Coil-Globule Collapse of Polystyrene Chains in Tetrahydrofuran-Water

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Coil-Globule Collapse of Polystyrene Chains in Tetrahydrofuran-Water Mixtures Tatiana Igorevna Morozova, and Arash Nikoubashman J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b10603 • Publication Date (Web): 29 Jan 2018 Downloaded from http://pubs.acs.org on February 3, 2018

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The Journal of Physical Chemistry

Coil-Globule Collapse of Polystyrene Chains in Tetrahydrofuran-Water Mixtures Tatiana I. Morozova and Arash Nikoubashman∗ Institute of Physics, Johannes Gutenberg University Mainz, Staudingerweg 7, 55128 Mainz, Germany E-mail: [email protected]

∗

To whom correspondence should be addressed

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ACS Paragon Plus Environment

The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract We study the coil and globule states of a single polymer chain in solution by performing Molecular Dynamics simulations with an united atom model. Specifically, we characterize the structural properties of atactic polystyrene chains with N = 20 to 150 monomers in tetrahydrofuran-water mixtures at varying mixing ratios. We find that the hydrophobic polymers form rather open coils when the mole fraction of water, XW , is roughly below 0.25, whereas the chains collapse into globules when XW & 0.75. We confirm the theoretically expected scaling laws for the radius of gyration, Rg , in these regimes, i.e. Rg ∝ N 3/5 and Rg ∝ N 1/3 for good and poor solvent conditions, respectively. For poor solvent conditions with XW = 0.75, we find a sizable fraction of residual tetrahydrofuran trapped inside the collapsed polymer chains with an excess amount located at the globule surface, acting as a protective layer between the hydrophobic polystyrene and the surrounding water-rich mixture. These findings have important implications for nanoparticle fabrication techniques where solvent exchange is exploited to drive polymer aggregation, since residual solvent can significantly influence the physical properties of the precipitated nanoparticles.

1

Introduction

Polymers in solution are ubiquitous in technology and nature, where they serve a wide range of purposes, for example as viscosity modifiers in the food and cosmetic industry 1 or for storing the genetic makeup in the form of DNA. The physical properties and functions of these macromolecules depend to a high degree on their conformation. For example, proteins in biological cells fold into specific structures based on the sequence of the constituent aminoacids and the local environment, 2,3 and it has been hypothesized that misfoldings can give rise to numerous pathological conditions. 4 Thus, it is crucial to understand and control the numerous factors which can influence the self-assembly of macromolecules. One important aspect that determines the conformation of a polymer dispersed in solution 2


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is the quality of the surrounding solvent. Under good solvent conditions, the chains attain a swollen coil-like configuration with a sizable amount of solvent located within the enclosed volume. At poor solvent conditions, however, the chains collapse into a compact globule to minimize the interface area with the surrounding liquid. This transition between extended and collapsed states has been the focus of numerous theoretical, 5–7 computational, 8–10 and experimental studies. 7,11–13 Here, the main goal usually was to elucidate the details of the coil-globule transition of a single chain, e.g. whether it is a first or second-order phase transition. At sufficiently high polymer concentrations, the coil-globule transition of the individual chains is usually accompanied by the aggregation of neighboring macromolecules. This behavior can be exploited for the controlled fabrication of polymeric colloids, 10,14–18 where the structure and composition of the aggregates can be tuned through, e.g., the employed polymers, additives, and mixing conditions. However, the underlying microscopic mechanisms responsible for the particle formation are still unclear, as the precipitation process typically takes place on nanometer length scales and millisecond time scales, making experimental characterization of this process difficult. Recent computational efforts have shed some light on the precipitation process, 10,19–21 but relied on a rather coarse-grained description of the polymer solution to render the simulations computationally tractable. Polymers were modeled as generic bead-spring chains, and mixing of the good and poor solvent was mimicked by a gradual change of the effective monomer-solvent interactions. Although such simplified models can capture a substantial amount of the relevant physics, they suffer from a certain ambiguity of the interaction parameters, and hence inevitably miss some effects which are related to the specific chemistry of the materials. Further, the grouping of good and poor solvent molecules into effective particles precludes situations where the distribution of solvent molecules becomes heterogeneous, which can be for instance the case when good solvent is enclosed in the precipitated nanoparticles. To better understand the coil-globule transition and the effect of solvent trapping, we

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performed computer simulations of a single polystyrene (PS) chain in tetrahydrofuran-water mixture at various mixing ratios. We employed an united atom (UA) model to reproduce the experimental system as realistically as computationally feasible. We show that the coilglobule transition occurs when the mole fraction of water, XW , is roughly above 0.75. At such poor solvent conditions, we find a sizable fraction of residual tetrahydrofuran (THF) trapped inside the collapsed polymer chains. Further, an excess amount of THF is located at the surface of the globule, acting as a protective layer between the hydrophobic polystyrene and the surrounding water-rich mixture. The rest of this manuscript is organized as follows. In Section 2 we describe in detail the employed simulation model and method, while we present and discuss our findings in Section 3. In Section 4 we provide our conclusions and outlook.

2

Simulation Details and Methods

In order to model PS chains in solution as accurately as computationally possible, we opted for an UA model where a small group of adjacent atoms is represented by one effective bead. For instance, hydrogen and carbon atoms in a methyl group are treated as a single interaction site, which is a reasonable approximation for molecular systems where the intermolecular motion is much more important than the intramolecular one. Such an approach can save valuable computation time compared to fully atomistic models, while preserving molecular detail to a greater extent than more coarse-grained models. In particular, we decided to use the transferable potentials for phase equilibria-united atom (TraPPE-UA) force fields for describing the THF molecules 22 and atactic PS chains. 23,24 We modeled the water molecules using the three-site SPC/Fw model with flexible bonds and point charges. 25 In what follows, we will provide a detailed description of the simulation model and method. We have included the employed simulation parameters as supporting information (SI), for the sake of brevity. Figure 1 shows a schematic representation of a THF molecule and a PS monomer, together

4


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with the employed grouping into pseudo-atoms. Specifically, the CHx groups are represented by one interaction site: for a THF molecule, each CH2 group and each oxygen atom is modeled as a single bead (five in total per molecule), while for a PS monomer, each CH3 , CH2 , CH, Caro and CHaro group is represented by a single bead (eight in total per monomer).

Figure 1: Schematic representation of a (a) THF molecule and a (b) PS monomer, highlighting the employed grouping into pseudo-atoms. Partial charges for a THF molecule are also shown, where e is the elementary charge. For the simulations containing polymer chains, starting configurations were generated using a modified version of the program Assemble!, 26 and we performed three independent runs for each polymer length and mixture composition. The edge length of the cubic simulation box was chosen to be at least 2.5 times the radius of gyration of the dispersed polymer, Rg , to avoid self-interactions. Typically, the simulation box contained approximately 2, 000 solvent molecules for the shortest PS chains (N = 20), and up to 32, 000 solvent molecules for the longest polymers (N = 150). The simulation time, τsim , was chosen to be several Zimm relaxation times of the dispersed polymer chain, 27 τZimm = ηRg3 /kB T . As a reference, we chose τsim = 10 ns for the shortest chains in pure water (τZimm = 0.22 ns), and τsim = 40 ns for the longest chains in THF-water mixtures with XW = 0.75 (τZimm = 14.3 ns). All systems were studied at room temperature (T = 298 K) and atmospheric pressure (P = 101325 Pa), as in recent flash nanoprecipitation experiments involving PS homopolymers. 10,28 Molecular Dynamics (MD) simulations were carried out on Graphical Processing Units (GPUs) using the HOOMD-blue software package. 29–31 The equations of motion were integrated using the velocity Verlet algorithm 32,33 with a timestep of ∆t = 1.0 fs. A 5



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Nosé-Hoover thermostat was employed for simulations at N V T conditions, 34,35 while the Martyna-Tobias-Klein equations of motion were employed to achieve N P T conditions. 36

3 3.1

Results and Discussions Structural and dynamic properties of THF-water mixtures

We first performed MD simulations of THF-water mixtures at different compositions without PS chains, to test our model and to characterize the properties of the liquid. In particular, we studied the following mole fractions of water, XW = 0, 0.3, 0.63, 0.9, and 1. In order to determine the dynamic and static properties of pure THF (XW = 0) and pure water (XW = 1), we conducted simulations with 1, 200 molecules in the simulation box. For the THF-water mixtures, the simulation box typically contained ∼ 4, 000 solvent molecules, distributed between the water and THF molecules according to XW . We conducted three independent runs at each state point to compute averages and standard deviations. We determined the mass density ρ at room temperature and ambient pressure by conducting N P T simulations for 30 ns. The dynamic viscosity was then calculated in the N V T ensemble using start configurations obtained from the previous N P T simulations. Here, we employed the Green-Kubo relation for computing the shear viscosity: V η= kB T

Z

∞

dt hPαβ (t)Pαβ (0)i ,

(1)

0

where Pαβ are the Cartesian components of the stress tensor. We improved the sampling by using all five independent components of the traceless stress tensor, 37 i.e. Pxy , Pxz , Pyz , (Pxx − Pyy )/2, and (Pyy − Pzz )/2, and computed η from the averaged stress autocorrelation function of each trajectory. We have plotted our simulation results for all investigated XW in Fig. 2, where we also included experimental measurements 38,39 as a reference. The mass densities determined from the simulations are always within 1 % of the experimental value,

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whereas the viscosities deviate by up to 20 % with respect to the experimental measurements. Note that the discrepancy between η from simulations and experiments is almost negligible for XW ≥ 0.9, since the employed SPC/Fw water model has been parametrized to faithfully reproduce the dynamic properties of water. 25 Experiments Simulations

ρ[g/cm3 ]

1.00 0.95 0.90

(a) 0.85

1.5

η [cp]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0

0.5

(b) 0.0

0.2

0.4

0.6

0.8

1.0

XW Figure 2: (a)Mass density ρ and (b) dynamic viscosity η from experiments 38,39 and simulations for THF-water mixtures as a function of water mole fraction XW . All data reported for systems at room temperature (T = 298 K) and under atmospheric pressure (P = 101325 Pa). We also verified that the THF and water molecules remained mixed during the simulation by visual inspection and by computing the intermolecular radial pair distribution function g(r) between the centers of mass of THF and water molecules [g(r) for THF-THF and water-water can be found in the SI]. The g(r) data shown in Fig. 3 reveals the presence of two pronounced solvation shells of water around THF molecules at r ≈ 3 ˚ A and r ≈ 5 ˚ A, respectively. For larger distances r & 10 ˚ A, we find g(r) ≈ 1, which indicates homogeneous

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mixing of the two liquids. Similar results were obtained in previous united atom 40 and fully atomistic 41,42 simulations of THF-water systems. Note that the radial pair distribution function between the centers of mass of THF and water molecules did not change significantly when a polymer was dispersed in the mixture (graphs not shown for the sake of brevity).

XW = 0.30 XW = 0.63 XW = 0.90

2.0

gCM (r)THF−W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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1.5

1.0

0.5

0.0

0

4

8

12

16

˚ r[A] Figure 3: Intermolecular radial pair distribution function, gCM (r)THF−W , between center of masses of THF and water molecules at different water mole fractions XW .

3.2

Polystyrene in THF-water mixtures

We studied the properties of a single PS chain in THF-water mixtures at XW = 0 (pure THF), XW = 0.25, XW = 0.50, XW = 0.75 and XW = 1 (pure water). The length of the PS chain was set to N = 20, 50, 100 and 150 monomers, which corresponds to a molecular weight of M = 2.1, 5.2, 10.4 and 15.5 kg/mol respectively. Figure 4 shows simulation snapshots of a PS chain with 150 monomers dispersed in pure THF and water, respectively. In THF, the polymer is in an extended state, whereas in water the hydrophobic chain collapses to a compact globule to minimize the interface with the surrounding poor solvent. In order to quantify the size and shape of the dispersed polymers, we computed the radius of gyration 8


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tensor:

Gαβ =

NUA 1 X mi ∆ri,α ∆ri,β , M i=1

(2)

where NUA is the number of united atoms in a polymer chain, M is the total polymer mass, ∆ri,α is the position of united atom i relative to the polymer center of mass rcm , while α and β are the components in the Cartesian x, y, and z direction. The polymer radius of gyration is then given by Rg2 = Gxx + Gyy + Gzz .

Figure 4: Simulation snapshots of a PS chain with 150 monomers (M = 15.5 kg/mol) dispersed in (a) pure THF and (b) pure water. The scale bar indicates 10 ˚ A. Snapshots rendered using Visual Molecular Dynamics 1.9.2. 43 We have plotted in Fig. 5(a) Rg as a function of N for various solvent compositions, and it is clear that the polymers are collapsed when they are dispersed in solutions with a high water content (XW ≥ 0.75). In contrast, the chains are rather swollen when the 9



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mixtures consist of mostly THF (XW ≤ 0.25). We found that the radius of gyration scaled as Rg ∝ N ν , with νpoor = 0.27 ± 0.06 and νgood = 0.67 ± 0.12 for our simulations in pure water and pure THF, respectively. These scaling exponents are in good agreement with theoretical considerations, which predict a value of νpoor = 1/3 for a densely packed globular configuration in a poor solvent, and νgood = 3/5 for good solvent conditions set by the balance between excluded volume and conformational entropy. 44 Our simulation results for PS in pure THF also agree well with previous experiments, where a scaling exponent of νgood = 0.56 ± 0.02 was found. 45 Furthermore, the computed Rg values in pure THF are in quantitative agreement with experimental measurements of low M PS in toluene [open symbols in Fig. 5(a)], which is also a good solvent for PS. 46,47 At intermediate solvent qualities, the intermolecular interactions between polymer chain segments and coordinated solvent molecules become similar (Θ-conditions), and the dispersed chain effectively acts like an ideal chain with νΘ = 1/2. From the simulation data shown in Fig. 5(a), we expect that Θ-conditions should be achieved roughly in the range 0.25 . XW . 0.50. Experimental measurements of PS in THF-water mixtures revealed that the Θ-composition for PS in THF-water mixtures is XW ≈ 0.3, 48 which lies in the range estimated from our simulations. Note that the effect of solvent quality becomes considerably less pronounced for short chains, since the inherent stiffness of the PS impedes the full collapse of neighboring segments. (The chains with N = 20 consist of approximately 3 Kuhn segments. 44,49 ) A similar trend has also been observed in experiments for low M PS using a wide range of good solvents. 45,50 The shape anisotropy of the dispersed polymer can be quantified through: κ2 = 1 − 3

λ1 λ2 + λ1 λ3 + λ2 λ3 (λ1 + λ2 + λ3 )2

(3)

where λ1 , λ2 , and λ3 are the three eigenvalues of the instantaneous gyration tensor G. The shape anisotropy parameter attains a value of κ2 = 0 for a fully isotropic shape (sphere)

10


30

Experiments, toluene Pure THF XW = 0.25 XW = 0.50 XW = 0.75

20

Pure water

40

(a)

10 20

50

100

150

(b)

0.6

κ2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


˚ Rg [A]

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0.4

0.2 Ideal chain 0.0

20

50

100

150

N Figure 5: (a) Radius of gyration Rg and (b) relative shape anisotropy parameter κ2 of a PS chain as a function of chain length N for various water mole fractions XW . The solid lines in (a) are power law fits for pure water (poor solvent) and pure THF (good solvent) with exponents νpoor = 0.27±0.06 and νgood = 0.67±0.12, respectively. Open symbols correspond to experimental Rg measurements of PS in toluene. 46,47 The solid line in (b) shows the value of κ2 for an ideal chain. 51,52

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and κ2 = 1 for an infinitely thin rod. For a randomly coiling chain in three dimensions, a value of κ2 ≈ 0.39 is expected in the limit of N → ∞ due to conformational fluctuations of the macromolecule. (Note, however, that the eigenvalues of the average gyration tensor hGi become identical for sufficiently long sampling and thus exhibit no anisotropy.) Figure 5(b) shows κ2 vs. N for various solvent qualities, and we can see that κ2 is close to zero for collapsed chains in pure water and XW = 0.75, as expected. For good and Θ-solvent conditions, we find κ2 ≈ 0.4, which is in line with the theoretical prediction for randomly coiling chains. 51,52 To quantify the shape of polymers in solution, it is also instructive to consider the single chain structure factor S(q)

S(q) =

1 X exp (−iq · rij ) N i,j

(4)

where q is the scattering wave vector. For isotropic systems, the structure factor depends only on the magnitude of q, so that we can simplify Eq. (4) to

S(q) =

1 X sin (qrij ) . N i,j qrij

(5)

The number of monomers in the scattering volume 1/q 3 depends on the size and conformation of the polymer. Theoretical considerations suggest S(q) ∝ q −1/ν for self-similar objects such as polymers at intermediate wave vectors 1/Rg < q < 1/b (monomer size b), whereas one would expect S(q) ∝ q −4 for a compact spherical particle. 44 We have plotted in Fig. 6 the structure factor S(q) of a PS chain with N = 150 monomers in pure THF as well as in pure water. (The curves for N = 100 are qualitatively similar and have been omitted for the sake of brevity.) For small wave vectors q 1/Rg , we find S(q) ≈ N as expected for dilute systems. At intermediate q, the structure factor decays with a power law. For good solvent conditions, we find S(q) ∝ q −5/3 , which corroborates our measurements of the Flory exponent νgood = 0.67 ± 0.12 [cf. Fig. 5(a)]. In contrast, the scattering function decays as S(q) ∝ q −4 12


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under poor solvent conditions. To facilitate the interpretation of the simulation data, we have included in Fig. 6 also the analytical solution of S(q) for a solid spherical particle with the same geometric radius as the collapsed globule, 44 i.e. a = Rg (5/3)1/2 ≈ 20.3 ˚ A. Indeed, for q . 0.2 ˚ A

−1

the simulation data precisely follow the theoretical S(q) prediction, suggesting

a fully collapsed poor solvent conformation of the PS chain. As q increases, the simulation data start to deviate from the analytical solution, due to the internal structuring of the globule (multiple scatterers vs. single scatterer). At high q, we can identify the microscopic structures of the PS chain. The peak of S(q) at q ≈ 2 ˚ A

−1

corresponds to the nearest

neighbor distance of bonded monomers, and is therefore visible under both good and poor −1

solvent conditions. In pure water, we can identify an additional maximum at q ≈ 0.9 ˚ A , which stems from the packing of monomers in the interior of the globule. Next, we investigated the localization of the THF and water molecules with respect to the dispersed PS chain. To this end, we computed the radial distribution function between the center of mass of the dispersed polymer chain and the center of mass of a solvent molecule, gCM (r)PS−THF/W . Here, we focus on the cases with N = 100 and N = 150 monomers, as these chain lengths exhibit the strongest dependence on solvent quality (cf. Fig. 5). It is clear from the data shown in Figs. 7(a) and (c) that the THF molecules are almost homogeneously distributed in the system, irrespective of solvent composition XW and chain length N ; for the cases with XW = 0.75, we can identify a slight depletion of THF molecules in the polymer core (r ≈ 0) and a weak accumulation at the polymer shell (r ≈ Rg ). In contrast, the distribution of water molecules depends much more strongly on the solvent composition. For XW = 0.25 and 0.5, we find a sizable fraction of water inside the PS coils, which increases monotonically with r until it reaches its bulk value at r ≈ 1.5 Rg [see Figs. 7(b) and (d)]. At first, this behavior might seem implausible because of the hydrophobic nature of PS, but it can be rationalized when one considers the open and coil-like conformation of the dispersed PS chains at these conditions; although some water molecules are located near the polymer’s center of mass, they do not necessarily need to be

13



Pure THF Pure water Analytical solution

150 100

S(q)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10

q −5/3

q −4

1

0.1

1.0

2.5

˚ q [1/A] Figure 6: Structure factor S(q) of a single PS chain with N = 150 monomers in pure THF and pure water. The solid lines indicate the theoretically expected behavior for good [S(q) ∝ q −5/3 ] and poor solvent conditions [S(q) ∝ q −4 ], respectively. The dashed line shows the analytic S(q) for a spherical particle.

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in close contact with the contour of the polymer (see discussion below). For XW = 0.75 and 1, there is almost no water inside the collapsed polymer globules [gCM (r)PS−W ≤ 0.5 for r ≤ Rg ], and the bulk value is reached at r ≈ Rg after a relatively sharp transition. (Note that the geometric radius of a spherical globule is slightly larger than its radius of gyration 44 ). XW = 0.50 XW = 0.75

Pure THF XW = 0.25

1.5

gCM (r)PS−THF

gCM (r)PS−THF

1.5

1.0

0.5

XW = 0.50 XW = 0.75

Pure THF XW = 0.25

1.0

0.5

(a)

(c)

0.0

0.0 XW = 0.25 XW = 0.50

XW = 0.75

1.5

Pure water

gCM (r)PS−W

1.5

gCM (r)PS−W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.0

0.5

XW = 0.25 XW = 0.50

XW = 0.75 Pure water

1.0

0.5

(b) 0.0 0.0

0.5

1.0

(d) 1.5

0.0 0.0

r/Rg

0.5

1.0

1.5

r/Rg

Figure 7: Radial distribution functions gCM (r) between the centers of mass of the dispersed PS chain and THF/water molecules for various solvent mixtures. Data for N = 100 shown in (a) and (b), while data for N = 150 shown in (c) and (d). The shaded regions correspond to the measurement uncertainty determined from the standard error of the independent simulations. To study in more detail the distribution of THF and water molecules in the immediate vicinity of the dispersed PS chain, we also computed the radial distribution function between monomers and solvent molecules, gCM (r)S−THF/W . Figure 8(a) shows the distribution of THF molecules, and we can identify a rather sharp peak at r ≈ 6 ˚ A which becomes more distinct with increasing XW . This peak is followed by a sequence of less pronounced minima and maxima, reflecting the layering of solvent particles. From Fig. 8(b) we can further see that 15



g(r) between PS monomers and water molecules reaches unity only at distances r & Rg [cf. Fig. 5(a)], due to the hydrophobic nature of the PS chain. Interestingly, for the THFwater mixtures, the relative amount of water around a PS monomer decreases slightly with increasing water content, while the local amount of THF increases. These findings suggest the formation of a (thin) protective layer between the hydrophobic polymer and the aqueous surrounding [cf. Figs. 7(a) and (c)].

gCM (r)S−THF

1.5

(a)

1.0 Pure THF XW = 0.25 XW = 0.50 XW = 0.75

0.5

0.0

0

1.5

gCM (r)S−W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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10

20

30

(b)

XW = 0.25 XW = 0.50 XW = 0.75

1.0

40

Pure water

0.5

0.0

0

10

20

30

40

˚ r[A] Figure 8: Radial distribution function gCM (r) between the centers of mass of the monomers of a PS chain (N = 150) and THF/water molecules for various solvent mixtures. The shaded regions correspond to the measurement uncertainty determined from the standard error of the independent simulations. In order to quantify the enrichment/depletion of THF and water molecules around the PS chain, we computed the local density along the polymer contour. To this end, we first 16


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constructed a fictitious worm-like chain along the PS backbone, with thickness set to the interaction range of the dispersion force (rc = 16 ˚ A for THF and rc = 10 ˚ A for water). Then, we identified all THF and water molecules lying in this region, and computed the size of the search volume through Monte Carlo integration. Figure 9 shows the local number densities of THF (ρTHF ) and water (ρW ) molecules normalized by their bulk value, ρ0THF and ρ0W , respectively. For THF, we find that ρTHF is slightly below ρ0THF for most mixing fractions irrespective of chain length, which we attribute to the higher solubility of THF in THF than PS in THF. Interestingly, the simulations with the largest water fraction (XW = 0.75) indicate a small but significant enrichment of THF along the contour of the PS chain, which is consistent with the g(r) data shown in Figs. 7 and 8. Figure 9(b) shows the normalized local number density of water molecules along the PS contour, from which we can identify several key trends. First, ρW /ρ0W decreases with increasing chain length N for all investigated XW , because the water molecules are primarily located at the shell of the polymer coil [cf. Figs. 7(b) and (d)] and the surface to volume ratio of the polymer decreases with increasing polymerization. Second, the relative fraction of adsorbed water molecules decreases with increasing XW for the THF-water mixtures, which seems counterintuitive at first sight. This behavior can be explained, however, when we consider that the polymer is much more collapsed at high water concentrations, and thus fewer water molecules can penetrate the core of the polymer [see Fig. 5(a)]. Finally, we briefly investigated the dynamics of the coil-globule transition, by conducting a simulation in pure water where the dispersed polymer (N = 100) was initialized in an extended coil configuration. Once we started the simulation, the hydrophobic macromolecule tightened into a compact globule [cf. Fig. 4(b)], and we estimated the characteristic collapse time τc by fitting the time evolution of Rg through a sigmoid function. 10 Here, we found τc = 3.0 ± 0.1 ns, which is comparable to the Zimm relaxation time of the entire polymer (τZimm ≈ 5 ns). However, this finding should be taken with a grain of salt, because of the limited sample size here; there are numerous pathways from the coil state to the collapsed

17



ρTHF /ρ0THF

1.00 0.75 0.50 0.25 0.00

XW = 0.75

Pure THF XW = 0.25 XW = 0.50

Pure water (a)

1.00

ρW /ρ0W

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

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0.75 0.50 0.25 (b)

0.00 50

100

150

N

Figure 9: Number densities of (a) THF and (b) water molecules around a polymer chain with N monomers, normalized by the bulk value.

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one, and studying the coil-globule collapse dynamics in detail requires advanced sampling methods, such as Markov state modeling, 53 so that all relevant transitions can be captured efficiently. This aspect will be the focus of a future publication.

4

Conclusions

We studied the conformation of single polystyrene chains with various degrees of polymerization, N , dispersed in a tetrahydrofuran/water mixture using Molecular Dynamics simulations of an united atom model. We systematically varied the mixing ratio of the two liquids to tune the solvent conditions from good (pure tetrahydrofuran) to poor (pure water). In good solvents, the macromolecules exhibited an open coil-like conformation, whereas the chains collapsed to compact globules in the poor solvents. We measured the radius of gyration, Rg , and found that the simulation data followed the theoretically expected trends, i.e. Rg ∝ N 3/5 and Rg ∝ N 1/3 for good and poor solvent conditions, respectively. Further, we investigated the spatial distribution of the tetrahydrofuran and water molecules with respect to the polystyrene chains. We discovered that the location of water molecules depended strongly on the specific solvent composition, whereas the tetrahydrofuran molecules were almost homogeneously distributed in the system, irrespective of solvent composition and chain length. Interestingly, we identified a small mixing regime (water mole fraction XW = 0.75), where a sizable fraction of residual tetrahydrofuran molecules was trapped inside the collapsed polymers, with an excess amount located at the globule-solvent interface, serving as a protective layer between the hydrophobic polystyrene and the surrounding water-rich mixture. Further, our simulations provide important insights for developing coarse-grained models for polymers in solvent mixtures: when the distribution of the good and poor solvent can become heterogeneous (as it is case in our investigated systems), then a coarse-grained description where multiple solvent molecules of different species are lumped into a single

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effective solvent bead does not capture the entire physical picture anymore, since effects such as solvent trapping cannot be replicated. 54 In addition, our simulations lay the foundation for accurately studying the dynamic pathways of the coil-globule transition for polymers in poor solvents. Based on these atomistic simulations, we plan to develop enhanced coarsegrained polymer models, which not only reproduce the structural polymer properties, but also the internal polymer dynamics and its hierarchies.

Supporting Information Simulation details and force field parameters (pdf).

Acknowledgments We thank Prof. F. M¨ uller-Plathe for helpful discussions. We further acknowledge funding from the German Research Foundation (DFG) under the project number NI 1487/2-1 and SFB TRR146, and also acknowledge the computing time granted on the supercomputer Mogon at Johannes Gutenberg University Mainz (hpc.uni-mainz.de).

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Graphical TOC Entry

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Coil-Globule Collapse of Polystyrene Chains in Tetrahydrofuran-Water

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