Numerical Investigation of Polymer Coated Nanoporous Gold

Nanoporous metals represent a fascinating class of materials. They consist of a bi-continuous three-dimensional network of randomly intersecting pores and ligaments where the ligaments form the skeleton of the structure. The open-pore structure allows for applying a thin electrolytic coating on the ligaments. In this paper, we will investigate the stiffening effect of a polymer coating numerically. Since the coating adds an additional difficulty for the discretization of the microstructure by finite elements, we apply the finite cell method. This allows for deriving a mesh in a fully automatic fashion from the high resolution 3D voxel model stemming from the 3D focused ion beam-scanning electron microscope tomography data of nanoporous gold. By manipulating the voxel model in a straightforward way, we add a thin polymer layer of homogeneous thickness numerically and study its effect on the macroscopic elastic properties systematically. In order to lower the influence of the boundary conditions on the results, the window method, which is known from homogenization procedures, is applied. In the second part of the paper, we fill the gap between numerical simulations and experimental investigations and determine real material properties of an electrolytic applied polypyrrole coating by inverse computations. The simulations provide an estimate for the mechanical properties of the ligaments and the polymeric coating and are in accordance with experimental data.


Introduction
Nanoporous gold (NPG) represents a fascinating class of nanoporous metals. An overview over their morphologies and mechanical properties can be found, for example, in [1,2] and [3,4]. NPG consists of a three-dimensional network of randomly intersecting pores. Ligaments form the skeleton of the structure. Their diameter can be controlled by altering dealloying conditions, thus allowing for examining the impact of the ligament size on the macroscopic mechanical properties [5][6][7][8]. The open-pore structure of NPG enables creating gold-polymer composites (NPG-composites) either by fully infiltrating the porous metal by epoxy resins [9,10] or by adding a thin electrolytic coating layer of a conjugated polymer (such as polyaniline or polypyrrole) on top of the ligaments [11][12][13][14]. The latter strategy has resulted in novel hybrid materials with improved mechanical behavior along with substantially enhanced functional properties [14]. The suggested potential application of such nanocomposites as mechanical actuators requires a detailed understanding of the impact of the polymer layer on the effective mechanical response. This study thus aims to elucidate the effects of a 1.2 mm × 1.5 mm, which were subsequently annealed in argon atmosphere for recovery (2 h at 800 • C, infrared furnace behr IRF 10, behr Labor Technik, Düsseldorf, Germany). The dealloying was carried out in a three-electrode electrochemical cell at a constant potential 0.75 V vs. a Ag/AgCl pseudo reference electrode in 1 M HClO 4 electrolyte (60% HClO 4 , ACS grade, Merck, Darmstadt, Germany) at room temperature. A coiled Ag wire (99.9985%, Alfa Aesar, Kandel, Germany) was used as the counter electrode. Afterwards, the as-dealloyed samples were electrochemically reduced by cyclic voltammetry cycles (15 cycles within a potential range of −0.5 V...1.0 V vs. the Ag/AgCl pseudo in 1 M HClO 4 prepared from Suprapur grade HClO4, Merck), rinsed with ultrapure water, and dried in air. PPy was deposited into bulk NPG specimens by the electropolymerization of pyrrole monomer (pyrrole, 99%, extra pure, ACROS Organics) in a lithium perchlorate acetonitrile solution solution (LiClO 4 , 99.99%, Sigma-Aldrich, Darmstadt, Germany, acetonitrile, LiChrosolv, Merck), following a synthesis protocol described in [14]. The resulting PPy was thus doped with  . Before the electropolymerization, NPG was annealed in air (500 • C for 30 min) to coarsen the pores. This was necessary to facilitate diffusion of pyrrole monomers in the porous network in order to achieve a uniform PPy coating throughout the bulk NPG samples. To obtain a PPy layer with a predefined thickness and to keep the nanoporosity, the total electrodeposition time (800 s) was optimized accordingly. All electrochemical measurements were performed using a potentiostat PGSTAT 302N (Metrohm). Finally, the NPG-PPy composites were cleaned with ultrapure water and dried at ambient conditions. Figure 1 shows a typical microstructure of the resulting NPG-PPy composites after the electropolymerization (characterized by a high resolution scanning electron microscope Zeiss Supra 55VP FEG SEM, Jena, Germany). Investigation of the fracture surfaces of the PPy-coated NPG (obtained through cleavage with a scalpel) revealed an average ligament size of ≈220 ± 30 nm and ≈50 ± 8 nm PPy layer. ImageJ (version 1.49v, National Institutes of Health, Bethesda, MD, USA) was used to analyze the sizes of gold ligaments and PPy coatings in SEM images by averaging 20 measurements.

Compression Tests
Before uniaxial compression with several unloading-reloading sequences were undertaken, the NPG and PPy-coated NPG bulk specimens were cleaned and dried. Afterwards, they were subjected to a similar testing protocol as employed in [15]. Linear segments of the first unload-reload cycles were used to compute the effective Young's moduli. The mechanical experiments were carried out on a calibrated universal testing machine Zwick Z010 TN, (Ulm, Germany) at a constant engineering strain rate of 10 −4 s −1 and ambient conditions. A sample displacement tracking in a longitudinal direction during compression was achieved by a laser extensometer (Zwick laserXtens). The displacement data served to determine the true strains. We presented the experimental stress-strain diagrams in true stress-true strain coordinates. Because of the insignificant cross-section area variation of NPG upon compressive loading [32,33], engineering and true stresses practically coincide.

Modeling and Simulation of Nanoporous Gold Based on the Finite Cell Method
This section is intended to give a short introduction into the FCM which was applied for the simulation of NPG. A subsection is related to the window method that describes the approach to find homogenized elastic properties for the NPG samples.
The finite cell method [34,35] relies on the fictitious domain approach in combination with the high-order finite element method. The well known weak form of the equilibrium conditions in the current configuration reads: where ϕ describes the mapping between the reference and the current configuration, σ is the Cauchy stress tensor, b denotes the volumetric loads and t the prescribed traction acting on the Neumann boundary Γ N of the body Ω. The current density is denoted by ρ and η are the test functions or virtual displacements. After introducing the indicator function α and reformulating Equation (1) for an extended domain Ω e , we obtain the following form which is the basis for the FCM: The modified weak form allows to extend any arbitrarily complex shaped geometry by a fictitious domain Ω f (Ω e − Ω) such that it obtains a simple shape. In general, this is just the bounding box of the underlying geometry. Now, a simple discretization of the extended domain Ω e can be derived by a Cartesian grid that does not conform with the boundary of the original problem-see Figure 2. The equivalence with the original problem is enforced by the discontinuous indicator function: that penalizes the fictitious domain. Both formulations are equivalent for q = ∞ but in fact we choose q ∈ [5, .., 12] to avoid ill-conditioning of the problem. By this little modification, the effort of mesh generation is considerably reduced due to the rectangular shape of the extended domain and shifted towards the integration of the weak form that involves a discontinuity caused by the indicator function. The quadrature of the weak form can be carried out by an adaptive Gaussian quadrature scheme in a straightforward and efficient way that only involves point-membership-tests based on the underlying geometric models. To this end, every integration point has to be tested whether it lies in the physical domain. This feature makes the FCM flexible so that it can directly operate on implicitly defined geometries, i.e., voxel-based data generated by computer tomography or B-rep models [36].

Adaptive Gaussian Quadrature
The discretization of the weak form in Equation (2) leads to a set of finite cells that can be classified into three groups. Cells that are completely placed within the fictitious domain do not contribute to the weak form (α = 0) and can be disregarded. For cells fully lying in the physical part of the domain (full cells) α = 1 holds, so they are equivalent to standard finite elements and can be integrated in the standard fashion. Special care has to be taken for cells placed over the boundary of the original domain (cut cells) that exhibit a discontinuous integrand due to the indicator function α. The adaptive Gaussian quadrature scheme subdivides the integration domain Ω of these cells recursively along the discontinuity into smaller sub-cells Ω sc ( Figure 3) until a predefined refinement level is reached. The integrand can then be computed as a composed Gaussian quadrature on the level of the sub-cells: where r i are the well-known Gauss-Legendre points and ω i the corresponding weights. To organize the refinement process space trees are used, e.g., quadtrees in 2D ( Figure 3) and octrees in 3D problems. Since α (X(ξ (r i ))) det J sc (r i ) ω r i and ξ (r i ) are constant during the computation, they can be stored as the resulting weights and coordinates of the integration points belonging to the adaptive integration scheme. When considering nonlinear problems, the solution of the nonlinear set of algebraic equations by means of the Newton-Raphson method requires the linearization with respect to the unknown displacements ∆u-see, for example, [37]. To simplify the notation, we do not take follower loads and volume loads into account. The resulting linear equation system that is solved within each iteration k for every loadstep i reads :

Reference configuration
where K k−1 is the global tangent stiffness matrix that stems from the discretization of the linearization of the weak form and λ i is the accumulated load factor which serves to apply the load increment-wise. The global matrices are computed by assembling the tangent stiffness matrix and the load vector of every cell c: where the tangent stiffness matrix, the internal and the external force vector of cell c read: Here, B presents the strain-displacement matrix containing the derivatives of the high-order shape functions, and C is the spatial elasto-plastic tangent modulus that is consistent with the radial return mapping scheme applied to integrate the elastoplastic material model. The computation of the external load vector f ext,c requires integrating the product of the shape functions and the traction vector over the surface of the corresponding cell. In Equation (7), q ,α and q ,β denote the tangent vectors of the surface of the cell in question, where α and β are two of the three local coordinates of the related cell.

Von Mises Plasticity
The mathematical formulation of plasticity [38] in the small strain case relies on an additive split of the small strain tensor ε into an elastic ε e and a plastic contribution ε p : The constitutive relation for the stress is given by the derivative of the strain energy function ψ with respect to the elastic strains ε e : where λ, µ are the Lamé constants and 1 is the second order identity tensor. In order to define the onset of yielding, we use the von Mises yield function: where s is the deviatoric part of the stress tensor and J 2 denotes its second invariant. The evolution of the plastic strains is described by the equation: whereγ is the incremental plastic multiplier. The hardening variable ε p is related to the incremental plastic strain by

The Window Method
The computation of the effective material properties for heterogeneous materials calls for a high computational effort due to the requirement of large representative volume elements (RVE). Therefore, the analysis of such structures is only possible for smaller testing volume elements (TVE) and thus the influence of boundary conditions on the results is large. The window method [39] lowers this influence by embedding the TVE into a domain (window) with homogeneous material and applying displacement boundary conditions to the surface of the window as depicted in Figure 4.  This approach is known from analytical homogenization schemes and called the self-consistency method-see [40]. The window models the surrounding material that a real RVE would be embedded in and reduces the influence of the boundary conditions on the results such that tighter bounds for the effective properties are obtained-see [41].
In order to achieve self-consistency, the window material has to have the same effective material properties as the embedded microstructure. It is unknown at the beginning of the computations and therefore an iterative procedure is required to determine the effective material properties and adjust the material of the window in every step.
In the following, we assume that the material behavior is linear elastic and inclusions or coatings are perfectly bonded to the surrounding material. An anisotropic material requires solving six linear independent loadcases j = 1, . . . , 6 where every loadcase determines one column of the elasticity matrix. To this end, we define the macroscopic small strain projection tensor ε P and disturb a single component of it in each loadcase by a small increment. The projection rule: imposes the related displacements ∆ū on the surface of the window Γ w . Here, ∆X is the branch vector that points from the center of the TVE to the location of the applied displacements. After solving the mechanical problem, the averaged stresses σ m and strains ε m are computed in a post-processing step. The discrete form of the averaged stress reads: where ⊗ is the dyadic product and r (i) m are the nodal forces related to the nodes n nodes on the surface Γ m of the spatial discretization of the microstructure. The computation of the average stress in Equation (14) might yield an unsymmetric stress tensor which is therefore symmetrized in a subsequent step. The computation of the averaged strains in its discrete form reads: where u  (14) and (15), V m represents the volume of the microstructure and A m is the inner surface area of the window. After solving a loadcase one column of the effective elasticity matrix, C eff is obtained by numerical differentiation of the averaged stress quantities with respect to the related applied macroscopic strains. Applying the Voigt notation for the strains and stresses, the effective elasticity matrix reads: where i corresponds to the stress component and j = 1, . . . , 6 denotes the corresponding load case. The basic load case is denoted by index 1 and can be related to a case with no loads. In order to obtain a symmetric elasticity matrix, we again symmetrize C eff . To adjust the window material in a self-consistent way, the elasticity matrix of the window C w is updated for the next iteration by the actual computed effective properties [42]-see Figure 5. The iterative procedure can be accelerated by the Aitken's ∆ 2 -method [43] and is considered to be converged when the difference between the effective properties of two consecutive steps falls below a prescribed tolerance e C,Frob ≤ TOL in terms of the Frobenius norm, Compute initial material properties: Apply analytical rule of mixture Pre-processing: Set boundary conditions Set window material properties Solve Problem by FCM: Average strain and stress field: Do post-processing for six loadcases j = 1, .., 6 Compute effective properties by numerical differentiation: Check convergence criteria: Apply Aitken's ∆ 2 -method: Figure 5. The homogenization procedure.

Numerical and Experimental Investigation of the Influence of Coating on Nanoporous Gold
This section is separated into two parts. In the first part of this section, we investigate the geometrical stiffening effect caused by a polymeric coating. To this end, we consider macroscopic samples of nanoporous gold with and without a coating and analyze it under the assumption that we can apply bulk material values for the gold ligaments and the polymer coating. To find out the orthotropic material parameters of such samples and to quantify the stiffening effect caused by the coating, we perform a homogenization by means of the window method.
The second part of this section is related to the comparison of numerical computations on NPG-PPy-composites with a ligament diameter of d lig = 421 nm and a coating thickness of t c = 107.16 nm − 138 nm to experimental investigations on NPG-PPy samples with a ligament diameter of d lig = 220 nm and a PPy coating thickness of t c = 50 nm. Under the assumption of self-similarity of these structures, which was shown in [28,44], we aim to solve the inverse problem, meaning to find the material parameters of the polymer coating such that numerically and experimentally obtained stress-strain curves coincide. The self-similarity properties of NPG were also studied in [45].

Linear Elastic Properties
The starting point for our investigation of the elastic properties of nanoporous gold with a mean ligament diameter of d lig = 421 nm is a voxel model of a representative sample-see Figure 6a. The data was obtained by 3D focused ion beam-scanning electron microscope (FIB-SEM) tomography undertaken by [28,46]. The solid fraction of the sample is 32%. For details on the tomography and data analysis in terms of structural and mechanical properties, see [28,29,44]. The model preparation requires only little pre-processing effort. Disconnected regions-made visible by gray color-bear no loads and are removed from the model. In order to detect these isolated regions, we make use of a seed-fill algorithm [47]. In a second step, we extracted a sub-sample from the full model that is small enough to be analyzed on a computer in a reasonable time and large enough to carry all characteristic information of the microstructure (see Figure 6b). We discretize the sub-sample by finite cells where each cell contains four voxels per direction-see Figure 7a. The refinement of the spatial discretization is carried out by increasing the polynomial degree of the shape functions (p-refinement) and not by reducing the element size (h-refinement). The application of the window method requires to embed the sample of nanoporous gold into a so-called window as depicted in Figure 7b. Determining all elastic properties of the microstructure requires solving different load-cases. The window serves to apply the corresponding displacement boundary conditions not directly to the microstructure but to the window and by this lowers the influence of the boundary conditions. The window layer itself is composed of one layer of finite cells and is adjustable in thickness to model different types of boundary conditions. For a large window thickness, the boundary conditions resemble a Neumann boundary, while, for a small window thickness, they correspond to Dirichlet boundary conditions. By changing the window thickness (t w ∈ {0.357 µm, 0.714 µm, 1.072 µm, 1.423 µm}) and its material properties, the surroundings of the sub-sample are modeled in a self consistent way and different results are obtained for different sizes of the window. The procedure regarding how to adjust the material properties is explained in more detail in Section 3.3. The material properties of the bulk gold material applied in the simulations chosen according to [15] are given in Table 1. The window method allows for finding the full anisotropic material properties of the sample as explained in Section 3.3. Assuming an orthotropic material behavior, the directional Young's moduli, E 1 , E 2 , E 3 , the shear moduli, G 12 , G 23 , G 13 , and the Poisson ratios ν 12 , ν 23 , ν 31 , can be determined from a least squares fit of the obtained elasticity matrix. The results of a first investigation of the elastic properties are given in Figures 8-10, where the convergence of the elastic parameters against the degrees of freedom for a uniform increase of the polynomial degree of the shape functions of all cells is plotted. The error-bars include the results obtained for different sizes of the window. With an increasing spatial resolution, the differences in the results obtained for different window sizes are more or less reduced and will be neglected in the further considerations. In summary, the obtained results are in the same range of the data already presented in [46] for a sample of the same size.    For our second investigation, we apply a homogeneous polymer coating of t c = 4 [voxel] (Figure 11b), resp. t c = 6 [voxel] (Figure 11c) to the voxel model depicted in Figure 11a. The voxel spacing is not equal in each direction, so the coating thickness varies from t c = 71.44 nm to 92 nm, resp. from t c = 107.16 nm to 138 nm. A Cartesian mesh with 85,388, respectively 117,683 cells, is generated for the composites in the same fashion as for the nanoporous gold microstructure. The interface between the different materials is not resolved by cell interfaces, but considered during the integration of the stiffness matrix. To this end, we apply the adaptive Gaussian quadrature as explained in Section 3.1 to capture the material interface by means of an octree refinement. For this study, we assume the properties of epoxy, which will also serve as starting values for the calibration of the polymer properties of PPy in Section 4.2. The material properties for the polymer used in the second investigation are taken from [46] and listed in Table 2. Table 2. Elastic material properties of polymer coating applied for the homogenization.

Parameter Value Unit
Young's modulus

Young's modulus [MPa]
Degrees of freedom ×10 6 By comparing the Young's modulus for the finest discretization (p = 4) in Figure 13, it is observed that the application of a polymer layer significantly increases the stiffness of the composite by up to a factor of 10, although the thickness of the polymer coating is below the ligament diameter and the Young's modulus is only 6% of gold.
If the coating was not a polymer but had the bulk properties of gold, one could simply estimate that, under bending dominated deformation of cylindrical ligaments [15], the geometrical increase in stiffness would scale with the area moment of inertia by a factor of (r+t c ) 4 r 4 ≈ 6.71. In tensile loading conditions, the increase in geometrical stiffness would scale by the area of the ligament (r+t c ) 2 r 2 ≈ 2.59. When considering that the real coating is a polymer which is more compliant than gold, the real stiffening effect would be even less than given by these two estimates. However, both scaling relations still underestimate the real increase in stiffness. This indicates that the dominant stiffening effect of the additional coating has other origins among which the forming of new connections between neighboring ligaments is one mechanism-see Figure 14.

Investigation of Elastoplastic Properties
In a next step, we extend our investigations to the nonlinear regime by also considering the plastic behavior of the NPG-PPy nanocomposite. In order to relate the computations to experiments, we consider experimentally obtained true stress-strain curves on nanoporous gold samples with a ligament diameter of d lig = 220 nm and NPG samples coated by a PPy layer with a thickness of t c = 50 nm-see Figure 15. An important assumption that allows us to apply the tomography data of a NPG sample with d lig = 421 nm to a smaller ligament diameter is the self-similarity of the NPG which holds for structures with a ligament diameter d lig ≥ 100 nm. The self-similarity of NPG has been intensively investigated for a large range of ligament sizes in [28,44]. The deformation behavior of the non-coated NPG under compression is comparable to previous studies of NPG specimens with similar ligament size [8,9,33]. Here, we also find an absence of the initial elastic regime and an early yielding along with weak work hardening at higher strains. In contrast, an NPG-PPy nanocomposite exhibits the enhanced flow stress and work hardening, although the elastic-plastic transition is still ill defined. The effective elastic moduli as inferred from the first unloading cycles for both samples differ by less than 100 MPa-see Table 3. The presence of the t c = 50 nm thick PPy layer marginally stiffens NPG, by about 20%. The influence of the polymer coating on the plastic flow, however, is more significant. For instance, at 0.15 strain, there is a nearly 4-fold increase in the flow stress of the nanocomposite.
In order to fit the numerical stress-strain curve to the one of the NPG-composite, we follow a two-step approach. In a first step, we adjust the properties of the nanoporous gold such that we meet the experimental curve of nanoporous gold. In a second step, we take the fitted properties of the nanoporous gold and apply bulk polymer material properties of epoxy as starting values and adjust the polymer parameters until we meet the experimental NPG-composite stress-strain curve.
The values for the macroscopic yield stress as well as the macroscopic elastic modulus determined from the first unloading are given in Table 3. To have a unique definition for the yield stress, we took the offset yield point where 0.2% plastic deformation occurs. In order to simulate the coating, we added a homogeneous layer of polymer with t c = 107.16 nm − 138 nm-see Figure 11c. These values correspond to the same ratio t c vs. d lig as for the experimentally tested samples. As a starting point, we take bulk values for the material properties from [46] which are listed in Table 4 and apply them to the sample of nanoporous gold that is shown in Figure 11a. In order to analyze the anisotropic behavior of the nanoporous gold, we apply three different compressive loadings in 1-, 2-and 3-direction by clamping the lower side of the sample and prescribing a displacement on the upper side. The discretization consists of n c = 130,114 cells with a size of 71.44 nm × 71.44 nm × 92 nm. Figure 16 shows the comparison of the numerical results of this first investigation to the experimental results. Since the slopes of the elastic unloading fit best for the 1-direction and also fit to the results for the homogenization, we consider this loading direction for our further investigations.
In a next step, we change the initial yield stress σ 0 and the hardening modulus of the gold, such that the numerical stress-strain curve fits the experimental one. Figure 17 shows the stress-strain curve after fitting the material parameters of the gold. The Young's modulus of the sample is E = 451.56 MPa and the initial yield stress σ 0 = 1.48 MPa, yielding a sufficiently close fit to the experimental values listed in Table 3.
The fitted material parameters for gold are given in Table 5. The initial yield stress compares very well with the values determined by [10] for ligaments of 200 nm which ranged from 60 MPa to 100 MPa, but the hardening modulus is much higher by about an order of magnitude compared to the values found by [15,28] who used a work hardening rate of 1000 MPa.  In a next step, we analyze a coated sample using the material parameters of the nanoporous gold obtained from the fit together with material parameters for the polymer for which we use properties of epoxy as starting values-see Table 6. Applying these material parameters, the polymer has a much more pronounced stiffening effect on the stress-strain curve than determined from the experiments-see Figure 18a.
Therefore, in a next step, we also adjust the material parameters of the polymer in our numerical computations, such that the stress-strain curve fits to the experiments-see Figure 18b. For the fitted material parameters, which are given in Table 7, the computed Young's modulus of the macroscopic sample is E = 1307.71 MPa and the initial yield stress is R p0,2 = 3.06 MPa. Both are still slightly higher than the values measured from the experimental stress-strain curve. In order to fully align the numerical computations with the experiments, a further reduction of the Young's modulus and the initial yield stress of the polymer coating is required. However, the further reduction of these material parameters yields to a very poor conditioning of the resulting stiffness matrix which causes convergence problems of the Newton-Raphson method.  Our numerical investigations of nanoporous gold samples with a ligament diameter of d lig = 421 nm and a polymer coating of t c = 107.16 nm − 138 nm indicate an enormous increase of the macroscopic stiffness by a factor of ten if a polymer coating such as epoxy resin with bulk material values could be applied-see Figure 13. In line with our expectations from the experimental results, the increase in stiffness entailed by coating the gold ligaments with the softer polymer such as PPy is ten times smaller. Our numerical computations capture the stress-strain behavior of the NPG-PPy nanocomposite and predict the mechanical properties of PPy. The Young's modulus and yield stress of PPy which need to be applied in the numerical computations in order to match the stress-strain curve of the experiments have been found to be 200 MPa and 2 MPa, respectively. The values match well to the experimental data available in the literature, where the elastic moduli have been reported in the range of 100-4300 MPa [48][49][50][51][52][53][54] and the yield strength between 2.4-5.3 MPa [49]. Significant variations in the mechanical parameters of the bulk PPy in the literature have been attributed to the sensitivity of the PPy morphology and structure to synthesis conditions [55]. Among other factors, a type of the dopant counterions used for electropolymerization can strongly affect PPy mechanical characteristics. For the perchlorate anion employed as a dopant during synthesis of PPy in our work, the reported values of elastic modulus are in the range 100-300 MPa [48], and agree well with the predictions from our simulations. The good agreement underlines the high sensitivity of the approach for calibration of the material parameters for PPy using the numerical model. Starting the calibration with much larger values taken from bulk epoxy, the solution rapidly converged to the much lower and much more realistic values of PPy within a few iterations. In this way, there is now a set of material parameters for PPy available that includes the yield stress and hardening rate for the given synthesis conditions on NPG.

Conclusions
In this paper, we investigated the potential stiffening effect of an infiltrated epoxy resin and an electrolytic applied polypyrrole polymer on the elastic and elastoplastic properties of NPG-composites. For the infiltrated epoxy resin, we could determine the anisotropic elasticity matrix by employing the window method and computing a full set of transversal isotropic material properties by a least-square fit. By this procedure, we could show that the Young's moduli increase significantly due to new connections formed between the ligaments when a coating is applied. The obtained results are in good agreement with already existing data. In a second investigation, we analyzed the influence of an electrolytic applied polypyrrole coating by direct comparisons with experiments on NPG with a ligament size of d lig = 200 nm. Due to the self similarity of these structures, we could reuse our geometric model with a ligament diameter of d lig = 421 nm. After fitting the material properties of a pure NPG sample to meet the experimentally obtained stress-strain curve, we created a numerical NPG-composite by manipulating the voxel model. After applying a coating with the same t c vs. d lig ratio as was used for the experiments, we could analyze the properties of an electrolytic applied PPy coating. Due to the sensitivity of PPy with respect to its synthesis condition, the focus was on inverse computations that served to determine the elastoplastic properties of PPy. The final numerically determined parameters for PPy are in good agreement with already existing data of PPy obtained under similar synthesis conditions. With this, it is demonstrated that the Finite Cell Method is an efficient alternative to common Finite Element approaches to predict the elastic-plastic deformation behavior of NPG and polymer coated NPG. This approach has the big advantage that the model can be set-up in a straightforward way based on tomography data without the usual challenge to discretize the complex structure of NPG by a surface conforming mesh.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: FCM Finite cell method NPG Nanoporous gold RVE Representative volume element TVE Testing volume element PPy Polypyrrole