Article pubs.acs.org/JPCB
Cite This: J. Phys. Chem. B 2017, 121, 9753-9759
The Effects of Terminal Groups on Elastic Asymmetries in Hybrid Molecular Materials Joseph A. Burg and Reinhold H. Dauskardt* Department of Materials Science and Engineering, Stanford University, Stanford, California, United States S Supporting Information *
ABSTRACT: An asymmetric elastic modulus is a recently discovered and unexpected property of hybrid molecular materials that has significant implications for their underlying thermomechanical reliability. Elastic asymmetries are inherently related to terminal groups in the molecular structure, which limit network connectivity. Terminal groups sterically interact to stiffen the network in compression, while they disconnect the network and interact significantly less in tension. Here we study the importance of terminal group molecular weight and size (OH, methyl, vinyl, and phenyl) on the resulting elastic asymmetries and find that increasing the terminal group size actually leads to even larger degrees of asymmetry. As a result, we develop a molecular design criterion to predict how molecular structure affects the mechanical properties, a vital step toward integrating hybrid molecular materials into emerging nanotechnologies.
1. INTRODUCTION
terminal groups in the underlying molecular architecture with asymmetric elastic behavior. We define an asymmetric elastic modulus, which is derived from atomic- and molecular-scale phenomena, as having different compressive, EC, and tensile, ET, moduli and report values in terms of the ratio EC/ET, the degree of elastic asymmetry. Elastic and thermal expansion asymmetries are related to the existence of terminal chemical groups, which reduce the network connectivity. Terminal groups sterically interact to stiffen the network in compression, whereas they disconnect the network and interact significantly less in tension. The strength and stiffness of these generally brittle materials scale with density, which is controlled by the underlying molecular structure and network connectivity.8−11 However, designing hybrid glasses to maintain high levels of mechanical strength and stiffness remains a significant challenge that can limit their integration into emerging nanotechnologies. Before more mechanically robust hybrids can be designed, their underlying thermomechanical asymmetries must be understood. To accurately model hybrid molecular materials, we use molecular dynamics-based simulated annealing with LAMMPS to generate large, distortion-free networks.12,13 The molecular materials investigated include an ethane-bridged oxycarbosilane (Et-OCS) with a variety of terminal groups: methyl (Et-OCSmethyl), vinyl (Et-OCS-vinyl), and phenyl (Et-OCS-phenyl),
Hybrid organic−inorganic materials possess novel properties and functionalities by combining organic and inorganic species at the molecular scale. Due to the presence of terminal chemical groups that limit network connectivity, hybrid molecular materials can have a marked asymmetric elastic modulus1 (wherein the compressive, EC, and tensile, ET, moduli are different), which has significant implications for thermomechanical reliability2 and can ultimately limit their integration into emerging nanotechnologies including dielectrics in microelectronics, antireflective (AR) coatings, protective coatings in flexible electronics, and molecular sieves for biosensing.3−7 Hybrid materials may contain a range of network terminations that vary in molecular weight and size. These terminal groups affect the material’s density, which controls their interaction distance and thus may also affect the resulting elastic asymmetries. In this work, we explore a range of terminal groups (OH, methyl, vinyl, and phenyl) using a consistent molecular architecture to study their effects on the elastic asymmetries. Since the hybrid material density decreases with increasing terminal group size, it was expected that molecular architectures with large terminal groups should result in reduced elastic asymmetries. Contrary to this expectation, we show that hybrid molecular materials with large terminal groups are dense enough to allow significant steric interactions, and as a result, we show that increasing the terminal group size actually increases the degree of elastic asymmetry even further. Thus, we develop a new design principle that connects more complex © 2017 American Chemical Society
Received: September 27, 2017 Revised: October 4, 2017 Published: October 4, 2017 9753
DOI: 10.1021/acs.jpcb.7b09615 J. Phys. Chem. B 2017, 121, 9753−9759
Article
The Journal of Physical Chemistry B
represent mSi as a function of both metrics. For the Et-OCS model, mSi is given by
which represent state-of-the-art molecular-reinforced hybrid films (Figure 1a).14 Each model has a range of terminal OH
mSi = 1 + 3q
(3)
while, for the Et-OCS-methyl, Et-OCS-vinyl, and Et-OCSphenyl models, mSi is given by mSi = 1 + 2.5q
For a given condensation degree, the network connectivity and resulting elastic properties can be significantly degraded by introducing terminal groups into the molecular structure. It is critical to distinguish between chemical coordination number and mean silicon network connectivity, mSi. Chemical coordination number refers to the number of bonds (covalent, ionic, etc.) in the first coordination sphere of an atom. In our case, these are covalent bonds. Regardless of the molecular model in our work, all silicon atoms have a chemical coordination number of four with four covalent bonds. However, for amorphous materials, chemical coordination is not sufficient to describe the elastic properties due to the possibility of terminal chemical groups that decrease the stiffness of the network. Thus, constraint and bond percolation theory are used to model the number of degrees of translational freedom and bond constraints for each atom in the covalent network.18−24 In our work, mSi is the average number of bond constraints for a silicon atom in the covalent molecular network. Constraint and bond percolation theories have been shown to predict observed material properties in addition to the increase in material properties with increasing bond constraints (eq 1).17,25−31 We simulated the bulk modulus by incrementally applying hydrostatic pressure and measuring the resulting volumetric strain (the dilatation of the simulation cell) and subsequently convert the bulk modulus to the elastic modulus for an isotropic material. Defining stress and pressure from an atomistic perspective is difficult and thus has been a topic of controversy.32,33 The Cauchy stress was developed for the continuum scale, while the virial stress was developed for the atomic scale. In the present work, we consider the ensemble average of the virial stress to be equivalent to the Cauchy stress. Using our simulated annealing technique, we generated distortion-free networks for Et-OCS, Et-OCS-methyl, Et-OCSvinyl, and Et-OCS-phenyl and measured the elastic properties. We show that increasing the size of the terminal groups decreases the density and stiffness of the material (Figure 1a). We plot the stiffness in terms of the average elastic modulus, Eavg, which is the average of the compressive, EC, and tensile, ET, moduli. Our results are consistent with the literature, wherein the elastic properties of hybrid materials increase with density.3,14,34 We approximated the volume of each terminal group based on each atom’s van der Waals radius and show that, for increasing terminal group volume, Eavg decreases (Figure 1b). This is consistent with results where increased size of terminal groups degrades the mechanical properties of hybrid molecular materials.35 2.2. The Asymmetric Elastic Modulus. Hybrid molecular materials can have a marked asymmetric elastic modulus that is due to terminal chemical groups that reduce network connectivity: terminal groups sterically interact to stiffen the network in compression, while they interact less and disconnect the network in tension. We show an example visualization of terminal phenyl groups that can sterically interact in
Figure 1. Increasing the size of terminal groups degrades the elastic properties. (a) The average elastic modulus, Eavg, is plotted with respect to density for the Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, and Et-OCS-phenyl models at a condensation degree of q = 0.75. (b) The average elastic modulus, Eavg, is plotted with respect to the volume of the terminal group on each molecular structureOH (Et-OCS), methyl (Et-OCS-methyl), vinyl (Et-OCS-vinyl), and phenyl (Et-OCSphenyl)at a condensation degree of q = 0.75. The error bars denote one standard deviation of the mean of three trials in parts a and b.
groups depending on the network connectivity. Unlike existing methods for generating model hybrid molecular materials,15,16 we do not define the network topology prior to structural relaxation in addition to having the ability to control the network connectivity.
2. RESULTS AND DISCUSSION 2.1. Molecular Architecture and Elastic Properties. In hybrid molecular materials, network connectivity is a fundamental parameter that controls the elastic properties and depends on the condensation degree, q, and the molecular structure.1,8 The condensation degree has a maximum value of 1 and is a measure of the fraction of possible Si−O−Si bonds that form. However, it ignores the influence that bridging and terminal groups have on the network connectivity, which are accounted for in the molecular structure. Thus, we define the Si−X−Si (where X = O or C−C) connectivity, p, to equal the fraction of bridging Si−X−Si bonds per silicon atom that are present compared to fully connected SiO2. This metric provides an absolute measure of network connectivity. The network connectivity of a typical hybrid material without built-in terminal groups using NMR is p ∼ 0.86.8 The elastic modulus, E, and bulk modulus, K, have a superlinear dependence on the network connectivity8,17 E , K ∝ (p − pc )n
(1)
where n is greater than 1 and pc is the percolation threshold. We define the mean Si network connectivity, mSi, as mSi = CNSi × p
(4)
(2)
where CNSi is the silicon network coordination and CNSi = 4 for the molecular models used in this study. The mean silicon network connectivity, mSi, is the number of silicon nearest neighbors that a silicon atom is connected to via various linear bridges such as oxygen, methylene, ethylene, or other short organic chains. Importantly, network connectivity describes the material’s structure and is based on the bonding within the molecular network. Since mSi depends on the condensation degree, q, and intrinsic molecular structure, we can explicitly 9754
DOI: 10.1021/acs.jpcb.7b09615 J. Phys. Chem. B 2017, 121, 9753−9759
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The Journal of Physical Chemistry B compression given that they are situated close enough together (Figure 2a).
Γmax =
To predict the degree of elastic asymmetry, we previously developed a model that computes the contribution of steric interactions to the stiffness of the material.1 First, we find terminal groups that can sterically interact (i.e., terminal group clusters) and bound the number of steric interactions in each cluster. At a minimum, each terminal group interacts with only one neighboring group (a linear chain model, γmin), while, at maximum, all pairs of terminal groups in a cluster interact (a maximum interaction model, γmax) γi = ni − 1 (5) min
ni(ni − 1) (6) 2 where ni is the number of terminal group atoms in the ith cluster. We then define stiffening coefficients (Γmin, Γmax) that sum all the steric interactions and provide bounds on the influence of steric interactions on the degree of elastic asymmetry max
=
Γmin =
β ∑i γi
min
+ NT − M + NB NB
max
+ NT − M + NB NB
(8)
where β is the relative stiffness between a steric interaction and a molecular bond in the network, NT−M is the number of steric interactions between terminal groups and any other repulsive interaction in the surrounding matrix, and NB is the number of molecular bonds in the network. For p < 0.75, we previously found β ∼ 0.33, while, for p > 0.75, β ∼ 1.0.1 The steric interactions between terminal groups (γmin and γmax) mainly contribute to Γmin and Γmax, while NT−M has a marginal contribution. Fitting the set of stiffening coefficients bounds the degree of elastic asymmetry, Γmin < EC/ET < Γmax.1 2.3. Dependence of Elastic Asymmetries on Network Connectivity. We have shown that network connectivity is a fundamental parameter that controls the elastic properties and degree of elastic asymmetry in hybrid molecular materials. In Figure 2b, we plotted the elastic modulus against the Si−X−Si connectivity, p, for the model glasses (Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, Et-OCS-phenyl) and show that each model displays an asymmetric elastic modulus. Since the molecular architectures of the Et-OCS-methyl, Et-OCS-vinyl, and EtOCS-phenyl have built in terminal groups, the maximum network connectivity, pmax = 0.875, and percolation threshold, pc ∼ 0.475, are different from Et-OCS, where pmax = 1.0 and pc ∼ 0.6 (eqs 1−4). The superlinear relationship between network connectivity and the elastic modulus is observed for each model network, and we see that the elastic properties decrease with increasing terminal group size. Since the rigid regions of the network do not percolate below pc, the compressive and tensile moduli are the same at pc. At values above pc, the network has percolated and the degree of elastic asymmetry, EC/ET, increases to a maximum value as high as ∼31, ∼36, ∼39, and ∼66% for the Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, and Et-OCS-phenyl models, respectively (Figure 2c). As the network achieves full connectivity, the degree of asymmetry decreases, since more terminal groups become isolated and interact less. Importantly, increasing the size of the terminal groups increases the maximum degree of elastic asymmetry (Figure 2d). This key result informs new materials design: the size of terminal groups in the intrinsic molecular structure will affect the underlying elastic asymmetries, which ultimately affects the thermomechanical reliability. To predict the degree of elastic asymmetry for each model network, we computed stiffening coefficients (Γmin and Γmax) on the equilibrated structures (i.e., at zero loading and atmospheric pressure) for each trial. In our model, we assume that steric interactions enhance the modulus in compression and have no effect in tension. In Figure 3, we show a leastsquares fit of the stiffening coefficients (Γmin in red and Γmax in blue) for each model, which bound the degree of elastic asymmetry: Γmin < EC/ET < Γmax. An important consideration when establishing the connection between terminal group size and degree of elastic asymmetry is that the material must be dense enough for terminal groups to sterically interact. Since the models in our study (Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, and Et-OCSphenyl) have the same molecular architecture in terms of the ethylene-bridge organosilicate, the densities have a similar scaling with network connectivity (Figure S1). As a result, the terminal groups are close enough to sterically interact. If, for example, large terminal groups (such as phenyl groups) were integrated into a flexible, low-density molecular architecture,
Figure 2. Larger terminal groups result in higher degrees of elastic asymmetry. (a) A visual representation of terminal phenyl groups in the Et-OCS-phenyl matrix. (b) The relation between network connectivity, p, and the elastic modulus under both compressive and tensile loading for the Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, and EtOCS-phenyl models. The percolation threshold, pc, and maximum network connectivity, pmax, are pc = 0.475 and pmax = 0.875 (illustrated by the dashed vertical line) for the Et-OCS-methyl, Et-OCS-vinyl, and Et-OCS-phenyl models and pc = 0.6 and pmax = 1.0 for the Et-OCS model. (c) The degree of elastic asymmetry, EC/ET, as a function of network connectivity, p, for each molecular model. (d) The maximum degree of elastic asymmetry, EC/ET, with respect to the volume of the terminal group in each molecular model: OH (Et-OCS), methyl (EtOCS-methyl), vinyl (Et-OCS-vinyl), and phenyl (Et-OCS-phenyl). The error bars denote one standard deviation of the mean of three trials in parts b−d.
γi
β ∑i γi
(7) 9755
DOI: 10.1021/acs.jpcb.7b09615 J. Phys. Chem. B 2017, 121, 9753−9759
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The Journal of Physical Chemistry B
Figure 3. The degree of elastic asymmetry with respect to network connectivity can be modeled by counting the number of steric interactions. The degree of elastic asymmetry, EC/ET, as a function of network connectivity, p, lies between bounding models defined by a minimum stiffening coefficient, Γmin (red), and a maximum stiffening coefficient, Γmax (blue), for the (a) Et-OCS, (b) Et-OCS-methyl, (c) Et-OCSVinyl, and (d) Et-OCS-phenyl models. The error bars denote one standard deviation of the mean of three trials in parts a−d.
Figure 4. Nanoscale porosity decreases the degree of elastic asymmetry. (a1) A visual representation of a porogen molecule. (a2, a3) The porogen molecules form a network within the surrounding molecular matrix. (a4) The porogen molecules are removed from the matrix (mimicking porogen burnout) to introduce nanoscale porosity. (b) Introducing nanoscale porosity decreases both the compressive and tensile moduli for the Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, and Et-OCS-phenyl models. (c) The terminal groups for each respective molecular model increasingly situate at internal pore surfaces for increasing volume percentage porosity. (d) The degree of elastic asymmetry, EC/ET, as a function of volume percentage porosity for the molecular model. The error bars denote one standard deviation of the mean of three trials in parts b−d.
then the density would be such that the terminal groups would not necessarily be close enough or confined enough to sterically interact. A key distinction between intrinsically terminated (Et-OCSmethyl, Et-OCS-vinyl, and Et-OCS-phenyl) and nonintrinsically terminated (Et-OCS) molecular models is that the degree of asymmetry in the nonintrinsically terminated models theoretically trends to EC/ET = 1.0 at p = pmax, since no terminal OH groups exist in the molecular network. However, for the intrinsically terminated models, the terminal groups are built into the molecular architecture, and as a result, the degree of asymmetry is not necessarily EC/ET = 1.0 at p = pmax. This is evident from the value of EC/ET at connectivities near pmax, wherein the intrinsically terminated models (Figure 3b−d) are much more asymmetric than the nonterminating models (Figure 3a) at similar connectivity values near pmax. 2.4. Nanoscale Porosity Decreases Elastic Asymmetry Regardless of Molecular Architecture. At nanoscale dimensions, porosity is a fundamental structural parameter that contributes to the free volume of surrounding atoms. While the incorporation of nanoporosity has application benefits such as a reduced dielectric constant or a controlled refractive index, it also severely degrades the elastic modulus, which can ultimately limit the integration of these materials into emerging technologies.36−39 Hence, it is vital to study how porosity not only affects the elastic modulus but also the elastic asymmetries, which are fundamental to the thermomechanical reliability.2 To model nanoporous materials, we perform a simulated anneal procedure with the addition of model porogen molecules (Figure 4a1), which form a percolated network (Figure 4a2 and a3). After a low energy structure is achieved, the porogen molecules are removed to mimic porogen burnout
(Figure 4a4). We introduced nanoscale porosity into the EtOCS, Et-OCS-methyl, Et-OCS-vinyl, and Et-OCS-phenyl structures and computed the moduli under compressive and tensile loading, and see that porosity degrades the mechanical properties (Figure 4b). For each molecular model, the Si−X−Si network connectivity remained constant regardless of volume percentage porosity (Figure S2a), while the densities each follow as similar scaling with volume percentage porosity (Figure S2b). Terminal groups preferentially situate at internal pore surfaces and deplete the density of terminal groups within the surrounding hybrid matrix (Figure 4c). The OH terminations in each model network (Et-OCS, Et-OCS-methyl, Et-OCSvinyl, Et-OCS-phenyl) follow similar probabilities of situating at internal pore surfaces with increasing volume percentage porosity, and ∼80% of terminal OH groups are on internal pore surfaces at 40% porosity (Figure S3a). However, depending on the molecular architecture, precursors with larger terminal groups have a higher probability of the terminal groups situating on a pore surface with increasing volume percentage porosity (Figure S3b). Thus, larger terminal groups increase the probability that the group will situate at an internal pore surface (Figure S4). With increasing volume percentage porosity, the number of possible steric interactions decreases (regardless of the precursor size), which decreases the degree of elastic asymmetry, EC/ET, for each model network (Figure 4d). We computed the stiffening coefficients (Γmin, Γmax) from eqs 7 and 8, respectively, and show that our model for the degree of 9756
DOI: 10.1021/acs.jpcb.7b09615 J. Phys. Chem. B 2017, 121, 9753−9759
Article
The Journal of Physical Chemistry B
many more. For further studies and the successful integration into current and emerging device technologies, it is vital to understand and control the underlying asymmetric thermomechanical properties in hybrid molecular materials.
elastic asymmetry nicely bounds the simulated values in the nanoporous Et-OCS, Et-OCS-methyl, Et-OCS-vinyl, and EtOCS-phenyl networks (Figure 5). At low volume percentage
4. MATERIALS AND METHODS 4.1. Interatomic Potential. While a number of interatomic potentials have been developed to study sol−gel silicate materials and organosilicate molecules, they struggle to assemble highly connected, dense hybrid networks with controlled network connectivities.40−44 We use a three-body potential to generate the model Et-OCS, Et-OCS-methyl, EtOCS-vinyl, and Et-OCS-phenyl hybrid materials.45 Hydrogen atoms were modeled implicitly using a united atom approach. For the bonded precursors, we used harmonic potentials to model the bonds and angles and the OPLS and COMPASS potentials to model the dihedral angles (Supporting Information).46,47 The torsional parameters were determined by fitting to rotational energy profiles obtained from ab initio molecular orbital calculations at the RHF/6-31G level using Gaussian (Figure S5). The potential parameters are provided in Table S1−S6. To allow Si−O bonds to form during the simulations, we parametrized a Stillinger−Weber potential,48 since we could control the Si−O bond length with the two-body term and the O−Si−O, Si−O−Si, and O−Si−C bond angles with the threebody term.8 To calibrate our potential, we synthesized hybrid glasses and used the NMR spectra to match the condensation degrees and network connectivities, which we describe elsewhere.8 The resulting radial distribution functions (for bond lengths) and bond angle distributions show good quantitative agreement with experimental values. The equilibrated densities in our simulations are within 5% of the experimental values. Finally, the average elastic modulus for the Et-OCS model was verified using SAWs.8,34,49 4.2. Model Network Connectivity. We vary the O:Si ratio within the simulation cell to control the network connectivity for each model network (Figure S6). The minimum energy configuration of the interatomic potential was achieved when all possible Si−O bonds were formed. When the O:Si ratio exceeds the stoichiometric ratio for a fully condensed network (e.g., 1.5 for Et-OCS), then nonbridging oxygen (i.e., terminal OH) atoms are present, which reduces the network connectivity. 4.3. Simulated Annealing. The simulated annealing procedure was done with NPT molecular dynamics simulations in LAMMPS13 and involved linearly cooling the simulation cell from 12,000 to 6000 K over 10,000 1 fs timesteps and then quenching the simulation cell from 6000 to 300 K over 100,000 1 fs timesteps. Networks with increased network connectivity had a higher density than less connected networks. The pressure profile was chosen such that the final pressure-free density was within 5% of the experimental values.8,34 The initial pressure was 9 GPa for the simulations and linearly decreased during the simulation until atmospheric pressure was reached. The simulation cell dimensions of the equilibrated structures varied from 4.8 to 7.2 nm. 4.4. Bulk Modulus. Using a NPT molecular dynamics simulation, hydrostatic pressure was incrementally applied to the simulation cell, where each step was averaged over 20,000 1 fs timesteps. The bulk modulus was determined by the slope of a least-squares fit of the average pressure with respect to the average volumetric strain. The simulation cell was loaded with pressures between (0 GPa, 0.5 GPa) and (−0.5 GPa, 0 GPa)
Figure 5. The degree of elastic asymmetry with respect to nanoscale porosity can be modeled by counting the number of steric interactions. The degree of elastic asymmetry, EC/ET, as a function of volume percentage porosity lies between bounding models defined by a minimum stiffening coefficient, Γmin (red), and a maximum stiffening coefficient, Γmax (blue), for the (a) Et-OCS, (b) Et-OCS-methyl, (c) Et-OCS-vinyl, and (d) Et-OCS-phenyl models. The error bars denote one standard deviation of the mean of three trials in parts a−d.
porosities (
