# Structure and Dynamics of Highly Attractive Polymer Nanocomposites in the Semi-Dilute Regime: The Role of Interfacial Domains and Bridging Chains

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{inter}, defined as the minimum surface-to-surface separation between neighboring nanoparticles. Clearly, when d

_{inter}becomes comparable to the size of polymer chains, interesting phenomena should be expected due to the interplay between adsorbed interfacial layers, alterations in polymer conformation, and development of an underlying microscopic structure (topological network) by bridging chains that link different nanoparticles. Baeza et al. [15] examined the collective dynamics of poly(2-vinyl pyridine) (P2VP) melts filled with spherical silica nanoparticles and observed the transition to a gel-like dynamics for nanoparticle loadings above a critical value where d

_{inter}becomes comparable to the size of polymer chains; this favors the formation of a bridge-like network in the nanocomposite, with nanoparticles acting as nodes and P2VP chains as strands connecting different nanoparticles. The authors also reported that at extreme loadings of the matrix in silica nanoparticles, the temperature dependence of microscopic dynamics changes from a Williams-Landel-Ferry (WLF) to an Arhenius-like one. This was attributed to the fact that at such large volume fractions, the polymer bound layers that develop around nanoparticles start to percolate, thus dominating the network-like character of the nanocomposite at medium or lower silica concentrations.

_{inter}becomes smaller than the polymer Kuhn length).

_{3}- to OH- terminal units.

_{60}) molecules of size considerably smaller than that of silica, the polymer density in the vicinity of C

_{60}increased significantly.

_{p}/R

_{g}ratio (R

_{p}stands for the nanoparticle radius and R

_{g}for the polymer radius-of-gyration) affects the conformational properties and found that, when R

_{p}/R

_{g}< 1, polymer chains significantly expand [43], especially at higher nanoparticle volume fractions.

_{inter}(e.g., by increasing the silica volume fraction) favors the formation of a bridge-like network, since more chains can adsorb on the nanoparticles or more chains can create bridges between nanoparticles.

## 2. Systems Studied and Simulation Details

^{−1}) terminated with hydroxyl groups (i.e., PEG chains). Eight different model systems were examined, denoted as systems 1−8, with system 1 corresponding to the pure PEG melt and systems 2−8 to the PEG-silica nanocomposites. The molecular characteristics of these systems (number of PEG chains and total number of interacting units, nanoparticle diameter, nanoparticle volume fraction, and surface concentration of nanoparticles in silanol groups) and abbreviations concerning their notation can be found in Table 1. For example, with the name d5_v15_PEG we mean the nanocomposite that consists of PEG chains (i.e., chains terminated with hydroxyl groups) and contains silica nanoparticles with diameter d = 5 nm at volume fraction v = 15 v/v%. Detailed information concerning the construction of the amorphous silica nanoparticles and their surface concentration in silanol moieties (SiOH, Si(OH)

_{2}and Si(OH)

_{3}groups) can be found in ref. [21]. Initial structures were built by placing one silica particle in the simulation box and filling the remaining volume with PEG chains at the desired volume fraction v (in the range from 15 to 35 v/v%). By construction, our polymer nanocomposites are characterized by an ideal dispersion of nanoparticles in the polymer matrix, thus phenomena related with non-uniformities in nanoparticle dispersion or with nanoparticle agglomeration are absent. To examine the effect of the nature of the terminal groups on the structure of the network formed, we also simulated a system that is identical to the d5_v15_PEG one in Table 1, except that the hydroxyl terminal groups in the host polymer chains have been replaced by methoxy ones (system d5_v15_PEO in Table 1). A typical atomistic configuration from the atomistic simulations with system 4 (system d9_v15_PEG in Table 1) is shown in Figure 1.

## 3. Results

#### 3.1. Local Density

^{−3}, which agrees well with previously reported experimental [55] and simulation [53] data. At the same thermodynamic conditions, all nanocomposites are characterized by higher densities whose values vary linearly with the concentration of the nanocomposite in silica nanoparticles. At fixed silica volume fraction, on the other hand, the density of the nanocomposites seems to be independent of the size of nanoparticles.

^{−3}). More specifically, the local polymer mass density increases rapidly with increasing distance from the silica surface, reaches a maximum at around 1.8 Å, and then levels off approaching asymptotically the density of pure PEG at the same conditions. For systems d5_v15_PEG and d13_v35_PEG, at even larger distances from the nanoparticle surface, the mass density ρ decreases further. This happens because in these systems, the distance between neighboring silica nanoparticles is smaller (Table 3), thus the probability of finding another nanoparticle (instead of bulk polymer mass) next to a reference nanoparticle is higher. An interesting point in all cases discussed in Figure 2 is that the polymer mass density starts to increase at a distance r smaller than r = 0 (denoting the surface of the nanoparticle), which can be explained by the fact that the silica nanoparticles used in our simulations contain cavities on their surface that are filled by PEG segments, thus allowing for polymer mass to be observed even at distances smaller than the average nanoparticle radius.

#### 3.2. Structure and Conformation of Adsorbed and Free PEG Chains

_{pol-pol}), and hydrogen bonds that form between silica and matrix PEG chains (denoted as HB

_{pol-sil}). In all cases, the number of hydrogen bonds computed is normalized with the total number of polymer chains in the system.

_{pol-pol}in the nanocomposites decreases compared to their value in bulk PEG (= 0.69 hydrogen bonds per chain) due to the preference of the PEG terminal hydroxyl groups to form hydrogen bonds with the silica oxygen atoms (and not with other polymer chains in the matrix). This tendency is more pronounced in the systems with smaller nanoparticles and/or higher silica nanoparticle concentrations. An explanation for this is the higher fraction of adsorbed PEG chains in these systems (Table 3), implying that more PEG chains come into contact with silica nanoparticles. On the other hand, the population of HB

_{pol-sil}increases with decreasing nanoparticle diameter and increasing silica content (which is compatible with the larger number of polymer chains coming into contact with silica).

_{COCC}) and OCCO (φ

_{OCCO}) along their backbone. The respective MD predictions are presented in Figure 3 and Figure 4, with parts (a) and (b) showing the dependence (for dihedrals in adsorbed and free chain segments, respectively) on silica size, and parts (c) and (d) showing the corresponding dependence on silica loading. Dihedral angles referring to mixed (adsorbed and free) sequences of atoms were not considered in our analysis.

#### 3.3. Static Structure Factor

_{α}and x

_{β}denote the number fraction of α-type and β-type atoms in the system, f

_{α}and f

_{β}the respective scattering factors, and g

_{αβ}(r) the total pair distribution function. The values of the scattering factors used in the present work for carbon, hydrogen, and oxygen (the atoms appearing in a PEG chain) can be found in ref [58]. By taking the Fourier transform of the function H(r),

#### 3.4. Single Chain form Factor

_{ch}and N denote the number of PEG chains in the system and the number of atoms in each chain, f

_{i}and f

_{j}the respective scattering factors, and r

_{ij}the distance between atoms i and j along the same chain. The brackets on the right-hand side indicate a configurational average over all PEG chains in the nanocomposite (adsorbed and free). For the pure PEG melt, the MD results for the single chain form factor can be compared with the analytical expression derived by Burchard and Kajiwara [60] for the random-flight model, which assumes randomly oriented segments along the chain:

_{K}denotes the number of Kuhn segments per chain and b the Kuhn length (equal to 5.95 Å for the PEG matrix chains studied here). For very long Gaussian chains, the single chain form factor is also described by the Debye function [61]:

_{g}denotes the mean radius of gyration for the chains. The single chain form factor calculated by the MD simulations (Figure 6a) is in excellent agreement with the predictions of the random-flight model. Although the Debye equation is valid for quite long chains, its predictions are also in good agreement with the MD results except from some slight deviations at higher q values.

^{2}P(q) curves versus q. Figure 6b shows the dependence on silica diameter and Figure 6c the dependence on silica volume fraction. No deviations are observed at low q values. At higher q values, on the other hand, deviations are observed which are correlated with the fraction of adsorbed PEG chains on the silica nanoparticles and the resulting alterations in the structure of adsorbed polymer chains. The deviations become more pronounced as the size of the silica nanoparticles decreases (at fixed volume fraction) or as their volume fraction in the nanocomposite increases (i.e., as the relative fraction of adsorbed PEG chains increases).

#### 3.5. Network of Nanoparticle Bridging Chains

_{inter}. The values of d

_{inter}for each of the six nanocomposite melts studied in the present work have been listed in Table 3. Clearly, with decreasing nanoparticle size and increasing volume fraction, d

_{inter}decreases, thus favoring the formation of a network between nanoparticles.

_{b}) formed as a function of silica diameter and silica volume fraction (the results are shown normalized with the available surface nanoparticle area for adsorption in each melt, see third column in Table 3). Because of the use of only one nanoparticle in our simulations, to compute the number of polymer bridges we identified those PEG chains that adsorb simultaneously to the nanoparticle inside the primary cell and to its images in the 26 neighboring cells. According to Figure 7a, a decrease in the size of the silica nanoparticle causes a significant increase in the number of bridges formed in the melt; this implies that, at fixed silica loading, smaller particles enhance network formation, which agrees perfectly with the experimental studies for P2VP/silica nanocomposites [16]. The same holds if one increases the volume fraction (Figure 7b) because of the smaller d

_{inter}values characterizing nanocomposites containing higher silica concentrations, thereby making it easier for PEG chains to extend out from a given nanoparticle and to adsorb to another one.

_{inter}for each nanocomposite system (Table 3), whereas the dotted lines denote the $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ values of PEG chains in their own melt (Table 5). Surprisingly, apart from those nanocomposites where d

_{inter}is smaller than the $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ of the pure PEG (systems 2, 3, 6 and 7), and thus a network is expected to develop, PEG chains appear to form bridges even in the nanocomposites where $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ is considerably larger than its bulk value (e.g., systems 4 and 5). In all cases, bridging PEG chains assume highly extended conformations as they are simultaneously adsorbed on different nanoparticles, a behavior which is more pronounced in the systems containing larger nanoparticles or characterized by lower nanoparticle concentrations (i.e., when d

_{inter}increases). Interestingly, for the system with the larger silica nanoparticle and the lowest volume fraction (d13_v15_PEG), the value of $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ of bridging chains is almost two times larger compared to the unperturbed polymer size. Typical atomistic snapshots of the network of bridging chains formed in systems 2 and 4 are depicted in Figure 9, revealing the denser structure of the network in the nanocomposite with the smaller-sized particles (i.e., d = 5 nm).

_{b}becomes almost five times smaller when methoxy groups serve as terminal units instead of hydroxyl groups (Figure 7c). This is clearly related with the different adsorption mechanisms of the two different polymers, as thoroughly discussed in a recent study [21]. Hydroxyl-terminated PEG chains tend to adsorb laterally on the silica surface thus forming graft-like conformations, which makes it easier for them to extend to longer distances and form a bridge with another silica nanoparticle. Methoxy-terminated chains, on the other hand, adsorb tightly onto the silica nanoparticle along their entire contour (they practically lie on the nanoparticle surface), a behavior which prohibits contacts with other nanoparticles. However, upon examining the size of bridging chains, it appears that, within statistical error, the value of $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ of PEG chains in the two nanocomposites is the same, implying that at fixed nanoparticle size and volume fraction and irrespective of the end units of the matrix chains, these should extend by the same factor in order to adsorb to another silica nanoparticle. That this nanoparticle network formation is less favored in the case of the d5_v15_PEO nanocomposite is further implied by the strong fluctuations seen in the $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$-vs.-t curve in Figure 8c for this system, which is due to the smaller number of PEO chains that bridge silica nanoparticles compared to PEG ones under exactly the same conditions of loading of the nanocomposite in nanoparticles (see also Figure 7c). To the best of our knowledge, the dependence of network formation on the nature of polymer terminal units has not been discussed in the literature before. The corresponding dependence on the MW of the polymer matrix chains will be the subject of a future contribution.

#### 3.6. Orientational Relaxation of Polymer Chains

**u**

_{ee}(t)·

**u**

_{ee}(0)⟩. The decay of ⟨

**u**

_{ee}(t)·

**u**

_{ee}(0)⟩ for the pure PEG melt and its silica-based nanocomposites as a function of nanoparticle size and nanoparticle loading is presented in Figure 10 and Figure 11, respectively. Part (a) of the Figures depicts the ACF curves for the entire population of PEG chains in the various nanocomposites; parts (b) and (c), on the other hand, present the respective ACF functions separately for adsorbed and free chains in the given system. In all cases, the corresponding relaxation curve of PEG chains in their own melt is shown with the green curve. From Figure 10a it becomes clear that the presence of the silica nanoparticles considerably slows down the orientational dynamics of polymer chains due to their strong attractive interactions with the nanoparticles. Decreasing the size of nanoparticles or increasing their concentration in the melt further slows down dynamics, which is related to the higher fraction of adsorbed PEG chains in the nanocomposites with smaller-sized nanoparticles or higher nanoparticle volume fractions (Table 3). As expected, orientational deceleration is more pronounced in the case of adsorbed PEG chains (Figure 10b); for these chains, the corresponding ACF curves deviate significantly from the corresponding bulk behavior as they tend to approach a plateau at long times implying a too slow relaxation to be tracked ergodically by the MD simulation. As far as the orientational relaxation of non-adsorbed chains is concerned (Figure 10c), although full relaxation in all cases is observed, the corresponding curves are above that corresponding to the pure polymer. This is more pronounced in the nanocomposites where nanoparticles have a smaller diameter or are present in the melt in higher concentrations, and is attributed to the polymer network formed that severely constrains the orientational motion of non-adsorbed PEG chains.

#### 3.7. Diffusive Behavior of Polymer Chains

#### 3.8. Dynamic Structure Factor

_{ch}denotes the total number of chains in the system, q is the magnitude of the scattering vector

**q**, f

_{i}

_{,n}and f

_{i}

_{,m}are the scattering factors of atoms n and m along the same chain i, R

_{i}

_{,nm}(t) is the magnitude of the displacement vector

**R**

_{i}

_{,nm}(t) =

**R**

_{i}

_{,n}(t) −

**R**

_{i}

_{,m}(t) between atoms n and m on chain i, and the brackets denote a configurational average. MD simulation results for the ratio S(q,t)/S(q,0) for the bulk PEG melt and its silica-based nanocomposite melts studied here are shown in Figure 14 revealing significant differences between the various systems, with the S(q,t)/S(q,0) spectra for the nanocomposites being well above those for the pure melt and decaying much slower. This is obvious not only for the small q value examined (q = 0.04 Å

^{−1}, open squares in Figure 14) reflecting the diffusive behavior of PEG chains, but also for the large one (q = 0.15 Å

^{−1}, open circles in Figure 14) reflecting local dynamics. The slow-down in the dynamics (practically at all length scales examined) is more pronounced in the nanocomposites with the smaller silica nanoparticles (Figure 14a) or the higher silica concentration (Figure 14b) (all other properties kept the same), which is fully consistent with the conclusions drawn from the analysis of orientational and diffusive dynamics already discussed in previous sections.

#### 3.9. Self-Intermediate Scattering Function

_{s}(q,t), which focuses on time-dependent spatial variations of the dynamics of single atoms. The self-intermediate scattering function, which is experimentally measured by adopting incoherent neutron scattering techniques, reflects the impact of interactions of single PEG atoms with the silica nanoparticles on their dynamics. From MD, F

_{s}(q,t) is calculated through the equation:

_{i}

_{,n}(t) is the magnitude of the displacement vector

**R**

_{i}

_{,n}(t) =

**R**

_{i}

_{,n}(t) −

**R**

_{i}

_{,n}(0) of atom n on chain i. The MD simulation results for F

_{s}(q,t) for the simulated nanocomposites are shown in Figure 15, including a direct comparison with those for the pure PEG melt. Qualitatively, the results are very similar to those for S(q,t)/S(q,0), revealing again strong deviations in the dynamics between nanocomposites and pure PEG melt. More specifically, the F

_{s}(q,t) curves for all nanocomposites are above the corresponding ones for the bulk PEG melt and decay much slower. This deceleration in the dynamics is true for both values of q studied: q = 0.04 Å

^{−1}(open squares in Figure 15) and q = 0.15 Å

^{−1}(open circles in Figure 15), reflecting the impact of PEG chain-silica nanoparticle interactions on the single atom dynamics at long and short scales, respectively. Dynamics is more suppressed in the nanocomposites containing smaller nanoparticles (Figure 15a) at fixed loading or nanoparticles (of given diameter) at higher concentrations (Figure 15b). These behaviors are consistent with all previous conclusions and can be explained by the respective alterations in the fraction of absorbed polymer chains and the number of PEG chains bridging different nanoparticles. The characteristic long relaxation time is defined as the time τ for which F

_{s}(q,τ)=1/e [13], and is denoted with the horizontal dashed orange line in Figure 15a,b. For q = 0.15 Å

^{−1}, in particular, Figure 15b indicates that dynamics slows down at long times; the F

_{s}(q,t) curves do not seem to drop to zero, which must be associated with the nanoparticle network that forms in the melt, mediated by bridging PEG chains.

#### 3.10. Thermal Expansion Coefficient and Isothermal Compressibility

_{p}) defined as:

_{T}) defined as:

_{p}and the red-colored ones to k

_{T}. Part (a) of the figure shows the dependence on nanoparticle diameter and part (b) the dependence on nanoparticle loading.

_{P}and k

_{T}predicted from the simulations are 9.07 × 10

^{−4}K

^{−1}and 8.73 × 10

^{−4}MPa

^{−1}, respectively, which agree quite well with experimentally measured ones under similar conditions: a

_{P}= 7.6 × 10

^{−4}K

^{−1}[62] and k

_{T}= 7.7 × 10

^{−4}MPa

^{−1}[55]. For the PEG-silica nanocomposites, our simulations reveal a significant decrease in both a

_{P}and k

_{T}, under all circumstances. Both thermodynamic quantities show a strong dependence on silica concentration with their values decreasing almost linearly with increasing nanoparticle volume fraction (Figure 16b). On the other hand, the dependence on the size of the silica nanoparticles (Figure 16a) is milder: a

_{P}is not affected much, whereas k

_{T}exhibits a mild decrease with decreasing nanoparticle diameter. As shown in Table 3, a decrease in the size of the silica nanoparticles causes an increase in the amount of PEG adsorbed on silica. These adsorbed regions are characterized by higher densities than the bulk ones (Figure 2) and are less compressible, and this explains why k

_{T}decreases slightly with decreasing nanoparticle diameter.

## 4. Conclusions

_{s}(q,t), quantities that can be accessed through state-of-the-art experimental techniques, for all simulated melts (pure PEG and PEG-silica nanocomposites).

_{P}remains practically constant while that of k

_{T}exhibits a small increase.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Typical atomistic configuration in box-relative coordinates of the d9_v15_PEG system (system 4 in Table 1) containing 750 PEG chains with degree of polymerization N = 41 and one spherical silica nanoparticle of diameter d = 9 nm in a cubic simulation cell with dimensions equal to 138 Å × 138 Å × 138 Å. Carbon, oxygen, hydrogen and silica atoms are shown with gray, red, white, and yellow color, respectively.

**Figure 2.**MD predictions (T = 413 K and p = 1 atm) for the variation of PEG mass density with radial distance from the surface of the silica nanoparticle in the studied nanocomposites, as a function of silica size (

**a**), and silica concentration (

**b**). The horizontal green dashed line indicates the density of pure PEG at the same thermodynamic conditions, while the perpendicular orange dashed line at r = 0 indicates the surface of the nanoparticle.

**Figure 3.**Distribution of the COCC backbone dihedral as obtained from the present npT MD simulations at T = 413 K and p = 1 atm for: (

**a**) adsorbed, and (

**b**) free chains in systems 2–4 (dependence on nanoparticle size). Parts (

**c**,

**d**) show the same distributions but for systems 5–7 (dependence on silica loading). In all cases, the green curve corresponds to the result obtained for the pure PEG melt (system 1) at the same thermodynamic conditions.

**Figure 5.**Simulation predictions for the static structure factor S(q) (T = 413 K and p = 1 atm) and dependence on: (

**a**) silica diameter, and (

**b**) silica volume fraction. In all cases, the green curve corresponds to the result obtained for the pure PEG melt (system 1) at the same thermodynamic conditions.

**Figure 6.**(

**a**) MD predictions for the single chain form factor, P(q), of pure PEG melt (T = 413 K and p = 1 atm), and the respective predictions provided by the random-flight model and the Debye equation. (

**b**) Kratky plots (q

^{2}P(q) versus q) for all systems studied (pure PEG and PEG-silica nanocomposites) and dependence on silica nanoparticle size. (

**c**) Kratky plots (q

^{2}P(q) versus q) for all systems studied (pure PEG and PEG-silica nanocomposites) and dependence on the volume fraction of nanoparticles.

**Figure 7.**Number of polymer bridges per silica nm

^{2}from the present npT MD simulations (T = 413 K and p = 1 atm) as a function of: (

**a**) nanoparticle diameter (systems 2–4), and (

**b**) nanoparticle volume fraction (systems 5–7). (

**c**) Time evolution of the number of bridges formed per silica nm

^{2}in the d5_v15_PEG and d5_v15_PEO nanocomposites.

**Figure 8.**Time evolution of the end-to-end distance $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ of bridging chains as a function of: (

**a**) nanoparticle size (systems 2–4), (

**b**) nanoparticle volume fraction (systems 5–7), and (

**c**) type of PEG chain terminal units (systems 2 and 8), as obtained from the present npT MD simulations (T = 413 K and p = 1 atm). The solid line in each graph represents the simulation data, the horizontal dashed line indicates the d

_{inter}value for the corresponding nanocomposite, and the green horizontal dotted line indicates the $\langle {R}_{\mathrm{ee}}^{2}\rangle {}^{0.5}$ value of PEG chains in their pure melt at the same thermodynamic conditions (Table 5).

**Figure 9.**Typical atomistic snapshots of the nanoparticle network formed in: (

**a**) the d5_v15_PEG, and (

**b**) the d9_v15_PEG nanocomposite (T = 413 K and p = 1 atm). With yellow, red, and white we show the silica atoms. With blue we indicate images of the silica nanoparticle in neighboring cells. Non-bridging chains are shown in red, white, and cyan, and bridging ones in green.

**Figure 10.**Decay of the ACF of the chain end-to-end unit vector in time (T = 413 K and p = 1 atm) in systems 1–4 (dependence on silica size). The results refer to: (

**a**) all chains, (

**b**) only adsorbed chains, and (

**c**) only free chains in the corresponding nanocomposites.

**Figure 12.**Mean-square displacement of the chains centers-of-mass as a function of time for: (

**a**) all PEG chains, (

**b**) only adsorbed PEG chains, and (

**c**) only free PEG chains (T = 413 K and p = 1 atm) in systems 1–4 (dependence on silica size).

**Figure 14.**MD-predicted S(q,t)/S(q,0)-vs.-t spectra (T = 413 K and p = 1 atm) and dependence on: (

**a**) silica size, and (

**b**) silica concentration. Results are shown for q = 0.04 Å

^{−1}(open squares) and q = 0.15 Å

^{−1}(open circles).

**Figure 15.**MD-predicted F

_{s}(q,t)-vs.-t spectra (T = 413 K and p = 1 atm) and dependence on: (

**a**) silica size, and (

**b**) silica concentration. Results are shown for q = 0.04 Å

^{−1}(open squares) and q = 0.15 Å

^{−1}(open circles). The orange dashed line indicates the value 1/e of the self-intermediate scattering function (attained when time equals the characteristic relaxation time).

**Figure 16.**MD predictions for the thermal expansion coefficient a

_{P}(black color) and isothermal compressibility k

_{T}(red color) of the simulated nanocomposites, and dependence on nanoparticle size (

**a**), and nanoparticle volume fraction (

**b**).

System | Abbreviation | Number of PEG Chains | Silanol Concentration (OH nm ^{−2}) | Volume Fraction (v/v%) | Silica Nanoparticle Diameter (nm) | Total Number of Interacting Atoms |
---|---|---|---|---|---|---|

1 | PEG | 1000 | - | - | - | 126,000 |

2 | d5_v15_PEG | 130 | 3.6 | 15 | 5 | 21,160 |

3 | d7_v15_PEG | 354 | 3.6 | 15 | 7 | 57,636 |

4 | d9_v15_PEG | 750 | 3.6 | 15 | 9 | 122,201 |

5 | d13_v15_PEG | 1799 | 4.2 | 15 | 12.8 | 305,887 |

6 | d13_v25_PEG | 975 | 4.2 | 25 | 12.8 | 202,063 |

7 | d13_v35_PEG | 600 | 4.2 | 35 | 12.8 | 154,813 |

8 | d5_v15_PEO | 130 | 3.6 | 15 | 5 | 21,940 |

System | ρ (g cm^{−3}) |
---|---|

PEG | 1.015 ± 0.001 |

d5_v15_PEG | 1.247 ± 0.004 |

d7_v15_PEG | 1.241 ± 0.002 |

d9_v15_PEG | 1.240 ± 0.002 |

d13_v15_PEG | 1.264 ± 0.001 |

d13_v25_PEG | 1.437 ± 0.001 |

d13_v35_PEG | 1.602 ± 0.001 |

**Table 3.**MD predictions for the fraction of adsorbed and free chains in the simulated nanocomposites (systems 2–7) at T = 413 K and p = 1 atm. In the third column of the table, we report the available adsorption surface area per PEG chain. The fourth column denotes the inter-particle distance d

_{inter}in each nanocomposite examined.

^{a}Data obtained from ref. [21].

System | Fraction of | Available Adsorption Surface per PEG Chain (nm^{2} chain^{−1}) | Interparticle Distance d_{inter} (nm) | |
---|---|---|---|---|

Adsorbed PEG Chains | Free PEG Chains | |||

d5_v15_PEG | 0.779 ± 0.021 | 0.221 ± 0.021 | 0.597 | 2.61 |

d7_v15_PEG | 0.601 ± 0.013 | 0.399 ± 0.013 | 0.434 | 3.63 |

d9_v15_PEG | 0.424 ± 0.008 | 0.576 ± 0.008 | 0.339 | 4.67 |

d13_v15_PEG | 0.346 ± 0.006 ^{a} | 0.654 ± 0.006 ^{a} | 0.285 | 5.68 |

d13_v25_PEG | 0.598 ± 0.005 | 0.402 ± 0.005 | 0.526 | 2.88 |

d13_v35_PEG | 0.781 ± 0.006 | 0.219 ± 0.006 | 0.855 | 1.20 |

**Table 4.**MD simulation predictions for the number of hydrogen bonds formed in the simulated PEG-silica nanocomposites per PEG chain present in the melt (T = 413 K and p = 1 atm). For the nanocomposites, separate results are reported for HB

_{pol-pol}and HB

_{pol-sil}.

System | Hydrogen Bonds Per Chain | ||
---|---|---|---|

Total | Polymer-Polymer (HB_{pol-pol}) | Polymer-Silica (HB_{pol-sil}) | |

PEG | 0.690 | 0.690 | - |

d5_v15_PEG | 1.416 | 0.559 | 0.857 |

d7_v15_PEG | 1.258 | 0.568 | 0.690 |

d9_v15_PEG | 1.073 | 0.621 | 0.451 |

d13_v15_PEG | 1.072 | 0.614 | 0.458 |

d13_v25_PEG | 1.394 | 0.557 | 0.838 |

d13_v35_PEG | 1.742 | 0.497 | 1.245 |

**Table 5.**MD predictions for the $\langle {R}_{\mathrm{ee}}^{2}\rangle $ of PEG chains (T = 413 K and p = 1 atm). As also explained in the text, for the nanocomposites, we also report the separate $\langle {R}_{\mathrm{ee}}^{2}\rangle $ values for adsorbed and free PEG chains.

^{a}Data obtained from ref. [21].

System | $\langle {R}_{\mathrm{ee}}^{2}\rangle $(Å^{2}) | ||
---|---|---|---|

Adsorbed PEG Chains | Free PEG Chains | All PEG Chains | |

PEG | - | - | 1454 ± 31 ^{a} |

d5_v15_PEG | 1518 ± 36 | 1379 ± 68 | 1479 ± 30 |

d7_v15_PEG | 1580 ± 29 | 1380 ± 41 | 1501 ± 25 |

d9_v15_PEG | 1556 ± 55 | 1415 ± 47 | 1475 ± 38 |

d13_v15_PEG | 1587 ± 42 ^{a} | 1443 ± 23 ^{a} | 1487 ± 18 ^{a} |

d13_v25_PEG | 1544 ± 40 | 1395 ± 55 | 1482 ± 30 |

d13_v35_PEG | 1495 ± 38 | 1346 ± 93 | 1432 ± 34 |

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**MDPI and ACS Style**

Skountzos, E.N.; Karadima, K.S.; Mavrantzas, V.G.
Structure and Dynamics of Highly Attractive Polymer Nanocomposites in the Semi-Dilute Regime: The Role of Interfacial Domains and Bridging Chains. *Polymers* **2021**, *13*, 2749.
https://doi.org/10.3390/polym13162749

**AMA Style**

Skountzos EN, Karadima KS, Mavrantzas VG.
Structure and Dynamics of Highly Attractive Polymer Nanocomposites in the Semi-Dilute Regime: The Role of Interfacial Domains and Bridging Chains. *Polymers*. 2021; 13(16):2749.
https://doi.org/10.3390/polym13162749

**Chicago/Turabian Style**

Skountzos, Emmanuel N., Katerina S. Karadima, and Vlasis G. Mavrantzas.
2021. "Structure and Dynamics of Highly Attractive Polymer Nanocomposites in the Semi-Dilute Regime: The Role of Interfacial Domains and Bridging Chains" *Polymers* 13, no. 16: 2749.
https://doi.org/10.3390/polym13162749