Article pubs.acs.org/Macromolecules
Direct Measurement of Rubber Interphase Stiffness Philip F. Brune,*,† Gregory S. Blackman,† Ted Diehl,† Jeffrey S. Meth,† Don Brill,† Yuefei Tao,† and John Thornton‡ †
DuPont Engineering Research & Technology and DuPont Central Research and Development, E.I. DuPont de Nemours & Co., Inc., Wilmington, Delaware 19805, United States ‡ Bruker Nano Surfaces, Bruker Corporation, Billerica, Massachusetts 01821, United States S Supporting Information *
ABSTRACT: One of the factors that is supposed to be responsible for exceptional properties in polymer nanocomposites is the interphase. The interphase is a region surrounding each nanoparticle where the polymer chains are somehow influenced by the presence of the inorganic surface nearby. It is postulated that due to the high surface area of the nanoparticles that even with a relatively small thickness of interphase around the particle, an appreciable amount of the nanocomposite is in fact this modified interphase material. Despite decades of indirect evidence and speculation about the interphase, there is very little direct evidence, visualization, or detailed measurements of its properties. In this paper we create a strongly coupled system between the surface of a silicon wafer and covalently attached low-Tg elastomer molecules via silane coupling and thiol−ene click chemistry. Then, using advanced scanning probe microscopy techniques coupled with detailed finite element modeling, we determine the true extent of the interphase for this system and measure its mechanical stiffness. We find an interphase of 40 nm in extent, composed of a region of tightly bound rubber with thickness less than 10 nm and shear modulus greater than 250 MPa, and one of loosely bound rubber with thickness around 30 nm and shear modulus around 7 MPa, as compared to the neat polymer shear modulus of 0.3 MPa. ranges from ∼10× stiffer than the hydrodynamic value at small strain amplitudes (∼0.1%) to ∼2× stiffer at larger (∼50%) strain amplitudes.8 The action of nongeometric mechanisms is needed to account for the significant additional measured stiffness. Two principal categories of explanation have been proposed: filler−filler interactions9 and filler−rubber interactions.10 The latter category more completely describes the observed phenomenologyparticularly the thermal activation and occurrence below the percolation thresholdand is therefore the focus of much work in the field, including the present paper.11,12 A central component of many filler−matrix theories is the existence of an “interphase” region of rubber that surrounds filler particles and has a mechanical response that is strongly altered by physical and/or chemical interactions with the filler surface. The rubber interphase is generally assumed to exhibit a spatial gradient response: the rubber that is closer to the filler surface interacts more strongly, and exhibits a stiffer response, than rubber further away from the surface.13 The direct measurement of rubber interphase material modulus, and its spatial variation, is the subject of this paper.
1. INTRODUCTION Polymer composites are important engineering materials. Their increasing prevalence is due in part to tunable performance envelopes dictated by variable material compositions. Central to the continued advance of polymer composite performance is furthering “rational design”, which hinges on the ability to deterministically connect material constituents with system behavior, i.e. to accurately predict end-use property entitlements from constituent ingredients. Nanoscale fillers challenge rational design, as their presence can induce composite behavior that differs from theoretical expectations, many of which emerge from larger scale continuum assumptions.1,2 Filled rubber is perhaps the first material system wherein such a discrepancy was documented.3 Specifically, the “Payne effect” describes how, for rubbers filled with appreciable quantities (>∼5 vol %) of nanoscale carbon black and/or silica, the measured linear viscoelastic moduli stronglyand, for the most part, reversiblydepend upon the applied dynamic strain amplitude.4,5 This behavior defies continuum predictions, which predict that the incorporation of elastic (effectively rigid) filler into linearly viscoelastic rubber would produce a similarly linear viscoelastic composite response.6 In particular, the expected storage modulus would approach the constant hydrodynamic (or geometric) value of (1 + 2.5φ)G′0, with φ the filler volume fraction and G′0 the storage modulus of the unfilled rubber.7 Instead, the measured composite response © XXXX American Chemical Society
Received: April 5, 2016 Revised: May 26, 2016
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DOI: 10.1021/acs.macromol.6b00689 Macromolecules XXXX, XXX, XXX−XXX
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interphase material to the rigid surface. Indentation into interphase material is resisted by the stiffness of the material and its nearby connection to the rigid surface. This is evident in Figure 1B, which shows a perturbation of the stress field due to the presence of the wall, even though the probe is not directly encountering it. By comparison, the stress field from an indent farther away from the wall shows a symmetric stress field that is unaffected by the boundary conditions (Figure 1C). Together, the probe and boundary condition effects require careful mechanical simulation to unambiguously separate material modulus effects from the structural contributions inherent to the nanoindentation measurement technique with a nearby rigid surface. Consideration of probe and boundary condition effects motivate our technical approach and will be revisited throughout the paper. Our work builds upon two preceding works, touched upon briefly here and considered in more detail in the Discussion section. The first combined quantitative AFM and nanostructural modeling to successfully obtain a first measurement of the filled rubber interphase size and stiffness from samples filled with carbon black nanoparticles.14 This work demonstrated the feasibility of such a measurement using AFM and also established the complexities that must be addressed to confirm the measurement validity. Several of those complexities originated from the use of particle-filled samples. The second work analyzed a more tractable film−substrate sample that facilitated direct measurement of interphase material.15 We used an experimental sample of similar configuration, as detailed in the next section. The paper has two main parts: high-resolution experimental measurements of structural stiffness across film−substrate cross sections with an atomic force microscope (AFM) and highfidelity finite element analysis (FEA) to determine material modulus from the experimental responses. The AFM measured the structural stiffness in terms of the force and displacement applied to the sample at various spatial locations. The FEA model then calculated the gradient in material modulus that in conjunction with the boundary condition applied by the substratewas necessary to reproduce the measured variation in structural stiffness. The paper is organized as follows: section 2 describes the experimental materials and methods, section 3 presents the experimental results, section 4 describes the computational data reduction methods, section 5 presents the data reduction results, section 6 discusses the experimental and data reduction results, and section 7 concludes the paper.
Direct mechanical measurement of the interphase is complicated by its nanoscale dimensions and proximity to a rigid (filler) surface. The small interphase size requires nanoscale spatial resolution. While this is tractable with nanoindentation, the necessary presence of the rigid surface nearby the interphase materialexacerbated by its vertical orientation in our experimental sample (Figure 1)introduces
Figure 1. (A) FEA model Von Mises stress field illustrating the geometric or probe effect, where the AFM probe tip physically encounters the substrate boundary. (B) FEA model Von Mises stress field illustrating the “boundary condition effect”, where the nearby mechanical attachment to the substrate stiffens the sample, visible via the perturbation in the stress field. (C) FEA model Von Mises stress field showing a regular indent, i.e., one that is not affected by probe or boundary condition effect, as is evident in the unperturbed stress field.
2. EXPERIMENTAL MATERIALS AND METHODS 2.1. Experimental Materials. The sample used for the AFM experiments consisted of an elastomer film coupled to a silicon substrate via a silane monolayer. The strong coupling was achieved using thiol-containing silane coupling agents and ultraviolet (UV)initiated thiol−ene chemistry to covalently attached the elastomer molecules to the clean silicon surface (Figure 2). In particular, a mercapto-type silane was selected on the basis of its functionality with the selected experimental materials, which are detailed in the following paragraph.16,17 Once assembled, the sample (Figure 3A) was freezefractured to expose a cross section that contained silicon, interphase rubber (elastomer), and far-field rubber (Figure 3B). This facilitated direct measurement of interphase mechanical properties. The sample was fabricated as follows. First, a 3-mercaptopropyltrimethoxysilane monolayer was deposited on top of the silicon wafer via the vacuum vapor deposition method. Next, a rubber polymer solution (50 mg/mL) in toluene was spin-coated with a UV curative (Irgacure 651, 5 wt % in solution) onto the silicon substrate at 3000
two difficulties in interpreting results. First is the geometric or probe effect, which refers to the direct physical interaction between the probe and the rigid surface, as shown in Figure 1A. In this case, the probe encounters the rigid wall as it indents into the near-interface material, which confounds the measurement of the indented material with stiffness contributions from the rigid surface. Second is the boundary condition effect, or wall effect, which describes the influence of the mechanical attachment of B
DOI: 10.1021/acs.macromol.6b00689 Macromolecules XXXX, XXX, XXX−XXX
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rpm for 40 s. The rubber polymerhenceforth referred to as “rubber”was a low styrene, medium-high vinyl polymerized styrene−butadiene copolymer (SLF 16S42 from Goodyear) with molecular weight of 445 kDa and polydispersity of 2.25. The sample was cured under applied UV for 20 min. The cross section was created by cleaving under liquid nitrogen via direct break along the ⟨100⟩ plane and then sonicated in water for 2 min, rinsed in water, and dried under nitrogen. The sample had a rubber film thickness of 1860 nm, determined by stylus profilometry. 2.2. Experimental Methods. A Bruker Dimension ICON AFM was operated at room temperature in “force ramp” mode at lower excitation frequencies ( 65 nm). We compared this to a model-calculated value of 0.08 N/m. This value was determined by an FEA simulation in which a high-fidelity representation of the experimental probe tip was indented to a depth of 20 nm in a rubber block with uniform (static) shear modulus of 0.33 MPa. The modulus value used in the FEA simulation was the storage modulus from DMA of the rubber polymer (Figure S1). The somewhat higher experimental value can be accounted for by the additional stiffness from the larger contact area engaged by the snap-on event; this issue is examined further in the next section. The bounding results credibly satisfied the two spatially limiting checks, giving confidence that the intermediate decayby a factor of 4in measured structural stiffness emerged from the combination of a spatial gradient in interphase material modulus and the decreasing influence of the structural boundary condition from the attachment to the substrate of the rubber. The goal of the subsequent FEA modeling section was to definitively separate these two factorsmaterial modulus and structural boundary conditionand determine to what extent a gradient in material modulus was necessary to account for the experimentally measured structural stiffness data.
Figure 5. Selected force−displacement curves and linear fits, from force ramps at different X-coordinates in the sample, based on coordinate system from Figure 4.
slope versus the X-coordinate of the measurement point (Figure 6). The coordinate system origin was located at the
4. COMPUTATIONAL DATA REDUCTION METHODS Our inverse data reduction process used a high-fidelity FEA model to determine the gradient in modulus of the rubber interphase materialproperly separated from the boundary condition effect of the substratethat was implied by the measured change in structural stiffness. Details of the FEA model setup, execution, and output processing appear in the following sections. 4.1. FEA Model General Setup. The FEA model combined standard modeling considerations with AFM-specific ones, with the latter detailed in three following subsections. The experimental sample was modeled as a rectangular solid defined by linear dimensions in X−Y−Z of 250−250−120 nm (Figure 7). These dimensions were selected to make the model size relatively large with respect to the contact radius and indentation depth, while remaining small enough to reduce model run time. The selected model dimensions were verified to produce a structural stiffness equivalent to that from a sample of infinite width (Y-dir) and depth (Z-dir). A “contact zone” of the sample with refined mesh size (and tied to a more coarsely meshed base) was used to improve simulation of the contact interaction between the sample and probe tip. The contact interaction was implemented as hard contact with no separation allowed and a friction coefficient of 1.0 to represent the strong adhesion between probe and sample. Fully fixed
Figure 6. Structural stiffness from AFM force ramps plotted as a function of X-coordinate on the sample. The silicon substrate and farfield rubber regions of roughly constant stiffness surround the intermediate region, where a decay in stiffness is observed due to the combined effects of interphase material, probe effect, and wall or boundary condition effect.
interface between the substrate and film. Examining the stiffness results and some additional measures as a function of spatial location on the measurement path (Figure S2), we estimated the interface location to be a distance of 50 nm away from the farthest measurement point in the silicon (Figure 6). Two checks were applied to the experimental stiffness data in Figure 6. First were the results in the silicon substrate. From the experiments, a mean structural stiffness value of 0.377 N/m with standard deviation of 0.013 N/m was obtained for the 24 data points that clearly lay in the silicon region (Figure 6, x < −2 nm). We expected these to approach a value of 0.380 N/m. E
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subsection that quantifies the probe and boundary condition effects for our experimental conditions follows the probe tip section. A final subsection describes running the model and postprocessing its output for comparison with experimental results. 4.2. Probe Tip Geometry. The FEA model incorporated a close reproduction of the actual probe tip used in the AFM experiment. This was necessary to ensure that the model results correctly balanced the stiffness due to modulus contributions and contact area, respectively. If the conventional hemispherical approximation were used, the probe tip in the model would be too large. This would make the modulus predicted by the model smaller as a result of the excessive contact area, which would necessitate a lower modulus to produce the same structural stiffness. A polycrystalline titanium standard (Bruker part number RS) and built-in software for tip qualification was used to evaluate the probe tip shape. This tip qualification sample had a series of sharp edges that allowed the reconstruction of the threedimensional shape of the probe tip (Figure 8).23 This
Figure 7. FEA model of rubber sample. Dimensions in X, Y, Z are 250, 250, 120 nm. Fully fixed boundary conditions were applied to the Y−Z plane at x = 0 to represent the attachment to the silicon substrate and to the X−Y plane at z = 120 to approximate the presence of additional rubber material.
boundary conditions were applied to the front face in the Y−Z plane to model the attachment of the rubber to the silicon via the coupling agent. This represents the silicon as completely rigid and its adhesion with the rubber film as perfect. The first approximation is justified by the compliance of the rubber material and cantilever structure; there is no quantitative difference between a model that directly includes the silicon substrate and one that indirectly represents it as a fully fixed boundary condition. The second approximation is justified by the relative continuity of the experimental structural stiffness results near the interface (Figure 6); imperfect adhesion would have been evident as a much sharper decrease in the stiffness values measured close to the rubber−silicon interface. Fully fixed boundary conditions were also applied on the bottom face in the X−Y plane (Figure 7) to approximate the compliance from the additional rubber material. Ideally, this would have been implemented via an elastic foundation, but the expected gradient in material modulus complicated such an implementation. Instead, the fully fixed condition proved an effective approximation because it was far enough away from the indentation location that it had negligible influence on the local indentation response (e.g., Figure 1C). We focused on the equivalent static response of the viscoelastic rubber film at the excitation frequency (∼70 Hz) of the AFM force ramps. The rubber material was therefore modeled as hyperelastic, with the neo-Hookean strain energy density function.21 This represented the rubber elasticity as a function of the shear modulus and bulk modulus, with the latter being considered infinite for incompressibility, and sufficiently captured the material response up to the moderately large peak (∼50%) strains applied by the indentations. The rubber was modeled everywhere as fully incompressible (Poisson ratio ν = 0.5) with hybrid linear hexahedral reduced integration finite elements (C3D8RH in Abaqus22). The probe tip was modeled as a discrete rigid surface with three-dimensional four-noded rigid elements (R3D4 in Abaqus22) controlled by a reference point tied to the end of the cantilever representation. The above parameters established a “baseline” model. This model was, however, insufficient for accurately simulating the experimental force ramps. AFM-specific considerations were necessary. As detailed in the following subsections, these consisted of representations of the actual probe tip geometry, the equivalent cantilever kinematics and stiffness, and an approximate adhesive contact behavior. Additionally, a
Figure 8. (A) Height image from tip calibration sample, with estimated tip profiles at two different vertical heights shown at right. (B) Equivalent probe diameter as a function of vertical height, based on tip calibration data. (C) Front view (X−Z plane) of probe tip in FEA model. (D) Isometric view of probe tip in FEA model.
representation only needed to capture the probe tip surface that was engaged by the sample in the experiments. The vertical distance was limited to 20 nm, a conservative maximum value based on experimental data for Zind in the far-field rubber region. This probe tip geometry included the 12° angle of inclination of the cantilever, as it was present when the tip calibration sample was evaluated. 4.3. Quantifying the Probe and Boundary Condition Effects. The actual probe tip was used with the experimentally measured indentation depths (Zind, Figure 3) to quantify the F
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Macromolecules range of the probe and boundary condition effects (Figure 1). A three-dimensional FEA model was created of the actual probe tip and an effectively infinite sample with one edge fully constrained to represent the mechanical attachment to the substrate. The indentation depths (Zind) from the experimental force ramp results ranged from a minimum depth of 2 nm from material near the interface to a maximum depth of 12 nm from the far-field rubber. A series of models were run that included the substrate, probe tip, and indentation depths, in order to quantify the length of the probe and boundary effects for our experimental conditions. The probe tip was indented to the minimum depth of 2 nm at various distances from the interface, ranging from x = 2 nm to x = 100 nm. The same sequence was then repeated for the maximum depth of 12 nm. First, the deformed shapes were examined to quantify the probe effect. Models where the applied indent caused the probe tip to overlap the fixed boundary (e.g., Figure 1A) were judged to have a probe effect, and no force data were recorded. For the minimum indentation depth, there was probe effect for x < 4 nm. For the maximum indentation depth, the threshold was x < 14 nm. These findings indicated that the probe effect was not present for most of the experimental data. Second, the calculated indentation forces from the models were used to quantify the boundary condition effect. Only forces from models at points beyond the probe effect thresholds were considered: x ≥ 4 nm for the minimum indent depth of 2 nm and x ≥ 14 nm for the maximum indent of 12 nm. The maximum force at each point was divided by the indentation depth to determine the indentation stiffness, Kind. The indentation stiffness was converted to structural stiffness Kss by combining it in series with the experimental cantilever spring constant Kc via the relation Kss = KindKc/(Kind + Kc). Finally, the stiffness ratio was obtained by normalizing the structural stiffness values by the structural stiffness at x = 100 nm, i.e., far away from the boundary. The stiffness ratio was then plotted as a function of distance from the interface (Xcoordinate), shown in Figure 9. For both indent depths, the stiffness ratios had converged to the far-field steady state at distances beyond 50 nm. The stiffness ratio results showed that the boundary condition effect was not sufficient to explain the increased structural stiffness measured in the interphase region (Figure 6). There, the peak experimentally measured stiffness was 3.5 times larger than the far-field values. By comparison, the boundary condition effect contributed a 1.5 times increase, as calculated from the minimum indentation depth. 4.4. Equivalent Cantilever Kinematics. Experimentally, cantilever deflection was used to calculate the force in the AFM experiments. This deflection causes both a rotation and translation of the probe tip, an effect that is accentuated by the cantilever’s 12° angle of inclination from horizontal. Although the deflection and resulting rotation and translation were small with respect to the cantilever size (length = 115 μm), they were significant with respect to the probe tip size (diameter ∼5 nm). In the stiffest regions of the sample, the horizontal (X-direction) movement of the probe tip was roughly equal to the vertical (Z-direction) indentation depth. This influenced where material was engaged in terms of the spatial modulus gradient as well as the contributions made by this motion to the structural stiffness response, i.e., the primary model output. The cantilever kinematics was thus a crucial component in the extraction of material modulus information.24
Figure 9. (A) Plot of stiffness ratio vs X-coordinate for the minimum zind = 2 nm and maximum zind = 12 nm indentation depths. The first data point for each curve defines the threshold between the probe effect and boundary condition effect for that indentation depth. (B) Von Mises stress field from minimum indentation depth case at x = 6 nm, showing perturbation of stress field due to boundary condition effect. (C) Von Mises stress field from minimum indentation depth case at x = 16 nm. No perturbation is visible in the stress distribution at the chosen contour scale, while a moderate value is observed for the stiffness ratio.
Including cantilever kinematics, while essential for accuracy, was not straightforward. The considerable difference in length scales between the nanometer-sized probe tip and the micrometer-sized cantilever severely impaired FEA model convergence behavior and discouraged direct representation of the cantilever. To solve this problem, we used an indirect representation via a superelement. This modeled the cantilever as an effective stiffness that controlled the relative movement between two points: the cantilever base where Z-disp was applied and the base of the probe tip (Figure 10). The relative motion of these two points sufficiently captured cantilever kinematics without harming convergence behavior. 4.5. Adhesion and Contact Area. The final AFM-specific consideration for the FEA model was adhesion. The nanometer-scale curvature of the probe tip and size of the applied Zdisp meant that the force arising from probe−sample adhesion was similar in magnitude to the contact normal force. This was evident in the magnitude of the mean snap-on force (−1.2 nN) with respect to the applied set point force (1.8 nN) (Figure 11A, point ii vs point iv). The main consequence of the adhesive force was an increase in the initial contact area as the probe tip snapped onto the sample surface. This can be seen through a comparison with nonadhesive contact from the FEA model, which predicts a significant nonlinear portion in the early stages of load-up as an increasing amount of the tip G
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Figure 10. Sequence of images showing multiscale approach to representing the cantilever and probe tip: (A) Micron-scale cantilever superelement, showing 12° angle of incline, with total length of 115 μm. The mechanical behavior of the cantilever is represented as an effective stiffness that controls the motion between the cantilever base point and the probe tip base point. (B−D) Zoomed-in images of sample model section and probe tip. Model dimensions are the same as in Figure 7.
Figure 11. Loading protocol to establish snap-on contact area. (A) Experimental data for AFM force ramp from load-up curve. (B) FEA model force−displacement response. Key states during the measurement are labeled as points i−iv and illustrated in the equivalent FEA model state as follows: (i) probe starts atop sample in zero-force state; (ii) back-to-zero displacement applied in +Z direction; (iii) snap-on displacement applied in −Z direction; (iv) total Z-disp applied in +Z direction.
Figures 11A and 11B as points i−iv. The FEA model cantilever started at zero force, with the probe tip atop the surface (Figure 11B, i). First, the back-to-zero Z-disp (Zbtz) was applied to the cantilever base to indent the probe tip into the sample (Figure 11B, ii). The contact interaction was defined so that all contact area accumulated was retained throughout the simulation. Next, a “snap-on” displacement (3.3 nm) was applied to retract the cantilever in the negative-Z direction (Figure 11, iii). This displacement was equal to the ratio of the snap-on (adhesion)
engages the sample (Figure 11B). No such nonlinear region was evident in the experimental AFM force curves, indicating that significant contact area had already been established via the adhesive interaction between probe tip and sample surface. This outcome was captured in the FEA model via a protocol developed to approximately incorporate the additional contact area accumulated due to adhesion. The four-step protocol was defined according to experimental data, with the equivalent states in the experiment and model responses identified in H
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Macromolecules force (−1.2 nN) and the cantilever spring constant (0.36 N/m) from the experiment, and returned the cantilever to near-zero force, thus establishing a new starting point for the FEA model. The total Z-disp (Ztot) was then applied atop this new reference point (Figure 11B, iv) to simulate the complete force curve with the extra contact area established by adhesive snap-on. This method gave excellent results by capturing the experimentally measured linear structural stiffness values (Figure 11A, Kexp) for the stiffer sample locations (x < 50 nm) where contact area was created by the probe snapping down. In the softer regions (x ≥ 50 nm), material compliance permitted the film surface to “jump up” in addition to the probe snapping down, further increasing the contact area. For these cases, the FEA loading protocol was an acceptable approximation. 4.6. FEA Model Execution and Output. The FEA model was used to simulate the structural stiffness of five target points from selected locations of the interphase portion of the measurement path (Figure 12). The loading at each target point was defined according to the measured loading conditions from the AFM experiments at that point, which included the displacements defining the snap-on protocol and the subsequent total Z-disp (Table 1). All displacements were applied to the cantilever base. Values of Ztot ranged from 9 nm in the material very near to the interface to 20 nm in the farfield material (Table 1). The force in the model was calculated as the product of the cantilever deflection Zdefl (as illustrated in Figure 3) and the spring constant from the experiment (0.36 N/m), an approach that was consistent with the AFM experiments. This force was plotted versus Z-disp to cast the FEA model response in the same terms as the experimental data collected from the AFM force ramps (e.g., Figure 5). The structural stiffness from the FEA models was obtained from the force versus Z-disp response (Figure 13A) as the first derivative of the force with respect to the Z-disp (Figure 13B). Only the last 30% of the curve was used in order to partially account for changes in contact area in the earlier parts of the curve, particularly for the far-field regions (e.g., x = 68 nm). From this section of the structural stiffness versus Z-disp curve, the model-predicted structural stiffness was computed as the mean value, and the standard deviation was taken as a measure of model error.
Figure 12. (A) Structural stiffness vs X-coordinate, with area targeted for FEA modeling highlighted. (B) Isolated area targeted for FEA modeling, with exponential fit to stiffness data. The five target points for simulation are indicated. (C) FEA model showing setup for second target point at x = 16 nm. (D) Force versus Z-disp response from FEA model, showing the section used to calculate the model-predicted structural stiffness, KFEA.
5. COMPUTATIONAL DATA REDUCTION RESULTS 5.1. Constant Modulus. We first used the FEA models to assess how well simulations using constant (as opposed to spatially varying) values for the rubber modulus described the experimental results. Three shear modulus values were used: a high of 250, a medium of 7.5, and a low of 0.5 MPa (Figure 14A). The values were selected based on trial FEA simulations of various locations from the experimentally measured response. The constant modulus models did not accurately describe the experimental results (Figure 14B). Two trends were observed. First, for the high and medium modulus values, the spatially uniform material modulus generally produced a spatially constant structural stiffness. For both cases, there was a moderate increase in the predicted structural stiffness with X-coordinate (Figure 14B), in sharp contrast with the experimental response. The increase in stiffness was due to the larger contact area that was engaged by the greater applied Zdisp values (Table 1). With a small exception for the “medium” modulus case between x = 8 and x = 16 nm, the increase in
Table 1. Applied Z-Disp Values Used for the Snap-On Protocol in the FEA Model, As Shown in Figure 11, for Each of the Five Target Points; “Delta Disp” Is the Difference of ZBTZ and the Snap-On Displacement, While “End Disp” Is the Sum of the Delta Disp and Ztot x [nm]
snap-on force [nN]
snap-on disp [nm]
ZBTZ [nm]
delta disp [nm]
Ztot [nm]
end disp [nm]
8 16 32 50 68
1.2
3.3
3.7 4.1 5.2 6.7 10.7
0.4 0.8 1.9 3.4 7.4
8.7 12.3 13.9 15.2 20.3
9.1 13.1 15.8 18.6 27.7
structural stiffness from the enlarged contact area was sufficient to counteract the decrease in stiffness from the boundary condition effect (Figure 9). I
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Figure 13. (A) Force vs Z-disp curves from the FEA model at different X-coordinates. The last 30%, in terms of Z-disp of the curves, was used for the stiffness calculation and is shown with dashed lines. (B) Structural stiffness vs Z-disp curves obtained from the force− displacement curves in (A). Structural stiffness was calculated pointwise as the first derivative of force with respect to Z-disp.
Figure 14. (A) Constant modulus definitions, shown on logarithmic scale. Target points identified with markers. (B) Structural stiffness response of constant modulus FEA models at the simulated target points. The exponential fit to the experimental response, as defined in Figure 12B, is shown as the solid red line.
The low modulus cases showed a stronger influence of the changing contact area (Figure 14B). The simulated structural stiffness had a larger value at the target point closest to the interface (x = 8 nm), caused by the boundary condition effect (Figure 9). At the intermediate target points (x = 16, 32, and 50 nm), the simulated structural stiffness dropped to values (∼0.05 N/m) below the experimentally measured minimum value (∼0.08 N/m). The relatively smaller applied Z-disp values (Table 1) engaged a smaller contact area, reducing the structural stiffness. At the far-field target point (x = 68 nm), the structural stiffness increased back to the far-field experimental value, as the Z-disp values engaged a contact area that more closely approximated the experiments. 5.2. Exponential Modulus Gradient. Next, an exponential modulus gradient was evaluated. The exponential curve was defined based in part on the results from the previous constant modulus cases (Figure 14). Pairings of modulus and Xcoordinate that reasonably predicted the experimental response (Figure 14B) were taken as the basis for the curve, with intermediate points added to refine the discretization for the Abaqus implementation, which linearly interpolated between
points. The result is shown in Figure 15A. The exponential modulus gradient more accurately described the experimental results (Figure 15B) and was used as a basis for further optimization. 5.3. Best-Fit Modulus Gradient. The best-fit modulus (Figure 16A) was determined by iteratively adjusting the spatial gradient of the material modulus. The modulus gradient was set to a trial distribution based on the previous simulation results, and the structural response at each of the five target points was simulated. The model-predicted structural stiffness value at each target point was compared with the experimentally measured values from the curve fit, and the material modulus gradient was adjusted accordingly. This process was repeated until a modulus gradient was identified for which the model stiffness prediction at each target point agreed to the third decimal place with the experimentally measured values (Figure 16B). In general, the predicted structural stiffness at a given Xcoordinate depended on the distribution of the modulus as a function of X-coordinates up to and slightly beyond that point in space. For example, at the x = 16 nm target point, the J
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Figure 16. (A) Best-fit modulus gradient, shown on logarithmic scale. The points used to discretize the curve for implementation in Abaqus are shown as filled circles, while the target points are shown with triangular markers. (B) Structural stiffness response of the FEA model with best-fit modulus gradient, simulated at the five target points. The exponential fit to the experimental response is shown as the solid red line.
Figure 15. (A) Exponential modulus gradient, shown on logarithmic scale. The points used to discretize the curve for implementation in Abaqus are shown as filled circles, while the target points are shown with triangular markers. (B) Structural stiffness response of the FEA model with exponential gradient, simulated at the five target points. The exponential fit to the experimental response is shown as the solid red line.
6. DISCUSSION We begin by revisiting discussion points from the experimental and modeling results. First is the quantitative assessment of the structural stiffness measurements from the AFM force curves (Figure 6). The bounding results agreed well with FEA model predictions. In the silicon substrate, the mean experimentally measured stiffness of 0.377 N/m almost exactly matched the FEA model prediction of 0.380 N/m. In the far-field rubber, the mean experimentally measured stiffness of 0.10 N/m was ∼20% larger than the FEA model prediction of 0.08 N/m. We attribute the difference to a larger contact area in the experiments, caused by the adhesive interaction between probe tip and sample. The increased contact area reduced the contact pressure, resulting in less indentation than expected, which translated to an increased apparent stiffness. The protocol we developed to approximate the adhesive contact behavior (Figure 11) worked well in the stiffer region of the sample but provided a coarser approximation in the more
distribution of modulus over the domain of x = 0, 20 nm influenced the predicted structural stiffness; the response was relatively insensitive to the distribution of modulus values beyond x = 20 nm. Thus, the iterative fitting process started at the target point closest to the interface (x = 8 nm) and proceeded outward (increasing X-coordinate). Around 20 manual iterations were required to converge. The best-fit FEA model results revealed a strong spatial gradient in material shear modulus that can be divided into three regions (Figure 14). Close to the interface (x < 10 nm) the modulus exceeded 100 MPa, at intermediate distances (10 < x < 50 nm) it approached 10 MPa, and far away (x > 50 nm) it reached a far-field value of 0.5 MPa. The FEA modeling results clearly show that the nearly 4× change in experimentally measured structural stiffness could only be accurately reproduced by significant spatial changes in material modulus. K
DOI: 10.1021/acs.macromol.6b00689 Macromolecules XXXX, XXX, XXX−XXX
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Wood et al. used nanoindentation on a variety of rubber polymer substrate combinations, with the indentation applied to the free surface of uncleaved film−substrate samples, i.e., to the top surface of the sample illustrated in Figure 3A.13 An FEA model was used to determine the increase in apparent Young’s modulus contained in the experimental results. Again there is no direct comparison with the current work; while several samples were composed of solution styrene−butadiene rubbers and silicon substrates, no coupling agent was used, and the reported modulus ratios were much lower, in some cases below unity. This qualitatively matches our experience; we attempted to fabricate samples without coupling agent for comparison, but the film did not survive the cleavage step, due to the weak surface interaction. While our work primarily addresses the experimental detection of the interphase, it is worthwhile to discuss the molecular causes that could create the spatially varying modulus. We first establish the length scale of the rubber polymer as an entangled network. We use Treloar’s entropic description of rubber elasticity, N = G/kT, to obtain the crosslink density N from the DMA storage modulus results (Figure S2).25 From this, we obtain a distance d between cross-links of 2.9 nm for the uncured rubber polymer, using d = (N/2)−1/3. This suggests a minimal effect of the Irgacure cross-linker in the far-field rubber because the modulus from that region (0.5 MPa) indicates a comparable cross-link density to that of the uncured rubber polymer analyzed via DMA. A more pronounced difference is predicted from the two higher modulus regions. The modulus from the intermediate region with G ∼ 7 MPa would give a distance between cross-links of 1.05 nm, while in the region with the highest modulus values with G ∼ 250 MPa, the predicted distance between cross-links shrinks by nearly an order of magnitude (with respect to the far-field value) to 3.2 Å. Further information about the length scales involved can be examined in terms of the polymer chains. For the styrene− butadiene copolymer used here, we can estimate the radius of gyration and end-to-end distance using well-known molecular correlations.26 From this, we estimate the unperturbed radius of gyration of the chain ⟨Rg2⟩1/2 ∼ 18−25 nm, with an end-to-end distance ⟨R02⟩1/2 ∼ 44−62 nm. Thus, the extent of the interphase is approximately 2Rg. However, when chains are grafted to a surface, as they are here, we expect their structure to be perturbed, forming a brush layer near the surface. This brush layer is expected to be stiffer and thus possess a higher modulus.27 Using the formulas from Fujii et al.,27 one can estimate the shear modulus of the brush layer to be 5−10 MPa, which is not inconsistent with the FEA derived value of 6−8 MPa in the intermediate region of the interphase. To understand the higher modulus in the near-surface region, one would posit that the cross-link density of the material is higher here due to a localized presence of the Irgacure UV cross-linker. The mobility of the chains in this region must be restricted to the point where the polymer is almost glassy near the surface. Unfortunately, more characterization work would need to be performed to prove these hypotheses. The grafting density would need to be known, and the structure of the chains would need to be measured with neutron scattering. That is beyond the scope of our work but could be explored in the future. However, even with these crude approximations, the values for the FEA moduli are reasonable, and the spatial extent of the interphase is in line
compliant region, which is exactly where the far-field bounding calculation was done. The moderate difference in far-field response can also be attributed to rate effects. The FEA model that generated the prediction used a material modulus based on the DMA storage modulus results (Figure S1), which were collected at 10 Hz, while the AFM force ramps used an equivalent excitation frequency of around 70 Hz. It is expected that the modulus increases with increasing frequency. Together, the bounding results give confidence in the quantitative accuracy of the AFM structural stiffness measurements. Between the substrate and far-field bounding regions, the probe and wall effects are key components in interpreting measurements of interphase properties. Our approach to the probe effect ensured that we did not erroneously attribute interphase stiffness to the probe tip physically encountering the substrate. We used several measurements from the AFM data (Figure S2) to conservatively define the interface location and then quantified the region susceptible to probe effect (Figure 9) using an FEA model with an accurate three-dimensional representation of the experimental probe tip and applying the experimentally measured maximum and minimum indentation depths. Finally, in determining the material response via FEA simulations of the experiment, we selected the closest simulation point (x = 8 nm) to be unambiguously beyond the probe effect threshold of x = 4 nm for the shallow indents in this stiffer portion of the interphase. At each step of the data analysis and data reduction process, we used conservative approaches to ensure that probe effect artifacts were not present. Properly accounting for the boundary condition effect in the measured stiffness data was the primary motivation for the FEA modeling approach to data reduction. Previous quantitative AFM experiments by Cheng et al. on similarly configured film− substrate samples with PMMA as the polymer used FEA models to identify a “substrate effect” (i.e., boundary condition or wall effect) region for x < 15 nm, in which modulus data were discarded.15 This approach sacrificed data in the region where interphase properties are likely to be the most pronounced. Our approach of directly simulating the experimental conditions allowed us to retain this data, which revealed an order-of-magnitude increase in interphase modulus (Figure 16A). In comparing our results with other previous work aimed at characterizing the filled rubber interphase, we find some commonalities but no direct comparison. Qu et al. used torsional harmonic AFM on a cross-linked hydrogenated nitrile butadiene rubber (HNBR) filled with low volumes (∼5 vol %) of N330 carbon black nanoparticles.14 A similar FEA-based approach was employed to define the spatial extents of the probe and boundary condition effects, and this information was integrated with AFM measurements and micromechanical effective particle modeling to obtain an interphase with thickness of 19 ± 8 nm and average Young’s modulus of 53 ± 11 MPa over the interphase thickness extending beyond the probe/boundary effect uncertainty limit of 6 nm from the particle surface.14 We compare that modulus estimate with an average Young’s modulus of 93.4 MPa from our results, calculated as the integral average of the shear modulus curve in Figure 16A from 6 to 50 nm, assuming a Poisson ratio of 0.5. The average modulus values from the two studies are similar, despite the differences in experimental materials, sample morphologies, and analytical methodologies. L
DOI: 10.1021/acs.macromol.6b00689 Macromolecules XXXX, XXX, XXX−XXX
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numerous helpful discussions, to W. Wysock, N. Tassi, and L. Zhang for assistance with sample manufacture, to C. Rasmussen for SEC analysis of the rubber polymer, to J. Howe for DMA of the rubber polymer, and to J. Li for assisting with AFM.
with the size scales of the polymer chains and network, respectively. Finally, we return to filled rubber phenomenology. Previous work on bound rubber in carbon black filled elastomers identified “immobilized” or “tightly bound” rubber that is chemisorbed and “loosely bound” rubber that is physically adsorbed and/or mechanically entangled.28 These two categories of bound rubber have also been identified in styrene−butadiene rubber filled with silane-coupled silica.29 Our measurements have identified and quantified a significant difference in modulus between these two layers. The interphase thickness and modulus results we obtained more than account for gap between the hydrodynamic prediction and the frequently measured values of composite modulus in reasonably filled (∼20 vol %) rubber nanocomposites at small macrostrains.7,8 It is possible that the thickness values of
