An MD simulation study to the indentation size effect of polystyrene and polyethylene with various indenter shapes and loading rates

Applied Surface Science 492 (2019) 579–590

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An MD simulation study to the indentation size effect of polystyrene and polyethylene with various indenter shapes and loading rates ⁎

Chao Penga, Fanlin Zenga, , Bin Yuana, Youshan Wangb,

T

⁎

a

Department of Astronautic Science and Mechanics, Harbin Institute of Technology, Harbin 150001, People's Republic of China National Key Laboratory of Science and Technology on Advanced Composites in Special Environment, Center for Composite Materials, Harbin Institute of Technology, Harbin 150001, People's Republic of China

b

A R T I C LE I N FO

A B S T R A C T

Keywords: MD simulation Indentation size effect Polystyrene Polyethylene

Several nanoindentation MD simulations were conducted to investigate the indentation size effect (ISE) of Polystyrene (PS) and Polyethylene (PE) at nano-scale using different indenters and loading rates. A strain gradient elasticity model in which the ISE is related to the Frank energy caused by finite bending stiffness and neighboring interactions of chains was imported to describe the ISE in this work. Two large-scale molecular substrates for PS and PE were constructed to compare the ISE performance during indentation simulations. The spherical indenter was firstly considered, and an uptrend of the calculated hardness and modulus with increasing indenter radius was found for both PS and PE. A conical indenter consisting of 25 virtual rigid spheres used in LAMMPS was designed. Considering different tip sizes and loading rates, indentation simulations using these conical indenters were conducted. The uptrend of hardness curves at the initial indentation stage was also found in conical cases. Through fitting the hardness curves, the strengthening effects of tip size and loading rate were certified, especially for PS. At nano-scale, the dominant effect of benzene ring component on ISE performance was confirmed. The downtrend of ISE was found to exist at nano-scale for PE although it was not obviously found in experimental cases and the contributing factors for this kind of ISE were discussed.

1. Introduction Indentation size effect (ISE) has been observed and researched for crystalline materials at the micro- and nano-scales by various researchers [1–10]. However, such size dependent behavior was also found in nanoindentation (NI) [11–14], micro beam bending [15,16] for polymers and its composites [17]. There is a need for reliable characterization and prediction of ISE in polymer for reasons of (i) nonoindentation experiment has been a conveniently and most used method to test the mechanical properties of polymers and its composites with small dimensions, (ii) the technological applications such as MEMS [18], sensing elements [19] of such low dimension polymeric materials are critical and enormous, (iii) the fundamental advancement of polymer physic [20]. Researchers have made many achievements on understanding the ISE mechanisms for polymers. In the context of continuum, strain gradient theories by adding gradients in the stain or higher gradients in the displacements into the classical continuum mechanics play an important role in explaining size dependent deformation for polymers [11,12,15,21–26]. Based on molecular yielding theory [27] and the

⁎

introduced geometrically necessary kinks which is similar to the notion of geometrically necessary dislocations in metals [3], Lam et al. [11,12,26] built a strain gradient plasticity model for plastic ISE of glassy polymers. However, this model is not applicable for elastic ISE in polymers such as silicone since it was modeled by plastic deformation only. As for elastic ISE, early strain gradient elasticity theory was proposed by Mindlin [22,28,29], but it contains higher order gradients so that not applicable even in simple cases. A simpler strain gradient elasticity theory was proposed by Yang et al. [23] and Lam et al. [15] in which an additional equilibrium relation is developed to govern the behavior of the couples and the number of independent elastic length scale parameters is reduced. Although this model successfully predicts the size dependence of bending rigidity during bending tests of micro beams using nanoindenter, it is phenomenologically derived and thus can't link the constitutive equations to the physical microstructure of materials. Nikolov et al. [25] suggested a micromechanically derived elastic gradient model which attributes the size dependent elastic deformation in polymers to Frank energy [30] which is always present as long as the chains possess finite bending stiffness and neighboring interactions between each other. Han [31] subsequently extended it to

Corresponding authors. E-mail addresses: [email protected] (F. Zeng), [email protected] (Y. Wang).

https://doi.org/10.1016/j.apsusc.2019.06.173 Received 4 April 2019; Received in revised form 13 June 2019; Accepted 17 June 2019 Available online 18 June 2019 0169-4332/ © 2019 Elsevier B.V. All rights reserved.

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Fig. 1. The diagrammatic sketch of the deformation in network model of cross-linked wormlike chains.

2. Strain gradient elasticity and the corresponding ISE model

elasto-plastic materials and deduced a hardness model similar to rule suggested by Chong and Lam [12] for epoxy and PC but physically explained the length scale parameter cl as an expression of the characteristic length l corresponding to Frank energy. Han used the proposed hardness model to quantitatively compare the ISE performance of experimental data [60] of several polymers and found its explicit effectiveness in describing the ISE performance of these polymers. According to the quantitative comparison [32], the benzene ring which induces the finite bending stiffness of chains plays a dominant part in causing the ISE, because the ISE exists in all polymers with benzene rings but not in polymers without benzene rings such as PE and PTFE. However, the experiment itself has many influences on ISE such as 1) the limitations of experimental resolution [20], 2) surface effects of the specimen [33,34], 3) the shape of the indenter [35,36] and 4) differences in the material properties through the load depth [37,38]. These interfering factors become especially noticeable at nano-scale, here therefore come several questions including 1) whether the ISE performances predicted by the elastic gradient model by Nikolov and Han exist or not at nano-scale without these interfering factors, 2) if they exist, how the ISE will be present and which factors have caused these kinds of ISE performances, 3) how the accurately controllable factors for instance the indenter shape and loading rate at nano-scale will influence these ISE performances, 4) how effective are their hardness models proposed at micro-scale in predicting and describing these ISE performances and what are the appropriate hardness models targeting differently derived ISE. The present work aims to figure out these questions. At nano-scale the indentation MD simulation conducted in this work is an effective method to eliminate the experimental deviation, which has been approached by various researchers [39–41]. We chose two typical polymers PS (polystyrene) and PE (polyethylene) which has the benzene ring component in one and not in the other to verify the discovery in Ref. [32]. Additionally, since it is very convenient to control the loading rates of indenter and construct the shapes of indenter in MD simulation, we investigate the influence of loading rates and tip blunting on ISE by controlling the displacement at each integral step and using indenters with different tip angles respectively. The present work is organized as follows. The stain gradient elasticity and its corresponding extension to elasto-plastic polymers are reviewed, followed by the indentation MD simulation of PS and PE consisting of construction of simulation models and indentation simulation using different spherical indenters. Then the indentation simulations using the defined conical indenter in this work were conducted. Tips with different sizes are considered and the influence of the tip size is quantitatively discussed using the proposed hardness model by Nikolov et al. [25] and Han [31]. We also conducted simulations under different loading rate and the corresponding effect of loading rate is discussed.

The differences in indentation size effect can be quantitatively assessed by the fit of hardness models. The hardness model proposed by Nikolov et al. [25] and Han [31] show their great validities in predicting the ISE at micro-scale for glassy polymers with benzene rings such as epoxy and PC. Therefore, the questions that which factors have caused and how they influence the ISE performances at nano-scale can be assessed by verifying whether the proposed hardness model is effective or not when fitting with hardness at nano-scale. In order to illustrate the rationality of this kind of verification, this section displays in detail how the Frank energy is derived and how it contributes to the ISE predicted in the consequently constructed hardness model. In the elastic gradient model, Nikolov et al. [25] attributed the size effects coming from rotational gradients to Frank elasticity suggested by de Gennes and Prost [30]. Liu and Fredrickson [42], Fukuda [43,44] and Yokoyama [44] found that Frank elasticity is always presented as long as the chains possess finite bending stiffness and neighboring interactions. Then Nikolov et al. considered a network of cross-linked wormlike chains which can represent real polymer chains as long flexible rods with finite elastic bending stiffness and described the network model as a diagram schematically shown in Fig. 1. In the network model, a given point represents a cross-link point or entanglement in polymer and a given unit vector n0 emanating from the cross-link point represents a sub-chain segment connecting two crosslink points. In amorphous polymers, the vector field n0(r) represents the undeformed network which has a non-zero Frank energy due to the strong spatial fluctuations caused by the numerous topological defects around the cross-link point. Even though, any deviation of the vector field n0(r) under the network stretching, network bending and/or twisting of deformation gradient F would give rise to distortion Frank energy [30] and cause n0(r) turn into the deformed vector field n(r) which is also unit vector. The change in the distortion Frank energy during this process is expressed as:

∆WF =

K K 〈ni, j ni, j〉 − 〈n 0(i, j) n 0(i, j) 〉 2 2

(1)

where K is the average Frank elastic constant. In the right polar decomposition of the deformation gradient, F = R · U, R has the relation Rij ≈ δij + ωij with the spin tensor ω for small rotations and U can be treated as δij since both n0 and n are unit vectors, so the deformed vector n is expressed as ni = Fi, jn0(j) ≈ n0(j) + ωijn0(j). Thus the gradient ni, j is written as:

ni, j = n 0(i, j) + ωik, j n 0(k ) + ωik n 0(k, j)

(2)

Considering the changed Frank energy ΔWF must be rotationally invariant and vanish in the undeformed state, ΔWF depends only on the gradient of ω and should be its quadratic form. Then the Frank energy 580

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H = H0 + HF. The first component H0 is the hardness in the absence of ISE and expressed as H0 = F0/A where A and F0 are respectively the contact area and the indentation force without ISE and both approximately proportional to h2. The second component HF is the hardness due to the rotational gradients related to Frank elasticity. Through assuming a case in which a concentrated force F on a linear elastic and isotropic half space, the Frank deduced hardness HF is approximately expressed as:

at the deformed state under these constrains will be K ( 〈n 0(i, j) n 0(i, j) 〉 + 〈ωik, j n 0(k ) ωik, j n 0(k ) 〉 ) . The expression of ΔWF was easily 2 written as:

∆WF =

K 〈ωik, j n 0(k ) ωik, j n 0(k ) 〉 2

(3)

In small rotations, ω is an anti-symmetric second-order tensor and has the matrix component form:

HF = α 2l 2μ (1 − ν 2)2h−γ

Ω3 − Ω2 ⎤ ⎡ 0 0 Ω1 ⎥ ω = ⎢− Ω3 ⎢ Ω − Ω 0 ⎥ 2 1 ⎣ ⎦

where μ and ν are respectively shear modulus and Poisson ratio, α and γ are correction parameters. The addition of H0 and HF yields:

Through analyzing the matrix expression of the gradient of the product ω ∙ n0 and eliminating the zero components, the scalar product 〈ωik, jn0(k)ωik, jn0(k)〉 is expressed as:

〈ωik, j n 0(k ) ωik, j n 0(k ) 〉 =

2 2 Ωi, j Ωi, j ≡ χ ij χ ij 3 3

c γ H = H0 ⎡1 + ⎛ l ⎞ ⎤ h⎠ ⎦ ⎝ ⎣

K χ χ 3 ij ij

(4)

(5)

c H = H0 ⎡1 + l ⎤. h⎦ ⎣

where the phenomenologically derived tensor χ is finally related to the physical Frank elasticity energy. In the conventional couple stress elasticity theory suggested by Yang et al. [23] the simplest form of deformation energy density is given by:

1 W = λεij εij + μ (εij εij + l 2χijS χijS ) 2

∼ 1 K λεij εij + μεij εij + χijS χijS 2 3

cl =

∼ K 3μ

Finally, two hardness equations are achieved and used to assess the ISE performances in the present simulation. It should be noted a reasonable verified result should be that the employed hardness equation is at least more effective for PS than for linear PE, because the hardness model itself is based on the cross-linked network framework and has verified the dominance of benzene ring in causing the ISE of polymers with this component. Additionally, Eq. (12) should be firstly considered when used to verify the effectiveness because comparing with Eq. (12) which has the macro hardness H0 and the length scale parameter cl ∼ micromechanically related to Frank constant K , Eq. (11) has an additional parameter γ only playing a role of correction. Thus the correction parameter γ in Eq. (11) can be regarded as a reflection of intricate influences except for benzene ring, but the exact relationship between them is worthy of further investigation. In the present work, Eq. (12) is used to conduct the verification process by comparing the fitting results of H0 and cl with their experimental counterparts and meanwhile analyze the influence of tip blunting on the ISE performance for both PS and PE. Eq. (11) is used to investigate the dependency of loading rate on the ISE later by giving a more obvious fitting result.

(7)

(8)

Han [31] extended Eq. (7) to elasto-plastic application by explaining that the Frank energy is independent of the classical notion of elastic or plastic deformation energy. The Frank energy is therefore added to the total deformation energy W through:

W = W p + W e + ∆WF

α 2l 2μ (1 − ν 2)2 . H0

(13) ∼ Considering Eq. (8), the Frank constant K has a positive proportional relationship with cl: ∼ cl ∝ K . (14)

(6)

In conventional Frank elasticity theory, the Frank strain energy doesn't contribute to the total deformation energy [25,30]. But the Frank energy is now added in the total energy and considering the assumption of rod-like molecules which does not refer to conventional ∼ cases, so the K is the effective Frank constant different from K in Eq. (1). Comparing Eqs. (6) and (7), the characteristic length l can be re∼ placed by the effective Frank constant K in the form as:

l=

(12)

Comparing Eq. (12) with Eq. (10), the length scale parameter cl can be deduced as:

where λ and μ are the lame constants, l is a characteristic length and χijS is the symmetric component of χ considering the anti-symmetric part of χ does not contribute to deformation energy. Assuming that the form of deformation energy density of isotropic polymer is similar to Eq. (6), and also considering that the symmetric component of the Frank energy ΔWF expressed by Eq. (5) accounts for rotational gradients, the deformation energy density can be written as [25]:

W=

(11)

where the scalar γ within the interval [0, 2] and the length scale parameter cl character the ISE. It has the similar dependence form of H versus h as Refs. [3,13]. Han [32] examined the dependence of ISE on the Frank energy by fitting indentation experiments data of different polymers with Eq. (11) since this hardness model is established by adding the Frank energy WIF to the total indentation work. When used, the parameter γ takes value 1 and Eq. (11) becomes:

where the second order curvature tensor χij derived from classical strain gradient theories was defined by Fleck and Hutchinson [45] as the gradient of rotation vector Ω. With this equation, the changed Frank energy ΔWF becomes:

∆WF =

(10)

3. Simulation methodology

(9)

3.1. Simulation models

During nanoindentation, the plastic deformation would cause the incomplete release of ΔWF when the loads are removed. The stored ΔWF inside polymer would in turn increase the area under the loading curve and consequently cause a higher indentation force F. Therefore, according to the indentation hardness H = F/A, a higher hardness versus h is presented. In order to develop the terminal relation of hardness versus h reflecting ISE, the total hardness H is assumed to be split through

A nanoindentation simulation system consists of a substrate and an indenter. The construction of substrates used in our simulation considers the following points according to the introduction mentioned above. Firstly, the size of each substrate in three directions should be big enough to avoid the cross interactions of indenters among different simulation cells under the periodic boundary condition. Secondly, the substrate should contain an upper surface without surface roughness. 581

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annealing simulation, the top carbon layer was moved away and a surface without roughness was obtained. The bottom layer retained and would be treated as a fixed boundary during the indentation process later [47]. We ultimately got the PE substrate sample shown in Fig. 3(a), inside which the PE chains fill in the confined layer more homogeneously than Fig. 2(a). The PS substrate shown in Fig. 3(b) experienced the same melting and annealing simulation as PE. The density of PS substrate reached about 1.0 g/cm3 and the temperature was 300 K at last, which is lower than the glass-transition temperature (Tg) of PS. As for the indenter, a virtual spherical rigid one is usually used in typical NI simulations. A repulsive force described by F(r) = − K (r − R)2 for r < R, and F(r) = 0 for r ≥ R is acted on the substrate by the indenter. In this study, K = 1 eV/Å3 is the force constant for interactions between the indenter and the substrate, r is the distance from the atom to the center of the indenter and R is the indenter radius. In both cases, the indenter was firstly moved toward (along -z axis) the substrate until it touched the upper surface of substrate as shown in Fig. 4.

Besides, there should be no inhomogeneous distribution of molecular chains or defects inside the substrate. In this work, we built substrates with the Amorphous Cell module in Materials Studio software packages from Accelrys Inc. However, we can only obtain a relatively small amorphous polymer cell in every task due to limitation of computing resources and the ability of this software. In order to obtain enough substrate size which is suitable to nanoindentation simulation, we imported the relatively small cell into LAMMPS [46] and then replicated it to the wanted size. For PE substrate, a single PE molecular chain with a polymerization degree of 100 was generated firstly. Then 176 chains (105,952 atoms) were packed into a cubic cell to form an amorphous PE cell using the Amorphous Cell module in Materials Studio software. Then the initial PE configuration data was imported to LAMMPS as input data. In LAMMPS, the cell was replicated in 3 directions and a bigger cell (316.165 Å × 316.165 Å × 479.776 Å) containing 3168 chains (1,907,136 atoms) was obtained. In the extended space above and under the substrate, two diamond layers were built respectively. The construction process of PS substrate is the same as PE substrate. The polymerization degree of PS chain was 80, and a total of 600 PS chains (769,200 atoms) were packed into a 228.475 Å × 228.475 Å × 265.116 Å confined layer. They are shown in Fig. 2(a) and (b) respectively. As is shown in Fig. 2, there exist many defects inside the two substrates and the upper surfaces are also rough. To get a more homogeneous configuration and smooth surface for each substrate, an annealing simulation within NVE ensemble for each cell was performed. The substrate atoms are Newtonian atoms, whose motions obey classic Newton's second law. The Newton's equation of motion was integrated by velocity-Verlet algorithm with a time step of 0.5 fs in both cases. Periodic boundary conditions were imposed in the 3 directions and a rescaling algorithm was set to adjust the temperature via modifying the atom velocities. The PCFF force field was employed to describe the interactions between atoms. In PE case, after fixing the top layer and bottom layer, the PE cell was heated gradually to 600 K and kept at 600 K for 100 ps, allowing the solid to melt. The substrate was then annealed at a constant rate over 100 ps to 200 K, which is lower than the glass-transition temperature (Tg) of PE. During the annealing process, the top layer was moved toward –z direction as a rigid body at a constant rate to compress the substrate volume and ultimately allow the substrate density reach about 0.95 g/cm3. After the melting and

3.2. Simulation details After the simulation system was built, the indentation simulation was conducted through displacement controlling. During the loading stage, the indenter was moved vertically down at a rate of 10 Å/ps until a certain depth equal to the radius of the indenter was achieved. It should be noted that the indentation rates used in our simulation is larger than the typical rates in NI experiments. It is difficult to employ a typical experiment rate in our simulation since bridging the large time/ scale gaps between simulation and experiment still remains one of the outstanding challenges in this field. Nevertheless, the range of indentation rate in this study is in accordance with previous MD simulations of NI. The unloading process was then conducted by vertically moving the indenter upward at the same rate as loading. After the indenter was moved above the upper surface of substrate, the unloading process and the whole indentation simulation ended. During the whole loading and unloading process, the bottom carbon layer was treated as a fixed boundary and the neighbored atoms as thick as 10 Å were also frozen to prevent ‘sample drift’. Then the rest of the simulation system was subjected to NVE ensemble and a rescaling algorithm was set to

Fig. 2. (a) The primary cell of polyethylene substrate and (b) the primary cell of polystyrene substrate. 582

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Fig. 3. (a) The PE substrate without surface toughness and anisotropy and (b) the PS substrate without surface toughness and anisotropy.

Therefore, besides using spherical indenters, we also conducted several indentation simulations using the constructed conical indenters. We firstly conducted 5 indentation simulations respectively for PS and PE using the 5 conical indenters at loading rate 1 nm/ps. Then we conducted indentation simulations for PS and PE respectively at a series of loading rates, 0.2 nm/ps, 0.4 nm/ps, 0.6 nm/ps, 0.8 nm/ps and 1 nm/ ps, using the conical indenter with bluntness 0.9. Except for no unloading process, the same boundary condition and thermodynamic conditions as set for sphere indenters were used in these simulations. The maximum loading depth for each indentation is 8 nm. 4. Results and discussions 4.1. The load-depth curves and ISE performance of modulus and hardness under spherical indenters Fig. 6 shows the load-depth curves of PE and PS under spherical indenters. As is shown in the picture, the PS load on each curve increases rapidly with the increasing depth till it reaches a pop-in and the curve extends smoothly during this initial stage. It has been found that there is only elastic deformation inside the substrate during this stage [50–52]. After the elastic stage, the load continues to increase but at a lower rate till the end of indentation and the curve presents the serrated shape as a result of a large number of load droppings. We have certified this kind of serrated shape of curves is caused by the alternant occurrence of elastic and plastic deformation inside the substrate in our previous work [53]. The PE load-depth curves also consist of the smoothly rising elastic stage and the serrated plastic stage. But unlike the upward trend of PS curves, the overall trend of PE curves after the elastic stage is downward. Through comparing the movement of molecular chains inside the two materials, we investigated why the PE loads decrease during the plastic stage while PS loads still increase during the same stage. We computed the atomic-level strain tensor [54] of PS and PE at the loading depth of 1 nm, 4 nm and 7 nm respectively, taking the configuration at the 0th step as reference configuration in each computing case. As shown in Fig. 7, it is found that there is hardly any difference of strain tensor distribution between PS and PE at the depth 1 nm, which also belongs to the elastic stage. But for configurations belonging to plastic stage, whether at depth 4 nm or 7 nm, the atomic-level strain of PE is much stronger than PS and the region affected by the indenter inside PE substrate is larger than PS substrate. It proves during the plastic stage the movement of chains inside PE substrate is easier than PS substrate under the same loading condition. It is consistent with Ref. [32] that the flexible and linear chains in PE substrate make it more easily to adjust to minimize the conformation energy. Evidently, the benzene ring makes the slipping of PS chains more difficult so that the PS substrate can withstand greater loading force. We computed the hardness of the two materials to inspect their ISE performance under the spherical indenters. In typical NI experiment,

Fig. 4. The nanoindentation simulation system of PE.

keep the temperature at 200 K for PE and 300 K for PS respectively via modifying the atom velocities. In order to study ISE, we considered an indenter radius series of 10 Å, 20 Å, 30 Å, 40 Å, 50 Å, 60 Å, 70 Å for both PE and PS. The loading sites were distributed within a circular region with a radius of 30 Å centered on the center of the upper surface and they were at least 15 Å apart from each other to avoid cross interactions. However, as subsequently explained in the discussion section, we can only obtain the ISE performance of modulus and hardness at scales less than 2 nm if spherical indenter is used. In order to investigate the ISE performance at larger scales, we developed a conical indenter where the cone angle can be changed. This conical indenter schematically shown in Fig. 5(a) is composed of 25 external common tangent spherical virtual indenters where the common difference of z-coordinates between the centers of two adjacent spheres is invariably set as 5 Å. The minimum sphere has a constant radius of 5 Å and we define different cone angles through varying the common difference of sphere radii between two adjacent spheres. We use sin(A) to denote the bluntness extent of this conical indenter as it is exactly the ratio of common difference of sphere radii to common difference of z-coordinates of sphere centers. We finally constructed 5 conical indenters where the bluntness values are 0.5, 0.6, 0.7, 0.8, 0.9 respectively. It should be mentioned that the indenter with bluntness 0.9 is close to indenters used in FEM simulations [48,49] in which the angle A of 70.3° gives the same nominal contact area per unit depth as a Berkovich indenter; i.e. A = 24.5h2, where A is the nominal contact area and h is the indentation depth. The conical indenter with bluntness 0.5 is shown in Fig. 5(b) where the spheres are close enough together to simulate a cone. 583

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Fig. 5. (a) The geometric drawing of conical indenter and (b) the conical indenter consists of 25 virtual rigid spheres where the common difference of z-coordinates of sphere centers is 5 Å and the common difference of sphere radii is 2.5 Å.

It is known that the load data in initial elastic stage is well approximated by Hertz fitting [47,57–59]:

the hardness and modulus are achieved via the Oliver and Pharr method [55]. However, the O-P method is unsuitable for our computation since it is derived only for experimental curve. In NI experiment, the unloading curve is usually well approximated by power law relation and based on this approximation the contact stiffness S defined as the slope of the upper portion of the unloading curve during the initial stages of unloading is used for calculating the contact area. But it can be seen on Fig. 8 that for example the unloading curves of PS and PE under “7 nm” indenter do not conform to the power law and the slopes at the right outset of unloading tend to ∞ on both unloading curves though. The unloading curves under other indenters also exhibit the same trend. Not considering the unloading curve, the hardness can still be computed by Pmax where Pmax is the peak load at the end of initial elastic Ac stage [40,56]. The contact area Ac can be directly decided by Ac = π(2r − h) where r is indenter radius and h is the contact depth recorded at end of initial elastic stage [47]. The obtained hardness H is depicted in Fig. 9 which shows the hardness increases with the increasing indenter radius for both substrates.

F=

4 1 ∗ 3 r 2E h2 3

(15)

∗

where E is the effective modulus which can be expressed by:

(1 − νi2 ) 1 (1 − ν 2) = + E∗ E Ei

(16)

where Ei and νi are the modulus and Poisson ratio of indenter, and ν is the Poisson ratio of the substrate. The Poisson ratios 0.35 for PS and 0.4 for PE were taken as a medium of values found in the literature [60]. The assumption for rigid indenter makes Eq. (16) become:

E = E ∗ (1 − ν 2)

(17)

Through fitting the initial elastic load-depth curves of PS and PE under spherical indenters with Eq. (15) and then taking the obtained E∗ to Eq. (17), we achieve the moduli plotted in Fig. 10. It can be seen that the moduli present the same trend as hardness in the initial elastic.

Fig. 6. (a) The load-depth curves of PS under spherical indenters with different radii, (b) The load-depth curves of PE under the same indenters. 584

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Fig. 7. (a), (b) and (c) The atomic-level strain tensor maps of PS at loading depth of 1 nm, 4 nm and 7 nm respectively and (d), (e) and (f) the strain tensor maps of PE at the corresponding loading depth. Each figure is the central slice as thick as 3 nm of the corresponding substrate after the strain tensor are calculated.

4.2. The influence of indenter cone angle on ISE performance Fig. 11 shows the load-depth curves at loading rate of 1 nm/ps under conical indenters with bluntness defined previously. It shows that for the same loading depth an indenter with larger bluntness will cause larger load in both PS and PE cases, meaning the indenter bluntness has an evident influence on the load-depth curves. But comparing Fig. 11(a) and (b) we can find all curves in PS case show an uptrend during the whole loading depth while curves in PE case show a downtrend during the deeper depths especially for indenters with larger bluntness. It is consistent with simulation result using spherical indenters that the higher bending stiffness caused by benzene ring components in PS chains make it exhibit more rigidity against indentation. Fig. 12 shows the corresponding hardness curves which are obtained by directly dividing the load curves by contact area curves calculated according to the geometric equation of the corresponding conical indenter [53]. In both cases, all curves initially increase with the increasing loading depth till they reach the respective hardness peaks at about 0.5 nm. It is consistent with the uptrend of hardness curves in the initial elastic stage under spherical indenter, considering meanwhile the fact that the tip of the conical indenter is a sphere with radius of 0.5 nm. Then they seesaw ever lower until hardness finds some new peaks. This is because as the indenter penetrates down, the tip effect of the cone makes the deformation of substrate turns to be plastic. What should be really noteworthy is the downtrend of hardness curves in both cases, which is similar to the ISE performance in experiments. To quantitatively compare the ISE performance in this stage, we fitted the harness curves at the decreasing stage with Eq. (12) derived for experimental data. The employed hardness data and the corresponding fitting curves are depicted in Fig. 13(a) for PS and Fig. 14(a) for PE, respectively. The corresponding fitted parameters were H0 and cl displayed in Table 1 for PS and Table 2 for PE, respectively. In PS case, as is shown in Fig. 13(a), the hardness curves at this stage are well fitted with Eq. (12). It means that the decreasing hardness is consistent with the prediction by Nikolov et al. [25] and Han [31] that the downtrend of ISE comes from Frank energy related to finite bending stiffness induced by benzene ring component and interactions between neighboring chains. This kind of consistency also presents the validity of their hardness models proposed based on the network framework of cross-linked wormlike chains. The bluntness of indenter obviously

Fig. 8. The unloading curves and Hertz fitting curves of PE and PS under “7 nm” indenter.

It should be noted that all the hardness and moduli are computed within the indentation depth no more than 2 nm according to Fig. 6 although the indenter radius can reach as much as 7 nm. For each indenter radius, H and E are only once computed at the end of elastic stage, so the variation of H and E can be practically seen as increasing with the increasing indentation depth at this stage. Similar ISE performance of the hardness at this scale has been observed in our previous work for indentation molecular mechanics (MM) simulation of PE [53]. Due to the initial contact between the indenter and a handful of molecular branched chains extending out of the microcosmically defective surface, for smaller indentation depth there are less substrate atoms which can enter into the effects radius of indenter, and consequently a smaller hardness is induced [53]. As to modulus, despite of different indenter radii, for the same substrate, all load curves in Fig. 6 have the same indentation depth within which elastic strain occurs. For a fixed indentation depth, a smaller indenter radius means a larger strain. When the strain is large enough, the substrate (partially) enters the plastic zone and thus the E decreases [61].

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influences the ISE performance—the bigger the cone angle of the indenter is, the more easily the hardness will decrease. It can be also seen from the data in Table 1 that the macroscopic hardness H0 decreases and characteristic length cl increases with the increasing loading depth, which also denotes the strengthening influence of indenter bluntness on ISE performance. Moreover, the calculated H0 and cl are very close to the experimental ones (0.265 and 30.1, respectively) when the bluntness is 0.9. Considering in indentation experiments the cone angle A of 70.3°gives the same nominal contact area per unit depth as a Berkovich indenter, our results sufficiently support the use of a conical indenter with a conical angle of 70.3°in molecular simulation of nanoindentation. But in PE case the fitting result of Eq. (12) to the hardness data is not good. The influence of indenter bluntness on the ISE performance of PE is nonlinear. It can be even seen from Table 2 that the calculated H0 and cl for bluntness 0.9 become extremely small and large, respectively, which is clearly unreasonable. Nevertheless, Fig. 14(a) shows that the downward ISE of PE still exists at nano-scale although this phenomenon is not so much obvious in continuum experiments [32]. In fact, a larger scale Y-axis was used in their articles when they depicted the hardness curves of PE so that the curves look more horizontal. The seeming nonexistence of downtrend of ISE in continuum experiments is also attributed to that the ISE of PE is too trivial, because according to the stain gradient model the ISE derived from the Frank energy is only related to neighboring interactions between chains when the benzene ring component is absent. The absence of benzene ring component can also explain why the proposed hardness model is ineffective to predict the ISE performance of linear polymer. A newly depicted curve of the experimental data in Fig. 14(b) shows that there obviously exists the downward ISE of PE at scale smaller than 800 nm. Both the simulation result and the experimental data demonstrate that the ISE still exists for PE, just less obvious than other substances.

Fig. 9. The hardness calculated at the end of initial elastic stage of load curve of PE and PS, respectively. An indenter radius series of 10 Å, 20 Å, 30 Å, 40 Å, 50 Å, 60 Å, 70 Å for both PE and PS are considered.

4.3. The influence of the loading rates on ISE performance The load-depth curves in Fig. 15 produced by different loading rates show that the loading rate has significant influence. Under high speed indentation, the substrate has less time to relax and dissipate the punch of indenter so that larger load was induced. Due to more dissipation, load-depth curves under low loading rates in both cases grow very slowly. Especially in PE case the curve for rate 0.2 nm/s shows a downtrend between the interval 2.5 nm to 4.0 nm. It is natural that torsion and bond stretching during the process lead to the rise in the total energy because of the “macroscopic” isotropy S of the random-coil chains [40]. But in PE case, low loading rate and simple linear

Fig. 10. The modulus calculated by Hertz fitting of load curve of PE and PS, respectively. An indenter radius series of 10 Å, 20 Å, 30 Å, 40 Å, 50 Å, 60 Å, 70 Å for both PE and PS are considered.

Fig. 11. (a) The load-depth curves of PS and (b) the load-depth curves of PE. In both cases, the simulations were conducted under conical indenters with loading rate of 1 nm/ps and different defined bluntness: 0.5, 0.6, 0.7, 0.8, 0.9. 586

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Fig. 12. The hardness of PS (a) and PE (b) calculated by dividing the load by contact area and every loading depth.

Fig. 13. (a) The solid lines are fits of hardness at the decreasing stage in Fig. 12(a) with Eq. (12), which denotes that the indenter bluntness has a linear strengthening influence on the nano- ISE performance in PS and that the downtrend of nano- ISE mainly comes from Frank energy related to finite bending stiffness induced by benzene ring; (b) Experimental hardness data of PS by Briscoe [60].

Fig. 14. (a) The solid lines are fits of hardness at the decreasing stage in Fig. 12(b) with Eq. (12), which denotes that the indenter bluntness has a nonlinear influence on the nano- ISE performance in PE and that the downtrend of nano- ISE is caused by intricate factors including the trivial Frank energy related to neighboring interactions between chains; (b) experimental hardness data of PE by Briscoe [60].

molecular structure lead to that the torsion angles are more easily set to the “ordered” configuration in the “disordered” amorphous structure during limited time gap, and consequently a lower load is caused [62]. The fitting of hardness data shown in Fig. 16 at the decreasing stage make it obviously to investigate the influence of loading rates on ISE performance. Since the fitting result with Eq. (12) is not so much good, we fitted these hardness data with the suggested Eq. (11). In both cases, lower loading rates make the fitting curves more easily to approach the horizontal line. For PS substrate, loading rate has the same evident and almost linearly strengthening influence on the ISE performance as the indenter bluntness, which means the indentation deformation of PS at this scale is not only strongly related to the shape of indenter but also rate dependent. Comparing Fig. 16(b) to Fig. 14(a), Eq. (11) gives a better fitting result, which means the hardness model with more parameters is more suitable for linear PE due to the multiple and intricate influencing factors to the nano- ISE. It can also be seen the influence of loading rate on ISE performance of PE becomes more linear and distinguishable than indenter shape does. This means the indentation deformation of PE at this nano-scale is rate dependent rather than indenter shape dependent, although this dependency is not so linear and distinguishable

Table 1 The fitted macroscopic hardness H0 and the length scale parameter cl of PS. Indenter Indenter bluntness: Indenter bluntness: Indenter bluntness: Indenter bluntness: Indenter bluntness: Fit of experimental

0.5 0.6 0.7 0.8 0.9 data by Han [32]

H0 (Gpa)

cl (nm)

3.04480653 2.35375866 1.77532066 0.9430112 0.20578589 0.265

−0.09039123 0.90027911 1.91570113 5.68257793 26.31618474 30.1

Table 2 The fitted macroscopic hardness H0 and the length scale parameter cl of PE. Indenter Indenter bluntness: Indenter bluntness: Indenter bluntness: Indenter bluntness: Indenter bluntness: Fit of experimental

587

0.5 0.6 0.7 0.8 0.9 data by Han [32]

H0 (Gpa)

cl (nm)

1.10014497 1.58192564 1.61715974 1.39355877 7.30794091e-06 0.058

4.23114416 2.56770225 2.52723519 3.59043767 1.18318472e+06 0

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Fig. 15. (a) The load-depth curves of PS and (b) the load-depth curves of PE. In both cases, the simulations were conducted under conical indenters with the bluntness of 0.9 and different loading rates: 0.2 nm/ps, 0.4 nm/ps, 0.6 nm/ps, 0.8 nm/ps, 1 nm/ps.

comparing as PS curves. Such increased hardness for a fixed indentation depth at higher loading rates has also been observed by Ref [62,63]. Comparing Fig. 16(a) to Fig. 16(b), all curves in PE case enter the horizontal stage more early than in PS case, and the lower loading rate strengthens this effect, for instance, the curve of 0.2 nm/ps becomes horizontal even at depth of 3.5 nm. This is because without benzene rings and cross-linking the linear chains in PE make it more flexibly to dissipate the indentation work, so that longer loading time (lower loading rate) definitely strengthens this dissipation effect. This is also the reason why the indenter bluntness and loading rate have less effect on ISE of PE since comparing to PS case the easily induced deformation of PE makes the factors effecting the deformation of PE in nano-scale multiple and intricate.

•

5. Conclusions

•

The indentation size effect in PS and PE has been studied at nanoscale through MD simulations using spherical and conical indenters with different loading rates. From the simulation results we can draw the following conclusions.

• The load-curves of both PS and PE using spherical indenters consist

of obvious Hertz elastic stage and plastic stage. But the tip of conical indenter makes the substrate more easily to enter the plastic zone so

that all load-curves contain almost no elastic stage. During plastic stage, the curves in both PS and PE cases present the serrated shape, which has been certified as the result of alternant occurrence of elastic and plastic deformation inside the substrate. The flexible and linear chains in PE substrate make it more easily to adjust to minimize the conformation energy and thus the load curves of PE show a downtrend. Using spherical indenters, the calculated hardness at the end of elastic stage increases with the increasing indenter radius in both PS and PE case. The modulus calculated via Hertz fit of the elastic stage has the same ISE performance as hardness within the depth no more than 2 nm. This kind of ISE performance which has not be found in indentation experiments is attributed to the complex initial contact between the indenter and the substrate surface, and the easily plastic deformation of substrate under smaller spherical indenter, respectively. Using the defined conical indenter, the calculated hardness curves show that in both PS and PE cases the ISE consists of the uptrend and downtrend. The uptrend of ISE is consistent with the indentation using spherical indenter and can be explained for the same reason. The downtrend of ISE in PS case mainly comes from Frank energy related to finite bending stiffness induced by benzene ring component. The downtrend of ISE was not found in the indentation test of PE due to its trivial magnitude but found in our nano-scale

Fig. 16. (a) The solid lines are fits of hardness of PS with Eq. (11), which denotes that the loading rate has a linear strengthening influence on the nano- ISE performance in PS; (b) The solid lines are fits of hardness of PE with Eq. (11) giving a better fitting result, which means the hardness model with more parameters is more suitable for linear PE due to the multiple and intricate influencing factors to the nano- ISE. In each case, the hardness data were calculated by dividing the loaddepth curves at the decreasing stage in Fig. 15 by contact area curves. 588

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•

•

•

indentation simulation. In the absence of benzene ring this ISE in PE case is attributed to Frank energy related to interactions between neighboring chains. The indenter bluntness has a linearly and evidently strengthening effect on ISE performance of PS. Moreover, the fitting results sufficiently support the use of a conical indenter with a conical angle of 70.3°in molecular simulation of nanoindentation. As to PE, although the downtrend of ISE does not seem to exist when the experimental data was shown schematically using large scale axes, it was also found at nano-scale and nonlinearly influenced by indenter bluntness, however, in a smaller magnitude than PS. The loading rate also has a strengthening effect on ISE of PS and PE. In PE case the longer loading time makes the linear chains in PE more completely to dissipate the indentation work so that all curves in PE case enter the horizontal stage more early than in PS case, and the larger loading rate strengthens this effect. Based on the same explanation, the indenter bluntness and loading rate have less effect on ISE of PE. Thus the factors effecting the deformation of PE in nano-scale are multiple and intricate. The fitting results also show the great validity of both Eq. (11) and Eq. (12) in predicting the downtrend ISE of glassy cross-linked PS both in micro- and nano-scale. This is reasonable because the used hardness models were proposed based on the network framework of cross-linked wormlike chains. But in PE case the trivial ISE induced by neighboring interactions between linear chains make the hardness models relatively ineffective in predicting the ISE performance. Considering the fact that Eq. (11) with an extra parameter γ has more predictive power than Eq. (12), a hardness model with more parameters probably concerning factors such as formula weight, degree of polymerization, surface energy and crystalline orientation of crystalline polymers should be developed for PE.

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