A Method To Determine Θ Condition of a Polymer Solution

Jun 12, 2018 - The internal motions of narrowly distributed flexible polymer chains in extremely dilute solutions were studied by dynamic laser light ...
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A Method To Determine Θ Condition of a Polymer Solution Xiaoqing Ming† and Chi Wu*,†,‡ †

Department of Chemistry, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong The Hefei National Laboratory of Physical Science at Microscale, Department of Chemical Physics, The University of Science and Technology of China, Hefei, Anhui 230026, China

‡

ABSTRACT: The internal motions of narrowly distributed flexible polymer chains in extremely dilute solutions were studied by dynamic laser light scattering (LLS). We focused on the temperature-dependent relative intensity of the internal motions AI/(AI + AD), where AI and AD are the scattered light intensities, respectively, related to the internal motions and diffusive relaxation and obtainable from their related peak areas when ⟨Rg2⟩q2 > 1, where ⟨Rg2⟩ and q are the radius of gyration and the scattering vector, respectively. The results from four different polymer solutions reveal a maximum or turning point of AI/(AI + AD) near the theta temperature (Θ), confirming that the thermal energy can excite more internal motions when two-body and higher-order interactions diminish. Our study illustrates that dynamic LLS can be used to determine the Θ condition in a better way, in which only one dilute solution is needed in comparison with a set of polymer solutions previously required.

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internal motions.11 To reduce the interchain interaction, it is also necessary to use very low polymer concentrations.12,13 In the past 40 years, the internal dynamics of linear flexible polymer chains has also been experimentally studied,14−25 but most of these studies were focused on the scaling behavior of the first cumulant Γ(q), the asymptotic behavior of the reduced first cumulant Γ* (= [η0Γ(q)/kBTq3]q→∞), and the characteristic relaxation time τ1 of the first normal mode. Previously, we had studied the effect of the solution temperature on the relative intensity contribution of the internal motions, AI/(AI + AD), and found that there exists a turning point around the pseudoideal state, i.e., the Flory Θ temperature.26,27 The qualitative explanation or hypothesis is as follows. As the temperature approaches Θ from the good solvent side, the two-body interaction diminishes so that both the additional excluded volume and the entropic elastic effects decrease to zero if the Θ state is used as a reference point.26 Therefore, the chain should become softer and more deformable as T approaches Θ. On the other hand, in the poor solvent region (i.e., T < Θ), the intersegment interaction becomes stronger so that the chain shrinks in comparison with its size at the Θ state, which suppresses the fluctuation of individual segments. Therefore, more internal motions of a polymer chain should be excited near its Θ state. In principle, the relative contribution of the internal motions should reach a maximum when the chain can be modeled as a Kuhn equivalent freely jointed one and the Kuhn segments exhibit a selfintersecting random walk.

INTRODUCTION Physically for a flexible polymer chain in a dilute solution, its individual segments are agitated by the thermal energy (kBT) to undergo the Brownian motions so that their center of gravity diffuses and the segments move randomly and relatively to the center of gravity. These motions can be described by a linear combination of different normal modes with corresponding characteristic relaxation frequencies. The zeroth-order normal mode is related to the translational diffusive relaxation, while other higher-order normal modes are attributed to the internal motions.1 The internal motions have been studied by dynamic laser light scattering (LLS).2−7 The spectrum of the scattered light contains the contributions from the relaxations of the translational diffusion and those internal motions when the observation length 1/q is comparable or shorter than the rootmean-square average radius of gyration (⟨Rg2⟩1/2, or simply written as Rg), where q = 4πn sin(θ/2)/λ0 is the scattering vector with n, θ, and λ0 being the solvent refractive index, the scattering angle, and the wavelength in a vacuum, respectively.8,9 Theoretically, Pecora,5 de Gennes,4,10 and Dubois-Violette10 used the Rouse−Zimm model to analyze the dynamic structure factor S(q,t) that is measurable in dynamic LLS. In this model, S(q,t) is mathematically described as a sum of multiple exponential terms related to translational diffusion and internal motions. Accordingly, the intensity contributions of the translational diffusive and internal modes, denoted as AD and AI, in the spectrum of scattered light have been predicted for different strengths of the hydrodynamic interaction.2,3 It was shown that the relative intensity contribution from the internal motions, i.e., AI/(AI + AD), is a function of the observation length (1/q) for a given chain in the range of qRg ≥ 1. Therefore, it is necessary to use long chains in studying the © XXXX American Chemical Society

Received: March 15, 2018 Revised: June 6, 2018

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DOI: 10.1021/acs.macromol.8b00559 Macromolecules XXXX, XXX, XXX−XXX

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where x = (qRg)2, L(ω,Γ) is a frequency ω-normalized Lorentzian distribution, centered at the frequency (ω0) of the incident light, and Γ is the line width (the half-width at halfheight), namely

In the current study, using dynamic LLS, we further studied four different kinds of dilute polymer solutions to further check whether the above explanation/hypothesis is correct. Namely, how the relative contribution of the internal motions AI/(AI + AD) varies with the solution temperature T at different scaled observation lengths, x = (qRg)2. Note that the Θ state is defined as the vanishing of the intrachain two-body interaction, but practically the Θ state is determined from the concentration dependence (i.e., the interchain interaction) of a physical property, such as the scattered light intensity, the solution viscosity, or the osmotic pressure, presumably because we had no experimental method to probe the intrachain two-body interaction directly before. Therefore, we think that the validation of the above hypothesis, i.e., the existence of a maximum or turning point of AI/(AI + AD) for different polymer solutions near their Θ points, should lead to a novel method to determine the Θ state of a polymer solution, which is not only closer to its original theoretical definition but also easier than the previous time-consuming concentrationdependent measurements because only a single diluted solution is required.

L(ω , Γ) =

THEORETICAL BACKGROUND It was shown by Pecora that when an infinitely dilute solution is illuminated by a coherent and monochromatic laser light beam, the spectral distribution S(q,ω) of a flexible polymer chain due to the translational diffusion and internal motions can be formulated as28 1 2π

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(1)

where ω is the angular frequency difference between the scattered and the incident light and D is the chain’s translational diffusion coefficient. The Fourier transform of S(q,ω) is as follows: S(q , t ) =

1 N2

N

N

∑ ∑ e−iq[r (0) − r (t)] l

m

(2)

l=0 m=0

Table 1. Weight-Average Molar Mass and Polydispersity of Polymers Used

where rl(0) and rm(t) are the position vectors of the lth segment at time 0 and the mth segment at time t, respectively, and the center of gravity of the chain is chosen as the reference point for the position vectors. Equation 2 expresses the interference of the scattered light from N chain segments. Therefore, all the spatial and temporal information related to intrachain relaxations (internal motions) are comprised in S(q,t). Note that for a sufficiently dilute polymer solution, the interference of different chains can be practically ignored. Incorporating the Oseen−Kirkwood−Riseman hydrodynamic interaction into the chain’s bead-and-spring model, Perico, Piaggio, and Cuniberti (PPC)29,30 showed that the ensemble average of S(q,ω) leads to

∑ P1(x , α)L(ω , q2D + Γα) α=1

N

+

N

∑ ∑ P2(x , α , β)L(ω , q2D + Γα + Γβ) α=1 β=1 N

+

N

polymer

Mw (g/mol)

Mz/Mw

solvent (Θ)

PMMA PIB PS PVAC

1.05 4.6 1.3 2.3

× × × ×

1.16 1.25 1.05 1.30

acetonitrile (Θ = 45.0 °C) isoamyl isovalerate (Θ = 27.0 °C) cyclohexane (Θ = 34.5 °C) 3-heptanone (Θ = 29.0 °C)

107 106 107 107

For each polymer sample, a stock solution was gravimetrically prepared and homogenized by gently shaking twice per day at ca. 50 °C for several days. For dynamic LLS, each stock solution was diluted to different desired concentrations, C (g/mL) = (0.5−5.0) × 10−5, by its corresponding solvent and then clarified at a temperature higher than its corresponding Θ using a 0.45 μm hydrophobic Millipore filter. The optical LLS cell (17 × 60 × 9.5 mm3) was used and sealed by a green melamine resin cap with an F217/PTFE linear. Each dilute polymer solution was kept at its clarification temperature before the LLS measurements. Laser Light Scattering (LLS). A modified commercial LLS spectrometer (ALV/DLS/SLS-5022F) equipped with a multitau digital time correlator (ALV5000) and a cylindrical 22 mW UNIPHASE He−Ne laser (λ0 = 632.8 nm) was used. The incident beam was vertically polarized with respect to the scattering plane. The details of the LLS instrumentation and theory can be found elsewhere.33 Briefly, in static LLS, the excess absolute time-averaged

N

⟨S(q , ω)⟩ = P0(x)L(ω , q2D) +

EXPERIMENTAL METHODS

Samples and Solution Preparation. Poly(methyl methacrylate) (PMMA) was prepared in water using CaCO3 as a suspension agent and 2,2′-azobis(2-methylpropionitrile) as an initiator. Polystyrene (PS) sample was synthesized in tetrahydrofuran using high-vacuum anionic polymerization with sodium naphthalene as an initiator. Polyisobutylene (PIB) was purchased from Sigma-Aldrich. Poly(vinyl acetate) (PVAC) was synthesized in water using sodium dodecyl sulfate as an emulsifying agent and potassium peroxysulfate as an initiator and subsequently alcoholyzed and esterified. All the polymer samples were further precipitation-fractionated.31 The weight-average molar mass of each polymer sample was characterized using static LLS. The polydispersity index was estimated from the relative average line width measured in dynamic LLS.32 The molecular characteristics of all the polymer samples used are listed in Table 1.

∞

∫−∞ exp(−iωt − q2Dt )S(q , t ) dt

(4)

and Pn is the contribution of each Lorentzian to the line-width distribution G(Γ) of the scattered light. P0(x) represents the contribution from the translational diffusive relaxation, P1(x,α) the first-order contribution from the αth internal mode, P2(x,α,β) the second-order contribution from both the αth and βth internal modes, and so on. When x ≪ 1, the observation length 1/q is much longer than the chain dimension so that the internal motions become invisible in dynamic LLS; namely, P0(x) becomes dominant. As x increases, the interference among the light scattered from different segments starts to contribute to the scattered light intensity; i.e., the higher-order terms Pn with n ≥ 1 become more and more important and have to be considered in analysis. Nowadays, a modern dynamic LLS instrument can generate S(q,t) from the measured intensity−intensity time correlation function G(2)(t) and S(q,ω) is easily obtainable.

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S(q , ω) =

1 2Γ 2π (ω − ω0)2 + Γ 2

N

∑ ∑ ∑ P3(x , α , β)L(ω , q2D + Γα + Γβ + Γγ) + ... α=1 β=1 γ=1

(3) B

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Macromolecules scattered light intensity of a dilute polymer solution, known as the excess Rayleigh ratio Rvv(θ), at a polymer concentration C and a scattering angle θ is related to the weight-average molar mass Mw, the mean-square average radius of gyration ⟨Rg2⟩, and the second virial coefficient A2 as

⎞ KC 1 ⎛⎜ 1 ≅ 1 + q2⟨R g 2⟩⎟ + 2A 2C ⎠ R w(θ) Mw ⎝ 3

square of the mass of a scattering object, very sensitive to any interchain association.39 The inset in Figure 1 shows the hydrodynamic radius distributions f(Rh) of PMMA in MeCN at 49.2 and 37.6 °C, where ⟨Rh⟩ = 64.8 and 56.9 nm, respectively. The narrow peaks indicate that PMMA used is very narrowly distributed; on the other hand, it further shows no interchain aggregation even in the poor solvent region.40 Note that the average scattered light intensity in the collapsed state remains a constant up to 116 h. The shrinking of individual PMMA chains without any interchain association lays a solid ground for further studies of the internal motions of individual PMMA chains at different temperatures, especially below Θ. Figure 2 shows how individual PMMA chains shrink as the solution temperature T decreases from 49.2 to 37.6 °C; namely,

(5)

where K = 4π n (dn/dC) /NAλ0 , a constant for a given polymer solution/dispersion, and q = 4πn sin(θ/2)/λ0, the scattering vector with dn/dC, NA, and λ0 being the specific refractive index increment, the Avogadro number, and the light wavelength in vacuo, respectively. In dynamic LLS, each measured intensity−intensity time correlation function G(2)(q,t) is related to the normalized electric field−field time correlation function g(1)(q,t) as 2 2

2

4

G(2)(q , t ) = ⟨I(q , 0)I(q , t )⟩ = A[1 + β |g(1)(q , t )|2 ]

(6)

where A ≡ ⟨I(0)⟩ is the measured baseline; 0 ≤ β ≤ 1, a spatial coherent constant, depends on the instrumental detection optics. The value of β reflects the signal-to-noise ratio of a dynamic LLS instrument. |g(1)(q,t)| ∼ S(q,t), related to the characteristic line-width distribution G(Γ) as34 2

|g(1)(q , t )| =

∫0

∞

G(Γ)e−Γt dΓ

(7) (1)

Therefore, the Laplace inversion of each measured g (q,t) yields one G(Γ). In this study, the CONTIN algorithm in the digital time correlator was used with PROB1(α) = 0.5 and the zeroth-order regularization for the data analysis.35−37 For a pure diffusive relaxation,38 (Γ/q2)q→0,c→0 leads to the translational diffusion coefficient D or further to the hydrodynamic radius Rh by the Stokes−Einstein equation: Rh = kBT/6πη0D, where kB, T, and η0 are the Boltzmann constant, the absolute temperature, and the solvent viscosity, respectively.

Figure 2. Solution temperature T dependence of average radius of gyration ⟨Rg⟩ and hydrodynamic radius ⟨Rh⟩ of poly(methyl methacrylate) in acetonitrile, where C = 5.00 × 10−5 g/mL. Inset shows solution temperature dependence of characteristic ratio ⟨Rg⟩/ ⟨Rh⟩.

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RESULTS AND DISCUSSION Figure 1 shows a Berry plot of (KC/Rvv)0.5 vs q2 for PMMA in MeCN at different solution temperatures T. The decrease of

⟨Rg⟩ decreases from 89.5 to 76.8 nm while ⟨Rh⟩ changes much smaller from 64.8 to 56.7 nm because ⟨Rg⟩ is more sensitive to the chain segment distribution in space and ⟨Rh⟩ is affected by the less chain draining as the chain shrinks. The inset in Figure 2 shows the corresponding temperature-dependent ratio of ρ = ⟨Rg⟩/⟨Rh⟩ of PMMA in MeCN. ⟨Rg⟩/⟨Rh⟩ reflects the chain conformation, decreasing from 1.505 to 0.774 as a linear chain changes its conformation from a self-avoiding random walk coil to a collapsed uniform globule.39 In the temperature range measured, ⟨Rg⟩/⟨Rh⟩ only slightly decreases to ∼1.4, revealing that individual PMMA chains remain its random coil conformation so that we can measure the relative contribution of the internal motions around the Θ point. Figure 3 shows typical Dq2-scaled line width distributions G(Γ/Dq2) of PMMA in acetonitrile at T = 44.9 °C. When x ≪ 1, the line-width distribution has only one narrowly distributed peak since the internal motions are invisible. When 1/q is comparable to ⟨Rg⟩, the contributions of the internal motions appears as an additional small Peak. While the total scattered light intensity decreases with increasing x, Figure 3 demonstrates that the total intensity is composed of a larger decrease in the intensity of the slow diffusion peak combined with an increase in the intensity of the fast internal mode. As x increases further, the scaled diffusion peak in G(Γ/Dq2) remains centered on Γ/Dq2. The fast internal mode peak in G(Γ/Dq2) shifts to the left, reflecting the q-independent contributions to Γ/Dq2 from the internal motions. It was shown that for a linear flexible coiled chain S(q,t) mainly contains the contributions from the translational

Figure 1. Scattering vector (angular) dependence of time-average scattered light intensity (Rayleigh ratio) of poly(methyl methacrylate) in acetonitrile at different solution temperatures T, where C = 5.00 × 10−5 g/mL and inset shows corresponding hydrodynamic radius distributions f(Rh) at θ = 15°.

the slope with T from 49.2 to 37.6 °C leads to the decrease of ⟨Rg2⟩1/2 (simply written as ⟨Rg⟩ hereafter) from 89.5 to 76.8 nm. The extrapolations of (KC/Rvv)0.5 to q = 0 at different temperatures result in a similar intercept, i.e., a similar weightaverage molar mass, revealing no interchain association at lower measured temperatures even when the chain shrinks. This is because the scattered light intensity is proportional to the C

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Figure 3. Dq2-scaled line width distributions G(Γ/Dq2) of poly(methyl methacrylate) in acetonitrile at different values of x [= (q⟨Rg⟩)2], where T = 44.9 °C (close to Θ) and small peaks related to internal motions are enlarged 20 times for a better view.

Figure 4. Solution temperature T dependence of line width distributions G(Γ/Dq2) of poly(methyl methacrylate) in acetonitrile at θ = 150°, where small peaks related to internal motions are enlarged for a better view.

diffusion and the first four internal motions in either the freedraining41 or the non-free-draining29,30 limit. At x > 1, S(q,t) can be rewritten as ∞

S(q , t ) =

∑ Pne

Γn/Dq2 =

where Γ = Γ0 = Dq , Γ = Γ0(1 + 2Γ1/Dq ), Γ = Γ0(1 + Γ2/Dq2), Γ(3) = Γ0(1 + 4Γ1/Dq2), and Γ(4) = Γ0(1 + 2Γ2/Dq2), and the numerical values of Pn in the range of 1 ≤ x ≤ 10 were reported by PPC.29,30 In the Zimm model,42 the line width of the nth-order normal mode (n ≥ 1) is related the eigenvalue λn′ = π2n3/2[1/2 − 1/(4πn)], the solution temperature T, the molar mass M, the solvent viscosity η0, and the intrinsic viscosity [η] as

Γn =

(1)

2

(2)

0.293RTλn′ Mη[η]

=

5.52λn′ NAR h 3 (qR h)2 M[η]

(12)

where Γn/Dq is proportional to 1/Rh at a given observation length 1/q since NARh3/M[η] is a constant for a given polymer solution. However, there is a discrepancy at T = 40.5 °C when acetonitrile becomes a poor solvent; namely, the peak related to the internal motions moves to the right (slower). In this case, it is difficult to give an explanation based solely on eq 12. We think that it is due to stronger intersegment interaction in the poor solvent when the PMMA chain shrinks at T = 40.5 °C. Three other studied polymer solutions also have similar behaviors. Figure 5 shows typical angular dependence of the average characteristic line widths of two peaks in the line width

(8) 2

xM[η] 2

−Γ(n)t

n=0 (0)

5.52λn′NAR g 2R h

2

(9) 2

Dividing the both sides by Dq and converting q to x, we have Γn/Dq2 =

0.293λn′R g 2RT xDMη0[η]

(10)

Using the Stokes−Einstein equation Rh = kBT/6πη0D, we can rewrite eq 10 as 2

Γn/Dq =

5.52λn′NAR g 2R h xM[η]

5.52λn′(R g /R h)2 NAR h 3 = x M[η] (11)

Figure 5. Angular dependence of average line widths ⟨Γ⟩ of two peaks related to translational diffusion and internal motions in line width distribution of poly(methyl methacrylate) in acetonitrile, where C = 5.00 × 10−5 g/mL, T = 44.9 °C, and dashed line shows ⟨Γ⟩I and ⟨Γ⟩D have a similar angular dependence when q2Rg2 < 2.

where Rg/Rh characterizes the chain conformation, and M[η] is proportional to the hydrodynamic volume of a flexible linear chain and widely used in the size exclusion chromatography for the universal calibration.43 At a given T, both Rg/Rh and NARh3/ M[η] are constants for a given polymer solution so that eq 11 indicates that the average peak position Γn/Dq2 decreases as x increases, which explains the peak shifting to the left in Figure 3. Figure 4 shows typical temperature dependence of Dq2scaled line width distributions G(Γ/Dq2) of PMMA in MeCN at θ = 150° or 1/q = 38.8 nm. Relatively, the fast translational diffusion peak is located around Γ/Dq2 = 1, nearly independent of T. In the good solvent region, as the solvent quality decreases from 49.2 to 44.9 °C, the peak related to the internal motions shifts to the left (faster), which is explainable on the basis of a variation of eq 11, namely

distribution of PMMA in acetonitrile. As expected, the line width of the translational diffusive relaxation is a linear function of q2. Note that the average line width ⟨Γ⟩I (related to internal motions) is also linearly dependent on q2 in the range of x < 2, consistent with the previous prediction:1,9,15,24 ⟨Γ⟩I = 2/τc + Dq2. The intercept of the dashed line at x = 0 leads to τc, i.e., τ1 = (1.6 ± 0.2) × 10−4 s. In the Rouse−Zimm model, the longest internal relaxation time at Θ is related to the solvent viscosity η0, the intrinsic viscosity [η], and the solution temperature as D

DOI: 10.1021/acs.macromol.8b00559 Macromolecules XXXX, XXX, XXX−XXX

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Macromolecules τ1 =

summarizes the plateau values of Γ* of other three kinds of polymer solutions. To account for such a discrepancy between the experimental observations and theoretical prediction, various explanations were proposed in the literature, including that η0 should be replaced with a local solvent viscosity ηeff;47 the internal friction and hydrodynamic screening should be included in the Rouse−Zimm model;48 some energetically favored internal motions might not be observed by dynamic LLS;12,13 and the chain stiffness and the local conformation should be considered.49 We would like to attribute it to the inability of thermal energy to excite all the predicted higherorder internal modes, especially when the chains are shrunken in the poor solvent. Figure 7 shows a typical angular dependence of relative contribution, AI/(AI + AD), of scattered light intensity related

Mη0[η] A1RT

(13) 2

where A1 for the free-draining limit and nondraining limit with3 and without preaverage44 is 0.822, 1.184, and 0.574, respectively. The corresponding calculated values of τ1 are 0.23 ms (Rouse model, τR), 0.16 ms (Zimm model with preaverage, τZ), and 0.32 ms (Zimm model without preaverage, τ′Z) for PMMA in acetonitrile. Together with other three types of polymer solutions, all the values of τ1 are summarized in Table 2. The measured results (τexpt) are not far away from those Table 2. Summary of Longest Internal Relaxation Time (τ1) and Plateau Values of Γ* (= [η0Γ(q)/kBTq3]q→∞) of Different Flexible Linear Polymer Chains in Solutions τ1 (ms) polymer

η0 (cP)

[η] (mL g−1)

τR

τZ

τ′Z

τexpt

Γ*

PMMA PIB PS PVAC

0.295 1.290 0.776 0.664

159 240 328 446

0.23 0.69 1.57 3.29

0.16 0.48 1.09 2.29

0.32 0.99 2.35 4.72

0.16 0.20 1.50 1.20

0.047 0.057 0.052 0.050

predicted values, but we are not able to differentiate them even it seems that those from the nondraining Zimm model with preaverage are generally closer to our measured ones. Figure 6 shows typical angular dependence of η0Γ(q)/kBTq3 of PMMA in acetonitrile at different solution temperatures,

Figure 7. (qRg)2-dependence of relative contribution, AI/(AI + AD), of scattered light intensity related to internal motions of polystyrene in cyclohexane, where T = 34.5 °C, C = 1.00 × 10−5 g/mL, and AI and AD are intensity contributions (peak area in line width distribution) of internal motions and translational diffusive relaxation, respectively.

to the internal motions of polystyrene in cyclohexane at Θ. As expected, light probes more internal motions of a linear flexible polymer chain as 1/q becomes smaller and smaller, which explains the observed increase of AI/(AI + AD) up to 20% for the highest available scattering angle. In the current study, other types of polymer solutions also show a similar trend. Figure 8 shows the reduced temperature (1 − Θ/T) dependence of the relative contribution, AI/(AI + AD), of the scattered light intensity related to the internal motions of PMMA, PIB, PS, and PVAC at (qRg)2 = 2.5, 4.0, and 5.0, respectively. Note that Rg is a function of the solution

Figure 6. (qRg)2 dependence of reduced first cumulant η0Γ(q)/kBTq3 of poly(methyl methacrylate) in acetonitrile at different solution temperatures, where C = 5.00 × 10−5 g/mL, where T is the absolute temperature and kB is the Boltzmann constant.

where Γ(q) is obtained from the cumulant fitting of G(2)(q,t). When (qRg)2 < 1, η0Γ(q)/kBTq3 decreases inversely proportional to q because the line width of the translational diffusive relaxation is proportional to q2. As qRg further increases, the decrease of η0Γ(q)/kBTq3 slows down and approaches a plateau. Such a scaling of the first cumulant Γ(q) with q3 has been predicted by the nondraining bead−spring model for flexible linear polymer chains in infinitely dilute solutions.10 The plateau of Γ*, i.e., [η0Γ(q)/kBTq3]q→∞, in poor solvent (37.6 °C) is clearly lower than those in good solvents, revealing that the internal motions are suppressed due to the chain contraction. Note that at 44.9 °C (Θ), Γ* = 0.047, smaller than 0.053 and 0.071 predicated in the nondraining model with a preaveraging Oseen tensor for the theta and good solvents, respectively, and 0.071 and 0.0625 predicted without the preaverage in the Rouse−Zimm model.45,46 Table 2 also

Figure 8. Reduced temperature (1 − Θ/T) dependence of relative intensity contribution of internal motions, AI/(AI + AD), at different values of x = (qRg)2, where concentrations of poly(methyl methacrylate) in acetonitrile, polyisobutylene in isoamyl isovalerate, polystyrene in cyclohexane, and poly(vinyl acetate) in 3-heptanone are 5.00, 4.50, 1.00, and 0.50 × 10−5 g/mL, respectively. E

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Chinese University of Hong Kong. Dr. Ming is fully responsible for all the experimental data presented in this manuscript.

temperature, as shown in Figure 1, and the relative intensity contribution of the internal motions is related to both Rg and q, as shown in Figure 7. Therefore, we have to fix the relative length, i.e., x = (qRg)2, in order to compare AI/(AI + AD) at different solution temperatures. As x increases, more internal motions were detectable because of shorter observation lengths 1/q. Lowering the solution temperature generally results in a smaller AI/(AI + AD) for a given x, but the data are noisy in the good solvent range. However, it is clear that in the poor solvent region AI/(AI + AD) decreases with the solution temperature; i.e., the internal motions are suppressed as the chain shrinks. In other words, AI/(AI + AD) reaches a maximum or turning point at one temperature (T*) that is close to the corresponding Θ reported in the literature.31 There is no quantitative theory about T*, but it is physically reasonable; namely, the thermal energy can excite more internal modes of a linear polymer chain at its pseudoideal free-joint state or practically at its Θ state due to the diminishing of the additional excluded volume interaction and entropic elasticity.26,27

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CONCLUSION The current systematic study of the internal dynamics of four different flexible linear polymer chains near their corresponding Θ temperatures confirms that the relative contribution of the scattered light intensity related to internal motions, AI/(AI + AD), reaches a maximum or a turning point of temperature dependence of AI/(AI + AD) at a certain T* that is close to Θ, which supports our previous hypothesis that the thermal energy can excite more internal motions when a polymer chain approaches its pseudoideal freely jointed state because the excluded volume and the entropic elasticity diminish if using the Θ state as a reference point. Our results reveal that we can f irst use dynamic light scattering to measure the line width distribution of the scattered light at a sufficiently high scattering angle, i.e., the observation length (1/q) is shorter than the chain dimension (Rg); then calculate the relative intensity contribution of the internal motions, AI/(AI + AD); and f inally determine the Θ state from the maximum or turning point (T*) of the temperature dependence of AI/(AI + AD). Such a novel method requires only one dilute solution rather than a set of at least five polymer concentrations in most of conventional methods. More importantly, it measures the intrachain segment−segment interaction instead of the interchain chain−chain interaction so that the Θ state determined in this way is theoretically closer to its original definition; namely, the second-order segment−segment intrachain interaction diminishes in a solution. We are now confident of using T* as Θ; namely, a novel method of determining the Θ state of a polymer solution has been firmly established.

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REFERENCES

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (C.W.). ORCID

Xiaoqing Ming: 0000-0002-8290-5094 Chi Wu: 0000-0002-5606-4789 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is financially supported by the National Natural Scientific Foundation of China Projects (51773192) and the F

DOI: 10.1021/acs.macromol.8b00559 Macromolecules XXXX, XXX, XXX−XXX

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DOI: 10.1021/acs.macromol.8b00559 Macromolecules XXXX, XXX, XXX−XXX