## INTRODUCTION

The interest in describing the transition of a polymer chain from the globular state to the extended coil state arose historically in connection with the problem of denaturation of protein globules.1 The nature of the coil-to-globule transition, as well as the features of globule conformations in regular homo-polymers, outline a frame of reference for analogous studies of biopolymers.2, 3 Aspects of the same physics influence a variety of phenomena with applications in responsive materials,4–6 hetero-polymer and protein folding,7 native DNA packing, polymer network collapse,8 and interpolymer complexes.

Most experiments focused on the mechanism of coil-to-globule transition have been carried out using solutions of polystyrene in cyclohexane9–11 or poly(*N*-isopropyl acrylamide), PNIPAAm, in water.12–17 A number of investigations focused on poly(methyl methacrylate) in appropriate Θ solvents18–20 or on solutions of macromolecules containing polar groups or even of polyelectrolytes in water.21 Various experimental methods, such as static and quasi-elastic light scattering, small angle neutron scattering, and viscometry, have been used for the determination of polymer dimensions. Monitoring individual chain dimensions upon coil-to-globule collapse is, unfortunately, extremely difficult, since at finite concentrations and in poor solvents a multi-chain precipitant may be formed. Utilizing, however, fairly monodisperse very high molecular weight polymer samples [of *O*(10^{7}) g/mol] and extremely low concentrations, the researchers were able to probe the coil-to-globule transition of single chains in a relatively wide temperature range near the phase transition temperature but still in the thermodynamically stable regime. Besides, the two-stage kinetics of single chain collapse has been monitored using ultra-thin capillary-tube cells; a similar behavior was suggested theoretically as well.22, 23

The van der Waals interaction driving the coil-to-globule transition in an organic solvent is very weak; thus, the individual chains can hardly reach the single chain globule state before the interchain aggregation (demixing) occurs.24 This is why it has been very difficult to observe fully collapsed thermodynamically stable single chain globules in those systems. Aqueous solutions of temperature responsive polymers like PNIPAAm, on the other hand, allow the observation of thermodynamically stable globular states of single chains. Actually, the dependence of the chain dimensions on temperature for the aqueous PNIPAAm solutions and the observed hysteresis were attributed to the coil-to-globule and globule-to-coil transitions involving four distinct “thermodynamically stable” states:16 the random coil, the crumpled coil, the molten globule, and the fully collapsed globule. These states were differentiated based on the value of the ratio of the radius of gyration, *R*_{G}, to the hydrodynamic radius, *R*_{H}. The molten coil is thought to have a higher-density core relatively to a rough lower-density surface leading to a *R*_{G}/*R*_{H} ratio smaller than the value 0.774 predicted for a sphere with uniform density; this value is recovered for the fully collapsed state. Note that this latter, more compact, conformation was only observed in aqueous systems whose thermodynamics is dominated by hydrophobic interactions, that is, interactions much stronger than the van der Waals type.

From the point of view of statistical thermodynamics, a concise definition of a polymer globule was proposed by Lifshitz:25 “A globule is the state of a macromolecule in which it has a definite, thermodynamically reliable, spatial structure, i.e., the density fluctuations in a globule are less than the density itself and the range of density correlations remains finite as the chain length tends to infinity (*N*→∞).” An important complementary (though not as encompassing as Lifshitz's) definition of a globule is Volkensthein's:1 a polymer globule is a dense three-dimensional structure from which almost all solvent molecules are ousted by the chain monomers. In contrast, a coil is the state of a macromolecule where density fluctuations are of the same order as the density itself, and the range of correlations is comparable to the coil dimensions. Therefore, the range of correlations diverges when *N*→∞ (criticality).26–28

On approach to the Θ-temperature from a good solvent, a coil gradually shrinks and turns into a globule, where attraction dominates the interactions among chain links. A large polymer globule consists of a dense homogeneous nucleus and a relatively thin surface layer, a “fringe.” Thermodynamics imposes the shape (spherical) of the globule. Furthermore, it settles the globule's size at such a value that the osmotic pressure of the polymer reaches zero in the interior of the globule.

The simplest approximation capable of describing large well-formed globules is associated with the volume (i.e., the uniform density “core”) dominant contribution to the free energy (*F*) and the neglect of the contribution originating from the surface layer. This is the so-called “volume approximation,”26 where the conformational entropy of the fringe is totally disregarded. This simplification, which is correct in the limit of very long chains, supplies an estimate for the Θ-temperature.29 However, the determination of the transition temperature for finite chains requires a balance of volume and surface contributions. The finite *N* transition temperature, *T*_{tr}(*N*) differs from *Θ* by a factor of *O*(*N*^{−1/2}).26 Furthermore, as this is a “finite” transition, there exists a range of temperatures Δ*T*(*N*) around *T*_{tr} within which thermal fluctuations suffice to induce spontaneous jumps from coil-like (i.e., extended) to globularlike (i.e., compact) conformations and vice versa. Δ*T*(*N*) is also of *O*(*N*^{−1/2}).26 At any rate in the vicinity of the Θ-point, a third order virial expansion suffices for the quantitative description of the transition. At lower temperatures, many body effects may alter the transition features quantitatively but not qualitatively.

The character of the coil-to-globule transition depends in an essential way on chain stiffness. For finite stiff chains, the transition is very sharp and akin to a precursor of a first order phase transition, while for flexible chains, it is smoother and assumes features of a second order phase transition. For stiff chains, that is *υ*/*a*^{3} ≪ 1, Δ*T*/*Θ* ≪ |τ_{tr}|, where *a* is the monomer length, *υ* is the excluded volume of the monomer and *τ*_{tr} is the reduced transition temperature, *τ*_{tr} =(*T*_{tr} − *Θ*)/*Θ*. In this case, the coil-to-globule transition proceeds in a relatively narrow temperature interval, clearly separated and substantially removed from the Θ-point. Conversely, for flexible chains (*υ*/*a*^{3} ∼ 1), Δ*T*/*Θ* ∼ |τ_{tr}, the coil-to-globule transition proceeds relatively smoothly and the transition interval includes the Θ-point.

The above are mathematically contained in the functional form of the dependence of the macromolecule's swelling parameter, *α* = *R*_{G}/*R* (the ratio of the mean radius of gyration of a chain over the mean radius of gyration of an ideal chain) on the relevant variables, which are reduced temperature *τ* =(*T* − *Θ*)/*Θ* and chain length *N*. The exact functional form of this dependence encompasses the effect of chain stiffness; however, its scaling behavior is independent of stiffness (“universal”).26

Grosberg and Kuznetsov in a series of four articles30 studied the characteristics of a single polymer chain and the interactions between chains (two globules at a distance *z* from each other) in a poor solvent, employing a mean field approach. They compared their theoretical results with experiments and computer simulations and found that, generally, mean field theory provides a satisfactory description of the coil-to-globule transition.

Several computer simulation studies have focused on the effect of stiffness on chain configurations in different solvent qualities. These studies have identified regions where a polymer chain exists as a coil, as a spherical globule, and as a toroidal globule. Some of these regions are not separated by sharply defined boundaries, but rather wide two-state coexistence regions are encountered, where intermediate metastable structures, such as rods and disks, toruses and “tennis racquets” were also observed.31–35 In general, it appears that the coil-to-globule transition for semiflexible chains proceeds through a series of long-lived metastable dynamical intermediates.

In most simulation studies, a narrow range of relatively low molecular weights was considered, using both Monte Carlo36–38 and molecular dynamics techniques.39, 40 Finite chain length effects on the coil-to-globule transition of stiff chains were studied using the bond fluctuation model,41 where chain lengths up to 200 monomer units were simulated. A major finding was the lack of any noticeable dependence of the coil-to-globule transition temperature on chain stiffness in the thermodynamic limit.

An alternative and promising approach was introduced by Imbert et al.42 Relying on numerical simulation, these authors presented a phenomenological model of the statistics of a self avoiding walk. It appears that this effort constitutes a “theoretically-based shortcut” of the “generalized ensemble” approaches43 that can be productively used for the problem in hand. This approach incorporates, though not rigorously, the influence of chain topology (self-avoidance constraints) and energetics.

The consensus from numerous theoretical and simulation studies30–33, 36, 39–42 is that, for flexible chains as *N*→∞, the coil-to-globule transition is a second order phase transition described by tricritical field theory.44–46 However, simulation studies have revealed additional features of the precursor finite transitions. Monte Carlo simulations in four dimensions, where mean-field theory should be accurate, indicate that the approach to the tricritical point is dominated by the build up of first order like singularities masking the second order nature of the transition (e.g., two clear peaks in the distribution of the internal energy47, 48). This happens for finite chain lengths while, on approach to the infinite length limit, simulation data follow closely the predictions of the Lifshitz–Grosberg–Khokhlov theory applied to high dimensions.46

Most computer simulation studies focused on the effect of chain stiffness and were limited to rather short chains. Furthermore, simulation studies of flexible chains have revealed certain phenomena, which, albeit anticipated,26, 46 go beyond the framework of existing theories. Intraglobular liquid–solid and solid–solid phase transitions were reported in refs. 39, 49, and 50.

Even though simulations of high molecular weights are time consuming, they incorporate crucial information for the characteristics of the passage from the coil to the globular state. In this work, we present simulations of polymer chains on a cubic lattice and under conditions of varying solvent quality. Our studies span a wide range of chain lengths (*N* = 20–10,000 lattice units) and have focused on quantitative tests of mean field predictions and on certain counterintuitive features of the finite precursors of the coil-to-globule transition in flexible homolpolymers, specifically on the unexpected size-evolution of the globule (vs *N*) and also on unexpectedly high value of *N* required to attain poor solvent scaling (*R*_{G}, or *R*_{EE} ∼ *N*^{1/3}, as required by the “volume approximation”26, 27, 30).

The cubic lattice model with only mild nearest neighbor attractions was not selected merely on the grounds of convenience and computational efficiency. Its choice served other purposes as well, that is, to guarantee flexibility, to disallow any condensed phase–condensed phase (i.e., liquid–solid and solid–solid) intraglobular phase transitions, as well as to exclude any specific, more complex interactions (e.g., H-bonding) and permit only the simplest cohesive interactions of the van der Waals type. Therefore, the aim of this research was to perform computer experiments of the “purest” type on the coil-to-globule transition of very flexible homopolymers, which would facilitate a direct comparison with theoretical predictions.

The rest of this article has been organized as follows: The following section contains a description of the simulation method and the Results section is a presentation of the results. Further discussion of the findings and their implications, as well as a summary of the broad picture, are contained in the last section.