Molecular Dynamics simulations of polymeric systems confined in thin films

University of Pisa

Department of Physics

Master Thesis

Molecular Dynamics

simulations of polymeric

systems confined in thin films

Candidate Supervisor

Matteo Becchi Prof. Dino Leporini

Preface

If a liquid is cooled avoiding crystallization it shows an enormous slowing down of the molecular dynamics, becoming what is said a supercooled liquid. At a certain temperature Tg the dynamics timescales exceed the observation

time and we obtain a liquid out of equilibrium, called glass. A glass has the mechanical response of a solid (very high viscosity), but still exhibits the microscopic disordered structure of a liquid.

Polymeric systems are especially useful to study supercooled liquid near the glass transition Tg. In fact, they present several different length scales,

some related to the internal structure of the monomers, others to the entire chain. When cooled, the presence of different length scales leads to phenom-ena of geometric frustration, and the polymeric melt is not able to crystallise in an ordered phase, thus becoming a supercooled liquid.

Two important quantities that characterise the dynamics of a polymeric melt are the mean squared displacement (MSD) of the monomers and the structural relaxation time τα. The MSD is defined as the average

displace-ment of the monomers from a certain time t0 to the time t0 + t. Near the

glass transition, the MSD shows a plateau region which is index of the cage dynamics, the rattling motion of the monomer in the cage of his neighbours. The structural relaxation time τα is the typical timescale in which the

density fluctuation of the melt relax. It can be determined as the time at which the incoherent part of the intermediate scattering function [1] drops below the value 1/e.

Larini et al [2] found a universal scaling between the cage dynamics (of the order of the picosecond) and the structural relaxation of the liquid (up to 100 s at the transition temperature Tg) in many experimental and simulative

systems. They show that the value of the MSD in the plateau region hu2i and the structural relaxation time τα satisfy the relation

log τα ∝

a hu2_i+

b hu2_i2

where a and b are constants that are universal for the simulative model v

used for polymers, and depends on a single parameter for the experimental systems.

On the other hand, many researches have investigated the effects of the confinement on the dynamics of supercooled systems, caused by finite-size effects and interaction between the liquid and the interfaces (see, for instance, Bashnagel et al [3]). In this thesis work, we investigate polymeric melts confined in thin film, that is, systems that are infinite in two dimensions but have a finite thickness (from few nm up to ∼ 100 nm) in the third. In particular, we focus on the simulation of supported films, where one of the surfaces is in contact with a substrate, while the other is free. These systems are of particular interest because they can be easily obtained experimentally.

This work is articulated in the following way.

• In chapter 1 the fundamental concepts about the glass transition and the physics of polymers are described. We see how the dimensional confinement affects the behaviour near the glass transition, and we define the physical properties of interest used in the thesis.

• In chapter2the Molecular Dynamics simulations of polymeric systems are described, and the coarse-grained model used is shown, with atten-tion to the relaatten-tion with real systems.

Chapters 3 and 4contain the original part of the thesis.

• In chapter 3 a specific kind of confined system is analysed. We simu-late polymeric melt confined in thin films supported on an atomic and attractive substrate, varying the temperature and the thickness of the film. We found that this system show a strong modulation of the linear density in the confined direction: the monomers position themselves in well-defined layers parallel to the substrate. Then we measured hu2_i

and τα for the single layers. These quantities show a strong variation

with the distance from the substrate, i.e. with the layers in which they are measured. Surprisingly, it seems that the universal scaling between cage dynamics and structural relaxation, despite being lost for the film as a whole, can be fully recovered if the measures are performed on the single layers.

• In chapter 4 the same analysis are performed on a smooth and struc-tureless substrate, leading to completely different results. Some pos-sible interpretation of these differences are proposed, and further in-vestigations are suggested, varying the nature and strength of the con-finement, in order to achieve a deeper understanding of the scaling in confined systems.

Contents

Aknowledgements iii

Preface v

1 Introduction 1

1.1 The glass transition . . . 1

1.2 Polymers. . . 6

1.3 Confinement . . . 9

1.4 Quantities of interest . . . 11

1.4.1 Radial pair distribution function g(r) . . . 11

1.4.2 Static structure factor S(q) . . . 12

1.4.3 Mean Squared Displacement of the particles . . . 13

1.4.4 Incoherent Intermediate Scattering Function . . . 14

1.4.5 Van Hove function . . . 15

1.5 Scaling between relaxation and vibrational dynamics . . . 16

2 Molecular Dynamics simulations of polymers 19 2.1 Molecular Dynamics . . . 19

2.2 Equations of motion . . . 20

2.3 Periodic boundary conditions . . . 21

2.4 The canonical ensamble . . . 23

2.5 The bead-spring model . . . 24

2.5.1 Lennard-Jones units and mapping to real units . . . . 25

2.6 Simulation protocol . . . 26

3 Effects of confinement on the dynamics 31 3.1 Molecular Dynamics simulations of confined polymeric systems 31 3.2 Rough substrate . . . 33

3.2.1 Details of the model . . . 33

3.2.2 Static properties . . . 34

3.2.3 Dynamical properties . . . 39

3.3 Layer analysis . . . 41

3.3.1 Methods . . . 42

3.3.2 Results . . . 46

4 Failures and perspectives 51 4.1 Smooth substrate . . . 51

4.1.1 Details of the model . . . 51

4.1.2 Results . . . 52

4.2 Interpretation and future perspectives . . . 56

Conclusions 59

Monitoring of thermodynamic quantities 61

Chapter 1

Introduction

1.1

The glass transition

When a liquid is cooled down below its melting temperature Tm, it usually

undergoes a first order phase transition and becomes a crystal, a solid with long-range order. The process of crystallization consists in two major events: nucleation and crystal growth [4].

The nucleation is the formation of an initial cluster of molecules with the lattice spatial ordering. This process requires a certain work to be done, because the decrease in free energy due to the order of the new phase is compensated by the energy due to the formation of an interface between the nucleus and the surrounding liquid. The energy needed for this work can be obtained from spontaneous thermal fluctuation. If the liquid is cooled at a higher rate than the characteristic crystallization rate, the crystallization can be avoided [5], obtaining a metastable phase called supercooled liquid. In figure1.1we can see the volume V of a liquid as a function of the temperature T . If the crystallisation process takes place, there is a first order phase transition at T = Tm, and the volume has an instantaneous drop (in T ).

Instead, if crystallisation is avoided, the volume keep decreasing without phase transitions, and the supercooled phase is obtained. To see how this is possible, we have to find the temperature dependence of the time needed to crystallise.

It can be shown [5] than the minimum free energy needed to the formation of a nucleus of n molecules is

Wmin = σA + n [µs− µl] (1.1)

where σ is the surface tension, A the nucleus surface, and µs and µl are the

chemical potentials in the solid and liquid phase respectively. The first term 1

Figure 1.1: The drop in V (T ) at T = TM is index of crystallisation, a first order

phase transition. If crystallisation is avoided, we obtain a supercooled liquid and, with further cooling, we reach Tg, obtaining a glass, a disordered systems that

display solid mechanical features. Picture taken from Napolitano et al [6]

in the equation 1.1 is positive, and the second negative (because µs < µl), so

this expression can be written, as a function of the nucleus radius r, as

Wmin(r) = ar2− br3 (1.2)

with a, b > 0. As shown in figure 1.2, this expression has the shape of a potential barrier, and there is a value r∗ that maximise Wmin. Thermal

fluctuation at least of size Wmin(r∗) are needed for the nucleation process to

became spontaneous.

Finding the maximum of the equation 1.2 and substituting a = 4πσ and b = 4π∆µ_3v

s , where vs is the specific volume in the crystalline phase, we obtain

Wmin(r∗) = 16π 3  vsσ3/2 −∆µ 2 (1.3)

1.1 The glass transition 3 0 r*

r

0 W min(r*)

W

min

(

r)

Figure 1.2: Minimum free energy Wmin(r) = ar2− br3 needed to the creation of a

nucleus, as a function of the nucleus radius r. The function displays a maximum at r∗ = 2a/3b.

and the nuclei density will then be proportional to a Boltzmann factor ρ(r∗) ∝ exp  −Wmin(r ∗₎ kBT  (1.4) To find the temperature dependence of this expression, we need ∆µ(T ) (we can neglect the temperature dependence of vs and σ). In order to do

this, we can make a Taylor expansion of the difference in free energy between the crystalline and the liquid phase near Tm

∆G = Gs− Gl' ∂G ∂T     s P,N − ∂G ∂T     l P,N ! (T − Tm) = ∆S(T − Tm) (1.5)

where ∆S = Sl− Ssis the difference between the entropies in the two phases.

From this we obtain

∆µ = ∆G n = ∆S n (T − Tm) = ∆h Tm (T − Tm) (1.6)

∆h being the specific heath of fusion. Finally we have

Wmin(r∗) = 16π 3  vsσ3/2 ∆h Tm Tm− T 2 (1.7)

However, the energetic barrier is not the only obstacle to the crystalliza-tion. In fact, the growing of the nucleus requires that molecules of the fluid come in contact with it, and this can be made difficult by a high viscos-ity. The diffusion coefficient D of the fluid has a temperature dependence in the form D ∝ e−∆E/kBT _{(where ∆E is a constant energy), so the rate of} formation of a critical nucleus J can be written as

J (T ) ∝ ρ(r∗)e−∆E/kBT _{∝ exp}  −∆E kBT −16π 3 v2 sσ3 ∆h2_k B T2 m T (Tm− T )2  (1.8) This expression can be simplified introducing the adimensional parame-ters  ≡ ∆E/kBTm α ≡ 16π 3 v2_sσ3 ∆h2_k BTm θ = T /Tm so that J (T ) ∝ exp  − θ − α θ(1 − θ)2  (1.9) Finally, we need to relate this crystallization rate J to the crystallization time τC. In the hypothesis that the linear growth velocity of the nucleus

u = dr_dt is constant, the volume fraction crystallised after a time t is

φ(t) = Z t 0 J4π 3 " Z t0 0 udt00 #3 dt0 = π 3J u 3_t4 _(1.10)

From this, if we fix a certain φ at which we said that the system is crystallised, we have τC =  3φ πJ u3 1/4 (1.11) and because of J ∝ u we find

τC ∝ exp   θ + α θ(1 − θ)2  (1.12) The function J (T ) is plotted in figure 1.3. As we can see, the crystallization rate tends to zero for T ∼ Tm (because of the difficulty in overcome the

energy barrier) but also for T  Tm, because the high viscosity obstacles the

nucleus growth. There is then a maximum J , and if the cooling rate is faster than the crystal growth, crystallization can be avoided.

1.1 The glass transition 5 0 0.2 0.4 0.6 0.8 1

T/T

m 10-7 10-6 10-5

J

Figure 1.3: The crystallisation rate J , as a function of T /Tm, shows a maximum.

A fundamental feature of the liquid in the supercooled phase is that its typical relaxation times increase sharply with small decreases in tempera-ture. The typical time scale τα on which density fluctuations relax is of the

order of the picosecond when T ∼ Tm. Cooling down at a high rate to avoid

crystallization, this time scale grows up to being comparable with the ex-perimental time, changing of 14 order of magnitude [7]. The temperature at which this time scale reach the conventional value of 100 s is called glass transition temperature Tg, and the supercooled liquid at this point is called

glass.

Because of the slowing down of the relaxation time is followed by a cor-responding increasing of the viscosity η of the fluid, at the glass transition this quantity become so high (η ∼ 103 P) that the glass exhibits the me-chanical features of a solid, though it still shows the disordered microscopic structure of a liquid. It is important to stress that the glass transition is not a real phase transition. In fact, the glass temperature Tg depends on the

cooling rate and, in general, on the history of the system. Its definition is, however, quite robust, thanks to the exponential increasing of the relaxation time when Tg is approached.

An important characteristic of glass dynamics is that the temperature dependence of τα cannot, in general, be expressed with an Arrhenius law

τα ∼ exp

 EA

kBT



where EA is an activation energy, positive and temperature independent. In

fact, the experiments [8] show that, for many systems, EA increases upon

cooling. The data are usually fitted using the Vogel-Fulcher-Tammann ex-pression τα ∼ exp  TA T − T0 

where TA is a positive, temperature independent constant.

1.2

Polymers

Polymeric materials are characterised by having molecules built up by the repetition of a certain numbers of fundamental units, called monomers, gen-erally connected by covalent bonds. This number can eventually be very large, up to several thousands or more. There is an enormous number of different polymers around us, from hydrocarbons and plastic materials to biological molecules like proteins or nucleic acids. A polymer’s molecule is characterised [9] by its configuration and conformation. The configuration is the succession of chemical elements along the polymeric chain; a distinction can be made between omopolymers, composed entirely by the same monomer, and eteropolymers, where different kind of monomers are repeated following some scheme or in a random order. The conformation is the spatial organ-isation of the chain: many possibilities exist, like linear, star, comb, and others. This enormous variety in the polymers’ structure is the cause of their enormous importance in nature as in technological applications.

Because of their very complex structure, many different length scales are involved in the dynamics of polymers, some connected with the internal structure of the single monomer, others with the dimension of the entire chain. This competition between different scales gives rise to phenomena of geometric frustration: at low temperature, the monomers can not easily position themselves in an ordered configuration because of the limit imposed by the different length scales. So the more regular polymers result in a two phases structure, where some regions are crystalline and others liquid; for the more complex polymers crystallization is instead completely avoided [10]. Polymers are thus very good glass former, and can be used to study the glass transition.

1.2 Polymers 7

Figure 1.4: Different kinds of polymers. In the left panel are shown several conformations; in the right panel several configurations.

The subjects of this work are simple linear omopolymers with fixed num-ber of monomers n + 1. The variables used to characterise the chain confor-mation in this systems are (see figure 1.5):

• the n bond vectors bi = ri+1− ri, i = 1, ..., n, where ri is the position

of the i-th monomers of the chain; • the n − 1 bond angles cos(θi) = bi

·bi+1

|bi||bi+1|, i = 1, ..., n − 1; • the end-to-end vector

Ree = n

X

i=1

bi

from which it can be obtain the end-to-end distance R2_ee≡ |Ree|2 =

n

X

i,j=1

bi· bj (1.13)

One of the most simple and useful model for a linear omopolymer is the freely jointed model, in which all the vectors bi have the same length b

and they are disposed independently from each other. Within this model, the polymer can be considered an n-steps random walk, and the averaged magnitude of the end-to-end distance is then

hR2_eei = nb2 (1.14)

The flexibility of any polymer is characterised [9] by the length scale of segments in which the chain must be decomposed in order for these segments

Figure 1.5: A simple model for a linear omopolymer.

to be freely jointed. This length scale is known as the Kuhn length lk and is

defined as the ratio

lk =

hR2 eei

L (1.15)

where L = nb is called contour length of the chain.

For fully flexible chains with averaged bond length lb the end-to-end

dis-tance is hR2

eei = nl2b, and then the Kuhn length is just

lk =

nl2 b

nlb

= lb

which means that the real bonds are freely jointed. In the opposite limit case of rigid rod-like polymers, chains are fully stretched, so that hR2

eei = (nlb)2 and lb = (nlb)2 nlb = L

If a polymeric chain is found in a solvent, the interaction between the monomers tent to swell the chain, and R2

eeis larger than in the freely jointed

case. But in a polymeric melt this effect is compensated by the interaction with the monomers of the other chains, and the freely jointed model is often excellent [11].

The complex structure of polymers brings about the presence of multiple relaxation phenomena, each characterised by some typical time scale. An example is the structural relaxation, that is the escape process of monomers from the cage of nearest neighbours inside which they are confined to vibrate for an average time τα.

Another important relaxation process is that of chains losing memory of their initial conformation. This process is characterised by the

autocorrela-1.3 Confinement 9

tion function Cee(t) of the end-to-end vector Ree

Cee(t) = 1 M M X p=1 Rp_ee(t) · Rp_ee(0) (Rpee(0))2 (1.16)

where the p index runs over all the chains of the system, and M is the total number of chains. This correlation function decreases exponentially in time, with a characteristic time-scale τee given by

Cee(τee) = 0.1 (1.17)

It is assumed [12] that in a polymer melt without permanent chain entangle-ments, τee is the maximum relaxation time, as the chain-scale rearrangement

represents the slowest motion in absence of collective modes.

1.3

Confinement

An important idea [3,13] proposed to understand the glass transition is that near Tg there is a correlation length ξ that grows. In the Adam-Gibbs theory

[14], ξ is the mean size of a cooperative rearranging region (CRR), a group of particle that can rearrange themselves in a different configuration. In other words, one can think that when the temperature is cooled down near Tg, the

mobility of the monomers decrease strongly because they are caged. So, in order to a monomer to move, its neighbours also have to move, and a global rearrangement of particles is possible only if a great number of particles participate in this process. If the size ξ of the CRR grows, the motion of the monomers will be more difficult, and a Tg it will requires macroscopically

long time.

There is an (apparently) easy way to investigate this hypothesis. If a polymeric system is confined along one or more dimensions to a size that is smaller than the length ξ, the size of the CRR is bounded, and the dynamics should be faster than in the unconfined case [15]. This is an interesting idea also because, from an experimental point of view, a great number of such confined systems can be created and studied [16–19]. Also several computer simulation [13,20,21] and theoretical approaches [22, 23] have been created to investigate the effects of the confinement on the glass transition. Studies have been performed on simple liquids [24, 25], molecular liquids [18, 26], silica [27] and polymers [13,16,17]; many different type of geometry can be used to confine the liquids, like three dimensional cavities [28], pores [18,26], nanosized fillers embedded in glass-forming liquids [29,30] and thin films [13,

In fact, it is found [31, 32] that the confinement, starting from 100 nm, up to few nm, largely modify the glass transition temperature Tg. However,

the interpretation of these results is not so simple. In fact, it is clear [3, 21] that the behaviour of confined polymeric melt differs from the bulk case not only because of the spatial confinement. There are at least three classes of effects that have to be taken in consideration.

Interfacial effects The particles near the surfaces feel very different forces than the ones situated in the interior of the system. These effects obviously depend on the type of surface with which the liquid is in contact. Surfaces usually used in literature in studying confined glass-forming liquids are of three different kinds:

• Free surfaces. In this case the mobility of the particles is greatly in-creased because their motion near the surface is very less constrained than in the bulk;

• Smooth surfaces. These are smooth potential walls that create a force only in the direction perpendicular to the surface. Also in this case the mobility of the particles is generally enhanced because along the direction of the surface they are free to move without constraints. • Rough surfaces. Here the mobility of the system is usually decreased,

because the particles interact with the surface that has a very low (possibly null) mobility.

Effects on the structure If the surface has a structure, this structure can “propagate” in the liquid because of the interactions. This effect is stronger near the surfaces and gradually weaker moving away from them, because of the disordered nature of the liquid.

Density fluctuations The presence of a rigid surface tends to align the particles to it. This can cause a modulation in the linear density along the direction perpendicular to the surface, that also in this case will be stronger near the surface.

All these effects make it very difficult to understand what behaviour is due to the spatial confinement and what is not. A great amount of studies [6, 21, 33, 34] has been done in order to characterise the behaviour of the liquids confined varying the nature of the confinement. Another interesting possibility [24] is to create the confinement surfaces by freezing amorphous

1.4 Quantities of interest 11

configurations of the same liquid under confinement. In this way structural changes are avoided and the relaxation dynamics should be affected by the pure effect of confinement.

Among all the different possibilities, thin films have a particularly simple geometry to study. Moreover, they can be easily produced by spincoating, an experimental technique consisting in pouring drops of a dilute solution onto a plate spinning at several thousands rotations per minute, followed by an annealing step above Tg to remove residual solvent. This technique allows

a fine control of the thickness h of the film, by changing the concentration and the rotational speed [6]. At last, they are used extensively in techno-logical applications such as coatings, adhesives, barrier layers and packaging materials [35].

1.4

Quantities of interest

In this section the quantities of interest to characterise the static and dy-namical properties of liquids near the glass transition are defined.

1.4.1

Radial pair distribution function g(r)

For a system of non-interacting particles, the average number of particles found in a shell of thickness dr and radius r centred in a certain particle is 4πρr2dr, where ρ is the average density of the liquid.

If the system is interacting, the density will depend on the distance r, and can be written [1] ρ(r) = ρg(r), where g(r) is the radial pair distribution function, written as:

g(r) = 1 N (N − 1) * _N X i=1 N X j6=i δ(|rij| − r) + (1.18)

with N number of particles and rij = ri − rj, i, j labelling the different

particles. Then, in general the average number of particles found in a shell of thickness dr and radius r centred in a certain particle (see figure 1.6) is

dN (r) = 4πρr2g(r)dr (1.19)

A typical radial pair distribution function for a liquid is shown in fig-ure 1.7. The function is null for r smaller than the radius of the particle, because of the repulsive interactions, then it presents a series of peaks, cor-responding to the different shells of neighbours that surround the central

Figure 1.6: The physical meaning of the g(r). The reference system is centred in the red particle, and the blue particles are in shell between r and r + dr.

particle. For increasing r the peaks are less defined because correlations are lost and the liquid appears isotropic, and

lim

r→∞g(r) = 1

1.4.2

Static structure factor S(q)

The static structure factor is defined as

S(q) = 1

Nhρqρ−qi (1.20)

where ρq is the Fourier transform of the local density ρ(r). If the system is

isotropic, S(q) will be a function only of q = |q|.

This is an important quantity because it can be determined experimen-tally from measurement of the cross-section for scattering of neutrons or X-rays by the fluid, and it is easily related [1] to the pair distribution func-tion:

S(q) = 1 + ρ Z

1.4 Quantities of interest 13

Figure 1.7: Radial pair distribution function measured on liquid argon. The ripple at small r are artefacts of the data analysis. Picture taken from [36].

So, via the measurement of S(q), the g(r) can also be measured.

1.4.3

Mean Squared Displacement of the particles

The mean squared displacement (MSD) of the particles is defined as

hr2_{(t)i =} 1 N * _N X i=1 |ri(t) − ri(0)| 2 + (1.22)

where the index i runs over all the N particles of the system, and ri(t) is the

position of the i−th monomer at the time t.

The MSD for some typical liquids near the glass transition is shown in figure1.8. At very short time (with respect to the typical interaction times of the system), the graph in bilogaritmic scale is a straight line with slope 2 [37], because at this time the particles still are not influenced by the interaction with the surrounding neighbours, and perform a ballistic motion with u2 _{∝ t}2_.

On the other limit, at long time the graph is a straight line with slope 1, because the particle performs a free diffusive motion with u2 ∝ t.

For low enough temperature we can see an intermediate regime between the ballistic and the stochastic motion, in which the MSD is almost constant. In this rattling motion the particle try to escape from the cage made up by

Figure 1.8: Mean squared displacement (here indicated with h∆r2(t)i) for the particles of a simulated liquid, for different temperatures T . The appearance of a plateau region approaching the glass transition temperature is evident. Picture taken from [38].

his neighbours, but because their low mobility this requires a certain time. This time interval is sometimes referred as the plateau region. The value of the MSD in this region can be used to quantify the mobility of the system.

1.4.4

Incoherent Intermediate Scattering Function

To quantify the structural relaxation process (also called α−relaxation), the incoherent part of the intermediate scattering function (ISF) can be used [21]. The ISF is defined as

F (q, t) = 1 N S(q) * _N X i,j=1 e−iq·[ri(t)−rj(0)] + (1.23)

where q = |q| and S(q) is the static structure factor, that normalise F (q, 0) = 1. The incoherent (o self-) part of the ISF is then

Fs(q, t) = 1 N * _N X i=1 e−iq·[ri(t)−ri(0)] + (1.24) still normalised to Fs(q, 0) = 1.

1.4 Quantities of interest 15

Figure 1.9: Self-intermediate scattering function Fs(q, t) for the particles of a

simulated liquid, for different temperatures T . The wave-vector corresponds to the first peak in the static structure factor. Picture taken from [39]

This function can be seen as the overlap between the positions of the par-ticles at the time 0 and at the time t on a length scale 2π_q . The characteristic time of the structural relaxation τα is defined via Fs(˜q, τα) = 1/e, where ˜q is

the position of the first peak of the static structure factor S(q), corresponding to the first shell of neighbours. In figure 1.9 we can see that for the systems with lower temperature, close to the glass transition, a plateau region appears at intermediate times shifting the α−relaxation at higher times. As in the MSD (subsection 1.4.3), this is index of the rattling motion of the particles in the cage of the neighbours.

1.4.5

Van Hove function

The van Hove function is useful to characterise the distribution of the dis-placement of the particles as function of time. It is defined as

G(r, t) = 1 N * _N X i=1 N X j=1 Z δ [r − rj(t) + ri(0)] + (1.25)

which can be easily rewritten [1] as G(r, t) = 1

where ρ = N/V . The van Hove function is naturally separated in two parts G(r, t) = Gs(r, t) + Gd(r, t) where Gs(r, t) = 1 N * _N X i=1 Z δ [r − ri(t) + ri(0)] +

is the self part and

G(r, t) = 1 N * _N X i=1 N X j6=i Z δ [r − rj(t) + ri(0)] +

the distinct part. The meaning of the van Hove function is that G(r, t)dr is the number of particles j in the region dr around r at the time t, given that there was a particle i at the position r = 0 at time t = 0. The self part of the van Hove at fixed t is the distribution of the displacement of the particles.

At t = 0, Gs(r, 0) = δ(r) and Gd(r, 0) = ρg(r). As the time increases,

the self part broadens in a bell-shaped curve, while the peaks of the distinct part becomes gradually less evident.

1.5

Universal scaling between structural

re-laxation and vibrational dynamics

Measuring the MSD and the ISF of a system, we are able to determine two important quantities that characterised the dynamics of the system: the amplitude hu2_{i of the MSD during the rattling motion within the cage of}

neighbours, and the structural relaxation time τα.

Larini et al propose [2] a universal correlation between these two quan-tities, extending to glasses the Lindemann melting criterion for crystalline solids. According to the aperiodic crystal structure model [40], the Hall-Wolynes equation holds

τα ∝ exp  a2 2hu2_i  (1.27) where a is the displacement needed to a particle to overcome the energy barrier created by the surrounding neighbours.

To generalise this equation, one can adopt a distribution p(a2_{) of the}

1.5 Scaling between relaxation and vibrational dynamics 17

distribution for p(a2_{), with mean a}2 _{and variance σ}2

a2, and averaging the equation 1.27 over this distribution, we find

τα ∝ exp a2 2hu2_i+ σ2 a2 8hu2_i2 ! (1.28)

Larini et al show [2, 41–43] that this relation holds for a great number of both simulative and experimental systems. Moreover, the parameters a2 _and

σ_a22 have universal values if hu2i is rescaled, for each system, with the value hu2

gi at its glass transition temperature Tg. The main goal of this thesis is

to investigate how this relation is modified if instead of bulk systems we use polymer confined in thin films.

Chapter 2

Molecular Dynamics

simulations of polymers

2.1

Molecular Dynamics

Numerical simulations are an important tools to study systems near the glass transition. In fact, they can be used to test a model from which analytic pre-dictions are too difficult to obtain (because the equation can not be solved exactly, or not even written because of the great number of degrees of free-dom). The simulation based on the model is performed, and the results are compared with the experimental ones. On the other hand, once the accuracy of the model has been tested, the simulations allow us to explore aspect of the systems that are difficult to investigate experimentally.

In Molecular Dynamics (MD) simulations, the equations of motion which govern the time evolution of the system are explicitly solved. We consider a system composed by a certain number N of particle, that interact with each other and eventually are subject to an external potential. The model is fixed by knowing the total potential U ({ri}) acting on the particle i, where {ri}

is the set of coordinates for all the particles that make up the system. From that, the equations of motion for each particle

( ˙ri = _∂p∂H i ˙ p_i = −∂H_∂r i (2.1) are written down, H being the Hamiltonian of the system and {p_i} the conjugate momenta of the coordinates. In MD simulations, equations 2.1

are solved numerically at discrete times, multiples of a chosen timestep δt. So, the time evolution of the system is performed and information like the positions and the velocities of all particles are stored at any desired time.

From these information all the static and dynamical features of the system can be computed.

A great advantage of the Molecular Dynamics simulations, with respect to the Monte Carlo technique [44], is that the simulation time can be put in correspondence with the physical time of the system’s evolution, and this is necessary to study dynamics of supercooled liquids. Instead Monte Carlo simulations generate a trajectory in phase space which samples from a cho-sen statistical ensamble; this trajectory is not physical, though it allows to explore configurations of the system that, with a physical trajectory, would be reached only after excessively long computational times.

MD simulation can be performed at various level of resolution. Simulating a molecule from a quantum point of view, solving Shrodinger’s equation for all the electrons and the nuclei, is computationally very expansive, because it requires a time step of ∼ 10−17 s [3]. Given the current computer power, a system of N = 100 nuclei can be simulated for about 1 ns. This time is barely sufficient to equilibrate the system at high temperature.

A solution for this problem is to use an atomistic approach. The electronic degrees of freedom can be replaced by effective potential for the bond lengths and angles between the atoms. This atomistic approach allows one to use a time step of about 10−15 s, making it possible the simulation of 100 nuclei for ∼ 100 ns. A problem with this method, however, is that the result of the simulation depends drastically on the values that are given to the parameters of the potentials between the atoms, so a difficult fine-tuning of this parameters is required [45].

A further simplification is permissible if we are not interested in studying the properties of specific polymers. The strong slowing down of the polymer approaching the glass transition is a quite universal process, so it is possible to integrate over the fast degrees of freedom (vibration of bond lengths and angles) and use a coarse grained model [46]. This model contains only the fundamental feature of the molecule; in the case of linear polymers, these are the chain connectivity and the excluded-volume interaction and the attrac-tion between different monomers. The entire monomer, or even groups of more monomers, are considered as a single object, and these object interact with some kind of effective potential.

2.2

Equations of motion

The equation 2.1 are 6N differential equations of first order (where N is the total number of particles), or equivalently 3N equations of second order in the variables {ri}. The implementation of differential equations in computer

2.3 Periodic boundary conditions 21

code requires an algorithm to discretize the equations; the software used in this work uses the Verlet algorithm [44]. Choosing a timestep δt small with respect to the typical times of the dynamics of the system, the position of the particle i at the time t + δt and at the time t − δt can be approximated, up to terms of order O(δt4), by a Taylor’s expansion

ri(t + δt) ' ri(t) + δtvi(t) + 1 2δt 2 ai(t) ri(t − δt) ' ri(t) − δtvi(t) + 1 2δt 2_a i(t)

where the acceleration ai(t) is obtained by the force

−∂U ({rj(t)}) ∂ri(t)

divided by the mass mi of the particle. Summing up this two equations, one

obtains

ri(t + δt) ' 2ri(t) − ri(t − δt) + δt2ai(t) (2.2)

The velocities, that are not needed to compute the trajectory of the system, can be easily obtained at each timestep with

vi(t) =

ri(t + δt) − ri(t − δt)

2δt (2.3)

up to term of order O(δt2_).

It is important to notice that δt cannot be too small, otherwise the max-imum period of time that one is able to simulate becomes too short.

2.3

Periodic boundary conditions

The number of particle that are under simulation is usually N  NA. The

size of the system is in fact limited by the time necessary to the evaluation, at each step of the simulation, of the forces among each pair of particles. If each particle interact with all the others, this number of forces to evaluate is proportional to N2_{. In the (common) case of short-range interactions, the}

potential can usually be cut at a certain distance from the particle without affecting significantly the evolution of the system. However, also in this case we need to evaluate at each timestep a number of forces proportional to N , and this fact limits N to be of the order of some thousands.

This is a problem because, if N is small, a relevant fraction of the to-tal particles are locate on the surfaces of the system, and experience quite different forces from the particles in the bulk, causing finite size effects.

Figure 2.1: A two dimensional periodic system. Picture taken from [1]

This problem can be overcome by imposing periodic boundary conditions. The particles simulated are placed inside, for instance, a cubic box, that is repeated in all the direction infinite times. During the simulation, if a particle moves, all its images in all the other boxes move in the same way (figure2.1). It is not necessary to store the coordinates of all the (infinite) images, only the particles in the central box. When a particle leaves the box crossing a boundary, the attention may be switched to the image just entering. In this way, there are no boundaries, and no particles on the surface.

However, it is not obvious, a priori, that such a system have to behave in the same way of a real macroscopic system. In fact, this is in general not true. If the interactions between particles are long-ranged (i.e., proportional to r−ν, where ν is less than the dimensionality of the system), the symmetry caused by the repetition of boxes will be imposed to the system that should be isotropic, because each particle interact with its images in the other boxes. Moreover, the fluctuation over wavelengths bigger than the box size L are suppressed with this kind of boundary conditions.

But if the interactions are short ranged and the box size L is taken suffi-ciently bigger than the typical interaction length, and long-wavelength fluc-tuations are not relevant in the phenomenon under study, is reasonable to expect that periodic boundary conditions allow to simulate a macroscopic system avoiding finite size effects. However, this assumption has to be veri-fied for each new system we want to investigate.

2.4 The canonical ensamble 23

2.4

The canonical ensamble

MD simulation are usually performed in the microcanonical ensamble, with constant energy and number of particles. However, before taking measures on the system, is necessary to equilibrate it, to minimise the dependence of the results by the initial configuration. The equilibration has to be performed in the canonical ensamble, with a fixed temperature T , while the energy is allowed to fluctuate. The system has then to be posed in thermal contact with a thermostat.

In MD simulation this can be done via the Nos´e-Hoover extended phase space technique [47]. In this method, a fictitious variable s is added to the system (obtaining an extended system), and an Hamiltonian is imposed such that the projection of the extended system (evolved in a microcanonical ensamble) will be the real system evolved in a canonical ensamble.

If (q0_i, p0_i, t0) are the coordinates in the real system, we introduce virtual variables (q_i, p_i, t) related to the real one via

   q0_i = q_i p0_i = p_i/s dt0 = dt/s (2.4)

and we impose the Hamiltonian

H =X i |p_i|2 2mis2 + U (q) + p 2 s 2Q + gkBT ln s (2.5) where ps is the conjugate momentum of s, Q is a parameter that act like a

mass for the variable s, kB is the Boltzmann constant, T is the temperature

set for the reservoir and g is a constant that have to be set equal to 3N − 1 to obtain the canonical ensamble.

If a partition function is built for this extended system in the micro-canonical ensamble, and is then projected on the real system, the partition function obtained is, up to a multiplicative factor, the partition function for the real system in the canonical ensamble with temperature T . Looking to the equations of motion for the real variables, we can see that we are intro-ducing a viscous force that tend to modify the velocity distribution in order to match with the desired distribution a T :

dp0_i dt0 = − ∂U ∂q0_i − p 0 i ps Q (2.6)

It is also possible to desire to evolve the system at constant pressure P instead of constant volume V . The system must here be coupled to an

external isotropic piston (barostat) that compresses or expands the system in response to fluctuations of the instantaneous internal pressure, in order to keep the latter equal in average to the externally applied pressure. In order to reproduce the volume fluctuations induced by the piston (compres-sions and expan(compres-sions), volume is introduced in the expanded phase space as an independent dynamical variable, together with its conjugate momentum. The equations of motion used in LAMMPS are build up via the combination of the method of Nos´e-Hoover chains for thermostatting and the Martyna-Tobias-Klein (MTK) method for barostatting. The physics involved is the same as in the Nos´e-Hoover equations of motion, and we refer to [48] for more technical details.

2.5

The bead-spring model

A coarse grained model that is commonly used in the literature to study of glass-forming polymers is the bead-spring model [3, 46, 49]. In this model, each chain contains a number n of monomers. Each pair of non-bonded monomers belonging to the same or different chains interacts via a truncated and shifted Lennard-Jones (LJ) potential, that mimics the Wan Der Waals interaction: VLJ(r) = ( 4h σ_r
12− σ r
6i + Vc for r ≤ rc 0 else (2.7)

where σ is the zero-crossing distance for the potential, − is the energy in the minimum (situated at r = 21/6_{σ) and the constant V}

c is added to assure

the continuity of the potential in rc. Note that the potential is continuous in

rc but the forces are not. The equation 2.7 is plotted in figure2.2.

Each pair of first neighbour bonded monomers along a chain interacts instead via an harmonic potential

VEL(r) = k(r − r0)2 (2.8)

Notice that in this model there are two different characteristics length scales, σ and r0. This is important because if they are different from each other, it

is very difficult for the system to crystallise, even if the number of monomers per chain n is small, and the supercooled phase can be reached [46].

Finally, it is possible to introduce a chain stiffness Vb(θ) = kθ(1 − cos θ)

2.5 The bead-spring model 25 σ∗

r

-ε 0

V

LJ

(

r)

Figure 2.2: The truncated and shifted Lennard-Jones potential. Note that the minimum is located in σ∗= 21/6σ.

that depends on the angle θ formed by three subsequent atoms in the chain. When this potential is absent the chain is said to be fully flexible. In all this work, simulations are performed with fully flexible chains.

2.5.1

Lennard-Jones units and mapping to real units

The equation2.7, together with the mass of a monomer m, defines the char-acteristic scales of the system: the energy , the length σ and the time τ =

q

σ2_m

 . A system of units can be choose in which the masses are

mea-sured in m, the lengths in σ and the energies in  (or the temperatures in /kB, where the Boltzmann constant is set equal to 1). This choice presents

several advantages:

• the parameters of the simulation are of the order of unit, instead of the very small number associated with the microscopic scale, and the results of the numeric simulation are easily manipulated;

• the equations of motion are simplified because some of the constant are put equal to one;

• the results of the simulation are expressed with universal formulas in adimensional units, that can be easily mapped in real units for different systems.

Physical quantity Unit Value for Ar Value for PS length σ 3.4 · 10−10 m 9.7 · 10−10 m energy  1.65 · 10−21 J 6.74 · 10−21 J mass m 6.69 · 10−26 kg 5.8 · 10−25 kg time (σ2_m/)1/2 _{2.17 · 10}−12 _{s 9 · 10}−12 _s velocity (/m)1/2 _{1.57 · 10}2 _{m/s 1.08 · 10}2 _m/s force /σ 4.85 · 10−12 N 6.95 · 10−12 N temperature /kB 120 K 490 K pressure /σ3 _{4.20 · 10}7 _N/m2 _{7.38 · 10}6 _N/m2

Table 2.1: Typical experimental values for the liquid Argon and for Polystyrene are reported compared to the reduced LJ units used in MD simulations.

In fact, it is clearly important to be able to map these reduced units into real units, to make a comparison with the experimental systems under study. Typical values for these reduced units are reported in table2.1 in the case of liquid Argon [50] and of Polystyrene [51].

2.6

Simulation protocol

In this work simulations are carried out with the open-source LAMMPS software (lammps.sandia.gov/, [52]) , while the post processing analyses are performed with our own code in C. A coarse grained model (the bead-spring model, illustrates in the subsection 2.5) is used for the polymers.

Initial configuration

The simulation starts from a random configuration of the monomers’ po-sitions. To do this, the simulation box is defined, and periodic boundary conditions are imposed in all the directions.

To generate the configuration, the position of the first monomer of each chain is randomly chosen. Then, the second monomer is added, choosing its coordinates in a sphere around the first one, with the constraint of minimum distance for all the non-bonded monomers. The procedure is iterated until all the monomers are placed in the box.

The generated configuration could be non physical, mainly because of the overlapping of bonds. This problem is overcome by minimising the total potential

2.6 Simulation protocol 27 U ({ri}) = N X i,j VLJ(rij) + X i,j VEL(rij) (2.9)

(the second summation is performed over the pairs of bonded monomers), moving each monomers in the local minimum of the potential. This minimi-sation is first performed at a high T and with a decreased chain rigidity k, so that is easier to disentangle eventually overlapped bonds. Then a second minimisation, with the values of T and k used for the actual study of the sys-tem, is performed, obtaining a physically acceptable configuration, without residual non physical monomer positions (such as overlapping bonds) caused by the initial random generating algorithm.

Because of the small numbers of particle, it is hard to eliminate the dependence of the results on the initial configuration. So, for each system, several different initial configurations (runs) are prepared and evolved, and the final results are averaged over the different runs. One can see that the error obtained from this average is bigger than the errors due to the numerical solution of the equations of motion at discrete times, so it will be taken as estimate of the uncertainty on the results.

Equilibration

Before taking measurements on the system, it has to equilibrate, in order to avoid every dependence on the initial configuration. To do this, a random Gaussian distribution of velocities

f (vi) = r m 2πkBT exp  − mv 2 i 2kBT 

is generated for the monomers (i labels the three components of v), and the system is evolved in the canonical ensamble at the desired T , using the Nos´e-Hoover technique.

In particular, in our simulations the systems is first evolved at a tempera-ture high with respect to the interaction energy between monomers (T = 1.5), and with a low value of the spring constant of the bonding potential (k = 50) in order to remove eventual entanglement between the polymeric chains. Af-ter this, the system is equilibrated at the desired temperature.

This second phase has to last a longer time than the larger relaxation time of the system, τee, taken as the mean decorrelation time of the

end-to-end vector of the chains Ree. The corresponding correlation function Cee(t)

is then monitored during the equilibration; τee is defined via Cee(τee) = 0.1,

In this step, as in every other step of the simulation, the timestep for the evolution used is δt = 0.003 (in Lennard-Jones reduced units). In real units, it corresponds to about 10−14− 10−15_{s. This value has been chosen, in}

agreement with previously performed simulations [2] and with the LAMMPS documentation [52] , because it is much smaller than all the characteristic times of the phenomena under study, but it is still sufficiently big to allow the simulation of a long enough time to reveal the dynamical features we are interested in, in a reasonable computational time (from a few days to a few weeks).

Measurements

Measurements are taken evolving the system in the microcanonical ensamble. Two precautions are needed. First of all, the final results are averaged over the different runs, but, because of the coupling with the thermostat and the barostat, at the end of the equilibration they are at different T and V . To minimise this problem, the systems are evolved (in the N P T ensamble) for a small number of steps, and among these the one with the smallest variation in volume among the different runs is chosen. Then the same is done for the temperature, evolving the systems in the N V T ensamble.

Secondly, in order to avoid transient effects due to the changing between canonical and microcanonical ensambles, the system is evolved for a certain amount of time in the second one, and only after this the measurements start. Positions and velocities for each monomers are saved every ∆t, and from this information all the desired functions are calculated, using our own code written in C. In order to cover several order of magnitude in time, data are taken with different ∆t. We take data first each ∆t = δt, until a tmax = 2.25;

then, we take data each ∆t = 333δt, until a tmax = 750; finally, we take

data each ∆t = 6666δt (or, for the slower systems, ∆t = 133333δt), until a time of tmax = 15000 (or, for the slower systems, tmax = 300000). All the

times are here expressed in reduced LJ units. In real units, the timestep δt corresponds to ∼ 0.003 ps, and the maximum time tmax covered by a

simulation is ∼ 0.3 µs. Data analysis

In the following the algorithm used to compute the most important quantities are described.

The algorithm used to obtain the g(r) (subsection 1.4.1) computes, for many different configurations of the system, the distance between each pair of particles, and divides this distance in bins I(ri), that count the pairs with

2.6 Simulation protocol 29

distance between ri and ri+ ∆r. Then the g(r) is calculated via the equation

1.19:

g(ri) =

I(ri)

4πr2_∆r

averaging this quantity on all the different configurations. Obviously, the maximum possible r depends on the size of the simulation box L. It is important, to avoid spurious effects due to the finite size of the simulation, that g(r) is close enough to 1 at the edges of the box.

The MSD (subsection1.4.3) and the ISF (subsection1.4.4) are calculated exactly, having stored the positions of all the monomers at all times, and then averaged over the different runs.

Finally, the self part of the van Hove function (subsection 1.4.5) at a certain time ¯t is computed measuring the displacements of all the particles from t = 0 to ¯t and making an histogram, binning the possible displacements from r = 0 to a chosen rmax. In order to average this quantity, if the system’s

configurations are stored each δt from t = 0 to t = tmax, the histogram is

made for each possible pair (t1; t1+ ¯t), with t1 > 0 and t1+ ¯t < tmax, and

Chapter 3

Effects of confinement on the

dynamics

This chapter reports the original results obtained in this thesis. The main question to which we want to answer is if (and how) the universal scaling between the structural relaxation and vibrational dynamics, found in bulk polymeric systems, is affected by the confinement in a thin film.

In this chapter we show that the universal scaling seems lost if we consider a polymeric melt confined in thin supported films with a rough substrate; however, the universal scaling can be recovered if the analysis is performed not on the entire system, but on the single layers that made up the film.

In section 3.1 the protocol for the simulation of the systems in analysis is described. Section 3.2 presents the details and the results of the analysis on the whole rough-supported film, while the analysis of the single layers is presented in section 3.3.

3.1

Molecular Dynamics simulations of

con-fined polymeric systems

The simulation of polymeric systems confined in a thin film presents several differences with respect to the bulk one, described in section 2.6. In the following, the simulation protocol is discussed. The confinement will be ap-plied in the z direction, so that the film lays in the xy plane. Moreover, the systems simulated are supported films, meaning that one of the surfaces is in contact with a substrate, while the other is free.

Differences between the bulk and the confined case are mainly of two kinds. The system is no more isotropic, but has a preferential direction (the

z direction in our case), and some of the functions used for the system’s analysis have to be modified to work with a confined system.

First of all, the simulation box is not cubic. The size in the x and y direc-tion are chosen in the same way as in the bulk case, and periodic boundary conditions are applied, so that the system is virtually infinite in these direc-tions, but finite boundary conditions are applied in the z direction. When the initial configuration is created, all the box is filled with the monomers. The simulation is performed allowing the z-sides of the box to move, follow-ing the monomers with the higher and the lower value of the z coordinate. This is necessary to avoid the loss of particles, that would otherwise come out of the simulation box and be lost.

This procedure requires to pay attention during the first phase of thermal-isation, that is performed at high temperature (T = 1.5) in order to eliminate eventual entanglement among the polymeric chains. At this temperature the chains have enough thermal energy to escape from the Lennard-Jones poten-tial of the other chains; so, to avoid the evaporation of the film, it has to be confined between two potential walls.

After this phase, the two walls are removed, and the substrate chosen for the simulation is created near one of the surfaces; the other surface is left free. The system is then thermalised at fixed N and T . There is no need to add terms to the Hamiltonian to control the system’s pressure: the free surface feels a null pressure, and the thickness of the film changes so that also the internal pressure of the polymer is zero. Note that, in this way, the film thickness h depends on the thermodynamic quantities N and T ; at fixed T , the thickness can be changed via changing N .

As in the bulk case, the equilibration is protracted until a time t = 3τee

has passed, where τeeis defined looking at the end-to-end correlation function

via Cee(τee) = 0.1. The end-to-end correlation of a simulated polymeric film

during the equilibration is shown in figure 3.1. The different lines refer to the different runs. Time is expressed in reduced units. One can see that the threshold value Cee = 0.1 is reached, for all the runs, at about τee= 105, and

the system is then equilibrated until 3τee.

After the equilibration, measurements are taken in the microcanonical ensamble, keeping N and E fixed. The pressure of the system is always zero, thanks to the free surface. In the analysis of the results, it is important to notice that one of the main quantities used to characterise the static structure of the system, the radial pair distribution function g(r), has to be re-defined in a different way. In fact, the g(r) defined in subsection 1.4.1 requires the homogeneity and isotropy of the system, that are lost in thin films. However, the same function can be used considering only the monomers in a slice of the film parallel to the xy plane. In this way, we have a function g(r, z, ∆z),

3.2 Rough substrate 33 103 104 105

t

0 0.1 0.2

C

ee

(

t)

Figure 3.1: End-to-end correlation function Cee(t) for several different runs, for a

simulated polymeric film with N = 3000 monomers at T = 0.49. All the quantities are expressed in reduced units. We can determine τee' 105.

where z is the position of the slice and ∆z its thickness. If z and ∆z are chosen carefully, the slice can be considered homogeneous and isotropic, and the pair correlation function contains information on the polymer’s structure inside it.

3.2

Rough substrate

In this section the details of the model used and the results of the analysis performed on the rough-supported films are described.

3.2.1

Details of the model

With rough substrate we mean that the polymeric melt is in touch with an atomic lattice that mimic a crystalline solid. In this work the substrate is created tethering substrate atoms to the sites of a square lattice with spacing σs = 0.9 · 21/6σ with an harmonic potential

Vth(ri) = kth|ri− ri0|2 (3.1)

where ri is the position of the atom and ri0 is the position of the lattice site.

Figure 3.2: A configuration of a simulative polymeric melt (pink monomers) sup-ported on a rough substrate (blue beads). Periodic boundary conditions are ap-plied on the lateral directions. The surface on top of the film is free.

interaction kth the rigidity of the substrate can be varied. According to [21],

in all the following investigations we have set kth = 100. The interaction

between monomers is the potential 2.7, that in reduced units is written

VLJ(r) = ( 4h 1_r
12− 1 r
6i + Vc for r ≤ rc 0 else (3.2)

Finally the atoms of the substrate interact with the monomers via a standard Lennard-Jones potential: Vsp(r) = ( 4sp h σsp r
12 − σsp r
6i + Vc for r ≤ rc 0 else (3.3)

and we set σsp = σ = 1 and sp =  = 1. Still according to [21], we set for the

harmonic potential between connected monomers k = 555.5 and r0 = 0.9.

Note that r0 is chosen different from σ in order to avoid crystallization.

All the following investigations are done with polymeric chains of n = 3 monomers.

3.2.2

Static properties

Several quantities can be measured in order to characterise the static prop-erties of the film.

3.2 Rough substrate 35 0.85 0.9 0.95 1

r

0 5 10 15 20

P

(r

)

Figure 3.3: Bond length distribution for a simulated film with N = 3000 monomers at T = 0.49. The minimum of the bonding potential is at r0 = 0.9. The

distribu-tion is normalised to one.

The bond length distribution is the probability distribution of the dis-tance between two bonded monomers. As it can be seen in figure 3.3, the distribution is peaked around a value a bit smaller than the minimum value of the bonding potential r0 = 0.9, and has a full width at half maximum of

δr = 0.048.

The angular distribution function is the probability distribution of the angles formed by one central monomer and two other monomers situated at a distance between r and r + ∆r from the first one. In the following figures the values of r and ∆r are chosen in such a way to include the first shell of neighbours of the central monomer (r = 0.8, ∆r = 0.5). In figure 3.4

the angular distribution functions for simulated polymeric film of different thicknesses are plotted. Some of the different peaks can be easily related to particular favoured dispositions of the monomers. For instance, the peak at cos θ ' 0.5, corresponding to an angle θ ' π₃ between the monomers, corre-sponds to the situation in which three monomers are placed in the vertex of an equilateral triangle, each one in the minimum of the Lennard-Jones po-tential created by the other two. The peak at θ ' −0.5, instead, corresponds to the situation where the three monomers form an angle of 2π₃ , i.e. they are, together with another monomer, in the vertex of a rhombus with angle of 2π₃ and π₃. The small peak at cos θ ' 0.7 correspond to an angle of θ ' π₄. This

-1 -0.5 0 0.5 1

cosθ

0 0.2 0.4 0.6 0.8 1

P(

θ)

N=2001

N=3000

N=3999

Figure 3.4: Angular distribution function for simulated polymeric film at T = 0.50 and different thicknesses. The distribution is normalised to one.

situation can be understood considering that the lattice of the substrate is square: these are monomers forced at this angle by the interaction with the substrate. In fact, this peak is absent in bulk systems or in film supported on a smooth substrate.

The linear density along the ˆz direction ρ(z) is the average number of monomers in a slice of the film parallel to the xy plane situated at z. This quantity, that is obviously constant in a bulk, isotropic system, shows an enormous modulation under confinement (see figure3.5): near the substrate, monomers are placed in well-defined layers. Going away from the substrate, this strong modulation is lost, because of the disordered nature of the poly-meric melt. It is natural to imagine that the dynamics of the monomers in the layers near the substrate is different from that of those far away. This is the main motivation to the layer analysis that will be discussed in the section 3.3.

In figure3.6the linear density along the z direction is plotted for different numbers of monomers N (figure3.6a) and temperature T (figure3.6b). From the upper panel we can determine the thickness of the film, considering that the film ends when the linear density goes under 0.1. The resulting thickness is h = 5, 7.5 and 10 (in reduced units) for systems with a number of monomers N = 2001, 3000 and 3999 respectively. This corresponds, in real units, to a thickness of 5 − 10 nm.

3.2 Rough substrate 37 0 1 2 3 4 5 6 7 8 9 z 0 1 2 3 4 5 6 ρ ( z)

Substrate

Polymer

Figure 3.5: Linear density along ˆz for a system with N = 3000 monomers and T = 0.49. The red peak, at z = 0, is the substrate (the vertical axis is cut at y = 6 to magnify the density of the polymeric melt, ρ(0) ' 15, not shown). The dashed line is the linear density of the equivalent bulk system at T = 0.49 and P = 0.

From the lower panel we can see that the density modulation is a little bit stronger for the systems with lower temperature. This makes sense, because the reduced mobility of the monomers at lower temperature helps to keep them confined in the layers.

The radial pair distribution function (see subsection1.4.1) of a polymeric melt presents a characteristic feature: the first peak is split into different peaks, that correspond to the different length scales present in the polymer. In the case of our systems, where two different length scales are introduced (see subsection 2.5), the peak is split in the two parts shown in figure 3.7

(black line). The first peak, at r = 0.9 (using reduced Lennard-Jones units), corresponds to the minimum r0 of the bonding potential. The second, at

r = 21/6_{, corresponds to the minimum σ}∗ _{of the Lennard-Jones interaction}

between all the pairs of non-bonded monomers. In figure 3.7 we can also see that, with the exception of the monomers nearest to the substrate, the structure in the xy plane does not seem to be strongly affected by the con-finement. Instead, the monomers nearest to the substrate (blue line) show a stronger variation in the g(r): the curve is significantly different from 1 also at the boundaries of the simulation box, and some structure seems to emerge in the second peak. These are both signs that some sort of long-range

0 1 2 3 4 5 6 7 8 9 10 11

z

0 1 2 3 4 5

ρ

(

z

)

N3999

N3000

N2001

(a) 0 1 2 3 4 5 6 7 8 9

z

0 1 2 3 4 5

ρ

(

z

)

T=0.48

T=0.49

T=0.50

(b)

Figure 3.6: Linear density along z for rough-supported films with various N (upper panel) and T (lower panel). The substrate is located at z = 0.

3.2 Rough substrate 39 0 1 2 3 4 5 6 7 8 9 10

r

0 1 2 3 4

g

(r

)

1

2

3

Bulk

Layer

Figure 3.7: Planar g(r, z, ∆z) measured in slices that include the three layers nearest to the substrate (figure3.5) of a film with N = 3000 monomers at T = 0.49. ∆z = 0.9. The black line is the g(r) of a bulk system at the same temperature.

order is imposed on these part of the film, because of the interaction with the substrate.

3.2.3

Dynamical properties

The dynamics of the melt is affected by the rough confinement in a complex way. In fact, there are two effects that lead in opposite directions. The free surface causes an enhanced dynamics, speeding up the motion of monomers; this tends to increase the MSD of the monomers at the plateau hu2i, and to decrease the structural relaxation time τα. Vice versa, the interaction

with the rough substrate causes a strong slowing down of the dynamics, increasing τα and decreasing hu2i. So, it is difficult to determine a priori if

the mean value of τα and hu2i for the film will be increased or decreased. In

figure 3.8 we can see the MSD and the self part of the ISF for different film thickness, at fixed T . It seems that the cage dynamics is weakly speeded up with decreasing thickness (figure 3.8a), but the relaxation time τα also

increases with decreasing thickness (figure 3.8b). This two quantities show then opposite trends under confinement.

This fact makes it impossible for the universal scaling illustrated in sub-section 1.5 to hold for these systems. In fact, for a bulk polymeric melt the

0.001 0.01 0.1 1 10 100 1000 10000

t

0.0001 0.001 0.01 0.1 1 10 100

〈

r

2

(

t)

〉

N=2001

N=3000

N=3999

1 0.1

(a) Mean squared displacement for different thickness, T=0.49. The insert mag-nifies the region around t = 1.

0.01 0.1 1 10 100 1000 10000

t

0 0.2 0.4 0.6 0.8 1

F

S

(q,t

)

N=2001

N=3000

N=3999

(b) Self-part of the intermediate scattering function at q = ˜q (the first peak of the structure factor) for different thickness, T = 0.49.

Figure 3.8: Both the MSD in the plateau and the relaxation time τα are weakly

3.3 Layer analysis 41 11 12 13 14 15 16 17 18

1/〈u

2

〉

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2

Log(

τ

α

)

h = 5

h = 7.5

h = 10

Figure 3.9: log(τα) versus 1/hu2i, for films with different temperatures and

thick-ness h. The black line is the universal scaling (equation1.28).

relation between log(τα) and 1/hu2i has the parabolic form of equation 1.28.

If hu2_{i increases with the confinement (i.e., 1/hu}2_{i decreases) and τ} α also

increases (i.e., also log(τα) increases), the point



1

hu2_i, log(τα)



cannot stay on the same parabola 1.28.

We have determined the two quantities hu2_{i (the mean squared}

displace-ment of the monomers at t = 1) and τα (the structural relaxation time) for

the systems described in subsection3.2.1, for different numbers of monomers N and temperatures T . In figure 3.9 these quantities are plotted, together with the parabola 1.28of the universal scaling (black line). We can see that, as said, the universal scaling relation does not hold: the points are not on the parabola. Moreover, a general trend can be seen: the thinner the film, the more the points deviates from the bulk case.

3.3

Layer analysis

The strong modulation of the linear density in the z direction shown in figure 3.5 suggests that the dynamics of the monomers could be strongly influenced by their z coordinate. In fact, especially near the substrate, the monomers are divided in well defined layers; the density between one layer and one another is zero, or really close to zero. It is natural to imagine that