Addressing the challenges of modeling the scattering from bottlebrush polymers in solution

Abstract

Small-angle scattering measurements of complex macromolecules in solution are used to establish relationships between chemical structure and conformational properties. Interpretation of the scattering data requires an inverse approach where a model is chosen and the simulated scattering intensity from that model is iterated to match the experimental scattering intensity. This raises challenges in the case where the model is an imperfect approximation of the underlying structure, or where there are significant correlations between model parameters. We examine three bottlebrush polymers (consisting of polynorbornene backbone and polystyrene side chains) in a good solvent using a model commonly applied to this class of polymers: the flexible cylinder model. Applying a series of constrained Monte-Carlo Markov Chain analyses demonstrates the severity of the correlations between key parameters and the presence of multiple close minima in the goodness of fit space. We demonstrate that a shape-agnostic model can fit the scattering with significantly reduced parameter correlations and less potential for complex, multimodal parameter spaces. We provide recommendations to improve the analysis of complex macromolecules in solution, highlighting the value of Bayesian methods. This approach provides richer information for understanding parameter sensitivity compared to methods which produce a single, best fit.

1 |. INTRODUCTION

Characterization of the solution-state properties of macromolecules is one of the fundamental means of evaluating the relationship between structural parameters and conformation, which in turn controls properties ranging from self-assembly to rheology. Small-angle X-ray scattering and neutron (small-angle neutron scattering [SANS]) scattering are the workhorse methods for performing this characterization. Modern instrumentation allows rapid, routine collection of data from solution samples and, as a result, much of the challenge in utilizing these techniques resides with the proper interpretation and modeling. Small-angle scattering data are typically modeled with an inverse-iterative approach, where simulated data from a real- or Fourier-space model is iteratively fit to a set of experimental data. The parameters of the model are then interpreted to describe the structure and conformation of the measured system. Unfortunately, due to information loss associated with measuring the Fourier-space scattering intensity rather than the scattering amplitude (i.e., the phase problem), scattering analyses are plagued by lack of uniqueness and parameter correlation issues.^[1,2] Compounding this problem, scattering data from macromolecules in dilute solution tend to have few well-defined features, and generally consist of a series of power law relationships between the scattered intensity, I, and the magnitude of the momentum transfer vector, q. Different models often result in fits of equivalent quality, making it unclear which model best represents the true macromolecular shape. This becomes more problematic as the complexity of the target structure increases. Scattering measurements of biologically relevant molecules are a classic example of this problem, where highly complex molecules such as proteins result in deceptively simple scattering patterns. The biological community tackles this problem by combining scattering measurements with complementary data and extensive molecular modeling to constrain the structure.^[3–5] This combined approach is powerful, but the specialized expertise needed to correctly integrate simulations into the analysis of scattering data is still a barrier to its widespread adoption.

Synthetic bottlebrush polymers, which consist of a linear backbone with densely grafted side chains, are a class of complex macromolecules which have received renewed attention in recent years.^[6–8] The side chains impart additional bending rigidity to the backbone, and as a result, bottlebrush polymers adopt extended conformations compared to their linear counterparts. There have been a large number of efforts to characterize the behavior of these molecules with solution scattering measurements.^[9–19] These systems are modeled primarily with the flexible cylinder model, which is a natural choice given the anticipated extended shape of the polymer. The flexible cylinder model provides direct access to the persistence length (l_p), which quantifies the stiffness of the polymer, and other important measures of the chain dimensions. In this article, we will discuss the underlying challenges with utilizing the flexible cylinder model to interpret scattering from these systems and recommend an approach which will improve the reliability of comparisons across different sets of data. Improving understanding of graft polymer conformations will provide new insight into the design of bottlebrush polymers in the many contexts where molecular shape is important, including drug delivery,^[20] biological imaging,^[21] and interfacial modification.^[22]

Bottlebrush polymers consist of a backbone with densely grafted side chains, as described in Figure 1a. The addition of the side chains restricts the conformation of the backbone on intermediate length scales, resulting in an increase in l_p. Solution scattering represents the most straightforward method for developing relationships between the polymer structure and persistence length. The scattering is typically interpreted using variations on the flexible cylinder model, which is schematically represented in Figure 1b.^{[11,13,19,23]} In this work, in order to illustrate issues with the flexible cylinder model, we conducted SANS measurements on a set of polynorbornene-graft-polystyrene bottlebrushes. The synthesis of these bottlebrushes via grafting-through ring-opening metathesis polymerization has been reported elsewhere.^[24] The measurements were conducted in d₈toluene (a good solvent for both the backbone and side chains) at volume fractions ϕ ≤ 0.005.

FIGURE 1.

(a) Schematic of the general structure of a bottlebrush polymer, where the relevant parameters are the degrees of polymerization of the backbone (N_BB) and side chains (N_Sc) and the distance between grafting points (σ), which is a function of the backbone chemistry and grafting density. (b) Schematic of the bottlebrush polymer as seen by the scattering measurement, indicating the persistence length (l_p), length per monomer (l_m), and radii of gyration for the major (R_g,2), and minor axes (R_g,1). (Arrows imply the directions of the components of R_g and do not represent the magnitudes.) R represents the radius of the cylinder in the flexible cylinder model

The first sample examined had a backbone degree of polymerization N_BB = 105 and one side chain of degree of polymerization N_SC = 40 per backbone unit (i.e., 100% grafting density). This sample is denoted PNB₁₀₅-g₁₀₀-PS₄₀. The dispersity of the entire bottlebrush was Ð = 1.03, as measured by size exclusion chromatography. SANS data for PNB₁₀₅-g₁₀₀-PS₄₀ were first fit to the flexible cylinder model. In this model, the structure is represented by a set of freely jointed cylinders. The flexible cylinder form factor (S_fc) was developed from empirical fits to a series of Monte-Carlo simulations of semiflexible linear chains targeted toward wormlike micelles (by utilizing systems where l_p was at least 10× the monomer size) and is shown in Equation (1).^[25] The model is parameterized with the length of the cylinder (L_c) and the persistence length (l_p, which is often represented in the model through the Kuhn length, λ = 2l_p). The scattering is calculated from the combination of the form factors from a flexible chain (S_chain) and a rigid rod (S_Rod), combined through the crossover function (χ) and the correction to the crossover function (Γ) (see the original publication [ref. [25]] for details on these functions). Information on the cross-sectional structure of the polymer is added according to Equation (2), by multiplying S_fc by the Fourier transform of the chosen radial cross section (S_xs).

$$\begin{gathered} S_{fc}\left(q,L_c,l_p\right)=[\left(1-\chi \left(q,L_c,l_p\right)\right)S_{\text{chain}}\left(q,L_c,l_p\right) \\ +\chi \left(q,L_c,l_p\right)S_{\text{Rod}}\left(q,L_c\right)]\text{Γ}\left(q,L_c,l_p\right) \end{gathered}$$ (1)

$$S\left(q,L_c,l_p,\text{R}\right)=S_{fc}\left(q,L_c,l_p\right)S_{xs}(q,R)$$ (2)

To illustrate some of the issues that have emerged in the literature due to the use of this model, we will examine a commonly used metric, l_m, the unit length of the flexible cylinder (L_C) per backbone monomer, defined in Equation (3). It is important to note that L_C is a fit parameter in the flexible cylinder model, and not necessarily equivalent to the contour length of the bottlebrush backbone (L_B) which can in principle be calculated from the chemical structure and the molar mass. l_m values from the literature for both polynorbornene-based backbones and vinyl-based backbones are shown as a function of N_BB in Figure 2a,b, respectively. For polynorbornene-based backbones, the upper limit for this value is expected to be ≈0.6 nm, based on the bond lengths for one norbornene repeat unit. Figure 2a shows that this value is widely distributed from 0.03 to 0.27 nm. The variation is particularly significant for shorter polymers, which in the limit of extremely short backbones can be considered stars. The star-to-bottlebrush transition is expected to occur around N_BB ≈ 10, but may shift with side chain length.^[26] Because l_m effectively represents the tortuosity of the bottlebrush backbone along the contour of the flexible cylinder model (as described by the red line in Figure 1b), it may be expected to vary somewhat based on the side chain chemistry, length, solvent quality, backbone length, and solution concentration (which can have a significant effect on the size of the macromolecule),^[27,28] but the extent of the variation shown here brings into question the reliability of interstudy comparisons. A similar variation in this value between studies appears for the vinyl-based backbones, with l_m ranging from 0.05 to 0.4 nm. Two measurements result in values that exceed the physical limit of l_m ≈ 0.27 nm for a vinyl backbone. To cite one example of the variation between studies, one group examined poly(methacrylate) backbones with PS side chains, and the l_m values calculated from their results are on the order of 0.17 nm (green triangles on Figure 2).^[13] A similar study on another system with a vinyl-based backbone—featuring a poly(hydroxyethyl methacrylate) backbone and poly(n-butyl acrylate) side chains—results in values on the order of 0.25 nm (red circles on Figure 2).^[12] Both groups used similar models to interrogate the scattering profiles (flexible cylinders utilizing a Gaussian radial decay for the cross-sectional contributions), and both systems were in a good solvent. In both cases, the backbones were sufficiently long to minimize side chain effects on the determination of l_m. The chemistries are similar enough that it would be reasonable to expect similar l_m values between these two studies, but they diverge significantly across all the samples measured. This wide range demonstrates how challenges in consistent implementation of these models acts as a barrier to comparisons between studies.

FIGURE 2.

Literature data on the length per monomer (l_m) as a function of the backbone degree of polymerization (N_BB) for polynorbornene-based^[17–19] (a) and vinyl-based^[11–16,29] (b) grafted polymers. The dashed line in Part (b) indicates the physical limit for l_m based on the fully extended trans zigzag conformation and the vinyl bond lengths. It should be noted that in some cases these values were not explicitly calculated in the publications and were determined according to Equation (3). A table with these values is provided in the SI. The blue highlighted region indicates the N_BB where the star-to-bottlebrush transition is expected to occur^[26]

$$l_m=\frac{L_c}{\left(N_{BB}+2N_{Sc}\right)}$$ (3)

The origin of this variation lies primarily with the flexible cylinder model and the fitting approaches used to evaluate the scattering data. In this work, fits were conducted using the flexible cylinder model with the directed evolution Monte-Carlo Markov Chain (MCMC) algorithm (DREAM) in SasView.^[30] The DREAM algorithm,^[31] along with other similar MCMC-based algorithms, provides a means of sampling complex parameter spaces.^[32–37] Briefly, MCMC algorithms utilize a series of directed “walkers” in parameter space, and each step of the walker represents a set of model parameters. Random perturbations to each parameter in the model are generated, and the change in the goodness of fit (GF) is compared to the previous step in the chain. A better GF is always accepted while worse GF values are accepted or rejected based on the Metropolis acceptance criteria. The primary benefit of the DREAM algorithm is that it is nongradient-based and therefore can navigate the rough solution space of complex models, such as the flexible cylinder model, and more easily find the global minima. This contrasts with gradient methods such as the commonly used Levenberg–Marquardt algorithm, which perform poorly in complex solution spaces unless coupled with global search strategies. Furthermore, the DREAM algorithm is Bayesian in nature and therefore allows users to not only calculate parameter distributions of arbitrary type (e.g., normal, lognormal, Boltzmann) but also the cross-correlation statistics between model parameters. This is critical for understanding parameter relationships in inverse problems where they are often correlated, particularly for feature-poor data.

The best fit to the data using the DREAM algorithm with the flexible cylinder model is shown in Figure 3a, with the correlation maps between L_c, R, and λ shown in the inset. All three sets of correlation plots show diagonal elliptical patterns, indicating strong correlations between the model parameters. Uncorrelated parameters would result in a circular pattern in this representation. The magnitude of these correlations was further explored by conducting a series of fits, where L_c was held constant and λ and R were allowed to vary. The contour length was iterated from 400 to 850 Å; this range was chosen to capture the reasonable bounds of the structure, where the upper length was just short of the backbone contour length including the terminal side chains, L_B. The values of λ, R and the corresponding GF for each value of L_c are shown in Figure 3c. This analysis shows two results which help explain the variation in the literature data for parameters calculated using the flexible cylinder model. The first is that L_c and λ are highly correlated; the value of λ can shift over twofold depending on the value of L_c. This problem is compounded by the shallowness and multimodality of the GF space. Distinct minima occur at 500, 550, and 820 Å, with the shallow space ranging from 475 to 725 Å. Both the multimodality and the shallow minima are likely the result of the parameter correlations. These correlations emerge in the scattering profile because the contributions from L_c and λ can emerge at similar regions of q space. In this region, a visual comparison of the simulated and experimental curves would suggest that any of the model values yield a plausible fit to the experimental data. This demonstrates that local gradient-based optimization tools will always provide an incomplete representation of the parameter space compared to Bayesian approaches which interrogate parameter distributions. In this case, there are several possible L_c that could have been chosen if local gradient-based optimization methods had been utilized. Converting L_c to l_m results in values from 0.22 to 0.61 nm, consistent with the magnitude of variation seen in the literature data shown in Figure 2b. The combination of the parameter correlations and shallow GF space demonstrates that, for bottlebrush polymers, the analysis of scattering data utilizing the unconstrained flexible cylinder model is unlikely to yield consistent results.

FIGURE 3.

Fits to the scattering data from the PNB₁₀₅^-g₁₀₀^-PS₄₀ bottlebrush (N_BB = 105 and N_Sc = 40) at a volume fraction of 0.005 in d8-toluene. (a) Utilizing the flexible cylinder model, correlations between all three structural parameters (λ, R, L_c) are shown in the inset. (b) Best fit using the generalized Guinier–Porod (GGP) model; correlations between R_g,2, R_g,1, and s₁ are shown in the inset of ref. [38]. In both cases, the data range used to fit the scattering is truncated above q = 0.1 Å⁻¹ where the scattering is dominated by the background and does not influence the structural parameters in the model. (c) Results of the parameter mapping performed by holding L_c constant and allowing λ and R to vary during fits with the flexible cylinder model. (d) Results of the correlation mapping from the GGP model, holding R_g,₂ constant and allowing the other parameters to vary. Vertical arrows in (c) and (d) correspond to the fits shown in Parts (a) and (b)

The challenges with utilizing the flexible cylinder model have been previously discussed in the literature, but no compelling alternative has been proposed. Zhang et al. utilized a Holtzer analysis in an attempt to extract l_m and, from that, constrain L_c.^[13] The Holtzer analysis plots (I(q)*q/c)/S_xs(q,R) versus q, thereby scaling the intensity by the concentration and dividing out the radial cross section. In this form, for large q, there is expected to be a plateau value where the mass/length can be extracted and used to calculated l_m. The accuracy of this approach for bottlebrush polymers is unclear, particularly given that it is dependent on the choice of the form of the radial cross section. Zhang et al. assumed a Gaussian cross section in their analysis, but Hsu et al. conducted simulations which suggested that an alternative form (Equation (4)) better represented the radial density profile. In Equation (4), r₁ and r₂ are length scales for exponential decay close to and far from the backbone, respectively, and x₁ and x₂ are scaling parameters.^[39] While it is perhaps more rigorous, this functional form further obfuscates the analysis of the scattering data. Examination of the data from our system using the Holtzer analysis and a Gaussian radial cross-section did not yield a satisfactory high-q plateau for extraction of the mass/length for any of the samples examined in this study.

$$\rho (r)=\frac{\sigma }{1+{\left(\frac{r}{r_1}\right)}^{x_1}}{e}^{\left[-{\left(\frac{r}{r_2}\right)}^{x_2}\right]}$$ (4)

Shape-agnostic models provide an alternative to structurally informed models such as the flexible cylinder. One example is the generalized Guinier–Porod (GGP) model, which is described in Equations (5)–(7).^[38] In this model, different length scales are represented by Guinier regions interconnected by power law regimes. Crossover points (Q₁, Q₂) between different regions and scaling factors (G₁, G₂) are calculated internally to ensure smooth transitions between different length scales. The model also includes shape factors (s₁, s₂) which are related to the dimensionality (d = 3-s); a value of 0 represents spheres, 1 represents rods, and 2 represents plates or lamellae. R_g,₂ represents the long radius of gyration (a proxy for the cylinder length) and R_g,₁ represents short radius of gyration (a proxy for the cylinder radius), their relationship to the bottlebrush structure is shown schematically in Figure 1b.

$$I(Q)=\frac{G_2}{Q^{5_2}}\exp \left(-\frac{Q^2{R}_{\text{g},2}^2}{3-s_2}\right)\text{for}Q\le Q_2$$ (5)

$$I(Q)=\frac{G_1}{Q^{s_1}}\exp \left(-\frac{Q^2{R}_{g,1}^2}{3-s_1}\right)for{Q}_2\le Q\le Q_1$$ (6)

$$I(Q)=\frac{D}{Q^d}forQ\ge Q_1$$ (7)

The structural parameters from the GGP model suffer from fewer correlation issues compared to the flexible cylinder, as shown in the inset of Figure 3b, and yield equivalent GF values. The two R_g parameters are only weakly correlated, and the correlation between R_g,1 and s₁ is significantly weaker than the correlations in the flexible cylinder model as shown in the insets of Figure 3. The weak correlations in this model largely emerge due to the distinct q regions where each R_g contributes to the scattering profile. From this fit, R_g,2 was found to be 89.1 ± 1.8 Å, and R_g,1 was found to be 25.8 ± 0.7 Å. Uncertainties represent 95% confidence intervals derived from the sample populations from the DREAM algorithm. The shape parameters s₂ and s₁ were 0.08 ± 0.04 and 1.31 ± 0.03, respectively, consistent with a rod-like shape.^[38] The only parameter of interest not directly interpretable from the GGP model is l_p. A simple estimate can be obtained from R_g,2 using Equation (8), as long as a value for L_c is provided.^[40] Using the contour length of the backbone provides a means of calculating l_p using a value that is readily determinable from standard characterization. Using this approach, l_p for PNB₁₀₅-g₁₀₀-PS₄₀ in toluene was found to be 47.8 ± 1.0 Å. More robust methods of extracting l_p could be developed by mapping simulation results onto the R_g values using approaches similar to Dutta et al.^[41] The magnitude of the parameter correlations in the GGP model was obtained by mapping the GF space relative to R_g,2, using an identical approach to the flexible cylinder mapping, and the results are shown in Figure 3d. The parameter space is greatly simplified relative to the flexible cylinder model, with only a single, well-defined minimum. This demonstrates that while parameter correlations do exist, their magnitude is mild. To demonstrate that this method can apply to a range of architectures the same analysis was applied to two longer bottlebrushes, PNB₂₀₈-g₅₂-PS₄₀ and PNB₄₈₂-g₁₀₀-PS₆₂ and the results of this analysis are discussed in detail in the SI. A table summarizing the results from GGP fits to the three samples is shown in the Supplemental Information. A reduced l_p is observed for PNB₂₀₈-g₅₂-PS₄₀ compared to PNB₁₀₅-g₁₀₀-PS₄₀ (38.5 ± 3.4 Å vs. 47.8 ± 1.0 Å) due to the sharp reduction in the grafting density, consistent with the expected trend. The fits to PNB₄₈₂-g₁₀₀-PS₆₂ resulting in an l_p of 89.6 ± 7.3 Å, again consistent with the expected trend due to the longer side chains. PNB₂₀₈-g₅₂-PS₄₀ shows a wide, shallow minima around the best fit to R_g,2, this highlights the importance of using this type of Bayesian parameter mapping for interrogating the sensitivity of any model to the given data. While this fit still lacks the multimodality of the flexible cylinder model, there is considerable uncertainty in the major R_g. Both samples show similar correlation issues with the flexible cylinder model, while these are markedly reduced for the GGP model. For the largest and most highly extended molecule (PNB₄₈₂-g₁₀₀-PS₆₂), the flexible cylinder model underpredicts the scattering intensity at low q (Figure S4) indicating that in this case the model is incapable of fully reproducing the scattering data.

$$R_g=\left(\frac{L_B{l}_p}{3}-l_p^2+\frac{2l_p^3}{L_B}-2\left[1-e^{-L_B/l_p}\right]\frac{l_p^4}{L_B^2}\right)$$ (8)

The issues discussed here demonstrate the challenges in using the flexible cylinder model to fit the scattering from bottlebrush polymers in solution. To be clear, it is not the inherent physics of the model or the interpretation of the bottlebrush as a flexible cylinder which are problematic; instead, the difficulty arises in unambiguously fitting the data. The multimodal parameter space of the flexible cylinder model, which is a product of the significant correlations between model parameters, results in significant impediments toward utilizing this model to compare values between different studies. One reason the inconsistencies in the data are problematic is that, since the advent of machine learning, data mining approaches have the potential to reveal underlying trends in parameter relationships but require well-established methodologies for producing comparable data. We recommend that future solution studies use the shape-agnostic GGP model as a basis for interrogating the conformation of bottlebrush and similar polymers. The flexible cylinder model has the advantage of directly connecting parameters of interest (λ, l_p, R) to the scattering data, but should be used with caution. Utilizing the approach presented here to map out the parameter space and ensure that the results presented represent a true global minimum will go a long way to improving its reliability. Additionally, it would be beneficial to couple either model with complementary measurements or simulations to more fully inform the scattering analysis and either eliminate correlations or constrain model parameters. Dynamic light scattering, for example, can provide information on the hydrodynamic radius, complementing the information gained through neutron or X-ray scattering.^[42] Finally, constraining L_c in the flexible cylinder model based on the backbone contour length, estimated from the chemical structure of the monomer and the molar mass, will provide a more consistent basis for comparisons. Contrast variation is another approach that could be used to constrain the modeling. In neutron scattering, contrast matching is a well-established technique, but contrast matching the brush from the backbone would require selective deuteration, a significant synthetic challenge. Soft X-rays are a relatively new tool that has demonstrated significant potential for constraining difficult scattering problems through contrast variation, but there are few examples on systems in solution.^[43–46] Utilizing simulations to constrain the scattering analysis is a feasible alternative to additional measurements, having already been demonstrated by Ahn et al., who used atomistic simulations of short bottlebrushes to assist with the modeling of solution scattering data.^[17] Atomistic modeling could provide direct relationships between the contour length of the bottlebrush backbone and the length of the equivalent flexible cylinder, providing a reliable constraint for that model. Unfortunately, atomistic modeling of molecules of the size described here is still a significant endeavor. There have been recent advances in mapping complex simulations to coarse grained simulations, a route which provides hope for ultrahigh molecular mass molecules.^[41]

Equally important to the choice of model is the approach for analyzing and interpreting the data. The Bayesian based approach utilized here provides a complete picture of variations within a parameter space and the results show the importance of a thorough exploration of the parameter correlation landscape for even relatively simple models. While this type of analysis would be computationally prohibitive in the past, modern capabilities make it routine. Reporting results from scattering analysis without describing parameter correlations lead to an incomplete understanding of the model’s accuracy. This is particularly true for systems yielding scattering data with few defining features. While this manuscript focused on one type of complex architecture, the principles used to interrogate the structure could be readily extended to other classes of materials such as dendrimers.^[47,48] The shape-agnostic model would be particularly useful for such a system where it might be hard to anticipate the solution conformation. The process outlined can be readily applied to a wide range of scattering problems using existing tools, enhancing understanding of polymer conformations toward improved molecular design.