I. Introduction

The vitreous state of matter is ubiquitous in modern technology. For example, waveguides composed of pure, glassy silica provide the infrastructure for high-speed fiber optic networks.¹ In the pharmaceutical industry, glassy matrices composed of sugar and water are used to store and preserve labile proteins.² Most polymeric engineering plastics are amorphous solids, and their manufacture serves as a major driving force in the chemical industry. Bulk metallic glasses have been shown to exhibit levels of mechanical strength substantially higher than even the most carefully processed conventional metals and ceramics.^3,4 Combined with their relatively low electrical conductivity, high corrosion resistance, and low density, this particular family of glasses offers exciting possibilities for high-performance materials in the future. Despite its widespread technological utility and promising prospects, the vitreous state remains the most poorly understood state of matter at the molecular level.

Although much theoretical work remains to be done to explain the microscopic mechanisms underlying the extraordinary viscosity slow-down that is the hallmark of the glass transition, a number of practical routes to the glassy state are known, ranging from vapor deposition to ion implantation and cold compression of crystals.^5-7 Historically, the most common route to the glassy state is to supercool a liquid fast enough to avoid crystallization. The resulting material lacks long-range crystalline order, and its properties are closely related to those of its precursor, the supercooled liquid. However, what distinguishes a glass from a liquid is its ability to resist shear deformation. In other words, glasses exhibit proportionality between stress and deformation: they deform reversibly under shear and possess a shear modulus, much like crystalline solids. Unlike crystalline materials, and because they possess inherently liquidlike disorder, glasses tend to exist in bulk form as homogeneous solids devoid of microscopic defects and microstructural heterogeneities that their polycrystalline analogues possess, and which facilitate void nucleation or crack propagation. The important consequence of this is that amorphous solids can exhibit surprisingly high mechanical strength.

Ultimate mechanical strength is an important material property to consider in any engineering application. Experimental determination of a material's ultimate strength is commonly done dynamically, by deforming a specimen at a fixed rate of strain until mechanical failure occurs.^8,9 The ultimate strength is the stress at the point of failure. Because the results are sensitive to the experimental parameters, such as strain rate and specimen size, several measurements are usually performed to obtain reliable statistics. However, destructive testing can be potentially expensive, especially for materials that are difficult to produce, and it is thus desirable to predict their ultimate mechanical properties by nondestructive means. Theory and simulation can play a useful role in this respect. Because most practical theories for glasses, in particular those describing their mechanical properties, remain phenomenological,¹⁰ computer simulation presents itself as a useful alternative not only for predicting the properties of these materials, but also for aiding in the development of rigorous, microscopic theory.

In this spirit, there has been considerable effort devoted toward predicting the ultimate tensile strength of engineering materials. Because their atomic positions can be determined experimentally, crystalline solids have historically been the focus of such studies. Traditionally, the study of material strength in a lattice has been formulated as a stability problem.¹¹ More recently, at the microscopic-length scale, significant effort has been devoted toward predicting the tensile and shear strength of crystalline solids using sophisticated ab initio and quantum mechanical simulations.^12-17 Although much of this work, in addition to classical simulation studies, has provided important insight into failure mechanisms operating in ideal solids,^18-26 there still remains significant disagreement between computational predictions and experimental measurement.¹³

Because glasses possess history-dependent structure and are hence not in equilibrium, they pose difficulties for theoretical and computational modeling not encountered with crystals. Polymeric systems have received special attention due to their industrial prominence.^27-29 Ab initio molecular dynamics studies have focused on the rupture strength of the carbon-carbon bond in simple n-alkane chains in various conformations but are limited to systems of only a few molecules.^30-33 Kinetic Monte Carlo methods,^34,35 which are capable of incorporating a variety of failure mechanisms, such as bond rupture and chain slippage, have proven quite useful in modeling the actual experimental tensile strength protocol, but at the outset assume rate laws for the microscopic mechanisms included in the simulation. In such an approach, because the microscopic structure is unknown, a very coarse-grained representation of the amorphous network is employed.^36-38

Since its introduction, the energy landscape formalism^39,40 has provided valuable insight into the molecular-level phenomena occurring in supercooled liquids and glasses.^41,42 However, only in recent years has the connection between the landscape and the mechanical properties of glasses been demonstrated.^43-45 In particular, it is the so-called equation of state of the energy landscape that bridges the gap between the two, and it is this thermodynamic construction that can be exploited to predict the ultimate strength of glassy materials. Thus far, investigations of the ultimate isotropic tensile strength of glasses have been limited to monatomic fluids,^43,46 water,⁴⁷ and simple hydrocarbons.⁴⁸ What remains unknown is the connection between molecular architecture and mechanical strength. Elucidation of this relationship requires systematic study, and as a starting point, we have performed such a study for the n-alkane glasses. Interest here in the n-alkanes is motivated by the fact that these simple linear hydrocarbons exhibit a rich range of thermodynamic behavior as a function of the simplest molecular parameter, namely chain length n. For example, surface-freezing phenomena^49-53 and the even-odd effect⁵⁴ in their freezing temperatures have recently received special attention. Their critical pressure and mass density exhibit maxima at n = 2 and 6,^55-57 respectively, and display power-law dependence for large n.⁵⁸ Furthermore, the n-alkanes serve as a convenient starting point for studying amorphous polymeric systems.

In this paper, we present results from a systematic study of the ultimate isotropic tensile strength of the n-alkane glasses using the energy landscape formalism. The most striking result of this work is that the strength within this family of glasses is maximized at n = 3. The outline of this paper is as follows. Section II reviews the landscape formalism and its relation to the mechanical strength of glasses. Section III describes the simulation model for the n-alkanes and methods used in this study. Results and discussion are presented in Section IV. Finally, Section V contains the main conclusions and lists the open questions suggested by this work.

II. Energy Landscape/Inherent Structure Formalism

The potential energy hypersurface or energy landscape, (r^N), is simply the relationship between a system's total potential energy and its degrees of freedom, and it naturally contains all the physics that govern its dynamics and thermodynamics. Although the complexity of this surface precludes its detailed description in all but the simplest systems, significant simplification can be achieved by configurational mapping to topologically important features in the landscape, namely potential energy minima. By definition, these minima are mechanically stable, and provided that every point in configuration space (with the possible exception of a zero-measure subset) can be mapped uniquely to a minimum, they represent the mechanically stable atomic packings that a system inherently samples as it explores configuration space. Accordingly, these local potential energy minima are called inherent structures, and configurations that map to the same minimum are collectively referred to as a basin of attraction. Given a unique configurational mapping process, it can be rigorously shown that the thermodynamics of glasses and supercooled liquids can be described in terms of the distribution of inherent structure energies;³⁹ at the same time, dynamics are related to transitions between basins.⁵⁹ Within the energetic hierarchy of this formalism, it naturally follows that crystalline solids are systems occupying the lowest-lying energy basins; glasses are systems trapped within amorphous inherent structures whose energies are higher than that of the crystal; and deeply supercooled liquids are systems that move infrequently between amorphous basins.

Mathematically, for a system of N atoms in the canonical ensemble, the simplest configurational mapping takes each atom i along a steepest descent path^39,40

The connection between the landscape and macroscopic properties of a glass is provided by the equation of state of the energy landscape, that is to say average inherent-structure pressure P_IS as a function of bulk density.⁴⁴ It is precisely this thermodynamic construction that is exploited here as a method to determine the ultimate isotropic tensile strength of a glass. Because a glass is a liquid trapped within a potential energy minimum or inherent structure, the equation of state of the energy landscape physically corresponds to the equation of state of a liquid in the T = 0 limit. In the canonical ensemble, construction of the landscape equation of state involves taking configurations from a thermally equilibrated simulation at some temperature and density, minimizing their potential energies by eq 1, and then calculating the average inherent-structure pressure. This procedure is repeated at other densities, while maintaining the same equilibration temperature. Although the equilibration temperature affects the average inherent-structure energy, that is the average basin depth the system samples, the equation of state of the landscape (pressure vs density) is largely insensitive to it as long as the equilibration temperature is sufficiently high (i.e., greater than the critical temperature). A notable exception is water whose energy landscape is usually studied under conditions where the liquid is supercooled and therefore exhibits slow inter-basin dynamics.^47,61,62 For illustrative purposes, the complete equation of state of the energy landscape for n-hexane is shown in Figure 1. It is clear that the landscape equation of state depends nontrivially on density and its shape resembles the isotherms of analytical equations of state for liquids. However, there are two important distinctions. First, although points along an isotherm in a real equation of state are directly connected by a continuum of thermally equilibrated states, points along the equation of state of the landscape are only indirectly connected to each other via the thermally equilibrated simulation runs from which they were obtained. Second, points along the "unstable" portion of the landscape EOS are mechanically stable (i.e., they represent local-minimum energy configurations), whereas corresponding points in analytical equations of state are unphysical (i.e., experimentally unattainable). These subtleties aside, the equation of state of the landscape exhibits behavior that is characteristic of a real equation of state. For example, the density at which the average inherent-structure energy E_IS reaches a minimum coincides with the density at which the average inherent-structure pressure P_IS is zero.⁴⁸ It should be emphasized that although the phrase "equation of state of an energy landscape" has other meanings in the literature,⁴⁵ our usage of the phrase in this paper follows the original definition,⁴⁴ namely the average inherent structure pressure as a function of bulk density. Physically, this construction can be interpreted as the T = 0 isotherm of a liquid, in the sense that the system is equilibrated under conditions where it can sample all energy minima, but its properties are studied upon removal of the kinetic energy.

Figure 1 Equation of state of the energy landscape for n-hexane. This zero-temperature isotherm can generally be divided into three density regions A, B, and C. In region A, PIS > 0. In region B, PIS < 0. In region C, the inherent structures are fissured and the packings are spatially inhomogeneous.

For systems studied to date,^43,46-48 the equation of state of the landscape can generally be subdivided into three density regions, as shown in Figure 1. The high-density region A is characterized by compressed inherent structures. At intermediate densities, region B, inherent structures are under isotropic tension. The pressure here decreases with decreasing density until a minimum in pressure is reached. As in region B, inherent structures in region C are also under tension, but the pressure here increases with decreasing density and eventually vanishes as density approaches zero. The density at which the minimum pressure, _S, is attained is called the Sastry density, _S, and separates region B from C.

Both P_S and _S are characteristic material properties and have special physical significance. Statistical geometric analyses of inherent structures in atomic and simple molecular fluids have shown that they are spatially heterogeneous below _S, that is to say inherent structures below this limiting density consist of densely packed regions of molecules coexisting with voids or cracks.^43,48 Thus, the Sastry density represents the lowest density at which a homogeneous glass can exist and hence corresponds to the maximally stretched glass. Therefore, the isotropic tension at this density, -P_S, is the maximum tension that a homogeneous glass can sustain prior to fracture, otherwise known as its ultimate isotropic tensile strength. These arguments strongly suggest that the Sastry point is the T = 0 limit of the superheated liquid spinodal,^43,44 and represents a fundamental limit to homogeneous glass formation. For monatomic fluids equilibrated above their critical temperature, the Sastry point bears a simple yet intriguing relationship to the critical point. The ratio of the Sastry density to critical density is roughly 3, whereas the ratio of the ultimate isotropic tensile strength to critical pressure is approximately -27.⁴² An important aspect of this work is to investigate whether such simple relationships hold for molecular fluids.

The equation of state of the energy landscape is not only the means by which to study the mechanical properties of glasses within the landscape formalism. Alternatively, one can probe their mechanical behavior by directly deforming the inherent structures themselves.^27-29,63 By using a series of steps consisting of incremental deformations and subsequent energy minimizations under constant strain, the mechanical response of glasses under various loading conditions can be investigated in the low-temperature limit. Although the majority of these studies have focused on how the underlying landscape changes as a result of structural rearrangements that accompany macroscopic strain, in particular, the relationship between mechanical yielding and the disappearance of energy minima,^60,64,65 such an approach can, in principle, also be used to determine the ultimate mechanical strength of glasses.⁶⁶

We have studied the ultimate isotropic strength of the n-alkane glasses using two methods. In the first, the ultimate isotropic strength is determined by constructing the equation of state of the energy landscape and locating the Sastry point. In the second method, high-density inherent structures are processed by a sequence of incremental isotropic expansions and energy minimization steps. This series of steps is repeated until the system fractures, that is to say a minimum in pressure as a function of density is reached. A priori, it is unclear how the fracture properties determined by the two processes relate to each other, and hence an important part of this investigation is to clarify this relationship.

III. Simulation Model and Methods

In this work, the n-alkanes are modeled as fully flexible chains of united atoms with each interaction site corresponding to a CH₄ molecule or either a -CH₃ or -CH₂- group. For a linear chain of united atoms of length n, its conformation is completely specified by the set of n-1 bond lengths d, n-2 bond angles , n-3 dihedral angles , and three Euler angles (, , and ) describing the relative orientation of the first two bonds in the chain. This set constitutes the chain's generalized coordinates. In conjunction with the position of the first united atom within the chain, the spatial coordinates of each interaction site can be specified completely by 3n degrees of freedom.

Bond stretching and bending are described by harmonic potentials^67,68

The above model for the n-alkanes was developed and optimized by Errington and Panagiotopoulos^69,71 to fit experimental liquid-vapor equilibrium data. There are two key features to the model. First, the methyl and methylene groups differ in size and possess different energy well depths. In Table 1, it seems counterintuitive that a methyl group should be smaller than a methylene group, but this is actually consistent with other force fields designed for simulating n-alkanes.^67,72 Second, adjacent united atoms within the same chain overlap by virtue of specification of the equilibrium bond lengths. It has been shown that these are essential requirements in order for a united atom model to reproduce the maxima in critical pressure and density as a function of chain length.⁵⁸ The Buckingham exponential-6 potential was cut and shifted so that the force vanishes smoothly at 15 Å. This was necessary because the energy minimization procedure requires a continuous force for numerical stability. Although this modification changes the potential slightly from its original form, it is expected that deviations from the experimental properties for which the parameters were optimized will be systematic (in the same direction) for all carbon numbers.

To construct the equation of state of the energy landscape for the n-alkanes, either configurational-biased Monte Carlo (CBMC)^73,74 or multiple time-step molecular dynamics (MTS MD)⁷⁵ simulation was used to generate thermally equilibrated configurations in the canonical ensemble for energy minimization. These more sophisticated simulation methods were required to deal efficiently with the stiff degrees of freedoms in the chains. The choice of simulation was based on computational convenience. A system consisting of 1600-2000 united atoms in a cubic simulation box with periodic boundary conditions was used in all cases. All species were equilibrated at approximately 1.5× the experimental critical temperature, to ensure that the system could in principle sample the overwhelming majority of local energy minima in the landscape. The Monte Carlo simulations consisted of an equilibration period of two million trials followed by a production period in which configurations were saved every 200 000 trials. In the molecular dynamics simulations, the equilibration period lasted for 200 ps using a large and small time step of 2.0 and 0.5 fs, respectively. Temperature was thermostated using a Nose-Hoover chain of length five, and the equations of motion were integrated using an explicit reversible multiple time step integrator as developed by Martyna et al.⁷⁶ Configurations were saved every 10 ps for energy minimization. For each species, thirty configurations per state point were generated, with the exception of n = 48, where forty configurations were saved to obtain better statistical sampling. Energy minimization was performed using the conjugate gradient method⁷⁷ and was terminated when the relative fractional energy difference between the previous and current step was less than 1 × 10^-8.

Investigation of ultimate tensile strength via direct inherent structure deformation was only performed for n-alkanes of length n < 48. High-density inherent structures corresponding to P_IS = 0 were taken as the starting configurations. Each of these was expanded isotropically stepwise by 5 kg/m³ and then its potential energy was minimized. This process was repeated until a minimum in pressure as a function of density was reached. For each chain length, results were averaged over fifteen such runs.

IV. Results and Discussion

We have studied the ultimate isotropic tensile strength of the n-alkane glasses by constructing their landscape equations of state. Before proceeding, we first demonstrate that the Sastry point also corresponds to a condition of fracture for complex molecular fluids, such as those investigated here, and not just for atomic systems.⁴³ Representative inhomogeneous and homogeneous inherent structures for n = 8 from both sides of the Sastry density are shown in Figures 2 and 3, respectively. It is clear that the fractured inherent structure, Figure 2, contains significant cavity space, which not surprisingly resembles a crack in the system, whereas the homogeneous inherent structure, Figure 3, is devoid of such defects. The connection between the Sastry point and mechanical strength can be demonstrated quantitatively by statistical geometric analysis of the void space⁷⁸ present in the inherent structures as a function of density. In Figure 4, the average inherent-structure pressure and void fraction are plotted as a function of density for n = 8. As in the case for simple fluids, high density inherent structures are spatially homogeneous above _S and inhomogeneous below. More importantly, as the system is expanded isotropically, Figure 4 clearly shows that the density at which cavities begin to form is practically coincident with the density at which the minimum in pressure occurs, namely the Sastry density. Notice that the size of these initial cavities is of the order of atomic dimensions. Because harmonic bonding potentials are used here, bond rupture is precluded as a viable fracture mechanism, and therefore, fracture in this case is driven solely by the relative displacements of individual molecules. The loss of cohesion below _S is directly related to the creation of void space. The physical picture that emerges from these observations is that the Sastry point corresponds to the maximally stretched molecular glass, whereas the pressure at this point is its ultimate isotropic tensile strength, or the maximum tension that can be sustained just prior to fracture. Although results have only been presented for n = 8, identical behavior is observed for all chain lengths investigated in this work.

	Figure 2 Representative fractured inherent structure for n-octane ( < S). Dark spheres correspond to methyl groups, and the lighter ones are methylene groups.
	Figure 3 Representative homogeneous inherent structure for n-octane ( > S). Dark spheres correspond to methyl groups, and the lighter ones are methylene groups.
	Figure 4 Equation of state of the energy landscape (circles) and the average void volume fraction (squares) in the inherent structures as a function density for n-octane, n = 8.

The equations of state of the energy landscape for the n-alkanes of chain length n = 1, 2, 3, 4, 6, 8, 16, 24, and 48 are shown in Figure 5. All of the curves have similar shapes and extend significantly into the negative pressure region where the system is under isotropic tension. More importantly, notice that pressure on the y-axis is given in units of kbar, which is the same order of magnitude as the tensile strength of steel alloys.⁸ There are two important trends to glean from Figure 5, namely the chain length dependences of _S and -P_S. For clarity, these two quantities are plotted separately as a function of inverse chain length in Figures 6 and 7.

	Figure 5 Equations of state of the energy landscape for the n-alkanes studied in this work, n = 1, 2, 3, 4, 6, 8, 16, 24, and 48.
	Figure 6 Relationship between the Sastry density and inverse chain length for the n-alkanes studied in this work. The line is a guide to the eye.
	Figure 7 Relationship between the ultimate isotropic tensile strength determined by the landscape equation of state and inverse chain length for the n-alkanes studied in this work. The line is a guide to the eye.

Figure 6 shows that the Sastry density initially increases with chain length but then reaches a plateau value at approximately n = 16. This nontrivial chain length dependence is due to the fact that adjacent united atoms within the same chain are actually interpenetrating spheres by virtue of the bonding constraints in the simulation model. Consequently, the volume occupied by m unbonded united atoms is larger than that of m bonded ones. Because the Sastry density is the lowest density at which a mechanically stable amorphous solid can exist absent of any void space, this limiting density should at least initially increase with carbon number. Beyond some intermediate chain length, the homogeneous amorphous packing problem that is associated with the Sastry density becomes insensitive to carbon number. In other words, in the long chain limit where n is large, a system composed of chains of length n is indistinguishable from a system composed of chains of length n+1 at least from a molecular packing point of view. This naturally gives rise to an asymptotic approach to a limiting value of _S in the limit of large n.

In Figure 7, the ultimate isotropic tensile strength, -P_S, is plotted as a function of inverse chain length. There are two striking features to this plot. The first is the tensile strength maximum at n = 3, which is reminiscent of the maximum in critical pressure for the n-alkanes at n = 2. The second is that the isotropic tensile strength decreases with increasing carbon number beyond n = 3. In contrast, we note that the average inherent structure energy per molecule at the Sastry point (E_IS(_S) < 0) decreases monotonically with carbon number throughout the entire range of chain lengths studied. The tensile strength results are quite remarkable because physical intuition would lead one to believe that it should in fact increase with carbon number simply due to chain entanglements. Although the degree of chain entanglement is unknown in these systems, work by Saitta and Klein³² suggests that systems composed of united atom n-alkanes are capable of exhibiting some degree of topological chain entanglement starting at n = 9 even though they do not display rheological entanglement dynamics. It has been found experimentally that the ultimate tensile strength for amorphous low density, linear polyethylene under uniaxial tension decreases with molecular weight.⁷⁹ In reference to previous work on the ultimate isotropic tensile strength of simple hydrocarbons,⁴⁸ the complex chain length dependence found here is consistent with the observation that chains of length n = 2 and 5 possess very similar tensile strength.

That the ultimate isotropic tensile strength should be maximized at n = 3 is not obvious. Because fracture is initiated by voids,^8,9 a statistical geometric analysis of the n-alkane glasses at their Sastry density was performed to see if the defects that drive mechanical failure could provide physical insight into the origin of the tensile strength maximum. Details of the geometric algorithm, which allows calculation of the volume, surface area, and connectivity of voids, have been given elsewhere.⁷⁸ The average void fraction and defect density are plotted as a function of inverse chain length in Figures 8 and 9, respectively. It is interesting to see that the void fraction at the Sastry density, Figure 8, behaves non-monotonically with chain length. Although this quantity does not correlate perfectly with tensile strength, it too displays an extremum at small chain length. However, the defect density, which is defined as the number of cavities per unit volume at the Sastry point, correlates quite well with tensile strength, exhibiting a minimum near n = 3. The observation that the void fraction and defect density at the Sastry point correlate with tensile strength suggests that molecular packing effects play a basic role in determining the relative ultimate mechanical strength of n-alkane glasses. This should not be entirely surprising because the manner in which molecules pack together directly affects their propensity to form voids and cavities which can lead to failure.

	Figure 8 Average void fraction in the n-alkane inherent structures at their respective Sastry densities plotted as a function inverse carbon number. The line is a guide to the eye.
	Figure 9 Defect density in the n-alkane inherent structures at their respective Sastry densities plotted as a function of inverse carbon number. The line is a guide to the eye.

We have also determined the ultimate isotropic tensile strength by directly deforming n-alkane inherent structures for carbon numbers n = 1, 2, 3, 4, 6, 8, 16, and 24. Recall that the computational procedure consists of a series of isotropic expansion and energy minimization steps and, thus, closely resembles the experimental protocol one would use to measure isotropic tensile strength. These deformation curves are shown in Figure 10. In this case too, the ultimate isotropic tensile strength corresponds to the minimum in pressure, and the corresponding density is likewise analogous to the Sastry density. Statistical geometric analysis also shows that this density signals the sudden emergence of void space as the system is expanded isotropically. Although the deformation curves and the equations of state of the landscape have similar shapes, the deformation curves are noticeably sharper in the vicinity of the pressure minimum, which is indicative of a brittle, catastrophic failure mode. Interestingly, the density at which fracture occurs, as determined by both constructions, is the same. We note that although each curve in Figure 10 represents an average over fifteen runs, the density at which each run fractured occurred over a range of values around the Sastry density. Moreover, this density range increased with chain length. In Figure 11, the tensile strengths calculated by the two methods are plotted simultaneously for comparison. The direct deformation process gives rise to a slightly greater tensile strength than that obtained by isochoric energy minimization (the equation of state of the landscape). Because the initial configuration of the deformation process is itself an inherent structure, the intuitive expectation is that the process is restricted at the outset to sample lower-lying energy basins. This is in contrast to the isochoric quench procedure, in which higher-energy minima are accessible during the high-temperature equilibration simulation that connects points along the equation of state of the energy landscape. To the extent that these lower-lying basins are closer to the ground state, and hence more crystal-like than the higher-energy ones, they represent "stronger" states.⁴⁸ This difference aside, the deformation curves and the landscape equation of state both predict the same nonmonotonic chain length dependence for the tensile strength, with a maximum at n = 3. For completeness, the defect densities at fracture for the two computational protocols are shown in Figure 12. Again, results generated by the two processes exhibit the same dependence on chain length. Also, in accord with the expectation that the defect density should correlate with material strength, the defect density at fracture by direct deformation is less than that observed in the landscape equation of state. As before, this correlation suggests that the nonmonotonic chain length dependence is related to molecular packing effects. In Appendix A, a simple theory that captures this nontrivial behavior and exposes the underlying physics is presented.

	Figure 10 Inherent structure deformation curves for the n-alkanes of chain length n = 1, 2, 3, 4, 6, 8, 16, and 24.
	Figure 11 Comparison of the ultimate isotropic tensile strength determined by the landscape equation of state (circles) and direct inherent structure deformation (squares) versus inverse carbon number. Lines are guides to the eye.
	Figure 12 Comparison of the defect density at fracture for the equation of state of the energy landscape construction (circles) and the direct inherent structure deformation curve (squares). Lines are guides to the eye.

Because the extent of chain entanglement increases with chain length, the intuitive expectation is that ultimate tensile strength should increase with carbon number. We again point to the work of Klein and co-workers^30-33 who have investigated the effect of topological constraints, including chain entanglements and self-knots, on the rupture strength of the unsaturated carbon-carbon bond in n-alkane molecules ranging from n = 9 to n = 35. Their work suggests that chains of length greater than n = 8 should contain topological interchain entanglements, even though they might not exhibit rheological entanglement dynamics. Although the topological problem of identifying chain entanglements at the atomistic level remains unsolved,^37,80-86 visual inspection of the configurations for the longer chains, n = 16, 24, and 48, confirms that the molecules tend to adopt curled or folded conformations and tend to intertwine, indicating there is some degree of entanglement present. However, results from the landscape equations of state and the deformation curves show a counterintuitive trend, namely that the tensile strength decreases with chain length beyond n = 3. The disagreement between intuition and the results presented here stems from differences between the experimental and computational (this work) protocols used to measure ultimate strength. Experimentally, tensile strength tests are performed while deforming the specimen at a constant strain rate. The measurement is therefore a dynamical one. Thus, what is being measured is actually a time-dependent response to a perturbation. Because chain entanglements only affect the dynamic response of a system,⁸⁷ one should expect experimental measurements of tensile strength to increase with chain length if the time scale for chain relaxation is longer than that of the imposed strain rate. However, the computational methods employed here, namely construction of the landscape equations of state and the deformation curves, provide only static measures of material strength. It is important to emphasize that the equation of state of the energy landscape is a purely thermodynamic construction that corresponds to the zero-temperature isotherm of a liquid.⁴⁴ Ultimate tensile strength determined via this construct is based solely on thermodynamic stability. Although the inherent structure deformation curves were constructed using a protocol that more closely resembles an experimental isotropic tensile strength test, the procedure does not involve any dynamic component because after each deformation, the potential energy of the resulting structure is minimized, allowing the system to attain a mechanically stable condition. In other words, the inherent structure deformation curves correspond to the stress-strain curves one would generate in an experimental measurement using an infinitely slow strain rate, in the low-temperature limit. Because the computational procedures used in this work determine ultimate tensile strength within a purely thermodynamic framework, chain entanglements should not be expected to influence the predicted material strength. In fact, it might be more appropriate to refer to material strength measured within the inherent structure formalism as the material's intrinsic mechanical strength.

We now point out another interesting trend that can be seen in Figures 5 and 10. Looking at the equation of state of the energy landscape, Figure 5, it can be seen that the curves become smoother with increasing chain length, particularly on the low density side just after fracture. Although the deformation curves, Figure 10, are fairly sharp near the fracture point, the curvature clearly decreases with carbon number. This effect is a reflection of increasing material toughness, that is to say resistance to mechanical failure in the presence of a defect,⁸⁸ with chain length. In Figure 13, the evolution of void space beyond the Sastry density is shown for the isochoric energy minimization procedure. Notice the systematic decrease in slope with increasing chain length, indicating that systems composed of short chains are particularly susceptible to the creation of void space. The same qualitative trends are observed for the direct deformation process. Physically, the data point to the fact that systems composed of small molecules are unable to accommodate "internal" void space and simply come apart catastrophically. In contrast, as the density is lowered below the Sastry point, chain molecules partially unbind to create void space, but the voids thus created are not catastrophic. The constraints associated with chemical bonding serve as "stitching points" that hold the system together. Consequently, because the number of "stitching points" increases with chain length, it becomes increasingly difficult to create cavity space beyond that already present at the Sastry point.

Figure 13 Evolution of void space beyond the Sastry density for the equation of state of the energy landscape of the n-alkanes.

It has been shown that the Sastry point for simple fluids practically coincides with the T = 0 limit of the superheated liquid spinodal and, hence, bears a simple relationship to the critical point.^42,44,48 In Figure 14, the ratios of the Sastry density to experimental critical density and the ultimate isotropic tensile strength to experimental critical pressure are plotted as a function of chain length. The ratios for united atom methane (_S/_C = 2.89, -P_S/P_C = 34) are close to the corresponding values for the Lennard-Jones fluid (_S/_C = 2.76, -P_S/P_C = 39.8). Although the experimental critical density and pressure of the n-alkanes both depend nonmonotonically on chain length, it is interesting to see that both Sastry-to-critical point ratios increase monotonically with carbon number and appear to diverge in the infinitely long chain limit. Vega and co-workers^89,90 have shown that the critical density and pressure of the n-alkanes scale as n^-3/2 in the long-chain limit. Because the Sastry density reaches a plateau value, the ratio _S/_C should indeed diverge in the limit of large n. However, based on our tensile strength calculations, it is not clear a priori how the ratio -P_S/P_C should behave in the infinitely long chain limit. We can deduce that the scaling exponent for -P_S as a function of n is less than 3/2.

Figure 14 Ratios of the Sastry density to the experimental critical density (squares) and ultimate isotropic tensile strength to the experimental critical pressure (circles) as a function of inverse chain length for the n-alkanes. Lines are guides to the eye.

The chain length dependence of the maximum in ultimate isotropic tensile strength, -P_S, within the family of n-alkane glasses is a surprising result and has yet to be confirmed experimentally. Microscopic theories designed to capture the liquid-vapor phase behavior of simple molecular fluids, such as that of Sanchez-Lacombe,⁹¹ do not predict complex chain length dependence of the pressure in the T = 0 limit of the superheated liquid spinodal and thus the underlying physics is not at all obvious. To elucidate the underlying physics, it is useful to examine the relationship between tensile strength -P_S and the cohesive energy density ² which is defined thermodynamically to be

Figure 15 Correlation between ultimate isotropic tensile strength determined by the isochoric quench process and cohesive energy density for the n-alkanes studied in this work.

In Appendix A, a simple theory capable of reproducing the maximum in cohesive energy, and therefore tensile strength, is presented. Here, we briefly summarize its key points. Notice in eq 18 that the ultimate isotropic tensile strength has been decomposed into energetic and geometric contributions. Each is treated separately in the theory. The energetic contribution to the cohesive energy density is calculated by making an analogy to the conceptual framework of lattice models where it is assumed each lattice site is occupied by a chain segment. Invoking a simple mean field approximation, the average intermolecular interaction energy per molecule can be calculated while accounting for different characteristic interaction energies for the methyl and methylene groups. The geometric contribution to the cohesive energy density, or the total volume of the system, is approximated as a simple summation over all molecular volumes, accounting for the fact that adjacent pairs of united atoms within the same molecule overlap (i.e., they are interpenetrating spheres by virtue of chemical bonding). Within this simple framework, the theory is not only able to reproduce a maximum in cohesive energy density at n = 3, but also predicts that the cohesive energy density, and therefore tensile strength, decreases with chain length beyond n = 3 and approaches a finite asymptotic value as n . Furthermore, when extended to calculate the Sastry density, the theory predicts the same qualitative behavior as observed in the simulations, namely that _S initially increases with carbon number and then reaches a plateau value.

As shown in Appendix A, these simple arguments allow the cohesive energy density to be written in the following general form

V. Conclusions

A systematic study of the ultimate isotropic tensile strength of n-alkane glasses up to n = 48 has been performed. We have shown that the equation of state of the energy landscape and the inherent structure deformation curve provide two independent measures of the isotropic tensile strength for these glasses. In particular, it is the pressure minimum in both curves that corresponds to the maximum isotropic tension that a system can sustain prior to fracture. Although both methods predict the same density at which fracture occurs, the deformation curve systematically predicts a higher tensile strength than the landscape equation of state simply because the process is biased to sample basins of lower energy. This difference aside, both constructions exhibit similar trends with respect to the Sastry density and tensile strength dependence on chain length. The Sastry density initially increases with carbon number and then reaches a plateau value at some intermediate carbon number. More interesting, however, is the nonmonotonic chain length dependence of the ultimate isotropic tensile strength with a maximum at n = 3, which is reminiscent of the maximum in critical pressure at n = 2 for this homologous series. By relating the cohesive energy to pressure in the zero-temperature limit, it can be shown by means of a simple theory that the observed chain length dependence of the Sastry point is driven by a competition between intermolecular energetic and intramolecular packing effects. Although the results seem to contradict intuitive expectations regarding the connection between mechanical strength and molecular structure, in particular, the notion that strength should increase with chain length, the crucial distinction between the actual experimental protocol and computational methods used here is the following. Experimental measurements inevitably are determined while the sample is deformed at an externally imposed strain rate, and the measured strength is therefore a reflection of the dynamic response of the system to a perturbation. It should therefore be expected that features such as chain entanglements and even cross-links increase the experimentally observed tensile strength. In this work, mechanical strength has been determined within a purely static framework and ultimate strength corresponds to a limit of mechanical stability. Because the computational protocols used in this work correspond to an experimental measurement using an infinitely slow strain rate, points of chain entanglements are always allowed to relax, and thus do not influence the intrinsic tensile strength.

From an interesting theoretical perspective, we have shown that the relationship between the Sastry point and the critical point is more complex for molecular than for monatomic fluids. Both ratios, _S/_C and -P_S/P_C, diverge in the infinitely long chain limit. Combined with theoretical work by Vega and co-workers^89,90 showing that the critical density and critical pressure for the n-alkanes scale as n^-3/2, this behavior should be expected in light of the theory for tensile strength summarized in Section IV and presented in Appendix A which predicts an asymptotic approach to a limiting value as n .

A number of interesting issues are raised by this work and deserve further investigation. Experimentally, the ultimate strength of a material usually refers to its mechanical response under uniaxial loading. Shear strength is also an important mechanical property in engineering applications. These particular properties can be determined using the inherent structure formalism, and it would be interesting to see if they exhibit complex chain length dependence. We have also pointed out that the equations of state of the energy landscape and the inherent structure deformation curves become smoother with carbon number, particularly just after the point of fracture. This smoothing effect is believed to be associated with "stitching points" provided by longer chain molecules by virtue of chemical bonds which serve to hold the system together. The implication of this subtle trend is that amorphous solids composed of longer chains are mechanically tougher, that is to say less susceptible to mechanical failure when a void defect is present, than shorter, smaller molecules. Experimental studies of the chain-length dependence of toughness are necessary to confirm or disprove this prediction.

Appendix A: Mean-Field Theory for Cohesive Density

In this appendix, we present a simple mean-field theory capable of capturing the qualitative behavior of the cohesive energy density, and therefore tensile strength, as a function of chain length. The starting point for this theory is eq 18 which decomposes the cohesive energy density into purely energetic and geometric contributions, each of which is treated independently in what follows.

The total intermolecular interaction energy is calculated using a simple mean-field approximation. To facilitate such a calculation, we make an analogy to the conceptual framework of lattice models, where we crudely assume that each chain segment occupies a lattice site. All sites are assumed to be occupied because the system of interest is at its Sastry density, where the only voids are interstitial. Surrounding each site are z near-neighbors. In this work, we take the coordination number to be an adjustable parameter.⁹⁴ For a chain of length n > 2, there are two CH₃ groups and (n-2) CH₂ groups. Intermolecular methyl-methyl interactions have a characteristic energy ₁₁, and methylene-methylene interactions are characterized by an energy ₂₂. A geometric mean is used to define interactions between different united atoms

Because the Sastry point is the lowest density at which a homogeneous glass can exist, that is to say it corresponds to the onset of void formation, the total volume of the system is approximated as a simple summation over all molecular volumes. Implicit in this approximation is the assumption that although intermolecular overlaps do exist, the overlap volume occupies only a small fraction of the total system volume which is largely composed of the volume of the molecules themselves. The average molecular volume can be written in the following general form

Figure A1 Two-dimensional schematic setup for calculating the overlap volume between two interpenetrating spheres of radii r1 and r2, with centers separated a distance d from each other. The shaded region corresponds to the overlap volume in three dimensions.

It has already been pointed out in Section IV how packing effects relate to the behavior of the Sastry density as a function of carbon number. However, more intriguing is the ultimate tensile strength, and therefore cohesive energy density, maximum at n = 3. The underlying physics that give rise to this maximum can be elucidated by multiplying and dividing eq A11 by n

Acknowledgment

P.G.D. gratefully acknowledges the financial support of the U. S. Department of Energy, Division of Chemical Sciences, Geosciences, and Biosciences, Office of Basic Energy Sciences (Grant No. DE-FG02-87ER13714).