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Phys Rev E 2021 Jan;103(1-1):012501

Department of Physics, University of Prince Edward Island, 550 University Avenue, Charlottetown, Prince Edward Island C1A 4P3, Canada.

Polymers confined to a narrow channel are subject to strong entropic forces that tend to drive the molecules apart. In this study, we use Monte Carlo computer simulations to study the segregation behavior of two flexible hard-sphere polymers under confinement in a cylindrical channel. We focus on the effects of using polymers of different lengths. We measure the variation of the conformational free energy, F, with the center-of-mass separation distance, λ. The simulations reveal four different separation regimes, characterized by different scaling properties of the free energy with respect to the polymer lengths and the channel diameter, D. We propose a regime map in which the state of the system is determined by the values of the quantities N_{2}/N_{1} and λ/(N_{1}+N_{2})D^{-β}, where N_{1} and N_{2} are the polymer lengths, and where β≈0.64. The observed scaling behavior of F(λ) in each regime is in reasonable agreement with predictions using a simple theoretical model. In addition, we use MC dynamics simulations to study the segregation dynamics of initially overlapping polymers by measurement of the incremental mean first-passage time with respect to λ. For systems characterized by a wide range of λ in which a short polymer is nested within a longer one, the segregation dynamics are close to that expected for two noninteracting one-dimensional random walkers undergoing unbiased diffusion. When the free-energy gradient is large, segregation is rapid and characterized by out-of-equilibrium effects.

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http://dx.doi.org/10.1103/PhysRevE.103.012501 | DOI Listing |

January 2021

Phys Rev E 2020 Nov;102(5-1):052502

Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island, C1A 4P3, Canada.

Monte Carlo simulations are used to study the conformational behavior of a semiflexible polymer confined to cylindrical and conical channels. The channels are sufficiently narrow that the conditions for the Odijk regime are marginally satisfied. For cylindrical confinement, we examine polymers with a single knot of topology 3_{1}, 4_{1}, or 5_{1}, as well as unknotted polymers that are capable of forming S loops. We measure the variation of the free energy F with the end-to-end polymer extension length X and examine the effect of varying the polymer topology, persistence length P, and cylinder diameter D on the free-energy functions. Similarly, we characterize the behavior of the knot span along the channel. We find that increasing the knot complexity increases the typical size of the knot. In the regime of low X, where the knot/S-loop size is large, the conformational behavior is independent of polymer topology. In addition, the scaling properties of the free energy and knot span are in agreement with predictions from a theoretical model constructed using known properties of interacting polymers in the Odijk regime. We also examine the variation of F with the position of a knot in conical channels for various values of the cone angle α. The free energy decreases as the knot moves in a direction where the cone widens, and it also decreases with increasing α and with increasing knot complexity. The behavior is in agreement with predictions from a theoretical model in which the dominant contribution to the change in F is the change in the size of the hairpins as the knot moves to the wider region of the channel.

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http://dx.doi.org/10.1103/PhysRevE.102.052502 | DOI Listing |

November 2020

Phys Rev E 2019 Jul;100(1-1):012504

Department of Physics, University of Prince Edward Island, 550 University Avenue, Charlottetown, Prince Edward Island, Canada C1A 4P3.

Monte Carlo simulations are used to study the translocation of a polymer into a cavity. Modeling the polymer as a hard-sphere chain with a length up to N=601 monomers, we use a multiple-histogram method to measure the variation of the conformational free energy of the polymer with respect to the number of translocated monomers. The resulting free-energy functions are then used to obtain the confinement free energy for the translocated portion of the polymer. We characterize the confinement free energy for a flexible polymer in cavities with constant cross-sectional area A for various cavity shapes (cylindrical, rectangular, and triangular) as well as for tapered cavities with pyramidal and conical shape. The scaling of the free energy with cavity volume and translocated polymer subchain length is generally consistent with predictions from simple scaling arguments, with small deviations in the scaling exponents likely due to finite-size effects. The confinement free energy depends strongly on cavity shape anisometry and is a minimum for an isometric cavity shape with a length-to-width ratio of unity. Entropic depletion at the edges or vertices of the confining cavity are evident in the results for constant-A and pyramidal cavities. For translocation into infinitely long cones, the scaling of the free energy with taper angle is consistent with a theoretical prediction employing the blob model. We also examine the effects of polymer bending rigidity on the translocation free energy for cylindrical cavities. For isometric cavities, the observed scaling behavior is in partial agreement with theoretical predictions, with discrepancies arising from finite-size effects that prevent the emergence of well-defined scaling regimes. In addition, translocation into highly anisometric cylindrical cavities leads to a multistage folding process for stiff polymers. Finally, we examine the effects of crowding agents inside the cavity. We find that the confinement free energy increases with crowder density. At constant packing fraction the magnitude of this effect lessens with increasing crowder size for a crowder-to-monomer size ratio ≥1.

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http://dx.doi.org/10.1103/PhysRevE.100.012504 | DOI Listing |

July 2019

Phys Rev E 2019 Jun;99(6-1):062503

Department of Physics, University of Prince Edward Island, 550 University Avenue, Charlottetown, Prince Edward Island, C1A 4P3, Canada.

We describe a simple Monte Carlo simulation method to calculate the free-energy cost of localizing a single monomer of a polymer confined to a cavity. The localization position is chosen to be on the inside surface of the confining cavity. The method is applied to a freely jointed hard-sphere polymer chain confined to cavities of spherical and cubic geometries. In the latter case, we consider localization at a corner and at the center of a face of the confining cube. We consider cases of end-monomer localization both with and without tethering of the other end monomer to a point on the surface. We also examine localization of monomers at arbitrary positions along the contour of the polymer. We characterize the dependence of the free energy on the cavity size and shape, the localization position, and the polymer length. The quantitative trends can be understood using standard scaling arguments and use of a simple theoretical model. The results are relevant to those theories of polymer translocation that focus on the importance of the free-energy barrier as the translocation process requires an initial localization of a monomer to the position of a nanopore.

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http://dx.doi.org/10.1103/PhysRevE.99.062503 | DOI Listing |

June 2019

Soft Matter 2018 Aug;14(30):6360-6373

Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island C1A 4P3, Canada.

Monte Carlo computer simulations are used to study the segregation behaviour of two polymers under cylindrical confinement. Using a multiple-histogram method, the conformational free energy, F, of the polymers was measured as a function of the centre-of-mass separation distance, λ. We examined the scaling of the free energy functions with the polymer length, the length and diameter of the confining cylinder, the polymer topology (i.e. linear vs. ring polymers), and the packing fraction and size of mobile crowding agents. In the absence of crowders, the observed scaling of F(λ) is similar to that predicted using a simple model employing the de Gennes blob model and the approximation that the free energy of overlapping chains in a tube is equal to that of two isolated chains each in a tube of half the cross-sectional area. Simulations were used to test the latter approximation and reveal that it yields poor quantitative predictions. This, along with generic finite-size effects, likely gives rise to the discrepancies between the predicted and measured values of scaling exponents for F(λ). For segregation in the presence of crowding agents, the free energy barrier generally decreases with increasing crowder packing fraction, thus reducing the entropic forces driving segregation. However, for fixed packing fraction, the barrier increases as the crowder/monomer size ratio decreases.

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http://dx.doi.org/10.1039/c8sm01062e | DOI Listing |

August 2018

J Chem Phys 2015 May;142(17):174903

Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island C1A 4P3, Canada.

Monte Carlo simulations are used to study the translocation of a polymer into and out of an ellipsoidal cavity through a narrow pore. We measure the polymer free energy F as a function of a translocation coordinate, s, defined to be the number of bonds that have entered the cavity. To study polymer insertion, we consider the case of a driving force acting on monomers inside the pore, as well as monomer attraction to the cavity wall. We examine the changes to F(s) upon variation in the shape anisometry and volume of the cavity, the polymer length, and the strength of the interactions driving the insertion. For athermal systems, the free energy functions are analyzed using a scaling approach, where we treat the confined portion of the polymer to be in the semi-dilute regime. The free energy functions are used with the Fokker-Planck (FP) equation to calculate mean translocation times, as well as translocation time distributions. We find that both polymer ejection and insertion are faster for ellipsoidal cavities than for spherical cavities. The results are in qualitative agreement with those of a Langevin dynamics study in the case of ejection but not for insertion. The discrepancy is likely due to out-of-equilibrium conformational behaviour that is not accounted for in the FP approach.

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http://dx.doi.org/10.1063/1.4919642 | DOI Listing |

May 2015

J Chem Phys 2014 Oct;141(16):164902

Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island C1A 4P3, Canada.

Monte Carlo simulations are used to study the behavior of two polymers under confinement in a cylindrical tube. Each polymer is modeled as a chain of hard spheres. We measure the free energy of the system, F, as a function of the distance between the centers of mass of the polymers, λ, and examine the effects on the free energy functions of varying the channel diameter D and length L, as well as the polymer length N and bending rigidity κ. For infinitely long cylinders, F is a maximum at λ = 0, and decreases with λ until the polymers are no longer in contact. For flexible chains (κ = 0), the polymers overlap along the cylinder for low λ, while above some critical value of λ they are longitudinally compressed and non-overlapping while still in contact. We find that the free energy barrier height, ΔF ≡ F(0) - F(∞), scales as ΔF/k(B)T ∼ ND(-1.93 ± 0.01), for N ⩽ 200 and D ⩽ 9σ, where σ is the monomer diameter. In addition, the overlap free energy appears to scale as F/k(B)T = Nf(λ/N; D) for sufficiently large N, where f is a function parameterized by the cylinder diameter D. For channels of finite length, the free energy barrier height increases with increasing confinement aspect ratio L/D at fixed volume fraction ϕ, and it decreases with increasing ϕ at fixed L/D. Increasing the polymer bending rigidity κ monotonically reduces the overlap free energy. For strongly confined systems, where the chain persistence length P satisfies D ≪ P, F varies linearly with λ with a slope that scales as F'(λ) ∼ -k(B)TD(-β)P(-α), where β ≈ 2 and α ≈ 0.37 for N = 200 chains. These exponent values deviate slightly from those predicted using a simple model, possibly due to insufficiently satisfying the conditions defining the Odijk regime. Finally, we use Monte Carlo dynamics simulations to examine polymer segregation dynamics for fully flexible chains and observe segregation rates that decrease with decreasing entropic force magnitude, f ≡ |dF/dλ|. For both infinite-length and finite-length channels, the polymers are not conformationally relaxed at later times during segregation.

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http://dx.doi.org/10.1063/1.4898714 | DOI Listing |

October 2014

J Chem Phys 2014 May;140(18):184904

Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown,Prince Edward Island C1A 4P3, Canada.

Brownian dynamics (BD) simulations are used to study the translocation dynamics of a coarse-grained polymer through a cylindrical nanopore. We consider the case of short polymers, with a polymer length, N, in the range N = 21-61. The rate of translocation is controlled by a tunable friction coefficient, γ0p, for monomers inside the nanopore. In the case of unforced translocation, the mean translocation time scales with polymer length as <τ1> ∼ (N - Np)(α), where Np is the average number of monomers in the nanopore. The exponent approaches the value α = 2 when the pore friction is sufficiently high, in accord with the prediction for the case of the quasi-static regime where pore friction dominates. In the case of forced translocation, the polymer chain is stretched and compressed on the cis and trans sides, respectively, for low γ0p. However, the chain approaches conformational quasi-equilibrium for sufficiently large γ0p. In this limit the observed scaling of <τ1> with driving force and chain length supports the Fokker-Planck (FP) prediction that <τ> ∝ N/fd for sufficiently strong driving force. Monte Carlo simulations are used to calculate translocation free energy functions for the system. The free energies are used with the FP equation to calculate translocation time distributions. At sufficiently high γ0p, the predicted distributions are in excellent agreement with those calculated from the BD simulations. Thus, the FP equation provides a valid description of translocation dynamics for sufficiently high pore friction for the range of polymer lengths considered here. Increasing N will require a corresponding increase in pore friction to maintain the validity of the FP approach. Outside the regime of low N and high pore friction, the polymer is out of equilibrium, and the FP approach is not valid.

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http://dx.doi.org/10.1063/1.4874976 | DOI Listing |

May 2014

J Chem Phys 2013 May;138(17):174902

Monte Carlo (MC) simulations are used to study the dynamics of polymer translocation through a nanopore in the limit where the translocation rate is sufficiently slow that the polymer maintains a state of conformational quasi-equilibrium. The system is modeled as a flexible hard-sphere chain that translocates through a cylindrical hole in a hard flat wall. In some calculations, the nanopore is connected at one end to a spherical cavity. Translocation times are measured directly using MC dynamics simulations. For sufficiently narrow pores, translocation is sufficiently slow that the mean translocation time scales with polymer length N according to <τ> ∝ (N - N(p))(2), where N(p) is the average number of monomers in the nanopore; this scaling is an indication of a quasi-static regime in which polymer-nanopore friction dominates. We use a multiple-histogram method to calculate the variation of the free energy with Q, a coordinate used to quantify the degree of translocation. The free energy functions are used with the Fokker-Planck formalism to calculate translocation time distributions in the quasi-static regime. These calculations also require a friction coefficient, characterized by a quantity N(eff), the effective number of monomers whose dynamics are affected by the confinement of the nanopore. This was determined by fixing the mean of the theoretical distribution to that of the distribution obtained from MC dynamics simulations. The theoretical distributions are in excellent quantitative agreement with the distributions obtained directly by the MC dynamics simulations for physically meaningful values of N(eff). The free energy functions for narrow-pore systems exhibit oscillations with an amplitude that is sensitive to the nanopore length. Generally, larger oscillation amplitudes correspond to longer translocation times.

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http://dx.doi.org/10.1063/1.4803022 | DOI Listing |

May 2013

J Chem Phys 2013 Jan;138(2):024906

Department of Physics, University of Prince Edward Island, 550 University Ave., Charlottetown, Prince Edward Island, C1A 4P3, Canada.

Monte Carlo simulations are used investigate the properties of the free energy barrier associated with polymer translocation through a nanopore. We employ a multiple-histogram method to calculate the variation of the free energy with Q, a coordinate used to quantify the degree of translocation. The system is modeled as a flexible hard-sphere chain that translocates through a cylindrical hole in a hard flat wall. Some calculations were carried out for nanopores connected to a spherical cavity at one or both ends. Attractive monomer-nanopore interactions and a linear driving force through the nanopore were also included in some calculations. The properties of the free energy functions for short polymers were studied upon variation in all of the key system parameters, including polymer length, the nanopore dimensions, the strengths of the attractive, and driving force interactions. The results were analyzed using a simple theoretical model, whose only free parameter is the confinement free energy per link for monomers inside the nanopore. Generally, the results are in excellent quantitative agreement with the model. One notable feature of the free energy functions is the presence of oscillations whose amplitude increases with decreasing pore radius. These oscillations are due to the nature of the variation with Q of the orientational entropy of bonds at the two edges of the pore. A simple model was constructed to account for dependence of the oscillation amplitude and period on the system parameters. We propose that the theoretical models developed here can be used to make quantitatively accurate predictions of translocation free energy functions for very long polymers using simulation data acquired for short polymers.

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http://dx.doi.org/10.1063/1.4774118 | DOI Listing |

January 2013

Cell Biochem Biophys 2013 May;66(1):29-36

Department of Physics, University of Prince Edward Island, Charlottetown, PE, C1A 4P3, Canada.

In this study, we employ the discontinuous molecular dynamics simulation method to investigate the collapse properties of a single heteropolymer chain in an explicit solvent. Solvent density ρ, fraction of hydrophobic monomers n H (defined as the ratio of the number of hydrophobic monomers to the total number of monomers) and a hydrophobicity parameter λ (which controls the energy mismatch between the monomers and solvent particles) were systematically varied to examine their role in polymer collapse. The average static structure factor of the polymer was used to find the so-called θ-point characterizing the state of an ideal chain. Phase diagrams of ρ versus λ for the coil-globule transition were mapped out for different values of n H. Increasing the fraction of hydrophobic monomers n H, solvent density ρ, and hydrophobicity parameter λ were all shown to aid in stabilizing the globule phase. In an effort to explore scaling behaviour of the coil-globule phase diagram as a function of n H, and to investigate whether the phase boundaries for different n H collapsed on to one universal curve, we rescaled λ by n H (δ) λ; we determined δ = 1.72, in contrast to mean-field predictions of δ = 2.0.

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http://dx.doi.org/10.1007/s12013-012-9394-6 | DOI Listing |

May 2013

Phys Rev E Stat Nonlin Soft Matter Phys 2010 Nov 30;82(5 Pt 1):051931. Epub 2010 Nov 30.

Department of Physics, University of Prince Edward Island, Charlottetown, Prince Edward Island, Canada C1A 4P3.

We have used Brownian dynamics simulations to study the dynamics of a bead-and-spring polymer subject to a flashing ratchet potential. To elucidate the role of hydrodynamic (HD) interactions, simulations were carried out for the cases where HD interactions are present and when they are absent. The average speed of the polymer and its conformational properties were examined upon variation in the polymer length, N, and the ratchet spatial period, L. Two distinct dynamical regimes were evident. In the low-N/high-L regime, the velocity decreases with increasing N, and center-of-mass diffusion is a key part of the motional mechanism. By contrast, in the high-N /low-L regime, the velocity is insensitive to variation in N, and motion is achieved via the coupling of internal modes to the cycling of the ratchet potential. The location of the regimes is correlated with the average conformational state of the polymer. Incorporating HD interactions increases the average polymer velocity for all polymer lengths and ratchet spatial periods considered. The dynamical behavior of polymers in the low-N/high-L regime can be understood using simple a theoretical model that yields quantitatively reasonable predictions of the polymer velocity.

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http://dx.doi.org/10.1103/PhysRevE.82.051931 | DOI Listing |

November 2010

J Chem Phys 2009 Jun;130(24):244902

The coil-globule transition of a polymer in a solvent has been studied using Monte Carlo simulations of a single chain subject to intramolecular interactions as well as a solvent-mediated effective potential. This solvation potential was calculated using several different theoretical approaches for two simple polymer/solvent models, each employing hard-sphere chains and hard-sphere solvent particles as well as attractive square-well potentials between some interaction sites. For each model, collapse is driven by variation in a parameter which changes the energy mismatch between monomers and solvent particles. The solvation potentials were calculated using two fundamentally different methodologies, each designed to predict the conformational behavior of polymers in solution: (1) the polymer reference interaction site model (PRISM) theory and (2) a many-body solvation potential (MBSP) based on scaled particle theory introduced by Grayce [J. Chem. Phys. 106, 5171 (1997)]. For the PRISM calculations, two well-studied solvation monomer-monomer pair potentials were employed, each distinguished by the closure relation used in its derivation: (i) a hypernetted-chain (HNC)-type potential and (ii) a Percus-Yevick (PY)-type potential. The theoretical predictions were each compared to results obtained from explicit-solvent discontinuous molecular dynamics simulations on the same polymer/solvent model systems [J. Chem. Phys. 125, 194904 (2006)]. In each case, the variation in the coil-globule transition properties with solvent density is mostly qualitatively correct, though the quantitative agreement between the theory and prediction is typically poor. The HNC-type potential yields results that are more qualitatively consistent with simulation. The conformational behavior of the polymer upon collapse predicted by the MBSP approach is quantitatively correct for low and moderate solvent densities but is increasingly less accurate for higher densities. At high solvent densities, the PRISM-HNC and MBSP approaches tend to overestimate, while the PRISM-PY approach underestimates the tendency of the solvent to drive polymer collapse.

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http://dx.doi.org/10.1063/1.3153350 | DOI Listing |

June 2009

J Chem Phys 2006 Nov;125(19):194904

Department of Physics, University of Prince Edward Island, 550 University Avenue, Charlottetown, Prince Edward Island C1A 4P3, Canada.

Discontinuous molecular dynamics simulations were used to study the coil-globule transition of a polymer in an explicit solvent. Two different versions of the model were employed, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambda(theta) separating the coil from the globule state. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. Additionally, we explore for each model the effects of varying the range of the attractive interactions on the phase boundary separating the coil and globule phases. The results are analyzed in terms of a simple Flory-type theory of the collapse transition.

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http://dx.doi.org/10.1063/1.2388270 | DOI Listing |

November 2006

J Chem Phys 2006 May;124(18):184905

Department of Physics, University of Prince Edward Island, 550 University Avenue, Charlottetown, Prince Edward Island, C1A 4P3, Canada.

Molecular dynamics simulations were used to study the conformational dynamics of a bead-spring model polymer in an explicit solvent under good solvent conditions. The dynamics of the polymer chain were investigated using an analysis of the time autocorrelation functions of the Rouse coordinates of the polymer chain. We have investigated the variation of the correlation functions with polymer chain length N, solvent density rho, and system size. The measured initial decay rates gamma(p) of the correlation functions were compared with the predictions from a theory of polymer dynamics which uses the Oseen tensor to describe hydrodynamic interactions between monomers. Over the range of chain lengths considered (N = 30-60 monomers), the predicted scaling of gamma(p) proportional to N(-3nu) was observed at high rho, where nu is the polymer scaling exponent. The predicted gamma(p) are generally higher than the measured values. This discrepancy increases with decreasing rho, as a result in the breakdown in the conditions required for the Oseen approximation. The agreement between theory and simulation at high rho improves considerably if the theoretical expression for gamma(p) is modified to avoid sum-to-integral approximations, and if the values of (R(p)2), which are used in the theory, are taken directly from the simulation rather than being calculated using approximate scaling relations. The observed finite-size scaling of gamma(p) is not quantitatively consistent with the theoretical predictions.

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http://dx.doi.org/10.1063/1.2194903 | DOI Listing |

May 2006

J Chem Phys 2005 Jan;122(2):024905

Department of Physics, University of Prince Edward Island, 550 University Avenue, Charlottetown, Prince Edward Island, C1A 4P3, Canada.

Molecular dynamics simulations are used to study the coil-globule transition for a system composed of a bead-spring polymer immersed in an explicitly modeled solvent. Two different versions of the model are used, which are differentiated by the nature of monomer-solvent, solvent-solvent, and nonbonded monomer-monomer interactions. For each case, a model parameter lambda determines the degree of hydrophobicity of the monomers by controlling the degree of energy mismatch between the monomers and solvent particles. We consider a lambda-driven coil-globule transition at constant temperature. The simulations are used to calculate average static structure factors, which are then used to determine the scaling exponents of the system in order to determine the theta-point values lambdatheta separating the coil from the globule states. For each model we construct coil-globule phase diagrams in terms of lambda and the particle density rho. The results are analyzed in terms of a simple Flory-type theory of the collapse transition. The ratio of lambdatheta for the two models converges in the high density limit exactly to the value predicted by the theory in the random mixing approximation. Generally, the predicted values of lambdatheta are in reasonable agreement with the measured values at high rho, though the accuracy improves if the average chain size is calculated using the full probability distribution associated with the polymer-solvent free energy, rather than merely using the value obtained from the minimum of the free energy.

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http://dx.doi.org/10.1063/1.1830435 | DOI Listing |

January 2005