## Thesis etd-12072011-120258 |

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Thesis type

Tesi di dottorato di ricerca

Author

CHULKIN, OLEKSANDR

URN

etd-12072011-120258

Thesis title

Microscopic Dynamics of Polymer Melts: Numerical Simulations with Coarse-Grained Models

Academic discipline

FIS/03

Course of study

FISICA APPLICATA

Supervisors

**tutor**Prof. Leporini, Dino

Keywords

- cage correlations
- Debye-Waller factor
- density-temperature scaling
- Glass transition
- molecular dynamics
- polymers

Graduation session start date

16/12/2011

Availability

Full

Summary

Every liquid in principle can transform into a glass if it is cooled or compressed fast enough. Glass

is one of the most important artificial materials utilized by man. Numerous applications of glass

materials and glass transition phenomenon include obtaining of the optimal wet-skid resistance by

using the tread rubber with high glass transition temperature, enhancement of the ballistic

penetration resistance of armor coatings obtained by energy dissipation associated with an impact induced

transition of the coating to the glassy state, the material selection for acoustic tiles on

submarines allowing to undergo the glass transition at the frequency of the active sonar at seawater

temperatures thus dampening the sonar, arterial walls restoring, preservation of food,

highly soluble pharmaceuticals. So, the study of glassy state tends to be one of the most promising

and interesting tasks in chemistry in our days. This fact was my main motivating factor to study the

structural and dynamical characteristics of the polymer melt approaching to glass transition from above

in group of Prof. Dino Leporini in Pisa University.

The particles(in this Thesis the word “particle” signifies the most elementary rigid part of the system

under investigation. In general, it can be either molecule for the system containing the rigid molecules,

or atom for the case of flexible molecules. As far, as in this Thesis the polymer melt is the system

that I am investigating, the word “particle” in context of my research signifies the monomer) spend

increasing periods of time, rattling in cages formed by their neighbors while the liquid, consisting of

these particles, approaches the glass transition. Finally, the particles are relaxed from their cages -

this process is called the structure relaxation. The main aim of this thesis is the investigation of the

microscopic dynamics of the particles trapped in these cages and the study of the possible ways to

connect this fast dynamics in cage to slow macroscopic dynamics associated with structure relaxation.

First direction of research is based on the density-temperature scaling of the one of the most important

characteristics of the cage dynamics - the Debye-Waller factor(that is the amplitude of the particle’s

rattling in the cage, see the section 1.2 for the details). The density-temperature scaling has become

very popular in the recent years due to the improvement of the technical equipment of the experiments,

that allowed to change both density and temperature of the system in simultaneous and prompt way.

The density-temperature scaling of the structure relaxation time for single and multi-component liquids

having different interaction potentials was introduced and investigated in this thesis from the different

points of view, including the pressure-energy correlations and potential energy landscapes with citations

of the correspondent sources (see the section 3.1). Recent research of the Prof. Leporini group has

presented the universal scaling between the structure relaxation time and Debye-Waller factor(see the

subsection 3.1.5). Nevertheless, the research of the density-temperature scaling of the Debye-Waller factor has been missing so far and I tried my best to improve this situation and to show that the

density-temperature scaling is valid for the microscopic processes, too. In the terms of the second

direction of investigation I studied several functions of cage correlation for the polymer melt. This

study should have allowed me to better understand the processes taking place in the cages or “shells”

surrounding the tagged particle and to discover the new types of connection between these microscopic

processes from one side and the macroscopic processes, like e.g. the structure relaxation, from another

side. Results of this study could help to develop the new ways of understanding of the universal scaling

between the structure relaxation time and Debye-Waller factor. Furthermore, it is interesting to study

the cage correlation functions of the polymer melt because the polymers have been never studied in

this way so far. Previous researches of the correlation functions dealt with the binary mixtures,

hard spheres and disks, etc. I also have to say that this study is based on introduction of different

correlation functions including the self-correlation functions of displacement that have not been studied

in context of the liquids state research so far.

As to the meaning of the results of the thesis, I may state the following:

• the density-temperature scaling of the Debye-Waller factor is evidenced(see the section 3.1 and

the Chapter number 4). The values of the scaling exponent γ are consistent with the predictions

from the study of Lennard-Jones liquids. Data of all the polymers in the study, that have different

molecular masses and interaction potential, collapse the straight lines of Debye-Waller factor vs

TV γ plot, where any straight line is uniquely defined by the molecular mass and parameters of

the interaction potential. These lines cross in one “universal” point.

• the cage correlation functions, describing the time evolution of the neighbor cage of the given

particle(see the subsection 3.2.1 and section 5.1) and following from the immediate physical interpretation,

represent the alternative but perfectly equivalent instrument for the description of

the structure relaxation, compared to the more rigorous intermediate scattering function.

• the analysis of the spatial correlation of displacement(see the subsection 3.2.2 and section 5.2)

evidences the link between the well-known static properties of the system and the dynamic properties,

represented by the direction and modulus correlation functions of displacement. Origin of

this correlation is not perfectly clear and needs further investigation.

• analysis of the time correlations of displacement (see the section 5.3) shows that the directionality

of motion rather than the displacement modulus seems to connect the fast microscopic and slow

macroscopic dynamics. The direction of the displacement of a particle at the time scales of cage

regime determines in general the direction of the particle’s motion even at much longer times.

As far, as the outlook of the Thesis is concerned, I have to point out that the further investigation,

generalization and improvement of these results could be useful. The results of current Thesis allow to

construct only several very narrow and rickety bridges between the fast microscopic and slow macroscopic

dynamics. These narrow links should be united with the other existing ones and with those that

will be created in future, thus forming the new theories, methods and approaches. Among the possible

ways of developing of the results of this Thesis I can point to either the prospect of the study of the

influence of TV γ on the behavior of the cage correlation functions, uniting the two main directions of

research of this thesis, could be the most promising direction of further study, or to the aforementioned

possible research of the reasons of the correlation between the static and dynamic(represented by the

direction and modulus correlation functions of displacement) properties of the system. Results and

outlook of the Thesis are discussed in the Conclusions in more detailed way.

My thesis consists of five chapters. First chapter of my PhD thesis presents the introduction to glass

transition phenomenon (section 1.1); covers the information about the static structure and relaxation in

liquids and introduces the functions that describe these processes (section 1.2); and describes the glass

transition in polymers with particular attention to the similarities and differences of the glass transition

processes in simple liquids and polymers(section 1.3), thus explaining why the polymers usually are the

good glass-formers.

Then, in the second chapter, covering the methods of the computer simulations of polymers, there

follow:

• the review of numerous methods of molecular dynamics and (a bit) of monte-carlo simulations of

the glass-forming polymers(section 2.1), allowing to better understand the future ways of development

of the simulation studies of the polymers and to enrich the arsenal of a modern scientist

with new powerful computational methods enabling the simulations to become more quick and

effective

• separate section (2.2) describing the numerical methods and MD model (developed by Cristiano

De Michele, a former member of the Leporini group) used in the MD simulations runs during the

work over the fourth and fifth chapters

Third chapter contains the theoretical background for the fourth and fifth chapters.

• the theoretical introduction to the popular aspect of density-temperature scaling of the relaxation

time in numerous classes of simple liquids and polymers (section 3.1). Theoretical basis of the

scaling and its connection to the inverse power law is discussed in the subsection 3.1.1, the numerous

ways of approximation of the scaling exponent γ are presented in the subsection 3.1.2. The

speculations upon the relation between the density-temperature scaling and the pressure-energy

correlations in liquids are presented in the subsection 3.1.3; the connection of the temperaturedensity

scaling to the fragility and potential energy landscapes is discussed in the subsection 3.1.4.

Section ends with results of Leporini group (subsection 3.1.5) obtained just before beginning of

my work over the PhD thesis. These results allowed to ascertain the universal scaling between the

fast microscopic and the slow macroscopic dynamics and also urged me to explore the possibility

of temperature-density scaling not only of the slow macroscopic dynamics reflected in such macroscopic

parameter as the relaxation time, but also of the microscopic dynamics closely connected

to the Debye-Waller factor.

• section 3.2, consisting of the introduction to the functions regarding the cages surrounding the

tagged atom (subsection 3.2.1) and the basic theoretical information about the spatial correlation

functions (subsection 3.2.2)

The original results of the research of temperature-density scaling of the Debye-Waller factor are

presented into the fourth chapter.

The fifth chapter reports an original investigation of the correlation functions of supercooled polymeric

melt. The peculiarities of motion of the atoms in the supercooled liquid, especially when it

approaches the glass transition, are always of great interest and importance, because the understanding

of the laws of this motion could be crucial for prediction of various properties of materials close to their

glass transition. The correlation functions give us a rather detailed picture of the motion of the atoms.

In the section 5.1 I explored the correlation functions regarding the cages surrounding the tagged atom.

The description of the program, calculating the neighbor list and cage correlation functions (already

introduced in subsection 5.4.1), follows in subsection 5.1.1. The results of the run of this program using

the input data from the simulations of our MD model, already described in section 2.2, are presented in

subsection 5.1.2. In section 5.2 there follows the research of the spatial correlation functions. The structure

of this subsection is similar to the previous one with the introductory part of the corresponding

functions being presented in subsection 3.2.2, program description in subsection 5.2.1 and the results of

the run in subsection 5.2.2. In the section 5.3 I introduced the original self-correlation functions of displacement

in subsection 5.3.1. The program, calculating these functions, is described in the subsection

5.3.2, the results of run - in the subsection 5.3.3.

Finally there follow the conclusions, appendix and bibliography.

is one of the most important artificial materials utilized by man. Numerous applications of glass

materials and glass transition phenomenon include obtaining of the optimal wet-skid resistance by

using the tread rubber with high glass transition temperature, enhancement of the ballistic

penetration resistance of armor coatings obtained by energy dissipation associated with an impact induced

transition of the coating to the glassy state, the material selection for acoustic tiles on

submarines allowing to undergo the glass transition at the frequency of the active sonar at seawater

temperatures thus dampening the sonar, arterial walls restoring, preservation of food,

highly soluble pharmaceuticals. So, the study of glassy state tends to be one of the most promising

and interesting tasks in chemistry in our days. This fact was my main motivating factor to study the

structural and dynamical characteristics of the polymer melt approaching to glass transition from above

in group of Prof. Dino Leporini in Pisa University.

The particles(in this Thesis the word “particle” signifies the most elementary rigid part of the system

under investigation. In general, it can be either molecule for the system containing the rigid molecules,

or atom for the case of flexible molecules. As far, as in this Thesis the polymer melt is the system

that I am investigating, the word “particle” in context of my research signifies the monomer) spend

increasing periods of time, rattling in cages formed by their neighbors while the liquid, consisting of

these particles, approaches the glass transition. Finally, the particles are relaxed from their cages -

this process is called the structure relaxation. The main aim of this thesis is the investigation of the

microscopic dynamics of the particles trapped in these cages and the study of the possible ways to

connect this fast dynamics in cage to slow macroscopic dynamics associated with structure relaxation.

First direction of research is based on the density-temperature scaling of the one of the most important

characteristics of the cage dynamics - the Debye-Waller factor(that is the amplitude of the particle’s

rattling in the cage, see the section 1.2 for the details). The density-temperature scaling has become

very popular in the recent years due to the improvement of the technical equipment of the experiments,

that allowed to change both density and temperature of the system in simultaneous and prompt way.

The density-temperature scaling of the structure relaxation time for single and multi-component liquids

having different interaction potentials was introduced and investigated in this thesis from the different

points of view, including the pressure-energy correlations and potential energy landscapes with citations

of the correspondent sources (see the section 3.1). Recent research of the Prof. Leporini group has

presented the universal scaling between the structure relaxation time and Debye-Waller factor(see the

subsection 3.1.5). Nevertheless, the research of the density-temperature scaling of the Debye-Waller factor has been missing so far and I tried my best to improve this situation and to show that the

density-temperature scaling is valid for the microscopic processes, too. In the terms of the second

direction of investigation I studied several functions of cage correlation for the polymer melt. This

study should have allowed me to better understand the processes taking place in the cages or “shells”

surrounding the tagged particle and to discover the new types of connection between these microscopic

processes from one side and the macroscopic processes, like e.g. the structure relaxation, from another

side. Results of this study could help to develop the new ways of understanding of the universal scaling

between the structure relaxation time and Debye-Waller factor. Furthermore, it is interesting to study

the cage correlation functions of the polymer melt because the polymers have been never studied in

this way so far. Previous researches of the correlation functions dealt with the binary mixtures,

hard spheres and disks, etc. I also have to say that this study is based on introduction of different

correlation functions including the self-correlation functions of displacement that have not been studied

in context of the liquids state research so far.

As to the meaning of the results of the thesis, I may state the following:

• the density-temperature scaling of the Debye-Waller factor is evidenced(see the section 3.1 and

the Chapter number 4). The values of the scaling exponent γ are consistent with the predictions

from the study of Lennard-Jones liquids. Data of all the polymers in the study, that have different

molecular masses and interaction potential, collapse the straight lines of Debye-Waller factor vs

TV γ plot, where any straight line is uniquely defined by the molecular mass and parameters of

the interaction potential. These lines cross in one “universal” point.

• the cage correlation functions, describing the time evolution of the neighbor cage of the given

particle(see the subsection 3.2.1 and section 5.1) and following from the immediate physical interpretation,

represent the alternative but perfectly equivalent instrument for the description of

the structure relaxation, compared to the more rigorous intermediate scattering function.

• the analysis of the spatial correlation of displacement(see the subsection 3.2.2 and section 5.2)

evidences the link between the well-known static properties of the system and the dynamic properties,

represented by the direction and modulus correlation functions of displacement. Origin of

this correlation is not perfectly clear and needs further investigation.

• analysis of the time correlations of displacement (see the section 5.3) shows that the directionality

of motion rather than the displacement modulus seems to connect the fast microscopic and slow

macroscopic dynamics. The direction of the displacement of a particle at the time scales of cage

regime determines in general the direction of the particle’s motion even at much longer times.

As far, as the outlook of the Thesis is concerned, I have to point out that the further investigation,

generalization and improvement of these results could be useful. The results of current Thesis allow to

construct only several very narrow and rickety bridges between the fast microscopic and slow macroscopic

dynamics. These narrow links should be united with the other existing ones and with those that

will be created in future, thus forming the new theories, methods and approaches. Among the possible

ways of developing of the results of this Thesis I can point to either the prospect of the study of the

influence of TV γ on the behavior of the cage correlation functions, uniting the two main directions of

research of this thesis, could be the most promising direction of further study, or to the aforementioned

possible research of the reasons of the correlation between the static and dynamic(represented by the

direction and modulus correlation functions of displacement) properties of the system. Results and

outlook of the Thesis are discussed in the Conclusions in more detailed way.

My thesis consists of five chapters. First chapter of my PhD thesis presents the introduction to glass

transition phenomenon (section 1.1); covers the information about the static structure and relaxation in

liquids and introduces the functions that describe these processes (section 1.2); and describes the glass

transition in polymers with particular attention to the similarities and differences of the glass transition

processes in simple liquids and polymers(section 1.3), thus explaining why the polymers usually are the

good glass-formers.

Then, in the second chapter, covering the methods of the computer simulations of polymers, there

follow:

• the review of numerous methods of molecular dynamics and (a bit) of monte-carlo simulations of

the glass-forming polymers(section 2.1), allowing to better understand the future ways of development

of the simulation studies of the polymers and to enrich the arsenal of a modern scientist

with new powerful computational methods enabling the simulations to become more quick and

effective

• separate section (2.2) describing the numerical methods and MD model (developed by Cristiano

De Michele, a former member of the Leporini group) used in the MD simulations runs during the

work over the fourth and fifth chapters

Third chapter contains the theoretical background for the fourth and fifth chapters.

• the theoretical introduction to the popular aspect of density-temperature scaling of the relaxation

time in numerous classes of simple liquids and polymers (section 3.1). Theoretical basis of the

scaling and its connection to the inverse power law is discussed in the subsection 3.1.1, the numerous

ways of approximation of the scaling exponent γ are presented in the subsection 3.1.2. The

speculations upon the relation between the density-temperature scaling and the pressure-energy

correlations in liquids are presented in the subsection 3.1.3; the connection of the temperaturedensity

scaling to the fragility and potential energy landscapes is discussed in the subsection 3.1.4.

Section ends with results of Leporini group (subsection 3.1.5) obtained just before beginning of

my work over the PhD thesis. These results allowed to ascertain the universal scaling between the

fast microscopic and the slow macroscopic dynamics and also urged me to explore the possibility

of temperature-density scaling not only of the slow macroscopic dynamics reflected in such macroscopic

parameter as the relaxation time, but also of the microscopic dynamics closely connected

to the Debye-Waller factor.

• section 3.2, consisting of the introduction to the functions regarding the cages surrounding the

tagged atom (subsection 3.2.1) and the basic theoretical information about the spatial correlation

functions (subsection 3.2.2)

The original results of the research of temperature-density scaling of the Debye-Waller factor are

presented into the fourth chapter.

The fifth chapter reports an original investigation of the correlation functions of supercooled polymeric

melt. The peculiarities of motion of the atoms in the supercooled liquid, especially when it

approaches the glass transition, are always of great interest and importance, because the understanding

of the laws of this motion could be crucial for prediction of various properties of materials close to their

glass transition. The correlation functions give us a rather detailed picture of the motion of the atoms.

In the section 5.1 I explored the correlation functions regarding the cages surrounding the tagged atom.

The description of the program, calculating the neighbor list and cage correlation functions (already

introduced in subsection 5.4.1), follows in subsection 5.1.1. The results of the run of this program using

the input data from the simulations of our MD model, already described in section 2.2, are presented in

subsection 5.1.2. In section 5.2 there follows the research of the spatial correlation functions. The structure

of this subsection is similar to the previous one with the introductory part of the corresponding

functions being presented in subsection 3.2.2, program description in subsection 5.2.1 and the results of

the run in subsection 5.2.2. In the section 5.3 I introduced the original self-correlation functions of displacement

in subsection 5.3.1. The program, calculating these functions, is described in the subsection

5.3.2, the results of run - in the subsection 5.3.3.

Finally there follow the conclusions, appendix and bibliography.

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