# The Use of Scattering Data in the Study of the Molecular Organisation of Polymers in the Non-Crystalline State

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{i}, b

_{j}are the coherent scattering lengths of atoms i and j, respectively, N is the total number of atoms and r

_{ij}is the position vector between atoms i and j. The angular brackets indicate the thermal average and Q is the scattering vector commonly expressed as $Q=4\pi \mathrm{sin}\theta /\lambda $, where $2\theta $ is the scattering angle and $\lambda $ is the incident wavelength [3,4]. In a neutron scattering experiment over a broad Q range, we can obtain information about bond lengths and segmental interactions at distances of approximately 0.1–100 Å. The experimentally observed structure factor S(Q) is defined as the spatial Fourier transform of the atomic pairwise correlation functions

_{o}the average density of the system and g(r) the pair distribution function that quantifies the probability density of an atom existing at a distance

**r**from the origin. In the case of multiple chemical species, both S(Q) and g(r) can be split into partial terms. The experimentally observed structure factor arises from a wide range of structural correlations in the bulk. From this expression, we can see that structural correlations can be separated into those arising from correlations within the chain (intrachain) and those arising by different chains (interchain) [2,5,6]. The interchain correlations provide information on the different ways polymeric segments arrange themselves in the bulk. The intrachain contribution to the scattering can be extracted from the observed structure factor at values of the momentum transfer $Q\ge 2-2.5$Å

^{−1}, using atomistic modelling techniques [6]. In Figure 1, a schematic representation of the different structural regimes accessible for neutron and X-ray scattering can be seen.

## 2. Diffraction and Experimental Requirements

## 3. Use of Partial Structure Factors

_{αβ}the partial structure factors with weights determined by the products of the atomic fractions c

_{α}, c

_{β}and the scattering length. The Kronecker delta δ

_{αβ}is introduced to avoid multiple counting of similar items. According to this, the partial structure factor can be related to the appropriate partial distribution function g

_{αβ}(

**r**) by

_{0}, D

_{3}, D

_{5}and D

_{8}, with the number indicating the level of deuteration with respect to the monomer unit [2]. D

_{3}involves the deuteration of the skeletal chain, whereas D

_{5}refers to the deuteration of the side group. The existence of different deuteration levels leads to significant changes in the observed structure factor, something that is not visible with X-rays. Further manipulation of these functions leads to a set of partial correlation functions (see Figure 2).

_{8}. This effect has been attributed to the correlation between polymer backbones, and the peak is fairly broad and weak, indicating very limited backbone correlations. Such a feature is much weaker than the corresponding features in simple polymers like polyethylene [26] or poly(tetrafluoroethylene) [27], thus concluding that no substantial chain correlations are present in glassy polystyrene [2]. These limited backbone correlations further indicate that correlations between the bulky side groups dominate the structure, a model previously proposed on the basis of X-ray scattering analysis [28].

## 4. Local Mixing in Polymer Blends

_{A}, c

_{B}the concentrations of segments A and B, respectively, and ${F}_{A}\left(\mathrm{Q}\right)$, ${F}_{B}\left(\mathrm{Q}\right)$ the molecular form factors for the given segments. Here, we must note that these expressions are valid for samples that are isotropic in which the local mixing is also isotropic. Extracting the partial structure factors from such an expression requires a series of isostructural blends where either composition of the deuteration level (that alters the molecular form factors) is systematically varied. This approach can only be carried out with neutron scattering methods, as X-ray diffraction is insensitive to the isotope content, thus confirming that the blends are largely at least isostructural.

## 5. Coupling Diffraction with Modelling

^{−1}for the 20 °C sample is indicative of extensive trans sequences along the backbone [35] with the interchain peak maximum occurring at Q = 1.5 Å

^{−1}in comparison with Q = 1.35 Å

^{−1}for the sample irradiated in the melt, suggesting a closer packing of chains. From this superficial analysis of the structure factor, we can deduce that the sample irradiated at 20 °C probably contains some small clusters of parallel segments in the trans configuration while the sample irradiated in the melt is representative of a more random coil. In other words, these two structure factors represent two totally different models of orientation correlations despite the experimentally observed broad similarities. This example is quite informative and underlines the need for detailed, robust, and accurate models and the level of sensitivity of the different structural parameters on the overall structure factor.

## 6. Real vs. Reciprocal Space Functions—The Reverse Monte Carlo Method

_{E}(Q). The calculation of the scattering from a model S

_{C}(Q) can be obtained using Debye’s relationship [6].

_{i}, b

_{j}are the neutron scattering lengths [3,4] of atoms i and j, respectively, and r

_{ij}is the equivalent interatomic distance. We have chosen to make the comparison using the reciprocal- instead of the real-space function due to the simplicity of separating inter and intrachain contributions.

#### The RMC Method

_{α}is the concentration of nuclei of atom α and ${n}_{\alpha \beta}^{{C}_{o}}\left(r\right)$ is the number of nuclei of type β located at a distance between r and r + dr from the nuclei of type α, averaged over all nuclei of type α. By performing a Fourier transform on ${n}_{\alpha \beta}^{{C}_{o}}\left(r\right)$, we can obtain the partial structure factor ${F}_{\alpha \beta}^{{C}_{o}}\left(Q\right)$ and the total structure factor ${S}^{{C}_{o}}\left(Q\right)$:

^{2}test:

_{i}value needs to be larger than $2\pi /L$ with L corresponding to the minimum dimension of the configuration under investigation.

^{2}test:

^{2}is typically used due to simplicity [6,13,31]. It can be shown that for a structural system with only pairwise forces, the RMC procedure leads to an exact solution.

#### RMC Variations

## 7. The Role of the Scattering Data

#### 7.1. Intrachain Correlations

^{2}results are clustered around the same mean value, indicating the robustness and limitations of the method.

#### 7.2. The Case of Selenium

#### 7.2.1. First-Order Probabilities

#### 7.2.2. Conditional Probabilities

#### 7.2.3. Predicted Models

^{2}test for the helix is 0.0044 ± 0.0001 compared with 0.0065 ± 0.0001 of the ‘random coil’ being at a value 50% larger. Comparison of the structural parameters extracted with the technique presented here with values reported in the literature can be seen in Table S2 in the Supportive Information.

^{2}test of the order of 0.008 significantly larger than the value obtained for the helical chain. Inclusion of conditionality in the probability matrix yields a picture of almost equally probable –trans+ and +trans– sequences with a significantly lower statistical test value of 0.0025 ± 0.0001.

#### 7.2.4. Chain Length

^{2}test in an iterative manner, changing the number of atoms in the model each time. We concluded that based on the scattering data, the diffraction pattern stays almost unaffected by the number of atoms in the chain above N > 100. This can be seen in Figure 12 where results of the comparison between similar models of different chain lengths are plotted against the experimentally observed scattering curve. Based on this information, we can say that the model is not sensitive enough to distinguish between models of very large chains. It indicates though that the chains have to be at least 100 atoms long to represent in an adequate way the scattering.

#### 7.2.5. Rings

## 8. Time-Resolved Crystallisation

## 9. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic representation of the different correlations available by the use of diffraction methods in the study of amorphous polymers.

**Figure 2.**Structure factors (left) and radial distribution functions (right) for selectively deuterated glassy polystyrene. The level of deuteration can be assessed by the number next to the label B (see text for full details). (a) indicates the total correlation function for system D

_{8}and (b) the partial correlation function g

_{bb}describing the correlations involving only the backbone atoms. Both graphs were reproduced from reference [2].

**Figure 3.**Experimentally observed structure factors of protonated and deuterated 1,2 and 1,4 polybutadiene (left) and blends consisting of these components (right). In both graphs, the term d or h indicate the deuterated or protonated system, respectively, and the term 1,2 or 1,4 indicate the stereoregularity of the polybutadiene chain, respectively.

**Figure 4.**Comparison of the experimentally observed structure factor for a blend of deuterated 1,2 (d12) and 1,4 (d14) polybutadiene (d12/d14) with an artificial structure factor (d12/d14 addition) comprising the sum of the experimentally observed structure factors of the two individual components weighted by their relevant concentration. See text for more detailed discussion.

**Figure 5.**Partial structure factors for the different correlations derived from experimental data on blends of 1,2 and 1,4 polybutadiene. 1,2-1,2 (red), 1,2-1,4 (blue) and 1,4-1,4 (black) indicate the correlations between segments of 1,2 and 1,4 polybutadiene respectively. See text for more detailed discussion.

**Figure 6.**Experimentally observed structure factors from samples of deuterated polyethylene prepared with different methods. The solid line indicates the sample irradiated in the melt (160 °C), and the dashed line indicates the sample irradiated at room temperature (20 °C). On the right-hand side, the diffraction of the semi-crystalline deuterated polyethylene can be seen.

**Figure 7.**Comparison between the experimentally observed structure factor (solid line) and that calculated from a model (dashed line) for 1,2 polybutadiene at 250 K (

**A**,

**B**). In (

**B**), the areas where the different structural parameters have the most impact on the structure factor is indicated. In (

**C**), a surface plot of the statistical test as a function of the C-C and C=C bond lengths can be seen for a model of 1,4 polybutadiene. In (

**D**), the final result of the best-fit statistical test for the different models of 1,2 polybutadiene can be seen.

**Figure 8.**Total structure factor of vitreous selenium obtained from neutron scattering data. In the inset, the radial distribution function of the same scattering curve can be seen.

**Figure 9.**Schematic representation of vitreous selenium indicating the various structural parameters used in the modelling procedure.

**Figure 11.**Comparison between the experimentally obtained structure factor (solid line) for vitreous selenium and the scattering pattern obtained from a model of (

**A**) a distorted helix, (

**B**) a flexible chain, (

**C**) six-member rings and (

**D**) eight-member rings (open circles).

**Figure 12.**Comparison between the different χ

^{2}test results for a model of selenium based on the disordered helix as a function of chain length (indicated by the number of atoms in the model) and the different areas of the experimentally observed structure factor. Q

_{max}= 30 Å

^{−1}indicates the use of the entire scattering curve, Q

_{max}= 5 Å

^{−1}indicates the use of the first peak of the scattering curve and Q

_{max}= 10 Å

^{−1}indicates the use of the first two peaks of the scattering curve. In all cases, we can see that the statistical test becomes invariant for a model of at least 100 selenium atoms.

**Figure 13.**Effect of the carbon–deuterium bond on the overall picture of the diffraction pattern. The best-fit distance of 1.13 Å (short-dashed line) shows a significant resemblance to the experimental data (solid line) when compared with two artificial C-D values of 1 Å (dotted line) and 1.5 Å (short dashed line). All the curves have been shifted for clarity. The image has been reproduced with permission from Reference [6] © American Chemical Society.

**Figure 14.**Experimentally observed structure factor (left) and radial distribution function (right) as obtained from a simultaneous small and wide angle neutron scattering experiment in NIMROD. The different length scales associated with the different regions of the functions is highlighted. The inset in the right figure indicates the time evolution of the radial distribution function with temperature.

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Gkourmpis, T.; Mitchell, G.R.
The Use of Scattering Data in the Study of the Molecular Organisation of Polymers in the Non-Crystalline State. *Polymers* **2020**, *12*, 2917.
https://doi.org/10.3390/polym12122917

**AMA Style**

Gkourmpis T, Mitchell GR.
The Use of Scattering Data in the Study of the Molecular Organisation of Polymers in the Non-Crystalline State. *Polymers*. 2020; 12(12):2917.
https://doi.org/10.3390/polym12122917

**Chicago/Turabian Style**

Gkourmpis, Thomas, and Geoffrey R. Mitchell.
2020. "The Use of Scattering Data in the Study of the Molecular Organisation of Polymers in the Non-Crystalline State" *Polymers* 12, no. 12: 2917.
https://doi.org/10.3390/polym12122917