Simulia Living Heart Mechanics

arXiv:cond-mat/0502303v1 [cond-mat.stat-mech] 12 Feb 2005

Abstract

We use molecular simulations to explore how sample dimensions and interfacial properties impact some generic aspects of the mechanical and structural behavior of nanoconfined materials. Specifically, we calculate the strain-dependent proper- ties of minimum-energy thin-film particle configurations (i.e., inherent structures) confined between attractive, parallel substrates. We examine how the relationship between the transverse strain and the stress tensor (the equation of state of the energy landscape) depends on the properties of the film and substrate. We find that both film thickness and film-substrate attractions influence not only the mechanical properties of thin films, but also the shape and location of the “weak spots” where voids preferentially form in a film as it is strained beyond its point of maximum ten- sile stress. The sensitivity of weak spots to film properties suggests that nanoscale materials may be intrinsically vulnerabile to specific mechanisms of mechanical fail- ure.

Key words: Nanoscale films; Mechanical failure; Potential energy landscape; Molecular simulations.

Introduction

Materials confined to very small spatial dimensions behave differently than in the bulk. In addition to showing quantum confinement effects, they also dis- play thermodynamic, kinetic, and mechanical limits of stability that depend on sample size, shape, and the characteristics of their interfaces. Specific exam- ples of property modifications include surface-induced phase transitions, shifts of the bulk glass transition, and interface-mediated modes of mechanical fail- ure (Gelb et. al, 1999; Forrest and Dalnoki-Veress, 2001; Hutchinson and Suo, 1992). Unfortunately, because molecular-scale processes in highly inhomoge- neous environments are difficult to resolve experimentally, a mechanistic pic- ture for precisely how nanoconfinement impacts stability has been slow to develop. This presents a practical barrier to the design of technological ap- plications, in particular those relying on solid-state nanostructures to exhibit mechanical integrity over a broad range of conditions.

In this Article, we study an elementary model system that sheds new light on how sample dimensions and interfacial properties can influence the mechanical behavior of nanoconfined materials. Specifically, we use molecular simulations to calculate the strain-dependent properties of mechanically-stable films of particles confined between attractive, parallel substrates. Although analogous studies have been carried out for models of isotropic materials, this is, to ∗Corresponding author. Tel.: +1-512-471-6308; fax: +1-512-471-7060

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our knowledge, the first systematic investigation of the relationship between the transverse strain and the pressure tensor of the inherent structures (min- imum potential energy configurations) of highly inhomogeneous films. Our main finding is that both sample dimensions and substrate attractions sub- stantially influence not only the mechanical properties of thin films, but also the morphology and location of “weak spots” where voids preferentially form in a film as it is strained beyond its point of maximum tensile stress. Although the precise role that these weak spots play in dynamic deformation processes is presently unknown, they appear intimately linked to material failure by quasistatic tensile deformation. Moreover, the sensitivity of weak spots to film properties suggests that nanoscale materials may be intrinsically vulnerabile to specific mechanisms of mechanical failure.

Since plastic deformation and failure are inherently dynamic events, and since the molecular-scale rearrangements that accompany them in amorphous mate- rials are still poorly understood, molecular dynamics (MD) simulations would appear to represent an ideal theoretical tool for their investigation. In fact, MD simulation studies over the past decade have been instrumental in gaining insights into deformation processes in polymeric and small-molecule materials (Falk and Langer, 1998; Gersappe and Robbins, 1999; Rottler and Robbins, 2001, 2003; Stevens, 2001; Gersappe, 2002; Capaldi et al., 2002; Varnik et al., 2004; Van Workum and de Pablo, 2003; Yoshimoto et al., 2004). These in- sights have facilitated the interpretation of experiments and have aided in the introduction of simple theories for viscoplastic flow (Falk and Langer, 1998).

On the other hand, despite recent advancements in algorithms for long-time dynamics, MD simulations are still limited to accessing relatively short time and length scales. Thus, it is currently computationally prohibitive to use MD

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to exhaustively explore the effects that sample dimensions and interfacial con- ditions have on the mechanisms of mechanical failure, even for simple model systems. The development of alternative methods for probing the molecular- scale origins of failure in glasses is still essential.

One complimentary approach is to calculate how the properties of a mate- rial’s mechanically-stable inherent structures depend on the state of strain. This strategy has typically been implemented in one of two ways to investi- gate deformation, tensile strength, and failure of amorphous solids. The first

Method

(Mott et al., 1993; Hutnik et al., 1993; Malandro and Lacks, 1997, 1998, 1999; Lund and Schuh, 2003a,b; Maloney and Lemaˆitre, 2004a,b) calls for subjecting an ensemble of inherent structures, created at a prescribed state of strain or stress, to an athermal quasistatic deformation process consisting of alternating steps of small affine deformation followed by potential energy minimization. Malandro and Lacks (1997, 1998, 1999) have used this protocol to investigate the connection between strain-induced plastic rearrangements in amorphous materials and the annihilation of minima on the material’s po- tential energy landscape. A similar implementation has also been used by Maloney and Lemaˆitre (2004a,b) to study the energy fluctuations associated with amorphous plasticity and the behavior of elastic constants near the onset of material failure.

Alternatively, one can generate collections of inherent structures at each macro- scopic strain state of interest by mapping equilibrium particle configurations from high-temperature simulations to their local potential energy minima (Stillinger and Weber, 1982). This procedure, which we adopt here, has been primarily employed by Debenedetti, Stillinger, and collaborators (Sastry et al., 1997; Roberts et al., 1999; Utz et al., 2001; Shen et al., 2002) to determine

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how the inherent structure pressure of ‘bulk’ glass-formers depends on density, a relationship that has been termed the equation of state of an energy land- scape (EOSEL) (Debenedetti et al., 1999). Several trends have emerged from simulated EOSELs that give insights into the mechanical properties of amor- phous solids. For example, inherent structures formed with densities higher than the material’s Sastry density ρS (shown in Fig 1a) are structurally ho- mogeneous, whereas “weak spots” (Sastry et al., 1997) contained in lower- density equilibrium configurations develop into fissures or voids upon energy minimization. Thus, ρS is a material property that represents the minimum density for which mechanically-stable solid structures can remain structurally homogeneous and void free. Moreover, the corresponding isotropic tension −pIS(ρS) is the maximum amount that an inherent structure of that material can sustain prior to rupture. Note that ρS obtained from the EOSEL is concep- tually similar to the density of maximum isotropic tensile stress obtained from a quasistatic expansion process, and, in fact, recent simulations (Shen et al., 2002) show quantitative agreement between these two densities for a number of model materials.

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Probing Vulnerabilities to Failure in Nanoscale Films The EOSELs of ultrathin films have been recently predicted by an approximate mean-field theory (Mittal et al., 2004; Truskett and Ganesan, 2003), but, to our knowledge, they have never been determined via molecular simulation.

One naturally expects far richer behavior in thin films than in bulk materials because interfacial interactions render them anisotropic and statistically inho- mogeneous. One consequence, illustrated in Fig. 1b, is the emergence of two

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distinct versions of the curve shown in Fig. 1a, representing the density depen-

Is) Components Of The Inherent

structure pressure tensor. This additional dimension of the EOSEL prompts several new questions pertaining to the film’s intrinsic vulnerabilities to me- chanical failure. For example, how do film thickness and substrate attractions affect the directionality of mechanical failure (i.e., which component of the stress tensor will exhibit a maximum at a smaller value of strain)? Moreover, is structural failure (i.e., void formation) initiated at the strain of maximal stress as it is in bulk materials (Sastry et al., 1997; Debenedetti et al., 1999)? If so, do film thickness and substrate attractions strongly influence the location

And The Morphology Of The Corresponding Voids?

To investigate these issues, we calculated the EOSEL of model films com- prising N = 864 Lennard-Jones (LJ) particles (truncated and shifted at r = 2.5 (Sastry et al., 1997)) confined between parallel substrates that in- teract with them through an effective 9-3 LJ potential,

(1)

Here, z is the distance between substrate and particle center, ρwσ3

W = 0.988,

and σw = 1.0962 (Ebner and Saam, 1977). All quantities reported in this work are implicity nondimensionalized by the standard combinations of length and energy scales provided by the effective diameter σ and the well-depth ǫ of the LJ film particles, respectively. Periodic boundary conditions were applied in the x and y directions to simulate a system of infinite transverse extent.

Additional simulations (not shown here) using N = 1024 and N = 2048 particles were also carried out to verify that the results were insensitive to N.

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The effects of film thickness and strength of the film-substrate attractions were examined by simulating nine films characterized by the permutations of the following substrate separations L ∈{5, 7.5, 10} and film-substrate well depths ǫw ∈{0.2ǫfw, ǫfw, 5ǫfw}, where ǫfw = 1.2771 (Ebner and Saam, 1977).

The inherent structures required for the EOSEL analysis were obtained by mapping high-temperature (T = 2.5) equilibrium configurations from a se- ries of fluid film densities in the range ρ = 0.2 −1.0 to their local poten- tial energy minima using LBFGS (Liu and Nocedal, 1989), a limited-memory quasi-Newton multidimensional optimization routine. The equilibrium config- urations were generated by Monte Carlo simulations in the canonical ensemble.

For every film type and strain state, 100 equilibrium configurations (separated by 500 Monte Carlo cycles) were selected for minimization. The components of the inherent structure pressure tensor were calculated both by the global virial equation (Born and Huang, 1954) and by spatially averaging the lo- cal Irving and Kirkwood (1950) expressions. The two approaches produced indistinguishable results.

For the structural analysis of the inherent structure configurations, void space was defined to be the volume of film comprising all points that lie both a dis- tance d > 1 from any particle center and d > (σw +1)/2 from either substrate.

The void probability is then simply the fractional film volume available for insertion of an additional particle center without creating “overlap” with an existing particle or substrate. For each strain state, the average void probabil- ity was calculated by performing 106 trial insertions of hard-core test particles of unit diameter. These data were binned according to normal position z to determine the film’s inhomogeneous void probability profiles. The meth- ods described above for both generating and analyzing our thin-film struc-

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tures follow directly from established techniques for studying bulk materials, and the interested reader can find more information in recent review articles by Debenedetti et al. (1999, 2001).

As a final point, we discuss the main rationale for choosing the LJ model for our initial investigation. First, it is a microscopic model that shows qualita- tively realistic mechanical and structural behavior (Sastry et al., 1997), but it also is simple enough to allow a systematic investigation of the effects of inter- faces and confinement on its properties. Moreover, since the bulk LJ system is already well characterized (Sastry et al., 1997; Debenedetti et al., 1999), any confinement-induced deviations should be relatively straightforward to recognize and interpret. Finally, a knowledge of these deviations will provide a useful basis for understanding future simulations on materials with richer molecular attributes.

Results And Discussion

In this section, we study how the specific properties of our model LJ films im- pact their vulnerability to various modes of mechanical failure. To accomplish this, we investigate how sample dimensions and film-substrate attractions af- fect the strain dependencies of the stress tensor and the void-space morphology of thin-film inherent structures. The locations and shapes of the voids or ‘weak spots’ that appear in the films when they are strained beyond their maximal stress states provide markers for the corresponding mechanisms of failure. We discuss how these mechanisms arise from structural inhomogeneities in the film and the properties of the film-substrate interfaces.

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Fig. 2a shows the simulated EOSEL of an ultrathin film (L = 5) confined between ‘weakly attractive’ substrates (ǫw = 0.2ǫfw). Note that p⊥

Is, Or, Equivalently, The Max-

imum normal stress corresponds to a smaller strain than does the maximum transverse stress. Thus, from a mechanical perspective, the inherent structure film can be considered more vulnerable to failure in the normal direction when subjected to plane strain. Fig. 2a also illustrates that, as is observed in the bulk (Sastry et al., 1997; Debenedetti et al., 1999), the probability of finding voids in the film becomes non-neglible for densities below the maximal stress state. This connection between mechanical and structural failure is examined more closely in the void probability profiles and film images of Fig. 2b and 2c, respectively. As can be seen, the main structural effect of transverse strain is the formation of planar voids where the film detaches from one of the sub- strates, indicating an adhesive mechanism for mechanical failure.

One natural question to ask is, do substrate attractions noticeably affect the film’s mechanical and structural vulnerabilities to failure? Fig. 3 shows the EOSEL for an ultrathin film (L = 5) with ‘strongly attractive’ substrates (ǫw = 5ǫfw). Comparison of Fig. 2a and 3a shows that, at least for ultrathin films (L = 5), there seem to be some features of the EOSEL that are indepen- dent of the strength of the substrate attractions. For example, the film confined between strongly attractive substrates also shows mechanical vulnerability to

Value Of Density Than P∥

IS. Moreover, Fig. 3a shows a similar correspondence between the attainment of maximal stress and the initial appearance of voids in the film. However, the key difference can be found in the location of the ‘weak spots’ where strain-induced voids appear. As is evidenced by the void

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profiles and film images of Fig. 3b and 3c, the strongly attractive substrates promote a cohesive rather than adhesive failure mechanism, indicated by the formation of planar voids between the molecular layers in the center of the film. This mode of failure results in films with much larger tensile strength and toughness than films that fail adhesively.

The main physical result from Fig. 2 and 3 is that strain-induced local stresses in highly inhomogeneous films, dictated by the relative strengths of the inter- particle and film-substrate attractions, introduce specific types of vulnerabil- ities to failure (e.g., adhesive or cohesive failure). These vulnerabilities have important implications for the material properties of the films. Interestingly, there is also preliminary evidence to suggest that such energetic vulnerabili- ties may strongly influence the actual mechanisms of dynamic failure events.

In particular, recent MD simulations of polymeric thin films confined between solid substrates (Gersappe and Robbins, 1999) have found that the transition from adhesive to cohesive mechanisms of failure is essentially independent of temperature and, moreover, that it is primarily controlled by the strength of substrate attractions. The fact that both dynamic and EOSEL-type simula- tion studies provide a similar physical picture is intriguing. However, more sys- tematic investigations of the energetic and entropic contributions to dynamic failure events in model systems will be necessary to assess the generality of these observations.

To explore the impact of thickness L on the mechanical and structural prop- erties of nanoscale films, we consider two additional cases. The first, shown in Fig. 4a, is the EOSEL of a thicker film (L = 10) confined between strongly attractive substrates (ǫw = 5ǫfw). Note that considerable asymmetry between

Is And P∥

IS is still apparent in thicker films, with the former component at-

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taining a larger magnitude of maximum tensile stress at a slightly smaller value of plane strain. This behavior reflects the fact that film particles are strongly attracted to the substrates, making it favorable for the film to acco- modate plane strain by tearing itself apart internally. However, the proximity of the strains of maximal normal and tensile stress in Fig. 4a (as compared to Fig. 3a) suggests that one can expect differences between the structural mechanisms of failure for thicker versus ultrathin films.

To interrogate the microscopic origins of these differences, we look to the void space analysis. As can be seen by comparing Fig. 4b and 4c to Fig. 3b and 3c, cohesive failure does show qualitatively different structural consequences in ultrathin and thicker films. While ultrathin films fail by forming planar voids between molecular layers parallel to the substrates, thicker films generally fail by forming voids that are ‘local’ and more isotropic (more balanced in the transverse and normal directions), not so different from the “bubble-like” voids that form from weak spots in bulk materials (Sastry et al., 1997; Shen et al., 2002). Note that this local cohesive failure mechanism produces a film with a much smaller tensile strength than the ultrathin film of Fig. 3.

The second thicker-film case (L = 10) that we examine is the one confined between neutral substrates (ǫw = ǫfw). What type of failure mechanism is expected to prevail for this film? Fig. 4 shows that in thicker films, like in bulk materials, there is a strong tendency to cavitate upon expansion due local cohesive stresses. However, since the substrates are not nearly as attractive in the neutral wall film, it is not clear at the outset whether cavitation will be controlled by the cohesive stresses in the normal direction (as in Fig. 4) or in the transverse direction.

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The EOSEL shown in Fig. 5a displays only slight asymmetry in the normal and transverse pressure tensor components, indicating that the two aforementioned mechanisms for failure (normal cohesive and transverse cohesive) are in close competition. In contrast to the results of Fig. 4a, p∥

Is In The Neutral Wall Film

exhibits its minimum at a density slightly larger than does p⊥

Is. This Suggests

that there is a preference for the transverse cohesive mechanism for failure. To see the structural consequences of this type of failure, we examine the void probability profiles and the film images of Fig. 5b and 5c, respectively.

Fig. 5b shows that while voids can appear anywhere in the film, they show a modestly higher tendency for forming near the neutral substrates. The film configurations of Fig. 5c confirm that the strain-induced voids that form in the thicker film are also “bubble-like” like in Fig. 4c, as opposed to the planar voids observed in the ultrathin films of Fig. 2 and 3.

Finally, we point out there is an additional structural consequence of trans- verse cohesive failure. Since the initiating voids have a modest preference for forming near substrates with weak attractions, and since they grow in size upon the application of strain, they can ultimately lead to a secondary adhe- sive failure event. This simply means that it may not be easy to distinguish between adhesive and transverse cohesive modes of failure “after the fact”, i.e., by only examining the final structure of the failed sample. In contrast, we do not find a similar strain-induced cascading of failure events for thin films confined between strongly attractive substrates.

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