ARTICLE pubs.acs.org/JPCC
Bending of Layered Silicates on the Nanometer Scale: Mechanism, Stored Energy, and Curvature Limits Yao-Tsung Fu,† Gregory D. Zartman,† Mitra Yoonessi,‡ Lawrence F. Drummy,§ and Hendrik Heinz*,† †
Department of Polymer Engineering, University of Akron, Akron, Ohio 44325, United States Ohio Aerospace Institute, 21000 Brookpark Road, Cleveland, Ohio 44135, United States § Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright-Patterson AFB, Ohio 45433, United States ‡
bS Supporting Information ABSTRACT: Bending and failure of aluminosilicate layers are common in polymer matrices although mechanical properties of curved layers and curvature limits are hardly known. We examined the mechanism of bending, the stored energy, and failure of several clay minerals. We employed molecular dynamics simulation, AFM data, and transmission electron microscopy (TEM) of montmorillonite embedded in epoxy and silk elastin polymer matrices with different weight percentage and different processing conditions. The bending energy per layer area as a function of bending radius can be converted into force constants for a given layer geometry and is similar for minerals of different cation exchange capacity (pyrophyllite, montmorillonite, mica). The bending energy increases from zero for a flat single layer to ∼10 mJ/m2 at a bending radius of 20 nm and exceeds 100 mJ/ m2 at a bending radius of 6 nm. The smallest observed curvature of a bent layer is 3 nm. Failure proceeds through kink and split into two straight layers of shorter length. The mechanically stored energy per unit mass in highly bent aluminosilicate layers is close to the electrical energy stored in batteries. Molecular contributions to the bending energy include bond stretching and bending of bond angles in the mineral as well as rearrangements of alkali ions on the surface of the layers. When embedded in polymers, average radii of curvature of aluminosilicates exceed hundreds of nanometers. The small fraction of highly bent layers (400 nm. Layers with very large radii of curvature (rC > 400 nm) were categorized as flat. (5) We prepared histograms of number count versus radius of curvature and identified typical bending radii in TEM images of the given types of composites (Figure S3 in Supporting Information). As an alternative to the visual approach, the analysis of bending radii through automated image processing is, at least in principle, possible. The inclusive, automated recognition of curved features and their processing in even intervals on the local scale into radii of curvature is challenging, however, and would require extensive controls to yield results of comparable quality.
3. BENDING MECHANISM AND ENERGY BY SIMULATION Bending of clay mineral layers can be caused by vertical and out-of-plane forces acting on the layers (Figure 3). Bending also
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occurs when the in-plane compression of single layers surpasses a certain threshold.38,47 In the simulation, the aluminosilicate layers adapt well to the radius of curvature prescribed by the carbon wraps (Figure 3). At larger radius of curvature, the edges slightly straighten out, which can also be observed in TEM micrographs (Section 4). Bending of the aluminosilicate layers involves primarily local changes in bond angles and bond lengths (Figure 3). The position of superficial alkali ions also shifts on the scale of picometers to tens of picometers when the radius of bending decreases, related to changes in local surface geometry. These changes are reflected in the energy of bending (Figure 4a), and in more detail in the bonded and nonbonded contributions (Figure 4b). The possibility of surface reconstruction by lateral migration of cations agrees with previous quantitative findings.14,59 Interestingly, only minor changes in structure and layer energy were identified to reach radii of curvature >20 nm, and much more significant changes occur for radii of curvature below this value. The CEC of the mineral does not have a major influence on the deformation mechanism and the bending energy, although some differences can be seen. Mica with the highest cation density leads to the highest bending energy and is closely followed by pyrophyllite, which contains no interlayer cations (Figure 4a). Montmorillonite with an intermediate cation density (Figure 1) contains more free volume between the bonded aluminosilicate layer and the rigid carbon wrap, which enables better accommodation of the bending strain, as indicated by the comparatively lower bending energies for the smallest computed radii of curvature of 6 nm. The analysis of contributions to the bending energy shows that the energy increase upon bending is mediated by bonded interactions including angle bending and bond stretching, whereas the nonbonded interactions including Coulomb and van der Waals become more favorable upon bending and reduce the bending strain (Figure 4b). In particular, small lateral movements of alkali ions on the surface increase attractive Coulomb interactions and can compensate some of the strain energy in montmorillonite at higher CEC (143 meq/100 g). For radii of curvature >20 nm, the bending energy typically fluctuates on the order of (10 mJ/m2 for a single layer or (50 kcal/mol for a 35.2 nm2 area (Figure 4a). This area contains ∼2000 polar covalent SiO and AlO bonds, and the bending energy per bond corresponds to (0.025 kcal/mol. This is several orders of magnitude lower than bond energies of ∼100 kcal/ mol61 and even the concerted deformation of 101 to 102 bonds falls within thermal fluctuations at room temperature (1 RT = 0.6 kcal/mol at 298 K). Therefore, single aluminosilicate layers may assume local radii of curvature larger than 20 nm without significant activation energies, consistent with observations in TEM images5053 and the observation of undulatory motion of single aluminosilicate layers at room temperature in large-scale simulations.38,41,42,47 For smaller bending radii on the order of 10 nm, the bending energy of a single aluminosilicate layer increases to between 50 and 100 mJ/m2. For bending radii 150 nm thickness35,63 using an AFM tip to bend the layers in a channel between two supports.35,63 In agreement with the deformation of proportionally more angles and bonds, force constants increased with the number of stacked layers. However, reported force constants also depend on the length, width, and deflection (radius of curvature) of the sample or of the deformed region of the layers.35 In contrast, the bending energy per layer area Eb (Figure 4) is independent from the layer geometry because Eb is simply a function of the number of layers and of the bending radius. Force constants of bending k can be derived from the second derivative of Eb for a layer of given length, l, width, w, and deflection, dz, perpendicular to the layer plane. k ¼ lw
∂2 Eb ∂z2
ð2Þ
We can use elementary geometry to express the perpendicular deflection z of the midpoint of the curved layer by the radius of curvature rC and the nominal length l of the layer. The bent layer spans an angle j of length l = jrC as part of a circle (0 < l < πrC), and we obtain for the perpendicular deflection z l z ¼ rC 1 cos ð3Þ 2rC The resulting force constants k cover a wide range from 0.001 to 10 N m1 for single aluminosilicate layers as a function of layer 22296
dx.doi.org/10.1021/jp208383f |J. Phys. Chem. C 2011, 115, 22292–22300
No No No No No Yes No Yes kinked layers found
Epoxy resin consisted of bisphenol-F epoxy (Epon 862) and aromatic diamine (Epikure W). The clay mineral was organically modified montmorillonite of CEC 143 meq/100 g (Nanocor I.30E); see also refs 50 and 51. b Clay mineral was organically modified montmorillonite of CEC 91 meq/100 g (Cloisite 6A and 30A); see also ref 52. c Ref 53.
a
; Yes
150 150
; Yes ; No
250
>400 No ; No >400 Yes >400 No
200 100200
>400 Yes
250300
>500
150
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exfoliation RC 800
; 100200
300400 400600
100200
100200
300400 agglomeration
exfoliation
layer length (nm)
distribution and
intercalation intercalation
distribution and distribution and
intercalation exfoliation
distribution and
intercalation
agglomeration, partial distribution, intercalation,
exfoliation intercalation
agglomeration, partial distribution, intercalation, morphology
exfoliation
5 wt % 4 wt % 2 wt % non-extruded, 6 wt % extruded, 6 wt % non-extruded, 6 wt % extruded, 6 wt % nanocomposite
silk elastin with montmorillonite (solution blending)c epoxy with low CEC organoclayb epoxy with high CEC organoclaya
Table 1. Morphology, Average Length, and Bending Radius of Aluminosilicate Layers in Epoxy and Silk Nanocomposites
8 wt %
The Journal of Physical Chemistry C
length, width, and radius of curvature. (See the Supporting Information.) For example, deflection of a single layer of 94 nm 94 nm size to a bending radius of 30 nm yields force constants ∼0.3 N m1, and a different 13 nm 2.7 nm size at a bending radius of 30 nm yields ∼0.1 N m1. The results agree with nanoindentation measurements of 0.15 to 0.4 N m1 per aluminosilicate layer (1 nm thick).35 The values are also similar to carbon nanotubes (0.42.5 N m1)64 and graphene sheets (0.2 to 0.4 N m1),65 yet the considerable dependence of the spring constant k on layer dimensions must be taken into account: (1) When a 2.7 nm wide layer would be doubled in width, w, and equally deflected, twice as many bonds and angles are deformed and the spring constant doubles. (2) When the length of the layer l = rCj increases for the same radius of curvature, the force constant decreases because the increase in vertical deflection is proportionally higher than the increase in layer length. (3) When the radius of curvature approaches small values keeping constant length and width, the force constant increases because the bending energy increases steeply with vertical deflection z. Therefore, force constants are intensive (system-specific) quantities, and bending energies per layer area are extensive (system-unspecific) quantities with broader applicability. In addition to force constants, the bending stiffness D1 of layered silicates according to plate theory has been previously reported.39,4143,45 The bending stiffness D1 is defined by the Young modulus in the x direction E1, the layer thickness h, and the Poisson ratios ν12, ν21 in x and y directions43 D1 ¼
E1 h3 12ð1 v12 v21 Þ
ð4Þ
However, the bending stiffness is the same for any bending radius and provides no insight into layer energy and layer stability because the in-plane moduli E1, E2 and Poisson ratios ν12, ν21 are material constants that do not change over a wide range of stress.48 Also, the underlying approximation of layered silicates as ideal (smooth) plates is debatable because of the presence of rough surfaces with cations (Figure 1). The bending stiffness D1 is subject to a sensitive third-power dependence on the layer thickness h and varies between (1.1 and 3.2) 1017 J for pyrophyllite, montmorillonite, and mica. (See the Supporting Information.) This is mostly a result of an increase in thickness of single layers from 0.919 to 1.3 nm, whereas we have seen that the bending energy per layer area is similar among all four minerals and depends on the radius of curvature (Figure 4). In summary, reported force constants and bending stiffness are consistent with our data but have limitations in applicability. The bending energy per layer area Eb describes curved layers in a more general way. It takes the curvature and the number of stacked layers into account and does not depend on layer dimensions. Derivatives with respect to vertical deflection yield force constants and local stresses66,67 for a given layer geometry. 4.3. Weight Percentage and Layer Curvature in Composites. In nanocomposites with
