Scaling Exponent and Effective Interactions in Linear and Cyclic

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Article Cite This: Macromolecules 2019, 52, 4579−4589

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Scaling Exponent and Effective Interactions in Linear and Cyclic Polymer Solutions: Theory, Simulations, and Experiments Thomas E. Gartner, III,† Farihah M. Haque,‡ Aila M. Gomi,§ Scott M. Grayson,*,‡ Michael J. A. Hore,*,§ and Arthi Jayaraman*,†,∥ Department of Chemical and Biomolecular Engineering, and ∥Department of Materials Science and Engineering, University of Delaware, Newark, Delaware 19716, United States ‡ Department of Chemistry, Tulane University, New Orleans, Louisiana 70118, United States § Department of Macromolecular Science and Engineering, Case Western Reserve University, Cleveland, Ohio 44106, United States Downloaded via IDAHO STATE UNIV on July 18, 2019 at 12:54:06 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

†

S Supporting Information *

ABSTRACT: Cyclic polymers have garnered increasing attention in the materials community as lack of free-chain ends in cyclic polymers results in significant differences in structure, thermodynamics, and dynamics compared to their linear counterparts. Yet, key open questions remain about how cyclic polymer chain conformations and effective interactions in solution change as a function of solvent quality, ring closure chemistry, and the presence/absence of common synthetic impurities. We use coarse-grained molecular dynamics simulations, polymer reference interaction site model theory, and small-angle neutron-scattering experiments on polystyrene in d-cyclohexane to demonstrate how linear and cyclic polymer chain configurations, scaling, and effective interactions are influenced by solvent quality and polymer concentration. We find that the balance between the available intraversus interchain contacts in solution dictate the trends in cyclic polymer size scaling and effective polymer−solvent and polymer−polymer interactions; these results are largely insensitive to the ring closure chemistry and the presence of linear or cyclic dimer impurities. This study provides the broader polymer science and engineering community fundamental insights into how synthesis, purification, and assembly of cyclic polymers in solvent(s) impact the resulting chain structure and polymer solution thermodynamics. “topological excluded volume”) introduced by the cyclic topological constraint.25 These effects could be thought of as cyclic polymers experiencing an effectively “better” solvent condition than linear polymers. A key measure of polymer chain conformations in solutions is the Flory exponent (ν), which describes the scaling of chain size (i.e., the radius of gyration, Rg) with molecular weight (N).

1. INTRODUCTION Cyclic polymers are of significant fundamental and practical interest in a variety of natural and synthetic systems. For instance, plasmid and mitochondrial DNA are often cyclic.1 The lack of reptation in cyclic polymers makes them promising candidates as rheology modifiers2−4 and alters their biodistribution and blood circulation.5 Cyclic block polymers have also been explored as next-generation lithographic templates because of their smaller domain spacing relative to linear polymers.6,7 These preferred properties of cyclic polymers over analogous linear polymers stem from their topological differences (i.e., lack of chain ends), which result in vast changes in chain size and scaling, interchain interactions, packing, thermal properties, and dynamics in both melts and solutions.2,3,8−21 In particular, in solutions and/or during solvent processing approaches, understanding solvent−cyclic polymer interactions and contrasting them to established trends of linear polymers is vital to capitalize on the unique properties that cyclic polymers may offer for biological and industrially relevant material applications such as polymer synthesis, separation/purification, drug delivery, and so forth. Previous studies on cyclic polymer solutions have shown that cyclic polymers have a lower θ temperature22−24 and higher dissolution limits than their linear counterparts because of an additional intrachain repulsion (sometimes referred to as © 2019 American Chemical Society

Rg ∝ Nν

(1)

Experiments26,27 have shown that ν is higher for cyclic polymers than linear polymers in a near-θ solvent (νcyclic = 0.53 and νlinear = 0.5 for cyclic and linear polystyrene (PS), respectively, in d-cyclohexane at 40 °C), but in a good solvent, νcyclic and νlinear are similar (near 0.58).26 An additional complication is that ν as defined in eq 1 is strictly valid in the limit of infinite N, and for cyclic polymers, very long chains are required to reach large-N asymptotic scaling behavior.12 Therefore, experimental and simulation studies probing cyclic polymers at finite molecular weight may be exploring crossover scaling regimes where the apparent ν is different from that expected at infinite N. Similarly, even simple properties such as Received: March 25, 2019 Revised: May 28, 2019 Published: June 13, 2019 4579

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Macromolecules the ratio of the Rgs of analogous cyclic and linear polymers have been the topic of much debate. Ideal Gaussian chains have a “g-factor” ⟨Rg,cyclic2⟩/⟨Rg,linear2⟩ = 0.528 while a variety of computational and experimental methods have found g-factors ranging from 0.5 to 0.6 depending on the technique and solvent conditions8,29−34 (see ref 26 for a thorough discussion). Recent small-angle neutron-scattering (SANS) experiments on cyclic and linear PS in d-cyclohexane26 determined that the g-factor in θ solvents has a molecularweight dependence and ranges from 0.557 to 0.730. It is hypothesized that sample purity may also contribute to some of these discrepancies (up to 10−20% linear impurities may have been present in some of the earliest cyclic polymer samples);35 we will return to the issue of sample purity below. These experimental efforts only examined such conformational properties individually in θ and good solvents, yet many industrially relevant processes (e.g., synthesis, separation/ purification, and self-assembly/flash nanoprecipitation) occur over a range of solvent qualities. Thus, it is necessary to clarify some of these outstanding questions in the literature and map how the cyclic polymer chain size, scaling, and solution thermodynamics change for intermediate solvent conditions (i.e., other than good and θ solvents). There are a few previous simulation efforts that have looked at cyclic polymers at varying solvent quality using implicit solvent models, where solvent effects are mimicked through effective polymer− polymer potentials. These studies help clarify some of the questions raised above.36−38 However, the translation of a given solvent quality to effective polymer−polymer interactions is likely different for cyclic and linear architectures; thus, if one models similar polymer−polymer interactions for cyclic and linear polymers, it may not be representative of the same solution condition. Another fundamental issue of significance to the eventual application of cyclic polymers is the impact of practical considerations related to the cyclic polymer synthesis. To evaluate the effective impact(s) of the cyclic architecture on its physical properties, it is crucial to use high-purity cyclic polymers. One route to synthesize topologically pure samples is via the copper(I)-catalyzed alkyne-azide cycloaddition (CuAAC) “click” cyclization approach.39 Although it is known that the CuAAC “click” reaction is an extremely successful method of cyclization because of the highly rapid coupling of the azide and alkyne functional groups, recent work has also demonstrated that the linear α-alkynyl-ω-azidopolystyrene can undergo trace amounts of uncatalyzed alkyne-azide dimerization events, resulting in a population of chains of double the desired molecular weight (Mn).40 However, detailed mass spectrometry experiments analyzing the dimers (7.3% at 50 days for PS Mn = 3700 g/mol) have shown that post-cyclization, all linear dimers are converted to cyclic dimers, confirming that the CuAAC “click” cyclization approach results in a cyclic product of very high topological purity. Additionally, it does not appear that any linear dimers are formed in appreciable quantity ( 0.09 kBT because polymer chains phase-separate from the solvent in the MD simulations at those conditions. 2.2. Experimental Section. 2.2.1. Synthetic Protocols. 2.2.1.1. α-Propargyl-ω-bromo-polystyrene (Scheme 1 1). To synthesize the α-propargyl-ω-bromo-polystyrene precursor, we charge a round-bottom flask with PMDETA (43.6 mg, 0.252 mmol, 1.05 equiv), styrene (2.50 g, 24.0 mmol, 100 equiv), and the initiator propargyl 2-bromoisobutyrate (49.2 mg, 0.240, 1.00 equiv). Upon two freeze/pump/thaw cycles, we refreeze the flask, add CuBr (34.0 mg, 0.240 mmol, 1.00 equiv), and then further degas the flask by a final pump/thaw cycle. After warming to room temperature, we place the reaction mixture into a preheated 100 °C oil bath and allow to stir under nitrogen for 5 and 6 h to synthesize the 11k and 14k polymer, respectively. We then cool the reaction mixture to room temperature and purify by extraction from water into dichloromethane, followed by three washes with a saturated aqueous ammonium chloride solution. We collect and dry the organic layer with sodium sulfate, eff

ee

points in that q-range are included in the fit. Then, we choose the candidate fit with the highest coefficient of determination (R2 statistic) value to calculate ν. We also compare the ν calculated in this manner with ν obtained by fitting the entire P(q) with a theoretical scattering form factor as often done in the literature.49 For the linear polymers, ν from our fitting procedure and ν from fits to the entire form factor49 agree closely, but we find that the available 4582

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detection. We prepare samples by making stock solutions in THF of matrix (20 mg/mL), polymer analyte (2 mg/mL), and an appropriate cation source (2 mg/mL). We mix the stock solutions in a 10/2/1 ratio (matrix/analyte/cation), deposit onto the MALDI target plate, and allow to evaporate via the dried droplet method. We use trans-2[3-(4-tert-butylphenyl)-2-methyl-2-propenylidene]malononitrile and sodium trifluoroacetate as the matrix and cation, respectively. We calibrate MALDI-ToF MS data against SpheriCal dendritic calibrants from Polymer Factory (Stockholm, Sweden). We calculate numberaverage molecular weight (Mn) and dispersity (Đ) of the resultant spectra using Polytools software (see Supporting Information Table S1). 2.2.3. SANS Measurements. We perform SANS measurements on the NGB 30 m SANS instrument at the NIST Center for Neutron Research (Gaithersburg, MD) with a variable temperature sample block. We dissolve ∼11 000 g/mol linear and cyclic PS and ∼14 000 g/mol linear and cyclic PS in d-cyclohexane at 1 wt % and collect scattered neutron intensities over the temperature range 30 °C ≤ T ≤ 70 °C to probe a range of solvent qualities. We use two sample-todetector distances of 1 and 4 m to cover a range of q from 0.01 to 0.4 Å−1. The scattering variable q is expressed in terms of the neutron wavelength and scattering angle as q = (4π/λ)sin(θ/2), where the neutron wavelength λ = 6 Å−1. We report absolute scattered intensities corrected for scattering from the empty sample cell, for detector efficiency, and the presence of background radiation using standard methods.55 For all samples at 70 °C (1/T ≈ 0.0029 K−1), bubbles are formed in the sample cell because of the proximity to the boiling point of cyclohexane (≈80 °C), resulting in a “flare” in the scattering pattern. Because it does not represent scattering from the sample and masks the scattering intensity from the polymer, the flare was removed by excluding the affected region when radially averaging the two-dimensional scattering pattern. Although this correction does not alter the shape of the scattering curve, it results in an artificially lower absolute scattering intensity. Although the determination of the scaling exponent is not affected by the lower scattering, incorrect scattering intensities lead to incorrect values of the effective Flory− Huggins parameter at 70 °C (χeff RPA). Our experimentally reported value of the χeff RPA at 70 °C is an estimate based on the values at lower temperatures. The complete set of SANS data is contained in the Results and Discussion section and the Supporting Information. 2.2.4. RPA Analysis of SANS Results. The scattering intensity predicted by the RPA49,50,54 for a polymer + solvent system is given by ÄÅ É−1 eff Ñ ÅÅ ÑÑ 2χRPA ÑÑ 1 1 2Å Å ÑÑ + B I(q) = Δρ ÅÅÅ + − ÑÑ ÅÅ NvmϕpP(q) v (1 ) v v − ϕ s m s Ñ p ÅÇ ÑÖ (15)

Scheme 1. Synthesis of Linear and Cyclic PS

followed by precipitation into methanol to yield the polymer as a white solid. 2.2.1.2. α-Propargyl-ω-azido-polystyrene (Scheme 1 2). In a typical reaction, we dissolve the α-propargyl-ω-bromo-polystyrene 1 (2.46 g, 0.164 mmol, 1 equiv) in dimethylformamide (10 mL). We add sodium azide (1.07 g, 16.4 mmol, 100 equiv) to the reaction mixture and stir under ambient conditions for 24 h using a blast shield. We then dilute the reaction solution with dichloromethane (20 mL), followed by washing with deionized water (3 × 10 mL). We collect and dry the organic layer over anhydrous magnesium sulfate. After filtration, we remove the solvent in vacuo resulting in a ∼90% yield. 2.2.1.3. Cyclic PS (Scheme 1 3). We add the α-propargyl-ω-azidopolystyrene 2 (39.7 mg g, 0.004 mmol, 0.001 equiv) and 100 mL of dichloromethane to a two-neck round-bottom flask. We add a solution of PMDETA (0.415 g, 2.4 mmol, 0.66 equiv) in DCM (150 mL) to a second two-neck round bottom flask. We perform two freeze/pump/thaw cycles on the flasks containing the linear polymer and PMDETA solutions. Upon an additional freeze of the PMDETA solution, we add CuBr (0.315 g, 2.2 mmol, 0.61 equiv) to the frozen solution. We reseal the flask, refill with N2, pump, and thaw. Upon both flasks reaching room temperature, we fill a gastight syringe with the polymer solution and add to the CuBr/PMDETA solution via a syringe pump at a rate of 2.0 mL/min. Once the polymer is completely added to the CuBr/PMDETA solution, we stir the mixture for an additional 2 h to ensure complete cyclization. We open the reaction mixture to air and wash with several portions of saturated aqueous ammonium chloride, until the organic and aqueous layers are colorless. We collect and dry the organic layer over anhydrous magnesium sulfate. After filtration, we remove the solvent in vacuo to yield a white powder with a yield of 90%. 2.2.1.4. Impurity Considerations. The polymers synthesized herein are of substantially increased molecular weight compared to that of the dimerization study mentioned in the Introduction section40 and therefore the chance of dimerization is significantly reduced. Furthermore, the bromine is only converted to an azide directly prior to the preparation of the cyclic polymers to ensure well-defined samples. Finally, the α-alkynyl-ω-bromo-linear precursor is used as the linear analogue during SANS experiments to avoid potential unwanted dimerization. 2.2.2. Analytical Protocols. We collect the gel permeation chromatography (GPC) data from a Waters model 1515 isocratic pump (Milford, MA) with tetrahydrofuran (THF) as the mobile phase with a 1 mL/min flow rate with columns heated at a constant 30 °C by a column oven. We operate this system with a set of two columns in series from Polymer Laboratories Inc. consisting of PSS SDV analytical linear M (8 × 300 mm) and PSS SDV analytical 100 Å (8 × 300 mm) columns. We use a model 2487 differential refractometer detector as a refractive index detector. We calibrate the instrument with Polystyrene ReadyCal Standards from Waters. We collect mass spectral data using a Bruker−Daltonics matrixassisted laser desorption ionization time-of-flight (MALDI-ToF) Autoflex III mass spectrometer in linear mode with positive ion

where Δρ is the difference in the scattering length densities of the polymer and solvent, N is the polymer degree of polymerization, vm and vs are the volumes of the monomer and solvent particles, respectively, χeff RPA is the effective Flory−Huggins parameter for the polymer−solvent system, ϕP is the polymer volume fraction, and B is the incoherent background. We determine the volume fraction of the polymer in the solution and molecular weight from scattering at T = 30 °C, which are in quantitative agreement with the values used in the simulations, as well as from GPC and MALDI-ToF measurements of the polymers. For the case of a linear polymer, the form factor P(q) for chains with excluded volume is well-known49 and expressed in terms of the lower incomplete gamma function, γ, as P(q) =

1 1 i1 y i1 y γ jjj , U zzz − γ jjj , U zzz νU1/2ν k 2ν { νU1/ ν k ν { 2 2

(16)

2ν

where U = q b N /6. For a cyclic polymer with excluded volume (i.e., ν ≠ 0.5), there is no analytical solution to the form factor, and the form factor must be evaluated numerically according to50

∫0

P(q) = 2 4583

1

ds(1 − s)exp[− s 2ν(1 − s 2ν)U ]

(17)

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Macromolecules We fit SANS data to the RPA using SasView (http://www.sasview. org), with ν and χeff as floating variables. We evaluate the integral at 1000 points in the interval s ∈ [0, 1] to ensure numerical accuracy.

tional and experimental results for each architecture at one concentration. If each CG polymer bead represents one monomer of PS and each CG solvent bead represents one molecule of d-cyclohexane, the N = 108, ϕP = 0.007 simulation corresponds to the ∼11 000 g/mol, 1.0 wt % solution in the experiments. Increasing the computational model χPS is equivalent to decreasing the experimental temperature for a standard upper-critical solution temperature (UCST) polymer−solvent pair, for example, PS and cyclohexane. Figure 1a shows the computational ν obtained by fitting the slope of the intermediate scaling regime of P(q) from simulations (see Materials and Methods and Supporting Information Figures S1−S3 for an example P(q) and discussion of our fitting procedure). From the main plot shown in Figure 1a at ϕP = 0.007 (solid lines), as model χPS increases (i.e., as the solvent quality decreases), νlinear decreases from the known good solvent value (ν ≈ 0.58) to θ solvent (ν ≈ 0.50) and below. At all model χPS explored, νcyclic is higher than νlinear. In the good solvent regime (lower values of χPS), the slope of ν versus χPS is similar for cyclic and linear polymers, but as the solvent quality worsens, the cyclic and linear ν deviate, with νcyclic becoming less sensitive to model χPS in near-θ solvents. In Figure 1b, the experimental ν obtained by fitting the SANS I(q) demonstrates that PS11k and PS14k samples show equivalent scaling behavior, with νcyclic being higher than νlinear as observed in our simulation results. Comparison of computational and experimental νlinear shows that the temperature range 30−70 °C (i.e., 1/T ≈ 0.0033− 0.0029 K−1) corresponds to the χPS range 0.06 < χPS < 0.09 kBT. Using this mapping between T and model χPS, we also plot the computational results for χPS = 0.07 and 0.08 kBT in Figure 1b (black and dark blue symbols), which demonstrate quantitative agreement between the simulations and SANS experiments over the available temperature range. I(q) profiles for the PS11k and PS14k samples are plotted in Figure 2. Note that in Figure 2b, the scattering intensities at the various temperatures do not overlap because of the formation of bubbles in the sample cell at T = 50 and 70 °C. Although the decrease in intensity does not affect the determination of the scaling exponent, it interferes with estimation of the effective Flory−Huggins parameter (see Supporting Information Figure S6). In all other systems, bubbles formed at T = 70 °C. The upturn in scattering intensity at low-q for Figure 2d may be due to aggregates or bubbles in solution, this upturn also does not affect the determination of the scaling exponent. For the systems that exhibited a low-q upturn, the range of the fit was limited to between 0.03 < q < 0.4 Å−1. The results given in Figures 1 and 2 reveal nontrivial changes in the cyclic chain scaling as solvent quality transitions from good to θ solvent, illustrating a competition between shrinking chain dimensions due to unfavorable polymer− solvent interactions balanced by a topological excluded volume effect that disfavors full-chain collapse. These factors balance near the θ condition to result in the plateau in νcyclic. We also note that the scaling behavior noted herein agrees with ν values found by measuring linear and cyclic Rg over a wide range of molecular weights in both good and θ solvents, thus lending credence to our P(q)-based approach for determining the ν for finite-N chains.26 In contrast to the computational ν results obtained using a one-monomer-per-CG-bead model (Figure 1a main plot), the inset in Figure 1a shows the computational ν results obtained

3. RESULTS AND DISCUSSION In Figure 1, we present the scaling exponent of linear and cyclic polymers obtained from theory and simulations (Figure

Figure 1. Linear and cyclic polymer scaling exponent. (a) Scaling exponent, ν, from simulations of linear (black squares) and cyclic (blue circles) polymers at ϕP = 0.007 (solid symbols) and ϕP = 0.25 (open symbols) as a function of solvent quality, χPS, using the “one monomer per CG bead” model. Inset plot shows results obtained from a coarser “one Kuhn segment per CG bead” model. (b) ν from SANS experiments on l-PS11k (orange squares), c-PS11k (light-blue circles), l-PS14k (orange diamonds), and c-PS14k (light-blue hexagons) in d-cyclohexane at varying temperatures, T, with lines drawn to guide the eye. Computational results at ϕP = 0.007 and model χPS = 0.07 and 0.08 kBT are also replotted in (b) for comparison.

1a,b) and SANS experiments (Figure 1b). In the theory and simulations, we study linear and cyclic polymer solutions of chain length N = 108 at two polymer volume fractions, ϕP = 0.007 and 0.25, and over a range of solvent qualities, 0.0 ≤ χPS ≤ 0.11 kBT. χPS = 0 denotes the case where nonbonded polymer−polymer, polymer−solvent, and solvent−solvent LJ interactions are identical and increasing model χPS denotes decreasing solvent quality. In the experiments, the molecular weights of the two linear (l-) and cyclic (c-) PS samples are ∼11 000 and ∼14 000 g/mol, as characterized by MALDI-ToF MS and GPC (see Materials and Methods and Supporting Information). The MALDI-ToF MS data confirm that the cyclic polymer is of approximately the same molecular weight as the linear precursor while the GPC shows a substantial reduction in the molecular size, as expected because of the architectural change from linear to cyclic.56 The polymer samples used in the experiments are hereafter referred to as lPS11k, c-PS11k, l-PS14k, and c-PS14k. To validate the CG model and computational methodology, we compare the computa4584

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Figure 2. SANS Profiles. SANS intensities, I(q), for (a) l- and (b) c-PS11k in d-cyclohexane at ϕP = 0.007 (1 wt %), (c) l-, and (d) c-PS14k in dcyclohexane at ϕP = 0.007 (1 wt %).

using a more commonly used “Kuhn segment” model. If each CG bead represents one Kuhn segment of PS57 and each solvent bead maps to multiple molecules of d-cyclohexane, the PS11k sample would correspond to bead-spring chains of N ≈ 20. Interestingly, we find that while this coarser CG model captures the expected trend in νlinear, the values of νcyclic are much higher than the experimental values and are not sensitive to changes in model χPS. This result demonstrates the importance/impact of excluded volume interactions in controlling the cyclic chain conformations and scaling. We conjecture that the failure of this Kuhn segment model (using LJ nonbonded interactions) is due to overestimation of the excluded volume between segments in the cyclic polymers such that cyclic chains are not able to expand/contract with changing solvent quality. Thus, for the molecular weights studied herein, a monomer-per-bead model is the best choice to probe the experimentally relevant scaling behavior as a function of solvent quality. A more detailed discussion of the results from this model is shown in Supporting Information Figure S4. Using the experimentally validated one-monomer-per-bead CG model, we calculate the ν at high ϕP (Figure 1a dotted lines) and find that νlinear is less sensitive to changes in solvent quality than νcyclic, such that the linear and cyclic solutions at ϕP = 0.25 have similar scaling near the θ condition. Essentially, the chain connectivity/architecture becomes less important as the solvent quality decreases, and the resulting solvent-induced monomer crowding and monomer−monomer packing at concentrations above the overlap concentration dominate the chain conformations. In Figure 3, we plot the chain dimensions from simulations as characterized by the Rg and g-factor. In good solvents, chain

Figure 3. Chain dimensions from simulations. (a) Linear (black squares) and cyclic (blue circles) polymer radius of gyration (⟨Rg2⟩0.5) as a function of model χPS and (b) cyclic−linear g-factor, ⟨Rg,cyclic2⟩/ ⟨Rg,linear2⟩, from simulations. In both panels, solid symbols denote ϕP = 0.007 and open symbols denote ϕP = 0.25.

Rg (Figure 3a) decreases as ϕP increases, in agreement with the recent work on semidilute cyclic polymer solutions.18 In 4585

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At high ϕP, chain architecture has little impact, and the linear and cyclic χeff are similar over the entire model χPS range. The above trends can be interpreted via the local packing/ availability of intra- and intermolecular polymer−polymer and polymer−solvent contacts for linear and cyclic architectures, as shown in Supporting Information Figure S5. For instance, at ϕP = 0.007, essentially all of the available polymer− polymer contacts are intrachain. The local density of monomers within a cyclic chain is higher than in a linear chain because of the smaller cyclic Rg, thus, as model χPS increases, more favorable intrachain contacts are available. This decreases the enthalpic penalty for solvating the cyclic chains and causes the lower cyclic χeff. However, at ϕP = 0.25, both intra- and intermolecular contacts are abundant, making linear and cyclic χeff similar. Some of this intra- and intermolecular packing can also be seen visually in the inset images in Figure 4 and the simulation snapshots in Figure S5. We also compare the PRISM χeff with that obtained from the SANS results in Supporting Information Figure S6. To understand polymer packing as mediated by the χeff described in Figure 4, we calculate the effective polymer− polymer interactions (B2) using PRISM theory (Figure 5). At

poorer solvents, however, the chain Rg actually increases with increasing ϕP because of the larger potential number of favorable polymer−polymer contacts. The average number of intra- and intermolecular polymer−polymer contacts is shown in Supporting Information Figure S5. Unsurprisingly, at higher ϕP, the total number of nonbonded polymer−polymer contacts is higher, which results in a more favorable enthalpic environment for the chains and the larger Rg. Given the debate/disagreement in the literature about the cyclic−linear g-factor, we calculate the computational g-factor and find that the values range from 0.54 to 0.58 for varying model χPS and ϕP (Figure 3b), which matches recent experimental work that suggests that the g-factor is significantly higher than 0.5.26 Interestingly, there are two distinct slopes in the ϕP = 0.007 results; this change in slope occurs at roughly the same model χPS as the plateau in the ν for the same system (Figure 1). Thus, the balance between solvent-induced chain collapse and topological excluded volume affects both the size of the polymer chains and their size scaling. The subtle interplay between solvent quality and ϕP in determining the value of the g-factor may also explain some of the considerable disagreement in the literature, as small differences in sample preparation may result in divergent quantitative behavior. We note that we do not estimate the experimental g-factor from the SANS data because of two factors which complicate the experimental determination of Rg. First, the cyclic P(q) has no analytical form for cases where ν ≠ 0.5, and unlike the case of a linear polymer, we are unaware of any closed-form expression to relate the N and ν of a cyclic polymer to its Rg. Second, because the PS/d-cyclohexane system exhibits a UCST behavior, the scattering intensity decreases as T increases. If polymer−solvent interactions are not taken into account (e.g., by the RPA), estimations of Rg (e.g., using the Guinier approximation) will yield incorrect values. In Figure 4, we present the effective polymer−solvent interactions as a function of polymer architecture and solvent

Figure 5. Polymer−polymer second virial coefficient. B2 between monomers for linear (black squares) and cyclic (blue circles) polymers at ϕP = 0.007 (solid symbols) and ϕP = 0.25 (open symbols).

low ϕP (solid lines), effective interchain repulsion denoted by positive B2 or effective interchain attraction denoted by negative B2 are stronger for linear chains. The θ point (as defined by the polymer−polymer B2 = 0) occurs at a worse solvent condition (higher model χPS) for cyclic compared to linear, in agreement with prior experimental and simulation results.22−24,38 Also, the model χPS that results in linear polymer B2 = 0 loosely matches the model χPS that results in ν = 0.5 (Figure 1), thus consistency in two different definitions of the θ point is maintained. We also plot the B2 for ϕP = 0.25 conditions (dotted lines). We note that the definition of B2 used herein as the long length scale limit of the total correlation function (eq 14) is strictly valid only in the limit of dilute polymer concentration. However, in other regimes of polymer concentration, we can still use this “B2-like” quantity to examine the effective interactions between chains. Effective polymer−polymer interactions in the semidilute regime are dominated by the correlation hole effect,58 which, as seen in Figure 5, lowers the sensitivity of B2 to changes in solvent quality or architecture. The results shown in Figures 4 and 5 demonstrate the subtle and delicate effects that ϕP and model χPS have on effective polymer−solvent and polymer−polymer

Figure 4. Polymer−solvent effective interactions. PRISM polymer− solvent χeff for linear (black squares) and cyclic (blue circles) polymers at ϕP = 0.007 (solid symbols) and ϕP = 0.25 (open symbols). Inset images are representative chain configurations at model χPS = 0.07 kBT and ϕP = 0.007.

quality by plotting the polymer−solvent χeff versus model χPS from PRISM theory. At low ϕP and high model χPS, the cyclic polymer−solvent χeff is more favorable than linear polymer− solvent χeff, which matches with previous experimental and computational studies that show a θ point depression in cyclic polymer solutions.22−24,38 Surprisingly, at low model χPS, the polymer−solvent χeff is similar for linear and cyclic polymers. 4586

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Figure 6. Effect of linker chemistry and impurity content. (a,b) ν from simulations (a) and PRISM polymer−solvent χeff (b) for linear (squares) and cyclic (circles) polymers at ϕP = 0.007 as a function of linker chemistry: no linker (black and blue solid lines), linker-prefers-polymer (cyan dotted lines), linker-neutral (magenta dotted lines), and linker-prefers-solvent (green dotted lines). Inset plot in (a) is zoomed-in to the region 0.07 < χPS < 0.09 kBT for clarity. (c,d) ν from simulations (c) and PRISM polymer−solvent χeff (d) for linear (squares) and cyclic (circles) polymers at ϕP = 0.007 as a function of impurity content: no impurities (black and blue solid lines), 5% linear (green open squares), 10% linear (cyan open squares), 5% cyclic dimer (green open circles), and 10% cyclic dimer (cyan open circles) impurities. (e) ν from SANS experiments on l-PS11k (orange circles), l-PS14k (orange hexagons), c-PS11k (light blue circles), c-PS14k (light blue hexagons), c-PS14k with 1% l-PS14k (gray hexagons), and c-PS14k with 5% l-PS14k (green hexagons), with lines drawn to the pure linear and cyclic results to guide the eye. (f) Representative chain configurations of the linker-bead chains and the cyclic dimer impurities at model χPS = 0.07 kBT and ϕP = 0.007, with the linker beads rendered in gray.

and are in quantitative agreement with the scaling exponents obtained for pure cyclic solutions. These combined results suggest that the structure and effective interactions in cyclic polymer solutions are less sensitive to impurity content, as opposed to some recent work on polymer melt rheology which demonstrated significant deviations as small amounts of linear impurities are introduced.2,3

thermodynamics in these solutions; such a mapping is essential to tailor a particular desired solvent quality or set of polymer− polymer interactions. In Figure 6, we examine the effects of explicitly including the linker chemistry or undesired impurities in the simulations/ theory, as well as SANS results for linear/cyclic blends. We use three generic choices to model linker−polymer and linker− solvent interactions, as described in Materials and Methods, to capture linker and end group chemistries that could be preferential to the polymer, preferential to the solvent, or neutral to both polymer and solvent. Figure 5a,b shows that both the ν and the χeff shift systematically at a given model χPS depending on the linker chemistry. When the linker prefers the solvent, the chains expand slightly, increasing ν; conversely, when the linker prefers the polymer, ν decreases. Explicitly including the linker decreases the polymer−solvent χeff in all cases. However, for both the chain conformations and effective interactions, the differences caused by the presence of linker beads are similar for linear and cyclic polymers, so the linker (regardless of chemistry) appears to be unimportant in determining the relative behavior of the cyclic and linear architectures. Figure 5c,d shows that in the simulations/theory, 5 and 10% linear and 5 and 10% cyclic dimer impurities have no observable impact on the chain scaling or effective polymer−solvent interactions. Using SANS, we also examine the effect of linear impurities on cyclic chain scaling. Shown in Figure 6e is ν as a function of inverse temperature for 1 wt % cyclic polymer solutions containing 1 and 5% linear precursor chains with an identical molecular weight (the SANS I(q) for these systems are shown in Supporting Information Figure S7). Analysis of the scattering data confirms that for 1 and 5% linear impurities, the scaling exponents do not change appreciably

4. CONCLUSIONS In this paper, we show a unified picture of the effects of solvent quality, linker/end group chemistry, and presence of impurities on cyclic polymer chain conformations and solution thermodynamics. Previous work, though careful and valuable, has in most cases shown only a subset of the polymer solution parameter space. We demonstrate a powerful combined simulation-theory-experiment approach, where we use computationally intensive explicit solvent MD simulations to capture the effects of changing solution environment on the chain configurations, use computationally fast PRISM theory calculations to access long length scale and high-precision structural correlation functions that enable the calculation of effective interactions, and validate the computational results through comparisons to SANS. We find that the linear versus cyclic architecture plays a significant role in effective polymer− solvent and polymer−polymer thermodynamics because of the changes in available intra- and intermolecular contacts, even when local monomer−solvent interactions are identical. Additionally, cyclic−linear g-factor and cyclic polymer Flory exponent show nonmonotonic trends as solvent quality changes, particularly near the θ condition. We attribute these effects to a balance between unfavorable solvent−polymer 4587

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Macromolecules

agreement no. DMR-1508249. F.M.H. and S.M.G. thank the National Science Foundation for funding the MALDI-ToF MS instrumentation (MRI 0619770) at Tulane University, the Smart MATerials Design, Analysis, and Processing consortium (SMATDAP) under cooperative agreement IIA-1430280, and the Joseph H. Boyer Professorship for support of this work.

interactions and the topological excluded volume in the cyclic architecture. The quantitative agreement in experimental and simulation theory scaling exponent (over the accessible experimental temperature range) is remarkable and provides validation for the chosen CG polymer and solvent model. Interestingly, if one had set out to conduct the above computational study using the more popular model that maps one Kuhn segment per CG bead,57 the experimental chain configurations and scaling would not be captured in the simulations. Finally, for the purposes of determining effective interactions and chain scaling behavior, our results demonstrate that the effects of the end group/linker chemistry and common synthetic impurities (e.g., linear polymers or cyclic dimers) can be largely ignored. Overall, this effort allows us to map trends in how solvent quality, architecture, and concentration cooperate/compete to produce effective polymer−solvent and polymer−polymer interactions and chain configurations/scaling, vital for optimizing any polymer synthesis, assembly, purification, or separation procedures that might occur in solution. Furthermore, it may shed light on important biophysical processes such as the packing of DNA in chromatins.59

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.macromol.9b00600. Additional simulation data, comparison of computational and experimental RPA χeff, PRISM χeff calculation for multicomponent systems, experimental SANS profiles, and polymer molecular weight data (PDF)

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REFERENCES

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] (S.M.G.). *E-mail: [email protected] (M.J.A.H.). *E-mail: [email protected] (A.J.). ORCID

Thomas E. Gartner, III: 0000-0003-0815-1930 Farihah M. Haque: 0000-0002-5029-7442 Scott M. Grayson: 0000-0001-6345-8762 Michael J. A. Hore: 0000-0003-2571-2111 Arthi Jayaraman: 0000-0002-5295-4581 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS T.E.G. and A.J. acknowledge funding from the National Science Foundation (NSF DMR-1609543) and the use of computational resources from the University of Delaware (Farber cluster) and the Extreme Science and Engineering Discovery Environment (XSEDE) Stampede cluster (allocation MCB100140), which is supported by NSF grant ACI1548562. M.J.A.H. and A.M.G. acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the neutron research facilities used in this work. Access to NGB 30m SANS was provided by the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology and the National Science Foundation under 4588

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