Dynamic Feature of Incipient Polymer Collapse below the Theta Point

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Dynamic Feature of Incipient Polymer Collapse below the Theta Point Yicen Liu,†,‡ Yibing Dai,†,‡ and Xiaofei Xu*,† †

Center for Soft Condensed Matter Physics and Interdisciplinary Research and ‡College of Physics, Optoelectronics and Energy, Soochow University, Suzhou, Jiangsu 215006, China ABSTRACT: We study the dynamics of a polymer chain with gradually changing the solvent quality from good to poor by dissipative particle dynamics simulation. We find several spectral modes related to internal motions of intrachain interaction. Approaching the coil-to-globule transition point, all fast modes of spectrum ω > 1 (ns)−1 disappear. There is only a slow mode at ω ≈ 0.66 (ns)−1. Moreover, the spectral density at this slow mode reaches a maximum value at the transition point. We suggest that, at the transition point, the chain conformation relaxes to the most probable distribution only by the slow mode. There is a critical slowing down of internal motion with passing through the transition point.

related to the incipient polymer collapse.6 The general experimental method to explore polymer chain dynamics is by dynamic laser light scattering (LLS), to measure the intensity−intensity time correlation function of the scattered light.2 The internal motion can be resolved into a series of normal modes with different frequencies, which contribute to the spectrum of scattered light. We could conclude the dynamic behaviors of polymer from the spectral density distribution. Results by LLS show that the mode of internal motion is observable only as qRg ∼ 1,11 where q is the wavenumber (the length of the wave vector), and Rg is the gyration radius of the polymer chain. Measurements of the diffusivity of internal motion also indicate that the dynamics would be slowing down passing through the theta point.12 The contribution of internal motion to the spectrum reaches a maximum value near the theta point.3 The method of LLS requires calculation of the ensemble average for the dynamic structure factor,13 by some empirical approaches14−16 such as the Rouse−Zimm model.17,18 In this model, chain segments diffuse by Brownian motion under the effect of chain connection and hydrodynamic interactions.19,20 Akcasu and co-workers21−24 developed a useful approach to obtain the spectral information from the dynamic structure factor. Because of the strict requirements in these experiments,25,26 it is also impossible to prepare many independent samples. Thus, it is hard to explore theta point dynamics systematically by experiment. In contrast to the experimental method, molecular simulation is a straightforward approach to study the system of this problem.27−31 It is easy to prepare

I. INTRODUCTION Polymer behaves as an ideal chain at the theta point in a solution, which separates good solvent quality from poor solvent quality.1 There are several ways to define the theta point in equilibrium thermodynamics, including the method by gyration radius1 and the second-order viral coefficient.2 For the same species, the theta point measured by those nondynamic definitions may vary in a wide range.3,4 The reason is that, for a flexible polymer, the internal motion (all the spatial and temporal information related to intrachain interaction) would change the conformation all the time. The conformation fluctuates to reach the most probable distribution under the effect of thermal energy.5 Therefore, the dynamic behaviors of the polymer chain are the key factor to determining the condition of the theta point. How to define the theta point from dynamic behaviors remains an open problem in polymer physics.5 As the solvent quality turns from good to poor, the polymer chain condenses from a coil to a globular state. The theta point is just the critical point of the coil-to-globule transition at the limit of infinite chain length. For a polymer with a short chain length, the coil-to-globule transition point lies below the theta point. In this region, the chain conformation is at a state of incipient collapse: the state between a coil and a compact globule. There is a great theoretical and practical interest to study the dynamics in this subtheta region,6 which is also insightful to understanding the theta point dynamics. Theoretical description of the regime below the theta point includes a variational method by the mean field approach7,8 or self-consistent-field theory,9 and the renormalization group theory of using the thermal blob model10 or bead−spring model.6 Existing theory predicted a critical slowing down of the relaxation of internal motion in the contractive regime, which is © XXXX American Chemical Society

Received: August 1, 2017 Revised: September 10, 2017 Published: September 19, 2017 A

DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B many independent samples. The time evolution for the position of a polymer segment is a direct output. We could explore the dynamic behaviors in a direct and efficient way by molecular simulation. Most references in the literature explored the dynamic process for the coil-to-globule transition.27,28,32−35 Few of them paid attention to the dynamic differences between theta state and nontheta state. While molecular dynamics simulation is the best method to study this problem,28,34,35 we chose the dissipative particle dynamics (DPD) simulation. DPD could account for hydrodynamic interactions, could treat solvent particles explicitly, and is also computationally efficient. We could handle a large system (chain length >200) and reach long time evolution (∼2000 ns) for the dynamics. In this work, the water molecules are coarse-grained by one spherical particle, and the polymer is described by the bead−spring model. With fixing the interaction energies of polymer−polymer (ϵPP) and solvent−solvent (ϵSS), we change the cross interaction of polymer−solvent (ϵPS) to mimic various solvent qualities. By gradually changing the solvent quality from good to poor, we explore how the dynamics behavior changes upon passing from the theta point to the collapsed globular state.

f ijC = ϵijw(rij)riĵ f ijD = −γw 2(rij)(riĵ ·vij)riĵ f ijR

⎧1 − r /rc , r ≤ rc w(r ) = ⎨ r > rc ⎩ 0, ⎪

⎪

dvi = fi dt

where n is the total number of solvent molecules. The solvent density determined in this way gives the same surrounding environment for polymer of different N. We fix ρs at 1rc−3. Therefore, n may be 72 000, 192 000, 256 000, and 600 000 for various chain lengths and box sizes. We run the simulation by the DPD package in LAMMPS using a modified velocity-Verlet algorithm with λ = 0.65 (relaxation factor) and time step Δt = 0.02τ, where τ is the time scale. The τ value is determined by fitting the diffusion constant of solvent with that of real water at physical temperature T = 298 K.27 One could get τ ≈ 0.15 ns. In DPD, physical temperature T can only be determined in this indirect way. The reduced temperature defined by38

(1)

T* =

∑ (f ijC

+ f ijD + f ijR )

j≠i

⟨|vi|2 ⟩/3 kT

also needs to be specified, where ⟨...⟩ is an average over all particles. We fix it at T* = 1. We connect the DPD model and the real atomistic model as follows. First, one solvent bead corresponds to three water molecules, which specifies the rc value. Second, we run the simulation of a pure solvent system, and fit the simulated selfdiffusion coefficient to the experimental value at T = 298 K.27 This specifies the time scale of the simulation (τ = 0.15 ns) and the interaction energy strength between solvent beads (ϵSS = 25kT/rc). Third, we assume the polymer segment has the same size as that of a solvent bead and also fix the interaction energy strength between polymer segments at ϵPP = 25kT/rc. Then, we change the cross interaction between the polymer segment and solvent bead (ϵPS) to mimic various solvent qualities. The larger ϵPS is, the more repulsion is between a polymer segment and a solvent particle, which corresponds to a poorer quality for the solvent. Because we are studying the effect of solvent quality on chain dynamics, the molecular model and assumptions should be reasonable. The dynamics of the polymer chain is described by the dynamic structure factor

(2)

in which vi and f i are the velocity and force of bead i, respectively. The force on each bead contains the sum contributions of a conservative force f Cij , a dissipative force f Dij , and a random force f Rij : fi =

(5)

We perform the simulation in cubic boxes with periodic boundaries in three dimensions. In order to eliminate the effect of finite box size, we vary box sizes with different chain lengths: N = 80 in a box of L = 30rc, N = 120 and 160 in L = 40rc, and N = 240 in L = 50rc. For different N, the solvent density is specified such that every monomer of the chain is surrounding by the same number of solvent molecules per cross-sectional area in a box, which is measured by n ρs = 2 L Nrc

The spring constant of the bond is KS = 80kT/rc2, where k is the Boltzmann constant. rc is the cutoff radius of the bead for molecular interaction, which is the length scale in this work. From the volume of three water molecules, we could estimate that rc ≈ 0.646 44 nm. The equilibrium spring length is req = 0.86rc. We denote the position of bead i by ri and the vector from bead j to bead i by rij = rj − ri. In eq 1, r̂ij = r̂ij/rij is the unit vector of rij and rij = |rij|. The velocity and position of each bead follow Newton’s equations: dri = vi dt

= σηijw(rij)riĵ

where ϵij is the interaction strength between beads i and j. γ is a viscosity coefficient, and σ2 = 2γkT. vij = vj − vi is the relative velocity between beads i and j. ηij is a Gaussian random variable. w is a weight function of

II. SIMULATION METHOD AND THEORY We consider one single homopolymer chain (P) immersed in a solvent (S) of water. The system is simulated by the dissipative particle dynamics.36,37 The molecular model and simulation method used here are the same as those used by Guo et al.27 The molecular units of both components are coarse-grained as spherical particles. The polymer is modeled as a flexible chain of N identical segments. The masses and sizes of both molecular units are assumed to be the same. The solvent molecules are coarse-grained as one spherical particle with the size of three real water molecules.27 The consecutive segments (segments i and i + 1) in the polymer chain are connected by a bond of spring force. FiS, i + 1 = KS(ri , i + 1 − req)rî , i + 1

(4)

(3)

Every term in eq 3 is given by the following: B

DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B S(q , t ) =

1 N

N

∑ exp[iq·(ri(t ) − rj(0))] i,j=1

(6)

where q is the wave vector. The angle brackets represent the ensemble average. We calculate the ensemble average over 505 independent samples. For each sample, we collect the data after the system reaches the dynamic equilibrium by running enough time (about 100 ns). All the spatial and temporal information related to intrachain is incorporated by eq 6. It contains the information on both internal motion and translational motion. If the wave vector q is homogeneous in space, we could use S(q, t) with q = |q|. This is the most common case for application in experiment and simulation. We use the numerical method developed by Moe and Ediger39 to calculate S(q, t). The spectral density distribution of internal motion is defined by the Fourier transformation of S(q, t):13 S(̃ q , ω) =

∫0

+∞

S(q , t )e−iωt dt

(7)

where ω is a real number having a unit of time−1. S̃ may be a complex number. We usually use the norm form |S̃| for application. |S̃| is a function of ω with periodic 2π, and is also symmetric between intervals [0, π] and [π, 2π]. Therefore, we can just limit ω in the range [0, π]. It should be mentioned that eq 7 is different from the one in dynamic LLS. Dynamic LLS uses the inverse Laplace transformation to obtain the spectra from the electric field− field time correlation function.3 In comparison with dynamic LLS, we could obtain the segment position ri(t) directly. There is no approximation to calculate the S(q, t) and its ensemble average. It becomes easy to obtain the spectral information by performing Fourier transformation from S(q, t).

Figure 1. Scaled mean square radius of gyration as a function of solvent quality.

err =

∫0

t0

err(t ) dt

where err(t) is the standard deviation of 505 independent samples at time t, and t0 = 750 ns. There are strong fluctuations for Rg approaching the phase transition point from the coil state. Because of the strong internal motion, this fluctuation effect is particularly great at finite chain length. As a result, the behavior of the ensemble average may not follow the ideal scaling behavior in eq 8. It becomes hard for those curves to meet at one point at the theta point. In fact, the four profiles shown in Figure 1 cannot meet at the same crossing point. Note that we show the data in four panels to give a better view for the error bar size. It should be mentioned that Guo et al.27 locate the theta point for the same system at ϵPS ≈ 27kT/rc by the Rg method. We believe that their method is qualitative as it does not consider the fluctuation effect, particularly for short chain cases. The Rg method fails because of its nondynamic feature. We try to understand theta point dynamics from the dynamic structure factor S(q, t). For the convenience of discussion, we limit the chain length at N = 240. The coil-to-globule transition point of N = 240 lies at ϵPS ≈ 28kT/rc, the corresponding temperature at which is lower than the temperature at the theta point. However, the physics should be similar for the two cases. We first test our simulation by comparing the S(q, t) behavior with the Brownian theory; see Figure 2. In Brownian theory, the polymer chain is treated as a whole particle with neglecting all the internal motions. If the transition probability is Gaussian,40 the dynamic structure factor can be written as

III. RESULTS AND DISCUSSION We first reexamine the dependence of gyration radius on solvent quality for various chain lengths. As the solvent quality turns from good to poor, polymer−solvent interaction becomes unfavorable. The chain condenses from a coil to a globule state. In Figure 1, all the data reveal this phase transition. The phase transition point is an N-dependent quantity. The theta point is just the critical temperature of the phase transition at the limit of infinite N. In equilibrium thermodynamics, the theta point can be identified by the scaling behavior of chain dimension. The method can be by the scaling of either end-to-end distance32,33 or gyration radius:1 ⎧ N1/5 , good ⎪ ⎪ ⟨R g 2⟩/N ∼ ⎨1, θ ⎪ ⎪ N −1/3 , poor ⎩

1 t0

(8)

At the theta point, ⟨Rg2⟩/N is independent of chain length. Therefore, it is possible to locate the theta point by searching the meeting point for those curves of ⟨Rg2⟩/N versus ϵPS (i.e., solvent quality) for various chain lengths. However, this is the ideal case in theory. In practice, the situation is affected by the fluctuation of chain conformation near the phase transition point. Figure 1 shows these fluctuations. We estimate the error of each data by the mean standard deviation over the time interval [0, t0].

S(q , t ) = exp( −Dq2t )

(9)

D is the long-time diffusion coefficient D = lim

t →∞

⟨(r(t ) − r(0))2 ⟩ 6t

(10)

where r(t) and r(0) are the position of center of mass for the chain at time t and 0, respectively. C

DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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Figure 3. Dynamic structure factor of polymer chain (N = 240) in water solvent as a function of time. The magnitude of the scattering vector is fixed at q = 0.09rc−1. The mean standard deviations are (a) err = 0.02512, (b) err = 0.03180, (c) err = 0.03336, and (d) err = 0.05969.

Figure 2. Dynamic structure factor of polymer chain (N = 240) in water solvent as a function of time. The value of q is at 0.09rc−1. The dashed lines are the prediction by eq 9. The mean standard deviations for the red lines are (a) err = 0.02512 and (b) err = 0.03031.

poor solvent, the polymer chain behaves as a whole nanoparticle. At ϵPS = 35kT/rc, S(q, t) decays quickly to a value less than 10−6 at t ≈ 600 ns. For long-time diffusion, there is a fluctuation on the scale of 10−5, due to the internal motion. The internal motion inside a compact globule is not frozen. S(q, t) successfully captures different behaviors in various solvent qualities. These behaviors should have different contributions to the spectral information. It would be clearer to understand the differences if we observe the spectral data by S(q, ω). Figure 4 shows the spectral density distribution for the data in Figure 3. There are several local maxima in these profiles. We believe that each of them corresponds to one

In Figure 2, q is fixed at 0.09rc−1 so that the internal motion is observable (i.e., such that qRg ∼ 1).41 At the coil state (ϵPS = 25kT/rc), the profile of S(q, t) is an exponential decay curve. The decay rate of the DPD curve is slower than that of Brownian theory, because DPD simulation could capture more details of internal motion. At the globule state, the DPD curve almost coincides with the curve of Brownian theory as t < 100 ns. This means that treating the globular chain as a whole particle is a good approximation for short-time diffusion. S(q, t) by DPD simulation shows a peak at t ≈ 250 ns. In a very poor solvent condition (ϵPS = 60kT/rc), the polymer chain condenses as a compact globule, but the dynamics inside the globule is not frozen. The internal motion of the chain is still large. The same behavior has been observed by another simulation method.29 Note that experiments by dynamic LLS cannot measure the internal motion inside a compact globule directly, because laser light is hard to penetrate and scatter in a compact globule. As t > 400 ns, S(q, t) decreases to zero because of the translational motion of the globule. There are more interesting details if we observe the data in a logarithm scale. S(q, t) shows different behaviors at different solvent qualities. Figure 3 shows S(q, t) at four solvent qualities: one in good solvent (ϵPS = 25kT/rc), two near the transition point (ϵPS = 27, 28kT/rc), and one in poor solvent (ϵPS = 35kT/rc). At ϵPS = 25kT/rc, the polymer chain is at the coil state. The chain changes its conformation freely under the effect of thermal energy, and also translationally diffuses due to the collision of water solvents. S(q, t) decays to the magnitude of 10−5 at t ≈ 800 ns and then fluctuates in the range [0, 10−3) for long-time diffusion. Approaching the transition point, the translational motion becomes weak and the internal motion starts to be important. At ϵPS = 27kT/rc, S(q, t) has a long tail and cannot touch zero value even at t = 1500 ns. At ϵPS = 28kT/rc, S(q, t) cannot decrease to a value below 10−3. At the transition point, the contribution by translational motion is very weak, and the contribution by internal motion reaches a maximum value. At

Figure 4. Spectral density distribution for polymer chain (N = 240) in water solvent. The magnitude of the scattering vector is fixed at q = 0.09rc−1. The mean standard deviations over ω are (a) err = 0.001961, (b) err = 0.002322, (c) err = 0.002814, and (d) err = 0.005256. D

DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry B Table 1. Spectral Density at ω ≈ 0.66 (ns)−1 ϵPS [kT/rc] |S̃| ϵPS [kT/rc] |S̃|

25.0 1.61826 ± 0.00113 29.0 1.66153 ± 0.00151

26.0 1.62154 ± 0.00111 30.0 1.64812 ± 0.00136

mode of internal motion. The mode of a sample is the element that occurs most often in statistics. Chain conformation relaxes to the most probable distribution by the frequency at these modes. At good or poor solvent conditions, four or five modes give contributions to the chain dynamics. Approaching the transition point, all fast modes disappear. There is only a slow mode at ω ≈ 0.66 (ns)−1 (see the red arrows). This means that chain conformation relaxes only by this slow mode. Moreover, the spectral density at this slow mode reaches a maximum value; see the data in Table 1. Experiments by dynamic LLS also obtained similar behavior.3 The contribution of the peak related to internal motion in the line-width distribution makes a turning (i.e., reaches a maximum) near the theta point. By measuring the spectral density at the peak (or the slow mode in this work), we could identify the theta point accurately. Figures 3 and 4 indicate that the internal motion would be slowing down passing through the transition point. An experiment by dynamic LLS observed the same behavior.12 The diffusivity of internal motion reaches a minimum value nearby the theta point. Theory by renormalization group description also predicted similar behavior.6 Approaching the contractive point, the largest mode approaches zero, which is related to the incipient molecular collapse. This indicates a critical slowing down of relaxation of internal motion in the contractive regime, which leads to a very slow decay of the S(q, t). Figure 4 shows that all fast modes disappear approaching the transition point, which agrees with the previous theoretical predictions.6 But it is worth noting that the fast mode observed in dynamic LLS is related to the translational diffusion of individual chains. They should never disappear at any solvent conditions. Therefore, these fast modes in Figure 4 must be different from the fast mode observed in experiments. It is insightful to know how S(q, ω) is at the limit case. We consider two limit systems: one only has internal motion and the other one only has translational motion. Ideal coupled oscillator is an example of the previous system, which is an ideal and simplified model for lattice structure.42 For the onedimensional case, the beads in the oscillator are connected equidistantly by a harmonic spring force; see Figure 5. All beads

27.0 1.64887 ± 0.00139 31.0 1.64057 ± 0.00139

28.0 1.68375 ± 0.00117 32.0 1.63462 ± 0.00141

Figure 6. (a) Spectral density distribution of ideal coupled oscillators (N = 10). The frequency is ω = π/2, and A = 0.01906rc. q = 0.5rc−1. (b) Spectral density distribution of polymer chain (N = 240) in water solvent by Brownian theory. All the internal motions are neglected. q = 0.09rc−1 and ϵPS = 25kT/rc.

opposite direction. The time evolution of the jth bead follows the equation xj(t ) = A sin(ωt + jπ ) + xj0

(11)

where ω is the frequency and A is the amplitude. is the position of bead j at t = 0. We fix ω = π/2 and A = 0.01906rc. This A value is picked as the average moving distance of water molecules in ice structure at T = 205 K by molecular dynamics simulation.43,44 As is shown, the spectral information at ω = π/ 2 (see the red arrow in Figure 6a) can be clearly captured by the profile. The data also reveal several minor modes of internal motion. Figure 6b shows the other limit case which only has translational motion: the spectral information on the polymer chain (N = 240) in water solvents predicted by Brownian theory (eq 9). As we discussed above, all internal motions are neglected, so there is only translational diffusion. As is shown, it is a smooth curve of exponential decay. The dynamics of polymer in solvent is a combination of internal motion and translational motion. Although the model in Figures 5 and 6a is quite different from the polymer, we could expect that S(q, ω) is at the limit of only internal motion. This is helpful to understanding the real dynamics of the polymer chain. In conclusion, the spectral density distribution is a smooth curve of exponential decay at the limit of only translational motion, and is a curve having many local maximum points at the limit of only internal motion. Any deviation from the smooth curve should originate in the internal motion. Therefore, we can conclude that the local maximum points in Figure 4 correspond to the modes of internal motion. We can identify the coil-to-globule transition point by measuring the spectral density at the slow mode. x0j

Figure 5. Schematic of ideal coupled oscillators in one dimension. The distance between two consecutive beads is rc. Each bead oscillates only in the x direction.

do simple harmonic motion in the x direction around its equilibrium position. The amplitude of the oscillation is selected to mimic the movement of a lattice site. Because there is no translational motion, the internal motion should account for the spectral information completely. Figure 6a shows the spectral density distribution for an example of 10 connected oscillators (N = 10). At t = 0, the beads are equidistant by rc. The consecutive two beads always move in an

IV. CONCLUSIONS We study the dynamics of the polymer chain with gradually changing the solvent quality from good to poor by dissipative E

DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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(5) Dai, Z.; Ngai, T.; Wu, C. Internal motions of linear chains and spherical microgels in dilute solution. Soft Matter 2011, 7, 4111−4121. (6) Douglas, J. F.; Freed, K. F. Polymer contraction below the Θ point: A renormalization group description. Macromolecules 1985, 18, 2445−2454. (7) Moore, M. A. Theory of polymer coil-globule transition. J. Phys. A: Math. Gen. 1977, 10, 305−314. (8) Kholodenko, A. L.; Freed, K. F. Coil globule transition: Comparison of field theoretic and conformational space formulations. J. Phys. A: Math. Gen. 1984, 17, 2703. (9) Wang, R.; Wang, Z.-G. Theory of polymers in poor solvent: Phase equilibrium and nucleation behavior. Macromolecules 2012, 45, 6266−6271. (10) Akcasu, A. Z.; Han, C. C. Molecular weight and temperature dependence of polymer dimension in solution. Macromolecules 1979, 12, 276−280. (11) Yang, C.; Kizhakkedathu, J. N.; Brooks, D. E.; Jin, F.; Wu, C. Laser-light-scattering study of internal motions of polymer chains grafted on spherical latex particles. J. Phys. Chem. B 2004, 108, 18479− 18484. (12) Nishio, I.; Swislow, G.; Sun, S.-T.; Tanaka, T. Critical density fluctuations within a single polymer chain. Nature 1982, 300, 243− 244. (13) Pecora, R. Doppler shifts in light scattering. II. Flexible polymer molecules. J. Chem. Phys. 1965, 43, 1562−1564. (14) Kramer, O.; Frederick, J. E. Effect of molecular weight on Rayleigh line spectrum of polystyrene in 2-butanone. Macromolecules 1972, 5, 69−75. (15) King, T. A.; Knox, A.; McAdam, J. D. G. Internal motion in chain polymers. Chem. Phys. Lett. 1973, 19, 351−354. (16) McAdam, J. D. G.; King, T. A. Polymer dynamics in solution from Rayleigh line-profile spectroscopy. Chem. Phys. 1974, 6, 109− 116. (17) Rouse, P. E. A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 1953, 21, 1272− 1280. (18) Zimm, B. H. Dynamics of polymer molecules in dilute solution: viscoelasticity, flow birefringence and dielectric loss. J. Chem. Phys. 1956, 24, 269−278. (19) Perico, A.; Piaggio, P.; Cuniberti, C. Dynamics of chain molecules. I. Solutions to the hydrodynamic equation and intrinsic viscosity. J. Chem. Phys. 1975, 62, 4911−4918. (20) Perico, A.; Piaggio, P.; Cuniberti, C. Dynamics of chain molecules. II. Spectral distribution of the light scattered from flexible macromolecules. J. Chem. Phys. 1975, 62, 2690−2695. (21) Akcasu, Z.; Gurol, H. Quasielastic scattering by dilute polymer solutions. J. Polym. Sci., Polym. Phys. Ed. 1976, 14, 1−10. (22) Akcasu, Z.; Higgins, J. S. Quasielastic scattering of neutrons from freely jointed polymer chains in dilute solutions. J. Polym. Sci., Polym. Phys. Ed. 1977, 15, 1745−1756. (23) Akcasu, Z.; Benmouna, M. Concentration effects on the dynamic structure factor in polymer solutions. Macromolecules 1978, 11, 1193−1198. (24) Han, C. C.; Akcasu, Z. Dynamic light scattering of dilute polymer solutions in the nonasymptotic q region. Macromolecules 1981, 14, 1080−1084. (25) Wu, C.; Wang, X. Globule-to-coil transition of a single homopolymer chain in solution. Phys. Rev. Lett. 1998, 80, 4092−4094. (26) Wu, C.; Zhou, S. First observation of the molten globule state of a single homopolymer chain. Phys. Rev. Lett. 1996, 77, 3053−3055. (27) Guo, J.; Liang, H.; Wang, Z.-G. Coil-to-globule transition by dissipative particle dynamics simulation. J. Chem. Phys. 2011, 134, 244904. (28) Chang, R.; Yethiraj, A. Solvent effects on the collapse dynamics of polymers. J. Chem. Phys. 2001, 114, 7688−7699. (29) Jentzsch, C.; Werner, M.; Sommer, J.-U. Single polymer chains in poor solvent: Using the bond fluctuation method with explicit solvent. J. Chem. Phys. 2013, 138, 094902.

particle dynamics simulation. The dynamic structure factor S(q, t) shows different behaviors at different solvent qualities. At good solvent, S(q, t) (at qRg ∼ 1) decays exponentially and fluctuates due to the contribution of internal motion. Near the coil-to-globule transition point, S(q, t) decays very slowly because of the strong internal motion. At poor solvent, S(q, t) decays to zero quickly and then fluctuates for long-time diffusion. The globular chain behaves as a whole nanoparticle, whose dynamics is not frozen. The chain dynamics of internal motion would be slowing down passing through the transition point. This result agrees well with previous theoretical and experimental predictions in the literature.6,12 We find several spectral modes in S̃(q, ω), the Fourier transformation of S(q, t), that are related to internal motions of intrachain interaction. Approaching the coil-toglobule transition point, all fast modes at the spectrum ω > 1 (ns)−1 disappear. There is only a slow mode at ω ≈ 0.66 (ns)−1. Moreover, the spectral density at this slow mode reaches a maximum value at the transition point. We suggest that, at the transition point, the chain conformation relaxes to the most probable distribution only by the slow mode. We can identify the transition point by measuring the spectral density at this slow mode. Because the time evolution of particle position is a direct output, this method is useful in particular for molecular simulation. At the limit of infinite chain length, the coil-to-globule transition point of dilute polymer solution is very close to the theta point. Therefore, we expect that our results should also be valid for theta point dynamics. The theta point measured by nondynamic definitions may vary in a wide range.3,4 How to define the theta point of polymer solutions from dynamic behaviors remains an open problem in polymer physics. This work provides a possible answer by using the dynamic structure factor. This work focuses on dilute polymer solution. We suggest further work in the systems of semidilute and concentrated solutions.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +86-150-5147-8694. ORCID

Xiaofei Xu: 0000-0003-2459-7748 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Prof. Zhen-Gang Wang for introducing this topic of theta point dynamics to us, and for helpful comments and discussions to this work. Y.L. thanks Dr. Lei Wang for help in building the simulation model. This work was supported by the National Natural Science Foundation of China under Grants 21404078 and 21674077.

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REFERENCES

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DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.jpcb.7b07637 J. Phys. Chem. B XXXX, XXX, XXX−XXX