
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
Crystal field theory (CFT) is applicable to molecules in geometries other than octahedral. In octahedral complexes, the lobes of the dx2−y2 and dz2 orbitals point directly at the ligands. For tetrahedral complexes, the d orbitals remain in place, but with only four ligands located between the axes. None of the orbitals points directly at the tetrahedral ligands. However, the dx2−y2 and dz2 orbitals (along the Cartesian axes) overlap with the ligands less than the dxy,...
01:19Routh-Hurwitz Criterion II
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first column of the Routh...
01:17Ziegler–Natta Chain-Growth Polymerization: Overview
Ziegler–Natta polymerization is another form of addition or chain‐growth polymerization used for synthesizing linear polymers over branched polymers. The catalyst used for polymerization is the Ziegler–Natta catalyst, named after Karl Ziegler and Giulio Natta, who developed it in 1953. This catalyst is an organometallic complex of titanium tetrachloride and triethyl aluminum, with the active form of the catalyst being an alkyl titanium compound. Using the Ziegler–Natta catalyst, high molecular...