Monte Carlo study of the coil-to-globule transition of a model polymeric system


  • Anastassia N. Rissanou,

    1. Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 711 10 Heraklion Crete, Greece
    2. Department of Physics, University of Crete, 710 03 Heraklion Crete, Greece
    Current affiliation:
    1. Institute of Physical Chemistry, National Center for Scientific Research “Demokritos”, 15310 Aghia Paraskevi Attikis, Greece
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  • Spiros H. Anastasiadis,

    Corresponding author
    1. Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 711 10 Heraklion Crete, Greece
    2. Department of Physics, University of Crete, 710 03 Heraklion Crete, Greece
    3. Department of Chemical Engineering, Aristotle University of Thessaloniki, 541 24 Thessaloniki, Greece
    • Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 711 10 Heraklion Crete, Greece
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  • Ioannis A. Bitsanis

    Corresponding author
    1. Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 711 10 Heraklion Crete, Greece
    • Institute of Electronic Structure and Laser, Foundation for Research and Technology-Hellas, 711 10 Heraklion Crete, Greece
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Monte Carlo computer simulations of single, flexible, self-avoiding chains on a cubic lattice have been performed upon conditions of increasing segment–segment cohesive energy (deteriorating solvent quality). The simulations spanned a wide range of chain lengths (20–10,000, i.e., up to molecular weights of a few millions) and cohesive energies (0.0–0.45kBT, i.e., from athermal to very poor solvents). The chain length dependence of the chain size in poor solvents was characterized by a wide plateau of almost null growth for intermediate chain lengths. This plateau was linked to the development of the incipient constant density core, while genuine power law dependence (1/3) was not reached even for the longest chains and poorest solvents simulated here. The mere appearance of a core required substantial chain lengths (higher than 1000; molecular weights of a few hundred thousands), while short chains underwent a gradual densification devoid of any qualitative changes in the density distribution. Sufficiently long chains became more but not quite spherical and underwent a reasonably sharp second order phase transition. The findings were generally in agreement with predictions of mean-field theory and with the use of the standard scaling variables, despite slight inconsistencies. Nevertheless, the results stress the fact that short chains never form a constant density core and that core-dominance on the globule's properties (“volume approximation”) is only valid for extraordinarily long chains [molecular weight of O(109)], an effect linked to the relatively diffuse nature of the surface layer and originating from chain connectivity in conjunction with spherical geometry. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 3651–3666, 2006


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(107) 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, RG, to the hydrodynamic radius, RH. The molten coil is thought to have a higher-density core relatively to a rough lower-density surface leading to a RG/RH 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, Ttr(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 ΔT(N) around Ttr within which thermal fluctuations suffice to induce spontaneous jumps from coil-like (i.e., extended) to globularlike (i.e., compact) conformations and vice versa. ΔT(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/a3 ≪ 1, ΔT/Θ ≪ |τtr|, where a is the monomer length, υ is the excluded volume of the monomer and τtr is the reduced transition temperature, τtr =(TtrΘ)/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/a3 ∼ 1), ΔT/Θ ∼ |τtr, 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 = RG/Rmath image (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 =(TΘ)/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

equation image(1)

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 (RG, or REEN1/3, 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.


Various Monte Carlo methods on a cubic lattice were utilized to simulate isolated polymer chains as a function of “solvent” quality. The original configuration bias Monte Carlo (CBMC)51 method and a modification of CBMC developed by some of us52 was utilized in the thermal regions where it is effective (good to Θ solvent conditions), while Monte Carlo with local moves and one-segment reptation (“slithering snake”) was used in poor solvent conditions, where large-scaled movements are not efficient. The “internal” CBMC53 method, a combined “rebridging”54 CBMC methodology convenient for our lattice systems, resulted in unacceptable attrition at poor solvent conditions.

A simple cubic lattice was considered with unit length that of the monomer length (a) and the chain was contained in a simulation box with side at least five times larger than the chain radius of gyration. Periodic boundary conditions were applied in the all three directions. The chains were fully flexible in the sense that there was no potential penalty for bending between adjacent bond vectors. The excluded volume was enforced through the requirement of self-avoidance. Solvent “quality” was quantified via an attractive (cohesive) energy (−E, in units of thermal energy kBT), effective between nonsuccessive segments occupying adjacent lattice sites. In this context, the polymer–solvent (i.e., polymer–void) and solvent–solvent (i.e., void–void) interaction energies were zeroed. This is a rather drastic assumption, which, however, should not alter either the nature, or the qualitative features of the coil-to-globule transition.

Chains up to N = 10,000 segments were simulated. This would correspond to molecular weights of a few millions for most real flexible polymers. Certain memory problems, which appeared for chain lengths beyond N = 3000, were overcome using external routines for bit manipulation. Specifically, these external routines made feasible the storing of the occupancy information in binary form (i.e., 1 bit), therefore considerably reducing “occupancy matrix” memory requirements.

The initial configuration for each run was independently generated utilizing a growth and equilibration method at athermal (E = 0.0) conditions.55, 56 Simulations of systems with large N and large E are more demanding in terms of the number of Monte Carlo steps required for equilibration (the dependence on chain size goes roughly as N2, whereas Monte Carlo moves have a lower degree of acceptance in the more compact conformations obtained for large E). Equilibration was monitored primarily through the time evolution of RG and total energy. In the most demanding case, simulated (N = 10,000, E = 0.45) equilibration was reached after about 106 Monte Carlo cycles. Beyond that point, no systematic trend in the evolution of the aforementioned quantities was detected and there fluctuations were well within the limits of what might be expected from finite size effects. After equilibration, multiple chain configurations were stored at a proper frequency for data analysis.


We start the presentation of the simulation results with a study of the effect of solvent quality (E) on the N-dependence of chain size. Figure 1(a) shows our data on the radius of gyration, RG, of isolated polymer chains as a function of chain length N for a wide range of E-values (E = 0.0 corresponds to athermal solvent conditions and increasing E signifies deterioration of solvent quality). RG values in Figure 1(a) represent the average over several thousand decorrelated chain configurations collected after equilibration. Simulation data for chain lengths ranging from 20 to 10,000 segments are shown in Figure 1(a).

Figure 1.

(a) Chain radius of gyration as a function of chain length N for different values of the energy parameter E corresponding to different solvent qualities. (b) RG/(Na2/6)1/2 as a function of N for different values of E. N-independence indicates the E that corresponds to the Θ-condition, EΘ. [Color figure can be viewed in the online issue, which is available at]

In athermal solvent conditions (E = 0.0), the RG versus N data follow closely a RGNν power law dependence with the expected exponent ν = 0.59 for athermal solvent conditions. When the energy parameter E increases, which for our model is exactly equivalent to decreasing temperature T, the data continue to follow power law dependencies with apparent exponents ν intermediate between 0.59 and 0.5 that gradually decrease toward the value of 0.5, which signifies the Θ solvent conditions. Clearly, the intermediate exponent values reflect transient effects related to the increasingly higher Ns required to obtain the genuine scaling behavior (ν = 0.59 for any E below that corresponding to Θ-conditions27).

For our model, the Θ-solvent conditions are realized for EΘ = 0.28 [EΘE(T = Θ)]. This value agrees closely (to within two significant digits) with the Θ-point for chains on a cubic lattice, determined by earlier simulation studies.57–59 For this value of E, a plot of RG/(Na2/6)1/2 versus N [Fig. 1(b)] shows hardly any N-dependence for N > 500. This method of estimating the Θ-point is expected theoretically46 to carry an inherent error of O(N−1), due to the index dependence of the (renormalized) 2nd virial coefficient (“end effects”). As it will be seen later, the identification of the Θ-point will be used to test correlations inside the transition zone of width of order O(N−1/2). Therefore, the level of accuracy in the determination of the Θ-point by the approach illustrated in Figure 1 is appropriate for the study of such correlations.

The intersect with the vertical axis in Figure 1(b) furnishes a value for the characteristic ratio C of 1.66 for the fully flexible self-avoiding chains on a simple cubic lattice modeled in this work. C is larger than one, despite the fully flexible nature of our model. This is clearly and solely the outcome of excluded volume interactions, which preclude multiple occupancy of lattice sites and, more specifically, do not allow chain back-folding.

For energy parameter values higher than EΘ, that is, for poor solvent conditions, a peculiar dependence of the radius of gyration RG on the polymer chain length N is detected. For E = 0.33 and higher, RG exhibits an almost N-independent regime for intermediate chain lengths (a plateaulike region) before it eventually starts increasing again with chain length. The anticipated RGN1/3 dependence (for a compact object in a bad solvent) is only gradually approached for high values of E and very high Ns (i.e., for E = 0.33, a cohesive energy far removed from the Θ-threshold, the 1/3 exponent has not been reached even for N = 10,000, i.e., a molecular weight of a few millions). Moreover, the range of N, where the plateaulike region occurs, depends on solvent quality; it tends to be more extended at the vicinity of the Θ-point.

The aforementioned features of the N-dependence of the chain radius of gyration reappear in a more pronounced fashion in the chain length dependence of the end-to-end distance, REE, under poor solvent conditions, as illustrated in Figure 2 for three different energy parameter values (E = 0.33, 0.37, and 0.45, all in the poor solvent regime). A plateaulike region is clearly observed for both values of E, whereas the range of N-values spanning the plateau is broader for the lower E, that is, the one closer to EΘ. It will become evident below that this apparently peculiar dependence of RG and REE on N is due to the changing shape of the chain from that of a coil toward that of a more compact conformation, which would finally evolve to a globule in sufficiently poor solvents and for sufficiently long chains. As it will be illustrated later, this behavior is consistent with the mean field theoretical description of the coil-to-globule transition of Grosberg et al.26, 30 The REE dependence on N under poor solvent conditions has been studied in the past by Grassberger and Hegger.57 In particular, the presence of a weak minimum like the one of our E = 0.37 data was also reported in this work. It should be noted that the statistical uncertainty of the data in Figures 1 and 2 is, at most, comparable to the size of the symbols.

Figure 2.

Chain end-to-end distance as a function of chain length N for three values of the energy parameter E, which correspond to the poor solvent regime. [Color figure can be viewed in the online issue, which is available at]

The density distribution of chain segments as a function of the distance from the center of mass of the chain can provide quantitative information on the development of the compact chain configuration when solvent quality deteriorates. Figure 3(a,b) depicts the chain length dependence of the segment density profiles for two energy parameter values E = 0.37 [Fig. 3(a)] and 0.45 [Fig. 3(b)] both in the poor solvent regime. The simulation data are shown for chain lengths ranging from 20 to 10,000 segments. The most important feature in Figure 3 is the development of a high density homogeneous core for sufficiently long chains. Core development is signaled by a qualitative change in the shape of the density distribution. It is the development of the dense region of constant density around the center of the mass of the chain that signifies the development of a globulelike structure for high Ns.

Figure 3.

Segment density distribution as a function of the distance from the center of mass of the polymer chain for various chain lengths N from 20 to 10,000 and for energy parameters E = 0.37 (a) and E = 0.45 (b) both in the poor solvent regime. [Color figure can be viewed in the online issue, which is available at]

One can roughly distinguish three regimes with respect to chain length. For very short chains, an increase of N leads to an increase in the density values throughout the profile, which, nevertheless, remain relatively low. For intermediate Ns, the density increases more sharply in the central region of the chain (small r), reflecting the development of the precursor of the core-to-be. For still higher Ns, the compact core of constant density60 has been formed and expands further with increasing N, whereas the surface layer, which even for the longest chains studied in this work still constitutes a substantial portion of the chain, has acquired its final internal structure and is merely displaced outwards by chain growth.

The development of the core with increasing chain length is quantified in Figure 4, where the density near the center of the chain (i.e., the first two lattice sites adjacent to the chain center of mass) is plotted as a function of chain length for the two E values of Figure 3. For short chains, where no core exists, the density in the inner part of the chain simply increases with increasing N. The density increases sharply for intermediate chain lengths until it saturates to a constant value beyond a certain large N, indicating that the core has reached its limiting density. The N value at which saturation is reached, as well as the final core density, depends on the energy parameter E as can be seen in Figure 4. It is worth noting that the N-range of the abrupt density increase (for example, for 200 ≤ N ≤ 1000 in the E = 0.37 curve of Fig. 4) coincides with the N-range of the plateau in Figures 1 and 2. Once the core is formed, the density values in the inner part of the “globule” are quite high, and, in sufficiently poor solvents, the core attains a meltlike state.

Figure 4.

Core density as a function of chain length for two values of the energy parameter E, which correspond to the poor solvent regime (E = 0.37 and E = 0.45). [Color figure can be viewed in the online issue, which is available at]

The portion of the chain outside the core forms the surface layer; an appropriate measure of surface layer thickness, δ, is given by the following formula:

equation image(2)

The δ-values estimated for two different values of E in the poor solvent regime are listed in Table 1. In both cases, the surface layer thickness δ increases very weakly for high N values and exhibits a clear trend toward saturation. Table 1 also contains a comparison between δ and chain radius (i.e., RG). It is clear from these data that the surface layer still constitutes a considerable part of the chain, even for the longest chains studied in our work. Characteristically, even for the longest chains studied here (N = 10,000), 70 and 75% of chain segments belong to the surface layer for E = 0.45 and 0.37, respectively. It should be pointed out that N = 10,000 (cubic lattice segments) corresponds to a molecular weight of millions for most real polymers and that the long chains studied here are among the longest ever studied via simulations in connection with the coil-to-globule transition. Based on the data of Figures 3 and 4 and assuming that the core density has saturated and the surface layer profile has become molecular weight independent, we predict that a chain of about 2 × 106 cubic lattice units (i.e., a polymer of about 6 × 108 in molecular weight) for E = 0.45 would be required for the “fringe” to have become a “secondary,” yet not entirely insignificant factor; in plain terms to contain a “mere” 10% of chain segments. Closer to the Θ-point (E = 0.37), the aforementioned estimates are doubled.

Table 1. Thickness of the Surface Layer of the Compact Structures Formed in the Poor Solvent Regime
NSurface Layer Thickness, δ/aReduced Surface Layer Thickness, δ/RG
E = 0.37E = 0.45E = 0.37E = 0.45

Therefore, the development of a homogeneous high density core requires very high Ns and the neglect of the surface layer contribution (“volume approximation”26) would be justified only for extraordinarily long chains. The transition range, that is, the N-range for core formation, is shifted to higher Ns as the cohesive energy E approaches EΘ. This is illustrated in Figures 5 and 6. Figure 5 shows the segment density profiles for two different chain lengths, N = 100 (a) and N = 3000 (b) and for several different energy parameter values corresponding to an athermal solvent (E = 0.0), a Θ-solvent (E = 0.38), and three poor solvents (E = 0.33, E = 0.37, and E = 0.45). An abrupt change in the shape of the density profile is observed on going from a good (or even a Θ) solvent to a poor solvent for the high N = 3000, whereas there is only a gradual change for the low N = 100 for the same values of the energy parameter E. For N = 3000, a clear core develops at E = 0.45, whereas there is no indication of a corelike nucleus for N = 100 (not even for E = 0.45). In other words, N = 100 is not a suitable chain length for the observation of the characteristic features of the coil-to-globule transition, at least for E values that are not extraordinarily larger than EΘ. The above comments are corroborated by the snapshots of respective chain configurations (N = 100 and 3000) in the limits of an athermal and a very poor solvent, which are shown in Figure 6. The N = 100 chain remains relatively loose even in its shrunk state at E = 0.45 [Fig. 6(a)] in contrast to the compact almost spherical-like configuration of the N = 3000 chain [Fig. 6(b)].

Figure 5.

Density distribution as a function of the distance from the center of mass of the polymer chain for three different energy parameters E = 0.00 (athermal solvent), E = 0.28 (Θ-solvent), and E = 0.33, 0.37, and 0.45 (poor solvents) and for two chain lengths N = 100 (a) and N = 3000 (b). [Color figure can be viewed in the online issue, which is available at]

Figure 6.

(a) A three-dimensional snapshot of a chain with N = 100. The extended (dark) configuration represents a chain in an athermal solvent (E = 0.0), whereas the more compact (light) configuration represents a chain in a bad solvent with E = 0.45. (b) A three-dimensional snapshot of a chain with N = 3000. The extended (dark) configuration represent a chain in an athermal solvent (E = 0.0), whereas the compact (light) configuration represents a chain in a bad solvent with E = 0.45. [Color figure can be viewed in the online issue, which is available at]

A systematic shape analysis31 quantifies further important features of chain conformations. In general, the shape of an object can be characterized by two structural parameters K1 and K2, which are defined as suitable ratios of the three principal moments L(a) (a = x, y, z) of the gyration tensor, X:

equation image(3)
equation image(4)
equation image(5)

where rcm is the center of mass of the chain, ri is the position of the i-th segment, and ω(a) and L(a) (a = x, y, z) are the eigenvectors and eigenvalues of X). The quantities K1 and K2 assume well known values for typical shapes: for a perfect sphere K1 = K2 = 1, for an infinitely thin rod K1 = 0 and K2 = 1, whereas for an infinitely thin disc K1 = K2 = 0.5. K1 and K2 have been employed successfully for the identification of quite nonstandard shapes (e.g., torroids).31

Figure 7 contains histograms of the shape parameters K1 and K2 (shaded) for E = 0.45 and for three different chain lengths, N = 100 (a), N = 500 (b) and N = 10,000 (c). Most of the weight of the histogram of K1 for N = 100 is in a broad peak at a quite small value K1 ≈ 0.2–0.3, while the corresponding peak of the K2 histogram occurs close to 0.9. Increasing the chain length, a shift of the K1 peak is observed toward larger values, almost 0.5 for N = 500 and 0.7 for N = 10,000, whereas the K1 peak is more or less stable near 0.9. This analysis indicates a state close to a uniaxial prolate ellipsoid for the small N values and an evolution to a much more spherical shape with increasing N. An analysis based on more familiar shape descriptors, like the “asphericity,” “acylindricity,” and “shape anisotropy” factors,61, 62 (Table 2) also demonstrates the gradual evolution from the shape of a uniaxial prolate ellipsoid to that of a sphere with increasing chain length. All this is corroborative to the earlier discussion for the chain length dependence of the development of compact configurations of a polymer chain in a poor solvent and clarifies further the issue of when a dense configuration can be characterized as a globule.

Figure 7.

Histograms of the structural parameters K1 and K2 (shaded) indicating the shape of the chain, for E = 0.45 and for three different chain lengths N = 100 (a), N = 500 (b), and N = 10,000 (c). [Color figure can be viewed in the online issue, which is available at]

Table 2. Asphericity, Acylindricity, and Shape Anisotropy of Chains62, 63 in a Poor Solvent (E = 0.45)
NAsphericityAcylindricityShape Anisotropy

The transition from an extended coil to a compact globule has been theoretically discussed in terms of the effect of solvent quality (via changes in temperature) on the size of the chain expressed by the swelling parameter (α), which is defined as the ratio of the radius of gyration of the chain in a particular solvent (or temperature) over the radius of gyration of an ideal chain (i.e., at Θ conditions),

equation image(6)

where 〈Rmath imagemath image is the unperturbed radius of gyration obtained at the Θ-condition for EΘ = 0.28 (as discussed in relation to Fig. 1). Figure 8(a) shows the dependence of the swelling parameter α on the reduced distance from the Θ-condition; this is expressed via an effective reduced temperature τ = (TΘ)/Θ, which in the context of our model is expressed as τ = (1/E − 1/EΘ)/(1/EΘ), where EΘ = 0.28 is the energy parameter at the Θ-condition. One can clearly observe the transition from the coil-like extended conformation for positive values of τ (in the good to Θ solvent regime) toward a globularlike compact chain configuration for negative values of τ (in the poor solvent regime). It is clear that the actual transition is completed below the Θ-temperature (in the present parameterization, for E > EΘ and τ < 0, whereas the transition region is fairly broad and its width decreases as the molecular weight increases. Indeed, mean field theoretical predictions by Grosberg et al.26, 30 indicate that, for finite chain lengths, the transition takes place at a reduced transition temperature τtr,

equation image(7)

where C is the third virial coefficient, b is the temperature coefficient of the second virial coefficient B(T) ≅ bτ (B < 0) and a is the segment length. This indicates that the transition occurs at a temperature that is removed below the Θ temperature, whereas this temperature difference depends on N−1/2, that is, it could be substantial for small Ns, and it coincides with the Θ temperature only in the limit of N→∞. Furthermore, the transition is “finite,” that is, it takes place over a range of temperature ΔT; the width of ΔT is predicted to scale with τN1/2. Thus, it was predicted that the transition occurs further below the Θ condition when the molecular weight is lower, whereas the width of the transition region decreases as the chain length increases. The findings in Figure 8(a) are in very good agreement with both these predictions.

Figure 8.

(a) Swelling parameter α = 〈RG21/2 / 〈RG2math image as a function of the effective reduced temperature τ = (1/E − 1/EΘ)/(1/EΘ) for different chain lengths. (b) Swelling parameter α as a function of reduced temperature τ multiplied by N1/2. (c) The cube of the swelling parameter α multiplied by |τ| N1/2 as a function of the reduced quantity |τ| N1/2 in the poor solvent regime (|τ| denotes the absolute value of τ). The dashed lines in (a) and (b) denote the Θ-condition.

Moreover, Grosberg et al.26, 30 pointed out that the general characteristics of the coil-to-globule transition should depend on a normalized reduced distance from the Θ condition, τN1/2. As noted in eq 1, the values of the swelling parameter α for various chain lengths and attractive energy values should cohere on a master curve when plotted as α versus τN1/2. This master curve is depicted in Figure 8(b), where an excellent superposition of the data is observed in the poor solvent regime below the Θ condition (i.e., for negative values of τ). It is noted that according to eq 7, the exact value of τN1/2 at the transition depends on the virial coefficients b and C. Furthermore, it should be reminded that the actual shape of the master curve of Figure 8(b) is model-dependent. Most importantly, it depends on the flexibility of the polymer chain31 quantified by the ratio.

Finally, in the limit of τ→−∞ and α→0, that is, in the limit of extremely poor solvents deeply below the transition, and very long chains, where the volume approximation should hold and the shape of the chain is that of a well-formed globule with a steplike density distribution, the swelling parameters α must follow the scaling law.30

equation image(8)

The simulation data for various energy parameter values EEΘ are shown in Figure 8(c), where the reduced quantity α3 |τ| N1/2 is plotted versus the reduced distance from the Θ point |τ| N1/2. The data appear to approach a constant value only for very high values of the reduced parameter |τ| N1/2 (≫10), whereas there are significant deviations at lower values due to fact that a compact globule has not yet formed. These discrepancies illustrate, once more, the fact that the volume approximation is not yet valid even for the longest chains studied in this work, which are certainly among the longest ever studied in connection with the coil-to-globule transition. Furthermore, this observation is consistent with the comments made earlier regarding the data of Table 1 (surface layer thickness).

In a related note, Grassberger and Hegger57 have argued that the critical exponents describing the size of the chain at Θ-temperature are of mean field type, but there are logarithmic corrections to this behavior, which are different from those predicted by field theoretical methods. In a plot similar to Figure 8(c), based on their data, they observed a superposition only for small values of |τ| N1/2, and claimed that the lack of superposition provides information for the nature of the required corrections to mean field theory. Our data in Figure 8(c) are consistent with the argument put forth in ref. 57, however they are not accurate enough to serve as a quantitative test of it.

Finally, the simulation data were utilized to probe the thermodynamics of the coil-to-globule transition via energy and specific heat calculations. For this reason, the average energy per chain segment, U/N, has been determined as a function of the cohesive energy E (effectively inverse reduced temperature) for different chain lengths. In Figure 9(a), the average energy is plotted as a function of E for four different Ns. The change in the slope of the curve, which becomes abrupt for longer chains, is a signature of a second order phase transition as is the finite jump in the specific heat [Fig. 9(b)]. The specific heat in our parameter space is calculated via eq 9.

equation image(9)
Figure 9.

Average energy per chain segment (a) and specific heat calculated according to eq 9 (b) as a function of solvent quality for four different chain lengths. [Color figure can be viewed in the online issue, which is available at]

The jump in the specific heat becomes more pronounced as the chain length increases, which stresses the fact that the transition is second order in the limit of infinite molecular weights. Studies on shorter chains, elsewhere focused (e.g., crystallization),39 merely reported the jump of Figure 9(b) as a “shoulder.”


In this manuscript, several aspects of the coil-to-globule transition for isolated, flexible polymer chains were studied. The investigation utilized Monte Carlo computer simulations on a cubic lattice and extended to quite substantial chain lengths (up to N = 10,000 lattice segments) corresponding to molecular weights of a few million.

The dependence of the radius of gyration of flexible chains on the chain molecular weight was determined over a wide range of chain lengths from N = 20 to N = 10,000. An unexpected dependence of RG on N has been established in the poor solvent regime, where the coil-to-globule transition occurs. Specifically, and for all EEΘ (poor solvents) values studied here, a broad RG plateau was observed for intermediate chain lengths [Fig. 1(a)].

In the case of the end-to-end distance even a weak minimum appeared close to the Θ cohesive energy (Fig. 2). We consider the existence of a minimum57 in the end-to-end distance versus N curve as a manifestation of the inappropriateness of using this quantity (instead of RG) as a measure of chain size under poor solvent conditions. Theoretical analyses30, 63 have pointed this out in the context of developing63 an optimal Flory-type theory for the coil-to-globule transition.

The plateau region in the RG curve [Fig. 1(a)] coincided with the range of Ns over which an inner shoulder appears in the segment radial density (Fig. 3). In slightly more quantitative terms, the unexpected, even null, growth of chain size with chain mass (N) reflects the uneven densification of the chain central portion. The central shoulder gradually saturates to a central constant density core (Figs. 3 and 4).

The transition region is broadest in the vicinity of the Θ-point and shrinks as the solvent quality deteriorates [see Fig. 5(b)]. This is in qualitative agreement with the usage of τ N1/2 as a reduced variable. It should be pointed out however, that although fairly short chains [e.g., NO(100)] undergo a broad transition, they never exhibit the qualitative signature of a globule, namely, a constant density region around the chain's center [Fig. 5(a)].

Upon further increase of N, the constant density core expands and the “fringe,” that is, the surface layer, is pushed further out, while preserving its own density distribution. This leads to an upturn of the RG versus N curve with an apparent exponent that will eventually reach the expected value of 1/3 for purely geometric reasons. However, the term “fringe” is a misnomer, even for the longest chains studied in this work (N = 10,000), as the surface layer still accounts for about 1/2 of the chain RG (Table 1) and contains more than 2/3 of its segments. Therefore, flexible chains with molecular weight of a few million hardly qualify for the application of the “volume approximation.”26 Indeed, as was discussed in the previous section, it would take a flexible polymer with molecular weight of O(109) and at a temperature at least 50 °C below TΘ for the surface layer to be literally a fringe (i.e., to contain a “mere” 10% of the segments).

In a related note, the data in Figure 8(b) (expansion parameter versus τ N1/2) provide considerable quantitative support for the usage of τ N1/2 as a reduced distance from the transition point. The failure of the data in Figure 8(c) to reach a clear plateau illustrates once more that the regime of validity of the “volume approximation” was not reached.

One cannot fail to notice a discrepancy between the collapse of the data in Figure 8(b) and the much poorer superposition of data Figure 8(c). In plain terms, if α is a function of τ N1/2 only, then α3|τ| N1/2 should also depend only on the same variable and the data in Figure 8(c) should also follow a master curve. Grassberger and Hegger57 have argued that this paradox is related to the fact that, while the critical exponents describing the size of the chain at Θ-temperature are of mean field type, there are logarithmic corrections to mean field behavior, which are different from those predicted by field theoretical methods. As pointed out earlier, our data in Figure 8(c) are not accurate enough to serve as a quantitative test of this argument.

A systematic quantification of chain shape (Fig. 7, and Table 2) elucidated further the broad picture of “coil-to-globule” transitions. The prolate ellipsoid shape, characteristic of chains in good and Θ-solvents, persisted in the poor solvent regime for short chains. Longer chains, however, when they crossed their transition point, assumed much more spherical shapes (shift of K1 peak toward 1.0 as N increases in Fig. 7). These shapes, nevertheless, remain noticeably different from that of a perfect sphere. In particular, K1 displays a broad peak around the value of 0.7 for N = 10,000, which has indeed shifted substantially and peaked considerably compared to the almost flat peak of the N = 100 chains, yet it differs considerably from the terminal value of 1.0, characteristic of a macroscopic globule and required by thermodynamics (surface area minimization).

Despite the fact that the surface layer contribution remains substantial even for our longest chains, the 2nd order nature of the coil-to-globule transition is apparent in Figure 9 (for the flexible chains studied here). The transition is reasonably sharp for the longest chains, while it is hardly noticeable for short chains (N = 100).

Summarizing, one can distinguish four different patterns of E-induced transitions in order of increasing N. It is noted that the parameter space (N, E) spanned by our simulations overlapped only with the first three:

  • A gradual broad transition with a fairly uniform densification of all chain portions; the qualitative signature of this regime is that the density profile never ceases to be concave [Figs. 5(a) and 6(a)].

  • A somewhat narrower transition, at intermediate Ns, characterized by the development of a “shoulder” in the inner portion of the density distribution [i.e., the appearance of an inflection point in the density profile; Figs. 3 and 5(b)]. This N-range coincides with the range of the plateau in the RG (and REE) versus N curve.

  • A fairly sharp transition for values of N in the few thousands (molecular weights of a several hundred thousands) signified by the formation of a constant density core [Figs. 3, 4, and 5(b)]. In this N-range, the inner portion of the chain has features identical with those of a truly macroscopic globule, however the surface layer still contributes very heavily to the shrunk chain's properties.

  • A genuinely sharp transition for extraordinarily high values of N (in the several millions; molecular weights of a several hundred millions), where the surface layer is literally a “fringe”, the density profile almost step-shaped and the “volume approximation” is quantitatively accurate.

While it is certainly correct that the usage of τ N1/2 leads to self-similar patterns exhibited by chains of different N, it is also correct that not every N can span all four patterns by manipulating τ. The reasons for this are both of a priori and of practical nature. First of all, τ N1/2 for a certain N is bounded by N1/2. This absolute limitation renders the third regime, let alone the fourth, inaccessible to very short chains (e.g., N = 100, or molecular weight in the few tens of thousands). Chains that are short never form a core. Therefore, they never undergo a genuine “coil-to-globule” transition, even with allowance of it being substantially “disfigured” by strong side-effects (surface layer). Secondly, there exist practical upper limits to the value of E, or lower limits, to the value of T. For instance, E = 0.45 would correspond to a T at least 100 °C below TΘ; the corresponding τ in this case is merely 0.38. Considering this as a practical upper limit for the (negative) cohesive energy E, one is lead to the conclusion that the fourth regime is not accessible by chains with NO(104) (molecular weight higher than 106). In actuality, the fourth regime (“step-like” density profile or core-dominance) for flexible chains would only be accessible by macromolecules with molecular weights of O(109).

Clearly, the extremely high degrees of polymerization required to reach the range of validity of the “volume approximation” result partly from a trivial factor, namely the spherical (3-d) geometry. However, chain connectivity exaggerates this trend. Specifically, even at Es substantially removed from the “critical point” (EΘ), surface layer thickness, although N independent, is considerable (6–8 lattice units according to Table 1) and much larger than the width of the corresponding surface layer of condensed “disconnected monomers” (1–2 lattice units far from the critical point). This state of affairs reflects the fact that the polymeric Θ-temperature is much higher than the critical temperature of the “disconnected monomer” gas.26, 46 For two spherical objects with surface layer widths δchain (chain) and δmonomers (disconnected “monomers”), the “range of validity of the volume approximation” will be shifted upwards (mass-wise) by a factor of O([δchain/δmonomer]3), that is, a factor of about 30.

All the above stress the importance of proper accounting of configurational entropy in any theory63 of the coil-to-globule transition that will be applied to polymer chains of typical molecular weights. In a less direct way, the extremely slow approach toward the range of validity of the volume approximation argues in favor of theories of the Lifshitz type30, 46 (with the full density distribution, ρ(r), being used as the order parameter), as opposed to Flory-type theories,29, 63, 64 which employ a scalar order parameter (expansion factor α).

A direct comparison with experiments is due and can be illuminating, despite already stated differences between the physics of our model and those of experimental systems. polystyrene9–11 and poly(methyl methacrylate)18–20 solutions are characterized by mild, van der Waals type monomer–solvent interactions. Experimental data on polystyrene yield curves of the swelling parameter α versus τ [such as Fig. 8(a)] qualitatively, and even semiquantitatively, similar to our simulation data [α ≈ 0.2, for comparable τ and N; Fig. 8(b)].9 A similar comment can be made regarding poly(methyl methacrylate) data,20 if an appropriate rescaling of MW1/2 (in the experiments20) versus N1/2 on a cubic lattice is made (a factor of about 20).

Several experimental studies focused on aqueous solutions of PNIPAAm.12, 13 PNIPAAm was selected for bio-relevance reasons, for technical reasons and (in our opinion) because it allowed an efficient charting of the poor solvent regime. The last feature is intimately related to the marginal character20 of PNIPAAm (i.e., between flexible and semiflexible chains) in the context of “quantitative mean-field theory.”30 This feature, that is, a relatively low value of the (estimated) 3rd virial coefficient, narrows the transition range considerably. Clearly, hydrogen-bonding interactions dominate PNIPAAm behavior in solution (as manifested by the appearance of a lower, rather than upper, critical solution temperature). Our findings are in qualitative agreement with PNIPAAm data on the more sensitive measure of α3|τ| N1/2 versus |τ| N1/2 plots [maximum location and plateau region in Fig. 8(c)], if sufficient allowance is made for PNIPAAm local rigidity.

The issue of “molten” versus “compact” globules has arisen intensely in the context of PNIPAAm studies.14–17 Our data on RH/RG (RH is the hydrodynamic radius) clearly testify in favor of a “molten globule.” Specifically, the values of RH/RG (1.65 for E = 0.45 and 1.93 for E = 0.37; both for N = 10,000) determined from our simulations are consistent with the picture presented in this and the last section. Note that our calculation for RH is based on the nondraining assumption. In our opinion, the more “compact” globules reported in ref. 16, compared with those in this study, originate from one of the following two reasons: (i) effectively higher molecular weights combined with the exaggerating factor of PNIPAAm local rigidity, which narrows the transition region and corresponds to effective severe overcooling even 10 or 20 °C below the PNIPAAm Θ-point, or (ii) a (thermodynamic?) transition caused by hydrogen-bonding.

It should be mentioned, however, that not all experimental data on PNIPAAm point to the direction of a “compact” globule as the final outcome of the coil-to-globule transition.20 Our simulation data, and thermodynamic equilibrium theories, are not consistent with a “molten” versus “compact” globule transition scenario. Both our data and theories clearly predict a gradual “compactification” of the globule with deteriorating solvent quality.

Simulation models that permit intraglobular phase transitions39, 49, 50 have revealed such a transition from a liquidlike globule with a dense core (at liquidlike densities) and a rather broad surface layer to a solidlike (crystalline) globule with an even denser crystalline core and a narrower surface layer. In the aforementioned simulation studies, it was postulated that the experimentally reported “molten” to “compact” globule transition14–17 has a similar basis. As we discussed earlier, our model was specifically selected to render such transitions impossible. Therefore, our results do not supply any evidence in favor, or against the intraglobular crystallization scenario in PNIPAAm.

The lattice Monte Carlo simulations of Hu37 suggested a different explanation for the formation of a “molten” globule, namely intraglobular microphase separation, where the dense core plays the role of the condensed phase, while the surface layer corresponds to the gaseous phase. The distinguishing feature of this situation from a regular globule with core and surface layer lies in the discontinuity (jump change) of the density distribution at the contact surface between core and periphery (surface layer). This phenomenon has been anticipated theoretically26, 46 for temperatures below the critical point of the “disconnected monomer gas,” that is, substantially below the Θ-point. We did not observe any such discontinuities of ρ(r) in this study, primarily because it takes severe (bordering to the unrealistic) overcooling for microphase separation to occur for chains of molecular weight higher than 105.26, 46 Furthermore, as discussed earlier, the PNIPAAm data can also be explained otherwise.

The interest (and the wording itself) on “molten globules” derives from the term used for a special state of protein molecules with a native secondary structure but a poorly defined tertiary structure and which is considered to be a metastable intermediate of the protein folding–unfolding process.65 Although much of the original work on the homopolymer coil-to-globule transition46 (and references therein) was motivated by a perceived analogy with protein unfolding–folding, it is currently fully realized that, for meaningful comparisons, one has to bring modeling at least up to the level of heteropolymers.66 Nevertheless, the clarification of the basic physics supplied by doing “exact” statistical mechanics with models that allow direct comparisons with theory, besides motivating theoretical improvements, identifies the applications for which such modeling is relevant. In the case of homopolymers, these applications are the ones related with polymer network collapse, where the “volume approximation” is certainly a reasonable starting point.8, 14 Our findings, however, which quantify the internal structure of the globule may prove useful in designing materials for applications sensitive to the degree of “porosity” of the globule periphery (and that of the whole globule), like controlled release of pharmaceutical drugs, or contaminants.4 Simulation efforts on dendrimeric supramolecules are promising.67

Even in the context of cubic lattice models that simplify the comparison with theories and exclude complicating, yet real, factors inducing phenomena beyond the existing theoretical framework, remain “uncompleted tasks.” Specifically, for flexible polymers, according to the criterion of ref. 30, a detailed charting of the Θ-region (T or E-wise) is needed. A satisfactory interpolation formula linking the poor solvent regime (weak fluctuations) to the Θ-solvent region (strong-fluctuations/criticality) does not exist in the context of Lifshitz-type theories30 and lacks any fundamental basis in Flory-type theories.63, 64 A Wang-Landau43 energy-histogram approach is required for this purpose and is currently underway. Analogous studies on chains adsorbed from poor solvent are also in progress. A recent simulation study,50 with a different focus from ours, has elucidated the potential of this approach in addressing profound issues of the coil-to-globule transition, like the system-specific realization of the (theoretically inevitable)27, 28 tricritical nature of the homo-polymer coil-to-globule transition.


Part of this research was sponsored by NATO's scientific Affairs Division (Science for Peace Program), by the Greek General Secretariat of Research and Technology, and by the European Union (G5RD-CT-2002-00834).