Elasticity and scalings in amorphous solids and thin polymer films

Department of Physics

Ph.D. graduate school in Physics

Ph.D. Thesis

Elasticity and scalings in

amorphous solids and thin

polymer films

Candidate Supervisor

Andrea Giuntoli Prof. Dino Leporini

Coordinator of the Ph.D. graduate school Prof. Marco Sozzi

1 Introduction 3

1.1 The glass transition . . . 3

1.2 Polymers. . . 7

1.3 Molecular Dynamics simulations . . . 8

1.4 Outline of the Thesis . . . 13

I

Solidification and Rheology

15

2.2 Methods and models . . . 21

2.3 Polymorphs characterization . . . 23

2.4 Conclusions and perspectives. . . 38

3 Role of polymer connectivity under elastic shear deforma-tions 39 3.1 Research context . . . 39

3.2 Methods and models . . . 40

3.3 Structural features and shear deformations . . . 43

3.3.1 Comparison of the atomic structures . . . 43

3.3.2 Elastic response . . . 49

3.4 Conclusions and perspectives. . . 55

4 Nematic ordering effect on the elastic properties 56 4.1 Research context . . . 56

4.2 Methods and models . . . 58

4.3 Structural analysis . . . 60

4.3.1 Equilibrium liquid phase . . . 60

4.3.2 Quench and solid phase . . . 65

4.4 Elastic properties and scalings . . . 78 1

4.5 Conclusions and perspectives. . . 86

II

Confined liquids

88

5.2 MD simulation of confined systems . . . 91

5.2.1 Rough substrate . . . 93

5.2.2 Smooth substrate . . . 97

5.3 Dynamical properties of polymeric films and scalings . . . 101

5.4 Conclusions and perspectives. . . 108

6 Outlook 110

Introduction

1.1

The glass transition

A liquid cooled down below its melting temperature Tmelt usually undergoes

a first order phase transition and the liquid turns to a crystal. However, the crystallization is a complex phenomenon involving the nucleation of an initial organized structure and the crystal growth to reach long scale order. If the cooling down of the liquid is fast enough with respect to the time scales involved with the phase transition, it is possible to avoid crystallization and keep the system in a metastable phase, named the supercooled phase.

If the system is further cooled, eventually molecules will rearrange so slowly that they cannot adequately sample configurations in the observed time-scales, at which the liquid’s structure appears frozen. This falling out of equilibrium happens abruptly approaching a transition temperature and the resulting material is classified as a glass, see Fig 1.1. Supercooled liquids and glasses have a very rich phenomenology which is still far from being completely understood [1–3].

In the supercooled phase the typical time scales increase sharply and they become larger than the experimental observation times at the glass transition temperature Tglass, or Tg. Fig. 1.2 shows how around the melting

temper-ature Tmelt the typical time scales τα on which density fluctuations relax

and the system flows at a molecular level are of the order of the picosecond whereas at Tg the typical time scales become of the order of 102s, 14 orders

of magnitude larger [5, 6].

The increasing of the relaxation time τα is accompanied by an increase

of the shear viscosity η of the liquid. For a glass-forming liquid the typical viscosity value is of the order of 1013_{P just above the glass transition. To}

understand how large this value is, remember that at the melting point a 3

Figure 1.1: Left: typical temperature dependence of first order thermodynamic quantities such as enthalpy, entropy, volume, etc. Right: schematic plot of the free energy barrier between a liquid (shallower minimum) and a crystalline state (deeper minimum). At T = Tmelt a liquid can either crystallize (c) or remain

in the metastable state of a supercooled liquid (a). As the cooling is carried on, the supercooled liquid eventually falls out of equilibrium (b). This happens at the glass transition temperature Tg, where the relaxation time exceeds the

experimental observation times. For lower temperatures the system falls out of equilibrium forming a glass. Adapted from [4].

liquid’s viscosity seldom exceeds η ∼ 10−1P . A system with η = 1013_P

appears mechanically solid on the experimental time scale, but it still exhibits the typical structural features and lacking long range order of a liquid. The glass transition temperature marks the point at which the relaxation time of the liquid τα exceeds the available experimental time τex. It is important

to underline that the glass transition is not an actual phase transition, as Tg

depends on the experimental protocol and in particular on the cooling rate and experimental times. Still, the peculiar characteristics of a glass and the sharp increase of the relaxation time and viscosity make the definition of Tg

quite robust from an operative point of view. In particular, the robustness of the definition is granted by the exponential increase of the relaxation time and viscosity when the glass transition is approached. Tg is often defined as

the temperature at which the relaxation time is equal to 102_s.

Figure 1.2: Comparison of the diffusivity and viscosity between a typical liquid (such as the water at room temperature) and a typical glass. Note the huge increase of the relaxation time from 10−12s to 102s. Reprinted from [5].

viscous liquid state is reached, glass formation is unavoidable upon continued cooling. Any sort of molecular liquid [7], polymer [8], bio-material [9], metal [10] or molten salt [11] can form a glass. In many cases a rapid cooling is necessary to avoid the crystallization (critical cooling rate up to ∼ 106_K/s

for many metallic glasses [12]). A good glass former is a liquid which is easily supercooled, i.e. characterized by very low rates of crystal nucleation and growth at all temperatures [13]. Good glass formers are organic and ionic liquids, silicates, polymers and also some metallic glasses [14].

Despite the universality of the glass transition, different liquids show a variety of behaviors in the temperature dependence of relaxation times and viscosity approaching Tg. Close to Tg, the viscosity of liquids like silica (SiO2)

exhibits an Arrhenius temperature dependence η(T ) ∼ exp ∆E

kBT



(1.1) where ∆E represents a temperature independent barrier of energy, to be overcome by thermal fluctuations [2]. Other liquids, like many polymers, are characterized by a more dramatic increase in the viscosity with decreasing temperature. The Angell plot [15] shown in Fig. 1.3displays the temperature

Figure 1.3: Angell plot showing the viscosity as a function of the inverse temper-ature normalized at the glass transition tempertemper-ature Tg for different substances.

An Arrhenius behavior results in a straight line in this plot. This is typical of strong glass-formers. Non-Arrhenius increase of the viscosity corresponds to frag-ile glass-formers. Reprinted from [2].

dependence of the viscosity of a wide variety of supercooled liquids at atmo-spheric pressure. Defining Tg as the temperature at which viscosity reaches

the value of η(Tg) = 1013P , all the curves are normalized to the same value at

Tg. Two different behaviors are displayed in the Angell plot dividing liquids

into strong and fragile glass formers. The dynamics of strong glass formers is characterized by a clear Arrhenius behavior, and the corresponding curves of log η(Tg

T ) in Fig. 1.3 are linear. The other family of curves, to which the vast

majority of glass-forming polymers belongs, corresponds to fragile liquids, for which the viscosity increases much more strongly upon cooling toward Tg.

These non-Arrhenius data are often fitted [2] by the Vogel-Fulcher-Tammann (VFT) expression: η(T ) ∼ exp  A T − T0  (1.2) where A is a temperature independent constant. Eq. 1.2implies a divergence of the viscosity at the finite temperature T0, a prediction that cannot be

verified because the system is supposed to fall out of equilibrium as T0 is

of fragile liquids is still an actively investigated question, and many other models exist (see for example [16–18]).

While crystal solids are easily classified by their structure, in liquids and glasses this is not possible, as they lack long-range positional ordering and the molecular arrangement is complex and strongly depending upon the cooling history. The Angell plot offers a way to classify liquids through their fragility, which is defined as m = ∂ log η ∂Tg T        T =Tg (1.3)

The van der Waals molecular liquids, such as o-terfenile OT P and toluene C7H8 are the classical fragile (m = 70 ÷ 150) systems. The strong

glass-formers (m = 17÷35) are instead characterized by strong covalent directional bonds, forming space-filling networks (like silica SiO2 and germanium

diox-ide GeO2). Hydrogen bonded materials (like glycerol or propylene glycols)

present an intermediate level of fragility (m = 40 ÷ 70).

1.2

Polymers

Polymeric materials are everywhere in nature. From cellulose to the DNA molecules, from plexiglass in our homes to the nylon in the clothes, polymers exploit their high functional specificity in various applications. This extreme adaptability in properties makes this class of materials attractive for technical applications and industrial research.

Polymers [19–21] are macro-molecules made up by repetition of funda-mental units, called monomers, connected by covalent bonds. One reason for the abundance of applications of polymeric materials is the diversity in the chemical structure and composition ranging from simple linear homopoly-mers to branched polyhomopoly-mers or more complex structures. Polymer chains can be classified by their conformation and configuration. The conformation of a polymer chain is its spatial organization. Many different conformations exist: linear, star, comb etc. Some of them are reported in the left part of Fig. 1.4. On the other hand, the configuration is the succession of chemical elements along the chain. A distinction can be made between homopolymers, consisting in chains of a single repeated unit, and copolymers, i.e. chains of monomers of different species, arranged in an alternating, random, block or other ways (right part of Fig. 1.4).

Polymer science is attested to begin in 1920s [22], and the interest has been growing year after year also from a theoretical point of view, because of their many scaling properties [23]. Being very large molecules, polymers

Figure 1.4: Classification of different types of polymers. Differences can be found both in the chain architecture (left panel) and in the chain constitution (right panel).

are very complex systems to study, especially due to the many diverse length scales involved in their dynamics, from the internal structure of a single monomer to the length and structure of the entire chain. The competi-tion between chain-scale and monomer-scale spatial disposicompeti-tion gives rise to phenomena of geometrical frustration: at low temperature, the tendency of monomers to assume positions corresponding to an ordered configuration is opposed by the constraints imposed by the conformation of the chains. For polymers with regular enough chain structure, this usually results in two phase structures, in which crystalline and amorphous regions alternate [19]. For those polymers which have a strongly irregular chain structure, such as atactic and random copolymers, crystallization is completely inhibited, and in general even for those which own an ordered ground state the ordering is kinetically hard to achieve. For this reason polymers are usually good glass formers.

1.3

Molecular Dynamics simulations

Molecular Dynamics simulations have proven to be a powerful tool to study molecular liquids at a microscopic level [24–27]. Addressing the problem of Molecular Dynamics from first principles is an extremely complex task be-cause the Schroedinger’s equation of the entire system (atoms and electrons), should be solved. When biophysical systems are studied, for instance, the system often includes a large amount of solvent molecules, which usually do not participate in the reaction but influence it [28]. This is a formidable

task, due to the huge amount of degrees of freedom (DoFs) of the system and to the complexity of the equations. For this reason quantum mechanics computer simulation can currently be performed for relatively small systems, containing at most ∼ 102 _{atoms on workstations, up to several thousands on}

high performance computing parallel systems [29]. On the other hand, such a complexity may not be relevant for the investigation of the phenomena related to the glass transition and simplified models may be used as long as the core features of the physical systems are reproduced.

A way to reduce the computational cost is to eliminate the bottleneck of the calculation, namely the electronic DoFs, and with them the quantum mechanic treatment of the system. In fact, atomic DoFs can in most cases be described by classical dynamics by means of the so called atomistic (or all-atom) empirical models. Further simplification can be done through a coarse graining procedure: the system is considered to a less-than-atomistic resolution, in which ”beads” representing groups of atoms interact with each other. Clearly the number of DoFs of the system can be greatly reduced, proportionally to the size of the bead considered, which roughly correspond to the resolution chosen. Fig. 1.5 reports a typical Coarse Graining procedure for a polypeptide from its atomistic representation. The Coarse Grained

Figure 1.5: Coarse graining procedure from the atomistic to a Coarse Grained model in a typical polypeptide (protein). From the all-atom representation (A) to the backbone atoms of the polypeptide chain (B) to the Coarse Grained model (C) with reduced DoFs [30].

level of resolution is commonly used in the simulation of atomic or polymeric liquids in the study of phenomena related to the glass transition. Beside the advantage of being able to simulate larger systems for longer time scales, a strong argument in favor of the Coarse Grained resolution level in the study of polymeric liquids is the universality of the glass transition, with many

related phenomena that are not affected by the chemical details [23].

The reduction of DoFs implies that some of the interactions must be implicitly treated and empirical interactions must be used. If, as it is usu-ally done, the interactions are represented by analytical functions, these will include a number of adjustable parameters. The set of those functions is generally called empirical Force Field (FF). The fewer the explicit DoFs left, the more empirical the interactions. In the present work a monomer is con-sidered as a single interacting bead and beads are consecutively connected to form a linear polymer chain. The connections between subsequent beads

Figure 1.6: Main interactions between monomers in the model under study. The listed interactions (left column) are physically described in terms of their corre-sponding degrees of freedom (central column) and the typical analytical functions associated to them (right colum).

are referred to as bonds and each couple of monomers in the system is either bonded or non-bonded. The typical interactions introduced in this kind of models called bead-spring models are visually represented in Fig. 1.6. In this work bonded monomers interact via an harmonic potential Ub_{(r) = k(r −r}

0)2

depending on the distance r between the monomers with varying stiffness k and equilibrium bond length r0. For some of the simulations a chain stiffness

is also introduced Ubending = kθ(1 − cos θb) depending on the angle θ formed

is absent the stiffness is neglected and the chain is considered fully-flexible. Non-bonded monomers interact via a Lennard-Jones potential mimicking the Van der Waals interactions [29].

ULJ(r) = ε "  σ∗ r 12 − 2 σ ∗ r 6# + Ucut (1.4)

for r ≤ rc = 2.5 σ and zero otherwise, where σ∗ = 21/6σ is the position

of the potential minimum with depth ε. The value of the constant Ucut is

chosen to ensure that ULJ_{(r) is continuous at r = r} c.

It is important to note that up to now no physical scale has yet been introduced by the model. They are in fact defined by the Lennard-Jones interactions. If σ is taken as the fundamental length scale of the system,  as the fundamental energy scale and the mass of monomers m as the fundamen-tal mass scale, then a system of reduced units called Lennard-Jones units can be defined where lengths are measured in units of σ, temperatures in units of ε/kB (with kB the Boltzmann constant set equal to 1) and times τM D in units

of σpm/ε. The reduced units notation allows Coarse Grained simulations the express their results with universal expression in adimensional units. It is obviously important to being then able to map the reduced units results to the experimental systems under investigation. Typical values for these reduced units are reported in Table 1.1 in the case of liquid Argon [31] and of Polystyrene [32].

Physical quantity Unit Value for Ar Value for PS

length σ 3.4 · 10−10m 9.7 · 10−10m energy ε 1.65 · 10−21J 6.74 · 10−21J mass m 6.69 · 10−26Kg 5.8 · 10−25Kg time (σ2m/ε)1/2 2.17 · 10−12s 9 · 10−12s velocity (ε/m)1/2 1.57 · 102_{m/s 1.08 · 10}2_m/s force ε/σ 4.85 · 10−12N 6.95 · 10−12N pressure ε/σ3 _{4.20 · 10}7_N/m2 _{7.38 · 10}6_N/m2 temperature ε/kB 120 K 490 K

Table 1.1: Typical experimental values for the liquid Argon and for Polystyrene are reported compared to the reduced LJ units used in MD simulations.

Once the Force Field associated with the DoFs is defined, the next and final step is the implementation of the interactions in the equations of mo-tion in computer code using more or less complex algorithms to extract the

dynamical evolution of the system in time. The simplest method is the im-plementation of deterministic Newton equations, in the form

miR¨i = Fi Fi = −∇RiV (R1, ..., Rn) (1.5)

This is a deterministic system of 3N equations, and in the case of the con-sidered model N is the number of monomers of the system. Implementation of differential equations in computer code requires the introduction of algo-rithms to discretize the equations, such as the the Verlet algorithm [33, 34] reported in equation 2.15

Ri(t + 4t) = 2Ri(t) − Ri(t − 4t) +

Fi(t) 4 t2

mi

+ O(4t4) (1.6)

where the 4t is the time step of the simulation, which is arbitrary and decided on various considerations [29]: it must be small enough to produce little error in the single time step evolution, yet not too small to reduce the computational efficiency (due to the large number of operation required).

After all the parameters have been chosen, the time evolution of the system is performed and quantities such as the positions and velocities of the monomers are stored at any desired time. From these informations all the dynamical or static features of the system can be ’measured’ after the post-processing analysis. In this work simulations are carried out with the open-source LAMMPS [35] software, while the post processing analyses are performed with our own code.

Each simulation is prepared starting from an initial random distribution for monomers positions and velocities. The equilibration of the system is then performed in the canonical ensemble, where the number of particle is constant and specific variables are added to the Hamiltonian of the system [36] to simulate external control on the temperature (thermostats) and, if needed, the pressure (barostats) of the system, so that they can be considered adjustable parameters of the simulation. Measurements are then taken after the equilibration. More details on the specific preparation of each system is given in the next chapters. Each simulation is usually repeated many times (runs) with the same thermodynamical conditions but starting from different initial configurations to obtain a meaningful statistical ensemble of data to average over. This is done because of the limited sample (typically some thousands of monomers) which can be far from the thermodynamic limit N = ∞.

1.4

Outline of the Thesis

MD simulations of supercooled polymeric liquids at the Coarse Grained res-olution aim to find connections between the static and morphological prop-erties of the system with its dynamics. The investigation of meaningful cor-relations between the many space- and time-scales involved in the dynamics of such systems can help us to understand more deeply the physics behind the glass transition, both in terms of more refined models and experiments. This Thesis work tackles two main areas of current interest in the field of polymer research and the physics of the glass transition by means of Coarse Grained Molecular Dynamics simulations.

Chapters 2-4 face the problem of the rheological properties of polymeric structures in the solid phase, glassy or semicrystalline. In Chapter 2 the spontaneous crystallization of the polymer liquid studied with the present model is observed and investigated. In Chapter 3 the elastic properties un-der shear deformation of semicrystalline solids are studied. Polymeric solids are compared with atomic structures (with no polymer bonds) to determine the role of connectivity. Chapter 4 extends the investigation of the elastic response to different polymeric systems to study the presence of a universal correlation between the elastic shear modulus and the yielding stress of the solids under deformation.

Chapter5investigates the problem of confinement. When a liquid is con-fined, new length scales arise and finite-size effects as well as border effects are introduced into the system. The study of confined liquids is thus moti-vated by new ideas for practical implementations and the search of new hints that can shed light on the complex phenomena related to the glass tran-sition. This work aims to investigate the presence of the important scaling laws related to the relaxation phenomena of supercooled liquids in a confined environment. The effect of confinement on these universal scalings helps to shed light on their nature, which is still debated.

Each Chapter presents a brief introductory paragraph to contextualize the work, motivated by the research performed in recent literature. The original results of my PhD are then presented and conclusions with open questions and perspectives are reported. This Thesis work resulted in the following papers:

ˆ A. Giuntoli, S. Bernini, D. Leporini, Bond disorder, frustration and polymorphism in the spontaneous crystallization of a polymer melt ; J. Non-Cryst. Sol. 453 (2016) 88-93

ˆ A. Giuntoli, N. Calonaci, S. Bernini, D. Leporini, Effect of nematic ordering on the elasticity and yielding in disordered polymeric solids;

J. Polym. Sci. Part B 00 (2017) 1-10

ˆ A. Giuntoli, D. Leporini, Role of polymer connectivity in the elastic properties of amorphous glasses and semicrystalline structures; sub-mitted to the J. Phys. Chem. Solids

ˆ A. Giuntoli, D. Leporini, Confinement effects on the universal scalings between α- and β-relaxations and elastic properties in polymer films above the glass transition; to be submitted

Solidification and Rheology

Crystallization plays an important role in many areas of different sci-entific fields, ranging from biology to engineering and physics. Still, many microscopic details of the phenomenon are unknown, despite the abundance of related results both experimental and theoretical [37–47]. The process of crystallization consists in two major events: nucleation and crystal growth. Nucleation is the stage at which some atoms or molecules of the liquid begin to gather in clusters characterized by an ordered structure, which are the nuclei of the crystalline phase. This process is driven by spontaneous density fluctuations and requires a certain amount of work to be done, since the free energy gain due to the inner order of the new phase competes with the energy loss required for the formation of an interface between the crystal and liquid phases.

The minimum reversible work needed for the formation of a crystal em-bryo of n molecules in an incompressible liquid, under conditions of constant temperature T and pressure P is

Wmin = σA + n [µ0(T, P ) − µ(T, P )] (1.7)

as derived in [48], where σ is the surface tension, A the inter-facial area between the embryo and the bulk phase, µ0 and µ the chemical potentials in the embryo and bulk phase respectively. The first term is positive and surface dependent, while the second is negative and depends on volume, so that Eq. 1.7 can be written

Wmin(r) = br2− cr3 (1.8)

where b and c are positive constants, and r is the radius of the crystal embryo, assumed to be spherical. Therefore, the work required for nucleation has the form of an energy barrier depending on the size of the nucleus, so that a critical size r∗ which maximizes Wmin(r) can be identified as that at which

nucleation has become effective. Nucleation is thus a thermally activated process. When a critical-sized nucleus has formed, it then begins to grow spontaneously.

The number of critical-sized embryos formed at a temperature T ≤ TM

is expected to have a Boltzmann distribution ρn(T ) ∼ exp  −Wmin(r ∗_{, T )} kBT  (1.9) where kB is the Boltzmann constant. However, nucleation cannot be

de-scribed only by equilibrium thermodynamics. When a nucleus has formed, additional atoms or molecules must be transported on it, in order for criti-cal size to be reached. This process has a kinetic nature. Since the ability

Figure 1.7: Surface and volume contributions to the energy barrier representing the work needed for the formation of a crystal embryo of radius r. Adapted from [48].

of transporting matter from a point to another of a system is characterized by its diffusion coefficient D, the rate J of formation of a critical nucleus is expected to have the form

J (T ) ∼ ρn(T ) · D(T ) (1.10)

i.e. to be proportional both to the probability of a nucleus to be formed, and to the rate at which matter is transported on it.

At fixed temperature and rate J , a certain amount of time t(φ) is needed for the growth of a fraction φ of crystal phase. For instance, if the critical nucleus is assumed to be spherical and expanding with constant velocity u, at any time this fraction can be estimated as

φ(t) = Z t 0 J4 3π " Z t0 0 udt00 #3 dt0 = π 3J u 3_t4 _(1.11)

so that the time needed for a crystal fraction φ to grow is t(φ) =  3φ πJ u3 1/4 (1.12)

If the system is cooled down beyond the freezing point (T ≤ TM), the cooling

rate required to obtain a fraction φ of crystal phase can be estimated as | ˙T | ∼ TM − T

t(φ) (1.13)

The critical cooling rate under which complete crystallization is achieved is obtained by imposing φ = 1 in Eq. 1.13:

| ˙T |critical = (TM − T )

 πJu3

3 1/4

(1.14) If the cooling process is faster than crystal growth, i.e. for cooling rates | ˙T | > | ˙T |critical, crystallization can be hindered, completely or in part. It

is important to notice that this can happen even if the crystal phase is the energetically favored one. The resulting situation is that of supercooled liq-uids eventually going out of equilibrium into glassy states as exposed in the general introduction.

For complex liquids intermediate situations may obviously arise in which the solidification of glassification of the system happens during the evolution of the system towards an ideal and energetically favored crystalline structure. In particular, polymeric liquids are systems in which the structural features, namely the chain connectivity, cause serious hindrance to the homogeneous crystallization of the sample, and the formation of mesophases with complex structures and intermediate structural characteristics is commonly observed. In silico simulations of such systems provide great insight on this problem and have proven to be an invaluable tool in the analysis of the crystallization under controlled conditions [39].

The problem of frustrated crystallization, extremely interesting per se, is also of the utmost importance in the investigation of the rheological prop-erties and elastic response of semicrystalline and amorphous solids to exter-nally driven deformations, which poses interesting challenges to the material science research field. The understanding of the microscopic behavior of com-plex materials concurring to the macroscopic effects detectable during their deformation is of particular interest both for the importance of its techni-cal applications and for the absence of a full theoretitechni-cal comprehension of it [49–52].

Elasticity theories [53–56] predict that small deformations cause solid ma-terials to respond linearly. The internal stress of the system increases with the externally applied strain. In the assumption of linear behavior of the sys-tem, the slope of the stress-strain curve during the deformation determines the elastic modulus G. This definition holds even for amorphous solids with

non linear elastic behavior, if a suitable small strain regime is considered [51, 57]. Upon increasing strain, amorphous solids show complex and far from linear behavior [58–60]. When a critical yield strain c is reached,

cor-responding to the shear yield strength τY, the transition from the (reversible)

elastic state to the (irreversible) plastic state takes place [50, 61, 62]. In an ideal elasto-plastic body (Hooke-St.Venant) τY is the maximum stress [50].

The absence of a long-range order in amorphous systems is the main cause [49] of highly non linear phenomena in the plastic regime. The heterogeneity and protocol-dependent nature of these effects [63, 64] must be taken into consideration when results from different solids are compared. In this regard, an accurate investigation of the relation between the structural properties of the solids and its elastic behavior is important both for a deeper under-standing of the elastic and plastic phenomena and for being able to predict and engineer the macroscopic rheological properties of a solids controlling its solidification.

The spontaneous crystallization of polymeric liquids at a finite tempera-ture T < Tmeltis investigated in Chapter2, while Chapter4reports a detailed

analysis of the solidification process of fully-flexible and stiff polymer chains quenched from the liquid phase to temperature T = 0. The rheological re-sponse to external deformations of the polymeric solids is investigated in Chapter 3 on a single set of polymeric solids to assess the role of the poly-mer connectivity and in Chapter4exploring the polymer model by changing connectivity parameters to observe universal behaviors.

The investigation is performed in the athermal regime T = 0 which brings several advantages. At finite temperatures, the local motion of particles is a complex superposition of thermal and of deformation induced movements and the identification of specific plastic events associated with the defor-mation is not possible [51]. The athermal regime allows to isolate the two phenomena by removing the entropic effects caused by thermal motion. This is of particular relevance when the simulation tries to mimic solids with finite but low temperature, such that the expected contribution of thermal agita-tion is small. Also, simulaagita-tions are limited by the timescales they can access, smaller than the experimental ones by orders of magnitude. Moreover, al-though simulations permit to access detailed information about the structure and dynamics, the microscopic motions are very blurred at finite tempera-ture, so that it turns out to be quite difficult in practice to extract relevant information [51]. The athermal regime is thus a useful limit to investigate the mechanical response of solids.

Spontaneous formation of

polymer crystals

2.1

Research context

Many simulations have been performed aiming to observe the crystalliza-tion of polymers and to characterize the structural order reached by the crystalline sample under various conditions [65–73]. Recently, Monte Carlo (MC) simulations of polymer melts made by linear chains of tangent hard-sphere monomers [74–76], i.e. with bond length equal to monomer diameter, have been performed to study spontaneous crystallization. The resulting crystallized structures have been interpreted as a distribution of the most densely packed structures: face-centered cubic (Fcc) and hexagonal close packed (Hcp) lattices. Hcp and Fcc were selected as ideally ordered struc-tures because they are known to be the primary competing alternatives in dense systems of hard spheres in the presence of a single length scale [74]. MC simulations, differently from Molecular Dynamics (MD) simulations, may fail to account for the arrest into metastable intermediate phases [75] which hin-der the evolution towards the thermodynamically stable phase [77]. MD simulations of a polymer melt of chains with soft monomers, promoting the crystallization by equal bond length and equilibrium non-bonded separation, have been performed with the aim of comparing the crystalline structures obtained by cooling down to zero temperature with the highly packed Fcc and Hcp lattices [78,79]. Still, the route towards the closest packing of poly-mers is hindered by allowing length-scale competition of the bonding and the non-bonding interactions, as recently proven in a MD study of the crystal-lization triggered by confinement due to Fcc walls, where structures similar to body-centered cubic (Bcc) are observed [80].

Polymorphism, the presence of different crystal structures of the same molecule, is a well-known phenomenon in molecular crystals [81]. In particle systems the crystal structure depends on the steepness of the repulsive part of the interacting potential with hard and soft repulsions favoring Fcc and Bcc ordering respectively [82]. To date, the selection mechanism of polymorphs is elusive. One widely used criterion is the Ostwald step rule, stating that in the course of transformation of an unstable, or metastable state, into a stable one the system does not go directly to the most stable conformation but prefers to reach intermediate stages having the closest free energy to the initial state [77, 81,83, 84]. Alternatives are reported [85].

In this chapter the isothermal spontaneous crystallization of an unbounded polymeric system is studied via MD simulation of fully-flexible linear chains, i.e. bond-bending and bond-torsions potentials are not present. The em-phasis is on the global and local order of the crystalline phase with respect to the pristine supercooled liquid where crystallization started. To this aim, specific order parameters will be used for their characterization [86, 87]. A distinctive feature of the model is the presence of two different length scales, namely the bond length b and the distance σ∗ where the minimum of the non-bonding potential, the Lennard-Jones (LJ) pair potential, is located. It is known that the competition of two incommensurate length scales favors frustration in the self-assembly of ordered structures from an initial disor-dered state, like in molecular crystallization [88]. Frustrated crystallization of polymers has been reviewed [89]. The role played in the crystallization behavior (including its absence) by frustration, where there is an incompati-bility between the preferred local order and the global crystalline order, has been highlighted [90]. We expect different responses to frustration from the putative crystalline structures at finite temperature, i.e. Fcc, Hcp and Bcc lattices. In fact, not all the atoms in the first neighbors shell of a Bcc lattice are at the same distance, as in the Fcc and Hcp lattices. It is known that the mechanical stability of the Bcc structure is lower than in closed packed structures as Fcc [82].

The chapter is organized as follows: In Sec. 2.2 the polymer model is detailed and the simulation details are provided. The results are presented and discussed in Sec. 2.3. Finally, the conclusions are drawn in Sec. 2.4.

2.2

Methods and models

We consider a coarse-grained polymer model of Nc= 50 linear, unentangled

chains with M = 10 monomers per chain. The total number of monomers is N = 500. The chains are fully-flexible, i.e. bond-bending and bond-torsions

potentials are not present. Non-bonded monomers at distance r interact via the truncated Lennard-Jones (LJ) potential:

ULJ(r) = ε "  σ∗ r 12 − 2 σ ∗ r 6# + Ucut (2.1)

for r ≤ rc= 2.5 σ and zero otherwise, where σ∗ = 21/6σ is the position of the

potential minimum with depth ε. The value of the constant Ucut is chosen

to ensure that ULJ(r) is continuous at r = rc. Henceforth, all quantities are

expressed in terms of reduced units: lengths in units of σ, temperatures in units of ε/kB (with kB the Boltzmann constant) and time τM D in units of

σpm/ε where m is the monomer mass. We set m = kB = 1. The bonding

interaction is described by an harmonic potential Ub [91]:

Ub(r) = k(r − r0)2 (2.2)

The parameters k and r0 have been set to 2500 ε/σ2 and 0.97 σ respectively

[92]. Given the high stiffness of the bonding interaction, b = 0.97 ± 0.02. Notice that the bond length and the minimum of the non-bonding potential are different, b 6= σ∗ ' 1.12. Periodic boundary conditions are used. The study was performed in the N P T ensemble (constant number of particles, pressure and temperature). The integration time step is set to ∆t = 0.003 time units [93, 94]. The samples were initially equilibrated at temperature T = 0.7 and pressure P = 4.7.

To ensure that an equilibrium state is reached, the end-to-end correlation function Cee(t) is defined as:

Cee(t) = 1 Nc Nc X p=1 Rp_ee(t) · Rp_ee(0) R2 ee (2.3) where Rp

ee(t) is the vector joining the first and the last monomers of the chain

p at time t, and R2 ee is defined by: R2_ee = 1 Nc Nc X p=1  Rp_ee 2 (2.4)

At the initial time t = 0 the equality Cee(0) = 1 holds by definition and

it decreases with passing time as the chains rearrange with respect to their initial position. We define the average reorientation time τee of the end-end

vector of the chain by the relation Cee(τee) = 0.1 and consider the system in

liquid state lasted at least 3τee. After equilibration, we started production

runs waiting for the spontaneous crystallization of the system. We analyzed the initial equilibrated liquid state, the development of the solid phase and the final crystalline state. 56 starting liquid runs were used with different random monomer positions and velocities. 42 runs underwent crystallization, while 14 of them failed to crystallize in a reasonable amount of time.

2.3

Polymorphs characterization

0 5e+05 1e+06 1.5e+06 2e+06 2.5e+06 450 460 470 480

V

0 5e+05 1e+06 1.5e+06 2e+06 2.5e+06

t

U

Figure 2.1: Volume (top) and potential energy (bottom) drops due to the spon-taneous crystallization occurring in a single run (blue curve). They are compared to the typical fluctuations occurring in the metastable liquid (red curve).

In the NPT ensemble (isothermic and isobaric conditions) the volume of the system is free to change and in equilibrium conditions it fluctuates around a mean value. The same can be observed for the energy of the system. The spontaneous crystallization of the polymeric system from the equilibrium liquid phase is characterized by a sudden drop of the mean value of both the volume V and the potential energy Up during the time evolution

of the system, as shown in Fig. 2.1 for a sample run undergoing crystal-lization compared to a stable run in the equilibrated liquid phase. After the

crystallization, fluctuations around a mean value can again be observed. The steepness of the volume and energy drops is distinctive of the first order phase transition associated with the passage from the liquid to the solid phase.

445 450 455 460 465 470

V

U

Crystals

Liquids

Figure 2.2: Correlation plot between the average volume and the energy of all the crystalline (blue) and liquid (red) states under study. Note the large region spanned by the different crystalline states signaling polymorphism.

Fig. 2.2 reports the volume V and potential energy Up values of all the

56 liquid samples in the equilibrium phase before the crystallization and the 42 crystalline samples after the phase transition. It can be observed that the mean values of both the volume and the potential energy of the crystals span a wider range with respect to the the metastable liquid. This is evidence of polymorphism due to different kinetic pathways leading to crystallization in more than one, metastable, ordered form [81]. In our polymer melt polymorphism is contributed, with respect to the corresponding nonbonded -atomic liquid, by the chain connectivity and the presence of incommensurate length scales involving the bonding and the non-bonding potentials.

While atoms in the liquid phase are expected to be randomly distributed, a partial ordering in the atomic arrangement can be observed in the obtained crystalline structures, as can be noted from Fig. 2.3 in which a typical snapshot of a crystalline system is reported. On the other hand, no increased order of the polymer bonds can be observed as they appear to be randomly oriented. The complex nature of the solid phases formed by our samples thus requires an accurate structural analysis of the obtained polymorphs, which is reported in the following. The description of the selection mechanism of the polymorph is beyond the purpose of the present work.

As initial step, we study the monomer mobility during the transition from the liquid to the polymorph. To this aim, we resort to the self-part of the

Figure 2.3: Snapshot of a polymer crystalline structure without (left) or with (right) the explicit representation of the polymer bonds. A partial ordering of the atomic positions can be observed, while the bonds orientation appears to be randomly distributed. Bonds across the boundary conditions of the simulation box are not represented.

van Hove function Gs(r, t) [96] defined as

Gs(r, t) = 1 N N X j=1 δhr + rj(0) − rj(t) i (2.5)

where rj(t) is the vector position of the j-th monomer at time t. In isotropic

liquids, the van Hove function depends on the modulus r of r. The inter-pretation of Gs(r, t) is direct. The product Gs(r, t) · 4πr2 is the probability

that the monomer is at a distance between r and r + dr from the initial position after a time t. We observe that the crystallization completes, on average, within τcry ' (1.2 ± 0.1) · 104 time units, corresponding to ' 26 ± 2

ns in Argon units [31]. For the different crystallization paths we evaluate the displacement distribution in a time τcry of: i) the initial liquid in metastable

equilibrium, ii) the liquid during crystallization and iii) the final polymorph. The results are in Fig. 2.4. During the crystallization monomers displace nearly as far as in the liquid. Starting from liquid states with nearly identical displacement distribution, see Fig. 2.4 (top), each subsequent crystallization path and final polymorph is characterized by a different displacement dis-tribution. Polymorphs have little mobility. Their displacement distribution exhibits a bimodal structure with a large peak corresponding to the rattling

0 0.2 0.4 0.6 0.8 1 1.2 1.4

r

4

π

r

G

(r,

τ

)

_Crystal

4

π

r

G

(r,

τ

)

_Liquid

Crystallization

Figure 2.4: Displacement distribution of the monomers in a time τcry during

different stages of the crystallization. The average time to start and complete the crystallization is τcry= 12000. Top: initial supercooled liquids in metastable

equi-librium (red), selected supercooled liquid states undergoing crystallization (green). Different curves with same color refer to different crystallization paths. Bottom: final polymorphs. The small peak at r ∼ 1 signals the presence of monomers jumping of about one diameter. The initial liquid states exhibit nearly the same displacement distribution. The displacement distribution during the crystalliza-tion process is only mildly narrower than the one of the melt. The different poly-morphs have distinct displacement distributions and lower mobility than the liquid. The inset of the bottom panel shows the two polymorphs with highest and lowest mobility, respectively. The jump process is largely suppressed in the less mobile polymorph.

motion of the monomer within the cage of the first neighbors and a sec-ondary peak due to monomers displacing by jumps with size comparable to their diameter. The inset of Fig. 2.4 shows the displacement distribution of the polymorphs with the highest and the lowest mobility and evidences that the jump process is largely suppressed in the less mobile polymorph.

arrangements of both the monomer radial distribution and the angular dis-tribution of adjacent bonds in a chain. We introduce the radial disdis-tribution function g(r) defined as:

g(r) = 1 N (N − 1) N X i=1 N X i6=j δ(|rij| − r) (2.6)

where N is the total number of the particles and |rij| is the distance between

Figure 2.5: Physical meaning of the radial distribution function g(r). The red particle is our reference particle and the blue particles are within the circular shell (included between r and r + dr) dotted in orange.

the i-th particles and the j-th one. The factor N (N − 1) is the total number of pairs in the system. Intuitively, the radial distribution function of a single particle is a measure of the probability to find a particle at distance r away from it. The g(r) defined in eq. 2.6 is the average of the single particle distribution over all the monomers of the system. Fig. 2.6 reports the g(r) function of all the liquid and crystal samples. The g(r) shows characteristic fluctuations around the value g(r) = 1 which would be reached in an infinite system for a large enough value of r. The range of the plot is limited by the dimensions of simulation box. Each fluctuation of the g(r) is associated with a shell of monomers around the reference particle. A double peak can clearly be observed in the first shell of neighbors (r . 1.35) corresponding to bonded and non-bonded monomers for r ∼ 0.97 and r ∼ 1.07 respectively. The split

1 1.5 2 2.5 3 3.5

r

g(r)

Crystal

Liquid

bonded monomers

non bonded

first neighbors

Figure 2.6: Radial pair distribution function g(r) of all the crystalline and liquid states. On increasing the distance r from the tagged central monomer, the first, sharp peak corresponds to the bonded monomers, whereas the other ones signal the different neighbor shells. Notice that the broad features of the g(r) of the crystalline and the liquid states are quite similar within the first neighbor shell r . 1.35, while the difference increases starting from the second shell. Also, the ordered states exhibit sharper and, due to polymorphism, more widely distributed features than the disordered ones.

of the first neighbors peak is caused by the difference between the bond b and Lennard-Jones σ∗ length scales. The behavior of liquids and polymorphs is not markedly different in the fist shell of neighbors, while more pronounced differences can be observed starting from the second neighbors shell r & 1.35. The polymorphs also exhibit sharper, and more widely distributed, features among different samples. As already noted in previous plots, this is caused by the metastable condition reached by the polymorphic structures.

The distribution P (cos θb) of the angle θb between adjacent bonds in a

chain is now considered. The angle θb between two consecutive bonds is

defined as:

(cos θb)i = −

bi· bi+1

|bi| |bi+1|

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4

cosθ

P(cos

θ)

Crystal

Liquid

Figure 2.7: Distribution P (cos θb) of the angle θb between adjacent bonds in a

chain. The peaks occur at θb ≈ 70◦, 122◦, 180◦, corresponding to three consecutive

monomers which are folded - with the two non-consecutive monomers in contact (r ∼ σ∗) -, partially folded, and aligned, respectively [87]. The crystal samples show sharper peaks and a wider distribution among different sample runs due to the polymorphism.

where bi is the bond vector bi = ri− ri−1 related to the bonded monomers

with coordinates ri and ri−1. For each of the analyzed configurations, the

ob-tained values are binned in a histogram I [cos θb] of N bins of width ∆(cos θb).

The probability density of finding a triplet of bonded monomers to form an angle of cosine included between cos θb and (cos θb + ∆(cos θb)) is then

ob-tained via normalization of I [cos θb] by the total number of bonded triplets

P (cos θb) =

I [cos θb]

Nc(M − 2)

(2.8) where M and Nc are respectively the number of monomers in a polymer

chain, and the total number of chains. Average on the configurations is then carried out. Fig. 2.7 reports the bond angle distribution function for the polymeric liquids and polymorphs. P (cos θb) is not markedly different in

and more widely distributed, features, including a larger fraction of aligned bonds.

In order to study more rigorously the structural order of the systems, we resort to the order parameters defined by Steinhardt et al. [86]. One considers in a given coordinate system the polar and azimuthal angles θ(rij)

and φ(rij) of the vector rij joining the i-th central monomer with the j-th one

belonging to the neighbors within a preset cutoff distance rcut = 1.2 σ∗ ' 1.35

[86]. rcut is a convenient definition of the first coordination shell size [97].

The vector rij is usually referred to as a “bond” and has not to be confused

with the actual chemical bonds of the polymeric chain. To define a global measure of the order in the system, one then introduces the quantity:

¯ Qglob_lm = 1 Nb N X i=1 nb(i) X j=1 Ylm[θ(rij), φ(rij)] (2.9)

where nb(i) is the number of bonds of i-th particle, N is the total number of

particles in the system, Ylm denotes a spherical harmonic and Nb is the total

number of bonds: Nb = N X i=1 nb(i) (2.10)

The global orientational order parameter Qglob_l is defined by [86]:

Qglob_l = " 4π (2l + 1) l X m=−l | ¯Qglob_lm |2 #1/2 (2.11)

The above quantity is invariant under rotations of the coordinate system and takes characteristic values which can be used to quantify the kind and the degree of rotational symmetry in the system [86]. In the absence of large-scale order, the bond orientation is uniformly distributed around the unit sphere and Qglob_l is rather small since it vanishes as ∼ N_b−1/2 [98].

These bond-orientational order parameters were introduced as useful tools to investigate the growing order and presence of orientational symmetries in three-dimensional complex fluids and disordered solids. They are a natural generalization of the two-dimensional hexatic order parameter [99] used to investigate two-dimensional anisotropic fluids in which sixfold hexatic order had been found [86]. The spherical harmonics of even order l = 4, 6, are of particular relevance, as they are related to the cubic and icosahedral symme-tries found in known crystalline structures. The Qglob₆ parameter is also very sensitive to any kind of crystallization, increasing significantly when order appears [86, 100, 101] and has been used extensively in the literature.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 -5.5 -5.4 -5.3 -5.2 -5.1

U

p

Q

V

L

L

C

C

Figure 2.8: Correlation plot between the global order parameter Qglob₆ and both the potential energy (top) and volume (bottom). Both liquid states (red) and polymorphs (blue) are shown. Polymorphs span an extended region of the plot with larger global order than the liquid.

Fig. 2.8 shows the correlations between the volume V , the potential energy Up and the global order parameter Qglob6 for both the liquids and the

polymorphs. We see that liquid states have rather small Qglob₆ with narrow distributions of V , Up and Qglob6 . The polymorphs have much larger Q

glob 6

values with wider distributions of V , Up and Qglob6 . A local orientational

parameter Qloc

l can also be defined. We define the auxiliary quantity

¯ Qloc_lm(i) = 1 nb(i) nb(i) X j=1 Ylm[θ(rij), φ(rij)] (2.12)

The local order parameter Qloc_l is defined as [86]:

Qloc_l = 1 N N X i=1 " 4π (2l + 1) l X m=−l | ¯Qloc_lm(i)|2 #1/2 (2.13) In general Qloc l ≥ Q glob

l . In the presence of ideal order, all the particles have

Figure 2.9: Bcc-, Fcc- and Hcp-lattices at T = 0. Fcc and Hcp structures are closely packed and each atom is surrounded by twelve neighbors at the distance a. In the Bcc structure each atom has neighbors at two different distances: a and 2a/√3.

An interesting feature of the order parameters is that they exhibit spe-cific values for different kind of orders [86]. This suggests to plot their values to gain insight into the global and the local order of the polymorphs and the liquid and to compare them to known crystalline structures. Namely, a comparison with Fcc, Hcp and Bcc ideal lattices was performed. A rep-resentation of the first shell of neighbors of these structures is reported in Fig. 2.9. The ideal structures were created arranging N = 2000 non-bonded monomers in the Bcc lattice and N = 4000 non-bonded monomers in the Fcc and Hcp lattices at zero temperature T = 0, i.e. with no thermal mo-tion involved. The different number of atoms is due to the number of atoms for unit cell (2 in the Bcc and 4 in the Fcc and Hcp), so that each system contains exactly 1000 unit cells. The closest particles are spaced by the av-erage monomer-monomer distance of the polymer system, a = 1.07 (see Fig.

2.6). Thermal fluctuations are also included introducing a Lennard-Jones interaction potential between the atoms and raising the temperature of the crystals to T = 0.3, 0.7 to observe the deformations caused by thermal agita-tion. This is particularly important for the Bcc lattice, which possesses two characteristic first neighbors distances a and √2a

3, see Fig. 2.9. This causes

the Bcc lattice to be less stable [82] than the Fcc and Hcp lattices, as can be observed in the following.

Fig. 2.10 reports the correlation plots Qglob_l vs Qloc_l for l = 4, 6. The polymeric liquid and crystalline samples are reported, as well as the atomic crystals in the ideal lattice (Fcc, Hcp and Bcc) configuration at T = 0 and at T = 0.3, 0.7, when thermal fluctuations are added. Considering the polymer

0 0.05 0.1 0.15 0.2

Q

Q

Q

Q

L

Bcc

Fcc

C

Hcp

L

C

Bcc

Hcp

Fcc

Figure 2.10: Correlation plots between the local and the global parameters with l = 4 (top) and l = 6 (bottom) for polymorphs (blue dots) and liquids (red dots). The dashed line is the bisectrix. The diamonds mark the ideal Bcc- (black), Fcc-(green) and Hcp- (magenta) atomic crystals at T = 0 with Qglob_l = Qloc_l . The black, green and magenta dots mark the same crystals at T = 0.3, 0.7. Differently from the Bcc lattice, the global and the local order of the Fcc and Hcp lattices are negligibly affected by the temperature, signaling more thermal stability. In each plot the black dot closest to the polymorph region is the Bcc lattice at the same temperature (T = 0.7).

systems, we see that Qglob_l is higher in the polymorphs than in the liquids, whereas Qloc_l is nearly the same. This corroborates the idea that even though the frustration caused by the polymer connectivity hinders the formation of a perfectly ordered structure at the monomer level, the polymorphs show an increased global order with respect to the liquid phase. The ideal lattices at T = 0 (green, magenta and black diamonds in Fig. 2.10) have Qglob_l = Qloc

l as

expected. When thermal fluctuations are introduced, the order parameters of both Fcc and Hcp atomic lattices show little changes, whereas the ones of the Bcc crystal are much more sensitive. This is consistent with the known lower stability of the Bcc crystal [82]. It can also be noticed that the polymorphs position themselves in a region close to the one of the Bcc structure at the

0 0.1 0.2 0.3 0.4 0.5 0.6

Q

Q

Q

Q

L

Bcc

Fcc

C

Hcp

L

_C

Bcc

_Hcp

Fcc

Figure 2.11: Correlation plots Qglob₄ vs. Qglob₆ (top) and Qloc₄ vs. Qloc₆ (bottom). All the symbols are used as in Fig. 2.10. In each plot the black dot closest to the polymorph region is the Bcc lattice at the same temperature (T = 0.7).

same temperature (T = 0.7). This is a piece of evidence that our melt of fully-flexible chains crystallizes spontaneously into a distorted Bcc-like structure. Fig. 2.11 presents different pairs of order parameters. It is seen that the pair of global parameters (top panel) are more informative and confirm the conclusions drawn by the analysis of Fig. 2.10, i.e. the structures of the polymorphs approach the Bcc structure at the same temperature (T = 0.7). Further support to the conclusion that fully-flexible chains crystallize in a Bcc-like structure is provided by the angular distribution of the monomers belonging to specific shells of neighbors surrounding a central monomer. To this aim, we consider the angle αjk between rij and rik where the vector rij

joins the i-th central monomer with the j-th one which is rij apart. Our

quantity of interest is the Angular Distribution Function ADF (cos αjk) [102]

of the monomers with rmin ≤ rij, rik ≤ rmax. A visual representation of the

Angular Distribution Function is given in Fig. 2.12.

Fig. 2.13 shows the ADF for the polymorphs and the liquids restricted to the first and the second shells, respectively. As a reference, the ADF

Figure 2.12: Definition of α. All the couples j − k of green monomers inside the shell centered on the red monomer i are considered. The angle α is defined as the angle between the vectors joining the central monomer i with the j − th and k − th ones.

distributions of Bcc, Fcc and Hcp crystals at T = 0.7 are also plotted. On the basis of the radial distribution function in Fig. 2.6 (top), the boundaries of the shells are taken as: rmin = 0.8, rmax = 1.35 (first shell) and rmin = 1.35,

rmax = 2.2 (second shell). It is seen that the ADF distributions of the

different polymorphs are distinct. They differ from both the liquid and the reference atomic crystals in the first shell. The ADF distributions of the different polymorphs are in excellent agreement with the Bcc ADF in the second shell. It is worth noting that the ADF of the liquid is nearly flat in the second shell for cos α < 0, signaling the large loss of anisotropy beyond the first shell.

Finally, more insight is provided by the study of the static structure factor S(q) defined as: S(q) = * 1 N N X k=1 N X j=1 eiq·(rk−rj) + (2.14) where rkand rj are the vector positions of the k-th and j-th monomer

re-spectively, the sum goes over all N monomers in the system and the brackets h. . .i denote the average over all vectors q of modulus q. The static structure

-0.9 -0.6 -0.3 0 0.3 0.6 0 1 2 3

ADF(cos

α

)

cosα

ADF(cos

α

)

Figure 2.13: Angular distribution function (ADF) in the first (0.8 < r < 1.35, top panel) and second (1.35 < r < 2.2, bottom panel) shells of the liquids and the polymorphs. As reference, the distributions of Bcc, Fcc and Hcp crystals at T = 0.7 are also plotted. Different polymorphs exhibit distinct ADFs. No clear conclusion is drawn by comparing the ADF of the crystalline polymers and the ones of the atomic Bcc, Fcc and Hcp crystals in the first shell. Instead, the ADF of the crystalline polymers and the Bcc crystal at T = 0.7 is almost identical in the second shell.

factor is a measure of the density response of a system, initially in equilib-rium, to a weak, external perturbation of wavelength 2π/q. When the probe is a beam of neutrons, S(q) is proportional to the total scattered intensity in a direction determined by the momentum transfer ~q between beam and sample [96]. Use of such a probe provides an experimental mean to determine the radial distribution function of a liquid. In fact, in a homogeneous system the static structure factor can be expressed in term of the radial distribution function g(r) by:

3

6

9

12

15

18

21

24

q

0

2

4

6

8

10

S(q)

Crystal

Bcc

Fcc

Hcp

6

9 12 15 18 21

0

2

4

6

8

10

_Crystal

Bcc T=0

Figure 2.14: Static structure factor S(q) of polymer crystals and atomic Fcc, Hcp and Bcc lattices at T = 0.7. Fcc and Hcp lattices preserve well defined peaks in the distribution despite the thermal fluctuations introduced. The Bcc lattice is deformed and follows the distribution of the polymer crystals. Inset: the peaks of the S(q) distribution of the polymorphs are consistent with the peaks of the ideal Bcc lattice (dashed lines).

S(q) = 1 + ρ Z +∞ −∞ g(r)eiq·rdr (2.15) = 1 + 4πρ Z +∞ 0 g(r)sin qr qr r 2_{dr (2.16)}

Fig. 2.14shows the static structure factor S(q) of polymer crystals at T = 0.7 compared to the ideal Fcc, Hcp and Bcc lattices at the same temperature. Despite the high level of noise caused by the complexity of the measured quantity, it can be observed that Fcc and Hcp structures preserve sharp, well defined peaks in the S(q) characteristic of a regular crystalline structure. The deformed Bcc structure (black line in Fig. 2.14) at T = 0.7 does not present such sharp peaks and follows closely the polymeric samples (blue

lines). Also, the inset shows that the polymorphs have more pronounced peaks of the S(q) where the characteristic peaks of the ideal Bcc lattice (dashed lines) are located, as also observed in previous studies [80].

2.4

Conclusions and perspectives

The isothermal, isobaric spontaneous crystallization of a supercooled polymer melt is investigated by MD simulation of an ensemble of fully-flexible linear chains. Frustration is introduced via two incommensurate length scales, i.e. the bond length and the minimum of the non-bonding potential, resulting in marked polymorphism with considerable bond disorder, distortions of both the local packing and the global monomer arrangements. The crystalliza-tion process involves monomer displacements as large as a few diameters. Jump processes are detected in the polymorphs. The analyses in terms of: i) orientational order parameters characterizing the global and the local order and ii) the angular distribution of the next-nearest neighbors of a monomer reach the conclusion that the polymorphs are arranged in distorted Bcc-like lattices.

Considering future works, it could be interesting to investigate more closely the crystallization process to understand the atomic mechanism asso-ciated with the spontaneous nucleation and crystal growth of the polymorphs in this model. The computational resources available only allowed for the study of a small system of N = 500 monomers, as the crystallization of big-ger samples could not be observed in reasonable times (order of the month on the PhD student time scales). The study of systems with different sizes could provide interesting insights on the universality of the process and the spatial arrangement of the polymorphic regions inside the sample.

Role of polymer connectivity

under elastic shear

deformations

3.1

Research context

The structural features and dynamics of polymer solids are guided by intra-molecular and inter-intra-molecular forces. The energy barriers to plastic flow in glassy polymers are governed by inter-molecular interactions [49] and the finite temperature elastic modulus of glassy polymers is also mainly deter-mined by inter-molecular forces in fully flexible systems [103]. It can then be concluded that the comparison between atomic and fully flexible polymeric systems regarding the elastic response under deformations show some similar-ities [104] and some macroscopic features such as the τY/G ratio measured in

experimental atomic glasses [105] and in simulations of fully flexible polymer glasses [106] is comparable. On the other hand a primary role in determining the structure of a polymer solid upon cooling from the liquid phase is played by the intra-molecular interactions [74–76, 78–80, 101] and the microscopic details of the solid structure is of great importance to determine its elastic properties [63, 64].

The results reported in this chapter aim to determine the role of the polymer connectivity in the elastic response of semicrystalline solids. The polymorphs described in Chapter 2 are considered and compared to atomic systems with similar features except for the presence of polymer bonds. The elastic properties of the solids are measured following the same protocol, described in Sec. 3.2, so that comparisons between the results let us assess the specific contribution of polymer connectivity and atomic arrangement in

the solid phase to the measured values of critical yield strain c, shear yield

strength τY and elastic modulus G. The comparison of polymer and atomic

crystals with the same atomic structure shows that the presence of polymer connectivity causes a noticeable increase both in the yield stress τY and in

the elastic modulus G.

3.2

Methods and models

The polymer samples considered are as in Chapter 2: Nc = 50 linear

fully-flexible unentangled chains with M = 10 monomers per chain. The poly-morphs obtained with a spontaneous crystallization of the polymer melts at temperature T = 0.7 and pressure P = 4.7 are taken in this chapter for the study of their elastic properties. Non-bonded monomers at distance r interact via the truncated Lennard-Jones (LJ) potential

ULJ(r) = ε "  σ∗ r 12 − 2 σ ∗ r 6# + Ucut (3.1)

and the bonded monomers interaction is described by an harmonic potential Ub_:

Ub(r) = k(r − r0)2 (3.2)

with k and r0 parameters set to 2500 ε/σ2 and 0.97 σ respectively [92]. In

addition to the polymeric samples, atomic liquids have also been simulated in order to compare systems with and without polymer bonds in the study of the elastic properties. For the atomic samples we consider systems of N = 500 atoms interacting with the truncated Lennard-Jones potential reported in equation 3.1, while bonded interactions are obviously absent. As in chapter

2 all quantities are expressed in term of reduced units: lengths in units of σ, temperatures in units of ε/kB (with kB the Boltzmann constant) and time

τM D in units of σpm/ε where m is the monomer mass. We set m = kB = 1.

Periodic boundary conditions are used. The integration time step is set to ∆t = 0.003 time units [93, 94].

42 polymer semicrystalline structures are considered. Additionally, 64 atomic liquid runs were equilibrated with starting temperature T = 1.5 and pressure P = 20.0. These thermodynamic values have been chosen to ensure a comparable initial density with respect to the polymer samples in the liquid phase. After an initial equilibration, 51 runs have crystallized into two well defined classes. 17 runs formed Fcc-like atomic crystals with a high struc-tural order and 34 run formed Bcc-like atomic crystals similar to our polymer crystals. See Sec. 3.3 for a detailed discussion. The remaining 13 runs have

reached different metastable conformations after the crystallization and have been neglected in this study for clarity reasons.

After the spontaneous crystallization at finite temperature, all the systems

One of three considered deformations (xy, xz, yz)

Figure 3.1: Graphical representation of the simple shear deformation applied to the simulation box in one of the three possible directions. The volume of the simulation box is not modified by the simple shear deformation procedure.

are quenched to zero temperature and pressure T = 0.01, P = 0 in a single simulation step, corresponding to a quench rate of dT /dt ∼ 102_{. The quench}

is faster then any time scale of the system and can be considered instanta-neous, so that the atomic configuration is frozen in place. A low temperature equilibration in the N P T ensemble at T = 0.001 and P = 0 is performed to allow atoms to reach a local metastable equilibrium with small adjustments. After the quench, simple shear deformations of the T = 0, P = 0 solids were performed via the Athermal Quasi-Static (AQS) protocol described in details in [51]. An infinitesimal strain increment ∆ε = 10−5 · L is applied to a simulation box of side L containing the sample, after which the system is allowed to relax in the local potential energy minimum via a suitable minimization algorithm. The procedure is repeated until a total strain of ∆εtot = 15%L. A visual representation of the applied deformation is reported