Since Stockmayer [32] predicted collapse of the single polymer chain below the Θ point, many theoretical investigations on coil-globule transition have been conducted [9]. Birshtein and Pryamitsyn [33] developed a theory based on the simple scheme of Flory [2] and derived an equation for the quantitative comparison between theory and experiment. In the Flory-type theory, the expansion or shrinkage of a polymer chain under different conditions is represented by the expansion factor α = R/R ₀, where R is a characteristic polymer dimension and R ₀ is R in the ideal state (at the Θ point). Experimentally, the characteristic size of a polymer chain depends on the measurement technique. For example, it is the mean-square radius of gyration <S²> for the static light scattering (SLS) experiment but the hydrodynamic radius <R _h> for the dynamic light scattering (DLS) experiment. According to the theoretical approach of Flory, polymer free energy can be written as
where F _el and F _int are the contributions of the deviation of the conformational entropy from the ideal chain and of the excluded-volume interactions, respectively. The latter is calculated by mean-field approximation with virial expansion:
Here, V, n, vτ = v(1 − Θ/T), and w are the volume occupied by a polymer chain, the segment density in the volume V, and the second and third virial coefficients, respectively. With respect to the entropy term, F _el ~ kTα ² is expected for the good solvent condition. For the poor solvent condition, Birshtein et al. [33] derived an equation for F _el as
where γ is a constant and α _S is the expansion factor defined as α _S ² = <S²>/<S²>₀. By using Eqs. (2.1), (2.2), and (2.3), an equation of the expansion factor α _S is derived in the form of
where M is the molecular weight of the polymer. The parameters B and C in Eq. (2.4) are related explicitly to v and w in Eq. (2.2), respectively. Thus, experimental data can be readily compared with Eq. (2.4) to obtain the parameters of the second (vτ) and third (w) virial coefficients of segment interactions.

Among the experimentally observable characteristic of chain dimensions, only the radius of gyration was considered in the Flory-type theory of Birshtein and Pryamitsyn. Grosberg and Kuznetsov [34–36] developed a theory (GK theory) of the coil-globule transition according to a more rigorous and complex scheme derived by Lifshitz et al. [37]. In the GK theory, both the radius of gyration <S²>^1/2 and the hydrodynamic radius <R _h> have been considered, and a comparison between the theoretical and experimental results can be made quantitatively.

There have been only a few experimental studies of the coil-globule transition of a polymer chain in solution. This is because there are numerous practical problems to overcome, particularly those associated with intermolecular aggregation and the subsequent macroscopic phase separation, which may occur, in most cases, before chain collapse finishes and obscures coil-globule transition. To minimize the effects of aggregation, most experiments have been carried out with highly diluted polymer solutions. For example, light scattering experiments have been performed on polystyrene (PS) in cyclohexane, which is a typical UCST-type system, at extremely low concentrations (<10⁻⁷ g/cm³), and coil-globule transition was observed [38]. Because of difficulties in the experiments such as the low signal-to-noise ratio due to the very low concentrations, data for the coil-globule transition of PS in cyclohexane obtained by different groups seem contradictory [39].

Several groups [40–42] have observed the coil-globule transition of PNIPAM in water, which shows an LCST-type phase behavior, by means of static light scattering (SLS) and dynamic light scattering (DLS) techniques. In their studies, SLS and DLS measurements were carried out on dilute solutions of PNIPAM in water below and above the Θ temperature (30.6 °C), and the temperature dependences of the radius of gyration <S²>^1/2 and the hydrodynamic radius <R _h> were determined.

Kubota et al. [40] carried out experiments on PNIPAM samples with weight-average molecular weights M _w ranging from 1.6 × 10⁶ to 2.5 × 10⁷ at concentrations from 1 × 10⁻⁵ to 2 × 10⁻⁴. The dimensions of the polymer chains showed a sharp decrease just above the Θ temperature. The data obtained under the poor solvent condition were limited because at temperatures 1–2 K higher than the Θ temperature, the solution became metastable where scattered light intensity increased gradually, suggesting that chain aggregation was in progress.

Wu and Zhou [41] also observed chain collapse above the Θ temperature in dilute solutions of narrowly distributed (M _w /M _n  < 1.05), high molecular weight PNIPAM with M _w  = 1.1 × 10⁷ and 1.2 × 10⁷ at a concentration of 5 × 10⁻⁶ g/cm³. They observed intermolecular aggregation above ca. 32 °C, where they noted that individual PNIPAM chains first collapsed to densely packed single-chain globules that were stable for a limited period (~ 10² min), which were then thermodynamically driven to aggregate. In this kinetically stable two-phase region, the dimensions of the polymer chain were measured up to ca. 33 °C.

Wang et al. [42] studied the coil-globule transition of PNIPAM using an extremely dilute (7 × 10⁻⁷ g/cm³) solution of narrowly distributed (M _w /M _n  < 1.05) PNIPAM with M _w  = 1.3 × 10⁷. Because the solution was thermodynamically stable up to ca. 39 °C, the dimensions of the polymer chain in the highly collapsed state were measured at temperatures far above Θ.

From Eq. (2.4), it is conjectured that the coil-globule transition of PNIPAM is represented by the chain expansion factor α _S as a function of τM _w ^1/2, independent of the molecular weight M _w . Figure 2.3 shows the temperature dependence of the chain dimensions of PNIPAM in water in a plot of α _S versus τM _w ^1/2 made from data cited from different works [40–42]. The sharp decrease in α _S above the Θ temperature indicates the coil-globule transition of PNIPAM. In Fig. 2.3, the coil-globule transition curves observed by the different groups are almost the same, independent of the molecular weight of the sample. Figure 2.4 shows the same data as in Fig. 2.3, plotted as α _S ³ τM ^1/2 versus τM ^1/2 for closer inspection. Figure 2.4 shows that the data obtained by the different groups are in rather good agreement except that measured by Kubota et al. [40] for the sample with the highest molecular weight M _w  = 2.5 × 10⁷. They estimated the polydispersity of this sample as M _w /M _n  = 1.3 from μ ₂/<Γ>² determined from the correlation function of DLS data; the discrepancy of the data of this sample from the others would be due to effects of the molecular weight distribution.

In Figs. 2.3 and 2.4, the data for the kinetically stable two-phase region obtained by Wu and Zhou [41] are shown with the data for the thermodynamically stable phase. The data of the kinetically stable region are almost in accord with those obtained by Wang et al. [42] with an extremely dilute solution in the thermodynamically stable region, suggesting that the equilibrium size of a globule can be estimated if the measurement is carried out when chain collapse has finished before the intermolecular aggregation starts, even in a kinetically stable two-phase region.

Figure 2.5 shows the data obtained by Wang et al. [42] in the plots of α _S and α _h (=<R _h>/<R _h>₀) as a function of τM _w ^1/2, showing that α_h also decreases sharply above the Θ temperature owing to the coil-globule transition. In Fig. 2.5, the chain dimensions in terms of α _S decrease by approximately sixfold; the decrease in α _h is much less in this temperature range, i.e., ~3.5 times. This difference between the changes in the radius of gyration <S²>^1/2 and the hydrodynamic radius <R _h> is understandable because they are defined in different ways. <S²>^1/2 is related to the actual space occupied by the polymer chain, whereas <R _h> is the radius of an equivalent hard sphere that has an identical diffusion coefficient as the polymer chain in the solution. When a polymer chain is in an extended coil form under good solvent conditions, the polymer chain is nearly free draining, so that <R _h> is much smaller than <S²>^1/2. At the Θ point, the polymer chain becomes nondraining so that the ratio <S²>^1/2/<R _h> becomes smaller. According to the data of Wang et al., <S²>^1/2/<R _h> had a nearly constant value of 1.50 at T < Θ and decreased markedly from 1.50 to 0.56 just above Θ. At higher temperatures, <S²>^1/2/<R _h> increased and reached 0.77. The values of 1.50 and 0.77 are close to 1.504 and (3/5)^1/2, which are predicted for a random coil at the Θ point and a uniform hard sphere, respectively [43]. The value of 0.56 being less than (3/5)^1/2 was interpreted to indicate the formation of a molten globule, i.e., a globule with many small stressed loops on the surface. The loops are so small that their contribution to <S²>^1/2 is negligible, but they are not draining, which leads to a larger <R _h>. The average chain density ρ can be estimated using ρ = M _w /[N _A(4/3)π<R _h>³], and ρ was estimated to be 0.34 g/cm³ in the globule region, indicating that the globule contains 66 % water in its hydrodynamic volume.

In Fig. 2.5, the experimental data are compared with those calculated from the following equation [42]:
which is obtained by replacing τ in Eq. (2.4) with (−τ) in order to reproduce the LCST behavior. The coil-globule transition curve calculated using Eq. (2.5) coincides with the experimental data near the Θ temperature, but there is a deviation between the calculated and experimental results in the globular region. The deviation would be attributed to the complex mechanism of LCST; Equation (2.5) based on the simple excluded-volume theory may not be applicable to LCST-type polymers. A more elaborate theory in which cooperativity in hydration is taken into account has recently been developed [44, 45] for representing coil-globule transition of LCST-type polymers. According to Eq. (2.5), the plot of α _S ³ τM ^1/2 approaches an asymptotic value of (C′/B′) in the poor solvent limit (τM ^1/2 → ∞). However, Fig. 2.4 shows that after reaching the minimum, α _S ³ τM ^1/2 starts to increase with τM ^1/2. This behavior can be explained by a finite compressibility of a collapsed globule [46]. Although the globule contains 66 % water inside, the observed <S²>^1/2 in the globule state is independent of temperature, which causes the increase in α _S ³ τM ^1/2 in a high-temperature limit.

Nakata [14] has studied chain collapse below the Θ temperature using PMMA solutions of UCST-type phase behaviors. He carried out SLS measurements with moderately dilute solutions (~10⁻⁴ g/cm³). Although the concentrations were higher than those used in the studies of PS in cyclohexane [38, 39], he observed coil-globule transition to be not affected by intermolecular aggregation. This is because chain aggregation occurs very slowly in PMMA solutions. For example, the rate of chain aggregation in a dilute solution of PMMA in acetonitrile (AcN) is more than one hundred times lower than in that of PS in cyclohexane [15]. When the temperature of a dilute solution of PMMA in AcN is changed from the Θ temperature to a lower temperature, the mean-square radius of gyration <S²> of a single PMMA chain in the dilute solution decreases with time and reaches equilibrium before the polymer chains start to aggregate. The coil-globule transition curve is obtained by plotting the equilibrium values of <S²> against temperature [15].

Figure 2.6(a) shows the coil-globule transition curves for PMMA in AcN [15], isoamyl acetate (IAA) [16], and a mixed solvent of t-butyl alcohol (TBA) + water (2.5 vol%) [17], in the plots of α _S ² = <S²>/<S²>₀ versus τM ^1/2. The experiments employed PMMA with the molecular weights M _w of 6.4 × 10⁶ and 1.1 × 10⁷ in AcN [15], 2.4 × 10⁶ in IAA [16], and 1.6 × 10⁶, 2.8 × 10⁶, 4.0 × 10⁶, and 1.2 × 10⁷ in TBA + water [17]. The data points for different molecular weights construct a single curve for each polymer-solvent system. The coil-globule transition is sharp in TBA + water and broad in AcN. For PMMA in TBA + water, α _S ² in the globular region reaches 0.055, from which ρ is estimated to be 0.24 g/cm³.

The coil-globule transition curves were well represented by Eq. (2.4). The solid curves in Fig. 2.6 were depicted with the values of the parameters B and C shown in Table 2.1. The values of B and C of the other systems are also shown in Table 2.1 [15, 18]. Figure 2.6b shows the same data as those in Fig. 2.6a plotted as −α ³ τM ^1/2 versus −τM ^1/2. As is predicted in Eq. (2.4), the plot of −α ³ τM ^1/2 approaches an asymptotic value of (C/B) in the poor solvent limit (τM ^1/2 → −∞).

In Eq. (2.4), B and C stem from the second (vτ) and third (w) virial coefficients among monomers, respectively. According to Grosberg and Kuznetsov [34], B = 1.09v(N/M)^1/2/a ³ and C = 3.55w/a ⁶, where a and N are the monomer size and number, respectively. Using the equations, v and w were estimated and their values are given in Table 2.1. The virial coefficients, vτ and w, in the free-energy expansion in Eq. (2.2) of monomer interactions can be related to the binary β ₂ (=β ₀ τ) and the ternary cluster integral β ₃ among monomers through vτ = β ₂/2 and w = β ₃/6, respectively. For the solutions of PMMA in AcN, β ₀ and β ₃ have been estimated from the experimental values of the virial coefficients A ₂ and A ₃ for dilute solutions as β ₀ = 3.5 × 10⁻²³ cm³ and β ₃ = 4.8 × 10⁻⁴⁵ cm⁶, respectively [47, 48], and v and w corresponding to β ₀ and β ₃ are 1.75 × 10⁻²³ cm³ and 0.8 × 10⁻⁴⁵ cm⁶, respectively, which are comparable to those in Table 2.1. The coil-globule transition curve for PMMA in AcN has been analyzed on the basis of a more rigorous theory formulated by Grosberg and Kuznetsov (GK theory) [34], and v and w are estimated to be 0.88 × 10⁻²³ cm³ and 0.83 × 10⁻⁴⁵ cm⁶, respectively, which are closer to those due to A ₂ and A ₃ [15]. Thus, the shape of the coil-globule transition curves for the different polymer-solvent systems in Fig. 2.6 can be explained in terms of the excluded-volume interactions between monomers, which are characterized by v and w.

In the GK theory, both the radius of gyration <S²> and the hydrodynamic radius <R _h> are considered [34, 35]. Figure 2.7 shows the coil-globule transition curve in the plot of α _S ² against τM _w ^1/2 for PMMA in the mixed solvent of TBA + water (2.5 vol%) together with data of α_h ² obtained by a preliminary DLS experiment. The solid and dashed curves are obtained for α _S ² and α _h ² according to the GK theory [34, 35]. The theoretical values of α _h ² are smaller than the experimental data, which would be attributed to the breakdown of the supposition that all monomers are equally bathed by the solvent even in the dense globular state [36]. The coil-globule transition of PMMA studied by viscometry [49, 50] and DLS [51, 52] has been reported by several groups.

The previous section focused on the equilibrium chain dimensions as a function of temperature. When a polymer solution quenched from the Θ temperature to temperatures of poor solvent conditions, the chain dimensions will decrease with time until reaching their equilibrium value. The kinetics of chain collapse has been studied theoretically by means of two main approaches: phenomenological models [53–57] and computer simulations [58–64]. Phenomenological models show the dependence of the characteristic time of chain collapse on chain length, which can be verified by experimental observations.

A phenomenological theory of the kinetics of chain collapse was first considered by de Gennes [53]. A chain was assumed to occupy a sausage-shaped domain, and its chain collapse was modeled as a lateral swelling and a longitudinal contraction of the domain. In phenomenological models, the balance of driving and dissipative forces is considered to give scaling laws. In de Gennes’s theory, driving free energy is related to interfacial free energy, and the dissipation is attributed to hydrodynamic flow. The characteristic time τ _col of the chain collapse is predicted to show the dependence of the number N of monomers as τ _col ~ N ².

The topological constraints due to the nonphantomness of the chain and possible self-entanglement formation were not considered in de Gennes’s model. On the basis of these concepts, Grosberg et al. [54] developed a two-stage chain collapse model: a fast crumpling of the unknotted polymer chain to a crumpled globule and the subsequent slow knotting due to the reptation-like chain motion of the crumpled globule to a more compact globule. The characteristic times τ _crum and τ _eq for the first and second stages are written as τ _crum ~ N ² and τ _eq ~ N ³, respectively.

A different view was later proposed by Buguin et al. [55]. In their model, a polymer chain is assumed to form a shape consisting of globular clusters (“pearls”) of a dense phase connected by strings immediately after the quench (“pearl-necklace” model), as demonstrated by computer simulations [58, 59]. The collapse occurs in two stages: the coalescence of “pearls” to form a semi-compact state and the subsequent Ostwald ripening of the solvent “bubbles” remaining inside the semi-compact state. Two scenarios were considered for the first stage. In one, the driving energy is free energy of transfer of monomers in the strings into the “pearls” of the dense phase (transfer free energy), and the associated dissipation is due to hydrodynamic flow, yielding the characteristic time τ _col ~ N ^1/2. In another, the chain is envisioned as two pearls connected by a string of monomers, and the relation τ _col ~ N ^4/3 is obtained based on the assumption that the driving energy and the associated dissipation mechanism are the transfer free energy and the Stokes drag force on the “pearls,” respectively. For the second stage, the characteristic time τ _comp ~ N is obtained.

Several authors developed phenomenological models based on the “pearl-necklace” picture but obtained different scaling relations of collapse time depending on their assumptions. Klushin [56] described a self-similar coalescence stage in the collapse of a necklace of clusters. In this model, the driving free energy is the transfer free energy, and the dissipation is due to the Stokes drag force; the scaling relation τ _col ~ N ^0.93 is obtained in the nondraining limit. Halperin and Goldbart [57] considered the early stages of the chain collapse in detail. In the first stage, nascent droplets of a dense phase are formed, with little effect on the configuration of the bridges between the droplets. The nascent droplets then grow by accreting monomers from the bridges, thus causing the bridges to stretch. During these two stages, the overall dimensions of the chain decrease only slightly. In the third stage, the collapse of the “pearl-necklace” occurs, which causes the shrinkage of the overall dimensions of the chain. The characteristic times of the three stages scale as N ⁰, N ^1/5, and N ^6/5, respectively.

Experimentally, the first attempt to measure chain collapse kinetics was made by Chu et al. for PS in cyclohexane [65]. They observed two stages in chain collapse kinetics with characteristic times of τ _crum = 357 s and τ _eq = 323 s, using the dynamic light scattering of PS solution in a capillary tube with a thin wall (0.01 mm). They attributed the observed two steps to the fast crumpling and the slow knotting proposed by Grosberg et al. [54], respectively. The observed characteristic times are much longer than those predicted on the basis of the de Gennes’s theory [53]. In their experiments, although such a thin wall scattering cell was used, it still took at least 120 s for the temperature jump to reach its equilibrium, which is comparable to the observed chain collapse time. Moreover, interchain association occurred within 100 s after the temperature jump, so that collapsed single-chain globules coexisted with multichain aggregates in a solution.

Wu and Zhou [41] attempted to observe the chain collapse kinetics of PNIPAM using DLS after temperature jumps from 30.6 to 31.8 °C (thermodynamically stable region) and from 30.6 to 33.0 °C (kinetically stable two-phase region). In their experiments, the temperature jump took ~400 s. In the case of the temperature jump to 33.0 °C, the collapse kinetics was too fast to be monitored. The two-stage chain collapse kinetics was observed for the temperature jump to 31.8 °C, with the time scales for the two stages estimated to be ~50 and ~300 s, respectively.

Xu et al. [66] used a stopped-flow device to study the transition kinetics of a narrowly distributed PNIPAM chain randomly labeled with pyrene (PNIPAM-Py). It has been shown that PNIPAM chains exhibit coil-globule transition in a mixture of methanol and water at a specific composition, although both pure methanol and pure water are good solvents for PNIPAM [67]. The kinetics of coil-globule transition after mixing equal volumes of PNIPAM-Py in methanol and water was observed by recording the time dependence of the ratio of excimer emission intensity to monomer emission intensity (I_E/I_M). The time dependence of I _E/I _M was represented by a double exponential function with two distinct characteristic times (τ _fast ~ 12 ms and τ _slow ~270 ms). The observed τ _fast was in reasonable agreement with the value estimated by de Gennes’s theory [53].

Ye et al. [68] studied the collapse kinetics of linear PNIPAM homopolymer chains in dilute aqueous solutions using a fast infrared laser heating pulse and water-soluble 8-anilino-1-naphthalenesulfonic acid ammonium salt (ANS) in solution as a fluorescence probe owing to its sensitivity to its hydrophobic/hydrophilic surroundings. ANS fluorescence intensity increased rapidly after the temperature jump caused by the heating pulse. The time dependence of fluorescence intensity was represented by a double exponential function with two characteristic times τ _fast ~ 0.1 ms and τ _slow ~ 0.8 ms. In the range of the molecular weight between M _w  = 1.2 × 10⁶ and 2.3 × 10⁷, τ _fast is independent of chain length, and τ _slow shows a weak dependence on chain length. The first and second stages characterized by τ _fast and τ _slow could be attributed to the formation of droplets of the dense phase on the chain (~N ⁰) and the growth of the dense phase by accreting monomers from strings (~N ^1/5), respectively, as proposed in the theory of Halperin and Goldbart [57].